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\begin{document}
\title{\bf The tight bound for the strong chromatic indices of claw-free subcubic graphs
\thanks{Supported by NSFC 11771080.}}
\author{Yuquan Lin and Wensong Lin\footnote{Corresponding author. E-mail address: [email protected]}\\
{\small School of Mathematics, Southeast University, Nanjing 210096, P.R. China}}
\date{}
\maketitle
\vspace*{-1cm} \setlength\baselineskip{6.5mm}
\begin{abstract}
Let $G$ be a graph and $k$ a positive integer. A strong $k$-edge-coloring of $G$ is a mapping $\phi: E(G)\to \{1,2,\dots,k\}$ such that for any two edges $e$ and $e'$ that are either adjacent to each other or adjacent to a common edge, $\phi(e)\neq \phi(e')$. The strong chromatic index of $G$, denoted as $\chi'_{s}(G)$, is the minimum integer $k$ such that $G$ has a strong $k$-edge-coloring. Lv, Li and Zhang [Graphs and Combinatorics 38 (3) (2022) 63] proved that if $G$ is a claw-free subcubic graph other than the triangular prism then $\chi_s'(G)\le 8$. In addition, they asked if the upper bound $8$ can be improved to $7$. In this paper, we answer this question in the affirmative. Our proof implies a linear-time algorithm for finding strong $7$-edge-colorings of such graphs. We also construct infinitely many claw-free subcubic graphs with their strong chromatic indices attaining the bound $7$.\\
\noindent{\bf Keywords:} strong edge coloring; strong chromatic index; claw-free; subcubic graph. \\
\end{abstract}
\section{Introduction}
Let $G=(V(G), E(G))$ be a finite undirected simple graph.
For $v\in V(G)$, let $N(v)=\{u\in V(G): uv\in E(G)\}$ denote the open neighborhood of $v$ and $d(v)=|N(v)|$ be the degree of $v$. Let $\Delta(G)=\max\limits_{v\in V(G)}d(v)$ denote the maximum degree of $G$.
For convenience, we use the abbreviation $[1,n]$ for $\{1,2, \dots, n\}$, where $n$ is any positive integer.
Let $e$ and $e'$ be two edges of $G$. If $e$ and $e'$ are adjacent to each other, we say that the distance between $e$ and $e'$ is $1$, and if they are not adjacent but both of them are adjacent to a common edge, we say they are at distance $2$. Given a positive integer $k$, a {\em strong $k$-edge-coloring} of $G$ is a mapping $\phi: E(G)\to [1,k]$ such that for any two edges $e$ and $e'$ that are at distance $1$ or $2$, $\phi(e)\neq \phi(e')$. The {\em strong chromatic index} of $G$, denoted by $\chi'_{s}(G)$, is the minimum integer $k$ such that $G$ has a strong $k$-edge-coloring.
The concept of strong edge coloring, first introduced by Fouquet and Jolivet \cite{FJ1983}, can be used to model the conflict-free channel assignment in radio networks \cite{R1997,NKGB2000}. In 1985, Erd\H{o}s and Ne\v{s}et\v{r}il \cite{E1988,EN1989} proposed the following conjecture about the upper bound of $\chi'_{s}(G)$ in term of $\Delta(G)$, which if true, is the best possible.
\begin{conjecture}\label{Conj-EN}
(Erd\H{o}s and Ne\v{s}et\v{r}il \cite{E1988,EN1989})
If $G$ is a graph with maximum degree $\Delta(G)$, then
\begin{equation*}
\chi'_{s}(G)\le\begin{cases}
\begin{array}{cl}
\dfrac{5}{4}\Delta(G)^{2},& \text{if}\ \Delta(G) \ \text{is even,} \\
\dfrac{5}{4}\Delta(G)^{2}-\dfrac{1}{2}\Delta(G)+\dfrac{1}{4},& \text{if}\ \Delta(G)\ \text{is odd.}
\end{array}
\end{cases}
\end{equation*}
\end{conjecture}
The conjecture is clearly true for $\Delta(G)\le 2$. The case $\Delta(G)=3$ was verified by Andersen \cite{A1992} in 1992, and independently by Hor\'{a}k, Qing, and Trotter \cite{HQT1993} in 1993. Furthermore, if $G$ is a subcubic planar graph then $\chi'_s(G)\le 9$ \cite{KLRSWY2016}. For $\Delta(G)=4$, an upper bound of $23$ was proved by Hor\'{a}k \cite{H1990} in 1990. It was improved to $22$ by Cranston \cite{C2006} in 2006 and more recently Huang, Santana and Yu \cite{HSY2018} obtained the upper bound $21$. Recall that the conjectured bound is $20$ for $\Delta(G)=4$. Although the general case when $\Delta(G)=4$ has not been solved yet, the bound $20$ holds for some special cases. In 1990, Faudree, Schelp, Gy{\'a}rf{\'a}s and Tuza \cite{FSGT1990} proved that $\chi'_{s}(G)\le 4\Delta(G) + 4$ for any planar graph $G$, which implies that Conjecture \ref{Conj-EN} holds for planar graphs with maximum degree $4$. And this bound for planar graphs with maximum degree $4$ was further improved to $19$ by Wang, Shiu, Wang and Chen \cite{WSWC2018} in 2018.
In 2015, Bensmail, Bonamy and Hocquard \cite{BBH2015} studied the upper bound on $\chi'_{s}(G)$ in terms of maximum average degree for the class of graphs with maximum degree $4$, where they gave some sufficient conditions under which Conjecture \ref{Conj-EN} is true. In 2018, Lv, Li and Yu \cite{LLY2018} strengthened Bensmail, Bonamy and Hocquard's results. And they proved that for any graph $G$ with $\Delta(G)=4$, if there are two vertices of degree $3$ whose distance is at most $4$, then $\chi'_{s}(G)\le 20$.
For graphs with maximum degree $5$, Zang \cite{Z2015} gave the upper bound 37 (recall that the conjectured bound is 29), which is the only progress as we know. And for larger $\Delta(G)$, the problem is widely open. In 1997, Molloy and Reed \cite{MR1997} used probabilistic techniques to prove that $\chi'_{s}(G)\le 1.998\Delta(G)^{2}$ for sufficiently large $\Delta(G)$. And an improvement to $1.93\Delta(G)^{2}$ was provided by Bruhn and Joos \cite{BJ2015} in 2015. Recently, Bonamy, Perrett and Postle \cite{BPP2022} further strengthened this bound to $1.835\Delta(G)^{2}$. The current best known upper bound is $1.772\Delta(G)^{2}$ which was showed by Hurley, de Joannis de Verclos and Kang \cite{HdK2021} in 2021.
A graph is called {\em claw-free} if it has no induced subgraph isomorphic to the complete bipartite graph $K_{1,3}$. In 2020, Debski, Junosza-Szaniawski and {\'S}leszy{\'n}ska-Nowak \cite{DJS2020} presented the following upper bound for the strong chromatic indices of claw-free graphs.
\begin{theorem}\label{claw-free}
(D{e}bski, Junosza-Szaniawski and {\'S}leszy{\'n}ska-Nowak \cite{DJS2020})
For any claw-free graph $G$ with maximum degree $\Delta(G)$, $\chi'_{s}(G)\le \frac{9}{8}\Delta(G)^{2}+\Delta(G)$.
\end{theorem}
A graph with maximum degree less than or equal to $3$ is called a {\em subcubic} graph. In 2022, Lv, Li and Zhang \cite{LLZ2022} proved that, for any claw-free subcubic graph $G$ other than the triangular prism, $\chi'_{s}(G)\le 8$. Please see Figure \ref{fig:3-prism} for the triangular prism (also called the $3$-prism). Notice that the $3$-prism is a claw-free cubic graph with its strong chromatic index being equal to $9$. In the same paper, the authors left the problem whether this bound can be improved to $7$. This paper solves this problem and the main result is the following theorem.
\begin{theorem}\label{main}
Let $G$ be a claw-free subcubic graph. If each component of $G$ is not isomorphic to the triangular prism, then $\chi'_{s}(G)\leq 7$.
\end{theorem}
\noindent{\bf Remark 1.}
Recall that when $\Delta (G)= 3$, the upper bound in Conjecture \ref{Conj-EN} is $10$, the
upper bound in Theorem \ref{claw-free} is $\frac{105}{8}$, and the upper bound proved by Lv, Li and Zhang \cite{LLZ2022} is $8$, while our bound in Theorem \ref{main} is $7$. \\
\noindent{\bf Remark 2.} A graph $G_0$ on five vertices with strong chromatic index $7$ was presented in \cite{LLZ2022} (see Figure \ref{fig:G0}). This shows the sharpness of the upper bound $7$. In fact, there are other claw-free subcubic graphs with their strong chromatic indices being equal to $7$. It is not difficult to verify that the graph $H_{0}$ shown in Figure \ref{fig:H0} has the strong chromatic index $7$. Therefore, by Theorem \ref{main}, any claw-free subcubic graph containing $H_0$ as a subgraph has the strong chromatic index $7$. This implies that there are infinitely many claw-free subcubic graphs with their strong chromatic indices attaining the upper bound $7$.
\begin{figure}
\caption{The $3$-prism}
\label{fig:3-prism}
\caption{The graph $G_0$}
\label{fig:G0}
\caption{The graph $H_{0}
\label{fig:H0}
\end{figure}
\noindent{\bf Remark 3.} We would like to point out that our proof of Theorem \ref{main} implies a linear-time algorithm that can produces a strong $7$-edge-coloring of any claw-free subcubic graph other than the $3$-prism.\\
The remainder of this paper is dedicated to the proof of Theorem \ref{main} which is organized as follows. In Section 2, after stating some definitions and notations, we explain how to order the edges of a connected claw-free subcubic graph and apply the greedy algorithm to obtain a partial strong $7$-edge-coloring of the graph. Section 3 consists of a series of lemmas, each of which shows that a particular partial strong $7$-edge-coloring of a claw-free subcubic graph can be extended to a strong $7$-edge-coloring of the whole graph. Finally, Section 4 summarizes our works and suggests some future research directions.
\section{Preliminaries and notations}
Let $G$ be a connected claw-free subcubic graph not isomorphic to the $3$-prism. To prove Theorem \ref{main}, we shall first find a subgraph of $G$ and construct a strong $7$-edge-coloring of this subgraph, and then try to extend the coloring to the whole graph $G$. The subgraph we find should have the property that any two of its edges at distance greater than $2$ must also be at distance greater than $2$ in $G$. We next introduce some notations and preliminary facts that we will use in our proofs.
We use $\alpha$, $\beta$, $\gamma$ to denote colors and $\phi$, $\psi$, $\sigma$ to denote edge colorings. Given two distinct edges $e$ and $e'$ of $G$, we say that $e$ {\em sees} $e'$ in $G$ if they are distance $1$ or $2$ apart. An edge coloring of a graph $G$ is {\em good}, if it is a strong edge coloring of $G$ using at most $7$ colors. A {\em good partial coloring} of a graph $G$ is a good coloring $\phi$ of some subgraph $H$ of $G$ such that $\phi(e)\neq \phi(e')$ if $e$ and $e'$ see each other in $G$.
Let $\phi$ be a good partial coloring of $G$. We say that $e$ {\em sees} a color $\alpha$ in $\phi$, if $e$ sees an edge $e'$ for which $\phi(e) = \alpha$. For $e\in E(G)$, let $F_{\phi}(e)$ denote the set of colors that $e$ sees in $\phi$. In addition, let $\bar{E}_{\phi}$ denote the set of edges in $G$ not already assigned colors by $\phi$ and $\bar{G}_{\phi}$ denote the subgraph of $G$ induced by $\bar{E}_{\phi}$. For $e\in \bar{E}_{\phi}$, let $A_{\phi}(e)$ denote the set of colors that $e$ does not see in $\phi$. It is clear that $A_{\phi}(e)=[1,7]\setminus F_{\phi}(e)$ for any $e\in \bar{E}_{\phi}$.
In order to construct a good partial coloring of a claw-free subcubic graph $G$, we first apply the similar method in \cite{WL2008} to order the edges of $G$, and then use {\em the greedy algorithm} to color them in the order, left only a few particular edges uncolored.
Suppose $S$ is a nonempty subset of $V(G)$. For a vertex $v\in V(G)$, the distance from $v$ to $S$, denoted by $d_S(v)$, is equal to $\min\limits_{w\in S}\{d(v,w)\}$, where $d(v,w)$ is the distance between $v$ and $w$ in $G$. Let $I$ be the maximum distance from a vertex of $G$ to $S$. For $i=0,1,\dots,I$, let $D_i=\{v\in V(G):\ d(v,S)=i\}$. A mapping $d_S$ from $E(G)$ to nonnegative real numbers is defined as: for any edge $e$ with two end vertices $u$ and $v$, $d_S(e)=\frac{1}{2} (d_S(u)+ d_S(v))$. Suppose $R=(e_{k_1},e_{k_2},\dots,e_{k_m})$ is an ordering of the edges of $G$. For any two integers $i$ and $j$ in $[1,m]$, if $i<j$ implies $d_S(e_{k_i})\geq d_S(e_{k_j})$, then we say that the edge ordering $R$ of $G$ is {\em compatible} with the mapping $d_S$. It is clear that any edge joins two vertices that are either in the same $D_{i}$, or one in $D_{i}$ and the other in $D_{i+1}$ for some $i$, and that each vertex in $D_{i}$ with $i \ge 1$ is adjacent to at least one vertex in $D_{i-1}$. Let $e=xy$ be an edge with $d_S(x)\le d_S(y)$. If $d_S(e)\ge 1$, then $e$ is adjacent to an edge $e'=xz$ with $d_S(z)=d_S(x)-1$.
We are now ready to produce a good partial coloring of a claw-free subcubic graph.
\begin{lemma}\label{greedy}
Let $G$ be a connected claw-free subcubic graph other than the $3$-prism and $S$ a nonempty subset of $V(G)$. The greedy algorithm, coloring the edges of $G$ in an order $R$ that is compatible with the mapping $d_S$, will produce a good partial coloring of $G$ with only edges in $\{e\in E(G): d_S(e)<1\}$ being left uncolored.
\end{lemma}
\begin{pf}
Let $e=xy$ be an edge with $d_S(e)\ge 1 $. Suppose $d_S(x) \le d_S(y)$. Let $e'=xz$ be a neighbor of $e$ with $d_S(z) = d_S(x)-1$. It is clear that all edges incident to the vertex $z$ are behind the edge $e$ in the order $R$. And so, when $e$ is going to be colored by the greedy algorithm, all edges incident to $z$ are not colored.
If $d(x)=2$ or $d(x)=3$ and the third neighbor $w$ of $x$ satisfies $d_S(w)=d_S(x)-1$, then $e$ sees at most $5$ colors at that moment, implying that $e$ can be colored properly. Thus we assume that $d(x)=3$ and the third neighbor $w$ of $x$ satisfies $d_S(w) \ge d_S(x)$.
If $d_S(y)=d_S(x)\ge 1$, then $y$ has a neighbor $y'$ with $d_S(y')=d_S(z)$ (It is possible that $z=y'$). As $G$ is a claw-free subcubic graph, it is easy to check that $e$ sees at most $6$ colors (please refer to Figure \ref{fig:greedy} (a) and (b)). Thus $e$ can be colored properly.
\begin{figure}
\caption{The illustrations of Lemma \ref{greedy}
\label{fig:greedy}
\end{figure}
If $d_S(y)=d_S(x)+1\ge 2$, then $d_S(w)=d_S(x)$ or $d_S(w)=d_S(x)+1$. Again, because $G$ is a claw-free subcubic graph, $e$ sees at most $6$ colors (please refer to Figure \ref{fig:greedy} (c) and (d)) and so $e$ can be colored properly. Therefore, the lemma holds.
\end{pf}
In the proof of Theorem \ref{main}, we will often choose some subset $S$ of $V(G)$. With this subset $S$, by Lemma \ref{greedy}, we can obtain a good partial coloring $\phi$ of $G$ with only edges $e$ with $d_S(e)<1$ being uncolored. And then our task is to extend $\phi$ to a good coloring of the whole graph $G$.
There are two main techniques in our proofs, one is using Hall's theorem \cite{H1935}, the other is the modified caterpillar tree method which is slightly different from the caterpillar tree method used by Hor\'{a}k, Qing and Trotter \cite{HQT1993} in proving the upper bound $10$ for the strong chromatic indices of cubic graphs.
By Hall's theorem \cite{H1935}, $\{A_{\phi}(e):e\in\bar{E}_{\phi}\}$ has a system of distinct representatives (abbreviated SDR) if and only if $|\cup_{e\in M}A_{\phi}(e)|\ge |M|$, for every $M\subseteq \bar{E}_{\phi}$. Whenever $\{A_{\phi}(e):e\in\bar{E}_{\phi}\}$ has a SDR, $\phi$ can be easily extended to a good coloring of $G$. In this situation, we will say that we can obtain a good coloring of $G$ by SDR.
\section{The proof of Theorem \ref{main} }
It is sufficient to prove Theorem \ref{main} for connected graphs.
Let $G$ be a connected claw-free subcubic graph not isomorphic to the $3$-prism.
If $G$ is isomorphic to the complete graph $K_{4}$ or the graph $K_{4}^{\Delta}$ (i.e., the graph obtained from $K_{4}$ by replacing each vertex with a $3$-cycle, as shown in Figure \ref{fig:K_{4}}), then it is easy to check that $\chi'_{s}(G)\le 7$. Please refer to Figure \ref{fig:K_{4}} for a good coloring of $K_{4}^{\Delta}$. Thus in this section, we always assume that $G$ is a connected claw-free subcubic graph that is not isomorphic to any graph in $\{\mbox{the $3$-prism}, K_{4}, K_{4}^{\Delta}\}$.
\begin{figure}
\caption{The two graphs $K_{4}
\label{fig:K_{4}
\end{figure}
\begin{lemma}\label{lemma:degree-1}
If $G$ has a vertex of degree $1$, then $\chi'_{s}(G)\leq 7$.
\end{lemma}
\begin{pf}
Let $v_0$ be a vertex of degree $1$ in $G$ and $e_0$ be the edge incident with $v_0$.
Set $S=\{v_0\}$. Then, by Lemma \ref{greedy}, $G-v_{0}$ has a good coloring.
Since $e_{0}$ sees at most $5$ edges in $G$, we can extend this good coloring of $G-v_{0}$ to $G$.
\end{pf}
\begin{lemma}\label{lemma:degree-2}
If $G$ has a vertex of degree $2$, then $\chi'_{s}(G)\leq 7$.
\end{lemma}
\begin{pf}
Let $v_0$ be a vertex of degree $2$ in $G$ with two neighbors $v_{1}$ and $v_{2}$. If $v_{1}v_{2}\in E(G)$ or $d(v_{1})=2$, then put $S=\{v_0\}$. By Lemma \ref{greedy}, $G-v_{0}$ has a good coloring $\phi$ with $|A_{\phi}(v_{0}v_{1})|\ge 2$ and $|A_{\phi}(v_{0}v_{2})|\ge 1$. We can obtain a good coloring of $G$ by SDR easily. So by symmetry, we may assume that $v_{1}v_{2}\notin E(G)$ and $d(v_{1})=d(v_{2})=3$. Let $N(v_{1})=\{v_{0},u_{1},u_{1}'\}$ and $N(v_{2})=\{v_{0},u_{2},u_{2}'\}$. Since $G$ is a claw-free subcubic graph, we must have $u_1u_1'\in E(G)$ and $u_2u_2'\in E(G)$. This implies that $\{u_1,u_1'\}=\{u_2,u_2'\}$ or $\{u_1,u_1'\}\cap \{u_2,u_2'\}=\emptyset$.
If $\{u_1,u_1'\}=\{u_2,u_2'\}$ then $G$ is isomorphic to $H_{1}$ (please see Figure \ref{fig:H_{1}H_{2}H_{3}} for $H_{1}$), and so $\chi'_{s}(G)= 7$. Thus we assume that $\{u_1,u_1'\}\cap \{u_2,u_2'\}=\emptyset$. If $d(u_{1})=2$, then the two neighbors of $u_1$ are $v_1$ and $u_1'$. By setting $S=\{u_1\}$, similar to the argument in the previous paragraph, we can get a good coloring of $G$. Thus we assume $d(u_{1})=3$. Symmetrically, we may also assume that $d(u_{1}')=d(u_{2})=d(u_{2}')=3$. Let $w_{1},w_{1}',w_{2},w_{2}'$ denote the third neighbors of $u_{1},u_{1}',u_{2},u_{2}'$, respectively. Denote by $E(\{u_1,u_1'\},\{u_2,u_2'\})$ the set of edges with one vertex in $\{u_1,u_1'\}$ and the other in $\{u_2,u_2'\}$. If $|E(\{u_1,u_1'\},\{u_2,u_2'\})|=2$, then $G$ is isomorphic to $H_2$ (see Figure \ref{fig:H_{1}H_{2}H_{3}}) and we clearly have $\chi'_{s}(G)\le 7$. We next deal with the remaining two cases $|E(\{u_1,u_1'\},\{u_2,u_2'\})|=1$ and $|E(\{u_1,u_1'\},\{u_2,u_2'\})|=0$.
\begin{figure}
\caption{The three graphs $H_{1}
\label{fig:H_{1}
\end{figure}
{\bf Case 1}. $|E(\{u_1,u_1'\},\{u_2,u_2'\})|=1$.
Without loss of generality, assume that $u_{1}u_{2}\in E(G)$. If $u_{1}'$ and $u_{2}'$ are adjacent to a common neighbor, then $G$ must be isomorphic to $H_{3}$ (see Figure \ref{fig:H_{1}H_{2}H_{3}}), implying that $\chi'_{s}(G)\le 7$. Thus, we assume $u_{1}'u_{2}'\notin E(G)$ and $w_{1}'\neq w_{2}'$.
Set $S=\{v_{0},v_{1},v_{2},u_{1},u_{2}\}$. By Lemma \ref{greedy}, we get a good partial coloring $\phi$ of $G$ with nine edges uncolored (please see Figure \ref{fig:degree2_case1} for the names of the uncolored edges of $G$). Observe that $|A_{\phi}(e_{3})|\ge5$, $|A_{\phi}(e_{i})|=6$ for $i=1,2,4,5$ and $|A_{\phi}(f_{j})|\ge 4$ for $j=1,2,3,4$.
\begin{figure}
\caption{Case 1 in the proof of Lemma \ref{lemma:degree-2}
\label{fig:degree2_case1}
\end{figure}
Let $\alpha$ be a color in $A_{\phi}(f_{1})\cap A_{\phi}(f_{3})$. By coloring $f_{1}$ and $f_{3}$ with the same color $\alpha$, we extend $\phi$ to a new good partial coloring $\psi$ of $G$, in which $|A_{\psi}(e_{3})|\ge4$, $|A_{\psi}(e_{i})|\ge 5$ for $i=1,2,4,5$ and $|A_{\psi}(f_{j})|\ge 3$ for $j=2,4$. Observe that $A_{\psi}(e_{5})\cap A_{\psi}(f_{2}) \neq \emptyset$, we can further extend $\psi$ to another good partial coloring $\sigma$ of $G$ by coloring $e_{5}$ and $f_{2}$ with the same color $\beta\in A_{\psi}(e_{5})\cap A_{\psi}(f_{2})$. Then we have $|A_{\sigma}(e_{3})|\ge3$, $|A_{\sigma}(e_{i})|\ge 4$ for $i=1,2,4$ and $|A_{\sigma}(f_{4})|\ge 2$.
Now, if $A_{\sigma}(e_{2})\cap A_{\sigma}(f_{4}) \neq \emptyset$, we can color $e_{2}$ and $f_{4}$ with the same color $\gamma\in A_{\sigma}(e_{2})\cap A_{\sigma}(f_{4})$ and then color $e_{3},e_{4},e_{1}$ by SDR. Otherwise, we greedily color $f_4, e_3,e_4,e_1,e_2$ in this order.
{\bf Case 2}. $|E(\{u_1,u_1'\},\{u_2,u_2'\})|=0$.
In this case, $w_{1},w_{1}', w_{2}, w_{2}'$ are not necessarily distinct. However, this does not affect the following arguments. Please see Figure \ref{fig:degree2_case2} for the names of vertices and edges. Set $S=\{v_{0},v_{1},v_{2}\}$. By Lemma \ref{greedy}, there is a good partial coloring $\phi$ of $G$ with six uncolored edges $e_1,e_2,f_1,f_2,f_3,f_4$. It is clear that $|A_{\phi}(e_{1})|=|A_{\phi}(e_{2})|=4$ and $|A_{\phi}(f_{j})|\ge 2$ for $j=1,2,3,4$.
\begin{figure}
\caption{Case 2 in the proof of Lemma \ref{lemma:degree-2}
\label{fig:degree2_case2}
\end{figure}
We now make two easy but useful observations. One is that $A_{\phi}(f_{1})\cup A_{\phi}(f_{2})\subseteq A_{\phi}(e_{1})=[1,7]\setminus F_{\phi}(e_{1})$ and $A_{\phi}(f_{3})\cup A_{\phi}(f_{4})\subseteq A_{\phi}(e_{2})=[1,7]\setminus F_{\phi}(e_{2})$, where $F_{\phi}(e_{1})=\{\phi(g_{1}),\phi(h_{1}),\phi(h_{2})\}$ and $F_{\phi}(e_{2})=\{\phi(g_{2}),\phi(h_{3}),\phi(h_{4})\}$. The other is that, for any $i\in\{1,2\}$ and any $j\in\{3,4\}$, $f_{i}$ and $f_{j}$ do not see each other.
As $|A_{\phi}(e_{1})|=|A_{\phi}(e_{2})|=4$, $A_{\phi}(e_{1})\cap A_{\phi}(e_{2})\neq \emptyset$. According to the value of $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|$, we divide the proof into the following four subcases.
{\bf Subcase 2.1}. $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=1$. ($|A_{\phi}(e_{1})\cup A_{\phi}(e_{2})|=7$.)
In this case, let's explain why $\phi$ can be extend to a good coloring of $G$ by greedily coloring $f_{1},f_{2},f_{3},f_{4},e_{1}$ and $e_{2}$ in this order. First, $f_{1},f_{2},f_{3},f_{4}$ can be colored properly. Secondly, since $A_{\phi}(f_{3})\cup A_{\phi}(f_{4})\subseteq A_{\phi}(e_{2})$ and $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=1$, there is at least one color available for $e_{1}$ when $e_{1}$ is to be colored. Finally, since $A_{\phi}(f_{1})\cup A_{\phi}(f_{2})\subseteq
A_{\phi}(e_{1})$ and $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=1$, after $e_{1}$ is colored, at least one color is available for $e_{2}$.
{\bf Subcase 2.2}. $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=2$. ($|A_{\phi}(e_{1})\cup A_{\phi}(e_{2})|=6$.)
Let $A_{\phi}(e_{1})\cap A_{\phi}(e_{2})=\{\alpha_{1},\alpha_{2}\}$. If $\{\alpha_{1},\alpha_{2}\}\cap (A_{\phi}(f_{1})\cup A_{\phi}(f_{2}))\cap (A_{\phi}(f_{3})\cup A_{\phi}(f_{4}))\neq\emptyset$, without loss of generality, assume $\alpha_{1} \in A_{\phi}(f_{1})\cap A_{\phi}(f_{3})$. Now, a good coloring of $G$ can be obtained by first coloring $f_{1}$ and $f_{3}$ with $\alpha_{1}$ and then greedily coloring $f_{2},f_{4},e_{1},e_{2}$ in this order. Thus, we assume that $\{\alpha_{1},\alpha_{2}\}\cap (A_{\phi}(f_{1})\cup A_{\phi}(f_{2}))\cap (A_{\phi}(f_{3})\cup A_{\phi}(f_{4}))=\emptyset$.
If $\{\alpha_{1},\alpha_{2}\}\cap (A_{\phi}(f_{1})\cup A_{\phi}(f_{2})) =\{\alpha_{1},\alpha_{2}\}\cap (A_{\phi}(f_{3})\cup A_{\phi}(f_{4})) =\emptyset$, then it is easy to see that $\phi$ can be extended to a good coloring of $G$. Therefore, by symmetry, we may assume that $\alpha_{1}\in A_{\phi}(f_{1})$ and $\alpha_{1}\notin A_{\phi}(f_{3})\cup A_{\phi}(f_{4})$.
Suppose that $A_{\phi}(f_{1})\cup A_{\phi}(f_{2})\neq \{\alpha_{1},\alpha_{2}\}$. If $\alpha_{1}\in A_{\phi}(f_{2})$, then there exists a color $\beta\in (A_{\phi}(f_{1})\cup A_{\phi}(f_{2})) \setminus \{\alpha_{1},\alpha_{2}\}$. We can color $f_{1}$ and $f_{2}$ to get a new good partial coloring $\psi$ of $G$ so that $\{\psi(f_{1}),\psi(f_{2})\}=\{\alpha_{1},\beta\}$. And then we extend $\psi$ to a good coloring of $G$ by greedily coloring $f_{3},f_{4},e_{2},e_{1}$ in this order. If $\alpha_{1}\notin A_{\phi}(f_{2})$, then there exists a color $\beta\in A_{\phi}(f_{2}) \setminus \{\alpha_{1},\alpha_{2}\}$. A good coloring of $G$ can be obtained by coloring $f_{1}$ with $\alpha_{1}$, $f_{2}$ with $\beta$ and then coloring the remaining edges in the order $f_{3},f_{4},e_{2},e_{1}$.
Now we assume $A_{\phi}(f_{1})\cup A_{\phi}(f_{2}) = \{\alpha_{1},\alpha_{2}\}$.
Recall that $A_{\phi}(f_{3})\cup A_{\phi}(f_{4}) \subseteq A_{\phi}(e_{2})$,
we must have $A_{\phi}(f_{3})= A_{\phi}(f_{4}) = A_{\phi}(e_{2})\setminus \{\alpha_{1},\alpha_{2}\}$.
Notice that $g_{1}$ sees a color $\alpha$ in $\phi$ if and only if $f_{1}$ or $f_{2}$ also sees $\alpha$ in $\phi$, we can recolor $g_{1}$ with $\alpha_{1}$ to obtain a new good partial coloring of $G$, which we refer to it as $\psi$. Observe that $A_{\psi}(f_{1})= A_{\psi}(f_{2}) = \{\phi(g_{1}),\alpha_{2}\}$, $A_{\psi}(e_{1})=(A_{\phi}(e_{1})\setminus \{\alpha_{1}\}) \cup \{\phi(g_{1})\}$, $A_{\psi}(f_{3})= A_{\psi}(f_{4}) =A_{\phi}(f_{3})$ and $A_{\psi}(e_{2})= A_{\phi}(e_{2})$.
Now, we can get a good coloring of $G$ by coloring $f_{1}$ with $\phi(g_{1})$, $f_{2}$ with $\alpha_{2}$, $f_{3}$ and $f_{4}$ with the two colors in $A_{\phi}(e_{2})\setminus \{\alpha_{1},\alpha_{2}\}$, $e_{2}$ with $\alpha_{1}$ and $e_{1}$ with a color in $A_{\phi}(e_{1})\setminus \{\alpha_{1},\alpha_{2}\}$.
{\bf Subcase 2.3}. $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=3$. ($|A_{\phi}(e_{1})\cup A_{\phi}(e_{2})|=5$.)
Let $A_{\phi}(e_{1})\cap A_{\phi}(e_{2})=\{\alpha_{1},\alpha_{2},\alpha_{3}\}$, $A_{\phi}(e_{1})=\{\alpha_{1},\alpha_{2},\alpha_{3},\beta\}$ and $A_{\phi}(e_{2})=\{\alpha_{1},\alpha_{2},\alpha_{3},\gamma\}$.
Recall that $A_{\phi}(f_{1})\cup A_{\phi}(f_{2})\subseteq A_{\phi}(e_{1})$ and
$A_{\phi}(f_{3})\cup A_{\phi}(f_{4})\subseteq A_{\phi}(e_{2})$, as $|A_{\phi}(f_{j})|\ge 2$, we must have
$A_{\phi}(f_{j})\cap\{\alpha_{1},\alpha_{2},\alpha_{3}\}\neq \emptyset$ for each $j=1,2,3,4$.
We have two subcases to deal with.
{\bf Subcase 2.3.1}. $(A_{\phi}(f_{1})\cup A_{\phi}(f_{2}))\cap (A_{\phi}(f_{3})\cup A_{\phi}(f_{4}))\neq \emptyset$.
W.l.o.g., let $\alpha_{1} \in A_{\phi}(f_{1}) \cap A_{\phi}(f_{3})$.
By assigning the color $\alpha_{1}$ to $f_{1}$ and $f_{3}$ and coloring $f_{2}$ (resp. $f_{4}$) with a color in $A_{\phi}(f_{2})\setminus \{\alpha_1\}$ (resp. $A_{\phi}(f_{4})\setminus \{\alpha_1\}$), we extend $\phi$ to a new good partial coloring $\psi$. Observe that $|A_{\psi}(e_{1})|\ge 1$, $|A_{\psi}(e_{2})|\ge 1$
and $|A_{\psi}(e_{1})\cup A_{\psi}(e_{2})|=|(A_{\phi}(e_{1})\cup A_{\phi}(e_{2}))\setminus\{\alpha_{1},\psi(f_{2}),\psi(f_{4})\}|\ge 2$. Therefore, we can further extend $\psi$ to a good coloring of $G$ by SDR.
{\bf Subcase 2.3.2}. $(A_{\phi}(f_{1})\cup A_{\phi}(f_{2}))\cap (A_{\phi}(f_{3})\cup A_{\phi}(f_{4}))= \emptyset$.
In this subcase, it is clear that $|A_{\phi}(f_{1})\cup A_{\phi}(f_{2})|=2$ or $|A_{\phi}(f_{3})\cup A_{\phi}(f_{4})|=2$.
W.l.o.g., we assume that $|A_{\phi}(f_{1})\cup A_{\phi}(f_{2})|=2$ (i.e., $A_{\phi}(f_{1})= A_{\phi}(f_{2})$), $\alpha_{1}\in A_{\phi}(f_{1})\cup A_{\phi}(f_{2})$ and $\alpha_{3}\in A_{\phi}(f_{3})\cup A_{\phi}(f_{4})$.
If $A_{\phi}(f_{1})= A_{\phi}(f_{2})= \{\alpha_{1},\alpha_{2}\}$, then $A_{\phi}(f_{3})= A_{\phi}(f_{4})= \{\alpha_{3},\gamma\}$.
Recall that $g_{1}$ sees a color $\alpha$ in $\phi$ if and only if $f_{1}$ or $f_{2}$ also sees $\alpha$ in $\phi$.
Thus, we can always modify $\phi$ by recoloring $g_{1}$ with $\alpha_{1}$ to obtain a new good partial coloring $\psi$ of $G$, in which $A_{\psi}(f_{3})=A_{\psi}(f_{4})=\{\alpha_{3},\gamma\}$ and
$A_{\psi}(e_{2})=\{\alpha_{1},\alpha_{2},\alpha_{3},\gamma\}$.
Now, if $\phi(g_{1})=\gamma$, we have
$A_{\psi}(f_{1})=A_{\psi}(f_{2})=\{\alpha_{2},\gamma\}$ and
$A_{\psi}(e_{1})=\{\alpha_{2},\alpha_{3},\beta,\gamma\}$, and we are back to Subcase 2.3.1.
While if $\phi(g_{1})\neq\gamma$,
we have
$A_{\psi}(f_{1})=A_{\psi}(f_{2})=\{\alpha_{2},\phi(g_{1})\}$ and
$A_{\psi}(e_{1})=\{\alpha_{2},\alpha_{3},\beta,\phi(g_{1})\}$, implying $|A_{\psi}(e_{1})\cap A_{\psi}(e_{2})|=2$. And we are back to Subcase 2.2.
If $ A_{\phi}(f_{1})= A_{\phi}(f_{2})=\{\alpha_{1},\beta\}$, by symmetry,
we may assume that $A_{\phi}(f_{3})\cup A_{\phi}(f_{4})\neq \{\alpha_{2},\alpha_{3}\}$
(i.e., $\{\alpha_{3},\gamma\} \subseteq A_{\phi}(f_{3})\cup A_{\phi}(f_{4})$).
Also, we can recolor $g_{1}$ with $\alpha_{1}$ to get a new good partial coloring $\psi$ of $G$, in which $\{\alpha_{3},\gamma\}\subseteq A_{\psi}(f_{3})\cup A_{\psi}(f_{4})$ and
$A_{\psi}(e_{2})=\{\alpha_{1},\alpha_{2},\alpha_{3},\gamma\}$.
Now, if $\phi(g_{1})=\gamma$,
we have $A_{\psi}(f_{1})=A_{\psi}(f_{2})=\{\beta,\gamma\}$ and
$A_{\psi}(e_{1})=\{\alpha_{2},\alpha_{3},\gamma,\beta\}$, and we are back to Subcase 2.3.1.
And if $\phi(g_{1})\neq\gamma$,
then we have
$A_{\psi}(f_{1})=A_{\psi}(f_{2})=\{\beta,\phi(g_{1})\}$ and
$A_{\psi}(e_{1})=\{\alpha_{2},\alpha_{3},\beta,\phi(g_{1})\}$.
As $|A_{\psi}(e_{1})\cap A_{\psi}(e_{2})|=2$, we are back to Subcase 2.2.
{\bf Subcase 2.4}. $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=4$. ($|A_{\phi}(e_{1})\cup A_{\phi}(e_{2})|=4$.)
Let $A_{\phi}(e_{1})= A_{\phi}(e_{2})=\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\}$.
Erasing the colors of $g_{1}$ and $g_{2}$ in $\phi$ yields another good partial coloring $\psi$ of $G$.
If $|A_{\psi}(g_{1})|= |A_{\psi}(g_{2})|=1$, then it is straightforward to check that $A_{\phi}(f_{1})\cap A_{\phi}(f_{2})=\emptyset$, $A_{\phi}(f_{1})\cup A_{\phi}(f_{2})=A_{\phi}(e_{1})$, $A_{\phi}(f_{3})\cap A_{\phi}(f_{4})=\emptyset$ and $A_{\phi}(f_{3})\cup A_{\phi}(f_{4})=A_{\phi}(e_{2})$.
Now based on $\phi$, we can always color $f_{1},f_{2},f_{3},f_{4}$ to obtain a new good partial coloring $\sigma$ with $\{\sigma(f_{1}),\sigma(f_{2})\}=\{\sigma(f_{3}),\sigma(f_{4})\}$. Consequently, $e_{1}$ and $e_{2}$ can be colored properly.
Now, w.l.o.g., we assume that $|A_{\psi}(g_{1})| \ge 2$. In this situation, it is not difficult to verify that
$A_{\psi}(g_{1})\cap \{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\}\neq \emptyset$.
W.l.o.g., let $\alpha_{1}\in A_{\psi}(g_{1})$.
By recoloring the edge $g_1$ with the color $\alpha_1$ in $\phi$, we obtain a new good partial coloring $\sigma$ of $G$, in which $A_{\sigma}(e_{1})=\{\phi(g_{1}),\alpha_{2},\alpha_{3},\alpha_{4}\}$ and $A_{\sigma}(e_{2})=\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\}$.
Obviously, $|A_{\sigma}(e_{1}) \cap A_{\sigma}(e_{2})|=3$ and so we are back to Subcase 2.3.
As we have exhausted all cases, the lemma follows.
\end{pf}
\begin{lemma}\label{lemma:cut vertex}
Let $G$ be a connected claw-free cubic graph. If $G$ has a cut vertex, then $\chi'_{s}(G)\leq 7$.
\end{lemma}
\begin{pf}
Let $v_0$ be a cut vertex of $G$ and $N(v_{0})=\{u_{0},v_{1},v_{2}\}$.
As $G$ is a claw-free cubic graph, there is exactly one edge in $G$ among the vertices in $N(v_{0})$.
Without loss of generality, let $v_{1}v_{2}\in E(G)$.
Let $G_{1}$ be the component of $G-v_{0}$ containing $u_{0}$ and $G_{2}$ the component of $G-v_{0}$ containing $v_{1}v_{2}$. Please see Figure \ref{fig:cut vertex} for the names of vertices and edges in $G$.
\begin{figure}
\caption{A graph $G$ with a cut vertex $v_{0}
\label{fig:cut vertex}
\end{figure}
By Lemma \ref{lemma:degree-2}, $G_{1}$ (resp. $G_{2}$) has a good coloring, say $\phi_{1}$ (resp. $\phi_{2}$).
We may assume that $\{\phi_{1}(f_{1}),\phi_{1}(f_{2}),\phi_{1}(f_{3})\}=\{\phi_{2}(h_{1}),\phi_{2}(h_{2}),\phi_{2}(h_{3})\}$ as otherwise we can permute the colors among the edges of $G_{1}$.
Combining $\phi_{1}$ and $\phi_{2}$ yields a good partial coloring of $G$, calling it $\phi$.
It is clear that $|A_{\phi}(e_{0})|=2$, $|A_{\phi}(e_{1})|\ge 2$ and $|A_{\phi}(e_{2})|\ge 2$.
If $|A_{\phi}(e_{0})\cup A_{\phi}(e_{1})\cup A_{\phi}(e_{2})|\ge 3$, then we can obtain a good coloring of $G$ by SDR.
Thus we assume that
$A_{\phi}(e_{0})= A_{\phi}(e_{1})= A_{\phi}(e_{2})=\{\alpha_{1},\alpha_{2}\}$. Let $\beta=\phi(f_{4})$.
Permute the two colors $\alpha_{1}$ and $\beta$ in $E(G_{1})$.
After the permutation, we have $A_{\phi}(e_{0})=\{\alpha_{2},\beta\}$ and $A_{\phi}(e_{1})= A_{\phi}(e_{2})=\{\alpha_{1},\alpha_{2}\}$.
Since $|A_{\phi}(e_{0})\cup A_{\phi}(e_{1})\cup A_{\phi}(e_{2})|\ge 3$, we complete the proof by SDR.
\end{pf}
\begin{lemma}\label{lemma:$4$-cycle with a chord}
Let $G$ be a $2$-connected claw-free cubic graph.
If $G$ contains a $4$-cycle with a chord, then $\chi'_{s}(G)\le 7$.
\end{lemma}
\begin{pf}
Let $C=v_{1}v_{2}v_{3}v_{4}v_{1}$ be a $4$-cycle in $G$ with $v_{1}v_{3}\in E(G)$.
Because $G$ is not isomorphic to $K_{4}$, $v_{2}v_{4}\notin E(G)$.
Let $u_{2}$ and $u_{4}$ denote the third neighbor of $v_{2}$ and $v_{4}$, respectively.
Since $G$ is a $2$-connected claw-free cubic graph, it is not difficult to see that $u_{2}\neq u_{4}$ and $u_{2}u_{4}\notin E(G)$.
If $N(u_{2})\cap N(u_{4})\neq \emptyset$, then $G$ is isomorphic to $H_{4}$ (as shown in Figure \ref{fig:H_{4}}) and so $\chi'_{s}(G)\le 7$.
Thus we assume that $N(u_{2})\cap N(u_{4})= \emptyset$. Please see Figure \ref{fig:H_{4}} for the the names of vertices and edges in $G$.
\begin{figure}
\caption{The graph $H_{4}
\label{fig:H_{4}
\end{figure}
Set $S=\{v_{1},v_{2},v_{3},v_{4}\}$. By Lemma \ref{greedy}, we get a good partial coloring $\phi$ of $G$, in which
$|A_{\phi}(e_{i})|=5$ for $i\in [1,4]$, $|A_{\phi}(e_{5})|=7$ and $|A_{\phi}(f_{1})|=|A_{\phi}(f_{2})|=2$.
If $|A_{\phi}(e_{1})\cup A_{\phi}(e_{3})|\ge 6$, then a good coloring of $G$ can be easily obtained by SDR.
Thus we assume that $|A_{\phi}(e_{1})\cup A_{\phi}(e_{3})|=5$.
Observe that $A_{\phi}(e_{1})=A_{\phi}(e_{2})$ and $A_{\phi}(e_{3})=A_{\phi}(e_{4})$,
we may assume that $A_{\phi}(e_{1})=A_{\phi}(e_{2})=A_{\phi}(e_{3}) =A_{\phi}(e_{4})=[1,5]$.
And so $\{\phi(g_{1}),\phi(g_{2})\}=\{\phi(g_{3}),\phi(g_{4})\}=\{6,7\}$.
Erasing the colors of $g_{1}$ and $g_{2}$ in $\phi$ yields a good partial coloring of $G$, calling it $\psi$.
Now, if there exists some color $\alpha\in A_{\psi}(g_{1})\setminus \{6,7\}$,
we color $g_{1}$ with $\alpha$ and $g_{2}$ with $\phi(g_{2})$ to get a new good partial coloring $\sigma$ of $G$, in which $A_{\sigma}(e_{1})=([1,5]\setminus \{\alpha\})\cup \{\phi(g_{1})\}$ and $ A_{\phi}(e_{3})=[1,5]$.
Clearly, $|A_{\sigma}(e_{1})\cup A_{\sigma}(e_{3})|\ge 6$.
So we can extend $\sigma$ to a good coloring of $G$ by SDR.
Thus we assume $A_{\psi}(g_{1})= \{6,7\}$.
Symmetrically, we may also assume $A_{\psi}(g_{2})=\{6,7\}$.
Notice that $h_{1}$ sees a color $\alpha$ in $\psi$ if and only if $g_{1}$ or $g_{2}$ also sees $\alpha$ in $\psi$,
we can recolor $h_{1}$ with $\phi(g_{1})$ in $\psi$. By further coloring $g_{1}$ with $\phi(h_{1})$ and $g_{2}$ with $\phi(g_{2})$, we get a new good partial coloring $\sigma$ of $G$ with $A_{\sigma}(e_{1})=([1,5]\setminus \{\phi(h_{1})\})\cup \{\phi(g_{1})\}$ and $ A_{\phi}(e_{3})=[1,5]$. Again we have $|A_{\sigma}(e_{1})\cup A_{\sigma}(e_{3})|\ge 6$ and so complete the proof by SDR.
\end{pf}
\begin{lemma}\label{lemma:induced 4-cycle}
Let $G$ be a $2$-connected claw-free cubic graph.
If $G$ contains an induced $4$-cycle, then $\chi'_{s}(G)\le 7$.
\end{lemma}
\begin{pf}
Let $C=v_{1}v_{2}v_{3}v_{4}v_{1}$ be an induced $4$-cycle in $G$.
As $G$ is a claw-free cubic graph not isomorphic to the $3$-prism, $u_{1}u_{2}\notin E(G)$.
Let $N(u_{1})=\{v_{1},v_{2},w_{1}\}$ and
$N(u_{2})=\{v_{3},v_{4},w_{2}\}$ (Please refer to Figure \ref{fig:induced 4cycle}).
Since $G$ is $2$-connected, $w_{1}w_{2}\notin E(G)$.
\begin{figure}
\caption{A graph $G$ with an induced $4$-cycle $C=v_{1}
\label{fig:induced 4cycle}
\end{figure}
Setting $S=\{v_{1},v_{2},v_{3},v_{4}\}$, by Lemma \ref{greedy}, we get a good partial coloring $\phi$ of $G$,
in which $|A_{\phi}(e_{i})|=6$ for $i=1,3$,
$|A_{\phi}(e_{i})|\ge 5$ for $i=2,4$ and
$|A_{\phi}(f_{j})|=4$ for $j=1,2,3,4$.
Observe that there exists some color $\alpha \in A_{\phi}(f_{1})\cap A_{\phi}(f_{3})$,
by coloring $f_{1}$ and $f_{3}$ with $\alpha$, we extend $\phi$ to another good partial coloring $\psi$ of $G$,
in which $|A_{\psi}(f_{2})|=|A_{\psi}(f_{4})|=3$,
$|A_{\psi}(e_{1})|=|A_{\psi}(e_{3})|=5$ and
$|A_{\psi}(e_{2})|=|A_{\psi}(e_{4})|\ge 4$.
Now, if there exists some $\beta \in A_{\psi}(f_{2})\cap A_{\psi}(f_{4})$,
then we can color $f_{2}$ and $f_{4}$ with $\beta$ and greedily color $e_{2},e_{4},e_{1},e_{3}$ in this order to
get a good coloring of $G$.
Otherwise, we have
$|A_{\psi}(f_{2})\cup A_{\psi}(f_{4})|=6$,
and so the remainning six egdes can be colored properly by SDR. Thus the lemma holds.
\end{pf}
At present time, the only case we need to deal with is that $G$ is a $2$-connected claw-free cubic graph without $4$-cycles. It is clear that each vertex is exactly on one $3$-cycle. Let $\Delta_{1}$ and $\Delta_{2}$ be two $3$-cycles in $G$. Then $\Delta_{1}$ and $\Delta_{2}$ are vertex-disjoint and there is at most one edge from $V(\Delta_{1})$ to $V(\Delta_{2})$. Moreover, it is not difficult to see that $G$ has no induced odd cycles of length at least $5$ and each induced even cycle has length at least $6$.
\begin{lemma}\label{lemma:induced 2p-cycle}
If $G$ is a $2$-connected claw-free cubic graph without $4$-cycles, then $\chi'_{s}(G)\le 7$.
\end{lemma}
\begin{pf}
Choose a minimum induced even cycle $C$ in $G$ and label the vertices of $C$ as
$v_{1},v_{2},\ldots,v_{2p}$ and the edges $e_{i}=v_{i-1}v_{i}$ for $i=2,3,\ldots,2p$ and $e_{1}=v_{2p}v_{1}$.
Clearly, $2p\ge6$.
As $G$ is a claw-free cubic graph and has no $4$-cycles, we may assume that $v_{2j-1}$ and $v_{2j}$ have the common neighbor $u_{2j}$ for each $j\in[1,p]$ and $u_{2},u_{4},\ldots,u_{2p}$ are distinct.
Since $C$ is a minimum induced even cycle in $G$, $u_{2i}u_{2j}\notin E(G)$ for any $i,j\in[1,p]$.
Let $w_{2j}$ denote the third neighbor of $u_{2j}$ for each $j\in[1,p]$.
Also, $w_{2},w_{4},\ldots,w_{2p}$ are distinct because $G$ is claw-free and cubic.
As for the names of vertices and edges in $G$, please refer to Figure \ref{fig:induced 6cycle}.
\begin{figure}
\caption{A graph $G$ with an induced $2p$-cycle $(p\ge3)$}
\label{fig:induced 6cycle}
\end{figure}
Notice that if $2p\ge 8$, then $w_{2j}w_{2j+2}\notin E(G)$ for any $j\in[1,p-1]$ and $w_{2}w_{2p}\notin E(G)$.
And if $2p=6$, then there is at most one edge in the subgraph of $G$ induced by $\{w_{2},w_{4},w_{6}\}$ since $G$ is not isomorphic to $K_{4}^{\Delta}$.
So in either cases, we may assume that $w_{2}w_{4}\notin E(G)$. This implies that the four edges $h_{1},h_{2},h_{3},h_{4}$ are distinct.
Let $G'$ be the graph obtained from $G$ by deleting $v_{1},v_{2},\ldots,v_{2p},u_{2},u_{4},u_{6}$
and adding a new edge $w_{2}w_{4}$.
Observe that $G'$ is a claw-free subcubic graph and each component of $G'$ has a vertex of degree less than $3$, by Lemmas \ref{lemma:degree-1} and \ref{lemma:degree-2}, $G'$ has a good coloring $\phi$.
Ignoring $w_{2}w_{4}$ in $\phi$ yields a good partial coloring of $G$.
Let $\alpha=\phi(w_{2}w_{4})$. And let $c_{i}=\phi(h_{i})$ for each $i\in[1,2p]$.
Then $\alpha,c_{1},c_{2},c_{3},c_{4}$ are distinct.
As $|A_{\phi}(g_{6})|\ge2$, there exists a color $\beta\in A_{\phi}(g_{6})\setminus\{\alpha\}$.
Without loss of generality, assume that $c_{3}\neq \beta$.
Now based on $\phi$, we color $g_{2},g_{4}$ and $e_{6}$ with the same color $\alpha$,
color $g_{6}$ with $\beta$ and $e_{5}$ with $c_{3}$.
Observe that $f_{5}$ sees at most $5$ colors at this moment,
we can color $f_{5}$ with some color $\gamma\neq c_{4}$.
This yields a new coloring $\psi$, which is indeed a good partial coloring of $G$.
Please refer to Figure \ref{fig:Tn1} and Figure \ref{fig:Tn2} for this coloring.
\begin{figure}
\caption{$\psi$, $\bar{G}
\label{fig:Tn1}
\end{figure}
\begin{figure}
\caption{$\psi$, $\bar{G}
\label{fig:Tn2}
\end{figure}
In the following, we will construct a good coloring $\sigma$ of $\bar{G}_{\psi}$
so that $\sigma(e)\in A_{\psi}(e)$ for every $e\in\bar{E}_{\psi}$.
Then, this coloring $\sigma$ combining with the coloring $\psi$ forms a good coloring of $G$, and we are done.
If $2p\ge 8$, then it is straightforward to check that
$|A_{\psi}(e_{1})|\ge 5$, $|A_{\psi}(e_{2})|=6$, $|A_{\psi}(e_{3})|=5$,
$|A_{\psi}(e_{4})|=4$, $|A_{\psi}(e_{7})|\ge 2$, $|A_{\psi}(e_{i})|\ge 5$ for each $i\in \{8,9,\dots,2p\}$,
$|A_{\psi}(f_{4})|=3$, $|A_{\psi}(f_{6})|\ge 1$, $|A_{\psi}(f_{7})|\ge 3$,
and $|A_{\psi}(f_{i})|=4$ for each $i\in \{1,2,3,8,9,\dots,2p\}$. (Please refer to Figure \ref{fig:Tn1}).
And if $2p=6$, it is clear that
$|A_{\psi}(e_{1})|=3$, $|A_{\psi}(e_{2})|=6$, $|A_{\psi}(e_{3})|=5$, $|A_{\psi}(e_{4})|=4$,
$|A_{\psi}(f_{4})|=3$, $|A_{\psi}(f_{6})|\ge 1$, and $|A_{\psi}(f_{i})|=4$ for each $i\in \{1,2,3\}$.
(Please refer to Figure \ref{fig:Tn2}).
Before we go further to the next step, we make two observations which are simple but helpful in the following arguments (please refer to Figure \ref {fig:Property} for the illustrations of Observation A and Observation B).\\
\noindent{\bf Observation A}: $A_{\psi}(f_1)=A_{\psi}(f_2)=[1,7]\setminus \{\alpha,c_1,c_2\}\subseteq A_{\psi}(e_2) = [1,7]\setminus \{\alpha\}$.\\
\noindent{\bf Observation B}: $A_{\psi}(e_4)=A_{\psi}(f_4)\cup \{c_4\}$, $A_{\psi}(f_3)=A_{\psi}(f_4)\cup \{\gamma\}$, and $A_{\psi}(e_3)=A_{\psi}(f_4)\cup \{c_4,\gamma\}$.
\begin{figure}
\caption{Observetion A and Observetion B}
\label{fig:Property}
\end{figure}
Now, if $2p\ge 8$, we color $f_{6},e_{7},f_{7},e_{8},f_{8} \ldots,e_{2p},f_{2p},e_{1},f_{1},e_{2},f_{2},e_{3},f_{3},f_{4}$ in this order greedily
to get a good partial coloring $\sigma$ of $\bar{G}_{\psi}$, where only $e_{4}$ is uncolored.
And if $2p=6$, we color $f_{6},e_{1},f_{1},e_{2},f_{2},e_{3},f_{3},f_{4}$ in this order greedily
to get a good partial coloring $\sigma$ of $\bar{G}_{\psi}$, with only $e_{4}$ being uncolored.
For simplicity, let $a_{i}=\sigma(e_{i})$ and $b_{j}=\sigma(f_{j})$ for edges $e_{i},f_{j}\in \bar{E}_{\psi}$ that are colored in $\sigma$ (please refer to Figure \ref{fig:Tn1_color0}).
\begin{figure}
\caption{the good partial coloring $\sigma$ of $\bar{G}
\label{fig:Tn1_color0}
\end{figure}
If $A_{\psi}(e_4)\setminus \{a_2,b_2,a_3,b_3,b_4\}\not=\emptyset$, then the edge $e_4$ can be colored properly and we are done. We therefore assume that $A_{\psi}(e_4)\subseteq \{a_2,b_2,a_3,b_3,b_4\}$.
If $\gamma\not\in \{a_2,b_2,a_3\}$, then we may assume that $b_3=\gamma$ as otherwise we can recolor the edge $f_3$ with the color $\gamma$ to make $b_3=\gamma$. By Observation B, $\gamma\not\in A_{\psi}(f_4)$ and $\gamma\not\in A_{\psi}(e_4)$. It follows that there exists a color $b_4^*$ in $A_{\psi}(f_4)\setminus \{a_3,b_3,b_4\}$ and $A_{\psi}(e_4)=\{a_2,b_2,a_3,b_4\}$. Now, we can recolor $f_4$ with $b_4^*$ and then color $e_4$ with $b_4$. This results in a good coloring of $\bar{G}_{\psi}$. Therefore we assume that $\gamma\in \{a_2,b_2,a_3\}$.
It follows from the assumption $\gamma\in \{a_2,b_2,a_3\}$ that $b_3\not=\gamma$.
Since $b_3\in A_{\psi}(f_3)$, by Observation B, $b_3\in A_{\psi}(f_4)$.
Therefore we must have $\{b_3,b_4\}\subset A_{\psi}(f_4)$.
The remainder of the proof is divided into two cases according to
whether $a_3$ is in $A_{\psi}(f_4)$ or not.
{\bf Case 1.} $a_3\not\in A_{\psi}(f_4)$.
Since $a_3\in A_{\psi}(e_3)$ and, by Observation B, $A_{\psi}(e_3)=A_{\psi}(f_4)\cup \{c_4,\gamma\}$, it is clear that $a_3\in \{c_4,\gamma\}$. Let $A_{\psi}(f_4)=\{b_3,b_4, \alpha^*\}$. Then $A_{\psi}(e_4)=\{c_4,b_3,b_4, \alpha^*\}$. If $c_4$ (or $\alpha^*$) is not contained in $\{a_2,b_2,a_3\}$, then we can color the edge $e_4$ with $c_4$ (or $\alpha^*$) and obtain a good coloring of $\bar{G}_{\psi}$.
Thus we assume that $\{c_4,\alpha^*\} \subset \{a_2,b_2,a_3\}$. This together with $\gamma\in \{a_2,b_2,a_3\}$ implies that $\{a_2,b_2,a_3\}=\{c_4,\alpha^*,\gamma\}$. As $a_3\in\{c_4,\gamma\}$, $\alpha^*$ must be in $\{a_2,b_2\}$. Now we can recolor the edge $f_4$ with $\alpha^*$ and then color $e_4$ with $b_4$. This completes the coloring of $\bar{G}_{\psi}$.
{\bf Case 2.} $a_3\in A_{\psi}(f_4)$. (So $A_{\psi}(f_4)=\{a_3,b_3,b_4\}$.)
In this case, by Observation B, $A_{\psi}(e_4)=A_{\psi}(f_4)\cup \{c_4\}=\{c_4,a_3,b_3,b_4\}$ and $a_3\not=\gamma$. Since $A_{\psi}(e_4)\subseteq \{a_2,b_2,a_3,b_3,b_4\}$, $c_4$ must be in $\{a_2,b_2\}$.
As $\gamma \in \{a_2,b_2,a_3\}$ and $a_3\not=\gamma$, $\gamma\in\{a_2,b_2\}$.
Therefore, $\{a_2,b_2\}=\{c_4,\gamma\}$.
Now we erase the colors of $f_{3}$ and $f_{4}$ in $\sigma$ to get a new good partial coloring of $\bar{G}_{\psi}$, where the only uncolored edges are $f_{3}$, $f_{4}$ and $e_{4}$.
This coloring is still denoted by $\sigma$.
We next extend $\sigma$ to a good coloring of $\bar{G}_{\psi}$.
If $a_{2}\in A_{\psi}(f_{2})$, then, by Observation A, we also have $a_{2}\in A_{\psi}(f_{1})$.
Notice that $b_{1},a_{2}\notin \{a_{2p},b_{2p},a_{1},b_{2},a_{3}\}$, we exchange the colors of $f_{1}$ and $e_{2}$ in $ \sigma$.
If $a_{2}=c_{4}$ and $b_{2}=\gamma$ (so $b_{1}\neq c_{4}$), we color $e_{4}$ with $c_{4}$.
It follows that there is at least one color available for $f_{3}$ and
there are at least two colors available for $f_{4}$.
Hence we can color $f_{3}$ and $f_{4}$ properly.
If $a_{2}=\gamma$ and $b_{2}=c_{4}$ (so $b_{1}\neq \gamma$), we color $f_{3}$ with $\gamma$.
Then $e_{4}$ has at least one available color and $f_{4}$ has at least two available colors.
Therefore, coloring $e_{4}$ and then $f_{4}$ gives a good coloring of $\bar{G}_{\psi}$.
If $a_{2}\notin A_{\psi}(f_{2})$,
then there exists some color $b_{2}^{*}$ in $A_{\psi}(f_{2})\setminus \{a_{1},b_{1},b_{2}\}$.
We recolor $f_{2}$ with $b_{2}^{*}$.
Now, if $a_{2}=c_{4}$ and $b_{2}=\gamma$, we color $f_{3}$ with $\gamma$.
Recall that $\gamma\not\in A_{\psi}(e_4)$,
there is at least one color available for $e_{4}$ and there are at least two colors available for $f_{4}$.
Thus $e_4$ and $f_4$ can be colored properly.
If $a_{2}=\gamma$ and $b_{2}=c_{4}$, we recolor $e_{3}$ with $c_{4}$.
Then both $e_{4}$ and $f_{3}$ have at least two available colors and $f_{4}$ has three available colors.
It follows that we can obtain a good coloring of $\bar{G}_{\psi}$ by SDR.
\end{pf}
Combining all lemmas in this section, Theorem \ref{main} holds.
\section{Final Remarks}
Let $G$ be any connected claw-free subcubic graph not isomorphic to the triangular prism.
This paper proves that $\chi'_{s}(G)\leq 7$. And this upper bound $7$ is sharp.
In addition, our proof implies a linear-time
algorithm for finding a strong $7$-edge-coloring for such a graph.
If $G$ is a connected claw-free cubic graph not isomorphic to the triangular prism, then it is easy to see that $\chi'_{s}(G)\geq 6$. Therefore, for such graph $G$, $\chi'_{s}(G)\in\{6,7\}$. Let $H$ be a connected cubic graph. Denote by $H^{\Delta}$ the graph obtained from $H$ by replacing each vertex with a $3$-cycle. It is clear that $H^{\Delta}$ is a connected claw-free cubic graph.
We end this paper by asking the following three questions.\\
\noindent {\bf Question 1}: Let $k\ge 3$ be an integer and $H$ the $k$-prism. Is it true that $\chi'_{s}(H^{\Delta})=6$?\\
\noindent {\bf Question 2}: Is it possible to characterize all connected claw-free cubic graphs $G$ with $\chi'_{s}(G)=6$?\\
\noindent {\bf Question 3}: Is the strong list-chromatic index of any claw-free subcubic graph other than the triangular prism at most 7?\\
\noindent{\bf Declaration of competing interest}
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
\end{document} |
\begin{document}
\title{\textbf{Two spectral extremal results for graphs with given order and rank}}
\author{
\small Xiuqing Li,\ \ Xian'an Jin\footnote{Corresponding author},\ \ Chao Shi,\ \ Ruiling Zheng\\[0.2cm]
\small School of Mathematical Sciences, Xiamen University,\\
\small Xiamen, Fujian 361005, P. R. China\\[0.2cm]
\small E-mails: [email protected]; [email protected];\\
\quad \quad \ [email protected]; [email protected]}
\date{}
\begin{abstract}
The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum or minimum spectral radius over certain families of graphs. Monsalve and Rada [Extremal spectral radius of graphs with rank 4, Linear Algebra Appl. 609 (2021) 1–11] obtained the extremal graphs with maximum and minimum spectral radii among all graphs with order $n$ and rank $4$. In this paper, we first determine the extremal graph which attains the maximum spectral radius among all graphs with any given order $n$ and rank $r$, and further determine the extremal graph which attains the minimum spectral radius among all graphs with order $n$ and rank $5$.
\vskip0.2cm
\noindent{\bf Keywords:} Rank of graphs; Extremal graphs; Maximum spectral radius; Minimum spectral radius
\vskip0.2cm
\end{abstract}
\maketitle
\section{Introduction}
Graphs considered in the paper are all simple, connected and undirected. Let $G=(V(G),E(G))$ be a graph. For $v \in V(G)$, the degree $d(v)$ is the cardinality of the neighborhood $N_{G}(v)$ (or $N(v)$ for short) of $v$ in $G$. Let $A(G)$ be the adjacency matrix of $G$. The characteristic polynomial of a graph $G$ is the determinantal expansion of $xI-A(G)$, denoted by $\phi(G,x)$. According to the famous Perron-Frobenius theorem, the largest eigenvalue $\rho(G)$ of $A(G)$ is exactly the spectral radius of $G$ and there is a unique positive unit eigenvector corresponding to $\rho(G)$, called the principal eigenvector of $G$.
Let $G$ be a graph with vertex set $V(G)=\{v_{1},v_{2},\dots,v_{k}\}$ and $\textbf{m}=(n_{1},n_{2},\dots,n_{k})$ be a vector of positive integers. Denote by $G \circ \textbf{m}$, the graph obtained from $G$ by replacing each vertex $v_{i}$ with an independent set $V_{i}$ with $n_{i}$ vertices $v_{i}^{1}, v_{i}^{2}, \dots, v_{i}^{n_{i}}$ and joining each vertex in $V_{i}$ with each vertex in $V_{j}$ if and only if $v_{i}v_{j} \in E(G)$. The resulting graph $G \circ \textbf{m}$ is said to be obtained from $G$ by multiplication of vertices by Chang, Huang and Yeh in \cite{01}. Further, let $G$ be a graph of order $k$, we define $M_{n}(G)$ to be the set of all graphs $G \circ (n_{1},n_{2},\dots,n_{k})$ with $\sum_{i=1}^{k}n_{i}=n$. Moreover, for a given set of graphs $\{H_{1},\dots,H_{l}\}$, we denote the set $\bigcup_{i=1}^{l}M_{n}(H_{i})$ by $M_{n}(H_{1},\dots,H_{l})$.
Let $G$ be a connected graph of order $n$ and $R(G)$ be its rank. Sciriha \cite{04} proved that $R(G)=i$ if and only if $G \in M_{n}(K_{i})$ for $i=2,3$, where $K_i$ is the complete graph of order $i$. Chang, Huang and Yeh \cite{01,05} characterized the set of all connected graphs with rank 4 and 5, respectively. They obtained the set of connected graphs of order $n$ and rank 5 is $$M_{n}(G_{1}, G_{2}, \dots, G_{24}),$$
where the graphs $G_{1}, G_{2}, \dots, G_{24}$ are shown in Figure \ref{Fig.1.1}.
\begin{figure}
\caption{Reduced graphs of rank 5.}
\label{Fig.1.1}
\end{figure}
For a given class of graphs $\mathscr{G}$, there are many results on characterizing the extramal graphs with maximum and minimum spectral radius among $M_{n}(\mathscr{G})$. For example, in \cite{06}, Stevanovi\'{c}, Gutman and Rehman determined the extremal graphs with the maximum and minimum spectral radii in $M_{n}(K_{p})$. Monsalve and Rada \cite{07} obtained the extremal graphs with maximum and minimum spectral radii among all connected graphs of order $n$ and rank $4$. In the same article, they conjectured that in $M_{n}(P_{k})$, $P_{k}\circ (1,\dots,1,\lfloor \frac{n-k+2}{2} \rfloor,\lceil \frac{n-k+2}{2} \rceil,1,\dots,1)$ and $P_{k}\circ (\lfloor \frac{n-k+2}{2} \rfloor,1,\dots,1,$ $\lceil \frac{n-k+2}{2} \rceil)$ attain the maximum and minimum spectral radius, respectively, and $C_{k}\circ (\lfloor \frac{n-k+2}{2} \rfloor,\lceil \frac{n-k+2}{2} \rceil,1,\dots,1)$ attains the maximum spectral radius in $M_{n}(C_{k})$. Recently, Lou, Zhai \cite{02} and Sun, Das \cite{03} independently proved the above conjectures on the extremal graphs with the maximum spectral radius in $M_{n}(P_{k})$ and $M_{n}(C_{k})$ by using different techniques, and they independently constructed a class of graphs disproving the conjecture on the minimum spectral radius in $M_{n}(P_{k})$.
The Tur\'{a}n graph $T(n,r)$ is the complete $r$-partite graph on $n$ vertices where its part sizes are as equal as possible. In this paper, we first determine the extremal graph that attains the maximum spectral radius with any given order and rank, and obtain:
\begin{theorem}\label{1}
$T(n,r)$ is the unique extremal graph that attains the maximum spectral radius among all graphs of order $n$ and rank $r$.
\end{theorem}
However, it seems that it is a difficult task to find the extremal graph that attains the minimum spectral radius with given order and rank. In this paper, we focus on graphs with order $n$ and rank $5$, and obtain:
\begin{theorem}\label{2}
The extremal graph that attains the minimum spectral radius among all connected graphs of order $n$ and rank 5 is:
\begin{itemize}
\item $G_{7}=C_{5}$, for $n=5$;
\item $G_{1} \circ(1,1,1,1,n-4)$, for $6\leq n \leq 10$;
\item $G_{10} \circ(1,1,1,1,1,n-5)$, for $n=11$;
\item $G_{10} \circ (1,1,1,1,k,n-k-4)$, where $k=\lfloor \frac{6n-37-\sqrt{24n+1}}{18} \rfloor$ or $\lceil \frac{6n-37-\sqrt{24n+1}}{18} \rceil$, for $n \geq 12$.
\end{itemize}
\end{theorem}
\section{The proof of Theorem \ref{1}}
We will use the following results to prove Theorem \ref{1}.
\begin{theorem}\cite{01} \label{2.1}
Suppose that $G$ and $H$ are two graphs. If $H \in M_{n}(G)$, then $R(H)=R(G)$.
\end{theorem}
\begin{theorem}\cite{08} \label{2.2}
Let $T(n,r)$ be the $r$-partite Tur\'{a}n graph of order n. If $G$ is a $K_{r+1}$-free graph of order $n$, then $\rho(G)<\rho(T(n,r))$ unless $G=T(n,r)$.
\end{theorem}
\begin{proof}[\textbf{Proof of Theorem \ref{1}}]
Let $G$ be a graph of order $n$ and rank $r$. We claim that $G$ is a $K_{r+1}$-free graph. Otherwise, since $K_{r+1}$ is a subgraph of $G$, selecting the rows and columns corresponding to the vertices in $K_{r+1}$ can obtain a nonzero minor of order $r+1$ of $A(G)$, i.e.,
\begin{align} \det \left(\begin{array}{cccc}
0 &1&\cdots &1 \\
1 &0&\cdots &1 \\
\vdots &\vdots &\ddots &\vdots \\
1&1 &\cdots &0 \\
\end{array}\right)_{(r+1)\times(r+1)}=(-1)^{r}\cdot r \neq 0.\notag
\end{align}
Therefore, we have $R(G)\geq r+1$, a contradiction. Since $T(n,r)=K_r \circ (\lceil \frac{n}{r} \rceil,\dots,\lceil \frac{n}{r} \rceil,\lfloor \frac{n}{r} \rfloor,\dots,\lfloor \frac{n}{r} \rfloor) \in M_{n}(K_r)$, by Theorem \ref{2.1}, we have $R(T(n,r))=R(K_r)=r$. By Theorem \ref{2.2}, we obtain $\rho(G)<\rho(T(n,r))$ unless $G=T(n,r)$.
\end{proof}
\section{The proof of Theorem \ref{2}}
In this section, we focus on the extremal graph that has the minimum spectral radius among all connected graphs of order $n$ and rank 5. We firstly outline our proof for Theorem \ref{2}.
{\bf Step 1.} We first apply a result of Monsalve and Rada in \cite{07} to prove that the extremal graph with minimum spectral radius belongs to $M_{n}(G_{1},G_{7}$, $G_{10})$.
{\bf Step 2.} Then, using the method of comparing characteristic polynomials, we characterize the extremal graph with minimum spectral radius in $M_{n}(G_{1})$, $M_{n}(G_{7})$ and $M_{n}(G_{10})$, respectively.
{\bf Step 3.} Next, for $n\geq 12$, we compare the spectral radii of these three types of extremal graphs by some well-known results and obtain that the extremal graph of order $n$ and rank $5$ with minimum spectral radius is $G_{10}\circ(1,1,1,1,k,n-4-k)$ for some integer $k$. Further, we determine $k \in \{ \lfloor \frac{6n-37-\sqrt{24n+1}}{18} \rfloor, \lceil \frac{6n-37-\sqrt{24n+1}}{18} \rceil \}$.
{\bf Step 4.} Finally, for $5\leq n\leq 11$, we obtain the extremal graphs by calculating directly the spectral radii of the extremal graphs in $M_{n}(G_{1})$, $M_{n}(G_{7})$ and $M_{n}(G_{10})$, respectively.
\subsection{Step 1}
We begin with recalling a well-known result.
\begin{theorem}\cite{09} \label{3.1}
If $H$ is a proper subgraph of a connected graph $G$, then $\rho(H) <\rho(G)$.
\end{theorem}
In \cite{07}, Theorem \ref{3.1} is used to prove the following results.
\begin{theorem}\cite{07} \label{3.2}
Let $G$ be a connected graph with $k$ vertices and $\textbf{m}=(n_{1},n_{2},\dots,n_{k})$ a vector of positive integers. If $v_{1}v_{2} \in E(G)$, then $$\rho((G-v_{1}v_{2}) \circ \textbf{m}) < \rho(G \circ \textbf{m}).$$
\end{theorem}
\begin{theorem}\cite{07} \label{3.3}
Let $G$ be a connected graph with $k$ vertices and $\textbf{m}=(n_{1},n_{2},\dots,n_{k})$ a vector of positive integers. If $v_{i}v_{j} \notin E(G)$ and $N(v_{i}) \subsetneq N(v_{j})$, then
$$\rho(G \circ (n_{1},\dots,n_{i},\dots,n_{j},\dots,n_{k})) < \rho(G \circ (n_{1},\dots,n_{i}-1,\dots,n_{j}+1,\dots,n_{k})).$$
\end{theorem}
By Theorem \ref{3.2}, we obtain the following proposition.
\begin{proposition}\label{sets}
Let $G$ be the extremal graph with minimum spectral radius among all connected graphs of order $n$ and rank 5. Then $G \in M_{n}(G_{1}, G_{7}, G_{10})$.
\end{proposition}
\begin{proof}
Let $\textbf{m}_{1}=(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}),$ $\textbf{m}_{2}=(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}),$ $\textbf{m}_{3}=(n_{1}, n_{2},$ $n_{3}, n_{4}, n_{5}, n_{6}, n_{7})$ and $\textbf{m}_{4}=(n_{1}, n_{2}, n_{3},$ $n_{4}, n_{5}, n_{6}, n_{7}, n_{8})$ be arbitrary vectors of positive integers with $\sum_{i=1}n_{i}=n$.
As a consequence of Theorem \ref{3.2}, we have
\begin{align*}
&\rho(G_{1} \circ \textbf{m}_{1}) < \rho(G_{i} \circ \textbf{m}_{1}), i=2,3,4,5,6,8,\\
&\rho(G_{10} \circ \textbf{m}_{2}) < \rho(G_{j} \circ \textbf{m}_{2}), j=11,12,13,14,15.\\
\end{align*}
Thus,
$$G \in M_{n}(G_{1}, G_{7}, G_{9}, G_{10}, G_{16}, G_{17}, G_{18}, G_{19}, G_{20},G_{21},G_{22}, G_{23}, G_{24}).$$
Let $H_{1}=G_{1} \circ (1, 1, 1, 1, 2)$, $H_{2}=G_{10} \circ (1, 1, 1, 1, 1, 2)$, $H_{3}=G_{10} \circ (1, 1, 1, 1, 2, 1)$ and $H_{4}=G_{10} \circ (1, 1, 1, 1, 1, 3)$, as shown in Figure \ref{Fig.3.1}.
Obiviously,
\begin{itemize}
\item $H_{1}$ is the spanning proper subgraph of $G_{9}$;
\item $H_{2}$ is the spanning proper subgraph of $G_{i},i \in \{16,17,18,19,21,22\}$;
\item $H_{3}$ is the spanning proper subgraph of $G_{20}$;
\item $H_{4}$ is the spanning proper subgraph of $G_{j},j \in \{23,24\}$.
\end{itemize}
Therefore, it follows from Theorem \ref{3.2} that
\begin{align*}
&\rho(G_{1} \circ \textbf{m}_{2}')=\rho(H_{1} \circ \textbf{m}_{2})<\rho(G_{9} \circ \textbf{m}_{2}),\\
&\rho(G_{10} \circ \textbf{m}_{3}')=\rho(H_{2} \circ \textbf{m}_{3})<\rho(G_{i} \circ \textbf{m}_{3}), i=16,17,18,19,21,22,\\
&\rho(G_{10} \circ \textbf{m}_{3}'')=\rho(H_{3} \circ \textbf{m}_{3})<\rho(G_{20} \circ \textbf{m}_{3}),\\
&\rho(G_{10} \circ \textbf{m}_{4}')=\rho(H_{4} \circ \textbf{m}_{4})<\rho(G_{j} \circ \textbf{m}_{4}), j=23, 24,
\end{align*}
where $\textbf{m}_{2}'=(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}+n_{6}),$ $\textbf{m}_{3}'=(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}+n_{7}),$ $\textbf{m}_{3}''=(n_{1}, n_{2}, n_{3}, n_{4},n_{5}+n_{7},n_{6})$ and $\textbf{m}_{4}'=(n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}+n_{7}+n_{8}).$
Hence, $G \in M_{n}(G_{1}, G_{7}, G_{10})$.
\begin{figure}
\caption{The graphs $H_{i}
\label{Fig.3.1}
\end{figure}
\end{proof}
\subsection{Step 2}
In this subsection we characterize the extremal graphs with minimum spectral radii in $M_{n}(G_{1}), M_{n}(G_{7})$ and $M_{n}(G_{10})$, respectively. To accomplish this, let's introduce some classic results in spectral graph theory.
\begin{definition}\cite{10} \label{3.4}
Let $A$ be an $n \times n$ real matrix whose rows and columns are indexed by $X=\{1,2,\dots,n\}$. We partition $X$ into ${X_{1},X_{2},\dots,X_{k}}$ in order and rewrite $A$ according to the partition ${X_{1},X_{2},\dots,X_{k}}$ as follows:
\begin{align} A=\left(\begin{array}{ccc}
A_{1,1} &\cdots &A_{1,k} \\
\vdots &\ddots &\vdots \\
A_{k,1} &\cdots &A_{k,k} \\
\end{array}\right),\notag
\end{align}
where $A_{i,j}$ is the block of $A$ formed by rows in $X_{i}$ and the columns in $X_{j}$. Let $b_{i,j}$ denote the average row sum of $A_{i,j}$. Then the matrix $B=[b_{i,j}]$ will be called the \textbf{quotient matrix} of the partition of $A$. In particular, when the row sum of each block $A_{i,j}$ is constant, the partition is called an \textbf{equitable partition}.
\end{definition}
\begin{theorem}\cite{10} \label{3.5}
Let $A \geq 0$ be an irreducible square matrix, $B$ be the quotient matrix of an equitable partition of $A$. Then the spectrum of $A$ contains the spectrum of $B$ and $\rho(A)=\rho(B).$
\end{theorem}
\begin{theorem}\cite{11} \label{3.6}
Let $G$ and $H$ be two connected graphs such that $\phi(H,x)>\phi(G,x)$ for $x\geq \rho(G)$. Then $\rho(H)<\rho(G)$.
\end{theorem}
\begin{theorem}\cite{09} \label{cm}
Let $K_{n_{1},n_{2},\dots,n_{k}}$ be the complete multipartite graph of order $n$. Then
$$\phi(K_{n_{1},n_{2},\dots,n_{k}},x)=x^{n-k}(1-\sum_{i=1}^{k}\frac{n_{i}}{x+n_{i}})\prod_{i=1}^{k}(x+n_{i}).$$
\end{theorem}
The following Propositions \ref{g1}, \ref{g10} and \ref{g7} give the extremal graph which attains the minimum spectral radius in $M_{n}(G_{1})$, $M_{n}(G_{10})$ and $M_{n}(G_{7})$, respectively.
\begin{proposition}\label{g1}
The extremal graph in $M_{n}(G_{1})$ which attains minimum spectral radius is of the form
$$G_{1} \circ (1, 1, 1, k, n-k-3),$$
where $1 \leq k\leq \frac{n-3}{2}.$
\end{proposition}
\begin{proof}
Since $N(v_{5})=\{v_{3}\} \subsetneq N(v_{1})$ and $v_{1}v_{5} \notin E(G_{1})$, then by Theorem \ref{3.3} we have
$$\rho(G_{1} \circ (1,n_{2},n_{3},n_{4},n_{5}+n_{1}-1)) \leq \rho(G_{1} \circ (n_{1},n_{2},n_{3},n_{4},n_{5})),$$
with equality if and only if $n_{1}=1$. It follows that the extremal graph in $M_{n}(G_{1})$ which attains minimum spectral radius is of the form $F=G_{1} \circ (1,n_{2},n_{3},n_{4},n_{5})$.
Then $V(F)$ can be naturally partitioned into $5$ parts:
$$\{V_{1},V_{2},V_{3},V_{4},V_{5}\},$$
where $V_{i}=\{v_{i}^{1}, \dots, v_{i}^{n_{i}}\}, i=1,2,3,4,5$. Obviously, this partition of $A(F)$ is equitable and the corresponding quotient matrix B is
\begin{align} B=\left(\begin{array}{ccccc}
0 &n_{2} &n_{3} &n_{4} &0 \\
1 &0 &0 &n_{4} &0 \\
1 &0 &0 &0 &n_{5} \\
1 &n_{2} &0 &0 &0 \\
0 &0 &n_{3} &0 &0\\
\end{array}\right).\notag
\end{align}
Then the characteristic polynomial of the quotient matrix $B$ is:
\begin{equation}
\begin{split}
\phi(B,x)&=x^{5}-(n_{2}+n_{3}+n_{4}+n_{2}n_{4}+n_{3}n_{5})x^{3}-2n_{2}n_{4}x^{2}+\\
&(n_{2}n_{3}n_{4}+n_{2}n_{3}n_{5}+n_{3}n_{4}n_{5}+n_{2}n_{3}n_{4}n_{5})x+2n_{2}n_{3}n_{4}n_{5}.
\end{split}\notag
\end{equation}
Since $R(A(F))=5$, by Theorem \ref{3.5} we have $\phi(F,x)=x^{n-5} \phi(B,x)$ and $\rho(F)=\rho(A(F))=\rho(B)$.
Note that $G_{1} \circ (1,n_{2},n_{3},n_{4},n_{5}) \cong G_{1} \circ (1,n_{4},n_{3},n_{2},n_{5})$. Therefore, without loss of generality, we suppose that $n_{4} \geq n_{2}.$
\textbf{Claim 1.} $n_{2}=1.$
Assume $n_{2} \geq 2$, let $F_{1}=G_{1} \circ (1,n_{2}-1,n_{3},n_{4}+1,n_{5})$ then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{1},x)-\phi(F,x)\\
&=x^{n-5}(n_{4}-n_{2}+1)(x^{3}+2x^{2}-(n_{3}+n_{3}n_{5})x-2n_{3}n_{5})\\
&=x^{n-5}(n_{4}-n_{2}+1)\left(x(x^{2}-n_{3}(n_{5}+1))+2(x^{2}-n_{3}n_{5})\right).
\end{split} \notag
\end{equation}
Since $n_{4} \geq n_{2}$, we have $n_{4}-n_{2}+1 >0$. It is clear that $K_{n_{3},n_{5}+1}$ is a proper subgraph of $F$, we obtain $\rho(F)>\rho(K_{n_{3},n_{5}+1})=\sqrt{n_{3}(n_{5}+1)}$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{1})<\rho(F)$ which contradicts to the extremality of $F$.
\textbf{Claim 2.} $n_{3}=1.$
Now $F=G_{1} \circ (1,1,n_{3},n_{4},n_{5})$, we claim that $n_{5} \geq n_{3}$. If not, let $F_{2}=G_{1} \circ (1,1,n_{5},n_{4},n_{3})$, then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{2},x)-\phi(F,x)=x^{n-4}(x^{2}-n_{4})(n_{3}-n_{5}).
\end{split}\notag
\end{equation}
Since $n_{3}>n_{5}$, we have $n_{3}-n_{5} >0$. It can be seen that $K_{n_{4},2}$ is a proper subgraph of $F$, we obtain $\rho(F)>\rho(K_{n_{4},2})=\sqrt{2n_{4}}$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{2})<\rho(F)$, a contradiction. Therefore $n_{5} \geq n_{3}$.
Next, we assume $n_{3} \geq 2$, let $F_{3}=G_{1} \circ (1,1,n_{3}-1,n_{4},n_{5}+1)$ then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{3},x)-\phi(F,x)\\
&=x^{n-5}\left((n_{5}-n_{3}+1)(x^{3}-(2n_{4}+1)x-2n_{4})+x(x^{2}-n_{4})\right).
\end{split}\notag
\end{equation}
Since $n_{5} \geq n_{3}$, we have $n_{5}-n_{3}+1>0$. It is clear that $K_{n_{4},1,1}$ is a proper subgraph of $F$, by Theorem \ref{cm}, we obtain $\rho(F)>\rho(K_{n_{4},1,1})=(\sqrt{8n_{4}+1}+1)/2$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{3})<\rho(F)$, which contradicts to the extremality of $F$.
\textbf{Claim 3.} $n_{5} \geq n_{4}.$
Now $F=G_{1} \circ (1,1,1,n_{4},n_{5})$. Otherwise, let $F_{4}=G_{1} \circ (1,1,1,n_{5},n_{4})$ then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{4},x)-\phi(F,x)=x^{n-3}(x+2)(n_{4}-n_{5}).
\end{split}\notag
\end{equation}
Since $n_{4} > n_{5}$ and $\rho(F)>0$, then $r(x)>0$ for $x \geq \rho(F)$. By Theorem \ref{3.6}, we have $\rho(F_{4})<\rho(F)$ which contradicts to the extremality of $F$, thus $n_{5} \geq n_{4}$.
From above three claims, we conclude that the extremal graph with minimum spectral radius in $M_{n}(G_{1})$ is of the form $G_{1} \circ (1, 1, 1, k, n-k-3)$, where $1 \leq k\leq (n-3)/2$.
\end{proof}
Similarly, we characterize the extremal graph with minimum spectral radius in $M_{n}(G_{10})$.
\begin{proposition}\label{g10}
The extremal graph in $M_{n}(G_{10})$ which attains minimum spectral radius is of the form
$$G_{10} \circ (1, 1, 1, 1, k, n-k-4),$$
where $1 \leq k\leq \frac{n-4}{2}.$
\end{proposition}
\begin{proof}
By Theorem \ref{3.3}, we have
$$\rho(G_{10} \circ (1,n_{2},n_{3},n_{4},n_{5},n_{6}+n_{1}-1)) \leq \rho(G_{10} \circ (n_{1},n_{2},n_{3},n_{4},n_{5},n_{6})),$$
with equality if and only if $n_{1}=1$. Thus, we may suppose that the extremal graph in $M_{n}(G_{10})$ which attains minimum spectral radius is of the form $F=G_{10} \circ (1,n_{2},n_{3},n_{4},n_{5},n_{6})$.
Similarly, we obtain
\begin{align} B=\left(\begin{array}{cccccc}
0 &n_{2} &n_{3} &n_{4} &0 &0 \\
1 &0 &n_{3} &0 &n_{5} &0 \\
1 &n_{2} &0 &0 &n_{5} &0 \\
1 &0 &0 &0 &0 &n_{6} \\
0 &n_{2} &n_{3} &0 &0 &0 \\
0 &0 &0 &n_{4} &0 &0\\
\end{array}\right),\notag
\end{align}
is the quotient matrix of an equitable partition of $A(F)$. The characteristic polynomial of the quotient matrix $B$ is:
\begin{equation}
\begin{split}
\phi(B,x)&=x(x^{5}-(n_{2}+n_{3}+n_{4}+n_{2}n_{3}+n_{2}n_{5}+n_{3}n_{5}+n_{4}n_{6})x^{3}\\
&-(2n_{2}n_{3}+2n_{2}n_{3}n_{5})x^{2}+(n_{2}n_{3}n_{4}+n_{2}n_{4}n_{5}+n_{2}n_{4}n_{6}\\
&+n_{3}n_{4}n_{5}+n_{3}n_{4}n_{6}+n_{2}n_{3}n_{4}n_{6}+n_{2}n_{4}n_{5}n_{6}+n_{3}n_{4}n_{5}n_{6})x\\
&+2n_{2}n_{3}n_{4}n_{5}+2n_{2}n_{3}n_{4}n_{6}+2n_{2}n_{3}n_{4}n_{5}n_{6}).
\end{split}\notag
\end{equation}
Since $R(A(F))=5$, by Theorem \ref{3.5} we have $\phi(F,x)=x^{n-6} \phi(B,x)$ and $\rho(F)=\rho(A(F))=\rho(B)$.
Note that $G_{10} \circ (1,n_{2},n_{3},n_{4},n_{5},n_{6}) \cong G_{10} \circ (1,n_{3},n_{2},n_{4},n_{5},n_{6})$. Therefore, without loss of generality, we suppose that $n_{3} \geq n_{2}.$
\textbf{Claim 1.} $n_{2}=1.$
Assume $n_{2} \geq 2$, let $F_{1}=G_{10} \circ (1,n_{2}-1,n_{3}+1,n_{4},n_{5},n_{6})$ then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{1},x)-\phi(F,x)\\
&=x^{n-5}(n_{3}-n_{2}+1)(x^{3}+2(1+n_{5})x^{2}-(n_{4}+n_{4}n_{6})x-2n_{4}n_{5}\\
&-2n_{4}n_{6}-2n_{4}n_{5}n_{6})\\
&=x^{n-5}(n_{3}-n_{2}+1)(x(x^{2}-n_{4}(n_{6}+1))+2n_{5}(x^{2}-n_{4}(n_{6}+1))\\
&+2(x^{2}-n_{4}n_{6})).
\end{split}\notag
\end{equation}
Since $n_{3} \geq n_{2}$, we have $n_{3}-n_{2}+1 >0$. It is clear that $K_{n_{4},n_{6}+1}$ is a proper subgraph of $F$, we obtain $\rho(F)>\rho(K_{n_{4},n_{6}+1})=\sqrt{n_{4}(n_{6}+1)}$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{1})<\rho(F)$ which contradicts to the extremality of $F$.
\textbf{Claim 2.} $n_{3}=1.$
Now $F=G_{10} \circ (1,1,n_{3},n_{4},n_{5},n_{6})$, we claim that $n_{5} \geq n_{3}$. If not, let $F_{2}=G_{10} \circ (1,1,n_{5},n_{4},n_{3},n_{6})$, then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{2},x)-\phi(F,x)=x^{n-5}(n_{3}-n_{5})(x^{2}-n_{4}n_{6})(x+2).
\end{split}\notag
\end{equation}
Since $n_{3} > n_{5}$, we have $n_{3}-n_{5} >0$. It can be seen that $K_{n_{4},n_{6}}$ is a proper subgraph of $F$, we obtain $\rho(F)>\rho(K_{n_{4},n_{6}})=\sqrt{n_{4}n_{6}}$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{2})<\rho(F)$, a contradiction. Therefore $n_{5} \geq n_{3}$.
Next, we assume $n_{3} \geq 2$, let $F_{3}=G_{10} \circ (1,1,n_{3}-1,n_{4},n_{5}+1,n_{6})$ then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{3},x)-\phi(F,x)\\
&=x^{n-5}(x+2)\left((n_{5}-n_{3}+2)(x^{2}-n_{4}(n_{6}+1))+n_{4}\right).
\end{split}\notag
\end{equation}
Since $n_{5} \geq n_{3}$, we have $n_{5}-n_{3}+2>0$. It is clear that $K_{n_{4},n_{6}+1}$ is a proper subgraph of $F$, we obtain $\rho(F)>\rho(K_{n_{4},n_{6}+1})=\sqrt{n_{4}(n_{6}+1)}$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{3})<\rho(F)$ which contradicts to the extremality of $F$.
\textbf{Claim 3.} $n_{4}=1.$
Now $F=G_{10} \circ (1,1,1,n_{4},n_{5},n_{6})$, we claim that $n_{6} \geq n_{4}$. If not, let $F_{4}=G_{10} \circ (1,1,1,n_{6},n_{5},n_{4})$, then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{4},x)-\phi(F,x)=x^{n-5}(n_{4}-n_{6})(x^{2}-x-2n_{5})(x+1).
\end{split}\notag
\end{equation}
Since $n_{4} > n_{6}$, we have $n_{4}-n_{6} >0$. It can be seen that $K_{n_{5},1,1}$ is a proper subgraph of $F$, we obtain $\rho(F)>\rho(K_{n_{5},1,1})=(\sqrt{8n_{5}+1}+1)/2$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{4})<\rho(F)$, a contradiction. Therefore $n_{6} \geq n_{4}$.
Next, we assume $n_{4} \geq 2$, let $F_{5}=G_{10} \circ (1,1,1,n_{4}-1,n_{5},n_{6}+1)$ then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{5},x)-\phi(F,x)\\
&=x^{n-5}(x+1)\left((n_{6}-n_{4}+2)(x^{2}-x-2n_{5}-2)+2\right).
\end{split}\notag
\end{equation}
Since $n_{6} \geq n_{4}$, we have $n_{6}-n_{4}+2>0$. It is clear that $H \circ(n_{5},1,1,1)$ is a proper subgraph of $F$, where $H$ is shown in Figure \ref{Fig.3.2}, we obtain $\rho(F)>\rho(H \circ(n_{5},1,1,1))=(\sqrt{8n_{5}+9}+1)/2$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{5})<\rho(F)$, which contradicts to the extremality of $F$.
\begin{figure}
\caption{The graph $H$.}
\label{Fig.3.2}
\end{figure}
\textbf{Claim 4.} $n_{6} \geq n_{5}.$
Now $F=G_{10} \circ (1,1,1,1,n_{5},n_{6})$. Otherwise, let $F_{6}=G_{10} \circ (1,1,1,1,n_{6},n_{5})$ then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{6},x)-\phi(F,x)=x^{n-4}(x+1)^{2}(n_{5}-n_{6}).
\end{split}\notag
\end{equation}
Since $n_{5}>n_{6}$ and $\rho(F)>0$, then $r(x)>0$ for $x \geq \rho(F)$, by Theorem \ref{3.6}, we have $\rho(F_{6})<\rho(F)$ which contradicts to the extremality of $F$, thus $n_{6} \geq n_{5}$.
It follows from above four claims that the extremal graph with minimum spectral radius in $M_{n}(G_{10})$ is of the form $G_{10} \circ (1, 1, 1, 1, k, n-k-4),$ where $1 \leq k\leq (n-4)/2$.
\end{proof}
Next we determine the extremal graph with minimum spectral radius in $M_{n}(G_{7})$.
\begin{proposition}\label{g7}
The extremal graph in $M_{n}(G_{7})$ which attains minimum spectral radius is
$$G_{7} \circ (\lceil \frac{n-3}{2} \rceil, 1, \lfloor \frac{n-3}{2} \rfloor, 1,1).$$
\end{proposition}
\begin{proof}
Suppose that the extremal graph in $M_{n}(G_{7})$ which attains minimum spectral radius is of the form $F=G_{7} \circ (n_{1},n_{2},n_{3},n_{4},n_{5})$. Similarlly, we obtain
\begin{align} B=\left(\begin{array}{ccccc}
0 &n_{2} &0 &0 &n_{5} \\
n_{1} &0 &n_{3} &0 &0 \\
0 &n_{2} &0 &n_{4} &0 \\
0 &0 &n_{3} &0 &n_{5} \\
n_{1} &0 &0 &n_{4} &0\\
\end{array}\right),\notag
\end{align}
is the quotient matrix of an equitable partition of $A(F)$ and the characteristic polynomial of $B$ is:
\begin{align*}
\phi(B,x)&=x^{5}-(n_{1}n_{2}+n_{2}n_{3}+n_{1}n_{5}+n_{3}n_{4}+n_{4}n_{5})x^{3}+(n_{1}n_{2}n_{3}n_{4}+\\
&n_{1}n_{2}n_{3}n_{5}+n_{1}n_{2}n_{4}n_{5}+n_{1}n_{3}n_{4}n_{5}+n_{2}n_{3}n_{4}n_{5})x-2n_{1}n_{2}n_{3}n_{4}n_{5}.
\end{align*}
Since $R(A(F))=5$, by Theorem \ref{3.5} we have $\phi(F,x)=x^{n-5} \phi(B,x)$ and $\rho(F)=\rho(A(F))=\rho(B)$.
Without loss of generality, we may suppose that $n_{1}=\text{max}\{n_{i},i=1,2,3,4,5\}$ and $n_{2} \leq n_{5}$, then we have the following claims.
\textbf{Claim 1.} $n_{2} \leq n_{3}$ and $n_{5} \leq n_{4}$.
Suppose that $n_{2}>n_{3}$. Let $F_{1}=G_{7} \circ (n_{1},n_{3},n_{2},n_{4},n_{5})$, then
\begin{equation*}
r(x)=\phi(F_{1},x)-\phi(F,x)=x^{n-2}(n_{1}-n_{4})(n_{2}-n_{3}).
\end{equation*}
Since $n_{1}=\text{max}\{n_{i},i=1,2,3,4,5\}$, we have $n_{1}\geq n_{4}$. And if $n_{1}=n_{4}$, then $F_{1} \cong F$. Thus, without loss of generality, we may suppose that $n_{1}>n_{4}$.
Since $n_{2}>n_{3}$, $n_{1}>n_{4}$ and $\rho(F)>0$, then $r(x)>0$ for $x \geq \rho(F)$. By Theorem \ref{3.6}, we have $\rho(F_{1})<\rho(F)$ which contradicts to the extremality of $F$.
Similarly, we obtain $n_{5} \leq n_{4}$.
\textbf{Claim 2.} $n_{4} \leq n_{3}$.
Suppose to the contrary that $n_{4}>n_{3}$. Let $F_{2}=G_{7} \circ (n_{1},n_{2},n_{4},n_{3},n_{5})$, then
\begin{equation*}
r(x)=\phi(F_{2},x)-\phi(F,x)=x^{n-2}(n_{2}-n_{5})(n_{3}-n_{4}).
\end{equation*}
Since $n_{5}\geq n_{2}$ and if $n_{5}=n_{2}$, then $F_{2} \cong F$. Thus without loss of generality we may suppose that $n_{5}>n_{2}$.
Since $n_{4}>n_{3}$, $n_{5}>n_{2}$ and $\rho(F)>0$. Then $r(x)>0$ for $x \geq \rho(F)$. By Theorem \ref{3.6}, we have $\rho(F_{2})<\rho(F)$ which contradicts to the extremality of $F$.
From above two claims, we have $n_{1}\geq n_{3}\geq n_{4}\geq n_{5}\geq n_{2}$. Next, we will prove $n_{2}=n_{4}=n_{5}=1$ and $n_{1}-n_{3}\leq 1$.
\textbf{Claim 3.} $n_{2}=n_{5}$
Assume $n_{2}<n_{5}$, let $F_{3}=G_{7} \circ (n_{1}+n_{5}-n_{2},n_{2},n_{3},n_{4},n_{2})$ then
\begin{align*}
r(x)&=\phi(F_{3},x)-\phi(F,x)\\
&=x^{n-5}(n_{5}-n_{2})((n_{1}-2n_{2}+n_{4})x^{3}-(n_{1}-n_{2})((n_{3}n_{4}+n_{2}n_{3}+n_{2}n_{4})x\\
&-2n_{2}n_{3}n_{4})) \\
&\geq x^{n-5}(n_{5}-n_{2})(n_{1}-n_{2})\left(x^{3}-(n_{3}n_{4}+n_{2}n_{3}+n_{2}n_{4})x+2n_{2}n_{3}n_{4}\right) \\
&=x^{n-5}(n_{5}-n_{2})(n_{1}-n_{2})g(x).
\end{align*}
Since $n_{1}\geq n_{3} \geq n_{4} \geq n_{5}>n_{2}\geq1$, we have $n_{1}-n_{2}>0$ and $n_{5}-n_{2}>0$. It is clear that $K_{n_{3},n_{2}+n_{4}}$ is a proper subgraph of $F$, we obtain $\rho(F)>\rho(K_{n_{3},n_{2}+n_{4}})=\sqrt{n_{3}(n_{2}+n_{4})}$.
Since $g(\sqrt{n_{3}(n_{2}+n_{4})})>0$ and $\sqrt{n_{3}(n_{2}+n_{4})}>\sqrt{(n_{3}(n_{2}+n_{4})+n_{2}n_{4})/3}$, where $\sqrt{\left(n_{3}(n_{2}+n_{4})+n_{2}n_{4}\right)/3}$ is the largest root of $g'(x)$, we have $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{3})<\rho(F)$ which contradicts to the extremality of $F$.
Note that $G_{7} \circ (n_{1},n_{2},n_{3},n_{4},n_{5}) \cong G_{7} \circ (n_{1},n_{2},n_{4},n_{3},n_{5})$ when $n_{2}=n_{5}$, therefore without loss of generality we may suppose that $n_{3} \geq n_{4}.$
\textbf{Claim 4.} $n_{4}=1.$
Assume $n_{4}\geq 2$, let $F_{4}=G_{7} \circ (n_{1},n_{2},n_{3}+1,n_{4}-1,n_{5})$ then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{4},x)-\phi(F,x)\\
&=x^{n-5}(x-n_{5})(n_{3}-n_{4}+1)(x^{2}+n_{5}x-2n_{1}n_{5}).
\end{split}\notag
\end{equation}
Since $n_{1} \geq n_{3}\geq n_{4}\geq n_{5}=n_{2}$, we have $n_{3}-n_{4}+1>0$. It can be seen that $K_{n_{1},2n_{5}}$ is a proper subgraph of $F$, we obtain $\rho(F)>\rho(K_{n_{1},2n_{5}})=\sqrt{2n_{1}n_{5}}>n_{5}$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{4})<\rho(F)$ which contradicts to the extremality of $F$, therefore $n_{4}=1$ and hence $n_{2}=n_{5}=1$.
\textbf{Claim 5.} $n_{1}-n_{3}\leq 1.$
Now $F=G_{7} \circ (n_{1},1,n_{3},1,1)$. Assume $n_{1}\geq n_{3}+2$, let $F_{5}=G_{7} \circ (n_{1}-1,1,n_{3}+1,1,1)$ then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{5},x)-\phi(F,x)=x^{n-5}(3x-2)(n_{1}-n_{3}-1).
\end{split}\notag
\end{equation}
Since $n_{1} \geq n_{3}+2$, we have $n_{1}-n_{3}-1>0$. It is clear that $K_{n_{1},2}$ is a proper subgraph of $F$, we obtain $\rho(F)>\rho(K_{n_{1},2})=\sqrt{2n_{1}}>1$, then $r(x)>0$ for $x \geq \rho(F)$.
Thus, by Theorem \ref{3.6}, we have $\rho(F_{5})<\rho(F)$ which contradicts to the extremality of $F$, therefore $n_{1}-n_{3}\leq 1$ and hence $n_{1}=\lceil(n-3)/2 \rceil, n_{3}=\lfloor (n-3)/2 \rfloor.$
From above five claims, we obtain $G_{7} \circ (\lceil \frac{n-3}{2} \rceil, 1, \lfloor \frac{n-3}{2} \rfloor, 1,1)$ attains the minimum spectral radius in $M_{n}(G_{7})$.
\end{proof}
\subsection{Step 3}
We first prove that the extremal graph with minimum spectral radius in $M_{n}(G_{1},G_{7})$ must be in $M_{n}(G_{1})$ by the following lemma.
\begin{lemma}\label{3.7}
For $n \geq 8$, we have $\rho(G_{1} \circ (1,1,1,\lfloor \frac{n-3}{2} \rfloor, \lceil \frac{n-3}{2} \rceil )) < \rho(G_{7} \circ (\lceil \frac{n-3}{2} \rceil, 1, \lfloor \frac{n-3}{2} \rfloor, 1,1))$.
\end{lemma}
\begin{proof}
Let $F_{1}=G_{1} \circ (1,1,1,\lfloor \frac{n-3}{2} \rfloor, \lceil \frac{n-3}{2} \rceil )$ and $F_{2}=G_{7} \circ (\lceil \frac{n-3}{2} \rceil, 1, \lfloor \frac{n-3}{2} \rfloor, 1,1)$.
For $8\leq n \leq 12$, we use the MATLAB software to calculate the spectral radii of $F_{i}$ for $i=1,2$, as shown in the Table \ref{tab1}.
\begin{table}[H]\tiny
\centering
\caption{$\rho(F_{i})$.}
\label{tab1}
\resizebox{\textwidth}{!}{
\begin{tabular}{ccc}
\hline
\ \ \ \ \ \ \ \ \ \ $n$ \ \ \ \ \ \ \ \ \ \ & \ \ \ \ \ \ \ \ \ \ $\rho(F_{1})$ \ \ \ \ \ \ \ \ \ \ & \ \ \ \ \ \ \ \ \ \ $\rho(F_{2})$ \ \ \ \ \ \ \ \ \ \ \\ \hline
8 & 2.7676 & 2.9764 \\
9 & 3.1474 & 3.2176 \\
10 & 3.1713 & 3.4630 \\
11 & 3.5047 & 3.6737 \\
12 & 3.5223 & 3.8879 \\ \hline
\end{tabular}}
\end{table}
So let us assume that $n\geq 13$.
\textbf{Case 1.} $n-3=2k$ is even.
In this case, $F_{1}=G_{1} \circ (1,1,1,k,k)$ and $F_{2}=G_{7} \circ (k,1,k,1,1)$, then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{1},x)-\phi(F_{2},x)=x^{n-5}\left((k-1)x^{3}-2kx^{2}-k^{2}x+4k^{2}\right).
\end{split}\notag
\end{equation}
It can be seen that $K_{2k,1}$ is a proper subgraph of $F_{2}$, we obtain $\rho(F_{2})>\rho(K_{2k,1})=\sqrt{2k}$. Since $n \geq 13$, we have $r(\sqrt{2k})>0$ and $\sqrt{2k}>(2k+k\sqrt{3k+1})/3(k-1)$. Since $(2k+k\sqrt{3k+1})/3(k-1)$ is the largest root of $r'(x)$, we obtain $r(x)>0$ for $x \geq \rho(F_{2})$.
Thus by Theorem \ref{3.6}, we have $\rho(F_{1})<\rho(F_{2})$.
\textbf{Case 2.} $n-3=2k+1$ is odd.
In this case, $F_{1}=G_{1} \circ (1,1,1,k,k+1)$ and $F_{2}=G_{7} \circ (k+1,1,k,1,1)$, then
\begin{equation}
\begin{split}
r(x)&=\phi(F_{1},x)-\phi(F_{2},x)=x^{n-5}k\left(x^{3}-2x^{2}-(k+1)x+4k+\right).
\end{split}\notag
\end{equation}
It is clear that $K_{2k+1,1}$ is a proper subgraph of $F_{2}$, we obtain $\rho(F_{2})>\rho(K_{2k+1,1})=\sqrt{2k+1}$. Since $n \geq 13$, we have $r(\sqrt{2k+1})>0$ and $\sqrt{2k+1}>(2+\sqrt{3k+7})/3$. Since $(2+\sqrt{3k+7})/3$ is the largest root of $r'(x)$, we obtain $r(x)>0$ for $x \geq \rho(F_{2})$.
Thus by Theorem \ref{3.6}, we have $\rho(F_{1})<\rho(F_{2})$.
\end{proof}
Next we prove the extremal graph with minimum spectral radius in $M_{n}(G_{1},G_{10})$ must be in $M_{n}(G_{10})$. We need the following theorem.
\begin{theorem}\cite{12} \label{3.8}
Let $G$ be a graph with m edges and n vertices. Then $\rho(G) \leq \sqrt{2m-n+1}$, with equality if and only if $G$ is isomorphic to the star $K_{1,n-1}$ or the complete graph $K_{n}$.
\end{theorem}
\begin{lemma} \label{3.9}
Let $G_{1} \circ(1, 1, 1,k,n-k-3)$ be the extremal graph with minimum spectral radius in $M_{n}(G_{1})$ for $n \geq 12$. Then $2\leq k \leq \frac{n-3}{2}$.
\end{lemma}
\begin{proof}
We denote $F_k=G_{1} \circ(1, 1, 1,k,n-k-3)$ for convenience. By Proposition \ref{g1}, we have $1\leq k \leq (n-3)/2$.
For $12 \leq n \leq 18$, we use the MATLAB software to calculate the spectral radii of $F_{k}$, as shown in the Table \ref{tab2}, where the minimum spectral radius is bolded.
\begin{table}[H]\scriptsize
\centering
\caption{$\rho(F_k)$.}
\label{tab2}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|ccccccc}
\hline
\diagbox{$n$}{$\rho(F_k)$}{$k$} & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline
12 & 3.0751 & \textbf{3.0649} & 3.2427 & 3.5223 & \textbackslash{} & \textbackslash{} & \textbackslash{} \\
13 & 3.2229 & \textbf{3.1791} &3.2951 & 3.5443& 3.8231& \textbackslash{} & \textbackslash{} \\
14 & 3.3668 & \textbf{3.3013} &3.3616 &3.5722& 3.8368& \textbackslash{} & \textbackslash{} \\
15 & 3.5064&\textbf{3.4274} &3.4422 & 3.6076& 3.8535 &4.1131& \textbackslash{} \\
16 & 3.6418 & 3.5544& \textbf{3.5353}& 3.6526 & 3.8742& 4.1243 & \textbackslash{} \\
17 & 3.7731 & 3.6807 & \textbf{3.6377} & 3.7088& 3.8998& 4.1376 & 4.3813 \\
18 &3.9006 & 3.8053& \textbf{3.7459} & 3.7767& 3.9318 &4.1536& 4.3906\\ \hline
\end{tabular}}
\end{table}
For $n \geq 19$, note that $F_{1}=G_{1}\circ (1,1,1,1,n-4), F_{2}=G_{1}\circ (1,1,1,2,n-5)$, let \bm{$x$} be the principal eigenvector of $F_{2}$ and $x_{i}$ correspond to vertices in $V_{i}$ for $i=1,2,3,4,5.$ By $\rho(F_{2})\bm{x}=A(F_{2})\bm{x}$, we have
\begin{align}
\rho(F_{2})x_{1}&=x_{2}+x_{3}+2x_{4}, \label{eq3.1}\\
\rho(F_{2})x_{2}&=x_{1}+2x_{4}, \label{eq3.2}\\
\rho(F_{2})x_{3}&=x_{1}+(n-5)x_{5}, \label{eq3.3}\\
\rho(F_{2})x_{4}&=x_{1}+x_{2}, \label{eq3.4}\\
\rho(F_{2})x_{5}&=x_{3}, \label{eq3.5}
\end{align}
From $(\ref{eq3.1})$-$(\ref{eq3.3})$, we have
\begin{equation*}
\rho(F_{2})(x_{3}-x_{1}-x_{2})=x_{1}+(n-5)x_{5}-x_{2}-x_{3}-2x_{4}-x_{1}-2x_{4},
\end{equation*}
multiplying $\rho(F_{2})$ on both sides, by ($\ref{eq3.4}$) and ($\ref{eq3.5}$), yields
\begin{equation*}
\rho(F_{2})^{2}(x_{3}-x_{1}-x_{2})=(n-5)x_{3}-\rho(F_{2})x_{3}-\rho(F_{2})x_{2}-4(x_{1}+x_{2}),
\end{equation*}
then
\begin{equation} \label{eq3.6}
(\rho(F_{2})^{2}-\rho(F_{2})-4)(x_{3}-x_{1}-x_{2})=(n-9-2\rho(F_{2}))x_{3}+\rho(F_{2})x_{1}.
\end{equation}
Since $n \geq 19$ and $K_{n-5,1}$ is a proper subgraph of $F_{2}$, we have $\rho(F_{2})>\rho(K_{n-5,1})=\sqrt{n-5}>3$, thus $\rho(F_{2})^{2}-\rho(F_{2})-4>0$. By Theorem \ref{3.8} and $n\geq 19$, we obtain $\rho(F_{2})<\sqrt{2m(F_{2})-n+1}=\sqrt{2(n+1)-n+1}=\sqrt{n+3}<(n-9)/2$, therefore $n-9-2\rho(F_{2})>0$. Since \bm{$x$} is the principal eigenvector of $F_{2}$, we have $x_{i}>0$.
Thus, it follows from (\ref{eq3.6}) that $x_{3}-x_{1}-x_{2}>0$.
Now we have
\begin{equation}
\begin{split}
\rho(F_{1})-\rho(F_{2}) &\geq \bm{x}^{T}A(F_{2})\bm{x}-\bm{x}^{T}A(F_{1})\bm{x} \\
&=2x_{4}x_{3}-2x_{4}(x_{1}+x_{2}) \\
&=2x_{4}(x_{3}-x_{1}-x_{2})>0.
\end{split}\notag
\end{equation}
Therefore, $\rho(F_{1})>\rho(F_{2})$, which means $k\geq 2$.
\end{proof}
Now we prove that $\rho(G_{10} \circ (1,1,1,1,k-1,n-k-3))<\rho(G_{1} \circ(1, 1, 1,k,n-k-3))$ for $k\geq 2$ and $n\geq 12$ by using a well-known operation.
\begin{theorem}\cite{08} \label{3.10}
Let $v_{1}, v_{2}$ be two vertices of a connected graph $G$ and let $\{u_{1}, u_{2}, \dots, u_{t}\} \subseteq N(v_{1})\setminus N(v_{2})$. Let $G'$ be the graph obtained from $G$ by rotating the edge $v_{1}u_{i}$ to $v_{2}u_{i}$ for $i=1,2,\dots,t$. If $x_{v_{1}} \leq x_{v_{2}}$, where \textbf{x} is the principal eigenvector of $G$, then $\rho(G')>\rho(G)$.
\end{theorem}
\begin{lemma}\label{3.11}
For $k\geq 2$ and $n\geq 12$, we have $\rho(G_{10} \circ (1,1,1,1,k-1,n-k-3))<\rho(G_{1} \circ(1, 1, 1,k,n-k-3))$.
\end{lemma}
\begin{proof}
Let $F_{1}=G_{1} \circ(1, 1, 1,k,n-k-3)$ and $F_{2}=G_{10} \circ (1,1,1,1,k-1,n-k-3)$. Let \bm{$x$} be the principal eigenvector of $F_{2}$ and $x_{i}$ correspond to vertices in $V_{i}$ for $i=1,2,3,4,5,6$.
Let us first suppose that $x_{3} \geq x_{1}$, then by Theorem \ref{3.10} we have $\rho(F_{2})<\rho(F')$, where $F'$ is obtained from $F_{2}$ by rotating the edge $v_{1}v_{4}$ to $v_{3}v_{4}$. Since $F' \cong F_{1}$, we obtain $\rho(F_{2})<\rho(F_{1})$.
Now, suppose that $x_{3} < x_{1}$. Since $F'' \cong F_{1}$, where $F''$ is obtained from $F_{2}$ by rotating the edge $v_{3}v_{5}^{i}$ to $v_{1}v_{5}^{i}$ for $i=1,2,\dots,k-1$, we have $\rho(F_{2})<\rho(F'')=\rho(F_{1})$.
Thus, we complete the proof of the Lemma.
\end{proof}
Now we know that the extremal graph of order $n$ and rank $5$ with minimum spectral radius is $G_{10} \circ (1,1,1,1,k,n-4-k)$ for some integer $k$ with $1\leq k\leq \frac{n-4}{2}$ when $n\geq 12$.
For convenience, we set $F_n(i)=G_{10} \circ (1,1,1,1,i,n-4-i)$ and $\mathcal{F}=\{F_n(i):1\leq i\leq \frac{n-4}{2}\}$. It is only remained to find the extremal graph with minimum spectral radius in $\mathcal{F}$.
\begin{theorem} \cite{10} \label{3.12}
Let $A$ be an $n \times n$ nonnegative matrix. Then the largest eigenvalue $\rho(A) \geq \bm{x}^{T}A\bm{x}$ for any unit vector $\bm{x}$, with equality if and only if $A\bm{x}=\rho(A)\bm{x}$.
\end{theorem}
\begin{lemma}\label{3.13}
Let $\alpha=\frac{6n-37-\sqrt{24n+1}}{18}$ and $n\geq 12$. Then for $1\leq i\leq \frac{n-4}{2}$, we have
$$\rho(F_n(i))>\min\{\rho(F_n(\lfloor \alpha \rfloor)), \rho(F_n(\lceil \alpha \rceil))\}$$
unless $i=\lfloor\alpha \rfloor$ or $\lceil \alpha \rceil$.
\end{lemma}
\begin{proof}
Let $\rho_i=\rho(F_n(i))$. Our aim is to prove that $\rho_i<\rho_{i-1}$ if $2\leq i\leq \lfloor \alpha\rfloor$ and $\rho_i<\rho_{i+1}$ if $\lceil \alpha\rceil\leq i\leq \frac{n-6}{2}$.
Let $\bm{x}_i$ be the principal eigenvector of $F_n(i)$ and $x_{j}^{i}$ correspond to vertices in $V_{j}$ for $j=1,2,3,4,5,6$. Then by $\rho_i\bm{x}_i=A(F_n(i))\bm{x}_i$ we have
\begin{align}
\rho_ix_{1}^{i}&=x_{2}^{i}+x_{3}^{i}+x_{4}^{i}, \label{eq3.7}\\
\rho_ix_{2}^{i}&=x_{1}^{i}+x_{3}^{i}+ix_{5}^{i}, \label{eq3.8}\\
\rho_ix_{3}^{i}&=x_{1}^{i}+x_{2}^{i}+ix_{5}^{i}, \label{eq3.9}\\
\rho_ix_{4}^{i}&=x_{1}^{i}+(n-i-4)x_{6}^{i}, \label{eq3.10}\\
\rho_ix_{5}^{i}&=x_{2}^{i}+x_{3}^{i}, \label{eq3.11}\\
\rho_ix_{6}^{i}&=x_{4}^{i}, \label{eq3.12}
\end{align}
By $(\ref{eq3.8})$ and $(\ref{eq3.9})$, we have
\begin{align*}
&\rho_i(x_{2}^{i}-x_{3}^{i})=x_{3}^{i}-x_{2}^{i} \text{, i.e.,}\\
&(\rho_i+1)(x_{2}^{i}-x_{3}^{i})=0,
\end{align*}
which implies that
\begin{align}\label{eq3.13}
x_{2}^{i}=x_{3}^{i}.
\end{align}
Therefore, by (\ref{eq3.7}) and (\ref{eq3.10})-(\ref{eq3.13}), we have
\begin{align*}
x_{5}^{i}=\frac{2x_{2}^{i}}{\rho_i}&=x_{1}^{i}-\frac{x_{4}^{i}}{\rho_i} \\
&=x_{1}^{i}-x_{6}^{i}\\
&=\rho_ix_{4}^{i}-(n-i-4)x_{6}^{i}-x_{6}^{i}\\
&=\rho_i^{2}x_{6}^{i}-(n-i-4)x_{6}^{i}-x_{6}^{i}\\
&=(\rho_i^{2}-n+i+3)x_{6}^{i},
\end{align*}
and from (\ref{eq3.7})-(\ref{eq3.8}) and (\ref{eq3.11})-(\ref{eq3.13}), we have
\begin{align*}
x_{6}^{i}=\frac{x_{4}^{i}}{\rho_i}&=x_{1}^{i}-\frac{2x_{2}^{i}}{\rho_i} \\
&=x_{1}^{i}-x_{5}^{i}\\
&=(\rho_i-1)x_{2}^{i}-ix_{5}^{i}-x_{5}^{i}\\
&=\frac{\rho_i(\rho_i-1)}{2}x_{5}^{i}-ix_{5}^{i}-x_{5}^{i}\\
&=\frac{1}{2}(\rho_i^{2}-\rho_i-2i-2)x_{5}^{i}.
\end{align*}
Hence, we obtain that
\begin{align}\label{eq3.14}
\begin{cases}
(\rho_i^{2}-n+i+3)(\rho_i^{2}-\rho_i-2i-2)=2,\\
\rho_i^{2}-n+i+3>0,\\
\rho_i^{2}-\rho_i-2i-2>0.\\
\end{cases}
\end{align}
Note that, if we let
\begin{align*}
\begin{cases}
\rho_i^{2}-n+i+3=1,\\
\rho_i^{2}-\rho_i-2i-2=2,
\end{cases}
\end{align*}
then we have
\begin{align*}
\begin{cases}
\rho_i=\sqrt{n-i-2},\\
\rho_i=\frac{1+\sqrt{8i+17}}{2}.
\end{cases}
\end{align*}
By calculation, we can find that $i=\alpha=(6n-37-\sqrt{24n+1})/18$ is the only solution of $\sqrt{n-i-2}=(1+\sqrt{8i+17})/2$. Since $i \in \mathbb{N}$, we will complete the proof by classifying the value of $i$.
\textbf{Case 1.} If $2\leq i\leq \lfloor \alpha\rfloor$.
We have $\sqrt{n-i-2} \geq (1+\sqrt{8i+17})/2$. We claim that $(1+\sqrt{8i+17})/2 \leq \rho_{i}\leq \sqrt{n-i-2}$. Suopose that $\rho_{i}<(1+\sqrt{8i+17})/2$. By (\ref{eq3.14}), we have $0<\rho_i^{2}-n+i+3<1$ and $0<\rho_i^{2}-\rho_i-2i-2<2$. Then $(\rho_i^{2}-n+i+3)(\rho_i^{2}-\rho_i-2i-2)<2$, a contradiction. Suopose that $\rho_{i}>\sqrt{n-i-2}$. By (\ref{eq3.14}), we obtain that $\rho_i^{2}-n+i+3>1$ and $\rho_i^{2}-\rho_i-2i-2>2$. Then $(\rho_i^{2}-n+i+3)(\rho_i^{2}-\rho_i-2i-2)>2$, a contradiction.
Thus we have $(1+\sqrt{8i+17})/2 \leq \rho_{i}\leq \sqrt{n-i-2}$. This induces that $\rho_i^{2}-n+i+3\leq 1$ and $\rho_i^{2}-\rho_i-2i-2\geq 2$, which lead to $x_{6}^{i}\geq x_{5}^{i}$. Therefore
\begin{equation}\label{eq3.15}
\begin{split}
&\rho_{i-1}-\rho_{i}\\
\geq & \bm{x_{i}}^{T}A(F_n(i-1))\bm{x_{i}}-\bm{x_{i}}^{T}A(F_n(i))\bm{x_{i}}\\
= &2x_{5}^{i}(x_{4}^{i}-x_{2}^{i}-x_{3}^{i})\\
= &2\rho_{i}x_{5}^{i}(x_{6}^{i}-x_{5}^{i})\\
\geq&0.
\end{split}
\end{equation}
Now we only need to prove $\rho_{i-1}\neq \rho_{i}$. Suppose that $\rho_{i-1}=\rho_{i}$, then $\rho_{i-1}=\bm{x_{i}}^{T}A(F_n(i-1))\bm{x_{i}}$. By Theorem \ref{3.12}, we have
\begin{align*}
\rho_{i-1}x_{4}^{i}=x_{1}^{i}+(n-i-4)x_{6}^{i}+x_{5}^{i},
\end{align*}
and since
\begin{align*}
\rho_{i}x_{4}^{i}=x_{1}^{i}+(n-i-4)x_{6}^{i},
\end{align*}
we obtain $0=(\rho_{i-1} -\rho_{i})x_{4}^{i}=x_{5}^{i}$, which contradicts to the definition of the principal eigenvector.
Therefore, from (\ref{eq3.15}) we have $\rho_{i-1}>\rho_{i}$ for $2\leq i\leq \lfloor \alpha\rfloor$.
\textbf{Case 2.} If $\lceil \alpha\rceil\leq i\leq \frac{n-6}{2}$.
We have $\sqrt{n-i-2} \leq (1+\sqrt{8i+17})/2$. Similarly, by (\ref{eq3.14}), we conclude that $ \sqrt{n-i-2}\leq \rho_{i}\leq (1+\sqrt{8i+17})/2$. This induces that $\rho_i^{2}-n+i+3\geq 1$ and $\rho_i^{2}-\rho_i-2i-2\leq 2$, which lead to $x_{5}^{i}\geq x_{6}^{i}$, therefore
\begin{equation}\label{eq3.16}
\begin{split}
&\rho_{i+1}-\rho_{i}\\
\geq & \bm{x_{i}}^{T}A(F_n(i+1))\bm{x_{i}}-\bm{x_{i}}^{T}A(F_n(i))\bm{x_{i}}\\
=&2x_{6}^{i}(x_{2}^{i}+x_{3}^{i}-x_{4}^{i})\\
= &2\rho_{i}x_{6}^{i}(x_{5}^{i}-x_{6}^{i})\\
\geq&0.
\end{split}
\end{equation}
Similarly, we have $\rho_{i+1}\neq \rho_{i}$. Using this, from (\ref{eq3.16}), we obtain $\rho_{i+1}> \rho_{i}$ for $\lceil \alpha\rceil \leq i\leq \frac{n-6}{2}$.
Therefore, the proof of Lemma is completed.
\end{proof}
\subsection{Step 4}
It only remains for the case that $5\leq n \leq 11$. Applying Proposition \ref{g1}, \ref{g10} and \ref{g7}, we obtain the extremal graphs with minimum spectral radius in $M_{n}(G_{1})$, $M_{n}(G_{7})$ and $M_{n}(G_{10})$, respectively. And then calculate their spectral radii by using MATLAB, as shown in Table \ref{tab3}, where the extremal graphs and the minimum spectral radii are bolded.
\begin{table}[H]\Huge
\centering
\caption{The extremal graph with minimum spectral radius in $M_{n}(G_{1}),M_{n}(G_{7}),M_{n}(G_{10}).$}
\label{tab3}
\resizebox{\textwidth}{!}{
\begin{tabular}{ccccccc}
\hline
\multirow{2}{*}{n} &
\multicolumn{2}{c}{$M_{n}(G_{1})$} &
\multicolumn{2}{c}{$M_{n}(G_{7})$} &
\multicolumn{2}{c}{$M_{n}(G_{10})$} \\
\cline{2-7}
& Extremal graph & Spectral radius& Extremal graph & Spectral radius& Extremal graph & Spectral radius\\ \hline
5 & $G_{1} \circ(1,1,1,1,1)$ & 2.2143 & \bm{$G_{7} \circ(1,1,1,1,1)$} & \textbf{2.0000} & $\backslash$ & $\backslash$ \\
6 & \bm{$G_{1} \circ(1,1,1,1,2)$} & \textbf{2.2784} & $G_{7} \circ(2,1,1,1,1)$ & 2.3912 & $G_{10} \circ(1,1,1,1,1,1)$ & 2.6544 \\
7 & \bm{$G_{1} \circ(1,1,1,1,3)$} & \textbf{2.3686} & $G_{7} \circ(2,1,2,1,1)$ & 2.6813 & $G_{10} \circ(1,1,1,1,1,2)$ & 2.6751 \\
8 & \bm{$G_{1} \circ(1,1,1,1,4)$} & \textbf{2.4860} & $G_{7} \circ(3,1,2,1,1)$ & 2.9764 & $G_{10} \circ(1,1,1,1,1,3)$ & 2.7033 \\
9 & \bm{$G_{1} \circ(1,1,1,1,5)$} & \textbf{2.6239} & $G_{7} \circ(3,1,3,1,1)$ & 3.2176 & $G_{10} \circ(1,1,1,1,1,4)$ & 2.7448 \\
10 & \bm{$G_{1} \circ(1,1,1,1,6)$} & \textbf{2.7724} & $G_{7} \circ(4,1,3,1,1)$ & 3.4630 & $G_{10} \circ(1,1,1,1,1,5)$ & 2.8060 \\
11 & $G_{1} \circ(1,1,1,1,7)$ & 2.9243 & $G_{7} \circ(4,1,4,1,1)$ & 3.6737 & \bm{$G_{10} \circ(1,1,1,1,1,6)$} & \textbf{2.8915} \\ \hline
\end{tabular}}
\end{table}
By Table \ref{tab4}, we obtain that when $5\leq n \leq 11$, the extremal graph with minimum spectral radius of rank $5$ is:
\begin{itemize}
\item $G_{7}=C_{5}$, for $n=5$;
\item $G_{1} \circ(1,1,1,1,n-4)$, for $6\leq n \leq 10$;
\item $G_{10} \circ(1,1,1,1,1,n-5)$, for $n=11$.
\end{itemize}
\section{Concluding remarks}
In the last case of Theorem \ref{2}, we obtain that $k \in \{ \lfloor \frac{6n-37-\sqrt{24n+1}}{18} \rfloor$ $,\lceil \frac{6n-37-\sqrt{24n+1}}{18} \rceil\}$. When $12\leq n \leq 23$, we use the MATLAB software to calculate the spectral radii of the graphs in $\mathcal{F}=\{F_n(i):1\leq i\leq \frac{n-4}{2}\}$, as shown in the Table \ref{tab4}, where the minimum spectral radius is bolded. It demonstrates that $k=\lfloor \frac{6n-37-\sqrt{24n+1}}{18} \rfloor$ or $\lceil \frac{6n-37-\sqrt{24n+1}}{18} \rceil$ depends on $n$.
\begin{table}[H]
\centering
\caption{$\rho(F_{n}(i))$.}
\label{tab4}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|ccccccccc|c}
\hline
\diagbox{$n$}{$\rho(F_{n}(i))$}{$i$} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9& $\frac{6n-37-\sqrt{24n+1}}{18}$\\ \hline
12 & \textbf{3} & 3.1370 & 3.4319& 3.7362 & \textbackslash{} & \textbackslash{} & \textbackslash{} & \textbackslash{} & \textbackslash{}&1 \\
13 & \textbf{3.1239} & 3.1818 & 3.4431& 3.7404& \textbackslash{} & \textbackslash{} & \textbackslash{} & \textbackslash{} & \textbackslash{}& 1.2949\\
14 & 3.255 &\textbf{3.2470}&3.4588&3.7457&4.0278& \textbackslash{} & \textbackslash{} & \textbackslash{} & \textbackslash{} & 1.5912\\
15 & 3.3894& \textbf{3.3347} & 3.4817 & 3.7525& 4.0308& \textbackslash{} & \textbackslash{} & \textbackslash{} & \textbackslash{} & 1.8889 \\
16 & 3.5227& \textbf{3.4402}& 3.5160& 3.7616& 4.0344& 4.2979 & \textbackslash{} & \textbackslash{} & \textbackslash{} & 2.1877 \\
17 & 3.6539 & \textbf{3.5563} & 3.5674 & 3.7743& 4.0389& 4.3001 & \textbackslash{} & \textbackslash{} & \textbackslash{} & 2.4876 \\
18 &3.7824 & 3.6770& \textbf{3.6394} & 3.7926& 4.0446 &4.3027& 4.5506& \textbackslash{} & \textbackslash{} & 2.7884 \\
19 & 3.9079& 3.7889& \textbf{3.7303}&3.8199& 4.0523 & 4.3058 &4.5522& \textbackslash{} & \textbackslash{} & 3.0901 \\
20 & 4.0303 & 3.9201& \textbf{3.8338} & 3.8612& 4.0628 & 4.3097& 4.5542& 4.7888& \textbackslash{} & 3.3927 \\
21 &4.1498&4.0396& 3.9439&\textbf{3.9211}& 4.0779& 4.3147 & 4.5565 & 4.7900& \textbackslash{} & 3.6960\\
22 &4.2663 & 4.1570 & 4.0564 & \textbf{4} & 4.1002 & 4.3213& 4.5593 & 4.7915& 5.0146 & 4\\
23 &4.3801 & 4.2721 & 4.1694 & \textbf{4.0929 } & 4.1341 & 4.3303 & 4.5627 & 4.7933& 5.0157 & 4.3047\\ \hline
\end{tabular}}
\end{table}
It is a natural problem to determine the extramal spectral radii of the graphs of order $n$ and rank $r$. By Theorem \ref{1}, we know that the maximum spectral radius of all connected graphs of order $n$ and rank $r$ is $\rho(T(n,r))$. Feng et al. gave the spectral radius of $T(n,r)$ in \cite{13}.
\begin{theorem}\cite{13}
Let $T(n,r)$ be a Tur\'{a}n graph. Then
$$\rho(T(n,r))=\frac{1}{2}\left(n-2\lfloor \frac{n}{r} \rfloor-1+\sqrt{(n+1)^{2}-4(n-r\lfloor \frac{n}{r} \rfloor)\lceil \frac{n}{r} \rceil}\right)\leq n-\lfloor \frac{n}{r} \rfloor$$
with the last equality if and only if $T(n,r)$ is regular.
\end{theorem}
Further, we obtain a sharp upper and lower bound for the spectral radius of the extremal graph $G$ which attains the minimum spectral radius among all connected graphs of order $n\geq 12$ and rank $5$. By Theorem \ref{2}, we know that
\begin{align*}
\rho(G)= \ \text{min} \ \{\rho(F_n(\lfloor \alpha\rfloor)), \rho(F_n(\lceil \alpha\rceil)) \},
\end{align*}
where $\alpha=\frac{6n-37-\sqrt{24n+1}}{18}$.
From the proof of Lemma \ref{3.13}, we have
\begin{align*}
\frac{1+\sqrt{8\lfloor \alpha\rfloor +17}}{2} \leq \rho(F_n(\lfloor \alpha\rfloor) \leq \sqrt{n-\lfloor \alpha\rfloor-2},\\
\sqrt{n-\lceil \alpha\rceil-2} \leq \rho(F_n(\lceil \alpha\rceil) \leq \frac{1+\sqrt{8\lceil \alpha \rceil +17}}{2}.
\end{align*}
Therefore, we obtain that
\begin{align*}
\rho(G) \geq \text{min} \{ \frac{1+\sqrt{8\lfloor \alpha\rfloor +17}}{2}, \sqrt{n-\lceil \alpha\rceil-2}\},
\end{align*}
and
\begin{align*}
\rho(G) \leq \text{min} \{\sqrt{n-\lfloor \alpha\rfloor-2}, \frac{1+\sqrt{8\lceil \alpha \rceil +17}}{2}\}.
\end{align*}
In general, the problem of determining the minimum spectral radius of all connected graphs with order $n$ and rank $r$ deserves further study.
\section*{Declaration of compting interest}
There is no competing interest.
\end{document} |
\begin{document}
\title{{\large Tur\'{a}n numbers of complete $3$-uniform Berge-hypergraphs}}
\author{\small L. Maherani$^{\textrm{a}}$, M. Shahsiah$^{\textrm{b,c}}$ \\
\footnotesize $^{\textrm{a}}$ Department of Mathematical Sciences,
Isfahan
University of Technology,\\ \footnotesize Isfahan, 84156-83111, Iran\\
{\small $^{\textrm{b}}$Department of Mathematics, Alzahra University,}\\
{\small P.O. Box 1993891176, Tehran, Iran}\\
{\small $^{\textrm{c}}$School of Mathematics, Institute for Research in Fundamental Sciences (IPM),}\\
{\small P.O. Box 19395-5746, Tehran, Iran }\\
\footnotesize {[email protected],
[email protected]}}
\date {}
\footnotesize\maketitle
\begin{abstract}\rm{}
\footnotesize
Given a family $\mathcal{F}$ of $r$-graphs, the Tur\'{a}n number of $\mathcal{F}$ for a given
positive integer $N$, denoted by $ex(N,\mathcal{F})$, is the maximum number of edges of an $r$-graph on $N$ vertices
that does not contain any member of $\mathcal{F}$ as a subgraph. For given $r\geq 3$, a complete $r$-uniform Berge-hypergraph, denoted by { ${K}_n^{(r)}$}, is an $r$-uniform hypergraph of order $n$ with the core sequence $v_{1}, v_{2}, \ldots ,v_{n}$ as the vertices and distinct edges $e_{ij},$ $1\leq i<j\leq n,$ where every $e_{ij}$ contains both $v_{i}$ and $v_{j}$. Let $\mathcal{F}^{(r)}_n$ be the family of complete $r$-uniform Berge-hypergraphs of order $n.$ We determine precisely $ex(N,\mathcal{F}^{(3)}_{n})$ for $n \geq 13$. We also find the extremal hypergraphs avoiding $\mathcal{F}^{(3)}_{n}$.
\\{ {Keywords}:{ \footnotesize Tur\'{a}n number, Extremal hypergraph, Berge-hypergraph.
}}
\noindent
\\{\footnotesize {AMS Subject Classification}: 05C65, 05C35, 05D05.}
\end{abstract}
\small
\section{\normalsize{Introduction}}
A {\it hypergraph} $\mathcal{H}$ is a pair $\mathcal{H}=(V,E)$, where $V$ is a finite non-empty set (the set of
vertices) and $E$ is a collection of distinct non-empty subsets of $V$ (the set of edges). We denote by $e(\mathcal{H})$ the number of edges of $\mathcal{H}.$
An {\it $r$-uniform
hypergraph} or {\it $r$-graph} is a hypergraph such that all its edges have size $r$. A {\it complete $r$-uniform hypergraph} of
order $N$, denoted by { $\mathcal{K}_N^r$}, is a hypergraph consisting of
all the $r$-subsets of a set $V$ of cardinality $N$. For a family $\mathcal{F}$ of $r$-graphs, we say that the hypergraph $\mathcal{H}$ is $\mathcal{F}$-free if $\mathcal{H}$ does not contain any member of $\mathcal{F}$ as a subgraph.
Given a family $\mathcal{F}$ of $r$-graphs, the {\it Tur\'{a}n number} of $\mathcal{F}$ for a given
positive integer $N$, denoted by $ex(N,\mathcal{F})$, is the maximum number of edges of an $\mathcal{F}$-free $r$-graph on $N$ vertices.
An $\mathcal{F}$-free $r$-graph $\mathcal{H}$ on $N$ vertices is {\it extremal hypergraph} for $\mathcal{F}$ if $e(\mathcal{H})= ex(N,\mathcal{F})$. These are natural generalizations of the classical Tur\'{a}n number for
$2$-graphs \cite{turan}.
For given $n, r \geq 2$, let $\mathcal{H}^{(r)}_n$ be the family of $r$-graphs $F$ that have at most ${n \choose 2}$ edges, and have some set $T$ of size $n$ such that every pair of vertices in $T$ is contained in some edge of $F$.
Let the $r$-graph $H^{(r)}_n \in \mathcal{H}^{(r)}_n$ be obtained from the complete $2$-graph $\mathcal{K}_{n}^{2}$ by enlarging each edge with a new set of $r-2$ vertices. Thus $H^{(r)}_n$ has $(r-2){n \choose 2}+n$ vertices and ${n \choose 2}$ edges. For given $n \geq 5$ and $r\geq 3$, a {\it complete $r$-uniform Berge-hypergraph} of order $n$, denoted by { ${K}_n^{(r)}$}, is an $r$-uniform hypergraph with the core sequence $v_{1}, v_{2}, \ldots ,v_{n}$ as the vertices and ${n \choose 2}$ distinct edges $e_{ij},$ $1\leq i<j\leq n,$ where every $e_{ij}$ contains both $v_{i}$ and $v_{j}$. Note that a complete $r$-uniform Berge-hypergraph is not determined uniquely as there are no constraints on how the $e_{ij}$'s intersect outside $\{v_{1}, v_{2}, \ldots ,v_{n}\}$.
\\
Extremal graph theory is that area of combinatorics
which is concerned with finding the largest, smallest, or otherwise optimal
structures with a given property. There is a long history in
the study of extremal problems concerning hypergraphs. The first such result is due to Erd\H{o}s, Ko and Rado \cite{Erdos-ko-rado}.\\
In contrast to the graph case, there are comparatively few known results on
the hypergraph Tur\'{a}n problems. In the
paper in which Tur\'{a}n proved his classical theorem on the extremal numbers
for complete graphs \cite{turan}, he posed the natural question of determining the Tur\'{a}n number of the complete $r$-uniform hypergraphs. Surprisingly, this
problem remains open in all cases for $r > 2$, even up to asymptotics. Despite the lack of progress on the Tur\'{a}n problem for dense hypergraphs, there are considerable results on certain sparse hypergraphs.
Recently, some interesting results were
obtained on the exact value of extremal number of paths and cycles in hypergraphs.
F{\"u}redi et al. \cite{loose pathI} determined the extremal number of $r$-uniform loose paths of length $n$ for $r\geq 4$ and large $N$.
They also conjectured a similar result for $r = 3$. F{\"u}redi and Jiang \cite{loose cycleI} determined the extremal function of loose cycles of length $n$ for $r\geq 5$ and large $N.$ Recently, Kostochka et al. \cite{loose paths and cycles} extended these results to $r=3$ for loose paths and $r=3,4$ for loose cycles. Gy{\H{o}}ri et al. \cite{Berge paths} found the extremal numbers of $r$-uniform hypergraphs avoiding Berge paths of length $n$. Their results substantially extend earlier results of Erd\H{o}s and Gallai \cite{Erdos-Gallai} on extremal number of paths in graphs. Let $\mathcal{C}_{n}^{(r)}$
denote the family of $r$-graphs that are Berge cycles of length $n$. Gy{\H{o}}ri and Lemons \cite{Berge cyclesI, Berge cycles2}
showed that for all $r \geq 3$ and $ n \geq 3$, there exists a positive constant $c_{r,n},$ depending on $r$ and $n,$ such that
$$ex(N,\mathcal{C}_{n}^{(r)}) \leq c_{r,n} N^{1+\frac{1}{\lfloor \frac{n}{2} \rfloor}}.$$
Let $N$, $n$, $r$ be integers, where $N\geq n >r$ and $r \geq 2$. Also let $ T_r(N,n-1)$ be the complete $r$-uniform $(n-1)$-partite hypergraph with $N$ vertices and $n-1$ parts $V_1,V_2,...,V_{n-1}$ whose partition sets differ in size by at most 1. Suppose that $t_r(N,n-1)$ denotes the number of edges of $T_r(N,n-1)$. If $N=\ell(n-1)+j$, where $\ell \geq 1$ and $1 \leq j\leq n-1$, then it is straightforward to see that
$$t_r(N,n-1)= \sum _{i=0}^{r} \ell ^{r-i} {j \choose i}{n-1-i \choose r-i}.$$
In 2006, Mubayi \cite{mubay} showed that the unique largest $\mathcal{H}_{n}^{(r)}$-free $r$-graph on $N$ vertices is $T_r(N,n-1)$.
Settling a conjecture of Mubayi in \cite{mubay}, Pikhurko \cite{pikh} proved that there exists $N_0$ so that the
Tur\'{a}n numbers of ${H}_{n}^{(r)}$ and $\mathcal{H}_{n}^{(r)}$ coincide for all $N>N_0.$
Let $\mathcal{F}^{(r)}_n$ be the family of complete $r$-uniform Berge-hypergraphs of order $n.$ Because ${H}_{n}^{(3)} \in \mathcal{F}_{n}^{(3)}$, the Pikhurko's result \cite{pikh} implies that $ex (N,\mathcal{F}_{n}^{(3)}) \leq t_3(N,n-1)$ for sufficiently large $N$.
In this paper, for $N \geq 13$, we show that $ex (N,\mathcal{F}_{n}^{(3)})=t_3(N,n-1)$ and $ T_3(N,n-1)$ is the unique extremal hypergraph for $\mathcal{F}_{n}^{(3)}$. More precisely, we prove the following theorem.\\
\begin{theorem}\label{main}
Let $N,n$ be integers so that $N\geq n\geq 13$. Then
$$ex(N,\mathcal{F}_n^{(3)})=t_3(N,n-1).$$
Furthermore, the unique extremal hypergraph for $\mathcal{F}_n^{(3)}$ is $T_3(N,n-1)$.
\end{theorem}
First we show that $ex (N,\mathcal{F}_{n}^{(r)}) \geq t_r(N,n-1)$. To see that, consider an arbitrary sequence $v_1,v_2,...,v_n$ of the vertices of $ T_r(N,n-1)$. By the pigeonhole principle, there exists some part $V_h$, $1 \leq h \leq n-1$, in $T_r(N,n-1)$ containing at least two vertices of this sequence. Since every edge of $ T_r(N,n-1)$ includes at most one vertex of each part $V_i$, $1 \leq i \leq n-1$, This sequence can not be the core sequence of a $K_{n}^{(r)}$. Hence $T_r(N,n-1)$ is $\mathcal{F}_{n}^{(r)}$-free and
\begin{equation}\label{lbound}
ex (N,\mathcal{F}_{n}^{(r)}) \geq t_r(N,n-1), \ \ \ \ \ \ r\geq 3.
\end{equation}
Therefore, in order to clarify Theorem \ref{main}, it suffices to show that $ex (N,\mathcal{F}_{n}^{(3)}) \leq t_3(N,n-1)$ and $T_3(N,n-1)$ is the only $\mathcal{F}_{n}^{(3)}$-free hypergraph with $N$ vertices and $t_3(N,n-1)$ edges. Here, we give a proof by induction on the number of vertices. More precisely, we prove Theorem \ref{main} in three steps. First, we show that Theorem \ref{main} holds for $N=n$ (see Theorem \ref{n}). Then, in Theorem \ref{l=1}, we demonstrate that it is true for $n\leq N \leq 2n-2.$ Finally, using Theorem \ref{n} and Theorem \ref{l=1}, we show that the desired holds for all $N\geq n$ (Section 3).\\
\noindent
\textbf{Conventions and Notations:}
For an $r$-uniform hypergraph $\mathcal{H}=(V,E)$, the complement hypergraph of $\mathcal{H}$, denoted by $\mathcal{H}^c$, is the hypergraph on $V$ so that $E(\mathcal{H}^c)= {V \choose r}\setminus E$. Also we say that $X\subseteq V$ is an independent set of $\mathcal{H}$ if for any pair $ v,v' \in X$, there is no edges in $E$ containing both of $v$ and $v'$. For $U\subseteq V$ we denote by $\mathcal{H}[U]$ the subgraph of $\mathcal{H}$ induced by the edges of $U$. For $U, W \subseteq V$, The hypergraph $\mathcal{H}[U,W]$ is the subgraph of $\mathcal{H}$ induced by the edges of $\mathcal{H}$ intersecting both $U$ and $W.$ For a vertex $v \in V$, the degree of $v$ in $\mathcal{H}$, denoted by $d_\mathcal{H}(v)$, is the number of edges in $\mathcal{H}$ containing $v$. Also $\mathcal{H}-v$ is the subhypergraph of $\mathcal{H}$ obtained by deleting of $v$ and all the edges containing it.\\
\section{Preliminaries }
In this section, we present some
results that will be used in the follow up section. Let $\mathcal{A}=\{A_{1},A_{2},...,A_{n}\}$ be a family of subsets of a set $X$. A system of distinct representatives,
or SDR, for the family $\mathcal{A}$, is a set $\{a_1,a_2,...,a_n\}$ of elements of $X$ satisfying
two following conditions:
\begin{itemize}
\item[$\bullet$] $a_i \in A_i$ \ \ \ \ \ \ $i=1,...,n$,
\item[$\bullet$] $a_i \neq a_j$ \ \ \ \ \ \ $i \neq j$.
\end{itemize}
\begin{lemma}\label{sdr}
Let $U=\{u_1,u_2,...,u_m\}$, $m\geq 5$ and $x \notin U$. Also, let $\mathcal{A}=\{A_{1},A_{2},...,A_{m}\}$ be a family of sets so that $\vert A_{1}\vert \leq \vert A_{2}\vert \leq ... \vert A_{m}\vert $ and
$A_{i} \subseteq \{ B :\ B=\{x, u_i ,u_k \},\ k\neq i \}$ for $ 1\leq i \leq m.$
If $\mathcal{A}$ has no SDR, then $$\vert \bigcup _{i=1}^{m} A_{i} \vert \leq {m-1 \choose 2}$$ and equality holds if and only if $A_{1}= \emptyset$ and
$$A_{i} = \{ B :\ B=\{x, u_i ,u_k \},\ k\neq 1,i \} \ \ \ \ \ \ 2\leq i \leq m.$$
\end{lemma}
\noindent\textbf{Proof. } Since $\mathcal{A}$ contains no SDR, using the Hall's theorem \cite{hall}, for some $q$, $1\leq q\leq m$, we have $\vert \bigcup _{i=1}^{q} A_{i} \vert \leq q-1$. So $\vert \bigcup _{i=1}^{m} A_{i} \vert \leq f(q),$ where $f(k)= k-1 + {m-k \choose 2}$, for $1 \leq k \leq m$. On the other hand, one can easily see that $f(1)>f(k)$, for $2\leq k \leq m$. Therefore
$$\vert \bigcup _{i=1}^{m} A_{i} \vert \leq f(1)={m-1 \choose 2}$$ and the equality holds if and only if $A_{1}= \emptyset$ and
$$A_{i} = \{ B :\ B=\{x, u_i ,u_k \},\ k\neq 1,i \} \ \ \ \ \ \ 2\leq i \leq m.$$
$
\blacksquare$\\
In order to state our main results we need some definitions. Let $\mathcal{H}=(V,E)$ be an $r$-uniform hypergraph, where $V=\{v_1,v_2,...,v_n\}$ and $E=\{e_1,e_2,...,e_m\}$.
We denote by $B(\mathcal{H}),$ the bipartite graph with parts $X$ and $Y$
so that $X=\{v_iv_k :\ i< k\ \ {\rm and} \ \ v_i,v_k \in V(\mathcal{H})\}$, $Y=E(\mathcal{H})$ and $v_iv_k$ is adjacent to $e_h$ if and only if $\{v_i,v_k \}\subseteq e_h$, for every $v_iv_k\in X$ and $e_h \in Y$. For every $v_iv_k\in X$, $d_{B(\mathcal{H})}(v_iv_k)$ is the number of edges in $B(\mathcal{H})$ containing $v_i v_k$.
A matching of $X$ in $B(\mathcal{H})$ is matching that saturates all vertices of $X$. Note that, every matching of $X$ in $B(\mathcal{H})$ is equivalent to a complete $r$-uniform Berge-hypergraph with core sequence $v_1,v_2,...,v_n$. \\
\noindent Now, we demonstrate that Theorem \ref{main} holds for $N=n$.
\begin{theorem}\label{n}
Let $n\geq 13$ be an integer. The hypergraph $T_3(n,n-1)$ is the only $\mathcal{F}_{n}^{(3)}$-free hypergraph with $n$ vertices and $ex(n,\mathcal{F}_{n}^{(3)})$ edges.
\end{theorem}
\noindent\textbf{Proof. }Assume that $\mathcal{H}$ is an $\mathcal{F}_{n}^{(3)}$-free hypergraph with $n$ vertices and $ex(n,\mathcal{F}_{n}^{(3)})$ edges. Let $V(\mathcal{H})=\{v_1,v_2,...,v_n\}$. First, suppose that there is a vertex $v\in V(\mathcal{H})$, say $v_n$, so that $d_{\mathcal{H}}(v_n) \leq {n-2 \choose 2}$. Therefore
\begin{equation}\label{up1}
e(\mathcal{H}) =d_{\mathcal{H}}(v_n) +e(\mathcal{H}-v_n) \leq {n-2 \choose 2} +{n-1 \choose 3}=t_3(n,n-1).
\end{equation}
\noindent So by (\ref {lbound}) and (\ref{up1}), we have
$$ex(n,\mathcal{F}_{n}^{(3)}) = t_3(n,n-1).$$
Therefore $d_{\mathcal{H}}(v_n) ={n-2 \choose 2}$ and $e( \mathcal{H}-v_n )={n-1 \choose 3}$. So $\mathcal{H}-v_n \cong \mathcal{K}_{n-1}^3$ and clearly there is a copy of $K_{n-1}^{(3)}$ with the core sequence $v_1,v_2,...,v_{n-1}$ in $\mathcal{H}-v_n$.
Set $x=v_n$, $U=\{v_1,v_2,...,v_{n-1}\}$ and $\mathcal{A}=\{A_{1},A_{2},...,A_{n-1}\}$, where $$A_{i} = \{ B :\ B\in E(\mathcal{H}),\ \{x,v_{i}\} \subseteq B\} \ \ \ \ \ \ 1\leq i\leq n-1.$$
Note that $d_{\mathcal{H}}(v_n)=\vert \bigcup _{i=1}^{n-1} A_{i} \vert= {n-2 \choose 2}.$
Since $ \mathcal{H}$ is $\mathcal{F}_{n}^{(3)}$-free and there is a copy of $K_{n-1}^{(3)}$ in $ \mathcal{H}-v_n$, $\mathcal{A}$ has no SDR.
Now, using Lemma \ref{sdr}, we have $\mathcal{H}\cong T_3(n,n-1)$.\\*
Now suppose that for every vertex $v\in V(\mathcal{H})$, $d_{\mathcal{H}}(v) \geq {n-2 \choose 2}+1$. Set $G=B(\mathcal{H})$. So we may assume that $G=[X,Y]$, where
$$X=\{u_{ik}=v_iv_k :\ i< k\ \ {\rm and} \ \ v_i,v_k \in V(\mathcal{H})\}$$ and $Y=E(\mathcal{H})$. Since, by (\ref{lbound}), $|Y|\geq {n-1 \choose 3} +{n-2 \choose 2},$ we have $\vert X\vert \leq \vert Y\vert$. Let $X=X_1 \cup X_2$, where $X_{1}=\{u \in X :\ d_{G}(u)\leq 4\}$ and $X_2 =X\setminus X_1$.
Recall that every matching of $X$ in $G$ is equivalent to a $K_{n}^{(3)}$ in $\mathcal{H}$.
We have two following cases.\\
\noindent{\bf Case 1.} $X_1 =\emptyset$.\\
Since for every $y\in Y$ and $u\in X$, we have $d_{G}(y)=3$ and $d_{G}(u)\geq 5$, the Hall's theorem \cite{hall} guarantees the existence of a matching of $X$, a contradiction.\\\\
\noindent{\bf Case 2.} $X_1 \neq \emptyset$.\\
Let $X_1 = \{v_{i_1}v_{i'_1},v_{i_2}v_{i'_2},...,v_{i_t}v_{i'_t}\}$. We show that the following claim holds.
\begin{emp}\label{pairdisj1}
The elements of $X_1$ are pairwise disjoint.
\end{emp}
\noindent\textbf{Proof of Claim \ref{pairdisj1}}. Suppose to contrary that for $2 \leq s \leq t$, $ \{wv_{i'_1},wv_{i'_2},...,wv_{i'_s}\} \subseteq X_1$. So $ d_{\mathcal{H}}(w) \leq f(s),$ where $f(k)= 4k + {n-k-1 \choose 2}$ is a function on $k$, $2 \leq k \leq t \leq n-1$. Using $n\geq 13$, it is straightforward to see that the absolute maximum of $f(k)$ occurs in point $k=2$. Hence $$ d_{\mathcal{H}}(w) \leq f(2) = 8 + {n-3 \choose 2}.$$
Since $ 8 + {n-3 \choose 2} < {n-2 \choose 2}+1$ for $n\geq 13$, we have $ d_{\mathcal{H}}(w) < {n-2 \choose 2}+1$. That is a contradiction to our assumption.
$
\square$\\
\noindent Since for every vertex $v \in V(\mathcal{H})$, we have $ d_{\mathcal{H}}(v) \geq {n-2 \choose 2}+1$, so for any two vertices $x,y \in V(\mathcal{H})$, there is at least one edge in $E(\mathcal{H})$ containing both of $x$ and $y$.
So $d_{G}(v_{i_l}v_{i'_l}) \geq 1$ for every $1 \leq l \leq t$. On the other hand, by Claim \ref{pairdisj1}, the elements of $X_1$ are pairwise disjoint. Therefore $G$ contains a matching $M_1$ of $X_1$. Suppose that $G'=[X_2,Y']$ is the subgraph of $G$ so that $Y' \subset Y$ is obtained by deleting the vertices of $M_1$. Note that for every $u \in X_2$ and $y \in Y'$, we have $d_{G'}(u) \geq 3$ and $d_{G'}(y) \leq 3$. Therefore the Hall's theorem \cite{hall} implies the existence of a matching $M_2$ of $X_2$ in $G'$. This is a contradiction, since $M_1 \cup M_2$ is a matching of $X$ in $G$. This contradiction completes the proof.
$
\blacksquare$
\begin{theorem}\label{l=1}
Let $n\geq 13$ and $N,n$ be integers so that $n\leq N \leq 2n-2$.
Also, let $\mathcal{H}$ be an $\mathcal{F}_{n}^{(3)}$-free hypergraph with $N$ vertices and $ex(N,\mathcal{F}_{n}^{(3)})$ edges. Then $ e(\mathcal{H}) = t_3(N,n-1)$ and $\mathcal{H}\cong T_3(N,n-1)$.
\end{theorem}
\noindent\textbf{Proof. } Let $N=n-1+j$, where $1\leq j \leq n-1$. We apply induction on $j$. Using Theorem \ref{n}, the basic step $j=1$ is true. For the induction step, let $j >1$. Set
$$d={n-2 \choose 2}+(j-1)(n-3)+{j-1 \choose 2}.$$
First suppose that there is a vertex $x \in V(\mathcal{H})$ so that $ d_{\mathcal{H}}(x) \leq d$. So using the induction hypothesis, we have
$$e( \mathcal{H}) = d_{\mathcal{H}}(x) + e( \mathcal{H}-x) \leq d+t_3(N-1,n-1) =t_3(N,n-1).$$
Therefore by (\ref{lbound}), we conclude that $e(N,\mathcal{F}_{n}^{(3)}) =t_3(N,n-1)$. Hence $d_{\mathcal{H}}(x) =d$ and $e( \mathcal{H}-x) = t_3(N-1,n-1)$. So, using the induction hypothesis, $\mathcal{H}-x \cong T_3(N-1,n-1)$. Hence we may assume that $\mathcal{H}-x$ is a complete $3$-uniform $(n-1)$-partite hypergraph with parts $V_1,V_2,...,V_{n-1}$, where
$$
V_i = \left\lbrace
\begin{array}{ll}
\{v_i,x_i\} & \ \ \ \ 1\leq i \leq j-1,
\\
\{v_i\} & \ \ \ \ j\leq i \leq n-1.
\end{array}
\right.
$$
Let $\mathcal{H}'$ be the induced subgraph of $\mathcal{H}-x$ on $\{v_1,v_2,...,v_{n-1}\}.$ According to the construction of $\mathcal{H}-x$, we have $\mathcal{H}'\cong \mathcal{K}_{n-1}^3$ and so there is a copy of
$K_{n-1}^{(3)}$ with core sequence $v_1,v_2,...,v_{n-1}$ in $ \mathcal{H}'$. Set $U=\{v_1,v_2,...,v_{n-1}\}$ and $\mathcal{A}=\{A_{1},A_{2},...,A_{n-1}\}$, where for $1\leq i\leq n-1$,
$$A_{i} = \{ e \in E(\mathcal{H}) : \ e=\{x,v_{i},v_k\}, \ \ k\neq i\}.$$
\noindent For a vertex $v \in V(\mathcal{H})$, we denote by $E_v$ the set of edges of $\mathcal{H}$ containing $v$. Clearly we have
\begin{equation}\label{dx}
d_{\mathcal{H}}(x)= |E_x| = |E_1|+ |E_2|+ |E_3|,
\end{equation}
where
$$E_i= \{e \in E_x:\ \ \vert e \cap \{x_1,x_2,...,x_{j-1}\}\vert =i-1\}, \ \ \ \ \ \ 1\leq i \leq 3.$$
\noindent We have the following claim.
\begin{emp}\label{Ex}
\noindent
\begin{itemize}
\item[{\rm (i)}] $\vert E_1\vert \leq {n-2 \choose 2}.$
\item[{\rm (ii)}] $ \vert E_2\vert \leq (j-1)(n-3).$
\item[{\rm(iii)}] $\vert E_3\vert \leq {j-1 \choose 2}.$
\end{itemize}
\end{emp}
\noindent\textbf{Proof of Claim \ref{Ex}}. (i)
Clearly $\vert E_1\vert =\vert \bigcup _{i=1}^{n-1} A_{i} \vert$. If $\mathcal{A}$ contains an SDR, then $x,v_1,v_2,...,v_{n-1}$ is the core sequence of a copy of $K_n^{(3)}$ in $\mathcal{H}$, a contradiction. So, using Lemma \ref{sdr},
$$\vert E_1\vert =\vert \bigcup _{i=1}^{n-1} A_{i} \vert \leq {n-2 \choose 2}.$$
\noindent (ii) For $1 \leq k \leq j-1$, set $$B_k=\{e \in E_2:\ \{x,x_k\}\subseteq e\}.$$
We demonstrate that for $1 \leq k \leq j-1$, $\vert B_k \vert \leq n-3$ and so
$$\vert E_2\vert =\vert \bigcup _{k=1}^{j-1} B_k \vert \leq (j-1)(n-3).$$
Because of the similarity, it suffices to show that $\vert B_1 \vert \leq n-3$. Suppose not. So $\vert B_1\vert \geq n-2.$ On the other hand, the construction of $\mathcal{H}-x$ and the fact that $\mathcal{F}_{n}^{(3)}\nsubseteq \mathcal{H}$ imply that every edge in $E_x$ contains at most one vertex of each $V_i,$ for $1\leq i \leq n-1.$ Hence $|B_1|=n-2$ and
$$B_1=\{\{x,x_1,v_2\}, \{x,x_1,v_3\},...,\{x,x_1,v_{n-1}\}\}.$$
In this case, there is no edge in $E(\mathcal{H})\setminus B_1$ containing both of $x$ and $v_i,$ for $2 \leq i \leq n-1$. To see it, suppose that $f=\{x,v_2,u\}\in E(\mathcal{H})\setminus B_1$. Let $\mathcal{H}''$ be the induced subgraph of $\mathcal{H}-x$ on $\{x_1,v_2,...,v_{n-1}\}.$ By the construction of $\mathcal{H}-x$, we have $\mathcal{H}''\cong \mathcal{K}_{n-1}^3$ and so
$\mathcal{H}''$ contains a $K_{n-1}^{(3)}$, say $\mathcal{K}'$. Hence $x, x_1,v_2,...,v_{n-1}$ represents the core sequence of a $K_{n}^{(3)}$ in $\mathcal{H}$ with the following edge assignments. Set $e_{xx_1}=\{x,x_1,v_2\}$, $e_{xv_2}=f$, $e_{xv_i}=\{x,x_1,v_i\}$ for $3 \leq i \leq n-1$ and other edges are selected from $E(\mathcal{K}')$. That is a contradiction to our assumption. Therefore the set of edges in $\mathcal{H}$ containing $x$ and $v_1$ is a subset of the following set:
$$S=\{\{x,v_1,x_2\}, \{x,v_1,x_3\},...,\{x,v_1,x_{j-1}\}\}.$$ Hence
$$d_{\mathcal{H}}(x) \leq |B_1|+ |S|+{j-1 \choose 2}=
(n-2)+(j-2)+{j-1 \choose 2} < d.$$ This contradiction demonstrates that $\vert B_1 \vert \leq n-3$ and so $\vert E_2\vert \leq (j-1)(n-3)$.\\
\noindent (iii) This case is trivial.
$
\square$\\
\noindent Since $d_{\mathcal{H}}(x)=d$, using (\ref{dx}) and Claim \ref{Ex}, we have
\begin{equation}\label{ddd}
\vert E_1\vert = {n-2 \choose 2},\ \ \ \vert E_2\vert = (j-1)(n-3),\ \ \ \vert E_3\vert = {j-1 \choose 2}.
\end{equation}
Since $\vert E_1\vert = {n-2 \choose 2}$, using the proof of part (i) of Claim \ref{Ex} and Lemma \ref{sdr}, for some $1 \leq i' \leq n-1$, $A_{i'} = \emptyset$ and
$$A_{i} = \{ e \in E(\mathcal{H}) : \ e=\{x,v_{i},v_l\}, \ \ l\neq i,i'\}, \ \ \ \ \ \ 1\leq i \leq n-1\ \ {\rm and} \ \ i\neq i'.$$
If $j \leq i' \leq n-1$, using (\ref{ddd}), we have $\mathcal{H} \cong T_3(N,n-1)$. Hence we may assume that for some $1 \leq i' \leq j-1$, say $i'=1$, ${A}_1 = \emptyset$. By considering the sets $E_1$ and $E_2$ and using (\ref{ddd}), it can be shown that $\mathcal{H}[x,x_1,v_2,...,v_{n-1}]\cong \mathcal{K}_{n}^{3}$ and so it contains a copy of $K_{n}^{(3)}$. This contradiction completes the proof of the theorem.
Now we may assume that for every vertex $x \in V(\mathcal{H})$, $d_{\mathcal{H}}(x) \geq d+1$. Set $G=B(\mathcal{H})$. So we may assume that $G=[X,Y]$, where $$X=\{u_{ik}=v_iv_k :\ i< k\ \ {\rm and}\ \ v_i,v_k \in V(\mathcal{H})\}$$ and $Y=E(\mathcal{H})$. Since, by (\ref{lbound}), $|Y|\geq \sum _{i=0}^{3} \ell ^{3-i} {j \choose i}{n-1-i \choose 3-i},$ we have $\vert X\vert \leq \vert Y\vert$. Recall that every matching of $X$ in $G$ is equivalent to a ${K}_{N}^{(3)}$ in $\mathcal{H}$. Let $X=X_1 \cup X_2 \cup X_3$, where
\begin{eqnarray*}
X_{1}&=&\{u \in X : d_{G}(u)=0\},\\
X_{2}&=&\{u \in X : 1 \leq d_{G}(u)\leq 4\},\\
X_{3}&=&\{u \in X : d_{G}(u)\geq 5\}.
\end{eqnarray*}
We have one of the following cases:\\
\noindent{\bf Case 1.} $X_1 \cup X_2 =\emptyset.$\\
In this case, the Hall's theorem \cite{hall} guarantees the existence of a matching of $X$ in $G$. That is a contradiction.\\\\
\noindent{\bf Case 2.} $X_1 \cup X_2 \neq \emptyset.$\\
Let $X_1 \cup X_2 = \{v_{i_1}v_{i'_1},v_{i_2}v_{i'_2},...,v_{i_t}v_{i'_t}\}$. First we show that the following claim holds.
\begin{emp}\label{pairdisj2}
The elements of $X_1 \cup X_2$ are pairwise disjoint.
\end{emp}
\noindent\textbf{Proof of Claim \ref{pairdisj2}}. Suppose to contrary that for some $2 \leq s \leq t$, $ \{wv_{i'_1},wv_{i'_2},...,wv_{i'_s}\} \subseteq X_1 \cup X_2$. So we have $ d_{\mathcal{H}}(w) \leq f(s),$ where $f(k)= 4k + {n+j-k-2 \choose 2}$ is a function on $k$, $2 \leq k \leq t \leq N-1$. It is straightforward to see that the absolute maximum of $f(k)$ occurs in point $k=2$. Hence $$ d_{\mathcal{H}}(w) \leq f(2) = 8 + {n+j-4 \choose 2}.$$
On the other hand, $ 8 + {n+j-4 \choose 2} < d+1$ for $n\geq 13$. That contradiction completes the proof of our claim.
$
\square$\\
Also we have the following claim.
\begin{emp}\label{sizex2}
$\vert X_1 \vert \leq j-1$.
\end{emp}
\noindent\textbf{Proof of Claim \ref{sizex2}}. Suppose not. Therefore we may assume that $\{v_{i_1}v_{i'_1},v_{i_2}v_{i'_2},...,v_{i_j}v_{i'_j}\}\subseteq X_1.$ Set $L=\{v_{i_2},v_{i_3},...,v_{i_j}\}$. We have $E_{v_{i_1}}= F_1 \cup F_2 \cup F_3$, where
$$F_k =\{e \in E_{v_{i_1}} : \vert e \cap L\vert =k-1\},\ \ \ \ \ \ 1 \leq k \leq 3.$$
Since, using Claim \ref{pairdisj2}, the elements of $X_1 $ are pairwise disjoint, the elements of $L$ are distinct. So, an easy computation shows that $\vert F_1\vert \leq {n-2 \choose 2}$, $\vert F_2\vert \leq (j-1)(n-3)$ and $\vert F_3\vert \leq {j-1 \choose 2}$. Therefore $$d_{\mathcal{H}}(v_{i_1})= \vert E_{v_{i_1}}\vert \leq {n-2 \choose 2}+(j-1)(n-3)+{j-1 \choose 2}=d.$$ This contradiction completes the proof of this claim.
$
\square$\\
Using the definition of $X_2,$ for every $u_{ik}=v_iv_k\in X_2,$ we have $d_{G}(u_{ik}) \geq 1.$
On the other hand, by Claim \ref{pairdisj2}, the elements of $X_2$ are pairwise disjoint. Therefore $G$ contains a matching $M_1$ of $X_2$ in $G$.
Suppose that $G'=[X_3,Y']$ is the induced subgraph of $G$ so that $Y'\subseteq Y$ is obtained by deleting the vertices of $M_1$. Note that for every $u \in X_3$ and $y \in Y'$, we have $d_{G'}(u) \geq 3$ and $d_{G'}(y) \leq 3$. So the Hall's theorem \cite{hall} guarantees the existence of a matching $M_2$ of $X_3$ in $G'$. Now, using Claim \ref{sizex2}, we may suppose that $X_1= \{v_{i_1}v_{i'_1},v_{i_2}v_{i'_2},...,v_{i_t}v_{i'_t}\}$, where $t\leq j-1$. Set $V'= V(\mathcal{H})\setminus \{v_{i'_1},v_{i'_2},...,v_{i'_t}\}$. Clearly
$\vert V'| \geq n$ and $M_1 \cup M_2$ induces a matching of $X_2\cup X_3$ in $G$. As every matching of $X_2\cup X_3$ in $G$ is equivalent to a
$K_{\vert V' \vert}^{(3)}$ in $\mathcal{H}[V']$, we have a copy of $K_{n}^{(3)}$ in $\mathcal{H}$. This is a contradiction to our assumption.
$
\blacksquare$\\
\section{proof of Theorem \ref{main}}
Let $\mathcal{H}$ be an $\mathcal{F}_n^{(3)}$-free hypergraph with $N$ vertices and $ex(N,\mathcal{F}_n^{(3)})$ edges. Also let $N=\ell (n-1)+j,$ where $\ell\geq 1$ and $1\leq j \leq n-1.$
We use induction on $\ell$ to show that $ex(N,\mathcal{F}_n^{(3)})=t_3(N,n-1)$. Using Theorem \ref{l=1}, the basic step $\ell =1$ is true. Now suppose that $\ell >1$. Since at least one $K_{n}^{(3)}$ is made by adding one edge to $\mathcal{H}$, we deduce that $\mathcal{H}$ contains a $K_{n-1}^{(3)}$. Let $\mathcal{K}$ be such a $K_{n-1}^{(3)}$ in $\mathcal{H}$ with the core sequence $v_1,v_2,...,v_{n-1}$ so that $e(\mathcal{H}[v_1,v_2,...,v_{n-1}])\cap e(\mathcal{K})$ is maximum.
Let $\mathcal{H}_1=\mathcal{H}[V_1]$, $\mathcal{H}_2=\mathcal{H}[V_2]$ and $\mathcal{H}_3=\mathcal{H}[V_1 ,V_2]$, where $V_1 =V(\mathcal{K})=\{v_1,v_2,...,v_{n-1}\}$ and $V_2=V(\mathcal{H})\setminus V_1$. Also let $N'= \vert V_2\vert = (\ell -1)(n-1)+j$, where $\ell >1$ and $1\leq j \leq n-1$. Set
$$\mathcal{H}_3 ^{\vartriangle}= \{e \in E(\mathcal{H}_3) : \ \vert e \cap V_1 \vert =1\ \ {\rm and} \ \ \vert e \cap V_2 \vert =2\},$$
$$\mathcal{H}_3 ^{\triangledown}= \{e \in E(\mathcal{H}_3) :\ \vert e \cap V_1 \vert =2 \ \ {\rm and} \ \ \vert e \cap V_2 \vert =1\}.$$
Note that $E(\mathcal{H}_3)=\mathcal{H}_3 ^{\vartriangle} \cup \mathcal{H}_3 ^{\triangledown}$.
So
\begin{equation}\label{eh}
e(\mathcal{H}) =e(\mathcal{H}_1) +e(\mathcal{H}_2) + \vert \mathcal{H}_3 ^{\vartriangle}\vert + \vert \mathcal{H}_3 ^{\triangledown}\vert.
\end{equation}
By the induction hypothesis, we have
\begin{equation}\label{eh2}
e(\mathcal{H}_2) \leq t_3(N',n-1)=\sum _{i=0}^{3} (\ell-1) ^{3-i} {j \choose i}{n-1-i \choose 3-i}.
\end{equation}
Moreover,
\begin{equation}\label{eh3}
\vert \mathcal{H}_3^{\vartriangle} \vert \leq t_2(N',n-1).
\end{equation}
To see that, let $G$ be a graph on $V_2$ so that the vertices $u$ and $v$ of $V_2$ are adjacent in $G$ if and only if there exists the edge $\{x,u,v\} \in \mathcal{H}_3 ^{\vartriangle}$, for some $x \in V_1$. If there is a $K_n$ in $G$, then we can find a $K_n^{(3)}$ in $\mathcal{H}$, a contradiction. Therefore, by Tur\'{a}n's theorem \cite{turan}, we have $\vert \mathcal{H}_3^{\vartriangle} \vert \leq t_2(N',n-1)$.\\
\noindent Now we show that $e(\mathcal{H}_1)+\vert \mathcal{H}_3 ^{\triangledown}\vert \leq {n-1 \choose 3}+N'{n-2 \choose 2}$.
For this purpose, set $$\mathcal{B}_1 ^{\triangledown}= \{ e \in \mathcal{H}_3 ^{\triangledown} :\ e \in E(\mathcal{K}) \}$$ and $\mathcal{B}_2 ^{\triangledown}= \mathcal{H}_3 ^{\triangledown}\setminus \mathcal{B}_1 ^{\triangledown}$. Clearly, we have
\begin{eqnarray}\label{B2}
\vert \mathcal{B}_2 ^{\triangledown}\vert \leq N'{n-2 \choose 2}.
\end{eqnarray}
To see that, choose an arbitrary vertex $u \in V_2$. Set $x=u$ and $U=V_1=\{v_1,v_2,...,v_{n-1}\}$ and $\mathcal{A}_u=\{A_{1}^u,A_{2}^u,...,A^u_{n-1}\}$, where
$$A^u_{i} = \{ e \in \mathcal{B}_2 ^{\triangledown} :\ \{u,v_{i}\} \subset e\}.$$
If $\mathcal{A}_u$ contains an SDR, then $u,v_1,v_2,...,v_{n-1}$ is the core sequence of a copy of $K_n ^{(3)}$ in $\mathcal{H}$, a contradiction. So using Lemma \ref{sdr}, we have $\vert \bigcup _{i=1}^{n-1} A^u_{i} \vert \leq {n-2 \choose 2}$. Since $u$ is choosed as an arbitrary vertex of $V_2$, Thus $\vert \mathcal{B}_2 ^{\triangledown}\vert \leq N'{n-2 \choose 2} $. Now we demonstrate that
\begin{eqnarray}\label{eh1h3}
e(\mathcal{H}_1) +\vert \mathcal{B}_1 ^{\triangledown}\vert \leq {n-1 \choose 3}.
\end{eqnarray}
To see this, Suppose that $\vert \mathcal{B}_1 ^{\triangledown}\vert =t$. If $t\leq e(\mathcal{H}_1 ^c)$, then we are done. So we may assume that $e(\mathcal{H}_1 ^c) \leq t-1$. On the other hand, clearly $\mathcal{K}_{n-1}^3$ contains a copy of ${K}_{n-1}^{(3)}$. Therefore, by the maximality of $\mathcal{K}$, at most $t-1$ edges of $\mathcal{K}$ are not in $E(\mathcal{H}_1)$. This is a contradiction to the assumption that $\vert \mathcal{B}_1 ^{\triangledown}\vert =t$.
Therefore by (\ref{B2}) and (\ref{eh1h3}), we have
\begin{eqnarray}\label{H1H3}
e(\mathcal{H}_1)+\vert \mathcal{H}_3 ^{\triangledown}\vert \leq {n-1 \choose 3}+N'{n-2 \choose 2}.
\end{eqnarray}
Now set
$$B= {n-1 \choose 3}+N'{n-2 \choose 2} + t_3(N',n-1) +t_2(N',n-1).$$
Hence by (\ref{eh}),(\ref{eh2}),(\ref{eh3}) and (\ref{H1H3}), we have
\begin{eqnarray*}
\vert E(\mathcal{H})\vert \leq B &=& {n-1 \choose 3}+((\ell -1)(n-1)+j){n-2 \choose 2} + (\ell -1)^{3}{n-1 \choose 3}
+ j(\ell -1)^{2}{n-2 \choose 2}\\
&+& (\ell -1) (n-3){j \choose 2} +{j \choose 3} + (\ell -1)^{2}{n-1 \choose 2}
+ j(\ell -1)(n-2)
+ {j \choose 2}.
\end{eqnarray*}
\noindent To demonstrate that $ex(N, \mathcal{F}_n^{(3)}) \leq t_3(N,n-1)$, it suffices to show that
$$B \leq t_3(N,n-1) = \ell^{3}{n-1 \choose 3} + j\ell^{2}{n-2 \choose 2} + \ell (n-3){j \choose 2} +{j \choose 3}.$$
By simplifying the above inequality, it suffices to show that
\begin{eqnarray*}
3\ell {n-1 \choose 3} + (\ell -1)^2{n-1 \choose 2}+j(\ell -1)(n-2)\\ + {j \choose 2} +(\ell -1)(n-1){n-2 \choose 2} +2j{n-2 \choose 2}\\
\leq 3 \ell^{2} {n-1 \choose 3}+2j\ell {n-2 \choose 2}+{j \choose 2}(n-3).
\end{eqnarray*}
But the above inequality is certainly true since $n \geq 13$, $\ell >1$ and $j \geq 1$ imply
\begin{itemize}
\item[$\bullet$]
${j \choose 2} \leq {j \choose 2}(n-3)$.
\item[$\bullet$]
$ j(\ell -1)(n-2) +2j{n-2 \choose 2} \leq 2j\ell {n-2 \choose 2}$.
\item[$\bullet$]
$3\ell {n-1 \choose 3}+ (\ell -1)^2{n-1 \choose 2}+(\ell -1)(n-1){n-2 \choose 2} \leq 3 \ell^{2} {n-1 \choose 3}$.
\end{itemize}
So $$ex(N, \mathcal{F}_n^{(3)}) \leq t_3(N,n-1)$$ and the equality follows by inspection of (\ref{lbound}). Therefore,
\begin{itemize}
\item[(i)] $e(\mathcal{H}_{2}) =t_3(N',n-1)$.
\item[(ii)] $\vert \mathcal{H}_3^{\vartriangle} \vert = t_2(N',n-1)$.
\item[(iii)] $\vert \mathcal{B}_2 ^{\triangledown}\vert = N'{n-2 \choose 2}$.
\item[(iv)] $e(\mathcal{H}_1) +\vert \mathcal{B}_1 ^{\triangledown}\vert = {n-1 \choose 3}$.
\end{itemize}
In the sequel, we demonstrate that $\mathcal{H}\cong T_3(N,n-1)$. Since $e(\mathcal{H}_{2}) =t_3(N',n-1)$, the induction hypothesis implies that $\mathcal{H} _{2}\cong T_3(N',n-1)$. Therefore $\mathcal{H}_2$ is a complete $3$-uniform $(n-1)$-partite hypergraph on $N'$ vertices whose partition sets differ in size by at most 1.
Assume that $U_1,U_2,...,U_{n-1}$ are the partition sets of $\mathcal{H}_2$. Recall that $U=V_1=\{v_1,v_2,...,v_{n-1}\}$ and for every $u \in V_2$, $\mathcal{A}_u=\{A_{1}^u,A_{2}^u,...,A^u_{n-1}\}$, where
$A^u_{i} = \{ e \in \mathcal{B}_2 ^{\triangledown} :\ \{u,v_{i}\} \subset e\}$ and $\vert \bigcup _{i=1}^{n-1} A^u_{i} \vert \leq {n-2 \choose 2}$.
Since $\vert \mathcal{B}_2 ^{\triangledown}\vert = N'{n-2 \choose 2}$,
$$\vert \bigcup _{i=1}^{n-1} A^{u}_{i} \vert = {n-2 \choose 2},\ \ \ \ \ \ \forall \ u \in V_2.$$
So using Lemma \ref{sdr}, there exists $1 \leq q_u \leq n-1$, so that $A^{u}_{q_{u}}=\emptyset$ and for every $1 \leq i \leq n-1$ and $i \neq q_u$, we have
$$A_{i}^{u} = \{ e \in \mathcal{B}_2 ^{\triangledown} :\ \{u,v_{i}\} \subset e\}=\{ \{u,v_i,v_k\}:\ k \neq i, q_u \}.$$
On the other words,$$\bigcup _{i\neq q_u} A_i^{u}= \{ \{u,v_l,v_k\}:\ v_l ,v_k \in V_1,\ l,k \neq q_u\}.$$
So we can partition the vertices of $V_2$ into $n-1$ parts $U'_1,U'_2,...,U'_{n-1}$, so that for every $x \in U'_m$, $A^{x}_m = \emptyset$ and
$$\bigcup _{i\neq m} A_i^{x}= \{ \{x,v_l,v_k\}:\ v_l ,v_k \in V_1,\ l,k \neq m\}.$$
Now we show that for $1\leq i\leq n-1$, $U'_i$ is an independent set in $\mathcal{H}$. Suppose not. By symmetry we may assume that for two vertices $x,y \in U'_1$, the edge $\{x,y,z\} \in E(\mathcal{H})$. It can be shown that $x,y,v_2,v_3,...,v_{n-1}$ represents the core sequence of a copy of $K_n^{(3)}$ in $\mathcal{H}$ with the following edge assignments. Set $e_{xy}=\{x,y,z\}$, $e_{xv_i}\in A_i^x$ for $2\leq i\leq n-1$, $e_{yv_i}\in {A}_i^y$ for $2\leq i\leq n-1$ and $e_{v_iv_{i'}}\in E(\mathcal{K})$ for $2\leq i,i'\leq n-1$. Hence $U'_i$'s, $1 \leq i \leq n-1$, are independent sets in $\mathcal{H}$.\\
Therefore $\{U_1,U_2,...,U_{n-1}\}=\{U'_1,U'_2,...,U'_{n-1}\}$. With no loss of generality, we may suppose that
$$U_i =U'_i \ \ \ \ \ \ \ \ {\rm for}\ \ 1\leq i\leq n-1.$$
Now we demonstrate that for $1\leq i\leq n-1$, $U_i\cup \{v_i\}$ is an independent set in $\mathcal{H}$. Suppose to the contrary that for some $1\leq h\leq n-1$, $U_h \cup \{v_h\}$ is not independent set. So for some $u_h \in U_h$, $f=\{u_h,v_h,w\}\in E(\mathcal{H})$. Since $U_h$ is an independent set in $\mathcal{H}$, $w\notin U_h$. Choose the vertices $x_1,x_2,...,x_{n-1}$ so that $x_h=u_h$ and
$$x_i \in U_i\ \ \ \ \ \ \ \ {\rm for} \ \ i\neq h.$$
Since $\mathcal{H}[\{x_1,x_2,...,x_{n-1}\}]\cong \mathcal{K}_{n-1}^3$, $x_1,x_2,...,x_{n-1}$ is the core sequence of a ${K}_{n-1}^{(3)}$, say $\mathcal{K}'$, in $\mathcal{H}$. Thus $x_1,x_2,...,x_{n-1},v_h$ represents the core sequence of a ${K}_n^{(3)}$ in $\mathcal{H}$ with the following edge assignments. Set $e_{x_hv_h}=f$, $e_{v_hx_i}\in A_h^{x_i}$ for $i\neq h$ and $e_{x_i x_{i'}}\in E(\mathcal{K}')$ for $i,i' \neq h$.
\noindent Therefore for $1\leq i \leq n-1$, $W_i=U_i \cup \{v_i\}$ is an independent set in $\mathcal{H}$. So $\mathcal{H}$ is an $n-1$-partite hypergraph with parts $W_1,W_2,...,W_{n-1}$ whose partition sets differ in size by at most 1. Since $e(\mathcal{H})=t_3(N,n-1)$, we deduce that $\mathcal{H}\cong T_3(N,n-1)$.
$
\blacksquare$\\
\footnotesize
\end{document} |
\begin{document}
\title
{Locally upper bounded poset-valued maps and stratifiable spaces}
\thanks{}
\author{Ying-Ying Jin}
\address{(Y.Y. Jin)
School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, P.R. China}
\email{[email protected]}
\author{Li-Hong Xie}
\address{(L.H. Xie) School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, P.R. China}
\email{[email protected]}
\author{Han-Biao Yang$^*$}
\address{(H.B. Yang) School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, P.R. China}
\email{[email protected]}
\thanks{* The corresponding author}
\thanks{$^1$Supported by NSFC (Nos. 11526158, 11601393).}
\subjclass[2000]{54D20; 54D40; 54D45; 54E18; 54E35; 54H11}
\keywords{Locally upper bounded poset-valued maps; Stratifiable spaces; Semi-stratifiable spaces; MCP;
MCM; Lower semi-continuous (l.s.c.); Upper semi-continuous (u.s.c.).}
\begin{abstract}
In this paper, we characterize stratifiable (or semi-stratifiable) spaces, and monotonically countably paracompact (or monotonically countably metacompact) spaces by
expansions of locally upper bounded semi-continuous poset-valued maps.
These extend earlier results for real-valued Locally bounded functions.
\end{abstract}
\maketitle
\section{Introduction}
Throughout this paper, let $\mathbb{R}$ the set of all real numbers, and $\mathbb{N}$ set of all natural numbers. All topological spaces are assumed to be $T_1-$spaces.
J. Mack characterized \cite{JM} countably paracompact spaces with
locally bounded real-valued functions as follows:
\begin{theorem}(\cite{JM})
A space $X$ is countably paracompact if and only if for each locally bounded
function $h:X \rightarrow\mathbb{R} $ there exists a locally bounded l.s.c. function $g:X \rightarrow \mathbb{R}$ such that $|h|\leq g$.
\end{theorem}
C.R. Borges \cite{Bo} introduced definitions called stratifiable spaces and semi-stratifiable spaces.
\begin{definition}\cite{Bo}
A space $X$ is said to be stratifiable if, to
each open set $U$, one can assign an increasing sequence $(U_n)_{n\in\mathbb{N}}$, called a
stratification of $X$, of open subsets of $X$ such that
\begin{enumerate}
\item $ \overline{U_n}\subseteq U$ for each $n\in\mathbb{N}$;
\item $\bigcup_{n\in\mathbb{N}}U_n=U$;
\item if $ U\subseteq V$, then $ U_n\subseteq V_n$ for each $n\in\mathbb{N}$.
\end{enumerate}
$X$ is said to be semi-stratifiable, if to each open set $U$, one can assign a sequence of closed subsets $(U_n)_{n\in\mathbb{N}}$
such that (2) and (3) above hold.
\end{definition}
Recall that a space $X$ is said to be {\it perfect} \cite{En} if to each open set $U$ of $X$, one can assign an increasing sequence of closed subsets $(U_n)_{n\in\mathbb{N}}$ such that (2) above holds.
A perfect space $X$ is said to be {\it perfectly normal} if $X$ is normal.
It is well known that a space is stratifiable if and only if it is monotonically normal and semi-stratifiable.
C. Good, R. Knight and I. Stares \cite{GK} and C. Pan \cite{Pa} introduced a monotone version of countably paracompact spaces, called monotonically countably paracompact spaces (MCP) and monotonically cp-spaces, respectively, and it was
proved in \cite[Proposition 14]{GK} that both these notions are equivalent.
\begin{definition}\cite{GK}\label{def2.4}
A space $X$ is said to be monotonically countably metacompact (MCM) if there is an operator $U$ assigning to each decreasing sequence $(D_j)_{j\in\mathbb{N}}$
of closed sets with empty intersection, a sequence of open sets $U((D_j))=(U(n,(D_j)))_{n\in\mathbb{N}}$ such that
\begin{enumerate}
\item $D_n \subseteq U(n,(D_j))$ for each $n\in\mathbb{N}$;
\item $\bigcap_{n\in\mathbb{N} }U(n,(D_j))=\emptyset$;
\item given two decreasing sequences of closed sets $(F_j)_{j\in \mathbb{N}}$ and $(E_j)_{j\in \mathbb{N}}$ such that $F_n \subseteq E_n$ for each $n\in\mathbb{N}$, then $U(n,(F_j))\subseteq U(n,(E_j))$ for each $n\in \mathbb{N}$.
\end{enumerate}
$X$ is said to be monotonically countably paracompact (MCP) if, in addition,
$(2') \bigcap_{n\in \mathbb{N}}\overline{U(n,(D_j))}=\emptyset$.
\end{definition}
Many insertion results present some classic characterizations of topological spaces, such as stratifiable spaces, monotonically countably paracompact spaces and others.
T. Kubiak \cite{TK} investigated monotonically normal spaces by the monotonization of insertion properties.
P. Nyikos and C. Pan \cite{Pa} and C. Good and I. Stares \cite{GS} respectively gave a characterization of stratifiable spaces by the monotonizations of insertion properties.
Also, C. Good, R. Knight and I. Stares \cite{GK} characterized monotonically countably paracompact spaces by the insertions of semi-continuous functions.
By extending the insertion properties of real-valued maps, K. Yamazaki \cite{KY} introduced the notion of local boundedness for set-valued mappings and described MCP spaces by expansions of locally bounded set-valued mappings.
L.H. Xie, P.F. Yan\cite{XH} gave some characterizations of stratifiable, semi-stratifiable by expansions of set-valued mappings.
K. Yamazaki \cite{Ya}, Y.Y. Jin, L.H. Xie, H.W. Yue \cite{JX} considered the locally upper bounded maps with values in the ordered topological vector spaces and provided new monotone insertion theorems.
The following theorems were proved in \cite[Theorem 2.4]{KY}, \cite[Theorem 3.1 and Theorem 3.2]{Ya} and \cite[Theorem 3.1 and Theorem 3.2]{XH}.
\begin{theorem}(\cite{KY})
For a space $X$, the
following statements are equivalent:
\begin{enumerate}
\item $X$ is MCP (resp. MCM);
\item for every metric space $Y$, there exists a preserved order operator $\Phi$ assigning to each
locally bounded set-valued mapping $\varphi: X \rightarrow \mathcal {B}(Y)$, a locally bounded l.s.c. (resp. a l.s.c.) set-valued mapping $\Phi(\varphi): X \rightarrow \mathcal {B}(Y)$ such that $\varphi\subseteq \Phi(\varphi)$;
\end{enumerate}
where $\mathcal {B}(Y)$ is the set of all nonempty closed bounded sets of $Y$.
\end{theorem}
\begin{theorem}(\cite{Ya})
Let $X$ be a topological space and $Y$ an ordered topological vector space with a positive interior point.
Then, the following conditions are equivalent:
\begin{enumerate}
\item $X$ is MCP (resp. MCM).
\item There exists an operator $\Phi$ assigning to each locally upper bounded map $f:X\rightarrow Y$, a locally upper bounded lower semi-continuous (resp. a lower semi-continuous) map $\Phi(f):X\rightarrow Y$ with $f\leq\Phi(f)$ such that $\Phi(f)\leq\Phi(f')$ whenever $f\leq f'$.
\end{enumerate}
\end{theorem}
\begin{theorem}(\cite{XH})
For a space $X$, the following statements are equivalent:
\begin{enumerate}
\item $X$ is perfectly normal (resp. stratifiable);
\item for every space $Y$ having a strictly increasing closed cover $\{B_n\}$, there exists an operator $\Phi$ (resp. a preserved order operator $\Phi$) assigning to each
set-valued mapping $\varphi: X \rightarrow \mathcal {F}(Y)$, a l.s.c. set-valued mapping $\Phi(\varphi): X \rightarrow \mathcal {F}(Y)$ such that $\Phi(\varphi)$ is locally bounded at each $x\in U_\varphi$ and that $\varphi\subseteq \Phi(\varphi)$;
\end{enumerate}
\end{theorem}
\begin{theorem}(\cite{XH})
For a space $X$, the following statements are equivalent:
\begin{enumerate}
\item $X$ is perfect (resp. semi-stratifiable);
\item for every space $Y$ having a strictly increasing closed cover $\{B_n\}$, there exists an operator $\Phi$ (resp. a preserved order operator $\Phi$) assigning to each set-valued mapping $\varphi: X \rightarrow \mathcal {F}(Y)$, a l.s.c. set-valued mapping $\Phi(\varphi): X \rightarrow \mathcal {F}(Y)$ such that $\Phi(\varphi)(x)$ is bounded at each $x\in U_\varphi$ and that $\varphi\subseteq \Phi(\varphi)$;
\end{enumerate}
\end{theorem}
The purpose of this paper is to generalize real-valued locally bounded functions to locally upper bounded maps with values into some bi-bounded complete and bi continuous posets, which are not necessarily vector spaces or spaces with strictly increasing closed covers, by using the way-below relation $\ll$ and the way-above relation $\ll_d$.
This provides some advantage to the real-valued and set-valued cases.
Indeed, the range $\mathbb{R}$ with the total order can be extended to spaces $P$ with the partial order.
Inspired by Theorem 1.2, Theorem 1.3, Theorem 1.4 and Theorem 1.5,
another purpose of this paper is to characterize stratifiable (or semi-stratifiable) spaces, and monotonically countably paracompact (or monotonically countably metacompact) spaces by expansions of locally upper bounded poset-valued maps along the same lines.
At the last part, we also consider monotone
poset-valued insertions on monotonically normal and monotonically countably paracompact spaces.
Throughout this paper,
all the undefined topological concepts can be found in \cite{En}.
\section{Basic facts and definitions}
In this section, some definitions are restated and some basic facts are listed.
Also, some notions are introduced which seem to be convenient
though they may be found in the references.
\begin{lemma}\label{lem2.2}
For a space $X$, the following statements are equivalent:
\begin{enumerate}
\item $X$ is semi-stratifiable (resp. stratifiable);
\item there is an operator $F$
assigning to each increasing sequence of open sets $(U_{j})_{j\in
\mathbb{N}}$, an increasing sequence of closed sets
$(F(n,(U_{j})))_{n\in\mathbb{N}}$ such that
\begin{enumerate}
\item [(i)] $U_{n}\supseteq
F(n,(U_{j}))$ for each $n\in\mathbb{N}$;
\item [(ii)] $\bigcup_{n\in
\mathbb{N}}F(n,(U_{j}))=\bigcup_{n\in\mathbb{N}}U_{n}$ (resp. (ii)' $\bigcup_{n\in
\mathbb{N}}Int F(n,(U_{j}))=\bigcup_{n\in\mathbb{N}}U_{n}$);
\item [(iii)] given two increasing sequences of open sets
$(U_{j})_{j\in\mathbb{N}}$ and $(G_{j})_{j\in\mathbb{N}}$ such that
$U_{n}\subseteq G_{n}$ for each $n\in\mathbb{N}$, then
$F(n,(U_{j}))\subseteq F(n,(G_{j}))$ for each
$n\in\mathbb{N}$.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
From De Morgan's laws it follows easily that conditions (2) in Theorem 2.2 and (2) in \cite[Theorem 3.6, 3.7]{XY} are equivalent.
\end{proof}
It is well known that semi-stratiffiable (stratiffiable) spaces
are naturally monotone versions of perfect (perfectly normal) spaces.
We can easily obtain the following result without proof.
\begin{lemma}\label{the2.4}
For a space $X$, the following statements are equivalent:
\begin{enumerate}
\item $X$ is perfect (resp. perfectly normal);
\item there is an operator $F$
assigning to each increasing sequence of open sets $(U_{j})_{j\in
\mathbb{N}}$, an increasing sequence of closed sets
$(F(n,(U_{j})))_{n\in\mathbb{N}}$ such that
\begin{enumerate}
\item [(i)] $U_{n}\supseteq
F(n,(U_{j}))$ for each $n\in\mathbb{N}$;
\item [(ii)] $\bigcup_{n\in
\mathbb{N}}F(n,(U_{j}))=\bigcup_{n\in\mathbb{N}}U_{n}$ (resp. $\bigcup_{n\in
\mathbb{N}}Int F(n,(U_{j}))=\bigcup_{n\in\mathbb{N}}U_{n}$);
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{lemma}\label{lem2.5}
A space $X$ is said to be monotonically countably metacompact (MCM) (resp. monotonically countably paracompact (MCP)) if and only if there is an operator $F$ assigning to each increasing sequence $(U_j)_{j\in\mathbb{N}}$
of open sets of $X$ satisfying $\bigcup_{i\in \mathbb{N}}U_i=X$, a sequence of closed sets $F((U_j))=(F(n,(U_j)))_{n\in\mathbb{N}}$ such that
\begin{enumerate}
\item $U_n \supseteq F(n,(U_j))$ for each $n\in\mathbb{N}$;
\item $\bigcup_{n\in\mathbb{N} }F(n,(U_j))=X$ (resp. (2)' $\bigcup_{n\in \mathbb{N}}Int F(n,(U_j))=X$);
\item given two increasing sequences of open sets $(U_j)_{j\in \mathbb{N}}$ and $(G_j)_{j\in \mathbb{N}}$ such that $U_n \subseteq G_n$ for each $n\in\mathbb{N}$, then $F(n,(U_j))\subseteq F(n,(G_j))$ for each $n\in \mathbb{N}$.
\end{enumerate}
\end{lemma}
\begin{proof}
From De Morgan's laws it follows easily that conditions (1), (2) and (3) are equivalent to Definition 2.4.
\end{proof}
In the rest of this section, let us recall some definitions and terminology from \cite{Gi, Ke}.
Let $P=(P,\leq)$ be a poset.
For $a, b \in P$, the symbol $[a, b]$ stands for $\{y\in P:a \leq y\leq b\}$.
A subset $A$ of $P$ is said to be {\it directed} (resp. {\it filtered}) if $A$ is nonempty and for every $x, y\in A$ there exists $z\in A$ such that $x\leq z$ and $y\leq z$ (resp. $z\leq x$ and $z\leq y$).
For a subset $A$ of $P$, $\bigvee A$(resp. $\bigwedge A$) stands for the sup (resp. inf) of $A$, if exists.
For $x, y\in P$, $x$ is {\it way below} $y$, in symbol $x \ll y$, if for all directed subset $D$ of $P$ with $\bigvee D$, the relation $y\leq\bigvee D$ always implies the existence of $d\in D$ with $x \leq d$.
For $x, y\in P$, $x$ is {\it way above} $y$, in symbol $y\ll_d x$, if for all filtered subset $F$ of $P$ with $\bigwedge F$, the relation $\bigwedge F\leq y$ always implies the existence of $f\in F$ with $f\leq x$.
In a lattice $L$, $x \ll y$ (resp. $x \ll_d y$) if and only if for every subset $A$ of $L$ with $\bigvee A$ (resp. $\bigwedge A$) the relation $y\leq \bigvee A$ (resp. $\bigwedge A \leq y$) always implies the existence of a finite subset $B$ of $A$ such that $x \leq \bigvee B$ ($\bigwedge B\leq x$).
Note that the way-above relation $\ll_d$ is precisely the dual relation of the way below relation of $P^{op}$, i.e. $x \ll_d y\Leftrightarrow y\ll^{op}x$.
An element $x$ of $P$ is {\it isolated from below} (resp. {\it isolated from above}) if $x \ll x$ (resp. $x \ll_dx$).
Clearly, an element $x$ of $P$ is isolated from above iff $x$ is isolated from below in $P^{op}$.
On $\mathbb{R}$, it is clear that $x \ll y$ if and only if $x <y$ if and only if $x \ll_dy$.
On the unit interval $[0, 1]$ of $\mathbb{R}$, we have that $x \ll y$ if and only if $x <y$ or $x =y=0$, and that $x \ll_dy$ if and only if $x <y$ or $x =y=1$.
A poset $P$ is {\it continuous} if $\{u \in P:u \ll x\}$ is directed and $x =\bigvee\{u \in P:u\ll x\}$ for all $x \in P$.
A poset $P$ is {\it dually continuous} if $\{u\in P:x \ll_du\}$ is filtered and $x =\bigwedge\{u \in P:x \ll_du\}$ for all $x \in P$, in other words, if $P^{op}$ is continuous.
A poset $P$ is called bicontinuous if $P$ is continuous and dually continuous.
As is pointed out in \cite{Ke}, the way-below relation in a bicontinuous poset need not be the opposite of the way-above relation in the sense that $x \ll y$ does not imply $x \ll_dy$.
A {\it bicontinuous lattice} $P$ is a lattice which is bicontinuous as a poset.
It should be noted that we do not require the completeness of $P$ in our definition of $P$ being bicontinuous, where a {\it complete} lattice $P$ is a poset in which every subset has the sup and the inf.
We call a poset $P$ {\it lower-bounded complete} (resp. {\it upper-bounded complete}) if every non-empty subset $A$ of $P$ with a lower bound (resp. an upper bound) has the inf (resp. sup).
When $P$ is lower-bounded complete and upper-bounded complete, we call $P$ {\it bi-bounded complete}.
Note that every bounded complete domain in the sense of [5] is bi-bounded complete.
For maps $f, g:X\rightarrow P$ into a poset $P$, the symbol $f\ll g$ (resp. $f\ll_d g$, $f\leq g$) stands for $f(x)\ll g(x)$ (resp. $f(x)\ll_d g(x)$, $f(x) \leq g(x)$) for each $x\in X$.
For a point $z$ and a pair of points $\langle y,y'\rangle$ of a poset $P$, $z$ is an {\it interpolated point} of $\langle y,y'\rangle$ if $y\ll_d z\ll y'$.
A pair $\langle f,g\rangle$ of maps $f, g:X\rightarrow P$ has {\it interpolated points pointwise} if $\langle f(x), g(x)\rangle$ has an interpolated point for each $x\in X$.
For a subset $B$ of a poset $P$ and $y, y'\in P$, the pair of points $\langle y,y'\rangle$ has interpolated points on $B$ if there exists $z\in B$ such that $z$
is an interpolated point of $\langle y,y'\rangle$.
A pair $\langle f,g\rangle$ of maps $f, g:X\rightarrow P$ has interpolated points pointwise on B if $\langle f(x), g(x)\rangle$ has interpolated points on $B$ for each $x\in X$.
For a non-empty subset $A$ of $X$, a pair $\langle f,y\rangle$ (resp. $\langle y,f\rangle$) of a map $f: X\rightarrow P$ and a point $y\in P$ has {\it interpolated points of $A$} if $\langle f(x),y\rangle$ (resp. $\langle y,f(x)\rangle$) has interpolated points for each $x\in A$.
For a non-empty subset $A$ of $X$, a pair $\langle f,g\rangle$ of maps $f,g:X\rightarrow P$ has {\it interpolated points of $A$} if $\langle f(x),g(x)\rangle$ has interpolated points for each $x\in A$.
We define $G_{f,g}=\{x\in X: \langle f(x),g(x)\rangle \text{~has an interpolated point~}\}.$
See \cite{En} and \cite{Gi} for undefined terminology.
\begin{lemma}\label{lem2.6}\cite{Gi}
For a poset $P$, the following statements hold.
\begin{enumerate}
\item $x \ll_dy$$\Rightarrow$$x \leq y$;
\item $u \leq x \ll_dy\leq v\Rightarrow u \ll_dv$;
\item $z\ll_dx$ and $z\ll_dy$$\Rightarrow$$z\ll_dx \wedge y$, whenever $x \wedge y$ exists.
\end{enumerate}
If $P$ is a dually continuous poset, (4) below also holds.
(4) $x\ll_dy$ $\Rightarrow$ $\exists$ $z\in P$ s.t. $x \ll_dz\ll_dy$.
\end{lemma}
For a subset $A$ of a topological space $X$ and $x\in X$, $\overline{A}$ stands for the closure of $A$ and $\mathcal{N}_x$ is the set of all neighborhoods of $x$.
Let $f:X\rightarrow P$ be a map from a topological space $X$ to a poset $P$ (which is not assumed to be a topological space), and $x \in X$. Set
$$\mathcal{N}_{x*}(f) = \{N \in \mathcal{N}_x : f(N) \text{~has the inf}\}, \text{~and~} \mathcal{N}^*_x (f) = \{N \in \mathcal{N}_x : f(N) \text{~has the sup}\}.$$
We call that $f$ {\it admits} $f_*(x)$ if $\mathcal{N}_{x*}(f)\neq\emptyset$ and $\{\bigwedge f(N):N\in\mathcal{N}_{x*}(f)\}$ has the sup, and then we define $f_*(x)=\bigvee\{\bigwedge f(N):N\in\mathcal{N}_{x*}(f)\}$.
Also, $f$ {\it admits} $f^*(x)$ if the set $\mathcal{N}^*_x(f)\neq\emptyset$ and $\{\bigvee f(N) :N\in \mathcal{N}_x^*(f)\}$ has the inf, and then we define
$f^*(x) =\bigwedge\{\bigvee f(N):N\in \mathcal{N}_{x*}(f)\}$.
A map $f:X\rightarrow P$ is {\it lower semi-continuous} (resp. {\it upper semi-continuous}) at $x$ if $f$ admits $f_*(x)$ (resp. $f^*(x)$) and $f(x) =f_*(x)$ (resp. $f(x) =f^*(x)$).
A map $f:X\rightarrow P$ is {\it lower semi-continuous} (resp. {\it upper semi-continuous}) if $f$ is lower (resp. upper) semi-continuous at every $x\in X$.
Since $P$ is not assumed to be complete, for $A \subset P$, $\bigwedge A$ or $\bigvee A$ does not necessarily exist.
If there is no confusion, we simply express $f_*(x) =\bigvee_{N\in\mathcal{N}_x}\bigwedge f(N)$ and $f^*(x) =\bigwedge _{N\in \mathcal{N}_x}\bigvee f(N)$ for each $x \in X$ (if exists).
It is defined in \cite{En} that a real-valued function $f:X\rightarrow \mathbb{R} $ is lower semi-continuous if $\{x:f(x)>r\}$ is open for each $r\in \mathbb{R}$
(namely, for each $x\in X$ and each $\varepsilon>0$ there exists a neighborhood $O_x$ of $x$ such that $f(x')>f(x)-\varepsilon$ for each $x'\in O_x$).
A real-valued function $f:X\rightarrow\mathbb{R}$ is upper semi-continuous if $-f$ is lower semi-continuous.
Note that this definition coincide with the above definition of semi-continuous maps with values into ordered topological vector spaces $Y$ for $Y=\mathbb{R}$.
\begin{proposition}\cite{Yama}\label{prop2.7}
Let $P$ be a poset, $x\in X$ and $f:X\rightarrow P$ a map. Consider the following conditions:
\begin{enumerate}
\item [(1)] $\{\bigwedge f(N):N\in\mathcal{N}_{x*}(f)\}$ (resp.) $\{\bigvee f(N):N\in\mathcal{N}_{x}^*(f)\}$ has the sup (resp. inf);
\item [(2)] $\mathcal{N}_{x*}(f)\neq\emptyset$ (resp. $\mathcal{N}_{x}^*(f)\neq\emptyset$ );
\item [(3)] $\mathcal{N}_{x*}(f)$ (resp. $\mathcal{N}_{x}^*(f)$) is a neighborhood base of $x$.
\end{enumerate}
Then, the following statements (a), (b), (c) and (d) hold.
\begin{enumerate}
\item [(a)] If P is lower-bounded (resp. upper-bounded) complete, (1)$\Rightarrow$(2) and (2)$\Rightarrow$(3) hold.
\item [(b)] If P is a cdcpo (resp. cfcpo), (3)$\Rightarrow$(1) holds.
\item [(c)] If $P$ is bi-bounded complete, the conditions (1), (2)and (3)are equivalent, that is, $f$ admits $f_*(x)$ (resp.$f^*(x)$) whenever either one of (1), (2 )and (3)
holds
\item [(d)] If $P$ is bi-bounded complete and $f(X)$ has a lower (resp. an upper) bound, (2) holds, thus, $f$ admits $f_*$ (resp. $f^*$).
\end{enumerate}
\end{proposition}
\begin{proposition}\cite{Yama}\label{prop2.8}
Let $P$ be a poset, $x\in X$ and $f:X\rightarrow P$ a map. Consider the following conditions:
Let $X$ be a topological space, $P$ a poset, and assume that $f$ admits $f_*$ and $\mathcal{N}_{x*}(f)$ is a
neighborhood base of $x$ for each $x\in X$. For a map $f: X\rightarrow P$, consider the following conditions:
\begin{enumerate}
\item [(1)] f is lower semi-continuous;
\item [(2)] $\{x\in X: a\ll f(x)\}$ is open for each $a\in P$;
\item [(3)] $\{x\in X: f(x)\leq a\}$ is closed for each $a\in P$.
\end{enumerate}
Then, $(1)\Rightarrow(3)$ always holds. If P is continuous, $(1)\Leftrightarrow(2)$ holds.
\end{proposition}
\begin{proposition}\cite{Yama}\label{prop2.9}
For a topological space $X$ and a bi-bounded complete, continuous (resp. dually continuous) poset $P$,
if $f:X\rightarrow P$ is lower (resp. upper)semi-continuous, then $\{x\in X: a \ll f(x)\}$ (resp. $\{x\in X: f(x) \ll_d a\}$ )is open for each $a\in P$.
The converse holds if, in addition, $\mathcal{N}_{x*}(f)\neq\emptyset$ (resp. $\mathcal{N}_{x}^*(f)\neq\emptyset$).
\end{proposition}
We call a point $z_0$ of a poset $P$ is
a $\ll_d$-{\it increasing} $\ll-${\it limit point} \cite{Yama} if there exists a sequence $\{y_i:i\in\mathbb{N}\}\subset P$ such that
$y_i\ll_d y_{i+1}$, $y_i\ll y_0 (i\in \mathbb{N})$ and $y_0=\bigvee_{i\in\mathbb{N}}y_i$.
Also, $y_0$ of $P$ is a $\ll-${\it decreasing} $\ll_d-$ {\it limit point} if there exist $y_i\in P (i\in\mathbb{N})$ such that $y_{i+1}\ll y_i$, $y_0\ll_d y_i (i\in\mathbb{N})$ and $y_0=\bigwedge_{i\in\mathbb{N}}y_i$.
We introduce the following:
\begin{definition}\cite{Yama}
For a poset $(P,\leq)$ and $y, z\in P$, $y$ is a lower bound (resp. upper bound) of $f:X\rightarrow P$ if $y$ is a lower (resp. upper) bound of $f(X)$.
\end{definition}
\begin{definition}
For a topological space $X$ and a bi-bounded complete, dually continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$, a map $f:X\rightarrow P$ is called locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ if for every $x\in X$ there exist $n\in\mathbb{N}$ and a neighborhood $O_x$ of $x$ such that $f(x')\leq y_n$ for each $x'\in O_x$.
\end{definition}
For a mapping $g:$ $X\rightarrow P$, define$$U_g=\{x\in X: g(x) \text{~is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ ~} \}.$$
Clearly, $U_g$ is an open set in $X$.
$$F_g=\{x\in X: \exists n\in\mathbb{N} \text{~such that~} g(x)\leq y_n\}.$$
\section{main results}
\begin{theorem}\label{the3.1}
Let $P$ be a bi-bounded complete, continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
Then $X$ is stratifiable if and only if
there exists an operators $\Phi$ assigning to each map $g:X\rightarrow P$ with $F_g\neq\emptyset$,
a l.s.c. map $\Phi(g):X\rightarrow P$ such that
$g\leq \Phi(g)$, $\Phi(g)$ is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ at each $x\in U_g$ and that $\Phi(g)\leq \Phi(g')$ whenever $g\leq g'$.
\end{theorem}
\begin{proof}
Assume that $X$ is a stratifiable.
There exists an operator $F$ satisfying (i), (ii)' and (iii) in Lemma \ref{lem2.2}.
Let $P$ be a bi-bounded complete, continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
For each map $g:X\rightarrow P$ with $F_g\neq\emptyset$ and each $n\in \mathbb{N}$, define
$$ U_n(g)=Int\{x\in X: g(x)\leq y_n\} \quad\quad \quad\quad(\ref{the3.1}.1)$$
Then we have $U_g=\bigcup_{n\in \mathbb{N}}U_n(g).$
In fact, for each $x\in U_g$, then there exists an open neighborhood $O$ of $x$ such that $g(x')\leq y_i $ for some $i\in \mathbb{N}$ and each $x'\in O$, which implies that $ x\in U_i(g)$. It implies that $U_g\subseteq\bigcup_{n\in \mathbb{N}}U_n(g).$
On the other hand, take any $x\in \bigcup_{n\in \mathbb{N}}U_n(g)$.
Then there is $U_j(g)$ such that $x\in U_j(g)$, and therefore, there exists an open neighborhood $O$ of $x$ such that $ g(x')\leq y_j$ for each $x'\in O$.
It implies that $x\in O\subseteq U_g$.
Hence, $F((U_j(g)))=(F(n,(U_j(g))))_{n\in\mathbb{N}}$ is a sequence of closed subsets of $X$ such that
$$F(n,(U_j(g)))\subset U_n(g) \text{~for each~} n\in\mathbb{N};\quad\quad \quad\quad(\ref{the3.1}.2)$$
$$\bigcup_{n\in\mathbb{N}}Int F(n, (U_j(g)))=\bigcup_{n\in \mathbb{N}}U_n(g);\quad\quad \quad\quad(\ref{the3.1}.3)$$
$$F(n,(U_j(g)))\subset F(n+1,(U_j(g))), n\in\mathbb{N};\quad\quad \quad\quad(\ref{the3.1}.4)$$
Thus, we can define $\Phi(g):X\rightarrow P$ as follows:
$$\Phi(g)(x)=\left\{
\begin{array}{rcl}
y_1 & & {x\in F(1,(U_j(g)))}\\
y_{n+1} & & x\in F(n+1,(U_j(g)))\setminus F(n,(U_j(g)))\\
y_0 & &x\in X\setminus \bigcup_{n\in\mathbb{N}}F(n, (U_j(g)))\\
\end{array} \right. \quad\quad \quad\quad(\ref{the3.1}.5)$$
It is obvious that $\Phi(g)(x)\leq y_0$ and $\Phi(g)$ has a lower bounded $y_1$.
To show $g\leq\Phi(g)$. For each $x\in X\setminus U_g$, $g(x)\leq y_0=\Phi(g)(x)$ is obvious. Let $x\in U_\varphi $.
Then, $\Phi(\varphi)(x)=y_{i}$ for some $i\in \mathbb{N}$, and therefore, $x\in F(i,(U_j(g)))\setminus F(i-1,(U_j(g)))$.
Since $g(x)\leq y_{i}=\Phi(g)(x)$ by $x\in F(i,(U_j(g)))\subset U_{i}(g)=Int\{x\in X: g(x)\leq y_{i}\}$.
If $x\in F(1,(U_j(g)))$, it is obvious that $g(x)\leq y_{1}=\Phi(g)(x)$. Thus, we have $g\leq\Phi(g)$.
To show that $\Phi(g):X\rightarrow P$ is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ at each $x\in U_g$. Take any $x\in U_g$,
by (\ref{the3.1}.3), there exists $n\in\mathbb{N}$ such that $x\in Int F(n+1, (U_j(g)))\setminus Int F(n, (U_j(g)))$.
Consider the neighborhood $O_x=Int F(n+1, (U_j(g)))$ of $x$.
For each $x'\in O_x$, it follows from the definition of $\Phi(g)$ that $\Phi(g)(x')\leq y_{n+1}$.
This completes the
proof that $\Phi(g)$ is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ at each $x\in U_g$.
Next we show $\Phi(g)$ is l.s.c..
For each $x\in \bigcup_{n\in\mathbb{N}} F(n, (U_j(g)))$, there exists some $m\in\mathbb{N}$ such that $x\in F(m+1, (U_j(g)))\setminus F(m, (U_j(g)))$.
We consider the neighborhood $O_x=X\setminus F(m, (U_j(g)))$ of $x$.
For each $x'\in O_x$, we can get $\Phi(g)(x')\geq y_{m+1}=\Phi(g)(x)$.
Therefore, $\bigwedge\Phi(g)(O_x)=y_{m+1}$ and $O_x\in \mathcal{N}_{x*}(\Phi(g))$.
This provides that $\Phi(g)(x)$ admits $\Phi(g)_*(x)$ because of (c) of Proposition \ref{prop2.7}.
We have
$$\Phi(g)_*(x)=\bigvee_{N\in\mathcal{N}_{x*}(\Phi(g))}\bigwedge\Phi(g)(N)\geq\bigwedge_{x'\in O_x}\Phi(g)(x')=y_{m+1}=\Phi(g)(x).$$
Hence, $\Phi(g)_*(x)=\Phi(g)(x)$ for each $x\in X$, that is, $\Phi(g)(x)$ is l.s.c. at $x$.
For each $x\in X\setminus\bigcup_{n\in\mathbb{N}} F(n, (U_j(g)))$, we consider the neighborhood $V=X\setminus\bigcup_{n\in\mathbb{N}} F(n, (U_j(g)))$ of $x$.
For each $x'\in V$, we can get $\Phi(g)(x')= y_{0}=\Phi(g)(x)$.
Therefore, $\bigwedge\Phi(g)(V)=y_{o}$ and $V\in \mathcal{N}_{x*}(\Phi(g))$.
This provides that $\Phi(g)(x)$ admits $\Phi(g)_*(x)$ because of (c) of Proposition \ref{prop2.7}.
We have
$$\Phi(g)_*(x)=\bigvee_{N\in\mathcal{N}_{x*}(\Phi(g))}\bigwedge\Phi(g)(N)\geq\bigwedge_{x'\in O_x}\Phi(g)(x')=y_{0}=\Phi(g)(x).$$
Hence, $\Phi(g)_*(x)=\Phi(g)(x)$ for each $x\in X$, that is, $\Phi(g)(x)$ is l.s.c. .
Finally, let $g':X\rightarrow P$ be a map with $g\leq g'$.
Then
$$\{x\in X: g'(x)\leq y_n\}\subseteq \{x\in X: g(x) \leq y_n\}$$
and hence, $U_n(g)\supseteq U_n(g')$ for each $n\in\mathbb{N}$.
Therefore we have $F(n,(U_j(g)))\supseteq F(n,(U_j(g')))$ for each $n\in\mathbb{N}$, and
$$\bigcup_{n\in\mathbb{N}}Int F(n, (U_j(g)))=\bigcup_{n\in \mathbb{N}}U_n(g)$$
$$\bigcup_{n\in\mathbb{N}}Int F(n, (U_j(g')))=\bigcup_{n\in \mathbb{N}}U_n(g')$$
For each $x\in \bigcup_{n\in \mathbb{N}}U_n(g')$,
there exists $n\in \mathbb{N}$ such that $x\in F(n+1, (U_j(g')))\setminus F(n, (U_j(g')))$.
That is $x\in F(n+1, (U_j(g')))\subseteq F(n+1, (U_j(g)))$.
By (\ref{the3.1}.5), we can get $\Phi(g)(x)\leq y_{n+1}=\Phi(g')(x)$.
$\Phi(g)(x)\leq y_0= \Phi(g')(x)$ is obvious whenever $x\in X\setminus\bigcup_{n\in \mathbb{N}}U_n(g')$,
which proves the necessity.
Conversely, let $(U_j)_{j\in\mathbb{N}}$ be a sequence of increasing open subsets of $X$.
Define a map $g_{((U_j))}:X\rightarrow P$ by:
$$g_{((U_j))}(x)=\left\{
\begin{array}{rcl}
y_1 & & {x\in U_{1}}\\
y_{n+1} & & {x\in U_{n+1}\backslash \texttt{}U_{n}}\\
y_0 & & {x\in X\setminus\bigcup_{n\in\mathbb{N}}U_n}\\
\end{array} \right. \quad\quad \quad\quad(\ref{the3.1}.6)$$
Then, we have $U_{g_{((U_j))}}=\bigcup_{n\in\mathbb{N}}U_n$, where
$$U_{g_{((U_j))}}=\{x\in X: g_{((U_j))} \text{~is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ at~}x \}.$$
By the assumption, there exist an operators $\Phi$ assigning to each $g_{((U_j))}$ with an upper bound,
a l.s.c. map $\Phi(g_{((U_j))}):X\rightarrow P$ such that
$g_{((U_j))}\leq \Phi(g_{((U_j))})$, $\Phi(g_{((U_j))})(x)$ is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ at each $x\in$ and $\Phi(g)\leq \Phi(g')$ whenever $g\leq g'$.
For each sequence $(U_j)_{j\in\mathbb{N}}$ of increasing open subsets of $X$, define
$$F(n,(U_j))=\{x\in X: \Phi(g_{((U_j))})(x)\leq y_n\}\quad\quad \quad\quad(\ref{the3.1}.7)$$
We can get that $F(n,(U_j))$ is closed, by (a) of Proposition \ref{prop2.7} and Proposition \ref{prop2.8}.
It suffices to show the operator $F$ satisfies (i), (ii)' and (iii) of Lemma \ref{lem2.2}.
To see $U_n \supseteq F(n,(U_j))$ for each $n\in\mathbb{N}$ and let $x\in F(n,(U_j))$.
Then, $g_{((U_j))}(x)\leq\Phi(g_{((U_j))})(x)\leq y_n$ and $g_{((U_j))}(x)\leq y_n$, and thus $x\in U_{m}\setminus U_{m-1}$ and $m\leq n$ by (\ref{the3.1}.1).
So we have $x\in U_m\subset U_n$.
Hence $U_n \supseteq F(n,(U_j))$ holds.
In additional, $\Phi(g_{((U_j))})$ is l.s.c., so $F(n,(U_j))$ is a closed set of $X$ for each $n\in \mathbb{N}$. This shows that the condition (i) is satisfied.
To show (2)', note that $\Phi(g_{((U_j))})$ is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ at each $x\in U_{g_{((U_j))}}$.
Then, for each $x\in U_{g_{((U_j))}}$, there exists an open neighborhood $O$ of $x$ such that $\Phi(g_{((U_j))})(x')\leq y_{n_0}$ for some $n_0\in \mathbb{N}$ and each $x'\in O$.
It implies that $x\in IntF(n,(U_j))$.
Hence, $\bigcup_{n\in\mathbb{N}}Int F(n, (U_j(g)))=U_{g_{((U_j))}}=\bigcup_{n\in \mathbb{N}}U_n(g)$.
To show (3),
let $((G_j))$ be an increasing sequence of open subsets of $X$ such that $(U_j)\preceq (G_j)$.
Since $U_n\subseteq G_n$ for each $n\in \mathbb{N}$, it follows from (\ref{the3.1}.6) that $g_{((G_j))}(x)\leq g_{((U_j))}(x)$.
Hence, we have $\Phi(g_{((G_j))})\leq\Phi(g_{((U_j))})$.
Furthermore,
$F(n,(U_j))=\{x\in X: \Phi(g_{((U_j))})(x)\leq y_n\}\subseteq \{x\in X: \Phi(g_{((G_j))})(x)\leq y_n\}=F(n,(G_j))$ for each $n\in\mathbb{N}$,
which implies that $F((U_j))\preceq F((G_j))$.
Thus, $X$ is a stratifiable space.
\end{proof}
\begin{theorem}\label{the3.2}
Let $P$ be a bi-bounded complete, continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
Then $X$ is semi-stratifiable if and only if
there exists an operators $\Phi$ assigning to each map $g:X\rightarrow P$ with an upper bound $y_0$,
a l.s.c. map $\Phi(g):X\rightarrow P$ such that
$g\leq \Phi(g)$, $\Phi(g)$ is upper bounded at each $x\in U_g$ and that $\Phi(g)\leq \Phi(g')$ whenever $g\leq g'$.
\end{theorem}
\begin{proof}
Assume that $X$ is a semi-stratifiable.
There exists an operator $F$ satisfying (i), (ii) and (iii) in Lemma \ref{lem2.2}.
Let $P$ be a bi-bounded complete, continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
For each map $g:X\rightarrow P$ and each $n\in \mathbb{N}$, define $U_n(g)$ as (\ref{the3.1}.1) in Theorem \ref{the3.1}.
Then we have $U_g=\bigcup_{n\in \mathbb{N}}U_n(g).$
Define $\Phi(g):X \rightarrow P$ as (\ref{the3.1}.5) in Theorem \ref{the3.1}.
We only show that $\Phi(g)(x)$ is upper bounded at each $x\in U_g$, since the other properties of $\Phi(g)$ are proved in Theorem \ref{the3.1}.
Take any $x\in U_g$, by (ii) of lemma \ref{lem2.2}, we have $x\in \bigcup_{n\in \mathbb{N}}U_n(g)=\bigcup_{n\in\mathbb{N}}F(n, (U_j(g)))$, and therefore, there exists $k\in \mathbb{N}$ such that $x\in F(k, (U_j(g)))\setminus F(k-1, (U_j(g)))$.
It implies that $\Phi(g)(x)=y_k$.
This completes the
proof that $\Phi(g)(x)$ is upper bounded at each $x\in U_g$.
Conversely, let $(U_j)_{j\in\mathbb{N}}$ be a sequence of increasing open subsets of $X$.
Define a map $g_{((U_j))}:X\rightarrow P$ as (\ref{the3.1}.6) in Theorem \ref{the3.1}.
Then, we have $U_{g_{((U_j))}}=\bigcup_{n\in\mathbb{N}}U_n$, where
$$U_{g_{((U_j))}}=\{x\in X: g_{((U_j))} \text{~is locally upper bounded at~}x \}.$$
By the assumption, there exist an operators $\Phi$ assigning to each $g_{((U_j))}$ with an upper bound,
a l.s.c. map $\Phi(g_{((U_j))}):X\rightarrow P$ such that $\Phi(g_{((U_j))})$ is bounded at each $x\in U_{g_{((U_j))}}$,
$g_{((U_j))}\leq \Phi(g_{((U_j))})$, and that $\Phi(g)\leq \Phi(g')$ whenever $g\leq g'$.
For each sequence $(U_j)_{j\in\mathbb{N}}$ of increasing open subsets of $X$ and each $n\in \mathbb{N}$, define the operator $F$ as (\ref{the3.1}.7) in Theorem \ref{the3.1}.
It suffices to show that the operator $F$ satisfies (i), (ii) and (iii) of Lemma \ref{lem2.2}.
We can get that $F(n,(U_j))$ is closed, by (a) of Proposition \ref{prop2.7} and Proposition \ref{prop2.8}.
It suffices to show the operator $F$ satisfies (i), (ii) and (iii) of Lemma \ref{lem2.2}.
The proof that the operator $F$ satisfies (i) and (iii) of Lemma \ref{lem2.2} is as same as Theorem \ref{the3.1}, so we only shows that the operator $F$ satisfies (ii) of Lemma \ref{lem2.2}.
To show (ii), note that $\Phi(g_{((U_j))})$ is upper bounded at each $x\in U_{g_{((U_j))}}$.
Then, for each $x\in U_{g_{((U_j))}}$, there exists $n_0\in \mathbb{N}$ such that $\Phi(g_{((U_j))})(x)\leq y_{n_0}$.
It implies that $x\in F(n_0,(U_j))$.
Hence, $\bigcup_{n\in\mathbb{N}} F(n, (U_j(g)))=U_{g_{((U_j))}}=\bigcup_{n\in \mathbb{N}}U_n(g)$.
Thus, $X$ is a semi-stratifiable space.
\end{proof}
\begin{theorem}\label{the3.3}
Let $P$ be a bi-bounded complete, dually continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
Then $X$ is MCP if and only if
there exists an operators $\Phi$ assigning to each map $g:X\rightarrow P$ which is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ with a lower bound,
a l.s.c. map $\Phi(g):X\rightarrow P$ which is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ such that
$g\leq \Phi(g)$, and $\Phi(g)\leq \Phi(g')$ whenever $g\leq g'$.
\end{theorem}
\begin{proof}
Suppose that $X$ is MCP and $F$ is any operator that satisfies conditions (1), (2)' and (3) of Lemma \ref{lem2.5}.
Let $g:X\rightarrow P$ be a locally upper bounded map.
For each $n\in \mathbb{N}$, we define
$$ U_n(g)=Int\{x\in X: g(x)\leq y_n\} \quad\quad \quad\quad(\ref{the3.3}.1)$$
Then, $\{U_n(g):n\in\mathbb{N}\}$ is a increasing sequence of open subsets of $X$ because of Proposition \ref{prop2.9}.
It is clear that $\bigcup_{n\in \mathbb{N}}U_n(g)=X$.
Hence, $F((U_j(g)))=(F(n,(U_j(g))))_{n\in\mathbb{N}}$ is a sequence of closed subsets of $X$ such that
$$F(n,(U_j(g)))\subset U_n(g) \text{~for each~} n\in\mathbb{N};\quad\quad \quad\quad(\ref{the3.3}.2)$$
$$\bigcup_{n\in\mathbb{N}}Int F(n, (U_j(g)))=X;\quad\quad \quad\quad(\ref{the3.3}.3)$$
$$F(n,(U_j(g)))\subset F(n+1,(U_j(g))), n\in\mathbb{N};\quad\quad \quad\quad(\ref{the3.3}.4)$$
Thus, we can define $\Phi(f):X\rightarrow P$ as follows:
$$\Phi(g)(x)=\left\{
\begin{array}{rcl}
y_1 & & {x\in F(1,(U_j(g)))}\\
y_{n+1} & & x\in F(n+1,(U_j(g)))\setminus F(n,(U_j(g)))\\
\end{array} \right. \quad\quad \quad\quad(\ref{the3.3}.5)$$
It is obvious that $\Phi(g)(x)\leq y_0$ and $\Phi(g)$ has a lower bounded $y_1$.
Let us show that $g(x)\leq\Phi(g)(x)$ for each $x\in \bigcup_{n\in\mathbb{N}}F(n, (U_j(g)))$.
Also there exists $n\in\mathbb{N} $ such that $x\in F(n+1,(U_j(g)))\setminus F(n,(U_j(g)))$.
Then $g(x)\leq y_{n+1}=\Phi(g)(x)$ by $x\in F(n+1,(U_j(g)))\subset U_{n+1}(g)=Int\{x\in X: g(x)\leq y_{n+1}\}$.
If $x\in F(1,(U_j(g)))$, it is obvious that $g(x)\leq y_{1}=\Phi(g)(x)$. Thus, we have $g\leq\Phi(g)$.
To show that $\Phi(g):X\rightarrow P$ is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$, let $x\in X$.
By (\ref{the3.3}.3), there exists $n\in\mathbb{N}$ such that $x\in Int F(n+1, (U_j(g)))\setminus Int F(n, (U_j(g)))$.
Consider the neighborhood $O_x=Int F(n+1, (U_j(g)))$ of $x$.
For each $x'\in O_x$, it follows from the definition of $\Phi(g)$ that $\Phi(g)(x')\leq y_{n+1}$.
This completes the
proof that $\Phi(g)$ is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$.
Next we show $\Phi(g)$ is l.s.c..
For each $x\in \bigcup_{n\in\mathbb{N}} F(n, (U_j(g)))$, there exists some $m\in\mathbb{N}$ such that $x\in F(m+1, (U_j(g)))\setminus F(m, (U_j(g)))$.
We consider the neighborhood $O_x=X\setminus F(m, (U_j(g)))$ of $x$.
For each $x'\in O_x$, we can get $\Phi(g)(x')\geq y_{m+1}=\Phi(g)(x)$.
Therefore, $\bigwedge\Phi(g)(O_x)=y_{m+1}$ and $O_x\in \mathcal{N}_{x*}(\Phi(g))$.
This provides that $\Phi(g)(x)$ admits $\Phi(g)_*(x)$ because of (c) of Proposition \ref{prop2.7}.
We have
$$\Phi(g)_*(x)=\bigvee_{N\in\mathcal{N}_{x*}(\Phi(g))}\bigwedge\Phi(g)(N)\geq\bigwedge_{x'\in O_x}\Phi(g)(x')=y_{m+1}=\Phi(g)(x).$$
Hence, $\Phi(g)_*(x)=\Phi(g)(x)$ for each $x\in X$, that is, $\Phi(g)(x)$ is l.s.c..
Finally, let $g':X\rightarrow P$ be a map with $g\leq g'$.
Then
$$\{x\in X: g'(x)\leq y_n\}\subseteq \{x\in X: g(x) \leq y_n\}$$
and hence, $U_n(g)\supseteq U_n(g')$ for each $n\in\mathbb{N}$.
Therefore we have $F(n,(U_j(g)))\supseteq F(n,(U_j(g')))$ for each $n\in\mathbb{N}$, and
$$\bigcup_{n\in\mathbb{N}}Int F(n, (U_j(g)))=\bigcup_{n\in \mathbb{N}}U_n(g)$$
$$\bigcup_{n\in\mathbb{N}}Int F(n, (U_j(g')))=\bigcup_{n\in \mathbb{N}}U_n(g')$$
For each $x\in \bigcup_{n\in \mathbb{N}}U_n(g')$,
there exists $n\in \mathbb{N}$ such that $x\in F(n+1, (U_j(g')))\setminus F(n, (U_j(g')))$.
That is $x\in F(n+1, (U_j(g')))\subseteq F(n+1, (U_j(g)))$.
By (\ref{the3.3}.5), we can get $\Phi(g)(x)\leq y_{n+1}=\Phi(g')(x)$,
which proves the necessity.
Conversely, let $(U_j)_{j\in\mathbb{N}}$ be an increasing open cover of $X$.
Define a map $g_{((U_j))}:X\rightarrow P$ by:
$$g_{((U_j))}(x)=\left\{
\begin{array}{rcl}
y_1 & & {x\in U_{1}}\\
y_{n+1} & & {x\in U_{n+1}\backslash \texttt{}U_{n}}\\
\end{array} \right. \quad\quad \quad\quad(\ref{the3.3}.1)$$
Then, $g_{((U_j))}$ is locally upper bounded. In fact, for each $x\in X$ there exists $n\in \mathbb{N}$ such that $x\in U_{n+1}\setminus U_n$.
Consider the neighborhood $O_x=U_{n+1}$ of $x$.
For each $x'\in O_x$, it follows from the definition of $g_{((U_j))}$ that $g_{((U_j))}(x')\leq y_{n+1}$.
This shows that $g_{((U_j))}$ is locally upper bounded.
By the assumption, there exist an operators $\Phi$ assigning to each locally upper bounded map $g:X\rightarrow P$ with a lower bound,
a l.s.c. map $\Phi(g):X\rightarrow P$ which is locally upper bounded about $(y_i)_{j\in\mathbb{N}}$ such that
$g(x)\leq \Phi(g)$, and $\Phi(g)\leq \Phi(g')$ whenever $g\leq g'$.
For each sequence $(U_j)_{j\in\mathbb{N}}$ of increasing open cover of $X$, define
$$F(n,(U_j))=\{x\in X: \Phi(g_{((U_j))})(x)\leq y_n\}\quad\quad \quad\quad(\ref{the3.3}.2)$$
We can get that $F(n,(U_j))$ is closed, by (a) of Proposition \ref{prop2.7} and Proposition \ref{prop2.8}.
It suffices to show the operator $F$ satisfies (1), (2)' and (3) of Lemma \ref{lem2.5}.
To show (1) and (2), let $(U_j)_{j\in\mathbb{N}}$ be an increasing open cover of $X$.
To see $U_n \supseteq F(n,(U_j))$ for each $n\in\mathbb{N}$ and let $x\in F(n,(U_j))$.
Then, $g(x)\leq\Phi(g_{((U_j))})(x)\leq y_n$ and $g(x)\leq y_n$, and thus $x\in U_{m}\setminus U_{m-1}$ and $m\leq n$ by (\ref{the3.3}.1).
So we have $x\in U_m\subset U_n$.
Hence $U_n \supseteq F(n,(U_j))$ holds.
To show (2)', let $x\in X$, take a neighborhood $O_x$ of $x$ and $n\in\mathbb{N}$ such that $\Phi(g_{((U_j))}(x')\leq y_n$ for each $x'\in O_x$, that is, $x\in Int F(n,(U_j))$,
which shows that $\bigcup_{n\in\mathbb{N} }IntF(n,(U_j))=X$.
To show (3),
let $((G_j))$ be an increasing sequence of open subsets of $X$ such that $(U_j)\preceq (G_j)$.
Since $U_n\subseteq G_n$ for each $n\in \mathbb{N}$, it follows from (\ref{the3.3}.1) that $g_{((G_j))}(x)\leq g_{((U_j))}(x)$.
Hence, we have $\Phi(g_{((G_j))})\leq\Phi(g_{((U_j))})$.
Furthermore,
$F(n,(U_j))=\{x\in X: \Phi(g_{((U_j))})(x)\leq y_n\}\subseteq \{x\in X: \Phi(g_{((G_j))})(x)\leq y_n\}=F(n,(G_j))$ for each $n\in\mathbb{N}$,
which implies that $F((U_j))\preceq F((G_j))$. This shows that $U$ satisfies (3) of Lemma \ref{lem2.5}.
So $X$ is MCP.
\end{proof}
\begin{theorem}\label{the3.4}
Let $P$ be a bi-bounded complete, dually continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
Then $X$ is MCM if and only if
there exists an operators $\Phi$ assigning to each map $g:X\rightarrow P$ which is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ with a lower bound,
a l.s.c. map $\Phi(g):X\rightarrow P$ such that
$g\leq \Phi(g)\ll y_0$ and $\Phi(g)\leq \Phi(g')$ whenever $g\leq g'$.
\end{theorem}
\begin{proof}
The proof is obtained by a modification of that of Theorem \ref{the3.3}. So, we only show the outline of the proof.
Suppose that $X$ is MCM and $F$ is any operator that satisfies conditions (1), (2) and (3) of Lemma \ref{lem2.5}.
Let $g:X\rightarrow P$ be locally upper bounded about $(y_j)_{j\in\mathbb{N}}$.
For each $n\in \mathbb{N}$, we define
$$ U_n(g)=Int\{x\in X: g(x)\leq y_n\} \quad\quad \quad\quad(\ref{the3.4}.1)$$
Then, $\{U_n(g):n\in\mathbb{N}\}$ is a increasing sequence of open subsets of $X$ because of Proposition \ref{prop2.9}.
It is clear that $\bigcup_{n\in \mathbb{N}}U_n(g)=X$.
Thus, we can define $\Phi(g):X\rightarrow P$ as follows:
$$\Phi(g)(x)=\left\{
\begin{array}{rcl}
y_1 & & {x\in F(1,(U_j(g)))}\\
y_{n+1} & & x\in F(n+1,(U_j(g)))\setminus F(n,(U_j(g)))\\
\end{array} \right. \quad\quad \quad\quad(\ref{the3.4}.2)$$
It is obvious that $\Phi(g)(x)\ll y_0$ and $\Phi(g)$ has a lower bounded $y_1$.
Then, $\Phi(g):X\rightarrow P$ is l.s.c. such that $g\leq\Phi(g)$.
For a locally upper bounded map $g':X\rightarrow P$ with $g\leq g'$, we have $\Phi(g)\leq \Phi(g')$, which proves the necessity.
Conversely, let $(U_j)_{j\in\mathbb{N}}$ be an increasing open cover of $X$.
Define a map $g_{((U_j))}:X\rightarrow P$ by:
$$g_{((U_j))}(x)=\left\{
\begin{array}{rcl}
y_1 & & {x\in U_{1}}\\
y_{n+1} & & {x\in U_{n+1}\backslash \texttt{}U_{n}}\\
\end{array} \right. \quad\quad \quad\quad(\ref{the3.4}.3)$$
Then, $g_{((U_j))}$ is locally upper bounded.
Set $F(n,(U_j))=\{x\in X: \Phi(g_{((U_j))})(x)\leq y_n\}$ for each $n\in\mathbb{N}$.
Then, $U_n \supseteq F(n,(U_j))$ for each $n\in\mathbb{N}$.
Now, let us show that $\bigcup_{n\in\mathbb{N} }F(n,(U_j))=X$.
Let $x\in X$. Since $\Phi(g)\ll y_0=\bigvee\{y_i:i\in\mathbb{N}\}$, and $\{y_i:i\in\mathbb{N}\}$ are directed, there exists $i\in\mathbb{N}$ such that $\Phi(g)(x)\leq y_i$.
That is, $x\in F(i,(U_j))$, which shows that $\bigcup_{n\in\mathbb{N} }F(n,(U_j))=X$.
For an increasing sequence $((G_j))$ of open subsets of $X$ such that $(U_j)\preceq (G_j)$, we have $F((U_j))\preceq F((G_j))$.
So $X$ is MCM.
\end{proof}
\section{Other results}
By analogy with Theorems 3.1 through 3.4, we can prove the following Theorems, which extend some earlier results.
\begin{theorem}
Let $P$ be a bi-bounded complete, dually continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
Then $X$ is countably paracompact if and only if
there exists an operators $\Phi$ assigning to each map $g:X\rightarrow P$ which is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ with a lower bound,
a l.s.c. map $\Phi(g):X\rightarrow P$ which is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ such that
$g\leq \Phi(g)$.
\end{theorem}
\begin{theorem}
Let $P$ be a bi-bounded complete, dually continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
Then $X$ is countably metacompact. if and only if
there exists an operators $\Phi$ assigning to each map $g:X\rightarrow P$ which is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ with a lower bound,
a l.s.c. map $\Phi(g):X\rightarrow P$ such that
$g\leq \Phi(g)\ll y_0$.
\end{theorem}
Recall that the stratifiable (semi-stratifiable) spaces is the monotone versions of the perfectly normal (perfect) spaces.
We get the similar results for perfectly normal (perfect) spaces as follows.
\begin{theorem}
Let $P$ be a bi-bounded complete, continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
Then $X$ is perfectly normal if and only if
there exists an operators $\Phi$ assigning to each map $g:X\rightarrow P$ with $F_g\neq\emptyset$,
a l.s.c. map $\Phi(g):X\rightarrow P$ such that
$g\leq \Phi(g)$ and that $\Phi(g)$ is locally upper bounded about $(y_i)_{i\in\mathbb{N}}$ at each $x\in U_g$.
\end{theorem}
\begin{theorem}
Let $P$ be a bi-bounded complete, continuous poset $P$ with a $\ll_{d}$-increasing $\ll$-limit point $y_0$.
Then $X$ is perfect if and only if
there exists an operators $\Phi$ assigning to each map $g:X\rightarrow P$ with an upper bound $y_0$,
a l.s.c. map $\Phi(g):X\rightarrow P$ such that
$g\leq \Phi(g)$ and that $\Phi(g)$ is upper bounded at each $x\in U_g$.
\end{theorem}
The proofs of Theorem 4.1, 4.2, 4.3 and 4.4 follow in the same way as Theorem 3.1, 3.2, 3.3 and 3.4.
T. Kubiak \cite[Theorem 2.5]{TK} and E. Lane and C. Pan (see \cite{LN} ) gave the characterizations of monotonically normal spaces by monotone insertions of real-valued functions.
From viewpoints of Theorem \ref{the3.3}, it is natural to ask about monotone
poset-valued insertions on monotonically normal and monotonically countably paracompact spaces.
\begin{definition}\cite{Yama}
A poset $P$ endowed with a topology is called sup-continuous if $\bigvee: J_2\rightarrow P$ is continuous, where $J_2=\{\langle x,y\rangle\in P\times P:\exists x\vee y\in P\}$ is endowed with the subspace topology of the product space $P \times P$.
Dually,
a poset $P$ endowed with a topology is called inf-continuous if $\bigwedge:M_2\rightarrow P$ is continuous, where $M_2=\{\langle x,y\rangle\in P\times P:\exists x\wedge y\in P\}$ is endowed with the subspace topology of the product space $P \times P$.
\end{definition}
A topological poset is a sup-and inf-continuous poset.
Now, define $J_1=M_1=P$ and $J_n=\{\langle x_1,x_2,\cdots, x_n\rangle\in P^n:\exists \bigvee_{i=1}^nx_i\in P\}$ and $M_n=\{\langle x_1,x_2,\cdots,x_n \rangle\in P^n:\exists \bigwedge_{i=1}^nx_i\in P\}$, endowed with the subspace topology of $P^n$, for each $n\in\mathbb{N}$ with $n\geq3$.
\begin{proposition}\label{prop4.6}\cite{Yama}
If $P$ is an upper-bounded (resp. lower-bounded) complete and sup-continuous (resp. inf-continuous) poset, then $\bigvee:J_n\rightarrow P$ (resp. $\bigwedge:M_n\rightarrow P$) is continuous for each $n\in\mathbb{N}$.
\end{proposition}
\begin{theorem}\label{the4.7}
Let $X$ be monotonically normal and monotonically countably paracompact and $P$ be a bi-bounded complete, dually continuous poset with a $\ll_{d}$-increasing $\ll$-limit point $y_0$
such that $y_i\ll_d y_{i+1}$, $y_i\ll y_0$ $(i\in \mathbb{N})$ and $y_0=\bigvee_{i\in \mathbb{N}}y_i$.
Let $g:X\rightarrow P$ be an u.s.c. map with a lower bound $\perp g$.
Assume that $g\leq y_0$, and $\langle g, y_0\rangle$ has interpolated points on $\{y_n; n\in \mathbb{N}\}$
such that $\langle g(x),y_0\rangle$ has interpolated points $y_{j}(x)$ and $y_{k}(x)$ on $\{y_n; n\in \mathbb{N}\}$ with $y_{j}(x)\leq y_{k}(x)$ and a monotone increasing path $\varphi_x:[0,1]\rightarrow [y_j(x),y_0]$
from some lower point of $y_k(x)$ for each $x\in X$ to $y_0$.
Then, there exist an operator $\Phi$ assigning to each u.s.c. map $g:X\rightarrow P$,
a continuous map $\Phi(g):X\rightarrow P$ such that
$g(x)\ll_{d}\Phi(g)(x)\ll y_0$ for each $x\in X$ and that
$\Phi(g)\geq \Phi(g')$ whenever $g\leq g'$.
\end{theorem}
\begin{proof}
There exists an operator $F$ satisfying (1), (2)' and (3) in Lemma \ref{lem2.5}.
Let $g:X\rightarrow P$ be a u.s.c. map where $g\leq y_0$, and $\langle g, y_0\rangle$ has interpolated points on $\{y_n: n\in \mathbb{N}\}$.
For each $n\in \mathbb{N}$, we define
$$ U_n=\{x\in X: g(x)\ll_dy_n\} \quad\quad \quad\quad(\ref{the4.7}.1)$$
Then, $\{U_n:n\in\mathbb{N}\}$ is a increasing sequence of open subsets of $X$ because of Proposition \ref{prop2.9}.
It is clear that $\bigcup_{n\in \mathbb{N}}U_n=X$.
In fact, for each $x\in X$, there exists $n\in\mathbb{N}$ such that $g(x)\ll_d y_n\ll y_0$.
It provides that $x\in U_n$.
Hence, $F((U_j))=(F_n)_{n\in\mathbb{N}}$ is a sequence of closed subsets of $X$ such that
$$F_n\subset U_n \text{~for each~} n\in\mathbb{N};$$
$$\bigcup_{n\in\mathbb{N}}Int F_n=X;$$
$$F_n\subset F_{n+1}, n\in\mathbb{N};$$
Similarly, $F((Int F_{n}))=(E_n)_{n\in\mathbb{N}}$ is a sequence of closed subsets of $X$ such that
$$E_n\subset Int F_{n} \text{~for each~} n\in\mathbb{N};$$
$$\bigcup_{n\in \mathbb{N}}IntE_n=\bigcup_{n\in\mathbb{N}}Int F(n, (U_j))=X;$$
$$E_n\subset E_{n+1}, n\in\mathbb{N};$$
And $F((Int E_{n}))=(L_n)_{n\in\mathbb{N}}$ is a sequence of closed subsets of $X$ such that
$$L_n\subset Int E_{n} \text{~for each~} n\in\mathbb{N};$$
$$\bigcup_{n\in \mathbb{N}}IntL_n=\bigcup_{n\in\mathbb{N}}Int E(n, (U_j))=X;$$
$$L_n\subset L_{n+1}, n\in\mathbb{N};$$
Let $H_n=U_n\setminus L_{n-1}$, $G_n=IntF_n\setminus E_{n-1}$ for each $n\in \mathbb{N}$ and $H_1=U_1$, $G_1=IntF_1$.
It is obvious that $\{H_n:n\in\mathbb{N}\}$ and $\{G_n:n\in\mathbb{N}\}$ are locally finite open covers of $X$ such that $\overline{G_n}\subset H_n\subset U_n$ for each $n\in\mathbb{N}$.
Since $x$ is monotone normal, take a continuous function $j_n:X\rightarrow [0,1]$ by
$$j_n(x)=\left\{
\begin{array}{rcl}
0 & & {x\in \overline{G_n}}\\
1 & & {x\in X\setminus H_n}\\
\end{array} \right. \text{~for each~} n\in\mathbb{N}.\quad\quad \quad\quad(\ref{the4.7}.2)$$
For each $n\in\mathbb{N}$, there exists $t_n\in P$ and a continuous monotone increasing map $\varphi_n:[0,1]\rightarrow [y_n,y_0]\subset P$ such that
$$ \varphi_n(0)=t_n, \text{~and~} \varphi_n(1)=y_0, y_n\leq t_n\leq y_{n+1}.\quad\quad \quad\quad(\ref{the4.7}.3)$$
Define a continuous map $k_n:X\rightarrow[y_n,y_0]\subset P$ by
$$k_n=\varphi_n\circ j_n \text{~for each~} n\in\mathbb{N}.\quad\quad \quad\quad(\ref{the4.7}.4)$$
Thus, we can define a continuous map $\Phi(g):X\rightarrow P$ as follows:
$$\Phi(g)(x)=\bigwedge_{n\in\mathbb{N}}k_n(x)\quad\quad \quad\quad(\ref{the4.7}.5)$$
for each $x\in X$. Now, to show that $\Phi(g)$ is defined, we set $\delta_x=\{n\in \mathbb{N}:x\in H_n\}$ for each $x\in X$.
For each $n\notin \delta_x$, $k_n(x)=\varphi_n\circ j_n(x)=\varphi_n(1)=y_0$.
For each $n\in \delta_x$, $k_n(x)=\varphi_n\circ j_n(x)\geq y_n$ because of the range of $\varphi_n$ is $[y_n,y_0]$.
It follows from $H_n\subset U_n$ that
$$g(x)\ll_dy_n\leq k_n(x) \text{~for each~} n\in \delta_x.$$
This shows that $\{k_n(x):n\in \delta_x\}$ and $\{y_n:n\in \delta_x\}$ have an lower bound $g(x)$, thus $\bigwedge_{n\in\delta_x}k_n(x)$ and $\bigwedge_{n\in\delta_x}y_n$ exist.
Hence, by (1) of \cite[Proposition I-1.2 (3)]{Gi}, we obtain that
$$g(x)\ll_d\bigwedge_{n\in\delta_x}y_n\leq\bigwedge_{n\in\delta_x}k_n(x)=\bigwedge_{n\in\delta_x}k_n(x)\wedge y_0=\Phi(g)(x).$$
Therefore, $\Phi(g)$ is defined, and we also have $g\ll_d \Phi(g)$.
Next, to show $\Phi(g)\ll y_0$, let $x\in X$.
Since $\{G_n:n\in\mathbb{N}\}$ is a cover of $X$, take $n'\in\mathbb{N}$ so as to satisfy $x\in G_{n'}$.
Then, it follows from $G_{n'}\subset U_{n'}$ that
$$\Phi(g)(x)\leq k_{n'}(x)=\varphi_{n'}\circ j_{n'}(x)=\varphi_{n'}(0)=t(n')\leq y_{n'}\ll y_0.$$
Thus, $\Phi(g)\ll y_0$ holds, because of (2) of Lemma \ref{lem2.6}.
Finally, to show that $\Phi(g)$ is continuous. Let $x\in X$, and take a neighborhood $O_x$ of $x$ and a finite subset $\delta'_x$ of $\mathbb{N}$ such that $O_x\bigcap H_n\neq \emptyset$ for each $n\in \delta'_x$.
Then, $\delta_y\subset \delta'_x$ for each $y\in\delta_y$.
Since $\Phi(g)(y)$ can be re-expressed as $\Phi(g)(y)=\bigwedge_{n\in\delta'_x}k_n(y)\wedge y_0=\bigwedge_{n\in\delta'_x}k_n(y)$ for each $y\in O_x$, we have that $\langle k_n(y)\rangle_{n\in\delta'_x}\in M_{|\delta'_x|}$ for each $y\in O_x$.
This means $(\Delta_{n\in\delta'_x}k_n)(O_x)\subset M_{|\delta'_x|}$, where $\Delta_{n\in\delta'_x}k_n$ is the diagonal of mappings $\{k_n:n\in\delta'_x\}$.
Hence, on $O_x$, $\Phi(g)$ is the composition $\bigwedge\circ(\Delta_{n\in\delta'_x}k_n)$ of $\Delta_{n\in\delta'_x}k_n$ and $\bigwedge:M_{|\delta'_x|}\rightarrow P$.
By Proposition \ref{prop4.6}, $\Phi(g)$ is continuous at $x$.
Finally, let $g':X\rightarrow P$ be a map with $g\leq g'$.
Then
$$\{x\in X: g'(x)\ll_dy_n\}\subseteq \{x\in X: g(x)\ll_dy_n\}$$
and hence, $U_n\supseteq U'_n$ for each $n\in\mathbb{N}$.
Therefore we have $F_n\supseteq F'_n$, $E_n\supseteq E'_n$ and $L_n\supseteq L'_n$ for each $n\in\mathbb{N}$, and
$$\bigcup_{n\in \mathbb{N}}U'_n=\bigcup_{n\in\mathbb{N}}Int F'_n=\bigcup_{n\in\mathbb{N}}Int E'_n=\bigcup_{n\in\mathbb{N}}Int L'_n$$
For each $x\in X$, there exists $n\in \mathbb{N}$ such that $x\in G'_n=IntF'_n\setminus E'_{n-1}$.
That is $x\in IntF'_n\subseteq IntF_n$.
By (\ref{the4.7}.2), (\ref{the4.7}.3), (\ref{the4.7}.4) and the monotonicity of $\varphi_n(x)$ we can get $k_n(x)\geq t_n=\varphi'_n(0)=\varphi'_n\circ j'_n(x)=k'_n(x)$.
This implies that $\Phi(g)\geq \Phi(g')$ whenever $g\leq g'$.
This completes the proof.
\end{proof}
For a map $f:$ $X\rightarrow P$ and a point $y\in P$, we define
$$U_{f,y}=\{x\in X: \langle f(x),y\rangle \text{~has an interpolated point~}\}$$ and $U_{y,f}=\{x\in X: \langle y,f(x)\rangle\text{~has an interpolated point~}\}$.
We have known that monotonically normal and monotonically countably paracompact spaces are stratifiable, then we have the following question:
\begin{question}
Let $X$ be stratifiable and $P$ be a bi-bounded complete, dually continuous poset with a $\ll_{d}$-increasing $\ll$-limit point $y_0$
such that $y_i\ll_d y_{i+1}$, $y_i\ll y_0$ $(i\in \mathbb{N})$ and $y_0=\bigvee_{i\in \mathbb{N}}y_i$.
Do the following conditions exist?
Let $g:X\rightarrow P$ be an u.s.c. map with a lower bound $\perp g$.
Assume that $g\leq y_0$, and $\langle g, y_0\rangle$ has interpolated points on $\{y_n; n\in \mathbb{N}\}$
such that $\langle g(x),y_0\rangle$ has interpolated points $y_{j}(x)$ and $y_{k}(x)$ on $\{y_n; n\in \mathbb{N}\}$ with $y_{j}(x)\leq y_{k}(x)$ and a monotone increasing path $\varphi_x:[0,1]\rightarrow [y_j(x),y_0]$
from some lower point of $y_k(x)$ for each $x\in X$ to $y_0$.
Then, there exist an operator $\Phi$ assigning to each u.s.c. map $g:X\rightarrow P$,
a continuous map $\Phi(g):X\rightarrow P$ such that $g(x)\leq\Phi(g)(x)\leq y_0$,
$g(x)\ll_{d}\Phi(g)(x)\ll y_0$ for each $x\in U_{g,y_0}$ and that
$\Phi(g)\geq \Phi(g')$ whenever $g\leq g'$.
\end{question}
\vskip0.9cm
\end{document} |
\begin{document}
\
\maketitle
\centerline{\scshape L.~Boulton$^1$}
{\footnotesize \centerline{Departmento de Matem\'aticas,
Universidad Sim\'on Bol\'\i var} \centerline{Apartado 89000,
Caracas 1080-A, Venezuela} \centerline{email: [email protected]}}
\begin{abstract}
Let $H$ be the discrete Schr\"odinger operator \linebreak
$Hu(n):=u(n-1)+u(n+1)+v(n)u(n)$, $u(0)=0$ acting on $\le (\mathbb{Z}^+)$
where the potential $v$ is real-valued and $v(n)\to 0$ as $n\to
\infty$. Let $P$ be the orthogonal projection onto a closed linear
subspace $\mathcal{L} \subset \le (\mathbb{Z}^+)$. In a recent paper E.B. Davies
defines the second order spectrum ${\rm Spec}_2(H,\mathcal{L})$ of $H$
relative to $\mathcal{L}$ as the set of $z \in \mathbb{C}$ such that the
restriction to $\mathcal{L}$ of the operator $P(H-z)^2P$ is not invertible
within the space $\mathcal{L}$. The purpose of this article is to
investigate properties of ${\rm Spec}_2(H,\mathcal{L})$ when $\mathcal{L}$ is
large but finite dimensional. We explore in particular the
connection between this set and the spectrum of $H$. Our main
result provides sharp bounds in terms of the potential $v$ for the
asymptotic behaviour of ${\rm Spec}_2(H,\mathcal{L})$ as $\mathcal{L}$ increases
towards $\le(\mathbb{Z}^+)$.
{\rm e}nd{abstract}
\section{Introduction} \label{s1}
Let the discrete Schr\"odinger operator $H$ be defined by
\[
Hu(n):=u(n+1)+u(n-1)+v(n)u(n),\qquad u(0)=0
\]
acting on $\le (\mathbb{Z}^+)$ where $v:\mathbb{Z}^+ \longrightarrow \mathbb{R}$.
In \cite{SECR}, E.B.~Davies investigated the concept of resonance
of $H$ associated to a closed subspace \linebreak $\mathcal{L}\subset \le (\mathbb{Z}^+)$.
Let $P$ be the orthogonal projection onto $\mathcal{L}$.
He called the isolated points of the second order spectrum of $H$
\begin{equation} \label{e19}
\spec[2]{H;\mathcal{L}} := \{ \lambda \in \mathbb{C} \,:\, P(H-\lambda)^2P|\mathcal{L}\
\mathrm{is\ not\ invertible}\},
{\rm e}nd{equation}
resonances of $H$ relative to $\mathcal{L}$. This definition is motivated
by graphical and numerical connections between other notions of
resonance adopted to this context, and isolated points of
$\spec[2]{H;\mathcal{L}}$ (cf. \cite[section 9]{SECR}). This was observed
by Davies in the case when the dimension of $\mathcal{L}$ is low ($\leq
30$) and $v$ has finite support.
In \cite{SECR} Davies also pointed out that $\spec[2]{H;\mathcal{L}}$
contains information about the spectrum $\spec{H}$ of $H$.
In \cite{GHORS}
E.~Shargorodsky developed this idea for general linear operators
$T$. This began the study of second order spectra as a projection
method for localizing the spectrum. Among other interesting
results on geometrical properties of $\mathrm{Spec}_2$,
Shargorodsky showed that for any bounded self-adjoint operator
$T$,
\begin{equation} \label{e18}
\bigcup \lim_{k\to \infty} \spec[2]{T;\mathcal{L}_k} \cap \mathbb{R} =\spec{T}
{\rm e}nd{equation}
where the union is taken over the set of all sequences $(\mathcal{L}_k)$
of subspaces of the domain of $T$ such that the orthogonal
projection $P_k$ onto $\mathcal{L}_k$ converges strongly to the identity,
see \cite[theorem 21]{GHORS}. Hence the second order spectra of
$H$ might be useful for approximate computation of $\spec{H}$ when
$v$ is bounded.
The purpose of the present paper is to continue investigating the
connection between the spectrum and second order spectra of $H$.
We present a detailed description of
\[
\spec[2]{H;k}:=\spec[2]{H;\mathcal{L}_k}
\]
for large $k$ when
\begin{equation*}
\mathcal{L}_k:=\mathrm{span} \{\delta_1,\ldots,\delta_k\}, \qquad
\delta_j(n):=\left\{\begin{array}{cc} 0 & n\not=j \\ 1 &n=j
{\rm e}nd{array} \right.
{\rm e}nd{equation*}
and there exist constants $a,r>0$ such that
\begin{equation} \label{e2}
|v(n)|\leq \frac{a}{n^{r+1}} \qquad \qquad n=1,2,\ldots
{\rm e}nd{equation}
As $v$ is real-valued and bounded, $H$ is a bounded self-adjoint
operator so we are in the situation covered by Shargorodsky's
results. Since {\rm e}qref{e2} holds for finite rank $v$, our
discussion includes the potentials investigated by Davies in
\cite{SECR}.
By construction, $\mathcal{L}_k$ is $k$-dimensional. Then
the quadratic operator pencil in {\rm e}qref{e19} is a $k\times k$ matrix and
\begin{equation*}
\spec[2]{H;k} = \{ \lambda \in \mathbb{C} \,:\, \det[P_k(H-\lambda)^2P_k|\mathcal{L}_k]=0 \}.
{\rm e}nd{equation*}
It is easy to see that the determinant at the right hand side is
a polynomial in $\lambda$ of degree at most $2k$ so that
$\spec[2]{H;k}$ consists exclusively of at most
$2k$ relative resonances of $H$.
Since our interest is in how the $\spec[2]{H;k}$ are related to
the spectrum of $H$, we describe the latter. The classical theory
allows us to find $\spec{H}$ under the hypothesis {\rm e}qref{e2} quite
efficiently. For this we decompose
\[
H=H_0+V
\]
where $H_0$ is $H$ without potential and
$ Vu(n):= v(n)u(n)$. The spectrum of $H_0$ is pure
absolutely continuous and equal to the interval $[-2,2]$. Since
\begin{align*}
\trac{V} & =
\sum_{n=1}^\infty |\dotp{V\delta_n,\delta_n}| = \sum_{n=1}^\infty
|v(n)| \\ & \leq \sum_{n=1}^\infty \frac{a}{n^{r+1}} < +\infty,
{\rm e}nd{align*}
$V$ is a trace class operator. Then, by virtue of Pearson's theorem,
\[
\spec[\mathrm{ess}]{H}=\spec[\mathrm{ac}]{H}=[-2,2]
\]
so $\spec{H}$ consists of the interval $[-2,2]$ together with a
discrete set of isolated eigenvalues of finite multiplicity which
can only accumulate at $\pm 2$. A recent adaptation by F.~Luef and
G.~Teschl of the Sturm-Liouville theorem
to the discrete context, cf. \cite{LuTe}, shows that in fact
if $r>1$ in ${\rm e}qref{e2}$, then $H$ has
only a finite number of eigenvalues.
The results we will discuss in the forthcoming sections are motivated by
the following proposition.
\begin{proposition} \label{t3}
Let $V=V^{\ast}$ be a compact operator acting on $\le (\mathbb{Z}^+)$.
Then for all $z\not\in \spec{H_0+V}$ there exists $\tilde{k}>0$
such that
\[
z \not \in \spec[2]{H_0+V;k}, \qquad k\geq \tilde{k}.
\]
{\rm e}nd{proposition}
In other words, roughly speaking
\begin{equation} \label{e12}
S:=\lim_{k\to \infty} \spec[2]{H;k} \subseteq \spec{H}.
{\rm e}nd{equation}
Notice that we are less restrictive on the requirements for $V$.
In comparing with {\rm e}qref{e18}, this result says that our
particular choice of $\mathcal{L}_k$ does not lead to spurious points in
approximating $\spec{H}$. In section \ref{s2} we will show
proposition \ref{t3} using an argument that will be crucial for
our latter work. It consists in considering $\spec[2]{H;k}$ as the
perturbed spectrum of a convenient Toeplitz matrix.
Our main results are theorem~\ref{t7} and corollary~\ref{t8} of section~\ref{s4}.
There we give a precise meaning to the limit in {\rm e}qref{e12} and provide
sharp estimates on the rate of convergence in terms of $v$ when it satisfies
{\rm e}qref{e2}.
The proof of theorem~\ref{t7} requires to develop some results on
the stability of Toeplitz operators with smooth symbols. This is
carried out in section~\ref{s3}. It will become clear from the
beginning that these techniques can be extended to more general
situations, hence we give a general treatment to the topic.
Section~\ref{s3} is closely related to chapter~2 of the monograph
\cite{BoSi} on large Toeplitz matrices. Since we failed to find a
reference for theorem~\ref{t4} and corollary~\ref{t6} we include
proofs of both.
We complete our discussion by considering in detail some numerical examples in
section~\ref{s5}. These are closely related to those of \cite[section~9]{SECR}.
\section{The finite section method} \label{s2}
The finite section method for perturbed Toeplitz operators was
developed by I.~Gohberg, I.A.~Feldman and H.~Widom in the 1970's
(see \cite{GF} and \cite{W}). Through this paper we follow
closely their ideas, in particular we follow the notation and
procedures of the introductory text on the subject by A.~B\"otcher
and B.~Silbermann, \cite{BoSi}. This section is devoted to discuss
the basic notation and results in the Gohberg-Feldman-Widom
approach which will be the core of our latter work. This will lead
us to a proof of proposition \ref{t3}.
Below and elsewhere we identify the truncation
\[
P_kTP_k|\mathcal{L}_k:
\mathcal{L}_k \longrightarrow \mathcal{L}_k
\]
of any linear operator $T:\le(\mathbb{Z}^+)\longrightarrow \le(\mathbb{Z}^+)$,
with its matrix representation acting on $\mathbb{C}^k$. We will often
denote $T_k:=P_kTP_k|\mathcal{L}_k$. The following definition is taken
from \cite[chapter 2]{BoSi}: we say that the sequence
$\{P_kTP_k\}$ is stable (with constants $k_0$ and $M$) if and only
if, there exists $k_0>0$ such that $T_k$ is invertible for all $k
\geq k_0$ and
\begin{equation*}
\sup_{k \geq k_0} \| T_k^{-1} \| \leq M
<\infty.
{\rm e}nd{equation*}
Notice that if $\{P_k(H-z)^2P_k\}$ is stable then $z\in \mathbb{C}$ falls
outside $\spec[2]{H;k}$ for large $k$.
If $T$ is a Toeplitz operator with continuous symbol, then
$\{P_kTP_k\}$ is stable if and only if $T$ is invertible (cf.
\cite[theorem 2.11]{BoSi}). Furthermore (see corollary~\ref{t6}
below or \cite[theorem 2.16]{BoSi}), if $\{P_kTP_k\}$ is stable,
$K$ is compact and $T+K$ is invertible, then $\{P_k(T+K)P_k\}$ is
stable.
The following argument is crucial to out later work. Decompose
\begin{equation} \label{e1} \begin{aligned}
(H-z)^2 & = (H_0-z)^2+(H_0-z)V+V(H_0-z) +V^2 \\
& = T(\beta)+K(z)
{\rm e}nd{aligned} {\rm e}nd{equation}
where $T(\beta)$ is the Toeplitz operator whose symbol is
\begin{align*}
\beta(t;z)&:= t^{-2}-2zt^{-1}+(2+z^2)-2zt+t^2 \\
& = (t^{-2}-zt^{-1}+1)(t^{2}-zt+1) \\
& =(t^{-1}-z+t)^2,\\
t\in & \,\mathbb{T}:=\{{\rm e}^{i\vartheta}:-\pi<\vartheta\leq \pi\}
{\rm e}nd{align*}
and
\begin{align*}
K(z)& :=(H_0-z)V+V(H_0-z) +V^2-|\delta_1\rangle \langle \delta_1| \\
& = H_0V+VH_0+V^2-2zV-|\delta_1\rangle \langle \delta_1|
{\rm e}nd{align*}
is compact. According to the above, the stability of
$\{P_kT(\beta)P_k\}$ implies stability of $\{P_k(H-z)^2P_k\}$ when
$z\not \in \spec{H}$. This is the key argument in the proof of
proposition \ref{t3}.
\textbf{\underline{Proof} [of proposition \ref{t3}].} The symbol
\begin{equation*}
\beta({\rm e}^{i\vartheta};z) = (2\cos\vartheta -z)^2, \qquad -\pi < \vartheta \leq \pi.
{\rm e}nd{equation*}
Then $\beta(\mathbb{T};z)$ is an open curve without loops so therefore, by
virtue of Gohberg's theorem on the spectrum of Toeplitz operators
with continuous symbol (see for instance
\cite[theorem~1.17]{BoSi}), $\mathrm{Spec}\,T(\beta)=\beta(\mathbb{T};z)$.
Thus $T(\beta)$ is invertible if and only if $z\not\in
[-2,2]$. For all such $z$, $\{P_kT(\beta)P_k\}$
is stable. Clearly $(H-z)^2$ is invertible if and only if
$z\not\in \spec{H}$. Hence,
\[
\{P_k[T(\beta)+K(z)]P_k\}=\{P_k(H-z)^2P_k\}
\]
is stable for all $z \not \in \spec{H}$. \hspace*{\fill} $\, \blacksquare$
\section{Stability of Toeplitz operators with smooth symbol} \label{s3}
In this section $T(\alpha)$ denotes the Toeplitz operator associated
to the symbol $\alpha:\mathbb{T} \longrightarrow \mathbb{C}$. If $q>0$, $W^q$ stands for the set of
continuous $\alpha$ such that
\[
\sum_{n\in\mathbb{Z}} |n|^q|\widehat{\alpha}(n) |<\infty
\]
where $\widehat{\alpha}(n)$ denote here and elsewhere the
(non-normalized) Fourier coefficients of $\alpha$. It is well known
that $W^q$ is a Banach algebra with pointwise algebraic operations
and norm $\|\alpha\|:=\sum (1+|n|)^q|\widehat{\alpha}(n)|$. According
to Wiener's theorem (cf. \cite{wie}), if $\alpha\in W^q$ and $0\not
\in \alpha(\mathbb{T})$, then \linebreak $\alpha^{-1}\in W^q$.
Throughout this section we will assume that
\begin{itemize}
\item[i)] The symbol $\alpha\in W^q$ for some $q>1$.
\item[ii)] The sequence $\{P_kT(\alpha)P_k\}$ is stable with
constants denoted by $k_0$ and $M$: that is $T_k(\alpha)$ is
invertible for all $k \geq k_0$ and \linebreak $\sup_{k \geq k_0}
\| T_k^{-1}(\alpha) \| \leq M$.
{\rm e}nd{itemize}
The continuity of $\alpha$ and stability of $\{P_kT(\alpha)P_k\}$
guarantee that $T(\alpha)$ is invertible. Under these conditions it
is well known (cf. \cite[pp.32-34]{BoSi}) that $T^{-1}_k(\alpha)$
converges in the strong operator topology to $T^{-1}(\alpha)$.
Therefore it is legitimate to estimate the columns of the matrix
of $T^{-1}(\alpha)$ from the columns of $T_k^{-1}(\alpha)$ for large
$k$. The following result states that if we not only assume
continuity of the symbol but the stronger condition i), the error
in the estimation of the first $k/2$ columns of $T^{-1}(\alpha)$
from $T_k^{-1}(\alpha)$ is $O(k^{q-\varepsilon})$.
\begin{theorem} \label{t4}
Let $T(\alpha)$ satisfy i)-ii). Then for all $1\leq p< q$ there
exists a constant $c>0$ independent of $k$, such that
\[
\|T_k^{-1}(\alpha)\delta_j-T^{-1}(\alpha)\delta_j\| \leq \frac{c}{k^p}
\]
provided $k\geq k_0$ and $1\leq j\leq k/2$.
{\rm e}nd{theorem}
Below we will compute an explicit upper bound for $c$ in terms of
$\alpha$, $M$ and $\|T^{-1}(\alpha)\|=:\tilde{M}$.
This theorem improves \cite[theorem~2.15]{BoSi}. It allows us to
show a version of the result mention in section~\ref{s2} on
stability of compact perturbation of stable sequences, when the
perturbed operator $T$ is Toeplitz, its symbol satisfies i) and
the perturbation $K$ is a trace class band matrix. This is the
content of corollary \ref{t6}. Let
\begin{align*}
K_{j,l}&=\dotp{K\delta_l,\delta_j}\qquad \qquad & l,j=1,2,\ldots \\
K_{j,l}&=0 & l\leq0 \mathrm{\ or\ } j\leq0 .
{\rm e}nd{align*}
We assume that $K$ satisfies the following hypotheses,
\begin{itemize}
\item[a)] There exists $L> 0$ such that $K_{j,l}=0$
for $| j-l|> L$.
\item[b)] There exist $b>0$ and $r>0$, such that
\begin{equation*}
K_{j,l}\leq \frac{b}{j^{r+1}}, \qquad \qquad l,j=1,2,\ldots
{\rm e}nd{equation*}
{\rm e}nd{itemize}
\begin{corollary} \label{t6}
Let $T(\alpha)$ satisfy i)-ii) and let $K$ satisfy a)-b). Fix
\linebreak $1\leq p <q$, let $\rho:=\min\{r,(2p-1)/2\}$ and let
$c$ be as in theorem~\ref{t4}. If $T(\alpha)+K$ is invertible, then
$P_k(T(\alpha)+K)P_k|\mathcal{L}_k$ is invertible provided
\begin{align}
k \geq & \nonumber \\ \, \max& \left\{
\left(\frac{4L\left[c\sqrt{3}\|K\|+b 2^{r}\sqrt{\pi}(M+\tilde{M})\right]
\|(T(\alpha)+K)^{-1}T(\alpha)\|}{\sqrt{6}}\right)^{1/\rho},k_0\right\}\label{e11}.
{\rm e}nd{align}
{\rm e}nd{corollary}
\textbf{\underline{Proof}.}\ Throughout the proof we assume $1\leq p<q$ and $k> k_0$. By
virtue of ii),
\[
I+T^{-1}(\alpha)K=T^{-1}(\alpha)(T(\alpha)+K)
\]
is invertible. Let
\[
1/\varepsilon:=\|(I+T^{-1}(\alpha)K)^{-1}\|=\|(T(\alpha)+K)^{-1}T(\alpha)\|.
\]
Then for all $u\in \le (\mathbb{Z}^+)$,
\begin{align*}
\varepsilon \|P_ku \|&\leq \|(I+T^{-1}(\alpha)K)P_ku\| \\
& \leq \|(I+T_k^{-1}(\alpha)P_kK)P_ku\| +
\|T^{-1}_k(\alpha)P_kK-T^{-1}(\alpha)K\|
\,\|P_ku\|.
{\rm e}nd{align*}
The operator $K$ acts on $u$ as follows,
\[
Ku=\sum_{n=1}^\infty \sum _{m=-L}^{L} K_{n,n+m} u(n+m) \delta_n.
\]
Let $R_k:=T^{-1}_k(\alpha)P_k-T^{-1}(\alpha)$. Then
\begin{align*}
\|R_kKu\| & = \left\|\sum_{n=1}^\infty \sum _{m=-L}^{L} K_{n,n+m}
u(n+m)R_k\delta_n \right\| \\
& \leq \sum_{m=-L}^{L} \sum_{n=1}^\infty
|u(n+m)|\,|K_{n,n+m}|\,\|R_k \delta_n\|.
{\rm e}nd{align*}
For each $m=-L,\ldots,L$,
\begin{align*}
\sum_{n=1}^\infty &|u(n+m)| \,|K_{n,n+m}|\,\|R_k \delta_n\| \leq \\
& \leq \left( \sum_{n=1}^\infty |u(n+m)|^2 \right)^{1/2}
\left(\sum_{n=1}^\infty |K_{n,n+m}|^2\|R_k \delta_n\|^2\right)^{1/2}
\\ & \leq \|u\| \left( \sum_{n=1}^{k/2} |K_{n,n+m}|^2\|R_k \delta_n\|^2
+\sum_{n=k/2+1}^{\infty} |K_{n,n+m}|^2\|R_k \delta_n\|^2
\right)^{1/2}.
{\rm e}nd{align*}
By virtue of theorem \ref{t4},
\begin{align*}
\sum_{n=1}^{k/2} |K_{n,n+m}|^2\|R_k \delta_n\|^2 & \leq
\frac{c^2\sum_{n=1}^{k/2} |K_{n,n+m}|^2}{k^{2p}} \\
& \leq \frac{c^2\|K\|^2\sum_{n=1}^{k/2}1}{k^{2p}} =
\frac{c^2\|K\|^2}{2k^{2p-1}}.
{\rm e}nd{align*}
By virtue of a)-b) and by the definition of $R_k$,
\begin{align*}
\sum_{n=k/2+1}^{\infty} |K_{n,n+m}|^2\|R_k \delta_n\|^2 & \leq
\|R_k\|^2 \sum_{n=k/2+1}^{\infty} |K_{n,n+m}|^2 \\
& \leq (M+\tilde{M})^2 \sum_{n=k/2+1}^{\infty} |K_{n,n+m}|^2 \\ &
\leq \frac{b^22^{2r}(M+\tilde{M})^2 \sum_{n=k/2+1}^{\infty}
\left[(k/2)^r/n^{r+1}\right]^2}{k^{2r}} \\ &
\leq \frac{b^22^{2r}(M+\tilde{M})^2 \sum_{n=k/2+1}^{\infty}
1/n^2}{k^{2r}} \\ & \leq
\frac{b^22^{2r}(M+\tilde{M})^2 \pi
}{6k^{2r}}.
{\rm e}nd{align*}
Then, if $k$ satisfies {\rm e}qref{e11},
\[
\|T^{-1}_k(\alpha)K-T^{-1}(\alpha)K\|
\leq 2L\frac{c\sqrt{3}\|K\|+b2^{r}\sqrt{\pi}
(M+\tilde{M})}{\sqrt{6}k^{\rho}}\leq \varepsilon /2.
\]
For all such $k$,
\begin{align*}
(\varepsilon/2)\|P_kv\| & \leq \|(I+T_k^{-1}(\alpha)P_kK)P_kv\| \\
& \leq \|T_k^{-1}(\alpha)\|\,\|(T_k(\alpha)+P_kK)P_kv\| \\
& =\|T_k^{-1}(\alpha)\|\,\|P_k(T(\alpha)+K)P_kv\| \\
& \leq M \|P_k(T(\alpha)+K)P_kv\|,
{\rm e}nd{align*}
so that $P_k(T(\alpha)+K)P_k|\mathcal{L}_k$ is injective and thus
invertible. Notice that $\dim \mathcal{L}_k$ is finite. \hspace*{\fill} $\, \blacksquare$
In section \ref{s4} we will employ this corollary to estimate sharp
bounds for the error in the limit {\rm e}qref{e12}.
We now prove theorem~\ref{t4}. For this purpose we use the
following convention: $Q_k:=I-P_k$,
\[
W_ku(n):=\left\{
\begin{array}{cc} u(k-n+1) & 1\leq n \leq k \\
0 &n>k {\rm e}nd{array} \right.,
\]
$\tilde{\alpha}:=\alpha(t^{-1})$, $H(\alpha)$ is the Hankel operator
generated by $\alpha$ (cf. \cite[p.13]{BoSi}),
\[
\Lambda(\alpha):=T^{-1}(\alpha)-T(\alpha^{-1})
\]
and
\[
B_k:=P_kT^{-1}(\alpha)P_k+W_k\Lambda(\tilde{\alpha})W_k.
\]
Assumption ii) and the fact that $\alpha$ is continuous, ensure that
$\Lambda(\alpha)$ is well defined. The following results are due to
Widom, but a proof can be found in \cite[pp.40-42]{BoSi}: {\rm e}mph{If
$\alpha \in \Le[\infty] (\mathbb{T})$ and $T(\alpha)$ is invertible, then
\begin{itemize}
\item[1)] $\Lambda(\alpha)=T^{-1}(\alpha)H(\alpha)H(\tilde{\alpha}^{-1})
=H(\alpha^{-1})H(\tilde{\alpha})T^{-1}(\tilde{\alpha})$,
\item[2)] $T_k(\alpha)B_k = P_k-P_kT(\alpha)Q_k\Lambda(\alpha)P_k-
W_kT(\tilde{\alpha}) Q_k \Lambda(\tilde{\alpha})W_k. $
{\rm e}nd{itemize}}
\textbf{\underline{Proof} [of theorem \ref{t4}].} Throughout the
proof we assume that \linebreak $k\geq k_0$ and $1\leq j\leq k/2$.
Decompose
\begin{gather*}
T_k^{-1}(\alpha) = B_k+C_k, \\
C_k:=T_k^{-1}(\alpha)-B_k = T_k^{-1}(\alpha)(P_k-T_k(\alpha)B_k).
{\rm e}nd{gather*}
Then
\begin{equation} \label{e5}
\begin{aligned}
T_k^{-1}(\alpha)\delta_j-T^{-1}&(\alpha)\delta_j =
[T^{-1}_k(\alpha)-P_kT^{-1}(\alpha)P_k]\delta_j+
[P_kT^{-1}(\alpha)P_k-T^{-1}(\alpha)]\delta_j\\
& = [B_k+C_k-P_kT^{-1}(\alpha)P_k]\delta_j+
[P_kT^{-1}(\alpha)P_k-T^{-1}(\alpha)]\delta_j \\
& =C_k\delta_j+W_k\Lambda(\tilde{\alpha})W_k\delta_j+
[P_kT^{-1}(\alpha)P_k-T^{-1}(\alpha)]\delta_j.
{\rm e}nd{aligned}
{\rm e}nd{equation}
Fix $1\leq p <q$. In order to find the parameter $c$, we will
estimate the norm of the three terms in the sum at the end of
{\rm e}qref{e5}. For this, let
\begin{gather*}
c_1:=\sum_{n\in\mathbb{Z}}|n|^p|\widehat{\alpha^{-1}}(n)|, \\
c_2:=\left(\sum_{n\in\mathbb{Z}}|n|^{2p}|\widehat{\alpha^{-1}}(n)|^2\right)^{1/2}.
{\rm e}nd{gather*}
Since $\alpha\in W^q$ and $0\not \in \alpha(\mathbb{T})$, both $c_1$ and hence
$c_2$ are finite. By definition of $Q_k$ and $H(\alpha^{-1})$,
\begin{align*}
\|Q_kH(\alpha^{-1})\| & \leq \|\alpha^{-1}(\cdot)-\sum_{n\leq k-1}
\widehat{\alpha^{-1}}(n){\rm e}^{in(\cdot)} \|_\infty \\
& \leq \sum_{|n| \geq k}|\widehat{\alpha^{-1}}(n)| \\
& \leq\frac{ \sum_{|n| \geq k}|n|^p|\widehat{\alpha^{-1}}(n)|}{k^p}
\leq \frac{c_1}{k^p}.
{\rm e}nd{align*}
Hence by virtue of 1),
\begin{equation} \label{e5.5}
\|Q_k\Lambda(\alpha)\| \leq \frac{c_1\|H(\tilde{\alpha})T^{-1}(\tilde{\alpha})\|}{k^p}
\leq \frac{c_1\tilde{M}\|\alpha\|_\infty}{k^p}.
{\rm e}nd{equation}
In a similar manner one can show
\[
\|Q_k\Lambda(\tilde{\alpha})\| \leq
\frac{c_1\tilde{M}\|\tilde{\alpha}\|_\infty}{k^p}=
\frac{c_1\tilde{M}\|\alpha\|_\infty}{k^p}.
\]
We estimate the norm of $C_k$. By virtue of 2),
\begin{align*}
C_k&=-T^{-1}_k(\alpha)(T_k(\alpha)B_k-P_k) \\
& = T^{-1}_k(\alpha)(P_kT(\alpha)Q_k\Lambda(\alpha)P_k+W_kT(\tilde{\alpha}) Q_k
\Lambda(\tilde{\alpha})W_k).
{\rm e}nd{align*}
Thus
\begin{equation} \label{e6}
\begin{aligned}
\|C_k\| &\leq \|T^{-1}_k(\alpha)\|\,\|P_kT(\alpha)Q_k\Lambda(\alpha)P_k +
W_kT(\tilde{\alpha}) Q_k\Lambda(\tilde{\alpha}) W_k\| \\
&\leq M\left(\|\alpha\|_\infty\|Q_k\Lambda(\alpha)\|+
\|\tilde{\alpha}\|_\infty\|Q_k\Lambda(\tilde{\alpha})\|\right) \\
&\leq \frac{2Mc_1\tilde{M}\|\alpha\|^2_\infty}{k^p}.
{\rm e}nd{aligned}
{\rm e}nd{equation}
The second term is
\begin{equation} \label{e7}
\begin{aligned}
\|W_k\Lambda(\tilde{\alpha})W_k\delta_j\| &\leq
\|\Lambda(\tilde{\alpha})\delta_{k-j+1}\| \\
& = \|T^{-1}(\tilde{\alpha})H(\tilde{\alpha})H(\alpha^{-1})\delta_{k-j+1}\|
\\
& \leq \tilde{M} \|\alpha\|_\infty \|H(\alpha^{-1})\delta_{k-j+1}\| \\
& = \tilde{M} \|\alpha\|_\infty \left(
\sum_{n\geq k-j}|\widehat{\alpha^{-1}}(n)|^2 \right)^{1/2} \\
& \leq \tilde{M} \|\alpha\|_\infty \left(
\sum_{n\geq k/2}|\widehat{\alpha^{-1}}(n)|^2 \right)^{1/2} \\
&\leq \frac{2^p\tilde{M} \|\alpha\|_\infty \left( \sum_{n\geq
k/2}n^{2p}|\widehat{\alpha^{-1}}(n)|^2\right)^{1/2}}{k^{p}} \\ &\leq
\frac{2^p\tilde{M} \|\alpha\|_\infty c_2}{k^{p}}
{\rm e}nd{aligned}
{\rm e}nd{equation}
The third term is
\begin{equation} \label{e8}
\begin{aligned}
\|[P_kT^{-1}(\alpha)P_k-T^{-1}(\alpha)]\delta_j\| & =
\|P_kT^{-1}(\alpha)\delta_j-T^{-1}(\alpha)\delta_j\| \\
& = \|Q_kT^{-1}(\alpha)\delta_j\| \\
& = \|Q_k[\Lambda(\alpha)+T(\alpha^{-1})]\delta_j\| \\
& \leq \|Q_k\Lambda(\alpha)\delta_j\|+\|Q_kT(\alpha^{-1})\delta_j\| \\
& \leq \frac{c_1\|\alpha\|_\infty \tilde{M}}{k^p}+
\|Q_kT(\alpha^{-1})\delta_j\|\\
& =\frac{c_1\|\alpha\|_\infty \tilde{M}}{k^p} +
\left(\sum_{n\geq k-j+1} |\widehat{\alpha^{-1}}(n)|^2\right)^{1/2}
\\ & \leq \frac{c_1\|\alpha\|_\infty \tilde{M}}{k^p} +
\left(\sum_{n\geq k/2} |\widehat{\alpha^{-1}}(n)|^2\right)^{1/2}
\\& \leq \frac{c_1\|\alpha\|_\infty \tilde{M}}{k^p} +
2^p\left(\sum_{n\geq k/2} (n/k)^{2p}|\widehat{\alpha^{-1}}(n)|^2
\right)^{1/2}
\\& \leq \frac{c_1\|\alpha\|_\infty \tilde{M}+2^pc_2}{k^p}
{\rm e}nd{aligned}
{\rm e}nd{equation}
Hence the conclusion of theorem~\ref{t4} can be recovered from
{\rm e}qref{e5}, {\rm e}qref{e6}, {\rm e}qref{e7} and {\rm e}qref{e8} by putting
\[
c=\tilde{M}\|\alpha\|_\infty(2Mc_1\|\alpha\|_\infty+2^pc_2+c_1)+2^p c_2. \hspace*{\fill} $\, \blacksquare$d
\]
\section{The second order spectra of $H$} \label{s4}
We are now ready to state and prove the main results of this paper.
These roughly say that for all $k$ large enough,
$\spec[2]{H;k}$ is contained in a small neighbourhood
of $\spec{H}$ with diameter of order some negative powers of $k$.
As we previously mentioned, applying corollary~\ref{t6}
to the decomposition {\rm e}qref{e1} will play a crucial part into the proofs.
\begin{theorem} \label{t7}
If {\rm e}qref{e2} holds, then there exists a constant $b(r)>0$
independent of $z$, such that
\[
\spec[2]{H;k} \subset \{z\in \mathbb{C}\,:\, \dist{z,\mathrm{Spec}\,H} <
b(r)/k^{2r/(2r+35)}\},
\]
for all $k$.
{\rm e}nd{theorem}
This automatically implies:
\begin{corollary} \label{t8}
Suppose that $r>0$ can be chosen arbitrarily large for $v$ in
{\rm e}qref{e2}. Then for all $0<q<1$ there exists a constant $b(q)>0$
independent of $z$, such that
\[
\spec[2]{H;k} \subset \{z\in \mathbb{C}\,:\, \dist{z,\mathrm{Spec}\,H} <
b(q)/k^q\},
\]
for all $k$.
{\rm e}nd{corollary}
The rest of this section is devoted to proving theorem~\ref{t7}.
We first introduce some further notation. Let
\[
d(z):=\dist{z,[-2,2]}\qquad \mathrm{and} \qquad
\tilde{d}(z):=\dist{z,\mathrm{Spec}\,H}.
\]
Then $d(z)\geq \tilde{d}(z)$. For $z\not \in [-2,2]$, we denote by
$T(\beta^{-1})$ the Toeplitz operator whose symbol is
\[
\beta^{-1}(t;z) =(t^{-2}-2zt^{-1} +(2+z^2)-2zt+t^2 )^{-1}.
\]
Since $\beta^{-1}(\mathbb{T};z)$ is an open curve and $0\not \in
\beta^{-1}(\mathbb{T};z)$, then $T(\beta^{-1})$ is invertible. Put
$\tilde{M}_2:=\|T^{-1}(\beta^{-1})\|$.
We break the proof into various steps. The aim of these steps is
to show that for all $R>0$, there exists $\tilde{b}>0$ independent
of $z$, such that $P_k(H-z)^2P_k$ is invertible provided
\[
z\not \in \spec{H}, \qquad \tilde{d}(z)\leq R
\]
and
\[
k\geq \frac{\tilde{b}}{[\tilde{d}(z)]^{(2r+35)/2r}}.
\]
Since the $\spec[2]{H;k}$ are bounded uniformly in $k$ (cf.
\cite[theorem~18]{SECR}), this assertion implies theorem~\ref{t7}.
Below we assume without further mention that $z\not \in \spec{H}$
and $\tilde{d}(z)\leq R$. The various constants $b_j$ that appear
below are independent of $z$ but they may depend on $R$, $p$ or
$V$.
\underline{Step 1}: we estimate $\tilde{M}$ and $\tilde{M}_2$ for
the symbol under consideration. When $z\not\in [-2,2]$,
\[
\zetaa_{\pm}:=\frac{z\pm \sqrt{z^2-4}}{2}\not =0
\]
are such that $\zetaa_+=\zetaa_-^{-1}$ and $|\zetaa_\pm|\not=1$. One
of these numbers is inside the unit disk whereas the other is
outside it. Denote by $\zetaa$ the one that is outside and put
\[
\betaa_{\pm}(t;z):=\frac{1}{\zetaa}(t^{\pm 1}-\zetaa)^2.
\]
Then
\[
\beta(t;z)=\beta_+(t;z)\beta_-(t;z),
\]
$\betaa_+^{\pm 1}$ can be extended analytically inside the unit
disk and $\betaa_-^{\pm 1}$ can be extended analytically outside
it. This is the so called Wiener-Hopf factorization of $\beta$, so
that (see for instance \cite[theorem~1.15]{BoSi})
\[
T^{-1}(\beta)=T(\beta^{-1}_+)T(\beta_-^{-1}).
\]
Hence
\[
\begin{aligned}
\tilde{M} & = \|T^{-1}(\beta)\| \\
& \leq \|T(\beta_+^{-1})\|\,\|T(\beta_-^{-1})\|
=\|\beta^{-1}_+\|_\infty\|\beta^{-1}_-\|_\infty \\
& = |\zetaa|^2 \sup_{t\in \mathbb{T}} |t-\zetaa|^{-2}\sup_{t\in \mathbb{T}}
|1/t-\zetaa|^{-2} \\&=|\zetaa|^2
\sup|t-\zetaa|^{-4} \\
&\leq \frac{b_0}{d(z)^4}.
{\rm e}nd{aligned}
\]
By writing the corresponding Wiener-Hopf factorization for
$\beta^{-1}$ we obtain in a similar manner
\[
\begin{aligned}
\tilde{M}_2 & \leq \|T(\beta_+)\|\,\|T(\beta_-)\| \\
& = \|\beta_+\|_\infty \|\beta_-\|_\infty = \frac{1}{|\zetaa|^2}
\sup_{t\in \mathbb{T}} |t-\zetaa|^{2}\sup_{t\in \mathbb{T}}
|1/t-\zetaa|^{2} \leq b_1.
{\rm e}nd{aligned}
\]
\underline{Step 2}: we now want to estimate the constants $c_1$
and $c_2$ in the proof of theorem~\ref{t4}, in terms of $z$.
Notice that since $\beta$ is a trigonometric polynomial, $\beta$ and
hence $\beta^{-1}$ belong to $W^p$ for all $p=1,2,\ldots$. It is
easy to show that
\[
\frac{{\rm d}^p}{{\rm d} \vartheta^p}(\beta^{-1})({\rm e}^{i\vartheta}) =
\sum _{r=3}^{p+2} \frac{\phi_r(\vartheta)}{(2\cos \vartheta-z)^r},
\]
where the $\phi_r(\vartheta)$ are smooth, bounded and independent of
$z$. Then
\begin{align*}
c_2& =
\left(\sum_{n\in\mathbb{Z}}|n|^{2p}|\widehat{\beta^{-1}}(n)|^2\right)^{1/2}
= \left(\sum_{n\not =0}|\widehat{(\beta^{-1})^{(p)}}(n)|^2\right)^{1/2}
\\ & \leq \|(\beta^{-1})^{(p)}\|_{\Le(-\pi,\pi)}
= b_2\left( \int_{-\pi}^{\pi} |(\beta^{-1})^{(p)}({\rm e}^{i\vartheta})|^2
{\rm d} \vartheta \right)^{1/2} \\ &\leq b_2\left( \int_{-\pi}^{\pi} \left(\sum _{r=3}^{p+2}
\left|\frac{\phi_r(\vartheta)}
{(2\cos \vartheta-z)^r}\right|\right)^2
{\rm d} \vartheta \right)^{1/2} \\
& \leq b_3 \sum_{r=3}^{p+2} \frac{1}{d(z)^r} \leq
\frac{b_4}{d(z)^{p+2}}.
{\rm e}nd{align*}
Similarly
\begin{align*}
c_1 & =\sum_{n\in\mathbb{Z}}|n|^{p}|\widehat{\beta^{-1}}(n)| =
\sum_{n\in\mathbb{Z}} \frac{|n|^{p+1}}{|n|}|\widehat{\beta^{-1}}(n)| \\
& \leq \left(\sum_{n\not=0}\frac{1}{n^2}\right)^{1/2}
\left(\sum_{n\not=0}|n|^{2p+2}|\widehat{\beta^{-1}}(n)|^2
\right)^{1/2} \\
&= \sqrt{2\pi/6}\left(\sum_{n\not =0}|\widehat{(\beta^{-1})^{(p+1)}}(n)|^2\right)^{1/2}
\leq \frac{b_5}{d(z)^{p+3}}.
{\rm e}nd{align*}
\underline{Step 3}: estimation of $M$. This only makes sense for
$k$ large enough.
\begin{lemma} \label{t10}
For all $R>0$ and $1<q<2$, there exists constants $b_6,\,b_7>0$
independent of $z$, such that $T_k(\beta)$ is invertible and
\begin{equation} \label{e20}
\|T_k^{-1}(\beta)\| \leq \frac{b_6}{d(z)^8},
{\rm e}nd{equation}
provided
\[
z\not\in [-2,2], \qquad d(z)\leq R
\]
and
\[
k\geq \frac{b_7}{[d(z)]^q}.
\]
{\rm e}nd{lemma}
\textbf{\underline{Proof}.}\ By virtue of \cite[lemma 2.9]{BoSi}, we know that
$T_k(\beta)$ is invertible if and only if
\[
Q_kT^{-1}(\beta)Q_k|\mathrm{Ran}\, Q_k
\]
is invertible, and in this case
\begin{align*}
T_k^{-1}(\beta)P_k=& \\ P_kT^{-1}(\beta)&P_k-P_kT^{-1}(\beta)Q_k
( Q_kT^{-1}(\beta)Q_k|\mathrm{Ran}\, Q_k)^{-1}Q_kT^{-1}(\beta)P_k.
{\rm e}nd{align*}
The truncation
\begin{equation} \label{e13}
Q_kT^{-1}(\beta)Q_k|\mathrm{Ran}\, Q_k =
Q_kT(\beta^{-1})Q_k|\mathrm{Ran}\, Q_k+Q_k\Lambda(\beta)Q_k|\mathrm{Ran}\, Q_k,
{\rm e}nd{equation}
where
\[
\Lambda(\beta)=T^{-1}(\beta)-T(\beta^{-1})
\]
as in section \ref{s2}. The matrix of
$Q_kT(\beta^{-1})Q_k|\mathrm{Ran}\, Q_k$ is the same matrix
$T(\beta^{-1})$. Then, since $0\not \in \beta^{-1}(\mathbb{T})$, the former
is invertible and
\[
\left(Q_kT(\beta^{-1})Q_k|\mathrm{Ran}\, Q_k\right)^{-1}
\]
has the same matrix as $T^{-1}(\beta^{-1})$. Thus
\[
\|(Q_kT(\beta^{-1})Q_k|\mathrm{Ran}\, Q_k)^{-1}\|=\|T^{-1}(\beta^{-1})\|.
\]
Let $p=1,2,\ldots$. By virtue of {\rm e}qref{e5.5},
\[
\|Q_k\Lambda(\beta)Q_k\| \leq
\frac{c_1\tilde{M}\|\beta\|_\infty}{k^p}\, \qquad k\geq 1.
\]
According to the steps 1 and 2,
\[
2c_1\tilde{M}\|\beta\|_\infty \tilde{M}_2 \leq
\frac{b_8}{d(z)^{p+7}}.
\]
Then, if
\begin{equation*}
k\geq \frac{b_8^{1/p}}{d(z)^{(p+7)/p}},
{\rm e}nd{equation*}
\begin{align*}
\|(Q_kT(\beta^{-1})Q_k|\mathrm{Ran}\, Q_k)^{-1}&Q_k\Lambda(\beta)Q_k\| \\
&\leq
\|(Q_kT(\beta^{-1})Q_k|\mathrm{Ran}\, Q_k)^{-1}\|\,\|Q_k\Lambda(\beta)Q_k\| \\
&\leq
\frac{\|T^{-1}(\beta^{-1})\|c_1\tilde{M}\|\beta\|_\infty}{k^p}\\
&=\frac{c_1\tilde{M}\|\beta\|_\infty \tilde{M}_2}{k^p}\leq 1/2.
{\rm e}nd{align*}
By virtue of {\rm e}qref{e13}, for all such $k$,
$Q_kT^{-1}(\beta)Q_k|\mathrm{Ran}\, Q_k$ and so $T_k(\beta)$ are
invertible, and
\begin{align*}
\|T^{-1}_k(\beta)\| & \leq
\|T^{-1}(\beta)\|+2\|T^{-1}(\beta)\|^2\tilde{M}_2 \\
& \leq \tilde{M}+2\tilde{M}^2\tilde{M}_2 \leq
\frac{b_0}{d(z)^4}+\frac{2b_0b_1}{d(z)^8}.
{\rm e}nd{align*}
Take $p$ large enough to complete the proof. \hspace*{\fill} $\, \blacksquare$
Below $M$ denotes the right hand side {\rm e}qref{e20}.
\underline{Step 4}: We are now ready to complete the proof of
theorem~\ref{t7}. For this we use corollary \ref{t6}. According to
{\rm e}qref{e1},
\[
P_k(H-z)^2P_k =T_k(\beta)+P_kK(z)P_k,
\]
where
\[
K(z)=H_0V+VH_0+V^2-2zV-|\delta_1\rangle \langle \delta_1|.
\]
It is easy to see that the matrix entries of $K(z)$ are
\begin{align*}
\dotp{K(z)\delta_n,\delta_n} & = \delta_1(n) +v^2(n)-2zv(n) \\
\dotp{K(z)\delta_n,\delta_{n+1}} & = v(n)+v(n+1) \\
\dotp{K(z)\delta_{n+1},\delta_n} & = v(n)+v(n+1) \\
\dotp{K(z)\delta_l,\delta_m}& =0 \qquad \mathrm{if}\ |l-m|\geq 2.
{\rm e}nd{align*}
By virtue of {\rm e}qref{e2}, this ensures that $K(z)$ satisfies
conditions a)-b) of section~\ref{s3}.
Fix $p=(2r+1)/2$. According to steps 1-3, the constant $c$ of
theorem~\ref{t4} for this particular symbol is
\begin{align*}
c&=\tilde{M}\|\beta\|_\infty(2Mc_1\|\beta\|_\infty+2^pc_2+c_1)+2^p c_2 \\
& \leq \frac{b_{10}}{d(z)^{4}}\left(\frac{b_{11}}{d(z)^{p+11}}
+\frac{b_{12}}{d(z)^{p+2}}
+\frac{b_{13}}{d(z)^{p+3}}\right) + \frac{b_{14}}{d(z)^{p+2}} \\
& \leq \frac{b_{15}}{d(z)^{p+15}} \leq
\frac{b_{15}}{\tilde{d}(z)^{p+15}}.
{\rm e}nd{align*}
Also $\|K(z)\|\leq b_{16}$ and
\begin{align*}
\|(T(\beta)+K(z))^{-1}T(\beta)\| &\leq
\|(H-z)^{-1}\|^2\|\beta\|_\infty \\
& \leq \frac{(2+|z|)^2}{\tilde{d}(z)^2} \leq
\frac{b_{17}}{\tilde{d}(z)^2}.
{\rm e}nd{align*}
Then
\begin{align*}
[c\sqrt{3}\|K\|+b 2^{r}\sqrt{\pi}(M+\tilde{M})]&
\|(T(\beta)+K)^{-1}T(\beta)\| \leq \\
& \leq \frac{b_{18}}{\tilde{d}(z)^{(2p+34)/2}}=
\frac{b_{18}}{\tilde{d}(z)^{(2r+35)/2}}.
{\rm e}nd{align*}
Thus, by virtue of corollary~\ref{t6} and lemma~\ref{t10}, there
exists $\tilde{b}>0$ such that $P_k(H-z)^2P_k$ is invertible
provided
\[
z\not\in \spec{H}, \qquad \tilde{d}(z)\leq R
\]
and
\[
k \geq \frac{\tilde{b}}{[\tilde{d}(z)]^{(2r+35)/2r}}.
\]
This completes the proof of theorem~\ref{t7}. \hspace*{\fill} $\, \blacksquare$
In principle one can follow track of the constants
$b_0,\ldots,b_{18}$ for particular potentials. Nonetheless one
should be aware that $\tilde{b}$ increases its value faster than
$2^r$ as $r$ increases towards infinity.
\section{Numerical examples} \label{s5}
In this final section we discuss corollary~\ref{t8} for rank one
potentials from the numerical point of view. In particular we show
some computations of second order spectrum for this kind of
potentials and compare these with $\spec{H}$ which in this case
can be found explicitly. The results below extend those of
\cite[example 20]{SECR}.
Everywhere in this section, we assume that $v$
is such that
\begin{equation} \label{e15}
v(n)= \left\{ \begin{array}{lc} a & n=j \\
0 & \mathrm{otherwise,} {\rm e}nd{array} \right.
{\rm e}nd{equation}
for some $j=3,4,\ldots$ and $a>2$. In the proof of the following
result we use some properties of the transfer matrix associated to
the difference equation
\begin{equation} \label{e14}
Hu(n)=\lambda u(n), \qquad \qquad u(0)=0,\, u(1)=1.
{\rm e}nd{equation}
See \cite{LaSi} and \cite{NaYa} for some recent results on the
relationship between spectral properties of $H$ and the transfer
matrix for slow decaying potentials.
\begin{proposition} Let $v$ be as in {\rm e}qref{e15}. Then the discrete
spectrum of $H$ consists of an isolated eigenvalue $\lambda_a$, such
that
\[
a<\lambda_a<a+2/a.
\]
{\rm e}nd{proposition}
\textbf{\underline{Proof}.}\ Since $a$ is positive, any eigenvalue $\lambda_a$ of $H$
should be $\lambda_a>2$. Let $\lambda>2$ be a solution of the
eigenvalues equation {\rm e}qref{e14} where $u(n)$ is a sequence with
(at the moment) no constraint of growth at infinity. For all
$n\geq 1$,
\[
\begin{vect} u(n) \\ u(n+1) {\rm e}nd{vect} = \begin{matr2} 0 & 1 \\
-1 & \lambda - v(n) {\rm e}nd{matr2} \begin{vect} u(n-1) \\ u(n)
{\rm e}nd{vect}.
\]
Let
\[
T:= \begin{matr2} 0 & 1 \\ -1 & \lambda
{\rm e}nd{matr2}.
\]
Then for all $n\not= j$,
\begin{equation} \label{e16}
\begin{vect} u(n) \\ u(n+1){\rm e}nd{vect}= T \begin{vect} u(n-1) \\ u(n)
{\rm e}nd{vect}.
{\rm e}nd{equation}
The Jordan decomposition for $T$ yields
\[
T=U\begin{matr2} \mu_+ & 0 \\ 0 & \mu_- {\rm e}nd{matr2} U^{-1},
\]
where $\mu_\pm(\lambda){\rm e}quiv \mu_\pm:=(\lambda\pm\sqrt{\lambda^2-4})/2$,
\[
U=\begin{matr2} 1 & 1 \\ \mu_+ & \mu_- {\rm e}nd{matr2} \qquad
\mathrm{and} \quad U^{-1}=(\mu_- -\mu_+)^{-1}
\begin{matr2} \mu_- & -1 \\ -\mu_+ & 1 {\rm e}nd{matr2}.
\]
By applying inductively {\rm e}qref{e16} for $n$ from $1$ up to the
step $j-1$, we obtain
\begin{align*}
\begin{vect} u(j-1) \\ u(j) {\rm e}nd{vect} & =
T^{j-1} \begin{vect} u(0) \\ u(1) {\rm e}nd{vect} =
U \begin{matr2} \mu_+^{j-1} & 0 \\ 0 & \mu_-^{j-1} {\rm e}nd{matr2} U^{-1}
\begin{vect} 0 \\ 1 {\rm e}nd{vect} \\
& = \frac{1}{\mu_- -\mu_+} \begin{vect} \mu_-^{j-1}-\mu_+^{j-1} \\
\mu_-^{j}-\mu_+^{j} {\rm e}nd{vect}.
{\rm e}nd{align*}
Then
\begin{align*}
\begin{vect} u(j) \\ u(j+1) {\rm e}nd{vect} & = \begin{matr2}
0 & 1 \\ -1 & \lambda-a {\rm e}nd{matr2} \begin{vect} u(j-1) \\ u(j) {\rm e}nd{vect}
\\ & = \frac{1}{\mu_- -\mu_+} \begin{vect} \mu_-^{j}-\mu_+^{j} \\
(\lambda-a)(\mu_-^{j}-\mu_+^{j})-(\mu_-^{j-1}-\mu_+^{j-1})
{\rm e}nd{vect}.
{\rm e}nd{align*}
Hence, for all $n>j$
\begin{equation} \label{e17}
\begin{aligned}
\begin{vect} u(n) \\ u(n+1){\rm e}nd{vect} & = T^{n-j} \begin{vect} u(j) \\ u(j+1)
{\rm e}nd{vect} \\
& = U \begin{matr2} \mu_+^{n-j} & 0 \\ 0 & \mu_-^{n-j} {\rm e}nd{matr2}
U^{-1} \begin{vect} u(j) \\ u(j+1) {\rm e}nd{vect} \\
& = U \begin{vect} \mu_+^{n-j} w_1 \\ \mu_-^{n-j}w_2 {\rm e}nd{vect},
{\rm e}nd{aligned}
{\rm e}nd{equation}
where
\begin{gather*}
w_1(\lambda){\rm e}quiv w_1:= \frac{\mu_-u(j) - u(j+1)}{\mu_- - \mu_+} \qquad
\mathrm{and} \\
w_2(\lambda){\rm e}quiv w_2:= \frac{u(j+1)-\mu_+u(j)}{\mu_- - \mu_+}.
{\rm e}nd{gather*}
If $u$ is an eigenvector in $\le (\mathbb{Z}^+)$, then necessarily
$u(n)\to 0$ as $n\to \infty$. Hence
\[
\mu_{+} ^{n-j}w_{1} \to 0 \qquad \mathrm{and} \qquad
\mu_{-} ^{n-j}w_{2} \to 0
\]
as $n\to \infty$. Since $\lambda >2$, then $\mu_+>1$ and $0<\mu_-<1$.
Notice that $w_1$ and $w_2$ are independent of $n$. Thus in order
for $u\in \le (\mathbb{Z}^+)$, necessarily $w_1=0$.
By substituting $u(j)$ and $u(j+1)$ in the identity for $w_1$,
\[
w_1= \frac{\mu_-(\mu_-^j-\mu_+^j)-(\lambda-a)
(\mu_-^j-\mu_+^j)+(\mu_-^{j-1}-\mu_+^{j-1})}{(\mu_- - \mu_+)^2}.
\]
Notice also that $(\mu_+ / \mu_-)=\mu_+^2$. Then, a
straightforward computation shows that $w_1=0$, if and only if
$q(\mu_+)=0$ where
\[
q(x):=x^{2j+1}-ax^{2j}-x^{2j-1}+a.
\]
It is easy to see that $q'(x)$ vanishes, if and only if $x=0$ or
\[
x=x_{\pm}:=\frac{2ja\pm\sqrt{(2ja)^2+4(2j+1)(2j-1)}}{4j+2}.
\]
Furthermore
\begin{itemize}
\item[1)] For all $j\geq 3$, $x_-<0$ and $x_+>1$,
\item[2)] $q(0)=a>0$,
\item[3)] $q(1)=0$ and
\item[4)] $q'(x)<0$ for $0<x<x_+$.
{\rm e}nd{itemize}
Hence, it is not difficult to see that $q(x)$ has only one root
$x_a$ in the interval $(1,\infty)$. Since $q(a)=a-a^{2j-1}<0$ and
\[
q(a+1/a)=\frac{a^2+a^4+(a+1/a)^{2j}}{a+a^3}>0,
\]
necessarily $a<x_a<a+1/a$. Now,
\[
\mu_+(\lambda)=x_a,
\]
if and only if
\[
\lambda=\lambda_a:=x_a+1/x_a.
\]
Therefore $\lambda_a$ is the only possible eigenvalue of $H$ and
\[
a<\lambda_a<a+2/a.
\]
Finally we must show that $\lambda_a$ is indeed an eigenvalue of $H$.
By virtue of {\rm e}qref{e17}, for all $n>j$
\[
u(n)=\mu_-^{n-j}(\lambda_a)w_2(\lambda_a).
\]
Since $w_2(\lambda_a)\not =0$ is fixed in $n$ and $0<\mu_-<1$, $u$
is an $\le (\mathbb{Z}^+)$ eigenvector of $H$ as we required. \hspace*{\fill} $\, \blacksquare$
In the third column of table \ref{tb2} we show the value of
$\lambda_a$ for selected $a$ and $j$. We found the roots of the
polynomial $q(x)$ using the internal algorithm ``roots'' that the
package Matlab provides. More complete data can be found without
difficulty by this method.
\begin{table}[t]
\begin{tabular} {|c|c|c|c|} \hline $a$ & $j$ & $\lambda_a$
& Estimation of $\lambda_a$ using $\spec[2]{H;60}$ \\ \hline 3 & 3
&3.60362098809610& 3.60362098809609 - 0.00000008894009$i$
\\ 3 & 6& 3.60554979388607 & 3.60554979388608 -
0.00000005798745$i$\\ 3 & 9 & 3.60555127432257 &
3.60555134832088\\ 3 & 12 & 3.60555127546311 & 3.60555133471164
\\ \hline 3 & 5& 3.60553511316735 & 3.60553516591671 \\ 6 & 5 &
6.32455524824927 & 6.32455533827497 \\ 9 & 5 & 9.21954445506085 &
9.21954445506086 - 0.00000009147845$i$ \\ 12 & 5 &
12.16552506041796& 12.16552524742606
\\
\hline
{\rm e}nd{tabular}
\caption{$\lambda_a$ for selected values of $a$ and
$k$.} \label{tb2}
{\rm e}nd{table}
In figures~\ref{f4} and \ref{f5} we show $\spec[2]{H;60}$ for the
first four and the last four pairs $a,j$ in table~\ref{tb2}
respectively. We found the data for this and all the other
pictures of second order spectra in this paper, by adapting in
Matlab the Maple algorithm ``reson25'' included in the appendix-a
of the electronic version of \cite{SECR}. Notice that there is
always a point in the second order spectrum which approximates the
isolated eigenvalue $\lambda_a$. We reproduce the coordinates of
these points in the forth column of table~\ref{tb2}. Nonetheless
some of these values have imaginary part different from zero, in
all cases the real part of the approximation of $\lambda_a$ is
accurate up to the $6^{\mathrm{th}}$ digit. This is confirmed by
further numerical experiments.
From figures~\ref{f4} and \ref{f5} we can also say something about
the shape of the second order spectra of $H$. Since $V$ is of rank
one, roughly speaking $\spec[2]{H;60}$ should not be too far from
the set
\[
\{z\in \mathbb{C}\,:\, \det T_{60}(\beta) =0 \}
\]
which is shown in figure~\ref{f1} (the matrix is tridiagonal and
it is not very large so any of the standard computer packages can
produce the data for this picture). According to
corollary~\ref{t8} the convergence of this sets to the interval
$[-2,2]$ is $o(k^{-q})$ for all $0<q<1$ as $k\to \infty$. Compare
with figures~\ref{f4} and \ref{f5}. Notice that in the latter
there are some perturbed points very close to the real axis inside
the elliptical region which comprises most of the second order
spectrum. Further numerical experiments confirm that they increase
in number as $j$ increases. Their real part are related to the
non-real roots of $q(x)$. These non-real roots of $q(x)$ are
generalized eigenvalues which according to Davies \cite{SECR} can
be regarded as (absolute) resonances of $H$. It would be
interesting to explore further how these resonances are related to
$\spec[2]{H;k}$.
In figure~\ref{f2} we consider $\spec[2]{H;k}$ for $a=3$ and $j=3$
as $k$ increases. Notice that the convergence to the continuous
spectrum seems to be much slower than the convergence to the
eigenvalue. The point $\phi_k\in \spec[2]{H;K}$ which is furthest
from $\spec{H}$ seems to have real part approximately equal to
$0$. Figure~\ref{f3} is a log-log graph of the imaginary part of
$\phi_k$ when $k$ varies from 100 to 1500. The slope of the line
is very close to -1. This suggests that corollary~\ref{t8} fails
for $q > 1$. In table~\ref{tb1}, the slope of the line in
figure~\ref{f3} changes from test point to test point. This
suggests that the convergence to the spectrum might be of the
order $\log^\alpha(k)/k$ for some $\alpha>0$.
\begin{table}
\begin{tabular} {|c|>{\PBS\centering}m{1.6in}|} \hline
k & Slope between $k$ and $k+100$.
\\ \hline100 & -0.8494 \\ 200 & -0.8590 \\ 300 & -0.8647 \\ 400 &-0.8688
\\ 500 &-0.8719 \\600 & -0.8744 \\700 & -0.8765 \\800 &-0.8782 \\
900 & -0.8798 \\1000& -0.8811\\1100& -0.8823\\ 1200&-0.8834 \\
1300 &-0.8844
\\1400 & -0.8853 \\ \hline
{\rm e}nd{tabular}
\caption{Slope between the test points $k$ and
$k+100$ in figure~\ref{f3}.} \label{tb1}
{\rm e}nd{table}
{\samepage {\scshape Acknowledgments.} The author wishes to thank
Prof.~E.B.~Davies for his kind interest in this research and for
so many useful comments. He also wishes to thank
Prof.~S.~Marcantognini, Prof.~M.~Mor\'an, Dr.~E.~Shargorodsky and
Dr.~S.~Yakovlev for interesting discussions. Many thanks to the
referee for pointing out and suggesting how to correct a mistake
in an earlier version of this paper.}
\begin{thebibliography}{99}
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``Spectral enclosures and complex resonances for general
self-adjoint operators'', {\rm e}mph{LMS J. Comput. Math.} 1 (1998)
42-74.
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{\rm e}mph{Theory of $H^p$ spaces}, Pure and Applied Mathematics Vol.
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{\rm e}mph{Convolution equations and projection methods for their
solution}, American Mathematical Society, Providence, 1974.
\bibitem{LaSi}{\scshape Y. Last, B. Simon},
``Eigenfunctions, transfer matrices, and absolutely continuous
spectrum of one-dimensional Schr\"odinger operators'',
{\rm e}mph{Invent. math.} 135 (1999) 329-367.
\bibitem{LuTe} {\scshape F.~Luef, G.~Teschl},
``On the number of eigenvalues of Jacobi operators'',
Preprint (2001), {\rm e}mph{math.SP/0109213}.
\bibitem{NaYa}{\scshape S.N.~Naboko, S.I.~Yakovlev},
``The discrete Schr\"odinger operator. The point spectrum lying in
the continuous spectrum'', {\rm e}mph{St. Petersburgh Math. J.} 4
(1993) 559-568.
\bibitem{GHORS}{\scshape E. Shargorodsky},
``Geometry of higher order relative spectra and projection
methods'', {\rm e}mph{J. Oper. Theo.} 44 (2000) 43-62.
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determinants. II'', {\rm e}mph{Adv. in Math} 21 (1976) 1-29.
{\rm e}nd{thebibliography}
\begin{figure}[t]
\begin{picture}(250,250)(75,150) \special{psfile=op5f3.eps
hoffset=0 voffset=0 hscale=65 vscale=65}
{\rm e}nd{picture}
\caption{Points where $\det T_{60}(\beta)$ vanishes.} \label{f1}
{\rm e}nd{figure}
\begin{figure}[b]
\begin{picture}(250,250)(75,150) \special{psfile=op5f4.eps
hoffset=0 voffset=0 hscale=65 vscale=65}
{\rm e}nd{picture}
\caption{$\spec[2]{H;60}$ selected values of $a$ and $j$
corresponding to table \ref{tb2}.} \label{f4}
{\rm e}nd{figure}
\begin{figure}[t]
\begin{picture}(250,250)(75,150) \special{psfile=op5f5.eps
hoffset=0 voffset=0 hscale=65 vscale=65}
{\rm e}nd{picture}
\caption{$\spec[2]{H;60}$ selected values of $a$ and $j$
corresponding to table \ref{tb2}.} \label{f5}
{\rm e}nd{figure}
\begin{figure}[b]
\begin{picture}(250,250)(75,150) \special{psfile=op1f2.eps
hoffset=0 voffset=0 hscale=65 vscale=65}
{\rm e}nd{picture}
\caption{$\spec[2]{H;k}$ for $a=3$, $j=3$ and three values of
$k$.} \label{f2}
{\rm e}nd{figure}
\begin{figure}[t]
\begin{picture}(250,250)(75,140) \special{psfile=op1f1.eps
hoffset=0 voffset=0 hscale=65 vscale=65}
{\rm e}nd{picture}
\caption{Log-log plot of the maximum distance between
$\spec[2]{H;k}$ and $\spec{H}$ for $a=3$ and $j=3$ as $k$
increases.} \label{f3}
{\rm e}nd{figure}
{\rm e}nd{document} |
\begin{document}
\twocolumn[
\begin{center}
{\LARGE
Second-level randomness test based on the Kolmogorov-Smirnov test
}
{\large
Akihiro Yamaguchi$^{1}$ and Asaki Saito$^{2}$ \par
}
\end{center}
\hspace{1.5cm} $^1$ Fukuoka Institute of Technology, 3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka 811-0295, Japan \\
\hspace{1.5cm} $^2$ Future University Hakodate, 116-2 Kamedanakano-cho, Hakodate, Hokkaido 041-8655, Japan \\
\begin{center}
\abstract
\end{center}
We analyzed the effect of the deviation of the exact distribution of the p-values from the uniform distribution on the Kolmogorov-Smirnov (K-S) test that was implemented as the second-level randomness test.
We derived an inequality that provides an upper bound on the expected value of the K-S test statistic when the distribution of the null hypothesis differs from the exact distribution.
Furthermore, we proposed a second-level test based on the two-sample K-S test with an ideal empirical distribution as a candidate for improvement.
\\
\\
{\bf Keywords}: Kolmogorov-Smilrov test, uniformity of p-value, chaotic true orbits
\\
\\
\\
]
\section{Introduction}
Several randomness test suites have been proposed as evaluation methods for random or pseudorandom number generators (PRNGs)\cite{NIST,TestU01}, in which randomness is tested at two levels.
The first-level test is an individual test that yields p-values as well as pass or fail results for each tested sequence, and the second-level test evaluates the results of the first-level tests.
As one of the second-level tests, the uniformity of p-values obtained by the first-level test was tested using the goodness-of-fit test.
However, it is known that the exact distribution of p-values differs from the uniform distribution depending on the first-level test\cite{Pareschi, Haramoto, Y15}.
For the $\chi^2$ test adopted as one of the second-level tests in the test suite NIST SP80022, the effect of this difference on the test results was analyzed, and upper limits of sample size (number of tested sequences) were proposed by F.~Pareschi et. al\cite{Pareschi}, by H.~Haramoto\cite{Haramoto}, and \cite{Y15}.
Pareschi et. al. also considered adopting the Kolmogorov-Smirnov (K-S) test as a second-level test \cite{Pareschi}, but their analysis was limited to the case where first-level tests were based on the binomial distribution.
In this study, we adopt the K-S test as the second-level test, without restricting the nature of the first-level tests.
We analyze the effect of the deviation of the exact distribution of p-values from the uniform distribution on $[0, 1]$, which is usually assumed by the null hypothesis of randomness.
Therefore, we derive an inequality that provides an upper bound on the expected value of the K-S test statistic.
The obtained inequality is numerically examined for a toy distribution of p-values and some of the practical first-level tests in NIST SP800-22.
This inequality also allows us to estimate the maximal sample sizes required to pre-empt a high probability of incorrectly identifying an ideal generator as non-random.
To improve the second-level test, we propose using the K-S test based on the empirical distribution of p-values generated by the first-level test results of ideal random sequences.
In practice, we propose using pseudorandom sequences obtained from the chaotic true orbits of the Bernoulli map\cite{Saito16,Saito18} as a substitute for such ideal random sequences.
\section{Second-level randomness test based on the K-S test}
Using the K-S test, we can test the goodness-of-fit between the empirical distribution and the reference distribution, or between two empirical distributions.
Let $p=\{ p_i \in [0,1] | i=1,2,\cdots,m\}$ be the $m$ p-values obtained by the first-level randomness test.
The empirical distribution with $m$ samples is defined as
\begin{equation}
\label{gpm}
G_{p,m}(x) = (1/m)\ \#\{1\leq i \leq m | p_i \leq x \},
\end{equation}
where $0\leq x \leq 1$ and $\#\{\cdot\}$ denotes the number of elements in a set $\{\cdot\}$.
Let the null hypothesis $H_0$ be $p_1,\cdots,p_m$ from the reference distribution $F$.
The reference distribution $F$ is usually assumed to be a uniform distribution $F_{unif}(x) = x \ (x \in [0,1])$.
However, there are some cases in which the exact distribution of the p-value is different from $F_{unif}$ depending on the first-level randomness test\cite{Pareschi}.
The test statistic of the one-sample K-S test with reference distribution $F$ is defined as follows.
\begin{equation}
\label{defDF}
D_F = \sqrt{m} \cdot \sup_{x\in [0,1]} |G_{p,m}(x)-F(x)| .
\end{equation}
The null hypothesis $H_0$ is accepted if
\begin{equation}
\label{oneks}
D_F \leq K(\alpha),
\end{equation}
where $K(\alpha)$ is the boundary value for the significant level $\alpha$.
This boundary value can be approximated as $K(\alpha)\simeq \sqrt{-(1/2) \log{(\alpha/2)}}$ for a large $m$ and small $\alpha$\cite{Press}.
The boundary values for $\alpha = 0.01$ and $0.0001$ are given by $K(0.01) \simeq 1.628$ and $K(0.0001) \simeq 1.949$, respectively.
\section{Inequality for the expected value of test statistic}
Let $G$ be the exact distribution for $G_{p,m}$.
The test statistic of the K-S test with the exact reference distribution $G$ is defined as
\begin{equation}
D_G = \sqrt{m} \cdot \sup_{x\in [0,1]} |G_{p,m}(x)-G(x)| .
\end{equation}
The distribution of $D_G$ asymptotically obeys the Kolmogorov distribution under the null hypothesis if the exact reference distribution $G$ is continuous.
If the distribution of p-values of the first-level test is discrete, $G$ is not continuous but is a piecewise constant.
Following Pareschi et. al. [3], we also assume that the distribution of $D_G$ still obeys the Kolmogorov distribution, even if $G$ is piecewise constant.
In the following, we analyze the difference between the expected values of the test statistics $D_F$ and $D_G$ under this assumption.
Applying the triangle inequality to the right-hand side of Equation \eqref{defDF}, we obtain
\begin{align}
D_F & = \sqrt{m}\cdot \sup_{x\in[0,1]} |G_{p,m}(x)-G(x)+G(x)-F(x)| \nonumber \\
& \leq \sqrt{m}\cdot \sup_{x\in[0,1]} |G_{p,m}(x)-G(x)| \nonumber \\
& \ \ \ \ \ +\sqrt{m}\cdot \sup_{x\in[0,1]} |G(x)-F(x)| \nonumber \\
& = D_G +\sqrt{m}\cdot d \ ,
\label{inequality0}
\end{align}
where
\begin{equation}
d = \sup_{x\in[0,1]} |G(x)-F(x)| \ .
\label{def_d}
\end{equation}
This $d$ is a constant determined by the reference distribution $F$ and the exact distribution $G$ for the first-level test.
Considering the expectation with respect to the direct product of the measure determined by $G$ for inequality \eqref{inequality0}, we obtain the inequality
\begin{equation}
E[D_F] - E[D_G] \leq \sqrt{m}\cdot d \ .
\label{inequality1}
\end{equation}
It is known that the expected value $E[D_G]$ converges to the constant
\begin{equation}
\mu = \sqrt{\pi / 2}\cdot \ln{2} = 0.868\cdots.
\end{equation}
when $m\rightarrow \infty$, and the constant $\mu$ is independent of $G$ \cite{Marsaglia} .
Inequality \eqref{inequality1} implies that the difference $E[D_F] -\mu$ has an upper bound of $\sqrt{m}\cdot d$.
Note that for the $\chi^2$ test, the difference between the expected value of the test statistic based on the reference distribution that differs from the exact distribution and that based on the exact distribution is proportional to $m$\cite{Matsumoto}.
From this perspective, the K-S test is regarded as more robust to increasing sample size $m$ than the $\chi^2$ test, because the difference in the test statistics is proportional to $\sqrt{m}$ for the K-S test.
However, for the same reason, the power of the K-S test is expected to be lower than that of the $\chi^2$ test.
Furthermore, the safety of the randomness test was evaluated using inequality \eqref{inequality1}.
If the difference $\Delta$ is admissible for $E[D_F]-E[D_G]$, the maximum sample size within the difference $\Delta$ is given by $(\Delta/d)^2$.
\begin{table*}[h]
\caption{Results of the K-S test based second-level randomness tests}
\label{table1}
\begin{tabular}{r|l|ccrr|ccrr}
\hline
& &
\multicolumn{4}{p{4.5cm}|}{
(a) The one-sample K-S test with the uniform distribution
}&
\multicolumn{4}{p{4.5cm}}{
(b) The two-sample K-S test with the empirical distribution
} \\
No. & Test name & \multicolumn{2}{c}{p-value}& \multicolumn{2}{c|}{Pass Rate } & \multicolumn{2}{c}{p-value}& \multicolumn{2}{c}{Pass Rate } \\
& & mean & SD & {\tiny $\alpha\!=\!0.01 $} & {\tiny $\alpha\!=\!0.0001$ }& mean & SD & {\tiny $\alpha\!=\!0.01$} & {\tiny $\alpha\!=\!0.0001$}\\
\hline
\hline
1& Frequency\,Test&0.033&0.025&8/10&10/10&0.510&0.234&10/10&10/10\\
2& Block\,Frequency\,Test &0.499&0.303&10/10&10/10&0.511&0.329&10/10&10/10\\
3& Runs\,Test&0.374&0.259&10/10&10/10&0.489&0.327&10/10&10/10\\
4&Longest\,Run\,of\,Ones\,Test&0.000&0.000&0/10&0/10&0.594&0.252&10/10&10/10\\
5& Binary\,Matrix\,Rank\,Test&0.000&0.000&0/10&0/10&0.504&0.321&10/10&10/10\\
6& Discrete\,Fourier\,Transform\,Test&0.000&0.000&0/10&0/10&0.618&0.354&10/10&10/10\\
7& Non-overlapping\,Template\,Matching\,Test\,(1)&0.394&0.314&10/10&10/10&0.656&0.303&10/10&10/10\\
8& Overlapping\,Template\,Matching\,Test&0.000&0.000&0/10&0/10&0.636&0.260&10/10&10/10\\
9& Maurer's\,"Universal\,Statistical"\,Test&0.000&0.000&0/10&0/10&0.439&0.240&10/10&10/10\\
10& Linear\,Complexity\,Test&0.064&0.121&6/10&10/10&0.489&0.291&10/10&10/10\\
11& Serial\,Test\,(1)&0.415&0.131&10/10&10/10&0.520&0.170&10/10&10/10\\
12& Approximate\,Entropy\,Test&0.000&0.000&0/10&0/10&0.394&0.347&9/10&10/10\\
13& Cumulative Sums Test (1)&0.089&0.138&7/10&10/10&0.409&0.247&10/10&10/10\\
\hline
\end{tabular}
\end{table*}
\section{Two-sample K-S test with ideal empirical distribution}
A simple method to improve the K-S test based second-level test involves the use of the statistic $D_G$ instead of $D_F$ if the exact distribution $G$ is known for the target first-level test.
In this case, we can obtain test statistics without the error effect.
However, it is not always possible to compute the exact distribution for a given first-level test.
Therefore, as another method, we examine a method that uses the empirical distribution of p-values obtained from the first-level test for ideal random sequences as the reference distribution.
Let $q = \{q_i \in [0,1] | i = 1,2,\cdots,m'\} $ be the $m'$ p-values obtained by the first-level test for ideal or nearly ideal random sequences.
By the definition, the distribution of $q$ obeys $G$.
Similar to Equation \eqref{gpm}, the empirical distribution of $q$ is defined as
\begin{equation}
G_{q,m'}(x) = (1/m')\ \#\{1\leq i \leq m' | q_i \leq x \}.
\end{equation}
By using the two-sample K-S test, the goodness-of-fit between the empirical distribution $G_{p,m}$ and $G_{q,m'}$ is also tested as a second-level randomness test.
The test statistic of this two-sample K-S test is defined as
\begin{equation}
D_{G_{q,m'}} = \sqrt{\frac{m\cdot m'}{m+m'}} \cdot \sup_{x\in [0,1]} |G_{p,m}(x)-G_{q,m'}(x)| .
\end{equation}
For the two-sample K-S test, the null hypothesis
that $p_1,\cdots,p_m$, and $q_1,\cdots,q_m'$ are from the same exact distribution $G$ is accepted if
\begin{equation}
\label{twoks}
D_{G_{q,m'}} \leq K(\alpha)
\end{equation}
for the significance level $\alpha$.
In this study, we propose to construct an empirical distribution $G_{q,m'} $ using the chaotic true orbit of the Bernoulli map\cite{Saito16, Saito18}.
The dynamical system given by the Bernoulli map is defined as
\begin{equation}
x_{i+1} = 2 x_i \mod 1,
\end{equation}
where $x_i \in [0,1)$ and $i=0,1\cdots$.
By providing an irrational algebraic number as an initial state $x_0$, we can generate a chaotic true orbit $x_i$ with infinite precision.
Then, we can obtain the binary sequence $\varepsilon = \varepsilon_0, \varepsilon_1, \cdots$ by assigning
\begin{equation}
\varepsilon_i = \left\{
\begin{array}{cc}
0 & (x_i < 1/2) \\
1 & (x_i \geq 1/2)
\end{array} \right. .
\end{equation}
This binary sequence $\varepsilon$ corresponds to the binary expansion of the initial state $x_0$.
See \cite{Saito16} and \cite{Saito18} for mathematical support for the good statistical qualities of $\varepsilon$.
\section{Numerical results}
\subsection{Examples of second-level tests based on the K-S test}
As a first numerical experiment, two second-level tests based on the K-S test were applied to some of the first-level tests in NIST SP800-22.
One second-level test was based on the one-sample K-S test with the reference distribution $F_{unif}$, and the second is the second-level test based on the two-sample K-S test with the empirical distribution that was separately prepared.
We performed these second-level tests ten times, wherein, for each second-level test, we used the p-values obtained by applying the first-level test to $m = 10^6$ sequences with length $n = 10^6$.
The tested sequences were generated by the Mersenne twister-based PRNG.
The empirical distribution $G_{q,m'}$ used as a reference was constructed based on the results of the first-level tests for the PRNG based on the chaotic true orbit of the Bernoulli map with $m'=10^7$ and $n=10^6$.
The results of the one-sample K-S test and the two-sample K-S test are shown in columns (a) and (b) of Table 1, respectively.
Here, the mean and the standard deviation of the ten obtained p-values, and the pass rate of the number of passes divided by ten are shown for each randomness test.
For the first-level tests of Nos. 7, 11, and 13, which consist of several tests, the result for one test is only shown as an example.
The random excursions test and the random excursions variant test were excluded because the number of obtained p-values varied depending on the tested sequences.
The results of the one-sample K-S test with a uniform distribution completely failed for the first-level tests of Nos. 4, 5, 6, 8, 9, and 12.
However, almost all the results of the two-sample K-S test with the empirical distribution were successful.
These results suggest an improvement in the second-level test using the two-sample K-S test with the empirical distribution constructed using high-quality PRNG.
\subsection{Examination of the derived inequality}
To examine the inequality \eqref{inequality1}, we numerically analyze the difference between test statistics $D_F$ and $D_G$ for a particular distribution $G$ under the reference distribution $F=F_{unif}$.
As a toy model, we consider the exact distribution $G_e$, which is a piecewise linear function, given by
\begin{equation}
G_e(x) = \left\{
\begin{matrix}
(1+2e)x & x\in [0,1/2] \\
(1-2e)x+2e & x\in (1/2,1]
\end{matrix}
\right. \ ,
\end{equation}
where $|e|<1$ .
The graph of $G_e$ is shown in Fig. \ref{Fig1}.
The constant $d$ in Equation \eqref{def_d} for $G_e$ and $F_{unif}$ is equal to $e$.
For a given sample size $m$ and constant parameter $d$, we randomly generate $p_1, p_2, \cdots, p_m \in [0,1]$ that obeys the distribution $G_e$ and calculate $D_{F}$ and $D_{G}$ for $10^4$ times.
Then, we obtain the mean values $\overline{D_F}$ and $\overline{D_G},$ and $\Delta_m = \overline{D_{F}}-\overline{D_{G}}$, respectively.
In Fig. \ref{Fig2}, $\Delta_m$ (circles) and $\sqrt{m}\cdot d$ (solid line) are shown for the cases $e=d=10^{-1}$ and $10^{-4}$.
Here, ten samples of $\Delta_m$ are plotted for each $m$.
As a result, $\Delta_m$ is less than $\sqrt{m}\cdot d$ for both cases and converges to $\sqrt{m}\cdot d$ with increasing $m$ for $e=d=10^{-1}$.
This result is consistent with the inequality \eqref{inequality1}.
\subsection{Safe sample sizes for the frequency test and the binary matrix rank test}
Here, we analyze the frequency test and the binary matrix rank test shown in Table 1 as examples.
The frequency test was analyzed by Pareschi et. al. as an example of tests based on binomial distribution.
As a different example, we analyzed the binary matrix rank test based on the trinomial distribution.
The binary matrix rank test also failed for the one-sample K-S test with a uniform distribution.
For these two tests, we calculated the exact distributions for the sequence length $n=10^6$ and obtained the exact value of the constant $d$\cite{Y18}.
The statistics $\overline{D_{F}}$ and $\overline{D_G}$, and their difference $\Delta_m$ were also calculated from the test results shown in column (a) of Table 1.
Results are shown in Table 2.
The range of the standard error of the mean (SEM) is also shown.
The difference $\Delta_m$ is less than $\sqrt{m}\cdot d$ for both tests, and these results are consistent with the inequality \eqref{inequality1}.
For the safety of these tests, we can obtain the maximum sample size for the given admissible difference $\Delta$ of the expected values of $D_F$ and $D_G$, as mentioned in Section 3.
For example, if $\Delta=0.1628$, which is 10\% of the boundary value $K(0.01)$, is admissible, the maximum sample size is $15,703$ for the frequency test and $1,693$ for the binary matrix rank test.
Furthermore, the sample size $m=10^3$, which is the recommended parameter of NIST SP800-22, is safe if $\sqrt{m}\cdot d \simeq 0.025$ is admissible for the frequency test, and $\sqrt{m}\cdot d \simeq 0.153$ is admissible for the binary matrix rank test.
\begin{figure}
\caption{ Examined distribution of $G_e$ and $F_{unif}
\label{Fig1}
\end{figure}
\begin{figure}
\caption{ Differene $\Delta_m = \overline{D_{G}
\label{Fig2}
\end{figure}
\begin{table}[t]
\caption{The difference between the mean values $\overline{D_F}$ and $\overline{D_G}$ for the frequency test and the binary matrix rank test}
\renewcommand{1}{1.5}
\begin{tabular}{ccc}
\hline
& Frequency\,Test & Binary\,Matrix\,Rank\,Test \\
\hline
\hline
$\overline{D_F}$ & 1.319$\pm$0.114 & 5.082$\pm$0.107 \\
$\overline{D_G}$ & 0.863$\pm$0.057 & 0.840$\pm$0.093 \\
$\Delta_m\!=\!\overline{D_F}\!-\!\overline{D_G}$ & 0.456$\pm$0.100 & 4.242$\pm$0.120 \\
\hline
\hline
$\sqrt{m}\cdot d$ & 0.798 & 4.860 \\
\hline
\end{tabular}
\renewcommand{1}{1}
\end{table}
\section{Conclusion}
In this work we derived an inequality that provides the upper bound on the difference of the expected values of the test statistics for the K-S test based second-level randomness test.
The derived inequality was numerically examined and consistent results were obtained.
In addition, we examined the second-level test that uses the two-sample K-S test with the nearly ideal empirical distribution constructed from the PRNG based on the chaotic true orbit for several randomness tests in NIST SP800-22.
These results are expected to prove useful for evaluating the safety of the randomness test using the K-S test.
We intend to perform an analysis of the other goodness-of-fit tests, such as the Cr\'{a}mer-von-Mises test and the Anderson-Darling test, in future work.
\end{document} |
\begin{document}
\title{Generalized Framework for Nonlinear Acceleration}
\begin{abstract}
Nonlinear acceleration algorithms improve the performance of iterative methods, such as gradient descent, using the information contained in past iterates. However, their efficiency is still not entirely understood even in the quadratic case. In this paper, we clarify the convergence analysis by giving general properties that share several classes of nonlinear acceleration: Anderson acceleration (and variants), quasi-Newton methods (such as Broyden Type-I or Type-II, SR1, DFP, and BFGS) and Krylov methods (Conjugate Gradient, MINRES, GMRES). In particular, we propose a generic family of algorithms that contains all the previous methods and prove its optimal rate of convergence when minimizing quadratic functions. We also propose multi-secants updates for the quasi-Newton methods listed above. We provide a Matlab code implementing the algorithm.
\end{abstract}
\section{Introduction}
Consider the simple fixed-point iteration
\[
x_{i+1} = g(x_i)
\]
which produces a sequence of points $\{x_0,x_1,\ldots,x_N\}$. In most cases this converges to the fixed-point $x^*$,
\[
x^* = g(x^*).
\]
This setting is quite generic, for example in the case of optimization $g$ can be a gradient step on an objective function $f$ and $x^*$ is its minimizer. The sequence $\{x_i\}$ converges to $x^*$ at a certain speed, but ideally, we would like the procedure to be as fast as possible. For example, in optimization the accelerated gradient method \cite{nesterov2013introductory}
\[
\begin{cases}
x_{i+1} & = y_i-h\nabla f(y_i) \\
y_{i+1} & = (1+\beta) x_{i+1}-\beta x_i
\end{cases}
\]
converges faster to the optimum $x^*$ than gradient method, provided good constants $h$ and $\beta$. In practice, those constants may be hard to estimate, especially $\beta$ which depends on the strong convexity parameter of the objective function, whose estimation is still a challenge \citep{fercoq2016restarting}.
Some nonlinear acceleration algorithms such as Anderson Acceleration share the same idea of Nesterov's acceleration. It combines the gradient step with a linear combination of previous iterates as follows,
\begin{eqnarray}
x_{i} & = & g(y_{i-1}), \label{eq:family_algo}\\
y_{i} & = & \textstyle \sum_{k=1}^{i} \alpha_k^{(i)} x_k, \nonumber
\end{eqnarray}
where the vector $\alpha^{(i)}$ is function of the iterates, thus changing over time. As the coefficients $\alpha$ depend on the iterates $x_i$, the acceleration is thus called \textit{nonlinear}. Its main drawbacks is the lack of convergence guarantees, and in fact it has been showed that Anderson Acceleration is unstable when $g(x)$ is not a deterministic linear function \cite{scieur2016regularized}. The same paper proposes a regularized version of Anderson acceleration, whose rate of convergence is asymptotically optimal even in the presence of noise \citep{scieur2017nonlinear} or when the Jacobian of $g$ is non-symmetric \cite{scieur2018nonlinearb,bollapragada2018nonlinear}.
Other techniques such as Quasi-Newton methods schemes, popular in optimization, approximate the Newton step using the matrix $H \approx (\nabla^2 f(x_i))^{-1}$ as follow,
\[
x_{i+1} = x_i - H\left( h \nabla f(x_i)\right).
\]
This can be extended to fixed-point iteration by coupling a fixed-point step with a Quasi-Newton step,
\begin{eqnarray*}
x_{i} & = & g(y_{i-1}), \\
y_{i} & = & y_{i-1} - H(x_i-y_{i-1}).
\end{eqnarray*}
Such matrix $H$ can be found using several formulas. The simplest ones are Broyden Type-I and Type-II updates \cite{broyden1965class}, and the most popular is certainly BFGS or of DFP \cite{nocedal2006nonlinear}. There also exists the symmetric rank-one update which has been rediscovered many time in many different fields.
Finally, we study Krylov subspace techniques such as the Conjugate Gradient method and GMRES \citep{saad1986gmres}. These algorithms minimize some error function using a Krylov basis, usually updated with orthonormal vectors to ensure stability. Their primary usage is solving large systems of linear equations and optimizing quadratic functions.
The optimal convergence rate of Krylov methods is well-known when the fixed-point operator $g$ is a linear mapping, and works of \citep{scieur2018nonlinearb} show similar performance for Anderson Acceleration. For quasi-Newton methods, the results are less clear, even for quadratic objectives with two variables. For example, DFP and BFGS algorithms may converge poorly without line-search \citep{powell1986bad}.
When the function $g$ is nonlinear, it is unclear how fast those methods converge. In particular, the bad theoretical rates of convergence (if any) does not match the usual good numerical performance. The lack of robustness of nonlinear acceleration algorithms can explain this phenomenon since instability issues are known for some of them \citep{powell1977restart,johnson1988modified,scieur2016regularized}.
With recent result from \cite{scieur2016regularized,scieur2017nonlinear,scieur2018nonlinear}, it is now possible to have nonlinear acceleration techniques that achieve an asymptotically optimal rate of convergence even in the presence of stochastic noise. However, because the analysis of nonlinear acceleration methods is independent of each other, we unify the analysis to identify the central argument of nonlinear acceleration.
Several results linked some acceleration methods to each other. For example \cite{fang2009two} propose a general family of Broyden methods, including Type-I or Type-II updates, as well as Anderson mixing. \citet{walker2011anderson} show the link between Anderson and GMRES. However, the study does not include common schemes, such as BFGS and DFP.
\textbf{Contributions}. In this paper, we propose the Generalized Nonlinear Acceleration algorithm which mixes Anderson acceleration and quasi-Newton methods. In function of its parameters, it can produce the same steps than Anderson Acceleration, Broyden Type-I or Type-II, DFP, BFGS, SR-$k$ (symmetric rank $k$ update) or even conjugate gradients and GMRES. We give the proof of its (optimal) rate of convergence when applied to a linear function $g$, in the metric $\|\cdot\|_W = \|W^{1/2}\cdot\|_2$ where $W$ is positive definite. We derive the multi-secant updates for DFP and BFGS and extends the SR-1 to SR-$k$ updates, then analyze connections with quasi-Newton methods. We show equivalences between our algorithm and with multi-secant updates of the estimate of the Hessian. We also investigate the links with Krylov methods and propose another way to generalize CG for solving a nonlinear system of equations (or for minimizing non-quadratic functions).
\textbf{Paper Organization.} TODO
\subsection{Notations and Assumptions} \label{sec:notations}
This paper studies a way to accelerate the convergence of the family of algorithms \eqref{eq:family_algo} when $g$ is linear, i.e.,
\begin{eqnarray}
g(x) & = & G(x-x^*) + x^*.\nonumber
\end{eqnarray}
Usually, such mapping is written as $Ax+b$ since $x^*$ is not explicitly known. However this notation is more convenient for our theoretical analysis. In particular, we study the following family of algorithm, that alternates between one fixed-point iteration and one linear combination step,
\begin{eqnarray}
x_{i} & = & G(y_{i-1}-x^*) + x^*, \label{eq:linear_algo}\\
y_{i} & = & \textstyle \sum_{k=1}^{i} \alpha_k^{(i)} x_k.\nonumber
\end{eqnarray}
Thorough this paper, we always assume that
\begin{itemize}
\item $G$ is a symmetric definite positive matrix, whose spectrum is bounded by $0 \preceq G \preceq (1-\kappa) I$ with $\kappa < 1$. Usually, $\kappa$ is close to zero and often refers to be the inverse of a condition number.
\item $\textstyle \sum_{k=1}^{i} \alpha_k^{(i)} = 1$, to have a consistent algorithm \cite{scieur2017integration}. It ensures $y_i = x^*$ when all $x_k$ are replaced by $x^*$.
\item The last coefficient $\alpha_i^{(i)}$ is nonzero. Intuitively, if this coefficient is equal to zero, we waste the last call of $g$, making that iteration useless.
\end{itemize}
\paragraph{Polynomial Notation} \citet{scieur2018nonlinearb} shows that \eqref{eq:linear_algo} is equivalent to a sequence of polynomials,
\begin{eqnarray}
x_{i} & = & G(y_{i-1}-x^*) + x^*, \label{eq:poly_algo}\\
y_{i} & = & p_i(G) (y_0-x^*)+x^*,\nonumber
\end{eqnarray}
where $p_i(G)$ is a polynomial of degree \textit{exactly} $i$, whose coefficients sum to one.
\subsection{Residual and Rate of Convergence}
We define the residual
\begin{equation}
r_i = x_i-y_{i-1}. \label{eq:def_residual}
\end{equation}
In this paper, we often refer to the link between the residual \eqref{eq:def_residual} and $y_i-x^*$. In particular,
\begin{equation}
r_i = g(y_{i-1})-y_{i-1} = (G-I)(y_{i-1}-x^*). \label{eq:link_residual}
\end{equation}
We can write this relation under the matrix form. Let the matrices, assumed to be \textbf{full column rank},
\begin{eqnarray}
X & = & [x_1,x_2,\ldots,x_N], \nonumber \\
Y & = & [y_0,y_1,\ldots,y_{N-1}], \label{eq:matrix_form}\\
X^* & = & x^*\textbf{1}^T_N = [x^*,\ldots,x^*] \;\;(\text{$N$ times}). \nonumber
\end{eqnarray}
In this case, the relation \eqref{eq:link_residual} becomes
\begin{equation}
R = (G-I)(Y-X^*). \label{eq:link_residual_matrix}
\end{equation}
We now bound the performance of algorithm \eqref{eq:linear_algo} or equivalently \eqref{eq:poly_algo} in the \textit{weighted Euclidean norm}
\begin{equation}
\| v \|_W = v^TWv, \quad W\succ 0. \label{eq:def_metric}
\end{equation}
Using \eqref{eq:poly_algo} and \eqref{eq:link_residual}, the norm \eqref{eq:def_metric} of $r_{i+1}$ can be bounded by
\begin{eqnarray}
\| r_{i+1} \|_W \hspace{-2ex}& = &\hspace{-2ex} \|(G-I)p_i(G)(y_0-x^*)\|_W = \| p_i(G) r_1\|_W \nonumber\\
& \leq & \hspace{-2ex}\| p_i(G) \|_2 \|r_1\|_W. \label{eq:norm_residual}
\end{eqnarray}
This means the performance of the algorithm \eqref{eq:linear_algo} can be summarized by the study of $\| p_i(G) \|_2$. Ideally, we would like to find the smallest polynomial to ensure fast convergence. In this paper, we propose the Generalized Nonlinear Acceleration algorithm that finds the best polynomial in \eqref{eq:norm_residual} at each iteration to ensure good convergence speed.
\section{Generalized Nonlinear Acceleration}
\label{sec:generic_acc_method}
The Generalized Nonlinear Acceleration Algorithm \ref{algo:gna} combines the ideas from Anderson Acceleration and quasi-Newton methods. In short, it combines linearly iterates that have been refined by a preconditioner $P$. This aims to minimizes the residual \eqref{eq:def_residual} in the weighted Euclidean norm\eqref{eq:def_metric} defined by the weights matrix $W\succ 0$.
\begin{algorithm}[htb]
\caption{Generalized Nonlinear Acceleration}
\label{algo:gna}
\begin{algorithmic}
\STATE {\bfseries Data:} Matrices $X$ and $Y$ of size $d\times N$.
\STATE {\bfseries Parameters:} Weight matrix $W\succ 0$, Preconditioner $P$.\\
\hrulefill
\STATE \textbf{1.} Compute matrix of residual $R = X-Y$.
\STATE \textbf{2.} Solve
\begin{equation}
\gamma_W = \frac{(R^TWR)^{-1}\textbf{1}_N}{\textbf{1}_N^T(R^TWR)^{-1}\textbf{1}_N} = \mathop{\rm argmin}_{\gamma:\textbf{1}^T\gamma = 1} \| R\gamma \|_W. \label{eq:gw}
\end{equation}
\STATE \textbf{3.} Perform the extrapolation
\begin{equation}
y^{\text{extr\,}} = (Y-PR)\gamma_W. \label{eq:gna_step}
\end{equation}
\end{algorithmic}
\end{algorithm}
In Algorithm \ref{algo:gna}, the parameters $P$ and $W$ are user-defined. In the next section, we discuss two standard way to choose $W$. Later in the paper, we show that the choice of $P$ algorithm \eqref{algo:gna} can produce steps that are identical to existing nonlinear acceleration algorithms. For now, we can consider for simplicity that $P$ is the scaled identity $\beta I$.
The coefficients $\gamma_W$ \eqref{eq:gw}, when $W=I$, correspond to Lemma 2.4 in \citep{scieur2016regularized}. With a minimal adaptation of the proof, this extends to arbitrary $W\succ 0$. We now quickly discuss of two "classical" choices for the weight matrix $W$.
\subsection{Choice of \texorpdfstring{$W$}{W}}\label{sec:choice_w}
We briefly discuss two possible choices of the weight matrix $W$. The first one is the simple case where $W=I$, the second one is $W=(G-I)^{-1}$.
\subsubsection{Case where \texorpdfstring{$W=I$}{W=I}}
In the first case, $W=I$ simply recover the classical Anderson Acceleration \citep{anderson1965iterative}, when $P=\beta I$ (where $\beta \neq 0$ is a scalar). This algorithm is known to minimize the residual of the extrapolation, achieving in the worst case an optimal rate of convergence \citep{scieur2016regularized}.
\subsubsection{Case where \texorpdfstring{"$W=(G-I)^{-1}$"}{"W=(G-I){-1}"}}
The case when $W=(G-I)^{-1}$ looks impossible at first sight, as it requires the inversion of $(G-I)$. However, the next proposition shows we do not need to use it explicitly.
\begin{proposition} \label{prop:sol_good_anderson}
Let any symmetric positive definite matrix $W$ that satisfies
\begin{equation}
WR=Y-X^*. \label{eq:cond_inv_w}
\end{equation}
For example, $W=(G-I)^{-1}$. Then, the coefficients $\gamma_W$ defined in Algorithm \ref{algo:gna} can be computed using the formula
\begin{equation}
\gamma_W = \frac{(Y^TR)^{-1}\textbf{1}_N}{\textbf{1}_N^T(Y^TR)^{-1}\textbf{1}_N}.
\end{equation}
\end{proposition}
The proof can be found in Appendix \ref{prop:sol_good_anderson_proof}.
\section{Computational Complexity of GNA}
\label{sec:complexity_appendix}
The next proposition gives the computational complexity of Algorithm \ref{algo:gna} when matrices are updated properly. In short, for standard choices of $W$ and $P$, Algorithm \ref{algo:gna} take $O(Nd)$ operation per call. The memory size $N$ is small in practice, thus computing $y^{\text{extr\,}}$ is as costly as the step \eqref{eq:linear_algo}.
\begin{proposition}
Assume we already have computed, using $N$ iterates from \eqref{eq:linear_algo},
\[
WR, \quad (R^TWR)^{-1}\textbf{1}, \quad PR.
\]
Let $y^+$ and $r^+$ be the new iterate and residual. The update
\[
WR^+, \quad ((R^+)^TWR^+)^{-1}\textbf{1}, \quad PR^+.
\]
costs at most $O_W+O_P+O(Nd+N^2)$ operations, where $O_W$ and $O_P$ are upper bounds on the number of operations for $Wr^{+}$ and $PR^+$.
\end{proposition}
\begin{proof}
First, we update the matrix $R^TWR$ to $[R,r^+]^TW[R,r^+]$. After expansion,
\[
[R,r^+]^TW[R,r^+] = \begin{bmatrix}
R^TWR & R^TWr^+ \\
(r^+)^TWR & (r^+)^TWr^+
\end{bmatrix}
\]
Since $WR$ and $R^TWR$ are known, the complexity of the update is equal to $O(O_W)$ (computation of the bottom-right element) plus $O(Nd)$.
The second step produces the coefficients $\gamma_W$ by solving the system of equations $([R,r^+]^TW[R,r^+])^{-1}\textbf{1}_N$. The size of the system grows in $N$, which means $O(N^{3})$ operation to solve it. However, we update the matrix $R^TWR$ with a rank two matrix. With the Woodbury matrix identity (Appendix \ref{sec:woodbury}) it is possible to update the previous solution $\gamma_W$, reducing the complexity to $O(N^2)$.
Finally, the last step consists in forming the extrapolation. First, it takes $O(dN)$ operation to form $Y\gamma_W$ and $R\gamma_W$. Then, we apply the preconditioner $P$ to $R\gamma_W$, which takes $O(O_p)$ operations.
The total complexity is thus bounded by
\[
O_W+O_P+O(Nd+N^2).
\]
\end{proof}
In practice, $N$ is much smaller than $d$, and usually $N\in [5,20]$ in most application. In addition, the matrix $W$ is usually used implicitly (see section \ref{sec:choice_w}), so $O_W=O(1)$. Finally, the preconditioner $P$ corresponds to the sum of the identity matrix and a rank $N$ matrix in most nonlinear acceleration algorithms, thus $O_P=O(N^2d)$. Under those conditions, the computational complexity of GNA is bounded by $O(N^2d)$ where $N^2$ is small compared to $d$. This property is desirable as it is \textit{as expansive as} computing a new residual $r^+$.
\section{Rate of Convergence}
\subsection{Chebyshev Acceleration}
We have seen that the norm of the polynomial \eqref{eq:norm_residual} quantifies the rate of convergence of algorithm \eqref{eq:poly_algo},
\[
\| r_{i+1} \|_W \leq \|p_i(G)\|_2 \| r_1\|_W,
\]
where $p_i$ is a polynomial of degree exactly equal to $i$ whose coefficients sums to one. The best polynomial of this class is a rescaled Chebyshev polynomial \cite{golub1961chebyshev}, which achieves the optimal rate
\begin{equation}
\textstyle \| r_{i+1} \|_W \leq \frac{\xi^i}{1+\xi^{2i}} \|r_1\|_W, \quad \xi = \frac{1-\sqrt{\kappa}}{1+\sqrt{\kappa}} \label{eq:cheby_rate}
\end{equation}
where $\|G\|_2\leq 1-\kappa$. This requires the knowledge of $\kappa$, usually referring to the inverse of a condition number and thus unknown in practice. In addition, Chebyshev acceleration is not adaptive to the initial point $r_1$. The next section shows that Algorithm \ref{algo:gna} achieves the same bound without the knowledge of $\kappa$.
\subsection{Optimal Rate of Convergence of (offline) GNA}
The next Theorem shows Algorithm \ref{algo:gna} implicitly solves
\[
\textstyle \min_p \| p(G) r_1 \|_W \quad \text{s.t.} \,\, p(1) = 1,\,\, \deg(p) \leq N-1.
\]
This strategy is optimal, in the sense that it recovers \eqref{eq:cheby_rate} up to a constant that depends on the preconditionner $P$.
\begin{theorem}\label{thm:rate_conv}
Let $\{(x_1,y_0),\ldots, (x_N,y_{N-1})\}$ be pairs generated by \eqref{eq:linear_algo} (or equivalently \eqref{eq:poly_algo}) where the matrix $G$ is symmetric and $\|G\|\leq 1-\kappa$. Let $y^{\text{extr\,}}$ be the output of Algorithm \ref{algo:gna}. Consider the residual of the extrapolation
\[
r^{\text{extr\,}} = g(y^{\text{extr\,}})-y^{\text{extr\,}}.
\]
Then, the norm $\|r^{\text{extr\,}}\|_W$ is bounded by
\begin{eqnarray}
\hspace{-3ex}\| r^{\text{extr\,}} \|_{W} \hspace{-2ex} & \leq & \hspace{-2ex} \| I-(G-I)P \|_2 \hspace{-1ex} \min_{\substack{ \deg(p)\leq N-1\\ p(1)=1}} \|p(G)r_1\|_W \label{eq:opti_polynomial} \\
& \leq & \hspace{-2ex} \| I-(G-I)P \|_2 \frac{\xi^{N-1}}{1+\xi^{2(N-1)}} \| r_1 \|_{W} \label{eq:rate_gna}
\end{eqnarray}
where $\xi$ is defined in \eqref{eq:cheby_rate} and $W$ is a positive definite. In addition, after at most $d$ steps, $r^{\text{extr\,}} = 0$.
\end{theorem}
The proof can be found in Appendix \ref{thm:rate_conv_proof}. This theorem shows that it is possible to ensure the optimal rate of convergence by post-processing the iterates using Algorithm \eqref{algo:gna} for any norm defined by $W$. Also, a finite number of iteration are required to reach $x^*$.
Intuitively, it would be more efficient to inject at each iteration the output of GNA. The next section studies this strategy and gives a sufficient condition to ensure an optimal rate of convergence.
\subsection{Online GNA}
As discussed in the previous section, instead of computing $y^{\text{extr\,}}$ on the side, we can inject the extrapolated point directly in \eqref{eq:linear_algo} as following,
\begin{eqnarray}
x_{i} & = & G(y_{i-1}-x^*) + x^*, \label{eq:online_algo}\\
y_{i} & = & y^{\text{extr\,}}.\nonumber
\end{eqnarray}
However, it is not clear from Algorithm \eqref{algo:gna} that $y^{\text{extr\,}}$ satisfies the special structure \eqref{eq:poly_algo}, in particular we need to ensure that the extrapolation can be written as a polynomial in the matrix $G$, whose degree is exactly equal to $i-1$ and its coefficients sums to one. The next proposition gives a simple condition on $P$ to ensure those two properties.
\begin{proposition} \label{prop:online_accel}
Consider the output of Algorithm \ref{algo:gna} after $N$ iterations of \eqref{eq:poly_algo}. If we have, for $\beta \neq 0$,
\[
y^{\text{extr\,}} = (Y-PR)\gamma_W = Y\tilde\gamma + \beta R\gamma_{\tilde W},
\]
where $\tilde\gamma$ is an arbitrary vector whose coefficients sum to one, and $\tilde W$ a positive definite matrix, then $y^{\text{extr\,}} - x^*= p_{N-1}(G)(y_0-x^*)$, where $p_N$ is a polynomial of degree $N-1$ whose coefficients sum to one.
\end{proposition}
The proof can be found in Appendix \ref{prop:online_accel_proof}. In short, this proposition shows that whatever the coefficients that combine the $y_i$'s, as long as we use some "optimal" coefficients for combining the residuals, the extrapolation satisfies the structure \eqref{eq:poly_algo}. By using this argument recursively, \eqref{eq:online_algo} produces iterates with the same structure than \eqref{eq:poly_algo}. In combination with Theorem \ref{thm:rate_conv}, this proves the optimality of the rate of convergence of online Generalized Nonlinear Acceleration algorithm.
It remains to fix the value of $P$. The next sections show some choices of the preconditioner that correspond to existing nonlinear acceleration algorithms.
\section{Connections with Anderson Acceleration}
We begin with nonlinear acceleration algorithms that form a polynomial that minimizes \eqref{eq:norm_residual} at each iteration. The link between GNA and this class of algorithm is quite straightforward as they share the same ideas.
The Anderson Acceleration \citep{anderson1965iterative}, Minimal Polynomial Method \cite{cabay1976polynomial} and Mesina Method \cite{mesi77} are different variant of the same algorithm. They solve, at each iteration,
\[
\gamma_{\text{Anderson}} = \mathop{\rm argmin}_{\gamma:\textbf{1}^T\gamma = 1} \|R\gamma\|_2.
\]
Anderson Acceleration and Mesina method use the same formula, while MPE method uses a slightly different approach giving the same result. Then, Anderson acceleration uses $\gamma_{\text{Anderson}}$ to combines the previous iterates using a so-called \textit{mixing} parameter $\beta \neq 0$ as following,
\[
y^{\text{extr\,}} = (Y-\beta R) \gamma_{\text{Anderson}}.
\]
Clearly, this corresponds to a special case of Algorithm \ref{algo:gna} where $W=I$ and $P=\beta I$.
Recently, \citet{fang2009two} introduce the type-I (or "Good") Anderson Acceleration. They used the Broyden \mbox{Type-I} update to derive the formula
\begin{equation}
\textstyle \gamma_{\text{Good Anderson}} = \frac{(Y^TR)^{-1}\textbf{1}}{\textbf{1}^T(Y^TR)^{-1}\textbf{1}}.\label{eq:formula_good_anderson}
\end{equation}
Then, it combines the previous iterates as follow,
\[
y^{\text{extr\,}} = (Y-\beta R) \gamma_{\text{Good Anderson}}.
\]
This formula comes from the analogy between Anderson and Broyden methods, so it is unclear why \eqref{eq:formula_good_anderson} is a good candidate for minimizing the norm of the residual \eqref{eq:norm_residual}. Besides, the rate of convergence was not specified.
With our results, we see that equation \eqref{eq:formula_good_anderson} corresponds to \eqref{eq:gw} when $W=(G-I)^{-1}$. This means equation \eqref{eq:formula_good_anderson} is the solution of \eqref{eq:gw}. With the application of Theorem \ref{thm:rate_conv}, we directly obtain the rate of convergence of the Good Anderson Acceleration. Indeed, using GNA with $W=(G-I)^{-1}$ and $P=\beta I$ produces the same extrapolated point, thus leading to an optimal rate of convergence.
\section{Connections with Quasi-Newton (qN)}
We quickly introduce the idea of quasi-Newton methods. First, consider Newton methods step
\[
y_{\text{Newton}} = y_{N-1}-(G-I)^{-1}r_N.
\]
Using equation \eqref{eq:link_residual}, we have $y_{Newton} = x^*$. However it needs $O(d^3)$ operations to invert the matrix $(G-I)$.
Unlike the Newton method, the qN step is cheaper but does not converge in one iteration. It uses an approximation $H \approx (G-I)^{-1}$ and performs
\begin{equation}
y_{\text{q-Newton}} = y_{N-1}-Hr_N, \label{eq:qn_step}
\end{equation}
where the matrix $H$ satisfies \textit{secant equations}
\[
H(r_{i+1}-r_{i}) = (y_i-y_{i-1}) \quad \forall i=1...N-1.
\]
In the matrix form, this gives
\begin{equation}
HRC=YC, \quad \textbf{1}^TC = 0, \;\; \textbf{rank}(C) = N-1.\label{eq:secant_equation}
\end{equation}
For example, $C$ can be formed using $N-1$ different vectors $[\ldots 0,1,-1,0,\ldots]^T$. Indeed, the secant equation still holds if $H$ is replaced by $(G-I)$. Some qN methods estimates instead the matrix $J\approx G-I$ then take its inverse $H=J^{-1}$. In this case, $J$ follows
\begin{equation}
RC=JYC, \quad \textbf{1}^TC = 0, \;\; \textbf{rank}(C) = N-1. \label{eq:secant_equation_2}
\end{equation}
Any $H$ (or $J^{-1}$) that satisfies the secant equation \eqref{eq:secant_equation} is a candidate for the qN step. There exist several strategies to choose one particular matrix. In the next section, we introduce the generalized quasi-Newton step, then we investigate further more classical schemes.
\subsection{Generalized qN Method}
We introduce here the Generalized qN scheme, which performs the extrapolation
\begin{equation}
y_{\text{Generalized qN}} = (Y-HR)\gamma, \quad \forall\gamma^T\textbf{1}=1,\label{eq:generalized_qn}
\end{equation}
where $\gamma$ is an arbitrary vector whose entries sum to one and $H$ satisfies the secant equation \eqref{eq:secant_equation}. Proposition \ref{prop:invariance_gamma} shows that \eqref{eq:generalized_qn} is invariant in the choice of $\gamma$. For illustration, classical schemes use $\gamma = [0,\ldots,0,1]$.
The solution of the secant equation \eqref{eq:secant_equation} is given by
\begin{equation}
H = YC(RC)^{\dagger} + H_0RC(RC)^{\dagger}, \label{eq:general_solution}
\end{equation}
where $(RC)^{\dagger}$ is the generalized inverse of $RC$ and $H_0$ is arbitrary (it can be viewed as the initialization). There exist a large number of generalized pseudo-inverses, and choosing one of them corresponds to one qN method.
Some qN schemes minimize the distance between $H$ and $H_0$ in, for example, the \textit{weighted Frobenius norm}
\begin{equation}
\|A\|_M = \text{Trace}(M^{1/2} A^T M A M^{1/2}), \quad M\succ 0. \label{eq:weighted_frobenius}
\end{equation}
Other methods impose in addition some constraints on the matrix $H$, for example symmetry.
Despite the different approach of qN and GNA, writing the qN method under the form of \eqref{eq:generalized_qn} makes the links straightforward: \eqref{eq:generalized_qn} is equivalent to run Algorithm \ref{algo:gna} with $P=H$ and $W$ arbitrary. Using Theorem \ref{thm:rate_conv}, this means that using \textit{any} matrix that satisfies the secant equation \eqref{eq:secant_equation} in the qN step \eqref{eq:generalized_qn} lead to an optimal rate of convergence.
However, it is unclear if a given qN method meets the requirements of Proposition \ref{prop:online_accel}. In addition, the computation of \eqref{eq:general_solution} may be complicated in some cases. Finally, it is unclear how bad is the constant factor in Theorem \ref{thm:rate_conv}.
In the next section, we show that specific choices of $W$ (and thus $\gamma_W$) simplify the term $HR\gamma$ in \eqref{eq:generalized_qn} for some known qN schemes. It leads to more explicit update rules that meet the assumptions of Proposition \ref{prop:online_accel}.
\subsection{Broyden Methods}
\subsubsection{(Multi-Secant) Type-I Broyden}
The Type-I Broyden method estimates the matrix $G-I$ first, then invert it. Among all the possible solutions of \eqref{eq:secant_equation_2} it chooses the closest to the initialization $J_0$ in the Frobenius norm. Here, we consider the general case in the weighted Frobenius norm, i.e.,
\[
\textstyle J = \mathop{\rm argmin}_J \| J-J_0 \|_M,\quad JYC=RC,
\]
then $H=J^{-1}$ using the Woodbury identity (See Appendix \ref{sec:woodbury}). The explicit formula for $H$ is derived in Appendix \ref{sec:good_broyden}. After simplification, injecting $H$ in \eqref{eq:generalized_qn} gives
\begin{eqnarray}
& & y_{\text{Type-I Broyden}} = (Y-J_0^{-1}R)\gamma_{\tilde M} \label{eq:broyden_step} \\
& & \text{Where} \;\;\; \tilde M = (G-I)^{-1}M^{-1}J_0^{-1}. \label{eq:variable_metric}
\end{eqnarray}
This corresponds to Algorithm \ref{algo:gna} with parameters $P = J_0^{-1}$ and $W = \tilde M$. At this state, this is unclear if $\tilde M$ is symmetric definite definite, so Theorem \ref{thm:rate_conv} does not apply in this situation. If we consider the simple case where $M=I$ and $J_0^{-1} = \beta I$ with $\beta \neq 0$, \eqref{eq:broyden_step} simplifies into
\begin{equation}
y_{\text{Type-I Broyden}} = (Y-\beta R)\gamma_{(G-I)^{-1}}. \label{eq:broyden_type1_step}
\end{equation}
This corresponds to Algorithm \ref{algo:gna} with $P = \beta I$ and $ W = \beta (G-I)^{-1}$. With these parameters Theorem \eqref{thm:rate_conv} shows that the Broyden Type-I method is an optimal algorithm for minimizing the residual in the norm $\|\cdot\|_{(G-I)^{-1}}$. In addition, the step meets the structure of Proposition \ref{eq:online_algo}, so online acceleration is possible.
\subsubsection{(Multi-Secant) Type-II Broyden}
The Type-II Broyden method estimates directly $(G-I)^{-1}$ by choosing $H$ as close as possible to the initialization $H_0$ in the Frobenius norm. Again, we give here the update in the weighted Frobenius norm \eqref{eq:weighted_frobenius} (see Appendix \ref{sec:bad_broyden} for more details). We have
\[
\textstyle H = \mathop{\rm argmin}_H \|H-H_0\|_{M^{-1}} \quad s.t. \;\; HRC=YC.
\]
After simplification, injecting $H$ in \eqref{eq:generalized_qn} gives
\[
y_{\text{Type-II Broyden}} = (Y-H_0R)\gamma_M.
\]
This time, since $M$ is symmetric positive definite, Theorem \ref{thm:rate_conv} directly applies, but this is not the case with Proposition \eqref{eq:online_algo}. However if we assume $H_0 = \beta I$ with $\beta \neq 0$, online acceleration is possible since
\begin{equation}
y_{\text{Type-II Broyden}} = (Y-\beta R)\gamma_{I}. \label{eq:broyden_type2_step}
\end{equation}
\subsubsection{Observations}
Usually, the matrices $J_0$ and $H_0$ correspond to the previous estimate and are updated with rank-one matrices. However, for the Type-I method, it is unclear if the method has an optimal rate of convergence, as \eqref{eq:variable_metric} may not be symmetric positive definite. Thus Theorem \ref{thm:rate_conv} does not apply.
When the initialization is a scaled identity matrix, The Broyden Type-I \eqref{eq:broyden_type1_step} and Broyden Type-II \eqref{eq:broyden_type2_step} steps are equivalent up to the weight matrix $M$. For example, the Broyden Type-2 updates with $M=(G-I)^{-1}$ is identical to Broyden Type-I with $M=I$. In this case, they are also equivalent to Anderson Acceleration.
\subsection{Symmetric Methods}
We introduce the multi-secant symmetric updates. This setting was studied by \cite{schnabel1983quasi} for DFP and BFGS (not SR-$k$), but not for the general weighted Frobenius norm. It requires that \eqref{eq:secant_equation_2} admits a symmetric solution, or equivalently that \citep{don1987symmetric,hua1990symmetric}
\[
(YC)^TRC = (RC)^T(YC).
\]
This may not be the true when $g$ in \eqref{eq:family_algo} is nonlinear, but in our setting \eqref{eq:linear_algo} the symmetric solution exists since
\[
(YC)^TRC = (YC)^T(G-I)YC =(RC)^T(YC).
\]
\subsubsection{(Multi-Secant) DFP}
The DFP Algorithm \cite{davidon1991variable,fletcher2013practical} finds the closest \textit{symmetric} matrix to $J_0$ in the weighted Frobenius norm defined by the matrix $(G-I)^{-1}$ that solves \eqref{eq:secant_equation_2}. In Appendix \ref{sec:dfp} we consider the general case
\begin{equation}
\textstyle J = \mathop{\rm argmin}_J \| J-J_0 \|_M,\;\; JYC=RC,\,\, J=J^T, \label{eq:optim_dfp}
\end{equation}
where $M$ is arbitrary. The explicit formula (see Appendix \ref{sec:dfp}) is more complicated than Broyden updates, so we directly jump to the standard and simpler case where
\[
M=(G-I)^{-1}, \quad J_0^{-1} = \beta I.
\]
Injecting the solution of \eqref{eq:optim_dfp} in the qN step \eqref{eq:generalized_qn} gives
\begin{equation*}
y_{\text{DFP}} = Y\gamma_I-\left(\beta I + YC \left( (YC)^TRC \right)^{-1}(YC)^T\right)R\gamma_I
\end{equation*}
This is equivalent to Algorithm \ref{algo:gna} with parameters $W=I$ and $P=\beta I + YC \left( (YC)^TRC \right)^{-1}(YC)^T$. This step can be written in the form of Propopsition \eqref{prop:online_accel},
\[
Y\underbrace{\big((I-C ( (YC)^TRC )^{-1}(YC)^TR\big)\gamma_I}_{=\tilde \gamma} + \beta R\gamma_I.
\]
By Proposition \ref{prop:online_accel} and Theorem \ref{thm:rate_conv}, online acceleration is possible if $\beta \neq 0$ with an optimal convergence rate.
\subsubsection{(Multi-Secant) BFGS}
We now introduce the BFGS algorithm. The idea is similar to the DFP formula, but it updates the approximation \mbox{$(G-I)^{-1}$} directly. Again, solving the general case
\[
\mathop{\rm argmin}_H \| H-H_0 \|_{M^{-1}} \quad s.t. \;\; HRC=YC,\;\; H=H^T
\]
leads to complicated formula (see Appendix \ref{sec:bfgs} for details). We directly jump to the standard case where
\[
M=(G-I)^{-1},\quad H_0=\beta I.
\]
Injecting $H$ in the generalized qN step \eqref{eq:generalized_qn}gives
\begin{eqnarray*}
y_{\text{BFGS}} & = & Y\gamma_{(G-I)^{-1}} - \beta R\gamma_{(G-I)^{-1}}\\
& + & \beta YC\big( (YC)^TRC \big)^{-1}(RC)^T R\gamma_{(G-I)^{-1}}.
\end{eqnarray*}
This corresponds to Algorithm \ref{algo:gna} with parameters
\[
W=(G-I)^{-1},\,\, P=\beta \big(I-YC\big( (YC)^TRC \big)^{-1}(RC)^T\big).
\]
With the initialization $H_0=\beta I$ we can use online acceleration (Proposition \ref{prop:online_accel}). Unlike DFP, the usage of $\beta$ here is similar to \eqref{eq:broyden_type1_step} and \eqref{eq:broyden_type2_step}.
\subsubsection{SR-\texorpdfstring{$1$}{1} and SR-\texorpdfstring{$k$}{k}}
We now introduce the symmetric rank $k$ (SR-$k$) update, whose SR-$1$ is a particular member. The method starts with an initialization $H_0$, then looks to a symmetric, low-rank update $H-H_0$ such that $H$ satisfies the secant equation \eqref{eq:secant_equation}. More formally, SR-$k$ solves
\[
H = \min \text{rank}(H-H_0) \quad \text{s.t.}\;\; HRC=YC,\;\; H=H^T.
\]
Its explicit solution is given in Appendix \ref{sec:srk},and injecting this matrix in the generalized qN step \eqref{eq:generalized_qn} gives
\begin{equation}
y_{\text{SR-}k} = (Y-H_0R)\gamma_{(G-I)^{-1}-H_0}. \label{eq:srk_step}
\end{equation}
The vector $\gamma_{(G-I)^{-1}-H_0}$ is computed using the formula
\[
\gamma_{(G-I)^{-1}-H_0} = \frac{ \left(Y^TR-R^TH_0R\right)^{-1}\textbf{1}}{\textbf{1}^T\left(Y^TR-R^TH_0R\right)^{-1}\textbf{1}}.
\]
We have here a direct correspondence between the SR-$k$ update and Algorithm \ref{algo:gna} with
\[
P = H_0,\quad W = (G-I)^{-1}-H_0.
\]
Under the condition that $(G-I)^{-1} \succ H_0$, Theorem \eqref{thm:rate_conv} applies, so the SR-$k$ updates has an optimal rate of convergence. If $H_0=\beta I$, by proposition \ref{prop:online_accel} the SR-$k$ algorithm \eqref{eq:srk_step} can be used online.
\subsection{Comparison with Previous Work} \label{sec:single_secant}
We showed, with Theorem \ref{thm:rate_conv}, that qN methods with multi-secant updates converge with a rate similar to Chebyshev acceleration \eqref{eq:cheby_rate}. In addition, after at most $d$ iterations they reach exactly $x^*$. In practice, quasi-Newton methods are used with a single secant update, so our convergence result may not apply, as \eqref{eq:secant_equation} may not hold for all secants.
There exist some convergence results for single secant update of some quasi-Newton methods. For example, \citet{gay1979some} shows that Broyden Type-I and Type-II update converges to $x^*$ after $2d$ steps. \citet{nocedal2006nonlinear} show that BFGS and DFP with \textit{with exact line-search} converges after $d$ steps, but it does not hold for inexact line search or unitary step size. Moreover, the rate of convergence for the first $N$ iterations is still unknown, and in fact can be particularly bad even for a quadratic problem with two variables \citep{powell1986bad}.
Surprisingly, the SR-1 method converges after $d$ iterations \cite{nocedal2006nonlinear} because it satisfies all previous secant equations. Here, with Theorem \ref{thm:rate_conv} we refine the result by specifying its rate on quadratics for any $N<d$.
\textit{A priori}, there is no reason to use single-secant updates for optimizing quadratics as they are as costly as the multi-secant update, and their theoretical properties are apparently weaker. This may not be true numerically, so we compare the two approaches in Appendix \ref{sec:num_experiments}.
\section{Connections with Krylov Methods}
We investigate connections with Krylov methods, used for solving linear systems $Ax=b$ where $A$ is positive definite, or equivalently minimizing a quadratic function. In the case where $g(x)$ is a linear function \eqref{eq:linear_algo}, we can also use Krylov methods to find an extrapolation $y^{\text{extr\,}}$.
Krylov methods start with an approximation $x_0$, associated to the residual $r_1$, then generate the \textit{Krylov subspace}
\begin{eqnarray}
\mathcal{K}_{N} & = & \text{span}\{ \underbrace{[r_1,Gr_1,G^2r_1,\ldots, G^{N-1}r_1]}_{=K_N}\} \nonumber \\
& = & \{q(G)r_1: \deg(q) \leq N-1\}. \label{eq:krylov_subspace}
\end{eqnarray}
As the \textit{Krylov matrix} $K_N$ can be ill-conditioned in practice, Krylov subspace methods maintain an orthonormal basis to $\mathcal{K}_N$ to ensure better numerical stability.
After creating the subspace, Krylov methods minimize some error function $e(x)$, like the norm of the residual, under the constraint that the next iterate belongs to $x_0+\mathcal{K}_N$,
\begin{equation}
y^{\text{extr\,}} = \min_{x\in x_0 + \mathcal{K}_N} e(x). \label{eq:krylov_method}
\end{equation}
Different error functions lead to different Krylov methods.
\subsection{MINRES and GMRES}
In the case of GMRES (or equivalently MINRES, since we work with symmetric matrices), the iterations are \textit{not} coupled. The algorithm creates a basis $K_N$ of $\mathcal{K}_N$ using Arnoldi (MINRES, \cite{paige1975solution}) or Lancoz (GMRES, \cite{saad1986gmres}) then computes
\[
y_{\text{GMRES}} = \mathop{\rm argmin}_{x\in x_0 + \mathcal{K}_{N-1}} \| x-g(x) \|_2.
\]
We see the iterates belongs to $\mathcal{K}_{N-1}$ rather than $\mathcal{K}_{N}$. This can be explained by the fact that the residual of $y_{\text{GMRES}}$ belongs to $\mathcal{K}_{N}$.
This corresponds to \eqref{eq:krylov_method} with $e(x)=\|\cdot\|_2$. In Appendix \ref{sec:gmres}, we show that
\[
y_{\text{GMRES}} \hspace{-0.5ex}=\hspace{-0.5ex} x^* + p^*(G)(x_0-x^*), \;\; p^* \hspace{-0.5ex}=\hspace{-1.5ex} \mathop{\rm argmin}_{\substack{\deg(p)\leq N-1,\\p(1)=1.}}\hspace{-1.5ex} \|p(G)r_1\|_2.
\]
Even if $y_{\text{GMRES}}$ belongs to $x_0+\mathcal{K}_{N-1}$, the algorithm uses $\mathcal{K}_{N}$ to compute the polynomial $p^*$. We can cast GMRES into an instance of GNA using $P=0$, $W = I$. Indeed,
\[
y_{\text{GNA}} = Y\gamma_I = x^* + (Y-X^*)\gamma_I= x^* + (G-I)^{-1}R\gamma_I
\]
Appendix \ref{sec:gmres} shows that $R\gamma_I = p^*(G)r_1$, thus
\[
y_{\text{GNA}} = y_{\text{GMRES}} \;\; \text{when } W=I,\;\; P=0.
\]
Thus, GNA and GMRES / MINRES are equivalent. Since $P=0$, by Proposition \ref{prop:online_accel}, these algorithms are not suitable for online acceleration.
\subsection{Conjugate Gradient}
The Conjugate Gradient in an online acceleration algorithm, with is usually represented under the form of a two-step recurrence. Conceptually, CG builds a new iterate whose residual is orthogonal to the search space \eqref{eq:krylov_subspace}. This can be written under the form of \eqref{eq:krylov_method},
\[
y_{\text{CG}} = \mathop{\rm argmin}_{y \in x_0 + \mathcal{K}_{N}} \|(K_N)^T(G-I)(y-x^*)\|_2
\]
Appendix \ref{sec:cg} shows that $y_{\text{CG}}$ is obtained using GNA with
\begin{equation}
P = R(R^T(G-I)R)^{-1}R^T \quad \text{and} \quad W \text{ arbitrary}. \label{eq:gna_cg_step}
\end{equation}
Here we see that we need to multiply $R$ with $G-I$, which implies two call to $g(x)$ per iteration. Appendix \ref{sec:cg} shows how to avoid this problem using the linearity of $g$ by deducing the residual of $y_{CG}$. With this strategy, the matrix $R^TR$ is diagonal and $R^T(G-I)R$ tri-diagonal. The solution can thus be updated using a two-step recurrence.
\textbf{Conjugate Broyden Method} Using \eqref{eq:gna_cg_step} with $W=I$, the iterate of GNA simplifies into
\begin{equation}
y_{\text{GNA}} = Y\gamma_{I} + \beta^* R\gamma_{G-I}, \;\; \textstyle \beta^* = \frac{\textbf{1}^T(R^T(G-I)R)^{-1}\textbf{1}}{\textbf{1}^T(R^TR)^{-1}\textbf{1}}. \label{eq:conjugate_broyden}
\end{equation}
In Appendix \ref{sec:cg}, we show that $\beta^*$ can be found by solving
\[
\textstyle \beta^* = \mathop{\rm argmin}_{\beta,z} f(z), \;\; z = Y\gamma_{I} + \beta R\gamma_{G-I},
\]
where $f(z)$ is the quadratic objective function whose gradient step is represented by $g$. This algorithm a possible extension of nonlinear conjugate gradient. The computation of $\gamma_{G-I}$ requires one Hessian-vector operation, or an approximation of $G-I$. Then, one line-search on the objective function $f$ is needed to compute $\beta$.
Compared to other nonlinear CG algorithm, it requires more memory ($O(N)$ rather than the two previous iterations), but this looks more stable as it does not relies on too much on the linearity of $g$. We present some numerical experiments on nonlinear functions in Appendix \ref{sec:num_experiments}.
\section{Numerical Experiments}
We briefly compare several instances of GNA for optimizing a quadratic function. We minimize the linear regression
\[
\min_x \frac{1}{2} \left(\| Ax-b \|^2_2 + \lambda \|x\|_2^2\right)
\]
using the fixed-step gradient method for strongly convex functions \cite{nesterov2013introductory}, combined with GNA using online acceleration \eqref{eq:online_algo}.
We use data sets \texttt{Rand} and \texttt{Madelon} on all several instances of GNA algorithm. For dataset \texttt{Rand}, $A$ is generated randomly using Gaussian distribution and the vector $y$ is full of ones. Its size is $50 \times 25$. We use $N = \infty$, i.e., $N$ grows linearly with $i$. This experiment highlights the results of Theorem \ref{thm:rate_conv}, i.e., optimal rate and convergence after $d$ iterations of GNA.
The second dataset, \texttt{Madelon} comes from \citet{guyon2003design}, and posses 2000 data point in dimension 500. Here, we fix $N_{\max}=20$. This experiment illustrates the efficiency of previous methods when used with limited memory.
We distinguish the case with and without line-search on the $\beta$, the mixing parameter in Anderson acceleration, or the initialization $H_0=\beta I$ in qN methods. Without line-search, the parameter is fixed to $-1$. The starting point $x_0$ is full of ones and $\lambda$ is set such that $\kappa = 10^{-6}$. The experiments are presented in Figure \ref{fig:num_experiments}. Since GMRES and CG are a bit different (the line search is necessary, and they do not \eqref{eq:linear_algo}), we do not compare them in this section. We performed more advanced numerical experiments in Appendix \ref{sec:num_experiments}.
\begin{figure}
\caption{Comparison between different nonlinear acceleration algorithm. Top row: \texttt{Rand}
\label{fig:num_experiments}
\end{figure}
We see in Figure \ref{fig:num_experiments} that, when used with full memory, all algorithms behave the same. This is still true when GNA is used with limited memory and without line-search, with a slight advantage to BFGS and Bad Broyden/Anderson method. With line-search, BFGS and Bad Broyden/Anderson methods are superior to the others, which explains why BFGS is more used in practice.
\mathop{\bf cl}earpage
\mathop{\bf cl}earpage
\appendix
\section{Missing Proofs}
\subsection{Proof of Proposition \ref{prop:sol_good_anderson}}
\label{prop:sol_good_anderson_proof}
\begin{proposition}
Let the symmetric positive definite matrix $W$ satisfies \eqref{eq:cond_inv_w}. Then, the coefficients $\gamma_W$ defined in Algorithm \ref{algo:gna} can be computed using the formula
\begin{equation}
\gamma_W = \frac{(Y^TR)^{-1}\textbf{1}_N}{\textbf{1}_N^T(Y^TR)^{-1}\textbf{1}_N}.
\end{equation}
\end{proposition}
\begin{proof}
We start with \eqref{eq:gw}. If we expand the objective function we have
\[
\gamma^T (Y-X^*)(G-I)^{-1}(Y-X^*) \gamma.
\]
However, $\gamma^TX^* = x^*$ since $\gamma^T\textbf{1} = 1$. If we drop the constant terms, the objective function becomes
\[
\gamma^T Y^T(G-I)Y \gamma + 2\gamma^TY(G-I)x^*.
\]
Now, consider the Lagrangian function $\mathcal{L}(\gamma,\pi) $,
\[
\gamma^T Y^T(G-I)Y \gamma + 2\gamma^TY(G-I)x^* + 2\pi(\gamma^T\textbf{1}-1).
\]
Its derivative over $\gamma$ gives
\[
\frac{\nabla_\gamma \mathcal{L}(\gamma,\pi)}{2} = Y^T(G-I)Y \gamma + Y(G-I)x^* + \pi\textbf{1}
\]
Using $X^*\gamma=x^*$, we can simplify it into
\[
\frac{\nabla_\gamma \mathcal{L}(\gamma,\pi)}{2} = Y^T(G-I)(Y-X^*) \gamma + \pi\textbf{1}.
\]
Finally, since $(G-I)^{-1}(Y-X^*) = R$ we have
\[
\frac{\nabla_\gamma \mathcal{L}(\gamma,\pi)}{2} = Y^TR\gamma + \pi\textbf{1}.
\]
If we equal the derivative to zero,
\[
\gamma = -\pi(Y^TR)^{-1}\textbf{1}.
\]
Finally, using the constraint that $\gamma^T\textbf{1}=1$, we have
\[
\pi = \frac{-1}{\textbf{1}^T(Y^TR)^{-1}\textbf{1}}.
\]
\end{proof}
This proposition can be easily extended to the general case there $W=(G-I)^{-1}W_1+W_2$, assuming $(G-I)^{-1}W_1+W_2$ symmetric positive definite. In this case,
\begin{equation}
\gamma_{W} = \frac{(Y^T W_1 R + R^TW_2R)^{-1}\textbf{1}}{\textbf{1}^T(Y^T W_1 R + R^TW_2R)^{-1}\textbf{1}}. \label{eq:sol_good_anderson_extended}
\end{equation}
\subsection{Proof of Theorem \ref{thm:rate_conv}}
\begin{theorem}\label{thm:rate_conv_proof}
Let $\{(x_1,y_0),\ldots, (x_N,y_{N-1})\}$ be pairs generated by \eqref{eq:linear_algo} (or equivalently \eqref{eq:poly_algo}) where the matrix $G$ is symmetric and $\|G\|\leq 1-\kappa$. Let $y^{\text{extr\,}}$ be the output of Algorithm \ref{algo:gna}. Consider the residual of the extrapolation
\[
r^{\text{extr\,}} = g(y^{\text{extr\,}})-y^{\text{extr\,}}.
\]
Then, the norm $\|r^{\text{extr\,}}\|_W$ is bounded by
\begin{eqnarray*}
\| r^{\text{extr\,}} \|_{W} \leq \| I-(G-I)P \|_2 \frac{\xi^{i-1}}{1+\xi^{2(i-1)}} \| r_1 \|_{W}
\end{eqnarray*}
where $\xi$ is defined in \eqref{eq:cheby_rate} and $W$ is a positive definite.
\end{theorem}
\begin{proof}
We start with the extrapolation step
\[
y^{\text{extr\,}} = (Y-PR)\gamma_W.
\]
Since for any vector $v$ such that $\textbf{1}^Tv = 1$ we have
\begin{equation}
\textstyle X^*v = (\sum_jv_j) x^* = x^* \label{eq:xstar_ones},
\end{equation}
combining \eqref{eq:xstar_ones} with \eqref{eq:link_residual_matrix} gives
\begin{equation}
y^{\text{extr\,}}-x^* = ((G-I)^{-1}-P)R\gamma_W. \label{eq:thm1temp1}
\end{equation}
In addition, the residual $r^{\text{extr\,}}$ can be written
\begin{equation}
r^{\text{extr\,}} = g(y^{\text{extr\,}})-y^{\text{extr\,}} = (G-I)(y^{\text{extr\,}}-x^*). \label{eq:thm1temp2}
\end{equation}
Combining equations \eqref{eq:thm1temp1} and \eqref{eq:thm1temp2} gives
\[
r^{\text{extr\,}} = (I-(G-I)P)R\gamma_W.
\]
Its norm becomes
\begin{eqnarray*}
\|r^{\text{extr\,}}\|_W & = & \|(I-(G-I)P)R\gamma_W\|_W \\
& \leq & \|(I-(G-I)P)\|_2\|R\gamma_W\|_W.
\end{eqnarray*}
By definition,
\[
\gamma_W = \mathop{\rm argmin}_\gamma \|R\gamma\|_W \quad \text{subject to} \;\; \gamma^T\textbf{1} = 1.
\]
Using the structure of $R$, we have for some polynomial $p_{\gamma_W}$ that $R\gamma_W = p(G)r_1$. This means
\[
\|R\gamma_W\|_W = \min_{p:p\in\mathcal{P}_{N-1},p(1)=1} \|p(G)r_1\|_W,
\]
where $\mathcal{P}_{N-1}$ is the space of polynomials of degree at most equal to one. Finally,
\begin{eqnarray*}
& & \min_{p:p\in\mathcal{P}_{N-1},p(1)=1} \|p(G)r_1\|_W \\
& \leq & \|r_1\|_W\min_{p:p\in\mathcal{P}_{N-1},p(1)=1} \|p(G)\|_2.
\end{eqnarray*}
The last term is bounded by \eqref{eq:cheby_rate}, concluding the proof.
\end{proof}
\subsection{Proof of Proposition \ref{prop:online_accel}}
\label{prop:online_accel_proof}
\begin{proposition} \label{prop:invariance_gamma}
Consider the output of Algorithm \ref{algo:gna} after $N$ iterations of \eqref{eq:poly_algo}. If we can ensure
\[
y^{\text{extr\,}} = (Y-PR)\gamma_W = Y\tilde\gamma + X\gamma_{\tilde W},
\]
then $y^{\text{extr\,}} - x^*= p_{N-1}(G)(y_0-x^*)$, where $p_N$ is a polynomial of degree $N-1$ whose coefficients sum to one.
\end{proposition}
\begin{proof}
First, consider $X\gamma_{\bar W}$. We showed it can be written as a polynomial of degree $1$, whose coefficients sum to one, with nonzero leading coefficient. Since $Y\bar \gamma$ is, for any value of gamma, at most of degree $N-1$, the whole expression still have a nonzero leading coefficient, and thus remains of degree $N$. Finally, because $\bar\gamma$ sum to zero, we end with a polynomial of degree $N$, whose coefficients sum to one.
\end{proof}
\section{Elements of Matrix Theory}
\subsection{Woodbury Matrix Identity}
\label{sec:woodbury}
The Woodbury matrix identity \citep{woodbury1950inverting} computes $(A+USV)^{-1}$ using a update on $A^{-1}$. The formula is
\begin{equation}
(A+USV)^{-1} = A^{-1}-A^{-1}U(S^{-1}+VA^{-1}U)^{-1}VA^{-1}.
\end{equation}
\subsection{Symmetric Solution of a System of Linear Equations}
\label{sec:symmetric_solution}
The results in \citep{don1987symmetric,hua1990symmetric} show the symmetric solution of a system of equation. We give here the statement of Theorem 2 in \cite{don1987symmetric}.
\begin{theorem} (Don, 1987) Consider the system of linear equations
\[
AX=B, \quad X=X^T.
\]
Then, the system admits a symmetric solution if and only if
\[
AA^{\thicksim} B=B \qquad \text{and} \qquad AB^T=BA^T.
\]
If such condition is satisfied, the explicit formula for $X$ reads
\[
A^{\thicksim}B + (I-A^{\thicksim}A)(A^+B)^T + (I-A^{\thicksim}A)^T Z (I-A^{\thicksim}A),
\]
where $Z$ is an arbitrary symmetric matrix ($Z=0$ gives the minimum norm solution) and $A^{\thicksim}$ is a minimum-norm generalized reflexive pseudo-inverse of $A$, which means
\[
AA^{\thicksim}A = A \qquad , \qquad A^{\thicksim}AA^{\thicksim} = A^{\thicksim} \qquad \text{and} \qquad A^{\thicksim}A = (A^{\thicksim}A)^T.
\]
\end{theorem}
This theorem is useful when solving the optimization problem present in BFGS and DFP. As a corollary, we have that the solution of the transposed system
\[
XA=B
\]
is given by
\begin{eqnarray}
X & = & BA^+ + (BA^+)^T(I-AA^+) + (I-AA^+)^TZ(I-AA^+), \label{eq:symmetric_solution}\\
& = & (BA^+)^T + (I-AA^+)^TBA^+ + (I-AA^+)^TZ(I-AA^+), \nonumber
\end{eqnarray}
under the condition that
\[
BA^+A=B \qquad \text{and} \qquad A^TB=B^TA,
\]
where $A^+$ is a generalized reflexive pseudo-inverse,
\begin{equation}
AA^{+}A = A \qquad \text{and} \qquad A^{+}AA^{+} = A^{+}. \label{eq:reflexive_pseudo_inverse}
\end{equation}
Here, the result is a bit more general, as $A^+$ does not need to be minimal-norm.
\section{Explicit Formulas for Quasi-Newton Methods}
\subsection{Generalized Quasi-Newton Step}
Before analyzing each qN method, we introduce the generalized quasi-Newton step. We are looking for an approximation $H\approx (G-I)^{-1}$, where $H$ satisfies the secant equation
\[
HRC = YC.
\]
After $N$ iterations usual qN methods perform the step
\[
y^{\text{extr\,}} = y_{N-1}-Hr_{N}.
\]
Here, we are more interested in the \textit{generalized quasi-Newton} (GqN) step
\begin{equation}
y_{\text{Generalized qN}} = (Y-HR)\gamma, \quad \forall\gamma^T\textbf{1}=1
\label{eq:generalized_qn2}
\end{equation}
The next proposition shows that the choice of $\gamma$ \textit{does not change $y_{GqN}$}.
\begin{proposition}
Let $H$ satisfies the secant equation \eqref{eq:secant_equation}. Then, the generalized qN step \eqref{eq:generalized_qn} is invariant in $\gamma$.
\end{proposition}
\begin{proof}
Consider two different vectors $\gamma_1$ and $\gamma_2$, whose entries sum to one. Then,
\begin{equation}
(Y-HR)\gamma_1 = (Y-HR)\left(\gamma_2+(\gamma_1-\gamma_2)\right) \label{eq:difference_gamma}
\end{equation}
Let $c = \gamma_1-\gamma_2$. Since the entries of $\gamma_1$ and $\gamma_2$ sum to one, the entries of $c$ sum to zero. In particular, because $C$ is full column rank, this means there exist a vector $v$ such that $Cv=c$. We can rewrite \eqref{eq:difference_gamma} into
\begin{eqnarray*}
(Y-HR)\gamma_1 & = & (Y-HR)\left(\gamma_2+Cv\right)\\
& = & (Y-HR)\gamma_2+(YC-HRC)v.
\end{eqnarray*}
Since $H$ satisfies the secant equation \eqref{eq:secant_equation}, we have $(YC-HRC)=0$ and
\[
(Y-HR)\gamma_1 = (Y-HR)\gamma_2.
\]
Because the choice of $\gamma_1$ and $\gamma_2$ was arbitrary, this prove the proposition.
\end{proof}
In this section, we use this proposition to simplify the qN steps, in particular when $\gamma=\gamma_W$ \eqref{eq:gw} for some $W$. The previous results also hold when $H=J^{-1}$ and $JYC=RC$.
\mathop{\bf cl}earpage
\subsection{(Multi-Secant) Type-I Broyden}
\label{sec:good_broyden}
\subsubsection{Generalized Multi-Secant Broyden Type-I Update}
The multi-secant Broyden method solves
\[
J = \mathop{\rm argmin}_J \| J-J_0 \|_{M} \qquad s.t. \;\; JYC=RC
\]
First, we parametrize $J$ to satisfy the constraints automatically. Let the matrix $P_M$ such that
\[
P_M YC = YC, \qquad (I-P_M)YC = 0, \qquad P_MM(I-P_M)^T = 0.
\]
For example, we can take
\[
P_M = YC\left((YC)^TM^{-1}YC\right)^{-1}(YC)^{T}M^{-1}.
\]
Now, without loss off generality, we divide $J$ into
\[
J = J_1P_M + J_2(I-P_M).
\]
The optimization problem becomes
\[
J = \mathop{\rm argmin}_{J=J_1P_M + J_2(I-P_M)} \| (J_1-J_0)P_M + (J_2-J_0)(I-P_M) \|_{M} \qquad s.t. \;\; J_1YC=RC
\]
We can expand the objective into
\begin{eqnarray*}
& & \| (J_1-J_0)P_M + (J_2-J_0)(I-P_M) \|_{M}\\
& = & \text{Trace}\left(M^{1/2}\left((J_1-J_0)P_M + (J_2-J_0)(I-P_M)\right)M\left((J_1-J_0)P_M + (J_2-J_0)(I-P_M)\right)^TM^{1/2}\right)
\end{eqnarray*}
Using the property that $P_MM(I-P_M)^T = 0$,
\begin{eqnarray*}
\| (J_1-J_0)P_M + (J_2-J_0)(I-P_M) \|_{M} & = & \| (J_1-J_0)P_M \|_M + \| (J_2-J_0)(I-P_M) \|_M
\end{eqnarray*}
Using the expression of $P_M$ and the constraint $J_1YC=RC$,
\begin{eqnarray*}
(J_1-J_0)P_M & = & (J_1-J_0)YC\left((YC)^TM^{-1}YC\right)^{-1}(YC)^{T}M^{-1}\\
& = & (J_1YC-J_0YC)\left((YC)^TM^{-1}YC\right)^{-1}(YC)^{T}M^{-1} \\
& = & (RC-J_0YC)\left((YC)^TM^{-1}YC\right)^{-1}(YC)^{T}M^{-1}
\end{eqnarray*}
As this term is constant, we remove it from the optimization problem, which becomes
\begin{eqnarray*}
J & = & \mathop{\rm argmin}_{J=J_1P_M + J_2(I-P_M)} \| (J_1-J_0)P_M \|_M + \| (J_2-J_0)(I-P_M) \|_M \qquad s.t. \;\; J_1YC=RC\\
& = & \mathop{\rm argmin}_{J=J_1P_M + J_2(I-P_M)} \| (J_2-J_0)(I-P_M) \|_M
\end{eqnarray*}
Clearly, the optimal solution is obtained when $J_2=J_0$. Finally,
\begin{eqnarray*}
J & = & J_1P_M + J_2(I-PM)\\
& = & RC\left((YC)^TMYC\right)^{-1}(YC)^{T}M + J_0(I-PM)\\
& = & J_0 + (RC-J_0YC)\left((YC)^TM^{-1}YC\right)^{-1}(YC)^{T}M^{-1}
\end{eqnarray*}
Now we use the Woodbury matrix identity (Appendix \ref{sec:woodbury}),
\begin{eqnarray*}
J^{-1} & = & J_0^{-1} - J_0^{-1}(RC-J_0YC) + \left((YC)^TM^{-1}YC + (YC)^{T}M^{-1}J_0^{-1}(RC-J_0YC)\right)^{-1}(YC)^{T}M^{-1}J_0^{-1}\\
& = & J_0^{-1} + (YC-J_0^{-1}RC) + \left( (YC)^{T}M^{-1}J_0^{-1}RC\right)^{-1}(YC)^{T}M^{-1}J_0^{-1}
\end{eqnarray*}
This gives the generalized multi-secant update for the Type-I Broyden method.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Multi-Secant Type-I Broyden Matrix Update}
\begin{eqnarray*}
J^{-1} = J_0^{-1} + (YC-J_0^{-1}RC) + \left( (YC)^{T}M^{-1}J_0^{-1}RC\right)^{-1}(YC)^{T}M^{-1}J_0^{-1}
\end{eqnarray*}
\end{minipage}}
Now, consider the generalized qN step \eqref{eq:generalized_qn}
\[
y_{\text{Broyden Type-I}} = (Y-J^{-1}R)\gamma.
\]
By Proposition \ref{prop:invariance_gamma}, this step is invariant in $\gamma$. In particular, if $\gamma = \gamma_W$ \eqref{eq:gw} with $W=(G-I)^{-1}M^{-1}J_0^{-1}$ (assuming this matrix symmetric positive definite) we can simplify the qN step. Indeed, using the formula \eqref{eq:sol_good_anderson_extended} we have
\[
\gamma_{(G-I)^{-1}M^{-1}J_0^{-1}} = \frac{(Y^TM^{-1}J_0^{-1}R)^{-1}\textbf{1}}{\textbf{1}^T(Y^TM^{-1}J_0^{-1}R)^{-1}\textbf{1}}.
\]
Injecting those coefficients into the generalized qN step gives
\[
(Y-J^{-1}R)\gamma_{(G-I)^{-1}M^{-1}J_0^{-1}} = (Y-J_0 R)\gamma_{(G-I)^{-1}M^{-1}J_0^{-1}}.
\]
Indeed, because by definition $C^T1=0$, the term highlighted in red in the equation below equal zero,
\begin{eqnarray*}
& & J^{-1}R\gamma_{(G-I)^{-1}M^{-1}J_0^{-1}} \\
& = & \left(J_0^{-1} + (YC-J_0^{-1}RC) + \left( (YC)^{T}M^{-1}J_0^{-1}RC\right)^{-1}\textcolor{red}{(YC)^{T}M^{-1}J_0^{-1}R}\right)\frac{\textcolor{red}{(Y^TM^{-1}J_0^{-1}R)^{-1}\textbf{1}}}{\textbf{1}^T(Y^TM^{-1}J_0^{-1}R)^{-1}\textbf{1}}.
\end{eqnarray*}
This step is a special instance of Algorithm \ref{algo:gna} as shown in the following box.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Nonlinear Acceleration: Multi-Secant Type-I Broyden Step}
\begin{itemize}
\item Set $W = (G-I)^{-1}M^{-1}J_0^{-1}$, compute $\gamma_W$ with formula \eqref{eq:sol_good_anderson_extended} (Assuming $W$ symmetric positive definite).
\item Set $P=J_0^{-1}$.
\end{itemize}
\end{minipage}}
\subsubsection{Simple Multi-Secant Broyden Type-I Update}
In the standard case where $M^{-1}=I$ and $J_0^{-1}=\beta I$, it gives the following update.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Standard Multi-Secant Type-I Broyden Matrix Update}
\begin{eqnarray*}
J^{-1} = \beta I + (YC-\beta IRC) + \left( (YC)^{T}RC\right)^{-1}(YC)^{T}
\end{eqnarray*}
\end{minipage}}
In this case, the call to Algorithm \ref{algo:gna} is more straightforward.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Nonlinear Acceleration: Simple Multi-Secant Type-I Broyden Step}
\begin{itemize}
\item Set $W = (G-I)^{-1}$, compute $\gamma_W$ with proposition \eqref{prop:sol_good_anderson}.
\item Set $P=\beta I$, where $\beta$ is a nonzero scalar.
\end{itemize}
\end{minipage}}
As the structure of this GNA step matches Proposition \ref{prop:online_accel}, this instance is eligible for online acceleration.
\mathop{\bf cl}earpage
\subsection{(Multi-Secant) Type-II Broyden}
\label{sec:bad_broyden}
\subsubsection{Generalized Multi-Secant Broyden Type-II Update}
The multi-secant Type-II Broyden update is similar to the Type-I. We start with
\[
H = \mathop{\rm argmin}_H \| H-H_0 \|_{M^{-1}} \qquad s.t. \;\; HRC=YC
\]
To be coherent with the units, we use $M^{-1}$ as we estimate the inverse of $(G-I)$. Using the matrix
\[
P_M = RC((RC)^TMRC)^{-1}(RC)^TM,
\]
we decompose $H$ into
\[
H = H_1P_M + H_2(I-P_M).
\]
The optimization problem becomes
\[
H = \mathop{\rm argmin}_H \| (H_1-H_0)P_M \|_{M^{-1}}+\| (H_2-H_0)(I-P_M) \|_{M^{-1}} \qquad s.t. \;\; HRC=YC
\]
The left term is constant since
\[
H_1P_M = \underbrace{H_1RC}_{=YC}\left((RC)^TMRC\right)^{-1}(RC)^TM.
\]
The optimization problem is thus optimal when $H_2=H_0$. Finally, the update reads
\[
H = H_1P_M + H_2(I-P_M) = H_0 + (RC-H_0YC)\left((RC)^TMRC\right)^{-1}(RC)^TM.
\]
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Multi-Secant Type-II Broyden Matrix Update}
\begin{eqnarray*}
H = H_0 + (RC-H_0YC)\left((RC)^TMRC\right)^{-1}(RC)^TM.
\end{eqnarray*}
\end{minipage}}
Now, consider the generalized qN step \eqref{eq:generalized_qn}. Using this update and $\gamma=\gamma_W$ with $W=M$ gives
\[
y_{\text{Broyden Type-II}} = \left(Y-H_0R + (RC-H_0YC)\left((RC)^TMRC\right)^{-1}\textcolor{red}{(RC)^TMR}\right)\frac{\textcolor{red}{(R^TWR)^{-1}\textbf{1}}}{\textbf{1}^T(R^TMR)^{-1}\textbf{1}}
\]
Since $C^T\textbf{1}=0$, the term in red simplifies and the updates reads
\[
y_{\text{Broyden Type-II}} = (Y-H_0R)\gamma_M.
\]
We summarize in the next box the call to GNA to produce the same step than Broyden Type-II.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Nonlinear Acceleration: Generalized Multi-Secant Type-I Broyden Step}
\begin{itemize}
\item Set $W = M$, compute $\gamma_W$ with equation \eqref{eq:gw}.
\item Set $P=H_0$
\end{itemize}
\end{minipage}}
\subsubsection{Simple Multi-Secant Broyden Type-II Update}
In the simple case, $M=I$ and $H_0=\beta I$ where $H_0$ is a nonzero scalar. We summarize the simple Broyden update and its GNA call in the boxes below.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Multi-Secant Type-II Broyden Matrix Update}
\begin{eqnarray*}
H = \beta I + (RC-\beta YC)\left((RC)^TRC\right)^{-1}(RC)^T.
\end{eqnarray*}
\end{minipage}}
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Nonlinear Acceleration: Simple Multi-Secant Type-II Broyden Step}
\begin{itemize}
\item Set $W = I$, compute $\gamma_W$ with equation \eqref{eq:gw}.
\item Set $P=\beta I$
\end{itemize}
\end{minipage}}
\mathop{\bf cl}earpage
\subsection{(Multi-Secant) DFP}
\label{sec:dfp}
\subsubsection{Generalized Multi-Secant DFP update}
We start with
\[
J = \mathop{\rm argmin}_J \| J-J_0 \|_M \qquad s.t. \;\; JYC=RC,\,\, J=J^T.
\]
We parametrize $J$ as follow, using equation \eqref{eq:symmetric_solution},
\[
J = RC(YC)^+ + (RC(YC)^+)^T(I-YC(YC)^+) + (I-YC(YC)^+)^TZ(I-YC(YC)^+)
\]
In particular, we choose as pseudo-inverse
\[
(YC)^+ = (YC)^+_M = \left((YC)^TM^{-1}(YC)\right)^{-1}(YC)^TM^{-1}.
\]
Clearly we see that $(YC)^+_M$ satisfies \eqref{eq:reflexive_pseudo_inverse}. We also simplify the expression by writing
\[
P_M = YC(YC)^+_M.
\]
This way we ensure that $J$ is symmetric an satisfies the secant equation. The optimization problem becomes
\[
J = \mathop{\rm argmin}_J \left\| RC(YC)^+ + (RC(YC)^+)^T(I-P_M) + (I-P_M)^TZ(I-P_M) -J_0 \right\|_M
\]
First, we notice that
\[
P_M M (I-P_M^T) = 0, \qquad YC^+ = YC^+P_M .
\]
Using these properties and the definition of the weighted Frobenius norm \eqref{eq:weighted_frobenius} we have
\[
J = \mathop{\rm argmin}_J \left\| \left(RC(YC)^+ - J_0\right) P_M \right\|_M + \left\| \left( (RC(YC)^+_M)^T + (I-P_M)^TZ - J_0 \right)(I-P_M)\right\|_M
\]
Using again the same ideas on the right term gives
\[
J = \mathop{\rm argmin}_J \left\| \left(RC(YC)^+_M - J_0\right) P_M \right\|_M + \left\| \left((RC(YC)^+)^T P_M^T J_0 \right)(I-P_M)\right\| + \left\| (I-P_M)^T \left(Z - J_0 \right)(I-P_M)\right\|_W
\]
As $Z$ is the only free parameter, we have $Z = J_0$ and
\begin{equation}
J_{\text{DFP}} = RC(YC)^+_M + (RC(YC)^+_M)^T(I-P_M) + (I-P_M)^TJ_0(I-P_M) \label{eq:jdfp}
\end{equation}
To invert $J$ we use two times the Woodbury matrix identity (Appendix \ref{sec:woodbury}). First, consider the temporary matrix $T$
\begin{eqnarray*}
T & = & (RC(YC)^+)^T + (I-P_M)^TJ_0 \\
& = & J_0 + W^{-1}(YC)\left((YC)^TM^{-1}(YC)\right)^{-1}\left((RC) - J_0(YC) \right)^T
\end{eqnarray*}
The approximation $J$ becomes
\begin{eqnarray*}
J & = & T + \left( RC(YC)^+_M-T \right)P_M \\
& = & T + (RC-TYC)\left((YC)^TM^{-1}(RC)\right)^{-1}(YC)^TM^{-1}
\end{eqnarray*}
Then, using the Woodbury matrix identity we have
\begin{eqnarray*}
J^{-1} & = & T^{-1} - T^{-1}(RC-TYC) \left( (YC)^TM^{-1} YC + (YC)^TM^{-1} T^{-1}(RC-TYC) \right)^{-1}(YC)^TM^{-1}T^{-1} \\
& = & T^{-1} + (YC-T^{-1}RC) \left( (YC)^TM^{-1} T^{-1}RC \right)^{-1}(YC)^TM^{-1}T^{-1}
\end{eqnarray*}
It remains to compute $T^{-1}$ using again the Woodbury matrix identity,
\begin{eqnarray*}
T^{-1} = J_0^{-1}+J_0^{-1}M^{-1}(YC)\left((RC)^TJ_0^{-1}M^{-1}(Y C)\right)^{-1}\left((YC) - J_0^{-1}(RC) \right)^{T}.
\end{eqnarray*}
We summarize the update in the following box.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Multi-Secant DFP Matrix Update}
\begin{eqnarray*}
J^{-1} = T^{-1} + (YC-T^{-1}RC) \left( (YC)^TM^{-1} T^{-1}RC \right)^{-1}(YC)^TM^{-1}T^{-1}
\end{eqnarray*}
where
\begin{eqnarray*}
T^{-1} & = & J_0^{-1}+J_0^{-1}M^{-1}(YC)\left((RC)^TJ_0^{-1}M^{-1}(Y C)\right)^{-1}\left((YC) - J_0^{-1}(RC) \right)^{T}.
\end{eqnarray*}
\end{minipage}}
\subsubsection{Simple Multi-Secant DFP update}
In the (much) simpler case where $M=(G-I)^{-1}$ and $J_0^{-1} = \beta I$, we have
\begin{eqnarray*}
T^{-1} & = & \beta I+(RC)\left((RC)^T(R C)\right)^{-1}\left((YC) - \beta(RC) \right)^{T},\\
J^{-1} & = & T^{-1} + (YC-T^{-1}RC) \left( (RC)^TT^{-1}RC \right)^{-1}(RC)^TT^{-1}.
\end{eqnarray*}
We can simplify the expression by looking at $(RC)^TT^{-1}$,
\begin{eqnarray*}
(RC)^TT^{-1} & = & \beta (RC)^T+(RC)^T(RC)\left((RC)^T(R C)\right)^{-1}\left((YC) - \beta(RC) \right)^{T}\\
& = & (YC)^T.
\end{eqnarray*}
The expression of $J^{-1}$ becomes simpler,
\[
J^{-1} = T^{-1} + (YC-T^{-1}RC) \left( (YC)^TRC \right)^{-1}(YC)^T.
\]
We can simplify even more since
\begin{eqnarray*}
T^{-1}RC & = & \beta RC+(RC)\left((RC)^T(R C)\right)^{-1}\left((YC) - \beta(RC) \right)^{T}RC\\
& = & RC\left((RC)^T(R C)\right)^{-1}(YC)^TRC.
\end{eqnarray*}
Multiplying the equation above by $\left( (YC)^TRC \right)^{-1}(YC)^T$ gives
\begin{eqnarray*}
T^{-1}RC\left( (YC)^TRC \right)^{-1}(YC)^T & = & RC\left((RC)^T(R C)\right)^{-1}(YC)^TRC\left( (YC)^TRC \right)^{-1}(YC)^T, \\
& = & RC\left((RC)^T(R C)\right)^{-1}(YC)^T.
\end{eqnarray*}
We update the expression of $J^{-1}$ using this result,
\[
J^{-1} = T^{-1} + YC \left( (YC)^TRC \right)^{-1}(YC)^T - RC\left((RC)^T(RC)^T\right)^{-1}(YC)^T.
\]
If we replace $T^{-1}$ in $J^{-1}$, we have
\begin{eqnarray*}
J^{-1} & = & \beta I+(RC)\left((RC)^T(R C)\right)^{-1}\left((YC) - \beta(RC) \right)^{T}\\
& & + YC \left( (YC)^TRC \right)^{-1}(YC)^T - RC\left((RC)^T(R C)\right)^{-1}(YC)^T,\\
& = & \beta \left(I - (RC)\left((RC)^T(R C)\right)^{-1}(RC)\right) + YC \left( (YC)^TRC \right)^{-1}(YC)^T
\end{eqnarray*}
This gives the standard DFP matrix update, summarized below.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Standard Multi-Secant DFP Matrix Update}
\begin{eqnarray*}
J^{-1} = \beta \left(I - (RC)\left((RC)^T(R C)\right)^{-1}(RC)^T\right) + YC \left( (YC)^TRC \right)^{-1}(YC)^T.
\end{eqnarray*}
\end{minipage}}
The generalized qN step \eqref{eq:generalized_qn} reads
\[
y_{\text{DFP}} = (Y-J^{-1}_{DFP}R)\gamma
\]
By proposition \ref{prop:invariance_gamma}, the choice of $\gamma$ does not impact the result. We pick in particular
\[
\gamma = \gamma_{I} = \frac{(R^TR)^{-1}\textbf{1}}{\textbf{1}^T(R^TR)^{-1}\textbf{1}}.
\]
Because $C^T\textbf{1} = 0$, we have
\[
(RC)^TR\gamma_{I} = C^TR^T\frac{(R^TR)^{-1}\textbf{1}}{\textbf{1}^T(R^TR)^{-1}\textbf{1}} = 0.
\]
Using this relation, the formula of the step becomes simpler,
\[
y_{\text{DFP}} = Y\gamma_I-\left(\beta I + YC \left( (YC)^TRC \right)^{-1}(YC)^T\right)R\gamma_I
\]
We have a perfect match with the structure of Algorithm \ref{algo:gna}.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Nonlinear Acceleration: Multi-Secant DFP}\\
\begin{itemize}
\item Set $W=I$ and compute $\gamma_W$ using formula \eqref{eq:gw}.
\item Set $P = \beta I + YC \left( (YC)^TRC \right)^{-1}(YC)^T $, where $\beta$ is a nonzero scalar.
\end{itemize}
\end{minipage}}
\paragraph{About the choice of $W$.} We had two possible choices of $W$ to simplify the expression of the qN step. For example, $W=(G-I)^{-1}$ leads to
\begin{eqnarray*}
J^{-1}_{DFP}R\gamma_{(G-I)^{-1}} & = & \beta \left(I - (RC)\left((RC)^T(R C)\right)^{-1}(RC)^T\right)R\gamma_{(G-I)^{-1}},\\
& = & \beta Rv,\qquad v = \left(I - C\left((RC)^T(R C)\right)^{-1}(RC)^T(RC)^T \right)\gamma_I
\end{eqnarray*}
The major problem with this simplification is that it is unclear if the coefficient $v_N$, associated with $R_N$, in non-zero. This means we potentially lose the structure of \eqref{eq:poly_algo} as we do not satisfy the assumptions of Proposition \ref{prop:online_accel}.
\mathop{\bf cl}earpage
\subsection{(Multi-Secant) BFGS}
\label{sec:bfgs}
The Multi-Secant BFGS update follows the same reasoning that DFP method. It suffices to take the update of $J_{DFP}$ \eqref{eq:jdfp}, swap the matrices $Y$ and $R$ and replacing $M^{-1}$ by $M$ and this gives the multi-secant BFGS update, summarized in the box below.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Multi-Secant BFGS Matrix Update}
\begin{eqnarray*}
H = YC(RC)^+_M + (YC(RC)^+_M)^T(I-P_M) + (I-P_M)^TH_0(I-P_M)
\end{eqnarray*}
where
\begin{eqnarray*}
(RC)_M^+ & = & \left((RC)^TMRC\right)^{-1}(RC)^TM \\
P_M & = & RC(RC)_M^+
\end{eqnarray*}
\end{minipage}}
\subsubsection{Simple Multi-Secant DFP update}
Usually, the BFGS update is used with parameters $M=(G-I)^{-1}$ and $H_0=\beta I$ where $\beta$ is a nonzero scalar. We have the following simplifications.
\begin{eqnarray*}
(RC)_M^+ & = & \left((YC)^TRC\right)^{-1}(YC)^T, \\
P_M & = & RC\left((YC)^TRC\right)^{-1}(YC)^T.
\end{eqnarray*}
In particular, the term $(YC(RC)^+_M)^T(I-P_M)$ is equal to zero as
\begin{eqnarray*}
(YC(RC)^+_M)^T(I-P_M) & = & YC\left((YC)^TRC\right)^{-1}\textcolor{red}{(YC)^T} \left( I- \textcolor{red}{RC\left((YC)^TRC\right)^{-1}}(YC)^T\right), \\
& = & 0.
\end{eqnarray*}
The update becomes
\begin{eqnarray*}
H & = & YC(RC)^+_M + \beta (I-P_M)^T(I-P_M),\\
& = & \beta (I-P_M)^T + \left(YC-\beta RC\right)(RC)^+_M
\end{eqnarray*}
Replacing $(RC)_M^+$ and $P_M$ gives the "standard" multi-secant BFGS update.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Standard Multi-Secant BFGS Matrix Update}
\begin{eqnarray*}
H = \beta \left(I-YC\left( (YC)^TRC \right)^{-1}(RC)^T\right) + \left(YC-\beta RC\right)\left((YC)^TRC\right)^{-1}(YC)^T
\end{eqnarray*}
\end{minipage}}
This time, we use the parameters $\gamma_W$ \eqref{eq:gw} in the generalized qN step \eqref{eq:generalized_qn} with $W=(G-I)^{-1}$. This simplifies the qN step since
\[
(YC)^TR\gamma_{(G-I)^{-1}}=C^TY^TR\frac{(Y^TR)^{-1}\textbf{1}}{\textbf{1}^T(Y^TR)^{-1}\textbf{1}} = 0.
\]
Using this result, \eqref{eq:generalized_qn} becomes
\[
(Y-H_{\text{BFGS}}R)\gamma_{(G-I)^{-1}} = \left(Y-\beta \left(I-YC\left( (YC)^TRC \right)^{-1}(RC)^T\right)R\right)\gamma_{(G-I)^{-1}}.
\]
Again, this matches perfectly the structure of Algorithm \eqref{algo:gna}, whose parameters are summarized below.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Nonlinear Acceleration: Multi-Secant BFGS}\\
\begin{itemize}
\item Set $W=(G-I)^{-1}$ and compute $\gamma_W$ using Proposition \eqref{prop:sol_good_anderson}.
\item Set $P = \beta \left(I-YC\left( (YC)^TRC \right)^{-1}(RC)^T\right) $, where $\beta$ is a nonzero scalar.
\end{itemize}
\end{minipage}}
\mathop{\bf cl}earpage
\subsection{SR-\texorpdfstring{$1$}{1} and SR-\texorpdfstring{$k$}{k}}
\label{sec:srk}
Here, we want to update the matrix $H_0$ with a symmetric matrix, with the lowest possible rank, so that the update $H$ satisfies the secant equation \eqref{eq:secant_equation}. In other words, we want to solve
\[
H = \mathop{\rm argmin}_H \textbf{rank} \left( H-H_0 \right), \quad HRC=YC, \quad (H-H_0) = (H-H_0)^T.
\]
We can writte the above problem as follow,
\[
H = \mathop{\rm argmin}_H \textbf{rank} \left( \Delta \right), \quad \Delta RC=YC-H_0RC, \quad \Delta = \Delta^T.
\]
where $\Delta = H-H_0$. Using the solution to a symmetric system \eqref{eq:symmetric_solution}, we get
\[
\Delta = \left(YC-H_0RC\right)(RC)^+ + (\left(YC-H_0RC\right)(RC)^+)^T(I-(RC)(RC)^+) + (I-(RC)(RC)^+)^TZ(I-(RC)(RC)^+),
\]
Clearly, we can easily reduce the rank if $Z = 0$. It remains to find the right pseudo inverse $(RC)^+$ which minimizes the rank of
\[
\Delta = \left(YC-H_0RC\right)(RC)^+ + (\left(YC-H_0RC\right)(RC)^+)^T(I-(RC)(RC)^+).
\]
In particular, choosing
\[
(RC)^+ = \left( \left(YC-H_0RC\right)^TRC \right)^{-1}\left(YC-H_0RC\right)^T
\]
leads to the symmetric rank $k$ update
\[
\Delta = \left(YC-H_0RC\right)\left( \left(YC-H_0RC\right)^TRC \right)^{-1}\left(YC-H_0RC\right)^T.
\]
This leads to the SR-$k$ update below.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Symmetric Rank-$k$ Update}
\begin{eqnarray*}
H = H_0 + \left(YC-H_0RC\right)\left( \left(YC-H_0RC\right)^TRC \right)^{-1}\left(YC-H_0RC\right)^T.
\end{eqnarray*}
\end{minipage}}
The sr-$k$ step reads
\[
y_{\text{sr}-k} = \left(Y-\left(H_0 + \left(YC-H_0RC\right)\left( \left(YC-H_0RC\right)^TRC \right)^{-1}\left(YC-H_0RC\right)^T\right)R\right)\gamma.
\]
Using
\[
\gamma_{[(G-I)^{-1}+H_0]} =\frac{ \left(R^T\left((G-I)^{-1}+H_0\right)R\right)^{-1}\textbf{1}}{\textbf{1}^T\left(R^T\left((G-I)^{-1}+H_0\right)R\right)^{-1}\textbf{1}}
\]
we have
\[
HR\gamma_{[(G-I)^{-1}+H_0]} = H_0R\gamma_{[(G-I)^{-1}+H_0]}.
\]
Notice that $\gamma$ can be simplified using \eqref{eq:sol_good_anderson_extended},
\[
\gamma_{[(G-I)^{-1}+H_0]} =\frac{ \left(Y^TR-R^TH_0R\right)^{-1}\textbf{1}}{\textbf{1}^T\left(Y^TR-R^TH_0R\right)^{-1}\textbf{1}}.
\]
The call to GNA is straightforward and summarized in the box below.
\fbox{\begin{minipage}{0.9\linewidth}
\textbf{Generalized Nonlinear Acceleration: Symmetric Rank-$k$}
\begin{itemize}
\item Set $W = (G-I)^{-1}+H_0$ and compute $\gamma_W$ using \eqref{eq:sol_good_anderson_extended}.
\item Set $P = H_0$ where $H_0$ is symmetric and $(G-I)\succ H_0$.
\end{itemize}
\end{minipage}}
\section{Explicit Formulas for Krylov-subspace Methods}
\subsection{GMRES}
\label{sec:gmres}
In the case of GMRES (or equivalently MINRES, since we work with symmetric matrices), the iterations are \textit{not} coupled. The algorithm creates a smart basis $K_N$ of $\mathcal{K}_N$ using Arnoldi (MINRES, \cite{paige1975solution}) or Lancoz (GMRES, \cite{saad1986gmres}) then computes
\[
y_{\text{GMRES}} = \mathop{\rm argmin}_{x\in x_0 + \mathcal{K}_{N-1}} \| x-g(x) \|_2.
\]
Here, we see that the iterates belongs to $\mathcal{K}_{N-1}$ rather than $\mathcal{K}_{N}$.
In this section, we show that $y_{\text{GMRES}} = y_{\text{GNA}}$ when $W=I$ and $P=0$. In particular, we show that
\begin{equation}
y_{\text{GMRES}} = x^* + p^*(G)(x_0-x^*) \quad \text{and} \quad y_{\text{GNA}} = x^* + p^*(G)(x_0-x^*), \quad \text{where} \;\; p^* = \mathop{\rm argmin}_{\substack{p : p(1)=1,\\ \deg(p) \leq N}} \| p(G)r_1)\|_2. \label{eq:link_gmres_gna}
\end{equation}
\subsubsection{GMRES and \texorpdfstring{$p^*$}{p*}}
First, we start with the definition of the GMRES iterate. Indeed,
\[
y_{\text{GMRES}} = \mathop{\rm argmin}_{x\in x_0 + \mathcal{K}_{N-1}} \| x-g(x) \|_2 = \mathop{\rm argmin}_{x\in x_0 + \mathcal{K}_{N-1}} \| (G-I)(x-x^*) \|_2
\]
Using \eqref{eq:krylov_subspace}, we have
\begin{eqnarray}
y_{\text{GMRES}} = x_0 + q^*(G)r_1, \;\; q^* & = & \mathop{\rm argmin}_{q:\deg(q) \leq N-2} \| (G-I)(x_0+q(G)r_0-x^*) \|_2 \label{eq:ygmres} \\
& = & \mathop{\rm argmin}_{q:\deg(q) \leq N-2} \| (I+(G-I)q(G))r_0 \|_2\nonumber
\end{eqnarray}
Instead of optimizing over all polynomial $q$ of degree $\leq N-2$, we optimize over $p$ of degree at most $N-1$ whose coefficients sum to one since
\[
\min_{\substack{
p=(I+(G-I)q(G)),\\\deg(q) \leq N-1
}} \| p(G)r_1)\|_2 = \min_{\substack{p : p(1)=1,\\ \deg(p) \leq N}} \| p(G)r_1)\|_2.
\]
Let $p^*$ be the optimal polynomial. We can deduce $q^*$ from $p^*$ using
\[
q^*(G) = (G-I)^{-1}(p^*(G) - I)
\]
If we replace $q^*$ in \eqref{eq:ygmres} by the expression above, we have
\begin{eqnarray*}
y_{\text{GMRES}} & = & x_0 + (G-I)^{-1}(p^*(G) - I)r_1,\\
& = & x^* + (G-I)^{-1}p^*(G)r_1,\\
& = & x^* + p^*(G)(G-I)^{-1}r_1,\\
& = & x^* + p^*(G)(x_0-x^*).
\end{eqnarray*}
This prove the first part of \eqref{eq:link_gmres_gna}.
\subsubsection{GNA and \texorpdfstring{$p^*$}{p*}}
We start with the definition of the GNA iterate when $W=I$ and $P = 0$,
\begin{equation}
y_{\text{GNA}} = (Y-PR) \gamma_W = Y\gamma_I, \qquad \gamma_I = \mathop{\rm argmin}_{\gamma : \gamma^T\textbf{1} = 1} \| R\gamma \|_2. \label{eq:gna_gmres_iteration}
\end{equation}
Since $R$ is a basis of $\mathcal{K}_N$, we have that $R\gamma = p(G)r_1$. In addition, because $r_i$, the i-th column of $R$, can be written as
\[
r_i = (G-I)(y_i-x^*) = (G-I)p_i(G)(x_0-x^*) = p_i(G) r_1,
\]
where $p_i(G)$ is defined in \eqref{eq:poly_algo} and $p_i(1) = 1$, we have that for all $p :\deg(p)\leq N-1,\, p(1) = 1$, the exist coefficients $\gamma : \gamma^T1 = 1$ such that
\[
p(G) = \sum_{i=1}^{N} \gamma_i p_i(G) r_1 = \sum_{i=1}^{N} \gamma_ir_i = R\gamma.
\]
By consequence,
\[
R\gamma_I = p^*(G)r_1, \qquad \gamma_I = \mathop{\rm argmin}_{\gamma : \gamma^T\textbf{1} = 1} \| R\gamma \|_2, \qquad p^* = \mathop{\rm argmin}_{p:\deg(p)\leq N-1, p(1)=1} \| p(G)r_1 \|_2
\]
We can inject this solution in \eqref{eq:gna_gmres_iteration},
\begin{eqnarray*}
y_{\text{GNA}} & = & Y\gamma_I,\\
& = & x^* + (G-I)^{-1}R\gamma_I,\\
& = & x^* + (G-I)^{-1}p^*(G)r_1,\\
& = & x^* + p^*(G)(x_0-x^*).
\end{eqnarray*}
This prove the second part of \eqref{eq:link_gmres_gna}.
\subsection{Conjugate Gradient}
\label{sec:cg}
TBA
\section{Numerical Experiments}
\label{sec:num_experiments}
TBA
\end{document} |
\begin{document}
\title{On certain definite integrals and infinite series}
\author{Ernst Eduard Kummer, Dr. of Mathematics}
\date{}
\maketitle
\begin{abstract}
We provide a translation of E. E. Kummer's paper "De integralibus quibusdam definitis and seriebus infinitis" \cite{1} \footnote{This paper was translated from Kummer's Latin original "De integralibus quibusdam definitis and seriebus infinitis" by Alexander Aycock}.
\end{abstract}
The definite integrals, that I undertake to treat now, are very closely connected to the infinite series, that I treated in a commentary of this journal on the hypergeometric series, volume $XV$ page 138 \cite{2} and the following, which, so that they can be represented in a simpler way, I will denote by these functions:\\
\begin{alignat*}{9}
&1. && 1 && + \frac{\alpha \cdot x}{\beta \cdot 1} && + \frac{\alpha(\alpha+1) \cdot x^2}{\beta(\beta+1) \cdot 1 \cdot 2} && +\frac{\alpha(\alpha+1)(\alpha+2) \cdot x^3}{\beta(\beta+1)(\beta+2) \cdot 1 \cdot 2 \cdot 3} && +\cdots \cdot && =\varphi{(\alpha, \beta, x)}, && && \\
&2. && 1 && + \frac{x}{\alpha \cdot 1} && + \frac{x^2}{\alpha(\alpha +1) \cdot 1 \cdot 2} && + \frac{x^3}{\alpha(\alpha +1)(\alpha +2) \cdot 1 \cdot 2 \cdot 3} && + \cdots \cdot && =\psi{(\alpha,x)}, && && \\
&3.~~ && 1 && - \frac{\alpha \cdot \beta}{1 \cdot x} && + \frac{\alpha(\alpha+1) \beta(\beta+1)}{1 \cdot 2 \cdot x^2} && - \frac{\alpha(\alpha+1)(\alpha+2) \beta(\beta+1)(\beta+2)}{1 \cdot 2 \cdot 3 \cdot x^3}&& +\cdots \cdot && = \chi{(\alpha,\beta,x)}. && && \\
\end{alignat*}
From this the transformations, found at the cited place, of the series can also be exhibited in this way:\\
\begin{alignat*}{9}
&4.~~~~~~ && \varphi{(\alpha, \beta, x)} && = e^{x} \cdot \varphi{(\beta-\alpha, \beta, -x)},&& && && && && \\
&5. && \psi{(\alpha,x)} && = e^{\pm2\sqrt{x}}\varphi{(\alpha-\frac{1}{2},2\alpha-1, \pm4\sqrt{x})}, && && && && \\
\end{alignat*}
which is the same formula as
\[
~~~~~~~~~~~~~~~~6.~~~ \varphi{(\alpha,2\alpha,x)}= e^{\frac{x}{2}}\psi({\alpha+\frac{1}{2},\frac{x^2}{16}})
\]
and
\[
7. ~~~~ \chi{(\alpha, \beta, x)}= \frac{x^{\alpha} \Pi(\beta-\alpha-1)}{\Pi(\beta-1)}\varphi{(\alpha, \alpha-\beta+1,x)}+\frac{x^{\beta} \Pi(\alpha-\beta-1)}{\Pi(\alpha-1)}\varphi{(\beta, \beta-\alpha+1,x)}
\]
After having prepared these things, I want to settle the question about this integral at first
\[
~~~~~~~~~~~~~~~~8. ~~~ y= \int_0^{\infty} u^{\alpha-1} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u,
\]
from where it follows
\[
\frac{\diff{d}y}{\diff{d}x}= -\int_0^{\infty} u^{\alpha-2} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u, ~~\frac{\diff{d}^2y}{\diff{d}x^2}=\int_0^{\infty} u^{\alpha-3} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u,
\]
it is by differentiating of the quantity $u^{\alpha-1} \cdot e^{-u} \cdot e^{-\frac{u}{x}}$:
\[
~~~~~~~~~~~~~~~~~~~~~~~~~\diff{d}(u^{\alpha-1} \cdot e^{-u} \cdot e^{-\frac{u}{x}})
\]
\[
= - u^{\alpha-1} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u+(\alpha-1) u^{\alpha-2} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u+x \cdot u^{\alpha-3} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u,
\]
and by integration between the boundaries $0$ and $\infty$:
\[
0= -\int_0^{\infty} u^{\alpha-1} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u+(\alpha-1)\int_0^{\infty} u^{\alpha-2} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u
\]
\[
~~~~~~~~~~~~~~~~~~~~~~~~~+x \int_0^{\infty} u^{\alpha-3} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u,
\]
or, what is the same,
\[
~~~~~~~~~~~~~~~~~~~9. ~~~ 0= y+(\alpha-1)\frac{\diff{d}y}{\diff{d}x}-x\frac{\diff{d}^2y}{\diff{d}x^2},
\]
The complete integral of this equation is easily found by means of series, that we denoted by the function $\psi$,
\[
~~~~~~~~~~~~~~~~10. ~~~~ y= A \cdot \psi(1-\alpha, x)+B \cdot x^{\alpha} \psi(1+\alpha,x),
\]
where $A$ and $B$ are arbitrary constants. From there this expression for the presented integral follows
\[
~~~~~\int_0^{\infty} u^{\alpha-1} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u = A \cdot \psi(1-\alpha, x)+B \cdot x^{\alpha} \psi(1+\alpha,x).
\]
The determination of the constant $A$ is easy; for, if we suppose the quantity $\alpha$ to be positive, and put $x=0$, we have
\[
~~~~~~~~~~~~~~~~~~~~~~~\int_0^{\infty} u^{\alpha-1} \cdot e^{-u} \diff{d}u=A
\]
or
\[
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~A=\Pi(\alpha-1).
\]
To determine the constant $B$ in the same way, the integral $y$ has to be transformed by the substitution $u=\frac{x}{v}$, whence it is
\[
~~~~~~~~\int_0^{\infty} u^{\alpha-1} \cdot e^{-u} \cdot e^{-\frac{u}{x}} \diff{d}u =x^{\alpha}\int_0^{\infty} v^{-\alpha-1} \cdot e^{-v} \cdot e^{-\frac{v}{x}} \diff{d}v,
\]
after having used this integral transformation, equation (11.) is converted into this one:
\[
~~~\int_0^{\infty} v^{-\alpha-1} \cdot e^{-v} \cdot e^{-\frac{v}{x}} \diff{d}v= A \cdot x^{-\alpha} \psi(1-\alpha,x)+B \cdot \psi(1+\alpha,x),
\]
hence, if we suppose the quantity $\alpha$ to be negative and put $x=0$, we have
\[
~~~~~~~~~~~~~~~~~~~~~~~\int_0^{\infty} v^{-\alpha-1} \cdot e^{-v} \diff{d}v=B
\]
or
\[
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~A=\Pi(-\alpha-1),
\]
after having finally substituted which values of the constants, it is:
\[
~~~~~~~~12. ~~~ \Pi(\alpha-1) \psi(1-\alpha, x)+\Pi(-\alpha-1) x^{\alpha} \psi(1+\alpha,x).
\]
From this determination of the constants certain doubts are to be removed, which can arise from the fact, that the one constant was found, after having put $\alpha>0$, the determination of the other constant on the other hand requires the opposite assumption. But it is nevertheless clear, that these conditions would have been superfluous, if, while determining the constants, we had not used the value $x=0$, but any other positive values, and the values of the constants would not have been other ones. Moreover it is to be beared in mind, that formula (12.) is only valid, if $x$ is a positive quantity, otherwise the integral would become infinite; but if $x$ is positive, this integral has a finite value, whatever the quantity $\alpha$ is, positive or negative.\\
From this formula (12.) one can deduce another integral, which is expressed by two series of the kind $\varphi(\alpha, \beta, x)$. By putting $xv$ in the place of $x$, by multiplying by $e^{-v} \cdot v^{\beta-1} \cdot \diff{d}v$ and integrating between the boundaries $0$ and $\infty$, it is
\[
\int_0^{\infty} \int_0^{\infty} u^{\alpha-1} \cdot e^{-u} \cdot v^{\beta-1} \cdot e^{-v} \cdot e^{-\frac{xv}{u}} \diff{d}u \diff{d}v= \Pi(\alpha-1) \int_0^{\infty} v^{\beta-1} \cdot e^{-v} \cdot \varphi(1-\alpha,xv) \diff{d}v
\]
\[
~~~~~~~~~~~~~~~+\Pi(-\alpha-1)x^{\alpha}\int_0^{\infty} v^{\alpha+\beta-1} \cdot e^{-v} \cdot \varphi(1+\alpha,xv) \diff{d}v,
\]
the integrations with respect to the variable $v$ are easily executed; or it is
\begin{alignat*}{9}
&&& \int_0^{\infty} v^{\beta-1} e^{-v} \psi(1-\alpha,xv)\diff{d}v && = \Pi(\beta-1)\varphi(\beta,1-\alpha,x),&& && && && && \\
& && \int_0^{\infty} v^{\alpha+\beta-1} e^{-v} \psi(1+\alpha,xv)\diff{d}v && = \Pi(\alpha+\beta-1)\varphi{(\alpha+\beta,1+\alpha, x)}, && && && && \\
& && \int_0^{\infty} v^{\beta-1} e^{-v} \cdot e^{-\frac{xv}{u}}\diff{d}v && = \frac{\Pi(\beta-1)}{(1+\frac{x}{u})^{\beta}}, && && && && \\
\end{alignat*}
whence
\[
\int_0^{\infty} \int_0^{\infty} u^{\alpha-1} \cdot e^{-u} \cdot v^{\beta-1} \cdot e^{-v} \cdot e^{-\frac{xv}{u}} \diff{d}u \diff{d}v= \Pi(\beta-1)\int_0^{\infty} \frac{u^{\alpha-1} \cdot e^{-u} \diff{d}u}{(1+\frac{x}{u})^{\beta}},
\]
which integral, by putting $ux$ in the place of $u$, is changed into
\[
~~~~~~~~~~~~~~~~~~~~~~~~\Pi(\beta-1)x^{\alpha} \int_0^{\infty} \frac{u^{\alpha+\beta-1}\cdot e^{-ux} \diff{d}u}{(1+u)^{\beta}}
\]
after having substituted which values, we finally have
\[
~~~~~~~~~~~~~~~~~~~~~~~~\Pi(\beta-1)x^{\alpha} \int_0^{\infty}\frac{u^{\alpha+\beta-1} \cdot e^{-ux} \diff{d}u}{(1+u)^{\beta}}
\]
\[
=\Pi(\alpha-1)\Pi(\beta-1)\varphi(\beta,1-\alpha,x)+\Pi(-\alpha-1)\Pi(\alpha+\beta-1)x^{\alpha} \varphi(\alpha+\beta, 1+\alpha,x),
\]
which formula, by changing $\alpha$ into $\alpha-\beta$, obtains this more convenient form
\[
~~~~~~~~~~~~~~~~~~13. ~~~~ \Pi(\beta-1)x^{\alpha} \int_0^{\infty} \frac{u^{\alpha+\beta-1}\cdot e^{-ux} \diff{d}u}{(1+u)^{\beta}}
\]
\[
\frac{\Pi(\alpha-\beta-1)}{\Pi(\alpha-1)}x^{\beta} \cdot \varphi(\beta, \beta-\alpha+1,x)+\frac{\Pi(\beta-\alpha-1)}{\Pi(\beta-1)}x^{\alpha} \cdot \varphi(\alpha, \alpha-\beta+1,x).
\]
Because the one part of this equation, after having interchanged the quantities $\alpha$ and $\beta$, remains the same, it has to be
\[
14. ~~~~\frac{x^{\alpha}}{\Pi(\alpha-1)} \int_0^{\infty} \frac{u^{\alpha-1}\cdot e^{-ux}\cdot \diff{d}u}{(1+u)^{\beta}}=\frac{x^{\beta}}{\Pi(\beta-1)} \int_0^{\infty} \frac{u^{\beta-1}\cdot e^{-ux} \cdot \diff{d}u}{(1+u)^{\alpha}}.
\]
If the transformation, that equation (7.) contains, is applied to formula (13.), it is
\[
~~~~~~~~~~~~~~~15. ~~~\frac{x^{\alpha}}{\Pi(\alpha-1)} \int_0^{\infty} \frac{u^{\alpha-1}\cdot e^{-ux}\cdot \diff{d}u}{(1+u)^{\beta}}=\chi(\alpha, \beta, x).
\]
Because the series $\chi(\alpha, \beta, x)$ belongs to the class of semiconvergent series, it seems to be neccessary, that formula (15.) is reeinforced by a proof, from which it becomes clear at the same time, that by computation of a certain number of the first terms of this series, the proximate value of this integral is found. For this purpose I use the known equation
\[
1-\frac{\beta}{1}z+\frac{\beta(\beta+1)}{1 \cdot 2}z^2-\cdots \cdot (-1)^{k-1}\frac{\beta(\beta+1)\cdots \cdot (\beta+k-2)}{1 \cdot 2 \cdots \cdot (k-1)}z^{k-1}
\]
\[
=\frac{1}{(1+z)^{\beta}}-\frac{(-1)^{k}\beta(\beta+1)\cdots \cdot (\beta+k-1)}{1 \cdot 2 \cdots \cdot (k-1)}z^{k} \int_0^1 \frac{(1-u)^k \diff{d}u}{(1+zu)^{\beta+k}},
\]
it is by putting $z=\frac{v}{x}$, by multiplaying by $v^{\alpha-1} \cdot e^{-v} \cdot \diff{d}v$, then by integrating from $v=0$ to $v=\infty$ and dividing by $\Pi(\alpha-1)$
\[
16. ~~~ 1-\dfrac{\alpha \cdot \beta}{1 \cdot x}+\dfrac{\alpha(\alpha+1)\beta(\beta+1)}{1 \cdot 2 \cdot x^2}- \cdots \cdot (-1)^{k-1}\dfrac{\alpha(\alpha+1) \cdots \cdot (\alpha+k-2)\beta(\beta+1) \cdots \cdot (\beta+k-2)}{1 \cdot 2 \cdot 3 \cdots \cdot (k-1) \cdot x^{k-1}}
\]
\[
=\frac{1}{\Pi(\alpha-1)}\int_0^{\infty} \frac{v^{\alpha-1} \cdot e^{-v} \cdot \diff{d}v}{(1+\frac{v}{x})^{\beta}}-\tfrac{(-1)^k\beta(\beta+1) \cdots \cdot (\beta+k-1)}{\Pi(\alpha-1)1 \cdot 2 \cdot 3 \cdots \cdot (k-1) \cdot x^{k}} \int_0^1 \int_0^{\infty} \frac{(1-u)^{k-1} \cdot v^{\alpha+k-1} \cdot e^{-v} \cdot \diff{d}v \cdot \diff{d}u}{(1+\frac{uv}{x})^{\beta+k}},
\]
this double integral along with its coefficients indicates the error, that is committed, if the integral
\[
\frac{1}{\Pi(\alpha-1)}\int_0^{\infty} \frac{v^{\alpha-1} \cdot e^{-v} \cdot \diff{d}v}{(1+\frac{v}{x})^{\beta}}, ~~~~ \text{or, what is the same} ~~~~ \frac{x^{\alpha}}{\Pi(\alpha-1)}\int_0^{\infty} \frac{v^{\alpha-1} \cdot e^{-vx} \cdot \diff{d}v}{(1+v)^{\beta}}
\]
is computed by the first terms of that series, whose number is $k$, if $k$ is so large, that $\beta+k$ is positive, that quantity, we called the error, changes the sign at the same time as $k$ is converted into $k+1$, or, if a certain number of terms of that series is computed, this sum is either larger or smaller than the desired integral, but if the subsequent term of the series is added, this new sum is smaller than the desired integral, if the sum was larger, if that sum was smaller. Therefore the sums, which that series gives, are alternately too large and too small, and it becomes clear, that the proximate value is found, if the computation is extended to the smallest terms of the semiconvergent series. The same thing can be demonstrated from equation (16.) in this way.\\
Of course it is for positive $\beta+k$:
\[
\int_0^1 \int_0^{\infty} \frac{(1-u)^{k-1} \cdot v^{\alpha+k-1} \cdot e^{-v} \cdot \diff{d}v \diff{d}u}{(1+\frac{uv}{x})^{\beta+k}}<\int_0^1 \int_0^{\infty} (1-u)^{k-1} \cdot v^{\alpha+k-1} \cdot e^{-v} \cdot \diff{d}v \diff{d}u
\]
and
\[
~~~~~~\int_0^1 \int_0^{\infty}(1-u)^{k-1} \cdot v^{\alpha+k-1} \cdot e^{-v} \cdot \diff{d}v \diff{d}u= \frac{\Pi(\alpha+k-1)}{k},
\]
so the error, which is expressed by that double integral, is always smaller than
\[
~~~~~~~~~~~~~~~~~~~~\frac{\beta(\beta+1) \cdots \cdot (\beta+k-1)\Pi(\alpha+k-1)}{1 \cdot 2 \cdot 3 \cdots \cdot (k-1) \cdot \Pi(\alpha-1) x^{k}},
\]
because which is the first term neglected, it follows, that the error is always smaller than that term of the series, to which the summation is extended.\\[2mm]
After having put $\beta=1- \alpha$, equation (15.) is converted into this one:
\[
~~~~~~~~~~~~~~~~~~~~~~~~~\frac{x^{\alpha}}{\Pi(\alpha-1)} \int_0^{\infty} (u+u^2)^{\alpha-1} \cdot e^{-ux} \cdot \diff{d}u
\]
\[
~~~~~~~=\frac{\Pi(2\alpha-2)}{\Pi(\alpha-1)}x^{1-\alpha} \cdot e^{\frac{x}{2}}\cdot \varphi(1-\alpha,2-2\alpha,x)+ \frac{\Pi(-2\alpha)}{\Pi(-\alpha)}x^{\alpha} \cdot \varphi(\alpha, 2\alpha,x),
\]
after having transformed which series by means of formula (6.), it is
\[
~~~~~~~~~~~~~~~~~~~~~~~~~\frac{x^{\alpha}}{\Pi(\alpha-1)} \int_0^{\infty} (u+u^2)^{\alpha-1} \cdot e^{-ux} \cdot \diff{d}u
\]
\[
~~~~~~~=\frac{\Pi(2\alpha-2)}{\Pi(\alpha-1)}x^{1-\alpha} \cdot e^{\frac{x}{2}}\cdot \psi(\frac{3}{2}-\alpha,\frac{x^2}{16})+ \frac{\Pi(-2\alpha)}{\Pi(-\alpha)}x^{\alpha} \cdot \psi(\frac{1}{2}+\alpha,\frac{x^2}{16}),
\]
if further $x$ is changed into $4\sqrt{x}$ and $\alpha$ into $\alpha+\frac{1}{2}$, we have by a few reductions
\[
~~~~~~~~17. ~~~~~ \frac{2^{2\alpha+1} \cdot \sqrt{\pi} \cdot x^{\alpha} \cdot e^{-2\sqrt{x}}}{\Pi(\alpha-\frac{1}{2})} \int_0^{\infty} (u+u^2)^{\alpha-\frac{1}{2}} e^{-4u\sqrt{x}} \diff{d}u
\]
\[
~~~~~~~~~~~=\Pi(\alpha-1)\psi(1-\alpha,x)+\Pi(-\alpha-1)x^{\alpha}\psi(1+\alpha,x),
\]
from there it follows by comparision to formula (12.)
\[
\int_0^{\infty} u^{\alpha-1} \cdot e^{-u} \cdot e^{-\frac{x}{u}} \cdot \diff{d}u=\frac{2^{2\alpha+1} \cdot \sqrt{\pi} \cdot x^{\alpha} \cdot e^{-2\sqrt{x}}}{\Pi(\alpha-\frac{1}{2})} \int_0^{\infty} (u+u^2)^{\alpha-\frac{1}{2}} e^{-4u\sqrt{x}} \diff{d}u,
\]
from this formula, or if you like it better, from formula (12.), after having put $\alpha=\frac{1}{2}$, this very simple value of the integral is easily deduced
\[
~~~~~~~~~~~~18.~~~~ \int_0^{\infty} e^{-u^2} \cdot e^{-\frac{x}{u^2}} \cdot \diff{d}u = \frac{\sqrt{\pi}}{2} \cdot e^{-2\sqrt{x}}.
\]
The integrals, the we just found, have muliple applications in Analyisis, for the sake of an example in the integration of the Riccati equation, that, by means of easy substitutions can be changed into the form of equation (9.); I will not sprend more time on these things, but will rather also settle the question about other similar integrals, whose first I chose to be this one:
\[
~~~~~~~~~~~~19.~~~ z= \int_0^{\frac{\pi}{2}} \cos^{\alpha-1} v\cdot \cos(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v.
\]
I always suppose the quantity $x$ to be positive, because its negative sign can be transferred to the quantity $\beta$. By dfferentiation of the quantity
\[
~~~~~~~~~~~~~~~~~~~~~~~\cos^{\alpha-1} v\cdot \sin(\frac{1}{2}x \tan{v}+\beta v)
\]
it is
\[
\diff{d} (\cos^{\alpha-1} v\cdot \sin(\frac{1}{2}x \tan{v}+\beta v))= \cos^{\alpha-2} v\cdot \sin v \cdot \sin(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v
\]
\[
~~~~~~~~~~~~~~~+(\frac{x}{2\cos^2 v}+\beta) \cos^{\alpha-1} v \cos(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v,
\]
and by integrating between the boundaries $v=0$ and $v=\frac{\pi}{2}$
\begin{alignat*}{9}
&20.~~~~&& 0 && = -(\alpha-1)\int_0^{\frac{\pi}{2}}\cos^{\alpha-2} v\cdot \sin v \cdot \sin(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v&& && && && && \\
& && && +\frac{x}{2}\int_0^{\frac{\pi}{2}}\cos^{\alpha-3} v\cdot \cos(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v && && && && \\
& && && +\beta\int_0^{\frac{\pi}{2}}\cos^{\alpha-1} v\cdot \cos(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v, && && && && \\
\end{alignat*}
it is further
\begin{alignat*}{9}
& \frac{\diff{d}z}{\diff{d}x}&&~~~~= \frac{1}{2} \int_0^{\frac{\pi}{2}}\cos^{\alpha-2} v\cdot \sin v \cdot \sin(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v,&&\\
& \frac{\diff{d}^2z}{\diff{d}x^2}&&~~~~= -\frac{1}{4} \int_0^{\frac{\pi}{2}}\cos^{\alpha-3} v\cdot \sin v \cdot \cos(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v,&&\\
\end{alignat*}
therefore
\[
~~~~~~~~~~~z-\frac{\diff{d}^2z}{\diff{d}x^2}= \int_0^{\frac{\pi}{2}}\cos^{\alpha-3} \cdot \cos(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v,
\]
after having substituted which values, equation (20.) is converted into this one
\[
~~~~~~~~~~~21. ~~~~ 0= (x+2 \beta)z+4(\alpha-1)\frac{\diff{d}z}{\diff{d}x}-4x \frac{\diff{d}^2z}{\diff{d}x^2},
\]
whose complete integral is:
\[
~~~~~~~~y= A \psi(\frac{\beta-\alpha+1}{2}, 1-\alpha, x)+Bx^{\alpha} \psi(\frac{\beta+\alpha+1}{2}, 1+\alpha, x),
\]
and because $z= e^{-\frac{x}{2}} \cdot y$, it is
\[
~~~~~~~~~~~22. ~~~~~ \int_0^{\frac{\pi}{2}}\cos^{\alpha-1} \cdot \cos(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v
\]
\[
~~~~= A \cdot \psi(\frac{\beta-\alpha+1}{2}, 1-\alpha, x)+Bx^{\alpha} \psi(\frac{\beta+\alpha+1}{2}, 1+\alpha, x).
\]
The determination of the constant $A$ is easily obtained by putting $x= \infty$, if $a$ is a positive quantity, the determination of the other constant on the other hand requires peculiar artifices; we will obtain both constants by this method: Let us multiply equation (22.) by $x^{\lambda-1} e^{-\frac{x}{2}} \diff{d}x$ and integrate between the boundaries $x=0$ and $x= \infty$, whereafter it is
\begin{alignat*}{9}
&&& && \int_0^{\infty} \int_0^{\frac{\pi}{2}}\cos^{\alpha-1} v\cdot x^{\lambda-1} e^{-\frac{x}{2}} \cos(\frac{1}{2}x \tan{v}+\beta v)\diff{d}v \diff{d}x&& && && && && \\
& && && =A \int_0^{\infty} x^{\lambda-1} \cdot e^{-x} \psi(\frac{\beta-\alpha+1}{2}, 1-\alpha,x)\diff{d}x && && && && \\
& && && =B \int_0^{\infty} x^{\lambda+\alpha-1} \cdot e^{-x} \psi(\frac{\beta+\alpha+1}{2}, 1+\alpha,x)\diff{d}x. && && && && \\
\end{alignat*}
The values of all of these integrals can be expressed by known functions, for it is
\[
~~~~~~~~~\int_0^{\infty} x^{c-1} \cdot e^{-x} \cdot \psi(a,b,x)\diff{d}x = \Pi(c-1)F(c,a,b,1),
\]
where $F$ denotes the known hypergeometric series, by which, after having expressed it by the the function $\Pi$, it is
\[
\int_0^{\infty} x^{c-1} \cdot e^{-x} \cdot \psi(a,b,x)\diff{d}x = \frac{\Pi(c-1)\Pi(b-1)\Pi(b-a-c-1)}{\Pi(b-a-1)\Pi(b-c-1)},
\]
further it is
\[
\int_0^{\infty} x^{\lambda-1} \cdot e^{-\frac{x}{2}} \cos{(\frac{1}{2}x \tan{v}+\beta v)}\diff{d}x= 2^{\lambda} \Pi(\lambda-1) \cos^{\lambda} v \cdot \cos{(\lambda+\beta)v},
\]
whose value is expressed by means of the function $\Pi$ in this way
\[
~~~~~~~~~~~~~~~~~~~~~~~~\frac{\pi \cdot \Pi(\lambda-1)\Pi(\alpha+\lambda-1)}{2^{\alpha} \Pi(\frac{\alpha-\beta-1}{2})\Pi(\frac{\alpha+\beta-1}{2}+\lambda)},
\]
after having substituted all which values, equation (23.) is converted into this one:
\[
~~~~~~~~~~~~~~~~~~~~~~~~\frac{\pi \cdot \Pi(\lambda-1)\Pi(\alpha+\lambda-1)}{2^{\alpha} \Pi(\frac{\alpha-\beta-1}{2})\Pi(\frac{\alpha+\beta-1}{2}+\lambda)}
\]
\[
=A \frac{\Pi(\lambda-1)\Pi(-\alpha)\Pi(-\frac{\alpha+\beta+1}{2}-\lambda)}{\Pi(-\frac{\alpha+\beta+1}{2})\Pi(-\alpha-\lambda)}+B\frac{\Pi(\alpha+\lambda-1)\Pi(\alpha)\Pi(-\frac{\alpha+\beta+1}{2}-\lambda)}{\Pi(\frac{\alpha-\beta-1}{2})\Pi(-\lambda)},
\]
this equation is easily reduced to this mre convenient form
\[
~~~~~~~~~\frac{\pi \cdot \cos(\frac{\alpha+\beta}{2})\pi}{2^{\alpha} \Pi(\frac{\alpha-\beta-1}{2})}= \frac{A \cdot \Pi(-\alpha)\sin(\alpha+\lambda)\pi}{\Pi(-\frac{\alpha+\beta+1}{2})}+\frac{B \cdot \Pi(\alpha)\sin \lambda \pi}{\Pi(-\frac{\alpha-\beta-1}{2})},
\]
which, because it holds for any arbitrary value of the quantity $\lambda$, is converted into these two
\begin{alignat*}{9}
&& \frac{\pi \cos{\frac{\alpha+\beta}{2}\pi}}{2^{\alpha} \Pi(\frac{\alpha-\beta-1}{2})}&&=& \frac{A \cdot \sin {\alpha \pi} \Pi(-\alpha)}{\Pi(-\frac{\alpha+\beta+1}{2})},&&\\
&& -\frac{\pi \sin{\frac{\alpha+\beta}{2}\pi}}{2^{\alpha} \Pi(\frac{\alpha-\beta-1}{2})}&&=& \frac{A \cdot \cos {\alpha \pi} \Pi(-\alpha)}{\Pi(-\frac{\alpha+\beta+1}{2})}&&+\frac{B \cdot \Pi(\alpha)}{\Pi(\frac{\alpha-\beta-1}{2})},\\
\end{alignat*}
from which the values of the constants $A$ and $B$ are easily found
\[
~~~~~A= \frac{\pi \cdot \Pi(\alpha-1)}{2^{\alpha} \Pi(\frac{\alpha-\beta-1}{2})\Pi(\frac{\alpha+\beta-1}{2})}, ~~~~~~~ B= -\frac{\pi \cdot \cos(\frac{\alpha-\beta}{2})\pi}{2^{\alpha} \cdot \sin{\alpha \pi} \Pi(\alpha)},
\]
after having finally substituted which values of the constants in equation (22.), it is
\[
~~~~~~~~~~24. ~~~~ \int_0^{\frac{\pi}{2}} \cos^{\alpha-1} v \cdot \cos(\frac{1}{2}x \tan{v} +\beta v)\diff{d}v
\]
\[
=\frac{\pi \cdot \Pi(\alpha-1)e^{-\frac{x}{2}} \cdot \psi(\frac{\beta-\alpha+1}{2},1-\alpha,x)}{2^{\alpha} \cdot \Pi(\frac{\alpha-\beta-1}{2})\Pi(\frac{\alpha+\beta-1}{2})}-\frac{\pi \cdot \cos{\frac{\alpha-\beta}{2}\pi} \cdot x^{\alpha} \cdot e^{-\frac{x}{2}} \psi(\frac{\beta+\alpha+1}{2},1+\alpha,x)}{2^{\alpha} \sin{\alpha \pi} \Pi(\alpha)}.
\]
Very simple special cases of this formula are:
\begin{alignat*}{9}
&&27. ~~~~ &&\int_0^{\frac{\pi}{2}} \cos^{\alpha-1} v \cdot \cos(x \tan{v} -(\alpha+1)v)\diff{d}v &=\frac{\pi \cdot x^{\alpha} \cdot e^{-x}}{\Pi(\alpha)},&&\\
&&28. ~~~~ &&\int_0^{\frac{\pi}{2}} \cos^{\alpha-1} v \cdot \cos(x \tan{v} +(\alpha+1)v)\diff{d}v &=0, \\
\end{alignat*}
of which the one is obtained, after having put $\beta=-\alpha-1$, the other, after having put $\beta=\alpha+1$. From the connected formulas (25.) and (26.) also these ones follow:
\begin{alignat*}{9}
&&27. ~~~~ &&\int_0^{\frac{\pi}{2}} \cos^{\alpha-1} v \cdot \cos(x \tan{v})\cos(\alpha+1)v\cdot \diff{d}v &=\frac{\pi \cdot x^{\alpha} \cdot e^{-x}}{2\Pi(\alpha)},&&\\
&&28. ~~~~ &&\int_0^{\frac{\pi}{2}} \cos^{\alpha-1} v \cdot \sin(x \tan{v})\sin(\alpha+1)v\cdot \diff{d}v &=\frac{\pi \cdot x^{\alpha} \cdot e^{-x}}{2\Pi(\alpha)}. \\
\end{alignat*}\\
The formulas (25.) and (26.) agree with the formula, found by the Ill. Laplace, which others later demonstrated in other ways, confer this journal's volume $XIII$, p. 231, where the Cl. Liouville \cite{3}, by the method of differentiation, found for arbitrary parameters
\[
~~~~~~~~~~~~~~~~~~~~~~\int_{-\infty}^{\infty} \frac{e^{\alpha\sqrt{-1}} \cdot \diff{d}\alpha}{(x+\alpha \sqrt{-1})^{\mu}}= \frac{2\pi \cdot e^{-x}}{\Gamma(\mu)}.
\]
Formula (24.) gives another very simple integral, after having put $\beta=\alpha-1$
\[
~~~~~~~~29. ~~~ \int_0^{\frac{\pi}{2}} \cos^{\alpha-1} v \cdot \cos{(x\tan{v}+(\alpha-1)v)}\diff{d}v= \frac{\pi e^{-x}}{2}.
\]
The two series, that are contained in the one part of equation (24.), after having put $\beta$, become $\varphi(\frac{1-\alpha}{2}, 1-\alpha,x)$ and $\varphi(\frac{1+\alpha}{2}, 1+\alpha,x)$, and they can be, by means of formula (6.), transformed into series of the kind $\psi$. After having done the transformations, if one changes $\alpha$ into $2\alpha$ and $x$ into $4\sqrt{x}$, this formula emerges
\[
~~~~~~~~30. ~~~~~ \frac{2\Pi(\alpha-\frac{1}{2})}{\sqrt{\pi}} \int_0^{\frac{\pi}{2}} \cos^{2\alpha-1} v \cdot \cos(2\sqrt{x} \tan{v})\diff{d}v
\]
\[
~~~~~~~= \Pi(\alpha-1)\psi(1-\alpha,x)+\Pi(-\alpha-1) \cdot x^{\alpha} \cdot \psi(1+\alpha,x),
\]
from this, by comparison to formula (12.), it is
\[
31. ~~~~ \frac{2\Pi(\alpha-\frac{1}{2})}{\sqrt{\pi}} \int_0^{\frac{\pi}{2}} \cos^{2\alpha-1} v \cdot \cos(2\sqrt{x} \tan{v})\diff{d}v= \int_0^{\infty} u^{\alpha-1} \cdot e^{-u} \cdot e^{-\frac{x}{u}} \cdot \diff{d}u.
\]\\
Likewise the connection of the two integrals can be demonstrated, that are contained in the equations (13.) and (24.); for this formula (24.), if $\alpha-\beta$ is put in the place of $\alpha$ and $\alpha+\beta-1$ in the place of $\beta$ and multiplied by $\frac{1}{\pi}\Pi(-\beta) \cdot 2^{\alpha} \cdot e^{\frac{x}{2}} \cdot x^{\beta}$, obtains this form
\[
32. ~~~\frac{2\Pi(-\beta)\cdot e^{\frac{x}{2}}\cdot x^{\beta}}{\pi}\int_0^{\frac{\pi}{2}} (2\cos v)^{\alpha-\beta-1} \cdot \cos(\frac{1}{2}x \tan{v} +(\alpha+\beta-1)v)\diff{d}v
\]
\[
=\frac{\Pi(\alpha-\beta-1)}{\Pi(\alpha-1)}x^{\beta} \psi(\beta, \beta-\alpha+1,x)+\frac{\Pi(\beta-\alpha-1)}{\Pi(\beta-1)}x^{\alpha} \psi(\alpha, \alpha-\beta+1,x),
\]
after having compared which to formula (13.), it is seen to be
\[
~~~~~~~~~~~~~~~~~~~~33. ~~~~ \int_0^{\infty} \frac{u^{\beta} \cdot e^{-ux} \cdot \diff{d}u}{(1+u)^{\alpha}}
\]
\[
~~~~~~=\frac{2 \cdot e^{x}{2}}{\sin{\beta \pi}}\int_0^{\frac{\pi}{2}} (2\cos v)^{\alpha-\beta-1} \cdot \cos(\frac{1}{2}x \tan{v} +(\alpha+\beta-1)v)\diff{d}v,
\]
moreover, since the one part of equation (32.) can be transformed by means of formula (7.), it is
\[
34. ~~~~ \frac{2\Pi(-\beta)e^{\frac{x}{2}}\cdot x^{\beta}}{\pi}=\int_0^{\frac{\pi}{2}} (2\cos v)^{\alpha-\beta-1} \cdot \cos(\frac{1}{2}x \tan{v} +(\alpha+\beta-1)v)\diff{d}v= \chi(\alpha, \beta,x).
\]
We will also treat this more general integral in the same way
\[
~~~~~~~~~~~~~~y= \int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} \cdot \cos(x\tan{v}+\gamma v)\diff{d}v
\]
and we will chose the cases, in which it can be expressed by the series mentioned above. We also suppose the quantity $x$ always to be positive in this integral, because it is possible, to transfer its negative sign to the quantity $\gamma$. By differentiating the formula $\sin^{\alpha} v \cdot \cos^{\beta} \cdot \cos(x\tan{v}+\gamma v)$, and integrating from $u=0$ to $u=\frac{\pi}{2}$ thereafter, it is
\begin{alignat*}{9}
& && 0 && = \alpha \int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta+1} \cdot \cos(x\tan{v}+\gamma v)\diff{d}v&& && && && && \\
& && && -\beta \int_0^{\frac{\pi}{2}} \sin^{\alpha+1} v \cdot \cos^{\beta-1} \cdot \cos(x\tan{v}+\gamma v)\diff{d}v && && && && \\
& && && -x \int_0^{\frac{\pi}{2}} \sin^{\alpha} v \cdot \cos^{\beta-2} \cdot \sin(x\tan{v}+\gamma v)\diff{d}v && && && && \\
& && && -\gamma \int_0^{\frac{\pi}{2}} \sin^{\alpha} v \cdot \cos^{\beta} \cdot \sin(x\tan{v}+\gamma v)\diff{d}v, && && && && \\
\end{alignat*}
from this equation, if the integrals are expressed by $y$ and its differentials, this differential equation of third order is easily deduced
\[
~~~~~~~~~~ 35. ~~~ 0= \alpha y +(\gamma+x)\frac{\diff{d}y}{\diff{d}x}+(\beta-2)\frac{\diff{d}^2y}{\diff{d}x^2}-x\frac{\diff{d}^3y}{\diff{d}x^3},
\]
if one puts
\[
~~~~~~~~~~~~~~~~36. ~~~~ A_0+A_1x+A_2x^2+A_3x^3+\cdots,
\]
the conditional equations are easily found, that have to hold between the coefficients of this series, that this series satisfies the differential equation
\begin{alignat*}{9}
& ~~~~~~~&& \alpha &&A_0 &&+\gamma \cdot 1 \cdot A_1 && -1 \cdot 2 \cdot (2-\beta)A_2, &&\\
& && (\alpha +1)&& A_1 &&+\gamma \cdot 2 \cdot A_2 && -2 \cdot 3 \cdot (3-\beta)A_3, &&\\
\end{alignat*}
and in general
\[
~~~ 37. ~~~~~ (\alpha+k)A_k+\gamma \cdot (k+1) A_{k+1}- (k+1)(k+2)(k+2-\beta)A_{k+2}.
\]
If one puts in the same way
\[
~~~~~~~~~~~~~~ 38. ~~~~ y= x^{\beta}(B_0+B_1x+B_2x^2+B_3x^3+\cdots),
\]
one finds these relations for the coefficients
\begin{alignat*}{9}
& ~~~~~~~&& &&+\gamma \cdot \beta \cdot B_0 &&-\beta(\beta+1) \cdot 1 \cdot B_1, &&\\
& && (\alpha +\beta)B_0&& +\gamma(\beta+1)B_1 &&+(\beta+1)(\beta+2) \cdot 2 \cdot B_2, &&\\
\end{alignat*}
and in general
\[
39. ~~~ (\alpha+\beta+k)B_k+\gamma(\beta+k+1)B_{k+1}-(\beta+k+1)(\beta+k+2)(k+2)B_{k+2},
\]
from this it is clear, that the complete integral of equation (35.) is
\[
~~~~~ 40. ~~~ y= A_0+A_1x+A_2x^2+\cdots+x^{\beta}(B_0+B_1x+B_2x^2+\cdots),
\]
for, by the equations (37.), two of the quantities $A_0$,$A_1$,$A_2$ etc, and by the equations (39.) one of the quantities $B_0$,$B_1$,$B_2$ etc. remain arbitrary, so that this integral contains three arbitrary constants. Therefore, if the integral mentioned above is resubstituted, it is
\[
~~~~~~~~~~~~~~~~ \int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} \cdot \cos(x\tan{v}+\gamma v)\diff{d}v
\]
\[
~~~~~~~~~~~=A_0+A_1x+A_2x^2+\cdots+x^{\beta}(B_0+B_1x+B_2x^2+\cdots).
\]
From the relations of the coefficients it is easily seen, that these series and this general integral are higher transcendentals than those, that we undertake to treat here; but they nevertheless coincide with those in certain special cases. At first, if we suppose $\gamma= \alpha+\beta$, it follows from the equations (39.)
\begin{alignat*}{9}
&~~~~~~~~~&& B_1 &&=\frac{\alpha+\beta}{1(1+\beta)}B_0, &&\\
& && B_2 &&=\frac{(\alpha+\beta)(\alpha+\beta+1)}{1\cdot 2(1+\beta)(2+\beta)}B_0, &&\\
& && B_3 &&=\frac{(\alpha+\beta)(\alpha+\beta+1)(\alpha+\beta+2)}{1\cdot 2 \cdot 3(1+\beta)(2+\beta)(3+\beta)}B_0, &&\\
& &&\text{etc.}&& ~~~~~~~~~~~~\text{etc.}&&\\
\end{alignat*}
Further, if $\beta$ is positive, after having put $x=0$, it follows from equation (41.)
\[
A_0= \int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \cos((\alpha+\beta)v \cdot \diff{d}v= \frac{\cos{\frac{\alpha \pi}{2}} \Pi(\alpha-1)\Pi(\beta-1)}{\Pi(\alpha-\beta-1)},
\]
if equation (41.) is differentiated with respect to $x$ in the same way and one puts $x=0$ afterwards, it is
\[
A_1= -\int_0^{\frac{\pi}{2}} \sin^{\alpha} v \cdot \cos^{\beta-2} v \cdot \sin((\alpha+\beta)v \cdot \diff{d}v= -\frac{\cos{\frac{\alpha \pi}{2}} \Pi(\alpha)\Pi(\beta-2)}{\Pi(\alpha-\beta-1)},
\]
therefore it is
\[
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~A_1=\frac{\alpha}{1(1-\beta)}A_0,
\]
from there it easily follows from the equations (37.)
\begin{alignat*}{9}
&~~~~~~~~~~~~~~~~~~~~~~~~~&& A_2 &&=\frac{\alpha(\alpha+1)A_0}{1 \cdot 2(1-\beta)(2-\beta)}, &&\\
& && A_1 &&=\frac{\alpha(\alpha+1)[\alpha+2)A_0}{1 \cdot 2 \cdot 3(1-\beta)(2-\beta)(3-\beta)}, &&\\
& &&\text{etc.}&& ~~~~~~~~~~~~\text{etc.}&&\\
\end{alignat*}
Therefore these two series, by which we expressed our integral, belong to the class of series, that we denoted by $\varphi$ above, in this case $\gamma=\alpha+\beta$, and formula (41.) is converted into this one:
\[
~~~~~~~~~~~~~~~~ \int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} \cdot \cos(x\tan{v}+(\alpha+\beta) v)\diff{d}v
\]
\[
~~~=\frac{\cos{\frac{\alpha \pi}{2}}\Pi(\alpha-1)\Pi(\beta-1)}{\Pi(\alpha+\beta-1)} \varphi(\alpha, 1-\beta,x)+B_0x^{\beta}\varphi(\alpha+\beta,1+\beta,x).
\]
In the determination of the constant $B_0$ we will use the same method as above in the determination of the constants of equation (22.). By multiplying by $x^{\lambda-1} \cdot e^{-x} \diff{d}x$ and integrating between the boundaries $0$ and $\infty$, it is
\[
~~~~~~~~~~\Pi(\lambda-1) \int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta+\lambda-1} \cdot \cos(\alpha+\beta+\lambda) v\diff{d}v
\]
\[
=\frac{\cos{\frac{\alpha \pi}{2}}\Pi(\alpha-1)\Pi(\beta-1)\Pi(\lambda-1)}{\Pi(\alpha+\beta-1)}F(\lambda, \alpha, 1-\beta,1)+ B_0 \Pi(\beta+\lambda-1)F(\lambda+\beta, \alpha+\beta, 1+\beta,1),
\]
and after having expressed the hypergeometric series along with the integral by the function $\Pi$, it is
\[
~~~~~~~~~~~~~~~~~~~\frac{\cos{\frac{\alpha \pi}{2}}\Pi(\lambda-1)\Pi(\alpha-1)\Pi(\beta+\lambda-1)}{\Pi(\alpha+\beta+\lambda-1)}
\]
\[
= \frac{\cos{\frac{\alpha \pi}{2}}\Pi(\lambda-1)\Pi(\alpha-1)\Pi(\beta+\lambda-1)\Pi(-\beta)\Pi(-\beta-\alpha-\alpha)}{\Pi(\alpha+\beta+\lambda-1)\Pi(-\alpha-\beta)\Pi(-\beta-\lambda)}
\]
\[
~~~~~~~~~~~~~~~~+B_0\frac{\Pi(\beta+\lambda-1)\Pi(\beta)\Pi(-\beta-\alpha-\lambda)}{\Pi(-\alpha)\Pi(-\beta)},
\]
after some reductions the quantity $\lambda$, because it has to, vanishes completely, and this very simple value of the constant $B_0$ emerges
\[
~~~~~~~~~~~~~~~~~~~~~~~~~~~B_0 = \cos{\frac{\alpha \pi}{2}}\Pi(-\beta-1),
\]
after having finally substituted which value, we have
\[
~~~~~~ 42. ~~~ \int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \cos(x\tan{v}+(\alpha+\beta)v)\diff{d}v
\]
\[
=\frac{\cos{\frac{\alpha \pi}{2}}\Pi(\alpha)\Pi(\beta-1)}{\Pi(\alpha+\beta-1)}\varphi(\alpha,1-\beta,x)+x^{\beta}\cos{\frac{\alpha \pi}{2}}\Pi(-\beta-1)\varphi(\alpha+\beta,1+\beta,x).
\]
and after having compared these formulas to each other, one sees the connection of the two integrals
\begin{alignat*}{9}
& 43. ~~~~ && ~&& \cos{\frac{\alpha \pi}{2}}\int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \sin(x\tan{v}+(\alpha+\beta)v)\diff{d}v\\
& &&=&& \sin{\frac{\alpha \pi}{2}}\int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \cos(x\tan{v}+(\alpha+\beta)v)\diff{d}v,\\
\end{alignat*}
which formula can also be exhibited in this way
\[
~~~ 44. ~~~ \int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \sin(x\tan{v}+(\alpha+\beta)v-\frac{\alpha \pi}{2})\diff{d}v=0
\]
The special case of formula (42.), in which $\alpha=0$, is worth to be noted
\[
~~~~~~~~~~~~~~ 45. ~~~~ \int_0^{\frac{\pi}{2}} \frac{\cos^{\beta-1} \cdot \sin(x \tan{v}+\beta v}{\sin{v}}\diff{d}v= \frac{\pi}{2},
\]
of which the Cl. Liouville found the special case, corresponding to the value $x=0$, in this journal, volume $XIII$, page 232 \cite{3}. Moreover, having compared the formulas (42.) and (13.), one sees the connection of this integral to those, that we treated above, without any difficulty
\[
~~~~~~~~~46. ~~~~~\frac{\cos{\frac{\alpha \pi}{2}} \Pi(\alpha-1)}{\Pi(\alpha+\beta-1)}x^{\beta} \int_0^{\infty} \frac{u^{\alpha+\beta-1}\cdot e^{-ux}\diff{d}u}{(1+u)^{\alpha}}
\]
\[
~~~~~=\int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \cos(x\tan{v}+(\alpha+\beta)v)\diff{d}v.
\]
Another case, in which the series of formula (41.) are converted into series denoted by the character $\psi$, is $\gamma= -\alpha-\beta$, for in this case formula (41.) is easily found by the same method as above, to be converted into this one:
\[
~~~~~~~~~=\int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \cos(x\tan{v}-(\alpha+\beta)v)\diff{d}v
\]
\[
= \frac{\cos{\frac{\alpha \pi}{2}}\Pi(\alpha-1) \Pi(\beta-1)}{\Pi(\alpha+\beta-1)}\varphi(\alpha, 1-\beta, -x)+B_0x^{\beta} \varphi(\alpha+\beta, 1+\beta, -x),
\]
but in this case the constant $B_0$ obtains another value, that we find by muliplying by $x^{\alpha+\beta} \cdot e^{-x} \diff{d}x$ and by integrating between the boundaries $x=0$ and $x=\infty$, after having done those integrations, it is
\[
~~~~~~~~~~~\Pi(\alpha+\beta-1) \int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\alpha+2\beta-1}v \cdot \diff{d} v
\]
\[
~~~~~~~=\cos{\frac{\alpha \pi}{2}} \Pi(\alpha-2) \Pi(\beta-1)F(\alpha+\beta, \alpha, 1-\beta, -1)
\]
\[
~~~~~~~~~~+B_0 \Pi(\alpha+2\beta-1)F(\alpha+2\beta, \alpha+\beta, 1+\beta, -1),
\]
these hypergeoemtric series, whose forth element is $=-1$, can also be expressed by the function $\Pi$ by means of the formula
\[
~~~~~~~~~~~~~~~F(\alpha, \beta, \alpha-\beta+1, -1) = \frac{2^{-\alpha} \sqrt{\pi} \Pi(\alpha-\beta)}{\Pi(\frac{\alpha}{2}-\beta)\Pi(\frac{\alpha-1}{2})},
\]
that I proved in the commentary about the hypergeometric series in this journal volume $XV$ page 135 \cite{2}. From this, if that integral and the hypergeometric series are expressed by the function $\Pi$, it emerges after certain easy reductions:
\[
~~~~~~~~~~~~~~~~~~~~~~~B_0= \cos(\frac{\alpha}{2}+\beta)\pi \cdot \Pi(-\beta-1),
\]
and after having substituted the value of the constant, it is:
\[
~~~ 47. ~~~~\int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \cos(x\tan{v}-(\alpha+\beta)v)\diff{d}v
\]
\[
= \frac{\cos{\frac{\alpha \pi}{2}}\Pi(\alpha-1) \Pi(\beta-1)}{\Pi(\alpha+\beta-1)}\varphi(\alpha, 1-\beta, -x)+x^{\beta}\cos(\frac{\alpha}{2}+\beta)\pi \cdot \Pi(-\beta-1) \varphi(\alpha+\beta, 1+\beta, -x).
\]
A similar formula is deduced from this one, by changing $\alpha$ into $\alpha-1$, $\beta$ into $\beta+1$ and differentiating with respect to the variable $x$
\[
~~~ 48. ~~~~\int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \sin(x\tan{v}-(\alpha+\beta)v)\diff{d}v
\]
\[
= -\frac{\sin{\frac{\alpha \pi}{2}}\Pi(\alpha-1) \Pi(\beta-1)}{\Pi(\alpha+\beta-1)}\varphi(\alpha, 1-\beta, -x)-x^{\beta}\sin(\frac{\alpha}{2}+\beta)\pi \cdot \Pi(-\beta-1) \varphi(\alpha+\beta, 1+\beta, -x).
\]
These formulas (47.) and (48.) can easily be combined in two ways like this, that they obtain these simpler forms:
\[
~~~ 49. ~~~~\int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \sin(x\tan{v}-(\alpha+\beta)v+(\frac{\alpha}{2}+\beta)\pi)\diff{d}v
\]
\[
~~~~~~~~~~~~~~~~~~~~~~= \frac{\pi \Pi(\alpha-1)\varphi(\alpha, 1-\beta,-x)}{\Pi(-\beta)\Pi(\alpha+\beta-1)},
\]
\[
~~~ 50. ~~~~\int_0^{\frac{\pi}{2}} \sin^{\alpha-1} v \cdot \cos^{\beta-1} v \cdot \sin(x\tan{v}-(\alpha+\beta)v+\frac{\alpha \pi}{2})\diff{d}v
\]
\[
~~~~~~~~~~~~~~~~~~~~~~= \frac{\pi x^{\beta}}{\Pi(\beta)}\varphi(\alpha+\beta, 1+\beta, -x).
\]
In all these integrals, that were treated here, as we already mentioned earlier, $x$ has to be a positive quantity, but if $x$ is supposed to be negative, all sums found would be wrong; in this the integral of equation (50.) is worth to be noted, that, for positive $x$, is equal to that series, but vanishes for negative $x$, confer equation (44.).\\[5mm]
Legnica, in the month of April, 1837
\end{document} |
\begin{document}
\begin{abstract}
We study regularity of solutions $u$ to $\overlineerline\partial u=f$ on a relatively compact $C^2$ domain $D$ in a complex manifold of dimension $n$, where $f$ is a $(0,q)$ form. Assume that there are either $(q+1)$ negative or $(n-q)$ positive Levi eigenvalues at each point of boundary $\partial D$.
Under the necessary condition that a locally $L^2$ solution exists on the domain,
we show the existence of the solutions on the closure of the domain that gain $1/2$ derivative when $q=1$ and $f$ is in the H\"older-Zygmund space $\Lambda^r(\overlineerline D)$ with $r>1$. For $q>1$, the same regularity for the solutions is achieved when $\partial D$ is either sufficiently smooth or of $(n-q)$ positive Levi eigenvalues everywhere on $\partial D$.
\epsilonnd{abstract}
\maketitle
\setcounter{thm}{0}\setcounter{equation}{0}
\section{Introduction}\lambdabel{sec1}
Let $D$ be a relatively compact domain in a complex manifold $X$ of dimension $n$. We say that $D$ satisfies the condition $a_q$ if $\partial D\in C^2$ and its Levi form has either $(q+1)$ negative or $(n-q)$ positive eigenvalues at each point on $\partial D$. We are interested in the regularity of solutions $u$ to the $\overline\partial$-equation $\overline\partial u=f$ on $D$. We will study the case where $f$ is a $V$-valued $(0,q)$ form for a holomorphic vector bundle $V$ on $X$. Denote by $\Lambda_{(0,q)}^r(D, V)$ the space of $V$-valued $(0,q)$ forms whose coefficients are in H\"older-Zygmund space $\Lambda^r(D)$ (see definition in Section~\ref{h-space}).
To ensure the existence of solutions, we impose a minimum requirement that there is an $L_{loc}^2$ solution $u_0$ on $D$ and seek a solution of better regularity.
Our main results are the following.
\th{regsol} Let $r\in(1,\infty)$ and $q\geq 1$.
Let $D $ be a relatively compact domain with $C^3$ boundary in a complex manifold $X$ satisfying the condition $a_q$. Let $V$ be a holomorphic vector bundle on $X$.
Then there exists a bounded
linear $\overline\partial$-solution operator $H_q\colon \Lambda_{(0,q)}^r(D, V)\cap\overline\partial L^2_{loc}(D)\to \Lambda_{(0,q-1)}^{r+1/2}(D, V)$,
provided $(a)$
$q=1$ or $\partial D$ has $(n-q)$ positive Levi eigenvalues at each point on $\partial D$; or $(b)$
$\partial D\in\Lambda^{r+\f{5}{2}}$.
\epsilonth
Note that $H_q$ is independent of $r$ and it provides a smooth ($C^\infty$) linear $\overline\partial$-solution operator for smooth forms in the two cases.
When $\partial D\in C^2$, \rt{regsol} (a) provides a satisfactory regularity result for $\overline\partial$-solutions in the H\"older-Zygmund spaces for $q=1$. For $q>1$, we have the following.
\th{regsol+} Let $q\geq 2$ and keep notations in \rta{regsol} with $\partial D\in C^2$. Suppose $4 \leq r< \infty$. Then there exists a bounded
linear $\overline\partial$-solution operator $H_q^{r}\colon \Lambda_{(0,q)}^r(\overline D, V)\cap \overline\partial L^2_{loc}(D)\to \Lambda_{(0,q-1)}^{r-3}(\overline D)$. Furthermore, $H_q^{r}$ maps $C_{(0,q)}^\infty(\overline D, V)\cap \overline\partial L^2_{loc}(D) $ into $ C_{(0,q-1)}^{\infty}(\overline D)$
\epsilonth
We first state some closely related results on $\overline\partial$-solutions $u$ to $\overline\partial u=f$ on strictly pseudoconvex domains $D$ in ${\bf C}^n$: After work of Lieb-Grauert~~\cite{MR273057} and Kerzman~~\cite{MR0281944}, Henkin-Romanov~~\cite{MR0293121} achieved the sharp $C^{1/2}$ solutions for $f\in L^\infty$ by integral formulas. The $C^{k+1/2}$ solutions for $f\in C^k$ $(k\in{\bf N})$ was obtained by Siu~~\cite{MR330515} for $(0,1)$ forms and by Lieb-Range~~\cite{MR597825} for forms with $q\geq1$ when $\partial D$ is sufficiently smooth boundary.
For $\partial D\in C^2$, Theorem~\ref{regsol} and analogous result for a homotopy formula were proved in~~\cite{MR4289246} through the construction of a homotopy formula. These results were extended by Shi~~\cite{MR4244873} to a weighted Sobolev spaces with a gain less than $1/2$ derivative and by Shi-Yao~~\citetes{https://doi.org/10.48550/arxiv.2107.08913, https://doi.org/10.48550/arxiv.2111.09245} to $H^{s+1/2,p}$ space gaining $1/2$ derivative for $s>1/p$ when $\partial D\in C^2$ and for $s\in{\bf R}$ when $\partial D$ is sufficiently smooth. It is worthy to point out that Shi-Yao achieved the first regularity result for negative order $s$. Furthermore, Gong-Lanzani~~\cite{MR4289246} obtained $\Lambda^{r+1/2}$ (with $r>1$) regularity gaining $1/2$ derivative on strongly ${\bf C}$-linear convex $C^{1,1}$ domains $D$.
Next, we mention results on $\overline\partial$ solutions on $a_q$ domains in complex manifolds:
The basic estimates for the Cauchy-Riemann operator were proved by Morrey~~\cite{MR0099060} for $(0,1)$-forms
and by Kohn~~\citetes{MR0153030,MR0208200} for forms of any type on strongly pseudoconvex manifolds with smooth boundary. These results lead to the regularity of the $\overline\partial$-Neumann operator for strictly pseudoconvex manifolds and more general for compact manifolds whose boundary satisfies property $Z(q)$. Kohn proved that property $Z(q)$ is satisfied by strongly pseudoconvex manifolds with smooth boundary and H\"ormander proved that the $Z(q)$ condition is satisfied by the condition $a_q$ on $D$ when $\partial D\in C^3$. When $\partial D\in C^\infty$, sharp regularity results for $\overline\partial$ solutions were obtained by Greiner-Stein~~\cite{MR0499319} for $(0,1)$ forms and Beals-Greiner-Stanton~~\cite{MR886418} for $(0,q)$ forms under condition $Z(q)$ through the study of the regularity of $\overline\partial$-Neumann operator in $L^{k,p}$ and $\Lambda^r$ spaces.
Condition $a_q$ also ensures the stability of the solvability of the $\overline\partial$-equation on $(0,q)$ forms; namely if $f=\overline\partial u_0+\tilde f$ on $D$ while $\tilde f$ is a $\overline\partial$-closed form on a larger domain and $u_0$ has the regularity in the sought-after class, then $\tilde f=\overline\partial\tilde u$ for some $L^2$ form $\tilde u$. This stability is useful to obtain regularity for $\overline\partial$ solutions as shown by Kerzman~~\cite{MR0281944}; first one seeks regularity for $u_0$ without solving the $\overline\partial$-equation. Then $u_0+\tilde u$ provides a desired solution based on regularity of $u_0$ and the interior regularity of $\tilde u$ from the elliptic theory on systems of partial differential equations.
To prove our results, we will use integral formulas to obtain local solutions near each boundary point of $D$. We then use the Grauert bumping method as in~~\cite{MR0281944} to construct $\tilde f$. To provide background for our results, let us mention regularity results for $\overline\partial$-solutions on the transversal intersection of domains in ${\bf C}^n$. Range-Siu~~\cite{MR338450} obtained $C^{1/2-\epsilonpsilon}$ estimate with any positive $\epsilon$ for a real transversal intersection of sufficiently smooth strictly pseudoconvex domains. For higher order derivatives,
J. Michel~~\cite{MR928297} obtained $C^{k+1/2-\epsilon}$ estimate for $\overline\partial$ solutions on a certain intersection of smooth strictly pseudoconvex domains. J.~Michel and Perotti~~\cite{MR1038709} extended the result to real transversal intersection of strictly pseudoconvex domains with sufficiently smooth boundary.
We should also mention that the local version of \rt{regsol} was proved by Laureate-Thiébaut and Leiterer ~\cite{MR1207871} when $\partial D\in C^\infty$ and $k\in {\bf N}$.
Ricard~~\cite{MR1992543} obtained regularity for concave wedge with $C^{k+2}$ boundary and convex wedges with $C^2$ boundary. The reader is referred to
Barkatou~~\cite{MR1888228} and
Barkatou-Khidr~~\cite{MR2844676} for further results in this direction.
However, all existing integral formulas for $\overline\partial$ solutions, including ours, require boundary to be sufficiently smooth when concavity is present. On the other hand, it is well-known that concavity of the domains is useful; the classical Hartogs' theorem says that a holomorphic function on a $1$-concave domain extends to a holomorphic function across the boundary. Therefore, on a $1$-concave domain in a complex manifold, all $\overline\partial$-solutions for $(0,1)$ forms must have the same regularity regardless the smoothness of the boundary of the domain. In Section 7 we will successfully implement this idea to prove \rt{regsol} (i) with $q=1$.
When $q>1$, not all $\overline\partial$ solutions have the same regularity.
To prove \rt{regsol+}, we first derive an estimate for the solution operator when $\partial D$ is sufficiently smooth. Then we apply the Nash-Moser iteration methods by solving the $\overline\partial$-equation on the subdomains $D_k\subset D_{k+1}$ that have smooth boundary and in the limit, we obtain a desired solution on the closure of $D=\cup D_k$.
We organize the paper as follows.
In Section 2, we formulate an approximate local homotopy formula on a suitable neighborhood of a boundary point $\zeta_0\in\partial \Omega$. In Sections 3 and 4. we derive (genuine) local homotopy formulas for $\overline\partial$-closed $(0,q)$ forms near $(n-q)$ convex and $(q+1)$ concave boundary points of an $a_q$ domain. There we follow approaches developed in Lieb-Range~~\cite{MR597825} and Henkin-Leiterer~~\cite{MR986248}.
While a homotopy formula for forms that are not necessary $\overline\partial$-closed can be derived for $(n-q)$ convex points without extra conditions, such a formula on the concave side of the boundary turns out to be subtle. We will need an extra negative Levi eigenvalues, i.e. $(q+2)$ negative Levi eigenvalues. It is not clear if a local homotopy formula exists without this stronger negativity condition. Such phenomena already occurs to strictly pseudoconvex hypersurfaces of dimension $5$ in work of Webster~~\cite{MR995504} on the local CR embeddings and concave compact CR manifolds in work of Polykov~~\cite{MR2088929} on global CR embeddings.
Section 5 contains some elementary facts on H\"older-Zygmund spaces, where we derive an equivalence characterization on the H\"older-Zygmund norms solely relying on a version of Hardy-Littlewood lemma. This allows us to simplify previous work.
Section 6 contains the main local estimates for the homotopy operator that appear in the local homotopy formula. One of main purposes of the section is to derive precise estimates that reflex the convexity of the H\"older-Zygmund norms; see \re{hqfr12-c} for strictly $(n-q)$ convex $C^2$ domains and \re{hqfr12} for strictly $(q+1)$-concave domains. We emphasize that the estimates do not require the forms to be $\overline\partial$-closed; in fact, the estimates hold for $(q+1)$ concave boundary points, although we don't know if a homotopy formula exists. These estimates immediately give us the desired regularity stated in \rt{regsol} for local solutions.
In Section 7 we show how Hartogs' extension can be used to study the regularity of $\overline\partial$ solutions for $(0,1)$ forms.
A local version of \rt{regsol} (a) for $q=1$ is proved in this section.
In Section 8 we show the existence of global solutions with the desired regularity by using local solutions in Sections 6 and 7 and the interior regularity of elliptic systems. We also derive a global estimate for $\overline\partial$-solutions.
Using this global estimate, we employ the Nash-Moser smoothing operator to prove a detailed version of \rt{regsol+} in Section 9.
The paper has two appendices. In Appendix A, we recall the existing regularity on the signed distance function near a $C^2$ hypersurface in a Riemannian manifold. In Appendix B, we describe a stability result of solvability of $\overline\partial$-equation on $a_q$ domains with $C^2$ boundary using results in H\"ormander~~\cite{MR0179443}.
\setcounter{equation}{0}
\section{A local approximate
homotopy formula}
Let $X$ be a complex manifold of dimension $n$.
Let $D$ be a relatively compact domain in $X$ defined by $\rho<0$, where $\rho$ is a $C^2$ defining function with $d\rho(\zeta)\neq0$ when $\rho(\zeta)=0$. Following Henkin-Leiterer~~\cite{MR986248}, we say that $\partial D$ is \epsilonmph{strictly $q$-convex} at $\zeta\in\partial D$, if the Levi-form $L_\zeta\rho$, i.e. the restriction of the complex Hessian $H_\zeta\rho$ on $T_\zeta^{1,0}(\partial D)$, has at least $q$ positive eigenvalues. We says that $\partial D$ is \epsilonmph{strictly $q$-concave} at $\zeta\in\partial D$, if $L_\zeta\rho$ has at least $q$ negative eigenvalues. Thus a domain is strictly pseudoconvex if and only if it is strictly $(n-1)$-convex. Following H\"ormander~~\cite{MR0179443}, we say that $D$ satisfies the \epsilonmph{condition $a_q$} if the Levi-form $L_\zeta\rho$ has at least either $(q+1)$ negative eigenvalues or $(n-q)$ positive eigenvalues for every $\zeta\in\partial D$.
To see a domain satisfying the condition $a_q$,
let ${\bf P}^n$ be the complex project space. Let $B_{q}^r\subset{\bf P}^n$ be defined by
\begin{gather}{}
B_{q}^r\colon |z^{q}|^2+\cdots+ |z^n|^2< r|z^0|^2+\cdots+ r|z^{q-1}|^2
\epsilonnd{gather}
with $1\leq q\leq n$ and $r>0$. Then $B^r_{q}$ is both strictly $(n-q)$-convex and $(q-1)$-concave. In general, when $\partial D$ is Levi non-degenerate, the condition $a_q$ for $\partial D$ is equivalent to the number of negative Levi eigenvalue not being $q$ at every point $\zeta\in\partial D$. Thus $B^{r_2}_{q_1} \setminus\overline{ B^{r_1}_{q_2}}$ satisfies the condition $a_q$ if $r_2>r_1$, $q_2\geq q_1$, and
\begin{gather}{} q\neq q_1-1, q_2-1.
\epsilonnd{gather}
\begin{defn}\lambdabel{pck} Let $k\geq1$ and $r>1$. A relatively compact domain $D$ in a $C^k$ manifold $X$ is {\it piecewise smooth} of class $C^k$
(resp. $\Lambda^r$), if for each $\zeta\in\partial D$, there are $C^k$ (resp. $\Lambda^r$) functions $\rho_1,\dots, \rho_\epsilonll$ defined on a neighborhood $U$ of $\zeta$ such that $D\cap U=\{z\in U\colon\rho_j(z)<0, j=1,\dots, \epsilonll\}$, $\rho_j(\zeta)=0$ for all $j$, and
$$
d\rho_{1}\wedge\cdots\wedge d\rho_{\epsilonll} \neq0.
$$
\epsilonnd{defn}
The main purpose of this section is to construct an approximate homotopy formula on a subdomain $D'$ of $D$, where $\partial D'$ and $\partial D$ share a piece of boundary containing a given point $\zeta_0\in\partial D$. The $D'$ will have the form $D^{12}=D^1\cap D^2$ where $D^1=D$ and $D^2$ is a ball in suitable coordinates for $\partial D$.
Since the construction of $D'$ is local, we assume that $D$ is contained in ${\bf C}^n$.
We will need Leray maps, following notation in ~\cite{MR986248}.
\begin{defn} Let $D$ be a domain in ${\bf C}^n$. Let $S\subset {\bf C}^n\setminus D$ be a $C^1$ submanifold or an open subset in ${\bf C}^n$. We say that $g( z,\zeta )$ is a \epsilonmph{Leray map} on $D\times S$ if $g\in C^1(D\times S)$ and
\epsilonq{leray-map}
g( z,\zeta )\cdot(\zeta-z)\neq 0, \quad \zeta\in S, \quad z\in D.
\epsilonnd{equation}
Throughout the paper, we use
\epsilonq{g0}
g_0( z,\zeta )=\overline\zeta-\overline z.
\epsilonnd{equation}
\epsilonnd{defn}
Let $g^j \colon D\times S^j\to{\bf C}^n$ be $C^1$ Leray mappings for $j=1,\dots,\epsilonll$.
Let $w=\zeta-z$. Define \begin{gather}n
\omegaega^i=\f{1}{2\pi i}\f{g^i\cdot dw}{g^i\cdot w},
\quad
\Omegaega^i=\omegaega^i\wedge(\overline\partial\omegaega^i)^{n-1},\\
\Omegaega^{01}=\omegaega^0\wedge\omegaega^1\wedge\sum_{\begin{align}pha+\beta=n-2}
(\overline\partial\omegaega^0)^{\begin{align}pha}\wedge(\overline\partial\omegaega^1)^{\beta}.
\epsilonnd{gather*}
Here both differentials $d$ and $\overline\partial$ are in $z,\zeta$ variables.
In general, define
\begin{gather}n
\Omegaega^{1\cdots \epsilonll}=\omegaega^{g_1}\wedge\cdots\wedge\omegaega^{g_\epsilonll}\wedge\sum_{\begin{align}pha_1+\cdots+\begin{align}l_\epsilonll=n-\epsilonll}
(\overline\partial\omegaega^{g_1})^{\begin{align}pha_1}\wedge\cdots(\overline\partial\omegaega^{g_\epsilonll})^{\begin{align}l_\epsilonll}.
\epsilonnd{gather*}
Decompose $\Omega^{\bigcdot}=\sum\Omega_{0,q}^{\bigcdot}$,
where $\Omega_{0,q}^{\bigcdot}$
has type $(0,q)$ in $z$. Hence $\Omega^{i_1,\dots, i_\epsilonll}_{0,q}$ has type $(n,n-\epsilonll-q)$ in $\zeta$. Set
$\Omega^{\bigcdot}_{0,-1}=0$. The Koppelman lemma says that
$$
\overline\partial_z\Omega^{1\cdots \epsilonll}_{0,q-1}+\overline\partial_\zeta\Omega^{1\cdots \epsilonll}_{0,q}=\sum(-1)^{j}\Omega_{0,q}^{1\cdots\hat j\cdots\epsilonll}.
$$
See Chen-Shaw~~\citete{MR1800297}*{p.~263} for a proof. We will use special cases
\begin{gather}
\lambdabel{kop1}\overline\partial_\zeta\Omega_{0,q}^1+\overline\partial_z\Omega_{0,q-1}^1=0,
\\
\lambdabel{kop2}\overline\partial_\zeta\Omega^{01}_{0,q}+\overline\partial_z\Omega_{0,q-1}^{01}=-\Omega_{0,q}^1+\Omega_{0,q}^0,
\\
\lambdabel{kop2}\overline\partial_\zeta\Omega^{012}_{0,q}+\overline\partial_z\Omega_{0,q-1}^{012}=-\Omega_{0,q}^{12}+\Omega_{0,q}^{02}-\Omega^{01}_{0,q}.
\epsilonnd{gather}
Here each identity holds in the sense of distributions on the set where the kernels are non-singular.
To integrate on submanifolds of ${\bf C}^n$, let us see how a sign changes when an exterior differentiation interchanges with integration. Following~~\citete{MR1800297}*{p.~263}, define
$$
\int_{y\in M} u(x,y)dy^J\wedge dx^I = \left \{\int_{y\in M}u (x,y)dy^J\right\}dx^I
$$
for a function $u$ on a manifold $M$ with boundary. For the exterior differential $d_x$, we have
\begin{gather}\lambdabel{checksign}
d_x\int_{y\in M}\phi(x,y) =(-1)^{\dim M}\int_{y\in M}d_x\phi(x,y).
\epsilonnd{gather}
Stokes' formula has the form
\begin{gather}
\int_{y\in\partial M}\phi(x,y)\wedge \psi(y)=\int_{y\in M}\Bigl\{ d_y \phi(x,y)\wedge \psi(y)+
(-1)^{\deg \phi} \phi(x,y)\wedge d\psi(y)\Bigr\},
\lambdabel{checksign+}
\epsilonnd{gather}
where $\deg\phi$ is the total degree of $\phi$ in $(x,y)$.
Throughout the paper, we use notation $$D^{1\dots\epsilonll}:=D^1\cap\cdots\cap D^\epsilonll,$$
which is a relatively compact piecewise $C^1$ domain by \rd{pck}. We choose orientations so that
$$
\int_{D^{1\dots\epsilonll}}\, df=\sum_{i=1}^\epsilonll\int_{\overline D^{1\dots\epsilonll}\cap\partial D^i}f.
$$
Then we define
\epsilonq{d12s}
\partial{D^{12}}=S^1\cup S^2,\quad S^i=\overline{D^{12}}\cap \partial D^i, \quad \partial S^1=S^{12}, \quad \partial S^2=S^{21}.
\epsilonnd{equation}
Thus Stokes' formula has the following special cases
$$
\int_{D^1\cap D^2}df=\int_{S^1}f+\int_{S^2}f,\quad
\int_{S^1}df=\int_{S^{12}}f, \quad
\int_{S^2}df=\int_{S^{21}}f.
$$
We introduce integrals on domains and lower-dimensional sets:
\begin{gather} \lambdabel{defnLR}
R_{D, q}^{i_1\dots i_\epsilonll}f(z)=\int_{D}\Omega^{i_1\dots i_\epsilonll}_{0,q}(z,\zeta)\wedge f(\zeta), \quad
L_{i_1\cdots i_\mu, q}^{j_1\dots j_\nu}f=\int_{S^{i_1\cdots i_\mu}}\Omegaega_{0,q}^{j_1\cdots j_\nu}\wedge f.
\epsilonnd{gather}
When $f$ is said to have type $(0,q)$, we write $R_{D,q}f$, $L_{\bigcdot,q}^{\bigcdot}f$ as $R_{D}f$, $L_{\bigcdot}^{\bigcdot}f$, respectively.
Let $E_D\colon C^0(\overline D)\to C^0_0({\bf C}^n)$ be the Stein extension operator such that
\epsilonq{prop-E}
|Eu|_{{\bf C}^n,r}\leq C_r(D)|u|_{D,r}
\epsilonnd{equation}
where the H\"older-Zygmund norm is defined in Section~\ref{h-space}. Note that the extension exists for any bounded Lipschitz domain $D$. See~~\cite{MR3961327} for a proof of the extension property and references therein.
The main purpose of this section is to
derive the following approximate homotopy formula on a piecewise $C^2$ domain.
\pr{hf} Let $D^{12}\subset{\bf C}^n$ be a bounded piecewise $C^2$ domain. Let $S^1,S^2$ be given by \rea{d12s}. Let $U^1\subset{\bf C}^n\setminus D^{1}$ be a bounded piecewise $C^1$ domain such that $\partial U^1=S^1\cup S^1_+$ with $S^1_+=\partial U^1\setminus S^1$.
Suppose that $g^1(z,\zeta)$ is a $C^1$ Leray map on $D^{12}\times \overline{U^1}$ and $g^2$ is a $C^1$ Leray map on $D^{12}\times S^2$.
Let $ f$ be a $(0,q)$-form such that $ f$ and $\overline\partial f$ are in $C^1(\overline{ D^{12}})$. Then on $D^{12}$ we have
\begin{gather}\lambdabel{tsqf}
f=L_{1,q}^1f+L_{2,q}^2f+L_{12, q}^{12}f+\overline\partial H_q f+H_{q+1}\overline\partial f,\quad \text{if $q\geq1$},\\
\lambdabel{tsqf+}
f=L_{1,0}^1f+L_{2,0}^2f+L_{12,0}^{12}f+H_1\overline\partial f, \quad \text{if $q=0$}
\epsilonnd{gather}
where
\begin{align}{}
\lambdabel{hqf}
H_q f&:=H^{(1)}_q f+ H_q^{(2)}f,
\\
\lambdabel{hq1} H^{(1)}_q f&:=R_{U_1\cup D^{12}, q-1}^0 E f+R_{U_1,q-1 }^{01}[\overline\partial,E] f, \quad q>0,
\\
\lambdabel{hq2}
H^{(2)}_qf&:=-R^1_{U_1,q-1}Ef+L_{1^+,q-1}^{01} Ef +L_{2,q-1}^{02} f+L_{12,q-1}^{012}f,\\
L^{01}_{1^+, q-1}Ef&:=\int_{S^1_+}\Omegaega^{01}_{0,q-1}\wedge Ef,
\\
H_0 f&:=\int_{\partial D^{12}}\Omega_{0,0}^1 f-\int_{U_1}\Omega_{0,0}^1\wedge E\overline\partial f=\int_{U_1}\Omega_{0,0}^1\wedge [\overline\partial, E] f.
\lambdabel{H0f}
\epsilonnd{align}
\epsilonnd{prop}
\begin{rem}Formula \re{tsqf} is called an \epsilonmph{approximate homotopy formula} due to the presence of the boundary integrals $L^1_1,L^{2}_2,L^{12}_{12}$. We will get rid of these boundary integrals under further conditions on the Levi-form of $\partial D^1$.
\epsilonnd{rem}
\begin{proof}
We first consider case $q\geq1$. We recall the Bochner-Martinelli-Koppelman formula~~\citete{MR1800297}*{p.~273} and a version for piecewise $C^1$ domains~~\cite{MR986248}*{Thm 3.12, p.~53}:
\begin{align}\lambdabel{BM}
f(z)&=\overline\partial_z\int_{D^{12}}\Omega_{0,q-1}^0(z,\zeta)\wedge f(\zeta)+\int_{D^{12}}\Omega_{0,q}^0(z,\zeta)
\wedge\overline\partial f\\
\nonumber &\quad +\int_{\partial{D^{12}}}\Omega_{0,q}^0(z,\zeta)\wedge f(\zeta).
\epsilonnd{align}
Here and in what follows, we assume $z\in D^{12}$.
To apply Stokes' formula for piecewise smooth set $\partial D^{12}$, we use notation in \re{kop1}-\re{kop2} and rewrite the last term as
\begin{align}n
\int_{\partial {D^{12}}}\Omega_{0,q}^0(z,\zeta)\wedge f(\zeta)
&=\int_{S^1}\Omega_{0,q}^0(z,\zeta)\wedge f(\zeta)+\int_{S^2}\Omega_{0,q}^0(z,\zeta)\wedge f(\zeta).
\epsilonnd{align*}
Using Koppelman's lemma and Stokes' formula for $S^1$ with $\partial S^1=S^{12}$, we obtain
\begin{align}n
\int_{S^1}&\Omega_{0,q}^0(z,\zeta)\wedge f(\zeta)=
L_1^1f(z)
+\int_{S^1}\left(\overline\partial_\zeta\Omega_{(0,q)}^{01}(z,\zeta)+\overline\partial_z\Omegaega^{01}_{0,q-1}\right)\wedge f(\zeta)
\\
&=L_1^1f(z)
+L^{01}_{12}f(z)-\int_{S^1}\Omega_{(0,q-1)}^{01}(z,\zeta)\wedge\overline\partial_\zeta f(\zeta)-
\overline\partial_z\int_{S^1}\Omega_{(0,q-1)}^{01}(z,\zeta)\wedge f(\zeta).
\epsilonnd{align*}
Analogously, we get
\begin{align}
L_2^0f=L_2^2f+L_{21}^{01}f
-L_{2}^{02}\overline\partial f -
\overline\partial_zL_2^{02} f.
\nonumber
\epsilonnd{align}
Using $L^{02}_{21}f=-L^{02}_{12}f$, we get
$$
L^{01}_{12}f+L_{21}^{02}f=-L_{12}^{12}f+\int_{S^{12}}\overline\partial_z\Omega^{012}_{0,q-1}\wedge f+\int_{S^{12}}\overline\partial_\zeta\Omega^{012}_{0,q}\wedge f.
$$
Applying Stokes' theorem to last term and using $\partial(S^1\cap S^2)=\epsilonmptyset$, we obtain from \re{checksign+}
$$
L^{01}_{12}f+L_{21}^{02}f=-L_{12}^{12}f+\overline\partial L_{12}^{012}f
+L_{12}^{012}\overline\partial f.
$$
This shows that
\begin{align}\lambdabel{BM+}
f(z)&=-
\overline\partial_z\int_{S^1}\Omega_{0,q-1}^{01}(z,\zeta)\wedge f(\zeta)+\overline\partial_z\int_{D^{12}}\Omega_{0,q-1}^0(z,\zeta)\wedge f(\zeta)\\ \nonumber&\quad-\int_{S^1}\Omega_{0,q}^{01}(z,\zeta)\wedge\overline\partial_\zeta f(\zeta)
+\int_{D^{12}}\Omega_{0,q}^0(z,\zeta)
\wedge\overline\partial f\\
\nonumber &\quad-L_{12}^{12}f + \overline\partial L_{12}^{012}f+L_{12}^{012}\overline\partial f
-L_2^{02}\overline\partial f -\overline\partial L_2^{02}f.
\epsilonnd{align}
Next, we transform both integrals on $S^1$ into volume integrals using Stokes' formula. Here we need to modify the methods in Lieb-Range~~\citete{MR597825} and~~\cite{MR3961327}, since $S^1$ is not boundary free.
With orientations, we have $\partial U_+= S^1_+-S^1$.
By Stokes' formula and
~\cite{MR3961327}*{(2.12)}, we have
\begin{align}\lambdabel{keyidsim}
&-\int_{\zeta\in S^1}\Omega_{(0,q-1)}^{01}( z,\zeta )\wedge f(\zeta)+ L^{01}_{1^+}E f(z)=\\
&
\nonumber{\bf Q}uad {\bf Q}uad\int_{U_1 }
\Omega_{(0,q-1)}^{01}(z,\zeta)\wedge\overline\partial E f(\zeta) +
\int_{U_1 }
\Omega_{0,q-1}^{0}(z,\zeta)\wedge E f(\zeta)\\
&\nonumber{\bf Q}uad
{\bf Q}uad-\int_{U_1 }
\Omega_{0,q-1}^{1}(z,\zeta)\wedge E f(\zeta)+
\int_{U_1 }
\overline\partial_z\Omega_{(0,q-2)}^{01}(z,\zeta)\wedge E f(\zeta).
\epsilonnd{align}
This shows that
\begin{align}\lambdabel{keyidsim}
R^0_{D^{12}}f-L_{1}^{01} f&=- L^{01}_{1^+}E f(z)+ R_{U_1 }
^{01}\overline\partial E f\\
&\quad +
R_{U_1\cup D^{12} }
^{0} E f-R_{U_1 }^
{1} E f+\overline\partial
R_{U_1 }
^{01} E f.\nonumber
\epsilonnd{align}
After applying $\overline\partial$, the last term will be dropped.
This shows that
\begin{align}\lambdabel{keyidsim+}
\overline\partial R^0_{D^{12}}f- \overline\partial L_1^{01}f &=-\overline\partial L^{01}_{1^+}Ef-
\overline\partial
R_{U_1}^{1} E f\\
&\quad+\overline\partial R_{U_1 }
^{01}\overline\partial E f+
\overline\partial R_{U_1\cup D^{12} }
^{0} E f(\zeta).
\nonumber\epsilonnd{align}
We apply \re{keyidsim} in which $f$ is replaced by $\overline\partial f=E\overline\partial f$ to obtain
\begin{align}\lambdabel{keyidsimb}
R_{D^{12}}^0\overline\partial f-L_{1}^{01}\overline\partial f&=-L_{1+}^{01}E\overline\partial f
+R_{U_1 }
^{01}\overline\partial E\overline\partial f +
R_{U_1\cup D^{12} }
^{0} E\overline\partial f\\
&\nonumber \quad-R_{U_1 }
^{1} E\overline\partial f +
\overline\partial R_{U_1 }
^{01} E\overline\partial f(\zeta).
\epsilonnd{align}
We can pair the last term with the second last term in \re{keyidsim+} to form the desired commutator $[E,\overline\partial] f$. Finally, we write $\overline\partial E\overline\partial f=[E,\overline\partial]\overline\partial f$.
This completes the proof of \epsilonqref{tsqf}. The above proof is still valid for \epsilonqref{tsqf+} (case $q=0$), as the Koppelman lemma holds with $\Omega_{0,-1}=0$.
Strictly speaking, the above computation is only valid when $\partial D\in C^3$, since the Koppelman lemma can be verified easily when all Leray maps $W_j\in C^2$. When $\partial D^i\in C^2$, one can still verify the integral formula on the domain $D^1\cap D^2$ by smoothing $g^j$. For instance, see ~\citetes{MR3961327, MR4289246} for details.
\epsilonnd{proof}
\setcounter{equation}{0}
\section{A local homotopy formula for
$(n-q)$ convex configuration}
The main purpose of this section is to construct a local homotopy formula near a boundary point of strictly $(n-q)$ convex.
In \rp{hf}, we have derived a local approximate homotopy formula \re{tsqf} for a $(0,q)$ form $f$:
$$ f=L_1^1f+L_2^2f+L_{12}^{12}f+\overline\partial H_q f+H_{q+1}\overline\partial f.
$$
To obtain a genuine local homotopy formula, we will show that the boundary integrals $L^1_1f, L^2_2f, L^{12}_{12}f$ vanish when the boundary is $(n-q)$ convex and the Leray mappings $g^1,g^2$
are chosen appropriately.
The constructions in this section and the next are inspired by Henkin-Leiterer~~\cite{MR986248}.
Throughout the paper let $B_r=\{z\in{\bf C}^n\colon|z|<r\}$ be the ball of radius $r$.
We first transform a $(n-q)$-convex domain $D$ into a new form $D^1$.
\le{convex-rho}Let $D\subset U$ be a domain defined by a $C^2$ function $\rho^0<0$ satisfying $\nabla\rho^0\neq0$ at each point of $U\cap\partial D$.
Suppose that $\partial D$ is $(n-q)$-convex at $\zeta\in U$.
\bpp
\item There is a local biholomorphic mapping $\psi$ defined in an open set $U$ containing $\zeta$ such that $\psi(\zeta)=0$ while $D^1:=\psi(U\cap D)$ is defined by
\epsilonq{qconv-nf}
\rho^1(z)=-y_{n}+\lambda_1|z_1|^2+\cdots+\lambda_{q-1}|z_{q-1}|^2+|z_{q}|^2+
\cdots+|z_{n}|^2+R(z)<0,
\epsilonnd{equation}
where $|\lambda_j|\leq1$ and $R(z)=o(|z|^2)$. There exists $r_1>0$ such that the boundary $\partial\psi(U\cap D)$ intersects the sphere $\partial B_r$ transversally when $0<r<r_1$. Furthermore, the function $R$ in \rea{qconv-nf} is in $C^a(B_{r_1})$ $($resp. $\Lambda^a(B_{r_1}))$,
when $\rho^0\in C^a(U)$ with $a\geq2$ $($resp. $\Lambda^a(U)$ with $a>2)$.
\item Let $\psi$ be as above. There exists $\deltata(D)>0$ such that if $\tilde D$ is defined by $\tilde\rho^0<0$ and $\|\tilde\rho^0-\rho^0\|_2<\deltata(D)$, then $\psi(U\cap\tilde D)$ is given by
\epsilonq{qconv-nf-t}
\tilde\rho^1(z)=-y_{n}+\lambda_1|z_1|^2+\cdots+\lambda_{q-1}|z_{q-1}|^2+|z_{q}|^2+
\cdots+|z_{n}|^2+\tilde R(z)<0
\epsilonnd{equation}
with $|\tilde R-R|_{B_{r_1},a}\leq C_a|\tilde\rho^0-\rho^0|_{U,a}$ for $a>2$ and $\|\tilde R-R\|_{B_{r_1},a}\leq C_a\|\tilde\rho^0-\rho^0\|_{U,a}$ for $a\geq2$.
There exists $r_1>0$ such that the boundary $\partial\psi(U\cap \tilde D)$ intersects the sphere $\partial B_{r_2}$ transversally when $r_1/2<r_2<r_1$.
\epsilonpp
Here $\deltata(D)$ depends on the modulus of continuity of $\partial^2\rho^0$.
\epsilonnd{lemma}
\begin{rem} For the rest of paper, we will refer to $(\tilde D^1,\tilde\rho^1)$ as $(D^1, \rho^1)$ indicating the various estimates are \epsilonmph{uniform} in $\tilde\rho$ or \epsilonmph{stable} under small $C^2$ perturbations of $\partial D$. We refer to $B_{r_2}$ as $D_{r_2}^2$ or $D^2$ and set $\rho^2=|\zeta|^2-r_2^2$ with restriction $r_1>r_2>r_{1}/2$. We still refer to $\psi(U)$ as $U$.
\epsilonnd{rem}
\begin{proof}
(a) Let $D$ be defined by $\rho^0<0$. We may assume that $\zeta=0$. Permuting coordinates yields $\rho^0_{z_n}\neq0$. Let $\tilde z_n=2\rho^0_\zeta\cdot(\zeta-z)-\sum \rho^0_{\zeta_j\zeta_k}(\zeta_j-z_j)(z_k-\zeta_k)$ and $\tilde z'=z'$. Then $\rho^0(z)=\rho_1(\tilde z)$, where the new domain has a defining function
$$
\rho_1(z)=-y_n+\sum a_{j\overline k}z_j\overline z_k+o(|z|^2).
$$
Choose a nonsingular matrix $U$ so that with $U\tilde z=z$ and $\tilde z_n=z_n$. The new defining function $\rho_2(\tilde z)=\rho_1(z)$ has the form $$\rho_2(z)=-y_n+\sum_{j<q}\lambda_j|z_j|^2+\sum_{j=q}^{n-1}|z_{j}|^2 +\sum_{j=1}^{n}\operatorname{Re}\{a_jz_j\overline z_n\}+o(|z|^2),
$$
where $a_n$ is a real constant. Setting $\rho_3=\rho_2+\mu\rho_2^2$ with $\mu=a_{n}+1$, we get
$$
\rho_3(z)=-y_n+\sum_{j<q}\lambda_j|z_j|^2+\sum_{j=q}^{n}|z_{j}|^2+\operatorname{Re}\Bigl\{\mu z_n^2+\sum_{j=1}^{n-1}a_jz_j\overline z_n\Bigr\}+o(|z|^2).
$$
Using new coordinates $\tilde z_n=z_n-i\mu z_n^2+iz_n\sum_{j=1}^{n-1}a_j$ and $\tilde z'=z'$,
we get
$$
\rho_4(z)=-y_n+\lambda_1|z_1|^2+\cdots+\lambda_{q-1}|z_{q-1}|^2+|z_{q}|^2+
\cdots+|z_{n}|^2+y_n\operatorname{Re}\{\sum_{j=1}^{n}b_jz_j\}+o(|z|^2).
$$
We get $|\lambda_j|<1$ after a dilation. Then $\rho_4+\rho_4\operatorname{Re}\{\sum b_jz_j\}$, renamed as $\rho^1$, has the form \re{qconv-nf}. As usual, the implicit function theorem can be proved by using the inverse mapping theorem~~\cite{MR0385023}*{pp.~224-225}. Then the inverse mapping theorem for Zygmund spaces~~\cite{GG} yields the desired smoothness of $R$.
The details are left to the reader. The transversality of $\partial D^1$ and $\partial D^2_{r_2}$ also follows from the computation below.
(b) The above construction of $\psi$ is explicit in $\rho^0$ with the exception of the linear change of coordinates $z\to Uz$ that is fixed for all small perturbations $\tilde\rho^0$ of $\rho^0$. Thus, it is easily to check that $\|\tilde R-R\|_a\leq C_a\|\tilde\rho^0-\rho^0\|_a$ for $a\geq2$ and $|\tilde R-R|_a\leq C_a|\tilde\rho^0-\rho^0|_a$ for $a>2$.
We want to show that $\nabla \tilde\rho^1$ is not proportional to $\nabla\rho^2$ on the common zero set of $\tilde\rho^1,\rho^2$. Suppose that
$\nabla\rho^2=\mu\nabla\tilde\rho^1$
when $\tilde\rho^1(z)=\rho^2(z)=0$. We get $2y_n=\mu (-1+2 y_n+\tilde R_{y_n})$. When $r<1/4$ and $\deltata(D)$ are sufficient small, by $|z|=r$ we obtain $|-1+2 y_n+\tilde R_{y_n}|<1/2$. Hence $-\mu^{-1}y_n\in(1/4,3/4)$ as
\epsilonq{}
\|\tilde R\|_{2}<1/C.
\epsilonnd{equation} For $j<n$, we have
$2y_j=2\mu\tilde\rho_{y_j}$. This shows that $|y_j|\leq C|y_n|$. Also, $|x_k|\leq C|y_n|$. Thus $\tilde\rho^1(z)=0$ implies $|y_n|\leq C'|y_n|^2+|\tilde R(z)|$. In view of $C'|y_n|<1/2$, we get
$$
|z|\leq C\|R\|_2|z|^2+C\delta(D).
$$
By choosing $\delta(D)$ depending on $r_1$, we get $|z|<r_1^2/C$. The latter contradicts the vanishing of $\rho^2(z)=r^2-|z|^2$ since $r>r_1/2$.
\epsilonnd{proof}
Recall that our original domain $D$ is normalized as $D^1$. We now fix notation. Let $(D^1,U,\phi,\rho^1)$ be as in \rl{convex-rho}. Thus $\rho^1$ is given by \re{qconv-nf} (or \re{qconv-nf-t}). Recall that
\epsilonq{rho2}
\rho^2=|z|^2-r^2_2
\epsilonnd{equation}
where $0<r_2<r_1$ and $r_1/2<r_2<r_1$ for \rl{convex-rho} (a), (b). Let us define
\begin{gather}{}\lambdabel{d12}
D^1\colon\rho^1<0, \quad D^2\colon\rho^2<0, \quad D^{12}=D^1\cap D^2,\\
\lambdabel{s12}
\partial D^{12}=S^1\cup S^2, \quad S^i\subset\partial D^i,\\
\lambdabel{g02}
g^2( z,\zeta )=\partial_{\zeta}\rho^2=\overline\zeta.
\epsilonnd{gather}
It is well-known that
\epsilonq{W2}
| g^2(z,\zeta)\cdot(\zeta-z)|>0, \quad (z,\zeta)\in D^2\times\partial D^2.
\epsilonnd{equation}
\le{}Let $(D^1,U,\phi,\rho^1)$ be as in \rla{convex-rho}. Define
\epsilonq{}\lambdabel{g1}
g^{1}_j( z,\zeta )=\begin{cases}
\mathbb{D}D{\rho^1}{\zeta_j},&q\leq j\leq n,\\
\mathbb{D}D{\rho^1}{\zeta_j}+(\overline\zeta_j-\overline z_j),& 1\leq j<q.
\epsilonnd{cases}
\epsilonnd{equation}
Then for $\zeta,z\in U$ and by shrinking $U$ if necessary, we have
\begin{gather}\lambdabel{W1-dist}
2\operatorname{Re}\{ g^1( z,\zeta )\cdot(\zeta-z)\}\geq \rho^1(\zeta)-\rho^1(z)+\f{1}{2}|\zeta-z|^2, \\
|g^1(z,\zeta)\cdot(\zeta-z)|\geq 1/C_*, \quad \forall\zeta\in S^{12}, |z|<\deltata_0(D).\lambdabel{g1z}\epsilonnd{gather}
\epsilonnd{lemma}
\begin{proof} We have
$
\operatorname{Re}\{ g^1( z,\zeta )\cdot(\zeta-z)\}=\operatorname{Re}\{ \rho^1( z,\zeta )\cdot(\zeta-z)\}+\sum_{j<q}|\zeta_j-z_j|^2.
$
By Taylor theorem, we have
\epsilonq{taylor}
\rho^1(z)-\rho^1(\zeta)=2\operatorname{Re}\{\rho^1_{\zeta}\cdot(z-\zeta)\}+
H_\zeta\rho^1(z-\zeta)+\operatorname{Re}\{\rho^1_{\zeta_j\zeta_k}(z_j-\zeta_j)(z_k-\zeta_k)\}+R( z,\zeta ).
\epsilonnd{equation}
Note that $H_\zeta\rho$, restricted to $(z_q,\cdots, z_n)$, is a positive definite quadratic form. Also, $\rho_{\zeta_j\zeta_k}$ and second-order derivatives of $R$ are small. We can show that
\begin{align}n\rho^1(z)-\rho^1(\zeta)
&\geq2\operatorname{Re}\{\rho^1_{\zeta}\cdot(z-\zeta)\}+\sum_{j\geq q}|\zeta_j-z_j|^2-c|\zeta-z|^2
\epsilonnd{align*}
where $c<1/2$. Now \re{W1-dist} follows from \re{qconv-nf}. By \re{taylor} and \re{W1-dist} for case \re{qconv-nf}, we also get \re{W1-dist} for the $\tilde\rho^1$ in
\re{qconv-nf-t} when $\|\tilde\rho^1-\rho^1\|_2\leq C\delta(D)$ is small. Note that \re{g1z} follows from \re{W1-dist}.
\epsilonnd{proof}
\begin{defn}\bpp \item As in~~\cite{MR986248}, the $(U,D^1,\psi,\rho^1, \rho^2)$ in \rl{convex-rho} (a) and \re{rho2} is called an {\it $(n-q)$-convex configuration}. \rl{convex-rho} (b) is referred to as the {\it stability} of the configuration. In brevity, we call $(D^1,D^2_{r_2})$ an $(n-q)$-convex configuration.
\item The $g^1,g^2$ given by \re{g02} and \re{g1} are called the {\it canonical Leray maps} for the $(n-q)$-convex configuration $(D^1,D^2_{r_2})$.
\epsilonpp
\epsilonnd{defn}
Note that $g^2(z,\zeta)$ is holomorphic in $z$ and $g^1(z,\zeta)$ is anti-holomorphic in merely $q-1$ variables of $z$. Checking the types, we have
\begin{gather}
\lambdabel{type-2}
\Omegaega_{0,k}^{2}( z,\zeta )=0, \quad k\geq1,\quad \overline\partial_z\Omega^2_{0,0}(z,\zeta)=0;\\
\lambdabel{type-1}
\Omegaega_{0,k}^{1}( z,\zeta )=0, \quad k\geq q;\quad \overline\partial_z\Omega_{0,q-1}^1( z,\zeta )=0;
\\
\lambdabel{type-12}
\Omegaega_{0,k}^{12}( z,\zeta )=0;\quad k\geq q;\quad \overline\partial_z\Omega_{0,q-1}^{12}( z,\zeta )=0.
\epsilonnd{gather}
By \re{type-2}-\re{type-12}, we have
\begin{gather}\lambdabel{L110}
L_i^if(z)=\int_{S^i}\Omega_{0,q}^i(z,\zeta)\wedge f(\zeta)=0, \ i=1,2; \quad L_{12}^{12}f=0;\\
\overline\partial_z\int_{U_1}\Omega_{0,q-1}^1(z,\zeta)\wedge Ef=0, \quad \int_{U_1}\Omega_{0,q}^1(z,\zeta)\wedge E\overline\partial f=0.
\epsilonnd{gather}
This shows that given by \re{hq2}, the $H^{(2)}_q$ can be written as
\epsilonq{nhq2}
H^{(2)}_qf=L_{1^+}^{01} Ef +L_{2}^{02} f+L_{12}^{012}f.
\epsilonnd{equation}
Therefore, we have obtained the following local homotopy formula.
\begin{thm}\lambdabel{hf-c} Let $ 0<q\leq n$.
Let $(D^1,D_{r_2}^2)$ be a $(n-q)$-convex configuration with Leray maps \rea{g02}-\rea{g1}.
Suppose that $U^1\subset{\bf C}^n\setminus\overline{D^1}$, $\partial U^1=S^1\cup S^1_+$ and $S_+^1\cap D^{12}=\epsilonmptyset$.
Suppose that $ f$ is a $(0,q)$ form such that $f$ and $\overline\partial f$ are in $C^1(\overline {D^{12}})$. Then on $D^{12}$
\begin{gather}\lambdabel{tsqf-c}
f= \overline\partial H_q f+ H_{q+1}\overline\partial f
\epsilonnd{gather}
with $H_q=H_q^{(1)}+H_q^{(2)}$. Here $H^{(1)}_q$ is defined by \rea{hq1} and
\begin{align}{}
\lambdabel{Hqv}
H_q^{(2)} f&:= \int_{S^1_+}\Omegaega_{0,q}^{01}\wedge Ef+\int_{ S^2}\Omega^{02}_{0,q-1}\wedge f+\int_{S^{1}\cap S^{2}}\Omegaega_{0,q}^{012}\wedge f.
\epsilonnd{align}
\epsilonnd{thm}
To be used later we remark that the integral kernel in $H^{(2)}_q$ is smooth, since $S^1_{+}, S^2$ and $ S^{12}$ do not intersect small neighborhoods of the origin in $\partial D^1$. Therefore, terms in $H^{(2)}_q$ can be estimated easily, while the main term $H_q^{(1)}$ in $H_q$ will be estimated in section~\ref{sec:1form}.
\setcounter{equation}{0}
\section{A local $\overline\partial$ solution operator for
$(q+1)$-concave configuration}
We recall again from \rp{hf} the approximate local homotopy formula for a $(0,q)$ form $f$:
\epsilonq{f=l11} f=L_1^1f+L_2^2f+L_{12}^{12}f+\overline\partial H_q f+H_{q+1}\overline\partial f.
\epsilonnd{equation}
As the strictly $(n-q)$-convex case, we will show that $L^1_1f, L^2_2f$ vanish when the boundary is $(q+1)$-concave and the Leray mappings $g_1,g_2$
are chosen appropriately. However, the boundary integral $L^{12}_{12}f$ may not vanish. We will show that this term is $\overline\partial$-closed for a $\overline\partial$-closed $f$, and this allows us to use a third Leray mapping to transform it into a genuine $\overline\partial$ solution operator
$f=\overline\partial H_qf$ for possibly different $H_q$.
Thus, the $(q+1)$ concavity is sufficiently to construct a $\overline\partial$ solution operator.
The presence of $L_{12}^{12}f$ will lead to a subtlety.
For a local homotopy formula for forms which are not necessarily $\overline\partial$-closed, we however need an {\it extra} negative Levi eigenvalue, which will be assumed at the end of the section.
The following is a restatement of \rl{convex-rho}, by considering the complement $(D^1)^c$ and $-\rho_0$ where $\rho_0$ defines $D^1$. We include the last assertion on $D^1_*$, which holds obviously. The $D^1_*$ has $C^\infty$ boundary and it will be useful to obtain a sharp regularity for $\overline\partial u=f$ when $f$ are $(0,1)$ forms and $\partial D$ is merely $C^2$.
\le{concave-rho}Let $D\subset U$ be a $C^2$ domain.
Suppose that $U\cap\partial D$ is strictly $(q+1)$-concave at $\zeta\in U\cap\partial D$. Let $D,\tilde D$ be domains in $U$ defined by $\rho^0<0,\tilde\rho^0<0$ respectively. There is a biholomorphic mapping $\psi$ with $\psi(\zeta)=0$ such that $D^1=\psi(D\cap U)$ and $\tilde D^1=\psi(\tilde D\cap U)$ are defined by $\rho^1<0$ and $\tilde\rho^1<0$ with
\begin{gather}\lambdabel{rho1-v}
\rho^1(z)=-y_{q+2}-|z_1|^2-\cdots-|z_{q+2}|^2+\lambda_{q+3}|z_{q+3}|^2+
\cdots+\lambda_n|z_n|^2+R(z),\\
\lambdabel{rho1-t-v}
\tilde\rho^1(z)=-y_{q+2}-|z_1|^2-\cdots-|z_{q+2}|^2+\lambda_{q+3}|z_{q+3}|^2+
\cdots+\lambda_n|z_n|^2+\tilde R(z).
\epsilonnd{gather}
Here $|\lambda_j|<1$ for $j>q+2$. Also assertions $(a),(b)$ in \rla{convex-rho} are valid.
Furthermore, $D^1\subset\psi(U)\colon\rho^1<0$ contains $D^1_*\subset B_r\colon\rho^1_*<0$ for
\epsilonq{rho1*}
\rho^1_*= -y_{q+2}-\f{1}{2}|z_1|^2-\cdots-\f{1}{2}|z_{q+2}|^2+2|z_{q+3}|^2+
\cdots+2|z_n|^2.
\epsilonnd{equation}
\epsilonnd{lemma}
As in \rl{convex-rho}, we rename $\psi(U)$ as $U$.
When $\rho^1$ has the form \re{rho1-v}, as in~~\cite{MR986248} define
\epsilonq{}\lambdabel{HL2pg81}
g^1_{j}( z,\zeta )=\begin{cases}
\mathbb{D}D{\rho^1}{z_j},&1\leq j\leq q+2,\\
\mathbb{D}D{\rho^1}{z_j}+\overline z_j-\overline \zeta_j,& q+3\leq j\leq n.
\epsilonnd{cases}
\epsilonnd{equation}
Note that this kind of Leray mapping was first used by Hortmann~~\cite{MR422688, MR627759} for strictly concave domains.
Then we have
\epsilonq{W1s-dist}
-2\operatorname{Re}\{g^1( z,\zeta )\cdot(\zeta-z)\}\geq \rho(\zeta)-\rho(z)+|\zeta-z|^2/C.
\epsilonnd{equation}
An essential difference between the $q$-convex and $(q+1)$ concave cases is that $g^1( z,\zeta )$ is no longer $C^\infty$ in $z$ when $\partial D$ is only finitely smooth. A useful feature is that $g^1(z,\zeta)$ is holomorphic in $\zeta_1,\dots, \zeta_{q+2}$.
As $(n-q)$-convex case, we use
$
D^2_{r_2}\colon
\rho^2(z):=|z|^2-r_2^2<0.
$
We still take Leray maps $g^0(z,\zeta)=\overline\zeta-\overline z$ and
$
g^2=(\f{\partial\rho^2}{\partial\zeta_1},\dots, \f{\partial\rho^2}{\partial\zeta_n})=\overline \zeta.
$
Denote by $\text{deg}_\zeta$ the degree of a form in $\zeta$. We get
\begin{gather}
\lambdabel{Ca-type-1}
\text{deg}_\zeta\, \Omegaega_{0,*}^{1}\leq 2n-q-2.
\epsilonnd{gather}
Therefore, we still have \re{L110}. Thus \re{f=l11} becomes
\epsilonq{f=l11+} f=L_{12}^{12}f+\overline\partial H_q f+H_{q+1}\overline\partial f.
\epsilonnd{equation}
However, unlike the $(n-q)$ convex case,
$
L_{12}^{12}f=\int_{S^{12}} \Omega^{12}_{0,q}\wedge f
$
may not be identically zero. Let us try to transform this integral via Stokes' formula. We intersect $D^1\cap D^2$ with a third domain
\epsilonq{defD3}
D^3\colon \rho^3<0, \quad 0\in D^3
\epsilonnd{equation}
where
\epsilonq{rho3}
\rho^3=-y_{q+2}+\sum_{j=q+3}^n3|z_j|^2-r^2_3
\epsilonnd{equation}{}
with $r_3>0$.
Define
\begin{gather}
\lambdabel{defW-3}
g^{3}_j( z,\zeta )=\begin{cases}
0,&1\leq j<q+2,\\
\f{i}{2},&j= q+2,\\
3(\overline\zeta_j+\overline z_j),& q+3\leq j\leq n.
\epsilonnd{cases}
\epsilonnd{gather}
We can verify
\epsilonq{HH}
\operatorname{Re}\{g^3(z,\zeta)\cdot(\zeta-z)\}=\rho^3(\zeta)-\rho^3(z).
\epsilonnd{equation}
Then
we have the following.
\le{y123} Let $\rho^i, g^i$ be defined by \rea{rho1-v}, \rea{HL2pg81}, \rea{rho2}, \rea{g02}, \rea{rho3}, \rea{defW-3} for $i=1,2,3$.
\bpp
\item
There exists $r_1\in(0,1/6)$ such that $\partial D^1,\partial D^2_{r_2},\partial D^3_{r_3}$ are pairwise in the transversal position when $0<C_nr_3<r_2<r_1$ and $r_1<2r_2$.
\item Let $\tilde\rho^0,\tilde\rho^1$ be as in \rla{y123}.
If $\delta(D)$ is sufficiently small, $\|\tilde \rho^0-\rho^0\|_2<\delta(D)$, and $1/{C_n'}<C_nr_3<r_2<r_1$ and $r_2<r_1/2$, then $\partial\tilde D^1,\partial D^2_{r_2},\partial D^3_{r_3}$ are also pairwise in general position.
\item Let $r_1,r_2,r_3$ be as in $(b)$. Then
\begin{gather}\lambdabel{gizC}\partial\tilde D^1\cap\partial D^2_{r_2}\cap D^3_{r_3}=\epsilonmptyset,\quad
\partial\tilde D^1\cap\partial D^3_{r_3}\cap\partial D^2_{r_2}=\epsilonmptyset,\\
|g^i(z,\zeta)\cdot(\zeta-z)|\geq 1/C, \quad \forall(z,\zeta)\in B_{r_4} \times \overline{D^2_{r_2}\setminus(D^1\cup D^3_{r_3})}, \ i=0,1,2,3,
\lambdabel{gicZ+}\\
S^{12}\subset \overline{D^2_{r_2}\setminus(D^1\cup D^3_{r_3})}.
\lambdabel{gicZ=}
\epsilonnd{gather}
\epsilonpp
\epsilonnd{lemma}
\begin{proof}
$(a)$ Suppose $\nabla\rho^1(z)=\mu\nabla\rho^3(z)$, and $\rho^1(z)=\rho^3(z)=0$. We have $-1+R_{y_{q+2}}=\mu$. This shows that $-3/4<\mu<-1/2$. We also have $z_j+o(|z|)=0$ for $j=1,\dots q+1$,
$x_{q+2}+o(|z|)=0$, and
$\lambda_jz_j+o(|z|)=3\mu z_j$
for $j>q+2$. The latter implies that $|z_j|=o(|z|)$ since $|3\mu|-|\lambda_j|>1/4$. Hence, $\rho^1(z)=0$ yields $y_{q+2}=o(|z|)$. This shows that $z=0$, which contradicts $\rho^3=r_3>0$.
To show $\partial D^2_{r_2}$ and $\partial D^3_{r_3}$ intersect transversally, suppose that at an intersection point $z$ we have $\nabla\rho^2=\mu\nabla\rho^3$. We first get $2y_{q+2}=\mu(-1+R_{y_{q+2}})$. Hence $1/4<-\mu^{-1}y_{q+2}<3/4$ and $|\mu|<4|y_{q+2}|$. Then $\nabla\rho^2=\mu\nabla\rho^3$ implies that $|z|\leq C_n|y_{q+2}|$. We get $|y_{q+2}|\geq r_2/{C_n}$ and $|y_{q+2}|\leq r_3+|z|^2\leq r_3+C_n^2|y_{q+2}|^2$. Then $2r_3\geq|y_{q+2}|\geq r_2/{C_n}$, a contradiction to the assumption $r_3<r_2/C$.
That $\partial D_{r}^1,\partial D_{r_2}^2$ intersect transversally is proved in \rl{convex-rho}.
$(b)$ We leave the details to the reader.
$(c)$ Suppose $\rho^1(z)=0$ and $\rho^3(z)<0$. Then we have
$$
|z_1|^2+\cdots+|z_{q+2}|^2+\sum_{j>q+2}(\lambda_j+3)|z_j|^2-\tilde R(z)<r_3^2.
$$
This shows that $|z|<2r_3$. We obtain the first identity in \re{gizC}. The proof of the second identity in \re{gizC} is similar.
We now verify \re{gicZ+}. The cases for $i=0,2$ are trivial. Case $i=1$ is proved in \re{g1z}. To verify \re{W1s-dist} for case $i=3$, by \re{HH}
we have
$$
\operatorname{Re}\{g^3(\zeta,z)\cdot(\zeta-z)\}=\rho^3(\zeta)-\rho^3(z).
$$
Note that $D^3$ is defined by $\rho^3<0$. Thus for $\zeta\not\in D^3_{r_3}$ and $z\in D^3_{r_3}$, we have $r^3(\zeta)-r^3(z)>0$. When $r_4<r_3/C$, $B_{r_4}$ is contained in $D^3_{r_3}$. This shows that when $z\in B_{r_4}$ and $\zeta\not\in D^3_{r_3}$ we have $\rho^3(\zeta)-\rho^3(z)>1/C$.
Now it is easy to see that $\operatorname{Re}(g^3(\zeta,z)\cdot(\zeta-z))>1/C$ when $|z|<1/C$.
Finally, \re{gicZ=} follows from \re{gizC}.
\epsilonnd{proof}
\begin{defn}\lambdabel{ccav} The $(D^1, U,\phi,\rho^1)$ in \rla{concave-rho} $(a)$
is called a {\it $(q+1)$-concave configuration}, while \rl{concave-rho} $(b)$ is referred as the {\it stability} of the configuration. In brevity, we call $(D^1,D^2_{r_2},D^3_{r_3})$ in \rla{concave-rho}, a $(q+1)$-concave configuration, in which $\partial D^1,\partial D^2_{r_2},\partial D^3_{r_3}$ intersect pairwise transversally when $1/{C_n'}<C_nr_3<r_2<r_1$ and $r_2<r_1/2$.
The $g^1,g^2,g^3$ in \rea{HL2pg81}, \rea{g02} and \rea{defW-3} are called the \epsilonmph{standard Leray maps} of the configuration.
\epsilonnd{defn}
So far, we have been following Henkin-Leiterer~~\cite{MR986248}. We could derive a homotopy formula on $D^1\cap D^2_{r_2}\cap D^3_{r_3}$ as in~~\cite{MR986248}. However, since we only need a local homotopy formula near a boundary point,
we now departure from the approach in~~\cite{MR986248}. Let us still use the approximate homotopy formula on $D^1\cap D^2_{r_2}$. Modify it only for $z\in D^1\cap D^2_{r_2}\cap B_{r_4}$ using mainly \re{gizC}-\re{gicZ+} to derive a homotopy formula on this smaller domain. Thus our starting point is still the approximate homotopy formula \re{f=l11}. We will however use Koppelman's lemma for $g^1,g^2,g^3$ on the set $S^{12}$.
Note that the anti-holomorphic differentials $d\overline\zeta_j,d\overline z_k$ appear in $\Omega^3$ as a wedge product in some of
$$
d(\overline\zeta_j+\overline z_j), \quad q+3\leq j\leq n.
$$
Consequently, $d\overline z_j$ and $d\overline\zeta_j$, having the \epsilonmph{same} index, cannot appear in $\Omega_{0,q}^3$ simultaneously. Therefore
\begin{gather}\lambdabel{type-3}
\text{deg}_\zeta\, \Omega^3_{0,\epsilonll}( z,\zeta )\leq n+([n-(q+3)+1]-\epsilonll)= 2n-q-\epsilonll-2, \quad\forall\epsilonll.
\epsilonnd{gather}
We now derive a result analogous to ~\cite{MR986248}*{p. 122, Lemma 13.6 $(iii)$} but for different boundary integrals $L^{\bigcdot}_{12}$.
\begin{lemma} Let $0<q\leq n-2$. Let $(D^1,D^2_{r_2},D^3_{r_3})$ be a $(q+1)$ concave configuration with Leray maps $(g^1,g^2,g^3)$. Then
\begin{gather}
\lambdabel{db13=0}
\Omega^{13}_{0,\epsilonll}( z,\zeta )=0,\quad\epsilonll<q;\quad \overline\partial_\zeta\Omega^{13}_{0,q}( z,\zeta )=0.
\epsilonnd{gather}
\bpp\item
Suppose that $f$ is $\overline\partial$-closed $(0,q)$ form on $D_1$ and $f$ is in $C^1(\overline D_1)$. Then on $D^1\cap D_{r_2}^2\cap D_{r_3}^3$,
\begin{gather}
\lambdabel{dL2312}L_{12}^{13}f(z)=0,\\
L_{12}^{12}f=-L_{12}^{23}f+\overline\partial L_{12}^{123}f+L_{12}^{123}\overline\partial f.
\lambdabel{L1212}
\epsilonnd{gather}
\item If $(D^1,D_{r_2}^2,D_{r_3}^3)$ is a $(q+2)$-concave configuration and $0<q\leq n-3$, then \rea{dL2312} and \rea{L1212} are valid for any $(0,q)$ forms $f\in C^1(\overline D_1)$.
\epsilonpp
\epsilonnd{lemma}
\begin{proof}To verify
\re{db13=0}, we note that $\Omega^{13}_{0,q}( z,\zeta )$ has type $(0,q)$ in $z$. It has type $(n,n-2-q)$ in $\zeta$ and it is holomorphic in $\zeta_1,\dots, \zeta_{q+2}$. After taking $\overline\partial_\zeta$, it has type $(n,n-q-1)$ in $\zeta$ for anti-holomorphic variables $\zeta_{q+3}, \dots, \zeta_n$. However, the number of these anti-holomorphic variables is $<n-(q+3)+2=n-q-1$. We have verified \re{db13=0}.
$(a)$
To verify \re{dL2312}, we need an approximation theorem by Henkin-Leiterer~~\cite{MR986248}*{Lemma 12.5 (iii), p. 122}. Fix $z\in B_{r_4}$. By \re{gicZ+}, we know that $\Omega^{13}$ is a continuous $(n,n-q-2)$ in $\zeta\in K:=\overline{D^2\setminus(D^1\cup D^3)}$. Consequently, we can find a sequence $\omega^\nu_z$ of $\overline\partial_\zeta$-closed continuous $(n,n-q-2)$-forms on $U\supset D^2$ such that $\omega^\nu_z$ is a sequence of $\overline\partial$-closed $(n,n-q-2)$ forms on $U$ converges to $\Omega^{13}(z,\cdot)$ uniformly on $K$. Using a standard smoothing, we may assume that $\omega^\nu_z$ are $C^1$ in $\zeta\in D^2$, $\overline\partial$-closed and approximate $\Omega^{13}(z,\cdot)$ uniformly on $K$. By \re{gicZ=}, $S^{12}$ is contained in $K$. Applying Stokes' formula for $\omega^\nu_z$
contained in $K$ and we obtain for $z\in D^1\cap D^2\cap B_{r_4}$,
$$
L_{12}^{13}f=\lim_{\nu\to\infty}\int_{S^{12}}\omega^\nu_z\wedge f=\lim_{\nu\to\infty}\int_{S^1}\omega^\nu_z\wedge \overline\partial_\zeta f=0.
$$
Now \re{L1212} follows from
Koppelman's lemma:
$$
\Omegaega_{0,q}^{12}=\Omegaega_{0,q}^{13}-\Omegaega_{0,q}^{23}+\overline\partial_\zeta \Omegaega_{0,q}^{123}+\overline\partial_z \Omegaega_{0,q-1}^{123}.
$$
$(b)$
Note that when $(D^1,D^2,D^3)$ is a $(q+2)$ concave configuration. We have $L^{13}_{12}f=0$ for any $(0,q)$ forms $f$ that are not necessarily $\overline\partial$-closed by \re{db13=0} that now holds for $\epsilonll<q+1$. We get the desired conclusion immediately.
\epsilonnd{proof}
\begin{remark}\lambdabel{no-control}
In the proof for case $(i)$, we do not have any control on $\omega^\nu$ outside $\overline {D_{r_2}^2\setminus(D^1\cup D_{r_3}^3)}$ other than the uniform convergence. Therefore, in $(a)$ it is crucial that $f$ is $\overline\partial$-closed.\epsilonnd{remark}
We recall the homotopy formula
$$
f=\overline\partial T_{q,B_{r_4}}f+T_{q+1,B_{r_4}}\overline\partial f
$$
on the ball $B_{\delta}$ centered at the origin with radius $\delta$.
We now transform $L_{12}^{23}f$ in \re{L1212}.
The following is analogous to ~\citete{MR986248}*{Lemma 13.7, p. 125} for $L_{23}^{23}$.
\begin{lemma} Let $1\leq q\leq n-2$. Let $(D^1,D^2_{r_2},D^3_{r_3})$ be a $(q+1)$-concave configuration. Let $L_{\bigcdot}^{\bigcdot}$ be defined by \rea{defnLR}.
For a $(0,q)$ form $f$, we have
\begin{gather}{}\overline\partial L^{23}_{12}f=\int_{S^{12}}\Omega^{23}_{0,q+1}\wedge \overline\partial f,\\
L^{23}_{12}f=\overline\partial T_{B_{r_4},q}L^{23}_{12}f+T_{B_{r_4}, q+1}L_{12}^{23}\overline\partial f, \quad \text{on $B_{r_4}$},\\
\lambdabel{MqL}
L_{12}^{23}\colon Z_{0,q}(C^0(\overline D))\to Z_{0,q}(C^0(\overline D^2\cap \overline D^3)\cap C^\infty(D^2\cap D^3)).
\epsilonnd{gather}
\epsilonnd{lemma}
\begin{proof}By \re{gicZ+} the form $\Omega^{23}$ is smooth in $z\in B_{r_4}$ and $\zeta\in S^{12}$. We have
$$
\overline\partial_\zeta\Omega_{0,q+1}^{23}+\overline\partial_z\Omega_{0,q}^{23}=\Omega_{0,q+1}^2-\Omega_{0,q+1}^3.
$$
By \re{type-2}, $\Omega_{0,q+1}^2=0$. Thus $\int_{S^{12}}\Omegaega_{0,q+1}^2( z,\zeta )\wedge f=0$.
By \re{type-3}, the degree of $f(\zeta)\wedge \Omega^3_{0,q+1}( z,\zeta )$ is less than $ 2n-3$, which is less than $\dim (S^1\cap S^2)$.
This shows that
\epsilonq{}
\int_{S^{12}}\Omega_{q+1}^3( z,\zeta )\wedge f(\zeta)=0, \quad q>0.
\epsilonnd{equation}
By Stokes' formula and $\partial(S^{12})=\epsilonmptyset$, we obtain
$$
\overline\partial L^{23}_{12}f=-\int_{S^{12}}\overline\partial_\zeta\Omega_{0,q+1}^{23}\wedge f=\int_{S^{12}} \Omega_{0,q+1}^{23}\wedge\overline\partial f.
\qedhere
$$
\epsilonnd{proof}
In summary, we have the following local $\overline\partial$-solution operator for the concave case.
\th{cchf-closed}Let $1\leq q\leq n-2$. Let $(D^1,D^2_{r_2},D^3_{r_3})$ be a $(q+1)$-concave configuration. Let $f$ be a $\overline\partial$-closed $(0,q)$ form on $D^{12}$. On $D^{12}\cap B_{r_4}$, we have \begin{gather}{}
f=\overline\partial H_qf
\lambdabel{tsqf+-cv-closed}
\epsilonnd{gather}
with $H_q=H_q^{(1)}+H_q^{(2)}+H_q^{(3)}$. Here $ H^{(1)}_q$ and $H^{(2)}_q$ are given by \rea{hq1}, \rea{nhq2} and
\begin{align} H_{q}^{(3)}f=
L_{12}^{123}f+T_{B_{r_4}, q}L^{23}_{12} f.
\epsilonnd{align}
\epsilonth
Although it is not used in this paper, for potential applications, it is worthy to state the following local homotopy formula if we have an extra negative Levi eigenvalue: if $\partial D^1$ is strictly $(q+2)$ concave, then
$
L_{12}^{23}f=0
$
by \re{db13=0} for a $(0,q)$-form $f$.
Therefore, we have the following.
\th{cchf} Let $1\leq q\leq n-3$. Let $(D^1,D^2_{r_2},D^3_{r_3})$ be a $(q+2)$-concave configuration. Let $D^{12}=D^1\cap D^2_{r_2}$. Let $f$ be a $(0,q)$ form on $D^{12}$. On $D^{12}\cap B_{r_4}$, we have \begin{gather}{}
f=\overline\partial H_qf+ H_{q+1}\overline\partial f
\epsilonnd{gather}
with $H_q=H_q^{(1)}+H_q^{(2)}+H_q^{(3)}$. Here $ H^{(1)}_q$ and $H^{(2)}_q$ are given by \rea{hq1}, \rea{nhq2} and
\begin{align} H_{q}^{(3)}f=
L_{12}^{123}f+T_{B_{r_4},q}L^{23}_{12} f.
\epsilonnd{align}
\epsilonth
As in the convex case, the integral kernels in $H^{(2)}_q, H^{(3)}_q$ have smooth kernels, since $S^1_{+}, S^2, S^{12}$ do not intersect small neighborhood of the origin in $\partial D^1$. There is another main difference between the convex and concave cases. The kernel for the latter is only $\Lambda^{m-1}$ away from the singularity when $\partial D\in \Lambda^{m}$. Anyway $H_q^{(1)}$ is the main term to be estimated.
\setcounter{equation}{0}
\section{H\"{o}lder-Zygmund spaces and a Hardy-Littlewood lemma}\lambdabel{h-space}
In this section, we recall the H\"{o}lder-Zygmund norms and indicate how a Hardy-Littlewood lemma can be used to derive the estimates. This version of Hardy-Littlewood lemma allows us to simplify some estimation in~~\cite{MR3961327}.
Let $\Omega$ be a domain in ${\bf R}^n$. Denote by $\|u\|_{D,a}$ the H\"older norm on a domain $D$ for $a\geq0$. Define $\mathbb{D}el_hf(x)=f(x+h)-f(x)$ and
$\mathbb{D}el^2_h f(x)=f(x+2h)+f(x)-2f(x+h)$.
Following~~\cite{MR2250142}*{Defn. 1.120, p.~76},
define the space ${\Lambda}^r(\Omega)$ for $r>0$, to be the set of functions $f$ with finite norm
\epsilonq{firstLambdar}
|f|_{{\Lambda}^r(\Omega)}=\sup_{
x, x-h,x+h\in\Omega, h\neq0;|\begin{align}l|<r}\left\{|D^\begin{align}l f(x)|+ \f{|\mathbb{D}el^2_hD^\begin{align}l f(x)|}{|h|^{r-|\begin{align}l|}}\right\}.
\epsilonnd{equation}
Denote by $\Lambda_{loc}^r(\Omega)$ the space of functions $f$ such that $f\in\Lambda^r(\Omega')$ for any relatively compact subdomains $\Omega'$
of $\Omega$.
\le{HL}Let $0\leq\beta<1$.
Let $ D\subset{\bf R}^n$ be a bounded and connected Lipschitz domain. Suppose that $f$ is in $C_{loc}^{2}( D)$ and
$$
|\partial^{2}f(x)|\leq A\operatorname{dist}(x,\partial D)^{\beta-1}.
$$
Fix $x_0\in D$. Then
$
|f|_{ D;{1+\beta}}\leq C_0(|f(x_0)|+|\nabla f(x_0)|)+C_\beta A,
$
where constants $C_0,C_\beta$ depend on a finite set of Lipschitz graph defining functions of $\partial D$.
\epsilonnd{lemma}
\begin{proof}The proof is standard for $0<\beta<1$.
Suppose $\beta=0$. The assumption implies that the second-order derivatives of $f$ is bounded on each compact subset of $ D$ and it suffices to estimate $\mathbb{D}el_h^2f(x)$ when $x$ is close to the boundary and $h\in{\bf R}^n$ is small.
Take a boundary point of $\partial D$. We may assume that $x_0=0$. By definition, we may assume that $ D$ is defined by $x_n>g(x')$ where $g$ is a Lipschitz function satisfying
$|g(\tilde x')-g(x')|\leq L|\tilde x'-x'|$ with $L>1$. Then
$$
\operatorname{dist}(y,\partial D)\geq (y_n-g(x'))/L.
$$
Suppose that $x,x-h, x+h$ are in $ D$ and close to the origin.
We first consider the special case when $x,h$ satisfy
\epsilonq{spcs}
x+th\in D, \quad \operatorname{dist}(x+th,\partial D)\geq |h|/L, \quad \forall t\in[0,2].
\epsilonnd{equation}
Set $u(s)=f(x+h+sh)+f(x+h-sh)-2f(x+h)$. Thus $u(0)=u'(0)=0$ and $|u''(s)|\leq C_nLA|h|$. This shows that $|u(1)|\leq C_nLA|h|$.
For the general case, set $\tilde h=(0',3L|h|)$. When $y\in D$ is close to the origin and $t\in[0,2]$, we have
\begin{gather}\lambdabel{yavn}
y+t\tilde h\in D,\quad y+\tilde h+th\in D,\\
\operatorname{dist}(y+\tilde h+th,\partial D)\geq |h|.\lambdabel{yavn+}
\epsilonnd{gather}
Decompose
\begin{align}\lambdabel{thedecom}
f(x+2h)&+f(x)-2f(x+h)\\
\nonumber
&=2f(x+2h+\tilde h)+2f(x+\tilde h)-4f(x+h+\tilde h)\\
\nonumber
&\quad - f(x+2\tilde h)-f(x+2h+2\tilde h)+2f(x+h+2\tilde h)\\
\nonumber
&\quad + f(x)+f(x+2\tilde h)-2f(x+\tilde h)\\
\nonumber
&\quad+ f(x+2h)+f(x+2h+2\tilde h)-2f(x+2h+\tilde h)\\
\nonumber
&\quad - 2f(x+h)-2f(x+h+2\tilde h)+4f(x+h+\tilde h).
\epsilonnd{align}
We estimate each row on the right-hand side of \re{thedecom}. Let us denote by $[a,b]$ the line segment connecting two points $a,b$ in ${\bf R}^n$.
By \re{yavn} and \re{yavn+}, we have
$$
[x+\tilde h,x+\tilde h+2h]\subset D, \quad
\operatorname{dist}(x+\tilde h+t h,\partial D)\geq|h|,
$$
for $t\in[0,2]$.
Therefore, we can estimate the first row using the estimation for the special case \re{spcs} in which $x$ is replaced by $x+\tilde h$. The second row is estimated similarly. For the third row, by \re{yavn+} and $x\in D$, we get
$$
\operatorname{dist}(x+\tilde h+s\tilde h,\partial D)\geq (1+s)|h|\geq (1-|s|)| h|, \quad s\in[-1,1].
$$
Take $u(s)=f(x+\tilde h+s\tilde h)+f(x+\tilde h-s\tilde h)-2f(x+\tilde h)$. This yields $|u''(s)|\leq C_n AL| h|/{(1-s)}$ for $s\in[0,1]$ and
$|u(1)|\leq C_nAL| h|$ by
$
u(1)=\int_{0}^1(1-s)u''(s)\, ds$.
This gives us the desired estimate for the third row. The last two rows in \re{thedecom} can be estimated similarly.
\epsilonnd{proof}
We remark that the decomposition \re{thedecom} shows the following.
\begin{prop}\lambdabel{equiv-norms}
Let $ D$ be a bounded Lipschitz-graph domain in ${\bf R}^n$.
Set
$$
D_{h}=\{x\in D\colon [x,x+2h]\subset D\}, \quad \forall h\in{\bf R}^n.
$$
The norm ~\rea{firstLambdar} is equivalent to
$$
|f|_{\widetilde{\Lambda}^r( D)}=\sup_{x\in D}|f(x)|+\sum_{|\begin{align}l|<r}\sup_{h\in{\bf R}^N\setminus\{0\}}\sup_{x\in D_{h,2}}\f{|\mathbb{D}el^2_hD^\begin{align}l f(x)|}{|h|^{r-|\begin{align}l|}}.
$$
In other words, $c|f|_{\widetilde{\Lambda}^r( D)}\leq
|f|_{{\Lambda}^r( D)}\leq C|f|_{\widetilde{\Lambda}^r( D)}$. Here one can take $c=1$ and $C$ a constant depending on finitely many graph Lipschitz functions defining $\partial D$.
\epsilonnd{prop}
There are other equivalent norms for $\Lambda^r$.
For other equivalent norms of $\Lambda^r$, see
~\cite{MR0487423}*{Thm. 1} and ~\cite{MR3961327}.
\setcounter{thm}{0}\setcounter{equation}{0}
\section{$\f{1}{2}$-gain estimates for local homotopy operators}\lambdabel{sec:1form}
In this section, we derive the estimates for homotopy operators. We will give precise estimates which are potentially useful for applications. We remark that the local estimates do not require the forms to be $\overline\partial$ closed.
We first consider the $(n-q)$ convex case. In this case the result is essentially in~~\cite{MR3961327}. We simplified proof by using \rl{HL} which is applicable for $C^2$ domains.
\begin{thm}\lambdabel{conv-est} Let $r\in(1,\infty)$ and $1\leq q\leq n-1$.
Let $(D^1,D^2)$ be a $(n-q)$-convex configuration.
The homotopy operator $H_q$ in \rta{hf-c} satisfies
\epsilonq{hqfr12-c}
|H_q\varphi|_{D^{1}\cap D^2_{r_3},r+1/2}\leq C( \partial\rho^1, \partial^2\rho^1)|\varphi|_{D^{12},r}, \quad r_1/2<r_3<3r_2/4.
\epsilonnd{equation}
\epsilonnd{thm}
\begin{proof}
We now derive our main estimate. We will also simplify the proof in~~\cite{MR3961327} by using~\rl{HL}. Recall the homotopy operator
$$
H_{q}\varphi=(H^{(1)}_q+H^{(2)}_q)\varphi
$$
with
$$
H^{(1)}_q\varphi=R_{U_1\cup D^{12}}^0 E \varphi+R_{U_1 }^{01}[\overline\partial,E] \varphi.
$$
Near the origin, each term in $H^{(2)}_q\varphi$ given by \re{Hqv} is a linear combination of integrals of the form
$$ Kf(z):=\int_{S^I}\f{A(\partial\rho^1_\zeta,\partial^2_\zeta\rho^1,\zeta, z) f(\zeta )}{(g^1\cdot(\zeta-z))^a(g^2\cdot(\zeta-z))^b |\zeta-z|^{2c}}\, dV
$$
where $f$ is the coefficients of $\varphi$, $a,b,c$ are integers. Note that $S^I$ is one of $S^1_+, S^{12}, S^2$. Therefore, the kernel of $K$ is smooth for $z$ close to the origin. We have
$
| Kf|_{r+1/2}\leq C_r\|f\|_{0}.
$
Here and in what follows $C_r$ denotes a constant depending on $\rho,\partial\rho,\partial^2\rho$.
We now estimate the main term $H^{(1)}\varphi$.
Decompose it as
\epsilonq{dbECV}
H^{(1)}\varphi=\int_{D^{12}\cup U^1}\Omega_{0,q}^{0}(z,\zeta)\wedge E\varphi(\zeta)+ \int_{\mathcal U\setminus D}\Omega_{0,q}^{01}(z,\zeta)\wedge[\overline\partial,E]\varphi(\zeta).
\epsilonnd{equation}
Denote the first integral by $K_1\varphi$. By estimates on Newtonian potential~~\cite{MR999729}, we have
$$
|K_1\varphi|_{r+1/2}\leq C|\varphi|_{r-1/2}.
$$
Note that the above is proved in~~\cite{MR999729} when $r$ is not an integer. When it is an integer, the estimate follows the interpolation for the Zygmund spaces.
The last integral in \re{dbE} can be written as a linear combination of
\begin{gather}\lambdabel{defnKfCV}
K_2f:=\int_{\mathcal U\setminus D} f(\zeta )\f{A(\partial\rho^1_\zeta,\partial^2_\zeta\rho^1,\zeta, z)N_{1}(\zeta-z )}
{\Phi^{n-j}(z,\zeta)|\zeta -z |^{2j}}\, dV(\zeta), \quad 1\leq j<n,\\
\Phi(z,\zeta)=g^1(z,\zeta)\cdot(\zeta-z)
\epsilonnd{gather}
where $f$ is a coefficient of the form $[\overline\partial,E]\varphi$ and hence
\epsilonq{}
|f|_{r-1}\leq C|\varphi|_r.
\epsilonnd{equation}
Here and in what follows, $N_m(\zeta)$ denotes a monomial of $\zeta$ with degree $m\geq0$. Note that $f$ vanishes on $\overline D$.
Fix $\zeta_0\in\partial D_1$. We first choose local coordinates such that $s_1(\zeta),s_t(\zeta),t(\zeta)=(t_3,\dots, t_{2n})(\zeta)$ vanishing at $\zeta_0$, $D^1$ is defined by $s_1<0$, and
\begin{gather}\lambdabel{LbPhi}
|\Phi(z,\zeta)|\geq c_*(d(z)+ s_1(\zeta)+|s_2(\zeta)|+|t(\zeta)|^2),\\
|\Phi(z,\zeta)|\geq c_*|\zeta-z|^2,\quad |\zeta-z|\geq c_* |t(\zeta)|.
\lambdabel{LbPhi+}
\epsilonnd{gather}
Let $r=k+\begin{align}l$ with integer $k\geq1$ and $0<\begin{align}l\leq1$.
Consider first the case that $0<\begin{align}l<1/2$. By fundamental theorem of calculus, we have $|f(\zeta)|\leq C|f|_{r}\operatorname{dist}(\zeta,\partial D_1)^{r-1}$. We have
$|\partial^{k+1}Kf(z)|\leq C_r|f|_{r-1}I(z)$, where
\epsilonq{dk+2}
I(z):=\int_{[0,1]\times[-1,1]^{2n-1}}\f{s_1^{r-1}d s_1ds_2dt}{(d(z)+ s_1+|s_2|+|t|^2)^{(n-j)+a}(s_1+|s_2|+|t|)^{2j+b-1}}
\epsilonnd{equation}
with $a+b=k+1$ and $1\leq j<2n$. The worst term occurs when $j=n-1$ and $a=k+1$. Therefore, using polar coordinates for $t(\zeta)$ we obtain $I(z)\leq C \tilde I(z)$ for
$$
\tilde I(z):= \int_{[0,1]^3}\f{s_1^{r-1}d s_1ds_2dt}{(d(z)+ s_1+s_2+t^2)^{k+2}}.
$$
We have $\tilde I(z)\leq
\hat I_\begin{align}l(z)$ for
\begin{align}
\hat I_\begin{align}l(z):= \int_{t=0}^1\int_{s_1=0}^1\int_{s_2=0}^1 \f{s_1^{\begin{align}l-1}\, ds_1ds_2dt}{(d(z)+ s_1+s_2+t^2)^2}
\leq Cd(z)^{(\begin{align}l+1/2)-1}
\epsilonnd{align}
because $\begin{align}l-1/2<0$. Here we leave the proof of last inequality to the reader, or see similar arguments in the proof of \rl{henint} below.
When $\begin{align}l+1/2\geq1$, we use $|\partial^{k+2}Kf(z)|\leq |f|_{r-1} I(z)$ defined by \re{dk+2} in which $a+b=k+2$. Then $I(z)\leq C\tilde I _1 (z)$ for
\epsilonq{}
\tilde I _1 (z):= \int\f{s_1^{r-1}d s_1ds_2dt}{(d(z)+ s_1(\zeta)+|s_2(\zeta)|+|t(\zeta)|^2)^{k+3}}.
\epsilonnd{equation}
We now have $\tilde I_1(z)\leq\hat I_{\begin{align}l-1}(z)\leq d(z)^{(\begin{align}l+1/2)-2}$ because $(\begin{align}l+1/2)<2$.
\epsilonnd{proof}
\le{henint}Let $\beta,\mu_1\in[0,\infty)$, $\lambda\in{\bf R}$, and $0<\delta<1$. Suppose
\epsilonq{betap}
\beta':=\beta-\lambda+1+\min\{ 0, (\lambda-\mu_1+1)/2-\epsilon\}<0
\epsilonnd{equation}
and $\epsilon>0$. Then
\epsilonq{the-int}
\int_0^1\int_0^1\int_{t\in[0,1]^{m}}\f{ s_1^\beta(\delta+s_1+s_2+|t|^2)^{-1-\mu_1}}{(\delta+s_1+s_2+|t|)^{m-1+\lambda-\mu_1}
}\, ds_1ds_2dt
<
\begin{cases}
C\delta^{\beta'},& \beta'<0;\\
C, &\beta'>0.
\epsilonnd{cases}
\epsilonnd{equation}
\epsilonnd{lemma}
\begin{proof} We consider the integral in the following regions.
(i) $s_2>\max\{s_1,\deltata,|t|\}$. On this region the integral is less than
$$
\int_{s_2>\delta}\int_{0<s_1<s_2}\int_{|t|<s_2}\frac{s_1^{\beta}}{s_2^{m+\lambda }}\, dt ds_1ds_2\leq C\delta^{\beta-\lambda +2}
$$
if $\beta<\lambda -2$. Also the integral bounded by a constant if $\beta>\lambda-2$. The same bounds can be obtained for the integral on regions $(ii)$ $s_1>\max\{\deltata,s_2,|t|\}$ and $(iii)$ $\delta>\max\{s_1,s_2,|t|\}$.
$(iv).$ $|t|^2>\min\{\delta,s_1,s_2\}$. On this region, the integral is less than
$$
\int_{\rho>\sqrt\delta}\int_{s_1<{\rho}^2}\int_{s_2<{\rho}^2}\frac{s_1^{\beta}\, ds_1ds_2d\rho}{\rho^{2(1+\mu_1)+\lambda -\mu_1}}\leq \int_{\rho>\sqrt\delta}\rho^{2\beta-\mu_1-\lambda }\, d\rho<C\delta^{\beta-(\mu_1+\lambda -1)/2}.
$$
$(v). \ |t|^2<\delta+s_1+s_2<|t|$. On this region, the integral is less than
\epsilonq{triple}
\int_{(s_1,s_2)\in[0,1]^2}\int_{ \delta+s_1+s_2}^{\sqrt{\delta+s_1+s_2}} \frac{s_1^{\beta}\, d\rho ds_1ds_2}{(\delta+s_1+s_2)^{1+\mu_1}\rho^{\lambda -\mu_1}}.
\epsilonnd{equation}
Suppose $\lambda-\mu_1>1$. The latter is less than
$$
\int_{(s_1,s_2)\in[0,1]^2} \frac{s_1^{\beta}\, ds_1ds_2}{(\delta+s_1+s_2)^{\lambda }}<C\delta^{\beta-\lambda +1},
$$
for $\beta-\lambda +1<0$. When $\beta-\lambda +1\geq0$, the integral \re{triple} is less than
$$
\int_{(s_1,s_2)\in[0,1]^2} \frac{s_1^{\beta}(|\ln (\delta+s_1+s_2)|+(\delta+s_1+s_2)^{\f{\mu_1-\lambda +1}{2}})}{(\delta+s_1+s_2)^{1+\mu_1}}\, ds_1ds_2,
$$
which is less than $C_\epsilon\delta^{\beta-(\mu_1+\lambda -3)/2-\epsilon}$
if
$$
\beta-(\mu_1+\lambda -3)/2-\epsilon<0.
$$
Thus we can take $\beta'=\min\{\beta-\lambda +1, \beta-(\mu_1+\lambda -3)/2-\epsilon\}$, which is \re{betap}. We left the reader to check that when $\beta'>0$ the integral \re{the-int} is bounded above by a constant.
\epsilonnd{proof}
We need the following
\begin{gather}{}
\lambdabel{Zcov}
|uv|_a\leq C(|u|_a\|v\|_0+\|u\|_0|v|_a),\quad a\in(0,\infty);\\
\lambdabel{Hcov}
\|u\|_{a+b}\|v\|_{c+d}\leq C_{a,b,c,d}(\|u\|_{a+b+d}\|v\|_d+\|u\|_a\|v\|_{c+b+d})
\epsilonnd{gather}
for $a,b,c,d\in[0,\infty)$. For \re{Zcov}, see ~\cite[Thm 2.86, p. 104]{MR2768550}; for \re{Hcov}, see~~\citete{MR2829316}.
\begin{thm}\lambdabel{concave-est} Let $r\in(1,\infty)$ and $1\leq q\leq n-2$. Let $(D^1,D^2,D^3)$ be a $(q+1)$-concave configuration.
The homotopy operators $H_q$ in Theorems~$\ref{cchf-closed}$ and $\ref{cchf}$ satisfy
\begin{gather}\lambdabel{hqfr12}
|H_q\varphi|_{D^{1}\cap D^2_{r_4}, r+1/2}\leq C_r(\partial\rho,\partial^2\rho) (|\rho^1|_{r+5/2}\|\varphi\|_{1}+ |\varphi|_{D^{12},r})
\epsilonnd{gather}
for $r_1/2<r_4<3r_3/4.$ Moreover, $C_r(\partial\rho,\partial^2\rho)$ is upper-stable under a small $C^2$ perturbation of $\rho$
\epsilonnd{thm}
\begin{proof}
We now derive our main estimates. Recall the homotopy operator
$$
H_{q}\varphi=(H^{(1)}_q+H^{(2)}_q+H^{(3)}_q)\varphi
$$
where $\varphi$ has type $(0,q)$ and
$$
H^{(1)}_q\varphi=R_{U_1\cup D^{12}}^0 E \varphi+R_{U_1 }^{01}[\overline\partial,E] \varphi.
$$
Near the origin, each term in $H^{(2)}_q\varphi, H^{(3)}_q\varphi$ is a linear combination of integrals of the form
$$
K_0f_0(z)=A(\partial^2_z\rho^1)\tilde K_0f_0
$$
with $f_0$ being a coefficient of $\varphi$, and $A(\partial^2_z\rho)$ being a polynomial in derivatives of $\rho$ of order at most two. Here $\tilde K_0$, which involves only $\partial\rho^1$,
is defined by
$$\tilde K_0f_0(z):=\int_{S^I} f_0(\zeta )(g^1\cdot(\zeta-z))^a(g^2\cdot(\zeta-z))^b(g^3\cdot(\zeta-z))^c|\zeta-z|^{2d}(\operatorname{Re}\zeta,\operatorname{Im}\zeta)^e\, dV
$$
where $a,b,c,d$ are negative integers and $S^I$ is one of $S^1_+, S^{12}, S^2$. Therefore, for the $g^i$ that appear in the kernel, we have
$$
|g^i\cdot(\zeta-z)|\geq c_0
$$
when $|z|$ is sufficiently small and $\zeta\in S^I$. By \re{Zcov}, we have
$
|\tilde K_0f_0|_{r+1/2}\leq C_r|\rho^1|_{r+3/2}\|f_0\|_{0}
$
and
\begin{gather}{}
|K_0f_0|_{r+1/2}\leq C|\rho^1|_{r+5/2}\|\tilde K_0f_0\|_{0}+C |\tilde K_0f_0|_{r+1/2}.
\epsilonnd{gather}
Here and in what follows $C_r$ denotes a constant depending on $\rho,\partial\rho,\partial^2\rho$.
Therefore, we have
$$
|K_0f_0|_{r+1/2}\leq C_r|\rho^1|_{r+5/2}\|f\|_0.
$$
We now estimate the main term $H^{(1)}\varphi$. Decompose it as
\epsilonq{dbE}
H^{(1)}\varphi=\int_{D^{12}\cup U^1}\Omega_{0,q}^{0}(z,\zeta)\wedge E\varphi(\zeta)+ \int_{\mathcal U\setminus D}\Omega_{0,q}^{01}(z,\zeta)\wedge[\overline\partial,E]\varphi(\zeta).
\epsilonnd{equation}
Denote the first integral by $K_1\varphi$. By estimates on Newtonian potential, we have
$$
|K_1\varphi|_{r+1/2}\leq C|\varphi|_{r-1/2}.
$$
The last integral in \re{dbE} can be written as a linear combination of
\begin{gather}\lambdabel{defnKf}
K_2f(z):=\partial_z^2\rho^1\tilde K_2f
\epsilonnd{gather}
where $f$ is a coefficient of the form $[\overline\partial,E]\varphi$ and hence
\epsilonq{}
|f|_{r-1}\leq C|\varphi|_r.
\epsilonnd{equation}
Also $\partial^2\rho^1$ indicates a derivative of order at most two; and
\begin{gather}{}
\tilde K_2f:=\int_{\mathcal U\setminus D} f(\zeta )\f{A( z,\zeta )N_{1}(\zeta-z )}
{\Phi^{n-j}(z,\zeta)|\zeta -z |^{2j}}\, dV(\zeta), \quad 1\leq j<n,\\
\Phi(z,\zeta)=g^1(z,\zeta)\cdot(\zeta-z).
\lambdabel{defnKf+}
\epsilonnd{gather}
Here and in what follows, $N_m(\zeta)$ denotes a monomial of $\zeta$ with degree $m\geq0$. Also, $A( z,\zeta )$ is a monomial in $ z,\zeta $. Note that $f$ vanishes on $\overline D$.
We have
$$
|K_2f|_{r+1/2}\leq C_r(|\rho^1|_{r+5/2}\|\tilde K_2f\|_0+ |\tilde K_2f|_{r+1/2}).
$$
We have $\|\tilde K_2f\|_{1/2}\leq C\|f\|_0$, by the estimate in \rt{conv-est}. For later purpose we note that this also gives us
\epsilonq{C0est}
\|K_2f\|_0\leq C\|f\|_0.
\epsilonnd{equation}
The rest of the proof is devoted to the proof of
\begin{gather}\lambdabel{tK2f}
|\tilde K_2f|_{r+1/2}\leq C_r( \|\rho^1\|_{r+2}\|f\|_0+ |f|_{r-1}), \quad r>1.
\epsilonnd{gather}
Then combining above estimates yields the proof for \re{hqfr12}.
A technical difficulty to prove \re{tK2f} is that we do not have a version of \re{Hcov} for Zygmund norms. Therefore, we must use \re{Hcov} as a substitute to treat Zygmund norms. The computation is tedious and our main observation is that the kernel of $\tilde K$ involves only $\partial^1\rho$ instead of $\partial^2\rho$. Therefore, \re{Hcov} is still good enough to derive \re{tK2f} which is a crude estimate.
Let $r=k+\begin{align}l$ with integer $k\geq1$ and $0<\begin{align}l\leq1$.
In the following cases, we will apply \rl{henint} several times. For clarity, we will call values $(\beta,\beta',\lambda)$ as $(\beta_i,\beta_i',\lambda_i)$ when we use the lemma.
(i) $0<\begin{align}l<1/2$. Recall that the kernel of $K_2$ involves only first-order derivatives of $\rho^1$ in $z$-variables and it does not involve $\zeta$-derivatives of $\rho^1$. The expansion of $\Phi^1$ contains some $\zeta_j-z_j$. Since $\rho^1\in \Lambda^{k+\begin{align}l+5/2}\subset C^{k+2}$, we therefore express $\partial_z^{k+1}\tilde K_2f$ as a sum of
\epsilonq{rhotimesK}
K_{\mu,\nu}^{(k+1)}f:= \partial^{1+\nu'_1}_z\rho^1\cdots\partial_z^{1+\nu'_{\mu_1}}\rho^1K_\mu f
\epsilonnd{equation}
with
\epsilonq{Kmuf}
K_\mu f(z):=\int_{U^1}\f{f(\zeta)N_{1-\mu_0+\mu_1-\nu_1''-\cdots-\nu_{\mu_1}''+\mu_2}(\zeta-z)}
{(\Phi^1(z,\zeta))^{n-j+\mu_1}|\zeta-z|^{2j+2\mu_2}}dV.
\epsilonnd{equation}
Here $1\leq j<n$,
$\nu_i''=0,1$, and
\epsilonq{sumk1}
\mu_0+\mu_1+\mu_2+\sum(\nu'_i+\nu''_i)\leq k+1.
\epsilonnd{equation}
To estimate \re{rhotimesK}, we use the facts that $z\in D$ and $\zeta\in U$ and
$$
C|\zeta-z|\geq |\Phi^1(\zeta,z)|\geq \operatorname{dist}(z,\partial D)+\operatorname{dist}(\zeta,\partial D)+|\operatorname{Im}\Phi^1(\zeta,z)|+|\zeta-z|^2.
$$
Consequently, the worst term for $K_\mu f$ occurs when $j=n-1$, which we now assume.
We want to show that
\epsilonq{Kmuf}
|K_\mu f(z)|\leq |f|_{\beta_1}\operatorname{dist}(z)^{\beta_1'}, \quad\beta_1':=\begin{align}l-1/2
\epsilonnd{equation}
where $\beta_1\geq0$ is to be specified.
The case all $\nu'_i=0,1$ can be estimated as in \rt{conv-est},
for $\beta_1=\mu_0+\mu_1+\mu_2-1+\begin{align}l\leq r-1$.
Thus we may assume that all $\nu'_i\geq2$ for $1\leq i\leq\mu_1'$, $\nu'_i=0,1$ for $i>\mu_1'$, and $\mu_1'>0$. Define
\epsilonq{defla}
\lambda:=\mu_0+\mu_2+\sum\nu_i''.
\epsilonnd{equation}
We have $\mu_1\geq\mu_1'\geq1$.
By \rl{henint} with
$
\beta_1'=\begin{align}l-1/2<0,
$
we need to find $\beta_1\geq0$ satisfying
\epsilonq{defbp}
\beta_1'\leq\beta_1-\lambda+1+\min\{0,(\lambda-\mu_1+1)/2-\epsilon\}<0.
\epsilonnd{equation}
Thus we take
\epsilonq{defbet}
\beta_1=\max\{0,\begin{align}l-3/2+\lambda -\min\{0,(\lambda -\mu_1+1)/2-\epsilon\}\}.
\epsilonnd{equation}
We obtain
\begin{align}n{}
|K_{\mu,\nu}^{(k+1)} f(z)|&\leq C\|\rho^1\|_{1+\nu_1}\cdots\|\rho^1\|_{1+\nu_{\mu_1}}\|f\|_{\beta_1}
\operatorname{dist}(z,\partial\Omega)^{\begin{align}l-1/2}.
\epsilonnd{align*}
For $i>\mu_1'$, we use $\|\rho^1\|_{1+\nu'_i}\leq \|\rho^1\|_2$.
By \re{Hcov}, we have
\epsilonq{s52f}\|\rho^1\|_{1+\nu'_1}\cdots\|\rho^1\|_{1+\nu'_{\mu_1'}}\|f\|_{\beta_1}\leq C(\|\rho^1\|_2)\{ \|\rho^1\|_{s_1+2}\|f\|_0+ \|f\|_{s_1}\}
\epsilonnd{equation}
for $s_1:=\beta_1+\sum_{i\leq\mu_1'}(\nu'_i-1)=\beta_1-\mu_1'+\sum\nu'_i$.
Recall that $\mu_1\geq\mu_1'\geq1$. We have
\epsilonq{beta=0}
s_1=\beta_1-\mu_1'+\sum\nu_i'\leq k+1+\beta_1-\mu_1'-\mu_1.
\epsilonnd{equation}
Thus $s_1<r-1$ when $\beta_1=0$. Suppose $\beta_1>0$. Then
$s_1=\begin{align}l-3/2+\lambda -\mu'_1
+\sum\nu_i'-\min\{0,(\lambda -\mu_1+1)/2-\epsilon\}$.
When $\lambda >\mu_1-1$, we have
\epsilonq{s1La-}
s_1\leq \begin{align}l-3/2+\lambda -\mu_1'+\sum_{i\leq\mu_1'}\nu'_i <r-1.
\epsilonnd{equation}
When $0\leq\lambda \leq\mu_1-1$, in view of $\mu_1\geq\mu'_1\geq1$ we get
\begin{align}\lambdabel{s1La}
s_1&\leq \begin{align}l-3/2+\lambda -\mu'_1-(\lambda -\mu_1+1)/2+\sum\nu'_i+\epsilon\\
&\leq r-\mu_1'-\mu_1/2-\lambda/2 +\epsilon< r-1.\nonumber
\epsilonnd{align}
$(iii)\ 1/2<\begin{align}l\leq1$. In this case we can take an extra derivative since $\rho^1\in\Lambda^{k+\begin{align}l+5/2}\subset C^{k+3}$. Write
$\partial_z^{k+2}\tilde K_2f$ as a sum of
$$
K_{\mu,\nu}^{(k+2)}f:=\partial^{1+\nu'_1}_z\rho^1\cdots\partial_z^{1+\nu'_{\mu_1}}\rho^1K_\mu f
$$
where $K_\mu f$ is defined by \re{Kmuf}
with $1\leq j<n$ and
\epsilonq{mu12}
\mu_0+\mu_1+\mu_2+\sum(\nu'_i+\nu''_i)\leq k+2.
\epsilonnd{equation}
As before, the worst term occurs for $j=n-1$ in $K_\mu$, which is assumed now. We need to show that
\epsilonq{}
| K_{\mu,\nu}^{(k+2)}f(z)|\leq C |f|_{\beta_2} \operatorname{dist}(z,\partial\Omega)^{\beta_2'}, \quad\beta_2'=\begin{align}l-3/2
\epsilonnd{equation}
for $\beta_2$ to be specified.
The case that all $\nu_i'\leq1$ can be estimated as in the proof of \rt{conv-est}, for $\beta_2=r-1$. Suppose now $\nu_i'\geq2$ for $i\leq \mu_1'$ with $\mu_1'\geq1$, and $\nu_i'>0$ for $i>\mu_1'$. Recall that $\lambda$ is defined by \re{defla}. In particular, $ \lambda\leq k+1$.
With $\beta_2'=\begin{align}l-3/2$, we take
\epsilonq{defbet+}
\beta_2=\max\{0,\begin{align}l-5/2+\lambda-\min\{0,(\lambda-\mu_1+1)/2-\epsilon\}\}.
\epsilonnd{equation}
Set $s_2:=\beta_2-1+\sum_{i\leq\mu_1'}(\nu'_i-1)$. When $\beta_2=0$, we have $s_2<r-1$ by analogue of \re{beta=0}. Suppose $\beta_2>0$. We have $s_2=\begin{align}l-7/2+\lambda-\mu_1-\min\{0,(\lambda-\mu_1+1)/2-\epsilon\}
+\sum_{i\leq\mu_1'}\nu'_i$.
Using $\mu_1\geq\mu'_1\geq1$, we can verify \re{s52f} where $s_1$ is replaced by $s_2< r-1$.
$(iii)\ \begin{align}l=1/2$. We need to estimate $|K_1f|_{r+1/2}$ with $r+1/2=k+1$. Recall that $\rho^1\in\Lambda^{k+3}$. We write $\partial^kK_1f$ as
a sum of $K_{\mu, \nu}^{(k)}f$ defined by \re{Kmuf}
with $1\leq j<n$ and
\epsilonq{sumk1half}
\mu_0+\mu_1+\mu_2+\sum(\nu'_i+\nu''_i)\leq k.
\epsilonnd{equation}
As before, the worst term occurs for $j=n-1$ in $K_\mu$, which is assumed now.
Consider case $\mu_1\geq1$. Then $\lambda\leq k-1$. By \re{Zcov}, we have
\begin{align}\lambdabel{halfcase}
|K_{\mu, \nu}^{(k)}f|_1
&\leq
|\partial^{1+\nu'_1}\rho^1\cdots\partial^{1+\nu'_{\mu'_1}}\rho^1|_1\|K_\mu f\|_0\\
&\quad
+\|\partial^{1+\nu'_1}\rho^1\cdots\partial^{1+\nu'_{\mu'_1}}\rho^1\|_0|K_\mu f|_1=:I+II.
\nonumber
\epsilonnd{align}
We first estimate $I$. We have $\|K_\mu f\|_0\leq\|K_\mu f\|_{\beta_3'}\leq C\|f\|_{\beta_3}$ for $\beta_3'=\epsilon$, $\beta_3\geq0$ and
$$
\beta_3'\leq\beta_3-\lambda+1+\min\{ 0, (\lambda-\mu_1+1)/2-\epsilon\}>0.
$$
We take
$$
\beta_3=\max\{0,\lambda-1-\min\{0,(\lambda-\mu_1+1)/2+\epsilon\}+\epsilon\}.
$$
We have $\|\partial^{1+\nu'_1}\rho^1\cdots\partial^{1+\nu'_{\mu'_1}}\rho^1\|_1\|f\|_{\beta_3}\leq C\|\rho^1\|_{s_3+2}\|f\|_0+ C\|f\|_{s_3}$ with
$$
s_3:=\beta_3-\mu_1'+\sum_{i\leq\mu_1'}\nu_i'.
$$
By analogue of \re{beta=0}, we have $s_3<r-1$ when $\beta_3=0$. Suppose $\beta_3>0$. Then
$$
s_3= \lambda-1-\mu_1'+\sum\nu_i'-\min\{0,(\lambda-\mu_1+1)/2-\epsilon\}+\epsilon <r-1,
$$
which is verified by analogue of \re{s1La-}-\re{s1La}.
To estimate $II$, by \rl{henint}, we have $|K_\mu f|_1\leq \|f\|_{\beta_4}$ for $\beta_4\geq0$ and
$$
-1\leq \beta_4-\lambda+1+\min\{ 0, (\lambda-\mu_1+1)/2\}.
$$ Then
$\beta_4=\max\{0,\lambda-2-\min\{ 0, (\lambda-\mu_1+1)/2-\epsilon\}\}$. We obtain
$$
\|\partial^{1+\nu'_1}\rho^1\cdots\partial^{1+\nu'_{\mu'_1}}\rho^1\|_1\|f\|_{\beta_4}\leq \|\rho^1\|_{s_4+2}\|f\|_0+ \|f\|_{s_4}
$$
with
$
s_4:=\beta_4-\mu_1'+\sum_{i\leq\mu_1'}\nu_i'+\epsilon.
$
When $\beta_4=0$, we get $s_4<r-1$. Suppose $\beta_4>0$. Then
$$
s_4=\lambda-2-\min\{ 0, (\lambda-\mu_1+1)/2-\epsilon\} -\mu_1'+\sum_{i\leq\mu_1'}\nu_i'+\epsilon.
$$
We can verify that $s_4<r-1$ when $\lambda<\mu_1-1$. When $\lambda\geq\mu_1-1$, we also have
$$
s_4=\lambda-2- (\lambda-\mu_1+1)/2 -\mu_1'+\sum_{i\leq\mu_1'}\nu_i'+2\epsilon\leq k-5/2-\mu_1/2-\mu_1'+2\epsilon<r-1.
$$
Consider the case $\mu_1=0$.
Write $\partial_z^2K_\mu f$ as a sum of
$$
\partial_z^3\rho^1K_{\tilde\mu+\mu}f,\quad
\quad \partial^2\rho^1K_{\tilde\mu+\mu}f.
$$
For the last term, by interpolation, we have
$$
|K_{\tilde\mu+\mu}f(z)|\leq |f|_{r-1}d(z)^{-1}.
$$
This gives us the estimate for $\partial^2\rho^1K_{\tilde\mu+\mu}f$.
For the first term, \rl{henint} implies that
$
|K_{\tilde\mu+\mu}f(z)|\leq C\|f\|_{\beta_5} d(z)^{-1}
$
for $\beta_5\geq0$ and
$$
\beta_5:=\max\{0, (\lambda+\tilde\lambda)+\tilde\mu_2-2-\min\{ 0, (\lambda+\tilde\lambda-\mu_1-\tilde\mu_1+1)/2+\epsilon\}\}.
$$
We have $\tilde\mu_1=1$, $\tilde\mu_0=\tilde\mu_2=\tilde\nu_1''=0$, $\tilde\lambda=0$. Thus ${\beta_5}=\max\{0,\lambda-2-\min\{ 0, \lambda/2-\epsilon\}\}$. When $\beta_5=0$, we obtain $|\partial_z^3\rho^1 K_{\tilde\mu+\mu}(z)|\leq C\|\rho^1\|_3\|f\|_0d(z)^{-1}$, while $\|\rho^1\|_3\|f\|_0\leq C |\rho^1|_{r+5/2}\|f\|_0.$ When $\beta_5>0$,
we have $\lambda>2$ and $\beta_5 \leq \lambda-2\leq k-2<r-5/2+\epsilon'$ for a small $\epsilon'>0$. Therefore,
\begin{align}n
\|\rho^1\|_3\|f\|_{r-5/2+\epsilon'}&\leq C\|\rho^1\|_{r+1/2+\epsilon'}\|f\|_0+C\|\rho^1\|_{2}\|f\|_{r-3/2+\epsilon'}\\
&\leq C|\rho^1|_{r+5/2}\|f\|_0+ C|f|_{r-1}. \qedhere
\epsilonnd{align*}
\epsilonnd{proof}
\setcounter{thm}{0}\setcounter{equation}{0}
\section{An estimate of $\overline\partial$ solution for $(0,1)$ forms via Hartogs's theorem}\lambdabel{sec:1form}
In this section we obtain the regularity of functional $\overline\partial$-solutions for $(0,1)$-forms can be achieved via Hartogs' theorem for concavity domains that require merely $C^2$ boundary. Here we need $2$-concavity. We will only prove the regularity for local solutions.
Let $D\subset U$ be defined by $\rho<0$. Suppose that $U\cap \partial D$ is strictly $2$-concave. For each $\zeta\in\partial D$, we can apply local biholomorphic map $\psi_\zeta$ such that $\rho~\citerc\psi_\zeta=a_\zeta \rho_\zeta$ has the form
\epsilonq{}
\rho_\zeta(z)=-y_n-3|z_1|^2-3|z_{2}|^2+\sum_{j>2}\lambda_j|z_j|^2+o(|z|^2).
\epsilonnd{equation}
Then $D_\zeta:=\psi_\zeta (D)$ contains $ D_\epsilon=\{z\in\mathbb{D}elta_\epsilon^n\colon\tilde\rho(z)<0\}$, where
\epsilonq{}
\tilde\rho(z)=-y_n-2|z_1|^2-2|z_{2}|^2+2\sum_{j=3}^n|z_j|^2.
\epsilonnd{equation}
Both $D_\zeta, D_\epsilon$ share a Hartogs's subdomain $H_\epsilon=\{z\in\mathbb{D}elta_\epsilon^n\colon\hat\rho(z)<0\}$, where
$$
\hat \rho=-y_n- |z_1|^2+\sum_{j=2}^n (1+\lambda_j)|z_j|^2.
$$
Note that $\partial D\cap\partial H_\epsilon=\{0\}$. We want to show that if $f\in\Lambda_r(\overline D)$, any solution $u$ to $\overline\partial u=f$ on $ D$ is in $\Lambda_{r+1/2}$ on $D\cap B_\epsilon(\zeta)$ for $\zeta\in\partial D$ and small $\epsilon$.
We remark that the $C^{1/2}$ estimate in~~\citete{MR986248}*{Thm. 14.1} seems to require $\partial D\in C^{5/2}$ to repeat the proof of ~~\citete{MR986248}*{Thm. 9.1}, while the latter needs $\rho\in C^2$ for the estimates.
Let us first produce a solution $u_0\in C^0(\overline D)$ for $\partial D\in C^2$. For instance, we can take the solution in~~\citete{MR986248}*{Thm. 13.10, p. 127} or our solutions with the $C^0$ estimate for $C^2$ boundary given by \re{C0est}.
Since $\partial\tilde D_\epsilon$ is smooth, we have a solution $u$ on $\tilde D_\epsilon$ such that $\overline\partial u=f$ and $ u\in\Lambda_{r+1/2}$.
Then $u_0- u$ admits a holomorphic extension $h$ to
\epsilonq{}
\hat D_\epsilon:= D_\epsilon\cup\mathbb{D}elta_\deltata^n.
\epsilonnd{equation}
On $D_\epsilon$, we now know that $u_0\in \Lambda^{r+1/2}(D_\epsilon)$. We can write
$$\partial_{x_j}u_0=\partial_{x_j} u+\partial_{x_j}h.
$$
We have $C^0$ estimates for $u_0, u$ and $h$. By Cauchy formula, we have
$$
h(z)=\f{1}{2\pi}\int_{|\zeta_1|=1}\frac{h(\zeta_1,z')}{\zeta_1-z_1}\, d\zeta_1.$$
This gives estimate for higher order derivatives of $\partial_{z_1}^k h$. Similarly, we can obtain higher order derivatives of $\partial_{v}h$ for any directions $v$ that are small perturbations of the unit vector $(1,0,\dots, 0)$. Now these small perturbations span all unit vectors. Therefore, we can get the desired estimates for partial derivatives of order $m <r+1/2$, where $m=[r+1/2]$ if $r+1/2$ is not an integer, or $m=r-1/2$. Thus $\begin{align}l=r+1/2-m\in(0,1]$.
Next we need to estimate the $\Lambda^\begin{align}l$ norms of $m$-th derivatives of $u_0$.
Set $v=\partial^mu_0$. It remains to estimate
$
v(z+w)+v(z-w)-2v(z)$ when $ z,z\pm w\in D.
$
We may assume that $z$ is sufficiently close to the origin and $|w|$ is small. Let $z^*\in\partial D$ be the closed point to $z$.
Suppose $\delta=|w|$ is small. Let $\tilde w=\deltata(z-z^*)/{|z-z^*|}$. Since $\partial D_{z^*}$ is tangent to $\partial D$ at $z^*$, then $z+t\tilde w$ and $z\pm w+t'\tilde w\in D_{z^*}$ for $t\in(0,2)$ and $t'\in[1,2]$.
We now use decomposition \re{thedecom} and get
\begin{align}\lambdabel{thedecom+}
v(z+w)&+v(z-w)-2v(z)\\
\nonumber
&=2v(z+w+ \tilde w )+2v(z-w+\tilde w)-4v(z+w+ \tilde w )\\
\nonumber
&\quad - v(z-w+2 \tilde w )-v(z+w+2 \tilde w )+2v(z+2 \tilde w )\\
\nonumber
&\quad + v(z-w)+v(z-w+2 \tilde w )-2v(z-w+ \tilde w )\\
\nonumber
&\quad+ v(z+w)+v(z+w+2\tilde w)-2v(z+w+ \tilde w )\\
\nonumber
&\quad - 2v(z)-2v(z+2\tilde w)+4v(z+ \tilde w ).
\epsilonnd{align}
We can estimate each row because the triple points in each row are in some smooth domain $D_\zeta$ for some $\zeta$.
We have obtained the H\"older ratio estimate for $v$. This finishes the proof of \rt{regsol} $(a)$ for its local version.
\setcounter{thm}{0}\setcounter{equation}{0}
\section{Proof of \rt{regsol} via canonical solutions}\lambdabel{sec1}
The proof of regularity of the solutions from local to global uses some standard approaches. See Kerzman~~\cite{MR0281944} for the case when $D$ is a domain in ${\bf C}^n$. We will also derive an global estimate reflecting the norm convexity and this estimate will be used in the next section to prove \rt{regsol+}.
We start with the following.
\le{}Let $ D\subset X$ be a domain defined by a $C^2$ function $\rho<0$ and let $D_a$ be defined by $\rho<a$. Let $\rho_t=S_t\rho$, where $S_t$ is the Moser smoothing operator. Suppose that $\partial D$ is an $q_q$ domain. Let $ D^t_a$ be defined by $\rho_t<a$. There exists $t_0=t_0(\partial^2\rho)>0$ and $C>1>c>0$ such that if $0\leq t<t_0$, then
\begin{gather}
\partial D^t_{-t}\subset D_{-ct}\setminus D_{-Ct} \quad \partial D^t_{t}\subset D_{Ct}\setminus D_{ct}
\epsilonnd{gather}
while $ D_{b}$ and $ D^t_{b}$ still satisfy the condition $a_q$ for $b\in(-t_0,t_0)$.
\epsilonnd{lemma}
\begin{proof}Let $\rho_t=S_t\rho$. We have
$
\|\rho_t-\rho\|_0\leq Ct^2\|\rho\|_2.
$
This shows that
\epsilonq{C2t2}
\operatorname{dist}(\partial D^t_s,\partial D_s)<C_2t^2.
\epsilonnd{equation}
When $|s|,|s'|$ are sufficiently small and $s'>s$, we also have
$$
c_1(s'-s)\leq \operatorname{dist}( D_s,\partial D_{s'})\leq C_1(s'-s),\quad c_1(s'-s)\leq \operatorname{dist}( D^t_s,\partial D^t_{s'})\leq C_1(s'-s).
$$
Thus \re{C2t2} implies that
\begin{align}
\operatorname{dist}(\partial D^t_{t}, D)&\geq \operatorname{dist}(\partial D_{t}, D)-\operatorname{dist}(\partial D^t_{t},\partial D_{t})\\
&\geq c_1t-C_2t^2>c_1t/2.
\nonumber
\epsilonnd{align}
Suppose $L_\zeta\rho$ has $(q+1)$ negative Levi eigenvalues bounded above by $-\lambda$ or $(n-q)$ positive Levi eigenvalues bounded below by $\lambda$ for $\zeta\in U$, where $U$ is neighborhood of $\partial D$ and $\lambda$ is positive number. We have $\|\rho_t-\rho\|_{C^2(U)}\leq \epsilon$ when $t<t_0$ and $\zeta\in U$. We find a subspace $W$ of $T_\zeta^{1,0}\partial D$ dimension $q+1$ such $L\rho(\zeta,v)\leq-\lambda$ for $v$ in the unit sphere of $W$. Projecting $W$ onto $\tilde W\subset T_{\tilde \zeta}^{(1,0)}\partial D^t_a$ when $\tilde\zeta\in \partial D^t_a$ is sufficiently close to $\zeta$ and $t$ is close to zero. Then $\dim W\geq q+1$ and $L\rho(\tilde\zeta,v)\leq-\lambda/2$ for $\tilde v$ in the unit sphere of $\tilde W$.
One can also verity that if $L_\zeta\rho$ has at least $(n-q)$ positive eigenvalues, so is $L_{\tilde\zeta}\rho^t$ when $\tilde\zeta$ is sufficiently close to $\zeta$. Therefore, $D^t_a$ still satisfies the condition $a_q$.\epsilonnd{proof}
We know formulate the main result of this paper in details.
\th{regsol-full} Let $r\in(0,\infty]$ and $q\geq 1$.
Let $D\colon\rho<0 $ be a relatively compact domain with $C^2$ boundary in a complex manifold $X$ satisfying the condition $a_q$. Let $V$ be a holomorphic vector bundle of finite rank over $X$.
Then there exists a
linear $\overline\partial$ solution operator $H_q\colon \Lambda_{(0,q)}^r(D, V)\cap\overline\partial L^2_{loc}(D,V)\to \Lambda_{(0,q-1)}^{r'}(D, V)$ satisfying the following
\bppp
\item When
$q=1$ or $\partial D$ is strictly $(n-q)$ convex, we have $|H_qf|_{r+1/2}\leq C_r(\rho,\partial\rho,
\partial^2\rho)|f|_r$
\item When $q>1$ and $r'=r+1/2$ and $\partial D\in\Lambda^{r+\f{5}{2}}$, we have
\epsilonq{}|H_qf|_{r+1/2}\leq C_r(\rho,\partial\rho,
\partial^2\rho) ( |\rho|_{r+5/2}\|f\|_1+|f|_r).\epsilonnd{equation}
\item In both cases, $H_qf\in C^\infty(\overline D)$ when $f\in C^\infty(\overline D)$.
\epsilonppp
Furthermore, the constant $C_r(\rho,\partial\rho,\partial^2\rho)$ is upper-stable under small $C^2$ perturbations of $\rho$; more precisely there exists $\epsilon>0$ such that if $\|\tilde\rho-\rho\|_2<\epsilon$, then
\epsilonq{defupst}
C_r(\tilde\rho,\partial\tilde\rho,\partial^2\tilde\rho)<C_rC_r(\rho,\partial\rho,\partial^2\rho).
\epsilonnd{equation}
We emphasize that $C_r(\rho,\partial\rho,\partial^2\rho)$ involves an unknown constant that is $C_*$ from \rta{3.4.6}.
\epsilonth
\begin{rem}
The stability of estimates on $\overline\partial$ solutions has been discussed extensively in literature; see Greene-Krantz~~\cite{MR644667} for strictly pseudoconvex domains in ${\bf C}^n$, Lieb-Michel~~\cite{MR1900133} strictly pseudoconvex domains with smooth boundary in a complex manifold. The stability in terms \re{defupst} is called upper-stability in Gan-Gong~~\cite{GG} where the reader can find a version of lower stability and its use.
\epsilonnd{rem}
\begin{proof}The proof is a combination of the following: the local regularity results obtained, Grauert's bumping method, the stability of solvability of the $\overline\partial$-equation after the bumping is applied ~\cite{MR0179443}*{Thm.~3.4.1} (see \rt{3.4.6} for the vector bundle version), and the interior estimates of $\overline\partial$-solutions on Kohn's canonical solution.
We will complete the proof in three steps.
\noindent{\epsilonm Step 1. Reduction to interior regularity.}
Let $D$ be a relatively compact subset of $ \mathcal U$, defined by $\rho<0$ in $\mathcal U$ in ${\bf C}^n$ with $|\rho|_{C^{5/2}(\mathcal U)}<\infty$ and $\nabla \rho\neq0$ on $\partial D$. For each $p\in\partial D$, we have a configuration $(D,U_p,\psi_p,\rho^1_p)$. Consider the domain
\begin{gather}{}
\omega_p':=\psi_p^{-1}(D^1_p\cap D^2_{r_2/2}).
\epsilonnd{gather}
Let $\chi\geq0$ be a smooth cut-off function with compact support in $B_{r_2}$ such that $\chi$ equals $1$ on $B_{r_2/2}$. Thus $\omega_p'\cap D$ is contained in $\psi_p^{-1}(D^{12}_p)$. On $D^{1}_p\cap D^2_{r_2}$, we solve the $\overline\partial$ equation $\overline\partial u=(\psi_p^{-1})^*f$ with $u\in \Lambda^{r}(D^{1}_p)$. Then $f_1=f-\psi_p^*\overline\partial(\chi u_p)$ is still in $\Lambda^r$ as
$$
(\psi_p^{-1})^*f-\overline\partial(\chi u_p)=(1-\chi)(\psi_p^{-1})^*f-\overline\partial\chi\wedge u_p.
$$
In fact, setting $f_1=0$ on $X\setminus D$, we have $f_1\in \Lambda^r(D\cup \tilde B_{r_2/2}(p))$ for $\tilde B_{r_2/2}(p)=\psi_p^{-1}(B_{r_2/2})$.
Let $D_p$ be defined by $\rho_p:=\rho-\epsilon\chi~\citerc\psi_p<0$. When $\epsilon$ is sufficiently small, $D_p$ satisfies the condition $a_q$, while
$$
D\cup\omega_p\subset D_p\subset D\cup \tilde B_{r_2/2}(p).
$$
Here $\omega_p$ is an open set containing $p$. Note that the size of $\omega_p$ can be chosen uniformly in $\tilde\rho^0$ when $\|\tilde\rho^0-\rho^0\|_2<\deltata$, which depends on the modulus of continuity of $\partial^2\rho^0$ and the $\epsilonpsilon$.
As in~~\cite{MR1900133,MR3848426}, we find finitely many $p_1,\dots, p_m\in\partial D$ so that $\{\omega_{p_1},\dots, \omega_{p_m}\}$ covers $\partial \Omega$ and $\sum\chi~\citerc\psi_{p_j}>0$ on $\partial\Omega$. With $\rho_0=\rho$, $D_0=D$ and $\epsilon>0$, set
\epsilonq{rhoj=}
\rho_j=\rho_{j-1}-\epsilon\chi~\citerc\psi_{p_j}
\epsilonnd{equation}
and $D_j:=(D_{j-1})_{p_j}\colon\rho_j<0$ for $j\geq1$. We have $D_j\setminus D_{j-1}\subset \tilde B_{r_2}(p_j)$. Also, $D_{j}$ contains $ D\cup \omega_{p_j}$ and $D_j\subset D_{j+1}$. Hence $D_{m}$ contains $\overline D$. Finally, we should choose a small $\epsilon$ that $\|\rho_j-\rho\|_2$ are sufficiently small for $j=1,\dots, m$ in order to apply the stability results in Lemmas~\ref{convex-rho} and \ref{concave-rho}.
Using the $\overline\partialar$-solution operator $T_j$ for the configuration $(\psi_{p_j}D_j,D_j^2)$ with $\overline\partialar T_j(\psi_{p_j}^{-1})^*f_{j}=f_{j}$, we define
$$
f_{j+1}=f_j-\overline\partial\{\psi_{p_j}^*(\chi ( T_j\psi_{p_j}^{-1})^*f_{j})\}.
$$
Then $f_{m}\in \Lambda^r(D_m)$ is $\overline\partialar$ closed on $D_{m}$.
We remark that $f\mapsto u_m$ is a linear operator $\mathcal G_D\colon \Lambda^r(D)\cap \ker\overline\partial\to \Lambda^r(D_m)\cap\ker\overline\partialar$, and $\mathcal G_D$ is independent of $r$.
We write
\epsilonq{ftilde-f}
f=\overline\partial u_m+f_m, \quad u_m=\sum \psi_{p_j}^*(\chi ( T_j\psi_{p_j}^{-1})^*f_{j}.
\epsilonnd{equation}
Therefore, we focus on the $\overline\partial$ equation for a {\it fixed} $a_q$ domain $\Omega$ such that
\epsilonq{Om*}
\Omega_{-c_*}\subset\tilde D\subset\Omega=D_{c_*}\subset\tilde D_m
\epsilonnd{equation}
where $\tilde D$ is any small $C^2$ perturbations of $D$ depending on $c_*$ which is a small positive number. We will apply \rt{3.4.6} to this domain $\Omega$ with $c_*$ bing as in \rt{3.4.6}. In what follows, $\tilde D, \tilde D_m$ is denoted by $D, D_m$ respectively.
We also need to estimates the norms for $f_m$. We have
\begin{align}n
\|u_{j+1}\|_0&\leq C(D_j)\|f_j\|_0,\\
|u_{j+1}|_{r+1/2}&\leq C(D_j)(|f_j|_r+|\rho_j|_{r+5/2}\|f_j\|_1),\\
|f_{j+1}|_{r}&\leq C(D_j)(|f_j|_{r}+ |\rho_j|_{r+5/2}\|f_j\|_1).
\epsilonnd{align*}
By \re{rhoj=}, we have $|\rho|_{r+5/2}\leq C_{j,r}(1+|\rho|_{r+5/2})\leq 2C_{j,r}|\rho|_{r+5/2}$. Thus,
\begin{gather}\lambdabel{f_test}
\|f_m\|_0\leq C\|f\|_0, \quad |f_{m}|_{r}\leq C(D)|f|_{r}+C(D_j)|\rho|_{r+5/2}\|f\|_1,\\
\|u_m\|_0\leq \|f\|_0, \quad |u_{m}|_{r}\leq C(D)|f|_{r}+C(D_j)|\rho|_{r+5/2}\|f\|_1.
\epsilonnd{gather}
\noindent{\epsilonm Step 2. Smoothing for interior regularity.}
To obtain the interior regularity, we will use regularity in Sobolev spaces. We need to avoid the loss in H\"older exponent from the Sobolev embedding. To this end, we will again use a partition of unity to overcome the loss.
We can make $f_m$ to be $C^\infty$ on any relatively compact subdomain $U'$ of $U$ via local solutions as follows. Fix $x_0\in D$. We solve $\overline\partial u=f_m$ on an open set $ D$ containing $x_0$. Let $\chi$ be a smooth function with compact support in $\omega$ such that $\chi=1$ on a neighborhood $\omega'$ of $x_0$. Then $\tilde f=f_m-\overline\partial(\chi u)=(1-\chi)f_m+\overline\partial\chi\wedge u$ is still in $\Lambda^r$, while $\tilde f=0$ on $\omega'$. In particular, $\tilde f\in C^\infty(\omega')$. Repeating this finitely many times, we can find $\tilde u\in\Lambda^{r+1}( D)$ with compact support in $ D$ such that
\epsilonq{tffm}
\tilde f=f_m-\overline\partial \tilde u\in C^\infty( D').
\epsilonnd{equation}
We can also obtain
$$
|\tilde f|_{\Lambda^{r'}( D')}\leq C_{r}|f_m|_{\Lambda^r}, \quad |\tilde u|_{\Lambda^{r+1}( D')}\leq C_{r'}|f_m|_{\Lambda^r}
$$
for any relatively compact subset $ D'$ of $ D$ and any $r'>r$.
We may assume that $ D'$ is a smooth domain satisfying the condition $a_q$. Furthermore, the defining function $\rho'$ of $D'$ satisfies $\|\rho'\|_a\leq C_a\|\rho\|_2$. Rename $ D', \rho'$ as $ D,\rho$.
\noindent{\epsilonm Step 3. Interior regularity with estimates.} Let $\varphi=e^{\tauu\rho}$, where $\rho=-\operatorname{dist}_{\partial\Omega}$.
Let $L_{p,q}( \Omega,V,\varphi)$ be the space of $V$-valued $(p,q)$ forms $f$ on $ \Omega$ such that
$$
\|f\|_{\varphi}^2:=\int_ \Omega |f(x)|^2e^{-\varphi(x)}\, d\upsilon(x)<\infty,
$$
where $d\upsilon$ is a volume form on $X$. Write $\|f\|_\varphi$ as $\|f\|$ when $\varphi=0$.
As mentioned early, we want to apply ~\cite{MR0179443}*{Thm.~3.4.6}, and in fact \rt{3.4.6} for the vector bundle version to
$
\Omega=D_{c_*}.
$
Then $\varphi$ satisfies condition $a_q$ on $\Omega_{c_*}\setminus\Omega_{-c*}$.
Let $\varphi_k=\chi_k(\varphi)$. Fix $k$ to be sufficiently large and let $\tilde\varphi=\varphi_k$.
We now consider $T_{q}=\overline\partial$ as densely defined from $ L_{(0,q-1)}^2(\Omega,\tilde\varphi)$ into $ L_{(0,q)}^2( \Omega,\tilde\varphi)$ and $T_{q}^*$ its adjoint. Let $f_m$ be the $(0,q)$ form derived in Step 1. By \rt{3.4.6}, we find a solution $u_0$ satisfying
$
\overline\partial u_0=f_m
$ on $\Omega$ and
$$
\|u_0\|_{\varphi_{k_*}}\leq C_*\|f^m\|_{\varphi_{k_*}}.
$$
For the estimate, we need Kohn's canonical solutions. By ~\cite{MR0179443}*{Thm.~1.1.1}, $R_{T_{q}^*}$ is also closed and $R_{T_q^*}=N_{T_q}^\perp$.
We now apply the decomposition
\begin{gather}
u_0=u+h, \quad u\in N_{T_q}^\perp, \quad h\in N_{T_q}.
\epsilonnd{gather}
Thus, $u=(T^\varphi_q)^*v$ and
\epsilonq{uvark}
\|u\|_{\varphi_{k_*}}\leq\|u_0\|_{\varphi_{k_*}}\leq C_*\|f_m\|_{\varphi_{k_*}}.
\epsilonnd{equation}
In particular, in the sense of distributions, we have
$
\varphitheta_q^\varphi v=u$ on $D$. Here $\varphitheta_q^\varphi$ is the formal adjoint (acting on test forms) of $\overline\partial_q$ in the $L^2$ spaces with weight $\varphi$.
Since $\varphitheta^\varphi_{q-1}\varphitheta^\varphi_q=0$, we get in the sense of distributions
\epsilonq{keyid}
\theta^\varphi_{q-1}u=0.
\epsilonnd{equation}
We now use the system of elliptic equations for $u$ to derive the \epsilonmph{interior} estimates, using \re{uvark}.
Let $\|\cdot\|:=\|\cdot \|_{\Omega;k,p}$ denote the norm for space $W^{k,p}(\Omega)$.
In local holomorphic coordinates $z$, we have for $u=\sum u_{J}d\overline z^J$
\begin{gather}
\overline\partial u=\sum\f{\partial u_J}{\partial \overline z_k}d\overline z_k\wedge d\overline z^J,\quad
\varphitheta^\varphi_{q-1} u=-\sum \Bigl(\f{\partial u_{jK}}{\partial z_j}+c_{jJK}(\partial\varphi)u_J\Bigr)\, d\overline z^K.
\epsilonnd{gather}
Write $\varphitheta^\varphi$ as $\varphitheta$ when $\varphi=0$.
Let $\chi$ be a smooth function with support in a ball $B_R$ of radius $R$ centered at a point in $x_0\in\Omega$. Let $\tilde u=\chi u$. By~~\citete{MR1045639}*{Lemma 4.2.3, p.~86}, we obtain
$$
\|\overline\partial\tilde u\|_{0,2}+\|\varphitheta \tilde u\|_{0,2}\leq C(\|f\|_{0,2}+\|u\|_{0,2})\leq 2C'C_*\|f\|_{0,2}.
$$
This shows that for any relatively compact subset $\Omega'$ of $\Omega$, we have
$$
\|u\|_{\Omega';{1,2}}\leq C(\Omega',\Omega)C_*\|f\|_{0,2}.
$$
Recall the Sobolev compact embedding of $L^{j}_q$ in $L^{j+1}_p$ for $q=\f{p}{1-\f{p}{2n}}$ when $1\leq p<2n$.
For the following, we take $p_0=2$ and fix any $2<q<\infty$. We then fix $p_1,\dots, p_{n^*}$ such that $p_{n^*-1}<2n$, $p_{j+1}\leq \f{p_j}{1-\f{p}{2n}}$ and $p_{n^*}>q$.
Let $\Box_{\tilde\varphi}= \overline\partialar_{q-1}\varphitheta^\varphi_{q-1}+\varphitheta_q^\varphi\overline\partialar_{q}$. Here $\mathbb{D}elta_g=-\sum g^{jk}\f{\partial^2}{\partial z^j\partial \overline z^k}$ has smooth coefficients and it independent of $\varphi$. The principal part $\mathbb{D}elta_g$ of $\Box^\varphi$ is diagonal and elliptic, where $g$ is the smooth hermitian metric $M$ (see~~\cite{MR2109686}*{pp. 154, 160}). In the sense of distribution, we have
\begin{gather}{}
\mathbb{D}elta_gu= b(\partial\tilde\varphi)\partial f+c_1(\partial\tilde\varphi)\partial u+ c_0(\partial^2\tilde\varphi)u.
\epsilonnd{gather}
Let $\chi$ be a smooth function with support in $B_R$ of radius $R$ centered at a point in $x_0\in\Omega$. Without loss of generality, we may assume $x_0=0$.
Let $\tilde u=\chi u$. Then as a weak solution, $\tilde u$ satisfies
\begin{gather}\lambdabel{Dgtu}
\mathbb{D}elta_g\tilde u= \tilde f, \quad \tilde f:=\chi f+\chi b(\partial\tilde\varphi)\partial f+(\chi c_1(\partial\tilde\varphi)+\chi_1)\partial u+ (\chi c_0(\partial^2\tilde\varphi)+\chi_0)u.
\epsilonnd{gather}
Here we recall two interior estimates on systems of elliptic equations from Morrey~~\cite{MR0202511}*{Thm 6.4.4., p. 246}:
\begin{gather}{}\lambdabel{sobe}
\|\tilde u\|_{k+2,p}\leq C \|\mathbb{D}elta_g\tilde u\|_{k,q}+C_R\|\tilde u\|_{0,1},\quad 1<p<\infty;\\
\|\tilde u\|_{k+2+\begin{align}l}\leq C\|\mathbb{D}elta_g\tilde u\|_{k+\begin{align}l}+C_R\|\tilde u\|_{Lip}, \quad \operatorname{supp}\tilde u\subset B_R
\lambdabel{holde}
\epsilonnd{gather}
provided the right-hand sides are finite.
By Sobolev inequality, $\|\tilde u\|_{1,p_1}\leq C_1\|\tilde u\|_{2,2} \leq C_2\|f\|_{1,2}\leq C_3\|f\|_{1,p_1}$. Repeating this bootstrapping argument, we can show that for any $q<\infty$, we have $\|\tilde u\|_{\Omega';1,q}\leq \|f\|_{1,q}$.
By \re{sobe}, we get $\|\tilde u\|_{\Omega';2,q}\leq C \|f\|_{1,q}$.
Recall Sobolev inequality $C^{k,\begin{align}l}\subset L^{k+1, q}$ for $\begin{align}l=1-\f{2n}{q}>0$. Thus we have
\epsilonq{est1}
\|\tilde u\|_{\Omega';1+\begin{align}l}\leq C(\Omega',\Omega)\|f\|_{\Omega;1}
\epsilonnd{equation}
for any $\begin{align}l<1$. Using \re{holde}, we get
\begin{gather}{}
\|\tilde u\|_{\Omega';k+2+\begin{align}l}\leq CC_*(\|\tilde f\|_{\Omega';k+\begin{align}l}+|f|_{\Omega;0,2}).
\epsilonnd{gather}
Next, we prove by induction that
\epsilonq{uk1}
\|u\|_{\Omega';k+1+\begin{align}l}\leq CC_*(\|\varphi\|_{k+1+\begin{align}l}\|f\|_0+\|f\|_{k+\begin{align}l}).
\epsilonnd{equation}
By \re{est1}, the above holds for $k=1$. Suppose the above hold and we want to verify it when $k$ is replaced by $k+1$. We have for $\tilde f$ in \re{Dgtu}
\begin{align}n{}
\|\tilde f\|_{\Omega';k+\begin{align}l}&\leq C\|f\|_{\Omega';k+1+\begin{align}l}
+CC_*\|\varphi\|_{\Omega';k+2+\begin{align}l}(\|f\|_{\Omega';0}+\|u\|_{\Omega';0} ) +C\|u\|_{\Omega';k+1+\begin{align}l}.
\epsilonnd{align*}
By \re{uk1}, we get $\|\tilde f\|_{\Omega';k+\begin{align}l}\leq C\|f\|_{\Omega';k+1+\begin{align}l}
+C\|\varphi\|_{\Omega';k+2+\begin{align}l}\|f\|_{\Omega';0}$. Then \re{uk1} yields
$$\|\tilde u\|_{\Omega';k+2+\begin{align}l}\leq CC_*\|\varphi\|_{k+2+\begin{align}l}\|f\|_0+C\|f\|_{k+1+\begin{align}l}.
$$
This gives us \re{uk1} with $k,\Omegaega'$ being replaced by $k+1$ and any open set $\Omegaega''$ on which $\chi=1$.
\epsilonnd{proof}
We conclude this section with an isomorphism theorem on cohomology groups with bounds.
Let $Z^r_{(0,q)}(\Omega,V)$ be the space of $\overline\partial$-closed forms on $\Omega$ of class $\Lambda^r$. Define $B_{(0,q)}^{r,r'}(\Omega,V)=\Lambda_{(0,q)}^r(\Omega,V)\cap\overline\partial\Lambda_{(0,q-1)}^{r'}(\Omega,V)$ and
$B_{(0,q)}^{r,loc}(\Omega,V)=\Lambda_{(0,q)}^r(\Omega,V)\cap\overline\partial L^{2}_{loc}(\Omega,V)$.
Define $\overline H^{r,r'}_{(0,q)}(\Omega, V)=Z^{r}_{(0,q)}(\Omega,V)/{B^{r,r'}_{(0,q)}(\Omega,V)}$ and $\overline H^{r,loc}_{(0,q)}(\Omega, V)=Z^{r}_{(0,q)}(\Omega,V)/{B^{r,loc}_{(0,q)}(\Omega,V)}$.
\begin{thm}Let $\Omega$ be relatively compact $a_q$ domain in $X$ and let $V$ be a holomorphic vector bundle on $X$. There exists $c>0$ such that the restriction $\overline H_{(0,q)}^{r,r'}(\Omega, V)\to\overline H^{r,loc}_{(0,q)}(\Omega_c, V)$ is an isomorphism for the following cases.
\bppp
\item $r\in(1,\infty)$, $r'=r+1/2$ and $q=1$.
\item $r=r'=\infty$.
\item $r\in(1,\infty)$, $r'=r+1/2$ and $\partial \Omega$ is strictly $(n-q)$ convex.
\item $\partial \Omega\in \Lambda^{r+5/2}$, $r>1$ and $r'=r+1/2$.
\item $\partial D\in \Lambda^s$ with $s\geq7/2$, $r>s+5/2$, and $r'=r+1/{50}$.
\epsilonppp
\epsilonth
\begin{proof}The injectivity follows from the stability result proved in the appendix and our regularity results including \rt{nash-moser-c12} $(iii)$ below for case $(v)$. The surjectivity can be obtained by the Grauert bumping method which is valid as $r'\geq r$ and the same regularity results.
\epsilonnd{proof}
We remark that the isomorphism of the restriction $\overline H_{(0,q)}^{0,1/2}(\Omega, V)\to\overline H^{0,0}_{(0,q)}(\Omega_c, V)$ is proved in ~\citete{MR986248}*{Thm. 12.14, Thm. 15.12}, provided $\Omega$ is either $(n-q)$ strictly convex with $\partial\Omega\in C^2$ or it is $(q+1)$ concave with $\partial\Omega$ in $C^{5/2}$.
\setcounter{thm}{0}\setcounter{equation}{0}
\section{Proof of \rt{regsol+} via a Nash-Moser method}\lambdabel{sec:NM}
In this section, we will prove \rt{regsol+} for $q\geq2$ when the domains have negative Levi eigenvalues by using the Nash-Moser smoothing operators. Our approach was inspired by a method of
Dufresnoy~~\cite{MR526786} for the $\overline\partial$-equation on a compact set that can be approximated from outside by strictly pseudoconvex domains of which the Levi eigenvalues are well controlled.
It is interested that V. Michel~~\cite{MR1198845} showed that if the number of \epsilonmph{non-negative} Levi eigenvalues of $\partial \Omega$ is \epsilonmph{exact} $n-q'$ near $z_0\in\partial\Omega$, then $\overline\partial u=f_{0,q}$ has a solution in $C^\infty(U\cap\Omega)$ when $\partial\Omega\in C^2$ for all $q\geq q'$. When $\partial\Omega\in C^4$, there is (a possibly different) solution $u$ in $C^\infty(\overline\Omega\cap U)$. For pseudoconvex domains with $C^2$ boundary in ${\bf C}^n$, the $C^\infty$ regularity of $\overline\partial$ solutions under suitable assumptions on the Levi-form has been proved by Zampieri~~\citetes{MR1757879, MR1749685} and Baracco-Zampieri~~\citetes{MR2145559, MR2178735}. The reader is referred to the thesis of Yie~~\cite{MR2693230} for the
global regularity of $\overline\partial$ solutions with $\partial D\in C^4$. When $\partial D\in C^\infty$ additionally, the existence of $u$ was proved by Kohn~~\cite{MR344703}. Michel-Shaw~~\citete{MR1675218}
obtained smooth regularity of the $\overline\partial$ solutions on annulus domain $D_1\setminus \overline{D_2}$ where $D_1$ is a pseudoconvex domain with piecewise smooth boundary and $D_2$ is the intersection of bounded pseudoconvex domains.
The $H^{s}$ solutions was proved by Harrington~~\cite{MR2491606} for pseudoconvex domains $D$ with
$\partial D\in C^{k-1,1}$, $k>s+1/2, k\geq2$, and $s\geq0$.
As observed in~~\cite{MR3961327}, to the author's best knowledge it remains an open problem if $C^\infty(\overline D)$ solutions $u$ to $\overline\partial u=f$ exist on a bounded pseudoconvex domain $D$ in ${\bf C}^n$ with $C^2$ boundary.
We now state a detailed version of \rt{regsol+}.
\begin{thm}\lambdabel{nash-moser-c12}Let $q>1$. Let $D$ be a relatively compact domain with $C^s$ boundary in a complex manifold $X$ satisfying the condition $a_q$. Let $V$ be a holomorphic vector bundle on $X$. Suppose $r>s+5/2$. Define $\hat r$ as follows.
\bppp
\item When $s=2$ and $r>26/7$, set $\hat r>r-{19}/{7}$.
\item When $2< s< 7/2$, set $$\hat r=r+\f{1}{2}-\f{r}{r-1}\f{r+5/2-s}{(s-1)r-5/2}.$$
\item When $s\geq 7/2$, set
$$\hat r= r+\f{1}{2}-\f{1}{s-1}\cdot\f{1}{1-\f{5}{2(s-1)r}}\geq r+\f{1}{50}.
$$
\epsilonppp
For $r'<\hat r$, there exists a linear $\overline\partial$ solution operator
$$ H_{q}^{r,r'}\colon \Lambda_{(p,q)}^r(D, V)\cap\overline\partial L^2_{loc}(D,V)\to \Lambda_{(p,q-1)}^{r'}(D, V)
$$
satisfying
$
|H_q^{r,r'}f|_{r'}\leq C_{r,r'}(D)|f|_r.
$
Furthermore, $C_{r,r'}(D)$ is stable under small $C^2$ perturbations of $D$, and $H_{q}^{r,r'}f\in C^\infty(\overline D)$ if $f\in \Lambda_{(p,q)}^\infty(D, V)\cap\overline\partial L^2_{loc}(D,V)$ additionally.
\epsilonnd{thm}
\begin{proof}
It suffices to prove the theorem when $r,r+1/2,r',r'+1/2$ are not integers, by replacing $r$ by a smaller number and $r'$ by a larger number. This allows us to identity the Zygmund spaces with the H\"older spaces for these orders. This also allows us to use the Taylor theorem and the global estimates for smooth domains.
Let $D$ be a $(q+1)$ concave domain with $C^{2}$ boundary. Suppose that $f\in C^r$.
Using Moser smoothing operator $S_t$, we
define
\GA{\tilde S_tu=S_tE_{D^\epsilon}u.
}
Since $t_1$ is larger than $\epsilon_1$, in what follows we will identify $f$ with $Ef$ to define $S_{t_1}f$.
We have
\GA{\lambdabel{tildeS}
\|\tilde S_tu-u\|_{D^\epsilon,a}=\|S_tE_{D^\epsilon}u-E_{D^\epsilon}u\|_{D^\epsilon,a}\leq C_{a,b}t^{b-a}\|u\|_{D^\epsilon,b},\\
\lambdabel{tildeS+}
|\tilde S_tu|{b}\leq C_{a,b}t^{a-b}\|u\|_{D^\epsilon,a}.
}
Assume that $\partial\Omega\in C^{s}$ with
$$
s\geq 2, \quad r\in(s+5/2,\infty).
$$
As in Yie~~\cite{MR2693230}, we apply the above to the defining function $\rho$ of $D$ by setting
$$
\rho_{t_1}=\rho \ast\chi_{t_1}, \quad t_1=c_*\epsilon_1^{1/s}, \quad \tilde D^{\epsilon_1}=\{\rho _{t_1}<-\epsilon_1\}.
$$
We now have
\begin{gather}
\|\rho_{t_i}-\rho\|_{0}\leq C_s t^s_i\|\rho\|_{s},\lambdabel{rho1-1-25} \\
\|\rho_{t_i}-\rho\|_{2}\leq \epsilon_*t_i^{s-2}\|\rho\|_2,\lambdabel{rho1-2c-25} \quad t_i<t^*(\partial^s\rho),\\
\|\rho_{t_i}\|_b\leq C_{b,s}t_i^{s-b}\|\rho\|_{s}, \quad b\geq s.
\lambdabel{rho1-3-25}
\epsilonnd{gather}
We also have $t^{s}_1=c^{s}_*\epsilon_1$. When $c_*$ is sufficiently small, we have by \re{rho1-1-25}
$$
c\epsilon_1<\operatorname{dist}(D^{\epsilon_1},\partial D)<c'\epsilon_1, \quad c'<1,
$$
where $c,c'$ are independent of $c_*$.
We emphasize that $t^*(\partial^2\rho)$ in \re{rho1-2c-25}
depends on the {\it modulus of continuity} of $\partial^2\rho$.
This however does not cause any difficulty for domains satisfying the condition $a_q$, because the Levi eigenvalues do not decay towards the boundary. This is decisive for the Nash-Moser iteration to succeed in our proof for $C^2$ domains.
By \re{rho1-3-25}, we have a solution operator satisfying
$$
|H_q\varphi|_{D^{1}\cap D^2_{\rho_4}, r+1/2}\leq C_r(\partial\rho,\partial^2\rho) ( |\varphi|_{D^{12},r}+|\rho^1|_{r+5/2}\|\varphi\|_{1}).
$$
Therefore, we obtain
\begin{gather}\lambdabel{v1t}
|v_1|_{D^{\epsilon_1},r+1/2}\leq C_j |f_1|_{D^{\epsilon_1},r}+C_rt_1^{-\f{1}{2}-r}\|f_1\|_{D^{\epsilon_1},1}, \quad r\in(1,\infty).
\epsilonnd{gather}
Smoothing with a different parameter $\epsilon$, we define $w_1=S_{\epsilon_1}\tilde v_1$ on ${\bf C}^n$. On $D$ define
$$
f_2=f_1-\overline\partial w_1.
$$
We iterate this.
Set $\epsilon_i=\epsilon_{i-1}^d$ with $d>1$. We also find a solution $u_i$ on $D^{\epsilon_i}$ so that
$$
f_i=\overline\partial v_i,\quad \text{on $D^{\epsilon_i}$}.
$$
Define $\tilde v_i=E_{D_{\epsilon_i}}v_i$, $w_i=S_{t_i}\tilde v_i$ and $f_{i+1}=f_i-\overline\partial w_i$. Then $u:=\sum w_i$ is the desired solution to $\overline\partial u=f_1$ on $D$, provided $f_j$ tends to zero on $D$ as $j\to\infty$.
We have
$$
\overline\partial v_2= f_1-\overline\partial w_1, \quad \text{on $D^{\epsilon_2}$}.
$$
On $D^{\epsilon_1}$, $\tilde v_1=E_{D^{\epsilon_1}}v_1=v_1$ and hence $\overline\partial v_1=\overline\partial\tilde v_1$. Therefore
$$
f_2= f_1-\overline\partial w_1=\overline\partial(\tilde v_1-w_1)=\overline\partial(\tilde v_1-S_{\epsilon_1}\tilde v_1),\quad \text{on $D^{\epsilon_1}$}.
$$
We have
\AL{\|w_1-\tilde v_1\|_{D^{\epsilon_1},a}=\|S_{\epsilon_1}\tilde v_1-\tilde v_1\|_{D^{\epsilon_1},a}\leq C_{b,a}\epsilon_1^{b-a}\|\tilde v_1\|_{b}.
}
Hence by Taylor's theorem, we get
\AL{\lambdabel{b2t}
\| f_2\|_{D,1}&\leq C_j\sum_{k=1}^{[b]-1}\epsilon_1^{k-1}\| f_2\|_{D^{\epsilon_1},k}+C_j\epsilon_1^{b-1}\| f_2\|_{D,b}\\
&= C_j\sum_{k=1}^{[b]-1}\epsilon_1^{k-1}\|\overline\partial\tilde v_1-\overline\partial w_1\|_{D^{\epsilon_1},k}+C_b\epsilon_1^{b-1}\| f_2\|_{D,b}
\nonumber\\
&\leq C_j\sum_{k=1}^{[b]-1}\epsilon_1^{k-1}\epsilon_1^{b-k-\f{1}{2}}\|v_1\|_{D^{\epsilon_1},b+\f{1}{2}}+C_b\epsilon_1^{b-1}\| f_2\|_{D,[b]}\nonumber\\
&= C'_b \epsilon_1^{b-\f{3}{2}}\|v_1\|_{D^{\epsilon_1},b+\f{1}{2}}+C_b\epsilon_1^{b-1}\| f_2\|_{D,b}.\nonumber
}
By analogy of \re{v1t} and \re{b2t}, we have
\AL{\lambdabel{vii-25}
\|v_{i}\|_{D^{\epsilon_{i}},r+1/2}&\leq C_r \| f_i\|_{D^{\epsilon_i},r}+C_rt_i^{s-\f{5}{2}-r}\| f_i\|_{D^{\epsilon_i},1}, \quad r\in(1,\infty)\\
\lambdabel{bi+-25}
\| f_{i+1}\|_{D,1}
&\leq C_r \epsilon_i^{r-\f{3}{2}}\|v_i\|_{D^{\epsilon_i},r+\f{1}{2}}+C_r\epsilon_i^{r-1}\| f_{i+1}\|_{D,r}.
}
Thus the $r$-norm is estimated by
\AL{\lambdabel{m-norm--25}
\| f_{i+1}\|_r\leq \| f_i\|_r+C_r\|\tilde v_i\|_{r+1}\leq \| f_i\|_r+C_r\epsilon_i^{-1/2}\|v_i\|_{r+1/2}.
}
Therefore, by \re{vii-25}, we obtain
\AL{\lambdabel{m-norm-25}
\| f_{i+1}\|_r\leq 2 C_r \epsilon_i^{-1/2}( \| f_i\|_{D^{\epsilon_i},r}+ t_i^{s-\f{5}{2}-r}\| f_i\|_{D^{\epsilon_i},1}), \quad r\in(1,\infty)^*.
}
Here $(1,\infty)^*=(1,\infty)\setminus({\bf N}\cup\f{1}{2}{\bf N})$.
We now define
\epsilonq{Bi1-25}
B_{i+1}=\hat C^2_rt_{i}^{-s/2}B_{i},
\epsilonnd{equation}
where $B_{0}$ is fixed, depending on $r$, so that
\AL{
\max\left\{\| f_0\|_r, t_0^{s-5/2-r}\| f_0\|_{1}, \|v_0\|_{r+1/2}\right\}\leq B_{0}.
}
By induction, let us show that when $r\in(1,\infty)^*$,
\AL{\lambdabel{bi-25}
\| f_i\|_r&\leq B_{i},\\
\lambdabel{bi0-25}
t_i^{s-5/2-r}\| f_i\|_1&\leq\hat C_r B_i,\\
\lambdabel{vi-25}
\|v_i\|_{r+\frac{1}{2}}&\leq \hat C^2_rB_i.
}
Suppose that the three inequalities hold. We want to verify them when $i$ is replaced by $i+1$.
Clearly, $\re{bi-25}_{i+1}$ follows from \re{m-norm-25}, \re{Bi1-25}, \re{bi-25}, and $\hat C_r\geq 4C_r$. By \re{bi+-25}, \re{vi-25} and $\re{bi-25}_{i+1}$, we obtain
\AL{
t_{i+1}^{s-5/2-r}\| f_{i+1}\|_1&\leq C_r t_{i+1}^{s-5/2- r}\epsilon_i^{r-\f{3}{2}}\|v_i\|_{r+\f{1}{2}}+C_r
t_{i+1}^{s-5/2- r}\epsilon_i^{r-1}\| f_{i+1}\|_r\\
&\leq C''_rt_{i+1}^{ s-5/2-r}t_i^{sr-\f{3s}{2}}\hat C_r^2B_i+C_rt_{i+1}^{ s-5/2 -r}\epsilon_i^{r-1}B_{i+1}
\nonumber\\
\nonumber&= C'_r c_*^{-2r}t_{i+1}^{ s-5/2- r}t_i^{sr-s} B_{i+1}+C_rc_*^{-2r}t_{i+1}^{ s-5/2- r}t_i^{sr-s}B_{i+1},
}
which gives us $\re{bi0-25}_{i+1}$, provided $\hat C_r>(C_r'+C_r)c_*^{-2r}$ and
\epsilonq{condd-25}
-d(r+5/2-s)+sr-s\geq0.
\epsilonnd{equation}
The latter is assumed now.
Then $\re{vi-25}_{i+1}$ follows from $\re{vii-25}_{i+1}$, $\re{bi-25}_{i+1}$ and $\re{bi0-25}_{i+1}$ and $\hat C_r>2C_r$.
By interpolation, we get
$$
| f_i|_{1-\theta+\theta r}\leq C_{r,\theta}| f_i|_1^{1-\theta}| f_i|_r^\theta\leq C_{r,\theta}t_i^{(1-\theta)(r+5/2-s)}B_i.
$$
We have
$$
t_{i}=t_{1}^{d^{i-1}}, \quad
B_i=\hat C_r^{i-1}(t_1\cdots t_{i-1})B_1=\hat C_r^{i-1}t_1^{-\f{d^{i-1}-1}{d-1}}B_1\leq \hat C_r^{i-1}t_i^{-\f{1}{d-1}}B_1.
$$
Therefore, $| f_i|_{1-\theta+\theta r}\leq C_r^{i-1}B_1t_i^{\lambda}$ converges rapidly if \re{condd-25} holds and
\epsilonq{la1--25}
\lambda:=(1-\theta)(r+5/2-s)-\f{1}{d-1}>0.
\epsilonnd{equation}
By \re{vii-25}, we have for $\theta r\in(0,\infty)^*$
\begin{align}\lambdabel{vii-+-25}
|v_{i}|_{D^{\epsilon_{i}},3/2-\theta+\theta r}&\leq C_r | f_i|_{D^{\epsilon_i},1-\theta+\theta r}+C_rt_i^{s-7/2+\theta-\theta r}| f_i|_{D^{\epsilon_i},1}\\
&\leq B_1C_r\hat C_r^{i-1} t_i^{\lambda}+C_r\hat C_r^it_i^{r-1+\theta-\theta r-\f{1}{d-1}}B_1.
\nonumber\epsilonnd{align}
This shows that $|v_j|_{\f{3}{2}-\theta+\theta r}\leq \hat C_r^{i}t_i^{\lambda_*}$ converges rapidly if
\epsilonq{}\lambdabel{mula}
\lambda_*:= (1-\theta)r_* -\f{1}{d-1}>0, \quad r_*:=r-1+\min\{0, 7/2-s\}.
\epsilonnd{equation}
We want to maximize $\theta r$ or $\theta$.
Now, $\lambda_*>0$ implies that
$$
\theta <1 -\f{1}{(d-1)r_*}.
$$
The latter is an increasing function of $d$. Assume
\epsilonq{m1d1-25}
1 -\f{1}{(d-1)r_*}>0.
\epsilonnd{equation}
We now specify the parameters. Note that \re{condd-25} and \re{m1d1-25} are equivalent to
\epsilonq{existd*}
1+\f{1}{r_*}<d<d_*,\quad d_*:=\f{s(r-1)}{r+5/2-s}.
\epsilonnd{equation}
Under the above restriction on $d$, $\f{3}{2}-\theta+\theta r$ has maximum value
\epsilonq{hatrval}
\hat r=r+\f{1}{2}-\f{r}{(d_*-1)r_* }.
\epsilonnd{equation}
Then $|v_i|_{r'}<C^i_{r'}t_i^{\lambda'}$ with $\lambda'>0$ where $\lambda'$ depends on $r'$ and $r'<\hat r$. Consequently the solution $u=\sum S_{t_i}E_{D^{\epsilon_i}}v_i$ to $\overline\partial u=f$ is also in $\Lambda^{r'}(\overline D)$.
We now compute the value of $\hat r$.
When $2\leq s<7/2$, we have $r_*=r-1$ and there exists $d$ satisfying \re{existd*}. We have
$$
\hat r=r+\f{1}{2}-\f{r}{r-1}\f{r+5/2-s}{(s-1)r-5/2}.
$$
For $s=2$ and $r>s+5/2$, we get $\hat r>r-\f{19}{7}.$
When $s\geq7/2$ and $r> s+5/2\geq 6$, we have $r_*=r-s+5/2$. One can check that \re{existd*} is satisfied.
Then
$$
\hat r= r+\f{1}{2}-\f{1}{s-1}\f{1}{1-\f{5}{2(s-1)r}}\geq r+\f{1}{2}-\f{1}{\f{5}{2}}\f{1}{1-\f{1}{r}}> r+\f{1}{50}.
$$
Finally, we show that if $f\in C^\infty$, then there is a solution $u\in C^\infty$ on $\omega\cap\overline D$. For notations, we fix $r,r'$ and rename them as $r_0,r_0'$. Thus we have found solutions $\sum w_i$ with
$$
|w_i|_0\leq t_i^{r_0+\f{1}{2}}B_i, \quad i=0,1,\dots.
$$
Here we have fixed $d\in(1,2),\theta,t_0$ so that
$$
(1-\theta_0)r_0 -\f{1}{d-1}>0, \quad -d(r_0+\f{1}{2})+2r_0\geq0, \quad r_0-\f{1}{d-1}\geq0.
$$
For any $r>r_0$ satisfying $r,r+\f{1}{2}\not\in{\bf N}$, we choose $B_0$ which depends on $r$ so that \re{bi-25}-\re{vi-25} hold for $i=0$. We define $B_{i+1}$ by \re{Bi1-25}. For the above choice of $\hat C_m$, the proof shows that \re{bi-25}-\re{vi-25} hold for all $i$ since \re{condd-25} and \re{mula}-\re{m1d1-25} hold for the fixed $\theta,d$ (depending on $r_0,r_0'$) and the $r$. This shows that $|v_i|_{1-\theta+\theta r+1/2}$ converges rapidly. Therefore, $u:=\sum w_i\in\Lambda^{1-\theta+\theta r+1/2}$. We concluded that $u\in C^\infty(\overline D)$.
\epsilonnd{proof}
\appendix
\setcounter{thm}{0}\setcounter{equation}{0}
\section{Distance function to $C^2$ boundary}\lambdabel{sec:vb}
To relax the boundary condition $\partial\Omega\in C^3$ to $\partial\Omega\in C^2$, we need regularity for the distance functions to the boundary of the domains. The following is proved in Li-Nirenberg~~\cite{MR2305073} for $\partial\Omega\in C^{k,\begin{align}l}$ for $k\geq 2$ and $0<\begin{align}l\leq1$ and stated in Spruck~~\cite{MR2351645} for $C^2$ case and proved in Crasta-Malusa~~\cite{MR2336304} for $C^2$ boundary.
We provide a proof here, including a stability property for our purpose. See Gilbarg-Trudinger~~\cite{MR1814364} and Krantz-Park~~\cite{MR614221} when domains are in ${\bf R}^N$.
\begin{prop}[Li-Nirenberg~~\cite{MR2305073}, Spruck~~\cite{MR2351645}]\lambdabel{LNS}
Let $s\in[2,\infty]$. Let $M$ be
a smooth Riemannian manifold $M$. Let $\Omega\subset M$ be a bounded domain with $C^s$ (resp. $\Lambda^s$ with $s>2$) boundary. Let $\rho$ be the signed distance function of $\partial\Omega$. There is $\delta=\delta(\Omega)>0$ so that $\rho\in C^s$ (resp. $\Lambda^s$ with $s>2$) in $B_\deltata(\partial\Omega)=\{x\in M\colon\operatorname{dist}(x,\partial\Omega)<\delta\}$. Furthermore, $\delta(\Omega)$ is upper-stable under small $C^2$ perturbations of $\partial\Omega$.
\epsilonnd{prop}
\begin{proof}
We are given a smooth Riemannian metric on $M$. Let $N$ be a subset of $M$. Let $\begin{gather}mma_p\colon [0,d]\to M$ be a geodesic connecting $\begin{gather}mma(0)=p\in M\setminus\overline N$ and $p^*=\begin{gather}mma(d)\in\overline N$. Suppose that $\begin{gather}mma$ is normal, i.e. $|\begin{gather}a_p'|=1$ and length $|\begin{gather}mma_p|$ of $\begin{gather}mma_p$ equals $\operatorname{dist}(p,N)$. Then $$|\begin{gather}mma_p([t,t'])|=t'-t, \quad \operatorname{dist}(\begin{gather}mma_p(t),N)=d-t.
$$
Thus if $N$ is a $C^1$ hypersurface near $p^*\in M$, then $\begin{gather}mma_p$ is orthogonal to $N$ at $p^*$.
We recall some facts about geodesic balls; see~~\cite{MR1138207}*{Chap.~3, Sect.~4}:
a) For $p\in M$, there is $0<r(p)\leq\infty$ so that the geodesic ball $B:=B_r(p)$, centered at $p$ with radius $r$, is strictly geodesic convex for $0<r<r(p)$. Specifically, any two points $p_0,p_1\in B$ are connected by a unique shortest normal geodesic in $M$ and the geodesic is contained in $B$. Here the uniqueness is up to a reparameterization $t\to \pm t+c$.
b) $\partial B$ is a smooth compact hypersurface.
Let $K$ be a compact set in $M$. By a), we have $r(K):=\inf_{p\in K}r(p)>0$. We can cover $K$ by finitely many open sets $U_i$ and choose coordinate chart $x_i$ on $U_i$ such that if a normal geodesic $\begin{gather}mma$ is contained in $K$ then $|(x_i~\citerc\begin{gather}mma)^{(j)}(t)|\leq C_j(K)$ wherever $x_i~\citerc\begin{gather}mma(t)$ is defined.
This implies that if $p\in N$, and $N\cap B$ is closed in $B$, then any point $q\in B_{r/2}(p)$ is connected to a point $q^*\in N\cap B$ via a geodesic $\begin{gather}mma_q$ in $B$ with length $\operatorname{dist}(q,N)$. However, $q^*$ may not be unique.
Let us use the above facts for $N=\partial\Omega$. Set $r_0=r(\partial\Omega)$. Let $\nu$ be the unit inward normal of $\partial\Omega$ with respect to the Riemann metric and choose sign so that $\rho<0$ in $\Omega$. Then $\nu$ is $C^{s-1}$ on $\partial\Omega$. Let $\begin{gather}mma(t,p)$ be the geodesic through $p\in\partial\Omega$ with $\partial_t\begin{gather}mma(0,p)=\nu(p)$ and $|t|<r_0$. Then $\begin{gather}mma$ is $C^{s-1}$ on $(-r_0,r_0)\times\partial \Omega$. Since $s\geq2$, $\partial_t\begin{gather}mma(0,p)$ is non zero, normal to $\partial\Omega$, and $\begin{gather}mma$ fixes $\{0\}\times\partial\Omega$ pointwise, then the Jacobian of $\begin{gather}mma$ is non-singular. By the inverse mapping theorem, there exists a unique solution $( \tilde\rho,P)\in{\bf R}\times\partial\Omega$ satisfying
\epsilonq{gPx}
\begin{gather}mma( \tilde\rho(x),P(x))=x, \quad |\tilde\rho(x)|<r_1
\epsilonnd{equation}
for $x\in B_{r_2}(\partial\Omega)$. Here $0<r_2<r_1<r_0$ and $r_1,r_2$ are sufficiently small. Furthermore, $\tilde\rho,P$ are in $C^{s-1}(B_{r_2}(\partial\Omega))$.
We want to show that $\tilde\rho=\rho$ on $B_{r_2}(\partial\Omega)$. Fix $x\in B_{r_2}(\partial\Omega)$ and take $x^*\in\partial\Omega$ with $\operatorname{dist}(x,x^*)=\rho(x)(=\operatorname{dist}(x,\partial\Omega))$. Since $r_2<r(\partial\Omega)$, then $x$ is connected to the center $x^*$ by a geodesic $\tilde\begin{gather}a$ in the geodesic ball $B_{r_2}(x^*)$. Since $\operatorname{dist}(x,x^*)=\rho(x)$, then $\tilde\begin{gather}mma$ is orthogonal to $\partial\Omega$ at $x^*$. Then $\tilde\begin{gather}mma$ must be contained in the normal geodesic $\begin{gather}mma(\cdot,x^*)$ with tangent vector $\nu(x^*)$. Next we choose the parametrization of $\tilde \begin{gather}a$ so that $\tilde\begin{gather}mma'(0)=\nu(x^*)$. We get
$$
\begin{gather}mma(\rho( x),x^*)=\tilde \begin{gather}mma(\rho(x))=x, \quad\tilde\begin{gather}mma(0)=x^*=\begin{gather}mma(0,x^*),\quad
|\rho(x)|=\operatorname{dist}(x,x^*)<r_2.
$$
By the uniqueness of solution to \re{gPx}, we conclude that $\tilde\rho(x)=\rho(x)$ (and $x^*=P(x)$, $|\tilde\rho(x)|<r_2$).
Next we verify $\rho\in C^s$. Recall that the vector field $X_1(x):=\partial_t\begin{gather}mma(\rho(x),P(x))$ is $C^{s-1}$ in $x=\begin{gather}mma(\rho(x),P(x))$. Fix $x_0\in B_{r_2}(\partial\Omega)$. In a small neighborhood $U$ of $x_0$ in $B_{r_2}(\partial\Omega)$, we use Gram-Schmidt orthogonalization to find pointwise linearly independent vector fields $X_2,\dots, X_n$ of class $C^{s-1}$ that are orthogonal to $X_1$. We already know that $\rho=\tilde\rho\in C^{s-1}\subset C^{1}$. This allows us to compute a directional derivative of $\rho$ via any $C^1$ curve that is tangent to the direction. Since $\begin{gather}mma$ is normal, then $X_1\rho=1$. Let $j>1$ and we want to show that $X_j\rho=0$. At $x\in U\setminus\partial\Omega$, Gauss lemma says that $X_2,\dots, X_n$ are tangent to the smooth geodesic sphere $\partial B_{|\rho(x)|}(P(x))$. We have $|\rho(x)|=\operatorname{dist}(x,\partial\Omega)>0$ and
$$
|\rho(q)|=\operatorname{dist}(q,\partial\Omega)\leq \operatorname{dist}(q,P(x))=|\rho(x)|, \quad \forall q\in\partial B_{|\rho(x)|}(P(x)).
$$
Hence on this geodesic sphere, the $C^{s-1}$ function $\rho$ attains a local extreme at $x$. This shows that $X_j\rho=0$ on $U\setminus\partial\Omega$. By continuity, $X_j\rho(x)=0$ on $U$.
Therefore all $X_i\rho$ are $C^\infty$ functions on $U$. As observed by Spruck~~\cite{MR2305073}, since all $X_i$ are $C^{s-1}$, then $\rho\in C^{s}(U)$. Therefore, $\rho\in C^{s}(B_{r_2}(\partial\Omega))$.
Finally, the stability of $\operatorname{dist}(\cdot, \partial\Omega)$ near $\partial\Omega$ is a consequence of the geodesic equations of the second-order ODE system. We leave the details to the reader.
\epsilonnd{proof}
\setcounter{thm}{0}\setcounter{equation}{0}
\section{Stability of $L^2$ solutions on pseudoconvex manifolds with $C^2$ boundary satisfying condition $a_q$}\lambdabel{sec:vb}
Let $\Omega$ be a relatively compact domain in a complex manifold $X$. Let $V$ be a holomorphic vector bundle on $X$. Let $f$ be a $V$-valued $(0,q)$ form on $\Omega$. Suppose that $\overline\partial u=f$ can be solved on $\Omega$ for some $u\in L^2_{loc}(\Omega)$ and $f=\tilde f+\overline\partial v$ for $v\in L_{loc}^2(\Omega)$ and $\tilde f$ is a $\overline\partial$ closed form on a larger domain $\Omega'$ containing $\overline\Omega$. We want to know if there exists a neighborhood $\tilde\Omega$ of $\overline\Omega$, that is independent of $f$
such that $\tilde f=\overline\partial \tilde u$ for some $\tilde u\in L^2_{loc}(\tilde\Omega)$. If such a domain $\tilde \Omega$ exists, we say the solvability of the $\overline\partial$-equation on $\Omega$ is \epsilonmph{stable}.
This stability is proved by H\"ormander~~\cite{MR0179443} when $\Omega$ is an $a_q$ domain and $V$ is the trivial bundle and by Andreotti-Vesentini~~\cite{MR0175148}*{Lemma 29, p. 122} for vector bundles on domains that are strictly $(n-q)$ convex with smooth boundary. For completeness, we sketch a proof for the case of vector bundle. We also take this opportunity to relax the boundary condition $\partial\Omega\in C^3$ to the minimum $C^2$ smoothness and we also formulate a stability for the $L^2$ bounds of the $\overline\partial$ equation on $a_q$ domains.
For the reader's convenience, we will give our statements for the vector bundle case the references in~~\cite{MR0179443} for a reader to locate them easily.
We fix smooth hermitian metrics on $X$ and $V$. Cover $\overline \Omega $ by finitely open sets $U_j$ of $X$. We assume that each $U_j$ is biholomorphic to the unit ball in ${\bf C}^n$
by a coordinate map $z_j$ which is biholomorphic on $\overline{U_j}$.
In what follows, we will denote by $U$ one of $U_j'$s or their subdomains.
Let $\{e_1,\dots, e_m\}$ be a smooth unitary basis of $V$ on $U$. Let $u=\sum u^\mu e_\mu\in C^1_{(0,q-1)}(\Omega,V,loc)$. We have
\epsilonq{}
\overline\partial u=Au +Ru, \quad A(u^\nu e_\nu)=(\overline\partial u^\nu) e_\nu,
\epsilonnd{equation}
where $Au^\nu=\overline\partial u^\nu$ is as in~~\cite{MR0179443} for the scalar case, and $Ru$ involves no derivatives of $u$, i.e. $Ru$ is of {\it order zero} in $u$ with smooth coefficients.
Therefore, the principal part $A$ of $\overline\partial$
is locally {\it diagonal}.
This is an important property so that the proofs for the scalar case can be carried out to the vector bundles case without difficulty.
Let $\omega_1,\dots, \omega_n$ be unitary smooth $(1,0)$ forms on $X$. For a $V$-valued $(p,q)$-forms $f=\sum f^\nu e_\nu$, define
$$
\left|\sum f^\nu e_\nu(x)\right|=\sum_\nu{\sum_{I,J}}^{\prime}|f^\nu_{I,J}(x)|^2, \quad f^\nu= {\sum_{I,J}}^{\prime}f^\nu_{I,J}\omega^I\wedge\overline\omega^J.
$$
We take the volume form on $X$ as
$$
d\upsilon=
(\sqrt{-1})^n\omega^1\wedge\cdots\wedge \omega^n\wedge\overline\omega^1\wedge\cdots\wedge\overline\omega^n.
$$
For $q\geq0$, let $L_{(p,q)}^2(\Omega ,V,loc)$ and $\Lambda^r_{(p,q)}(\Omega ,V)$ be the spaces of $V$-valued $(p,q)$ forms of which the coefficients on $U$ are in $L^2_{loc}(\Omega\cap U)$, $\Lambda^r({ \Omega\cap U})$, respectively. Let $\mathcal D^{(p,q)}(\Omega ,V)$ be the space of smooth $V$-valued $(p,q)$ forms of which the coefficients are in $\mathcal D(\Omega )$, i.e. smooth functions with compact support. Let $\mathcal D'_{(p,q)}(\Omega ,V)$ be the space of $V$-valued $(p,q)$ forms of which the coefficients are distributions in $\Omega $.
If $\varphi$ is a real $L^\infty$ function in $\Omega$, let $L_{(p,q)}^2(\Omega ,V,\varphi)$ be the space of sections of $V$-valued $(p,q)$ forms satisfying
$$
\|f\|_{\varphi}^2=\int_\Omega |f(x)|^2e^{-\varphi(x)}\, d\upsilon(x)<\infty.
$$
We will write $\jq{\cdot,\cdot}_\varphi$ the induced hermitian product and $\|\cdot\|_\varphi$ the norm on $L^2_{(p,q)}(\Omega ,V, \varphi)$. The latter is independent of $\varphi$ as sets, and
the norms are equivalent for all weights $\varphi\in L^\infty$.
Throughout the appendix, we assume $q\geq1$. The operator $\overline\partial$ defines a linear, closed, densely defined operator
$$
T\colon L^2_{(p,q-1)}(\Omega, V ,\varphi)\to L_{(p,q)}^2(\Omega ,V,\varphi)
$$
while $Tu=f$ holds if $\overline\partial u=f$ in the sense of distributions.
We abbreviate $
T=T_{q}, S=T_{q+1}.
$
We will write $T_q$ for $T$ if needed. Note that the domain $D_T$ and range $R_T$ are independent of $\varphi\in L^\infty$.
For $f\in L_{0,q}^2(\Omega,V,\varphi)$, write $v=T^* f$
if
$
\jq{u,v}_\varphi=\jq{\overline\partial u, f}_\varphi
$
for all $u\in D_{T}$.
Throughout the section, we assume that $\partial\Omega\in C^2$.
By \rp{LNS},
$\Omega$ has a $C^2$ defining function $\rho$ in $X$ satisfying
\epsilonq{2dr=1}
2|\overline\partial\rho|=1 \quad \text{on $\partial\Omega$}.
\epsilonnd{equation}
We also assume that $\varphi\in Lip(\Omega)$. Then $D_{T^*}$ is independent of $\varphi$, while $R_{T^*}$ depends on $\varphi$.
\begin{rem}
As in~~\cite{MR0179443}, $|T^*f|_\psi$ is {\it always} referred to as the dual with respect to $\psi$, where $\psi$ will be chosen appropriately. For clarity, we write $T_\psi^*$ for $T^*$ when $\psi$ needs to be specified.
\epsilonnd{rem}
Using integration by parts, we can verify that if $f\in C^1_{(p,q)}(\overline \Omega,V )$ has compact support in $U\cap\overline \Omega $, then $f\in D_{T^*}$ if and only if
\epsilonq{bvc}
\sum_{j=1}^nf^\nu_{I,jK}\mathbb{D}D{\rho}{\omega^j}=0, \quad\text{on $U\cap\partial \Omega $}, \quad \nu=1,\dots, m.
\epsilonnd{equation}
Define
$
\mathcal D^1_{T^*}(\overline\Omega):=C^1(\overline\Omega,V)\cap D_{T^*}.$
We have from~~\cite{MR0179443}*{p. 148}
$$
T^* f=B f +R^*f, \quad B f:=(B f^\nu)e_\nu,
\quad\text{on $U$}
$$
with
$$
B f^\nu=-\sum_j{\sum_{J,K}}'\delta_j f^\nu_{I,jK}\omega^I\wedge\overline\omega^K.
$$
Thus $R^*$, {\it independent} of $\varphi$, is an operator of the zero-th order with smooth coefficients, and the $B$ is {\it diagonal} and its principle part is also independent of $\varphi$. Thus the boundary condition is principle and zero-th order.
In summary, we have
\pr{density}Let $\Omega$ be a relatively compact $C^2$ domain in $X$ and let $\varphi\in Lip(\Omega)$.
Then $D_{T^*}$ is independent of $\varphi$. Let $\psi\in L^\infty(\Omega)$ be a real function.
\bppp
\item For all $f=\sum f^\nu e_\nu\in C_{(p,q)}^1(U\cap\overline\Omega,V )$,
\begin{gather}\lambdabel{deco}
\left|\|Sf\|_\psi^2-\|Af\|_\psi^2\right|\leq C(\Omega)\|f\|_\psi^2.
\epsilonnd{gather}
\item
For all $f=\sum f^\nu e_\nu\in C_{(p,q)}^1(\overline \Omega,V )\cap D_{T^*}$ which have compact support in $U\cap\overline \Omega $,
\begin{gather}\lambdabel{deco+} \left|\|T^*f\|_\psi^2- \|B f\|_\psi^2\right|\leq C(\Omega)\|f\|_\psi^2.
\epsilonnd{gather}
\item $\mathcal D^1_{T^*}(\overline\Omega)$ is dense in $D_{T^*}\cap D_{S}$ w.r.t. the graph norm $f\to |f|_\psi+|Sf|_\psi+|T^*f|_\psi$.
\epsilonppp
Furthermore, the constants $C(\Omega)$, independent of $\varphi,\psi$, depend only on the diameter of $\Omega$.
\epsilonnd{prop}
Here the last assertion follows from ~\citete{MR0179443}*{p.~121}.
We also have
\begin{prop}[~\citete{MR0179443}*{eq.~(3.1.9)}]\lambdabel{MKH} Let $\Omega$ be a relatively compact $C^2$ domain in $X$. Let $\rho$ be the signed distance function of $\partial\Omega$. Let $\varphi\in C^{1,1}(\Omega)$.
For all $f\in C^1_{p,q}(\overline \Omega,V )$ with compact support in $U\cap\overline\Omega$, we have
\begin{align}\lambdabel{t4}
\|Af\|_\varphi^2+\|B f\|_\varphi^2&=\sum_{\nu=1}^{m}\|Af^\nu \|_\varphi^2+\|B f^\nu \|_\varphi^2\\
&=\sum_{\nu=1}^{m}(Q_1+Q_2+t_1+t_2+t_3+t_4)
(f^\nu,f^\nu)\nonumber\\
&=:(Q_1+Q_2+t_1+t_2+t_3+t_4)
(f,f),
\nonumber
\\
Q_1(f,f)&:=\sum_{I,J}\sum_j\int_{U\cap \Omega }\left|\mathbb{D}D{f_{I,J}}{\overline\omega^j}\right|^2e^{-\varphi}\, d\upsilon,\\
Q_2(f,f)&:= \sum_{I,K}\sum_{k,j}\int_{U\cap \Omega }\varphi_{j\overline k}f_{I,jK}\overline{ f_{I,kK}}e^{-\varphi}\, d\upsilon,
\epsilonnd{align}
\begin{align}
t_1(f,f)&:=\sum_{I,K}\sum_{k,j}\int_{U\cap\partial \Omega }\left(f_{I,jK}\mathbb{D}D{\rho}{\omega^j}\overline{\delta _kf_{I,kK}}-
f_{I,jK}\mathbb{D}D{\rho}{\overline\omega^k}\overline{\mathbb{D}D{f_{I,kK}}{\overline\omega^j}}\right)e^{-\varphi}\, d\sigma,\\
t_2(f,f)&:=\sum_{I,K}\sum_{k,j}\int_{U\cap \Omega }\left(f_{I,jK}\overline\sigma_j\overline{\delta _kf_{I,kK}}-
f_{I,jK}\sigma_k\overline{\mathbb{D}D{f_{I,kK}}{\overline\omega^j}}\right)e^{-\varphi}\, d\upsilon,\\
t_3(f,f)&:=\sum_{I,K}\sum_{i,j,k}\int_{U\cap \Omega } f_{I,jK}\overline c_{jk}^i\overline{\delta_if_{I,kK}} \, d\upsilon,\\
t_4(f,f)&:=-\sum_{I,K}\sum_{i,j,k}\int_{U\cap\partial \Omega } f_{I,jK} c_{jk}^i\overline{\delta_if_{I,kK}} \, d\upsilon.
\epsilonnd{align}
\epsilonnd{prop}
\begin{proof} The proof for $f\in C^2$ and $\varphi\in C^2$ is in~~\cite{MR0179443}. The case for $f\in C^1$ can be obtained by $C^1$ approximation of $C^2$ forms as in~~\cite{MR0179443}*{p.~101}.
We remark that computation from $C^2$ forms does not require the forms to be in $D_{T^*}$ and this allows us to apply the formulas or estimates to $C^1$ forms that are in the domain of $T^*$.
\epsilonnd{proof}
Let $\varphi$ be a $C^2$ real function defined in a neighborhood of $z_0\in X$. Let $1\leq q<n$. Recall from ~\cite{MR0179443}*{Def.~3.3.4} that $\varphi$ satisfies {\it the condition $a_q$} at $z_0$,
if \epsilonq{dpsi0}
\nabla\varphi(z_0)\neq0\epsilonnd{equation}
and the Levi-form $L_{z_0}\varphi$, the restriction of $H_{z_0}\varphi(t):=\sum\mathbb{D}D{^2\varphi}{\omega_j\overline\omega_k}(z_0)t_j\overline t_k$ on $T_{z_0}'\varphi:=\{t\in{\bf C}^n\colon\sum t_j\mathbb{D}D{\varphi}{\omega_j}(z_0)=0\}$,
has at least $q+1$ negative or at least $n-q$ positive eigenvalues.
Let $\mu_1(z)\leq\mu_2(z)\leq\cdots\leq\mu_{n-1}(z)$ be the eigenvalues of the Levi form $L_{z_0}\varphi$ and let $\lambda_1(z)\leq\lambda_2(z)\leq\cdots\leq\lambda_n(z)$ be the eigenvalue of the hermitian form $H_\zeta\varphi(t):=\sum\mathbb{D}D{^2\varphi}{\omega_j\overline\omega_k}t_j\overline t_k$. Recall that the minimum-maximum principle for the eigenvalues says that
$$
\lambda_j(z)=\min_{\dim V=j}\max_{v\in V, |v|=1}\{H_z\varphi (v)\}.
$$
Thus $\lambda_1(z)\leq\mu_1(z)\leq\cdots\leq\mu_{n-1}(z)\leq \lambda_n(z)$.
Let $r^-=\max(-r,0)$ for a real $r$. Then at $z_0$, condition $a_q$ is valid if and only if
$$
\mu_1+\cdots+\mu_q+\sum_{j=1}^{n-1}\mu_j^->0.
$$
If $\psi<\psi(z_0)$ is strictly pseudoconvex at $z_0$ then $\psi$ satisfies the $a_q$ condition for $q=0,\dots, n-1$.
Recall from~~\cite{MR0179443}*{Def. ~3.3.2} that $\psi$ satisfies {\it the condition $A_q$} at $z_0$, if
\re{dpsi0} holds
and
$$
\lambda_1(z_0)+\cdots+\lambda_q(z_0)+\sum_{j=1}^{n-1}\mu_j^-(z_0)>0.
$$
When needed, we denote the above eigenvalues $\lambda_j,\mu_k$ by $\lambda_j(z_0,\varphi),\mu_k(z_0,\varphi)$.
Let us prove the following estimate for \epsilonmph{weighted} eigenvalues.
\begin{lemma}[~\citete{MR0179443}*{Lem.~3.3.3}]\lambdabel{3.3.3}
Suppose that $\varphi$ satisfies the condition $a_q$ at $\zeta$. Then $e^{\tauu\varphi}$ satisfies the condition $A_q$. More specifically, there exists $c(\varphi)>0$ and $\tauu_0(\varphi)$ such that for $
\tauu>\tauu_0$
\epsilonq{newAq}e^{-\tauu\varphi(\zeta)}\left \{ \lambda_1(\zeta,e^{\tauu\varphi})+\cdots+\lambda_q(\zeta,e^{\tauu\varphi})+\sum_{j=1}^{n-1}\mu_j^-(\zeta,e^{\tauu\varphi})\right\}
>c(\varphi)\tauu.
\epsilonnd{equation}
Furthermore, $c(\varphi), \tauu_0(\varphi)$ are stable under small $C^2$ perturbation of $\varphi$.
\epsilonnd{lemma}
\begin{proof}The proof in~~\citete{MR0179443} uses a proof-by-contradiction argument. For stability, we need a direct proof. We have
$$\lambda_1(z_0)+\cdots+\lambda_q(z_0)+\sum_{j=1}^{n-1}\mu_j^-(z_0)\geq
\lambda_1(z_0)+\cdots+\lambda_q(z_0)+\sum_{j=1}^{n-1}\mu_j^-(z_0).
$$ For $t$, decompose $t\cdot\f{\partial}{\partial \zeta}=t'+t''$ where $t'$ is in the complex tangent space $T'_\zeta \varphi$ and $t''$ is in its orthogonal complement. We have
$$
\tilde H^\tauu_\zeta(t):=\tauu^{-1} e^{-\tauu\varphi}He^{\tauu\varphi}_\zeta(t)= H_\zeta\varphi(t)+
\tauu|\partial\varphi(\zeta)|^2|t''|^2.
$$
Restricted on $T_\zeta'$, the above is still the Levi form $L_\zeta\varphi$ of which the eigenvalues are $\mu_1\leq\cdots\leq\mu_{n-1}$.
Let $\lambda_1(\tauu),\dots,\lambda_n(\tauu)$ be the eigenvalues of the above quadratic form. We still have
$\lambda(\tauu)\leq\mu_1\leq\cdots\leq\mu_n\leq\lambda_n(\tauu)$. For any $\delta>0$, we choose $\tauu_0$ so that
$$
\tauu|\partial\varphi(\zeta)|^2\delta^2>\lambda_{n}(0)+1, \quad \forall\tauu>\tauu_0.
$$
Then $\lambda_1(\tauu)\geq H^\tauu_\zeta\varphi(t)\geq \mu_1-\epsilon$ when $\delta$ is sufficiently small. Analogously, we get $\lambda_j(\tauu)\geq \mu_j-\epsilon$ for $j=1,\dots, n-1$ when $\delta$ is sufficiently small. We can choose $\epsilon$ depending on
$
\mu_1+\cdots+\mu_q+\sum_{j=1}^{n-1}\mu_j^-$
and modulus of continuity of $\partial^2\varphi$ to obtain \re{newAq}.
\epsilonnd{proof}
\begin{thm}[~\citete{MR0179443}*{Thm.~3.3.1}]\lambdabel{3.3.1}
Let $\Omega$ be a relatively compact $C^2$ domain in $X$. Let $z_0\in\Omega$. Suppose that $\varphi\in C^{2}(\overline\Omega)$. Then
\epsilonq{eq3.3.1}
\tauu \|f\|^2_{\tauu\varphi}\leq C_\varphi^*\left\{ \|T_{\tauu\varphi}^*f\|_{\tauu\varphi}^2+\|S f\|^2_{\tauu\varphi} + |f|_{\tauu\varphi}^2\right\}
\epsilonnd{equation}
holds for some neighborhood $U\subset\Omega$ of $z_0$, some $C_\varphi,\tauu_\varphi$, and all $\tauu>\tauu_\varphi$ and all $f\in C_{(p,q)}^1(\overline \Omega,V ) $ with compact support in $U\cap\overline\Omega$, if and only if the hermitian form $\sum\varphi_{jk}(z_0)t_j\overline t_k$ on ${\bf C}^n$ has either at least $q+1$ negative or at least $n-q+1$ positive eigenvalues.
Furthermore, we can take
$$
C_\varphi^*=\f{C(\Omega)}{\min_{z_0\in \Omega\setminus\Omega_c}(\sum_1^{n-1}\mu^-_j(z_0,\varphi)+\sum_{j=1}^q\mu_j(z_0,\varphi))}
$$
where $\mu_1(z_0,\varphi)\leq \cdots\leq \mu_{n-1}(z_0,\varphi)$ are eigenvalues of
$L_{z_0}\varphi$
with respect the hermitian metric on $X$, while $U$ depends on the modulus of continuity of $\partial^2 \varphi$. The constants $C_\varphi^*, \tauu_\varphi$ are stable under $C^2$ perturbation of $\partial\Omega$.
\epsilonth
\begin{proof}Take any $g\in C^1_{(p,q)}(\overline\Omega)\cap D_{T^*}$ with compact support in $U\cap\overline\Omega$.
Apply \re{eq3.3.1} to $f=ge_1$, which is actually proved in~~\citete{MR0179443} for the $g$; see (3.3.4), (3.3.5), (3.3.6) in~~\cite{MR0179443}. By \re{deco} we get
\begin{align}n
\tauu \|g\|^2_{\tauu\varphi}&\leq C_\varphi(\|T^*(ge_1)\|_{\tauu\varphi}^2+\|\overline\partial (ge_1)\|^2_{\tauu\varphi})\\
&\leq C_\varphi(\|T^*g\|_{\tauu\varphi}^2+\|\overline\partial g\|^2_{\tauu\varphi}+C_1\|g\|^2_{\tauu\varphi}),
\epsilonnd{align*}
where $C_\varphi$ depends on the eigenvalues of $\varphi$ and $C_1$ is independent of $\tauu$ and $\varphi$. Assume further that $\tauu>2C_\varphi C_1$. Then we get \re{eq3.3.1} in which $f,C_\varphi$ are replaced by $g, 2C_0$.
Note that the constant $C$ in \re{deco} is independent of $\tauu$. By ~\citete{MR0179443}*{Thm 3.3.1}, we get the eigenvalue condition. Assume that the eigenvalue condition holds. Then \re{eq3.3.1} holds when $f$ is
replaced by $f^\nu$ for each $\nu$. By \re{deco} again, we get \re{eq3.3.1} by adjusting $\tauu_0$ and $C_0$.
\epsilonnd{proof}
\begin{thm}[~\citete{MR0179443}, Thm.~$3.3.5$]\lambdabel{3.3.5}Let $\Omega$ be a relatively compact $C^2$ domain in $X$. Let $\varphi$ satisfy condition $A_q$ at $z_0\in\overline\Omega$. If $z_0\in\partial\Omega$ assume further that $\varphi<\varphi|_{\partial\Omega}=\varphi(z_0)$ in $\Omega$. Then there are a neighborhood $U$ of $z_0$
and a constant $C_\varphi^*$ such that for all convex increasing function $C^2$ function $\chi$ in ${\bf R}$ we have
\epsilonq{eq3.3.11}
\int\chi'(\varphi)|f|^2e^{-\chi(\varphi)}\, d\upsilon\leq C^*_\varphi(\|T^*f\|_{\chi(\varphi)}^2+
\|\overline\partial f\|_{\chi(\varphi)}^2+\|f\|_{\chi(\varphi)}^2)
\epsilonnd{equation}
for all $f\in C_{(p,q+1)}^1(\overline \Omega,V )\cap D_{T^*}$ with compact support in $U\cap\overline\Omega$.
\epsilonth
\begin{proof}We apply the scalar version of the result as in the proof of \rt{3.3.1}. The proof in ~\citete{MR0179443} is valid via $C^1$ density.
\epsilonnd{proof}
By partition of unity, the above yields the following.
\begin{prop}[~\citete{MR0179443}*{Prop.~3.4.4}]\lambdabel{3.4.4} Let $\Omega$ be a relatively compact $C^2$ domain in $X$.
Let $\varphi<0$ in $\Omega$ and vanish in $\partial\Omega$ with $\varphi\in C^2(\overline\Omega)$. Let $\Omega_a=\{z\in\Omega\colon\varphi(z)<a\}$. Suppose that $\varphi$ satisfies condition $A_q$ in $\overline\Omega\setminus\Omega_{-c}$ for some $c>0$. Then there are a compact subset $K$ of $\Omega_{-c}$ and a constant $C_\varphi^*$ such that for all convex increasing function $\chi\in C^2({\bf R})$
\epsilonq{eq3.4.2}
\int_{\Omega\setminus K}\chi'(\varphi)|f|^2e^{-\chi(\varphi)}\, d\upsilon\leq C^*_\varphi(\|T^*f\|_{\chi(\varphi)}^2+
\|S f\|_{\chi(\varphi)}^2+\|f\|_{\chi(\varphi)}^2)
\epsilonnd{equation}
holds for all $f\in C^1_{(p,q)}(\overline\Omega, V)\cap D_{T^*}$.
\epsilonnd{prop}
\begin{thm}[~\citete{MR0179443}*{Thm.~3.4.1}]\lambdabel{3.4.1} Let $\Omega$ be a relatively compact $C^2$ domain in $X$. Suppose that $\partial\Omega$ satisfies the condition $a_q$. Fix $C^2$ defining function $\rho$ of
$\Omega$ such that $\rho$ is the signed distance function to $\partial\Omega$ and fix
$\varphi=e^{\lambda\rho}$ with $\lambda$ sufficiently large. Then there exist compact subset $K$ of $\Omega$ and constant $\tauu_\varphi$
such that if $\tauu>\tauu_\varphi$
and $ f\in D_{S}\cap D_{T^*}\cap L^2_{p,q}(\Omega,V)$ we have
\epsilonq{eq3.4.1}
\int_{\Omega\setminus K}|f|^2e^{-\tauu\varphi}\leq\|T^*f\|_{\tauu\varphi}^2+\|S f\|_{\tauu\varphi}^2+\int_K|f|^2e^{-\tauu\varphi}\, d\upsilon.
\epsilonnd{equation}
The latter implies that
$R_{T}$ is closed and finite codimensional in $N_{S}$.
\epsilonth
\begin{proof}
Here we need to go through the proof of ~\citete{MR0179443}*{Thm.~3.4.1}.
There is a compact set $K$ in $\Omega$ such that
\epsilonq{tOms-}
\tauu\int_{\Omega\setminus K}|f|^2e^{-\tauu\varphi}\, d\upsilon\leq C^*_\varphi(\|T^*f\|^2_{\tauu\varphi}+\|Sf\|^2_{\tauu\varphi}+\|f\|^2_{\tauu\varphi}),
\epsilonnd{equation}
where $ C^*_\varphi$ is independent of $\tauu$. The above is proved in ~\citete{MR0179443}*{Thm.~3.4.1} when $V$ is trivial. Thus it also holds for any $V$ by \re{deco} and \re{deco+}.
We get
\re{eq3.4.1} for $f\in C^1(\overline\Omega, V)\cap D_{T^*}$ when $\tauu>2 C^*_\varphi$. By the density theorem, it holds for $f\in D_{T^*}\cap D_S$. The proof for the other direction in ~\citete{MR0179443}*{Thm.~3.4.1} is valid without any change.
\epsilonnd{proof}
So far, all the constants in the estimates are stable under $C^2$ perturbations of the domain $\Omega$ and these constants are explicit to some extent. The next constant is however not explicit since it comes from a proof by contradiction. Nevertheless, it leads no essential difficulty in our applications.
Fix $\begin{gather}a>2C_\varphi^*$, where $C_\varphi^*$ is in \re{eq3.4.1}.
Let $\chi_k\in C^2$ be an increasing sequence of convex increasing functions such that
\epsilonq{chik}
\chi_k(\tauu)=\begin{gather}mma\tauu,\quad \text{when $\tauu<c$}; {\bf Q}uad \chi'_k(\tauu)\to\infty, \quad \text{as $k\to\infty, \ \tauu>c$}.
\epsilonnd{equation}
Set $\varphi_k=\chi_k(\varphi)$. Note that $\varphi_k\in C^2(\overline\Omega)$. We have the following.
\begin{prop}[~\citete{MR0179443}*{Prop.~3.4.5}]\lambdabel{3.4.5}Let $\Omega, \Omega_{-c}, \varphi$ satisfy the hypotheses in \rpa{3.4.4}. In particular, $\varphi$ satisfies condition $A_q$ in $\overline\Omega\setminus\Omega_{-c}$.
There exist constants $C_*$ and $k_*$, depending on $\varphi,c,\begin{gather}a$, and the sequence $\chi_k$ such that for $k>k_*$
\epsilonq{eq3.4.4a} \|f\|^2_{\varphi_k}\leq C_*(\|T^*f\|_{\varphi_k}^2+\|S f\|^2_{\varphi_k})
\epsilonnd{equation}
provided $f\in D_{T^*}\cap D_{S}\cap L_{(p,q)}^2(\Omega,V)$ with $q\geq1$ and
\epsilonq{eq3.4.4b}
\int_{\Omega_{-c}}\ip{f,g}e^{-\begin{gather}a\varphi}\, d\upsilon=0, \quad \forall g\in N_{(p,q)}(\Omega_{-c}, V,\begin{gather}a\varphi).
\epsilonnd{equation}
Furthermore, $k_*, c, C_*$ are stable under $C^2$ perturbation of $\partial\Omega$ in the sense defined in \rta{3.4.6} $(ii)$ below.
\epsilonnd{prop}
\begin{proof}Fix $\begin{gather}a>2C^*_\varphi$ for the constant $C^*_\varphi$ in \rea{eq3.4.2}.
Define
\epsilonq{Npq}
N_{(p,q)}(\Omega_{-c}, V,\begin{gather}a\varphi):=N_{S_c}\cap N_{T_c^*},
\epsilonnd{equation}
where $T_c^*$ is the adjoint of $T_c=\overline\partial\colon L^2_{(p,q-1)}(\Omega_{-c},V,\begin{gather}a\varphi)\to L^2_{(p,q)}(\Omega_{-c},V,\begin{gather}a\varphi)$, while $S_c$ is the operator $\overline\partial\colon L^2_{(p,q)}(\Omega_{-c},V,\begin{gather}a\varphi) \to L^2_{(p,q+1)}(\Omega_{-c},V,\begin{gather}a\varphi)$.
Assume that estimate \re{eq3.4.4a} is false. Then
we can find $f_k\in D_{T^*}\cap D_{S}$ such that
\begin{gather}\lambdabel{eq3.4.5a}
\|f_k\|_{\varphi_k}=1, \quad \|T^*f_k\|_{\varphi_k}+\| Sf_k\|_{\varphi_k}<1/k,\\
\lambdabel{eq4.4.5aa}
\int_{\Omega_{-c}}\ip{f_k,g}e^{-\begin{gather}a \varphi}\, d\upsilon=0, \quad \forall g\in N_{(p,q)}(\Omega_{-c}, V,\begin{gather}a\varphi).
\epsilonnd{gather}
By the density theorem, we may assume that $f_k\in C_{(p,q)}^1(\overline \Omega,V )\cap D_{T^*}$, while \re{eq3.4.5a} still holds and \re{eq4.4.5aa} is, however, weakened to
\epsilonq{eq3.4.5b}
\left|\int_{\Omega_c}\ip{f_k,g}e^{-\begin{gather}a\varphi}\, d\upsilon\right|<1/k, \quad \forall g\in N_{(p,q)}(\Omega_{-c}, V,\begin{gather}a\varphi), \quad |g|_{\Omega_{-c},\begin{gather}mma\varphi}\leq1.
\epsilonnd{equation}
(Compare \re{eq3.4.5a} and \re{eq3.4.5b} with ~\citete{MR0179443}*{eq.~(3.4.5)}.) Here we used $\varphi_k=\begin{gather}mma\varphi$ on $\Omega_{-c}$ and Cauchy-Schwarz inequality
$$
\left|\int_{\Omega_{-c}}\ip{f_k,g}e^{-\begin{gather}a\varphi}\, d\upsilon\right|=\left|\int_{\Omega_{-c}}\ip{f_k-f,g}e^{-\begin{gather}a\varphi}\, d\upsilon\right| \leq C\left\{\int_{\Omega}|f_k-f|^2e^{-\varphi_k}\, d\upsilon\right\}^{1/2}.
$$
The rest of proof in~~\citete{MR0179443} was stated for $f_k\in C^2$. However the arguments are valid for $f_k\in C^1$ without any change. We will not repeat here.
\epsilonnd{proof}
\begin{thm}[~\citete{MR0179443}*{Thm.~3.4.6}]\lambdabel{3.4.6}Let $\Omega$ be a relatively compact $C^2$ domain in $X$. Let $\Omega_{-c}, \varphi, k,\varphi_k$ be as in \rpa{3.4.4}. Let $V$ be a holomorphic vector bundle in $X$. Assume that $\varphi$ satisfies the condition $a_q$ in $\overline \Omega\setminus\Omega_{-c}$. There exist $k_*$ and $C_*$ satisfying the following.
\bppp\item If $f\in L^2_{(p,q)}(\Omega,V)$
and the equation
$\overline\partial u_0=f$ has a solution $u_0$ in $L^2(\Omega_{-c},V)$, then
it has a solution $u$ in $L^2_{(p,q-1)}(\Omega,V)$. In other words, the restriction $\overline H_{(p,q)}(\Omega, V)\to\overline H_{(p,q)}(\Omega_{-c}, V)$ is injective. Moreover,
\epsilonq{uCs}
\|u\|_{\varphi_k}\leq C_*\|f\|_{\varphi_k}, \quad k\geq k_*.
\epsilonnd{equation}
where $C_*, c, k_*$ are the constants in \rea{eq3.4.4a}.
\item Furthermore, $C_*,c, k_*$ are stable under $C^2$ perturbations of $\partial\Omega$ in the following sense: Fix $c_*:=c$. If $\Omega,\tilde\Omega$ have $C^2$ defining functions $\rho,\tilde \rho$ such that $\|\tilde\rho-\rho\|_2<\deltata$, then we have
\epsilonq{Omcs}
\Omega_{-c_*}\subset \tilde\Omega_{-c_*/2}\subset\Omega
\epsilonnd{equation}
for some $c_*>0$. If $\overline\partial u=f$ with $f\in L^2(\Omega)$ admits a solution $u_0\in L^2_{loc}(\tilde\Omega_{-c_*/2})$ then there is a solution $u\in L^2(\Omega_{c_*})$ such that $\overline\partial u=f$ on $\tilde\Omega$.
Furthermore,
$$
|u|_{\Omega,\varphi_k}\leq C_k^*|f|_{\Omega_{c_*},\varphi_k}, \quad k\geq k_*.
$$
\epsilonppp
Therefore, $C_k^*, c_*, k_*$ are independent of $\tilde\Omega$ and $\tilde\rho$, and $\delta$ depends on $c_*$ and $\varphi$.
\epsilonth
\begin{proof}The proof is identical to that of ~\citete{MR0179443}*{Thm.~3.4.6}, using \rp{3.4.5}.
\epsilonnd{proof}
We should mention that there is a detailed study in Lieb-Michel~~\cite{MR1900133}*{Chapt.~VIII, Sect.~8} on the stability of estimates for the $\overline\partial$-Neumann operator on $\Omega_c$, when $\Omega$ is a strictly pseudoconvex manifold with smooth boundary. In our case, we must treat a slightly more general situation where $\tilde \Omega$ can be any $C^2$ perturbations of $\Omega$.
We emphasize that we do not know if $C_k^*, k_*$ are stable under $C^2$ perturbations in the sense of \re{defupst}. Nevertheless, using Grauert's bumping method for $a_q$ domains, \rt{3.4.6} suffices our purposes.
\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{doi:#1}}
\newcommand{\arxiv}[1]{\href{https://arxiv.org/pdf/#1}{arXiv:#1}}
\def\MR#1{\relax\ifhmode\unskip\spacefactor3000 \space\fi
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{MR#1}}
\nocite{}
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\epsilonnd{biblist}
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\epsilonnd{document} |
\begin{document}
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\title{The parallel versus branching recurrences in computability logic\thanks{Supported by the NNSF $(60974082)$ of China.}}
\author{Wenyan Xu \ and \ Sanyang Liu}
\date{}
\maketitle
\begin{abstract} This paper shows that the basic logic induced by the parallel recurrence $\pst$ of Computability Logic (i.e., the one in the signature $\{\neg,\wedge,\vee,\pst,\pcost\}$) is a proper superset of the basic logic induced by the branching recurrence $\st$ (i.e., the one in the signature $\{\neg,\wedge,\vee,\st,\cost\}$). The latter is known to be precisely captured by the cirquent calculus system {\bf CL15}, conjectured by Japaridze to remain sound---but not complete---with $\pst$ instead of $\st$. The present result is obtained by positively verifying that conjecture. A secondary result of the paper is showing that $\pst$ is strictly weaker than $\st$ in the sense that, while $\st F$ logically implies $\pst F$, vice versa does not hold.
\end{abstract}
\noindent {\em MSC}: primary: 03B47; secondary: 03B70; 68Q10; 68T27; 68T15.
\
\noindent {\em Keywords}: Computability logic; Cirquent calculus; Interactive computation; Game semantics; Resource semantics.
\section{Introduction}\label{ssintr}
{\em Computability logic} (CoL), introduced by G. Japaridze \cite{Jap03,Japfin}, is a formal theory of interactive computational problems, understood as games between a machine and its environment (symbolically named as $\top$ and $\bot$, respectively). Formulas in it represent such problems; logical operators stand for operations on them; ``truth" means existence of an algorithmic solution, i.e. $\top$'s effective winning strategy; and validity is understood as truth under every particular interpretation of atoms.
Among the most important operators of CoL are {\em recurrence operators}, in their overall logical spirit reminiscent of the exponentials of linear logic. Recurrences, in turn, come in several flavors, two most natural and basic sorts of which are {\em parallel recurrence} $\pst$ and {\em branching recurrence} $\st$, together with their duals $\pcost,\cost$ defined by $\pcost F=\neg\pst\neg F$ and $\cost F=\neg\st\neg F$. Ample intuitive discussions and elaborations on the two sorts of recurrences and the relations between them were given in \cite{Japtur,Japfour,Japsep,Mosc}. However, finding syntactic characterizations of the logic induced by recurrences had been among the greatest challenges in CoL until the recent work \cite{Japtam,Japtam2}, where a sound and complete axiomatization, called {\bf CL15}, for the basic $(\neg,\wedge,\vee,\st,\cost)-$fragment of computability logic was constructed.\footnote{The soundness part was proven in \cite{Japtam}, and the completeness part in \cite{Japtam2}.} At the same time, the logical behavior of parallel recurrence $\pst$ still remains largely ununderstood. It is not even known whether the set of principles validated by $\pst$ is recursively enumerable. The present paper brings some initial light into this otherwise completely dark picture. It shows that the set of principles validated by $\pst,\pcost$ in combination with the basic operations $\neg,\wedge,\vee$ is a proper superset of the set of those validated by $\st,\cost$. This is achieved by positively settling Conjecture 6.3 of \cite{Japtam}, according to which {\bf CL15} continues to be sound---but not complete--- with $\pst$ and $\pcost$ instead of $\st$ and $\cost$. Further, to make our investigation of
the relationship between $\pst$ and $\st$ more complete, at the end of the paper we also prove that $\pst$ is strictly weaker than $\st$ in the sense that, while $\st F$ logically implies $\pst F$ (as shown in \cite{Japtur}), vice versa does not hold.
{\bf CL15} is a system built in {\em cirquent calculus}. The latter is a
refinement of sequent calculus. Unlike the more traditional proof theories that manipulate tree-like objects (formulas, sequents, hypersequents, etc.), cirquent calculus deals with graph-style structures called {\em cirquents} (the term is a combination of ``CIRcuit'' and ``seQUENT''), with its main characteristic feature being allowing to explicitly account for possible {\em sharing} of subcomponents between different subcomponents. The approach was introduced by Japaridze
in \cite{Cirq} as a new deductive tool for CoL and was further developed in \cite{Cirdeep,fromto,wenyan1,wenyan2} where a number of advantages of this novel sort of proof theory were revealed, such as high expressiveness, flexibility and efficiency.
In order to make this paper reasonably self-contained, in the next section we reproduce the basic concepts from \cite{Japfin,Japtam} on which the later parts of the paper will rely. An interested reader may consult \cite{Japfin,Japtam} for the associated motivations, detailed explanations and examples.
\section{Preliminaries}
The letter $\wp$ is used as a variable ranging over $\{\top,\bot\}$, with $\neg\wp$ meaning $\wp$'s adversary. A {\bf move} is a finite string over standard keyboard alphabet. A {\bf labmove} is a move prefixed (``labeled'') with $\top$ or $\bot$. A {\bf run} is a finite or infinite sequence of labmoves, and a {\bf position} is a finite run. Runs are usually delimited by ``$\langle$" and ``$\rangle$", with $\langle\rangle$ thus denoting the {\bf empty run}. For any run $\Gamma$, $\neg\Gamma$ is the same as $\Gamma$, with the only difference that every label $\wp$ is changed to $\neg\wp$.
A {\bf game}\footnote{The concept of a game considered in CoL is more general than the one defined here, with games in the present sense called {\em constant games}. Since we (for simplicity) only consider constant games in this paper, we omit the word ``constant" and just say ``game".} is a pair $A=({\bf Lr}^{A},{\bf Wn}^{A})$, where: (1) ${\bf Lr}^{A}$ is a set of runs satisfying the condition that a finite or infinite run $\Gamma$ is in ${\bf Lr}^{A}$ iff so are all of $\Gamma$'s nonempty finite initial segments.\footnote{This condition can be seen to imply that
the empty run $\langle\rangle$ is always in ${\bf Lr}^{A}$.} If $\Gamma\in{\bf Lr}^{A}$, then $\Gamma$ is said to be a {\bf legal run} of $A$; otherwise $\Gamma$ is an {\bf illegal run} of $A$. A move $\alpha$ is a {\bf legal move} for a player $\wp$ in a position $\Phi$ of $A$ iff $\langle\Phi,\wp\alpha\rangle\in{\bf Lr}^{A}$; otherwise $\alpha$ is an {\bf illegal move}. When the last move of the shortest illegal initial segment of $\Gamma$ is $\wp$-labeled, $\Gamma$ is said to be a $\wp${\bf -illegal run} of $A$. (2) ${\bf Wn}^{A}$ is a function that sends every run $\Gamma$ to one of the players $\top$ or $\bot$, satisfying the condition that if $\Gamma$ is a $\wp$-illegal run of $A$, then ${\bf Wn}^{A}\langle\Gamma\rangle=\neg\wp$. When ${\bf Wn}^{A}\langle\Gamma\rangle=\wp$, $\Gamma$ is said to be a $\wp${\bf -won} run of $A$.
The game operations dealt with in the present paper are $\neg$ (negation), $\vee$ (parallel disjunction), $\wedge$ (parallel conjunction), $\pst$ (parallel recurrence), $\pcost$ (parallel corecurrence), $\st$ (branching recurrence) and $\cost$ (branching corecurrence). Intuitively, $\neg$ is a role switch operator: $\neg A$ is the game $A$ with the roles of $\top$ and $\bot$ interchanged ($\top$'s legal moves and wins become those of $\bot$, and vice versa). Both $A\wedge B$ and $A\vee B$ are games playing which means playing the two components $A$ and $B$ simultaneously (in parallel). In $A\wedge B$, $\top$ is the winner if it wins in both components, while in $A\vee B$ winning in just one component is sufficient. Next, $\pst A$ is nothing but the infinite parallel conjunction $A\wedge A\wedge A\wedge\ldots$, and $\pcost A$ is nothing but the infinite parallel disjunction $A\vee A\vee A\vee\ldots$. Finally,
both $\st A$ and $\cost A$ are games playing which means simultaneously playing a continuum of copies (or ``threads") of $A$. Each copy/thread is denoted by an infinite bitstring and vice versa, where a {\bf bitstring} is a finite or infinite sequence of bits 0,1. Making a move $w.\alpha$, where $w$ is a finite bitstring, means making the move $\alpha$ simultaneously in all threads of the form $wy$. In $\st A$, $\top$ is the winner iff it wins in all threads of $A$, while in $\cost A$ winning in just one thread is sufficient. Again, it should be pointed out that the above is just a very brief and incomplete intuitive characterization. See \cite{Japfin} for more.
Let $\Gamma$ be a run and $\alpha$ be a move. The notation
\begin{center}
$\Gamma^{\alpha}$
\end{center}
will be used to indicate the result of deleting from $\Gamma$ all moves (together with their labels) except those that look like $\alpha\beta$ for some move $\beta$, and then further deleting the prefix ``$\alpha$" from such moves. For instance, $\langle\top 1.\alpha, \bot 2.\beta, \top 1.\gamma, \bot 2.\delta\rangle^{1.}=\langle\top\alpha, \top\gamma\rangle$.
Let $\Omega$ be a run and $x$ be an infinite bitstring. The notation
\begin{center}
$\Omega^{\preceq x}$
\end{center}
will be used to indicate the result of deleting from $\Omega$ all moves (together with their labels) except those that look like $u.\beta$ for some move $\beta$ and some finite initial segment $u$ of $x$, and then further deleting the prefix ``u." from such moves. For instance, $\langle\bot 10.\alpha, \top 111.\beta, \bot 1.\gamma, \bot 00.\alpha\rangle^{\preceq 111\ldots}=\langle\top\beta, \bot\gamma\rangle$.
The earlier-outlined intuitive characterizations of the game operators are captured by the following formal definition.
Below, $A$, $A_1$, $A_2$ are arbitrary games, $\alpha$ ranges over moves, $i\in\{1,2\}$, $u$ ranges over positive integers identified with its decimal representation, $w$ ranges over finite bitstrings, $x$ ranges over infinite bitstrings, $\Gamma$
is an arbitrary run, and $\Omega$ is any legal run of the game that is being defined.
\\
1. $\neg A$ ({\bf negation}) is defined by:
{\bf (i)} $\Gamma\in{\bf Lr}^{\neg A}$ iff $\neg\Gamma\in{\bf Lr}^{A}$.
{\bf (ii)} ${\bf Wn}^{\neg A}\langle\Omega\rangle=\top$ iff ${\bf Wn}^{A}\langle\neg\Omega\rangle=\bot$.
\\
2. $A_1\wedge A_2$ ({\bf parallel conjunction}) is defined by:
{\bf (i)} $\Gamma\in{\bf Lr}^{A_1\wedge A_2}$ iff every move of $\Gamma$ is $i.\alpha$ for some $i$,$\alpha$ and, for both $i$, $\Gamma^{i.}\in{\bf Lr}^{A_i}$.
{\bf (ii)} ${\bf Wn}^{A_1\wedge A_2}\langle\Omega\rangle=\top$ iff, for both $i$, ${\bf Wn}^{A_i}\langle\Omega^{i.}\rangle=\top$.
\\
3. $A_1\vee A_2$ ({\bf parallel disjunction}) is defined by:
{\bf (i)} $\Gamma\in{\bf Lr}^{A_1\vee A_2}$ iff every move of $\Gamma$ is $i.\alpha$ for some $i$,$\alpha$ and, for both $i$, $\Gamma^{i.}\in{\bf Lr}^{A_i}$.
{\bf (ii)} ${\bf Wn}^{A_1\vee A_2}\langle\Omega\rangle=\top$ iff, for some $i$, ${\bf Wn}^{A_i}\langle\Omega^{i.}\rangle=\top$.
\\
4. $\pst A$ ({\bf parallel recurrence}) is defined by:
{\bf (i)} $\Gamma\in \legal{\psti A}{}$ iff every move of $\Gamma$ is $u.\alpha$ for some $u$ and $\alpha$ and, for each such $u$, $\Gamma^{u.}\in\legal{A}{}$.
{\bf (ii)} $\win{\psti A}{}\seq{\Omega}= \top$ iff, for all $u$, $\win{A}{}\seq{\Omega^{u.}}= \top$.
\\
5. $\pcost A$ ({\bf parallel corecurrence}) is defined by:
{\bf (i)} $\Gamma\in \legal{\pcosti A}{}$ iff every move of $\Gamma$ is $u.\alpha$ for some $u$ and $\alpha$ and, for each such $u$, $\Gamma^{u.}\in\legal{A}{}$.
{\bf (ii)} $\win{\pcosti A}{}\seq{\Omega}= \top$ iff, for some $u$, $\win{A}{}\seq{\Omega^{u.}}= \top$.
\\
6. $\st A$ ({\bf branching recurrence})\footnote{The present version of branching (co)recurrence was introduced recently in \cite{Japface}. It is different from yet equivalent to (in all relevant respects) the older version found in \cite{Jap03,Japfin}. The same applies to $\cost$.} is defined by:
{\bf (i)} $\Gamma\in{\bf Lr}^{\sti A}$ iff every move of $\Gamma$ is $w.\alpha$ for some $w$,$\alpha$ and, for all $x$, $\Gamma^{\preceq x}\in{\bf Lr}^{A}$.
{\bf (ii)} ${\bf Wn}^{\sti A}\langle\Omega\rangle=\top$ iff, for all $x$, ${\bf Wn}^{A}\langle\Omega^{\preceq x}\rangle=\top$.
\\
7. $\cost A$ ({\bf branching corecurrence}) is defined by:
{\bf (i)} $\Gamma\in{\bf Lr}^{\costi A}$ iff every move of $\Gamma$ is $w.\alpha$ for some $w$,$\alpha$ and, for all $x$, $\Gamma^{\preceq x}\in{\bf Lr}^{A}$.
{\bf (ii)} ${\bf Wn}^{\costi A}\langle\Omega\rangle=\top$ iff, for some $x$, ${\bf Wn}^{A}\langle\Omega^{\preceq x}\rangle=\top$.
In what follows, we explain---formally or informally---several additional concepts relevant to our proofs.
(1) {\bf Static games}: CoL restricts its attention to a special yet very wide subclass of games termed ``static". Intuitively, static games are interactive tasks where the relative speeds of the players are irrelevant, as it never hurts a player to postpone making moves. A formal definition of this concept can be found in \cite{Japfin}, which we will not reproduce here as nothing in this paper relies on it. The only relevant for us fact, proven in \cite{Jap03,Japfin,Japface}, is that the class of static games is closed under the operations $\neg,\wedge,\vee,\pst,\pcost,\st,\cost$ (as well as any other game operations studied in CoL).
(2) {\bf EPM}: CoL understands $\top$'s effective strategies as interactive machines. Several sorts of such machines have been proposed and studied in CoL, all of them turning out to be equivalent in computing power once we exclusively consider static games. In this paper we only use one sort of such machines, called the {\em easy-play machine} ({\bf EPM}). It is a kind of a Turing machine with the additional capability of making moves, and has two tapes\footnote{Often there is also a third tape called the {\em valuation tape}. Its function is to provide values for the variables on which a game may depend. However, as we remember, in this paper we only consider constant games --- games that do not depend on any variables. This makes it possible to safely remove the valuation tape (or leave it there but fully ignore), as this tape is no longer relevant.}: the ordinary read/write {\em work tape}, and the read-only {\em run tape}. The run tape serves as a dynamic input, at any time (``{\bf clock cycle}") spelling the current position: every time one of the players makes a move, that move---with the corresponding label---is automatically appended to the content of this tape. The machine can make a (one single) move at any time, while its environment can make an (at most one) move only when the machine explicitly allows it to do so (this sort of an action is called {\bf granting permission} ).\footnote{In the more basic sort of machines
called {\em hard-play machines} ({\bf HPM}), the environment can make any number of moves at any time (needing no ``permission'' for that). It is known (\cite{Jap03,Japfin}) that the two sorts of machines win the same static games.}
(3) {\bf Strategies}:
Let ${\cal M}$ be an EPM. A {\em configuration} of ${\cal M}$ is a full description of the current state of the machine, the contents of its two tapes, and the locations of the corresponding two scanning heads. The {\em initial configuration } is the configuration where ${\cal M}$ is in its start state and both tapes are empty. A configuration $C'$ is said to be an {\em successor} of a configuration $C$ if $C'$ can legally follow $C$ in the standard sense, based on the (deterministic) transition function of the machine and accounting for the possibility of nondeterministic updates of the content of the run tape through environment's moves. A {\bf computation branch} of ${\cal M}$ is a sequence of configurations of ${\cal M}$ where the first configuration is the initial configuration, and each other configuration is a successor of the previous one. Each computation branch $B$ of ${\cal M}$ incrementally spells a run $\Gamma$ on the run tape, which is called the {\bf run spelled by} $B$. Subsequently, any such run $\Gamma$ will be referred to as a {\bf run generated by} ${\cal M}$. A computation branch $B$ of ${\cal M}$ is said to be {\bf fair} iff, in it, permission has been granted infinitely many times. An {\bf algorithmic solution} ({\bf $\top$'s winning strategy}) for a given game $A$ is understood as an EPM ${\cal M}$ such that, whenever $B$ is a computation branch of ${\cal M}$ and $\Gamma$ the run spelled by $B$, $\Gamma$ is a $\top$-won run of $A$, where $B$ should be fair unless $\Gamma$ is a $\bot$-illegal run of $A$. When the above is the case, we say that ${\cal M}$ {\bf wins} $A$.
Now about formulas and the underlying semantics. We have some fixed set of syntactic objects, called {\bf atoms}, for which $P$, $Q$, $R$ will be used as metavariables.
A {\bf formula} is built from atoms in the standard way
using the connectives $\neg$,$\vee$,$\wedge$,$\pst$,$\pcost$,$\st$,$\cost$, with $F\rightarrow G$ understood as an
abbreviation for $\neg F\vee G$ and $\neg$ limited only to atoms, where $\neg\neg F$ is understood as $F$, $\neg(F\wedge G)$ as $\neg F\vee\neg G$, $\neg(F\vee G)$ as $\neg F\wedge\neg G$, $\neg\pst F$ as $\pcost\neg F$, $\neg\pcost F$ as $\pst\neg F$, $\neg\st F$ as $\cost\neg F$, and $\neg\cost F$ as $\st\neg F$. A {\bf $(\neg,\wedge,\vee,\pst,\pcost)$-formula} is one not containing $\st$,$\cost$. Similarly, a {\bf $(\neg,\wedge,\vee,\st,\cost)$-formula} is one not containing $\pst$,$\pcost$. An {\bf interpretation} is a function $^*$ that sends every atom $P$ to a static game $P^*$, and extends to all formulas by seeing the logical connectives as the same-name game operations.
A formula $F$ is {\bf uniformly valid} iff there is an EPM ${\cal M}$, called a {\bf uniform solution} of $F$, such that, for every interpretation $^*$, ${\cal M}$ wins $F^*$.\footnote{Another sort of validity studied in CoL is multiform validity. A formula $F$ is {\bf multiformly valid} iff, for every interpretation $^*$, there is a machine that wins $F^*$. Since uniform validity is stronger than multiform validity, all soundness-style results that we are going to establish about uniform validity automatically extend to multiform validity as well. Partly for this reason, in this paper we will be exclusively interested in uniform validity.}
Throughout the rest of this paper, unless otherwise specified or suggested by the context, by a
``formula'' we will always mean a $(\neg,\wedge,\vee,\pst,\pcost)$-formula.
As noted in section 1, {\bf CL15} is built in {\em cirquent calculus}, whose formalism goes beyond formulas. Namely, a {\bf cirquent} is a triple $C=(\vec{F},\vec{U},\vec{O})$ where: (1) $\vec{F}$ is a nonempty finite sequence of formulas, whose elements are said to be the {\bf oformulas} of $C$. Here the prefix ``o" is used to mean a formula together with a particular occurrence of it in $\vec{F}$. For instance, if $\vec{F}=\langle G, H, H\rangle$, then the cirquent has three oformulas while only two formulas. (2) Both $\vec{U}$ and $\vec{O}$ are nonempty finite sequences of nonempty sets of oformulas of $C$. The elements of $\vec{U}$ are said to be the {\bf undergroups} of $C$, and the elements of $\vec{O}$ are said to be the {\bf overgroups} of $C$. Again, two undergroups (resp. overgroups) may be identical as sets (have identical {\bf contents}), yet they count as different undergroups (resp. overgroups) because they occur at different places in $\vec{U}$ (resp. $\vec{O}$). (3) Additionally, every oformula is required to be in at least one undergroup and at least one overgroup.
Rather than writing cirquents as ordered tuples in the above style, we prefer to represent them through (and identify them with) {\bf diagrams}. Below is such a representation for the cirquent that has four oformulas $E, F, G, H$, three undergroups $\{E,F\}$, $\{F\}$, $\{G,H\}$ and three overgroups $\{E,F,G\}$, $\{G\}$, $\{H\}$.
\begin{center}
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Each group in the diagram is represented by (and identified with) a $\bullet$, where the {\bf arcs} (lines connecting the $\bullet$ with oformulas) are pointing to the oformulas that the given group contains.
There are ten inference rules in {\bf CL15}. Below we reproduce those rules from \cite{Japtam} with $\st$ and $\cost$ rewritten as $\pst$ and $\pcost$, respectively. To semantically differentiate the two versions of {\bf CL15} (when necessary), we may use the name ${\bf CL15}(\st)$ for the system that understands (and writes) the recurrence operator as $\st$, and use ${\bf CL15}(\pst)$ for the system that understands (and writes) the recurrence operator as $\pst$.
{\bf Axiom (A):} Axiom is a ``rule" with no premises. It introduces the cirquent
\ \ \ \ \ \ \ $(\langle\neg F_1,F_1,\ldots,\neg F_n,F_n\rangle, \langle\{\neg F_1,F_1\},\ldots,\{\neg F_n,F_n\}\rangle, \langle\{\neg F_1,F_1\},\ldots,\{\neg F_n,F_n\}\rangle)$,\\
where $n$ is any positive integer, and $F_1,\ldots,F_n$ are any formulas. All rules other than Axiom take a single premise.
{\bf Exchange (E):} This rule comes in three versions: {\bf Undergroup Exchange}, {\bf Oformula Exchange} and {\bf Overgroup Exchange}.
The conclusion of Oformula Exchange is obtained by interchanging in the premise two adjacent oformulas $E$ and $F$, and
redirecting to $E$ (resp. $F$) all arcs that were originally pointing to $E$ (resp. $F$). Undergroup (resp. Overgroup) Exchange is the same, with the only difference that the objects interchanged are undergroups (resp. overgroups).
{\bf Duplication (D):} This rule comes in two versions: {\bf Undergroup Duplication} and {\bf Overgroup Duplication}. The conclusion of Undergroup Duplication is obtained by replacing in the premise some undergroup $U$ with two adjacent undergroups whose contents are identical to that of $U$. Similarly for Overgroup Duplication.
{\bf Merging (M):} The conclusion of this rule can be obtained from the premise by merging any two adjacent overgroups $O_1$ and $O_2$ into one overgroup $O$, and including in $O$ all oformulas that were originally contained in $O_1$ or $O_2$ or both.
{\bf Weakening (W):} For the convenience of description, we explain this and the remaining rules in the bottom-up view. The premise of this rule is obtained by deleting in the conclusion an arc between some undergroup $U$ with $\geq 2$ elements and some oformula $F$; if $U$ was the only undergroup containing $F$, then $F$ should also be deleted, together with all arcs between $F$ and overgroups; if such a deletion makes some overgroups empty, then they should also be deleted.
{\bf Contraction (C):} The premise of this rule is obtained by replacing in the conclusion an oformula $\pcost F$ by two adjacent oformulas $\pcost F$ and $\pcost F$, and including both of them in exactly the same undergroups and overgroups in which the original $\pcost F$ was contained.
{\bf Disjunction introduction ($\vee$):} The premise of this rule is obtained by replacing in the conclusion an oformula $E\vee F$ by two adjacent oformulas $E$ and $F$, and including both of them in exactly the same undergroups and overgroups in which the original $E\vee F$ was contained.
{\bf Conjunction introduction ($\wedge$):} According to this rule, if a cirquent (the conclusion) has an oformula $E\wedge F$, then the premise
can be obtained by splitting the original $E\wedge F$ into two adjacent oformulas $E$ and $F$, including both of them in exactly the same overgroups in which the original $E\wedge F$ was contained, and splitting every undergroup $\Gamma$ that originally contained $E\wedge F$ into two adjacent undergroups $\Gamma^{E}$ and $\Gamma^{F}$, where $\Gamma^{E}$ contains $E$ (but not $F$), and $\Gamma^{F}$ contains $F$ (but not $E$), with all other ($\neq E\wedge F$) oformulas of $\Gamma$ contained by both $\Gamma^{E}$ and $\Gamma^{F}$.
{\bf Recurrence introduction ($\pst$):} The premise of this rule is obtained by replacing in the conclusion an oformula $\pst F$ by $F$, with all arcs unchanged, and inserting a new overgroup $\Gamma$ that contains $F$ as its {\em only} oformula.
{\bf Corecurrence introduction ($\pcost$):} The premise of this rule is obtained by replacing in the conclusion an oformula $\pcost F$ by $F$, with all arcs unchanged, and additionally including $F$ in any (possibly zero) number of the already existing overgroups.
Below we provide illustrations for all rules, in each case an abbreviated name of the rule standing next to the horizontal line separating the premise from the conclusion. Our illustration for the axiom (the ``{\bf A}" labeled rule) is a specific cirquent where $n=2$; our illustrations for all other rules are merely examples chosen arbitrarily. Unfortunately, no systematic ways for schematically representing cirquent calculus rules have been elaborated so far. This explains why we appeal to examples instead.
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\end{picture}
\end{center}
The above are all ten rules of {\bf CL15$(\pst)$}. A {\bf CL15$(\pst)$-proof} (or simply a {\bf proof}) of a cirquent $C$ is a sequence $\langle C_1,\ldots,C_n\rangle$ of cirquents, where $n\geq 1$, such that $C_n=C$, $C_1$ is an axiom, and $C_{i}$ ($1< i\leq n$) follows from $C_{i-1}$ by one of the rules of {\bf CL15$(\pst)$}. For any formula $F$, the expression $F^{\clubsuit}$ is used to denote the cirquent $(\langle F\rangle,\langle\{F\}\rangle,\langle\{F\}\rangle)$. Then a {\bf CL15$(\pst)$-proof} (or simply a {\bf proof}) of a formula $F$ is stipulated to be a proof of the cirquent $F^{\clubsuit}$. A formula or cirquent $X$ is {\bf provable}, symbolically {\bf CL15}$(\pst)\vdash X$, iff it has a proof.
As mentioned, {\bf CL15}$(\st)$ is the same as {\bf CL15}$(\pst)$, only with $\st$,$\cost$ instead of $\pst$,$\pcost$.
\begin{theorem}\label{niu}
{\em (Japaridze \cite{Japtam,Japtam2})} A $(\neg,\wedge,\vee,\st,\cost)$-formula is uniformly valid iff it is provable in {\bf CL15}$(\st)$.
\end{theorem}
\section{A semantics of cirquents}
To prove the soundness of {\bf CL15}$(\pst)$, we need to extend the earlier-described semantics from formulas to cirquents.
\begin{notation}
Let $\Gamma$ be a run, $a$ be a positive integer, and $\vec{x}=x_1,\ldots,x_n$ be a nonempty sequence of $n$ {\em positive integers}. We will be using the notation
\begin{center}
$\Gamma^{[a;\vec{x}]}$
\end{center}
to mean the result of
\begin{itemize}
\item deleting from $\Gamma$ all moves (together with their labels) except those that look like $a;u_1,\ldots,u_n.\beta$ for some move $\beta$ and some sequence of $n$ {\em natural numbers} $u_1,\ldots,u_n$ satisfying the condition that whenever $u_i\neq 0$ $(i\in\{1,\ldots,n\})$, $u_i=x_i$, and
\item then further deleting the prefix ``$a;u_1,\ldots,u_n.$" from such moves.
\end{itemize}
For instance, $\langle\bot 1;1,1.\alpha, \top 1;1,2.\beta, \bot 1;1,0.\gamma, \bot 2;1,0.\delta\rangle^{[1;1,2]}=\langle\top\beta, \bot\gamma\rangle$.
\end{notation}
\begin{definition}
Let $^*$ be an interpretation, and $C=(\langle F_1,\ldots,F_k\rangle,\langle U_1,\ldots,U_m\rangle,\langle O_1,\ldots,O_n\rangle)$ be a cirquent. Then $C^*$ is the game defined as follows, where $\Gamma$ is an arbitrary run and $\Omega$ is any legal run of $C^*$.
\\
{\bf (i)} $\Gamma\in {\bf Lr}^{C^*}$ iff the following two conditions are satisfied:
\begin{itemize}
\item Every move of $\Gamma$ looks like $a;\vec{u}.\alpha$, where $\alpha$ is some move, $a\in\{1,\ldots,k\}$, and $\vec{u}=u_1,\ldots,u_n$ is a sequence of $n$ natural numbers such that, for every $j\in\{1,\ldots,n\}$, we have $u_j=0$ iff the overgroup $O_j$ does not contain the oformula $F_a$.
\item For every $a\in\{1,\ldots,k\}$ and every sequence $\vec{x}$ of $n$ positive integers, $\Gamma^{[a;\vec{x}]}\in {\bf Lr}^{F_a^{*}}$.
\end{itemize}
{\bf (ii)} ${\bf Wn}^{C^*}\langle\Omega\rangle=\top$ iff, for every $i\in\{1,\ldots,m\}$ and every sequence $\vec{x}$ of $n$ positive integers, there is an $a\in\{1,\ldots,k\}$ such that the undergroup $U_i$ contains the oformula $F_a$ and ${\bf Wn}^{F_a^*}\langle\Omega^{[a;\vec{x}]}\rangle=\top$.
\end{definition}
\begin{remark}\label{feb13a}
Intuitively, any legal run $\Omega$ of $C^*$ consists of parallel plays of countably infinite copies of each of the games $F_{a}^{*}$ ($1\leq a\leq k$). To every sequence $\vec{x}$ of $n$ positive integers corresponds a copy of $F_a^*$, and $\Omega^{[a;\vec{x}]}$ is the run played in that copy. We shall simply say {\bf the copy $\vec{x}$} of $F_a^*$ to mean the copy of $F_a^*$ which corresponds to the sequence $\vec{x}$. Now, consider a given undergroup $U_i$. $\top$ is the winner in $U_i$ iff, for every sequence $\vec{x}$ of $n$ positive integers, there is an oformula $F_a$ in $U_i$ such that $\Omega^{[a;\vec{x}]}$ is won by $\top$. Finally, $\top$ wins the overall game $C^*$ iff it wins in all undergroups of $C$. In fact, overgroups can be seen as generalized $\pst$s, with the only main difference that the former can be shared by several oformulas; undergroups can be seen as generalized disjunctions, with the only main difference that the former may have shared arguments with other undergroups.
\end{remark}
We say that a cirquent $C$ is
{\bf uniformly valid} iff there is an EPM $\cal M$, called a {\bf uniform solution} of $C$, such that, for every interpretation $^*$, $\cal M$ wins $C^*$.
\section{Main results}
\begin{lemma}\label{apr14a}
There is an effective function $f$ from EPMs to EPMs such that, for every EPM ${\cal M}$, formula $F$ and interpretation $^*$, if ${\cal M}$ wins $\pst F^*$, then $f({\cal M})$ wins $F^*$.
\end{lemma}
\begin{proof}
Our proof here almost literally follows the proof of Lemma 9.1 of \cite{Japtam}. It is known that affine logic proves $\pst P\rightarrow P$. At the same time, according to Theorem 37 of \cite{Japfin}, affine logic is sound with respect to uniform validity. So, the formula $\pst P\rightarrow P$ is uniformly valid. This almost immediately implies that there is an EPM ${\cal N}_0$ such that ${\cal N}_0$ wins $\pst F^*\rightarrow F^*$ for any formula $F$ and interpretation $^*$. Furthermore, by Proposition 21.3 of \cite{Jap03}, there is an effective procedure that, for any pair $({\cal N},{\cal M})$ of EPMs, returns an EPM $h({\cal N},{\cal M})$ such that, for any static games $A$ and $B$, if ${\cal N}$ wins $A\rightarrow B$ and ${\cal M}$ wins $A$, then $h({\cal N},{\cal M})$ wins $B$. So, let $f({\cal M})$ be the function satisfying $f({\cal M})=h({\cal N}_0,{\cal M})$. Then $f({\cal M})$ wins $F^*$.
\end{proof}
\begin{lemma}\label{apr14b}
There is an effective function $g$ from EPMs to EPMs such that, for every EPM ${\cal M}$, formula $F$ and interpretation $^*$, if ${\cal M}$ wins $(F^{\clubsuit})^*$, then $g({\cal M})$ wins $F^*$.
\end{lemma}
\begin{proof}
Again, it should be acknowledged that the present proof very closely follows
the proof of Lemma 9.2 of \cite{Japtam}, even though there are certain differences.
Every legal move of $(F^{\clubsuit})^*$ looks like $1;u.\alpha$ for some positive integer $u$ and move $\alpha$, while the corresponding legal move of $(\pst F)^*$ simply looks like $u.\alpha$, and vice versa. Consider an arbitrary EPM ${\cal M}$ and an arbitrary interpretation $^*$. Below we show the existence of an effective function $f$ such that, if ${\cal M}$ wins $(F^{\clubsuit})^*$, then (the strategy) $f({\cal M})$ wins $(\pst F)^*$.
We construct an EPM $f({\cal M})$ that plays $(\pst F)^*$ by simulating and mimicking a play of $(F^{\clubsuit})^*$ (called the {\bf imaginary play}) by ${\cal M}$ as follows. Throughout simulation, $f({\cal M})$ grants permission whenever the simulated ${\cal M}$ does so, and feeds its environment's response---in a slightly modified form described below---back to the simulated $\cal M$ as the response of ${\cal M}$'s imaginary adversary (this detail of simulation will no longer be explicitly mentioned later in similar situations). Whenever the environment makes a move $u.\alpha$ for some positive integer $u$ and move $\alpha$, $f({\cal M})$ translates it as the move $1;u.\alpha$ made by the imaginary adversary of ${\cal M}$, and ``vice versa":
whenever the simulated ${\cal M}$ makes a move $1;u.\alpha$ for some positive integer $u$ and move $\alpha$ in the imaginary play of $(F^{\clubsuit})^*$, $f({\cal M})$ translates it as its own move $u.\alpha$ in the real play of $(\pst F)^*$.
The effect achieved by $f({\cal M})$'s strategy can be summarized by saying that it synchronizes every copy of $F^*$ in the real play of $(\pst F)^*$ with the ``same copy" of $F^*$ in the imaginary play of $(F^{\clubsuit})^*$.
Let $\Gamma$ be an arbitrary run generated by $f({\cal M})$, and $\Omega$ be the corresponding run in the imaginary play of $(F^{\clubsuit})^*$ by ${\cal M}$. From our description of $f({\cal M})$ it is clear that the latter never makes illegal moves unless its environment or the simulated ${\cal M}$ does so first. Hence we may safely assume that $\Gamma$ is a legal run of $(\pst F)^*$ and $\Omega$ is a legal run of $(F^{\clubsuit})^*$, for otherwise either $\Gamma$ is a $\bot$-illegal run of $(\pst F)^*$ and thus $f({\cal M})$ is an automatic winner in $(\pst F)^*$, or $\Omega$ is a $\top$-illegal run of $(F^{\clubsuit})^*$ and thus ${\cal M}$ does not win $(F^{\clubsuit})^*$. Now, it is not hard to see that, for any positive integer $x$, we have $\Gamma^{x.}=\Omega^{[1;x]}$. Therefore, $f({\cal M})$ wins $(\pst F)^*$ as long as ${\cal M}$ wins $(F^{\clubsuit})^*$.
Finally, in view of Lemma \ref{apr14a}, the existence of function $g$ satisfying the promise of the present lemma is obviously guaranteed.
\end{proof}
A rule of {\bf CL15}$(\pst)$ (other than Axiom) is said to be {\bf uniform-constructively sound} iff there is an effective procedure that takes any instance $(A,B)$ (i.e. a particular premise-conclusion pair) of the rule, any EPM ${\cal M}_A$ and returns an EPM ${\cal M}_B$ such that, for any interpretation $^*$, whenever ${\cal M}_A$ wins $A^*$, ${\cal M}_B$ wins $B^*$. Axiom is uniform-constructively sound iff there is an effective procedure that takes any instance $B$ of (the ``conclusion" of) Axiom and returns a uniform solution ${\cal M}_B$ of $B$.
\begin{theorem}\label{mainth1}
All rules of {\bf CL15}$(\pst)$ are uniform-constructively sound.
\end{theorem}
\begin{proof}
In what follows, $A$ is the premise of an arbitrary instance of a given rule of {\bf CL15}$(\pst)$, and $B$ is the corresponding conclusion, except the case of Axiom where we only have $B$. We will prove that each rule of {\bf CL15}$(\pst)$ is uniform-constructively sound by showing that an EPM ${\cal M}_B$ can be constructed effectively from an arbitrary EPM ${\cal M}_A$ such that, for whatever interpretation $^*$,
whenever ${\cal M}_A$ wins $A^{\ast}$, ${\cal M}_B$ wins $B^{\ast}$. Since an interpretation $^{\ast}$ is never relevant in such proofs, we may safely omit it, writing simply $A$ instead of $A^{\ast}$ to represent a game. Next, in all cases the assumption that ${\cal M}_A$ wins $A$ will be implicitly made, even though it should be pointed out that the construction of ${\cal M}_B$ never depends on this assumption. Correspondingly, it will be assumed that ${\cal M}_A$ never makes illegal moves. Further, as in the proof of Lemma \ref{apr14b}, we shall always implicitly assume that ${\cal M}_B$'s adversary never makes illegal moves either. To summarize, when analyzing ${\cal M}_B$, ${\cal M}_A$ and the games they play, we safely pretend that illegal runs never occur.
{\bf (1)}\ Assume that $B$ is an axiom with $2n$ oformulas. An EPM ${\cal M}_B$ that wins $B$ can be constructed as follows. It keeps granting permission. Whenever the environment makes a move $a;\vec{w}.\alpha$, where $1\leq a\leq 2n$ and $\vec{w}$ is a sequence of $n$ natural numbers, ${\cal M}_B$ responds by the move $b;\vec{w}.\alpha$, where $b=a+1$ if $a$ is odd, and $b=a-1$ if $a$ is even. Then, for any run $\Gamma_B$ of $B$ generated by ${\cal M}_B$ and any sequence $\vec{x}$ of $n$ positive integers , we have $\Gamma_B^{[a;\vec{x}]}=\neg\Gamma_B^{[b;\vec{x}]}$. It is obvious that $\Gamma_B$ is a $\top$-won run of $B$, so that ${\cal M}_B$ wins $B$.
{\bf (2)}\ Assume that $B$ follows from $A$ by Overgroup Exchange, where the $i$'th ($i\geq 1$) and the $(i+1)$'th overgroups of $A$ have been swapped when obtaining $B$ from $A$. The EPM ${\cal M}_B$ works by simulating and mimicking ${\cal M}_A$ as follows. Let $n$ be the number of overgroups of either cirquent, and $a$ be a positive integer not exceeding the number of oformulas of either cirquent. For any move (by either player) $a;\vec{w_1},u_1,u_2,\vec{w_2}.\alpha$ of the real play of $B$, where $\vec{w_1}$ and $\vec{w_2}$ are any sequences of $i-1$ and $n-i-1$ natural numbers, respectively, and $u_1,u_2$ are two natural numbers, ${\cal M}_B$ translates it as the move $a;\vec{w_1},u_2,u_1,\vec{w_2}.\alpha$ (by the same player) of the imaginary play of $A$, and vice versa, with all other moves not reinterpreted. Let $\Gamma_B$ be any run generated by ${\cal M}_B$, and $\Gamma_A$ be the corresponding imaginary run generated by ${\cal M}_A$. It is obvious that, for any sequence $\vec{x}$ of $n$ positive integers, $\Gamma_B^{[a;\vec{x}]}=\Gamma_A^{[a;\vec{y}]}$, where $\vec{y}$ is the result of swapping in $\vec{x}$ the $i$'th and $(i+1)$'th integers. Hence ${\cal M}_B$ wins $B$ (because ${\cal M}_A$ wins $A$).
In the case of Oformula Exchange, a similar method can be used to construct ${\cal M}_B$, with the only difference that the reinterpreted objects are the occurrences of two adjacent oformulas rather than the occurrences of two adjacent overgroups.
As for Undergroup Exchange, its conclusion, as a game, is the same as its premise. So, the machine ${\cal M}_B={\cal M}_A$ does the job.
In the subsequent clauses, as in the preceding one, without any further indication, $\Gamma_B$ will stand for an arbitrary run of $B$ generated by ${\cal M}_B$, and $\Gamma_A$ will stand for the run of $A$ generated by the simulated machine ${\cal M}_A$ in the corresponding scenario.
{\bf (3)}\ Assume $B$ is obtained from $A$ by Weakening. If no oformula of $B$ was deleted when moving from $B$ to $A$, then ${\cal M}_B$ works exactly as ${\cal M}_A$ does and succeeds, because every $\top$-won run of $A$ is also a $\top$-won run of $B$ (but not necessarily vice versa). If, when moving from $B$ to $A$, an oformula $F_a$ of $B$ was deleted, then ${\cal M}_B$ can be constructed as a machine that works by simulating and mimicking ${\cal M}_A$. What ${\cal M}_B$ needs to do during its work is to ignore the moves within $F_a$, and play exactly as ${\cal M}_A$ does in all other oformulas. Again, it is obvious that every $\top$-won run of $A$ is also a $\top$-won run of $B$, which means that ${\cal M}_B$ wins $B$ as long as ${\cal M}_A$ wins $A$.
{\bf (4)}\ Since Exchange has already been proven to be uniform-constructively sound, in this and the remaining clauses of the present proof, we may safely assume that the oformulas and overgroups affected by a rule are at the end of the corresponding lists of objects of the corresponding cirquents.
Assume $B$ follows from $A$ by Contraction, and the contracted oformula $\pcost F$ is at the end of the list of oformulas of $B$. Let $a$ be the number of oformulas of $B$, and let $b=a+1$. Thus, the $a$'th oformula of $B$ is $\pcost F$, and the $a$'th and $b$'th oformulas of $A$ are $\pcost F$ and $\pcost F$. Next, let $n$ be the number of overgroups in either cirquent.
As always, we let ${\cal M}_B$ be an EPM that works by simulating and mimicking ${\cal M}_A$. Namely, let $\vec{w}$ be any sequence of $n$ natural numbers. If the moves take place within the oformulas other than $\pcost F$, then nothing should be reinterpreted. If the moves take place in $\pcost F$, then we have:
\begin{itemize}
\item For any move $a;\vec{w}.u.\alpha$ (by either player) in the real play of $B$, where $u=2k-1$ for some $k\in\{1,2,3,\ldots\}$, ${\cal M}_B$ translates it as the move $a;\vec{w}.k.\alpha$ (by the same player) of the imaginary play of $A$, and vice versa.
\item For any move $a;\vec{w}.v.\alpha$ (by either player) in the real play of $B$, where $v=2m$ for some $m\in\{1,2,3,\ldots\}$, ${\cal M}_B$ translates it as the move $b;\vec{w}.m.\alpha$ (by the same player) of the imaginary play of $A$, and vice versa.
\end{itemize}
Below we will show that ${\cal M}_B$ wins $B$, i.e., ${\cal M}_B$ is the winner in every undergroup of $B$.
Let $U_i^{B}$ be any $i$'th undergroup of $B$ and $U_i^{A}$ be the corresponding $i$'th undergroup of $A$, and let $\vec{x}$ be any sequence of $n$ positive integers. Since ${\cal M_A}$ wins $A$, $U_i^{A}$ is won by ${\cal M}_A$. So, for the sequence $\vec{x}$, there is an oformula $F_j$ ($1\leq j\leq b$) in $U_i^{A}$ such that $\Gamma_A^{[j;\vec{x}]}$ is a $\top$-won run of $F_j$. Next, if such $F_j$ is not one of the two contracted oformulas $\pcost F$ and $\pcost F$, then, for $\vec{x}$, the corresponding oformula $F_j$ of $B$ is also won by ${\cal M}_B$, i.e. $\Gamma_B^{[j;\vec{x}]}$ is a $\top$-won run of $F_j$, because ${\cal M}_B$ plays in the copy $\vec{x}$ of $F_j$ exactly as ${\cal M}_A$ does. This means that $U_i^{B}$ is won by ${\cal M}_B$. If such $F_j$ is one of the two contracted oformulas $\pcost F$ and $\pcost F$, below let us assume that $F_j$ is the left $\pcost F$, with the case of the right $\pcost F$ being similar. Then there is a positive integer $w$ such that the $w$'th component $F$ of the copy $\vec{x}$ of the left $\pcost F$ is won by ${\cal M}_A$, i.e. $(\Gamma_A^{[j;\vec{x}]})^{w.}$ is a $\top$-won run of $F$. But, according to the above description, ${\cal M}_B$ plays in the $(2w-1)$'th component $F$ of the copy $\vec{x}$ of $\pcost F$ in $B$ exactly as ${\cal M}_A$ plays in the $w$'th component $F$ of the copy $\vec{x}$ of the left $\pcost F$ in $A$, i.e. $(\Gamma_B^{[j;\vec{x}]})^{(2w-1).}=(\Gamma_A^{[j;\vec{x}]})^{w.}$. Therefore, $(\Gamma_B^{[j;\vec{x}]})^{(2w-1).}$ is a $\top$-won run of $F$, which means that $\Gamma_B^{[j;\vec{x}]}$ is a $\top$-won run of $\pcost F$ in $B$, and hence the $\pcost F$-containing undergroup $U_i^{B}$ is won by ${\cal M}_B$.
{\em Remark}\hspace{1pt}:\ In the remaining clauses, just as in the preceding one, when talking about playing, winning, etc. in $A$ (resp. $B$) or any of its components, it is to be understood in the context of $\Gamma_A$ (resp. $\Gamma_B$). Furthermore, if $A$ and $B$ have the same number $n$ of overgroups, then the context will additionally include some arbitrary but fixed sequence $\vec{x}$ of $n$ positive integers.
{\bf (5)}\ Undergroup Duplication does not modify the game associated with the cirquent, so we only need to consider Overgroup Duplication.
Assume $B$ is obtained from $A$ by Overgroup Duplication. We assume that the duplicated overgroup is at the end of the list of overgroups of $A$. Let $n+1$ be the number of overgroups of $A$. Thus, every legal move of $A$ (resp. $B$) looks like $a;\vec{w},u.\alpha$ (resp. $a;\vec{w},u_1,u_2.\alpha$), where $a$ is a positive integer not exceeding the number of oformulas of $A$, $\vec{w}$ is a sequence of $n$ natural numbers, and $u,u_1,u_2$ are natural numbers.
Let $f$ be some standard 1-to-1 correspondence from the set of all pairs of positive integers to the set of all positive integers.
As before, ${\cal M}_B$ works by simulating ${\cal M}_A$. Whenever ${\cal M}_A$ makes a move $a;\vec{w},0.\alpha$ in $A$, ${\cal M}_B$ makes the move $a;\vec{w},0,0.\alpha$ in the real play of $B$, and vice versa. Whenever ${\cal M}_A$ makes the move $a;\vec{w},u.\alpha$ in $A$ for some positive integer $u$, ${\cal M}_B$ makes the move $a;\vec{w},u_1,u_2.\alpha$ in $B$, where $u_1,u_2$ are integers with $f(u_1,u_2)=u$, and vice versa. Note that ${\cal M}_A$'s (legally) making a move $a;\vec{w},0.\alpha$ means that the $a$'th oformula $F_a$ of $A$ is not contained in the $(n+1)$'th overgroup $O_{n+1}$ that was duplicated when moving from $A$ to $B$, which, in turn, means that the corresponding $F_a$ of $B$ is contained in neither the $(n+1)$'th overgroup $O'_{n+1}$ nor the $(n+2)$'th overgroup $O'_{n+2}$ of $B$. Similarly, if ${\cal M}_A$ makes a move $a;\vec{w},u.\alpha$ for some positive integer $u$, then $F_a$ is contained in $O_{n+1}$ of $A$, and hence the corresponding $F_a$ of $B$ is contained in both $O'_{n+1}$ and $O'_{n+2}$ of $B$, with the case of $F_a$ being contained in $O'_{n+1}$ but not in $O'_{n+2}$ (or in $O'_{n+2}$ but not in $O'_{n+1}$) being impossible.
For every oformula $F_a$ of either cirquent, every sequence $\vec y$ of $n$ positive integers and any positive integers $x_1$ and $x_2$, we have $\Gamma_B^{[a;\vec{y},x_1,x_2]}=\Gamma_A^{[a;\vec{y},x]}$, where $x=f(x_1,x_2)$. So it is obvious that ${\cal M}_B$ wins $B$ as long as ${\cal M}_A$ wins $A$.
{\bf (6)}\ Assume $B$ follows from $A$ by Merging. Let us assume that $A$ has $n+2$ overgroups, and $B$ is the result of merging in $A$ the two adjacent overgroups $O_{n+1}$ and $O_{n+2}$. Then every legal move of $A$ (resp. $B$) looks like $a;\vec{w},u_1,u_2.\alpha$ (resp. $a;\vec{w},u.\alpha$), where $a$ is a positive integer not exceeding the number of oformulas in either cirquent, $\vec{w}$ is a sequence of $n$ natural numbers, and $u,u_1,u_2$ are natural numbers.
The EPM ${\cal M}_B$ works as follows.
If the $a$'th oformula of $A$ is neither in $O_{n+1}$ nor in $O_{n+2}$, then ${\cal M}_B$ interprets every move $a;\vec{w},0,0.\alpha$ made by ${\cal M}_A$ in the imaginary play of $A$ as the move $a;\vec{w},0.\alpha$ in the real play of $B$, and vice versa.
If the $a$'th oformula of $A$ is in $O_{n+1}$ but not in $O_{n+2}$, ${\cal M}_B$ interprets every move $a;\vec{w},v,0.\alpha$ ($v$ is a positive integer) made by ${\cal M}_A$ in the imaginary play of $A$ as the move $a;\vec{w},v.\alpha$ that ${\cal M}_B$ itself should make in the real play of $B$, and vice versa. Namely, ${\cal M}_B$ interprets every move $a;\vec{w},v.\alpha$ by its environment in the real play of $B$ as the move $a;\vec{w},v,0.\alpha$ by ${\cal M}_A$'s adversary in the imaginary play of $A$.
The case of the $a$'th oformula of $A$ being in $O_{n+2}$ but not in $O_{n+1}$ is similar.
Now assume that the $a$'th oformula of $A$ is in both $O_{n+1}$ and $O_{n+2}$. ${\cal M}_B$ interprets every move $a;\vec{w},v_1,v_2.\alpha$ by ${\cal M}_A$ in the imaginary play of $A$ as the move $a;\vec{w},v.\alpha$ in the real play of $B$, where $v_1,v_2,v$ are positive integers such that $v=f(v_1,v_2)$, with $f$ here standing for the pairing function explained in the preceding clause of this proof.
For every oformula $F_a$ of either cirquent, every sequence $\vec y$ of $n$ positive integers and any positive integer $x$, we have $\Gamma_B^{[a;\vec{y},x]}=\Gamma_A^{[a;\vec{y},x_1,x_2]}$, where $x_1,x_2$ are positive integers satisfying that $x_1=x$ (when $F_a$ is contained in $O_{n+1}$ but not $O_{n+2}$), or $x_2=x$ (when $F_a$ is contained in $O_{n+2}$ but not $O_{n+1}$), or $f(x_1,x_2)=x$ (when $F_a$ is contained in both $O_{n+1}$ and $O_{n+2}$, or is contained in neither of them). So it is obvious that ${\cal M}_B$ wins $B$ as long as ${\cal M}_A$ wins $A$.
{\bf (7)}\ In this and the remaining clauses of the present proof, we will limit our descriptions to what moves ${\cal M}_B$ needs to properly reinterpret and how, with any unmentioned sorts of moves implicitly assumed to remain unchanged.
Assume $B$ is obtained from $A$ by Disjunction Introduction. Let us assume that the last ($a$'th) oformula of $B$ is $E\vee F$, and the last two ($a$'th and $b$'th, where $b=a+1$) oformulas of $A$ are $E$ and $F$. We let ${\cal M}_B$ reinterpret every move $a;\vec{w}.\alpha$ (resp. $b;\vec{w}.\alpha$) by either player in the imaginary play of $A$ as the move $a;\vec{w}.1.\alpha$ (resp. $a;\vec{w}.2.\alpha$) by the same player in the real play of $B$, and vice versa.
Consider any undergroup $U_i^{B}$ of $B$, and let $U_i^{A}$ be the corresponding undergroup of $A$. As before, ${\cal M}_A$'s winning $A$ means that $U_i^{A}$ is won by ${\cal M}_A$, which, in turn, means that there is an oformula $G$ in $U_i^{A}$ that is won by ${\cal M}_A$. If $G$ is neither $E$ nor $F$, then the oformula $G$ of $B$ is also won by ${\cal M}_B$, because ${\cal M}_B$ plays in $G$ exactly as ${\cal M}_A$ does. Hence $U_i^{B}$ is won by ${\cal M}_B$. If $G$ is $E$, then its being $\top$-won means that ${\cal M}_B$ wins the $E$ component of $E\vee F$, because ${\cal M}_B$ plays in the $E$ component of $E\vee F$ exactly as ${\cal M}_A$ plays in $E$. Therefore, $E\vee F$ is won by ${\cal M}_B$, and hence so is the $E\vee F$-containing undergroup $U_i^{B}$. The case of $G$ being $F$ is similar.
{\bf (8)}\ Assume $B$ follows from $A$ by Conjunction Introduction. We also assume that the last ($a$'th) oformula of $B$ is $E\wedge F$, and the last two ($a$'th and $b$'th, where $b=a+1$) oformulas of $A$ are $E$ and $F$. As the case of Disjunction Introduction, ${\cal M}_B$ reinterprets every move $a;\vec{w}.\alpha$ (resp. $b;\vec{w}.\alpha$) by either player in the imaginary play of $A$ as the move $a;\vec{w}.1.\alpha$ (resp. $a;\vec{w}.2.\alpha$) by the same player in the real play of $B$, and vice versa.
Let $U_i$ be any undergroup of $B$. If $U_i$ does not contain $E\wedge F$, then the corresponding undergroup $V_i$ of $A$ contains neither $E$ nor $F$. In this case, $U_i$ is won by ${\cal M}_B$ for the same reason as in the preceding clause. If $U_i$ contains $E\wedge F$, then there are two undergroups $V_i^{E}$, $V_i^{F}$ of $A$ corresponding to $U_i$, where $V_i^{E}$ contains $E$ (but not $F$), and $V_i^{F}$ contains $F$ (but not $E$), with all other ($\neq E\wedge F$) oformulas of $U_i$ contained by both $V_i^{E}$ and $V_i^{F}$. Of course, both $V_i^E$ and $V_i^F$ are won by ${\cal M}_A$ because ${\cal M}_A$ wins the overall game $A$. This means that there is an oformula $G_1$ (resp. $G_2$) in $V_i^E$ (resp. $V_i^F$) such that ${\cal M}_A$ wins it. If at least one oformua $G\in\{G_1,G_2\}$ is neither $E$ nor $F$, then the corresponding oformula $G$ of $B$ is won by ${\cal M}_B$, because ${\cal M}_B$ plays in $G$ exactly as ${\cal M}_A$ does. Hence the $G$-containing undergroup $U_i$ of $B$ is won by ${\cal M}_B$. If $G_1$ is $E$ and $G_2$ is $F$, then ${\cal M}_A$ winning them means that ${\cal M}_B$ wins both the $E$ and the $F$ components of $E\wedge F$, because ${\cal M_B}$ plays in the $E$ (resp. $F$) component of $E\wedge F$ exactly as ${\cal M_A}$ does in $E$ (resp. $F$). Hence $E\wedge F$ is won by ${\cal M}_B$, and hence so is the $E\wedge F$-containing undergroup $U_i$.
{\bf (9)}\ Assume $B$ is obtained from $A$ by Recurrence Introduction. Namely, the last ($a$'th) oformula of $B$ is $\pst F$, and the last ($a$'th) oformula of $A$ is $F$. We further assume that the number of overgroups of $B$ is $n$, and thus the number of overgroups of $A$ is $n+1$. In what follows, $\vec{w}$ is any sequence of $n$ natural numbers, and $b$ is a positive integer not exceeding the number of oformulas of either cirquent.
If $b\neq a$, then ${\cal M}_B$ simply reinterprets every move $b;\vec{w},0.\alpha$ by either player in the imaginary play of $A$ as the move $b;\vec{w}.\alpha$ by the same player in the real play of $B$, and vice versa. If $b=a$, then ${\cal M}_B$ reinterprets, for any positive integer $u$, every move $a;\vec{w},u.\alpha$ by either player in the imaginary play of $A$ as the move $a;\vec{w}.u.\alpha$ by the same player in the real play of $B$, and vice versa.
Consider any undergroup $U_i^{B}$ of $B$. Let $\vec{x}=x_1,\ldots,x_n$ be any sequence of $n$ positive integers. ${\cal M}_A$'s winning $A$ means that $\Gamma_A$ is a $\top$-won run of $A$ and that the corresponding undergroup $U_i^{A}$ of $A$ is won by ${\cal M}_A$. Then, for any sequence $\vec{y}=x_1,\ldots,x_n,x$, where $x$ is any positive integer, there is an oformula $F_b$ in $U_i^{A}$ such that $\Gamma_A^{[b;\vec{y}]}$ is a $\top$-won run of $F_b$. If such $F_b$ is not the $a$'th oformula $F$, then, in the context of $\vec{x}$, the oformula $F_b$ of $B$ is also won by ${\cal M}_B$, i.e. $\Gamma_B^{[b;\vec{x}]}$ is a $\top$-won run of $F_b$, because ${\cal M}_B$ plays in the copy $\vec{x}$ of $F_b$ in $B$ exactly as ${\cal M}_A$ does in the copy $\vec{y}$ of $F_b$ in $A$. Hence $U_i^{B}$ is won by ${\cal M}_B$. If $F_b$ is the $a$'th oformula $F$, then, in the context of $\vec{x}$, the corresponding oformula $\pst F$ of $B$ is won by ${\cal M}_B$ as well, i.e. $\Gamma_B^{[a;\vec{x}]}$ is a $\top$-won run of $\pst F$. This is so because ${\cal M}_B$ plays in the $x$'th component $F$ of the copy $\vec{x}$ of $\pst F$ exactly as ${\cal M}_A$ does in the copy $\vec{y}$ of $F$ in $A$. Namely, $(\Gamma_B^{[a;\vec{x}]})^{x.}=\Gamma_A^{[a;\vec{y}]}$. Since $\Gamma_A^{[a;\vec{y}]}$ is a $\top$-won run of $F$, so is $(\Gamma_B^{[a;\vec{x}]})^{x.}$. Further, due to the arbitrariness of $x$, $\Gamma_B^{[a;\vec{x}]}$ is a $\top$-won run of $\pst F$. Therefore, the $\pst F$-containing undergroup $U_i^{B}$ is won by ${\cal }M_B$.
{\bf (10)}\ Finally, assume that $B$ is obtained from $A$ by Corecurrence Introduction. Let us assume that the last ($a$'th) oformula of $B$ is $\pcost F$, and the last ($a$'th) oformula of $A$ is $F$. And assume that $n$ $(n\geq 0)$ is the number of the {\em new} overgroups $U_j$ in which the $a$'th oformula $F$ was included when moving from $B$ to $A$. Let us further assume that all of such $n$ overgroups are at the end of the list of overgroups of either cirquent. In what follows, let $\vec{w}$ be any sequence of $m$ natural numbers, where $m$ is the total number of overgroups of either cirquent minus $n$. We construct the EPM ${\cal M}_B$ as follows.
Let $f$ be some standard injective function from the set of $n$-tuples $(u_1,\ldots,u_n)$ of positive integers onto the set of positive integers $u$. In its simulation routine, ${\cal M}_B$ reinterprets every move $a;\vec{w},u_1,\ldots,u_n.\alpha$ made by ${\cal M}_A$ in the imaginary play of $A$ as the move $a;\vec{w},0,\ldots,0.u.\alpha$ ($n$ occurrences of $0$ after $\vec{w}$) in the real play of $B$, where $u=f(u_1,\ldots,u_n)$. Whenever the environment makes a move $a;\vec{w},0,\ldots,0.v.\beta$ (also $n$ occurrences of $0$ after $\vec{w}$) for some positive integer $v$ in the real play of $B$, if there is no $n$-tuple $(u_1,\ldots,u_n)$ such that $v=f(u_1,\ldots,u_n)$, then ${\cal M}_B$ simply ignores it; if $v=f(u_1,\ldots,u_n)$, then ${\cal M}_B$ translates it as the move $a;\vec{w},u_1,\ldots,u_n.\beta$ by ${\cal M}_A$'s adversary in the imaginary play. Note that the above routine works as well in the case of $n=0$. Simply, $f()=c$ for some fixed positive integer $c$, ${\cal M}_B$ reinterprets every move $a;\vec{w}.\alpha$ made by ${\cal M}_A$ in $A$ as the move $a;\vec{w}.c.\alpha$ in $B$, and whenever the environment makes a move $a;\vec{w}.v.\beta$ in $B$, if $v\neq c$, ${\cal M}_B$ ignores it, and if $v=c$, ${\cal M}_B$ translates it as the move $a;\vec{w}.\beta$ by ${\cal M}_A$'s adversary in the imaginary play of $A$.
As usual, consider any undergroup $U_i^{B}$ of $B$, and let $\vec{x}=\vec{y},x_1,\ldots,x_n$ be any sequence of $(m+n)$ positive integers, where $\vec{y}$ is any sequence of $m$ positive integers. Then the corresponding undergroup $U_i^{A}$ of $A$ is won by ${\cal M}_A$, which, in turn, means that there is an oformula $F_b$ ($1\leq b\leq a$) in $U_i^{A}$ such that ${\cal M}_A$ wins it. If such $F_b$ is not the $a$'th oformula $F$, then the corresponding oformula $F_b$ of $B$ is also won by ${\cal M}_B$, because ${\cal M}_B$ plays in $F_b$ of $B$ exactly as ${\cal M}_A$ does in $F_b$ of $A$. Therefore, the $F_b$-containing undergroup $U_i^{B}$ is won by ${\cal M}_B$. If $F_b$ is the $a$'th oformula $F$, then the corresponding oformula $\pcost F$ of $B$ is won by ${\cal M}_B$ as well. This is so because ${\cal M}_B$ plays in at least one component $F$ of $\pcost F$ in $B$ exactly as ${\cal M}_A$ does in $F$ of $A$. Precisely, we have
$(\Gamma_B^{[a;\vec{y},x_1,\ldots,x_n]})^{x.}=\Gamma_A^{[a;\vec{y},x_1,\ldots,x_n]}$, where $x=f(x_1,\ldots,x_n)$. Thus the $\pcost F$-containing undergroup $U_i^{B}$ is won by ${\cal M}_B$.
\end{proof}
\begin{theorem}\label{mainth2}
Every cirquent provable in {\bf CL15}$(\pst)$ is uniformly valid.
Furthermore, there is an effective procedure that takes an arbitrary {\bf CL15}$(\pst)$-proof of an arbitrary cirquent $C$ and constructs a uniform solution of $C$.
\end{theorem}
\begin{proof}
Immediately from Theorem \ref{mainth1} by induction on the lengths of {\bf CL15}$(\pst)$-proofs.
\end{proof}
\begin{theorem}\label{mainth3}
For any formula $F$, if {\bf CL15}$(\pst)\vdash F$, then $F$ is uniformly valid.
Furthermore, there is an effective procedure which takes any {\bf CL15}$(\pst)$-proof of any formula $F$ and constructs a uniform solution of $F$.
\end{theorem}
\begin{proof}
Immediately from Theorem \ref{mainth2} and Lemma \ref{apr14b}.
\end{proof}
Below, a {\bf uniformly valid $(\neg,\wedge,\vee,\pst,\pcost)$-principle} means the result of replacing every occurrence of the operator $\pst$ (resp. $\pcost$) by the symbol $!$ (resp. $?$) in some uniformly valid $(\neg,\wedge,\vee,\pst,\pcost)$-formula. Similarly, a {\bf uniformly valid $(\neg,\wedge,\vee,\st,\cost)$-principle} means the result of replacing every occurrence of the operator $\st$ (resp. $\cost$) by the symbol $!$ (resp. $?$) in some uniformly valid $(\neg,\wedge,\vee,\st,\cost)$-formula. The reason for introducing these technical concepts is merely to make it possible to directly compare the otherwise syntactically nonidentical $(\neg,\wedge,\vee,\pst,\pcost)$-formulas with $(\neg,\wedge,\vee,\st,\cost)$-formulas.
\begin{theorem}\label{superset}
The set of uniformly valid $(\neg,\wedge,\vee,\pst,\pcost)$-principles is a proper superset of the set of uniformly valid $(\neg,\wedge,\vee,\st,\cost)$-principles.
\end{theorem}
\begin{proof} The fact that the set of uniformly valid $(\neg,\wedge,\vee,\pst,\pcost)$-principles is a {\em superset} of the set of uniformly valid $(\neg,\wedge,\vee,\st,\cost)$-principles is immediate from Theorems \ref{niu} and \ref{mainth3}. Furthermore, the former set is in fact a {\em proper} superset of the latter set because, as proven in \cite{Japsep}, the formula $P\wedge\pst(P\rightarrow P\wedge P)\rightarrow\pst P$ is uniformly valid while its counterpart $P\wedge\st(P\rightarrow P\wedge P)\rightarrow\st P$ is not.
\end{proof}
\section{A secondary result}
Japaridze \cite{Japfin,Japfour} claimed that $\st$ is strictly stronger than $\pst$ (and thus $\cost$ is strictly weaker than $\pcost$) in the sense that the formula $\st P\rightarrow\pst P$ is uniformly valid while its converse $\pst P\rightarrow\st P$ is not. The first part of this claim was proven in \cite{Japtur}, but the second part has never been verified. In order to make our investigation of the relationship between the two sorts of recurrences more comprehensive, below we provide such a verification.
\begin{theorem}\label{secondary}
The formula $\pst P\rightarrow\st P$ is not uniformly valid.
\end{theorem}
\begin{proof}
Let ${\cal M}$ be an arbitrary EPM, i.e. strategy of the machine $(\top)$. Below we construct a counterstrategy ${\cal C}$ such that, when the environment $(\bot)$ follows it, ${\cal M}$ loses $\pst P\rightarrow\st P$ with $P$ interpreted as a certain enumeration game. Here, an {\bf enumeration game} (\cite{Japsep}) is a game where any natural number, identified with its decimal representation, is a legal move by either player at any time (and there are no other legal moves). It should be noted that, as shown in \cite{Japtam2}, every enumeration game is static, and hence is a legitimate value of an interpretation $^*$ on any atom. Hence, due to the arbitrariness of ${\cal M}$, $\pst P\rightarrow\st P$ (i.e. $\pcost\neg P\vee\st P$) is not uniformly valid.
Since $P$ is going to be interpreted as an enumeration game and its legal moves are known even before we actually define that interpretation, in certain contexts we may identify formulas with games without creating any confusion. The work of ${\cal C}$ consists in repeating the following interactive routine over and over again (infinitely many times), where $i$ is the number of the iteration. In our description below, a {\em fresh number} means a natural number that has not yet been chosen in the play by either player as a move in any thread/copy of $P$.
LOOP($i$): Whenever permission is granted by the machine ${\cal M}$, make the move $2.w.u$, where $u$ is a fresh number and $w$ is the $i$th finite bitstring of the lexicographic list of all finite bitstrings.
Consider the run $\Delta$ generated by $\cal M$ in the scenario when its adversary follows the above counterstrategy. Let $\Omega=\Delta^{1.}$ and $\Gamma=\Delta^{2.}$. That is, $\Omega$ is the (sub)run that took place in the $\pcost\neg P$ component, and $\Gamma$ is the (sub)run that took place in the $\st P$ component. From some analysis of the work of LOOP, details of which are left to the reader, one can see that $\Gamma^{\preceq x_1}\neq\Gamma^{\preceq x_2}$ for any two different infinite bitstrings $x_1$ and $x_2$. Hence, as there are uncountably many infinite bitstrings while only countably many positive integers, there is an infinite bitstring $y$ such that, for every positive integer $v$, $\Omega^{v.}\neq\neg\Gamma^{\preceq y}$. Fix this $y$.
Now we select an interpretation $^*$ that interprets $P$ as the enumeration game such that, for any legal run $\Theta$ of the game $P$, $Wn^{P}\langle\Theta\rangle=\bot$ iff $\Theta=\Gamma^{\preceq y}$. We claim that ${\cal M}$ loses the overall game under this interpretation. First, it is obvious that ${\cal M}$ loses the game $P$ in the thread $y$, which means that it loses the $\st P$ component. Next, ${\cal M}$ also loses the $\pcost\neg P$ component because it loses every component $\neg P$ of $\pcost\neg P$. This is so because the run that took place in any component $\neg P$ of $\pcost\neg P$ is won by $\top$ iff it is $\neg\Gamma^{\preceq y}$, which, however, is impossible (due to the above analysis).
\end{proof}
An alternative albeit non-constructive and less direct proof of Theorem \ref{secondary} would rely on Theorem \ref{superset}. Namely, one could show that, if $\pst P\rightarrow\st P$ was uniformly valid and hence (in view of the already known fact of the uniform validity of the converse of this formula) $\st P$ and $\pst P$ were ``logically equivalent'', then they would induce identical logics, in the precise sense that the set of uniformly valid $(\neg,\wedge,\vee,\pst,\pcost)$-principles would coincide with the set of uniformly valid $(\neg,\wedge,\vee,\st,\cost)$-principles, contrary to what Theroem \ref{superset} asserts.
\end{document} |
\begin{document}
\title{On quantification of systemic redundancy in reliable systems
}
\titlerunning{On quantification of systemic redundancy in reliable systems}
\author{Getachew K. Befekadu}
\author{Getachew K. Befekadu \and Panos~J.~Antsaklis
}
\institute{G. K. Befekadu~ ({\large\Letter}\negthinspace) \at
Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA. \\
\email{[email protected]}
\and
P. J. Antsaklis \at
Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA. \\
\email{[email protected]}
}
\date{February 15, 2015}
\maketitle
\begin{abstract}
In this paper, we consider the problem of quantifying systemic redundancy in reliable systems having multiple controllers with overlapping functionality. In particular, we consider a multi-channel system with multi-controller configurations -- where controllers are required to respond optimally, in the sense of best-response correspondence, a {\it reliable-by-design} requirement, to non-faulty controllers so as to ensure or maintain some system properties. Here we introduce a mathematical framework, based on the notion of relative entropy of probability measures associated with steady-state solutions of Fokker-Planck equations for a family of stochastically perturbed multi-channel systems, that provides useful information towards a systemic assessment of redundancy in the system.
\keywords{Entropy \and Fokker-Planck equation \and Liouville equation \and quantification of systemic redundancy \and random perturbation \and relative entropy \and reliable systems}
\end{abstract}
\section{Introduction} \label{S1}
The notion of redundancy, which promotes robustness by ``backing-up" important functions of systems, together with its systemic quantification, has long been recognized as an essential design philosophy by researchers in different fields of studies (to mention a few, e.g., see \cite{KraP02}, \cite{Tau92} and \cite{Wag05} for related discussions in biological systems; see also \cite{CarD99}, \cite{KanB11}, \cite{SieS98} and \cite{RanLOT11} for related discussions in engineering systems). In this paper, we consider the problem of quantifying systemic redundancy in reliable systems having multiple controllers with overlapping functionality. To be more specific, we consider a multi-channel system with multi-controller configurations -- where controllers are required to respond optimally, a {\it reliable-by-design} requirement, to non-faulty controllers so as to maintain the stability of the system when there is a single failure in any of the control channels.
Here we introduce a mathematical framework, based on the notion of relative entropy of probability measures associated with steady-state solutions of the Fokker-Planck equations for a family of stochastically perturbed multi-channel systems, that provides useful information towards a systemic assessment of redundancy in the system. For such steady-state solutions of the Fokker-Planck equations, we establish a quantifiable redundancy measure by using the difference between the average relative entropy of the steady-state probability measures, with respect to any single controller failure in the system, and the entropy of the steady-state probability measure under a nominal operating condition, i.e., without any single controller failure in the system. Moreover, we determine the asymptotic behavior of the systemic redundancy measure in the multi-channel system as the random perturbation decreases to zero, where such an asymptotic result can be related to the solutions of controlled Liouville equations for the underlying original unperturbed multi-channel system.
It is worth mentioning that some interesting studies on systemic measures, based on information theory, have been reported in literature (e.g., see \cite{ToSE99}, \cite{LiDHKY12} or \cite{EdekG01} in the context of complexity, degeneracy and redundancy measures in biological systems; see \cite{Kit04} or \cite{Kit07} in the context of robustness in biological systems; and see also \cite{Bru83} or \cite{Cru12} for related discussion on the complexity measure for trajectories in dynamical systems). Moreover, such studies have also provided some useful information in characterizing or understanding the systemic measures (based on mutual information between appropriately partitioned input and output spaces) in systems with multiple subsystems/modules having overlapping functionality. Note that the rationale behind our framework follows in some sense the settings of these papers. However, to our knowledge, this problem has not been addressed in the context of reliable systems with multi-controller configurations having ``overlapping or backing-up" functionality, and it is important because it provides a framework for quantifying or gauging systemic redundancy measure in multi-channel systems, for example, when there is a single channel failure in the system.
This paper is organized as follows. In Section~\ref{S2}, we present some preliminary results that are useful for our main results. Section~\ref{S3} presents our main results, where we introduce a mathematical framework that provides useful information towards a systemic assessment of redundancy in reliable systems. This section also contains a result on the asymptotic behavior of the systemic redundancy measure in the multi-channel system as the random perturbation decreases to zero.
\section{Preliminary results} \label{S2}
Consider the following continuous-time multi-channel system
\begin{align}
\dot{x}(t) &= A x(t) + \sum\nolimits_{i=1}^N B_i u_i(t), \quad x(0)=x_0, \label{Eq1}
\end{align}
where $x \in \mathbb{R}^{d}$ is the state, $u_i \in \mathbb{R}^{r_i}$ is the control input to the $i$th-channel.
Let $S$ be a compact manifold in $\mathbb{R}^{d}$ and let $\rho_0(x) \triangleq \rho(0, x) > 0$, with $\int_{S} \rho_0(x)dx=1$, be an initial density function. Further, let $\psi$ be a smooth function $\psi \colon S \rightarrow \mathbb{R}^{+}$ having compact support, then the expected value of $\psi$ at some future time $t > 0$ is given by
\begin{align}
E \bigl\{ \psi(x) \bigr\} = \int_{S} \psi(x) \rho(t, x) dx. \label{Eq2}
\end{align}
Moreover, if we take the time derivative of the above equation and make use of integration by parts, then we will have
\begin{align}
\int_{S} \psi(x) \frac{\partial \rho(t, x)}{\partial t} dx = - \int_{S} \psi(x) \Bigl \langle \frac{\partial }{\partial x}, \Bigl (A x + \sum\nolimits_{i=1}^N B_i u_i\Bigr)\rho(t, x) \Bigr \rangle dx. \label{Eq3}
\end{align}
Since $\psi(x)$ is an arbitrary function, we can rewrite the above equation as a first-order partial differential equation (which is also known as the Liouville equation)
\begin{align}
\frac{\partial \rho(t, x)}{\partial t} = - \Bigl \langle \frac{\partial }{\partial x}, \Bigl (A x + \sum\nolimits_{i=1}^N B_i u_i\Bigr)\rho(t, x) \Bigr \rangle. \label{Eq4}
\end{align}
\begin{remark} \label{R1}
Here we remark that the above first-order partial differential equation describes how the density function $\rho(t, x)$ evolves in time (i.e., Equation~\eqref{Eq4} describes an evolution equation on $L_1\bigl(\mathbb{R}^{d}\bigr)$ under a flow defined by the deterministic system of \eqref{Eq1}). Furthermore, it is easy to verify that
\begin{align*}
\rho(t, x) > 0, \quad t \ge 0,
\end{align*}
and further it satisfies
\begin{align*}
\frac{d}{d t} \int_{S} \rho(t, x) dx = 0.
\end{align*}
\end{remark}
Notice that the partial derivative with respect to $x$ in \eqref{Eq4} depends on whether the input controls $u_i$ are expressed as open-loop functions (i.e., $u_i=u_i(t)$ for $i=1,2, \ldots, N$) or as closed-loop functions (i.e., $u_i=u_i(t,x)$ for $i=1,2, \ldots, N$); and, as a result of this, the solution for $\rho(t, x)$ depends on the type of input controls used in the system.
\begin{remark} \label{R2}
Recently, using a class of variational problems, the author in \cite{Broc07} has considered the Liouville equations that involve control terms. Moreover, such a formulation is useful for relating the behavior of the solutions of the Liouville equation to that of the behavior of the underlying differential equation of the system.
\end{remark}
In what follows, we recall some known results that will be used for our main results (e.g., see \cite{Csi67} or \cite{Les14}).
\begin{definition}\label{D1}
Let $\mu$ be a probability measure on $\mathbb{R}^d$ with respect to the density function $\rho(t, x)$ which satisfies the Liouville equation in \eqref{Eq4} starting from an initial density function $\rho_0(x_0)$. Then, the entropy of $\mu$ (with respect to Lebesgue measure) is defined by
\begin{align}
H\bigl(\mu\bigr) = - \int_{S} \rho(t, x) \log_2 \rho(t, x) dx, \quad t \ge 0. \label{Eq5}
\end{align}
\end{definition}
\begin{remark} \label{R3}
In general, we have the following inequality
\begin{align*}
- \int_{S} \rho_1(t, x) \log_2 \rho_1(t, x) dx \le - \int_{S} \rho_1(t, x) \log_2 \rho_2(t, x) dx,
\end{align*}
for any two density functions $\rho_1(t, x)$ and $\rho_2(t, x)$ that satisfy \eqref{Eq4} starting from $\rho_{1,0}(x_0)$ and $\rho_{2,0}(x_0)$, respectively.
\end{remark}
\begin{definition}\label{D2}
Let $\mu_1$ and $\mu_2$ be two probability measures on $\mathbb{R}^d$ with respect to the density functions $\rho_1(t, x)$ and $\rho_2(t, x)$ that satisfy the Liouville equation in \eqref{Eq4} starting from initial density functions $\rho_{1,0}(x_0)$ and $\rho_{2,0}(x_0)$, respectively. Then, the relative entropy of $\mu_2$ with respect to $\mu_1$ is defined by
\begin{align}
D\bigl(\mu_2 \,\Vert\, \mu_1\bigr) & = \int_{S} \rho_2(t, x) \log_2 \left (\frac{\rho_2(t, x)}{\rho_1(t, x)}\right) dx\notag \\
&= \int_{S} \Bigl(\rho_2(t, x) \log_2 \rho_2(t, x) - \rho_2(t, x) \log_2 \rho_1(t, x) \Bigr) dx, \quad t \ge 0. \label{Eq6}
\end{align}
\end{definition}
\begin{remark} \label{R4}
Note that the relative entropy (also called the Kullback-Leibler distance) $D\bigl(\mu_2 \,\Vert\, \mu_1\bigr)$, which measures the deviation of $\mu_2$ with respect to the probability measure $\mu_1$, is nonnegative, i.e., $D \bigl(\mu_2 \,\Vert\, \mu_1\bigr) \ge 0$, and $D\bigl(\mu_2 \,\Vert\, \mu_1\bigr) = 0$ if and only if $\mu_2 = \mu_1$.
\end{remark}
Next, consider the following stochastically perturbed multi-channel system
\begin{align}
dx^{\epsilon}(t) = A x^{\epsilon}(t) dt + \sum\nolimits_{i=1}^N B_i u_i(t) dt + \epsilon \sigma(x^{\epsilon}(t)) dW(t), \quad x^{\epsilon}(0)=x_0, \label{Eq7}
\end{align}
where
\begin{itemize}
\item[-] $x^{\epsilon}(\cdot)$ is an $\mathbb{R}^{d}$-valued diffusion process, $\epsilon > 0$ is a small parameter that represents the level of random perturbation in the system,
\item[-] $\sigma \colon \mathbb{R}^{d} \rightarrow \mathbb{R}^{d \times m}$ is Lipschitz continuous with the least eigenvalue of $\sigma(\cdot)\sigma^T(\cdot)$ uniformly bounded away from zero, i.e.,
\begin{align*}
\sigma(x)\sigma^T(x) \ge \kappa I_{d \times d} , \quad \forall x \in \mathbb{R}^{d},
\end{align*}
for some $\kappa > 0$,
\item[-] $W(\cdot)$ (with $W(0)=0$) is an $m$-dimensional standard Wiener process, and
\item[-] $u_i(\cdot)$ is a $U_i$-valued measurable control process to the $i$th-channel (i.e., an admissible control from the set $U_i \subset \mathbb{R}^{r_i}$) such that, for all $t > s$, $W(t) - W(s)$ is independent of $u_i(\nu)$ for $\nu > s$.
\end{itemize}
Note that the time evolution of the density function $\rho^{\epsilon}(t, x)$ associated with \eqref{Eq7} satisfies the following second-order partial differential equation (i.e., the Fokker-Planck equation)
\begin{align}
\frac{\partial \rho^{\epsilon}(t, x)}{\partial t} = - \Bigl \langle \frac{\partial} {\partial x},\, \Bigl( A x + \sum\nolimits_{i=1}^N B_i u_i\Bigr) \rho^{\epsilon}(t, x) \Bigr \rangle + & \frac{\epsilon^2}{2} \Bigl \langle \frac{\partial^2} {\partial x^2},\, \sigma(x)\sigma^T(x) \rho^{\epsilon}(t, x) \Bigr \rangle, \notag \\
& \quad\quad \rho^{\epsilon}(0, x) \,\, \text{is given}. \label{Eq8}
\end{align}
In this paper, among all solutions of the above Fokker-Planck equation, we will only consider the steady-state solution that satisfies the following stationary Fokker-Planck equation\footnote{For example, see \cite{Kif74} or \cite{VenFre70} for additional discussion on the limiting behavior of invariant measures for systems with small random perturbation.}
\begin{align}
0 = - \Bigl \langle \frac{\partial} {\partial x},\, \Bigl( A x + \sum\nolimits_{i=1}^N B_i u_i\Bigr) \rho_{\ast}^{\epsilon}(x) \Bigr \rangle + & \frac{\epsilon^2}{2} \Bigl \langle \frac{\partial^2} {\partial x^2},\, \sigma(x)\sigma^T(x) \rho_{\ast}^{\epsilon}(x) \Bigr \rangle, \notag \\
\rho_{\ast}^{\epsilon}(x) > 0, & \quad \int_{\mathbb{R}^d} \rho_{\ast}^{\epsilon}(x) dx = 1. \label{Eq9}
\end{align}
In the following section, i.e., Section~\ref{S3}, such a steady-state solution, together with the solution of the Liouville equation, will allow us to provide useful information towards a systemic assessment of redundancy in the reliable multi-channel systems with small random perturbation.
\section{Main results} \label{S3}
In what follows, we consider a particular class of stabilizing state-feedbacks that satisfies\footnote{$\operatorname{Sp}(A)$ denotes the spectrum of a matrix $A \in \mathbb{R}^{d \times d}$, i.e., $\operatorname{Sp}(A) = \bigl\{s \in \mathbb{C}\,\bigl\lvert \,\rank(A - sI) < d \bigr\}$.}
\begin{align}
\mathcal{K} \subseteq \Biggl\{\bigl(K_1, K_2, \ldots, K_N\bigr) & \in \prod\nolimits_{i=1}^N \mathbb{R}^{r_i \times d} \biggm \lvert \operatorname{Sp}\Bigl(A + \sum\nolimits_{i=1}^N B_{i} K_i\Bigr) \subset \mathbb{C}^{-} \,\, \& \notag \\
& \operatorname{Sp}\Bigl(A + \sum\nolimits_{i \neq j}^N B_{i} K_i\Bigr) \subset \mathbb{C}^{-},\,\, j = 1,2, \ldots, N \Biggr\}. \label{Eq10}
\end{align}
\begin{remark} \label{R5}
We remark that the above class of state-feedbacks is useful for maintaining the stability of the closed-loop system both when all of the controllers work together, i.e., $\bigl(A + \sum\nolimits_{i=1}^N B_{i} K_i\bigr)$, as well as when there is a single-channel controller failure in the system, i.e., $\bigl(A + \sum\nolimits_{i \neq j}^N B_{i} K_i\bigr)$ for $j=1,2, \ldots, N$. Moreover, such a class of state-feedbacks falls within the redundant/passive fault tolerant controller configurations with overlapping functionality, i.e., a reliable-by-design requirement in the system (e.g., see \cite{BefGA14} or \cite{FujBe09}).
\end{remark}
Note that, for such a class of state-feedbacks, the density functions $\rho^{(j)}(t, x)$ for $j = 0,1, \ldots, N$, satisfy the following family of Liouville equations
\begin{align}
\frac{\partial \rho^{(0)}(t, x)}{\partial t} = - \Bigl \langle \frac{\partial }{\partial x}, \Bigl (Ax + \sum\nolimits_{i=1}^N B_i K_i x \Bigr) \rho^{(0)}(t, x) \Bigr \rangle \label{Eq11}
\end{align}
and
\begin{align}
\frac{\partial \rho^{(j)}(t, x)}{\partial t} = - \Bigl \langle \frac{\partial }{\partial x}, \Bigl (Ax + \sum\nolimits_{i \neq j}^N B_i K_i x \Bigl) \rho^{(j)}(t, x) \Bigr \rangle, \,\, j=1,2, \ldots, N, \label{Eq12}
\end{align}
starting from an initial density function $\rho_0(x_0)$. Moreover, using the fact that the solution of
\begin{align}
\dot{x}(t) = \Bigl (A + \sum\nolimits_{i=1}^N B_i K_i \Bigr)x(t), \quad x(0)=x_0, \label{Eq13}
\end{align}
can be written as
\begin{align}
x(t) = \exp(A t) x(0)+ \int_{0}^t \exp(A (t - \lambda)) \sum\nolimits_{i=1}^N B_i K_i x(\lambda) d\lambda. \label{Eq14}
\end{align}
Then, the density function $\rho^{(0)}(t, x)$ (when $j=0$) corresponding to the Liouville equation in \eqref{Eq11}, with an initial density $\rho_0(x_0)$, is given by
\begin{align}
\rho^{(0)}(t, x) = \frac{1}{\exp(\operatorname{tr}A t)} \rho_0 \biggl(\exp(- A t)\biggl(x(t) &- \int_{0}^t \exp(A (t - \lambda)) \sum\nolimits_{i=1}^N B_i K_i x(\lambda) d\lambda \biggr ) \biggr). \label{Eq15}
\end{align}
Similarly, the density functions $\rho^{(j)}(t, x)$ (when $j=1,2, \ldots, N$) corresponding to the Liouville equations in \eqref{Eq12} are given by
\begin{align}
\rho^{(j)}(t, x) = \frac{1}{\exp(\operatorname{tr}A t)} \rho_0 \biggl(\exp(- A t)\biggl(x(t) &- \int_{0}^t \exp(A (t - \lambda))\sum\nolimits_{i \neq j}^N B_i K_i x(\lambda) d\lambda \biggr ) \biggr). \label{Eq16}
\end{align}
On the other hand, for a given random perturbation $(\sigma, \epsilon)$, the steady-state density functions $\rho^{(\epsilon, j)}(x) $ for $j = 0,1, \ldots, N$, satisfy the following family of stationary Fokker-Planck equations
\begin{align}
0 = - \Bigl \langle \frac{\partial} {\partial x},\, \Bigl ( A x + \sum\nolimits_{i=1}^N B_i K_i x \Bigr) \rho^{(\epsilon, 0)}(x) \Bigr \rangle + & \frac{\epsilon^2}{2} \Bigl \langle \frac{\partial^2} {\partial x^2},\, \sigma(x)\sigma^T(x) \rho^{(\epsilon, 0)}(x) \Bigr \rangle, \notag \\
\rho^{(\epsilon, 0)}(x) > 0, \quad & \int_{\mathbb{R}^d} \rho^{(\epsilon, 0)}(x) dx = 1, \label{Eq17}
\end{align}
and
\begin{align}
0 = - \Bigl \langle \frac{\partial} {\partial x},\, \Bigl ( A x + \sum\nolimits_{i \neq j}^N B_i K_i x \Bigr ) \rho^{(\epsilon, j)}(x) \Bigr \rangle + & \frac{\epsilon^2}{2} \Bigl \langle \frac{\partial^2} {\partial x^2},\, \sigma(x)\sigma^T(x) \rho^{(\epsilon, j)}(x) \Bigr \rangle, \notag \\
\rho^{(\epsilon, j)}(x) > 0, \quad \int_{\mathbb{R}^d} \rho^{(\epsilon, j)}(x) dx = 1,& \quad j = 1, 2, \ldots, N. \label{Eq18}
\end{align}
\begin{remark} \label{R6}
Note that $\sigma(x)$ is Lipschitz continuous, with the least eigenvalue of $\sigma(\cdot)\sigma^T(\cdot)$ uniformly bounded away from zero. Further, if it is twice differentiable on $\mathbb{R}^d$, the uniqueness for smooth steady-state solutions for \eqref{Eq17} and \eqref{Eq18} depends on the behavior of the original unperturbed multi-channel system as well as on the type of input controls used in the system.
\end{remark}
The following proposition provides a condition on the uniqueness of the solutions for the stationary Fokker-Planck equations.
\begin{proposition} \label{P1}
Suppose that there exists at least one $N$-tuple stabilizing state-feedbacks that satisfies the conditions in \eqref{Eq10}. Then, there exist unique smooth density functions $\rho^{(\epsilon, j)}(x)$ for $j = 0,1, \ldots, N$, with respect to $(\sigma, \epsilon)$, corresponding to the stationary Fokker-Planck equations in \eqref{Eq17} and \eqref{Eq18}.
Furthermore, the relative entropy of $\mu^{(\epsilon, i)}$ for $i = 1, 2, \ldots, N$, with respect to $\mu^{(\epsilon, 0)}$ (i.e., probability measures associated with the density functions $\rho^{(\epsilon, i)}(x)$ and $\rho^{(\epsilon, 0)}(x)$, respectively) is finite, i.e.,
\begin{align}
D\bigl(\mu^{(\epsilon, i)} \,\Vert\, \mu^{(\epsilon, 0)}\bigr) < +\infty. \label{Eq19}
\end{align}
\end{proposition}
\begin{proof}
Suppose that $\sigma(x)$ is twice differentiable on $\mathbb{R}^d$. Let the $N$-tuple of state-feedbacks $\bigl(K_1, K_2, \ldots, K_N \bigr)$ satisfy the conditions in \eqref{Eq10}. Then, the origin is a stable equilibrium point for the original unperturbed multi-channel system in \eqref{Eq1} (with respect to this particular set of state-feedbacks). Moreover, for sufficiently small $\epsilon > 0$, there exists a Lyapunov function $V(x) > 0$, with $\lim_{\vert x \vert \rightarrow \infty} V(x) = +\infty$, such that\footnote{Note that the assumption of a common Lyapunov function $V(x)$ is not necessary in \eqref{Eq20} and \eqref{Eq21}.}
\begin{align}
\Bigl \langle \Bigl ( A x + \sum\nolimits_{i=1}^N B_i K_i x\Bigl) \frac{\partial} {\partial x},\, V(x)\Bigr \rangle + \frac{\epsilon^2}{2} \Bigl \langle \sigma(x)\sigma^T(x) \frac{\partial^2} {\partial x^2},\, V(x)\Bigr \rangle < - \eta, \label{Eq20}
\end{align}
and
\begin{align}
\Bigl \langle \Bigl ( A x + \sum\nolimits_{i \neq j}^N B_i K_i x \Bigr ) \frac{\partial} {\partial x},\, V(x)\Bigr \rangle + \frac{\epsilon^2}{2} \Bigl \langle \sigma(x)\sigma^T(x) \frac{\partial^2} {\partial x^2},\, V(x)\Bigr \rangle < - \eta, \notag \\
j =1,2, \dots, N, \label{Eq21}
\end{align}
for all $x \in \mathbb{R}^d \backslash \{0\}$ and for some constant $\eta > 0$. Then, from Theorem~2.1 and Theorem~5.7 in \cite{BogKrB09}, the stationary measures $\mu^{(\epsilon, j)}$ for $j =0, 1, \dots, N$, of the Fokker-Planck equations in \eqref{Eq17} and \eqref{Eq18} uniquely admit smooth density functions $\rho^{(\epsilon, j)} \in C^{\infty}(\mathbb{R}^d)$ for $j =0, 1, \dots, N$, with $\rho^{(\epsilon, j)} > 0$ on $\mathbb{R}^d$, i.e.,
\begin{align*}
\mu^{(\epsilon, j)}(dx) = \rho^{(\epsilon, j)}(x) dx, \quad j =0, 1, \dots, N.
\end{align*}
Furthermore, the measure $\mu^{(\epsilon, i)}$ for $i \in \{1, 2, \ldots, N\}$, is absolutely continuous with respect to $\mu^{(\epsilon, 0)}$ (i.e., $\mu^{(\epsilon, i)} \ll \mu^{(\epsilon, 0)}$, $i = 1, 2, \ldots, N$) and, as a result, the relative entropy of $\mu^{(\epsilon, i)}(x)$ with respect to $\mu^{(\epsilon, 0)}(x)$ satisfies the following
\begin{align*}
D\bigl(\mu^{(\epsilon, i)} \,\Vert\, \mu^{(\epsilon, 0)}\bigr) < +\infty, \quad i = 1, 2, \ldots, N.
\end{align*}
This completes the proof.
\end{proof}
Next, let us define the systemic redundancy $r_{(\sigma, \epsilon)} \in \mathbb{R}$ (with respect to the random perturbation $(\sigma, \epsilon)$) as follows
\begin{align}
r_{(\sigma, \epsilon)} = \frac{1}{2N} \sum\nolimits_{i=1}^N D\bigl(\mu^{(\epsilon, i)} \,\Vert\, \mu^{(\epsilon, 0)}\bigr) - H\bigl(\mu^{(\epsilon, 0)}\bigr), \label{Eq22}
\end{align}
where $\bigl(1/2N\bigr) \sum\nolimits_{i=1}^N D\bigl(\mu^{(\epsilon, i)} \,\Vert\, \mu^{(\epsilon, 0)}\bigr)$ represents the average relative entropy of probability measures with respect to any single failure in the control channels.
\begin{remark} \label{R7}
Here we remark that $r_{(\sigma, \epsilon)}$ provides useful information in characterizing the systemic redundancy (i.e., with respect to the state-space and partitioned input spaces) in the system with multi-controller configurations having overlapping functionality, i.e., a requirement to maintain the stability of the system when there is a single failure in any of the control channels.
\end{remark}
Then, we have the following result on the asymptotic property of the systemic redundancy measure $r_{(\sigma, \epsilon)}$ as the random perturbation decreases to zero (i.e., as $\epsilon \rightarrow 0$).
\begin{corollary}\label{C1}
Suppose that Proposition~\ref{P1} holds, then the redundancy $r_{(\sigma, \epsilon)}$ satisfies the following asymptotic property
\begin{align}
r_{(\sigma, \epsilon)} \rightarrow r_{\infty} \quad \text{as} \quad \epsilon \rightarrow 0, \label{Eq23}
\end{align}
where
\begin{align}
r_{\infty} &= \lim_{ t \rightarrow \infty} \Biggl[\underbrace{\frac{1}{2N} \sum\nolimits_{i=1}^N D\bigl(\mu^{(i)} \,\Vert\, \mu^{(0)}\bigr) - H\bigl(\mu^{(0)}\bigr)}_{\substack{\triangleq \, r_t, ~ t \ge 0}} \Biggr], \label{Eq24}
\end{align}
and $\mu^{(j)}$, $j=0, 1, \ldots, N$, are probability measures with respect to the density functions $\rho^{(j)}(t, x)$, $j=0, 1, \ldots, N$, respectively, that satisfy the Liouville equations in \eqref{Eq11} and \eqref{Eq12} starting from an initial density function $\rho_0(x_0)$.
\end{corollary}
\begin{remark} \label{R8}
The proof is based on the idea of comparing the solutions of the Liouville equations to that of the steady-state solutions of the Fokker-Planck equations as $\epsilon \rightarrow 0$, and thus it is omitted.
\end{remark}
\begin{remark} \label{R9}
Note that a closer look at Equation~\eqref{Eq22} (see also Equations~\eqref{Eq23} and \eqref{Eq24} above) shows that $r_{(\sigma, \epsilon)} > r_t$, $\forall t \ge 0$, with respect to some perturbation $(\sigma, \epsilon)$ and further if $\epsilon_1 \le \epsilon_2$, then $r_{(\sigma, \epsilon_1)} \le r_{(\sigma, \epsilon_2)}$.
\end{remark}
\end{document} |
\begin{eqnarray}gin{document}
\begin{eqnarray}gin{CJK*}{GB}{gbsn}
\title{Edge state, bound state and anomalous dynamics in the Aubry-Andr\'{e}-Haper system coupled to non-Markovian baths}
\author{H. T. Cui (´Þº£ÌÎ)$^ {a,b}$}
\email{[email protected]}
\author{H. Z. Shen (ÉòºêÖ¾)$^ {b}$}
\email{[email protected]}
\author{M. Qin (ÇØÃ÷)$^{a}$}
\author{X. X. Yi (ÒÂѧϲ)$^{b}$}
\email{[email protected]}
\affiliation{$^a$ School of Physics and Optoelectronic Engineering, Ludong University, Yantai 264025
, China}
\affiliation{$^b$ Center for Quantum Sciences, Northeast Normal University, Changchun 130024, China}
\date{\today}
\begin{eqnarray}gin{abstract}
Bound states and their influence on the dynamics of an one-dimensional tight-binding system subject to environments are studied in this paper. We identify specifically three kinds of bound states. The first is a discrete bound state (DBS), of which the energy level exhibits a gap from the continuum. The DBS exhibits the similar features of localization as the edge states in the system and thus can suppress the decay of system. The second is a bound state in the continuum (BIC), which can suppress the system decay too. It is found that the BIC is intimately connected to the edge mode of the system since both of them show almost the same features of localization and energy. The third one displays a large gap from the continuum and behaves extendible (not localized). Moreover the population of the system on this state decays partly but not all of them does. This is different from the two former bound states. The time evolution of a single excitation in the system is studied in order to illustrate the influence of the bound states. We found that both DBS and BIC play an important role in the time evolution, for example, the excitation becomes localized and not decay depending on the overlap between the initial state and the DBS or BIC. Furthermore we observe that the single excitation takes a long-range hopping when the system falls into the regime of strong localizations. This feature can be understood as the interplay of system localizations and the bath-induced long-range correlation.
\end{abstract}
\maketitle
\end{CJK*}
\section{introduction}
In experiments, the environmental effect is unavoidable. A typical example is solid-state quantum devices, which are frequently disturbed by thermal as well as nonthermal environments. This stimulates the study of open quantum systems. In addition to exploring environmental effects in quantum devices and shedding light on the boundary between quantum and classical world, the study on open quantum systems may provide a paradigm to interpret how an open system equilibrates with its surroundings. Especially the localization-delocalization phase transition has been studied intensively in many-body systems with disorders \cite{ai, mbl}, and the quantum many-body scarred state has been found responsible for the breakdown of thermalization \cite{quantumscar}\cite{thermalization} when there is no disorder in systems.
Recently bound states that decay exponentially with small rates have been reported in open systems \cite{john, kofman, bic}. These bound states stem from the shift of system energy levels, induced by the emitted photon that pushes the level beyond the cut-off frequency of the environment \cite{kofman}. As a result of the appearance of energy gap, the bound states become robust against environment induced decays, and they can prevent quantum systems thermalising since the excitations on these states do not equilibrate. The appearance of bound states is a general feature of open quantum systems, independent of the fine structure of the systems. Thus it provides a general way for systems to prevent decoherence.
In fact, the recent experimental explorations of localization-delocalization transition in cold atomic gas suffer from atom-atom collisions and imperfect trapping \cite{exp-quasidisorder, luschen2017}. The collisions and imperfection can be modeled as environments and the localized phase would become unstable \cite{luschen2017} due to their influences. On the theoretical side, it was shown that the system exhibits a stretched exponential decay when coupled to a Markovian bath \cite{opendisorder}, then the localization is destroyed and the system is equilibrated finally. Despite these progresses in this direction, the effect of bound states on the dynamics of open system as well as on the localization remains unexplored.
In this paper, we will examine the bound states and the dynamics of an open system. For concreteness, we consider a one-dimensional tight-binding atomic chain with onsite modulation and being coupled to a bosonic bath. The Hamiltonian of such a chain is
\begin{eqnarray}\label{hs}
H_S=\sum_{n=1}^N \left(c^{\dagger}_{n} c_{n+1} +c^{\dagger}_{n+1} c_{n} \right) + \Delta\cos(2\pi \begin{eqnarray}ta n +\phi)c^{\dagger}_n c_{n},\nonumber \\
\end{eqnarray}
where $N$ is the length of atomic chain. $c_n (c^{\dagger}_n)$ is the annihilation (creation) operator of excitation at the $n$-th atomic site. $\begin{eqnarray}ta$ can be either rational or not, which characterizes two distinct cases. For $\begin{eqnarray}ta=p/q$ with $p$ and $q$ being coprime (commensurate case), the edge mode can occur because of nontrivial topological phase in $H_S $ \cite{lang2012}, which depicts the localization of excitation at boundary. When $\begin{eqnarray}ta$ is a Diophantine number \cite{syj99} (incommensurate case), $H_S$ corresponds to the Aubry-Andr\'{e}-Haper (AAH) model \cite{aah}, in which a delocalization-localization phase transition happens when $\Delta=2$. Recently it has been demonstrated that AAH model shows the correspondence to a two-dimensional quantum Hall system \cite{kraus}. Thus the topological edge mode can be found, in which the excitation would be localized at boundary \cite{kraus}. Moreover AAH model can be realized in cold atomic gas, and the experimental exploration of the delocalization-localization phase transition has been implemented \cite{exp-quasidisorder}.
The bath and its coupling to the atomic chain are respectively depicted by the following Hamiltonians,
\begin{eqnarray}
H_B&=& \sum_k \omega_k b_k^{\dagger}b_k;\nonumber\\
H_{int}&=& \sum_{k, n}\left(g_k b_k c_n^{+} + g_k^* b_k^{\dagger} c_n\right)\nonumber,
\end{eqnarray}
where $b_k (b_k^{\dagger})$ is the bosonic annihilation (creation) operator of the $k$-th mode of bath, and the frequency $\omega_k\geq 0( \forall k)$ consists of a continuum. $g_k$ characterizes the coupling strength between the lattice site and the $k$-th mode of bath. We assume that the coupling is so weak that the rotating-wave approximation (RWA) can be applied in $H_{int}$. Then the total Hamiltonian is
\begin{eqnarray}
H=H_S +H_B + H_{int}.
\end{eqnarray}
Since there is no particle interaction in $H_S$, the following discussion is restricted to the case of a single excitation, i.e. $\sum_nc^{\dagger}_n c_{n} + \sum_k b_k^{\dagger}b_k=1 $. In this case the bound state can be determined exactly, and the population dynamics can also be evaluated exactly. Although the particle interaction is important, we do not try to touch it in the current study since it would make the discussion complicated and ambiguous.
The remainder of this paper is organized as follows. In Sec. II the definition of bound state is presented. Interestingly a special discrete bound state can be found outside of the continuum $\omega_k$, which does not decay and displays similar localization as the edge mode in $H_S$. However, there also exists a single bound state with very small energy, which is extended and has a certain probability of spontaneous emission. In Sec. III, the population evolution dynamics is calculated, especially focusing on the interplay of bound state and localization in system. It is found that the discrete bound state (DBS) is predominant for the population evolution dynamics. Depending on the overlap of initial state and DBS, the excitation could become localized against spontaneous emission. Moreover, the bound state in the continuum (BIC) can also be identified by finding similar influence on the population evolution dynamics as the discrete one. The occurrence of BIC could be attributed to the nontrivial topology in $H_S$ \cite{yang, yao17}. In Sec.IV the interplay of disorder-induced localization and bath-induced long-range hopping is studied. We note that the hopping of excitation can occur over long-range atomic sites, even if the system is localized strongly. However it is suppressed greatly when DBS or BIC appears. In Sec.V the long-time behavior of evolution is studied. We observe a very slow decay of excitation in incommensurate case, even if the initial state overlaps with DBS or BIC. However this feature is not found in commensurate case. Finally conclusion is presented in Sec. VI.
\section{Bound state in open systems}
The bound state in open systems is defined as the discrete energy level of the total Hamilitonian \cite{fain}. As for the continuous spectrum $\omega_k > 0$, the bound state can be determined only by finding the negative solutions to the Schr\"{o}dinger equation
\begin{eqnarray}\label{bs}
H\ket{\psi_E}= E \ket{\psi_E}.
\end{eqnarray}
When $E>0$ the solutions could be obtained only for specific $\omega_k$, which thus constitute a continuum. It is the conventional wisdom that the state with frequency inside the continuum would leak and radiate out to infinity. However, a bound state in the continuum (BIC) can be found inside the continuum and coexists with extended states, but remains perfectly confined without any radiation\cite{bic}. Physically the occurrence of BIC can be attributed to the level resonance \cite{bic}. However, it is shown recently that BIC can also be found in the system with nontrivial topology\cite{yang}. In order to avoid confusion, we refer to the discrete bound state (DBS) as the discrete solution to Eq. \eqref{bs}. With respect that BIC can be identified only by the population evolution dynamics, as shown in Appendix C, the following discussion in this section is only suitable for DBS.
For a single excitation, $\ket{\psi_E}$ can be expressed generally as
\begin{eqnarray}\label{psie}
\ket{\psi_E}&=& \left(\sum_{n=1}^N \alpha_n \ket{1}_n\ket{0}^{\otimes (N-1)} \right)\otimes \ket{0}^{\otimes M} + \nonumber \\ &&\ket{0}^{\otimes N} \otimes \left(\sum_{k=1}^{M} \begin{eqnarray}ta_k \ket{1}_k\ket{0}^{\otimes (M-1)} \right),
\end{eqnarray}
where $\ket{1}_n = c_n^{\dagger}\ket{0}_n$ denotes the occupation of the $n$-th lattice site, $\ket{0}_k$ is the vacuum state of $b_k$ and $\ket{1}_k=b_k^{\dagger}\ket{0}_k$, and $M$ denotes the number of modes of bath. Substituting Eq. (\ref{psie}) into Eq. (\ref{bs}), one obtains
\addtocounter{equation}{1}
\begin{eqnarray}gin{align}\label{bsea}
&\left(\alpha_{n+1} + \alpha_{n-1}\right) +\Delta \cos(2\pi \begin{eqnarray}ta n +\phi)\alpha_n + \sum_{k=1}^{M} g_k \begin{eqnarray}ta_k = E \alpha_n; \tag{\theequation a} \\
&\omega_k \begin{eqnarray}ta_k + g_k^*\sum_{n=1}^N \alpha_n= E \begin{eqnarray}ta_k. \tag{\theequation b}\label{bseb}
\end{align}
According to Eq. (\ref{bseb}),
\begin{eqnarray}
\begin{eqnarray}ta_k=\frac{ g_k^*}{E-\omega_k}\sum_{n=1}^N \alpha_n.
\end{eqnarray}
Substitute the expression of $\begin{eqnarray}ta_k$ into Eq. (\ref{bsea}), and then
\begin{eqnarray}
\left(\alpha_{n+1} + \alpha_{n-1}\right) &+&\Delta \cos(2\pi \begin{eqnarray}ta n +\phi)\alpha_n +\nonumber \\
&& \left(\sum_{k=1}^{M} \frac{ \left|g_k\right|^2}{E-\omega_k}\right) \sum_{n=1}^N \alpha_n= E \alpha_n.\nonumber
\end{eqnarray}
As for the continuous spectrum $\omega_k$,
\begin{eqnarray}\label{int}
\sum_{k=1}^{M} \frac{ \left|g_k\right|^2}{E-\omega_k} \rightarrow \int_0^{\infty} \frac{J(\omega)}{E-\omega} \text{d}\omega,
\end{eqnarray}
where the spectral density $J(\omega)= \sum_{k=1}^{M} \left|g_k\right|^2 \delta\left(\omega-\omega_k\right)$. Then one has
\begin{eqnarray}\label{boundeqn}
\left(\alpha_{n+1} + \alpha_{n-1}\right) &+& \Delta \cos(2\pi \begin{eqnarray}ta n +\phi)\alpha_n +\nonumber \\
&& \int_0^{\infty} \text{d}\omega\frac{J(\omega)}{E-\omega} \sum_{n=1}^N \alpha_n = E \alpha_n.
\end{eqnarray}
With respect that the integrals Eq. (\ref{int}) is divergent for $E>0$, the solutions to Eq. \eqref{boundeqn} can be acquired only for $E<0$. Physically the last term at the left hand of Eq. \eqref{boundeqn} characterizes a homogenous hopping of excitation over atomic sites. As will be displayed in Sec. IV, the interplay of this effective long-range correlation and localization in system will impose a significant effect on the population evolution dynamics.
For concreteness, the spectral function is chosen as
\begin{eqnarray}\label{j}
J(\omega)= \eta \omega \left(\frac{\omega}{\omega_c}\right)^{s-1}e^{-\omega/\omega_c},
\end{eqnarray}
where $\eta$ characterizes the coupling strength between system and bath. The bath can be classified as sub-Ohmic ($s<1$), Ohmic ($s=1$) and super-Ohmic ($s>1$) \cite{leggett}. Physically Eq. \eqref{j} characterizes the damping movement of electrons in a potential, and thus provides a general picture for the dissipation of excitation in system. When disorder exists, it is expected that the competition between localization and the bath-induced dissipation would have a major influence on the dynamics of excitation. So the choice for $J(\omega)$ is suitable for the current interest. As for $s$, it is shown in Appendix A that the discrete solutions to Eq. \eqref{boundeqn} show negligible dependence on the value of $s$, except for the ground state. Thus the following discussion is restricted to the case of $s=1$. $\omega_c$ is the cutoff frequency of the bath spectrum, beyond which the spectral density starts to fall off. Hence, it determines a regime of frequency in bath, which is predominant for dissipation. In general the value of $\omega_c$ depends on specific environment. However as shown in Appendix A, $\omega_c$ shows a negligible effect on the solutions to Eq. \eqref{boundeqn}, except for the ground state. Hence $\omega_c=10$ is chosen in order to ensure $\Delta/\omega_c <1$ \cite{leggett}. In addition, an exceptional case can be found for the minimal solution $E_0$, which exhibits heavy dependence on the size of system and the properties of the bath. Thus the level $E_0$ would show distinct behavior.
Eq. \eqref{boundeqn} constitutes a linear system of equations for variable $\alpha_n$. The values of $E$ can be determined by finding out the zero points of determinant of coefficient matrix. However, noting that $E$ is also involved in the integrals, one thus has to appeal to numerics. Our evaluation shows that there are $N$ negative solutions to $E$ at most. Consequently as for large $N$, these solutions could constitute a band. Actually we find that the band is significantly overlapped with that in $H_S$ for $E\leq 0$. This feature can be attributed to the weak system-bath couplings: The bath cannot provide enough energy for the transition between different bands. It is difficult to determine the continuous spectrum $E$ in numerics. As a consequence we try to find the discrete $E$ in band gap, which is more tractable in numerics and meaningful in physics. Moreover it is expected that the discrete solution would be related intimately with the edge model in $H_S$ and thus could be stable against decoherence. So the remaining discussion in this section would focus on the discrete solutions occurring in gap instead. The terminology of DBS is designated as the special solution in this place. For this purpose, two situations are discussed respectively: commensurate ($\begin{eqnarray}ta=1/3$) and incommensurate ($\begin{eqnarray}ta=\left(1 + \sqrt{5}\right)/2$) cases, in which DBS behaves differently.
\begin{eqnarray}gin{figure}[t]
\center
\includegraphics[width=8cm]{f1.pdf}
\caption{(Color online)(a)Plots of the energy levels for $H_S$ with $\begin{eqnarray}ta=1/3$ and $\Delta=2$ (blue point) and the discrete bound states (red empty circle) when $E<0$. The labels $n=1 (99)$ denote the site, at which excitation is localized; (b) The plots of IPR (in blue empty circle) and $d$ (in red solid triangle) for DBS in the panel (a). The labels $E_1$, $E_2$ and $E_3$ denote the levels of DBS by increscent order. $N=99$, $s=1$, $\eta=0.1$ and $\omega_c=10$ are chosen for all plots.}
\label{fig:com-be}
\end{figure}
\subsection{Commensurate case: $\begin{eqnarray}ta=1/3$}
When $\begin{eqnarray}ta=p/q$ ($p$ and $q$ being coprime), the spectrum of $H_S$ consists of $q$ bands. As an exemplification, the spectrum of $H_S$ are demonstrated for $\begin{eqnarray}ta=1/3$ $\Delta=2$ under open boundary in Fig. \ref{fig:com-be}(a) (solid points). The edge mode, plotted by the discrete solid points in gap, depicts the localization of excitation at ends. In contrast the state in band is extended. By solving Eq. \eqref{boundeqn} three discrete solutions at most can be found in gap when $E<0$, which are highlighted by red empty circles in Fig.\ref{fig:com-be}(a). It is evident that two different features can be observed for these solutions. One is the DBS that has nearly the same energy as the edge mode in $H_S$. We find that it exhibits similar localization as the edge state, and thus could be considered as the renormalization of edge state. The other is the DBS that has different energy from the edge mode. We find that it is extended instead, as shown by the inverse participation ratio (IPR) $\text{IPR}=\sum_n \left|\alpha_n\right|^4$ in Fig. \ref{fig:com-be}(b), and thus comes from the transition of the state in band.
The unnormalized probability of spontaneous emission defined as
\begin{eqnarray}
d= \sum_k \left|\begin{eqnarray}ta_k\right|^2= \left|\sum_{n=1}^N\alpha_n\right|^2\int_0^{\infty} \frac{J(\omega)}{(E-\omega)^2} \text{d}\omega,
\end{eqnarray}
is calculated for all DBSs, as shown in Fig.\ref{fig:com-be}(b) by $\log d$. It is clear that $d$ has an amplitude not larger than $\sim 10^{-2}$. This picture means that DBS is robust against spontaneous emission.
However a single special solution $E_0 \sim 23.13$ to Eq. \eqref{boundeqn} can be found, for which the corresponding $\text{IPR}\sim1/99\approx 0.01$ and the probability of spontaneous emission is $\sim 0.405$. Furthermore we also find that $E_0$ is almost independent of $\phi$ and $\Delta$. For example, $E_0\sim -23.13$ for $\Delta=1$ and $-23.356< E_0<-23.31$ for $\Delta=4$. Instead it shows significant dependence on the system size $N$ and the properties of bath, as shown in Appendix A. It thus means that this special bound state is extended, and characterizes strong entanglement between the system and bath.
\begin{eqnarray}gin{figure*}
\center
\includegraphics[width=15cm]{f2.pdf}
\caption{(Color online) Plots of the levels (blue solid point) of $H_S$ with $\begin{eqnarray}ta= \left(1 + \sqrt{5}\right)/2$ for (a) $\Delta=1$, (b) $\Delta=2$, (c) $\Delta=4$ and DBS for $E<0$ (red empty-circle). The other parameters are same to those in Fig. \ref{fig:com-be}. The label $n=1 (99)$ denotes the site, occupied by excitation in edge mode and DBS.}
\label{fig:incom-be}
\end{figure*}
\subsection{Incommensurate case: $\begin{eqnarray}ta= \left(1 + \sqrt{5}\right)/2$}
The localization-delocalization phase transition can occur when $\begin{eqnarray}ta$ is a Diophantine number \cite{syj99}. With respect that the Diophantine number can be approached infinitely by rational numbers, the system is actually quasi-periodic, which induces a fractal structure in band as shown in Fig.\ref{fig:incom-be}. Furthermore there is a critical point $\Delta =2$ in $H_S$, which separates the delocalized phase ($\Delta<2$) from the localized phase ($\Delta>2$). In the delocalized phase all eigenstates tend to be extended. In contrast they show strong localization in localized phase. The in-gap edge state can also be found under open boundary condition since $H_S$ is equivalent to a two-dimensional Hofstadter model \cite{kraus}.
As for concreteness, $\begin{eqnarray}ta= \left(1 + \sqrt{5}\right)/2$ is chosen. By solving Eq. \eqref{boundeqn}, DBS can be decided exactly, which is highlighted by red empty circles in Fig. \ref{fig:incom-be} for $\Delta=1, 2, 4$ respectively. It is evident that there are two main gaps as well as several mini gaps. Although some discrete solutions may be found in the mini gaps, the following discussion will focus on the solutions in the two main energy gaps since the fractal bands are meaningless in physics. It should be pointed out that we do not try to discuss the variance of critical point because of the coupling to the bath. So the following discussion for $\Delta=2$ is just to show the influence of quasi-disorder.
An interesting feature in this case is that the discrete solution in main gap shows an apparent correspondence to the edge mode. This phenomenon could be attributed to the robustness of quasi-disorder against dissipation. So there is no transition occurring for the state in band, and the edge mode is renormalized as the DBS. In addition the localization in DBS is enhanced with the increment of $\Delta$, as shown by IPR in Fig.\ref{fig:incom-bs} in Appendix B. The corresponding $d$ also tends to be disappearing, which implies that the spontaneous emission of excitation is suppressed greatly.
Similar to the commensurate case, a single special solution $E_0$ can also be found. For instance we find that $E_0\sim -23.13, -23.17$ for $\Delta=1, 2$ and $\sim -23.351 <E_0 < \sim-25.32 $ for $\Delta=4$. Moreover the corresponding $\text{IPR}\approx 0.01$ and $d \approx 0.4$, independent of $\Delta$ and $\phi$.
\subsection{Further Discussion}
In conclusion the DBS can always be found in gap, which is connected intimately with the edge mode in $H_S$. A common property for DBS is the disappearing spontaneous emission, and thus the excitation can be preserved in system against decoherence. While the DBS shows one-to-one correspondence to the edge mode in the incommmensurate case, an additional DBS can be found in commensurate case, which has distinct energy from the edge mode and behaves extended instead, as shown in Fig. \ref{fig:com-be}(a). This phenomena can be attributed to the quasi-disorder in $H_S$, which makes the system stable against the transition induced by the coupling to a bath. In addition, we also find that the corresponding IPR is smaller than 1. The reason is the competition between the disorder-induced localization and the coupling-induced long-range correlation that makes the excitation hop in different sites. A detailed discussion for IPR can be found in Appendix B.
Another common picture is the existence of a special bound state $E_0$, which is extended and has a probability of spontaneous emission $\sim 0.4$. Moreover this special state exhibits strong dependence on the properties of bath and the system size $N$. Consequently $E_0$ characterizes the equilibrium between localization and dissipation, and thus be useless for the storage of quantum information.
\section{time evolution}
The population evolution of single excitation in system is discussed in this section, in order to demonstrate the strong influence of bound state. The evolution equation is written as
\begin{eqnarray}\label{evolution}
\mathbbm{i}\frac{\partial }{\partial t}\alpha_n(t)&=& \left[\alpha_{n+1}(t) + \alpha_{n-1}(t)\right]+ \Delta \cos(2\pi \begin{eqnarray}ta n +\phi)\alpha_n(t) \nonumber\\
&&- \mathbbm{i} \sum_{n=1}^N \int_0^t \text{d}\tau\alpha_n(\tau)f(t-\tau),
\end{eqnarray}
where $\mathbbm{i}$ is the imaginary unit, and the memory kernel $f(t-\tau)= \frac{\eta}{\omega_c^{s-1}} \frac{\Gamma(s+1)}{\left[\mathbbm{i}(t-\tau) + 1/\omega_c\right]^{s+1}}$ is responsible for dissipation. Because of involved integrals, numerical evaluation has to be implemented to find out $\alpha_n(t)$. Our way is to rewrite the integrals as a summation with suitable step length. Then by solving Eq. (\ref{evolution}) iteratively, $\alpha_n(t)$ can be determined finally.
Formally when the bound state occurs, $\ket{\psi(t)}$ can be decomposed into two parts, i.e.
\begin{eqnarray}\label{psit}
\ket{\psi(t)}= \sum \alpha_b\ket{\psi_b} e^{-\mathbbm{i}E_b t} + \int\text{d}E_c \alpha(E_c)e^{-\mathbbm{i}E_c t } \ket{\psi_c}.
\end{eqnarray}
The summation is over all bound states $\ket{\psi_b}$ with energy $E_b$, which means unitary evolution and thus is responsible for the robustness of excitation. While the integrals over the continuum $E_c$ is responsible for the decay of excitation, which tends to be vanish after a long time. As a result the bound states will determine completely the final state of system. In order to highlight the effect of DBS or BIC, we choose the initial state $\ket{\psi(t=0)}=\sum_n \alpha_n(0)\ket{n}$ with a single excitation located at atomic site $n_0=1$ and $n_0=99$ respectively. The corresponding revival probability of excitation $\left|\alpha_1(t)\right|^2$ and $\left|\alpha_{99}(t)\right|^2$ are calculated, as well as the corresponding $\text{IPR}_{1(99)}$. Three distinct behaviors can be found for the population evolution of single excitation. First the excitation becomes localized at its initial site. Second the excitation can hop to a different site from its initial one. Thirdly the evolution is dissipative and excitation could be absorbed finally by bath.
\begin{eqnarray}gin{figure}
\center
\includegraphics[width=8cm]{f3.pdf}
\caption{(Color online) The evolution of survival probability $\left|\alpha_n\right|^2 (n=1,99)$ (solid line) and the corresponding $\text{IPR}_n$ (dashed line) for a single excitation initially at $n_0=1$ (left column) or $n_0=99$ (right column). $\begin{eqnarray}ta=1/3$ and $\Delta=2$ are chosen, and the other parameters are same to those in Fig.\ref{fig:com-be}. }
\label{fig:com-evolution}
\end{figure}
\begin{eqnarray}gin{figure}
\center
\includegraphics[width=6cm]{f4.pdf}
\caption{(Color online) The plots of survival probability for different $\Delta$ when $\begin{eqnarray}ta=1/3$ and $\phi=-\pi$. The other parameters are same to those in Fig. \ref{fig:com-evolution}. }
\label{fig:com-delta}
\end{figure}
\subsection{Commensurate case: $\begin{eqnarray}ta=1/3$}
Five different cases are plotted in Fig. \ref{fig:com-evolution}. For $\phi=-\pi$ two DBSs can be found when $E<0$, as shown in Fig. \ref{fig:com-be}(a); One is overlapped with the edge state and shows strong localization at site $n=99$. Whereas the other is extended. It is clear that the survival probability $\left|\alpha_{99}\right|^2$ shows a stable oscillation around 0.5 for excitation located initially at site $n_0=99$, as shown in Fig.\ref{fig:com-evolution}(b1). This oscillation stems from the interference of two DBSs, that can be affirmed by measuring the frequency of oscillation. As shown in Fig.\ref{fig:com-evolution}(b1), the period of oscillation is $T=16.77$. Then the frequency $\omega=2\pi/T=0.3747$, which is closed to the energy difference $\delta E= 0.3768$ of the two DBSs. The slight difference comes from the computational error. However, $\left|\alpha_{1}\right|^2$ for $n_0=1$ displays a rapid decay, as shown in Fig.\ref{fig:com-evolution}(a1). The same features can also be found for IPR (dashed line in Fig.\ref{fig:com-evolution}). The observation implies that DBS would determine completely the population evolution: When the initial state is overlapped with DBS, the excitation can be preserved with a large probability. While if not, the information of initial state would be erased completely. So in this sense the edge state would be renormalized as a DBS. It should be pointed out that the weak fluctuation of survival probability for $t >\sim 180$ comes from the accumulation of computational error in solving Eq. \eqref{evolution} iteratively.
Similar phenomena can also be observed for $\phi=0.5\pi$, in which there are three DBSs, as shown in Fig. \ref{fig:com-be}(a). Two of them show similar localization as the edge states. The third behaves extended instead. It is noted that $\left|\alpha_{1}\right|^2$ shows a stable oscillation with period $T=32.11$ because of the interference of the two lowest DBSs, with the energy difference $\delta E= 0.1953$. At the same time $\left|\alpha_{99}\right|^2$ becomes stable when the initial state is overlapped with the DBS, which shows strong localization at $n=99$. An interesting situation is $\phi=0.66\pi$: There are two DBSs with localizations at $n=1$ and $n=99$ respectively. They are closed to each other in energy as shown in Fig.\ref{fig:com-be}(a). Consequently a stable oscillation can be found for both $\left|\alpha_{1}\right|^2$ and $\left|\alpha_{99}\right|^2$ because of the interference, as shown in Fig.\ref{fig:com-be}(a5) and (b5). As will be discussed in next section, this interference induces an end-to-end hopping of excitation.
A special case happens for $\phi=0$, in which there is no DBS when $E<0$. In contrast to the rapid decay of $\left|\alpha_{1}\right|^2$, a stable evolution can be noted for the excitation initially located at $n_0=99$, as shown in Fig.\ref{fig:com-be}(a3) and (b3). This phenomenon can be attributed to the occurrence of BIC \cite{bic}, as shown in Appendix C. Generally BIC is induced by the level resonance \cite{bic}. However in the present discussion BIC could be understood by the nontrivial topology in $H_S$ \cite{yang, yao17}. It is clear that both DBS and BIC manifest similar influence on the population evolution dynamics. Another exemplification of BIC can be found when $\phi=-0.3\pi$. Under this circumstance, there are two edge states in $H_S$ when $E>0$ with the localization at $n=1$ and $n=99$ respectively. Consequently both $\left|\alpha_{1}\right|^2$ and $\left|\alpha_{99}\right|^2$ show stable evolution, as shown in Fig.\ref{fig:com-evolution} (a4) and (b4).
The localization is enhanced with the increment of $\Delta$, as shown by $\left|\alpha_{99}\right|^2$ in Fig. \ref{fig:com-delta} for $\phi=-\pi$. At the same time the decay of $\left|\alpha_{1}\right|^2$ also becomes stretched slightly. This feature can be attributed to the trapping effect of on-site potential.
\begin{eqnarray}gin{figure}[t]
\center
\includegraphics[width=8.5cm]{f5.pdf}
\caption{(Color online) The plots of survival probability $\left|\alpha_{1(99)}\right|^2$ for excitation initially at site $n_0=1$ (left column) and $n_0=99$ (right column) versus $t$ when $\Delta=1, 2,4$. $\begin{eqnarray}ta=\left(1 + \sqrt{5}\right)/2$ are chosen for these plots. The other parameters are same to those in Fig. \ref{fig:com-evolution} }
\label{fig:incom-evolution}
\end{figure}
\subsection{Incommensurate: $\begin{eqnarray}ta= \left(1 + \sqrt{5}\right)/2$ }
Two distinct phases can be identified in this case: delocalized phase ($\Delta<2$), in which the system is extendible, and the localized phase ($\Delta>2$), in which the system displays strong localization. As exemplifications, the cases of $\phi=-\pi$ and $\phi=0.4\pi$ are studied in details, for which there is a DBS and a BIC with localization at site $n=99$ and $1$ respectively, as shown in Fig. \ref{fig:incom-be}. As expected, the stable evolution can be found for excitation located initially at $n_0=99$ or $n_0=1$, as shown in Fig. \ref{fig:incom-evolution} (a2) and (b1). Furthermore we also note that although the survival probability is enhanced with the increment of $\Delta$, a strange feature can be found in Fig. \ref{fig:incom-evolution}(b1), where $\left|\alpha_{1}\right|^2$ declines smoothly when $\Delta=4$. This abnormal feature will be discussed alone in Sec. V.
However The picture becomes different when the initial state is not overlapped with any DBS or BIC. For example, the survival probability $\left|\alpha_{1}\right|^2$ for $\phi=-\pi$ exhibits a rapid decay when $\Delta=1$. However when $\Delta=2, 4$, a significant recurrence can be found for $\left|\alpha_{1}\right|^2$, as shown in Fig. \ref{fig:incom-evolution} (a1). This feature could be attributed to the influence of the bound states other than DBS and BIC. As stated in Sec. II, the solutions to Eq. \eqref{boundeqn} other than the discrete ones in gap, constitute the band, which become more localized with the increment of $\Delta$. Consequently when the initial state is overlapped substantially with the states in band, the interference of states thus would induce the temporal revival of $\left|\alpha_{1}\right|^2$. This explanation can be verified in further by noting that the recurrence is absent in commensurate case and for $\Delta=1$, in which the states in band are extended or delocalized. Similar picture can also be found for $\left|\alpha_{99}\right|^2$ when $\phi=0.4\pi$, as shown in Fig. \ref{fig:incom-evolution} (b2).
\subsection{Further Discussion}
It is evident that the bound state is predominant in the population evolution. Dependent on the overlap of initial state and DBS or BIC, the survival probability of excitation can become stable against dissipation. For both commensurate and incommensurate cases, the excitation can be preserved in system with a large probability if the initial state is overlapped with DBS or BIC. In contrast if not, two different features would be obtained in our discussion. When $H_S$ is commensurate or in delocalized phase ($\Delta<2$), the population evolution is dissipative. However in the localized phase of $H_S$ ($\Delta>2$), it can show a recurrence due to the strong localization of $H_S$, which cannot be destroyed completely by coupling to a bath.
An interesting question is the excitation dynamics when there is no DBS or BIC. As shown by the integrals in \eqref{boundeqn}, an effective long-range correlation in atomic sites is inspired by the coupling to bath, which is responsible for the dissipation of excitation. However the quasi-disorder in $H_S$ tends to localize the excitation in the system. Hence it is expected that the interplay of the long-range correlation and the localization induced by quasi-disorder would inspire exotic dynamics of excitation. In the next section, we shed light on the influence of this interplay.
\begin{eqnarray}gin{figure}
\center
\includegraphics[width=8.5cm]{f6a.pdf}
\includegraphics[width=8.5cm]{f6b.pdf}
\caption{(Color online) The evolution of survival probability for $\phi=-0.3\pi$ (a) and $\phi=0.7\pi$ (b) when $\Delta=4$, as well as $\text{IPR}_{1(99)}$. The other settings are same to those in Fig.\ref{fig:incom-evolution}. }
\label{fig:incom-hopping}
\end{figure}
\begin{eqnarray}gin{figure}
\center
\includegraphics[width=8.5cm]{f7a.pdf}
\includegraphics[width=8.5cm]{f7b.pdf}
\caption{(Color online) The plots for the distribution $\left|\alpha_{n}\right|^2 $ for $\begin{eqnarray}ta=1/3$ (a) and $\begin{eqnarray}ta=\left(1+\sqrt{5}\right)/2$ (b) versus $t$. The selected $\phi$ and $\Delta$ are presented in the plot labels. The other settings are same to those in Fig.\ref{fig:com-evolution} for all plots. }
\label{fig:incom-hopping2}
\end{figure}
\section{The long-range hopping of excitation}
In order to demonstrate the effect of effective long-range correlation and quasi-disorder, the cases $\phi=-0.3\pi$ and $0.7\pi$ are inspected for $\Delta=4$. There is no DBS or BIC under these circumstances as shown in Fig.\ref{fig:incom-be}. The survival probability and the corresponding distribution of excitation in system are plotted in Fig.\ref{fig:incom-hopping}. It is clear that the occupation probabilities of the excitation located on some sites becomes pronounced, except for the initial one. Meanwhile the evolution of IPR also becomes complex. This phenomenon is a result of the interplay of the quasi-disorder and the effective long-range correlation: The long-range correlation is devoted to the hopping and dissipation of excitation. Whereas, the quasi-disorder tends to trap and preserve the excitation against dissipation. Consequently at some moment the excitation is kept as some site with a significant probability, where the on-site potential is stronger.
However we find that the hopping could be restrained greatly when DBS or BIC appears. As an exemplification, we examine the case of $\phi=0$ when $\begin{eqnarray}ta=\left(1+\sqrt{5}\right)/2$, in which there is a BIC with localization at site $n=99$. It is found for $n_0=1$ that the distribution $\left|\alpha_{n}\right|^2 (n\neq 1)$ becomes pronounced at some sites with the increment of $\Delta$, as shown in Fig. \ref{fig:incom-hopping2} (b). However for $n_0=99$, it is clear from Fig.\ref{fig:incom-hopping2}(b) that $\left|\alpha_{n}\right|^2 (n\neq 99)$ tends to disappear even for $\Delta=4$. The phenomenon originates from the strong localization of DBS or BIC, which is protected by the nontrivial topology in $H_S$.
This picture can also be noted in commensurate case. As shown in Fig. \ref{fig:incom-hopping2} (a) for $\phi=0.66\pi$ when $\begin{eqnarray}ta=1/3$, a hopping of excitation can be found \emph{only} between sites $n=1$ and $n=99$. In contrast, it is absent when there is only one DBS, as shown for $\phi=0$ in Fig. \ref{fig:incom-hopping2} (a).
\section{The Long-time behavior}
\begin{eqnarray}gin{figure}
\center
\includegraphics[width=6cm]{f8.pdf}
\caption{(Color online) The long-time feature of survival probability for excitation with $\begin{eqnarray}ta=1/3$ (a) and $\begin{eqnarray}ta=\left(1+\sqrt{5}\right)/2$ (b, c). The chosen initial states, as well as $\phi$ and $\Delta$, are presented in the plot labels. $N=99$, $s=1$, $\eta=0.1$ and $\omega_c=10$ are chosen for all plots. }
\label{fig:longtime}
\end{figure}
Although we claim that DBS or BIC could determine the steady behavior of system, an exception can be found. As for $\phi=0.4\pi$ and $-\pi$ with $\begin{eqnarray}ta=\left(1+\sqrt{5}\right)/2$, the survival probability of excitation declines very slowly when $\Delta=4$, even if the initial state is overlapped with DBS or BIC, as shown in Fig. \ref{fig:longtime} (b) and (c). We find that this declination cannot be attributed to computational errors. In contrast it does not occur for $\begin{eqnarray}ta=1/3$, as shown in Fig. \ref{fig:longtime} (a), as well for $\Delta=1$ when $\begin{eqnarray}ta=\left(1+\sqrt{5}\right)/2$ shown in Fig. \ref{fig:longtime} (b) and (c). For these two cases, the system is extendible or in delocalized phase. For longer time evolution, the numerical evaluation becomes exhaustive and thus unreliable because of the accumulation of computational error.
Unfortunately we cannot determine the reason for this declination because of the difficulty of deciding all bound states. As for this phenomena is absent when the system is extendible or delocalized, a possible understanding might be the influence of the bound states in band. These states also become much localized with the increment of disorder in $H_S$. As a consequence they would show non-negligible contribution to the evolution after a long time.
\section{Conclusion}
In conclusion, the bound states and their influence on the population evolution are investigated in a one-dimensional tight-binding atomic chain. Each site of the chain is coupled to an environment and all sites share a common environment. By solving the Schr\"{o}dinger equation in the limit of a single excitation, three special kinds of bound states are identified. The first is the DBS, which corresponds to a single negative eigen-energy with finite gap from the continuum. It is concluded from the calculations that the system on DBS does not decay, and it has similar localization features to the edge mode of the system. An additional DBS is found in the gap when the system is commensurate, which is extendible and can be understood as the bath-induced transition of the state in band. The situation changes when the system is incommensurate due to the intrinsic localization in system, which prevents the system being excited due to its couplings to the environment.
The second is a bound state in continuum, which is connected intimately to the edge mode with positive energy and also exhibits zero decay rate. The robustness of BIC could be attributed to the nontrivial topology of the system. The third is a single special bound state of the lowest energy. Different from the first two bound states, it is extendible and displays a certain probability to decay. Moreover it depends sharply on the size of system and the properties of the bath.
The time evolution of a single excitation is simulated in order to explore the influence of the bound states. It is concluded that the bound states are predominant for the population evolution. When the system is extendible or delocalized, the excitation becomes stable against dissipation provided the initial state overlaps with DBS or BIC. However if the overlapping is zero, the evolution is dissipative and the information of initial states will be erased finally. The situation changes for incommensurate systems with strong quasi-disorders (for example, $\Delta=4$), the occupation probabilities of the excitation decrease slowly, even if the initial state overlaps with DBS or BIC. Furthermore a significant recurrence of survival probabilities for the excitation can be found when the initial state overlaps with neither DBS nor BIC. These two features may be understood as the interplay between localizations in the system and the effective long-range correlation induced by the bath. Another important consequence of this interplay is the long-range hopping of the single excitation of the system, which makes the excitation hop to a different site from an initial one. We note that the hopping can also happen between two localized DBSs in commensurate cases, as shown for $\phi=0.66\pi$ in Fig. \ref{fig:com-evolution}(a6) and (b6).
An open question is the effect of interactions between atoms on the prediction. It is known that the competition between interactions and disorders is responsible for the many-body localization transition in AAH model \cite{iyer13}. Recall that the interatomic interaction might destroy the localization, the edge mode in the system could be changed. Moreover, the bound states in open systems amount to an effective trap potential \cite{shi16, shi18}, which prevents excitation from decaying. Hence when the interatomic interactions are involved, the competition between the effective trapping and interatomic interactions would intrigue interesting feature. When the trapping is predominant, the excitation could be preserved in the system. Otherwise, the excitation dissipates. Due to the complicated and involved calculation for multi-excitation bound states \cite{shi16}, we left the related discussion in the future work.
\section*{Acknowledgments}
HZS acknowledges the support of National Natural Science Foundation of China (NSFC) under Grant No.11705025. MQ acknowledges the support of National Natural Science Foundation of China (NSFC) under Grant No. 11805092. HTC and XXY acknowledges the financial support of NSFC under Grants No. 11775048, No. 11534002 and No. 11947405.
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\widetext
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\section*{Appendix A}
In this appendix, the exact solutions to Eq. (\ref{bs}) is presented for $\begin{eqnarray}ta=p/q$ under periodic boundary condition. Assume $N=Lq$, and then $H_S$ can be written as
\begin{eqnarray}
H_S&= &\sum_{x=1}^L \left(c_1^{\dagger}, c_2^{\dagger},\cdots, c_q^{\dagger}\right)_x \left(\begin{eqnarray}gin{array}{cccc} \Delta\cos\left(\frac{2\pi p}{q} +\phi\right) & J & 0 & \cdots \\
J & \Delta\cos\left(\frac{4\pi p}{q} +\phi\right) & J & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & \cdots & J & \Delta\cos\left(\phi\right) \end{array}\right)\left(\begin{eqnarray}gin{array}{c} c_1 \\ c_2\\ \vdots \\ c_q\end{array}\right)_x +\nonumber\\
&&\sum_{x=1}^L \left(c_1^{\dagger}, c_2^{\dagger},\cdots, c_q^{\dagger}\right)_x \left(\begin{eqnarray}gin{array}{cccc}
0 & 0& \cdots & 0 \\ \vdots & \ddots & \vdots & \vdots\\
1 & 0 &\cdots & 0
\end{array}\right)\left(\begin{eqnarray}gin{array}{c} c_1 \\ c_2\\ \vdots \\ c_q\end{array}\right)_{x+1} + h.c.
\end{eqnarray}
By Fourier transformation $c_x= \frac{1}{\sqrt{L}} \sum_{\lambda=1}^L a_{\lambda} e^{i2\pi \lambda x/L}$, then
\begin{eqnarray}
H_S= \sum_{\lambda=1}^L \left(a_1^{\dagger}, a_2^{\dagger},\cdots, a_q^{\dagger}\right)_{\lambda} \left(\begin{eqnarray}gin{array}{ccccc} \Delta\cos\left(\frac{2\pi p}{q} +\phi\right) & J & 0 & \cdots & e^{i2\pi q \lambda/L}\\
J & \Delta\cos\left(\frac{4\pi p}{q} +\phi\right) & J & 0& \cdots \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
e^{-i2\pi q \lambda /L}& 0 & \cdots & J & \Delta\cos\left(\phi\right) \end{array}\right)\left(\begin{eqnarray}gin{array}{c} a_1 \\ a_2\\ \vdots \\ a_q\end{array}\right)_{\lambda}.\nonumber
\end{eqnarray}
As for $H_{int}$,
\begin{eqnarray}
H_{int}&=&\sum_{k, x}g_k b_k \left(c_1^{\dagger}, c_2^{\dagger},\cdots, c_q^{\dagger}\right)_x+ g_k^* b_k^{\dagger} \left(\begin{eqnarray}gin{array}{c} c_1 \\ c_2\\ \vdots \\ c_q\end{array}\right)_x \nonumber\\
&&\Rightarrow \frac{1}{\sqrt{L}}\sum_{k}g_k b_k \left(a_1^{\dagger}, a_2^{\dagger},\cdots, a_q^{\dagger}\right)_{\lambda=0}+ g_k^* b_k^{\dagger} \left(\begin{eqnarray}gin{array}{c} a_1 \\ a_2\\ \vdots \\ a_q\end{array}\right)_{\lambda=0}.
\end{eqnarray}
Considering a single excitation for $\lambda=0$, the eigenfunction $\ket{\psi}_E$ can be written as
\begin{eqnarray}
\ket{\psi}_E= \left(\sum_{n=1}^q \alpha_n a^{\dagger}_n \ket{0}_n \right)\otimes \ket{0}^{\otimes M} +\ket{0}^{\otimes q} \otimes \left(\sum_{k=1}^{M} \begin{eqnarray}ta_k b_k\ket{0}_k\ket{0}^{\otimes (M-1)} \right).
\end{eqnarray}
As for $p=1$ and $ q=3$, substitute $\ket{\psi}_E$ into Eq. (\ref{bs}) and eliminate the degree of freedom of bath. One can obtain the equation
\begin{eqnarray}
E^3- 3 d(E) E^2 - \left(3 + 6 d(E) + \frac{3}{4}\Delta^2\right)E - \left(2 + 3d(E) - \frac{3a}{4}\Delta^2 + \frac{\Delta^3}{4}\cos3\phi\right)=0,\nonumber
\end{eqnarray}
where $d(E)=\frac{1}{L} \int_0^{\infty} \frac{J(\omega)}{E-\omega} \text{d}\omega $. By solving the above equation, three relations can be found
\begin{eqnarray}\label{solution}
E_0&=& \sqrt{4\left[1 +d(E_0)\right]^2 + \Delta^2} \cos\left(\theta_{\phi}+\frac{2\pi}{3}\right) +d(E_0);\nonumber\\
E_1&=& \sqrt{4\left[1 +d(E_1)\right]^2 + \Delta^2} \cos\left(\theta_{\phi}+\frac{4\pi}{3}\right) +d(E_1);\nonumber\\
E_2&=& \sqrt{4\left[1 +d(E_2)\right]^2 + \Delta^2} \cos\theta_{\phi} +d(E_2),
\end{eqnarray}
where
\begin{eqnarray}
\theta_{\phi}= \frac{1}{3}\arccos \left\{\frac{\left[1 +d(E)\right]^3 + \frac{1}{8}\Delta^3\cos3\phi}{\sqrt[3]{\left[1 +d(E)\right]^2 + \Delta^2/4}}\right\}.
\end{eqnarray}
$E_0, E_1, E_2$ correspond to three real solutions, which are plotted for different parameters by blue dashed lines in Fig.\ref{fig:appendix}. We find that $E_0$ shows significant dependence on the properties of the bath and the system size $L$, and thus it is extensive. In contrast, both $E_1$ and $E_2$ are determined completely by the properties of system, and thus are intensive.
Actually the three levels $E_0$ and $E_1, E_2$ characterize the main feature of the bound state in main text. $E_0$ corresponds to the minimal solution to Eq. \eqref{boundeqn}, which is extended and has a finite probability of spontaneous emission. However $E_1$ and $E_2$ have correspondence to DBS. In Fig.\ref{fig:s}, the evolution of excitation initially at $n_0=1, 99$ are plotted for different $s$. It is apparent that the survival probability is insensitive to the value of $s$.
\begin{eqnarray}gin{figure}
\center
\includegraphics[width=12cm]{fa1.pdf}
\includegraphics[width=12cm]{fa2.pdf}
\caption{(Color online) The plots of the numerical solutions to Eq. (\ref{solution}) versus the different parameters in system and bath. $\eta=0.1$ is chosen for all plots. }
\label{fig:appendix}
\end{figure}
\begin{eqnarray}gin{figure}[t]
\center
\includegraphics[width=15cm]{fa3.pdf}
\caption{(Color online) The plots of survival probability $\left|\alpha_{1(99)}\right|^2$ for different $s$, when the excitation is initially at $n_0=1$ and $n_{0}=99$ respectively. $N=99$ $\eta=0.1, \omega_c=10$ for all plots. }
\label{fig:s}
\end{figure}
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\section*{Appendix B}
The inverse participation ratio (IPR) is a general measure of the localization of state. For state $\ket{\psi}=\sum_{n=1}^N \alpha_n \ket{n}$, where $\ket{n}$ denotes the occupation of the $n$-th site, and $N$ is the number of site, IPR is defined as
\begin{eqnarray}
\text{IPR}_{\psi}=\sum_{n=1}^N \left|\alpha_n\right|^4.
\end{eqnarray}
IPR has the minimum $1/N$ only if $\left|\alpha_n\right|^2=1/N$ for any $n$, which means that the distribution of excitation is uniform, and thus $\ket{\psi}$ is extended. While IPR has the maximum 1 only if $\left|\alpha_n\right|^2=1$ for a special $n$, which means that excitation can appear only at site $n$, and thus $\ket{\psi}$ is localized completely.
\begin{eqnarray}gin{figure}
\center
\includegraphics[width=5.5cm]{fb1.pdf}
\includegraphics[width=5.5cm]{fb2.pdf}
\includegraphics[width=5.5cm]{fb3.pdf}
\caption{(Color online) Plots of IPR (blue empty symbols) and $d$ (red-solid symbols) for DBS when $\begin{eqnarray}ta=\left(1+\sqrt{5}\right)/2$ and $\Delta=1, 2, 4$, respectively. The parameters are chosen as the same in Fig. \ref{fig:incom-be}. The labels of $E_1$, $E_2$ and $E_3$ denote the levels of DBS, plotted in Fig. \ref{fig:incom-be}, in increscent order. }
\label{fig:incom-bs}
\end{figure}
In Fig. \ref{fig:incom-bs}, IPR and corresponding $d$ are plotted for different $\Delta$s when $\begin{eqnarray}ta=\left(1+\sqrt{5}\right)/2$. It is clearly concluded that the DBS is localized. Moreover IPR is enhanced with the increment of $\Delta$, which means that the system becomes more localized.
\begin{eqnarray}gin{figure}[t]
\center
\includegraphics[width=15cm]{fb4.pdf}
\caption{(Color online) The site-distribution $\left|\alpha_n\right|^2$ for DBS when $\phi=-\pi$ versus $\Delta=1,2,4$. $N=99$, $s=1, \eta=0.1, \omega_c=10$ are chosen for all plots.}
\label{fig:distribution}
\end{figure}
We note that the IPR of DBS is always smaller than 1. The reason is the interplay between the localization, which tends to localize the excitation in system, and the effective long-range correlation in atomic sites, which tends to delocalize the excitation instead. In Fig. \ref{fig:distribution}, the distribution $\left|\alpha_n\right|^2$ of excitation in DBS is shown for different $\Delta$ when $\phi=-\pi$ as an exemplification. When $\begin{eqnarray}ta=1/3$, there is two DBSs. One corresponds to the renormalized edge state, and thus show strong localization. The other comes from the transition of state in band, and thus is extended. It is clear for the former that the distribution becomes much pronounced at end site $n=99$ with the increment of $\Delta$, as shown in the upper row in Fig. \ref{fig:distribution}. However for the latter it tends to be multipeaked with the increment of $\Delta$, as shown in the middle row in Fig. \ref{fig:distribution}. As for $\begin{eqnarray}ta=\left(1+\sqrt{5}\right)/2$, the value of $\Delta$ characterizes the strength of disorder in system. Thus it is not surprising that the localization of DBS is enhanced with the increment of $\Delta$, as shown by the bottom row in Fig.\ref{fig:distribution}.
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\section*{Appendix C}
In this appendix, we demonstrate the existence of BIC analytically. For this purpose, we first diagonalize the system Hamiltonian as $H_S=\sum_{i=1}^{N} \epsilon_i \eta^{\dagger}_i\eta_i$, where $\eta_i=\sum_n \gamma_{in}^{*}c_n$. The array $\left(\gamma_{i1}, \gamma_{i2},\cdots, \gamma_{iN}\right)^{T}$ denotes the $i$-th eigenstate of Eq. (\ref{hs}). Then the total Hamiltonian can be rewritten as
\begin{eqnarray}
H=\sum_{i=1}^{N} \epsilon_i \eta^{\dagger}_i\eta_i + \sum_k \omega_k b_k^{\dagger}b_k + \sum_{i,k}g^*_{ik} \eta_i b_k^{\dagger} + g_{ik} \eta^{\dagger}_i b_k.
\end{eqnarray}
where $ g_{ik}=g_k\sum_n\gamma_{in}$. For a arbitrary state $\ket{\psi(t)}=\left(\sum_i \alpha_i(t)\eta_i^{\dagger}\ket{0}_i\right)\ket{0}^{\otimes M} + \ket{0}^{\otimes N}\left(\sum_k \begin{eqnarray}ta_k(t)b_k^{\dagger}\ket{0}_k\right)$, the evolution equation can be written as
\begin{eqnarray}
\mathbbm{i}\frac{\partial \alpha_i(t)}{\partial t}&=&\alpha_i(t)\epsilon_i -\mathbbm{i}\left(\sum_n\gamma^*_{in}\right) \sum_j \left(\sum_n\gamma_{jn}\right)\int_0^{\tau}\text{d}\tau \alpha_j(t)\sum_k\left|g_k\right|^2e^{- \mathbbm{i}\omega_k t}\nonumber\\
&=&\alpha_i(t)\epsilon_i -\mathbbm{i}\left(\sum_n\gamma^*_{in}\right) \sum_j \left(\sum_n\gamma_{jn}\right)\int_0^{\tau}\text{d}\tau \alpha_j(t)\int_0^{\infty}J(\omega)e^{- \mathbbm{i}\omega t},\nonumber
\end{eqnarray}
where we have assumed that the excitation is located initially in system, and thus $\begin{eqnarray}ta_k(0)=0$. By Laplace transformation $G_i(z)=\int_0^{\infty}\text{d}t \alpha_i(t) e^{-zt}$, the equation above can be rewritten as
\begin{eqnarray}\label{appendixB}
\left(\mathbbm{i}z - \epsilon_i \right)G_i(z)- \Sigma(z)\left(\sum_n\gamma^*_{in}\right) \sum_j \left(\sum_n\gamma_{jn}\right)G_j(z)= \mathbbm{i} \alpha_i(0),
\end{eqnarray}
where $\Sigma(z)= \int_0^{\infty}\frac{J(\omega)}{\mathbbm{i}z-\omega}$ is the self-energy. Then we obtain a linear system of equations for $G_i(z)$, for which the solution can be expressed as
\begin{eqnarray}
G_i(z)= \frac{Det(B_i)}{Det(A)}.
\end{eqnarray}
The element of coefficients matrix $A$ is $A_{ij}= \left(\mathbbm{i}z - \epsilon \right)\delta_{ij}- \Sigma(z)\left(\sum_n\gamma^*_{in}\right)\left(\sum_n\gamma_{jn}\right)$, $B_i$ denotes the modified $A$ with the $i$-th column replaced by $\left(\alpha_1(0),\alpha_2(0), \cdots, \alpha_N(0)\right)^T$.
\begin{eqnarray}gin{figure}
\center
\includegraphics[width=5.5cm]{fc1.pdf}
\includegraphics[width=5.5cm]{fc2.pdf}
\includegraphics[width=5.5cm]{fc3.pdf}
\caption{(Color online) Plots of $Det(A)$ for $\begin{eqnarray}ta=(1+\sqrt{5})/2, \phi=0.4\pi$ when $\Delta=1, 2, 4$ respectively. $N=99$, $s=1, \eta=0.1, \omega_c=10$ are chosen for all plots. The region highlighted by dark-pink color, denotes the main energy gap in Fig.\ref{fig:incom-be} }
\label{fig:appendixC1}
\end{figure}
Then the BIC corresponds to a pole of $G_i(z)$ with $\mathbbm{i}z>0$, which can be determined by seeking the solutions to $Det(A)=0$. However because of the involved term $\left(\sum_n\gamma^*_{in}\right)\left(\sum_n\gamma_{jn}\right)$, the result would be different from $\epsilon_i$, as shown in Fig.\ref{fig:appendixC1}. This feature is different from the single qubit case \cite{bic, longhi}, in which BIC is due to the level resonance. This phenomenon can be explained by the level shift, induced by the coupling to a bath. As an example, $Det(A)$ is plotted for positive $\mathbbm{i}z$ for different $\Delta$ in Fig. \ref{fig:appendixC1}. For these plots, the integral $\Sigma(z)$ is expressed by its principle value. It is clear that a discrete zero point can be found, as shown in Fig. \ref{fig:appendixC1}. Furthermore We find that the positive energy for the discrete zero points are slightly different from the edge mode, which are $0.80462, 1.10176$ and $1.82622$ for $\Delta=1, 2 ,4$ respectively. Besides of the discrete one, there are many continuous zero points, which construct a band.
\begin{eqnarray}gin{figure}
\center
\includegraphics[width=6cm]{fc4.pdf}
\caption{(Color online) The evolution of the edge mode as the initial state, occurring for $\phi=0$ when $\begin{eqnarray}ta=1/3, \Delta=2$, is plotted by fidelity. $N=99$, $s=1, \eta=0.1, \omega_c=10$ are chosen for this plot. }
\label{fig:appendixC2}
\end{figure}
Now we will show the correspondence to BIC for discrete zero point. By inverse Laplace transformation, $\alpha_i(t)$ can be determined. We choose the initial state as the edge state at $\phi=0$ when $\begin{eqnarray}ta=1/3, \Delta=2$ as an exemplification, which corresponds the $47$-th eigenstate in $H_S$. Then one can find by inverse Laplace transformation of $G_{47}(z)$ that the contribution of the discrete zero point at $\mathbbm{i}z=2.30752$ is $\sim 0.9876 e^{-\mathbbm{i}2.30752 t} \eta^{\dagger}_{47}\ket{0}$. Furthermore we exactly study the evolution dynamics dominated by Eq. (\ref{evolution}) with the edge mode as the initial state. As depicted in Fig. \ref{fig:appendixC2}, the fidelity $\left|\inp{\psi_{edge}}{\psi(t)}\right|^2$ shows a stable oscillation around $0.9876^2\sim 0.975$. Similar observation can be found for the other edge states. Thus we have demonstrated that the discrete poles of $G_i(z)$ characterizes the occurrence of BIC.
\end{document} |
\begin{document}
\title{Towards Identification of Relevant Variables in the observed Aerosol Optical Depth Bias between MODIS and AERONET observations}
\author[1]{N. K. Malakar }
\author[1]{ D. J. Lary}
\author[1]{D. Gencaga}
\author[2]{\\A. Albayrak}
\author[2]{ J. Wei}
\affil[1]{Department of Physics, University of Texas at Dallas}
\affil[2]{Goddard Earth Sciences DISC, NASA Goddard Space Flight Center\\
http://disc.gsfc.nasa.gov}
\maketitle
\begin{abstract}
Comparison of the aerosol optical depth values from two datasets, observed by satellite remote sensing Moderate Resolution Imaging Spectroradiometer (MODIS), and globally distributed Aerosol Robotic Network (AERONET), show that there are biases between the two data products.
In this paper, we present a general framework to identify the possible factors influencing the bias, which might be associated with the measurement conditions such as the solar and sensor zenith angles, the solar and sensor azimuth, scattering angles, and surface reflectivity at the various measured wavelengths, etc. Specifically, we performed analysis for remote sensing MODIS Aqua-Land data set, and used machine learning technique, neural network in this case, to perform multivariate regression between the ground-truth and the training data sets.
Finally, we used mutual information between the observed and the predicted values as the measure of similarity. The set is then identified as the most relevant set of variables. The search consists of a brute force method as we have to consider all possible combinations of the regressors to the neural network. The computations involves a huge number crunching exercise, and we implemented it by writing a job-parallel program.
\end{abstract}
\section*{Introduction}
Atmospheric aerosols play an important role in earth's climate system \cite{pachauri2007ipcc}, and can also pose harmful effects on human health when inhaled. Therefore, accurately characterizing the global aerosol distribution is valuable for many reasons.
The total light extinction caused by aerosols over a vertical column in the atmosphere of unit cross section is known as the aerosol optical depth (AOD) \cite{amt2012}. Much effort has been placed in observing AOD from space and ground-based instruments \cite{Holben92, kaufman_operational_1997, Torres98, holben_aeronet_1998, Mishchenko99, ichoku2002, remer_global_2008, remer_modis_2005, chu2002validation, Liu2004}.
A comparison between the AOD measurements made by MODIS and AERONET shows that there are biases between the two observations. Figure \ref{fig:aodBias} shows the distribution of the bias between the AOD measured by MODIS instrument at 550 nm. In this paper, we will delineate our attempts to understand or explain the factors behind the bias. We performed a comprehensive brute force search for all possible combination of variables, and used neural network as one of the machine learning toolbox to predict the AOD values. Moreover, we used mutual information between the predicted and observed variables as the measure of correlation between them. We found the set which reproduced the AOD, producing the highest value of mutual information with respect to the AERONET data set. The set is then identified as the most relevant set of variables for training the machine learning algorithm. The method of identification of relevant set of variable is a more general problem, which could be applied to similar multi-variate problems. Therefore, the presented example should be viewed as a test study. However, the framework and implementation will find its use for general purpose search for relevant variables.
\begin{figure}
\caption{The distribution of bias with respect to the AOD at 550 nm. Although the bias at high AOD values are higher, the bias at the low AOD values are also important.}
\label{fig:aodBias}
\end{figure}
\section*{Data and Methodology}
The data used in this study were derived from Multi-sensor Aerosol Products Sampling System (MAPSS), which derives the data from multiple sources such as original MODIS and AERONET datasets and provides level-2 aerosol scientific data sets \cite{amt2012, ichoku2002}.
MAPSS provides a consistent and uniform sampling of aerosol products by identifying the MODIS data pixels within approximately 27.5 km of the AERONET sites. We analyzed the MODIS-aqua land data set. Readers interested in the details of the MAPSS are directed to \cite{amt2012}. In the present paper we present the analysis of only the MODIS-Aqua land data set.
We applied a machine learning technique to predict the AOD values given the other components as the regressors to the learning module. The method developed in this paper is constructed with plug-and-play in mind. The machine-learning module can be any regression tools of choice. In our case, we simply choose, as an example, Neural networks (NN) method. NNs are widely used in pattern recognition, machine learning and artificial intelligence. In addition, NNs have found many applications in other fields such as geoscience, remote sensing, oceanography, etc.
While training the NN as the regression tool, we regard an observation data set as the product of n input variables, say $\{x_1, x_2, x_3, ... , x_n\}$, feeding into the regression machine. Therefore, the observed output variable, AOD, is some function of these input variables. In terms of neural networks, as shown in figure \ref{fig:nnets}, the output of the $k^{th}$ neuron can be written as the weighted sum of inputs
\begin{equation}
y_k = \varphi \left( \sum_{j=1}^n w_{kj} x_j \right),
\end{equation}
where $\varphi$ is the transfer function, $w_{kj}$ represents the weight from unit $j$ to unit $k$, and $x_j$ represents the $n$ input variables to the neuron.
During training, the NN weights are adjusted appropriately to learn the data. The learning and adjustments of the weights are inspired by the synaptic learning behavior of neurons.
The learning module is explored with all possible combinations of the input variables.
\begin{figure}
\caption{A cartoon of neural network, showing the inputs, hidden layers and output layers. The machine learning module in this case consists of a neural network. }
\label{fig:nnets}
\end{figure}
The following variables have been used as the regressor to the neural network. We constructed all possible set of variables, this came out to be total of 32,781 possible combinations.
\begin{enumerate}
\item Aerosol optical depth at 550 nm (AOD0550)
\item Aerosol optical depth at 470 nm (AOD0470)
\item Aerosol optical depth at 660 nm (AOD0660)
\item Mean reflectance at 470 nm (mref0470)
\item Mean reflectance at 550 nm (mref0550)
\item Surface reflectance at 660 nm (surfre0660)
\item Surface reflectance at 470 nm (surfre0470)
\item Surface reflectance at 2100 nm (surfre2100)
\item Cloud fraction from land aerosol cloud mask (cfrac)
\item Quality assurance (QAavg)
\item Solar zenith angle (SolarZenith)
\item Solar azimuth angle (SolarAzimuth)
\item Viewing zenith angle (SensorZenith)
\item Sensor azimuth angle (SensorAzimuth)
\item Scattering angle (ScatteringAngle)
\end{enumerate}
For each combination set, one at a time, we trained the NN with AERONET AOD as the target variable. Once training phase is completed, we then predicted the AERONET AOD values from the trained network.
The NN algorithm used a feed-forward back propagation algorithm with a hidden layer having 200 nodes. The training was done by the Levenberg-Marquardt algorithm with mean-squared error as the performance factor. We used the Matlab toolbox. We randomly split the training data set into three portions. The first $80\%$ portion is used to train the NN weights using an iterative process so that for each iteration, the root mean square (RMS) error of the neural network is computed by using the second $10\%$ portion of the data. We used the RMS error to determine the convergence of our training. When the training is complete, we use the final $10\%$ of the data as the validation data set.
Once the neural network training is completed, we have obtained a mapping between the set of input and output variables. Therefore, the most relevant set of variables is the one that can best reproduce the target data. We explored all combinations of variables, trained each of the combination, and compared which provided the fit of the observed AERONET AOD data. The end product is the result of regression between the input variables to the NN and the observed AERONET AOD.
The measure of similarity between predicted and observed AOD is measured by mutual information, which is described in next section. Since the learning module constructs a mapping between the set of input and output variables, the most relevant set of variables is the one that can best reproduce the target data.
\section*{Mutual Information}
The Correlation coefficient (Pearson's correlation) is a widely used measure of dependence between two variables, and represents the normalized measure of the strength of their linear relationships.
The correlation coefficient $\rho_{X, Y}$ between two random variables $X$ and $Y$ with expected values $\mu_X$ and $\mu_Y$ and standard deviations $\sigma_X$ and $\sigma_Y$ is defined as
\begin{eqnarray}
\rho_{X,Y}={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over \sigma_X\sigma_Y}
\end{eqnarray}
where, E is the expected value operator, cov means covariance, and, $\rho$ a widely used alternative notation for Pearson's correlation.
The correlation coefficient is defined only if both of the standard deviations are finite and both of them are nonzero. The correlation coefficients range from -1 to 1. The correlation coefficient values close to 1 (or -1) suggest that there is a positive (or negative) linear relationship between the data columns, whereas the values close to or equal to 0 suggest there is no linear relationship between the data columns. It can only be applied to the cases of linear relationship between two variables.
Mutual information quantifies the mutual dependence between two variables by taking into account all of the characteristics of the variables in the Probability Distribution Function (PDF). Mutual information is defined as follows in discrete form:
\begin{equation}
I(X,Y) = \sum_{x \in X} \sum_{y \in Y} p(x,y) \log \frac{p(x,y)}{p(x) p(y)},
\end{equation}
which is a special case of a measure called Kullback-Leibler divergence \cite{kullback_information_1997, cover_elements_2006}.
If $X$ and $Y$ are statistically independent, then
\begin{equation}
p(x, y) =p(x) p(y).
\end{equation}
In this case, the mutual information becomes 0, showing independency. A proper mapping of the form
\begin{equation}
\delta(X,Y) = \sqrt{1- e^{-2I(X,Y)} }
\end{equation}
normalizes the measure of general correlation as depicted by the MI \cite{joe_relative_1989, granger_using_1994, dionisio_mutual_2004}.
In the case when X and Y are normally distributed,
\begin{equation}
(X,Y) ~ \sim \mathcal{N}(\mu, K)
\end{equation}
where,
$K = (\sigma^2, ~ \rho \sigma^2; \rho \sigma^2, ~ \sigma^2).$
Then, the mutual information reduces to
\begin{equation}
I(X,Y) = -\frac{1}{2} \log (1- \rho^2) .
\end{equation}
So, that,
\begin{eqnarray}
\delta(X,Y) = \sqrt{1- e^{-2I(X,Y)} } = |\rho(X,Y)|. \end{eqnarray}
This relation shows the generality of the normalized correlation measure.
There are several methods to estimate MI from data \cite{moon1995estimation, Reshef16122011, DenizME12}. We applied the Variable Bin Width Histogram Approach \cite{darbellay1999estimation, mutingCode} to compute the MI between the observed and predicted AOD. Higher values indicate better agreement between the observed and predicted set, and thus are the best indicators of the input variables needed to assess a relevant set of variables.
\section*{Results and Conclusions}
We applied machine learning technique, specifically neural network method in supervised learning mode, and explored all possible combination of variables. By the brute force exploration of all possible combinations, we found the best set of variables by comparing a measure of correlation between the predicted and observed variables. The measure of agreement between the observed data and the predicted data was obtained by using the mutual information. Therefore, we could identify the most relevant set of variables which could result into the maximum correlation between the predicted and observed data.
Figure \ref{fig:nnresults} shows the best result we obtained as a result of the brute force search. It shows a comparison between the AERONET AOD, which was used as the target set for NN training, and the predicted AOD as the result of the NN training. The mutual information and correlation coefficients values for the predicted and observed AOD values from the variable-set as regressor has been presented in table \ref{table1}. For brevity we only show the first 15 rows of the table. The best prediction set had the MI value of $0.771$ to the AERONET AOD, and was result of an input set consisting of the following variables as regressors: AOD at 470 nm, and AOD at 660 nm, mean reflectance at 470 nm, and mean reflectance at 550 nm, surface reflectance at 660nm, 470 nm, 2100 nm, cloud fraction, quality assurance values, solar zenith angle, solar azimuth angle, zenith angle, sensor azimuth angle and scattering angle.
\begin{figure}
\caption{ The distribution of the AERONET AOD and predicted AOD after the NN training. The mutual information is $0.771$, and the correlation coefficient is $0.927$. This particular product was obtained by the best relevant set of variables as determined by the highest mutual information value between the two variables.}
\label{fig:nnresults}
\end{figure}
There are several developments which could further benefit the methodology of finding relevant variables. In the future studies we will use mutual information with error bars. In the future work we will present cross examination of multiple machine learning techniques to explain the bias-correction using the framework developed in this paper.
We presented a brute force search method, which could be useful in many other cases involving multivariate exploration. This could be useful in finding the most relevant set of factors to get insights from physical data.
\begin{landscape}
\begin{table}
\label{table1}
\caption{The top 15 results of brute force search is presented in the tabular format. The regressor variables are explained in the text.}
\begin{tabular}{rrr}
\hline
Combination & Mutual Information (MI)& Corr corrcoeff ($\rho$)\\ \hline
$2,3,4,5,6,7,8,9,10,11,12,13,14,15$ &0.771&0.927\\
$1,2,4,5,6,7,8,10,11,12,13,15$&0.769&0.926\\
$1,2,3,4,5,6,8,10,11,12,13,14,15$&0.768&0.926\\
$1,2,4,5,6,8,9,10,11,12,13,14,15$&0.766&0.926\\
$1,2,3,4,5,6,7,8,9,10,12,13,15$&0.765&0.926\\
$1,2,4,5,7,8,10,12,13,14,15$&0.764&0.925\\
$1,2,4,5,6,7,8,9,10,11,12,13,14$&0.762&0.921\\
$2,3,4,5,6,7,8,10,11,12,13,14$&0.761&0.924\\
$1,3,4,5,6,7,8,10,11,12,13,14,15$&0.760&0.924\\
$1,2,4,5,7,8,10,11,12,13,14,15$&0.759&0.925\\
$1,2,4,5,6,7,10,11,12,13,14,15$&0.759&0.925\\
$1,3,4,5,6,7,8,9,10,11,12,13,15$&0.756&0.924\\
$1,2,4,5,6,8,10,11,12,13,15$&0.756&0.921\\
$1,2,4,5,7,8,10,11,12,13,15$&0.755&0.923\\
$2,3,4,5,7,8,9,10,11,12,13,15$&0.755&0.924\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\end{document} |
{kf e}gin{document}
\title{Efficient Fully Sequential Indifference-Zone Procedures Using Properties of Multidimensional Brownian Motion Exiting a Sphere}
\author{A.B.~Dieker \\
Columbia University \\
New York, NY 10027 \\
Seong-Hee Kim\\
Georgia Institute of Technology \\
Atlanta, GA 30332-0205}
\maketitle
{kf e}gin{abstract}
We consider a ranking and selection (R\&S) problem with the goal to
select a system with the largest or smallest expected performance
measure among a number of simulated systems with a pre-specified
probability of correct selection. Fully sequential procedures take one
observation from each survived system and eliminate inferior systems
when there is clear statistical evidence that they are inferior. Most
fully sequential procedures make elimination decisions based on sample
performances of each possible pair of survived systems and exploit the
bound crossing properties of a univariate Brownian motion. In this
paper, we present new fully sequential procedures with elimination
decisions that are based on sample performances of all competing
systems.
Using properties of a multidimensional Brownian motion exiting a sphere,
we derive heuristics that aim to achieve a given target probability of correct selection.
We show that in practice the new procedures significantly outperform a
widely used fully sequential procedure. Compared to BIZ, a recent fully-sequential procedure that uses
statistics inspired by Bayes posterior probabilities, our procedures have better
performance under difficult mean or variance configurations but similar performance under easy mean configurations.
\noindent {\em Subject classification:} Simulation, Ranking and
Selection, Fully Sequential, Multidimensional Brownian Motion, Sphere
\end{abstract}
\par{kf a}selineskip=24pt
\section{Introduction}
\label{sec:intro}
Ranking and selection (R\&S) is one of the classical and well-studied problems in the operations research literature. It aims to find the best system among a number of systems for which noisy performance information is accessible through simulation. In this paper, we assume that the best system is one with the largest or smallest expected performance, which is known as the finding-the-best problem.
There are at least three approaches for the finding-the-best problem: the indifference-zone (IZ) approach, the Bayesian approach, and the optimal computing budget allocation (OCBA) approach. Hong et al.\ (2014) and Kim and Nelson (2011) provide a brief review of each approach. For more information on the Bayesian approach, see Chick (2006) and Chen et al.\ (2014). When there is a fixed computing budget until a decision is made, the OCBA approach provides an efficient way to find the best system, see for example Chen and Lee (2011).
In this paper we study an indifference-zone (IZ) procedure, where the decision maker specifies a difference worth detecting called the IZ parameter.
Among procedures that take the IZ approach, Rinott (1978) is one of the earliest procedures. It is a two-stage procedure and does not have any elimination step for clearly inferior systems. Nelson et al.\ (2001) also propose a two-stage procedure but their procedures can eliminate systems after the first stage if there is statistical evidence that they are inferior. Therefore the latter procedure is more efficient than Rinott's procedure in terms of the number of observations needed until a decision is made. On the other hand, fully-sequential IZ procedures take one observation from competing systems and eliminate inferior systems as additional observations become available. They carry the risk of incorrectly eliminating the best system due to stochastic noise in the performance measurements.
Examples of fully-sequential IZ procedures are the KN procedures from Kim and Nelson (2001), which are widely used as they are available in leading commercial simulation software.
KN's parameters are chosen to control the probability of eliminating the best system.
Since this probability is intractable, the procedures instead rely on a Bonferroni-type lower bound on the worst-case probability of incorrect selection, which corresponds to the best system having a mean performance that exceeds the means of the other systems by exactly the IZ parameter;
this setup is known as the slippage configuration (SC).
Particularly when the number of systems is large, this lower bound tends to be a poor approximation for the worst-case probability of correct selection.
As discussed in Wang and Kim (2011), the result is that KN procedures tend to take many more observations than necessary to control the probability of incorrect selection, and are thus inefficient in that sense.
{kf e}gin{figure}[th!]
{kf e}gin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\linewidth]{projectionKN}
\caption{KN; $\min_{i<j} |x_i-x_j|$}
\end{subfigure}
{kf e}gin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\linewidth]{projectionBIZ}
\caption{BIZ; $\min_i e^{x_i}/(e^{x_1}+e^{x_2}+e^{x_3})$}
\end{subfigure}
{kf e}gin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\linewidth]{projectionDK}
\caption{This paper; $\sum_{i<j} (x_i-x_j)^2$}
\end{subfigure}
\caption{
Contours of the screening statistics with $k=3$ competing systems for three procedures.
The form of the screening statistic is given below each figure.
Also depicted are each of the three possible drifts (mean sample paths) under the slippage configuration (SC).
Since the screening statistics do not change when adding a constant to each coordinate,
we have plotted the plane $\{x\in \mathbb{R}^3:x_1+x_2+x_3=0\}$ with its intersections of the contours.}
\label{fig:1}
\end{figure}
The primary contribution of this paper is to develop a new family of IZ procedures that does not suffer from the
inefficiencies caused by the use of the Bonferroni bound.
The screening statistic used in an IZ procedure gives rise to contours, and a system is eliminated when
the vector of cumulative sums of performance measurements hits such a contour, see Figure~\ref{fig:1}.
Controlling PICS is done by analyzing hitting behavior of
Brownian motion
to set the `radius' of the contour.
It is the second ingredient where the Bonferroni bound is invoked for KN procedures,
since it is analytically intractable to study properties of a Brownian motion hitting KN contours, see Figure~\ref{fig:1}(a).
An important recently developed family of IZ ranking and selection procedures is the Bayes-inspired indifference zone (BIZ)
procedures from Frazier (2014). An example of the resulting elimination contours is given in Figure~\ref{fig:1}(b).
BIZ procedures circumvent the use of the Bonferroni bound, which results in dramatic improvements
over the KN procedure especially when the number of competing systems is large.
We see in Figure~\ref{fig:1}(b) that the three possible drifts under SC (one for each of the systems being the best)
hit the elimination contour at its closest point to the origin.
Thus, possibly sample paths that deviate significantly from their mean sample paths require a larger number of observations from the
competing systems than those that are close to their mean
sample paths.
This paper is a first investigation towards efficient procedures with spherical elimination contours, see Figure~\ref{fig:1}(c).
Using path length as a proxy for the number of observations needed, with spherical elimination contours
all points on the contour are equally close to the origin, and this could perhaps lead to faster elimination.
Setting the radius of the contours given a target probability of correct selection is facilitated by some analytic results
about a multidimensional Brownian motion hitting hyperspheres.
Like BIZ, we do not need to appeal to the Bonferroni bound to exert control over the probability of correct selection.
However, our procedure is ultimately heuristic since we found it intractable
to control the multi-stage probability of correct selection; we leave this as an open problem.
Experimental results show that estimated probability of correct selection is all higher or close to the target confidence level for all cases tested including SC. Our procedures also significantly outperform KN. On the other hand, our procedures perform better than BIZ under difficult mean or variance configurations while they perform similarly compared to BIZ under easy mean configurations. More specifically, when variances are unknown and unequal across systems with a slippage mean configuration, our procedures show up to 30\% savings compared to the BIZ procedures in terms of the number of replications needed until a decision is made. Under easier scenarios where means spread out over systems unlike SC, our procedures perform similar to the BIZ procedures.
To extend our procedures to unknown and unequal variances, we use multiple tricks. These tricks are standard but render any statistically valid approach (including BIZ) heuristic. To handle unknown variances, we update variance estimates as the procedures advance. Kim and Nelson (2006) show that variance update enables a procedure to be treated as if variances are known in an appropriate limit.
To handle unequal variances we use a heuristic approach which essentially changes the sampling frequency of each system hoping to approximately equalize variances across systems.
Preliminary work related to this work is published in the Winter Simulation Conference proceedings which include Kim and Dieker (2011) and Dieker and Kim (2012, 2014). The first two papers consider only three systems with known variances. Dieker and Kim (2014) give a procedure for a general number of systems but require known and equal variances. Moreover, the spheres that play a crucial role in the procedure all have the same radius and the procedure performs worse than KN when the means of the systems are spread out evenly. In the procedures presented in the present paper, the radii of the spheres vary as the number of survived systems decreases, outperforming KN in all scenarios; and a version of our procedure can handle unknown and unequal variances.
When there exists a finite simulation budget or a tight deadline in time, OCBA and Bayesian procedures are shown to be highly efficient and very useful in practice. Branke et al.\ (2007), Chen and Lee (2011) and Powell and Ryzhov (2012) provide a good review of OCBA and Bayesian ranking and selection procedures and provide extensive empirical results. As our primary goal is to investigate the impact of different shapes of continuation regions in IZ ranking and selection procedures,
we only compare our procedures with IZ procedures. Specifically, two state-of-art IZ procedures, KN and BIZ, are considered.
The paper is organized as follows. Section~\ref{sec:notation} defines our problem and introduces notation. Section~\ref{sec:procedure} proposes new fully-sequential procedures. Section~\ref{sec:stat} explains the statistics that we use for elimination decisions and the properties of our statistics. In Section~\ref{sec:approx}, we provide justifications for our procedures and approximations in order to set the parameter values of the procedures.
Experimental results are presented in Section~\ref{sec:exp}, followed by conclusions in Section~\ref{sec:conclusion}.
\section{Problem and Notation}
\label{sec:notation}
This section introduces our notation and assumptions and defines the problem.
We assume there are $k$ systems ($k \ge 2$).
Let $X_{ij}$ represent the $j$th observation from system $i$ for $i=1,\ldots,k$ and $j=1,2,\ldots$. Then the mean and variance of the outputs from system $i$ are defined as $\mu_{i}={\rm E}[X_{ij}]$ and $\sigma_i^2 = {\rm Var}[X_{ij}]$, respectively.
We want to find the system with the largest mean $\mu_i$.
Throughout the paper, we assume that the following assumptions hold:
{kf e}gin{assumption} \label{assump:normal}
\[
X_{ij} \widesim[2]{\mbox{iid\ }}
\mybold{N}(\mu_i, \sigma_i^2), \;\;\;\; j=1,2,\ldots,
\]
where $\widesim[2]{\mbox{iid\ }}$ represents `are
independent and identically distributed as' and $\mybold{N}(\mu_i, \sigma_i^2)$ denotes the
normal distribution with mean $\mu_i$ and
variance $\sigma_i^2$. Moreover, $X_{ij}$ and $X_{i^\prime j}$ are independent for any $i \neq i^\prime$ and $j=1,2,\ldots$.
\end{assumption}
{kf e}gin{assumption}\label{assump:sc}
$\mu_1 \leq \mu_2 \leq \ldots \leq \mu_{k-1} \leq \mu_k - \delta$ for $\delta \in \mathbb{R}^+$.
\end{assumption}
Assumption~\ref{assump:normal} implies that observations from each
system are marginally \mbox{iid\ } normally distributed and systems are simulated independently (note that this rules out common random numbers).
Without loss of generality, we assume that system $k$ is the true best system. Assumption~\ref{assump:sc} assumes that the mean of the true best system $k$ is at least $\deltaelta$ better than any alternative system. The parameter $\delta$ is a user-specified parameter known as the IZ parameter.
We aim to devise a method that observes systems sequentially and eliminates clearly inferior systems from further consideration. The method stops once only one system remains, and this system
is declared as the best system.
Additional notation is needed for later sections:
{kf e}gin{eqnarray*}
n & \equiv & \mbox{the current number of observations or the current stage number};\\
I & \equiv & \mbox{set of competing systems at the $n$th stage};\\
{kf X}ir &\equiv& \frac{1}{n}\sum_{j=1}^n{X_{ij}}\mbox{, the sample mean of system $i$ based on the first $n$ observations};\\
\boldsymbol{X}_I(n) &\equiv& \mbox{$ |I| \times 1$ vector of $\sum_{j=1}^n X_{ij}$ for $i \in I$};\\
\hat{\sigma}^2_i(n) &\equiv& \mbox{sample variance of system $i$ from $X_{i1}, \ldots, X_{in}$} \mbox{ which is } {1 \over n-1} \sum_{j=1}^n (X_{ij} - {kf X}ir)^2;\\
A^T & \equiv& \mbox{the transpose of a matrix $A$};\\
\delta_{|I|}^2 &\equiv& {\delta^2} {|I|-1 \over |I|}.\\
\end{eqnarray*}
\section{${\cal DK}$ Procedures}
\label{sec:procedure}
In this section, we provide the descriptions of our new procedures. We present ${\cal DK}_1$ for known and equal variances and extend it to unknown but equal variances, resulting in ${\cal DK}_2$. Then ${\cal DK}_3$ is presented for unknown and unequal variances.
\subsection{Equal and Known Variances}
We first consider a case where variances are equal across all systems and known so that $\sigma_i^2 = \sigma^2$ for any system $i$.
Suppose $x\in\mathbb{R}^s$ and $I \subset \{1,\ldots, k\}$ and define a function ${\cal S}_I(x)$ as follows:
\[
{\cal S}_I(x) = {1 \over \sigma^2} \sum_{i\in I} (x_i - {kf a}r{x})^2
\]
where ${kf a}r{x} = {1 \over s} \sum_{i \in I} x_i$.
The ${\cal DK}_1$ procedure for equal and known variances is as follows:
{kf e}gin{center}
\fbox{
{kf e}gin{minipage}{6.5in}
{kf e}gin{center}\textbf{The ${\cal DK}_1$ Procedure}\end{center}
{kf e}gin{hangref}
\item {kf Setup:} Select the nominal level $1-\alpha$ and the IZ
parameter $\delta$. Set $I = \{1, 2, \ldots, k\}$ and choose $\eta_{|I|}$ (which will be discussed in Section~\ref{sec:approx}). Take one observation from each
system. Set $n=1$ and go to {kf Calculation}.
\item {kf Calculation:}
Calculate ${\cal S}_I(\boldsymbol{X}_I(n))$.
\item {kf Screening:} If ${\cal S}_I(\boldsymbol{X}_I(n)) \ge \left({ \sigma \cdot \eta_{|I|} \over \deltaelta_{|I|}}\right)^2$, then eliminate the system with the smallest ${kf a}r{X}_{i}(n)$ among $i\in I$. Update $I$ by removing the eliminated system and go back to {kf Calculation}.
Otherwise, go to {kf Stopping Rule}.
\item {kf Stopping Rule:} If $|I| = 1$, stop and declare the surviving system as the best. Otherwise, take one more observation for all $i \in I$, set $n=n+1$, and go to {kf Calculation}.
\end{hangref}
\end{minipage}
}
\end{center}
In Section~\ref{sec:stat}, we show that ${\cal S}_I(x)$ calculates the squared distance of the point $x$ orthogonally projected onto a hyperplane $\{x:\sum_{i\in I} x_i = 0\}$ and that the screening rule in ${\cal DK}_1$ implies that we have an open (infinite) cylinder as our continuation region with a radius depending on $\eta_{|I|}$ and $\deltaelta_{|I|}$. When $x$ is located outside the cylinder, elimination occurs and the screening rule is checked again with updated parameters (without obtaining additional observations), i.e., a lower dimensional cylinder. We only obtain new observations (i.e., move to the next stage) if no more elimination occurs for a given number of observations.
\subsection{Unknown but Equal Variances}
We present a straightforward variant of ${\cal DK}_1$ for unknown but equal variances, $\sigma^2$.
As the variance parameter $\sigma^2$ is unknown, it needs to be estimated. Let $\hat{\sigma}_i^2(n)$ represent sample variance of system $i$.
The pooled variance estimator $\hat{\sigma}_p^2(n)$ is defined as follows:
\[
\hat{\sigma}^2_p(n) = {1 \over |I|} \sum_{i \in I} \hat{\sigma}^2_i(n).
\]
Then our statistic is modified to
\[
{\cal S}_I'(x) = {1 \over \hat{\sigma}^2_p(n)} \sum_{i \in I} (x_i - {kf a}r{x})^2
\]
and $\hat{\sigma}^2_i(n)$ and $\hat{\sigma}^2_p(n)$ need to be updated in the [Stopping Rule] step after additional observations are obtained. Then the ${\cal DK}_2$ procedure is defined below.
{kf e}gin{center}
\fbox{
{kf e}gin{minipage}{6.5in}
{kf e}gin{center}\textbf{The ${\cal DK}_2$ Procedure}\end{center}
{kf e}gin{hangref}
\item {kf Setup:} Select the nominal level $1-\alpha$ and the IZ
parameter $\delta$. Set $I = \{1, 2, \ldots, k\}$ and choose $\eta_{|I|}$. Take $n_0 \ge 2$ observations from each
system and calculate $\hat{\sigma}^2_i(n_0)$ and $\hat{\sigma}^2_p(n_0)$. Set $n=n_0$ and go to {kf Calculation}.
\item {kf Calculation:}
Calculate ${\cal S}_I'(\boldsymbol{X}_I(n))$.
\item {kf Screening:} If ${\cal S}_I'(\boldsymbol{X}_I(n)) \ge \left({\hat{\sigma}_p(n) \cdot \eta_{|I|} \over \deltaelta_{|I|}}\right)^2$, then eliminate the system with the smallest ${kf a}r{X}_{i}(n)$ among $i\in I$. Update $I$ by removing the eliminated system and go back to {kf Calculation}.
Otherwise, go to {kf Stopping Rule}.
\item {kf Stopping Rule:} If $|I| = 1$, stop and declare the surviving system as the best. Otherwise, take one more observation for all $i \in I$; set $n=n+1$; and update $\hat{\sigma}^2_i(n)$ for $i\in I$ and $\hat{\sigma}^2_p(n)$. Then go to {kf Calculation}.
\end{hangref}
\end{minipage}
}
\end{center}
\subsection{Unknown and Unequal Variances}
This subsection extends the ${\cal DK}_1$ procedure to handle unknown and unequal variances, resulting in ${\cal DK}_3$. The main idea is to make the sampling frequency of each system proportional to the variance parameter of the system, which eventually leads to equal variances. This approach is similar to the one in Frazier (2014).
Let $n_i$ denote the number of observations system $i$ have received so far. In ${\cal DK}_1$, $n_i = n$ for any system $i \in I$ but in ${\cal DK}_3$, $n_i \le n$.
Also let $W_i(n) = \sum_{j=1}^{n_i} X_{ij}/n_i$ and $\boldsymbol{W}_I(n)$ represent a $|I| \times 1$ vector of $W_i(n)$ for $i\in I$.
Then
\[
{\cal S}_I''(x) = {1 \over \hat{\lambda}^2} \sum_{i \in I} \left( x_i - {kf a}r{x} \right)^2
\]
where
\[
\hat{\lambda}^2 = {\sum_{i\in I} \hat{\sigma}^2_i(n_i) \over \sum_{i\in I} n_i}.
\]
We can now describe Procedure ${\cal DK}_3$.
{kf e}gin{center}
\fbox{
{kf e}gin{minipage}{6.5in}
{kf e}gin{center}\textbf{The ${\cal DK}_3$ Procedure}\end{center}
{kf e}gin{hangref}
\item {kf Setup:} Select the nominal level $1-\alpha$ and the IZ
parameter $\delta$. Also select a constant $B_z$. Set $I = \{1, 2, \ldots, k\}$ and choose $\eta_{|I|}$. Take $n_0$ observations from each
system and calculate $W_i(n_0)$, $\hat{\sigma}^2_i(n_0)$ and $\hat{\lambda}^2$. Set $n=n_0$ and $n_i=n_0$ for $i\in I$, and go to {kf Calculation}.
\item {kf Calculation:}
Calculate ${\cal S}_I''(\boldsymbol{W}_I(n))$.
\item {kf Screening:} If ${\cal S}_I''(\boldsymbol{W}_I(n)) \ge \left({\hat{\lambda} \cdot \eta_{|I|} \over \deltaelta_{|I|}}\right)^2$, then eliminate the system with the smallest ${kf a}r{X}_{i}(n)$ among $i\in I$. Update $I$ by removing the eliminated system and go back to {kf Calculation}.
Otherwise, go to {kf Stopping Rule}.
\item {kf Stopping Rule:} If $|I| = 1$, stop and declare the surviving system as the best. Otherwise,
let $z = \mbox{arg}\min {n_i \over \hat{\sigma}^2_i(n_i)}$ for $i\in I$.
For each $i \in I$,
{kf e}gin{itemize}
\item calculate
\[
\Delta_i = \left\lceil { \hat{\sigma}^2_i(n_i) \cdot { n_z + B_z \over \hat{\sigma}^2_z(n_z)}} \right\rceil;
\]
\item if $\Delta_i > n_i$, then take $(\Delta_i - n_i)$ observations.
\end{itemize}
Set $n=n+1$ and $n_i =\max(n_i, \Delta_i)$; and update $\hat{\sigma}^2_i(n_i)$ for all $i\in I$ and $\hat{\lambda}^2$. Then go to {kf Calculation}.
\end{hangref}
\end{minipage}
}
\end{center}
Frazier (2014) recommends $B_z= 1$. The parameter $\eta_{|I|}$ needs to be chosen carefully so that the actual probability of correct selection is at least $1-\alpha$. In the next section, we derive some analytical results for the ${\cal DK}_1$ procedure and then discuss how to choose $\eta_{|I|}$.
\section{Statistics for Screening}
\label{sec:stat}
The canonical choice for fully sequential procedures is to use $\sum_{j=1}^n (X_{ij} - X_{\ell j})$ for every $i\neq \ell$ as observed statistics and to eliminate a system whenever the statistics exit a so-called continuation region defined by two parallel lines such as $(-a, a)$ for a constant $a>0$ or a function $h(n)>0$ such as $(-h(n), h(n))$. Kim and Nelson (2014) use a triangular shaped continuation region defined by a decreasing linear function $h(n)$. Note that traditional continuation regions are defined in a two-dimensional space. Our procedures use different statistics based on a quadratic form and our continuation region is an open cylinder.
Consider $x\in\mathbb{R}^s$ and $I \subset \{1,\ldots, k\}$ with $I=\{i_1,\ldots,i_s\}$.
Furthermore let $\Gamma$ represent the covariance matrix of $(X_{i_1 j}, X_{i_2 j}, \ldots, X_{i_s j})^T$,
\[
\Gamma = {kf e}gin{bmatrix}
\sigma_{i_1}^2 & 0 & 0 & \cdots & 0 \\
0 & \sigma_{i_2}^2 & 0 & \cdots & 0\\
0 & 0 & \sigma_{i_3}^2 & \cdots &0\\
\vdots & \vdots & \vdots & \deltadots & \vdots\\
0 & \cdots & \cdots & \cdots & \sigma_{i_s}^2\\
\end{bmatrix}
\]
and let $V$ represent an $s-1$ by $s$ matrix given by
\[
V = {kf e}gin{bmatrix}
1 & 0 & \cdots& 0 & -1\\
0 & 1 & \cdots& 0 & -1\\
\vdots & \vdots & \deltadots & \vdots & \vdots\\
0 & 0 & \cdots & 1 & -1\\
\end{bmatrix}.
\]
Then our statistic ${\cal S}_I(x)$ is defined as
{kf e}gin{equation}
\label{eq:defSI}
{\cal S}_I(x) \equiv (Vx)^T(V \Gamma V^T)^{-1}(Vx)=\left[
{kf e}gin{array}{c}
x_{i_1} - x_{i_s} \\
\vdots\\
x_{i_{s-1}} - x_{i_s}
\end{array}\right]^T (V \Gamma V^T)^{-1} \left[
{kf e}gin{array}{c}
x_{i_1} - x_{i_s} \\
\vdots\\
x_{i_{s-1}} - x_{i_s}
\end{array}\right]
\end{equation}
and our continuation region is related to this quadratic form.
From the definition of ${\cal S}_I$ it may seem that ${\cal S}_I$ is complicated to calculate and that it depends on the order in which its elements are listed.
The following lemma is useful in deriving a simpler form of ${\cal S}_I(x)$ which allows us to argue that ${\cal S}_I(x)$ only depends on the set $I$, so not on the order of the elements in $I$. The proof is given in the appendix.
{kf e}gin{lemma}
\label{lem:S}
Suppose $x\in\mathbb{R}^s$ and $I \subset \{1,\ldots, k\}$ with $I=\{i_1,\ldots,i_s\}$.
If $\Pi = \Gamma V^T (V \Gamma V^T)^{-1} V$, then
\[
{\cal S}_I(x)={\cal S}_I(\Pi x).
\]
\end{lemma}
{kf e}gin{figure}[th!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{3Dball01}
\caption{Projection inside ball: no elimination}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{3Dball02}
\caption{Projection outside ball: elimination}
\end{subfigure}
\caption{Projected points on plane $y_1 + y_2 + y_3 = 0$ and elimination rules. A ball (with radius 7 here) is also visible.}
\label{fig:3d}
\end{figure}
The above lemma holds for $\Gamma$ regardless of whether it has equal diagonal elements. The matrix $\Pi$ is a (non-orthogonal) projection matrix with range $R = \{ y \in \mathbb{R}^s: \sum_{i\in I} y_i/\sigma_i^2 = 0 \}$ and null space $N = \{ \alpha (1, \ldots, 1): \alpha \in \mathbb{R} \}$, i.e., when $\Pi$ is applied to a vector then a multiple of $(1 , \ldots, 1)$ is subtracted from this vector so the result lies in $R$. It becomes an orthogonal projection matrix when $\sigma_i^2 = \sigma^2$ for all $i \in I$, since the null space $N$ is orthogonal to the range $R$ in that case. This lemma implies that the value of our statistic at any $x$ equals the value of our statistic at the projected point on the plane determined by $\sum_{i\in I} y_i/\sigma_i^2=0$ (or $\sum_{i\in I} y_i=0$ for equal variances). Since the null space and range do not change if the order of the elements in $I$ changes, Lemma~\ref{lem:S} shows that the quadratic form $\cal S_I$ remains the same if the indices are ordered differently, i.e., both in $\Gamma$ and in $x$. In Section~\ref{sec:approx} this lemma is used to make the elimination decision depend on the IZ parameter $\deltaelta$ only and not on the unknown mean parameter. Using the above lemma, we next derive a simpler form of ${\cal S}_I(x)$ when the variances are equal.
As an aside, it may be tempting to think that ${\cal S}_I(x) = x^T V^T(V^T)^{-1}\Gamma^{-1} V^{-1}Vx=x^T\Gamma^{-1}x$ in view of (\ref{eq:defSI}),
but this is incorrect since $V$ is not invertible.
{kf e}gin{corollary}
\label{cor:S}
Suppose $x\in\mathbb{R}^s$ and $I \subset \{1,\ldots, k\}$ with $I=\{i_1,\ldots,i_s\}$.
If $\sigma_i^2 = \sigma^2$, then
\[
{\cal S}_I(x) = \frac1{\sigma^2}\frac 1{|I|} \sum_{i < \ell \atop i, \ell \in I} (x_{i} - x_{\ell})^2
= {1 \over \sigma^2} \sum_{i\in I} (x_i - {kf a}r{x})^2
\]
where ${kf a}r{x} = {1 \over s} \sum_{i \in I} x_i$.
\end{corollary}
Our elimination decision rule takes the form ${\cal S}_I(x) \ge r^2$ for $r\in \mathbb{R}^+$. Let $x'$ denote the orthogonal projection of $x$ on the plane with $\sum_{i=1}^s y_i=0$. From Lemma~\ref{lem:S}, we know that ${\cal S}_I(x)$ is equal to ${\cal S}_I(x')$. As $x'$ lies on the hyperplane $\{y:\sum_{i=1}^s y_i=0\}$, we know that ${kf a}r{x'} = 0$. From the second equality of ${\cal S}_I(x)$ in Corollary~\ref{cor:S}, it is easy to see that ${\cal S}_I(x')$ becomes simply the squared distance between $x'$ and the origin. Therefore our elimination decision rule implies that no elimination occurs and sampling continues when the projected point $x'$ is inside a sphere as in Figure~\ref{fig:3d}(a); but one system with the smallest value is eliminated when the projected point $x'$ is outside the sphere as in Figure~\ref{fig:3d}(b).
One may wonder why our elimination rule only considers the largest set $I$ but not any subset $J$, thinking that the screening statistics ${\cal S}_J(x_J)$ for $J \subseteq I$ may be larger than ${\cal S}_I(x_I)$. This would mean that even if an elimination does not occur with set $I$, an elimination might be possible for a subset $J$. However, the following lemma implies that our elimination rule for the largest set $I$ actually verifies elimination for all $2^{|I|}-1$ nonempty subsets $J\subseteq I$ by showing that we always get the largest screening statistics with set $I$.
{kf e}gin{lemma}
\label{lem:Smon}
Suppose $J \subseteq I\subseteq \{1,\ldots,k\}$.
Then ${\cal S}_J(x_J)\le {\cal S}_I(x_I)$ for all $x\in\mathbb{R}^k$.
\end{lemma}
\section{Proofs and Approximations}
\label{sec:approx}
This section presents an approximation for the probability of incorrect selection
under ${\cal DK}_1$, which assumes known and equal variances $\sigma^2$.
We use these approximations in lieu of possibly conservative bounds in order to choose
the parameters $\eta_2,\ldots,\eta_k$ of ${\cal DK}_1$, thus bypassing a main source of inefficiencies.
In the course of the presentation, we explain how we choose the parameters $\eta_2,\ldots,\eta_{k}$ of our procedure.
The event of incorrect selection can be partitioned according to when the best system is eliminated.
If the best system is eliminated first, then we say that the level of elimination is $1$.
Similarly, if the second system to be eliminated is the best system, then we say that the level of elimination is $2$.
Thus, the possible levels of incorrect elimination are $1,\ldots,k-1$. The key building block for our approximation scheme is an approximation for the probability of incorrect selection at the first elimination level, which we discuss in Section~\ref{subsec:immediateelim}.
Other levels of incorrect elimination are studied in Section~\ref{subsec:otherelim}.
With this, we devise a procedure for choosing the parameter $\eta_{|I|}$ for ${\cal DK}_1$. We then explain how $\eta_2,\ldots,\eta_k$ for ${\cal DK}_1$ are related to parameters for ${\cal DK}_2$ and ${\cal DK}_3$ in Section~\ref{subsec:unknown}.
In the continuous analog of our problem, the discrete observation window is replaced with a continuous one.
The analog of the random walk $\boldsymbol{X}_{\{1,\ldots,k\}}(n)$ is $\sigma B(t)$, where $B(t)$ is a standard
Brownian motion in $\mathbb{R}^k$ with drift $( \mu,\ldots,\mu,\mu+\deltaelta) \times 1/\sigma$.
Throughout this section, we study this continuous problem as a proxy for the discrete problem, and the results we state for the
${\cal DK}_1$ algorithm are to be understood as for its continuous analog.
{kf e}gin{lemma}
\label{lem:deltasigma}
For fixed $\eta_k,\ldots,\eta_2$ and $\ell\in\{2,\ldots,k\}$, the probability of eliminination at level $\ell$ in ${\cal DK}_1$ is constant as a
function of $\deltaelta$ and $\sigma$.
In particular, the probability of incorrect selection in ${\cal DK}_1$ does not depend on $\deltaelta$ or $\sigma$.
\end{lemma}
{kf e}gin{proof}
Consider $\sigma B(t) + \mu {kf o}ldsymbol{1} t + \deltaelta w t$ instead of $\boldsymbol{X}_{\{1,\ldots,k\}}(n)$, where $B(\cdot)$ is a standard Brownian motion in $\mathbb{R}^N$ with $B(0)=0$, $w=(0,\ldots,0,1)$, and ${kf o}ldsymbol{1}=(1,\ldots,1)$.
Since the screening statistic uses the projection of this process on the hyperplane $\{x:\sum_i x_i=0\}$, i.e., the `sample mean' is subtracted,
we may assume without loss of generality that $\mu=0$ and replace $w$ by its projected version $v=(1/k,\ldots,1/k,-(k-1)/k)$.
Suppose that we are given some $x\in \mathbb{R}^N$ and an $N$-dimensional set $S$. We set
\[
\tau_S = \inf\left\{t\ge 0: \frac {\sigma^2}\deltaelta x + \sigma B(t)+ \deltaelta vt \in \frac{\sigma^2}\deltaelta S\right\},
\]
so that
{kf e}gin{eqnarray*}
\Pr(\tau_S<\infty)
&=&\Pr\left(\exists t' \ge 0: \frac {\sigma^2}\deltaelta x + \sigma B\left(\frac{\sigma^2}{\deltaelta^2}t' \right)+\frac {\sigma^2} \deltaelta vt' \in \frac{\sigma^2}\deltaelta S\right)\\
&=&\Pr\left(\exists t' \ge 0: \frac {\sigma^2}\deltaelta x +\frac{\sigma^2}{\deltaelta}B(t')+\frac {\sigma^2} \deltaelta vt' \in \frac{\sigma^2}\deltaelta S\right)\\
&=&\Pr\left(\exists t' \ge 0: x + B(t')+ vt' \in S\right),
\end{eqnarray*}
where the first equality follows from rescaling time and the second from the Brownian scaling property.
This argument extends to the hitting location, i.e.,
$\Pr(\tau_S<\infty, \frac {\sigma^2}\deltaelta x + \sigma B(\tau_S)+ \deltaelta v \tau_S \in \frac {\sigma^2}\deltaelta dy)$.
In particular, the hitting location scales with $\sigma^2/\deltaelta$.
Elimination at level $\ell$ amounts to successively hitting appropriate regions of sets of the form
{kf e}gin{eqnarray*}
\left\{x: {\cal S}_I(x_I)\ge \frac{k \sigma^2\eta^2 }{(k-1) \deltaelta^2}\right\}
&=& \left\{x: {1 \over \sigma^2} \sum_{i\in I} (x_i - {kf a}r{x})^2\ge \frac{k \sigma^2\eta^2 }{(k-1) \deltaelta^2}\right\}\\
&=& \left\{x: \sum_{i\in I} (x_i - {kf a}r{x})^2\ge \frac{k \sigma^4\eta^2 }{(k-1) \deltaelta^2}\right\} \\
&=& \frac{\sigma^2}{\deltaelta} \left\{x: \sum_{i\in I} (x_i - {kf a}r{x})^2\ge \frac{k \eta^2 }{k-1}\right\},
\end{eqnarray*}
where we used Corollary~\ref{cor:S}. Such sets are of the form $(\sigma^2/\deltaelta) S$, and
the successive hitting locations scale with $\sigma^2/\deltaelta$.
By the strong Markov property and the calculation in the first part of this proof,
this means that the elimination probability does not depend on $\deltaelta$ or $\sigma$.
\end{proof}
\subsection{Immediate (Level 1) Elimination of the Best System}
\label{subsec:immediateelim}
Our approximation for the probability of eliminating system $k$ first is
based on an asymptotic analysis as the number of systems $k$ goes to infinity.
Our results use the commonly employed idea of (i) considering the slippage configuration (SC) where $\mu_1 = \cdots = \mu_{k-1} = \mu_k-\deltaelta = \mu$ and (ii) replacing the (discrete) Gaussian observation sequence with
a (continuous) Brownian motion.
Throughout this section we use the following notation. For a given a vector $x\in\mathbb{R}^k$,
we define
\[
{\rm E}_k(x) = \frac 1k \sum_{i=1}^k x_i,\quad
{\rm Var}_k(x) = \frac 1k \sum_{i=1}^k x_i^2 - {\rm E}_k(x)^2.
\]
The $\cal {DK}$ algorithms require evaluating ${\cal S}_{\{1,\ldots,k\}}$ at $\boldsymbol{X}_{\{1,\ldots,k\}}(n)$, and by
Lemma~\ref{lem:S} this equals (up to $1/\sigma^2$) the squared norm of $\boldsymbol{X}_{\{1,\ldots,k\}}(n) - \overline{\boldsymbol{X}_{\{1,\ldots,k\}}(n)}$,
which corresponds to $\sigma B(t)-\sigma{\rm E}_k(B(t))$. (We abuse notation and interpret subtraction of a constant as elementwise subtraction.)
The following lemma specifies the probabilistic behavior of this process,
and it is important to note that it is free of the unknown mean parameter $\mu$.
{kf e}gin{lemma}
$B(t)-{\rm E}_k(B(t))$ has drift $(-1/k,\ldots,-1/k,(1-1/k))\times \deltaelta/\sigma$ and is a standard Brownian motion in the $(k-1)$-dimensional
hyperplane
\[
H=\left\{x\in\mathbb{R}^k:\sum_{i=1}^k x_i=0\right\}.
\]
\end{lemma}
{kf e}gin{proof}
The claim that $B(t)-{\rm E}_k(B(t))$ takes values in $H$ is evident.
We can write $B(1)-{\rm E}_k(B(1)) = ({\text{id}}_k - {kf o}ldsymbol{1}_k {kf o}ldsymbol{1}_k^T /k) B(1)$,
where ${\text{id}}_{k}$ is the $k \times k$ identity matrix and ${kf o}ldsymbol{1}_k$ is the $k \times 1$ vector of ones.
Therefore, the covariance matrix of $B(1)-{\rm E}_k(B(1))$ is
\[
({\text{id}}_k - {kf o}ldsymbol{1}_k {kf o}ldsymbol{1}_k^T /k) \times ({\text{id}}_k - {kf o}ldsymbol{1}_k {kf o}ldsymbol{1}_k^T /k) = ({\text{id}}_k - {kf o}ldsymbol{1}_k {kf o}ldsymbol{1}_k^T /k).
\]
This matrix has one eigenvalues 0 (with corresponding eigenvector ${kf o}ldsymbol{1}_k$) and 1 (with corresponding eigenspace $H$). Therefore it acts as the identity on $H$ and it is degenerate on the complement. \end{proof}
Setting $r ={\sigma \eta_{k} \over \deltaelta_k}$, we define a $(k-1)$-dimensional sphere in $H$ by
\[
C=\left\{x\in \mathbb{R}^k: \sum_{i=1}^k x_i = 0, \|x\| = r\right\}.
\]
Elimination of the best system can be formulated as $B(t)-{\rm E}_k(B(t))$ hitting $C$ in the region
\[
E_k = \{x\in C: x_k = \min(x_1,\ldots,x_k) \}.
\]
{kf e}gin{figure}[t!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{planeC}
\caption{Sphere $C$ (circle here) on hyperplane $H$}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
$C=\left\{x\in \mathbb{R}^k: \sum_{i=1}^k x_i = 0, \|x\| = r\right\}$
$E_k = \{x\in C: x_k = \min(x_1,\ldots,x_k) \}$
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{CurveEk}
\caption{Region $E_k$ (red) on hyperplane $H$ }
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{CurveEk2}
\caption{Region $E_k$ (red) with planes $x_1=x_3$ and $x_2=x_3$ on hyperplane $H$}
\end{subfigure}
\caption{Graphical depiction of $C$ and $E_k$ for $k=3$.}
\label{fig:region}
\end{figure}
Plane $H$ is shown in Figure~\ref{fig:region} when $k=3$. The blue curve in Figure~\ref{fig:region}(a) shows $C$ when $k=3$ and the red curve in Figure~\ref{fig:region}(b) shows $E_k$, which is a part of $C$ divided by planes $x_1=x_3$ and $x_2=x_3$ as shown in Figure~\ref{fig:region}(c).
We now state the main result of this section.
{kf e}gin{lemma}
\label{lem:level1error}
Let $k\ge 3$.
Suppose that $Z_1,\ldots, Z_k$ are \mbox{iid\ } standard normal.
The probability that the process $B(t)-{\rm E}_k(B(t))$
first hits $C$ in the part $E_k$ where the best system $k$ gets eliminated equals
{kf e}gin{equation}
\label{eq:probfirstelim}
\frac{\int_{-r}^r e^{{\frac{\deltaelta_k}{\sigma}} y } d_y\Pr(Z_k=\min(Z_1,\ldots,Z_k), r(Z_k-{\rm E}_k(Z))\le y \sqrt{(k-1){\rm Var}_k(Z)} )}
{\left( {\eta_k \over 2} \right)^{-\nu} \Gamma(\nu+1)I_\nu\left({ \eta_k }\right)},
\end{equation}
where $\nu=(k-3)/2$,
$\Gamma$ stands for the Gamma function, and $I_\nu$ for the modified Bessel function of the first kind.
\end{lemma}
{kf e}gin{proof}
Writing $\zeta$ for the drift of $B(t)-{\rm E}_k(B(t))$, then
the hitting place of $B(t)-{\rm E}_k(B(t))$ on $C$ has density $f$ with respect to the uniform distribution $u_C$ on $C$.
Here $u_C$ should be interpreted as a volume element on $C$ in the terminology of differential geometry,
and by rotational invariance it has a `simulation interpretation' as the distribution of
\[
X=\frac{r(Z_1-{\rm E}_k(Z),\ldots,Z_k-{\rm E}_k(Z))}{\sqrt{k{\rm Var}_k(Z)}},
\]
where $Z$ is a standard normal vector in $\mathbb{R}^k$.
The density $f$ with respect to $u_C$ is given by
(e.g., Rogers and Pitman (1981))
\[
f(x) = \frac{e^{\langle\zeta, x\rangle}}{\int_C e^{ {\langle\zeta, w \rangle}} u_C(dw)}, \quad x\in C.
\]
This distribution is known as the von Mises distribution.
According to Rogers and Pitman (1981), for any $\mu\in\mathbb{R}^k$ with $\sum_i \mu_i=0$,
\[
\int_C e^{\langle \mu, w \rangle} u_C(dw) = (\|\mu\| r/2)^{-\nu} \Gamma(\nu+1) I_\nu(\|\mu\| r),
\]
where $\nu = (k-3)/2$.
Therefore, the denominator can be written as
\[
\int_C e^{ {\langle\zeta, w \rangle}} u_C(dw) =\left( { {\deltaelta_k \over \sigma} r \over 2} \right)^{-\nu} \Gamma(\nu+1)I_\nu\left({ {\deltaelta_k \over \sigma} r}\right)=\left( { \eta_k \over 2} \right)^{-\nu} \Gamma(\nu+1)I_\nu\left({ \eta_k}\right)
\]
because $\|\zeta\|=\deltaelta\sqrt{(k-1)/k}/\sigma=\deltaelta_k/\sigma$ and $(\deltaelta_k / \sigma) r = \eta_k$.
Note that larger values of $B_k(t) -{\rm E}_k(B(t))$ are more likely than smaller values when the process hits $C$, which
should be expected because system $k$ is the best one.
The probability of eliminating the best system in level 1 equals
{kf e}gin{eqnarray*}
\int_{E_k}f(x)u_C(dx) &=& {\rm E}[\mathbbm{1}(X\in E_k) f(X)] \\
&=& {\rm E}[\mathbbm{1}(X_k=\min(X_1,\ldots,X_k)) f(X)] \\
&=& \frac{{\rm E}[\mathbbm{1}(X_k=\min(X_1,\ldots,X_k)) e^{\langle \zeta, X\rangle}]}{\int_C e^{\langle\zeta, w\rangle} u_C(dw)} \\
&=& \frac{\int_{-r}^r e^{\frac{\deltaelta_k}\sigma y} d_y\Pr(X_k=\min(X_1,\ldots,X_k), \langle \zeta,X\rangle\le \frac{\deltaelta_k}\sigma y)}{\int_C e^{\langle\zeta, w\rangle} u_C(dw)},
\end{eqnarray*}
where $X$ has a uniform distribution on $C$ (see the beginning of this proof) and $\mathbbm{1}$ stands for the indicator function.
Since $\langle \zeta,x\rangle = \frac\deltaelta\sigma x_k$ for $x\in H$, the sought probability equals
\[
\frac{\int_{-r}^r e^{\frac{\deltaelta_k}\sigma y} d_y\Pr(Z_k=\min(Z_1,\ldots,Z_k), \frac \deltaelta \sigma r (Z_k-{\rm E}_k(Z))/\sqrt{k{\rm Var}_k(Z)}\le \frac{\deltaelta_k}\sigma y)}{\int_C e^{\langle\zeta, w\rangle} u_C(dw)},
\]
as claimed.
\end{proof}
The preceding lemma yields a Monte Carlo method for calculating the probability of immediate elimination of the best system.
Indeed, it states that this probability equals
{kf e}gin{equation}
\label{eq:simnew}
\frac{{\rm E}\left[\exp\left(\eta_k \frac{Z_k-{\rm E}_k(Z)}{\sqrt{(k-1){\rm Var}_k(Z)}}\right); Z_k=\min(Z_1,\ldots,Z_k)\right]}
{\left( {\eta_k \over 2} \right)^{-\nu} \Gamma(\nu+1)I_\nu\left({ \eta_k }\right)},
\end{equation}
for \mbox{iid\ } standard normal $Z_1,\ldots,Z_k$.
However, for large $k$, such a Monte Carlo method is not efficient and we instead approximate the level 1 probability (\ref{eq:probfirstelim})
by replacing several of its components by asymptotic approximations.
For instance, as $k\to\infty$, the random variables ${\rm E}_k(Z)$ and ${\rm Var}_k(Z)$ converge in distribution to 0 and 1, respectively, by the
strong law of large numbers. The rate of convergence is relatively fast (order $1/\sqrt{k}$ by the central limit theorem).
We, therefore, approximate those variables by
their deterministic asymptotic approximations. The term with the minimum is slightly more complicated.
Writing
\[
c_k = \sqrt{2\log k} - \frac{\log\log k + \log(4\pi)}{2\sqrt{2\log k}},
\]
$\min(Z_1,\ldots,Z_{k-1})+c_{k-1}$ converges in distribution to 0. For example, see Example 3.3.29 in Embrechts, Kluppelberg and Mikosch (1997). The rate of convergence is relatively slow (order $1/\sqrt{2\log k}$), so we use an approximation based on the fact that
\[
\sqrt{2\log k}(\min(Z_1,\ldots,Z_{k-1})+c_{k-1})
\]
converges in distribution to $-G$ where $G$ is a Gumbel distributed random variable which is equal in distribution to
$-\log(-\log(U))$ where $U$ is standard uniformly distributed.
Even when the central limit theorem is used for the sum instead of the law of large numbers,
the minimum and sum are asymptotically independent (e.g., Chow and Teugels 1978). This motivates the approximation, for $ y\in(-r,r)$,
{kf e}gin{eqnarray*}
\lefteqn{\Pr(Z_k=\min(Z_1,\ldots,Z_k), r(Z_{k}-{\rm E}_k(Z))\le y\sqrt{(k-1){\rm Var}_k(Z)} ) }\\&\approx&
\Pr(Z_k\le \min(Z_1,\ldots,Z_{k-1}), rZ_{k}\le y\sqrt{(k-1)} ) \\&\approx&
\Pr(Z_{k} \le {-G}/\sqrt{2\log k} -c_{k-1}, rZ_{k} \le y \sqrt{k-1}),
\end{eqnarray*}
where $Z_{k}$ and $G$ are independent.
We are now ready to formulate our approximation for (\ref{eq:probfirstelim}), and we first assume $G$ is a given constant.
{kf e}gin{lemma}\label{lem:level1approx}
For fixed $a \in \mathbb{R}$, we have
{kf e}gin{eqnarray*}
\lefteqn{
\int_{-r}^r e^{\frac{\deltaelta_k}\sigma y} d_y\Pr(Z_{k}\le -a /\sqrt{2\log k} -c_{k-1}, rZ_{k}/\sqrt{k-1} \le y)}\\ &=&
\exp\left(\frac{\eta_k^2}{2(k-1)}\right) \left[ \Phi\left(\min\left( \max\left(-\sqrt{k-1},\frac{ -a}{\sqrt{2\log k}} -c_{k-1}\right),\sqrt{k-1}\right) - \frac{ \eta_k}{{\sqrt{k-1}}} \right) - \Phi\left( -\sqrt{k-1} - \frac{ \eta_k}{{\sqrt{k-1}}} \right) \right].
\end{eqnarray*}
where $\Phi(\cdot)$ is the cumulative distribution function (cdf) of the standard normal random variable.
\end{lemma}
{kf e}gin{proof}
Letting $Y$ be a centered Gaussian variable with variance $r^2/(k-1)$. For any $\kappa\in\mathbb{R}$, we then have
{kf e}gin{eqnarray*}
\lefteqn{
\int_{-r}^r e^{(\deltaelta_k/\sigma) y} d_y\Pr(Z_{k}\le \kappa, rZ_{k}/\sqrt{k-1} \le y)}\\
&=& \int_{-r}^{r\min(\max(-1,{\kappa/\sqrt{k-1}}),1)} e^{(\deltaelta_k/\sigma) y} d\Pr(Y\le y) \\
&=& \int_{-r}^{r\min(\max(-1,{\kappa/\sqrt{k-1}}),1)} e^{(\deltaelta_k/\sigma) y} \frac{\sqrt{k-1}}{r\sqrt{2\pi}} \exp\left({-\frac {(k-1)y^2}{2r^2}}\right) dy \\
&=& e^{\frac{ (\deltaelta_k/\sigma)^2 r^2}{2(k-1)}} \frac{\sqrt{k-1}}{\sqrt{2\pi} r} \int_{-r}^{r\min(\max(-1,{ \kappa/\sqrt{k-1}}),1)} \exp\left({-\frac {\left(y - \frac{(\deltaelta_k/\sigma) r^2}{{(k-1)}}\right)^2}{2r^2/(k-1)}}\right)dy\\
&=& e^{\frac{ \eta_k^2}{2(k-1)}} \frac{\sqrt{k-1}}{\sqrt{2\pi} r} \int_{-r}^{r\min(\max(-1,{\kappa/\sqrt{k-1}}),1)} \exp\left({-\frac {\left(y - \frac{(\deltaelta_k/\sigma) r^2}{{(k-1)}}\right)^2}{2r^2/(k-1)}}\right)dy \\
&=& e^{\frac{ \eta_k^2}{2(k-1)}} \left[ \Phi\left( \min(\max(-\sqrt{k-1},\kappa),\sqrt{k-1}) - \frac{ (\deltaelta_k / \sigma) r}{{\sqrt{k-1}}} \right) - \Phi\left( -\sqrt{k-1} - \frac{ (\deltaelta_k / \sigma) r}{{\sqrt{k-1}}} \right) \right]\\
&=&e^{\frac{ \eta_k^2}{2(k-1)}} \left[ \Phi\left(\min(\max(-\sqrt{k-1},\kappa),\sqrt{k-1}) - \frac{ \eta_k}{{\sqrt{k-1}}} \right) - \Phi\left( -\sqrt{k-1} - \frac{ \eta_k}{{\sqrt{k-1}}} \right) \right],
\end{eqnarray*}
as claimed.
\end{proof}
In summary, we approximate the probability of first eliminating the best system by
{kf e}gin{equation}
\label{eq:approximationlevelI}
\frac{\exp\left(\frac {\eta_k^2}{2(k-1)}\right) \left[ {\rm E}\Phi\left( \min\left(\max\left(-\sqrt{k-1},\frac{ -G}{\sqrt{2\log k}} -c_{k-1}\right),\sqrt{k-1}\right) - \frac{ \eta_k}{{\sqrt{k-1}}} \right)- \Phi\left(-\sqrt{k-1} - \frac{\eta_k }{\sqrt{k-1}}\right) \right]}
{\left(\eta_k /2\right)^{-\nu} \Gamma(\nu+1)I_\nu(\eta_k)}.
\end{equation}
The expectation in (\ref{eq:approximationlevelI}) can be estimated through either Monte Carlo by generating Gumbel random variates or numerical integration from 0 to 1 which is the range of a random number $U$. Both can be done fast but we use the latter method because it is faster and free of sampling error. In \label{subsec:otherelim} we explain in more detail how the numerical integration is performed.
{kf e}gin{remark}
\label{rem:shane}
Shane Henderson communicated to us that the numerator in (\ref{eq:simnew}) can be written as
\[
{\rm E}\left[\exp\left(\eta_k \frac{\min(Z_1,\ldots,Z_k) - {\rm E}_k(Z)}{\sqrt{(k-1){\rm Var}_k(Z)}}\right); Z_k = \min(Z_1,\ldots,Z_k)\right],
\]
and since $\min(Z_1,\ldots,Z_k)$, ${\rm E}_k(Z)$, and ${\rm Var}_k(Z)$ do not change when the elements of $Z$ are permuted, this equals
\[
\frac 1k \sum_{i=1}^k
{\rm E}\left[\exp\left(\eta_k \frac{\min(Z_1,\ldots,Z_k) - {\rm E}_k(Z)}{\sqrt{(k-1){\rm Var}_k(Z)}}\right); Z_i = \min(Z_1,\ldots,Z_k)\right] =
\frac 1k {\rm E}\left[\exp\left(\eta_k \frac{\min(Z_1,\ldots,Z_k) - {\rm E}_k(Z)}{\sqrt{(k-1){\rm Var}_k(Z)}}\right)\right].
\]
Using the approximations ${\rm E}_k(Z)\approx 0$, ${\rm Var}_k(Z)\approx 1$, $\min(Z_1,\ldots,Z_k) \approx -G/\sqrt{2\log(k)} - c_k$ as before,
the numerator in (\ref{eq:simnew}) can be approximated by
\[
\frac 1k {\rm E}\left[\exp\left(-\eta_k \frac{G}{\sqrt{2(k-1)\log(k)}} - \frac{\eta_kc_k}{\sqrt{k-1}}\right)\right] =
\frac 1k e^{-\eta_k c_k/\sqrt{k-1}} \Gamma\left( 1+ \frac{\eta_k}{\sqrt{2(k-1) log(k)}}\right),
\]
since ${\rm E}[e^{-\gamma G}] = {\rm E}[Y^\gamma] = \int_0^\infty x^\gamma e^{-x} dx = \Gamma(\gamma+1)$ for a standard exponentially distributed random variable $Y$.
We do not use this approximation in the remainder of this paper, since experiments have shown that it leads to higher PCS than (\ref{eq:approximationlevelI}).
\end{remark}
\subsection{Other Level Errors}
\label{subsec:otherelim}
For level $\ell$ errors for $\ell=2,3,\ldots, k-1$, the number of survived systems $|I|$ is $|I|=k-\ell+1$ and it is natural to replace $k$ with $|I|$ in (\ref{eq:approximationlevelI}) as follows: {kf e}gin{equation}
\label{eq:raweta}
\frac{\exp\left(\frac {\eta_{|I|}^2}{2(|I|-1)}\right) \left[ {\rm E}\Phi\left( \min\left(\max\left(-\sqrt{|I|-1},\frac{ -G}{\sqrt{2\log |I|}} -c_{|I|-1}\right),\sqrt{|I|-1}\right) - \frac{ \eta_{|I|}}{{\sqrt{|I|-1}}} \right)- \Phi\left(-\sqrt{|I|-1} - \frac{\eta_{|I|} }{\sqrt{|I|-1}}\right) \right]}
{\left(\eta_{|I|} /2\right)^{-\nu} \Gamma(\nu+1)I_\nu(\eta_{|I|})}
\end{equation}
where $\nu = (|I|-3)/2$.
In our procedure,
$\eta_{|I|}$ is calculated as the solution to $(\ref{eq:raweta}) = {kf e}ta_\ell$ for $0< {kf e}ta_\ell < \alpha$. We let $P_k(\ell/k,{kf e}ta_\ell)$ represent level $\ell$ error, the probability of incorrectly eliminating the best system at level $\ell$ when $\eta_{|I|}$ is calculated with target ${kf e}ta_\ell$. Note that it does not depend on $\deltaelta$ or $\sigma$ by Lemma~\ref{lem:deltasigma}.
The probability of incorrect selection (PICS) of ${\cal DK}_1$ is
\[
{\rm PICS} = \sum_{\ell=1}^{k-1} P_k(\ell/k,{kf e}ta_\ell).
\]
Let ${kf e}ta_0 = \alpha/(k-1)$. If $P_k(\ell/k, {kf e}ta_0)$ for $\ell=1,\ldots,k-1$ are all approximately equal to ${kf e}ta_0$, then the overall PICS would be approximately equal to $\alpha$. For large $k$, the analysis in Section~\ref{subsec:immediateelim} ensures that $\eta_k$, the solution to $(\ref{eq:approximationlevelI}) = {kf e}ta_0$, would result in the level 1 error approximately equal to ${kf e}ta_0$. For other level errors, we do not have control over the error probability but we propose an approximation.
For the derivation of (\ref{eq:approximationlevelI}), it is critical that the starting point of the corresponding Brownian motion is the origin. For levels $\ell > 1$, we start at a random point from the previous level and thus we do not necessarily have $P_k(\ell/k, {kf e}ta_0) \approx {kf e}ta_0$ if we let $\eta_{|I|}$ be the solution to $(\ref{eq:raweta}) = {kf e}ta_0$, unless we discard all observations from previous levels. This is not desirable because too many observations would be wasted. Instead, we seek for a heuristic way to determine $\eta_{|I|}$ under the following assumption:
{kf e}gin{assumption}\label{assump:strong}
For $0< {kf e}ta_\ell<\alpha$, $\ell = 1, 2, \ldots, k-1$ and ${kf e}ta_0 = \alpha/(k-1)$,
{kf e}gin{enumerate}
\item $P_k( \ell/k, {kf e}ta_\ell) \approx {kf e}ta_\ell \cdot q_k(\ell/k)$; and
\item If ${kf e}ta_\ell = {kf e}ta_0$ for all $\ell$, then the probability that an incorrect selection (ICS) event occurs at standardized level $\ell/k$ is approximately $\int_{\ell -1 \over k-1}^{\ell \over k-1}g(w)dw$ for a density function $g(\cdot)$ in [0,1].
\end{enumerate}
\end{assumption}
Assumption~\ref{assump:strong}.1 is effectively a first-order Taylor approximation under appropriate differentiability assumptions because $\lim_{{kf e}ta\deltaownarrow 0}P_k(\ell/k,{kf e}ta)=0$. This assumption implies that for small ${kf e}ta_\ell$, the level $\ell$ error is approximately linear in ${kf e}ta_\ell$. For example, if ${kf e}ta_\ell$ decreases in half for level $\ell$, then the level $\ell$ error is expected to be cut in half.
We have empirical evidence for Assumption~\ref{assump:strong}.2. To test Assumption~\ref{assump:strong}.2, we made one million replications and recorded standardized levels where ICS occurred for each experimental setting. Then a kernel density estimator is fitted to the data using Matlab with a normal kernel and support $[0,1]$. A bandwidth was chosen by Matlab, which is known to be optimal for the normal kernel. Figure~\ref{fig:ratio} shows kernel density estimates of standardized levels $\ell/k$ of having ICS for $k=75, 150, 500$ and $1000$ and $\alpha = 0.05$ and $0.10$ when $\deltaelta = 0.3$ and $\sigma^2 = 1$. Note that the specific choice of $\deltaelta$ and $\sigma$ does not matter in view of Lemma~\ref{lem:deltasigma}. From the figure, one can see that the shapes of kernel estimates for various $k$ are similar.
{kf e}gin{figure}[h!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{kernel90}
\caption{ $\alpha = 0.10$}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{kernel95}
\caption{$\alpha = 0.05$}
\end{subfigure}
\caption{Kernel density estimates on standardized levels where an incorrect selection occurs for various $k$ when $\deltaelta =0.3$, and $\sigma^2 =1$.}\label{fig:ratio}
\end{figure}
{kf e}gin{figure}[t!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{beta90}
\caption{ $\alpha = 0.10$ and $g(w) \propto w^{0.19805}(1-w)^{0.30662}$}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{beta95}
\caption{ $\alpha = 0.05$ and $g(w) \propto w^{0.2317}(1-w)^{0.39658}$}
\end{subfigure}
\caption{ Kernel density estimates and beta density estimates for $g(w)$ when $k=1000$ and $\deltaelta = 0.3$.}
\label{fig:actualfitted}
\end{figure}
To approximate $g(w)$ for $0<w<1$, we use $k=1000$ rather than $k=75$ because $k=75$ gives sparse points in $[0,1]$. In addition, as kernel estimates are not stable on the boundary 0 and 1, we further fit it using a beta distribution to get a smooth function especially close to the boundary points, assuming
\[
g(w) \approx {1 \over {\rm Beta}(A, B)} \; w^{A-1} \; (1 - w)^{B-1} \quad \mbox{for} \quad 0<w<1 \mbox{ and } A,B \in \mathbb{R}
\]
where Beta$(A,B) = \int_0^1 t^{A-1}(1-t)^{B-1} dt$. Figure~\ref{fig:actualfitted} shows the fitted beta densities for $\alpha = 0.05$ and $\alpha = 0.10$, respectively. Both beta densities are very similar.
Once $g(w)$ is approximated, $\eta_{|I|}$ which ensures the probability of correct selection (PCS) of ${\cal DK}_1$ can be calculated as follows:
{kf e}gin{description}
\item [Step 1:] Calculate the constant $m_\ell$ as follows:
{kf e}gin{eqnarray*}
m_\ell &= & \frac{ G\left( \ell \over k-1\right) - G\left( \ell - 1 \over k-1\right) } {G\left(1 \over k-1\right)}
\end{eqnarray*}
where $G(w) = \int_0^w g(t) dt$, the cdf of $g(w)$.
\item [Step 2:] Set ${kf e}ta_\ell = {{kf e}ta_0 \over m_\ell}$ and calculate $\eta_{|I|}$ from $(\ref{eq:raweta}) = {kf e}ta_\ell$.
\end{description}
The relative magnitude between $G\left(1 \over k-1\right)$ and $G\left( \ell \over k-1\right) - G\left( \ell - 1 \over k-1\right)$ can be interpreted as the relative magnitude between level 1 error and level $\ell$ error when $\eta_{|I|}$ is calculated from $(\ref{eq:raweta}) = {kf e}ta_0$. As we know that level 1 error is ${kf e}ta_0$,
\[
G\left(1 \over k-1\right): G\left( \ell \over k-1\right) - G\left( \ell - 1 \over k-1\right) \approx {kf e}ta_0: P_k(\ell/k, {kf e}ta_0).
\]
Therefore the level $\ell$ error is expected to be inflated by $m_\ell = \frac{ G\left( \ell \over k-1\right) - G\left( \ell - 1 \over k-1\right) } {G\left(1 \over k-1\right)}$ compared to target ${kf e}ta_0$. If we adjust ${kf e}ta_\ell = {kf e}ta_0/m_\ell$, then $P_k(\ell/k, {kf e}ta_\ell) \approx {kf e}ta_0$ by Assumption~\ref{assump:strong}.1, which in turns implies that the overall PICS is approximately equal to $\alpha$.
Figure~\ref{fig:adjustedlevel} shows estimated level errors $\hat{P}_k (\ell/k, {kf e}ta_0/m_\ell)$ for $\alpha =$ 5\% and 10\% when $k=512$ with $\deltaelta = 0.3$ and $\sigma^2=1$ and one million replications. One can see that the level errors do not show a beta shape as in Figure~\ref{fig:actualfitted}. Instead the ratios between level errors and ${kf e}ta_0$ fluctuate around one for the two values of $\alpha$, which empirically supports Assumption~\ref{assump:strong}.1. Also, it shows that the function we found $g(w)$ for $\alpha=5\%$ and $k=1000$ seems to work well for other popular choices of $\alpha$, including $\alpha= 10\%$.
{kf e}gin{figure}[t!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{pcs90a}
\caption{ $1-\alpha = 0.90, {\rm PCS}=0.907$}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{pcs95a}
\caption{ $1-\alpha = 0.95, {\rm PCS}=0.950$}
\end{subfigure}
\caption{Ratios between $\hat{P}_{k} (\ell/k, {kf e}ta_0/m_\ell)$ and ${kf e}ta_0$ for $\alpha=10\%$ and $5\%$ when $k=512$, $\deltaelta = 0.3$ and $\sigma^2 =1$.}
\label{fig:adjustedlevel}
\end{figure}
To search for $\eta_{|I|}$ for given target ${kf e}ta_\ell$, a bisection search is used and this requires estimating the expectation in (\ref{eq:raweta}). Instead of using Monte Carlo by generating Gumbel random variates $G$, we use numerical integration in the range of a standard uniform random variable $U$, $[0,1]$ using one million intervals on function $f(u)$ defined as follows:
\[
f(u) \equiv \Phi\left(\min\left(\max\left(-\sqrt{|I|-1},\frac{ \log(-\log u)}{\sqrt{2\log |I|}} -c_{|I|-1}\right),\sqrt{|I|-1}\right) - \frac{ \eta_{|I|}}{{\sqrt{|I|-1}}}\right) \quad \mbox{for} \quad 0 < u < 1,
\]
$f(0) \equiv \lim_{u\rightarrow 0} f(u)$ and $f(1) \equiv \lim_{u\rightarrow 1}f(u)$. When $u\rightarrow 0$ or $u \rightarrow 1$, $\log(-\log U)$ converges to either $\infty$ or $-\infty$ but the minimum and maximum functions inside $\Phi(\cdot)$ in $f(u)$ ensure a finite number is returned.
Then
{kf e}gin{eqnarray*}
& {\rm E}\Phi\left( \min\left(\max\left(-\sqrt{|I|-1},\frac{ \log(-\log u)}{\sqrt{2\log |I|}} -c_{|I|-1}\right),\sqrt{|I|-1}\right) - \frac{ \eta_{|I|}}{{\sqrt{|I|-1}}} \right)\\[6pt]
&\approx {1\over 1000000}\left[ {1 \over 2} f(0) + \sum_{j=1}^{999999} f( j/1000000) + {1\over 2} f(1) \right].
\end{eqnarray*}
The parameter $\eta_{|I|}$ is searched using the deterministic bisection method when the numerical integration is used.
Note that the above approximation is based on the assumption that $|I|$ is large. When $|I|$ is small, say $|I|<10$, (\ref{eq:raweta}) does not work well. Instead we use (\ref{eq:simnew}) which requires a Monte Carlo simulation with $|I|$ number of iid standard normal random variables. When a Monte Carlo simulation is used, there is a chance that a deterministic bisection method may fail due to simulation error. Therefore when $|I|<10$, we employee a probabilistic bisection algorithm (Section 1.5 of Waeber (2013)) is used. The stochastic bisection algorithm stops when the returned median of a posterior distribution in the current search iteration is within 0.001 of the median from the previous iteration. A sequential test of power one which determines the sign of the objective function is implemented with parameters $r_0=50000$ and $\gamma = 0.01$. The sequential test stops either when $m$ reaches 1000 or when its test statistics exit $(-k_m, k_m)$ where $k_m$ is from equation (B.6) of Waeber (2013).
Since $\eta_{|I|}$ only depends on $\alpha$ and $k$, a table can be made for popular choices of $\alpha$ such as 5\% and 10\% and $k=2,3,\ldots, 10000$. Then the values of $\eta_{|I|}$ can be read from the table while running our procedures. Table~\ref{tab:eta} in the appendix shows the values of $\eta_{|I|}$ for a few selected values of $k$ when $\alpha = 10\%$.
\subsection{Justification of Procedures for Unknown Variances}
\label{subsec:unknown}
In this subsection, we discuss why ${\cal DK}_2$ and ${\cal DK}_3$ should be expected to work for unknown variances as well.
For unknown variances, it is natural to replace variance parameters in ${\cal DK}_1$ to their estimated values. In general, it is not sufficient to replace the variance parameter with its estimated value to keep the statistical validity. It is critical to account for the variability in the estimated parameter especially when variances are estimated only once based on an initial $n_0$ observations. Kim and Nelson (2006) and Wang and Kim (2011) show that if variance estimators are updated on the fly in a procedure as more observations are obtained, then the procedure converges to the known variance case under some appropriate asymptotic regime. In the light of these results, we employ a variance updating scheme in ${\cal DK}_2$ and ${\cal DK}_3$ to avoid the difficulty of accounting for the variability in the estimated variance parameters but without any claim for the asymptotic validity in this paper.
When the decision maker believes that the variances across systems are equal (but unknown), then the natural estimator for $\sigma^2$ is the pooled variance estimator
\[
\hat{\sigma}^2_p(n) = {1 \over |I|} \sum_{i \in I} \hat{\sigma}^2_i(n).
\]
As we update $\hat{\sigma}^2_p(n)$ as more observations become available, the estimator converges to $\sigma^2$ and thus it is expected that ${\cal DK}_2$ works similarly to ${\cal DK}_1$.
When variances are unknown and unequal, we use similar arguments as in Frazier (2014). Let $n_i = \gamma \sigma_i^2 n$ for some $\gamma >0$ and thus the number of samples obtained by stage $n$ for system $i$ is proportional to its variance $\sigma_i^2$.
Then
\[
{\sum_{j=1}^{n_i} X_{ij} \over \gamma \sigma_i^2} \sim N\left( {n_i \over \gamma \sigma_i^2} \mu_i, {n_i \over \gamma^2 \sigma_i^2} \right) = N\left( n \mu_i, {n \over \gamma} \right) \approx B_{(\mu_i, 1/\gamma)}(t)
\]
where $B_{(\mu_i, 1/\gamma)}(t)$ is a Brownian motion with drift $\mu_i$ and variance $1/\gamma$. The $ {\sum_{j=1}^{n_i} X_{ij} \over \gamma \sigma_i^2}$ have equal variance as long as $n_i = \gamma \sigma_i^2 n$ and thus we can apply ${\cal DK}_1$ to ${\sum_{j=1}^{n_i} X_{ij} \over \gamma \sigma_i^2}$. Note that when $n_i = \gamma \sigma_i^2 n$,
\[
n_i \lambda^2 = n_i {\sum_{i\in I} \sigma_i^2 \over \sum_{i \in I} n_i} = \sigma_i^2
\]
where
\[
\lambda^2 = {\sum_{i\in I} \sigma^2_i(n_i) \over \sum_{i\in I} n_i}.
\]
Then
\[
{\sum_{j=1}^{n_i} X_{ij} \over \gamma \sigma_i^2} = {\sum_{j=1}^{n_i} X_{ij} \over \gamma n_i \lambda^2} = {W_i(n) \over \gamma \lambda^2}.
\]
Finally, the screening rule in ${\cal DK}_1$ is
\[
\frac{ \sum_{i\in I} \left( {W_i(n) \over \gamma \lambda^2} - {1 \over |I|} \sum_{i\in I} {W_i(n) \over \gamma \lambda^2}\right)^2} { 1/\gamma }
\ge {1 \over \gamma} \left({\eta_{|I|} \over \deltaelta_{|I|}}\right)^2
\]
which is equivalent to
\[
{1 \over \lambda^4} \sum_{i\in I} \left( W_i(n) - {1 \over |I|} \sum_{i\in I} W_i(n) \right)^2 \ge \left({\eta_{|I|} \over \deltaelta_{|I|}}\right)^2
\]
or
{kf e}gin{equation} \label{eq:screening3}
{1 \over \lambda^2} \sum_{i\in I} \left( W_i(n) - {1 \over |I|} \sum_{i\in I} W_i(n) \right)^2 \ge \left({\lambda \cdot \eta_{|I|} \over \deltaelta_{|I|}}\right)^2
\end{equation}
When $\lambda^2$ is replaced with its estimator $\hat{\lambda}^2$ in (\ref{eq:screening3}), we get the same elimination rule in the ${\cal DK}_3$ procedure, which is
\[
{\cal S}_I''(\boldsymbol{W}_I(n)) \ge \left({ \hat{\lambda} \cdot \eta_{|I|} \over \deltaelta_{|I|}}\right)^2.
\]
\section{Experiments}
\label{sec:exp}
In this section, we compare the performance of ${\cal DK}$ procedures with KN and BIZ.
For unknown variances, we use the KN procedure as originally described in Kim and Nelson (2001) with $c=1$ and $n_0=30$ and Algorithm 2 of Frazier (2014) with $B_z=1$ and $n_0=30$. For known variances, we use KN with $h^2 = 2 \eta$ where $\eta = - \ln{ \left(2 {\alpha\over k-1}\right)}$ and $n_0=1$, which is same as the ${\cal P}$ procedure in Wang and Kim (2011), and Algorithm 1 of Frazier (2014).
Throughout this section, KN and BIZ refer procedures for known variances while KN-UNK and BIZ-UNK refer procedures for unknown variances.
The number of systems $k$ varies over
\[
k\in \{ 2,3,4,5,6,7,8,16,32, 64, 128, 256, 512, 1024, 2048, 4096, 8192\}.
\]
For the mean, we consider two mean configurations, namely slippage configuration (SC) and monotonic decreasing mean configuration (MDM); and for variances, we consider three variance configurations called Equal, INC, and DEC. Thus we have total six configurations: SC-Equal, MDM-Equal, SC-INC, SC-DEC, MDM-INC and MDM-DEC.
We use same parameter settings for mean, variances, $\deltaelta$ and $\alpha$ as in Frazier (2014). Table~\ref{tab:conf} gives all six mean-variance configurations and other parameter settings.
{kf e}gin{table}
{kf e}gin{center}
\caption{Mean and variance configurations}\label{tab:conf}
{kf e}gin{tabular}{ccccc} \hline
Configuration & Means& Variances& $\deltaelta$ & $\alpha$\\ \hline
SC-Equal & $\mu = [\deltaelta, 0, \ldots, 0]$ & $\sigma^2 = 100$ & 1 & 0.1\\
MDM-Equal & $\mu_i = -\deltaelta i$ & $\sigma^2 = 100$ & 1 & 0.1\\
SC-INC & $\mu = [\deltaelta, 0, \ldots, 0]$ & $\sigma_i^2 = 25 \left( 1 + 3 {i-1 \over k-1}\right)^2$ & 1 & 0.1\\
SC-DEC & $\mu = [\deltaelta, 0, \ldots, 0]$ & $\sigma_i^2 = 25 \left( 1 + 3 {k-i \over k-1}\right)^2$ & 1 & 0.1\\
MDM-INC & $\mu_i = -\deltaelta i$ & $\sigma_i^2 = 25 \left( 1 + 3 {i-1 \over k-1}\right)^2$ & 1 & 0.1\\
MDM-DEC & $\mu_i = -\deltaelta i$ & $\sigma_i^2 = 25 \left( 1 + 3 {k-i \over k-1}\right)^2$ & 1 & 0.1\\ \hline
\end{tabular}
\end{center}
\end{table}
When calculating $\eta_{|I|}$ for ${\cal DK}$ procedures, we take logs to avoid numerical overflows and underflows in the denominator, since the Gamma term can be very large and the Bessel term can be very small. When $\ell=k-1$ or only two systems are survived, we use $\eta_2 = -\ln(2{kf e}ta_\ell)$.
The nominal confidence level is set to $1-\alpha = 0.9$. Estimated probability of correct selection (PCS) and an average number of observations per system until a decision is made (REP/$k$) are reported based on 10,000 macro replications. Standard errors for estimated PCS are approximately 0.003.
\subsection{${\cal DK}_1$ with Known and Equal Variances}
When variances are known and equal, we compare ${\cal DK}_1$ with KN and BIZ. Figure~\ref{fig:knownequal} shows REP/$k$ and PCS under SC and MDM configurations.
Procedure ${\cal DK}_1$ significantly outperforms KN under both SC and MDM. When $k$ is large, ${\cal DK}_1$ is more than three times better than KN in terms of REP/$k$.
On the other hand, the performances of BIZ and ${\cal DK}_1$ are very similar under the slippage configuration in terms of both REP/$k$ and PCS. When $k$ is small, ${\cal DK}_1$ spends a slightly more number of observations than BIZ but its probability of correct selection is slightly higher than BIZ. For large $k$, their performances are very close in both measures. Under the monotonic decreasing mean configuration, ${\cal DK}_1$ achieves PCS greater than the nominal value 90\% and clearly outperforms KN. However, BIZ achieves PCS close to the nominal value 90\% than ${\cal DK}_1$ and spends slightly fewer but very similar number of observations than ${\cal DK}_1$.
{kf e}gin{figure}[t!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{KnownEqualSC_REP}
\caption{ SC-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{KnownEqualSC_PCS}
\caption{ SC-PCS}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{KnownEqualMDM_REP}
\caption{ MDM-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{KnownEqualMDM_PCS}
\caption{ MDM-PCS}
\end{subfigure}
\caption{REP/$k$ and PCS for ${\cal DK}_1$ when variances are known and equal with $1-\alpha = 0.9$
}
\label{fig:knownequal}
\end{figure}
\subsection{${\cal DK}_2$ and ${\cal DK}_3$ with Unknown but Equal Variances}
When variances are unknown but a decision maker knows that variances across systems are equal, ${\cal DK}_2$ or ${\cal DK}_3$ can be used.
Figure~\ref{fig:unknownequal} compares performances of ${\cal DK}_2$ with those of KN-UNK and BIZ-UNK. As in the case of known and equal variances, ${\cal DK}_2$ significantly outperforms KN-UNK. Compared to BIZ-UNK, ${\cal DK}_2$ achieves slightly higher PCS and spends fewer number of observations for large $k$ under the slippage configurations. Then under the monotonic decreasing configuration, PCS is higher in ${\cal DK}_2$ and uses slightly more observations for small $k$ and then similar number of observations for large $k$.
{kf e}gin{figure}[tb!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{UnknownEqualSC_REP}
\caption{ SC-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{UnknownEqualSC_PCS}
\caption{ SC-PCS}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{UnknownEqualMDM_REP}
\caption{ MDM-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{UnknownEqualMDM_PCS}
\caption{ MDM-PCS}
\end{subfigure}
\caption{REP/$k$ and PCS for ${\cal DK}_2$ when variances are unknown but equal with $1-\alpha = 0.9$
}
\label{fig:unknownequal}
\end{figure}
In reality, it is impossible to know in advance whether variances across systems are equal. In fact, equal variances across systems rarely hold. Thus we also consider ${\cal DK}_3$. Our experiments show that ${\cal DK}_3$ actually spends slightly fewer observations than ${\cal DK}_2$ while achieving similar PCS. Figure~\ref{fig:unknownequalvaryingfreq} compares ${\cal DK}_3$ with KN-UNK and BIZ-UNK. Graphs in Figure~\ref{fig:unknownequalvaryingfreq} show similar tendency as those in Figure~\ref{fig:unknownequal}.
{kf e}gin{figure}[tb!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{UnknownEqualVaryingFreqSC_REP}
\caption{SC-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{UnknownEqualVaryingFreqSC_PCS}
\caption{SC-PCS}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{UnknownEqualVaryingFreqMDM_REP}
\caption{ MDM-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{UnknownEqualVaryingFreqMDM_PCS}
\caption{ MDM-PCS}
\end{subfigure}
\caption{REP/$k$ and PCS for ${\cal DK}_3$ when variances are unknown but equal with $1-\alpha = 0.9$
}
\label{fig:unknownequalvaryingfreq}
\end{figure}
\subsection{${\cal DK}_3$ with Unknown and Unequal Variances}
Finally, we consider unknown and unequal variances. Figure~\ref{fig:incdec_SC} compares the three procedures under the slippage configuration with increasing and decreasing variances while Figure~\ref{fig:incdec_MDM} compares them under the MDM configuration with increasing and decreasing variances.
The efficiency of ${\cal DK}_3$ compared to KN-UNK is more obvious. When $k=8192$, ${\cal DK}_3$ is more than four times better than KN-UNK under SC-INC and six times better under SC-DEC in terms of REP/$k$ while achieving PCS close to 90\%. ${\cal DK}_3$ spends up to 30\% fewer observations than BIZ-UNK under the slippage configuration.
Interestingly, under the MDM configuration with increasing variances, ${\cal DK}_3$ significantly outperforms both KN-UNK and BIZ-UNK, showing up to 63\% savings in the number of observations compared to BIZ-UNK. But ${\cal DK}_3$ uses slightly more observations than BIZ-UNK under decreasing variances.
{kf e}gin{figure}[tb!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{SCINC_REP}
\caption{SC-INC-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{SCINC_PCS}
\caption{SC-INC-PCS}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{SCDEC_REP}
\caption{SC-DEC-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{SCDEC_PCS}
\caption{SC-DEC-PCS}
\end{subfigure}
\caption{REP/$k$ and PCS for ${\cal DK}_3$ when variances are unknown and unequal with $1-\alpha = 0.9$
}
\label{fig:incdec_SC}
\end{figure}
{kf e}gin{figure}[h!]
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{MDMINC_REP}
\caption{ MDM-INC-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{MDMINC_PCS}
\caption{ MDM-INC-PCS}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{MDMDEC_REP}
\caption{MDM-DEC-REP}
\end{subfigure}
{kf e}gin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{MDMDEC_PCS}
\caption{MDM-DEC-PCS}
\end{subfigure}
\caption{REP/$k$ and PCS for ${\cal DK}_3$ when variances are unknown and unequal with $1-\alpha = 0.9$
}
\label{fig:incdec_MDM}
\end{figure}
Overall, ${\cal DK}$ procedures achieve PCS close to the nominal value for all settings we tested and they outperform KN significantly while performing similarly to BIZ under easy mean configurations but outperforming it under difficult mean configurations especially with unknown and unequal variances.
\section{Conclusions}
\label{sec:conclusion}
We present new fully-sequential procedures whose continuation regions are derived exploiting the properties of multidimensional Brownian motions, which is the first work in the literature. Our procedures deliver a probability of correct selection close to the nominal level. Compared to the existing state-of-art fully-sequential IZ procedure KN, the proposed procedures show a tight worst-case probability of incorrect selection under the slippage configuration and significant savings in the number of observations needed until a decision is made. Compared to BIZ, our procedures perform better for a large number of systems under difficult mean configurations, but tend to spend slightly more observations for small $k$ but similar number of observations for large $k$ under easier mean configurations except the increasing-variances case.
\section*{References}
{kf e}gin{hangref}
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\item Kim, S.-H., and B. L. Nelson. 2001. “A Fully Sequential Procedure for Indifference-Zone Selection in
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\item Powell, W. B. and Ryzhov, I. O. 2012. ``Ranking and selection''. In Chapter 4 in {\it Optimal
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\end{hangref}
\setcounter{table}{0}
\renewcommand\thefigure{A.\arabic{figure}}
\renewcommand\thetable{A.\arabic{table}}
\section*{Appendix}
{kf e}gin{proof}[Proof of Lemma~\ref{lem:S}]
{kf e}gin{eqnarray*}
{\cal S}_I(\Pi x) & = & (V \Pi x)^T (V \Gamma V^T)^{-1} (V \Pi x)\\
&=& (V \Gamma V^T (V \Gamma V^T)^{-1}V x)^T (V \Gamma V^T)^{-1} (V \Gamma V^T (V \Gamma V^T)^{-1}V x)\\
&=& (Vx)^T (V \Gamma V^T)^{-1} (Vx) = {\cal S}_I(x).
\end{eqnarray*}
\end{proof}
{kf e}gin{proof}[Proof of Corollary~\ref{cor:S}]
We first derive an explicit expression for $(V \Gamma V^T)^{-1}$. Without loss of generality, assume that $I=\{1,\ldots,s\}$. Then by noting that $V\Gamma V^T$ is the covariance matrix of $Vx$, we get
\[
Vx = \left[
{kf e}gin{array}{c}
x_{1} - x_{s} \\
\vdots\\
x_{s-1} - x_s
\end{array}\right] \quad \quad \mbox{ and } \quad \quad
V\Gamma V^T = \left[ {kf e}gin{array}{ccccc}
{\sigma_1^2 + \sigma_s^2} & \sigma_s^2 & \cdots & \cdots & \sigma_s^2 \\
\sigma_s^2 & \sigma_2^2 + \sigma_s^2 & \sigma_s^2 & \cdots & \sigma_s^2\\
\vdots & & \deltadots & &\vdots \\
\vdots & & & \deltadots & \sigma_s^2 \\
\sigma_s^2 & \cdots & \cdots &\sigma_s^2 & \sigma_{s-1}^2 + \sigma_s^2
\end{array}
\right].
\]
For equal variances,
\[
V\Gamma V^T = \sigma^2 \left[ {kf e}gin{array}{ccccc}
2 & 1 & \cdots & \cdots & 1 \\
1 & 2 &1 & \cdots & 1\\
\vdots & & \deltadots & &\vdots \\
\vdots & & & \deltadots & 1 \\
1 & \cdots & \cdots &1 & 2
\end{array}
\right] = \sigma^2 \left( {\text{id}}_{s-1} + {kf o}ldsymbol{1}_{s-1} {kf o}ldsymbol{1}_{s-1}^T\right)
\]
where ${\text{id}}_{s}$ is the $s \times s$ identity matrix and ${kf o}ldsymbol{1}_s$ is the $s \times 1$ vector of ones.
By the Sherman-–Morrison formula,
{kf e}gin{eqnarray}
(V\Gamma V^T)^{-1} & = & {1 \over \sigma^2} \left( {\text{id}}_{s-1}^{-1} - \frac{ {\text{id}}_{s-1}^{-1} {kf o}ldsymbol{1}_{s-1} {kf o}ldsymbol{1}_{s-1}^T {\text{id}}_{s-1}^{-1}}{1 + {kf o}ldsymbol{1}^T {\text{id}}_{s-1}^{-1} {kf o}ldsymbol{1}} \right) \nonumber \\
&=& {1 \over \sigma^2} \left( {\text{id}}_{s-1} - \frac{{kf o}ldsymbol{1}_{s-1} {kf o}ldsymbol{1}_{s-1}^T}{1 + (s-1)} \right) \nonumber \\
&=& {1 \over \sigma^2} {1 \over s} \left( s \cdot {\text{id}}_{s-1} - {kf o}ldsymbol{1}_{s-1} {kf o}ldsymbol{1}_{s-1}^T \right). \label{eqn:inv}
\end{eqnarray}
Then we have
{kf e}gin{eqnarray*}
{\cal S}_I(x) & = & {1 \over \sigma^2} {1 \over s} \left[
{kf e}gin{array}{c}
x_{1} - x_{s} \\
\vdots\\
x_{s-1} - x_{s}
\end{array}\right]^T
\left( s \cdot {\text{id}}_{s-1} - {kf o}ldsymbol{1}_{s-1} {kf o}ldsymbol{1}_{s-1}^T \right)
\left[
{kf e}gin{array}{c}
x_{1} - x_{s} \\
\vdots\\
x_{s-1} - x_{s}
\end{array}\right] \\
&=& {1 \over \sigma^2}{1 \over s} \left\{ (s-1) \sum_{i=1}^{s-1}(x_i-x_s)^2 - 2 \sum_{1 \le i < \ell <s} (x_i-x_s)(x_\ell-x_s)\right\}\\
& = & {1 \over \sigma^2} {1 \over s} \sum_{i < \ell \atop i, \ell \in I} (x_i-x_\ell)^2,
\end{eqnarray*}
which shows the first equality in the corollary because $|I|=s$.
Now we show the second equality of the corollary.
From (\ref{eqn:inv}),
\[
V^T(V \Gamma V^T)^{-1} V = {1 \over \sigma^2}{1\over s} ( s \cdot {\text{id}}_s - {kf o}ldsymbol{1}_s {kf o}ldsymbol{1}_s^T)
\quad \quad \mbox{and} \quad \quad
\Pi = \Gamma V^T (V \Gamma V^T)^{-1} V = {1 \over s} ( s \cdot {\text{id}}_s - {kf o}ldsymbol{1}_s {kf o}ldsymbol{1}_s^T).
\]
Then
\[
\Pi x = {1 \over s} ( s \cdot {\text{id}}_s - {kf o}ldsymbol{1}_s {kf o}ldsymbol{1}_s^T) x = \left[
{kf e}gin{array}{c}
x_1 - {kf a}r{x} \\
\vdots\\
x_s - {kf a}r{x}
\end{array}\right].
\]
Finally,
{kf e}gin{eqnarray*}
{\cal S}_I(\Pi x)& =& (V \Pi x)^T(V \Gamma V^T)^{-1} (V \Pi x)\\
&=& (\Pi x)^T [ V^T (V \Gamma V^T)^{-1} V] (\Pi x)\\
&=& {1 \over \sigma^2} {1 \over s}\left[
{kf e}gin{array}{c}
x_1 - {kf a}r{x} \\
\vdots\\
x_s - {kf a}r{x}
\end{array}\right]^T
( s \cdot {\text{id}}_s - {kf o}ldsymbol{1}_s {kf o}ldsymbol{1}_s^T)
\left[
{kf e}gin{array}{c}
x_1 - {kf a}r{x} \\
\vdots\\
x_s - {kf a}r{x}
\end{array}\right] \\
&=& {1 \over \sigma^2} \left[
{kf e}gin{array}{c}
x_1 - {kf a}r{x} \\
\vdots\\
x_s - {kf a}r{x}
\end{array}\right]^T\left[
{kf e}gin{array}{c}
x_1 - {kf a}r{x} \\
\vdots\\
x_s - {kf a}r{x}
\end{array}\right] \\
&=& {1 \over \sigma^2} \sum_{i=1}^s (x_i - {kf a}r{x})^2.
\end{eqnarray*}
\end{proof}
{kf e}gin{proof}[Proof of Lemma~\ref{lem:Smon}]
It suffices to prove the claim for $|I| = |J| +1$.
By relabeling systems if necessary, it suffices to prove the claim with $J= \{1,\ldots,s\}$ and $I = \{1,\ldots,s+1\}$.
We set
\[
H_{s+1} = \left\{ (x_1, x_2, \ldots, x_{s+1})^T: \sum_{i=1}^{s+1} x_{i}=0 \right\},\quad\quad
Q_{s} = \left\{ (x_1, x_2, \ldots, x_{s+1})^T: \sum_{i=1}^{s} x_{i}=0, x_{s+1}=0 \right\}.
\]
By the second equality of Corollary~\ref{cor:S}, it suffices to show that for $x\in \mathbb{R}^{s+1}$,
{kf e}gin{equation}
\label{eq:Sineq1}
{\cal S}_I(x) \ge \frac 1{\sigma^2} \sum_{i=1}^s (x_i-{kf a}r x_s)^2,
\end{equation}
where ${kf a}r x_s = (x_1+\cdots+x_s)/s$.
To see that this holds, we define $\Psi_s$ on $H_{s+1}$ as the matrix that projects orthogonally on $Q_s$, i.e.,
$\Psi_s x = (x_1-{kf a}r x_s,\ldots,x_s-{kf a}r x_s,0)$.
By Lemma~\ref{lem:S} and (\ref{eqn:inv}), we have, for $x\in\mathbb{R}^{s+1}$,
\[
{\cal S}_I (x) = {1 \over \sigma^2} {1 \over (s+1)} \left[
{kf e}gin{array}{c}
x_{1} - x_{s+1} \\
\vdots\\
x_{s} - x_{s+1}
\end{array}\right]^T
\left( (s+1) \cdot {\text{id}}_{s} - {kf o}ldsymbol{1}_{s} {kf o}ldsymbol{1}_{s}^T \right)
\left[
{kf e}gin{array}{c}
x_{1} - x_{s+1} \\
\vdots\\
x_{s} - x_{s+1}
\end{array}\right]
\]
This representation immediately yields that
\[
{\cal S}_I(\Psi_s x) = \frac 1{\sigma^2} \sum_{i=1}^s (x_i-{kf a}r x_s)^2.
\]
Since projecting decreases any quadratic form, this establishes (\ref{eq:Sineq1}).
\end{proof}
{kf e}gin{table}[h!]
\caption{$\eta_{|I|}$ when $\alpha = 10\%$} \label{tab:eta}
{kf e}gin{center}
{\tiny\renewcommand{.8}{.8}
\resizebox{!}{.35\paperheight}{
{kf e}gin{tabular}{|c|ccccccccc|} \hline
$|I|$ & $k=64$ & $k=32$ & $k=16$ & $k=8$ & $k=7$ & $k=6$ & $k=5$ & $k=4$ & $k=3$ \\ \hline
64 & 6.05042 & & & & & & & & \\
63 & 6.79306 & & & & & & & & \\
62 & 7.07127 & & & & & & & & \\
61 & 7.24002 & & & & & & & & \\
60 & 7.35712 & & & & & & & & \\
59 & 7.44289 & & & & & & & & \\
58 & 7.50694 & & & & & & & & \\
57 & 7.55688 & & & & & & & & \\
56 & 7.59517 & & & & & & & & \\
55 & 7.62437 & & & & & & & & \\
54 & 7.64624 & & & & & & & & \\
53 & 7.66164 & & & & & & & & \\
52 & 7.67146 & & & & & & & & \\
51 & 7.67755 & & & & & & & & \\
50 & 7.67896 & & & & & & & & \\
49 & 7.67697 & & & & & & & & \\
48 & 7.67216 & & & & & & & & \\
47 & 7.66385 & & & & & & & & \\
46 & 7.65204 & & & & & & & & \\
45 & 7.63885 & & & & & & & & \\
44 & 7.62218 & & & & & & & & \\
43 & 7.60345 & & & & & & & & \\
42 & 7.58269 & & & & & & & & \\
41 & 7.55989 & & & & & & & & \\
40 & 7.53440 & & & & & & & & \\
39 & 7.50692 & & & & & & & & \\
38 & 7.47817 & & & & & & & & \\
37 & 7.44679 & & & & & & & & \\
36 & 7.41350 & & & & & & & & \\
35 & 7.37764 & & & & & & & & \\
34 & 7.34060 & & & & & & & & \\
33 & 7.30173 & & & & & & & & \\
32 & 7.26040 & 4.61250 & & & & & & & \\
31 & 7.21664 & 5.16401 & & & & & & & \\
30 & 7.17116 & 5.35848 & & & & & & & \\
29 & 7.12400 & 5.46804 & & & & & & & \\
28 & 7.07454 & 5.53579 & & & & & & & \\
27 & 7.02218 & 5.57870 & & & & & & & \\
26 & 6.96829 & 5.60356 & & & & & & & \\
25 & 6.91162 & 5.61553 & & & & & & & \\
24 & 6.85224 & 5.61724 & & & & & & & \\
23 & 6.79024 & 5.61064 & & & & & & & \\
22 & 6.72631 & 5.59634 & & & & & & & \\
21 & 6.65867 & 5.57542 & & & & & & & \\
20 & 6.58867 & 5.54845 & & & & & & & \\
19 & 6.51518 & 5.51603 & & & & & & & \\
18 & 6.43833 & 5.47773 & & & & & & & \\
17 & 6.35826 & 5.43468 & & & & & & & \\
16 & 6.27395 & 5.38576 & 3.55536 & & & & & & \\
15 & 6.18617 & 5.33234 & 3.96549 & & & & & & \\
14 & 6.09394 & 5.27360 & 4.09377 & & & & & & \\
13 & 5.99752 & 5.20970 & 4.15224 & & & & & & \\
12 & 5.89658 & 5.13991 & 4.17580 & & & & & & \\
11 & 5.79086 & 5.06446 & 4.17568 & & & & & & \\
10 & 5.68014 & 4.98455 & 4.15873 & & & & & & \\
9 & 5.46038 & 4.84321 & 4.13041 & & & & & & \\
8 & 5.26699 & 4.68543 & 4.02733 & 2.83446 & & & & & \\
7 & 5.05352 & 4.50849 & 3.90131 & 3.05966 & 2.66510 & & & & \\
6 & 4.80933 & 4.29929 & 3.74380 & 3.04936 & 2.85635 & 2.47348 & & & \\
5 & 4.52132 & 4.04717 & 3.54574 & 2.95465 & 2.81492 & 2.62431 & 2.25053 & & \\
4 & 4.16553 & 3.73733 & 3.28628 & 2.78163 & 2.67081 & 2.53302 & 2.34537 & 1.98200 & \\
3 & 3.67921 & 3.30305 & 2.91240 & 2.49134 & 2.40324 & 2.29859 & 2.16417 & 1.97851 & 1.63182 \\
2 & 2.81738 & 2.52053 & 2.21620 & 1.89611 & 1.83132 & 1.75468 & 1.66093 & 1.54027 & 1.37146 \\ \hline
\end{tabular}}}
\end{center}
\end{table}
\end{document} |
\begin{document}
\title{Raman Adiabatic Transfer of Optical States}
\author{Jürgen Appel, K.-P. Marzlin, A. I. Lvovsky}
\affiliation{Institute for Quantum Information Science, University of
Calgary, Calgary, Alberta T2N 1N4,
Canada\footnote{\url{http://www.iqis.org/}}}
\date{\today}
\begin{abstract}
We analyze electromagnetically induced transparency and light
storage in an ensemble of atoms with multiple excited levels
(multi-$\Lambda$ configuration) which are coupled to one of the
ground states by quantized signal fields and to the other one via
classical control fields. We present a basis transformation of
atomic and optical states which reduces the analysis of the system
to that of EIT in a regular 3-level configuration. We demonstrate
the existence of dark state polaritons and propose a protocol to
transfer quantum information from one optical mode to another by an
adiabatic control of the control fields.
\end{abstract}
\pacs{03.67.-a, 32.80.Qk, 42.50.Gy}
\maketitle
\section{Introduction} \label{sec:Introduction}
Electromagnetically induced transparency (EIT) is a quantum
interference effect occurring when a weak signal light field and a
stronger control field both interact with an ensemble of atoms with
$\Lambda$-shaped energy level configuration
\cite{harris97:_elect,arimondo96:_coher}. The quantum probabilities
for an excitation of the atoms by both light fields interfere
destructively, so that no excitation takes place and the normally
highly opaque medium becomes transparent for the signal field.
EIT in atomic media attracts great interest due to its possible
applications in nonlinear optics and quantum information processing.
In particular, high sensitivity to the two-photon resonance condition
leads to a steep dispersion for the signal field which therefore
experiences a greatly reduced group velocity. The demonstration of
such an effect in an ultra-cold atomic
gas~\cite{hau99:_light_speed_reduction} and hot atomic
vapor~\cite{scully99:_ultras_group_veloc_enhan_nonlin} and the
subsequent stopping of
light~\cite{hau01:_coherent_optical_information_storage,lukin01:_storag_light_atomic_vapor}
by an adiabatic process make this system appealing as a candidate for
a quantum optical memory device.
Of further interest are double- and multi-$\Lambda$ configurations
that contain two or more excited levels and are excited by several
control fields. Nonlinear effects such as four-wave mixing
\cite{deng03:_inhib,merriam00:_effic_lambd}, phase conjugation
\cite{hemmer96:_optic_raman}, and amplification without inversion
\cite{kocharovskaya90:_amplif,Scully90_LukinRev} have been
investigated for strong fields applied to both sides of the~$\Lambda$
\cite{lukin00:_advan_atomic_molec_optic_physic}. If, on the other
hand, the control fields couple to the same ground state, and the
signal fields to the other (\fref{fig:multilambda1}), the behavior of
the system with respect to the signal fields is analogous to regular
EIT, but with given control fields EIT is experienced by only one
specific linear combination of signal modes
\cite{cerboneschi95:_trans_lambd,cerboneschi96:_propag_lambd,korsunsky99:_phase,liu04:_dynam_symmet_its_applic_in,lukin00:_advan_atomic_molec_optic_physic}
whereas others get absorbed. The action of the atomic sample on the
signal fields is analogous to that of an interferometer followed by
absorption of all but one output modes. Raczy\'{n}ski and Zaremba
\cite{raczynski02:_contr,raczynski04:_elect} investigated formation of
dark-state polaritons
\cite{fleischhauer02:_quantum_memory_for_photons} as well as storage
of light in a double-$\Lambda$ system.
Most of the existing work on EIT in multilevel systems was done with
classical fields. An expansion into the quantum domain was undertaken
by Liu {\it et al.} who derived an expression for a dark state with
multiple quantum signal fields in stationary
modes~\cite{liu04:_dynam_symmet_its_applic_in,li04:_manip_synch_optic_signal_doubl}.
However, to our knowledge, no full quantum EIT/light storage formalism
has been developed for propagating optical fields in this system. In
the present paper, we bridge this gap by elaborating a basis
transformation for both atomic and optical states which reduces
multi-level EIT to the well investigated EIT in a regular $\Lambda$
scheme. In addition, we show that by an adiabatic change of the
control fields, a transfer of quantum optical states between different
signal modes or their linear combinations can be implemented. This
procedure resembles stimulated Raman adiabatic passage
(STIRAP)~\cite{bergmann98:_stirap}, but applies to optical rather than
atomic states and can be useful for routing and distribution of
optically encoded information in classical and quantum communication.
\section{Discrete Field Modes} \label{sec:Discrete_Mode_Field}
\begin{figure}
\caption{Multi $\Lambda$-system: $Q$ excited states $\ket{A_q}
\label{fig:multilambda1}
\end{figure}
In order to better understand EIT in a multi-$\Lambda$ ensemble, we
first focus on a simplified system with discrete (non-propagating,
such as in a cavity) quantized field modes before we generalize our
treatment to propagating wave packets.
We consider an ensemble of $N$ multi-$\Lambda$-configuration atoms
(\fref{fig:multilambda1}). Each of the excited states
$\left\{\ket{A_{q}}\right\}_{q=1,\dotsc, Q}$ is coupled to the two
ground states $\ket{B}$ and $\ket{C}$ by a quantized signal field
$\hat a_{q}$ and a classical control field~$\Omega_{q}$, respectively.
All the signal beams are detuned from the optical resonance by the
same amount $\delta=E_q-E_B-\hbar \nu_q$; the detuning of the control
beams is $\Delta=E_q-E_C-\hbar \omega_q$, where $\nu_q$ and $\omega_q$
are the respective laser frequencies.
Let $\hat \sigma^j_{\alpha \beta}=\ketbra[jj]{\alpha}{\beta}$ be the
flip operator between the states $\ket{\beta}$ and $\ket{\alpha}$ of
the $j$-th atom. When all fields are resonant ($\delta=\Delta=0$) the
dynamics of this system is described by the interaction Hamiltonian
\begin{equation}
\label{eq:1}
\hat H_\sub{int} = -\hbar \sum \limits_{j=1}^N \sum \limits_{q=1}^Q \left( g_q \hat a_q \, \hat \sigma^j_{a_qb} + \Omega_q(t) \, \hat \sigma^j_{a_qc} \right)+ \text{H.a.}
\end{equation}
in the co-rotating frame. Here $\hat a_q$ is the photon annihilation
operator of the $q$-th mode and $g_q$ describes the \mbox{vacuum} Rabi
frequency of that transition which is assumed to be equal for all the
atoms (Dicke limit). $\Omega_q(t)$~is the slowly varying Rabi
frequency of the according classical control field.
Let $\ket{\mathbf C^k}$ denote the totally symmetric state with $k$
atoms in state $\ket{C}$ and all others in state $\ket{B}$.
\begin{equation}
\label{eq:2}
\ket{\mathbf C^k} = \frac{1}{\sqrt{\binom{N}{k}}} \sum \limits_{1 \le j_1 < \dotsb < j_k \le N} \!\!\!\!\!\! \ket{B_1,\dotsc, C_{j_1},\dotsc, C_{j_k},\dotsc, B_N}
\end{equation}
By analogy to ref.~\cite{li04:_manip_synch_optic_signal_doubl}, it
then can be shown that the states
\begin{multline}
\label{eq:3}
\!\!\!\!\ket{D,n} \! = \! \left[ \sum \limits_{j=1}^{N} \hat
\sigma^j_{CB} -\sum \limits_{q=1}^{Q} \left(\frac{\Omega_q}{g_q}
{\hat a^\dagger}_q\right)\right]^n
\!\!\!\ket{\mathbf{C}^0,(0,\dotsc,0)}
\end{multline}
are dark states: they are eigenstates of the interaction Hamiltonian
with zero eigenvalue. Here the $(n_1,\ldots,n_q)$-part denotes the
state of the quantized light field in Fock representation.
\subsection{Adiabatic transfer of optical states}
If one of the control fields is strong ($\Omega_i\gg g_i\sqrt{N})$)
while others vanish, the dark state takes the form
\begin{equation}
\label{eq:4}
\ket{D,n} \xrightarrow{\Omega_{k\ne i}\to0} \underset{\quad{i\textrm{th mode}}}{\ket{\mathbf C^0,(0,\dotsc,n,\dotsc,0)}};
\end{equation}
all photons gather in the according signal field mode. If all
controls are slowly switched off the dark state adiabatically changes
to
\begin{equation}
\label{eq:5}
\ket{D,n} \xrightarrow{\Omega_1=\dotsb=\Omega_Q=0} \ket{\mathbf C^n,(0,\dotsc,0)},
\end{equation} so the quantum optical state carried by the $i$th mode is converted to a coherent collective ground state
superposition \cite{fleischhauer02:_quantum_memory_for_photons}.
Suppose now that while the system is in the state (\ref{eq:4}),
another control field $\Omega_j$ is turned on. In this case, by
adiabatic following, the state of the system will convert to
\begin{equation} \label{interm}
\ket{D,n}\xrightarrow{\Omega_{k\ne i,k\ne j}=0} \left(\frac{\Omega_i}{g_i} {\hat a^\dagger}_i+\frac{\Omega_j}{g_j} {\hat a^\dagger}_j\right)^n\ket{\mathbf C^0,(0,\dotsc,0)}.
\end{equation}
If $\Omega_i$ is then slowly turned off, the quantum state of the
$i$th optical mode will be transferred to the $j$th mode completely:
\begin{equation}
\label{eq:7}
\ket{D,n} \xrightarrow{\Omega_{k\ne j}\to 0} \underset{\quad{j\textrm{th mode}}}{\ket{\mathbf C^0,(0,\dotsc,n,\dotsc,0)}}.
\end{equation}
We see that, by varying the control fields, the quantum state can be
transferred to any other optical mode or their coherent superposition.
We call this procedure \emph{Raman adiabatic transfer of optical
states} (RATOS) by analogy to the well known STIRAP technique which
permits transfer of population between atomic states by means of
adiabatic interaction with light \cite{bergmann98:_stirap}. In RATOS,
on the other hand, quantum states are transferred between optical
states by adiabatic interaction with atoms.
The above treatment is valid for the case of discrete, non-propagating
modes, e.g. in a cavity. In the practical case of a propagating field,
photons first travel through an atom-free environment, then couple
into an EIT medium, experience RATOS while in transfer, and finally
leave the medium. In order to understand the propagation dynamics, the
theory must be reformulated in terms of dark-state polaritons akin to
ref.~\cite{fleischhauer02:_quantum_memory_for_photons}. This is our
task for the remainder of the paper.
One specific question that needs to be addressed is whether RATOS can
be applied to optical fields that are initially (prior to coupling
into an EIT system) not in a dark state in the sense of \fref{eq:3}.
An example is both modes $i$ and $j$ containing one photon while the
remaining modes are in the vacuum state. Can one choose control fields
in such a way that these photons are losslessly coupled into an EIT
medium, and if not, what minimum loss can one expect? More generally,
what are the possibilities of quantum optical state engineering in a
multi-level EIT environment?
\section{Mapping to a single-\texorpdfstring{$\Lambda$}{Lambda} system}
Our approach is to develop a basis transformation of the atomic and
optical states that will map a multi-$\Lambda$-system to a normal EIT
(single-$\Lambda$) scheme, thus providing an intuitive understanding
for the optical properties of the system.
\subsection{Changing the atomic basis}
Consider one atom with a multi-$\Lambda$ level structure as in
\fref{fig:multilambda1}. In the rotating wave frame the Hamiltonian
reads
\begin{equation}
\label{eq:10}
\begin{split}
\hat H(t)
& = -\hbar \frac{\delta}{2} \ketbra{B}{B} - \hbar \frac{\Delta}{2} \ketbra{C}{C} \\
& \quad -\hbar \sum \limits_{q=1}^{Q} \Bigl( g_q \hat a_q
\ketbra{A_q}{B} + \Omega_q(t) \ketbra{A_q}{C} \Bigr) \enspace +
\text{H.a.},
\end{split}
\end{equation}
which in the absence of the quantized fields reduces to
\begin{equation}
\label{eq:11}
\hat H_0 =-\hbar \frac{\delta}{2} \ketbra{B}{B} -\hbar \left(\frac{\Delta}{2} \ket{C} + \Omega \ket{EB}\right)\bra{C} \quad +\text{H.a.,}\\
\end{equation} where
\begin{equation}
\label{eq:12}
\ket{EB} = \sum \limits_{q=1}^Q \frac{\Omega_q}{\Omega} \ket{A_q} \text{ and}\\
\end{equation} and
\begin{equation}
\label{eq:13}
\Omega = \sqrt{\sum \limits_{q=1}^Q |\Omega_q|^2}.
\end{equation}
$\hat H_0$ possesses $Q$ eigenstates of zero eigenvalue, one of them
obviously being $\ket{B}$. The others are superpositions of excited
states $\ket{A_q}$ that are orthogonal to the ``excited bright state''
$\ket{EB}$ and thus not coupled by the control fields.
A basis spanning this subspace can be explicitly constructed by an
unitary Householder reflection \cite{lehoucq96:_comput}
\begin{gather}
\label{eq:15}
\hat U= \sigma \ketbra{u}{u}-\mathbf{1}
\text{ with } \sigma=\braket{A_Q}{EB}+1=\frac{\Omega_Q}{\Omega}+1 \\
\text{and } \ket{u}=\tfrac{1}{\sigma} (\ket{A_Q}+\ket{EB}) \nonumber
\end{gather}
so that $\ket{EB} = \hat U \ket{A_Q}$ and
\begin{gather}
\label{eq:16}
\begin{split}
\ket{ED_q} \equiv \hat U \ket{A_q} = \frac{\Omega_q^*}{\Omega_Q^*+
\Omega} \Bigl( \ket{A_Q} + \ket{EB} \Bigr) - \ket{A_q} \\
\text{for } q=1,\dotsc,Q-1.
\end{split}
\end{gather}
In this basis the interaction Hamiltonian then reads
\begin{gather}
\label{eq:17}
\begin{split}
\hat H
& = - \hbar \, \frac{\Delta}{2} \ketbra{C}{C} - \hbar \, \frac{\delta}{2} \ketbra{B}{B} - \hbar \, \Omega \ketbra{EB}{C} \\
& \quad -\hbar \sum \limits_{q=1}^{Q} \sum \limits_{r=1}^{Q-1}
g_q \braket{ED_{r}}{A_q} \hat a_q \ketbra{ED_{r}}{B} \\
& \quad -\hbar \sum \limits_{q=1}^{Q} g_q \braket{EB}{A_q} \hat
a_q \ketbra{EB}{B} \quad + \text{H.a.}
\end{split}
\end{gather}
As expected, the states $\ket{ED_q}$ do not undergo any interaction
with the classical fields $\Omega_q$ at all.
This can be interpreted physically by understanding that the phases
and amplitudes of the excited states $\ket{A_q}$ are such that the
probability amplitudes for a transition from the states $\ket{ED_q}$
to $\ket{C}$ interfere destructively, akin to dark states in a normal
EIT scheme --- hence we call the $\ket{ED_q}$ ``excited dark states''.
Also in close analogy to EIT, the ground state $\ket{C}$ is coupled to
only one particular superposition $\ket{EB}$ of the excited states
(the ``excited bright state''), where the transition probabilities
interfere constructively.
However, each of the weak quantized optical modes $\hat a_q$ couples
the ground state $\ket{B}$ to all of the states $\ket{EB},
\ket{ED_1},\dotsc,\ket{ED_{Q-1}}$ (see~\fref{fig:multilambda2}). If
the excited states $\ket{A_q}$ have a short lifetime, so do the states
$\ket{ED_q}$. Hence, in general, light in the modes $\hat a_q$ would
not experience EIT; the photons would get absorbed, exciting the atom
to the $\ket{ED_q}$ levels which decay due to spontaneous emission. In
the next subsection we show, however, that there exists a linear
superposition of signal states which does not couple to
$\ket{ED_q}$'s, thus enabling EIT in this system.
\begin{figure}
\caption{Multi $\Lambda$-system in the ``excited dark-state'' basis: The classic fields $\Omega_q$
drive only the $\ket{EB}
\label{fig:multilambda2}
\end{figure}
\subsection{Changing the optical basis}
We are looking for a new set of quantized field mode operators $\hat
b_q$ defined by the unitary transform $\hat W$: $\hat a_q=\sum_{s=1}^Q
W_{qs} \, \hat b_{s}$, so that one of the new operators $\hat b_Q$
does not couple to any of the ``excited dark-states'' $\ket{ED_q}$, in
other words
\begin{equation}
\label{eq:18}
\sum \limits_{q=1}^{Q} g_q \braket{ED_r}{A_q} W_{qQ} \,\, \hat b_Q= 0 \quad \text{for all } r \neq Q.
\end{equation}
Since $\sum_{q=1}^{Q} \Omega_q \braket{ED_r}{A_q}=\braket{ED_r}{EB}=0$
we can choose
\begin{equation}
\label{eq:19}
W_{qQ} =\frac{1}{R} \frac{\Omega_q}{g_q} \quad \text{ with } R = \sqrt{\sum_{q=1}^{Q} \left|\frac{\Omega_q}{g_q}\right|^2}
\end{equation}
as a solution for \fref{eq:18} and fix the other components of $\hat
W$ by constructing it as a Householder reflection in a fashion
analogous to \fref{eq:15}:
\begin{gather}
\label{eq:20}
W = \gamma \vec w \vec w^\dagger -\mathbf{1}
\text{ with } \gamma= \frac{1}{R} \frac{\Omega_Q}{g_Q} +1 \\
\text{and } \vec w = \frac{1}{\gamma} \left(\vec e_Q + \frac{1}{R}
\sum \limits_{q=1}^{Q} \frac{\Omega_q}{g_q} \vec e_q\right)
\nonumber.
\end{gather}
In this new atomic and photonic basis the Hamiltonian reads
\begin{gather}
\label{eq:21}
\begin{split}
\hat H & = - \hbar \left(
\frac{\Delta}{2} \ketbra{C}{C} + \Omega \ketbra{EB}{C} \right) \\
& \quad - \hbar \left(\frac{\delta}{2} \ketbra{B}{B} + g \, \hat b_Q \ketbra{EB}{B} \right) \\
& \quad -\hbar \sum \limits_{q=1}^{Q-1} \hat b_q \left( g_q^{EB}
\ketbra{EB}{B}
+ \sum \limits_{r=1}^{Q-1} g_q^{ED_r} \ketbra{ED_r}{B} \right) \\
& \quad +\text{H.a.}
\end{split}
\end{gather}
The first two terms correspond exactly to the Hamiltonian of a
traditional \mbox{$\Lambda$-system} $(\ket{B}\leftrightarrow \ket{EB}
\Leftrightarrow \ket{C})$. The quantized field mode
\begin{equation}
\label{eq:22}
\hat b_Q = \frac{1}{R} \sum \limits_{q=1}^Q \frac{{\Omega_q}^*}{{g_q}^*} \hat a_q
\end{equation}
couples $\ket{EB}$ to $\ket{B}$ with strength $g=\tfrac{\Omega}{R}$
and detuning $\delta$ whereas, among all excited atomic states, only
$\ket{EB}$ is coupled to $\ket{C}$ by the classical field
mode~$\Omega$ detuned by~$\Delta$.
We see that weak signal pulses in the $\hat b_Q$ mode interact with
the atoms of a multi-\mbox{$\Lambda$-medium} in a fashion completely
analogous to pulses propagating through the well understood standard
EIT system.
In addition, we have the modes
\begin{gather}
\label{eq:23}
\hat b_q = \frac{1}{R + \frac{\Omega_Q}{g_Q}} \frac{\Omega_q}{g_q}
\Bigl( \hat a_Q + \hat b_Q \Bigr) - \hat a_q, \qquad \qquad q \neq Q
\end{gather}
each coupling to the excited bright state $\ket{EB}$ and also to the
(absorbing) excited dark states $\ket{ED_q}$ (\fref{fig:multilambda3})
with strengths $g_q^{EB}$ and $g_q^{ED_r}$ (whose explicit form is not
of interest). These modes do not experience EIT.
\begin{figure}
\caption{The multi $\Lambda$-system after basis transformation
of both atomic and optical Hilbert spaces: The classical
fields~$\Omega_q$ drive only the $\ket{EB}
\label{fig:multilambda3}
\end{figure}
\section{Propagating Fields}
\subsection{Dark-state polaritons}
The preceding section demonstrated that by a unitary transformation in
both the atomic states and the quantized field modes the
multi-$\Lambda$-system can be mapped to the well known standard
EIT-scheme. In order to apply the dark-state polariton formalism to
this system, we need to derive the wave propagation (Maxwell-Bloch)
equation for the field $\tilde b_Q$. For reference, we first rewrite
the main definitions of Ref.
\cite{fleischhauer02:_quantum_memory_for_photons} in our notation.
We introduce the atomic operators
\begin{equation}
\label{eq:25}
\tilde \sigma_{\alpha,\beta}^{(j)} = \ketbra[jj]{\alpha}{\beta} e^{i \frac{\omega_{\alpha \beta}}{c} z_j}
\end{equation}
acting on the $j$-th atom located at position $z_j$, with
$\omega_{\alpha \beta}$ being the laser frequency. Assuming that the
transition energies of the quantized fields are well separated, the
electric field can be decomposed into components each interacting only
with their respective transition:
\begin{equation}
\label{eq:26}
\hat E(z,t) = \sum \limits_{q=1}^Q \hat E_q(z,t), \qquad \hat E_q \textrm{ coupling } \ket{B} \leftrightarrow \ket{A_q}.
\end{equation}
We now define the slowly varying field operators $\tilde a_q(z,t)$ by
the positive frequency parts of our field components:
\begin{equation}
\label{eq:27}
{{\hat E_q}^+}(z,t) = \tilde a_q(z,t) \sqrt{\frac{\hbar \nu_q}{2 \varepsilon_0 V}} \enspace \exp\left[{i \frac{\nu_q}{c}(z-ct)}\right].
\end{equation}
To describe the evolution of the atomic variables, we can assume that
the quantum amplitude of the atomic variables does not depend strongly
on the position. By introducing a ``smearing kernel''~$s$ with
$\int_0^L s(z) \,dz=\frac{L}{N}$ and a zero-centered support with a
width that is large compared to the average distance of two atoms but
small in relation to the medium length $L$, we obtain the mean-field
operators
\begin{equation}
\label{eq:30}
\tilde \sigma_{\alpha,\beta}(z) = \sum_{j=1}^N s(z-z_j) \, \tilde \sigma_{\alpha,\beta}^{(j)},
\end{equation}
so that, assuming $\Delta=\delta=0$,
\begin{gather}
\label{eq:31}
\begin{split}
\hat H(t) & = - \hbar \frac{N}{L} \int \limits_0^L \sum \limits_{q=1}^Q \Bigl[ \,g_q \, \tilde \sigma_{A_q,B}(z) \tilde a_q(z,t) \\
& \quad + \tilde \sigma_{A_q,C}(z) \Omega_q(t) \Bigr] \,dz \qquad
+\text{H.a.}
\end{split}
\end{gather}
Performing mappings $\hat U$ and $\hat W$ on atoms and light, the
Hamiltonian transforms as follows:
\begin{gather}
\label{eq:32}
\begin{split}
\hat H_\sub{int} & = - \hbar \frac{N}{L} \int \limits_0^L dz \Biggl( \quad g \, \tilde \sigma_{EB,B} \, \tilde b_Q + \tilde \sigma_{EB,C} \, \Omega \\
& +\sum \limits_{q=1}^{Q-1} \tilde b_q \Bigl( g_q^B \tilde
\sigma_{EB,B} + \sum \limits_{r=1}^{Q-1} g_q^{ED_r} \tilde
\sigma_{ED_r,B} \Bigr) + \text{H.a.} \Biggr).
\end{split}
\end{gather}
The Maxwell-Bloch equations for the individual fields are
\begin{equation}
\label{eq:33a}
\Bigl( \frac{\partial}{\partial t} + c \frac{\partial}{\partial z} \Bigr) \tilde a_q = i N g_i^* \tilde \sigma_{B,A_Q}.
\end{equation}
Performing summation of eqs. (\ref{eq:33a}) over $q$'s with weights
$\Omega_q^*/g_q^*$ and utilizing relations (\ref{eq:12}) and
(\ref{eq:22}), we find
\begin{equation}
\label{eq:33}
\Bigl( \frac{\partial}{\partial t} + c \frac{\partial}{\partial z} \Bigr) \tilde b_Q = i N g \tilde \sigma_{B,EB}.
\end{equation}
In other words, if there is no light in the modes $\tilde b_{q\neq Q}$
and no atoms are in the excited states, the propagation of mode
$\tilde b_Q$ in a multilevel EIT setting is fully equivalent to that
in a single-$\Lambda$ system defined by \fref{fig:multilambda3}.
Similarly to ref.~\cite{fleischhauer02:_quantum_memory_for_photons},
one can define the dark-state polariton for this system. Upon entering
the medium an incoming light pulse in the EIT mode forms a
polariton~$\hat \Psi$, a superposition of an electromagnetic wave in
the $\tilde b_Q$ mode and a collective atomic excitation $\tilde
\sigma_{EB,C}$ which generates an eigenstate of eigenvalue zero of the
interaction Hamiltonian.
\begin{gather}
\begin{split}
\label{eq:34}
\hat \Psi = \cos \theta(t) \, \tilde b_Q - \sin \theta(t) \sqrt{N}
\, \tilde \sigma_{B,C}
\end{split} \\
\begin{split}
\tan \theta(t) = \frac{\sqrt{N}}{R(t)}
\end{split}
\end{gather}
By changing the classical control fields' parameter~$R$ the character
of this polariton (whether it is more optical ($\theta\approx 0$) or
has a stronger atomic component ($\theta \approx \tfrac{\pi}{2}$)) can
be changed.
\subsection{Incoupling and Slowdown}
The susceptibility for the EIT mode \cite{scully97:_quant_optic} is
proportional to
\begin{equation}
\label{eq:35}
\chi_Q \propto N g^2 \frac{\Delta - \delta}{(\Delta - \delta)(\delta + i \frac{\gamma}{2}) + \Omega^2}.
\end{equation} where
$\gamma$ is the spontaneous decay rate of the excited bright-state
$\ket{EB}$. So for a signal beam in precise two-photon resonance
($\Delta=\delta$) the refractive index is one: no back-reflection or
absorption of a signal entering and passing through the medium occurs.
This also holds for pulses as long as their bandwidth is significantly
smaller than the EIT transparency window
\begin{equation}
\label{eq:36}
\textrm{FWHM} = \frac{\gamma}{2} \left(\sqrt{\left(\frac{4 \Omega}{\gamma}\right)^2+1}-1\right).
\end{equation}
If the effective Rabi frequency~$\Omega$ is small compared
to~$\gamma$, the transparency window is narrow and \fref{eq:35}
predicts a strong dispersion. This leads to a strongly reduced group
velocity $v_g$ for the polariton wave
\begin{equation}
\label{eq:37}
v_g = \frac{c}{1+n_g}, \quad n_g=\frac{N}{R^2}.
\end{equation}
\section{RATOS}
\subsection{The procedure}
Based on this formalism we now describe a protocol for transfer of
quantum information between optical modes (Raman adiabatic transfer of
optical states, RATOS).
If the intensities of the control fields are changed slowly, the
eigenstates follow the new conditions adiabatically~\footnote{Strictly
speaking, time variation of $W_{Qq}$s leads to a geometric phase
which we will explore in a subsequent publication.}. The dark-state
polariton as eigenstate of zero interaction energy is thus preserved
-- however its mode composition and propagation velocity can be
controlled by the parameters $\{\Omega_q\}$ of the strong control
fields according to \fref{eq:22}.
This allows for transfer of quantum information from an optical mode
$\tilde a_i$ to another mode $\tilde a_j$:
\begin{itemize}
\item First only one strong control field~$\Omega_i$ is switched on.
The medium then exhibits electromagnetically induced transparency
for the $\tilde b_Q=\tilde a_i$ mode.
\item An incoming quantum pulse in the $\tilde a_i$ mode can enter the
EIT medium without absorption or reflexion since at two-photon
resonance the refractive index for the signal field is 1. The pulse
experiences a reduction of the group velocity according to
\fref{eq:37}.
\item This slowdown also leads to a spatial compression: the pulse
gets shorter in length, which helps in keeping the size of the
medium reasonably small.
\item Once the pulse is completely inside the medium, the control
field~$\Omega_i$ is replaced by another field~$\Omega_j$
adiabatically. Assuming the mixing angle $\theta$ is kept constant,
the polariton changes its characteristics as follows:
\begin{equation}
\begin{split}
\label{eq:44}
\hat \Psi_{t=-\infty} & = \cos \theta\, \tilde a_i - \sqrt{N} \sin{\theta} \, \tilde \sigma_{C,EB} \\
& \to \cos \theta \, \tilde a_j - \sqrt{N} \sin{\theta} \,
\tilde \sigma_{C,EB} = \hat \Psi_{t=\infty}
\end{split}
\end{equation}
and all photons are now in the $j$-th mode.
\item A pulse with a different frequency but in the same optical
quantum state as the original pulse exits the medium in mode $\tilde
a_j$.
\end{itemize}
RATOS might find applications as an optical switch to route optical
quantum information. If in the end not one but several control fields
are present, the incoming pulse is split into optical modes with
different frequencies.
We now review a few recently published procedures for transferring
optical information via atomic transitions that are related to the one
developed above. Zibrov et al.~\cite{zibrov02:_trans} used the
double~$\Lambda$ system formed by the fine structure splitting of
$^{87}$Rb. They first couple in a light pulse resonant to one of the
fine transition lines and store it via an adiabatic turn-off of the
control field. Later on they retrieve it with a control field tuned
to the other fine structure transition. Matsko {\it et al.} and Peng
{\it et al.} \cite{matsko01:_nonad,peng05:_pulse} investigate
transferring a light pulse to another mode using storage in a
single-$\Lambda$ system. By interchanging the roles of the control and
signal modes in the retrieving process, the pulse is retrieved at the
frequency of the original control field. The main difference of RATOS
with respect to these proposals is that it offers a way to extend this
transfer to multiple modes (and even to their coherent superpositions)
and that an intermediate storing of the pulse is not necessary.
\section{Quantum state engineering}
Now we also can answer the question to which extent RATOS can be used
for engineering of optical quantum states. The only mode that can
losslessly enter a multilevel EIT sample is that associated with the
operator $\hat b_Q$ which is a linear combination of individual mode
operators $\{\hat a_q\}$. However, by choosing amplitudes and phases
of the control fields one can adjust the coefficients of the linear
combination.
The linear transformation $W(\{\Omega_1,\ldots,\Omega_Q\}):\ \tilde
a_q\to \tilde b_q$ of the fields at the cell entrance can be
visualized as an interferometer, i.e. a sequence of linear optical
elements such as beam splitters and mirrors (\fref{fig:equiv}). While
this transformation does not by itself represent any physical process,
the modes $\tilde b_q$ do have a physical meaning as only one of them
is able to propagate through the cell due to EIT; the rest get
absorbed. While the mode $\tilde b_Q$ is traversing the cell, the
control fields may change adiabatically so at the cell output, when
the propagating modes convert back to $\tilde a_q$'s, the
interferometer, defined by $W(\{\Omega'_1,\ldots,\Omega'_Q\})$, may be
completely different. As a result, optical states can be transferred
among different input and output modes~$\tilde a_q$.
As an example, we consider a double-$\Lambda$-system ($Q=2$) with two
control fields such that $\tfrac{\Omega_1}{g_1} =
\tfrac{\Omega_2}{g_2}$. Then the incoming light fields can be
decomposed into the orthogonal modes $\tilde b_{1/2} =
\frac{1}{\sqrt{2}} \left(\tilde a_2 \mp \tilde a_1 \right)$. Light in
the mode $\tilde b_Q=\tilde b_2$ sees EIT and is subject to the RATOS
process. Light in the mode $\tilde b_1$ however couples to both
excited states $\ket{ED_1}$ and $\ket{EB}$. This leads to absorption;
due to spontaneous emission excitations of this mode will be scattered
away or decay into the EIT mode. This agrees with Raczy\'nsky's and
Zaremba's predictions for a classical double-$\Lambda$-system
\cite{raczynski04:_elect}.
Suppose this system is irradiated by an optical pulse which contains
exactly one photon in each mode. The optical mode associated with this
pulse consists to equal parts of the EIT-mode~$\tilde b_Q=\tilde b_2$
and the absorbing mode~$\tilde b_1$.
\begin{equation}
\label{eq:45}
\tilde a_1^\dagger \tilde a_2^\dagger \ket{0} = \frac{1}{2} \left( {\tilde b_2^\dagger}\,^2 - {\tilde b_1^\dagger}\,^2 \right) \ket{0}.
\end{equation}
For this reason, only with $50\%$ probability will both photons be
coupled into the medium and get fused into the EIT mode $\tilde b_2$;
with equal probability they will experience absorption. So in this
setup the double-$\Lambda$ medium does not perform better than an
ordinary beam splitter: here the Hong-Ou-Mandel effect
\cite{hong87:_measur} also provides a $50\%$ probability for the two
photons to fuse into a specific mode. Furthermore, it is clear that no
combination of control fields would make the atomic system fully
transparent for the state (\ref{eq:45}), so this state cannot be
coupled into the EIT medium without loss.
In summary, a multi-$\Lambda$ medium is equivalent to a linear optical
system with a built-in storage device and with multiple input and
output modes which differ in frequency (\fref{fig:equiv}).
\begin{figure}
\caption{Linear optical circuit equivalent to a multi-$\Lambda$ configuration. The phase shifts and reflectivities of the input (output)
combining mirrors are determined by the phases and amplitudes of
the classical control fields during the in-coupling (out-coupling)
process. In this model, the acousto-optical modulators (AOM) at
cell entrances and exits bring the input fields to the same
frequency so they become indistinguishable when handled by the
interferometers.}
\label{fig:equiv}
\end{figure}
\section{conclusions}
We have extended a full quantum treatment of the
electromagnetically-induced transparency to multi-$\Lambda$ systems.
An explicit form of an unitary mapping is presented that relates the
dark states to the effects observed in a standard EIT scheme. Most of
the properties of this well investigated system can be transferred and
extended to systems with multiple excited levels.
The mapping provides a physical explanation for the existence of the
decay sensitive $\ket{ED_q}$ states and the according bright-state
modes $\hat b_{q\neq Q}$.
EIT in a multi-$\Lambda$ scheme might be useful for multiplexing and
routing of optical quantum information as well as for the preparation
of multi-mode entangled quantum states. Its application to
quantum-optical engineering is however limited by its equivalence to a
linear-optical setup with a built-in storage capability.
We thank M.~Fleischhauer, B.~Brezger, A.~Raczy\'nsky, J.~Zaremba, and
B.~Sanders for helpful discussions. This work was supported by NSERC,
CIAR, and CFI.
\end{document} |
\mathbf{b}egin{document}
\title{{\mathbf{b}f Instantaneous Quantum Computation} }
\mathbf{a}uthor{
Dan~Shepherd\footnote{[email protected], [email protected]}~~and~Michael~J.~Bremner
\\
\mathbf{s}mall\it Department of Computer Science, University of
Bristol,\\ \mathbf{s}mall \it Woodland Road, Bristol, BS8 1UB, United Kingdom.
}
\mathbf{d}ate{December 13th, 2008}
\maketitle
\mathbf{d}ef\mathbf{k}et#1{|\,#1\,\rangle}
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\mathbf{d}ef\mathbf{b}raket#1#2{\mathbf{l}angle\, #1\,|\,#2\,\rangle}
\mathbf{d}ef\mathbf{k}etbra#1#2{\mathbf{k}et{#1}\mathbf{b}ra{#2}}
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\mathbf{d}ef\hbox to \hsize{
\rule[5pt]{2.5cm}{0.5pt}
}{\hbox to \hsize{
\rule[5pt]{2.5cm}{0.5pt}
}}
\mathbf{d}ef \mathbf{p}ar
\noindent \mathbf{e}mph{[This Section Not Yet Complete]} \mathbf{p}ar
{ \mathbf{p}ar
\noindent \mathbf{e}mph{[This Section Not Yet Complete]} \mathbf{p}ar
}
\mathbf{d}ef\mathbf{c}omment#1{}
\mathbf{d}ef\mathbf{s}et#1{\{ #1\}}
\mathbf{d}ef\mathbbm{P}rob#1{\mbox{Prob}(#1)}
\mathbf{d}ef\modulus#1{\mathbf{l}eft| #1 \right|}
\newcommand{\QED}{\nopagebreak\hspace*{\fill}\mbox{\rule[0pt]{1.5ex}{1.5ex}}}
\newcommand{\mathbf{q}ed}{\mbox{\rule[0pt]{1.5ex}{1.5ex}}}
\newcommand{\half}{\mbox{$\textstyle \frac{1}{2}$} }
\mathbf{d}ef\indicator#1{\mathbf{l}eft\{ \mathbf{p}hantom{\mathbf{b}ig|} #1 \mathbf{p}hantom{\mathbf{b}ig|}\right\}}
\mathbf{d}ef\mathbbm{Z}{\mathbbm{Z}}
\mathbf{d}ef\mathbbm{R}{\mathbbm{R}}
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}
\newtheorem{lemma}{Lemma}
\newtheorem{example}{Example}
\newtheorem{property}{Property}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{conjecture}{Conjecture}
\mathbf{d}ef\textbf{Proof~:~}{\textbf{Proof~:~}}
\mathbf{d}ef\mathcal{C}{\mathcal{C}}
\mathbf{d}ef\mathbbm{C}{\mathbbm{C}}
\mathbf{d}ef\mathbbm{E}{\mathbbm{E}}
\mathbf{d}ef\mathbbm{F}{\mathbbm{F}}
\mathbf{d}ef\mathbbm{P}{\mathbbm{P}}
\mathbf{d}ef\mathbf{a}{\mathbf{a}}
\mathbf{d}ef\mathbf{b}{\mathbf{b}}
\mathbf{d}ef\mathbf{c}{\mathbf{c}}
\mathbf{d}ef\mathbf{d}{\mathbf{d}}
\mathbf{d}ef\mathbf{e}{\mathbf{e}}
\mathbf{d}ef\mathbf{k}{\mathbf{k}}
\mathbf{d}ef\mathbf{l}{\mathbf{l}}
\mathbf{d}ef\mathbf{p}{\mathbf{p}}
\mathbf{d}ef\mathbf{q}{\mathbf{q}}
\mathbf{d}ef\mathbf{s}{\mathbf{s}}
\mathbf{d}ef\mathbf{w}{\mathbf{w}}
\mathbf{d}ef\mathbf{x}{\mathbf{x}}
\mathbf{d}ef\mathbf{X}{\mathbf{X}}
\mathbf{d}ef\mathbf{y}{\mathbf{y}}
\mathbf{d}ef\mathbf{Y}{\mathbf{Y}}
\mathbf{d}ef\mathbf{z}{\mathbf{z}}
\mathbf{d}ef\mathbf{z}ero{\mathbf{0}}
\mathbf{d}ef\mathbf{1}{\mathbf{1}}
\mathbf{d}ef\mathbbm{S}{\mathbbm{S}}
\mathbf{d}ef\mathcal{B}{\mathcal{B}}
\mathbf{d}ef\mathbf{e}ps{\varepsilon}
\mathbf{d}ef\mathbf{e}q#1{=_{\mathbf{p}hantom|_{\!\!#1}}}
\mathbf{d}ef\mathbf{p}l#1{+_{\mathbf{p}hantom|_{\!\!#1}}}
\mathbf{d}ef\mi#1{-_{\mathbf{p}hantom|_{\!\!#1}}}
\mathbf{d}ef\om#1{\omega_{\mathbf{p}hantom|_{\!\!#1}}}
\mathbf{b}egin{abstract}
We examine theoretic architectures and an abstract model for a restricted class of quantum computation, called here \mathbf{e}mph{temporally unstructured (``instantaneous'') quantum computation} because it allows for essentially no temporal structure within the quantum dynamics.
Using the theory of binary matroids, we argue that the paradigm is rich enough to enable sampling from probability distributions that cannot, classically, be sampled from efficiently and accurately.
This paradigm also admits simple interactive proof games that may convince a skeptic of the existence of truly quantum effects.
Furthermore, these effects can be created using significantly fewer qubits than are required for running Shor's Algorithm.
\mathbf{e}nd{abstract}
\mathbf{d}ef\mathbf{IQP}{\mathbf{IQP}}
\mathbf{d}ef\mathcal{B}PP{\mathbf{BPP}}
\mathbf{d}ef\mathcal{B}QP{\mathbf{BQP}}
\mathbf{s}ection{Introduction}
\mathbf{s}ubsection{Mathematical motivation}
It has often been said that underlying the power of quantum computers is the close connection between the computational model and the way we represent dynamics in quantum systems. This connection is implicit in the standard circuit model, where we require a universal gate set for an $n$-logical-qubit processor to be capable of simulating the dynamics of the $n$-qubit unitary group $SU(2^n)$. While there are many equivalent models of (universal) quantum computing, and not all of them explicitly `generate' the special unitary group on $n$ qubits, they each simulate (to within some pre-defined precision) operations drawn from \mathbf{e}mph{some} non-abelian unitary group on a set of qubits. Our approach in this paper departs from this well trod path, by focussing almost exclusively on an abelian subgroup of the unitary group. This approach is much more restrictive in the kinds of computation allowed, leading to a computational paradigm that lies somewhere between classical and universal quantum computing.
The non-abelian nature of quantum circuit elements is undoubtedly a crucial feature of universal quantum computing; for example, it imposes a clear physical limitation to the time-ordering of the gates in a circuit. In the standard model of quantum computation, the only circuits that can be performed in a single ``time-step'' are those composed only of single-qubit gates and two-qubit gates that act on disjoint sets of qubits. We often refer to such circuits as depth-1 circuits.
When an abelian group is being used for the gates within a circuit, that circuit need not be depth-1 in the sense just described, though it will nonetheless be essentially devoid of temporal structure, since the order of the gates is immaterial.
Physically, the quantum circuit model can be interpreted as applying a controlled sequence of unitary operations, which can in turn be thought of as a sequence of Hamiltonian evolutions. If any two consecutive gates in a sequence commute with one another, then their order in the sequence can be freely interchanged, or equivalently, their Hamiltonians can simply be combined additively, which corresponds to simultaneous evolution. When \mathbf{e}mph{all} gates commute, a single simultaneous Hamiltonian evolution describes the dynamics, whose terms are the individual gates.
\mathbf{s}ubsection{Physical motivation}
How can we tell when we have successfully built a quantum computer? Given that tomography quickly becomes difficult as the number of qubits in a system grows, it is pertinent to ask if there is a simpler way of verifying the success of a quantum computation. One way, which has already been attempted in several experiments, \mathbf{e}mph{e.g.} \mathbf{c}ite{lit:Van01, lit:Lan07, lit:Lu07, lit:Tame06}, would be to use the prototype quantum computer to find the solution to a problem which we think is difficult to solve on a classical computer.
For instance, the following scenario is generic. Alice is a skeptic, she doesn't believe that Bob has a quantum device at his disposal. Fortunately, she is relatively certain that classical computers can't efficiently find the prime factors of a large integer, whereas quantum computers can \mathbf{c}ite{lit:Sh94} (although many qubits may be required for a convincing demonstration). So she issues a challenge to Bob~: she chooses a large number for which she cannot find the prime factors and sends it to Bob. If Bob then sends back the prime factors of her number within a reasonable time period, she can easily convince herself that Bob must have had a quantum device at his disposal.
This scenario in particular is one which has been used in attempts to verify the success of several small-scale quantum computers \mathbf{c}ite{lit:Van01, lit:Lan07, lit:Lu07}, though of course the numbers used were too small to be considered hard to factor classically. Unfortunately, so far as we know, Shor's factoring algorithm is a relatively difficult quantum algorithm to perform. It is well known that it can be implemented in a circuit model using polynomial circuit depth and linear circuit width, or logarithmic depth with a larger width \mathbf{c}ite{lit:CW00}, or even with constant depth if arbitrarily wide `fanout' gates are allowed \mathbf{c}ite{lit:Hoy02}. In either case, we'd apparently require a fully universal set of quantum gates, and more than a thousand logical qubits, for a convincing demonstration.
In this paper, we use abelian dynamics to suggest a two-party protocol (with classical message-passing), which could be used to test remotely a quantum device, which we believe is physically far less complex than factorization.
We conjecture that it is classically infeasible to simulate the quantum process in our protocol, and that our protocol is simpler to implement than all known versions of Shor's algorithm, not requiring anything like a universal gate set.
\mathbf{s}ubsection{Guide to the paper}
We introduce the paradigm $\mathbf{IQP}$, which stands for ``Instantaneous Quantum Polytime''. It is a restricted model of quantum computation, which can also be thought of in terms of an oracle for computation. Here `polytime' means that the process is bound to consume at most a polynomial amount of resource in any reasonable model of quantum computation, while `instantaneous' means that the algorithmic process itself is to contain no inherent temporal structure.
We give formal definitions, and argue that there are non-trivial applications for $\mathbf{IQP}$. In particular, a two-party interactive protocol game is described and discussed.
These are our main points, to bear in mind throughout the paper~:
\mathbf{b}egin{itemize}
\item
We define a restricted model of quantum computation, called $\mathbf{IQP}$ (section \ref{sect:defs}).
\item
We present several different quantum architectures (section~\ref{sect:architectures}) that can render computations in the $\mathbf{IQP}$ model, suggesting that it is a `lowest common denominator' of some `natural' ideas for computing based on `abelian' notions.
\item
Although we know of no specific architecture where the Hamiltonians and measurements involved in the $\mathbf{IQP}$ paradigm really are genuinely `easy' to implement, nonetheless there is a clear sense in which the mathematics that underlies the computation is `easy'. Perhaps the ``Graph State'' architecture (section~\ref{sect:architectures}) gives the clearest example of a practical computing idea.
\item
We argue and conjecture that the probability distributions generated in the $\mathbf{IQP}$ paradigm (section \ref{sect:defs}) are not only classically hard to sample from approximately, but that there are actual polynomial-time protocols (section~\ref{sect:protocol}) that can be completed using an $\mathbf{IQP}$ oracle that (we believe) can't be completed classically in polynomial time.
\item
We provide an explicit example of such a protocol, involving two parties. The purpose of the protocol is simply for one party to prove to the other that they are capable of approximating a multi-qubit output distribution having characteristics that match the output distribution of a particular $\mathbf{IQP}$ process. In the protocol, Alice designs a problem with a `hidden' property, and sends it to Bob; Bob runs the problem through his $\mathbf{IQP}$ oracle several times, and sends the classical outputs back to Alice; Alice then uses the secret `hidden' property to assess whether there is good evidence for Bob having used a real $\mathbf{IQP}$ oracle. This is all done in section~\ref{sect:protocol}.
\item
We make a pragmatic analysis of our suggested protocol, showing how to fine-tune its parameters in order to make plausible the conjecture that it really can't be `faked' classically (section~\ref{sect:heuristics}).
\item
By analogy, this protocol is to quantum computation what Bell experiments are to quantum communication~: the simplest known `proof' of a distinctly quantum phenomenon. (Of course, since there is no mathematical proof published to date of a separation between the power of quantum computation and classical computation, we still have to rely on certain computational hardness conjectures.)
\item
Despite the existence of protocols apparently requiring an $\mathbf{IQP}$ oracle, we are unable to find any \mathbf{e}mph{decision language} in $\mathcal{B}PP^\mathbf{IQP}$ that is not in $\mathcal{B}PP$. There seems to be a sense in which the paradigm isn't able to `compute new information' (section~\ref{sect:heuristics}).
\item
If there should one day be architectures that can implement $\mathbf{IQP}$ oracles---even though full-blown universal quantum computing remain an unresolved engineering challenge---then our protocol may be an important demonstrator of the power of quantum mechanics for quantum computing.
\mathbf{e}nd{itemize}
Much of the mathematics used depends significantly on the theory of binary matroids and binary linear codes, and so we spend some time in section~\ref{sect:defs} recalling some basic definitions. Readers interested in the main construction (the interactive game) should start at section~\ref{sect:protocol} and dip back into the earlier definitions where needed. Pure mathematicians might prefer to start at section~\ref{sect:mathythingo}; cryptographers might particularly appreciate section~\ref{sect:heuristics}; whereas physicists may prefer section~\ref{sect:architectures}.
\mathbf{s}ection{The $\mathbf{IQP}$ paradigm} \mathbf{l}abel{sect:defs}
In this section, we define what we call the ``X-program'' architecture, and use it to define the notion of $\mathbf{IQP}$ oracle. The rest of the paper depends heavily on these definitions. Note that the X-program architecture is not particularly `physical', but is easier to work with than the more physically relevant architectures discussed in section~\ref{sect:architectures}.
\mathbf{s}ubsection{X-programs} \mathbf{l}abel{sect:Xprogs}
Recall that a Pauli $X$ operator acts on a single qubit, exchanging $\mathbf{k}et0$ with $\mathbf{k}et1$, \mathbf{e}mph{i.e.} $X = \mathbf{k}etbra01 + \mathbf{k}etbra10$. One can also think of $X$ as a Hamiltonian term, since $X \mathbf{p}ropto \mathbf{e}xp( i\frac\mathbf{p}i2 X )$. An X-program is essentially a Hamiltonian that is a sum of products of $X$s acting on different qubits.
In this architecture, allow for a set of $n$ qubits, initialised into the pure separable computational basis state $\mathbf{k}et{\mathbf{z}ero}$.
The \mathbf{e}mph{X-program} is specified as a (polysize) list of pairs $(\theta_\mathbf{p}, \mathbf{p}) \in [0,2\mathbf{p}i] \times \mathbbm{F}_2^n$, so $\theta_\mathbf{p}$ is an angle and $\mathbf{p}$ is a string of $n$ bits.
Each such program element (pair) is interpreted as the action of a Hamiltonian on the qubits indicated by $\mathbf{p}$, applied for action\footnotemark{} $\theta_\mathbf{p}$~:
the Hamiltonian to apply is made up from a product of Pauli $X$ operators on the indicated qubits, and naturally these all commute.
\footnotetext{action = the integral of energy over time.}
This means that---in principle---the program elements could all be applied simultaneously~: their time ordering is irrelevant.
The measurement to be performed, once all the Hamiltonians have been applied, is simply a computational-basis measurement, and the program \mathbf{e}mph{output} is simply that measurement result, regarded as a (probabilistic) sample from the vectorspace $\mathbbm{F}_2^n$.
Combining this together, we see that the probability distribution for such an output is
\mathbf{b}egin{equation} \mathbf{l}abel{eqn:dist1}
\mathbbm{P}(\mathbf{X}=\mathbf{x}) ~=~ \mathbf{l}eft| \mathbf{b}ra\mathbf{x} ~\mathbf{e}xp\mathbf{l}eft(~\mathbf{s}um_\mathbf{p} i\theta_\mathbf{p} \mathbf{b}igotimes_{j:p_j=1} X_j~\right)~ \mathbf{k}et{\mathbf{z}ero^n} \right|^2.
\mathbf{e}nd{equation}
The \mathbf{e}mph{output string} here is labelled $\mathbf{x}$. The random variable $\mathbf{X}$ here (and throughout) codifies this probability distribution of classical output samples.
For almost all of our purposes, we are only interested in the case where the $\theta_\mathbf{p}$ action values are the same for every term. When that condition applies, and the value $\theta$ is specified, the entire X-program can be represented using a $poly(n)$-by-$n$ binary matrix, with each row corresponding to a term in the Hamiltonian.
For example, the $7$-by-$4$ binary matrix
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:exampleP}
P &=& \mathbf{l}eft( \mathbf{b}egin{array}{cccc}
1& 0& 0& 0 \\
1& 1& 0& 0 \\
0& 1& 1& 0 \\
1& 0& 1& 1 \\
0& 1& 0& 1 \\
0& 0& 1& 0 \\
0& 0& 0& 1
\mathbf{e}nd{array} \right)
\mathbf{e}nd{eqnarray}
would represent the $4$-qubit Hamiltonian
\mathbf{b}egin{eqnarray}
H_{P,\theta} &=&
\theta \mathbf{c}dot ( X_1 + X_1X_2 + X_2X_3 + X_1X_3X_4 + X_2X_4 + X_3 + X_4 ).
\mathbf{e}nd{eqnarray}
\mathbf{s}ubsection{$\mathbf{IQP}$ oracle}
\mathbf{b}egin{definition}
On input the explicit description of an X-program, we define an $\mathbf{IQP}$ oracle to be any computational method that efficiently returns a sample string from the probability distribution as given at line~(\ref{eqn:dist1}).
\mathbf{e}nd{definition}
For a formal definition of what is meant by an $\mathbf{IQP}$ oracle, let it be any device that interfaces to a probabilistic Turing machine via an `oracle tape', so that if the oracle tape holds a description of a particular X-program ($P,\theta$ in the `constant action' case) at the time when the Turing machine calls its `implement oracle' instruction, then in unit time (or perhaps in time polynomial in the length of the description of $P$, \mathbf{e}mph{i.e.} polynomial in $n$), a bitstring sample in $\mathbbm{F}_2^n$ from the probability distribution at line~(\ref{eqn:dist1}) is created and written to the oracle tape, and control is passed back to the Turing machine to continue processing.
We write the \mathbf{e}mph{overall paradigm}---of classical computation augmented by this oracle---as $\mathbf{BPP^{IQP}}$, to denote the fact that classical randomised polytime pre- and post-processing is usually to be considered allowed in a simulation, and to denote the fact that we don't much care which of several quantum architectures might be being used to supply the `\textbf{IQP}-power' of sampling from probability distributions of the form at line~(\ref{eqn:dist1}).
This notation is not necessarily supposed to indicate a particular class of \mathbf{e}mph{decision languages} as such, but rather a particular class of computations.
The interactive proof games in section~\ref{sect:protocol} require the Prover to have access to an $\mathbf{IQP}$ oracle, and to access it a polynomial number of times, though these calls may be made in parallel and without precomputation.
\mathbf{b}egin{lemma} \mathbf{l}abel{lem:thing}
The probability distribution given at line~(\ref{eqn:dist1}) is equivalent to the one given below.
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:distpaths}
\mathbbm{P}(\mathbf{X}=\mathbf{x}) &=& \mathbf{l}eft|~ \mathbf{s}um_{ \mathbf{a} ~:~ \mathbf{a} \mathbf{c}dot P = \mathbf{x}}
~~\mathbf{p}rod_{\mathbf{p} ~:~ a_\mathbf{p} = 0} \mathbf{c}os \theta_\mathbf{p}
\mathbf{p}rod_{\mathbf{p} ~:~ a_\mathbf{p} = 1} i \mathbf{s}in \theta_\mathbf{p} ~\right|^2.
\mathbf{e}nd{eqnarray}
\mathbf{e}nd{lemma}
\textbf{Proof~:~}
Let $P$ denote the $k$-by-$n$ binary matrix whose rows are the $\mathbf{p}$ vectors of the X-program under consideration.
Then using the fact that the Hamiltonian terms in an X-program all commute, we can think of the quantum amplitudes arising in an X-program implementation as a sum over paths,
\mathbf{b}egin{eqnarray}
\mathbf{l}efteqn{ \mathbf{b}ra{\mathbf{x}}~ \mathbf{p}rod_\mathbf{p}
\mathbf{l}eft( \mathbf{c}os \theta_\mathbf{p} ~+~ i \mathbf{s}in \theta_p \mathbf{b}igotimes_{j:p_j=1} X_j \right)
~\mathbf{k}et{\mathbf{z}ero^n} } \nonumber \\
&=&
\mathbf{b}ra{\mathbf{x}} ~\mathbf{s}um_{ \mathbf{a} \in \mathbbm{F}_2^k}
~\mathbf{p}rod_{\mathbf{p} ~:~ a_\mathbf{p} = 0} \mathbf{c}os \theta_\mathbf{p}
\mathbf{p}rod_{\mathbf{p} ~:~ a_\mathbf{p} = 1} i \mathbf{s}in \theta_\mathbf{p}
~\mathbf{p}rod_{j=1}^n X_j^{(\mathbf{a} \mathbf{c}dot P)_j} ~~\mathbf{k}et{\mathbf{z}ero^n},
\mathbf{e}nd{eqnarray}
and hence derive a new form for the probability distribution accordingly.
\mathbf{q}ed
\mathbf{s}ubsection{Binary matroids and Linear binary codes}
Before proceeding to the main topics of the paper, it behooves us to establish the link that these formulas have with the (closely related) theories of binary matroids and binary linear codes.
\mathbf{b}egin{definition}
A linear binary code, $\mathcal{C}$, of length $k$ is a (linear) subspace of the vectorspace $\mathbbm{F}_2^k$, represented explicitly.
The elements of $\mathcal{C}$ are called \mathbf{e}mph{codewords}, and the Hamming weight $wt(c) \in [0..k]$ of some $c \in \mathcal{C}$ is defined to be the number of 1s it has. The rank of $\mathcal{C}$ is its rank as a vectorspace.
\mathbf{e}nd{definition}
Linear binary codes are frequently presented using \mathbf{e}mph{generator matrices}, where the columns of the generator matrix form a basis for the code. If $P$ is a generator matrix for a rank $r$ code $\mathcal{C}$, then $P$ has $r$ columns and the codewords are $\{ P \mathbf{c}dot \mathbf{d}^T ~:~ \mathbf{d} \in \mathbbm{F}_2^r \}$.
There are many different, isomorphic, definitions for matroids, \mathbf{c}ite{lit:matroids}.
We shall adopt the following definition.
\mathbf{b}egin{definition}
A $k$-point binary matroid is an equivalence class of matrices defined over $\mathbbm{F}_2$, where each matrix in the equivalence class has exactly $k$ rows, and two matrices are equivalent (written $M_1 \mathbf{s}im M_2$) when for some ($k$-by-$k$) permutation matrix $Q$, the column-echelon reduced form of $M_1$ is the same as the column-echelon reduced form of $Q \mathbf{c}dot M_2$. Here we take column-echelon reduction to delete empty columns, so that the result is full-rank. Hence the rank of a matroid is the rank of any of its representatives.
\mathbf{e}nd{definition}
Less formally, this means that a binary matroid is like a matrix over $\mathbbm{F}_2$ that doesn't notice if you rearrange its rows, if you add one of its columns into another (modulo 2), or if you duplicate one of its columns. This means that a matroid is like the generator matrix for a linear binary code, but it doesn't mind if it contains redundancy in its spanning set (\mathbf{e}mph{i.e.} has more columns than its rank) and it doesn't care about the actual order of the zeroes and ones in the individual codewords.
To be clear, when thinking of a matrix such as $P$ in line~(\ref{eqn:exampleP}), we are simultaneously thinking of its \mathbf{e}mph{columns} as the elements of a spanning set for a \mathbf{e}mph{code}, and its \mathbf{e}mph{rows} as the points of a corresponding \mathbf{e}mph{matroid}.
Because one cannot express a matroid independently of a representation, we consistently conflate notation for the matrix $P$ with the matroid $P$ that it represents.
Perhaps the main structural feature of a binary matroid is its \mathbf{e}mph{weight enumerator polynomial}.
\mathbf{b}egin{definition} \mathbf{l}abel{def:WEP}
If the $k$ rows of binary matrix $P$ establish the points of a $k$-point matroid, then the weight enumerator of the matroid is defined to be the weight enumerator of the $k$-long code $\mathcal{C}$ spanned by the columns of $P$, which in turn is defined to be the bivariate polynomial
\mathbf{b}egin{eqnarray}
WEP_\mathcal{C}(x,y) &=& \mathbf{s}um_{\mathbf{c} \in \mathcal{C}} x^{wt(\mathbf{c})} y^{k-wt(\mathbf{c})}.
\mathbf{e}nd{eqnarray}
\mathbf{e}nd{definition}
This is well-defined, because the effect of choosing a different matrix $P$ that represents the same binary matroid simply leads to an isomorphic code that has the same weight-enumerator polynomial as the original code $\mathcal{C}$.
\mathbf{s}ubsection{Bias in probability distributions}
\mathbf{b}egin{definition} \mathbf{l}abel{def:bias}
If $\mathbf{X}$ is a random variable taking values in $\mathbbm{F}_2^n$, and $\mathbf{s}$ is any element of $\mathbbm{F}_2^n$, then the \mathbf{e}mph{bias} of $\mathbf{X}$ in direction $\mathbf{s}$ is simply the probability that $\mathbf{X} \mathbf{c}dot \mathbf{s}^T$ is zero, \mathbf{e}mph{i.e.} the probability of a sample being orthogonal to $\mathbf{s}$.
\mathbf{e}nd{definition}
Let us now consider an X-program on $n$ qubits that has constant action value $\theta$, whose Hamiltonian terms are specified by the rows of matrix $P$, as discussed earlier.
Then we can use lemma~\ref{lem:thing} to obtain the following expression of bias, for any binary vector $\mathbf{s} \in \mathbbm{F}_2^n$~:
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:walshpaths}
\mathbbm{P}(\mathbf{X} \mathbf{c}dot \mathbf{s}^T=0)
&=& \mathbf{s}um_{\mathbf{x} ~:~ \mathbf{x} \mathbf{c}dot \mathbf{s}^T = 0} ~
\mathbf{l}eft|~ \mathbf{s}um_{ \mathbf{a} ~:~ \mathbf{a} \mathbf{c}dot P = \mathbf{x}} ~
(\mathbf{c}os \theta)^{k-wt(\mathbf{a})} (i \mathbf{s}in \theta)^{wt(\mathbf{a})}
~\right|^2.
\mathbf{e}nd{eqnarray}
Since it would obviously be nice to interpret this expression as the evaluation of a weight enumerator polynomial, we are led to define $P_\mathbf{s}$ to be the submatrix of $P$ obtained by deleting all rows $\mathbf{p}$ for which $\mathbf{p} \mathbf{c}dot \mathbf{s}^T = 0$, leaving only those rows for which $\mathbf{p} \mathbf{c}dot \mathbf{s}^T = 1$. We call\footnotemark{} the number of rows remaining $n_\mathbf{s}$.
\footnotetext{$n_\mathbf{s}$ is here being used for the length of the code $\mathcal{C}_\mathbf{s}$ in deference to the usual practice of reserving the letter $n$ for code lengths. This $n_\mathbf{s}$ is counting a number of rows, and should not be confused with the $n$ used earlier for counting a number of columns.}
This in turn leads to the code $\mathcal{C}_\mathbf{s}$ being the span of the columns of $P_\mathbf{s}$, and likewise a submatroid\footnotemark{} is correspondingly defined.
\footnotetext{also called a \mathbf{e}mph{matroid minor}}
\mathbf{b}egin{theorem} \mathbf{l}abel{thm:bias}
When considering constant-action X-programs, the bias expression $\mathbbm{P}(\mathbf{X} \mathbf{c}dot \mathbf{s}^T=0)$ for the random variable $\mathbf{X}$ introduced at line~(\ref{eqn:dist1}) depends only on the action value $\theta$ and (the weight enumerator polynomial of) the $n_\mathbf{s}$-point matroid $P_\mathbf{s}$, as defined above.
Moreover, if $\mathcal{C}_\mathbf{s}$ is a binary code representing the matroid $P_\mathbf{s}$, then the following formula\footnotemark{} expresses the bias~:
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:walshcode}
\mathbbm{P}(\mathbf{X} \mathbf{c}dot \mathbf{s}^T=0)
&=& \mathbbm{E}_{\mathbf{c} \mathbf{s}im \mathcal{C}_\mathbf{s}} \mathbf{l}eft[~ \mathbf{c}os^2\mathcal{B}igl(~ \theta( n_\mathbf{s} ~-~ 2 \mathbf{c}dot wt(\mathbf{c}) ) ~\mathcal{B}igr) ~\right].
\mathbf{e}nd{eqnarray}
\mathbf{e}nd{theorem}
\footnotetext{Subscripts on expectation operators indicate a variable ranging uniformly over its natural domain.}
\textbf{Proof~:~}
See the appendix for a proof.
\mathbf{q}ed
To recap, this means that if we run an X-program using the action value $\theta$ for all program elements, then the probability of the returned sample being orthogonal to an $\mathbf{s}$ of our choosing (`orthogonal' in the $\mathbbm{F}_2$ sense of having zero dot-product with $\mathbf{s}$) depends only on $\theta$ and on the (weight enumerator polynomial of the) linear code obtained by writing the program elements $\mathbf{p}$ as rows of a matrix and ignoring those that are orthogonal to $\mathbf{s}$.
There is a definition in the literature for \mathbf{e}mph{weighted matroids}, which in this context would correspond to allowing different $\theta$ values for different terms in the Hamiltonian of an X-program. While mathematically (and physically) natural, such considerations would not help with the clarity of our presentation.
We emphasise at this point the value of theorem~\ref{thm:bias}~: it means that for any direction $\mathbf{s} \in \mathbbm{F}_2^n$, the bias of the output probability distribution from an X-program $(P,\theta)$ in the direction $\mathbf{s}$ depends \mathbf{e}mph{only} on $\theta$ and the rows of $P$ that are \mathbf{e}mph{not} orthogonal to $\mathbf{s}$, and not at all on the rows of $P$ that \mathbf{e}mph{are} orthogonal to $\mathbf{s}$.
Moreover, the bias in direction $\mathbf{s}$ depends \mathbf{e}mph{only} on the \mathbf{e}mph{matroid} $P_\mathbf{s}$, and not on the particular \mathbf{e}mph{matrix} $P_\mathbf{s}$ that represents it. That is, directional bias (definition~\ref{def:bias}) is a matroid invariant.
Note that whenever $A$ is an $n$-by-$n$ invertible matrix over $\mathbbm{F}_2$, then
$\mathbf{p} \mathbf{c}dot \mathbf{s}^T ~=~ \mathbf{p} \mathbf{c}dot A \mathbf{c}dot A^{-1} \mathbf{c}dot \mathbf{s}^T ~=~
(\mathbf{p} \mathbf{c}dot A) \mathbf{c}dot (\mathbf{s} \mathbf{c}dot A^{-T})^T$,
so any invertible column operation on matrix $P$ accompanies an invertible change of basis for the set of directions of which $\mathbf{s}$ is a member. Note also that appending or removing an all zero column to $P$ has the effect of including or excluding a qubit on which no unitary transformations are performed.
Thus if $P_\mathbf{s}$ is a submatroid of $P$ by point-deletion, as described earlier, then if the invertible column transformation $A$ is applied to the matrix $P$ that represents the matroid $P$, then the same \mathbf{e}mph{matroid} that was formerly called $P_\mathbf{s}$ is still a submatroid, but now it is represented by the matrix $P_{\mathbf{s} \mathbf{c}dot A^{-T}}$.
Likewise, appending or removing a column of zeroes to $P$ necessitates an extra zero be appended or removed from any $\mathbf{s}$ that serves as a direction for indicating a submatroid.
This is purely an issue of representation, and we consider that intuition about these objects is aided by taking an `abstractist' approach to the geometry.
\mathbf{s}ubsection{Entropy, and trivial cases} \mathbf{l}abel{sect:entropy}
Because it will be useful later, we will define the R\'enyi entropy (collision entropy) of a random variable, before exemplifying theorem~\ref{thm:bias} and proceeding with the main construction of the paper.
\mathbf{b}egin{definition}
The collision entropy, $S_2$, of a discrete random variable, $\mathbf{X}$, measures the randomness of the sampling process by measuring the likelihood of two (independent) samples being the same. It is defined by
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:Renyi}
2^{-S_2} &=& \mathbf{s}um_\mathbf{x} \mathbbm{P}(\mathbf{X}=\mathbf{x})^2
~~=~~ \mathbbm{E}_\mathbf{s} \mathbf{l}eft[~ \mathcal{B}igl(~ 2\mathbbm{P}( \mathbf{X} \mathbf{c}dot \mathbf{s}^T = 0 ) - 1 ~\mathcal{B}igr)^2 ~\right].
\mathbf{e}nd{eqnarray}
\mathbf{e}nd{definition}
And so there are a few `easy cases' for our $\mathbf{X}$ random variable of lemma~\ref{lem:thing} that should be highlighted and dismissed up front~:
\mathbf{b}egin{lemma}
For a constant-action X-program, if $\theta$ is\mathbf{l}dots
\mathbf{b}egin{itemize}
\item
\mathbf{l}dots a multiple of $\mathbf{p}i$, then the returned sample will always be $\mathbf{z}ero$.
The collision entropy will be zero.
\item
\mathbf{l}dots an odd multiple of $\mathbf{p}i/2$, then the returned sample will always be $\mathbf{s}um_{\mathbf{p} \in P} \mathbf{p}$.
The collision entropy is zero.
\item
\mathbf{l}dots an odd multiple of $\mathbf{p}i/4$, then the collision entropy need not be zero, but the probability distribution will be classically simulable to full precision.
\mathbf{e}nd{itemize}
\mathbf{e}nd{lemma}
\textbf{Proof~:~}
In the first case, considering line~(\ref{eqn:distpaths}), there is then a $\mathbf{s}in(\mathbf{p}i)=0$ factor in every term of the probability, except where $\mathbf{x}=\mathbf{z}ero$.
In the second case, considering again line~(\ref{eqn:distpaths}), there is then a $\mathbf{c}os(\mathbf{p}i/2)=0$ factor in every term, except where all the $\mathbf{p}$ vectors are summed together to give $\mathbf{x}$. The same can also be deduced from theorem~\ref{thm:bias}, which implies that $\mathbf{x}$ will be surely orthogonal to $\mathbf{s}$ exactly when $n_\mathbf{s}$ is even, \mathbf{e}mph{i.e.} exactly when an even number of rows of $P$ are \mathbf{e}mph{not} orthogonal to $\mathbf{s}$, \mathbf{e}mph{i.e.} exactly when $\mathbf{s}um_{\mathbf{p} \in P} \mathbf{p}$ \mathbf{e}mph{is} orthogonal to $\mathbf{s}$.
For the third case, if $\theta$ is an odd multiple of $\mathbf{p}i/4$, then all the gates in the program would be Clifford gates. By the Gottesman-Knill theorem there is then a classically efficient method for sampling from the distribution, by tracking the evolution of the system using stabilisers, \mathbf{e}mph{etc.}
\mathbf{q}ed
For other sufficiently different values of the action parameter, classical intractibility becomes a plausible conjecture (\mathbf{e}mph{cf.} \mathbf{c}ite{lit:SWVC08, lit:SWVC2}).
In particular, the remainder of this paper will specialise to the case $\theta = \mathbf{p}i/8$, since we are able to make all our points about the utility of $\mathbf{IQP}$ even with this restriction.
\mathbf{s}ection{Interactive protocol} \mathbf{l}abel{sect:protocol}
One would naturally like to find some `use' for the ability to sample from the probability distribution that arises from a temporally unstructured quantum polytime computation; a `task' or `proof' that can be completed using \mathbf{e}mph{e.g.} an X-program, which could not be completed by purely classical means.
In this section we develop our main construction; a two-player interactive protocol game in which a Prover uses an $\mathbf{IQP}$ oracle simply to demonstrate that he does have access to an $\mathbf{IQP}$ oracle.
\mathbf{s}ubsection{At a glance}
There are three aspects of design involved in specifying an actual ``Alice \& Bob'' game~:
\mathbf{b}egin{itemize}
\item[A)] a code/matroid construction, for Alice to select a problem, to send to Bob,
\item[B)] an architecture or technique by which Bob to take samples from the $\mathbf{IQP}$ distribution of the challenge he receives, to send back to Alice,
\item[A')] an hypothesis test for Alice to use to verify (or reject) Bob's attempt.
\mathbf{e}nd{itemize}
We have already defined the X-program architecture, and in section~\ref{sect:architectures} we discuss some alternatives that Bob might like to try.
In section~\ref{sect:heuristics} we make an analysis of some classical cheating strategies for Bob, in case he can't lay his hands on a quantum computer.
The details of precisely how to make a good hypothesis test are omitted from this paper for the sake of brevity, but sourcecode is available on our website (see section~\ref{sect:challenge}).
Alice plays the role of the Challenger/Verifier, while Bob plays the role of the Prover.
Alice uses secret random data to obfuscate a `causal' matroid $P_\mathbf{s}$ inside a larger matroid $P$, and the latter she publishes (as a matrix) to Bob. Bob interprets matrix $P$ as an X-program to be run several times, with $\theta = \mathbf{p}i/8$. He collects the returned samples, and sends them to Alice. Alice then uses her secret knowledge of `where' in $P$ the special $P_\mathbf{s}$ matroid is hidden, in order to run a statistical test on Bob's data, to validate or refute the notion that Bob has the ability to run X-programs.
This application is perhaps the simplest known protocol, requiring (say) $\mathbf{s}im 200$ qubits, that could be expected to convince a skeptic of the existence of some \mathbf{e}mph{computational} quantum effect. The reason for this is that there seems to be no classical method to fake even a \mathbf{e}mph{classical transcript} of a run of the interactive game between Challenger and Prover, without actually \mathbf{e}mph{being} (or subverting the secret random data of) the classical Challenger.
\mathbf{s}ubsection{Concept overview}
Consider therefore the following game, between Alice and Bob.
Alice, also called the Challenger/Verifier, is a classical player with access to a private random number generator.
Bob, also called the Prover, is a supposedly quantum player, whose goal is to convince Alice that he can access an $\mathbf{IQP}$ oracle, \mathbf{e}mph{i.e.} run X-programs. The rules of this game are that he has to convince her simply by sending classical data, and so in effect Bob offers to act as a remote $\mathbf{IQP}$ oracle for Alice, while Alice is initially skeptical of Bob's true $\mathbf{IQP}$ abilities.
\mathbf{s}ubsubsection{Alice's challenge}
The game begins with Alice choosing some code $\mathcal{C}_\mathbf{s}$ that has certain properties amenable to her analysis.
She chooses the code $\mathcal{C}_\mathbf{s}$ in such a way that there is a $\theta$ for which the (quantum) expectation value at line~(\ref{eqn:walshcode}) of theorem~\ref{thm:bias} is somewhere well within $(\frac12, 1)$, and for which the corresponding expectation value that arises from the best-known classical approaches to `cheating' (\mathbf{e}mph{e.g.} presumably the one at line~(\ref{eqn:classfromcode}) of section~\ref{sect:classical}, in case $\theta=\mathbf{p}i/8$) is significantly smaller.
She then finds a matrix $P_\mathbf{s}$ whose columns generate the code (not necessarily as a basis), and ensures that there is some $\mathbf{s}$ that is not orthogonal to any of the rows of $P_\mathbf{s}$. The vector $\mathbf{s}$ should be thought of not as a structural property of the code $\mathcal{C}_\mathbf{s}$, but as a `locator' that can be used to `pinpoint' $P_\mathbf{s}$ even after is has later been obfuscated.
\mathbf{e}mph{Obfuscation} of $P_\mathbf{s}$ is achieved by appending arbitrary rows that \mathbf{e}mph{are} orthogonal to $\mathbf{s}$. This gives rise to matrix $P$. The matroid $P$ has $P_\mathbf{s}$ as a submatroid, in the sense that removal of the correct set of rows will recover $P_\mathbf{s}$.
Alice publishes to Bob a representation of matroid $P$ that hides the structure that she has embedded. Random row permutations are appropriate, and reversible column operations likewise leave the matroid invariant (though the latter will affect $\mathbf{s}$ and must therefore be tracked by Alice).
\mathbf{s}ubsubsection{Bob's proof}
Bob, being $\mathbf{BPP^{IQP}}$-capable by hypothesis, may interpret the published $P$ as an X-program, to be run with the (constant) action set to $\theta = \mathbf{p}i/8$ (say). He will be able to generate random vectors which independently have the correct bias in the (unknown to him) direction $\mathbf{s}$, \mathbf{e}mph{i.e.} the correct probability of being orthogonal to Alice's secret $\mathbf{s}$, in accordance with theorem~\ref{thm:bias}. Although he may still be entirely unable to recover this $\mathbf{s}$ from such samples, he nonetheless can send to Alice a list of these samples as proof that he is $\mathbf{BPP^{IQP}}$-capable.
Note that Bob's strategy is error-tolerant, because if each run of the \textbf{IQP} algorithm were to use a `noisy' $\theta$ value, then the overall proof that he generates will still be valid, providing the noise is small and unbiased and independent between runs. Note also that he can manage several runs in one oracle call, if desired, simply by concatenating the matrix $P$ with itself diagonally. That is to say, we even avoid classical temporal structure (\mathbf{e}mph{adaptive feed-forward}) on Bob's part.
\mathbf{s}ubsubsection{Alice's verification}
Since Alice knows the secret value $\mathbf{s}$, and can presumably compute the value $\mathbbm{P}(\mathbf{X} \mathbf{c}dot \mathbf{s}^T=0)$ from the code's weight enumerator polynomial (see theorem~\ref{thm:bias} and recall that she is free to choose any $\mathcal{C}_\mathbf{s}$ that suits her purpose), it is not hard for her to use a hypothesis test to confirm that the samples Bob sends are \mathbf{e}mph{commensurate} with having been sampled independently from the same distribution that an X-program generates.
That is to say, Alice will not try to test whether Bob's data \mathbf{e}mph{definitely fits the correct $\mathbf{IQP}$ distribution,} but she will ensure that it has the particular characteristic of a strong bias in the secret direction $\mathbf{s}$. This enables her to test the null hypothesis that Bob is cheating, from the alternative hypothesis that Bob has non-trivial quantum computational power.
\mathbf{e}mph{This requires belief in several conjectures on Alice's part.}
She must believe that there is a classical separation between quantum and classical computing; in particular that $\mathbf{IQP}$ is not classically efficiently approximately simulable---at least she must believe that Bob doesn't know any good simulation tricks.
She must believe that her problem is hard---at least she should believe that the problem of identifying the location of $P_\mathbf{s}$ within $P$ is not a $\mathcal{B}PP$ problem---on the assumption that the matroid $P_\mathbf{s}$ is known.
If he passes her hypothesis test, Bob will have `proved' to Alice that he ran a quantum computation on her program, provided she is confident that there is no feasible way for Bob to simulate the `proof' data classically efficiently, \mathbf{e}mph{i.e.} provided she has performed her hypothesis test correctly against a plausibly best null hypothesis.
In particular, Alice should ensure a large collision entropy for the true ($\mathbf{IQP}$) distribution, since she will want to remove all `short circuits' (\mathbf{e}mph{i.e.} all the empty rows and all the duplicate rows) from Bob's data, before testing it, to make a test that is both fair and efficient. Otherwise it would be too easy for Bob to generate a set of data that has a strong bias in very many directions simultaneously; and it would be tedious for Alice to confirm that he has not cheated in this way if she did not remove the short circuits.
\mathbf{s}ubsubsection{Significance}
This kind of interactive game could be of much significance to validation of early quantum computing architectures, since it gives rise to a simple way of `tomographically ascertaining' the actual presence of at least \mathbf{e}mph{some} quantum computing, modulo some relatively basic complexity assumptions.
In this sense it is to quantum computation what Bell violation experiments are to quantum communication.
\footnote{We have serendipitously identified a construction for which the probability gap---quantum $85.4\%$ over classical $75\%$---precisely matches the gap available in Bell's inequalities. See lemma~\ref{lem:QRCode}.}
Of course, this protocol really comes into its own when the architecture being tested happens to have the undesirable engineering feature of being unable to sustain long-term quantum coherence, and therefore perhaps only ever being capable of shallow-depth computation. Unfortunately, the prescription for X-programs given in section~\ref{sect:defs} requires Hamiltonian terms that act across potentially hundreds of qubits, and the alternative architectures discussed in section~\ref{sect:architectures} have similar physical drawbacks that still make this paradigm extremely challenging for today's engineering.
Note that this `testing concept' does not use the $\mathbf{IQP}$ paradigm to \mathbf{e}mph{compute any data that is unknown to everyone}, nor does it directly provide Bob with any `secret' data that could be used as a witness to validate an $\mathbf{NP}$ language membership claim. Its only effect is to provide Bob with data that he \mathbf{e}mph{can't} use for any purpose other than to pass on to Alice as a `proof' of $\mathbf{IQP}$-capability. It is an open problem to find something more commonly associated with computation---perhaps deciding a decision language, for example---that can be achieved specifically by the $\mathbf{BPP^{IQP}}$ paradigm.
\mathbf{s}ubsection{Recommended construction method} \mathbf{l}abel{sect:recommend}
Here is a specific example of a construction methodology (with implicit test methodology) for Alice, which we conjecture to be asymptotically secure (against cheating Prover) and efficient (for both Prover and Verifier).
\mathbf{s}ubsubsection{Recipe for codes}
The family of codes that we suggest Alice should employ within the context of the game outlined above are the \mathbf{e}mph{quadratic residue codes}. These will be shown to have the significant property that there is a non-negligible gap between the quantum- and best-known-classical-approximation expectation values for the bias in the secret direction, both of which are significantly below 1. (The bias for a truly quantum-enabled Bob has already been defined in theorem~\ref{thm:bias}, in terms of the weight-enumerator polynomial of the causal code. For a classically cheating Bob, we discuss the best-known classical strategies and their biases in section~\ref{sect:classical}.)
Consider a quadratic residue code over $\mathbbm{F}_2$ with respect to the prime $q$, chosen so that $q+1$ is a multiple of eight. The rank of such a code is $(q+1)/2$, and the length is $q$.
A quadratic residue code is a cyclic code, and can be specified by a single cyclic generator. There are several ways of defining these, but the simplest definition is to take the codeword that has a 1 in the $j$th place if and only if the Legendre symbol\footnotemark{} $\mathbf{l}eft(\frac{j}{q}\right)$ equals 1, \mathbf{e}mph{i.e.} if and only if $j$ is a non-zero quadratic residue modulo $q$.
\footnotetext{This Legendre symbol is equivalent, modulo $q$, to $j^{(q-1)/2}$.}
For example, if $q=7$ (the smallest example) then the non-zero quadratic residues modulo $q$ are $\{1,2,4\}$, and so the quadratic residue code in question is the rank-$4$ code spanned by the various rotations of the generator $(0,1,1,0,1,0,0,0)^T$. A basis for this code is found in the columns of the matrix at line~(\ref{eqn:exampleP}).
\mathbf{b}egin{lemma} \mathbf{l}abel{lem:QRCode}
When $q$ is a prime and 8 divides $q+1$, then there is a unique quadratic residue code $\mathcal{C}$ (up to isomorphism) of length $q$ over $\mathbbm{F}_2$, having rank $(q+1)/2$, and it satisfies
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:QRCodestats1}
\mathbbm{E}_{\mathbf{c} \mathbf{s}im \mathcal{C}} \mathbf{l}eft[~ \mathbf{c}os^2\mathcal{B}igl(~ \frac\mathbf{p}i8 ( q ~-~ 2 \mathbf{c}dot wt(\mathbf{c}) ) ~\mathcal{B}igr) ~\right]
&=& \mathbf{c}os^2( \mathbf{p}i/8 ) ~~=~~ 0.854\mathbf{l}dots
\mathbf{e}nd{eqnarray}
Moreover, it also satisfies
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:QRCodestats2}
\mathbbm{P}\mathcal{B}igl(~ \mathbf{c}_1^T \mathbf{c}dot \mathbf{c}_2 = 0 ~|~ \mathbf{c}_1, \mathbf{c}_2 \mathbf{s}im \mathcal{C} ~\mathcal{B}igr)
&=& 3/4 ~~=~~ 0.75,
\mathbf{e}nd{eqnarray}
which is relevant to certain classical strategies (section~\ref{sect:classical}).
\mathbf{e}nd{lemma}
\textbf{Proof~:~}
The proof of the \mathbf{e}mph{rank} of the code and its uniqueness are well established results from classical coding theory \mathbf{c}ite{lit:macwilliams}. Other classical results of coding theory include that quadratic residue codes are a parity-bit short of being self-dual and doubly even. That is, the extended quadratic code, with length $q+1$, obtained by appending a single parity bit to each codeword, has every codeword weight a multiple of 4 and every two codewords orthogonal.
For line~(\ref{eqn:QRCodestats1}) this means that the (unextended) code has codeword weights which, modulo 4, are half the time 0 and half the time $-1$. On putting these values into the left side of the formula, we immediately obtain the right side.
For line~(\ref{eqn:QRCodestats2}) this means that in the (unextended) code, any two codewords are non-orthogonal if and only if they are both odd-parity, which happens a quarter of the time~: from which the formula follows.
\mathbf{q}ed
The corollary here is that if Alice uses one of these codes for her `causal' $\mathcal{C}_\mathbf{s}$, then if Bob runs a series of X-programs (with constant $\theta = \mathbf{p}i/8$) described by the (larger) matrix $P$, the data samples he recovers should be orthogonal to the hidden $\mathbf{s}$ about $85.4\%$ of the time (\mathbf{e}mph{cf}. theorem~\ref{thm:bias}); whereas if Bob tries to cheat using the classical strategy outlined in section~\ref{sect:heuristics}, then his data samples will tend to be orthogonal to the hidden $\mathbf{s}$ only about $75\%$ of the time (\mathbf{e}mph{cf}. lemma~\ref{lem:classprob}).
Alice's hypothesis test therefore basically consists in measuring this single characteristic, after having filtered duplicate and null data samples from Bob's dataset.
We would conjecture that Bob has no pragmatic way of boosting these signals, at least not without feedback from Alice, or exponential resources.
Note that \mathbf{e}mph{with} exponential time on his hands, Bob could choose to simulate classically an $\mathbf{IQP}$ oracle, in order to obtain a dataset with a bias in direction $\mathbf{s}$ that is approximately $85.4\%$. Alternatively, he could consider every possible $\mathbf{s}$ in turn, and test to see whether the matroid obtained by deleting rows orthogonal to his guess is in fact correspondent to a quadratic residue code, assuming he knew that this had been Alice's strategy.
\mathbf{s}ubsubsection{Recipe for obfuscation}
Having chosen $q$ as outlined above, and constructed a $q$-by-$(q+1)/2$ binary matrix generating a quadratic residue code, Alice needs to obfuscate it. The easiest way to manage this process is not to start with a particular secret $\mathbf{s}$ in mind, but rather to recognise the obfuscation problem as a \mathbf{e}mph{matroid} problem, proceeding as follows~:
\mathbf{b}egin{itemize}
\item
Append a column of 1s to the matrix~: this does not change the code spanned by its columns since the all-ones (full-weight) vector is always a codeword of a quadratic code.
Other redundant column codewords may also be appended, if desired.
\item
Append many (say $q$) extra rows to the matrix, each of which is random, subject to having a zero in the column lately appended. This gives rise to a $2q$-point matroid, and ensures that there now \mathbf{e}mph{is} an $\mathbf{s}$ such that the causal submatroid (quadratic residue matroid) is defined by non-orthogonality of the rows to that $\mathbf{s}$.
\item
Reorder the rows randomly. This has no effect on the matroid that the matrix represents, nor on the hidden causal submatroid. Nor does it affect $\mathbf{s}$, the `direction' in which the sumbatroid is hidden.
\item
Now column-reduce the matrix. There is no (desirable) structure within the particular form of the matrix before column-reduction, nothing that affects either codes or matroids. Echelon-reduction provides a canonical representative for the overall matroid, while stripping away any redundant columns that would otherwise cost an unnecessary qubit, when interpreted as an X-program. By providing a canonical representative, it closes down the possibility that information in Alice's original construction of a basis for her causal code might leak through to Bob, which might be useful to him in guessing $\mathbf{s}$. Rather more importantly, this reduction actually serves to \mathbf{e}mph{hide} $\mathbf{s}$. (We can be sure by zero-knowledge reasoning that this hiding process is random~: echelon reduction is canonical and therefore supervenes any column-scrambling process, including a random one.)
\item
Finally, one might sort the rows, though this is unnecessary. The resulting matrix is the one to publish. It will have at least $(q+1)/2$ columns, since that is the rank of the causal submatroid hidden inside.
\mathbf{e}nd{itemize}
\mathbf{s}ubsubsection{Mathematical problem description} \mathbf{l}abel{sect:mathythingo}
This method of obfuscation amounts to---mathematically speaking---a situation whereby for each suitable prime $q$, we start by acknowledging a particular (public) $q$-point binary matroid $Q$, \mathbf{e}mph{viz} the one obtained from the QR-Code of length $q$. Then an `instance' of the obfuscation consists of a published $2q$-point (say) binary matroid $P$, and there is to be a hidden ``obfuscation'' subset $O$ such that $Q = P\mathbf{b}ackslash O$;
and the practical instances occur with $P$ chosen effectively at random, subject to these constraints. (One could choose to make $O$ bigger than $q$ points if that were desired.) This has the feel of a fairly generic hidden substructure problem, so it seems likely that it should be \textbf{NP}-hard to determine the location of the hidden $Q$, given $P$ and the appropriate promise of $Q$'s existence within.
More syntactically, we should like to prove that it is \textbf{NP}-complete to decide the related matter of \mathbf{e}mph{whether or not} $P$ is of the specified form, given only a matrix for $P$. Clearly this problem is in \textbf{NP}, since one could provide $Q$ \mathbf{e}mph{in the appropriate basis} as witness. We conjecture this problem to be \textbf{NP}-complete.
\mathbf{b}egin{conjecture} \mathbf{l}abel{conj:NPc}
The language of matroids $P$ that contain a quadratic-residue code submatroid $Q$ \mathbf{e}mph{by point deletion}, where the size of $Q$ is at least half the size of $P$, is \textbf{NP}-complete under polytime reductions.
\mathbf{e}nd{conjecture}
These sorts of conjecture are apparently independent of conjectures about hardness of classical efficient $\mathbf{IQP}$ simulation, since they indicate that \mathbf{e}mph{actually identifying the hidden data} is hard, even for a universal quantum computer.
Even should this conjecture prove false, we know of no reason to think that a quantum computer would be much better than a classical one at finding the hidden $Q$, notwithstanding Grover's quadratic speed-up for exhaustive search.
One might compare the structure of conjecture~\ref{conj:NPc} to that of the following important \mathbf{e}mph{theorem} from graph theory~:
\mathbf{b}egin{proposition}
The language of graphs $G$ that contain a complete graph $K$ \mathbf{e}mph{by vertex deletion}, where the size of $K$ is at least half the size of $G$, is \textbf{NP}-complete under polytime reductions.
\mathbf{e}nd{proposition}
This is a classic result, see \mathbf{e}mph{e.g.} \mathbf{c}ite{lit:Papa}, where the problem in different guises is called `Clique' and `Independent Set' and `Node Cover'.
\mathbf{s}ubsection{Challenge} \mathbf{l}abel{sect:challenge}
It seems reasonable to conjecture that, using the methodology described, with a QR-code having a value $q \mathbf{s}im 500$, it is very easy to create randomised Interactive Game challenges for $\mathbf{BPP^{IQP}}$-capability, whose distributions have large entropy, which should lead to datasets that would be easy to validate and yet infeasible to forge without an $\mathbf{IQP}$-capable computing device (or knowledge of the secret $\mathbf{s}$ vector). We propose such challenges as being appropriate `targets' for early quantum architectures, since such challenges would essentially seem to be the simplest ones available (at least in terms of inherent temporal structure and number of qubits) that can't apparently be classically met.
Accordingly, we have posted on the internet (http://quantumchallenges.wordpress.com) a \$25 challenge problem, of size $q=487$, to help motivate further study.
This challenge website includes the source code (C) used to make the challenge matrix, and also the source code of the program that we will use to check candidate solutions, excluding only the secret seed value that we used to randomise the problem.
\mathbf{s}ection{Heuristics} \mathbf{l}abel{sect:heuristics}
Next we address in more detail the reasons for thinking the problem classically intractable, and also give an accounting of our failure to find a \mathbf{e}mph{decision language} for proving the worth of $\mathbf{IQP}$.
\mathbf{s}ubsection{Hardness of strong simulation}
In support of the supposed complexity of this paradigm, Terhal and Divincenzo \mathbf{c}ite{lit:TD02}, and Aaronson \mathbf{c}ite{lit:Aa04}, have already showed that it is \textbf{PP-complete} to \mathbf{e}mph{strongly simulate} the generic probability distributions that arise hence.
\mathbf{b}egin{lemma}
It is $\mathbf{P^{GapP}}$-hard to determine the numerical value of $\mathbbm{P}(\mathbf{X}=\mathbf{z}ero)$ (as defined in section~\ref{sect:defs}, line~(\ref{eqn:dist1})) to within exponential precision, for arbitrary matroids.
\mathbf{e}nd{lemma}
\textbf{Proof~:~}
Let $Ker_L(P) = \{ \mathbf{a}^T : \mathbf{a} \mathbf{c}dot P = \mathbf{z}ero \}$ denote the linear code for which $P$ is a parity-check matrix, and note from line~(\ref{eqn:distpaths}) and definition~\ref{def:WEP} that the probability in question is a function of the weight-enumerator polynomial of this code. Specifically,
\mathbf{b}egin{eqnarray}
\mathbbm{P}(\mathbf{X}=\mathbf{z}ero)
&=& \mathbf{l}eft|~ WEP_{Ker_L(P)}(~ \mathbf{c}os \theta, ~i \mathbf{s}in \theta ~) ~\right|^2.
\mathbf{e}nd{eqnarray}
By varying $\theta$ over the range $(0, \mathbf{p}i/2)$, accurate values of $\mathbbm{P}(\mathbf{X}=\mathbf{z}ero)$ would enable the recovery of the (integral) coefficients of the weight-enumerator polynomial of $Ker_L(P)$, which by choice of $P$ may be set to be any appropriately sized linear binary code we please. The recovery of arbitrary weight-enumerator polynomials is $\mathbf{P^{GapP}}$-hard \mathbf{c}ite{lit:vyalyi}.
\mathbf{q}ed
\mathbf{s}ubsection{Background}
There has been a wide range of work into discovering restricted models of quantum computation which \mathbf{e}mph{are} classically simulable. For example, quantum circuits generating limited forms of entanglement, with classical simulations based on analysing matrix product states or contracting tensor networks; these circuits have a particularly constrained `circuit-topology', which leads to their simplicity (see \mathbf{c}ite{lit:Mar05} for a summary of known results). There is no particular circuit-topology imposed in our Z-network architecture (discussed in section~\ref{sect:architectures}), so it seems unlikely that the same methods would apply here.
Other positive classical simulability results include the stabiliser circuits of the Gottesman-Knill theorem and various matchgate constructions (see \mathbf{c}ite{lit:Val02, lit:Jo08, lit:SWVC08, lit:SWVC2} and references therein). These constructions differ significantly from our Z-networks in terms of the underlying algebra, the group generated by the set of allowable gates.
H\o{}yer and Spalek \mathbf{c}ite{lit:Hoy02} have shown that Shor's algorithm for Integer Factorisation can indeed be performed within a \mathbf{e}mph{constant} number of timesteps on a Graph State processor (discussed in section~\ref{sect:architectures}), though their constructions offer no reason to believe that that constant might be smaller than, say, $\mathbf{s}im 100$; and of course, a general methodology for reducing the inherent time-complexity of oracle-dependent quantum search algorithms is known to be impossible, due to lower bounds on Grover's algorithm. Dan Browne \mathbf{c}ite{lit:Browne06} wrote about \mathbf{e}mph{CD-decomposability}, which is the first rigorous treatment that we know of that explicitly links Graph State temporal depth with commutativity of Hamiltonian terms used to simulate a Graph state computation.
Dan Simon \mathbf{c}ite{lit:Si97} wrote about algorithms that use nothing more than an oracle and a Hadamard transform, and which therefore could be described as `temporally unstructured'. However, his notion of `oracle' was one tailored for a universal quantum architecture, being essentially an arbitrarily complex general unitary transformation, and since there is no natural notion of one of these within our `temporally unstructured' paradigm, there is no real sense in which Simon's algorithm can count as an example of an algorithm within the $\mathbf{BPP^{IQP}}$ framework. In particular, Simon's oracle implements a unitary that does \mathbf{e}mph{not} commute with the Hadamard transform.
\mathbf{s}ubsection{Conjectures, implicit and explicit}
It is possible to form various hardness conjectures about the classical simulation of these $\mathbf{IQP}$ probability distributions.
For a randomly chosen X-program $P$ of a given width $n$, it seems likely that the associated $\mathbf{IQP}$ distribution would be exponentially close to flat random. Conditioned on its \mathbf{e}mph{not} being random, there is no particular reason to think it would be approximately efficiently classically samplable.
Here is an example of one such conjecture, though the precise details are not important to our arguments.
\mathbf{b}egin{conjecture} \mathbf{l}abel{conj:sample}
There exists a distribution $\mathcal{D}$ on the set of X-programs, for which no classical Turing machine can gain a non-negligible $\Omega(1/poly)$ advantage in deciding whether or not the distribution associated to an X-program chosen randomly from $\mathcal{D}$ is exponentially close in trace distance to the uniform distribution.
\mathbf{e}nd{conjecture}
This particular hardness conjecture is not quite what we really \mathbf{e}mph{require}, but it gives an example of a plausible conjecture about classical simulation, and implies that for almost any X-program of interest, there is a certain \mathbf{e}mph{event} (subset of output possibilities) whose probability will (probably) be estimated wrongly by your favourite classical polynomial-time event-probability-estimating device.
(Many similar conjectures sound equally plausible, in an area where almost nothing is known for sure.)
The point to emphasise in context of our two-player interactive protocol games is that it is not unreasonable for Alice to \mathbf{e}mph{believe} that Bob can have no classical cheating strategy \mathbf{e}mph{so long as} none such has been published nor proven to exist; and so our protocol may still serve as a demonstration (if not a proof) of a genuinely quantum computing phenomenon, despite the lack of proof of any simulation conjecture.
Another conjecture implicit in Alice's ability to make a fair hypothesis test---so that Bob will indeed have a good chance of passing the test when he does have an $\mathbf{IQP}$ oracle (or approximate version of one), but will stand little chance of faking a proof if relying on guesswork and (known) classical techniques---is one that ensures that Alice's X-programs really do incorporate a non-negligible amount of entropy. Although we have little to go on besides scant simulation evidence from small examples, we want to make a conjecture that collision entropy is close to maximal within at least one relevant family of random constant-action X-programs.
\mathbf{b}egin{conjecture} \mathbf{l}abel{conj:entropy}
The expected collision entropy of the probability distribution of a randomly selected X-program of width $n$, with constant action $\mathbf{p}i/8$, scales as $n - O(1)$ with the size of the program.
\mathbf{e}nd{conjecture}
This conjecture is perhaps not directly relevant to the `hardness' of the $\mathbf{IQP}$ paradigm itself, but merely relevant to our game construction.
Note, for example, that since arbitrary---or random---obfuscation rows are used in the construction of the matrix $P$ in the construction of section~\ref{sect:protocol}, it follows that there will be much about the random variable $\mathbf{X}$ that is arbitrary---or `typical' in some vague sense---to the point that if one were sure that the only structure of significance were the hidden `causal' code $\mathcal{C}_\mathbf{s}$, one could hope to approximate the distribution for $\mathbf{X}$, using knowledge of $\mathbf{s}$, by sampling uniformly at random (no biases) and applying a post-filter to create a bias in direction $\mathbf{s}$ of the required strength. This gives some context for conjecture~\ref{conj:entropy}.
\mathbf{s}ubsection{Classical approximations} \mathbf{l}abel{sect:classical}
Rather than speculate at this stage on which of the very many possible conjectures may or may not be true, we instead turn back to an examination of the mathematical structures underpinning the probability distributions in question.
Suppose we wish to construct a probability distribution that arises from some purely classical methods, which can be used to approximate our $\mathbf{IQP}$ distribution. Our motivation here is to check whether any purported application for an $\mathbf{IQP}$ oracle might not be efficiently implemented without any quantum technology. We proceed using the relatively \mathbf{e}mph{ad hoc} methods of linear differential cryptanalysis.
\mathbf{s}ubsubsection{Directional derivatives}
For the case $\theta = \mathbf{p}i/8$, we will need to consider only second-order derivatives. The same sort of method will apply to the case $\theta = \mathbf{p}i/2^{d+1}$ using $d$th order derivatives, but the presentation would not be improved by considering that general case here.
In terms of a binary matrix/X-program $P$, proceed by defining
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:def_f}
f &:& \mathbbm{F}_2^n ~\rightarrow~ \mathbbm{Z}/16\mathbbm{Z}, \nonumber \\
f(\mathbf{a}) &\mathbf{e}quiv& \mathbf{s}um_{\mathbf{p} \in P} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T} \mathbf{p}mod{16},
\mathbf{e}nd{eqnarray}
and notate discrete directional derivatives as
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:deriv}
f_\mathbf{d}(\mathbf{a}) &\mathbf{e}quiv& f(\mathbf{a}) - f(\mathbf{a}\oplus\mathbf{d}) \mathbf{p}mod{16}.
\mathbf{e}nd{eqnarray}
Consider also the \mathbf{e}mph{second} derivatives of $f$, given by
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:deriv2}
f_{\mathbf{d},\mathbf{e}}(\mathbf{a})
&\mathbf{e}quiv& f_\mathbf{e}(\mathbf{a}) ~-~ f_\mathbf{e}(\mathbf{a}\oplus\mathbf{d}) \mathbf{p}mod{16} \nonumber \\
&\mathbf{e}quiv& 2\mathbf{s}um_{\mathbf{p} \in P_\mathbf{e}} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T}
\mathbf{l}eft( 1 ~-~ (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{d}^T} \right) \mathbf{p}mod{16} \nonumber \\
&\mathbf{e}quiv& 4\!\!\!\!\mathbf{s}um_{\mathbf{p} \in P_\mathbf{d} \mathbf{c}ap P_\mathbf{e}} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T} \mathbf{p}mod{16} \nonumber \\
&\mathbf{e}quiv& 4\!\!\!\!\mathbf{s}um_{\mathbf{p} \in P_\mathbf{d} \mathbf{c}ap P_\mathbf{e}} ~\mathbf{p}rod_{j:p_j=1} \mathcal{B}igl(~ 1 ~-~ 2a_j ~\mathcal{B}igr) \mathbf{p}mod{16} \nonumber \\
&\mathbf{e}quiv& \mathbf{s}um_{\mathbf{p} \in P_\mathbf{d} \mathbf{c}ap P_\mathbf{e}} \mathbf{l}eft(~ 4 ~+~ 8\!\!\!\mathbf{s}um_{j~:~p_j=1} \!\!a_j ~\right) \mathbf{p}mod{16},
\mathbf{e}nd{eqnarray}
each of which is quite patently a linear function in the bits $(a_1, \mathbf{l}dots, a_n)$ of $\mathbf{a}$, as a function with codomain the ring $\mathbbm{Z}/16\mathbbm{Z}$, regardless of the choice of directions $\mathbf{d},\mathbf{e}$.
\mathbf{b}egin{lemma}
With $f$ defined as per line~(\ref{eqn:def_f}), and $\mathbf{X}$ the random variable of lemma~\ref{lem:thing}, for all~$\mathbf{s}$,
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:piby8}
\mathbbm{P}( \mathbf{X} \mathbf{c}dot \mathbf{s}^T = 0 )
&=& \mathbbm{E}_\mathbf{a} \mathbf{l}eft[ \mathbf{c}os^2\mathcal{B}igl(~ \frac\mathbf{p}i8 \mathbf{c}dot f_\mathbf{s}(\mathbf{a}) ~\mathcal{B}igr) \right],
\mathbf{e}nd{eqnarray}
and so the $\mathbf{IQP}$ probability distribution (in the case $\theta=\mathbf{p}i/8$) may be viewed as a function of $f$ rather than as a function of $P$.
\mathbf{e}nd{lemma}
\textbf{Proof~:~}
Starting from the proof of theorem~\ref{thm:bias}, line~(\ref{eqn:wooo}),
\mathbf{b}egin{eqnarray}
\mathbbm{P}(\mathbf{X} \mathbf{c}dot \mathbf{s}^T = 0)
&=& \frac12\mathbf{l}eft(~ 1 ~+~ \mathbbm{E}_\mathbf{a} \mathbf{l}eft[ e^{ i\theta \mathbf{s}um_\mathbf{p} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T} \mathcal{B}igl(1 - (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{s}^T}\mathcal{B}igr) } \right]
~\right) \nonumber \\
&=& \frac12\mathbf{l}eft(~ 1 ~+~ \mathbbm{E}_\mathbf{a} \mathbf{l}eft[ \mathbf{e}xp\mathbf{l}eft( \frac{i\mathbf{p}i}8 \mathbf{b}igl( f(\mathbf{a}) - f(\mathbf{a}\oplus\mathbf{s}) \mathbf{b}igr) \right) \right] ~\right) \nonumber \\
&=& \frac12\mathbf{l}eft(~ 1 ~+~ \mathbbm{E}_\mathbf{a} \mathbf{l}eft[ \mathbf{c}os\mathcal{B}igl(~ \frac\mathbf{p}i8 \mathbf{c}dot f_\mathbf{s}(\mathbf{a}) ~\mathcal{B}igr) \right] ~\right).
\mathbf{e}nd{eqnarray}
The second line above is obtained immediately from the first, using the definition of $f$.
The third line follows because the expression is real-valued.
The conclusion follows from a basic trigonometric identity, and linearity of the expectation operator.
\mathbf{q}ed
And so \mathbf{e}mph{if} there is a hidden $\mathbf{s}$ such that $\mathbbm{P}( \mathbf{X} \mathbf{c}dot \mathbf{s}^T = 0 )=1$, \mathbf{e}mph{then} that implies $f_\mathbf{s}(\mathbf{a}) \mathbf{e}quiv 0 \mathbf{p}mod{16}$ for all $\mathbf{a}$.
This is essentially a non-oracular form of the kind of function that arises in applications of Simon's Algorithm \mathbf{c}ite{lit:Si97}, with $\mathbf{s}$ playing the role of a \mathbf{e}mph{hidden shift}.
One could find linear equations for such an $\mathbf{s}$ if it exists, because it would follow immediately that $f_{\mathbf{d},\mathbf{e}}(\mathbf{s}) = f_{\mathbf{d},\mathbf{e}}(\mathbf{z}ero)$ for any directions $\mathbf{d},\mathbf{e}$, which is by line~(\ref{eqn:deriv}) equivalent with
\mathbf{b}egin{eqnarray}
\mathbf{l}eft( \mathbf{s}um_{\mathbf{p} \in P_\mathbf{d} \mathbf{c}ap P_\mathbf{e}} \!\!\mathbf{p} \right) \mathbf{c}dot \mathbf{s}^T &=& 0.
\mathbf{e}nd{eqnarray}
\mathbf{s}ubsubsection{Classical sampling}
To make use of this specific second-order differential property, we need to analyse the probability distribution that a classical player can generate efficiently from it.
Proceed by defining a new probability distribution for a new random variable $\mathbf{Y}$, as follows~:
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:classprob}
\mathbbm{P}(\mathbf{Y}=\mathbf{y})
&=& \mathbbm{P}_{\mathbf{d},\mathbf{e}} \mathbf{l}eft(~ \mathbf{s}um_{\mathbf{p} \in P_\mathbf{d} \mathbf{c}ap P_\mathbf{e}} \!\!\mathbf{p} ~=~ \mathbf{y} ~\right).
\mathbf{e}nd{eqnarray}
This may be classically rendered, simply by choosing $\mathbf{d},\mathbf{e} \in \mathbbm{F}_2^n$ independently with a uniform distribution, and then returning the sum of all rows in $P$ that are not orthogonal to either $\mathbf{d}$ or~$\mathbf{e}$.
\mathbf{b}egin{lemma} \mathbf{l}abel{lem:classprob}
The classical simulable distribution on the random variable $\mathbf{Y}$ defined in line~(\ref{eqn:classprob}) satisfies
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:classfromcode}
\mathbbm{P}( \mathbf{Y} \mathbf{c}dot \mathbf{s}^T = 0 )
&=& \mathbbm{P}\mathcal{B}igl(~ \mathbf{c}_1^T \mathbf{c}dot \mathbf{c}_2 = 0 ~~|~~ \mathbf{c}_1,\mathbf{c}_2 \mathbf{s}im \mathcal{C}_\mathbf{s} ~\mathcal{B}igr) \\
&=& \frac12\mathbf{l}eft(~ 1 ~+~ 2^{-rank(~ P_\mathbf{s}^T \mathbf{c}dot~ P_\mathbf{s} ~)} ~\right),
\mathbf{e}nd{eqnarray}
and so the bias of $\mathbf{Y}$ in direction $\mathbf{s}$ is a function of the matroid $P_\mathbf{s}$.
\mathbf{e}nd{lemma}
\textbf{Proof~:~}
Starting from line~(\ref{eqn:classprob}),
\mathbf{b}egin{eqnarray}
\mathbbm{P}( \mathbf{Y} \mathbf{c}dot \mathbf{s}^T = 0 )
&=& \mathbf{s}um_{\mathbf{y} ~:~ \mathbf{y} \mathbf{c}dot \mathbf{s}^T = 0} ~~ \mathbbm{P}_{\mathbf{d},\mathbf{e}} \mathbf{l}eft(~ \mathbf{s}um_{\mathbf{p} \in P_\mathbf{d} \mathbf{c}ap P_\mathbf{e}} \!\!\mathbf{p} ~=~ \mathbf{y} ~\right) \\
&=& \mathbbm{P}_{\mathbf{d}, \mathbf{e}} \mathbf{l}eft(~ \mathbf{s}um_{\mathbf{p} \in P_\mathbf{d} \mathbf{c}ap P_\mathbf{e}} \!\!\mathbf{p} \mathbf{c}dot \mathbf{s}^T ~=~ 0 ~\right) \nonumber \\
&=& \mathbbm{P}_{\mathbf{d}, \mathbf{e}} \mathbf{l}eft(~ wt(~ P\mathbf{c}dot\mathbf{d}^T ~\mathbf{w}edge~ P\mathbf{c}dot\mathbf{e}^T ~\mathbf{w}edge~ P\mathbf{c}dot\mathbf{s}^T ~) \mathbf{e}quiv 0 \mathbf{p}mod2 ~\right) \nonumber \\
&=& \mathbbm{P}_{\mathbf{d}, \mathbf{e}} \mathbf{l}eft(~ wt(~ P_\mathbf{s}\mathbf{c}dot\mathbf{d}^T ~\mathbf{w}edge~ P_\mathbf{s}\mathbf{c}dot\mathbf{e}^T ~) \mathbf{e}quiv 0 \mathbf{p}mod2 ~\right) \nonumber \\
&=& \mathbbm{P}_{\mathbf{d}, \mathbf{e}} \mathbf{l}eft(~ \mathbf{d}\mathbf{c}dot P_\mathbf{s}^T \mathbf{c}dot P_\mathbf{s}\mathbf{c}dot\mathbf{e}^T = 0 ~\right). \nonumber
\mathbf{e}nd{eqnarray}
The \mathbf{e}mph{wedge operator} $\mathbf{w}edge$ here denotes the logical \mathbf{e}mph{AND} between binary column-vectors.
The first line of the lemma follows from the obvious substitutions $\mathbf{c}_1 = P_\mathbf{s} \mathbf{c}dot \mathbf{d}^T$, $\mathbf{c}_2 = P_\mathbf{s} \mathbf{c}dot \mathbf{e}^T$.
The second line follows because unimodular actions on the left or right of a quadratic form (such as $(P_\mathbf{s}^T \mathbf{c}dot P_\mathbf{s})$) affect neither its rank nor the probabilities derived from it; so it suffices to consider the cases where it is in Smith Normal Form, \mathbf{e}mph{i.e.} diagonal, which are trivially verified.
Since this expression is patently invariant under invertible linear action on the right and permutation action on the left of $P_\mathbf{s}$, it too is a matroid invariant.
\mathbf{q}ed
\mathbf{s}ubsubsection{Inter-relation}
Thus we have established some kind of correlation between random variables $\mathbf{X}$ and $\mathbf{Y}$.
\mathbf{b}egin{theorem} \mathbf{l}abel{thm:unitbound}
In the established notation, for X-programs with fixed $\theta=\mathbf{p}i/8$,
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:unitbound}
\mathbbm{P}( \mathbf{X} \mathbf{c}dot \mathbf{s}^T = 0 ) = 1
~~&\mathbbm{R}ightarrow&~~ \mathbbm{P}( \mathbf{Y} \mathbf{c}dot \mathbf{s}^T = 0 ) = 1.
\mathbf{e}nd{eqnarray}
\mathbf{e}nd{theorem}
\textbf{Proof~:~}
By theorem~\ref{thm:bias}, the antecedent gives, for all $\mathbf{c} \in \mathcal{C}_\mathbf{s}, ~n_\mathbf{s} \mathbf{e}quiv 2 wt(\mathbf{c}) \mathbf{p}mod8$, where $n_\mathbf{s}$ is again the length of the code $\mathcal{C}_\mathbf{s}$.
This entails that every codeword in $\mathcal{C}_\mathbf{s}$ has the same weight modulo 4, including the null codeword, so $\mathcal{C}_\mathbf{s}$ must be doubly even\footnotemark{}.
\footnotetext{\mathbf{e}mph{Doubly even} just means that every codeword has weight a multiple of 4.}
It is easy to see that doubly even linear codes are self-dual.\footnotemark{}
\footnotetext{\mathbf{e}mph{Self-duality} just means that the dual code---consisting of all words orthogonal to every codeword---is equal to the code.}
By lemma~\ref{lem:classprob}, the consequent is obtained.
\mathbf{q}ed
The only counterexamples to the \mathbf{e}mph{converse} implication seem to occur in the trivial cases whereby the binary matroid $P_\mathbf{s}$ has circuits of length 2, \mathbf{e}mph{i.e.} where $P_\mathbf{s}$ has repeated rows.
Note that if $\mathbbm{P}( \mathbf{X} \mathbf{c}dot \mathbf{s}^T = 0 )$ were equal to 1 precisely, then by making a list of samples from $\mathbf{IQP}$ runs, storing them in a matrix, and performing Gaussian Elimination to recover the kernel of the matrix, it would be straightforward to compute the hidden $\mathbf{s}$. However, theorem~\ref{thm:unitbound} shows that this is exactly the condition required for being able to compute $\mathbf{s}$ via purely classical means. For this reason, it seems hard to find decision languages that plausibly lie in $\mathbf{BPP^{IQP}} \mathbf{b}ackslash \mathbf{BPP}$.
This random variable $\mathbf{Y}$ is the `best classical approximation' that we have been able to find for $\mathbf{X}$.
(The intuition is that it captures all of the `local' information in the function $f$, which is to say all the `local' information in the matroid $P$, so that the only data left unaccounted for and excluded from use within building this classical distribution is the `non-local' matroid information, which is readily available to the quantum distribution via the magic of quantum superposition.)
There seems to be no other sensible way of processing $P$ (or $f$) classically, to obtain useful samples efficiently, though it also seems hard to make any rigorous statement to that effect.
\mathbf{b}egin{conjecture} \mathbf{l}abel{conj:best}
The classical method defined in this section, yielding random variable $\mathbf{Y}$, is asymptotically classically optimal (when comparing worst-case behaviour and restricting to polynomial time) for the simulation of $\mathbf{IQP}$ distributions arising from constant-action $\theta=\mathbf{p}i/8$ X-programs.
\mathbf{e}nd{conjecture}
This conjecture lends credence to the design methodology of section~\ref{sect:protocol}.
\mathbf{s}ubsection{Future work}
We might also recommend the further study of matroid invariants through quantum techniques, or perhaps the invariants of \mathbf{e}mph{weighted} matroids, since they seem to be the natural objects of $\mathbf{IQP}$ computation as hitherto circumscribed.
This would seem to be fertile ground for developing examples of things that only genuine quantum computers can achieve.
Note that if it weren't for the correlation described in theorem~\ref{thm:unitbound}, then it would be possible to conceive of a mechanism whereby an $\mathbf{IQP}$-capable device could compute an actual secret or witness to something (\mathbf{e}mph{e.g.} learn $\mathbf{s}$), so that the computation wouldn't require two rounds of player interaction to achieve something non-trivial. Yet as it stands, it is an open problem to suggest tasks for this paradigm involving no communication nor multi-party concepts.
\mathbf{s}ection{Architectures} \mathbf{l}abel{sect:architectures}
In this section we sketch two more architectures for implementing $\mathbf{IQP}$ oracles. These architectures are probably more physically feasible than X-programs.
\mathbf{s}ubsection{Z-networks}
The network- or circuit-model of computation is perhaps the most familiar one.
Programs for the first architecture we call ``Z-networks'', since the program is most easily described as a network of gates on an array of qubits, where the allowed gate-set includes just the Controlled-Not gate from any qubit to any qubit and the single-qubit gate that implements the Pauli $Z$ Hamiltonian\footnotemark{} for some time.
\footnotetext{$Z = \mathbf{k}etbra00 - \mathbf{k}etbra11$, on a single qubit.}
Although this Z-network architecture \mathbf{e}mph{does} have a notion of temporal structure---because it is important the order in which the gates of the network are carried out---nonetheless it is useful for our analysis because it turns out to have effectively the same computational power as the X-program architecture under some basic assumptions, and the Lie group structure underpinning the kinds of transformation allowable within the Z-network architecture is particularly easy to work with.
On the understanding that $n$ qubits are initialised into $\mathbf{k}et{\mathbf{z}ero}$ in the computational basis and ultimately measured in the computational basis, it is well known that the gate-set consisting of Controlled-Not gates together with \mathbf{e}mph{all single-qubit rotations} is universal for \textbf{BQP}, and the Lie group generated by this gate-set generates the whole of $SU(2^n)$, modulo global phase.
However, by the term ``Z-network'' we mean explicitly to limit \mathbf{e}mph{the single-qubit gates} to being those which implement $e^{i \theta Z}$ for some action $\theta$, so that the Lie group spanned by the gate-set (represented in the computational basis) consists of unitary matrices that are supported by permutation matrices, \mathbf{e}mph{i.e.} those unitaries that have just one non-zero entry per row. Any such unitary can be factored into a diagonal matrix followed by a permutation matrix. (In all cases, global phase is to be regarded as physically irrelevant, and may be `quotiented out' from the groups in question.)
We can describe groups by giving generator sets for them.
The group containing \mathbf{e}mph{all even} permutations and all diagonal elements (modulo global phase) is
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:FullLiegroup}
\mathbf{l}eft<~ \mbox{Toffoli, C-Not, } X, ~e^{i\theta Z} ~\right> &=& \mathbf{l}eft<~ \mbox{any even permutation}, ~\mbox{any diagonal} ~\right>.
\mathbf{e}nd{eqnarray}
This `qualifies' as a Z-network group; indeed, all Z-network groups are to be a subgroup of this one. But for the purposes of comparison with X-programs and the $\mathbf{IQP}$ paradigm, it suffices to consider the much simpler group given by
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:Liegroup}
\mathbf{l}eft<~ \mbox{C-Not, } X, ~e^{i\theta Z} ~\right> &=& \mathbf{l}eft<~ \mbox{any linear permutation}, ~\mbox{any diagonal} ~\right>.
\mathbf{e}nd{eqnarray}
This latter group does not apparently contain (efficiently) the dynamics of classical computation.
(The $X$ gate is necessary to enable the construction of all diagonals, but one might prefer to replace implementation of $X$ by the availability of an ancilla $\mathbf{k}et1$ qubit, so that a C-Not gate can simulate an $X$ gate. In the language of complexity theory, (\ref{eqn:FullLiegroup}) might be said to stand in relation to~(\ref{eqn:Liegroup}) as $\mathbf{P}$ stands to $\mathbf{\mathbf{b}igoplus L}$.)
One can see how to build a variety of constructions within the group at line~(\ref{eqn:Liegroup}), using the specified gate-set. \mathbf{e}mph{E.g.} by conjugating $e^{i\theta Z_1}$ with two C-Not gates one can create an $e^{i\theta Z_1 Z_2}$ composite unitary, and similarly $e^{i\theta Z_1 Z_2 Z_3}$ can be created, \mathbf{e}mph{etc}.
\mathbf{s}ubsubsection{Reductions between Z-networks and X-programs}
This neat mathematical structure (\mathbf{e}mph{i.e.} as at line~(\ref{eqn:Liegroup})) enables probability distributions of the kind at line~(\ref{eqn:dist1}) to be simulated.
\mathbf{b}egin{lemma}
A Z-network can always be designed to simulate an X-program efficiently.
\mathbf{e}nd{lemma}
\textbf{Proof~:~}
Simply by initialising the input qubits to the Z-network to $\mathbf{k}et{+^n}$ in the Hadamard basis, and measuring output in the same Hadamard basis, the simulation of a given X-program, $P$, proceeds by constructing a small network of CNot gates and an $\mathbf{e}xp(i \theta_\mathbf{p} Z)$ gate for each row $\mathbf{p}$ of $P$. Details omitted for brevity.
\mathbf{q}ed
Conversely,
\mathbf{b}egin{lemma}
An X-program can efficiently simulate a given Z-network, provided that that Z-network uses Hadamard-basis input, C-Nots, $X$ gates, and $e^{i\theta Z}$ gates only, and outputs in the Hadamard basis.
\mathbf{e}nd{lemma}
\textbf{Proof~:~}
The required reduction just associates one X-program element $(\theta_\mathbf{p}, \mathbf{p})$ to each $e^{i\theta Z}$ gate, setting $\theta_\mathbf{p} \mathbf{l}eftarrow \theta$ and specifying $\mathbf{p}$ according to the location of the $e^{i\theta Z}$ gate \mathbf{e}mph{and} the totality of C-Not gates to the left of that gate. A final piece of simple post-processing is needed after the measurement phase of the X-program, to account for the C-Nots in the Z-network, but this post-processing simply consists in applying the same C-Nots (with directions reversed) on the classical measurement outcomes. (This is because moving from the Hadamard basis to the computational basis has the effect of reversing the direction of C-Not gates.)
\mathbf{q}ed
The point of these reductions is to highlight the sense in which the group associated to simple Z-networks stands in the same relation to the set of X-programs as the `full' $SU(2^n)$ Lie group stands to the set of proper full-blown quantum algorithms.
\mathbf{s}ubsection{Graph-programs}
Programs for the second of these architectures we call ``Graph-programs'', since the program is most easily described as the construction of a graph state followed by a series of measurements of the qubits in the graph state in various bases \mathbf{c}ite{lit:Browne06}.
Graph state computing architectures are popular candidates for scalable fully universal quantum processors \mathbf{c}ite{lit:Raus01, lit:Raus03}.
Here we are concerned not with universal architectures, but with the appropriate restriction to `unit time' computation. So, unlike universal graph state computation, our Graph-programs do not admit adaptive feed-forward, which is to say that all measurement angles must be known and fixed at compile-time, so that all measurements can be made simultaneously once the graph state has been built. In this sense, the `depth' of a Graph-program is 1.
A graph state has qubits that are initially devoid of information, but which are entangled together according to the pattern of some pre-specified graph.
A graph state can be constructed without inherent temporal complexity, perhaps even prepared in a single computational time-step, because there is no implicit reason requiring one edge of the graph to be prepared before any other.
(It is still fair to argue that the circuit-depth of the process that generates a graph state is linear in the valency of the graph, but that is not a measure of `inherent' temporal complexity.)
We will show how Graph-programs can simulate the output of X-programs if a little trivial classical post-processing of the measurement results is allowed.
A Graph-program is taken to be an undirected (usually bipartite) graph with labelled and distinguished vertices.
The vertex set is denoted $V$, of cardinality $n$, and for each $v \in V$ there is an element of $SU(2)$ labelling it; $R_v \in SU(2)$.
The edge set is denoted $E$.
To implement the program, a qubit is associated with each vertex and is initialised to the state $\mathbf{k}et+$ in the Hadamard basis.
Then a Controlled-$Z$ Pauli gate is applied between each pair of qubits whose vertices are a pair in $E$. Since these Controlled-$Z$ gates commute, they may be applied simultaneously, at least in theory.
Finally, each vertex qubit $v$ is measured in the direction prescribed by its label $R_v$, returning a single classical bit. Clearly the order of measurement doesn't matter, because the measurement direction is \mathbf{e}mph{prescribed} rather than \mathbf{e}mph{adaptive}.
Hence a sample from $\mathbbm{F}_2^n$ (a bit-string) is thus generated as the total measurement result.
Combining this together, we see that the probability distribution for such an output is
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:dist2}
\mathbbm{P}(\mathbf{X}=\mathbf{x})
&=& \mathbf{l}eft| \mathbf{b}ra\mathbf{x} ~\mathbf{p}rod_{v \in V} R_v
\mathbf{c}dot \!\!\!\!\mathbf{p}rod_{(u,v) \in E}\!\!\!\frac{1 + Z_u + Z_v - Z_u Z_v}2
~ \mathbf{k}et{+^n} \right|^2.
\mathbf{e}nd{eqnarray}
Here the measurement has been written using the notation of the computational basis, with an appropriate (passive) rotation immediately prior.
\mathbf{s}ubsubsection{Graph-programs and X-programs}
Having described three architectures, we've indicated that the X-programs, characterised by the formulation at line~(\ref{eqn:dist1}), are in some natural sense the `lowest common denominator' amongst the architectures (and unitary groups) of interest.
\mathbf{b}egin{lemma}
A Graph-program can always be designed to simulate an X-program efficiently.
\mathbf{e}nd{lemma}
\textbf{Proof~:~}
Suppose we're given an X-program, written $\{~ (\theta_\mathbf{p}, \mathbf{p}) ~:~ \mathbf{p} \in P \mathbf{s}ubset \mathbbm{F}_2^n ~\}$. Then it is straightforward to simulate it on a Graph State architecture, as follows.
Let $V$ be the disjoint union of $[1..n]$ and $P$, so that the graph state used to simulate the program will have one \mathbf{e}mph{primal} qubit for each qubit being simulated, plus one \mathbf{e}mph{ancilla} qubit for each program element $\mathbf{p}$, the total cardinality of $V$ being polynomial in $n$, by hypothesis.
Let $(j,\mathbf{p}) \in E$ exactly when the $j$th component of $\mathbf{p}$ is a 1. In this way, the resulting graph is bipartite, linking primal qubits to those ancill\mathbf{a}e{} that they have to do with. Let $R_j$ be the Hadamard element ($H$) for all primal qubits, so that all primal qubits are measured in the Hadamard basis. Let $R_\mathbf{p} = \mathbf{e}xp( i\theta_\mathbf{p} X )$, so that every ancilla qubit is measured in the $(YZ)$-plane at an angle specified by the corresponding program element.
If the resulting Graph-program is executed, it will return a sample vector $\mathbf{x} \in \mathbbm{F}_2^{n+\#P}$ for which the $n$ bits from the primal qubits are correlated with the $\#P$ bits from the ancill\mathbf{a}e{} in a fashion which captures the desired output, (though these two sets separately---marginally---will look like flat random data.) To recover a sample from the desired distribution, we simply apply a classical Controlled-Not gate from each ancilla bit to each neighbouring primal bit, according to $E$, and then discard all the ancilla bits.
One can use simple circuit identities to check that this produces the correct distribution of line~(\ref{eqn:dist1}) precisely.
\mathbf{q}ed
\mathbf{s}ubsection{Physical comparison}
An X-program uses only `one timeslice', but presumes an arbitrarily large gatespan (interaction length), up to $n$, the number of qubits in the X-program.
The Z-network reduction is physically preferable because it uses small gates (gatespan = 2) to simulate an X-program. However, it has network depth that is not constant; rather, it is likely to be quadratic in $n$, in general, unless one is careful to make optimisations when `compiling' the network.
The Graph-program reduction uses more qubits than $n$, but has better depth properties than the Z-network architecture. Again, only small gates are used.
It seems natural to suggest that the `true temporal cost' of implementing a Graph program would either be one process for measurement plus either one process or $degree$ processes for initialising the graph state, where $degree$ denotes the vertex-degree of the underlying graph.
The number of qubits required in the simulation of an X-program is also slightly larger, since ancilla qubits are used.
The worst case in our interactive protocol application would therefore require about $5q/2$ qubits, rather than $q/2$, though the vertex-degree would no bigger than the number of qubits.
The kinds of graph called for in our Graph-program reduction are not the usual cluster state graphs (regular planar lattice arrangement) used in measurement-based quantum computation.
The bipartite graphs described in the reduction will usually be far from planar for generic X-programs, having a relatively high genus. This means that the graph cannot be `laid out' on a plane without edges crossing.
\mathbf{s}ection*{Acknowledgements}
We would like to thank Tobias Osborne, Richard Jozsa, Ashley Montanaro, Dan Browne, Scott Aaronson, and Richard Low for useful discussions and suggestions. We also acknowledge the support of the EC-FP6-STREP network QICS.
\mathbf{s}ection*{Appendix}
A proof of theorem~\ref{thm:bias}.
Throughout, the variable $\mathbf{p}$ ranges over the rows of the binary matrix $P$, which are the program elements of an X-program.
Derive line~(\ref{eqn:walshcode}) from line~(\ref{eqn:dist1}) in the case that the value $\theta$ is constant.
\mathbf{b}egin{eqnarray}
\mathbbm{P}(\mathbf{X}=\mathbf{x})
&=& \mathbf{l}eft| \mathbf{b}ra\mathbf{x} ~\mathbf{e}xp\mathbf{l}eft(~\mathbf{s}um_\mathbf{p} i\theta_\mathbf{p} \mathbf{b}igotimes_{j:p_j=1} X_j~\right)~ \mathbf{k}et{\mathbf{z}ero^n} \right|^2 \nonumber \\
&=& \mathbf{l}eft| 2^{-n}\mathbf{s}um_\mathbf{a} (-1)^{\mathbf{x} \mathbf{c}dot \mathbf{a}^T}\mathbf{b}ra\mathbf{a} ~~\mathbf{e}xp\mathbf{l}eft(~\mathbf{s}um_\mathbf{p} i\theta_\mathbf{p} \mathbf{b}igotimes_{j:p_j=1} Z_j~\right)~~
\mathbf{s}um_\mathbf{b} \mathbf{k}et{\mathbf{b}} \right|^2 \nonumber \\
&=& \mathbf{l}eft| ~\mathbbm{E}_{\mathbf{a}}~ \mathbf{l}eft[ (-1)^{\mathbf{x} \mathbf{c}dot \mathbf{a}^T} ~\mathbf{e}xp\mathbf{l}eft(~i\theta \mathbf{s}um_{\mathbf{p}} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T} ~\right)
\right] ~\right|^2 \nonumber \\
&=& \mathbbm{E}_{\mathbf{a},\mathbf{d}}~ \mathbf{l}eft[ (-1)^{\mathbf{x} \mathbf{c}dot \mathbf{d}^T} ~\mathbf{e}xp\mathbf{l}eft(~ i\theta \mathbf{s}um_{\mathbf{p}} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T}
\mathcal{B}igl(1 - (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{d}^T}\mathcal{B}igr) ~\right) \right].
\mathbf{e}nd{eqnarray}
On the second line we made a change of basis, so as to replace the Pauli $X$ operators with Pauli $Z$ ones.
\mathbf{b}egin{eqnarray} \mathbf{l}abel{eqn:wooo}
\mathbbm{P}(\mathbf{X} \mathbf{c}dot \mathbf{s}^T = 0)
&=& 2^{n} ~\mathbbm{E}_{\mathbf{x}} \mathbf{l}eft[~ \{\mathbf{x}\mathbf{c}dot\mathbf{s}^T=0\} \mathbf{c}dot \mathbbm{P}( \mathbf{X} = \mathbf{x} ) ~\right] \nonumber \\
&=& 2^{n} ~\mathbbm{E}_{\mathbf{a},\mathbf{d},\mathbf{x}}
\mathbf{l}eft[ \frac{(1+(-1)^{\mathbf{x}\mathbf{c}dot\mathbf{s}^T})}2 ~(-1)^{\mathbf{x} \mathbf{c}dot \mathbf{d}^T}
~e^{ i\theta \mathbf{s}um_\mathbf{p} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T} \mathcal{B}igl(1 - (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{d}^T}\mathcal{B}igr) } \right] \nonumber \\
&=& 2^{n} ~\mathbbm{E}_{\mathbf{a},\mathbf{d}}
\mathbf{l}eft[ \frac{\mathcal{B}igl( \{\mathbf{d}=\mathbf{z}ero\} + \{\mathbf{d}=\mathbf{s}\} \mathcal{B}igr)}2
~e^{ i\theta \mathbf{s}um_\mathbf{p} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T} \mathcal{B}igl(1 - (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{d}^T}\mathcal{B}igr) } \right] \nonumber \\
&=& \frac12\mathbf{l}eft( 1 ~+~ \mathbbm{E}_\mathbf{a} \mathbf{l}eft[ e^{ i\theta \mathbf{s}um_\mathbf{p} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T} \mathcal{B}igl(1 - (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{s}^T}\mathcal{B}igr) } \right]
\right).
\mathbf{e}nd{eqnarray}
These transformations are conceptually simple but notationally untidy.
\mathbf{b}egin{eqnarray}
2 \mathbf{c}dot \mathbbm{P}(\mathbf{X} \mathbf{c}dot \mathbf{s}^T = 0) - 1
&=& \mathbf{s}um_j e^{ij\theta} ~\mathbbm{E}_{\mathbf{a},\mathbf{p}hi}
\mathbf{l}eft[ e^{i\mathbf{p}hi\mathbf{l}eft( -j ~+~ \mathbf{s}um_\mathbf{p} (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T} \mathcal{B}igl(1 - (-1)^{\mathbf{p} \mathbf{c}dot \mathbf{s}^T}\mathcal{B}igr) ~\right)} \right] \nonumber \\
&=& \mathbf{s}um_j e^{ij\theta} ~\mathbbm{P}_{\mathbf{a}}
\mathbf{l}eft(~ j ~=~ 2\!\!\!\!\!\!\mathbf{s}um_{\mathbf{p}~:~\mathbf{p}\mathbf{c}dot\mathbf{s}^T=1} \!\!(-1)^{\mathbf{p} \mathbf{c}dot \mathbf{a}^T} ~\right) \nonumber \\
&=& \mathbf{s}um_j e^{ij\theta} ~\mathbbm{P}
\mathbf{l}eft(~ j = 2 (~ n_\mathbf{s} - 2 \mathbf{c}dot wt( \mathbf{c} ) ~) ~~|~~ \mathbf{c} \mathbf{s}im \mathcal{C}_\mathbf{s} ~\right) \nonumber \\
&=& \mathbf{s}um_w \mathbf{c}os(~ 2\theta(n_\mathbf{s} - 2 w) ~) \mathbf{c}dot \mathbbm{P}\mathbf{l}eft(~ w = wt( \mathbf{c} ) ~~|~~ \mathbf{c} \mathbf{s}im \mathcal{C}_\mathbf{s} ~\right).
\mathbf{e}nd{eqnarray}
Here we have used the standard Fourier decomposition of a periodic function, and used the fact that the function is known to be real. The variable substitution at the third line was $\mathbf{c} = P_\mathbf{s} \mathbf{c}dot \mathbf{a}^T$, understood in the correct basis. At the fourth line it was $w = (2n_\mathbf{s}-j)/4$.
\mathbf{b}egin{eqnarray*}
\mathbbm{P}(\mathbf{X} \mathbf{c}dot \mathbf{s}^T = 0)
&=& \mathbf{s}um_{w=0}^{n_\mathbf{s}} \mathbf{c}os^2(~ \theta(n_\mathbf{s} - 2 w) ~) \mathbf{c}dot \mathbbm{P}\mathbf{l}eft(~ w = wt( \mathbf{c} ) ~~|~~ \mathbf{c} \mathbf{s}im \mathcal{C}_\mathbf{s} ~\right) \\
&=& \mathbbm{E}_{\mathbf{c} \mathbf{s}im \mathcal{C}_\mathbf{s}} \mathbf{l}eft[~ \mathbf{c}os^2\mathcal{B}igl(~ \theta( n_\mathbf{s} ~-~ 2 \mathbf{c}dot wt(\mathbf{c}) ) ~\mathcal{B}igr) ~\right]. ~~~~~~~~~~~~~~~~~~~~~~~ \mathbf{q}ed
\mathbf{e}nd{eqnarray*}
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\mathbf{e}nd{document} |
\begin{document}
\title{Speeding Up SMT-Based\\ Quantitative Program Analysis}
\titlerunning{Speeding Up SMT-Based Quantitative Program Analysis}
\author{Daniel J. Fremont \and Sanjit A. Seshia}
\institute{University of California, Berkeley\\\email{[email protected]\\[email protected]}}
\authorrunning{Fremont and Seshia}
\maketitle
\begin{abstract}
Quantitative program analysis involves computing numerical quantities
about individual or collections of program executions. An example of
such a computation is quantitative information flow analysis, where
one estimates the amount of information leaked about secret data
through a program's output channels. Such information can be quantified
in several ways, including channel capacity and (Shannon) entropy. In
this paper, we formalize a class of quantitative analysis problems
defined over a weighted control flow graph of a loop-free program. These
problems can be solved using a combination of path enumeration, SMT solving, and model counting. However, existing methods can
only handle very small programs, primarily because the number of
execution paths can be exponential in the program size. We show how
path explosion can be mitigated in some practical cases by taking
advantage of special branching structure and by novel algorithm
design. We demonstrate our techniques by computing the channel capacities of the
timing side-channels of two programs with extremely large numbers of
paths.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Quantitative program analysis involves computing numerical quantities
that are functions of individual or collections of program
executions. Examples of such problems include computing worst-case or
average-case execution time of programs, and
quantitative information flow, which seeks to compute the amount of
information leaked by a program. Much of the work in this area has
focused on extremal quantitative analysis problems --- that is,
problems of finding worst-case (or best-case) bounds on
quantities. However, several problems involve not just finding
extremal bounds but computing functions over multiple (or all)
executions of a program. One such example, in the general area of
quantitative information flow, is to estimate the entropy or channel
capacity of a program's output channel. These quantitative analysis
problems are computationally more challenging, since the number of
executions (for terminating programs) can be very large, possibly
exponentially many in the program size.
In this paper, we present a formalization and satisfiability modulo
theories (SMT) based solution to a family of quantitative
analysis questions for deterministic, terminating programs.
The formalization is covered in detail in
Section~\ref{sec:model}, but we present some basic intuition here.
This family of problems can be defined over
a weighted graph-based model of the program.
More specifically, considering the
program's control flow graph, one can ascribe weights to nodes or
edges of the graph capturing the quantity of interest (execution time,
number of bits leaked, memory used, etc.) for basic blocks. Then, to obtain the
quantitative measure for a given program path, one sums up the
weights along that path. Furthermore, in order to count the number of
program inputs (and thus executions) corresponding to a program path, one can perform model
counting on the formula encoding the path condition. Finally, to
compute the quantity of interest (such as entropy or channel capacity)
for the overall program, one combines the quantities and model counts
obtained for all program paths using a prescribed formula.
The obvious limitation of the basic approach sketched above is that,
for programs with substantial branching structure, the number of
program paths (and thus, executions) can be exponential in the program size.
We address this problem in the present paper with two ideas.
First, we show how a certain type of ``confluent''
branching structure which often occurs in real programs can be
exploited to gain significant performance enhancements. A common
example of this branching structure is the presence of a conditional
statement inside a for-loop, which leads to $2^N$ paths for $N$ loop
iterations.
In this case, if the branches are proved to be ``independent'' of each
other (by invoking an SMT solver), then one can perform model
counting of individual branch conditions rather than of entire path
conditions, and then cheaply aggregate those model counts.
Secondly, to compute a quantity such as channel capacity, it is not necessary
to derive the entire distribution of values over all paths.
For this case, we give an efficient algorithm to compute all the values attained by a given quantity (e.g. execution time) over all possible paths ---
i.e., the support of the distribution --- which runs in time polynomial in the sizes of the program and the support.
Our algorithmic methods are particularly tuned to the
analysis of timing side-channels in programs.
Specifically, we apply our ideas to computing
the channel capacity of timing side-channels for two standard programs which
have far too many paths for previous techniques to handle.
Our techniques enable the use of SMT methods in a new application, namely quantitative program analyses such as assessing the feasibility of side-channel attacks. While SMT methods are used in other program verification problems with exponentially-large search spaces, na\"ive attempts to use them to compute statistics like those we consider do not circumvent path explosion. The optimizations that form our primary contributions are essential in making feasible the application of SMT to our domain.
To summarize, the main contributions of this paper include:
\begin{itemize}
\item a method for utilizing special branching structure to reduce the number of model counter invocations needed to compute the distribution of a class of quantitative measures from potentially exponential to linear in the size of the program, and
\item an algorithm which exploits this structure to compute the support of such distributions in time polynomial in the size of the program and the support.
\end{itemize}
The rest of the paper is organized as follows. We present background
material and problem definitions in Sec.~\ref{sec:model}. Algorithms
and theoretical results are presented in Sec.~\ref{sec:algos-theory}.
Experimental results are given in Sec.~\ref{sec:expts} and we conclude
in Sec.~\ref{sec:concl}.
\comment{
consumptions, runtimes, memory requirements, etc. as a result of
allowing execution to proceed along paths with different
statements. Understanding these variations is important in a number of
domains, such as performance characterization or assessment of
vulnerability to side-channel attacks. As an example of the latter,
there has been work in quantitative information flow (QIF) on
automatically finding bounds on how much information an adversary can
learn by observing the overall runtime of a program, where the timing
variation is due either to cache accesses \cite{cacheaudit} or control
flow \cite{KB}. In the case of control flow, these techniques
ultimately require enumerating all execution paths of the
program. Since the number of paths can be exponential in the number of
branch points, these methods are only practical for programs with very
few conditionals.
In this paper, we propose two abstract problems which subsume these
types of analyses of resource consumption variation due to control
flow. In general path explosion makes the solving of these problems
difficult, but we
}
\section{Background and Problem Definition}
\label{section-preliminaries}
\label{sec:model}
We present some background material in Sec.~\ref{sec:prelim} and the
formal problem definitions in Sec.~\ref{sec:probdef}.
\subsection{Preliminaries}
\label{sec:prelim}
We assume throughout that we are given a loop-free deterministic program
$F$ whose input is a set of bits $I$. Our running example for $F$ will be the standard algorithm for modular exponentiation by repeated squaring, denoted \texttt{modexp}, where the base and modulus are fixed and the input is the exponent. Usually \texttt{modexp} is written with a loop that iterates once for each bit of the exponent. To make \texttt{modexp} loop-free we unroll its loop, yielding for a 2-bit exponent the program shown on the left of Figure \ref{figure-modexp}. Lines \ref{line-modexp-conditional1}--\ref{line-modexp-endloop1} and \ref{line-modexp-conditional2}--\ref{line-modexp-endloop2} correspond to the two iterations of the loop.
\begin{figure}
\caption{Unrolled pseudocode and CFG for \texttt{modexp}
\label{line-modexp-conditional1}
\label{line-modexp-endloop1}
\label{line-modexp-conditional2}
\label{line-modexp-endloop2}
\label{figure-modexp}
\end{figure}
To describe the execution paths of $F$ we use the formalism introduced by
McCabe~\cite{basis-paths}. Consider the control-flow graph (CFG) of
$F$, where there is a vertex for each basic block, conditionals
having two outgoing edges. For example, since 2-bit \texttt{modexp} has two conditionals, its CFG (shown in Figure \ref{figure-modexp}) has two vertices with outdegree 2. We call such vertices \emph{branch points}, and denote the set of them by $B$. Which edge out of a branch point $b \in B$ is taken depends on the truth of its \emph{branch condition} $\condition{b}$, the condition in the corresponding conditional statement. In Figure \ref{figure-modexp}, the branch condition for the first branch point is $(e \& 1) = 1$: if this holds, then edge $e_3$ is taken, and otherwise edge $e_2$ is taken. We model the finite-precision semantics of programs, variables being represented as bitvectors, so that the branch conditions can be expressed as bitvector SMT formulae. Since these conditions can depend on the result of prior computations (e.g. the second branch condition in Figure \ref{figure-modexp}), the corresponding SMT formulae include constraints encoding how those computations proceed. Then each formula uniquely determines the truth of its branch condition given an assignment to the input bits. When necessary, these formulae can be bit-blasted into propositional SAT formulae for further analysis (e.g. model counting).
For convenience we add a dummy vertex to the CFG which
has an incoming edge from all sink vertices. Since $F$ is
loop-free the CFG is a DAG, and each execution of $F$
corresponds to a simple path from the source to the (now unique)
sink. Given such a path $P$, we write $\branches{P}$ for the set of branch points where $P$ takes the
right of the two outgoing edges, corresponding to making $\condition{b}$ true. If there are $N$ edges then these paths can be viewed as vectors
in $\{0,1\}^N$, where each coordinate specifies whether the
corresponding edge is taken. For example, in Figure \ref{figure-modexp} path \textbf{A} corresponds to the vector $(1,0,1,1,1,1,0,0,1)$ under the given edge labeling. This representation allows us to speak meaningfully about linear combinations of paths, as long as the result is in $\{0,1\}^N$. A \emph{basis} of the set of paths is defined by analogy to vector spaces to be a minimal set of paths from which all paths can be obtained by taking linear combinations. In Figure \ref{figure-modexp}, the paths \textbf{A}, \textbf{B}, and \textbf{C} form a basis, as the only other path through the CFG can be expressed as $\mathbf{A} + \mathbf{B} - \mathbf{C}$.
Now suppose we are given an integer weight for each basic block of
$F$, or equivalently for each vertex of its CFG.\footnote{Note that our formalism and approach can be made
to work with rational weights, but we focus here on applications for
which integer weights suffice.}
We define the \emph{total weight} $\wt{P}$ of an execution
path $P$ of $F$ to be the sum of the weights of all basic
blocks along $P$. Note that we get the same value if the weight of
each vertex is moved to all of its outgoing edges (obviously excluding
the dummy sink), and we sum edge instead of vertex weights --- thus $\wt{\cdot}$ is a linear function.
Since $F$ is deterministic, each input $x \in \{0,1\}^{I}$ triggers a unique
execution path we denote $\path{x}$, and so has a well-defined total
weight $\wt{x} = \wt{\path{x}}$.
\subsection{Problem Definition}
\label{sec:probdef}
We consider in this paper the following problems:
\begin{problem} \label{problem-distribution}
Picking $x \in \{0,1\}^{I}$ uniformly at random, what
is the distribution of $\wt{x}$?
\end{problem}
and the special case:
\begin{problem} \label{problem-values}
What is the support of the distribution of $\wt{x}$, i.e. what is the set $\wt{\{0,1\}^{I}} = \{ \wt{x} \: | \: x \in \{0,1\}^{I} \}$?
\end{problem}
One way to think about these problems is to view the weight of a basic
block as some quantity or resource, say execution time or energy, that the block
consumes when executed. Then Problem \ref{problem-distribution} is to
find the distribution of the total execution time or energy
consumption of the program.
Computing or estimating this distribution is useful in a range of applications
(see~\cite{seshia-acmtecs12}).
We consider here a quantitative information flow (QIF) setting,
with an adversary who tries to recover $x$ from $\wt{x}$. In the
example above, this would be a timing side-channel attack scenario
where the adversary can only observe the total execution time of the
program. Given the distribution of $\wt{x}$, we can compute any of
the standard QIF metrics such as {\em channel capacity} or {\em
Shannon entropy} measuring how much information is leaked
about $x$. For deterministic programs, the
channel capacity\footnote{Sometimes called the
\emph{conditional min-entropy} of $x$ with respect to $\wt{x}$, since
for deterministic programs with a uniform input distribution they are
the same \cite{qif-foundations}.} is simply the (base 2) logarithm of
the number of possible observed values~\cite{qif-foundations}. Thus to
compute the channel capacity we do not need to know the full
distribution of $\wt{x}$, but only how many distinct values it can
take --- hence our isolation of Problem~\ref{problem-values}. As we
will see, this special case can sometimes be solved much more rapidly
than by computing the full distribution.
We note that the general problems above can be applied to a variety of
different types of resources. On platforms where the execution time of
a basic block is constant (i.e. not dependent on the state of the
machine), they can be applied to timing analysis. The weights could
also represent the size of memory allocations, or the number of writes
to a stream or device. For all of these, solving Problems
\ref{problem-distribution} and \ref{problem-values} could be useful
for performance characterization and analysis of side-channel
attacks.
\section{Algorithms and Theoretical Results}
\label{sec:algos-theory}
The simplest approach to Problem \ref{problem-distribution} would be
to execute program $F$ on every $x \in \{0,1\}^{I}$, computing
the total weight of the triggered path and eventually obtaining the
entire map $x \mapsto \wt{x}$. This is obviously impractical when
there are more than a few input bits, and is wasteful because often
many inputs trigger the same execution path. A more refined approach
is to enumerate all execution paths, and for each path compute how
many inputs trigger it. This can be done by expressing the branch
conditions corresponding to the path as a bitvector or propositional formula and
applying a \emph{model counter} \cite{gomes-modelcountbookch2009} (this idea was used in \cite{BKR} to count how many inputs led to a given output, although with a linear integer
arithmetic model counter). If the number of paths is much less than
$2^{\abs{I}}$, as is often the case, this approach can be
significantly more efficient than brute-force input
enumeration. However, as noted above the number of paths can be
exponential in the size of $F$, in which case this approach
requires exponentially-many calls to the model counter and therefore is also
impractical.
A prototypical example of path explosion is our running example \texttt{modexp}. For
an $N$-bit exponent, there are $N$ conditionals, and all possible
combinations of these branches can be taken, so that there are $2^N$
execution paths. This makes model counting each path infeasible, but
observe that the algorithm's branching structure has two special
properties. First, the conditionals are \emph{unnested}: the two paths
leading from each conditional always converge prior to the next
one. Second, the branch conditions are \emph{independent}: they depend
on different bits of the input. Below we show how we can use these
properties to gain greater efficiency, yielding Algorithms
\ref{algorithm-distribution} and \ref{algorithm-values} for Problems
\ref{problem-distribution} and \ref{problem-values} respectively.
\subsection{Unnested Conditionals}
\label{sec:unnested}
If $F$ has no nested conditionals, its CFG has an
``$N$-diamond'' form like that shown in Figure \ref{figure-modexp}
(the number of basic blocks within and between the ``diamonds'' can
vary, of course --- in particular, we do not assume that the ``else'' branch of a conditional is empty, as is the case for \texttt{modexp}). This type of structure naturally arises when
unrolling a loop with a conditional in the body, as indeed is the case
for \texttt{modexp}. Verifying that there are no nested conditionals is
a simple matter of traversing the CFG.
With unnested conditionals, there is a one-to-one correspondence
between execution paths and subsets of $B$, given by $P
\mapsto \branches{P}$. For any $b \in B$, we write
$\basispath{b}$ for the path which takes the left edge at every branch
point except $b$ (i.e. makes every branch condition false except for
that of $b$ --- of course it is possible that no input triggers this
path). We write $B_{\mathrm{none}}$ for the path which always takes the left edge at each branch point. For example, in Figure \ref{figure-modexp} if the conditionals on lines \ref{line-modexp-conditional1} and \ref{line-modexp-conditional2} correspond to branch points $a$ and $b$ respectively, then $\mathbf{A} = \basispath{a}$, $\mathbf{B} = \basispath{b}$, and $\mathbf{C} = B_{\mathrm{none}}$. In general, $B_{\mathrm{none}}$ together with the paths $\basispath{b}$ form a basis for the set of all
paths. In fact, for any path $P$ it is easy to see that
\begin{equation}
\label{equation-path-rep}
P = \left( \sum_{c \in \branches{P}} \basispath{c} \right) - \left( \abs{\branches{P}} - 1 \right) B_{\mathrm{none}} \enspace.
\end{equation}
This representation of paths will be useful momentarily.
\subsection{Independence}
\label{sec:independence}
Recall that an input variable of a Boolean function is a \emph{support
variable} if the function actually depends on it, i.e.~the two
cofactors of the function with respect to the variable are not
equivalent. For each branch point $b \in B$, let
$\support{b} \subseteq I$ be the set of input bits which are support variables of
$\condition{b}$. We make the following definition:
\begin{definition}
Two conditionals $b, c \in B$ are \emph{independent} if $\support{b} \cap \support{c} = \emptyset$.
\end{definition}
Independence simply means that there are no common support variables,
so that the truth of one condition can be set independently of the
truth of the other.
To compute the supports of the branch conditions and check
independence, the simplest method is to iterate through all the input
bits, checking for each one whether the cofactors of the branch
condition with respect to it are inequivalent using an SMT query in
the usual way. This can be substantially streamlined by doing a simple
dependency analysis of the branch condition in the source of
$F$, to determine which input variables are involved in its
computation. Then only input bits which are part of those variables need be
tested (for example, in Figure~\ref{figure-modexp} both branch conditions depend only on the input variable $e$, and if there were other input variables the bits making them up could be ignored). This procedure is outlined as Algorithm
\ref{algorithm-supports}. Note that as indicated in Sec.~\ref{sec:prelim}, the formula $\phi$ computed in line~\ref{supports-smt} encodes the semantics of $F$ so that the truth of $C_b$ (equivalently, the satisfiability of $\phi$) is uniquely determined by an assignment to the input bits. For lack of space, the proofs of Lemma \ref{lemma-algorithm-supports} and the other lemmas in this section are deferred to the Appendix.
\begin{algorithm}
\caption{FindConditionSupports($F$)}
\label{algorithm-supports}
\begin{algorithmic}[1]
\STATE Compute CFG of $F$ and identify branch points $B$
\IF{there are nested conditionals}
\RETURN \texttt{FAILURE}
\mathbb{E}NDIF
\FORALL{$b \in B$}
\STATE $\support{b} \leftarrow \emptyset$ \COMMENT{these are global variables}
\STATE $\phi \leftarrow$ SMT formula representing $\condition{b}$ \label{supports-smt}
\STATE $V \leftarrow$ input bits appearing in $\phi$
\FORALL{$v \in V$}
\IF{the cofactors of $\condition{b}$ w.r.t. $v$ are not equivalent}
\STATE $\support{b} \leftarrow \support{b} \cup \{ v \}$
\mathbb{E}NDIF
\mathbb{E}NDFOR
\mathbb{E}NDFOR
\RETURN \texttt{SUCCESS}
\end{algorithmic}
\end{algorithm}
\begin{lemma}
\label{lemma-algorithm-supports}
Algorithm \ref{algorithm-supports} computes the supports $\support{b}$ correctly, and given an SMT oracle runs in time polynomial in $\abs{F}$ and $\abs{I}$.
\end{lemma}
If all of the conditionals of $F$ are pairwise independent,
then $I$ can be partitioned into the pairwise disjoint sets
$\support{b}$ and the set of remaining bits which we write
$S_{\mathrm{none}}$. For any $b \in B$, the truth of
$\condition{b}$ depends only on the variables in $\support{b}$, and we
denote by $\truecount{b}$ the number of assignments to those variables
which make $\condition{b}$ true. Then we have the following formula
for the probability of a path:
\begin{lemma}
\label{lemma-path-prob}
Picking $i \in \{0,1\}^{I}$ uniformly at random, for any path
$P$, the probability that the path corresponding to input $i$ is $P$
is given by
\begin{equation*}
\Pr \left[ \path{i} = P \right] = \left[ 2^{\abs{S_{\mathrm{none}}}} \left( \prod_{b \in \branches{P}} \truecount{b} \right) \left( \prod_{b \in B \setminus \branches{P}} \left( 2^{\abs{\support{b}}} - \truecount{b} \right) \right) \right] / 2^{\abs{I}} \enspace.
\end{equation*}
\end{lemma}
Lemma \ref{lemma-path-prob} allows us to compute the probability of
any path as a simple product if we know the quantities
$\truecount{b}$. Each of these in turn can be computed with a single
call to a model counter, as done in Algorithm
\ref{algorithm-distribution}.
\begin{algorithm}
\caption{FindWeightDistribution($F, weights$)}
\label{algorithm-distribution}
\begin{algorithmic}[1]
\IF{FindConditionSupports($F$) = \texttt{FAILURE} }
\RETURN \texttt{FAILURE}
\mathbb{E}NDIF
\IF{the sets $\support{b}$ are not pairwise disjoint}
\RETURN \texttt{FAILURE}
\mathbb{E}NDIF
\FORALL{$b \in B$}
\STATE $\truecount{b} \leftarrow$ model count of $\condition{b}$ over the variables in $\support{b}$
\mathbb{E}NDFOR
\STATE $dist \leftarrow$ constant zero function
\FORALL{execution paths $P$}
\STATE $p \leftarrow$ probability of $P$ from Lemma \ref{lemma-path-prob}
\STATE $dist \leftarrow dist [ \wt{P} \mapsto dist(\wt{P}) + p ]$
\mathbb{E}NDFOR
\RETURN $dist$
\end{algorithmic}
\end{algorithm}
\begin{theorem}
Algorithm \ref{algorithm-distribution} correctly solves Problem
\ref{problem-distribution}, and given SMT and model counter oracles
runs in time polynomial in $\abs{F}$, $\abs{I}$, and the
number of execution paths of $F$. The model counter is only
queried $\abs{B}$ times.
\end{theorem}
\begin{proof}
Follows from Lemmas \ref{lemma-algorithm-supports} and \ref{lemma-path-prob}.
\end{proof}
Algorithm \ref{algorithm-distribution} improves on path enumeration by
using one invocation of the model counter per branch point, instead of
one invocation per path. In total the algorithm may still take
exponential time, since we need to compute the product of Lemma
\ref{lemma-path-prob} for each path, but if model counting is
expensive there is a substantial savings.
Further savings are possible if we restrict ourselves to Problem
\ref{problem-values}. For this, we want to compute the possible values of
$\wt{x}$ for all inputs $x$. This is identical to the set of possible
values $\wt{P}$ for all {\em feasible} paths $P$ (the paths
that are executed by some input). Thus,
we do not need to know the probability
associated with each individual path,
but only which paths are feasible and which are not. Lemma \ref{lemma-path-prob} implies that all paths are feasible (unless some $T_b = 0$ or $T_b = 2^{|S_b|}$, corresponding to a conditional which is identically false or true; then $S_b = \emptyset$, so we can detect and eliminate such trivial conditionals), and this leads to
\begin{lemma}
\label{lemma-submultiset-sums}
Let $D$ be the multiset of differences $\wt{\basispath{b}} -
\wt{B_{\mathrm{none}}}$ for $b \in B$. Then the possible values of
$\wt{i}$ over all inputs $i \in \{0,1\}^{I}$ are the possible values of
$\wt{B_{\mathrm{none}}} + D^+$, where $D^+$ is the set of sums of
submultisets of $D$.
\end{lemma}
To use Lemma \ref{lemma-submultiset-sums} to solve Problem
\ref{problem-values}, we must find the set $D^+$. The brute-force
approach of enumerating all submultisets is obviously impractical
unless $D$ is very small. We cannot hope to do better than exponential
time in the worst case\footnote{Although we note that for channel
capacity analysis we only need $\abs{D^+}$ and not $D^+$ itself, and
there could be a faster (potentially even polynomial-time) algorithm
to find this value.}, since $D^+$ can be exponentially larger than
$D$. However, in many practical situations $D^+$ is not too much
larger than $D$. This is because the paths $\basispath{b}$ often have
similar weights, so the variation $V = \max D - \min D$ is small and we can apply the following lemma:
\begin{lemma}
\label{lemma-sumset-size}
If $V = \max D - \min D$, then $\abs{D^+} = O(V \abs{D}^2)$.
\end{lemma}
Small differences between weights are exploited by Algorithm
\ref{algorithm-sums}, which as shown in the Appendix
computes $D^+$ in $O(\abs{D} \abs{D^+})$
time. By Lemma \ref{lemma-sumset-size}, the algorithm's runtime is
$O(|D| \cdot V |D|^2) = O(V \abs{D}^3)$, so it is very efficient when $V$ is small. The
essential idea of the algorithm is to handle one element $x \in D$ at
a time, keeping a list of possible sums found so far sorted so that
updating it with the new sums possible using $x$ is a linear-time
operation. For simplicity we only show how positive $x \in D$ are
handled, but see the analysis in the
Appendix for the general case.
\begin{algorithm}
\caption{SubmultisetSums($D$)}
\label{algorithm-sums}
\begin{algorithmic}[1]
\STATE $sums \leftarrow (0)$ \label{countsums-init}
\FORALL{$x \in D$} \label{outer-loop}
\STATE $newSums \leftarrow (sums[0])$ \label{countsums-newsums-init}
\STATE $i \rightarrow 1$ \COMMENT{index of next element of $sums$ to add to $newSums$}
\FORALL{$y \in sums$} \label{countsums-inner-loop}
\STATE $z \leftarrow x + y$ \label{countsums-z}
\WHILE{$i < \mathsf{len}(sums)$ \AND $sums[i] < z$} \label{countsums-scanloop}
\STATE $newSums.\mathsf{append}(sums[i])$ \label{countsums-oldsum}
\STATE $i \leftarrow i + 1$
\mathbb{E}NDWHILE
\STATE $newSums.\mathsf{append}(z)$ \label{countsums-newsum}
\IF{$i < \mathsf{len}(sums)$ \AND $sums[i] = z$} \label{countsums-duplicate}
\STATE $i \leftarrow i + 1$
\mathbb{E}NDIF
\mathbb{E}NDFOR
\STATE $sums \leftarrow newSums$ \label{countsums-update}
\mathbb{E}NDFOR
\RETURN $sums$ \label{countsums-return}
\end{algorithmic}
\end{algorithm}
Using Algorithm \ref{algorithm-sums} together with Lemma
\ref{lemma-submultiset-sums} gives an efficient algorithm to solve
Problem \ref{problem-values}, outlined as Algorithm
\ref{algorithm-values}. This algorithm has runtime polynomial in the
size of its input and output.
\begin{algorithm}
\caption{FindPossibleWeights($F, weights$)}
\label{algorithm-values}
\begin{algorithmic}[1]
\IF{FindConditionSupports($F$) = \texttt{FAILURE} }
\RETURN \texttt{FAILURE}
\mathbb{E}NDIF
\IF{the sets $\support{b}$ are not pairwise disjoint}
\RETURN \texttt{FAILURE}
\mathbb{E}NDIF
\STATE Eliminate branch points with $\support{b} = \emptyset$ (trivial conditionals)
\STATE $D \leftarrow$ empty multiset
\FORALL{$b \in B$}
\STATE $d \leftarrow \wt{\basispath{b}} - \wt{B_{\mathrm{none}}}$
\STATE $D \leftarrow D \cup \{ d \}$
\mathbb{E}NDFOR
\STATE $D^+ \leftarrow \text{SubmultisetSums}(D)$
\RETURN $\wt{B_{\mathrm{none}}} + D^+$
\end{algorithmic}
\end{algorithm}
\begin{theorem}
Algorithm \ref{algorithm-values} solves Problem \ref{problem-values}
correctly, and given an SMT oracle runs in time polynomial in
$\abs{F}$, $\abs{I}$, and
$\abs{\wt{\{0,1\}^{I}}}$.
\end{theorem}
\begin{proof}
Clear from Lemmas \ref{lemma-algorithm-supports} and
\ref{lemma-submultiset-sums}, and the analysis of Algorithm
\ref{algorithm-sums} (see the Appendix).
\end{proof}
\subsection{More General Program Structure}
As presented above, our algorithms are restricted to loop-free programs which have only unnested, independent conditionals. However, our techniques are still helpful in analyzing a large class of more general programs. Loops with a bounded number of iterations can be unrolled. Unrolling the common program structure consisting of a for-loop with a conditional in the body yields a loop-free program with unnested conditionals. If the conditionals are pairwise independent, as in the \texttt{modexp} example, our methods can be directly applied. If the number of dependent conditionals, say $D$, is nonzero but relatively small, then each of the $2^D$ assignments to these conditionals can be checked for feasibility with an SMT query, and the remaining conditionals can be handled using our algorithms. If many conditionals are dependent then checking all possibilities requires an exponential amount of work, but we can efficiently handle a limited failure of independence. An example where this is the case is the Mersenne Twister example we discuss in Sec.~\ref{sec:expts}, where 2 out of 624 conditionals are dependent. A small level of conditional nesting can be handled in a similar way. In general, when analyzing a program with complex branching structure, our methods can be applied to those regions of the program which satisfy our requirements. Such regions do frequently occur in real-world programs, and thus our techniques are useful in practice.
\section{Experiments}
\label{sec:expts}
As mentioned in Sec.~\ref{section-preliminaries}, Problem
\ref{problem-values} subsumes the computation of the channel capacity
of the timing side-channel on a platform where basic blocks have
constant runtimes. To demonstrate the effectiveness of our techniques,
we use them to compute the timing channel capacities of two real-world
programs on the \emph{PTARM} simulator \cite{ptarm}. The tool
\emph{GameTime}~\cite{seshia-tacas11} was used to generate SMT formulae
representing the programs, and to interface with the simulator to
perform the timing measurements of the basis paths. SMT formulae for
testing cofactor equivalence were generated and solved using \emph{Z3}
\cite{z3}. Model counting was done by using Z3 to convert SMT queries
to propositional formulae, which were then given to the model counter
\emph{Cachet} \cite{cachet}. Raw data from our experiments can be obtained at \url{http://math.berkeley.edu/~dfremont/SMT2014Data/}.
The first program tested was the \texttt{modexp} program already
described above, using a 32-bit exponent. With $2^{32}$ paths,
enumerating and model counting all paths is clearly infeasible. Our
new approach was quite fast: finding the branch supports, model
counting\footnote{We note that for this program, each branch condition
had only a single support variable, and thus we have $\truecount{b} =
1$ automatically without needing to do model counting.}, and running
Algorithm \ref{algorithm-sums} took only a few seconds, yielding a timing channel capacity of just over 8 bits. In fact, although the number of paths is very large, the per-path cost of Algorithm \ref{algorithm-distribution} is so low that we were able to compute \texttt{modexp}'s entire timing distribution with it in 23 hours (effectively analyzing more than 50,000 paths per second). The distribution is shown in Figure \ref{figure-modexp-distribution}.
\begin{figure}
\caption{Timing distribution of 32-bit \texttt{modexp}
\label{figure-modexp-distribution}
\end{figure}
The second program we tested was the state update function of the
widely-used pseudorandom number generator the Mersenne Twister
\cite{mersenne-twister}. We tested an implementation of the most
common variant, \verb|MT19937|, which is available at
\cite{mt-implementation}. On every 624th query to the generator,
\verb|MT19937| performs a nontrivial updating of its internal state,
an array of 624 32-bit integers. We analyzed the program to see how
much information about this state is leaked by the time needed to do
the update. The relevant portion of the code has $2^{624}$ paths and
thus would be completely impossible to analyze using path
enumeration. With our techniques the analysis became feasible: finding
the branch supports took 54 minutes, while Algorithm \ref{algorithm-sums} took only 0.2 seconds because there was a high level of uniformity across the
path timings. The channel capacity was computed to be around 9.3
bits. We note that among the 624 branch conditions there are two which
are not independent. Thus all four truth assignments to these
conditions needed to be checked for feasibility before applying our
techniques to the remaining 622 conditionals.
\section{Conclusions}
\label{sec:concl}
We presented a formalization of certain quantitative program analysis
problems that are defined over a weighted control-flow graph
representation. These problems are concerned with understanding how a
quantitative property of a program is distributed over the space of program
paths, and computing metrics over this distribution.
These computations rely on the ability to solve a set of satisfiability
(SAT/SMT) and model counting problems.
Previous work along these
lines has only been applicable to small programs with very few conditionals,
since it typically depends on enumerating all execution paths and the
number of these can be exponential in the size of the program. We
investigated how in certain situations where the number of paths is
indeed exponential, special branching structure can be exploited to
gain efficiency. When the conditionals are unnested and independent,
we showed how the number of expensive model counting calls can be
reduced to be linear in the size of the program, leaving only a very
fast product computation to be done for each path. Furthermore, a
special case of the general problem, which for example is sufficient
for the computation of side-channel capacities, can be solved avoiding
exponential path enumeration entirely. Finally, we showed the
practicality of our methods by using them to compute the timing
side-channel capacities of two commonly-used programs with very large
numbers of paths.
\appendix
\section{Proofs}
\subsection{Lemmas from Sec. \ref{sec:algos-theory}}
\newtheorem*{lemma:AlgorithmSupports}{Lemma \ref{lemma-algorithm-supports}}
\begin{lemma:AlgorithmSupports}
Algorithm \ref{algorithm-supports} computes the supports $\support{b}$ correctly, and given an SMT oracle runs in time polynomial in $\abs{F}$ and $\abs{I}$.
\end{lemma:AlgorithmSupports}
\begin{proof}
Correctness is obvious. Computation of the CFG can clearly be done in
time linear in $\abs{F}$, and likewise for finding nested
conditionals (say by doing a DFS and keeping track of the nesting
level). Generating SMT representations of the branch conditions and
doing dependency analyses on them can be done in time polynomial in
$\abs{F}$. In the worst-case scenario where every input bit
appears in every branch condition, checking all the cofactor
equivalences requires $\abs{B} \abs{I}$ calls to the
SMT solver (generating the SMT query for a single cofactor equivalence
obviously takes time linear in $\abs{F}$). So the algorithm
runs in time polynomial in $\abs{F}$ and $\abs{I}$.
\end{proof}
\newtheorem*{lemma:PathProb}{Lemma \ref{lemma-path-prob}}
\begin{lemma:PathProb}
Picking $i \in \{0,1\}^{I}$ uniformly at random, for any path
$P$, the probability that the path corresponding to input $i$ is $P$
is given by
\begin{equation*}
\Pr \left[ \path{i} = P \right] = \left[ 2^{\abs{S_{\mathrm{none}}}} \left( \prod_{b \in \branches{P}} \truecount{b} \right) \left( \prod_{b \in B \setminus \branches{P}} \left( 2^{\abs{\support{b}}} - \truecount{b} \right) \right) \right] / 2^{\abs{I}} \enspace.
\end{equation*}
\end{lemma:PathProb}
\begin{proof}
We show that the product in square brackets is the number of $i \in
\{0,1\}^{I}$ such that $\path{i} = P$. Since the conditionals of
$F$ are unnested, $\path{i} = P$ iff $i$ makes $\condition{b}$
true for exactly those $b \in \branches{p}$. To specify $i$ we must
give its values on $S_{\mathrm{none}}$, on the sets $\support{b}$ for $b
\in \branches{p}$, and on the sets $\support{b}$ for $b \in
B \setminus \branches{p}$. On $S_{\mathrm{none}}$ the bits may
have any value, since they do not affect any of the branch conditions,
giving the first factor of the product. On $\support{b}$ for $b \in
\branches{p}$ the bits must be set to make $\condition{b}$ true, and
by definition there are $\truecount{b}$ ways of doing this, giving the
second factor. Finally, on $\support{b}$ for $b \in B
\setminus \branches{p}$ the bits must be set to make $\condition{b}$
false, and there are $2^{\abs{\support{b}}} - \truecount{b}$ ways of
doing this, giving the third factor.
\end{proof}
\newtheorem*{lemma:SubmultisetSums}{Lemma \ref{lemma-submultiset-sums}}
\begin{lemma:SubmultisetSums}
Let $D$ be the multiset of differences $\wt{\basispath{b}} -
\wt{B_{\mathrm{none}}}$ for $b \in B$. Then the possible values of
$\wt{i}$ over all inputs $i \in \{0,1\}^{I}$ are the possible values of
$\wt{B_{\mathrm{none}}} + D^+$, where $D^+$ is the set of sums of
submultisets of $D$.
\end{lemma:SubmultisetSums}
\begin{proof}
By Lemma \ref{lemma-path-prob}, unless there is a ``fake'' branch
point whose condition is identically true or false ($\abs{\support{b}}
= 0$), every path has nonzero probability and is therefore feasible (we can detect and eliminate fake branch points when we
compute the condition supports).
Now for any path $P$, by Equation \ref{equation-path-rep} and
linearity of the weight function we have
\begin{align}
\label{equation-weight}
\wt{P} &= \left( \sum_{c \in \branches{P}} \wt{\basispath{c}} \right) - \left( \abs{\branches{P}} - 1 \right) \wt{B_{\mathrm{none}}} \nonumber \\
&= \wt{B_{\mathrm{none}}} + \sum_{c \in \branches{P}} \left( \wt{\basispath{c}} - \wt{B_{\mathrm{none}}} \right) \enspace.
\end{align}
Since the conditionals of $F$ are unnested, for every $B'
\subseteq B$ there is some path $P$ such that $B' =
\branches{P}$. Thus by Equation \ref{equation-weight} the possible
values of $\wt{P}$ are all sums of elements of $D$ shifted by
$\wt{B_{\mathrm{none}}}$.
\end{proof}
\newtheorem*{lemma:SumsetSize}{Lemma \ref{lemma-sumset-size}}
\begin{lemma:SumsetSize}
If $V = \max D - \min D$, then $\abs{D^+} = O(V \abs{D}^2)$.
\end{lemma:SumsetSize}
\begin{proof}
We may assume $D$ has at least two elements --- list these in
increasing order as $d_1, \dots, d_n$. Letting $\tilde{D}$ be the
multiset of the values $\tilde{d}_i = d_i - d_1$, we have $0 =
\tilde{d}_1 \le \dots \le \tilde{d}_n = V$. Now for any $s \in
\tilde{D}^+$ we have $0 \le s \le \sum_i \tilde{d}_i \le (n-1)V$, and
so $\abs{\tilde{D}^+} \le (n-1)V + 1$. Finally, observe that for any
$0 \le k \le n$ and indices $1 \le i_1 < \dots < i_k \le n$, we have
$d_{i_1} + \dots + d_{i_k} = d'_{i_1} + \dots + d'_{i_k} + k
d_1$. Therefore we have $\abs{D^+} \le \abs{\tilde{D}^+} (n+1) \le
(n+1) \left[ (n-1)V + 1 \right] = O(V \abs{D}^2)$.
\end{proof}
\subsection{Analysis of Algorithm \ref{algorithm-sums}}
\label{section-sums-analysis}
Algorithm \ref{algorithm-sums} can easily be adapted to handle
arbitrary integers by removing any occurrences of 0 in $D$ and altering the inner loop so that when $x < 0$, we
enumerate $sums$ and build up $newSums$ from right to left instead of
from left to right. Since the unmodified algorithm is simpler to state
and slightly faster (having one less conditional in the outer loop),
we restrict our analysis to that case. Note however that if only the
\emph{size} of $D^+$ is needed and not its elements (as when computing
channel capacity), we can apply the following lemma:
\begin{lemma}
If $D$ is a multiset of integers, and $T$ is $D$ with the absolute
value applied to all of its elements, then $\abs{D^+} = \abs{T^+}$.
\end{lemma}
\begin{proof}
We show that if $D = R \cup \{ x \}$ and $\tilde{D} = R \cup \{ -x
\}$, then $D^+ = \tilde{D}^+ + x$. This suffices to prove the general
case, since flipping the signs on all negative elements of $D$ one by
one and applying the above result each time shows that $D^+$ is $T^+$
shifted by some constant.
Take any $y \in D^+$. If $y$ can be written as a sum of elements of
$R$, then letting $y'$ be the same sum plus $-x$ we have $y' \in
\tilde{D}^+$ and thus $y = y' + x \in \tilde{D}^+ + x$. Otherwise, $y$
equals $x$ plus some sum of elements of $R$, and we let $y'$ be the
latter sum. Then $y' \in \tilde{D}^+$, and again $y = y' + x \in
\tilde{D}^+ + x$. So $D^+ \subseteq \tilde{D}^+ + x$. Conversely, take
any $y \in \tilde{D}^+ + x$, so that $y$ is $x$ plus a sum of elements
of $\tilde{D}$. If this sum contains $-x$, then $y$ is just equal to a
sum of elements of $R$, and so is in $D^+$. Otherwise, $y$ is $x$ plus
a sum of elements of $R$, and so again is in $D^+$. Therefore
$\tilde{D}^+ + x \subseteq D^+$, and so $D^+ = \tilde{D}^+ + x$.
\end{proof}
So if we only need $\abs{D^+}$, as a preprocessing step we can take
the absolute value of all elements of $D$ to ensure they are
positive (removing 0), and then apply the unmodified Algorithm
\ref{algorithm-sums}. This was done in our experiments. Now we prove
\begin{theorem}
\label{theorem-algorithm-sums}
Algorithm \ref{algorithm-sums} is correct, and has worst-case runtime $\Theta(\abs{D} \abs{D^+})$.
\end{theorem}
\begin{proof}
We prove that if $sums$ is a list of distinct nonnegative integers
sorted in increasing order, the body of the loop on line
\ref{outer-loop} results in $sums$ being updated to include all
integers of the form $s + x$ for $s$ in $sums$, still in increasing
order and with no duplicates. Since $sums$ is initially set to be the
list with the single element $0$ on line \ref{countsums-init}, it will
follow by induction that on line \ref{countsums-return} the list
$sums$ is $D^+$ sorted in increasing order. So the algorithm returns
$D^+$, and is correct.
For notational convenience we will sometimes refer to $sums$ and
$newSums$ as sets. We need to show that at line
\ref{countsums-update}, $newSums$ lists the set $sums \cup (sums + x)$
in increasing order without duplicates. It is clear from lines
\ref{countsums-newsums-init}, \ref{countsums-oldsum}, and
\ref{countsums-newsum} that $newSums$ is contained in $sums \cup (sums
+ x)$, and from line \ref{countsums-newsum} that $sums + x$ is
contained in $newSums$. From the conditions on lines
\ref{countsums-scanloop} and \ref{countsums-duplicate}, we see that
while $i$ is less than $\mathsf{len}(sums)$, it is only incremented
when $sums[i]$ has been added to $newSums$, either by line
\ref{countsums-oldsum} or by line \ref{countsums-newsum} if $sums[i] =
z$. When $y$ is the last, and thus the unique largest, element of
$sums$, either $z = x + y$ is larger than every element of $sums$ or
equal to $y$ (since $x$ is nonnegative). In either case, the loop at
line \ref{countsums-scanloop} will repeat until $i =
\mathsf{len}(sums)$. Therefore, since $newSums$ starts with $sums[0]$
from line \ref{countsums-newsums-init}, at line \ref{countsums-update}
every element of $sums$ will be in $newSums$, and so $newSums$ lists
the set $sums \cup (sums + x)$. By the conditions on line
\ref{countsums-scanloop}, no value of $z$ is added to $newSums$ unless
all smaller values of $sums$ and $sums + x$ have already been added,
since $sums$ is in increasing order. Furthermore, if $z$ equals some
value in $sums$, say with index $i$, then the check on line
\ref{countsums-duplicate} ensures that $i$ is incremented so that the
value $z$ is only added once to $newSums$ (and since $z \ge x > 0$, we have $z > sums[0]$ and thus the value added to $newSums$ on line \ref{countsums-newsums-init} is not duplicated). Therefore at line
\ref{countsums-update}, $newSums$ is in increasing order and has no
duplicates, as desired.
From our work above, we see that in every iteration of the loop on
line \ref{outer-loop} the variable $i$ is incremented until $i =
\mathsf{len}(sums)$ and no further. Therefore in every such iteration,
the loop on line \ref{countsums-scanloop} takes
$O(\mathsf{len}(sums))$ time in total for all of its iterations. Since
the body of the loop on line \ref{countsums-inner-loop} takes constant
time excluding the loop on line $\ref{countsums-scanloop}$, every
iteration of the loop on line \ref{outer-loop} takes
$O(\mathsf{len}(sums))$ time. In every iteration $\mathsf{len}(sums)$
is bounded above by $\abs{D^+}$, since $sums$ never gets shorter and
after the last iteration has length exactly $\abs{D^+}$. Since there
are exactly $\abs{D}$ iterations of the loop on line \ref{outer-loop},
the entire algorithm runs in $O(\abs{D} \abs{D^+})$ time. If $D$
consists of $n$ copies of 1, it is easy to see that $sums$ grows
linearly from length 1 to length $n+1$, so that the algorithm runs in
$\Omega(n^2) = \Omega(\abs{D} \abs{D^+})$ time.
\end{proof}
\end{document} |
\begin{document}
\footnotetext{(M.B.A, G.E.M., R.V.) IMECC - UNICAMP, Departamento de Matematica, Rua Sérgio Buarque de
Holanda, 651, 13083-970 Campinas-SP, Brazil \\ \textit{E-mail addresses: [email protected], [email protected], [email protected] }}
\title{Chain Recurrence and Positive Shadowing in Linear Dynamics}
\begin{abstract}
We study positive shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even on normed vector spaces. We show that for linear operators there is only one chain recurrent set, and this set is a closed invariant subspace. We prove that every chain transitive linear dynamical system with positive shadowing property is frequently hypercyclic and, as a corollary, we obtain that every positive shadowing hypercyclic linear dynamical system is frequently hypercyclic.
\end{abstract}
\textit{Keywords: Chain recurrence, positive shadowing, frequently hypercyclic, non wandering set.}
\section{Introduction}
Linear Dynamics is the study of linear operators on topological vector spaces. It is relatively simple to describe the dynamical behavior of any linear operator on a finite dimensional vector space. However, when the dimension of the vector space is infinite it turns out that the dynamical behavior of linear operators becomes very rich. To grasp how rich the dynamics can be we mention Feldman's result \cite{feldman} in which he proves that there is a linear operator $T:X \rightarrow X$ on a Banach space $X$ which ``contains'' all topological dynamical systems on compact spaces. More precisely, given a continuous map $f:M \rightarrow M$ on a compact metric space $M$, there is a compact $T$-invariant subset $Y$ in $X$ such that $f$ is conjugate to $T|_Y$.
In linear dynamics one is often interested in the study of the behavior orbits of a system. For instance, one might ask whether a system has a dense orbit. In topological dynamics a point whose orbit is dense is called a \emph{transitive point}, while in linear dynamics we call such a point a \emph{hypercyclic vector}. A system which has a hypercyclic vector is called \textit{hypercyclic system}. Linear Dynamics is not solely influenced by Dynamical Systems, it is, of course, also influence by Functional Analysis. That is why definitions are not necessarily following those typically found in compact dynamics. A final remark of the importance of linear dynamics is the classical Functional Analysis open problem the ''Invariant Subspace Problem" (see \cite{dynamicsoflinearoperators}) which can be formulated in terms of closure of orbits of points, i.e. from a Dynamical System point of view.
There has been a recent effort to use tools that have helped dynamicists understand compact dynamical systems in the context of linear dynamical systems. As examples of such effort we may cite \cite{Messaoudi} and \cite{bernardes2020shadowing} that applied the concept of shadowing and hyperbolicity for linear dynamical systems, we may also cite \cite{brian} that used entropy in the study of translation operators. The main focus of this paper is to use the concepts of positive shadowing and chain recurrence in the study of linear dynamical operators.
We may say that both shadowing and chain recurrence study how the system responds to pseudo trajectories. In words a pseudo trajectory (or chain) is almost a piece of orbit from the system. The difference from an actual orbit is that at each interaction of the dynamics there is a possible offset added to the result. This definition of chain occurs naturally in computational dynamics where given a dynamical system almost any orbit calculated in a computer will be a pseudo orbit, since few computer operations are error free. The formal definitions of chain recurrence and positive shadowing will be given in their respective sections.
In this manuscript $X$ will always be a normed vector space (frequently a Banach space). We will denote by $\mathbb{K}$ the field over $X$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{C}$. The symbol $\mathbb{N}$ denotes the set of natural numbers including $0$, that is, $\mathbb{N}=\{0,1,2,\ldots\}$.
We will now define some of the terms used in this article. Let $Y$ be a topological space and $f: Y \to Y$ a continuous function, $f$ is said to be \textbf{transitive} (or topologically transitive) if, for any pair of non-empty open sets $U, V \subset Y$, there is a natural number $N > 0$ such that $T^N U \cap V \neq \emptyset$. $f$ is said to be \textbf{topologically mixing} if, for any pair of non-empty open sets $U, V \subset Y$, there is $N \in \mathbb{N}$ such that, for every $n > N$, $f^{n}(U) \cap V \neq \emptyset$. It is a consequence of Birkhoff's Transitivity Theorem \cite{Birkhoff} that a linear dynamical system $(X,T)$, with $X$ a separable Banach space and $T : X \to X$ a linear operator, is transitive if, and only if, is hypercyclic.
An operator is frequently hypercyclic if it has a vector whose orbit visits each open set with a positive lower density. Formally, the \textbf{lower density} of a subset of natural numbers $A$ is defined by
\[\underline{dens}(A):=\liminf_{N\rightarrow\infty}\frac{\#(A\cap[1,N])}{N},\] where $\#(B)$ denotes the cardinality of the set $B$. A linear operator $T:X\rightarrow X$ on separable metric space $X$ is \textbf{frequently hypercyclic} if there is $x\in X$ such that
\[\underline{dens}(\{n\in\mathbb{N}\;:\;T^n(x)\in V\})>0\] for every non-empty open subset $V$ of $X$.
Our main result is
\begin{maintheorem} \label{gymstheorem}
Let $X$ be a separable Banach space, and $T : X \to X$ an operator with both chain transitivity and positive shadowing property. Then $T$ is topologically mixing and frequently hypercyclic.
\end{maintheorem}
Since hypercyclic systems are chain transitive, then an immediate consequence of the previous theorem is the following corollary.
\begin{corollary} \label{hyperimpliesfreq.hyper}
Let $X$ be a separable Banach space. If $T : X \to X$ is hypercyclic and has the positive shadowing property then $T$ is frequently hypercyclic and topologically mixing.
\end{corollary}
We also prove
\begin{maintheorem}\label{theo:nonwandering} \label{nonwanderingfreq}
Let $X$ be a normed vector space and $T$ a bounded linear operator on $X$ that has the positive shadowing property. Let $\Omega$ be the non wandering set of $T$. Then the following holds:
\begin{enumerate}
\item $\Omega$ is the chain recurrent set of $T$;
\item $\Omega$ is a closed and invariant subspace of $X$;
\item Suppose further that $X$ is a separable Hilbert space and $T$ is self-adjoint. Then $T|_{\Omega} : \Omega \to \Omega$ is topologically mixing and frequently hypercyclic, in particular either $\Omega=\{0\}$ or $\Omega$ is infinite dimensional.
\end{enumerate}
\end{maintheorem}
In section 2 chain recurrent systems are defined and some elementary new results for the linear dynamical setting are obtained. In section 3 we define the concept of positive shadowing and use its synergy with chain recurrence to obtain the above theorems among other smaller results. The last section brings some open questions that emerged during the elaboration of this text to motivate future research.
\section{The Chain Recurrent Subspace}
Let $(Y,d)$ be a metric space and $f: Y \rightarrow Y$ a continuous function. We say that a finite sequence $\{x_0, x_1,\ldots, x_n\}$ is an \textbf{$\epsilon$-chain} with $\epsilon>0$ if $n\in\mathbb{N}\setminus\{0\}$ and $d(f(x_i),x_{i+1})< \epsilon$ for every $ 0 \leq i < n$. Given two points $x,y\in Y$, we write $x\mathcal{R}y$ if the following holds:
\begin{center}
given $\epsilon>0$ there is an $\epsilon$-chain beginning in $x$ and ending in $y$, $\{x_0=x,x_1,\ldots, x_{n-1},x_n=y\}$, and another beginning in $y$ and ending in $x$, $\{y_0=y, y_1,\ldots, y_{m-1},y_m=x\}$.
\end{center} The set $CR(f)=\{x\in Y\;:\;x\mathcal{R}x\}$ is called \textbf{chain recurrent set}. A point $x\in CR(f)$ is called a \textbf{chain recurrent point}. Restricted to $CR(f)$, the relation $\mathcal{R}$ is an equivalence relation. We will say that a dynamical system $(X,f)$ is \textbf{chain transitive} if $x\mathcal{R} y$ for every $x,y\in Y$.
As an example of chain transitive system one may easily see that the identity operator on any normed vector space is chain transitive. Also any hypercyclic operator on any normed vector space is chain transitive as well. In compact dynamics it is a relevant problem to find how many different chain recurrent classes there are in the space. The next two results state that a linear dynamical system has only one chain recurrent class, which we will simply call the chain recurrent set.
\begin{theorem} \label{spanofchainrecurrent}
Let $X$ be a normed vector space and $T$ be a bounded linear operator acting on $X$. If $x \in X$ is chain recurrent, then every point of span$[x]$ is chain recurrent and span$[x]$ is contained in only one recurrent class.
\end{theorem}
\begin{proof}
Since $x$ is chain recurrent, given $\eps> 0$ there is an $\eps$-chain \[\{x_0=x, x_1, \ldots, x_n=x\},\] that is, $\|Tx_0 - x_1 \| < \eps,\ldots,\| T x_{n-1} - x \| < \eps$. One can readily verify that if $|\lambda| \in (0,1]$
then the finite sequence $\{\lambda x_0, \lambda x_1, \ldots, \lambda x_n\}$ is a finite $\eps$-chain that begins and ends in $\lambda x$. Therefore $\lambda x$ is chain recurrent for any $|\lambda| \in (0,1]$. We know that $0$ is always chain recurrent, since $T(0)=0$, therefore $\lambda x$ is chain recurrent for $|\lambda| \in [0,1]$.
The case $|\lambda| > 1$ is analogous: given $\eps > 0 $ we know that there is an $\eps/ |\lambda|$-chain starting and ending in $x$, given by $\{x_0, x_1, \ldots ,x_n\}$, it is not hard to see that the chain given by $\{\lambda x_0, \lambda x_1, \ldots , \lambda x_n\}$ is an $\eps$-chain starting and ending in $\lambda x$.
We have proved above that every point of the span$[x]$ is chain recurrent. It remains to prove that span$[x]$ belongs to just one recurrent class. Given $\lambda \in \mathbb{K}$ and $\eps>0$, we want to find an $\eps$-chain that begins in $x$ and ends in $\lambda x$. Let $k \in \mathbb{N}$ be such that $\displaystyle\dfrac{\|x-\lambda x\|}{k}<\frac{\eps}{2}$. For each $j\in\{0,\ldots, k\}$, consider
$$
\displaystyle x^j=\left(1-\frac{j}{k}\right)x+\frac{j}{k}\lambda x \in \mbox{span}[x].
$$
We proved above that $x^j$ is chain recurrent. Now we can choose for each $j\in\{0,1,\ldots, k-1\}$ an $\eps/2$-chain $\{x_0^j=x^j, x_1^j,\ldots, x_{{n_j}-1}^j, x_{n_j}^j=x^j\}$. The sequence $$\{x_0^0=x, x_1^0,\ldots, x_{n_0-1}^0, x_0^1=x^1, x_1^1,\ldots, x_{n_1-1}^1,\ldots, x_0^{k-1}=x^{k-1},x^{k-1}_1,\ldots, x_{n_{k-1}-1}^{k-1}, x^k=\lambda x\}$$ is an $\eps$-chain from $x$ to $\lambda x$. Indeed, it is enough to show that $$\|Tx_{n_j-1}^j-x^{j+1}\|<\eps,\;\forall\;j\in\{0,1,\ldots, k-1\}.$$ This follows from the fact that
$$\|x^j-x^{j+1}\| = \left\|\left(1-\frac{j}{k}\right)x+\frac{j}{k}\lambda x - \left(1-\frac{j+1}{k}\right)x-\frac{j+1}{k}\lambda x\right\| = \frac{1}{k}\|x-\lambda x\|<\frac{\eps}{2}$$ and
$$\|Tx_{n_j-1}^j-x^{j+1}\|\leq \|Tx_{n_j-1}^j - x^j\|+\|x^j-x^{j+1}\|<\dfrac{\eps}{2}+\dfrac{\eps}{2}= \eps,$$ for all $j\in\{0,\ldots k-1\}$.
We now need to do the inverse path, go from $\lambda x$ to $x$. We saw that $\lambda x$ is chain recurrent. Then, if $\lambda\neq 0$ we can apply the previous argument to create an $\eps$-chain from $y=\lambda x$ to $\lambda'y=x$ with $\lambda'=\frac{1}{\lambda}$. If $\lambda=0$, that is, $\lambda x=0$, consider ${\lambda'}\neq 0$ such that $\|{\lambda'}x\|\leq\epsilon/2$ and use the previous argument to create an $\epsilon/2$-chain, $\{x_0=\lambda'x,x_1,\ldots, x_{n-1}, x_n= x\}$ which begins in $\lambda'x$ and ends in $x$. This yields an $\eps$-chain, $\{0,x_0=\lambda'x,x_1,\ldots, x_{n}=x\}$, starting in the origin and ending in $x$.
\end{proof}
\begin{corollary} \label{onerecurrenceclass}
If $X$ is a normed vector space and $T$ is a bounded linear operator acting on $X$, then $T$ has only one chain recurrent class.
\end{corollary}
\noindent \textbf{Proof:} It is clear that $0$ is chain recurrent for any linear operator $T$, therefore the chain recurrent set of $T$ is non empty. Now, due to the previous result, every chain recurrent class has the origin of $X$ in common. Since these chain recurrent classes have a common point they are the same.
\qed
In view of Corollary \ref{onerecurrenceclass}, we will refer to the chain recurrent class of an operator as the chain recurrent set. This corollary says that a bounded linear operator is chain transitive if every point of the space is chain recurrent. Notice that the chain recurrent set of any operator is non-empty since the origin is always contained in this set.
Let $Y$ be a set and $f : Y \to Y$ a function. A subset $A$ of $Y$ is \textbf{invariant} for $f$ if $f(A) \subset A$. The next corollary tell us that the chain recurrent set is a closed invariant subspace.
\begin{corollary} \label{chainspace}
The chain recurrent set is a closed and invariant subspace. If $T$ is invertible, $T(CR(T))= CR(T)$.
\end{corollary}
\noindent \textbf{Proof: } Theorem \ref{spanofchainrecurrent} tell us that $CR(T)$ is closed under scalar multiplication. Let $x,y \in CR(T)$, $\epsilon > 0$ and $\{x_0=x,x_1,\ldots,x_n=0\}$, $\{y_0=y,y_1,\ldots,y_m=0\}$ be two $\epsilon/2$-chains that go from $x$ to $0$ and from $y$ to $0$ respectively. We may suppose $m>n$ so $$\{x+y=x_0+y_0,x_1+y_1, \ldots , x_n + y_n, 0 + y_{n+1} , \ldots ,0 + y_{m} = 0\}$$ is an $\epsilon$-chain that connects $x+y$ to $0$. A similar idea may be used to show that one can go from zero to $x+y$ with an $\epsilon$-chain. Therefore $x+y$ is chain recurrent.
The fact that the chain recurrent set is closed and invariant is true for topological dynamical systems in general. But we include the proof of these facts for the reader's convenience.
To show that $CR(T)$ is closed consider $\{x_n\}$ a sequence in $CR(T)$ converging to a point $x$ in $X$. Given $\epsilon > 0$ choose $x_n$ such that $\|x_n - x\| < \epsilon/2$ and $\|Tx_n-Tx\|<\eps/2$ and let $\{y_0=x_n,y_1,\ldots,y_N=x_n\}$ be an $\epsilon/2$-chain that goes from $x_n$ to $x_n$. It is immediate to see that $\{x,y_1,\ldots, y_{N-1},x\}$ is an $\epsilon$-chain that goes from $x$ to $x$. Therefore, $x\in CR(T)$.
We now prove the invariance. Given $x \in CR(T)$, there is an $\epsilon/(2 \max \{\|T\|,1\})$-chain, $\{x_0=x,x_1,\ldots,x_N=x\}$, that goes from $x$ to $x$. Define the chain $\{y_0=Tx, y_1=x_2 , y_2=x_3 , \ldots, y_{N-1} = x_N =x,y_N=Tx\}$. The only difficult step is to prove that $\|Ty_0 - y_1\|$ is smaller than $\eps$. But we have that
$$\begin{array}{rcl}
\|Ty_0 - y_1\|&=&\|Ty_0 - x_2\|\\
\\
&=&\|T^2 x -Tx_1 + Tx_1 - x_2\|\\
\\
&\leq& \|T^2 x -Tx_1\| + \|Tx_1 - x_2\|\\
\\
&\leq&\|T\| \, \|T x -x_1 \| + \|Tx_1 - x_2\|\\
\\
&<&\epsilon.
\end{array}
$$
Suppose now that $T$ is invertible. To show that $T(CR(T))=CR(T)$ it is enough to prove that $T^{-1}(CR(T))\subset CR(T)$ since it is already proven that $T(CR(T))\subset CR(T)$. Let $x\in CR(T)$ and $\eps>0$. Then, there is an $\epsilon/(2 \max \{\|T^{-1}\|,1\})$-chain, $\{x_0=x,x_1,\ldots,x_N=x\}$, that goes from $x$ to $x$. The finite sequence $\{y_0=T^{-1}x, y_1=x_0=x , y_2=x_1 , \ldots, y_{N-1} = x_{N-2}, y_{N}=T^{-1}x\}$ is an $\eps$-chain from $T^{-1}x$ to $T^{-1}x$, since
\[\begin{array}{rcl}
\|T(y_{N-1})-y_N\| & = &\|T(x_{N-2})-T^{-1}(x)\|\\
\\
& \leq &\|T(x_{N-2})-x_{N-1}\|+\|x_{N-1}-T^{-1}(x)\|\\
\\
&\leq&\|T(x_{N-2})-x_{N-1}\|+\|T^{-1}\|\|T(x_{N-1})-x\|\\
\\
&<&\eps.
\end{array}\]
\qed
The next propositions give examples of operators that are chain transitive and operators that are not.
\begin{proposition} \label{unitaryimplieschaintransitivity}
Let $X$ be a normed vector space and $T:X \to X$ a bounded linear operator which is a surjective isometry (or equivalently, T is invertible and $\|T^{-1}\| = \|T\| = 1$), then $T$ is chain transitive. In particular, if $X$ is an inner product space, every unitary operator $T:X \to X$ is chain transitive.
\end{proposition}
\noindent \textbf{Proof:} Let $x\in X$ and $\epsilon > 0$. We will show that $x$ is a chain recurrent point. Choose $n \in \mathbb{N}$ such that $\dfrac{\|x\|}{n} < \dfrac{\epsilon}{2}$. Define the sequence
$$\begin{array}{rcl}
x_0 & = & x,\\
\\
x_1& = &T(x) + \dfrac{T^{-n+1}(x)}{n} - \dfrac{T(x)}{n},\\
\\
x_2 & = &T(x_1) + \dfrac{T^{-n+2}(x)}{n} - \dfrac{T^2(x)}{n} \;\;=\;\; T^2(x) + \dfrac{2T^{-n+2}(x)}{n} - \dfrac{2T^2(x)}{n},\\
&\vdots&\\
x_k &= &T(x_{k-1}) + \dfrac{T^{-n+k}(x)}{n} - \dfrac{T^k(x)}{n} \;\;=\;\; T^k(x) + \dfrac{kT^{-n+k}(x)}{n}- \dfrac{kT^k(x)}{n},\\
\end{array}$$
for every $1 \leq k \leq n$. Notice that $x_n = x$, and that
$$
\|x_k - T(x_{k-1})\| = \left|\left| \dfrac{T^{-n+k}(x)}{n} - \dfrac{T^k(x)}{n}\right|\right| \leq \left| \left| \dfrac{T^{-n+k}(x)}{n} \right| \right| +\left| \left| \dfrac{T^{k}(x)}{n} \right| \right| < \epsilon
$$
for $1 \leq k \leq n$. Therefore, $\{x_0,x_1,\ldots,x_n\}$ is an $\eps$-chain which begins and ends in $x$. This means that $x$ is a chain recurrent point.
\qed
If $X$ is a normed vector space, a map $T: X \to X$ is \textbf{recurrent} if, for every $x \in X$, and for every open set $U \subset X$, with $x \in U$, there is some $n \in \mathbb{N} \setminus \{0\}$, such that $T^n(U) \cap U \neq \emptyset$. Clearly any recurrent operator is also chain transitive, but there are operators that are chain transitive and not recurrent. Indeed let $X = \ell_p(\mathbb{Z})$ for $1 \leq p \leq \infty$ and $T: X \to X$ be the shift $T(e_i) = e_{i-1}$. Then by the above proposition $T$ is chain transitive. For $x = e_0$ there is no $n>0$ such that $T^n(B(x,1/2)) \cap B(x,1/2) \neq \emptyset$, therefore $T$ is not recurrent.
Let $X$ be a normed vector space. We say that a bounded linear operator $T: X \to X$ is a \textbf{proper contraction} if $\|T\| < 1$ and a \textbf{contraction} if $\|T\| \leq 1$. We say that $T$ is a \textbf{proper dilation} if $T$ is invertible and $\|T^{-1}\|<1$ and a \textbf{dilatation} if $\|T^{-1}\| \leq 1$. An operator $T$ on a Banach space $B$ is said to be \textbf{hyperbolic} \cite{hyperbolic} if there is a splitting
$$
B = B_s \oplus B_u, \hspace{1cm} T = T_s \oplus T_u,
$$
where $B_s$ and $B_u$ are closed $T$-invariant linear subspaces of $B$, $T_s = T |_{B_{s}}$ is a proper contraction, and $T_u = T |_{B_u}$ is a proper dilation. It is common in the literature to assume that hyperbolic operators are invertible, but in this manuscript such assumption is not needed and therefore we do not assume it.
\begin{proposition}\label{CRtrivial}
Let $X$ be a normed vector space and $T : X \to X$ a linear operator. If $T$ is a proper contraction, then the chain recurrent set of $T$ is the origin.
\end{proposition}
\noindent \textbf{Proof: } Recall that $0$ is always in the chain recurrent set. Let $x\neq 0$, define $\epsilon = (\|x\| - \|Tx\|-\delta)(1-\|T\|)$ and choose a small enough $\delta >0$ such that $\eps > 0$. Consider $\{x_0=x,x_1,\ldots,x_N\}$ an $\epsilon$-chain that starts in $x$. Thus, we have that
$$\begin{array}{rcl}
\|x_{N}\| &=&
\|x_{N} + (Tx_{N-1} - Tx_{N-1}) + \cdots + (T^{N-1}x_1 - T^{N-1}x_1) + (T^{N}x_0 - T^{N}x_0) \|\\
\\
&=&\|(x_{N} - Tx_{N-1}) + T(x_{N-1} - Tx_{N-2}) + \cdots + T^{N-1}(x_1 - Tx) + T^{N}x\|\\
\\
&\leq & \|(x_{N} - Tx_{N-1})\| + \|T\| \, \|x_{N-1} - Tx_{N-2}\| + \cdots + \|T^{N-1}\| \, \|x_1 - Tx\| + \|T^{N}x\|\\
\\
&\leq&\|T^N x\| + \dfrac{\epsilon}{1-\|T\|}\\
\\
&=&\|x\|-(\|Tx\|-\|T^Nx\|+\delta)\\
\\
&<& \|x\|,
\end{array}
$$
since $\|T^Nx\|\leq\|Tx\|$. Therefore there is no $\epsilon$-chain that starts in $x$ and finishes in $x$.
\qed
\begin{lemma} \label{refereelemma}
If $T$ is an invertible operator on the normed space $X$, then $CR(T)=CR(T^{-1})$.
\end{lemma}
\noindent \textbf{Proof:}
Let $x\in CR(T)$, $\epsilon>0$ and $\{x_0=x,\ldots, x_n=x\}$ an $\epsilon/\|T^{-1}\|$-chain for $T$. It is then straightforward that $\{y_0=x, y_1=x_{n-1},\ldots, y_{n-1}=x_1,y_n=x\}$ is an $\epsilon$-chain for $T^{-1}$ from $x$ to $x$. This means that $CR(T)\subset CR(T^{-1})$ and interchanging the roles of $T$ and $T^{-1}$ we get the conclusion.
\qed
\begin{corollary} \label{properdilation}
If $T$ is a proper dilation, then $CR(T)=\{0\}$.
\end{corollary}
\noindent \textbf{Proof}
It follows immediately from the above lemma and Proposition \ref{CRtrivial}.
\qed
The next theorem is interesting in itself and will provide two corollaries. Corollary \ref{decompositionchainrecurrentset} will aid us to prove that $CR(T)=\{0\}$ when $T$ is hyperbolic. In \cite{fabricio} it is proved something weaker, that $T$ is not chain recurrent when $T$ is hyperbolic. Corollary \ref{chainrecurrentselfadjoint} will aid in the proof of Theorem B.
\begin{theorem}\label{CRprojection}
Let $T$ be a bounded operator on a Banach space $X$ such that $X=M\oplus N$ where $M,N$ are $T$-invariant closed subspaces of $X$. Then $CR(T)\cap M=CR(T|_M)$.
\end{theorem}
\noindent \textbf{Proof:} It is clear that $CR(T|_{M})\subset CR(T) \cap M$. We will show that $CR(T)\cap M\subset CR(T|_{M})$. By Theorem 2.10 of \cite{Brezis}, there is $\alpha>0$ such that \begin{equation}\label{brezisequation}\|m\|\leq\alpha\|m+n\| \mbox{ and }\|n\|\leq\alpha\|m+n\|\end{equation} whenever $m\in M$ and $n\in N$. Given $x\in CR(T)\cap M$ and $\eps>0$ there is an $\eps/\alpha$-chain, $\{x,x_1,x_2,\ldots,x_{n-1},x\}$, from $x$ to $x$. For each $i\in\{1,2,\ldots,n-1\}$ there are $z_i\in M$ and $w_i\in N$ such that $x_i=z_i+w_i$. We shall show that $\{x,z_1,z_2,\ldots,z_{n-1},x\}$ is an $\eps$-chain in $M$ from $x$ to $x$, which will complete the proof. Since $M$ and $N$ are $T$-invariant and $X=M\oplus N$, by (\ref{brezisequation}) we have that
\[\begin{array}{l}
\|Tx-z_1\|\leq \alpha\|Tx-(z_1+w_1)\|<\alpha\dfrac{\eps}{\alpha}=\eps,\\
\\
\|Tz_i-z_{i+1}\|\leq \alpha\|T(z_i+w_i)-(z_{i+1}+w_{i+1})\|<\alpha\dfrac{\eps}{\alpha}=\eps,\;\;\forall\; i\in\{1,2,\ldots,n-2\}, and\\
\\
\|Tz_{n-1}-x\|\leq \alpha\|T(z_{n-1}+w_{n-1})-x\|<\alpha\dfrac{\eps}{\alpha}=\eps.
\end{array}\] This guarantees that $\{x,z_1,z_2,\ldots,z_{n-1},x\}$ is an $\eps$-chain in $M$ beginning and ending in $x$, that is, $x\in CR(T|_{M})$.
\qed
\begin{corollary}\label{decompositionchainrecurrentset}
Let $T$ be a bounded operator on a Banach space $X$. Suppose that $X = M \oplus N$, where $M$ and $N$ are closed $T$-invariant subspaces of $X$. Then $CR(T)=CR(T|_M)\oplus CR(T|_N)$.
\end{corollary}
\noindent \textbf{Proof:} Proposition \ref{chainspace} and the fact that $M$ and $N$ are closed and invariant subspaces of $X$ give us that $CR(T) \cap M$ and $CR(T) \cap N$ are closed invariant subspaces of $X$ and therefore of $CR(T)$. We know that every element $x \in CR(T)$ can be written in a unique manner as $x=m+n$, with $m \in M$ and $n \in N$. Following a similar reasoning as in the proof of Theorem \ref{CRprojection}, we have that both $m$ and $n$ are chain recurrent. Therefore $m \in CR(T) \cap M$ and $n \in CR(T) \cap N$, since $x$ is arbitrary, this implies that
\begin{equation} \label{projecaodoCR}
CR(T) = (CR(T) \cap M) \oplus (CR(T) \cap N) .
\end{equation}
By Theorem \ref{CRprojection} the above expression may be written as
$$
CR(T) = CR(T|_M) \oplus CR(T|_N).
$$
\qed
\begin{corollary}\label{chainrecurrentselfadjoint}
Let $T$ be a self-adjoint bounded operator on a Hilbert space $X$. Then the chain recurrent set of $T|_{CR(T)}$ coincides with $CR(T)$. In particular, $T|_{CR(T)}$ is chain transitive.
\end{corollary}
\noindent \textbf{Proof:} Since $X$ is a Hilbert space and $CR(T)$ is a closed space, then $X=CR(T)\oplus (CR(T))^\perp$. We have already seen that $CR(T)$ is invariant for $T$ which implies that $(CR(T))^\perp$ is invariant for $T^*=T$. By Theorem \ref{CRprojection} $CR(T|_{CR(T)}) = CR(T)$.
\qed
The following results are immediate consequences of Corollary \ref{decompositionchainrecurrentset}.
\begin{corollary}
Let $T$ be a hyperbolic operator on a Banach space $X$ then $CR(T) = \{0\}$.
\end{corollary}
\noindent \textbf{Proof:} It follows immediately from the corollaries \ref{decompositionchainrecurrentset}, \ref{properdilation} and proposition \ref{CRtrivial}.
\qed
\begin{corollary}\label{CRforsubspaces}
Let $T$ be an operator on a Banach space $X$. Suppose that $X = M \oplus N$, where $M$ and $N$ are closed $T$-invariant subspaces of $X$. Then $T$ is chain transitive if, and only if, $T|_M$ and $T|_N$ are both chain transitive.
\end{corollary}
The following example shows that $CR(T)$ can be non-trivial.
\begin{example} \label{refereeexample}
Let $X$ be a non-trivial normed space, $T:X\rightarrow X$ a proper contraction and $I:X\rightarrow X$ the identity operator. The operator
\[T\times I:X\times X\rightarrow X\times X,\] where $X\times X$ is endowed with any of the typical product norms, satisfies that $CR(T\times I)=\{0\}\times X$.
\end{example}
The next result, which is used in the last section of this paper, asserts that chain transitivity is preserved under Cartesian product. This is an obvious consequence of Corollary \ref{decompositionchainrecurrentset} when one assumes that the spaces are Banach.
\begin{proposition} \label{multiplechainrecurrence}
Let $T_1, T_2, ..., T_k$ be bounded chain transitive operators on normed vector spaces $X_1, X_2,..., X_k$ respectively. Then the product $T_1 \times ... \times T_k :X_1\times ... \times X_k \rightarrow X_1 \times ... \times X_k$ is chain transitive.
\end{proposition}
\noindent \textbf{Proof:} Consider any of the typical product norms in the product space. We will prove the case when $k=2$, the proof of the general case follows by induction. Let $(x,y) \in X_1 \times X_2$. Given $\eps>0$, consider an $\eps$-chain $\{x_0=x,x_1,\ldots,x_n=0\}$ that connects $x$ with $0$. We can see that \[\{(x_0,y),(x_1,T_2(y)),(x_2,{T_2}^2(y)),\ldots,(x_n, {T_2}^n(y))\}\] is an $\eps$-chain that connects $(x,y)$ with $(0,{T_2}^n(y))$. Following the same reasoning we are able to create an $\eps$-chain that connects $(0,{T_2}^n(y))$ with $(0,0)$.
To go from $(0,0)$ to $(x,y)$ we consider $\eps/2$-chains $\{x_0=0,x_1,\ldots,x_n=x\}$ in $X_1$ and $\{y_{0}=0,y_1,\ldots,y_m=y\}$ in $X_2$ that go from $0$ to $x$ and from $0$ to $y$ respectively. If $n=m$, the sequence $\{(x_0,y_0), (x_1,y_1),\ldots, (x_n,y_n)\}$ is an $\eps$-chain beginning in $(0,0)$ and ending in $(x,y)$. Assuming $n>m$, then the finite sequence $$\{(0,0), (x_1,0), (x_2,0), \ldots,(x_{n-m},0),(x_{n-m+1},y_{1}), (x_{n-m+2},y_{2}), \ldots, (x_n,y_m) = (x,y)\}$$ is an $\eps$-chain connecting $(0,0)$ to $(x,y)$.
One can see that this finite sequence is an $\eps$-chain that connects $(0,0)$ with $(x,y)$.
\qed
\section{Chain Recurrence and Positive Shadowing}
Let $(Y,d)$ be a metric space, $f : Y \to Y$ a continuous function, $\delta>0$ and $\{x_n\}_{n \in \mathbb{N}}$ a sequence in $Y$. We say that $\{x_n\}_{n \in \mathbb{N}}$ is a \textbf{positive $\delta$-pseudo orbit} of $f$ if $d(f(x_n),x_{n+1}) \leq \delta$ for all $n \in \mathbb{N}$. The function $f$ have the \textbf{positive shadowing property} if for each $\epsilon > 0$ there is $\delta > 0$ such that every $\delta$-pseudo orbit is $\epsilon$-shadowed by an $x \in Y$, i.e., there is $x \in Y$ such that
$$
d(x_n,f^{n}(x)) < \epsilon \, \, \text{ for all }n \in \mathbb{N}.
$$
It is easy to see that the identity map does not have the shadowing property, on the other hand it is also easy to see that proper contractions and proper dilations have the shadowing property. More generally, any hyperbolic operator on a Banach space has the shadowing property \cite{Messaoudi}. In \cite{bernardes2020shadowing} one necessary and sufficient condition for the weighted shift to have the shadowing property is obtained. To illustrate the positive shadowing property we present two examples bellow. The first one is an operator that has an eigenvalue equal to $1$ (and therefore it is not hyperbolic) and has positive shadowing. The second example is an invertible contraction that does not have the positive shadowing property.
\begin{example} \label{eigen1andshadowing}
In the space $\ell_p(\mathbb{N})$, for $1 \leq p \leq \infty$, consider the operator $T: \ell_p(\mathbb{N}) \to \ell_p(\mathbb{N})$ given by $T(e_0) = e_0$, and $T(e_i) = 2 e_{i - 1}$ if $i \geq 1$, where $\{e_i\}_{i\in\mathbb{N}}$ is the canonical basis. Note that there is an eigenspace associated with the eigenvalue $1$. This operator has positive shadowing. To see this consider the operator $Se_{i} = e_{i+1}/2$ for every $i \in \mathbb{N}$. Note that $\|S\|=1/2$ and that $S$ is a right inverse for $T$. Given any $\delta$-pseudo orbit $\{x_n\}_{n \in \mathbb{N}}$ note that the point
$$
x = x_0 + S(x_1 - Tx_0) + S^{2}(x_2 - Tx_1) + \cdots
$$
is in $\ell_p(\mathbb{N})$ and $2 \delta$-shadows the pseudo-orbit. Therefore $T$ has positive shadowing.
It is not hard to see that $T$ is transitive and therefore is chain transitive when $1 \leq p < \infty$. Indeed let $U$ and $V$ be two non-empty open subsets of $\ell_{p}(\mathbb{N})$. Since sequences with a finite number of non null elements are dense in $\ell_{p}(\mathbb{N})$ let $x=(x_0,x_1,\ldots,x_n,0,0,\ldots) \in U$ and $y=(y_0,y_1,\ldots,y_m,0,0,\ldots) \in V$. Consider
$$
r = \sum_{i=0}^{n} x_i 2^{i}
$$
and choose $k,l \in \mathbb{N}$ such that
$$
z = (
\underbrace{
\underbrace{x_0,x_1,\ldots,x_n,0,0,\ldots, 0 , -\dfrac{r}{2^k}}_{k \text{ positions}}, 0, \ldots, 0, \dfrac{y_{0}}{2^{l}}}_{l \text{ positions}}, \dfrac{y_{1}}{2^{l+1}}, \ldots, \dfrac{y_{m}}{2^{l+m}},0, 0, \ldots ) \in U.
$$
Hence $T^lz = y$ and so $T^{l} U \cap V \neq \emptyset$, therefore $T$ is transitive. Since $T$ has positive shadowing and is transitive, then Theorem A gives us that $T$ is topologically mixing and frequently hypercyclic.
\end{example}
\begin{example}
Consider the space of real Lebesgue integrable functions in $[1/2,1]$, $L_1([1/2,1])$, and the operator $T : L_1([1/2,1]) \to L_1([1/2,1])$, given by $T(f)(x)=f(x)x$ for every $x \in [1/2,1]$. Notice that $T$ is invertible. We will prove that this operator does not have the shadowing property, that is, for each $\delta>0$ there is a $\delta$-pseudo orbit which cannot be $\eps$-shadowed for any $\eps>0$. Given $\delta>0$ consider the sequence $\{f_n^\delta\}_{n\in\mathbb{N}}$ where
\[f_n^\delta(x)=\delta(x^{n-1}+x^{n-2}+\cdots+x+1).\] This sequence is a $\delta$-pseudo orbit, since
\[\begin{array}{rcl}\|T(f_n^\delta)-f_{n+1}^\delta\|_1 & = & \|\delta(x^{n-1}+x^{n-2}+\cdots+x+1)x - \delta(x^n+x^{n-1}+\cdots+x+1)\|_1\\
\\
&=&\|\delta(x^{n}+x^{n-1}+\cdots+x^2+x) - \delta(x^n+x^{n-1}+\cdots+x+1)\|_1\\
\\
&=& \|\delta\|_1 \;=\;\displaystyle\int_{1/2}^1\delta dx\;=\;\dfrac{\delta}{2}\;<\;\delta.
\end{array}\]
Let us show that $\{f_n^\delta\}$ previously defined cannot be $\eps$-shadowed for any $\eps>0$. Indeed, for any $g\in L^1([1/2,1])$ we have that
$$\begin{array}{rcl}
\|T^{n}g - f_n^{\delta}\|_1& = &
\|x^n g(x) - \delta (x^{n-1} + \cdots + 1)\|_1\\
\\
&\geq&| \, \|x^ng(x) \|_1 - \|\delta (x^{n-1} + \cdots + 1)\|_1 \, |.
\end{array}
$$
Notice that since $x<1$ almost surely, then $x^n g(x) \to 0$ almost surely as $n \to \infty$. Therefore, by the Dominated Convergence Theorem, we have that $\|x^ng(x) \|_1 \to 0$ as $n \to \infty$ and so this term is bounded. The second term in the above sum is not bounded. Indeed,
$$\begin{array}{rcl}
\displaystyle\|\delta (x^{n-1} + \cdots + 1)\|_1 &=& \displaystyle\delta \int_{1/2}^{1} x^{n-1} + \cdots + 1dx\\
\\
&= &\displaystyle
\delta \left[ \dfrac{x^n}{n} + \dfrac{x^{n-1}}{n-1} + \cdots + x \right]_{1/2}^1\\
\\
&= &\displaystyle\delta \left( \dfrac{1}{n} + \dfrac{1}{n-1} + \cdots + 1 - \dfrac{1}{2^n} \dfrac{1}{n} - \dfrac{1}{2^{n-1}} \dfrac{1}{n-1} - \cdots - \dfrac{1}{2} \right)\\
\\
&=& \displaystyle \delta \sum_{k=1}^{n} \dfrac{1}{k}\left( 1 - \dfrac{1}{2^k} \right),
\end{array}
$$
using a comparison test with the harmonic series, for instance the limit comparison test \cite{limit}, one can readily see that the right side goes to $+\infty$ as $n \to + \infty$. This ensures that $\|T^ng-f_n\|_1\rightarrow\infty$, therefore the $\delta$-pseudo orbit $\{f_n^\delta\}_{n\in\mathbb{N}}$ cannot be $\eps$-shadowed for any $\eps>0$.
\end{example}
It turns out that chain transitivity and positive shadowing have an interesting synergy that allows us to obtain the most important results of this paper. Chain transitivity allows us to connect different points of the space with $\epsilon$-chains and the shadowing property tell us that there will be a point that shadows such chain.
\begin{theorem} \label{positiveshadowingmixing}
Let $X$ be a normed vector space, and $T : X \to X$ a bounded linear operator with both chain transitive and positive shadowing property. Then $T$ is topologically mixing.
\end{theorem}
\noindent \textbf{Proof:} Let $U$, $V$ be non-empty open subsets of $X$. Let $x$ be a vector in $U$, and $y$ a vector in $V$ and let $\lambda > 0$ be such that $B(x,\lambda) \subset U $ and $B(y,\lambda) \subset V$. Let $\epsilon = \lambda/2$ and let $\delta>0$ be associated with this $\epsilon$ from the positive shadowing property. Since $T$ is chain transitive, there is a $\delta$-chain that goes from $x$ to the origin, $\{x_0,x_1,\ldots,x_n\}$, and a $\delta$-chain that goes from the origin to $y$, $\{y_0,y_1,\ldots,y_m\}$. Then we have that
$$
\{x, x_1, x_2, \ldots, x_n, \underbrace{0, 0, \ldots ,0}_{\text{total of } k \text{ zeroes}} ,y_1, \ldots, y_{m-1}, y, Ty, T^2y, \ldots\}
$$
is a positive $\delta$-pseudo orbit for every $k \in \mathbb{N}$, and therefore there is a $z_k \in X$ that $\epsilon$-shadows such pseudo orbit. Notice that $z_k \in U$ and $T^{n+m+k}z_k \in V$ for every $k \in \mathbb{N}$, hence $T^{n+m+k}U \cap V \neq \emptyset$ for every $k \in \mathbb{N}$.
\qed
The above Theorem together with Proposition \ref{multiplechainrecurrence} provide the following corollary.
\begin{corollary}
If $T_i : X_i \to X_i$ for $1\leq i \leq n$ is a finite family of linear dynamical systems such that, for each $i$, $(T_i,X_i)$ satisfies the hypothesis of the Theorem \ref{positiveshadowingmixing}. Then $T_1 \times \ldots \times T_n : X_1 \times \ldots X_n \to X_1 \times \ldots X_n$ is topologically mixing.
\end{corollary}
The next lemma allows us to obtain subsets of $\mathbb{N}$ that are big enough to have positive lower density and at the same time are spread enough apart. This lemma is crucial in the proof of our main result, Theorem \ref{gymstheorem}.
\begin{lemma}\cite[Lemma 6.19]{dynamicsoflinearoperators}\label{freqhypercycliclemma}
Let $\{N_p\}_{p \geq 1}$ be any sequence of positive real numbers. Then one can find a sequence $\{\Delta_p\}$ of pairwise disjoint subsets of $\mathbb{N}$ such that
\begin{enumerate}
\item Each set $\Delta_p$ has positive lower density;
\item $\min(\Delta_p) \geq N_p$ and $|n-m| \geq N_p + N_q$; whenever $n \neq m$ and $(n,m) \in \Delta_p \times \Delta_q$.
\end{enumerate}
\end{lemma}
\noindent \textbf{Proof of Theorem \ref{gymstheorem}:}
The conclusion of being topologically mixing is a direct consequence of Theorem \ref{positiveshadowingmixing}, we only need to prove that $T$ is frequently hypercyclic. Let $\{x_p\}_{p \in\mathbb{N}}$ be a countable dense set of vectors in $X$. For every $p \in \mathbb N$, let $\epsilon_p = 1/2^p$. By the positive shadowing property for each $p\in\mathbb{N}$ there is a $\delta_p>0$ such that every $\delta_p$-pseudo orbit is $\epsilon_p$ shadowed by some point of $X$.
For each $x_p$ in the dense countable set let $N_p$ be the size of an $\delta_p/2$-chain that connects $0$ to $x_p$ and then to $0$ again (by the size of a chain we mean its cardinality). By completing with zeroes, we may suppose $\{N_p\}_{p \in \mathbb{N}}$ is a strictly increasing sequence of even numbers of the form $N_p=2R_p$, with $R_p\in\mathbb{N}\setminus\{0\}$ and ${1}/{R_p}<{\delta_p}/{4}$ for every $p\in\mathbb{N}$. We may also suppose that it takes half of the chain to reach $x_p$ from $0$ and other half to go from $x_p$ to $0$. Formally, our chain has the general form
\begin{equation} \label{z0pseudoorbit}
\{x_0^p=0,x_1^p, \ldots, x_{R_p}^p= x_p,\ldots, x_{N_p-1}^p = 0\}.
\end{equation}
For the sequence $\{N_p=2R_p\}_{p\in\mathbb{N}}$, we associate a sequence of subsets $\{\Delta_p\}_{p\in\mathbb{N}}$ of $\mathbb{N}$ given by the Lemma \ref{freqhypercycliclemma}.
Bellow we describe an induction procedure to define a sequence of vectors $\{z_p\}_{p\in\mathbb{N}}$ of $X$ satisfying the following properties
\begin{itemize}
\item[(a)] $\|z_p\|<\dfrac{1}{2^p}$ for every $p \in \mathbb{N}$;
\item[(b)] if $m \in \Delta_p$ then $\displaystyle \left\|\sum_{0\leq q\leq p}T^{m+R_p}(z_q)-x_p\right\|<\frac{1}{2^p}$ for every $p \in \mathbb{N}$;
\item[(c)] given $n,p \in \mathbb{N}$ with $n \notin \{ m, m+1, \ldots, m+ N_p-1: m \in \Delta_p\}$, then $\| T^{n} (z_p) \| < \dfrac{1}{2^p}$.
\end{itemize}
\noindent Since we assume $T^0 z_p=z_p$, the reader will notice that Property (a) is a consequence of Property (c), but we decided to make Property (a) explicit for the sake of clarity. Assume for now that such a sequence $\{z_p\}_{p \in \mathbb{N}}$ exists. In this case the vector \[z=\sum_{p\in\mathbb{N}}z_p\] is well defined, since Property (a) implies $\|z\|=\|\sum z_p\|\leq\sum1/2^p=2$. We claim that $z$ is a frequently hypercyclic vector for $T$. Indeed, let $V$ be a non-empty open subset of $X$. Consider $w\in V$ and $\lambda>0$ such that $B(w,\lambda)\subset V$, and let $q_0\in\mathbb{N}$ be such that $1/2^{q_0}<\lambda/2.$ By the density of $\{x_p\}_{p\in\mathbb{N}}$, we can choose $x_p\in B(w,\lambda/2)$ with $p> q_0$. For each $m\in \Delta_p$, we have by Properties (b), (c) and item 2 of Lemma \ref{freqhypercycliclemma} that
$$
\|T^{m+R_p}(z)-x_p\| \leq \left\|\sum_{0\leq q\leq p}T^{m+R_p}(z_q)-x_p\right\|+\left\|\sum_{q>p}T^{m+R_p}(z_q)\right\| \leq
$$
$$
\frac{1}{2^p} + \sum_{q>p} \left\|T^{m+R_p}(z_q)\right\| \leq
\frac{1}{2^p}+\frac{1}{2^{p+1}}+\frac{1}{2^{p+2}}+\cdots = \frac{1}{2^{p-1}} < \dfrac{\lambda}{2}.
$$
Thus $T^{m+R_p}(z)\in B(x_p,\lambda/2)\subset V$ for all $m\in\Delta_p$, that is,
\[\underline{\emph{dens}}(\{n\in\mathbb{N}\;:\;T^n(z)\in V\})\geq \underline{\emph{dens}}(\Delta_p+R_p){=}\underline{dens}(\Delta_p)>0,\] where
$\Delta_p+R_p=\{m+R_p\;:\;m\in\Delta_p\}$ (see \cite{lowerdensity} for equality above). Proving, therefore, that $z$ is a frequently hypercyclic vector for $T$ and consequently $T$ is frequently hypercyclic.
We now obtain the vectors $z_p$ with Properties (a), (b) and (c). We use an induction procedure, as such the first step is to obtain $z_0$. For this end, consider the sequence $\{\gamma_n^0\}_{n\in\mathbb{N}}$ given by
\[\gamma_n^0=\left\{\begin{array}{ll}
x_k^{0} & \mbox{if } n=m+k \mbox{ with } m\in\Delta_0 \mbox{ and } k\in\{0,1,\ldots, N_0-1\}\\
0 &\mbox{if } n\in\mathbb{N}\setminus\{m,m+1,\ldots,m+N_0-1\;:\;m\in\Delta_0\}.
\end{array}\right.\] By the definition of \eqref{z0pseudoorbit} we have that $\{\gamma_n^0\}_{n\in\mathbb{N}}$ is a $\delta_0$-pseudo orbit. Thus the positive shadowing property guarantees the existence of $z_0\in X$ such that
\[\|T^n(z_0)-\gamma_n^0\|<\eps_0, \mbox{ for every }n\in\mathbb{N}.\] This immediately implies that $\|z_0\|=\|T^0(z_0)-0\|=\|T^0(z_0)-\gamma_0^0\|<\eps_0$, and therefore $z_0$ satisfies Property (a). Notice that for each $m\in\Delta_0$, \[\|T^{m+R_0}(z_0)-\gamma_{m+R_0}^0\|=\|T^{m+R_0}(z_0)-x_{R_0}^0\|=\|T^{m+R_0}(z_0)-x_0\|<\eps_0\] which implies that $z_0$ satisfies Property (b). For every $n\in\mathbb{N}\setminus\{m,\ldots, m+N_0-1\;:\;m\in\Delta_0\}$, $\|T^n(z_0)-0\|=\|T^n(z_0)\|<\eps_0\leq 1$ and therefore $z_0$ satisfies Property (c).
As part of the induction procedure we will now assume that we have $z_1,z_2,\ldots,z_{p-1}$ and will now obtain $z_p$, $p\geq1$. For this end, define the sequence
$\{\gamma_n^p\}_{n\in\mathbb{N}}$ as follows:
\[\gamma_n^p=\left\{\begin{array}{ll}
x_k^{p}-\dfrac{k}{R_p}T^{n}(z_0+z_1+\ldots+z_{p-1})& \mbox{if } n=m+k \mbox{ with } m\in\Delta_p\\
& \mbox{and } k\in\{0,1,\ldots,R_p\}\\
x_k^{p}- \dfrac{N_p-k}{R_p}T^n(z_0+z_1+\ldots+z_{p-1})& \mbox{if } n=m+k \mbox{ with } m\in\Delta_p\\
&\mbox{and } k\in\{R_p+1,\ldots, N_p-1\}\\
\\
0& \mbox{if }n\in\mathbb{N}\setminus\{m,\ldots, m+N_p-1\}\\
&\mbox{and } m\in\Delta_p.\\
\end{array}\right.\]
We shall see bellow that $\{\gamma_n^p\}_{n\in\mathbb{N}}$ is a $\delta_p$-pseudo orbit. This will guarantee the existence of a $z_p$ which $\eps_p$-shadows $\{\gamma_n^p\}$, that is,
\[\|T^n(z_p)-\gamma_n^p\|<\eps_p=\frac{1}{2^p}.\]
\noindent Since $\gamma_0^p=0$ follows that $\|z_p\|=\|T^0(z_p)-0\|<\eps_p=1/2^p$. Thus, $z_p$ satisfies the Property (a).
\noindent Notice that with $z_p$ obtained this way we have, for each $m\in\Delta_p$, that \[\|T^{m+R_p}(z_0+z_1+\ldots+z_p)-x_p\|<\eps_p=\frac{1}{2^p}.\] Indeed, since $z_p$ shadows $\{\gamma_{n}^p\}$
\[\begin{array}{rcl}
\epsilon_p>\|T^{m+R_p}(z_p)-\gamma_{m+R_p}^p\|&=&\|T^{m+R_p}(z_p)-x_p+T^{m+R_p}(z_0+\ldots+z_{p-1})\|\\
&=&\|T^{m+R_p}(z_0+\ldots+z_p)-x_p\|.\\
\end{array}\]
Therefore, Property (b) is assured. Since $\gamma_{n}^p=0$ if $n \notin \{ m, m+1, \ldots, m+ N_p-1: m \in \Delta_p\}$ we have Property (c).
The last step remaining is to show that $\{\gamma_n^p\}_{n\in\mathbb{N}}$ is a $\delta_p$-pseudo orbit. First notice that from the definitions of $\Delta_p$ and $\Delta_q$ and item (2) from Lemma \ref{freqhypercycliclemma} follows that $\{k,\ldots, k+N_p\;:\;k\in\Delta_p\} \cap \{m,\ldots, m+N_q\;:\;m\in\Delta_q\} = \emptyset$ whenever $p\neq q$. This fact and Property (c) imply that if $n \in \{m,\ldots, m+N_p-1\;:\;m\in\Delta_p\}$, then
\begin{equation}\label{soma}
\|T^{n}(z_0+z_1+\ldots+z_{p-1})\|=\left\|\sum_{0\leq q<p}T^n(z_q)\right\|\leq \sum_{0\leq q<p}\|T^n(z_q)\|\leq\sum_{0\leq q<p}\frac{1}{2^q}<2.
\end{equation} It is obvious that if $n,n+1\in\mathbb{N}\setminus\{m,\ldots,m+N_p-1\;:\;m\in\Delta_p\}$, then \[\|T(\gamma_n^p)-\gamma_{n+1}^p\|=0<\delta_p.\] If $n$ or $n+1$ belongs to $\{m,\ldots, m+N_p-1\;:\;m\in\Delta_p\}$, we have $5$ possibilities:
\noindent\emph{Case 1:} $n\in\mathbb{N}\setminus\{m,\ldots, m+N_p-1\;:\;m\in\Delta_p\}$ and
$n+1\in\Delta_p$.
In this case, we have
\[\gamma_n^p=0 \mbox{ and }\gamma_{n+1}^p=x_0^p-\frac{0}{R_p}T^{n+1}(z_0+\cdots+z_{p-1})=0.\]
Hence,
\[\|T(\gamma_n^p)-\gamma_{n+1}^p\|=0<\delta_p.\]
\noindent\emph{Case 2:} $n=m+k$ with $m\in\Delta_p$ and $k\in\{0,\ldots, R_p-1\}.$
In this case, we have \[\gamma_n^p=x_k^{p}-\dfrac{k}{R_p}T^n(z_0+\cdots+z_{p-1}) \mbox{ and } \gamma_{n+1}^p=x_{k+1}^p-\frac{k+1}{R_p}T^{n+1}(z_0+\cdots+z_{p-1}).\] Then,
\[\begin{array}{rcl}
\|T(\gamma_n^p)-\gamma_{n+1}^p\| &=&\displaystyle \left\|T(x_k^{p}) - \dfrac{k}{R_p}T(T^n(z_0+\cdots+z_{p-1}))\right. -\\
\\
&&\displaystyle \;\;\;- \left.\left(x_{k+1}^p-\dfrac{k+1}{R_p}T^{n+1}(z_0+\cdots+z_{p-1})\right)\right\|\\
\\
&\leq &\displaystyle \|T(x_k^{p})-x_{k+1}^p\|+\frac{1}{R_p}\|T^{n+1}(z_0+\cdots+z_{p-1})\|\\
\\
&<&\dfrac{\delta_p}{2}+\dfrac{\delta_p}{4}\cdot 2\\
\\
& < &\delta_p.
\end{array}\] The second inequality is assured by Equation (\ref{soma}).
\noindent\emph{Case 3:} $n=m+R_p$ with $m\in\Delta_p$.
In this case, \[\gamma_n^p=x_{R_p}^p-\dfrac{R_p}{R_p}T^n(z_0+\cdots+z_{p-1})=x_p-T^n(z_0+\cdots+z_{p-1}) \mbox{ and }\]\[\gamma_{n+1}^p = x_{R_p+1}^p-\dfrac{N_p-(R_p+1)}{R_p}T^{n+1}(z_0+\cdots+z_{p-1}).\] Thus,
\[\begin{array}{rcl}
\|T(\gamma_n^p)-\gamma_{n+1}^p\| &=& \displaystyle\left\|T(x_p)-T^{n+1}(z_0+\cdots+z_{p-1})-x_{R_p+1}^p+\frac{R_p-1}{R_p}T^{n+1}(z_0+\cdots+z_{p-1})\right\|\\
\\
&\leq&\|T(x_p)-x_{R_p+1}^p\|+\dfrac{1}{R_p}\|T^{n+1}(z_0+\cdots+z_{p-1})\|\\
\\
&<&\dfrac{\delta_p}2+\dfrac{\delta_p}{4}\cdot 2\;\;\;\;\;\;\;\;\mbox{( by Eq. (\ref{soma}))}\\
\\
&=&\delta_p.\\
\end{array}\]
\noindent\emph{Case 4:} $n=m+k$ with $m\in\Delta_p$ and $k\in\{R_p+1, R_p+2,\ldots, N_p-2\}$. It is analogous to case 2.
\noindent\emph{Case 5:} $n=m+N_p-1$ with $m\in\Delta_p$.
In this case, we have
\[\gamma_n^p=x_{N_p-1}^p-\frac{N_p-(N_p-1)}{R_p}T^n(z_0+\cdots+z_{p-1})=0-\dfrac{1}{R_p}T^n(z_0+\cdots+z_{p-1}) \mbox{ and }\gamma_{n+1}^p=0.\] By Equation (\ref{soma})
\[\|T(\gamma_n^p)-\gamma_{n+1}^p\|=\dfrac{1}{R_p}\|T^{n+1}(z_0+\cdots+z_{p-1})\|<\frac{\delta_p}{4}\cdot 2<\delta_p.\]
\noindent This concludes that $\{\gamma_n^p\}_{n\in\mathbb{N}}$ is a $\delta_p$-pseudo orbit.
\qed
\begin{corollary} \label{hyperimpliesfreq.hyper}
Let $X$ be a separable Banach space. If $T : X \to X$ is hypercyclic (or recurrent) and has the positive shadowing property then $T$ is frequently hypercyclic and topologically mixing.
\end{corollary}
\begin{corollary} \label{prodct of shadowing and CR}
If $T_i : X_i \to X_i$, for $1\leq i \leq n$, is a finite family of linear dynamical systems such that, for each $i$, $(T_i,X_i)$ satisfies the hypothesis of the Theorem \ref{gymstheorem}, then $T_1 \times \ldots \times T_n : X_1 \times \ldots X_n \to X_1 \times \ldots X_n$ is frequently hypercyclic and topologically mixing.
\end{corollary}
\noindent \textbf{Proof:} It follows from Theorem \ref{gymstheorem}, Proposition \ref{multiplechainrecurrence}, Theorem \ref{positiveshadowingmixing} and the fact that the shadowing property is preserved under cartesian product.
\qed
Consider a topological dynamical system $(X,T)$. By an \textbf{invariant measure with full support} for $(X,T)$ we mean a measure $\mu$ defined over the Borelian $\sigma$-algebra of $X$ such that $\mu(X)=1$, $\mu(U)>0$ for any open set $U \subset X$ and $\mu(T^{-1}A) = \mu(A)$ for any mensurable set $A$.
\begin{corollary}
Let $X$ be a separable and reflexive Banach space and $T : X \to X$ a chain transitive map that has the positive shadowing property then there is an invariant measure with full support for $T$.
\end{corollary}
\noindent \textbf{Proof:} It follows from Theorem \ref{gymstheorem} and the fact that frequently hypercyclic operators on reflexive Banach spaces admit an invariant measure with full support \cite{invariant}.
\qed
\textbf{Devaney chaotic} systems are those that are transitive and have a dense set of periodic points \cite{devaney2}. The next corollary gives a suficient condition for a system to have this property.
\begin{corollary}
Let $X$ be a separable Banach space. If $T : X \to X$ has a dense set of periodic points and has the positive shadowing property then $T$ is Devaney chaotic.
\end{corollary}
\noindent \textbf{Proof:} Since $CR(T)$ is closed, and the set of periodic points is dense, then $X=CR(T)$. Therefore, Theorem A implies that $T$ is frequently hypercyclic and topologically mixing.
\qed
\begin{corollary}
If $X$ is a separable Hilbert space and $T$ is an unitary operator, then $T$ does not have the positive shadowing property.
\end{corollary}
\noindent \textbf{Proof:} By proposition \ref{unitaryimplieschaintransitivity} unitary operators are chain transitive, but since $\|T\| = 1$ they may never be hypercyclic. Therefore, by Theorem A these operators cannot have positive shadowing.
\qed
Given a topological dynamical system $(Y,f)$, a point in $x \in Y$ is said to be a \textbf{non-wandering point} if, for every open set $B$ that contains $x$, and for every $N \in \mathbb{N}$, there is $n>N$, such that $f^{n}(B) \cap B \neq \emptyset$. If $(X,T)$ is a linear dynamical system, linearity implies that the origin is always a non wandering point of $X$. We call the set of non wandering points of \textbf{non wandering set}. The complement of the non wandering set is the \textbf{wandering set}.
It is not difficult to see that if $T$ is recurrent, then every $x \in X$ is non wandering, and if $T$ has a dense set of non wandering points, then $T$ is recurrent.
The next result can be easily proven (it is similar to Proposition 6 of \cite{Messaoudi}), but we state it since it is needed in the proof of Theorem B.
\begin{proposition} \label{shadowingforsubspaces}
Let $T$ be an operator on a Banach space $X$. Suppose that $X = M \oplus N$, where $M$ and $N$ are closed $T$-invariant subspaces of $X$. Then $T$ has positive shadowing property if, and only if, $T|_M$ and $T|_N$ both have positive shadowing property.
\end{proposition}
\noindent \textbf{Proof of Theorem \ref{theo:nonwandering}:} It is easy to see that the non-wandering set is contained in the set of chain recurrent elements. For the contrary inclusion if $x$ is chain recurrent, and $U$ is an open set that contains $x$, then there is a pseudo-orbit that goes from $x$ to $x$ and then to $x$ again and so on, and the shadowing property tells us that such pseudo orbit can be shadowed by a real orbit. This proves that $x$ is a non-wandering point. Since $\Omega$ is equal to the chain recurrent set, then Corollary \ref{chainspace} tells us that $\Omega$ is a closed and invariant subspace of $X$. This allows us to obtain the first two conclusions.
By item 1, we have $\Omega=CR(T)$. By Corollary \ref{chainspace}, $\Omega$ is $T$-invariant. This implies that $\Omega^\perp$ is $T^*$-invariant, consequently, it is $T$-invariant, because $T=T^*$. Since $X$ is a Hilbert space and $\Omega$ is a closed subspace of $X$ then $X=\Omega\oplus\Omega^{\perp}$. Hence, by Proposition \ref{shadowingforsubspaces}, $T|_{\Omega}$ has positive shadowing property. Corollary \ref{chainrecurrentselfadjoint} guarantees that $T|_{\Omega}$ is chain transitive. Since $T|_{\Omega} : \Omega \to \Omega$ has positive shadowing property and is chain transitive, Theorem \ref{gymstheorem} gives that $T|_{\Omega}$ is topologically mixing and frequently hypercyclic.
\qed
\section{Open Questions}
In this section we leave some open questions for the reader. Question \ref{refereequestion} was kindly offered to us by the anonymous referee. Originally, this question only addressed the shadowing property, but we decided to expand it for chain recurrence as well.
\begin{question}
Is there any simple criterion to decide if a system is chain recurrent?
\end{question}
Since every hypercyclic operator is chain recurrent, then Kitai's Hypercyclic Criterion \cite{dynamicsoflinearoperators} is a simple criterion for chain recurrence. But since there are operators that are chain recurrent but not hypercyclic, e.g. the identity operator, it would be nice to find a tighter criterion.
\begin{question}
Is there any simple criterion to decide if a system has positive shadowing?
\end{question}
This question is not new and was addressed by other authors for the shadowing property. It seems that the notion of generalized hyperbolicity \cite{generalized} captures much of the essence of the shadowing property and could be equivalent to it. In view of the results presented in this text positive shadowing might be a more relevant property than shadowing itself, and therefore worthy of a similar search for an equivalent notion. Example \ref{eigen1andshadowing} shows that when $T$ has a right inverse that is a proper contraction, then $T$ will have positive shadowing.
\begin{question} \label{refereequestion}
Let $X$ be a normed vector space and $T:X \to X$ a linear operator with the positive shadowing property (and or chain recurrence), and let $Y \subset X$ be a closed invariant subspace. Under which hypotheses one may guarantee that the operator $T|_Y$ has the shadowing property (and or chain recurrence)?
\end{question}
Proposition \ref{shadowingforsubspaces} and Corollary \ref{CRforsubspaces} provide partial answers to this question. New results in this direction may provide a proof of the last item of Theorem B under more general assumptions than $X$ being Hilbert and $T$ being self-adjoint.
\begin{question}
Are the conclusions of Theorem A strong enough to imply the hypothesis?
\end{question}
More precisely does every frequently hypercyclic and topologically mixing linear dynamical system have positive shadowing? At the moment, the authors have no plausible argument that favors such conclusion, but we are unable to provide a counter-example to this claim. Based on Corollary \ref{prodct of shadowing and CR}, one may also ask a weaker version of this question, which is: if $(X,T)$ is a linear dynamical system such that any finite product of $T \times \ldots \times T$ is topologically mixing and frequently hypercyclic, then would this imply that $T$ has positive shadowing?
\noindent \textit{Acknowledgements: We would like to thank Prof. Udayan B. Darji for helpful comments on an earlier version of this text. We would also like to show our deep appreciation for the careful review and interesting suggestions given by the anonymous referee, which include among others, lemma \ref{refereelemma} and its proof, Corollary \ref{properdilation}, example \ref{refereeexample} and question \ref{refereequestion}. M.B.A. was supported by CAPES. R.V. was partially supported by CNPq.}
\end{document} |
\begin{document}
\title{Quantum and Classical Dynamics of a BEC in a Large-Period Optical Lattice}
\author{J.~H.~Huckans} \email{[email protected]} \affiliation{Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland, Gaithersburg, Maryland, 20899-8424, USA}
\author{I.~B.~Spielman} \affiliation{Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland, Gaithersburg, Maryland, 20899-8424, USA}
\author{B.~Laburthe~Tolra} \affiliation{\it Laboratoire de Physique des Lasers, Universit\'e de Paris 13, 93430 Villetaneuse, France}
\author{W.~D.~Phillips}
\author{J.~V.~Porto}
\affiliation{Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland, Gaithersburg, Maryland, 20899-8424, USA}
\date{\today}
\begin{abstract}
We experimentally investigate diffraction of a $^{87}$Rb Bose-Einstein condensate from a 1D optical lattice. We use a range of lattice periods and timescales, including those beyond the Raman-Nath limit. We compare the results to quantum mechanical and classical simulations, with quantitative and qualitative agreement, respectively. The classical simulation predicts that the envelope of the time-evolving diffraction pattern is shaped by caustics: singularities in the phase space density of classical trajectories. This behavior becomes increasingly clear as the lattice period grows.
\end{abstract}
\pacs{67.85.Hj, 67.85.Jk, 03.75.Kk}
\maketitle
\subsection{Introduction}
Modern atom optics~\cite{adams, burnett}, in particular the diffraction of atoms by standing waves of light~\cite{moskowitz,gould} provides a dramatic demonstration of the wave nature of atoms. The advent of Bose-Einstein condensates (BEC), with their extremely narrow momentum distribution, has made resolving diffraction components straightforward in these systems~\cite{ovchinnikov}. Most atom diffraction experiments with BECs focused on the regime where the diffracting standing wave has relatively few ``bound'' states~\cite{boundstate} or where the diffraction produces relatively few diffraction orders. Under such conditions, the wave nature of the atoms is essential for describing the behavior of the system. By contrast, when the optical potential has many bound states the quantum system can be well described by classical particle trajectories. This latter regime has been theoretically investigated with such trajectories~\cite{berry1}, as well as from a matter wave perspective~\cite{bernhardt,janicke}.
In 1994, Janicke and Wilkens calculated the diffraction of cold atoms from a lattice potential and predicted a dramatic collapse and revival of the number of diffracted orders as the atoms coherently evolve in the lattice~\cite{janicke}. Here we present the first experimental observation of this collapse and revival in both the quantum and classical regimes. The envelope of the time-evolving diffraction pattern for a sinusoidal potential is dominated by caustics: singularities in the phase space density of classical trajectories~\cite{berry1}. We investigate the quantum and classical regimes of atom wave diffraction by applying an optical standing wave (an optical lattice) to a BEC, and measuring the time evolution of the momentum distribution for a range of lattice periods.
While the phenomenon of atom diffraction is quantum mechanical, some early experiments used a classical trajectory approach to describe the observed channeling~\cite{salomon} and focussing~\cite{timp, mcclelland} of atoms by optical standing waves. These experiments used thermal beams of atoms with a large momentum spread and measured the atomic position distribution within the standing wave. Another early experiment used an optical standing wave to diffract laser-cooled atoms and observed the growth and subsequent collapse of the width of the diffraction pattern in the quantum regime~\cite{kunze}. Later, the collapse and revival of a few diffraction orders was observed in the diffraction of a BEC from a lattice with only a few bound states~\cite{denschlag}. Here we extend this earlier work by measuring the time-evolution of a BEC's momentum distribution in lattice potentials for a range of lattice periods (see Fig. 1). We observe several collapses and revivals in both quantum and classical regimes. We compare our results to the predictions of a single-particle quantum simulation and find excellent agreement. We also develop a classical model which reproduces essential features of our data and provides physical insight into the evolution of the momentum distribution.
\subsection{Experimental Procedure}
We create a static 1D optical lattice at the intersection of two laser beams. In the plane-wave approximation, the electric field of each beam is $\vec{E}(\vec{r},t) = \hat e E_0 e^{i (\vec k \cdot \vec r - \omega t)} + c.c.$ where $\hat e$ is the polarization vector. The combined electric field for both beams creates an optical potential for the atoms given by
\begin{equation}
\label{latticepotential}
U(z) = U_0\sin^2(\kappa_{\rm L}z).
\end{equation}
For a two-level system $U_0 = \hbar\Omega_0^2/\delta$, and $\kappa_{\rm L} \equiv |\vec k_1 -\vec k _2| /2 = \pi/d$ is one-half of the magnitude of the reciprocal lattice vector, $M$ is the atomic mass, $d$ is the lattice period, $\Omega_0$ is the resonant, single-beam Rabi frequency, and $\delta<0$ is the detuning of the laser from atomic resonance (In this experiment, $U_0$ is nominally 30~$E_{\rm R}$ where $E_{\rm R} = \hbar^2 k^2/2M$ is the single-photon recoil energy.). We vary $\kappa_{\rm L}$ by changing the angle between $\vec k_1$ and $\vec k_2$ and define $\hat{z}$ parallel to $\vec k_1 - \vec k_2$. Near the energy minimum of each lattice site, the potential is nearly harmonic, $U_{\rm ho}(z) \approx M \omega_{\rm ho}^2 z^2 /2$, where $\omega^2_{\rm ho} = 2U_0\pi^2/Md^2$. We choose the detuning $\left|\delta\right| \gg (\Omega_0,\Gamma$) (where $\Gamma$ is the natural linewidth of the atomic transition) so spontaneous emission is negligible for our experiment durations.
\begin{figure*}
\caption{Concatenated absorption images of diffraction patterns, showing the evolving momentum distribution at four lattice periods $d$. The momentum is scaled by $k_{\rm max}
\label{Figure:figex1}
\end{figure*}
We produce a nearly pure BEC with $N_{\mbox{\scriptsize tot}} = 5 \times 10^{4}$ to $14 \times 10^{4}$ atoms in the $(F,m_{F})=(1,-1)$ hyperfine state of $^{87}$Rb~\cite{peil03}. We use a Ioffe-Pritchard trap with an oscillation frequency of $\nu_{\rm z} =8.2$~Hz in the weak direction and $\nu_{\rm x} = \nu_{\rm y} = 24$~Hz in the tight directions. The optical lattice is suddenly applied to the magnetically-trapped atoms for a time $T_{\rm pulse}$. The lattice periods $d$ used here, along with other relevant experimental parameters, are listed in Table I~\cite{stddev}.
\begin{table}[hbt]
\label{Experimental Data table}
\begin{tabular}{l|c|c|c|c}
& (a) & (b) & (c) & (d) \\
\hline
$d$ ($\mu$m) & 1.80(2) & 3.5(1) & 6.5(1) & 9.3(1) \\
$U_0$ ($E_{\rm R}$) & 33(1) & 26(2) & 32(3) & 29(3) \\
$N_{\rm tot}$ ($10^4$) & 12(2) & 14(5) & 4(1) & 5(3) \\
$D_{\rm TF}$ ($\mu$m) & 55(6) & 57(6) & 45(4) & 46(5) \\
$D_{\rm TF}/d$ & 31(3) & 16(2) & 7(1) & 5.0(5) \\
\end{tabular}
\caption{Experimental parameters for the four lattice periods investigated. The Thomas-Fermi diameters $D_{\rm TF}$ along $\hat z$ were calculated using the measured atom numbers and known scattering length and trap frequencies. Lattice depths were obtained by measuring the maximum kinetic energy of an atom during its evolution in a lattice. The number of occupied lattice sites is approximately $D_{\rm TF}/d$.}
\end{table}
The lattice beams, linearly polarized perpendicular to the plane defined by the crossed beams, derive from a Ti:Sapphire laser operating at $\lambda=810$~nm (detuned below both $5S \rightarrow 5P$ transitions at 795~nm and 780~nm). We constructed an ``accordion'' lattice allowing us to continuously vary the period of the diffracting potential~\cite{huckansthesis}. Rotation of a galvanometer-controlled mirror causes the relative angle of the two beams to change while maintaining their intersection at the BEC. The $e^{-2}$ radius of each beam is $\approx$~200~$\mu$m. The lattice is turned on abruptly ($\lesssim 500\ {\rm ns}$) to its nominal depth of 30 $E_{\rm R}$, held constant for a variable time $T_{\rm pulse}$ and then turned off abruptly. This constitutes a ``pulse'' of the lattice potential. Immediately after the pulse, we release the atoms by turning off the magnetic trap in $\approx$~250~$\mu$s. After the atom cloud expands for 20.2 ms, we record the spatial distribution of the atoms using resonant absorption imaging~\cite{cornell}. Each image approximates the momentum distribution at the time of release. Figure 1 shows a concatenated series of such images as a function of $T_{\rm pulse}$ at four different lattice periods. Together, the images reveal the evolving momentum distribution for each lattice period, which collapse and revive with the characteristic features predicted in Ref.~\cite{janicke}.
\subsection{Results and interpretation}
Figure 1a depicts the measured momentum distribution as a function of pulse duration for a BEC diffracted by a 1.8~$\mu$m lattice. For small $T_{\rm pulse}$, the position of the apparent edge of the momentum distribution grows linearly with $T_{\rm pulse}$. As expected, the distribution is composed of diffraction orders separated by $2\hbar \kappa_{\rm L}$. At early times, our data are consistent with the diffraction predicted using the Raman-Nath approximation. The Raman-Nath approximation can be viewed in a number of ways. For example, it neglects the kinetic energy term in the single-particle Hamiltonian during application of the pulse. This is equivalent to assuming that the only effect of the pulse is to impose a spatially periodic phase on the atomic wave function, with no effect on its amplitude profile. This implies that the atoms move by a distance small compared to the lattice period $d$ during the pulse. The Raman-Nath approximation is valid when
\
\begin{equation}
T_{\rm pulse} \ll t_{\rm RN} \equiv \frac {\hbar} {\sqrt {U_0 E_{\rm L}}} = \frac{T_{\rm ho}}{\pi},
\end{equation}
where $E_{\rm L} = \hbar^2 \kappa_{\rm L}^2/2M$ is the lattice recoil energy and $T_{\rm ho} = 2\pi/\omega_{\rm ho}$ is the harmonic oscillator period. (Note that the lattice recoil in general differs from the photon recoil, but becomes equal to it when the lattice beams are counterpropagating.) In this approximation, the fractional atomic population of the $n^{\rm th}$ diffraction order is $J_n^2(U_0 T_{\rm pulse}/2\hbar)$ where the $J_n$ are Bessel functions of the first kind.
\begin{figure}
\caption{Splitting of diffraction orders (1.8 $\mu$m lattice period) for pulse durations beyond the Raman-Nath regime. The effect is more pronounced the longer the pulse duration and two examples are indicated with white boxes.}
\label{Figure:figex2}
\end{figure}
Figure 1a shows that as the pulse duration increases beyond $t_{\rm RN}$, the apparent edge of the momentum distribution is bounded by a maximum momentum $\hbar k_{\rm max}$. We use this observed value of $\hbar k_{\rm max}$ to determine the lattice depth $U_0 = \hbar^2 k_{\rm max}^2/2M$. The numerical calculations verify the accuracy of this identification to within 3~$E_{\rm R}$. (The validity of the identification relies on the lattice being deep enough to support many bound states and the rise and fall times of the pulse being short compared to $T_{\rm ho}$.) Once the edge of the distribution reaches $k_{\rm max}$, it gradually fades and a new outgoing edge appears near $k=0$, shortly after $T_{\rm ho}/2$. Diffraction orders re-emerge as the new edge moves outward to higher momentum and the process approximately repeats. At each collapse, a large fraction of the population returns to the lowest orders. Collapses repeat at times which are approximate multiples of $T_{\rm ho}/2$ .
The collapses and revivals can be understood from a classical model where atoms initially at rest in the sinusoidal potential oscillate; those starting near the bottom of the potential oscillate at approximately $\omega_{\rm ho}$. As we will see below, the suddenness of each collapse results from the anharmonicity of the potential and can also be understood in our classical model. Because of the anharmonicity, no collapse is total, and there remains a sizable occupation of the higher momentum orders at the collapse point. This collapse and revival always occurs at times greater than $t_{\rm RN}$ and illustrates a complete breakdown of the Raman-Nath approximation.
Figures 1b-d depict the momentum evolution for longer lattice periods. While each evolution is similar, there are differences. For $d > 1.8 \mu$m, the diffraction orders overlap; indeed, mean-field-driven expansion of the individual orders can lead to such overlap for all times-of-flight. The apparent discreteness in the momentum distribution in Fig 1b, does not represent individual momentum orders but rather a modulated momentum-envelope over several unresolved orders. As the lattice period $d$ increases and the spacing between the momentum orders decreases, the momentum distribution appears as a continuos curve. Also, the point of collapse occurs increasingly close to $T_{\rm ho}/2$ as $d$ increases.
The quantum simulations shown in the right-hand column of Fig. 1 reproduce these features of our experiment. They depict the {\it in-situ} momentum distribution after $T_{\rm pulse}$. We numerically solve the time-dependent Gross-Pitaevskii (GP) equation during $T_{\rm pulse}$ including the mean-field interactions~\cite{nointeractions}. We assume that the solution factors into a time-independent radial wavefunction and a time-dependent axial wavefunction. We treat the axial component using a 1D GP equation with effective interaction strength $g_{\rm 1D} = 4 g_{\rm 3D} / 3\pi R_{\rm x} R_{\rm y}$ where $g_{\rm 3D} = 4\pi\hbar^2 a_s/M$ is the 3D interaction strength~\cite{spielman}. Here $R_{\rm x}$ and $R_{\rm y}$ are the Thomas-Fermi radii in the directions perpendicular to the lattice, and $a_s$ is the s-wave scattering length for our $^{87}$Rb atoms. Figure 1 shows the good agreement between these simulations and our experimental data. The simulation stops at the end of each pulse and we plot the momentum distribution (before the time-of-flight).
A detail of the experimental results, absent in the simulations, is the visible splitting of some diffraction orders for pulse durations beyond the Raman-Nath regime (see Fig. 2). The effect becomes more pronounced the longer the pulse duration. The effect implies spatial structure larger than the lattice spacing and it becomes more pronounced the longer the pulse duration. One possible explanation is that the approximate periodic translational symmetry of the lattice is gradually compromised as the inhomogeneous mean-field interaction increases over time due to the atoms accelerating radially toward the cloud center because of the dipole force arising from the laser beam profile. If this were the mechanism, it would explain why we do not see the order splittings in the simulation since radial dynamics are neglected.
Certain salient features reproduced by the numerical simulations can also be understood on the basis of simple arguments, both quantum and classical. In the following two sections, we first give a single-particle quantum mechanical argument which explains the collapse and revival periods and their deviations from $T_{\rm ho}/2$. We continue with a classical explanation for the sudden collapse of the higher momentum orders and their subsequent revival.
\subsection{Quantum Mechanical Analysis}
To understand the atomic evolution after sudden application of an optical lattice, it is useful to decompose the initial BEC state into the relevant eigenstates of the lattice potential with energies $E_n$. We assume an initial state with momentum eigenvalue zero, equivalent to an infinite-extent BEC, projected onto the Bloch states (with zero quasimomentum) of 1D sinusoidal lattices having various periods. Only even parity Bloch states are occupied since the initial wavefunction is symmetric. We calculate the projection onto all significantly occupied states and find that the bound states contain the vast majority of the population. Figure 3 shows the populations of all Bloch states up to the first unbound state, which include more than 99.5\% of the population.
\begin{figure}
\caption{Calculated projections of a uniform wavefunction onto even-parity, zero-quasimomentum Bloch states of a 30~$E_{\rm R}
\label{Figure:figex2}
\end{figure}
\begin{figure}
\caption{Calculated evolution of the probability density $|\Psi(z,t)|^2$ within a single site of a lattice (potential shown in red) with period $d = 1.8~\mu$m and depth 30~$E_{\rm R}
\label{Figure:figex3}
\end{figure}
The number of significantly occupied eigenstates grows with increasing lattice period. The increase of occupied bound states corresponds to the system becoming increasingly classical. In the deep-lattice limit, the number of bound states in a sinusoidal potential scales as $(U_0/E_{\rm L})^{1/2}$. Since the lattice recoil energy $E_{\rm L} \propto d^{-2}$, the number of bound states in a lattice grows linearly with the lattice period at fixed total depth. For example, a lattice with a depth of 30 $E_{\rm R}$ formed by two counter-propagating beams has a period $d= \lambda/2$, a depth of 30~$E_{\rm L}$, and 4 bound states. However, a 30 $E_{\rm R}$ lattice formed by two beams intersecting at 87 mrad $\approx 5^\circ$ has a period $d = 9.3~\mu$m, a depth of $15.8 \times 10^3~E_{\rm L}$, and $\approx 80$ bound states.
The phase of the $n^{\rm th}$ Bloch state evolves independently as $\omega_n t$ where $\omega_n = E_n/\hbar$. In a harmonic potential, the frequency differences would be multiples of the harmonic frequency and all even eigenstates would rephase with period $T_{\rm ho}/2$. The anharmonic potential of a single lattice site leads to a non-uniform spacing of the energy levels, so the system never perfectly rephases. Nevertheless, the wavefunction approximately rephases such that a large fraction of the population is in diffraction orders close to zero momentum at times close to $T_{\rm ho}/2$ . (See Fig.~4 where a wide spread in position corresponds to the collapse of the momentum distribution as seen in Fig.~1.)
\subsection{Classical Analysis}
Many aspects of this quantum mechanical system can be understood classically, in some cases quantitatively. We study the corresponding classical system by calculating the trajectories of an ensemble of particles initially at rest and distributed uniformly in a sinusoidal potential (see Fig. \ref{Figure:figex4}). Each trajectory reaches a turning point where the velocity returns to zero. For small amplitude oscillations in the sinusoidal potential, the motion is nearly harmonic and atoms reach these turning points approximately concurrently. Larger amplitude oscillations are increasingly anharmonic, leading to increasing times to reach the turning points. Figure \ref{Figure:figex4} displays several momentum-trajectories of initially motionless classical particles evolving in a sinusoidal potential~\cite{sinediffraction}.
\begin{figure}
\caption{Classical motion of particles in a sinusoidal potential. The curves
correspond to the trajectories of particles starting at different
points in the potential, each with zero initial momentum, $p_z=0$.
The smallest amplitude oscillations correspond to the smallest
period, $T_{\rm ho}
\label{Figure:figex4}
\end{figure}
Insight can be gained from this classical picture. For
example, the suddenness of the collapse can be understood~\cite{berry1} by referring to a single-well phase space portrait
plotting position vs. momentum as shown in Fig.~\ref{Figure:figex5}:
Starting with a uniform distribution at rest ($p_z=0$), the
distribution rotates clockwise about the origin. At the origin, the
period of rotation is $2\pi/\omega_{\rm ho}$. Points farther from
the origin rotate more slowly about the origin. The rotational
period diverges for atoms nearest the top of the sinusoidal
potential, $z=\pm d/2$. The classical analog of our measured
momentum distribution is the projection of this evolving phase space
distribution onto the momentum axis. The horizontal tangents of the
phase space distribution (indicated in the figure) project to
singularities in the momentum distribution. The locus of these
tangent points is referred to as a caustic. As the distribution
approaches the first turning point, the two horizontal tangents are
near the maximum momentum; there are no singularities near $p_z= 0$.
When the central part of the distribution crosses the turning point
(at $T_{\rm ho}/2$), another pair of caustics suddenly emerges from
the origin. This explains the asymmetry in the momentum evolution;
the caustics only appear after $T_{\rm ho}/2$. (The time evolution
is symmetric only for a perfectly harmonic potential.)
In a quantum or wave-optics system, diffraction softens the divergence of the underlying classical caustics. The sequence in Fig. 1a-d shows, for the first time in a matter wave system, a progression from diffractive caustics~\cite{thom, arnold1, arnold2,trinkaus} toward classical caustics: the diffractive structure is manifest for the shortest period lattice and nearly invisible in the longest.
\begin{figure}
\caption{Classical phase space evolution of atoms in a single well of a sinusoidal potential. Each black curve depicts the phase space distribution at a specific time. The lips of a single well are located at positions $z=\pm d/2$. The phase space evolution is characterized by fixed points at the lips and rotation approaching the harmonic frequency near $z=0$.}
\label{Figure:figex5}
\end{figure}
\subsection{Conclusion}
We measured the collapse and revival of the diffraction pattern of a BEC exposed to a pulsed 1D optical lattice. Our results are found to be in good agreement with the predictions of the time-dependent Gross-Pitaevskii equation. In addition, we employed a classical model that captures essential features and adds physical insight to the evolving diffraction pattern. For long lattice periods, bound states proliferate and we observed classical behavior in the long-time evolution of the momentum distribution. We captured, for the first time, the emergence of both diffractive and classical caustics in ultra-cold atom systems interacting with optical lattices.
We gratefully acknowledge helpful discussions with Paul Julienne, Ennio Arimondo, Mikkel Andersen, and Vincent Boyer. This work was partially supported by ARDA, NASA, NRL and IBS acknowledges the support of the NRC Postdoctoral Fellowship Program.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{A class of multifractal processes constructed using an embedded branching process}
\runtitle{Multifractal process with embedded branching process}
\begin{aug}
\author[A]{\fnms{Geoffrey} \snm{Decrouez}\ead[label=e1]{[email protected]}}
\and
\author[A]{\fnms{Owen Dafydd} \snm{Jones}\corref{}\ead[label=e2]{[email protected]}}
\runauthor{G. Decrouez and O. D. Jones}
\affiliation{University of Melbourne}
\address[A]{Department of Mathematics\\
\quad and Statistics\\
University of Melbourne\\
Parkville VIC 3010\\
Australia\\
\printead{e1}\\
\phantom{E-mail: }\printead*{e2}}
\end{aug}
\received{\smonth{10} \syear{2010}}
\revised{\smonth{10} \syear{2011}}
\begin{abstract}
We present a new class of multifractal process on $\mathbb{R}$,
constructed using an embedded branching process. The construction makes
use of known results on multitype branching random walks, and along the
way constructs cascade measures on the boundaries of multitype
Galton--Watson trees. Our class of processes includes Brownian motion
subjected to a continuous multifractal time-change.
In addition, if we observe our process at a fixed spatial resolution,
then we can obtain a finite Markov representation of it, which we can
use for on-line simulation. That is, given only the Markov
representation at step $n$, we can generate step $n+1$ in $O(\log n)$
operations. Detailed pseudo-code for this algorithm is provided.
\end{abstract}
\begin{keyword}[class=AMS]
\kwd[Primary ]{60G18}
\kwd[; secondary ]{28A80}
\kwd{60J85}
\kwd{68U20}.
\end{keyword}
\begin{keyword}
\kwd{Self-similar}
\kwd{multifractal}
\kwd{branching process}
\kwd{Brownian motion}
\kwd{time-change}
\kwd{simulation}.
\end{keyword}
\end{frontmatter}
\section{Introduction}
Information about the local fluctuations of a process $X$ can be
obtained using the local exponent $h_X(t)$, defined as~\cite{R03}
\[
h_X(t):= \liminf_{\varepsilon\rightarrow0} \frac{1}{\log\varepsilon
}\log
\sup_{|u-t|<\varepsilon} |X(u)-X(t)|.
\]
When $h_X(t)$ is constant all along the sample path with probability
$1$, $X$ is said to be monofractal.
In contrast, we can consider a class of processes whose exponents
behave erratically with time: each interval of positive length exhibits
a range of different exponents.
For such processes, it is, in practice, impossible to estimate $h_X(t)$
for all $t$, due to the finite precision of the data.
Instead, we use the Hausdorff spectrum $D(h)$, a global description of
its local fluctuations.
$D(h)$ is defined as the Hausdorff dimension of the set of points with
a given exponent $h$.
For monofractal processes, $D(h)$ degenerates to a single point at some
$h=H$ [so $D(H)=1$, and the convention is to set $D(h)=-\infty$ for
$h\neq H$].
When the spectrum is nontrivial for a range of values of $h$, the
process is said to be multifractal.
The term multifractal is also well defined for measures.
Let $B(x,r)$ be a ball centered at $x\in\mathbb{R}^n$ with radius $r$.
The local dimension of a finite measure $\mu$ at $x\in\mathbb{R}^n$ is
defined as
\[
\mathrm{dim}_{\mathrm{loc}}\mu(x) = \lim _{r\to0}\frac
{\log\mu
(B(x,r))}{\log r}.
\]
The Hausdorff spectrum $D(\alphapha)$ of a measure at scale $\alpha$ is then
defined as the Hausdorff dimension of the set of points with a given
local dimension $\alphapha$.
Measures for which the Hausdorff spectrum does not degenerate to a
point are called multifractal measures.
Constructions of multifractal measures date back to the $m$-ary
cascades of Mandelbrot~\cite{M74}, and the multifractal spectrum of
such measures can be found in, for example,~\cite{R03}.
A positive nondecreasing multifractal process can be obtained by
integrating a multifractal measure.
Other processes with nontrivial multifractal structure can be obtained
by using the integrated measure as a multifractal time change, applied
to monofractal processes such as fractional Brownian motion.
This is the basis of models such as infinitely divisible cascades
\cite{BM02,BM03,CRA05}.
Multifractals have a wide range of applications.
For example, the rich structure of network traffic exhibits
multifractal patterns~\cite{ABFRV02}, as does the stock market
\mbox{\cite{M97,M99}}.
Other applications include turbulence~\cite{SM88}, seismology \cite
{H01,TLM04} and imaging~\cite{RRCB00}, to cite but a few.
On-line simulation of multifractal processes is in general difficult,
because their correlations typically decay slowly, meaning that to
simulate $X(n+1)$ one requires $X(1), \ldots, X(n)$.
This is the same problem faced when simulating fractional Brownian
motion, where to simulate $X(n+1)$ one needs the whole covariance
matrix of $X(1), \ldots, X(n+1)$.
Some simple monofractal processes avoid this problem, for example,
$\alphapha$-stable or $M/G/\infty$ processes~\cite{C84}, but it remains a
real problem to find flexible multifractal models that can be quickly simulated.
We propose a new class of multifractal processes, called Multifractal
Embedded Branching Process (MEBP) processes, which can be efficiently
simulated on-line.
MEBP are defined using the crossing tree, an ad-hoc space--time
description of the process, and are such that the spatial component of
their crossing tree is a Galton--Watson branching process.
For any suitable branching process, there is a family of
processes---identical up to a continuous time change---for which the
spatial component of the crossing tree coincides with the branching process.
We identify one of these as the Canonical Embedded Branching Process
(CEBP), and then construct MEBP from it using a multifractal time change.
To allow on-line simulation of the process, the time change is
constructed from a multiplicative cascade on the crossing tree.
The simulation algorithm presents nice features since it only requires
$O(n\log n)$ operations and $O(\log n)$ storage to generate $n$ steps,
and can generate a new step on demand.\vadjust{\goodbreak}
To construct the time change we use here, we start by constructing a
multiplicative cascade on a \textit{multitype} Galton--Watson tree.
The cascade defines a measure on the boundary of the tree, whose
existence follows from known results for multitype branching random walks.
(See, e.g.,~\cite{L00} for the single-type case.)
To map the cascade measure onto $\mathbb{R}_+$, we use the so called
``branching measure'' on the tree, in contrast to the way this is
usually done, using a ``splitting measure.''
See Section~\ref{CEBPMEBP} for details and further background.
The MEBP processes constructed here include a couple of special cases
of interest.
We can represent Brownian motion as a CEBP, thus MEBP processes include
a subclass of multifractal time changed Brownian motions.
Such models are of particular interest in finance~\cite{M97,M99}.
In the special case when the number of subcrossings is constant and
equal to two (for the definition see Section~\ref{secEBP}), the CEBP
degenerates to a straight line, and the time change is just the
well-known binary cascade (see, e.g.,~\cite{B99,KP76,Mo96} and
references therein).
Although we do show that MEBP possess a form of discrete multifractal
scaling [see the discussion following equation (\ref{MFCmulteqn})], the
multifractal nature of MEBP processes is not studied in this paper.
We refer the reader to a coming paper for a full study of the
multifactal spectrum of MEBP~\cite{DHJ}.
In particular, it can be shown that CEBP processes are monofractal, and
that the multifractal formalism holds for MEBP processes, with a
nontrivial spectrum.
The monofractal nature of CEBP processes, together with an upper bound
of the spectrum of MEBP, was derived in the Ph.D. thesis of the first
author~\cite{Dec09}.
The paper is organized as follows.
First we recall the definition of the crossing tree and then construct
the CEBP process.
We then construct MEBP processes and give conditions for continuity.
Finally we provide an efficient on-line algorithm for simulating MEBP processes.
An implementation of the algorithm is available from the second
author's website~\cite{J}.
\section{CEBP and the crossing tree}
\label{secEBP}
Let $X\dvtx\mathbb{R}^+\rightarrow\mathbb{R}$ be a continuous process, with $X(0)=0$.
For $n\in\mathbb{Z}$ we define level $n$ passage times $T_{k}^{n}$ by putting
$T_0^n = 0$ and
\[
T_{k+1}^{n}= \inf\{t>T_{k}^{n}|X(t)\in2^{n}\mathbb{Z}, X(t)\not=
X(T_{k}^{n}) \}.
\]
The $k$th level $n$ (equivalently scale $2^n$) crossing $C_k^n$ is the
sample path from $T_{k-1}^{n}$ to $T_{k}^{n}$.
\[
C_k^n:= \{ (t,X(t))| T_{k-1}^n\leq t < T_k^n\}.
\]
When passing from a coarse scale to a finer one, we decompose each
level $n$ crossing into a sequence of level $n-1$ crossings.
To define the crossing tree, we associate nodes\vadjust{\goodbreak} with crossings, and the
children of a node are its subcrossings.
The crossing tree is illustrated in Figure~\ref{crossingtree}, where
the level 3, 4 and 5 crossings of a given sample path are shown.
\begin{figure}
\caption{A section of sample path and levels 3, 4 and 5 of its crossing
tree. In the top frame we have joined the points $T^n_k$ at each level,
and in the bottom frame we have identified the $k$th level $n$
crossing with the point $(2^n, T^n_{k-1}
\label{crossingtree}
\end{figure}
The crossing tree is an efficient way of representing a self-similar
signal, and can also be used for inference.
In~\cite{JS04} the crossing tree is used to test for self-similarity
and to obtain an asymptotically consistent estimator of the Hurst index
of a self-similar process with stationary increments, and in \cite
{JS05} it is used to test for stationarity.
In addition to indexing crossings be their level and position within
each level, we will also use a tree indexing scheme.
Let $\varnothing$ be the root of the tree, representing the first level 0
crossing.
The first generation of children (which are level $-1$ crossings, of
size $1/2$) are labeled by $i$, $1\leq i \leq Z_\varnothing$, where
$Z_\varnothing$ is the number of children of~$\varnothing$.
The second generation (which are level $-2$ crossings, of size $1/4$)
are then labeled $ij$, $1\leq j\leq Z_i$, where $Z_i$ is the number of
children of $i$.
More generally, a node is an element of $U = \bigcup_{n\geq0}\mathbb{N}^{n}$ and
a branch is a couple $(\mathbf{u},\mathbf{u} j)$ where $\mathbf{u}\in U$ and $j\in\mathbb{N}$.
The length of a node $\mathbf{i}=i_1, \ldots, i_n$ is $|\mathbf{i}|=n$, and the
$k$th element is $\mathbf{i}[k] = i_k$.
If $|\mathbf{i}|>n$, $\mathbf{i}|_n$ is the curtailment of $\mathbf{i}$ after $n$ terms.
Conventionally $|\varnothing|=0$, and $\mathbf{i}|_0=\varnothing$.
A tree $\Upsilon$ is a set of nodes, that is, a subset of $U$, such that:
\begin{itemize}
\item$\varnothing\in\Upsilon$;
\item if a node $\mathbf{i}$ belongs to the tree, then every ancestor node
$\mathbf{i}|_k$, $k\leq|\mathbf{i}|$, belongs to the tree;
\item if $\mathbf{u}\in\Upsilon$, then $\mathbf{u} j\in\Upsilon$ for $j = 1, \ldots,
Z_\mathbf{u}
$ and $\mathbf{u} j \notin\Upsilon$ for $j > Z_\mathbf{u}$, where $Z_\mathbf{u}$ is the
number of children of $\mathbf{u}$.
\end{itemize}
Let $\Upsilon_n$ be the $n$th generation of the tree, that is, the set of
nodes of length~$n$. (These are level $-n$ crossings, of size
$2^{-n}$.) Define\vspace*{1pt} $\Upsilon_\mathbf{i}= \{\mathbf{j}\in\Upsilon |
|\mathbf{j}|\geq|\mathbf{i}|$ and \mbox{$\mathbf{j}|_{|\mathbf{i}|}=\mathbf{i}\}$}. The boundary of
the tree is given by $\partial\Upsilon= \{\mathbf{i}\in\mathbb{N}^\mathbb{N} | \forall n\geq0,
\mathbf{i}|_n\in\Upsilon\}$. Let $\psi(\mathbf{i})$ be the position of node $\mathbf{i}$
within generation $|\mathbf{i} |$, so that crossing $\mathbf{i}$ is just
$C^{-|\mathbf{i}|}_{\psi(\mathbf{i})}$. The nodes to the left and right of $\mathbf{i}$,
namely $C^{-|\mathbf{i}|}_{\psi (\mathbf{i})-1}$ and $C^{-|\mathbf{i}|}_{\psi(\mathbf{i})+1}$,
will\vspace*{1pt} be denoted $\mathbf{i}-$ and $\mathbf{i}+$. In general, when we
have quantities associated with crossings, we will use tree indexing
and level/position indexing interchangeably. So $Z_\mathbf{i}=
Z^{-|\mathbf{i}|}_{\psi(\mathbf{i})}$, $T_\mathbf{i}= T^{-|\mathbf{i}|}_{\psi (\mathbf{i} )}$, etc.
At present\vspace*{1pt} our tree indexing only applies to crossings
contained within the first level 0 crossing; however, in Section
\ref{secextend} we will extend this notation to the whole tree.
Let $\alpha^n_k \in\{+, -\}$ be the orientation of $C^n_k$, $+$ for up
and $-$ for down, and let $A^n_k$ be the vector given by the
orientations of the subcrossings of $C^n_k$.
Let $D^n_k = T^n_k - T^n_{k-1}$ be the duration of $C^n_k$.
Clearly, to reconstruct the process we only need $\alpha^n_k$ and $D^n_k$
for all $n$ and $k$.
The $\alpha^n_k$ encode the spatial behavior of the process, and the
$D^n_k$ the temporal behavior.
Our definition of an EBP is concerned with the spatial component only.
\begin{definition}
A continuous process $X$ with $X(0) = 0$ is called an Embedded
Branching Process (EBP) process if for any fixed $n$, conditioned on
the crossing orientations $\alpha^n_k$, the random variables $A^n_k$ are
all mutually independent, and $A^n_k$ is conditionally independent of
all $A^m_j$ for $m > n$.
In addition we require that $\{A^n_k | \alpha^n_k = i\}$ are
identically distributed, for $i = +, -$.
That is, an EBP process is such that if we take any given crossing,
then count the orientations of its subcrossings at successively finer
scales, we get a (supercritical) two-type Galton--Watson process, where
the types correspond to the orientations.
\end{definition}
Subcrossing orientations have a particular structure.
A level $n$ up crossing is from $k2^n$ to $(k+1)2^n$, a down crossing
is from $k2^n$ to $(k-1)2^n$.
The level $n-1$ subcrossings that make up a level $n$ parent crossing
consist of \textit{excursions}\vadjust{\goodbreak} (up--down and down--up pairs) followed by a
\textit{direct crossing} (down--down or up--up pairs), whose direction
depends on the parent crossing: if the parent crossing is up, then the
subcrossings end up--up; otherwise, they end down--down.
Let $Z^n_k$ be the length of $A^n_k$, that is, the number of
subcrossings of $C^n_k$.
The number of up and down subcrossings will be written $Z^{n+}_k$ and
$Z^{n-}_k$, respectively.
Clearly, each of the $Z_k^n-2$ first entries of $A_k^n$ comes in pairs,
each pair being up--down or down--up. The last two components are either
the pair up--up or down--down, depending on~$\alphapha_k^n$.
Thus, given $\alpha^n_k = +$, we must have $Z^{n+}_k = \half Z^n_k + 1$
and $Z^{n-}_k = \half Z^n_k - 1$, and conversely given $\alpha^n_k = -$.
Let $\mathcalA$ be the space of possible orientations.
That is, $a \in\mathcalA$ consists of some number of pairs, $+-$ or $-+$,
then a single pair $++$ or $--$.
Given an EBP process, for the offspring type distributions we write
$p^+_{A}(a) = \mathbb{P}(A^n_k = a | \alpha^n_k = +)$ and $p^-_{A}(a) = \mathbb{P}
(A^n_k = a | \alpha^n_k = -)$, for $a \in\mathcalA$.
Let $\mu^+ = \mathbb{E}(Z^n_k | \alpha^n_k=+)$, $\mu^- = \mathbb{E}(Z^n_k | \alpha
^n_k=-)$ and $\mu= \half(\mu^+ + \mu^-)$, then the mean offspring
matrix is given by
\[
M:= \mathbb{E}\pmatrix{
(Z^{n+}_k|\alpha^n_k = +) & (Z^{n-}_k|\alpha^n_k = +) \cr
(Z^{n+}_k|\alpha^n_k = -) & (Z^{n-}_k|\alpha^n_k = -)}
=
\pmatrix{
\half\mu^+ + 1 & \half\mu^+ - 1 \cr
\half\mu^- - 1 & \half\mu^- + 1}.
\]
To proceed we need to make some assumptions about $p^\pm_{A}$.
\begin{assp}\label{AssGW}
$\mu^+, \mu^- > 2$ and $\mathbb{E}(Z^{ni}_k \log Z^{ni}_k | \alpha^n_k =
j) <
\infty$ for \mbox{$i, j = \pm$}.
\end{assp}
The first of\vspace*{1pt} these assumptions ensures that $M$ is
strictly positive with dominant eigenvalue $\mu> 2$, and corresponding
left eigenvector $(\half, \half)$. The corresponding\vspace*{1pt} right
eigenvector is $((\mu^+-2)/(\mu-2), (\mu ^--2)/(\mu-2))^T$. The second
assumption is the usual condition for the normed limit of a
supercritical Galton--Watson process to be nontrivial.
\begin{theorem}\label{CEBP}
For any offspring orientation distributions $p^\pm_{A}$ satisfying
Assumption~\ref{AssGW}, there exists a corresponding continuous EBP
process $X$ defined on~$\mathbb{R}_+$.
\end{theorem}
\begin{pf}
A version of this result can be found as Theorem 1 in~\cite{J04}, for
particular orientation distributions.
\begin{figure}
\caption{Construction of $X^0$, $X^{-1}
\label{Spmfig}
\end{figure}
\textsc{Step 1.}
We initially construct a single crossing from 0 to 1, with support $[0, T^0_1]$.
In step 2 we will extend the range to $\mathbb{R}$ and the support to
$[0,\infty)$.
$X$~is obtained as the limit as $n \to+\infty$, of a sequence of
random walks $X^{-n}$ with steps of size $2^{-n}$ and duration $\mu^{-n}$.
Put $X^0(0)=0$ and $X^0(1)=1$, so that the coarsest scale is $n=0$.
Given $X^{-n}$ we construct $X^{-(n+1)}$ by replacing the $k$th step of
$X^{-n}$ by a sequence of $Z_k^{-n}$ steps of size $2^{-(n+1)}$ and
duration $\mu^{-(n+1)}$.
If $\alpha^{-n}_k = i$, then the orientations $A^{-n}_k$ of the
subcrossings are distributed according to $p^i_{A}$.
For a given $n$ the $A^{-n}_k$ are all mutually independent, and, given
$\alpha^{-n}_k$, $A^{-n}_k$ is conditionally independent of all
$A^{-m}_j$, for $-m > -n$.\vadjust{\goodbreak}
Denote the (random) time that $X^{-n}$ hits 1 by
\[
T^{0,-n}_1 = \inf\{t | X^{-n}(t)=1 \}.
\]
We define $X^{-n}(t)$ for all $t \in[0, T^{0,-n}_1]$ by linear
interpolation, and set $X^{-n}(t) = 1$ for all $t > T^{0,-n}_1$.
The interpolated $X^{-n}$ have continuous sample paths, and we will
show that they converge uniformly on any finite interval, from which
the continuity of the limit process follows.
For any $m \leq n$, let $T_0^{-m,-n}=0$ and
\[
T_{k+1}^{-m,-n}= \inf\{t> T_k^{-m,-n} | X^{-n}(t)\in2^{-m}\mathbb{Z}
,X^{-n}(t)\neq X^{-n}(T_k^{-m,-n})\}.
\]
If $X^{-n}(T_k^{-m,-n})=1$, then set $T_{k+1}^{-m,-n}= \infty$.
By construction $X^{-n}(T^{-m,-n}_k) = X^{-m}(\mu^{-m}k)$, for all $k$
and $m \leq n$.
The duration of the $k$th level $-m$ crossing of $X^{-n}$ is
$D_k^{-m,-n}= T_k^{-m,-n}- T_{k-1}^{-m,-n}$.
A realization of $X^0$, $X^{-1}$ and $X^{-2}$ is given in
Figure~\ref{Spmfig}, with the associated crossing tree.
We use a branching process result to establish that the crossing
durations converge.
When we defined the crossing tree (see Figure~\ref{crossingtree}) we
started with a sample path and then defined generations of crossings:
taking the first crossing of size 1 as the root (level or generation
0), its subcrossings of size $1/2$ form the second generation (or
level), its subcrossings of size $1/4$ form the third generation, and
so on.
Each crossing can be up or down, so our tree has two types of nodes.
Here we are reversing that process.
That is, we are growing a tree using a two-type Galton--Watson process,
and from the tree, constructing a sample path.
The offspring distributions for our tree are just $p^\pm_A$.
Given the tree at generation $n$, we get an approximate sample path by
taking a sequence of up and down steps of size $2^{-n}$ and duration
$\mu^{-n}$, with directions taken from the node types of the tree.
We need to show that the sequence of sample paths, obtained as $n\to
\infty$, converges.
Consider the subtree descending from crossing $C^{-m}_k$.
Let $S^{+}_{-m,k}(n-m)$ and $S^{-}_{-m,k}(n-m)$ be the number of up and
down crossings of size $2^{-n}$ which are descended from the $k$th
crossing of size $2^{-m}$; then $\{(S^{+}_{-m,k}(n-m),
S^{-}_{-m,k}(n-m))\}_{n=m}^\infty$ is a two-type Galton--Watson process.
From Athreya and Ney~\cite{AN72}, Section~V.6, Theorems 1 and 2, we
have that as $n \to\infty$, $\mu^{m-n}(S^{+}_{-m,k}(n-m),
S^{-}_{-m,k}(n-m))$ converges almost surely and in mean to $(\half,
\half) W^{-m}_k$, where $W^{-m}_k$ is strictly positive, continuous and
$\mathbb{E}(W^{-m}_k | \alpha^{-m}_k = \pm) = (\mu^\pm- 2)/(\mu- 2)$.
Moreover, the distribution of $W^{-m}_k$ depends only on $\alpha^{-m}_k$,
and for any fixed $m$ the $W^{-m}_k$ are all independent.
Finally, since $S^{+}_{-m,k}(n-m) + S^{-}_{-m,k}(n-m) = \mu
^{n}D^{-m,-n}_k$, we have
\[
D_k^{-m,-n}\rightarrow\mu^{-m} W_k^{-m} \qquad\mbox{a.s. as
}n\rightarrow
\infty.
\]
Accordingly, let $T_k^{-m} = \sum_{j=1}^k \mu^{-m} W_j^{-m} = \lim
_{n\rightarrow\infty} T_k^{-m,-n}$.\vspace*{1pt}
Take any $\varepsilon>0$, $\delta>0$ and $T>0$.
To establish the a.s. convergence of the processes $X^{-n}$, uniformly
on compact intervals, we show that we can find a $u$ so that with
probability $1-\varepsilon$,
\begin{equation}\label{eqconti}
|X^{-r}(t)- X^{-s}(t)|\leq\delta\qquad\mbox{for all } r,s \geq u
\mbox{ and } t\in[0,T].
\end{equation}
Given $t\in[0,T]$, let $k=k(n,t)$ be such that
\[
T_{k-1}^{-n}\leq t < T_k^{-n}.
\]
For any $r,s \geq n$, the triangle inequality yields
\begin{eqnarray}\label{eqineqtri}
|X^{-r}(t)- X^{-s}(t)|&\leq&|X^{-r}(t)-X^{-r}(T_k^{-n,-r})|\nonumber\\[-8pt]\\[-8pt]
&&{}+ |X^{-s}(T_k^{-n,-s})-X^{-s}(t)|,\nonumber
\end{eqnarray}
since $X^{-r}(T_k^{-n,-r})=X^{-s}(T_k^{-n,-s})= X^{-n}(k\mu^{-n})$.
For any $u \geq n$ let $j=j(n,u)$ be the smallest $j$ such that
$T_j^{-n,-u}> T$.
As $u\rightarrow+\infty$, $j(n,u)\rightarrow j(n) < \infty$ a.s., so
for any $n$ we can choose $\varepsilon_0$ such that
\[
\mathbb{P}\Bigl(\min _{i\leq j(n)}\mu^{-n} W_i^{-n} \geq\varepsilon_0
\Bigr) \geq1-\varepsilon,
\]
and $u$ such that for all $q \geq u$,
\[
\mathbb{P}\Bigl(\max _{i\leq j(n)}|T_i^{-n,-q}-T_i^{-n}| <\varepsilon_0
\Bigr) \geq1-\varepsilon,
\]
which yields
\[
\mathbb{P}\Bigl(\max _{i\leq j(n)}|T_i^{-n,-q}-T_i^{-n}|
< \min _{i\leq j(n)}\mu^{-n}W_i^{-n} \Bigr) \geq1 -
\varepsilon.
\]
Thus, given $n$ we can find $u$ such that for all $q \geq u$, with
probability at least $1-\varepsilon$,
\[
T^{-n,-q}_{k-2}<t<T_{k+1}^{-n,-q} \qquad\mbox{for all }t\in[0,T].
\]
Now, since $X^{-q}(T_{k-2}^{-n,-q})= X^{-n}((k-2)\mu^{-n})$,
$X^{-q}(T_{k+1}^{-n,-q})= X^{-n}((k+1)\mu^{-n})$, and in three steps
$X^{-n}$ can move at most distance $3\cdot2^{-n}$, we have
\[
|X^{-q}(t)-X^{-q}(T_k^{-n,-q})|\leq3\cdot2^{-n}.
\]
Choosing $n$ large enough that $6\cdot2^{-n}\leq\delta$, we see that
(\ref{eqconti}) follows from (\ref{eqineqtri}).
Sending $\delta$ and $\varepsilon$ to $0$ shows that $X^{-n}$ converges to
some continuous limit process $X$ uniformly on all closed intervals
$[0, T]$, with probability $1$.
By construction, the duration of crossing $C^{-n}_k$ is $\mu^{-n}W^{-n}_k$.
\textsc{Step 2.}
Clearly the construction above can be used to generate any crossing
from 0 to $\pm2^n$.
Thus, to extend our construction from a single crossing to a process
$X(t)$ defined for all $t \in\mathbb{R}_+$, we proceed by constructing a
nested sequence of processes $\{X^{(n)}\}_{n=0}^\infty$, such that
$X^{(n)}$ is a crossing from 0 to $\pm2^n$, and the first level $n$
crossing of $X^{(n+1)}$ is precisely $X^{(n)}$.
To make this work, we just need to specify $\mathbb{P}(X^{(n)}(T^n_1) = 2^n)$
in a consistent manner.
Consider the orientation of the first crossing from 0 to $\pm2^n$ for
an EBP process.
Let $u = \mathbb{P}(\alpha^n_1 = + | \alpha^{n+1}_1 = +)$ and $v = \mathbb{P}(\alpha
^n_1 =
+ | \alpha^{n+1}_1 = -)$; then $u$ and $v$ are determined by $p^\pm
_{A}$, and
\begin{equation}\label{firstcrossingeqn}
a_n:= \mathbb{P}(\alpha^n_1 = +) = u a_{n+1} + v (1 - a_{n+1}) = v + (u-v) a_{n+1}.
\end{equation}
For $(u, v) \in[0, 1]^2 \setminus\{(1,0)\}$, we see that equation
(\ref{firstcrossingeqn}) has fixed point $a = v/(1-u+v) \in[0, 1]$.
Moreover, the only doubly infinite sequence $\{ a_n \}_{n = -\infty
}^\infty$ which satisfies (\ref{firstcrossingeqn}) and remains in
$[0, 1]$ is given by $a_n = a$ for all $n$.
Given this, it follows that $a_n = a$, and thus from Bayes's theorem
that $\mathbb{P}(\alpha^{n+1}=+ | \alpha^n_1=+) = u$ and $\mathbb{P}(\alpha^{n+1}=+
|
\alpha^n_1=-) = v$.
If $(u, v) = (1, 0)$, then any $a \in[0, 1]$ is possible, but
everything else goes through as before.
In this case the $\alpha^n_1$ are all the same, but may be of either type.
Construct $X^{(0)}$ as a crossing from 0 to 1 with probability $a$ [the
fixed point of~(\ref{firstcrossingeqn})], otherwise as a
crossing from 0 to $-1$.
Then, given $X^{(n)}$, construct $X^{(n+1)}$ as follows: first, put
$\alpha
^{n+1}_1 = +$ with probability $u$ if $\alpha^0_1=+$, with probability $v$
otherwise; second, generate $A^{n+1}_1$ conditional on $\alpha^{n+1}_1$
and $\alpha^n_1$; third, use $X^{(n)}$ as the first level $n$ crossing of
$X^{(n+1)}$; finally construct the remaining level $n$ crossings
conditional on $\alpha^{n}_2, \alpha^{n}_3, \ldots, \alpha^{n}_{Z^{n+1}_1}$.
Write $X$ for the limit of the $X^{(n)}$.
To complete our construction we just need to check that the process $X$
does not escape to $\pm\infty$ in finite time.
By construction, we have $T^n_1 = \inf\{ t | X(t) = \pm2^n\} =
\mu
^n W^n_1$, where $W^n_1$ is strictly positive, continuous, and has a
distribution depending only on the orientation $\alpha^n_1$.
Thus for any $T > 0$, $\mathbb{P}(T^n_1 < T) \to0$ as $n\to\infty$.
\end{pf}
\begin{theorem}\label{CEBP2}
Let $X$ be the EBP constructed in Theorem~\ref{CEBP}; then, for
each~$n$, conditioned on the crossing orientations $\alpha^n_k$, the
crossing durations $D^n_k$ are all mutually independent, and $D^n_k$ is
conditionally independent of all $A^m_j$ for $m > n$. Also, $\mathbb{E}(D^n_k
| \alpha^n_k = \pm) = \mu^{n} (\mu^\pm- 2)/(\mu- 2)$, and the
distribution of $\mu^{-n}D^n_k$ depends only on $\alpha^n_k$. Moreover, up
to finite-dimensional distributions, $X$~is the unique such EBP with
offspring orientation distributions $p^\pm_{A}$. That is,\vspace*{1pt} for any other
EBP process $Y$ with offspring orientation distributions $p^\pm_A$ and
crossing durations as above, we have $(X(t_1), \ldots, X(t_k))
\eqdist(Y(t_1), \ldots, Y(t_k))$ for any $0 \leq t_1 < t_2 < \cdots<
t_k$.
Accordingly, we call $X$ the Canonical EBP (CEBP) process with these
offspring distributions.
We also observe that $X$ is discrete scale-invariant: let $H=\log
2/\log
\mu$; then for all $c\in\{\mu^n, n\in\mathbb{Z}\}$,
\begin{equation}\label{eqholder}
X(t) \stackrel{\mathit{fdd}}{=} c^{-H} X(ct),
\end{equation}
where $\stackrel{\mathit{fdd}}{=}$ denotes equality for finite-dimensional
distributions.
$H = \log\mu/\log2$ is known as the Hurst index.
\end{theorem}
\begin{pf}
We retain the notation of Theorem~\ref{CEBP}.
For the process $X$, the dependence structure of the crossing durations
is clear from the construction.
To show\vspace*{1pt} uniqueness, let $Y$ be some other EBP process with offspring
orientation distributions $p^\pm_{A}$, and crossing durations
satisfying the conditions of the theorem statement.
We will make use of the same notation for the crossing times,
durations, orientations, etc. of $Y$ as for $X$, and rely on the
context to distinguish them.
For an EBP, the finite joint distributions of the orientations $A^n_k$
are determined completely by $p^\pm_{A}$, and thus are identical for
$X$ and $Y$.\vadjust{\goodbreak}
For the crossing durations of $Y$, note that for any $m \leq n$ and
$k$, we have
\begin{equation}\label{poinc}
\mu^m D^{-m}_k = \mu^{m-n} \sum_{j=\zeta(-m,-n,k)+1}^{\zeta(-m,-n,k+1)}
\mu^{n} D_{j}^{-n},
\end{equation}
where $\zeta(-m,-n,k)$ is such that $\zeta(-m,-n,k)+1$ is the index of
the first level $-n$ subcrossing of $C^{-m}_k$.
Thus by the strong law of large numbers, sending $n\to\infty$,
\begin{eqnarray*}
\mu^m D^{-m}_k
&=& \mu^{m-n}S^{+}_{-m,k}(n-m)
\Biggl[ \frac{\mu^n}{S^{+}_{-m,k}(n-m)}
\mathop{\sum_{j=\zeta(-m,-n,k)+1}}_{\alpha^{-n}_j=+}^{\zeta(-m,-n,k+1)}
D^{-n}_j \Biggr] \\[-2pt]
&&{}+ \mu^{m-n}S^{-}_{-m,k}(n-m)
\Biggl[ \frac{\mu^n}{S^{-}_{-m,k}(n-m)}
\mathop{\sum_{j=\zeta(-m,-n,k)+1 }}_{
\alpha ^{-n}_j=-}
^{\zeta(-m,-n,k+1)}
D^{-n}_j \Biggr] \\[-2pt]
&\stackrel{\mathbb{P}}\longrightarrow& \half W^{-m}_k \mu^n \mathbb{E}(D^{-n}_j | \alpha
^{-n}_j=+) +
\half W^{-m}_k \mu^n \mathbb{E}(D^{-n}_j | \alpha^{-n}_j=-) \\[-2pt]
&=& W^{-m}_k,
\end{eqnarray*}
where the distribution of $W^{-m}_k$ is completely determined by $p^\pm
_{A}$, and thus is the same for $X$ and $Y$.
Once we have the crossing orientations and the assumed dependence
structure of the crossing durations, the crossing distributions (for up
and down types) determine the joint distributions of the crossing times
$\{ T^n _k \}$.
Thus, for any $n$ and $k$, $\{ X(T^n_i) \}_{i=0}^k$ and $\{ Y(T^n_i) \}
_{i=0}^k$ are identically distributed.
Since any $t$ can be bracketed by a sequence of hitting times, $X$ and
$Y$ are identical up to finite-dimensional distributions.
That $X$ is discrete scale-invariant is a direct consequence of its
construction, since simultaneously scaling the state space by $2^k$ and
time space by $\mu^k$ does not change the distribution of $X$.
\end{pf}
\begin{remark}
From~\cite{H92} it is clear that Brownian motion is an example of a
CEBP process, where the offspring of any crossing consist of a
geometric ($1/2$) number of excursions, each up--down or down--up with
equal probability, followed by either an up--up or down--down direct crossing.
That is,\looseness=-1
\[
p^+_{A}(\cdots{+} {+}) = p^-_{A}(\cdots{-} {-}) = 2^{-(z+1)},
\]\looseness=0
where $\cdots$ represents a combination of $z$ pairs, each either $+ -$
or $- +$.
It follows that $\mathbb{P}(Z^n_k = 2x) = 2^{-x}$, independently of $\alpha^n_k$.
\end{remark}
\section{From CEBP to MEBP}
\label{CEBPMEBP}
In this section we construct Multifractal Embedded Branching processes
(MEBP processes) as time changed CEBP processes.\vadjust{\goodbreak}
Consider initially a single crossing of a CEBP $X$, from 0 to $\pm1$.
We constructed $X$ as the limit of a sequence of processes $X^{-n}$,
which take steps of size $2^{-n}$ and duration $\mu^{-n}$.
The crossing tree gives the number of subcrossings of each crossing.
If we add a weight of $1/\mu$ to each branch of the tree, then
truncating the tree at level $-n$, the product of the weights down any
line of descent is $\mu^{-n}$, which is the duration of any single
crossing by $X^{-n}$.
We generalize this by allowing the weights to be random, then defining
the duration of a crossing to be the product of the random weights down
the line of descent of the crossing.
The resulting process, $Y^{-n}$ say, can be viewed as a time-change of
$X^{-n}$, where the time-change is obtained from a multiplicative
cascade defined on a (two-type) Galton--Watson tree.
As for CEBP, we will initially construct a single level 0 crossing of
an MEBP, then extend the construction to $\mathbb{R}_+$.
We will retain the notation of Section~\ref{secEBP}, but note that we
will prefer the tree indexing scheme to the level/position indexing
scheme in what follows.
In particular, the number of level $-n$ up and down subcrossings of
node $\mathbf{i}$ in level $-m$ are denoted $S^+_\mathbf{i}(n-m)$ and $S^-_\mathbf{i}
(n-m)$, and, under Assumption~\ref{AssGW}, the almost sure limit and
mean limit of $\mu^{m-n}(S^+_\mathbf{i}(n-m), S^-_\mathbf{i}(n-m))$ is $(\half,
\half)W_\mathbf{i}$.
The duration of crossing $\mathbf{i}$ of the CEBP process $X$ is then $\mu
^{-m}W_{\mathbf{i}}$.
We assign weight $R_\mathbf{i}(j)$ to the branch $(\mathbf{i}, \mathbf{i} j)$.
$R_\mathbf{i}:= (R_\mathbf{i}(1), \ldots, R_\mathbf{i}(Z_\mathbf{i}))$ may depend on
$A_\mathbf{i}$,
but conditioned on $\alpha_\mathbf{i}$ must be independent of other nodes
$\mathbf{j}$
that are not descendants of $\mathbf{i}$.
For $r \in\mathbb{R}_+^{|a|}$, write $F^\pm_{R|a}(r) = \mathbb{P}(R_\varnothing(1)
\leq
r(1), \ldots, R_\varnothing(z) \leq r(z) | \alpha_\varnothing=\pm,
A_\varnothing=a, |a|=z)$ for the joint distribution of $R_\varnothing$,
conditioned on the crossing orientations $a$.
The weight attributed to node $\mathbf{i}$ is
\[
\rho_{\mathbf{i}} = \prod _{k=0}^{|\mathbf{i}|-1} R_{\mathbf{i}|_{k}}(\mathbf{i}[k+1]).
\]
That is, $\rho_\mathbf{i}$ is the product of all weights on the line of
descent from the root down to node $\mathbf{i}$.
We use the weights to define a measure, $\nu$, on the boundary of the
crossing tree.
The measure $\nu$ on $\partial\Upsilon$ is then mapped to a measure
$\zeta$
on $\mathbb{R}$, with which we define a chronometer $\mathcalM$ (a nondecreasing
process) by\vspace*{1pt} $\mathcalM(t) = \zeta([0, t])$.
The MEBP process is then given by $Y = X \circ\mathcalM^{-1}$, where $X$
is the CEBP.
The crossing trees of $X$ and $Y$ have the same spatial structure, but
have different crossing durations.
In Figure~\ref{EBPMEBP} we plot a realization of an MEBP process and
its associated CEBP.
The literature on multiplicative cascades is rather extensive.
For the existence of limit random measures and the study of the
properties of certain martingales defined on $m$-ary trees, one can
refer, for instance, to the works of Kahane and Peyri\`ere~\cite{KP76},
Barral~\cite{B99}, Liu and Rouault~\cite{LR00} and Peyri\`ere~\cite{P00}.
For results on random cascades defined on Galton--Watson trees, see,
for example, Liu~\cite{L99,L00}, and Burd and Waymire~\cite{BW00}.
To obtain the time-change process explicitly, the random measure
defined on the boundary of the tree is mapped to $\mathbb{R}_+$ then
integrated. Note that this mapping, given explicitly in Section
\ref{cascade}, differs from random partitions previously considered in
the literature. The usual approach is to use a ``splitting measure'' to
map the boundary of the tree to $[0, 1]$, then use the density of the
cascade measure with respect to the splitting measure; see, for
example,~\cite{P77,P79,R03}. Our approach can be thought of as using
the ``branching measure'' instead of a splitting measure. A~splitting
measure is constructed by splitting the mass associated with a given
node between its offspring, with no mass lost or gained. The branching
measure allocates mass according to the number of offspring, and is
only conserved in mean. We have taken the terminology of splitting and
branching measures from~\cite{L00}, Example 1.3. A multifractal study
of the measure we construct on $\mathbb{R}_+$ is given in a forth-coming paper
\cite{DHJ}.
\subsection{\texorpdfstring{The measure $\nu$}{The measure nu}}\label{measurenu}
To construct $\nu$, we use a well-known correspondence between
branching random walks and random cascades, in which the offspring of
individual $\mathbf{i}$ have types given by $A_\mathbf{i}$ and displacements
(relative to $\mathbf{i}$) given by $-{\log R_\mathbf{i}}$.
For background on multitype branching random walks, we refer the reader
to Kyprianou and Sani~\cite{KS01} and Biggins and Sani~\cite{BS05}.
Suppose $|\mathbf{i}|=m$ and $n \geq m$.
Define
\[
\mathcalW^\pm_\mathbf{i}(n-m, \theta) = \sum_{\mathbf{j}\in\Upsilon_n\cap\Upsilon_\mathbf{i},
\alpha_\mathbf{j}=\pm
} (\rho_\mathbf{j}/\rho_\mathbf{i})^\theta
\]
and for $i,j = \pm$,
\begin{eqnarray*}
m_{i,j}(\theta) &=& \mathbb{E}\bigl(\mathcalW^j_\varnothing(1, \theta) | \alpha
_\varnothing=i\bigr)
\\
&=& \mathbb{E}\biggl(\sum_{1\leq k\leq Z_\varnothing, \alpha_k=j}
R_\varnothing
(k)^\theta \Big| \alpha_\varnothing=i \biggr).
\end{eqnarray*}
Let $M(\theta) = (m_{i,j}(\theta) )_{i,j=\pm}$, and write
$m^n_{i,j}(\theta)$
for the $(i,j)$ entry of the $n$th power $M^n(\theta)$. Then it is
straight forward to check that $\mathbb{E}( \mathcalW^j_\mathbf{i}(n-m, \theta) |
\alpha
_\mathbf{i}=i) = m^{n-m}_{i,j}(\theta)$.
If we take constant weights equal to $1/\mu$, then $\mathcalW^\pm_\mathbf{i}(n-m,
1) = \mu^{m-n} S^\pm_\mathbf{i}$, in the notation of Theorem~\ref{CEBP}.
Let $\mu(\theta)$ be the largest eigenvalue of $M(\theta)$.
We make the following assumptions about $R_\varnothing$.
\begin{assp}\label{Ass1}
We suppose that $0 < R_\varnothing< \infty$ a.s., $M(\theta) < \infty$ in
an open neighborhood of 1, $\mu(1) = 1$ and $\mu'(1) < 0$.
In the case where the distribution of $R_\varnothing$ (and thus
$Z_\varnothing$) does not depend on $\alpha_\varnothing$, we assume in
addition that
\[
\mathbb{E}\Biggl( \sum_{j=1}^{Z_\varnothing} R_\varnothing(j) \log\sum
_{j=1}^{Z_\varnothing} R_\varnothing(j) \Biggr) < \infty.
\]
In the case where there is dependence on the crossing orientation
(type), we suppose that for some $\delta> 1$, $\mu(\delta) < 1$ and
\[
\mathbb{E}\Biggl( \Biggl( \sum_{j=1}^{Z_\varnothing} R_\varnothing(j)
\Biggr)^\delta \bigg| \alpha_\varnothing=i \Biggr) < \infty\qquad\mbox{for }
i =
\pm.
\]
\end{assp}
Note that if the weights are finite and strictly positive, then $M$ and
$\mu$ from the previous section are just $M(0)$ and $\mu(0)$, and from
Assumption~\ref{AssGW} we get $0 < M(\theta)$ for all $\theta\geq0$.
In the case where $R_\varnothing$ does not depend on $\alpha_\varnothing$, the
BRW simplifies to a single-type process, and the condition $\mu(1) = 1$
simplifies to $\mathbb{E}( \sum_{j=1}^{Z_\varnothing} R_\varnothing(j)
)
= 1$, which we recognize as a conservation of mass condition.
Left and right eigenvectors corresponding to $\mu(1)$ will be denoted
$\mathbf{u}= (u^+, u^-)$ and $\mathbf{v}= (v^+, v^-)^T$, normed so that $\mathbf{u}(1,
1)^T = 1$ and $\mathbf{u}\mathbf{v}= 1$.
The following lemma is a direct consequence of Biggins and Kyprianou
\cite{BK04}, Theorem 7.1, and Biggins and Sani~\cite{BS05}, Theorem 4.
\begin{lem}\label{BRWlem}
Under Assumptions~\ref{AssGW} and~\ref{Ass1}, $(\mathcalW^+_\mathbf{i}(n-m, 1),
\mathcalW^-_\mathbf{i}(n-m, 1))$ converges almost surely to $\mathbf{u}\mathcalW_\mathbf{i}$, for
some random variable $\mathcalW_\mathbf{i}$ such that the distribution of
$\mathcalW
_\mathbf{i}$ depends only on $\alpha_\mathbf{i}$, and $\mathbb{E}( \mathcalW_\mathbf{i} | \alpha
_\mathbf{i}=
i) = v^i$.
Moreover, for each $n$, conditioned on the crossing orientations $\alpha
_\mathbf{i}$, $\mathbf{i}\in\Upsilon_n$, the $\mathcalW_\mathbf{i}$ are mutually independent,
and $\mathcalW_\mathbf{i}$ is conditionally independent of $(A_\mathbf{j}, R_\mathbf{j})$ for
$|\mathbf{j}| < |\mathbf{i}|$.
For all nodes~$\mathbf{i}$,
\begin{equation}\label{eqE}
\mathcalW_\mathbf{i}= \sum _{j=1}^{Z_{\mathbf{i}}} R_\mathbf{i}(j) \mathcalW_{\mathbf{i} j}.
\end{equation}
\end{lem}
Note that in the case where $R_\varnothing$ does not depend on $\alpha
_\varnothing$, the right eigenvector $\mathbf{v}= (1, 1)^T$.
We can now define the measure $\nu$ on $\partial\Upsilonsilon$.
Recall $\Upsilon_\mathbf{i}= \{\mathbf{j}\in\Upsilon | |\mathbf{j}|\geq|\mathbf{i}|$
and
\mbox{$\mathbf{j}|_{|\mathbf{i}|}=\mathbf{i}\}$}, so $\partial\Upsilonsilon_{\mathbf{i}}$ contains all the nodes on the
boundary of the tree which have $\mathbf{i}$ as an ancestor.
We define $\nu(\partial\Upsilonsilon_\mathbf{i}) = \rho_\mathbf{i}\mathcalW_\mathbf{i}$.
By Carath\'eodory's extension theorem, we can uniquely extend $\nu$ to
the sigma algebra generated by these cylinder sets.
\subsection{\texorpdfstring{The measure $\zeta$ and time change $\mathcalM$}
{The measure zeta and time change $\mathcalM$}}\label{cascade}
The measure $\zeta$ is a mapping of $\nu$ from $\partial\Upsilonsilon$ to $[0,
W_\varnothing] \subset\mathbb{R}$.
By analogy with $m$-ary cascades, we call $\zeta$ a Galton--Watson
cascade measure on $[0, W_\varnothing]$.
As above, let $T^{-n}_k$ denote the $k$th level $-n$ passage time of
the CEBP process~$X$, and put
\[
\zeta((T^{-n}_{k-1}, T^{-n}_k]):= \nu(\partial\Upsilonsilon^{-n}_k)= \rho^{-n}_k
\mathcalW^{-n}_k.
\]
Putting $\zeta(\{0\}) = 0$, this gives us $\zeta([0, T^{-n}_k])$ for
all $n, k \geq0$.
For arbitrary $t \in(0, W_\varnothing]$, let $\mathbf{i}\in\partial\Upsilonsilon$ be such
that $t \in(T^{-n}_{\psi(\mathbf{i}|_n)-1}, T^{-n}_{\psi(\mathbf{i}|_n)}]$ for all
$n \geq0$.
Noting that $T^{-n}_{\psi(\mathbf{i}|_n)} = T_{\mathbf{i}|_n}$ is a nonincreasing
sequence, we define $\zeta([0, t]) = \lim_{n\to\infty}\zeta([0,
T^{-n}_{\psi(\mathbf{i}|_n)}])$.\vspace*{2pt}
We can now define $\mathcalM(t) = \zeta([0,t])$, and define the MEBP
process $Y$ (on $[0, \mathcalW_\varnothing]$) as
\[
Y = X \circ{\mathcal M}^{-1}.
\]
Here we take $\mathcalM^{-1}(t) = \inf\{s\dvtx\mathcalM(s) \geq t\}$, so that
it is well defined, even if $\mathcalM$ has jumps or flat spots.
Put $\mathcalT^{-n}_k = \mathcalM(T_{k}^{-n}) = \sum_{j=1}^k \rho^{-n}_j
\mathcalW
^{-n}_j$. Then $Y({\mathcal T}_{k}^{-n}) = X(T_{k}^{-n})$, so $\mathcalT
^{-n}_k$ is the $k$th level $-n$ crossing time for $Y$, and $\mathcalD
^{-n}_k = \rho^{-n}_k \mathcalW^{-n}_k$ the $k$th level $-n$ crossing duration.
Note that if we take constant weights equal to $1/\mu$, then $\mathcalT
^{-n}_k = T_{k}^{-n}$ and $Y=X$.
\begin{lem}
Under Assumptions~\ref{AssGW} and~\ref{Ass1}, $\mathcalM$ and $\mathcalM^{-1}$
are continuous.
That is, $\mathcalM$ has neither jumps nor flat spots.
\end{lem}
\begin{pf}
To show that $\mathcalM$ has no flat spots, it is enough to show that:
\begin{longlist}[(b)]
\item[(a)]
\[
\max_k \mu^{-n} W^{-n}_k \stackrel{\mathbb{P}}\longrightarrow0 \qquad\mbox{as } n\to\infty,
\]
\item[(b)]
\[
\mathcalW^{-n}_k > 0 \qquad\mbox{a.s. for each } n, k \geq0.
\]
\end{longlist}
Property (a) follows directly from Theorem 1 in~\cite{O80}, noting that
under Assumption~\ref{AssGW} $\mathbb{E}( W_\mathbf{i} | \alpha_\mathbf{i}=\pm) <
\infty$,
so that $\int_0^y x \,dF^\pm(x)$ is slowly varying, where $F^\pm(x) =
\mathbb{P}
( W_\mathbf{i}\leq x | \alpha_\mathbf{i}=\pm)$.
This is equivalent to saying that the measure $\bar{\nu}$, defined on
$\partial\Upsilonsilon$ by $\bar{\nu}(\partial\Upsilonsilon_\mathbf{i}) = \mu^{-|\mathbf{i}|}W_\mathbf{i}$, has no atoms.
To show (b), let $q_\pm= \mathbb{P}(\mathcalW_\varnothing=0 | \alpha_\varnothing
=\pm
)$, then note that since the weights $R_\varnothing> 0$, we have, from
(\ref{eqE}), that
\begin{equation}\label{eqE2}
q_i = f_{i}(q_+, q_-),
\end{equation}
where $f_{i}$ is the joint probability generating function of $Z^\pm
_\varnothing$ given $\alpha_\varnothing=i$.
[Note that $\bar{q}_\pm= \mathbb{P}(W_\varnothing=0 | \alpha_\varnothing
=\pm)$
satisfy the same equations.]
Since $(Z^i_\varnothing | \alpha_\varnothing=i) \geq2$ and $\mathbb{P}(Z^\pm
_\varnothing= 2 | \alpha_\varnothing=i) < 1$, we have for $(q_+, q_-)
\in
[0, 1]^2 \setminus\{ (0,0), (1,1) \}$, $f_i(q_+, q_-) < q_i$.
Thus the only solutions to (\ref{eqE2}) are $(0, 0)$ and $(1, 1)$, and
as $\mathbb{E}( \mathcalW_\varnothing | \alpha_\varnothing=\pm) > 0$, we get
$q_\pm= 0$.
$\mathcalM$ is continuous (has no jumps) if $\zeta$ has no atoms.
That is,
\begin{longlist}[(b*)]
\item[(a*)]
\[
\max_k \rho^{-n}_k \mathcalW^{-n}_k \stackrel{\mathbb{P}}\longrightarrow0 \qquad\mbox{as } n\to
\infty,\vadjust{\goodbreak}
\]
\item[(b*)]
\[
W^{-n}_k > 0 \qquad\mbox{a.s. for each } n, k \geq0.
\]
\end{longlist}
We prove (b*) in exactly the same way as (b).
Property (a*) is equivalent to saying that $\nu$ has no atoms.
In the case where the distribution of $R_\varnothing$ does not depend on
$\alpha_\varnothing$, the BRW embedded in the crossing tree is effectively
single-type, and (a*) is given by Liu and Rouault~\cite{LR97}, Theorem 6.
In the case where the distribution of $R_\varnothing$ does depend on
$\alpha
_\varnothing$, the approach of~\cite{LR97} generalizes only as far as the
end of their Lemma 13, at which point we require, for some $\lambda< 1$,
\begin{equation}\label{rightmosteqn}
\mathbb{E}\nu(\{ \mathbf{i}\dvtx\rho_{\mathbf{i}|_n} \geq\lambda^n\}) \to0
\qquad\mbox{as }
n \to\infty.
\end{equation}
However, this can be shown using some recent results of Biggins \cite
{B10}, as we now demonstrate.
In the notation of~\cite{B10}, consider a BRW with offspring types $\{
\sigma_i \} \eqdist\{ A_\varnothing(i) \}$ and displacements $\{ z_i \}
\eqdist\{ \log(R_\varnothing(i)\gamma) \}$, for some $\gamma> 1$.
Put
\[
\bar{m}_{i,j}(\theta) = \mathbb{E}\biggl( \sum_{1 \leq k \leq
Z_\varnothing,
A_\varnothing(k)=j} R_\varnothing(k)^\theta\gamma^\theta \Big| \alpha
_\varnothing=i \biggr)
\]
(this is $m_{i,j}$ in the notation of \cite
{B10}). Then the matrix $\bar{M}(\theta) = (\bar{m}_{i,j}(\theta
))_{i,j=\pm
}$ has maximum ``Perron--Frobenius'' eigenvalue $\kappappa(\theta) = \mu
(\theta) \gamma^\theta$.
From assumptions~\ref{AssGW} and~\ref{Ass1} it is clear that for some
$\theta> 0$, $\bar{M}(\theta)$ is finite, irreducible and primitive.
Let $\mathcalB^{(n)}_i$ be the rightmost particle of type $i$ in generation
$n$, that is,
\[
\mathcalB^{(n)}_\pm= \max_{\mathbf{i}\in\Upsilon_n, \alpha_\mathbf{i}=\pm} \log\rho
_\mathbf{i}+ n
\log\gamma.
\]
Then Proposition 5.6 of~\cite{B10} shows that
\[
\frac{\mathcalB^{(n)}_\pm}n \convergeas\Gamma(\kappappa^*),
\]
where $\kappappa^*(a) = \sup_{\theta\geq0}\{\theta a - \kappappa(\theta
)\}$
and $\Gamma(\kappappa^*) = \sup\{a\dvtx\kappappa^*(a) < 0\}$.
We have $\kappappa(0) = \mu(0)$, $\kappappa(1) = \gamma$, $\kappappa'(1) =
\mu
'(1)\gamma+ 1$, and for $\gamma$ large enough, $\kappappa(\theta) \to
\infty$ as $\theta\to\infty$, faster than linear.
$\Gamma(\kappappa^*)$ corresponds to the slope of the line that passes
through the origin and is tangent to $\kappappa^*$, from which it follows
that $\Gamma(\kappappa^*) < \gamma$ provided that $\kappappa'(1) \neq
\gamma
$, that is, provided $\mu'(1) \neq(\gamma- 1)/\gamma$.
But $\mu'(1) < 0$ and $\gamma> 1$ by assumption, so $\kappappa'(1) \neq
\gamma$, and we get
\[
\max_{\mathbf{i}\in\Upsilon_n, \alpha_\mathbf{i}=\pm} \frac{\log\rho_\mathbf{i}}n
\convergeas
\Gamma(\kappappa^*) - \gamma< 0.
\]
Equation (\ref{rightmosteqn}) follows immediately, completing the
proof of our lemma.
\end{pf}
\subsection{Extending the construction to $\mathbb{R}_+$}\label{secextend}
We can extend $Y$ from $[0, \mathcalW_\varnothing]$ to $\mathbb{R}_+$ in much the
same way we extended the CEBP $X$, by constructing a sequence of nested
processes $Y^{(n)}$, where $Y^{(n)}$ consists of a a single level $n$\vadjust{\goodbreak}
crossing from 0 to $\pm2^n$, and the first level $n$ crossing of
$Y^{(n+1)}$ is precisely $Y^{(n)}$.
As for the CEBP we need to specify $\mathbb{P}(Y^{(n)}(\mathcalT^n_1) = 2^n)$ in a
consistent manner, but we also need to scale the first crossing.
Construct $Y^{(0)}$ as a crossing from 0 to 1 with probability $a$ [the
fixed point of (\ref{firstcrossingeqn})], otherwise as a
crossing from 0 to $-1$.
Then, given $Y^{(n)}$, construct $Y^{(n+1)}$ as follows: first, put
$\alpha
^{n+1}_1 = +$ with probability $u$ if $\alpha^n_1=+$ and probability $v$
otherwise; second, generate $(A^{n+1}_1, R^{n+1}_1)$ conditional on
$\alpha
^{n+1}_1$ and $\alpha^n_1$; third, scale the weights $R^{n+1}_1$ by
$1/R^{n+1}_1(1)$; fourth, use $Y^{(n)}$ as the first level $n$ crossing
of $Y^{(n+1)}$; finally, construct the remaining level $n$ crossings
conditional on\vspace*{-2pt} $\alpha^n_2, \alpha^n_3, \ldots, \alpha^n_{Z^{n+1}_1}$.
Write $Y$ for the limit of the $Y^{(n)}$.
When constructing $Y^{(n+1)}$ we take $Z^{n+1}_1$ independent
processes, each constructed like $Y^{(n)}$, then scale the first by $1
= R^{n+1}_1(1)/R^{n+1}_1(1)$, the second by
$R^{n+1}_1(2)/R^{n+1}_1(1)$, and so on, before stitching them together.
When constructing the second and subsequent level $n$ crossings of
$Y^{(n+1)}$, we proceed exactly as for the construction of $Y^{(0)}$,
except for a spatial scaling of $2^n$ and a temporal scaling of $\prod
_{k=1}^n 1/R^{k}_1(1)$, noting that the $R^{k}_1(1)$ are taken from the
first level $n$ crossing, and are thus independent of the second and
subsequent level $n$ crossings.
Thus with this construction, the process $Y^{(n)}(t)$ is distributed as
$2^nY^{(0)}(t \rho^{-n}_1)$, where $\rho^{-n}_1$ is the weight given to
the first level $-n$ crossing of $Y^{(0)}$ (a product of $n$ weights,
from level $-1$ to $-n$).
To complete our construction, we just need to check that the process
$Y$ does not escape to $\pm\infty$ in finite time.
To see this note that the \textit{second} level $n$ crossing of
$Y^{(n+1)}$ is distributed as
\[
\frac{R^{n+1}_1(2)}{\prod_{k=1}^{n+1} R^{k}_1(1)} \mathcalW^n_2,
\]
where, conditioned on its orientation, $\mathcalW^n_2$ is equal in
distribution to the level 0 crossing of $Y^{(0)}$, and is independent
of $R^{k}_1(1)$ for $k = 1, \ldots, n+1$ and of $R^{n+1}_1(2)$.
We have already seen that $\mathcalW^n_2 > 0$ almost surely, and by
assumption, $R^{n+1}_1(2) > 0$, so it suffices to show that $\prod
_{k=1}^{n+1} R^{k}_1(1) \to0$ almost surely as $n\to\infty$.
Given the orientations $\alpha^k_1$, $k = 1, \ldots, n+1$, the weights
$R^k_1(1)$ are independent.
The sequence of orientations $\{ \alpha^k_1 \}_{k=1}^\infty$ form a
two-state ($+$ and $-$) Markov chain, with transition matrix
\[
\pmatrix{u & 1-u \cr v & 1-v}.
\]
Thus the product $R^1_1(1) R^2_1(1) \cdots$ can be written as a product
of independent random variables of the form
\[
C = \prod_{k=1}^U A_k \prod_{k=1}^V B_k,
\]
where $U \sim\operatorname{geom}(u)$, $V \sim\operatorname{geom}(1-v)$, $A_k \sim
(R^k_1(1)|\alpha^k_1=+)$, $B_k \sim(R^k_1(1)|\break \alpha^k_1(1)=-)$, and they are
all independent.
The product $\prod_{k=1}^{n+1} R^{k}_1(1)$ converges to zero if the sum
$\sum_{k=1}^{n+1} \log R^{k}_1(1)$ diverges to $-\infty$, which follows
almost surely from the strong law of large numbers, provided $\mathbb{E}\log C
= \frac1{1-u} \mathbb{E}\log A_1 + \frac1v \mathbb{E}\log B_1 < 0$ (assuming $u\neq
1$ and $v\neq0$).
That is, the process $Y$ is defined on $\mathbb{R}_+$ provided the following
assumption holds.
\begin{assp}\label{assInf}
If $u = \mathbb{P}(\alpha^n_1 = + | \alpha^{n+1}_1 = +) \neq1$ and $v = \mathbb{P}
(\alpha
^n_1 = \break + | \alpha^{n+1}_1 = -) \neq0$, then we suppose that
\[
\frac1{1-u} \mathbb{E}\bigl(\log R^n_1(1) | \alpha^n_1=+\bigr) + \frac1v \mathbb{E}\bigl(\log R^n_1(1)
| \alpha^n_1=-\bigr) < 0.
\]
If $u = 1$, then we require $\mathbb{E}(\log R^n_1(1) | \alpha^n_1=+) < 0$, and if
$v = 0$, then we require $\mathbb{E}(\log R^n_1(1) | \alpha^n_1=-) < 0$.
\end{assp}
To describe the crossing tree of the extended process $Y$, it is
convenient to extend the tree-indexing notation introduced earlier.
We do this by indexing nodes relative to a \textit{spine}, defined by the
first crossing at each level.
For any node in the tree, we can trace its ancestry back to the spine.
For any $n$ let $n\dvtx\varnothing$ be the node on level $n$ of the spine
and $\Upsilon_{n\dvtx\varnothing}$ the tree descending from that node.
Nodes in the tree $\Upsilon_{n\dvtx\varnothing}$ will be labeled $n\dvtx\mathbf{i}$, where
$\mathbf{i}$ is the node index relative to $n\dvtx\varnothing$.
Thus $n\dvtx\mathbf{i}$ is in level $n-|\mathbf{i}|$ of the crossing tree, and a
crossing previously labeled $\mathbf{i}$ is now labeled $0\dvtx\mathbf{i}$.
Note that this labeling is not unique, as $n\dvtx\mathbf{i}= (n+1)\dvtx1\mathbf{i}$.
Write $\rho_{n\dvtx\mathbf{i}}$ for the weight assigned to node $n\dvtx\mathbf{i}$, which
is given by
\[
\rho_{n\dvtx\mathbf{i}} = \cases{
\displaystyle \prod_{k=0}^{|\mathbf{i}|-1} R_{n\dvtx\mathbf{i}|_{k}}(\mathbf{i}[k+1]) \bigg/\prod_{k=0}^{(n
\wedge|\mathbf{i}|)-1} R_{(n-k)\dvtx\varnothing}(1), &\quad $n > 0$,\vspace*{2pt}\cr
\displaystyle \prod_{k=0}^{|\mathbf{i}|-1} R_{n\dvtx\mathbf{i}|_{k}}(\mathbf{i}[k+1]), &\quad $n =
0$,
\vspace*{2pt}\cr
\displaystyle \prod_{k=0}^{|\mathbf{i}|-1} R_{n\dvtx\mathbf{i}|_{k}}(\mathbf{i}[k+1]) \prod_{k=0}^{|n|-1}
R_{-k\dvtx\varnothing}(1), &\quad $n < 0$.}
\]
Here we have used the convention that $\prod_{k=0}^{-1} x_k = 1$, to
deal with the case $|\mathbf{i}|=0$.
Let $\mathcalW_{n\dvtx\mathbf{i}}$ be branching random walk limit associated with
crossing $n\dvtx\mathbf{i}$; see Lemma~\ref{BRWlem}. Then the duration of
crossing $n\dvtx\mathbf{i}$ is
\begin{equation}\label{Yxingeqn}
\mathcalD_{n\dvtx\mathbf{i}} = \rho_{n\dvtx\mathbf{i}} \mathcalW_{n\dvtx\mathbf{i}}.
\end{equation}
We summarize conditions for existence and continuity of $Y$ in the
theorem below.
\begin{theorem}\label{ThmMEBP1}
Suppose we are given subcrossing orientation distributions $p^\pm_{A}$
and weight distributions $F^\pm_{R|a}$, satisfying Assumptions \ref
{AssGW},~\ref{Ass1}\break and~\ref{assInf}.
Then there exists a continuous EBP process $Y$ with subcrossing
orientation distributions $p^\pm_{A}$ and crossing durations $\mathcalD
_{n\dvtx\mathbf{i}} \stackrel{\mathit{fdd}}{=} \rho_{n\dvtx\mathbf{i}} \mathcalW_{n\dvtx\mathbf{i}}$.
For each $n$, conditioned on the crossing orientations $\alpha^n_k$, the
random variables $\mathcalW^n_k$ are mutually independent, and $\mathcalW^n_k$
is conditionally independent of all $(A^m_j, R^m_j)$ for $m > n$.
Also, $\mathbb{E}(\mathcalW^n_k | \alpha^n_k = i) = v^i$, and the distribution of
$\mathcalW^n_k$ depends only on $\alpha^n_k$.
We call $Y$ the multifractal embedded branching process (MEBP) defined
by $p^\pm_A$ and $F^\pm_{R|a}$.
\end{theorem}
As a corollary of our construction we also obtain a novel
Galton--Watson cascade measure $\zeta$ on $\mathbb{R}_+$, constructed by
mapping the cascade measure $\nu$ from the boundary of the (doubly
infinite) tree to $\mathbb{R}_+$, using the measure $\bar{\nu}$ as a reference.
[Where $\bar{\nu}$ is defined on $\partial\Upsilonsilon$ by $\bar{\nu}(\partial\Upsilonsilon
_{n\dvtx\mathbf{i}}) =
\mu^{n} W_\mathbf{i}$.]
Mandelbrot, Fisher and Calvet~\cite{MFC97} described a class of
multifractal processes such that
\[
Y(at) \stackrel{\mathrm{fdd}}{=} M(a)Y(t)
\quad\mbox{and}\quad M(ab) \stackrel{d}{=}
M_1(a) M_2(b),
\]
where $M_1$ and $M_2$ are independent copies of $M$.
Write $A$ for $M^{-1}$, and then we can re-express the scaling rule for
$Y$ as
\begin{equation}\label{MFCmulteqn}
Y(A(a)t) \stackrel{\mathrm{fdd}}{=} a Y(t) \quad
\mbox{and}\quad A(ab) \stackrel{d}{=}
A_1(a)A_2(b),
\end{equation}
where $A_1$ and $A_2$ are independent copies of $A$.
When constructing our MEBP~$Y$, we noted that $Y^{(n)}(t)$ is
distributed as $2^nY^{(0)}(t \rho^{-n}_1)$.
More generally we have $Y^{(m+n)}(t) \stackrel{\mathrm{fdd}}{=} 2^nY^{(m)}(t
\rho
^{-n}_1)$, so sending $m\to\infty$ we get, for $n = 0, 1,
\ldots,$
\[
Y(t) \stackrel{\mathrm{fdd}}{=} 2^nY(t \rho^{-n}_1).
\]
This is close to the form (\ref{MFCmulteqn}) with $A(2^{-n}) = \rho
^{-n}_1 = \prod_{k=0}^{-n+1} R^k_1(1)$. The differences are that $A(a)$
is only defined for $a = 2^{-n}$, $n \in\mathbb{Z}_+$, and the product form
$A(ab) \stackrel{d}{=} A_1(a)A_2(b)$ does not quite hold because of the
dependence of $R^k_1$ on the orientation $\alpha^k_1$. [In fact, the
sequence $\{ (-\log\rho^{-n}_1, \alpha^{-n}_1) \}$ is Markov additive.]
Nonetheless, we recognize that MEBP processes possess a form of
discrete multifractal scaling. The full multifractal spectrum is
obtained in a forthcoming paper~\cite{DHJ}.
\section{On-line simulation}\label{secsim}
There are many ways we could make a multifractal time-change of a CEBP.
However, by defining the time-change via the crossing tree, we obtain a
fast on-line algorithm to simulate the process.
As before, we will suppose that we are given subcrossing orientation
distributions $p^\pm_A$ and weight distributions $F^\pm_{R|a}$,
satisfying Assumptions~\ref{AssGW},~\ref{Ass1} and~\ref{assInf}.
Let $Y$ be the corresponding MEBP. Then we will simulate the sequence
$\{(\mathcalT^0_k, Y(\mathcalT^0_k))\}$.
That is, we will simulate $Y$ at the \textit{spatial} scale of 1.
Given the multifractal nature of the process, the choice spatial scale
is not a restriction, as the process can be scaled to any desired resolution.
An immediate consequence of the definition of the crossing times $\mathcalT
^0_k$ is the following bound on $Y$:
\[
Y(t) \in\bigl(Y(\mathcalT^0_k)-1, Y(\mathcalT^0_k)+1\bigr) \qquad\mbox{for } t \in(\mathcalT
^0_k, \mathcalT^0_{k+1}).
\]
The basis of our simulation is a Markov process, which describes the
line of descent of the current level zero crossing, from the spine down
to level 0.
For $n \geq m$ and $k \geq0$ let $\kappa(m,n,k)$ be such that $C^m_k$ is
a subcrossing of $C^n_{\kappa(m,n,k)}$, and let $S^n_k \in\{1, \ldots,
Z^{n+1}_{\kappa(n,n+1,k)}\}$ be the position of $C^n_k$ within
$C^{n+1}_{\kappa(n,n+1,k)}$.
Using this notation, if $n\dvtx\mathbf{i}$ is the tree-index of $C^0_k$, then
for $0 \leq m \leq n-1$, $\mathbf{i}[n-m] = S^m_{\kappa(0,m,k)}$.
Let $\mathcalY^n(k) = (\kappa(0,n,k), S^n_{\kappa(0,n,k)}, Z^{n+1}_{\kappa
(0,n+1,k)}, A^{n+1}_{\kappa(0,n+1,k)},\break R^{n+1}_{\kappa(0,n+1,k)})$, which is
a description of the level $n$ super-crossing of $C^0_k$, and the
family it belongs to.
Let $N(k)$ be the smallest $n$ such that $\kappa(0,n+1,k) = 1$, and put
\[
\mathcalY(k) = \bigl(\mathcalY^0(k), \ldots, \mathcalY^{N(k)}(k)\bigr).
\]
\begin{lem}
$\mathcalY$ is a Markov process.
\end{lem}
\begin{pf}
We first show how to update $\mathcalY(k)$ to obtain $\mathcalY(k+1)$.
Let $M$ be the largest $m \leq N(k)$ such that
\[
S^n_{\kappa(0,n,k)} = Z^{n+1}_{\kappa(0,n+1,k)} \qquad\mbox{for } n = 0, \ldots
, m.
\]
That is, for all $m \leq M$ we have that $C^m_{\kappa(0,m,k)}$ is the last
level $m$ crossing in its family.
If $M = N(k)$, then $N(k+1) = N(k) + 1$, and $\mathcalY$ gains the
component $\mathcalY^{N(k+1)}(k+1)$.
Let $n = N(k+1)$. Then we have $\kappa(0,n,k+1) = 2$, $S^{n}_2 = 2$ and
$\kappa(0,n+1,k+1) = 1$.
The distribution of $(Z^{n+1}_1, A^{n+1}_1, R^{n+1}_1)$ depends on
$\mathcalY(k)$ only through $\alpha^{n}_1 = A^{n+1}_1(1)$, which is given by
$A^{n}_1(Z^{n}_1)$.
If $M < N(k)$, then for $n = M + 1$ we have $\kappa(0,n,k+1) = \kappa(0,n,k)
+ 1$, $S^n_{\kappa(0,n,k+1)} = S^n_{\kappa(0,n,k)} + 1$ and $\kappa
(0,n+1,k+1) =
\kappa(0,n+1,k)$.
Thus $Z^{n+1}_{\kappa(0,n+1,k+1)} = Z^{n+1}_{\kappa(0,n+1,k)}$,
$A^{n+1}_{\kappa
(0,n+1,k+1)} = A^{n+1}_{\kappa(0,n+1,k)}$, and $R^{n+1}_{\kappa(0,n+1,k+1)} =
R^{n+1}_{\kappa(0,n+1,k)}$.\vspace*{1pt}
For $n > M+1$, we have $\mathcalY^n(k+1) = \mathcalY^n(k)$.
For $n = M, \ldots, 0$, we generate $\mathcalY^n(k+1)$ recursively.
We have $\kappa(0,n,k+1) = \kappa(0,n,k)+1$, $S^{n}_{\kappa(0,n,k+1)}=1$, and
$\kappa(0,n+1,k+1) = \kappa(0,n+1,k)+1$.
The distribution of $(Z^{n+1}_{\kappa(0,n+1,k+1)}, A^{n+1}_{\kappa
(0,n+1,k+1)}, R^{n+1}_{\kappa(0,n+1,k+1)})$ is determined by $\alpha
^{n+1}_{\kappa(0,n+1,k+1)}$, that is, $A^{n+2}_{\kappa
(0,n+2,k+1)}(S^{n+1}_{\kappa(0,n+1,k+1)})$.
Thus $\{ \mathcalY^n(k+1) \}_{n=0}^M$ depends on $\mathcalY(k)$ only through
$\alpha^{M+1}_{\kappa(0,M+1,k+1)} = A^{M+2}_{\kappa(0,M+2,k+1)}(S^{M+1}_{\kappa
(0,M+1,k+1)}) =
A^{M+2}_{\kappa(0,M+2,k)}(S^{M+1}_{\kappa(0,M+1,k)+1})$.\vspace*{2pt}
That $\mathcalY$ is Markov follows from the conditional independence of the
$(Z^n_k, A^n_k$, $R^n_k)$ given the orientations $\alpha^n_k$.
\end{pf}
From $\mathcalY(k)$, we get the orientation $\alpha^0_k$ of $C^0_k$, and the
weights
\[
R^{n+1}_{\kappa(0,n+1,k)}\bigl(S^n_{\kappa(0,n,k)}\bigr)\qquad \mbox{for $n=0, \ldots
, N(k)$}.
\]
To calculate the crossing duration $\mathcalD^0_k$ we also need
$R^{n+1}_1(1)$, for $n=0, \ldots, \break N(k)$ and $\mathcalW^0_k$.
Keeping track of the spine weights $R^{n+1}_1(1)$ is no problem.
Calculating $\mathcalW^0_k$ is less straightforward.
We do have that the $\mathcalW^0_k$ are conditionally independent given the
$\alpha^0_k$, but we do not have an explicit formulation of the density of
$(\mathcalW^0_k | \alpha^0_k=\pm)$.
The simplest way to approximate the $\mathcalW^0_k$ is to generate a BRW
(using $p^\pm_A$ and $F^\pm_{R|a}$) for a fixed number of generations,
$m$ say, and sum the node weights across the final generation.
However, this is exactly the same as setting the $\mathcalW^0_k$ to be
constant, then scaling the resulting process by $2^{-m}$, so we will
just set $\mathcalW^0_k$ equal to its mean $v^{\alpha^0_k}$.
\begin{remark}
Writing $Y$ as $X \circ\mathcalM^{-1}$, where $X$ is the CEBP
corresponding to $Y$, we note that $X$ and $\mathcalM$ are, in general, dependent.
However, in the case where $X$ is Brownian motion, we can construct
$\mathcalM$ independently of $X$, simply by taking the orientations $\alpha
^0_k$ as i.i.d. random variables, equal to $+$ and $-$ with equal probability.
This is because for Brownian motion $X(T^0_k)$ is just a simple random walk.
In fact, in this case, there need not be any relation at all between
the crossing tree of $X$ and that used to construct~$\mathcalM$.
\end{remark}
\subsection{Pseudo-code}
We give pseudo code for simulating $\{(\mathcalT^0_k, Y(\mathcalT^0_k))\}$,
with the crossing durations $\mathcalD_{n\dvtx\mathbf{i}}$ approximated by $\rho
_{n\dvtx\mathbf{i}}\mathbb{E}(\mathcalW_{n\dvtx\mathbf{i}})$ (i.e., putting $\mathcalW_{n\dvtx\mathbf{i}} = v^i$,
where $i = \alpha_{n\dvtx\mathbf{i}}$).
Updating $\mathcalY(k)$ is handled by procedures \texttt{Expand} and
\texttt
{Increment}.
Procedure \texttt{Expand} checks if $M = N(k)$.
If so, it then generates the component $\mathcalY^{M+1}(k)$ and updates $N(k)$.
Assuming $M < N(k)$, procedure \texttt{Increment} updates $\mathcalY^n(k)$
to $\mathcalY^n(k+1)$ recursively, for $n = M+1, \ldots, 0$.
The actions of \texttt{Expand} and \texttt{Simulate} are illustrated in
Figure~\ref{figalgo}.
\begin{figure}\label{figalgo}
\end{figure}
Given sample position $Y(\mathcalT^0_{k})$, sample time $\mathcalT^0_{k}$ and
crossing state $\mathcalY(k)$, the procedure \texttt{Simulate} applies the
procedures \texttt{Expand} and \texttt{Increment}, calculates
$Y(\mathcalT
^0_{k+1})$, $\mathcalT^0_{k+1}$ and $\mathcalY(k+1)$, then increments $k$.
Procedure \texttt{Initialize} generates an initial $Y(\mathcalT^0_{1})$,
$\mathcalT^0_{1}$ and $\mathcalY(1)$ suitable for passing to \texttt
{Simulate}.
Recall that $u = \mathbb{P}( \alpha^{n+1}_1 = + | \alpha^n_1 = +)$ and $v =
\mathbb{P}(
\alpha^{n+1}_1 = + | \alpha^n_1 = -)$.
Here $\alpha^{N(k)+1}_1$ is given by
$A^{N(k)+1}_1(Z^{N(k)+1}_1)$.\vspace*{12pt}
\texttt{Procedure Expand} $\mathcalY(k)$\vspace*{2pt}
\hspace{1cm} \texttt{If} $S^{N(k)}_{\kappa(0,N(k),k)} = Z^{N(k)+1}_{\kappa(0,N(k)+1,k)}$
\texttt{Then}\vspace*{2pt}
\hspace{1.5cm} $\kappappa(0,N(k)+2,k) = 1$\vspace*{2pt}
\hspace{1.5cm} Generate $\alpha^{N(k)+2}_1$ using $u$, $v$ and
$\alpha^{N(k)+1}_1$\vspace*{2pt}
\hspace{1.5cm} Generate $(Z^{N(k)+2}_{\kappa(0,N(k)+2,k)}, A^{N(k)+2}_{\kappa(0,N(k)+2,k)},
R^{N(k)+2}_{\kappa(0,N(k)+2,k)})$\vspace*{2pt}
\hspace{2cm} using the distributions $p^i_A$ and $F^i_{R|a}$\vspace*{1pt}
\hspace{2cm} conditioned on the first offspring having orientation
$\alpha^{N(k)+1}_1$\vspace*{1pt}
\hspace{2cm} where $i = \alpha^{N(k)+2}_1 \in \{+, -\}$\vspace*{2pt}
\hspace{1.5cm} $S^{N(k)+1}_{\kappa(0,N(k)+1,k)} = 1$\vspace*{2pt}
\hspace{1.5cm} Store $R^{N(k)+2}_1(1)$\vspace*{1pt}
\hspace{1.5cm} $N(k) = N(k) + 1$\vspace*{1pt}
\hspace{1cm} \texttt{End If}
\texttt{End Procedure}\vspace*{12pt}
\texttt{Procedure Increment} $\mathcalY^n(k)$\vspace*{1pt}
\hspace{1cm} \# Assume that $C^{n-1}_k$ is at the end of a level $n$
crossing,\vspace*{1pt}
\hspace{1cm} \# so $S^{n-1}_{\kappa(0,n-1,k)} = Z^n_{\kappa(0,n,k)}$. This is always the case for
$n=0$\vspace*{2pt}
\hspace{1cm} $\kappappa(0,n,k+1)=\kappappa(0,n,k)+1$\vspace*{2pt}\vadjust{\goodbreak}
\hspace{1cm} \texttt{If} $S_{\kappappa(0,n,k)}^{n}=Z_{\kappappa(0,n+1,k)}^{n+1}$
\texttt{Then}\vspace*{2pt}
\hspace{1.5cm} \texttt{Increment} ${\mathcal X}^{n+1}(k)$\vspace*{1pt}
\hspace{1.5cm} $S^{n}_{\kappappa(0,n,k+1)}=1$\vspace*{1pt}
\hspace{1.5cm} Generate $(Z^{n+1}_{\kappappa(0,n+1,k+1)}, A_{\kappappa(0,n+1,k+1)}^{n+1},
R^{n+1}_{\kappappa(0,n+1,k+1)})$\vspace*{2pt}
\hspace{2cm} using the distributions $p^i_A$ and $F^i_{R|a}$\vspace*{1pt}
\hspace{2cm} where $i = A_{\kappappa(0,n+2,k+1)}^{n+2}(S^{n+1}_{\kappappa(0,n+1,k+1)}) \in
\{+,-\}$\vspace*{1pt}
\hspace{1cm} \texttt{Else}\vspace*{1pt}
\hspace{1.5cm} ${\mathcal X}^q(k+1)=\mathcalY^q(k)$ \texttt{for}
$q=n+1,\ldots,N(k)$\vspace*{2pt}
\hspace{1.5cm} $S^{n}_{\kappappa
(0,n,k+1)}=S^{n}_{\kappappa(0,n,k)}+1$\vspace*{2pt}
\hspace{1cm} \texttt{End If}
\texttt{End Procedure}\vspace*{12pt}
We apply procedure \texttt{Increment} to $\mathcalY^0(k)$, and then it is
recursively applied to all $\mathcalY^n(k)$ such that $C^q_{\kappa(0,q,k)}$ is
at the end of a level $q+1$ crossing for all $0 \leq q < n$.
$\mathcalY^n(k+1) = \mathcalY^n(k)$ for all $n$ larger than this.\vspace*{12pt}
\texttt{Procedure Simulate}
\hspace{1cm} \texttt{Expand} $\mathcalY(k)$\vspace*{1pt}
\hspace{1cm} \texttt{Increment} ${\mathcal X}^0(k)$\vspace*{1pt}
\hspace{1cm} Put $i = A^1_{\kappa(0,1,k+1)}(S^0_{k+1})$\vspace*{2pt}
\hspace{1cm} \texttt{If} $i = +$ \texttt{Then}\vspace*{1pt}
\hspace{1.5cm} $Y(\mathcalT^{0}_{k+1}) = Y(\mathcalT^{0}_{k}) + 1$\vspace*{2pt}
\hspace{1cm} \texttt{Else}\vspace*{1pt}
\hspace{1.5cm} $Y(\mathcalT^{0}_{k+1}) = Y(\mathcalT^{0}_{k}) - 1$\vspace*{2pt}
\hspace{1cm} \texttt{End If}\vspace*{1pt}
\hspace{1cm} $\mathcalT^0_{k+1} = \mathcalT^0_k + v^i \prod_{j=0}^{N(k+1)} ( R^{j+1}_{\kappappa(0,j+1,k+1)}( S^j_{\kappappa(0,j,k+1)}) /
R^{j+1}_1(1))$\vspace*{2pt}
\hspace{1cm} $k \leftarrow k + 1$\vspace*{1pt}
\texttt{End Procedure}\vspace*{12pt}
To initialize the algorithm, the procedure \texttt{Initialize} is used.
Recall that $(v^+, v^-)^T$ is the right $\mu(1)$-eigenvector of
$M(1)$.\vadjust{\goodbreak}\vspace*{12pt}
\texttt{Procedure Initialize} $\mathcalY(1)$\vspace*{1pt}
\hspace{1cm} $k = 1$, $N(1) = 0$, $\kappappa(0,0,1)=1$,
$\kappappa(0,1,1)=1$\vspace*{1pt}
\hspace{1cm} Put $\alpha^1_1 = i = +$ with probability $a$\vspace*{1pt}
\hspace{1cm} Generate $(Z^1_1, A^1_1, R^1_1)$ using the
distributions\vspace*{1pt}
\hspace{1.5cm} $p^i_A$ and $F^i_{R|a}$, with $i = \alpha^1_1$\vadjust{\goodbreak}
\hspace{1cm} $S^0_1 = 1$\vspace*{1pt}
\hspace{1cm} Store $R^1_1(1)$\vspace*{1pt}
\hspace{1cm} $\mathcalT^0_1 = v^i$\vspace*{1pt}
\hspace{1cm} \texttt{If} $i = +$ \texttt{Then} $Y(\mathcalT^0_1) = 1$ \texttt{Else} $Y(\mathcalT^0_1) = -1$ \texttt{End
If}\vspace*{1pt}
\texttt{End Procedure}\vspace*{12pt}
An implementation is available from the web page of Jones~\cite{J}.
An example of the type of signal obtained with this algorithm is given
in Figure~\ref{EBPMEBP}, where we have represented an MEBP process
\begin{figure}\label{EBPMEBP}
\end{figure}
with its corresponding CEBP.
$p^\pm_A$ and $F^\pm_{R|a}$ are described in the caption.
\subsection{Efficiency}
Consider the tree descending from crossing $C^{N(k)}_1$ down to level 0.
On average $C^{N(k)}_1$ has $\mu^{N(k)}$ level 0 subcrossings, so we
must have $N(k) = O(\log k)$.
At each step, the number of operations required by procedure \texttt
{Expand} is fixed [independent of $N(k)$], but we can go through
\texttt
{Increment} up to $N(k)$ times, so the number of operations required by
\texttt{Simulation} is of order $N(k)$.
Thus, to generate $n$ steps, we use $O(n \log n)$ operations, since
$\sum _{k=1}^n \log k = O(n\log n)$, and $O(\log n)$ storage.
The algorithm is on-line, meaning that given the current state [of size
$O(\log n)$] we can generate the next immediately [using $O(\log n)$
operations].
\section{Randomizing the starting point}
Crossing times are points where the behavior of the process can change,
spatially and temporally, and the higher the level, the more dramatic
this can be.
For MEBP processes, 0 is a crossing time for all levels, and because of
this we cannot expect MEBP to have stationary increments.
To avoid the problem of 0 being special, we would like to start the
process at a ``random'' time, as if the process had been running since
time immemorial and we just happened across it.
To make the idea of a ``random'' starting time more precise, let $Y$ be
an MEBP and $\{ Y^{(n)} \}$ the nested sequence of processes used to
construct~$Y$, where $Y^{(n)}$ is a single level $n$ crossing from $0$
to $\pm2^n$.
Choose a time $t$ uniformly in $[0, T^n_1] = [0, \mathcalD_{n\dvtx\varnothing}]$.
For any $\mathbf{i}\in\Upsilon_{n\dvtx\varnothing}$, the probability that $t$ is in
$C_{n\dvtx\mathbf{i}}$ is proportional to the crossing duration $\mathcalD_{n\dvtx\mathbf{i}
} =
\rho_{n\dvtx\mathbf{i}} \mathcalW_{n\dvtx\mathbf{i}}$.
That is, choosing $t$ is equivalent to choosing $n\dvtx\mathbf{j}\in\partial\Upsilonsilon
_{n\dvtx\varnothing}$ so that the probability that $n\dvtx\mathbf{j}|_{|\mathbf{i}|} =
n\dvtx\mathbf{i}$ is proportional to $\rho_{n\dvtx\mathbf{i}} \mathcalW_{n\dvtx\mathbf{i}}$.
It turns out that we can do exactly this using a size-biased measure
for a multitype branching random walk.
Size-biased measures for branching processes were introduced by Lyons,
Pemantle and Peres~\cite{LPP95} and generalized to branching random
walks by Lyons~\cite{L97}.
Kyprianou and Sani~\cite{KS01} then extended their construction to
multitype branching random walks.
Fix $n$, and for brevity write $\mathbf{i}$ for $n\dvtx\mathbf{i}$.
Let $\Omega$ be the space of marked trees, where the mark associated with
node $\mathbf{i}$ is $(-\log R_{\mathbf{i}|_{k-1}}(\mathbf{i}[k]), \alpha_\mathbf{i})$, writing
$k$ for~$|\mathbf{i}|$.
Let $\mathcalF$ be the $\sigma$-field generated by all finite truncations
of trees.
The offspring orientation distributions $p^\pm_A$ and weight
distributions $F^\pm_{R|a}$ induce a measure $\xi$ on $(\Omega, \mathcalF)$.
Let $\tilde{\Omega}$ be the space of trees with a distinguished line of
descent $\mathbf{i}\in\partial\Upsilonsilon$, called a spine, and $\tilde{\mathcalF}$ the
$\sigma
$-field generated by all finite truncations of trees with spines.
Kyprianou and Sani define a size-biased measure $\tilde{\pi}$ on
$(\tilde{\Omega}, \tilde{\mathcalF})$ such that
\begin{equation}\label{eqnsize-biased}
\int_{\mathbf{j}\in\partial\Upsilonsilon_\mathbf{i}} d\tilde{\pi}(\Upsilon, \mathbf{j}) = \frac{\rho
_\mathbf{i}\mathcalW
_\mathbf{i}}{v^{\alpha_\varnothing}} \,d\xi(\Upsilon).
\end{equation}
This is precisely what we want, and, remarkably, the measure can be
constructed using the original multitype branching walk, modified so
that the offspring generation down the spine is size-biased.
That is, rather than construct $Y^{(n)}$ and then choose a spine, we
can construct the process and the spine together.
Let $\mathbf{x}\in\partial\Upsilonsilon$ be the spine, and let $\tilde{p}^\pm_{A}$ and
$\tilde{F}^\pm_{R|a}$ be the offspring orientation and weight
distributions for nodes on the spine. Then from~\cite{KS01}, Section 2,
we have that
\begin{eqnarray*}
\tilde{p}^i_{A}(a) \tilde{F}^i_{R|a}(r)
&=& \mathbb{P}(A_{\mathbf{x}|_n} = a, R_{\mathbf{x}|_n} \leq r | \alpha_{\mathbf{x}|_n} =
i) \\
&\propto& p^i_{A}(a) \sum_{j=1}^{|a|} v^{a(j)} \int_{s \leq r} s(j)
F^i_{R|a}(ds).
\end{eqnarray*}
Note here that $s$ and $r$ are in $\mathbb{R}_+^{|a|}$.
Putting $r = \infty^{|a|}$ to get $\tilde{p}^i_{A}(a)$, and then
dividing out $\tilde{p}^i_{A}(a)$ to get $\tilde{F}^i_{R|a}(r)$,
gives us
\begin{eqnarray*}
\tilde{p}^i_{A}(a) &\propto& p^i_A(a) \sum_{j=1}^{|a|} v^{a(j)} \int
_{\mathbb{R}
_+^{|a|}} s(j) F^i_{R|a}(ds), \\
\tilde{F}^i_{R|a}(r) &\propto& \sum_{j=1}^{|a|} v^{a(j)} \int_{s
\leq
r} s(j) F^i_{R|a}(ds).
\end{eqnarray*}
That these are well defined follows from Assumption~\ref{Ass1}.
In the case where the offspring weights are i.i.d. with distribution
$F$, we get
\begin{eqnarray*}
\tilde{p}^i_{A}(a) &\propto& |a| p^i_A(a), \\
\tilde{F}^i_{R|a}(r) &\propto& \sum_{j=1}^{|a|} \int_0^{r(j)} s F(ds)
\prod_{i\neq j} F(r(i)).
\end{eqnarray*}
The first of these is clearly a size-biased version of $p^i_A$.
The second can be interpreted as conditioning on which offspring is on
the spine, then size-biasing the weight for that offspring.
For selecting the next node on the spine, we again have from \cite
{KS01}, Section~2, that
\[
\tilde{p}_{a,r}(j):= \mathbb{P}(\mathbf{x}[n+1] = j | A_{\mathbf{x}|_n}=a,
R_{\mathbf{x}
|_n}=r) \propto v^{a(j)} r(j).
\]
Kyprianou and Sani also also show that under $\tilde{\pi}$, the
sequence $\{ \alpha_{\mathbf{x}|_n} \}_{n=1}^\infty$ of orientations down the
spine is Markovian, with transition probabilities
\[
\pmatrix{v^+&0\cr
0&v^-}^{-1}
M(1)
\pmatrix{v^+&0\cr
0&v^-}.
\]
The stationary distribution is $(u^+v^+, u^-v^-)$, and so the reversed
chain (moving up the spine) has transition matrix
\begin{equation}\label{eqnspineorientation}
\pmatrix{u^+&0\cr
0&u^-}
^{-1}
M(1)^T
\pmatrix{u^+&0\cr
0&u^-},
\end{equation}
and the same stationary distribution as before.
Note that it follows from assumptions~\ref{AssGW} and~\ref{Ass1} that
$\mathbf{u}, \mathbf{v}> 0$.
\subsection{MEBP construction with random start}
We now show how, given an MEBP $Y\dvtx [0,\infty) \to\mathbb{R}$ generated by
$p^\pm_A$ and $F^\pm_{R|a}$, we can construct a shifted version,
$\tilde
{Y}\dvtx (-\infty, \infty) \to\mathbb{R}$, with a ``randomly'' chosen starting point.
Where unambiguous, we will use the same notation to describe $\tilde
{Y}$ as $Y$, and we will assume that assumptions~\ref{AssGW} and \ref
{Ass1} hold throughout.
As before, we start by constructing a crossing of size 1 (level 0).
Let $\mathbf{x}$ be the spine, which will be the line of descent
corresponding to time 0.
Accordingly, we will write $C^{-n}_0 = C_{\mathbf{x}|_n}$ for the level $-n$
spinal crossing.
Note that previously, the first crossing at level $-n$ was labeled 1,
and started at time 0.
For our new construction, time 0 will occur somewhere in the interior
of crossing $C^{-n}_0$, so crossing $C^{-n}_1$ will still be the first
full crossing to occur after time 0.
The generation $n$ (level $-n$) nodes in $\Upsilon_n$ are totally ordered
according to the rule $\mathbf{i}< \mathbf{j}$ if and only if, for some $m$,
$\mathbf{i}
|_m = \mathbf{j}|_m$ and $\mathbf{i}[m+1] < \mathbf{j}[m+1]$.
For $\mathbf{i}, \mathbf{j}\in\Upsilon_n$ let
\[
d(\mathbf{i}, \mathbf{j}) = \cases{
|\{ \mathbf{k}\dvtx \mathbf{i}< \mathbf{k}\leq\mathbf{j}\}|, &\quad $\mathbf{i}< \mathbf{j}$,\cr
0, &\quad $\mathbf{i}= \mathbf{j}$,\cr
-|\{ \mathbf{k}\dvtx \mathbf{i}> \mathbf{k}\geq\mathbf{j}\}|, &\quad $\mathbf{i}> \mathbf{j}$.}
\]
We will write $C^{-n}_{d(\mathbf{x},\mathbf{i})}$ for $C_{\mathbf{i}}$.
Set the orientation of $C^0_0$ to be $+$ with probability $u^+v^+$, and
then generate $(A^0_0, R^0_0)$ using $\tilde{p}^i_A$ and $\tilde
{F}^i_{R|a}$, where $i = \alpha^0_0$.
Choose $j \in\{1, \ldots, Z_\varnothing\}$ using $\tilde
{p}_{A_\varnothing
, R_\varnothing}$, and then put $\mathbf{x}|_1 = j$.
Subsequent generations are produced using $p^\pm_A$ and $F^\pm_{R|a}$
for nodes off the spine, and $\tilde{p}^\pm_A$ and $\tilde{F}^\pm
_{R|a}$ for the spinal node.
The spinal node in the next generation is chosen using $\tilde{p}_{a,r}$.
Crossing durations are defined as before; that is, $\mathcalD^{-n}_k =
\rho
^{-n}_k \mathcalW^{-n}_k$, where $\mathcalW_\mathbf{i}$ is the $\tilde{\pi
}$-a.s.
limit of $\sum_{\mathbf{j}\in\Upsilon_n\cap\Upsilon_\mathbf{i}} \rho_\mathbf{j}/\rho_\mathbf{i}$.
For $k \neq0$ (nodes off the spine) the convergence of this sequence
a.s. and in mean follows as before.
For $k = 0$ (nodes on the spine) a.s. convergence follows from
(\ref{eqnsize-biased}) and the fact that $\rho_{\mathbf{x}|_n}
\mathcalW
_{\mathbf{x}|_n} \in(0, \infty)$ $\xi$-a.s.
Given crossing durations, we define crossing times as follows.
Time 0 corresponds to the spine $\mathbf{x}$.
For any $m \geq0$, $\mathcalT^{-m}_0 > 0$ is the first time the process
starts a level $-m$ crossing:
\begin{eqnarray*}
\mathcalT^{-m}_0
&=& \lim_{n\to\infty} \sum_{\mathbf{i}\in\Upsilon_n, \mathbf{i}|_m=\mathbf{x}|_m,
\mathbf{i}> \mathbf{x}}
\rho_\mathbf{i}\mathcalW_\mathbf{i},\\
\mathcalT^{-m}_{k+1} &=& \mathcalT^{-m}_k + \rho^{-m}_{k+1} \mathcalW^{-m}_{k+1}
\qquad\mbox{for } k\geq0 ,\\[-2pt]
\mathcalT^{-m}_{-1}
&=& \lim_{n\to\infty} \sum_{\mathbf{i}\in\Upsilon_n, \mathbf{i}|_m=\mathbf{x}|_m,
\mathbf{i}< \mathbf{x}}
\rho_\mathbf{i}\mathcalW_\mathbf{i},\\[-2pt]
\mathcalT^{-m}_{-k-1} &=& \mathcalT^{-m}_{-k} - \rho^{-m}_{-k} \mathcalW^{-m}_{-k}
\qquad\mbox{for } k\geq1.
\end{eqnarray*}
We also put $\tilde{Y}(0) = 0$ and
\begin{eqnarray*}
\tilde{Y}(\mathcalT^{-m}_0)
&=& \lim_{n\to\infty} \sum_{\mathbf{i}\in\Upsilon_n, \mathbf{i}|_m=\mathbf{x}|_m,
\mathbf{i}> \mathbf{x}}
\alpha_\mathbf{i}2^{-n}, \\[-2pt]
\tilde{Y}(\mathcalT^{-m}_{k+1}) &=& \tilde{Y}(\mathcalT^{-m}_k) + \alpha
^{-m}_{k+1} 2^{-m} \qquad\mbox{for } k\geq0, \\[-2pt]
\tilde{Y}(\mathcalT^{-m}_{-1})
&=& \lim_{n\to\infty} \sum_{\mathbf{i}\in\Upsilon_n, \mathbf{i}|_m=\mathbf{x}|_m,
\mathbf{i}< \mathbf{x}}
\alpha_\mathbf{i}2^{-n}, \\[-2pt]
\tilde{Y}(\mathcalT^{-m}_{-k-1}) &=& \tilde{Y}(\mathcalT^{-m}_{-k}) + \alpha
^{-m}_{-k} 2^{-m} \qquad\mbox{for } k\geq1.
\end{eqnarray*}
So for $k\geq1$, $C^{-m}_k$ is from $\mathcalT^{-m}_k$ to $\mathcalT
^{-m}_{k+1}$, while for $k\leq0$ it is from $\mathcalT^{-m}_{k-1}$ to~$\mathcalT^{-m}_k$.
Let $\tilde{Y}^{(0)}$ be the level 0 crossing constructed above. We now
show how to extend the construction from $\tilde{Y}^{(n)}$ to
$\tilde{Y}^{(n+1)}$. Let $n\dvtx\mathbf{x}$ be the spine starting at level
$n$. First choose $\alpha^{n+1}_0 = i$ using the reversed Markov chain
\ref {eqnspineorientation}, then choose $(A^{n+1}_0, R^{n+1}_0)$ and
$(n+1)\dvtx\mathbf{x}[1] = j$ using $\tilde{p}^i_A$, $\tilde{F}^i_{R|a}$ and
$\tilde{p}_{a,r}$, all conditioned on $\alpha^n_0$, which is the
orientation of $(n+1)\dvtx\mathbf{x}[1]$. Put the $j$th level $n$ subcrossing
of $\tilde{Y}^{(n+1)}$, that is~$C^n_0$, equal to $\tilde{Y}^{(n)}$.
For the other level $n$ subcrossings, we use the construction of
Section~\ref{secextend}, and scale the $k$th subcrossing by
$R^{n+1}_0(k)/R^{n+1}_0(j)$. That is, we use the weights up the spine,
from level 0 to $n$, to rescale the process. Let $\tilde{Y}$ be the
limit of the $\tilde{Y}^{(n)}$.
To see that $\tilde{Y}(t)$ is defined for all $t \in\mathbb{R}$ we need two things.
First we note that from the form of $\tilde{p}_{a,r}$, with probability
1 we cannot have $n\dvtx\mathbf{x}[1]$ equal to 1 eventually, or equal to
$Z_{n\dvtx\varnothing}$ eventually.
That is, at all levels there will be crossings to the left and right of
the spinal crossing.
Second, we need to know that the scaling coming from the spine weights
grows to infinity, that is, $\prod_{k=1}^{n+1} R^k_0(k\dvtx\mathbf{x}[1]) \to
0$ a.s. as $n \to\infty$.
As noted above, the sequence of orientations up the spine is a Markov process.
Because the weights are conditionally independent given the
orientations, the sequence $(\sum_{k=1}^{n+1} \log R^k_0(k\dvtx\mathbf{x}[1]),
\alpha^{n+1}_0)$ is Markov additive.
Thus, $\sum_{k=1}^{n+1} \log R^k_0(k\dvtx\mathbf{x}[1]) \to-\infty$ a.s.,
equivalently $\prod_{k=1}^{n+1} R^k_0(k\dvtx\mathbf{x}[1]) \to0$ a.s.,
provided the expected increments of the sum are negative.
That is, provided the following assumption holds (this replaces
Assumption~\ref{assInf}).
\begin{assp}\label{assspinalweights}
Let $R^\pm$ be a random spinal weight, chosen according to $\tilde
{p}^\pm_A$, $\tilde{F}^\pm_{R|a}$ and $\tilde{p}_{a,r}$.
Then we assume that
\[
u^+v^+ \mathbb{E}\log R^+ + u^-v^- \mathbb{E}\log R^- < 0.\vadjust{\goodbreak}
\]
\end{assp}
It remains an open problem to show that the process $\tilde{Y}$ has
stationary increments.\vspace*{-2pt}
\subsection{On-line simulation}
To simulate $\tilde{Y}$ we need only modify procedures \texttt{Expand}
and \texttt{Initialie}.
Note that the spinal crossings are now counted as crossing 0 at each
level, so $N(k)$ is the smallest $n$ such that $\kappa(0,n+1,k) =
0$.\vspace*{11pt}
\texttt{Procedure Expand} $\mathcalY(k)$\vspace*{2pt}
\hspace{1cm} \texttt{While} $S^{N(k)}_{\kappa(0,N(k),k)} =
Z^{N(k)+1}_{\kappa
(0,N(k)+1,k)}$ \texttt{Do}\vspace*{2pt}
\hspace{1.5cm} $\kappappa(0,N(k)+2,k) = 0$\vspace*{2pt}
\hspace{1.5cm} Generate $\alpha^{N(k)+2}_0$ using $(u^+v^+, u^-v^-)$ and
$\alpha^{N(k)+1}_0$\vspace*{2pt}
\hspace{1.5cm} Generate $A^{N(k)+2}_0$, $R^{N(k)+2}_0$ and
$S^{N(k)+1}_0$\vspace*{2pt}
\hspace{2cm} using the distributions $\tilde{p}^i_A$, $\tilde
{F}^i_{R|a}$ and $\tilde{p}_{a,r}$\vspace*{2pt}
\hspace{2cm} conditioned on offspring $S^{N(k)+1}_0$ having orientation
$\alpha^{N(k)+1}_0$\vspace*{2pt}
\hspace{2cm} where $i = \alpha^{N(k)+2}_0 \in\{+, -\}$\vspace*{2pt}
\hspace{1.5cm} Store $R^{N(k)+2}_0(S^{N(k)+1}_0)$\vspace*{2pt}
\hspace{1.5cm} $N(k) = N(k) + 1$\vspace*{2pt}
\hspace{1cm} \texttt{End While}
\texttt{End Procedure}
\texttt{Procedure Initialize} $\mathcalY(0)$
\hspace{1cm} $k = 0$, $N(0) = 0$, $\kappappa(0,0,0)=0$,
$\kappappa(0,1,0)=0$\vspace*{2pt}
\hspace{1cm} Put $\alpha^1_0 = +$ with probability $u^+v^+$\vspace*{2pt}
\hspace{1cm} Generate $A^1_0$, $R^1_0$ and $S^0_0$ using the
distributions\vspace*{2pt}
\hspace{1.5cm} $\tilde{p}^i_A$, $\tilde{F}^i_{R|a}$ and $\tilde
{p}_{a,r}$, with $i = \alpha^1_0$\vspace*{2pt}
\hspace{1cm} Store $R^1_0(S^0_0)$\vspace*{2pt}
\hspace{1cm} $\mathcalT^0_0 = 0$, $Y(\mathcalT^0_0) = 0$\vspace*{2pt}
\texttt{End Procedure}\vspace*{-2pt}
\section*{Acknowledgments}
The authors are grateful for the many constructive comments received
from their anonymous referees.\vspace*{-2pt}
\printaddresses
\end{document} |
\begin{document}
\title[Physical interpretation of the Wigner rotations and relativistic quantum information]{Physical interpretation of the Wigner rotations and its implications for relativistic quantum information }
\author{Pablo L Saldanha$^{1,2}$ and Vlatko Vedral$^{1,3,4}$}
\address{$^1$ Department of Physics, University of Oxford, Clarendon Laboratory, Oxford, OX1 3PU, United Kingdom}
\address{$^2$ Departamento de F\'isica, Universidade Federal de Pernambuco, 50670-901, Recife, PE, Brazil}
\address{$^3$ Centre for Quantum Technologies, National University of Singapore, Singapore}
\address{$^4$ Department of Physics, National University of Singapore, Singapore}
\ead{[email protected]}
\begin{abstract}
We present a new treatment for the spin of a massive relativistic particle in the context of quantum information based on a physical interpretation of the Wigner rotations, obtaining different results in relation to the previous works. We are lead to the conclusions that it is not possible to define a reduced density matrix for the particle spin and that the Pauli-Lubanski (or similar) spin operators are not suitable to describe measurements where spin couples to an electromagnetic field in the measuring apparatus. These conclusions contradict the assumptions made by most of the previous papers on the subject. We also propose an experimental test of our formulation.
\end{abstract}
\pacs{03.65.Ta, 03.30.+p}
\maketitle
\section{Introduction}
The field of relativistic quantum information has recently emerged \cite{czachor97,peres02,alsing02,gingrich02,ahn03,terno03,terashima03,czachor03,li03,peres04,lee04,bartled05,kim05,czachor05,peres05,caban05,jordan06,lamata06,jordan07,landulfo09,dunningham09,caban10,friis10,choi11}, describing how relativistic particles behave in a regime where the nature and the number of particles do not change in the processes, such that not all the machinery of quantum field theory is necessary. This simplified view of the problems can shed light on many issues of relativistic quantum mechanics and may have applications in the near future if we use the spin of relativistic particles to encode quantum information. Latter the field has expanded to include non-inertial reference frames and general relativity effects, where the number of particles may not be conserved \cite{alsing03,fuentes05,alsing06,fuentes10,palmer11}.
Here we present a new treatment for the problem based on a physical interpretation of the Wigner rotations \cite{wigner39,weinberg}, which specify a momentum-dependent change of the spin state of a particle with a change of reference frame. We show that the Wigner rotations are consistent with the fact that different observers compute different quantization axes for a spin measurement, being a direct consequence of the dependence of the quantization axis of a spin measurement on the particle momentum. We are lead to the conclusion that it is not possible to make a momentum-spin separation of the system and to define a reduced density matrix for spin, as is done in many previous papers on the subject \cite{peres02,gingrich02,li03,peres04,bartled05,peres05,caban05,jordan06,lamata06,jordan07,landulfo09,dunningham09,friis10,choi11}. We also show that the use of the Pauli-Lubanski (or similar) spin operators to describe spin measurements, as in \cite{czachor97,ahn03,czachor03,lee04,kim05,czachor05,caban05,caban10,friis10}, depends on the coupling of the spin to a quantity that transforms as part of a 4-vector under the Lorentz transformations in the measuring apparatus. However, we do not know if such a coupling exists in nature. Our treatment assumes that spin couples to the electromagnetic field in the measuring apparatus, as in the Stern-Gerlach experiment, and it consequently makes different predictions for the expectation values of spin measurements in relation to the previous treatments. We also propose here an experimental test of our formulation.
\section{Physical interpretation of the Wigner rotations}
Weinberg's treatment \cite{weinberg} for the Wigner rotations for massive particles is reproduced in the Appendix. The conclusion is that representing the quantum state of a relativistic particle with mass $m$, 4-momentum $p=(p^0,\mathbf{p})$ and spin state $\phi$ as $|{p,\phi}\rangle$, with a change of reference frame represented by a homogeneous Lorentz transformation $\Lambda$, the particle state in the new frame is \cite{weinberg}
\begin{equation}\label{wigner}
U(\Lambda)|{p,\phi}\rangle=\sqrt{\frac{(\Lambda p)^0}{p^0}}\sum_{\phi'}D_{\phi,\phi'}(W)|{\Lambda p,\phi'}\rangle,
\end{equation}
where $U(\Lambda)$ is the corresponding unitary transformation and $\sqrt{(\Lambda p)^0/{p^0}}$ is a normalization factor. The particle 4-momentum in the new frame is $\Lambda p$. The Wigner rotation $W(\Lambda,p)\equiv L^{-1}(\Lambda p)\Lambda L(p)$, where $L(p)$ represents a Lorentz boost that transforms the 4-momentum $(m,0)$ in the 4-momentum $p=(p^0,\mathbf{p})$ and $L^{-1}$ is the inverse of $L$, changes the particle spin state via the matrix $D_{\phi,\phi'}(W)$. In this paper we are using a system of units in which the speed of light in vacuum is $c=1$.
We would now like to present a physical interpretation of the change of the particle spin state with the change of reference frame indicated by (\ref{wigner}). The particle spin is defined to be the angular momentum it has in its own rest frame. Let us consider a spin-1/2 particle. In the particle rest frame, where the 4-momentum is $(m,0)$ and a non-relativistic treatment can be used with a momentum-spin separation, the particle spin state can be described by the Bloch vector $\mathbf{r}\equiv \langle \phi|\hat{{\bsigma}}|\phi\rangle$, that represents the expectation value of the Pauli matrices $\hat{{\bsigma}}\equiv\hat{\sigma}_x\mathbf{\hat{x}}+\hat{\sigma}_y\mathbf{\hat{y}}+\hat{\sigma}_z\mathbf{\hat{z}}$. If we substitute in (\ref{wigner}) the labels $\phi$ by the labels $\mathbf{r}$, that contain the same amount of information for a spin-1/2 particle, the Wigner rotation will change the Bloch vector as a 3-dimensional rotation $\mathbf{r}'=R(W)\mathbf{r}$. So (\ref{wigner}) can be written as
\begin{equation}\label{wigner_r}
U(\Lambda)|{p,\mathbf{r}}\rangle=\sqrt{\frac{(\Lambda p)^0}{p^0}}|{\Lambda p,R(W)\mathbf{r}}\rangle.
\end{equation}
The treatment so far is standard, but now we ask the crucial question: how can we prepare the state $|{p,\mathbf{r}}\rangle$? According to quantum mechanics \cite{peres}, we have to measure the particle momentum, obtaining eigenvalues $p^i$ for each component, and measure spin with a quantization axis in the direction $\mathbf{r}$ in the particle rest frame obtaining an eigenvalue $+\hbar/2$ for the spin component. To measure the particle spin, we can use a Stern-Gerlach apparatus with a inhomogeneous magnetic field that points in the direction $\mathbf{r}$ in the particle rest frame. To find the magnetic field in the particle rest frame, we must apply the corresponding Lorentz transformation to the electromagnetic tensor of the apparatus field in the laboratory frame \cite{jackson}: $F^{(0)}=L^{-1}(p)F\tilde{L}^{-1}(p)$, where $\tilde{L}$ represents the transpose of $L$. Defining $\mathbf{b}(F)$ as an unitary vector in the direction of the magnetic field of the electromagnetic tensor $F$, we have
\begin{equation}\label{rbp}
\mathbf{r}=\mathbf{b}(L^{-1}(p)F\tilde{L}^{-1}(p)).
\end{equation}
Now let us consider the description of the process of preparation of the particle state by an observer in a reference frame obtained from a homogeneous Lorentz transformation $\Lambda$ acting on the previous frame. The electromagnetic tensor of the Stern-Gerlach apparatus in the new frame is $\Lambda F\tilde{\Lambda}$ and the particle 4-momentum is $\Lambda p$. So the Bloch vector $\mathbf{r}'$ in the direction of the magnetic field in the particle rest frame computed by this observer is $\mathbf{r}'=\mathbf{b}(L^{-1}(\Lambda p)\Lambda F\tilde{\Lambda}\tilde{L}^{-1}(\Lambda p))$. But the Wigner rotation is defined as $W(\Lambda,p)\equiv L^{-1}(\Lambda p)\Lambda L(p)$ and if we multiply both sides by $L^{-1}(p)$ from the right we obtain $WL^{-1}(p)=L^{-1}(\Lambda p)\Lambda$. So we have
\begin{equation}
\mathbf{r}'=\mathbf{b}(WL^{-1}(p) F\tilde{L}^{-1}(p)\tilde{W})=R(W)\mathbf{b}(L^{-1}(p)F\tilde{L}^{-1}(p))=R(W)\mathbf{r},
\end{equation}
in agreement with (\ref{wigner_r}). The Wigner rotation is a direct consequence of the dependence of the quantization axis of a spin measurement with the particle momentum, representing the fact that different observers compute different quantization axes for the measurement. For particles with spin higher than 1/2, the Bloch vector cannot be used to represent the spin state, but the physical interpretation of the Wigner rotation is the same.
\section{Expectation values of spin measurements on relativistic particles}
According to our formalism, the expectation value of the measurement of a spin component of a spin-1/2 particle prepared in the sate $|{p,\mathbf{r}}\rangle$ made by a Stern-Gerlach type measurement with electromagnetic field tensor $F$, considering eigenvalue $+1$ $(-1)$ if the spin is found aligned (anti-aligned) with the magnetic field in the particle rest frame, is
\begin{equation}\label{exp_val}
E(|{p,\mathbf{r}}\rangle,F)=\mathbf{r}\cdot \mathbf{b}(L^{-1}(p)F\tilde{L}^{-1}(p)).
\end{equation}
This occurs because in the particle rest frame we can use non-relativistic quantum mechanics to state that the expectation value of the measurement is the scalar product between the Bloch vector and an unitary vector in the direction of the magnetic field in this frame. In a recent work, Palmer \textit{et al.} treated Stern-Gerlach measurements on relativistic particles that are in momentum eigenstates obtaining equivalent results \cite{palmer11}. We see that the expectation value of the spin measurement depends explicitly on the particle momentum. Since it is not possible to measure the spin of a relativistic particle in an independent way from its momentum, a spin-momentum partition of the system is meaningless. As a consequence, it is not possible to define a reduced spin matrix for the system tracing out the momenta, as it was done in many previous treatments of the subject
\cite{peres02,gingrich02,li03,peres04,bartled05,peres05,caban05,jordan06,lamata06,jordan07,landulfo09,friis10,choi11}, since it is not possible to correctly predict the outcome statistics of a spin measurement without considering the particle momentum. Spin and momentum cannot be treated as independent variables if the particle has relativistic speeds.
To illustrate the impossibility of the definition of a reduced density matrix for a relativistic particle, consider the following example. A spin-1/2 particle in a superposition of different momenta is deflected up by a Stern-Gerlach apparatus. Since each momentum component have a different quantization axis for the measurement, there will be a correlation between the particle momenta and the particle spin after the measurement. So, if we trace out the particle momenta, we obtain a mixed reduced density matrix for the particle spin. Let us consider that the deflection of the particle is compensated by the application of electromagnetic fields such that the particle momentum distribution after the measurement is the same as before. If the particle spin is now measured by an identical apparatus, since each momentum component will have the same quantization axis as in the preparation procedure, we can state with 100\% certainty that the particle will be deflected up. But this characterizes a pure state of spin. The definition of a reduced density matrix for the spin of a relativistic particle leads to unavoidable paradoxes.
For consistency of the treatment, we must be able to describe the interaction energy between the particle spin and the Stern-Gerlach apparatus electromagnetic field in a covariant way. This would guarantee that observers in any inertial reference frame predict the same expectation values for the measurements. This is important because if one observer registers a detection of a particle in one particular detector, then all other observers must agree that the same detector has registered that particle. When the experiment is repeated many times and the data accumulated, all observers must agree with the outcomes statistics. In the particle rest frame, the interaction Hamiltonian is $\hat{H}_{SG}=-\alpha \hat{\mathbf{s}}\cdot\mathbf{B}_0$, where $\mathbf{B}_0$ is the magnetic field in the particle rest frame, $\hat{\mathbf{s}}=\hbar\hat{{\bsigma}}/2$ is the particle spin operator and $\alpha$ is the gyromagnetic ratio, $\hat{{\bmu}}_0=\alpha \hat{\mathbf{s}}$ being the particle magnetic dipole moment operator in the rest frame. The magnetic dipole moment ${\bmu}$ and the electric dipole moment $\mathbf{d}$ of a particle multiplied by $\gamma_v\equiv1/\sqrt{1-v^2}$, where $\mathbf{v}$ is the particle velocity and $v\equiv |\mathbf{v}|$, form an anti-symmetric tensor $D$ in the same way as the electromagnetic tensor $F$, with the substitutions $\mathbf{E}\rightarrow\gamma_v\mathbf{d}$ and $\mathbf{B}\rightarrow-\gamma_v{\bmu}$ \cite{penfield}. Since in the particle rest frame the magnetic and electric dipole moment operators are $\hat{{\bmu}}_0=\alpha \hat{\mathbf{s}}$ and $\hat{\mathbf{d}}_0=0$, in a reference frame where the particle has velocity $\mathbf{v}$ the interaction Hamiltonian can be written as
\begin{eqnarray}\label{U}
&&\hat{H}_{\mathrm{SG}}(p)=-\hat{{\bmu}}\cdot \mathbf{B}-\hat{\mathbf{d}}\cdot \mathbf{E}=-\frac{1}{2\gamma_v}\mathrm{Tr}(gFg\hat{D})\;,\\\nonumber
&&\mathrm{with}\;\;\hat{{\bmu}}=\alpha\left[\hat{\mathbf{s}}-\frac{\gamma_v}{\gamma_v+1}\mathbf{v}(\mathbf{v}\cdot\hat{\mathbf{s}})\right]\;\;\mathrm{and}\;\;
\hat{\mathbf{d}}=\alpha(\mathbf{v}\times\hat{\mathbf{s}}),
\end{eqnarray}
where $g$ is the diagonal matrix with elements $(1,-1,-1,-1)$ and $\mathrm{Tr}$ stands for the trace. $\hat{H}_{\mathrm{SG}}$ depends on the 4-momentum $p$, since the operators $\hat{{\bmu}}$ and $\hat{\mathbf{d}}$ depend on the particle velocity. The treatment is covariant because if we make a change of reference frame represented by a Lorentz transformation $\Lambda$, the interaction Hamiltonian in the new frame obeys
\begin{equation}
\gamma_{v'}\hat{H}_{\mathrm{SG}}'=-\frac{1}{2}\mathrm{Tr}(g\Lambda F\tilde{\Lambda}g\Lambda\hat{D}\tilde{\Lambda})=\gamma_{v}\hat{H}_{\mathrm{SG}},
\end{equation}
since $\tilde{\Lambda} g \Lambda=g$. We see that the spin operators encoded in the magnetic dipole moment operators must transform as part of a tensor under Lorentz transformations for the interaction to be described in a covariant way. The expectation value of spin measurements can then be written as $-\langle\hat{H}_{\mathrm{SG}}(p)\rangle/|\lambda(\hat{H}_{\mathrm{SG}}(p))|$ for each momentum component, being the same in all inertial reference frames, where $\langle\hat{H}_{\mathrm{SG}}(p)\rangle$ represents the expectation value of the interaction Hamiltonian and $|\lambda(\hat{H}_{\mathrm{SG}}(p))|$ the modulus of its eigenvalues. So we can write the operator related to a Stern-Gerlach spin measurement as
\begin{equation}\label{m_sg}
\hat{M}_{\mathrm{SG}}=\int d^3p\,|p\rangle\langle p|\otimes \frac{\hat{H}_{\mathrm{SG}}(p)}{|\lambda(\hat{H}_{\mathrm{SG}}(p))|}
\end{equation}
with a representation of states in the basis $|p\rangle\otimes|\phi(p)\rangle$ that, despite the notation, cannot have a momentum-spin separation for the reasons described before. The expectation value of the spin measurement can then be written as $E=\mathrm{Tr(\hat{M}_{\mathrm{SG}}\rho)}$, where $\rho$ represents the particle quantum state, being the same in all inertial reference frames.
In the relativistic quantum information literature, many authors use the Pauli-Lubanski (or similar) spin operators to describe spin measurements on relativistic particles \cite{czachor97,ahn03,czachor03,lee04,kim05,czachor05,caban05,caban10,friis10}. Their description is mathematically covariant, in the sense that the expectation values they obtain for the measurements are the same in all inertial reference frames \cite{lee04}, but none of the authors describe a physical system capable of performing the measurements. In a physical implementation, the operator related to the measurement must be written in terms of the interaction Hamiltonian between the spin and the measuring apparatus, like in (8). The Pauli-Lubanski spin operator $\hat{\mathbf{S}}$ is part of a 4-vector $\hat{W}=(\hat{W}^0,\hat{\mathbf{W}})\equiv(\hat{\mathbf{S}}\cdot\mathbf{p},p^0\hat{\mathbf{S}})$, where $p^0$ is the energy and $\mathbf{p}$ the momentum of the particle \cite{muirhead}. If we want to give a covariant description for the interaction of the Pauli-Lubanski spin with a measuring apparatus, such that the expectation value of a spin measurement be the same in all inertial reference frames, $\hat{W}$ must couple to a 4-vector quantity $(G^0,\mathbf{G})$ with an interaction Hamiltonian of the form $\hat{H}_\mathrm{PL}\propto \hat{W}^0G^0-\hat{\mathbf{W}}\cdot\mathbf{G}$. However, we don't know if such a coupling exists, and the physical implementation of the measurement depends on the existence of such coupling in nature. So the use of the Pauli-Lubanski (or similar) spin operators to describe measurements in \cite{czachor97,ahn03,czachor03,lee04,kim05,czachor05,caban05,caban10,friis10} may have consistency problems, and certainly is not suitable to describe measurements where spin couples to the electromagnetic field.
\section{Experimental proposal to test our formulation}
As a possible experimental test of our formulation, let us consider a neutral spin-1/2 particle that propagates with velocity $\mathbf{v}=v[\cos(\theta)\mathbf{\hat{x}}+\sin(\theta)\mathbf{\hat{y}}]$, having momentum $\mathbf{p}=m\mathbf{v}/\sqrt{1-v^2}$, and passes through two Stern-Gerlach apparatuses, the first one with an inhomogeneous magnetic field in the $\mathbf{\hat{x}}$ direction and the second one with an inhomogeneous magnetic field in the $\mathbf{\hat{y}}$ direction. Let us now compute the expectation value of the measurement of the second apparatus $E$ after the first one had yielded an eigenvalue $+1$ for the spin component. Using (\ref{rbp}) to find the spin state prepared by the first apparatus and (\ref{exp_val}) to find the expectation value of the measurement of the second apparatus we obtain
\begin{equation}\label{example}
E=\frac{-v^2\sin(\theta)\cos(\theta)}{\sqrt{[1-v^2\cos^2(\theta)][1-v^2\sin^2(\theta)]}}.
\end{equation}
Of course, the same result is obtained with the use of (\ref{m_sg}). We see that the expectation value tends to zero when $v$ is small, as it should be. Spin and momentum can then be treated independently, and in a non-relativistic scenario the expectation value of the measurement is the scalar product of unitary vectors in the direction of the apparatuses magnetic fields. However, if the particle has a relativistic speed this is not the case anymore. In figure \ref{fig1} we plot the expectation value for $\theta=\pi/4$ and $v$ between 0 and 1. When $v\rightarrow 1$, the expectation value tends to $-1$. This, of course, is surprising and would not be expected from a simple-minded analysis in which spin and momentum are assumed to behave independently. In fact, this example shows why, under general circumstances, we cannot treat the spin and the momentum of a relativistic particle as independent variables. If we trace out the particle momentum, we cannot correctly predict the expectation value of the spin measurement, even if the particle is in a momentum eigenstate.
\begin{figure}
\caption{Expectation value for a spin measurement on a particle with velocity $v(\mathbf{\hat{x}
\label{fig1}
\end{figure}
If we treat the same experiment using the Pauli-Lubanski formalism, hypothetically assuming that spin couples to 4-vectors $(0,\mathbf{G})$ in the laboratory frame, we find that the expectation value of the measurement of the second apparatus is $-E$ from (\ref{example}), with the opposite sign in relation to the Stern-Gerlach case. We obtain completely different results with the two formalisms when the particle has a relativistic velocity. The reason for the difference is illustrated in figure \ref{fig2} for $\theta=\pi/4$. If we have Stern-Gerlach apparatuses with magnetic fields $\mathbf{B}_1$ and $\mathbf{B}_2$ in the laboratory frame, in the particle rest frame the fields $\mathbf{B}_1'$ and $\mathbf{B}_2'$ will have the same components parallel to the particle velocity, but the orthogonal components will be multiplied by $\gamma_v$. So the fields $\mathbf{B}_1'$ and $\mathbf{B}_2'$ tend to point in the opposite direction as the particle velocity approaches the speed of light, as depicted in figure \ref{fig2}-(a). However, if spin couples to 4-vectors $(0,\mathbf{G}_1)$ and $(0,\mathbf{G}_2)$ in the laboratory frame, in the particle rest frame the fields $\mathbf{G}_1'$ and $\mathbf{G}_2'$ will have the same components parallel to the particle velocity, but the orthogonal components will be divided by $\gamma_v$. So the fields $\mathbf{G}_1'$ and $\mathbf{G}_2'$ tend to point in the same direction as the particle velocity approaches the speed of light, as depicted in figure \ref{fig2}-(b).
\begin{figure}
\caption{(a) Magnetic fields $\mathbf{B}
\label{fig2}
\end{figure}
\section{Conclusions}\label{sec:conc}
To summarize, we have provided a physical interpretation of the Wigner rotations in the quantum information context, that result from the fact that different observers compute different quantization axes for spin measurements. Based on that, we computed the expectation values of spin measurements made on relativistic spin-1/2 particles and concluded that it is not possible to measure the particle spin independently from its momentum, such that a momentum-spin partition of the system is actually completely meaningless. It is important to stress that our results for the expectation values of spin measurements should be valid for any measuring procedure in which spin couples to electromagnetic fields in the measuring apparatus, not only for the Stern-Gerlach apparatus. We also discussed that the spin operators must transform in the same way as the physical quantity to which they couple in the measuring apparatus in order to guarantee that all observers in inertial reference frames compute the same expectation value for the measurements. We presented an experimental proposal for the verification of our predictions, that we believe may be tested using an apparatus similar to the one in Sakai \textit{et al.} experiments \cite{sakai06}.
The use of the spin of massive relativistic particles as the carriers of information in quantum information protocols may soon become a reality, and the preset work sets a formalism to be used for such tasks. For instance, our treatment predicts important effects for the violation of Bell's inequalities with entangled relativistic particles \cite{saldanha11}.
\ack
The authors acknowledge Daniel R. Terno and Nicolai Friis for fruitful discussions. P.L.S. was supported by the Brazilian agencies CAPES, CNPq and FACEPE. V.V. acknowledges financial support from the National Research Foundation and Ministry of Education in Singapore and the support of Wolfson College Oxford.
\appendix
\setcounter{section}{1}
\section*{Appendix}
We reproduce here the treatment of Weinberg \cite{weinberg} to describe the Wigner rotations. We can define states of 4-momentum $p=(p^0,\mathbf{p})$ for a particle with mass $m$ as
\begin{equation}\label{psi_p}
|{p,\phi}\rangle\equiv\sqrt{\frac{k^0}{p^0}}U(L(p))|{k,\phi}\rangle,
\end{equation}
where $\phi$ denotes the spin state and $k=(m,0)$ is a standard 4-momentum, chosen to be in the particle rest frame. We are using a system of units in which the speed of light in vacuum is $c=1$. We have $p^\mu=\sum_\nu L^\mu_{\;\;\nu}(p)k^\nu$, or $p=L(p)k$, $L(p)$ being a standard boost, and $U(L(p))$ is the unitary transformation that transforms the state $|{k,\phi}\rangle$ in the state $|{p,\phi}\rangle$, $\sqrt{{k^0}/{p^0}}$ being a normalization factor.
If we make a change of the reference frame represented by a homogeneous Lorentz transformation $\Lambda$, the state in the new frame will be
\begin{eqnarray}\label{a2}\nonumber
U(\Lambda)|{p,\phi}\rangle&=&\sqrt{\frac{k^0}{p^0}}U(\Lambda L(p))|{k,\phi}\rangle\\&=&\sqrt{\frac{k^0}{p^0}}U(L(\Lambda p))
U(L^{-1}(\Lambda p)\Lambda L(p)) |{k,\phi}\rangle.
\end{eqnarray}
The transformation $W(\Lambda,p)\equiv L^{-1}(\Lambda p)\Lambda L(p)$, where $L^{-1}$ represents the inverse of $L$, leaves $k$ invariant, being a rotation denoted Wigner rotation, and we can write
\begin{equation}\label{a3}
U(W)|{k,\phi}\rangle=\sum_{\phi'}D_{\phi,\phi'}(W)|{k,\phi'}\rangle,
\end{equation}
the matrix $D_{\phi,\phi'}(W)$ realizing the Wigner rotation on the particle spin. Substituting (\ref{a3}) and (\ref{psi_p}) in (\ref{a2}) we arrive at (\ref{wigner}).
\section*{References}
\end{document} |
\begin{document}
\title[Koszul complexes and spectra]
{Koszul complexes and spectra of projective hypersurfaces with isolated singularities}
\dedicatory{To the memory of Egbert Brieskorn}
\author{Alexandru Dimca}
\address{Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France.}
\email{[email protected]}
\author{Morihiko Saito}
\address{RIMS Kyoto University, Kyoto 606-8502 Japan}
\email{[email protected]}
\begin{abstract}
For a projective hypersurface with isolated singularities, we generalize some well-known results in the nonsingular case due to Griffiths, Scherk, Steenbrink, Varchenko, and others. They showed, for instance, a relation between the mixed Hodge structure on the vanishing cohomology and the Gauss-Manin system filtered by shifted Brieskorn lattices of a defining homogeneous polynomial by using the V-filtration of Kashiwara and Malgrange. Numerically this implied an identity between the Steenbrink spectrum and the Poincar\'e polynomial of the Milnor algebra. In our case, however, we have to replace these with the pole order spectrum and the alternating sum of the Poincar\'e series of certain subquotients of the Koszul cohomologies respectively, and then study the pole order spectral sequence which does not necessarily degenerate at $E_2$. This non-degeneration is closely related with the torsion of the Brieskorn module which vanished in the classical case.
\end{abstract}
\maketitle
\centerline{\bf Introduction}
\par
\noindent
Let $f$ be a homogeneous polynomial in the graded ${\mathbf C}$-algebra $R:={\mathbf C}[x_1,\dots,x_n]$ where $\deg x_i=1$ and $n\geqslant 2$. Set $d=\deg f$. Consider the shifted Koszul complex
$${}^s\!K^{\ssb}_f:=K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}[n]\quad\hbox{with}\quad K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}=(\Omega^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\rm d}f\wedge).$$
Here $\Omega^j:={\mathcal G}amma({\mathbf C}^n,\Omega_{{\mathbf C}^n}^j)$ with $\Omega_{{\mathbf C}^n}^j$ algebraic so that the $\Omega^j$ are finite free graded $R$-modules, and the degree of $\Omega^j$ in ${}^s\!K^{\ssb}_f$ is shifted so that
$${}^s\!K^j_f=\Omega^{j+n}(jd)\,\,\,\hbox{(i.e.}\,\,\,{}^s\!K^j_{f,k}=\Omega^{j+n}_{jd+k})\,\,\,\hbox{for}\,\,\,j\in{\mathbf Z}.$$
In general the shift of degree by $p$ of a graded module $M$ will denoted by $M(p)$, where the latter is defined by $M(p)_k=M_{k+p}$. Since the dualizing complex for complexes of $R$-modules is given by $\Omega^n[n]$, we have the self-duality
$${\mathbf D}({}^s\!K^{\ssb}_f):={\mathbf R}{\rm Hom}_R({}^s\!K^{\ssb}_f,\Omega^n[n])={}^s\!K^{\ssb}_f(nd).$$
\par
In this paper we assume
$$\dim{\rm Sing}\,f^{-1}(0)\leqslant 1.
\leqno(A)$$
It is well-known, and is easy to show (see e.g. Remark~(1.9)(iv) below) that this implies
$$H^j({}^s\!K^{\ssb}_f)=0\quad\hbox{if}\,\,\,j\ne -1,0.$$
Define
$$M:=H^0({}^s\!K^{\ssb}_f),\quad N:=H^{-1}({}^s\!K^{\ssb}_f).$$
Let ${\mathbf m}=(x_1,\dots,x_n)\subset R$, the maximal graded ideal. Set
$$M':=H_{{\mathbf m}}^0M,\quad M'':=M/M'.$$
These are finitely generated graded $R$-modules having the decompositions $M=\bigoplus_k M_k$, etc.
In the isolated singularity case we have $M''=N=0$, and $M=M'$.
Generalizing a well-known assertion in the isolated singularity case, one may conjecture that the canonical morphism from $M'$ to the graded quotient of the pole order filtration on the Gauss-Manin system is injective, see Proposition~(3.5) below for a partial evidence.
This is closely related with Question~2 and Remark~(5.9) below, see also \cite{Ba1}, \cite{Ba2}, etc.
\par
Let $y:=\par
um_{i=1}^n\,c_ix_i$ with $c_i\in{\mathbf C}$ sufficiently general so that $\{y=0\}\subset{\mathbf C}^n$ is transversal to any irreducible component of ${\rm Sing}\,f^{-1}(0)$. Then $M'$ is the $y$-torsion subgroup of $M$, and $M''$, $N$ are finitely generated free graded ${\mathbf C}[y]$-modules of rank $\tau_Z$, where $\tau_Z$ is the total Tjurina number as in (0.4) below.
Note that there is a shift of the grading on $N$ by $d$ between this paper and \cite{DiSt1}, \cite{DiSt2}.
\par
Define the (higher) dual graded $R$-modules by
$$D_i(M):=\widetilde{E}xt_R^{n-i}(M,\Omega^n)\quad(i\in{\mathbf Z}),$$
and similarly for $D_i(N)$, etc. From the above self-duality of the Koszul complex ${}^s\!K^{\ssb}_f$, we can deduce the following duality (which is known to the specialists at least by forgetting the grading, see \cite{Pe}, \cite{vStWa} and also \cite{EyMe}, \cite{Se}, etc.):
\par
\noindent
{\bf Theorem~1.} {\it There are canonical isomorphisms of graded $R$-modules
$$\aligned D_0(M')=D_0(M)&=M'(nd),\\ D_1(M'')=D_1(M)&=N(nd),\\ D_1(N)&=M''(nd),\endaligned
\leqno(0.1)$$
and $D_i(M)$, $D_i(M')$, $D_i(M'')$, $D_i(N)$ vanish for other $i$.}
\par
This generalizes a well-known assertion in the isolated singularity case where $M''=N=0$.
Theorem~1 implies that $M'$, $M''$ and $N$ are Cohen-Macaulay graded $R$-modules of dimension $0$, $1$ and $1$ respectively (but $M$ itself is not Cohen-Macaulay). Moreover $M'$ is graded self-dual, and $M''$ and $N$ are graded dual of each other, up to a shift of grading.
\par
For $k\in{\mathbf Z}$, set
$$\mu_k=\dim M_k,\quad\mu'_k=\dim M'_k,\quad\mu''_k=\dim M''_k,\quad
\nu_k=\dim N_k.$$
Let $g:=\par
um_{i=1}^nx_i^d$, and $\gamma_k:=\dim(H^nK_g^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k=\dim\bigl(\Omega^n/\,\par
um_{i=1}^n\, x_i^{d-1}\Omega^n\bigr){}_k$, so that
$$\par
um_k\,\gamma_k\,t^k=(t^d-t)^n/(t-1)^n.
\leqno(0.2)$$
(Here $g$ can be any homogeneous polynomial of degree $d$ with an isolated singular point.)
It is known (see \cite{Di2}, \cite{DiSt1}, \cite{DiSt2}) that
$$\mu_k=\mu'_k+\mu''_k=\nu_k+\gamma_k\quad(k\in{\mathbf Z}),
\leqno(0.3)$$
since the Euler characteristic of a bounded complex is independent of its differential if the components of the complex are finite dimensional.
\par
By the first assertion of (0.1) together with (1.1.4) for $i=1$ and by (0.2), we get the following symmetries:
\par
\noindent
{\bf Corollary~1.}\quad $\mu'_k=\mu'_{nd-k},\quad\gamma_k=\gamma_{nd-k}\quad(k\in{\mathbf Z})$.
\par
Let $Z:=\{f=0\}\subset Y:={\mathbf P}^{n-1}$, and $\Sigma:={\rm Sing}\,Z$. The total Tjurina number $\tau_Z$ is defined by
$$\tau_Z:=\par
um_{z\in\Sigma}\,\tau_z\quad\hbox{with}\quad\tau_z:=\dim_{{\mathbf C}}{\mathcal O}_{Y,z}/(h_z,{\partial} h_z),
\leqno(0.4)$$
where $h_z$ is a local defining equation of $Z$ at $z$, and ${\partial} h_z$ is the Jacobian ideal of $h_z$ generated by its partial derivatives. By Theorem~1, $M''$ and $N$ are Cohen-Macaulay, and are dual of each other up to a shift of grading. Combining this with the graded local duality (1.1.4) for $i=1$ (see \cite{BrHe}, \cite{Ei}, etc.) together with (1.9.3) below, we get the following.
\par
\noindent
{\bf Corollary~2.}\quad $\mu''_k+\nu_{nd-k}=\tau_Z\quad(k\in{\mathbf Z})$.
\par
Here the calculation of the local cohomology in the local duality is not so trivial (see Remark~(1.7) below), and we can also use an exact sequence as in \cite[Prop.~2.1]{Sl} (see also \cite[Prop.~2.1.5]{Gro} and \cite{SaSl}, etc.) Note that Corollary~2 can also be deduced from Thm.~3.1 in \cite{Di2}, see Remark~(1.9)(i) below. By Corollaries~1 and 2 together with (0.3), we get the following.
\par
\noindent
{\bf Corollary~3.}\quad $\mu'_k=\mu_k+\mu_{nd-k}-\gamma_k-\tau_Z,\quad
\mu''_k=\tau_Z-\mu_{nd-k}+\gamma_k\quad(k\in{\mathbf Z})$.
\par
This means that $\mu'_k$ and $\mu''_k$ are essentially determined by $\mu_k$ and $\mu_{nd-k}$. Note that $\{\mu''_k\}$ and $\{\nu_k\}$ are weakly increasing sequences of non-negative integers. It is shown that $\{\mu'_k\}$ is log-concave in a certain case, see \cite{Sti}. Assuming ${\rm Sing}\,Z\ne\emptyset$, we have $\mu''_k=\nu_k=\tau_Z>0$ for $k\gg 0$, hence $M'',N$ are nonzero, although $M'$ may vanish, see Remark~(1.9)(iii) below. By Corollary~2 and (0.3) we get the following.
\par
\noindent
{\bf Corollary~4.}\quad$\gamma_k-\mu'_k=\mu''_k+\mu''_{nd-k}-\tau_Z\quad(k\in{\mathbf Z})$.
\par
Here a fundamental question seems to be the following.
\par
\noindent
{\bf Question~1.} Are both sides of the above equality non-negative?
\par
\noindent
This seems to be closely related to the subject treated in \cite{ChDi}, \cite{Di2}, \cite{DiSt1}, \cite{DiSt2}, etc.
We have a positive answer to Question~1 if $n=3$ and $\Sigma$ is a complete intersection in ${\mathbf P}^2$ (see \cite{Sti}) or if $f$ has type (I), where $f$ is called type (I) if the following condition is satisfied (and type (II) otherwise):
$$\mu''_k=\tau_Z\,\,\,\hbox{for}\,\,\,k\geqslant nd/2,\quad\hbox{i.e.}\quad
\nu_k=0\,\,\,\hbox{for}\,\,\,k\leqslant nd/2.
\leqno(0.5)$$
By the definition of $N$, the last condition in (0.5) cannot hold if there is a nontrivial relation of very low degree between the partial derivatives of $f$, e.g. in case $f$ is a polynomial of $n-1$ variables (or close to it), see Remark~(2.9) below. However, it holds in relatively simple cases, including the nodal hypersurface case by \cite[Thm.~2.1]{DiSt2}, see Remark~(2.10) below.
\par
In the type (I) case, we get the $\mu'_k$ by restricting to $k\leqslant nd/2$ (where $\mu'_k+\mu''_k=\mu_k=\gamma_k$ holds) if we know the $\mu''_k$. This can be done for instance in the following case.
\par
\noindent
{\bf Proposition~1.} {\it Assume $Z$ has only ordinary double points $z_1,\dots,z_{\tau_Z}$, and moreover the $z_i$ correspond to linearly independent vectors in ${\mathbf C}^n$ so that $\tau_Z=r\leqslant n$. Then
$$\aligned\mu''_k&=\begin{cases}\,0&(\,k<n\,),\\
\,1&(\,k=n\,),\\
\tau_Z&(\,k>n\,),\end{cases}\quad\quad
\nu_k=\begin{cases}\,0&(\,k<n(d-1)\,),\\
\tau_Z-1&(\,k=n(d-1)\,),\\
\tau_Z&(\,k>n(d-1)\,),\end{cases}\\
\mu'_k&=\begin{cases}\,0&(\,k\notin(n,n(d-1))\,),\\
\gamma_k-\tau_Z&(\,k\in(n,n(d-1))\,),\end{cases}\endaligned$$
where $(n,n(d-1))\subset{\mathbf R}$ denotes an open interval.}
\par
This follows from Lemma~(2.1) below together with Corollary~2 and (0.3). It can also be deduced from the results in \cite{Di2}, and seems to be closely related with \cite[Thm.~2]{DiSaWo}. The situation becomes, however, rather complicated if the number of singular points is large, see \cite{ChDi}, \cite{Di2}, \cite{DiSt1}, \cite{DiSt2}.
\par
Let ${\rm Sp}(f)=\sum_{\alpha}n_{f,\alpha}\,t^{\alpha}\in{\mathbf Q}[t^{1/d}]$ be the {\it Steenbrink spectrum} of $f$ (see \cite{St2}, \cite{St3}) which is normalized as in \cite{St2}. To study the relation with the Koszul cohomologies $M$, $N$ by generalizing the well-known assertion in the isolated singularity case where $M''=N=0$ and $M=M'$ (see \cite{St1} and also \cite{Gri}, \cite{SkSt}, \cite{Va}, etc.), we have to introduce the {\it pole order spectrum} ${\rm Sp}_P(f)$ by replacing the Hodge filtration $F$ with the pole order filtration $P$ in \cite{Di1}, \cite{Di3}, \cite{DiSa2}, \cite{DiSt1}. There are certain shifts of the exponents coming from the difference between $F$ and $P$. Here we have the inclusion $F\subset P$ in general, and the equality holds in certain cases (see \cite{Di3}).
We can calculate these spectra explicitly in the case $n=2$, see Propositions~(3.3) and (3.4).
The relation between the two spectra is, however, quite nontrivial in general (see for instance Example~(3.7) below).
\par
The reason for which we introduce ${\rm Sp}_P(f)$ is that it is related with the Poincar\'e series of $M$, $N$ as follows:
The differential of the de Rham complex $(\Omega^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d)$ induces a morphism of graded ${\mathbf C}$-vector spaces of degree $-d:$
$${\partial}d^{(1)}:N\to M,$$
i.e. preserving the degree up to the shift by $-d$. Let $H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ denote the Brieskorn module \cite{Bri} (in a generalized sense) which is a graded ${\mathbf C}$-module endowed with actions of $t$, $\dd_t^{-1}$, and $t{\partial}_t$, see (4.2) below. Let $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ be its $t$-torsion (or equivalently, $\dd_t^{-1}$-torsion) subspace. It has the kernel filtration $K_{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ defined by
$$K_i(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}:={\rm Ker}\,t^i\subset(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}\quad(i\geqslant 0).
\leqno(0.6)$$
The main theorem of this paper is as follows:
\par
\noindent
{\bf Theorem~2.} {\it There are inductively defined morphisms of graded ${\mathbf C}$-vector spaces of degree $-rd:$
$${\partial}d^{(r)}:N^{(r)}\to M^{(r)}\quad(r\geqslant 2),$$
such that $N^{(r)}$, $M^{(r)}$ are the kernel and the cokernel of ${\partial}d^{(r-1)}$ respectively, and are independent of $r\gg 0$ $($that is, ${\partial}d^{(r)}=0$ for $r\gg 0)$, and we have
$${\rm Sp}_P(f)=S(M^{(r)})(t^{1/d})-S(N^{(r)})(t^{1/d})\quad(r\gg 0),
\leqno(0.7)$$
where $S(M^{(r)})(t)$, $S(N^{(r)})(t)$ denote the Poincar\'e series of $M^{(r)}$, $N^{(r)}$ for $r\geqslant 2$.
\par
Moreover, there are canonical isomorphisms
$${\rm Im}\,{\partial}d^{(r)}={\mathcal G}r^K_{r-1}({\rm Coker}\,t)\quad(r\geqslant 2),
\leqno(0.8)$$
where $K_{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is the kernel filtration on $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$, and the right-hand side of $(0.8)$ is a subquotient of $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$. In particular, ${\partial}d^{(r)}$ vanishes for any $r\geqslant 2$ {\rm (}that is, $M^{(r)}=M^{(2)}$, $N^{(r)}=N^{(2)}$ for any $r\geqslant 2)$ if and only if $H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is torsion-free.}
\par
Note that ${\rm Ker}\,t^i$ in (0.6) and ${\rm Coker}\,t$ in (0.8) can be replaced respectively with ${\rm Ker}\,{\partial}_t^{-i}$ and ${\rm Coker}\,\dd_t^{-1}$ by using (4.2.2) below.
For the proof of Theorem~2 we use the spectral sequence associated with the pole order filtration on the algebraic microlocal Gauss-Manin complex (see (4.4.4) below), and the morphisms ${\partial}d^{(r)}$ are induced by the differentials ${\partial}d_r$ of the spectral sequence. (We can also use the usual Gauss-Manin complex instead of the microlocal one.) The last equivalent two conditions in Theorem~2 are further equivalent to the $E_2$-degeneration of the (microlocal) pole order spectral sequence, see Corollary~(4.7) below (and also \cite{vSt}). Moreover $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ is finite dimensional if and only if $Z$ is analytic-locally defined by a weighted homogeneous polynomial at any singular point, see Theorems~(5.2) and (5.3) below.
(In fact, the if part in the analytic local setting was shown in the second author's master thesis, see e.g. \cite[Thm.~3.2]{BaSa} and also \cite{vSt}.)
Here Theorem~(5.3) gives rather precise information about the kernel of ${\partial}d^{(1)}$.
This is a refinement of \cite[Thm.~2.4(ii)]{DiSt1}, and is used in an essential way in \cite{DiSa3}.
Theorem~(5.3) implies a sharp estimate for $\max\{k\,|\,\nu_k=0\}$ when $n=3$, see Corollary~(5.5) below.
This assertion is used in an essential way in \cite{DiSe}, and is generalized to the case $n>3$ in \cite[Theorem~9]{DiSa3} (see \cite{Di4} for another approach to the case $n>3$).
\par
In case $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}=0$, we can determine the pole order spectrum if we can calculate the morphism ${\partial}d^{(1)}:N\to M$, although the latter is not so easy in general unless the last conditions in Theorem~(5.3) are satisfied (see also Remark~(5.9) below). Note that the pole order spectral sequence was studied in \cite{vSt} from a slightly different view point in the (non-graded) analytic local case.
\par
For the moment there are no examples such that the singularities of $Z$ are weighted homogeneous and $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}\ne0$. We have the following.
\par
\noindent
{\bf Question~2.} Assume all the singularities of $Z$ are weighted homogeneous. Then, is $H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ torsion-free so that the pole order spectral sequence degenerates at $E_2$ and the equality (0.7) holds with $r=2$?
\par
We have a positive answer in certain cases; for instance, if $n=2$ or $1$ is not an eigenvalue of $T_z^d$ for any $z\in{\rm Sing}\,Z$ where $T_z$ is the monodromy of a local defining polynomial $h_z$ of $(Z,z)$, see Corollary~(5.4) below for a more general condition.
However, the problem is quite nontrivial in general.
In the above second case, Theorem~(5.3) actually implies the injectivity of ${\partial}d^{(1)}:N\to M$ (which is a morphism of degree $-d$), and we get the following.
\par
\noindent
{\bf Theorem~3.} {\it If $(Z,z)$ is weighted homogeneous and $1$ is not an eigenvalue of $T_z^d$ for any $z\in{\rm Sing}\,Z$, then $H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is torsion-free and we have}
$${\rm Sp}_P(f)=S(M)(t^{1/d})-S(N)(t^{1/d})\,t^{-1}.$$
\par
Here the second condition is satisfied if $1$ is not an eigenvalue of $T_z$ and moreover the order of $T_z$ is prime to $d$ for any $z\in{\rm Sing}\,Z$.
Note that the second assumption can be replaced with $H^{n-2}(f^{-1}(1),{\mathbf C})=0$ if Question~2 is positively solved in this case (see Remark~(5.9) below for a picture in the optimal case).
\par
The first author is partially supported by Institut Universitaire de France.
The second author is partially supported by Kakenhi 24540039.
\par
In Section~1 we prove Theorem~1 after reviewing graded local duality for the convenience of the reader. In Section~2 we explain some methods to calculate the Koszul cohomologies in certain cases. In Section~3 we recall some basics from the theory of spectra, and prove Proposition~(3.3), (3.4), and (3.5). In Section~4 we prove Theorem~2 after reviewing some facts from Gauss-Manin systems and Brieskorn modules. In Section~5 we calculate ${\partial}d^{(1)}$ in certain cases, and prove Theorems~(5.2) and (5.3).
\par
\par
\vbox{\centerline{\bf 1. Graded local cohomology and graded duality}
\par
\noindent
In this section we prove Theorem~1 after reviewing graded local duality for the convenience of the reader.}
\par
\noindent
{\bf 1.1.~Graded local duality.} Let $R={\mathbf C}[x_1,\dots,x_n]$, and ${\mathbf m}=(x_1,\dots,x_n)\subset R$. Set
$$\Omega^k={\mathcal G}amma({\mathbf C}^n,\Omega_{{\mathbf C}^n}^k)\quad(k\in{\mathbf Z}).
\leqno(1.1.1)$$
Here $\Omega_{{\mathbf C}^n}^k$ is algebraic, and $\Omega^k$ is a finite free graded $R$-module with $\deg x_i=\deg dx_i=1$.
\par
For a bounded complex of finitely generated graded $R$-modules $M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$, define
$$\aligned&D_i(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}):=\widetilde{E}xt_R^{n-i}(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},\Omega^n)= H^{-i}\bigl({\mathbf D}(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})\bigr)\\ &\hbox{with}\quad{\mathbf D}(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}):={\mathbf R}{\rm Hom}_R(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},\Omega^n[n]),\endaligned
\leqno(1.1.2)$$
where ${\mathbf D}(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})$ can be defined by taking a graded free resolution $P^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$.
\par
For a finitely generated graded $R$-module $M$, set
$$H_{{\mathbf m}}^0M:=\{a\in M\mid
{\mathbf m}^ka=0\,\,\,\hbox{for}\,\,\,k\gg 0\}.
\leqno(1.1.3)$$
Let $H^i_{{\mathbf m}}M$ be the cohomological right derived functors $(i\in{\mathbf N})$. These are defined by taking a graded injective resolution of $M$. We can calculate them by taking a graded free resolution of $M$ as is explained in textbooks of commutative algebra, see e.g. \cite{BrHe}, \cite{Ei}. Indeed, $H^i_{{\mathbf m}}R=0$ for $i\ne n$, and
$$H^n_{{\mathbf m}}R={\mathbf C}[x_1^{-1},\dots,x_n^{-1}]\hbox{$\frac{1}{x_1\dots x_n}$},$$
where the right-hand side is identified with a quotient of the graded localization of $R$ by $x_1\cdots x_n$. We then get the {\it graded local duality} for finitely generated graded $R$-modules $M$:
$$D_i(M)_k={\rm Hom}_{{\mathbf C}}((H^i_{{\mathbf m}}M)_{-k},{\mathbf C})\quad(k\in{\mathbf Z},\,i\geqslant 0),
\leqno(1.1.4)$$
see loc.~cit. (Indeed, this can be reduced to the case $M=R$ by the above argument.)
\par
\noindent
{\bf Remarks~1.2.} (i) The functors $H^i_{{\mathbf m}}$ and $D_i$ are compatible with the corresponding functors for non-graded $R$-modules under the forgetful functor, and moreover, the latter functors are compatible with the corresponding sheaf-theoretic functors as is well-known in textbooks of algebraic geometry, see e.g. \cite{Ha}. However, the information of the grading is lost by passing to the corresponding sheaf unless we use a sheaf with ${\mathbf C}^*$-action.
\par
(ii) If $M$ is a finitely generated graded $R$-module, then it is well-known that
$$D_i(M)=0\quad\hbox{for}\,\,\,i<0.
\leqno(1.2.1)$$
\par
\noindent
{\bf 1.3.~Spectral sequences.} For a bounded complex of finitely generated graded $R$-modules $M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$, we have a spectral sequence
$$'\!E_2^{p,q}(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})=D_{-p}(H^{-q}M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})\Longrightarrow D_{-p-q}(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}).
\leqno(1.3.1)$$
This can be defined for instance by taking graded free resolutions of $H^iM^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ and ${\rm Im}\,d^{\,i}$ for $i\in{\mathbf Z}$, and then extending these to a graded free resolution of $M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ by using the short exact sequences
$$0\to{\rm Im}\,d^{\,i-1}\to{\rm Ker}\,d^{\,i}\to H^iM^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to 0,\quad 0\to{\rm Ker}\,d^{\,i}\to M^i\to{\rm Im}\,d^{\,i}\to 0,$$
as is explained in classical books about spectral sequences. We can also construct (1.3.1) by using the filtration $\tau_{\leqslant -q}$ on $M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ as in \cite{De}.
\par
Applying (1.3.1) to ${\mathbf D}(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})$ and using ${\mathbf D}({\mathbf D}(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}))=M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$, we get
$$''\!E_2^{p,q}(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})=D_{-p}(D_q(M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}))\Longrightarrow H^{p+q}M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}.
\leqno(1.3.2)$$
\par
\noindent
{\bf Lemma~1.4.} {\it Let ${\mathcal S}\hskip-.5pt h(M)$ denote the coherent sheaf on $X:={\mathbf C}^n$ corresponding to a finitely generated graded $R$-module $M$. Then we have the following equivalence.}
$$\aligned H_{{\mathbf m}}^0M=M&\iff{\rm supp}\,{\mathcal S}\hskip-.5pt h(M)\subset\{0\}\\ &\iff\hbox{$M$ is finite dimensional over ${\mathbf C}$},\\ &\iff D_i(M)=0\,\,\,\hbox{for any}\,\,\,i\ne 0.\endaligned
\leqno(1.4.1)$$
\par
\noindent
{\it Proof.} This is almost trivial except possibly for the last equivalence. It can be shown by restricting to a sufficiently general point of the support of ${\mathcal S}\hskip-.5pt h(M)$ in case the support has positive dimension. Here we use the fact that the dual ${\mathbf D}({\mathcal S}\hskip-.5pt h(M))$ is compatible with the direct image under a closed embedding, and this follows from Grothendieck duality for closed embeddings as is well-known, see e.g. \cite{Ha}. This finishes the proof of Lemma~(1.4).
\par
The following is well-known, see \cite{BrHe}, \cite{Ei}, etc. We note here a short proof for the convenience of the reader.
\par
\noindent
{\bf Proposition~1.5.} {\it Let $M$ be a finitely generated $R$-module. Set $m:=\dim{\rm supp}\,{\mathcal S}\hskip-.5pt h(M)$. Then}
$$D_i(M)=0\,\,\,\hbox{for}\,\,\,i>m.
\leqno(1.5.1)$$
\par
\noindent
{\it Proof.} There is a complete intersection $Z$ of dimension $m$ in $X={\rm Spec}\,R$ such that $M$ is annihilated by the ideal $I_Z$ of $Z$, i.e. $M$ is an $R_Z$-module with $R_Z:=R/I_Z$, and $I_Z$ is generated by a regular sequence $(g_i)_{i\in[1,n-m]}$ of $R$ with $g_iM=0$. (Here $M$ is not assumed graded.) Set
$$\omega_Z=\widetilde{E}xt_R^{n-m}(R_Z,\Omega^n).$$
This is called the canonical (or dualizing) module of $Z$. We then get
$$D_i(M)=\widetilde{E}xt_{R_Z}^{-i}(M,\omega_Z[m]),
\leqno(1.5.2)$$
by Grothendieck duality for the closed embedding $i_Z:Z\hookrightarrow X$, see e.g. \cite{Ha}, etc. In fact, taking an injective resolution $G$ of $\Omega^n[n]$, one can show (1.5.2) by using the canonical isomorphism
$${\rm Hom}_{R_Z}(M,{\rm Hom}_R(R_Z,G))={\rm Hom}_R(M,G).$$
Since the right-hand side of (1.5.2) vanishes for $i>m$, the assertion follows.
\par
\noindent
{\bf Corollary~1.6.} {\it Let $M$ be a finitely generated graded $R$-module with $\dim{\rm supp}\,{\mathcal S}\hskip-.5pt h(M)=1$. Then we have a short exact sequence
$$0\to D_0(D_0(M))\to M\to D_1(D_1(M))\to 0,
\leqno(1.6.1)$$
together with}
$$D_0(D_1(M))=0,\quad D_1(D_0(M))=0.
\leqno(1.6.2)$$
\par
\noindent
{\it Proof.} By Lemma~(1.5) we get
$$''\!E_2^{p,q}(M)=0\quad\hbox{if}\quad(p,q)\notin[-1,0]\times[0,1].$$
So the spectral sequence (1.3.2) degenerates at $E_2$ in this case, and the assertion follows.
\par
\noindent
{\bf Remark~1.7.} Let $M$ be a graded $R$-module of dimension 1, i.e. $C:={\rm supp}\,{\mathcal S}\hskip-.5pt h(M)$ is one-dimensional. Let $I_M\subset R$ be the annihilator of $M$. Set ${\mathbf R}R:=R/I_M$. Let $y\in R$ be a general element of degree 1 whose restriction to any irreducible component of $C$ is nonzero. Set $R':={\mathbf C}[y]\subset R$. Let ${\mathbf m}m$, ${\mathbf m}'$ be the maximal graded ideals of ${\mathbf R}R$, $R'$. Let $H^i_{(R,{\mathbf m})}M$ denote $H^i_{{\mathbf m}}M$, and similarly for $H^i_{({\mathbf R}R,{\mathbf m}m)}M$, etc. (to avoid any confusion). There are canonical morphisms
$$(R,{\mathbf m})\to({\mathbf R}R,{\mathbf m}m)\leftarrow(R',{\mathbf m}'),$$
and they imply canonical morphisms
$$H^i_{(R,{\mathbf m})}M\leftarrow H^i_{({\mathbf R}R,{\mathbf m}m)}M\to H^i_{(R',{\mathbf m}')}M.
\leqno(1.7.1)$$
Indeed, any graded injective resolution of $M$ over ${\mathbf R}R$ can be viewed as a quasi-isomorphism over $R$ or $R'$, and we can further take its graded injective resolution over $R$ or $R'$, which induces the above morphisms.
\par
These morphisms are isomorphisms since they are isomorphisms by forgetting the grading as is well-known. (Note that the morphisms ${\rm Spec}\,R\leftarrow{\rm Spec}\,{\mathbf R}R\to {\rm Spec}\,R'$ are proper. Here it is also possible to use the graded local duality together with Grothendieck duality.) Using the long exact sequence associated with the local cohomology and the localization, we can show
$$H^1_{(R',{\mathbf m}')}M=M[y^{-1}]/M.
\leqno(1.7.2)$$
So we get the following canonical isomorphism (as graded $R'$-modules):
$$H^1_{{\mathbf m}}M=M[y^{-1}]/M.
\leqno(1.7.3)$$
This also follows from an exact sequence in \cite[Prop.~2.1]{Sl} (see also \cite[Prop.~2.1.5]{Gro} and \cite{SaSl}, etc.)
\par
\noindent
{\bf 1.8.~Proof of Theorem~1.} As is explained in the introduction, we have the self-duality
$${\mathbf D}({}^s\!K^{\ssb}_f)={}^s\!K^{\ssb}_f(nd),$$
which implies the isomorphisms of graded $R$-modules
$$D_i({}^s\!K^{\ssb}_f)=H^{-i}({}^s\!K^{\ssb}_f)(nd).
\leqno(1.8.1)$$
Consider the spectral sequence (1.3.1) for $M^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}={}^s\!K^{\ssb}_f$. By Lemma~(1.5) applied to $M$, $N$, this degenerates at $E_2$. Combining this with (1.8.1), we thus get
$$D_1(M)=N(nd),\quad D_0(N)=0,
\leqno(1.8.2)$$
together with a short exact sequence
$$0\to D_0(M)\to M(nd)\to D_1(N)\to 0.
\leqno(1.8.3)$$
By (1.6.2) in Corollary~(1.6) and Lemma~(1.5) applied to $M$, $N$, the proof of Theorem~1 is then reduced to showing that (1.8.3) is naturally identified, up to the shift of grading by $nd$, with
$$0\to M'\to M\to M''\to 0.
\leqno(1.8.4)$$
For this, it is enough to show
$$H^0_{{\mathbf m}}D_0(M)=D_0(M),\quad H^0_{{\mathbf m}}D_1(N)=0.
\leqno(1.8.5)$$
However, the first equality is equivalent to the vanishing of $D_i(D_0(M))$ for $i\ne 0$ by Lemma~(1.4), and follows from (1.6.2) in Corollary~(1.6) together with Lemma~(1.5) applied to $D_0(M)$. The second equality follows for instance from the local duality (1.1.4) for $i=0$ together with (1.6.2) in Corollary~(1.6) applied to $N$. Thus (1.8.5) is proved. This finishes the proof of Theorem~1.
\par
\noindent
{\bf Remarks~1.9.} (i) Corollary~2 can also be deduced from \cite[Thm.~3.1]{Di2}. Indeed, by the argument in Section~2 in loc.~cit., we can deduce
$${\rm def}_{k-n}\Sigma_f=\tau_Z -\mu_k'',
\leqno(1.9.1)$$
where ${\rm def}_{k-n}\Sigma_f$ is as in loc.~cit. Moreover, Thm.~3.1 in loc.~cit.\ gives
$${\rm def}_{k-n}\Sigma_f=\mu_{nd-k}-\gamma_{nd-k}=\nu_{nd-k}.
\leqno(1.9.2)$$
So Corollary~2 follows.
\par
(ii) It is well-known that
$$\dim_{{\mathbf C}}M''_k=\dim_{{\mathbf C}}M_k=\tau_Z\quad\hbox{if}\,\,\,k\gg 0.
\leqno(1.9.3)$$
Indeed, the first equality of (1.9.3) is trivial, and it is enough to show the last equality. Changing the coordinates, we may assume $x_n=y$, where $y$ is as in the introduction. On $\{x_n\ne 0\}\subset{\mathbf C}^n$, we have the the coordinates $x'_1,\dots,x'_n$ defined by $x'_j=x_j/x_n$ for $j\ne n$, and $x'_n=x_n$. Using these, we have $f(x)=x_n^dh(x')$, where $x'=(x'_1,\dots,x'_{n-1})$. This implies that the restriction of ${\mathcal S}\hskip-.5pt h(M)$ to the generic point of an irreducible component of the support of $M$ corresponding to $z\in Z$ has rank $\tau_z$ in the notation of the introduction. So (1.9.3) follows.
\par
(iii) Assume $\dim{\rm Sing}\,f^{-1}(0)=1$, i.e. $\Sigma={\rm Sing}\,Z\ne\emptyset$. Let $({\partial} f)\subset R$ denote the Jacobian ideal of $f$ (generated by the partial derivatives ${\partial} f/{\partial} x_i$ of $f$). Then the Jacobian ring $R/({\partial} f)$ (which is isomorphic to $M$ as a graded $R$-module up to a shift of grading) is a Cohen-Macaulay ring if and only if $M'=0$. Indeed, these are both equivalent to the condition that $M$ is a Cohen-Macaulay $R$-module (since $\tau_Z\ne 0$ and hence $M''\ne 0$). Here Grothendieck duality for closed embeddings is used to show the equivalence with the condition that $R/({\partial} f)$ is a Cohen-Macaulay ring. Note that $M'$ might vanish in general, for instance if $f$ is as in Example~(2.7) below or even in case $f=xyz$.
\par
(iv) Assume $\bigcap_{i=1}^mg_i^{-1}(0)
\subset{\mathbf C}^n$ has codimension $\geqslant r$, where $g_i\in R\,\,\,(i\in[1,m])$. Then there is a regular sequence $(h_j)_{j\in[1,r]}$ of $R$ with $h_j\in V:=\sum_{i=1}^m{\mathbf C}\,g_i$ by increasing induction on $r$ or $m$. This implies the vanishing of the cohomology of the Koszul complex:
$$H^kK^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}(R;g_1,\dots,g_m)=0\quad(k<r),$$
by using the $n$-ple complex structure of the Koszul complex as is well-known (see Remark~(v) below). In fact, we can replace the basis $(g_i)$ of the vector space $V$ so that a different expression of the Koszul complex can be obtained. (However, it is not always possible to choose $h_j$ so that $\sum_{i=1}^mRg_i=\sum_{j=1}^rRh_j$ even if $\bigcap_{i=1}^mg_i^{-1}(0)$ has pure codimension $r$ unless $(g_i)$ is already a regular sequence, i.e. $r=m$.)
\par
(v) For $g_i\in R\,\,(i\in[1,m])$, the Koszul complex $K^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}(R;g_1,\dots,g_m)$ can be identified with the associated single complex of the $m$-ple complex whose $(j_1,\dots,j_m)$-component is $R$ for $(j_1,\dots,j_m)\in[0,1]^m$, and $0$ otherwise, where its $i$-th differential ${\partial}d_i$ is defined by the multiplication by $g_i$ on $R$.
\par
(vi) Theorem~1 holds with ${\rm d}f$ in the definition of the Koszul complex replaced by a 1-form $\omega=\sum_{i=1}^n g_idx_i$ if the $g_i$ are homogeneous polynomials of degree $d-1$ such that $\bigcap_ig_i^{-1}(0)\subset{\mathbf C}^n$ is at most 1-dimensional. See \cite{Pe}, \cite{vStWa} for the (non-graded) analytic local case.
\par
\par
\vbox{\centerline{\bf 2. Calculation of the Koszul cohomologies}
\par
\noindent
In this section we explain some methods to calculate the Koszul cohomologies in certain cases.}
\par
\noindent
{\bf Lemma~2.1.} {\it Let $r$ be the dimension of the vector subspace of ${\mathbf C}^n$ generated by the one-dimensional vector subspaces corresponding to the singular points of $Z$. Then}
$$\mu''_n=1,\quad\mu''_{n+1}\geqslant r.$$
\par
\noindent
{\it Proof.} Let $\Sigma'$ be a subset of $\Sigma\,(={\rm Sing}\,Z)$ corresponding to linearly independent $r$ vectors of ${\mathbf C}^n$. Let $I_{\Sigma'}$ be the (reduced) graded ideal of $R$ corresponding to $\Sigma'$. There is a canonical surjection
$$M\to\M:=\Omega^n/I_{\Sigma'}\,\Omega^n.
\leqno(2.1.1)$$
The target is a free graded ${\mathbf C}[y]$-module of rank $r$, where $y$ is as in the introduction, and it has free homogeneous generators $w_i\,(i\in[1,r])$ with $\deg w_1=n$ and $\deg w_i=n+1$ for $i>1$. So the surjection (2.1.1) factors through $M''$, and the assertion follows.
\par
\noindent
{\bf Proposition~2.2.} {\it Let $f=f_1+f_2$ with $f_1\in{\mathbf C}[x_1,\dots,x_{n_1}]$, $f_2\in{\mathbf C}[x_{n_1+1},\dots,x_n]$ where $1<n_1<n-1$. Assume the dimensions of the singular loci of $f_1^{-1}(0)\subset{\mathbf C}^{n_1}$ and $f_2^{-1}(0)\subset{\mathbf C}^{n-n_1}$ are respectively $1$ and $0$. Then there are isomorphisms of graded $R$-modules
$$M'=M'_{(1)}\h{$\otimes$}_{{\mathbf C}}M'_{(2)},\quad M''=M''_{(1)}\h{$\otimes$}_{{\mathbf C}}M'_{(2)},\quad N=N_{(1)}\h{$\otimes$}_{{\mathbf C}}M'_{(2)},$$
and, setting $S(\mu):=\par
um_k\,\mu_k\,t^k\in{\mathbf Z}[[t]]$, etc., we have the equalities
$$S(\mu')=S(\mu'_{(1)})\,S(\mu'_{(2)}),\quad S(\mu'')=S(\mu''_{(1)})\,S(\mu'_{(2)}),\quad S(\nu)=S(\nu_{(1)})\,S(\mu'_{(2)}),$$
where $M'_{(i)}$, $M''_{(i)}$, $N_{(i)}$, and $\mu'_{(i),k}$, $\mu''_{(i),k}$, $\nu_{(i),k}\,\,(k\in{\mathbf Z})$ are defined for $f_i\,\,\,(i=1,2)$.}
\par
\noindent
{\it Proof.} Using the $n$-ple complex structure of the Koszul complex as in Remark~(1.9)(v), we get the canonical isomorphism
$${}^s\!K^{\ssb}_f={}^s\!K^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_1}\otimes_{{\mathbf C}}{}^s\!K^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_2},$$
where ${}^s\!K^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_1}$ is defined by using the subring ${\mathbf C}[x_1,\dots,x_{n_1}]$, and similarly for ${}^s\!K^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_2}$. Since $f_2^{-1}(0)$ has an isolated singularity, $K^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_2}$ is naturally quasi-isomorphic to $M'_{(2)}$. We get hence
$$M=M_{(1)}\h{$\otimes$}_{{\mathbf C}}M'_{(2)},\quad N=N_{(1)}\h{$\otimes$}_{{\mathbf C}}M'_{(2)}.$$
Moreover, the freeness of $M''_{(1)}\h{$\otimes$}_{{\mathbf C}}M'_{(2)}$ over ${\mathbf C}[y]$ can be shown by using an appropriate filtration of $M'_{(2)}$, where $y$ is as in the introduction. These imply that the following two short exact sequences are identified with each other:
$$\aligned 0\to M'\to&M\to M''\to 0,\\ 0\to M'_{(1)}\h{$\otimes$}_{{\mathbf C}}M'_{(2)}\to M_{(1)}&\h{$\otimes$}_{{\mathbf C}}M'_{(2)}\to M''_{(1)}\h{$\otimes$}_{{\mathbf C}}M'_{(2)}\to 0.\endaligned$$
So the assertion follows.
\par
For the proof of Proposition~(2.6) below, we need the following lemma. Essentially this may be viewed as a special case of Prop.~13 in \cite{ChDi}, see Remark~(2.5) below. We note here a short proof of the lemma using Corollaries~1 and 2 and (0.3) for the convenience of the reader.
\par
\noindent
{\bf Lemma~2.3.} {\it Assume $n=2$. Let $r$ be the number of the irreducible components of $f^{-1}(0)\subset{\mathbf C}^2$. Then $\tau_Z=d-r$, and we have for $k\in{\mathbf Z}$
$$\aligned\mu'_k&=\max(r-1-|d-k|,0),\\ \mu''_k&=(k-1)_{[0,\tau_Z]},\\ \nu_k&=(k-d-r+1)_{[0,\tau_Z]},\endaligned
\leqno(2.3.1)$$
where $x_{[\alpha,\beta]}$ for $x,\alpha,\beta\in{\mathbf Z}$ with $\alpha<\beta$ is defined by}
$$x_{[\alpha,\beta]}=\begin{cases}\alpha&\hbox{if}\,\,\,x\leqslant\alpha,\\ x&\hbox{if}\,\,\,\alpha\leqslant x\leqslant\beta,\\ \beta&\hbox{if}\,\,\,\beta\leqslant x.\end{cases}$$
\par
\noindent
{\it Proof.} We have the decomposition
$$f=\h{$\prod$}_{i=1}^r\,g_i^{m_i},$$
with $\deg g_i=1$ and $m_i\geqslant 1$. For $z\in{\mathbf P}^1$ corresponding to $g_i^{-1}(0)\subset{\mathbf C}^2$, we have
$$\tau_z=m_i-1,\quad\hbox{and hence}\quad\tau_Z=d-r.$$
Setting
$$f':=\h{$\prod$}_{i=1}^r\,g_i^{m_i-1},$$
we get
$$M''=\Omega^2/f'\,\Omega^2.$$
Indeed, the right-hand side is a quotient graded $R$-module of $M$, and is a free graded ${\mathbf C}[y]$-module of rank $\tau_Z$. Since $\deg f'=\tau_Z$, this implies
$$\mu''_k=(k-1)_{[0,\tau_Z]}.$$
Using Corollary~2, we then get
$$\nu_k=d-r-(2d-k-1)_{[0,\tau_Z]}=(k-d-r+1)_{[0,\tau_Z]}.$$
Here note that
$$\nu_k=0\quad\hbox{if}\,\,\,k\leqslant d.$$
For $n=2$ and $k\leqslant d$, we have
$$\gamma_k=\max(k-1,0).$$
By (0.3) we then get for $k\leqslant d$
$$\mu'_k=\gamma_k-\mu''_k=\max(k-1-\tau_Z,0).$$
The formula for $k\geqslant d$ follows by using the symmetry in Corollary~1. This finishes the proof of Lemma~(2.3).
\par
By an easy calculation we see that Lemma~(2.3) is equivalent to the following.
\par
\noindent
{\bf Corollary~2.4.} {\it With the notation and the assumption of Lemma~$(2.3)$, we have
$$\aligned S(\mu')&=S(1,r-1)\,S(d-r+1,d-1),\\ S(\mu'')&=S(1,\infty)\,S(1,d-r),\\ S(\nu)&=S(d+r-2,\infty)\,S(1,d-r),\endaligned
\leqno(2.4.1)$$
where $S(\mu')$ is as in Proposition~$(2.2)$, and $S(a,b)$ for $a\in{\mathbf N}$, $b\in{\mathbf N}\cup\{\infty\}$ is defined by}
$$S(a,b):=\par
um_{k=a}^b\,t^k\in{\mathbf Z}[[t]]\,\,\,\hbox{if}\,\,\,a\leqslant b,
\,\,\,\hbox{and}\,\,\,0\,\,\,\hbox{otherwise}.
\leqno(2.4.2)$$
\par
\noindent
{\bf Remark~2.5.} With the notation and the assumption of Corollary~(2.4), the following is shown in \cite[Example~14~(i)]{Di2} as a corollary of Prop.~13 (loc.~cit.)
$$S(\mu)=t^2(1-2t^{d-1}+t^{d+r-2})/(1-t)^2.
\leqno(2.5.1)$$
By Corollaries~2 and 3 together with (0.3), this is essentially equivalent to the equalities in (2.4.1). In fact, it seems rather easy to deduce (2.5.1) from (2.4.1). For the converse some calculation seems to be needed. (The details are left to the reader.)
\par
In case $n_1=2$, we can calculate $\mu'_{(1),k}$, $\mu''_{(1),k}$, $\nu_{(1),k}$ for $f_1$ by Lemma~(2.3), and get the following.
\par
\noindent
{\bf Proposition~2.6.} {\it Assume $f=f_1+f_2$ as in Proposition~$(2.2)$ with $n_1=2$. Let $r$ be the number of the irreducible components of $f_1^{-1}(0)\subset{\mathbf C}^2$. Then, under the assumption of Proposition~$(2.2)$, we have
$$\aligned S(\mu')&=S(1,r-1)\,S(d-r+1,d-1)\,S(1,d-1)^{n-2},\\ S(\mu'')&=S(1,\infty)\,S(1,d-r)\,S(1,d-1)^{n-2},\\ S(\nu)&=S(d+r-2,\infty)\,S(1,d-r)\,S(1,d-1)^{n-2},\endaligned$$
where $S(a,b)$ is as in $(2.4.2)$.}
\par
\noindent
{\it Proof.} The assertion follows from Corollary~(2.4) and Proposition~(2.2), since $S(\mu')$ in the isolated singularity case is invariant by $\mu$-constant deformation, and is given by (0.2).
\par
\noindent
{\bf Example~2.7.} Let $f$ be as in Theorem~1, and assume further
$$f\in{\mathbf C}[x_1,\dots,x_{n-1}]\subset{\mathbf C}[x_1,\dots,x_n].$$
Then $f$ has an isolated singularity at the origin of ${\mathbf C}^{n-1}$. Set
$$\gamma'_j:=\dim_{{\mathbf C}}\bigl(\Omega^{\prime\,n-1}/{\rm d}f\wedge\Omega^{\prime\,n-2}
\bigr){}_j\quad\hbox{with}\quad\Omega^{\prime\,k}:={\mathcal G}amma({\mathbf C}^{n-1},\Omega_{{\mathbf C}^{n-1}}^k).$$
We have the symmetry
$$\gamma'_j=\gamma'_{(n-1)d-j}.
\leqno(2.7.1)$$
In this case, we have $M'=0$, and
$$\mu_k=\mu''_k=\par
um_{j\leqslant k-1}\,\gamma'_j,\quad
\nu_k=\par
um_{j\leqslant k-d}\,\gamma'_j=\par
um_{j\geqslant nd-k}\,\gamma'_j,
\leqno(2.7.2)$$
where the last equality follows from the symmetry (2.7.1), and Corollary~2 is verified directly in this case.
\par
Equivalently, $\mu''_k=\mu_k$ and $\nu_k$ are given as follows:
$$\aligned S(\mu)&=S(1,\infty)\,S(1,d-1)^{n-1},\\ S(\nu)&=S(d,\infty)\,S(1,d-1)^{n-1},\endaligned
\leqno(2.7.3)$$
where $S(\mu)$, etc.\ are as in Proposition~(2.2), and the order of the coordinates are changed.
\par
\noindent
{\bf Example~2.8.} Assume $n,d\geqslant 3$. Let
$$f=x_1^ax_2^{d-a}+\par
um_{i=3}^n\,x_i^d\quad\hbox{with}\,\,\,0<a<d.
\leqno(2.8.1)$$
We can apply Proposition~(2.6) to this example. More precisely, the calculation of $\mu'_k$, $\mu''_k$ and $\nu_k$ are reduced to the case $n=2$ by Proposition~(2.2), where $n_1=2$ and
$$f_1=x_1^ax_2^{d-a},\quad f_2=\par
um_{i=3}^n\,x_i^d.$$
The calculation for $f_1$ follows from Lemma~(2.3) or Corollary~(2.4) where $r=2$. For instance, we get in the notation of Proposition~(2.2)
$$\mu'_{(1),k}=\begin{cases}1&\hbox{if}\,\,\,k=d,\\ 0&\hbox{if}\,\,\,k\ne d,\end{cases}$$
and hence
$$\mu'_k=\mu'_{(2),k+d}=\gamma''_{k+d}\quad(k\in{\mathbf Z}),$$
where $\gamma''_k$ is as in (0.2) with $n$ replaced by $n-2$. By Proposition~(2.6), we have
$$\aligned S(\mu')&=t^d\,S(1,d-1)^{n-2},\\ S(\mu'')&=S(1,\infty)\,S(1,d-2)\,S(1,d-1)^{n-2},\\ S(\nu)&=S(d,\infty)\,S(1,d-2)\,S(1,d-1)^{n-2},\endaligned
\leqno(2.8.2)$$
where $S(\mu')$, etc.\ are as in Proposition~(2.2).
\par
\noindent
{\bf Remark~2.9.} If there is a nontrivial relation of degree $k\leqslant d-2$ among the partial derivatives $f_i:={\partial} f/{\partial} x_i$, i.e. if there are homogeneous polynomials $g_i$ of degree $k\leqslant d-2$ with $\par
um_i\,g_if_i=0$ and $g_i\ne0$ for some $i$, then we have
$$\nu_{d+n+k-1}\ne 0,
\leqno(2.9.1)$$
and hence
$$\hbox{Condition~(0.5) does not hold if $(n-2)(d-2)\geqslant 2(k+1)$.}
\leqno(2.9.2)$$
Indeed, (2.9.1) follows from the definition $N:=H^{-1}({}^s\!K^{\ssb}_f)$ since $\deg f_i=d-1$.
\par
This applies to $f$ in Example~(2.7) with $k=0$ since $f_n=0$, and to $f$ in Example~(2.8) with $k=1$ since
$$(d-a)x_1\,f_1=ax_2\,f_2.$$
\par
\noindent
{\bf Remark~2.10.} Conditions~(0.5) hold in the nodal hypersurface case by \cite[Thm.~2.1]{DiSt2}. Indeed, it is shown there that
$$\hbox{$\nu_k=0\,$ if $\,k\leqslant(n_1+1)d\,$ with $\,n\,$ even or $\,k\leqslant(n_1+1)d-1\,$ with $\,n\,$ odd,}
\leqno(2.10.1)$$
where $n_1:=[(n-1)/2]$. (There is a difference in the grading on $N$ by $d$ between this paper and loc.~cit., and $n$ in this paper is $n+1$ in loc.~cit.)
\par
\par
\vbox{\centerline{\bf 3. Spectrum}
\par
\noindent
In this section we recall some basics from the theory of spectra, and prove Proposition~(3.3), (3.4), and (3.5).}
\par
\noindent
{\bf 3.1.~Hodge and pole order filtrations.} Let $f$ be a homogeneous polynomial of $n$ variables with degree $d$. It is well-known that there is a ${\mathbf C}$-local system $L_k$ ($k\in[1,d]$) of rank 1 on $U:=Y\setminus Z$ such that
$$H^j(U,L_k)=H^j(f^{-1}(1),{\mathbf C})_{\lambda}\quad\bigl(\lambda=\exp(-2\pi ik/d),
\,k\in[1,d]\bigr),
\leqno(3.1.1)$$
where $H^j(f^{-1}(1),{\mathbf C})_{\lambda}$ is the $\lambda$-eigenspace of the cohomology for the semisimple part of the monodromy, see e.g.\ \cite{Di1}, etc. (Note that monodromy in our paper means the one as a local system, see also \cite[Section 1.3]{BuSa}, etc.) Let ${\mathcal L}_k$ be the meromorphic extension of $L_k\h{$\otimes$}_{{\mathbf C}}{\mathcal O}_U$. This is a regular holonomic ${\mathbf D}c_Y$-module, and
$$H^j\bigl(Y,\Omega_Y^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}({\mathcal L}_k)\bigr)=H^j(f^{-1}(1),{\mathbf C})_{\lambda}\quad\bigl(\lambda=\exp(-2\pi ik/d),
\,k\in[1,d]\bigr),
\leqno(3.1.2)$$
where $\Omega_Y^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}({\mathcal L}_k)$ denotes the de Rham complex of ${\mathcal L}_k$. We have the Hodge and pole order filtrations $F_{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ and $P_{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ on ${\mathcal L}_k$ such that
$$F_i\subset P_i,
\leqno(3.1.3)$$
where the equality holds outside the singular points of $Z$, and
$$P_i{\mathcal L}_k=\begin{cases}{\mathcal O}_Y(id+k)&\hbox{if}\,\,\,i\geqslant 0,\\
\,0&\hbox{if}\,\,\,i<0,\end{cases}$$
see e.g.\ \cite[Section 4.8]{Sa4}. (Note that $F$ comes from the Hodge filtration of a mixed Hodge module.) Set $F^i=F_{-i}$, $P^i=P_{-i}$. They induces the Hodge and pole order filtrations on $\Omega_Y^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}({\mathcal L}_k)$ such that the $j$-th components of $F^i\,\Omega_Y^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}({\mathcal L}_k)$, $P^i\,\Omega_Y^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}({\mathcal L}_k)$ are respectively given by
$$\Omega_Y^j\h{$\otimes$}_{{\mathcal O}_Y}F^{i-j}{\mathcal L}_k,\quad
\Omega_Y^j\h{$\otimes$}_{{\mathcal O}_Y}P^{i-j}{\mathcal L}_k.$$
By the isomorphism (3.1.2) they further induce the Hodge and pole order filtrations on the Milnor cohomology $H^j(f^{-1}(1),{\mathbf C})$. Here $F$ coincides with the Hodge filtration of the canonical mixed Hodge structure. By using the Bott vanishing theorem, $H^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\bigl(Y,P^i\,\Omega_Y^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}({\mathcal L}_k)\bigr)$ can be calculated by the complex whose $j$-th component is
$${\mathcal G}amma(Y,\Omega_Y^j\h{$\otimes$}_{{\mathcal O}_Y}P^{i-j}{\mathcal L}_k)=
\begin{cases}{\mathcal G}amma\bigl(Y,\Omega_Y^j((j-i)d+k)\bigr)&\hbox{if}\,\,\,j\geqslant i,\\ 0&\hbox{if}\,\,\,j<i.\end{cases}$$
But it does not give a strict filtration, and it is not necessarily easy to calculate it.
\par
Note that the pole order filtration coincides with the one defined by using the Gauss-Manin system, see (4.4.7) and (4.5.7) below.
\par
\noindent
{\bf 3.2.~Spectrum.} For $f$ as in (3.1), the spectrum ${\rm Sp}(f)=\par
um_{\alpha\in{\mathbf Q}}\,n_{f,\alpha}\,t^{\alpha}$ is defined by
$$\aligned n_{f,\alpha}:=\par
um_j\,(-1)^{j-n+1}\dim{\mathcal G}r^p_F
\widetilde{H}^j(f^{-1}(1),{\mathbf C})_{\lambda}\\
\hbox{with}\quad p=\lfloor n-\alpha\rfloor,\,\,\lambda=\exp(-2\pi i\alpha),\endaligned
\leqno(3.2.1)$$
(see \cite{St2}, \cite{St3}). Here $\widetilde{H}^j(f^{-1}(1),{\mathbf C})$ is the reduced cohomology, and we have by definition
$$\lfloor\alpha\rfloor:=\max\{\,i\in{\mathbf Z}\mid i\leqslant\alpha\,\},\quad
\lceil\alpha\rceil:=\min\{\,i\in{\mathbf Z}\mid i\geqslant\alpha\,\}\quad(\alpha\in{\mathbf R}).
\leqno(3.2.2)$$
The {\it pole order spectrum} ${\rm Sp}_P(f)$ is defined by replacing $F$ with $P$.
\par
For $j\in{\mathbf N}$, we define ${\rm Sp}^j(f)=\par
um_{\alpha\in{\mathbf Q}}\,n^j_{f,\alpha}\,t^{\alpha}$ by
$$\aligned n^j_{f,\alpha}:=\dim{\mathcal G}r^p_F\widetilde{H}^{n-1-j}(f^{-1}(1),{\mathbf C})_{\lambda}\\
\hbox{with}\quad p=\lfloor n-\alpha\rfloor,\,\,\lambda=\exp(-2\pi i\alpha),\endaligned
\leqno(3.2.3)$$
so that
$${\rm Sp}(f)=\par
um_j\,(-1)^j\,{\rm Sp}^j(f).$$
Similarly ${\rm Sp}^j_P(f)=\par
um_{\alpha\in{\mathbf Q}}\,n^{P,j}_{f,\alpha}\,t^{\alpha}$ is defined by replacing $F$ with $P$.
\par
Set $Z:=\{f=0\}\subset Y:={\mathbf P}^{n-1}$. Let $\pi:(\widetilde{Y},{\mathbf Z}t)\to(Y,Z)$ be an embedding resolution, and $E_i$ be the irreducible components of ${\mathbf Z}t$ with $m_i$ the multiplicity of ${\mathbf Z}t$ at the generic point of $E_i$. Let $\alpha=k/d+q\in(0,n)$ with $k\in[1,d]$, $q\in[0,n-1]$. We have by \cite[1.4.3]{BuSa}
$$n^j_{f,\alpha}=\dim H^{q-j}\bigl(\widetilde{Y},\Omega_{\widetilde{Y}}^{n-1-q}(\log{\mathbf Z}t)\h{$\otimes$} _{{\mathcal O}}\,{\mathcal O}_{\widetilde{Y}}(-\ell\,\widetilde{H}+\par
um_i\,\lfloor\ell\,m_i/d\rfloor)E_i\bigr),
\leqno(3.2.4)$$
where $\ell:=d-k$, and $\widetilde{H}$ is the pull-back of a sufficiently general hyperplane $H$ of $Y$.
\par
In a special case we get the following.
\par
\noindent
{\bf Proposition~3.3.} {\it Assume $n=2$. Set $e:={\rm GCD}(m_i)$ with $m_i$ the multiplicities of the irreducible factors of $f$. Then, for $\alpha=k/d+q\in(0,2)$ with $k\in[1,d]$, $q=0,1$, we have
$$n^j_{f,\alpha}=\begin{cases}r-1+k-\par
um_i\,\lceil km_i/d\rceil&\hbox{if}\,\,\,j=0,\,q=0,\\
\max\bigl(-k-1+\par
um_i\,\lceil km_i/d\rceil,\,0\,\bigr)&\hbox{if}\,\,\,j=0,\,q=1,\\
1&\hbox{if}\,\,\,j=1,\,q=1,\,ke/d\in{\mathbf Z},\\
0&\hbox{otherwise},\end{cases}
\leqno(3.3.1)$$
where $\lceil\alpha\rceil$ is as in $(3.2.2)$.}
\par
\noindent
{\it Proof.} We have $\Omega_{{\mathbf P}^1}^1(\log Z)={\mathcal O}_{{\mathbf P}^1}(r-2)$ with $\widetilde{Y}=Y={\mathbf P}^1$, ${\mathbf Z}t=Z$, $\widetilde{H}=H$, Hence (3.2.4) in this case becomes
$$n^j_{f,\alpha}=\begin{cases}\dim H^0\bigl({\mathbf P}^1,\Omega_{{\mathbf P}^1}^1(\log Z) (-\ell+\par
um_i\,\lfloor\ell\,m_i/d\rfloor)\bigr)&\hbox{if}\,\,\,j=0,\,q=0,\\
\dim H^1\bigl({\mathbf P}^1,{\mathcal O}_{{\mathbf P}^1}(-\ell+\par
um_i\,\lfloor\ell\,m_i/d\rfloor)\bigr)&\hbox{if}\,\,\,j=0,\,q=1,\\
\dim H^0\bigl({\mathbf P}^1,{\mathcal O}_{{\mathbf P}^1}(-\ell+\par
um_i\,\lfloor\ell\,m_i/d\rfloor)\bigr)&\hbox{if}\,\,\,j=1,\,q=1,\\
0&\hbox{otherwise},\end{cases}$$
and then
$$n^j_{f,\alpha}=\begin{cases}r-1-\ell+\par
um_i\,\lfloor\ell\,m_i/d\rfloor&\hbox{if}\,\,\,j=0,\,q=0,\\
\max\bigl(\ell-1-\par
um_i\,\lfloor\ell\,m_i/d\rfloor,\,0\,\bigr)&\hbox{if}\,\,\,j=0,\,q=1,\\
\max\bigl(-\ell+1+\par
um_i\,\lfloor\ell\,m_i/d\rfloor,\,0\,\bigr)&\hbox{if}\,\,\,j=1,\,q=1,\\
0&\hbox{otherwise}.\end{cases}
\leqno(3.3.2)$$
Since $\par
um_i\,m_i=d$ and $e={\rm GCD}(m_i)$, we have
$$\ell>\par
um_i\,\lfloor\ell m_i/d\rfloor\iff\ell m_i/d\notin{\mathbf Z}\,\,(\exists\,i)\iff\ell e/d\notin{\mathbf Z}.$$
So (3.3.1) follows (since $\ell=d-k$). This finishes the proof of Proposition~(3.3).
\par
We note here an application of Theorem~2, Theorem~(5.3) and Corollary~(5.4) below.
(This will not be used in their proofs.)
\par
\noindent
{\bf Proposition~3.4.} {\it Assume $n=2$. Then ${\rm Sp}_P(f)={\rm Sp}_P^0(f)-{\rm Sp}_P^1(f)$ is given by
$${\rm Sp}^j_P(f)=\begin{cases}\par
um_k\,(\mu_k-\nu_{k+d})\,t^{k/d}+\bigl(\,t^{1/e}+\cdots+t^{(e-1)/e}\,\bigr)&\hbox{if}\,\,\,j=0,\\ t\,(\,t^{1/e}+\cdots+t^{(e-1)/e}\,)&\hbox{if}\,\,\,j=1,\end{cases}
\leqno(3.4.1)$$
with $\mu_k$, $\nu_k$ explicitly expressed in Lemma~$(2.3)$, and $e={\rm GCD}(m_i)$ as in Proposition~$(3.3)$.}
\par
\noindent
{\it Proof.} The pole order spectral sequence degenerates at $E_2$ by Corollary~(5.4) below. So the assertion is shown in the case $e=1$, since the last condition implies that $\nu_k^{(2)}=0$. In the general case it is well-known that
$$\widetilde{H}^0(f^{-1}(1),{\mathbf C})_{\lambdambda}=\begin{cases}{\mathbf C}&\hbox{if}\,\,\,\lambdambda^e=1\,\,\,\hbox{with}\,\,\,\lambdambda\ne 1,\\ \,0&\hbox{otherwise}\end{cases}.
\leqno(3.4.2)$$
By using Theorem~(5.3) and Lemma (2.3), this implies
$$N_{k+d}^{(2)}=\begin{cases}{\mathbf C}&\hbox{if}\,\,\,k=i\,(d/e)\,\,\hbox{with}\,\,\,i\in\{1,\dots,e-1\},\\ \,0&\hbox{otherwise},\end{cases}\leqno(3.4.2)$$
where $N^{(2)}\subset N$ is the kernel of ${\partial}d^{(1)}$. This gives also the information of the coimage of ${\partial}d^{(1)}$ which is a morphism of degree $-d$. So the correction terms for ${\rm Sp}_P^0(f)$ and ${\rm Sp}_P^1(f)$ coming form the non-vanishing of ${\partial}d^{(1)}$ are given respectively by
$$t^{1/e}+\cdots+t^{(e-1)/e}\quad\hbox{and}\quad t\,(\,t^{1/e}+\cdots+t^{(e-1)/e}\,).$$
So (3.4.1) follows. This finishes the proof of Proposition~(3.4).
\par
\noindent
{\bf Proposition~3.5.} {\it Assume $f=f_1+f_2$ as in Proposition~$(2.2)$ with $n_1=2$. Then, under the assumption of Proposition~$(2.2)$, we have
$$\mu'_k\leqslant n^0_{f,k/d},\quad\mu'_k\leqslant n^{P,0}_{f,k/d}\quad(k\in{\mathbf Z}),
\leqno(3.5.1)$$
where $n^0_{f,k/d}$, $n^{P,0}_{f,k/d}$ are as in $(3.2)$.}
\par
\noindent
{\it Proof.} The Thom-Sebastiani type theorem holds for ${\rm Sp}^0(f)$, ${\rm Sp}^0_P(f)$ under the assumption of Proposition~(2.2), see (4.9) below. So the assertion is reduced to the case $f=f_1$ with $n=2$.
The assertion for ${\rm Sp}^0_P(f)$ then follows from Proposition~(3.4) and Lemma~(2.3), where we may assume $r\geqslant 2$ since $\mu'_k=0$ otherwise.
By using Lemma~(2.3) and Proposition~(3.3) (more precisely, (3.3.2) for $q=0$ and (3.3.1) for $q=1$), the assertion for ${\rm Sp}^0(f)$ is reduced to the following trivial inequalities
$$\aligned r-1-(d-k)\leqslant r-1-\ell+\par
um_{i=1}^r\,\lfloor\ell\,m_i/d\rfloor&\quad(\ell\in[0,d-1],\,q=0),\\
r-1+d-(k+d)\leqslant -k-1+\par
um_{i=1}^r\,\lceil km_i/d\rceil&\quad(k\in[1,d-1],\,q=1),\endaligned$$
where $\ell=d-k$. (Note that $k$ in Lemma~(2.3) is $k+d$ in the case $q=1$.) This finishes the proof of Proposition~(3.5).
\par
\noindent
{\bf Remarks~3.6.} (i) If $f$ has an isolated singularity, the equality holds in (3.5.1), and $S(\mu')$ (with $t$ replaced by $t^{1/d}$) coincides with the spectrum ${\rm Sp}(f)$, see \cite{St1} and also \cite{Gri}, \cite{SkSt}, \cite{Va}, etc.
It would be interesting if (3.5.1) holds in a more general case.
\par
(ii) Let $f$ be as in (3.1). Assume $Z\subset{\mathbf P}^{n-1}$ has only isolated singularities. Let $\alpha'_f$ be the minimal of the exponents of the spectrum for all the singularities of $Z$ (see also Corollary~(5.5) below). Then the multiplicity $n_{f,\alpha}$ of the spectrum ${\rm Sp}(f)$ for $\alpha=p/d<\min(\alpha'_f,1)$ can be given by
$$n_{f,p/d}=\binom{p-1}{n-1}\quad(p/d<\min(\alpha'_f,1)).$$
This follows from a formula for multiplier ideals \cite[Prop.~1]{Sa4} together with \cite{Bu} (see also a remark before \cite[Cor.~1]{Sa4}). This equality holds also for the pole order spectrum since $\mu_p$ is at most the right-hand side of the equality and $F\subset P$ (and $\nu_p=0$ for $p<d$).
\par
\noindent
{\bf Example~3.7.} Let $f=(x^m+y^m)\,x^my^m$ ($m\geqslant 2$), where $d=3m$, $r=m+2$, $\tau_Z=2m-2$.
For $\alpha=k/3m+q$ with $k\in[1,3m]$, $q=0,1$, we have by Proposition~(3.3)
$$n_{f,\alpha}=\begin{cases}k+1-2\lceil k/3\rceil&\hbox{if}\,\,\,\alpha\in(0,1],\,\,\,q=0,\\
m-k-1+2\lceil k/3\rceil&\hbox{if}\,\,\,\alpha\in(1,2),\,\,\,q=1.\end{cases}
\leqno(3.7.1)$$
In fact, $m_i=1$ for $i\in[1,m]$, and $m_i=m$ for $i=m+1,m+2$. Here $e=1$ in the notation of Proposition~(3.3), and hence ${\rm Sp}^1(f)=0$, ${\rm Sp}(f)={\rm Sp}^0(f)$ (similarly for ${\rm Sp}_P(f)$).
\par
On the other hand, Lemma~(2.3) and Proposition~(3.4) imply that
$$n^P_{f,k/3m}=\mu^{(2)}_k=\mu'_k+\mu''_k-\nu_{k+3m}=\begin{cases}0&(\,k\leqslant 1\,),\\
k-1&(\,1\leqslant k\leqslant m+1\,),\\
m&(\,m+1\leqslant k\leqslant 3m-1\,),\\
4m+1-k\hbox{ }&(\,3m\leqslant k\leqslant 4m+1\,),\\
0&(\,4m+1\leqslant k\,).\end{cases}
\leqno(3.7.2)$$
(Note that $\mu^{(2)}_k+\mu^{(2)}_{k+3m}=m$ ($k\in[1,3m-1])$ and $\mu^{(2)}_{3m}=m+1$.)
In fact, we have by Lemma~(2.3)
$$\aligned\mu'_k&=\begin{cases}0&(\,k\leqslant 2m-1\,),\\
k-2m+1\hbox{ }&(\,2m-1\leqslant k\leqslant 3m\,),\\
4m+1-k&(\,3m\leqslant k\leqslant 4m+1\,),\\
0&(\,4m+1\leqslant k\,),\end{cases}\\
\mu''_k&=\begin{cases}0&(\,k\leqslant 1\,),\\
k-1&(\,1\leqslant k\leqslant 2m-1\,),\\
2m-2\quad\quad\!\hbox{ }&(\,2m-1\leqslant k\,),\end{cases}\\
\nu_{k+3m}&=\begin{cases}0&(\,k\leqslant m+1\,),\\
k-m-1\,\,\,\hbox{ }&(\,m+1\leqslant k\leqslant 3m-1\,),\\
2m-2&(\,3m-1\leqslant k\,).\end{cases}\endaligned$$
These formulas show that the relation between the Steenbrink spectrum ${\rm Sp}(f)$ and the pole order spectrum ${\rm Sp}_P(f)$ is rather complicated even for $n=2$ in general.
\par
\par
\vbox{\centerline{\bf 4. Gauss-Manin systems and Brieskorn modules}
\par
\noindent
In this section we prove Theorem~2 after recalling some facts from Gauss-Manin systems and Brieskorn modules.}
\par
\noindent
{\bf 4.1.~Graded Gauss-Manin complexes.} Let $f$ be a homogeneous polynomial in $R$ with degree $d$. In the notation of (1.1), the graded Gauss-Manin complex $C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ associated with $f$ is defined by
$$C_f^j:=\Omega^j[{\partial}_t]\quad(j\in{\mathbf Z}),$$
where ${\partial}_t$ has degree $-d$. This means that
$$\Omega^j\,{\partial}_t^p=\Omega^j(pd),$$
where $(pd)$ denotes the shift of the grading as in the introduction. Its differential ${\partial}d$ is defined by
$${\partial}d(\omega\,{\partial}_t^p)=({\partial}d\omega)\,{\partial}_t^p-({\rm d}f\wedge\omega)\,
{\partial}_t^{p+1}\quad\hbox{for}\,\,\,\omega\in\Omega^k.
\leqno(4.1.1)$$
where ${\partial}d\omega$ denotes the differential of the de Rham complex. It has a structure of a complex of ${\mathbf C}[t]\lambdangle{\partial}_t\rangle$-modules defined by
$$t(\omega\,{\partial}_t^p)=(f\omega)\,{\partial}_t^p-p\,\omega\,{\partial}_t^{p-1},\quad{\partial}_t (\omega\,{\partial}_t^p)=\omega\,{\partial}_t^{p+1}\quad\hbox{for}\,\,\,\omega\in\Omega^j.
\leqno(4.1.2)$$
The Gauss-Manin systems are defined by the cohomology groups $H^jC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\,\,(j\in{\mathbf Z})$. These are regular holonomic graded ${\mathbf C}[t]\lambdangle{\partial}_t\rangle$-modules. By the same argument as in \cite{BaSa}, we have
$$\hbox{The action of ${\partial}_t$ on $H^jC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is bijective for $j\ne 1$.}
\leqno(4.1.3)$$
\par
\noindent
{\bf 4.2.~Brieskorn modules.} Let $(A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d)$ be a graded subcomplex of the de Rham complex $(\Omega^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d)$ defined by
$$A_f^j:={\rm Ker}({\rm d}f\wedge:\Omega^j\to\Omega^{j+1}(d)).$$
The Brieskorn modules are graded ${\mathbf C}[t]\lambdangle{\partial}_tt\rangle$-modules defined by its cohomology groups
$$H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\quad(j\in{\mathbf Z}).$$
The actions of $t$, $\dd_t^{-1}$, ${\partial}_tt$ are respectively defined by the multiplication by $f$,
$$\aligned\dd_t^{-1}[\omega]=[{\rm d}f\wedge\widetilde{x}i]\quad\hbox{with}\quad{\partial}d\widetilde{x}i=\omega,\\
{\partial}_tt\,[\omega]=[{\partial}d\eta]\quad\hbox{with}\quad{\rm d}f\wedge\eta=f\omega,\endaligned$$
where $[\omega]$ denotes the cohomology class, see \cite{Bri}, \cite{BaSa}, etc. (In case $j=1$, we have to choose a good $\widetilde{x}i$ for the action of $\dd_t^{-1}$, see \cite{BaSa}.) Moreover, we have
$${\partial}_tt\,[\omega]=(k/d)[\omega]\quad\hbox{for}\quad[\omega]\in(H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k,
\leqno(4.2.1)$$
where $(H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k$ denotes the degree $k$ part. (This follows from the definition by using the contraction with the Euler vector field $\widetilde{x}i:=\par
um_i\,x_i\,{\partial}/{\partial} x_i$.) This implies
$$t\,[\omega]=(k/d)\,\dd_t^{-1}[\omega]\quad\hbox{for}\quad[\omega]\in(H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k.
\leqno(4.2.2)$$
Since $(H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k=0$ for $k\leqslant 0$, this implies that ${\rm Coker}\,t$ in Theorem~2 can be replaced with ${\rm Coker}\,\dd_t^{-1}$.
\par
There is a natural inclusion
$$A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\hookrightarrow C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}.$$
This is compatible with the actions of $t$, $\dd_t^{-1}$, ${\partial}_tt$ on the cohomology by definition. So (4.2.1) holds also for $\omega\in(H^jC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_j$, since the image of $H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ generates $H^jC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ over ${\mathbf C}[{\partial}_t]$. The last assertion is well-known in the analytic case, see e.g.\ \cite{BaSa}, and is reduced to this case by using the scalar extensions
$$R\hookrightarrow{\mathbf C}\{x_1,\dots,x_n\},\quad
{\mathbf C}[t]\hookrightarrow{\mathbf C}\{t\}.$$
\par
For $j\in{\mathbf Z}$, we then get
$$H^{j+1}(C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k=\begin{cases}H^j(f^{-1}(1),{\mathbf C})_{\lambda}&
\hbox{if}\,\,\,k/d\notin{\mathbf Z}_{\leqslant 0},\\
\widetilde{H}^j(f^{-1}(1),{\mathbf C})_{\lambda}&
\hbox{if}\,\,\,k/d\notin{\mathbf Z}_{\geqslant 1},\end{cases}
\leqno(4.2.3)$$
in the notation of (3.2), where $\lambda=\exp(-2\pi ik/d)$, see also \cite{Di1}.
\par
We have moreover
$${\rm Ker}(H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to H^jC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})=(H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor},
\leqno(4.2.4)$$
where the last term denotes the $t$-torsion subspace of $H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$, which coincides with the $\dd_t^{-1}$-torsion, and is annihilated by ${\partial}_t^{-p}$ for $p\gg 0$, see \cite{BaSa}.
\par
\noindent
{\bf 4.3.~Relation with the Koszul cohomologies.} Set
$$A_f^{\prime\,j}:={\rm d}f\wedge\Omega^{j-1}\buildrel\iota\over\hookrightarrow A_f^j\quad(j\in{\mathbf Z}).
\leqno(4.3.1)$$
Using the short exact sequence of complexes
$$0\to(A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d)\to(\Omega^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d)\to (A_f^{\prime\,\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d)[1]\to 0,$$
we get isomorphisms
$${\partial}:H^jA_f^{\prime\,\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\buildrel\sim\over\longrightarrow H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\quad(j\ne 1),
\leqno(4.3.2)$$
together with a short exact sequence
$$0\to{\mathbf C}\to H^1A_f^{\prime\,\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to H^1A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to 0.$$
By (4.3.1) and (4.3.2), we get an action of $\dd_t^{-1}$ on $H^jA_f^{\prime\,\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$, $H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ defined respectively by
$$\dd_t^{-1}:={\partial}^{-1}\,\raise.15ex\h{${\scriptstyle\circ}$}\, H^j\iota,\quad\dd_t^{-1}:=H^j\iota\,\raise.15ex\h{${\scriptstyle\circ}$}\,{\partial}^{-1}
\quad(j\ne 1).$$
\par
Let
$$(\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d):=(A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}/A_f^{\prime\,\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d).
\leqno(4.3.3)$$
The relation with the shifted Koszul complex $({}^s\!K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\rm d}f\wedge)$ in the introduction is given by
$$\K_f^{j+n}=H^j({}^s\!K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})(-jd)\quad(j\in[-n,0]).$$
By the short exact sequence of complexes
$$0\to(A_f^{\prime\,\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d)\buildrel\iota\over\to(A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d)\to(\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},{\partial}d)\to 0,$$
we get a long exact sequence
$$\to H^{j-1}\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to H^jA_f^{\prime\,\raise.15ex\h{${\scriptscriptstyle\bullet}$}}
\buildrel{\iota_j}\over\to H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to H^j\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to,
\leqno(4.3.4)$$
where the middle morphism $\iota_j:=H^j\iota$ can be identified by (4.3.2) with
$$\dd_t^{-1}:H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\,\,\,\hbox{if}\,\,\,j>1.$$
In particular we get for $j=n$
$$H^n\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}={\rm Coker}(\dd_t^{-1}:H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}).
\leqno(4.3.5)$$
\par
By the above argument, the $\dd_t^{-1}$-torsion of $H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ contributes to $H^{j-1}\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$, and we get in particular
$$\hbox{$H^cA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is torsion-free if $c$ is the codimension of ${\rm Sing}\,f^{-1}(0)\subset{\mathbf C}^n$.}
\leqno(4.3.6)$$
Note that $c=n-1$ under the assumption of the introduction. By Theorems~(5.2) and (5.3) below, the $\dd_t^{-1}$-torsion of $H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is finite dimensional if and only if all the singularities of $Z$ are weighted homogeneous.
\par
\noindent
{\bf 4.4.~Filtrations $P'$ and $G$.} There are two filtrations $P'$, $G$ on $C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ defined by
$$\aligned P'_p\,C_f^k&:=\h{$\bigoplus$}_{i\leqslant k+p}\,\Omega^k\,{\partial}_t^i,\\ G_p\,C_f^k&:=\bigl(\h{$\bigoplus$}_{i<p}\,\Omega^k\,{\partial}_t^i\bigr)\oplus A_f^p\,{\partial}_t^p.
\endaligned
\leqno(4.4.1)$$
These are exhaustive increasing filtrations. Set $P^{\prime\,p}=P'_{-p}$, $G^p=G_{-p}$. By definition, we have
$${\mathcal G}r^p_{P'}C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}=\sigma_{\geqslant p}\bigl(K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}((n-p)d)\bigr),
\leqno(4.4.2)$$
see \cite{De} for the truncation $\sigma_{\geqslant p}$. Let ${\rm Dec}\,P'$ be as in loc.~cit. Then we have
$$G={\rm Dec}\,P'.
\leqno(4.4.3)$$
Since the differential of $C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ respect the grading, we have the pole order spectral sequence in the category of graded ${\mathbf C}$-vector spaces
$$_{P'}E_1^{p,j-p}=H^j{\mathcal G}r^p_{P'}C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\Longrightarrow H^jC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},
\leqno(4.4.4)$$
with
$$_{P'}E_1^{p,j-p}=\begin{cases}\,0&\hbox{if}\,\,\,j<p,\\ A_f^p&\hbox{if}\,\,\,j=p,\\ H^jK_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}((n-p)d)&\hbox{if}\,\,\,j>p,\end{cases}
\leqno(4.4.5)$$
$$_{P'}E_2^{p,j-p}=\begin{cases}\,0&\hbox{if}\,\,\,j<p,\\ H^pA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}&\hbox{if}\,\,\,j=p,\\ H^j\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}((j-p)d)&\hbox{if}\,\,\,j>p,\end{cases}
\leqno(4.4.6)$$
where $\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is as in (4.3.3).
\par
Note that the degeneration at $E_2$ of the pole order spectral sequence is equivalent to the strictness of ${\rm Dec}\,P'$ by \cite{De}, and the latter condition is equivalent to the torsion-freeness of the $H^jA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ by using (4.2.4) and (4.4.3). The obtained equivalence seems to be known to the specialists (see e.g.\ \cite{vSt}), and the above argument may simplify some argument in loc.~cit.
\par
By the isomorphism (4.2.3) for $k\in[1,d]$, the filtration $P'$ on the left-hand side of (4.2.3) induces a filtration $P'$ on the right-hand side. This corresponds to the filtration $P$ by the isomorphism (3.1.2) up to the shift of the filtration by 1, and we get the isomorphisms
$$P^{\prime\,p+1}H^{j+1}(C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k\cong P^pH^j(f^{-1}(1),{\mathbf C})_{\lambda}\quad\bigl(\lambda=\exp(-2\pi ik/d),\,k\in[1,d]\bigr),
\leqno(4.4.7)$$
see \cite[Ch.~6, Thm.~2.9]{Di1} (and also \cite[Section 1.8]{DiSa2} in case $j=n-1$). By (3.1.3), we have the inclusions
$$F^p\subset P^p\quad\hbox{on}\,\,\,\,H^j(f^{-1}(1),{\mathbf C})_{\lambda},
\leqno(4.4.8)$$
Here it is possible to show (4.4.8) by calculating the direct image of $({\mathcal O}_X,F)$ by $f$ as a filtered ${\mathbf D}c$-module underlying a mixed Hodge module, see \cite{Sa1}, \cite{Sa2}, where a compactification of $f$ must be used. (The shift of the filtration by 1 comes from the direct image of ${\mathcal O}_X$ as a {\it left} ${\mathbf D}c$-module by the graph embedding of $f$.)
\par
The inclusion (4.4.8) implies some relation between the spectrum and the Poincar\'e series of the Koszul cohomologies via the spectral sequence (4.4.4), and the difference between $F^p$ and $P^p$ implies also their difference in certain cases, see also \cite{Di1}, \cite{Di3}, \cite{DiSt1}, etc.
\par
\noindent
{\bf 4.5.~Algebraic microlocal Gauss-Manin complexes.} For a homogeneous polynomial $f$, let ${\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_f$ be the algebraic microlocal Gauss-Manin complex (i.e. ${\mathbf C}t_f^j=\Omegaega^j[{\partial}_t,\dd_t^{-1}]$). The algebraic microlocal Gauss-Manin systems $H^j{\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_f$ are free graded ${\mathbf C}[{\partial}_t,\dd_t^{-1}]$-modules of finite type. Replacing $C^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_f$ with ${\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_f$ in (4.4.1) and (4.4.4), we have the filtrations $P'$, $G$ on ${\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_f$ together with the microlocal pole order spectral sequence
$$_{P'}\widetilde{E}_1^{p,j-p}=H^j{\mathcal G}r^p_{P'}{\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\Longrightarrow H^j{\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}},
\leqno(4.5.1)$$
where (4.4.3) holds again (i.e. $G={\rm Dec}\,P'$), and the last equalities of (4.4.5) and (4.4.6) hold for any $j,p\in{\mathbf Z}$, i.e.
$${}_{P'}\widetilde{E}_r^{p,j-p}=\begin{cases}H^jK_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}((n-p)d)&\hbox{if}\,\,\,r=1,\\
H^j\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}((j-p)d)&\hbox{if}\,\,\,r=2.\end{cases}
\leqno(4.5.2)$$
Moreover the last equality of (4.2.3) holds for any $k$, i.e.
$$H^{j+1}({\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k=\widetilde{H}^j(f^{-1}(1),{\mathbf C})_{\lambda}\quad\hbox{with}\quad\lambda=\exp(-2\pi ik/d),
\leqno(4.5.3)$$
(Note that the Gauss-Manin complex $C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ can be defined also as the single complex associated with the double complex having two differentials ${\partial}d$ and ${\rm d}f\wedge$, see \cite{Di1}, \cite{Di3}, etc.)
\par
Let $P',G$ denote also the induced filtrations on $H^j(C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})$, $H^j({\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})$ .There is a canonical inclusion
$$C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\hookrightarrow{\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}.$$
Set
$$\omega_0:={\rm d}f\in H^1(G_0C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})\,(=H^1A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}).$$
By the same argument as in \cite{BaSa}, it generates a free ${\mathbf C}[t]$-module for $p\in{\mathbf N}\cup\{\infty\}$
$${\mathbf C}[t]\omega_0\subset H^1(G_pC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}),$$
where $G_{\infty}C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}:=C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$.
Set
$$\widetilde{H}^j(G_pC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})=\begin{cases}H^j(G_pC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})&\hbox{if}\,\,\,j\ne 1,\\ H^j(G_pC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})/{\mathbf C}[t]\omega_0&\hbox{if}\,\,\,j=1.\end{cases}$$
Then the above inclusion induces the canonical isomorphisms
$$\widetilde{H}^j(G_pC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})\buildrel\sim\over\longrightarrow H^j(G_p{\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})\quad(p\in{\mathbf N}\cup\{\infty\},\,\,j\in{\mathbf Z}).
\leqno(4.5.4)$$
In fact, the assertion for $p=\infty$ follows from the same argument as in loc.~cit. This implies the assertion for $p\in{\mathbf N}$ by using the canonical morphism of long exact sequences
$$\begin{CD}@>>>\widetilde{H}^j(G_pC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})@>>>\widetilde{H}^j(C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})@>>>
H^j(C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}/G_pC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})@>>>\\
@. @VVV @VVV @|\\
@>>>H^j(G_p{\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})@>>>H^j({\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})@>>>
H^j({\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}/G_p{\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})@>>>\end{CD}$$
\par
From the canonical isomorphisms (4.5.4), we can deduce
$$G_p\widetilde{H}^j(C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})\buildrel\sim\over\longrightarrow G_pH^j({\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})={\partial}_t^pG_0H^j({\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})\quad(p\in{\mathbf N},\,\,j\in{\mathbf Z}).
\leqno(4.5.5)$$
This implies
$${\partial}_t:{\mathcal G}r^G_p\widetilde{H}^j(C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k\buildrel\sim\over\longrightarrow{\mathcal G}r^G_{p+1}\widetilde{H}^j(C_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{k-d}\quad(p\in{\mathbf N},\,\,j,k\in{\mathbf Z}).
\leqno(4.5.6)$$
Note that these hold with $G$ replaced by $P'$ by (4.4.3).
We then get by (4.4.7)
$$P^{\prime\,p+1}H^{j+1}({\mathbf C}t_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_k\cong P^p\widetilde{H}^j(f^{-1}(1),{\mathbf C})_{\lambda}\quad\bigl(\lambda=\exp(-2\pi ik/d),\,k\in[1,d]\bigr),
\leqno(4.5.7)$$
\par
\noindent
{\bf Proposition~4.6.} {\it With the notation of $(4.4)$ and $(4.5)$, there are canonical isomorphisms for $r\geqslant 2$
$$\aligned{\rm Im}({\partial}d_r:{}_{P'}E_r^{\,p-r,n-p+r-1}\to{}_{P'}E_r^{\,p,n-p})&=\begin{cases}\,0&\hbox{if}\,\,\,p>n,\\{\mathcal G}r^K_{r-1}(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}&\hbox{if}\,\,\,p=n,\\ {\mathcal G}r^K_{r-1}({\rm Coker}\,\dd_t^{-1})((n-p)d)&\hbox{if}\,\,\,p<n,\end{cases}\\
{\rm Im}({\partial}d_r:{}_{P'}\widetilde{E}_r^{\,p-r,n-p+r-1}\to{}_{P'}\widetilde{E}_r^{\,p,n-p})&={\mathcal G}r^K_{r-1}({\rm Coker}\,\dd_t^{-1})((n-p)d),\endaligned$$
where $K_{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is the kernel filtration, and ${\rm Coker}\,\dd_t^{-1}$ is a quotient of $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ as in Theorem~$2$.}
\par
\noindent
{\it Proof.} We first show the assertion for the microlocal pole order spectral sequence, i.e. for the second isomorphism. Since ${}_{P'}\widetilde{E}_r^{p,j-p}=0$ for $j>n$, the images of the differentials
$${\partial}d_r:{}_{P'}\widetilde{E}_r^{\,p-r,n-p+r-1}\to{}_{P'}\widetilde{E}_r^{\,p,n-p}\,\,\,(r\geqslant 2)$$
correspond to an increasing sequence of subspaces (with $p$ fixed):
$$\widetilde{I}_r^{\,p,n-p}\subset{}_{P'}\widetilde{E}_2^{\,p,n-p}=({\rm Coker}\,\dd_t^{-1})((n-p)d)\,\,\,(r\geqslant 2),
\leqno(4.6.1)$$
such that
$${\rm Im}\bigl({\partial}d_r:{}_{P'}\widetilde{E}_r^{\,p-r,n-p+r-1}\to{}_{P'}\widetilde{E}_r^{\,p,n-p}\bigr)=\widetilde{I}_r^{\,p,n-p}/\widetilde{I}_{r-1}^{\,p,n-p}\,\,\,(r\geqslant 2),$$
with $\widetilde{I}_1^{\,p,n-p}:=0$. Here ${\rm Coker}\,\dd_t^{-1}$ is a quotient of $H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ (and not $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor})$, and (4.3.5) is used for the last isomorphism of (4.6.1).
\par
By the construction of the spectral sequence (see e.g. \cite{De}), we have
$$\widetilde{I}_r^{\,p,n-p}=K_{r-1}({\rm Coker}\,\dd_t^{-1})((n-p)d),
\leqno(4.6.2)$$
where $K_{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is the kernel filtration defined just before Theorem~2.
(More precisely, $K_{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ defines a non-exhaustive filtration of $H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$, and its union is $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$.)
In fact, the left-hand side is given by the classes of $\omega\in\Omegaega^n$ such that there are
$$\eta_i\in\Omegaega^n\,\,(i\in[0,r-1])$$
satisfying
$$d\eta_0=\omega,\quad d\eta_{i+1}=df\wedge\eta_i\,(i\in[0,r-2]),\quad df\wedge\eta_{r-1}=0.$$
However, this condition is equivalent to that the class of $\omega$ in the Brieskorn module is contained in $K_{r-1}(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$.
(Note that $[df\wedge\eta_{r-2}]$ gives ${\partial}_t^{1-r}[\omega]$ and vanishes in $H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$.)
So the second isomorphism follows.
\par
The argument is essentially the same for the first isomorphism by replacing (4.6.2) with
$$I_r^{\,p,n-p}=\begin{cases}\,0&\hbox{if}\,\,\,p>n,\\ K_{r-1}(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}&\hbox{if}\,\,\,p=n,\\ K_{r-1}({\rm Coker}\,\dd_t^{-1})((n-p)d)&\hbox{if}\,\,\,p<n.\end{cases}$$
This finishes the proof of Proposition~(4.6).
\par
As a corollary of Proposition~(4.6), we get the following.
\par
\noindent
{\bf Corollary~4.7.} {\it The following three conditions are equivalent to each other$\,:$
\par
\noindent
$(a)$ The pole order spectral sequence $(4.4.4)$ degenerates at $E_2$.
\par
\noindent
$(b)$ The algebraic microlocal pole order spectral sequence $(4.5.1)$ degenerates at $E_2$.
\par
\noindent
$(c)$ The torsion subgroup $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ vanishes.}
\par
\noindent
{\bf 4.8.~Proof of Theorem~2.} By (4.5.7) the assertion follows from the second isomorphism in Proposition~(4.6) by choosing any $p\in{\mathbf Z}$, where the obtained isomorphism is independent of the choice of $p$ by using the bijectivity of the action of ${\partial}_t$. (It is also possible to use the first isomorphism in Proposition~(4.6) by choosing some $p<n$ although the independence of the choice of $p$ is less obvious unless the relation with the algebraic microlocal pole order spectral sequence is used.) This finishes the proof of Theorem~2.
\par
\noindent
{\bf 4.9.~Thom-Sebastiani type theorem for $P'$.} Let $f,f_1,f_2$ be as in Proposition~(2.2). In the notation of (4.5), we have a canonical isomorphism
$$({\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_f,P')=({\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_1},P')\otimes_{{\mathbf C}[{\partial}_t,\,{\partial}_t^{-1}]}({\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_2},P').$$
Assume $f_2$ has an isolated singularity at the origin as in Proposition~(2.2). Then
$$H^jGr^{P'}_k{\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_2}=0\quad(j\ne n_2,\,\,k\in{\mathbf Z}).$$
Hence $({\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_2},P')$ is strict, and we get a filtered quasi-isomorphism
$$({\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_2},P')\buildrel\sim\over\longrightarrow H^{n_2}({\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_2},P')[-n_2].$$
This implies a filtered quasi-isomorphism
$$({\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_f,P')\buildrel\sim\over\longrightarrow({\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_1},P')\otimes_{{\mathbf C}[{\partial}_t,\,{\partial}_t^{-1}]} H^{n_2}({\mathbf C}t^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_{f_2},P')[-n_2],
\leqno(4.9.1)$$
which is compatible with the action of $t$. More precisely, the action of $t$ on the left-hand side corresponds to $t\otimes id+id\otimes t$ on the right-hand side (since $f=f_1+f_2$).
\par
Combining (4.9.1) with (4.5.7), we get the Thom-Sebastiani type theorem for the pole order spectrum:
$${\rm Sp}_P(f)={\rm Sp}_P(f_1)\,{\rm Sp}_P(f_2),\quad{\rm Sp}_P^j(f)={\rm Sp}_P^j(f_1)\,{\rm Sp}_P^0(f_2)\quad(j\in{\mathbf N}),
\leqno(4.9.2)$$
assuming that $f_2$ has an isolated singularity as above so that ${\rm Sp}_P(f_2)={\rm Sp}_P^0(f_2)$, see \cite{SkSt} for the case where $f_1$ has also an isolated singularity.
Note that the Thom-Sebastiani type theorem holds for the Steenbrink spectrum by \cite{Sa5}.
\par
\noindent
{\bf Remarks~4.10.} (i) With the notation and the assumption of (4.9), the pole order spectral sequences degenerate at $E_2$ for $f$ if and only if they do for $f_1$. This follows from (4.9.1) together with Corollary~(4.7).
\par
(ii) The equivalence between the $E_2$-degeneration of the pole order spectral sequence (4.4.4) and the vanishing of $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ was shown in \cite{vSt} in the (non-graded) analytic local case.
\par
(iii) Assuming $\dim{\rm Sing}\,f^{-1}(0)=1$, we have by (4.3.4) the following exact sequence:
$$\aligned
0\to\widetilde{H}^{n-1}A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}(-d)&\,{\buildrel{\dd_t^{-1}}\over\longrightarrow}\,\widetilde{H}^{n-1}A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to H^{n-1}\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\\
\to H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}(-d)&\,{\buildrel{\dd_t^{-1}}\over\longrightarrow}\,H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to H^n\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}\to 0,\endaligned$$
where $\widetilde{H}^{n-1}A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ is defined by $H^{n-1}A_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}$ if $n\ne2$, and by its quotient by ${\mathbf C}[t]\omega_0$ if $n=2$. (For $\omega_0$, see the definition of $\widetilde{H}^j(G_pC_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})$ in (4.5).)
This exact sequence has sufficient information about the torsion subgroup $(H^n\K_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ to give another proof of Theorem~2.
\par
(iv) By forgetting the grading, Proposition~(4.6) and Corollary~(4.7) can be extended to the analytic local case where $f$ is a germ of a holomorphic function on a complex manifold with $\dim{\rm Sing}\,f^{-1}(0)=1$.
\par
The following will be used in the proof of Theorem~(5.2) below.
\par
\noindent
{\bf 4.11.~Multiplicity of the minimal exponent.}
Let $g$ be a germ of holomorphic function on a complex manifold $(Y,0)$ having an isolated singularity. We have the direct image ${\mathcal B}_g:={\mathcal O}_{Y,0}[{\partial}_t]$ of ${\mathcal O}_{Y,0}$ as a left ${\mathbf D}c_{Y,0}$-module by the graph embedding of $g$. (Note that it is an analytic ${\mathbf D}c$-module.) It has the Hodge filtration $F$ by the order of ${\partial}_t$ and the filtration $V$ of Kashiwara \cite{Ka} and Malgrange \cite{Ma}.
\par
Consider ${\mathcal G}r_V^{\alpha}({\mathcal B}_g,F)$ for $\alpha<1$. These underlie mixed Hodge modules supported at $0$, and are the direct images of filtered vector spaces by the inclusion $\{0\}\hookrightarrow Y$ as filtered ${\mathbf D}c$-modules. (This is shown by using \cite[Lemma 3.2.6]{Sa1} applied to any function vanishing at $0$.) So we get
$$\hbox{The ${\mathcal G}r^F_p{\mathcal G}r_V^{\alpha}{\mathcal B}_g$ are annihilated by ${\mathfrak m}_{Y,0}\subset{\mathcal O}_{Y,0}$ for $\alpha<1$,}
\leqno(4.11.1)$$
where ${\mathfrak m}_{Y,0}\subset{\mathcal O}_{Y,0}$ is the maximal ideal.
\par
Let ${\mathcal B}_gB:={\mathcal O}_{Y,0}[{\partial}_t,\dd_t^{-1}]$ be the algebraic microlocalization of ${\mathcal B}_g$. By \cite[Sections 2.1-2]{Sa2}, it has the Hodge filtration $F$ by the order of ${\partial}_t$ and also the filtration $V$ such that
$$\aligned{\partial}_t:F_pV^{\alpha}{\mathcal B}_gB&\buildrel\sim\over\longrightarrow F_{p+1}V^{\alpha-1}{\mathcal B}_gB\quad(\forall\,p,\alpha).\\({\mathcal G}r_V^{\alpha}{\mathcal B}_g,F)&\buildrel\sim\over\longrightarrow({\mathcal G}r_V^{\alpha}{\mathcal B}_gB,F)\quad(\alpha<1),\endaligned$$
Then (4.11.1) implies
$$\hbox{The ${\mathcal G}r^F_p{\mathcal G}r_V^{\alpha}{\mathcal B}_gB$ are annihilated by ${\mathfrak m}_{Y,0}\subset{\mathcal O}_{Y,0}$ for any $\alpha$.}
\leqno(4.11.2)$$
\par
Consider the (relative) de Rham complexes
$${\mathbf C}C:={\rm DR}_Y({\mathcal B}_g),\quad{\mathbf C}Ct:={\rm DR}_Y({\mathcal B}_gB).$$
Up to a shift of complexes, these are the Koszul complexes associated with the action of ${\partial}_{y_i}$ on ${\mathcal B}_g$ and ${\mathcal B}_gB$ where the $y_i$ are local coordinates of $Y$. It has the filtrations $F$ and $V$ induced by those on ${\mathcal B}_g$ and ${\mathcal B}_gB$. Here $V$ is stable by the action of ${\partial}_{y_i}$, but we need a shift for $F$ depending on the degree of the complexes ${\mathbf C}C$, ${\mathbf C}Ct$. By the above argument we have
$$H^j{\mathcal G}r_p^F{\mathcal G}r_V^{\alpha}{\mathbf C}Ct=H^j{\mathcal G}r_V^{\alpha}{\mathbf C}Ct=H^j{\mathcal G}r_p^F{\mathbf C}Ct=0\quad(j\ne 0),
\leqno(4.11.3)$$
where we also use the fact that ${\mathcal G}r_p^F{\mathbf C}Ct$ is the Koszul complex for the regular sequence $\{{\partial} g/{\partial} y_j\}$. These imply the vanishing of $H^jF_p{\mathcal G}r_V^{\alpha}{\mathbf C}Ct$, etc.\ for $j\ne 0$, and we get
$$\hbox{$({\mathbf C}Ct;F,V)$ is strict,}
\leqno(4.11.4)$$
by showing the exactness of some commutative diagram appearing in the definition of strict complex \cite{Sa1}.
\par
It is known that the filtration $V$ on ${\mathbf C}C$ is strict, and induces the filtration $V$ of Kashiwara and Malgrange on the Gauss-Manin system $H^0{\mathbf C}C$ (by using the arguments in the proof of \cite[Prop.~3.4.8]{Sa1}).
This assertion holds by replacing ${\mathbf C}C$ with ${\mathbf C}Ct$, since ${\mathbf C}C/V^{\alpha}{\mathbf C}C={\mathbf C}Ct/V^{\alpha}{\mathbf C}Ct$ for $\alpha\leqslant 1$ and $H^0{\mathbf C}C=H^0{\mathbf C}Ct$ (see e.g. \cite{BaSa}). Here we also get the canonical isomorphism
$$(H^0{\mathbf C}C,V)=(H^0{\mathbf C}Ct,V).
\leqno(4.11.5)$$
\par
Consider now $({\mathcal G}r^F_0{\mathbf C}Ct,V)$. This is a complex of filtered ${\mathcal O}_{Y,0}$-modules, and is strict. By the above argument we get the canonical isomorphism of filtered ${\mathcal O}_{Y,0}$-modules
$$H^0({\mathcal G}r^F_0{\mathbf C}Ct,V)=({\mathcal O}_{Y,0}/({\partial} g),V).
\leqno(4.11.6)$$
Combining this with (4.11.2), (4.11.4) and using ${\mathcal G}r_V^{\alpha}{\mathcal G}r^F_p{\mathbf C}Ct={\mathcal G}r^F_p{\mathcal G}r_V^{\alpha}{\mathbf C}Ct$, we get
$$\hbox{The ${\mathcal G}r_V^{\alpha}({\mathcal O}_{Y,0}/({\partial} g))$ are annihilated by ${\mathfrak m}_{Y,0}\subset{\mathcal O}_{Y,0}$ for any $\alpha$.}
\leqno(4.11.7)$$
In particular, the multiplicity of the minimal exponent is 1.
\par
\par
\vbox{\centerline{\bf 5. Calculation of ${\partial}d^{(1)}$.}
\par
\noindent
In this section we calculate ${\partial}d^{(1)}$ in certain cases, and prove Theorems~(5.2) and (5.3).}
\par
\noindent
{\bf 5.1.~Relation with the isolated singularities in ${\mathbf P}^{n-1}$.}
Let $\rho:\widetilde{X}\to X$ be the blow-up of the origin of $X:={\mathbf C}^n$. Let $y=\sum_ic_ix_i$ be as in the introduction (i.e. $(c_i)\in{\mathbf C}^n$ are sufficiently general). We may assume that
$$y=x_n,$$
replacing the coordinates $x_1,\dots,x_n$ of $X={\mathbf C}^n$. Let $\widetilde{X}'$ be the complement of the proper transform of $\{x_n=0\}$. It has the coordinates $\widetilde{x}_1,\dots,\widetilde{x}_n$ such that
$$\rho^*x_i=\begin{cases}\widetilde{x}_i\,\widetilde{x}_n&\hbox{if}\,\,\,i\ne n,\\
\widetilde{x}_n&\hbox{if}\,\,\,i=n.\end{cases}$$
Set $n':=n-1$. Define the complex $\spKf$ with $R$ and $f$ respectively replaced with
$${\mathbf C}[\widetilde{x}_1,\dots,\widetilde{x}_{n'}][\widetilde{x}_n,\widetilde{x}_n^{\,-1}]\quad\hbox{and}\quad f':=\rho^*f|_{\widetilde{X}'}=\widetilde{x}_n^d\,h(\widetilde{x}_1,\dots,\widetilde{x}_{n'}),$$
where $h(\widetilde{x}_1,\dots,\widetilde{x}_{n'}):=f(\widetilde{x}_1,\dots,\widetilde{x}_{n'},1)$, and the grading is given only by the degree of $\widetilde{x}_n$. This is compatible with ${}^s\!K^{\ssb}_f$ via $\rho^*$. We have the canonical graded morphism
$$H^j({}^s\!K^{\ssb}_f)\to H^j(\spKf),$$
in a compatible way with the differential ${\partial}d$. This morphism induces injective morphisms
$$N\hookrightarrow H^{-1}(\spKf),\quad M''\hookrightarrow H^0(\spKf),
\leqno(5.1.1)$$
where the image of $M'$ in $H^0(\spKf)$ vanishes. We have the inclusion
$$N^{(2)}_{p+d}\subset{\rm Ker}\bigl({\partial}d:H^{-1}(\spKf)\to H^0(\spKf)\bigr)\cap N_{p+d},
\leqno(5.1.2)$$
under the first injection of (5.1.1), and the equality holds if $M'_p=0$.
\par
Let $Y'$ be the complement of $\{\widetilde{x}_n=0\}$ in $Y:={\mathbf P}^{n'}$. Then
$$\widetilde{X}'=Y'\times{\mathbf C},$$
where $\widetilde{x}_1,\dots,\widetilde{x}_{n'}$ and $\widetilde{x}_n$ are respectively coordinates of $Y'$ and ${\mathbf C}$. Moreover $\spKf$ is quasi-isomorphic to the mapping cone of
$${\partial} f'/{\partial} \widetilde{x}_n=d\,\widetilde{x}_n^{\,d-1}h:(\Omegaega_{Y'}^{n'}/{\partial}h\wedge\Omegaega_{Y'}^{n'-1})[\widetilde{x}_n,\widetilde{x}_n^{\,-1}]\to (\Omegaega_{Y'}^{n'}/{\partial}h\wedge\Omegaega_{Y'}^{n'-1})[\widetilde{x}_n,\widetilde{x}_n^{\,-1}],$$
where $\Omegaega_{Y'}^j$ is identified with the group of global sections.
\par
Let $\{z_k\}$ be the singular points of the morphism $h:Y'\to{\mathbf C}$. These are all isolated singular points. (In fact, they are the union of the singular points of $\{h=c\}$ for $c\in{\mathbf C}$. But $\{h=c\}$ is the intersection of $\{f=c\}$ and $\{x_n=1\}$ in ${\mathbf C}^n$, and the intersection of its closure in ${\mathbf P}^n$ with the boundary ${\mathbf P}^{n'}={\mathbf P}^n\setminus{\mathbf C}^n$ is the intersection of $\{f=0\}$ and $\{x_n=0\}$ in ${\mathbf P}^{n'}$, which is smooth by hypothesis. So the assertion follows.)
\par
Since the support of the ${\mathbf C}[\widetilde{x}_1,\dots,\widetilde{x}_{n'}]$-module $\Omegaega_{Y'}^{n'}/{\partial}h\wedge\Omegaega_{Y'}^{n'-1}$ is $\{z_k\}$, we have the canonical isomorphism
$$\Omegaega_{Y'}^{n'}/{\partial}h\wedge\Omegaega_{Y'}^{n'-1}=\h{$\bigoplus$}_k\,\Omegaega_{h_k}^{n'}\quad \hbox{with}\quad\Omegaega_{h_k}^{n'}:=\Omegaega_{Y'_{\rm an},z_k}^{n'}/{\partial}h_k\wedge\Omegaega_{Y'_{\rm an},z_k}^{n'-1},$$
where $Y'_{\rm an}$ is the associated analytic space, and $h_k$ is the germ of an analytic function defined by $h$ at $z_k$.
\par
Let $z_k$ ($k\leqslant r_0$) be the singular points contained in $\{h=0\}$. These are the singular points of $Z:=f^{-1}(0)\subset{\mathbf P}^{n'}$ since $x_n$ is sufficiently general. Since $h_k$ is invertible for $k>r_0$, we have
$$H^n(\spKf)=\h{$\bigoplus$}_{k\leqslant r_0}\,(\Omegaega_{h_k}^{n'}/h_k\,\Omegaega_{h_k}^{n'})\wedge{\mathbf C}[\widetilde{x}_n,\widetilde{x}_n^{\,-1}]\,{\partial}d \widetilde{x}_n,
\leqno(5.1.3)$$
and there is a similar formula for $H^{n-1}(\spKf)$ (with ${\partial}d \widetilde{x}_n$ on the right-hand side deleted and $\wedge$ replaced by $\otimes$).
So the $z_k$ for $k>r_0$ may be forgotten from now on.
\par
Note that, via (5.1.1) and (5.1.3), we have for $p\gg 0$
$$M''\supset(\Omegaega_{h_k}^{n'}/h_k\,\Omegaega_{h_k}^{n'})\wedge{\mathbf C}[\widetilde{x}_n]\,\widetilde{x}_n^{\,p}\,{\partial}d \widetilde{x}_n.
\leqno(5.1.4)$$
\par
Take an element of pure degree $p$ of
$${\rm Ker}\bigl(h_k:\Omegaega_{h_k}^{n'}[\widetilde{x}_n,\widetilde{x}_n^{\,-1}]\to\Omegaega_{h_k}^{n'}[\widetilde{x}_n,\widetilde{x}_n^{\,-1}]\bigr)\quad(k\leqslant r_0).$$
It is represented by $\psi:=\frac{1}{d}\,\widetilde{x}_n^{\,p}\widetilde{x}i$ where $\widetilde{x}i\in\Omegaega_{Y'_{\rm an},z_k}^{n'}$ satisfies
$$h_k\,\widetilde{x}i={\partial}h_k\wedge\eta\quad\hbox{with}\quad\eta\in\Omegaega_{Y'_{\rm an},z_k}^{n'-1}.
\leqno(5.1.5)$$
The corresponding element of $H^{n'}(\spKf)$ is represented by
$$\psi':=\hbox{$\frac{1}{d}$}\,\widetilde{x}_n^{\,p}\widetilde{x}i+\widetilde{x}_n^{\,p-1}{\partial}d \widetilde{x}_n\wedge\eta.$$
Its image in $H^n(\spKf)$ by the differential ${\partial}d$ is given by
$${\partial}d[\psi']=\pm\bigl[\bigl(\hbox{$\frac{p}{d}$}\,\widetilde{x}i-{\partial}d\eta\bigr)\wedge \widetilde{x}_n^{\,p-1}{\partial}d \widetilde{x}_n\bigr],
\leqno(5.1.6)$$
and we have by (5.1.5)
$$[d\eta]={\partial}_tt\,[\widetilde{x}i]\quad\hbox{in}\,\,\,H''_{h_k}.
\leqno(5.1.7)$$
\par
Let $V$ be the $V$-filtration of Kashiwara \cite{Ka} and Malgrange \cite{Ma} on the Gauss-Manin system ${\mathcal G}_{h_k}$ indexed by ${\mathbf Q}$, see e.g.\ \cite{SkSt}. (It is closely related with the theory of asymptotic Hodge structure \cite{Va}.) We denote also by $V$ the induced filtration on the Brieskorn module $H''_{h_k}$ and also on $\Omegaega_{h_k}^{n'}$ via the canonical inclusion and the surjection
$${\mathcal G}_{h_k}\supset H''_{h_k}\to\Omegaega_{h_k}^{n'},$$
see \cite{Bri} for the latter. In this paper we index $V$ so that ${\partial}_tt-\alpha$ is nilpotent on ${\mathcal G}r_V^{\alpha}{\mathcal G}_{h_k}$.
\par
Let $\{\alpha_{h_k,j}\}$ be the exponents of $h_k$ counted with multiplicity; more precisely
$$\#\{j:\alpha_{h_k,j}=\alpha\}=\dim{\mathcal G}r^{\alpha}_V\Omegaega^{n'}_{h_k}\quad\hbox{and}\quad{\rm Sp}_{h_k}(t)=\par
um_i\,t^{\,\alpha_{h_k,j}}.
\leqno(5.1.8)$$
Here we may assume the $\alpha_{h_k,j}$ are weakly increasing (i.e. $\alpha_{h_k,j}\leqslant\alpha_{h_k,j+1}$) for each $k$. We have the symmetry $\{\alpha_{h_k,j}\}_j=\{n-\alpha_{h_k,j}\}_j$ (counted with multiplicity) by \cite{St2}.
\par
\noindent
{\bf Theorem~5.2.} {\it With the notation of $(5.1)$, assume $h_k$ is non-quasihomogeneous $($i.e. $h_k\notin({\partial} h_k))$ for some $k\leqslant r_0$. Then the kernel and cokernel of $d^{(1)}:N\to M$ $($i.e. $N^{(2)}$ and $M^{(2)}$ in Theorem~$2)$ and $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ are all infinite dimensional over ${\mathbf C}$.}
\par
\noindent
{\it Proof.} Since the minimal exponent $\alpha_{h_k,1}$ has multiplicity 1 (see (4.11)), we have
$$V^{>\alpha_{h_k,1}}\,\Omegaega_{h_k}^{n'}={\mathfrak m}_{Y,z_k}\Omegaega_{h_k}^{n'}\supset{\rm Ker}(h_k:\Omegaega_{h_k}^{n'}\to\Omegaega_{h_k}^{n'}).$$
Combined with (5.1.7), this implies for $\widetilde{x}i$ as in (5.1.5)
$$\bigl[\hbox{$\frac{p}{d}$}\,\widetilde{x}i-{\partial}d\eta\bigr]\in V^{>\alpha_{h_k,1}\,}\Omegaega_{h_k}^{n'}.
\leqno(5.2.1)$$
So the infinite dimensionality of $M^{(2)}$ follows from (5.1.4), (5.1.6) and (5.2.1). It implies the assertion for $N^{(2)}$ since the morphisms
$${\partial}d^{(1)}:N_{p+d}\to M_p$$
are morphisms of finite dimensional vector spaces of the same dimension for $p\gg 0$. The assertion for the torsion part $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ then follows from Theorem~2. This finishes the proof of Theorem~(5.2).
\par
\noindent
{\bf Theorem~5.3.} {\it With the notation of $(5.1)$, assume the $h_k$ are quasihomogeneous $($i.e. $h_k\in({\partial} h_k))$ for any $k\leqslant r_0$. Then the kernel and cokernel of $d^{(1)}:N\to M$ $($i.e. $N^{(2)}$ and $M^{(2)}$ in Theorem~$2)$ and $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ are finite dimensional over ${\mathbf C}$. More precisely we have
$$\nu^{(2)}_{p+d}:=\dim N^{(2)}_{p+d}\leqslant\#\bigl\{(k,j)\,\big|\,\,\alpha_{h_k,j}=\hbox{$\frac{p}{d}$}\,\,(k\leqslant r_0)\,\bigr\},
\leqno(5.3.1)$$
and the equality holds in the case where $\mu'_p=0$ and either $\nu_{p+d}=\tau_Z$ or all the singularities of $Z$ are ordinary double points.}
\par
\noindent
{\it Proof.} By Theorem~2, it is enough to show the inequality (5.3.1) together with the equality in the special case as above.
Take any $k\leqslant r_0$. In the notation of (5.1) there is a local analytic coordinate system $(y_1,\cdots,y_{n'})$ of $Y'$ around $z_k$ together with positive rational numbers $w_1,\dots,w_{n'}$ such that $h_k$ is a linear combination of monomials $\prod_iy_i^{m_i}$ with $\sum_iw_im_i=1$ (see \cite{SaK}). Then
$$v(h_k)=h_k\quad\hbox{with}\quad v:=\par
um_i\,w_i\,y_i\,{\partial}_{y_i}.$$
We will denote the contraction of ${\partial}d y_1\wedge\dots\wedge y_{n'}$ and $v$ by $\zeta$.
\par
Take a monomial basis $\{\widetilde{x}i_j\}$ of $\Omegaega_{h_k}^{n'}$, where monomial means that
$$\widetilde{x}i_j=\hbox{$\prod$}_i\,y_i^{m_{j,i}}\,{\partial}d y_1\wedge\dots\wedge{\partial}d y_{n'}\quad\hbox{with}\quad m_{j,i}\in{\mathbf N}.$$
Set
$$\eta_j:=\hbox{$\prod$}_i\,y_i^{m_{j,i}}\,\zeta,\quad w(\widetilde{x}i_j):=\par
um_i\,w_i(m_{j,i}+1).$$
Then
$${\rm d}f\wedge\eta_j=f\,\widetilde{x}i_j,\quad{\partial}d\eta_j=w(\widetilde{x}i_j)\,\widetilde{x}i_j.$$
So we get
$${\partial}_tt\,[\widetilde{x}i_j]=w(\widetilde{x}i_j)\,[\widetilde{x}i_j]\quad\hbox{in}\,\,\,H''_{h_k}.$$
In particular
$$w(\widetilde{x}i_j)=\alpha_{h_k,j},$$
by changing the ordering of the $\widetilde{x}i_j$ if necessary.
The inequality (5.3.1) then follows from (5.1.6) and (5.1.7) together with the inclusion (5.1.2).
In case the assumption after (5.3.1) is satisfied, we have the equality by using the remark after (5.1.2) together with the fact that $\alpha_{h_k,j}=n'/2$ if $z_k$ is an ordinary double point of $Z$.
This finishes the proof of Theorem~(5.3).
\par
\noindent
{\bf Corollary~5.4.} {\it With the hypothesis of Theorem~$(5.3)$, assume $n=2$ or more generally
$$\max\bigl\{\,\alpha_{h_k,j}\,\,\big|\,\,d\alpha_{h_k,j}\in{\mathbf N}\,\,\,(k\leqslant r_0)\,\bigr\}<1+\hbox{$\frac{n}{d}$},
\leqno(5.4.1)$$
$($for instance, $d\alpha_{h_k,j}\notin{\mathbf N}$ for any $j$ and $k\leqslant r_0)$. Then the pole order spectral sequences $(4.4.4)$ and $(4.5.1)$ degenerate at $E_2$, and $(H^nA^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}}_f)_{\rm tor}=0$.}
\par
\noindent
{\it Proof.} This follows from Theorem~(5.3) together with Corollary~(4.7) and Theorem~2 since ${\partial}d^{(r)}$ is a graded morphism of degree $-rd$.
\par
\noindent
{\bf Corollary~5.5.} {\it With the first hypothesis of Theorem~$(5.3)$, assume $n=3$. Let $\alpha'_f:=\min\{\alpha_{h_k,j}\}$ in the notation of $(5.1.8)$. Then}
$$\nu_{p+d}=0\quad\hbox{for}\quad p<d\alpha'_f.
\leqno(5.5.1)$$
\par
\noindent
{\it Proof.} Note first that $\alpha'_f\leqslant 1$ since $\dim Z=1$. Assume $\nu_{p+d}\ne 0$ with $p<d\alpha'_f$. Then the image of ${\partial}d^{(1)}:N_{p+d}\to M_p$ is nonzero by Theorem~(5.3). We get hence by Theorem~2
$$n'_{f,p/d}<\mu_p\leqslant\binom{p-1}{n-1},$$
where $n'_{f,p/d}$ is the coefficient of the pole order spectrum ${\rm Sp}_P(f)$ (i.e. ${\rm Sp}_P(f)=\par
um_{\alpha}\,n'_{f,\alpha}t^{\alpha}$). However, this contradicts Remark~(3.6)(ii). So Corollary~(5.5) follows.
\par
\noindent
{\bf Remarks~5.6.} (i) In Theorem~(5.3), the inequality (5.3.1) holds with the left-hand side replaced by the dimension of the kernel of the composition
$$N_{p+d}\buildrel{{\partial}d^{(1)}}\over\longrightarrow M_p\to M''_p.$$
In fact, (5.1.1) implies that (5.1.2) holds with $N_{p+d}^{(2)}$ replaced by this kernel.
\par
(ii) Corollary~(5.4) seems to be closely related with the short exact sequence in \cite[Thm.~1]{DiSa1}.
\par
(iii) If all the singularities of $Z$ are nodes, then $\alpha'_f=1$, and the estimation obtained by Corollary~(5.5) coincides with the one in \cite[Thm.~4.1]{DiSt1}, which is known to be sharp. It is also sharp for instance if the singularities are $A_1$ or $D_4$ (e.g. $f=(x^2-y^2)(x^2-z^2)(y^2-z^2)$).
\par
(iv) The proof of the finiteness of $(H^nA_f^{\raise.15ex\h{${\scriptscriptstyle\bullet}$}})_{\rm tor}$ can be reduced to the analytic local case by considering the formal completion where the direct sum is replaced with the infinite direct product and the convergent power series factors through the formal completion.
\par
(v) The argument in (5.1) can be extended to the analytic local case if there is a projective morphism of complex manifolds $\rho:\widetilde{X}\to X$ such that the restriction of $\rho$ over $X\setminus\{0\}$ is an isomorphism and the following two conditions are satisfied:
\par
\noindent
$(a)$ The proper transform of each irreducible component of ${\rm Sing}\,f$ transversally intersects $\rho^{-1}(0)_{\rm red}$ at a smooth point $z_k$, and $z_k\ne z_{k'}$ for $k\ne k'$,
\par
\noindent
$(b)$ We have $\rho^*f=\widetilde{x}_n^{\,a_k}h_k(\widetilde{x}_1,\dots,\widetilde{x}_{n-1})$ around each $z'_k$ where $\widetilde{x}_1,\dots,\widetilde{x}_n$ are local coordinates of $\widetilde{X}$ around $z_k$, $h_k$ is a germ of a holomorphic function of $n-1$ variables, and $a_k\in{\mathbf N}$.
\par
Note that condition~$(b)$ implies that $\rho^{-1}(0)_{\rm red}$ is locally defined by $\{\widetilde{x}_n=0\}$.
Let $y$ be a sufficiently general linear combination of local coordinates of $(X,0)$. Here we assume that $\rho$ factors through the blow-up along $0\in X$, and moreover the intersection of the exceptional divisor $E_0$ of the blow-up along $0$ with the proper transform of each irreducible component of ${\rm Sing}\,f$ is not contained in the hyperplane of $E_0$ defined by $h$ (by replacing $h$ if necessary). Then $\rho^*h=u_k\widetilde{x}_n^{b_k}$ with $b_k\in{\mathbf N}$ and $u_k$ an invertible function. Here we may assume $\rho^*h=\widetilde{x}_n^{b_k}$ by replacing $\widetilde{x}_n$, but the equality in condition~$(b)$ is replaced by $\rho^*f=u'_k\widetilde{x}_n^{\,a_k}h_k(\widetilde{x}_1,\dots,\widetilde{x}_{n-1})$ where $u'_k$ is an invertible function.
Then condition~$(b)$ can be replaced with
\par
\noindent
$(b)'$ The restriction of $\rho^*y$ to the proper transform ${\mathbf D}t$ of $f^{-1}(0)$ gives an analytically trivial deformation on a neighborhood of each $z'_k$ by replacing $\rho^*y:{\mathbf D}t\to{\mathbf C}$ with the normalization of the base change by an appropriate ramified covering of ${\mathbf C}$ if necessary.
\par
Under these assumptions, Theorem~(5.2) can be extended to the analytic local case where $h_k$ is as in condition~$(b)$ above. However, it does not seem easy to generalize Theorem~(5.3) unless $f$ admits a ${\mathbf C}^*$-action (or the arguments related to the grading are completely ignored).
\par
(vi) If $n=3$ and $Z$ has only ordinary double points as singularities, then the coefficients $n_{f,\alpha}$ of the Steenbrink spectrum for $\alpha\notin{\mathbf Z}$ are the same as that of a central hyperplane arrangement in ${\mathbf C}^3$ having only ordinary double points in ${\mathbf P}^2$. (Note that its formula can be found in \cite{BuSa}.)
In fact, the vanishing cycle sheaf $\varphi_{f,\ne 1}{\mathbf Q}_X$ is supported at the origin so that we have the symmetry of the coefficients $n_{f,\alpha}$ for $\alpha\notin{\mathbf Z}$. Moreover $n_{f,\alpha}$ for $\alpha<1$ can be obtained by Remark~(3.6)(ii), and $n_{f,\alpha}$ for $\alpha\in(1,2)$ can be calculated from the $n_{f,\alpha}$ for $\alpha\notin(1,2)$ by using the relation with the Euler characteristic of the complement of $Z\subset{\mathbf P}^2$. (The latter follows from (3.1.2).)
Note also that the $n_{f,\alpha}$ for $\alpha\in{\mathbf Z}$ can be obtained from the Hodge numbers of the complement of $Z\subset{\mathbf P}^{n-1}$.
\par
\noindent
{\bf Examples~5.7.}
We first give some examples where the assumptions of Corollary~(5.4) and the last conditions of Theorem~(5.3) are all satisfied, and moreover Remark~(5.6)(vi) can also be applied.
These are also examples of type (I) singularities (i.e. (0.5) is satisfied).
\par
\noindent
(i) $f=xyz\,\,\,$(three $A_1$ singularities in ${\mathbf P}^2$) $\,\,n=d=3.$
$$\begin{array}{cccccccccccccccc}
k\, &1 &2 &3 &4 &5 &6 &7 &8 &9 &\cdots \\
\gamma_k & & &1 &3 &3 &1\\
\mu'_k \\
\mu''_k & & &1 &3 &3 &3 &3 &3 &3 &\cdots\\
\mu_k & & &1 &3 &3 &3 &3 &3 &3 &\cdots\\
\nu_k & & & & & &2 &3 &3 &3 &\cdots\\
\mu^{\scriptscriptstyle(2)}_k & & &1\\
\nu^{\scriptscriptstyle(2)}_k & & & & & &2\\
{\rm Sp}_P & & &1 & & &-2\\
{\rm Sp} & & &1 & & &-2\\
\end{array}$$
\par
\noindent
(ii) $f=x^2y^2+x^2z^2+y^2z^2\,\,\,$(three $A_1$ singularities in ${\mathbf P}^2$) $\,n=3,\,d=4.$
$$\begin{array}{cccccccccccccccccc}
k\, &1 &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 &12 &\cdots&\raise-3mm\hbox{ }\\
\gamma_k & & &1 &3 &6 &7 &6 &3 &1\\
\mu'_k & & & & &3 &4 &3\\
\mu''_k & & &1 &3 &3 &3 &3 &3 &3 &3 &3 &3 &\cdots\\
\mu_k & & &1 &3 &6 &7 &6 &3 &3 &3 &3 &3 &\cdots\\
\nu_k & & & & & & & & &2 &3 &3 &3 &\cdots \\
\mu^{\scriptscriptstyle(2)}_k & & &1 &3 &4 &4 &3\\
\nu^{\scriptscriptstyle(2)}_k \\
{\rm Sp}_P & & &1 &3 &4 &4 &3 &0 &0\\
{\rm Sp} & & &1 &3 &3 &4 &3 &0 &1\\
\end{array}$$
\par
\noindent
(iii) $f=xyz(x+y+z)\,\,\,$(six $A_1$ singularities in ${\mathbf P}^2$) $\,n=3,\,d=4.$
$$\begin{array}{cccccccccccccccccc}
k\, &1 &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 &12 &\cdots&\raise-3mm\hbox{ }\\
\gamma_k & & &1 &3 &6 &7 &6 &3 &1\\
\mu'_k & & & & & &1 & & &\\
\mu''_k & & &1 &3 &6 &6 &6 &6 &6 &6 &6 &6 &\cdots\\
\mu_k & & &1 &3 &6 &7 &6 &6 &6 &6 &6 &6 &\cdots\\
\nu_k & & & & & & & &3 &5 &6 &6 &6 &\cdots \\
\mu^{\scriptscriptstyle(2)}_k & & &1 &3 &1 &1\\
\nu^{\scriptscriptstyle(2)}_k & & & & & & & &3\\
{\rm Sp}_P & & &1 &3 &1 &1 &0 &-3 &0\\
{\rm Sp} & & &1 &3 &0 &1 &0 &-3 &1\\
\end{array}$$
Here we have $\mu''_4=3$ by Lemma~(2.1), but the proof of $\mu''_5=6$ is not so trivial. In fact, if $\mu''_5<6$, then we have $\nu_{\,7}\ne 0$ by Corollary~2. However, this contradicts Corollary~(5.5).
\par
\noindent
{\bf Examples~5.8.}
(i) $f=x^2y^2+z^4\,\,\,$(two $A_3$ singularities in ${\mathbf P}^2$) $\,n=3,\,d=4.$
\par
\noindent
The calculation of this example does not immediately follow from Corollary~(5.4) since the last conditions of Theorem~(5.3) are not satisfied and Remark~(5.6)(vi) does not apply to this example.
This example can be calculated by using the Thom-Sebastiani type theorems in (2.2) and (4.9).
\par
$$\begin{array}{cccccccccccccccccc}
k\, &1 &2 &3 &4 &5 &6 &7 &8 &9 &10 &11 &12 &\cdots&\raise-3mm\hbox{ }\\
\gamma_k & & &1 &3 &6 &7 &6 &3 &1\\
\mu'_k & & & & &1 &1 &1\\
\mu''_k & & &1 &3 &5 &6 &6 &6 &6 &6 &6 &6 &\cdots\\
\mu_k & & &1 &3 &6 &7 &7 &6 &6 &6 &6 &6 &\cdots\\
\nu_k & & & & & & &1 &3 &5 &6 &6 &6 &\cdots \\
\mu^{\scriptscriptstyle(2)}_k & & &1 &1 &2 &1 &1\\
\nu^{\scriptscriptstyle(2)}_k & & & & & & &1 &1 &1\\\
{\rm Sp}_P & & &1 &1 &2 &1 &0 &-1 &-1\\
{\rm Sp} & & &1 &1 &2 &1 &0 &-1 &-1\\
\end{array}$$
We note the calculation in the case $f=x^2y^2$ for the convenience of the reader.
\par
\noindent
(ii) $f=x^2y^2\,\,\,$(two $A_1$ singularities in ${\mathbf P}^1$) $\,\,n=2,\,d=4.$
$$\begin{array}{cccccccccccccccc}
k\, &1 &2 &3 &4 &5 &6 &7 &8 &\cdots \\
\gamma_k & &1 &2 &3 &2 &1\\
\mu'_k & & & &1\\
\mu''_k & &1 &2 &2 &2 &2 &2 &2 &\cdots\\
\mu_k & &1 &2 &3 &2 &2 &2 &2 &\cdots\\
\nu_k & & & & & &1 &2 &2 &\cdots\\
\mu^{\scriptscriptstyle(2)}_k & &1 & &1\\
\nu^{\scriptscriptstyle(2)}_k & & & & & &1\\
{\rm Sp}_P & &1 &0 &1 &0 &-1\\
{\rm Sp} & &1 &0 &1 &0 &-1\\
\end{array}$$
\par
\noindent
{\bf Remark~5.9.} We have the $V$-filtration of Kashiwara and Malgrange on $N_p$, $M''_p$ by using the injections in (5.1.1). Assume all the singularities of $Z$ are weighted homogeneous. It seems that the following holds in many examples:
$$\dim{\mathcal G}r_V^{\alpha}N_{p+d}=\begin{cases}\nu_{p+d}^{(2)}=\nu_{p+d}^{(\infty)}=n^1_{f,\alpha+1}&\hbox{if}\,\,\,p/d=\alpha,\\ 0&\hbox{if}\,\,\,p/d<\alpha,\end{cases}
\leqno(5.9.1)$$
where $n^j_{f,\alpha}$ is as in (3.2.3), and $\nu_{p+d}^{(\infty)}:=\nu_{p+d}^{(r)}$ ($r\gg 0$).
Note that (5.9.1) would imply the $E_2$-degeneration of the pole order spectral sequence in Question~2.
\par
As for $M''_p$, (5.9.1) seems to correspond by duality to the following:
$$\dim{\mathcal G}r_V^{\alpha}M''_p=\begin{cases}n_{Z,\alpha}-n^1_{f,\alpha}&\hbox{if}\,\,\,p/d=\alpha,\\ n_{Z,\alpha}&\hbox{if}\,\,\,p/d>\alpha,\end{cases}
\leqno(5.9.2)$$
where $n_{Z,\alpha}:=\par
um_{k\leqslant r_0}\,n_{h_k,\alpha}$ with $n_{h_k,\alpha}$ defined for the isolated singularities $\{h_k=0\}$ ($k\leqslant r_0$) as in (3.2.1).
In fact, we have the symmetries
$$n_{Z,\alpha}=n_{Z,n-1-\alpha},\quad n^1_{f,\alpha}=n^1_{f,n-\alpha}\quad(\alpha\in{\mathbf Q}),$$
and it is expected that the duality isomorphism in Theorem~1 is compatible with the filtration $V$ on $N_p$, $M''_p$ in an appropriate sense so that we have the equality
$$\dim{\mathcal G}r_V^{\alpha}N_p+\dim{\mathcal G}r_V^{n-1-\alpha}M''_{nd-p}=n_{Z,\alpha}\quad(\alpha\in{\mathbf Q},\,p\in{\mathbf Z}),
\leqno(5.9.3)$$
giving a refinement of Corollary~2.
Note that (5.9.2) for $\alpha=p/d$ is closely related with \cite{Kl}.
\par
If the above formulas hold, these would imply a refinement of Corollary~5.5 (and also its generalization to the case $n>3$ in \cite[Theorem~9]{DiSa3}).
However, it is quite nontrivial whether (5.9.2) holds, for instance, even for $p/d>\alpha$, since this is closely related to the independence of the $V$-filtrations associated to various singular points of $Z$.
In the case where the Newton boundaries of $f$ are non-degenerate, the formula for $M''_p$ with $\alpha\leqslant 1$ seems to follow from the theories of multiplier ideals and microlocal $V$-filtrations.
\end{document} |
\begin{document}
\title{Specifying Prior Distributions in Reliability Applications}
\begin{abstract}
Especially when facing reliability data with limited information (e.g.,
a small number of failures), there
are strong motivations for using Bayesian inference methods.
These include the option to use information
from physics-of-failure or previous experience with a failure mode
in a particular material to specify an informative
prior distribution. Another advantage is the ability
to make statistical inferences without
having to rely on specious (when the number of failures is small)
asymptotic theory needed to justify
non-Bayesian methods. Users of non-Bayesian methods are faced with
multiple methods of constructing uncertainty intervals (Wald,
likelihood, and various bootstrap methods) that can give
substantially different answers when there is little information in
the data. For Bayesian inference, there is only one method of
constructing equal-tail credible intervals---but
it is necessary to provide a prior
distribution to fully specify the model.
Much work has been done to find default prior distributions
that will provide inference methods with good (and in some cases
exact) frequentist coverage properties. This paper reviews some of
this work and provides, evaluates, and illustrates principled
extensions and adaptations of these methods to the practical
realities of reliability data (e.g., non-trivial censoring).
\end{abstract}
\begin{keywords}
Bayesian inference, Default prior, Few failures, Fisher information matrix,
Jeffreys prior, Noninformative prior, Reference prior
\end{keywords}
\section{Background and Motivating Examples}
\subsection{Bayesian Methods in Reliability Applications}
The use of probability plotting and maximum likelihood (ML) methods
for the analysis of censored reliability data has matured over the
past 30 years. These methods appear in numerous textbooks and are
readily available in several widely-used commercial statistical software
packages (e.g., JMP, MINITAB, and SAS). More recently,
commercial statistical software that provides capabilities for doing Bayesian
estimation has become available (e.g., JMP and SAS). Bayesian
estimation requires the specification of a joint prior distribution
for the model parameters. The purpose of this paper is to provide
guidance on how prior specification should be done in reliability
applications. We focus on applications requiring a single
distribution (e.g., Weibull or lognormal). The basic ideas, however, can
be applied to more complicated models, as described in our
concluding remarks section.
\subsection{Motivating Examples}
\label{section:motivating.examples}
\subsubsection{Bearing cage field data}
Figure~\ref{figure:BearingCage.plots}(a) is an event plot of
bearing-cage fracture times for six failed units as well as running
times for 1,697 units that had accumulated various amounts of service
time without failing. The data and an analysis appear in
\citet{Abernethyetl1983}.
\begin{figure}
\caption{Bearing cage event plot~(a) and Weibull probability plot~(b).}
\label{figure:BearingCage.plots}
\end{figure}
These data represent a population of units that had been introduced
into service over time and the data are multiply censored
(censoring at multiple points in time). There
were concerns about the adequacy of the bearing-cage design.
Analysts wanted to use these initial data to decide if a redesign
would be needed to meet the design-life specification.
This requirement was that the 0.10 quantile of bearing life
(sometimes referred to as
B10) be at least 8,000 hours.
Figure~\ref{figure:BearingCage.plots}(b) is a Weibull probability
plot. Because of the small number of
failures, the confidence interval for the fraction failing at 8,000
hours is wide and deciding whether the reliability
goal is being met is difficult. A likelihood ratio confidence
interval for the fraction failing at 8,000 hours is $[0.026, \,\,
0.9999]$, which is not useful.
\subsubsection{Rocket motor field data}
This example was first presented in \citet{OlwellSorell2001}. The
US Navy had an inventory of approximately 20,000 missiles. Each
included a rocket motor---one of five critical components. These
missiles were subject, over time, to unmonitored thermal cycling due
to environmental variability during, storage, transit, and
take-off-landing cycles.
Only 1,940
of the missiles had actually been put into use over a period of
time up to 18 years subsequent to their manufacture. At their time
of use, 1,937 of these motors performed satisfactorily; but there
were three catastrophic launch failures. Responsible scientists and
engineers believed that these failures were caused by the thermal
cycling. In particular, it was believed that the thermal cycling
resulted in failed bonds between the solid propellant and the
missile casing.
The failures raised concern about the previously unanticipated
possibility of a sharply increasing failure rate over time (i.e.,
rapid wearout) as the motors aged and were subjected to thermal
cycling while in storage. If this were indeed the case, a
possible---but costly---remedial strategy might be to
replace aged rocket motors with new ones. Thus, to assess the
magnitude of the problem, it was desired to quantify the rocket-motor
failure probability as a function of the amount of thermal cycling
to which a motor was exposed and to obtain appropriate confidence
bounds around such estimates based on the results for the 1,940
rocket motors---assuming these to be a random sample from the larger
population (at least concerning their failure-time
distribution).
Because no information was directly available on the thermal cycling
history of the individual motors, the age of the motor (i.e., time
since manufacture) at launch was used as a surrogate. This was not
an ideal replacement because the thermal cycling rate, or rate of
accumulation of other damage mechanisms, varied across the
population of motors, depending on an individual missile's
age and environmental storage history.
Compared to a scale based on the number of thermal cycles, the
effect of using time since manufacture
is to increase the variability in the observed failure-time
response, as described in \citet{MeekerEscobarHong2009}. The failure
probability 20 years after manufacture was of particular interest.
The specific age at failure of each of the three failed motors was
not known---all that was known was that failure, in each case, had
occurred sometime before the time of launch---thus making the time
since manufacture at launch left-censored observations of the actual
failure times. Similarly, the information of (eventual) failure age
for the 1,937 successful motors is right-censored---all that is
known is that the time to the yet-to-occur failure exceeds the
calendar age at the time of launch. Thus, the available
rocket-motor field-performance data, contained only left- and
right-censored observations---but \textit{no} known exact failure
times (such data are known as ``current-status data'').
Figure~\ref{figure:RocketMotor.plots.ps}(a) is an event
plot that further illustrates the structure of the data.
Because failure times are only bounded (no exact failure times)
and because of the
very small number of known failures, the amount of information in
the data is severely limited. Nevertheless, it is possible to
estimate the Weibull parameters from these
data. Figure~\ref{figure:RocketMotor.plots.ps}(b) is a Weibull
probability plot of the data. The ML
estimates of the Weibull parameters are $\widehat{\eta}=21.23$
years and $\widehat{\beta}=8.126$.
A likelihood ratio confidence interval for the
fraction failing at 20 years is $[0.023, \,\, 0.9999]$, which,
again, is not useful.
For most failure mechanisms operating in the field, the Weibull
shape parameter $\beta$ will be less than 4. Using the surrogate
years since manufacturer in place of the unknown number of thermal
cycles will further increase the relative variability in the data
which would make $\beta$ even smaller. Thus the estimate
$\widehat{\beta}=8.126$ was surprisingly large.
\begin{figure}
\caption{Rocket motor event plot~(a) and Weibull probability plot~(b).}
\label{figure:RocketMotor.plots.ps}
\end{figure}
\subsection{Literature Review}
Books that focus on Bayesian methods for reliability data analysis
include~\citet{MartzWaller1982}, \citet{Hamadaetal2008}, and
\citet{LiuAbeyratne2019}. \citet{SanderBadoux1991} contains six contributions
that describe early work on the application of Bayesian methods
in different reliability applications.
Numerous papers describing particular applications of
Bayesian methods have appeared in the engineering and statistical
literature over the past 30 years.
This paper, primarily, focuses on finding noninformative or other
default prior distributions for reliability applications using a
single distribution. As described more fully in
Section~\ref{section:noninformative.prior}, in many reliability
applications that we have encountered there is a need to be
minimally informative so that the prior choice can be easily
defended. Section~\ref{section:noninformative.prior.literature}
reviews some of the literature on \textit{noninformative} prior
distributions.
There has, however, been much work done on eliciting informative
priors for reliability applications based on expert judgment. Here
we review some of this work. \citet{Lindley1983} describes
analytical methods for combining information (either location or
both location and scale) from several different sources that might
include expert opinion or other information sources. The paper
demonstrates the advantages of using $t$ distributions instead of
normal distributions for the underlying model being used to combine
the information into one probability distribution. As we mention
in Section~\ref{section:motivation.for.partially.informative.prior},
we have found location-scale-$t$ distribution to be useful for this
purpose. \citet{LindleySingpurwalla1986} use the conceptual ideas
from \citet{Lindley1983} in the application of combining expert
opinions to quantify the reliability of a multicomponent parallel
redundant system. \citet{VanNoortwijk_etal_1992} describe
procedures for combining expert information, based on discretized
life distributions at the elicitation
stage. \citet{CampodonicoSingpurwalla1995} describe how to use prior
information for the intensity function of a nonhomogeneous Poisson
point processes and illustrate the approach using two reliability
applications. \citet{WallsQuigley2001} describe an elicitation
process that allows for the identification and management of expert
bias. They apply the methods to reliability growth modeling.
\citet{BedfordQuigleyWalls2006} review how expert knowledge is used
to make needed assessments of reliability in applications involving
engineering system design.
\citet{Gutierrez-PulidoAguirre-TorresAndres2005} suggest specifying
fully informative prior distributions for a two-parameter
distribution by specifying intervals for the mean and standard
deviation or two quantiles for the failure-time distribution.
\citet{KaminskiyKrivtsov2005} suggest, for a Weibull distribution,
specifying fully informative joint prior distributions for the
parameters by specifying priors for two points on the cdf of
the Weibull distribution. \citet{KrivtsovFrankstein2017} extend the
method to other failure-time distributions.
\citet{MeyerBooker2001} provides a book-length guide to methods and
procedures for eliciting information from experts without focus on
any particular area of application. \citet{OHagan_etal2006} provide
a more technical book-length guide for eliciting information from
experts.
\subsection{Overview}
The remainder of this paper is organized as follows.
Section~\ref{section:single.distribution.reliability.models}
provides a brief review and defines
notation for reliability models, censoring, and likelihood.
Section~\ref{section:bayesian.methods.prior.information} briefly
introduces the basic concepts of Bayesian inference and explains
motivation for and mechanics of needed reparameterization. The
commonly used noninformative prior distributions are derived from
the Fisher information matrix (FIM). Section~\ref{section:lls.fim}
describes how to obtain the elements of the FIM for different kinds
of censoring. Section~\ref{section:noninformative.prior} reviews
the commonly used noninformative prior distributions and describes
extensions to \textrm{Type~1}{} and \textrm{Type~2}{} censoring.
Section~\ref{section:random.censoring.ij.prior} explains how the
results in Section~\ref{section:noninformative.prior} can be applied
in situations where there is random censoring.
Section~\ref{section:weakly.informative.prior} explains the
importance of weakly informative prior distributions for some
applications and illustrates the use of noninformative and weakly
informative priors in the examples. In most reliability
applications, engineers will have strong prior information on only
one parameter (e.g., the Weibull shape parameter).
Section~\ref{section:combining.informative.with.noninformative}
shows how to combine prior information for one parameter with a
noninformative or weakly informative prior for the other parameter
and applies the ideas to the examples.
Section~\ref{section:weibull.type.one.simulation} describes a
simulation that was conducted to study the coverage probabilities of
credible intervals computed under different noninformative priors.
Section~\ref{section:prior.sensitivity.analysis} suggests and
illustrates methods of doing prior distribution sensitivity
analysis. Section~\ref{section:concluding.remarks} gives
concluding remarks and suggests extensions and areas for future research.
\section{Review of Single Distribution Reliability Models, Censoring, and Likelihood}
\label{section:single.distribution.reliability.models}
This section briefly reviews the commonly used models, censoring, and methods
for fitting a single distribution to reliability data.
\subsection{Log-Location-Scale Distributions}
The most frequently used distributions for failure-time data are in
the log-location-scale family of distributions. A random variable
$T>0$ belongs to the log-location-scale family if
$Y=\log(T)$ is a member of the location-scale family. The cdf
for a log-location-scale distribution is
\begin{align}
\label{equation:lls.cdf}
F(t;\mu, \sigma)&=\Phi\left [\frac{\log(t)-\mu}{\sigma}
\right ], \,\, t > 0
\end{align}
and the corresponding pdf is
\begin{align}
\label{equation:lls.pdf}
f(t;\mu, \sigma)&=\frac{1}{ \sigma t} \phi
\left [
\frac{\log(t)-\mu}{\sigma} \right],
\end{align}
where $\Phi(z)$ and $\phi(z)=d\Phi(z)/dz$ are, respectively,
the cdf and pdf for the
particular standard
location-scale distribution.
The most common log-location-scale distributions are the
lognormal ($\Phi(z)=\Phi_{\textrm{norm}}(z)$ is the standard normal cdf),
Weibull ($\Phi(z)=\Phi_{\textrm{sev}}(z)=1-\exp[-\exp(z)]$) , Fr\'{e}chet
($\Phi(z)=\Phi_{\textrm{lev}}(z)=\exp[-\exp(-z)]$), and loglogistic
distributions ($\Phi(z)=\Phi_{\textrm{logis}}(z)=1/[1+\exp(-z)]$).
The hazard function is important in reliability theory and
applications and is defined by
\begin{align*}
h(t)&=\lim_{\Delta t \rightarrow 0} \frac{\Pr(
t < T\le
t+\Delta t \mid T > t)}{\Delta t}
= \frac{f(t)}{1-F(t)}.
\end{align*}
The hazard function is proportional to the probability of failing in the next small
interval of time, conditional on having survived to the beginning of
that interval. That is, for
small $\Delta t$,
\begin{align*}
\Pr(t < T \le t+\Delta t \mid
T > t) \approx h(t) \times \Delta t .
\end{align*}
\subsubsection{Example 1: The lognormal distribution}
If $T$ has a lognormal distribution, then $Y=\log(T) \sim \textrm{NORM}(\mu,
\sigma)$, where $\mu$ is the mean and $\sigma$ is the standard deviation of
the underlying normal distribution. For the lognormal distribution,
$\sigma$ is the shape parameter and $\exp(\mu)$ is the median (and a
scale parameter).
\subsubsection{Example 2: The Weibull distribution}
The Weibull cdf is often given as
\begin{align}
\label{equation:bayes.weibull.cdf}
\Pr(T \leq t;\eta,\beta ) &= F(t;\eta,\beta)=1-
\exp \left [-\left (\frac{t}{\eta} \right )^{\beta}
\right ], \,\, t > 0,
\end{align}
where $\beta$ is a shape parameter and $\eta$ is a scale
parameter---sometimes called ``characteristic life'' and is
approximately the 0.63 quantile of the distribution.
The Weibull cdf can also be expressed by (\ref{equation:lls.cdf})
with the parameters $\mu=\log(\eta)$ and $\sigma=1/\beta$. Although
results (e.g., from software) are typically presented in the more
familiar $(\eta, \beta)$ parameterization, it is common practice to
use the $(\mu, \sigma)$ parameterization for the development of
theory and software for the entire (log-)location-scale families and
this is especially true for regression models like those used for
accelerated testing \citep[e.g.,][Chapters 18 and
19]{MeekerEscobarPascual2022}.
\subsection{Quantities of Interest in Reliability Data Analysis}
In reliability applications, the usual distribution parameters are not
of primary interest. Instead, there is generally a need to estimate
failure probabilities (computed using (\ref{equation:lls.cdf})) at a
specified time or a
failure-time distribution quantile. The $t_{p}$
quantile for a distribution in the
log-location-scale family can be expressed as
$ t_{p}=\exp[\mu + \Phi^{-1}(p)\sigma]$.
These quantiles also play an important role in prior elicitation, as
described in Section~\ref{section:reparameterization}.
\subsection{Censoring}
\label{section:censoring.types}
Censoring is ubiquitous in the analysis of reliability and other
time-to-event data.
There are different kinds of censoring that arise in applications.
\begin{itemize}[itemsep=1mm, parsep=0pt]
\item
Right censoring arises when one or more units have not failed when the
data are analyzed and occur for different reasons.
\begin{itemize}[itemsep=1mm, parsep=0pt]
\item
Time (\textrm{Type~1}{}) censoring arises in life tests where the test ends
at a specified time. The number of failures is random (and could be zero).
\item
Failure (\textrm{Type~2}{}) censoring arises in life tests where the test ends
after a specified number of units have failed. The length of the test
is random. Such tests are not common in practice because of the need to adhere
to schedules.
\item
Multiple right censoring (many different censoring times) is common in
field data. Differing censoring times arise from some combination of
staggered entry, differing use rates (when time is measured in
amount of use since entering service), and competing risks (e.g., failure
modes unrelated to the one of primary interest).
\end{itemize}
\item
Interval censoring arises when failures are found at inspection
times. All that is known is that a failure occurred between the
most recent previous and the current inspection.
\item
Left censored observations arise when a failure has already occurred
at the first time a unit is observed and is equivalent to an
interval-censored observation that has zero as its beginning time.
\end{itemize}
An assumption of noninformative censoring \citep[e.g.,][pages
59--60]{Lawless2003} is generally required to use the common
methods for analyzing censored data.
\subsection{Log-likelihood}
\label{section:lls.likelihood}
For
data consisting of $n$ independent and identically distributed
(iid) exact failure times and right-censored observations,
with no explanatory variables, the likelihood is
\begin{align}
\nonumber
{L}( \textrm{DATA} | \mu,\sigma) &= \prod_{i=1}^{n}
{L}(\textrm{data}_{i} | \mu,\sigma)
\\
\label{equation:log.location.scale.likelihood}
&= \prod_{i=1}^{n}
\left\{ \frac{1}{\sigma t_{i}} \, \phi
\left[\frac{ \log(t_{i}) -\mu}{\sigma}
\right]
\right\}^{\delta_{i}}
\times
\left\{1-\Phi \left[\frac{ \log(t_{i}) -\mu}{\sigma}
\right] \right\}^{1-\delta_{i}},
\end{align}
where, for observation $i$, $\textrm{data}_{i}=(t_{i}, \delta_{i})$, $t_{i}$
is either a failure time or a right-censored time, $\delta_{i}=1$
for an exact failure time, and $\delta_{i}=0$ for a right-censored
observation. For interval-censored observations with lower and
upper interval endpoints $t_{L,i}$ and $t_{U,i}$, the factor on the
left in (\ref{equation:log.location.scale.likelihood}) is replaced
by
\begin{align}
\label{equation:interval.censoring.contribution}
{L}( \textrm{DATA} | \mu,\sigma) &= \prod_{i=1}^{n}
\left\{\Phi \left[\frac{ \log(t_{U,i}) -\mu}{\sigma}\right] - \Phi
\left[\frac{ \log(t_{L,i}) -\mu}{\sigma}\right] \right\}^{\delta_{i}}
\times
\left\{1-\Phi \left[\frac{ \log(t_{i}) -\mu}{\sigma}
\right] \right\}^{1-\delta_{i}}.
\end{align}
For a left-censored observation (an interval that starts at zero),
$\Phi[(\log(t_{L,i}) -\mu)/\sigma]$ in
(\ref{equation:interval.censoring.contribution}) is replaced by zero. For
more information about likelihoods for censored data, see
Chapters 2, 7, and 8 in \citet{MeekerEscobarPascual2022}. For many
purposes (e.g., computational and for the development of theory),
it is convenient to use the log-likelihood ${\cal L}( \textrm{DATA} | \mu,\sigma) =
\log[{L}( \textrm{DATA} | \mu,\sigma)]$.
\section{Using Bayesian Methods and Prior Information in Reliability
Applications}
\label{section:bayesian.methods.prior.information}
As we saw in the motivating examples in
Section~\ref{section:motivating.examples}, the data in reliability
applications often have few failures and thus contain little
information. Engineers, however, often have additional useful, but
imprecise information that can be combined with the limited
data. For example, if failures are caused by a wearout mechanism,
the hazard function would be increasing and thus the Weibull shape
parameter would be greater than one. Previous experience with
a particular failure mechanism in the same material may allow
bounding a Weibull or lognormal shape parameter more precisely.
On the other hand,
there may not be information that can be used to set an informative
prior distribution on a scale parameter. In such cases, the
informative prior for the shape parameter needs to be used in
conjunction with a noninformative or weakly information prior for
the scale parameter.
\subsection{Bayes' Theorem}
Bayes' theorem for continuous parameters in ${\boldsymbol{\theta}}$ can be
written as
\begin{equation}
\label{equation:bayes.theorem}
\pi({\boldsymbol{\theta}} | \textrm{DATA})=
\frac{{L}(\textrm{DATA} | {\boldsymbol{\theta}} ) \pi({\boldsymbol{\theta}}) }
{ \int {L}(\textrm{DATA} | {\boldsymbol{\theta}}^{\prime} ) \pi({\boldsymbol{\theta}}^{\prime} ) d {\boldsymbol{\theta}}^{\prime} }
\end{equation}
where the joint prior distribution $\pi({\boldsymbol{\theta}})$ quantifies the
available prior information about the unknown parameters in
${\boldsymbol{\theta}}$. The output of (\ref{equation:bayes.theorem}) is
$\pi({\boldsymbol{\theta}}|\textrm{DATA})$, the joint posterior distribution for
${\boldsymbol{\theta}}$, reflecting knowledge of ${\boldsymbol{\theta}}$ after the
information in the data and the prior distribution have been
combined. For (log-)location-scale distributions used here,
${\boldsymbol{\theta}}=(\mu, \sigma)$.
\subsection{Reparameterization}
\label{section:reparameterization}
In many situations, it is important to replace the traditional
parameters (e.g., $\mu$ and $\sigma$ for log-location-scale distributions)
with alternative parameters. For a
log-location-scale distribution, it is useful to replace the usual scale
parameter $\exp(\mu)$ with a particular quantile $t_{p_{r}}$ as an
alternative scale parameter for a value of $p_{r}$ that is chosen in
a purposeful manner. Doing this has important advantages for the
following reasons.
\begin{enumerate}[itemsep=1mm, parsep=0pt]
\item
Elicitation of a prior distribution is facilitated because the parameters have
practical interpretations and are familiar to practitioners.
\item
When prior knowledge is accumulated based on experience involving
heavy censoring the traditional parameters $\mu$ and $\sigma$ would
have strong correlation. Using $t_{p_{r}}$ and $\sigma$ for a
suitably specified value of $p_{r}$ would allow specifying the joint prior
density as a product of marginal densities.
\item
The numerical performance of MCMC algorithms is generally better when
strong correlation is avoided.
\end{enumerate}
A useful reparameterization for the Weibull distribution replaces
$\eta$ with a particular distribution quantile that could be
suggested by available data. For
heavily right-censored data from a high-reliability
product, this would be a lower-tail quantile of the failure-time
distribution. For example, if in certain applications one typically
sees 10\% of a population of units failing, then something like the
$0.05$ quantile would be a more appropriate scale parameter. Also,
it is easier to elicit prior information for such a small quantile
compared to the time at which a proportion 0.63 would fail.
The $p_{r}$ quantile of the Weibull
distribution is $t_{p_{r}}=\eta \left
[-\log(1-p_{r})\right ]^{1/\beta}$. Replacing $\eta$ with the
equivalent expression $\eta=t_{p_{r}}/[-\log(1-p_{r})]^{1/\beta}$ in
(\ref{equation:bayes.weibull.cdf}) provides a reparameterized version of
the Weibull distribution:
\begin{align}
\nonumber
\Pr(T \leq t;t_{p_{r}},\beta ) = F(t;t_{p_{r}},\beta)&=1-
\exp \left [-\left (\frac{t}{t_{p_{r}}/[-\log(1-p_{r})]^{1/\beta}} \right )^{\beta}
\right ]\\[0.5ex]
\label{equation:reparameterized.weibull.cdf}
&=1-\exp\left[\log(1-p_{r})\left(\frac{t}{t_{p_{r}}}\right)^{\beta}\right],
\,\, t > 0.
\end{align}
The latter expression shows that $t_{p_{r}}$ is an alternative scale parameter.
Especially when there is heavy censoring (i.e., only a small
fraction failing), estimation of $(t_{p_{r}}, \beta)$ will be more
stable than estimating $(\eta, \beta)$, for some appropriately
chosen value of $p_{r}$. Thus one could choose $p_{r}$, based on the
data, by taking the largest value of the nonparametric estimate of
$F(t)$ and dividing it by two, assuring that the parameter $t_{p_{r}}$
is within the data. Another alternative is to choose $p_{r}$ based
on engineering knowledge that would allow elicitation of an
informative or weakly informative prior distribution for
$t_{p_{r}}$. Note that it is possible to have two separate
reparameterizations (one for estimation and one for elicitation), as
a prior with one parameterization can be easily translated into a
prior for another parameterization. Usually, however, the
elicitation-motivated reparameterization will be sufficient for both
purposes.
Figure~\ref{figure:likelihood.contour.plots.reparameterization}
illustrates such reparameterizations for the bearing-cage and
rocket-motor data. For these examples, $p_{r}$ was chosen to provide a
well-behaved likelihood surface. The lognormal and other
log-location-scale distributions can be similarly reparameterized.
\begin{figure}
\caption{Likelihood contour plots for the bearing-cage (top) and
rocket motor data (bottom) in the traditional
$(\eta, \beta)$ parameterization (left) and in the
$(t_{0.005}
\label{figure:likelihood.contour.plots.reparameterization}
\end{figure}
\section{The Fisher Information Matrix}
\label{section:lls.fim}
As described in Section~\ref{section:noninformative.prior} and
Section~\ref{S.section:derivations.noninformative.priors} of the
appendix, the Fisher information matrix (FIM)
is used to define certain noninformative prior
distributions. This section shows how to compute the FIM for
(log-)location-scale distributions and different kinds of censoring.
\subsection{Scaled Fisher Information Matrix Elements}
\label{section:scaled.fim.elements}
The \textit{scaled} FIM elements are:
\begin{align}
\begin{split}
\label{equation:lls.fim}
f_{11}(z_{c}) &= \frac{\sigma^2}{n} E\left [-\frac{\partial^2 \log {L}(\mu,\sigma) }{\partial\mu^2} \right]
\\[2ex]
f_{12}(z_{c}) &= \frac{\sigma^2}{n} E\left
[-\frac{\partial^2\log {L}(\mu,\sigma) }{\partial\mu\partial\sigma} \right]
\\[2ex]
f_{22}(z_{c}) &= \frac{\sigma^2}{n} E\left
[-\frac{\partial^2 \log {L}(\mu,\sigma) }{\partial \sigma^2}\right ].
\end{split}
\end{align}
The $\sigma^2/n$ term cancels with a term $n/\sigma^2$ arising from
the expectations in (\ref{equation:lls.fim}) and thus these scaled
FIM elements depend only on $z_{c}$ and the assumed distribution (but
not on $n$ or $\sigma$).
For \textrm{Type~1}{} (time) censoring, $z_{c} = [\log(t_{c})-\mu]/\sigma$ is a
standardized censoring time where $t_{c}$ is the censoring time and
$p_{c}=\Phi(z_{c})$ is the expected fraction failing. For \textrm{Type~2}{} (failure)
censoring $z_{c} = \Phi^{-1}(r/n)$ where $r/n$ is the given fraction
failing. An algorithm to compute these elements is given by
\citet{EscobarMeeker1994} and implemented in the
function \texttt{lsinf} in the R package
\texttt{lsinf} \citep{Meeker2022}.
\subsection{Fisher Information for \textrm{Type~1}{} and \textrm{Type~2}{} Censored Samples}
\label{section:fisher.typeI.typeII.censored}
For a sample of $n$ iid observations, singly censored (i.e., all
censoring is at one time point) at time
$t_{c}$, the FIM for $(\mu, \sigma)$ is
\begin{align}
\label{equation:fisher-mu-sigma}
I_{(\mu, \sigma)} &= \frac{n}{\sigma^{2}}
\left[\begin{array}{lr}
f_{11}(z_{c}) & f_{12}(z_{c})\\[1ex]
\textrm{symmetric} & f_{22}(z_{c})
\end{array}\right].
\end{align}
With \textrm{Type~2}{} censoring, the scaled FIM $(\sigma^{2}/n)\, I_{(\mu,
\sigma)}$ depends on $r$, the known number of failures, and has no
unknown parameters and thus does not depend on any unknown parameters. With
\textrm{Type~1}{} censoring, the scaled FIM depends on $p_{c}=F(t_{c}; \mu, \sigma)$, the
unknown expected fraction failing at time $t_{c}$.
To simplify the presentation in the remainder of this
paper, when it is possible, we will suppress the dependency of the
$f_{ij}$ elements on $z_{c}$. That is, for example, we write
$f_{11}$ instead of $f_{11}(z_{c})$.
\subsection{Fisher Information Matrix for Randomly Censored Samples}
\label{section:fisher.random.censored}
Random censoring arises for different reasons such as staggered
entry, differing use rates in the population, and competing
risks. The competing risk model provides a convenient model to
describe or characterize random censoring. Suppose that $T$ is a
random failure time having a log-location-scale distribution with
parameters $(\mu, \sigma)$ and $C$ is a random censoring time for a
unit. Then the unit fails if $T \leq C$ and is censored if $T >
C$. If $\log(C)$ has a pdf $h(x)$, then, using conditional
expectation, as outlined in \citet{EscobarMeeker1998}, the FIM for a
sample of size $n$ is
\begin{align}
\label{equation.fim.random.censoring}
I_{(\mu, \sigma)} &= \frac{n}{\sigma^{2}}
\left[\begin{array}{lr}
\int_{-\infty}^{\infty}f_{11}(w)h(x)dx & \int_{-\infty}^{\infty}f_{12}(w)h(x)dx\\[1ex]
\textrm{symmetric} & \int_{-\infty}^{\infty}f_{22}(w)h(x)dx
\end{array}\right],
\end{align}
where $w=(x-\mu)/\sigma$.
\section{Noninformative Prior Distributions for Log-Location-Scale Distributions}
\label{section:noninformative.prior}
\subsection{Motivation}
In many (if not most) applications of Bayesian methods, there is a
desire to analyze data without using any information that might be
available to specify a prior distribution (sometimes known as
objective-Bayesian analysis). This would be the case, for example,
when interested parties might not agree on a subjective prior (e.g.,
engineers and managers or manufacturers and consumers) or when there
is a need to avoid having to defend an assumed prior distribution
(e.g., in legal or regulatory proceedings).
Practitioners may be unable to specify their belief using a
probability distribution due to lack of statistical expertise thus
requiring the use of a default prior.
In these situations, an
alternative is to use what is generically called a noninformative
prior distribution. Several different methods for specifying a
noninformative prior distribution have been suggested.
\subsection{Previous Work on Noninformative Prior Distributions}
\label{section:noninformative.prior.literature}
There is extensive literature concerning noninformative prior
distributions. Here we mention work most closely related to ours.
\citet{Bernardo1979} reviews the early literature and describes
criteria and methods for constructing reference prior
distributions. \citet{BergerBernardo1992a} extend previous work on
reference prior distributions, focusing on multiparameter models and
recommend a specification of order, based on parameter
importance. \citet{BergerBernardo1992b} provide an updated
literature review of this area and describe a general algorithm for
finding a reference prior for continuous multiparameter models with
a given parameter ordering. \citet{Sun1997} reviews earlier work
showing various conditions for second-order and third-order
probability matching
priors for two-parameter distributions and applies these results to
the Weibull distribution demonstrating that certain reference priors
meet the conditions. \citet{SunBerger1998} consider informative
priors when there is external information (e.g., for only one
parameter), with the idea of using a reference prior distribution
for other parameters, similar to what we suggest in
Section~\ref{section:combining.informative.with.noninformative}.
\citet{AbbasTang2015,AbbasTang2016} describe reference prior
distributions for the Fr\'{e}chet and loglogistic distributions.
\citet[Chapter 5]{GhoshDelampadyTapas2006} provide a summary of
methods and operational details for obtaining noninformative prior
distributions.
\subsection{Parameterization for Prior Distributions}
\label{subsection:param.for.priors}
In our log-location-scale distribution examples,
when specifying a prior distribution we will use the
parameterization $(t_{p_{r}}, \sigma)$ (or $(t_{p_{r}}, \beta=1/\sigma)$ for
the Weibull distribution) because these are the parameters that have
a practical interpretation, allowing elicitation of informative or
specifying weakly informative prior distributions, when needed. When using
algorithms to compute MCMC draws, however, we use the
$(y_{p_{r}}=\log(t_{p_{r}}), \log(\sigma))$ parameterization
because the expressions for the priors
tend to be simpler, MCMC algorithms work better in the unconstrained
parameter space, and plots of the posterior draws tend to be easier to
interpret on the log scales. Prior distributions for $(t_{p_{r}},
\sigma)$ are easily translated into priors for $(\log(t_{p_{r}}),
\log(\sigma))$. Examples are given in
Section~\ref{S.section:derivations.noninformative.priors} of the
appendix.
\subsection{A Fundamental Principle for Specifying Noninformative
Prior Distributions}
\label{section:noninformative.prior.fundamental.principle}
There is an important fundamental principle for specifying
noninformative prior distributions in situations where there is only a
small amount of information in the data corresponding to the desired
inference(s). The prior should put negligible
density in parts of the parameter space that are impossible or
clearly implausible and that would otherwise lead to nonnegligible posterior
probability in such parts of the parameter space. This is the
often-stated justification for the use of weakly informative priors
(Section~\ref{section:weakly.informative.prior}). For example, using
a prior $\pi[\log(t_{p_{r}}), \log(\sigma)]\propto 1$ (also known as
``flat'') in
situations with a small number of failures can result (because of
the diffuseness of the likelihood) in non-negligible posterior
probabilities in nonsensical regions of the parameter space.
We have developed our recommendations (summarized in
Section~\ref{section:recommended.lls.prior})
to be consistent with this principle.
\subsection{Jeffreys Prior Distributions}
\label{section:jeffreys.prior.distribution}
The Jeffreys prior can be derived as being proportional to
the square root of the determinant of the FIM
(defined in Section~\ref{section:fisher.typeI.typeII.censored}).
For models with one parameter (e.g., the exponential distribution or
the normal distribution with known standard deviation), the Jeffreys
prior distribution has been shown to provide results (e.g.,
credible, tolerance, and prediction intervals) that are the same as
or close to classical non-Bayesian methods for certain models.
The Jeffreys prior, even if there is more than one parameter, is
invariant to reparameterization. This means if you use
the square root of the determinant of the FIM
definition for a given parameterization you will get a Jeffreys
prior. For any other parameterization, the Jeffreys prior can be
obtained by finding the square root of the determinant
of the FIM in the new parameterization or by
using the transformation of a random variable method.
That is, either method will lead to the same Jeffreys prior distribution.
From (\ref{equation:lls.fim}),
the Jeffreys prior is defined as
\begin{equation}
\label{equation:jeffreys-mu-sigma}
\pi(\mu,\sigma)\propto\sqrt{\left|\text{I}_n(\mu,\sigma)\right|}=\frac{n}{\sigma^2}\sqrt{f_{11}f_{22}-f_{12}^2}.
\end{equation}
Here the $f_{ij}$ values are scaled elements of the FIM defined in
Section~\ref{section:scaled.fim.elements}, which depend on the
standardized censoring time $z_{c}$. For \textrm{Type~2}{} censoring or
complete data, $z_{c}=\Phi^{-1}(r/n)$ is a known constant, so that
the Jeffreys prior is $\pi(\mu,\sigma)\propto{1}/{\sigma^2}.$ Then,
using transformation of random variables, the Jeffreys prior for
the unrestricted parameterization is $\pi[\mu,\log(\sigma)]\propto1/\sigma$
for \textrm{Type~2}{} censoring (or complete data).
The priors for \textrm{Type~1}{} censoring are more complicated because the
$f_{ij}$ elements depend on the unknown parameters through
$z_{c}=(\log(t_{c})-\mu)/\sigma$ (more specifically, they depend on
$p_{\textrm{fail}}=\Phi(z_{c})$, the unknown expected fraction failing at
the censoring time $t_{c}$). For \textrm{Type~1}{} censoring, the Jeffreys
prior is
\[
\pi(\mu,\sigma)\propto\frac{1}{\sigma^2}\sqrt{f_{11}f_{22}-f_{12}^2}.
\]
Again, for the unrestricted parameterization, the \textrm{Type~1}{} censoring
Jeffreys prior is
\begin{align*}
\pi[\mu, \log(\sigma)] & \propto
\frac{1}{\sigma}\sqrt{f_{11}f_{22}-f_{12}^2}.
\end{align*}
For (log-)location-scale distributions (and other distributions with
more than one parameter), the Jeffreys priors have well-known
deficiencies \citep[][page 182]{Jeffreys1961}. Also, in models with
more than one parameter, Jeffreys priors
may not have the desirable classical properties of agreeing with reference
priors that are probability matching \citep[e.g.,][]{Sun1997}.
\subsection{Independence Jeffreys Prior Distributions}
\label{section:ij.prior}
The independence Jeffreys (IJ) prior (also
known as the modified Jeffreys prior) is obtained by finding the
Conditional Jeffreys (CJ) prior for each parameter, assuming that it is the only
unknown parameter, and then using the product of these conditional
priors as the joint prior, as if the parameters were independent
random variables (but notably, they are not independent).
In contrast to the Jeffreys prior, for
(log-)location-scale distributions,
the IJ prior distribution has an appealing
property. In particular, it provides,
for complete and \textrm{Type~2}{} censored data, the same exact inferences
(i.e., the credible/confidence intervals procedures with
coverage probabilities that are
the same as the nominal credible/confidence level) as
non-Bayesian pivotal-based methods (and approximately the same for
other kinds of censoring). This result is given for complete data in
\citet{DiCiccioKuffnerYoungAlastair2017} but is also true for
\textrm{Type~2}{} censored data \citep[e.g., page 565 in][]{Lawless2003}.
\subsubsection{\textrm{Type~2}{} censoring IJ prior distributions}
For
\textrm{Type~2}{} censoring or complete data, because $f_{11}$ and $f_{22}$
are known constants, the CJ prior for $\mu$ is
$\pi(\mu|\sigma)\propto(1/\sigma)\sqrt{f_{11}}\propto1$. The
CJ prior for $\sigma$ is
$\pi(\sigma|\mu)\propto(1/\sigma)\sqrt{f_{22}}\propto1/\sigma$. So,
the IJ prior for $(\mu,\sigma)$ under \textrm{Type~2}{} censoring is
$\pi(\mu,\sigma)\propto\pi(\mu|\sigma)\pi(\sigma|\mu)\propto1/\sigma$.
Again, using transformation of random variables, the CJ prior for
$\log(t_{p_{r}})$ given $\log(\sigma)$ is
$\pi[\log(t_{p_{r}})|\log(\sigma)]\propto 1$. The CJ prior for
$\log(\sigma)$ given $\log(t_{p_{r}})$ is $\pi[\log(\sigma)
|\log(t_{p_{r}})] \propto 1$. Thus the IJ prior is
$\pi[\log(t_{p_{r}}),\log(\sigma)]\propto 1$, which we (following
common usage) call ``flat.''
\subsubsection{\textrm{Type~1}{} censoring IJ prior distributions}
For \textrm{Type~1}{} censoring, the CJ prior for $\mu$, when $\sigma$ is a known
constant, is
$\pi(\mu|\sigma)\propto(1/\sigma)\sqrt{f_{11}}\propto\sqrt{f_{11}}$
and the CJ prior for $\sigma$ is
$\pi(\sigma|\mu)\propto(1/\sigma)\sqrt{f_{22}}$. So, the
IJ prior for $(\mu,\sigma)$ is
$$
\pi(\mu,\sigma)\propto\frac{1}{\sigma}\sqrt{f_{11}f_{22}}.
$$
Then,
using transformation of random variables, the
CJ prior for $\mu$ given $\log(\sigma)$ is
$\pi[\mu|\log(\sigma)] \propto \sqrt{f_{11}}$. The CJ
prior for $\log(\sigma)$ given $\mu$ is
\begin{align*}
\pi[\log(\sigma) | \mu] &\propto
\sqrt{f_{22}}.
\end{align*}
So, the IJ prior is
$\pi[\mu, \log(\sigma)] \propto \sqrt{f_{11}f_{22}}$.
\subsubsection{Results for other parameterizations}
The FIM for the parameterization $(\log(t_{p_{r}}),\log(\sigma))$ is
easily obtained by using the delta method on the inverse of the FIM
in the $(\mu, \sigma)$ parameterization given in
(\ref{section:fisher.typeI.typeII.censored}). Details are given in
Appendix
Section~\ref{section:priors.y.log.sigam.parameterization}. Then
expressions for the Jeffreys, CJ, and IJ priors are obtained by
following the same steps described earlier in this section and
leads, for example, to the IJ prior
\begin{align}
\nonumber
\pi[\log(t_{p_{r}}),\log(\sigma)] &\propto
\pi[\log(t_{p_{r}})|\log(\sigma)] \times \pi[\log(\sigma)|\log(t_{p_{r}})]\\[1ex]
\label{equation:IJ.typeI.yp.log.sigma}
& \propto
\sqrt{f_{11}\left\{f_{11}[\Phi^{-1}(p_{r})]^2-2f_{12}\Phi^{-1}(p_{r})+f_{22}\right\}},
\end{align}
where, again, the $f_{ij}$ values are scaled elements of the FIM
defined in Section~\ref{section:scaled.fim.elements}, which depend
on the standardized censoring time $z_{c}=[\log(t_{c})-\mu]/\sigma$
or $p_{c}=\Phi(z_{c})$.
Relative to flat priors, these IJ priors tend to be more
consistent with the fundamental principle
described in
Section~\ref{section:noninformative.prior.fundamental.principle},
and provide advantages over traditional noninformative priors. This is
explained in Section~\ref{section:implementing.ij.priors} and
demonstrated in Section~\ref{simulation.results.conclusions}.
\subsection{Reference Prior Distributions}
\label{section:reference.prior.distributions}
Another well-studied approach to finding a noninformative prior
distribution is to use a reference prior (e.g., \citet{Bernardo1979}
and \citet{BergerBernardo1992a}). Generally, a reference
prior is the prior distribution that maximizes the
Kullback-Leibler divergence
between the prior and the
expected posterior distribution.
An \textit{ordered reference prior} specifies the order of
importance of the parameters, and different reference priors can
arise, depending on the specified order of
importance of the parameters (or functions of the
parameters).
Interestingly, certain definitions of reference priors
lead exactly to the IJ priors (and others lead to the Jeffreys
prior).
As an example, with \textrm{Type~2}{} censoring or complete data, the IJ
prior for the log-location-scale distribution parameters $(t_{p_{r}},
\sigma)$ is (proportional to) $1/(t_{p_{r}}\sigma)$ and for
the parameterization $(\log(t_{p_{r}}), \log(\sigma))$, the IJ prior is flat (i.e.,
uniform over the entire $(\log(t_{p_{r}}), \log(\sigma))$ plane for any
$p_{r}$). As shown in
Section~\ref{S.section.parameterization.tp.sigma} of the
appendix, these IJ priors are also reference priors when
either $t_{p_{r}}$ or $\sigma$ is the parameter of first importance.
As described in
Section~\ref{S.section:reference.priors.for.typeI.censoring} of the
appendix, expressions for computing reference priors
are not readily available for \textrm{Type~1}{} censoring. However, given the
equivalence of IJ and ordered reference priors described above for
\textrm{Type~2}{} censoring, we expect that the IJ priors for \textrm{Type~1}{}
censoring (Section~\ref{section:ij.prior}) would provide a good
approximation to the corresponding reference priors.
\subsection{Improper Priors and Posteriors}
\label{section:improper.with.few.failures}
Noninformative priors are generally improper (i.e. their integrals
over the parameter space are not finite). Thus, when using such prior
distributions (e.g., flat, IJ, or CJ combined with a proper marginal
for the other parameter), it is important to assure that there is a
sufficient amount of information in the data that the posterior will
be proper. \citet{RamosRamosLouzada2020} give conditions under which
a Weibull posterior distribution will be proper for some improper
priors, suggesting that having two failures is sufficient to result
in a proper posterior. We did extensive numerical experiments with
flat and IJ prior distributions and simulated \textrm{Type~1}-censored
samples with different numbers of failures. These experiments
indicated that the posterior is proper and reasonably well behaved
when there are at least three failures. With only two failures and
a flat prior, however, even when an appropriate parameterization was
used, we were not able to find a stable sampler. For example, the
sampler would often return large numbers or infinite values of the
parameters, perhaps due to limitations in computer floating-point
representation of real numbers (we also encountered infinite values
in our experiments with three failures, but their occurrence was
relatively rare).
In summary, even if a proper posterior is assured theoretically with
two failures, there may be practical difficulties.
\subsection{A Summary of Noninformative Priors for
(Log-)Location-Scale Distributions}
\label{section:summary.of.noninformative.priors}
Table~\ref{table:summary.noninformative.priors} summarizes Jeffreys,
IJ, and reference noninformative prior distributions for
(log-)location-scale distributions using different
parameterizations.
Section~\ref{S.section:derivations.noninformative.priors} of the
appendix gives derivations of these priors.
The relationships and correspondences (some of which have been noted
previously in the literature) are interesting. For example,
\begin{itemize}[itemsep=1mm, parsep=0pt]
\item
For all parameterizations, the usual Jeffreys prior is the same as
the reference prior when no parameter-importance is specified
\citep[as noted by][page 1,350]{KassWasserman1996}.
\item
When the distribution parameters are defined as the shape parameter
$\sigma$ and a scale parameter (e.g., the traditional $\exp(\mu)$ or
the $p$ quantile $t_{p}$ for any $p$), the ordered reference prior
is the same as the the IJ prior, irrespective of the ordering. This
equivalence property also holds for one-to-one functions of the
individual parameters (e.g., when $\sigma$ is replaced by
$\log(\sigma)$).
\item
Related to the previous point, although IJ priors for
log-location-scale distributions are not in general invariant to
reparameterization (in the sense described in
Section~\ref{section:jeffreys.prior.distribution}), they are
invariant to one-to-one monotone reparameterizations of either or
both of the $(t_{p},\sigma)$ parameters. The proof of this result is
given in
Section~\ref{S.section:invariant.one.one.reparameterization} of the
appendix.
\item
This IJ/ordered-reference equivalence property does \emph{not} hold
when the distribution parameters are defined as the shape parameter
$\sigma$ and $\zeta_{e}=[\log(t_{e})-\mu)/\sigma]$ and $\sigma$ is
in second order.
\end{itemize}
\begin{sidewaystable}
\caption{Improper noninformative joint prior distributions
$\pi(\theta_{1},\theta_{2})$ for different parameterizations for
log-location-scale distributions based on \textrm{Type~2}{} and \textrm{Type~1}{}
censored data
\label{table:summary.noninformative.priors}}
\begin{tabular}{lcc}
\toprule
& \multicolumn{2}{c}{Type of censoring} \\
\cmidrule(l{3em}r{5.5em}){2-3}
Prior type and parameterization & \textrm{Type~2}{} & \textrm{Type~1}{} \\
\midrule
{\small Jeffreys ($\mu, \sigma$)} & {\small $1/\sigma^{2}$} & {\small $(1/\sigma^2)\sqrt{f_{11}f_{22}-f_{12}^2}$} \\[0.5ex]
{\small Jeffreys ($\log(t_{p_{r}}), \sigma$)} & {\small $1/\sigma^{2}$} & {\small $(1/\sigma^2)\sqrt{f_{11}f_{22}-f_{12}^2}$} \\[0.5ex]
{\small Jeffreys ($t_{p_{r}}, \sigma$)} & {\small $1/(t_{p_{r}}\sigma^{2})$ } & {\small $1/(t_{p_{r}}\sigma^2)\sqrt{f_{11}f_{22}-f_{12}^2}$ } \\[0.5ex]
{\small Jeffreys ($\log(t_{p}), \log(\sigma)$)} & {\small $1/\sigma$} & {\small $(1/\sigma)\sqrt{f_{11}f_{22}-f_{12}^2}$} \\[0.5ex]
{\small Jeffreys ($\zeta_{e}, \sigma$)} &{\small $1/\sigma$} & {\small $(1/\sigma)\sqrt{f_{11}f_{22}-f_{12}^2}$} \\[0.5ex]
{\small Jeffreys ($\zeta_{e}, \log(\sigma)$)} &{\small 1} & {\small $\sqrt{f_{11}f_{22}-f_{12}^2}$} \\[0.5ex]
{\small IJ ($\mu, \sigma$)} & {\small $1/\sigma$} & {\small $(1/\sigma)\sqrt{f_{11}f_{22}}$} \\[0.5ex]
{\small IJ ($\log(t_{p_{r}}), \sigma$)} & {\small $1/\sigma$} & {\small $(1/\sigma)\sqrt{f_{11}\left\{f_{11}[\Phi^{-1}(p_{r})]^2-2f_{12}\Phi^{-1}(p_{r})+f_{22}\right\}}$} \\[0.5ex]
{\small IJ ($t_{p_{r}}, \sigma$)} & {\small $1/(t_{p_{r}}\sigma)$ } & {\small $1/(t_{p_{r}}\sigma)\sqrt{f_{11}\left\{f_{11}[\Phi^{-1}(p_{r})]^2-2f_{12}\Phi^{-1}(p_{r})+f_{22}\right\}}$ } \\[0.5ex]
{\small IJ ($\log(t_{p_{r}}), \log(\sigma)$)} & {\small 1} & {\small $\sqrt{f_{11}\left\{f_{11}[\Phi^{-1}(p_{r})]^2-2f_{12}\Phi^{-1}(p_{r})+f_{22}\right\}}$} \\[0.5ex]
{\small IJ ($\zeta_{e}, \sigma$)} & {\small $1/\sigma$} & {\small $(1/\sigma)\sqrt{f_{11}[f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}]}$} \\[0.5ex]
{\small IJ ($\zeta_{e}, \log(\sigma)$)} & {\small 1} & {\small $\sqrt{f_{11}[f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}]}$} \\[0.5ex]
{\small Reference $(\log(t_{p_{r}}), \sigma)$} & {\small $1/\sigma^{2}$} & \\[0.5ex]
{\small Reference $(\{\log(t_{p_{r}}), \sigma\})$} & {\small $1/\sigma$} & \\[0.5ex]
{\small Reference $(\{\sigma, \log(t_{p_{r}}) \})$} & {\small $1/\sigma$} & \\[0.5ex]
{\small Reference $(t_{p_{r}}, \sigma)$} & {\small $1/(t_{p_{r}}\sigma^{2})$ } & \\[0.5ex]
{\small Reference $(\{t_{p_{r}}, \sigma\})$} & {\small $1/(t_{p_{r}}\sigma)$ } & \\[0.5ex]
{\small Reference $(\{\sigma, t_{p_{r}}\})$} & {\small $1/(t_{p_{r}}\sigma)$ } & \\[0.5ex]
{\small Reference $(\log(t_{p_{r}}), \log(\sigma)) $} & {\small $1/\sigma$ } & \\[0.5ex]
{\small Reference $( \{\log(t_{p_{r}}), \log(\sigma) \})$} & {\small 1} & \\[0.5ex]
{\small Reference $(\{\log(\sigma), \log(t_{p_{r}}) \} )$} & {\small 1} & \\[0.5ex]
{\small Reference $(\zeta_{e}, \sigma)$} &{\small $1/\sigma$} & \\[0.5ex]
{\small Reference $(\{\zeta_{e}, \sigma\})$} &{\small $\left[\sigma\sqrt{f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}} \right]^{-1}$} & \\[0.5ex]
{\small Reference $(\{\sigma, \zeta_{e}\})$} &{\small $1/\sigma$}& \\[0.5ex]
{\small Reference $(\zeta_{e}, \log(\sigma))$} & {\small 1} & \\[0.5ex]
{\small Reference $(\{\zeta_{e}, \log(\sigma)\})$} & {\small $\left[\sqrt{f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}} \right]^{-1}$} & \\[0.5ex]
{\small Reference $(\{\log(\sigma), \zeta_{e}\})$} & {\small 1} & \\[0.5ex]
\bottomrule
\end{tabular}
\begin{tablenotes}
\item[1] The scaled FIM elements $f_{11}$,
$f_{12}$, and $f_{22}$ depend on the standardized censoring time
$z_{c}=[\log(t_{c})-\mu]/\sigma$. Parameters shown within $\{\dots\}$
indicate parameter-importance order, if there is one. The parameter
$\zeta_{e}=[\log(t_{e})-\mu)/\sigma]$ where $t_{e}$ is the time at which a
failure probability is to be estimated. The three reference
priors for $(\mu, \sigma)$ are exactly the same as for
$(\log(t_{p_{r}}), \sigma)$ and are thus not presented here.
\end{tablenotes}
\end{sidewaystable}
\subsection{Implementing and Interpreting the IJ Prior Distributions}
\label{section:implementing.ij.priors}
Implementing the IJ priors (e.g.,
$\pi(\log(t_{p_{r}}), \log(\sigma)) \propto 1$ or
$\pi(\log(t_{p_{r}}), \sigma) \propto 1/\sigma$) that arise with
complete data or \textrm{Type~2}{} censoring is straightforward.
For \textrm{Type~1}{} censoring we describe
the IJ prior for $(\log(t_{p_{r}}),\log(\sigma))$
in (\ref{equation:IJ.typeI.yp.log.sigma}) as an example.
The prior is well defined over the entire
parameter space (e.g., the real plane for
$(\log(t_{p_{r}}),\log(\sigma))$. For the given $p_{r}$ and censoring
time $t_{c}$ (the only inputs needed),
$z_{c}=[\log(t_{c})-\mu]/\sigma$ (the argument for the
$f_{ij}$ values) is computed as a function of
$(\log(t_{p_{r}}),\log(\sigma))$ (i.e., by using $\mu =
\log(t_{p_{r}})- \Phi^{-1}(p_{r})\sigma$).
When specifying a reparameterization that replaces the usual scale
parameter $\exp(\mu)$ of a log-location-scale distribution with a
specific quantile (i.e., $t_{p_{r}}$), the specific value of $p_{r}$
is not critical and is usually chosen so that the likelihood
contours are well behaved (as illustrated in
Figure~\ref{figure:likelihood.contour.plots.reparameterization}). Due
to the invariance property of ML estimators, ML estimates of
$\sigma$, quantiles, and cdf values will not be affected by the
choice of $p_{r}$. In the case of Bayesian estimation, with a
noninformative prior distribution (e.g., the priors outlined in
Table~\ref{table:summary.noninformative.priors}) the invariance
property for the choice of $p_{r}$ will hold for \textrm{Type~2}{} censoring
and otherwise approximately. Extensive numerical experiments
suggest that the approximation is excellent with the IJ prior and
\textrm{Type~1}{} censoring.
Section~\ref{S.section:understanding.ij.prior.features} in the appendix provides a detailed description of the \textrm{Type~1}{}
censoring IJ prior distribution features and the reasons those
features arise. Here we provide a brief summary of that material.
Figure~\ref{figure:ijprior.density.examples} illustrates the general
shapes of the IJ priors for different values of $p_{r}$.
These improper IJ priors have been scaled to have a maximum
of 1.0 and thus we refer to them as relative densities.
\begin{figure}
\caption{Independence Jeffreys prior densities for $p_{r}
\label{figure:ijprior.density.examples}
\end{figure}
Generally, the IJ prior for small $\sigma$ and $t_{p_{r}} < t_{c}$
is flat at a level of 1.0.
For any small $\sigma$, as the value of $t_{p_{r}}$ crosses
$t_{c}$ from the left, there is a steep cliff, with the level of the density
dropping to near 0. Note that this is because, in bottom-right part
of the parameter space, the probability of getting even one failure
is negligible. Detailed examples are given in
Section~\ref{S.section:reasons.for.ij.prior.features} of the
appendix.
The importance of this result is that,
relative to other noninformative priors, the
IJ priors (and CJ priors when used in conjunction with an
informative or weakly information marginal prior for the
other parameter)
follow the fundamental principle described in
Section~\ref{section:noninformative.prior.fundamental.principle}.
For large values of $\sigma$, the IJ prior is approximately flat
at a level close to $p_{r}$.
\section{Random Censoring IJ Prior Distributions}
\label{section:random.censoring.ij.prior}
As described in Section~\ref{section:censoring.types}, field
reliability data
almost always result in multiply time-censored data (e.g.,
Figures~\ref{figure:BearingCage.plots}(a)
and~\ref{figure:RocketMotor.plots.ps}(a)). The random
censoring IJ prior, based on a competing risk model, can be obtained
from the random censoring FIM described in
Section~\ref{section:fisher.random.censored}. Suppose that $T$ is
the failure time for a unit but that it will not be observed if the
random censoring time $C < T$. There are two
situations to consider, described in the following subsections.
\subsection{IJ Prior Distributions for Limited-Time Random Censoring}
\label{section:limited.time.random.censoring}
High-reliability products or any product that has been in the field
for a small amount of time (e.g., a newly released smart phone model) will have a largest value of a
nonparametric estimate (e.g., Kaplan--Meier) of the marginal cdf of
$T$ that is considerably less than 1. This situation is
common and arises when data are analyzed at a particular data-freeze
date when only a small fraction of units in the field has failed. This is
similar to \textrm{Type~1}{} (time) censoring, except that there will be
some additional right-censored observations (e.g., due to staggered entry
into service) before the censoring time $t_{c}$.
The bearing cage field data (Section~\ref{section:motivating.examples})
provides an example of such data.
The rocket motor field data (also
Section~\ref{section:motivating.examples}) is different because all
three failures were left censored. The nonparametric ML
estimate of $F(t)$ jumps to 1.0 at the left-censored
observation at 16.5 years because this is larger than 16 years, the largest
right-censored observation (there is additional discussion of this
point at the end of Section~\ref{section:comparisons.bayes.noninformative}).
However, the similarity of the shapes of the
bearing cage and rocket motor likelihoods
(see Figure~\ref{figure:likelihood.contour.plots.reparameterization}),
suggests a much smaller effective $t_{c}$ for the rocket motor
data---perhaps 11 years.
It would be possible to define and compute an IJ prior for a
competing risk model describing multiple censoring
by using the scaled FIM elements inside the square
brackets in (\ref{equation.fim.random.censoring}) to replace the
$f_{ij}$ values in (\ref{equation:IJ.typeI.yp.log.sigma}). The pdf
of $\log(C)$, $h(x)$ would describe the pattern up to the censoring
time $t_{c}$, where all of the remaining mass would be
concentrated. The shape of the resulting IJ prior in this situation
will, however, be similar to the IJ prior for \textrm{Type~1}{} censoring,
described in Sections~\ref{section:ij.prior}
and~\ref{section:implementing.ij.priors} and thus those can be
used instead. We use this approach for our examples in
Sections~\ref{section:comparisons.ml.and.bayes}
and~\ref{section:comparisons.bayes.noninformative}.
\subsection{IJ Prior Distributions for Unlimited-Time Random Censoring}
For products that have a substantial amount of field experience and
are run until failure, after which they are replaced (e.g.,
single-use batteries), the largest value of the nonparametric estimate of
the marginal cdf of $T$ will typically be close to 1. An example of this kind
of data is given in the mechanical switch example in \citet{Nair1984},
where he focused on estimating the marginal distribution of failure mode A
and the occurrence of failure mode B resulted in the random censoring.
Again, it would be possible to define and compute an IJ prior for
this situation by using the scaled FIM elements inside the square
brackets in (\ref{equation.fim.random.censoring}) to replace the
$f_{ij}$ values in (\ref{equation:IJ.typeI.yp.log.sigma}). The pdf
of $\log(C)$, $h(x)$ could be obtained by looking at the
distribution of censoring times. The shape of the resulting IJ
prior in this situation will be approximately flat because there is
not a single censoring time that ends the failure-observation process.
\section{Motivation for Weakly Informative Prior Distributions}
\label{section:weakly.informative.prior}
\subsection{Potential difficulties with noninformative prior distributions}
The noninformative prior distributions described in
Section~\ref{section:noninformative.prior} have
appealing theoretical properties under certain \textit{specified conditions}
(e.g. (log-)location-scale distributions with complete data or
\textrm{Type~2}{} censoring). In most applications, such conditions will not
be met exactly. Noninformative prior distributions can put large
amounts of relative density at unreasonable (e.g., impossible) parts
of the parameter space. Then, with limited information in the data
(e.g., a small number of failures), a noninformative prior
distribution can strongly influence inferences and possibly
result in misleading conclusions. In such situations, especially, it is
better to use a prior distribution that rules out combinations of
parameters that are impossible or nonsensical.
\subsection{Previous Work on Weakly Informative Prior Distributions}
While there is a vast literature on noninformative or reference
prior distributions, less work has been done on weakly informative
prior distributions. Weakly informative priors are constructed to
be diffuse relative to the likelihood and known scale of the data.
However, as opposed to noninformative priors,
weakly informative priors put density
on reasonable values of the parameters while down weighting
nonsensical values. When there is a small amount of information in the
data (e.g., few failures) or if
fitting a complex model with many parameters, weakly informative
priors can help stabilize estimation whereas a noninformative prior
(e.g., a flat prior) can result in dispersed posterior distributions
with probability mass on extreme parameter values.
\cite{GelmanJakulinPittauSugelman2008}
recommend weakly informative priors for the
parameters of logistic and other regression models. More recently,
\cite{GelmanSimpsonBetancourt2017} provide a historical overview of the
different classes of Bayesian priors such as, uniform, Jeffreys,
reference, and weakly informative priors. The authors illustrate
the dangers of using flat or default priors and recommend weakly
informative prior distributions that are selected based on the data
and subject-specific domain. \cite{Lemoine2019} uses
simulation to demonstrate how noninformative priors can produce
spurious parameter estimates, and advocates for weakly informative
priors as the new default choice for Bayesian estimation.
\subsection{Weakly Informative Prior Distributions for
Log-Location-Scale Distributions}
\label{section:weakly.informative.priors.lls.distributions}
When there is little or no prior information about certain
parameters or when there is need to present an analysis
where results do not depend on subjective prior information, a commonly-used
alternative is to specify weakly informative marginal priors for
those parameters. Commonly-used weakly informative priors include a
normal distribution with a large variance for parameters that are
unrestricted in sign or a lognormal distribution
with a large value of the
shape parameter (log-standard-deviation) for parameters that
must be positive.
These choices can be motivated by the fact that a normal
distribution prior density with any mean will approach a flat prior
as the standard deviation of the normal distribution increases.
Correspondingly, a lognormal distribution prior $\pi(t)$
with any log-mean will be proportional to $1/t$ as the log-standard-deviation
increases. These results are illustrated in
Figure~\ref{figure:limiting.weakly.informative.plots} and proofs are
in
Section~\ref{S.section:limiting.results.weakly.informative.prior.distributions}
of the appendix.
\begin{figure}
\caption{Illustration that a normal distribution density approaches a
uniform (flat prior) distribution as $\sigma$ increases~(a) and that a lognormal
density in $t$
is proportional to $1/t$ for large values of $\sigma$~(b).}
\label{figure:limiting.weakly.informative.plots}
\end{figure}
The parameters for these weakly informative marginal prior
distributions can be chosen using knowledge about the scale of the
response (i.e., depending on the units of the response), what
values of the parameters are physically possible,
and knowledge based on previous experience and engineering
knowledge.
The center of these
distributions might be chosen in a conservative manner.
Instead of specifying the prior distribution parameters (especially
for the lognormal distribution), it is generally better (and much
easier for prior elicitation) to specify a
range that contains a large proportion of the probability
distribution. Here we use a 0.99 probability range which is defined
as the 0.005 and the 0.995 quantiles of the prior distribution. For
example, to
specify that the Weibull shape parameter has a lognormal prior distribution
with probability 0.99 between 1.5 and 5, we write $\beta \sim \textrm{$<$LNORM$>$}(1.5, 5)$.
The use of quantiles to elicit/specify prior distributions has been
discussed previously \citep[e.g.,
in][]{DeyLiu2007}. \citet{MeyerBooker2001} point out that
individuals tend to underestimate uncertainty. Thus, unless the
prior interval is based on quantitative information (e.g., interval
estimates on the same parameter from a previous study or studies) one might
want to ask for a 99\% interval and treat it as if it were a 95\%
interval. \citet{Mikkola.et.al2021} describe the current state of the art of
prior elicitation and provide an extensive literature review, including other
papers that use quantiles in elicitation.
\subsection{Comparisons of ML and Bayesian Estimation using
Noninformative or Weakly Informative
Prior Distributions}
\label{section:comparisons.ml.and.bayes}
\subsubsection{Bearing cage field data}
This is a continuation of the bearing cage examples in
Sections~\ref{section:motivating.examples}
and~\ref{section:reparameterization} where we compare ML and Bayes
estimation using alternative noninformative
priors. Figure~\ref{figure:BearingCage.ml.and.noninformative}
compares ML and Bayes estimation with a flat (on $\log(t_{0.10})$ and
$\log(\sigma)$)~(a) and ML and IJ priors~(b).
\begin{figure}
\caption{Bearing cage comparison of estimation results for ML and
flat prior~(a) and ML and IJ prior~(b).}
\label{figure:BearingCage.ml.and.noninformative}
\end{figure}
The IJ priors agree well with the ML results except for a little
deviation in the upper confidence bounds on $F(t)$ in the lower tail
of the distribution and the lower confidence bounds on $F(t)$ in the
upper tail of the distribution. The flat prior results are more
optimistic (smaller failure probabilities) in the upper tail of the
distribution (the region of interest).
\subsubsection{Rocket motor field data}
This is a continuation of the rocket motor examples in
Sections~\ref{section:motivating.examples}
and~\ref{section:reparameterization}.
\begin{figure}
\caption{Rocket motor comparison of estimation results comparing ML
and Bayes with flat prior for
$t_{p_{r}
\label{figure:RocketMotor.ml.weaklyinformative.cj}
\end{figure}
Although ML estimates and associated confidence intervals exist for
the rocket motor data, because the failures are left-censored
observations, using Bayes estimation with flat or IJ priors
apparently results in an improper posterior distribution. We avoid
this problem by using a weakly informative prior $\beta \sim
\textrm{$<$LNORM$>$}(0.2, 25)$ for the comparisons in this example (note
that this is an extremely wide range relative to values of $\beta$
seen in typical applications) that is approximately flat for the
$\log(\sigma)$ parameterization.
Figure~\ref{figure:RocketMotor.ml.weaklyinformative.cj}(a) compares
estimation results for ML and Bayes with a flat (uniform) prior for
$\log(t_{p_{r}})$. Figure~\ref{figure:RocketMotor.ml.weaklyinformative.cj}(b)
is similar but replaces the flat prior for $\log(t_{p_{r}})$ with a
CJ prior. The estimates using the CJ prior $\log(t_{p_{r}})$ in
Figure~\ref{figure:RocketMotor.ml.weaklyinformative.cj}(b) are
considerably closer to the ML estimates and less optimistic when
compared to the flat prior in
Figure~\ref{figure:RocketMotor.ml.weaklyinformative.cj}(a).
One might ask why the ML and Bayesian estimate of $F(t)$ in
Figure~\ref{figure:RocketMotor.ml.weaklyinformative.cj} differ so
much from the nonparametric estimate represented by the three
plotted points. The slope of the ML estimate of $F(t)$ is the same
as the ML estimate of $\beta$, as shown in
\citet[][Section~6.2.4]{MeekerEscobarPascual2022} and this is
approximately true for the Bayesian estimate. There were only three
failures and all were left censored. The nonparametric ML estimate
(NPMLE) of $F(t)$ for current-status data can be obtained by using
the ``Pool Adjacent Violators'' algorithm for the observed fraction
failing as a function of years. For the rocket motor data, the NPMLE
is a step function that jumps from 0 to $1/384=0.026$ at 8.5 years,
to $1/9=0.111$ at 14.2 years, and to 1.0 at 16.5 years. Then the
points are plotted at half of the jump height, as suggested in
\citet[][Section~3.3.1.2]{Lawless2003}. The left censoring implies
that these jumps occur after the (unknown) failure times, which
would result in a kind of upward bias in the plotted points,
relative to what would be plotted if the exact failure times had
been known.
\section{Combining Informative with Noninformative or Weakly Informative Prior Distributions}
\label{section:combining.informative.with.noninformative}
\subsection{Motivation for Partially Informative Prior Distributions}
\label{section:motivation.for.partially.informative.prior}
An important reason for using Bayesian methods is that they
provide a formal mechanism for including prior information (i.e.,
knowledge beyond that provided by the data) into the analysis.
When combining a CJ prior with an informative or
weakly informative prior our approach is similar to that suggested
in \citet{SunBerger1998}. For example, as described in
Section~\ref{section:ij.prior}, the CJ prior
$\pi[\log(t_{p_{r}}) \, |\log(\sigma)]$ can be combined with an
informative or weakly informative marginal prior $\pi(\log(\sigma))$
to give the joint prior $\pi[\log(t_{p_{r}}),\log(\sigma)]$. In our
software, for convenience, the prior is specified
in terms of the familiar Weibull shape
parameter $\beta=1/\sigma$ (or the lognormal shape parameter $\sigma$)
and then transformed into the marginal prior for $\log(\sigma)$.
Section~\ref{section:reparameterization} discussed the need for
reparameterization.
If there is prior information for one or more of the model
parameters and if the definition of the parameters (i.e., the particular
parameterization) has been chosen such that the information about
parameters is approximately mutually independent, then one can specify a
joint prior density as the product of marginal densities for
each parameter.
Informative marginal prior distributions can be used for those
parameters for which there is appreciable prior
information. As mentioned by \citet{MeyerBooker2001}, individuals
providing an informative marginal prior
distribution will typically feel more comfortable expressing their
knowledge about a parameter by using a symmetric distribution.
The normal distribution (truncated
below zero for a positive parameter) is a reasonable (approximately symmetric) choice. Then
noninformative (e.g., flat or CJ) or weakly
informative (e.g., normal with a large 99\% range) marginal prior
distributions can be specified for the other unrestricted parameters
(e.g., $\log(t_{p_{r}})$ or $\log(\sigma)$).
A useful generalization of the normal distribution is
the location-scale-$t$ (LST) distribution. That is, for a specified
degrees-of-freedom parameter $r_{d}>0$, there is a symmetric
LST distribution having tails that are heavier (or
much heavier) than the normal distribution. For large values of
$r_{d}$ (e.g., greater than 60), the LST distribution is
approximately the same as a normal distribution. For $r_{d}=1$,
the LST distribution is a Cauchy distribution.
\subsection{A Summary of Recommended Log-Location-Scale Prior Distributions}
\label{section:recommended.lls.prior}
Table~\ref{table:summary.llc.priors} provides a summary of the
recommended prior distributions for use with log-location-scale
distributions.
\begin{table}[!htbp]
\caption{Summary of recommended prior distributions for
log-location-scale distribution parameters
\label{table:summary.llc.priors}
}
\centering
\begin{tabular}{llc}
\toprule
Prior distributions for $t_{p_{r}}$ & Type of prior &Prior distribution inputs\\
\midrule
Lognormal for $t_{p_{r}}$ & \parbox{15em}{Informative\\Weakly informative}
&$(\mu_{\log(t_{p_{r}})}, \sigma_{\log(t_{p_{r}})} )$\\[2.5ex]
Truncated $(>0)$ normal for $t_{p_{r}}$ & \parbox{15em}{Informative}
&$(\mu_{t_{p_{r}}}, \sigma_{t_{p_{r}}} )$\\[2.5ex]
Log-location-scale-$t$ for $t_{p_{r}}$ & \parbox{15em}{Informative\\Weakly informative}
&$(\mu_{\log(t_{p_{r}})}, \sigma_{\log(t_{p_{r}})}, r_{d} )$\\[2.5ex]
Truncated $(>0)$ location-scale-$t$ for $t_{p_{r}}$ & \parbox{15em}{Informative}
&$(\mu_{t_{p_{r}}}, \sigma_{t_{p_{r}}}, r_{d} )$\\[2.5ex]
Flat for $\log(t_{p_{r}})$ & Noninformative &None\\[2.5ex]
\parbox{15em}{Conditional Jeffreys\\ for $\log(t_{p_{r}})|\log(1/\beta)$}
&Noninformative & $t_{c}$ and $p_{r}$\\[5ex]
\toprule
Prior distributions for $\beta=1/\sigma$ (or $\sigma$)&
Type of prior &Prior distribution inputs\\
\midrule
Lognormal for $\beta$ & \parbox{15em}{Informative\\Weakly informative}
& $(\mu_{\log(\beta)}, \sigma_{\log(\beta)})$\\[2.5ex]
Truncated $(>0)$ normal for $\beta$ &\parbox{15em}{Informative}
&$(\mu_{\beta}, \sigma_{\beta} )$\\[2.5ex]
Log-location-scale-$t$ for $\beta$ & \parbox{15em}{Informative\\Weakly informative}
& $(\mu_{\log(\beta)}, \sigma_{\log(\beta)}, r_{d} )$\\[2.5ex]
Truncated $(>0)$ location-scale-$t$ for $\beta$ &\parbox{15em}{Informative}
&$(\mu_{\beta}, \sigma_{\beta}, r_{d} )$\\[2.5ex]
Flat for $\log(1/\beta)$ & Noninformative &None\\[2.5ex]
\parbox{15em}{Conditional Jeffreys\\ for $\log(1/\beta)|\log(t_{p_{r}})$}
& Noninformative & $t_{c}$ and $p_{r}$\\
\bottomrule
\end{tabular}
\end{table}
Some comments on these prior distributions and Table~\ref{table:summary.llc.priors} are:
\begin{itemize}[itemsep=1mm, parsep=0pt]
\item
The CJ priors referenced in
Table~\ref{table:summary.llc.priors} are for \textrm{Type~1}{} censoring
(Section~\ref{section:ij.prior}) and limited-time random censoring
(Section~\ref{section:limited.time.random.censoring}).
\item
Table~\ref{table:summary.llc.priors} gives priors for the Weibull
failure-time distribution shape parameter $\beta=1/\sigma$.
The recommendations
are similar for the lognormal failure-time distribution with
shape parameter $\sigma$.
\item
As described in
Section~\ref{section:weakly.informative.priors.lls.distributions},
lognormal, truncated normal, log-location-scale-$t$, and truncated
location-scale-$t$ prior distributions for the interpretable
Weibull parameters $t_{p_{r}}$ and $\beta=1/\sigma$ are initially specified
by a 0.99 probability range. This range is then translated into
parameters for the marginal prior distributions for both $t_{p_{r}}$
and $\beta=1/\sigma$.
\item
Subsequently, the marginal priors for $t_{p_{r}}$ and
$\beta=1/\sigma$ are used to obtain (by standard methods for
obtaining the distribution of a transformation of random variables),
the marginal priors for $\log(t_{p_{r}})$ and
$\log(1/\beta)=\log(\sigma)$ that are used in the MCMC
computations. Details are given in
Section~\ref{S.section:log.truncated.normal.distributions} of the
appendix.
\item
For the CJ priors, the censoring time $t_{c}$ is
not a parameter, but it is a necessary input.
\item
The noninformative flat and CJ priors do not
require specification of any parameters and are thus given directly
in terms of the $\log(t_{p_{r}})$ or $\log(1/\beta)$
parameterization.
\item
When the CJ priors for
$\log(t_{p_{r}})|\log(1/\beta)$ and for
$\log(1/\beta)|\log(t_{p_{r}})$ are used together, the result is a
joint IJ prior for $(\log(t_{p_{r}}), \log(1/\beta))$. Note that
when these two densities are used together, the result is not a
joint distribution of independent random variables. This is the
reason that we use the name Independence Jeffreys (IJ).
\end{itemize}
\subsection{Comparisons of Bayesian Estimation using
Noninformative and Partially Informative
Prior Distributions}
\label{section:comparisons.bayes.noninformative}
This section returns to the two motivating examples, comparing
noninformative (or weakly informative) priors with partially
informative priors.
\subsubsection{Bearing cage field data}
This is a continuation of the bearing cage example in
Section~\ref{section:comparisons.ml.and.bayes} where ML estimates
were compared with Bayesian estimates based on
noninformative prior distributions.
For the bearing cage field data,
Figure~\ref{figure:BearingCage.noninformative.informative}
compares Bayesian estimation results for a noninformative prior (on
the left) and a partially informative prior (on the right).
\begin{figure}
\caption{Bearing cage Bayesian estimation results comparing
a noninformative prior (on the left) with a partially informative
prior (on the right). For each prior there is a Weibull
probability plot showing the point estimate and credible intervals
for $F(t)$ (top), draws from the bounded joint prior and likelihood contours
(middle), and draws from the joint posterior and likelihood contours
(bottom).}
\label{figure:BearingCage.noninformative.informative}
\end{figure}
For the noninformative prior, we used an IJ prior. For
the partially informative prior, we combined the CJ prior for $y_p$
given $\log(\sigma)$ (which, from
Section~\ref{subsubsection:ij.prior.yp.log.sigma} is
$\pi(y_p|\tau)\propto\sqrt{f_{11}}$) with an informative truncated
(to be positive) normal distribution prior
$\beta \sim \textrm{$<$TNORM$+>$}(1.5, 3)$.
The probability plot on the top-left in
Figure~\ref{figure:BearingCage.noninformative.informative} gives
Bayesian estimation results with the noninformative IJ prior
and they are similar to the ML results given on the right in
Figure~\ref{figure:BearingCage.plots}. The credible interval for
$F(8000)$ is $[0.03, \;\; 0.99992]$, which is not useful for
assessing whether the goal of fraction failing less than 0.10 has
been met.
The probability plot on the top-right gives Bayesian estimation
results with the partially informative prior, showing the improved
precision. A 95\% credible interval for $F(8000)$ is
$[0.15, \;\; 0.92]$ indicating clearly that the goal of
fraction failing of less than 0.10 has not been met.
The middle and bottom rows of plots show likelihood contours along
with prior and posterior draws, respectively.
As described more fully in
Section~\ref{S.section:implementation.using.bayes.without.tears} of
the appendix, prior draws were obtained by sampling
from versions of the noninformative priors that were bounded by a
large rectangle (much larger than the boundaries of the contour
plots where the draws are plotted) so that the resulting prior
distributions are proper.
These plots provide a
visualization of how constraining the values of the Weibull shape
parameter $\beta$ to those that are consistent with engineering
knowledge importantly improves precision for estimation.
\subsubsection{Rocket motor field data}
This is a continuation of the rocket motor example in
Section~\ref{section:comparisons.ml.and.bayes} where ML estimates
were compared with Bayesian estimates based on
weakly informative prior distributions.
For the rocket motor field data,
Figure~\ref{figure:RocketMotor.noninformative.informative} compares
Bayesian estimation results for a noninformative/weakly informative
prior (on the left) and a partially informative prior (on the right).
\begin{figure}
\caption{Rocket motor Bayesian estimation results comparing
a weakly informative/noninformative prior (on the left) with a
partially informative prior (on the right). For each prior there is a Weibull
probability plot showing the point estimate and credible intervals
for $F(t)$ (top), draws from the bounded joint prior and likelihood contours
(middle), and draws from the joint posterior and likelihood contours
(bottom).}
\label{figure:RocketMotor.noninformative.informative}
\end{figure}
As described in Section~\ref{section:comparisons.ml.and.bayes},
because of the limited amount of information in the three
left-censored observations, a weakly informative $\beta \sim
\textrm{$<$LNORM$>$}(0.2, 25)$ marginal prior was used for the
noninformative part of the comparison and, as in
Figure~\ref{figure:RocketMotor.ml.weaklyinformative.cj}(b), this was
paired with a noninformative CJ prior. \citet{OlwellSorell2001}
stated, ``It is unusual in Weibull analysis to get shape parameters
greater than 5.'' Also, because thermal cycling is a wearout mode,
$\beta>1$. Thus for the informative part of the comparison, the
weakly informative prior was replaced by a somewhat informative
$\beta \sim \textrm{$<$LNORM$>$}(1, 5)$.
The probability plot on the top-left in
Figure~\ref{figure:RocketMotor.noninformative.informative} gives
Bayesian estimation results with the noninformative/weakly
informative prior, and they are similar to the ML results given on
the right in Figure~\ref{figure:RocketMotor.plots.ps}. The credible
interval for $F(20)$ is $[0.007, \;\; 0.98]$, which does not
help assess whether there is a serious problem or not.
The probability plot on the top-right gives Bayesian estimation
results with the partially informative prior, showing considerably
better precision for estimating $F(20)$. A 95\% credible interval for
$F(20)$ is $[0.008, \;\; 0.16]$ which would help assess the
need for corrective action.
Similar to Figure~\ref{figure:BearingCage.noninformative.informative}
for the bearing cage example, the plots in the middle and bottom rows
of Figure~\ref{figure:RocketMotor.noninformative.informative} show
likelihood contours along with prior and posterior draws,
respectively. Again, these plots provide a visualization of how the
partially informative prior for the Weibull shape parameter $\beta$
improves estimation precision.
As mentioned in
Section~\ref{section:motivating.examples}, the ML estimate
$\widehat{\beta}=8.126$ which, as suggested by \citet{OlwellSorell2001}, is
physically unreasonable for Weibull field data.
Additionally, using years after manufacture as the surrogate
response for the unknown number and range of the thermal cycles has
the effect increasing variability (causing $\beta$ to be smaller).
\citet{OlwellSorell2001} also mention that it is well known
that the Weibull ML estimator for $\beta$ has serious upward bias
when there are few failures. The Bayesian estimate with the
noninformative/weakly informative prior $\widehat{\beta}=5.9$, is an
improvement but still physically unreasonable. Using the partially
informative prior, described above, gives a much more reasonable
$\widehat{\beta}=3.4$, but the change in slope results in the deviation
between the Bayesian and the nonparametric estimates of $F(t)$ seen
in Figure~\ref{figure:RocketMotor.noninformative.informative}(b).
\section{Simulation Study to Evaluate Alternative Noninformative Prior Distributions}
\label{section:weibull.type.one.simulation}
\subsection{Goals of the Simulation}
\label{section:goals}
As mentioned in Section~\ref{section:ij.prior}, for complete and
\textrm{Type~2}{} censored data from a (log-)location-scale distribution,
when using an IJ prior distribution, Bayesian credible intervals
have coverage probabilities that are the same as the nominal
credible/confidence level (i.e., giving ``exact'' interval procedures).
For \textrm{Type~1}{} and random censoring, this
result is approximate.
This section describes a simulation study to compare coverage
probabilities of credible intervals for the
flat and IJ noninformative prior distributions for the
most commonly used failure-time distributions (Weibull and
lognormal) under \textrm{Type~1}{} censoring. Here we focus on the Weibull
distribution simulation. Results for the lognormal distribution
simulation were similar and are given in
Section~\ref{S.section:lognormal.simulation.results.conclusions} of
the appendix. The rest of this section describes the
design of the study and the results.
\subsection{Simulation Factors}
\label{section:factors}
We simulated a life test where all units are put on test
simultaneously and are observed until a fixed censoring time,
$t_c$. Simulated failure times larger than $t_c$ are right-censored
at $t_c$ (i.e., \textrm{Type~1}{} censoring). Without loss of generality, we
use the Weibull parameters $\mu=0$ and $\beta=1/\sigma=1$. The
experimental factors for the simulation were:
\begin{itemize}[itemsep=1mm, parsep=0pt]
\item $\textrm{E}\big( r \big)$: the expected number of failures before $t_c$,
\item $p_{\textrm{fail}}$: the expected proportion of failures before time $t_c$, and
\item $\pi[\log(t_{p_{r}}), \sigma]$: the joint prior distribution for the
unknown parameters, where $p_{r}$ is the reparameterization
quantile and is always chosen to be $r/(2n)$, resulting in a
well-behaved likelihood (and posterior).
\end{itemize}
We use the expected number of failures instead of the sample size as
an experimental factor because it is a better measure of the
expected amount
of information in a data set and avoids a strong interaction that
arises when sample size and expected proportion censored are used as
factors.
This simulation is similar to that used in
\citet{JengMeeker2000} except that instead of comparing ten different
non-Bayesian confidence interval methods we compare \emph{the} Bayesian
method under different noninformative priors.
\subsection{Simulation Factor Levels}
\label{section:simulation.factor.levels}
In order to cover a range of situations and joint prior
specifications, we use the following levels of the factors:
\begin{itemize}[itemsep=1mm, parsep=0pt]
\item $\textrm{E}\big( r \big)$ = 10, 25, 35, 50, 75, 100,
\item $p_{\textrm{fail}}$ = 0.01, 0.05, 0.10, 0.50, and
\item $\pi[\log(t_{p_{r}}), \sigma]$ = flat and IJ.
\end{itemize}
For each factor-level combination, the sample size is computed as
$n=\textrm{E}(r)/p_{\textrm{fail}}$. The censoring time is computed as $t_c =
\text{exp}\left(\mu + \Phi^{-1}_{\textrm{sev}}(p_{\textrm{fail}}) \sigma
\right)$.
We simulate $t_i$, $i=1 \dots n$. Observations with $t_i >
t_c$ are coded as being censored at $t_c$.
As discussed in Section~\ref{section:improper.with.few.failures},
with fewer than three failures, the posterior is either improper
or poorly behaved. Thus we condition on at least 3 failures.
For each factor-level combination, we simulated 5,000 data sets.
As outlined in Section \ref{section:implementing.ij.priors},
the IJ prior $\pi[\log(t_{p_r}), \log(\sigma)]$
is specified in terms of the reparameterization
$(\log(t_{p_r}), \log(\sigma))$. We obtain posterior draws
for each simulated data set, for both flat
and IJ noninformative prior distributions.
\subsection{Estimation}
\label{section:simulation.estimation}
Similar to the other examples in this paper,
the Weibull distribution models were fit using codes based on the
{\tt R} package {\tt rstan} \citep{rstan}. {\tt R} package
\texttt{lsinf} was used to compute the scaled FIM needed for the IJ
priors. Implementation details are described in
Section~\ref{S.section:experiences.using.stan.with.ij.prior} of the
appendix.
Four chains were run for each estimation run, resulting in
10,000 draws after warmup and thinning. Before doing the production
simulation runs, extensive experiments were conducted to investigate
the performance of the {\tt rstan} NUTS sampler for our different
factor-level combinations.
Initial values for the NUTS sampler were obtained by generating
values from 99\% confidence intervals based on maximum likelihood estimates.
The Gelman-Rubin potential scale
reduction factor and close examination of select trace plots were
used to check for adequate mixing of the four chains
\citep{GelmanRubin1992}.
Performance of the prior distributions was evaluated in terms of
the error probabilities for each interval end point,
computed as the proportion of times the
computed 95\% credible interval lower (upper)
endpoint was greater than (was less than) the true value of the
quantiles being estimated. Evaluations were done for credible intervals
for the Weibull quantiles $t_{0.01}$, $t_{0.05}$,
$t_{0.10}$, and $t_{0.50}$.
\subsection{Weibull Distribution Simulation Results and Conclusions}
\label{simulation.results.conclusions}
Figures~\ref{figure:WeibullSimulationTwoSidedResults.small.pfail}
($p_{\textrm{fail}}=0.01$ and $p_{\textrm{fail}}=0.05$) and~\ref{figure:WeibullSimulationTwoSidedResults.large.pfail} ($p_{\textrm{fail}}=0.10$ and $p_{\textrm{fail}}=0.50$) summarize
the Weibull distribution simulation results with two-sided coverage probabilities.
\begin{figure}
\caption{Weibull distribution two-sided estimated coverage probabilities for $p_{\textrm{fail}
\label{figure:WeibullSimulationTwoSidedResults.small.pfail}
\end{figure}
\begin{figure}
\caption{Weibull distribution two-sided estimated coverage probabilities for $p_{\textrm{fail}
\label{figure:WeibullSimulationTwoSidedResults.large.pfail}
\end{figure}
Figures~\ref{figure:WeibullSimulationOneSidedResults.small.pfail}
($p_{\textrm{fail}}=0.01$ and $p_{\textrm{fail}}=0.05$)
and~\ref{figure:WeibullSimulationOneSidedResults.large.pfail}
($p_{\textrm{fail}}=0.10$ and $p_{\textrm{fail}}=0.50$) summarize the Weibull
distribution simulation results with one-sided error probabilities.
Section~\ref{S.section:lognormal.simulation.results.conclusions} of
the appendix gives similar results for the lognormal
distribution.
Note that if $\alpha_{L}$ is the error probability for the lower
endpoint of a credible interval and $\alpha_{U}$ is the error
probability for the upper endpoint, then the two-sided coverage
probability is $1 - \alpha_{L} -\alpha_{U}$.
\begin{figure}
\caption{Weibull distribution one-sided
estimated error probabilities for $p_{\textrm{fail}
\label{figure:WeibullSimulationOneSidedResults.small.pfail}
\end{figure}
\begin{figure}
\caption{Weibull distribution one-sided
estimated error probabilities for $p_{\textrm{fail}
\label{figure:WeibullSimulationOneSidedResults.large.pfail}
\end{figure}
When interpreting the results of the simulation, it is important to
keep in mind that, as mentioned in
Section~\ref{section:simulation.factor.levels}, within each of the
eight plots in
Figures~\ref{figure:WeibullSimulationTwoSidedResults.small.pfail}--\ref{figure:WeibullSimulationOneSidedResults.large.pfail}, the results
are based on the same set of 5,000 simulated data sets. Thus, for
example, both points in a plot for a particular value of
$\textrm{E}(r)$ and estimated quantiles tend to move together. Given the
nominal credible level of 0.95 for the intervals, the nominal error
probabilities for each tail are 0.025.
The standard error of the estimated coverage probabilities is
approximately $\sqrt{0.95(1-0.95)/5000}=0.003$.
The standard error
of the estimated error probabilities is approximately
$\sqrt{0.025(1-0.025)/5000}=0.002$. Also, recall that the simulation
results are conditional on observing at least three failures. The
probabilities of fewer than three failures when $\textrm{E}(r) = 10$ are
0.0027, 0.0023, 0.0019, and 0.0002, respectively, for
$p_{\textrm{fail}}=0.01$, 0.05 0.10, and 0.50.
Some observations from the two-sided estimated coverage
probabilities in Figures~\ref{figure:WeibullSimulationTwoSidedResults.small.pfail}
and~\ref{figure:WeibullSimulationTwoSidedResults.large.pfail} are:
\begin{itemize}[itemsep=1mm, parsep=0pt]
\item
The IJ coverage probabilities tend to be better or the same as flat almost everywhere.
\item
There can be a large departure from the nominal $0.95$ when
$\textrm{E}(r)=10$, perhaps partially because of the conditioning on $r \ge 3$.
\item
Both flat and IJ priors are close to nominal for $\textrm{E}(r) \ge 35$.
\end{itemize}
\noindent
Some observations from the one-sided estimated error
probabilities in Figures~\ref{figure:WeibullSimulationOneSidedResults.small.pfail}
and~\ref{figure:WeibullSimulationOneSidedResults.large.pfail} are:
\begin{itemize}[itemsep=1mm, parsep=0pt]
\item
As is common for approximate interval procedures the lower and upper
bound error probabilities are not the same. When one error probability (either lower or
upper) is above nominal, the other is below nominal resulting in
some cancellation that makes the two-sided coverage close to nominal.
\item
In most cases, the IJ performs better than flat (i.e., smaller error probabilities), but there are some
exceptions. Generally, the differences are not large.
\item
For the $p_{\textrm{fail}}=0.01$, the IJ priors have smaller error
probabilities for all levels of $\textrm{E}(r)$.
\item
The error
probabilities for the IJ and flat priors tend to get closer together
as $p_{\textrm{fail}}$ increases and especially for $p_{\textrm{fail}}=0.50$.
\end{itemize}
The results for the lognormal distribution
Section~\ref{S.section:lognormal.simulation.results.conclusions} of
the appendix are similar. From these simulation results, and
consistent with the comments at the end of
Section~\ref{section:implementing.ij.priors}, we conclude that the
IJ prior can provide useful improvement in performance in \textrm{Type~1}{}
(and similar) applications: particularly when there are few failures or a small fraction of failures.
\section{Prior Distribution Sensitivity Analysis for the Motivating Examples}
\label{section:prior.sensitivity.analysis}
As mentioned previously, with limited information in the data (e.g.,
a small number of failures due to censoring in reliability
applications), the choice of a prior distribution could have a strong
influence on inferences. When attempting to use weakly informative
prior distributions, it is useful to experiment with different
specifications to assess sensitivity.
The plots in Figure~\ref{figure:RocketMotorWinfSensitivity} show two
different $2 \times 2$ factorial comparisons.
\begin{figure}
\caption{Sensitivity analysis comparing different weakly informative
priors for the Weibull shape parameter $\beta$.}
\label{figure:RocketMotorWinfSensitivity}
\end{figure}
In both cases, the baseline prior is the log-location-scale-$t$
distribution with 60 degrees of freedom, denoted by $\beta \sim
\logLST{60}(0.20, 25)$, which is essentially equivalent to the $\beta
\sim \textrm{$<$LNORM$>$}(0.20, 25)$ used in the rocket motor example in
Section~\ref{section:comparisons.ml.and.bayes}.
The plot on the left varies the degrees of freedom between 5 and 60
(heavy versus lighter tails) and the 99\% quantile range for $\beta$
between $(0.20,
25)$ and $(0.10, 50)$ (smaller to larger range for the Weibull shape parameter).
The plot on the right varies the lower endpoint of the 99\% quantile
range for $\beta$ between
range between $0.10$ and $0.20$ and the upper endpoint between 25 and 50
(changing both the log-location-scale-$t$ prior median and shape
parameter). Note that the plots have two common factor-level
combinations.
There is not much difference among the different priors between years
11 and 14 where there is more information. Not surprisingly, when
extrapolating beyond 16 years, the point estimates and the upper
uncertainty bounds have what might be considered to be important
differences. In practice, one would consider the plausible ranges for
a parameter (even the narrower 99\% quantile range $(0.20, 25)$ for $\beta$ is
extremely wide relative to $\beta$ values typically encountered in
practice) and then perhaps choose a weakly informative prior that is
a compromise or somewhat conservative.
\section{Concluding Remarks and Areas for Future Research}
\label{section:concluding.remarks}
This paper provides guidance for setting prior distributions for
log-location-scale distributions used in reliability
applications. We applied our recommendations to
field data applications with complicated censoring
and derived new noninformative
priors that have good coverage-probability properties
that will be especially useful in small-information (e.g., few
failures) applications
where the use of methods based on
asymptotic theory can give misleading results.
The general
principles and prior distributions we suggest are applicable in many
other reliability models and in domains outside of reliability. For
example, in accelerated testing, engineers often have useful prior
information about the activation energy of a temperature-accelerated
failure mode or regression coefficients in other kinds of
acceleration models (but not the other parameters in the model).
\citet{XuFuTangGuan2015} describe the use of
noninformative prior distributions for accelerated test models, and
their methods could be extended to the more commonly used \textrm{Type~1}{}
censoring. Bayesian methods are also being used for other types of
reliability data, including degradation data,
fatigue S-N data, and recurrent events
data. Noninformative priors are needed for most, if not all, of the
model parameters in such applications.
The further development and implementation (e.g., in widely
available software) of default noninformative and partially
informative prior distributions is important for making the
advantages of Bayesian methods (described in our abstract) more
accessible to practitioners. For example, after selecting a failure-time
distribution, an analyst could be presented with options
to override default noninformative priors and specify
a weakly informative or an informative prior for each of the failure-time
distribution parameters. Such functionality would
allow users to easily perform a sensitivity analysis to see the
effect that various noninformative/weakly informative
priors have on the estimation results.
Once this type of software is available, other than additional
computational effort, there would be little justification for
recommending non-Bayesian model fitting in reliability applications.
\section{Acknowledgments}
Luis Escobar provided assistance in refining the proofs in
Section~\ref{S.section:limiting.results.weakly.informative.prior.distributions} of
the appendix and for suggesting simplifications for
the computation of the log-truncated-normal and
log-reciprocal-truncated-normal probability density functions
described in
Section~\ref{S.section:log.truncated.normal.distributions} of the
appendix.
Necip Doganaksoy, Luis Escobar,
Mike Hamada, Yili Hong, Qingpei Hu, Yew Meng Koh,
Larry Leemis, Peng Liu, Lu Lu, Jave Pascual, and Grant Reinman provided helpful
suggestions that improved the paper.
We would also like to thank the associate editor and the
anonymous reviewers who made many
comments and suggestions that helped us improve the paper.
\appendix
\part*{Appendix}
\section{Overview of the Appendix}
This appendix provides derivations, proofs,
additional detailed descriptions, and additional simulation results.
Section~\ref{S.section:derivations.noninformative.priors} shows how
to derive Jeffreys, Independence Jeffreys (IJ),
and reference priors for different parameterizations.
Section~\ref{S.section:understanding.ij.prior.features} describes
features of the \textrm{Type~1}{} censoring IJ priors
and the reasons that those features arise.
Section~\ref{S.section:experiences.using.stan.with.ij.prior}
describes some details of implementing a Conditional Jeffreys (CJ) or
IJ prior using Stan.
Section~\ref{section:weibull.type.one.simulation} in the main paper
gives results for a simulation using the Weibull
distribution. Simulations were also done for the lognormal
distribution and they are described in
Section~\ref{S.section:lognormal.simulation.results.conclusions}.
Section~\ref{S.section:careful.look.three.failures} looks carefully
at estimation results under different noninformative priors for two
data sets with only three failures,
based on two of the simulation
(Section~\ref{section:weibull.type.one.simulation}) factor-level combinations.
Section~\ref{S.section:limiting.results.weakly.informative.prior.distributions}
provides proofs for the limiting results for normal/lognormal weakly
informative prior distributions that are described in
Section~\ref{section:weakly.informative.priors.lls.distributions}.
Section~\ref{S.section:log.truncated.normal.distributions} gives
expressions for the log-truncated and log-reciprocal-truncated pdfs
that are used in our Stan implementation of truncated normal and truncated
location-scale-$t$ informative prior distributions.
\section{Derivations of the Noninformative Prior Distributions}
\label{S.section:derivations.noninformative.priors}
\subsection{Motivation for and Overview of the Derivations}
Section~\ref{section:summary.of.noninformative.priors} and
Table~\ref{table:summary.noninformative.priors} in the paper
provide a summary of noninformative prior distributions (Jeffreys,
IJ, and Reference) for different parameterizations of
(log-)location-scale distributions. The different parameterizations
are needed because
\begin{itemize}
\item
In reliability applications,
inferences are generally needed on alternative ``parameters'' such
as quantiles or failure probabilities.
\item
Priors may be elicited for parameters that differ from the
traditional parameters.
\item
MCMC computations may be conducted using a parameterization
that differs from the parameterization where the prior is
elicited/specified.
\end{itemize}
This section provides technical details and
derivations of these prior distributions.
Section~\ref{S.subsec:mu-sigma} shows how to compute the elements of
the Fisher information matrix (FIM) for the $(\mu, \sigma)$
parameterization. Section~\ref{S.section:jeffreys.priors} defines
and gives general methods for obtaining the Jeffreys, CJ, and IJ
priors for the $(\mu, \sigma)$ parameterization for \textrm{Type~2}{} and
\textrm{Type~1}{} censoring. Section~\ref{S.section-ref-prior} defines and
gives general methods for obtaining reference priors for the $(\mu,
\sigma)$ parameterization for \textrm{Type~2}{} censoring.
Sections~\ref{S.subsec:use-y_p-sigma}--\ref{S.section-zeta-log.sigma}
derive the noninformative priors for other parameterizations.
\subsection{The Fisher Information Matrix for $(\mu, \sigma)$}
\label{S.subsec:mu-sigma}
The FIM plays an important role in
computing noninformative priors such as Jeffreys, IJ, and reference priors.
Suppose the random variable $Y$ has a location-scale
distribution with cdf
\begin{equation*}
G(y;\mu,\sigma)=\Phi(z),
\end{equation*}
and pdf $g(y;\mu,\sigma)=\phi(z)/\sigma$, where $z=(y-\mu)/\sigma$.
$\Phi(\cdot)$ and $\phi(\cdot)$ are the standard cdf and pdf for the
particular distribution. We use the results in
\citet{EscobarMeeker1994}. For \textrm{Type~2}{} (failure) censoring after
$r$ out of $n$ failures, the scaled elements of the
FIM for the parameters $\mu$ and $\sigma$ are
\begin{equation}
\label{S.eq:fim-1}
\begin{split}
f_{11}(z_{c}) =& \ \frac{\sigma^2}{n}\textrm{E}\left[-\frac{\partial^2\log(\mathcal{L})}{\partial\mu^2}\right]=\Psi_0(z_{c}),\\
f_{12}(z_{c}) =& \ \frac{\sigma^2}{n}\textrm{E}\left[-\frac{\partial^2\log\mathcal{L}}{\partial\mu\partial\sigma}\right]=\Psi_1(z_{c}),\\
f_{22}(z_{c}) =& \ \frac{\sigma^2}{n}\textrm{E}\left[-\frac{\partial^2\log\mathcal{L}}{\partial\sigma^2}\right]=\Psi_2(z_{c}),
\end{split}
\end{equation}
where the largest
$n-r$ sample values are censored, $\Phi^{-1}(p)$ is the $p$-quantile
of $\Phi(\cdot)$, and $z_{c}=\Phi^{-1}(r/n)$.
Here $\mathcal{L}\equiv\mathcal{L}(\mu,\sigma)$ is the likelihood function
and
\[
\Psi_i(a)=\int_{-\infty}^{a}\left[1+xH(x)\right]^iH(x)^{2-i}\phi(x)dx,\quad(i=0,1,2)
\]
where
\[
H(x)=\frac{\phi^\prime(x)}{\phi(x)}+\frac{\phi(x)}{1-\Phi(x)},
\]
$\phi^\prime(x)$ is the derivative of $\phi(x)$ and $\textrm{E}$ is the
expectation with respect to $Y$.
For \textrm{Type~1}{} (time) censored data
with fixed right censoring time $y_c$, the $f_{jk}$ $(jk=11,12,22)$
are still given by (\ref{S.eq:fim-1}) but $z_{c}=(y_c-\mu)/\sigma$ and
the number of failures $0\leq r\leq n$ is random having a binomial
distribution with probability $p=\Phi(z_c)$. Then for either
\textrm{Type~1}{} or \textrm{Type~2}{} censoring, the FIM for
$(\mu,\sigma)$ is
\begin{equation}
\label{S.equation:fisher-mu-sigma}
\text{I}_n(\mu,\sigma)=\frac{n}{\sigma^2}
\begin{bmatrix}
\Psi_0(z_{c}) & \Psi_1(z_{c})\\
\textrm{symmetric} & \Psi_2(z_{c})
\end{bmatrix}=\frac{n}{\sigma^2}
\begin{bmatrix}
f_{11}(z_{c}) & f_{12}(z_{c})\\
\textrm{symmetric} & f_{22}(z_{c})
\end{bmatrix}
.
\end{equation}
To simplify the presentation in the remainder of this section we
suppress the dependency of the $f_{ij}$ elements on $z_{c}$. That
is, for example, we write $f_{11}$ instead of $f_{11}(z_{c})$.
\subsection{Jeffreys Priors and Independence Jeffreys Priors}
\label{S.section:jeffreys.priors}
\subsubsection{Jeffreys prior}
The Jeffreys priors for the $(\mu, \sigma)$ parameterization for
\textrm{Type~1}{} and \textrm{Type~2}{} censoring are derived in
Section~\ref{section:jeffreys.prior.distribution} of the main paper
and will not be repeated here.
\subsubsection{Independence Jeffreys priors}
\label{S.mu-sigma-ind-jeff-prior}
The Independence Jeffreys priors for the $(\mu, \sigma)$
parameterization for \textrm{Type~1}{} and \textrm{Type~2}{} censoring are derived in
Section~\ref{section:ij.prior} of the main paper and will not be
repeated here.
\subsection{Reference Priors}
\label{S.section-ref-prior}
\subsubsection{General description}
\label{S.subsubsec-ref-prior}
It is well known that Jeffreys prior has desirable properties for
one-parameter models, but
deficiencies for multi-parameter models. For example, for the
$\mathrm{Norm}(\mu,\sigma)$ distribution, the Jeffreys prior is
$\pi(\mu,\sigma)\propto1/\sigma^2$, which does not have the
desirable properties of the right invariant prior
$\pi(\mu,\sigma)\propto1/\sigma$, as described in \citet[][Chapter
5]{GhoshDelampadyTapas2006}. Also see \citet[][pages 182--184]{Jeffreys1961}.
\citet{Bernardo1979}
proposed an alternative to the Jeffreys prior, called a reference
prior. The basic idea is to find the prior that maximizes the
Kullback-Leibler (KL) divergence between the prior and the
expected posterior. Reference priors have been further studied and extended by
\citet{BergerBernardo1989,BergerBernardo1992a,
BergerBernardo1992b}. For a scalar parameter, the reference prior
is the same as the Jeffreys prior.
For multiple parameters, in the case where all parameters are of
same importance, the reference prior again leads to the Jeffreys
prior (cf.~\citealt{KassWasserman1996}, page 1350). But one can also allow
parameters to have different importance. For example, for a two
parameter vector $(\theta_1,\theta_2)$, $\theta_1$ might be of
primary importance.
In this section, we employ the general methods in
\citet{GhoshDelampadyTapas2006} to derive reference priors for \textrm{Type~2}{}
censored data. We first obtain the conditional prior for
$\theta_2$ given $\theta_1$, which is denoted by
$\pi(\theta_2|\theta_1)\propto\sqrt{\text{I}_{22}(\boldsymbol{\theta})}$,
where
$\text{I}_{22}(\boldsymbol{\theta})=\text{E}\left[-\partial^2\log\mathcal{L}(\boldsymbol{\theta})/\partial\theta_2^2\right]$.
The marginal prior for $\theta_1$, denoted by $\pi(\theta_1)$ is
defined as the maximizer of the KL-divergence between the prior
$\pi(\boldsymbol{\theta})=\pi(\theta_1)\pi(\theta_2|\theta_1)$ and
the expected posterior distribution. It should be noted that the
computing of a reference prior, including the KL-divergence, is done
over a sequence of compact sets $K_i$ that increase in size, where
the union $\bigcup\limits_{i=1}^{\infty}K_i$ is the parameter space.
A reference prior is computed on each compact set $K_i$, followed by
a limiting operation. This approach is used to avoid improper prior
distributions that would otherwise arise after the limiting operation. With two
parameters, we use increasing rectangles $K_i$.
\subsubsection{Reference priors for \textrm{Type~2}{} censoring}
For \textrm{Type~2}{} censoring (or complete data), we first consider the
reference priors for parameter ordering $\theta_{(1)}=\mu$ and
$\theta_{(2)}=\sigma$, where $\theta_{(1)}$ is more important than
$\theta_{(2)}$ (later this ordering is denoted by $\{\mu,\sigma\}$).
The conditional prior is
$\pi_i(\sigma|\mu)\propto\sqrt{f_{22}}/\sigma$ on the rectangle
$K_i=K_{1i}\times K_{2i}$. Because $\sqrt{f_{22}}$ is a constant,
the conditional prior becomes
$\pi_i(\sigma|\mu)=c_i/\sigma$, where $c_i$ is generic notation for
a normalizing constant. The marginal prior for $\mu$ on $K_i$ that
maximizes the KL-divergence is
\begin{equation}\label{S.eq:marginal-mu-max}
\pi_i(\mu)\propto\exp\left\{\int_{K_{2i}}\frac{1}{2}\log\left[h_1(\boldsymbol{\theta})\right]\pi_i(\sigma|\mu)d\sigma\right\},
\end{equation}
where
\begin{equation}\label{S.eq:h1-function}
h_1(\boldsymbol{\theta})=\frac{|\text{I}_n(\mu,\sigma)|}{\text{I}_{22}(\mu,\sigma)}=\frac{1}{\sigma^2}\frac{n(f_{11}f_{22}-f_{12}^2)}{f_{22}}.
\end{equation}
So, the marginal prior for $\mu$ on rectangle $K_i$
\begin{equation}\label{S.eq:marginal-simple-form}
\pi_i(\mu)\propto\exp\left\{\int_{K_{2i}}\frac{1}{2}\log\left(\frac{\textrm{const}}{\sigma^2}\right)\frac{c_i}{\sigma}d\sigma\right\}
\end{equation}
is a constant. Thus, $\pi_i(\mu)$ is flat on $K_i$ and the
reference prior for parameter ordering $\theta_{(1)}=\mu$,
$\theta_{(2)}=\sigma$ is
$\pi_i(\mu,\sigma)=\pi_i(\sigma|\mu)\pi_i(\mu)\propto1/\sigma$.
Therefore, the reference prior for parameter ordering $\{\mu,\sigma\}$ is
$\pi(\mu,\sigma)\propto1/\sigma$.
We denote such a prior by $\pi(\{\mu,\sigma\})$
to indicate the order of parameters.
For parameter ordering $\theta_{(1)}=\sigma$ and $\theta_{(2)}=\mu$,
the conditional prior is
$\pi_i(\mu|\sigma)\propto\sqrt{nf_{11}}/\sigma$. Given the value of
$\sigma$, the conditional prior for $\mu$ on $K_i$, which is denoted
by $\pi_i(\mu|\sigma)=c_i$, is flat as $f_{11}$ is a known constant.
The marginal prior for $\sigma$ on $K_{i}$ is
\[
\pi_i(\sigma)\propto\exp\left\{\int_{K_{1i}}\frac{1}{2}\log\left[h_2(\boldsymbol{\theta})\right]\pi_i(\mu|\sigma)d\mu\right\},
\]
where
$$
h_2(\boldsymbol{\theta})=\frac{|\text{I}_n(\mu,\sigma)|}{\text{I}_{11}(\mu,\sigma)}=\frac{1}{\sigma^2}\frac{n(f_{11}f_{22}-f_{12}^2)}{f_{11}}.
$$
So, the marginal prior for $\sigma$ on rectangle $K_i$
\[
\pi_i(\sigma)\propto\exp\left\{\int_{K_{1i}}\frac{1}{2}\log\left(\frac{\textrm{const}}{\sigma^2}\right)c_id\mu\right\}=\frac{c_i}{\sigma}.
\]
Thus, the joint prior on the rectangle $K_i$ is $\pi_i(\mu,\sigma)=c_i/\sigma$.
Then, as a limit of $\pi_i(\mu,\sigma)$, the reference prior for $\{\sigma,\mu\}$ is $\pi(\{\sigma,\mu\})\propto1/\sigma$.
\subsubsection{Reference priors for \textrm{Type~1}{} censoring}
\label{S.section:reference.priors.for.typeI.censoring}
Although we could use the same procedure to compute reference priors
for \textrm{Type~1}{} censoring data, the computations are usually intractable
and do not have closed forms. For \textrm{Type~1}{} censoring, $h_1(\boldsymbol{\theta})$
is no longer proportional to $1/\sigma^2$ because $f_{11}$,
$f_{12}$, and $f_{22}$ are no longer constant. Thus, the integral
in (\ref{S.eq:marginal-mu-max}) will not reduce to the simple integral
in (\ref{S.eq:marginal-simple-form}). Obtaining the analytical form
of the integral in~(\ref{S.eq:marginal-mu-max}) under \textrm{Type~1}{} censoring
is difficult and in the rest of this work, we only provide reference
priors for \textrm{Type~2}{} censoring (or complete) data. We will, however,
provide results for Jeffreys, CJ and IJ priors for \textrm{Type~1}{} censoring.
\subsection{Using the Parameterization $(y_p,\sigma)$}
\label{S.subsec:use-y_p-sigma}
\subsubsection{FIM for $(y_p,\sigma)$}
The $p$ quantile of a location-scale random variable
$Y$ is $y_p=\mu+\sigma\Phi^{-1}(p)$ so that
$z_{c}=\Phi^{-1}(r/n)$ for \textrm{Type~2}{} censoring and
$z_{c}\equiv(y_c-\mu)/\sigma=[y_c-y_p-\sigma\Phi^{-1}(p)]/\sigma$
for \textrm{Type~1}{} censoring.
The large-sample approximate covariance matrices of ML estimators
$(\widehat{\mu},\widehat{\sigma})$ and ML estimators
$(\widehat{y}_p,\widehat{\sigma})$ are, respectively, the inverse of
the FIMs $\text{I}_n^{-1}(\mu,\sigma)$ and
$\text{I}_n^{-1}(y_p,\sigma)$.
The large-sample approximate covariance matrix for
$(\widehat{y}_p,\widehat{\sigma})$ is $\text{I}^{-1}_n(y_p,\sigma)$
and can be computed from the large-sample approximate covariance matrix for
$(\widehat{\mu},\widehat{\sigma})$ using the delta method:
\[
\begin{split}
&\text{I}_n^{-1}[y_p(\mu,\sigma),\sigma(\mu,\sigma)]=\mathcal{J} \, \text{I}_n^{-1}(\mu,\sigma)\mathcal{J}^\prime\\[1ex]=&
\frac{\sigma^2}{n(f_{12}^2-f_{11}f_{22})}\times\begin{bmatrix}
\Phi^{-1}(p)\left[f_{12}-\Phi^{-1}(p)f_{11}\right]-f_{22}+\Phi^{-1}(p)f_{12}
& f_{12}-\Phi^{-1}(p)f_{11}\\ \textrm{symmetric} & -f_{11}
\end{bmatrix},
\end{split}
\]
where the Jacobian $\mathcal{J}$ is
\[
\mathcal{J}=
\begin{bmatrix}
\dfrac{\partial y_p}{\partial\mu} & \dfrac{\partial y_p}{\partial\sigma}\\
\dfrac{\partial\sigma}{\partial\mu} & \dfrac{\partial\sigma}{\partial\sigma}
\end{bmatrix}=
\begin{bmatrix}
1 & \Phi^{-1}(p) \\
0 & 1
\end{bmatrix}.
\]
Then the FIM for $(y_p,\sigma)$ is
\begin{equation}
\label{S.equation-page-5}
\text{I}_n(y_p,\sigma)=\frac{n}{\sigma^2}
\begin{bmatrix}
f_{11} & f_{12}-\Phi^{-1}(p)f_{11} \\
\textrm{symmetric} & f_{11}[\Phi^{-1}(p)]^2-2f_{12}\Phi^{-1}(p)+f_{22}
\end{bmatrix}.
\end{equation}
\subsubsection{Jeffreys prior}
\label{S.subsec-yp-logsigma}
The determinant of the FIM (\ref{S.equation-page-5}) is
\[
\left|\text{I}_n(y_p,\sigma)\right|=\frac{n^2}{\sigma^4}\left(f_{11}f_{22}-f_{12}^2\right).
\]
Because the determinant is the same as in
(\ref{equation:jeffreys-mu-sigma}), the Jeffreys prior for $(y_p,\sigma)$
is the same as $(\mu,\sigma)$ for both \textrm{Type~2}{} and \textrm{Type~1}{} censoring.
\subsubsection{IJ prior}
For \textrm{Type~2}{} censored or complete data, the elements of the FIM are
known constants and thus as $\pi(y_p|\sigma)\propto1$ and
$\pi(\sigma|y_p)\propto1/\sigma$. Then the IJ
prior is $\pi(y_p,\sigma)\propto{1}/{\sigma}$.
For \textrm{Type~1}{}
censoring, the CJ prior for $y_p$ is
\[
\pi(y_p|\sigma)\propto\sqrt{f_{11}},
\]
and the CJ prior for $\sigma$ is
\[
\pi(\sigma|y_p)\propto\frac{1}{\sigma}\sqrt{f_{11}[\Phi^{-1}(p)]^2-2f_{12}\Phi^{-1}(p)+f_{22}},
\]
which is the same as $\pi(\sigma|\mu)$
in Section~\ref{S.mu-sigma-ind-jeff-prior} when $p$ is chosen such
that $\Phi^{-1}(p)=0$.
Then the IJ prior for $(y_p,\sigma)$ is
\[
\begin{split}
\pi(y_p, \sigma)\propto&\,\pi(y_p|\sigma)\pi(\sigma|y_p)\\
\propto&\,\frac{1}{\sigma}\sqrt{f_{11}\left\{f_{11}[\Phi^{-1}(p)]^2-2f_{12}\Phi^{-1}(p)+f_{22}\right\}}.
\end{split}
\]
\subsubsection{Reference prior}
We consider \textrm{Type~2}{} censored or complete data.
For parameter ordering $\{y_p, \sigma\}$, the conditional Jeffreys prior
for $\sigma$ is $\pi(\sigma|y_p)\propto1/\sigma$.
The marginal prior for $y_p$ is
\[
\pi(y_p)\propto\exp\left\{\int\frac{1}{2}\log\left[\frac{|\textrm{I}_n(y_p,\sigma)|}{\mathrm{const}/\sigma^2}\right]\frac{1}{\sigma}d\sigma\right\}\propto 1.
\]
Thus the reference prior for $\{y_p,\sigma\}$ is
\[
\pi(\{y_p,\sigma\})\propto\frac{1}{\sigma}.
\]
For parameter ordering $\{\sigma, y_p\}$,
the conditional Jeffreys prior is $\pi(y_p|\sigma)\propto 1$.
The marginal prior for $\sigma$ is given by
\[
\pi(\sigma)\propto\exp\left\{\int\frac{1}{2}\log\left[\frac{\mathrm{I}_n(y_p,\sigma)}{\mathrm{const}/\sigma^2}\right]dy_p\right\}\propto\frac{1}{\sigma}.
\]
Thus the reference prior for $\{\sigma,y_p\}$ is
\[
\pi(\{\sigma,y_p\})\propto\frac{1}{\sigma}.
\]
\subsection{Using the Parameterization $(t_p, \sigma)$}
\label{S.section.parameterization.tp.sigma}
The $p$ quantile of a log-location-scale random variable $T$ is
$t_p=\exp(y_p)=\exp\left[\mu+\sigma\Phi^{-1}(p)\right]$.
\subsubsection{FIM for $(t_p, \sigma)$}
\label{S.section:fim.tp.sigma}
Using the delta method, the inverse of the FIM for $(t_p, \sigma)$ is
\[
\begin{split}
\text{I}^{-1}_n\left(t_p,\sigma\right)=&\mathcal{J} \, \text{I}^{-1}_n(\mu,\sigma)\mathcal{J}^\prime\\ =&\frac{\sigma^2}{n(f_{12}^2-f_{11}f_{22})}\times\begin{bmatrix}
t_p^2\left\{\Phi^{-1}(p)\left[f_{12}-\Phi^{-1}(p)f_{11}\right]-f_{22}+\Phi^{-1}(p)f_{12}\right\}
& t_p\left[f_{12}-\Phi^{-1}(p)f_{11}\right]\\ \textrm{symmetric} &
-f_{11}
\end{bmatrix},
\end{split}
\]
where the Jacobian $\mathcal{J}$ is
\[
\mathcal{J}=
\begin{bmatrix}
\dfrac{\partial t_p}{\partial\mu} & \dfrac{\partial
t_p}{\partial\sigma}\\ \dfrac{\partial\sigma}{\partial\mu}
& \dfrac{\partial\sigma}{\partial\sigma}
\end{bmatrix}=
\begin{bmatrix}
t_p & t_p\Phi^{-1}(p) \\
0 & 1
\end{bmatrix}.
\]
Then the FIM for $(t_p, \sigma)$ is
\[
\mathrm{I}_n(t_p,\sigma)=\frac{n}{\sigma^2}\begin{bmatrix}
\begin{array}{lr}
{f_{11}}/{t_p^2} &
{[f_{12}-f_{11}\Phi^{-1}(p)]}/{t_p}
\\ \textrm{symmetric} &
{f_{11}\Phi^{-1}(p)-2f_{12}\Phi^{-1}(p)+f_{22}}
\end{array}
\end{bmatrix}.
\]
\subsubsection{Jeffreys prior}
The determinant of the FIM is
\[
\mathrm{det}\left[\mathrm{I}_n(t_p,\sigma)\right]=n^2\frac{(f_{11}f_{12}-f_{12}^2)}{t_p^2\sigma^4}.
\]
For \textrm{Type~2}{} censoring, the Jeffreys prior is
\[
\pi(t_p,\sigma)\propto\frac{1}{t_p\sigma^2}.
\]
For \textrm{Type~1}{} censoring, the Jeffreys prior is
\[
\pi(t_p,\sigma)\propto\frac{1}{t_p\sigma^2}\sqrt{f_{11}f_{12}-f_{12}^2}.
\]
\subsubsection{IJ prior}
\label{S.section:ij.prior.tp.sigma}
The CJ prior for $t_p$ given $\sigma$ is
$\pi(t_p|\sigma)\propto\sqrt{f_{11}}/(t_p\sigma )$ and that for
$\sigma$ given $t_p$ is
\begin{align*}
\pi(\sigma|t_p) & \propto \frac{1}{\sigma}\sqrt{f_{11}\Phi^{-1}(p)-2f_{12}\Phi^{-1}(p)+f_{22}}.
\end{align*}
For \textrm{Type~2}{} censoring, we have $\pi(t_p|\sigma)\propto1/t_p$ and
$\pi(\sigma|t_p)\propto1/\sigma$. Then the IJ
prior is
\[
\pi(t_p,\sigma)\propto\frac{1}{t_p\sigma}.
\]
For \textrm{Type~1}{} censoring, we have
$\pi(t_p|\sigma)\propto\sqrt{f_{11}}/t_p$ and
\begin{align*}
\pi(\sigma|t_p) & \propto \frac{1}{\sigma} \sqrt{f_{11}\Phi^{-1}(p)-2f_{12}\Phi^{-1}(p)+f_{22}}.
\end{align*}
Then the IJ prior is
\[
\pi(t_p,\sigma)\propto\frac{1}{t_p\sigma}\sqrt{f_{11}[f_{11}\Phi^{-1}(p)-2f_{12}\Phi^{-1}(p)+f_{22}]}.
\]
\subsubsection{Reference prior}
We only consider complete or \textrm{Type~2}{} censored data.
For $\{t_p,\sigma\}$, the conditional Jeffreys prior for $\sigma$ is $\pi(\sigma|t_p)\propto1/\sigma$.
Then the marginal prior for $t_p$ is
\[
\pi(t_p)=\exp\left\{\frac{1}{2}\int\log\left[\frac{1/(t_p^2\sigma^4)}{1/\sigma^2}\right]\frac{1}{\sigma}d\sigma\right\}\propto\frac{1}{t_p}.
\]
The reference prior for $\{t_p,\sigma\}$ is
\[
\pi(\{t_p,\sigma\})\propto\frac{1}{t_p\sigma}.
\]
For $\{\sigma,t_p\}$, the conditional Jeffreys prior for $t_p$ is $\pi(t_p|\sigma)\propto1/t_p$.
The marginal prior for $\pi(\sigma)$ is
\[
\pi(\sigma)\propto\exp\left\{\frac{1}{2}\int\log\left[\frac{1/(t_p^2\sigma^4)}{1/(t_p^2\sigma^2)}\right]dy_p\right\}\propto\frac{1}{\sigma}.
\]
The reference prior for
$$
\pi(\{\sigma,t_p\})\propto\frac{1}{t_p\sigma}.
$$
\subsection{Using the Parameterization $(\log(t_{p_{r}}),\log(\sigma))$}
\label{section:priors.y.log.sigam.parameterization}
As described in Section~\ref{subsection:param.for.priors} of the
paper, $(\log(t_{p_{r}}),\log(\sigma))$ is the parameterization that we used for
our MCMC computations.
To simplify the notation in the derivation,
let $\tau\equiv\log(\sigma)$ and $y_p=\log(t_{p_{r}})$.
\subsubsection{FIM for $(y_p,\log(\sigma))$}
Then the inverse
of the FIM for $(y_p,\tau)$ is
\[
\begin{split}
&\text{I}_n^{-1}(y_p,\tau)=\mathcal{J} \, \text{I}_n^{-1}(\mu,\sigma)\mathcal{J}^\prime\\
=&\frac{1}{n\left[f_{12}^2-f_{11}f_{22}\right]}\times\\
&\begin{bmatrix}
\sigma^2\left\{\Phi^{-1}(p)\left[f_{12}-\Phi^{-1}(p)f_{11}\right]-f_{22}+\Phi^{-1}(p)f_{12}\right\} & \sigma\left[f_{12}-\Phi^{-1}(p)f_{11}\right]\\
\textrm{symmetric} & -f_{11}
\end{bmatrix},
\end{split}
\]
where the Jacobian is
\[
\mathcal{J}=
\begin{bmatrix}
\dfrac{\partial y_p}{\partial\mu} & \dfrac{\partial y_p}{\partial\sigma}\\
\dfrac{\partial\tau}{\partial\mu} & \dfrac{\partial\tau}{\partial\sigma}
\end{bmatrix}
=
\begin{bmatrix}
1 & \Phi^{-1}(p) \\
0 & 1/\sigma
\end{bmatrix}.
\]
Thus, the FIM for $(y_p,\tau)$ is
\[
\begin{split}
\text{I}_n(y_p,\tau)&=n
\begin{bmatrix}
f_{11}/{\sigma^2} & [f_{12}-\Phi^{-1}(p)f_{11}]/{\sigma}\\
\textrm{symmetric} & f_{11}[\Phi^{-1}(p)]^2-2f_{12}\Phi^{-1}(p)+f_{22} \\
\end{bmatrix}\\[1.5ex]
&=n\begin{bmatrix}
f_{11}/{\exp(2\tau)} & [f_{12}-\Phi^{-1}(p)f_{11}]/{\exp(\tau)}\\
\textrm{symmetric} & f_{11}[\Phi^{-1}(p)]^2-2f_{12}\Phi^{-1}(p)+f_{22} \\
\end{bmatrix}.
\end{split}
\]
\subsubsection{Jeffreys prior}
The determinant of the FIM is
\[
\left|\text{I}_n(y_p,\tau)\right|=\frac{n^2}{\exp(2\tau)}\left(f_{11}f_{22}-f_{12}^2\right)=\frac{n^{2}}{\sigma^2}\left(f_{11}f_{22}-f_{12}^2\right).
\]
Then, for \textrm{Type~2}{} censoring,
the Jeffreys prior is $\pi(y_p,\tau)\propto1/\sigma$
because the $f_{ij}$ elements are fixed constants.
This result also can be obtained directly
from Section~\ref{S.subsec-yp-logsigma} using the fact that the
Jeffreys prior is invariant to parameter transformation. For
example, we already know that $\pi(y_p,\sigma)\propto1/\sigma^2$.
Then the Jeffreys prior for parameters $(y_p,\tau)$ can be computed
as
\[
\pi(y_p,\tau)\propto\frac{1}{\exp(2\tau)}\left|
\begin{bmatrix}
\dfrac{\partial y_p}{\partial y_p} & \dfrac{\partial y_p}{\partial \tau} \\
\dfrac{\partial \sigma}{\partial y_p}& \dfrac{\partial \sigma}{\partial \tau} \\
\end{bmatrix}
\right|=\frac{1}{\exp(2\tau)}\left|\begin{bmatrix}
1 & 0\\
0 & \exp(\tau)
\end{bmatrix}\right|=\frac{1}{\exp(\tau)}=\frac{1}{\sigma}.
\]
For \textrm{Type~1}{} censoring, the Jeffreys prior is
$$
\pi(y_p,\tau)\propto\frac{1}{\exp(\tau)}\sqrt{f_{11}f_{22}-f_{12}^2}=\frac{1}{\sigma}\sqrt{f_{11}f_{22}-f_{12}^2}.
$$
\subsubsection{IJ prior}
\label{subsubsection:ij.prior.yp.log.sigma}
For \textrm{Type~2}{} censoring, the CJ prior for $y_p$
given $\tau$ is $\pi(y_p|\tau)\propto1$; the CJ
prior for $\tau$ given $y_p$ is $\pi(\tau|y_p)\propto 1$. Thus thee
IJ prior is $\pi(y_p,\tau)\propto 1$.
For \textrm{Type~1}{}
censoring, the CJ prior for $y_p$ given $\tau$ is
$\pi(y_p|\tau)\propto\sqrt{f_{11}}$; the CJ prior
for $\tau$ given $y_p$ is
$$
\pi(\tau|y_p)\propto\sqrt{f_{11}[\Phi^{-1}(p)]^2-2f_{12}\Phi^{-1}(p)+f_{22}}.
$$
So, the IJ prior is
\[
\begin{split}
\pi(y_p,\tau)\propto&\pi(y_p|\tau)\pi(\tau|y_p)\\[1ex]
\propto&\sqrt{f_{11}\left\{f_{11}[\Phi^{-1}(p)]^2-2f_{12}\Phi^{-1}(p)+f_{22}\right\}}.
\end{split}
\]
\subsubsection{Reference prior}
For \textrm{Type~2}{} censoring or complete data, the reference prior for
parameter ordering $\{y_p,\tau\}$
is computed by first deriving the conditional Jeffreys prior $\tau$
\[
\pi(\tau|y_p)\propto1;
\]
the marginal prior for $y_p$ is
\[
\pi(y_p)\propto\exp\left\{\frac{1}{2} \int \log\left[\frac{1/\exp(2\tau)}{1}\right]d\tau\right\}\propto1.
\]
Thus the reference prior for $\{y_p,\tau\}$ is given by
\[
\pi(\{y_p,\tau\})\propto1.
\]
For parameter ordering $\{\tau,y_p\}$, the conditional Jeffreys prior for $y_p$ is $\pi(y_p|\tau)\propto1$ and the marginal prior for $\tau$ is
\[
\pi(\tau)\propto\exp\left\{\frac{1}{2} \int \log\left[\frac{1/\exp(2\tau)}{1/\exp(2\tau)}\right]
\pi(y_p|\tau) dy_p\right\}\propto
1.
\]
The reference prior for $\{\tau,y_p\}$ is
\[
\pi(\{\tau, y_p\})\propto1.
\]
\subsection{Using the Parameterization $(\zeta_{e},\sigma)$}
\subsubsection{FIM for $(\zeta_{e},\sigma)$}
Let $\zeta_{e}=(y_e-\mu)/\sigma$, where $y_e>0$ is a given value. The
parameter $\zeta_{e}$ is important because
inferences are often needed for $\Pr(Y\leq y_e)=\Phi(\zeta_{e})$.
For \textrm{Type~2}{} censoring, $z_{c}=\Phi^{-1}(r/n)$; for \textrm{Type~1}{} censoring,
$z_{c}=(y-y_e+\sigma\zeta_{e})/\sigma$. The inverse of the FIM for
$(\zeta_{e},\sigma)$ is
\[
\begin{split}
\text{I}_n^{-1}[\zeta_{e}(\mu,\sigma),\sigma(\mu,\sigma)]&=\mathcal{J} \,
\text{I}_n^{-1}(\mu,\sigma)\mathcal{J}^\prime\\
&=\frac{1}{n(f_{12}^2-f_{11}f_{22})}\times
\begin{bmatrix}
\zeta_{e}[f_12-\zeta_{e} f_{11}]-f_{22}+\zeta_{e} f_{11} & \sigma[\zeta_{e} f_{11}-f_{12}] \\
\textrm{symmetric} & -\sigma^2f_{11}
\end{bmatrix}
\end{split}
\]
where the Jacobian $\mathcal{J}$ is
\[
\mathcal{J}=
\begin{bmatrix}
\dfrac{\partial\zeta_{e}}{\partial\mu} & \dfrac{\partial\zeta_{e}}{\partial\sigma} \\
\dfrac{\partial\sigma}{\partial\mu} & \dfrac{\partial\sigma}{\partial\sigma}
\end{bmatrix}=
\begin{bmatrix}
-\dfrac{1}{\sigma} & -\dfrac{\zeta_{e}}{\sigma}\\
0 & 1
\end{bmatrix}.
\]
By replacing $(\mu,\sigma)$ with $(\zeta_{e},\sigma)$ in
$\text{I}_n[\zeta_{e}(\mu,\sigma),\sigma(\mu,\sigma)]$, the FIM for
$(\zeta_{e},\sigma)$ is
\begin{equation}
\label{S.equation-psi}
\text{I}_n(\zeta_{e},\sigma)=\frac{n}{\sigma^2}
\begin{bmatrix}
\sigma^2f_{11} & \sigma(f_{11}\zeta_{e}-f_{12})\\
\textrm{symmetric} & f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}
\end{bmatrix}.
\end{equation}
\subsubsection{Jeffreys prior}
The determinant of the FIM (\ref{S.equation-psi}) is
\[
|\text{I}_n(\zeta_{e},\sigma)|=\frac{n^2}{\sigma^2}\left(f_{22}f_{11}-f_{12}^2\right).
\]
For \textrm{Type~2}{} censoring or complete data, the Jeffreys prior is
$
\pi(\zeta_{e},\sigma)\propto{1}/{\sigma}.
$
For \textrm{Type~1}{} censoring, the Jeffreys prior is
\[
\pi(\zeta_{e},\sigma)\propto\frac{1}{\sigma}{\sqrt{f_{22}f_{11}-f_{12}^2}}.
\]
\subsubsection{IJ prior}
For \textrm{Type~2}{} censoring using the appropriate elements of
(\ref{S.equation-psi}), the CJ prior for $\zeta_{e}$ given $\sigma$ is
$\pi(\zeta_{e}|\sigma)\propto1$; the prior for $\sigma$ given $\zeta_{e}$ is
$\pi(\sigma|\zeta_{e})\propto1/\sigma$. So, the IJ prior for
$(\zeta_{e}, \sigma)$ is
$\pi(\zeta_{e},\sigma)\propto{1}/{\sigma}. $
For \textrm{Type~1}{} censoring, the
CJ prior for $\zeta_{e}$ given $\sigma$
is $\pi(\zeta_{e}|\sigma)\propto\sqrt{f_{11}}$ and the
CJ prior for $\sigma$ given $\zeta_{e}$ is
$\pi(\sigma|\zeta_{e})\propto(1/\sigma)\sqrt{f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}}$.
So, the IJ prior for $(\zeta_{e},\sigma)$ is
\[
\pi(\zeta_{e},\sigma)\propto\frac{1}{\sigma}{\sqrt{f_{11}(f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22})}}.
\]
\subsubsection{Reference prior}
We consider the reference prior for \textrm{Type~2}{} censoring and complete
data.
For parameter ordering $\theta_{(1)}=\zeta_{e}$ and
$\theta_{(2)}=\sigma$, the conditional prior on
$K_i=K_{1i}\times K_{2i}$ is $\pi(\sigma|\zeta_{e})\propto1/\sigma$ and
it denoted by $\pi_i(\sigma|\zeta_{e})=c_i/\sigma$. Then the marginal
prior for $\zeta_{e}$ is
\begin{equation*}
\begin{split}
\pi_i(\zeta_{e})=&\,c_i\left(\int_{K_{2i}}\frac{1}{2}\log\left\{\frac{|\text{I}_n(\zeta_{e},\sigma)|}{[f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}]/\sigma^2}\right\}\frac{c_i}{\sigma}d\sigma\right)\\
=&\frac{c_i}{\sqrt{f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}}}.
\end{split}
\end{equation*}
Here $c_i$ is the generic notation for a normalizing constant.
Thus, the reference prior for $\{\zeta_{e},\sigma\}$ is
\begin{equation*}
\pi(\{\zeta_{e},\sigma\})\propto\frac{1}{\sigma\sqrt{f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}}}.
\end{equation*}
For parameter ordering $\{\sigma,\zeta_{e}\}$, the conditional
prior on $K_i=K_{1i}\times K_{2i}$ is $\pi(\zeta_{e}|\sigma)\propto
c_i$. The marginal prior for $\zeta_{e}$ is
\[
\pi_i(\sigma)\propto\exp\left\{\int_{K_{1i}}\frac{1}{2}\log\left[\frac{|\text{I}_n(\zeta_{e},\sigma)|}{f_{11}}\right]{c_i}d\zeta_{e}\right\}\propto\frac{1}{\sigma}.
\]
So, the corresponding reference prior is
$
\pi(\{\sigma,\zeta_{e}\})\propto{1}/{\sigma}
$.
\subsection{Using the Parameterization $(\zeta_{e},\log(\sigma))$}
\label{S.section-zeta-log.sigma}
\subsubsection{FIM for $(\zeta_{e},\log(\sigma))$}
We define $\tau=\log(\sigma)$; then the inverse of the FIM for
$(\zeta_{e},\tau)$ is
\[
\begin{split}
\text{I}^{-1}_n\left[\zeta_{e},\tau(\sigma)\right]=&\mathcal{J} \, \text{I}^{-1}_n(\zeta_{e},\sigma)\mathcal{J}^\prime\\
=&\frac{1}{n[f_{12}^2-f_{11}f_{22}]}\times
\begin{bmatrix}
\zeta_{e}[f_{12}-\zeta_{e} f_{11}]-f_{22}+\zeta_{e} f_{11} & \zeta_{e} f_{11}-f_{12} \\
\textrm{symmetric} & -f_{11}
\end{bmatrix},
\end{split}
\]
where
\[
\mathcal{J}=\begin{bmatrix}
1 & 0\\
0 & \dfrac{1}{\sigma}
\end{bmatrix}.
\]
Then the FIM for $(\zeta_{e},\tau)$ is
\begin{equation}\label{S.fim-psi-tau}
\text{I}_n(\zeta_{e},\tau)=n\begin{bmatrix}
f_{11} & \zeta_{e} f_{11}-f_{12}\\
\textrm{symmetric} & f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}
\end{bmatrix}.
\end{equation}
\subsubsection{Jeffreys prior}
The determinant of the FIM in (\ref{S.fim-psi-tau}) is
\[
\left|\text{I}_n(\zeta_{e},\tau)\right|=n^2\left[f_{11}f_{22}-f_{12}^2\right].
\]
For \textrm{Type~2}{} censoring or complete data, the Jeffreys prior is
$\pi(\zeta_{e},\tau)\propto1$.
For \textrm{Type~1}{} censoring data, the Jeffreys prior is
$$
\pi(\zeta_{e},\tau)\propto\sqrt{f_{11}f_{22}-f_{12}^2}.
$$
\subsubsection{IJ prior}
For \textrm{Type~2}{} censoring or complete data using the appropriate
elements of (\ref{S.fim-psi-tau}), the CJ prior for $\zeta_{e}$ is
$\pi(\zeta_{e}|\tau)\propto1$ and the CJ prior for $\tau$ is
$\pi(\tau|\zeta_{e})\propto1$; thus the IJ prior is
$\pi(\zeta_{e},\tau)\propto1$.
For \textrm{Type~1}{} censoring, the prior for
$\zeta_{e}$ is $\pi(\zeta_{e}|\tau)\propto\sqrt{f_{11}}$ and the prior for
$\tau$ is
$\pi(\tau|\zeta_{e})\propto\sqrt{f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}}$.
Then the IJ prior is
\[
\pi(\zeta_{e},\tau)\propto{\sqrt{f_{11}[f_{11}\zeta_{e}^2-2f_{12}\zeta_{e}+f_{22}]}}.
\]
\subsubsection{Reference prior}
We consider the reference prior for \textrm{Type~2}{} or complete data.
For parameter ordering $\{\zeta_e,\tau\}$, the CJ prior for $\tau$ is $\pi(\tau|\zeta_e)\propto1$ and the marginal distribution for $\zeta_e$ is
\[
\pi(\zeta_e)\propto\exp\left\{\frac{1}{2} \int \log\left[\frac{1}{f_{11}\zeta_e^2-2f_{12}\zeta_e+f_{22}}\right]d\tau\right\}\propto\frac{1}{\sqrt{f_{11}\zeta_e^2-2f_{12}\zeta_e+f_{22}}}.
\]
Thus the reference prior for parameter ordering $\{\zeta_e,\tau\}$ is
$$
\pi(\{\zeta_e,\tau\})\propto\frac{1}{\sqrt{f_{11}\zeta_e^2-2f_{12}\zeta_e+f_{22}}}.
$$
For $\{\tau,\zeta_e\}$, the CJ prior for $\zeta_e$ is $\pi(\zeta_e|\tau)\propto1$ and the marginal prior for $\tau$ is given by
\[
\pi(\tau)\propto\exp\left\{\frac{1}{2} \int \log\left[\frac{|\mathrm{I}|}{nf_{11}}\right]d\zeta_e\right\}\propto1.
\]
Then the reference prior for $\{\tau,\zeta_e\}$ is $\pi(\{\tau,\zeta_e\})\propto1$.
\subsection{\,Proof of invariance of IJ priors to one-to-one reparameterizations}
\label{S.section:invariant.one.one.reparameterization}
Here we show that IJ priors for log-location-scale distributions
are invariant (in the sense described in
Section~\ref{section:jeffreys.prior.distribution} of the paper) to
one-to-one monotone reparameterizations of either or both of the
$(t_{p},\sigma)$ parameters.
Suppose that the original parameters are $(t_p,\sigma)$. The
new parameters are
$\theta_1=A(t_p)$ and
$\theta_2=B(\sigma)$ where $A$ and $B$ are one-to-one monotone
transformations. Denote the derivatives by
\[
a=\frac{\mathrm{d} \theta_1}{\mathrm{d} t_p}\quad\text{and}\quad b=\frac{\mathrm{d} \theta_2}{\mathrm{d}\sigma}.
\]
Here $a$ is written as a function of $\theta_1$ as $a=a(\theta_1)$
and $b$ is written as a function of $\theta_2$ as $b=b(\theta_2)$.
The inverse of the FIM (large-sample approximate covariance matrix) under
parameterization $(t_p,\sigma)$ is (from
Section~\ref{S.section:fim.tp.sigma})
$$
\mathrm{I}^{-1}(t_p,\sigma)=\frac{\sigma^{2}}
{n\left(f_{12}^{2}-f_{11} f_{22}\right)} \times\left[\begin{array}{cc}
t_{p}^{2}\left\{\Phi^{-1}(p)
\left[f_{12}-\Phi^{-1}(p) f_{11}\right]-f_{22}+\Phi^{-1}(p)
f_{12}\right\} &
t_{p}\left[f_{12}-\Phi^{-1}(p) f_{11}\right] \\
\text { symmetric } & -f_{11}
\end{array}\right].
$$
Using the delta method, the inverse FIM (large-sample approximate covariance
matrix) for the $(\theta_1,\theta_2)$ parameterization
\[
\begin{split}
\mathrm{I}^{-1}(\theta_1,\theta_2)=&
\frac{\left[\sigma(\theta_2)\right]^2}
{n\left(f_{12}^{2}-f_{11} f_{22}\right)}\times\\
&\mathcal{J}\left[\begin{array}{cc}
\left[t_p(\theta_1)\right]^{2}
\left\{\Phi^{-1}(p)\left[f_{12}-\Phi^{-1}(p) f_{11}\right]-f_{22}+\Phi^{-1}(p) f_{12}\right\} & t_p(\theta_1)\left[f_{12}-\Phi^{-1}(p) f_{11}\right] \\
\text { symmetric } & -f_{11}
\end{array}\right]\mathcal{J}^{\prime}
\end{split},
\]
where
\[
\mathcal{J}=\begin{bmatrix}
a & 0\\
0 & b
\end{bmatrix}.
\]
Then the FIM for the $(\theta_1,\theta_2)$ parameterization is
\[
\begin{split}
\mathrm{I}(\theta_1,\theta_2)=&
\frac{n}{\left[t_p(\theta_1)\right]^2
\left[\sigma(\theta_2)\right]^2a^2b^2}\times\\
&\begin{bmatrix}
b^2f_{11} & ab\sigma(\theta_2)
\left[f_{12}-\Phi^{-1}(p)f_{11}\right] \\
\text{ symmetric } & -a^2
\left[t_p(\theta_1)\right]^2\left\{\Phi^{-1}(p)
\left[f_{12}-\Phi^{-1}(p)f_{11}\right]-f_{22}+\Phi^{-1}(p)f_{12}\right\} \\
\end{bmatrix}
\end{split}
\]
For the CJ priors, we have
\[
\pi(\theta_1|\theta_2)\propto
\frac{1}{at_p(\theta_1)}\quad\text{and}\quad\pi(\theta_2|\theta_1)
\propto\frac{1}{b\sigma(\theta_2)}.
\]
So, the IJ prior is
\[
\pi_1(\theta_1,\theta_2)\propto
\frac{1}{t_p(\theta_1)\sigma(\theta_2)}\frac{1}{ab}.
\]
To prove that the IJ prior is invariant to the transformations, we can
first perform the variable transformation on the IJ prior using
$(t_p,\sigma)$ and show that the prior after transformation is the
same as $\pi_1(\theta_1,\theta_2)$. From
Section~\ref{S.section:ij.prior.tp.sigma}, the IJ prior using
$(t_p,\sigma)$ is $1/(t_p\sigma)$. First replace $t_p$
and $\sigma$ with $t_p(\theta_1)$ and $\sigma(\theta_2)$, then
finish the transformation with the Jacobian
\[
\pi_2(\theta_1,\theta_2)\propto\frac{1}{t_p(\theta_1)\sigma(\theta_2)}|\mathcal{J}|,
\]
where
\[
|\mathcal{J}|=\left|\begin{bmatrix}
\dfrac{\partial t_p(\theta_1)}{\partial \theta_1} & \dfrac{\partial t_p(\theta_1)}{\partial \theta_2}\\
\dfrac{\partial \sigma(\theta_2)}{\partial \theta_1} & \dfrac{\partial \sigma(\theta_2)}{\partial \theta_2}
\end{bmatrix}\right|=\dfrac{1}{|\mathcal{J}^\ast|},
\]
and
\[
\mathcal{J}^\ast=\begin{bmatrix}
\dfrac{\partial \theta_1}{\partial t_p} & \dfrac{\partial \theta_1}{\partial \sigma}\\
\dfrac{\partial \theta_2}{\partial t_p} & \dfrac{\partial \theta_2}{\partial \sigma}
\end{bmatrix}=
\begin{bmatrix}
a & 0\\
0 & b\\
\end{bmatrix}
\quad\text{and}\quad|\mathcal{J}^\ast|=ab.
\]
Thus
\begin{align*}
\pi_2(\theta_1,\theta_2)\propto
\frac{1}{t_p(\theta_1)\sigma(\theta_2)}\frac{1}{ab},
\end{align*}
giving the desired result.
\section{Understanding the \textrm{Type~1}{} Censoring Independence Jeffreys Prior Distribution}
\label{S.section:understanding.ij.prior.features}
\subsection{Features of the \textrm{Type~1}{} Censoring Independence Jeffreys Prior
Distribution}
\label{S.section:ij.prior.features}
Figure~\ref{figure:ijprior.density.examples} shows contour plots of the
\textrm{Type~1}{} censoring IJ prior densities in
(\ref{equation:IJ.typeI.yp.log.sigma}) of the main paper for
various values of $p_{r}$. That is,
\begin{align}
\label{S.equation:IJ.typeI.yp.log.sigma}
\pi(\log(t_{p_{r}}),\log(\sigma)) &\propto
\sqrt{f_{11}\left\{f_{11}[\Phi^{-1}(p_{r})]^2-2f_{12}\Phi^{-1}(p_{r})+f_{22}\right\}}
\end{align}
where the $f_{ij}$ values are scaled elements of the FIM defined in
Section~\ref{section:scaled.fim.elements} of the main paper. These
scaled elements depend on the standardized censoring time
$z_{c}=[\log(t_{c})-\mu]/\sigma$ or or the expected fraction failing
$p_{c}=\Phi(z_{c})$. As described in
Section~\ref{section:implementing.ij.priors},
$\pi(\log(t_{p_{r}}),\log(\sigma))$ in
(\ref{section:implementing.ij.priors}) can be computed for any values
of $\log(t_{p_{r}}$ and $\log(\sigma)$ and
the only inputs needed are $t_{c}$ and $p_{r}$.
To make it easier to compare across different
inputs, the densities in
Figure~\ref{figure:ijprior.density.examples} were scaled to have a maximum
of 1.0 and thus we refer to them as relative densities.
We use the lognormal distribution for the
examples (because of the well-known interpretation of the usual
$(\mu, \sigma)$ parameters and the median $t_{0.50}=\exp(\mu)$. The
results are largely similar for the Weibull distribution when
presented in the $(t_{p_{r}}, \sigma)$ parameterization (where the
Weibull shape parameter is $\beta=1/\sigma$).
Without any important loss of generality, all of these examples
given here used $t_{c}=135$. Note that the density is computed as a
function of $\log(t_{p_{r}})$ and $\log(\sigma)$ but presented in
the plots (for
the sake of easier interpretation) using log axes for
$t_{p_{r}}$ and $\sigma$.
The prior joint relative densities in
Figure~\ref{figure:ijprior.density.examples} have some common
features. In particular,
\begin{itemize}
\item
For any given value of $\sigma$, $\pi(\log(t_{p_{r}}),\log(\sigma))$
is a decreasing function of $t_{p_{r}}$.
\item
For small $\sigma$ and $t_{p_{r}} < t_{c}$, the relative density is
approximately flat at a level of 1.0.
\item
Following from the previous point, the joint densities are improper
in the sense that the area under the density is infinite.
\item
For small $\sigma$ and $t_{p_{r}} > t_{c}$, the level of the density
is, relative to the flat region, negligible, for all values of
$p_{r}$.
\item
Following from the features in the previous two points, there is a
steep cliff as one crosses $t_{c}$ for small values of $\sigma$.
\item
For large values of $\sigma$ the level of the density is
approximately flat, but at a level that is approximately equal to $p_{r}$.
\item
With $p_{r}=0.99$ (or larger) the density is approximately flat
except in the ``Negligible density'' region of the parameter space.
\end{itemize}
The reasons for these features are given in
Section~\ref{S.section:reasons.for.ij.prior.features}.
\subsection{Reasons for the \textrm{Type~1}{} Censoring Independence Jeffreys Prior
Distribution Features}
\label{S.section:reasons.for.ij.prior.features}
Here we explore the reasons for the different IJ prior density shapes and
features in Figure~\ref{figure:ijprior.density.examples}.
Figure~\ref{S.figure:probability.of.failures} shows the probability of
having one or more failures in a \textrm{Type~1}-censored life test as a
function the parameters $t_{p_{r}}$ and $\sigma$ and sample size
$n$. This binomial probability can be computed as
\begin{align*}
\Pr(r>0) = 1 - \Pr(r=0) = 1 - [1 - \Pr(T < t_{c})]^{n}
\end{align*}
where
\begin{align}
\nonumber
\Pr(T < t_{c}) &= \Phi_{\textrm{norm}}\left [\frac{\log(t_{c}) - \mu}{\sigma}
\right ]\\
\label{S.equation:pr.t.le.tc}
&= \Phi_{\textrm{norm}}\left [\frac{\log(t_{c}) -
\log(t_{p_{r}})}{\sigma} + \Phi_{\textrm{norm}}^{-1}(p_{r})
\right ].
\end{align}
Figure~\ref{S.figure:probability.of.failures} shows that the
negligible-density region of the IJ prior densities in
Figure~\ref{figure:ijprior.density.examples} corresponds to that part of the
parameter space where the probability of having a failure is
negligible.
Every point in the $(t_{p_{r}}, \sigma)$ parameter space corresponds
to a particular lognormal cdf. Figure~\ref{S.figure:ijprior.explore}
provides a visualization of the relationship for the particular case
of $p_{r}=0.10$. The horizontal lines in the
probability plots on the right are at $p_{r}=0.10$. The vertical
lines are at the censoring time $t_{c}=135$. The horizontal
position of the colored squares in the density plots on the left
indicate the value of $t_{p_{r}}=$ 80, 90, 100, 120, 135, 160, and
180, controlling the horizontal location of the cdf in the plots on
the right. The vertical position of the squares indicate the value
of $\sigma=0.02, 0.4$, and $2.0$, which controls the slopes of the
cdfs on the lognormal scales on the right of Figure~\ref{S.figure:ijprior.explore}
(the slope of the lognormal
cdf is $1/\sigma$ on the linear axes underlying lognormal
probability scales, as described in
\citet[][Section~6.2.3]{MeekerEscobarPascual2022}).
In the top-right probability plot in
Figure~\ref{S.figure:ijprior.explore}, consider, for example, the cdf
on the far right with $t_{p_{r}}=180$ and $\sigma=0.02$ (corresponding
to the red square on the far right in the top-left contour plot). Substituting
these parameter values, $p_{r}=0.10$, and $t_{c}=135$
into (\ref{S.equation:pr.t.le.tc}) gives
\begin{align*}
\Pr(T < t_{c}) &= \Phi_{\textrm{norm}}\left [\frac{\log(135) -
\log(180)}{0.02} + \Phi_{\textrm{norm}}^{-1}(0.10)
\right ]= 1.298 \times 10^{-55}.
\end{align*}
If $t_{p_{r}}=135$ (corresponding to the cyan square in the top-left
contour plot), by definition of the distribution quantile, $\Pr(T <
t_{c})=0.10$ (for any value of $\sigma$) and if $t_{p_{r}}=120$ and
$\sigma=0.02$ (corresponding to the green square in the top-left
contour plot), $\Pr(T < t_{c})=0.999997$, showing the steepness of
the cliff.
\begin{figure}
\caption{Probability of one or more failures before the the
censoring time $t_{c}
\label{S.figure:probability.of.failures}
\end{figure}
\begin{figure}
\caption{IJ prior densities with points in the parameter space
indicated by the squares (on the left) and Lognormal cdfs on
lognormal probability scales for the different points (on the
right) for $\sigma=$ 0.02 (top), 0.4 (middle), and 2.0 (bottom).
}
\label{S.figure:ijprior.explore}
\end{figure}
\section{Implementation of and Experiences Using Stan with a CJ
or an IJ Prior Distribution}
\label{S.section:experiences.using.stan.with.ij.prior}
Section~\ref{section:ij.prior} of the main paper gives expressions
for the CJ priors for both $y_{p}=\log(t_{p})$ and $\log(\sigma)$ as
well as the the IJ priors. These are easy to compute given the
algorithms to compute the FIM elements described in
Section~\ref{section:lls.fim} and could be used in conjunction with
a standard MCMC method like the Metropolis--Hastings algorithm. In
our work, we used two alternative methods to compute posterior
draws. The one based on {\tt rstan} \citep{rstan} was used for our
examples and the simulation. Another that uses a simple rejection
method described in \citet{SmithGelfand1992} was used to check the
first method. The rest of this section describes some implementation
details and our experiences.
\subsection{Implementation Using Rstan}
\label{S.section:implementation.using.rstan}
We did not see a way to compute FIM elements within a Stan model. As
an alternative, we sent down (from R) vectors (length 200 was the default) of
values and then used a Stan-model function to compute the needed FIM
element values with linear interpolation. Expressions for the IJ
prior density were programmed directly in the model block of the
Stan model (as if they were part of the likelihood function).
These codes were
exercised extensively in preparation for the running of our simulation study. For
some factor-level combinations we noticed excessively long
run times (often with many treedepth exceedences) for some data
sets. The root cause was found to be extremely small stepsize values
obtained from the adaptation stage of the Stan NUTS sampler for an
IJ prior and a small number of failures (e.g., fewer than
ten). Because we never saw this behavior with flat priors (even with
only three failures), we suspect that the problems arose because of
our non-differentiable piecewise-linear approximation to the FIM
elements. We avoided the problems by disabling the NUTS adaptation and
specifying a stepsize. After some experimentation we found that, for
a given data set, the flat prior stepsize divided by 5 worked well
(i.e., fast sampling with few or no warnings).
Section~\ref{S.section:ThousandTestThreeFail} looks closely at
estimation results for a data set with three failures from a sample
of size 1000. The likelihood for these data has an interesting
funnel shape with a sharp point at the top (Weibull) or bottom
(lognormal). When running the Stan NUTS sampler for a flat prior, divergent
transitions were sometimes observed, although they were not concentrated in the
tip. Changing the default $\textrm{adapt}\_\textrm{delta}$ to 0.995
eliminated the divergent transitions.
\subsection{Implementation of Sampling from IJ/CJ Priors for Bayes
Without Tears Plots}
\label{S.section:implementation.using.bayes.without.tears}
\citet{SmithGelfand1992} describe a simple accept/reject posterior
sampling method that accepts points from the prior with a
probability equal to the value of the relative likelihood at the
point. As illustrated in
Figures~\ref{figure:BearingCage.noninformative.informative}
and~\ref{figure:RocketMotor.noninformative.informative} (also see
the numerous plots in Section~\ref{S.section:careful.look.three.failures}),
comparing plots of prior points and likelihood contours with
a similar plot of posterior points and likelihood contours provides
insight into how the prior and likelihood combine to produce a
posterior distribution.
It is easy to compute values of the IJ prior
using the expressions in Section~\ref{section:ij.prior} used, for
example, to compute the contours in
Figure~\ref{figure:ijprior.density.examples}. However, we saw no simple way to
sample from these improper priors. Instead we used Stan to sample
from the prior density, in a manner similar to what we describe in
Section~\ref{S.section:implementation.using.rstan}, but with no data
contributing to the likelihood.
Because the Stan NUTS sampler cannot
be used to sample from an improper distribution, sampling was
constrained to a large rectangle (much larger than the boundaries of
the contour plots where the draws are plotted) so that the
resulting prior distributions are proper.
\subsection{Software for the Examples and Simulations}
\label{S.section:software.examples.simulation}
All of the computing for this project was done using R functions
that are available in the R Package RSplida. A windows version of
RSplida is available from
https://wqmeeker.stat.iastate.edu/RSplida.zip and the echapters
folder in RSplida
includes a file of commands that can be used to run the examples in
this paper.
\section{Lognormal Distribution Simulation Results and Conclusions}
\label{S.section:lognormal.simulation.results.conclusions}
Section~\ref{simulation.results.conclusions} of the main paper
presents the simulation results, evaluating the credible interval
coverage probabilities using different noninformative priors
for the Weibull distribution. Here we present similar results for
the lognormal distribution.
Figures~\ref{figure:lognormalSimulationTwoSidedResults.small.pfail}
($p_{\textrm{fail}}=0.01$ and $p_{\textrm{fail}}=0.05$) and~\ref{figure:lognormalSimulationTwoSidedResults.large.pfail} ($p_{\textrm{fail}}=0.10$ and $p_{\textrm{fail}}=0.50$) summarize
the lognormal distribution simulation results with two-sided coverage probabilities.
\begin{figure}
\caption{Lognormal distribution two-sided estimated coverage probabilities for $p_{\textrm{fail}
\label{figure:lognormalSimulationTwoSidedResults.small.pfail}
\end{figure}
\begin{figure}
\caption{Lognormal distribution two-sided estimated coverage probabilities for $p_{\textrm{fail}
\label{figure:lognormalSimulationTwoSidedResults.large.pfail}
\end{figure}
Figures~\ref{figure:lognormalSimulationOneSidedResults.small.pfail}
($p_{\textrm{fail}}=0.01$ and $p_{\textrm{fail}}=0.05$)
and~\ref{figure:lognormalSimulationOneSidedResults.large.pfail}
($p_{\textrm{fail}}=0.10$ and $p_{\textrm{fail}}=0.50$) summarize the lognormal
distribution simulation results with one-sided error probabilities.
\begin{figure}
\caption{Lognormal distribution one-sided
estimated error probabilities for $p_{\textrm{fail}
\label{figure:lognormalSimulationOneSidedResults.small.pfail}
\end{figure}
\begin{figure}
\caption{Lognormal distribution one-sided
estimated error probabilities for $p_{\textrm{fail}
\label{figure:lognormalSimulationOneSidedResults.large.pfail}
\end{figure}
Similar to the results in
Section~\ref{simulation.results.conclusions}, observations from
Figures~\ref{figure:lognormalSimulationTwoSidedResults.small.pfail}--\ref{figure:lognormalSimulationOneSidedResults.large.pfail} are
\begin{itemize}
\item
With a few exceptions the IJ priors, when compared to flat, tend to result in
coverage probabilities closer to the nominal values.
\item
Taking into account MC error, the coverage probabilities for the IJ
and flat priors are close to the 0.95 nominal credible level for
$\textrm{E}(r) \geq 35$.
\end{itemize}
\section{A Careful Look at Examples with Three Failures}
\label{S.section:careful.look.three.failures}
This section looks carefully at Weibull and lognormal Bayesian
estimation results for two data sets with only three failures under
three different noninformative priors. The two data sets were chosen
based on the two extreme simulation factor-level combinations
described in Section~\ref{section:simulation.factor.levels} (sample
sizes 20 and 1000).
In some examples, the Bayesian estimate of $F(t)$ does not appear to
agree well with the nonparametric estimates. With \textrm{Type~1}{}
censoring, to a very high degree of approximation, the Weibull ML estimate
at the censoring time is $\widehat{F}(t_{c}) \approx r/n$
\citep[as shown in][]{Escobar2010}.
Thus there is an \textit{invisible pseudo data point} at
$(t_{c}, r/n)$ and this point is indicated in the probability plots
in this section with the symbol $\mathord{\scalebox{1.0}[1]{\scalerel*{\Box}{X}}}$. This invisible pseudo data point
has more influence than the other visible data points (smaller order
statistics have more variability).
When the Bayesian estimate of $F(t)$ does not agree well with
the points, this invisible pseudo data point will help explain why. We also
did experiments to assure that our conclusions in the section are
not sensitive to the location of the three failure times before the
censoring time (they are not).
The different point colors in the contour plots correspond to the
four different MCMC chains that were used in generating the points.
Recall that, as described in
Section~\ref{S.section:implementation.using.bayes.without.tears}
draws from an IJ/CJ priors are generated by using Stan.
Clusters of one color would indicate problems with the NUTS sampler,
but we do not see any such problems.
\subsection{Example of Three Failures from a Sample of Size of Twenty}
The results in this section correspond to the factor-level
combination $\textrm{E}\big( r \big)=10$ and $p_{\textrm{fail}}=0.50$ resulting in a
sample of size $n=20$ and a censoring time of 1. To illustrate an
extreme case, a simulated sample resulting in three failures was
chosen. The failures were at times 0.414, 0.586, and
0.684. The value of $p_{r}$ for reparameterization was
chosen to be $(3/20)/2=0.075$, resulting in a likelihood that is
reasonably well behaved (except for the funnel shape with a sharp point).
Figures~\ref{S.figure:small.sample.flat} and \ref{S.figure:small.sample.IJ}
give results for flat and IJ priors, respectively. It is interesting to
see how the priors and likelihood functions combine to produce the
posterior distribution and we can see the difference between the
posterior resulting from the flat and IJ priors.
\begin{figure}
\caption{Bayesian estimation results for data with $r=3$ and $n=20$
using the Weibull distribution (on the left) and the lognormal
distribution (on the right) showing an estimate of $F(t)$ on a
probability plot (top), draws from the bounded joint prior and likelihood contours
(middle), and posterior draws and likelihood contours (bottom) for
a flat prior.}
\label{S.figure:small.sample.flat}
\end{figure}
\begin{figure}
\caption{Bayesian estimation results for data with $r=3$ and $n=20$
using the Weibull distribution (on the left) and the lognormal
distribution (on the right) showing an estimate of $F(t)$ on a
probability plot (top), draws from the bounded joint prior and likelihood contours
(middle), and posterior draws and likelihood contours (bottom) for
an IJ prior. }
\label{S.figure:small.sample.IJ}
\end{figure}
\subsection{Example of Three Failures from a Sample of Size of One
Thousand}
\label{S.section:ThousandTestThreeFail}
The results in this section correspond to the
factor-level combination $\textrm{E}\big( r \big)=10$ and $p_{\textrm{fail}}=0.01$
resulting in a sample of size $n=1000$ and a censoring time of
0.0977. To illustrate an extreme, a simulated sample resulting in
three failures was chosen. The failures were at times 0.0526,
0.0825, and 0.0836. The value of $p_{r}$ for reparameterization was
chosen to be $(3/1000)/2=0.015$, resulting in a likelihood that is
reasonably well behaved (except for the funnel shape with a sharp point).
Figures~\ref{S.figure:large.sample.flat} and \ref{S.figure:large.sample.IJ}
give Weibull (on the left) and lognormal (on the right) estimation
results for flat and IJ priors, respectively.
\begin{figure}
\caption{Bayesian estimation results for data with $r=3$ and $n=1000$
using the Weibull distribution (on the left) and the lognormal
distribution (on the right) showing an estimate of $F(t)$ on a
probability plot (top), draws from the bounded joint prior and likelihood contours
(middle), and posterior draws and likelihood contours (bottom) for
a flat prior. }
\label{S.figure:large.sample.flat}
\end{figure}
\begin{figure}
\caption{Bayesian estimation results for data with $r=3$ and $n=1000$
using the Weibull distribution (on the left) and the lognormal
distribution (on the right) showing an estimate of $F(t)$ on a
probability plot (top), draws from the bounded joint prior and likelihood contours
(middle), and posterior draws and likelihood contours (bottom) for
an IJ prior. }
\label{S.figure:large.sample.IJ}
\end{figure}
\section{Proofs of Limiting Results for Weakly Informative Prior Distributions}
\label{S.section:limiting.results.weakly.informative.prior.distributions}
As mentioned in
Section~\ref{section:weakly.informative.priors.lls.distributions}
the normal (lognormal) distribution with a large standard
deviation (log standard deviation) is often use to specify
weakly informative prior distributions.
\subsection{Limit of a Normal Distribution as its Standard Deviation Increases}
As mentioned in
Section~\ref{section:weakly.informative.priors.lls.distributions} of
the paper, a normal distribution prior density with any mean will
approach a flat prior as the standard deviation of the normal
distribution increases. First we consider a $\textrm{NORM}(\mu, \sigma)$
distribution truncated outside of $\mu \pm A$ for a value $A>0$
\begin{align}
\label{S.equation:norm.to.flat.limit}
\lim_{\sigma \to \infty}\,
\frac{\dfrac{1}{\sigma}\,\phi_{\textrm{norm}}\left(\dfrac{x-\mu}{\sigma}\right)}
{
\Phi_{\textrm{norm}}\left(\dfrac{A}{\sigma}\right) -
\Phi_{\textrm{norm}}\left(-\dfrac{A}{\sigma}\right)}=
\frac{1}{2A},
\end{align}
for any $\mu$ and $ \mu-A \le x \le \mu+A$. We use this truncated
distribution so that the density remains proper in the limit.
Although it is possible to use L'Hospital's Rule to compute the
limit in (\ref{S.equation:norm.to.flat.limit}) directly, the needed
derivatives are complicated, making the proof lengthy. Here we take
an alternative simpler path. The denominator in
(\ref{S.equation:norm.to.flat.limit}) can be written as
\begin{align*}
\Phi_{\textrm{norm}}\left(\dfrac{A}{\sigma}\right) -
\Phi_{\textrm{norm}}\left(-\dfrac{A}{\sigma}\right)
=\int_{-A/\sigma}^{A/\sigma }\phi_{\textrm{norm}}(w)\, dw
\end{align*}
for $A>0$.
The mean value theorem for integrals says that
\begin{align*}
\int_{-A/\sigma}^{A/\sigma} \phi_{\textrm{norm}}(w)\, dw = \left[ -\frac{A}{\sigma}
-\left(\frac{A}{\sigma} \right)\right] \phi_{\textrm{norm}}(\zeta) = \frac{2A}{\sigma}
\phi_{\textrm{norm}}(\zeta),
\end{align*}
where $-A/\sigma \leq \zeta \leq A/\sigma $. Then
\begin{align*}
\frac{\dfrac{1}{\sigma}\,\phi_{\textrm{norm}}\left(\dfrac{x-\mu}{\sigma}\right)}
{\Phi_{\textrm{norm}}\left(\dfrac{A}{\sigma}\right) -
\Phi_{\textrm{norm}}\left(-\dfrac{A}{\sigma}\right)}&=
\frac{\dfrac{1}{\sigma}\,\phi_{\textrm{norm}}\left(\dfrac{x-\mu}{\sigma}\right)}
{ \dfrac{2A}{\sigma}
\phi_{\textrm{norm}}(\zeta) }=
\frac{1}{2A}\,\frac{\phi_{\textrm{norm}}\left(\dfrac{x-\mu}{\sigma}\right)}
{\phi_{\textrm{norm}}(\zeta) }.
\end{align*}
For large $\sigma$, both $-A/\sigma$ and $A/\sigma $ are
approximately zero, implying that $\zeta$ is approximately
zero. Of course, $(x-\mu)/\sigma$ will also be approximately
zero. Thus
\begin{align*}
\frac{1}{2A}\,\dfrac{\phi_{\textrm{norm}}\left(\dfrac{x-\mu}{\sigma}\right)}
{\phi_{\textrm{norm}}(\zeta) } \approx \frac{1}{2A}\,\dfrac{\phi_{\textrm{norm}}\left( 0 \right)}
{\phi_{\textrm{norm}}(0) } =\frac{1}{2A},
\end{align*}
giving the needed result.
\subsection{Limit of a Lognormal Distribution as its Log Standard Deviation Increases}
As mentioned in
Section~\ref{section:weakly.informative.priors.lls.distributions} of
the paper, a lognormal distribution
prior $f(t)$ with any log-mean will be proportional to $1/t$ as the
log standard deviation increases. We start by noting that, for any
values of $\mu$ and $t>0$, the
standard normal density has the limit
\begin{align*}
\lim_{\sigma \to \infty}\,
\phi_{\textrm{norm}}\left(\dfrac{\log(t)-\mu}{\sigma}\right)
=\lim_{\sigma \to \infty}\,\frac{1}{\sqrt{2 \pi}}\exp\left[-\frac{1}{2}\left(\frac{\log(t)-\mu}{\sigma}\right)^{2}\right]
=\frac{1}{\sqrt{2 \pi}} > 0.
\end{align*}
This implies that, for any fixed large value of $\sigma$, the lognormal density
\begin{align*}
\frac{1}{\sigma t}\phi_{\textrm{norm}}\left(\dfrac{\log(t)-\mu}{\sigma}\right)
\approx \frac{1}{\sigma t \sqrt{2 \pi}},
\end{align*}
giving the needed result.
\section{Log-Truncated and Log-Reciprocal-Truncated Distributions}
\label{S.section:log.truncated.normal.distributions}
\subsection{Motivation for the Distributions}
\label{S.section:motivation.log.truncated.distributions}
As mentioned in
Section~\ref{section:motivation.for.partially.informative.prior} of the main
paper, when specifying an informative prior
distribution for a positive parameter (like a log-location-scale
distribution shape parameter $\sigma$ or $\beta=1/\sigma$ or a
log-location-scale distribution quantile $t_{p}$), a normal
distribution truncated below zero (denoted by $\textrm{TNORM}$) is often
used. Although it is a slight abuse, to keep the notation simple and
standard, we employ $\mu$ and $\sigma$ to denote the parameters of
the $\textrm{TNORM}$ distribution. The pdf for the $\textrm{TNORM}(\mu, \sigma)$
random variable $T>0$ is
\begin{align}
\label{S.equation:tnorm.pdf}
\texttt{dtnorm}(t;\mu,\sigma)&=\left(\frac{1}{\sigma}\right)
\frac{\phi_{\textrm{norm}}\left ( \dfrac{t-\mu}{\sigma}\right)}
{1-\Phi_{\textrm{norm}}\left(-\dfrac{\mu}{\sigma}\right)}, \,\,\, t>0.
\end{align}
Informative (and perhaps weakly informative)
priors are specified in terms of the distributions for parameters
like $T=\sigma$, $T=\beta=1/\sigma$, or $T=t_{p}$.
Then expressions for the prior pdfs of the unconstrained parameters
$S=\log(\sigma)$, $S=\log(\beta)=\log(1/\sigma)$, or $S=\log(t_{p})$
are needed, as these transformed parameters are used in the MCMC sampling.
\subsection{Log-Truncated-Normal Distribution pdf}
\label{S.section:LTNORM}
If $T \sim \textrm{TNORM}(\mu, \sigma)$, the distribution of $S=\log(T)$ can
be obtained by using the standard random variable transformation
methods described, for example, in Chapter~2 of
\citet{CasellaBerger2002}. Specifically,
because $S$ is a monotone increasing function of $T$ and the inverse
of the transformation is $T=\exp(S)$, the pdf for $S$ is
\begin{align}
\nonumber
\textrm{dltnorm}(s;\mu,\sigma)&=\texttt{dtnorm}(\exp(s);\mu,\sigma) \times \exp(s)\\[1ex]
\label{S.equation:LTNORM.pdf}
&= \left(\frac{1}{\sigma}\right)
\frac{\phi_{\textrm{norm}}\left ( \dfrac{\exp(s)-\mu}{\sigma}\right)}
{1-\Phi_{\textrm{norm}}\left(-\dfrac{\mu}{\sigma}\right)} \times \exp(s),\\[1ex]
\nonumber
&= \left(\frac{1}{\sigma}\right)
\frac{\phi_{\textrm{norm}}\left (\dfrac{\mu-\exp(s)}{\sigma}\right)}
{\Phi_{\textrm{norm}}\left(\dfrac{\mu}{\sigma}\right)} \times \exp(s), \,\,\,-\infty<s<\infty.
\end{align}
We call this the log-truncated-normal (LTNORM) distribution.
The simpler second expression is obtained by using the symmetry relationships
$\Phi_{\textrm{norm}}(z)=1-\Phi_{\textrm{norm}}(-z)$ and $\phi_{\textrm{norm}}(z)=\phi_{\textrm{norm}}(-z)$.
Figure~\ref{S.figure:transformtnorm}(a) shows pdfs for the LTNORM distribution.
\begin{figure}
\caption{pdfs for the log-truncated-normal distribution (a) and the
log-reciprocal-truncated-normal distribution (b).}
\label{S.figure:transformtnorm}
\end{figure}
\subsection{Log-Reciprocal-Truncated-Normal Distribution pdf}
\label{S.section:LRTNORM}
Similar to Section~\ref{S.section:LTNORM}, if $T \sim \textrm{TNORM}(\mu,
\sigma)$, then the distribution of $W=\log(1/T)=-\log(T)$ can, again, be
obtained by using the standard random variable transformation
methods. Specifically,
because $W$ is a monotone decreasing function of $T$ and the inverse
of the transformation is $T=\exp(-W)$, the pdf for $W$ is
\begin{align}
\nonumber
\textrm{dlrtnorm}(w;\mu,\sigma)&=\texttt{dtnorm}(\exp(-w);\mu,\sigma) \times \exp(-w)\\[1ex]
\label{S.equation:LRTNORM.pdf}
&= \left(\frac{1}{\sigma}\right)
\frac{\phi_{\textrm{norm}}\left ( \dfrac{\exp(-w)-\mu}{\sigma}\right)}
{1-\Phi_{\textrm{norm}}\left(-\dfrac{\mu}{\sigma}\right)} \times
\exp(-w),\\[1ex]
\nonumber
&= \left(\frac{1}{\sigma}\right)
\frac{\phi_{\textrm{norm}}\left ( \dfrac{\mu-\exp(-w)}{\sigma}\right)}
{\Phi_{\textrm{norm}}\left(\dfrac{\mu}{\sigma}\right)} \times \exp(-w) \,\,\,-\infty<w<\infty.
\end{align}
We call this the log-reciprocal-truncated-normal
(LRTNORM) distribution.
Figure~\ref{S.figure:transformtnorm}(b) shows pdfs for the LRTNORM distribution.
\subsection{Log-Truncated-Location-Scale-$t$
and Log-Reciprocal-Truncated-Location-Scale-$t$ Distribution pdfs}
As mentioned in
Section~\ref{section:motivation.for.partially.informative.prior}, a
useful generalization of the normal distribution is the
location-scale-$t$ (LST) distribution. Similar to what is described
in Section~\ref{S.section:motivation.log.truncated.distributions}, if
priors are specified for the positive log-location-scale
distribution parameters (e.g., $\sigma$, $\beta=1/\sigma$, or
$t_{p}$) using a truncated LST distribution, pdfs of the
unconstrained parameters (e.g., $S=\log(\sigma)$,
$S=\log(\beta)=\log(1/\sigma)$, or $S=\log(t_{p})$) are needed. This
is because the pdfs of these transformed parameters are used to
specify priors in the MCMC sampling. We refer to these distributions
as LTLST and LRLST. Expressions for the LTLST and LRLST pdfs are
obtained in the same manner as in Sections~\ref{S.section:LTNORM}
and~\ref{S.section:LRTNORM} except that the standard normal pdf and
cdf are replaced by their LST counterparts and these depend on the
specified degrees of freedom parameter. In particular, the standard
LST pdf is the Student's $t$ pdf:
\begin{align*}
\phi_{\textrm{lst}}(z;r_{d})&=\dfrac{\Gamma\left[(r_{d}+1)/2\right]}
{\Gamma(r_{d}/2)\, \sqrt{\pi r_{d}}}\,
\frac{1}{\left(1+{z^{2}}/{r_{d}}\right)^{(r_{d}+1)/2}}, \,\,\,
-\infty< z< \infty.
\end{align*}
The corresponding LST
cdf is $\Phi_{\textrm{lst}}(z;r_{d})$. Then, following the same path
used in~(\ref{S.equation:LTNORM.pdf})
and~(\ref{S.equation:LRTNORM.pdf}) (without giving all of the steps), the LTLST
and LRLST pdfs are
\begin{align*}
\textrm{dltlst}(s;\mu,\sigma,r_{d})&=\left(\frac{1}{\sigma}\right)
\frac{\phi_{\textrm{lst}}\left (\dfrac{\mu-\exp(s)}{\sigma};r_{d} \right)}
{\Phi_{\textrm{lst}}\left(\dfrac{\mu}{\sigma};r_{d} \right)} \times
\exp(s), \,\,\,-\infty<s<\infty,
\end{align*}
and
\begin{align*}
\textrm{dlrtlst}(w;\mu,\sigma,r_{d})&=\left(\frac{1}{\sigma}\right)
\frac{\phi_{\textrm{lst}}\left ( \dfrac{\mu-\exp(-w)}{\sigma};r_{d} \right)}
{\Phi_{\textrm{lst}}\left(\dfrac{\mu}{\sigma};r_{d} \right)} \times \exp(-w) \,\,\,-\infty<w<\infty.
\end{align*}
\begingroup
\setstretch{0.9}
\addcontentsline{toc}{section}{\protect\numberline{}References}
\endgroup
\end{document} |
\begin{document}
\title
{On Multivalued Fixed-Point Free Maps on $\mathbb R^n$}
\author{Raushan ~Z.~Buzyakova}
\address{Department of Mathematics and Statistics,
The University of North Carolina at Greensboro,
Greensboro, NC, 27402, USA}
\email{[email protected]}
\keywords{fixed point, hyperspace, multivalued function}
\subjclass{54H25, 58C30, 54B20}
\begin{abstract}{
To formulate our results let $f$ be a continuous multivalued map from
$\mathbb R^n$ to $2^{\mathbb R^n}$ and $k$ a natural number such that $|f(x)|\leq k$ for all $x$.
We prove that $f$ is fixed-point free if and only if its continuous extension
$\tilde f:\beta \mathbb R^n\to 2^{\beta \mathbb R^n}$ is fixed-point free.
If one wishes to stay within metric terms, the result
can be formulated as follows: $f$ is fixed-point free if and only
if there exists a continuous fixed-point free extension $\bar f: b\mathbb R^n\to 2^{b\mathbb R^n}$ for some
metric compactificaton $b\mathbb R^n$ of $\mathbb R^n$.
Using the classical notion of colorablity, we prove
that such an $f$ is always colorable. Moreover, a number of colors sufficient to paint
the graph can be expressed as a function of $n$ and $k$ only.
The mentioned results also hold if the domain is replaced by any closed subspace
of $\mathbb R^n$ without any changes in the range.
}
\end{abstract}
\maketitle
\markboth{Raushan Z. Buzyakova}{On Multivalued Fixed-Point Free Maps on $\mathbb R^n$}
{ }
\section{Introduction}\label{S:intro}
A series of topological results about fixed-point free maps are motivated by these two classical set-theoretical
statements (see, in particular, \cite{BE}):
\par
\noindent
{\it
S1. If $f:X\to X$ is a fixed-point free map, then there exists a finite cover $\mathcal F$ of $X$ such that $f(F)$ misses $F$
for each $F\in {\mathcal F}$; and
\par
\noindent
S2. Let ${\mathcal P}(X)$ be the set of all non-empty subsets of $X$ and let $f: X\to {\mathcal P}(X)$ be a map with the property that $x\not \in f(x)$. If there exists a natural number $k$ such that $|f(x)|\leq k$ for
all $x\in X$, then there exists a finite cover $\mathcal F$ of $X$ such that $F$ misses $\cup\{f(x):x\in F\}$
for each $F\in {\mathcal F}$.
}
\par
\noindent
One of
the first results of topological nature related to these statements was obtained by Katetov in \cite{K} by translating the first statement
into this form: {\it For a discrete space $X$, a map $f:X\to X$ is fixed-point free if and only if its
continuous extension $\tilde f:\beta X\to \beta X$ is
fixed-point free.} This topological version suggests that if one wants to have a similar criterion for a non-discrete space $X$ one has to at least demand
that elements of covers $\mathcal F$ in the statements under discussion be closed subsets of $X$. Thus, it is commonly accepted that when working with a topological space $X$ and a continuous map $f$ from a closed
subspace $A$ of $X$ to $X$,
any closed subset $F$ of $A$ that misses its image under $f$ is called a {\it color of $f$}. If there exists a finite cover ("coloring") of $X$ by
colors, $f$ is said to be {\it colorable}.
Katetov's paper \cite{K} and van Dowen's work \cite{D} have made a significant impact on topologists' interest in the topic. A number of interesting results of topological nature in the direction of the first statement have been published since the mentioned papers
(see, in particular, \cite[Section 3.2]{VM2} for references). In this paper we consider one of natural topological versions of the second statement for
Euclidean space $\mathbb R^n$ and its hyperspace $2^{\mathbb R^n}$.
In \cite{bc} it is proved that a continuous fixed-point free map from a closed subspace of $\mathbb R^n$ to $\mathbb R^n$
is colorable. In this paper we consider fixed-point free multivalued maps on $\mathbb R^n$ and its closed subspaces. To formulate our main results
we first introduce the necessary terminology related to multivalued maps.
For a topological space $X$, we use symbol $2^X$ to denote the space of all non-empty closed subsets of $X$ endowed
with the Vietoris topology and symbol ${\rm exp}_k(X)$ to denote the subspace of $2^X$ that consists of only those
$A\in 2^X$ for which $|A|\leq k$.
A multivalued map $f: X\subset Z\to 2^Z$ is {\it fixed-point free} if $x\not \in f(x)$ for every $x\in X$.
A closed set $F\subset X$ is a {\it color}
of a continuous map $f$ from $X\subset Z$ to $2^Z$ if $F$ misses $\cup \{f(x):x\in F\}$.
If $X$ can be covered by finitely many colors of $f$ then $f$ is said to be {\it colorable}.
The main results of the paper are Theorems \ref{thm:chromaticnumber} and \ref{thm:colorability}, which state that any continuous fixed-point free map from a closed subspace of $\mathbb R^n$ to ${\rm exp}_k(\mathbb R^n)$ is colorable and
there exists a formula that computes a number of colors sufficient for painting in terms of $n$ and $k$ only. Using the main result
we also show that a criterion similar to the Katetov's holds for multivalued maps as well. Namely, we show
that a continuous map $f$ from a closed subspace $X$ of $\mathbb R^n$ to ${\rm exp}_k(\mathbb R^n)$ is fixed-point free
if and only if its continuous extension $\tilde f: \beta X\to {\rm exp}_k(\beta\mathbb R^n)$ is fixed-point free.
To justify the requirement on sizes of $f(x)$ in our main results let us consider one simple example.
Define $f$ from $\omega\setminus \{0\}$ to the space of finite subsets of $\omega$
as follows: $f(n) =\{n+1,n+2,...,2n\}$. The map $f$ is continuous because $\omega$ and the space of finite subsets
of $\omega$ are discrete. Since for every $n\in \omega\setminus \{0\}$, all elements of $\{n+1, n+2,...,2n\}$ must be of different color
we conclude that $f$ is not colorable. This example justifies our requirement in the main results
that the set $\{|f(x)|: x\in X\}$ is bounded by a positive integer.
Before we dive into the technical part of the paper we would like to outline a short transparent argument
of the main results of the paper for a fixed-point free map $f:\mathbb R\to {{\rm exp}}_2(\mathbb R)$.
For this let $f_1(x) =\min f(x)$ and $f_2(x)=\max f(x)$.
Since $f$ is fixed-point free, the maps $f_1$ and $f_2$
are fixed-point free real-valued maps.
By \cite[Theorem 2.5]{b}, both functions are colorable. If one lets ${\mathcal F}_1$ and ${\mathcal F}_2$
be colorings of these maps then it is easy to verify that the family
$\{A\cap B: A\subset {\mathcal F}_1,\ B\subset {\mathcal F}_2\}$ is a coloring of $f$. If one wishes to extend
the argument for the case ${{\rm exp}}_3(\mathbb R)$ using a straightforward inductive approach, then one may find
oneself dealing with open colors or with a single-valued map with the domain being a proper subset of the range.
Existence of open colors
in a single-valued case can be easily deduced from the definition of colorability when one deals with
self-maps. However, if one works with maps from a subspace $X$ of $Y$ into $Y$, a work needs to be done.
Nevertheless, the presented argument can be extended for ${{\rm exp}}_k(\mathbb R)$ for any $k$ with
some more work and suitable references. Although our argument for any $n$ and $k$ that we present in the paper may seem different from the one just described, a closer look will reveal that
it carries the same idea hidden behind technical details naturally arising when dealing with higher dimensions.
Throughout the paper we will follow standard
notation and terminology as in \cite{Eng}.
\section{Results}\label{S:results}
For simplicity, but without loss of generality, most of our arguments related to $\mathbb R^n$ will be carried out
for $\mathbb R^5$. This will free the letter "n" for other purposes.
Since throughout our discussion we will juggle several spaces at a time we agree that by $\bar S$ we denote the
closure of $S$ in $\mathbb R^k$ (where the value of $k$ is always understood from the context) while $cl_X(P)$ will denote the closure of $P$ in $X$.
\par
\noindent
A standard neighborhood in $2^X$ will be denoted as
$$
\langle U_1,...,U_m\rangle = \{A\in 2^X: A\subset U_1\cup ...\cup U_m\ and\ U_i\cap A\not =\emptyset\ for \ all\ i=1,...,m\},
$$
where $U_1,...,U_m$ are open sets of $X$.
\par
\noindent
By ${{\rm exp}}_n(X)$ we denote the subspace $\{F\in 2^X: |F|\leq n\}$.
\par
\noindent
\begin{defin}\label{defin:brightcolor}
Let $f$ be a continuous map from $X\subset Z$ to $2^Z$. A closed set $F\subset X$ is a bright color
of $f$ if $F$ misses $cl_{Z}[\cup \{f(x):x\in F\}]$.
\end{defin}
\par
\noindent
\begin{pro}\label{pro:openbrightcolor}
Suppose $Z$ is normal, $X$ is closed in $Z$, $f:X\to 2^Z$ is continuous, and $F$ is a bright color of $f$.
Then there exists an open neighborhood $U$ of $F$ such that $cl_{X}(U)$ is a bright color of $f$.
\end{pro}
\begin{proof}
Since $F$ is a bright color of $f$ and $X$ is closed in $Z$,
the sets $F$ and $cl_Z(\cup \{f(x):x\in F\})$ are disjoint closed sets in $Z$. Since
$Z$ is normal, there exist
$V$ an open neighborhood of $cl_{Z}[\cup \{f(x):x\in F\}]$ in $Z$ and $W$ an open neighborhood of
$F$ in $Z$ such that $cl_Z(W)$ misses $cl_Z(V)$.
Consider the open set $\langle V\rangle$ in $2^Z$. Since $f$ is continuous and $f(F)\subset \langle V\rangle$
there exists an open neighborhood $U$ of $F$ in $X$ such that $U\subset W$ and $f(cl_Z(U))\subset \langle V\rangle$.
Thus, $cl_Z(\cup\{f(x):x\in cl_Z(U)\})$ is in $cl_Z(V)$. The latter misses $cl_Z(W)$ and therefore
$cl_Z(U)$. Therefore, $U$ is as desired.
\end{proof}
\par
Proposition \ref{pro:openbrightcolor} implies that if $X$ is closed in $\mathbb R^k$ and
$\mathcal F$ is an $m$-sized bright coloring of a continuous map $f:X\to {{\rm exp}}_n(\mathbb R^k)$
then there exists an $m$-sized open cover $\mathcal U$ of $X$ the closures of whose elements are
bright colors of $f$. We will use this observation throughout the paper without formally
referencing it.
\par
In the following discussion that leads to the main result we will restrict ourselves to
closed subspace of $\mathbb R^5$. This is done to avoid accumulation of too many variables.
All arguments are valid if one replace "$5$" with any natural number.
\par
\noindent
\begin{defin}\label{defin:statementS5n}
$S(5,n)$ denotes the following statement: there exists the smallest integer $K(5,n)$ such that every
continuous fixed-point free map $f$ from a closed subset $X\subset \mathbb R^5$ to ${{\rm exp}}_n(\mathbb R^5)$
is colorable by at most $K(5, n)$ many bright colors.
\end{defin}
\par
\noindent
\begin{lem}\label{fgh}
Suppose $f,g,h: X\subset Z\to 2^Z$ are maps; and $g$ and $h$ are colorable by at most $N_g$ and $N_h$
(bright) colors, respectively. If $f(x)\subset g(x)\cup h(x)$ for every $x\in X$ then
$f$ is colorable by at most $N_g\cdot N_h$ (bright) colors.
\end{lem}
\begin{proof}
Let $\mathcal G$ and $\mathcal H$ be bright colorings of $g$ and $h$ of sizes $N_g$ and $N_h$, respectively. Put
${\mathcal F}=\{G\cap H: G\cap H\not =\emptyset, H\in {\mathcal H}, G\in {\mathcal G}\}$.
Clearly, $|{\mathcal F}|\leq N_g\cdot N_h$.
Since ${\mathcal G}$ and $\mathcal H$ are closed covers of $X$, so is $\mathcal F$.
Fix $F=H\cap G$ in $\mathcal F$. Since, by hypothesis, $f(x)\subset g(x)\cup h(x)$, we conclude that
$$
cl_Z[\cup\{f(x):x\in H\cap G\}]\subset cl_Z[\cup\{g(x):x\in H\cap G\}]\cup cl_Z[\cup\{h(x):x\in H\cap G\}].
$$
Since $\mathcal G$ and $\mathcal H$ are bright colorings,
$cl_Z[\cup\{g(x):x\in H\cap G\}]$ and $cl_Z[\cup\{h(x):x\in H\cap G\}]$ miss $H\cap G$. Therefore, the left side
of the above set inclusion formula misses $H\cap G$ as well, meaning $H\cap G$ is a bright color for $f$.
\end{proof}
\par
In what follows, by $\pi_i$ we denote the projection of $\mathbb R^5$ onto its $i$-th coordinate axis.
The next two statements (Lemmas \ref{lem:allsamenumberofmax} and \ref{lem:largefirstprojection}) hold if we replace $\pi_1$ by $\pi_i$ for any $i\in \{1,...,n\}$.
However, we will carry out our arguments for $\pi_1$ for the already mentioned reason
of avoiding unnecessary variables.
\par
\noindent
\begin{lem}\label{lem:allsamenumberofmax}
Suppose $n>M\geq 1$; $A$ is closed in $\mathbb R^5$;
$f:A\to {{\rm exp}}_n(\mathbb R^5)\setminus {{\rm exp}}_{n-1}(\mathbb R^5)$ is continuous and fixed-point free;
$|\{y\in f(x): \pi_1(y) = \max \pi_1(f(x))\}|=M$ for all $x\in A$; and $S(5,n-1)$ is true.
Then $f$ is colorable by at most $[K(5,n-1)]^2$ bright colors.
\end{lem}
\begin{proof}
Define $g$ and $h$ from $A$ to ${{\rm exp}}_{n-1}(\mathbb R^5)$ as follows:
$$
g(x) = \{z\in f(x): \pi_1(z) =\max \pi_1 (f(x))\} \ {\rm and}\ h(x) = f(x)\setminus g(x).
$$
Since $f(x)$ is finite, $\max \pi_1(f(x))$ exists. Hence $g(x)$ is defined. Since
$|\{y\in f(x): \pi_1(y) = \max \pi_1(f(x))\}|=M$ and $1\leq M<n$, we conclude that $0<|f(x)\setminus g(x)|<n$ and
$0<|g(x)|<n$. Therefore, $g$ and $h$ are well defined functions from $A$ to ${{\rm exp}}_{n-1}(\mathbb R^5)$.
Observe that $f(x)= g(x)\cup h(x)$ for each $x$. Therefore, by Lemma \ref{fgh}, to reach the conclusion of
our lemma we need to show that $g$ and $h$ are colorable by at most $K(5,n-1)$ bright colors each.
Since we assume that $S(5, n-1)$ holds it suffices to show that $g$ and $h$ are continuous and fixed-point free.
The latter property follows from the facts that $f$ is fixed-point free and $f(x) = g(x)\cup h(x)$.
To prove continuity of $g$ and $h$, fix $x\in A$ and open neighborhoods ${\mathcal U}_{g(x)}$ and ${\mathcal V}_{h(x)}$
of $g(x)$ and $h(x)$ in ${{\rm exp}}_{n-1}(\mathbb R^5)$. We need to find an open neighborhood
of $x$ in $A$ whose image under $g(x)$ and $h(x)$ are in ${\mathcal U}_{g(x)}$ and ${\mathcal V}_{h(x)}$, respectively.
Without loss of generality we may assume that the selected neighborhoods are standard, that is, in the form
$$
{\mathcal U}_{g(x)}=\langle U_y: y\in g(x)\rangle\ {\rm and}\ {\mathcal V}_{h(x)}=\langle U_y: y\in f(x)\setminus g(x)\rangle,
$$
where $U_y$ is a fixed bounded open neighborhood of $y$ in $\mathbb R^5$ for each $y\in f(x)$. We may also assume that the following properties hold.
\begin{description}
\item[\rm P1] $\min \pi_1(\overline U_y)> \max \pi_1 (\overline U_z)$ whenever $y\in g(x)$ and $z\in f(x)\setminus g(x)$.
\item[\rm P2] $\overline U_y\cap \overline U_z =\emptyset$ for any distinct $y,z\in f(x)$.
\end{description}
The property P1 can be achieved since by the definition of $g$, the set $\pi_1(g(x))$ is a singleton and
its element is strictly greater than any element of
$\pi_1(f(x)\setminus g(x))$.
Put ${\mathcal W}_{f(x)} = \langle U_y:y\in f(x)\rangle$. Clearly, ${\mathcal W}_{f(x)}$ is an open neighborhood of $f(x)$.
By continuity of $f$, there exists an open $O$ of $x$ in $A$ such that
$f(O)\subset {\mathcal W}_{f(x)}$. To finish the proof of continuity of $g$ and $h$ it suffices to show
that $g(O)\subset {\mathcal U}_{g(x)}$ and $h(O)\subset {\mathcal V}_{h(x)}$. For this fix an arbitrary $x'\in O$.
By the choice of $O$, we have $f(x')\in {\mathcal W}_{f(x)}$.
Let $f(x') =\{z_1,...,z_n\}\in {\rm exp}_n(\mathbb R^5)\setminus {\rm exp}_{n-1}(\mathbb R^5)$ and
$\pi_1(z_i) = \max \pi_1(f(x'))$ for any $i=1,...,M$. By the lemma's condition on $M$, we have
$\pi_1(z_j)<\max \pi_1(f(x))$ for any $j=M+1,...,n$. By P1, we have
\begin{description}
\item[\rm P3] $z_i\in \cup \{U_y:y\in g(x)\}$ for any $i=1,...,M$.
\end{description}
By P2 and P3, we have
\begin{description}
\item[\rm P4] $z_j\in \cup \{U_y:y\in f(x)\setminus g(x)\}$ for any $j=M+1,...,n$.
\end{description}
By P2 and the fact that $f(x')\in {\mathcal W}_{f(x)}=\langle U_y:y\in f(x)\rangle$, we conclude that
each $U_y$,
participating in the definition of ${\mathcal W}_{f(x)}$, contains exactly one $z_i\in f(x')$.
By P3 and P4, we have
$\{z_1,...,z_M\}\in {\mathcal U}_{g(x)}$ and $\{z_{M+1},...,z_n\}\in {\mathcal V}_{h(x)}$.
Since $g(x')=\{z_1,...,z_M\}$ and $h(x')=\{z_{M+1},...,z_n\}$, we are done.
\end{proof}
\par
\noindent
\begin{lem}\label{lem:largefirstprojection}
Suppose $n>1$; $A$ is closed in $\mathbb R^5$;
$f:A\to {{\rm exp}}_n(\mathbb R^5)\setminus {{\rm exp}}_{n-1}(\mathbb R^5)$ is continuous and fixed-point free;
$|\{y\in f(x): \pi_1(y) = \max \pi_1(f(x))\}|>1$ for all $x\in A$; and $S(5,n-1)$ is true.
Then $f$ is colorable by at most $n\cdot [K(5,n-1)]^2$ bright colors.
\end{lem}
\begin{proof} For $m=1,...,n-1$, define $O_m\subset A$ as follows: $x\in O_m$ if and only if
$M_x=|\{y\in f(x):\pi_1(y)=\max \pi_1(f(x))\}|<n-m$.
\par
\noindent
{\it Claim. $O_m$ is open.}
\par
\noindent
To prove the claim, fix $x\in O_m$ and let ${\mathcal V}_{f(x)}=\langle V_y:y\in f(x)\rangle$
be an open neighborhood of $f(x)$ such that the following hold:
\begin{enumerate}
\item $V_y$ is a bounded neighborhood of $y$ for every $y\in f(x)$;
\item $V_y\cap V_z=\emptyset$ for any distinct $y,z\in f(x)$; and
\item $\min \pi_1(\overline V_y) >\max \pi_1(\overline V_z)$ whenever $\pi_1(y)=\max \pi_1(f(x))$
and $\pi_1(z)<\max \pi_1(f(x))$.
\end{enumerate}
It suffices to show now that $f^{-1}({\mathcal V}_{f(x)})\subset O_m$. For this pick $x'\in f^{-1}({\mathcal V}_{f(x)})$.
We have $f(x')\in {\mathcal V}_{f(x)}$. By (2) and (3), $M_{x'}\leq M_x$. Hence, $M_{x'}<n-m$. By the definition of $O_m$,
$x'\in O_m$. The claim is proved.
\par
We will construct our coloring inductively. For short put $K=[K(5,n-1)]^2$.
\par
\noindent
{\it Step 1.} Put $A_1 = A\setminus O_1$.
Thus, an element $x$ of $A$ is in $A_1$ if and only if $M_x\geq n-1$. Since $|\pi_1(f(x))|>1$ for every $x\in A$ we
conclude that $M_x=n-1$ for every $x\in A_1$. Therefore, by Lemma \ref{lem:allsamenumberofmax}, there exists a
$K$-sized open cover ${\mathcal U}_1$ of $A_1$ the closures of whose elements are bright colors for $f$.
\par
\noindent
{\it Assumption.} Assume for $m-1$ an open family ${\mathcal U}_{m-1}$ of size at most
$(m-1)\cdot K$ is defined and the following conditions are met:
\begin{description}
\item[\rm P1] $\overline U$ is a bright color of $f$ for every $U\in {\mathcal U}_{m-1}$; and
\item[\rm P2] If $M_x\geq n-(m-1)$ then $x\in \bigcup {\mathcal U}_{m-1}$.
\end{description}
\par
\noindent
{\it Step $m<n$.}
Put $A_m = A\setminus [O_m\cup (\bigcup {\mathcal U}_{m-1})]$.
By Claim, the set $A_m$ is closed. Pick any $x\in A_m$. Then $x\not\in O_m$, meaning that $M_x\geq n-m$. Also
$x\not\in \bigcup {\mathcal U}_{m-1}$, which, by P2, implies $M_x<n-(m-1)$. Thus, $n-m\leq M_x<n-m+1$. Therefore,
$M_x=n-m$. Therefore, by Lemma \ref{lem:allsamenumberofmax}, there exists a
$K$-sized open cover ${\mathcal V}_m$ of $A_m$ the closures of whose elements are bright colors for $f$.
Put ${\mathcal U}_m = {\mathcal U}_{m-1}\cup {\mathcal V}_m$. Clearly, the size of this family is at most $m\cdot K$.
Let us verify P1 and P2 for $m$. Property P1 holds since ${\mathcal U}_m$ is the union of two families
satisfying P1. For P2, observe that $M_x\geq n-m$ if and only if
$x\not\in O_m$. But ${\mathcal U}_m$ covers the complement of $O_m$.
\par
\noindent
The construction is complete. It suffices to show now that $A=\bigcup {\mathcal U}_{n-1}$.
For this pick $x\in A$. By the lemma's hypothesis, $M_x>1=n-(n-1)$. By P2, $x\in \bigcup {\mathcal U}_{n-1}$.
\end{proof}
\par
The base step in the proof of our main theorem uses the following statement (\cite[Proposition 2.9]{bc}):
{\it If $f$ is a continuous fixed-point free map from a closed subset $X$ of $\mathbb R^m$ to
$\mathbb R^m$ then $f$ is colorable by at most $m+3$ bright colors.}
\par
\noindent
\begin{thm}\label{thm:chromaticnumber}
There exists an integer $K(m,n)$ such that every continuous fixed-point free map from a closed
subspace $X$ of $\mathbb R^m$ into ${{\rm exp}}_n(\mathbb R^m)$ is colorable by at most $K(m,n)$ many
bright colors.
\end{thm}
\begin{proof}
As in previous statements, to avoid accumulation of extra variables, let us deal with $m=5$.
Thus, we need to show that $K(5,n)$ exists for every natural number $n$.
By \cite[Proposition 2.9]{bc}, $K(5,1)$ exists. Assume that $K(5,n-1)$ exists.
To prove the conclusion of the theorem for $n$, fix any fixed-point free continuous map
$f$ from a closed subspace $X$ of $\mathbb R^5$ into ${{\rm exp}}_n(\mathbb R^5)$.
Next define $L\subset X$ as follows: $x\in L$ if and only if $|f(x)|<n$.
Then $L$ is closed and the range of
$f|_L$ is a subset of ${{\rm exp}}_{n-1}(\mathbb R^5)$ (notice that $L$ can be empty). Therefore, by our inductive assumption,
there exists a $K(5,n-1)$-sized open cover ${\mathcal U}_L$ of $L$ the closures of whose elements are bright colors.
Put $E=X\setminus \bigcup {\mathcal U}_L$. Then $E$ is closed and $|f(x)|=n$ for every $x\in E$.
For $1\leq i\leq n$,
define $S_i$ as follows: $x\in S_i$ if and only if $x\in E$ and $|\pi_i(f(x))|=1$. Notice that $S_i$ can be empty.
Clearly, $S_i$ is closed.
Inductively, we will first cover
$\cup_{i\leq n} S_i =\{x\in E: |\pi_i(f(x))|=1\ for\ some\ i\}$ by bright colors and then we will cover the rest of $E$.
\par
\noindent
{\it Step 1.} Put $E_i =\cap_{j\not =i} S_j$. Notice that $E_i$ can be empty.
Since $|f(x)|=n$ for every $x\in E$ we conclude that
$|\pi_i(f(x))|=n$ for all $x\in E_i$. Since $n>1$, by
Lemma \ref{lem:allsamenumberofmax} (with $\pi_1$ replaced by $\pi_i$ and $M=1$), there exists a finite open cover ${\mathcal U}_1$ of $\cup_{i\leq n}E_i$ the closures of whose
elements are bright colors for $f$.
\par
\noindent
{\it Assumption.} Suppose an open finite family ${\mathcal U}_{k-1}$, where $1\leq k-1<n$,
is defined and the following hold:
\begin{description}
\item[\rm P1] For every at most $(k-1)$-sized subset $I$ of $\{1,...,n\}$ the inclusion $\cap_{j\not \in I}S_j \subset \bigcup {\mathcal U}_{k-1}$ holds;
\item[\rm P2] $\overline U$ is a bright color for every $U\in {\mathcal U}_{k-1}$.
\end{description}
\par
\noindent
{\it Step $k<n$.} For every $k$-sized $I\subset \{1,...,n\}$ put $E_I = [\cap_{j\not \in I}S_j]\setminus [\bigcup {\mathcal U}_{k-1}]$.
Pick $i^*\in I$. Then $|I\setminus \{i^*\}| = k-1$. By P1, the set $\bigcup {\mathcal U}_{k-1}$ contains
$\cap \{S_j:j\in I\setminus \{i^*\}\}$. Since $E_I$ misses $\bigcup {\mathcal U}_{k-1}$, we conclude that
$|\pi_{i^*}(f(x))|>1$ for every $x$ in $E_I$. By Lemma \ref{lem:largefirstprojection}, there exists a finite open cover
${\mathcal U}_I$ of $E_I$ the closures of whose elements are bright colors. Put
${\mathcal U}_k =[\cup \{{\mathcal U}_I: I\subset \{1,...,n\},\ |I|=k\}]\cup {\mathcal U}_{k-1}$
Properties P1 and P2 clearly hold for $k$. The construction is complete.
\par
\noindent
Let us show that $\{x\in E: |\pi_i(f(x))|=1\ for\ some\ i\}$ is covered by ${\mathcal U}_{n-1}$. For this pick any $z$ in this set.
Put $I_z = \{i\leq n: |\pi_i(f(x))|>1\}$. Clearly, $|I_z| <n$. Since $x\in E$ we conclude that $|f(x)|=n$.
Therefore, $I_z\not = \emptyset$. Therefore $z\in \cap_{j\not\in I_z} S_j$. By P1, $z\in \bigcup {\mathcal U}_{n-1}$.
To finish the proof we need to cover $E\setminus [\bigcup {\mathcal U}_{n-1}]$ by bright colors.
For this observe that $E\setminus [\bigcup {\mathcal U}_{n-1}]$ contains only those $x$ for which $|\pi_i(f(x))|=n>1$
for all $i$. Therefore, by Lemma \ref{lem:allsamenumberofmax} (with $M=1$), the set in question is covered by bright colors.
Since we always used Lemmas \ref{lem:allsamenumberofmax} and \ref{lem:largefirstprojection} to construct our coloring, the size of the coloring depends on
$n$ and number $5$ only. Thus, $K(5,n)$ exists.
\end{proof}
\par
A non-technical version of the above result is the following theorem, where $n$ and $k$ denote any natural
numbers.
\par
\noindent
\begin{thm}\label{thm:colorability}
Any continuous fixed-point free map from a closed subspace $X$ of $\mathbb R^k$ into ${{\rm exp}}_n(\mathbb R^k)$ is
brightly colorable.
\end{thm}
\par
Observe that if a continuous map $f:X\to 2^Z$ has the property that $f(x)$ is compact in $Z$ for all $x$
then there exists the continuous extension $\tilde f:\beta X\to 2^{\beta Z}$. For our next discussion
put ${\mathcal K}(X)=\{A\in 2^X: A\ is\ compact\}$.
In \cite{bc} it is proved that a continuous map $f$ from a closed subspace $X$ of $\mathbb R^k$ to $\mathbb R^k$
is fixed-point free if and only if its continuous extension $\tilde f: \beta X\to \beta \mathbb R^k$ is fixed-point free.
It is natural to ask if the corresponding statement holds for a multivalued map
$f: X\to {{\rm exp}}_n(\mathbb R^k)$ and its continuous extension $\tilde f: \beta X\to {{\rm exp}}_n(\beta \mathbb R^k)$. Observe first that
the continuous extension exists since ${{\rm exp}}_n(\beta \mathbb R^k)$ is compact. The affirmative answer to this question
follows from the next statement.
\par
\noindent
\begin{pro}\label{pro:ftobetaf}
Let $X$ be a closed subspace of a normal space $Z$. If $\mathcal F$ is a bright coloring
of a continuous map $f:X\to {\mathcal K}(Z)$, then $\{\beta F:F\in {\mathcal F}\}$ is a bright coloring
of $\tilde f:\beta X\to {\mathcal K}({\beta Z})$.
\end{pro}
\begin{proof}
Since $X$ is normal and $\mathcal F$ is a finite closed cover of $X$ the family $\{\beta F:F\in {\mathcal F}\}$
is a closed cover of $\beta X$. Therefore, we only need to show that $\beta F$ is a bright color for
$\tilde f$. Since $F$ is a bright color for $f$ the set $F$ misses $cl_Z(\cup \{f(x): x\in F\})$.
Since $F$ and $cl_Z(\cup \{f(x): x\in F\})$ are disjoint closed subsets of the normal space $Z$ we conclude that
$\beta F$ misses $cl_{\beta Z}(\cup \{f(x): x\in F\})$. Since $\tilde f$ is the continuous extension of $f$
we conclude that $cl_{\beta Z}(\cup \{\tilde f(x): x\in \beta F\})=cl_{\beta Z}(\cup \{f(x): x\in F\})$.
Thus $\beta F$ misses $cl_{\beta Z}(\cup \{\tilde f(x): x\in \beta F\})$, whence $\beta F$ is a bright color of $\tilde f$.
\end{proof}
\par
\noindent
\begin{thm}\label{thm:fpfcriterion}
Let $f$ be a continuous map from a closed subspace $X$ of $\mathbb R^k$ to
${{\rm exp}}_n(\mathbb R^k)$. Then $f$ is fixed-point free if and only if
$\tilde f:\beta X\to {{\rm exp}}_n(\beta \mathbb R^k)$ is fixed-point free.
\end{thm}
\begin{proof}
Sufficiency is obvious. To prove necessity, let $\mathcal F$ be a bright coloring of $f$.
By Proposition \ref{pro:ftobetaf}, $\{\beta F:F\in {\mathcal F}\}$ is a coloring for $\tilde f$.
Since $\{\beta F:F\in {\mathcal F}\}$ covers $\beta X$ and $f(\beta F)$ misses $\beta F$ for each
$F\in {\mathcal F}$, we conclude that $\tilde f$ does not fix any point.
\end{proof}
\par
Using spectral techniques it is observed in \cite[Corollary 3.5.7]{VM2} that if $f$ is a continuous colorable
self-map on a separable metric space $X$ then one can find a continuous fixed-point free extension
$\bar{f}:bX\to bX$, where $bX$ is a metric compactification of $X$ of the same dimensionality as that of $X$. Using the same technique
we will next outline a proof for the following metric version of Theorem \ref{thm:fpfcriterion} for natural numbers
$n$ and $k$.
\par
\noindent
\begin{thm}\label{thm:metricversion}
Let $f:\mathbb R^k\to {{\rm exp}}_n(\mathbb R^k)$ be continuous. Then $f$ is fixed-point free
if and only if there exists a continuous fixed-point free extension
$\bar f:b\mathbb R^k\to {{\rm exp}}_n(b\mathbb R^k)$, where $b\mathbb R^k$ is some metric compactification
of $\mathbb R^k$ of dimension $k$.
\end{thm}
\begin{proof}
Sufficiency is obvious. Let us outline a proof of necessity. By Theorem \ref{thm:fpfcriterion},
$\tilde f:\beta \mathbb R^k\to {{\rm exp}}_n(\beta\mathbb R^k)\subset 2^{\beta\mathbb R^k}$
is fixed-point free. By \v S\v epin spectral theorem \cite{S}, we can find spectra
$\{b_\alpha(\mathbb R^k), \pi^\gamma_\alpha, {\mathcal A}\}$
and $\{2^{b_\alpha(\mathbb R^k)}, p^\gamma_\alpha, {\mathcal A}\}$
with inverse limits $\beta \mathbb R^k$ and $2^{\beta \mathbb R^k}$, respectively, and a family of maps
$\{f_\alpha:\alpha\in {\mathcal A}\}$ such that
the following hold:
\begin{enumerate}
\item $b_\alpha(\mathbb R^k)$ is a metric compactification of $\mathbb R^k$ of dimension $k$ for all $\alpha$
\item $\pi^\gamma_\alpha$ and $p^\gamma_\alpha$ are identity maps on $\mathbb R^k$ and $2^{\mathbb R^k}$, respectively.
\item $f_\alpha\circ \pi_\alpha = p_\alpha\circ f$.
\end{enumerate}
Since $\tilde f$ is fixed-point free and $b_\alpha(\mathbb R^k)$ is compact for every $\alpha$
we may assume that $f_\alpha$ is fixed-point free for every $\alpha$.
By (2) and (3), each $f_\alpha$ coincides with $f$ on $\mathbb R^k$. Therefore, each
$\{f_\alpha, b_{\alpha}(\mathbb R^k)\}$ serves our purpose.
\end{proof}
\par
We would like to finish the paper by commenting on a number of colors sufficient to paint a given graph.
If one follows the argument of Theorem \ref{thm:chromaticnumber} or the argument for the reals outlined in
the introduction one will quickly see that the size of the coloring constructed for the case
${{\rm exp}}_n(\mathbb R^k)$ is at least $k^n$. Therefore, it is natural to wonder if an estimate
for the required number of colors can be represented as a polynomial of both $n$ and $k$.
\par
\end{document} |
\begin{document}
\title{A relation between Mirković-Vilonen cycles and modules over preprojective algebra of Dynkin quiver of type ADE}
\author{Zhijie Dong }
\date{}
\maketitle
\begin{abstract}
The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the reduceness conjecture in \cite{kamnitzer2016reducedness}.
\end{abstract}
\section{Introduction}
Let $\mathfrak{g}$ be simply-laced semisimple finite dimensional complex Lie algebra.
There are several modern constructions of irreducible representation of $\mathfrak{g}$. In this paper we consider two models which realize the crystal of the positive part $U(\mathfrak{n})$ of $U(\mathfrak{g})$. One is by Mirkovic-Vilonen (MV) cycles and the other is by the irreducible compotents of Lusztig's nilpotent variety $\Lambda$ of the preprojective algebra of the quiver $Q$ corresponding to $\mathfrak{g}$.
Baumann and Kamnitzer \cite{MR2892443} studied the relations between $\Lambda$ and MV polytopes. They associate an MV polytope $P(M)$ to a generic module $M$ and construct a bijection between the set of irreducible components of $\Lambda$ and MV polytopes compatible with respect to crystal structures.
Since MV polytopes are in bijection with MV cycles, Kamnitzer and Knutson launched a program towards geometric construction of the MV cycle $X(M)$ in terms of a module $M$ over the preprojective algebra.
Here we consider a version by Kamnizter, Knutson and Mirkovic: conjecturally, the ring of functions $\mathcal{O}(X(M)^{T})$ on the $T$ fixed point subscheme of the cycle $X(M)$ associted to $M$, is isomorphic to $H^{*}(Gr^{\Pi}(M))$, the cohomology ring of the quiver Qrassmannian of $M$.
In this paper I will construct a map from $\mathcal{O}(X(M)^{T})$ to $H^{*}(Gr^{\Pi}(M))$ and prove it is isomorphism for the case when $M$ is a representation of $Q$.
In section two I recall the definition of MV cycles, quivers, preprojective algebra and Lusztig's nilpotent variety and state the conjecture precisely.
In chapter three I describe the ring of functions on the $T$ fixed point subscheme of the intersection of closures of certain semi-infinite orbits (which is called "cycle" in this paper). A particular case of these intersections is a scheme theoretic version of MV cycles. We realize these cycles as the loop Grassmannian with a certain condition Y.
In section four I construct the map $\Psi$ from $\mathcal{O}(X(M)^{T})$ to $H^{*}(Gr^{\Pi}(M))$. Here, $\Psi$ maps certain generators of $\mathcal{O}(X(M)^{T})$ to Chern classes of tautological bundles over $Gr^{\Pi}(M)$. So we need to check that the Chern classes satisfy the relations of generators of $\mathcal{O}(X(M)^{T})$. We reduce this problem to a simple $SL_3$ case. In this case we have a torus action on $Gr^{\Pi}(M)$ so we could use localization in equivariant cohomology theory (GKM theory).
In chapter five I will prove $\Psi$ is an isomorphism in the case when $M$ is a representation of $Q$ of type A.
In chapter six I will state some consequences given the conjecture
(one of which is the reduceness conjecture).
This is a piece of a big project to relate $\mathcal{G}(G)$ and the quiver $Q$, see \cite{localivan}, and more recently \cite{ivan2}.
\section{Statement of the conjecture}
\subsection{Notation}
Let $G$ be a simply-laced semisimple group over complex numbers. Let I be the set of vertices in the Dynkin diagram of $G$. In this paper I will work over base field $k=\ensuremath{\mathbb{C}}$. We fix a Cartan subgroup $T$ of $G$ and a Borel subgroup $B\subset G$. Denote by $N$ the unipotent radical of $B$. Let $\varpi_i,i\in I$ be the fundamental weights. Let $X_*, X^{*}$ be the cocharacter, character lattice and $\langle\ ,\ \rangle$ be the pairing between them. Let W be the Weyl group. Let $e$ and $w_0$ be the unit and the longest element in $W$. Let $\alpha_i$ and $\check{\alpha}_i$ be simple roots and coroots. Let $\Gamma=\{w\varpi_i,w\in W ,i\in I\}$. $\Gamma$ is called the set of chamber weights.\\
Let $d$ be the formal disc and $d^{*}$ be the punctured formal disc. The ring of formal Taylor series is the ring of functions on the formal disc, $ \mathcal{O}=\{\sum_{n\geq 0}a_{n}t^{n}\}$. The ring of formal Laurent series is the ring of functions on the punctured formal disc, $\mathcal{K}=\{\sum_{n\geq {n_0}}a_{n}t^n\}$. \\
For $X$ a variety, let Irr($X$) be the set of irreducible components of $X$.
\subsection{MV cycles and polytopes}
For a group $G$, let $G_{\mathcal{K}}$ be the loop group of $G$ and $G_{\mathcal{O}}$ the disc group of $G$.
We define loop grassmannian $\mathcal{G}(G)$ as the left quotient $G_{\mathcal{O}} \setminus G_{\mathcal{K}}$ and view $\mathcal{G}(G)$ as an ind-scheme \cite{beilinson1991quantization}, \cite{zhu}.
An MV cycle is a certain finite dimentional
subscheme in $\mathcal{G}(G)$. For a cocharacter $\lambda \in X_*(T)$, we denote the point it determines in $\mathcal{G}(G)$ by $L_\lambda$.
For $w\in W$, let $N^{w}=wNw^{-1}$.
Define $
S^{w}_\lambda =L_\lambda N^{w}_\mathcal{K}. $
This orbit is an ind-subscheme of $\mathcal{G}(G)$ and is called semi-infinite orbit since it is of infinite dimension and codimension in $\mathcal{G}(G)$.\\
An irreducible component of $\overline {S_0^{e} \bigcap S^{w_0}_\lambda}$ is called an MV cycle of weight $\lambda$. Kamnizter \cite{kamnitzer2005} describes them as follows:
\begin{theorem}[\cite{kamnitzer2005}]
Given a collection of integers $(M_\gamma)_{\gamma \in \Gamma}$, if it satisfies edge inequalities, and certain tropical relations, put $\lambda_{w}=\sum_i M_{w \varpi_i} w \check{\alpha_i}.$\\Then$
\overline {\bigcap_{w\in W} S^{w}_{\lambda_{w}}} $ is an MV cycle, and each MV cycle arises from this way for the unique data $(M_{\gamma})$.
\end{theorem}
The data $(M_\gamma)_{\gamma \in \Gamma}$ determines a pseudo-Weyl polytope. It is called an MV polytope if the corresponding cycle $\overline {\bigcap_{w\in W} S^{w}_{\lambda_{w}}} $ is an MV cycle.
MV polytopes are in bijection with MV cycles.
Using this description, Kamnizter \cite{kcrystal} reconstruct the crystal structure for MV cycles.
\begin{prop}[\cite{kcrystal}]
MV polytopes have a crystal structure isomorphic to $B(\infty)$.
\end{prop}
\subsection{Objects on the quiver side}
Let $Q=\{I,E\}$ be a Dynkin quiver of type ADE,
where I is the set of vertices and E is the set of edges.
We double the edge set E by adding all the opposite edges. Let $E^*=\{a^* | a\in E\}$
where for $a:i\xrightarrow[]{}j, a^*=j\xrightarrow[]{}i$, also we define $s(a)=i,t(a)=j$. Define $\epsilon(a)=1 $ when $a\in E$, $\epsilon(a)=-1$, when $a\in E^*$.
Let $H=E\bigsqcup E^*$ and $\overline{Q}=\{I, H\}$.
The preprojective algebra $\Pi$ of $Q$ is defined as quotient of the path algebra by a certain ideal:
$$
\Pi_{Q}= k\overline{Q} /<\sum_{a\in H} \epsilon(a)aa^*>.\footnote{Since most time we fix Q so Q is omitted when there is no confusion.}$$
A $\Pi_{Q}-$module is the data of an $I$ graded vector space $\bigoplus_{i\in I} M_i$ and linear maps $\phi_a : M_{s(a)} \xrightarrow[]{} M_{t(a)}$ for each $a\in H$ satisfying the preprojective relations $\sum_{a\in H,t(a)=i} \epsilon(a)\phi_{a}\phi_{a*}=0$.
Given a dimension vector $d\in \ensuremath{\mathbb{N}}^I$, define $\Lambda(d)$ to be the variety of all representations of $\Pi$ on M for $M_i=k^{d_{i}}$.
\begin{prop}[\cite{lusztig1990canonical}, \cite{MR2892443}]
Irr$(\Lambda)$ has a crystal structure isomorphic to $B(\infty)$.
\end{prop}
\subsection{A conjectural relation between MV cycles and modules over the preprojective algebra}
Baumann and Kamnitzer found an isomorphism between the crystal structure of Irr$(\Lambda)$ and MV polytopes.
For each $\gamma\in \Gamma$, they define constructible funtion $D_{\gamma}: \Lambda(d) \xrightarrow[]{} \ensuremath{\mathbb{Z}}_{
{\geq0}}$\footnote{$\Lambda(d)$ and $D_{\gamma}$ do not depend on the direction of the edges in E.}.
For any $M\in\Lambda(d)$, the collection $ (D_{\gamma})_{\gamma\in \Gamma}$ satisfies certain edge inequalities hence determines a polytope which we denote by $P(M)$.
\begin{theorem}[\cite{MR2892443}]
When $M$ is generic, $P(M)$ is an MV-polytope and for $d=(d_i)_{i\in I}$ this gives a map from Irr($\Lambda(d))$ to the set of MV polytopes of weight $\sum_{i\in I} d_i\alpha_i$. This map is a bijection compatible with the crystal structures.
\end{theorem}
We have MV-cycles (in bijection with MV-polytopes) as the geometric object on the loop Grassmannian side.
In order to upgrade the relations geometrically,
Kamnitzer-Knutson consider the quiver Grassmannian on the quiver side.
The quiver Grassmannian $Gr^{\Pi}(M)$ of a $\Pi$-module $M$ is defined as the moduli of submodules of $M$.
It is a subscheme of the moduli of $k$-vector subspaces of $M$ which is product of usual grassmannian $\prod_{i\in I} Gr(M_i)$. Here we will only consider $Gr^{\Pi}(M)$ with its reduced structure, and actually just as a topological space.
As the case of usual grassmannian, the
quiver Grassmannian $Gr^{\Pi}(M)$ is disjoint union of Grassmannians of different dimension vectors. Denote $Gr_{e}^{\Pi}(M)$ by the moduli of submodule N of M of dimension vector $e$, we have
$Gr_{e}^{\Pi}(M)\subset \prod_{i\in I} Gr_{e_i}(M_i)$.\\
Given a module $M\in \Lambda(d)$, form the subscheme\footnote{We will call it cycle in this paper.} $X(M)$=$\bigcap_{w\in W} \overline{S^{w}_{\lambda_{w}}}$, where
$\lambda_w=\sum_{i\in I}-D_{-w\varpi_{i}}(M)w\check{\alpha{_i}}$.
T acts on $S^{w}_{\lambda_w}$ by multiplication, hence it also acts on the closure and the intersection $X(M)$.\\
\begin{conj}
The ring of functions on the $T$-fixed point subscheme of $X(M)$ is isomorphic to the cohomology ring of the quiver grassmannian of $M$
$$\mathcal{O} (X(M)^{T}) \xrightarrow[\sim]{\Psi} H^*(Gr^{\Pi}(M)).$$
More precisely, $X(M)^{T}$ is disjoint union of finite schemes $X(M)^{T}_{\nu}$ supported at $L_\nu$, $\nu\in X_{*}(T)$ and we can further identify two sides for each connnected component
$$\mathcal{O} (X(M)_{\nu}^{T}) \xrightarrow[\sim]{\Psi} H^*(Gr_{e}^{\Pi}(M)),
\text{ where } e_i=(\nu, \varpi_i).$$
\end{conj}
Remark: we define $X(M)$ as a scheme theoretic intersection of closures while MV-cycles have been defined as varieties (closure of intersections). We notice that $X(M)$ may be reducible even when $P(M)$ is an MV-polytope.
For an example, see the appendix\footnote{Not yet written, I will add this later on.}.
The former certainly contains the latter and a further hope is to relate the latter to some subvariety of the quiver Grassmannian.
\section{The T fixed point subscheme of the cycle}
We introduce some notation first.
It is known that the $T$-fixed point subscheme of the loop grassmannian of a reductive group $G$ is the loop grassmannian of the Cartan $T$ of $G$, i.e.,
$\mathcal{G}(G)^{T}=\mathcal{G}(T)$.
We indentify $T$ with $I$ copies of the multiplicative group by
$T\xrightarrow[\sim]{\prod\varpi_i} G_{m}^{I}$
and this gives $\mathcal{G}(T)\xrightarrow[\sim]{\prod\varpi_i} \mathcal{G}(G_m)^{I}$.
For $\mathcal{G}(G_m)$, we have
\begin{align}
&\mathcal{G}(G_m)=G_m(\mathcal{O}) \setminus G_m(\mathcal{K})\\
&=\{\text{unit} \in \mathcal{O}\} \setminus \{\text{unit} \in \mathcal{K}\}\\
&=t^{\ensuremath{\mathbb{Z}}}\cdot K_{-}
\end{align}
where $K_{-}$ is called the negative congruence subgroup (of $G_m$).
The $R$-points of $K_{-}$ can be described as: $$K_{-}(R)=\{a=(1+a_1t^{-1}+...+a_mt^{-m})| a_i \text{ is nilpotent in }R \}.$$
We define the degree function from $K_{-}(R)$ to $\ensuremath{\mathbb{Z}}_{\geq}$: deg$(a)=m$ if $a_m\neq 0$.
Then
$(\bigcap \overline{S^{w}_{\lambda_{w}}})^{T}$
is a subscheme of $\mathcal{G}(G)^{T}\cong (t^{\ensuremath{\mathbb{Z}}}\cdot K_{-})^{|I|}$.
\begin{theorem}
Let $(\lambda_w)_{w\in W}$ be a collection of cocharacters such that $\lambda_v \geq_{w} \lambda_w$\footnote{This notation is used in \cite{kamnitzer2005}, $\lambda_v \geq_{w} \lambda_w$ whenever $w^{-1}\lambda_v \geq w^{-1}\lambda_w$.} for all $w\in W$ in which case we know (\cite{kamnitzer2005}) that $(\lambda_w)_{w\in W}$ determines a pseudo-Weyl polytope. The integers $A_{w\varpi_i}$ are well defined by $A_{w\varpi_i}=(\lambda_w,w\varpi_i)$. The R-points of $
(\bigcap \overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$ is the subset of R-point of $(t^{\ensuremath{\mathbb{Z}}}\cdot K_{-})^{|I|}$ containing elements
$(t^{(\nu,\varpi_{i})}a_i) \in \prod (t^\ensuremath{\mathbb{N}} \cdot K_-)^{|I|}|$ subject to the degree relations: $$deg(\Pi_{i\in I} a_i^{(\gamma, \check{\alpha_i})}) \leq -A_{\gamma}+\sum (\gamma,\nu)\text{ for all } \gamma \in \Gamma\}.$$
\end{theorem}
\begin{proof}
We define loop grassmannian with a condition $Y$ and list the facts we need. For details, see \cite{ivan3}.
Let $G$ acts on scheme $Y$ and $y$ be a point in $Y$. Denote the stack quotient by $Y/G$.
Then $\mathcal{G}(G,Y)$ is the moduli of maps of pairs from $(d, d^{*})$ to $(Y/G, y)$.
When $Y$ is a point we recover $\mathcal{G}(G)$.
In general, $\mathcal{G}(G,Y)$ is the subfunctor of $\mathcal{G}(G)$ subject to a certain extension condition:$$
\mathcal{G}(G,Y)=G_{\mathcal{O}}\setminus \{g\in G_{\mathcal{K}}\ |\ d^{*}\xrightarrow[]{g}G\xrightarrow[]{o} Y \text{ extends to $d$}\}, \text{ where } o(g)=gy.$$\\
We can realize semi-infinite orbits and their closures as follows:
\begin{itemize}
\item $\mathcal{G}(G,G/N)=S_0$, where G acts G/N by left multiplication.
\item $\mathcal{G}(G,(G/N)^{aff})=\overline{S_0}$, where "aff" means affinization.
\item $\mathcal{G}(G\times T,(G/N)^{aff})_{red}=\bigsqcup \overline{S_{\lambda}}$, where "red" means the reduced subscheme. Here T acts on $G/N$ by left multiplication with the inverse and this extends to an action on $(G/N)^{aff}$.
\item $$ \mathcal{G}(G\times \prod_{w\in W} T_w, \prod_{w\in W} (G/N^w)^{aff})=\bigsqcup_{(\lambda_w)_{w\in W}} (\bigcap_{w\in W} \overline{S^{w}_{\lambda_{w}}})$$, We denote a copy of T corresponding to $w\in W$ by $T_w$.
\end{itemize}
A single cycle $\bigcap_{w\in W}\overline{ S^{w}_{\lambda_{w}}}$ can be written as the fiber product:
$$\bigcap_{w\in W}\overline{ S^{w}_{\lambda_{w}}}=\mathcal{G}(G\times \prod_{w\in W} T_w, \prod (G/N^w)^{aff}) \times_{\mathcal{G}(\prod_{w\in W} T_w)} (t^{\lambda_{w}})_{w\in W}.$$
In this fiber product, the morphism for the first factor is the second projection and
the morphism for the second factor is the inclusion of the single point $t^{\underline{\lambda}}=(t^{\lambda_{w}})_{w\in W}$.\\
For a reductive group G, we have $\mathcal{G}(G,Y)^{T}=\mathcal{G}(T,Y)$, where T is the cartan of G.
\\
So, the T fixed point subscheme is \\
$$(\bigcap_{w\in W}\overline{ S^{w}_{\lambda_{w}}})^T=\mathcal{G}(T\times \prod_{w\in W} T_w, \prod (G/N^w)^{aff}) \times_{\mathcal{G}(\prod_{w\in W} T_w)} t^{\underline{\lambda}}.$$ \\
In terms of the above extension condition, this fiber product is:
$$(\bigcap_{w\in W}\overline{ S^{w}_{\lambda_{w}}})^T
=T(\mathcal{O})\setminus \{ g\in T_{\mathcal{K}}, \text{ such that } d^{*}\xrightarrow[]{g,t^{\underline{\lambda}} } T\times T^W \xrightarrow[]{}\prod (G/N^w)^{aff} \text { extends to } d \}$$
This is the $T(\mathcal{O})$ quotient of the set of all $g\in T_{\mathcal{K}}$, such that $$ d^{*}\xrightarrow[]{g,t^{\lambda_{w}} } T\times T_w \xrightarrow[]{} (G/N^w)^{aff} \text { extends to } d \text{ for all }w\in W.
$$
For $\gamma\in W\cdot \varpi_i \subset \Gamma$ , we fix weight vectors $v_{\gamma} \text{ in the weight space} \
(V_{\varpi_i})_{\gamma} $ of $V_{\varpi_i}$. For each $w\in W$, we embed $G/N^w \text{ into } \bigoplus_{i\in I} V_{\varpi_i} \text{ by } g\mapsto (g\cdot v_{w\varpi_i})_{i\in I}$. Under this embedding, $(G/N^w)^{aff}$ is a closed subscheme in $\bigoplus_{i\in I} V_{\varpi_i}$.
\\
For $g\in T_{\mathcal{K}}$, $w\in W$, the composition $y_w(g)$ of the map : $$ d^{*}\xrightarrow[]{g,t^{\lambda_{w} }} T\times T_w \xrightarrow[]{} G/N^w \hookrightarrow \bigoplus V_{\varpi_i}$$is
$$y_{w}(g)=(g \cdot(t^{\lambda_{w}})^{-1})\sum_{i\in I} v_{w\varpi_i}= \sum_{i\in I} (w\varpi_i(g \cdot t^{-\lambda_w}))v_{w\varpi_i}.$$\\
This map extends to $d$ when for each $i\in I$, the coefficient of $v_{w\varpi_i}$ is in $\mathcal{O}$.
The coefficient of $v_{w\varpi_i}$ is
\begin{align}
w\varpi_i(g \cdot t^{-\lambda_w})=w\varpi_i(g)\cdot w\varpi_i (t^{-\lambda_w})=w\varpi_i(g)\cdot t^{-(w\varpi_i,\ \lambda_w)}\notag\\
=w\varpi_i(g) t^{-A_{w\varpi_i}}
=\gamma(g) z^{-A_\gamma} \text{ where } \gamma=w\varpi_i\notag.
\end{align}
It follows that
$$(\bigcap_{w\in W}\overline{ S^{w}_{\lambda_{w}}})^T=T(\mathcal{O})\setminus
\{g \in T(\mathcal{K});
\gamma(g) t^{-A_\gamma} \in \mathcal{O} \text{ for all } \gamma \in \Gamma\}.$$
and the description of the R-points of $
(\bigcap \overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$ in the theorem follows when we identify $\mathcal{G}(T)\xrightarrow[]{\prod\varpi_i} \mathcal{G}(G_m)^{I}=(t^{\ensuremath{\mathbb{Z}}}\cdot K_{-})^{I}$.
\\
\end{proof}
\subsection{Ring of functions on $
(\bigcap \overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$}
For an R-point $(t^{(\nu,\varpi_{i})}a_i)_{i\in I}$ of $
(\bigcap \overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$, let us write $a_i=1+a_{i1}t^{-1}+\cdots+ a_{im}t^{-m}$. When $\gamma=\varpi_i$, the degree inequality is deg$(a_i)\leq (\varpi_i,\nu)-A_{\varpi_i}$.
We can take the coefficients $a_{ij}$ to be the coordinate functions on $
(\bigcap \overline{S^{w}_{\lambda_{w}}})_{\nu}^{T}$. Since deg$(a_i)\leq (\varpi_i,\nu)-A_{\varpi_i}$
, there are finitely many $a_{ij}$s which generate the ring of functions on $\mathcal{O}((\bigcap \overline{ S^{w}_{\lambda_{w}}})_{\nu}^{T})$.\\
When we take inverse of $a_i$, it is computed in $K_{-}$ as $a_i^{-1}=1+\sum_{s\geq 0}(-1)^{i}(a_{i1}t^{-1}+\cdots+a_{im}t^{-m})^s$ and then expands in the form $\sum_{i}b_{ik} t^{-k}$, where $b_{ik}$ is the coefficient of $t^{-k}$ in $a_i^{-1}$.
$$deg(\Pi_{i\in I} a_i^{(\gamma, \check{\alpha_i})}) \leq -A_{\gamma}+\sum (\gamma,\nu)\text{ for all } \gamma \in \Gamma.$$
is equivalent to the condition that the coefficient of the term $t^{-1}$ to the power $-A_{\gamma}+\sum (\gamma,\nu)+1$ in
$(\Pi_{i\in I} a_i^{(\gamma, \check{\alpha_i})})$ is 0. These coefficients are polynomials of $a_{ij}$s.
Set $b_i=1+\sum_{k}b_{ik} t^{-k}=a_i^{-1}$,
add $b_{ij}$'s as generaters and also add the relations $a_{i}b_{i}=1$ for $i\in I$ which eliminate all $b_{ij}$'s.
For $\gamma \in \gamma$, let $\gamma_i=(\gamma, \check{\alpha_i})$.
Denote by $I_\gamma^{+}$ the subset of I containing all i such that $\gamma_i $ is positive and by $I_\gamma^{-}$ containing all i $\gamma_i$ is negative. Set $\gamma^{+}_i=\gamma_i$ when $\gamma_i$ is positive and $\gamma^{-}_i=-\gamma_i$ when $\gamma_i$ negative.
\begin{cor}
The ring of functions on $\mathcal{O}((\bigcap \overline{ S^{w}_{\lambda_{w}}})_{\nu}^{T})$ is generated by $a_{ij}$'s and $b_{ik}$'s, for $i\in I$. The relations are degree conditions:
$$ deg(\prod_{i\in I_{\gamma}^{+}}
a_{i}^{\gamma{_i}^{+}}\prod_{i\in I_{\gamma}^{-}}
b_{i}^{\gamma{_i}^{-}})\leq (\gamma,\nu)-A_{\gamma}$$ for each $\gamma\in \Gamma$ and conditions $a_{i}b_{i}=1$ for each i in I.
\end{cor}
\section{Construction of the map $\Psi$ from functions to cohomology}
\subsection{Map $\Psi$}
For $M\in \Lambda(d)$, to apply corollary 1 to $X(M)$, we set $A_\gamma=-D_{-\gamma}(M)$. Then
$$\mathcal{O}(X(M)_{\nu}^{T})
=k[a_{ij},b_{ik}]/I(M)$$
where $I(M)$ is the ideal generated by the degree conditions: $$ deg(\prod_{i\in I_{\gamma}^{+}}
(a_i)^{\gamma_i^+}\prod_{i\in I_{\gamma}^{-}}
(b_i)^{\gamma_{i}^{-}})\leq (\gamma,\nu)+D_{-\gamma}(M)$$ for each $\gamma\in \Gamma$ and the conditions $a_{i}b_{i}=1$ for each i in I.
The conjecture
$\mathcal{O} (X(M)_{\nu}^{T}) \cong H^*(Gr_{e}^{\Pi}(M))$,
where $e_i=(\nu, \varpi_i)$,
is now equivalent to
$$k[a_{ij},b_{ik}]/I(M)
\cong H^*(Gr_{e}^{\Pi}(M)).$$
The quiver Grassmannian $Gr_{e}^{\Pi}(M))$ is a subvareity of
$\prod_{i\in I} Gr_{e_{i}}(M_i)$ and we have on each $Gr_{e_{i}}(M_i)$ the tautological subbundle $S_i$ and quotient bundle $Q_i$. We pull back $S_i$ and $Q_i$ to $\prod_{i\in I} Gr_{e_{i}}(M_i)$ and denote their restrictions on $Gr_{e}^{\Pi}(M))$ still by $S_i$ and $Q_i$ by abusing notion.
For a rank n bundle E, denote the Chern class by $c(E)$ and the $i^th$ Chern class $c_i(E)$, where $c(E)=1+c_1(E)+\cdots+c_n(E) $.
We want to define the map $$\Psi: \mathcal{O} (X(M)_{\nu}^{T}) \xrightarrow[]{} H^*(Gr_{e}^{\Pi}(M)),
\text{ where } e_i=(\nu, \varpi_i),$$ by mapping the generators $a_{ij}$ to $ c_{j}(S_i)$ and
$b_{ij}$ to $c_{j}(Q_i)$.
\begin{theorem}
The map $\Psi$ described above is well defined.
\end{theorem}
\subsection{Two lemmas}
For the proof, we need two lemmas. Lemma 1 is the special case of theorem 4 when $Q$ is the quiver $1\xrightarrow[]{}2$
and $M$ is a $kQ$-module.
\begin{lemma}
Let $Q$ be the quiver $1\xrightarrow[]{}2$
and $M$ be $\ensuremath{\mathbb{C}}^{d_1}\xrightarrow[]{\phi} \ensuremath{\mathbb{C}}^{d_2} $.On $X=Gr^{\Pi}_{e}(M)$, we have $c_i(S_2\oplus Q_1)=0 $ when $i> e_2-e_1+\text{dim}(ker\phi)$.
\end{lemma}
Let $\phi_{ij}: M_i\xrightarrow[]{} M_j$ be the composition of $\phi_a$ where a travels over the unique no going-back path which links i and j.
Let $M_\gamma= \oplus_{i\in I_{\gamma}^{-}}M_i^{\gamma^{-}} \xrightarrow[]{\phi_{\gamma}=\oplus\phi_{ij}}\oplus_{i\in I_\gamma^{+}}M_i^{\gamma^{+}}$ be the module over $k(1\xrightarrow[]{}2)$.
\begin{lemma}
For a $\Pi$-module $M$ and any chamber weight $\gamma$, we have
$$dim(ker\phi_{\gamma})= D_{-\gamma}(M).$$
\end{lemma}
Lemma 2 is a property of $D_{\gamma}$ and will be proved in the appendix.
\subsection{Proof of theorem 4 from lemmas in $\mathsection$ 4.2}
\begin{proof}[Proof of theorem 4]
We prove the theorem can be reduced to lemma 1.
For each $\gamma \in \Gamma$,we have to prove the degree inequilities carry over to Chern classes:
$$\Psi(\prod_{i\in I_{\gamma}^{+}} t_i^{\gamma_{i}^+}\prod_{i\in I_{\gamma}^{-}} s_i^{\gamma_{i}^{-}})=\prod_{i\in I_{\gamma}^{+}} c(S_i)^{\gamma_{i}^+}\prod_{i\in I_{\gamma}^{-}} c(Q_i)^{\gamma_{i}^{-}}\leq D_{w_0\gamma}(M)+(\nu,\gamma). $$
Define a map $\Phi$ from $Gr^{\Pi}(M)$ to $Gr^{k(1\xrightarrow[]{}2)}(M_\gamma)$: for $N\in Gr^{\Pi}(M)$,
$\Phi(N)=\oplus_{i\in I_{\gamma}^{-}}N_i \xrightarrow[]{\phi_{\gamma}}\oplus_{i\in I_\gamma^{+}}N_i$. We have
$$\Phi^{*}(c(S_2)c(Q_1))=c(\Phi^{*}(S_2))c(\Phi^{*}(Q_1))= c(\oplus_{i\in I_{\gamma}^{+}}S_i^{\gamma_{i}^{+}})c(\oplus_{i\in I_{\gamma}^{-}}Q_i^{\gamma_{i}^{-}})$$
$$=\prod_{i\in I_{\gamma}^{+}} c(S_i)^{\gamma_{i}^{+}}\prod_{i\in I_{\gamma}^{-}} c(Q_i)^{\gamma_{i}^{-}}.$$
Apply lemma1 to $M_{\gamma}$ we have $$deg(c(Q_1)c(S_2)) \leq dimker(\phi_{\gamma})+\sum_{i\in I_{\gamma}^{+}}\gamma_i e_i-\sum_{i\in I_{\gamma}^{-}}\gamma_i e_i$$
$$=dimker(\phi_{\gamma})+\sum_{i\in I}\gamma_i e_i=dimker(\phi_{\gamma})+(\gamma,\nu).$$
Then the theorem follows by lemma 2.
\end{proof}
Chern class vanishes in certain degree when the bundle contains a trivial bundle of certain degree but the desired trivial bundle in $Q_1\oplus S_2$ does not exist.
The idea is to pass to T-equivariant cohomology. Over $X^T$ which is just a union of isolated points we will decompose $Q_1\oplus S_2$ into the sum of the other two bundles $E_1$ and $E_2$ pointwisely, where $E_{1}$ will play the role of trivial bundle. Although there is no bundle over X whose restriction is $E_2$, there exist T-equivariant cohomology class in $H_{T}^{*}(X)$ whose restriction on $X^T$ is the T-equivariant Chern class of $E_2$.
\subsection{Recollection of GKM theory}
We first recall some facts in T-equivariant cohomoloy theory.
We follow the paper \cite{tymoczko2005introduction}.
Denote a n-dimensional torus by $T$, topologically $T$ is homotopic to$ (S^1)^n$. Take $ET$ to be a contractible space with a free
$T$-action.
Define $BT$ to be the quotient $ET/T$.
The diagonal action of $T$ on $X \times ET$ is free, since the action
on $ET$ is free. Define $X \times_T ET$ to be the quotient
$(X \times ET) / T$.
We define the equivariant cohomology of $X$ to be
\[H^*_T(X) = H^*(X \times_T ET).\]
When $X$ is
a point and $T=G_m$,
$$
H^*_T(X) = H^*(\textup{pt} \times_T ET) =
H^*(ET/T)=H^*(BT)=
H^*(\mathbb{CP}^{\infty})\cong k[t].$$ When $T=(S^1)^n$,
\begin{align}
H^*(pt)=k[t_1,\cdots,t_n]\cong S(\mathfrak{t}^*).
\end{align}
So we can identify any class in $H^*(pt)$ as a function on the lie algebra $\mathfrak{t}$ of $T$.
The map $X \xrightarrow[]{}pt$ allows us to pull back each class in
$H^*_T(\textup{pt})$ to $H^*_T(X)$, so $H^*_T(X)$ is a
module over $H^*_T(\textup{pt})$.
Fix a projective variety $X$ with an action of $T$. We say that $X$
is equivariantly formal with respect to this $T$-action if
$E^2=E^\infty$ in the spectral sequence associated to the fibration
$X \times_T ET \longrightarrow BT$.
When $X$ is equivariantly formal with respect to $T$,
the ordinary cohomology of $X$ can be reconstructed from its
equivariant cohomology. Fix an inclusion map
$X\xrightarrow[]{}X \times_T ET$, we have
the pull back map of cohomologies: $H^*(X \times_T ET\xrightarrow[]{i} H^*(X)$.
The kernel of $i$ is $\sum_{s=1}^{n} t_s\cdot H^*_T(X)$, where $t_s$ is the generator of $H^*_T(pt)$ (see (4)) and we view it as an element in $H^*_T(X)$ by pulling back the map $X\xrightarrow[]{}pt$. Also $i$ is surjective so
$H^*(X) = H^*_T(X)/ ker(i)$.
If in addition $X$ has finitely many fixed points and finitely many one-dimensional orbits,
Goresky, Kottwitz, and MacPherson show that
the combinatorial data encoded in the graph of fixed points
and one-dimensional orbits of $T$ in $X$
implies a particular algebraic characterization of $H^*_T(X)$.
\begin{theorem}[GKM, see \cite{tymoczko2005introduction}, \cite{goresky1997equivariant}]
Let $X$ be an algebraic variety with a $T$-action with respect to which
$X$ is equivariantly formal, and which has
finitely many fixed points and finitely
many one-dimensional orbits. Denote the one-dimensional orbits $O_1$, $\ldots$, $O_m$. For each
$i$, denote the the $T$-fixed points of $O_i$ by $N_i$ and $S_i$ and denote the
stabilizer of a point in $O_i$ by $T_i$. Then the map
$H^{*}_{T}(X)\xrightarrow[]{l}H^{*}_{T}(X^T)=\oplus_{p_{i}\in X^{T}}H^*_{T}(p_{i})$ is injective
and its image is
$$\left\{f=(f_{p_1}, \ldots, f_{p_m})
\in \bigoplus_{\textup{fixed pts}} S(t^*): f_{N_i}|_{\mathfrak{t}_i} =
f_{S_i}|_{\mathfrak{t}_i} \textup{ for each }i=1,\ldots,m \right\}.$$
Here $\mathfrak{t}_i$ is the lie algebra of $T_i$.
\end{theorem}
\subsection{Affine paving of $Gr^{\Pi}_e(M)$ when $M$ is a representation of $Q$ of type A}
\begin{defi}[\cite{affine} 2.2]
We say a space $X$ is paved by affines if $X$ has an order partition into disjoint $X_1,X_2,\cdots$ such that each finite union $\bigcup_{i=1}^j X_i$ is closed in X and each $X_i$ is an affine space.
\end{defi}
A space with an affine paving has odd cohomology vanishing.
\begin{prop}[\cite{affine}, 2.3]
Let $X=\bigcup X_i$ be a paving by a finite number of affines with
each $X_i$ homeomorphic to $\ensuremath{\mathbb{C}}^{d_i}$. The cohomology groups
of $X$ are given by $H^{2k}(X) = \bigoplus_{\{i\in I \ |\ d_i = k\}} \mathbb{Z}$.
\end{prop}
The main observation is the following lemma.
\begin{lemma}
Let $M$ be a representation of Q, where Q is of type A with all edges in E pointing to the right. Then the quiver Grassmannian $Gr^{\Pi}_e(M)$
is paved by affines for any dimension vector e.
\end{lemma}
We need a sublemma first.
\begin{sublemma}
Suppose X is paved by $X_i$'s. Let $Y\subset X $ be a subspace. if for each i, $Y_i=X_i \bigcap Y$ is $\emptyset$ or affine then $Y=\bigcup Y_i$ is an affine paving.\\
\end{sublemma}
\begin{proof}
$\bigcup_{i\leq j} Y_i=\bigcup_{i\leq j} (X_i \bigcap Y)=(\bigcup_{i\leq j} X_i) \bigcap Y$ is closed in $Y$ since $\bigcup_{i\leq j} X_i$ is closed in $X$.
\end{proof}
\begin{proof}[Proof of lemma 3]
Let $V=\oplus M_i$ be the underlying vector space and $\phi=\oplus_{a\in H} \phi_a$ be the nilpotent operator on $V$.
We adopt the notations in \cite{shimomura}.
Let $n=dimV$ and $d=\sum e_i$.
Let $C^{\phi}_\alpha =\{(v_1,...v_d)\in C_{\alpha}, \phi v_i=v_j $ if $\alpha$ contains
$\ytableaushort[\alpha_]{ij}$
$,(1\leq i <j<d)\}$.
We know that $Gr_{d}(k^n)^{\phi}=\bigsqcup C_{\alpha}^{\phi}$.
We want to show $S^{\phi}_\alpha \bigcap Gr^{\Pi}_{e}(M)$ is affine.
Take $x\in S^{\phi}_\alpha \bigcap Gr^{\Pi}_{e}(M)$, from 1.10 in \cite{shimomura}, $x=v_1\wedge...\wedge v_d$ where
$x= \wedge(\phi^{h}w_i : i's $ are the initial numbers of $\alpha$ and $\phi^{h}w_i\ne 0).$\\
We now show that
for $x=v_1\wedge\cdots \wedge v_d \in Gr^{\Pi}_{e}(M)$, where $v_i=e_i+ \sum_{j\neq i} x_i(j)e_j $, if $e_i\in M_t$, we have $v_i \in M_t$. Conversely, if for each i, there exists t such that $v_i \in M_t$, $x\in Gr^{\Pi}_{e}(M)$.Denote this t determined uniquely by i as t(i).\\
Since $span(v_1,...,v_d)$ is a direct sum of some $N_i\subset M_i$, we have $Pr_t(v_i) \in span(v_1,...,v_d)$,
where $Pr_t$ is the projection from V to $M_t$,
and so $Pr_t(v_i)=\sum a_{p}v_{p}$. Comparing the coefficient of $e_p$, by the definition of $C_{\alpha}$, we have $a_p=0$ for $p\neq i$.
So we have $Pr_t(v_i)=v_i$, which implies $v_i\in M_t$. \\
Note that $\{v_1,\cdots,v_d\}$ is determined by $w_i$ where i is an initial number(and vice versa).\\
We have $w_i\in M_{t(i)}$.
Denote $l(i)$ be the number on the left of i in the d-tableaus. If i is the leftmost, set $l(i)$ to be $\emptyset$, and set $e_{\emptyset}=0$. Write $w_i= e_i +\sum x_{ij}e_j$, where $e_j\in M_{t_i}$,
we have $\phi^{r}(w_i)= e_{l^{r}(i)} +\sum x_{ij}e_{l^{r}(j)}$. Since
M is kQ-module, we have $l^{r}(i)=l^{r}(j)$ hence $v_{i_{r}}\in M_{t(i_{r})}$ and $x\in Gr^{\Pi}_e(M)$. So we have
$S^{\phi}_\alpha \bigcap Gr^{\Pi}_{e}(M)$ is affine.
Apply lemma 3, we are done.
\end{proof}
\subsection{Proof of lemma 1}
\begin{proof}
For $M$ given by $\ensuremath{\mathbb{C}}^{d_{1}}\xrightarrow[]{\phi} \ensuremath{\mathbb{C}}^{d_{2}}$ and a choice of $e=(e_1, e_2)$,
denote $X=Gr^{\Pi}_e(M)$.
First, we define a torus action on $X$.
Let $I=ker\phi $. Choose a basis $e_1,e_2,\cdots ,e_s $ of $I$ and extend it to a basis $e_1,\cdots ,e_s,e_{s+1},\cdots,e_t$ of $M_1$. Let $J$ be span$\{e_{s+1},\cdots,e_t)\}$ so the image of $J$ is span $\{f_{s+1},\cdots,f_t\}$. We extend the basis $\{f_i=\phi(e_i)\}$ of the image of $J$ to a basis ($f_{s+1},\cdots,f_t,f_{t+1},\cdots,f_r)$ of $M_2$. Let K=span$\{f_{t+1},...f_r\}$.
we have $M_1=I\oplus J$ and $M_2=\phi(J)\oplus K$.\\
Let $\mathcal{I}=\{1,\cdots,s\}$, $\mathcal{J}=\{s+1,\cdots,t\}$ and $ \mathcal{L}=\{t+1,\cdots,r\}$.
Let tori $T_{I}=G_m^{\mathcal{I}},T_{J}=G_m^{\mathcal{J}} ,T_{L}=G_m^{\mathcal{L}}$
act on $I,J\cong \phi(J)$, $K$
by multiplication compotentwisely (For instance, $T_{I}$ acts on I by $(t_1,\cdots,t_s)\sum a_{i}e_i =\sum a_{i}t_{i}e_i$ and on $J,K$ trivially). Hence they act on $M_1=I\oplus J$ and $M_2=\phi(J) \oplus K$.
This induces an action of $T=T_{I}\times T_{J}\times T_{K}$ on $Gr^{\Pi}_e(M)$.
By lemma 3, $Gr^{\Pi}_e(M)$ is paved by affines so by proposition 3 it has odd cohomology vanishing therefore the spectral sequence associated to the fibration
$X \times_T ET \longrightarrow BT$ converges at $E^2$ and X is equivariantly formal.
Denote by $f$ the forgetful map $H^*_{T}(X)\xrightarrow[]{f} H^*(X)$. From $\mathsection 4.4 $ we have $
ker(f)=\sum_1^{dimT} t_s H^*_{T}(X)$.
Since $c^i(S_2\oplus Q_1)=f(c^{i}_{T}(S_2\oplus Q_1))$,
it suffices to prove $c^{i}_{T}(S_2\oplus Q_1))\in ker(f)$
when $i>e_2-e_1+dimI$.
To use GKM theorem, we need to know the one dimensional orbits and T-fixed points of X.
First, we see what $X^T$ is.
For a point $p=(V_1, V_2)$ in $X$, in order to be fixed by $T$, $V_1$ and $V_2$ need to be spanned by some of basis vectors $e_i$ and $f_i$.
For a subset $S$ of $\mathcal{I}$ (resp. $\mathcal{J}$), we denote by $e_{S}$ (resp. $f_{S}$) the span $\{e_i|i\in S\}$ (resp. span$\{f_i|i\in S\}$). The T-fixed points in $X$ consist of all $V=(V_1, V_2)$, such that $V_1=e_{A\bigcup B}, V_2=f_{C\bigcup D}$, for some
$A\subset I,B\subset C\subset J$ and $D\subset K$.
For any point $p=(V_1, V_2)$ in $X^T$, let $V_1=e_{A\bigcup B}, V_2=f_{C\bigcup D}$.
Over $p$, $Q_1=(I\oplus J)/ e_{A\bigcup B}$ is isomorphic to $e_{(\mathcal{I} \setminus A)\oplus (\mathcal{J} \setminus B) }$
(The restriction of a $T$-equivariant bundle to a T-fixed point is just a $T$-module).
So over $X^T$, we can decompose $S_2\oplus Q_1$ as follows:
$$S_2\oplus Q_1 \cong e_{(\mathcal{I} \setminus A)\oplus (\mathcal{J} \setminus B)}\oplus f_{(C\bigcup D)}=(e_{(\mathcal{I} \setminus A)\oplus (C \setminus B)}\oplus f_{D}) \oplus (e_{\mathcal{J} \setminus C}\oplus f_{C}).$$
Denote the bundle over $X^{T}$ whose fiber over each point p is $e_{(\mathcal{I} \setminus A)\oplus (C \setminus B)}\oplus f_{D})$ by $E_1$ and the bundle over $X^{T}$ whose fiber over p is $e_{(\mathcal{J} \setminus C)}\oplus f_{C}$ by $E_2$.
We now use localization. Denote by $l$ the map
$H^{*}_{T}(X)\xrightarrow[]{l}H^{*}_{T}(X^T)=\oplus_{p\in X^T} H^*(p)$. From GKM theory $l$ is injective, so the condition $c^{i}_{T}(S_2\oplus Q_1))\in ker(f)$ is equivalent to
$l(c^{i}_{T}(S_2\oplus Q_1))\in l(ker(f))$.
We have
\begin{align}
l(ker(f))=l(\sum_{s=1}^{dimT} t_s H^*_{T}(X))=\sum_{s=1}^{dimT} \underbrace{(t_s,\cdots, t_s)}_\text{the number of T-fixed points in X} l(H^*_{T}(X)).
\end{align}
By functorality of Chern class,
$l(c^{i}_{T}(S_2\oplus Q_1))=c^{i}_{T}(S_2|_{X^T}\oplus Q_1|_{X^T}))$.\\
We compute the $T$-equivariant Chern class over $X^{T}$. For\footnote{We always denote S and Q but indicate over which space we are considering.} each p,
$$c_{T}^p (S_2\oplus Q_1)=c_{T}^p (E_2\oplus E_1)=\sum_{i} c_{T}^{p-i} (E_1) c_{T}^i (E_2)=\sum_{i\geq 1}c_{T}^{p-i} (E_1) c_{T}^i (E_2) .$$ The last equality holds since $c_{T}^{p} (E_2)=0$ when $p> dimE_2=dimI+e_2-e_1$.
Now to show $c_{T}^p (S_2\oplus Q_1)\in l(ker(f))$, It suffices to show that $c_{T}^{p-i} (E_1) c_{T}^i (E_2)\in l(ker(f))$, for any $i$.
The action of T on $E_2$ is actually the same on each T-fixed point. And at each point, $c_{T}^i (E_2)$ is the $i^{th}$ elementary symmetric polynomial of $t_s, 1\leq s\leq dimT$. So by (5), it suffices to show that $c_{T}^{i} (E_1)\in l(H_{T}^{*}(X))$.
Now we will see what 1-dimensional orbits are. Take an orbit
$O_i$, in order to be 1 dimensional its closure must contain two fixed points. Let $\overline{O_i}=O_i\bigcup \{N_i\}\bigcup \{S_i\}$, where
$N_i=(e_{A\bigcup B}, f_{C\bigcup D})$ and $S_i=(e_{A^{\prime}\bigcup B^{\prime}}, f_{C^{\prime}\bigcup D^{\prime}})$ are the fixed points.
$O_i$ is one dimensional whenever either $A\bigcup B$ and $A^{\prime}\bigcup B^{\prime}$ differ by one element with $C\bigcup D=C^{\prime}\bigcup D^{\prime}$ or $C\bigcup D$ and $C^{\prime}\bigcup D^{\prime}$ differ by one element with $A\bigcup B=A^{\prime}\bigcup B^{\prime}$ . In the first case, we have some $s\in A\bigcup B$ and $s^{\prime} \in A^{\prime}\bigcup B^{\prime}$, such that $A\bigcup B \setminus s =A^{\prime}\bigcup B^{\prime} \setminus s^{\prime}$.
Notice that the annihilator for the lie algebra $\mathfrak{t}_i$ in $S(t^*)$ is generated by $t_s-t_{s^\prime}$, so by theorem 5, the condition along $O_i$ for an element $h\in H^*_{T}(X^T)$ to be in $im(l)$ is $$(t_s-t_{s^{\prime}})\ |\ (h_{N_i}-h_{S_i}).$$
But we have
$$c_{T}(E_1)|_{N_i}-c_{T}(E_1)|_{S_i}=\\
(1+t_{s{\prime}})\prod_{i\in \mathcal{I}\bigcup C \setminus (A\bigcup B)\setminus \{s^{\prime}\}} (1+t_i)-
(1+t_s)\prod_{i\in \mathcal{I}\bigcup C \setminus (A^{\prime}\bigcup B^{\prime})\setminus \{s\}} (1+t_i).$$
Note that $\mathcal{I}\bigcup C \setminus (A\bigcup B)\setminus \{s^{\prime}\} = \mathcal{I} \bigcup C \setminus (A^{\prime}\bigcup B^{\prime})\setminus \{s\}$,
so $t_s-t_{s^{\prime}} $ divides $c_{T}(E_1)|_{N_i}-c_{T}(E_1)|_{S_i}$.
We conclude that $c_{T}^{i} (E_1)\in l(H_{T}^{*}(X))$.\\
The other case is similar.
\end{proof}
\section{Proof of isomorphism when $M$ is a representation of $Q$ of type A}
We first prove that $\Psi$ is surjective.
\begin{lemma}
(a) Denote $Y=\prod Gr_{e_i} (k^{d_i})$ and $X=Gr^{\Pi}_{e}(M)$. Then $Y\setminus X $ is paved by affines.\\
(b) $\Psi$ is surjective.
\end{lemma}
\begin{proof}
(a).
Let a be the number where the Young diagram of $\phi_{Y}$ has $a^{th}$ row as the first row from the bottom that does not have one block. For example, in the left diagram, a=4.
$$\ydiagram{4,3,2,2,1,1,1}\xrightarrow[]{} \ydiagram{4,3,2,1,1,1,1,1}$$
Define $\phi^{\prime}$ be the operator of $V$ that corresponds to the diagram by moving the left most block A of the $a^{th}$ row to the bottom in the diagram of $\phi_{Y}$.
Let $M^{\prime}$ be the corresponding module and $X^{\prime}$ be $Gr_e{M^{\prime}}$.\\
We claim that $X^{\prime}\setminus X$ is paved by affines.\\
By lemma 4, we have $X=\bigsqcup_{\alpha \in I } C_{\alpha}$, where $I^{\prime}$ is the set of all semi-standard young tableau in $\lambda$. Also we have $Y^{\prime}=\bigsqcup_{\alpha \in I^{\prime} } C^{\prime}_{\alpha}$, where $I$ is the set of all semi-standard young tableau in $\lambda^{\prime}$.
If $\alpha$ contains block A, $\alpha$ is s.s in $\lambda$ implies $\alpha^{\prime}$ is s.s in $\lambda^{\prime}$.
If $\alpha$ does not contain block A , $\alpha$ also does not contain any block in that row, so $\alpha$ is still s.s in $\lambda^{\prime}$. So $I\subset I^{\prime}$.
\\
For $\alpha$ that contains block A, there are two types.
Let E be the set of $\alpha$ that contains block A and some other block in the row of A.
Let F be the set of $\alpha$ that contains block A but no other block in the row of A.
Let G be the set of $\alpha $ that does not contain block A.
So we have $I=E\bigsqcup F\bigsqcup G=F\bigsqcup (E\bigsqcup G)$.\\
Take $\alpha \in F$, in $\lambda$,
the block A in $\alpha$ is not initial so
the vector indexed by A is determined by the initial vector.
In $\lambda^{\prime}$, A is the last block so the vector indexed by A is the basis vector indexed by block A. In both case the vector indexed by A has been determined, so $C_{\alpha}=C_{\alpha}^{\prime}$ when $\alpha \in F.$
For $\alpha \in E\bigcup G$, let $s(\alpha) $ be the tableau of the same relative position in $\lambda^{\prime}$ as $\alpha$ in $\lambda$. Then $C_{\alpha}=C^{\prime}_{s(\alpha)}$. Since s is a bijection between $E\bigsqcup G$ and $E^{\prime}\bigsqcup G^{\prime}$ we have $\bigsqcup_{\alpha \in E\bigsqcup G} C_{\alpha}=\bigsqcup_{\alpha \in E^{\prime}\bigsqcup G^{\prime}} C^{\prime}_{\alpha}$ .\\
Then we have $X^{\prime}\setminus X=\bigsqcup_{\alpha \in I^{\prime}\setminus I} C^{\prime}_{\alpha}$ is paved by affines.
We can do this procedure step by step until $X^{\prime}$ becomes Y, so we are done.
(b). By lemma 1, X is paved. With part(a) , we have the homology map from X to Y is injective hence $\Psi$ is surjective (see 2.2 in\cite{affine}).
\end{proof}
We want to prove the two sides of $\Psi$ have the same dimension as k-vector spaces and actually we will prove it for a more general setting.
\begin{defi}
For a $\Pi$-mod $M$
Let $V=\oplus M_i$ be the underlying vector space and $\phi=\oplus_{a\in H} \phi_a$ be the nilpotent operator on $V$.
We say $M$ is $I$-compatible if there is a Jordan basis $\{v_{ij}\}$ of V such that each $v_{ij}$ is contained in some $M_r$.
\end{defi}
\begin{defi}
For a $\Pi$-module $M$, if the Young diagram of the associated operator $\phi$ has one raw, we call $M$ one direction module.
\end{defi}
\begin{prop}
If $M$ is I-compatible, it is a direct sum of one direction module with multiplicities.
\end{prop}
\begin{lemma}
If M is I-compatible, the dimension of two sides of $\Psi$ have the inequality:
dim $k[a_{ij},b_{ik}]/I(M,e))\leq \chi (Gr^{\Pi}_e(M))$, where $\chi$ is the Euler character.
We denote the ring on the left by $R(M,e)$.
\end{lemma}
First we state a lemma due to Caldero and Chapoton.
\begin{lemma}[see prop 3.6 in \cite{caldero2004cluster}]
For $\Pi$-module $M,N$, we have
$$\chi(Gr_{g}(M\oplus N)=\sum_{d+e=g} \chi(Gr_{d}(M))\chi(Gr_{e}(N)).$$
\end{lemma}
The following two lemmas are proved after the proof of lemma 5.
\begin{lemma}
$R(M,e)/ <b_{1(d_1-e_1)}> \cong R(M^{\prime},e)$.
\end{lemma}
\begin{lemma}
$b_{1(d_1-e_1)}R(M,e)$ is a module over $R(M^{\prime\prime},e-\sum_{i \in I} \alpha_i)$.
\end{lemma}
\begin{proof}[Proof of lemma 5]\footnote{The rest of this chapter is not well-written and should be revised.}
We index the basis vector according to the Young diagram of $\phi$ as before but slightly different: $e_{ij}$ corresponds to the block of $i^{th}$ row (from up to down) and $j^{th}$ column (from right to left, which is the difference from before and this will cause the problem that two blocks in the same column but different row have different j but we will fix an i sooner so will not be of trouble). so $\phi(e_{ij})=e_{i(j-1)}$.\\
The basis vector in $M_1$ appears in the first column or in the last column. If it lies in the last, we can take the dual to make it in the first. So,
there exist i such that $e_{i1}\in M_1$. \\Let $\lambda^{\prime}$ be the young diagram removing the block of $e_{i1}$ from the original one and $M^{\prime}$ be the corresponding module.
Let $\lambda^{\prime \prime}$ be the young diagram removing $i^{th}$ row and $M^{\prime \prime}$ be the corresponding module.
Apply lemma 6, we have
$$\chi(Gr^{\Pi}_e(M))=\chi(Gr_{e}(M^{\prime}))+\chi(Gr_{e-\sum \alpha_i}(M^{\prime \prime})).$$
We count the dim of $R(M,e)$ by dividing it into two parts.\\$$
dimR(M,e)=dim b_{1(d_1-e_1)} R(M,e)+ dim R(M,e)/ b_{1(d_1-e_1)} R(M,e).$$
By lemma 7 and 8, since $b_{1(d_1-e_1)}R(M,e)$ is acyclic, \\
$$dim b_{1(d_1-e_1)}R(M,e) \leq dimR(M^{\prime\prime},e-\sum_{i \in I} \alpha_i).$$So
$dim R(M,e)=dim b_{1(d_1-e_1)} R(M,e))+ dim R(M,e)/ b_{1(d_1-e_1)} R(M,e)\\
\leq dim R(M^{\prime\prime},e-\sum_{i \in I} \alpha_i)
+dim R(M^{\prime},e)=\chi (Gr_{e}(M^{\prime})+ \chi (Gr_{(e-\sum_{i \in I} \alpha_i) } (M^{\prime \prime}))=
\chi (Gr^{\Pi}_{e}(M))$.
\end{proof}
Now we prove lemma 7 and 8.
\begin{proof}[Proof of lemma 7]
Recall $I(M)=\text{deg}\prod_{i\in I{+}}
(t_i)^{\gamma_i}\prod_{i\in I{-}}
(s_i)^{\gamma_i}\leq (\gamma,\nu)+D_{-\gamma}(M) $
We denote $v(M,\gamma,e)=(\gamma,\nu)+D_{w_{0}\gamma}(M)$.
The difference between $I(M)$ and $I(M)^{\prime}$ only occurs when $\gamma= -\varpi_1$. In this case $v(M^{\prime},-\varpi_1,e)=v(M,-\varpi_1,e)-1$. The degree of $s_1$ goes down by by 1, meaning we have one more vanishing condition which is $b_{1(d_1-e_1)}=0$.
\end{proof}
\begin{proof}[Proof of lemma 8]
In order to define a module structure on $b_{1(d_1-e_1)}R(M,e)$, we lift the element in $R(M^{\prime\prime},e-\sum_{i \in I} \alpha_i)$ to $R(M,e)$ (since the former is a quotient of the latter) and let it act on $b_{1(d_1-e_1)}R(M,e)$ by multiplication.
We denote J to be the degree $v(M,\gamma,e)$ part of $<\prod_{i\in I{+}}
(t_i)^{\gamma_i}\prod_{i\in I{-}}
(s_i)^{\gamma_i},\gamma\in \Gamma>$.\\
We need to check it is independent of the choice of the lift: $$b_{1(d_1-e_1)}J \subset I(M,e), \textup{ where }
I(M,e)=J\oplus I(M^{\prime\prime},e-\sum_{i \in I} \alpha_i).$$
Denote the module corresponding to $i^{th}$ row P.
We have $M=M^{\prime\prime}\oplus P$. Then
$v(M,\gamma,e)-v(M^{\prime\prime},\gamma,\sum_{i \in I} \alpha_i)=v(P,\gamma,\sum_{i \in I} \alpha_i)$.\\
We claim that $v(P,\gamma,\sum_{i \in I} \alpha_i)=0 \ or \ 1$
and is 0 when $\gamma_{1}=1$.\\
This is a direct calculation.
When $\gamma_{1}=-1$, $t_{1j}$ appears in each summand $\prod_{i\in I{+}}
(t_i)^{\gamma_i}\prod_{i\in I{-}}
(s_i)^{\gamma_i},\gamma\in \Gamma$. Let one of the summand be
$t_{1j}k$.\\
$s_{1(d_1-e_1)} t_{1j} k=-\sum_{p+q=d_1-e_1+j} s_{1p}t_{1q}k
=-s_{1(d_1-e_1+j-q)}\sum_{q>j} t_{1q}k$.
We have $\sum_{q>j} t_{1q}k$ is in $I(M,e)$ since this is of degree larger than $v(M,\gamma,e)$.\\
When $\gamma_1 =0$,\\
we claim that when
$v(P,\gamma,\sum_{i \in I} \alpha_i)$ is 1 , we have $\gamma-\varpi_1 \in \Gamma$. \\
Then we want to show
$s_{1(d_1-e_1)}\prod_{i\in I{+}}
(t_i)^{\gamma_i}\prod_{i\in I{-}}
(s_i)^{\gamma_i}$ is in $I(M,e)$.
Let k is a summand of degree $v(M,\gamma,e)$
part of $I(M,e)$.
We want to show $s_{1(d_1-e_1)}k$ is of degree $v(M,\gamma-\varpi_1,e)+1$.
So we need $v(M,\gamma,e)+d_1-e_1\geq v(M,\gamma-\varpi_1,e)+1$. By the
dimension description, the image of $\Phi_{\gamma-\varpi_1}$ is
at least 1 dimensional bigger than the image of $\Phi_{\gamma}$ since $\phi_{1m}(e_{i1})= e_{im}$ is in the image but for $\gamma-\varpi_1$ (since $\phi_{1m}$ is not a summand of $\phi$) the projection of $img(\phi)$ on $V_m$ is zero, where m is the smallest number in $\Gamma_+$.
\end{proof}
\begin{theorem}
$\Psi$ is an isomorphism when M is a kQ-module.
\end{theorem}
\begin{proof}
by lemma 4, $\Psi$ is surjective so dim $k[a_{ij},b_{ik}]/I(M,e))\geq \chi (Gr^{\Pi}_e(M))$ and by lemma 5 this is an equality so the theorem follows.
\end{proof}
\section{A consequence of this conjecture}
In section 3, we defined $\mathcal{G}(G,Y)$ as moduli of maps of between pairs form $(d,d^*)$ to $(G/Y,pt)$.
This is actually a local version of (fiber at a closed point c) the global loop Grassmannian with a condition Y to a curve $C$, $\mathcal{G}^{C}(G,Y)$.
To a curve C, define $\mathcal{G}^{C}(G,Y)$ over the ran space $\mathcal{R}_C$ with the fiber at $E\in \mathcal{R}_C$:
$$\mathcal{G}^{C}(G,Y)_{E}=^{def} map[(C,C-E), (G/Y, pt)].$$
Denote the map from $\mathcal{G}^{C}(G,Y)$ to $\mathcal{R}_C$ remembering the singularities by $\pi$.
One can ask if $\pi$ is (ind) flat for any $G$ and $(Y,pt)$.
The case we are concerned is when $G^{\prime}=G\times \prod_{w} T_w$ and $Y=\prod_{w}(G/N^{w})^{aff}.$
Let $c\in C, \underline{\lambda_w}, \underline{\mu_w}\in X_*(T)^{W}.$
In particular, we restrict $\mathcal{G}^{C}(G^{\prime},Y)$ to $C\times c$ and denote the image under projection from $\mathcal{G}(G^{\prime}$ to $\mathcal{G}(G)$ by $X$.
We have $X$ is a closed subscheme of $Gr_{G, X\times c}$.
Explicitly, an $R$-point of $Gr_{G, X\times c}$ consists of the following data
\begin{itemize}
\item $x: specR\xrightarrow[]{} C$. Let $\Gamma_x$
be the graph of x. Let $\Gamma_c$ be the graph of the constant map taking value c.
\item
$\beta$ a G-bundle on spec$R\times C$.
\item
A trivialization $\eta: \beta_0 \xrightarrow[]{\eta} \beta $ defined on $specR\times C- (\Gamma_x\bigcup \Gamma_c)$.
\end{itemize}
An $R$-point of $X$ over $C\times c$ consists of an $R$-point of $Gr_{G, X\times c}$ subject to the condition:
For every $i\in I$,
the composition
$$\eta_i : \beta_0 \times^{G} V(\varpi_i) \xrightarrow[]{} \beta \times^{G} V(\varpi_i) \xrightarrow[]{} \beta \times^{G} V(\varpi_i) \otimes \mathcal{O}(\langle \gamma, \lambda_w \rangle \cdot \Gamma_x + \langle \gamma, \mu_w \rangle \cdot \Gamma_c) .$$
is regular on all of $specR\times C$.
We can show the fiber over a closed point other than $c$ is $\bigcap \overline{S^{w}_{\lambda_w}}\times \bigcap \overline{S^{w}_{\mu_w}}$ and the fiber over $c$ is $\bigcap \overline{S^{w}_{\lambda_w+\mu_w}}$.
\begin{cor}[Given the conjecture]
The T-fixed point subscheme of this family is flat.
\end{cor}
\begin{proof}
$dim \mathcal{O}((\overline{ \bigcap S^{w}_{\lambda_w+\mu_w}}^{T})_{\nu})=dim H^*(Gr^{\Pi}_e (M))=\footnote{By lemma 6, and given the conjecture, Euler character is the same as total cohomology.}\sum_{e_1+e_2=e} dim H^*(Gr^{\Pi}_{e_1} (M_1) dim H^*(Gr^{\Pi}_{e_2} (M_2)= \sum_{\nu_1 +\nu_2=\nu } \mathcal{O}(\overline {\bigcap S^{w}_{\lambda_w}}^{T}_{\nu_1}) \cdot \mathcal{O}(\overline{ \bigcap S^{w}_{\mu_w}}^{T}_{\nu_2})
=dim \mathcal{O}(
\bigsqcup_{\nu_1 +\nu_2=\nu } \overline{ \bigcap S^{w}_{\lambda_w}}^{T}_{\nu_1}\times \overline{ \bigcap S^{w}_{\mu_w}}^{T}_{\nu_2}=
dim (\overline{ \bigcup S^{w}_{\lambda_w}}^{T}\times \overline{ \bigcap S^{w}_{\lambda_w}}^{T})_{\nu}.$
\end{proof}
\begin{conj}
T-fixed subschemes flatness imply flatness.
\end{conj}
We take $\lambda_w=-w_0 \lambda+w\lambda$, then
$\overline{\bigcap S^{w}_{\lambda_w}}=\overline{Y^{\lambda}}$. In this case the conjecture is proved to be true.
This flatness is mentioned in \cite{kamnitzer2016reducedness} remark 4.3 and will reduce the proof of reduceness of $\overline{Y^{\lambda}}$ to the case when $\lambda$ is $\varpi_i$ for each $i\in I$.
\section{Appendix}
For an expression $s_{i_{m}}\cdots s_{i_{1}}$ of an element w in W, we say it is j-admissible if $\langle \alpha_{i_a},s_{i_{a-1}}\cdots s_{i_{1}}\varpi_j\geq0$ for any $a\leq m$.
\begin{lemma}
For any element $w\in W$, any reduced expression of $w$ is j-admissible.(since we will fix an j, we will omit j and just say admissible).
\end{lemma}
\begin{proof}
Since we are in the ADE case,
\begin{align}
s_i \varpi_i&=-\varpi_i+\sum_{h \text{ is adjacent to i }} \varpi_h. \\
s_i \varpi_h&= \varpi_h, \textup{ for $h\ne i$.}
\end{align}
We use induction on the length of $w$.
Suppose lemma holds when $l(w)\leq m$.
Take a reduced expression of $w \in W$ with length $m+1$ :
$w=s_{i_{m+1}}\cdots s_{i_{1}}$.
Suppose this expression is not admissible,
we have
$\langle \alpha_{i_{m+1}},s_{i_{m}}\cdots s_{i_{1}}\varpi_j\rangle \leq0$.
Since $\langle \alpha_{i_{m+1}}, \varpi_j \rangle \geq0$, and by (6),(7) $$\langle \alpha_{i_{m+1}}, s_t \gamma \rangle \geq \langle \alpha_{i_{m+1}}, \gamma \rangle. $$
unless $t=i_{m+1}$, there must exists $k$ such that $i_k=i_{m+1}$.Let k be the biggest number such that $i_{k}=i_{m+1}$.\\
In the case there is no element in the set $\{i_m,\cdots ,i_{k+1}\}$ is adjacent to $i_{m+1}$ in the Coxeter diagram, $s_{i_{m+1}}$ commutes with $s_{i_m}\cdots s_{i_{k+1}}$. Therefore
$s_{i_{m+1}} s_{i_{m}}\cdots s_{i_{k+1}} s_{i_{m+1}}=s_{i_{m+1}} s_{i_{m+1}}
s_{i_{m}}\cdots s_{i_{k+1}}=s_{i_{m}}\cdots s_{i_{k+1}}$
so the $w=s_{i_{m+1}}\cdots s_{i_{1}}$ is not reduced, contradiction.\\
In the case where for some $t$, $i_{t}$ is adjacent to $i_{m+1}$, we will show we can reduce to the case we have only one such t.
Suppose we have at least two elements $i_{t_1},i_{t_2}\cdots ,i_{t_h}$ such that they are all adjacent to $i_{m+1}$.
Since $\langle \alpha_{i_{m+1}},s_{i_{m+1}}\cdots s_{i_{k}}\cdots s_{i_1}\varpi_j\rangle \leq0$ and $h>1$, by (6), (7), we must have some $i_{u_1}, i_{u_2}$ such that they are adjacent to $i_{m+1}$.
Since one point at most has 3 adjacent points we must have some $i_{u_x}=i_{t_y}.$
Let $p=i_{u_x}=i_{t_y}$. Using $s_p s_{i_{m+1}} s_p=s_{i_{m+1}} s_p s_{i_{m+1}}$ we can move $s_{i_{m+1}}$ in front of $s_{t_y}$ so the number h is reduced by 1.
We could do this procedure until h=1.
In this case we can rewrite the sequence before $s_{i_{k-1}}$ using the braid relation between $i_{m+1}$ and $i_t$:
$$s_{i_m+1}\cdots s_{i_{t}} \cdots s_{i_k}=s_{i_m+1}\cdots s_{i_{t}} \cdots s_{{i_m+1}}= \cdots s_{i_{m+1}} s_{i_{t}} s_{i_{m+1}} \cdots=
\cdots s_{i_{t}} s_{i_{m+1}} s_{i_{t}}\cdots.$$
Set $\beta=s_{i_{k-1}}\cdots s_{i_{1}}\varpi_{j}$.
By induction hypothesis, $s_{i_{m+1}} s_{i_{t}} s_{i_{m+1}} \cdots $and $s_{i_{t}} s_{i_{m+1}} s_{i_{t}}\cdots $ are admissible.
So $\langle \alpha_{i_t}, \beta \rangle \geq0$ and $\langle \alpha_{i_{m+1}}, \beta \rangle \geq0$. Again using (6) and (7) we have $\langle s_{i_{t}} s_{i_{m+1}}\beta, \alpha_{i_{m+1}}\rangle\geq0$.
Then $\langle s_{i_{m+1}}\cdots s_{i_{1}}\varpi_j, \alpha_{i_{m+1}}\rangle
=\langle s_{i_{t}} s_{i_{m+1}}\beta, \alpha_{i_{m+1}}\rangle\geq0$, contradicts with $s_{i_{m+1}}\cdots s_{i_{1}}$ is not admissible.
\end{proof}
\begin{proof}[Proof of lemma 2]
Set $F_0=\{\varpi_j\}$.
Let $F_{m}$ be the set which contains all $w\varpi_j$,
where $l(w)\leq m$.
We use induction. Suppose lemma 2 holds for $\gamma\in F_{m}$, we will prove lemma holds when $\gamma\in F_{m+1}.$
For any $\gamma=w\varpi_j \in F_{m+1}$, by lemma 1, w has a reduced admissible expression: $w=s_{i_{m+1}}\cdots s_{i_{1}}$. Denote $i_{m+1}$ by i and $\beta$ by $s_{i_{m}}\cdots s_{i_{1}}\varpi_j$. So
$\gamma=s_{i}\beta$, $\beta \in F_{m}$.
Since $s_{i_{m+1}}\cdots s_{i_{1}}$ is admissible,
$\langle \beta, \alpha_i \rangle\geq 0$. Therefore $\langle \gamma, \alpha_i \rangle=
\langle s_i\beta, \alpha_i \rangle=-
\langle \beta, \alpha_i \rangle
\leq 0$ and we can apply prop 4.1 in \cite{MR2892443}.
Then $D_{\gamma}(M)=D_{s_{i}(s_{i}\gamma)}(M)=D_{s_{i}\gamma}(\Sigma_{i}M)$ , where $\Sigma_i$ is the reflection functor defined in section 2.2 in \cite{MR2892443}.\\
Let $A=\{j\ |\ j \text{ is adjacent to i, } j\in I\}$ and
$M_{A}=\oplus_{s\in A} M_s$.
The $i^{th}$ component of $\Sigma_{i}M$
is the kernel of the map $\xi$ (Still see section 2.2 in \cite{MR2892443} for the definition of $\xi$) from $M_{A}$ to $M_{i}$.
Since $\beta\in F_{m}$, by induction hypothesis, we can apply this lemma to the case where $\gamma$ is taken to be $\beta$ and the module $M$ is $\Sigma_{i}M$.
Recall we denote by $I_\gamma^{+}$ the subset of I containing all i such that $\langle \gamma, \check{\alpha_i}\rangle$ is positive and by $I_\gamma^{-}$ containing all i $\langle \gamma, \check{\alpha_i}\rangle$ is negative.
Denote $A_{+}=\{j\ |\ j \text{ is adjacent to i, } j\in I_{\gamma}^{+}\}$ and $A_{-}=A\setminus A_{+}.$
For a multiset $S$, let $M_S=\oplus M_s^{m(s)}$.
Regarding $I_{\gamma}^{-}$ as a multiset by setting $m(i)=\gamma_i^-$, we can rewrite
$\oplus_{i\in I_{\gamma}^{-}}M_{i}^{\gamma^{-}}$ as $
M_{I^{-}_{\gamma}}$, similarly $\oplus_{i\in I_{\gamma}^{+}}M_{i}^{\gamma^{+}}$ as $
M_{I^{+}_{\gamma}}$.
Consider the case when $\langle \gamma, \alpha_i \rangle =-1$.
We have $I_{\beta}^{+}=I_{s_i\gamma}^{+}=(I_{\gamma}^{+}\setminus A_{+})\bigcup \{i\}$ and $I_{\beta}^{-}=I_{s_i\gamma}^{-}=(I_{\gamma}^{-}\setminus \{i\}) \bigcup A_{-}$ as multisets. Therefore
$D_{s_{i}\gamma}(\Sigma_{i}M)$ is the dimension of the kernel the natural map (which is $\phi_{\beta}$) from $M_{I^{+}_{\gamma}\setminus A_{+}}\oplus ker(M_{A}\xrightarrow[]{\xi} M_{i})$ to $M_{I^{-}_{\gamma} \setminus\{i\}}\oplus M_{A_{-}}$. This is equal to the dimension of the kernel of the natural map from $M_{I^{+}_{\gamma}\setminus A_{+}}\oplus ker(M_{A_{+}}\xrightarrow[]{\xi} M_{i})$ to
$M_{I^{+}_{\gamma}\setminus \{i\}}$, which is just $ker(M^{+}_{I_{\gamma}}\xrightarrow[]{\phi_{\gamma}} M_{I^{+}_{\gamma}})$. The case when $\langle \gamma, \alpha_i \rangle=-2$ is similar.
\end{proof}
University of Massachusetts, Amherst, MA.\\
E-mail address: [email protected]
\end{document} |
\betaegin{document}
\muaketitle
\betaegin{abstract}
Working under $AD$, we investigate the length of prewellorderings given by the iterates of ${\mathcal{M}}_{2k+1}$, which is the minimal proper class mouse with $2k+1$ many Woodin cardinals. In particular, we answer some questions from \cite{Hjorth01} (the discussion of the questions appears in the last section of \cite{HjorthD}).
\end{abstract}
\betaaselineskip=24pt
In recent years, there have been many interactions between inner model theory and descriptive set theory. While the connection between the two areas was established early on in 1960s, the bulk of modern interactions go back to the work of Martin, Steel and Woodin carried out in late 80s and early 90s. In particular, Steel's computation of ${\rm{HOD}}^{L(\muathbb{R})}$ below $\Theta$ (see \cite{SteelHod}), Woodin's subsequent computation of ${\rm{HOD}}^{L(\muathbb{R})}$ (see \cite{WoodinHod}) and Woodin's computation of ${\rm{HOD}}^{L[x][g]}$ (largely unpublished) have been of crucial importance for the results that followed\footnote{Here, $x$ is a real and letting $\kappa$ be the least inaccessible of $L[x]$, $g\sigmaubseteq Coll(\omega, <\kappa)$ is $L[x]$-generic.}.
In this paper, we investigate the prewellordering associated with the directed system generated by ${\mathcal{M}}_{2k+1}$ where $k\in \omega$. Our intended application is the computation of the sup of the lengths of $\Game^{2k+1}(\omega\cdot n-\utilde{{\mathcal{P} }i}^1_1)$-prewellorderings. We show that the sup is $\kappaappa^1_{2k+3}$. This generalizes Hjorth's computation of $\Game^{1}(\omega\cdot n-\utilde{{\mathcal{P} }i}^1_1)$-prewellorderings. See \rsec{the main theorem} for the statement of the main theorem of this paper.
All the descriptive set theoretic notions that we will need come from \cite{Moschovakis} and and the inner model theoretic notions come from \cite{Zeman}.
{\hbox{\fiverm th}}etaextbf{Acknowledgments.} The results of this paper were proven in Berlin during the Spring of 2006 while the author was visiting his advisor John Steel. I am grateful to John Steel for introducing me to inner model theory and for bringing the questions considered in this paper to my attention. I also thank Farmer Schlutzenberg for very motivational conversations during Fall of 2006. Finally, I express my deepest gratitude to the referee for providing long list of fundamental improvements.
\sigmaection{On descriptive set theory}
We assume $AD$ throughout this paper. As is customary with descriptive set theorists, we let $\muathbb{R}$ be the Baire space $\omega^\omega$. We let $u_n$ be the $n$th uniform indiscernible and $s_n=\lambdaangle u_i: i\lambdaeq n\rangle$. We let $s_0=\emptyset$. Under $AD$, $u_n=\alphaleph_n$ (see \cite{Kanamori}).
Recall that for $x\in \muathbb{R}$,
\betaegin{center}
$C_{2n}(x)=\{ y\in \muathbb{R}: y$ is $\Delta^1_{2n}(x)$ in a countable ordinal $\}$
\end{center}
and
\betaegin{center}
$Q_{2n+1}(x)=\{ y\in \muathbb{R}: y$ is $\Delta^1_{2n+1}(x)$ in a countable ordinal $\}$.
\end{center}
The definitions of $C_{2n}$ and $Q_{2n+1}$ given above are actually theorems as these are not the original definitions of these objects. The first equality is due to Harrington and Kechris (see \cite{HarKech}) and the second one is due to Kechris, Martin and Solovay (see \cite{Qtheory}).
Following \cite{Moschovakis}, we let {\hbox{\fiverm th}}etaextit{pointclass} stand for any collection of sets of reals (that is, we are not requiring closure under the set theoretic operations). If $\Gamma$ is a pointclass then $\betareve{\Gamma}$ is the dual pointclass and $\Delta_\Gamma=\Gamma\cap \betareve{\Gamma}$.
A relation $\lambdaeq$ is a {\hbox{\fiverm th}}etaextit{prewellordering}\index{prewellordering} if it is transitive, reflexive, connected and wellfounded. Given a set of reals $A$, $\phi$ is a norm on $A$ if $\phi: A\rightarrow {\rm{Ord}}$. For each norm $\phi$ on $A$, we let $\lambdaeq^\phi$ be the binary relation on $A$ given by $x\lambdaeq^\phi y$ iff $\phi(x)\lambdaeq \phi(y)$. Then $\lambdaeq^\phi$ is a prewellordering of $A$. The opposite is true as well, given a prewellordering $\lambdaeq$ of $A$ there is an associated norm $\phi$ defined on $A$ such that $\lambdaeq=\lambdaeq^{\phi}$. If $\Gamma$ is a pointclass then $\phi$ is a $\Gamma$-norm if there are relations $\lambdaeq_{\Gamma}^\phi\in \Gamma$ and $\lambdaeq_{\betareve{\Gamma}}^\phi\in \betareve{\Gamma}$ such that for every $y\in dom(\phi)$ and for any $x\in \muathbb{R}$,
\betaegin{center}
$[x\in dom(\phi) \wedge \phi(x)\lambdaeq \phi(y)]\lambdaeftrightarrow x \lambdaeq^\phi_\Gamma y\lambdaeftrightarrow x\lambdaeq^\phi_{\betareve{\Gamma}} y$.
\end{center}
If $\Gamma$ is a pointclass, we let
\betaegin{center}
$\delta(\Gamma)=\sigmaup\{ \lambdaeq^* : \lambdaeq^*\in \Gamma$ and $\lambdaeq^*$ is a prewellordering $\}$.
\end{center}
A sequence of norms $\vec{\phi}=\lambdaangle \phi_i : i<\omega\rangle$ on $A$ is a {\hbox{\fiverm th}}etaextit{scale}\index{scale} on $A$ if whenever $\lambdaangle x_i: i<\omega\rangle\sigmaubseteq A$ is a sequence of reals converging to $x$ such that for each $i$ the sequence $\lambdaangle \phi_i(x_k) : k<\omega\rangle$ is eventually constant then $x\in A$ and for each $i$, $\phi_i(x)\lambdaeq \lambda_i$ where $\lambda_i$ is the eventual value of $\lambdaangle \phi_i(x_k) : k<\omega\rangle$. We write $x_i\rightarrow x (mod \vec{\phi})$ if $\lambdaangle x_i: i<\omega\rangle$ converges to $x$ in the above sense. $\vec{\phi}$ is a $\Gamma$-scale\index{$\Gamma$-scale} on $A$ if there are relations $R\in \Gamma$ and $S\in \betareve{\Gamma}$ such that for all $y\in A$, for any $x\in \muathbb{R}$ and for any $n<\omega$
\betaegin{center}
$[x\in A \wedge \phi_n(x)\lambdaeq \phi_n(y)]\lambdaeftrightarrow R(n, x, y) \lambdaeftrightarrow S(n, x, y)$.
\end{center}
We say $\Gamma$ has the prewellordering property if every set in $\Gamma$ has a $\Gamma$-norm. We say $\Gamma$ has the scale property if every set in $\Gamma$ has a $\Gamma$-scale. For more on prewellordering property and scale property see \cite{Moschovakis}.
Suppose $\kappaappa$ is a cardinal. $T\sigmaubseteq \cup_{n<\omega}\omega^n{\hbox{\fiverm th}}etaimes\kappaappa^n$ is a tree if whenever $s\in T$ then $s\restriction i\in T$ for any $i<lh(s)$. For $(x, f)\in \omega^\omega{\hbox{\fiverm th}}etaimes\kappaappa^\omega$ is a branch of $T$ if $(x\restriction i, f\restriction i)\in T$ for any $i<\omega$. $[T]$ is the set of branches of $T$. $p[T]$ is the projection of $[T]$ on the first coordinate, i.e., $x\in p[T]$ iff there is $f\in \kappaappa^\omega$ such that $(x, f)\in T$.
A set of reals $A$ is $\kappaappa$-Suslin\index{$\kappaappa$-Suslin} if there is a tree $T\sigmaubseteq \cup_{n<\omega}\omega^n{\hbox{\fiverm th}}etaimes\kappaappa^n$ such that $A=p[T]$. $A$ is Suslin if it is $\kappaappa$-Suslin for some $\kappaappa$. Given a scale $\vec{\phi}$ on $A$ one can construct a tree $T$ such that $p[T]=A$. More precisely, let $T$ be the set of pairs $(s, f)$ such that there is some real $x\in A$ such that $s{\rm lh}d x$ and $f(i)=\phi_i(x)$ for each $i<lh(f)$. Given a tree $T$ such that $p[T]=A$, one can get a scale $\vec{\phi}$ on $A$ by considering the leftmost branches of $T$ (see \cite{Moschovakis}). Thus, carrying a scale and being Suslin are equivalent.
Finally, we say that $\kappaappa$ is a {\hbox{\fiverm th}}etaextit{Suslin cardinal}\index{Suslin cardinal} if there is a set of reals $A$ which is $\kappaappa$-Suslin but $A$ is not $\eta$-Suslin for any $\eta<\kappa$. We let $S(\kappaappa)$\index{$S(\kappa)$} be the pointclass of $\kappaappa$-Suslin sets. It is not hard to show that $S(\kappaappa)$ is closed under projections (see \cite{Moschovakis}). For more on trees and Suslin sets see \cite{Moschovakis}. For a complete characterization of Suslin cardinals see \cite{Jackson}.
Under $AD$, for each $n$ and real $z$, ${\mathcal{P} }i^1_{2n+1}(z)$ and ${\mathcal{S}}igma^1_{2n+2}(z)$ have the scale property. The sup of $\utilde{{\mathcal{P} }i}^1_{2n+1}$ prewellorderings and $\utilde{{\mathcal{S}}igma}_{2n+2}$ prewellorderings play an important role in descriptive set theory. Following \cite{Moschovakis}, we let
\betaegin{center}
$\delta^1_{2n+1}=\delta(\utilde{{\mathcal{P} }i}^1_{2n+1})=\delta({\mathcal{P} }i^1_{2n+1})$
\end{center}
and
\betaegin{center}
$\delta^1_{2n}=\delta(\utilde{{\mathcal{S}}igma}^1_{2n})$.
\end{center}
It turns out that under $AD$,
\betaegin{center}
$\delta^1_{2n}=(\delta_{2n+1}^1)^+$
\end{center}
and $\delta^1_{2n+1}$ is a successor cardinal whose predecessor is denoted by $\kappaappa^1_{2n+1}$ (see \cite{Moschovakis}). It is shown in \cite{Moschovakis} that
\betaegin{center}
$\utilde{{\mathcal{S}}igma}^1_{2n+3}=S(\kappaappa^1_{2k+1})$.
\end{center}
Also, $\kappaappa^1_3=\alphaleph_\omega$, $\delta^1_3=\alphaleph_{\omega+1}$ and $\delta^1_4=\alphaleph_{\omega+2}$.
$\Game$ is the {\hbox{\fiverm th}}etaextit{game quantifier}. Recall that given a set of reals $A\sigmaubseteq \muathbb{R}^2$ we let $\Game A$ be the set
\betaegin{center}
$x\in \Game A\mathrel{\leftrightarrow} \exists x_0\forall x_1\exists x_2 \forall x_3 \cdot \cdot \cdot \exists x_{2n} \forall x_{2n+1} \cdot \cdot \cdot ( ( x, \lambdaangle x_i: i<\omega\rangle) \in A)$.
\end{center}
Here, the quantifiers range over $\omega$. Equivalently,
\betaegin{center}
$\Game A =\{ x \in \muathbb{R} : $ player $I$ has a winning strategy in $G_{A_x}\}$.
\end{center}
where $A_x=\{ y: (x, y)\in A\}$. A set is $\omega\cdot n-\utilde{{\mathcal{P} }i}^1_1$ if there is a sequence $\lambdaangle A_\alpha : \alpha< \omega \cdot n\rangle\sigmaubseteq \utilde{{\mathcal{P} }i}^1_1$ such that
\betaegin{center}
$x\in A\mathrel{\leftrightarrow}$ the least $\alpha$ such that $x\nuot \in A_\alpha$ is odd.
\end{center}
Equivalently sets in $\omega\cdot n-\utilde{{\mathcal{P} }i}^1_1$ constitute the first $\omega\cdot n$ levels of the {\hbox{\fiverm th}}etaextit{difference hierarchy} for $\utilde{{\mathcal{P} }i}^1_1$.
\sigmaection{On inner model theory}
Recall that if ${\mathcal{M}}$ is a premouse then $\muathcal{G}({\mathcal{M}}, \kappa)$ is the two player iteration game that has $<\kappa$ moves (see \cite{OIMT}). In this game, player $I$ plays the successor steps which amounts to choosing an extender and applying it to the earliest model it makes sense to apply. Player II plays limit stages and her job is to choose a well-founded cofinal branch of the resulting iteration tree. II wins if all the models produced in the game are well founded. ${\mathcal{S}}igma$ is then called a $\kappa$-iteration strategy for ${\mathcal{M}}$ if it is a winning strategy for player $II$.
If ${\mathcal{M}}$ is a mouse\footnote{The reader should consult \cite{OIMT} for the definition of a mouse.} and $\xi\lambdaeq o({\mathcal{M}})$, then we let ${\mathcal{M}}||\xi$ be ${\mathcal{M}}$ cutoff at $\xi$, i.e., we keep the predicate indexed at $\xi$. We let ${\mathcal{M}}|\xi$ be ${\mathcal{M}}||\xi$ without the last predicate. We say $\xi$ is a cutpoint of ${\mathcal{M}}$ if there is no extender $E$ on ${\mathcal{M}}$ such that $\xi\in({\rm cp }(E), lh(E)]$. We say $\xi$ is a strong cutpoint if there is no $E$ on ${\mathcal{M}}$ such that $\xi\in[{\rm cp }(E), lh(E)]$.
If $\T$ is an iteration tree, i.e., a play of the game, then, following the notation of \cite{FSIT}, $\T$ has the form
\betaegin{center}
$\T=\lambdaangle T, deg, D, \lambdaangle E_\alpha, {\mathcal{M}}^*_{\alpha+1}| \alpha+1<\eta\rangle\rangle$.
\end{center}
Recall that $D$ is the set of {\hbox{\fiverm th}}etaextit{dropping} points. Recall also that if $\eta$ is limit then
\betaegin{center}
$\vec{E}(\T)= \cup_{\alpha<\eta}(\deltaot{E}^{{\mathcal{M}}_\alpha\restriction lh(E_\alpha)})$,\\
${\mathcal{M}}(\T)=\cup_{\alpha<\eta}{\mathcal{M}}_\alpha\restriction lh(E_\alpha)$,\\
$\delta(\T)=\sigmaup_{\alpha<\eta} lh(E_\alpha)$.
\end{center}
If $b$ is a branch of $\T$ then ${\mathcal{M}}^\T_b$ is the branch model of the tree. Then if $\alpha\lambdaeq_T\beta$ then $i_{\alpha, \beta}^\T:{\mathcal{M}}^*_\alpha\rightarrow {\mathcal{M}}_\beta^\T$ is the iteration map if $[\alpha, \beta]_\T\cap D=\emptyset$ and $i_{\alpha, b}^\T:{\mathcal{M}}^*_\alpha\rightarrow {\mathcal{M}}_b^\T$ is the iteration map if $\alpha\in b$ and $b-\alpha\cap D =\emptyset$. {\hbox{\fiverm th}}etaextit{In this paper, all iteration trees are normal. We will refer to the general iterations as stacks of normal trees.}
It is by now a standard fact that if $b$ and $c$ are cofinal branches of $\T$ on ${\mathcal{M}}$ and ${\mathcal R}={\mathcal{M}}_b^\T\cap {\mathcal{M}}_c^\T$ then ${\mathcal R}\vDash ``\delta(\T)$ is Woodin" (see \cite{OIMT}). Moreover, if ${\mathcal{ Q}}$ is a mouse over ${\mathcal{M}}(\T)$ (this in particular means that ${\mathcal{ Q}}$ has no extender overlapping with $\delta(\T)$) such that ${\mathcal{ Q}}\vDash ``\delta(\T)$ is Woodin" yet there is a counterexample to Woodiness of $\delta(\T)$ in $L_1({\mathcal{ Q}})$ then there is at most one cofinal branch $b$ of $\T$ such that ${\mathcal{ Q}}{\hbox{\fiverm th}}etarianglelefteq{\mathcal{M}}^\T_b$ (see \cite{OIMT}). The following lemma, which builds upon the proof of the aforementioned fact is one of the most important ingredients available to us and will be used in this paper many times. It is essentially due to Martin and Steel, see Theorem 2.2 of \cite{IT}.
\betaegin{lemma}[Uniqueness of branches]\lambdaanglebel{uniqueness of branches} Suppose ${\mathcal{M}}$ is a mouse and $\T$ is an iteration tree on ${\mathcal{M}}$ of limit length. Suppose $s$ is a cofinal subset of $\delta(\T)$. Then there is at most one cofinal branch $b$ such that there is $\alpha\in b$ with the property that $i^\T_{\alpha, b}$ exists and $s\sigmaubseteq ran(i^{\T}_{\alpha, b})$.
\end{lemma}
\betaegin{proof}
Towards a contradiction, suppose there are two cofinal branches $b$ and $c$ such that for some $\alpha, \beta$, both $i^\T_{\alpha, b}$ and $i^\T_{\beta, c}$ exist and $s\sigmaubseteq ran(i^\T_{\alpha, b})\cap ran(i^\T_{\beta, c})$. Without loss of generality we can assume that $\alpha$ and $\beta$ are the least ordinals with this property, $\alpha\lambdaeq \beta$ and that $b$ and $c$ diverge at $\alpha$ or earlier, i.e., if $\gamma$ is the least ordinal in $b\cap c$ then $\gamma\lambdaeq \alpha$. By \cite{IT}, we can assume that $b$ is the downward closure of $\lambdaangle \alpha_n: n<\omega\rangle$, $c$ is the downwards closure of $\lambdaangle \beta_n: n<\omega\rangle$, $\alpha_0=\alpha$ and $\beta_0=\beta$. Let then $\xi$ be the least ordinal in $ran(i^\T_{\alpha, b})\cap ran(i^\T_{\beta, c})$. Let $n$ be the least such that ${\rm cp }(i^\T_{\alpha_n, b})>\xi$. This means that ${\rm cp }(E^\T_{\alpha_{n+1}-1})>\xi$ and that $lh(E^\T_{\alpha_n})<\xi$. By the proof of Theorem 2.2 of \cite{IT}, this means that for some $m\geq 1$, $\xi \in [{\rm cp }(E^\T_{\beta_m-1}), lh(E^\T_{\beta_m-1}))$. This then implies that $\xi\nuot \in ran(i^\T_{\beta_{m-1}, c})$, which is a contradiction.
\end{proof}
The proof of \rlem{uniqueness of branches} gives the following as well.
\betaegin{lemma}\lambdaanglebel{partial agreement} Suppose ${\mathcal{M}}$ is a mouse and $\T$ is an iteration tree on ${\mathcal{M}}$ of limit length. Suppose $b, c$ are two cofinal branches of $\T$ such that $i^\T_b$ and $i^\T_c$ exist. Suppose that for some $\alpha$,
\betaegin{center}
$i^\T_b(\alpha)=i^\T_c(\alpha)<d(\T)$.
\end{center}
Then $i^\T_b\restriction \alpha = i^\T_c\restriction \alpha$. Moreover, if $\xi\in b$ is the least such that ${\rm cp }(E_\xi^\T)>i^\T_b(\alpha)$ then $b\cap \xi=c\cap \xi$.
\end{lemma}
If ${\mathcal{M}}$ is a mouse and $\T$ is a tree then we say $\T$ is above $\eta$ if all extender used in $\T$ have critical point $>\eta$. If ${\mathcal{S}}igma$ is an $(\omega_1, \omega_1)$-iteration strategy for ${\mathcal{M}}$ and ${\vec{\mathcal{T}}}$ is a stack of trees on ${\mathcal{M}}$ according ${\mathcal{S}}igma$ with last model ${\mathcal{N}}$ then we let ${\mathcal{S}}igma_{{\mathcal{N}}, {\vec{\mathcal{T}}}}$ be the strategy of ${\mathcal{N}}$ induced by ${\mathcal{S}}igma$. We say ${\mathcal{S}}igma$ has the Dodd-Jensen property if whenever ${\mathcal{N}}$ is an iterate of ${\mathcal{M}}$ via ${\mathcal{S}}igma$ and $\pi:{\mathcal{M}}\rightarrow {\mathcal{W} }{\hbox{\fiverm th}}etarianglelefteq {\mathcal{N}}$ is (fine structural) embedding then the iteration from ${\mathcal{M}}$ to ${\mathcal{N}}$ doesn't drop, ${\mathcal{W} }={\mathcal{N}}$ and if $i:{\mathcal{M}}\rightarrow {\mathcal{N}}$ is the iteration embedding then for every $\alpha$, $i(\alpha)\lambdaeq \pi(\alpha)$. If ${\mathcal{S}}igma$ has the Dood-Jensen property and ${\vec{\mathcal{T}}}$ and ${\vec{\mathcal{U}}}$ are two stacks on ${\mathcal{M}}$ with last model ${\mathcal{N}}$ such that $i^{\vec{\mathcal{T}}}$ and $i^{{\vec{\mathcal{U}}}}$ exist then $i^{{\vec{\mathcal{T}}}}=i^{{\vec{\mathcal{U}}}}$ and ${\mathcal{S}}igma_{{\mathcal{N}}, {\vec{\mathcal{T}}}}={\mathcal{S}}igma_{{\mathcal{N}}, {\vec{\mathcal{U}}}}$. Lastly, we let
\betaegin{center}
$I({\mathcal{M}}, {\mathcal{S}}igma)=\{ {\mathcal{N}} :$ there is a stack ${\vec{\mathcal{T}}}$ on ${\mathcal{M}}$ according to ${\mathcal{S}}igma$ with last model ${\mathcal{N}}$ and $i^{{\vec{\mathcal{T}}}}$ exists $\}$.
\end{center}
\sigmaubsection{${\mathcal{S}}$-constructions}
Here we introduce ${\mathcal{S}}$-constructions which were first introduced in \cite{Selfiter} where they were called $P$-constructions. Such constructions are due to Steel and hence, we change the terminology and call them ${\mathcal{S}}$-constructions. These constructions allow one to translate mice over some set $A$ to mice over some set $B$ provided $A$ and $B$ are somehow close. The complete proof of the following proposition is essentially the proof of Lemma 1.5 of \cite{Selfiter}.
\betaegin{proposition}\lambdaanglebel{s-constructions prop}
Suppose ${\mathcal{M}}$ is a sound mouse and $\delta$ is a strong cutpoint cardinal of ${\mathcal{M}}$. Suppose further that ${\mathcal{N}}\in {\mathcal{M}}|\delta+1$ is such that $\delta\sigmaubseteq {\mathcal{N}}\sigmaubseteq H_\delta^{\mathcal{M}}$ and there is a partial ordering $\muathbb{P}\in L_{\omega}[{\mathcal{N}}]$ such that whenever ${\mathcal{ Q}}$ is a mouse over ${\mathcal{N}}$ such that $H_\delta^{\mathcal{ Q}}={\mathcal{N}}$ then ${\mathcal{M}}|\delta$ is $\muathbb{P}$-generic over ${\mathcal{ Q}}$. Then there is a mouse ${\mathcal{S}}$ over ${\mathcal{N}}$ such that ${\mathcal{M}}|\delta$ is generic over ${\mathcal{S}}$ and ${\mathcal{S}}[{\mathcal{M}}|\delta]={\mathcal{M}}$.
\end{proposition}
It is clear what ${\mathcal{S}}$ must be. Because $\muathbb{P}$ is a small forcing with respect to the critical points of the extenders of ${\mathcal{M}}$ that have indices bigger than $\delta$, all such extenders can be put on a sequence of some mouse over ${\mathcal{N}}$. This is exactly what $S$-constructions do. An $S$-construction of ${\mathcal{M}}$ over ${\mathcal{N}}$ is a sequence of ${\mathcal{N}}$-mice $\lambdaangle {\mathcal{S}}_\alpha, \betaar{{\mathcal{S}}}_\alpha: \alpha\lambdaeq \eta\rangle$ such that
\betaegin{enumerate}
\item ${\mathcal{S}}_0=L_{\omega}[{\mathcal{N}}]$,
\item if ${\mathcal{M}}|\delta$ is generic over $\betaar{{\mathcal{S}}}_\alpha$ for a forcing in $L_{\omega}[{\mathcal{N}}]$ then $\betaar{{\mathcal{S}}}_\alpha[{\mathcal{N}}]={\mathcal{M}}|(\omega{\hbox{\fiverm th}}etaimes\alpha)$ and
\betaegin{enumerate}
\item if ${\mathcal{M}}||(\omega{\hbox{\fiverm th}}etaimes \alpha)$ is active then ${\mathcal{S}}_\alpha$ is the expansion of $\betaar{{\mathcal{S}}_\alpha}$ by the last extender of ${\mathcal{M}}||(\omega{\hbox{\fiverm th}}etaimes \alpha)$ and $\betaar{{\mathcal{S}}}_{\alpha+1}=rud({\mathcal{S}}_\alpha)$,
\item if ${\mathcal{M}}||(\omega{\hbox{\fiverm th}}etaimes \alpha)$ is passive then ${\mathcal{S}}_\alpha=\betaar{{\mathcal{S}}_\alpha}$ and $\betaar{{\mathcal{S}}}_{\alpha+1}=rud({\mathcal{S}}_\alpha)$,
\end{enumerate}
\item if $\lambda$ is limit then $\betaar{S}_\lambda=\cup_{\alpha<\lambda}{\mathcal{S}}_\alpha$.
\end{enumerate}
By the proof of Lemma 1.5 of \cite{Selfiter}, the ${\mathcal{S}}$-construction described in 1-3 cannot fail as long as the hypothesis of 2 holds. Thus, we always have a last model of ${\mathcal{S}}$-construction which might be some $\betaar{{\mathcal{S}}}_\alpha$ instead of ${\mathcal{S}}_\alpha$.
\betaegin{definition}\lambdaanglebel{s(n)} We let ${\mathcal{S}}^{\mathcal{M}}({\mathcal{N}})$ be the last model of the ${\mathcal{S}}$ construction done over ${\mathcal{N}}$.
\end{definition}
Then by the proof of Lemma 1.5 of \cite{Selfiter}, ${\mathcal{S}}[{\mathcal{M}}|\delta]{\hbox{\fiverm th}}etarianglelefteq{\mathcal{M}}$. Moreover, if the hypothesis of 2 never fails then in fact, ${\mathcal{S}}[{\mathcal{M}}|\delta]={\mathcal{M}}$. It also follows that ${\mathcal{S}}$ inherits whatever iterability ${\mathcal{M}}$ has above $\delta$. The method of ${\mathcal{S}}$-constructions is a very useful inner model theoretic tool. A particularly important application for us is the following lemma.
\betaegin{lemma}\lambdaanglebel{s-constructions give q-structures} Suppose ${\mathcal{M}}\vDash ZFC-Powerset$ is a mouse and $\eta$ is a strong cutpoint non-Woodin cardinal of ${\mathcal{M}}$. Suppose $\gamma>\eta$ is a cardinal of ${\mathcal{M}}$ and ${\mathcal{N}}=L[\vec{E}]^{{\mathcal{M}}|\gamma}$. Suppose $L_{\omega}({\mathcal{N}}|\eta)\vDash ``\eta$ is Woodin". Let $\lambdaangle {\mathcal{S}}_\alpha, \betaar{{\mathcal{S}}}_\alpha :\alpha<\nuu\rangle$ be the ${\mathcal{S}}$-construction of ${\mathcal{M}}|(\eta^+)^{\mathcal{M}}$ over ${\mathcal{N}}|\eta$. Then for some $\alpha<\nuu$, ${\mathcal{S}}_\alpha\vDash ``\eta$ isn't Woodin".
\end{lemma}
\betaegin{proof}
Let ${\mathcal{S}}$ be the last model of the ${\mathcal{S}}$-construction of ${\mathcal{M}}|(\eta^+)^{\mathcal{M}}$ over ${\mathcal{N}}|\eta$. Suppose $\eta$ is a Woodin cardinal of ${\mathcal{S}}$. Then ${\mathcal{M}}|\eta$ is generic for the $\eta$-generator version of the extender algebra of $L_{\omega}({\mathcal{N}}|\eta)$. we also have that ${\mathcal{M}}|\eta$ is generic over ${\mathcal{S}}$ for the $\eta$-generator version of the extender algebra at $\eta$ and hence, ${\mathcal{S}}[{\mathcal{M}}|\eta]={\mathcal{M}}|(\eta^+)^{\mathcal{M}}$. Thus, $\eta$ isn't Woodin in ${\mathcal{S}}[{\mathcal{M}}|\eta]$. Let $f:\eta\rightarrow \eta$ be the function in ${\mathcal{M}}$ witnessing that $\eta$ isn't Woodin. Then because the $\eta$-generator version of extender algebra is $\eta$-cc, there is $g\in {\mathcal{S}}$ which dominates $f$. Let $E\in \vec{E}^{\mathcal{S}}$ be the extender that witnesses that $\eta$ is Woodin for $g$. Then if $E^*$ is the background extender of $E$ then $E^*$ witnesses the Woodiness of $\eta$ for $f$ in ${\mathcal{M}}$, contradiction!
\end{proof}
Before moving on, we set up one last notation. Given a model $M$ of a fragment of $ZFC$ with a unique Woodin cardinal, we let $\muathbb{B}^M$ be the extender algebra of $M$ at its unique Woodin cardinal. If $G\sigmaubseteq \muathbb{B}^M$ then we let $x_G$ be the set naturally coded by $G$.
\sigmaection{Descriptive inner model theory}\lambdaanglebel{dimt}
We let ${\mathcal{M}}_n$ be the minimal proper class mouse with $n$ Woodin cardinals. ${\mathcal{M}}_n^\#$ is the minimal mouse with last extender and with $n$ Woodin cardinals. Clearly, ${\mathcal{M}}_n$ is the result of iterating the last measure of ${\mathcal{M}}_n^\#$ through the ordinals. We let ${\mathcal{M}}_0=L$. In \cite{PWOIM}, Steel and Woodin computed the descriptive set theoretic complexity of the reals of ${\mathcal{M}}_n$. They showed that
\betaegin{center}
$C_{2n+2}(x)=\muathbb{R}^{{\mathcal{M}}_{2n}(x)}$
\end{center}
and
\betaegin{center}
$Q_{2n+3}(x)=\muathbb{R}^{{\mathcal{M}}_{2n+1}(x)}$.
\end{center}
We let
\betaegin{displaymath}
S_n(x) = \lambdaeft\{
\betaegin{array}{lr}
C_{n+2}(x) :& n \ is\ even \\
Q_{n+2}(x) :& n\ is\ odd \
\end{array}
\right.
\end{displaymath}
It is then clear that
\betaegin{center}
$S_n(x)=\muathbb{R}^{{\mathcal{M}}_n(x)}$.
\end{center}
Using standard techniques, we can now define $S_n(a)$ for any countable set $a$. More precisely, $b\in S_n(a)$ if for comeager many $g\sigmaubseteq Coll(\omega, a)$ letting $x_g$ be the real coding $a$ and $y_g$ be the real coding $b$ then $y_g\in S_n(x_g)$.
We also let ${\mathcal{M}}_\omega$ be the minimal proper class mouse with $\omega$ Woodin cardinals and ${\mathcal{M}}_\omega^\#$ be the minimal mouse with $\omega$ Woodin cardinals and with a last extender. Then ${\mathcal{M}}_\omega$ is the result of iterating the last measure of ${\mathcal{M}}_\omega^\#$ through the ordinals. The following theorems ara what allow us to use inner model theoretic tools to investigate descriptive set theoretic objects. The proofs of these results can be found in \cite{OIMT}.
\betaegin{theorem}[Woodin] Suppose ${\mathcal{M}}_\omega^\#$ exists and is $\omega_1$-iterable. Then $AD^{L(\muathbb{R})}$ holds.
\end{theorem}
\betaegin{theorem}[Steel-Woodin]\lambdaanglebel{mouse capturing} Suppose ${\mathcal{M}}_\omega^\#$ exists and is $\omega_1$-iterable. Let $\Gamma=({\mathcal{S}}igma^2_1)^{L(\muathbb{R})}$. Then for every countable transitive set $a$,
\betaegin{center}
$C_{\Gamma}(a)=\muathbb{R}^{{\mathcal{M}}_\omega(a)}=\cup \{\muathbb{R}^{\mathcal{N}}: L(\muathbb{R})\vDash {\mathcal{N}}$ is a sound $\omega_1$-iterable $a$-mouse such that for some $n<\omega$, $\rho_n({\mathcal{N}})=a\}$.
\end{center}
\end{theorem}
Let ${\mathcal{S}}igma$ be the canonical iteration strategy of ${\mathcal{M}}_\omega$. Let
\betaegin{center}
$\muathcal{F}=\{ {\mathcal{P} }:$ there is a ${\mathcal{S}}igma$-iterate ${\mathcal{N}}$ of ${\mathcal{M}}_\omega$ such that ${\mathcal{P} }={\mathcal{N}}|(\nuu^{+\omega})^{\mathcal{N}}$ where $\nuu$ is a successor cardinal of ${\mathcal{N}}$ which is less than the least ${\mathcal{N}}$-cardinal which is strong to the least Woodin of ${\mathcal{N}}\}$.
\end{center}
Then it follows from \rthm{mouse capturing} that for every ${\mathcal{P} }\in \muathcal{F}$, ${\mathcal{S}}igma_{\mathcal{P} }\in L(\muathbb{R})$. To see this, notice that whenever $\T$ is a tree on ${\mathcal{P} }$ of limit length and $b$ is a well-founded branch then ${\mathcal{ Q}}(b, \T)$ exists. Now, if $\T$ is according to ${\mathcal{S}}igma_{\mathcal{P} }$ and $b={\mathcal{S}}igma_{\mathcal{P} }(\T)$ then it follows from \rthm{mouse capturing} that ${\mathcal{ Q}}(b, \T)$ has an iteration strategy in $L(\muathbb{R})$. Thus, $L(\muathbb{R})$ can uniquely identify $b$. The details of such arguments appear in Section 7 of \cite{OIMT}.
We can define $\lambdaeq_{\muathcal{F}}$ on $\muathcal{F}$ by ${\mathcal{P} }\lambdaeq_{\muathcal{F}} {\mathcal{ Q}}$ iff there is $\alpha$ such that ${\mathcal{ Q}}|\alpha\in I({\mathcal{P} }, {\mathcal{S}}igma_{\mathcal{P} })$. Notice that if ${\mathcal{P} }\lambdaeq_{\muathcal{F}}{\mathcal{ Q}}$ and $\alpha$ is such that ${\mathcal{ Q}}|\alpha\in I({\mathcal{P} }, {\mathcal{S}}igma_{\mathcal{P} })$ then for some $\nuu<\alpha$, $\alpha=(\nuu^{+})^{\mathcal{ Q}}$. If ${\mathcal{P} }\lambdaeq_{\muathcal{F}}{\mathcal{ Q}}$ then we let $i_{{\mathcal{P} }, {\mathcal{ Q}}}:{\mathcal{P} }\rightarrow {\mathcal{ Q}}|\alphalpha$ be the iteration embedding.
Notice that $\lambdaeq_{\muathcal{F}}$ is directed and hence, we can let ${\mathcal{M}}_\infty$ be the direct limit of $(\muathcal{F}, \lambdaeq_{\muathcal{F}})$. We then have that
\betaegin{theorem}[Steel, \cite{SteelHod}]\lambdaanglebel{steel's thing} $L(\muathbb{R})\vDash {\mathcal{M}}_\infty=V_{\deltaelta}^{\rm{HOD}}$
where $\delta=\delta(\utilde{{\mathcal{S}}igma}^2_1)$.
\end{theorem}
Woodin extended this result to compute the full ${\rm{HOD}}$ of $L(\muathbb{R})$. We refer the reader to \cite{WoodinHod} for more on Woodin's work on ${\rm{HOD}}^{L(\muathbb{R})}$. It is important to note that the existence of ${\mathcal{M}}_\omega^\#$, which is a tiny bit stronger than $AD^{L(\muathbb{R})}$, is unnecessary and all the results in this paper can be proved only from $AD^{L(\muathbb{R})}$. Nevertheless, it is convenient and aesthetically more pleasant to assume that ${\mathcal{M}}_\omega^\#$ exists and we will do so whenever we wish. Experts will have no problem seeing how to remove this assumption. We refer the reader to \cite{OIMT} for an expanded version of this short summary of inner model theory. \cite{OIMT} also proves most of the results stated in this section without assuming the existence of ${\mathcal{M}}_\omega^\#$ but just $AD^{L(\muathbb{R})}$.
\sigmaection{The main theorem}\lambdaanglebel{the main theorem}
By a result of Martin (see \cite{Martin83}) and Neeman (see \cite{Neeman95}), for $k\geq 1$, a set of reals $A$ is $\Game^{k}(\omega\cdot n-\utilde{{\mathcal{P} }i}^1_1)$ iff there is $m\in \omega$, a real $z$ and a formula $\phi$ such that
\betaegin{center}
$x\in A\mathrel{\leftrightarrow} {\mathcal{M}}_{k-1}(x, z)\vDash \phi[x, z, s_m]$.
\end{center}
We let $\Gamma_{k, m}(z)$ be the set of reals $A$ such that there is a formula $\phi$ such that, letting $s_m$ be the sequence of the first $m$ uniform indiscernibles,
\betaegin{center}
$x\in A\mathrel{\leftrightarrow} {\mathcal{M}}_{k-1}(x, z)\vDash \phi[x, z, s_m]$.
\end{center}
We let $\Gamma_{k, m}=\Gamma_{k, m}(0)$ and $\utilde{\Gamma}_{k, m}=\cup_{z\in \muathbb{R}}\Gamma_{k, m}(z)$. Also, we let $\utilde{\Gamma}_k=\cup_{m<\omega}\utilde{\Gamma}_{k, m}$.
In \cite{Hjorth01}, Hjorth computed the sup of the lengths of $\utilde{\Gamma}_{1, m}$-prewellorderings. He showed that
\betaegin{center}
$\delta(\utilde{\Gamma}_{1, m})\lambdaeq u_{m+2}$.
\end{center}
and therefore,
\betaegin{center}
$\kappaappa^1_3=\alphaleph_\omega=\delta(\utilde{\Gamma}_{1})$\footnote{It is not hard to see that the standard prewellordering of the $\{x^\# : x\in \muathbb{R}\}$ has length $\kappaappa^1_3$, i.e, let $\phi(n, m, x^\#)={\hbox{\fiverm th}}etaau_n^{L[x]}(x, s_m)$ where $\lambdaangle {\hbox{\fiverm th}}etaau_n:n<\omega\rangle$ is some enumeration of the terms in the appropriate language. Then $\phi$ has length $u_\omega=\kappa^1_3$ and for each $m$ letting $\phi_m$ be the prewellordering given by $\phi_m(n, x^\#)=\phi(n, m. x^\#)$, we have that $\phi_m\in \Gamma_{1, m+1}$. Thus, we indeed have an equality.}.
\end{center}
In this paper, assuming $AD$, we compute $\delta(\utilde{\Gamma}_{k, m})$. First let
\betaegin{center}
$a_{k,m}=\delta(\utilde{\Gamma}_{k, m})$.
\end{center}
Here is our main theorem.
\betaegin{theorem}[Main Theorem]\lambdaanglebel{main theorem} Assume $AD$ and let $k$ be an integer. Then
\betaegin{center}
$\sigmaup_{m<\omega} a_{2k+1, m} =\kappa^1_{2k+3}$.
\end{center}
\end{theorem}
We will prove the theorem using directed systems of mice. Our proof relies on a generalization of Woodin's analysis of ${\rm{HOD}}^{L[x][g]}$. The proof is divided into subsections. The proof presented here suggests further applications of the directed systems in descriptive set theory and we will end with a discussion of projects that are left open. We start with introducing the direct limit associated with ${\mathcal{M}}_{n}$'s.
\sigmaection{The directed system associated to ${\mathcal{M}}_{n}$.}
In this section, we analyze the length of the prewellordering given by the iterates of ${\mathcal{M}}_{2n+1}$. As it turns out, the even case, i.e., the prewellordering associated to ${\mathcal{M}}_{2n}$s, doesn't give much beyond the results of \cite{OIMT}. Nevertheless, we make all the definitions for arbitrary $n$. The prewellordering associated with the iterates of ${\mathcal{M}}_{n+1}$ that we are interested in is the following.
For any iterate ${\mathcal{P} }$ of ${\mathcal{M}}_{n+1}$ we let $\delta^{\mathcal{P} }$ be the least Woodin of ${\mathcal{P} }$. Let ${\mathcal{S}}igma$ be the canonical iteration strategy of ${\mathcal{M}}_{n+1}$. If ${\mathcal{P} }\in I({\mathcal{M}}_{n+1}, {\mathcal{S}}igma)$ and ${\mathcal{ Q}}\in I({\mathcal{P} }, {\mathcal{S}}igma_{\mathcal{P} })$ then we let $i_{{\mathcal{P} }, {\mathcal{ Q}}}$ be the iteration embedding. We then define a prewellordering $R^+_n$ of the set
\betaegin{center}
$\{ ({\mathcal{P} }, \alpha) : {\mathcal{P} }\in I({\mathcal{M}}_{n+1}, {\mathcal{S}}igma) \wedge \alpha<\delta^{\mathcal{P} }\}$
\end{center}
by $({\mathcal{P} }, \alpha)R^+_n ({\mathcal{ Q}}, \beta)$ iff ${\mathcal{ Q}}\in I({\mathcal{P} }, {\mathcal{S}}igma_{\mathcal{P} })$ and $i_{{\mathcal{P} }, {\mathcal{ Q}}}(\alpha)\lambdaeq \beta$. Clearly $R^+_n$ is a prewellordering. One problem with $R^+_n$ is that it is a prewellordering of uncountable objects and hence, cannot be regarded as a prewellordering of the reals. Here is how one can find an equivalent prewellordering of countable objects.
We let ${\mathcal{W} }_n={\mathcal{M}}_{n+1}|(\delta^{+\omega})^{{\mathcal{M}}_{n+1}}$ and define the equivalent of $R^+_n$ on the set
\betaegin{center}
$\muathcal{J}_n^+=\{(P, \alpha): {\mathcal{P} }\in I({\mathcal{W} }_n, {\mathcal{S}}igma_{{\mathcal{W} }_n})\wedge\alpha<\delta^{\mathcal{P} }\}$.
\end{center}
We set $({\mathcal{P} }, \alpha)R_n^+ ({\mathcal{ Q}}, \beta)$ iff ${\mathcal{ Q}} \in I({\mathcal{P} }, {\mathcal{S}}igma_{{\mathcal{P} }})$ and $i_{{\mathcal{P} }, {\mathcal{ Q}}}(\alpha)\lambdaeq \beta$. It is not hard to see that $R^+_n$ is essentially the old $R^+_n$. Two questions then immediately come up: 1. What is the length of $R_n^+$? and 2. What is the complexity of $R_n^+$? It is not hard to find an upper bound for $R_n^+$.
\betaegin{lemma} $\card{R^+_n}<\delta^1_{n+3}$.
\end{lemma}
\betaegin{proof}
Here is the outline of the proof. Because $x\rightarrow {\mathcal{M}}_n^\#(x)$ is a ${\mathcal{P} }i^1_{n+2}$ (see \cite{PWOIM}), we get that $\muathcal{J}_n^+$ is ${\mathcal{S}}igma^1_{n+3}({\mathcal{W} }_n)$ (i.e., ${\mathcal{S}}igma^1_{n+3}(x)$ for any code $x$ of ${\mathcal{W} }_n$). The complexity essentially comes from the fact that we require $i_{{\mathcal{P} }, {\mathcal{ Q}}}$ be the correct iteration embedding and to say that we need to refer to $x\rightarrow {\mathcal{M}}_n^\#(x)$ operator.
\end{proof}
To prove our main theorem we need to somehow internalize $R^+_n$ to ${\mathcal{M}}_n(x)$ where $x$ is any real coding ${\mathcal{W} }_n$. Notice that ${\mathcal{M}}_n(x)$ doesn't know the strategy of ${\mathcal{W} }_n$ and hence, it doesn't know how to define its own version of $R^+_n$. We will define an enlargement of $R^+_n$ which ${\mathcal{M}}_n(x)$ can define and we will show that the enlargement has the same length as $R^+_n$.
We now start introducing concepts that we will need in order to internalize $R^+_n$ to ${\mathcal{M}}_n(x)$. Most of these concepts have their origins in Woodin's unpublished work on ${\rm{HOD}}^{L[x][g]}$. Various sources have expositions of similar concepts. For example, \cite{WoodinHod} has most of what we need excepts for the full hod limit. None of these concepts appeared for projective mice such as ${\mathcal{M}}_n$ and here we take a moment to develop these ideas. We start with suitability. First recall the $S_n$ operator from \rsec{dimt}.
\betaegin{definition}[$n$-suitable]\lambdaanglebel{$n$-suitable}
${\mathcal{P} }$ is \emph{$n$-suitable} if there is $\delta$ such that
\betaegin{enumerate}
\item ${\mathcal{P} }\vDash ZFC-Replacement$,
\item ${\mathcal{P} }\vDash ``\delta$ is the only Woodin cardinal",
\item $o({\mathcal{P} })=\sigmaup_{i<\omega}(\delta^{+i})^{\mathcal{P} }$,
\item for every strong cutpoint cardinal $\eta$ of ${\mathcal{P} }$, $S_n({\mathcal{P} }|\eta)={\mathcal{P} }|(\eta^+)^{\mathcal{P} }$.
\end{enumerate}
\end{definition}
If ${\mathcal{P} }$ is $n$-suitable then we let $\delta^{\mathcal{P} }$ be the $\delta$ of \rdef{$n$-suitable}. Clearly ${\mathcal{W} }_n$ is a $n$-suitable premouse. Moreover, if ${\mathcal{ Q}}\in I({\mathcal{W} }_n, {\mathcal{S}}igma_{{\mathcal{W} }_n})$ then ${\mathcal{ Q}}$ is $n$-suitable because $i_{{\mathcal{W} }_n, {\mathcal{ Q}}}$ can be lifted to $i:{\mathcal{M}}_{n+1}\rightarrow {\mathcal{M}}_n({\mathcal{ Q}})$. Sometimes we will just say that ${\mathcal{P} }$ is $n$-suitable implying that it is $n$-suitable for some $n$.
To approximate the iteration strategy of ${\mathcal{W} }_n$ inside ${\mathcal{M}}_n(x)$, the notion of $s$-iterability is used. We now work towards introducing it.
Given an iteration tree $\T$ on an $n$-suitable ${\mathcal{P} }$, we say $\T$ is {\hbox{\fiverm th}}etaextit{correctly guided} if for every limit $\alpha<lh(\T)$, if $b$ is the branch of $\T\restriction \alpha$ chosen by $\T$ and ${\mathcal{ Q}}(b, \T\restriction \alpha)$ exists then ${\mathcal{ Q}}(b, \T\restriction \alpha){\hbox{\fiverm th}}etarianglelefteq {\mathcal{M}}_n({\mathcal{M}}(\T\restriction \alpha))$. $\T$ is {\hbox{\fiverm th}}etaextit{short} if there is a well-founded branch $b$ such that $\T^\frown \{{\mathcal{M}}^\T_b\}$ is correctly guided. $\T$ is {\hbox{\fiverm th}}etaextit{maximal} if $\T$ is not short.
Suppose ${\mathcal{P} }$ is $n$-suitable. We say $\lambdaangle \T_i, {\mathcal{P} }_i : i<m\rangle$ is a {\hbox{\fiverm th}}etaextit{finite correctly guided stack} on ${\mathcal{P} }$ if
\betaegin{enumerate}
\item ${\mathcal{P} }_0={\mathcal{P} }$,
\item ${\mathcal{P} }_i$ is $n$-suitable and $\T_i$ is a correctly guided tree on ${\mathcal{P} }_i$ below $\delta^{{\mathcal{P} }_i}$,
\item for every $i$ such that $i+1< m$ either $\T_i$ has a last model and $i^\T$-exists or $\T$ is maximal, and
\betaegin{enumerate}
\item if $\T_i$ has a last model then ${\mathcal{P} }_{i+1}$ is the last model of $\T_i$,
\item if $\T_i$ is maximal then ${\mathcal{P} }_{i+1}={\mathcal{M}}_n({\mathcal{M}}(\T_i))|(\delta(\T_i)^{+\omega})^{{\mathcal{M}}_n({\mathcal{M}}(\T_i))}$.
\end{enumerate}
\end{enumerate}
We say ${\mathcal{ Q}}$ is the last model of $\lambdaangle \T_j, {\mathcal{P} }_j : i<k\rangle$ if one of the following holds:
\betaegin{enumerate}
\item $\T_{k-1}$ has a last model and ${\mathcal{ Q}}$ is the last model of $\T_{k-1}$,
\item $\T_{k-1}$ is short and there is a cofinal well-founded branch $b$ such that ${\mathcal{ Q}}(b, \T)$ exists and is iterable and ${\mathcal{ Q}}={\mathcal{M}}^\T_b$,
\item $\T_{k-1}$ is maximal and
\betaegin{center}
${\mathcal{ Q}}={\mathcal{M}}_n({\mathcal{M}}(\T_{k-1}))|(\delta(\T_{k-1})^{+\omega})^{{\mathcal{M}}_n({\mathcal{M}}(\T_{k-1}))}$.
\end{center}
\end{enumerate}
We say ${\mathcal{ Q}}$ is a {\hbox{\fiverm th}}etaextit{correct iterate} of ${\mathcal{P} }$ if there is a correctly guided finite stack on ${\mathcal{P} }$ with last model ${\mathcal{ Q}}$.
Suppose ${\mathcal{P} }$ is $n$-suitable and $s=\lambdaangle \alpha_0, ...,\alpha_m\rangle$ is a finite sequence of ordinals. Then we let $T^{\mathcal{P} }_{s, k}\sigmaubseteq [((\delta^{\mathcal{P} })^{+k})^{\mathcal{P} }]^{<\omega}{\hbox{\fiverm th}}etaimes \omega$ be the set
\betaegin{center}
$(t, \phi)\in T^{\mathcal{P} }_{s, k} \mathrel{\leftrightarrow} \phi$ is ${\mathcal{S}}igma_1$ and ${\mathcal{M}}_n({\mathcal{P} })\vDash \phi[t, s]$.
\end{center}
\betaegin{center}
$\gamma^{\mathcal{P} }_{s}=Hull^{\mathcal{P} }_1( \{ T^{\mathcal{P} }_{s, i} : i\in \omega \} )\cap \delta^{\mathcal{P} }$.
\end{center}
Notice that
\betaegin{center}
$\gamma^{\mathcal{P} }_{s}=Hull^{\mathcal{P} }_1(\gamma_s^{\mathcal{P} }\cup \{ T^{\mathcal{P} }_{s, i} : i\in \omega\})\cap \delta^{\mathcal{P} }$.
\end{center}
Let
\betaegin{center}
$H_{s}^{\mathcal{P} } =Hull^{\mathcal{P} }_1(\gamma^{\mathcal{P} }_{s}\cup \{ T^{\mathcal{P} }_{s, i} : i\in \omega \})$.
\end{center}
If $s=s_m$, then we let $\gamma_{m}^{\mathcal{P} }=\gamma_{s_m}^{\mathcal{P} }$ and $H_{m}^{\mathcal{P} }=H_{s_m}^{\mathcal{P} }$.
The following is not hard to show.
\betaegin{lemma}\lambdaanglebel{ggs sup up}
$\sigmaup_{n<\omega}\gamma_n^{\mathcal{P} }=\delta^{\mathcal{P} }$.
\end{lemma}
\betaegin{proof}
Suppose not. Let $\gamma=\sigmaup_{n<\omega}\gamma_n^{\mathcal{P} }$. Let $X=Hull_1^{\mathcal{P} }(\gamma\cap \{ T_{s_m, i}^{\mathcal{P} } : m, i\in \omega\})$. Let ${\mathcal{N}}$ be the collapse of $X$ and let $\pi:{\mathcal{N}}\rightarrow {\mathcal{P} }$ be the inverse of the collapsing map. We have that for each $m, i$ there is $S_{m, i}\in {\mathcal{N}}$ such that $\pi(S_{m, i})=T_{s_m, i}^{\mathcal{P} }$. We have that $\gamma=\delta^{\mathcal{S}}$. Notice that for each $i$, $\cup_{m\in \omega} S_{m, i}$ is a complete and consistent theory and if ${\mathcal R}$ is its model then ${\mathcal R}$ is essentially the hull of ordinals $<(\gamma^{+i})^{\mathcal{N}}$ and $\omega$ many indiscernibles. Moreover, we have that $\pi$ can be extended to $\pi^*:{\mathcal R} \rightarrow {\mathcal{M}}_n({\mathcal{P} })$. This implies that ${\mathcal R}$ is well-founded and therefore, it has to be ${\mathcal{M}}_n({\mathcal{N}}|(\gamma^{+i})^{\mathcal{N}})$. This shows that ${\mathcal{M}}_n({\mathcal{N}}|\gamma)\vDash ``\gamma$ is Woodin" which implies that ${\mathcal{P} }\vDash ``\gamma$ is Woodin". This is a contradiction as $\delta^{\mathcal{P} }$ is the least Woodin of ${\mathcal{P} }$.
\end{proof}
\betaegin{definition}[$s$-iterability]\lambdaanglebel{s-iterability} Suppose ${\mathcal{P} }$ is $n$-suitable and $s=\lambdaangle \alpha_i :i< l\rangle$ is an increasing finite sequence of ordinals. ${\mathcal{P} }$ is \emph{$s$-iterable} if whenever $\lambdaangle \T_k, {\mathcal{P} }_k : k<m\rangle$ is a finite correctly guided stack on ${\mathcal{P} }$ with last model ${\mathcal{ Q}}$
then there is a sequence $\lambdaangle b_k: k< m\rangle$ such that
\betaegin{enumerate}
\item for $k<m-1$,
\betaegin{displaymath}
b_k = \lambdaeft\{
\betaegin{array}{lr}
\emptyset :& \T_k \ has \ a \ successor \ length \\
cofinal\ well-founded\\ branch\ such\ that\ {\mathcal{M}}^\T_{b_k}={\mathcal{P} }_\kappa :& \T_k \ is \ maximal
\end{array}
\right.
\end{displaymath}
\item if $\T_{m-1}$ has a successor length then $b_{m-1}=\emptyset$, if $\T_{m-1}$ is short then $b_{m-1}$ is the unique cofinal well-founded branch such that ${\mathcal{ Q}}(b_{m-1}, \T_{m-1})$ exists and is iterable, and if $\T_{m-1}$ is maximal then $b_{m-1}$ is a cofinal well-founded branch,
\item letting
\betaegin{displaymath}
\pi_k = \lambdaeft\{
\betaegin{array}{lr}
i^{\T_k} :& \T_k \ has \ a \ successor \ length \\
i_{b_k}^{\T_k} :& \T_k \ is \ maximal
\end{array}
\right.
\end{displaymath}
and $\pi=\pi_{m-1}\circ \pi_{m-2}\circ\cdot \cdot\cdot \pi_0$ then for every $l$
\betaegin{center}
$\pi(T^{\mathcal{P} }_{s, l})=T^{\mathcal{ Q}}_{s, l}$.
\end{center}
\end{enumerate}
\end{definition}
Suppose ${\mathcal{P} }$ is $n$-suitable, $s=\lambdaangle \alpha_i :i< l\rangle$ is an increasing finite sequence of ordinals and ${\vec{\mathcal{T}}}=\lambdaangle \T_k, {\mathcal{P} }_k : k<m\rangle$ is a correctly guided finite stack on ${\mathcal{P} }$ with last model ${\mathcal{ Q}}$. We say $\vec{b}=\lambdaangle b_k: k< m\rangle$ witness $s$-iterability for ${\vec{\mathcal{T}}}=\lambdaangle \T_k, {\mathcal{P} }_k : k< m\rangle$ if 2 above is satisfied. We may also say that $\vec{b}$ is an $s$-iterability branch for ${\vec{\mathcal{T}}}$. We then let
\betaegin{displaymath}
\pi_{{\vec{\mathcal{T}}}, \vec{b}, k} = \lambdaeft\{
\betaegin{array}{lr}
i^{\T_k} :& \T_k \ has \ a \ successor \ length \\
i_{b_k}^{\T_k} :& \T_k \ is \ maximal
\end{array}
\right.
\end{displaymath}
and $\pi_{{\vec{\mathcal{T}}}, \vec{b}}=\pi_{{\vec{\mathcal{T}}}, \vec{b}, m-1}\circ \pi_{{\vec{\mathcal{T}}}, \vec{b}, m-2}\circ\cdot \cdot\cdot \pi_{{\vec{\mathcal{T}}}, \vec{b}, 0}$.
Suppose now that $\vec{b}$ and $\vec{c}$ are two $s$-iterability branches for ${\vec{\mathcal{T}}}$. Then using \rlem{partial agreement}, it is easy to see that $\pi_{{\vec{\mathcal{T}}}, \vec{b}}\restriction H_{s}^{\mathcal{P} }=\pi_{{\vec{\mathcal{T}}}, \vec{c}}\restriction H_s^{\mathcal{P} }$. Lets record this as a lemma.
\betaegin{lemma}[Uniqueness of $s$-iterability embeddings] \lambdaanglebel{uniqueness of s-iterability embeddings} Suppose ${\mathcal{P} }$ is $n$-suitable, $s$ is a finite sequence of ordinals and ${\vec{\mathcal{T}}}$ is a finite correctly guided stack on ${\mathcal{P} }$. Suppose $\vec{b}$ and $\vec{c}$ are two $s$-iterability branches for ${\vec{\mathcal{T}}}$. Then
\betaegin{center}
$\pi_{{\vec{\mathcal{T}}}, \vec{b}}\restriction H_{s}^{\mathcal{P} }=\pi_{{\vec{\mathcal{T}}}, \vec{c}}\restriction H_s^{\mathcal{P} }$.
\end{center}
Moreover, if ${\vec{\mathcal{T}}}$ consists of just one normal tree $\T$, ${\mathcal{ Q}}$ is the last model of $\T$ and $b$ and $c$ witness $s$-iterability for $\T$ then if $\xi\in b$ is the least such that ${\rm cp }(E^\T_\xi)>\gamma_s^{\mathcal{ Q}}$ then $b\cap \xi =c\cap \xi$. \end{lemma}
If ${\mathcal{P} }$ is $s$-iterable and $\T$ is a normal correctly guided tree then we let $b^\T_s=\cap \{b : b$ witnesses the $s$-iterability of ${\mathcal{P} }$ for $\T\}$. Here is how $s$-iterability is connected to iterability. Suppose ${\mathcal{P} }$ is $n$-suitable. We say ${\mathcal{P} }$ has a {\hbox{\fiverm th}}etaextit{correct $\omega_1$-iteration strategy} if it has an $\omega_1$-iteration strategy ${\mathcal{S}}igma$ such that whenever $\T$ is a correctly guided tree of limit length and $b={\mathcal{S}}igma(\T)$ then $\T^\frown {\mathcal{M}}^\T_b$ is correctly guided.
\betaegin{lemma}
Suppose ${\mathcal{P} }$ is $n$-suitable and for every $m$, ${\mathcal{P} }$ is $s_m$-iterable. Then ${\mathcal{P} }$ has a correct iteration strategy.
\end{lemma}
\betaegin{proof}
Let $\T$ be a correctly guided tree. If $\T$ is short then using $s_m$-iterability there must be a branch $b$ of $\T$ such that ${\mathcal{ Q}}(b, \T)$-exists and is iterable. In this case we define ${\mathcal{S}}igma(\T)=b$. Suppose now $\T$ is maximal with last model ${\mathcal{ Q}}$. Then for each $m$, let $b_m=b^\T_{s_m}$. Notice that $b_m\sigmaubseteq b_{m+1}$. Also, because $\sigmaup_{m\in \omega}\gamma_m^{\mathcal{ Q}}=\delta^{\mathcal{ Q}}$, we have that if $b=\cup_{m\in \omega} b_m$ then $b$ is a cofinal branch. We claim that ${\mathcal{M}}^\T_b={\mathcal{ Q}}$. Let ${\mathcal R}={\mathcal{M}}^\T_b$. For all we know ${\mathcal R}$ may not be well-founded. But notice that if $R_m=i^\T_b(H_m^{\mathcal{P} })$ then there is $\pi_m: {\mathcal R}_m\rightarrow_{{\mathcal{S}}igma_1} H_m^{\mathcal{ Q}}$. This is because $i_b^\T\restriction \gamma_m^{\mathcal{P} }=\pi_{\T, b_m}\restriction \gamma^{\mathcal{P} }_m$ where $b_m$ is any cofinal well-founded branch witnessing $s$-iterability of ${\mathcal{P} }$ for $\T$. It then follows that if $\pi=\cup_{m\in \omega} \pi_m$ then $\pi:\cup_{m\in \omega} {\mathcal R}_m \rightarrow {\mathcal{ Q}}$ and because $\cup_{m\in \omega}{\mathcal R}_m={\mathcal R}$, we have that ${\mathcal R}$ is well-founded. Because for each $i$ and $m$, $T^{\mathcal{ Q}}_{s_m, i}\in ran(\pi)$, using the proof of \rlem{ggs sup up}, we get that ${\mathcal R}$ is $n$-suitable and hence, ${\mathcal R}={\mathcal{ Q}}$ and $\pi=id$. In this case, then, we define ${\mathcal{S}}igma(\T)=b$. It follows from our construction that ${\mathcal{S}}igma$ is a correct iteration strategy.
\end{proof}
Notice that, if ${\mathcal{P} }$ is $s$-iterable, ${\vec{\mathcal{T}}}$ is a correctly guided finite stack on ${\mathcal{P} }$, and $\vec{b}$ witnesses $s$-iterability of ${\mathcal{P} }$ for ${\vec{\mathcal{T}}}$, then even though $\pi_{{\vec{\mathcal{T}}}, \vec{b}}\restriction H_{s}^{\mathcal{P} }$ is independent of $\vec{b}$ it may very well depend on ${\vec{\mathcal{T}}}$. This observation motivates the following definition.
\betaegin{definition}[Strong $s$-iterability]\lambdaanglebel{strong s-itearbility} Suppose ${\mathcal{P} }$ is $n$-suitable and $s$ is a finite sequence of ordinals. Then ${\mathcal{P} }$ is strongly $s$-iterable if ${\mathcal{P} }$ is $s$-iterable and whenever ${\vec{\mathcal{T}}}=\lambdaangle \T_j , {\mathcal{P} }_j : j< u\rangle$ and ${\vec{\mathcal{U}}}=\lambdaangle {\mathcal{U}}_j, {\mathcal{ Q}}_j : j < v\rangle$ are two correctly guided finite stacks on ${\mathcal{P} }$ with common last model ${\mathcal{ Q}}$, $\vec{b}$ witnesses $s$-iterability for ${\vec{\mathcal{T}}}$ and $\vec{c}$ witnesses $s$-iterability for ${\vec{\mathcal{U}}}$ then
\betaegin{center}
$\pi_{{\vec{\mathcal{T}}}, \vec{b}}\restriction H_s^{\mathcal{P} } = \pi_{{\vec{\mathcal{U}}}, \vec{c}}\restriction H_s^{\mathcal{P} }$.
\end{center}
\end{definition}
Are there $s$-iterable ${\mathcal{P} }$'s? Of course there must be, as otherwise we wouldn't define them, and here is an argument that shows it. Suppose not. Let $s=\lambdaangle \alpha_k : k< l\rangle$. Using the fact that there are no $s$-iterable ${\mathcal{P} }$'s, we can then get $\vec{B}=\lambdaangle B_k : k<\omega\rangle$ such that $B_k=\lambdaangle \T^k_j, {\mathcal{P} }^k_j , {\mathcal{ Q}}_k: j< m_k\rangle$ and
\betaegin{enumerate}
\item ${\mathcal{P} }^0_0={\mathcal{W} }_n$ and ${\mathcal{P} }^{k+1}_0={\mathcal{ Q}}_k$,
\item for every $k$, $\lambdaangle \T^k_j, {\mathcal{P} }^k_j : j< m_k\rangle$ is a correctly guided finite stack on ${\mathcal{P} }^k_0$ with last model ${\mathcal{ Q}}_k$,
\item whenever $\lambdaangle b^k_j: k<\omega \wedge j<k< m_k \rangle$ is such that
\betaegin{enumerate}
\item for $j<m_k-1$,
\betaegin{displaymath}
b^k_j = \lambdaeft\{
\betaegin{array}{lr}
\emptyset :& \T^k_j \ has \ a \ successor \ length \\
cofinal\ well-founded\ branch\\ such\ that\ {\mathcal{M}}^\T_{b^k_j}={\mathcal{P} }^k_j :& \T^k_j \ is \ maximal
\end{array}
\right.
\end{displaymath}
\item if $\T^k_{m_k-1}$ has a successor length then $b^k_{m_k-1}=\emptyset$, if $\T^k_{m_k-1}$ is short then $b^k_{m_k-1}$ is the unique cofinal well-founded branch such that ${\mathcal{ Q}}(b^k_{m_k-1}, \T^k_{m_k-1})$ exists and is iterable, and if $\T^k_{m_k-1}$ is maximal then $b^k_{m_k-1}$ is a cofinal well-founded branch,\footnote{Notice that $s$-iterability cannot fail because we cannot find correct branches for short trees as long as we start with ${\mathcal{P} }_0={\mathcal{W} }_n$.}
\end{enumerate}
then letting $\vec{b}_k=\lambdaangle b^k_j: j< m_k\rangle$, for some $m$ and every $k$,
\betaegin{center}
$\pi_{{\vec{\mathcal{T}}}_k, \vec{b}_k}(T^{{\mathcal{P} }^k_0}_{s, m})\nuot = T^{{\mathcal{ Q}}_k}_{s, m}$.
\end{center}
\end{enumerate}
Let then $\lambdaangle b^k_j: k<\omega \wedge j<m_k \rangle$ be the sequence of branches given by ${\mathcal{S}}igma_{{\mathcal{W} }_n}$. Then clearly for every $k$, $\pi_{{\vec{\mathcal{T}}}_k, \vec{b}_k}$'s extend to
\betaegin{center}
$\pi_k:{\mathcal{M}}_n({\mathcal{P} }^k_0)\rightarrow {\mathcal{M}}_n({\mathcal{ Q}}_k)$.
\end{center}
Let $\beta>\muax(s)$ be a uniform indiscernible and let $t=s^\frown \lambdaangle\beta\rangle$. Suppose that for some $k$, $\pi_k(t)=t$. Notice that for every $m$, $T^{{\mathcal{P} }^k_0}_{s, m}$ can be defined by
\betaegin{center}
$(t, \phi)\in T^{{\mathcal{P} }^k_0}_{s, m} \mathrel{\leftrightarrow} \phi$ is ${\mathcal{S}}igma_1$ and ${\mathcal{M}}_n({\mathcal{P} }^k_0)|\beta\vDash \phi[t, s]$.
\end{center}
Hence, because we are assuming $\pi_k(t)=t$, we get that $\pi_{{\vec{\mathcal{T}}}_k, \vec{b}_k}(T^{{\mathcal{P} }^k_0}_{s, m}) = T^{{\mathcal{ Q}}_k}_{s, m}$.
Therefore, we must have that $t<_{lex} \pi_k(t)$. Let ${\mathcal{ Q}}$ be the direct limit of $\lambdaangle {\mathcal{M}}_n({\mathcal{ Q}}_k): k< \omega \rangle$ under the maps $\sigmaigma_{k, l}= \pi_l\circ \pi_{l-1}\circ\cdot\cdot\cdot \pi_k$ and let $\pi^{*}_k: {\mathcal{M}}_n({\mathcal{ Q}}_k) \rightarrow {\mathcal{ Q}}$ be the embedding given by the direct limit construction. Now if $t_k=\pi^{*}_k(t)$, then $\lambdaangle t_k : k<\omega\rangle$ is a $\lambdaeq_{lex}$-decreasing sequence of finite sequences of ordinals. Because $\pi^*_k$'s are iteration embeddings according to ${\mathcal{S}}igma_{{\mathcal{W} }_n}$, we get a contradiction. This completes the proof that for every $s$ there is an $s$-iterable $n$-suitable ${\mathcal{P} }$.
\betaegin{lemma}\lambdaanglebel{s-iterability lemma}
For every $s\in Ord^{<\omega}$ and $n\in \omega$ there is an $s$-iterable $n$-suitable ${\mathcal{P} }$. Moreover, for any $n$-suitable ${\mathcal{ Q}}$ there is a normal correctly guided tree $\T$ with last model ${\mathcal{P} }$ such that ${\mathcal{P} }$ is $s$-iterable.
\end{lemma}
\betaegin{proof}
We have already shown that there is an $s$-iterable $n$-suitable ${\mathcal{P} }$. It is then the second clause that needs a proof. Fix a $n$-suitable ${\mathcal{ Q}}$ and let ${\mathcal{P} }$ be $s$-iterable. Comparing ${\mathcal{P} }$ and ${\mathcal{ Q}}$ produces our desired $\T$.
\end{proof}
Is there a strongly $s$-iterable ${\mathcal{P} }$? The proof we have just given shows that there is. Indeed, using the proof given above we have ${\mathcal{P} }$ which is $s$-iterable and is a ${\mathcal{S}}igma_{{\mathcal{W} }_n}$-iterate of ${\mathcal{W} }_n$. Moreover, if ${\rm{L}}ambda={\mathcal{S}}igma_{\mathcal{P} }$ then the branches witnessing $s$-iterability can be taken to be those given by ${\rm{L}}ambda$. It then easily follows from the Dodd-Jensen property of ${\rm{L}}ambda$ that ${\mathcal{P} }$ is strongly $s$-iterable.
\betaegin{lemma}[Strongly $s$-iterability lemma]\lambdaanglebel{strongly s-iterability lemma}
For every $s$ there is a strongly $s$-iterable ${\mathcal{P} }$. Moreover, for any $n$-suitable ${\mathcal{ Q}}$ there is normal correctly guided stack $\T$ with last model ${\mathcal{P} }$ such that ${\mathcal{P} }$ is strongly $s$-iterable.
\end{lemma}
\betaegin{proof}
We have already shown that there is a strongly $s$-iterable ${\mathcal{P} }$. It is then the second clause that needs a proof. Fix a $n$-suitable ${\mathcal{ Q}}$ and let ${\mathcal{P} }$ be a strongly $s$-iterable. Comparing ${\mathcal{P} }$ and ${\mathcal{ Q}}$ produces our desired $\T$.
\end{proof}
If ${\mathcal{P} }$ is strongly $s$-iterable and ${\vec{\mathcal{T}}}$ is a correctly guided finite stack on ${\mathcal{P} }$ with last model ${\mathcal{ Q}}$ then we let
\betaegin{center}
$\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s}:H_s^{\mathcal{P} }\rightarrow H_s^{\mathcal{ Q}}$
\end{center}
be the embedding given by any $\vec{b}$ which witnesses the $s$-iterability of ${\vec{\mathcal{T}}}$, i.e., fixing $\vec{b}$ which witnesses $s$-iterability for ${\vec{\mathcal{T}}}$,
\betaegin{center}
$\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s} =\pi_{{\vec{\mathcal{T}}}, \vec{b}}\restriction H_s^{\mathcal{P} }$.
\end{center}
Clearly, $\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s}$ is independent of ${\vec{\mathcal{T}}}$ and $\vec{b}$.
Notice that ${\mathcal{W} }_n$ is strongly $s_m$-iterable for every $m$. Moreover, if ${\vec{\mathcal{T}}}$ is any correctly guided stack on ${\mathcal{W} }_n$ with last model ${\mathcal{ Q}}$ then $\pi_{{\mathcal{W} }_n, {\mathcal{ Q}}, s_m}$ agrees with the correct iteration embedding, i.e., if $i:{\mathcal{W} }_n\rightarrow {\mathcal{ Q}}$ is the iteration embedding according to the canonical iteration strategy of ${\mathcal{W} }_n$ then
\betaegin{center}
$\pi_{{\mathcal{W} }_n, {\mathcal{ Q}}, s_m}=i\restriction H_m^{{\mathcal{W} }_n}$.
\end{center}
Moreover, since $\cup_{m<\omega}H_m^{{\mathcal{W} }_n}={\mathcal{W} }_n$, we get that
\betaegin{center}
$\cup_{m<\omega}\pi_{{\mathcal{W} }_n, {\mathcal{ Q}}, s_m} = i$.
\end{center}
This is how we will approximate ${\mathcal{S}}igma$ inside ${\mathcal{M}}_n(x)$.
Next let
\betaegin{center}
$\muathcal{F}^+_n=\{ {\mathcal{P} }: {\mathcal{P} }\in I({\mathcal{W} }_n, {\mathcal{S}}igma_{{\mathcal{W} }_n})$ as witnessed by some finite stack $\}$.
\end{center}
We let $\lambdaeq^+_n$ be a prewellording of $\muathcal{F}^+_n$ given by ${\mathcal{P} }\lambdaeq^+_n{\mathcal{ Q}}$ iff ${\mathcal{ Q}}\in I({\mathcal{P} }, {\mathcal{S}}igma_{\mathcal{P} })$ as witnessed by a finite stack. We then let ${\mathcal{M}}_{\infty, n}^+$ be the direct limit of $(\muathcal{F}^+_n, \lambdaeq^+_n)$ under the iteration maps $i_{{\mathcal{P} }, {\mathcal{ Q}}}$. Notice that $\card{R^+_n}=\delta^{{\mathcal{M}}_\infty^+}$. We let $\delta^+_{ \infty, n}=\delta^{{\mathcal{M}}_{\infty, n}^+}$.
We also let
\betaegin{center}
$\muathcal{I}_n=\{ ({\mathcal{P} }, s): {\mathcal{P} }$ is $n$-suitable, $s\sigmaubseteq Ord^{<\omega}$ and ${\mathcal{P} }$ is strongly $s$-iterable $\}$.
\end{center}
and
\betaegin{center}
$\muathcal{F}_n=\{ H^{\mathcal{P} }_s: ({\mathcal{P} }, s)\in \muathcal{I}_n\}$.
\end{center}
We define $\lambdaeq_n$ on $\muathcal{I}_n$ by: $({\mathcal{P} }, s)\lambdaeq_n ({\mathcal{ Q}}, t)$ iff ${\mathcal{ Q}}$ is a correctly guided iterate of ${\mathcal{P} }$ and $s\sigmaubseteq t$. Is $\lambdaeq_n$ directed? The answer is of course yes and to see that fix $({\mathcal{P} }, s), ({\mathcal{ Q}}, t)\in \muathcal{I}_n$. Then we have ${\mathcal R}$ which is strongly $s\cup t$-iterable. Let ${\mathcal{S}}$ be the result of comparing ${\mathcal{P} }, {\mathcal{ Q}}$ and ${\mathcal R}$. Then $({\mathcal{S}}, s\cup t)\in \muathcal{I}_n$ and
\betaegin{center}
$({\mathcal{P} }, s)\lambdaeq_n({\mathcal{S}}, s\cup t)$ and $({\mathcal{ Q}}, t)\lambdaeq_n ({\mathcal{S}}, s\cup t)$.
\end{center}
We can then form the direct limit of $(\muathcal{F}_n, \lambdaeq_n)$ under the maps $\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s}$. We let ${\mathcal{M}}_{\infty, n}$ be this direct limit. It is clear that ${\mathcal{M}}_{\infty, n}^+$ is well-founded. However, it is not at all clear that ${\mathcal{M}}_{\infty, n}$ is well-founded. We show that not only ${\mathcal{M}}_{\infty, n}$ is well-founded but that it is also the same as ${\mathcal{M}}_{\infty, n}^+$.
Before we continue, we fix some notation. If ${\mathcal{P} }\in I({\mathcal{W} }_n, {\mathcal{S}}igma_{{\mathcal{W} }_n})$, then we let $i_{{\mathcal{P} }, \infty}:{\mathcal{P} }\rightarrow {\mathcal{M}}_{\infty, n}^+$ be the iteration map. For $({\mathcal{P} }, s)\in \muathcal{I}_n$, we let $\pi_{{\mathcal{P} }, \infty, s}$ be the direct limit embedding acting on $H_s^{\mathcal{P} }$.
\betaegin{lemma}\lambdaanglebel{wellfoundness} ${\mathcal{M}}_{\infty, n}={\mathcal{M}}_{\infty, n}^+$.
\end{lemma}
\betaegin{proof}
To show the equality, we define a map $\pi:{\mathcal{M}}_{\infty, n} \rightarrow_{{\mathcal{S}}igma_1} {\mathcal{M}}_{\infty, n}^+$ and show that $\pi$ is the identity. Let $x\in {\mathcal{M}}_{\infty, n}$. Let $({\mathcal{P} }, s_m)\in \muathcal{I}_n$ be such that for some $y\in H_m^{\mathcal{P} }$, $\pi_{{\mathcal{P} }, \infty, s_m}(y)=x$ and ${\mathcal{P} }$ is a normal correct iterate of ${\mathcal{W} }_n$. Then we let
\betaegin{center}
$\pi(x)=i_{{\mathcal{P} }, \infty}(x)$.
\end{center}
First we need to see that $\pi$ is independent of the choice of ${\mathcal{P} }$. Let then $({\mathcal{P} }, s_p)\in \muathcal{I}_n$ and $({\mathcal R}, s_q)\in \muathcal{I}_n$ be such that there are $y\in H_p^{\mathcal{P} }$ and $z\in H_q^{\mathcal R}$ such that $\pi_{{\mathcal{P} }, \infty, s_p}(y)=\pi_{{\mathcal R}, \infty, s_q}(z)=x$ and both ${\mathcal{P} }$ and ${\mathcal R}$ are normal iterates of ${\mathcal{W} }_n$. Let ${\mathcal{ Q}}$ be the outcome of comparing ${\mathcal{P} }$ and ${\mathcal R}$. Notice that we must have that
\betaegin{center}
$\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s_p}(y)=\pi_{{\mathcal R}, {\mathcal{ Q}}, s_q}(z)$.
\end{center}
It then follows that
\betaegin{center}
$i_{{\mathcal{ Q}}, \infty}(\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s_p}(y))=i_{{\mathcal{ Q}}, \infty}(\pi_{{\mathcal R}, {\mathcal{ Q}}, s_q}(z))$.
\end{center}
and hence, $\pi$ is independent of the choice of ${\mathcal{P} }$. A similar argument shows that $\pi$ is a ${\mathcal{S}}igma_1$-elementary and this much is enough to conclude that ${\mathcal{M}}_{\infty, n}$ is well-founded. But we can in fact show that $\pi=id$. For this, fix $x\in {\mathcal{M}}_{\infty, n}^+|\delta^{{\mathcal{M}}_{\infty, n}^+}$. Let $Q$ be such that there is $y\in {\mathcal{ Q}}$ such that $x=i_{{\mathcal{ Q}}, \infty}(y)$. Let $s_m$ be such that $y\in H_{m}^{{\mathcal{ Q}}}$. Then if $z=\pi_{{\mathcal{ Q}},\infty, s_m}(y)$ then $\pi(z)=x$. This shows that $\pi\restriction \delta^{{\mathcal{M}}_{\infty, n}}+1=id$.
Now fix ${\mathcal{P} }$ and let $T^\infty_{m, l}=i_{{\mathcal{P} }, \infty}(T^{\mathcal{P} }_{m, l})$. We clearly have that $T^\infty_{m, l}\in ran(\pi)$.
Let then $S_{m, l}\in {\mathcal{M}}_{\infty, n}$ be such that $\pi(S_{m,l})=\T_{m,l}^\infty$. Now, let ${\mathcal{N}}={\mathcal{M}}_{\infty, n}$. Then for each $l$, $\cup_{m<\omega}S_{m, l}$ is a prescription for constructing a model with $n$ Woodin cardinals over ${\mathcal{N}}|(\delta^{+l})^{\mathcal{N}}$. Moreover, if $K$ is this model then $K$ is the ${\mathcal{S}}igma_1$-hull of ordinals $<(\delta^{+l})^{\mathcal{N}}$ and $\omega$ indiscernibles. Because of $\pi$, it follows that $K={\mathcal{M}}_n^\#({\mathcal{N}}|(\delta^{+l})^{\mathcal{N}})$. This then inductively implies that for every $l$, ${\mathcal{N}}|(\delta^{+l})^{\mathcal{N}}={\mathcal{S}}|(\delta^{+l})^{\mathcal{S}}$ where ${\mathcal{S}}={\mathcal{M}}_{\infty, n}^+$. Hence, $\pi$ has to be the identity.
\end{proof}
Before moving on, notice that everything we have done in this section relativizes to arbitrary real $x$. For any real $x$, we can define $\muathcal{J}_{x, n}^+$, $\muathcal{I}_{x, n}$, $\muathcal{F}^+_{x, n}$, $\muathcal{F}_{x, n}$, $\lambdaeq_{x, n}^+$, $\lambdaeq_{x, n}$, ${\mathcal{M}}_{\infty, x, n}^+$, and ${\mathcal{M}}_{\infty, x, n}$. We will then again have that ${\mathcal{M}}_{\infty, x, n}^+={\mathcal{M}}_{\infty, x, n}$ and $\delta^{{\mathcal{M}}_{\infty, x, n}}<\delta^1_{n+3}$. We let $\delta_{\infty, x, n}=\delta^{{\mathcal{M}}_{\infty, x, n}}$ and also for $s\in Ord^{<\omega}$, we let
\betaegin{center}
$\gamma_{\infty, s, x, n}=\sigmaup(\pi_{{\mathcal{P} }, \infty, s}"\gamma^{\mathcal{P} }_s)$
\end{center}
where $({\mathcal{P} }, s)\in \muathcal{I}_{x, n}$. Clearly $\gamma_{\infty, s, x, n}$ is independent of the choice of ${\mathcal{P} }$. We also let
\betaegin{center}
$\muathcal{J}_{n, s, z}=\{ ({\mathcal{P} }, \alpha) : ({\mathcal{P} }, s)\in \muathcal{I}_{n, z} \wedge \alpha<\gamma_s^{\mathcal{P} }\}$.
\end{center}
We let $R_{n, s, z}$ be a prewellordering of $\muathcal{J}_{n, s, z}$ given by $({\mathcal{P} }, \alpha)R_{n, s, z}({\mathcal{ Q}}, \beta)$ if ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{P} }$ and $\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s}(\alpha)\lambdaeq \beta$. We also let ${\mathcal{W} }_z={\mathcal{M}}_{n+1}(z)|(\delta^{+\omega})^{{\mathcal{M}}_{n+1}(z)}$ where $\delta$ is the least Woodin of ${\mathcal{M}}_{n+1}(z)$. We let ${\mathcal{S}}igma_z$ be the strategy of ${\mathcal{M}}_{n+1}(z)$ restricted to stacks on ${\mathcal{W} }_z$. We now move to internalizing the direct limit construction to ${\mathcal{M}}_n(x)$ where $x$ is any real coding ${\mathcal{W} }_n$.
\sigmaubsection{Internalizing the directed system}\lambdaanglebel{internalizing the directed system}
Fix a real $x$ that codes ${\mathcal{W} }_n$ and let $\delta$ be the least Woodin of ${\mathcal{M}}_n(x)$. We will work with this $x$ until the end of this subsection. Notice that because ${\mathcal{M}}_{n}(x)|\delta$ is closed under $S_n$ operator, if $\T\in {\mathcal{M}}_n(x)|\delta$ is a short tree on ${\mathcal{W} }_n$ then if $b$ is such that $\T^\frown {\mathcal{M}}^\T_b$ is correctly guided then in fact $b\in {\mathcal{M}}_n(x)|\delta$. Thus, ${\mathcal{S}}igma_{{\mathcal{W} }_n}\restriction\{\T\in {\mathcal{M}}_n(x)|\delta: \T$ is short$\}$.
How about maximal trees? We claim that ${\mathcal{S}}igma_{{\mathcal{W} }_n}\restriction \{ \T\in {\mathcal{M}}_n(x)|\delta: \T$ is maximal$\}$ is not in ${\mathcal{M}}_n(x)$. To see this, assume otherwise. By a result of Neeman from \cite{Neeman02}, there is a normal iterate ${\mathcal{ Q}}\in HC^{{\mathcal{M}}_n(x)}$ of ${\mathcal{W} }_n$ via a tree of length $\omega$ such that there is some ${\mathcal{ Q}}$-generic $g\sigmaubseteq Coll(\omega, \delta^{\mathcal{ Q}})$ such that $g\in {\mathcal{M}}_n(x)$ and $x\in {\mathcal{ Q}}[g]$. But this is a contradiction as ${\mathcal{ Q}}$ is essentially a real in ${\mathcal{M}}_n(x)$ while $\muathbb{R}^{{\mathcal{M}}_n(x)}=S_n(x)\sigmaubseteq {\mathcal{ Q}}[g]$.
Nevertheless, in the case of $n=0$, Woodin used $s$-iterability to track the iteration strategy of ${\mathcal{W} }_n$ inside ${\mathcal{M}}_n(x)$. We do that here for an arbitrary $n$. For the purpose of keeping the notation simple, while working in this subsection we let ${\mathcal{M}}={\mathcal{M}}_n(x)$ and $\delta$ be the least Woodin of ${\mathcal{M}}$.
Notice that the notions such as suitable, short tree, maximal tree, correctly guided finite stack and etc are all definable over ${\mathcal{M}}$. This is because all these notions refer to the $S_n$ operator and ${\mathcal{M}}|\delta$ is closed under the $S_n$ operator. For instance, we have that ${\mathcal{ Q}}\in {\mathcal{M}}|\delta$, ${\mathcal{ Q}}$ is suitable iff ${\mathcal{M}}\vDash ``{\mathcal{ Q}}$ is suitable". Notice, however, that $s$-iterability presents a difficulty as it is not immediately clear how to say ``a suitable ${\mathcal{P} }$ is $s$-iterable" inside ${\mathcal{M}}$. When $n=0$ and $s=\lambdaangle a_j : j< l\rangle$, one can just make do with \rdef{s-iterability}. This is because the ``guiding sets", $T^{\mathcal{P} }_{s, i}$, can be identified inside $L[x]$. In general, this doesn't seem to work because we need to correctly identify $T^{\mathcal{P} }_{s, i}$. If $\beta>\muax(s)$ is a uniform indiscernible then to identify $T^{\mathcal{P} }_{s, i}$ inside ${\mathcal{M}}$, it is enough to identify ${\mathcal{M}}_n({\mathcal{P} })|\beta$ inside ${\mathcal{M}}_n(x)$. This is because
\betaegin{center}
$(t, \phi)\in T^{\mathcal{P} }_{s, m} \mathrel{\leftrightarrow} \phi$ is ${\mathcal{S}}igma_1$ and ${\mathcal{M}}_n({\mathcal{P} })|\beta\vDash \phi[t, s]$.
\end{center}
We then solve the problem by dropping to a smaller set of ``good" ${\mathcal{P} }$'s. This new set of good ${\mathcal{P} }$'s will nevertheless be dense in the old one. To start, we fix $\kappa<\delta$ which is an inaccessible strong cutpoint cardinal of ${\mathcal{M}}$ such that ${\mathcal{M}}\vDash ``\kappa$ is a limit of strong cutpoint cardinals".
We let
\betaegin{center}
$\muathcal{G}_{\kappa}=\{ {\mathcal{P} } \in {\mathcal{M}}|\kappa: {\mathcal{P} }$ is suitable and ${\mathcal{M}}\vDash ``$ for some strong cutpoint $\eta$, $\delta^{\mathcal{P} }=\eta^+$ and ${\mathcal{M}}|\eta$ is generic over ${\mathcal{P} }$ for $\delta^{\mathcal{P} }$-generator version of the extender algebra at $\delta^{\mathcal{P} }"\}$.
\end{center}
If ${\mathcal{P} }\in \muathcal{G}_{\kappa}$ then we let $\eta_{\mathcal{P} }$ be the ordinal witnessing that ${\mathcal{P} }\in \muathcal{G}_{\kappa}$. Recall the definition of ${\mathcal{S}}^{\mathcal{M}}({\mathcal{N}})$ (see \rdef{s(n)}).
\betaegin{lemma}\lambdaanglebel{good points} Suppose ${\mathcal{P} }\in {\mathcal{M}}$ is suitable and such that for some strong cutpoint $\eta$ of ${\mathcal{M}}$, ${\mathcal{P} }|\delta^{\mathcal{P} }\sigmaubseteq {\mathcal{M}}|(\eta^+)^{\mathcal{M}}$ and ${\mathcal{M}}|\eta$ is generic over ${\mathcal{P} }$ for the $\delta^{\mathcal{P} }$-generator version of the extender algebra. Then ${\mathcal{P} }\in \muathcal{G}_\kappa$ and ${\mathcal{S}}^{\mathcal{M}}({\mathcal{P} })={\mathcal{M}}_n({\mathcal{P} })$.
\end{lemma}
\betaegin{proof}
Notice that using the ${\mathcal{S}}$-constructions, we can rearrange ${\mathcal{M}}|(\eta^{+\omega})^{{\mathcal{M}}}$ as ${\mathcal{P} }[{\mathcal{M}}|\eta]$ (see \rprop{s-constructions prop}).
Hence, $\delta^{\mathcal{P} }=(\eta^+)^{\mathcal{M}}$.
But then ${\mathcal{S}}^{{\mathcal{M}}}({\mathcal{P} })[{\mathcal{M}}|\eta]={\mathcal{M}}$. This means that ${\mathcal{S}}^{{\mathcal{M}}}({\mathcal{P} })$ is the hull of ordinals $<\delta^{\mathcal{P} }$ and the class of indiscernibles. But this is exactly what ${\mathcal{M}}_n({\mathcal{P} })$ is: it is the unique proper class mouse over ${\mathcal{P} }$ with $n$ Woodin cardinals which is the hull of a club class of indiscernibles.
\end{proof}
Let ${\mathcal{P} }\in \muathcal{G}_{\kappa}$ and $s=\lambdaangle \alpha_j : j< l\rangle$.
\betaegin{definition}
We then write ${\mathcal{M}}\vDash ``{\mathcal{P} }$ is $s$-iterable below $\kappa$" if whenever ${\vec{\mathcal{T}}}=\lambdaangle \T_j, {\mathcal{P} }_j : j<k\rangle\in {\mathcal{M}}|\kappaappa$ is a correctly guided finite stack on ${\mathcal{P} }$ with last model ${\mathcal{ Q}}$ such that ${\mathcal{ Q}}\in \muathcal{G}_{\kappa}$ and whenever $g\sigmaubseteq Coll(\omega, \card{{\mathcal{P} }\cup {\mathcal{ Q}}})$ is ${\mathcal{M}}$-generic, there is $\vec{b}=\lambdaangle b_j : j<k\rangle\in {\mathcal{M}}[G]$ such that for every $m$,
\betaegin{center}
$\pi_{{\vec{\mathcal{T}}}, \vec{b}}(T^{\mathcal{P} }_{s, m})=T^{\mathcal{ Q}}_{s, m}$
\end{center}
where $T^{\mathcal{P} }_{s, m}\sigmaubseteq [((\delta^{\mathcal{P} })^{+m})^{\mathcal{P} }]^{<\omega}{\hbox{\fiverm th}}etaimes \omega$ is defined by
\betaegin{center}
$(t, \phi)\in T^{\mathcal{P} }_{s, m} \mathrel{\leftrightarrow} \phi$ is ${\mathcal{S}}igma_1$ and ${\mathcal{S}}^{{\mathcal{M}}}({\mathcal{P} })\vDash \phi[t, s]$.
\end{center}
and $T^{\mathcal{ Q}}_{s, m}\sigmaubseteq [((\delta^{\mathcal{ Q}})^{+m})^{\mathcal{ Q}}]^{<\omega}{\hbox{\fiverm th}}etaimes \omega$ is defined by
\betaegin{center}
$(t, \phi)\in T^{\mathcal{ Q}}_{s, m} \mathrel{\leftrightarrow} \phi$ is ${\mathcal{S}}igma_1$ and ${\mathcal{S}}^{{\mathcal{M}}}({\mathcal{ Q}})\vDash \phi[t, s]$.
\end{center}
\end{definition}
Notice that in the light of \rlem{good points}, the definition just given indeed coincides with \rdef{s-iterability} for as long as we stay inside $\muathcal{G}_{\kappa}$. ${\mathcal{M}}\vDash ``{\mathcal{P} }$ is strongly $s$-iterable below $\kappa$" is defined similarly. Also, notice that even though the requirement that the sequence $\vec{b}$ exists in the generic extension cannot be dropped, the embedding $\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s}$ is in ${\mathcal{M}}$ as it is unique and hence, it is in all generic extensions.
We then let
\betaegin{center}
$\muathcal{I}_{\kappa}=\{({\mathcal{P} }, s) : {\mathcal{P} }\in \muathcal{G}_\kappa \wedge {\mathcal{M}}\vDash ``{\mathcal{P} }$ is strongly $s$-iterable below $\kappa"\}$.
\end{center}
and
\betaegin{center}
$\muathcal{F}_{\kappa}=\{ H^{\mathcal{P} }_s : ({\mathcal{P} }, s)\in \muathcal{I}_\kappa\}$.
\end{center}
Notice that the proof of \rlem{s-iterability lemma} can be used to show that for every $s$ there is ${\mathcal{P} }$ such that $({\mathcal{P} }, s)\in \muathcal{I}_\kappa$. More formally, we have the following:
\betaegin{lemma}\lambdaanglebel{internal s-iterability lemma} Suppose ${\mathcal{P} }\in \muathcal{G}_{\kappa}$ and $s$ is a finite sequence of ordinals. Then there is a normal correct iterate ${\mathcal{ Q}}$ of ${\mathcal{P} }$ such that $({\mathcal{ Q}}, s)\in \muathcal{I}_{\kappa}$
\end{lemma}
Clearly, $\muathcal{F}_{\kappa}\in {\mathcal{M}}$. We then define $\lambdaeq_{\kappa}$ on $\muathcal{I}_{\kappa}$ by: $({\mathcal{P} }, s)\lambdaeq_{\kappa} ({\mathcal{ Q}}, t)$ iff ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{P} }$ and $s\sigmaubseteq t$. It is not hard to see that $\lambdaeq_{\kappa}$ is directed.
\betaegin{lemma} $\lambdaeq_{\kappa}$ is directed
\end{lemma}
\betaegin{proof}
Fix $({\mathcal{P} }, s), ({\mathcal{ Q}}, t)\in \muathcal{I}_{\kappa}$. Then there is $({\mathcal R}, s\cup t)\in \muathcal{I}_{\kappa}$. Working in ${\mathcal{M}}$, simultaneously compare ${\mathcal{P} }, {\mathcal{ Q}}$ and ${\mathcal R}$ to get ${\mathcal{S}}^*\in {\mathcal{M}}|\kappaappa$. Let $\eta<\kappa$ be a strong cutpoint of ${\mathcal{M}}$ such that ${\mathcal{S}}^*\in {\mathcal{M}}|\eta$. Then iterate ${\mathcal{S}}^*$ to make ${\mathcal{M}}|\eta$-generic. This iteration produces ${\mathcal{S}}\in {\mathcal{M}}|\kappaappa$ such that $\delta^{\mathcal{S}}=(\eta^+)^{{\mathcal{M}}}$. It then follows that $({\mathcal{S}}, s\cup t)\in \muathcal{I}_{\kappa}$ and $({\mathcal{P} }, s), ({\mathcal{ Q}}, t) \lambdaeq_\kappa ({\mathcal{S}}, s\cup t)$.
\end{proof}
Let then ${\mathcal{M}}_{\infty, \kappa}$ be the direct limit of $(\muathcal{F}_{\kappa}, \lambdaeq_{\kappa})$ under the embeddings $\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s}$. We first claim that ${\mathcal{M}}_{\infty, \kappa}$ is well-founded.
\betaegin{lemma}\lambdaanglebel{internal wellfoundness} ${\mathcal{M}}_{\infty, \kappa}$ is well-founded.
\end{lemma}
\betaegin{proof} The proof is similar to the proof of \rlem{wellfoundness}. Let $\lambdaangle {\mathcal{P} }_\alpha: \alpha<\kappa\rangle\in {\mathcal{M}}$ be an enumeration of $\muathcal{G}_{\kappa}$. We construct a sequence $\lambdaangle {\mathcal{ Q}}^0_i, \T^0_i, {\mathcal{ Q}}^1_i, \T^1_i : i<\omega\rangle$ such that
\betaegin{enumerate}
\item ${\mathcal{ Q}}^0_0={\mathcal{W} }_n$ and $\T^l_i$ is a normal correctly guided tree on ${\mathcal{ Q}}^l_i$ for $l= 0, 1$,
\item ${\mathcal{ Q}}^1_{i}$ is the last model of $\T^0_i$ and ${\mathcal{ Q}}^0_{i+1}$ is the last model of $\T^1_i$,
\item for every $\alpha<\kappa$, there is $i<\omega$ such that ${\mathcal{ Q}}^0_i$ is a correct iterate of ${\mathcal{P} }_\alpha$,
\item ${\mathcal{ Q}}^0_i\in \muathcal{G}_{\kappa}$.
\end{enumerate}
To construct such a sequence, we first fix $\lambdaangle \eta_i : i<\omega\rangle$ such that $\sigmaup_{i<\omega}\eta_i=\kappa$. Suppose we have constructed $\lambdaangle {\mathcal{ Q}}^0_i, \T^0_i, {\mathcal{ Q}}^1_i, \T^1_i : i\lambdaeq k\rangle$. Let $\eta\in [\eta_i, \kappa)$ be a strong cutpoint of ${\mathcal{M}}$ such that $\lambdaangle {\mathcal{ Q}}^0_i, \T^0_i, {\mathcal{ Q}}^1_i, \T^1_i : i\lambdaeq k\rangle\in {\mathcal{M}}|\eta$. Thus, we actually have ${\mathcal{ Q}}^0_{k+1}$. Then let ${\mathcal{ Q}}^1_{k+1}$ be the result of simultaneously comparing all suitable ${\mathcal{P} }$'s such that ${\mathcal{P} }\in {\mathcal{M}}|\eta\cap \muathcal{G}_\kappa$. Notice that ${\mathcal{S}}$ is a normal correct iterate of every ${\mathcal{P} } \in {\mathcal{M}}|\eta\cap \muathcal{G}_\kappa$ including ${\mathcal{ Q}}^0_{k+1}$. Let then $\T^0_{k+1}$ be the normal correctly guided tree on ${\mathcal{ Q}}^0_{k+1}$ with last model ${\mathcal{ Q}}^1_{k+1}$. The problem is that ${\mathcal{ Q}}^1_{k+1}$ may not be in $\muathcal{G}_\kappa$. Let then $\nuu\in (\eta, \kappa)$ be a strong cutpoint of ${\mathcal{M}}$ such that ${\mathcal{ Q}}^1_{k+1}\in {\mathcal{M}}|\nuu$. Iterate ${\mathcal{ Q}}^1_{k+1}$ to make ${\mathcal{M}}|\nuu$ generic for the extender algebra. Let then $\T^1_{k+1}$ be the resulting tree on ${\mathcal{ Q}}^1_{k+1}$. Clearly ${\mathcal{ Q}}^1_{k+1}\in \muathcal{G}_\kappa$ and the resulting sequence $\lambdaangle {\mathcal{ Q}}^0_i, \T^0_i, {\mathcal{ Q}}^1_i , \T^1_i : i<\omega\rangle$ is as desired.
Let then $\sigmaigma_{j, k}=i_{{\mathcal{ Q}}^0_j, {\mathcal{ Q}}^0_k}$ and let ${\mathcal{ Q}}$ be the direct limit of $\lambdaangle {\mathcal{ Q}}^0_j, \sigmaigma_{j, k} : j<k<\omega\rangle$. Then the proof of \rlem{wellfoundness} can be used to show that in fact ${\mathcal{ Q}}={\mathcal{M}}_{\infty, \kappa}$.
\end{proof}
Next we show that $\delta^{{\mathcal{M}}_{\infty, \kappa}}=(\kappa^+)^{{\mathcal{M}}}$. For the purpose of keeping the notation nice, in this subsection we abuse the notation used in the previous subsection and whenever $({\mathcal{P} }, s)\in \muathcal{I}_\kappa$, we write $\pi_{{\mathcal{P} }, \infty, s}$ for the direct limit embedding. Thus, $\pi_{{\mathcal{P} }, \infty, s}$ is an embedding that acts on $H_s^{\mathcal{P} }$ and embeds it into the corresponding structure in ${\mathcal{M}}_{\infty, \kappa}$. For each $s\in Ord^{<\omega}$, let $\gamma_{\infty, s}=\sigmaup (\pi_{{\mathcal{P} }, \infty, s}"\gamma^{\mathcal{P} }_s)$ where $({\mathcal{P} }, s)\in \muathcal{I}_\kappa$. Clearly, $\gamma_{\infty, s}$ is independent of the choice of ${\mathcal{P} }$. Notice that $\delta^{{\mathcal{M}}_{\infty, \kappa}}=\sigmaup_{s\in Ord^{<\omega}} \gamma_{\infty, s}=\sigmaup_{m<\omega}\gamma_{\infty, s_m}$. Our proof uses an idea that originated in Hjorth's work.
\betaegin{lemma}
$\delta^{{\mathcal{M}}_{\infty, \kappa}}=(\kappa^+)^{{\mathcal{M}}}$.
\end{lemma}
\betaegin{proof}
First notice that for every $\alpha<\delta^{{\mathcal{M}}_\infty, \kappa}$ there is in ${\mathcal{M}}$ a surjective map $f:\kappaappa\rightarrow \alpha$. To see this, first fix $s$ such that $\alpha<\gamma_{\infty, s}$ and let $\lambdaangle ({\mathcal{P} }_\beta, \xi_\beta): \beta<\kappa\rangle$ be an enumeration of the set $\{ ({\mathcal{P} }, \xi) : ({\mathcal{P} }, s)\in \muathcal{I}_\kappa \wedge \xi<\gamma_s^{\mathcal{P} }\}$. Then let $f(\beta)=\pi_{{\mathcal{P} }_\beta, \infty, s}(\xi_\beta)$. Clearly $\alpha\sigmaubseteq ran(f)$ and $f$ is onto. This observation shows that $\delta^{{\mathcal{M}}_{\infty, \kappa}}\lambdaeq (\kappa^+)^{{\mathcal{M}}}$.
We therefore need to show that $\delta^{{\mathcal{M}}_{\infty, \kappa}}\nuot < (\kappa^+)^{{\mathcal{M}}}$. Suppose then $\delta^{{\mathcal{M}}_{\infty, \kappa}} < (\kappa^+)^{{\mathcal{M}}}$. We can then let $\lambdaeq^*\in {\mathcal{M}}$ be a well-ordering of $\kappa$ of length $\delta^{{\mathcal{M}}_{\infty, \kappa}}$. Without loss of generality we assume $\kappa$ is least such that $\delta^{{\mathcal{M}}_{\infty, \kappa}} <(\kappa^+)^{{\mathcal{M}}}$. It then follows that there is a formula $\phi$, a sequence $t\in [\kappa]^{<\omega}$ and an integer $m$ such that
\betaegin{center}
$\alpha\lambdaeq^*\beta \mathrel{\leftrightarrow} {\mathcal{M}}\vDash \phi[t, s_m, \alpha, \beta]$.
\end{center}
Now, fix $({\mathcal{P} }, s_m)\in \muathcal{I}_\kappa$ such that $t\sigmaubseteq \lambda$ where $\lambda$ is the least measurable cardinal of ${\mathcal{P} }$. Let ${\mathcal{N}}={\mathcal{M}}_n({\mathcal{P} })={\mathcal{S}}^{{\mathcal{M}}}({\mathcal{P} })$. We have that ${\mathcal{M}}|\eta_{{\mathcal{P} }}$ is generic over ${\mathcal{P} }$ for the extender algebra of $\delta^{\mathcal{P} }$. This means that ${\mathcal{N}}[{\mathcal{M}}|\eta_{\mathcal{P} }]$ can be reorganized as an $x$-mouse and in fact, ${\mathcal{N}}[{\mathcal{M}}|\eta_{\mathcal{P} }]={\mathcal{M}}$. This then means that there are conditions $p$ which force that ${\mathcal{N}}[G]$ can be reorganized via ${\mathcal{S}}$-constructions as a mouse over a real and such that in ${\mathcal{N}}[G]$, $\delta^{{\mathcal{M}}_{\infty, \kappa}}< (\kappa^+)^{{\mathcal{N}}[G]}$. Moreover, among those conditions there are also conditions that force that $\phi$ defines a well-ordering of $\kappa$ as above over ${\mathcal{N}}[G]$. Let then $D$ be the set of conditions $p$ of the extender algebra at $\delta^{\mathcal{P} }$ such that $p$ forces that
\betaegin{enumerate}
\item ${\mathcal{N}}[G]$ can be reorganized as a premouse over a real,
\item ${\mathcal{N}}[G]\vDash ``\delta^{{\mathcal{M}}_{\infty, \kappa}} < (\kappa^+)^{{\mathcal{N}}[G]}"$,
\item $\phi$ defines a well-ordering of $\kappa$ of length $(\delta^{{\mathcal{M}}_{\infty, \kappa}})^{{\mathcal{N}}[G]}$.
\end{enumerate}
We let ${\hbox{\fiverm th}}etaau$ be the name of the prewellordering given by $\phi$. Consider now the set $B$ of pairs $(p, \alpha)$ such that $p\in D$, $\alpha<\lambda$ and for some $\xi$, in $p$ forces that the rank of $\alpha$ . Notice that whenever $(p, \alpha)\in B$ and $G$ is ${\mathcal{P} }$-generic such that $p\in G$, $\alpha$ has a rank in the well-ordering given by $\phi$ over ${\mathcal{N}}[G]$. We can then for each $\alpha<\lambda$ choose a maximal antichain of conditions $p$ such that $(p, \alpha)\in B$ and for some $\xi$, $p$ forces that $\alpha$ has rank $\xi$ in the well-ordering given by $\phi$. Let $\muathcal{A}_\alpha$ be such an antichain and let $\muathcal{A}=\{ (p, \alpha) : p\in \muathcal{A}_\alpha\}$. Notice that without loss of generality we can assume that $\muathcal{A}\in H_{m+1}^{\mathcal{P} }$. We then let $\muathcal{A}^{\mathcal{P} }=\muathcal{A}$.
For $(p, \alpha)\in \muathcal{A}$ let $\xi_{p, \alpha}$ be the rank of $\alpha$ as forced by $p$. Define $\lambdaeq^{\mathcal{P} }$ on $\muathcal{A}$ by $(p, \alpha)\lambdaeq^{\mathcal{P} } (q, \beta)$ iff $\xi_{p, \alpha}\lambdaeq \xi_{q, \beta}$. Notice that $\card{\lambdaeq^{\mathcal{P} }}$ is independent of the choice of $\muathcal{A}_\alpha$'s and $\card{\lambdaeq^{\mathcal{P} }}< \gamma_{m+1}^{\mathcal{P} }$.
Define now a relation $R$ on the set $\{ (P, \xi) : {\mathcal{P} }\in \muathcal{G}_\kappa \wedge \xi<\gamma_{m+1}^{\mathcal{P} }\}$ given by
\betaegin{center}
$R(({\mathcal{P} }, \xi), ({\mathcal{ Q}}, \nuu))$ if whenever ${\mathcal R}$ is such that $({\mathcal{P} }, s_{m+1})\lambdaeq_\kappa ({\mathcal R}, s_{m+1})$ and $({\mathcal{ Q}}, s_{m+1})\lambdaeq_\kappa ({\mathcal R}, s_{m+1})$ then $i_{{\mathcal{P} }, {\mathcal R}, s_{m+1}}(\xi)\lambdaeq i_{{\mathcal{ Q}}, {\mathcal R}, s_{m+1}}(\nuu)$.
\end{center}
Clearly $R$ is well-founded and $\card{R}=\gamma_{\infty, s_{m+1}}$.
Fix now an $\alpha<\kappa$. We say that $({\mathcal{P} }, p)$ is a stable code for $\alpha$ if
\betaegin{enumerate}
\item $({\mathcal{P} }, s_{m+1})\in \muathcal{I}_\kappa$,
\item $(p, \alpha)\in \muathcal{A}^{\mathcal{P} }$, $\xi^{\mathcal{P} }_{p, \alpha}=\card{\alpha}_{\lambdaeq^*}$, and whenever ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{P} }$ such that ${\mathcal{ Q}}\in \muathcal{G}_\kappa$,
\betaegin{center}
$\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s_{m+1}}(\card{\alpha}_{\lambdaeq^*})=\card{\alpha}_{\lambdaeq^*}$,
\end{center}
\item if $G\sigmaubseteq \muathbb{B}^{\mathcal{P} }$ is a generic object such that $x_G={\mathcal{M}}|\eta_{\mathcal{P} }$ then $p\in G$.
\end{enumerate}
Notice that if $({\mathcal{P} }, p)$ is a stable code for $\alpha$ then $\xi_{p, \alpha}^{\mathcal{P} }=\card{\alpha}_{\lambdaeq^*}$. This is because of condition 3, i.e., if $G\sigmaubseteq \muathbb{B}^{\mathcal{P} }$ is the generic so that $x_G={\mathcal{M}}|\eta_{\mathcal{P} }$ then ${\mathcal{S}}(x_G)^{{\mathcal{M}}_n({\mathcal{P} })[G]}={\mathcal{M}}$, $p\in G$ and $(\card{\alpha}_{\lambdaeq^*})^{{\mathcal{S}}(x_G)^{{\mathcal{M}}_n({\mathcal{P} })[G]}}=\card{\alpha}_{\lambdaeq^*}$.
We claim that for every $\alpha$ there is a stable code for $\alpha$. Let $\xi=\card{\alpha}_{\lambdaeq^*}$. To see this, suppose not. Let then ${\mathcal{P} }$ be such that $({\mathcal{P} }, s_{m+1})\in \muathcal{I}_\kappa$, $\alpha<\lambda^{\mathcal{P} }$ and ${\mathcal{P} }$ is a correct iterate of ${\mathcal{W} }_n$. Then we can find $p\in {\mathcal{P} }$ such that $(p, \alpha)\in \muathcal{A}^{\mathcal{P} }$ and $({\mathcal{P} }, p)$ satisfies 1 and 3 above. If it satisfies 2 then we are done, and therefore, we assume that $({\mathcal{P} }, p)$ doesn't satisfy 2. Let then $({\mathcal{P} }_0, p_0)=({\mathcal{P} }, p)$ and let ${\mathcal{P} }_1$ witness the failure of 2. Thus, we have that $\xi=\xi^{\mathcal{P} }_{p, \alpha}$ and $i_{{\mathcal{P} }_0, {\mathcal{P} }_1, s_{m+1}}(\xi)>\xi$. But notice that there is $p_1\in {\mathcal{P} }_1$ such that $(p_1, \alpha)\in \muathcal{A}^{{\mathcal{P} }_1}$ and $\xi^{{\mathcal{P} }_1}_{p_1, \alpha}=\xi$. We then must have that $({\mathcal{P} }_1, p_1)$ doesn't satisfy condition 2 above and therefore, we get $({\mathcal{P} }_2, p_2)$ such that ${\mathcal{P} }_2\in \muathcal{G}_\kappa$ is a correct iterate of ${\mathcal{P} }_1$, $\pi_{{\mathcal{P} }_1, {\mathcal{P} }_2, s_{m+1}}(\xi)>\xi$ and $\xi^{{\mathcal{P} }_2}_{p_2, \alpha}=\xi$. In this fashion, by successively applying the failure of 2, we get a sequence $\lambdaangle {\mathcal{P} }_i: i<\omega\rangle$ such that for every $i$, ${\mathcal{P} }_{i}$ is a correct iterate of ${\mathcal{P} }_{i-1}$, for each $i$, ${\mathcal{P} }_{i}$ is a correct iterate of ${\mathcal{W} }_n$ and for $i\geq 0$,
\betaegin{center}
$\pi_{{\mathcal{P} }_i, {\mathcal{P} }_{i+1}, s_{m+1}}(\xi)>\xi$.
\end{center}
Let then ${\mathcal{ Q}}$ be the direct limit of $\lambdaangle {\mathcal{P} }_i, i_{{\mathcal{P} }_i, {\mathcal{P} }_j} : i<j<\omega\rangle$ and let $\sigmaigma_i:{\mathcal{P} }_i \rightarrow {\mathcal{ Q}}$ be the iteration embedding. Then because $\pi_{{\mathcal{P} }_i, {\mathcal{P} }_{i+1}, s_{m+1}}$'s agree with $i_{{\mathcal{P} }_i, {\mathcal{P} }_j}$, letting $\nuu_i=\sigmaigma_i(\xi)$ we get that $\lambdaangle \nuu_i : i<\omega\rangle$ is a decreasing sequence of ordinals, contradiction! Thus, there is indeed a stable code for $\alpha$.
Now, for each $\alpha<\kappa$ choose $({\mathcal{P} }_\alpha, p_\alpha)$ such that $({\mathcal{P} }_\alpha, p_\alpha)$ is a stable code for $\alpha$. Let $\nuu_\alpha=\card{(p, \alpha)}_{\lambdaeq^{{\mathcal{P} }_\alpha}}<\gamma^{{\mathcal{P} }_a}_{m+1}$. Then we claim that for any $\alpha, \beta<\kappa$, if $\alpha\lambdaeq^* \beta$ then $R(({\mathcal{P} }_\alpha, \nuu_\alpha), ({\mathcal{P} }_\beta, \nuu_\beta))$. Indeed, let ${\mathcal{ Q}}\in \muathcal{G}_\kappa$ be a common correct iterate of ${\mathcal{P} }_\alpha$ and ${\mathcal{P} }_\beta$. Let $\nuu=i_{{\mathcal{P} }_\alpha, {\mathcal{ Q}}, s_{m+1}}(\nuu_\alpha)$ and let $\zeta=i_{{\mathcal{P} }_\beta, {\mathcal{ Q}}, s_{m+1}}(\nuu_\beta)$. Let $\xi_\alpha=\card{\alpha}_{\lambdaeq^*}$ and $\xi_\beta=\card{\beta}_{\lambdaeq^*}$. We have that $i_{{\mathcal{P} }_\alpha, {\mathcal{ Q}}, s_{m+1}}(\alpha)=\alpha$, $i_{{\mathcal{P} }_\beta, {\mathcal{ Q}}, s_{m+1}}(\beta)=\beta$, $i_{{\mathcal{P} }_\alpha, {\mathcal{ Q}}, s_{m+1}}(\xi_\alpha)=\xi_\alpha$ and $i_{{\mathcal{P} }_\beta, {\mathcal{ Q}}, s_{m+1}}(\xi_\beta)=\xi_\beta$. Because $\xi_\alpha\lambdaeq \xi_\beta$, we have that
\betaegin{center}
$\card{(\pi_{{\mathcal{P} }_\alpha, {\mathcal{ Q}}, s_{m+1}}(p_\alpha), \alpha)}_{\lambdaeq^{\mathcal{ Q}}}\lambdaeq \card{(\pi_{{\mathcal{P} }_\beta, {\mathcal{ Q}}, s_{m+1}}(p_\beta), \beta)}_{\lambdaeq^{\mathcal{ Q}}}$.
\end{center}
Therefore, $\nuu\lambdaeq \xi$.
This shows that $\alpha \rightarrow ({\mathcal{P} }_\alpha, p_\alpha)$ is an order preserving map of $\lambdaeq^*$ into $R$ and hence,
\betaegin{center}
$\card{\lambdaeq^*}\lambdaeq \card{R}=\gamma_{\infty, s_{m+1}}<\delta^{{\mathcal{M}}_\infty, \kappa}$.
\end{center}
\end{proof}
We finish by remarking that the directed limit of ${\mathcal{M}}$ at $\kappa$ is invariant under small forcing. This means that if $\muathbb{P}\in {\mathcal{M}}|\kappa$ and $g\sigmaubseteq \muathbb{P}$ is ${\mathcal{M}}$-generic then one can, working inside ${\mathcal{M}}[g]$, construct a directed system, much like we did above, and show that the direct limit of this system is the same as ${\mathcal{M}}_{\infty, \kappa}$. This mainly follows from Woodin's generic comparison process. The idea has been explained in various places and because of this we will omit it. The idea is as follows. It is enough to show it for $g$'s that are generic for $Coll(\omega, \eta^+)$ where $\eta<\kappa$ is a strong cutpoint. One then fixes a strong cutpoint $\nuu<\kappa$ and performs a simultaneous comparison of all suitable pairs in ${\mathcal{M}}[g]|\nuu$. It is then shown that the tree on ${\mathcal{W} }_n$ is in fact in ${\mathcal{M}}$. This follows from the homogeneity of the forcing. Let then ${\mathcal{P} }$ be the last of this comparison. We then get that ${\mathcal{P} }\in {\mathcal{M}}$ and it dominates all the suitable mice in ${\mathcal{M}}[g]|\nuu$. This then easily implies that the directed system of ${\mathcal{M}}[g]$ is dominated by the one in ${\mathcal{M}}$, and hence, the direct limit of both systems must be the same. For more on the details of the generic comparison we refer the reader to \cite{CMI}, \cite{MSC} (Section 3.9) and \cite{PFA}.
\sigmaubsection{The full directed system.}
In this subsection, we will establish some lemmas that connect the directed system associated with ${\mathcal{M}}_\omega$ with the directed system associated with ${\mathcal{M}}_{2k+1}$. In particular, we will prove \rthm{woodins thing}, originally due to Woodin, which has been widely known yet has remained unpublished for many years. We do not know if the proof of \rthm{woodins thing} presented here is the same or similar to Woodin's original proof. Woodin's result gives a characterization of $\kappa^1_{2k+1}$ in terms of cardinals of ${\rm{HOD}}$. We remind our readers that we assume that ${\mathcal{M}}_\omega^\#$ exists. This assumption is made for aesthetic reasons. Readers familiar with the general theory can reduce the hypothesis to just $AD^{L(\muathbb{R})}$.
In what follows, we will use superscript $f$ to indicate that we are dealing with the full directed system, i.e., with the system associated with ${\mathcal{M}}_\omega^\#$. Notice that because of \rthm{steel's thing}, for $\eta<(\delta^2_1)^{L(\muathbb{R})}$, the notation ${\rm{HOD}}^{L(\muathbb{R})}|\eta$ makes sense.
Besides the proof of \rthm{woodins thing}, we will also prove \rlem{full limit is bounded} which we will use later on. When we talk about ${\rm{HOD}}$, we mean ${\rm{HOD}}^{L(\muathbb{R})}$. From now on until the end of the next subsection we fix $k\in \omega$. We will often omit superscripts or subscripts that usually would involve $k$ in them. By a standard Skolem hull argument done in ${\rm{HOD}}_z$, It follows from \rthm{steel's thing}, that there are many ${\rm{HOD}}$-cardinals $\nuu$ such that ${\mathcal{M}}_{2k}({\rm{HOD}}_z|\nuu)\vDash ``\nuu$ is Woodin". For each real $z$ let $\nuu_z$ be the least such $\nuu$.
Recall $\muathcal{F}$ of \rsec{dimt}. Next want to isolate a subset of $\muathcal{F}$ such that the direct limit of this subset will converge to ${\mathcal{M}}_{2k}({\rm{HOD}}|\nuu_0)|(\nuu_0^{+\omega})^{{\mathcal{M}}_{2k}({\rm{HOD}}|\nuu_0)}$. For each real $z$, let $\eta_z$ be the least cardinal of ${\mathcal{M}}_\omega(z)$ such that ${\mathcal{M}}_{2k}({\mathcal{M}}_\omega|\eta_z)\vDash ``\eta_z$ is Woodin". Then let ${\mathcal{W} }_z^f={\mathcal{M}}_{2k}({\mathcal{M}}_\omega(z)|\eta_z)|(\eta_z^{+\omega})^{{\mathcal{M}}_{2k}({\mathcal{M}}_\omega(z)|\eta_z)}$. We let ${\mathcal{S}}igma^f_z$ be the fragment of the $(\omega_1, \omega_1)$-strategy of ${\mathcal{M}}_\omega(z)$ that acts on stacks which are based on ${\mathcal{W} }_z^f$. Let
\betaegin{center}
$\muathcal{F}^{+,f}_z=\{ {\mathcal{P} }: {\mathcal{P} } \in I({\mathcal{W} }^f_z, {\mathcal{S}}igma^f_z)$ as witnessed by a finite stack $\}$.
\end{center}
Whenever ${\mathcal{P} }, {\mathcal{ Q}}\in \muathcal{F}^{+,f}_z$ and ${\mathcal{ Q}}\in I({\mathcal{P} }, ({\mathcal{S}}igma^f_z)_{\mathcal{P} })$, we will let $i_{{\mathcal{P} }, {\mathcal{ Q}}}^f:{\mathcal{P} }\rightarrow {\mathcal{ Q}}$ be the iteration embedding. Notice that in this notation we are omitting $z$ from subscripts and superscripts as it is usually clear what $z$ is. We hope this doesn't cause a confusion.
We can then define $\lambdaeq^f_z$ on $\muathcal{F}^{+, f}_z$ by ${\mathcal{P} }\lambdaeq^f_z{\mathcal{ Q}}$ iff ${\mathcal{ Q}}\in I({\mathcal{P} }, ({\mathcal{S}}igma^f_z)_{\mathcal{P} })$. We let ${\mathcal{M}}_{\infty, z}^{+, f}$ be the direct limit of $(\muathcal{F}^{+,f}_z, \lambdaeq^{+, f}_z)$ under the iteration maps $i_{{\mathcal{P} }, {\mathcal{ Q}}}^f$. We also let $i_{{\mathcal{P} }, \infty}^f:{\mathcal{P} } \rightarrow{\mathcal{M}}^{+, f}_{\infty, z}$ be the iteration map. Then clearly $\nuu_z=\delta^{{\mathcal{M}}_{\infty, z}^{+,f}}$.
Next we show that just like ${\mathcal{W} }_z$, $\muathcal{F}_z^{+, f}$ and $\lambdaeq^{+, f}_z$ can be internalized to ${\mathcal{M}}_{2k}(x)$ where $x$ codes ${\mathcal{W} }^f_z$. We first make the following definition.
\betaegin{definition}\lambdaanglebel{miserable drop} Suppose ${\mathcal{P} }$ is suitable and $\T$ is a normal tree on ${\mathcal{P} }$. We say $\T$ has a \emph{miserable drop} if there is $\alpha<lh(\T)$ and ordinal $\eta$ such that if
\betaegin{center}
${\mathcal{M}}=\cup \{{\mathcal{N}}: {\mathcal{M}}_\alpha^\T|\eta {\hbox{\fiverm th}}etarianglelefteq {\mathcal{N}}{\hbox{\fiverm th}}etarianglelefteq{\mathcal{M}}_\alpha^\T$ and $\eta$ is a strong cutpoint of ${\mathcal{N}}\}$
\end{center}
then the rest of $\T$ is a normal tree on ${\mathcal{M}}$ above $\eta$.
\end{definition}
\betaegin{lemma}\lambdaanglebel{no misearable drops} Suppose ${\mathcal{ Q}}, {\mathcal R}\in \muathcal{F}^{+,f}_z$. Let $\T$ on ${\mathcal{ Q}}$ and ${\mathcal{U}}$ on ${\mathcal R}$ be the trees constructed via the comparison process in which $II$ uses $({\mathcal{S}}igma^f_z)_{\mathcal{ Q}}$ on the ${\mathcal{ Q}}$-side and $II$ uses $({\mathcal{S}}igma^f_z)_{\mathcal R}$ on the ${\mathcal R}$ side. Then $\T$ and ${\mathcal{U}}$ have no miserable drops.
\end{lemma}
\betaegin{proof}\lambdaanglebel{no miserable drops}
Suppose towards a contradiction, $\T$ has a miserable drop. Let ${\mathcal{ Q}}^*$ be the last model of $\T$ and ${\mathcal R}^*$ be the last model of ${\mathcal{U}}$. Then $i^\T$ cannot exist and therefore, it follows from the comparison lemma that ${\mathcal R}^*\vartriangleleft {\mathcal{ Q}}^*$. Let $\alpha<lh(\T)$ be the largest such that there is a miserable drop in ${\mathcal{M}}_\alpha^\T$. Let $\eta$ be such that if
\betaegin{center}
${\mathcal{M}}=\cup \{{\mathcal{N}}: {\mathcal{M}}_\alpha^\T|\eta {\hbox{\fiverm th}}etarianglelefteq {\mathcal{N}}{\hbox{\fiverm th}}etarianglelefteq{\mathcal{M}}_\alpha^\T$ and ${\mathcal{N}}$ is a premouse over ${\mathcal{M}}_\alpha^\T|\eta\}$
\end{center}
then the rest of $\T$ is a tree on ${\mathcal{M}}$ above $\eta$. It then follows that $\eta\in {\mathcal R}^*$. Notice that $\eta$ is a strong cutpoint in ${\mathcal R}^*$ and by fullness of ${\mathcal R}^*$, ${\mathcal{M}}{\hbox{\fiverm th}}etarianglelefteq {\mathcal R}^*$. Because ${\mathcal{ Q}}^*$ is an iterate of ${\mathcal{M}}$ above $\eta$, we cannot have that ${\mathcal{M}}{\hbox{\fiverm th}}etarianglelefteq {\mathcal{ Q}}^*$, contradiction!
\end{proof}
Our next lemma shows that if ${\mathcal{P} }, {\mathcal{ Q}}\in \muathcal{F}_z^{+,f}$, then their comparison involves ${\mathcal{ Q}}$-structures that are below the $S_{2k}$-operator.
\betaegin{lemma}\lambdaanglebel{bound on q-structures} Suppose ${\mathcal{P} }, {\mathcal{ Q}}\in \muathcal{F}_z^{+,f}$. Let ${\mathcal R}$ be the result of their comparison and let $\T$ and ${\mathcal{U}}$ be the trees on ${\mathcal{P} }$ and ${\mathcal{ Q}}$ respectively that come from the comparison process. Then for every limit $\alpha$ such that $\alpha+1\lambdaeq lh(\T)$, if $b$ is the branch of $\T\restriction \alpha$ chosen in $\T$ and ${\mathcal{ Q}}(b, \T\restriction\alpha)$-exists then ${\mathcal{ Q}}(b, \T\restriction \alpha){\hbox{\fiverm th}}etarianglelefteq {\mathcal{M}}_{2k}({\mathcal{M}}(\T\restriction \alpha))$.
\end{lemma}
\betaegin{proof}
The reason for this is that the only way to produce normal trees with ${\mathcal{ Q}}$-structures that are beyond $S_{2k}$-operator is to do a miserable drop. To see that our claim is true, assume not, and fix $\alpha$ such that $\alpha+1\lambdaeq lh(\T)$ and if $b$ is the branch of $\T\restriction \alpha$ chosen in $\T$ such that ${\mathcal{ Q}}(b, \T\restriction\alpha)$-exists then ${\mathcal{ Q}}(b, \T\restriction \alpha)\nuot {\hbox{\fiverm th}}etarianglelefteq {\mathcal{M}}_{2k}({\mathcal{M}}(\T\restriction \alpha))$. It then follows that ${\mathcal{M}}_{2k}({\mathcal{M}}(\T\restriction \alpha))\vartriangleleft {\mathcal{ Q}}(b, \T\restriction \alpha)$ and therefore, ${\mathcal{M}}_{2k}({\mathcal{M}}(\T\restriction \alpha))\vDash ``\delta(\T\restriction \alpha)$ is Woodin". Notice that it follows from the comparison lemma and the minimality condition on ${\mathcal{P} }$ that $i^{\T\restriction \alpha}_b$ exists (i.e., there are no drops along $b$). This means that $\alpha+1< lh(\T)$. But then $lh(E^\T_\alpha)>\delta(\T\restriction\alpha)$. Because ${\mathcal R}$ agrees with ${\mathcal{M}}^\T_\alpha$ up to $lh(E^\T_\alpha)$ and ${\mathcal R}\vDash ``lh(E^\T_\alpha)$ is a cardinal", $\delta(\T\restriction \alpha)$ is a cardinal in ${\mathcal R}$ and moreover, ${\mathcal{M}}_{2k}({\mathcal R}|\delta(\T\restriction \alpha))\vDash ``\delta(\T\restriction \alpha)$ is Woodin". This means that $\delta(\T\restriction \alpha)=\delta^{\mathcal R}$.
Notice now that we must have that ${\rm cp }(E^\T_\alpha)\lambdaeq \delta(\T\restriction \alpha)$. To see this assume not. We then have that ${\rm cp }(E^\T_\alpha)>\delta(\T\restriction \alpha)$. But because $\delta(\T\restriction \alpha)=\delta^{\mathcal R}$, we have that there must be a miserable drop in $\T$ at stage $\alpha+1$ (as we must start iterating above $\delta(\T\restriction \alpha)$).
It now follows that ${\mathcal{M}}_\alpha^\T\vDash ``{\rm cp }(E^\T_\alpha)$ is a limit of cardinals $\eta$ such that ${\mathcal{M}}_{2k}({\mathcal{M}}_\alpha^\T|\eta)\vDash ``\eta$ is Woodin". Because of the agreement between ${\mathcal{M}}_\alpha^\T$ and ${\mathcal R}$, we get that there is an ${\mathcal R}$-cardinal $\eta<\delta^{\mathcal R}$ such that ${\mathcal{M}}_{2k}({\mathcal R}|\eta)\vDash ``\eta$ is Woodin". This is a contradiction.
\end{proof}
Using miserable drops, we can now define $s$-iterability for ${\mathcal{P} }\in \muathcal{F}^{+,f}_z$. First, given an iteration tree $\T$ on ${\mathcal{P} }$, we say $\T$ is {\hbox{\fiverm th}}etaextit{correctly guided} if $\T$ doesn't have miserable drops and for every limit $\alpha<lh(\T)$, if $b$ is the branch of $\T\restriction \alpha$ chosen by $\T$ and ${\mathcal{ Q}}(b, \T\restriction \alpha)$ exists then ${\mathcal{ Q}}(b, \T\restriction \alpha){\hbox{\fiverm th}}etarianglelefteq {\mathcal{M}}_{2k}({\mathcal{M}}(\T\restriction \alpha))$. $\T$ is {\hbox{\fiverm th}}etaextit{short} if there is a well-founded branch $b$ such that $\T^\frown \{{\mathcal{M}}^\T_b\}$ is correctly guided. $\T$ is {\hbox{\fiverm th}}etaextit{maximal} if $\T$ is not short. One can then proceed and define $s$-iterability as in \rdef{s-iterability}: the only difference is that we require that the trees in the stack be without miserable drops. We define $T_{s, m}^{\mathcal{P} }$, $\gamma_s^{\mathcal{P} }$ and $H_s^{\mathcal{P} }$ as before and we omit $z$ from superscripts and subscripts as that is really part of ${\mathcal{P} }$. Notice that
\betaegin{center}
$\sigmaup_{m\in \omega}\gamma_{s_m}^{\mathcal{P} }=\delta^{\mathcal{P} }$.
\end{center}
For ${\mathcal{P} }, {\mathcal{ Q}}\in \muathcal{F}^{+, f}_z$, we say ${\mathcal{ Q}}$ is a {\hbox{\fiverm th}}etaextit{correct iterate} of ${\mathcal{P} }$ if there is a correctly guided finite stack ${\vec{\mathcal{T}}}$ on ${\mathcal{P} }$ with last model ${\mathcal{ Q}}$.
Suppose now ${\mathcal{P} }$ and ${\mathcal{ Q}}$ are two correct iterates of ${\mathcal{W} }^f_z$. Then using the proof of \rlem{no miserable drops}, we can show that the comparison of ${\mathcal{P} }$ and ${\mathcal{ Q}}$ can be entirely, except possibly the very last step, be carried out in ${\mathcal{M}}_{2k}({\mathcal{P} }, {\mathcal{ Q}})$. That is, one can show that there are correctly guided trees $\T, {\mathcal{U}}\in {\mathcal{M}}_{2k}({\mathcal{P} }, {\mathcal{ Q}})$ such that $\T$ is on ${\mathcal{P} }$, ${\mathcal{U}}$ is on ${\mathcal{ Q}}$ and $\T$ and ${\mathcal{U}}$ have a common last model.
Using this observation and the results of \rsec{internalizing the directed system} one can internalize the directed system associated to ${\mathcal{W} }_z^f$. More precisely, suppose $x$ is a real coding ${\mathcal{W} }_z^f$ and $\kappa$ is an inaccessible strong cutpoint of ${\mathcal{M}}_{2k}(x)$ such that $\kappa$ is below the first Woodin of ${\mathcal{M}}_{2k}(x)$ and $\kappa$ is a limit of strong cutpoints, then one can form the direct limit of all correct iterates of ${\mathcal{W} }^f_z$ that are in ${\mathcal{M}}_{2k}(x)$. Notice that in \rsec{internalizing the directed system}, our internalization process didn't use ${\mathcal{W} }_z$ as a parameter in the definition. Here too we could make do without ${\mathcal{W} }_z^f$ but we don't it. Before we move on, let us then lay down the notation that is slowly evolving and becoming rather cumbersome.
\betaegin{enumerate}
\item We let $\muathcal{F}^{+,f}_z=\{ {\mathcal{P} }: {\mathcal{P} }$ is a correct iterate of ${\mathcal{W} }^f_z\}$, $\muathcal{J}^{+,f}_z=\{ ({\mathcal{P} }, \alpha) : {\mathcal{P} }\in \muathcal{F}^{+,f}_z \wedge \alpha<\delta^{\mathcal{P} }\}$, and ${\mathcal R}_z^{+,f}$ is the prewellordering defined on $\muathcal{J}^{+,f}_z$ by:
\betaegin{center}
$({\mathcal{P} }, \alpha)R^{+,f}_z ({\mathcal{ Q}}, \beta)$ iff ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{P} }$ and $i^f_{{\mathcal{P} }, {\mathcal{ Q}}}(\alpha)\lambdaeq \beta$.
\end{center}
We let $\lambdaeq^{+, f}_z$ be the prewellordering of $\muathcal{F}^{+, f}_z$ given by:
\betaegin{center}
${\mathcal{P} }\lambdaeq^{+,f}_z {\mathcal{ Q}}$ iff ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{P} }$.
\end{center}
\item We let $\muathcal{I}^f_z=\{ ({\mathcal{P} }, s): {\mathcal{P} }\in \muathcal{F}^f_z \wedge s\in Ord^{<\omega}\wedge {\mathcal{P} }$ is strongly $s$-iterable $\}$, $\muathcal{F}^{f}_z=\{ H_s^{\mathcal{P} } : ({\mathcal{P} }, s)\in \muathcal{I}^f_z\}$ and $\muathcal{J}^{f}_{z, s}=\{ ({\mathcal{P} }, \alpha) : {\mathcal{P} }\in \muathcal{F}^{+,f}_z \wedge \alpha<\gamma^{\mathcal{P} }_s\}$. We let $R^{f}_z$ be the prewellordering of $\muathcal{J}^{f}_z$ given by:
\betaegin{center}
$({\mathcal{P} }, \alpha)R^{f}_{z, s}({\mathcal{ Q}}, \beta)$ iff ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{P} }$ and $\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s}(\alpha)\lambdaeq \beta$.
\end{center}
We let $\lambdaeq^f_z$ be the prewellordering of $\muathcal{I}^f_z$ given by:
\betaegin{center}
$({\mathcal{P} }, s)\lambdaeq^f_z({\mathcal{ Q}}, t)$ iff ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{P} }$ and $s\sigmaubseteq t$. We have that $\lambdaeq^f_z$ is directed.
\end{center}
\item Given ${\mathcal{P} }$ and $s\in Ord^{<\omega}$ such that $({\mathcal{P} },s)\in \muathcal{I}^f_z$, if ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{P} }$ then we let $\pi^f_{{\mathcal{P} }, {\mathcal{ Q}}, s}:H_s^{\mathcal{P} }\rightarrow H_s^{\mathcal{ Q}}$ be the $s$-iterability embedding. $z$ will be clear from the context and hence, we omit it. Recall that we let $\pi_{{\mathcal{P} }, {\mathcal{ Q}}, s}:H_s^{\mathcal{P} }\rightarrow H_s^{\mathcal{ Q}}$ be the $s$-iterability embedding where ${\mathcal{P} }, {\mathcal{ Q}}$ are suitable ${\mathcal{P} }$ is $s$-iterable and ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{P} }$.
\item We let ${\mathcal{M}}_{\infty, z}^f$ be the direct limit of $(\muathcal{F}^f_z, \lambdaeq^f_z)$ under the maps $\pi^f_{{\mathcal{P} }, {\mathcal{ Q}}, s}$ and ${\mathcal{M}}^{+, f}_{\infty, z}$ be the direct limit of $(\muathcal{F}^{+, f}_z, \lambdaeq^{+, f}_z)$ under the iteration maps $i^f_{{\mathcal{P} }, {\mathcal{ Q}}}$. By the proof of \rlem{wellfoundness}, ${\mathcal{M}}^{+, f}_{\infty, z}={\mathcal{M}}_{\infty, z}^f$.
\item We let $\pi_{{\mathcal{P} }, \infty, s}^f:H_s^{\mathcal{P} }\rightarrow_{{\mathcal{S}}igma_1} {\mathcal{M}}_{\infty, z}^f$ and $\pi_{{\mathcal{P} }, \infty, s}:H_s^{\mathcal{P} }\rightarrow_{{\mathcal{S}}igma_1} {\mathcal{M}}_{\infty, z}$ be the corresponding iteration embeddings.
\item Recall that $\delta_{\infty, z}=\delta^{{\mathcal{M}}_{\infty, z}}$. We also let $\delta^f_{\infty, z}=\delta^{{\mathcal{M}}_{\infty, z}^f}$. Thus, $\delta^f_{\infty, z}=\nuu_z$ (this follows from \rthm{steel's thing}).
\item We let $\gamma^f_{\infty, s, z}=\sigmaup \pi^f_{{\mathcal{P} }, \infty, s}"\gamma_s^{\mathcal{P} }$ for some ${\mathcal{P} }$ such that $({\mathcal{P} }, s)\in \muathcal{I}^f_z$. Recall that $\gamma_{\infty, s, z}=\sigmaup \pi_{{\mathcal{P} }, \infty, s}"\gamma_s^{\mathcal{P} }$ for some ${\mathcal{P} }$ such that $({\mathcal{P} }, s)\in \muathcal{I}_z$.
\item We let ${\mathcal{M}}_{\infty, \kappa, z, x}^f$ be the direct limit of ${\mathcal{W} }^f_z$ constructed inside ${\mathcal{M}}_{2k}(x)$ at $\kappa$. Here $x$ codes ${\mathcal{W} }^f_z$ and $\kappa$ is an inaccessible strong cutpoint of ${\mathcal{M}}_{2k}(x)$ which is less than the first Woodin of ${\mathcal{M}}_{2k}(x)$ and is a limit of strong cutpoints of ${\mathcal{M}}_{2k}(x)$.
\item We let ${\mathcal{M}}_{\infty, \kappa, z, x}$ be the direct limit of ${\mathcal{W} }_{2k+1,z}$ constructed inside ${\mathcal{M}}_{2k}(x)$. Here $x$ codes ${\mathcal{W} }_{2k+1, z}$ and $\kappa$ is an inaccessible strong cutpoint of ${\mathcal{M}}_{2k}(x)$ which is less than the first Woodin of ${\mathcal{M}}_{2k}(x)$ and is a limit of strong cutpoints of ${\mathcal{M}}_{2k}(x)$.
\item We let ${\mathcal{M}}_{\infty, z, x}^f={\mathcal{M}}_{\infty, \kappa, z, x}^f$ and ${\mathcal{M}}_{\infty, z, x}={\mathcal{M}}_{\infty, \kappa, z, x}$ where $\kappa$ is the least inaccessible of ${\mathcal{M}}_{2k}(x)$.
\item $\pi^f_{{\mathcal{P} }, \infty, s, x}:{\mathcal{P} }\rightarrow {\mathcal{M}}_{\infty, z, x}^f$ and $\pi_{{\mathcal{P} }, \infty, s, x}:{\mathcal{P} }\rightarrow {\mathcal{M}}_{\infty,z, x}$ be the corresponding iteration embeddings.
\item If $a$ is a countable transitive set such that ${\mathcal{W} }_z\in a$ or ${\mathcal{W} }^f_z\in a$ then we let ${\mathcal{M}}_{\infty, \kappa, z, a}^f$, ${\mathcal{M}}_{\infty, \kappa, z, a}$, ${\mathcal{M}}_{\infty, z, a}^f$, ${\mathcal{M}}_{\infty, z, a}$, $\pi^f_{{\mathcal{P} }, \infty, s, a}$, and $\pi_{{\mathcal{P} }, \infty, s, a}$ be the corresponding objects.
\end{enumerate}
Our first lemma is that $R_{z}$ dominates $R_{z}^f$.
\betaegin{lemma}\lambdaanglebel{full limit is bounded} For every $z$ if $w$ is a real coding ${\mathcal{W} }_z^f$ then for every $m$, $\card{R^f_{z, s_m}}\lambdaeq \card{R_{w, s_m}}$.
\end{lemma}
\betaegin{proof}
Fix $z$, $w$ and $m$ as in the hypothesis.
We now construct an order preserving embedding $f:\card{R^f_{z, s_m}}\rightarrow \card{R_{z, s_m}}$.
Suppose ${\mathcal{P} }$ is such that $({\mathcal{P} }, s_m)\in\muathcal{I}_{w}$. By iterating if necessary, we get that there are conditions in the extender algebra of ${\mathcal{P} }$ that force that the generic object is a pair $({\mathcal{ Q}}, \alpha)\in \muathcal{J}^f_{z, s_m}$. The formula expressing this has ${\mathcal{W} }_z^f$ as a parameter and essentially says that ${\mathcal{ Q}}$ is a correct iterate of ${\mathcal{W} }_z^f$ and $\alpha<\gamma_m^{\mathcal{ Q}}$. Because if $G\sigmaubseteq Coll(\omega, \delta^{\mathcal{P} })$ is ${\mathcal{M}}_{2k}({\mathcal{P} })$-generic and $x_g\in {\mathcal{M}}_{2k}({\mathcal{P} })[g]$ is the real coding ${\mathcal{P} }|\delta^{\mathcal{P} }$ then we can form ${\mathcal{M}}_{\infty, z, x_g}^f$ \footnote{Notice that one can show via ${\mathcal{S}}$-constructions that ${\mathcal{M}}_{2k}({\mathcal{P} })[g]={\mathcal{M}}_{2k}(x)$.}, there are conditions $p$ in the extender algebra of ${\mathcal{P} }$ that decide values for $\pi^f_{\deltaot{{\mathcal{ Q}}}, \infty, \check{s}_m, x_g}(\check{\alpha})$ where $(\deltaot{{\mathcal{ Q}}}, \alpha)$ is the generic object containing $p$. Notice that the value of $\pi^f_{\deltaot{{\mathcal{ Q}}}, \infty, \check{s}_m, x_g}(\check{\alpha})$ is independent of $g$. We then let $\muathcal{A}^{\mathcal{P} }$ be a maximal antichain of conditions $p$ such that
\betaegin{enumerate}
\item $p$ forces that the generic object is a pair $({\mathcal{ Q}}, \alpha)\in \muathcal{I}^f_{z, s_m}$,
\item for some $\beta$, ${\mathcal{M}}_{2k}({\mathcal{P} })\vDash ``p\Vdash_{Coll(\omega, \delta^{\mathcal{P} })} \pi^f_{\deltaot{{\mathcal{ Q}}}, \infty, \check{s}_m, x_g}(\check{\alpha})=\check{\beta}"$.
\end{enumerate}
Notice that we can assume that $\muathcal{A}^{\mathcal{P} }\in H_{s_m}^{\mathcal{P} }$. For each $p\in \muathcal{A}^{\mathcal{P} }$ let $\beta_p$ be the witness for 2. We can then define $\lambdaeq^{\mathcal{P} }$ on $\muathcal{A}^{\mathcal{P} }$ by: $p\lambdaeq^{\mathcal{P} } q \mathrel{\leftrightarrow} \beta_p\lambdaeq \beta_q$. Notice that $\card{\lambdaeq^{\mathcal{P} }}<\gamma_{s_m}^{\mathcal{P} }$. We have that $p\lambdaeq^{\mathcal{P} } q$ iff ${\mathcal{M}}_{2k}({\mathcal{P} })\vDash (p, q) \Vdash ``$ if $\deltaot{G}=((\deltaot{{\mathcal{ Q}}}, \check{\alpha}), (\deltaot{R}, \check{\beta}))$ then $(\deltaot{Q}, \check{\alpha})R^f_{\check{z}, \check{s}_m} (\deltaot{{\mathcal R}}, \check{\beta})"$.
Fix now $({\mathcal{ Q}}, \alpha)\in \muathcal{I}^f_{z, s_m}$. We say $({\mathcal{P} }, p)$ is $({\mathcal{ Q}}, \alpha)$-stable if
\betaegin{enumerate}
\item $({\mathcal{ Q}}, \alpha)$ is generic for the extender algebra of ${\mathcal{P} }$ and $p\in G$ where $G\sigmaubseteq \muathbb{B}^{\mathcal{P} }$ is the generic object such that $x_G=({\mathcal{ Q}}, \alpha)$,
\item $p\in \muathcal{A}^{\mathcal{P} }$ and $\beta_p=\pi^f_{{\mathcal{ Q}}, \infty, s_m, {\mathcal{P} }[{\mathcal{ Q}}]}(\alpha)$,
\item whenever $({\mathcal R}, q)$ is such that ${\mathcal R}$ is a correct iterate of ${\mathcal{P} }$ such that $({\mathcal{ Q}}, \alpha)$ is generic over ${\mathcal R}$ for the extender algebra at $\delta^{\mathcal R}$ and letting $G\sigmaubseteq \muathbb{B}^{\mathcal R}$ be the generic such that $x_G=({\mathcal{ Q}}, \alpha)$, $q\in \muathcal{A}^{\mathcal R}\cap G$,
\betaegin{center}
$\beta_q=\pi_{{\mathcal{P} }, {\mathcal R}, s_m}(\beta_p)$.
\end{center}
Thus, $q=_{\lambdaeq^{\mathcal R}}\pi_{{\mathcal{P} }, {\mathcal R}, s_m}(p)$.
\end{enumerate}
We claim that for every $({\mathcal{ Q}}, \alpha)\in \muathcal{I}^f_{z, s_m}$ there is a $({\mathcal{ Q}}, \alpha)$-stable $({\mathcal{P} }, p)$. To see this assume not and fix $({\mathcal{ Q}}, \alpha)\in \muathcal{I}^f_{z, s_m}$ such that there is no $({\mathcal{ Q}}, \alpha)$-stable pair $({\mathcal{P} }, p)$. Let ${\mathcal{P} }_0$ be such that $({\mathcal{ Q}}, \alpha)$ is generic for the extender algebra of ${\mathcal{P} }_0$. Letting $G\sigmaubseteq \muathbb{B}^{\mathcal{P} }$ be the generic object such that $x_G=({\mathcal{ Q}}, \alpha)$, we have a unique condition $p_0\in \muathcal{A}^{\mathcal{P} }\cap G$. Because $({\mathcal{P} }_0, p_0)$ isn't $({\mathcal{ Q}}, \alpha)$-stable, there is ${\mathcal{P} }_1$ which is a correct iterate of ${\mathcal{P} }_0$ and is such that $({\mathcal{ Q}}, \alpha)$ is generic over ${\mathcal{P} }_1$ for the extender algebra at $\delta^{{\mathcal{P} }_1}$ and if $p_1\in \muathcal{A}^{{\mathcal{P} }_1}\cap H$ where $H\sigmaubseteq \muathbb{B}^{{\mathcal{P} }_1}$ is the ${\mathcal{P} }_1$-generic such that $x_H=({\mathcal{ Q}}, \alpha)$ then
\betaegin{center}
$\beta_{p_1}\nuot=\pi_{{\mathcal{P} }, {\mathcal R}, s_m}(\beta_{p_0})$
\end{center}
Let
\betaegin{center}
$i=_{def}i_{{\mathcal{M}}_{2k}({\mathcal{P} }_0), {\mathcal{M}}_{2k}({\mathcal{P} }_1)}\restriction {\mathcal{M}}^f_{\infty, z, {\mathcal{P} }_0}:{\mathcal{M}}^f_{\infty, z, {\mathcal{P} }_0} \rightarrow {\mathcal{M}}_{\infty, z, {\mathcal{P} }_1}^f$.
\end{center}
Then by Dodd-Jensen we have that
\betaegin{center}
$i(\pi^{f}_{{\mathcal{ Q}},\infty, s_m, {\mathcal{P} }_0}(\alpha))\geq \pi^f_{{\mathcal{ Q}},\infty, s_m, {\mathcal{P} }_1}(\alpha)$,
\end{center}
implying that
\betaegin{center}
$i(\beta_{p_0})\geq \beta_{p_1}$.
\end{center}
But because $i(\beta_{p_0})=\pi_{{\mathcal{P} }, {\mathcal R}, s_m}(\beta_{p_0})$ and $\beta_{p_1}\nuot= \pi_{{\mathcal{P} }, {\mathcal R}, s_m}(\beta_{p_0})$, we get that
\betaegin{center}
$\beta_{p_1}<i(\beta_{p_0})$.
\end{center}
Continuing this construction we get $\lambdaangle {\mathcal{P} }_k, p_k: k<\omega\rangle$ such that ${\mathcal{P} }_0$ is a correct iterate of ${\mathcal{W} }_w$, ${\mathcal{P} }_{k+1}$ is a correct iterate of ${\mathcal{P} }_k$ and $\beta_{p_{k+1}}< i_{{\mathcal{P} }_k, {\mathcal{P} }_{k+1}}(\beta_{p_k})$. Let then ${\mathcal{P} }$ be the direct limit of ${\mathcal{P} }_k$'s under the embeddings $i_{{\mathcal{P} }_k, {\mathcal{P} }_{k+1}}$ and let $\sigmaigma_k:{\mathcal{P} }_k\rightarrow {\mathcal{P} }$ be the direct limit embedding. Then letting $\xi_k=\sigmaigma_k(\beta_{p_k})$, we get that $\lambdaangle \xi_k : k\in \omega\rangle$ is a descending sequence of ordinals, contradiction.
For each $({\mathcal{ Q}}, \alpha)\in \muathcal{I}^f_{z, s_m}$ let $A_{{\mathcal{ Q}}, \alpha}=\{ ({\mathcal{P} }, p): ({\mathcal{P} }, p)$ is $({\mathcal{ Q}}, \alpha)$-stable $\}$. Let $B_{{\mathcal{ Q}}, \alpha}=\{ ({\mathcal{P} }, \xi ): \exists p ( ({\mathcal{P} }, p)\in A_{{\mathcal{ Q}}, \alpha} \wedge \card{p}_{\muathcal{A}^{\mathcal{P} }}=\xi)\}$. Then notice that if $({\mathcal{P} }_i, \xi_i)\in B_{{\mathcal{ Q}}_i, \alpha_i}$ for $i=0,1$ then
\betaegin{center}
$({\mathcal{ Q}}_0, \alpha_0)R^f_{z, s_m} ({\mathcal{ Q}}_1, \alpha_1)\mathrel{\leftrightarrow} ({\mathcal{P} }_0, \xi_0)R_{w, s_m} ({\mathcal{P} }_1, \xi_1)$
\end{center}
To see this, let ${\mathcal{P} }$ be a common correct iterate of ${\mathcal{P} }_0$ and ${\mathcal{P} }_1$ such that $({\mathcal{ Q}}_0, \alpha_0)$ and $({\mathcal{ Q}}_1, \alpha_1)$ are generic for the extender algebra of ${\mathcal{P} }$. Then let $G_i\sigmaubseteq \muathbb{B}^{{\mathcal{P} }}$ be the ${\mathcal{P} }$-generic such that $x_{G_i}=({\mathcal{ Q}}_i, \alpha_i)$ ($i=0, 1$). Let $p_i\in \muathcal{A}^{{\mathcal{P} }}\cap G_i$. Suppose now $i_{{\mathcal{P} }_0, {\mathcal{P} }}(p_0)\lambdaeq^{\mathcal{P} } i_{{\mathcal{P} }_1, {\mathcal{P} }}(p_1)$. Because of stability we have that
\betaegin{center}
$i_{{\mathcal{P} }_k, {\mathcal{P} }}(\beta_{p_k})=\pi_{{\mathcal{ Q}}_k, \infty, s_m, {\mathcal{P} }}(\alpha_\kappa)\ \ \ k=0,1$.
\end{center}
Because $i_{{\mathcal{P} }_k, {\mathcal{P} }}(\beta_{p_k})=\beta_{i_{{\mathcal{P} }_k, {\mathcal{P} }}(p_k)}$ ($k=0,1$) and $i_{{\mathcal{P} }_0, {\mathcal{P} }}(p_0)\lambdaeq^{\mathcal{P} } i_{{\mathcal{P} }_1, {\mathcal{P} }}(p_1)$, we get that
\betaegin{center}
$\pi_{{\mathcal{ Q}}_0, \infty, s_m, {\mathcal{P} }}(\alpha_0)\lambdaeq \pi_{{\mathcal{ Q}}_1, \infty, s_m, {\mathcal{P} }}(\alpha_1)$.
\end{center}
This then implies that
\betaegin{center}
${\mathcal{M}}_{2k}({\mathcal{P} })[({\mathcal{ Q}}_0, \alpha_0), ({\mathcal{ Q}}_1, \alpha_1)]\vDash ``({\mathcal{ Q}}_0, \alpha_0)R_{z, s_m}^f ({\mathcal{ Q}}_1, \alpha_1)$".
\end{center}
Hence, $({\mathcal{ Q}}_0, \alpha_0)R_{z, s_m}^f ({\mathcal{ Q}}_1, \alpha_1)$. The other direction is similar.
Let then $f: \card{R^f_{z, s_m}}\rightarrow \card{R_{w, s_m}}$ be given by $f(\nuu)=\eta$ if whenever $({\mathcal{ Q}}, \alpha)\in \muathcal{I}^f_{z, s_m}$ is such that $\card{({\mathcal{ Q}}, \alpha)}_{R^f_{z, s_m}}=\nuu$ then for any $({\mathcal{P} }, \beta)\in B_{{\mathcal{ Q}}, \alpha}$, $\card{({\mathcal{P} }, \beta)}_{R_{w, s_m}}=\eta$. The proof just used can be easily modified to show that $f$ is order preserving and hence, $\card{R^f_{z, s_m}}\lambdaeq \card{R_{w, s_m}}$.
\end{proof}
The proof of \rlem{full limit is bounded} can be used to prove the following.
\betaegin{corollary}\lambdaanglebel{corollary to boundness} For any $m\in \omega$ and $z, w\in \muathbb{R}$, if $z\lambdaeq_T w$ then $\card{R_{z, s_m}}\lambdaeq \card{R_{w, s_m}}$ and $\card{R}_{z, s_m}\lambdaeq \card{R^f_{z, s_m}}$.
\end{corollary}
Next, we prove Woodin's result. The proof presented here is due to the author. We are grateful to Woodin for letting us state and proof this very useful lemma.
\betaegin{theorem}[Woodin]\lambdaanglebel{woodins thing} Assume $AD+V=L(\muathbb{R})$. For $k\in \omega$, $\kappa^1_{2k+3}$ is the least cardinal $\delta$ of ${\rm{HOD}}$ such that \betaegin{center}
${\mathcal{M}}_{2k}({\rm{HOD}}|\delta) \vDash ``\delta$ is Woodin".
\end{center}
\end{theorem}
\betaegin{proof}
Again, we prove the theorem from the assumption that ${\mathcal{M}}_\omega^{\#}$ exists. However, readers familiar with the general theory surrounding this topic can reduce the hypothesis to just $AD^{L(\muathbb{R})}$. It easily follows from \rlem{no miserable drops} and the remarks following it that for each $z\in \muathbb{R}$, $\card{R^{+,f}_z} < \delta^1_{2k+3}$. To finish the proof of \rthm{woodins thing}, we need then to show that for all reals $z$, $\delta^f_{\infty, z}\lambdaeq \kappa^1_{2k+3}$ and that $\delta^f_{\infty, z}\geq \kappa^1_{2k+3}$. We start with the first.
Suppose that for some $z$, $\delta^f_{\infty, z} > \delta^1_{2k+3}$. Let $U\sigmaubseteq \muathbb{R}$ be the set
\betaegin{center}
$\{ (x, y) : y$ codes ${\mathcal{P} }i^1_{2k+2}$-iterable premouse ${\mathcal{M}}$ over $x$ such that ${\mathcal{M}}$ has $2k+1$ Woodins, proper initial segments of ${\mathcal{M}}$ are 2k+1-small and ${\mathcal{M}}$ has a last extender$\}$.
\end{center}
Then $U$ is ${\mathcal{P} }i^1_{2k+2}$ and we can let $T\sigmaubseteq \omega^{<\omega}{\hbox{\fiverm th}}etaimes \omega^{<\omega}{\hbox{\fiverm th}}etaimes (\kappa^1_{2k+3})^{<\omega}$ be a tree such that $p[T]=U$. It follows by \rthm{steel's thing} that for every $w$
\betaegin{center}
${\mathcal{M}}^f_{\infty, w}|\delta^f_{\infty, w}={\rm{HOD}}|\delta^f_{\infty, w}$.
\end{center}
Therefore, there is $w$ which codes ${\mathcal{W} }^f_z$ and is such that $T \in {\mathcal{M}}^f_{\infty, w}|\eta$ for some $\eta<\delta^f_{\infty, w}$ (because we are assuming that $\delta^f_{\infty, z}>\kappa^1_{2k+3}$ and by \rlem{full limit is bounded}, we have that $\delta^f_{\infty, z}\lambdaeq\delta^f_{\infty, w}$). Let then ${\mathcal{P} }\in \muathcal{F}^f_w$ be such that there is $S \in {\mathcal{P} }|\delta^{\mathcal{P} }$ such that $i^f_{{\mathcal{P} }, \infty}(S)=T$. We can fix $l$ such that $S \in H_l^{\mathcal{P} }$. Let $u$ be a real coding $({\mathcal{W} }^f_w, {\mathcal{P} })$. Let $S^*=\pi^f_{{\mathcal{P} }, \infty, s_l, {\mathcal{W} }_u}(S)$. We claim that ${\mathcal{M}}_{2k+1}(u)\vDash ``p[S^*_u]\nuot =\emptyset"$.
To see that ${\mathcal{M}}_{2k+1}(u)\vDash ``p[S^*_u]\nuot =\emptyset"$, fix a correct iterate ${\mathcal R}$ of ${\mathcal{P} }$ such that for some $y$ there is $h\in (\gamma_l^{\mathcal R})^\omega$ such that if $g=\pi^f_{{\mathcal R}, \infty, s_l}"h$ then $(u, y, g)\in [T]$. Notice that ${\mathcal{M}}_{2k+1}(u)={\mathcal{M}}_{2k}({\mathcal{W} }_u)$. Iterate ${\mathcal{W} }_u$ to make $({\mathcal R}, y)$ generic. Let ${\mathcal{ Q}}$ be this iterate. Let $\betaar{g}=\pi^f_{{\mathcal R}, \infty, s, {\mathcal{ Q}}[{\mathcal R}, y]}" h$. Then for every $k$, we must have that
\betaegin{center}
$(y\restriction k, \betaar{g}\restriction k)\in (\pi^f_{{\mathcal{W} }_w^f, \infty, s_l, {\mathcal{ Q}}}(S))_u=(\pi^f_{{\mathcal R}, \infty, s_l, {\mathcal{ Q}}[{\mathcal R}, y]}(S))_u$.
\end{center}
This means that $[(\pi^f_{{\mathcal{W} }_w^f, \infty, s_l, {\mathcal{ Q}}}(S))_u]\nuot =\emptyset$. By absoluteness we have that
\betaegin{center}
${\mathcal{M}}_{2k}({\mathcal{ Q}})\vDash [(\pi^f_{{\mathcal{W} }_w^f, \infty, s_l, {\mathcal{ Q}}}(S))_u]\nuot =\emptyset$.
\end{center}
It then follows by elementarity that
\betaegin{center}
${\mathcal{M}}_{2k+1}(u)\vDash ``p[S^*_u]\nuot =\emptyset"$.
\end{center}
It is, however, a well-known fact that there cannot be $y\in {\mathcal{M}}_{2k+1}(u)$ which codes a ${\mathcal{P} }i^1_{2k+2}$-iterable premouse ${\mathcal{M}}$ over $u$ such that the proper initial segments of ${\mathcal{M}}$ are 2k+1-small and ${\mathcal{M}}$ has $2k+1$-Woodins and a last extender.\footnote{One way to see this is to use a result from \cite{PWOIM}. It is shown there that $x\in {\mathcal{ Q}}_{2k+3}(u) \mathrel{\leftrightarrow} x$ is in every ${\mathcal{P} }i^1_{2k+2}$-iterable premouse ${\mathcal{M}}$ such that the proper initial segments of ${\mathcal{M}}$ are 2k+1-small and ${\mathcal{M}}$ has $2k+1$ Woodins and a last extender. Thus, if there was such a premouse ${\mathcal{M}}\in {\mathcal{M}}_{2k+1}(u)$ then as $Q_{2k+3}(u)=\muathbb{R}^{{\mathcal{M}}_{2k+1}(u)}$, ${\mathcal{M}}\in {\mathcal{M}}$, contradiction!} This contradiction shows that $\delta^f_{\infty, z}\lambdaeq \kappa^1_{2k+3}$.
To show that $\delta^f_{\infty, z}\geq \kappa^1_{2k+3}$, it is enough to show that $\delta_{\infty, 0} \geq \kappa^1_{2k+3}$. For this, we show that every ${\mathcal{P} }i^1_{2k+2}$-set is $\delta_{\infty, 0}$-Suslin. Let $\delta=\delta_{\infty, 0}$. To see that the universal ${\mathcal{P} }i^1_{2k+2}$-set is $\delta$-Suslin let ${\mathcal{ Q}}={\mathcal{M}}_{2k}^\#({\mathcal{M}}_{\infty, 0}|\delta)$. Notice that ${\mathcal{ Q}}$ has size $\delta$. Let $U$ be the universal ${\mathcal{P} }i^1_{2k+2}$-set. Let $\phi$ be ${\mathcal{P} }i^1_{2k+2}$ such that $x\in U\mathrel{\leftrightarrow} \phi(x)$. Let $T$ be the tree of attempts to construct a triple $( x, z, \pi )$ such that
\betaegin{enumerate}
\item $z$ codes a premouse ${\mathcal{M}}_z$,
\item $\pi: M_z\rightarrow {\mathcal{ Q}}$,
\item $x$ is generic over $M_z$ for the extender algebra at the least Woodin of ${\mathcal{M}}_z$,
\item $M_z[x]\vDash \phi[x]$.
\end{enumerate}
Let then $S=\{(s, f) : s\in \omega^{<\omega}$, $f\in [\delta]^{<\omega}$ and $f$ codes $f_0, f_1$ such that $(s, f_0, f_1)\in T\}$. Then, because ${\mathcal{M}}_{2k}(z)$ is ${\mathcal{P} }i^1_{2k+2}(z)$-correct, it is not hard to see that $p[S]=U$. This then completes the proof that $\delta^f_{\infty, z}=\kappa^1_{2k+3}$.
\end{proof}
As a corollary to \rlem{full limit is bounded}, we get the following.
\betaegin{corollary}
For every $z\in \muathbb{R}$, $\delta_{\infty, z}=\kappa^1_{2k+3}$.
\end{corollary}
\sigmaubsection{The proof of the main theorem}
In this subsection, we work towards the proof of \rthm{main theorem}. Recall that
\betaegin{center}
$a_{2k+1, m}=\sigmaup \{ \card{\lambdaeq^*} : \lambdaeq^*\in \utilde{\Gamma}_{2k+1, m}\}$\\
\end{center}
We let $\gamma_{\infty, m, x}=\gamma_{\infty, s_m, x}$ and $\gamma^f_{\infty, m, x}=\gamma^f_{\infty, s_m, x}$ and let
\betaegin{center}
$b_{2k+1, m}=\sigmaup_{x\in \muathbb{R}}\gamma_{\infty, m, x}$.
\end{center}
Notice that it follows from \rlem{full limit is bounded} that
\betaegin{center}
$b_{2k+1, m}=\sigmaup_{x\in \muathbb{R}}\gamma^f_{\infty, m, x}$.
\end{center}
It follows from \rthm{woodins thing} that
\betaegin{center}
$\kappa^1_{2k+3}=\sigmaup_{m\in \omega} b_{2k+1, m}$.
\end{center}
To make the notation as simple as possible, we fix an odd integer $2k+1$. We will omit it from various subscripts from now until the end of this subsection.
\betaegin{lemma}\lambdaanglebel{sups}
$a_{2k+1, m}\lambdaeq b_{2k+1, m+1}$.
\end{lemma}
\betaegin{proof}
Fix $m\in \omega$ and let $\lambdaeq^*\in\utilde{\Gamma}_{2k+1, m}$. Let $z^*, \phi$ be such that for all $x, y\in \muathbb{R}$,
\betaegin{center}
$x\lambdaeq^* y \mathrel{\leftrightarrow} {\mathcal{M}}_{2k}(z^*, x, y) \vDash \phi[z^*, x, y, s_{m}]$.
\end{center}
Suppose towards a contradiction that $\card{\lambdaeq^*}=\sigmaup_{x\in \muathbb{R}} \gamma_{\infty, x, m+1}$ (this may produce another real parameter, but we assume that it is already part of $z^*$).
First notice that for every $l$, $\sigmaup_{x\in \muathbb{R}} \gamma_{\infty, l, x}< \kappa^1_{2k+3}$. This is because if $\sigmaup_{x\in \muathbb{R}} \gamma_{\infty, l, x}=\kappa^1_{2k+3}$ then because ${\rm cf}(\kappaappa^1_{2k+3})=\omega$ (see \cite{Moschovakis}), there must be $x$ such that $\gamma_{\infty, l, x}=\kappa^1_{2k+3}$. But since $\delta_{\infty, x}>\gamma_{\infty, l, x}$, we get a contradiction. Thus, we can fix $z\in \muathbb{R}$ and $r\in \omega$ such that $z^*\lambdaeq_T z$ and $\gamma_{\infty, r, z} >\sigmaup_{x\in \muathbb{R}} \gamma_{\infty, m, x}$.
Following Hjorth (see \cite{Hjorth01}), using Moschovakis' coding lemma (see \cite{Moschovakis}), we get $w\in \muathbb{R}$ and a ${\mathcal{S}}igma^1_{2k+3}(w)$ set $B\sigmaubseteq \muathbb{R}^2$ such that $z\lambdaeq_T w$
\betaegin{enumerate}
\item if $(x, y)\in B$ then $x\in dom(\lambdaeq^*)$, $y\in dom(\lambdaeq_{z, r})$ and $\card{x}_{\lambdaeq^*}=\card{y}_{\lambdaeq_{z, r}}$,
\item for every $x\in dom(\lambdaeq^*)$ there is $y\in dom(\lambdaeq_{z,r})$ such that $(x, y)\in B$.
\end{enumerate}
Let $R$ be ${\mathcal{P} }i^1_{2k+2}(w)$ such that $(x, y)\in B \mathrel{\leftrightarrow} \exists u R(w, x, y, u)$. We now construct an embedding of $\lambdaeq^*$ into $R_{w, s_{m+1}}$. Let $A=\{ (x, y , u) : R(w, x, y, u)\}$. Notice that whenever $a$ is a countable transitive set, $\lambdaeq^*\cap \muathbb{R}^{{\mathcal{M}}_{2k}(a)}\in {\mathcal{M}}_{2k}(a)$. We will abuse our notation and write $\lambdaeq^*$ for $\lambdaeq^*\cap \muathbb{R}^{{\mathcal{M}}_{2k}(a)}$.
Given a suitable ${\mathcal{P} }$, there is a maximal antichain $\muathcal{A}\sigmaubseteq \muathbb{B}^{\mathcal{P} }$ such that if $p\in \muathcal{A}$ then for some $\alpha$, in ${\mathcal{M}}_{2k}({\mathcal{P} })$
\betaegin{enumerate}
\item $p\Vdash ``x_G=(x, y, u)\in A"$,
\item $p\Vdash``\Vdash_{Coll(\omega, \delta^{\mathcal{P} })} \card{x}_{\lambdaeq^*}=\alpha$".
\end{enumerate}
Notice that we can take $\muathcal{A}\in H_{m+1}^{\mathcal{P} }$. Let then $\muathcal{A}^{\mathcal{P} }$ be the least such maximal antichain. We can define $\lambdaeq^{\mathcal{P} }$ on $\muathcal{A}^{\mathcal{P} }$ as follows. Given $p\in \muathcal{A}$, let $\alpha_p$ be the ordinal $\alpha$ as in 2. Then for $p, q\in \muathcal{A}$, we let $p\lambdaeq^{\mathcal{P} } q$ iff $\alpha_p \lambdaeq \alpha_q$. Notice that $\card{\lambdaeq^{\mathcal{P} }}<\gamma_{m+1}^{\mathcal{P} }$. The remaining part of the proof is similar to the proof of \rlem{full limit is bounded}.
Given now an $x\in dom(\lambdaeq^*)$, a suitable ${\mathcal{P} }$ and $p\in \muathcal{A}^{\mathcal{P} }$ we say $({\mathcal{P} }, p)$ is $x$-stable if there is $(x, y, u)\in A$ which is generic over ${\mathcal{P} }$ for $\muathbb{B}^{\mathcal{P} }$ and
\betaegin{enumerate}
\item if $G\sigmaubseteq \muathbb{B}^{\mathcal{P} }$ is such that $x_G=(x, y, u)$ then $p\in G$,
\item whenever $({\mathcal R}, q)$ is such that ${\mathcal R}$ is a correct iterate of ${\mathcal{P} }$ such that some $(x, y^*, u^*)\in A$ is generic over ${\mathcal R}$ for $\muathbb{B}^{\mathcal R}$, and $q\in \muathcal{A}^{\mathcal R}\cap H$ where $H\sigmaubseteq \muathbb{B}^{\mathcal R}$ is the ${\mathcal R}$-generic such that $x_H=(x, y^*, u^*)$, then
\betaegin{center}
$\card{q}_{\lambdaeq^{\mathcal R}}=_{\lambdaeq^{\mathcal R}}\card{\pi_{{\mathcal{P} }, {\mathcal R}, s_{m+1}}(p)}$.
\end{center}
\end{enumerate}
We claim that for every $x\in dom(\lambdaeq^*)$ there is $x$-stable $({\mathcal{P} }, p)$. To see this, suppose not. First let $y, u$ be such that $(x, y, u)\in A$. Then let ${\mathcal{P} }$ be suitable such that $(x, y, u)$ is generic for $\muathbb{B}^{\mathcal{P} }$. There is then $p\in \muathcal{A}^{\mathcal{P} }$ such that if $G\sigmaubseteq \muathbb{B}^{\mathcal{P} }$ is ${\mathcal{P} }$-generic such that $x_G=(x, y, u)$ then $p\in G$. Let $\alpha=\alpha_{{\mathcal{P} }, p}$. Because $({\mathcal{P} }, p)$ isn't $x$-stable we must have that there is a correct iterate ${\mathcal R}$ of ${\mathcal{P} }$ such that some $(x, y^*, u^*)\in A$ is generic over ${\mathcal R}$ for $\muathbb{B}^{\mathcal R}$, and if $H$ is the generic such that $x_H=(x, y^*, u^*)$ and $q\in H\cap \muathcal{A}^{\mathcal R}$ then
\betaegin{center}
$\card{q}\nuot=_{\lambdaeq^{\mathcal R}}\card{\pi_{{\mathcal{P} }, {\mathcal R}, s_{m+1}}(p)}$.
\end{center}
Let $y$ code $({\mathcal{ Q}}, \beta)$ and let $y^*$ code $({\mathcal{ Q}}^*,\beta^*)$. Notice that $({\mathcal{ Q}}, \beta)=_{R_{z, r}}({\mathcal{ Q}}^*, \beta^*)$.
Let also
\betaegin{center}
$i=i_{{\mathcal{P} }, {\mathcal R}}\restriction {\mathcal{M}}_{\infty, z, {\mathcal{P} }}:{\mathcal{M}}_{\infty, z, {\mathcal{P} }} \rightarrow {\mathcal{M}}_{\infty, z, {\mathcal R}}$.
\end{center}
We have that
\betaegin{center}
$i\circ \pi_{{\mathcal{ Q}}, \infty, r, z, {\mathcal{P} }} : H_r^{\mathcal{P} }\rightarrow H_r^{{\mathcal{M}}_{\infty, z, {\mathcal R}}}$.
\end{center}
Because of Dodd-Jensen then we get that
\betaegin{center}
$i(\pi_{{\mathcal{ Q}}, \infty, r, z, {\mathcal{P} }}(\beta))\geq \pi_{{\mathcal{ Q}}^*, \infty, r, z {\mathcal R}}(\beta^*)$.
\end{center}
Notice that equality cannot hold. To see this, suppose $i(\pi_{{\mathcal{ Q}}, \infty, r, z, {\mathcal{P} }}(\beta))=\pi_{{\mathcal{ Q}}^*, \infty, r, z, {\mathcal R}}(\beta^*)$. We have that,
\betaegin{center}
${\mathcal{M}}_{2k}({\mathcal{P} })\vDash p\Vdash $ ``if $(x_G)_2=(\deltaot{{\mathcal{ Q}}}, \deltaot{\beta})$ then $\pi_{\deltaot{{\mathcal{ Q}}}, \infty, r, z, {\mathcal{P} }}(\deltaot{\beta})=\check{\xi}$"\footnote{Here we think of a real $x$ as coding a triple $(x_1, x_2, x_3)$.}.
\end{center}
where $\check{\xi}=\pi_{{\mathcal{ Q}}, \infty, r, z, {\mathcal{P} }}(\beta)$. We then have by elementarity that there is ${\mathcal R}$-generic $H\sigmaubseteq \muathbb{B}^{\mathcal R}$ such that $i_{{\mathcal{P} }, {\mathcal R}}(p)\in H$ and if $(x_H)_2=({\mathcal{S}}, \nuu)$ then $\pi_{{\mathcal{S}}, \infty, r, z, {\mathcal R}}(\nuu)=i_{{\mathcal{P} }, {\mathcal R}}(\xi)$. But since we are assuming that $i(\pi_{{\mathcal{ Q}}, \infty, r, z, {\mathcal{P} }}(\beta))=\pi_{{\mathcal{ Q}}^*, \infty, r, z, {\mathcal R}}(\beta^*)$, we must have that $({\mathcal{S}}, \nuu)=_{R_{z, r}}({\mathcal{ Q}}^*, \beta^*)$ and by the choice of $B$ we must have that $(x_H)_1=_{\lambdaeq^*} x$. This then implies that $i_{{\mathcal{P} }, {\mathcal R}}(p)=_{\lambdaeq^{\mathcal R}} q$, contradiction. Thus we must have that
\betaegin{center}
$i(\pi_{{\mathcal{ Q}}, \infty, z, w\oplus{\mathcal{P} }, r}(\beta))> \pi_{{\mathcal{ Q}}^*, \infty, z, w\oplus{\mathcal R}, r}(\beta^*)$.
\end{center}
Let then ${\mathcal{P} }_0={\mathcal{P} }$, $(x, y_0, u_0)=(x, y, u)$, ${\mathcal{P} }_1={\mathcal R}$ and $(x, y_1, u_1)=(x, y^*, u^*)$. Let $({\mathcal{ Q}}_0, \beta_0)$ be the pair coded by $y_0$ and let $({\mathcal{ Q}}_1, \beta_1)$ be the pair coded by $y_1$. Let $\xi_i=\pi_{{\mathcal{ Q}}_i, \infty, z, r, {\mathcal{P} }_i}(\beta_i)$ for $i=0,1$. It then follows from our discussion that $i_{{\mathcal{P} }_0, {\mathcal{P} }_1}(\xi_0)>\xi_1$.
By a repeated application of the argument used in the previous paragraph, we can get $\lambdaangle {\mathcal{P} }_l, ({\mathcal{ Q}}_l, \beta_l), \xi_l : l\in \omega\rangle$ such that
\betaegin{enumerate}
\item ${\mathcal{P} }_l\in \muathcal{F}_w$,
\item ${\mathcal{P} }_{l+1}$ is a correct iterate of ${\mathcal{P} }_l$,
\item $({\mathcal{ Q}}_l, \beta_l) \in \muathcal{I}_{w, r}$ and $(Q_l, \beta_l)$ is generic over ${\mathcal{P} }_l$ for $\muathbb{B}^{{\mathcal{P} }_l}$,
\item $\pi_{{\mathcal{ Q}}_l, \infty, r, z, {\mathcal{P} }_l}(\beta_l)=\xi_l$,
\item $i_{{\mathcal{P} }_l, {\mathcal{P} }_{l+1}}(\xi_l)>\xi_{l+1}$.
\end{enumerate}
Letting $\sigmaigma_{l, j}:{\mathcal{P} }_l\rightarrow {\mathcal{P} }_j$ be the iteration embedding, letting ${\mathcal{ Q}}$ be the direct limit of $\lambdaangle {\mathcal{P} }_l, \sigmaigma_{l, j} : l< j <\omega\rangle$ and letting $\sigmaigma_l:{\mathcal{P} }_l\rightarrow {\mathcal{ Q}}$ be the iteration embedding we get that
$\lambdaangle \sigmaigma_l(\xi_l) : l <\omega\rangle$ is a decreasing sequence of ordinals, contradiction. Thus, indeed, for every $x$ there is an $x$-stable $({\mathcal{P} }, p)$.
Let then for each $x$, $S_x$ be the set of $x$-stable $({\mathcal{P} }, p)$'s and let $\beta_{{\mathcal{P} }, p}=\card{p}_{\lambdaeq^{\mathcal{P} }}$. Using uniformization, we can choose $({\mathcal{P} }_x, p_x)\in S_x$. Notice now that
\betaegin{center}
$x\lambdaeq^*y\mathrel{\leftrightarrow} ({\mathcal{P} }_x, p_x)\lambdaeq_{w, m} ({\mathcal{P} }_y, p_y)$.
\end{center}
(To see this, let ${\mathcal R}$ be a common iterate of ${\mathcal{P} }_x$ and ${\mathcal{P} }_y$ such that for some $u, v, u^*, v^*\in \muathbb{R}$, $(x, u, v)$ and $(y, u^*, v^*)$ are generic over ${\mathcal R}$ for $\muathbb{B}^{\mathcal R}$. Then by $x$ and $y$ stability, we must have that $x\lambdaeq^*y$ holds if and only if $i_{{\mathcal{P} }_x, {\mathcal R}}(p_x)\lambdaeq^{\mathcal R} i_{{\mathcal{P} }_y, {\mathcal R}}(p_y)$.) We then have that $x\rightarrow ({\mathcal{P} }_x, p_x)$ is an order preserving map of $\lambdaeq^*$ into $R_{w, m+1}$. Therefore, $\card{\lambdaeq^*}\lambdaeq \card{R_{w, m+1}}\lambdaeq \sigmaup_{x\in \muathbb{R}}\gamma_{\infty, x, m+1}$, contradiction!
\end{proof}
We thus have that $\sigmaup_{m\in \omega} a_{2k+1,m}\lambdaeq \kappa^1_{2k+3}$. Notice that for each $m\in \omega$ and $w\in \muathbb{R}$, $R_{w, m}\in \Gamma_{2k+1, m+1}(w)$. Because we have that $\sigmaup_{m\in \omega}b_{2k+1,m}=\kappa^1_{2k+3}$, we easily get that $\sigmaup_{m\in \omega} a_{2k+1, m} = \kappa^1_{2k+3}$. This then finishes the proof of the Main Theorem.
\sigmaection{Some remarks}
First of all, it turns out that $b_{2k+1, m}$ is a cardinal for every $m$ and moreover, $b_{2k+1, m}<\kappa^1_{2k+3}$. Here is the proof.
\betaegin{lemma}\lambdaanglebel{as are cardinals} For every $m$, $b_{2k+1, m}<\kappa^1_{2k+3}$ and $b_{2k+1, m}$ is a cardinal.
\end{lemma}
\betaegin{proof}
We have that $b_{2k+1, m}< \kappa^1_{2k+3}$ because if for some $m$, $b_{2k+1, m}=\kappa^1_{2k+3}$ then because ${\rm cf}(\kappa^1_{2k+3})=\omega$, we can fix $x$ such that $\gamma_{\infty, s_m, x}=\kappa^1_{2k+3}$. But this contradicts \rthm{woodins thing}. Thus, we have that $b_{2k+1, m}<\kappa^1_{2k+3}$. Suppose no that for some $m$, $b_{2k+1, m}$ isn't a cardinal. Let $\kappa=\card{b_{2k+1, m}}$. Then $\kappa<b_{2k+1, m}$ and there is $A\sigmaubseteq \kappa$ such that $A$ codes a well-ordering of $\kappa$ of length $b_{2k+1, m}$. There is then a real $z$ such that $A\in {\rm{HOD}}_z$. We then can get $w$ such that $z\lambdaeq_T w$ and $\kappa<\gamma^f_{\infty, s_m, w}$. It follows that $A\in {\rm{HOD}}_w$ and in particular, $\gamma^f_{\infty, s_m, w}$ isn't a cardinal of ${\rm{HOD}}_w$. But clearly $\gamma^f_{\infty, s_m, w}$ is a cardinal of ${\rm{HOD}}_w$, contradiction.
\end{proof}
We do not know if $a_{2k+1, m}=b_{2k+1, m}$. A more interesting question that comes up naturally is what is the exact place of $b_{2k+1, m}$ in the sequence of $\alphaleph$'s. We conjecture that $a_{2k+1, 0}=\delta^1_{2k+2}$. One evidence for this is that by Hjorth's aforementioned result, $a_{1, 0}=u_2=\omega_2=\delta^1_2$. More generally, Jackson showed that the sup of the lengths of ${\mathcal{P} }i^1_{2k}$ prewellorderings is $\delta^1_{2k}$ and ${\mathcal{P} }i^1_{2k}$ is a subclass of $\Gamma_{2k+1, 0}$. The general question is open.
It seems to be possible to use the directed system associated with ${\mathcal{M}}_{2n+1}$ to prove Kechris-Martin kind of results for ${\mathcal{P} }i^1_{2k+3}$ (see \cite{KecMar}). In particular, one should be able to prove that ${\mathcal{P} }i^1_{2k+2}$ is closed under quantification over $\kappa^1_{2k+3}$. Another application should be the uniqueness of $L[T_{2k}]$, i.e., it should be possible to prove, using ideas from this paper, that $L[T_{2k}]$ is independent of the choice of the scale that produces $T_{2k}$. This would generalize Hjorth's theorem on the uniqueness of $L[T_2]$ (see \cite{Hjorth96}). It should also be possible to prove results like Solovay's $\Delta^1_3$-coding result (see \cite{Solovay}) for higher levels of projective hierarchy. The author, however, has no intuition on whether it is possible to use directed systems to carry out Jackson's analysis of projective ordinals. From an inner model theoretic point of view, Jackson's analysis remains a mystery.
\betaibliography{directsystem}
\betaibliographystyle{plain}
\end{document} |
\begin{document}
\title{Low degree Lorentz invariant polynomials as potential entanglement invariants for multiple Dirac spinors}
\author{Markus Johansson}
\affiliation{organization={ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology},
postcode={08860}, city={ Castelldefels (Barcelona)}, country={Spain}}
\date{\today}
\begin{abstract}
A system of multiple spacelike separated Dirac particles is considered and a method for constructing polynomial invariants under the spinor representations of the local proper orthochronous Lorentz groups is described. The method is a generalization of the method used in [Phys. Rev. A {\bf 105}, 032402 (2022), arXiv:2103.07784] for the case of two Dirac particles. All polynomials constructed by this method are identically zero for product states.
The behaviour of the polynomials under local unitary evolution that acts unitarily on any subspace defined by fixed particle momenta is described.
By design all of the polynomials have invariant absolute values on this kind of subspaces if the evolution is locally generated by zero-mass Dirac Hamiltonians. Depending on construction some polynomials have invariant absolute values also for the case of nonzero-mass or additional couplings.
Because of these properties the polynomials are considered potential candidates for describing the spinor entanglement of multiple Dirac particles, with either zero or arbitrary mass or additional couplings. Polynomials of degree 2 and 4 are derived for the cases of three and four Dirac spinors. For three spinors no non-zero degree 2 polynomials are found but 67 linearly independent polynomials of degree 4 are identified. For four spinors 16 linearly independent polynomials of degree 2 are constructed as well as 26 polynomials of degree 4 selected from a much larger number.
The relations of these polynomials to the polynomial spin entanglement invariants of three and four non-relativistic spin-$\frac{1}{2}$ particles are described.
Moreover, it is described how degree 4 polynomials for five spinors can be constructed and how degree 2 polynomials can be constructed for any even number of spinors.
\end{abstract}
\begin{keyword}Dirac particles \sep
multipartite entanglement \sep
polynomial invariants
\end{keyword}
\maketitle
\section{Introduction}
The Dirac equation was originally introduced as a relativistic description of the electron \cite{dirac2,dirac}. As such it is used in relativistic quantum mechanics \cite{bjorken}, quantum electrodynamics \cite{peskin,schwartz} and in relativistic quantum chemistry \cite{pykk}. It has subsequently also been used in the Standard Model to describe other leptons and quarks \cite{schwartz} and in the Yukawa model of hadrons to describe baryons \cite{yukawa}. For the case of zero mass the Dirac equation admits solutions with definite chirality, so called Weyl particles \cite{weyl}. Dirac-like equations are used also to describe Dirac and Weyl quasi-particles in graphene and other solid state and molecular systems as well as in photonic crystals \cite{semenoff,novos,lu,bohm,liu,Xu,pirie}.
Quantum entanglement is a feature of quantum mechanics that permits action at a distance \cite{epr,bell,chsh,bell2}, i.e., nonlocal causation between spacelike separated systems. The state of a composite quantum system with multiple spacelike separated subsystems is entangled if it is a superposition where some property of one subsystem is conditioned on properties of one or several other subsystems. In this case the state cannot be fully described by only local variables.
The presence of entanglement allows for phenomena that are impossible without nonlocal causation such as quantum teleportation \cite{bennett} or violation of a Bell inequality \cite{bell,chsh,svet}.
Entanglement in a system of multiple particles can in general exist in multiple forms that are not mutually convertible using local operations.
One often difficult problem in the theory of entanglement is the characterization of these different ways in which a multipartite system can be entangled \cite{popescu,carteret,coffman,toni,higuchi, dur,sud,wong,tarrach,verstraete2,luque,moor,toumazet}.
In non-relativistic quantum mechanics an extensively studied type of entanglement is that between the spins of spacelike separated spin-$\frac{1}{2}$ particles \cite{bell,bennett,popescu,carteret,coffman,toni,higuchi, dur,sud,wong,tarrach,verstraete2,luque,moor,toumazet,ghz,ekert,wootters,wootters2}.
A system of three or more such particles can be spin entangled in qualitatively different ways \cite{toni,higuchi,dur,tarrach,verstraete2,luque}. If two spin entangled states can be transformed into each other by physically allowed local unitary transformations they are said to be equivalently entangled with respect to local unitary transformations \cite{toni,higuchi,sud,tarrach,toumazet, ekert}. These transformations include the local unitary evolutions generated by the physically allowed Hamiltonian operators and the local changes of reference frame, such as spatial rotations. Equivalence with respect to larger groups of local transformations such as local $\mathrm{SL}(2,\mathbb{C})$, has also been considered \cite{dur,wong,verstraete2,luque,moor}. The number of parameters needed to describe the set of equivalence classes of this kind increases rapidly with the number of spin-$\frac{1}{2}$ particles \cite{popescu,carteret}.
One way to test if two states are inequivalently spin entangled with respect to a group of local transformations is to evaluate the polynomials in the state coefficients that are invariants of the action of the group \cite{hilbert,mumford,grassl}, the so called polynomial entanglement invariants. If the ratio of two such polynomials of the same degree takes different values for the different states they are necessarily inequivalently entangled. In particular, polynomials invariant under local $\mathrm{SL}(2,\mathbb{C})$ have been constructed for different numbers of non-relativistic spin-$\frac{1}{2}$ particles \cite{coffman,wong,luque,wootters,wootters2}. The polynomial invariants of this kind can be used to partially parametrize the set of spin entanglement equivalence classes \cite{popescu,carteret}.
In relativistic quantum mechanics the spinorial degree of freedom of a spin-$\frac{1}{2}$ particle is described by a four component Dirac spinor. Therefore the tools developed to describe the spin entanglement of non-relativistic spin-$\frac{1}{2}$ particles can not be used outside special cases.
This lead to a search for new tools and concepts suited for the description of entanglement of relativistic particles.
The entanglement and non-locality between Dirac spinors and other descriptions of entanglement between Dirac particles has been investigated and discussed previously in a number of works, see e.g. \cite{czachor,alsing,terno,adami,pachos,ahn,terno2,tera,tera2,mano,won,caban3,leon,caban,tessier,geng,delgado,moradi,caban2,spinorent}.
As in the case of non-relativistic spin entanglement one may consider the question of whether spinor entangled states are equivalent with respect to a given set of physically allowed local transformations. The issue of constructing polynomial invariants for this purpose was considered in Ref. \cite{spinorent} for the case of two Dirac spinors.
In this work we investigate the description of entanglement between the spinorial degrees of freedom of multiple Dirac particles. The conceptualization of spinor entanglement properties that has been introduced in Ref. \cite{spinorent} is considered also here.
We explore the same general idea as Ref. \cite{spinorent} for constructing polynomial Lorentz invariants but adapt it to the case of multiple Dirac spinors.
We describe how these polynomial Lorentz invariants can be used to partially characterize equivalence classes of spinor entangled states and how some of them reduce to the already known local $\mathrm{SL}(2,\mathbb{C})$ invariants \cite{coffman,wong,luque,moor} for the case of non-relativistic free particles in an energy eigenstate and for the case of Weyl particles.
As in Ref. \cite{spinorent} we make the assumption that for any number of spacelike separated Dirac spinors the state can be expanded in a basis that is formed from the tensor products of the local basis elements used to describe the individual spinors.
Given this assumption a method for constructing polynomials invariant under the spinor representation of the proper orthochronous Lorentz groups for any number of spinors is described.
These polynomials are also invariant, up to a U(1) phase, under local unitary transformations on a subspace spanned by the spinorial degrees of freedom that are generated by zero-mass Dirac Hamiltonians. Depending on construction some are invariant, up to a U(1) phase, also for local unitary transformations generated by arbitrary-mass Dirac Hamiltonians, and some for zero-mass Dirac Hamiltonians with a coupling to a Yukawa pseudo-scalar boson.
Therefore these polynomials are considered potential candidates for describing multi-spinor entanglement for different physical scenarios.
The computational difficulty of deriving polynomials of this kind increases with the number of spinors but polynomials of low degree can still be found with modest effort for the case of three and four Dirac spinors.
For the case of three Dirac spinors we derive polynomial invariants of degree 4 from which a set of 67 linearly independent polynomials can be selected. For four Dirac spinors we derive 16 linearly independent degree 2 polynomials.
We describe how to construct degree 4 polynomials for four spinors, but only calculate a few due to the large number of such polynomials. Further we describe how to construct degree 4 polynomials for five spinors but do not calculate them. Finally we describe how to construct degree 2 polynomials for any even number of spinors.
We discuss the relations of the derived polynomials to the Coffmann-Kundu-Wootters 3-tangle \cite{coffman} and the so called $2\times 2\times 4$ tangle described in Refs. \cite{moor,verstraete} for the case of three spinors and the relations to the polynomial invariants found by Luque and Thibon \cite{luque} for the case of four spinors. For the case of arbitrary even number of spinors we describe the relation of the degree 2 polynomials to the so called $N$-tangle introduced by Wong and Christensen \cite{wong}.
This work is organized as follows. In sections \ref{dir}-\ref{spen} the relevant background material is reviewed, the physical assumptions made are discussed and the tools used to construct the Lorentz invariant polynomials are described.
In particular, section \ref{dir} introduces the description of Dirac and Weyl particles and describes the fundamental assumptions made in this work. In section \ref{rep} we describe the spinor representation of the Lorentz group and the charge conjugation transformation. Section \ref{invariants} describes how to construct skew-symmetric bilinear forms that are invariant under the spinor representation of the local proper orthochronous Lorentz transformations. In section \ref{ham} the behaviour of the bilinear forms under local unitary evolution generated by Dirac-like Hamiltonians is described. Section \ref{hinn} describes the invariance groups of the bilinear forms and their relation to the physically allowed local operations. In Section \ref{spen} the conceptual framework used for characterizing entanglement properties with polynomial invariants is described.
Sections \ref{ent}-\ref{getto} contain the results.
In particular, section \ref{ent} describes the method for constructing candidate polynomial entanglement invariants. In section \ref{three} the case of three spinors is considered, in section \ref{four} the case of four spinors is considered, and in section \ref{five} the case of five spinors is considered. Section \ref{getto} describes the degree 2 polynomials for any even number of spinors. In Section \ref{lli} some general properties of polynomial invariants are reviewed in relation to the constructed polynomials and how they can be used to characterize spinor entanglement. Section \ref{diss} is the discussion and conclusions.
\section{Dirac spinors}\label{dir}
The Dirac equation was introduced in Ref. \cite{dirac2} as a relativistic description of a spin-$\frac{1}{2}$ particle, or Dirac particle. For a particle with mass $m$ and charge $q$ coupled to an electromagnetic four-potential $A_{\mu}(x)$ it can be expressed, with natural units $\hbar=c=1$, on the form
\begin{eqnarray}\label{vanilla}
\left[\sum_{\mu}\gamma^\mu(i\partial_{\mu}-qA_{\mu}(x)) -m\right]\psi(x)=0.
\end{eqnarray}
Here $\psi(x)$ is a four component Dirac spinor
\begin{eqnarray}\label{spinor}
\psi(x)\equiv
\begin{pmatrix}
\psi_0(x) \\
\psi_1(x)\\
\psi_2(x) \\
\psi_3(x) \\
\end{pmatrix},
\end{eqnarray}
where each component is a function of the four-vector $x$, and $\gamma^0,\gamma^1,\gamma^2,\gamma^3$ are $4\times 4$ matrices
satisfying the anticommutator relations
\begin{eqnarray}\label{anti}
\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2g^{\mu\nu}I,
\end{eqnarray}
where $g^{\mu\nu}$ is the Minkowski metric with signature $(+---)$. For a derivation of the Dirac equation see e.g. Ref. \cite{dirac2} or Ref. \cite{dirac} Ch. XI.
The matrices $\gamma^0,\gamma^1,\gamma^2,\gamma^3$ are not uniquely defined by Eq. (\ref{anti}) and can be chosen in different physically equivalent ways.
The choice we use here is the so called Dirac matrices or { \it gamma matrices} which are defined as
\begin{align}
\gamma^0&=
\begin{pmatrix}
I & 0 \\
0 & -I \\
\end{pmatrix},
&\gamma^1=
\begin{pmatrix}
0 & \sigma^1 \\
-\sigma^1 & 0 \\
\end{pmatrix},\nonumber\\
\gamma^2&=
\begin{pmatrix}
0 & \sigma^2 \\
-\sigma^2 & 0 \\
\end{pmatrix},
&\gamma^3=
\begin{pmatrix}
0 & \sigma^3 \\
-\sigma^3 & 0 \\
\end{pmatrix},
\end{align}
where $I$ is the $2\times 2$ identity matrix and $\sigma^1,\sigma^2,\sigma^3$ are the Pauli matrices
\begin{eqnarray}
I=
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix},\phantom{o}
\sigma^1=
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix},\phantom{o}
\sigma^2=
\begin{pmatrix}
0 & -i \\
i & 0 \\
\end{pmatrix},\phantom{o}
\sigma^3=
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}.
\end{eqnarray}
Two different choices of $4\times4$ matrices $\gamma^0,\gamma^1,\gamma^2,\gamma^3$ and $\gamma'^0,\gamma'^1,\gamma'^2,\gamma'^3$ that both satisfy the anticommutation relations in Eq. (\ref{anti}) are related by a similarity transformation.
By Pauli's Fundamental Theorem this similarity transformation is unique up to a constant factor.
\begin{ptheorem}
If for two sets of $4\times4$ matrices $\gamma^0,\gamma^1,\gamma^2,\gamma^3$ and $\gamma'^0,\gamma'^1,\gamma'^2,\gamma'^3$ we have that $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}I=\{\gamma'^\mu,\gamma'^\nu\}$, then there exist an $S\in \mathrm{GL}(4,\mathbb{C})$ such that $\gamma'^\mu=S\gamma^\mu S^{-1}$, and $S$ is unique up to a multiplicative constant.
\end{ptheorem}
\begin{proof}
See e.g. Ref. \cite{pauli} or Ref. \cite{messiah}.
\end{proof}
From here on we regularly suppress the four-vector dependence of the spinor $\psi(x)$ where it is not essential and write it as $\psi$.
The Dirac equation can be written on a form where the time derivative $\partial_0\psi$ is separated from the remaining expression
\begin{eqnarray}\label{ggh}
\left[-\sum_{\mu=1,2,3}\gamma^0\gamma^\mu(i\partial_{\mu}-qA_{\mu})+qA_0I +m\gamma^0\right]\psi=i\partial_0\psi,\nonumber\\
\end{eqnarray}
and the Dirac Hamiltonian $H_D$ can be identified in Eq. (\ref{ggh}) as
\begin{eqnarray}
H_D=-\sum_{\mu=1,2,3}\gamma^0\gamma^\mu(i\partial_{\mu}-qA_{\mu})+qA_0I +m\gamma^0.
\end{eqnarray}
We can identify two matrices with useful properties in the algebra generated by the gamma matrices that are frequently featured in the following.
The first is the matrix
\begin{eqnarray}
C\equiv i\gamma^1\gamma^3=
\begin{pmatrix}
-\sigma^2 & 0 \\
0 & -\sigma^2 \\
\end{pmatrix},
\end{eqnarray}
which has the property that for each gamma matrix $\gamma^\mu$ and its transpose $\gamma^{\mu T}$ we have that
\begin{eqnarray}\label{uub}
C\gamma^\mu C=\gamma^{\mu T}.
\end{eqnarray}
Moreover, $C$ is Hermitian and its own inverse, i.e., $C=C^\dagger=C^{-1}$.
The second is the matrix
\begin{eqnarray}
\gamma^5\equiv i\gamma^0\gamma^1\gamma^2\gamma^3=
\begin{pmatrix}
0 & I \\
I & 0 \\
\end{pmatrix},
\end{eqnarray}
which anticommutes with each of the $\gamma^\mu$
\begin{eqnarray}
\gamma^5\gamma^\mu+\gamma^\mu\gamma^5=0.
\end{eqnarray}
Moreover, $\gamma^5$ is real Hermitian and its own inverse, i.e., $\gamma^5=\gamma^{5\dagger}=\gamma^{5T}=(\gamma^{5})^{-1}$.
For the case of zero particle momentum and zero four-potential there is an invariant subspace defined by the projector $P_+=\frac{1}{2}(I+\gamma^0)$ and an invariant subspace defined by the projector $P_-=\frac{1}{2}(I-\gamma^0)$. The former subspace is an eigenspace of the Dirac Hamiltonian with eigenvalue $+m$ and is often identified with a non-relativistic free spin-$\frac{1}{2}$ particle. The latter subspace is an eigenspace of the Dirac Hamiltonian with eigenvalue $-m$ and is often identified with a non-relativistic free spin-$\frac{1}{2}$ antiparticle (See e.g. Ref. \cite{peskin} Ch. 3.5.). The spinors $\psi_+$ in the subspace defined by $P_+$ and the spinors $\psi_-$ in the subspace defined by $P_-$ are of the form
\begin{eqnarray}\label{weyn}
\psi_+=
\begin{pmatrix}
{\psi}_0 \\
{\psi}_1\\
0 \\
0 \\
\end{pmatrix},\phantom{o}
\psi_-=
\begin{pmatrix}
0 \\
0\\
{\psi}_2 \\
{\psi}_3 \\
\end{pmatrix}.
\end{eqnarray}
We can see in Eq. (\ref{weyn}) that spinors in both these subspaces have only two nonzero spinor components.
The Dirac equation for the case of zero mass was considered by Weyl in Ref. \cite{weyl}
\begin{eqnarray}\label{weyl}
\left[-\sum_{\mu=1,2,3}\gamma^0\gamma^\mu(i\partial_{\mu}-qA_{\mu})+qA_0I\right]\psi=i\partial_0\psi.
\end{eqnarray}
For the zero mass equation there is an invariant subspace defined by the projector $P_L=\frac{1}{2}(I-\gamma^5)$ called the left-handed chiral subspace, and an invariant subspace defined by the projector $P_R=\frac{1}{2}(I+\gamma^5)$ called the right-handed chiral subspace. Solutions to Eq. (\ref{weyl}) that belong to the right-handed subspace are called right-handed Weyl particles $\psi_R$ and solutions that belong to the left-handed subspace are called left-handed Weyl particles $\psi_L$. These have the form
\begin{eqnarray}\label{wey}
\psi_R=
\begin{pmatrix}
{\psi}_0 \\
{\psi}_1\\
{\psi}_0 \\
{\psi}_1 \\
\end{pmatrix},\phantom{o}
\psi_L=
\begin{pmatrix}
{\psi}_0 \\
{\psi}_1\\
-{\psi}_0 \\
-{\psi}_1 \\
\end{pmatrix}.
\end{eqnarray}
We can see in Eq. (\ref{wey}) that spinors in both the left- and right-handed chiral subspaces have only two independent spinor components.
Dirac or Weyl particles in solid state and molecular systems are quasiparticle excitations. As such their physical interpretation is fundamentally different from that of Dirac or Weyl particles in relativistic quantum mechanics. Even so, they can be described by four component spinors and their evolutions are generated by Dirac-like Hamiltonians.
In the 2D Dirac semimetal graphene the evolution of a Dirac particle is generated by a Hamiltonian that can be written on the form
\begin{eqnarray}\label{2d}
H_{2D}=iv_D\gamma^0\sum_{\mu=1,2}\gamma^\mu\partial_\mu-\mu_PI,
\end{eqnarray}
where $v_D$ is the Dirac velocity and $\mu_P$ is the deviation from the half-filling value of the chemical potential (See e.g. Ref. \cite{kotov} or \cite{vass}).
Similarly, in a 3D Dirac semimetal the Hamiltonian for a massless Dirac particle can be written on the form \cite{bohm}
\begin{eqnarray}\label{3d}
H_{3D}=iv_D\gamma^0\sum_{\mu=1,2,3}\gamma^\mu\partial_\mu.
\end{eqnarray}
Two examples of experimentally realized 3D Dirac semimetals are sodium bismuthide (Na$_3$Bi) \cite{fang,liu} and cadmium arsenide (Cd$_3$As$_2$) \cite{fang2,neupane,cava}.
\subsection{Describing solutions of the Dirac equation}
A solution to the Dirac equation or a Dirac-like equation can be expanded for any time $t$ in a set of basis modes $\phi_je^{i\bold{k}\cdot\bold{x}}$ as
\begin{eqnarray}\label{bass}
\psi(t,\bold{x})=\int_{\bold{k}}d\bold{k}\sum_{j}\psi_{j,\bold{k}}(t)\phi_je^{i\bold{k}\cdot\bold{x}},
\end{eqnarray}
where $\bold{k}$ is a wave three-vector, $\bold{x}$ is a spatial three-vector, the $\psi_{j,\bold{k}}(t)$ are complex numbers, and the $\phi_j$ are four spinors that form a basis for the spinorial degree of freedom
\begin{eqnarray}\label{basis}
{\phi_0}=
\begin{pmatrix}
1 \\
0 \\
0 \\
0 \\
\end{pmatrix},\phantom{o}
{\phi_1}=
\begin{pmatrix}
0 \\
1 \\
0 \\
0 \\
\end{pmatrix},\phantom{o}
{\phi_2}=
\begin{pmatrix}
0 \\
0 \\
1 \\
0 \\
\end{pmatrix},\phantom{o}
{\phi_3}=
\begin{pmatrix}
0 \\
0 \\
0 \\
1 \\
\end{pmatrix}.
\end{eqnarray}
On the space of these solutions Dirac introduced an inner product defined as
\begin{eqnarray}\label{inn}
(\psi(t),\varphi(t))=\int_{\bold{x}}d\bold{x}\psi^\dagger(t,\bold{x})\varphi(t,\bold{x}).
\end{eqnarray}
The Dirac inner product of two basis modes is in general not well defined. This is because the basis modes extend over all three-space, i.e., they are infinite plane waves. Since the support of these infinite plane waves is not bounded the integral in Eq. (\ref{inn}) does not converge for the inner products of basis modes with the same basis spinor $\phi_j$, i.e., inner products of the form $(\phi_je^{i\bold{k}\cdot\bold{x}},\phi_je^{i\bold{k'}\cdot\bold{x}})$ for any $\bold{k}$ and $\bold{k'}$.
Because of this non-convergence Dirac chose to disregard the definition of the inner product and instead ad hoc impose the orthogonality relations $(\phi_je^{i\bold{k}\cdot\bold{x}},\phi_le^{i\bold{k'}\cdot\bold{x}})=\delta_{jl}\delta(\bold{k}-\bold{k'})$ where $\delta_{jl}$ is the Kronecker delta and $\delta(\bold{k}-\bold{k'})$ is the Dirac delta (See Ref. \cite{dirac} Ch. IV {\S} 23). Subsequent works tried to find a mathematically meaningful interpretation of these orthogonality relations and this gave rise to the formalism of generalized eigenfunctions and rigged Hilbert spaces \cite{gelfand,maurin}.
In the rigged Hilbert space approach the momentum eigenmodes are not allowed as physical states. Instead only Schwartz functions are physically allowed. The Schwartz functions are the functions that are infinitely differentiable and such that for sufficiently large $\bold{x}$ the absolute values of the function and its derivatives to all orders decrease more rapidly than any inverse power of $|\bold{x}|$ (See e.g. Ref. \cite{stein}). Due to this rapid decrease of a Schwartz function outside a finite spatial region the Dirac inner product of any two Schwartz functions is well defined. Moreover, a Schwartz function can be said to be "localized" in a spatial region in the sense that it has negligible value outside it. Commonly used types of Schwartz functions are the Gaussian functions (See e.g. Ref. \cite{stein}) and the bump functions (See e.g. Ref. \cite{lee}).
Furthermore, a Schwartz function in spatial three-space is a Schwartz function also in momentum three-space. If the support of a Schwartz function in momentum space is contained in a sufficiently small closed ball it cannot be experimentally distinguished from the single point support of a momentum eigenmode (See \ref{opp} for a discussion).
A different approach to make the inner product well defined is to only consider modes in a finite spatial rectangular volume and impose periodic boundary conditions, so called box quantization (See e.g. Refs. \cite{mandl} and \cite{wightman}). Then there is only a countable set of allowed momentum modes and these satisfy the orthogonality relations $(\phi_je^{i\bold{k}\cdot\bold{x}},\phi_le^{i\bold{k'}\cdot\bold{x}})=V\delta_{\bold{k},\bold{k'}}\delta_{jl}$ where $V$ is the volume of the box and $\delta_{\bold{k},\bold{k'}}$ and $\delta_{jl}$ are Kronecker deltas. If the box is made sufficiently large the discrete set of $\bold{k}$ cannot be experimentally distinguished from a continuous set (See \ref{opp} for a discussion).
Note that box quantization is closely related to the introduction of a large length scale cutoff, i.e., an Infrared Cutoff (See e.g. Refs. \cite{wightman} and \cite{duncan}).
In this work we consider states with definite momenta as was done in References \cite{czachor,alsing,pachos,mano,caban3,caban,moradi,caban2,spinorent}.
Moreover, we use the momentum eigenbasis and assume that the modes are orthogonal and normalizable. As an exact description this is possible only if we assume boundary conditions that allow for definite momenta in a finite spatial volume as in the box quantization approach. In the rigged Hilbert space approach where such boundary conditions are not allowed it can only be an approximate description.
Thus we assume that the physical scenario is such that it is warranted to treat a particle as having both a definite momentum and being contained in a finite spatial volume, whether as an exact or approximate description.
With these qualifying remarks we can consider the subspace spanned by only spinorial degrees of freedom that is defined by a fixed momentum $\bold{k}$, i.e., the subspace spanned by the modes $\phi_je^{i\bold{k}\cdot\bold{x}}$ for the given fixed $\bold{k}$. On such a four-dimensional subspace we can consider the inner product $(\cdot,\cdot)_{\bold{k}}$ given by
\begin{eqnarray}
(\psi(t),\varphi(t))_{\bold{k}}=\psi^\dagger(t)\varphi(t).
\end{eqnarray}
\subsection{Describing multiple spacelike separated Dirac spinors}
In this work we consider a scenario with multiple spacelike separated indistinguishable Dirac particles.
As in Ref. \cite{spinorent} we introduce a number of laboratories and assume that each laboratory contains a single Dirac particle. Each laboratory is spacelike separated from the other laboratories and comes with its own local description of spacetime. We assume that each of the laboratories
use a Minkowski space for this purpose. Thus we describe each different particle as belonging to a different Minkowski space. These different Minkowski spaces could be either the same Minkowski space described by different spacelike separated observers or alternatively the different Minkowski tangent-spaces of different spacelike separated points in a curved spacetime described by General Relativity (See e.g. Ref. \cite{wald}).
As was done in References \cite{alsing,pachos,caban3,moradi,caban2,spinorent} we assume that for any number of spacelike separated particles that have not interacted their state can be described as a tensor product $\psi_1(t){\,\otimes\,}imes \psi_2(t){\,\otimes\,}imes \psi_3(t){\,\otimes\,}imes\dots$ of single particle states.
Each single particle state has its support contained in a single laboratory. Note that the tensor product structure used here is with respect to the laboratories and not the particles. We assume that the particles are indistinguishable and thus the state $\psi_1(t){\,\otimes\,}imes \psi_2(t){\,\otimes\,}imes \psi_3(t){\,\otimes\,}imes\dots$ expresses the presence of one particle in the single particle state $\psi_1(t)$ in the first laboratory, one particle in the single particle state $\psi_2(t)$ in the second laboratory, one particle in the single particle state $\psi_3(t)$ in the third laboratory and so on. The particles are not individuated and not given labels. Instead we refer to them by the laboratory they are found in.
Further, we assume that a basis for the multi-particle states can be constructed as the tensor products of the elements of the single particle bases $\phi_{j_1}e^{i\bold{k_1}\cdot\bold{x_1}}{\,\otimes\,}imes \phi_{j_2}e^{i\bold{k_2}\cdot\bold{x_2}}{\,\otimes\,}imes \phi_{j_3}e^{i\bold{k_3}\cdot\bold{x_3}}{\,\otimes\,}imes\dots$.
The assumption that the state of the particles can be described by such a tensor product structure is not trivial. The reason why this assumption is often made is that operations on any given particle can be made jointly with operations on any other spacelike separated particle, i.e., such operations commute. However, it is not known if a description in terms of commuting operator algebras is always equivalent to a description in terms of tensor product spaces \cite{navascues,tsirelson,werner}. This open question is called Tsirelson's Problem \cite{tsirelson}. Nevertheless, if the algebra of operations is finite dimensional for each observer however it has been shown that a tensor product structure can be assumed without loss of generality \cite{tsirelson,werner}. In particular this holds if the Hilbert space of the shared system is finite dimensional. Thus it holds for operations on a subspace defined by fixed particle momenta of a finite number of particles.
The solutions to the Dirac equation are for some purposes reinterpreted as operators. This is done for example in the context of relativistic Quantum Field Theory formalisms where the Dirac spinor is reinterpreted as an operator valued Dirac field acting on a Hilbert space. Such an operator can be made a bosonic operator by imposing equal-time canonical commutation relations
\begin{eqnarray}
&&[\psi_a(\bold{x}),\psi^{\dagger}_b(\bold{x'})]=\delta(\bold{x}-\bold{x'})\delta_{ab},\nonumber\\
&&{[}\psi_a(\bold{x}),\psi_b(\bold{x'}){]}=0,\nonumber\\
&&{[}\psi_a^{\dagger}(\bold{x}),\psi_b^{\dagger}(\bold{x'}){]}=0,
\end{eqnarray}
where $a,b$ label the spinor components, $\delta_{ab}$ is the Kronecker delta, and $\delta(\bold{x}-\bold{x'})$ is the Dirac delta (See e.g. Ref. \cite{peskin} Ch. 3.5.). Alternatively the operator can be made a fermionic operator by imposing equal-time canonical anticommutation relations
\begin{eqnarray}
&&\{\psi_a(\bold{x}),\psi^{\dagger}_b(\bold{x'})\}=\delta(\bold{x}-\bold{x'})\delta_{ab},\nonumber\\
&&{\{}\psi_a(\bold{x}),\psi_b(\bold{x'}){\}}=0,\nonumber\\
&&{\{}\psi_a^{\dagger}(\bold{x}),\psi_b^{\dagger}(\bold{x'}){\}}=0,
\end{eqnarray}
(See e.g. Ref. \cite{peskin} Ch. 3.5.).
If the support of two operator valued Dirac fields are spacelike separated the fields either commute, in the bosonic case, or anticommute, in the fermionic case. The property of commutation or anticommutation at spacelike separation ensures that causality is not violated, i.e., that no signals can be sent between spacelike separated events, and is one of the so called Wightman axioms for Quantum Field Theory (See e.g. Ref. \cite{all that} Ch. 3-1). This axiom originally formulated for Minkowski spacetime is straightforwardly generalized to curved spacetimes (See e.g. Ref. \cite{holl} for a discussion).
In this work we have assumed that the particles are all spacelike separated from each other.
We therefore assume that the spinors either commute or anticommute.
The only difference between imposing the commutation relations and imposing the anticommutation relations in this scenario is an overall sign that depends on the ordering of the operators. Such an overall sign has no physical meaning and is chosen by convention. Therefore we do not impose any bosonic or fermionic nature on the particles. For the purpose of this work it is irrelevant if the particles are are fermions or bosons. Note however that given some commonly made assumptions the spin statistics connection implies that Dirac particles must be fermions (See Ref. \cite{spin} and Ref. \cite{all that} Ch. 4-4).
Finally we comment on the case of distinguishable particles. If the particles are distinguishable they can be individuated based on intrinsic properties and given physically meaningful labels. These labels constitute a further degree of freedom that enlarges the Hilbert space. In a given laboratory in place of the single particle state $\psi_1(t)$ we would have a collection of single particle states $\psi_1(t)^J$ where $J$ indicate the particle species.
However if the assumption is made that in any given laboratory only one particle species can be found we can absorb the particle labels into the laboratory labels. With this restrictive assumption the Hilbert space is of a system of distinguishable particles is equivalent to the Hilbert space of a system of indistinguishable particles in the scenario considered in this work.
\section{Spinor representation of the Lorentz group and the charge conjugation}\label{rep}
In General Relativity a spacetime is described by a four-dimensional manifold and at every non-singular point one can define a four-dimensional tangent vector space. Any such tangent space is isomorphic to the Minkowski space (See e.g. Ref. \cite{wald}).
Here, as in Ref. \cite{spinorent}, we make the assumption that it is physically motivated to neglect the local curvature of spacetime and treat a Dirac particle as belonging to a Minkowski tangent space instead of the underlying spacetime manifold. Without this assumption the Dirac equation would have to be replaced by a curved spacetime counterpart such as that introduced by Weyl \cite{weyl} and Fock \cite{fock}.
A Lorentz transformation is a coordinate transformation on the local Minkowski tangent space to a spacetime point, but it also induces an action on the Dirac spinor in the point. This action is given by the spinor representation of the Lorentz transformation. Let $\Lambda$ be a Lorentz transformation and $S(\Lambda)$ be the spinor representation of $\Lambda$. Then the spinor transforms as $\psi(x)\to \psi'(x')=S(\Lambda)\psi(x)$ where $x'=\Lambda x$ (See e.g. Ref. \cite{zuber}), and the Dirac equation transforms as
\begin{eqnarray}
&&\left[\sum_{\mu}\gamma^\mu(i\partial_{\mu}-qA_{\mu}) -m\right]\psi(x)=0\nonumber\\
\to&&\left[\sum_{\mu,\nu}\gamma^\mu(\Lambda^{-1})^{\nu}_{\mu}(i\partial_{\nu}-qA_{\nu}) -m\right]S(\Lambda)\psi(x)=0.
\end{eqnarray}
The Lorentz invariance of the Dirac equation implies that
\begin{eqnarray}
S^{-1}(\Lambda)\gamma^\mu S(\Lambda)=\sum_\nu\Lambda^{\mu}_{\nu}\gamma^\nu.
\end{eqnarray}
The Lorentz group is a Lie group with four connected components. The connected component that contains the identity element is the proper orthochronous Lorentz group.
Likewise, the spinor representation of the Lorentz group is also a Lie group with four connected components. The connected component of this group that contains the identity element, the spinor representation of the proper orthochronous Lorentz group, can be generated by the exponentials of its Lie algebra.
This Lie algebra has six generators $S^{\rho\sigma}$ defined by
\begin{eqnarray}\label{gene}
S^{\rho\sigma}=\frac{1}{4}[\gamma^\rho,\gamma^\sigma]=\frac{1}{2}\gamma^\rho\gamma^\sigma-\frac{1}{2}g^{\rho\sigma}I,
\end{eqnarray}
where as before $g^{\rho\sigma}$ is the Minkowski metric with signature $(+---)$.
The spinor representations of spatial rotations are generated by $S^{12},S^{13}$, and $S^{23}$ while the spinor representations of the Lorentz boosts are generated by $S^{01},S^{02}$, and $S^{03}$.
By taking the exponential of an element in the Lie algebra a finite transformation can be obtained
\begin{eqnarray}
S(\Lambda)=\exp\left(\frac{1}{2}\sum_{\rho,\sigma} \omega_{\rho\sigma}S^{\rho\sigma}\right),
\end{eqnarray}
where the coefficients $\omega_{\rho\sigma}$ are real numbers. The spinor representation of any proper orthochronous Lorentz transformation
can be described as a product of such finite transformations. See e.g. Ref. \cite{zuber}.
The four connected components of the Lorentz group are related to each other by the parity inversion P and the time reversal T transformations. Likewise, the four connected components of the spinor representation of Lorentz group are related to each other by the spinor representations of the parity inversion P and the time reversal T transformations. These spinor representations of P and T are only defined up to a multiplicative U(1) factor that can be chosen in different physically equivalent ways.
The spinor representation of the parity inversion transformation can be chosen as
\begin{eqnarray}
S(\textrm{P})=\gamma^0.
\end{eqnarray}
The spinor representation of the time reversal transformation T involves the matrix $C$ and the complex conjugation of the spinor and can be chosen as $\psi \to C\psi^*$.
Besides the Lorentz group we can consider the charge conjugation transformation C, as well as the charge parity CP and charge parity time CPT transformation. As with P and T, the spinor representation of C is only defined up to a multiplicative U(1) factor that can be chosen in different ways.
The spinor representation of the charge conjugation, like the time reversal, involves complex conjugation of the spinor and can be chosen as $\psi \to i\gamma^2\psi^*$. It follows that the CP transformation is $\psi \to -i\gamma^0\gamma^2\psi^*= iC\gamma^5\psi^*$ and the CPT transformation is given by the matrix $-i\gamma^5$
\begin{eqnarray}
S(\textrm{CPT})=-i\gamma^5.
\end{eqnarray}
See e.g. Ref. \cite{bjorken} Ch. 5. In the following we use these choices of the spinor representations of the P, T and C transformations.
\section{Bilinear forms invariant under the spinor representation of the proper orthochronous Lorentz group}
\label{invariants}
A physical quantity that transforms under some representation of the Lorentz group is called a Lorentz covariant.
If it is also invariant under the action it is called a Lorentz invariant. A quantity that is invariant under the action of a representation of the proper orthochronous Lorentz group may not be invariant under the representation of the full Lorentz group but in the following we still refer to such a quantity as a Lorentz invariant for convenience.
Lorentz invariants can be constructed as bilinear forms on the Dirac spinors (See e.g. Ref. \cite{pauli}).
Given the properties of the matrix $C$ described in Eq. (\ref{uub}) and the form of the generators $S^{\rho\sigma}$ of the spinor representation of the proper orthochronous Lorentz group in Eq. (\ref{gene}) we have that the transpose $S^{\rho\sigma T}$ satisfies
\begin{eqnarray}
S^{\rho\sigma T}=\frac{1}{4}[\gamma^{\sigma T},\gamma^{\rho T}]=-\frac{1}{4}C[\gamma^{\rho},\gamma^{\sigma}]C=-CS^{\rho\sigma}C.
\end{eqnarray}
Thus it holds for any finite transformation $S(\Lambda)$ that $S(\Lambda)^TC=CS(\Lambda)^{-1}$.
This allows us to construct a Lorentz invariant bilinear form as
\begin{eqnarray}
\psi^TC\varphi,
\end{eqnarray}
where $\psi$ and $\varphi$ are Dirac spinors.
This bilinear form transforms as $\psi^TS(\Lambda)^TCS(\Lambda)\varphi=\psi^TCS(\Lambda)^{-1}S(\Lambda)\varphi=\psi^TC\varphi$ for any spinor representation $S(\Lambda)$ of a proper orthochronous Lorentz transformation. Moreover $\psi^TC\varphi$ is invariant under parity inversion P.
This follows since $\gamma^0=(\gamma^{0})^{T}=(\gamma^{0})^{-1}$ and $\gamma^0C=C\gamma^0$ and thus $\psi^TS(\textrm{P})^TCS(\textrm{P})\varphi=\psi^T\gamma^0C\gamma^0\varphi=\psi^TC\varphi$.
However, $\psi^TC\varphi$ is not invariant under the CPT transformation but changes sign.
This follows since $S(\textrm{CPT})=-i\gamma^{5}$ and $\gamma^5C=C\gamma^5$ and $\gamma^5=(\gamma^{5})^{T}=(\gamma^{5})^{-1}$ and thus $\psi^TS(\textrm{CPT})^TCS(\textrm{CPT})\varphi=-\psi^T\gamma^5C\gamma^5\varphi=-\psi^TC\varphi$.
Next we recall that the matrix $\gamma^5$ anti-commutes with all $\gamma^{\mu}$. Therefore we see that it commutes with any generator $S^{\rho\sigma}$
\begin{eqnarray}
[S^{\rho\sigma},\gamma^5]=\frac{1}{4}[\gamma^{\rho},\gamma^{\sigma}]\gamma^5-\frac{1}{4}\gamma^5[\gamma^{\rho},\gamma^{\sigma}]=0.
\end{eqnarray}
It follows that $\gamma^5$ commutes with the spinor representation of any proper orthochronous Lorentz transformation $S(\Lambda)\gamma^5=\gamma^5S(\Lambda)$.
Therefore, we can construct another Lorentz invariant bilinear form as
\begin{eqnarray}
\psi^TC\gamma^5\varphi.
\end{eqnarray}
This bilinear form transforms as $\psi^TS(\Lambda)^TC\gamma^5S(\Lambda)\varphi=\psi^TCS(\Lambda)^{-1}\gamma^5S(\Lambda)\varphi=\psi^TC\gamma^5S(\Lambda)^{-1}S(\Lambda)\varphi=\psi^TC\gamma^5\varphi$.
Moreover, $\psi^TC\gamma^5\varphi$ is not invariant under parity inversion but changes sign.
This follows since $\gamma^0\gamma^5=-\gamma^5\gamma^0$ and thus $\psi^TS(\textrm{P})^TC\gamma^5S(\textrm{P})\varphi=\psi^T\gamma^0C\gamma^5\gamma^0\varphi=-\psi^TC\gamma^5\varphi$.
Furthermore, $\psi^TC\gamma^5\varphi$ is not invariant under the CPT transformation but changes sign.
This follows since $S(\textrm{CPT})=-i\gamma^{5}$ and thus $\psi^TS(\textrm{CPT})^TC\gamma^5S(\textrm{CPT})\varphi=-\psi^T\gamma^5C\gamma^5\gamma^5\varphi=-\psi^TC\gamma^5\varphi$.
The two bilinear forms $\psi^TC\varphi$ and $\psi^TC\gamma^5\varphi$ are both skew-symmetric, i.e., $\psi^TC\varphi=-\varphi^TC\psi$ and $\psi^TC\gamma^5\varphi=-\varphi^TC\gamma^5\psi$ due to the anti-symmetry of $C$ and $C\gamma^5$ respectively. Thus $\psi^TC\psi=0$ and $\psi^TC\gamma^5\psi=0$. Moreover, the two bilinear forms are both non-degenerate.
If $\xi^TC\gamma^5\chi=0$ for all $\chi$ it follows that $\xi=0$, and if $\xi^TC\gamma^5\chi=0$ for all $\xi$ it follows that $\chi=0$. Similarly, if $\xi^TC\chi=0$ for all $\chi$ it follows that $\xi=0$, and if $\xi^TC\chi=0$ for all $\xi$ it follows that $\chi=0$.
Note that because the U(1) phases of the spinor representations of P, T and C have to be chosen the U(1) phases acquired by the two bilinear forms $\psi^TC\varphi$ and $\psi^TC\gamma^5\varphi$ under the $S(\textrm{P})$ transformation or under the $S(\textrm{CPT})$ transformation are determined by these choices. However the difference by a factor of $-1$ between the phase acquired by $\psi^TC\varphi$ under the $S(\textrm{P})$ transformation and the phase acquired by $\psi^TC\gamma^5\varphi$ under the $S(\textrm{P})$ transformation, is independent of these choices.
\section{Behaviour of the bilinear forms under unitary spinor evolution generated by Dirac-like Hamiltonians}\label{ham}
Here we consider a subspace defined by a fixed particle momentum $\bold{k}$, i.e., a subspace spanned by the four basis elements $\phi_je^{i\bold{k}\cdot\bold{x}}$ with the same $\bold{k}$. Furthermore, as was done in Ref. \cite{spinorent}, we consider an evolution that acts unitarily on such a subspace and is generated by a Hamiltonian operator $H$. For the subspace to be invariant under the evolution it is required that $(\phi_je^{i\bold{k}\cdot\bold{x}},H\phi_le^{i\bold{k'}\cdot\bold{x}})\propto\delta_{\bold{k},\bold{k'}}$. To have such unitary action on the subspace we consider evolution generated by Hamiltonians that do not depend on the spatial coordinate $\bold{x}$.
Given the restriction to evolutions generated by Hamiltonians without spatial dependence we can consider the behaviour of the bilinear forms $\psi^TC\varphi$ and $\psi^TC\gamma^5\varphi$ for such evolutions.
It can be shown \cite{spinorent} that the bilinear form $\psi^TC\varphi$ is invariant, up to a U(1) phase, under evolutions generated by any Hamiltonians on the form $H^{2,3}(t)+H^{0}(t)$ where
\begin{eqnarray}
H^{2,3}(t)=\gamma^\mu\gamma^\nu\phi_{\mu\nu}(t)+\gamma^\mu\gamma^\nu\gamma^\rho\kappa_{\mu\nu\rho}(t),
\end{eqnarray}
and $H^0(t)=f(t)I$. This follows from the relation $CH^{2,3}(t)=-(H^{2,3}(t))^TC$. Similarly it can be shown that the bilinear form $\psi^TC\gamma^5\varphi$ is invariant, up to a U(1) phase, under evolutions generated by any Hamiltonians on the form $H^{1,2}(t)+H^{0}(t)$ where
\begin{eqnarray}
H^{1,2}(t)=\gamma^\mu\eta_{\mu}(t)+\gamma^\mu\gamma^\nu\lambda_{\mu\nu}(t).
\end{eqnarray}
This follows from the relation $C\gamma^5H^{1,2}(t)=-(H^{1,2}(t))^TC\gamma^5$.
See \ref{hamm} for a derivation of how the two bilinear forms behave under these kinds of evolutions.
The Dirac Hamiltonian contains a first degree term in the gamma matrices, the mass term $m\gamma^0$, a second degree term, the generalized canonical momentum term $\sum_{\mu=1,2,3}\gamma^{0}\gamma^{\mu}(i\partial_{\mu}-qA_{\mu}(t))$, as well as a zeroth degree term, the coupling to the scalar potential $qA_0(t)I$. Thus the Dirac Hamiltonian for a massive particle is of the type $H^{1,2}(t)+H^{0}(t)$ while the Dirac Hamiltonian for a massless particle is on both the form $H^{1,2}(t)+H^{0}(t)$ and on the form $H^{2,3}(t)+H^{0}(t)$.
Apart from the terms in the standard Dirac Hamiltonian we can consider some additional or alternative Hamiltonian terms from different physical models. One example is a coupling to a scalar potential such as a Yukawa scalar boson $g\gamma^0\phi$ \cite{yukawa}, but such terms behave analogously to a mass term. A coupling to a pseudo-scalar potential such as a Yukawa pseudo-scalar boson $gi\gamma^0\gamma^5\phi$ on the other hand is third degree in the gamma matrices. Another kind of additional term is a coupling to a pseudo-vector potential $\sum_{\mu=1,2,3}\gamma^{0}\gamma^5\gamma^{\mu}(A_{\mu}(t))$ which is second degree in gamma matrices (See e.g. Ref. \cite{thaller}). A tensor term $i\gamma^{0}\sum_{\mu\nu=1,2,3}[\gamma^\mu,\gamma^\nu]F_{\mu\nu}(t)$ is third degree in gamma matrices. An example is an anomalous magnetic moment described by a Pauli-coupling $i\gamma^{0}\sum_{\mu\nu=1,2,3}[\gamma^\mu,\gamma^\nu](\partial_\mu A_\nu(t)-\partial_\nu A_\mu(t))$ to the electromagnetic tensor (See e.g. Ref. \cite{thaller} or \cite{das}).
A pseudo-tensor term $i\gamma^{0}\gamma^5\sum_{\mu\nu=1,2,3}[\gamma^\mu,\gamma^\nu]F_{\mu\nu}(t)$ is first degree in gamma matrices (See e.g. Ref. \cite{thaller}).
Finally we can consider a chiral coupling of electroweak type to a vector boson, e.g. $\sum_\mu g\gamma^0\gamma^\mu(I\pm\gamma^5)Z_{\mu}$ \cite{weinberg}, which has terms of degree 2 and 4 in the gamma matrices.
The bilinear forms ${\psi^T}C{\varphi}$ and ${\psi^T}C\gamma^5{\varphi}$ are not invariant under evolution generated by Hamiltonians that have both a mass term or a coupling to a scalar potential and also a coupling to a pseudo-scalar or a tensor term. Neither are they invariant under evolution generated by Hamiltonians that contain both a tensor and pseudo-tensor term.
Moreover, the bilinear forms ${\psi^T}C{\varphi}$ and ${\psi^T}C\gamma^5{\varphi}$ are not invariant under evolution generated by
Hamiltonians with chiral coupling of electroweak type to a vector boson.
The two bilinear forms ${\psi^T}C{\varphi}$ and ${\psi^T}C\gamma^5{\varphi}$ can be considered also in the context of Dirac or Weyl particles in solid state and molecular systems.
The Hamiltonian in Eq. (\ref{2d}) for Dirac particles in the 2D Dirac semimetal graphene \cite{kotov,vass}, and
the Hamiltonian in Eq. (\ref{3d}) for Dirac particles in 3D Dirac semimetals \cite{bohm} have only terms that are zeroth and second degree in the gamma matrices.
Therefore, for both these cases ${\psi^T}C\gamma^5{\varphi}$ and ${\psi^T}C{\varphi}$ are invariant up to a U(1) phase.
Terms that can be added to the Hamiltonians in this context are the Semenoff mass term $M_S\gamma^0\gamma^3$ \cite{semenoff} and the Haldane mass term $M_{H}\gamma^5\gamma^0\gamma^3$ \cite{haldane}. Both the Semenoff and Haldane mass terms are second degree in the gamma matrices, and thus ${\psi^T}C{\varphi}$ and ${\psi^T}C\gamma^5{\varphi}$ are still invariant, up to a U(1) phase, with these additions.
\section{The invariance groups of the bilinear forms}\label{hinn}
Here we consider the groups that preserve the bilinear form ${\psi^T}C\gamma^5{\varphi}$, up to a U(1) phase, the groups that preserve the bilinear form ${\psi^T}C{\varphi}$, up to a U(1) phase, and the groups that preserve both the bilinear forms up to a U(1) phase. The relations between these groups and the evolutions generated by Dirac-like Hamiltonians and the spinor representation of the proper orthochronous Lorentz group is also described. These groups and their relations to the evolutions generated by Dirac-like Hamiltonians and the spinor representation of the proper orthochronous Lorentz group were previously described in Ref. \cite{spinorent}.
As described in Section \ref{ham} the non-degenerate skew-symmetric bilinear form
${\psi^T}C\gamma^5{\varphi}$
is invariant, up to a U(1) phase, under any evolution generated by Dirac Hamiltonians. In greater generality it is invariant, up to a U(1) phase, under any evolution generated by Hamiltonians of the type $H^{1,2}(t)+H^0(t)$. These latter evolutions form a Lie group of real dimension 11 generated by the exponentials of the real Lie algebra spanned by the 11 skew-Hermitian matrices $i\gamma^0$, $\gamma^1$, $\gamma^2$, $\gamma^3$, $i\gamma^0 \gamma^1$, $i\gamma^0 \gamma^2$, $i\gamma^0 \gamma^3$, $\gamma^1 \gamma^2$, $\gamma^1 \gamma^3$, $\gamma^2 \gamma^3$ and $iI$. This group $G^{C\gamma^5}_U$ is isomorphic to $\mathrm{Sp}(2)\times\mathrm{ U}(1)$ where $\mathrm{Sp}(2)$ is the compact symplectic group of $4\times 4$ matrices (See e.g. Ref. \cite{hall} Ch. 1.2.8.). This follows since a symplectic group is defined as the set of linear transformations on a vector space over the complex numbers $\mathbb{C}$ that preserve a given non-degenerate skew-symmetric bilinear form. All such symplectic groups on the same vector space are isomorphic and their isomorphism class is defined as {\it the} symplectic group $\mathrm{Sp}(n,\mathbb{C})$, where $n$ is the dimension of the vector space. Moreover, the compact symplectic group $\mathrm{Sp}(n/2)$ is defined as the intersection of $\mathrm{Sp}(n,\mathbb{C})$ with $\mathrm{SU}(n)$, i.e., $\mathrm{Sp}(n/2)=\mathrm{Sp}(n,\mathbb{C})\cap \mathrm{SU}(n)$.
As described in Ref. \cite{spinorent} the set of unitary transformations that can be implemented by non-zero mass strongly continuous Dirac Hamiltonians is dense in the group $G^{C\gamma^5}_U$. Thus there is no continuous function that is invariant, up to a U(1) phase, under all evolutions generated by non-zero mass Dirac Hamiltonians that is not invariant, up to a U(1) phase, under $G^{C\gamma^5}_U$.
The invariance of ${\psi^T}C\gamma^5{\varphi}$, up to a U(1) phase, extends to a Lie group of real dimension 21 that contains $G^{C\gamma^5}_U$ as a subgroup. This larger group is generated by the exponentials of the real Lie algebra spanned by the 11 skew-Hermitian matrices $i\gamma^0$, $\gamma^1$, $\gamma^2$, $\gamma^3$, $i\gamma^0 \gamma^1$, $i\gamma^0 \gamma^2$, $i\gamma^0 \gamma^3$, $\gamma^1 \gamma^2$, $\gamma^1 \gamma^3$, $\gamma^2 \gamma^3$ and $iI$ and the 10 Hermitian matrices $\gamma^0$, $i\gamma^1$, $i\gamma^2$, $i\gamma^3$, $\gamma^0 \gamma^1$, $\gamma^0 \gamma^2$, $\gamma^0 \gamma^3$, $i\gamma^1 \gamma^2$, $i\gamma^1 \gamma^3$, and $i\gamma^2 \gamma^3$. This group $G^{C\gamma^5}$ is isomorphic to the group $\mathrm{Sp}(4,\mathbb{C})\times \mathrm{U}(1)$ where $\mathrm{Sp}(4,\mathbb{C})$ is the symplectic group of $4\times 4$ matrices (See e.g. Ref. \cite{hall} Ch. 1.2.4.). The group $G^{C\gamma^5}$ contains the spinor representation of the proper orthochronous Lorentz group. As described in Ref. \cite{spinorent} the group $G^{C\gamma^5}$ is the smallest Lie group that contains both $G^{C\gamma^5}_U$ and the spinor representation of the proper orthochronous Lorentz group as subgroups and any continuous function that is invariant, up to a U(1) phase, under both $G^{C\gamma^5}_U$ and the spinor representation of the proper orthochronous Lorentz group is invariant, up to a U(1) phase, under $G^{C\gamma^5}$.
Likewise, as described in Section \ref{ham} the non-degenerate skew-symmetric bilinear form
${\psi^T}C{\varphi}$
is invariant, up to a U(1) phase, under any evolution generated by zero-mass Dirac Hamiltonians. In greater generality it is invariant, up to a U(1) phase, under any evolution generated by Hamiltonians of the type $H^{2,3}(t)+H^0(t)$. These latter evolutions form a Lie group of real dimension 11 generated by the exponentials of the real Lie algebra spanned by the 11 skew-Hermitian matrices $\gamma^5\gamma^0$, $i\gamma^5\gamma^1$, $i\gamma^5\gamma^2$, $i\gamma^5\gamma^3$, $i\gamma^0 \gamma^1$, $i\gamma^0 \gamma^2$, $i\gamma^0 \gamma^3$, $\gamma^1 \gamma^2$, $\gamma^1 \gamma^3$, $\gamma^2 \gamma^3$ and $iI$. This group $G^C_U$ is isomorphic to $\mathrm{Sp}(2)\times\mathrm{ U}(1)$.
As described in Ref. \cite{spinorent} the set of unitary transformations that can be implemented by strongly continuous Dirac Hamiltonians with zero mass and a coupling to a Yukawa pseudoscalar boson is dense in the group $G^{C}_U$. Thus there is no continuous function that is invariant, up to a U(1) phase, under all evolutions generated by Dirac Hamiltonians with zero mass and a coupling to a Yukawa pseudoscalar boson that is not invariant, up to a U(1) phase, under $G^{C}_U$.
The invariance of ${\psi^T}C{\varphi}$, up to a U(1) phase, extends to a group of real dimension 21 that contains $G^{C}_U$ as a subgroup. This larger group is generated by the exponentials of the real Lie algebra spanned by the 11 skew-Hermitian matrices $\gamma^5\gamma^0$, $i\gamma^5\gamma^1$,$i\gamma^5\gamma^2$, $i\gamma^5\gamma^3$, $i\gamma^0 \gamma^1$, $i\gamma^0 \gamma^2$, $i\gamma^0 \gamma^3$, $\gamma^1 \gamma^2$, $\gamma^1 \gamma^3$, $\gamma^2 \gamma^3$ and $iI$ and the 10 Hermitian matrices $i\gamma^5\gamma^0$, $\gamma^5\gamma^1$, $\gamma^5\gamma^2$, $\gamma^5\gamma^3$, $\gamma^0 \gamma^1$, $\gamma^0 \gamma^2$, $\gamma^0 \gamma^3$, $i\gamma^1 \gamma^2$, $i\gamma^1 \gamma^3$, and $i\gamma^2 \gamma^3$. This group $G^C$ is isomorphic to the group $\mathrm{Sp}(4,\mathbb{C})\times \mathrm{U}(1)$, and contains the spinor representation of the proper orthochronous Lorentz group. As described in Ref. \cite{spinorent} the group $G^{C}$ is the smallest Lie group that contains both $G^{C}_U$ and the spinor representation of the proper orthochronous Lorentz group as subgroups and any continuous function that is invariant, up to a U(1) phase, under both $G^{C}_U$ and the spinor representation of the proper orthochronous Lorentz group is invariant, up to a U(1) phase, under $G^{C}$.
While the Lie groups $G^{C\gamma^5}_U$ and $G^C_U$ are isomorphic they are not identical and their intersection $G^{C\gamma^5}_U\cap G^C_U$ is a Lie group of real dimension 7 that contains evolutions generated by Hamiltonians with only second and zeroth degree terms in the gamma matrices. The spinor representations of the spatial rotations are also included in this group but not the spinor representations of the Lorentz boosts. As described in Ref. \cite{spinorent} the set of unitary transformations that can be implemented by zero mass Dirac Hamiltonians is dense in the group $G^{C\gamma^5}_U\cap G^C_U$. Thus there is no continuous function that is invariant, up to a U(1) phase, under all evolutions generated by zero mass Dirac Hamiltonians that is not invariant, up to a U(1) phase, under $G^{C\gamma^5}_U\cap G^C_U$.
Similarly, the Lie groups $G^{C\gamma^5}$ and $G^C$ are isomorphic but not identical and their intersection $G^{C\gamma^5}\cap G^C$ is a Lie group of real dimension 13 that also contains the evolutions generated by Hamiltonians with only second and zeroth degree terms in the gamma matrices and the spinor representations of the spatial rotations. Beyond this it contains non-unitary elements generated by the Hermitian second degree terms in the gamma matrices, which includes the spinor representations of the Lorentz boosts. As described in Ref. \cite{spinorent} the group $G^{C\gamma^5}\cap G^C$ is the smallest Lie group that contains both $G^{C\gamma^5}_U\cap G^C_U$ and the spinor representation of the proper orthochronous Lorentz group as subgroups and any continuous function that is invariant, up to a U(1) phase, under both $G^{C\gamma^5}_U\cap G^C_U$ and the spinor representation of the proper orthochronous Lorentz group is invariant, up to a U(1) phase, under $G^{C\gamma^5}\cap G^C$.
Besides the invariance of ${\psi^T}C\gamma^5{\varphi}$, up to a U(1) phase, under the connected Lie group $G^{C\gamma^5}$ and the invariance of ${\psi^T}C{\varphi}$, up to a U(1) phase, under the connected Lie group $G^C$ both the bilinear forms are invariant, up to a sign, under the 32 element finite group generated by the gamma matrices $\gamma^0,\gamma^1,\gamma^2,\gamma^3$, the so called {\it Dirac group} (See e.g. Ref. \cite{lomont}). As can be seen from Eq. (\ref{anti}) ${\psi^T}C{\varphi}$ is invariant under action by $\gamma^0$ but changes sign under action by $\gamma^1,\gamma^2$ and $\gamma^3$. Similarly we see from Eq. (\ref{anti}) that ${\psi^T}C\gamma^5{\varphi}$ is invariant under action by $\gamma^1,\gamma^2$ and $\gamma^3$ but changes sign under action by $\gamma^0$.
Note that if we choose a different physically equivalent set of matrices $\gamma'^0,\gamma'^1,\gamma'^2,\gamma'^3$ related to the gamma matrices $\gamma^0,\gamma^1,\gamma^2,\gamma^3$ by a similarity transformation $\gamma'^\mu=S\gamma^\mu S^{-1}$ where $S\in \mathrm{GL}(4,\mathbb{C})$ we can construct two skew-symmetric Lorentz invariant bilinear forms as ${\psi^T}(S^{-1})^TCS^{-1}{\varphi}$ and ${\psi^T}(S^{-1})^TC\gamma^5S^{-1}{\varphi}$.
The linear groups preserving these bilinear forms, up to a U(1) phase, are $SG^CS^{-1}$ and $SG^{C\gamma^5}S^{-1}$ respectively. Moreover, the set of evolutions generated by nonzero mass Dirac Hamiltonians is a dense subset of $SG_U^{C\gamma^5}S^{-1}$ and the set of evolutions generated by zero mass Dirac Hamiltonians with a coupling to a Yukawa pseudoscalar boson is a dense subset of $SG_U^{C}S^{-1}$.
\section{Describing spinor entanglement properties using polynomial invariants}\label{spen}
A general framework for describing entanglement properties of a composite system of non-relativistic particles was developed and described in References \cite{popescu} and \cite{carteret}. This general framework has been adapted to the case of Dirac spinors in Ref. \cite{spinorent} and it was described how Lorentz invariant homogeneous polynomials that are zero for product states and invariant, up to a U(1) phase, under physically allowed local unitary evolutions can be used to partially characterize inequivalent forms of spinor entanglement. Here we outline this conceptual framework previously described in Ref. \cite{spinorent} and describe the properties of polynomial invariants. We then discuss such Lorentz invariant homogeneous polynomials for the specific cases of physically allowed local unitary evolutions described in Section \ref{ham} and Section \ref{hinn}. Further, we consider the restrictions of these Lorentz invariant homogeneous polynomials to the case of zero momenta free particles in an energy eigenstate and to the case of Weyl particles. Finally we comment on the relation between the Lorentz invariant homogeneous polynomials and so called entanglement measures.
In general a system of multiple particles can be entangled in a variety of qualitatively different ways. The entanglement present in the system is typically considered to be preserved by changes of local reference frames and by local unitary evolution of the individual particles.
Therefore if two entangled states of the system are such that each of them can be transformed into the other deterministically through changes of local reference frames and local unitary evolution we may consider them to be equivalently entangled. Moreover, if two states are identical up to multiplication by a constant $c\in \mathbb{C}-\{0\}$ we may consider them to be equivalently entangled. Therefore, within the conceptual framework described in Ref. \cite{spinorent} two entangled states are considered equivalently entangled if they can be transformed into each other by changes of local reference frames, local unitary evolutions and multiplication by a constant, and inequivalently entangled otherwise.
Following the terminology of Ref. \cite{spinorent} we refer to the changes of local reference frames and the local unitary evolutions jointly as the {\it local reversible operations}.
In the general case one can identify a number of different properties of the entanglement and use these to distinguish between the inequivalent types of entanglement. By the above argument any such property describing the entanglement must be unchanged by all local reversible operations \cite{popescu,carteret,ben}. Moreover, no such entanglement property should be found in a product state, i.e., a state that can be created using only local resources.
For a system of multiple Dirac spinors we may following Ref. \cite{spinorent} identify the local reversible operations acting on the spinors as the
set of changes of local reference frames, i.e., the local spinor representations of the proper orthochronous Lorentz transformations and the set of local unitary spinor evolutions generated by the set of physically allowed Dirac Hamiltonians. Given this we have for a system of multiple Dirac particles with definite momenta three conditions that define a spinor entanglement property for pure states of such particles.
\begin{enumerate}
\item [(1)] Non-existence for any product state.
\item[(2)]Invariance under changes of local inertial reference frames, i.e., local Lorentz invariance.
\item [(3)] Invariance under local evolutions generated by physically allowed Dirac Hamiltonians that act unitarily on any fixed-momenta subspace.
\end{enumerate}
Next we can consider the the question of how to describe equivalence classes of states with the same entanglement properties.
A state $\psi_{ABC\dots}$ of a multipartite system belongs to the set $\mathcal{O}_{\psi_{ABC\dots}}$ that is made up of all states that can be obtained from $\psi_{ABC\dots}$ by local reversible operations. Following the terminology of Ref. \cite{spinorent} we call such a set $\mathcal{O}_{\psi_{ABC\dots}}$, an {\it orbit} of the local reversible operations. Any two different orbits are by definition disjoint and the Hilbert space of the system can be completely decomposed into the set of all such orbits. If the orbit $\mathcal{O}_{\psi_{ABC\dots}}$ is identical to the orbit $\mathcal{O}_{\phi_{ABC\dots}}$ up to element wise multiplication by a nonzero constant $c$, i.e., a map $m_c:\psi_{ABC\dots}\to c\psi_{ABC\dots}$ for some $c\in \mathbb{C}-\{0\}$, we consider them equivalent. We denote by $\tilde{\mathcal{O}}_{\psi_{ABC\dots}}$ the equivalence class of all orbits that are related to $\mathcal{O}_{\psi_{ABC\dots}}$ by the maps $m_{c}$ for all $c\in \mathbb{C}-\{0\}$.
Two states that belong to different equivalence classes are different with regard to some physical property that cannot be changed by any local reversible operation.
Therefore two entangled states $\psi_{ABC\dots}$ and $\phi_{ABC\dots}$ for which $\tilde{\mathcal{O}}_{\psi_{ABC\dots}}\neq\tilde{\mathcal{O}}_{\phi_{ABC\dots}}$ are by definition inequivalently entangled. The description of different kinds of entanglement in terms of inequivalence under local reversible operations has been considered for various systems of non-relativistic spin-$\frac{1}{2}$ particles (See e.g. References \cite{popescu,carteret,toni,higuchi,sud,tarrach,toumazet,ekert,grassl,kempe,lind2,car2}).
A method to characterize the different inequivalent types of entanglement in a system of particles is to construct parameters that distinguishes between different equivalence classes of entangled states. If we want such parameters to only distinguish between different inequivalent forms of entanglement and not between any other physical properties, we must require that these parameters are functions that take a constant value on any given equivalence class. Thus the parameters have to be invariant under any local reversible operation and invariant under multiplication of the state by any constant $c\in \mathbb{C}-\{0\}$.
A way to construct such parameters is to find a set of functions $f_i$ on the Hilbert space of the system that are invariant under local reversible operations with determinant 1 and that are homogeneous, i.e., satisfy $f_i(c\psi_{ABC\dots})=c^{k(i)}f_i(\psi_{ABC\dots})$ for $c\in \mathbb{C}-\{0\}$ where $k(i)$ is the degree of homogeneity of $f_i$.
Given two such homogeneous functions $f_i\neq0$ and $f_j\neq0$ the ratio $f_i^{k(j)}/f_j^{k(i)}$ has degree of homogeneity zero. This ratio is invariant under any local reversible operation as well as invariant under multiplication of a state by any constant $c\in \mathbb{C}-\{0\}$. Functions of this kind can therefore be used to parametrize the set of equivalence classes $\tilde{\mathcal{O}}_{\psi_{ABC\dots}}$.
If we also require that the homogeneous functions $f_i$ take the value zero for all product states they are witnesses of entanglement, i.e., any nonzero value of such a function implies that the state is entangled. This kind of functions are referred to here and in Ref. \cite{spinorent} as {\it entanglement invariants}.
One way to construct entanglement invariants is as homogeneous polynomials in the state coefficients.
The characterization of entanglement using such polynomial invariants has been considered previously for several different systems of non-relativistic spin-$\frac{1}{2}$ particles (See e.g. References \cite{coffman,sud,verstraete2,luque,toumazet,wootters,wootters2,grassl,verstraete,kempe,luq4,miyake1}).
If for two entangled states there exists some ratio of two polynomial entanglement invariants with degree of homogeneity zero that takes different values for the two states it follows that the two states belong to different equivalence classes and thus are inequivalently entangled. However, if all such ratios of polynomial entanglement invariants takes the same values for two states, this does not imply that the two states belong to the same equivalence class.
Thus a given set of polynomial invariants may be able to distinguish between some but not all inequivalent types of entanglement. We say that such a set of polynomial invariants only provides a {\it partial characterization} of the entanglement properties of the system.
In the general case it is impossible to construct a set of polynomial invariants that can distinguish between all equivalence classes due to the possibility of an equivalence class being in the closure of another equivalence class. Since a ratio of two polynomials with degree of homogeneity zero is a continuous function and constant on an equivalence class it cannot distinguish between the class and its closure. Thus polynomial entanglement invariants in general provide only a partial characterization of entanglement properties.
We can consider different cases of physically allowed local unitary evolutions for a system of Dirac spinors. This means we have to consider different cases of local reversible operations. In some cases the polynomial invariants are the same for two different cases of local reversible operations and in some cases they are different.
In particular the polynomial entanglement invariants constructed for different cases of local reversible operations can have either the same or alternatively different local invariance groups.
Here we identify four principal physically motivated cases of local invariance group acting on a single particle.
For the case where the local unitary evolution acting on a given particle is generated by Dirac Hamiltonians with a mass term and more generally by Hamiltonians of the form $H^{1,2}(t)+H^{0}(t)$ the group of local unitary evolutions acting on the particle is dense in $G_U^{C\gamma^5}$. Therefore, in this case any polynomial entanglement invariant is invariant, up to a U(1) phase, under $G_U^{C\gamma^5}$ since it is a continuous function. Moreover, any such polynomial entanglement invariant is invariant, up to a U(1) phase, also under $G^{C\gamma^5}$ acting on the particle since it is also Lorentz invariant and $G^{C\gamma^5}$ is the smallest Lie group that contains $G_U^{C\gamma^5}$ and the spinor representation of the proper orthochronous Lorentz group as subgroups, as described in Section \ref{hinn}.
For the case where the local unitary evolution acting on a given particle is generated by Dirac Hamiltonians with zero mass and a coupling to a Yukawa pseudoscalar boson and more generally by Hamiltonians of the form $H^{2,3}(t)+H^{0}(t)$ the group of local unitary evolutions acting on the particle is dense in $G^{C}_U$. Therefore, in this case any polynomial entanglement invariant is invariant, up to a U(1) phase, under $G_U^{C}$ since it is a continuous function. Moreover, any such polynomial entanglement invariant is invariant, up to a U(1) phase, also under $G^{C}$ acting on the particle since it is also Lorentz invariant.
For the case where the local unitary evolution acting on a given particle is generated by Dirac Hamiltonians with zero mass and a no additional couplings the group of local unitary evolutions acting on the particle is dense in $G^{C\gamma^5}_U\cap G^C_U$. Therefore, in this case any polynomial entanglement invariant is invariant, up to a U(1) phase, under $G^{C\gamma^5}_U\cap G^C_U$ since it is a continuous function. Moreover, any such polynomial entanglement invariant is invariant, up to a U(1) phase, also under $G^{C\gamma^5}\cap G^C$ acting on the particle since it is a Lorentz invariant.
Finally we consider case where the local unitary evolution on a given particle is generated by Dirac Hamiltonians
with both a mass term and a coupling to a Yukawa pseudoscalar boson where either the mass or the coupling to the Yukawa pseudoscalar boson is a free parameter.
For this case the group of local unitary evolutions acting on the particle is dense in the smallest Lie group that contains both $G^{C\gamma^5}_U$ and $G^C_U$ as subgroups. This group is U(4). Therefore, in this case any polynomial entanglement invariant is invariant, up to a U(1) phase, under U(4) since it is a continuous function. Moreover, any such polynomial entanglement invariant is invariant, up to a U(1) phase, also under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ acting on the particle since it is a Lorentz invariant.
From the above four cases we can see that a set of polynomial entanglement invariants for a system of $n$ Dirac particles is invariant, up to a U(1) phase, under a Lie group $G_1{\,\otimes\,}imes G_2{\,\otimes\,}imes\dots{\,\otimes\,}imes G_n$ where each $G_k$ is one out of $G^C$, $G^{C\gamma^5}$, $G^{C\gamma^5}\cap G^C$ and $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$. The local groups $G_k$ depend on the physically allowed local evolution.
\subsection{Polynomial entanglement invariants in the case of zero momenta free particles in an energy eigenstate and for the case of Weyl particles}
We here consider the restriction of a polynomial entanglement invariant to a subspace.
In particular we consider some physically motivated subspaces.
One such case is a system of Dirac particles with zero momenta that are not coupled to any four-potentials and that are in an eigenstate of the local Dirac Hamiltonians. The state of such a system belongs to a subspace invariant under local projections by $P_+=\frac{1}{2}(I+\gamma^0)$ or alternatively $P_-=\frac{1}{2}(I-\gamma^0)$. A system of particles of this kind is often identified with a system of non-relativistic free spin-$\frac{1}{2}$ particles or antiparticles.
Another such case is a system of Weyl particles, i.e., zero mass Dirac particles with definite chiralities. The state of a system of Weyl particles belongs to a subspace invariant under local projections by $P_R=\frac{1}{2}(I+\gamma^5)$ or alternatively $P_L=\frac{1}{2}(I-\gamma^5)$.
Any polynomial entanglement invariant restricted to a subspace reduces to a polynomial that is invariant under the subgroup of the local reversible operations with determinant one that preserve the subspace. Moreover, since
any polynomial entanglement invariant is invariant, up to a U(1) phase, under the local action of either $G^C$, $G^{C\gamma^5}$, $G^{C\gamma^5}\cap G^C$ or $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C}) $ the restriction of the invariant to a subspace is invariant, up to a U(1) phase, under the local action of a subgroup of one of these groups that preserve the subspace. For any of the local groups $G^C$, $G^{C\gamma^5}$, $G^{C\gamma^5}\cap G^C$ or $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C}) $ and any subspace invariant under $P_+$, $P_-$, $P_L$ or $P_R$ the subgroup that preserves the subspace is isomorphic to $\mathrm{U}(1)\times\mathrm{SL}(2,\mathbb{C})$.
Therefore, any polynomial entanglement invariant reduces on a subspace defined by local projectors $P_+$, $P_-$, $P_L$ or $P_R$ to a polynomial that is invariant under local action by a group isomorphic to $\mathrm{SL}(2,\mathbb{C})$ or reduces to zero.
For a system of two Dirac spinors the subspaces defined by local invariance under $P_R$ or $P_L$ and the subspaces defined by local invariance under $P_+$ or $P_-$ were considered in \cite{spinorent}.
The polynomial Lorentz invariants constructed in \cite{spinorent} were found to reduce on these subspaces either to zero or to a local $\mathrm{SL}(2,\mathbb{C})$ invariant polynomial, the Wootters concurrence \cite{wootters,wootters2}. The Wootters concurrence is a degree 2 polynomial in the state coefficients that was constructed to describe the spin entanglement of two non-relativistic spin-$\frac{1}{2}$ particles and other two-level systems. It generates the entire algebra of local $\mathrm{SL}(2,\mathbb{C})$ invariants for two non-relativistic spin-$\frac{1}{2}$ particles.
For higher numbers of particles a number of local $\mathrm{SL}(2,\mathbb{C})$ invariant polynomials have been constructed. For three non-relativistic spin-$\frac{1}{2}$ particles a polynomial of degree 4 that is invariant under local $\mathrm{SL}(2,\mathbb{C})$ is the Coffman-Kundu-Wootters 3-tangle \cite{coffman}. This polynomial generates the entire algebra of local $\mathrm{SL}(2,\mathbb{C})$ invariants for three non-relativistic spin-$\frac{1}{2}$ particles.
For a system of four non-relativistic spin-$\frac{1}{2}$ particles four polynomials of degrees 2, 4, 4, and 6 that are invariant under local $\mathrm{SL}(2,\mathbb{C})$ have been found by Luque and Thibon \cite{luque}. These four polynomials generate the entire algebra of local $\mathrm{SL}(2,\mathbb{C})$ invariants for four non-relativistic spin-$\frac{1}{2}$ particles.
Moreover, degree 2 polynomials invariant under local $\mathrm{SL}(2,\mathbb{C})$ for arbitrary even numbers of spin-$\frac{1}{2}$ particles have been described by Wong and Christensen \cite{wong}.
\subsection{Entanglement measures}
Here we briefly comment on a commonly used tool for describing entanglement, the so called {\it entanglement measures} \cite{entmes}. These are functions that are zero for product states and satisfy the condition of non-increase on average under local operations assisted by classical communication (LOCC)\cite{vidal}. This condition is called entanglement monotonicity and entanglement properties satisfying this condition can be quantified by the entanglement measures. The entanglement measures are related to entanglement invariants in that the condition of entanglement monotonicity implies invariance under local reversible operations.
However since classical communication between spacelike separated laboratories is not possible a different physical scenario where timelike or null paths connect the laboratories are required for LOCC. In this kind of scenario operations made in one laboratory are not prevented from influencing other laboratories along timelike or null curves by the causal structure of the spacetime. Therefore, other assumptions regarding causal influences between laboratories must be made for the notion of local operations to be meaningful. Given such assumptions one then needs to characterize the set of LOCC to identify entanglement properties that satisfy the condition of entanglement monotonicity. Considering this kind of physical scenario and characterizing the set of LOCC goes beyond the scope of this work and is left as an open problem.
\section{Constructing candidate entanglement invariants through tensor contractions}\label{ent}
We here describe a method for constructing polynomial entanglement invariants for a system of multiple Dirac spinors.
To do this we utilize the properties of the bilinear forms described in Sections \ref{invariants}, \ref{ham} and \ref{hinn} to construct polynomial entanglement invariants of the kind described in Section \ref{spen}.
In particular the two bilinear forms $\psi^T(x)C\gamma^5\varphi(x)$ and $\psi^T(x)C\varphi(x)$ are both pointwise Lorentz invariant. Furthermore, if $\psi$ and $\varphi$ belong to a subspace defined by fixed particle momenta and spanned by spinorial degrees of freedom both $\psi^T(x)C\gamma^5\varphi(x)$ and $\psi^T(x)C\varphi(x)$ are invariant, up to a U(1) phase, under any evolution that acts unitarily on the subspace and is generated by Dirac Hamiltonians and zero-mass Dirac Hamiltonians, respectively. Finally, both $\psi^T(x)C\psi(x)$ and $\psi^T(x)C\gamma^5\psi(x)$ are identically zero since they are skew-symmetric.
For the case where the physically allowed local unitary evolutions of the spinors are described by Dirac Hamiltonians with zero-mass or arbitrary mass, or more generally where the group of physically allowed local transformations belong to $G^C$ or to $G^{C\gamma^5}$, the bilinear forms $\psi^TC\varphi$ and $\psi^TC\gamma^5\varphi$, respectively, have the desired local invariance properties of an entanglement invariant as described in Section \ref{spen}. Below we show how polynomials with these invariance properties can be constructed for any number of Dirac spinors in a way that generalizes the method used for the case of two Dirac spinors in Ref. \cite{spinorent}.
The physical scenario as described in Section \ref{dir} is one where a number of spacelike separated observers each has their own laboratory holding a Dirac or Weyl particle. We give the first five observers the names Alice, Bob, Charlie, David, and Erin, respectively, for convenience. Then we let the particles be in a joint state and assume that operations by one observer on the shared system can be made jointly with the other observers operations, i.e., assume that operations made by different observers commute. Further we make the assumption that a tensor product structure can be used to describe the shared system and use the tensor products of local basis elements $\phi_{j_A}e^{i\bold{k_A}\cdot\bold{x_A}}{\,\otimes\,}imes \phi_{j_B}e^{i\bold{k_B}\cdot\bold{x_B}}{\,\otimes\,}imes\phi_{j_C}e^{i\bold{k_C}\cdot\bold{x_C}}{\,\otimes\,}imes\dots$ as a basis.
Then the state can be expanded in this basis as
\begin{eqnarray}
\psi_{ABC\dots}(t)=\sum_{\substack{j_A,j_B,j_C,\dots\\\bold{k_A},\bold{k_B},\bold{k_C},\dots} }\psi_{\substack{j_A,j_B,j_C,\dots\\\bold{k_A},\bold{k_B},\bold{k_C},\dots}}(t)\phi_{j_A}e^{i\bold{k_A}\cdot\bold{x_A}}{\,\otimes\,}imes \phi_{j_B}e^{i\bold{k_B}\cdot\bold{x_B}}{\,\otimes\,}imes \dots,
\end{eqnarray}
where $\psi_{j_A,j_B,j_C,\dots\bold{k_A},\bold{k_B},\bold{k_C},\dots}(t)$ are complex numbers.
Next we consider a state that belongs to a subspace where the momenta $\bold{k_A},\bold{k_B},\bold{k_C},\dots$ are fixed, i.e., a subspace spanned by only the spinorial degrees of freedom. Then we can suppress the indices $\bold{k_A},\bold{k_B},\bold{k_C},\dots$ in the description of the state and let $\psi_{jkl\dots}\equiv \psi_{j_A,k_B,l_C,\dots\bold{k_A},\bold{k_B},\bold{k_C},\dots}$.
We can arrange these coefficients $\psi_{jkl\dots}$ as a tensor, i.e., a multi-dimensional array, by letting the spinor basis indices $jkl\dots$ be the tensor component indices. Then the state coefficients $\psi_{jk}$ of a bipartite state form a two-dimensional tensor, i.e., a matrix, where $j$ and $k$ are the column and row indices respectively (See Ref. \cite{spinorent}). Likewise for a tripartite state the state coefficients $\psi_{jkl}$ form a three-dimensional tensor where $j$, $k$ and $l$ are the indices of the three different dimensions, respectively. See Fig. \ref{tretens} for an illustration. In this way an $n$-partite state corresponds to an $n$-dimensional tensor. Let us denote this tensor $\Psi^{ABC\dots}$, and its components $\Psi^{ABC\dots}_{jkl\dots}\equiv\psi_{jkl\dots}$.
\begin{figure}
\caption{Graphical representation of the three-dimensional tensor corresponding to the state of three Dirac spinors. \label{tretens}
\label{tretens}
\end{figure}
Transformations $S^A$ on Alice's part of the system act on the first index of $\Psi^{ABC\dots}$. Likewise, transformations $S^B$ on Bob's part of the system act on the second index, transformations $S^C$ on Charlies's part of the system act on the third index, and transformations on the $n$th observer's part act on the $n$th index
\begin{eqnarray}
\Psi^{ABC\dots}_{jkl\dots}\to\sum_{mno\dots} S^A_{jm}S^{B}_{kn}S^{C}_{lo}\dots\Psi^{ABC\dots}_{mno\dots} .
\end{eqnarray}
If we consider two copies of $\Psi^{ABC\dots}$ together with the matrix $C$ we can construct products of tensor components as $\Psi^{ABC\dots}_{jkl\dots}C_{jm}\Psi^{ABC\dots}_{mno\dots}$ and then take the sum over $j$ and $m$. This sum is invariant under the spinor representation of the proper orthochronous Lorentz group acting on Alices particle. This follows from the properties of the bilinear form $\psi^T C \varphi$ discussed in section \ref{invariants} since $\sum_{jm}\Psi^{ABC\dots}_{jkl\dots}C_{jm}\Psi^{ABC\dots}_{mno\dots}$ is a bilinear form for every fixed set of indices $kl\dots$ and $no\dots$.
In particular, for a transformation $S(\Lambda)$ in Alice's lab we have
\begin{eqnarray}
\sum_{jm}\Psi^{ABC\dots}_{jkl\dots}C_{jm}\Psi^{ABC\dots}_{mno\dots}\to &&\sum_{jmpq}\Psi^{ABC\dots}_{jkl\dots}S^T(\Lambda)_{jp}C_{pq}S(\Lambda)_{qm}\Psi^{ABC\dots}_{mno\dots}\nonumber\\
=&&\sum_{jm}\Psi^{ABC\dots}_{jkl\dots}C_{jm}\Psi^{ABC\dots}_{mno\dots}.
\end{eqnarray}
We refer to the construction $\sum_{jm}\Psi^{ABC\dots}_{jkl\dots}C_{jm}\Psi^{ABC\dots}_{mno\dots}$ as a {\it tensor sandwich contraction} where $C$ is being sandwiched between two copies of $\Psi^{ABC\dots}$.
Similarly we can make sandwich contractions over a pair of indices corresponding to any other observers particle with the matrix $C$ being sandwiched.
Furthermore, we can make sandwich contractions also with $C\gamma^5$ sandwiched between the copies of $\Psi^{ABC\dots}$.
Now consider an even number $n$ of copies of $\Psi^{ABC\dots}$. Next, for a given observer split the set of indices corresponding to that observer's particle into $n/2$ pairs. Then sandwich contract every such pair of indices with either $C$ or $C\gamma^5$ sandwiched inbetween the two copies of $\Psi^{ABC\dots}$, as described above. Then repeat this procedure for every other observer. The result is a scalar that is invariant under the spinor representation of the proper orthochronous Lorentz group in any lab. In particular it is a polynomial of degree $n$ in the state coefficients $\psi_{jkl\dots}$. Note that this procedure is very similar to Cayley's $\Omega$ process for constructing polynomial invariants under the special linear group \cite{omega}.
Due to the antisymmetry of $C$ and $C\gamma^5$ the polynomials constructed in this way are identically zero if $\Psi^{ABC\dots}$ can be factored as $\Psi^{A}{\,\otimes\,}imes\Psi^{BC\dots}$, i.e., if the state is a product state over the partition $A|BC\dots$. The same holds if any other observer's particle is in a product state with the rest of the particles.
On the other hand, a polynomial constructed in this way is not necessarily zero if the state is a product state over a partition that splits the particles into sets where each set has two or more particles. In this case the polynomial factorizes into polynomials defined on the different sets of particles in the partitioning.
All polynomials constructed in this way are invariant, up to a U(1) phase, under local unitary evolutions generated by zero-mass Dirac Hamiltonians, as follows from the discussion in Section \ref{ham}. This also holds for zero-mass Dirac Hamiltonians with additional terms that are second degree in gamma matrices such as a Semenoff mass term \cite{semenoff} or Haldane mass term \cite{haldane}. More generally all such polynomial are invariant, up to a U(1) phase, under the group $G^{C}\cap G^{C\gamma^5}$ acting locally on any spinor.
If all indices corresponding to a given observer's particle are sandwich contracted with the matrix $C$ sandwiched for each pair of indices, the resulting polynomial is invariant, up to a U(1) phase, also for evolutions of that spinor generated by zero-mass Dirac Hamiltonians with additional terms that are second degree in gamma matrices or third degree in gamma matrices such as a pseudo-scalar Yukawa term or a Pauli coupling. More generally the polynomial is invariant, up to a U(1) phase, under the group $G^C$ acting on the spinor.
If all indices corresponding to a given observer's particle are sandwich contracted with the matrix $C\gamma^5$ sandwiched for each pair of indices, the resulting polynomial is invariant, up to a U(1) phase, for evolutions of that spinor generated by arbitrary-mass Dirac Hamiltonians. More generally the polynomial is invariant, up to a U(1) phase, under the group $G^{C\gamma^5}$ acting on the spinor.
If the indices corresponding to a given observer's particle are sandwich contracted with zero or an even positive number of the matrix $C\gamma^5$ sandwiched, the resulting polynomial is invariant under P in the lab of the given observer. If the number of sandwiched $C\gamma^5$ is odd the polynomial changes sign under P.
The polynomials constructed in this way satisfy the requirements described in Section \ref{spen} and are therefore considered potential candidates for describing entanglement properties of a system of multiple Dirac spinors. Note that the absolute values are invariant under the local action of the Dirac group, i.e., the finite group generated by the gamma matrices $\gamma^0,\gamma^1,\gamma^2,\gamma^3$, and in particular invariant under the local P, CT and CPT transformations.
Note also that by using $C$ and $C\gamma^5$ to construct the invariants we have in essence used the T and CP transformations. Using such inversion transformations follows the same general idea of using "state inversion" transformations \cite{wootters2,uhlmann,rungta} as the construction of the Wootters concurrence for the case of two non-relativistic spin-$\frac{1}{2}$ particles.
The computational difficulty in constructing invariant polynomials through sandwich contractions rises sharply with the number of spinors and the number of copies of $\Psi^{ABC\dots}$ involved in the contractions, i.e., the degree of the polynomials.
However for the lowest polynomial degrees such invariants can be obtained with relatively small difficulty, in particular for three and four spinors. Furthermore, degree 2 polynomials can be obtained straightforwardly for all even numbers of spinors.
\section{The case of three spinors}\label{three}
For three Dirac spinors the state coefficients can be arranged as a $4\times 4\times 4$ tensor $\Psi^{ABC}$.
Transformations $S^A,S^B,S^C$ acting locally on Alice's, Bob's and Charlie's particle respectively are described as
\begin{eqnarray}
\Psi^{ABC}_{ijk}\to\sum_{lmn} S^A_{il}S^B_{jm}S^C_{kn}\Psi^{ABC}_{lmn}.
\end{eqnarray}
Lorentz invariants of degree 2 can be constructed as tensor sandwich contractions of the form $\sum_{ijklmn}X_{li}X_{mj}X_{nk}\Psi^{ABC}_{ijk}\Psi^{ABC}_{lmn}$ involving two copies of $\Psi^{ABC}$ where either $X=C$ or $X=C\gamma^5$ for each instance. However all the 8 degree 2 polynomials that can be constructed in this way are identically zero due to the antisymmetry of $C$ and $C\gamma^5$.
The next lowest degree is 4.
From the above follows that invariants of degree 4 that factorize as a product of two degree 2 polynomials are identically zero.
Thus it remains to consider the tensor sandwich contractions involving four copies of $\Psi^{ABC}$ that do not factorize into two degree 2 polynomials. There are four different such ways to pair up the tensor indices of the four copies. In writing these sandwich contractions we leave out the summation sign in the following, with the understanding that repeated indices are summed over. We also suppress the superscript $ABC$ of $\Psi$. The four different ways to contract the indices are
\begin{eqnarray}\label{tens}
I_a&=& X_{ij}X_{mk}X_{nl}\Psi_{jkl}\Psi_{qmn}X_{qr}X_{ps}X_{ut}\Psi_{rst}\Psi_{ipu},\nonumber\\
I_b&=& X_{ij}X_{mk}X_{nl}\Psi_{jkl}\Psi_{ipn}X_{qr}X_{ps}X_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_c&=& X_{ij}X_{mk}X_{nl}\Psi_{jkl}\Psi_{imu}X_{qr}X_{ps}X_{ut}\Psi_{rst}\Psi_{qpn},\nonumber\\
I_d&=&X_{ij}X_{mk}X_{nl}\Psi_{jkl}\Psi_{ipt}X_{qr}X_{ps}X_{ut}\Psi_{rsn}\Psi_{qmu}.
\end{eqnarray}
\begin{figure}
\caption{$I_a$}
\caption{$I_b$}
\caption{$I_c$}
\caption{$I_d$}
\caption{Graph representations of the four tensor sandwich contractions $I_a$, $I_b$, $I_c$, and $I_d$. Each of the four copies of $\Psi^{ABC}
\label{ris}
\end{figure}
The contraction $I_a$ is invariant with respect to a permutation of laboratories B and C. Similarly, the contraction $I_b$ is invariant with respect to a permutation of laboratories A and C and the contraction $I_c$ is invariant with respect to a permutation of laboratories A and B. The final contraction $I_d$ is invariant with respect to any permutation of the laboratories.
Moreover, the different tensor contractions can be represented as graphs which provides an additional way to understand them.
See Fig. \ref{ris} for a graph representation of $I_a$, $I_b$, $I_c$, and $I_d$.
For each way to pair up the tensor indices we consider $(2^6-2^3)/2+2^3=36$ ways to choose the $X$s as either $C$ or $ C\gamma^5$. There is $2^3$ such choices where for each lab both $X$s corresponding to the lab are identical and there is $2^6-2^3$ choices that are not of this kind. For the latter case however each such choice for $I_a$, $I_b$, $I_c$, and $I_d$ is equivalent to at least one other choice. This can be understood in terms of graph isomorphisms of the graph representations of $I_a$, $I_b$, $I_c$, and $I_d$ (See fig. \ref{ris}), and reduces the number of such choices that need to be considered to $(2^6-2^3)/2$.
Thus we can consider a total of $36\times 4=144$ ways to construct polynomials by this method. These 144 different tensor sandwich contractions were calculated and their linear independence tested. Due to the large number of terms in the polynomials they are not all given in a fully written out form, but a few selected examples are given fully written out in \ref{polllu}.
The resulting polynomials are not all linearly independent and not all unique but one can select a set of 67 linearly independent polynomials. In the following we use the abbreviated notation $C^5\equiv C\gamma^5$ in giving the formal expressions for the polynomials.
The polynomials can be divided into subsets based on their properties under parity inversion P in the different labs. This is useful since a polynomial cannot be linearly dependent on any polynomial with different behaviour under parity inversion P.
\subsection{Polynomials invariant under P in all labs}
There are 32 polynomials that are tensor sandwich contractions of the form in Eq. (\ref{tens}) and also invariant under P in Alice's, Bob's and in Charlie's lab. These Lorentz invariants are not all linearly independent, but a set of 23 linearly independent polynomials can be selected. Thus the polynomials span a 23 dimensional space. The 32 polynomials are
\begin{eqnarray}
I_{2a}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C^5_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{2b}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C^5_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{2c}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C^5_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{2d}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C^5_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{3a}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{qmn}C_{qr}C_{ps}C_{ut}\Psi_{rst}\Psi_{ipu},\nonumber\\
I_{3b}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{3c}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imu}C_{qr}C_{ps}C_{ut}\Psi_{rst}\Psi_{qpn},\nonumber\\
I_{3d}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{4a}&=&C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{4b}&=&C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{4c}&=&C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{4d}&=&C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{5a}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{5b}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{5c}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{5d}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{6a}&=&C_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{6b}&=&C_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{6c}&=&C_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{6d}&=&C_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{7a}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{7b}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{7c}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{7d}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{8a}&=&C_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{8b}&=&C_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{8c}&=&C_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{8d}&=&C_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{9a}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{9b}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{9c}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{9d}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu}.
\end{eqnarray}
The linear dependence of the polynomials can be described by the 9 equations
\begin{eqnarray}\label{grunt}
0&=& I_{5b} - I_{5c} + I_{5d} + I_{8b} - I_{8c} +I_{8d}, \nonumber\\
0&=& I_{3b} - I_{3c}+ I_{3d} + I_{9b} - I_{9c} +I_{9d},\nonumber\\
0&=& I_{5a}- I_{5c} - I_{5d} + I_{9a} - I_{9c}-I_{9d},
\end{eqnarray}
\begin{eqnarray}
0&=&I_{8d} + I_{8a} - I_{8b} + 2 I_{7a} - I_{7b} - I_{7c} + I_{6a} + I_{6b} -
2 I_{6c} + I_{4b} - I_{4c} + I_{4d},
\nonumber\\
0&=&I_{2a} + I_{2b} - 2 I_{2c} + I_{4a} + I_{4b} - 2 I_{4c} + I_{6a} + I_{6b} -
2 I_{6c} + I_{7a} + I_{7b} - 2 I_{7c},
\nonumber\\
0&=&2I_{2a}- I_{2b} - I_{2c} +2 I_{4a} - I_{4b} - I_{4c} +
2 I_{5a} - I_{5b} - I_{5c} +
2 I_{9a} - I_{9b} - I_{9c},
\nonumber\\
0&=&I_{2a}-2 I_{2b} + I_{2c} + I_{5a} - 2 I_{5b} + I_{5c} +
I_{7a} -2I_{7b} + I_{7c} + I_{8a} - 2 I_{8b} + I_{8c},
\nonumber\\
0&=&I_{3a} + I_{3b} - 2 I_{3c} + I_{5a} + I_{5b} - 2 I_{5c} + I_{8a} + I_{8b} -
2 I_{8c} + I_{9a} + I_{9b}- 2 I_{9c},
\nonumber\\
0&=&2 I_{3a} - I_{3b} - I_{3c} + 2 I_{6a} - I_{6b} - I_{6c} +
2 I_{7a} - I_{7b} - I_{7c} + 2 I_{8a} - I_{8b} - I_{8c}.
\end{eqnarray}
\subsection{Polynomials only invariant under P in Alice's and Charlie's labs}
There is 16 polynomials that are tensor sandwich contractions of the form in Eq. (\ref{tens}) and also invariant under P in Alice's and Charlie's lab, but not in Bob's lab. These Lorentz invariants are not all linearly independent, but a set of 8 linearly independent polynomials can be selected. Thus the polynomials span an 8 dimensional space. The 16 polynomials are
\begin{eqnarray}
I_{10a}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{10b}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{10c}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{10d}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{16a}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{16b}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{16c}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{16d}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{17a}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{17b}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{17c}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{17d}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{18a}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{18b}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{18c}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{18d}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu}.
\end{eqnarray}
The linear dependence of the polynomials can be described by the 8 equations
\begin{eqnarray}
I_{16a} - I_{16c} - I_{16d}&=&0,\nonumber\\
I_{10a} - I_{10c} - I_{10d}&=&0,\nonumber\\
I_{18a} - I_{18c} - I_{18d}&=&0,\nonumber\\
I_{17a} - I_{17c} - I_{17d}&=&0,
\end{eqnarray}
\begin{eqnarray}\label{gnu}
2 I_{10a} - I_{10b} - I_{10c} +
2 I_{18a} - I_{18b} - I_{18c}&=&0,\nonumber\\
I_{10a} + I_{10b} - 2 I_{10c} + I_{17a} + I_{17b}- 2I_{17c}&=&0,\nonumber\\
I_{16a} + I_{16b} - 2 I_{16c} + I_{18a} + I_{18b}- 2I_{18c}&=&0,\nonumber\\
2 I_{16a} - I_{16b} - I_{16c} + 2 I_{17a} - I_{17b} - I_{17c}&=&0.
\end{eqnarray}
\subsection{Polynomials only invariant under P in Alice's and Bob's labs}
There is 16 polynomials that are tensor sandwich contractions of the form in Eq. (\ref{tens}) and also invariant under P in Alice's and Bob's lab, but not in Charlie's lab. These Lorentz invariants are not all linearly independent, but a set of 8 linearly independent polynomials can be selected. Thus the polynomials span an 8 dimensional space. The 16 polynomials are
\begin{eqnarray}
I_{19a}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{19b}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{19c}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{19d}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{20a}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{20b}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{20c}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{20d}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{21a}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{21b}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{21c}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{21d}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{22a}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{22b}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{22c}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{22d}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu}.
\end{eqnarray}
The linear dependence of the polynomials can be described by the 8 equations
\begin{eqnarray}
I_{20a} - I_{20b} + I_{20d}&=&0,\nonumber\\
I_{19a} - I_{19b} + I_{19d}&=&0,\nonumber\\
I_{21a} - I_{21b} + I_{21d}&=&0,\nonumber\\
I_{22a} - I_{22b} +I_{22d}&=&0,
\end{eqnarray}
\begin{eqnarray}\label{anteloop}
2I_{20a}- I_{20b}- I_{20c} +2I_{21a}- I_{21b}- I_{21c}&=&0,\nonumber\\
I_{19a}- 2I_{19b}+I_{19c} +I_{21a}-2 I_{21b}+ I_{21c}&=&0,\nonumber\\
I_{20a}-2 I_{20b}+ I_{20c} +I_{22a}-2 I_{22b}+ I_{22c}&=&0,\nonumber\\
2I_{19a}- I_{19b}- I_{19c} +2I_{22a}- I_{22b}- I_{22c}&=&0.
\end{eqnarray}
\subsection{Polynomials only invariant under P in Bob's and Charlie's labs }
There are 16 polynomials that are tensor sandwich contractions of the form in Eq. (\ref{tens}) and also invariant under P in Bob's and Charlie's lab, but not in Alice's lab. These Lorentz invariants are not all linearly independent, but a set of 8 linearly independent polynomials can be selected. Thus the polynomials span an 8 dimensional space. The 16 polynomials are
\begin{eqnarray}
I_{23a}&=& C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{23b}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{23c}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{23d}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{24a}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{24b}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{24c}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{24d}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{25a}&=&C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{25b}&=&C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{25c}&=&C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{25d}&=&C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{26a}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{26b}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{26c}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{26d}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu}.
\end{eqnarray}
The linear dependence of the polynomials can be described by the 8 equations
\begin{eqnarray}
I_{23b} - I_{23c} + I_{23d}&=&0,\nonumber\\
I_{24b} - I_{24c} + I_{24d}&=&0,\nonumber\\
I_{26b} - I_{26c} + I_{26d}&=&0,\nonumber\\
I_{25b} - I_{25c} + I_{25d}&=&0,
\end{eqnarray}
\begin{eqnarray}\label{zebra}
I_{23a} -2 I_{23b} + I_{23c} +
I_{25a} -2 I_{25b} + I_{25c}&=&0,\nonumber\\
I_{24a} -2 I_{24b} + I_{24c} +
I_{26a} -2 I_{26b} + I_{26c}&=&0,\nonumber\\
I_{24a} + I_{24b} - 2 I_{24c} + I_{25a} + I_{25b}- 2I_{25c}&=&0,\nonumber\\
I_{23a} + I_{23b} - 2 I_{23c} + I_{26a} + I_{26b}- 2I_{26c}&=&0.
\end{eqnarray}
\subsection{Polynomials only invariant under P in Bob's lab }
There is 16 different ways to construct polynomials as tensor sandwich contractions of the form in Eq. (\ref{tens}) that are invariant under P in Bob's lab, but not in Alice's and Charlie's labs. The resulting Lorentz invariants are not all linearly independent and not all unique, but a set of 5 linearly independent polynomials can be selected. Thus the polynomials span a 5 dimensional space. The 16 polynomials are
\begin{eqnarray}
I_{27a}&=& C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{27b}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{27c}&=& C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{27d}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{28a}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{28b}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{28c}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{28d}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{29a}&=& C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{29b}&=& C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{29c}&=& C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{29d}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{30a}&=& C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{30b}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{30c}&=& C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{30d}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu}.
\end{eqnarray}
The linear dependence of the polynomials can be described by the 11 equations
\begin{eqnarray}\label{hippo}
I_{27a} &=&I_{29a},\nonumber\\
I_{28a} &=& I_{30a},\nonumber\\
I_{27c}&=&I_{29c},\nonumber\\
I_{28c} &=&I_{30c},\nonumber\\
I_{27a}-I_{27b}&=& I_{29b} - I_{29c},\nonumber\\
I_{28a}- I_{28b}&=& I_{30b} - I_{30c},\nonumber\\
\frac{1}{2}( I_{27c}- I_{27a})=I_{27d} =- I_{28d}&=&- I_{29d}=I_{30d}=\frac{1}{2}(I_{28a}-I_{28c} ).
\end{eqnarray}
In particular we see that all the polynomials on the form $I_d$ are equal up to a sign. Note that the equalities $I_{27a} =I_{29a}$, $I_{28a} = I_{30a}$, $I_{27c} = I_{29c}$, and $I_{28c} = I_{30c}$ can be understood as graph isomorphisms of the graphs corresponding to the tensor contractions (See Fig. \ref{ris}).
\subsection{Polynomials only invariant under P in Charlie's lab }
There is 16 different ways to construct polynomials as tensor sandwich contractions of the form in Eq. (\ref{tens}) that are invariant under P in Charlie's lab, but not in Alice's and Bob's labs. The resulting Lorentz invariants are not all linearly independent and not all unique, but a set of 5 linearly independent polynomials can be selected. Thus the polynomials span a 5 dimensional space. The 16 polynomials are
\begin{eqnarray}
I_{31a}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{31b}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{31c}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{31d}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{32a}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{32b}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{32c}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{32d}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{33a}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{33b}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{33c}&= &C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{33d}&=&C_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{34a}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{34b}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{34c}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{34d}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu}.
\end{eqnarray}
The linear dependence of the polynomials can be described by the 11 equations
\begin{eqnarray}\label{rhino}
I_{31a} &=& I_{34a},\nonumber\\
I_{32a} &=& I_{33a},\nonumber\\
I_{31b} &=& I_{34b},\nonumber\\
I_{32b} &=& I_{33b},\nonumber\\
I_{32c} - I_{32a}&=& I_{33b} - I_{33c},\nonumber\\
I_{31c} - I_{31a}&=& I_{34b} - I_{34c},\nonumber\\
\frac{1}{2}(I_{31a} - I_{31b})=I_{31d}=-I_{32d}&=&-I_{33d}=I_{34d}=\frac{1}{2}( I_{32b}- I_{32a} ).
\end{eqnarray}
In particular we see that all the polynomials on the form $I_d$ are equal up to a sign.
Note that the equalities $I_{31a} =I_{34a}$, $I_{32a} = I_{33a}$, $I_{31b} = I_{34b}$, and $I_{32b} = I_{33b}$ can be understood as graph isomorphisms of the graphs corresponding to the tensor contractions (See Fig. \ref{ris}).
\subsection{Polynomials only invariant under P in Alice's lab }
There is 16 different ways to construct polynomials as tensor sandwich contractions of the form in Eq. (\ref{tens}) that are invariant under P in Alice's lab, but not in Bob's and Charlie's labs. The resulting Lorentz invariants are not all linearly independent and not all unique, but a set of 5 linearly independent polynomials can be selected. Thus the polynomials span a 5 dimensional space. The 16 polynomials are
\begin{eqnarray}
I_{35a}&=& C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{35b}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{35c}&=& C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{35d}&=&C_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{36a}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{36b}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{36c}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{36d}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{37a}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C^5_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{37b}&=& C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{37c}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C^5_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{37d}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C^5_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{38a}&=& C_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{38b}&=& C_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{38c}&=& C_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{38d}&=&C_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu}.
\end{eqnarray}
The linear dependence of the polynomials can be described by the 11 equations
\begin{eqnarray}\label{giraffe}
I_{35b}& =& I_{38b},\nonumber\\
I_{36b} &=& I_{37b},\nonumber\\
I_{35c} &=& I_{38c},\nonumber\\
I_{36c} &=& I_{37c},\nonumber\\
I_{35a} - I_{35c}&=& I_{38b}-I_{38a},\nonumber \\
I_{36a}- I_{36c}&= & I_{37b}-I_{37a}, \nonumber\\
\frac{1}{2}(I_{35b} - I_{35c})=I_{35d} =- I_{36d}&=&I_{37d}=- I_{38d}=\frac{1}{2}( I_{37c} -I_{37b}).
\end{eqnarray}
In particular we see that all the polynomials on the form $I_d$ are equal up to a sign.
Note that the equalities $I_{35b} =I_{38b}$, $I_{36b} = I_{37b}$, $I_{35c} = I_{38c}$, and $I_{36c} = I_{37c}$ can be understood as graph isomorphisms of the graphs corresponding to the tensor contractions (See Fig. \ref{ris}).
\subsection{Polynomials not invariant under P in any lab }
There is 16 different ways to construct polynomials as tensor sandwich contractions of the form in Eq. (\ref{tens}) that are not invariant under P in any lab. The resulting Lorentz invariants are not all linearly independent and not all unique, but a set of 5 linearly independent polynomials can be selected. Thus the polynomials span a 5 dimensional space. The 16 polynomials are
\begin{eqnarray}
I_{11a}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{11b}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{11c}&=& C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{11d}&=&C^5_{ij}C^5_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{12a}&=& C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{12b}&=& C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{12c}&=& C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{12d}&=&C^5_{ij}C_{mk}C^5_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{14a}&=& C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{14b}&=& C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{14c}&=& C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{14d}&=&C^5_{ij}C^5_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu},
\end{eqnarray}
\begin{eqnarray}
I_{15a}&=& C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{pmn}C_{pr}C^5_{qs}C^5_{ut}\Psi_{rst}\Psi_{iqu},\nonumber\\
I_{15b}&=& C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipn}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rst}\Psi_{qmu},\nonumber\\
I_{15c}&=& C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{imp}C_{qr}C^5_{us}C^5_{pt}\Psi_{rst}\Psi_{qun},\nonumber\\
I_{15d}&=&C^5_{ij}C_{mk}C_{nl}\Psi_{jkl}\Psi_{ipt}C_{qr}C^5_{ps}C^5_{ut}\Psi_{rsn}\Psi_{qmu}.
\end{eqnarray}
The linear dependence of the polynomials can be described by the 11 equations
\begin{eqnarray}
I_{14a}&=&I_{12a},\nonumber\\
I_{11a}&=&I_{15a},\nonumber\\
I_{12b}&=&I_{11b},\nonumber\\
I_{14b}&=&I_{15b},\nonumber\\
I_{14c}&=&I_{11c},\nonumber\\
I_{12c}&=&I_{15c},\nonumber\\
I_{11d}=-I_{12d} &=&-I_{14d}=I_{15d}, \nonumber\\
I_{11a}+I_{12a}=I_{11c}&+&I_{12c}=I_{11b}+I_{14b}.
\end{eqnarray}
In particular we see that all the polynomials on the form $I_d$ are equal up to a sign.
Note that the equalities $I_{14a} =I_{12a}$, $I_{11a} = I_{15a}$, $I_{12b} = I_{11b}$, $I_{14b} = I_{15b}$, $I_{14c} = I_{11c}$, $I_{12c} = I_{15c}$, $I_{12d} = I_{14d}$, and $I_{11d} = I_{15d}$ can be understood as graph isomorphisms of the graphs corresponding to the tensor contractions (See Fig. \ref{ris}).
\subsection{Polynomials invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C}) $ up to a U(1) phase in one of the labs}
While the polynomials constructed as tensor sandwich contractions are designed to be invariant, up to a U(1) phase, under either $G^{C\gamma^5}$, $G^{C}$ or $G^{C}\cap G^{C\gamma^5}$ in any given lab, some linear combinations of the polynomials given above can be seen to be invariant, up to a U(1) phase, under both $G^{C}$ and $G^{C\gamma^5}$ in at least one lab.
For example the polynomial $I_{5b} - I_{5c} + I_{5d}$ is by construction invariant, up to a U(1) phase, under $G^{C\gamma^5}$ in Alice's lab and the polynomial $ I_{8b} - I_{8c} +I_{8d}$ is by construction invariant, up to a U(1) phase, under $G^{C}$ in Alice's lab, but the linear dependence relation $0= I_{5b} - I_{5c} + I_{5d} + I_{8b} - I_{8c} +I_{8d}$ given in Eq. (\ref{grunt}) implies that both the polynomial $I_{5b} - I_{5c} + I_{5d}$ and the polynomial $ I_{8b} - I_{8c} +I_{8d}$ are invariant, up to a U(1) phase, under both $G^{C}$ and $G^{C\gamma^5}$ in Alice's lab. Similar relations hold for many other linear combinations of the polynomials and other labs. Since the polynomials are continuous functions it follows that
any polynomial that is invariant, up to a U(1) phase, under both $G^{C}$ and $G^{C\gamma^5}$ is invariant, up to a U(1) phase, under the smallest Lie group that contains $G^{C}$ and $G^{C\gamma^5}$ as subgroups (See e.g. Ref. \cite{spinorent} Theorem 2 or Ref. \cite{wall} Ch. 1.3.3.). This Lie group is $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$.
For the polynomials invariant under P in all labs we have the following twelve linear dependence relations
\begin{eqnarray}
I_{5b} - I_{5c} + I_{5d} + I_{8b} - I_{8c} +I_{8d}&=&0, \nonumber\\
I_{3b} - I_{3c}+ I_{3d} + I_{9b} - I_{9c} +I_{9d}&=&0,\nonumber\\
I_{5a}- I_{5c} - I_{5d} + I_{9a} - I_{9c}-I_{9d}&=&0,\nonumber\\
I_{7a} - I_{7c} - I_{7d} - I_{6d} - I_{6c} + I_{6a}&=&0,\nonumber\\
I_{6b} - I_{6c} + I_{6d} + I_{4d} - I_{4c} + I_{4b}&=&0,\nonumber\\
I_{4a} - I_{4b} + I_{4d} + I_{9d} - I_{9b} + I_{9a}&=&0,\nonumber\\
I_{7a} - I_{7b} + I_{7d} + I_{8d} - I_{8b} + I_{8a}&=&0,\nonumber\\
I_{3a} - I_{3b} + I_{3d} + I_{6d} - I_{6b} + I_{6a}&=&0,\nonumber\\
I_{3a} - I_{3c} - I_{3d} - I_{8d} - I_{8c} + I_{8a}&=&0,\nonumber\\
I_{2a} - I_{2b} + I_{2d} + I_{5d} - I_{5b} + I_{5a}&=&0,\nonumber\\
I_{2a} - I_{2c} - I_{2d} - I_{4d} - I_{4c} + I_{4a}&=&0,\nonumber\\
I_{2b} - I_{2c} + I_{2d} + I_{7d} - I_{7c} + I_{7b}&=&0.
\end{eqnarray}
From these relations follows that the polynomials $I_{5b} - I_{5c} + I_{5d}$, $I_{8b} - I_{8c} +I_{8d}$, $I_{3b} - I_{3c}+ I_{3d}$, $I_{9b} - I_{9c} +I_{9d}$, $I_{6b} - I_{6c} + I_{6d}$, $I_{4d} - I_{4c} + I_{4b}$, $I_{2b} - I_{2c} + I_{2d}$, and $ I_{7d} - I_{7c} + I_{7b}$ are invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Alice's lab.
Moreover, the polynomials $I_{5a}- I_{5c} - I_{5d}$, $ I_{9a} - I_{9c}-I_{9d}$, $I_{7a} - I_{7c} - I_{7d}$, $- I_{6d} - I_{6c} + I_{6a}$,
$I_{3a} - I_{3c} - I_{3d}$, $- I_{8d} - I_{8c} + I_{8a}$,
$I_{2a} - I_{2c} - I_{2d}$, and $- I_{4d} - I_{4c} + I_{4a}$ are invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Bob's lab.
Finally, the polynomials $I_{4a} - I_{4b} + I_{4d}$, $I_{9d} - I_{9b} + I_{9a}$, $I_{7a} - I_{7b} + I_{7d}$, $ I_{8d} - I_{8b} + I_{8a}$, $I_{3a} - I_{3b} + I_{3d}$, $I_{6d} - I_{6b} + I_{6a}$, $I_{2a} - I_{2b} + I_{2d}$, and $I_{5d} - I_{5b} + I_{5a}$
are invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Charlie's lab.
For the polynomials only invariant under P in Alice's and Charlie's labs we can see from Eq. (\ref{gnu}) that the polynomials $I_{10a} + I_{10b} - 2 I_{10c}$, $ I_{17a} + I_{17b}- 2I_{17c}$, $I_{16a} + I_{16b} - 2 I_{16c}$, and $I_{18a} + I_{18b}- 2I_{18c}$
are invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Alice's lab.
Moreover, the polynomials $2 I_{10a} - I_{10b} - I_{10c}$, $2 I_{18a} - I_{18b} - I_{18c}$, $2 I_{16a} - I_{16b} - I_{16c}$, and $ 2 I_{17a} - I_{17b} - I_{17c}$
are invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Charlie's lab.
For the polynomials only invariant under P in Alice's and Bob's labs we can see from Eq. (\ref{anteloop}) that the polynomials $I_{19a}- 2I_{19b}+I_{19c}$, $ 2I_{21a}- I_{21b}- I_{21c}$, $I_{20a}-2 I_{20b}+ I_{20c} $, and $I_{22a}-2 I_{22b}+ I_{22c}$
are invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Alice's lab.
Moreover, the polynomials $2I_{20a}- I_{20b}- I_{20c}$, $2I_{21a}- I_{21b}- I_{21c}$, $2I_{19a}- I_{19b}- I_{19c}$, and $2I_{22a}- I_{22b}- I_{22c}$
are invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Bob's lab.
For the polynomials only invariant under P in Bob's and Charlie's labs we can see from Eq. (\ref{zebra}) that the polynomials $I_{24a} + I_{24b} - 2 I_{24c}$, $ I_{25a} + I_{25b}- 2I_{25c}$, $I_{23a} + I_{23b} - 2 I_{23c}$, and $ I_{26a} + I_{26b}- 2I_{26c}$
are invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Bob's lab.
Moreover, the polynomials $I_{23a} -2 I_{23b} + I_{23c}$, $I_{25a} -2 I_{25b} + I_{25c}$, $I_{24a} -2 I_{24b} + I_{24c}$, and $ I_{26a} -2 I_{26b} + I_{26c}$
are invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Charlie's lab.
For the polynomials only invariant under P in Bob's lab we can see from Eq. (\ref{hippo}) that the polynomial $I_{27d}$
is invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Bob's lab.
Similarly, for the polynomials only invariant under P in Charlie's lab we can see from Eq. (\ref{rhino}) that the polynomial $I_{31d}$
is invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Charlie's lab.
Finally, for the polynomials only invariant under P in Alice's lab we can see from Eq. (\ref{giraffe}) that the polynomial $I_{35d}$
is invariant under $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ in Alice's lab.
\subsection{Weyl particles}
We can consider the case where Alice's, Bob's and Charlie's particles are Weyl particles, i.e., have a definite chirality. Then the shared state of the spinors is invariant under some combination of projections $P_L^A$ or $P_R^A$ by Alice, $P_L^B$ or $P_R^B$ by Bob and, $P_L^C$ or $P_R^C$ by Charlie and the Lorentz invariants on the form $I_a$, $I_b$ and $I_c$ reduce, up to a sign, to $128\tau$ where
\begin{eqnarray}\label{tau}
\tau=&& (\psi_{011}\psi_{ 100} -\psi_{ 010}\psi_{ 101} -\psi_{ 001}\psi_{ 110} +\psi_{ 000}\psi_{ 111})^2 \nonumber\\
&&- 4 (\psi_{ 001} \psi_{ 010} -\psi_{ 000 }\psi_{ 011}) (\psi_{ 101}\psi_{ 110} -\psi_{ 100}\psi_{ 111}).
\end{eqnarray}
The polynomial $\tau$ is the Coffman-Kundu-Wootters 3-tangle \cite{coffman}, which is equal to the Cayley hyperdeterminant \cite{cayley} of the sub-tensor of $\Psi^{ABC}$ defined by the elements with only indices equal to $0$ or $1$.
The reduction of the polynomials to a form where all state coefficients have only two spinor basis indices $0$ and $1$ is due to the symmetry $\psi_{jkl}=(-1)^{|LA|}\psi_{(j-2) kl}=(-1)^{|LB|}\psi_{j (k-2)l}=(-1)^{|LC|}\psi_{j k(l-2)}$ of the shared state, where $|LA|=1$ if the state is invariant under $P_L^A$ and otherwise zero, $|LB|=1$ if the state is invariant under $P_L^B$ and otherwise zero, $|LC|=1$ if the state is invariant under $P_L^C$ and otherwise zero and $j$,$k$, and $l$ are defined modulo 4. Thus, for the case of Weyl particles the polynomials on the form $I_{a}$, $I_{b}$, and $I_{c}$ become essentially equivalent to the Coffman-Kundu-Wootters 3-tangle.
The invariants on the form $I_{d}$ do not reduce to the 3-tangle when Alice's, Bob's and Charlie's particles are Weyl particles, but instead reduce to zero.
However, we can consider also a scenario where only two of the particles are Weyl particles. For example let Alice's and Bob's particles be Weyl particles i.e., let the shared state be invariant under some combination of projections $P_L^A$ or $P_R^A$ by Alice and $P_L^B$ or $P_R^B$ by Bob but no condition is imposed on Charlie's particle. Then the polynomials on the form $I_{d}$ that are invariant under P in Charlie's lab , i.e., $I_{2d},I_{3d},I_{4d},I_{5d},I_{6d},I_{7d},I_{8d},I_{9d},I_{10d}$ $,I_{16d},I_{17d},I_{18d},I_{23d},I_{24d},
I_{25d},I_{26d},I_{31d},I_{32d},I_{33d},$ and $ I_{34d}$ reduce to $64V$, up to a sign, where
\begin{eqnarray}
V=
&&(\psi_{ 003} \psi_{012}-\psi_{ 002 }\psi_{ 013})(\psi_{ 101}\psi_{ 110}-\psi_{ 100}\psi_{ 111})\nonumber\\
&&+(\psi_{ 003}\psi_{ 011}-\psi_{ 001}\psi_{ 013})(\psi_{ 100}\psi_{ 112} -\psi_{ 102}\psi_{ 110})\nonumber \\
&&+(\psi_{ 003}\psi_{ 010}-\psi_{ 000 }\psi_{ 013})(\psi_{ 102}\psi_{ 111} -\psi_{ 101}\psi_{ 112})\nonumber\\
&&+ (\psi_{ 000 }\psi_{ 011}-\psi_{ 000 }\psi_{ 010})(\psi_{ 102}\psi_{ 113}-\psi_{ 103}\psi_{ 112 })\nonumber\\
&&+(\psi_{ 000 }\psi_{ 012}-\psi_{ 002 }\psi_{ 010})(\psi_{ 103}\psi_{ 111}-\psi_{ 101}\psi_{ 113})\nonumber\\
&&+(\psi_{ 001}\psi_{ 012}-\psi_{ 002 }\psi_{ 011})(\psi_{ 100}\psi_{ 113 } -\psi_{ 103}\psi_{ 110}),
\end{eqnarray}
while the remaining invariants on the form $I_{d}$ that are not invariant under P in Charlie's lab reduce to zero. The polynomial $V$, called the $2\times 2\times 4$ tangle, was described in Refs. \cite{verstraete,moor} and is invariant under $\mathrm{SL}(4,\mathbb{C})$ acting on Charlie's spinor. The cases where Bob's and Charlie's or Alice's and Charlie's particles are Weyl particles are completely analogous. For these cases the invariants on the form $I_{d}$ that are invariant under P in the lab that holds the non-chiral particle reduce to multiples of polynomials obtained from $V$ by permuting the state coefficient indices accordingly.
\subsection{Eigenspaces of the local Dirac Hamiltonians in the case of zero momenta and zero four-potentials}
We can consider the case of zero momentum and zero four-potential, sometimes described as the non-relativistic limit of a free particle. If Alice's, Bob's and Charlie's particles are all in this limit and also in an eigenstate of the local Dirac Hamiltonians the shared state is invariant under some combination of projections $P_+^A$ or $P_-^A$ by Alice, $P_+^B$ or $P_-^B$ by Bob, and $P_+^C$ or $P_-^C$ by Charlie. In this case only a $2\times 2\times 2$ subtensor of $\Psi^{ABC}$ is nonzero.
As a consequence the polynomials $I_{3a}$, $I_{3b}$ and $I_{3c}$ reduce, up to a sign and a relabelling of the indices, to $2\tau$ where $\tau$ is the Coffman-Kundu-Wootters 3-tangle \cite{coffman} given in Eq. (\ref{tau}). All other polynomials of degree 4 constructed above are zero for this case. In particular any polynomial obtained through tensor sandwich contractions where one or more contractions involve $C\gamma^5$ is zero in this case since $P_+C\gamma^5P_+=P_-C\gamma^5P_-=0$.
\subsection{Examples of tripartite spinor entangled states}
Here we consider a few examples of tripartite entangled states to illustrate how inequivalent forms of spinor entanglement are distinguished by the polynomials.
We can consider analogues of the tripartite entangled Greenberger-Horne-Zeilinger (GHZ)\cite{ghz} state for non-relativistic spin-$\frac{1}{2}$ particles.
One such state is
\begin{eqnarray}
\frac{1}{\sqrt{2}}({\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}).
\end{eqnarray}
For this state $|I_{3a}|=|I_{3b}|=|I_{3c}|=1/2$, but all the other degree 4 polynomials are identically zero. Similarly we can construct a state
\begin{eqnarray}
\frac{1}{\sqrt{2}}({\phi_2^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_2^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}).
\end{eqnarray}
for which $|I_{2a}|=|I_{2b}|=|I_{2c}|=1/2$, but all the other degree 4 polynomials are identically zero. Furthermore, one can construct GHZ-like states that involve more than two basis spinors. For example
\begin{eqnarray}
\frac{1}{\sqrt{2}}({\phi_2^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_2^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}).
\end{eqnarray}
for which $|I_{4a}|=|I_{4b}|=|I_{4c}|=1/2$, but all the other degree 4 polynomials are identically zero, and
\begin{eqnarray}
\frac{1}{\sqrt{2}}({\phi_2^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_3^B}{\,\otimes\,}imes{\phi_1^C}).
\end{eqnarray}
for which $|I_{5a}|=|I_{5b}|=|I_{5c}|=1/2$, but all the other degree 4 polynomials are identically zero. As for the above four examples, for any GHZ like state of this kind only one triplet of polynomials of the types $I_a,I_b$ and $I_c$ is non-zero, and these polynomials are invariant under P in all labs.
A generalization of the GHZ state with three terms is
\begin{eqnarray}
\frac{1}{{\sqrt{3}}}({\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}+{\phi_2^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_2^C}).
\end{eqnarray}
For this state $|I_{2a}|=|I_{2b}|=|I_{2c}|=|I_{3a}|=|I_{3b}|=|I_{3c}|=2/9$ and $|I_{11a}|=|I_{11b}|=|I_{11c}|=|I_{15a}|=|I_{12b}|=|I_{14c}|=1/9$ while all other degree 4 polynomials are identically zero.
Another generalization of the GHZ state with four terms is
\begin{eqnarray}
\frac{1}{{2}}({\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}+{\phi_2^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_2^C}+{\phi_3^A}{\,\otimes\,}imes{\phi_3^B}{\,\otimes\,}imes{\phi_3^C}).
\end{eqnarray}
For this state $|I_{2a}|=|I_{2b}|=|I_{2c}|=|I_{3a}|=|I_{3b}|=|I_{3c}|=|I_{4b}|=|I_{5c}|=|I_{6c}|=|I_{7a}|=|I_{8b}|=|I_{9a}|=1/4$ while all other degree 4 polynomials are identically zero.
An example of a state that is not on the GHZ form is
\begin{eqnarray}
\frac{1}{{2}}({\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_1^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_0^C}+{\phi_2^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}).
\end{eqnarray}
For this state $|I_{23a}|=|I_{23b}|=|I_{23c}|=1/4$ but all other degree 4 polynomials are identically zero.
This state was considered in Refs. \cite{miyake1,moor} although not in the context of Dirac spinors.
Another state not on the GHZ form is
\begin{eqnarray}
\frac{1}{{2}}({\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}+{\phi_0^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_2^C}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_3^C}).
\end{eqnarray}
for which $|I_{3a}|=|I_{6a}|=1/2$ and $|I_{3c}|=|I_{3d}|=|I_{6c}|=|I_{6d}|=1/4$ but all other degree 4 polynomials are identically zero. This state was considered in Refs. \cite{miyake,moor} although not in the context of Dirac spinors.
We can see that the eight examples above belong to eight different entanglement classes that can be discriminated by the polynomial invariants.
\section{The case of four spinors}\label{four}
For four Dirac spinors the state coefficients can be arranged as a $4\times 4\times 4\times 4$ tensor $\Psi^{ABCD}$.
Transformations $S^A,S^B,S^C$, and $S^D$ acting locally on Alice's, Bob's, Charlie's and David's particle, respectively, are described as
\begin{eqnarray}
\Psi^{ABCD}_{ijkl}\to\sum_{mnop}S^A_{im}S^B_{jn}S^C_{ko}S^D_{lp}\Psi^{ABCD}_{mnop}.
\end{eqnarray}
Lorentz invariants of degree 2 can be constructed as tensor contractions of the form
\begin{eqnarray}
\sum_{jklmnpqr}X_{nj}X_{pk}X_{ql}X_{rm}\Psi^{ABCD}_{jklm}\Psi^{ABCD}_{npqr},
\end{eqnarray}
involving two copies of $\Psi^{ABCD}$ where either $X=C$ or $X=C\gamma^5$ for each instance. Unlike the case of three particles the 16 degree 2 Lorentz invariants constructed this way are non-zero.
These 16 different polynomials were calculated and their linear independence tested.
Due to the large number of terms in the polynomials they are not all given in a fully written out form, but a few selected examples are given fully written out in \ref{polllu}.
In writing these contractions we leave out the summation sign in the following, with the understanding that repeated indices are summed over. We also suppress the superscript $ABCD$ of $\Psi$ and use the abbreviated notation $C^5\equiv C\gamma^5$.
The 16 polynomials of degree 2 are linearly independent and given by
\begin{eqnarray}\label{ghj}
H_a&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%=It_{4a}
H_b&=&C^5_{nj}C^5_{pk}C^5_{ql}C^5_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4b}
H_c&=&C_{nj}C_{pk}C_{ql}C^5_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4c}
H_d&=&C_{nj}C_{pk}C^5_{ql}C_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4d}
H_e&=&C^5_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4e}
H_f&=&C_{nj}C^5_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4f}
H_g&=&C_{nj}C_{pk}C^5_{ql}C^5_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4g}
H_h&=&C^5_{nj}C_{pk}C_{ql}C^5_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4h}
H_i&=&C_{nj}C^5_{pk}C_{ql}C^5_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4i}
H_j&=&C^5_{nj}C_{pk}C^5_{ql}C_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4j}
H_k&=&C_{nj}C^5_{pk}C^5_{ql}C_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4k}
H_l&=&C^5_{nj}C^5_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4l}
H_m&=&C^5_{nj}C^5_{pk}C^5_{ql}C_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4m}
H_n&=&C^5_{nj}C^5_{pk}C_{ql}C^5_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%= It_{4n}
H_o&=&C_{nj}C^5_{pk}C^5_{ql}C^5_{rm}\Psi_{jklm}\Psi_{npqr},\nonumber\\%=It_{4p}
H_p&=&C^5_{nj}C_{pk}C^5_{ql}C^5_{rm}\Psi_{jklm}\Psi_{npqr}.
\end{eqnarray}
The next lowest degree is 4.
There are 13 ways to pair the tensor indices of four copies of $\Psi^{ABCD}$ to create polynomials that do not factorize into two degree 2 polynomials
\begin{eqnarray}\label{cvn}
W_a&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{uvqr} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{npyz}\Psi_{wxst},\nonumber\\%m1222 m2033 m0301 m3110
W_b&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{uvqr} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{nxyz}\Psi_{wpst},\nonumber\\%m1122 m2333 m0211 m3000
W_c&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{nvqz} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{upsr}\Psi_{wxyt },\nonumber\\%m2020 m3232 m1101 m0313
W_d&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{nvqz} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{uxsr}\Psi_{wpyt },\nonumber\\%m2122 m3231 m0010 m1303
W_e&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{npst} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{uvqz}\Psi_{wxyr},\nonumber\\%m0003 m1131 m2210 m3322
W_f&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{upqt} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{nvsr}\Psi_{wxyz},\nonumber\\%m0222 m2331 m1013 m3100
W_g&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{upqt} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{nvsz}\Psi_{wxyr},\nonumber\\%m1221 m2332 m0013 m3100
W_h&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{upsr} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{nvyt}\Psi_{wxqz},\nonumber\\%m0212 m2333 m1021 m3100
W_i&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{nvsr} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{upyt}\Psi_{wxqz},\nonumber\\%m2112 m3233 m0021 m1300
W_j&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{nvqr} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{uxyt}\Psi_{wpsz},\nonumber\\%m2122 m3233 m0311 m1000
W_k&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{upqr} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{nvst}\Psi_{wxyz},\nonumber\\%m1222 m2333 m0111 m3000
W_l&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{npsr} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{uvyz}\Psi_{wxqt},\nonumber\\%m2212 m3323 m0030 m1101
W_m&=&X_{nj}X_{pk}X_{ql}X_{rm}\Psi_{jklm}\Psi_{npqt} X_{uw}X_{vx}X_{sy}X_{tz}\Psi_{uvsr}\Psi_{wxyz}.
\end{eqnarray}
We can categorize the tensor contractions in Eq. (\ref{cvn}) in terms of their permutation symmetries. The three contractions
$W_a$, $W_c$ and $W_f$ are each invariant with respect to permutations of two disjoint pairs of laboratories. The contraction $W_a$ is invariant with respect to permutations of AB and permutations of CD. The contraction $W_c$ is invariant with respect to permutations of AC and permutations of BD. The contraction $W_f$ is invariant with respect to permutations of AD and permutations of BC.
Six of the contractions $W_b$, $W_d$, $W_e$, $W_g$, $W_h$, and $W_i$ are each invariant with respect to permutations of a single pair of laboratories. The contraction $W_b$ is invariant with respect to permutations of CD while $W_d$ is invariant with respect to permutations of AC, $W_e$ with respect to AB, $W_g$ with respect to BC, $W_h$ with respect to BD, and $W_i$ with respect to AD. The final four contractions $W_j$, $W_k$, $W_l$, and $W_m$ are each invariant with respect to permutations of a triple of laboratories.
The contraction $W_j$ is invariant with respect to permutations of ACD, while $W_k$ is invariant with respect to permutations of BCD, $W_l$ with respect to ABD, and $W_m$ with respect to ABC.
Moreover, the different tensor sandwich contractions can be represented as graphs.
See \ref{graphs} for the graph representations of the tensor contractions in Eq. (\ref{cvn}).
For each pairing of tensor indices we can consider $(2^{8}-2^4)/2+2^4=136$ ways to choose the $X$s as either $C$ or $C^5$. There is $2^4$ choices where for each lab both $X$s corresponding to the lab are identical and there is $2^8-2^4$ choices that are not of this kind. For the latter case however each such choice for is equivalent to at least one other choice. This can be understood in terms of graph isomorphisms of the graph representations of the tensor contraction in Eq. (\ref{cvn}), and reduces the number of such choices that need to be considered to $(2^8-2^4)/2$.
Thus we can consider a total of $136\times 13=1768$ ways to construct polynomials by this method.
A complete list of polynomials will not be given here. Instead only a selection of 26 polynomials were computed and tested for linear dependence. Thirteen Lorentz invariant polynomials were constructed by choosing all $X$s in Eq. (\ref{cvn}) as $C$ and 13 polynomials were constructed by choosing all $X$s in Eq. (\ref{cvn}) as $C^5$.
These polynomials are invariant under P in all labs. When evaluating the possible linear dependencies of these polynomials on products of the degree 2 polynomials in Eq. (\ref{ghj}) we note that the square of any of these degree 2 polynomials is invariant under P in all labs while any product of two different degree 2 polynomials is not. Therefore only the squares need to be considered.
Evaluating the possible linear dependencies showed that the 26 degree 4 polynomials together with the 16 squares of the degree 2 polynomials form a 41 dimensional polynomial space. Thus there is a single equation describing the linear dependence.
Due to the large number of terms in the polynomials they are not all given in a fully written out form, but two selected examples are given fully written out in \ref{polllu}.
The 26 polynomials of degree four are
\begin{eqnarray}\label{ffr}
T_a&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{uvqr} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{npyz}\Psi_{wxst},\nonumber\\
T_b&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{uvqr} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{nxyz}\Psi_{wpst},\nonumber\\
T_c&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{nvqz} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{upsr}\Psi_{wxyt },\nonumber\\
T_d&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{nvqz} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{uxsr}\Psi_{wpyt },\nonumber\\
T_e&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{npst} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{uvqz}\Psi_{wxyr},\nonumber\\
T_f&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{upqt} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{nvsr}\Psi_{wxyz},\nonumber\\
T_g&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{upqt} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{nvsz}\Psi_{wxyr},\nonumber\\
T_h&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{upsr} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{nvyt}\Psi_{wxqz},\nonumber\\
T_i&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{nvsr} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{upyt}\Psi_{wxqz},\nonumber\\
T_j&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{nvqr} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{uxyt}\Psi_{wpsz},\nonumber\\
T_k&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{upqr} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{nvst}\Psi_{wxyz},\nonumber\\
T_l&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{npsr} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{uvyz}\Psi_{wxqt},\nonumber\\
T_m&=&C_{nj}C_{pk}C_{ql}C_{rm}\Psi_{jklm}\Psi_{npqt} C_{uw}C_{vx}C_{sy}C_{tz}\Psi_{uvsr}\Psi_{wxyz},
\end{eqnarray}
and
\begin{eqnarray}\label{ffr2}
Y_a&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{uvqr} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{npyz}\Psi_{wxst},\nonumber\\
Y_b&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{uvqr} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{nxyz}\Psi_{wpst},\nonumber\\
Y_c&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{nvqz} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{upsr}\Psi_{wxyt },\nonumber\\
Y_d&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{nvqz} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{uxsr}\Psi_{wpyt },\nonumber\\
Y_e&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{npst} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{uvqz}\Psi_{wxyr},\nonumber\\
Y_f&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{upqt} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{nvsr}\Psi_{wxyz},\nonumber\\
Y_g&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{upqt} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{nvsz}\Psi_{wxyr},\nonumber\\
Y_h&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{upsr} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{nvyt}\Psi_{wxqz},\nonumber\\
Y_i&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{nvsr} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{upyt}\Psi_{wxqz},\nonumber\\
Y_j&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{nvqr} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{uxyt}\Psi_{wpsz},\nonumber\\
Y_k&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{upqr} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{nvst}\Psi_{wxyz},\nonumber\\
Y_l&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{npsr} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{uvyz}\Psi_{wxqt},\nonumber\\
Y_m&=&C_{nj}^5C_{pk}^5C_{ql}^5C_{rm}^5\Psi_{jklm}\Psi_{npqt} C_{uw}^5C_{vx}^5C_{sy}^5C_{tz}^5\Psi_{uvsr}\Psi_{wxyz}.
\end{eqnarray}
The equation describing the linear dependence is
\begin{align}
H_b^2 - H_a^2=2(& T_a - T_b - T_c- T_d - T_e+ T_f - T_g
- T_h - T_i - T_j - T_k - T_l -
T_m \nonumber\\& - Y_a + Y_b + Y_c+ Y_d + Y_e - Y_f + Y_g +
Y_h + Y_i + Y_j + Y_k + Y_l + Y_m).
\end{align}
Note that the polynomial $H_a^2+2( T_a - T_b - T_c- T_d - T_e+ T_f - T_g
- T_h - T_i - T_j - T_k - T_l -
T_m )$ is invariant, up to a U(1) phase, under $G^{C}$ in all labs and the polynomial $H_b^2+2( Y_a - Y_b - Y_c- Y_d - Y_e + Y_f - Y_g -
Y_h - Y_i - Y_j - Y_k - Y_l - Y_m)$ is invariant, up to a U(1) phase, under $G^{C\gamma^5}$ in all labs. Therefore this equation implies that both these two polynomials are invariant, up to a U(1) phase, in all labs under the smallest Lie group that contains both $G^{C}$ and $G^{C\gamma^5}$ as subgroups (See Ref. \cite{spinorent} Theorem 2). The smallest Lie group that contains both $G^{C}$ and $G^{C\gamma^5}$ as subgroups is $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$.
\subsection{Weyl particles}
If Alice's, Bob's, Charlie's and David's particles are Weyl particles the shared state is invariant under some combination of projections $P_L^A$ or $P_R^A$ by Alice, $P_L^B$ or $P_R^B$ by Bob, $P_L^C$ or $P_R^C$ by Charlie, and $P_L^D$ or $P_R^D$ by David.
Thus the shared state of the spinors has the symmetry $\psi_{jklm}=(-1)^{|LA|}\psi_{(j-2) klm}=(-1)^{|LB|}\psi_{j (k-2)lm}=(-1)^{|LC|}\psi_{j k(l-2)m}=(-1)^{|LD|}\psi_{ jkl(m-2)}$, where $|LA|=1$ if the state is invariant under $P_L^A$ and otherwise zero, $|LB|=1$ if the state is invariant under $P_L^B$ and otherwise zero, $|LC|=1$ if the state is invariant under $P_L^C$ and otherwise zero, $|LD|=1$ if the state is invariant under $P_L^D$ and otherwise zero, and $j$,$k$,$l$, and $m$ are defined modulo 4.
In this case all the Lorentz invariants of degree 2 reduce to $16H$, up to a sign, where
\begin{eqnarray}\label{hpo}
H= &&\psi_{0000}\psi_{1111}-\psi_{0111}\psi_{1000} +\psi_{0110}\psi_{1001} +\psi_{0101}\psi_{1010} \nonumber\\
&&-\psi_{0100}\psi_{1011} + \psi_{ 0011}\psi_{1100} -\psi_{0010}\psi_{1101} -\psi_{0001}\psi_{1110},
\end{eqnarray}
is the degree 2 polynomial local $\mathrm{SL}(2,\mathbb{C})$ invariant for the case of four non-relativistic spin-$\frac{1}{2}$ particles described by Luque and Thibon in Ref. \cite{luque}, by Verstraete, Dehaene and De Moor in Ref. \cite{moor} and by Wong and Christensen in Ref. \cite{wong}.
Reference \cite{luque} also describe two degree 4 invariants $L$ and $M$ for the same scenario.
These are given by
\begin{eqnarray}\label{lpo}
L=&&\psi_{ 0011}\psi_{ 0110}\psi_{ 1001}\psi_{ 1100} -\psi_{ 0010}\psi_{ 0111}\psi_{1001}\psi_{ 1100}\nonumber\\&& -
\psi_{ 0011}\psi_{ 0101}\psi_{ 1010}\psi_{ 1100 }+\psi_{ 0001}\psi_{ 0111}\psi_{ 1010}\psi_{ 1100 }\nonumber\\&& +
\psi_{ 0010}\psi_{ 0101}\psi_{ 1011}\psi_{ 1100} -\psi_{ 0001}\psi_{ 0110}\psi_{ 1011}\psi_{ 1100 }\nonumber\\&&-
\psi_{ 0011}\psi_{ 0110}\psi_{ 1000}\psi_{ 1101} +\psi_{ 0010}\psi_{ 0111}\psi_{ 1000}\psi_{ 1101}\nonumber\\&& +
\psi_{ 0011}\psi_{ 0100}\psi_{ 1010}\psi_{ 1101} -\psi_{ 0000}\psi_{ 0111}\psi_{ 1010}\psi_{ 1101}\nonumber\\&& -
\psi_{ 0010}\psi_{ 0100}\psi_{ 1011}\psi_{ 1101} +\psi_{ 0000}\psi_{ 0110}\psi_{ 1011}\psi_{ 1101}\nonumber\\&& +
\psi_{ 0011}\psi_{ 0101}\psi_{ 1000}\psi_{ 1110} -\psi_{ 0001}\psi_{ 0111}\psi_{ 1000}\psi_{ 1110}\nonumber\\&& -
\psi_{ 0011}\psi_{ 0100}\psi_{ 1001}\psi_{ 1110} +\psi_{ 0000}\psi_{ 0111}\psi_{ 1001}\psi_{ 1110}\nonumber\\&& +
\psi_{ 0001}\psi_{ 0100}\psi_{ 1011}\psi_{ 1110} -\psi_{ 0000}\psi_{ 0101}\psi_{ 1011}\psi_{ 1110}\nonumber\\&& -
\psi_{ 0010}\psi_{ 0101}\psi_{ 1000}\psi_{ 1111} +\psi_{ 0001}\psi_{ 0110}\psi_{ 1000}\psi_{ 1111}\nonumber\\&& +
\psi_{ 0010}\psi_{ 0100}\psi_{ 1001}\psi_{ 1111} -\psi_{ 0000}\psi_{ 0110}\psi_{ 1001}\psi_{ 1111}\nonumber\\&& -
\psi_{ 0001}\psi_{ 0100}\psi_{ 1010}\psi_{ 1111} +\psi_{ 0000}\psi_{ 0101}\psi_{ 1010}\psi_{ 1111},
\end{eqnarray}
and
\begin{eqnarray}\label{mpo}
M =&& -\psi_{ 0101}\psi_{ 0110}\psi_{ 1001}\psi_{ 1010} +\psi_{ 0100}\psi_{ 0111}\psi_{ 1001}\psi_{ 1010}\nonumber\\&& +
\psi_{ 0101}\psi_{ 0110}\psi_{ 1000}\psi_{ 1011} -\psi_{ 0100}\psi_{ 0111}\psi_{ 1000}\psi_{ 1011}\nonumber\\&& +
\psi_{ 0011}\psi_{ 0101}\psi_{ 1010}\psi_{ 1100} -\psi_{ 0001}\psi_{ 0111}\psi_{ 1010}\psi_{ 1100}\nonumber\\&& -
\psi_{ 0010}\psi_{ 0101}\psi_{ 1011}\psi_{ 1100} +\psi_{ 0000}\psi_{ 0111}\psi_{ 1011}\psi_{ 1100}\nonumber\\&& -
\psi_{ 0011}\psi_{ 0100}\psi_{ 1010}\psi_{ 1101} +\psi_{ 0001}\psi_{ 0110}\psi_{ 1010}\psi_{ 1101}\nonumber\\&& +
\psi_{ 0010}\psi_{ 0100}\psi_{ 1011}\psi_{ 1101} -\psi_{ 0000}\psi_{ 0110}\psi_{ 1011}\psi_{ 1101}\nonumber\\&& -
\psi_{ 0011}\psi_{ 0101}\psi_{ 1000}\psi_{ 1110} +\psi_{ 0001}\psi_{ 0111}\psi_{ 1000}\psi_{ 1110}\nonumber\\&& +
\psi_{ 0010}\psi_{ 0101}\psi_{ 1001}\psi_{ 1110} -\psi_{ 0000}\psi_{ 0111}\psi_{ 1001}\psi_{ 1110 }\nonumber\\&&-
\psi_{ 0001}\psi_{ 0010}\psi_{ 1101}\psi_{ 1110} +\psi_{ 0000}\psi_{ 0011}\psi_{ 1101}\psi_{ 1110} \nonumber\\&&+
\psi_{ 0011}\psi_{ 0100}\psi_{ 1000}\psi_{ 1111} -\psi_{ 0001}\psi_{ 0110}\psi_{ 1000}\psi_{ 1111 }\nonumber\\&&-
\psi_{ 0010}\psi_{ 0100}\psi_{ 1001}\psi_{ 1111} +\psi_{ 0000}\psi_{ 0110}\psi_{ 1001}\psi_{ 1111 }\nonumber\\&&+
\psi_{ 0001}\psi_{ 0010}\psi_{ 1100}\psi_{ 1111} -\psi_{ 0000}\psi_{ 0011}\psi_{ 1100}\psi_{ 1111}.
\end{eqnarray}
The polynomials $L,M,$ and $H^2$ are linearly independent.
Much like the degree 2 invariants all reduce to a multiple of $H$ for the case of Weyl particles, the degree 4 invariants reduce to linear combinations of $H^2,L,$ and $M$.
The polynomials $T_a$ and $Y_a$ reduce to $512(H^2 - 2 L - 4 M)$, the polynomials $T_c$ and $Y_c$ reduce to $512( -H^2 - 4 L - 2 M)$ while $T_f$ and $Y_f$ reduce to $512 (H^2-2L +2M)$.
The polynomials $T_b,T_e,Y_b$ and $Y_e$ reduce to $1024 (-L - 2 M)$, the polynomials
$T_d,T_h,Y_d$ and $Y_h$ reduce to $1024 (2L + M)$, while $T_g,T_i,Y_g$ and $Y_i$ reduce to $1024 (-L +M)$. The polynomials $T_j,T_k,T_l,T_m,Y_j,Y_k,Y_l$, and $Y_m$ reduce to $512H^2$.
\subsection{Eigenspaces of the local Dirac Hamiltonians in the case of zero momenta and zero four-potentials}
We can consider the case of zero momentum and zero four-potential, sometimes described as the non-relativistic limit of a free particle. If Alice's, Bob's, Charlie's and David's particles are all in this limit and also in an eigenstate of the local Dirac Hamiltonians the shared state is invariant under some combination of projections $P_+^A$ or $P_-^A$ by Alice, $P_+^B$ or $P_-^B$ by Bob, $P_+^C$ or $P_-^C$ by Charlie, and $P_+^D$ or $P_-^D$ by David. In this case only a $2\times 2\times 2 \times 2$ subtensor of $\Psi^{ABCD}$ is nonzero.
For eigenspaces of the local Dirac Hamiltonians in this limit the polynomial $H_a$ reduces up to a relabelling of the indices to $2H$ where $H$ is the polynomial given in Eq. (\ref{hpo}). All the other degree 2 polynomials given in Eq. (\ref{ghj}) reduce to zero. This follows since $P_+C\gamma^5P_+=P_-C\gamma^5P_-=0$.
The degree 4 polynomials given in Eq. (\ref{ffr}) reduce up to a relabelling of the indices to linear combinations of $H^2$, $L$ and $M$ where $L$ is given in Eq. (\ref{lpo}) and $M$ is given in Eq. (\ref{mpo}).
The polynomial $T_a$ reduces up to a relabelling of the indices to $2 H^2- 4 L - 8 M $.
The polynomial $T_c$ reduces up to a relabelling of the indices to $- 2 H^2 - 8 L - 4 M$.
The polynomial $T_f$ reduces up to a relabelling of the indices to $2 H^2 -4 L+ 4 M $.
The polynomials $T_b$ and $T_e$ reduce up to a relabelling of the indices to $ - 4 L- 8 M$.
The polynomials $T_d$ and $T_h$ reduce up to a relabelling of the indices to $8 L+ 4 M $.
The polynomials $T_g$ and $T_i$ reduce up to a relabelling of the indices to $-4 L+ 4 M $.
Finally, the polynomials $T_j$, $T_k$, $T_l$ and $T_m$ reduce up to a relabelling of the indices to $ 2 H^2$.
Since we have that $P_+C\gamma^5P_+=P_-C\gamma^5P_-=0$ all polynomials of degree 4 given in Eq. (\ref{ffr2}) reduce to zero. Moreover, this property of $C\gamma^5$ implies that the polynomials in Eq. (\ref{ffr}) are the only polynomials on the form given in Eq. (\ref{cvn}) that do not reduce to zero in this case.
\subsection{Examples of fourpartite spinor entangled states}
Here we consider a few examples of fourpartite entangled states to illustrate how inequivalent forms of spinor entanglement are distinguished by the polynomials.
We can consider analogues of the four-partite entangled Greenberger-Horne-Zeilinger (GHZ)\cite{ghz} state for non-relativistic spin-$\frac{1}{2}$ particles.
One such state is
\begin{eqnarray}
\frac{1}{\sqrt{2}}({\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}).
\end{eqnarray}
For this state $|H_{a}|=1$, but all the other degree 2 polynomials are identically zero and $|T_a|=|T_c|=|T_f|=|T_j|=|T_k|=|T_l|=|T_m|=1/2$, while the other degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero.
Similarly we can construct a state
\begin{eqnarray}
\frac{1}{\sqrt{2}}({\phi_2^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_2^C}{\,\otimes\,}imes{\phi_2^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}).
\end{eqnarray}
for which $|H_{b}|=1$, but all the other degree 2 polynomials are identically zero and $|Y_a|=|Y_c|=|Y_f|=|Y_j|=|Y_k|=|Y_l|=|Y_m|=1/2$, while the other degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero.
Furthermore, one can construct analogues of the GHZ state that involve more than two basis spinors. One example is
\begin{eqnarray}
\frac{1}{\sqrt{2}}({\phi_0^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_3^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}).
\end{eqnarray}
for which $|H_{c}|=1$, but all the other degree 2 polynomials are identically zero and all the degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero. Another example is
\begin{eqnarray}
\frac{1}{\sqrt{2}}({\phi_0^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_3^C}{\,\otimes\,}imes{\phi_3^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_2^D}).
\end{eqnarray}
for which $|H_{d}|=1$, but all the other degree 2 polynomials are identically zero and all the degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero. As is the case for the four examples given here, for any GHZ like state of this kind only one of the polynomials of degree 2 is nonzero.
An example of a generalized GHZ state with three terms is
\begin{eqnarray}
\frac{1}{\sqrt{3}}({\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}+{\phi_2^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_2^C}{\,\otimes\,}imes{\phi_2^D}).
\end{eqnarray}
for which $|H_{a}|=|H_{b}|=2/3$, but all the other degree 2 polynomials are identically zero, and $|T_a|=|T_c|=|T_f|=|T_j|=|T_k|=|T_l|=|T_m|=|Y_a|=|Y_c|=|Y_f|=|Y_j|=|Y_k|=|Y_l|=|Y_m|=2/9$ while the other degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero. A generalized GHZ state with four terms is
\begin{align}
\frac{1}{{2}}(&{\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}\nonumber\\&+{\phi_2^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_2^C}{\,\otimes\,}imes{\phi_2^D}+{\phi_3^A}{\,\otimes\,}imes{\phi_3^B}{\,\otimes\,}imes{\phi_3^C}{\,\otimes\,}imes{\phi_3^D}).
\end{align}
for which $|H_{a}|=|H_{b}|=1$, but all the other degree 2 polynomials are identically zero, and $|T_a|=|T_c|=|T_f|=|T_j|=|T_k|=|T_l|=|T_m|=|Y_a|=|Y_c|=|Y_f|=|Y_j|=|Y_k|=|Y_l|=|Y_m|=1/4$ while the other degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero.
An example of a state not on the GHZ form is the analogue of the so called cluster state \cite{briegel}
\begin{align}
\frac{1}{{2}}(&{\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}-{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}\nonumber\\&+{\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}).
\end{align}
For this state all the degree 2 polynomials are zero and $|T_a|=|T_b|=|T_e|=1/2$, and $|T_c|=|T_d|=|T_f|=|T_g|=|T_h|=|T_i|=1/4$ while the other degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero.
Another analogue of the cluster state is
\begin{align}
\frac{1}{{2}}(&{\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}-{\phi_3^A}{\,\otimes\,}imes{\phi_3^B}{\,\otimes\,}imes{\phi_3^C}{\,\otimes\,}imes{\phi_3^D}\nonumber\\&+{\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_3^C}{\,\otimes\,}imes{\phi_3^D}+{\phi_3^A}{\,\otimes\,}imes{\phi_3^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}).
\end{align}
For this state all the degree 2 polynomials are zero and $|Y_a|=|Y_b|=|Y_e|=1/2$, and $|Y_c|=|Y_d|=|Y_f|=|Y_g|=|Y_h|=|Y_i|=1/4$ while the other degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero.
A further example of a state for which all degree 2 polynomials are zero is the state
\begin{align}
\frac{1}{{2}}(&{\phi_0^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_2^C}{\,\otimes\,}imes{\phi_2^D}+{\phi_2^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_3^C}{\,\otimes\,}imes{\phi_3^D}\nonumber\\&+{\phi_1^A}{\,\otimes\,}imes{\phi_3^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}+{\phi_3^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}).
\end{align}
Moreover, for this state $|T_a|=|Y_b|=1/4$ while the other degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero.
Likewise, all degree 2 polynomials are zero for the state
\begin{align}
\frac{1}{{2}}(&{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_2^C}{\,\otimes\,}imes{\phi_2^D}+{\phi_2^A}{\,\otimes\,}imes{\phi_3^B}{\,\otimes\,}imes{\phi_3^C}{\,\otimes\,}imes{\phi_3^D}\nonumber\\&+{\phi_0^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}+{\phi_3^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}),
\end{align}
and $|T_b|=|Y_k|=1/4$ while the other degree 4 polynomials in Eq. (\ref{ffr}) and in Eq. (\ref{ffr2}) are zero.
We can see that the ten examples above belong to ten different entanglement classes that can be discriminated by the polynomial invariants.
\subsection{Examples of bipartite entangled four spinor states}
The polynomials constructed as tensor sandwich contractions for the case of four spinors can be non-zero also for states that are not fourpartite entangled but only bipartite entangled.
An example of a state of four Dirac spinors that is a product state over the partitioning AB|CD but bipartite entangled on AB and CD is
\begin{align}
\frac{1}{{2}}(&{\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}\nonumber\\&+{\phi_0^A}{\,\otimes\,}imes{\phi_0^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_1^B}{\,\otimes\,}imes{\phi_0^C}{\,\otimes\,}imes{\phi_0^D}).
\end{align}
For this state $|H_a|=1$ while the other degree 2 polynomials are zero and $|T_a|=1$, $|T_b|=|T_e|=|T_j|=|T_k|=|T_l|=|T_m|=1/2$, and $|T_c|=|T_d|=|T_f|=|T_g|=|T_h|=|T_i|=1/4$ while the other degree 4 polynomials in Eq. (\ref{ffr}) and Eq. (\ref{ffr2}) are zero.
Similarly, the state
\begin{align}
\frac{1}{{2}}(&{\phi_0^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_2^C}{\,\otimes\,}imes{\phi_0^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_3^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_1^D}\nonumber\\&+{\phi_0^A}{\,\otimes\,}imes{\phi_3^B}{\,\otimes\,}imes{\phi_1^C}{\,\otimes\,}imes{\phi_0^D}+{\phi_1^A}{\,\otimes\,}imes{\phi_2^B}{\,\otimes\,}imes{\phi_2^C}{\,\otimes\,}imes{\phi_1^D}).
\end{align}
is a product state over the partitioning AD|BC but bipartite entangled on AD and BC.
For this state $|H_d|=1$ while the other degree 2 polynomials are zero and all degree 4 polynomials in Eq. (\ref{ffr}) and Eq. (\ref{ffr2}) are zero.
\section{The case of five spinors}\label{five}
For five Dirac spinors the state coefficients can be arranged as a $4\times 4\times 4\times 4\times 4$ tensor $\Psi^{ABCDE}$.
Transformations $S^A,S^B,S^C,S^D$, and $S^E$ acting locally on Alice's, Bob's, Charlie's, David's, and Erin's particle, respectively, are described as
\begin{eqnarray}
\Psi^{ABCDE}_{ijklm}\to\sum_{nopqr}S^A_{in}S^B_{jo}S^C_{kp}S^D_{lq}S^E_{mr}\Psi^{ABCDE}_{nopqr}.
\end{eqnarray}
Lorentz invariants of degree 2 can be constructed as tensor sandwich contractions involving two copies of $\Psi^{ABCDE}$, however all such polynomials are identically zero due to the antisymmetry of $C$ and $C\gamma^5$.
The next lowest degree is 4.
From the above follows that invariants of degree 4 that factorize as a product of two degree 2 polynomials are identically zero.
Thus it remains to consider the tensor sandwich contractions involving four copies of $\Psi^{ABCDE}$ that do not factorize into two degree 2 polynomials. There are 40 inequivalent such ways to pair up the tensor indices of the four copies. These are given in \ref{wides}.
For each pairing of tensor indices we can consider $(2^{10}-2^5)/2+2^5=528$ ways to choose the $X$s as either $C$ or $C\gamma^5$. This gives a total of 21120 ways to construct polynomials by this method. Due to the computational difficulty in constructing the polynomials and testing their linear independence no explicit examples are given here.
\section{Degree 2 polynomials for more than five spinors}\label{getto}
For $N$ Dirac spinors the state coefficients can be arranged as a $4\times 4\times \dots\times 4$ tensor $\Psi^{ABCD\dots N}$.
Transformations $S^A,S^B,S^C,\dots S^N$ acting locally on the particles, are described as
\begin{eqnarray}
\Psi^{ABCD\dots N}_{ijk\dots l}\to\sum_{mno\dots p}S^A_{im}S^B_{jn}S^C_{ko}\dots S^N_{lp}\Psi^{ABCD\dots N}_{mno\dots p}.
\end{eqnarray}
Lorentz invariants of degree 2 can be constructed as tensor sandwich contractions of the form
\begin{eqnarray}\label{hjl}
\sum_{jkl\dots mnpq \dots r}X_{nj}X_{pk}X_{ql}\dots X_{rm}\Psi^{ABC\dots N}_{jkl\dots m}\Psi^{ABC\dots N}_{npq\dots r},
\end{eqnarray}
involving two copies of $\Psi^{ABC\dots N}$ where either $X=C$ or $X=C\gamma^5$ for each instance. For odd $N$ these polynomials are zero due to the antisymmetry of $C$ and $C\gamma^5$. For every even $N$ on the other hand there is $2^N$ linearly independent polynomials on the form in Eq. (\ref{hjl}) corresponding to the different choices of $X$ as either $X=C$ or $X=C\gamma^5$.
If the particles of all observers are Weyl particles the shared state is invariant under some combination of projections, $P_L$ or $P_R$ for each observer. In this case the polynomials on the form in Eq. (\ref{hjl}) for even $N$ all reduce, up to a sign, to $2^N\tau_{1\dots N}$ where $\tau_{1\dots N}=\sum_{jkl\dots mnpq \dots r\in\{0,1\}}\epsilon_{nj}\epsilon_{pk}\epsilon_{ql}\dots \epsilon_{rm}\psi^{ABC\dots N}_{jkl\dots m}\psi^{ABC\dots N}_{npq\dots r}$ where the Levi-Civita antisymmetric symbol $\epsilon_{jk}$ is defined by $1=\epsilon_{10}=-\epsilon_{01},$ and $\epsilon_{00}=\epsilon_{11}=0$. The polynomial $\tau_{1\dots N}$ is the $N$-tangle introduced by Wong and Christensen in Ref. \cite{wong}.
In the case of zero momenta and zero four-potentials and with the state of the particles belonging to an eigenspace of the local Dirac Hamiltonians the polynomial where each $X=C$ reduces up to a relabelling of the indices to $\tau_{1\dots N}$ while the $2^N-1$ other degree 2 polynomials are zero.
The lowest nonzero case not already described in Section \ref{four} or in Ref. \cite{spinorent} is six Dirac spinors. For this case there is 64 linearly independent polynomials on the form in Eq. (\ref{hjl}) corresponding to the different choices of $X$ as either $X=C$ or $X=C\gamma^5$. For Weyl particles all these polynomials reduce, up to a sign, to $64\tau_{1\dots 6}$ where $\tau_{1\dots 6}$ is given by
\begin{eqnarray}
\tau_{1\dots 6}= &&\psi_{000000}\psi_{111111}-\psi_{011111}\psi_{100000} -\psi_{101111}\psi_{010000}\nonumber\\&&-\psi_{110111}\psi_{001000}-\psi_{111011}\psi_{000100}-\psi_{111101}\psi_{000010}\nonumber\\&&-\psi_{111110}\psi_{000001}+\psi_{001111}\psi_{110000}
+\psi_{010111}\psi_{101000}\nonumber\\&&+\psi_{011011}\psi_{100100}+\psi_{011101}\psi_{100010}+\psi_{011110}\psi_{100001}\nonumber\\&&+\psi_{100111}\psi_{011000}+\psi_{101011}\psi_{010100}+\psi_{101101}\psi_{010010}\nonumber\\&&+\psi_{101110}\psi_{010001}+\psi_{110011}\psi_{001100}
+\psi_{110101}\psi_{001010}\nonumber\\&&+\psi_{110110}\psi_{001001}+\psi_{111001}\psi_{000110}+\psi_{111010}\psi_{000101}\nonumber\\&&+\psi_{111100}\psi_{000011}-\psi_{111000}\psi_{000111}-\psi_{110100}\psi_{001011}\nonumber\\&&-\psi_{110010}\psi_{001101}-\psi_{110001}\psi_{001110}-\psi_{101100}\psi_{010011}\nonumber\\&&-\psi_{101010}\psi_{010101}-\psi_{101001}\psi_{010110}-\psi_{100110}\psi_{011001}\nonumber\\&&-\psi_{100101}\psi_{011010}-\psi_{100011}\psi_{011100}.
\end{eqnarray}
The polynomial $\tau_{1\dots 6}$ is the $6$-tangle of Wong and Christensen \cite{wong}.
In the case of zero momenta and zero four-potentials and with the state of the particles belonging to an eigenspace of the local Dirac Hamiltonians the polynomial where each $X=C$ reduces up to a relabelling of the indices to $\tau_{1\dots 6}$ while the 63 other degree 2 polynomials are zero.
\section{General properties of polynomial invariants of connected complex reductive Lie groups}\label{lli}
For a system of $n$ Dirac spinors any polynomial constructed by tensor sandwich contractions is invariant, up to a U(1) phase, under some Lie group $G_1{\,\otimes\,}imes G_2{\,\otimes\,}imes\dots{\,\otimes\,}imes G_n$ where each group $G_k$ is one out of $G^C$, $G^{C\gamma^5}$, $G^{C\gamma^5}\cap G^C$ and $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$. Here we consider some general properties of these groups and the polynomials invariant, up to a U(1) phase, under these groups.
For each of the groups $G^C$, $G^{C\gamma^5}$, $G^{C\gamma^5}\cap G^C$ and $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ we can consider its determinant one subgroup. The determinant one subgroup of $\mathrm{U}(1)\times\mathrm{SL}(4,\mathbb{C})$ is clearly
$\mathrm{SL}(4,\mathbb{C})$. Denote the determinant one subgroup of $G^C$ by $\mathrm{S}G^C$ and denote the determinant one subgroup of $G^{C\gamma^5}$ by $\mathrm{S}G^{C\gamma^5}$.
The group $\mathrm{S}G^C$ consists of all linear transformations that preserve the skew-symmetric bilinear form $\psi^TC\varphi$. The group $\mathrm{S}G^{C\gamma^5}$ consists of all linear transformations that preserve the skew-symmetric bilinear form $\psi^TC\gamma^5\varphi$.
Both $\mathrm{S}G^C$ and $\mathrm{S}G^{C\gamma^5}$ are isomorphic to $\mathrm{Sp}(4,\mathbb{C})$.
The group $\mathrm{S}G^{C\gamma^5}\cap \mathrm{S}G^C$ consists of all linear transformations that preserve both of the skew-symmetric bilinear forms $\psi^TC\varphi$ and $\psi^TC\gamma^5\varphi$ and is isomorphic to $\mathrm{SL}(2,\mathbb{C})\times\mathrm{SL}(2,\mathbb{C})$.
Next consider $G_U^C$ and $G_U^{C\gamma^5}$ and denote their determinant one subgroups by $\mathrm{S}G_U^C$ and $\mathrm{S}G_U^{C\gamma^5}$ respectively. The groups $\mathrm{S}G_U^C$ and $\mathrm{S}G_U^{C\gamma^5}$ are the maximal compact subgroups of $\mathrm{S}G^C$ and $\mathrm{S}G^{C\gamma^5}$, respectively, i.e., $\mathrm{S}G_U^C=\mathrm{S}G^C\cap \mathrm{U}(n)$ and $\mathrm{S}G_U^{C\gamma^5}=\mathrm{S}G^{C\gamma^5}\cap \mathrm{U}(n)$. Both $\mathrm{S}G_U^C$ and $\mathrm{S}G_U^{C\gamma^5}$ are isomorphic to $\mathrm{Sp}(2)$. The group $\mathrm{S}G_U^{C\gamma^5}\cap \mathrm{S}G_U^C$ is the maximal compact subgroup of $\mathrm{S}G^{C\gamma^5}\cap \mathrm{S}G^C$ and it is isomorphic to $\mathrm{SU}(2)\times\mathrm{SU}(2)$.
Let $\mathrm{Lie}(G)$ be the Lie algebra of a Lie group $G\subset\mathrm{GL}(n,\mathbb{C})$ and let $\mathrm{Lie}(K)$ be the Lie algebra of its maximal compact subgroup $K\equiv G\cap \mathrm{U}(n)$. We say that
$G$ is the complexification of $K$ if $\mathrm{Lie}(G)=\mathrm{Lie}(K)+i\mathrm{Lie}(K)$. The group $\mathrm{S}G^C$ is the complexification of $\mathrm{S}G_U^C$, the group $\mathrm{S}G^{C\gamma^5}$ is the complexification of $\mathrm{S}G_U^{C\gamma^5}$, the group $\mathrm{S}G^{C\gamma^5}\cap \mathrm{S}G^C$ is the complexification of $\mathrm{S}G_U^{C\gamma^5}\cap \mathrm{S}G_U^C$ and $\mathrm{SL}(4,\mathbb{C})$ is the complexification of $\mathrm{SU}(4)$.
A Lie subgroup of $\mathrm{GL}(n,\mathbb{C})$ that is the complexification of a compact Lie group is here called a {\it complex reductive group} following Onishchik and Vinberg (See Ref. \cite{vinberg} Ch. 5 {\S}2. 5$^{\circ}$). Note that this property is invariant under conjugation of the group by $S\in \mathrm{GL}(n,\mathbb{C})$, i.e., if a group $G\subset \mathrm{GL}(n,\mathbb{C})$ is complex reductive so is the group $SGS^{-1}$ defined by the elements $SgS^{-1}$ for $g\in G$. Thus the groups that preserve the bilinear forms ${\psi^T}(S^{-1})^TCS^{-1}{\varphi}$ and ${\psi^T}(S^{-1})^TC\gamma^5S^{-1}{\varphi}$ are complex reductive regardless of the choice of matrices $S\gamma^0S^{-1},S\gamma^1S^{-1},S\gamma^2S^{-1},S\gamma^3S^{-1}$ satisfying Eq. (\ref{anti}). However, since every compact Lie group can be represented as a subgroup of a unitary group (See e.g. Ref. \cite{nolan} Ch. 2.1.2), we can consider the case of compact subgroups of $\mathrm{U}(n)$ and their complexifications without loss of generality.
Any polynomial constructed by tensor sandwich contractions is invariant under some group $G_1{\,\otimes\,}imes G_2{\,\otimes\,}imes\dots{\,\otimes\,}imes G_n$ acting on $n$ Dirac spinors where each group $G_k$ is one out of the groups $\mathrm{S}G^C$, $\mathrm{S}G^{C\gamma^5}$, $\mathrm{S}G^{C\gamma^5}\cap \mathrm{S}G^C$ and $\mathrm{SL}(4,\mathbb{C})$. The tensor product of two connected complex reductive Lie groups is a connected complex reductive Lie group and thus $G_1{\,\otimes\,}imes G_2{\,\otimes\,}imes\dots{\,\otimes\,}imes G_n$ is a connected complex reductive Lie group. The algebra of polynomial invariants of a connected complex reductive Lie group have a number of useful general properties. We describe some of these properties in the following.
Let $G$ be a connected Lie subgroup of $\mathrm{GL}(n,\mathbb{C})$ that is the complexification of its maximal compact subgroup.
Then the algebra of polynomial invariants under $G$ and the algebra of polynomial invariants under its maximal compact subgroup are the same. To see this we consider the following theorem based on the so called Unitarian Trick \cite{hurw,weeyl}
\begin{theorem}\label{uii}
Let $G$ be a connected Lie subgroup of $\mathrm{GL}(n,\mathbb{C})$ and let $K\equiv G\cap \mathrm{U}(n)$. Assume that $G$ is the complexification of $K$.
Let $\mathcal{O}(\mathbb{C}^n)$ be the algebra over $\mathbb{C}$ of polynomials on $\mathbb{C}^n$, let $\mathcal{O}(\mathbb{C}^n)^G\subset \mathcal{O}(\mathbb{C}^n)$ be the subalgebra of polynomials that are invariant under $G$, i.e., $\mathcal{O}(\mathbb{C}^n)^G\equiv\{f\in\mathcal{O}(\mathbb{C}^n)|f(gx)=f(x) \phantom{t}\textrm{for all}\phantom{t} g\in G, x\in \mathbb{C}^n\}$, and let $\mathcal{O}(\mathbb{C}^n)^K\subset \mathcal{O}(\mathbb{C}^n)$ be the subalgebra of polynomials that are invariant under $K$, i.e., $\mathcal{O}(\mathbb{C}^n)^K\equiv\{f\in\mathcal{O}(\mathbb{C}^n)|f(gx)=f(x) \phantom{t}\textrm{for all}\phantom{t} g\in K, x\in \mathbb{C}^n\}$. Then $\mathcal{O}(\mathbb{C}^n)^G=\mathcal{O}(\mathbb{C}^n)^K$.
\end{theorem}
\begin{proof}
Let $F\in\mathcal{O}(\mathbb{C}^n)$ and $v\in \mathbb{C}^n$. For any $n\times n$ matrix $M$ consider the function on $\mathbb{R}\times \mathbb{C}^n$ defined by $F(e^{Mt}v)$. Then $\frac{d}{dt}F(e^{Mt}v)|_{t=0}=Mv\cdot\nabla F(v)$.
Since $F(e^{M(t+a)}v)=F(e^{Mt}e^{Ma}v)$ we see that $\frac{d}{dt}F(e^{Mt}v)|_{t=0}=0$ for all $v\in \mathbb{C}^n$ if and only if $\frac{d}{dt}F(e^{Mt}v)|_{t=a}=0$ for all $a\in \mathbb{R}$ and all $v\in \mathbb{C}^n$. This is equivalent to the condition $F(e^{Mt}v)=F(v)$ for all $t\in \mathbb{R}$ and all $v\in \mathbb{C}^n$.
Now consider the Lie algebra $\mathrm{Lie}(G)$. Since $G$ is connected every element $g\in G$ can be written as $g=e^{M_1}e^{M_2}\dots e^{M_l}$ for $M_k\in \mathrm{Lie}(G)$. Therefore the condition $\frac{d}{dt}F(e^{M_kt}v)|_{t=0}=M_kv\cdot\nabla F(v)=0$ for all $M_k\in \mathrm{Lie}(G)$ is equivalent to $F$ being invariant under $G$.
Next consider the Lie algebra $\mathrm{Lie}(K)$. Since $K$ is connected every element $g\in K$ can be written as $g=e^{M_1}e^{M_2}\dots e^{M_l}$ for $M_k\in \mathrm{Lie}(K)$. Therefore the condition $\frac{d}{dt}F(e^{M_kt}v)|_{t=0}=M_kv\cdot\nabla F(v)=0$ for all $M_k\in \mathrm{Lie}(K)$ is equivalent to $F$ being invariant under $K$.
If $\frac{d}{dt}F(e^{M_kt}v)|_{t=0}=M_kv\cdot\nabla F(v)=0$ for some $M_k\in \mathrm{Lie}(K)$ we see that $\frac{d}{dt}F(e^{iM_kt}v)|_{t=0}=iM_kv\cdot\nabla F(v)=0$.
Since $\mathrm{Lie}(G)=\mathrm{Lie}(K)+i\mathrm{Lie}(K)$ it follows that if $F$ is invariant under $K$ it is also invariant under $G$.
Moreover, since $K$ is a subgroup of $G$ we have that if $F$ is invariant under $G$ it is invariant under $K$.
Thus $\mathcal{O}(\mathbb{C}^n)^G=\mathcal{O}(\mathbb{C}^n)^K$.
See also e.g. Ref. \cite{dolgachev} for the special case of $\mathrm{SL}(2, \mathbb{C})$ and $\mathrm{SU}(2)$.
\end{proof}
Another property of a connected complex reductive Lie group is that the algebra of polynomial invariants of the group is finitely generated.
This is the content of the so called Hilbert's Finiteness Theorem \cite{hilbert,weeyl,nagata}
\begin{theorem}\label{fin}
Let $G$ be a connected complex reductive Lie subgroup of $\mathrm{GL}(n,\mathbb{C})$. Let $\mathcal{O}(\mathbb{C}^n)$ be the algebra over $\mathbb{C}$ of polynomials on $\mathbb{C}^n$ and let $\mathcal{O}(\mathbb{C}^n)^G\subset \mathcal{O}(\mathbb{C}^n)$ be the subalgebra of polynomials that are invariant under $G$, i.e., $\mathcal{O}(\mathbb{C}^n)^G\equiv\{f\in\mathcal{O}(\mathbb{C}^n)|f(gx)=f(x) \phantom{t}\textrm{for all}\phantom{t} g\in G, x\in \mathbb{C}^n\}$. Then $\mathcal{O}(\mathbb{C}^n)^G$ is finitely generated over $\mathbb{C}$.
\end{theorem}
\begin{proof}
See e.g. Ref. \cite{nolan} Ch. 3.1.4, Ref. \cite{wall} Ch. 5.1.1 or Ref. \cite{nagata}.
\end{proof}
This theorem implies that for any number $n$ of spacelike separated Dirac spinors and any group $G_1{\,\otimes\,}imes G_2{\,\otimes\,}imes\dots{\,\otimes\,}imes G_n$ where each $G_k$ is one out of $\mathrm{S}G^C$, $\mathrm{S}G^{C\gamma^5}$, $\mathrm{S}G^{C\gamma^5}\cap \mathrm{S}G^C$ and $\mathrm{SL}(4,\mathbb{C})$, a finite number of polynomials invariant under this group generate the algebra of all such invariants. Moreover, by Theorem \ref{uii} the same finitely generated algebra is the invariants of the maximal compact subgroup of $G_1{\,\otimes\,}imes G_2{\,\otimes\,}imes\dots{\,\otimes\,}imes G_n$. Note, however that this does not imply that such a finite number of polynomials can necessarily be constructed as tensor sandwich contractions.
If two orbits of a connected complex reductive Lie group are such that their closures do not overlap there exists a polynomial invariant of the group that distinguishes between the orbit closures.
\begin{theorem}
Let $G$ be a connected complex reductive Lie subgroup of $\mathrm{GL}(n,\mathbb{C})$. Let $\mathcal{O}(\mathbb{C}^n)$ be the algebra over $\mathbb{C}$ of polynomials on $\mathbb{C}^n$ and let $\mathcal{O}(\mathbb{C}^n)^G\subset \mathcal{O}(\mathbb{C}^n)$ be the subalgebra of polynomials that are invariant under $G$, i.e., $\mathcal{O}(\mathbb{C}^n)^G\equiv\{f\in\mathcal{O}(\mathbb{C}^n)|f(gx)=f(x) \phantom{t}\textrm{for all}\phantom{t} g\in G, x\in \mathbb{C}^n\}$. Let $Gx$ and $Gy$ be $G$-orbits for $x,y\in \mathbb{C}^n$ such that their closures $\overline{Gx}$ and $\overline{Gy}$ satisfy $\overline{Gx}\cap \overline{Gy}=\emptyset$. Then there exist $f\in\mathcal{O}(\mathbb{C}^n)^G$ such that $f_{|\overline{Gx}}=1$ and $f_{|\overline{Gy}}=0$.
\end{theorem}
\begin{proof}
See e.g. Ref. \cite{nolan} Ch. 3.1.4.
\end{proof}
Consider a set $\Sigma$ of states on which all polynomial invariants take constant values and that is not a subset of any other set on which all polynomial invariants take constant values.
A property of the polynomial invariants of a connected complex reductive Lie group is that
any such subset $\Sigma$ of states contains a unique closed orbit.
In particular this implies that every open orbit has one and only one closed orbit in its closure.
\begin{theorem}\label{tsw}
Let $G$ be a connected complex reductive Lie subgroup of $\mathrm{GL}(n,\mathbb{C})$. Let $\mathcal{O}(\mathbb{C}^n)$ be the algebra over $\mathbb{C}$ of polynomials on $\mathbb{C}^n$ and let $\mathcal{O}(\mathbb{C}^n)^G\subset \mathcal{O}(\mathbb{C}^n)$ be the subalgebra of polynomials that are invariant under $G$, i.e., $\mathcal{O}(\mathbb{C}^n)^G\equiv\{f\in\mathcal{O}(\mathbb{C}^n)|f(gx)=f(x) \phantom{t}\textrm{for all}\phantom{t} g\in G, x\in \mathbb{C}^n\}$.
Let $S_x=\{y\in \mathbb{C}^n| f(y)=f(x)\phantom{t}\textrm{for all}\phantom{t} f\in \mathcal{O}(\mathbb{C}^n)^G\}$.
Then each set $S_x$ contains a unique closed orbit of $G$. If $Gw$ is the unique closed orbit in $S_x$ and $v\in S_x$ then $Gw\subset \overline {Gv}$.
\end{theorem}
\begin{proof}
See e.g. Ref. \cite{nolan} Ch. 3.4.1.
\end{proof}
Theorem \ref{tsw} together with Theorem \ref{fin} implies that there exists a finite number $m$ of polynomials such that the set of $m$-tuples of values of these polynomials is in one-to-one correspondence with the set of closed orbits. Thus the polynomials provide coordinates for the set of closed orbits. In particular, for any group $G_1{\,\otimes\,}imes G_2{\,\otimes\,}imes\dots{\,\otimes\,}imes G_n$ where each $G_k$ is one out of $\mathrm{S}G^C$, $\mathrm{S}G^{C\gamma^5}$, $\mathrm{S}G^{C\gamma^5}\cap \mathrm{S}G^C$ and $\mathrm{SL}(4,\mathbb{C})$ a finite number of polynomial invariants distinguishes all closed orbits of $G_1{\,\otimes\,}imes G_2{\,\otimes\,}imes\dots{\,\otimes\,}imes G_n$. An open orbit on the other hand cannot be distinguished by such polynomials from the closed orbit in its closure. However it is an open question if a set of polynomials with these properties can be constructed as tensor sandwich contractions.
\section{Discussion and Conclusions}\label{diss}
In this work we have considered the problem of describing the spinor entanglement of a system of multiple Dirac particles with definite momenta held by spacelike separated observers. The general approach followed is the same as was considered in Ref. \cite{spinorent} for the case of two Dirac particles. We reviewed some properties of the Dirac equation, the spinor representation of the Lorentz group and the charge conjugation transformation, as well as some properties of Lorentz invariant skew-symmetric bilinear forms.
The assumption was made that the local curvature of spacetime can be neglected and each particle described as belonging to a Minkowski space. Beyond this, we assumed that the physical scenario is such that it is warranted to treat particle momentum eigenmodes as having a finite spatial extent.
Moreover, we assumed that the tensor products of the individual particle momentum eigenmodes can be used as a basis for the multi-particle states.
Given these assumptions and utilizing the properties of the skew-symmetric bilinear forms we described a method to construct polynomials in the state coefficients of the system of spacelike separated Dirac particles. This method is a generalization of the method used in Ref. \cite{spinorent} for the case of two Dirac particles.
The polynomials constructed by this method are invariant under the spinor representations of the local proper orthochronous Lorentz groups. Moreover, each such Lorentz invariant polynomial is identically zero for states where any one of the spinors is in a product state with the other spinors.
For the case of three and four Dirac spinors polynomials of degree 2 and 4 were constructed and their linear independence tested. For three spinors no non-zero degree 2 polynomials can be constructed, but a set of 67 linearly independent degree 4 polynomials were given. For four spinors 16 linearly independent degree 2 polynomials were derived and a further 26 polynomials of degree 4 were given. A larger number of degree 4 polynomials exist but was not derived due to the complexity of constructing a large number of such polynomials and testing their linear independence. For five spinors no nonzero degree 2 polynomial can be constructed. The degree 4 polynomials that can be constructed for five spinors using the method in this work were described but not computed. For any even number $N$ of Dirac spinors $2^N$ linearly independent polynomials of degree 2 can be constructed and these polynomials were described.
For the case of Weyl particles, i.e., particles with definite chirality, the Lorentz invariant polynomials for three Dirac spinors reduce to either a multiple of the Coffman-Kundu-Wootters 3-tangle \cite{coffman} or alternatively are identically zero. If only two of the particles are Weyl particles some of the polynomials reduce to the $2\times 2\times 4$ tangle described in Refs. \cite{verstraete,moor}. For the case of four Weyl particles the Lorentz invariant polynomials derived here reduce to linear combinations of the polynomials found by Luque and Thibon \cite{luque}. For the case of an even number $N$ of Weyl particles the degree 2 polynomials described here reduce to multiples of the $N$-tangle introduced by Wong and Christensen \cite{wong}.
In the case of zero particle momenta and zero four-potentials we considered the eigenspaces of the local Dirac Hamiltonians. These eigenspaces are often identified with non-relativistic free spin-$\frac{1}{2}$ particles and antiparticles. On these eigenspaces the polynomials constructed in this work all reduce to linear combinations of the previously known polynomials constructed for non-relativistic spin-$\frac{1}{2}$ particles or are identically zero. For three particles the polynomials either reduce to a multiple of the Coffman-Kundu-Wootters 3-tangle \cite{coffman} or are zero. For four particles the polynomials reduce to linear combinations of the polynomials found by Luque and Thibon \cite{luque} or are zero. For the case of an even number $N$ of particles the degree 2 polynomials constructed here reduce to the $N$-tangle of Wong and Christensen \cite{wong} or are zero. Since a system of Dirac particles can always be described in their respective rest frames, the previously known polynomials constructed for non-relativistic spin-$\frac{1}{2}$ particles can always be used for the case of free particles in eigenstates of the local Dirac Hamiltonians.
We considered evolutions generated by local Hamiltonians that act unitarily on the subspaces defined by fixed momenta, i.e., the subspaces spanned by the spinorial degrees of freedom. All polynomials derived using the method given here are invariant, up to a U(1) phase, under such local unitary evolution generated by zero-mass Dirac Hamiltonians and zero-mass Dirac Hamiltonians with additional terms that are second degree in Dirac gamma matrices such as a Semenoff mass term \cite{semenoff} or a Haldane mass term \cite{haldane}.
Some polynomials are by construction invariant, up to a U(1) phase, also under local unitary evolution generated by arbitrary-mass Dirac Hamiltonians and some are invariant, up to a U(1) phase, under local unitary evolution generated by zero-mass Dirac Hamiltonians with additional terms that are third degree in Dirac gamma matrices such as a Pauli coupling or a Yukawa pseudo-scalar coupling.
Because of these properties the polynomials constructed by the method in this work are considered potential candidates for describing the entanglement of the spinor degrees of freedom in a system of multiple Dirac particles with either zero or arbitrary mass, or zero-mass with an additional coupling such as a Yukawa pseudo-scalar. Polynomials of this kind can be used to partially characterize qualitatively inequivalent forms of spinor entanglement as described in Ref. \cite{spinorent}. Such a characterization is not complete since only the disjoint orbit-closures of the invariance groups of the polynomials can be distinguished by the polynomials. Therefore there exist inequivalently spinor entangled states that cannot be distinguished by any set of polynomials of this kind. Moreover, as described in Ref. \cite{spinorent}
for any set of polynomials that are invariant under the spinor representation of the proper orthochronous Lorentz group there exist spinor entangled states for which all the polynomials are zero.
The general properties of the algebras of Lorentz invariant polynomials that are invariant, up to a U(1) phase, under evolution generated locally by arbitrary-mass Dirac Hamiltonians, by zero-mass Dirac Hamiltonians, or by zero-mass Dirac Hamiltonians with a Yukawa pseudo-scalar coupling were discussed. Any such algebra is generated by a finite number of polynomials but if such a set of generators can be constructed by the method described in this work is not clear.
In Ref. \cite{spinorent} it was descried how the absolute values of Lorentz invariant polynomials for two Dirac spinors can be extended to functions on the set of states that are incoherent mixtures, i.e., mixed states, through convex roof extensions \cite{lima,wakker,uhlmannn}. Convex roof extensions can be made also for the case of the polynomials constructed for multiple Dirac spinors in this work. Therefore such convex roof extensions can provide a partial characterization of the different types of multi-spinor entanglement of incoherent mixtures. These convex roof extensions are by definition identically zero for all incoherent mixtures of product states, i.e., for all separable states.
The polynomial Lorentz invariants in this work were constructed to describe multi-spinor entanglement for the case of Dirac particles with definite momenta but it is an open question whether similar constructions can be made for the case without definite momenta.
Another open question is the description of entanglement in a scenario that allows for communication between the labs holding the particles.
In such a scenario one can consider multi-spinor entanglement properties that can be quantified by entanglement measures \cite{entmes}, i.e., multi-spinor entanglement properties that satisfy a condition of non-increase on average under any local operations assisted by classical communication \cite{vidal}.
More broadly one may consider if there are other conceptualizations of spinor entanglement properties that give a description that is complementary to the one given here.
\section*{Declaration of interests}
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
\section*{Acknowledgment}
The author thanks the anonymous referees for constructive comments that improved the work.
The author also thanks the anonymous referees for constructive comments on an earlier version of this work that motivated several major improvements.
Support
from the European Research Council Consolidator Grant QITBOX (Grant Agreement No. 617337), the Spanish MINECO
(Project FOQUS FIS2013-46768-P, Severo Ochoa grant SEV-
2015-0522), Fundaci\'o Privada Cellex, the Generalitat de Catalunya
(SGR 875) and the John Templeton Foundation is acknowledged.
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\appendix
\section{The use of real and rational numbers to quantify properties in experiments}\label{opp}
Any measurement that quantifies a property is an operational procedure that eventually concludes and results in an output number that is registered by an experimenter. In any experiment only finitely many such procedures can be performed and any memory storing the output numbers has finite capacity.
Any real number is by definition the limit of a Cauchy sequence of rational numbers \cite{cantor}. Such a sequence terminates only if the limit is a rational number itself. Thus if the limit is an irrational number the sequence does not terminate.
Consequently, the representation of an irrational number in a base-$n$ positional numeral system is a non-terminating and non-repeating sequence of digits for any $n$.
It follows that no memory with finite capacity can store an irrational number, and thus the output numbers from any given experiment is by necessity a finite set of rational numbers. In any finite set of rational numbers the elements are multiples of their greatest common divisor $q$.
Therefore, in any given experiment one cannot distinguish between the continuous spectrum of real numbers $\mathbb{R}$ and a discrete spectrum $nq$, where $n\in \mathbb{Z}$. Moreover, the experiment cannot distinguish a set $S\subset\mathbb{R}$ containing $mq$ for some $m\in \mathbb{Z}$ from $mq$ itself if $S$ does not contain any other multiple of $q$.
In particular, for any set of measured momentum vectors on the form $\bold{k}=(k_1,k_2,k_3)$ a rectangular box with periodic boundary conditions, and whose sides align with the directions corresponding to the components of the $\bold{k}$, can be found such that all the measured momenta are simultaneously allowed by the box dimensions. Alternatively, for any set of measured momenta each of the measured $\bold{k}$ can be matched to a Schwartz function with compact support in momentum space that contains $\bold{k}$ in such a way that no two of the Schwartz functions have overlapping support and no two of the measured momenta are in the support of the same Schwartz function.
\section{The bilinear forms ${\psi^T}C{\varphi}$ and ${\psi^T}C\gamma^5{\varphi}$ undergoing unitary spinor evolution generated by Dirac-like Hamiltonians}\label{hamm}
Here we show how to derive the properties of the bilinear forms ${\psi^T}C{\varphi}$ and ${\psi^T}C\gamma^5{\varphi}$ described in Section \ref{ham}.
We consider a subspace defined by a fixed particle momentum $\bold{k}$, i.e., a subspace spanned by the four basis elements $\phi_je^{i\bold{k}\cdot\bold{x}}$ with the same $\bold{k}$. Furthermore we assume that the evolution acts unitarily on such a subspace and is generated by a Hamiltonian operator $H$. For the subspace to be invariant under the evolution it is required that $(\phi_je^{i\bold{k}\cdot\bold{x}},H\phi_le^{i\bold{k'}\cdot\bold{x}})\propto\delta_{\bold{k},\bold{k'}}$. Therefore, to have such unitary action on the subspace we consider evolution generated by Hamiltonians that do not depend on the spatial coordinate $\bold{x}$.
We again consider the inner product on a subspace of this kind
\begin{eqnarray}
({\psi(t)},{\varphi(t)})_{\bold{k}}=\psi^{\dagger}(t)\varphi(t),
\end{eqnarray}
and assume that the time dependent Hamiltonian $H(s)$ is bounded and strongly continuous, i.e., for all ${\psi}$ and $s$ it holds that $\lim_{t\to s}||H(t){\psi}-H(s){\psi}||=0$ where $||\cdot||$ is the norm induced by the inner product. Then the evolution operator can be expressed as an ordered exponential as described in the following theorem.
\begin{theorem}
Let $t\in\mathbb{R}\to H(t)$ be a strongly continuous map into the bounded Hermitian operators on a Hilbert space $\mathcal{H}$. Then there exists an evolution operator $U(t,s)$ such that for all ${\psi}\in\mathcal{H}$ we have that
${\psi(t)}=U(t,s){\psi(s)}$ and $\partial_{t}U(t,s)=-iH(t)U(t,s)$. This evolution operator can be expressed as an ordered exponential
\begin{eqnarray*}
U(t,r)=&&\mathcal{T}_{\leftarrow}\{e^{-i\int_{r}^tH(s)ds}\}\nonumber\\
\equiv&&\sum_{n=0}^\infty(-i)^n\int_{r}^t\int_{r}^{s_n}\int_{r}^{s_{n-1}}\int_{r}^{s_{2}}H(s_n)\dots H(s_{1})ds_1\dots ds_{n-2} ds_{n-1} ds_{n},\nonumber\\
\end{eqnarray*}
and satisfies $U(t,t)=I$ and $U(r,s)U(s,t)=U(r,t)$.
\end{theorem}
\begin{proof}
See e.g. Ref. \cite{reed}.
\end{proof}
For an evolution operator on an ordered exponential form we can consider its conjugate transpose, complex conjugate, and transpose in the given basis,
\begin{eqnarray}
U(t,r)&&=\mathcal{T}_{\leftarrow}\{e^{-i\int_{r}^tH(s)ds}\},\nonumber\\
U(t,r)^\dagger &&=\mathcal{T}_{\rightarrow}\{e^{i\int_{r}^tH(s)ds}\},\nonumber\\
U(t,r)^*&&=\mathcal{T}_{\leftarrow}\{e^{i\int_{r}^t H^T(s)ds}\},\nonumber\\
U(t,r)^T&&=\mathcal{T}_{\rightarrow}\{e^{-i\int_{r}^t H^T(s)ds}\}.
\end{eqnarray}
Now assume that $X$ is a time independent matrix such that for all $s$ it holds that $XH(s)=-H(s)^TX$. Then it follows that $X\mathcal{T}_{\leftarrow}\{e^{-i\int_{0}^tH(s)ds}\}=\mathcal{T}_{\leftarrow}\{e^{i\int_{0}^tH(s)^Tds}\}X$, and the bilinear form $\psi^{T}X\varphi$ is invariant under the evolution generated by $H(s)$
\begin{eqnarray}
{\psi^T} U(t,0)^T XU(t,0){\varphi}={\psi^T}\mathcal{T}_{\rightarrow}\{e^{-i\int_{0}^t H^T(s)ds}\}
\mathcal{T}_{\leftarrow}\{e^{i\int_{0}^t H^T(s)ds}\}X{\varphi}={\psi^T}X{\varphi}.
\end{eqnarray}
Next we consider the possibilities $X=C$ and $X=C\gamma^5$.
For products of different numbers of distinct gamma matrices we have for $C$ that
\begin{eqnarray}\label{c}
(\gamma^\mu)^TC&=&C\gamma^\mu\nonumber\\
(\gamma^\mu\gamma^\nu)^TC&=&-C\gamma^\mu\gamma^\nu\nonumber\\
(\gamma^\mu\gamma^\nu\gamma^\rho)^TC&=&-C\gamma^\mu\gamma^\nu\gamma^\rho\nonumber\\
(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma)^TC&=&C\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma,
\end{eqnarray}
and for $C\gamma^5$ that
\begin{eqnarray}\label{c5}
(\gamma^\mu)^TC\gamma^5&=&-C\gamma^5\gamma^\mu\nonumber\\
(\gamma^\mu\gamma^\nu)^TC\gamma^5&=&-C\gamma^5\gamma^\mu\gamma^\nu\nonumber\\
(\gamma^\mu\gamma^\nu\gamma^\rho)^TC\gamma^5&=&C\gamma^5\gamma^\mu\gamma^\nu\gamma^\rho\nonumber\\
(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma)^TC\gamma^5&=&C\gamma^5\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma.
\end{eqnarray}
From these equations we can see that a Hamiltonian term that is second or third degree in the gamma matrices, i.e., a Hamiltonian term of the form
\begin{eqnarray}
H^{2,3}(t)=\gamma^\mu\gamma^\nu\phi_{\mu\nu}(t)+\gamma^\mu\gamma^\nu\gamma^\rho\kappa_{\mu\nu\rho}(t),
\end{eqnarray}
satisfies $CH^{2,3}(t)=-(H^{2,3}(t))^TC$. Similarly, a Hamiltonian term that is first or second degree in the gamma matrices, i.e., a Hamiltonian term of the form
\begin{eqnarray}
H^{1,2}(t)=\gamma^\mu\eta_{\mu}(t)+\gamma^\mu\gamma^\nu\lambda_{\mu\nu}(t),
\end{eqnarray}
satisfies $C\gamma^5H^{1,2}(t)=-(H^{1,2}(t))^TC\gamma^5$.
Any Hamiltonian term $H^0(t)=f(t)I$ that is zeroth degree in the gamma matrices, i.e., proportional to the identity matrix, is clearly its own transpose and commutes with both $C\gamma^5$ and $C$.
The Dirac Hamiltonian contains a first degree term in the gamma matrices, the mass term $m\gamma^0$, and a second degree term, the generalized canonical momentum term $\sum_{\mu=1,2,3}\gamma^{0}\gamma^{\mu}(i\partial_{\mu}-qA_{\mu}(t))$.
Beyond this, it has a zeroth degree term, the coupling to the scalar potential $qA_0(t)I$. However, any zeroth degree term $f(t)I$ can be removed from the Hamiltonian by a change of variables. If we define a new spinor as
$\psi'= e^{-i\theta(t)}\psi$ the new Hamiltonian $H'$ defined by $i\partial_t\psi'(t)=H'\psi'(t)$ is
$H'=H+\gamma^0\sum_\mu\gamma^\mu\partial_\mu \theta(t)$. This change of variables amounts to a change of local U(1) gauge (See e.g. Ref. \cite{griffiths}). For the choice $\theta(t)=-\int_{t_0}^tf(s)ds$ we see that since $f(t)-\partial_t \int_{t_0}^tf(s)ds=0$ the term proportional to the identity in $H'$ is identically zero. Except for the zeroth degree term the new Hamiltonian $H'$ in general contains terms with the same degrees in gamma matrices as $H$. It cannot acquire terms with degrees different from those of the terms in $H$. We can thus remove the zeroth degree term $qA_0(t)I$ from the Dirac Hamiltonian by choosing $\theta(t)=-\int_{t_0}^tqA_0(s)ds$. For this choice let $U'(t,t_0)$ be the evolution generated by $H'$. Then we can see that $\psi(t)=e^{-iq\int_{t_0}^tA_0(s)ds}\psi'(t)=e^{-iq\int_{t_0}^tA_0(s)ds}U'(t,t_0)\psi'(t_0)=e^{-iq\int_{t_0}^tA_0(s)ds}U'(t,t_0)\psi(t_0)$.
Therefore, as described in Ref. \cite{spinorent}, for an evolution $U_D(t,0)$ generated by Dirac Hamiltonians it holds for the bilinear form ${\psi^T}C\gamma^5{\varphi}$ that
\begin{eqnarray}\label{ghhh}
&&\psi^{T}U_D(t,0)^TC\gamma^5U_D(t,0)\varphi\nonumber\\
&&=e^{-2iq\int_{0}^tA_0(s)ds}{\psi^T}{U'}_D(t,0)^T C\gamma^5 U'_D(t,0){\varphi}\nonumber\\
&&=e^{-2iq\int_{0}^tA_0(s)ds}{\psi^T}C\gamma^5{\varphi}.
\end{eqnarray}
Similarly, for an evolution $U_W(t,0)$ generated by zero-mass Dirac Hamiltonians it holds for the bilinear form ${\psi^T}C{\varphi}$ that
\begin{eqnarray}\label{ujjjk}
&&\psi^{T}U_W(t,0)^TU_W(t,0)\varphi \nonumber\\
&&=e^{-2iq\int_{0}^tA_0(s)ds}{\psi^T}{U'}_W(t,0)^TCU'_W(t,0){\varphi}\nonumber\\
&&=e^{-2iq\int_{0}^tA_0(s)ds}{\psi^T}C{\varphi}.
\end{eqnarray}
Thus the bilinear form $\psi^TC\varphi$ is invariant, up to a U(1) phase, under evolutions generated by any Hamiltonians on the form $H^{2,3}(t)+H^{0}(t)$ and the bilinear form $\psi^TC\gamma^5\varphi$ is invariant, up to a U(1) phase, under evolutions generated by any Hamiltonians on the form $H^{1,2}(t)+H^{0}(t)$.
\section{Graph representations of the tensor sandwich contractions that yield degree 4 polynomials for four Dirac spinors}\label{graphs}
Here we give the graph representations of the tensor sandwich contractions in Eq. (\ref{cvn}). The contractions $W_a$, $W_c$, and $W_f$ that are each invariant with respect to permutations of two disjoint pairs of laboratories are given in Fig. \ref{ris3}. The contractions $W_b$, $W_d$, $W_e$, $W_g$, $W_h$, and $W_i$ that are each invariant with respect to permutations of a single pair of laboratories are given in Fig. \ref{ris2}. Finally, the contractions $W_j$, $W_k$, $W_l$, and $W_m$ that are each invariant with respect to permutations of a triple of laboratories are given in Fig. \ref{ris4}.
\begin{figure}
\caption{$W_a$}
\caption{$W_c$}
\caption{$W_f$}
\caption{Graph representations of the tensor sandwich contractions $W_a$, $W_c$, and $W_f$. Each of the four copies of $\Psi^{ABCD}
\label{ris3}
\end{figure}
\begin{figure}
\caption{$W_b$}
\caption{$W_d$}
\caption{$W_e$}
\caption{$W_g$}
\caption{$W_h$}
\caption{$W_i$}
\caption{Graph representations of the tensor sandwich contractions $W_b$, $W_d$, $W_e$, $W_g$, $W_h$, and $W_i$. Each of the four copies of $\Psi^{ABCD}
\label{ris2}
\end{figure}
\begin{figure}
\caption{$W_j$}
\caption{$W_k$}
\caption{$W_l$}
\caption{$W_m$}
\caption{Graph representations of the tensor sandwich contractions $W_j$, $W_k$, $W_l$, and $W_m$.
Each of the four copies of $\Psi^{ABCD}
\label{ris4}
\end{figure}
\section{Degree 4 polynomials for five spinors}\label{wides}
Here we give the 40 inequivalent ways to pair up the tensor indices of four copies of the $4\times 4\times 4\times 4\times 4$ tensor $\Psi^{ABCDE}$ that do not factorize into two degree 2 polynomials. In writing these sandwich contractions we leave out the summation sign with the understanding that repeated indices are summed over. We also suppress the superscript $ABCDE$ of $\Psi$. The 40 inequivalent ways to contract the indices are
\begin{eqnarray}\label{cvn2}
F_1=&&X_{gl} X_{hm}X_{in}X_{jo}X_{ku}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{sx}X_{ty}X_{pz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_2=&&X_{gl} X_{hm}X_{in}X_{jt}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{sx}X_{oy}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_3=&&X_{gl} X_{hm}X_{is}X_{jo}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{nx}X_{ty}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_4=&&X_{gl} X_{hr}X_{in}X_{jo}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{sx}X_{ty}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_5=&&X_{gq} X_{hm}X_{in}X_{jo}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rw}X_{sx}X_{ty}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_6=&&X_{gl} X_{hm}X_{in}X_{jt}X_{ku}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{sx}X_{oy}X_{pz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_7=&&X_{gl} X_{hm}X_{in}X_{jt}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{sx}X_{oy}X_{pu}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_8=&&X_{gl} X_{hm}X_{is}X_{jt}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{nx}X_{oy}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_9=&&X_{gl} X_{hm}X_{is}X_{jy}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{nx}X_{ot}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{10}=&&X_{gl} X_{hr}X_{is}X_{jo}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{nx}X_{ty}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{11}=&&X_{gl} X_{hr}X_{ix}X_{jo}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{ns}X_{ty}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{12}=&&X_{gq} X_{hr}X_{in}X_{jo}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{mw}X_{sx}X_{ty}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{13}=&&X_{gq} X_{hw}X_{in}X_{jo}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{mr}X_{sx}X_{ty}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{14}=&&X_{gq} X_{hm}X_{in}X_{jo}X_{ku}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rw}X_{sx}X_{ty}X_{pz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{15}=&&X_{gq} X_{hm}X_{in}X_{jo}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rw}X_{sx}X_{ty}X_{pu}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{16}=&&X_{gq} X_{hm}X_{in}X_{jt}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rw}X_{sx}X_{oy}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{17}=&&X_{gq} X_{hm}X_{in}X_{jy}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rw}X_{sx}X_{ot}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{18}=&&X_{gl} X_{hm}X_{is}X_{jo}X_{ku}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{nx}X_{ty}X_{pz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{19}=&&X_{gl} X_{hm}X_{is}X_{jo}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{nx}X_{ty}X_{pu}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{20}=&&X_{gl} X_{hr}X_{in}X_{jt}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{sx}X_{oy}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{21}=&&X_{gl} X_{hr}X_{in}X_{jy}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{sx}X_{ot}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{22}=&&X_{gq} X_{hm}X_{is}X_{jo}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rw}X_{ix}X_{ty}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{23}=&&X_{gq} X_{hm}X_{ix}X_{jo}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rw}X_{is}X_{ty}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{24}=&&X_{gl} X_{hr}X_{in}X_{jo}X_{ku}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{sx}X_{ty}X_{pz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{25}=&&X_{gl} X_{hr}X_{in}X_{jo}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{sx}X_{ty}X_{pu}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{26}=&&X_{gq} X_{hm}X_{in}X_{jy}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rw}X_{sx}X_{to}X_{up}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{27}=&&X_{gq} X_{hm}X_{ix}X_{jo}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rw}X_{sn}X_{ty}X_{up}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{28}=&&X_{gq} X_{hw}X_{in}X_{jo}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{lv} X_{rm}X_{sx}X_{ty}X_{up}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{29}=&&X_{gl} X_{hr}X_{in}X_{jy}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{sx}X_{to}X_{up}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{30}=&&X_{gl} X_{hr}X_{ix}X_{jo}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{sn}X_{ty}X_{up}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{31}=&&X_{gl} X_{hr}X_{ix}X_{jy}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{mw}X_{sn}X_{to}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{32}=&&X_{gl} X_{hm}X_{is}X_{jy}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{nx}X_{ty}X_{up}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{33}=&&X_{gl} X_{hw}X_{is}X_{jo}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rm}X_{nx}X_{ty}X_{up}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{34}=&&X_{gl} X_{hw}X_{is}X_{jy}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rm}X_{nx}X_{to}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{35}=&&X_{gl} X_{hm}X_{ix}X_{jt}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{sn}X_{oy}X_{up}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{36}=&&X_{gl} X_{hw}X_{in}X_{jt}X_{kz}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rm}X_{sx}X_{oy}X_{up}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{37}=&&X_{gl} X_{hw}X_{ix}X_{jt}X_{kp}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rm}X_{sn}X_{oy}X_{uz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{38}=&&X_{gl} X_{hm}X_{ix}X_{jy}X_{ku}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rw}X_{sn}X_{to}X_{pz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{39}=&&X_{gl} X_{hw}X_{in}X_{jy}X_{ku}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rm}X_{sx}X_{to}X_{pz}\Psi_{qrstu}\Psi_{vwxyz},\nonumber\\
F_{40}=&&X_{gl} X_{hw}X_{ix}X_{jo}X_{ku}\Psi_{ghijk}\Psi_{lmnop} X_{qv} X_{rm}X_{sn}X_{ty}X_{pz}\Psi_{qrstu}\Psi_{vwxyz}.\nonumber\\
\end{eqnarray}
For each pairing of tensor indices we can consider $(2^{10}-2^5)/2+2^5=528$ ways to choose the $X$s as either $C$ or $C\gamma^5$. This gives a total of 21120 tensor sandwich contractions that yield degree 4 polynomials for five spinors.
\section{Selected explicit polynomials}\label{polllu}
Here we give a few examples of explicitly written out polynomials. For the case of three Dirac spinors the polynomials $I_{2a},I_{2b},I_{2c}$ are given as examples of polynomials of the types $I_a,I_b$ and $I_c$ that are invariant under $G^{C\gamma^5}$, up to a U(1) phase, in all labs and invariant under P in all labs. The polynomials $I_{3a},I_{3b},I_{3c}$ are given as examples of polynomials of the types $I_a,I_b$ and $I_c$ that are invariant under $G^{C}$, up to a U(1) phase, in all labs and invariant under P in all labs. The polynomial $I_{3d}$ is given as an example of the polynomials of type $I_d$. The polynomial $I_{23a}$ is given as an example of a polynomial that is P invariant only in two of the labs, Bob's and Charlie's. The polynomial $I_{35a}$ is given as an example of a polynomial that is P invariant only in one lab, Alice's. Finally the polynomial $I_{11a}$ is given as an example of a polynomial that is not P invariant in any lab.
For the case of four Dirac spinors the polynomials $H_a,H_b,H_c$, and $H_d$ are given as examples of the degree 2 polynomials, and the polynomials $T_l$ and $Y_l$ are given as examples of the degree 4 polynomials.
We first give the polynomials $I_{2a},I_{2b},I_{2c}$ that are invariant under $G^{C\gamma^5}$, up to a U(1) phase, in all labs and invariant under P in all labs. Explicitly written out they are
\small
\begin{align}
I_{2a} = -2 (&\psi_{133}\psi_{200} - \psi_{132}\psi_{201} + \psi_{131}\psi_{202} - \psi_{130}\psi_{203} - \psi_{123}\psi_{210} +
\psi_{122}\psi_{211} - \psi_{121}\psi_{212} + \psi_{120}\psi_{213}\nonumber\\ &+ \psi_{113}\psi_{220} - \psi_{112}\psi_{221} +
\psi_{111}\psi_{222} - \psi_{110}\psi_{223} - \psi_{103}\psi_{230} + \psi_{102}\psi_{231} - \psi_{101}\psi_{232} +
\psi_{100}\psi_{233})^2\nonumber\\-
2 (&\psi_{033}\psi_{300} - \psi_{032}\psi_{301} + \psi_{031}\psi_{302} - \psi_{030}\psi_{303} - \psi_{023}\psi_{310} +
\psi_{022}\psi_{311} - \psi_{021}\psi_{312} + \psi_{020}\psi_{313}\nonumber\\ &+ \psi_{013}\psi_{320} - \psi_{012}\psi_{321} +
\psi_{011}\psi_{322} - \psi_{010}\psi_{323} - \psi_{003}\psi_{330} + \psi_{002}\psi_{331} - \psi_{001}\psi_{332} +
\psi_{000}\psi_{333})^2\nonumber\\ +
8 (&\psi_{113}\psi_{120} - \psi_{112}\psi_{121} + \psi_{111}\psi_{122} - \psi_{110}\psi_{123} - \psi_{103}\psi_{130} +
\psi_{102}\psi_{131} - \psi_{101}\psi_{132} + \psi_{100}\psi_{133})\nonumber\\\times (&\psi_{213}\psi_{220} - \psi_{212}\psi_{221} +
\psi_{211}\psi_{222} - \psi_{210}\psi_{223} - \psi_{203}\psi_{230} + \psi_{202}\psi_{231} - \psi_{201}\psi_{232} +
\psi_{200}\psi_{233})\nonumber\\ +
8 (&\psi_{013}\psi_{020} - \psi_{012}\psi_{021} + \psi_{011}\psi_{022} - \psi_{010}\psi_{023} - \psi_{003}\psi_{030} +
\psi_{002}\psi_{031} - \psi_{001}\psi_{032} + \psi_{000}\psi_{033}) \nonumber\\\times (&\psi_{313}\psi_{320} - \psi_{312}\psi_{321} +
\psi_{311}\psi_{322} - \psi_{310}\psi_{323} - \psi_{303}\psi_{330} + \psi_{302}\psi_{331} - \psi_{301}\psi_{332} +
\psi_{300}\psi_{333})\nonumber\\ +
4 (&\psi_{033}\psi_{200} - \psi_{032}\psi_{201} + \psi_{031}\psi_{202} - \psi_{030}\psi_{203} - \psi_{023}\psi_{210} +
\psi_{022}\psi_{211} - \psi_{021}\psi_{212} + \psi_{020}\psi_{213}\nonumber\\ &+ \psi_{013}\psi_{220} - \psi_{012}\psi_{221} +
\psi_{011}\psi_{222} - \psi_{010}\psi_{223} - \psi_{003}\psi_{230} + \psi_{002}\psi_{231} - \psi_{001}\psi_{232} +
\psi_{000}\psi_{233})\nonumber\\\times (&\psi_{133}\psi_{300} - \psi_{132}\psi_{301} + \psi_{131}\psi_{302} - \psi_{130}\psi_{303} -
\psi_{123}\psi_{310} + \psi_{122}\psi_{311} - \psi_{121}\psi_{312} + \psi_{120}\psi_{313}\nonumber\\ &+ \psi_{113}\psi_{320} -
\psi_{112}\psi_{321} + \psi_{111}\psi_{322} - \psi_{110}\psi_{323} - \psi_{103}\psi_{330} + \psi_{102}\psi_{331} -
\psi_{101}\psi_{332} + \psi_{100}\psi_{333})\nonumber\\ -
4 (&\psi_{033}\psi_{100} - \psi_{032}\psi_{101} + \psi_{031}\psi_{102} - \psi_{030}\psi_{103} - \psi_{023}\psi_{110} +
\psi_{022}\psi_{111} - \psi_{021}\psi_{112} + \psi_{020}\psi_{113}\nonumber\\ &+ \psi_{013}\psi_{120} - \psi_{012}\psi_{121} +
\psi_{011}\psi_{122} - \psi_{010}\psi_{123} - \psi_{003}\psi_{130} + \psi_{002}\psi_{131} - \psi_{001}\psi_{132} +
\psi_{000}\psi_{133})\nonumber\\\times (&\psi_{233}\psi_{300} - \psi_{232}\psi_{301} + \psi_{231}\psi_{302} - \psi_{230}\psi_{303} -
\psi_{223}\psi_{310} + \psi_{222}\psi_{311} - \psi_{221}\psi_{312} + \psi_{220}\psi_{313}\nonumber\\ &+ \psi_{213}\psi_{320} -
\psi_{212}\psi_{321} + \psi_{211}\psi_{322} - \psi_{210}\psi_{323} - \psi_{203}\psi_{330} + \psi_{202}\psi_{331} -
\psi_{201}\psi_{332} + \psi_{200}\psi_{333}),
\end{align}
\begin{align}
I_{2b} = -2 (&\psi_{123}\psi_{210}- \psi_{122}\psi_{211}+ \psi_{121}\psi_{212}- \psi_{120}\psi_{213}+ \psi_{113}\psi_{220}-
\psi_{112}\psi_{221}+ \psi_{111}\psi_{222}- \psi_{110}\psi_{223}\nonumber\\&- \psi_{023}\psi_{310}+ \psi_{022}\psi_{311}-
\psi_{021}\psi_{312}+ \psi_{020}\psi_{313}- \psi_{013}\psi_{320}+ \psi_{012}\psi_{321}- \psi_{011}\psi_{322}+
\psi_{010}\psi_{323})^2\nonumber\\ -
2 (&\psi_{133}\psi_{200}- \psi_{132}\psi_{201}+ \psi_{131}\psi_{202}- \psi_{130}\psi_{203}+ \psi_{103}\psi_{230}-
\psi_{102}\psi_{231}+ \psi_{101}\psi_{232}- \psi_{100}\psi_{233}\nonumber\\&- \psi_{033}\psi_{300}+ \psi_{032}\psi_{301}-
\psi_{031}\psi_{302}+ \psi_{030}\psi_{303}- \psi_{003}\psi_{330}+ \psi_{002}\psi_{331}- \psi_{001}\psi_{332}+
\psi_{000}\psi_{333})^2\nonumber\\+
8 (&\psi_{113}\psi_{210}- \psi_{112}\psi_{211}+ \psi_{111}\psi_{212}- \psi_{110}\psi_{213}- \psi_{013}\psi_{310}+
\psi_{012}\psi_{311}- \psi_{011}\psi_{312}+ \psi_{010}\psi_{313})\nonumber\\\times (&\psi_{123}\psi_{220}- \psi_{122}\psi_{221}+
\psi_{121}\psi_{222}- \psi_{120}\psi_{223}- \psi_{023}\psi_{320}+ \psi_{022}\psi_{321}- \psi_{021}\psi_{322}+
\psi_{020}\psi_{323})\nonumber\\ +
8 (&\psi_{103}\psi_{200}- \psi_{102}\psi_{201}+ \psi_{101}\psi_{202}- \psi_{100}\psi_{203}- \psi_{003}\psi_{300}+
\psi_{002}\psi_{301}- \psi_{001}\psi_{302}+ \psi_{000}\psi_{303})\nonumber\\ \times (&\psi_{133}\psi_{230}- \psi_{132}\psi_{231}+
\psi_{131}\psi_{232}- \psi_{130}\psi_{233}- \psi_{033}\psi_{330}+ \psi_{032}\psi_{331}- \psi_{031}\psi_{332}+
\psi_{030}\psi_{333})\nonumber\\+
4 (&\psi_{123}\psi_{200}- \psi_{122}\psi_{201}+ \psi_{121}\psi_{202}- \psi_{120}\psi_{203}+ \psi_{103}\psi_{220}-
\psi_{102}\psi_{221}+ \psi_{101}\psi_{222}- \psi_{100}\psi_{223}\nonumber\\&- \psi_{023}\psi_{300}+ \psi_{022}\psi_{301}-
\psi_{021}\psi_{302}+ \psi_{020}\psi_{303}- \psi_{003}\psi_{320}+ \psi_{002}\psi_{321}- \psi_{001}\psi_{322}+
\psi_{000}\psi_{323})\nonumber\\\times (&\psi_{133}\psi_{210}- \psi_{132}\psi_{211}+ \psi_{131}\psi_{212}- \psi_{130}\psi_{213}+
\psi_{113}\psi_{230}- \psi_{112}\psi_{231}+ \psi_{111}\psi_{232}- \psi_{110}\psi_{233}\nonumber\\&- \psi_{033}\psi_{310}+
\psi_{032}\psi_{311}- \psi_{031}\psi_{312}+ \psi_{030}\psi_{313}- \psi_{013}\psi_{330}+ \psi_{012}\psi_{331}-
\psi_{011}\psi_{332}+ \psi_{010}\psi_{333})\nonumber\\ -
4 (&\psi_{113}\psi_{200}- \psi_{112}\psi_{201}+ \psi_{111}\psi_{202}- \psi_{110}\psi_{203}+ \psi_{103}\psi_{210}-
\psi_{102}\psi_{211}+ \psi_{101}\psi_{212}- \psi_{100}\psi_{213}\nonumber\\&- \psi_{013}\psi_{300}+ \psi_{012}\psi_{301}-
\psi_{011}\psi_{302}+ \psi_{010}\psi_{303}- \psi_{003}\psi_{310}+ \psi_{002}\psi_{311}- \psi_{001}\psi_{312}+
\psi_{000}\psi_{313}) \nonumber\\\times (&\psi_{133}\psi_{220}- \psi_{132}\psi_{221}+ \psi_{131}\psi_{222}- \psi_{130}\psi_{223}+
\psi_{123}\psi_{230}- \psi_{122}\psi_{231}+ \psi_{121}\psi_{232}- \psi_{120}\psi_{233}\nonumber\\&- \psi_{033}\psi_{320}+
\psi_{032}\psi_{321}- \psi_{031}\psi_{322}+ \psi_{030}\psi_{323}- \psi_{023}\psi_{330}+ \psi_{022}\psi_{331}-
\psi_{021}\psi_{332}+ \psi_{020}\psi_{333}),
\end{align}
\normalsize
and
\small
\begin{align}
I_{2c} = -2 (&\psi_{132}\psi_{201}+ \psi_{131}\psi_{202}- \psi_{122}\psi_{211}- \psi_{121}\psi_{212}+ \psi_{112}\psi_{221}+
\psi_{111}\psi_{222}- \psi_{102}\psi_{231}- \psi_{101}\psi_{232}\nonumber\\&- \psi_{032}\psi_{301}- \psi_{031}\psi_{302}+
\psi_{022}\psi_{311}+ \psi_{021}\psi_{312}- \psi_{012}\psi_{321}- \psi_{011}\psi_{322}+ \psi_{002}\psi_{331}+
\psi_{001}\psi_{332})^2\nonumber\\ -
2 (&\psi_{133}\psi_{200}+ \psi_{130}\psi_{203}- \psi_{123}\psi_{210}- \psi_{120}\psi_{213}+ \psi_{113}\psi_{220}+
\psi_{110}\psi_{223}- \psi_{103}\psi_{230}- \psi_{100}\psi_{233}\nonumber\\&- \psi_{033}\psi_{300}- \psi_{030}\psi_{303}+
\psi_{023}\psi_{310}+ \psi_{020}\psi_{313}- \psi_{013}\psi_{320}- \psi_{010}\psi_{323}+ \psi_{003}\psi_{330}+
\psi_{000}\psi_{333})^2\nonumber\\+
8 (&\psi_{131}\psi_{201}- \psi_{121}\psi_{211}+ \psi_{111}\psi_{221}- \psi_{101}\psi_{231}- \psi_{031}\psi_{301}+
\psi_{021}\psi_{311}- \psi_{011}\psi_{321}+ \psi_{001}\psi_{331})\nonumber\\\times (&\psi_{132}\psi_{202}- \psi_{122}\psi_{212}+
\psi_{112}\psi_{222}- \psi_{102}\psi_{232}- \psi_{032}\psi_{302}+ \psi_{022}\psi_{312}- \psi_{012}\psi_{322}+
\psi_{002}\psi_{332})\nonumber\\ +
8 (&\psi_{130}\psi_{200}- \psi_{120}\psi_{210}+ \psi_{110}\psi_{220}- \psi_{100}\psi_{230}- \psi_{030}\psi_{300}+
\psi_{020}\psi_{310}- \psi_{010}\psi_{320}+ \psi_{000}\psi_{330})\nonumber\\\times (& \psi_{133}\psi_{203}- \psi_{123}\psi_{213}+
\psi_{113}\psi_{223}- \psi_{103}\psi_{233}- \psi_{033}\psi_{303}+ \psi_{023}\psi_{313}- \psi_{013}\psi_{323}+
\psi_{003}\psi_{333})\nonumber\\+
4 (&\psi_{132}\psi_{200}+ \psi_{130}\psi_{202}- \psi_{122}\psi_{210}- \psi_{120}\psi_{212}+ \psi_{112}\psi_{220}+
\psi_{110}\psi_{222}- \psi_{102}\psi_{230}- \psi_{100}\psi_{232}\nonumber\\&- \psi_{032}\psi_{300}- \psi_{030}\psi_{302}+
\psi_{022}\psi_{310}+ \psi_{020}\psi_{312}- \psi_{012}\psi_{320}- \psi_{010}\psi_{322}+ \psi_{002}\psi_{330}+
\psi_{000}\psi_{332})\nonumber\\\times (&\psi_{133}\psi_{201}+ \psi_{131}\psi_{203}- \psi_{123}\psi_{211}- \psi_{121}\psi_{213}+
\psi_{113}\psi_{221}+ \psi_{111}\psi_{223}- \psi_{103}\psi_{231}- \psi_{101}\psi_{233}\nonumber\\&- \psi_{033}\psi_{301}-
\psi_{031}\psi_{303}+ \psi_{023}\psi_{311}+ \psi_{021}\psi_{313}- \psi_{013}\psi_{321}- \psi_{011}\psi_{323}+
\psi_{003}\psi_{331}+ \psi_{001}\psi_{333})\nonumber\\ -
4 (&\psi_{131}\psi_{200}+ \psi_{130}\psi_{201}- \psi_{121}\psi_{210}- \psi_{120}\psi_{211}+ \psi_{111}\psi_{220}+
\psi_{110}\psi_{221}- \psi_{101}\psi_{230}- \psi_{100}\psi_{231}\nonumber\\&- \psi_{031}\psi_{300}- \psi_{030}\psi_{301}+
\psi_{021}\psi_{310}+ \psi_{020}\psi_{311}- \psi_{011}\psi_{320}- \psi_{010}\psi_{321}+ \psi_{001}\psi_{330}+
\psi_{000}\psi_{331}) \nonumber\\\times (&\psi_{133}\psi_{202}+ \psi_{132}\psi_{203}- \psi_{123}\psi_{212}- \psi_{122}\psi_{213}+
\psi_{113}\psi_{222}+ \psi_{112}\psi_{223}- \psi_{103}\psi_{232}- \psi_{102}\psi_{233}\nonumber\\&- \psi_{033}\psi_{302}-
\psi_{032}\psi_{303}+ \psi_{023}\psi_{312}+ \psi_{022}\psi_{313}- \psi_{013}\psi_{322}- \psi_{012}\psi_{323}+
\psi_{003}\psi_{332}+ \psi_{002}\psi_{333}).
\end{align}
\normalsize
The polynomials $I_{3a},I_{3b},I_{3c}$ are invariant under $G^C$, up to a U(1) phase, in all labs and invariant under P in all labs. Explicitly written out they are given by
\small
\begin{align}
I_{3a} = -2 (&\psi_{011}\psi_{ 100} -\psi_{ 010 }\psi_{ 101} +\psi_{ 013}\psi_{ 102} -\psi_{ 012}\psi_{ 103 }-\psi_{ 001} \psi_{ 110 }+
\psi_{ 000}\psi_{ 111} -\psi_{ 003}\psi_{ 112} +\psi_{ 002}\psi_{ 113}\nonumber\\ &+\psi_{ 031}\psi_{ 120} -\psi_{ 030}\psi_{ 121} +
\psi_{ 033}\psi_{ 122} -\psi_{ 032}\psi_{ 123 }-\psi_{ 021}\psi_{ 130} +\psi_{ 020}\psi_{ 131} -\psi_{ 023}\psi_{ 132} +
\psi_{ 022}\psi_{ 133})^2\nonumber\\
-
2 (&\psi_{211} \psi_{300} - \psi_{210} \psi_{301} + \psi_{213} \psi_{302} - \psi_{212} \psi_{303} - \psi_{201} \psi_{310} +
\psi_{200} \psi_{311} - \psi_{203} \psi_{312} + \psi_{202} \psi_{313}\nonumber\nonumber\\ &+ \psi_{231} \psi_{320} - \psi_{230} \psi_{321} +
\psi_{233} \psi_{322} - \psi_{232 }\psi_{323} - \psi_{221} \psi_{330} + \psi_{220} \psi_{331} - \psi_{223} \psi_{332} +
\psi_{222} \psi_{333})^2\nonumber\\
+
8 (&\psi_{ 001}\psi_{ 010} -\psi_{ 000}\psi_{ 011} +\psi_{ 003}\psi_{ 012} -\psi_{ 002}\psi_{ 013} +\psi_{ 021}\psi_{ 030 }-
\psi_{ 020}\psi_{ 031} +\psi_{ 023}\psi_{ 032} -\psi_{ 022}\psi_{ 033})\nonumber\\\times (&\psi_{ 101}\psi_{ 110} -\psi_{ 100} \psi_{ 111} +
\psi_{ 103}\psi_{ 112} -\psi_{ 102}\psi_{ 113} +\psi_{ 121}\psi_{ 130 }-\psi_{ 120 }\psi_{ 131} +\psi_{ 123}\psi_{ 132} -
\psi_{ 122}\psi_{ 133})\nonumber\\
+
8 (&\psi_{201} \psi_{210} - \psi_{200} \psi_{211} + \psi_{203} \psi_{212} - \psi_{202} \psi_{213} + \psi_{221 }\psi_{230} -
\psi_{220} \psi_{231} + \psi_{223 }\psi_{232 }- \psi_{222} \psi_{233})\nonumber\\\times (&\psi_{301} \psi_{310} - \psi_{300} \psi_{311} +
\psi_{303} \psi_{312} - \psi_{302} \psi_{313 }+ \psi_{321} \psi_{330} - \psi_{320} \psi_{331} + \psi_{323} \psi_{332} -
\psi_{322} \psi_{333})\nonumber\\
-
4 (&\psi_{ 111}\psi_{ 200} -\psi_{ 110}\psi_{ 201} + \psi_{113} \psi_{202} - \psi_{112} \psi_{203} - \psi_{101} \psi_{210} +
\psi_{100} \psi_{211} -\psi_{ 103}\psi_{ 212} +\psi_{ 102}\psi_{ 213} \nonumber\\&+\psi_{ 131}\psi_{ 220} - \psi_{130} \psi_{221} +
\psi_{133} \psi_{222} -\psi_{ 132}\psi_{ 223} -\psi_{ 121}\psi_{ 230} + \psi_{120} \psi_{231 }- \psi_{123} \psi_{232} +
\psi_{122} \psi_{233})\nonumber\\\times (&\psi_{011} \psi_{300} - \psi_{010} \psi_{301} + \psi_{013} \psi_{302} - \psi_{012} \psi_{303} -
\psi_{001} \psi_{310} + \psi_{000} \psi_{311} - \psi_{003} \psi_{312} + \psi_{002} \psi_{313}\nonumber\\ &+ \psi_{031} \psi_{320} -
\psi_{030} \psi_{321} + \psi_{033} \psi_{322} - \psi_{032} \psi_{323} - \psi_{021} \psi_{330} + \psi_{020} \psi_{331} -
\psi_{023} \psi_{332} + \psi_{022} \psi_{333})\nonumber\\ +
4 (&\psi_{011 }\psi_{200} - \psi_{010} \psi_{201} + \psi_{013} \psi_{202} - \psi_{012} \psi_{203} - \psi_{001} \psi_{210} +
\psi_{000} \psi_{211} - \psi_{003} \psi_{212} + \psi_{002} \psi_{213}\nonumber\\& + \psi_{031} \psi_{220} - \psi_{030} \psi_{221} +
\psi_{033} \psi_{222} - \psi_{032} \psi_{223} - \psi_{021} \psi_{230} + \psi_{020 }\psi_{231 }- \psi_{023} \psi_{232} +
\psi_{022} \psi_{233})\nonumber\\\times (&\psi_{111} \psi_{300} - \psi_{110 }\psi_{301} + \psi_{113} \psi_{302} - \psi_{112} \psi_{303} -
\psi_{101} \psi_{310} + \psi_{100} \psi_{311} - \psi_{103} \psi_{312} + \psi_{102} \psi_{313}\nonumber\\ &+ \psi_{131} \psi_{320} -
\psi_{130} \psi_{321} + \psi_{133} \psi_{322} - \psi_{132} \psi_{323} - \psi_{121} \psi_{330 }+ \psi_{120 }\psi_{331} -
\psi_{123} \psi_{332} + \psi_{122} \psi_{333}),
\end{align}
\begin{align}
I_{3b} = -2 (&\psi_{011}\psi_{100}- \psi_{010}\psi_{101}+ \psi_{013}\psi_{102}- \psi_{012}\psi_{103}+ \psi_{001}\psi_{110}-
\psi_{000}\psi_{111}+ \psi_{003}\psi_{112}- \psi_{002}\psi_{113}\nonumber\\&+ \psi_{211}\psi_{300}- \psi_{210}\psi_{301}+
\psi_{213}\psi_{302}- \psi_{212}\psi_{303}+ \psi_{201}\psi_{310}- \psi_{200}\psi_{311}+ \psi_{203}\psi_{312}-
\psi_{202}\psi_{313})^2\nonumber\\ -
2 (&\psi_{031}\psi_{120}- \psi_{030}\psi_{121}+ \psi_{033}\psi_{122}- \psi_{032}\psi_{123}+ \psi_{021}\psi_{130}-
\psi_{020}\psi_{131}+ \psi_{023}\psi_{132}- \psi_{022}\psi_{133}\nonumber\\ &+\psi_{231}\psi_{320}- \psi_{230}\psi_{321}+
\psi_{233}\psi_{322}- \psi_{232}\psi_{323}+ \psi_{221}\psi_{330}- \psi_{220}\psi_{331}+ \psi_{223}\psi_{332}-
\psi_{222}\psi_{333})^2\nonumber\\ +
8 (&\psi_{021}\psi_{120}- \psi_{020}\psi_{121}+ \psi_{023}\psi_{122}- \psi_{022}\psi_{123}+ \psi_{221}\psi_{320}-
\psi_{220}\psi_{321}+ \psi_{223}\psi_{322}- \psi_{222}\psi_{323})\nonumber\\\times (&\psi_{031}\psi_{130}- \psi_{030}\psi_{131}+
\psi_{033}\psi_{132}- \psi_{032}\psi_{133}+ \psi_{231}\psi_{330}- \psi_{230}\psi_{331}+ \psi_{233}\psi_{332}-
\psi_{232}\psi_{333})\nonumber\\ +
8 (&\psi_{001}\psi_{100}- \psi_{000}\psi_{101}+ \psi_{003}\psi_{102}- \psi_{002}\psi_{103}+ \psi_{201}\psi_{300}-
\psi_{200}\psi_{301}+ \psi_{203}\psi_{302}- \psi_{202}\psi_{303})\nonumber\\\times (&\psi_{011}\psi_{110}- \psi_{010}\psi_{111}+
\psi_{013}\psi_{112}- \psi_{012}\psi_{113}+ \psi_{211}\psi_{310}- \psi_{210}\psi_{311}+ \psi_{213}\psi_{312}-
\psi_{212}\psi_{313})\nonumber\\ -
4 (&\psi_{021}\psi_{110}- \psi_{020}\psi_{111}+ \psi_{023}\psi_{112}- \psi_{022}\psi_{113}+ \psi_{011}\psi_{120}-
\psi_{010}\psi_{121}+ \psi_{013}\psi_{122}- \psi_{012}\psi_{123}\nonumber\\& + \psi_{221}\psi_{310}- \psi_{220}\psi_{311}+
\psi_{223}\psi_{312}- \psi_{222}\psi_{313}+ \psi_{211}\psi_{320}- \psi_{210}\psi_{321}+ \psi_{213}\psi_{322}-
\psi_{212}\psi_{323})\nonumber\\\times (&\psi_{031}\psi_{100}- \psi_{030}\psi_{101}+ \psi_{033}\psi_{102}- \psi_{032}\psi_{103}+
\psi_{001}\psi_{130}- \psi_{000}\psi_{131}+ \psi_{003}\psi_{132}- \psi_{002}\psi_{133}\nonumber\\ &+ \psi_{231}\psi_{300}-
\psi_{230}\psi_{301}+ \psi_{233}\psi_{302}- \psi_{232}\psi_{303}+ \psi_{201}\psi_{330}- \psi_{200}\psi_{331}+
\psi_{203}\psi_{332}- \psi_{202}\psi_{333})\nonumber\\ +
4 (&\psi_{021}\psi_{100}- \psi_{020}\psi_{101}+ \psi_{023}\psi_{102}- \psi_{022}\psi_{103}+ \psi_{001}\psi_{120}-
\psi_{000}\psi_{121}+ \psi_{003}\psi_{122}- \psi_{002}\psi_{123}\nonumber\\&+ \psi_{221}\psi_{300}- \psi_{220}\psi_{301}+
\psi_{223}\psi_{302}- \psi_{222}\psi_{303}+ \psi_{201}\psi_{320}- \psi_{200}\psi_{321}+ \psi_{203}\psi_{322}-
\psi_{202}\psi_{323})\nonumber\\\times (&\psi_{031}\psi_{110}- \psi_{030}\psi_{111}+ \psi_{033}\psi_{112}- \psi_{032}\psi_{113}+
\psi_{011}\psi_{130}- \psi_{010}\psi_{131}+ \psi_{013}\psi_{132}- \psi_{012}\psi_{133}\nonumber\\ &+ \psi_{231}\psi_{310}-
\psi_{230}\psi_{311}+ \psi_{233}\psi_{312}- \psi_{232}\psi_{313}+ \psi_{211}\psi_{330}- \psi_{210}\psi_{331}+
\psi_{213}\psi_{332}- \psi_{212}\psi_{333}),
\end{align}
\normalsize
and
\small
\begin{align}
I_{3c} = -2 (&\psi_{011}\psi_{100} + \psi_{010}\psi_{101} - \psi_{001}\psi_{110} - \psi_{000}\psi_{111} + \psi_{031}\psi_{120} +
\psi_{030}\psi_{121} - \psi_{021}\psi_{130} - \psi_{020}\psi_{131}\nonumber\\ &+ \psi_{211}\psi_{300} + \psi_{210}\psi_{301} -
\psi_{201}\psi_{310} - \psi_{200}\psi_{311} + \psi_{231}\psi_{320} + \psi_{230}\psi_{321} - \psi_{221}\psi_{330} -
\psi_{220}\psi_{331})^2\nonumber\\ -
2 (&\psi_{013}\psi_{102} + \psi_{012}\psi_{103} - \psi_{003}\psi_{112} - \psi_{002}\psi_{113} + \psi_{033}\psi_{122} +
\psi_{032}\psi_{123} - \psi_{023}\psi_{132} - \psi_{022}\psi_{133}\nonumber\\& + \psi_{213}\psi_{302} + \psi_{212}\psi_{303} -
\psi_{203}\psi_{312} - \psi_{202}\psi_{313} + \psi_{233}\psi_{322} + \psi_{232}\psi_{323} - \psi_{223}\psi_{332} -
\psi_{222}\psi_{333})^2\nonumber\\ +
8 (&\psi_{012}\psi_{102} - \psi_{002}\psi_{112} + \psi_{032}\psi_{122} - \psi_{022}\psi_{132} + \psi_{212}\psi_{302} -
\psi_{202}\psi_{312} + \psi_{232}\psi_{322} - \psi_{222}\psi_{332})\nonumber\\\times (&\psi_{013}\psi_{103} - \psi_{003}\psi_{113} +
\psi_{033}\psi_{123} - \psi_{023}\psi_{133} + \psi_{213}\psi_{303} - \psi_{203}\psi_{313} + \psi_{233}\psi_{323} -
\psi_{223}\psi_{333})\nonumber\\ +
8 (&\psi_{010}\psi_{100} - \psi_{000}\psi_{110} + \psi_{030}\psi_{120} - \psi_{020}\psi_{130} + \psi_{210}\psi_{300} -
\psi_{200}\psi_{310} + \psi_{230}\psi_{320} - \psi_{220}\psi_{330})\nonumber\\\times (&\psi_{011}\psi_{101} - \psi_{001}\psi_{111} +
\psi_{031}\psi_{121} - \psi_{021}\psi_{131} + \psi_{211}\psi_{301} - \psi_{201}\psi_{311} + \psi_{231}\psi_{321} -
\psi_{221}\psi_{331})\nonumber\\ -
4 (&\psi_{012}\psi_{101} + \psi_{011}\psi_{102} - \psi_{002}\psi_{111} - \psi_{001}\psi_{112} + \psi_{032}\psi_{121} +
\psi_{031}\psi_{122} - \psi_{022}\psi_{131} - \psi_{021}\psi_{132}\nonumber\\& + \psi_{212}\psi_{301} + \psi_{211}\psi_{302} -
\psi_{202}\psi_{311} - \psi_{201}\psi_{312} + \psi_{232}\psi_{321} + \psi_{231}\psi_{322} - \psi_{222}\psi_{331} -
\psi_{221}\psi_{332})\nonumber\\\times (&\psi_{013}\psi_{100} + \psi_{010}\psi_{103} - \psi_{003}\psi_{110} - \psi_{000}\psi_{113} +
\psi_{033}\psi_{120} + \psi_{030}\psi_{123} - \psi_{023}\psi_{130} - \psi_{020}\psi_{133}\nonumber\\ &+ \psi_{213}\psi_{300} +
\psi_{210}\psi_{303} - \psi_{203}\psi_{310} - \psi_{200}\psi_{313} + \psi_{233}\psi_{320} + \psi_{230}\psi_{323} -
\psi_{223}\psi_{330} - \psi_{220}\psi_{333})\nonumber\\ +
4 (&\psi_{012}\psi_{100} + \psi_{010}\psi_{102} - \psi_{002}\psi_{110} - \psi_{000}\psi_{112} + \psi_{032}\psi_{120} +
\psi_{030}\psi_{122} - \psi_{022}\psi_{130} - \psi_{020}\psi_{132}\nonumber\\ &+ \psi_{212}\psi_{300} + \psi_{210}\psi_{302} -
\psi_{202}\psi_{310} - \psi_{200}\psi_{312} + \psi_{232}\psi_{320} + \psi_{230}\psi_{322} - \psi_{222}\psi_{330} -
\psi_{220}\psi_{332})\nonumber\\\times (&\psi_{013}\psi_{101} + \psi_{011}\psi_{103} - \psi_{003}\psi_{111} - \psi_{001}\psi_{113} +
\psi_{033}\psi_{121} + \psi_{031}\psi_{123} - \psi_{023}\psi_{131} - \psi_{021}\psi_{133}\nonumber\\ &+ \psi_{213}\psi_{301} +
\psi_{211}\psi_{303} - \psi_{203}\psi_{311} - \psi_{201}\psi_{313} + \psi_{233}\psi_{321} + \psi_{231}\psi_{323} -
\psi_{223}\psi_{331} - \psi_{221}\psi_{333}).
\end{align}
\normalsize
An example of a polynomial on the form $I_d$ is $I_{3d}$. It is given by $I_{3d}=4(Z_1+Z_2)$ where
\small
\begin{align}
Z_1=&(\psi_{001}\psi_{013}+ \psi_{021}\psi_{033}- \psi_{023}\psi_{031}- \psi_{003}\psi_{011}) (\psi_{102}\psi_{110}-
\psi_{100}\psi_{112}+ \psi_{122}\psi_{130}- \psi_{120}\psi_{132})\nonumber\\&
+ (\psi_{022}\psi_{031}+ \psi_{002}\psi_{011}- \psi_{001}\psi_{012}- \psi_{021}\psi_{032}) (\psi_{103}\psi_{110}-
\psi_{100}\psi_{113}+ \psi_{123}\psi_{130}- \psi_{120}\psi_{133})\nonumber\\&
+ (\psi_{003}\psi_{012}+ \psi_{020}\psi_{031}- \psi_{021}\psi_{030}- \psi_{002}\psi_{013}) (\psi_{101}\psi_{110}-
\psi_{100}\psi_{111}- \psi_{123}\psi_{132}+ \psi_{122}\psi_{133})\nonumber\\&
+ (\psi_{002}\psi_{033}+ \psi_{000}\psi_{031}- \psi_{003}\psi_{032}- \psi_{001}\psi_{030}) (\psi_{111}\psi_{120}-
\psi_{110}\psi_{121}+ \psi_{113}\psi_{122}- \psi_{112}\psi_{123})\nonumber\\&
+ (\psi_{000}\psi_{012}+ \psi_{020}\psi_{032}- \psi_{022}\psi_{030}- \psi_{002}\psi_{010}) (\psi_{103}\psi_{111}-
\psi_{101}\psi_{113}+ \psi_{123}\psi_{131}- \psi_{121}\psi_{133})\nonumber\\&
+ (\psi_{023}\psi_{030}+ \psi_{003}\psi_{010}- \psi_{020}\psi_{033}- \psi_{000}\psi_{013}) (\psi_{102}\psi_{111}-
\psi_{101}\psi_{112}+ \psi_{122}\psi_{131}- \psi_{121}\psi_{132})\nonumber\\&
+ (\psi_{021}\psi_{010}+ \psi_{023}\psi_{012}- \psi_{020}\psi_{011}- \psi_{022}\psi_{013}) (\psi_{101}\psi_{130}-
\psi_{100}\psi_{131}+ \psi_{103}\psi_{132}- \psi_{102}\psi_{133})\nonumber\\&
+ (\psi_{001}\psi_{010}+ \psi_{022}\psi_{033}- \psi_{000}\psi_{011}- \psi_{023}\psi_{032}) (\psi_{103}\psi_{112}-
\psi_{102}\psi_{113}- \psi_{121}\psi_{130}+ \psi_{120}\psi_{131})\nonumber\\&
+ (\psi_{011}\psi_{030}+ \psi_{013}\psi_{032}- \psi_{010}\psi_{031}- \psi_{012}\psi_{033}) (\psi_{101}\psi_{120}-
\psi_{100}\psi_{121}+ \psi_{103}\psi_{122}- \psi_{102}\psi_{123})\nonumber\\&
+ (\psi_{001}\psi_{020}+ \psi_{003}\psi_{022}- \psi_{000}\psi_{021}- \psi_{002}\psi_{023}) (\psi_{111}\psi_{130}-
\psi_{110}\psi_{131}+ \psi_{113}\psi_{132}- \psi_{112}\psi_{133})\nonumber\\&
+ (\psi_{121}\psi_{200}- \psi_{120}\psi_{201}+ \psi_{123}\psi_{202}- \psi_{122}\psi_{203}) (\psi_{330}\psi_{011}+
\psi_{332}\psi_{013}- \psi_{333}\psi_{012}- \psi_{010}\psi_{331})\nonumber\\&
+ (\psi_{131}\psi_{200}- \psi_{130}\psi_{201}+ \psi_{133}\psi_{202}- \psi_{132}\psi_{203}) (
\psi_{321}\psi_{010}+ \psi_{323}\psi_{012}- \psi_{320}\psi_{011}- \psi_{322}\psi_{013})\nonumber\\&
+ (\psi_{101}\psi_{200}- \psi_{100}\psi_{201}+ \psi_{103}\psi_{202}- \psi_{102}\psi_{203}) (\psi_{310}\psi_{011}+
\psi_{312}\psi_{013}- \psi_{313}\psi_{012}- \psi_{311}\psi_{010})\nonumber\\&
+ (\psi_{111}\psi_{210}- \psi_{110}\psi_{211}+ \psi_{113}\psi_{212}- \psi_{112}\psi_{213}) (\psi_{302}\psi_{003}+
\psi_{300}\psi_{001}- \psi_{303}\psi_{002}- \psi_{301}\psi_{000})\nonumber\\&
+ (\psi_{111}\psi_{201}- \psi_{101}\psi_{211}+ \psi_{131}\psi_{221}- \psi_{121}\psi_{231}) (\psi_{310}\psi_{000}+
\psi_{330}\psi_{020}- \psi_{320}\psi_{030}- \psi_{300}\psi_{010})\nonumber\\&
+ (\psi_{111}\psi_{202}- \psi_{101}\psi_{212}+ \psi_{131}\psi_{222}- \psi_{121}\psi_{232}) (\psi_{300}\psi_{013}+
\psi_{320}\psi_{033}- \psi_{330}\psi_{023}- \psi_{310}\psi_{003})\nonumber\\&
+ (\psi_{110}\psi_{202}- \psi_{100}\psi_{212}+ \psi_{130}\psi_{222}- \psi_{120}\psi_{232}) (\psi_{311}\psi_{003}+
\psi_{331}\psi_{023}- \psi_{321}\psi_{033}- \psi_{301}\psi_{013})\nonumber\\&
+ (\psi_{121}\psi_{210}- \psi_{120}\psi_{211}+ \psi_{123}\psi_{212}- \psi_{122}\psi_{213}) (\psi_{331}\psi_{000}+
\psi_{333}\psi_{002}- \psi_{330}\psi_{001}- \psi_{332}\psi_{003})\nonumber\\&
+ (\psi_{131}\psi_{210}- \psi_{130}\psi_{211}+ \psi_{133}\psi_{212}- \psi_{132}\psi_{213}) (\psi_{322}\psi_{003}+
\psi_{320}\psi_{001}- \psi_{323}\psi_{002}- \psi_{321}\psi_{000})\nonumber\\&
+ (\psi_{112}\psi_{202}- \psi_{102}\psi_{212}+ \psi_{132}\psi_{222}- \psi_{122}\psi_{232}) (\psi_{333}\psi_{023}+
\psi_{313}\psi_{003}- \psi_{303}\psi_{013}- \psi_{323}\psi_{033})\nonumber\\&
+ (\psi_{110}\psi_{200}- \psi_{100}\psi_{210}+ \psi_{130}\psi_{220}- \psi_{120}\psi_{230}) (\psi_{311}\psi_{001}+
\psi_{331}\psi_{021}- \psi_{321}\psi_{031}- \psi_{301}\psi_{011})\nonumber\\&
+ (\psi_{113}\psi_{201}- \psi_{103}\psi_{211}+ \psi_{133}\psi_{221}- \psi_{123}\psi_{231}) (\psi_{332}\psi_{020}+
\psi_{312}\psi_{000}- \psi_{302}\psi_{010}- \psi_{322}\psi_{030})\nonumber\\&
+ (\psi_{112}\psi_{201}- \psi_{102}\psi_{211}+ \psi_{132}\psi_{221}- \psi_{122}\psi_{231}) (\psi_{323}\psi_{030}+
\psi_{303}\psi_{010}- \psi_{333}\psi_{020}- \psi_{313}\psi_{000})\nonumber\\&
+ (\psi_{113}\psi_{200}- \psi_{103}\psi_{210}+ \psi_{133}\psi_{220}- \psi_{123}\psi_{230}) (\psi_{322}\psi_{031}+
\psi_{302}\psi_{011}- \psi_{312}\psi_{001}- \psi_{332}\psi_{021})\nonumber\\&
+ (\psi_{112}\psi_{200}- \psi_{102}\psi_{210}+ \psi_{132}\psi_{220}- \psi_{122}\psi_{230}) (\psi_{333}\psi_{021}+
\psi_{313}\psi_{001}- \psi_{303}\psi_{011}- \psi_{323}\psi_{031})\nonumber\\&
+ (\psi_{111}\psi_{230}- \psi_{110}\psi_{231}+ \psi_{113}\psi_{232}- \psi_{112}\psi_{233}) (
\psi_{300}\psi_{021}+ \psi_{302}\psi_{023}- \psi_{303}\psi_{022}- \psi_{301}\psi_{020})\nonumber\\&
+ (\psi_{101}\psi_{230}- \psi_{100}\psi_{231}+ \psi_{103}\psi_{232}- \psi_{102}\psi_{233}) (\psi_{313}\psi_{022}+
\psi_{311}\psi_{020}- \psi_{312}\psi_{023}- \psi_{310}\psi_{021})\nonumber\\&
+ (\psi_{131}\psi_{230}- \psi_{130}\psi_{231}+ \psi_{133}\psi_{232}- \psi_{132}\psi_{233}) (\psi_{320}\psi_{021}+
\psi_{322}\psi_{023}- \psi_{321}\psi_{020}- \psi_{323}\psi_{022})\nonumber\\&
+ (\psi_{111}\psi_{220}- \psi_{110}\psi_{221}+ \psi_{113}\psi_{222}- \psi_{112}\psi_{223}) (
\psi_{303}\psi_{032}+ \psi_{301}\psi_{030}- \psi_{300}\psi_{031}- \psi_{302}\psi_{033})\nonumber\\&
+ (\psi_{101}\psi_{220}- \psi_{100}\psi_{221}+ \psi_{103}\psi_{222}- \psi_{102}\psi_{223}) (\psi_{310}\psi_{031}+
\psi_{312}\psi_{033}- \psi_{313}\psi_{032}- \psi_{311}\psi_{030})\nonumber\\&
+ (\psi_{111}\psi_{203}- \psi_{101}\psi_{213}+ \psi_{131}\psi_{223}- \psi_{121}\psi_{233}) (\psi_{310}\psi_{002}+
\psi_{330}\psi_{022}- \psi_{320}\psi_{032}- \psi_{300}\psi_{012})\nonumber\\&
+ (\psi_{110}\psi_{203}- \psi_{100}\psi_{213}+ \psi_{130}\psi_{223}- \psi_{120}\psi_{233}) (\psi_{321}\psi_{032}+
\psi_{301}\psi_{012}- \psi_{311}\psi_{002}- \psi_{331}\psi_{022})\nonumber\\&
+ (\psi_{113}\psi_{203}- \psi_{103}\psi_{213}+ \psi_{133}\psi_{223}- \psi_{123}\psi_{233}) (\psi_{312}\psi_{002}+
\psi_{332}\psi_{022}- \psi_{322}\psi_{032}- \psi_{302}\psi_{012})\nonumber\\&
+ (\psi_{121}\psi_{220}- \psi_{120}\psi_{221}+ \psi_{123}\psi_{222}- \psi_{122}\psi_{223}) (\psi_{332}\psi_{033}+
\psi_{330}\psi_{031}- \psi_{333}\psi_{032}- \psi_{331}\psi_{030})\nonumber\\&
+ (\psi_{203}\psi_{212}+ \psi_{220}\psi_{231}- \psi_{221}\psi_{230}- \psi_{202}\psi_{213}) (\psi_{301}\psi_{310}-
\psi_{300}\psi_{311}- \psi_{323}\psi_{332}+ \psi_{322}\psi_{333})\nonumber\\&
+ (\psi_{201}\psi_{213}+ \psi_{221}\psi_{233}- \psi_{203}\psi_{211}- \psi_{223}\psi_{231}) (\psi_{302}\psi_{310}-
\psi_{300}\psi_{312}+ \psi_{322}\psi_{330}- \psi_{320}\psi_{332})\nonumber\\&
+ (\psi_{211}\psi_{230}- \psi_{210}\psi_{231}+ \psi_{213}\psi_{232}- \psi_{212}\psi_{233}) (\psi_{301}\psi_{320}-
\psi_{300}\psi_{321}+ \psi_{303}\psi_{322}- \psi_{302}\psi_{323})\nonumber\\&
+ (\psi_{223}\psi_{230}+ \psi_{203}\psi_{210}- \psi_{220}\psi_{233}- \psi_{200}\psi_{213}) (\psi_{302}\psi_{311}-
\psi_{301}\psi_{312}+ \psi_{322}\psi_{331}- \psi_{321}\psi_{332})\nonumber\\&
+ (\psi_{222}\psi_{231}+ \psi_{202}\psi_{211}- \psi_{221}\psi_{232}- \psi_{201}\psi_{212}) (\psi_{303}\psi_{310}-
\psi_{300}\psi_{313}+ \psi_{323}\psi_{330}- \psi_{320}\psi_{333})\nonumber\\&
+ (\psi_{200}\psi_{212}+ \psi_{220}\psi_{232}- \psi_{202}\psi_{210}- \psi_{222}\psi_{230}) (\psi_{303}\psi_{311}-
\psi_{301}\psi_{313}+ \psi_{323}\psi_{331}- \psi_{321}\psi_{333})\nonumber\\&
+ (\psi_{222}\psi_{233}+ \psi_{201}\psi_{210}- \psi_{223}\psi_{232}- \psi_{200}\psi_{211}) (\psi_{303}\psi_{312}-
\psi_{302}\psi_{313}- \psi_{321}\psi_{330}+ \psi_{320}\psi_{331})\nonumber\\&
+ (\psi_{200}\psi_{231}+ \psi_{202}\psi_{233}- \psi_{201}\psi_{230}- \psi_{203}\psi_{232}) (\psi_{311}\psi_{320}-
\psi_{310}\psi_{321}+ \psi_{313}\psi_{322}- \psi_{312}\psi_{323})\nonumber\\&
+ (\psi_{223}\psi_{212}+ \psi_{221}\psi_{210}- \psi_{220}\psi_{211}- \psi_{222}\psi_{213}) (
\psi_{301}\psi_{330}- \psi_{300}\psi_{331}+ \psi_{303}\psi_{332}- \psi_{302}\psi_{333})\nonumber\\&
+ (\psi_{203}\psi_{222}+ \psi_{220}\psi_{201}- \psi_{221}\psi_{200}- \psi_{202}\psi_{223}) (
\psi_{311}\psi_{330}- \psi_{310}\psi_{331}+ \psi_{313}\psi_{332}- \psi_{312}\psi_{333}),
\end{align}
\normalsize
and
\small
\begin{align}
Z_2=&\psi_{032}\psi_{323}(\psi_{112}\psi_{203}- \psi_{102}\psi_{213}+ \psi_{131}\psi_{220}- \psi_{130}\psi_{221}+
\psi_{133}\psi_{222}- \psi_{122}\psi_{233})\nonumber\\&
+ \psi_{022}\psi_{333}( \psi_{102}\psi_{213}-\psi_{112}\psi_{203}- \psi_{132}\psi_{223}+ \psi_{121}\psi_{230}-
\psi_{120}\psi_{231}+ \psi_{123}\psi_{232})\nonumber\\&
+ \psi_{010} \psi_{301}(\psi_{111}\psi_{200}+ \psi_{113}\psi_{202}- \psi_{112}\psi_{203}- \psi_{100}\psi_{211}+
\psi_{130}\psi_{221}- \psi_{120}\psi_{231})\nonumber\\&
+ \psi_{011}\psi_{300}(\psi_{110}\psi_{201}- \psi_{113}\psi_{202}+ \psi_{112}\psi_{203}- \psi_{101}\psi_{210}+
\psi_{131}\psi_{220}- \psi_{121}\psi_{230})\nonumber\\&
+ \psi_{012}\psi_{303}(\psi_{111}\psi_{200}- \psi_{110}\psi_{201}+ \psi_{113}\psi_{202}- \psi_{102}\psi_{213}+
\psi_{132}\psi_{223}- \psi_{122}\psi_{233})\nonumber\\&
+ \psi_{013}\psi_{302}(\psi_{110}\psi_{201}-\psi_{111}\psi_{200}+ \psi_{112}\psi_{203}- \psi_{103}\psi_{212}+
\psi_{133}\psi_{222}- \psi_{123}\psi_{232})\nonumber\\&
+ \psi_{001}\psi_{310}(\psi_{100}\psi_{211}-\psi_{111}\psi_{200}- \psi_{103}\psi_{212}+ \psi_{102}\psi_{213}-
\psi_{131}\psi_{220}+ \psi_{121}\psi_{230})\nonumber\\&
+ \psi_{031}\psi_{320}(\psi_{111}\psi_{200}- \psi_{101}\psi_{210}+ \psi_{130}\psi_{221}- \psi_{133}\psi_{222}+
\psi_{132}\psi_{223}- \psi_{121}\psi_{230})\nonumber\\&
+ \psi_{000}\psi_{311}(\psi_{101}\psi_{210}-\psi_{110}\psi_{201}+ \psi_{103}\psi_{212}- \psi_{102}\psi_{213}-
\psi_{130}\psi_{221}+ \psi_{120}\psi_{231}) \nonumber\\&
+ \psi_{021}\psi_{330}( \psi_{101}\psi_{210}-\psi_{111}\psi_{200}- \psi_{131}\psi_{220}+ \psi_{120}\psi_{231}-
\psi_{123}\psi_{232}+ \psi_{122}\psi_{233})\nonumber\\&
+ \psi_{020}\psi_{331}( \psi_{100}\psi_{211}-\psi_{110}\psi_{201}- \psi_{130}\psi_{221}+ \psi_{121}\psi_{230}+
\psi_{123}\psi_{232}- \psi_{122}\psi_{233})\nonumber\\&
+ \psi_{030}\psi_{321}(\psi_{110}\psi_{201}- \psi_{100}\psi_{211}+ \psi_{131}\psi_{220}+ \psi_{133}\psi_{222}-
\psi_{132}\psi_{223}- \psi_{120}\psi_{231})\nonumber\\&
+ \psi_{003}\psi_{312}(\psi_{100}\psi_{211}-\psi_{113}\psi_{202}- \psi_{101}\psi_{210}+ \psi_{102}\psi_{213}-
\psi_{133}\psi_{222}+ \psi_{123}\psi_{232})\nonumber\\&
+ \psi_{033}\psi_{322}(\psi_{113}\psi_{202}- \psi_{103}\psi_{212}- \psi_{131}\psi_{220}+ \psi_{130}\psi_{221}+
\psi_{132}\psi_{223}- \psi_{123}\psi_{232}) \nonumber\\&
+ \psi_{002}\psi_{313}(\psi_{101}\psi_{210}-\psi_{112}\psi_{203}- \psi_{100}\psi_{211}+ \psi_{103}\psi_{212}-
\psi_{132}\psi_{223}+ \psi_{122}\psi_{233})\nonumber\\&
+ \psi_{023}\psi_{332}(\psi_{103}\psi_{212}-\psi_{113}\psi_{202}- \psi_{133}\psi_{222}- \psi_{121}\psi_{230}+
\psi_{120}\psi_{231}+ \psi_{122}\psi_{233}).
\end{align}
\normalsize
An example of a polynomial that is invariant under P only in Bob's and Charlie's labs is $I_{23a}$. Written out it is
\small
\begin{align}
I_{23a} = -4 (&\psi_{101}\psi_{110} - \psi_{100}\psi_{111}+ \psi_{103}\psi_{112}- \psi_{102}\psi_{113}+
\psi_{121}\psi_{130} - \psi_{120}\psi_{131}+ \psi_{123}\psi_{132}- \psi_{122}\psi_{133}\nonumber\\&+\psi_{301}\psi_{310} - \psi_{300}\psi_{311}+ \psi_{303}\psi_{312}- \psi_{302}\psi_{313}+
\psi_{321}\psi_{330} - \psi_{320}\psi_{331}+ \psi_{323}\psi_{332}- \psi_{322}\psi_{333})\nonumber\\\times (&\psi_{011}\psi_{200} -
\psi_{010}\psi_{201}+ \psi_{013}\psi_{202}- \psi_{012}\psi_{203}- \psi_{001}\psi_{210}+ \psi_{000}\psi_{211}-
\psi_{003}\psi_{212}+ \psi_{002}\psi_{213}\nonumber\\&+ \psi_{031}\psi_{220} - \psi_{030}\psi_{221}+ \psi_{033}\psi_{222}-
\psi_{032}\psi_{223}- \psi_{021}\psi_{230}+ \psi_{020}\psi_{231}- \psi_{023}\psi_{232}+ \psi_{022}\psi_{233})\nonumber\\ -
4 (&\psi_{001}\psi_{010} - \psi_{000}\psi_{011}+ \psi_{003}\psi_{012}- \psi_{002}\psi_{013}+ \psi_{021}\psi_{030} -
\psi_{020}\psi_{031}+ \psi_{023}\psi_{032}- \psi_{022}\psi_{033}\nonumber\\&+\psi_{201}\psi_{210} - \psi_{200}\psi_{211}+ \psi_{203}\psi_{212}- \psi_{202}\psi_{213}+ \psi_{221}\psi_{230} -
\psi_{220}\psi_{231}+ \psi_{223}\psi_{232}- \psi_{222}\psi_{233})\nonumber\\\times (&\psi_{111}\psi_{300} - \psi_{110}\psi_{301}+
\psi_{113}\psi_{302}- \psi_{112}\psi_{303}- \psi_{101}\psi_{310}+ \psi_{100}\psi_{311}- \psi_{103}\psi_{312}+
\psi_{102}\psi_{313}\nonumber\\&+ \psi_{131}\psi_{320} - \psi_{130}\psi_{321}+ \psi_{133}\psi_{322}- \psi_{132}\psi_{323}-
\psi_{121}\psi_{330}+ \psi_{120}\psi_{331}- \psi_{123}\psi_{332}+ \psi_{122}\psi_{333})\nonumber\\ -
2 (&\psi_{011}\psi_{100} - \psi_{010}\psi_{101}+ \psi_{013}\psi_{102}- \psi_{012}\psi_{103}- \psi_{001}\psi_{110}+
\psi_{000}\psi_{111}- \psi_{003}\psi_{112}+ \psi_{002}\psi_{113}\nonumber\\&+ \psi_{031}\psi_{120} - \psi_{030}\psi_{121}+
\psi_{033}\psi_{122}- \psi_{032}\psi_{123}- \psi_{021}\psi_{130}+ \psi_{020}\psi_{131}- \psi_{023}\psi_{132}+
\psi_{022}\psi_{133}\nonumber\\&+\psi_{211}\psi_{300} - \psi_{210}\psi_{301}+ \psi_{213}\psi_{302}- \psi_{212}\psi_{303}-
\psi_{201}\psi_{310}+ \psi_{200}\psi_{311}- \psi_{203}\psi_{312}+ \psi_{202}\psi_{313}\nonumber\\&+ \psi_{231}\psi_{320} -
\psi_{230}\psi_{321}+ \psi_{233}\psi_{322}- \psi_{232}\psi_{323}- \psi_{221}\psi_{330}+ \psi_{220}\psi_{331}-
\psi_{223}\psi_{332}+ \psi_{222}\psi_{333}) \nonumber\\ \times (&\psi_{111}\psi_{200} - \psi_{110}\psi_{201}+ \psi_{113}\psi_{202}- \psi_{112}\psi_{203}-
\psi_{101}\psi_{210}+ \psi_{100}\psi_{211}- \psi_{103}\psi_{212}+ \psi_{102}\psi_{213}\nonumber\\&+ \psi_{131}\psi_{220} -
\psi_{130}\psi_{221}+ \psi_{133}\psi_{222}- \psi_{132}\psi_{223}- \psi_{121}\psi_{230}+ \psi_{120}\psi_{231}-
\psi_{123}\psi_{232}+ \psi_{122}\psi_{233}\nonumber\\&+\psi_{011}\psi_{300} - \psi_{010}\psi_{301}+ \psi_{013}\psi_{302}- \psi_{012}\psi_{303}-
\psi_{001}\psi_{310}+ \psi_{000}\psi_{311}- \psi_{003}\psi_{312}+ \psi_{002}\psi_{313}\nonumber\\&+ \psi_{031}\psi_{320} -
\psi_{030}\psi_{321}+ \psi_{033}\psi_{322}- \psi_{032}\psi_{323}- \psi_{021}\psi_{330}+ \psi_{020}\psi_{331}-
\psi_{023}\psi_{332}+ \psi_{022}\psi_{333}).
\end{align}
\normalsize
An example of a polynomial that is invariant under P only in Alice's lab is $I_{35a}$. Written out it is
\footnotesize
\begin{align}
I_{35a} = -
4 (&\psi_{001}\psi_{010}- \psi_{000}\psi_{011}+ \psi_{003}\psi_{012}- \psi_{002}\psi_{013}+ \psi_{021}\psi_{030}-
\psi_{020}\psi_{031}+ \psi_{023}\psi_{032}- \psi_{022}\psi_{033})\nonumber\\\times(&\psi_{113}\psi_{120}- \psi_{112}\psi_{121}+
\psi_{111}\psi_{122}- \psi_{110}\psi_{123}- \psi_{103}\psi_{130}+ \psi_{102}\psi_{131}- \psi_{101}\psi_{132}+
\psi_{100}\psi_{133})\nonumber\\ -
4 (&\psi_{013}\psi_{020}- \psi_{012}\psi_{021}+ \psi_{011}\psi_{022}- \psi_{010}\psi_{023}- \psi_{003}\psi_{030}+
\psi_{002}\psi_{031}- \psi_{001}\psi_{032}+ \psi_{000}\psi_{033})\nonumber\\\times (&\psi_{101}\psi_{110}- \psi_{100}\psi_{111}+
\psi_{103}\psi_{112}- \psi_{102}\psi_{113}+ \psi_{121}\psi_{130}- \psi_{120}\psi_{131}+ \psi_{123}\psi_{132}-
\psi_{122}\psi_{133})\nonumber\\ -
4 (&\psi_{201}\psi_{210}- \psi_{200}\psi_{211}+ \psi_{203}\psi_{212}- \psi_{202}\psi_{213}+ \psi_{221}\psi_{230}-
\psi_{220}\psi_{231}+ \psi_{223}\psi_{232}- \psi_{222}\psi_{233})\nonumber\\\times (&\psi_{313}\psi_{320}- \psi_{312}\psi_{321}+
\psi_{311}\psi_{322}- \psi_{310}\psi_{323}- \psi_{303}\psi_{330}+ \psi_{302}\psi_{331}- \psi_{301}\psi_{332}+
\psi_{300}\psi_{333})\nonumber\\ -
4 (&\psi_{213}\psi_{220}- \psi_{212}\psi_{221}+ \psi_{211}\psi_{222}- \psi_{210}\psi_{223}- \psi_{203}\psi_{230}+
\psi_{202}\psi_{231}- \psi_{201}\psi_{232}+ \psi_{200}\psi_{233})\nonumber\\\times (&\psi_{301}\psi_{310}- \psi_{300}\psi_{311}+
\psi_{303}\psi_{312}- \psi_{302}\psi_{313}+ \psi_{321}\psi_{330}- \psi_{320}\psi_{331}+ \psi_{323}\psi_{332}-
\psi_{322}\psi_{333})\nonumber\\-2 (&\psi_{033}\psi_{100}- \psi_{032}\psi_{101}+ \psi_{031}\psi_{102}- \psi_{030}\psi_{103}-
\psi_{023}\psi_{110}+ \psi_{022}\psi_{111}- \psi_{021}\psi_{112}+ \psi_{020}\psi_{113}\nonumber\\&+ \psi_{013}\psi_{120}-
\psi_{012}\psi_{121}+ \psi_{011}\psi_{122}- \psi_{010}\psi_{123}- \psi_{003}\psi_{130}+ \psi_{002}\psi_{131}-
\psi_{001}\psi_{132}+ \psi_{000}\psi_{133})\nonumber\\\times (&\psi_{011}\psi_{100}- \psi_{010}\psi_{101}+ \psi_{013}\psi_{102}-
\psi_{012}\psi_{103}- \psi_{001}\psi_{110}+ \psi_{000}\psi_{111}- \psi_{003}\psi_{112}+ \psi_{002}\psi_{113}\nonumber\\&+
\psi_{031}\psi_{120}- \psi_{030}\psi_{121}+ \psi_{033}\psi_{122}- \psi_{032}\psi_{123}- \psi_{021}\psi_{130}+
\psi_{020}\psi_{131}- \psi_{023}\psi_{132}+ \psi_{022}\psi_{133})\nonumber\\ -
2 (&\psi_{111}\psi_{200}- \psi_{110}\psi_{201}+ \psi_{113}\psi_{202}- \psi_{112}\psi_{203}- \psi_{101}\psi_{210}+
\psi_{100}\psi_{211}- \psi_{103}\psi_{212}+ \psi_{102}\psi_{213}\nonumber\\&+ \psi_{131}\psi_{220}- \psi_{130}\psi_{221}+
\psi_{133}\psi_{222}- \psi_{132}\psi_{223}- \psi_{121}\psi_{230}+ \psi_{120}\psi_{231}- \psi_{123}\psi_{232}+
\psi_{122}\psi_{233})\nonumber\\\times (&\psi_{033}\psi_{300}- \psi_{032}\psi_{301}+ \psi_{031}\psi_{302}- \psi_{030}\psi_{303}-
\psi_{023}\psi_{310}+ \psi_{022}\psi_{311}- \psi_{021}\psi_{312}+ \psi_{020}\psi_{313}\nonumber\\&+ \psi_{013}\psi_{320}-
\psi_{012}\psi_{321}+ \psi_{011}\psi_{322}- \psi_{010}\psi_{323}- \psi_{003}\psi_{330}+ \psi_{002}\psi_{331}-
\psi_{001}\psi_{332}+ \psi_{000}\psi_{333})\nonumber\\ -
2 (&\psi_{133}\psi_{200}- \psi_{132}\psi_{201}+ \psi_{131}\psi_{202}- \psi_{130}\psi_{203}- \psi_{123}\psi_{210}+
\psi_{122}\psi_{211}- \psi_{121}\psi_{212}+ \psi_{120}\psi_{213}\nonumber\\&+ \psi_{113}\psi_{220}- \psi_{112}\psi_{221}+
\psi_{111}\psi_{222}- \psi_{110}\psi_{223}- \psi_{103}\psi_{230}+ \psi_{102}\psi_{231}- \psi_{101}\psi_{232}+
\psi_{100}\psi_{233})\nonumber\\\times (&\psi_{011}\psi_{300}- \psi_{010}\psi_{301}+ \psi_{013}\psi_{302}- \psi_{012}\psi_{303}-
\psi_{001}\psi_{310}+ \psi_{000}\psi_{311}- \psi_{003}\psi_{312}+ \psi_{002}\psi_{313}\nonumber\\&+ \psi_{031}\psi_{320}-
\psi_{030}\psi_{321}+ \psi_{033}\psi_{322}- \psi_{032}\psi_{323}- \psi_{021}\psi_{330}+ \psi_{020}\psi_{331}-
\psi_{023}\psi_{332}+ \psi_{022}\psi_{333})\nonumber\\ +
2 (&\psi_{011}\psi_{200}- \psi_{010}\psi_{201}+ \psi_{013}\psi_{202}- \psi_{012}\psi_{203}- \psi_{001}\psi_{210}+
\psi_{000}\psi_{211}- \psi_{003}\psi_{212}+ \psi_{002}\psi_{213}\nonumber\\&+ \psi_{031}\psi_{220}- \psi_{030}\psi_{221}+
\psi_{033}\psi_{222}- \psi_{032}\psi_{223}- \psi_{021}\psi_{230}+ \psi_{020}\psi_{231}- \psi_{023}\psi_{232}+
\psi_{022}\psi_{233})\nonumber\\\times (&\psi_{133}\psi_{300}- \psi_{132}\psi_{301}+ \psi_{131}\psi_{302}- \psi_{130}\psi_{303}-
\psi_{123}\psi_{310}+ \psi_{122}\psi_{311}- \psi_{121}\psi_{312}+ \psi_{120}\psi_{313}\nonumber\\&+ \psi_{113}\psi_{320}-
\psi_{112}\psi_{321}+ \psi_{111}\psi_{322}- \psi_{110}\psi_{323}- \psi_{103}\psi_{330}+ \psi_{102}\psi_{331}-
\psi_{101}\psi_{332}+ \psi_{100}\psi_{333})\nonumber\\ +
2 (&\psi_{033}\psi_{200}- \psi_{032}\psi_{201}+ \psi_{031}\psi_{202}- \psi_{030}\psi_{203}- \psi_{023}\psi_{210}+
\psi_{022}\psi_{211}- \psi_{021}\psi_{212}+ \psi_{020}\psi_{213}\nonumber\\&+ \psi_{013}\psi_{220}- \psi_{012}\psi_{221}+
\psi_{011}\psi_{222}- \psi_{010}\psi_{223}- \psi_{003}\psi_{230}+ \psi_{002}\psi_{231}- \psi_{001}\psi_{232}+
\psi_{000}\psi_{233})\nonumber\\\times (&\psi_{111}\psi_{300}- \psi_{110}\psi_{301}+ \psi_{113}\psi_{302}- \psi_{112}\psi_{303}-
\psi_{101}\psi_{310}+ \psi_{100}\psi_{311}- \psi_{103}\psi_{312}+ \psi_{102}\psi_{313}\nonumber\\&+ \psi_{131}\psi_{320}-
\psi_{130}\psi_{321}+ \psi_{133}\psi_{322}- \psi_{132}\psi_{323}- \psi_{121}\psi_{330}+ \psi_{120}\psi_{331}-
\psi_{123}\psi_{332}+ \psi_{122}\psi_{333})\nonumber\\ -
2 (&\psi_{233}\psi_{300}- \psi_{232}\psi_{301}+ \psi_{231}\psi_{302}- \psi_{230}\psi_{303}- \psi_{223}\psi_{310}+
\psi_{222}\psi_{311}- \psi_{221}\psi_{312}+ \psi_{220}\psi_{313}\nonumber\\&+ \psi_{213}\psi_{320}- \psi_{212}\psi_{321}+
\psi_{211}\psi_{322}- \psi_{210}\psi_{323}- \psi_{203}\psi_{330}+ \psi_{202}\psi_{331}- \psi_{201}\psi_{332}+
\psi_{200}\psi_{333})\nonumber\\\times (&\psi_{211}\psi_{300}- \psi_{210}\psi_{301}+ \psi_{213}\psi_{302}- \psi_{212}\psi_{303}-
\psi_{201}\psi_{310}+ \psi_{200}\psi_{311}- \psi_{203}\psi_{312}+ \psi_{202}\psi_{313}\nonumber\\&+ \psi_{231}\psi_{320}-
\psi_{230}\psi_{321}+ \psi_{233}\psi_{322}- \psi_{232}\psi_{323}- \psi_{221}\psi_{330}+ \psi_{220}\psi_{331}-
\psi_{223}\psi_{332}+ \psi_{222}\psi_{333}).
\end{align}
\normalsize
An example of a polynomial that is not invariant under P in any lab is $I_{11a}$. Written out it is
\footnotesize
\begin{align}
I_{11a} = -2 (&\psi_{101}\psi_{110}- \psi_{100}\psi_{111}+ \psi_{103}\psi_{112}- \psi_{102}\psi_{113}+
\psi_{121}\psi_{130}- \psi_{120}\psi_{131}+ \psi_{123}\psi_{132}- \psi_{122}\psi_{133}\nonumber\\& + \psi_{301}\psi_{310}-
\psi_{300}\psi_{311}+ \psi_{303}\psi_{312}- \psi_{302}\psi_{313}+ \psi_{321}\psi_{330}- \psi_{320}\psi_{331}+
\psi_{323}\psi_{332}- \psi_{322}\psi_{333})\nonumber\\\times (&\psi_{033}\psi_{200}- \psi_{032}\psi_{201}+ \psi_{031}\psi_{202}-
\psi_{030}\psi_{203}- \psi_{023}\psi_{210}+ \psi_{022}\psi_{211}- \psi_{021}\psi_{212}+ \psi_{020}\psi_{213}\nonumber\\&+
\psi_{013}\psi_{220}- \psi_{012}\psi_{221}+ \psi_{011}\psi_{222}- \psi_{010}\psi_{223}- \psi_{003}\psi_{230}+
\psi_{002}\psi_{231}- \psi_{001}\psi_{232}+ \psi_{000}\psi_{233})\nonumber\\ +
2 (&\psi_{113}\psi_{120}- \psi_{112}\psi_{121}+ \psi_{111}\psi_{122}- \psi_{110}\psi_{123}- \psi_{103}\psi_{130}+
\psi_{102}\psi_{131}- \psi_{101}\psi_{132}+ \psi_{100}\psi_{133}\nonumber\\& + \psi_{313}\psi_{320}- \psi_{312}\psi_{321}+
\psi_{311}\psi_{322}- \psi_{310}\psi_{323}- \psi_{303}\psi_{330}+ \psi_{302}\psi_{331}- \psi_{301}\psi_{332}+
\psi_{300}\psi_{333})\nonumber\\\times (&\psi_{011}\psi_{200}- \psi_{010}\psi_{201}+ \psi_{013}\psi_{202}- \psi_{012}\psi_{203}-
\psi_{001}\psi_{210}+ \psi_{000}\psi_{211}- \psi_{003}\psi_{212}+ \psi_{002}\psi_{213}\nonumber\\&+ \psi_{031}\psi_{220}-
\psi_{030}\psi_{221}+ \psi_{033}\psi_{222}- \psi_{032}\psi_{223}- \psi_{021}\psi_{230}+ \psi_{020}\psi_{231}-
\psi_{023}\psi_{232}+
\psi_{022}\psi_{233})\nonumber\\ -
2 (& \psi_{001}\psi_{010}- \psi_{000}\psi_{011}+ \psi_{003}\psi_{012}- \psi_{002}\psi_{013}+ \psi_{021}\psi_{030}-
\psi_{020}\psi_{031}+ \psi_{023}\psi_{032}- \psi_{022}\psi_{033}\nonumber\\& \psi_{201}\psi_{210}- \psi_{200}\psi_{211}+
\psi_{203}\psi_{212}- \psi_{202}\psi_{213}+ \psi_{221}\psi_{230}- \psi_{220}\psi_{231}+ \psi_{223}\psi_{232}-
\psi_{222}\psi_{233} )\nonumber\\\times (&\psi_{133}\psi_{300}- \psi_{132}\psi_{301}+ \psi_{131}\psi_{302}- \psi_{130}\psi_{303}-
\psi_{123}\psi_{310}+ \psi_{122}\psi_{311}- \psi_{121}\psi_{312}+ \psi_{120}\psi_{313}\nonumber\\&+ \psi_{113}\psi_{320}-
\psi_{112}\psi_{321}+ \psi_{111}\psi_{322}- \psi_{110}\psi_{323}- \psi_{103}\psi_{330}+ \psi_{102}\psi_{331}-
\psi_{101}\psi_{332}+ \psi_{100}\psi_{333})\nonumber\\ +
2 (& \psi_{013}\psi_{020}- \psi_{012}\psi_{021}+ \psi_{011}\psi_{022}- \psi_{010}\psi_{023}- \psi_{003}\psi_{030}+
\psi_{002}\psi_{031}- \psi_{001}\psi_{032}+ \psi_{000}\psi_{033}\nonumber\\& + \psi_{213}\psi_{220}- \psi_{212}\psi_{221}+
\psi_{211}\psi_{222}- \psi_{210}\psi_{223}- \psi_{203}\psi_{230}+ \psi_{202}\psi_{231}- \psi_{201}\psi_{232}+
\psi_{200}\psi_{233} )\nonumber\\\times (&\psi_{111}\psi_{300}- \psi_{110}\psi_{301}+ \psi_{113}\psi_{302}- \psi_{112}\psi_{303}-
\psi_{101}\psi_{310}+ \psi_{100}\psi_{311}- \psi_{103}\psi_{312}+ \psi_{102}\psi_{313}\nonumber\\&+ \psi_{131}\psi_{320}-
\psi_{130}\psi_{321}+ \psi_{133}\psi_{322}- \psi_{132}\psi_{323}- \psi_{121}\psi_{330}+ \psi_{120}\psi_{331}-
\psi_{123}\psi_{332}+ \psi_{122}\psi_{333})\nonumber\\ - (&\psi_{011}\psi_{100}- \psi_{010}\psi_{101}+ \psi_{013}\psi_{102}- \psi_{012}\psi_{103}-
\psi_{001}\psi_{110}+ \psi_{000}\psi_{111}- \psi_{003}\psi_{112}+ \psi_{002}\psi_{113}\nonumber\\&+ \psi_{031}\psi_{120}-
\psi_{030}\psi_{121}+ \psi_{033}\psi_{122}- \psi_{032}\psi_{123}- \psi_{021}\psi_{130}+ \psi_{020}\psi_{131}-
\psi_{023}\psi_{132}+ \psi_{022}\psi_{133}\nonumber\\& + \psi_{211}\psi_{300}- \psi_{210}\psi_{301}+ \psi_{213}\psi_{302}-
\psi_{212}\psi_{303}- \psi_{201}\psi_{310}+ \psi_{200}\psi_{311}- \psi_{203}\psi_{312}+ \psi_{202}\psi_{313}\nonumber\\&+
\psi_{231}\psi_{320}- \psi_{230}\psi_{321}+ \psi_{233}\psi_{322}- \psi_{232}\psi_{323}- \psi_{221}\psi_{330}+
\psi_{220}\psi_{331}- \psi_{223}\psi_{332}+ \psi_{222}\psi_{333}) \nonumber\\\times (&\psi_{133}\psi_{200}- \psi_{132}\psi_{201}+
\psi_{131}\psi_{202}- \psi_{130}\psi_{203}- \psi_{123}\psi_{210}+ \psi_{122}\psi_{211}- \psi_{121}\psi_{212}+
\psi_{120}\psi_{213}\nonumber\\&+ \psi_{113}\psi_{220}- \psi_{112}\psi_{221}+ \psi_{111}\psi_{222}- \psi_{110}\psi_{223}-
\psi_{103}\psi_{230}+ \psi_{102}\psi_{231}- \psi_{101}\psi_{232}+ \psi_{100}\psi_{233}\nonumber\\& + \psi_{033}\psi_{300}-
\psi_{032}\psi_{301}+ \psi_{031}\psi_{302}- \psi_{030}\psi_{303}- \psi_{023}\psi_{310}+ \psi_{022}\psi_{311}-
\psi_{021}\psi_{312}+ \psi_{020}\psi_{313}\nonumber\\&+ \psi_{013}\psi_{320}- \psi_{012}\psi_{321}+ \psi_{011}\psi_{322}-
\psi_{010}\psi_{323}- \psi_{003}\psi_{330}+ \psi_{002}\psi_{331}- \psi_{001}\psi_{332}+
\psi_{000}\psi_{333})\nonumber\\ - (&\psi_{033}\psi_{100}- \psi_{032}\psi_{101}+ \psi_{031}\psi_{102}- \psi_{030}\psi_{103}-
\psi_{023}\psi_{110}+ \psi_{022}\psi_{111}- \psi_{021}\psi_{112}+ \psi_{020}\psi_{113}\nonumber\\&+ \psi_{013}\psi_{120}-
\psi_{012}\psi_{121}+ \psi_{011}\psi_{122}- \psi_{010}\psi_{123}- \psi_{003}\psi_{130}+ \psi_{002}\psi_{131}-
\psi_{001}\psi_{132}+ \psi_{000}\psi_{133}\nonumber\\& + \psi_{233}\psi_{300}- \psi_{232}\psi_{301}+ \psi_{231}\psi_{302}-
\psi_{230}\psi_{303}- \psi_{223}\psi_{310}+ \psi_{222}\psi_{311}- \psi_{221}\psi_{312}+ \psi_{220}\psi_{313}\nonumber\\&+
\psi_{213}\psi_{320}- \psi_{212}\psi_{321}+ \psi_{211}\psi_{322}- \psi_{210}\psi_{323}- \psi_{203}\psi_{330}+
\psi_{202}\psi_{331}- \psi_{201}\psi_{332}+ \psi_{200}\psi_{333})\nonumber\\\times ( & \psi_{111}\psi_{200}- \psi_{110}\psi_{201}+
\psi_{113}\psi_{202}- \psi_{112}\psi_{203}- \psi_{101}\psi_{210}+ \psi_{100}\psi_{211}- \psi_{103}\psi_{212}+
\psi_{102}\psi_{213}\nonumber\\&+ \psi_{131}\psi_{220}- \psi_{130}\psi_{221}+ \psi_{133}\psi_{222}- \psi_{132}\psi_{223}-
\psi_{121}\psi_{230}+ \psi_{120}\psi_{231}- \psi_{123}\psi_{232}+ \psi_{122}\psi_{233}\nonumber\\& + \psi_{011}\psi_{300}-
\psi_{010}\psi_{301}+ \psi_{013}\psi_{302}- \psi_{012}\psi_{303}- \psi_{001}\psi_{310}+ \psi_{000}\psi_{311}-
\psi_{003}\psi_{312}+ \psi_{002}\psi_{313}\nonumber\\&+ \psi_{031}\psi_{320}- \psi_{030}\psi_{321}+ \psi_{033}\psi_{322}-
\psi_{032}\psi_{323}- \psi_{021}\psi_{330}+ \psi_{020}\psi_{331}- \psi_{023}\psi_{332}+ \psi_{022}\psi_{333}).
\end{align}
\normalsize
Examples of the degree 2 polynomials for four Dirac spinors are $H_a,H_b,H_c$ and $H_d$. They are given by
\footnotesize
\begin{align}
H_a =2(&-\psi_{ 0111}\psi_{ 1000} + \psi_{ 0110}\psi_{ 1001} - \psi_{ 0113}\psi_{ 1002} +
\psi_{ 0112}\psi_{ 1003} + \psi_{ 0101}\psi_{ 1010} - \psi_{ 0100}\psi_{ 1011} + \psi_{ 0103}\psi_{ 1012} -
\psi_{ 0102}\psi_{ 1013}\nonumber\\& - \psi_{ 0131}\psi_{ 1020 }+ \psi_{ 0130}\psi_{ 1021} - \psi_{ 0133}\psi_{ 1022 }+
\psi_{ 0132}\psi_{ 1023} + \psi_{0121} \psi_{1030} - \psi_{0120} \psi_{1031} + \psi_{0123} \psi_{1032} -
\psi_{0122} \psi_{1033}\nonumber\\& + \psi_{0011} \psi_{1100} - \psi_{0010 }\psi_{1101} + \psi_{0013} \psi_{1102} -
\psi_{0012} \psi_{1103} - \psi_{0001} \psi_{1110 }+ \psi_{0000} \psi_{1111 }- \psi_{0003} \psi_{1112} +
\psi_{ 0002}\psi_{ 1113}\nonumber\\& + \psi_{ 0031}\psi_{ 1120} - \psi_{ 0030 }\psi_{ 1121} + \psi_{ 0033}\psi_{ 1122} -
\psi_{ 0032}\psi_{ 1123 }- \psi_{ 0021}\psi_{ 1130} + \psi_{ 0020}\psi_{ 1131} - \psi_{ 0023}\psi_{ 1132} +
\psi_{ 0022}\psi_{ 1133}\nonumber\\& - \psi_{0311} \psi_{1200} + \psi_{0310 }\psi_{1201} - \psi_{0313 }\psi_{1202} +
\psi_{0312} \psi_{1203} + \psi_{0301} \psi_{1210} - \psi_{0300} \psi_{1211} + \psi_{0303} \psi_{1212} -
\psi_{0302} \psi_{1213}\nonumber\\& - \psi_{0331 }\psi_{1220} + \psi_{0330} \psi_{1221} - \psi_{0333} \psi_{1222} +
\psi_{0332} \psi_{1223} + \psi_{0321} \psi_{1230} - \psi_{0320 }\psi_{1231 }+ \psi_{0323 }\psi_{1232 }-
\psi_{0322} \psi_{1233}\nonumber\\& + \psi_{0211} \psi_{1300} - \psi_{0210 }\psi_{1301} + \psi_{0213 }\psi_{1302} -
\psi_{0212} \psi_{1303} - \psi_{0201} \psi_{1310} + \psi_{0200} \psi_{1311} - \psi_{0203 }\psi_{1312} +
\psi_{0202} \psi_{1313}\nonumber\\& + \psi_{0231} \psi_{1320} - \psi_{0230} \psi_{1321 }+ \psi_{0233} \psi_{1322} -
\psi_{0232} \psi_{1323} - \psi_{0221} \psi_{1330 }+ \psi_{0220} \psi_{1331 }- \psi_{0223 }\psi_{1332} +
\psi_{0222} \psi_{1333}\nonumber\\& - \psi_{2111} \psi_{3000} + \psi_{2110} \psi_{3001 }- \psi_{2113 }\psi_{3002} +
\psi_{2112} \psi_{3003} + \psi_{2101} \psi_{3010} - \psi_{2100} \psi_{3011} + \psi_{2103} \psi_{3012} -
\psi_{2102} \psi_{3013}\nonumber\\& - \psi_{2131} \psi_{3020} + \psi_{2130 }\psi_{3021} - \psi_{2133} \psi_{3022} +
\psi_{2132} \psi_{3023} + \psi_{2121} \psi_{3030} - \psi_{2120} \psi_{3031 }+ \psi_{2123} \psi_{3032} -
\psi_{2122} \psi_{3033}\nonumber\\& + \psi_{2011} \psi_{3100} - \psi_{2010 }\psi_{3101 }+ \psi_{2013 }\psi_{3102} -
\psi_{2012} \psi_{3103} - \psi_{2001} \psi_{3110} + \psi_{2000 }\psi_{3111} - \psi_{2003} \psi_{3112} +
\psi_{2002} \psi_{3113}\nonumber\\& + \psi_{2031} \psi_{3120} - \psi_{2030} \psi_{3121} + \psi_{2033 }\psi_{3122 }-
\psi_{2032} \psi_{3123} - \psi_{2021} \psi_{3130 }+ \psi_{2020} \psi_{3131 }- \psi_{2023} \psi_{3132 }+
\psi_{2022} \psi_{3133}\nonumber\\& - \psi_{2311} \psi_{3200} + \psi_{2310} \psi_{3201} - \psi_{2313} \psi_{3202} +
\psi_{2312} \psi_{3203} + \psi_{2301} \psi_{3210 }- \psi_{2300 }\psi_{3211} + \psi_{2303} \psi_{3212} -
\psi_{2302} \psi_{3213}\nonumber\\& - \psi_{2331} \psi_{3220} + \psi_{2330} \psi_{3221} - \psi_{2333} \psi_{3222} +
\psi_{2332} \psi_{3223} + \psi_{2321} \psi_{3230 }- \psi_{2320} \psi_{3231} + \psi_{2323 }\psi_{3232} -
\psi_{2322} \psi_{3233}\nonumber\\& + \psi_{2211} \psi_{3300} - \psi_{2210} \psi_{3301} + \psi_{2213} \psi_{3302} -
\psi_{2212} \psi_{3303} - \psi_{2201} \psi_{3310} + \psi_{2200 }\psi_{3311 }- \psi_{2203 }\psi_{3312} +
\psi_{2202} \psi_{3313}\nonumber\\& + \psi_{2231} \psi_{3320} - \psi_{2230} \psi_{3321} + \psi_{2233} \psi_{3322} -
\psi_{2232} \psi_{3323} - \psi_{2221} \psi_{3330} + \psi_{2220 }\psi_{3331} - \psi_{2223 }\psi_{3332 }+
\psi_{2222 }\psi_{3333}),
\end{align}
\begin{align}
H_b =2( &\psi_{1333}\psi_{2000}- \psi_{1332}\psi_{2001}+ \psi_{1331}\psi_{2002}-
\psi_{1330}\psi_{2003}- \psi_{1323}\psi_{2010}+ \psi_{1322}\psi_{2011}- \psi_{1321}\psi_{2012}+
\psi_{1320}\psi_{2013}\nonumber\\&+ \psi_{1313}\psi_{2020}- \psi_{1312}\psi_{2021}+ \psi_{1311}\psi_{2022}-
\psi_{1310}\psi_{2023}- \psi_{1303}\psi_{2030}+ \psi_{1302}\psi_{2031}- \psi_{1301}\psi_{2032}+
\psi_{1300}\psi_{2033}\nonumber\\&- \psi_{1233}\psi_{2100}+ \psi_{1232}\psi_{2101}- \psi_{1231}\psi_{2102}+
\psi_{1230}\psi_{2103}+ \psi_{1223}\psi_{2110}- \psi_{1222}\psi_{2111}+ \psi_{1221}\psi_{2112}-
\psi_{1220}\psi_{2113}\nonumber\\&- \psi_{1213}\psi_{2120}+ \psi_{1212}\psi_{2121}- \psi_{1211}\psi_{2122}+
\psi_{1210}\psi_{2123}+ \psi_{1203}\psi_{2130}- \psi_{1202}\psi_{2131}+ \psi_{1201}\psi_{2132}-
\psi_{1200}\psi_{2133}\nonumber\\&+ \psi_{1133}\psi_{2200}- \psi_{1132}\psi_{2201}+ \psi_{1131}\psi_{2202}-
\psi_{1130}\psi_{2203}- \psi_{1123}\psi_{2210}+ \psi_{1122}\psi_{2211}- \psi_{1121}\psi_{2212}+
\psi_{1120}\psi_{2213}\nonumber\\&+ \psi_{1113}\psi_{2220}- \psi_{1112}\psi_{2221}+ \psi_{1111}\psi_{2222}-
\psi_{1110}\psi_{2223}- \psi_{1103}\psi_{2230}+ \psi_{1102}\psi_{2231}- \psi_{1101}\psi_{2232}+
\psi_{1100}\psi_{2233}\nonumber\\&- \psi_{1033}\psi_{2300}+ \psi_{1032}\psi_{2301}- \psi_{1031}\psi_{2302}+
\psi_{1030}\psi_{2303}+ \psi_{1023}\psi_{2310}- \psi_{1022}\psi_{2311}+ \psi_{1021}\psi_{2312}-
\psi_{1020}\psi_{2313}\nonumber\\&- \psi_{1013}\psi_{2320}+ \psi_{1012}\psi_{2321}- \psi_{1011}\psi_{2322}+
\psi_{1010}\psi_{2323}+ \psi_{1003}\psi_{2330}- \psi_{1002}\psi_{2331}+ \psi_{1001}\psi_{2332}-
\psi_{1000}\psi_{2333}\nonumber\\&- \psi_{0333}\psi_{3000}+ \psi_{0332}\psi_{3001}- \psi_{0331}\psi_{3002}+
\psi_{0330}\psi_{3003}+ \psi_{0323}\psi_{3010}- \psi_{0322}\psi_{3011}+ \psi_{0321}\psi_{3012}-
\psi_{0320}\psi_{3013}\nonumber\\&- \psi_{0313}\psi_{3020}+ \psi_{0312}\psi_{3021}- \psi_{0311}\psi_{3022}+
\psi_{0310}\psi_{3023}+ \psi_{0303}\psi_{3030}- \psi_{0302}\psi_{3031}+ \psi_{0301}\psi_{3032}-
\psi_{0300}\psi_{3033}\nonumber\\&+ \psi_{0233}\psi_{3100}- \psi_{0232}\psi_{3101}+ \psi_{0231}\psi_{3102}-
\psi_{0230}\psi_{3103}- \psi_{0223}\psi_{3110}+ \psi_{0222}\psi_{3111}- \psi_{0221}\psi_{3112}+
\psi_{0220}\psi_{3113}\nonumber\\&+ \psi_{0213}\psi_{3120}- \psi_{0212}\psi_{3121}+ \psi_{0211}\psi_{3122}-
\psi_{0210}\psi_{3123}- \psi_{0203}\psi_{3130}+ \psi_{0202}\psi_{3131}- \psi_{0201}\psi_{3132}+
\psi_{0200}\psi_{3133}\nonumber\\&- \psi_{0133}\psi_{3200}+ \psi_{0132}\psi_{3201}- \psi_{0131}\psi_{3202}+
\psi_{0130}\psi_{3203}+ \psi_{0123}\psi_{3210}- \psi_{0122}\psi_{3211}+ \psi_{0121}\psi_{3212}-
\psi_{0120}\psi_{3213}\nonumber\\&- \psi_{0113}\psi_{3220}+ \psi_{0112}\psi_{3221}- \psi_{0111}\psi_{3222}+
\psi_{0110}\psi_{3223}+ \psi_{0103}\psi_{3230}- \psi_{0102}\psi_{3231}+ \psi_{0101}\psi_{3232}-
\psi_{0100}\psi_{3233}\nonumber\\&+ \psi_{0033}\psi_{3300}- \psi_{0032}\psi_{3301}+ \psi_{0031}\psi_{3302}-
\psi_{0030}\psi_{3303}- \psi_{0023}\psi_{3310}+ \psi_{0022}\psi_{3311}- \psi_{0021}\psi_{3312}+
\psi_{0020}\psi_{3313}\nonumber\\&+ \psi_{0013}\psi_{3320}- \psi_{0012}\psi_{3321}+ \psi_{0011}\psi_{3322}-
\psi_{0010}\psi_{3323}- \psi_{0003}\psi_{3330}+ \psi_{0002}\psi_{3331}- \psi_{0001}\psi_{3332}+
\psi_{0000}\psi_{3333}),
\end{align}
\begin{align}
H_c = 2(&-\psi_{0113}\psi_{1000}+ \psi_{0112}\psi_{1001}- \psi_{0111}\psi_{1002}+
\psi_{0110}\psi_{1003}+ \psi_{0103}\psi_{1010}- \psi_{0102}\psi_{1011}+ \psi_{0101}\psi_{1012}-
\psi_{0100}\psi_{1013}\nonumber\\&- \psi_{0133}\psi_{1020}+ \psi_{0132}\psi_{1021}- \psi_{0131}\psi_{1022}+
\psi_{0130}\psi_{1023}+ \psi_{0123}\psi_{1030}- \psi_{0122}\psi_{1031}+ \psi_{0121}\psi_{1032}-
\psi_{0120}\psi_{1033}\nonumber\\&+ \psi_{0013}\psi_{1100}- \psi_{0012}\psi_{1101}+ \psi_{0011}\psi_{1102}-
\psi_{0010}\psi_{1103}- \psi_{0003}\psi_{1110}+ \psi_{0002}\psi_{1111}- \psi_{0001}\psi_{1112}+
\psi_{0000}\psi_{1113}\nonumber\\&+ \psi_{0033}\psi_{1120}- \psi_{0032}\psi_{1121}+ \psi_{0031}\psi_{1122}-
\psi_{0030}\psi_{1123}- \psi_{0023}\psi_{1130}+ \psi_{0022}\psi_{1131}- \psi_{0021}\psi_{1132}+
\psi_{0020}\psi_{1133}\nonumber\\&- \psi_{0313}\psi_{1200}+ \psi_{0312}\psi_{1201}- \psi_{0311}\psi_{1202}+
\psi_{0310}\psi_{1203}+ \psi_{0303}\psi_{1210}- \psi_{0302}\psi_{1211}+ \psi_{0301}\psi_{1212}-
\psi_{0300}\psi_{1213}\nonumber\\&- \psi_{0333}\psi_{1220}+ \psi_{0332}\psi_{1221}- \psi_{0331}\psi_{1222}+
\psi_{0330}\psi_{1223}+ \psi_{0323}\psi_{1230}- \psi_{0322}\psi_{1231}+ \psi_{0321}\psi_{1232}-
\psi_{0320}\psi_{1233}\nonumber\\&+ \psi_{0213}\psi_{1300}- \psi_{0212}\psi_{1301}+ \psi_{0211}\psi_{1302}-
\psi_{0210}\psi_{1303}- \psi_{0203}\psi_{1310}+ \psi_{0202}\psi_{1311}- \psi_{0201}\psi_{1312}+
\psi_{0200}\psi_{1313}\nonumber\\&+ \psi_{0233}\psi_{1320}- \psi_{0232}\psi_{1321}+ \psi_{0231}\psi_{1322}-
\psi_{0230}\psi_{1323}- \psi_{0223}\psi_{1330}+ \psi_{0222}\psi_{1331}- \psi_{0221}\psi_{1332}+
\psi_{0220}\psi_{1333}\nonumber\\&- \psi_{2113}\psi_{3000}+ \psi_{2112}\psi_{3001}- \psi_{2111}\psi_{3002}+
\psi_{2110}\psi_{3003}+ \psi_{2103}\psi_{3010}- \psi_{2102}\psi_{3011}+ \psi_{2101}\psi_{3012}-
\psi_{2100}\psi_{3013}\nonumber\\&- \psi_{2133}\psi_{3020}+ \psi_{2132}\psi_{3021}- \psi_{2131}\psi_{3022}+
\psi_{2130}\psi_{3023}+ \psi_{2123}\psi_{3030}- \psi_{2122}\psi_{3031}+ \psi_{2121}\psi_{3032}-
\psi_{2120}\psi_{3033}\nonumber\\&+ \psi_{2013}\psi_{3100}- \psi_{2012}\psi_{3101}+ \psi_{2011}\psi_{3102}-
\psi_{2010}\psi_{3103}- \psi_{2003}\psi_{3110}+ \psi_{2002}\psi_{3111}- \psi_{2001}\psi_{3112}+
\psi_{2000}\psi_{3113}\nonumber\\&+ \psi_{2033}\psi_{3120}- \psi_{2032}\psi_{3121}+ \psi_{2031}\psi_{3122}-
\psi_{2030}\psi_{3123}- \psi_{2023}\psi_{3130}+ \psi_{2022}\psi_{3131}- \psi_{2021}\psi_{3132}+
\psi_{2020}\psi_{3133}\nonumber\\&- \psi_{2313}\psi_{3200}+ \psi_{2312}\psi_{3201}- \psi_{2311}\psi_{3202}+
\psi_{2310}\psi_{3203}+ \psi_{2303}\psi_{3210}- \psi_{2302}\psi_{3211}+ \psi_{2301}\psi_{3212}-
\psi_{2300}\psi_{3213}\nonumber\\&- \psi_{2333}\psi_{3220}+ \psi_{2332}\psi_{3221}- \psi_{2331}\psi_{3222}+
\psi_{2330}\psi_{3223}+ \psi_{2323}\psi_{3230}- \psi_{2322}\psi_{3231}+ \psi_{2321}\psi_{3232}-
\psi_{2320}\psi_{3233}\nonumber\\&+ \psi_{2213}\psi_{3300}- \psi_{2212}\psi_{3301}+ \psi_{2211}\psi_{3302}-
\psi_{2210}\psi_{3303}- \psi_{2203}\psi_{3310}+ \psi_{2202}\psi_{3311}- \psi_{2201}\psi_{3312}+
\psi_{2200}\psi_{3313}\nonumber\\&+ \psi_{2233}\psi_{3320}- \psi_{2232}\psi_{3321}+ \psi_{2231}\psi_{3322}-
\psi_{2230}\psi_{3323}- \psi_{2223}\psi_{3330}+ \psi_{2222}\psi_{3331}- \psi_{2221}\psi_{3332}+
\psi_{2220}\psi_{3333}),
\end{align}
\normalsize
and
\footnotesize
\begin{align}
H_d = 2(&- \psi_{0131}\psi_{1000}+\psi_{0130}\psi_{1001}-\psi_{0133}\psi_{1002}+
\psi_{0132}\psi_{1003}+\psi_{0121}\psi_{1010}-\psi_{0120}\psi_{1011}+\psi_{0123}\psi_{1012}-
\psi_{0122}\psi_{1013}\nonumber\\&-\psi_{0111}\psi_{1020}+\psi_{0110}\psi_{1021}-\psi_{0113}\psi_{1022}+
\psi_{0112}\psi_{1023}+\psi_{0101}\psi_{1030}-\psi_{0100}\psi_{1031}+\psi_{0103}\psi_{1032}-
\psi_{0102}\psi_{1033}\nonumber\\&+\psi_{0031}\psi_{1100}-\psi_{0030}\psi_{1101}+\psi_{0033}\psi_{1102}-
\psi_{0032}\psi_{1103}-\psi_{0021}\psi_{1110}+\psi_{0020}\psi_{1111}-\psi_{0023}\psi_{1112}+
\psi_{0022}\psi_{1113}\nonumber\\&+\psi_{0011}\psi_{1120}-\psi_{0010}\psi_{1121}+\psi_{0013}\psi_{1122}-
\psi_{0012}\psi_{1123}-\psi_{0001}\psi_{1130}+\psi_{0000}\psi_{1131}-\psi_{0003}\psi_{1132}+
\psi_{0002}\psi_{1133}\nonumber\\&-\psi_{0331}\psi_{1200}+\psi_{0330}\psi_{1201}-\psi_{0333}\psi_{1202}+
\psi_{0332}\psi_{1203}+\psi_{0321}\psi_{1210}-\psi_{0320}\psi_{1211}+\psi_{0323}\psi_{1212}-
\psi_{0322}\psi_{1213}\nonumber\\&-\psi_{0311}\psi_{1220}+\psi_{0310}\psi_{1221}-\psi_{0313}\psi_{1222}+
\psi_{0312}\psi_{1223}+\psi_{0301}\psi_{1230}-\psi_{0300}\psi_{1231}+\psi_{0303}\psi_{1232}-
\psi_{0302}\psi_{1233}\nonumber\\&+\psi_{0231}\psi_{1300}-\psi_{0230}\psi_{1301}+\psi_{0233}\psi_{1302}-
\psi_{0232}\psi_{1303}-\psi_{0221}\psi_{1310}+\psi_{0220}\psi_{1311}-\psi_{0223}\psi_{1312}+
\psi_{0222}\psi_{1313}\nonumber\\&+\psi_{0211}\psi_{1320}-\psi_{0210}\psi_{1321}+\psi_{0213}\psi_{1322}-
\psi_{0212}\psi_{1323}-\psi_{0201}\psi_{1330}+\psi_{0200}\psi_{1331}-\psi_{0203}\psi_{1332}+
\psi_{0202}\psi_{1333}\nonumber\\&-\psi_{2131}\psi_{3000}+\psi_{2130}\psi_{3001}-\psi_{2133}\psi_{3002}+
\psi_{2132}\psi_{3003}+\psi_{2121}\psi_{3010}-\psi_{2120}\psi_{3011}+\psi_{2123}\psi_{3012}-
\psi_{2122}\psi_{3013}\nonumber\\&-\psi_{2111}\psi_{3020}+\psi_{2110}\psi_{3021}-\psi_{2113}\psi_{3022}+
\psi_{2112}\psi_{3023}+\psi_{2101}\psi_{3030}-\psi_{2100}\psi_{3031}+\psi_{2103}\psi_{3032}-
\psi_{2102}\psi_{3033}\nonumber\\&+\psi_{2031}\psi_{3100}-\psi_{2030}\psi_{3101}+\psi_{2033}\psi_{3102}-
\psi_{2032}\psi_{3103}-\psi_{2021}\psi_{3110}+\psi_{2020}\psi_{3111}-\psi_{2023}\psi_{3112}+
\psi_{2022}\psi_{3113}\nonumber\\&+\psi_{2011}\psi_{3120}-\psi_{2010}\psi_{3121}+\psi_{2013}\psi_{3122}-
\psi_{2012}\psi_{3123}-\psi_{2001}\psi_{3130}+\psi_{2000}\psi_{3131}-\psi_{2003}\psi_{3132}+
\psi_{2002}\psi_{3133}\nonumber\\&-\psi_{2331}\psi_{3200}+\psi_{2330}\psi_{3201}-\psi_{2333}\psi_{3202}+
\psi_{2332}\psi_{3203}+\psi_{2321}\psi_{3210}-\psi_{2320}\psi_{3211}+\psi_{2323}\psi_{3212}-
\psi_{2322}\psi_{3213}\nonumber\\&-\psi_{2311}\psi_{3220}+\psi_{2310}\psi_{3221}-\psi_{2313}\psi_{3222}+
\psi_{2312}\psi_{3223}+\psi_{2301}\psi_{3230}-\psi_{2300}\psi_{3231}+\psi_{2303}\psi_{3232}-
\psi_{2302}\psi_{3233}\nonumber\\&+\psi_{2231}\psi_{3300}-\psi_{2230}\psi_{3301}+\psi_{2233}\psi_{3302}-
\psi_{2232}\psi_{3303}-\psi_{2221}\psi_{3310}+\psi_{2220}\psi_{3311}-\psi_{2223}\psi_{3312}+
\psi_{2222}\psi_{3313}\nonumber\\&+\psi_{2211}\psi_{3320}-\psi_{2210}\psi_{3321}+\psi_{2213}\psi_{3322}-
\psi_{2212}\psi_{3323}-\psi_{2201}\psi_{3330}+\psi_{2200}\psi_{3331}-\psi_{2203}\psi_{3332}+
\psi_{2202}\psi_{3333}).
\end{align}
\normalsize
Examples of the degree four polynomials for four Dirac spinors are $T_l$ and $Y_l$. These are given by
\footnotesize
\begin{align}
T_l= 2 ( &\psi_{0111}\psi_{1000}- \psi_{0110}\psi_{1001}+ \psi_{0113}\psi_{1002}- \psi_{0112}\psi_{1003}-
\psi_{0101}\psi_{1010}+ \psi_{0100}\psi_{1011}- \psi_{0103}\psi_{1012}+ \psi_{0102}\psi_{1013}\nonumber\\&-
\psi_{0011}\psi_{1100}+ \psi_{0010}\psi_{1101}- \psi_{0013}\psi_{1102}+ \psi_{0012}\psi_{1103}+
\psi_{0001}\psi_{1110}- \psi_{0000}\psi_{1111}+ \psi_{0003}\psi_{1112}- \psi_{0002}\psi_{1113}\nonumber\\&+
\psi_{0311}\psi_{1200}- \psi_{0310}\psi_{1201}+ \psi_{0313}\psi_{1202}- \psi_{0312}\psi_{1203}-
\psi_{0301}\psi_{1210}+ \psi_{0300}\psi_{1211}- \psi_{0303}\psi_{1212}+ \psi_{0302}\psi_{1213}\nonumber\\&-
\psi_{0211}\psi_{1300}+ \psi_{0210}\psi_{1301}- \psi_{0213}\psi_{1302}+ \psi_{0212}\psi_{1303}+
\psi_{0201}\psi_{1310}- \psi_{0200}\psi_{1311}+ \psi_{0203}\psi_{1312}- \psi_{0202}\psi_{1313}\nonumber\\&+
\psi_{2111}\psi_{3000}- \psi_{2110}\psi_{3001}+ \psi_{2113}\psi_{3002}- \psi_{2112}\psi_{3003}-
\psi_{2101}\psi_{3010}+ \psi_{2100}\psi_{3011}- \psi_{2103}\psi_{3012}+ \psi_{2102}\psi_{3013}\nonumber\\&-
\psi_{2011}\psi_{3100}+ \psi_{2010}\psi_{3101}- \psi_{2013}\psi_{3102}+ \psi_{2012}\psi_{3103}+
\psi_{2001}\psi_{3110}- \psi_{2000}\psi_{3111}+ \psi_{2003}\psi_{3112}- \psi_{2002}\psi_{3113}\nonumber\\&+
\psi_{2311}\psi_{3200}- \psi_{2310}\psi_{3201}+ \psi_{2313}\psi_{3202}- \psi_{2312}\psi_{3203}-
\psi_{2301}\psi_{3210}+ \psi_{2300}\psi_{3211}- \psi_{2303}\psi_{3212}+ \psi_{2302}\psi_{3213}\nonumber\\&-
\psi_{2211}\psi_{3300}+ \psi_{2210}\psi_{3301}- \psi_{2213}\psi_{3302}+ \psi_{2212}\psi_{3303}+
\psi_{2201}\psi_{3310}- \psi_{2200}\psi_{3311}+ \psi_{2203}\psi_{3312}-
\psi_{2202}\psi_{3313})^2\nonumber\\ +
2 ( &\psi_{0131}\psi_{1020}- \psi_{0130}\psi_{1021}+ \psi_{0133}\psi_{1022}- \psi_{0132}\psi_{1023}-
\psi_{0121}\psi_{1030}+ \psi_{0120}\psi_{1031}- \psi_{0123}\psi_{1032}+ \psi_{0122}\psi_{1033}\nonumber\\&-
\psi_{0031}\psi_{1120}+ \psi_{0030}\psi_{1121}- \psi_{0033}\psi_{1122}+ \psi_{0032}\psi_{1123}+
\psi_{0021}\psi_{1130}- \psi_{0020}\psi_{1131}+ \psi_{0023}\psi_{1132}- \psi_{0022}\psi_{1133}\nonumber\\&+
\psi_{0331}\psi_{1220}- \psi_{0330}\psi_{1221}+ \psi_{0333}\psi_{1222}- \psi_{0332}\psi_{1223}-
\psi_{0321}\psi_{1230}+ \psi_{0320}\psi_{1231}- \psi_{0323}\psi_{1232}+ \psi_{0322}\psi_{1233}\nonumber\\&-
\psi_{0231}\psi_{1320}+ \psi_{0230}\psi_{1321}- \psi_{0233}\psi_{1322}+ \psi_{0232}\psi_{1323}+
\psi_{0221}\psi_{1330}- \psi_{0220}\psi_{1331}+ \psi_{0223}\psi_{1332}- \psi_{0222}\psi_{1333}\nonumber\\&+
\psi_{2131}\psi_{3020}- \psi_{2130}\psi_{3021}+ \psi_{2133}\psi_{3022}- \psi_{2132}\psi_{3023}-
\psi_{2121}\psi_{3030}+ \psi_{2120}\psi_{3031}- \psi_{2123}\psi_{3032}+ \psi_{2122}\psi_{3033}\nonumber\\&-
\psi_{2031}\psi_{3120}+ \psi_{2030}\psi_{3121}- \psi_{2033}\psi_{3122}+ \psi_{2032}\psi_{3123}+
\psi_{2021}\psi_{3130}- \psi_{2020}\psi_{3131}+ \psi_{2023}\psi_{3132}- \psi_{2022}\psi_{3133}\nonumber\\&+
\psi_{2331}\psi_{3220}- \psi_{2330}\psi_{3221}+ \psi_{2333}\psi_{3222}- \psi_{2332}\psi_{3223}-
\psi_{2321}\psi_{3230}+ \psi_{2320}\psi_{3231}- \psi_{2323}\psi_{3232}+ \psi_{2322}\psi_{3233}\nonumber\\&-
\psi_{2231}\psi_{3320}+ \psi_{2230}\psi_{3321}- \psi_{2233}\psi_{3322}+ \psi_{2232}\psi_{3323}+
\psi_{2221}\psi_{3330}- \psi_{2220}\psi_{3331}+ \psi_{2223}\psi_{3332}- \psi_{2222}\psi_{3333})^2\nonumber\\+
4 (&- \psi_{0121}\psi_{1010}+ \psi_{0120}\psi_{1011}- \psi_{0123}\psi_{1012}+ \psi_{0122}\psi_{1013}+
\psi_{0111}\psi_{1020}- \psi_{0110}\psi_{1021}+ \psi_{0113}\psi_{1022}- \psi_{0112}\psi_{1023}\nonumber\\&+
\psi_{0021}\psi_{1110}- \psi_{0020}\psi_{1111}+ \psi_{0023}\psi_{1112}- \psi_{0022}\psi_{1113}-
\psi_{0011}\psi_{1120}+ \psi_{0010}\psi_{1121}- \psi_{0013}\psi_{1122}+ \psi_{0012}\psi_{1123}\nonumber\\&-
\psi_{0321}\psi_{1210}+ \psi_{0320}\psi_{1211}- \psi_{0323}\psi_{1212}+ \psi_{0322}\psi_{1213}+
\psi_{0311}\psi_{1220}- \psi_{0310}\psi_{1221}+ \psi_{0313}\psi_{1222}- \psi_{0312}\psi_{1223}\nonumber\\&+
\psi_{0221}\psi_{1310}- \psi_{0220}\psi_{1311}+ \psi_{0223}\psi_{1312}- \psi_{0222}\psi_{1313}-
\psi_{0211}\psi_{1320}+ \psi_{0210}\psi_{1321}- \psi_{0213}\psi_{1322}+ \psi_{0212}\psi_{1323}\nonumber\\&-
\psi_{2121}\psi_{3010}+ \psi_{2120}\psi_{3011}- \psi_{2123}\psi_{3012}+ \psi_{2122}\psi_{3013}+
\psi_{2111}\psi_{3020}- \psi_{2110}\psi_{3021}+ \psi_{2113}\psi_{3022}- \psi_{2112}\psi_{3023}\nonumber\\&+
\psi_{2021}\psi_{3110}- \psi_{2020}\psi_{3111}+ \psi_{2023}\psi_{3112}- \psi_{2022}\psi_{3113}-
\psi_{2011}\psi_{3120}+ \psi_{2010}\psi_{3121}- \psi_{2013}\psi_{3122}+ \psi_{2012}\psi_{3123}\nonumber\\&-
\psi_{2321}\psi_{3210}+ \psi_{2320}\psi_{3211}- \psi_{2323}\psi_{3212}+ \psi_{2322}\psi_{3213}+
\psi_{2311}\psi_{3220}- \psi_{2310}\psi_{3221}+ \psi_{2313}\psi_{3222}- \psi_{2312}\psi_{3223}\nonumber\\&+
\psi_{2221}\psi_{3310}- \psi_{2220}\psi_{3311}+ \psi_{2223}\psi_{3312}- \psi_{2222}\psi_{3313}-
\psi_{2211}\psi_{3320}+ \psi_{2210}\psi_{3321}- \psi_{2213}\psi_{3322}+
\psi_{2212}\psi_{3323})\nonumber\\\times ( &\psi_{0131}\psi_{1000}- \psi_{0130}\psi_{1001}+ \psi_{0133}\psi_{1002}-
\psi_{0132}\psi_{1003}- \psi_{0101}\psi_{1030}+ \psi_{0100}\psi_{1031}- \psi_{0103}\psi_{1032}+
\psi_{0102}\psi_{1033}\nonumber\\&- \psi_{0031}\psi_{1100}+ \psi_{0030}\psi_{1101}- \psi_{0033}\psi_{1102}+
\psi_{0032}\psi_{1103}+ \psi_{0001}\psi_{1130}- \psi_{0000}\psi_{1131}+ \psi_{0003}\psi_{1132}-
\psi_{0002}\psi_{1133}\nonumber\\&+ \psi_{0331}\psi_{1200}- \psi_{0330}\psi_{1201}+ \psi_{0333}\psi_{1202}-
\psi_{0332}\psi_{1203}- \psi_{0301}\psi_{1230}+ \psi_{0300}\psi_{1231}- \psi_{0303}\psi_{1232}+
\psi_{0302}\psi_{1233}\nonumber\\&- \psi_{0231}\psi_{1300}+ \psi_{0230}\psi_{1301}- \psi_{0233}\psi_{1302}+
\psi_{0232}\psi_{1303}+ \psi_{0201}\psi_{1330}- \psi_{0200}\psi_{1331}+ \psi_{0203}\psi_{1332}-
\psi_{0202}\psi_{1333}\nonumber\\&+ \psi_{2131}\psi_{3000}- \psi_{2130}\psi_{3001}+ \psi_{2133}\psi_{3002}-
\psi_{2132}\psi_{3003}- \psi_{2101}\psi_{3030}+ \psi_{2100}\psi_{3031}- \psi_{2103}\psi_{3032}+
\psi_{2102}\psi_{3033}\nonumber\\&- \psi_{2031}\psi_{3100}+ \psi_{2030}\psi_{3101}- \psi_{2033}\psi_{3102}+
\psi_{2032}\psi_{3103}+ \psi_{2001}\psi_{3130}- \psi_{2000}\psi_{3131}+ \psi_{2003}\psi_{3132}-
\psi_{2002}\psi_{3133}\nonumber\\&+ \psi_{2331}\psi_{3200}- \psi_{2330}\psi_{3201}+ \psi_{2333}\psi_{3202}-
\psi_{2332}\psi_{3203}- \psi_{2301}\psi_{3230}+ \psi_{2300}\psi_{3231}- \psi_{2303}\psi_{3232}+
\psi_{2302}\psi_{3233}\nonumber\\&- \psi_{2231}\psi_{3300}+ \psi_{2230}\psi_{3301}- \psi_{2233}\psi_{3302}+
\psi_{2232}\psi_{3303}+ \psi_{2201}\psi_{3330}- \psi_{2200}\psi_{3331}+ \psi_{2203}\psi_{3332}-
\psi_{2202}\psi_{3333})\nonumber\\ +
4 ( &\psi_{0121}\psi_{1000}- \psi_{0120}\psi_{1001}+ \psi_{0123}\psi_{1002}- \psi_{0122}\psi_{1003}-
\psi_{0101}\psi_{1020}+ \psi_{0100}\psi_{1021}- \psi_{0103}\psi_{1022}+ \psi_{0102}\psi_{1023}\nonumber\\&-
\psi_{0021}\psi_{1100}+ \psi_{0020}\psi_{1101}- \psi_{0023}\psi_{1102}+ \psi_{0022}\psi_{1103}+
\psi_{0001}\psi_{1120}- \psi_{0000}\psi_{1121}+ \psi_{0003}\psi_{1122}- \psi_{0002}\psi_{1123}\nonumber\\&+
\psi_{0321}\psi_{1200}- \psi_{0320}\psi_{1201}+ \psi_{0323}\psi_{1202}- \psi_{0322}\psi_{1203}-
\psi_{0301}\psi_{1220}+ \psi_{0300}\psi_{1221}- \psi_{0303}\psi_{1222}+ \psi_{0302}\psi_{1223}\nonumber\\&-
\psi_{0221}\psi_{1300}+ \psi_{0220}\psi_{1301}- \psi_{0223}\psi_{1302}+ \psi_{0222}\psi_{1303}+
\psi_{0201}\psi_{1320}- \psi_{0200}\psi_{1321}+ \psi_{0203}\psi_{1322}- \psi_{0202}\psi_{1323}\nonumber\\&+
\psi_{2121}\psi_{3000}- \psi_{2120}\psi_{3001}+ \psi_{2123}\psi_{3002}- \psi_{2122}\psi_{3003}-
\psi_{2101}\psi_{3020}+ \psi_{2100}\psi_{3021}- \psi_{2103}\psi_{3022}+ \psi_{2102}\psi_{3023}\nonumber\\&-
\psi_{2021}\psi_{3100}+ \psi_{2020}\psi_{3101}- \psi_{2023}\psi_{3102}+ \psi_{2022}\psi_{3103}+
\psi_{2001}\psi_{3120}- \psi_{2000}\psi_{3121}+ \psi_{2003}\psi_{3122}- \psi_{2002}\psi_{3123}\nonumber\\&+
\psi_{2321}\psi_{3200}- \psi_{2320}\psi_{3201}+ \psi_{2323}\psi_{3202}- \psi_{2322}\psi_{3203}-
\psi_{2301}\psi_{3220}+ \psi_{2300}\psi_{3221}- \psi_{2303}\psi_{3222}+ \psi_{2302}\psi_{3223}\nonumber\\&-
\psi_{2221}\psi_{3300}+ \psi_{2220}\psi_{3301}- \psi_{2223}\psi_{3302}+ \psi_{2222}\psi_{3303}+
\psi_{2201}\psi_{3320}- \psi_{2200}\psi_{3321}+ \psi_{2203}\psi_{3322}-
\psi_{2202}\psi_{3323})\nonumber\\\times ( &\psi_{0131}\psi_{1010}- \psi_{0130}\psi_{1011}+ \psi_{0133}\psi_{1012}-
\psi_{0132}\psi_{1013}- \psi_{0111}\psi_{1030}+ \psi_{0110}\psi_{1031}- \psi_{0113}\psi_{1032}+
\psi_{0112}\psi_{1033}\nonumber\\&- \psi_{0031}\psi_{1110}+ \psi_{0030}\psi_{1111}- \psi_{0033}\psi_{1112}+
\psi_{0032}\psi_{1113}+ \psi_{0011}\psi_{1130}- \psi_{0010}\psi_{1131}+ \psi_{0013}\psi_{1132}-
\psi_{0012}\psi_{1133}\nonumber\\&+ \psi_{0331}\psi_{1210}- \psi_{0330}\psi_{1211}+ \psi_{0333}\psi_{1212}-
\psi_{0332}\psi_{1213}- \psi_{0311}\psi_{1230}+ \psi_{0310}\psi_{1231}- \psi_{0313}\psi_{1232}+
\psi_{0312}\psi_{1233}\nonumber\\&- \psi_{0231}\psi_{1310}+ \psi_{0230}\psi_{1311}- \psi_{0233}\psi_{1312}+
\psi_{0232}\psi_{1313}+ \psi_{0211}\psi_{1330}- \psi_{0210}\psi_{1331}+ \psi_{0213}\psi_{1332}-
\psi_{0212}\psi_{1333}\nonumber\\&+ \psi_{2131}\psi_{3010}- \psi_{2130}\psi_{3011}+ \psi_{2133}\psi_{3012}-
\psi_{2132}\psi_{3013}- \psi_{2111}\psi_{3030}+ \psi_{2110}\psi_{3031}- \psi_{2113}\psi_{3032}+
\psi_{2112}\psi_{3033}\nonumber\\&- \psi_{2031}\psi_{3110}+ \psi_{2030}\psi_{3111}- \psi_{2033}\psi_{3112}+
\psi_{2032}\psi_{3113}+ \psi_{2011}\psi_{3130}- \psi_{2010}\psi_{3131}+ \psi_{2013}\psi_{3132}-
\psi_{2012}\psi_{3133}\nonumber\\&+ \psi_{2331}\psi_{3210}- \psi_{2330}\psi_{3211}+ \psi_{2333}\psi_{3212}-
\psi_{2332}\psi_{3213}- \psi_{2311}\psi_{3230}+ \psi_{2310}\psi_{3231}- \psi_{2313}\psi_{3232}+
\psi_{2312}\psi_{3233}\nonumber\\&- \psi_{2231}\psi_{3310}+ \psi_{2230}\psi_{3311}- \psi_{2233}\psi_{3312}+
\psi_{2232}\psi_{3313}+ \psi_{2211}\psi_{3330}- \psi_{2210}\psi_{3331}+ \psi_{2213}\psi_{3332}-
\psi_{2212}\psi_{3333}),
\end{align}
\normalsize
and
\footnotesize
\begin{align}
Y_l=2 (&- \psi_{1323}\psi_{2010}+ \psi_{1322}\psi_{2011}- \psi_{1321}\psi_{2012}+ \psi_{1320}\psi_{2013}+
\psi_{1313}\psi_{2020}- \psi_{1312}\psi_{2021}+ \psi_{1311}\psi_{2022}- \psi_{1310}\psi_{2023}\nonumber\\&+
\psi_{1223}\psi_{2110}- \psi_{1222}\psi_{2111}+ \psi_{1221}\psi_{2112}- \psi_{1220}\psi_{2113}-
\psi_{1213}\psi_{2120}+ \psi_{1212}\psi_{2121}- \psi_{1211}\psi_{2122}+ \psi_{1210}\psi_{2123}\nonumber\\&-
\psi_{1123}\psi_{2210}+ \psi_{1122}\psi_{2211}- \psi_{1121}\psi_{2212}+ \psi_{1120}\psi_{2213}+
\psi_{1113}\psi_{2220}- \psi_{1112}\psi_{2221}+ \psi_{1111}\psi_{2222}- \psi_{1110}\psi_{2223}\nonumber\\&+
\psi_{1023}\psi_{2310}- \psi_{1022}\psi_{2311}+ \psi_{1021}\psi_{2312}- \psi_{1020}\psi_{2313}-
\psi_{1013}\psi_{2320}+ \psi_{1012}\psi_{2321}- \psi_{1011}\psi_{2322}+ \psi_{1010}\psi_{2323}\nonumber\\&+
\psi_{0323}\psi_{3010}- \psi_{0322}\psi_{3011}+ \psi_{0321}\psi_{3012}- \psi_{0320}\psi_{3013}-
\psi_{0313}\psi_{3020}+ \psi_{0312}\psi_{3021}- \psi_{0311}\psi_{3022}+ \psi_{0310}\psi_{3023}\nonumber\\&-
\psi_{0223}\psi_{3110}+ \psi_{0222}\psi_{3111}- \psi_{0221}\psi_{3112}+ \psi_{0220}\psi_{3113}+
\psi_{0213}\psi_{3120}- \psi_{0212}\psi_{3121}+ \psi_{0211}\psi_{3122}- \psi_{0210}\psi_{3123}\nonumber\\&+
\psi_{0123}\psi_{3210}- \psi_{0122}\psi_{3211}+ \psi_{0121}\psi_{3212}- \psi_{0120}\psi_{3213}-
\psi_{0113}\psi_{3220}+ \psi_{0112}\psi_{3221}- \psi_{0111}\psi_{3222}+ \psi_{0110}\psi_{3223}\nonumber\\&-
\psi_{0023}\psi_{3310}+ \psi_{0022}\psi_{3311}- \psi_{0021}\psi_{3312}+ \psi_{0020}\psi_{3313}+
\psi_{0013}\psi_{3320}- \psi_{0012}\psi_{3321}+ \psi_{0011}\psi_{3322}-
\psi_{0010}\psi_{3323})^2\nonumber\\+
2 (&- \psi_{1333}\psi_{2000}+ \psi_{1332}\psi_{2001}- \psi_{1331}\psi_{2002}+ \psi_{1330}\psi_{2003}+
\psi_{1303}\psi_{2030}- \psi_{1302}\psi_{2031}+ \psi_{1301}\psi_{2032}- \psi_{1300}\psi_{2033}\nonumber\\&+
\psi_{1233}\psi_{2100}- \psi_{1232}\psi_{2101}+ \psi_{1231}\psi_{2102}- \psi_{1230}\psi_{2103}-
\psi_{1203}\psi_{2130}+ \psi_{1202}\psi_{2131}- \psi_{1201}\psi_{2132}+ \psi_{1200}\psi_{2133}\nonumber\\&-
\psi_{1133}\psi_{2200}+ \psi_{1132}\psi_{2201}- \psi_{1131}\psi_{2202}+ \psi_{1130}\psi_{2203}+
\psi_{1103}\psi_{2230}- \psi_{1102}\psi_{2231}+ \psi_{1101}\psi_{2232}- \psi_{1100}\psi_{2233}\nonumber\\&+
\psi_{1033}\psi_{2300}- \psi_{1032}\psi_{2301}+ \psi_{1031}\psi_{2302}- \psi_{1030}\psi_{2303}-
\psi_{1003}\psi_{2330}+ \psi_{1002}\psi_{2331}- \psi_{1001}\psi_{2332}+ \psi_{1000}\psi_{2333}\nonumber\\&+
\psi_{0333}\psi_{3000}- \psi_{0332}\psi_{3001}+ \psi_{0331}\psi_{3002}- \psi_{0330}\psi_{3003}-
\psi_{0303}\psi_{3030}+ \psi_{0302}\psi_{3031}- \psi_{0301}\psi_{3032}+ \psi_{0300}\psi_{3033}\nonumber\\&-
\psi_{0233}\psi_{3100}+ \psi_{0232}\psi_{3101}- \psi_{0231}\psi_{3102}+ \psi_{0230}\psi_{3103}+
\psi_{0203}\psi_{3130}- \psi_{0202}\psi_{3131}+ \psi_{0201}\psi_{3132}- \psi_{0200}\psi_{3133}\nonumber\\&+
\psi_{0133}\psi_{3200}- \psi_{0132}\psi_{3201}+ \psi_{0131}\psi_{3202}- \psi_{0130}\psi_{3203}-
\psi_{0103}\psi_{3230}+ \psi_{0102}\psi_{3231}- \psi_{0101}\psi_{3232}+ \psi_{0100}\psi_{3233}\nonumber\\&-
\psi_{0033}\psi_{3300}+ \psi_{0032}\psi_{3301}- \psi_{0031}\psi_{3302}+ \psi_{0030}\psi_{3303}+
\psi_{0003}\psi_{3330}- \psi_{0002}\psi_{3331}+ \psi_{0001}\psi_{3332}-
\psi_{0000}\psi_{3333})^2\nonumber\\+
4 ( &\psi_{1323}\psi_{2000}- \psi_{1322}\psi_{2001}+ \psi_{1321}\psi_{2002}- \psi_{1320}\psi_{2003}-
\psi_{1303}\psi_{2020}+ \psi_{1302}\psi_{2021}- \psi_{1301}\psi_{2022}+ \psi_{1300}\psi_{2023}\nonumber\\&-
\psi_{1223}\psi_{2100}+ \psi_{1222}\psi_{2101}- \psi_{1221}\psi_{2102}+ \psi_{1220}\psi_{2103}+
\psi_{1203}\psi_{2120}- \psi_{1202}\psi_{2121}+ \psi_{1201}\psi_{2122}- \psi_{1200}\psi_{2123}\nonumber\\&+
\psi_{1123}\psi_{2200}- \psi_{1122}\psi_{2201}+ \psi_{1121}\psi_{2202}- \psi_{1120}\psi_{2203}-
\psi_{1103}\psi_{2220}+ \psi_{1102}\psi_{2221}- \psi_{1101}\psi_{2222}+ \psi_{1100}\psi_{2223}\nonumber\\&-
\psi_{1023}\psi_{2300}+ \psi_{1022}\psi_{2301}- \psi_{1021}\psi_{2302}+ \psi_{1020}\psi_{2303}+
\psi_{1003}\psi_{2320}- \psi_{1002}\psi_{2321}+ \psi_{1001}\psi_{2322}- \psi_{1000}\psi_{2323}\nonumber\\&-
\psi_{0323}\psi_{3000}+ \psi_{0322}\psi_{3001}- \psi_{0321}\psi_{3002}+ \psi_{0320}\psi_{3003}+
\psi_{0303}\psi_{3020}- \psi_{0302}\psi_{3021}+ \psi_{0301}\psi_{3022}- \psi_{0300}\psi_{3023}\nonumber\\&+
\psi_{0223}\psi_{3100}- \psi_{0222}\psi_{3101}+ \psi_{0221}\psi_{3102}- \psi_{0220}\psi_{3103}-
\psi_{0203}\psi_{3120}+ \psi_{0202}\psi_{3121}- \psi_{0201}\psi_{3122}+ \psi_{0200}\psi_{3123}\nonumber\\&-
\psi_{0123}\psi_{3200}+ \psi_{0122}\psi_{3201}- \psi_{0121}\psi_{3202}+ \psi_{0120}\psi_{3203}+
\psi_{0103}\psi_{3220}- \psi_{0102}\psi_{3221}+ \psi_{0101}\psi_{3222}- \psi_{0100}\psi_{3223}\nonumber\\&+
\psi_{0023}\psi_{3300}- \psi_{0022}\psi_{3301}+ \psi_{0021}\psi_{3302}- \psi_{0020}\psi_{3303}-
\psi_{0003}\psi_{3320}+ \psi_{0002}\psi_{3321}- \psi_{0001}\psi_{3322}+
\psi_{0000}\psi_{3323})\nonumber\\\times (&- \psi_{1333}\psi_{2010}+ \psi_{1332}\psi_{2011}- \psi_{1331}\psi_{2012}+
\psi_{1330}\psi_{2013}+ \psi_{1313}\psi_{2030}- \psi_{1312}\psi_{2031}+ \psi_{1311}\psi_{2032}-
\psi_{1310}\psi_{2033}\nonumber\\&+ \psi_{1233}\psi_{2110}- \psi_{1232}\psi_{2111}+ \psi_{1231}\psi_{2112}-
\psi_{1230}\psi_{2113}- \psi_{1213}\psi_{2130}+ \psi_{1212}\psi_{2131}- \psi_{1211}\psi_{2132}+
\psi_{1210}\psi_{2133}\nonumber\\&- \psi_{1133}\psi_{2210}+ \psi_{1132}\psi_{2211}- \psi_{1131}\psi_{2212}+
\psi_{1130}\psi_{2213}+ \psi_{1113}\psi_{2230}- \psi_{1112}\psi_{2231}+ \psi_{1111}\psi_{2232}-
\psi_{1110}\psi_{2233}\nonumber\\&+ \psi_{1033}\psi_{2310}- \psi_{1032}\psi_{2311}+ \psi_{1031}\psi_{2312}-
\psi_{1030}\psi_{2313}- \psi_{1013}\psi_{2330}+ \psi_{1012}\psi_{2331}- \psi_{1011}\psi_{2332}+
\psi_{1010}\psi_{2333}\nonumber\\&+ \psi_{0333}\psi_{3010}- \psi_{0332}\psi_{3011}+ \psi_{0331}\psi_{3012}-
\psi_{0330}\psi_{3013}- \psi_{0313}\psi_{3030}+ \psi_{0312}\psi_{3031}- \psi_{0311}\psi_{3032}+
\psi_{0310}\psi_{3033}\nonumber\\&- \psi_{0233}\psi_{3110}+ \psi_{0232}\psi_{3111}- \psi_{0231}\psi_{3112}+
\psi_{0230}\psi_{3113}+ \psi_{0213}\psi_{3130}- \psi_{0212}\psi_{3131}+ \psi_{0211}\psi_{3132}-
\psi_{0210}\psi_{3133}\nonumber\\&+ \psi_{0133}\psi_{3210}- \psi_{0132}\psi_{3211}+ \psi_{0131}\psi_{3212}-
\psi_{0130}\psi_{3213}- \psi_{0113}\psi_{3230}+ \psi_{0112}\psi_{3231}- \psi_{0111}\psi_{3232}+
\psi_{0110}\psi_{3233}\nonumber\\&- \psi_{0033}\psi_{3310}+ \psi_{0032}\psi_{3311}- \psi_{0031}\psi_{3312}+
\psi_{0030}\psi_{3313}+ \psi_{0013}\psi_{3330}- \psi_{0012}\psi_{3331}+ \psi_{0011}\psi_{3332}-
\psi_{0010}\psi_{3333})\nonumber\\ +
4 (&- \psi_{1313}\psi_{2000}+ \psi_{1312}\psi_{2001}- \psi_{1311}\psi_{2002}+ \psi_{1310}\psi_{2003}+
\psi_{1303}\psi_{2010}- \psi_{1302}\psi_{2011}+ \psi_{1301}\psi_{2012}- \psi_{1300}\psi_{2013}\nonumber\\&+
\psi_{1213}\psi_{2100}- \psi_{1212}\psi_{2101}+ \psi_{1211}\psi_{2102}- \psi_{1210}\psi_{2103}-
\psi_{1203}\psi_{2110}+ \psi_{1202}\psi_{2111}- \psi_{1201}\psi_{2112}+ \psi_{1200}\psi_{2113}\nonumber\\&-
\psi_{1113}\psi_{2200}+ \psi_{1112}\psi_{2201}- \psi_{1111}\psi_{2202}+ \psi_{1110}\psi_{2203}+
\psi_{1103}\psi_{2210}- \psi_{1102}\psi_{2211}+ \psi_{1101}\psi_{2212}- \psi_{1100}\psi_{2213}\nonumber\\&+
\psi_{1013}\psi_{2300}- \psi_{1012}\psi_{2301}+ \psi_{1011}\psi_{2302}- \psi_{1010}\psi_{2303}-
\psi_{1003}\psi_{2310}+ \psi_{1002}\psi_{2311}- \psi_{1001}\psi_{2312}+ \psi_{1000}\psi_{2313}\nonumber\\&+
\psi_{0313}\psi_{3000}- \psi_{0312}\psi_{3001}+ \psi_{0311}\psi_{3002}- \psi_{0310}\psi_{3003}-
\psi_{0303}\psi_{3010}+ \psi_{0302}\psi_{3011}- \psi_{0301}\psi_{3012}+ \psi_{0300}\psi_{3013}\nonumber\\&-
\psi_{0213}\psi_{3100}+ \psi_{0212}\psi_{3101}- \psi_{0211}\psi_{3102}+ \psi_{0210}\psi_{3103}+
\psi_{0203}\psi_{3110}- \psi_{0202}\psi_{3111}+ \psi_{0201}\psi_{3112}- \psi_{0200}\psi_{3113}\nonumber\\&+
\psi_{0113}\psi_{3200}- \psi_{0112}\psi_{3201}+ \psi_{0111}\psi_{3202}- \psi_{0110}\psi_{3203}-
\psi_{0103}\psi_{3210}+ \psi_{0102}\psi_{3211}- \psi_{0101}\psi_{3212}+ \psi_{0100}\psi_{3213}\nonumber\\&-
\psi_{0013}\psi_{3300}+ \psi_{0012}\psi_{3301}- \psi_{0011}\psi_{3302}+ \psi_{0010}\psi_{3303}+
\psi_{0003}\psi_{3310}- \psi_{0002}\psi_{3311}+ \psi_{0001}\psi_{3312}-
\psi_{0000}\psi_{3313})\nonumber\\\times (&- \psi_{1333}\psi_{2020}+ \psi_{1332}\psi_{2021}- \psi_{1331}\psi_{2022}+
\psi_{1330}\psi_{2023}+ \psi_{1323}\psi_{2030}- \psi_{1322}\psi_{2031}+ \psi_{1321}\psi_{2032}-
\psi_{1320}\psi_{2033}\nonumber\\&+ \psi_{1233}\psi_{2120}- \psi_{1232}\psi_{2121}+ \psi_{1231}\psi_{2122}-
\psi_{1230}\psi_{2123}- \psi_{1223}\psi_{2130}+ \psi_{1222}\psi_{2131}- \psi_{1221}\psi_{2132}+
\psi_{1220}\psi_{2133}\nonumber\\&- \psi_{1133}\psi_{2220}+ \psi_{1132}\psi_{2221}- \psi_{1131}\psi_{2222}+
\psi_{1130}\psi_{2223}+ \psi_{1123}\psi_{2230}- \psi_{1122}\psi_{2231}+ \psi_{1121}\psi_{2232}-
\psi_{1120}\psi_{2233}\nonumber\\&+ \psi_{1033}\psi_{2320}- \psi_{1032}\psi_{2321}+ \psi_{1031}\psi_{2322}-
\psi_{1030}\psi_{2323}- \psi_{1023}\psi_{2330}+ \psi_{1022}\psi_{2331}- \psi_{1021}\psi_{2332}+
\psi_{1020}\psi_{2333}\nonumber\\&+ \psi_{0333}\psi_{3020}- \psi_{0332}\psi_{3021}+ \psi_{0331}\psi_{3022}-
\psi_{0330}\psi_{3023}- \psi_{0323}\psi_{3030}+ \psi_{0322}\psi_{3031}- \psi_{0321}\psi_{3032}+
\psi_{0320}\psi_{3033}\nonumber\\&- \psi_{0233}\psi_{3120}+ \psi_{0232}\psi_{3121}- \psi_{0231}\psi_{3122}+
\psi_{0230}\psi_{3123}+ \psi_{0223}\psi_{3130}- \psi_{0222}\psi_{3131}+ \psi_{0221}\psi_{3132}-
\psi_{0220}\psi_{3133}\nonumber\\&+ \psi_{0133}\psi_{3220}- \psi_{0132}\psi_{3221}+ \psi_{0131}\psi_{3222}-
\psi_{0130}\psi_{3223}- \psi_{0123}\psi_{3230}+ \psi_{0122}\psi_{3231}- \psi_{0121}\psi_{3232}+
\psi_{0120}\psi_{3233}\nonumber\\&- \psi_{0033}\psi_{3320}+ \psi_{0032}\psi_{3321}- \psi_{0031}\psi_{3322}+
\psi_{0030}\psi_{3323}+ \psi_{0023}\psi_{3330}- \psi_{0022}\psi_{3331}+ \psi_{0021}\psi_{3332}-
\psi_{0020}\psi_{3333}).
\end{align}
\normalsize
\end{document} |
\begin{document}
\author{Pengcheng Yang}
\affiliation{School of Physics, International Joint Laboratory on Quantum Sensing and Quantum Metrology, Huazhong University of Science and Technology, Wuhan 430074, China}
\author{Min Yu}
\affiliation{School of Physics, International Joint Laboratory on Quantum Sensing and Quantum Metrology, Huazhong University of Science and Technology, Wuhan 430074, China}
\author{Ralf Betzholz}
\email{ralf\[email protected]}
\affiliation{School of Physics, International Joint Laboratory on Quantum Sensing and Quantum Metrology, Huazhong University of Science and Technology, Wuhan 430074, China}
\author{Christian Arenz}
\affiliation{Frick Laboratory, Princeton University, Princeton New Jersey 08544, USA}
\author{Jianming Cai}
\email{[email protected]}
\affiliation{School of Physics, International Joint Laboratory on Quantum Sensing and Quantum Metrology, Huazhong University of Science and Technology, Wuhan 430074, China}
\title{Complete Quantum-State Tomography with a Local Random Field}
\date{\today}
\begin{abstract}
Single-qubit measurements are typically insufficient for inferring arbitrary quantum states of a multiqubit system. We show that if the system can be fully controlled by driving a single qubit, then utilizing a local random pulse is almost always sufficient for complete quantum-state tomography. Experimental demonstrations of this principle are presented using a nitrogen-vacancy (NV) center in diamond coupled to a nuclear spin, which is not directly accessible. We report the reconstruction of a highly entangled state between the electron and nuclear spin with fidelity above 95\% by randomly driving and measuring the NV-center electron spin only. Beyond quantum-state tomography, we outline how this principle can be leveraged to characterize and control quantum processes in cases where the system model is not known.
\end{abstract}
\maketitle
\textit{Introduction.--} The ability to infer the full state of a quantum system is crucial for benchmarking and controlling emerging quantum technologies. In theory, this task can be accomplished by measuring an \emph{informationally complete}~\cite{busch1991informationally} set of observables, whose corresponding expectation values allow to reconstruct the quantum state of the system. In practice, measuring observables that are informationally complete typically requires access to each system component. While compressed sensing techniques can significantly improve the efficiency of reconstructing low-rank quantum states~\cite{candes2011probabilistic,gross2010quantum, flammia2012quantum,ohliger2013efficient, kalev2015quantum, shabani2011efficient, christandl2012reliable,riofrio2017experimental, steffens2017experimentally, cramer2010efficient, kalev2015quantum}, the problem of identifying an arbitrary state of a complex quantum system with limited measurement access (e.g., to a single qubit only) remains \cite{merkel2010random, smith2013quantum,Chantasri2019}. For example, one task of practical importance in the development of solid-state quantum devices~\cite{cai2013large,bradley2019, zhao2012sensing,Kolkowitz2012,Taminiau2012} is the complete characterization of coupled spin states. However, when nuclear spins are involved, access to the full system is limited due to their small magnetic moment. Even in settings where full access is currently possible (e.g., proof-of-principle few-qubit devices), this requirement becomes daunting as the complexity of the system (e.g., the number of qubits) grows.
A typical strategy for addressing these challenges is to create otherwise inaccessible observables. This can be accomplished by (i) deterministically applying unitary operations that transform an accessible observable into the desired inaccessible ones~\cite{silberfarb2005quantum, deutsch2010quantum, merkel2010random,liu2019pulsed}, typically via properly tailored classical fields, or (ii) randomly creating an informationally complete set of observables by approximating random unitary transformations through so-called unitary $t$-designs~\cite{flammia2012quantum, ohliger2013efficient}. However, both of these procedures can be highly demanding. While (i) does not require full system access, it does require identifying and accurately implementing the necessary classical fields; (ii), on the other hand, can be carried out with elementary gate operations, but typically necessitates full system access.
\begin{figure}
\caption{\label{fig1}
\label{fig1}
\end{figure}
Here, we provide a solution to the drawbacks of (i) and (ii) through the observation that a \emph{random} control field can create a random unitary evolution~\cite{banchi2017driven} when the system is \emph{fully controllable}, i.e., when there exist pulse shapes that, in principle, allow every unitary evolution to be created~\cite{d2007introduction}. We show that in this case, a randomly applied field (almost always) yields enough information in the measurement signal of any observable to reconstruct an arbitrary quantum state, provided the signal is long enough. Thus, for qubit systems that are fully controllable by addressing a single qubit, a local random pulse, that randomly "shakes" the total system, allows for the reconstruction of the full state of the qubit network by measuring only a \textit{single-qubit} observable (see Fig.~\ref{fig1}). We experimentally demonstrate this principle in a solid-state spin system in diamond, but due to its generality, the presented random-field-based tomography constitutes a broadly applicable strategy that can be readily adopted in a variety of partially-accessible systems.\\
\textit{Theory.--} Adopting the framework of~\cite{silberfarb2005quantum,merkel2010random}, we begin by developing the theory behind random-field quantum-state tomography. While the following assessment is completely general, for the sake of simplicity, we restrict ourselves to a single random field.
Consider a $d$-dimensional quantum system initially in an unknown state $\rho$, whose evolution is governed by a time-dependent Hamiltonian of the form
\begin{align}
\label{eq:Ham}
H(t)=H_{0}+f(t)H_{c},
\end{align}
that depends on a classical control field $f(t)$ steering the system. The time evolution of the expectation of an observable $M$ is then given by
\begin{equation}
\label{eq:evoobservable}
\langle M\rangle_{t}=\text{Tr}\{U^{\dagger}_{t}MU_{t}\rho\},
\end{equation}
where $U_{t}=\mathcal T\exp\{-i\int_{0}^{t}H(t^{\prime})dt^{\prime}\}$ is the time-evolution operator in units of $\hbar= 1$, with $\mathcal{T}$ indicating time ordering. We assume, without loss of generality, that $M$ is traceless. The quantum system is said to be fully controllable if there exist pulse shapes that allow for creating every unitary evolution. For unconstrained control fields this is guaranteed if and only if the dynamical Lie algebra $\mathfrak L=\text{Lie}(iH_{0},iH_{c})$ generated by nested commutators and real linear combinations of $H_{0}$ and $H_{c}$ spans the full space (i.e., $\mathfrak{u}(d)$ or $\mathfrak{su}(d)$ for traceless Hamiltonians)~\cite{d2007introduction}. Using the generalized Bloch-vector representation, we can write the initial state as $\rho=\frac{\mathds{1}}{d}+\sum_{m=1}^{d^{2}-1}r_{m}B_{m}$, where $\mathds{1}$ denotes the identity and $\mathbf{r}=({\rm Tr}\{\rho B_1\},\cdots,{\rm Tr}\{\rho B_{d^{2}-1}\})$ is the Bloch vector, with $\{B_{m}\}_{m=1}^{d^{2}-1}$ being a complete and orthonormal basis for traceless and Hermitian operators. This allows for \eqref{eq:evoobservable} to be expressed as $\langle M\rangle_{t}=\sum_{m=1}^{d^{2}-1} A_{t,m}r_{m}$, where $A_{t,m}=\text{Tr}\{U_{t}^{\dagger}MU_{t}B_{m}\}$. We assume that at times $t=n\Delta t$, with $n=1,\cdots, (d^{2}-1)$, the expectation $\langle M\rangle_{t}$ is measured, so that we obtain $d^{2}-1$ values, which are collected in the vector $\mathbf{y}\equiv(\langle M\rangle_{\Delta t},\cdots,\langle M\rangle_{(d^{2}-1)\Delta t})$, referred to as the measurement record. The measurement record is determined by the set of equations
\begin{align}
\label{eq:record}
\mathbf{y}=\mathcal M[f(t)]\mathbf{r},
\end{align}
where we have indicated, here, the explicit dependence of the matrix $\mathcal M\in \mathbb R^{(d^{2}-1)\times (d^{2}-1)}$, with entries given by $\mathcal M_{n,m}=A_{n\Delta t,m}$, on the control field $f(t)$. We call the measurement record \emph{informationally complete} if $\mathcal M$ is invertible, thereby allowing the state $\rho$ to be inferred via $\mathbf{r}=\mathcal M^{-1}\mathbf{y}$.
How can we ensure that the field and the measurement intervals chosen allow for inverting $\mathcal M$? It can be seen that if the system is not fully controllable, which is equivalent to the existence of symmetries~\cite{zimboras2015symmetry}, not every $\rho$ can be reconstructed~\cite{merkel2010random}. In contrast, for fully controllable quantum systems it is, in principle, possible to determine the pulses that create an informationally complete measurement record. For instance, this can be achieved through optimal-control algorithms designed to identify control fields that rotate $M$ into $\{B_{m}\}$, so that $\mathcal M$ is diagonal. However, optimal control typically depends on the availability of an accurate model. Moreover, it can be computationally expensive, and the designed pulses are often challenging to implement in the laboratory.
Fortunately, it was recently shown that for fully controllable systems a Haar-random unitary evolution (i.e., unitary transformations that are uniformly distributed over the unitary group~\cite{banchi2017driven}) is created when $f(t)$ is applied at random over an interval $[0,T_{*}]$ \cite{banchi2017driven}. The Haar-random time $T_{*}$ can be estimated from the time required to converge to a unitary $t$-design, which can be accomplished by mapping the expected evolution to the dynamics generated by a Lindbladian and finding its gap \cite{banchi2017driven}. Thus, a random field of length $(d^{2}-1)T_{*}$ along with measurements of the expectation of $M$ at time intervals $\Delta t=T_{*}$, yields row vectors of $\mathcal M$ that are statistically independent, due to the unitary invariance of the Haar measure. Furthermore, since the row vectors are uniformly distributed, with unit probability they are also linearly independent. This leads to the result that for almost all random pulse shapes, but a set of measure zero, the matrix $\mathcal M$ is invertible. Hence, almost all pulse shapes allow for reconstructing $\rho$ by measuring the expectation of any observable $M$. With further details found in the Supplemental Material~\cite{supp}, we summarize these findings in the following theorem. \\
\textit{Theorem.} For a $d$-dimensional fully controllable quantum system subject to a random field of length $t=(d^{2}-1)T_{*}$, with $T_{*}$ being the Haar-random time, the measurement record of any observable $M$ determined by \eqref{eq:record} with $\Delta t=T_{*}$ is almost always informationally complete. \\
\begin{figure*}
\caption{\label{fig2}
\label{fig2}
\end{figure*}
Since full controllability can often be obtained by acting with a single control $H_{c}$ on a part of the system only, e.g., a single qubit \cite{heule2010local, burgarth2010scalable, arenz2014control, zeier2011symmetry, schirmer2008global}, the appeal of this theorem is twofold: under the premise of full controllability, arbitrary quantum states can almost always be reconstructed (i) without the need for expensive numerical pulse designs and (ii) requiring only partial system access. Furthermore, full controllability of systems of the form~\eqref{eq:Ham} is a generic property, as almost all $H_{0}$ and $H_{c}$ generate the dynamical Lie algebra $\mathfrak{L}$~\cite{altafini2002controllability,wang2016subspace}. This leads to the general corollary: \\
\textit{Corollary.} Full quantum-state tomography of \textit{almost all} randomly-driven quantum systems of the form~\eqref{eq:Ham} is possible by reading out a single observable. \\
We remark that the above should be treated as a mathematical fact rather than a source of physical intuition. Nevertheless, it should be noted that in cases where full control is not achieved with a single field, adding additional control fields can be a straightforward approach for obtaining full controllability. In fact, if full system access is possible, in fully connected qubit networks two controls on each qubit are sufficient~\cite{zeier2011symmetry}. In general, a variety of algebraic tools and criteria \cite{d2007introduction,burgarth2009local}, as well as numerical algorithms \cite{zimboras2015symmetry, schirmer2001complete}, can be used to determine whether full control is achieved with the control field(s) at hand. Even in situations where full control is not achievable, as long as the state and the observable lie within the span of the dynamical Lie algebra, we expect random-field quantum-state tomography to succeed.\\
\textit{Experiment.--} In order to demonstrate the utility of the above principle, we experimentally perform the random-field tomography of a system of two interacting qubits ($d=4$). The solid-state spin system we employ is depicted in Fig.~\ref{fig2}(a) and consists of the electron spin of a nitrogen-vacancy (NV) center in diamond~\cite{doherty13}, coupled to the nuclear spin of a nearby $^{13}$C atom via hyperfine interaction. In the ground-state triplet, the NV center has the electronic sublevels $m_s=0,\pm1$, where the degeneracy between the $m_s=\pm1$ states is lifted by a magnetic field of strength $B\approx 504.7$~G along the NV axis. The first qubit is formed by the $m_s=0$ state, denoted by $\ket{\uparrow}_1$, and the $m_s=-1$ state, denoted by $\ket{\downarrow}_1$, of the electron spin [see lower panel of Fig.~\ref{fig2}(a)]. Likewise, for the second qubit we denote the $^{13}$C nuclear spin states with quantum numbers $m_I=\pm 1/2$ by $\ket{\uparrow}_2$ and $\ket{\downarrow}_2$, respectively. Furthermore, we represent the Pauli operators of the two qubits by $\sigma_j^\kappa$, for $j=1,2$ and $\kappa=x,y,z$, where $\ket{\uparrow}_j$ and $\ket{\downarrow}_j$ are the $\pm 1$ eigenstates of $\sigma_j^z$, respectively. In a rotating frame such a system is described by the Hamiltonian~\cite{supp}
\begin{equation}
\label{eq:drift}
H_0=\frac{\omega_1}{2}\sigma_1^z+\frac{\omega_2}{2}\sigma_2^z+\frac{\Omega_2}{2}\sigma_2^x+\frac{g_z}{2}\sigma_1^z\sigma_2^z+\frac{g_x}{2}\sigma_1^z\sigma_2^x.
\end{equation}
Since the gyromagnetic ratio of the nuclear spin is three orders of magnitude smaller than that of the electron spin, access to the system is effectively restricted to the electron spin, as a direct read out of the nuclear spin is extremely challenging. The electron spin is driven through a classical field, whose coupling to the electron spin is described by the control Hamiltonian
\begin{align}
\label{eq:control}
H_c=\frac{\Omega_1}{2} \sigma_1^x.
\end{align}
This control is achieved by applying a microwave field of frequency $\omega$, which is generated by an arbitrary waveform generator (AWG) and delivered to the sample through a copper microwave antenna, after being amplified by a microwave amplifier. The precise control over the AWG allows us to engineer the control field $f(t)$ with arbitrary amplitude modulations. The control field amplitude $\Omega_1$ is calibrated with the output power of the AWG by measuring the frequency of Rabi oscillations of the electron spin~\cite{supp}, i.e., for $f(t)\equiv 1$. We choose a microwave frequency $\omega/2\pi\approx 1455.5$~MHz, which, under the applied magnetic field, lies between the two allowed transitions between eigenstates of $H_0$~\cite{dreau2012}. The parameters in our experiment are $\{\omega_1,\omega_2,\Omega_1,\Omega_2,g_z,g_x\}/2\pi=\{-2.97,-6.46,7.91,-1.39,5.92,1.39\}$~MHz,
with minor variations~\cite{supp}, e.g., due to small drifts in the magnetic field between different runs of the experiment, which leads to imperfections in the state preparation.
A system described by the Hamiltonian~\eqref{eq:Ham}, with $H_{0}$ and $H_{c}$ defined in~\eqref{eq:drift} and \eqref{eq:control}, respectively, is fully controllable, as the dynamical Lie algebra spans $\mathfrak{su}(4)$. As an observable we choose the population of the electronic $m_s=0$ state, represented by $M=\sigma_1^z$, which can easily be read out by state-dependent fluorescence~\cite{doherty13}. To create a random control field we design random pulse shapes $f(t)$ based on a truncated Fourier series~\cite{banchi2017driven}
\begin{equation}
\label{eq:random_pulse}
f(t)=\sum_{j=1}^{K}F_j\cos(\nu_jt+\varphi_j),
\end{equation}
with uniformly-distributed random variables: amplitudes $F_j$ (fulfilling the normalization $\sum_{j=1}^KF_j=1$), frequencies $\nu_j/2\pi\in[0,4]$~MHz, and phases $\varphi_j\in[0,2\pi]$. Because of a limited coherence time, instead of using a single random pulse shape, in the experiment we use $d^{2}-1=15$ separate random pulses to create linearly independent rows of $\mathcal M$, thereby only evolving the system up to a time $\Delta t$ in each run. Throughout the remainder, we employ random pulses with $K=10$ Fourier components and a length of $\Delta t=0.7~\mu$s (see Supplemental Material Fig.~S1), which lies well below the coherence time of the microwave-driven system, and also allows for moderate levels of noise in the measurement record~\cite{supp}.
As a first check of the random-field tomography we reconstruct the state after the optical ground-state polarization with a 532~nm laser, i.e., with an empty preparation stage in Fig.~\ref{fig2}(b), which ideally leads to the pure state $\rho_{0}=\ket{\psi_0}\bra{\psi_0}$, with $\ket{\psi_0}=\ket{\uparrow}_1\ket{\uparrow}_2$~\cite{fischer2013}. To reconstruct this state we consecutively apply $15$ random pulses on the electron spin. One example random pulse shape $f(t)$ is shown in the central panel of Fig.~\ref{fig2}(b), with the corresponding full time trace of the expectation value $\langle M\rangle_t$ depicted below. In order to reconstruct the density operator from the obtained measurement record, we employ a least-square type minimization~\cite{supp}, using the last 10 data points of every random pulse [see Supplemental Material Fig.~S2(a)]. The resulting reconstructed density matrix yields $97.7\%$ fidelity with $\rho_0$ [see Supplemental Material Fig.~S2(b)].
In order to demonstrate the reconstruction of nontrivial states, as a first example, we randomly create a state by applying a preparation pulse of the form~\eqref{eq:random_pulse} with a duration of 0.8~$\mu$s [see Supplemental Material Fig.~S3(a)], after the initialization of the system into the state $\rho_{0}$. Since we have to preform 15 tomography pulses, and the slight drift in the experimental parameters leads to small differences in the states created from $\rho_0$ through the preparation stage before each of these pulses, the resulting state shows some impurity. The modulus of the reconstructed density matrix is shown in the upper panel of Fig.~\ref{fig2}(c) (gray bars). The reconstructed state shows a $96.1\%$ fidelity with the state ideally prepared (blue transparent bars) under the random pulse. The entanglement of this state, as quantified by the concurrence $C$, is given by $C=0.48$. As another example, we optimize a preparation pulse of the form~\eqref{eq:random_pulse} with a pulse length $1.8~\mu$s [see Supplemental Material Fig.~S3(b)] to create a highly entangled state of the two-qubit system. The modulus of the obtained density matrix, which also shows some impurity due to the preparation before each tomography pulse, is depicted in the lower panel of Fig.~\ref{fig2}(c). The reconstructed state has a concurrence $C=0.91$ and shows a fidelity of $94.9\%$ with the ideally prepared state. In the latter two cases the ideal states are obtained by numerical propagation of the initial state $\rho_0$ under the preparation pulses. \\
\textit{Discussion.--} We have shown that by randomly driving and measuring a single component of a multipartite quantum system, the quantum state of the total system can be reconstructed. This is a consequence of the fact that the data collected through expectation measurements of a single observable almost always contain enough information to reconstruct any state, provided the system is fully controllable and the randomly applied field is long enough. Based on this principle, we presented the successful experimental creation and reconstruction of composite states of an NV-center electron spin and a nuclear spin in diamond with high fidelities. The exponential overhead needed to reconstruct generic quantum states of qubit systems is reflected in the $d^{2}-1$ expectation measurements, as well as in in the length of the random pulse. However, numerical evidence presented in Fig.~S1 of the Supplemental Material suggests that often pulses much shorter than $(d^2-1)T_{*}$ can yield information completeness. Further, we remark that, for low-rank quantum states, we expect that the number of expectation measurements required can also be significantly reduced when random-field tomography is combined with compressed sensing methods~\cite{gross2010quantum, flammia2012quantum,ohliger2013efficient, kalev2015quantum, shabani2011efficient, christandl2012reliable,riofrio2017experimental, steffens2017experimentally, cramer2010efficient, kalev2015quantum}. It is also worth mentioning that in other settings, the knowledge of the full quantum state may not be necessary; instead, information carried in expectations of only certain many-body operators may be desired~\cite{guehne2002}. For example, this is the case in hybrid quantum simulation~\cite{mcclean2016theory}, where such expectation measurements are used by a classical co-processor to update a set of parameters governing the quantum simulation~\cite{kokail2019self, hempel2018quantum, peruzzo2014variational, kandala2017hardware,elben2019zoller}. We believe that a variant of the presented random-field approach could offer a way to extract the desired information with reduced overhead in accessing the system.
Besides full controllability, we also assumed knowledge of the model describing the controlled system. This assumption was needed to \emph{numerically} calculate the unitary evolution $U_{t}$ in~\eqref{eq:record}, which allowed for calculating $\mathcal M$. However, this assumption is not crucial, given that \emph{process tomography} can be performed without any prior knowledge of the model~\cite{burgarth2012quantum,blume2013robust}. That is, instead of numerically calculating $U_{t}$, the unitary evolution can \emph{experimentally} be determined. This can be achieved by additionally creating a complete set of states, for instance through randomly rotating the unknown state $\rho$. Since under the premise of full controllability uniformly-distributed states can be created through a random pulse shape, this implies that state and process tomography are possible by randomly driving and measuring a single system component without knowing system details. Therefore, the price is an increase in the number of expectation measurements needed, estimated to be $\mathcal O(d^{4})$~\cite{blume2013robust,hou2019}. However, the observation that no prior knowledge except full controllability is needed raises an interesting prospective: it is possible to fully control and read out a quantum system only based on measurement data~\cite{judson1992teaching,chen2018combining,li2017hybrid} by accessing merely part of the system~\cite{lloyd2004universal}. As such, under the premise of full controllability, a quantum computer/simulator can, in principle, be fully operated by processing classical data obtained from randomly driving and measuring a single qubit without knowing the physical hardware the quantum computer/simulator is made off.\\
\noindent
\textit{Acknowledgements.--} The authors thank R. Kosut, A. Magann, B. G. Taketani, and J. M. Torres for helpful comments. This work is supported by the National Natural Science Foundation of China (Grants No.~11950410494, No.~11574103, No.~11874024), the National Key R$\&$D Program of China (Grant No.~2018YFA0306600), and the Fundamental Research Funds for the Central Universities. Furthermore, C.A. is supported by the ARO (Grant No.~W911NF-19-1-0382).\\
\begin{thebibliography}{55}
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\bibitem [{\citenamefont {Cramer}\ \emph {et~al.}(2010)\citenamefont {Cramer},
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(\bibinfo {year} {2010})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Smith}\ \emph {et~al.}(2013)\citenamefont {Smith},
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{\bibfield {journal} {\bibinfo {journal} {Nat. Phys.}\ }\textbf {\bibinfo
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{NoStop}
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\bibitem [{\citenamefont {Zhao}\ \emph {et~al.}(2012)\citenamefont {Zhao},
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\citenamefont {Isoya}, \citenamefont {Markham}, \citenamefont {Twitchen},
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{Wrachtrup}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Sensing
single remote nuclear spins},}\ }\href {\doibase 10.1038/nnano.2012.152}
{\bibfield {journal} {\bibinfo {journal} {Nat. Nanotechnol.}\ }\textbf
{\bibinfo {volume} {7}},\ \bibinfo {pages} {657} (\bibinfo {year}
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\bibitem [{\citenamefont {Kolkowitz}\ \emph {et~al.}(2012)\citenamefont
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\bibitem [{\citenamefont {Taminiau}\ \emph {et~al.}(2012)\citenamefont
{Taminiau}, \citenamefont {Wagenaar}, \citenamefont {van~der Sar},
\citenamefont {Jelezko}, \citenamefont {Dobrovitski},\ and\ \citenamefont
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{\doibase 10.1103/PhysRevLett.109.137602} {\bibfield {journal} {\bibinfo
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\bibitem [{\citenamefont {Silberfarb}\ \emph {et~al.}(2005)\citenamefont
{Silberfarb}, \citenamefont {Jessen},\ and\ \citenamefont
{Deutsch}}]{silberfarb2005quantum}
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10.1103/PhysRevLett.95.030402} {\bibfield {journal} {\bibinfo {journal}
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\bibitem [{\citenamefont {Deutsch}\ and\ \citenamefont
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and measurement of atomic spins in polarization spectroscopy},}\ }\href
{\doibase 10.1016/j.optcom.2009.10.059} {\bibfield {journal} {\bibinfo
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\bibitem [{\citenamefont {Liu}\ \emph {et~al.}(2019)\citenamefont {Liu},
\citenamefont {Tian}, \citenamefont {Betzholz},\ and\ \citenamefont
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\bibitem [{\citenamefont {Paz-Silva}\ \emph {et~al.}(2019)\citenamefont
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\bibitem [{\citenamefont {Banchi}\ \emph {et~al.}(2017)\citenamefont {Banchi},
\citenamefont {Burgarth},\ and\ \citenamefont
{Kastoryano}}]{banchi2017driven}
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\bibitem [{\citenamefont {d'Alessandro}(2007)}]{d2007introduction}
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{NoStop}
\bibitem [{\citenamefont {Zimbor{\'a}s}\ \emph {et~al.}(2015)\citenamefont
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{Schulte-Herbr{\"u}ggen},\ and\ \citenamefont
{Burgarth}}]{zimboras2015symmetry}
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\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Burgarth}},\
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10.1103/PhysRevA.92.042309} {\bibfield {journal} {\bibinfo {journal} {Phys.
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\bibitem [{sup()}]{supp}
\BibitemOpen
\href@noop {} {}\bibinfo {note} {See Supplemental Material for more details,
including
Refs.~\cite{ohliger2013efficient,merkel2010random,doherty13,dreau2012,fischer2013,felton2009,yu2018pub,candes2011probabilistic}.}\BibitemShut
{Stop}
\bibitem [{\citenamefont {Heule}\ \emph {et~al.}(2010)\citenamefont {Heule},
\citenamefont {Bruder}, \citenamefont {Burgarth},\ and\ \citenamefont
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\bibinfo {author} {\bibfnamefont {V.~M.}\ \bibnamefont {Stojanovi{\'c}}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Local quantum control of
{H}eisenberg spin chains},}\ }\href {\doibase 10.1103/PhysRevA.82.052333}
{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo
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{2010})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Burgarth}\ \emph {et~al.}(2010)\citenamefont
{Burgarth}, \citenamefont {Maruyama}, \citenamefont {Murphy}, \citenamefont
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{\doibase 10.1103/PhysRevA.81.040303} {\bibfield {journal} {\bibinfo
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\bibitem [{\citenamefont {Arenz}\ \emph {et~al.}(2014)\citenamefont {Arenz},
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}\bibfield {title} {\enquote {\bibinfo {title} {Control of open quantum
systems: case study of the central spin model},}\ }\href {\doibase
10.1088/1367-2630/16/6/065023} {\bibfield {journal} {\bibinfo {journal}
{New J. Phys.}\ }\textbf {\bibinfo {volume} {16}},\ \bibinfo {pages} {065023}
(\bibinfo {year} {2014})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Zeier}\ and\ \citenamefont
{Schulte-Herbr{\"u}ggen}(2011)}]{zeier2011symmetry}
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10.1063/1.3657939} {\bibfield {journal} {\bibinfo {journal} {J. Math.
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(\bibinfo {year} {2011})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Schirmer}\ \emph {et~al.}(2008)\citenamefont
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{Pemberton-Ross}}]{schirmer2008global}
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10.1103/PhysRevA.78.062339} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev. A}\ }\textbf {\bibinfo {volume} {78}},\ \bibinfo {pages} {062339}
(\bibinfo {year} {2008})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Altafini}(2002)}]{altafini2002controllability}
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{pages} {2051} (\bibinfo {year} {2002})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Wang}\ \emph {et~al.}(2016)\citenamefont {Wang},
\citenamefont {Burgarth},\ and\ \citenamefont {Schirmer}}]{wang2016subspace}
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}\bibfield {title} {\enquote {\bibinfo {title} {Subspace controllability of
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10.1103/PhysRevA.94.052319} {\bibfield {journal} {\bibinfo {journal} {Phys.
Rev. A}\ }\textbf {\bibinfo {volume} {94}},\ \bibinfo {pages} {052319}
(\bibinfo {year} {2016})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Burgarth}\ \emph {et~al.}(2009)\citenamefont
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{NoStop}
\bibitem [{\citenamefont {Schirmer}\ \emph {et~al.}(2001)\citenamefont
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\bibinfo {author} {\bibfnamefont {A.~I.}\ \bibnamefont {Solomon}},\
}\bibfield {title} {\enquote {\bibinfo {title} {Complete controllability of
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\end{thebibliography}
\onecolumngrid
\begin{center}
\textbf{\large Supplemental Material: \\
Complete Quantum-State Tomography with a Local Random Field}
\end{center}
\twocolumngrid
\section{Proof of the theorem}
Here, we show that if at the measurement times $t=nT_{*}$, with $n=1,\cdots,(d^{2}-1)$, statistically independent Haar-random unitaries $U_{n}$ are created, then with probability 1 the matrix $\mathcal M$ with entries $\mathcal M_{n,m}=\text{Tr}\{U_{n}^{\dagger}MU_{n}B_{m}\}$ is invertible. This result should not be too surprising, since it is well known in the literature that observables created through Haar-random unitary operations are almost always informationally complete (see, e.g., \cite{ohliger2013efficient,merkel2010random}).
We first note that if $U_{n}$ is Haar random, then the Hermitian matrices $U_{n}^{\dagger}M U_{n}$ are independent and uniformly distributed within the set of Hermitian matrices, with the same spectrum as $M$. Equivalently, the row vectors $a_{n}=(\mathcal M_{n,1},\cdots, \mathcal M_{n,d^{2}-1})$ of $\mathcal M$ are independent and uniformly distributed within a vector space $V_{\text{Her}}$. For an explicit characterization of the measure induced on $V_{\text{Her}}$ we refer to~\cite{ohliger2013efficient}. We proceed by recalling a standard result from measure theory: the probability that choosing $d^{2}-1$ independent vectors $a_{n}$ uniformly random (on $V_{\text{Her}}$) are linearly independent is 1, which implies that $\mathcal M$ is invertible with probability 1. For completeness we give the proof below. Clearly, the first vector $a_{1}$ is linearly independent with probability 1. Now assume that the vectors $a_{1},\cdots, a_{d^{2}-2}$ are linearly independent with probability 1, i.e., assume that they span a $d^{2}-2$ dimensional subspace $V$, which intersects $V_{\text{Her}}$. Note that $V$ has measure zero within $V_{\text{Her}}$. Thus, the probability that $a_{d^{2}-1}\notin V$ is 1. Consequently, the probability that $\text{span}\{a_{1},\cdots,a_{d^2-1}\}=V_{\text{Her}}$ is 1, which completes the proof.
\section{System Hamiltonian}
In the NV-center ground-state manifold [the state with $^3A$ symmetry, see Fig.~2(a)] the Hamiltonian can be written as~\cite{doherty13,dreau2012}
\begin{equation}
\label{eq:Hgs}
H_\text{gs}=DS_z^2-\gamma_eBS_z+H_{\text N} -\gamma_{\text C}BI_z+\mathbf{S}\cdot\boldsymbol{\mathcal{A}}\cdot\mathbf{I},
\end{equation}
with the zero-field splitting $D/2\pi=2.87$~GHz. The electron spin-1 and $^{13}$C nuclear spin-1/2 operators are denoted by $\mathbf{S}$ and $\mathbf{I}$, and their respective gyromagnetic ratios are given by $\gamma_e/2\pi=-2.8~$MHz/G and $\gamma_{\text C}/2\pi=1.07~$kHz/G. The Hamiltonian $H_{\text N}$ describes the intrinsic nitrogen nuclear spin of the NV center, including its hyperfine interaction with the electron spin. The nitrogen nuclear spin is polarized into its $m_I=+1$ state during the optical ground-state polarization of the electron spin using a 532~nm green laser pulse~\cite{fischer2013} and remains therein afterward. The part $H_{\text{N}}$ thereby only entails an energy shift $\mathcal{A}_\text{N}/2\pi=\pm 2.16$~MHz of the electronic $m_s=\pm1$ states due to the hyperfine interaction~\cite{felton2009,dreau2012}. The hyperfine tensor $\boldsymbol{\mathcal{A}}$, on the other hand, describes the dipole-dipole interaction between the NV-center electron spin and the carbon nuclear spin. In a secular approximation and with an appropriate $x$ axis we can write this interaction as $\mathbf{S}\cdot\boldsymbol{\mathcal{A}}\cdot\mathbf{I}=\mathcal{A}_{zz}S_zI_z+\mathcal{A}_{zx}S_zI_x$.
Furthermore, the coupling of the electron spin to the microwave field can be described by the Hamiltonian $H_{\text{mw}}(t)=\sqrt{2}f(t)\Omega_1\cos(\omega t)S_x$ in the lab frame, where the factor $\sqrt{2}$ is included for convenience. In the subspace spanned by the $m_s=0$ and $m_s=-1$ states, for the electron-spin operators one can make the two substitutions $S_x\rightarrow\sigma_1^x/\sqrt{2}$ and $S_z\rightarrow(\sigma_1^z-\mathds{1}_1)/2$. Going to an interaction picture with respect to $\omega\sigma_1^z/2$ and performing a rotating-wave approximation, the system Hamiltonian $H_{\text{gs}}+H_{\text{mw}}(t)$ then takes the form~(4), with $\omega_1=\omega-D+\gamma_e B+\mathcal{A}_{\text N}$, $\omega_2=\gamma_nB-\mathcal{A}_{zz}/2$, $\Omega_2=-\mathcal{A}_{zx}/2$, $g_z=\mathcal{A}_{zz}/2$, and $g_x=\mathcal{A}_{zx}/2$. In our sample, we find a longitudinal hyperfine coupling $\mathcal{A}_{zz}/2\pi=(11.832\pm0.005)$~MHz and a transversal coupling $\mathcal{A}_{zx}/2\pi=(2.790\pm0.002)$~MHz. This was shown in electron-spin resonance (ESR) experiments~\cite{yu2018pub} and yields the parameters given in the main text. Drifts in the magnetic field and microwave amplitude between different runs of the experiment lead to
$\{\omega_1,\Omega_1\}/2\pi\in\{-2.974\pm0.247,7.910\pm0.129\}$~MHz, where in every separate run they can be determined up to a precision of a few kHz.
\section{Tomography using noisy data}
While in theory the measurement record $\mathbf{y}$ allows a perfect reconstruction of $\rho$, in practice the expectation measurements always contain noise described by some error $\boldsymbol{\epsilon}$, that adds to $\mathbf{y}$. A common way to reconstruct $\rho$ from noisy data $\tilde{\mathbf{y}}=\mathbf{y}+\boldsymbol{\epsilon}$ is to find the Bloch vector $\mathbf{r}$ that best explains the measured data $\tilde{\mathbf{y}}$. In order to do so, we solve the optimization problem
\begin{equation}
\label{eq:minimization}
\arg \min_{\mathbf{r}}\Vert\tilde{\mathbf{y}} -\mathcal M[f(t)]\mathbf{r} \Vert,
\end{equation}
such that $\rho\geq 0$ and $\text{Tr}\{\rho\}=1$. This optimization is done simultaneously for multiple measurement outcomes $\tilde{\mathbf{y}}^{(j)}$ and corresponding matrices $\mathcal{M}^{(j)}$, e.g., with $j=1,...,10$ in our reconstructions, as shown in Fig.~\ref{fig3}(a). The explicit operator basis $\{B_m\}_{m=1}^{d^2-1}$ that we employ for the representation of $\mathcal{M}$ and $\mathbf{r}$ in our state reconstructions is the standard Pauli-operator basis.
Note that since the error between the Bloch vector $\mathbf{r}$ corresponding to the noiseless measurement record and the one obtained from noisy data is given by $\Vert \mathcal M^{-1}\boldsymbol{\epsilon}\Vert$, pulse shapes minimizing $\Vert \mathcal M^{-1}\Vert$ allow for more robust state reconstruction, while pulses yielding a matrix $\mathcal M$ close to singular are more sensitive to measurement errors. More sophisticated statistical methods as well as different objective functions optimized over the control fields can be employed to design control fields that achieve the best performance for noise-robust quantum-state tomography. We remark here that it is well known that state reconstruction through randomly-created observables is already surprisingly robust against moderate levels of noise~\cite{candes2011probabilistic}. We study this robustness by analyzing $||\mathcal{M}^{-1}||$ for the experimental setting at hand. Therefore, in Fig.~\ref{fig3} we show the time evolution of the quantity $\mathcal{I}$, by which we denote the average of $||\mathcal{M}^{-1}||$ over 1000 realizations of the random pulses~(6), calculated numerically using the parameters of the experiment. The norm of the inverse saturates and the dashed red line indicates the pulse length we employ in our experimental state tomography, since we see no significant decrease in $\mathcal{I}$ after $0.7~\mu$s.
\begin{figure}
\caption{\label{fig3}
\label{fig3}
\end{figure}
\section{Initial-state reconstruction}
For the reconstruction of the density matrices we use the vectors $\tilde{\mathbf{y}}^{(j)}$ obtained from the last 10 data points of the expectation measurements under the 15 random pulses, along with the corresponding matrices $\mathcal{M}^{(j)}$ ($j=1,...,10$) and perform the minimization~\eqref{eq:minimization}. As mentioned in the main text, as a first check we perform the tomography of the state $\rho_0$ after the optical ground-state polarization, i.e., when the preparation stage in Fig.~2(b) is void. The corresponding measurement data and numerical propagation results are shown in Fig.~\ref{fig4}(a). The reconstructed density matrix is depicted as gray bars in Fig.~\ref{fig4}(b) and yields a $97.7\%$ fidelity with the ideal state $\rho_0$, which is indicated by blue transparent bars for comparison.
\begin{figure}
\caption{\label{fig4}
\label{fig4}
\end{figure}
\section{Preparation pulses}
The two non-trivial states we reconstructed in Fig.~2(c) are prepared using the microwave preparation stage [see top panel of Fig.~2(b)]. For the sake of completeness, the pulse shapes of the two preparation pulses are depicted in Fig.~\ref{fig5}. Here, Fig.~\ref{fig5}(a) shows the random pulse employed to create the state shown in the upper panel of Fig.~2(c). The pulse shape for the creation of the highly entangled state shown in the lower panel of Fig.~2(c) was obtained by numerical optimization of the truncated Fourier series~(6), in order to achieve the highest concurrence for a fixed preparation time of 1.8~$\mu$s, and is depicted in Fig.~\ref{fig5}(b).
\begin{figure}
\caption{\label{fig5}
\label{fig5}
\end{figure}
\section{Experimental parameters}
We calibrate the amplitude of the microwave driving field by measuring the frequency of Rabi oscillations of the electron spin, shown in Fig.~\ref{fig6}(a), for the given setting of the AWG and the microwave amplifier. Furthermore, we determine the coherence time $T^*_2$ of the electron spin by performing a free induction decay (FID) experiment, see Fig.~\ref{fig6}(b). Here, the FID is fitted (black line) according to $\langle\sigma_1^x\rangle_t=\exp\{-(t/T^*_2)^2\}$, yielding $T^*_2=0.86~\mu$s. However, under microwave driving the coherence time of the electron spin is significantly prolonged, as compared with the FID coherence time $T^*_2$. This is shown in Fig.~\ref{fig6}(c), where we performed a prolonged Rabi experiment. The data show no appreciable decay of the Rabi oscillations over a time span of 2.5~$\mu$s, which is the longest evolution time in our experiments. This indicates that under the microwave drive the system stays coherent sufficiently long for both the state preparation and the subsequent tomography.
\begin{figure}
\caption{\label{fig6}
\label{fig6}
\end{figure}
\end{document} |
\begin{document}
\title[Correction to A flexible construction\dots]{Correction to the
paper ``A flexible construction of equivariant Floer homology and applications''}
\author{Kristen Hendricks}
\address{Mathematics Department, Rutgers University\\
New Brunswick, NJ 08901}
^\text{th}anks{KH was supported by NSF grant DMS-1663778 and NSF CAREER grant DMS-1751857.}
\email{\href{mailto:[email protected]}{[email protected]}}
\author{Robert Lipshitz}
\address{Department of Mathematics, University of Oregon\\
Eugene, OR 97403}
^\text{th}anks{RL was supported by NSF grant DMS-1149800 (version 1) and
DMS-1810893 (revisions).}
\email{\href{mailto:[email protected]}{[email protected]}}
\author{Sucharit Sarkar}
^\text{th}anks{SS was supported by NSF grant DMS-1643401}
\address{Department of Mathematics, University of California\\
Los Angeles, CA 90095}
\email{\href{mailto:[email protected]}{[email protected]}}
\date{\today}
\begin{abstract}
We correct a mistake regarding almost complex structures on Hilbert
schemes of points in surfaces in~\cite{HEquivariant}. The error does
not affect the main results of the paper, and only affects the
proofs of invariance of equivariant symplectic Khovanov homology and
reduced symplectic Khovanov homology. We give an alternate
proof of the invariance of equivariant symplectic Khovanov homology.
\end{abstract}
\maketitle
\mathfrak{s}ection{The mistake}
At several points in Section 7 of our paper~\cite{HEquivariant} we assert that
an almost complex structure $j$ on the algebraic surface $S$ used to define
symplectic Khovanov homology induces an almost complex structure $\Hilb^n(j)$ on
the Hilbert scheme (or Douady space, following~\cite{Douady66}) $\Hilb^n(S)$ of length $n$ subschemes of $S$. If $j$ is a
complex structure this is true, but there is no known extension of the Hilbert
scheme of points in a complex manifold to the almost-complex case. (Indeed, even
the definition of the Hilbert scheme as a set depends on the complex structure.)
See also Voisin's paper~\cite{Voisin} for some interesting steps in this
direction and further discussion.
This (false) principle is used in a ``cylindrical'' formulation of
symplectic Khovanov homology in~\cite[Lemma 7.10]{HEquivariant}, which is
then used in the proof of stabilization invariance for symplectic Khovanov
homology in~\cite[Section 7.4.1]{HEquivariant}, equivariant symplectic
Khovanov homology in~\cite[Theorem 1.26]{HEquivariant}, and reduced
symplectic Khovanov homology in~\cite[Theorem 7.25]{HEquivariant}. (See
also Abouzaid-Smith's paper~\cite{AbouzaidSmith:arc-alg} for a more careful
cylindrical reformulation of the curves in symplectic Khovanov
homology in certain cases, and Mak-Smith's recent paper~\cite{MakSmith:cylindrical} for a more general cylindrical reformulation.)
Below, we give a corrected, weaker version of the offending Lemma
7.10, and an independent proof of equivariant stabilization invariance
(i.e.,~\cite[Theorem 1.26]{HEquivariant}). We have not been able to
correct the proof of stabilization invariance of reduced symplectic
Khovanov homology (i.e.~\cite[Theorem 7.25]{HEquivariant}). So, reduced
symplectic Khovanov homology is invariant under isotopies and
handleslides, but is only conjectured to be invariant under
stabilization.
\emph{Acknowledgments.} We thank Tomohiro Asano for pointing out our mistake,
Nick Addington for helpful conversations, and the referee for further
corrections and suggestions.
\mathfrak{s}ection{The corrected lemma}
Recall that the curves we consider in the cylindrical formulation of
symplectic Khovanov homology are maps
\[
(7.9)\qquad\!\!
\psi\co(X,\partial X)\to \bigl(\mathbb R\times[0,1]\times S,(\mathbb R\times\{0\}\times (\Genigma'_{A_1}\cup\dots\cup \Genigma'_{A_n}))\cup(\mathbb R\times\{1\}\times (\Genigma_{B_1}\cup\dots\cup \Genigma_{B_n}))\bigr),
\]
where $X$ is a Riemann surface with boundary and $2n$ boundary
punctures, $\psi$ is asymptotic to $\{-\infty\}\times[0,1]\times\mathbf x$
and $\{+\infty\}\times[0,1]\times \mathbf y$, and
$\pi_{\mathbb R\times[0,1]}\circ\psi$ is an $n$-fold branched covering.
The incorrect Lemma 7.10 introduced the following condition for these maps:
\begin{enumerate}[label=(YC)]
\item\label{item:YC} \ \ \ The map $
(\Id \times\Id \times i)\circ \psi \co (X\mathfrak{s}etminus B(\psi))\to \mathbb R\times[0,1]\times\mathbb C
$
is an embedding.
\end{enumerate}
The following is a corrected version of Lemma 7.10:
\noindent\textbf{Lemma 7.10${}^{\boldsymbol{\prime}}$} {\em
The set of holomorphic disks in the complex manifold $\mathfrak{s}sspace{n}$ connecting
$\mathbf x$ to $\mathbf y$ and which are transverse to the big diagonal $\Delta$ is in
bijection with the set of holomorphic maps as in Formula (7.9)
satisfying condition (YC).}
\begin{remark}
To emphasize, the corrected Lemma 7.10${}^{\prime}$ is about a
complex structure on $S$ and the induced complex structure on
$\mathfrak{s}sspace{n}$, not an almost complex structure on $S$ or
$\mathfrak{s}sspace{n}$. In particular, there is no assertion that the moduli
spaces in Lemma 7.10${}^{\prime}$ are transversely cut out. The
corrected Lemma 7.10${}^{\prime}$ is not used in the rest of this
note.
\end{remark}
\mathfrak{s}ection{Corrected proof of equivariant stabilization invariance}
\mathfrak{s}ubsection{Background}\label{sec:background}
\mathfrak{s}ubsubsection{Skein triangles}
The corrected proof of stabilization invariance is similar to our
proof of stabilization invariance for the equivariant Heegaard Floer
homology of branched double covers from~\cite[Theorem
1.24]{HEquivariant}. Here are the analogues for symplectic Khovanov
homology of the results about Heegaard Floer homology that proof used:
\begin{theorem}\cite{SeidelSmith6:Kh-symp}\cite{Waldron:KhSympMaps}\label{thm:SS-invt}
Symplectic Khovanov homology is a link invariant.
\end{theorem}
\begin{figure}
\caption{\textbf{Skein triangle via bridges.}
\label{fig:skein-bridges}
\end{figure}
\begin{theorem}\label{thm:skein-tri}\cite{AbouzaidSmith:KhSympKh}
Let $B$, $B'$, and $B''$ be collections of bridges which differ as
shown in Figure~\ref{fig:skein-bridges} in part of the diagram and
are small isotopic copies of each other in the rest of the
diagram. For any collection
of bridges $A$ there is an exact triangle
\[
\mathbf xymatrix{
& \mathit{HF}(\mathcal{K}_A,\mathcal{K}_B)\ar[dr] & \\
\mathit{HF}(\mathcal{K}_A,\mathcal{K}_{B''})\ar[ur] & & \mathit{HF}(\mathcal{K}_A,\mathcal{K}_{B'})\ar[ll]
}
\]
Further, there are elements
$\alpha\in{\mathit{CF}}(\mathcal{K}_B,\mathcal{K}_{B'})$,
$\beta\in{\mathit{CF}}(\mathcal{K}_{B'},\mathcal{K}_{B''})$, and
$\gamma\in{\mathit{CF}}(\mathcal{K}_{B''},\mathcal{K}_{B})$ so that the maps in the
exact triangle come from counting holomorphic triangles on
$(\mathcal{K}_A,\mathcal{K}_B,\mathcal{K}_{B'})$,
$(\mathcal{K}_A,\mathcal{K}_{B'},\mathcal{K}_{B''})$, and
$(\mathcal{K}_A,\mathcal{K}_{B''},\mathcal{K}_{B})$ with one corner at $\alpha$,
$\beta$, and $\gamma$, respectively.
\end{theorem}
\begin{proof}
Using ideas of Abouzaid-Ganatra, Abouzaid-Smith show that there is
an exact triangle of bimodules over the Fukaya category of
$\mathfrak{s}sspace{n}$ relating the identity bimodule $\Id$, the composition
of a cup and a cap, and the half-twist $\tau$; see~\cite[Proposition
7.4]{AbouzaidSmith:KhSympKh}. Evaluating these three bimodules on
$\mathcal{K}_B$ as the first object gives three one-sided
modules---$\mathcal{K}_B$, a module equivalent to $\mathcal{K}_{B'}$, and a
module equivalent to $\mathcal{K}_{B''}$. Since equivalences preserve
exact triangles, this implies that there is an exact triangle
relating $\mathcal{K}_B$, $\mathcal{K}_{B'}$, and $\mathcal{K}_{B''}$. In
particular, the maps $\mathcal{K}_B\to\mathcal{K}_{B'}$,
$\mathcal{K}_{B'}\to\mathcal{K}_{B''}$, and $\mathcal{K}_{B''}\to\mathcal{K}_B$ come
from elements $\alpha\in{\mathit{CF}}(\mathcal{K}_B,\mathcal{K}_{B'})$,
$\beta\in{\mathit{CF}}(\mathcal{K}_{B'},\mathcal{K}_{B''})$, and
$\gamma\in{\mathit{CF}}(\mathcal{K}_{B''},\mathcal{K}_{B})$, so that for any other
Lagrangian $L$, counting holomorphic triangles on
$(L,\mathcal{K}_B,\mathcal{K}_{B'})$ with a corner at $\alpha$ (respectively
on $(L,\mathcal{K}_{B'},\mathcal{K}_{B''})$ with a corner at $\beta$, on
$(L,\mathcal{K}_{B''},\mathcal{K}_{B})$ with a corner at $\gamma$) gives an
exact triangle relating $\mathit{HF}(L,\mathcal{K}_B)$, $\mathit{HF}(L,\mathcal{K}_{B'})$,
and $\mathit{HF}(L,\mathcal{K}_{B''})$. Taking $L=\mathcal{K}_A$ gives the result.
\end{proof}
\begin{remark}
For appropriate choices of diagrams, all of the
generators of ${\mathit{CF}}(\mathcal{K}_B,\mathcal{K}_{B'})$,
${\mathit{CF}}(\mathcal{K}_{B'},\mathcal{K}_{B''})$, and
${\mathit{CF}}(\mathcal{K}_{B''},\mathcal{K}_{B})$ are fixed by the $O(2)$-action.
\end{remark}
\mathfrak{s}ubsubsection{Projected domains}
While the arguments below take place in the Hilbert scheme, and avoid
a cylindrical formulation of symplectic Khovanov homology, we will
make use of the concept and some properties of projected domains
from~\cite[Section 7.1.3]{HEquivariant}. Fix a bridge diagram
$(A=\{A_i\},B=\{B_i\})$ for a link $L$. Choose a point $z_i$ in each
connected component of $\mathbb C\mathfrak{s}etminus (A\cup B)$. Each $z_i$ gives a
subvariety $\pi^{-1}(z_i)\times\Hilb^{n-1}(S)$ of $\Hilb^n(S)$
consisting of those length-$n$ subschemes where at least one point
lies over $z_i$. Fix a neighborhood $U_i$ of
$\pi^{-1}(z_i)\times\Hilb^{n-1}(S)$, small enough that the closure of
$U_i$ is disjoint from $\mathcal{K}_A\cup\mathcal{K}_B$.
Let $R$ denote the non-compact region of
$\mathbb C\mathfrak{s}etminus(A\cup B)$. Then $\pi^{-1}(R)\times\Hilb^{n-1}(S)$, the
set of length-$n$ subschemes where at least one point lies over $R$,
is an open subset of $\Hilb^n(S)$. Order the $z_i$ above so that $z_0\in R$.
We will only consider almost complex structures $J$ on $\mathfrak{s}sspace{n}$,
compatible with the symplectic form described in
Section~\ref{sec:convex} below, with the following three additional
properties:
\begin{enumerate}[label=(J-\arabic*)]
\item\label{item:J-first} The almost complex structure $J$ agrees with the standard
complex structure $\Hilb^n(j)$ on $U_i$. In particular, each
$(\pi^{-1}(z_i)\times\Hilb^{n-1}(S))\cap\mathfrak{s}sspace{n}$ is a
$J$-holomorphic submanifold.
\item The almost complex structure $J$ agrees with the standard
complex structure $\Hilb^n(j)$ on $\pi^{-1}(\overline{R})\times\Hilb^{n-1}(S)$.
\item\label{item:J-convex}\label{item:J-last} The almost complex structure $J$ agrees
with the standard complex structure $\Hilb^n(j)$ outside a compact
subset. (This is not implied by the previous restriction because the
fibers of $\pi$ are themselves non-compact.)
\end{enumerate}
(Compare~\cite[Definition 3.1]{OS04:HolomorphicDisks}.)
Since $(\pi^{-1}(z_i)\times\Hilb^{n-1}(S))\cap\mathfrak{s}sspace{n}$ is proper
and disjoint from $\mathcal{K}_A\cup\mathcal{K}_B$, each
$(\pi^{-1}(z_i)\times\Hilb^{n-1}(S))\cap\mathfrak{s}sspace{n}$ is dual to a
relative cohomology class
$PD[\pi^{-1}(z_i)\times\Hilb^{n-1}(S)]\in
H^2(\mathfrak{s}sspace{n},\mathcal{K}_A\cup\mathcal{K}_B)$. Given a Whitney disk $u$
for $(\mathcal{K}_A,\mathcal{K}_B)$, let $n_{z_i}(u)$ be the result of
evaluating the cohomology class
$PD[\pi^{-1}(z_i)\times\Hilb^{n-1}(S)]$ on $[u]$. The tuple
$(n_{z_i}(u))$ is the \emph{projected domain} of $u$. Sometimes, we
think of the projected domain as an element of
$H_2(\mathbb C\cup\{\infty\},A\cup B)$, where $n_{z_i}(u)$ is the coefficient
of the region containing $z_i$.
If $u$ is $J$-holomorphic then the projected domain of $u$ has the following properties:
\begin{enumerate}[label=(D-\arabic*)]
\item For each $i$, $n_{z_i}(u)\geq 0$. (Compare~\cite[Lemma
3.2]{OS04:HolomorphicDisks}.)
\item\label{item:domain-2} The value $n_{z_0}(u)=0$, and in fact $u(D^2)\cap (R\times\Hilb^{n-1}(S))=\varnothing$.
\item If $n_{z_i}(u)=0$ for each $i$ then $u$ is constant.
\end{enumerate}
The first two statements follow from positivity of intersections of
complex submanifolds. The third follows from the fact that any such
Whitney disk is homotopic to a constant disk, hence has zero area, and
hence is itself constant.
Finally, no non-constant, $J$-holomorphic Whitney disk is
entirely contained in the subspace
\[
\bigl(\pi^{-1}(\overline{R})\times\Hilb^{n-1}(S)\bigr)\cup \bigcup_i U_i
\]
where we have imposed constraints on $J$. So, the usual transversality
arguments for $J$-holomorphic curves (see, e.g.,~\cite[Chapter
3]{MS04:HolomorphicCurvesSymplecticTopology}) imply generic
transversality for (one parameter families of) almost complex
structures $J$ in this class.
\mathfrak{s}ubsubsection{Symplectic forms and convexity at infinity}\label{sec:convex}
In~\cite{HEquivariant}, we worked in the setting of symplectic
manifolds which are convex at infinity, in the sense
of~\cite{EliashbergGromov91:convex}. An alternative notion of
convexity comes from~\cite[Section 0.4]{Gromov85}: a symplectic
manifold $M$ is \emph{$I$-convex at infinity} (or simply
\emph{$I$-convex}) if there is an exhaustion of $M$ by open sets
$V_1\mathfrak{s}ubset V_2\mathfrak{s}ubset\cdots$ with $\overline{V_i}$ compact and such that if
$u\co D^2\to M$ is $I$-holomorphic with $u(\partial D^2)\mathfrak{s}ubset V_i$ then
$u(D^2)\mathfrak{s}ubset V_{i+1}$. For example, the maximum modulus theorem
implies that any affine variety is $I$-convex. Observe also that the
notion of a symplectic manifold being $I$-convex depends only on $I$
near infinity (i.e., outside a compact set). As noted
in~\cite{HLS:Lie}, the arguments from~\cite{HEquivariant} work for symplectic manifolds
which are $I$-convex at infinity for some $G$-invariant almost complex
structure $I$ defined outside a compact set.
In particular, the $I$-convex setting is convenient for
symplectic Khovanov homology. Let $I$ be the complex structure on
$\mathfrak{s}sspace{n}$ inherited from $\Hilb^n(S)$. The complex structure $I$
is $O(2)$-invariant and, since $\mathfrak{s}sspace{n}$ is an affine variety (see
also~\cite[Theorem 1.2]{Manolescu06:nilpotent}), $\mathfrak{s}sspace{n}$ is
$I$-convex. Inspired by Perutz's construction
in~\cite{Perutz07:HamHand}, in~\cite[Lemma
5.5]{AbouzaidSmith:arc-alg}, Abouzaid-Smith construct a K\"ahler form
$\omega'$ on $\Hilb^n(S)$ (with respect to $I$) whose restriction to
$\mathfrak{s}sspace{n}$ is exact and agrees with the product form outside a
neighborhood of the diagonal. (In the notation of~\cite[Lemma
5.5]{AbouzaidSmith:arc-alg}, we choose $\omega$ to be an exact
K\"ahler form on $S$.) We saw in~\cite[Lemma 4.24]{HLS:Lie} that
averaging $\omega'$ gives an $O(2)$-invariant K\"ahler form on
$\mathfrak{s}sspace{n}$ (still with respect to $I$) which is still exact and
still agrees with the product symplectic form outside a neighborhood
of the diagonal.
If $V$ is an open subset of $\mathbb C$, then
$\mathfrak{s}sspace{n}\cap \Hilb^n(\pi^{-1}(V))$ is also $I$-convex, for the
same reason that point~\ref{item:domain-2}, above, holds. In
particular, complex structures satisfying
condition~\ref{item:J-convex} satisfy the convexity-at-infinity
condition required for the constructions in~\cite{HEquivariant}.
\mathfrak{s}ubsection{General K\"unneth theorem for the freed Floer complex}
Another ingredient in our proof of Proposition~\ref{prop:equi-stab-inv} is
a K\"unneth theorem for equivariant symplectic Khovanov
homology. We give a general K\"unneth theorem for the equivariant
Floer complex in this section and specialize to the case of symplectic
Khovanov homology in the next section.
\begin{remark}
Throughout this section, the convexity assumption
from~\cite[Hypothesis 3.2]{HEquivariant} can be replaced with
$I$-convexity for some $G$-invariant almost complex structure $I$
defined outside a compact set. See Section~\ref{sec:convex} for
further discussion and references.
\end{remark}
\begin{theorem}\label{thm:external-Kunneth}
Suppose that $H$ acts on $(M,L_0,L_1)$ and $H'$ acts on
$(M',L'_0,L'_1)$, both satisfying~\cite[Hypothesis 3.2]{HEquivariant}.
Then the action of $H\times H'$ on $(M\times M',L_0\times L'_0,L_1\times L'_1)$
satisfies~\cite[Hypothesis 3.2]{HEquivariant} and there is a quasi-isomorphism
\[
\ECF[H\times H'](L_0\times L'_0,L_1\times L'_1)\mathfrak{s}imeq
\ECF[H](L_0,L_1)\otimes_{{\FF_2}} \ECF[H'](L'_0,L'_1)
\]
of chain complexes over ${\FF_2}[H\times H']$.
\end{theorem}
\begin{proof}
To keep notation simple, we will prove the result in the case that
$L_0\pitchfork L_1$ and $L'_0\pitchfork L'_1$; the extension to
non-transverse intersections is the same as~\cite[Section
3.6]{HEquivariant}.
Observe that if $\mathbf wt{J}$ (respectively $\mathbf wt{J}'$) is an eventually cylindrical
almost complex structure on $M$ (respectively $M'$) so that the moduli spaces
of $\mathbf wt{J}$-holomorphic Whitney disks in $M$ (respectively
$\mathbf wt{J}'$-holomorphic Whitney disks in $M'$) are transversely cut out then the
moduli space of $(\mathbf wt{J}\times\mathbf wt{J}')$-holomorphic Whitney disks in
$M\times M'$ are transversely cut out. More generally, given $k$-parameter
families $\mathbf wt{J}(t_1,\dots,t_k)$ and $\mathbf wt{J}'(t_1,\dots,t_k)$ of eventually
cylindrical almost complex structures on $M$ and $M'$, $t_i\in[0,1]$, the
moduli space of holomorphic Whitney disks with respect to the $k$-parameter
family $(\mathbf wt{J}\times\mathbf wt{J}')$ is tranvsersally cut out if the moduli spaces
with respect to $\mathbf wt{J}$ and $\mathbf wt{J}'$ and transversally cut out and intersect
transversally in $[0,1]^k$, in which case the moduli space with respect to
$(\mathbf wt{J}\times\mathbf wt{J}')$ is the fiber product, over $[0,1]^k$, of the moduli
spaces with respect to $\mathbf wt{J}$ and $\mathbf wt{J}'$.
Next, observe that $\mathscr{E}(H\times H')=(\mathscr{E} H)\times (\mathscr{E} H')$.
Now, fix sufficiently generic homotopy coherent diagrams
$F\co \mathscr{E} H\to \ol{\mathcal{J}}_M$ and
$F'\co \mathscr{E} H'\to \ol{\mathcal{J}}_{M'}$. Construct a homotopy coherent
diagram $FF'\co \mathscr{E}(H\times H')\to \ol{\mathcal{J}}_{M\times M'}$ as
follows. On objects, define $FF'(h,h')=F(h)\times F'(h')$. More generally, define
\begin{multline*}
FF'((f_n,f'_n),\dots,(f_1,f'_1))(t_1,\dots,t_{n-1})\\
=[F(f_n,\dots,f_1)(t_1,\dots,t_{n-1})]\times
[F'(f'_n,\dots,f'_1)(t_1,\dots,t_{n-1})].
\end{multline*}
Perturbing $F$ and $F'$ slightly, we may assume that the moduli spaces with respect to
the family of almost complex structures $F'(f'_n,\dots,f'_1)$ are transverse
to the moduli spaces with respect to $F(f_n,\dots,f_1)$, so $FF'$ is
sufficiently generic. Since $F$ and $F'$ were already generic, this perturbation does
not change the functors $G\co \mathscr{E} H\to \mathsf{Kom}$,
$G'\co \mathscr{E} H'\to \mathsf{Kom}$.
Given a
sequence of morphisms
$h_0\mathfrak{s}tackrel{f_1}{\longrightarrow}
h_1\mathfrak{s}tackrel{f_2}{\longrightarrow}\cdots\mathfrak{s}tackrel{f_k}{\longrightarrow}
h_k$ in $\mathscr{E} H$ and a sequence of morphisms
$h'_0\mathfrak{s}tackrel{f'_1}{\longrightarrow}
h'_1\mathfrak{s}tackrel{f'_2}{\longrightarrow}\cdots\mathfrak{s}tackrel{f'_\ell}{\longrightarrow}
h'_\ell$ in $\mathscr{E} H'$, a \emph{shuffle} of these sequences is a
sequence of morphisms
$(h_0,h'_0)\mathfrak{s}tackrel{g_0}{\longrightarrow}\cdots\mathfrak{s}tackrel{g_{k+\ell}}{\longrightarrow}(h_k,h'_\ell)$,
where each $g_i$ either has the form $(f_j,\Id)$ or
$(\Id,f'_j)$, and the morphisms $f_1,\dots,f_k$ appear in
order, once each, in this sequence, as do $f'_1,\dots,f'_\ell$. For
example, the three shuffles of
$h_0\mathfrak{s}tackrel{f_1}{\longrightarrow}h_1\mathfrak{s}tackrel{f_2}{\longrightarrow}h_2$
and $h'_0\mathfrak{s}tackrel{f'_1}{\longrightarrow}h'_1$ are
\begin{align*}
&(h_0,h'_0)\mathfrak{s}tackrel{f_1\times\Id}{\longrightarrow}(h_1,h'_0)\mathfrak{s}tackrel{f_2\times\Id}{\longrightarrow}(h_2,h'_0)\mathfrak{s}tackrel{\Id\times f'_1}{\longrightarrow} (h_2,h'_1)\\
&(h_0,h'_0)\mathfrak{s}tackrel{f_1\times\Id}{\longrightarrow}(h_1,h'_0)\mathfrak{s}tackrel{\Id\times f'_1}{\longrightarrow}(h_1,h'_1)\mathfrak{s}tackrel{f_2\times\Id}{\longrightarrow} (h_2,h'_1)\\
&(h_0,h'_0)\mathfrak{s}tackrel{\Id\times f'_1}{\longrightarrow}(h_0,h'_1)\mathfrak{s}tackrel{f_1\times\Id}{\longrightarrow}(h_1,h'_1)\mathfrak{s}tackrel{f_2\times\Id}{\longrightarrow} (h_2,h'_1).
\end{align*}
The shuffles correspond to permutations $\mathfrak{s}igma\in S_{k+\ell}$ so
that $\mathfrak{s}igma|_{\{1,\dots,k\}}$ and $\mathfrak{s}igma|_{\{k+1,\dots,k+\ell\}}$
are increasing.
Notice that if $(g_1,\dots,g_{k+\ell})$ is a shuffle then the moduli spaces of
index $-k-\ell+1$ with respect to $FF'(g_{k+\ell},\dots,g_{1})$ are empty unless
either $k=0$ or $\ell=0$. Indeed, the family
of almost complex structures $FF'(g_{k+\ell},\dots,g_{1})$
factors through a map to $[0,1]^{k+\ell-2}$,
so
Maslov index $1-k-\ell$ moduli spaces are empty. The exception is if $k=0$
(respectively $\ell=0$), in which case the moduli space is identified with the
moduli space of $F'(f'_\ell,\dots,f'_1)$-holomorphic disks (respectively
$F(f_k,\dots,f_1)$-holomorphic disks), multiplied by constant disks in the
other factor.
Let $G\co \mathscr{E} H\to \mathsf{Kom}$, $G'\co \mathscr{E} H'\to\mathsf{Kom}$, and
$GG'\co\mathscr{E} (H\times H')\to\mathsf{Kom}$ be the homotopy coherent diagrams
corresponding to $F$, $F'$, and $FF'$, respectively. With notation as
in~\cite[Definition 3.11]{HEquivariant}, define a map
\[
\eta\co (\hocolim G)\otimes (\hocolim G')\to \hocolim GG'
\]
by
\begin{multline*}
\eta\bigl( (f_k,\dots,f_1;\{0,1\}^{\otimes k};x)\otimes(f'_\ell,\dots,f'_1;\{0,1\}^{\otimes \ell};y) \bigr)
=\hspace{-2em}\mathfrak{s}um_{\text{shuffles }g_1,\dots,g_{k+\ell}}\hspace{-2em}\bigl(g_{k+\ell},\dots,g_1;\{0,1\}^{\otimes k+\ell};x\otimes y\bigr).
\end{multline*}
We verify that $\eta$ is a chain map. The terms arising from taking
the differential of $x$ or $y$ before or after applying $\eta$
clearly cancel in pairs, so we will ignore them from here on.
The remaining terms in $\eta\circ\partial$ are:
\begin{align*}
&\mathfrak{s}um_{i=1}^{k}\eta\bigl( (f_k,\dots,f_1;\{0,1\}^{\otimes i-1}\otimes 0\otimes\{0,1\}^{k-i};x)\otimes(f'_\ell,\dots,f'_1;\{0,1\}^{\otimes \ell};y) \bigr)\\
&\qquad\qquad\qquad+
\eta\bigl( (f_k,\dots,f_1;\{0,1\}^{\otimes i-1}\otimes 1\otimes\{0,1\}^{k-i}\otimes ;x)\otimes(f'_\ell,\dots,f'_1;\{0,1\}^{\otimes \ell};y) \bigr)\\
&\qquad+
\mathfrak{s}um_{i=1}^{\ell}\eta\bigl( (f_k,\dots,f_1;\{0,1\}^{\otimes k} ;x)\otimes(f'_\ell,\dots,f'_1;\{0,1\}^{\otimes i-1}\otimes 0\otimes\{0,1\}^{\otimes \ell-i} ;y) \bigr)
\\
&\qquad\qquad\qquad+\eta\bigl( (f_k,\dots,f_1;\{0,1\}^{\otimes k} ;x)\otimes(f'_\ell,\dots,f'_1;\{0,1\}^{\otimes i-1}\otimes 1\otimes\{0,1\}^{\otimes \ell-i} ;y) \bigr)\\
&=
\mathfrak{s}um_{i=1}^{k-1}\eta\bigl( (f_k,\dots,f_{i+1};\{0,1\}^{k-i};G(f_i,\dots,f_1)(\{0,1\}^{i-1}\otimes x)\otimes(f'_\ell,\dots,f'_1;\{0,1\}^{\otimes \ell};y) \bigr)\\
&\qquad\qquad\qquad+
\eta\bigl( (f_k,\dots,f_{i+1}\circ f_i,\dots, f_1;\{0,1\}^{\otimes k-1};x)\otimes(f'_\ell,\dots,f'_1;\{0,1\}^{\otimes \ell};y) \bigr)\\
&\qquad+ \eta\bigl( (f_{k-1},\dots, f_1;\{0,1\}^{\otimes k-1};x)\otimes(f'_\ell,\dots,f'_1;\{0,1\}^{\otimes \ell};y) \bigr)\\
&\qquad+
\mathfrak{s}um_{i=1}^{\ell}\eta\bigl( (f_k,\dots,f_1;\{0,1\}^{\otimes k} ;x)\otimes(f'_\ell,\dots,f'_{i+1};\{0,1\}^{\otimes \ell-i} ;G'(f'_i,\dots,f'_1)(\{0,1\}^{i-1}\otimes y) \bigr)
\\
&\qquad\qquad\qquad+\eta\bigl( (f_k,\dots,f_1;\{0,1\}^{\otimes k} ;x)\otimes (f'_\ell,\dots,f'_{i+1}\circ f'_i,\dots,f'_1;\{0,1\}^{\otimes \ell-1};y) \bigr)\\
&\qquad+\eta\bigl( (f_k,\dots,f_1;\{0,1\}^{\otimes k} ;x)\otimes (f'_{\ell-1},\dots,f'_1;\{0,1\}^{\otimes \ell-1};y) \bigr).
\end{align*}
Call the terms on the six lines type A1--A6.
The remaining terms in $\partial\circ\eta$ are
\begin{align*}
\mathfrak{s}um_{\text{shuffles }g_1,\dots,g_{k+\ell}}
&\mathfrak{s}um_{i=1}^{k+\ell-1}\bigl(g_{k+\ell},\dots,g_{i+1};\{0,1\}^{k+\ell-i};GG'(g_i,\dots,g_1)(\{0,1\}^{i-1}\otimes (x\otimes y))\bigr)\\
&\qquad+
(g_{k+\ell},\dots,g_{i+1}\circ g_i,\dots, g_1;\{0,1\}^{\otimes k+\ell-1};(x\otimes y))\\
&+(g_{k+\ell-1},\dots, g_1;\{0,1\}^{\otimes k+\ell-1};(x\otimes y)).
\end{align*}
Call the terms on the three lines type B1--B3.
\begin{itemize}
\item For type B1 terms, if some $g_j$, $1\leq j\leq i$, is of the
form $f\otimes\Id$ while another $g_{j'}$, $1\leq j'\leq i$, is of
the form $\Id\otimes f'$ then $GG'(g_i,\dots,g_1)=0$, because the
corresponding $FF'$-moduli spaces are empty. The remaining type B1
terms cancel with the type A1 and type A4 terms.
\item For type B2 terms, if $g_{i+1}=f_j\otimes \Id$ and
$g_i=\Id\otimes f_{j'}'$ then this term cancels with the
corresponding term of the shuffle $(g'_1,\dots,g'_{k+\ell})$ which
agrees with $(g_1,\dots,g_{k+\ell})$ except that
$g'_{i}=f_{j}\otimes \Id$ and $g'_{i+1}=\Id\otimes f'_{j'}$. The
remaining type B2 terms cancel with the type A2 and type A5 terms.
\item For the type B3 term, if $g_{k+\ell}$ is of the form
$(f_k,\Id)$, then it cancels with the type A3 term, and if
$g_{k+\ell}$ is of the form $(\Id,f'_\ell)$, then it cancels with
the type A6 term.
\end{itemize}
Thus, $\eta$ is a chain map.
Clearly, $\eta$ intertwines the actions of $H\times H'$.
For any objects $h\in H$ and $h'\in H'$, the diagram
\[
\mathbf xymatrix{
(\hocolim G)\otimes (\hocolim G') \ar[r]^-\eta& \hocolim GG'\\
G(h)\otimes G(h')\ar[r]_\cong \ar[u]^\mathfrak{s}imeq &GG'(h\times h')\ar[u]_\mathfrak{s}imeq
}
\]
commutes.
Since the bottom horizontal arrow is an isomorphism and the two vertical
arrows are quasi-isomorphisms, this implies that $\eta$ is a quasi-isomorphism
as well. This proves the result.
\end{proof}
\begin{corollary}\label{cor:internal-Kunneth}
Suppose that $H$ acts on both $(M,L_0,L_1)$ and $(M',L'_0,L'_1)$,
both satisfying~\cite[Hypothesis 3.2]{HEquivariant}. Endowing $(M\times M',L_0\times L_0',L_1\times L_1')$ with the diagonal action of $H$, there is a quasi-isomorphism
\[
\ECF[H](L_0\times L'_0,L_1\times L'_1)\mathfrak{s}imeq
\ECF[H](L_0,L_1)\otimes_{{\FF_2}} \ECF[H](L'_0,L'_1)
\]
as chain complexes over ${\FF_2}[H]$ with $H$ acting by the diagonal
action on the right hand side.
\end{corollary}
\begin{proof}
With notation as in the proof of Theorem~\ref{thm:external-Kunneth},
the diagonal map $H\to H\times H$ induces an inclusion map
$\Delta\co \mathscr{E} H\hookrightarrow \mathscr{E} H\times \mathscr{E} H$. Composing with the
functor $FF'$ gives a homotopy coherent $\mathscr{E} H$-diagram of
almost complex structures $(FF')\circ\Delta$. The
corresponding homotopy coherent diagram of chain complexes is
$(GG')\circ\Delta$. There is an induced map
\begin{equation}\label{eq:aux-Kunneth-map}
\hocolim_{\mathscr{E} H}[(GG')\circ \Delta]\to \hocolim_{\mathscr{E} H\times\mathscr{E} H} (GG').
\end{equation}
This map clearly respects the ${\FF_2}[H]$-module structure, and
using Theorem~\ref{thm:external-Kunneth}, the two terms are
quasi-isomorphic to $\ECF[H](L_0\times L'_0,L_1\times L'_1)$ and
$\ECF[H](L_0,L_1)\otimes_{{\FF_2}} \ECF[H](L'_0,L'_1)$ over
${\FF_2}[H]$. Since for any object $h$ of $\mathscr{E} H$ (i.e., element
$h\in H$) the inclusion of $G(h)\otimes G(h)$ into both
$\hocolim_{\mathscr{E} H\times\mathscr{E} H}(GG')$ and
$\hocolim_{\mathscr{E} H}[(GG')\circ \Delta]$ are quasi-isomorphisms, the
map~\eqref{eq:aux-Kunneth-map} is also a quasi-isomorphism. This
proves the result.
\end{proof}
\begin{corollary}\label{cor:internal-relaxed-Kunneth}
Suppose that $H$ acts (symplectically) on symplectic manifolds
$M,M',N$ and suppose there are $H$-invariant open subsets $V\mathfrak{s}ubset
M$, $V'\mathfrak{s}ubset M'$ and $U\mathfrak{s}ubset N$ containing $H$-invariant closed
Lagrangians $L_0,L_1\mathfrak{s}ubset V$, $L'_0,L'_1\mathfrak{s}ubset V'$, and
$K_0,K_1\mathfrak{s}ubset U$ such that the actions of $H$ on $(M,L_0,L_1)$,
$(M',L'_0,L'_1)$, $(N,K_0,K_1)$ all satisfy~\cite[Hypothesis
3.2]{HEquivariant}, and $(U,K_0,K_1)$ is identified
$H$-symplectically with the product $(V\times V',L_0\times
L'_0,L_1\times L'_1)$. As in the proofs of
Theorem~\ref{thm:external-Kunneth} and
Corollary~\ref{cor:internal-Kunneth}, suppose that there exist
systems of eventually cylindrical almost complex structures $F$ and
$F'$ for $M$ and $M'$ and extensions $\mathbf wt{(FF')\circ\Delta}$ of
$(FF')\circ\Delta$ from $U$ to all of $N$, so that $F$, $F'$, and $\mathbf wt{(FF')\circ\Delta}$ are
regular for the freed complexes $\ECF[H](L_0,L_1)$,
$\ECF[H](L'_0,L'_1)$, and $\ECF[H](K_0,K_1)$ and for which the
defining holomorphic strips all lie inside $V$, $V'$ and $U$. Then
there is a quasi-isomorphism
\[
\ECF[H](K_0,K_1)\mathfrak{s}imeq
\ECF[H](L_0,L_1)\otimes_{{\FF_2}} \ECF[H](L'_0,L'_1)
\]
as chain complexes over ${\FF_2}[H]$ with $H$ acting by the diagonal
action on the right hand term.
\end{corollary}
\begin{proof}
This follows immediately from Corollary~\ref{cor:internal-Kunneth}.
\end{proof}
\mathfrak{s}ubsection{The K\"unneth theorem for equivariant symplectic Khovanov homology}
In this section we prove an equivariant version of Waldron's K\"unneth
theorem for symplectic Khovanov homology in~\cite[Theorem
1.2]{Waldron:KhSympMaps}.
We will use the following elementary lemma:
\begin{lemma}\label{lem:elementary}
Let $p(z)$ be a complex polynomial which has simple roots and no
zeros in the open unit disk $D$. Then there is a smooth 1-parameter
family of complex polynomials $p_t(z)$, each with no zeros in $D$,
and only simple roots anywhere,
interpolating between $p(z)$ and the constant function $1$.
\end{lemma}
\begin{proof}
The proof is by induction on the degree $n$ of
$p(z)=a_0+\cdots+a_nz^n$. By multiplying by a path in $\mathbb C^\times$
from $1$ to $1/a_n$, we may assume $p(z)$ is monic. A monic
polynomial is uniquely determined by its roots.
So, there is a path from $p(z)$ to the polynomial
$(z-2)(z-3)\cdots(z-n-1)$, simply by moving all the roots outside
the unit disk. Next, let $q(z)=(z-2)(z-3)\cdots(z-n)$ and consider
the path of polynomials
\[
p_t(z)=[(1-t)z-n-1)]q(z).
\]
The roots of $p_t(z)$ are $2,3,\dots,n$ and $(n+1)/(1-t)$, all of
which lie outside the unit circle and are distinct. The polynomial
$p_1(z)$ has degree $(n-1)$, and all roots outside the unit
circle. By induction, this completes the proof. (Note that when
concatenating the paths in the different steps of the proof, one needs
to reparameterize the paths so the concatenation is smooth.)
\end{proof}
\begin{proposition}\label{prop:disjoint-union}
Given bridge diagrams $L$ and $L'$, there is a quasi-isomorphism of chain complexes over ${\FF_2}[D_{2^m}]$,
\[
\mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L\amalg L')\mathfrak{s}imeq \mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L)\otimes_{{\FF_2}}\mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L'),
\]
where the right-hand side has the diagonal action of $D_{2^m}$.
\end{proposition}
\begin{proof}
For symplectic Khovanov homology itself, this is~\cite[Theorem 1.2]{Waldron:KhSympMaps}.
For the freed Floer complex, we will deduce this result from
Corollary~\ref{cor:internal-relaxed-Kunneth} after choosing suitable
open sets $U$, $V$, and $V'$ and deforming the symplectic forms on
the open set $U$ to be a product (without losing control of the
holomorphic curves).
Let $b_1,\dots,b_{2n}$ be the endpoints of the bridges in $L$ (so
$n$ be the number of bridges in $L$) and $b_{2n+1},\dots,b_{2n+2n'}$
the endpoints of the bridges in $L'$. After an isotopy, we may
assume there are disjoint open disks $U_L,U_{L'}\mathfrak{s}ubset\mathbb C$, containing
$L$ and $L'$. Let
\begin{align*}
S_L&=\{(u,v,z)\in\mathbb C^3\mid u^2+v^2+(z-b_1)\cdots(z-b_{2n})=0\}\\
S_{L'}&=\{(u,v,z)\in\mathbb C^3\mid u^2+v^2+(z-b_{2n+1})\cdots(z-b_{2n+2n'})=0\}\\
S_{L\amalg L'}&=\{(u,v,z)\in\mathbb C^3\mid u^2+v^2+(z-b_1)\cdots(z-b_{2n+2n'})=0\}.
\end{align*}
Let $\mathbf wt{U}_L$ (respectively $\mathbf wt{U}_{L'}$) be the preimage of $U_L$
(respectively $U_{L'}$) in $S_{L\amalg L'}$. Let $V_L$ (respectively
$V_{L'}$) be the preimage of $U_L$ in $S_L$ (respectively the preimage
of $U_{L'}$ in $S_{L'}$).
By Point~\ref{item:domain-2} in Section~\ref{sec:background}, any
holomorphic Whitney disk in $\mathfrak{s}sspace{n+n'}$ lies in the subspace
$U\coloneqq [\Hilb^n(\mathbf wt{U}_L)\times \Hilb^{n'}(\mathbf wt{U}_{L'})]\cap\mathfrak{s}sspace{n+n'}$. If
we let $\nabla$ denote the subspace of $\Hilb^n(\mathbf wt{U}_L)$
(respectively $\Hilb^{n'}(\mathbf wt{U}_{L'})$) where the projection to $\mathbb C$
has length less than $n$ (respectively $n'$) then
\[
[\Hilb^n(\mathbf wt{U}_L)\times \Hilb^{n'}(\mathbf wt{U}_{L'})]\cap\mathfrak{s}sspace{n+n'}=[\Hilb^n(\mathbf wt{U}_L)\mathfrak{s}etminus\nabla]\times[\Hilb^{n'}(\mathbf wt{U}_{L'})\mathfrak{s}etminus\nabla].
\]
With respect to the restrictions of the averaged symplectic forms on
$\Hilb^{n+n'}(S_{L\amalg L'})$ from~\cite[Lemma 4.24]{HLS:Lie} (see
also Section~\ref{sec:convex}), this identification is a
symplectomorphism, and it is also biholomorphic with respect to the
standard complex structures.
Let
\begin{align*}
V&=\Hilb^n(V_L)\mathfrak{s}etminus\nabla \mathfrak{s}ubset\mathfrak{s}sspace{n}\\
V'&=\Hilb^{n'}(V_{L'})\mathfrak{s}etminus\nabla \mathfrak{s}ubset\mathfrak{s}sspace{n'}.
\end{align*}
By the general K\"unneth theorem for the freed Floer complex
(Corollary~\ref{cor:internal-relaxed-Kunneth}) it suffices to show
that the freed Floer complex of
$(\Genigma_{A_1}\times\cdots\times\Genigma_{A_n},\Genigma_{B_1}\times\cdots\times\Genigma_{B_n})$
inside $\Hilb^n(\mathbf wt{U}_L)\mathfrak{s}etminus\nabla$ is quasi-isomorphic to
their freed Floer complex inside $V$
(and similarly for $L'$).
By Lemma~\ref{lem:elementary}, the polynomials
$p_0(z)=(z-b_1)\cdots(z-b_{2n+2n'})$ and
$p_1(z)=(z-b_1)\cdots(z-b_{2n})$ can be connected by a smooth family
of polynomials $p_t(z)$, $t\in[0,1]$ whose roots in $U$ are exactly
$b_1,\dots,b_{2n}$, and so that all the roots of $p_t(z)$ are
simple. Let $S_t=\{(u,v,z)\in\mathbb C^3\mid u^2+v^2+p_t(z)=0\}$ and let
$V_t$ be the preimage of $U$ in $S_t$ ($t\in[0,1]$). The subspaces $V_t$ form a
smooth family of open complex surfaces. (In particular, each $S_t$
is smooth, since $p_t$ has only simple roots.)
Each $V_t$ contains Lagrangian spheres $\Genigma_{A_i}$ and
$\Genigma_{B_i}$ for $i=1,\dots,n$. Taking their Hilbert schemes
gives a smooth family of complex manifolds
$\Hilb^n(V_t)\mathfrak{s}etminus\nabla$. Abouzaid-Smith's construction of
their K\"ahler form $\omega'$ in~\cite[Lemma
5.5]{AbouzaidSmith:arc-alg} gives a smooth $1$-parameter family of
K\"ahler forms on $\Hilb^n(V_t)$, which restrict to
$\Hilb^n(V_t)\mathfrak{s}etminus\nabla$ as exact forms and which agree with
the product form outside a neighborhood of the diagonal, so
$\Genigma_{A_1}\times\cdots\times\Genigma_{A_n}$ and
$\Genigma_{B_1}\times\cdots\times\Genigma_{B_n}$ are Lagrangian. The
averaging construction from~\cite[Lemma 4.24]{HLS:Lie} then gives a
smooth family of $O(2)$-invariant K\"ahler forms for which
$\Genigma_{A_1}\times\cdots\times\Genigma_{A_n}$ and
$\Genigma_{B_1}\times\cdots\times\Genigma_{B_n}$ are still Lagrangian.
Let $V_t=V_0$ if $t<0$ and $V_t=V_1$ if $t>1$. The continuation map
from the freed Floer complex of
$(\Hilb^n(V_0)\mathfrak{s}etminus\nabla,
\Genigma_{A_1}\times\cdots\times\Genigma_{A_n},
\Genigma_{B_1}\times\cdots\times\Genigma_{B_n})$ to the freed Floer
complex of
$(\Hilb^n(V_1)\mathfrak{s}etminus\nabla,
\Genigma_{A_1}\times\cdots\times\Genigma_{A_n},
\Genigma_{B_1}\times\cdots\times\Genigma_{B_n})$ is defined by counting
$\mathbf wt{J}$-holomorphic sections of a bundle $E$ over $\mathbb R\times[0,1]$
whose fiber over $(t,s)$ is $V_t$, with boundary in the sub-bundle
$F\mathfrak{s}ubset E$ specified by
$\Genigma_{A_1}\times\cdots\times\Genigma_{A_n}$ and
$\Genigma_{B_1}\times\cdots\times\Genigma_{B_n}$ over $\mathbb R\times \{0\}$
and $\mathbb R\times\{1\}$, respectively. Here, the $\mathbf wt{J}$ are suitable
families of fiberwise almost complex structures $\mathbf wt{J}$ satisfying analogues
of~\ref{item:J-first}--\ref{item:J-last}.
We claim that the manifolds $\Hilb^n(V_t)\mathfrak{s}etminus\nabla$ are
uniformly $I$-convex in the following sense, which implies the
continuation maps are well-defined. Let $j_t$ be the complex
structure on $V_t$ and let $I_t=\Hilb^n(j_t)$ be the complex
structure on $\Hilb^n(V_t)$ inherited from $V_t$. Let $J_t$ be a
family of almost complex structures satisfying
conditions~\ref{item:J-first}--\ref{item:J-last} with respect to
$j_t$ and $I_t$. In particular, assume that $J_t$ agrees with $I_t$
outside a compact set $K_t$, so that $\bigcup_t K_t$ is also
compact. The almost complex structures $J_t$ give a fiberwise almost complex
structure on $E$. By uniform $I$-convexity we mean there is a
compact set $K'$, depending only on the $K_t$, so that if $u$ is a
$J_t$-holomorphic section of $(E,F)$ then the image of $u$ is
contained in $K'$.
Specifically, fix a family of $I_t$-holomorphic embeddings of the manifolds
$\Hilb^n(S_t)\mathfrak{s}etminus\nabla$ in $\mathbb C^N$ for some large $N$. Then
$K'$ is the intersection of $\bigcup_t\Hilb^n(V_t)\mathfrak{s}etminus\nabla$ with:
\begin{itemize}
\item $\Hilb^n(V'_t)$ where $V'_t$ is the preimage of a slightly
smaller open set $U'$ with $\overline{U'}\mathfrak{s}ubset U$, and
\item the polydisk $\{(z_1,\dots,z_n)\in\mathbb C^N\mid |z_i|\leq R\}$
where $R$ is large enough that this set contains the Lagrangians
and the compact sets $K_t$.
\end{itemize}
The fact that the image of a holomorphic curve $u$ lies in $K'$
follows from positivity of intersections (for the first term in the
intersection, as in~\ref{item:domain-2} above) and the maximum
modulus theorem (for the second term in the intersection).
As noted above, this convexity is enough to ensure that the proof of
invariance of the freed Floer complex~\cite[Proposition
3.28]{HEquivariant} applies. Hence, the freed Floer complexes in
$\Hilb^n(V_0)\mathfrak{s}etminus\nabla$ and $\Hilb^n(V_1)\mathfrak{s}etminus\nabla$
agree up to quasi-isomorphism, completing the proof.
\end{proof}
\mathfrak{s}ubsection{The basepoint action}
The last ingredient in the proof of equivariant stabilization
invariance is a module structure on symplectic Khovanov homology,
analogous to one on Khovanov homology. Fix a bridge diagram $L$ and a
basepoint $p\in L$ on one the $A$-arcs, say $A_i$. There is an
action of ${\FF_2}[X]$ on the symplectic Khovanov complex of $L$
defined as follows. Choose a preimage $p_e\in \Genigma_{A_i}$ lying over
$p\in A_i$. For any $q\in \Genigma_{A_i}$, let
$\mathcal{O}_q\mathfrak{s}ubset \mathcal{K}_A$ denote the codimension-2 subspace
where one of the coordinates is $q$. Then the action of $X$ counts
rigid holomorphic strips $u\co \mathbb R\times[0,1]\to \mathfrak{s}sspace{n}$ with
$u(0,0)\in \mathcal{O}_{p_e}$. (The fact that such moduli spaces with
a point constraint are transversely cut out is a straightforward
adaptation of~\cite[Theorem
3.4.1]{MS04:HolomorphicCurvesSymplecticTopology} to the relative
case.)
\begin{theorem}\label{thm:unknot-action-2}
The above action of ${\FF_2}[X]$ on the symplectic
Khovanov complex of $L$ satisfies the following properties:
\begin{enumerate}[label=(BP-\arabic*)]
\item\label{item:bp-square} Multiplication by $X^2$ is homotopic to
$0$, so symplectic Khovanov homology inherits an action of
${\FF_2}[X]/(X^2)$.
\item\label{item:bp-commute} If $p,p'\in L$ are different points then
multiplication by $X$ at $p$ and at $p'$ commute up to homotopy.
\item\label{item:bp-unknot} The symplectic Khovanov homology of the
$1$-bridge unknot is isomorphic to ${\FF_2}[X]/(X^2)$.
\item\label{item:bp-invariance} Up to homotopy, the chain maps
associated to changes of almost complex structures on
$\mathfrak{s}sspace{n}$, isotopies and handleslides of the $A$- and $B$-arcs
which do not move the point $p$, and diffeomorphisms (which may
move $p$), commute with multiplication by $X$.
\item\label{item:bp-skein} The maps in the skein exact triangle from
Theorem~\ref{thm:skein-tri} respect the action by
${\FF_2}[X]/(X^2)$ (where corresponding points $p$ are used for the
three diagrams).
\item\label{item:bp-move-bp} If $A_i,B_i,A_{i+1}$ are adjacent arcs
in a bridge diagram so that the interior of $B_i$ does not
intersect any $A$-arc and the interior of $A_i$ does not intersect
any $B$-arc---that is, if the configuration looks like
Figure~\ref{fig:stabilization}(b)---and if $p\in A_i$ and $p'\in
A_{i+1}$, then the actions by $p$ and $p'$ are homotopic.
\item\label{item:bp-Kunneth} If $L$ is a disjoint union of two
bridge diagrams $L_1\amalg L_2$ and $p$ is a basepoint on $L_1$
then the K\"unneth theorem
$\mathcal{C}_{\mathit{Kh},\mathit{symp}}(L)\mathfrak{s}imeq \mathcal{C}_{\mathit{Kh},\mathit{symp}}(L_1)\otimes_{\FF_2} \mathcal{C}_{\mathit{Kh},\mathit{symp}}(L_2)$
(from~\cite[Theorem 1.2]{Waldron:KhSympMaps} or the
non-equivariant version of Proposition~\ref{prop:disjoint-union})
intertwines the action of $X$ on $\mathcal{C}_{\mathit{Kh},\mathit{symp}}(L)$ and the action of
$X$ on $\mathcal{C}_{\mathit{Kh},\mathit{symp}}(L_1)\otimes_{\FF_2} \mathcal{C}_{\mathit{Kh},\mathit{symp}}(L_2)$ induced from the
action of $X$ on $\mathcal{C}_{\mathit{Kh},\mathit{symp}}(L_1)$. That is, for appropriate choices
of almost complex structures, $(m\otimes n)*_pX=(m*_pX)\otimes n$.
\end{enumerate}
\end{theorem}
\begin{proof}
Some of these properties (namely \ref{item:bp-square},
\ref{item:bp-commute}, \ref{item:bp-skein}, and a part of
\ref{item:bp-invariance}) are fairly standard (see,
e.g.,~\cite[Section 8l]{SeidelBook} or \cite[Section
3.9]{Perutz:Matching2}, and references therein) and hold for
Lagrangian Floer homology more generally in the absence of disk bubbles, so
we will only sketch the proofs of those properties and concentrate
on the properties that are specific to symplectic Khovanov
homology.
For Property~\ref{item:bp-square}, let $p'_e$ be a point in the
fiber over $p$ close to $p_e$. For a generic choice of almost
complex structure, if we choose $p'_e$ close enough to $p_e$ then
the actions induced by $p_e$ and $p'_e$ agree. Observe that
$O_{p_e}\cap O_{p'_e}=\varnothing$. Counting holomorphic bigons with
$u(0,0)\in O_{p_e}$ and $u(t,0)\in O_{p'_e}$ for some $t>0$ gives a
homotopy from multiplication by $X^2$ (corresponding to
$t\to\infty$) to $0$ (corresponding to $t=0$).
For Property~\ref{item:bp-commute}, let $*_pX$ and $*_{p'}X$ be the
actions at $p$ and $p'$, respectively. Counting holomorphic disks
with $u(0,0)\in O_{p_e}$ and $u(0,t)\in O_{p'_e}$, for any
$t\in\mathbb R$, gives a homotopy between $*_pX*_{p'}X$ (for
$t\to -\infty$) and $*_{p'}X*_pX$ (for $t\to +\infty$).
For Property~\ref{item:bp-unknot}, observe that, with respect to the
standard complex structure on $S$, there is an $S^1$-family of
holomorphic disks connecting the two generators of $\mathcal{C}_{\mathit{Kh},\mathit{symp}}(U)$, one
through each preimage of $p$ on $\Genigma_A=\mathcal{K}_A$. Indeed, the
subset $S_0=\{(u,v,z)\in S\mid v=0\}$ is bi-holomorphic to the
cylinder $\mathbb R\times S^1$, and $\Genigma_A\cap S_0$ and
$\Genigma_B\cap S_0$ are circles inside $S_0$ intersecting in two
points. There are two holomorphic disks in $S_0$, transversally cut
out in $S_0$, whose images under $S^1$ form a single family of
holomorphic disks passing through each preimage of $p$ on
$\Genigma_A$. It follows from a doubling argument and automatic
transversality as in~\cite{HLS97:GenericityHoloCurves} or,
equivalently, a doubling argument and~\cite[Lemma
3.3.1]{MS04:HolomorphicCurvesSymplecticTopology}, that these
holomorphic disks are transversally cut out in $S$. Next, consider
the involution $\tau\co S\to S$, $\tau(u,v,z)=(u,-v,z)$. Since the
holomorphic disks inside $\Fix(\tau)=S_0$ are transversally cut out,
we can perturb the complex structure slightly to a $\tau$-invariant
almost complex structure in which all holomorphic bigons are
transversally cut out (see, e.g.,~\cite[Section
5c]{KhS02:BraidGpAction}). Choose the preimage $p_e$ of $p$ to lie
in $S_0$. Then any holomorphic bigons not contained in $S_0$ passing
through $p_e$ come in pairs exchanged by $\tau$, and hence
contribute $0 \pmod{2}$ to the disk count.
Property~\ref{item:bp-invariance} is proved by considering
holomorphic disks with cylindrical-at-infinity complex structures
(for changes of complex structures or isotopies of the $A$- or
$B$-arcs) or holomorphic triangles (for handleslides) with a similar
point constraint. This in particular includes isotopies of $B$-arcs
that pass over $p$. For the case of a handleslide between $A$-arcs,
there is an additional complication, so we spell out that case. Let
$A'$ be a collection of bridges obtained from the $A$ bridges by a
handleslide, arranged in the plane so that the $A$- and $A'$-bridges
intersect only at their endpoints. Fix points $p\in A_i$ and
$p'\in A'_i$, where $A'_i$ is the $A'$-arc corresponding to $A_i$, which is either a small
translate of $A_i$, or, if $A_i$ is being handleslid, the result of
the handleslide. Let $p_e\in\Genigma_{A_i}$ over $p$, and
$p'_e\in\Genigma_{A'_i}$ over $p'$. We will focus on the case that $p$
is on the arc being handleslid; the other cases are similar but easier.
Consider the $1$-dimensional moduli space of holomorphic triangles
with boundary on $(\mathcal{K}_A,\mathcal{K}_{A'},\mathcal{K}_{B})$ with either a
point along the $A$-edge mapped to $\mathcal{O}_{p_e}$ or a
point along the $A'$-edge mapped to $\mathcal{O}_{p'_e}$, and the
corner at $(\mathcal{K}_A,\mathcal{K}_{A'})$ mapped to the generator
$1$. This moduli space has six kinds of ends:
\begin{enumerate}[label=(TM-\arabic*)]
\item Ends corresponding to a bigon for $(\mathcal{K}_A,\mathcal{K}_B)$ with
a point mapping to $\mathcal{O}_{p_e}$ and a holomorphic triangle. These
correspond to following the basepoint action for
$(\mathcal{K}_A,\mathcal{K}_B)$ by the isomorphism
$F\co \mathit{HF}(\mathcal{K}_A,\mathcal{K}_B)\to\mathit{HF}(\mathcal{K}_{A'},\mathcal{K}_{B})$.
\item Ends corresponding to a bigon for $(\mathcal{K}_{A'},\mathcal{K}_{B})$
with a point mapping to $\mathcal{O}_{p'_e}$ and a holomorphic triangle. These
correspond to following $F$ by the basepoint action for
$(\mathcal{K}_{A'},\mathcal{K}_B)$.
\item Ends corresponding to a bigon for $(\mathcal{K}_A,\mathcal{K}_B)$ and
a holomorphic triangle with a point constraint. These correspond
to $H\circ\partial$, where $H$ is a homotopy defined by counting rigid
triangles with a point constraint.
\item Ends corresponding to a bigon for $(\mathcal{K}_{A'},\mathcal{K}_{B})$
and a holomorphic triangle with a point constraint. These
correspond to $\partial\circ H$, where $H$ is a homotopy defined by
counting rigid triangles with a point constraint.
\item\label{item:TM-5} Ends corresponding to a bigon for
$(\mathcal{K}_A,\mathcal{K}_{A'})$ with a point mapping to
$\mathcal{O}_{p_e}$ and a holomorphic triangle.
\item\label{item:TM-6} Ends corresponding to a bigon for
$(\mathcal{K}_A,\mathcal{K}_{A'})$ with a point mapping to
$\mathcal{O}_{p'_e}$ and a holomorphic triangle.
\end{enumerate}
We need to show that the last two cases cancel. The triangles in the last two
cases are the same, so we need to know that the number of bigons in the two
cases agree (modulo 2). This count of bigons is exactly the basepoint action
for the bridge diagram $(A,A')$, with either a basepoint on $A_i$ or
$A'_i$. Since the differential on ${\mathit{CF}}(\mathcal{K}_A,\mathcal{K}_{A'})$ is trivial for
grading reasons, it suffices to verify that the two basepoint actions are
homotopic.
\begin{figure}
\caption{\textbf{Notation for proof of handleslide invariance.}
\label{fig:handleslide-notation}
\end{figure}
By the proof of Proposition~\ref{prop:disjoint-union} (in the
slightly simpler, non-equivariant case), it suffices to consider the
case that $A$ consists of only the two arcs involved in the
handleslide. Label the endpoints of these arcs $a_1,\dots,a_4$, so
that $a_1$ and $a_4$ are on the outer circle of $A\cup A'$. (See
Figure~\ref{fig:handleslide-notation}.) The generators of
${\mathit{CF}}(\mathcal{K}_A,\mathcal{K}_{A'})$ are $\{a_1,a_2\}$, $\{a_1,a_3\}$,
$\{a_4,a_2\}$, and $\{a_4,a_3\}$. Label the arcs so that
$\{a_1,a_2\}$ is the top-graded generator, denoted $1$ above. Let $A_1$ be the arc with
$\partial A_1=\{a_1,a_4\}$ and $A_2$ the arc with
$\partial A_2=\{a_2,a_3\}$.
Let $\tau\co \mathbb C\to\mathbb C$ be rotation by $\pi$. Arrange that $\tau$ exchanges
$A$ and $A'$, and in particular exchanges $A_i$ and $A'_i$. There is an induced
map $\tau\co \mathfrak{s}sspace{n}\to\mathfrak{s}sspace{n}$ which exchanges $\mathcal{K}_A$ and
$\mathcal{K}_{A'}$.
The basepoint $p$ lies on $A_1$. There is a corresponding basepoint
$\tau(p)$ on $A'_1$. Choose also a basepoint $q$ on $A_2$. We claim
that:
\begin{enumerate}
\item $\{a_1,a_2\}*_q X=\{a_1,a_3\}$ and $\{a_4,a_2\}*_q X=\{a_4,a_3\}$.
\item The coefficient of $\{a_1,a_3\}$ in $\{a_1,a_2\}*_p X$ is 0,
as is the coefficient of $\{a_1,a_3\}$ in
$\{a_1,a_2\}*_{\tau(p)}X$.
\item The coefficient of $\{a_4,a_2\}$ in $\{a_1,a_2\}*_{\tau(p)}X$ is the
same as the coefficient of $\{a_4,a_3\}$ in $\{a_1,a_3\}*_pX$.
\end{enumerate}
Together, these three claims imply the result. Indeed, our goal is to show that
$\{a_1,a_2\}*_pX=\{a_1,a_2\}*_{\tau(p)}X$. By the second claim,
$\{a_1,a_2\}*_pX=\epsilonilon\{a_4,a_2\}$ and
$\{a_1,a_2\}*_{\tau(p)}X=\delta\{a_4,a_2\}$ for some
$\epsilonilon,\delta\in \mathbb F_2$; our goal is to show that
$\epsilonilon=\delta$. By Property~\ref{item:bp-commute},
\[
\{a_1,a_2\}*_q X*_pX=\{a_1,a_2\}*_pX*_qX.
\]
For the left hand side, by the first claim, $\{a_1,a_2\}*_q
X=\{a_1,a_3\}$, and by the third claim,
$\{a_1,a_3\}*_pX=\delta\{a_4,a_3\}$. For the right hand side,
$\{a_1,a_2\}*_pX=\epsilonilon\{a_4,a_2\}$, and by the first claim,
$\epsilonilon\{a_4,a_2\}*_q X=\epsilonilon\{a_4,a_3\}$. Therefore, $\epsilonilon=\delta$.
It remains to verify the three properties. Recall from
Section~\ref{sec:background} that a Whitney disk in $\mathfrak{s}sspace{n}$
has a projected domain.
For the first claim, observe
that the projected domain of a holomorphic bigon counted for
$\{a_1,a_2\}*_q X$ is contained entirely in the bounded region of
$\mathbb C\mathfrak{s}etminus(A_2\cup A'_2)$. Thus, the proof of the K\"unneth
theorem shows that this computation is the same as in the $1$-bridge
unknot case, so this follows from
Property~\ref{item:bp-unknot}. Similarly, the second claim
follows from the fact that there is no projected domain compatible
with this action.
For the last claim we use the involution $\tau$. Given a
$J$-holomorphic Whitney disk $u\co \mathbb R\times[0,1]\to \mathfrak{s}sspace{n}$
for $(\mathcal{K}_A,\mathcal{K}_{A'})$ connecting $\{a_1,a_2\}$ to
$\{a_4,a_2\}$ (so $u(\mathbb R\times\{0\})\mathfrak{s}ubset\mathcal{K}_A$ and
$\lim_{s\to-\infty}u(s,t)=\{a_1,a_2\}$), there is a corresponding
$(\tau_*J)$-holomorphic disk
$\tau\circ u\co \mathbb R\times[0,1]\to \mathfrak{s}sspace{n}$ for
$(\mathcal{K}_{A'},\mathcal{K}_A)$ connecting $\{a_4,a_3\}$ to $\{a_1,a_3\}$
(that is, $(\tau\circ u)(\mathbb R\times\{0\})\mathfrak{s}ubset\mathcal{K}_{A'}$ and
$\lim_{s\to -\infty}(\tau\circ u)(s,t)=\{a_4,a_3\}$). There is a
holomorphic map $\mathfrak{s}igma\co \mathbb R\times[0,1]\to\mathbb R\times[0,1]$ given by
$\mathfrak{s}igma(s,t)=(1-s,-t)$ (rotation around the middle of the
strip). Then $\tau\circ u\circ \mathfrak{s}igma$ is a $(\tau_*J)$-holomorphic
Whitney disk for $(\mathcal{K}_A,\mathcal{K}_{A'})$ connecting $\{a_1,a_3\}$
to $\{a_4,a_3\}$. Further, $u$ passes through $\mathcal{O}_{\tau(p_e)}$ if
and only if $\tau\circ u\circ\mathfrak{s}igma$ passes through
$\mathcal{O}_{p_e}$. So, if we take $p'_e=\tau(p_e)$ then the
basepoint action using $p'_e$ and the almost complex structure $J$
agrees with the basepoint action using $p_e$ and the almost complex
structure $\tau_*(J)$. Since the basepoint action on homology is
independent of the choice of almost complex structure and the
differential on the Floer complex is trivial, it follows
that the coefficient of $\{a_4,a_2\}$ in $\{a_1,a_2\}*_{\tau(p)}X$ is the
same as the coefficient of $\{a_4,a_3\}$ in
$\{a_1,a_3\}*_{p}X$, as desired. This concludes the proof of
Property~\ref{item:bp-invariance}.
Property~\ref{item:bp-skein} follows from the same argument as the case of
$B$-handleslides in Property~\ref{item:bp-invariance}, by considering moduli
spaces of holomorphic triangles with a point constraint. (Because only one
edge of the triangle has a point constraint for the skein maps or a
$B$-handleslide, there are no degenerations analogous to types~\ref{item:TM-5}
and~\ref{item:TM-6}, making these cases easier than the case of
$A$-handleslides.)
\tikzstyle{aarc}=[draw={rgb,255:red,63;green,64;blue,150},line width=1pt,solid,-]
\tikzstyle{barc}=[draw={rgb,255:red,238;green,51;blue,56},line width=1pt,dash pattern={on 2pt off 1pt},-]
\tikzstyle{inter}=[anchor=center,circle,inner sep=0,outer sep=0,minimum width=5pt,fill={rgb,255:red,55;green,53;blue,53}]
\tikzstyle{outerr}=[outer sep=10pt,draw={rgb,255:red,55;green,53;blue,53},dash pattern={on 1pt off 1pt},line width=0.5pt,rounded corners]
\begin{figure}
\caption{\textbf{Moving the basepoint.}
\label{fig:move-basepoint}
\end{figure}
For Property~\ref{item:bp-move-bp}, we use the well-known trick of moving the
basepoint using handleslides and diffeomorphisms. Call a pair of adjacent
$A$ and $B$ arcs \emph{a small pair} if their interiors are disjoint
from all bridges, for example as in
Figure~\ref{fig:stabilization}(b). Figure~\ref{fig:move-basepoint}
shows how a small pair can be moved across an adjacent arc using
handleslides and a diffeomorphism. Repeating this move
once around the link component has the effect of moving the
basepoint from the $A$-arc in a small pair to the next $A$-arc.
Property~\ref{item:bp-Kunneth} is immediate from the definitions and
the proof of Proposition~\ref{prop:disjoint-union}.
\end{proof}
\mathfrak{s}ubsection{Proof of equivariant stabilization invariance}
\begin{figure}
\caption{\textbf{Stabilization and the skein sequence.}
\label{fig:stabilization}
\end{figure}
\begin{proposition}\label{prop:equi-stab-inv}
Let $(\{A_i\},\{B_i\})$ and $(\{A'_i\},\{B'_i\})$ be bridge diagrams
for a link $K$ which differ by a single stabilization, as in
Figure~\ref{fig:stabilization}(a,b). Then there is a quasi-isomorphism
$\mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(\{A_i\},\{B_i\})\mathfrak{s}imeq \mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(\{A'_i\},\{B'_i\})$ over ${\FF_2}[D_{2^m}]$.
\end{proposition}
\begin{proof}
Let $L$ be a link diagram as in Figure~\ref{fig:stabilization}(a),
$L_1$ as in Figure~\ref{fig:stabilization}(b), $L_0$ as in
Figure~\ref{fig:stabilization}(c), and $L'$ as in
Figure~\ref{fig:stabilization}(d). Our goal is to show that there is
a quasi-isomorphism
\[
\mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L)\mathfrak{s}imeq\mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L_1)
\]
over ${\FF_2}[D_{2^m}]$.
Let $p$ be a point in $L$ on the $A$-arc in the figure and $q$ a point
on $A_{n+1}$. By Proposition~\ref{prop:disjoint-union}, there is a
quasi-isomorphism
\[
\mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L_0)\mathfrak{s}imeq \mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L)\otimes\mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(U)
\]
over ${\FF_2}[D_{2^m}]$. On homology, this gives an isomorphism
\[
\mathit{Kh}Symp(L_0)\cong \mathit{Kh}Symp(L)\otimes\mathit{Kh}Symp(U).
\]
Theorem~\ref{thm:unknot-action-2} gives two actions of ${\FF_2}[X]/(X^2)$, one coming from the point $p$
and one from the point $q$, and this isomorphism respects the
actions. Write the action at $p$ (respectively $q$) as $*_p$
(respectively $*_q$). We claim that
\begin{equation}\label{eq:p-is-q-subset}
\mathit{Kh}Symp(L)\cong\{m\in\mathit{Kh}Symp(L)\otimes\mathit{Kh}Symp(U)\mid m*_pX=m*_qX\}.
\end{equation}
Indeed, from Properties~\ref{item:bp-unknot} and~\ref{item:bp-Kunneth}, $a\otimes 1+b\otimes X$ is an element of the right-hand side
if and only if $a*_pX=0$ and $a=b*_pX$, but the second equation
implies the first. So, this set is exactly
$\{(b*_pX\otimes 1)+(b\otimes X)\}$, which is isomorphic (as an
$\mathbb F_2$-vector space) to $\mathit{Kh}Symp(L)$.
Now, consider the skein exact triangle
\[
\cdots\to \mathit{Kh}Symp(L')\to \mathit{Kh}Symp(L_0)\to \mathit{Kh}Symp(L_1)\to\cdots.
\]
By Theorem~\ref{thm:SS-invt} and (the slightly simpler,
non-equivariant case of) Proposition~\ref{prop:disjoint-union},
this triangle is, in fact, a short exact sequence
\[
0\to \mathit{Kh}Symp(L')\to \mathit{Kh}Symp(L_0)\to \mathit{Kh}Symp(L_1)\to 0.
\]
Let $g_*\co \mathit{Kh}Symp(L_0)\to \mathit{Kh}Symp(L_1)$ be the map from this
sequence. Since Property~\ref{item:bp-move-bp} implies the actions at $p$ and $q$ are the same on
$\mathit{Kh}Symp(L_1)$, by Property~\ref{item:bp-skein}
the map $g_*$ sends the image of $(*_pX-*_qX)$ to $0$.
Hence, by exactness and comparing dimensions, $g_*$ sends
\[
\mathit{Kh}Symp(L_0)/\{m\in\mathit{Kh}Symp(L)\otimes\mathit{Kh}Symp(U)\mid m*_pX=m*_qX\}\cong \mathit{Kh}Symp(L)
\]
isomorphically to $\mathit{Kh}Symp(L_1)$.
There is a chain map over ${\FF_2}[D_{2^m}]$
\[
f\co \mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L)\to \mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L)\otimes_{\FF_2}\mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(U)
\]
induced by
$f(y)=y\otimes 1$. The induced map on homology is an isomorphism
from $\mathit{Kh}Symp(L)$ to
$\mathit{Kh}Symp(L)/\{m\in\mathit{Kh}Symp(L)\otimes\mathit{Kh}Symp(U)\mid m*_pX=m*_qX\}$.
Since the map $g_*$ is induced by counting holomorphic triangles
with one corner at a $D_{2^m}$-invariant intersection point, the map
$g_*$ is induced by a map $g\co \mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L_0)\to \mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L_1)$
(see, e.g.,~\cite[Proof of Proposition 3.25]{HEquivariant}). The
composition $g\circ f\co \mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L)\to \mathcal{C}_{\mathit{Kh}}^{\mathit{symp,free}}(L_1)$ is the
desired quasi-isomorphism.
\end{proof}
\begin{proof}[Proof of Theorem 1.26]
This is immediate from isotopy and handleslide invariance, verified
in our original proof of the theorem, and
Proposition~\ref{prop:equi-stab-inv}.
\end{proof}
\end{document} |
\begin{document}
\title{Tensor powers for non-simply laced Lie algebras\\
$B_2$-case }
\author{P P Kulish, V D Lyakhovsky and O V Postnova}
\author{P P Kulish$^1$, V D Lyakhovsky$^2$, and O V Postnova$^3$ \\
$^1$ Sankt-Petersburg Branch of \\
V A Steklov Mathematical Institute RAS\\
$^2$,$^3$ Sankt-Petersburg State University,\\
High Energy Physics and Elementary Particles Department\\
$^1$e-mail: [email protected],\\
$^2$ [email protected]\\
$^3$ [email protected]
}
\maketitle
\begin{abstract}
We study the decomposition problem for tensor powers of $B_2$-fundamental modules.
To solve this problem singular weight technique and injection fan algorithms are applied. Properties of multiplicity coefficients are formulated in terms of multiplicity functions.
These functions are constructed showing explicitly the dependence of multiplicity coefficients on the highest weight coordinates and the tensor power parameter. It is thus possible to study general properties of multiplicity coefficients for powers of the fundamental $B_2$- modules.
\end{abstract}
\section{Introduction}
\label{sec:Introduction}
Consider an analog of the Brauer centralizer algebras for the spinor groups
and define the subspaces of the tensor space $\left( \otimes ^{p}\mathbf{V}
^{n}\right) $ on which the symmetric group $S_{k}$ and $\mathrm{Spin}(n)$
act as the dual pair (in a direct product form). Here $\mathbf{V}^{n}$ is
the fundamental representation of $\mathrm{Spin}(n)$. Namely the
centralizer algebra of the orthogonal group in $\left( \otimes ^{p}\mathbf{V}
^{n}\right) $ is generated by the symmetric group $S_{k}$ and the
contractions and the immersions of the invariant form and is called the
Brauer centralizer algebras. To proceed further one needs the
list of $\mathrm{Spin}(n)$-irreducible subspaces in the decomposition $
\left( \otimes ^{p}\mathbf{V}^{n}\right) =\sum_{\mu \in P}m_{\mu }^{p}
\mathbf{V}^{\left( \mu \right) }$ ($P$ is the $\mathrm{Spin}(n)$ weight
space) and their multiplicities $m_{\mu }^{p}$. As far as we are interested
in an arbitrary power $p$ of the fundamental module $\mathbf{V}_{\mathrm{Spin
}\left( n\right) }^{n}$ our main problem is to find multiplicities of
submodules in a form of \textit{multiplicity functions} $M\left( \mu
,p\right) $ explicitly depending on the corresponding highest weight $\mu $
and the power parameter $p$.
There are numerous combinatorial studies of the problem \cite
{KirillovReshetikhin1996, Kleber1996, Kleber1998, Chari2000} and also series
of works dealing with fermonic formulas, some of them based on crystal basis
approach \cite{HatayamaKuniba1998,
HatayamaKuniba2000,HatayamaKuniba2001,NaitoSagaki2006}. On this way
important general results were obtained \cite{LeducRam1997,KulishManojlovicNagy2010,Kumar2010}.
On the other hand practical
computations with the corresponding formulas are scarcely possible for all
but the simplest examples. In most of these studies the simply laced
algebras are considered and as a rule the multiplicities formulas are
connected with complicated path countings.
We must mention also an algorithm for tensor product decompositions proposed
in \cite{Klimyk1968} and improved in \cite{KlimykSchmudgen1997}.\ It is used
in our investigations.
Summing up, we are to find multiplicities $m_{\mu }^{p}$ in the
decomposition $\left( \otimes ^{p}\mathbf{V}^{n}\right) =\sum_{\mu \in
P}m_{\mu }^{p}\mathbf{V}^{\left( \mu \right) }$ as a function of $\mu $\ and
$p$\ . To solve this problem we propose an algorithm based on singular
weights properties \cite{Feigin1986} and the injection fan technique \cite
{IlyinKulishLyakhovsky2009,LyakhNaz2011}. We study the multiplicities $
m_{\mu }^{p}$ formulated in terms of multiplicity functions $M_{\frak{g}
}\left( \mu ,p\right) $. The latter have the weight space $\mathcal{L}P$ for
the domain of definition. On the sublattice $P^{++}$ of dominant weights the
multiplicity function gives us the desired multiplicities, $M\left( \mu
,p\right) |_{\mu \in P^{++}}=m_{\mu }^{p}$. In this paper we shall show how
to adopt these tools to non-simply laced algebras and shall demonstrate how
they work by studying the tensor powers $\left( L_{B_{n}}^{\omega
_{i}}\right) ^{\otimes ^{p}}$ of the fundamental module $L_{B_{2}}^{\omega
_{i}}$ of $B_{2}$.
\section{Basic definitions and relations.}
$\frak{g}$ -- simple Lie algebra of the series $B_{n}$ ,
$L^{\mu }$\ \ -- the integrable module of $\frak{g}$ with the highest weight
$\mu $\ ;
$r$ -- the rank of the algebra $\frak{g}$ ;
$\Delta $ -- the root system; $\Delta ^{+}$ -- the positive root system for $
\frak{g}$ ;
$\mathcal{N}^{\mu }$ -- the weight diagram of $L^{\mu }$ ;
$W$ -- the Weyl group;
$C^{\left( 0\right) }$ -- the fundamental Weyl chamber, $\overline{C^{\left(
0\right) }}$ -- its closure;
$\rho $\ -- the Weyl vector;
$\epsilon \left( w\right) :=\det \left( w\right) $ , $w \in W$;
$\alpha _{i}$ -- the $i$-th simple root for $\frak{g}$ ; $i=0,\ldots ,r$ ;
$\omega _{i}$ -- the $i$-th fundamental weight for $\frak{g}$ ; $i=0,\ldots
,r$ ;
$L_{\frak{g}}^{\omega _{i}}$ -- the $i$-th fundamental module;
$\left\{ e^{i}\right\} _{\mid i=1,\ldots ,r}$ -- the natural Euclidean basis
of the weight space (the $e$-basis), $\left\{ v_{i}\right\} $ -- the
coordinates of a weight in the $e$-basis;
$P$ -- the weight lattice $\mathcal{L}P$ -- the weight space;
$Q$ -- the root lattice;
$\mathcal{E}$\ -- the group algebra of the group $P$ ;
$\Psi ^{\left( \mu \right) }:=\sum\limits_{w\in W}\epsilon (w)e^{w\circ (\mu
+\rho )-\rho }$ -- the singular element for the $\frak{g}$-module $L^{\mu }$;
$\widehat{\Psi ^{\left( \mu \right) }}$ -- the set of singular weights $\psi
\in P$ for the module $L^{\mu }$ with the coordinates $\left( \psi ,\epsilon
\left( w\left( \psi \right) \right) \right) \mid _{\psi =w\left( \psi
\right) \circ (\mu +\rho )-\rho }$;
$\mathrm{ch}\left( L^{\mu }\right) $ -- the formal character of $L^{\mu }$ ;
$\mathrm{ch}\left( L^{\mu }\right) =\frac{\sum_{w\in W}\epsilon (w)e^{w\circ
(\mu +\rho )-\rho }}{\prod_{\alpha \in \Delta ^{+}}\left( 1-e^{-\alpha
}\right) }=\frac{\Psi ^{\left( \mu \right) }}{\Psi ^{\left( 0\right) }}$ --
the Weyl formula;
$R:=\prod_{\alpha \in \Delta ^{+}}\left( 1-e^{-\alpha }\right) =\Psi
^{\left( 0\right) }$ -- the denominator;
$M_{\frak{g}}^{\omega _{i}}\left( \mu ,p\right) $ -- the multiplicity
function corresponding to the decomposition $\left( L_{\frak{g}}^{\omega
_{i}}\right) ^{\otimes ^{p}}=\sum m_{\mu }^{\left( i\right) p}L_{\frak{g}
}^{\mu }$ , $M_{\frak{g}}^{\omega _{i}}\left( \mu ,p\right) |_{\mu \in
P^{++}}=m_{\mu }^{\left( i\right) p}.$
\section{Some useful properties.}
\begin{lemma}
The projection $\Psi _{\downarrow \frak{g}}^{\left( \nu ,\xi \right) }$ of
the singular element $\Psi _{\frak{g}\oplus \frak{g}}^{\left( \nu ,\xi
\right) }$ for the irreducible representation $L^{\left( \nu _{1},\ldots
,\nu _{r}\right) }\otimes L^{\left( \xi _{1},\ldots ,\xi _{r}\right) }$ of
the direct sum $\frak{g}\oplus \frak{g}$ on the weight space of the diagonal
subalgebra $\frak{g}\rightarrow \frak{g}\oplus \frak{g}$. is equal to the
product
\begin{equation*}
\Psi _{\downarrow \frak{g}}^{\left( \nu ,\xi \right) }=\Psi _{\frak{g}}^{\nu
}\Psi _{\frak{g}}^{\xi },
\end{equation*}
Let $\left\{ \psi _{k}^{\left( \nu \right) }\mid \psi _{k}^{\left( \nu
\right) }\in \Psi ^{\left( \nu \right) },k=1,\ldots ,\#W\right\} _{\mid }$
and $\left\{ \psi _{p}^{\left( \xi \right) }\mid \psi _{p}^{\left( \xi
\right) }\in \Psi ^{\left( \xi \right) },p=1,\ldots ,\#W\right\} $ be the
sets of singular weights for the modules $L^{\left( \nu _{1},\ldots ,\nu
_{r}\right) }$ and $L^{\left( \xi _{1},\ldots ,\xi _{r}\right) }$
correspondingly then the set $\widehat{\Psi _{\downarrow \frak{g}}^{\left(
\nu ,\xi \right) }}$ consists of the weights $\left\{ \psi _{k}^{\left( \nu
\right) }+\psi _{p}^{\left( \xi \right) },\epsilon \left( w\left( \psi
_{k}^{\left( \nu \right) }\right) \right) \epsilon \left( w\left( \psi
_{p}^{\left( \xi \right) }\right) \right) \right\} .$
\end{lemma}
\begin{proof}
Let $\left\{ e^{1},e^{2},\ldots ,e^{r},e^{r+1},\ldots e^{2r}\right\} $ be
the weight space basis for $\mathcal{L}P\left( \frak{g}\oplus \frak{g}
\right) $, $\left( \nu _{1},\ldots ,\nu _{r},\xi _{1},\ldots ,\xi
_{r}\right) $ -- the coordinates for the highest weight $\left( \nu ,\xi
\right) $ naturally belonging to the space $\mathcal{L}P\left( \frak{g}
\oplus \frak{g}\right) $. The weights $v^{\left( \nu \right) }\in \mathcal{N}
^{\nu }$ , $u^{\left( \xi \right) }\in \mathcal{N}^{\xi }$\ and the singular
vectors $\psi _{k}^{\left( \nu \right) }$ and $\psi _{p}^{\left( \xi \right)
}$ also are lifted to the space $\mathcal{L}P\left( \frak{g}\oplus \frak{g}
\right) $
\begin{eqnarray*}
lv_{a}^{\left( \nu \right) } &\Rightarrow &\left( \nu _{a1}^{\left( \nu
\right) },\ldots ,\nu _{ar}^{\left( \nu \right) },0,\ldots ,0\right)
;lu_{b}^{\left( \xi \right) }\Rightarrow \left( 0,\ldots ,0,u_{b1}^{\left(
\eta \right) },\ldots ,u_{br}^{\left( \eta \right) }\right) \\
l\psi _{k}^{\left( \nu \right) } &\Rightarrow &\left( \psi _{k1}^{\left( \nu
\right) },\ldots ,\psi _{kr}^{\left( \nu \right) },0,\ldots ,0\right) ;l\psi
_{p}^{\left( \xi \right) }\Rightarrow \left( 0,\ldots ,0,\psi _{p1}^{\left(
\xi \right) },\ldots ,\psi _{pr}^{\left( \xi \right) }\right)
\end{eqnarray*}
The set $\left\{ lv_{a}+lu_{b}\mid a=1,\ldots ,\dim \left( L^{\nu }\right)
,b=1,\ldots ,\dim \left( L^{\xi }\right) \right\} $ forms the weight diagram
$\mathcal{N}_{\frak{g}\oplus \frak{g}}^{\left( \nu ,\xi \right) }$ of $L_{
\frak{g}\oplus \frak{g}}^{\left( \nu ,\xi \right) }$ . As far as for the
Weyl group $W_{\frak{g}\oplus \frak{g}}$ we have $W_{\frak{g}\oplus \frak{g}
}=W\times W$ and the Weyl vector is $\rho _{\frak{g}\oplus \frak{g}}=\left(
\rho ,\rho \right) $\ , the set of singular weights $\widehat{\Psi _{\frak{g}
\oplus \frak{g}}^{\left( \nu ,\xi \right) }}$ is formed by the vectors whose
first $2r$ coordinates are $\left\{ l\psi _{k}^{\left( \nu \right) }+l\psi
_{p}^{\left( \xi \right) }\mid k,p=1,\ldots ,\#W\right\} $ and the last one
is equal to the product $\epsilon \left( w\left( \psi _{k}^{\left( \nu
\right) }\right) \right) \epsilon \left( w\left( \psi _{p}^{\left( \xi
\right) }\right) \right) $ :
\begin{equation*}
\widehat{\Psi _{\frak{g}\oplus \frak{g}}^{\left( \nu ,\xi \right) }}=\left\{
l\psi _{k}^{\left( \nu \right) }+l\psi _{p}^{\left( \xi \right) },\epsilon
\left( w\left( \psi _{k}^{\left( \nu \right) }\right) \right) \epsilon
\left( w\left( \psi _{p}^{\left( \xi \right) }\right) \right) \mid
k,p=1,\ldots ,\#W\right\} .
\end{equation*}
The vector $\left( c_{1},\ldots ,c_{r},c_{r+1},\ldots ,c_{2r}\right) \in
\mathcal{L}P\left( \frak{g}\oplus \frak{g}\right) $ being projected to the
diagonal subalgebra weight space $\mathcal{L}P$ in the basis $\left\{ \left(
e^{1}+e^{r+1}\right) /2,\ldots ,\left( e^{r}+e^{2r}\right) /2\right\} $ has
the coordinates $\left( \left( c_{1}+c_{r+1}\right) ,\ldots ,\left(
c_{r}+c_{2r}\right) \right) $ . The latter means that
\begin{eqnarray*}
\widehat{\Psi _{\downarrow \frak{g}}^{\left( \nu ,\xi \right) }} &=&\left\{
\psi _{k}^{\left( \nu \right) }+\psi _{p}^{\left( \xi \right) },\epsilon
\left( w\left( \psi _{k}^{\left( \nu \right) }\right) \right) \epsilon
\left( w\left( \psi _{p}^{\left( \xi \right) }\right) \right) \right\} , \\
k,p &=&1,\ldots ,\#W.
\end{eqnarray*}
\end{proof}
Q.E.D.
One of the main tools to study the decomposition properties is the \textit{
injection fan }$\Gamma _{\frak{g}_{\mathrm{diag}}\rightarrow \frak{g}\oplus
\frak{g}}$. \cite{IlyinKulishLyakhovsky2009,LyakhNaz2011} . To use this
instrument we consider the decomposition of tensor products as a special
case of branching. The latter corresponds to the injection of the diagonal
subalgebra into the direct sum: $\frak{g}_{\mathrm{diag}}\rightarrow \frak{g}
\oplus \frak{g}$.
\begin{lemma}
The vectors of the injection fan $\Gamma $ for $\frak{g}_{\mathrm{diag}
}\rightarrow \frak{g}\oplus \frak{g}$ consists of the opposites to the
singular weights of the trivial module $L_{\frak{g}}^{\left( 0\right) }$
\begin{equation*}
\Gamma _{\frak{g}_{\mathrm{diag}}\rightarrow \frak{g}\oplus \frak{g}
}=-S\circ \widehat{\Psi ^{\left( 0\right) }}\backslash \left( 0,\ldots
,0\right) ,
\end{equation*}
(here S is the full reflection).
\end{lemma}
\begin{proof}
According to the definition \cite{IlyinKulishLyakhovsky2009} the vectors $
\gamma $ of the fan $\Gamma _{\frak{g}_{\mathrm{diag}}\rightarrow \frak{g}
\oplus \frak{g}}$ are fixed by the relation $1-\prod_{\alpha \in \left(
\Delta _{\frak{g}\oplus \frak{g}\downarrow \frak{g}_{\mathrm{diag}
}}^{+}\right) }\left( 1-e^{-\alpha }\right) ^{\mathrm{mult}_{\frak{g}\oplus
\frak{g}}\mathrm{\left( \alpha \right) -{mult}}_{\frak{g}_{\mathrm{diag}}}
\mathrm{\left( \alpha \right) }}=\sum_{\gamma \in \Gamma _{\frak{g}_{\mathrm{
diag}}\rightarrow \frak{g}\oplus \frak{g}}}s\left( \gamma \right) e^{-\gamma
}$. The projections of the\ $\frak{g}\oplus \frak{g}$ -roots to the diagonal
subalgebra obviously reproduce the set $\left\{ \alpha _{i}\right\}
_{i=0,\ldots ,r}$ in $\mathcal{L}P\left( \frak{g}\right) $ :
\begin{equation*}
\left.
\begin{array}{c}
\left( \alpha _{i},0\right) _{\downarrow \frak{g}} \\
\left( 0,\alpha _{i}\right) _{\downarrow \frak{g}}
\end{array}
\right\} =\alpha _{i}.
\end{equation*}
Thus $\mathrm{mult}_{\frak{g}}\left( \alpha \right) =2$ while $\mathrm{mult}
_{\frak{g}\oplus \frak{g}}\left( \alpha \right) =1$ and we have
\begin{eqnarray*}
\sum_{\gamma \in \Gamma _{\frak{g}_{\mathrm{diag}}\rightarrow \frak{g}\oplus
\frak{g}}}s\left( \gamma \right) e^{-\gamma } &=&1-\prod_{\alpha \in \left(
\Delta _{\frak{g}\oplus \frak{g}\downarrow \frak{g}}^{+}\right) }\left(
1-e^{-\alpha }\right) ^{\mathrm{mult}_{\frak{g}\oplus \frak{g}}\left( \alpha
\right) -\mathrm{mult}_{\frak{g}}\left( \alpha \right) }= \\
&=&1-\prod_{\alpha \in \left( \Delta ^{+}\right) }\left( 1-e^{-\alpha
}\right) .
\end{eqnarray*}
Q.E.D.
\end{proof}
\begin{lemma}
The singular element $\Psi _{\frak{g}}^{\left( \xi \right) }$ for the module
$L^{\mu }\otimes L^{\nu }$ can be presented in two equivalent forms:
\begin{equation}
\mathrm{ch}\left( L_{\frak{g}}^{\mu }\right) \Psi _{\frak{g}}^{\nu }=\Psi _{\frak{g}
}^{\mu }\mathrm{ch}\left( L_{\frak{g}}^{\nu }\right) .
\end{equation}
\end{lemma}
\begin{proof}
In the Weyl formula for $L^{\mu }\otimes L^{\nu }$,
\begin{equation*}
\mathrm{ch}\left( L^{\mu }\otimes L^{\nu }\right) _{\downarrow P_{\frak{g}
}^{+}}=\sum_{\xi \in P_{\frak{g}}^{++}}m_{\xi }^{\mu \nu }\mathrm{ch}\left(
L^{\xi }\right) ,
\end{equation*}
apply the result of Lemma 1:
\begin{equation*}
\left( \frac{\Psi _{\frak{g}\oplus \frak{g}}^{\left( \nu ,\xi \right) }}{
\Psi _{\frak{g}\oplus \frak{g}}^{0}}\right) _{\downarrow P_{\frak{g}}}=\frac{
\Psi _{\frak{g}}^{\mu }\Psi _{\frak{g}}^{\nu }}{\Psi _{\frak{g}}^{0}\Psi _{
\frak{g}}^{0}}=\sum_{\xi \in P_{\frak{g}}^{++}}m_{\xi }^{\mu \nu }\frac{\Psi
_{\frak{g}}^{\xi }}{\Psi _{\frak{g}}^{0}}.
\end{equation*}
\begin{equation*}
\left( \left( \Psi _{\frak{g}}^{0}\right) ^{-1}\Psi _{\frak{g}}^{\mu
}\right) \Psi _{\frak{g}}^{\nu }=\Psi _{\frak{g}}^{\mu }\left( \left( \Psi _{
\frak{g}}^{0}\right) ^{-1}\Psi _{\frak{g}}^{\nu }\right) =\sum_{\xi \in P_{
\frak{g}}^{++}}m_{\xi }^{\mu \nu }\Psi _{\frak{g}}^{\xi }.
\end{equation*}
Thus we have
\begin{equation}
\sum_{\xi \in P_{\frak{g}}^{++}}m_{\xi }^{\mu \nu }\Psi _{\frak{g}}^{\xi
}=\mathrm{ch}\left( L_{\frak{g}}^{\mu }\right) \Psi _{\frak{g}}^{\nu }=\Psi _{\frak{g}
}^{\mu }\mathrm{ch}\left( L_{\frak{g}}^{\nu }\right) .
\label{N-recursion}
\end{equation}
Q.E.D.
\end{proof}
Now put $\mu =\omega $, $\nu =\left( p-1\right) \omega $
\begin{equation}
\mathrm{ch}\left( L^{\left( \omega \right) }\right) \Psi ^{\left(
\otimes ^{\left( p-1\right) }\omega \right) }=\sum_{\xi \in
P}M_{\frak{g}}^{\omega _{1}}\left( \xi ,p\right) \Psi ^{\left( \xi
\right) }, \label{N-rec property}
\end{equation}
$M_{\frak{g}}^{\omega _{1}}\left( \xi ,p\right) $ defines the singular
element $\Psi ^{\left( \otimes ^{p}\omega \right) }$. On the other hand
these equations can be considered as a system of recurrent relations for the
multiplicity function $M_{\frak{g}}^{\omega _{1}}\left( \xi ,p\right) $ .
\begin{conjecture}
Let $\frak{g}=B_{2}$, $L^{\mu }$\ and $L^{\omega _{1}}=L^{vect}$ be the
highest weight modules with the highest weights $\mu $ and $\omega _{1}$
(the first fundamental weight). Then the tensor product decomposition $
\left( L^{\mu }\otimes L^{vect}\right) \downarrow _{\frak{g}_{\mathrm{diag}
}}=\oplus _{\gamma }L^{\gamma }$ is multiplicity free.
\end{conjecture}
\begin{proof}
According to Lemma 1 the projected singular element for a $\frak{g}\oplus
\frak{g}$ -module $L^{\mu }\otimes L^{vect}$ is $\Psi _{\frak{g}\oplus \frak{
g}\downarrow \frak{g}_{\mathrm{diag}}}^{\left( \mu ,vect\right) }=\Psi _{
\frak{g}}^{\mu }\Psi _{\frak{g}}^{vect}$ and the set of singular weights is
\begin{eqnarray*}
\widehat{\Psi _{\downarrow \frak{g}_{\mathrm{diag}}}^{\left( \mu
,vect\right) }} &=&\left\{ \psi _{k}^{\left( \mu \right) }+\psi _{p}^{\left(
vect\right) },\epsilon \left( w\left( \psi _{k}^{\left( \mu \right) }\right)
\right) \epsilon \left( w\left( \psi _{p}^{\left( vect\right) }\right)
\right) \right\} , \\
k,p &=&1,\ldots ,\#W.
\end{eqnarray*}
Suppose $\mu =\left( \mu _{1},\mu _{2}\right) $ is greater than $\omega
_{1}=\left( 1,0\right) $ and $\mu _{1}>\mu _{2}\geq 1$. The singular weights
of $L^{\mu }$ in the fundamental chamber and its nearest neighbours are
\begin{eqnarray*}
\left\{ \psi _{s}^{\left( \mu \right) }\right\} &=&\left( \mu _{1},\mu
_{2},\left( +1\right) \right) ,\left( \mu _{1},-\mu _{2}-1,\left( -1\right)
\right) ,\left( \mu _{2}-1,\mu _{1}+1,\left( -1\right) \right) \\
s &=&1,2,3
\end{eqnarray*}
They will give rise to the 24 weights of the type
\begin{eqnarray*}
\left\{ \psi _{s}^{\left( \mu \right) }+\psi _{p}^{\left( vect\right)
}\right\} &=&\left( \mu _{1},\mu _{2},\left( +1\right) \right) +\psi
_{p}^{\left( vect\right) }, \\
&&\left( \mu _{1},-\mu _{2}-1,\left( -1\right) \right) +\psi _{p}^{\left(
vect\right) }, \\
&&\left( \mu _{2}-1,\mu _{1}+1,\left( -1\right) \right) +\psi _{p}^{\left(
vect\right) } \\
s &=&1,2,3;p=1,\ldots ,\#W.
\end{eqnarray*}
Applying successively the fan $\Gamma _{\frak{g}_{\mathrm{diag}}\rightarrow
\frak{g}\oplus \frak{g}}$ to the set $\widehat{\Psi _{\downarrow \frak{g}_{
\mathrm{diag}}}^{\left( \mu ,vect\right) }}$ in the three selected chambers
(starting with the highest weight $\left( \mu _{1}+1,\mu _{2},\left(
+1\right) \right) $) we find the weights:
\begin{eqnarray*}
&&\left( \mu _{1},\mu _{2},\left( +1\right) \right) ,\left( \mu _{1}\pm
1,\mu _{2},\left( +1\right) \right) ,\left( \mu _{1},\mu _{2}\pm 1,\left(
+1\right) \right) , \\
&&\left( \mu _{1},-\mu _{2}-1,\left( -1\right) \right) ,\left( \mu _{1}\pm
1,-\mu _{2}-1,\left( -1\right) \right) ,\left( \mu _{1},-\mu _{2},\left(
-1\right) \right) ,\left( \mu _{1},-\mu _{2}-2,\left( -1\right) \right) , \\
&&\left( \mu _{2}-1,\mu _{1}+1,\left( -1\right) \right) ,\left( \mu _{2},\mu
_{1}+1,\left( -1\right) \right) ,\left( \mu _{2}-2,\mu _{1}+1,\left(
-1\right) \right) , \\
&&\left( \mu _{2}-1,\mu _{1}+2,\left( -1\right) \right) ,\left( \mu
_{2}-1,\mu _{1},\left( -1\right) \right) .
\end{eqnarray*}
Only the first 5 weights are in $\overline{C^{\left( 0\right) }}$ . They are
the highest singular weights for $\left( L^{\mu }\otimes L^{vect}\right)
_{\downarrow \frak{g}_{\mathrm{diag}}}$ and we have the decomposition:
\begin{equation*}
\left( L^{\mu }\otimes L^{vect}\right) _{\downarrow \frak{g}_{\mathrm{diag}
}}=L^{\mu }\oplus L^{\left( \mu _{1}+1,\mu _{2}\right) }\oplus L^{\left( \mu
_{1}-1,\mu _{2}\right) }\oplus L^{\left( \mu _{1},\mu _{2}+1\right) }\oplus
L^{\left( \mu _{1},\mu _{2}-1\right) }.
\end{equation*}
There are two special cases where the highest weights are on the borders of $
\overline{C^{\left( 0\right) }}$. For $\mu =n\omega _{1}=\left( n,0\right) $
and for $\mu =n\omega _{2}=\left( n/2,n/2\right) $ the same algorithm gives:
\begin{equation*}
\left( L^{\mu }\otimes L^{vect}\right) _{\downarrow \frak{g}_{\mathrm{diag}
}}=L^{\left( \mu _{1}+1,0\right) }\oplus L^{\left( \mu _{1}-1,0\right)
}\oplus L^{\left( \mu _{1},\mu _{2}+1\right) },
\end{equation*}
\begin{equation*}
\left( L^{\mu }\otimes L^{vect}\right) _{\downarrow \frak{g}_{\mathrm{diag}
}}=L^{\mu }\oplus L^{\left( \mu _{1}+1,\mu _{2}\right) }\oplus L^{\left( \mu
_{1},\mu _{2}-1\right) }.
\end{equation*}
correspondingly. Q.E.D.
\end{proof}
\section{Singular elements and fans. $B_{2}$-case}
For $\frak{g}=B_{2}$ , $r=2$ the simple roots in $e$-basis are $\alpha
_{1}=e_{1}-e_{2}, \alpha _{2}=e_{2}$, the fundamental weights are $\omega
_{2}=\frac{1}{2}\left( e_{1}+e_{2}\right) ,\omega _{1}=e_{1}$ and the
fundamental modules -- $L^{\omega _{2}}$(spinor) and $\dim L^{\omega
_{2}}=4,L^{\omega _{1}}$(vector), $\dim L^{\omega _{1}}=5$.
Consider the modules $(L^{\omega _{i}})^{\otimes p}|_{p\in Z_{+},i=1,2}$ and
the decompositions $(L^{\omega _{i}})^{\otimes p}=\sum_{\nu }m_{\nu
}^{\left( i\right) p}L^{\nu }.$
Our aim is to find multiplicities $m_{\nu }^{\left( i\right) p }$ as
functions of $\nu $\ and $p$. To solve the problem we propose to use the
\textit{singular elements formalism }\cite{Feigin1986}, the polynomial
dependence property that is a consequence of the relation (\ref{N-rec
property}) and the injection fan technique \cite{IlyinKulishLyakhovsky2009}
\cite{LyakhNaz2011}.
\subsection{Constructing the fan $\Gamma _{B_{2}^{\mathrm{diag}}\rightarrow
\oplus ^{p}B_{2}}$}
Consider the injection $B_{2}^{\mathrm{diag}}\rightarrow \oplus ^{p}B_{2}$.
The fan $\Gamma _{B_{2}^{\mathrm{diag}}\rightarrow \oplus ^{p}B_{2}}\equiv
\Gamma _{p}$ is the $\left( p-1\right) $-th tensor power of the trivial
module singular element that is of $\Psi _{B_{2}}^{\left( 0\right) }$.
\begin{conjecture}
Place the origin of the space $\mathcal{L}P$ at the end of the lowest weight
vector of the fan. The structure of the fan $\Gamma _{p}$ is as follows:
\begin{enumerate}
\item Along the line $p\alpha _{1}$ in the $k$-th root lattice point the
multiplicity is $\left( -1\right) ^{k}C_{p}^{k-1}$; $k=1,\ldots ,p+1.$
\item Each weight with the coordinates $\left( k-1,1-k\right) $ is an
origin of the set $S_{\left( k-1,1-k\right) }$ of singular weights described
below.
\item The set $S_{\left( k-1,1-k\right) }$ is composed of the tensor
product of the singular elements $\Psi _{x}^{\left[ 0\right] <}$\ and $\Psi
_{y}^{\left[ v\right] <}$of the trivial and vector lowest weight modules of
the algebras $A_{1}$ with the roots $x$ and $y$ correspondingly. The list of
modules for the set $S_{\left( k-1,1-k\right) }$ is completely defined by
its coordinates $\left( k-1,1-k\right) $:
\begin{equation*}
S_{\left( k-1,1-k\right) }=\left( \Psi _{e_{2}}^{\left[ v\right] <}\right)
^{\otimes \left( k-1\right) }\otimes \left( \Psi _{e_{1}}^{\left[ 0\right]
<}\right) ^{\otimes \left( k-1\right) }\otimes \left( \Psi _{e_{1}}^{\left[ v
\right] <}\right) ^{\otimes \left( p-k+1\right) }\otimes \left( \Psi
_{e_{2}}^{\left[ 0\right] <}\right) ^{\otimes \left( p-k+1\right) }.
\end{equation*}
\end{enumerate}
\end{conjecture}
\begin{proof}
Let $\Gamma _{p}$ be the fan with the properties described above. Remember
that the set $S_{\left( k-1,1-k\right) }$ has itself the multiplicity $
\left( -1\right) ^{k}C_{p}^{k-1}$. Multiply the fan $\Gamma _{p}$ by the
element $\Psi ^{\left( 0\right) }$, i.e. pass to the power $\left(
p+1\right) $. This means that the set $S_{\left( k-1,1-k\right) }$ will be
transformed to
\begin{equation*}
\left( -1\right) ^{k}C_{p}^{k-1}\left( \Psi _{e_{2}}^{\left[ v\right]
<}\right) ^{\otimes \left( k-1\right) }\otimes \left( \Psi _{e_{1}}^{\left[ 0
\right] <}\right) ^{\otimes \left( k-1\right) }\otimes \left( \Psi _{e_{1}}^{
\left[ v\right] <}\right) ^{\otimes \left( p-k+2\right) }\otimes \left( \Psi
_{e_{2}}^{\left[ 0\right] <}\right) ^{\otimes \left( p-k+2\right) }.
\end{equation*}
The set $S_{\left( k-2,2-k\right) }$ will become $S_{\left( k-1,1-k\right) }$
with the multiplicity $\left( -1\right) ^{k+1}C_{p}^{k-2}$. According to the
Pascal triangle property,
\begin{equation*}
\left( -1\right) ^{k}C_{p}^{k-1}+\left( -1\right) ^{k+1}C_{p}^{k-2}=\left(
-1\right) ^{k}C_{p+1}^{k-1},
\end{equation*}
the first set $S_{\left( 0,0\right) }$ takes the form
\begin{equation*}
\left( \Psi _{e_{1}}^{\left[ v\right] <}\right) ^{\otimes \left( p+1\right)
}\otimes \left( \Psi _{e_{2}}^{\left[ 0\right] <}\right) ^{\otimes \left(
p+1\right) },
\end{equation*}
while the last one becomes $S_{\left( p,p\right) }$,
\begin{equation*}
\left( -1\right) ^{p+2}\left( \Psi _{e_{2}}^{\left[ v\right] <}\right)
^{\otimes \left( p+1\right) }\otimes \left( \Psi _{e_{1}}^{\left[ 0\right]
<}\right) ^{\otimes \left( p+1\right) }.
\end{equation*}
Thus the structure of this product coinsides with the previously defined fan
$\Gamma _{q}$ with $q=p+1$. Q.E.D.
\end{proof}
An explicit expression for the fan $\Gamma _{p}$ is obtained by substituting
$\Psi _{e_{i}}^{\left[ 0\right] <}$ and $\Psi _{e_{i}}^{\left[ v\right] <}$
(as formal algebra elements) by their expressions in terms of a function
\begin{equation*}
\widehat{C}_{j}^{i}=\left\{
\begin{array}{c}
C_{j}^{i}\text{ \ for }0<i\leq j>0 \\
0\text{ \ \ \ \ \ \ \ \ \ \ \ \ otherwise}.
\end{array}
\right.
\end{equation*}
Finally we get
\begin{eqnarray}
\Gamma _{p} &=&\sum_{a,b}\gamma _{p}\left( a,b\right) e^{\left( a,b\right)
};\quad \left\{
\begin{array}{c}
a=k-1,\ldots ,3p-k-2, \\
b=1-k,\ldots ,p+k-2,
\end{array}
\right. \notag \\
\gamma _{p}\left( a,b\right)
&=&\sum_{k=1}^{p}\sum_{l_{k}=1}^{k}\sum_{m_{k}=1}^{p-k+1}e^{\left(
a,b\right) }\left( -1\right) ^{k+a+b-2\left( l_{k}+m_{k}\right) }\times
\notag \\
&&\times \widehat{C}_{p-1}^{k-1}\widehat{C}_{k-1}^{l_{k}-1}\widehat{C}
_{p-k}^{m_{k}-1}\widehat{C}_{p-k}^{b+k-3l_{k}+2}\widehat{C}
_{k-1}^{a-k-3m_{k}+4}, \label{fan-so5}
\end{eqnarray}
Here the fan is a function of the parameter $p$ and the coordinates $\left(
a,b\right) $ of the highest weight. The zero point has the multiplicity $
\left( -1\right) $. The result is that for any $\mu \in P$ the singular
weights diagram $\Psi \left( \sum_{\nu }m_{\nu }^{\left( i\right) p}L^{\nu
}\right) =\sum e^{\left( \mu \right) }\epsilon \left( w\left( \psi \left(
\mu \right) \right) \right) $ has the following fundamental property:
\begin{equation*}
\sum_{a,b}\gamma _{p}\left( a,b\right) \epsilon \left( w\left( \psi \left(
\mu +\left( a,b\right) \right) \right) \right) =0,
\end{equation*}
described by the fan $\Gamma _{p}$.
\subsection{Singular element for $\left( L^{\protect\omega _{2}}\right)
^{\otimes p}$ -- the spinor case}
Let us construct the singular element for the $p$-th power of the second
(spinor) fundamental module $L^{\omega _{2}}$ (coordinates of singular
weights here are half-integer and the $W$-invariant vector is $(0,0)$).
\begin{enumerate}
\item Define the system $S_{k}$ consisting of blocks enumerated by a pair
of indices $\left( l_{k},m_{k}\right) $ where $l_{k}=1,\ldots ,k+1$ and $
m_{k}=1,\ldots ,p-k+2$ and attached to the points $\left( \frac{p}{2}
-k+1-4\left( m_{k}-1\right) ,\frac{p}{2}+k-1-4\left( l_{k}-1\right) \right) $
. The multiplicities of these blocks are $\left( -1\right) ^{l_{k}+m_{k}-2}
\widehat{C}_{k}^{l_{k}-1}\widehat{C}_{p-k+1}^{m_{k}-1}$
\item Localize the systems $S_{k}$ along the line $(\frac{p}{2},\frac{p}{2}
)-p\alpha _{1}$: the first system $S_{1}$ has the origin at the point $(
\frac{p}{2},\frac{p}{2})$, the $k$-th -- at the point $\left( \frac{p}{2}
-k+1,\frac{p}{2}+k-1\right) $, the last one, $S_{p+1}$, -- at $\left( -\frac{
1}{2}p,\frac{3}{2}p\right) $. These systems have the multiplicities $\left(
-1\right) ^{k-1}C_{p}^{k-1}$; $k=1,\ldots ,p+1.$
\item The numbers $\left( \frac{p}{2}-k+1-4\left( m_{k}-1\right) ,\frac{p}{2
}+k-1-4\left( l_{k}-1\right) \right) $ are the coordinates of the upper
right corner of the $\left( l_{k},m_{k}\right) $-block. The blocks have the
structure dual to the structure of the system $S_{k}$ but the intervals in
the blocks are doubled. The weights in the block are enumerated by the
indices $\left( i_{k},j_{k}\right) $ where $j_{k}=1,\ldots ,k+1$ and $
i_{k}=1,\ldots ,p-k+2$.
\item Thus the $\left( l_{k},m_{k}\right) $-block in $S_{k}$ has the form:
\begin{equation*}
\sum_{i_{k}=1}^{p-k+2}\sum_{j_{k}=1}^{k+1}\left( -1\right) ^{i_{k}+j_{k}-2}
\widehat{C}_{k}^{j_{k}-1}\widehat{C}_{p-k+1}^{i_{k}-1}e^{\left( \frac{p}{2}
-k+1-4m_{k}-2j_{k}+7,\frac{p}{2}+k-1-4l_{k}-2i_{k}+5\right) }.
\end{equation*}
\item Now the system $S_{k}$ can be composed:
\begin{eqnarray*}
&&\sum_{i_{k},m_{k}=1}^{p-k+2}\sum_{j_{k},l_{k}=1}^{k+1}\left( -1\right)
^{l_{k}+m_{k}+i_{k}+j_{k}-4}\times \\
&&\times \widehat{C}_{k}^{l_{k}-1}\widehat{C}_{p-k+1}^{m_{k}-1}\widehat{C}
_{k}^{j_{k}-1}\widehat{C}_{p-k+1}^{i_{k}-1}e^{\left( \frac{p}{2}
-k+1-4m_{k}-2j_{k}+7,\frac{p}{2}+k-1-4l_{k}-2i_{k}+5\right) }.
\end{eqnarray*}
\item Finally, the singular element $\Psi ^{\left( \left( \omega
_{2}\right) ^{\otimes p}\right) }\equiv \Psi ^{\left( \left( s\right)
^{\otimes p}\right) }$ is fixed as
\begin{eqnarray*}
\Psi ^{\left( \left( s\right) ^{\otimes p}\right) }
&=&\sum_{k=1}^{p+1}\sum_{i_{k},m_{k}=1}^{p-k+2}\sum_{j_{k},l_{k}=1}^{k+1}
\left( -1\right) ^{l_{k}+m_{k}+i_{k}+j_{k}-4}\times \\
&&\times \widehat{C}_{k}^{l_{k}-1}\widehat{C}_{p-k+1}^{m_{k}-1}\widehat{C}
_{k}^{j_{k}-1}\widehat{C}_{p-k+1}^{i_{k}-1}\times \\
&&\times e^{\left( \frac{p}{2}-k+1-4m_{k}-2j_{k}+7,\frac{p}{2}
+k-1-4l_{k}-2i_{k}+5\right) }
\end{eqnarray*}
The singular multiplicities are functions of $p$ and $c,d$:
\begin{eqnarray*}
\psi ^{\left( \left( s\right) ^{\otimes p}\right) }\left( c,d\right)
&=&e^{\left( c,d\right)
}\sum_{k=1}^{p+1}\sum_{l_{k}=1}^{k}\sum_{m_{k}=1}^{p-k+2}\left( -1\right)
^{k+\frac{1}{2}\left( c-d\right) -\left( l_{k}+m_{k}\right) +1}\times \\
&&\times \widehat{C}_{p}^{k-1}\widehat{C}_{k}^{l_{k}-1}\widehat{C}_{k}^{
\frac{1}{2}\left( 4\left( 1-m_{k}\right) -k+c-\frac{1}{2}p+1\right) }\times
\\
&&\times \widehat{C}_{p-k+1}^{m_{k}-1}\widehat{C}_{k}^{\frac{1}{2}\left(
2-4m_{k}+k-d+\frac{1}{2}p+1\right) }.
\end{eqnarray*}
\end{enumerate}
\subsection{Singular element for $\left( L^{\protect\omega _{1}}\right)
^{\otimes p}$ -- the vector case}
The construction procedure in the vector case is analogous to that of the
spinor. It results in obtaining the expression:
\begin{eqnarray*}
\Psi ^{\left( \left( \omega _{1}\right) ^{\otimes p}\right) } &\equiv &\Psi
^{\left( \left( v\right) ^{\otimes p}\right)
}=\sum_{k=1}^{p+1}\sum_{j,n=0}^{k-1}\sum_{i,m=0}^{p-k+1}e^{\left(
2k+j+5m-4p-2,-2p+i-k+5n+1\right) }\times \\
&&\times \left( -1\right) ^{i+j+k+m+n-1}\widehat{C}_{p-k+1}^{m}\widehat{C}
_{k-1}^{n}\widehat{C}_{p-k}^{j}\widehat{C}_{p}^{k-1}\widehat{C}_{p-k+1}^{i}.
\end{eqnarray*}
The corresponding singular multiplicities function depending on $p$ and $c,d$
are
\begin{eqnarray}
\psi ^{\left( \left( v\right) ^{\otimes p}\right) }\left( c,d\right)
&=&\sum_{k=1}^{p+1}\sum_{l_{k}=1}^{k}\sum_{m_{k}=1}^{p-k+2}e^{\left(
c,d\right) }\times \notag \\
&&\times \left( -1\right) ^{k-d-c+p-4\left( l_{k}+m_{k}\right) +7}\times
\notag \\
&&\times \widehat{C}_{p}^{k-1}\widehat{C}_{k-1}^{l_{k}-1}\widehat{C}
_{p-k+1}^{m_{k}-1}\widehat{C}_{p-k+1}^{-d+2k-5\left( l_{k}-1\right) -2}
\widehat{C}_{k-1}^{p-c-2k-5\left( m_{k}-1\right) +2}.
\label{sing-el-vect-so5}
\end{eqnarray}
\subsection{Recursive procedure for the vector case}
To illustrate the recursive algorithm we perform calculations that must give
us the multiplicities $m_{\nu }^{\left( 1\right) p}$. The starting value is
always known -- this is the multiplicity of the highest weight $\nu =\left(
p,0\right) $ of $(L^{\omega _{1}})^{\otimes p}$ that equals $1$. Suppose we
have found the values of the multiplicity function $m_{\nu }^{\left(
1\right) p}$ for the 14 first weights:\\[1mm]
\begin{tabular}{||c||cccccc||}
\hline\hline
$c\setminus d$ & $-2$ & $-1$ & $0$ & $+1$ & $+2$ & $+3$ \\ \hline\hline
$p$ & $0$ & $-1$ & $+1$ & $0$ & $0$ & $0$ \\
$p-1$ & $1-p$ & $0$ & $0$ & $p-1$ & $0$ & $0$ \\
$p-2$ & & & & & $\frac{1}{2}p\left( p-3\right) $ & $0$ \\ \hline\hline
\end{tabular}\\[1mm]
Applying the fan $\Gamma _{p}$\ (\ref{fan-so5}) we find the multiplicity for
the next weight in the third line, it has the coordinates $\left(
p-2,1\right) $. The first line of the fan weights contributes the value
\begin{equation*}
\left( p-1\right) \times \frac{1}{2}p\left( p-3\right) ,
\end{equation*}
the second line --
\begin{equation*}
-\left( p-1\right) \left( p-2\right) \times \left( p-1\right)
\end{equation*}
and the third line --
\begin{eqnarray*}
&&\left( +\frac{1}{2}\left( p-1\right) \left( p-2\right) \left( p-3\right)
\right) \times \left( +1\right) + \\
&&+\left( \left( p-1\right) -\frac{1}{2}\left( p-1\right) \left( p-2\right)
\right) \times \left( -1\right)
\end{eqnarray*}
Now we must calculate the singular element contribution -- the value of the
singular weights function $\psi ^{\left( \left( v\right) ^{\otimes p}\right)
}\left( p-2,1\right) $ for the weight $\left( p-2,1\right) $. The ''singular
contribution'' is
\begin{equation*}
\psi ^{\left( \left( v\right) ^{\otimes p}\right) }\left( p-2,1\right)
=p\left( p-1\right) .
\end{equation*}
The multiplicity $m_{\left( p-2,1\right) }^{\left( 1\right) p}$ is the sum:
\begin{eqnarray*}
m_{\left( p-2,1\right) }^{\left( 1\right) p} &=&\left( p-1\right) \left(
\begin{array}{c}
\frac{1}{2}p\left( p-3\right) -\left( p-1\right) \left( p-2\right) +p \\
+\frac{1}{2}\left( p-2\right) +\frac{1}{2}\left( p-2\right) \left(
p-3\right) -1
\end{array}
\right) \\
&=&\frac{1}{2}\left( p-1\right) \left( p-2\right) .
\end{eqnarray*}
Notice that here we consider the line $\nu =\left( p-2,1\right) $ in the
space $P\times \mathbf{R}^{1}$. As a result we obtain polynomials
characterizing the $p$-dependence of the multiplicity for a fixed distance
between the highest weight and the weight $\nu $.
This example shows that the tools elaborated above (the injection fan and
singular elements) are effective in solving the reduction problem for tensor
power modules $(L^{\omega _{i}})^{\otimes p}$.
\section{Alternative approach.}
According to Lemma 3 the multiplicity coefficients have additional
recurrence properties generated by the fundamental module weights system $
N\left( L^{\left( \omega _{i}\right) }\right) $ :
\begin{equation}
\sum_{\mu \in P^{++}}m_{\mu }^{\left( i\right) p}\Psi ^{\left( \mu
\right) }=\mathrm{ch}\left( L^{\left( \omega _{i}\right) }\right)
\Psi ^{\left( \otimes ^{\left( p-1\right) }\omega _{i}\right) }.
\label{N-recurrence-fund}
\end{equation}
This relation can be decomposed using the multiplicity functions (defined on
$P$),
\begin{equation}
\sum_{\mu \in P}M^{\omega _{i}}\left( \mu ,p\right) e^{\mu
}=\mathrm{ch}\left( L^{\left( \omega _{i}\right) }\right) \Psi
^{\left( \otimes ^{\left( p-1\right) }\omega _{i}\right) },
\end{equation}
remember that $M^{\omega _{i}}\left( \mu ,p\right) \mid _{\mu \in \overline{
C^{\left( 0\right) }}}=m_{\mu }^{\left( i\right) p}$ . Thus instead of the
highest weight search for each singular element $\Psi ^{\left( \mu \right) }$
we use their anti-symmetry properties. This leads to the recurrent relation:
\begin{equation}
M^{\omega _{i}}\left( \mu ,p\right) =\sum_{\zeta \in N\left( L^{\left(
\omega _{i}\right) }\right) }n_{\zeta }\left( L^{\left( \omega _{i}\right)
}\right) M^{\omega _{i}}\left( \mu -\zeta ,p-1\right) ,
\label{N-rec-rel-omega-i}
\end{equation}
where $n_{\zeta }\left( L^{\left( \omega _{i}\right) }\right) =\mathrm{mult}
_{L^{\left( \omega _{i}\right) }}\left( \zeta \right) $. Obviously such
relations are especially useful when $\dim \left( L^{\left( \omega
_{i}\right) }\right) $ is small, thus for our needs (when the module $
L^{\left( \omega _{i}\right) }$ is fundamental with the trivial weights
multiplicities $n_{\zeta }\left( L^{\left( \omega _{i}\right) }\right) =1$)
the obtained recurrence must be effective.
Formula (\ref{N-rec-rel-omega-i}) tells us what happens when we pass from
the $(p-1)$-th power to the $p$-th. In particular for the spinor module $
L^{\omega _{2}}$ to find the value of $M^{\omega _{2}}\left( \mu
,p\right) $ the coordinates must be shifted by the vectors of
$\mathcal{N}\left( L^{\left( \omega _{2}\right) }\right) $
diagram:
\begin{equation}
M^{\omega _{2}}\left( \mu ,p\right) =\sum_{\zeta =N\left( L^{\left( \omega
_{2}\right) }\right) }M^{\omega _{2}}\left( \mu -\zeta ,p-1\right) .
\label{main rec rel}
\end{equation}
In the recurrence starting point the value of the multiplicity function is
known $M^{\omega _{2}}\left( p\omega _{2},p\right) =1$ and for all $\nu
>p\omega _{2}$ it has zero values. In the natural coordinates this means:
\begin{equation*}
M^{\omega _{2}}\left( \left( a,b\right) ,p\right) =\sum_{\lambda =\left(
a,b\right) -\left\{ \left( \frac{1}{2},\frac{1}{2}\right) \left( -\frac{1}{2}
,\frac{1}{2}\right) \left( \frac{1}{2},-\frac{1}{2}\right) \left( -\frac{1}{2
},-\frac{1}{2}\right) \right\} }M^{\omega _{2}}\left( \lambda ,p-1\right) .
\end{equation*}
For the vector module $L^{\omega _{1}}$ and its tensor powers we can
construct similar relations:
\begin{eqnarray}
M^{\omega _{1}}\left( \left( a,b\right) ,p\right) &=&\sum_{\zeta =N\left(
L^{\left( \omega _{1}\right) }\right) }M^{\omega _{1}}\left( \left(
a,b\right) -\zeta ,p-1\right) = \notag \\
&&\sum_{\lambda =\left( a,b\right) -\left\{ \left( 1,0\right) \left(
0,1\right) \left( -1,0\right) \left( 0,-1\right) ,\left( 0,0\right) \right\}
}M^{\omega _{1}}\left( \lambda ,p-1\right) ,
\end{eqnarray}
with the similar boundary condition
\begin{equation}
M^{\omega _{1}}\left( \left( p,0\right) ,p\right) =1.
\end{equation}
The obtained recurrence relations indicate an important property of $
M^{\omega _{i}}\left( \mu ,p\right) $:
\begin{conjecture}
The multiplicity function $M^{\omega _{i}}\left( \mu ,p\right) $
is a polynomial on $p$ over $\mathbf{Q}$ (rational numbers).
\end{conjecture}
Notice that when $\mu $ belongs to the correlated (different!) boundaries of
the area where the function is nontrivial the values $M^{\omega _{i}}\left(
\mu ,p\right) $ for $i=1$ and $i=2$ coinside.
\begin{conjecture}
The multiplicities $M^{\omega _{1}}\left( \mu ,p\right) $ of the ''upper
diagonal'' highest weights ($\mu =\left( p,0\right) -n\alpha _{1}$) for $
(L^{\omega _{1}})^{\otimes p}$ coinside with the multiplicities $M^{\omega
_{2}}\left( \mu ,p\right) $ of the ''upper line'' highest weights ($\mu
=\left( \frac{p}{2},\frac{p}{2}\right) -n\alpha _{2}$) for $(L^{\omega
_{2}})^{\otimes p}$.
\end{conjecture}
On the left boundary of the Weyl chamber $\overline{C^{\left( 0\right) }}$
the multiplicities $M^{\omega _{1}}\left( \mu ,p\right) $ are subject to the
presence of the ''reflected'' singular weights in the left adjacent Weyl
chamber. This observation is important because for an analogous boundary in
the spinor case the situation is different: the adjacent Weyl chamber had no
influence on the values of $M^{\omega _{2}}\left( \mu ,p\right) $ with $\mu
\in \overline{C^{\left( 0\right) }}$ (on the corresponding subdiagonal the
function has zero values). In particular this results in the following
property.
\begin{conjecture}
The ''second diagonal'' of the highest weights for $(L^{\omega
_{1}})^{\otimes p}$ starts with zero: $M^{\omega _{1}}\left( \left(
p-1\right) \omega _{1},p\right) =0.$
\end{conjecture}
\section{Solutions for recurrence relations}
We have found out that the multiplicity functions $M^{\omega _{i}}\left( \mu
,p\right) $ are subject to an infinite system of coupled algebraic equations
with simple and obvious boundary conditions. They can be solved step by step.
For example consider the Bratteli-like diagram for $B_{2}$ vector module $
L^{\omega _{1}}$ and let $p=1,\ldots ,6$. The maximal number of paths that
connect a point in the $p-1$ slice with the points in the $p$-th one is five.
\begin{figure}
\caption{Bratteli-like diagram for $B_{2}
\label{bratteli-3}
\end{figure}
Notice that the path counting procedure here is very complicated because of
the boundary effects. Such complexities grow up considerably if we try to
apply that counting procedures to algebras with higher rank. This fact
stimulates special interest to direct studies of the recurrence relations
systems. Moreover if the corresponding equations could be solved this will
give an explicit $p$-dependence of the multiplicity function -- the result
that scaresly could be achieved by combinatorial methods.
We can construct the solution for the recurrence equations successively and
the answer is limited only by the number of equations in the system solved .
This gives the explicit multiplicity dependence on $p$ but for a finite
number of successive weights. Thus having solved the first five equations
for the spinor case we get the following table of functions $M^{\omega
_{2}}\left( \left( a,b\right) ,p\right) $ (here the coordinates $\left(
a,b\right) $ are fixed by the relation $\mu =p\omega _{2}-a\alpha
_{2}-\left( b-1\right) \alpha _{1}$):\\[1mm]
\begin{tabular}{||c||cccccc|}
\hline\hline
$b\setminus a$ & 5/2 & \multicolumn{1}{||c}{2} & \multicolumn{1}{||c}{3/2} &
\multicolumn{1}{||c}{1} & \multicolumn{1}{||c}{1/2} & \multicolumn{1}{||c||}{
0} \\ \hline\hline
1 & & $\frac{1}{2}p\left( p-3\right) $ & & $p-1$ & & $1$ \\ \cline{2-7}
& 0 & & 0 & & 0 & \\ \cline{2-7}
2 & & $
\begin{array}{c}
\frac{1}{3}\left( p-1\right) \cdot \\
\cdot \left( p+1\right) \left( p-3\right)
\end{array}
$ & & $\frac{1}{2}p\left( p-1\right) $ & & 0 \\ \cline{2-7}
& 0 & & 0 & & 0 & \\ \cline{2-7}
3 & & $
\begin{array}{c}
\frac{1}{12}\left( p-1\right) \left( p-2\right) \cdot \\
\cdot \left( p-3\right) \left( p+2\right)
\end{array}
$ & & 0 & & \\ \hline
\end{tabular}
\\[2mm]
Correspondingly having solved the first fifteen equations for the vector
case we get the following table of functions $M^{\omega _{1}}\left( \left(
a,b\right) ,p\right) $, \\[1mm]
\noindent
\begin{tabular}{||c||c|c|c|c|}
\hline\hline
$b\setminus a$ & 0 & \multicolumn{1}{||c|}{1} & \multicolumn{1}{||c|}{2} &
\multicolumn{1}{||c||}{3} \\ \hline\hline
$p$ & $1$ & 0 & 0 & 0 \\ \hline
$p-1$ & $0$ & $p-1$ & 0 & 0 \\ \hline
$p-2$ & $\frac{1}{2}p\left( p-1\right) $ & $\frac{1}{2}\left( p-1\right)
\left( p-2\right) $ & $\frac{1}{2}p\left( p-3\right) $ & 0 \\ \hline
$p-3$ & ${\
\begin{array}{c}
\frac{1}{6}\left( p-1\right) \\
\left( p-2\right) \left( p-3\right)
\end{array}
}$ & ${\
\begin{array}{c}
\frac{1}{2}p\left( p-1\right) \\
\left( p-3\right)
\end{array}
}$ & ${\
\begin{array}{c}
\frac{1}{3}p\left( p-2\right) \\
\left( p-4\right)
\end{array}
}$ & ${\
\begin{array}{c}
\frac{1}{6}p\left( p-1\right) \\
\left( p-5\right)
\end{array}
}$ \\ \hline
\end{tabular}
\section{Weyl symmetry and solutions for recurrence equations}
When the algebra is simply laced, for example $\frak{g}=A_{n}$, the Weyl
symmetry was proven to be a highly effective tool to solve the set of
recurrences equations for the powers of the first fundamental module $
L_{A_{n}}^{\omega _{1}}$ \cite{KulLyakhPost2011}. In the simplest case $
\frak{g}=A_{1}$ the complete set of multiplicity functions for powers of an
arbitrary irreducible module were thus constructed.
In the case of $B_{2}$ the difficulties start when the vector fundamental
module is tensored. The recurrence equation can be solved successively, as
was shown in the previous section, but the complete solution for the
function $M^{\omega _{1}}\left( \left( a,b\right) ,p\right) $ was not found.
Nevertheless the recurrence property (\ref{main rec rel}) permits to
describe the general dependence of the multiplicities on one of the
coordinates. To see this consider the coordinates $(s,t)$ defined by the
relation $\mu =p\omega _{2}-t\alpha _{1}-\left( s-1\right) e_{1}$ . The $t$
-dependence will be explicitly described, but only for limited values of $
s=1,2,\ldots $. This description is based on the fact that the Conjecture 7
gives us an explicit answer to the ''first diagonal'' of multiplicities $
\left\{ m_{\left( 1,t\right) }^{\left( 1\right) p}|t=0,1,\ldots \right\} $.
Starting with this expression and using the relation (\ref{main rec rel})
reformulated for the ''diagonal lines'' of functions we can find explicit
expressions for any such line provided the previous lines are known:
\begin{equation*}
M^{\omega_1 }\left( \left(1,t\right),p\right)=M^{\omega_2 }\left(
\left(\frac{p}{2},\frac{p}{2}-t\right),p\right)=\frac{\Gamma
\left( p+1\right) \left( p+1-2t\right) }{\Gamma \left(
p+2-t\right) \Gamma \left( t+1\right) },
\end{equation*}
\begin{equation*}
M^{\omega_1 }\left( \left(2,t\right),p\right)=\frac{\Gamma \left(
p+1\right) \left( p-t\right) \left( p-2t\right) }{\Gamma \left(
p+2-t\right) \Gamma \left( t\right) \left( t+1\right) },
\end{equation*}
\begin{equation*}
M^{\omega_1 }\left( \left(3,t\right),p\right)=\frac{\Gamma \left(
p+1\right) \left( p-2t-1\right) }{2\Gamma \left( p-t\right) \Gamma
\left( t+1\right) },
\end{equation*}
\begin{eqnarray*}
M^{\omega_1 }\left( \left(4,t\right),p\right) &=&\frac{\Gamma
\left( p+1\right)
\left( p-2t-2\right) }{6\Gamma \left( p+1-t\right) \Gamma \left( t+3\right) }
\cdot \\
&&\cdot \left(
\begin{array}{c}
\left( t^{2}+6t+2\right) p^{2}-2\left( t+2\right) ^{2}\left( t+1\right) p+
\\
+t^{4}+4t^{3}+8t^{2}+8t+6
\end{array}
\right) ,
\end{eqnarray*}
\begin{eqnarray*}
M^{\omega_1 }\left( \left(5,t\right),p\right) &=&\frac{\Gamma
\left( p+1\right)
\left( p-2t-3\right) }{24\Gamma \left( p-t\right) \Gamma \left( t+3\right) }
\cdot \\
&&\cdot \left(
\begin{array}{c}
\left( t^{2}+11t+6\right) p^{2}-\left( 2t+4\right) \left( t+1\right) \left(
t+2\right) p+ \\
+\left( t^{4}+4t^{3}+8t^{2}+8t+6\right)
\end{array}
\right) ,
\end{eqnarray*}
\begin{eqnarray*}
M^{\omega_1 }\left( \left(6,t\right),p\right) &=&\frac{\Gamma
\left( p+1\right)
\left( p-2t-4\right) }{120\Gamma \left( p-t\right) \Gamma \left( t+4\right) }
\cdot \\
&&\cdot \left(
\begin{array}{c}
\left( t^{3}+21t^{2}+86t+36\right) p^{3}- \\
-\left( 3t^{4}+54t^{3}+309t^{2}+654t+276\right) p^{2}+ \\
+\left( 3t^{5}+45t^{4}+326t^{3}+1086t^{2}+1408t+516\right) p- \\
-\left( t+1\right) \left( t+3\right) \left(
t^{4}+8t^{3}+68t^{2}+208t+12\right)
\end{array}
\right) ,
\end{eqnarray*}
and so on. We see that beginning from $M_{B_2 }^{\omega_1 }\left(
\left(4,t\right),p\right)$ only some factor of the multiplicity
function can be presented as a product of simple binomials like
$\left( p-x\right) $. In the forthcoming publications we shall
discuss this property in details.$M^{\omega_1 }\left(
\left(4,b\right),p\right)$
\section{Conclusions}
The tensor powers decomposition algorithm based on singular
weights and injection fan technique was proven to be an effective
tool in multiplicity property studies. Its abilities were
demonstrated on tensor powers decompositions of $B_2$-fundamental
modules. This algorithm is universal and can be applied to
investigate decomposition properties in case of an arbitrary
simple Lie algebra and its arbitrary module.
As it was predicted in \cite{KulLyakhPost2011} in non-simply laced case the
Weyl symmetry properties are insufficient to provide the final solution
for the corresponding set of recurrence relations for multiplicity functions
(at least this appeared to be true for the vector fundamental modules).
Nevertheless (this was shown above in our studies of fundamental $B_2$-modules)
important properties of multiplicity coefficients for any highest weight $\nu$
can be found by constructing the functions
$M^{\omega _{i}}\left( \left( \nu \right) ,p\right) $ successively i.e. by
constructing the solution for a final part of the full set of recurrence relations.
\section{Acknowledgments}
Supported by the Russian Foundation for Fundamental Research grant N 09-01-00504
and the "Dynasty" Foundation.
\end{document} |
\begin{document}
\title{
Symblicit Exploration and Elimination\\for Probabilistic Model Checking
\thanks{
The authors are listed alphabetically.
This work was supported by
NWO VENI grant no.\ 639.021.754.}
}
\author{
Ernst Moritz Hahn\inst{1}
\and Arnd Hartmanns\inst{2}
}
\institute{
Queen's University Belfast, Belfast, UK
\and University of Twente, Enschede, The Netherlands}
\date{\today}
\maketitle
\begin{abstract}
Binary decision diagrams can compactly represent vast sets of states, mitigating the state space explosion problem in model checking.
Probabilistic systems, however, require multi-terminal diagrams storing rational numbers.
They are inefficient for models with many distinct probabilities and for iterative numeric algorithms like value iteration.
In this paper, we present a new ``symblicit'' approach to checking Markov chains and related probabilistic models:
We first generate a decision diagram that \emph{symb}olically collects all reachable states and their predecessors.
We then concretise states one-by-one into an exp\emph{licit} partial state space representation.
Whenever all predecessors of a state have been concretised, we \emph{eliminate} it from the explicit state space in a way that preserves all relevant probabilities and rewards.
We thus keep few explicit states in memory at any time.
Experiments show that very large models can be model-checked in this way with very low memory consumption.
\end{abstract}
\section{Introduction}
\label{sec:Introduction}
Many of the complex systems that we are surrounded by, rely on, and use every day are inherently probabilistic:
The Internet is built on randomised algorithms such as the collision avoidance schemes in Ethernet and wireless protocols, with the latter additionally being subject to random message loss.
Hard- and software in cars, trains, and airplanes is designed to be fault-tolerant based on mean-time-to-failure statistics and stochastic wear models.
Machine learning algorithms give recommendations based on estimates of the likelihoods of possible outcomes, which in turn may be learned from randomly sampled data.
Given a formal mathematical model of such a system, e.g.\ in the form of (a high-level description of) a discrete- or continuous-time Markov chain (DTMC or CTMC), probabilistic model checking can automatically compute (an approximation of) the value of a quantity of interest.
Such quantities include the probability to finally reach an unsafe state (a measure of reliability), the steady-state probability to be in a failure state (determining availability), the long-run average reward (measuring e.g.\ throughput or energy consumption), or the accumulated cost up to a certain set of states (where e.g.\ a job is complete).
The standard approach is to proceed in two phases:
First, \emph{explore} the state space, building a representation of the set of reachable states and the transitions connecting them.
Transitions are annotated with rational values for probabilities and rewards, which are usually represented as floating-point numbers.
Second, use an iterative numeric algorithm such as value iteration~\cite{Put94} or one of its sound variants~\cite{HM18,QK18,BKLPW17,HK19} to \emph{compute} the quantity of interest.
These algorithms in fact compute a value for every state that approximates the quantity starting from that state up to a prescribed error~$\epsilon$.
In contrast to classic functional model checking, which admits on-the-fly algorithms for e.g.\ reachability or LTL properties, probabilistic model checking is thus doubly affected by the state space explosion problem:
First, the entire state space must be stored in memory, including many numeric values.
Second, the numeric computation requires multiple vectors of values to be stored, and updates to be performed on them, for all states.
\paragraph{Current approaches} to mitigate the state space explosion problem in probabilistic model checking include the use of partial exploration and learning algorithms, bisimulation minimisation, and compact representations of the state space or value vectors by binary decision diagrams (BDDs).
They exploit different structural properties that only sometimes overlap.
The learning-based approaches~\cite{BCCFKKPU14,ABHK18} for reachability probabilities work well for models where a small initial subset of the state space determines most of the probability mass.
In such cases, which are not abundant among existing case studies~\cite{HHHKKKPQRS19}, they complete in a few seconds while exhaustive approaches run out of time or memory~\cite[Table~1]{BHH19}.
Bisimulation minimisation reduces the state space to a quotient according to a probabilistic bisimulation relation; see \cite[Sect.~5.1]{BHK19} for an overview.
It has been implemented in \tool{Storm}~\cite{DJKV17} and allows certain very large models to be checked efficiently; in general, its impact depends on the amount of bisimilar states in the given system.
Finally, BDDs~\cite{Lee59,Bry18} have a long history of use in (discrete-state) model checking~\cite{CG18} to compactly represent state spaces, in good cases reducing memory usage by orders of magnitude.
They work well when the state space is structured and exhibits symmetry, which is often the case for real-life case studies modelled by humans (as opposed to randomly generated examples).
In probabilistic model checking, however, numeric values from continuous domains are part of state spaces and must be encoded in the decision diagrams.
A binary encoding of their floating-point representation does not usually result in compact diagrams; instead, multi-terminal BDDs (MTBDDs), where each of the (unbounded number of) leaves represents one number, have been applied with some success, notably in the probabilistic model checker \tool{Prism}~\cite{KNP11}.
They however do not provide much compaction for models with many distinct probabilities or reward values due to the large number of leaves.
They also do not work well to represent the large vectors of values used in iterative numeric algorithms such as value iteration, which progress through many very different intermediate values for each state before converging to, but often not reaching, a fixpoint.
For this reason, \tool{Prism} defaults to its \textit{hybrid} engine, which uses MTBDDs for the state space but arrays of double-precision values for iteration.
Its fully symbolic \textit{mtbdd} engine only solves specific large structured models in reasonable runtime.
\paragraph{Our contribution}
is a new approach that combines (MT)BDDs, explicit state representations, and state elimination to tackle the problem of model-checking large probabilistic specifications.
Its novelty lies in
(1)~using BDDs precisely for those tasks that they work well for, and
(2)~using state elimination instead of the standard iterative algorithms in the computation phase.
We work with discrete-state probabilistic systems; in this paper, we use DTMC to explain our approach, but the same techniques apply directly to CTMC, Markov decision processes (MDP)~\cite{Put94}, and Markov automata (MA)~\cite{EHZ10}, too.
Our state space exploration performs a standard explicit-state breadth-first search, but we use a decision diagram instead of the standard hashset to store the set of visited states.
We do not store transitions, thus no continuous numeric values blow the diagram up.
However, we do count the number of predecessors of each state---thus we use an MTBDD.
Since this number is a discrete quantity with low variation in most models, the diagrams usually remain compact.
For the computation phase, we explore the state space again, this time creating a representation that includes transitions, but that is explicit.
During this exploration, we keep track of the number of explored predecessors of each state.
Whenever, for some fully-explored state~$s$, this number reaches the predecessor count given by the MTBDD, we apply \emph{state elimination}:
we remove $s$ from the (yet incomplete) explicit state space representation, and replace all of its incoming and outgoing transitions by direct transitions between the predecessors and successors of~$s$.
By redistributing the original transitions' probabilities and rewards in the right way, the quantity of interest remains unaffected.
This method of computation simultaneously avoids the iterative algorithms' convergence and precision issues~\cite{HM14} and keeps memory usage due to the explicit representation low:
on most models, most states are eliminated soon after they have been fully explored, thus only few need to be kept in memory at any time.
Upon termination, only the initial state and goal state(s) remain, and the value for the quantity of interest can be read off the transitions connecting them.
Two technical insights make our computation phase work well:
First, we use an explicit representation not only to avoid storing (continuous) probabilities and rewards in an MTBDD, but also because state elimination tends to create unstructured intermediate state spaces that would blow up any BDD representation.
Second, the precomputed predecessor count allows us to eliminate a state at precisely the moment after which we will not encounter it again in our search, avoiding costly re-explorations and re-eliminations.
\paragraph{Related work.}
State elimination stems from the classic reduction algorithm to convert a finite automaton into a regular expression~\cite{BM63}.
It was introduced to probabilistic model checking to solve parametric Markov chains~\cite{Daw04} and forms the core of the \tool{Param}~\cite{HHWZ10} and \tool{Prophesy}~\cite{DJJCVBKA15} tools.
For non-parametric models, it enables efficient computation of reward-bounded reachability probabilities~\cite{HH16}.
In this paper, we use it for non-parametric Markov chains and unbounded (infinite-horizon) properties.
In all of these settings, its effectiveness crucially depends on the order in which states are eliminated, which is determined by (configurable) heuristics.
Symblicit techniques have previously been used for long-run average properties~\cite{WBBHCHDT10}, based on bisimulation minimisation, and later expanded in related settings~\cite{BBRB17}.
A different form of elimination on strongly-connected components was used by Gui et al.\xspace \cite{GSSLD14} to accelerate the (explicit-state) computation of reachability probabilities via value iteration.
\section{Background}
\label{sec:Background}
\paragraph{Mathematical notions.}
$\ensuremath{\mathbb{R}}\xspacepluszero$ is the set of all non-negative, and $\ensuremath{\mathbb{R}}\xspaceplus$ the set of all positive, real numbers.
A (discrete) \emph{probability distribution} over $S$ is a function $\mu \in S \to [0, 1]$ with countable \emph{support} $\support{\mu} \mathrel{\vbox{\offinterlineskip\ialign{\hfil##\hfil\cr{\tiny \rm def}\cr\noalign{\kern0.30ex}$=$\cr}}} \set{ s \in S \mid \mu(s) > 0 }$ and $\sum_{s \in \support{\mu}} \mu(s) = 1$.
$\Dist{S}$ is the set of all probability distributions over $S$.
\subsection{Discrete-Time Markov Chains}
\label{sec:MarkovChains}
\begin{definition}
\label{def:MarkovChain}
A \emph{discrete-time Markov chain} (DTMC) is a tuple
$M = \tuple{S, s_I, P, R}$
consisting of a
finite set of \emph{states} $S$, an \emph{initial state} $s_I \in S$, a \emph{transition} function $P \colon \mathit{S} \to \Dist{S}$,
and a \emph{reward} function $R \colon S \to \ensuremath{\mathbb{R}}\xspacepluszero$.
\end{definition}
We also write a transition as $s \xtr{p} s'$ if $p = P(s)(s') > 0$.
A transition is uniquely identified by the two states it connects.
When in state $s$ of a DTMC, we delay for one time unit before jumping to the next state.
\emph{Continuous-time Markov chains} (CTMC) extend DTMC by additionally assigning a \emph{rate} $Q(s) \in \ensuremath{\mathbb{R}}\xspaceplus$ to every state.
Then the probability to delay for at most $t$ time units is
$1 - \mathrm{e}^{-Q(s) \cdot t}$,
i.e.\ the residence time follows the exponential distribution with rate $Q(s)$.
In both models, the probability to then move to state $s'$ is given by $P(s)$.
When staying for $t$ time units in state $s$, we incur a reward of $R(s) \cdot t$.
To simplify the presentation, we use DTMC throughout this paper, but mention the changes needed in definitions or algorithms to use CTMC, where appropriate.
\begin{figure}
\caption{DTMC $M_z$ for the Zeroconf protocol ($h=32, a=2^8,p=0.2,n=4$)}
\label{fig:ExampleDTMC}
\end{figure}
\begin{example}
\label{ex:MarkovCahin}
As a running example, we use a very abstract model of the Zeroconf protocol~\cite{BSHV03}, shown as DTMC $M_z$ in \Cref{fig:ExampleDTMC} (adapted from~\cite{GHS18}).
We draw transitions as arrows labelled with their probability.
Non-zero rewards are given next to the states.
$M^z$ has 7~states and 12~transitions.
The protocol is used by hosts joining a network to auto-configure a unique IP address.
A new host joining the network of $h = 32$ host starts in state \texttt{i}.
It selects an address uniformly at random from the space of $a = 2^8$ addresses.
The probability that the address is already in use is $\frac{h}{a} = \frac{1}{8}$.
The host checks $n = 4$ times whether its address is already in use.
If it is not, all checks will succeed, modelled by state \texttt{ok}, to which we moved with probability $1 - \frac{h}{a}$.
If it is, then states $n$ down to $1$ model the checks.
Each check can fail to give the correct negative result due to message loss with probability $p = 0.2$.
If all tests do so, then the host incorrectly believes that it has a unique address, in state~$\bot$.
Otherwise, it retries with a newly chosen address from state \texttt{i}.
We incur a reward of $1$ in state \texttt{i}, i.e.\ for every IP address we try.
The size of the DTMC can be blown up arbitrarily via parameter $n$.
\end{example}
In practice, higher-level modelling languages like \textsc{\mbox{Modest}}\xspace~\cite{HHHK13} or the \tool{Prism} language~\cite{KNP11} are used to specify larger DTMC.
The semantics of a DTMC is formally captured by its \emph{paths}:
\begin{definition}
Given a DTMC $M$ as above, a \emph{finite path} is a sequence
$\pi_\mathrm{fin} = s_0\, t_0\, s_1\, t_1 \dots s_n$
of states $s_i \in S$ and delays $t_i \in \ensuremath{\mathbb{R}}\xspaceplus$ where $P(s_i)(s_{i + 1}) > 0$ and $t_i = 1$ for all $i \in \set{ 0, \dots, n - 1 }$.
Let $|\pi_\mathrm{fin}| \mathrel{\vbox{\offinterlineskip\ialign{\hfil##\hfil\cr{\tiny \rm def}\cr\noalign{\kern0.30ex}$=$\cr}}} n$,
$\mathrm{last}({\pi_\mathrm{fin}}) \mathrel{\vbox{\offinterlineskip\ialign{\hfil##\hfil\cr{\tiny \rm def}\cr\noalign{\kern0.30ex}$=$\cr}}} s_n$,
$\mathrm{dur}(\pi_\mathrm{fin}) \mathrel{\vbox{\offinterlineskip\ialign{\hfil##\hfil\cr{\tiny \rm def}\cr\noalign{\kern0.30ex}$=$\cr}}} \sum_{i=0}^{n-1} t_i$,
and $\mathrm{rew}({\pi_\mathrm{fin}}) \mathrel{\vbox{\offinterlineskip\ialign{\hfil##\hfil\cr{\tiny \rm def}\cr\noalign{\kern0.30ex}$=$\cr}}} \sum_{i=0}^{n-1} t_i \cdot R(s_i)$.
$\Pi_\mathit{fin}$ is the set of all finite paths starting in $s_I$.
A \emph{path} is an analogous infinite sequence $\pi$, and $\Pi$ are all paths starting in $s_I$.
We define $s \in \pi \ensuremath{\Leftrightarrow}\xspace \exists\, i \colon s = s_i$.
Let $\pi_{\to m}$ for $m \in \ensuremath{\mathbb{N}}\xspace$ be the prefix of $\pi$ of length $m$, i.e.\ $|\pi_{\to m}| = m$, and let $\pi_{\to G}$ be the shortest prefix of $\pi$ that contains a state in $G \subseteq S$, or $\bot$ if $\pi$ contains no such state.
Let $\mathrm{rew}(\bot) \mathrel{\vbox{\offinterlineskip\ialign{\hfil##\hfil\cr{\tiny \rm def}\cr\noalign{\kern0.30ex}$=$\cr}}} \infty$.
\end{definition}
In CTMC, the $t_i$ can be arbitrary numbers in $\ensuremath{\mathbb{R}}\xspacepluszero$.
For $M$ as above, following the rules described below \Cref{def:MarkovChain} and the standard cylinder set construction~\cite{BK08}, we obtain a probability measure $\mathbb{P}_M$ on measurable sets of paths starting in $s_I$.
\begin{definition}
Given a set of goal states $G \subseteq S$, the \emph{reachability probability} w.r.t.\xspace $g$ is $\mathbb{P}(\diamond\: G) \mathrel{\vbox{\offinterlineskip\ialign{\hfil##\hfil\cr{\tiny \rm def}\cr\noalign{\kern0.30ex}$=$\cr}}} \mathbb{P}_M( \pi \in \Pi \mid \exists\, g \in G \colon g \in \pi)$.
Let $r_G \colon \Pi \to \ensuremath{\mathbb{R}}\xspacepluszero$ be the random variable defined by $r_G(\pi) = \mathrm{rew}(\pi_{\to G})$.
Then the \emph{expected reward} to reach $G$ is the expected value of $r_G$ under $\mathbb{P}_M$, written as $\mathbb{E}(\blackdiamond\: G)$.
Let $r_\mathit{lra} \colon \Pi \to \ensuremath{\mathbb{R}}\xspacepluszero$ be defined by $r_\mathit{lra}(\pi) = \lim_{i \to \infty} \mathrm{rew}(\pi_{\to i})/\mathrm{dur}(\pi_{\to i})$.
Then the \emph{long-run average reward} is the expected value of $r_\mathit{lra}$ under $\mathbb{P}_M$, written as $\mathbb{L}$.
\end{definition}
The steady-state probability $\mathbb{S}(S')$ of residing in a state in $S' \subseteq S$ is a special case of the long-run average reward where $R(s) = 1$ if $s \in S'$ and $0$ otherwise.
Whenever we consider a DTMC with a set of goal states $G$, we assume that they have been made absorbing, i.e.\ that for all $g \in G$ we have $P(g)(g) = 1$.
Given a CTMC, reachability probabilities and expected rewards can be computed on its \emph{embedded DTMC}, obtained by dividing all rewards by $Q(s)$; only for long-run averages do we need a dedicated treatment of the rates resp.\ residence times.
\begin{example}
For our Zeroconf example DTMC $M_z$ from \Cref{fig:ExampleDTMC}, we may want to compute the probability to eventually pick a unique address $\mathbb{P}(\diamond\: \set{\texttt{ok}})$, which will be just below $1$, and the expected number of addresses that we ever try $\mathbb{E}(\blackdiamond\: \set{ \texttt{ok}, \bot })$.
Note that $\mathbb{E}(\blackdiamond\: \set{ \texttt{ok} })$ is $\infty$ by definition since the set of paths that never reach state \texttt{ok} has positive probability.
\end{example}
\subsection{Binary Decision Diagrams}
\label{sec:BDDs}
\emph{Binary decision diagrams} (BDDs)~\cite{Lee59,Bry18} represent Boolean functions as rooted directed acyclic graphs.
They have two leaf nodes, \True and \False.
Every inner node is associated to one input bit, and has two children: the high (solid line) and low (dotted line) child.
On a path from the root to a leaf, every bit must occur at most one.
Such a path corresponds to the inputs in which bit $b_i$ is assigned to \True (\False) if we go from a node for $b_i$ to its high (low) child.
We typically order the bits on all paths, merge isomorphic subgraphs, and remove redundant nodes.
Then a BDD can represent many functions with few nodes.
In model checking, BDDs are used to represent sets of states (by assigning \True to the binary encoding of a state iff it is in the set) as well as the transition relation (by assigning \True to the binary encoding of a pair of states if they are connected by a transition).
In probabilistic model checking, however, we need to encode functions that map to rational numbers to encode transition probabilities, rewards, and the value vectors in value iteration.
Most tools represent them as 64-bit floating-point values, but the corresponding binary representation does not typically allow good compression with BDDs.
Symbolic probabilistic model checkers such as \tool{Prism}~\cite{KNP11} this use \emph{multi-terminal} BDDs (MTBDDs) with one leaf node per number.
Since a finite model only contains finitely many probabilities, or values for states, this approach is effective, but often not efficient:
for example, when performing value iteration on our example DTMC $M_z$ for $\mathbb{P}(\diamond\: \set{\texttt{ok}})$, we have to encode the following function after 5 iterations:\\[2pt]
\centerline{
$\set{ \texttt{i} \mapsto 0.99225, 4 \mapsto 0.9716, 3 \mapsto 0.966, 2 \mapsto 0.938, 1 \mapsto 0.784, \texttt{ok} \mapsto 1, \bot \mapsto 0 }$
}\\[2pt]
Observe that every state has a distinct value, thus the MTBDD offers no compression.
In practice, they only work well for very specific models with few distinct transition probabilities and rewards, and where the iterative numeric algorithms assign the same (intermediate) values to many states.
\begin{figure}
\caption{MTBDD counting the numbers of predecessors for all states of $M_z$}
\label{fig:ExampleMTBDD}
\end{figure}
\begin{example}
\label{ex:MTBDD}
\Cref{fig:ExampleMTBDD} shows an MTBDD mapping every state of $M_z$ to its number of predecessor states.
We have 7 states, thus use 3 bits for their encoding.
States $1$ through $4$ are encoded as that number, \texttt{i} is $5$ ($101_\mathrm{2}$), \texttt{ok} is $6$ (($110_\mathrm{2}$), and $\bot$ is $7$ ($111_\mathrm{2}$).
There is no (reachable) state encoded as $0$, thus we map $0$ to the extra $\bot$ leaf node---in this way, such an MTBDD can indicate that certain states are unreachable, or have not been explored yet.
Observe that the MTBDD representation achieves some compression by excluding two redundant nodes for bit~2.
If we scale the model up by increasing $n$, the compression would increase.
\end{example}
\subsection{State Elimination}
\label{sec:StateElimination}
\begin{figure}
\caption{DTMC state elimination}
\label{fig:Elimination}
\end{figure}
State elimination is a process by which a state of a DTMC is removed, i.e.\ transitions are modified such that it is no longer reachable from the initial state and it is removed from the state set $S$, in a way that preserves the values of all properties of interest.
We show the schematic of state elimination in \Cref{fig:Elimination}:
we eliminate a state $t$ by redistributing the probability to enter a self-loop onto its other outgoing transitions, then combine its incoming and outgoing transitions.
It is easy to see that this preserves the probabilities of all measurable sets of paths that pass through $t$ when projecting $t$ out from every path.
In particular, the paths that forever take the self-loop have probability mass zero, which is why we could eliminate the loop.
For rewards, the transformation only preserves the \emph{expected} reward values of sets of paths:
\begin{enumerate}
\item
In $t$, the expected number of times we take the self-loop is $\frac{p_c}{1 - p_c}$, thus the expected reward from passing through $t$ is $\frac{r_t \cdot p_c}{1 - p_c}$ (for the loop) plus $r_t$ (for taking one of the other outgoing transitions.
\item
Out of $s$, the probability to enter $t$ next is $p_a$, thus we multiply the expected reward of passing through $t$ by $p_a$ and add this to the reward of $s$.
\end{enumerate}
\Cref{alg:Eliminate} shows the pseudocode to perform state elimination on a DTMC stored in explicit data structures (i.e.\ hash sets for states, lists of transitions, etc.).
\begin{algorithm}[t]
\Function(\tcp*[f]{all explicit}){\texttt{Eliminate}$(\tuple{S, s_I, P, R}$, $s \in S$, $S_\mathit{keep} \subseteq S)$}{
\If(\tcp*[f]{if $s$ has a self-loop and other transitions:}){$s \in \support{P(s)} \wedge P(s) \,{<}\, 1$}{
\ForEach(\tcp*[f]{redistribute onto other transitions}){$s' \in \support{P(s)} \setminus \set{ s }$}{
$P(s)(s') := P(s)(s') / (1 - P(s)(s))$\tcp*{the self-loop's probability}
}
$R(s) := R(s) + R(s) \cdot {P(s)(s)}/({1 - P(s)(s)})$\tcp*{add the expected reward}
$P(s)(s) := 0$\tcp*{remove the self-loop}
}
\ForEach(\tcp*[f]{for every predecessor $s_\mathit{pre}$:}){$s_\mathit{pre} \in \set{ s'' \mid s \in \support{P(s'')} } \setminus \set{ s }$}{
$p := P(s_\mathit{pre})(s)$, $P(s_\mathit{pre})(s) := 0$\tcp*{remove the transition from $s_\mathit{pre}$ to $s$}
\ForEach(\tcp*[f]{then merge the transitions of $s$}){$s' \in \support{P(s)}$}{
$P(s_\mathit{pre})(s') := P(s_\mathit{pre})(s') + p \cdot P(s)(s')$\tcp*{into the transitions of~$s_\mathit{pre}$}
}
$R(s_\mathit{pre}) := R(s_\mathit{pre}) + p \cdot R(s)$\tcp*{merge the reward of $s$ into that of $s_\mathit{pre}$}
}
\If(\tcp*[f]{if $s$ is not needed: remove}){$s \notin S_\mathit{keep} \wedge P(s)(s) = 0$}{
$P := P \setminus \set{ s \mapsto P(s) }$, $R := R \setminus \set{ s \mapsto R(s) }$\tcp*{its transitions, reward,}
$S := S \setminus \set{ s }$\tcp*{and the state itself}
}
}
\caption{State elimination for probabilities and expected rewards}
\label{alg:Eliminate}
\end{algorithm}
\section{Symblicit Exploration and Elimination}
\label{sec:StateElimination}
As we explained in \Cref{sec:Introduction} and illustrated in \Cref{sec:BDDs}, many probabilistic models do not give rise to a compact BDD-based representation if the numeric values---probabilities, rewards, rates for CTMC---are included.
Furthermore, the standard iterative numeric algorithms like value iteration usually produce data that is hardly BDD-compressible.
In this section, we present a combined symbolic-explicit approach that uses MTBDDs in a way that usually avoids such problems, and that uses state elimination to calculate $\mathbb{P}$, $\mathbb{E}$, and $\mathbb{L}$ values without having to keep (values for all states of) the entire state space in memory.
\begin{algorithm}[t]
\Function(\tcp*[f]{explicit $s_I$, executable \texttt{P}, \texttt{R}, \texttt{G}}){\texttt{ExploreEliminate}$(s_I$, \texttt{P}, \texttt{R}, \texttt{G}$)$}{
$\hat{\mathit{pre}} := \texttt{Explore}(s_I$, \texttt{P}$)$\label{alg:ExploreEliminate:Explore}\tcp*{get predecessor count MTBDD for all states}
$\mathit{done} := \varnothing$, $\mathit{agenda} := \set{ s_I }$\label{alg:ExploreEliminate:Rest}\tcp*{$\mathit{done}$ stored as hash set, $\mathit{agenda}$ as queue}
$S := \set{ s_I }$, $P := \varnothing$, $R := \varnothing$, $\mathit{pre}' := \set{ s_I \mapsto 0 }$\tcp*{explicit sets and functions}
\While(\label{alg:ExploreEliminate:While}){$\mathit{agenda} \neq \varnothing$}{
$s := \text{next element of } \mathit{agenda}$, $\mathit{agenda} := \mathit{agenda} \setminus \set{ s }$\;
\ForEach(\tcp*[f]{explore $s$}){$s' \in \support{\texttt{P}(s)}$}{
$P(s)(s') := \texttt{P}(s)(s')$, $R(s) := \texttt{R}(s)$\;
\If{$s' \notin S$}{
$S := S \cup \set{ s }$, $\mathit{agenda} := \mathit{agenda} \cup \set{ s }$\;
$\mathit{pre}' := \mathit{pre}' \cup \set{ s' \mapsto 0 }$
}
\lIf{$s' \neq s$}{$\mathit{pre}'(s') := \mathit{pre}'(s') + 1$}
}
$\mathit{done} := \mathit{done} \cup \set{ s }$\label{alg:ExploreEliminate:FullyExplored}\tcp*{$s$ is now fully explored}
$E := \set{ s_e \mid s_e \in \{ s \} \,{\cup}\, \support{P(s)} \,{\cap}\, \mathit{done} }$\tcp*{collect just modified states}
$E := \set{ s_e \mid s_e \in E \wedge \hat{\mathit{pre}}(s_e) \,{=}\, \mathit{pre}'(s) }$\tcp*{with all predecessors explored}
\ForEach(\tcp*[f]{and eliminate them}){$s_\mathit{elim} \in E$}{
\texttt{Eliminate}$(\tuple{S, s_I, P, R}, s_\mathit{elim}, \set{ s_I })$\label{alg:ExploreEliminate:Eliminate}\;
\If(\tcp*[f]{cleanup}){$s_\mathit{elim} \notin S$}{
$\mathit{pre}' := \mathit{pre}' \setminus \set{ s_\mathit{elim} \mapsto \mathit{pre}'(s_\mathit{elim}) }$,
$\mathit{done} := \mathit{done} \setminus \set{ s_\mathit{elim} }$
}
}
}
\lIf{we compute a reachability probability}{\Return{
$\sum_{g \in \texttt{G}} P(s_I)(g)$\label{alg:ExploreEliminate:PReturn}
}}
\ElseIf{we compute an expected reward}{
\lIf(\tcp*[f]{we have $\mathbb{P}(\diamond\: \texttt{\emph{G}}) < 1$}){$\support{P(s_I)} \setminus \texttt{G} \neq \varnothing$}{\Return{$\infty$\label{alg:ExploreEliminate:EReturnInfty}}}
\lElse{\Return{$R(s_i) + \sum_{g \in \texttt{G}} P(s_I)(g) \cdot R(g)$\label{alg:ExploreEliminate:EReturnFinite}}}
}
}
\caption{Symblicit exploration-elimination for probabilities and expected rewards}
\label{alg:ExploreEliminate}
\end{algorithm}
The pseudocode of our approach is shown as function \texttt{ExploreEliminate} in \Cref{alg:ExploreEliminate}.
It uses functions \texttt{Explore} of \Cref{alg:Explore} and \texttt{Eliminate} of \Cref{alg:Eliminate}.
We typeset values that represent executable code in \texttt{monospace} font:
compact specifications in high-level modelling languages are typically compiled to or interpreted as functions that, given an explicit (bit string) representation of a state, enumerate its transitions (\texttt{P}), compute its reward (\texttt{R}), and return \True iff it is a goal state (\texttt{G}).
We mark variables storing symbolic data (i.e.\ BDDs or MTBDDs) with a $\hat{\mathit{hat}}$.
All other values typeset in $\mathit{italics}$ use explicit data structures such as bit strings for states, hash sets or queues of such bit strings, lists of transitions, etc.
\begin{algorithm}[t]
\Function(\tcp*[f]{explicit $s_I$, executable \texttt{P}}){\texttt{Explore}$(s_I$, \texttt{P}$)$}{
$\hat{\mathit{seen}} := \set{ s_I }$, $\mathit{agenda} := \set{ s_I }$\tcp*{$\mathit{seen}$ stored as BDD, $\mathit{agenda}$ as queue}
$\hat{\mathit{pre}} := \set{ s_I \mapsto 0 }$\tcp*{predecessor count, stored as MTBDD}
\While{$\mathit{agenda} \neq \varnothing$}{
$s := \text{next element of } \mathit{agenda}$, $\mathit{agenda} := \mathit{agenda} \setminus \set{ s }$\;
\ForEach{$s' \in \support{\texttt{P}(s)} \setminus \set{ s }$}{
\If{$s' \notin \hat{\mathit{seen}}$}{
$\hat{\mathit{seen}} := \hat{\mathit{seen}} \cup \set{ s }$, $\mathit{agenda} := \mathit{agenda} \cup \set{ s }$\;
$\hat{\mathit{pre}} := \hat{\mathit{pre}} \cup \set{ s' \mapsto 0 }$
}
$\hat{\mathit{pre}}(s') := \hat{\mathit{pre}}(s') + 1$\tcp*{$s$ is a previously-unseen predecessor of $s'$}
}
}
\Return{$\hat{\mathit{pre}}$}
}
\caption{Symbolic exploration via breadth-first search with predecessor counting}
\label{alg:Explore}
\end{algorithm}
Our first step, in line~\ref{alg:ExploreEliminate:Explore}, is to symbolically explore the set of reachable states by calling function \texttt{Explore}.
This function performs a standard breadth-first search, using a BDD for the set of visited states, and additionally constructs an MTBDD that counts the number of predecessors of each state like the one shown in \Cref{fig:ExampleMTBDD} for $M_z$.
In our implementation, $\mathit{seen}$ and $\mathit{pre}$ are actually managed in a single MTBDD as explained in \Cref{ex:MTBDD}.
We then, starting from line~\ref{alg:ExploreEliminate:Rest}, perform another exploration of the state space.
This time, however, we use explicit data structures, and we track the number of \emph{fully explored} predecessors for every state in hash table $\mathit{pre}'$.
A state is fully explored if its reward, all of its transitions, and all successor states, have been added to the explicit representations for $S$, $R$, and $P$.
We track the set of fully explored states in hash set $\mathit{done}$.
Whenever we are done visiting a state $s$ in this second exploration (i.e.\ in line~\ref{alg:ExploreEliminate:FullyExplored} and below), it has just become fully explored, and the fully-explored-predecessor count of its successors has changed.
We then check which of these changed states fulfils the criteria for being eliminated:
It must be fully explored (which only $s$ is for certain), and all of its predecessors must be fully explored (which we determine by comparing $\mathit{pre}'$ and $\hat{\mathit{pre}}$).
We call \texttt{Eliminate} on these states in line~\ref{alg:ExploreEliminate:Eliminate}.
In this way, if we indeed manage to eliminate most states soon after they have been explored, the explicit data structures---$S$, $P$, $R$, $\mathit{pre}'$, $\mathit{done}$, etc.---only track few states at any time and thus consume little memory.
The predecessor count in $\hat{\mathit{pre}}$ is crucial for being able to perform efficient elimination; without it, we would have to apply heuristics that could lead to states being eliminated that would later be explored as successors of other states again, leading to costly re-exploration and re-eliminations.
In \texttt{Eliminate}, if a state is part of the set $S_\mathit{keep}$, we still modify and ``redirect'' the transitions of its predecessors to go around this state, but we do not remove it from the state space.
We use this to avoid eliminating the initial state $s_I$.
We also do not eliminate states whose only transition is a self-loop: they do not have successors to which transitions could be redirected.
Elimination will thus eventually reduce each bottom strongly connected component (BSCC) of the DTMC to one such self-loop state.
Since we assume all goal states to only have a self-loop, each of them is a BSCC.
Once the outer loop of line~\ref{alg:ExploreEliminate:While} in \texttt{ExploreEliminate} finishes, \texttt{Eliminate} has been called for all states.
Every surviving state at this point is thus the result of eliminating a number of transient states plus a non-goal BSCC or a goal state, and has become a direct successor of the initial state.
We can then directly read the value of $\mathbb{P}(\diamond\: \texttt{G})$ from the transitions to the goal states (line~\ref{alg:ExploreEliminate:PReturn}).
Similarly, the value of $\mathbb{E}(\blackdiamond\: \texttt{G})$ can be derived directly from the remaining rewards, if it is not $\infty$ by definition (lines \ref{alg:ExploreEliminate:EReturnInfty}-\ref{alg:ExploreEliminate:EReturnFinite}).
\begin{figure}
\caption{Example for exploration with interleaved elimination on $M_z$}
\label{fig:ExampleAlg}
\end{figure}
\begin{example}
For our example DTMC $M_z$ of \Cref{fig:ExampleDTMC}, we have already shown the predecessor count MTBDD computed by \texttt{Explore} in \Cref{fig:ExampleMTBDD}.
Let us now step through the rest of \texttt{ExploreEliminate} on this model.
The partial state spaces that we consider in each step are shown in \Cref{fig:ExampleAlg}.
Fully explored states are drawn with solid outlines, all other states (i.e.\ those in $S$ but not in $\mathit{done}$) with dashed outlines.
In step~(1), we have just fully explored state \texttt{i}, i.e.\ we just executed line~\ref{alg:ExploreEliminate:FullyExplored} in the first iteration of the outer loop.
Since $\hat{\mathit{pre}}$ tells us that \texttt{i} still has unexplored predecessors, we cannot eliminate, and next explore \texttt{ok} in step~(2).
We then eliminate \texttt{ok}---its only predecessor \texttt{i} is fully explored---but since \texttt{ok} has just a single self-loop, the elimination has no effect.
In step~(3), we have just explored state $4$, which can now be eliminated.
The result is shown as step~(4).
We proceed in the same pattern in steps (5) through~(8).
Then, in step~(9), we fully explore state~$1$.
Now all predecessors of \texttt{i} are fully explored, and we can eliminate both $1$ and \texttt{i}.
For the sake of illustration, let us pick the more complicated ordering and eliminate \texttt{i} first.
The result is shown as step~(10).
Since \texttt{i} is the initial state, we keep it, but redirect all incoming transitions.
We also merge its rewards, which is why $1$ now has a non-zero reward.
Note that we show rationals in \Cref{fig:ExampleAlg}, but our implementation uses floating-point numbers.
Remember that, without $\hat{\mathit{pre}}$, we might have eliminated \texttt{i} too early; after any subsequent exploration of a state in $\set{1, \dots, n}$, we would then have to re-eliminate \texttt{i}.
We finally eliminate $1$ in step~(11) and explore state $\bot$ in step~(12).
At this point, the outer loop terminates; we read $\mathbb{P}(\diamond\: \set{\texttt{ok}}) = \frac{4375}{4376} \approx 0.999771$ and $\mathbb{E}(\blackdiamond\: \set{\texttt{ok}, \bot}) = 1 + \frac{119918}{119793} \approx 2.001043$.
Observe that, at any time, we kept at most 4 explicit states in memory.
We can arbitrarily increase the size of this model by increasing $n$, but will only ever need at most 4 states in memory.
\end{example}
\paragraph{Long-run average rewards.}
\begin{wrapfigure}[8]{r}{0.5\textwidth}
\begin{center}
\begin{tikzpicture}
\tikzstyle{nodestyle} = [draw, shape = circle, inner sep = 0pt, minimum size = 0.7cm];
\tikzstyle{arrow} = [-stealth, thick];
\tikzstyle{abovelabel} = [pos = 0.5, above];
\tikzstyle{belowlabel} = [pos = 0.35, below];
\def \spacing {0.78cm}
\def \bendangle {50}
\def \labelshift {(0.6, 0.1)}
\def \labelshift {(0.0, 0.0)}
\node (1) [nodestyle] {};
\node (foo) [left = 0.5cm of 1] {};
\draw [arrow] (foo)--(1);
\node (2) [nodestyle, shift={(0, -1.5)}] {};
\node (bar) [left = 0.5cm of 2] {};
\draw [arrow] (bar)--(2);
\node (3) [nodestyle, shift={(1.6, -1)}] {};
\node (4) [nodestyle, shift={(1.6, -2)}] {};
\node (5) [shift={(1.6, -1.5)}] {$\cdots$};
\path [-stealth, thick]
(2) edge [] node[left,near end] {$p_1$\ \ } (3);
\path [-stealth, thick]
(2) edge [] node[left,near end] {$p_n$\ } (4);
\path[-stealth, thick]
(1) edge [loop right] node {$1, r_u = u_{\overline{s}}, r_l = l_{\overline{s}}$} (1);
\path[-stealth, thick]
(3) edge [loop right] node {$1, r_u = u_1, r_l = l_1$} (3);
\path[-stealth, thick]
(4) edge [loop right] node {$1, r_u = u_n, r_l = l_n$} (4);
\draw [arrow] (foo)--(1);
\end{tikzpicture}
\caption{Computation of long-run averages.\label{fig:lra}}
\end{center}
\end{wrapfigure}
The algorithm we presented so far computed reachability probabilities and expected rewards.
For long-run average reward properties, there are no goal states.
In such a case, our state elimination procedure computes the \emph{recurrence reward} for each BSCC~\cite{GHS18b}.
To obtain the long-run average reward, we need to divide the recurrence reward for the rewards as given in the DTMC by the recurrence reward that we would obtain if all states had reward~$1$.
We can do so by straightforwardly extending \Cref{alg:ExploreEliminate,alg:Eliminate} to work on two reward structures $r_u$ and $r_l$ in parallel.
Upon termination of the outer loop in \texttt{ExploreEliminate}, we then have one of the two situations described at the end of Sect.~4 in~\cite{GHS18b}, and can again directly read off the value for our ($\mathbb{L}$-)property.
Consider \Cref{fig:lra}:
In the simpler case, the remaining model consists of the initial state ${\overline{s}}$ with a self-loop with probability one and $r_u = u_{\overline{s}}$, $r_l = l_{\overline{s}}$.
In this case, the average value is $\frac{u_{\overline{s}}}{l_{\overline{s}}}$.
In the other case, the remaining model consists of the initial state ${\overline{s}}$ which has a probability of $p_i$ to move to one of the other $n$ remaining states $s_i$ , $i = 1, \ldots, n$, which all have a self-loop with probability one and $r_u(s_i) = u_i$, $r_l(s_i) = l_i$.
In this case, the average value is $ \sum_{i=1,\ldots,n} p_i \frac{u_i}{l_i}$.
When computing long-run average rewards, the final value for $\mathbb{L}$ may be small, but the two recurrence rewards that we need to divide are often extremely large numbers beyond what can usefully be represented as 64-bit (i.e.\ double-precision) floating point numbers.
We thus implemented a variant of our algorithm that uses the GNU MPFR library (see \href{https://www.mpfr.org/}{mpfr.org}) for arbitrary-precision floating-point arithmetic, allowing us to use more than 64 bits.
We did not find this to significantly affect the performance of the overall approach.
\paragraph{Alternatives and optimisations.}
So far, we have assumed that we compute successors for each state explicitly and individually.
For the state elimination phase, doing so is indeed necessary.
However, for just exploring the states we could also compute the \emph{transition relation} as a BDD, and then use the transition relation to symbolically explore the set of reachable states.
This is the standard approach in model checkers such as PRISM~\cite{KNP11} and potentially faster than the semi-symbolic approach we have discussed.
Also, using the transition relation and according MTBDD operations (in particular sum-abstraction), the number of predecessors of each state can also be computed symbolically.
We have so far assumed that the reachable states and the number of predecessors are stored as (MT)BDDs.
An alternative to this approach is to store these numbers on secondary storage (e.g. hard disk) in a similar way as e.g. in~\cite{HartmannsH15}.
This approach would be useful for models the state space of which is not suitable to be stored as a BDD.
This might be the case because of lack of implicit symmetries or because the size of the representation of each state is not constant.
\section{Experimental Evaluation}
\label{sec:Experiments}
We have implemented a preliminary version of our method which we integrated as a plugin for the probabilistic model checker ePMC~\cite{HahnLSTZ14}.
For the analysis, we transform the model and property into C++ code so as to have a means to quickly compute successors of states, similar to the approach used in SPIN~\cite{Holzmann97}.
This C++ file is then appended with code so as to achieve the following:
In the first phase, we then explore the state space in a breadth-first manner where we explore each state explicitly but store sets of states as BDDs, using the BDD package CUDD~\cite{BaharFGHMPS97}.
In the second phase, we generate an MTBDD mapping all states to value $0$.
Then, we iterate over all reachable states, recompute their successors, and increment the value of these successors in the MTBDD by $1$ each time.
In the third phase, we execute the state elimination algorithm as discussed.
This C++ code is then compiled and run in a process separate from ePMC, such that we can measure the memory usage exactly
(the memory usage of ePMC itself is not of much interest, because it is rather small and about the same for any analysis).
In the following, we apply our tool on several case studies from the website of the probabilistic model checker PRISM.
All experiments were performed on a MacBook Pro with a 2.7 GHz Quad-Core Intel Core i7 processor and 16 GB 2133 MHz LPDDR3 RAM.
In the following tables, ``model states'' is the total number of states the model has for the given parameters, ``result'', is the value of the property computed, ``time'' is the total time of the analysis in seconds, ``exp. states'' (``exp. transitions'') are maximal number of states (transitions) being stored explicitly at the same time.
By ``peak mem'' we denote the maximal memory usage in MB of the analysis process.
\subsection{Simple Molecular Reactions: $\mathrm{Na} + \mathrm{Cl} \leftrightarrow \mathrm{Na}^+ + \mathrm{Cl}^-$}
This case study~\footnote{https://www.prismmodelchecker.org/casestudies/molecules.php} is a CTMC modelling the chemical reaction $\mathrm{Na} + \mathrm{Cl} \leftrightarrow \mathrm{Na}^+ + \mathrm{Cl}^-$.
The parameters of this case study are $\mathit{N1}$, the initial number of $\mathit{Na}$ molecules and $\mathit{N2}$, the initial number of $\mathit{Cl}$ molecules.
In Table~\ref{tab:perf-chem}, we consider the performance figures for the analysis of \verb+R=?[S]+ which describes the expected long-run average number of $\mathit{Na}$ molecules.
Here, we consider a starting configuration in which initially the number of $\mathit{Na}$ and $\mathit{Cl}$ is the same, that is, $\mathit{N1} = \mathit{N2}$.
\begin{table}
\begin{tabular}{rrrrrrrr}
\toprule
$\mathit{N1} {=} \mathit{N2}$ & model states & result & time & exp. states & exp. trans & peak mem\\
\cmidrule{1-1} \cmidrule{2-7}
10 & 11 & 2.2623e+01 & 8 & 5 & 5 & 22\\
100 & 101 & 2.3894e+01 & 8 & 5 & 5 & 22\\
1,000 & 1,000 & 2.4012e+01 & 7 & 5 & 5 & 22\\
10,000 & 10,001 & 2.4024e+01 & 7 & 5 & 5 & 25\\
100,000 & 100,001 & 2.4025e+01 & 10 & 5 & 5 & 28\\
1,000,000 & 1,000,001 & 2.4025e+01 & 32 & 5 & 5 & 27\\
10,000,000 & 10,000,001 & 2.4025e+01 & 258 & 5 & 5 & 31\\
100,000,000 & 100,000,001 & 2.4025e+01 & 2,609 & 5 & 5 & 29\\
1,000,000,000 & 1,000,000,001 & 2.4025e+01 & 18,807 & 5 & 5 & 25\\
\bottomrule
\end{tabular}
\caption{$\mathrm{Na} + \mathrm{Cl} \leftrightarrow \mathrm{Na}^+ + \mathrm{Cl}^-$ performance figures.
\label{tab:perf-chem}}
\end{table}
As we see, the model scales well for large numbers of molecules and accordingly large state spaces.
The memory usage grows only slowly with increasing model parameters, and the number of states and transitions required to be stored explicitly is constant.
\subsection{Bounded Retransmission Protocol}
The Bounded Retransmission Protocol~\cite{HSV94}\footnote{https://www.prismmodelchecker.org/casestudies/brp.php} is a file transmission protocol.
Files are divided into $N$ packages, each of which is transferred individually.
Data and confirmation packages are sent over unreliable channels, such that they might get lost.
Packages can only be resent a number of $\mathit{MAX}$ times.
In Table~\ref{tab:perf-brp}, we provide performance figures for the analysis of the property \verb+P=?[ F s=5 ]+, that is the probability that the sender does not eventually report a successful transmission.
``$N$'' and ``$\mathit{MAX}$'' are as discussed above, the other numbers are as in the previous case study.
\begin{table}
\begin{tabular}{rrrrrrrr}
\toprule
$N$ & $\mathit{MAX}$ & model states & result & time & exp. states & exp. trans & peak mem\\
\cmidrule{1-2} \cmidrule{3-8}
64 & 5 & 4,936 & 4.48e-08 & 9 & 12 & 29 & 25\\
64 & 10 & 9,101 & 1.05e-15 & 9 & 20 & 53 & 25\\
64 & 100 & 84,071 & 5.03e-153 & 9 & 140 & 517 & 25\\
64 & 1000 & 833,771 & 3.14e-1526 & 25 & 258 & 986 & 33\\
128 & 10 & 18,189 & 2.11e-15 & 9 & 20 & 53 & 25\\
128 & 100 & 168,039 & 1.01e-152 & 10 & 140 & 517 & 32\\
128 & 1000 & 1,666,539 & 6.28e-1526 & 38 & 514 & 1,978 & 36\\
256 & 10 & 36,365 & 4.21e-15 & 9 & 20 & 53 & 27\\
256 & 100 & 335,975 & 2.01e-152 & 15 & 150 & 517 & 28\\
256 & 1000 & 3,332,075 & 1.26e-1525 & 72 & 1,026 & 3,962 & 39\\
512 & 10 & 72,717 & 8.42e-15 & 9 & 20 & 53 & 29\\
512 & 100 & 671,847 & 4.03e-152 & 18 & 140 & 517 & 32\\
512 & 1000 & 6,663,147 & 2.51e-1525 & 140 & 1,340 & 5,167 & 42\\
1024 & 10 & 145,421 & 1.69e-14 & 10 & 20 & 53 & 29\\
1024 & 100 & 1,262,631 & 8.06e-152 & 29 & 140 & 517 & 30\\
1024 & 1000 & 13,325,291 & 5.02e-1525 & 280 & 1,340 & 5,167 & 43\\
2048 & 10 & 290,829 & 3.37e-14 & 12 & 20 & 53 & 27\\
2048 & 100 & 2,687,079 & 1.61e-151 & 50 & 140 & 517 & 30\\
2048 & 1000 & 26,649,579 & 1.00e-1524 & 552 & 1,340 & 5,167 & 41\\
4096 & 10 & 581,645 & 6.74e-14 & 17 & 20 & 53 & 27\\
4096 & 100 & 5,374,055 & 3.22e-151 & 95 & 140 & 517 & 28\\
4096 & 1000 & 53,298,155 & 2.01e-1524 & 1,151& 1,340 & 5,004 & 39\\
8192 & 10 & 1,163,277 & 1.35e-13 & 26 & 20 & 53 & 28\\
8192 & 100 & 10,748,007 & 6.45e-151 & 187 & 140 & 517 & 27\\
8192 & 1000 & 106,595,307 & 4.02e-1524 &2,385 & 1,340 & 5,004 & 40\\
16384 & 10 & 2,326,541 & 2.67e-13 & 48 & 20 & 53 & 27\\
16384 & 100 & 21,495,911 & 1.29e-150 & 392 & 140 & 517 & 28\\
16384 & 1000 & 213,189,611 & 8.03e-1524 &4,534 & 1,340 & 5,004 & 40\\
32768 & 10 & 4,653,069 & 5.39e-13 & 91 & 20 & 53 & 26\\
32768 & 100 & 42,991,719 & 2.58e-150 & 781 & 140 & 517 & 27\\
32768 & 1000 & 426,378,219 & 1.61e-1523 &9,039 & 1,340 & 5,004 & 41\\
65536 & 10 & 9,306,125 & 1.08e-12 & 20 & 54 & 156 & 25\\
65536 & 100 & 85,983,335 & 5.16e-150 & 140 & 503 & 1,362 & 29\\
131072 & 10 & 18,612,237 & 2.1572e-12 & 312 & 20 & 54 &26\\
131072 & 100 & 171,966,567 & 1.03e-149 &2,847 & 140 & 503 & 29\\
\bottomrule
\end{tabular}
\caption{Bounded Retransmission Protocol performance figures.\label{tab:perf-brp}}
\end{table}
Compared to the instances of the PRISM website, we have analysed instances with higher parameter numbers $N$ and $\mathit{MAX}$ because our focus was in the scalability of our method.
For comparison, for the first table entry we used the same parameters as the last table entry on the PRISM website.
As we see, we are able to handle instances with several million states with a low memory usage.
Even for higher parameter values for which the number of total states the model consists of is in the millions, we never use more than a few thousand explicit states and transitions and less than 100MB.
\subsection{Wireless Communication Cell}
This case study is a performance model of wireless communication cells~\cite{HMPT00}\footnote{https://www.prismmodelchecker.org/casestudies/cell.php}.
The parameter $N$ describes the number of channels in a cell.
We analyse the property \verb+R{"calls"}=? [ S ]+ which describes the average number of calls in the cell on the long run.
We provide performance figures in Table~\ref{tab:perf-cell}.
\begin{table}
\begin{tabular}{rrrrrrrr}
\toprule
$N$ & model states & result & time & exp. states & exp. trans & peak mem\\
\cmidrule{1-1} \cmidrule{2-7}
10,000 & 10,001 & 7.00e+01 & 7 & 5 & 5 & 24\\
100,000 & 100,001 & 7.00e+01 & 9 & 5 & 5 & 24\\
1,000,000 & 1,000,001 & 7.00e+01 & 24 & 5 & 5 & 25\\
10,000,000 & 10,000,001 & 7.00e+01 & 183 & 5 & 5 & 26\\
100,000,000 & 100,000,001 & 7.00e+01 & 1,731 & 5 & 5 & 25\\%!!
\bottomrule
\end{tabular}
\caption{Wireless Communication Cell performance figures.\label{tab:perf-cell}}
\end{table}
Also for this case study, the approach works fine in that the number of states and transitions to be stored is small as is the peak memory usage.
\subsection{Crowds Protocol}
The Crowds protocol~\cite{RR98}\footnote{https://www.prismmodelchecker.org/casestudies/crowds.php} is a means to allow anonymous web browsing.
To do so, messages are not directly sent, but forwarded to other users, who might either forward them again or send them to the destination.
By doing so, it is hard for attackers to decide whether the sender of a message is the original sender or is just forwarding the message.
The model version we consider has two parameters: $\mathit{TotalRuns}$ is the number of routing paths of the model instance, and $\mathit{CrowdSize}$ is the number of honest participants of the protocol.
We consider the property \verb+Pmax=?[true U (new & runCount=0 & observe0 > observe1 & ... &+\\\verb+observe0 > observe19)]+ which means that an attacker is eventually able to observe the true sender of a message more often than participants just forwarding the message and is thus able to guess the original sender.
In Table~\ref{tab:perf-crowds}, we provide performance figures.
\begin{table}
\begin{tabular}{rrrrrrrr}
\toprule
$\mathit{CrowdSize}$ & $\mathit{TotalRuns}$ & model states & result & time & exp. states & exp. trans & peak mem\\
\cmidrule{1-2} \cmidrule{3-8}
5 & 5 & 8,653 & 2.7884e-01 & 9 & 289 & 1,278 & 77\\
5 & 6 & 18,817 & 2.9791e-01 & 9 & 510 & 2,236 & 77\\
5 & 7 & 37,291 & 3.1812e-01 & 9 & 823 & 3,898 & 164\\
10 & 5 & 111,294 & 2.1662e-01 & 19 & 6,604 & 33,513 & 2487\\
10 & 6 & 352,535 & 2.3162e-01 & 106 & 17,824 & 89,120 & 10698\\
15 & 5 & 592,060 & 1.9674e-01 & 676 & 44,104 & 356,143 & 9240\\
\cmidrule{1-2} \cmidrule{3-8}
\end{tabular}
\caption{Crowds Protocol performance figures.\label{tab:perf-crowds}}
\end{table}
As we see, for this model the current implementation does not perform very well.
The reason is that too many states and transitions have to be stored explicitly at the same time, leading to a large memory overhead.
\subsection{Embedded Control System}
This case study~\cite{MCT94}\footnote{https://www.prismmodelchecker.org/casestudies/embedded.php} is an embedded control system which features a cyclic polling process.
If a certain component detects that more than a given number $\mathit{MAX\_COUNT}$ of cycles have been skipped due to issues,then the system is shut down for safety reasons.
We consider the property \verb+R{"danger"}=? [ F "down" ]+, which expresses the expected time the system is in an endangered state before it eventually has to be shut down.
We provide performance figures in Table~\ref{tab:perf-ecs}.
Again, the analysis method works fine, as the number of states and transitions stored explicitly is limited.
\begin{table}
\begin{tabular}{rrrrrrrr}
\toprule
$\mathit{MAX\_COUNT}$ & model states & result & time & exp. states & exp. trans & peak mem\\
\cmidrule{1-1} \cmidrule{2-7}
512 & 320,316 & 3.3454e-01 & 56 & 267 & 11,938 & 80\\
1024 & 639,804 & 3.3731e-01 & 107 & 267 & 11,938 & 80\\
2048 & 1,278,780 & 3.4283e-01 & 201 & 267 & 11,938 & 80\\
4096 & 2,556,732 & 3.5371e-01 & 400 & 267 & 11,938 & 80\\
8192 & 5,112,636 & 3.7485e-01 & 794 & 267 & 11,938 & 80\\
16384 & 10,224,444 & 4.1469e-01 & 1,579 & 267 & 11,938 & 80\\
32768 & 20,448,060 & 7.6563e-01 & 2,914 & 284 & 10,513 & 72\\
\bottomrule
\end{tabular}
\caption{Embedded Control System performance figures.\label{tab:perf-ecs}}
\end{table}
\section{Conclusion and Future Work}
\label{sec:Conclusion}
In this paper, we have discussed a new memory-efficient analysis method for properties of stochastic models.
Our method is widely applicable to a large variety of stochastic models and properties, though, for conciseness of presentation we have concentrated on DTMCs (with some of the case studies being CTMCs).
Experimental evidence has shown that our approach has the potential to analyse models with millions of states with just a few megabytes.
One advantage of our method is that we directly obtain a precise result:
Our current implementation is based on (variable-precision) floating-point numbers, and computation precision is limited by the properties of this representation.
We could however as well use exact (rational) arithmetic and would obtain exact values without any change in the core algorithms, since all intermediate and final values are rational, though at the cost of increased computation time and memory usage.
Alternatively, we could use interval arithmetic so as to obtain precise upper and lower bounds of the values computed, again, without any change in the algorithm, and with only moderate overhead.
This is in contrast to methods based on value iteration or state-space abstraction, for which special precaution is required to ensure this.
Our current implementation explores and eliminates states in a strict breadth-first search order.
Motivated by the problematic performance of the Crowds protocol case study, we also want to consider different search orders so as to improve the behaviour for models for which the strict breadth-first search order does not perform well.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{\large{\bf\uppercase{A Note on Efficient Performance Evaluation of the Cumulative Sum Chart and the Sequential Probability Ratio Test}}}
\author{Aleksey\ S.\ Polunchenko\corref{cor-author}}
\ead{[email protected]}
\ead[url]{http://www.math.binghamton.edu/aleksey}
\cortext[cor-author]{Address correspondence to A.S.\ Polunchenko, Department of Mathematical Sciences, State University of New York (SUNY) at Binghamton, 4400 Vestal Parkway East, P.O. Box 6000, Binghamton, NY 13902--6000, USA; Tel: +1 (607) 777-6906; Fax: +1 (607) 777-2450; Email:~\href{mailto:[email protected]}{[email protected]}}
\address{Department of Mathematical Sciences, State University of New York (SUNY) at Binghamton\\Binghamtom, NY 13902--6000, USA}
\begin{abstract}
We establish a simple connection between certain {\em in-control} characteristics of the CUSUM Run Length and their {\em out-of-control} counterparts. The connection is in the form of paired integral (renewal) equations. The derivation exploits Wald's likelihood ratio identity and the well-known fact that the CUSUM chart is equivalent to repetitive application of Wald's SPRT. The characteristics considered include the entire Run Length distribution and all of the corresponding moments, starting from the zero-state ARL. A particular {\em practical} benefit of our result is that it enables the in- and out-of-control characteristics of the CUSUM Run Length to be computed {\em concurrently}. Moreover, due to the equivalence of the CUSUM chart to a sequence of SPRTs, the ASN and OC functions of an SPRT under the null and under the alternative can {\em all} be computed {\em simultaneously} as well. This would double up the efficiency of any numerical method one may choose to devise to carry out the actual computations.
\end{abstract}
\begin{keyword}
Control charts\sep Cumulative Sum chart\sep Integral equations\sep Sequential analysis\sep Sequential Probability Ratio Test\sep Quality control.
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{sec:intro}
It cannot be disputed that Wald's~\cite{Wald:Book47} likelihood ratio identity is one of the fundamental methodological tools in all of {\em theoretical} sequential analysis. The powerful change-of-probability-measure technique essentially enabled the proof of nearly every classical result in the areas of sequential hypotheses testing and sequential (quickest) change-point detection: strong optimality of Wald's~\cite{Wald:Book47} Sequential Probability Ratio Test (SPRT) first proved by Wald and Wolfowitz~\cite{Wald+Wolfowitz:AMS1948} (see also~\cite{Wolfowitz:AMS1966}) and then also re-established, e.g., by Matthes~\cite{Matthes:AMS1963} and, notably, by Le Cam, whose proof may be found in~\cite{Lehmann:Book1959}; exact minimaxity (in the sense of Lorden~\cite{Lorden:AMS71}) of Page's~\cite{Page:B54} Cumulative Sum (CUSUM) ``inspection scheme'' established by Moustakides~\cite{Moustakides:AS86} (although an alternative, viz. game-theoretic, proof was also later offered by Ritov~\cite{Ritov:AS90}); exact Bayesian optimality of Shiryaev's~\cite{Shiryaev:SMD61,Shiryaev:TPA63} detection procedure shown in~\cite{Shiryaev:SMD61,Shiryaev:TPA63,Shiryaev:Book78}; exact maximum-probability-type optimality of the Shewhart's~\cite{Shewhart:JASA1925,Shewhart:Book1931} $\bar{X}$-chart proved in~\cite{Pollak+Krieger:SA2013} and in~\cite{Moustakides:SA2014}; and exact multi-cyclic optimality of the Shiryaev--Roberts procedure---to name a few; the Shiryaev--Roberts detection procedure emerged from the independent work of Shiryaev~\cite{Shiryaev:SMD61,Shiryaev:TPA63} and Roberts~\cite{Roberts:T59}---hence, the name,---and its multi-cyclic optimality was established in~\cite{Pollak+Tartakovsky:ISITA2008,Pollak+Tartakovsky:SS09} and in~\cite{Shiryaev+Zryumov:Khabanov2010}. For a recent survey of the state-of-the-art in {\em theoretical} sequential analysis, see, e.g.,~\cite{Polunchenko+Tartakovsky:MCAP2012} or~\cite{Tartakovsky+etal:Book2014}, and the references therein.
More recently, however, in~\cite{Polunchenko+etal:SA2014,Polunchenko+etal:ASMBI2014} the technique was put to a different, more {\em applicative} use: to improve the accuracy and efficiency of the numerical method the authors of these papers developed to compute the performance of the so-called Generalized Shiryaev--Roberts detection procedure; the Generalized Shiryaev--Roberts procedure was proposed in~\cite{Moustakides+etal:SS11} as a headstarted version of the classical Shiryaev--Roberts procedure, and the motivation to headstart the latter was drawn from the seminal work of Lucas and Crosier~\cite{Lucas+Crosier:T1982} where it was proposed to headstart the CUSUM chart. The aim of this work is to extend the ideas laid out in~\cite{Polunchenko+etal:SA2014,Polunchenko+etal:ASMBI2014} beyond the Generalized Shiryaev--Roberts procedure, viz. to the CUSUM chart and the SPRT; the possibility of such an extension was previously entertained in~\cite[Section~5]{Polunchenko+etal:SA2014}. Specifically, in this work we employ Wald's~\cite{Wald:Book47} likelihood ratio identity and establish a connection between a host of {\em in-control} characteristics of the CUSUM Run Length and their {\em out-of-control} counterparts. The connection is in the form of coupled integral (renewal) equations, and the derivation utilizes the well-known observation first made by Page~\cite{Page:B54} that the CUSUM chart is equivalent to repetitive application of the SPRT (with properly selected initial score and control bounds). The Run Length characteristics considered include the entire distribution and all of the corresponding moments, starting from the standard zero-state Average Run Length (ARL). On the {\em practical} side, the obtained connection enables {\em concurrent} evaluation of the in- and out-of-control characteristics of the CUSUM Run Length. This would double up the efficiency of any numerical method one may devise to compute the performance of the CUSUM chart (through solving the corresponding integral equations). The efficiency improvement would be of an even greater magnitude for the {\em two-sided} CUSUM chart, also proposed by Page~\cite[Section~3]{Page:B54}. Moreover, thanks to the observation first made by Page~\cite{Page:B54} that the CUSUM chart is equivalent to a sequence of SPRTs, the Average Sample Number (ASN) and the Operating Characteristic (OC) functions of an SPRT under the null and under the alternative can {\em all} be computed {\em simultaneously} as well, again with the aid of the main result obtained in the sequel. Hence, in a sense, this work is an attempt to bridge the gap between the theory and applications of sequential analysis.
It is worth recalling that the need to evaluate the performance of the CUSUM chart (or that of the SPRT, or any other control chart for that matter) {\em numerically} is dictated by the fact that the corresponding characteristics (e.g., the zero-state ARL, the ASN function, or the OC function) are governed by integral (renewal) equations that seldom allow for an analytical solution; cases where an analytic closed-form solution {\em is} possible are offered, e.g., in~\cite{Regula:PhDThesis1975,Gan:SS1992,Vardeman+Ray:T1985,Knoth:PhDThesis1995,DeLucia+Poor:IEEE-IT1997,Knoth:SA1998,Mazalov+Zhuravlev:PCS2002} for the CUSUM chart, in~\cite{Dvoretzky+etal:AMS1953,Albert:AMS1956,Kiefer+Wolfowitz:NRLQ1956,Schorr:U1967,Kohlruss:SA1994} for the SPRT, in~\cite{Novikov:TPA1990,Gan:JQT1998,Polunchenko+etal:ShLnkJAS2013} for the Exponentially Weighted Moving Average (EWMA) chart (introduced by Roberts~\cite{Roberts:T59}), and in~\cite{Pollak:AS85,Kenett+Pollak:IEEE-TR1986,Mevorach+Pollak:AJMMS91,Polunchenko+Tartakovsky:AS10,Tartakovsky+Polunchenko:IWAP10,Polunchenko+Tartakovsky:MCAP2012,Du+etal:SMTA2015} and~\cite[Chapter~4]{Du:PhD-Thesis2015} for the Generalized Shiryaev--Roberts procedure. Since control charts' performance evaluation is a persistent problem in applied sequential analysis (notably in quality control), numerical treatment of the corresponding integral equations has {\it de~facto} become a separate research field, and the literature on the subject is vast indeed. For a recent survey of the state-of-the-art in the field, see, e.g.,~\cite{Li+etal:JSCS2014}. By and large, two types of approaches can be distinguished: randomized (i.e., simulation) and deterministic. For specific examples, see, e.g.,~\cite{Page+Cox:MPCPhS1954,Brook+Evans:B1972,Champ+Rigdon:CommStat1991},~\cite{Moustakides+etal:SS11},~\cite{Polunchenko+etal:SA2014,Polunchenko+etal:ASMBI2014} and~\cite[Chapter~3]{Du:PhD-Thesis2015}. To get a clear picture as to the capabilities of a control chart in a concrete observations model, the chart's performance needs to be evaluated both in the in-control regime as well as in the out-of-control regime. The problem, however, is that the in- and out-of-control regime equations are usually treated {\em separately}, which is obviously inefficient. The reason, in part, is that (apparently) there is no a simple and explicit relationship between the in- and out-of-control regime equations. This work proves otherwise, if the chart of interest is either the CUSUM chart or the SPRT. Specifically, using the result obtained in the sequel, the {\em in-} and {\em out-of-control} characteristics of the CUSUM chart and those of the SPRT under the null and under the alternative can {\em all} be computed {\em simultaneously} and {\em irrespective} of which particular numerical method---whether randomized or deterministic---is used to carry out the actual calculations.
The rest of the paper is organized as follows. Section~\ref{sec:preliminaries} provides the necessary preliminary background. The centerpiece of the paper is Section~\ref{sec:main-results} which is where we first establish our main result and then also explain how exactly it can be used to compute the zero-state in- and out-of-control ARLs of the CUSUM chart and the SPRT's ASNs and OCs under the null and under the alternative---all in one run of whatever numerical method one may devise to perform the computations. Section~\ref{sec:conclusion} draws a line under the entire paper.
\section{Preliminaries}
\label{sec:preliminaries}
To fix ideas, suppose we wish to ``sense'' whether or not the common probability distribution function (pdf) of a ``live''-sampled series of independent observations $X_1,X_2,\ldots$ has changed from $f_0(x)$ initially to $f_1(x)\not\equiv f_0(x)$; in the quality control literature, the densities $f_0(x)$ and $f_1(x)$ are customarily referred to as the on- and off-target distributions, respectively. Since its inception, Page's~\cite{Page:B54} CUSUM ``inspection scheme'' has been just {\em the} tool for the job. The CUSUM scheme flags a alarm at sample number $\mathcal{C}_{h}\triangleq\min\{n\geqslant1\colon W_n\geqslant h\}$, where $h>0$ is a control limit (which is selected so as to achieve a desired level of the ``false positive'' risk), $\{W_n\}_{n\geqslant0}$ is the CUSUM statistic defined as $W_n\triangleq\max\{0,W_{n-1}+\log\LR_n\}$, $n\geqslant1$, with $W_0=0$, and $\LR_n\triangleq f_1(X_n)/f_0(X_n)$ is the instantaneous likelihood ratio (LR) for the $n$-th data point $X_n$. The (random) stopping time $\mathcal{C}_{h}$ is often referred to as the Run Length, for it literally is the length of a single run of the CUSUM statistic $\{W_n\}_{n\geqslant0}$ before it exits the strip $[0,h)$ through the control limit $h>0$. For simplicity, we shall assume here and throughout the paper that $\LR_1$ is absolutely continuous, although at an additional effort the case of purely nonarithmetic $\LR_1$ can be handled as well.
A more general version of the CUSUM chart, viz. one proposed by Lucas and Crosier~\cite{Lucas+Crosier:T1982}, assumes that the CUSUM statistic $\{W_n\}_{n\geqslant0}$ is started not off zero, but off a deterministic point $W_0=w\in[0,h)$. This point is a parameter referred to as either the headstart or the ``initial score''. Formally, the respective {\em Generalized} CUSUM Run Length is defined as
\begin{align}\label{eq:GenCS-T-def}
\mathcal{C}_{h}^w
&\triangleq
\min\{n\geqslant1\colon W_n^w\geqslant h\},\; h>0,
\end{align}
where the generalized CUSUM statistic $\{W_n^w\}_{n\geqslant0}$ admits the recurrence
\begin{align}\label{eq:GenCS-W-def}
W_n^w
&\triangleq
\max\{0,W_{n-1}^w+\log\LR_n\},\; n\geqslant1,\, W_0^w=w\in[0,h),
\end{align}
and it is apparent that setting $w=0$ reduces the Generalized CUSUM chart to the classical one introduced by Page~\cite{Page:B54}. From now on we shall concentrate exclusively on the Generalized CUSUM chart~\eqref{eq:GenCS-T-def}-\eqref{eq:GenCS-W-def}, although, for brevity and without loss of generality, we shall refer to it as simply the CUSUM chart. We note that, as a parameter of the chart, the headstart $w\in[0,h)$ directly affects the Run Length's characteristics, just as does the control limit $h>0$. The effect of the headstart on the performance of the CUSUM chart~\eqref{eq:GenCS-T-def}-\eqref{eq:GenCS-W-def} was thoroughly studied in~\cite{Lucas+Crosier:T1982}.
The two most popular metrics used to quantitatively assess the performance of the CUSUM chart are the zero-state in- and out-of-control ARLs, conventionally denoted as $\ARL_0(w;h)$ and $\ARL_1(w;h)$, respectively; the $0$ ($1$) in the subscript is to indicate that the pdf of the observations is assumed to be $f_0$ ($f_1$), so that the hypothesis in effect is the null hypothesis $H_0$ (the alternative hypothesis $H_1$, respectively). The two ARL metrics were introduced by Page~\cite{Page:B54} who formally defined them as $\ARL_i(w;h)\triangleq\EV_i[\mathcal{C}_{h}^w]$, $i=\{0,1\}$. It goes without saying that {\em both} ARLs are of interest, for either $\ARL_0(w;h)$ or $\ARL_1(w;h)$ alone does not tell the whole story as to the CUSUM Run Length's characteristics. However, given an initial score $w\in[0,h)$, a control limit $h>0$, and a particular observations model characterized by the densities $f_0(x)$ and $f_1(x)$, the evaluation of the ARLs is a major problem in applied sequential analysis, especially in quality control. To that end, a common practice in quality control has been to rely on the work of Page~\cite{Page:B54} who demonstrated that $\ARL_0(w;h)$ and $\ARL_1(w;h)$ satisfy certain integral (renewal) equations, which we state next.
For notational brevity, let $L_i(w;h)\triangleq\ARL_i(w;h)$, $i=\{0,1\}$. Then, according to Page~\cite{Page:B54}, we have
\begin{align}\label{eq:L-CS-eqn}
L_i(x;h)
&=
1+L_i(0;h)\,F_i(-x)+\int_{0}^{h} K_i(y-x)\,L_i(y;h)\,dy,\;i=\{0,1\},\; x\in[0,h),
\end{align}
where
\begin{align}\label{eq:Ki-def}
K_{i}(z)
&\triangleq
\frac{\partial}{\partial z}\mathbb{P}_{i}(\log\LR_{1}\leqslant z),\; i=\{0,1\},\; z\in\mathbb{R},
\end{align}
i.e., $K_i(z)$ is the pdf of the log-likelihood ratio (log-LR) under the hypothesis $H_i$, $i=\{0,1\}$, and
\begin{align}\label{eq:Fi-def}
F_i(z)
&\triangleq
\int_{-\infty}^{z}K_{i}(x)\,dx,\; i=\{0,1\},\; z\in\mathbb{R},
\end{align}
i.e., $F_i(z)$ is the cumulative distribution function (cdf) of log-LR under the hypothesis $H_i$, $i=\{0,1\}$; in the quality control literature $K_i(z)$ is sometimes referred to as the ``frequency function'' (under the hypothesis $H_i$, $i=\{0,1\}$). Equations~\eqref{eq:L-CS-eqn} are renewal equations, and for either $i=\{0,1\}$ can be derived merely by conditioning on the first observation $X_1$.
Depending on the particular observations model given by the pair of pdf-s $f_i(x)$, $i=\{0,1\}$, equations~\eqref{eq:L-CS-eqn} may be recognized as either Fredholm (linear) integral equations of the second kind, or as Volterra integral equations, or as {\em delayed} Volterra integral equations. Regardless, a closed-form analytic solution is rarely an option. Hence the equations are usually solved {\em numerically}, and, as we mentioned in the introduction, the quality control literature is rife with numerical methods to solve specifically equations~\eqref{eq:L-CS-eqn}, assuming that the densities $f_i(x)$, $i=\{0,1\}$, the control limit $h>0$, and the headstart $w\in[0,h)$ are all given.
One of the first numerical methods to treat integral (renewal) equations akin to equations~\eqref{eq:L-CS-eqn} dates back to the work of Page~\cite{Page+Cox:MPCPhS1954}, and is based on a technique now known as the Markov Chain Monte Carlo (MCMC) method---a truly pioneering idea at that time. With regard to equations~\eqref{eq:L-CS-eqn} specifically, Page~\cite{Page:B54} also observed that the CUSUM chart is equivalent to repetitive application of the SPRT with the same initial score (headstart) and control boundaries at $0$ and at $h>0$. This equivalence is significant, for it provides a way to link together the performance of the CUSUM chart and that of the underlying repeated SPRT.
To be more specific, recall that the SPRT with control boundaries $a$ and $b$ ($a\leqslant 0<b$, so that $a$ is the lower boundary and $b$ is the upper boundary) and initial score $w\in(a,b)$ is given by the stopping time
\begin{align*}
\mathcal{S}_{a,b}^w
&\triangleq
\min\{n\geqslant1\colon Z_n\not\in(a,b)\},
\end{align*}
where $Z_n\triangleq\sum_{i=1}^n\log\LR_i$ with $Z_0=w\in(a,b)$. Under either hypothesis $H_i$, $i=\{0,1\}$, the efficiency of the SPRT is customarily measured in terms of two functions: the ASN function and the OC function, defined, respectively, as $\ASN_i(w;a,b)\triangleq\EV_i[\mathcal{S}_{a,b}^w]$ and $\OC_i(w;a,b)=\mathbb{P}_i(Z_{\mathcal{S}_{a,b}^w}\leqslant a)$, $i=\{0,1\}$. The decision made by the SPRT at termination is either ``accept $H_0$'' if the statistic $\{Z_n\}_{n\geqslant1}$ exits the interval $(a,b)$ through the lower boundary $a$, or ``reject $H_0$'' (i.e., ``accept $H_1$'') if the statistic $\{Z_n\}_{n\geqslant1}$ exits the interval $(a,b)$ through the upper boundary $b$. If the terminal decision is ``accept $H_0$'' (``reject $H_0$'') then the SPRT is referred to as an acceptance test (rejection test, respectively). We also note that, by definition, the OC function is the probability that the SPRT will terminate at the lower boundary $a$, under the appropriate hypothesis $H_i$, $i=\{0,1\}$.
The significance of the aforementioned equivalence between the CUSUM chart and a sequence SPRTs can now be made more clear: it allows to express the ARLs of the former through the ASNs and OCs of the latter. Specifically, for notational convenience, put $N_i(x;a,b)\triangleq\ASN_i(x;a,b)$ and $P_i(x;a,b)\triangleq\OC_i(w;a,b)$ for $i=\{0,1\}$. Since for either $i=\{0,1\}$ the SPRTs are applied {\em independently}, the number of acceptance tests before the first rejection test is a geometrically-distributed random variable. As a result, for each hypothesis $H_i$, $i=\{0,1\}$, the ARL of the CUSUM chart and the ASN and OC functions of the SPRT turn out to be connected through the relation
\begin{align}\label{eq:ARL-CS-ASN-OC-SPRT}
L_{i}(x;h)
&=
N_{i}(0;0,h)\,\frac{P_{i}(x;0,h)}{1-P_{i}(0;0,h)}+N_{i}(x;0,h),\;\;i=\{0,1\},
\end{align}
where $x\in[0,h)$ and $h>0$. A detailed derivation of the foregoing formula may be found, e.g., in~\cite[p.~387]{Tartakovsky+etal:Book2014}. It is now evident that the problem of computing the ARLs of the CUSUM chart boils down to the problem of computing the ASNs and the OCs of the underlying SPRT, and the latter problem, in turn, consists in recovering $N_0(x;a,b)$, $N_1(x;a,b)$, $P_0(x;a,b)$, and $P_1(x;a,b)$. With regard to computing the latter four quantities, Page~\cite{Page:B54} proved that they satisfy the following integral equations:
\begin{align}\label{eq:SPRT-NP-int-eqn}
\begin{split}
N_{0}(x;a,b)
&=
1+\int_{a}^{b} K_{0}(y-x)\, N_{0}(y;a,b)\,dy,\\
P_{0}(x;a,b)
&=
F_{0}(a-x)+\int_{a}^{b} K_{0}(y-x)\, P_{0}(y;a,b)\,dy,\\
N_{1}(x;a,b)
&=
1+\int_{a}^{b} K_{1}(y-x)\, N_{1}(y;a,b)\,dy,\\
P_{1}(x;a,b)
&=
F_{1}(a-x)+\int_{a}^{b} K_{1}(y-x)\, P_{1}(y;a,b)\,dy,
\end{split}
\end{align}
where $K_{i}(z)$ and $F_{i}(z)$ for $i=\{0,1\}$ are as in, respectively,~\eqref{eq:Ki-def} and~\eqref{eq:Fi-def} above. Just as equations~\eqref{eq:L-CS-eqn}, each of the foregoing four equations can also be derived by conditioning on the first observation $X_1$.
More importantly, just as equations~\eqref{eq:L-CS-eqn}, the four equations~\eqref{eq:SPRT-NP-int-eqn} are again integral (renewal) equations, so that, again, just as equations~\eqref{eq:L-CS-eqn}, they can rarely be solved analytically, compelling one to resort to the numerical solution. To explain the general idea behind any numerical method to solve equations~\eqref{eq:SPRT-NP-int-eqn}, consider the following generic integral equation
\begin{align}\label{eq:gen-int-eqn}
u(x)
&=
v(x)
+
\int_{a}^{b}K(y-x)\,u(y)\,dy,
\end{align}
where the unknown function is $u(x)$, and the nonhomogeneous term $v(x)$ as well as the integral equation's kernel $K(z)$ are given. The generic integral equation~\eqref{eq:gen-int-eqn} can be easily turned into any one of the four equations~\eqref{eq:SPRT-NP-int-eqn} merely by appropriately choosing $v(x)$ and $K(z)$. Indeed, setting $K(z)=K_{i}(z)$ with $K_{i}(z)$ given by~\eqref{eq:Ki-def} and $v(x)\equiv 1$ for all $x\in[a,b]$ gives the equation for $N_{i}(x;a,b)$, $i=\{0,1\}$. Likewise, keeping $K(z)=K_{i}(z)$ but instead setting $v(x)=F_i(a-x)$ with $F_{i}(z)$ as in~\eqref{eq:Fi-def} gives the equation for $P_{i}(x;a,b)$, $i=\{0,1\}$. Since equation~\eqref{eq:gen-int-eqn} combines all of the four equations~\eqref{eq:SPRT-NP-int-eqn}, any methodology to solve equation~\eqref{eq:gen-int-eqn} can be quickly adapted to any one of the four equations~\eqref{eq:SPRT-NP-int-eqn}. The main step of any (deterministic) numerical method to solve the generic integral equation~\eqref{eq:gen-int-eqn} is to linearize the integral in the right-hand side. This linearization can be performed, e.g., by means of a quadrature scheme, or using an interpolation method of some sort. The end-result of the linearization is that the original equation is reduced to a system of linear equations $\boldsymbol{u}=\boldsymbol{v}+\boldsymbol{K}\,\boldsymbol{u}$ which is then solved for $\boldsymbol{u}$ by standard linear-algebraic methods. Here $\boldsymbol{v}\triangleq[v(x_1),v(x_2),\ldots,v(x_n)]^\top$ where $\{x_j\}_{1\leqslant j\leqslant n}$ is a set of {\it a~priori} chosen $n\geqslant1$ discrete partition points of the interval $[a,b]$, i.e., $a\leqslant x_1<x_2<\ldots<x_n\leqslant b$. The $n\times n$ matrix $\boldsymbol{K}$ is a discrete equivalent of the integral operator
\begin{align}\label{eq:int-op}
\mathcal{K}\circ u
&\triangleq
\int_{a}^{b} K(y-x)\,u(y)\,dy,
\end{align}
and the elements of $\boldsymbol{K}$ are computed off the actual kernel $K(z)$ using the partition points $\{x_i\}_{1\leqslant i\leqslant n}$. If the approximation of $\mathcal{K}$ by $\boldsymbol{K}$ is sufficiently accurate, then the system $\boldsymbol{u}=\boldsymbol{v}+\boldsymbol{K}\,\boldsymbol{u}$ has a unique solution $\boldsymbol{u}\triangleq[u_1,u_2,\ldots,u_n]$, and it is reasonable to expect this solution to be close to the column-vector $[u(x_1),u(x_2),\ldots,u(x_n)]^{\top}$ of the actual values of the unknown function $u(x)$ at the partition points $\{x_j\}_{1\leqslant j\leqslant n}$. It is straightforward to see that $\boldsymbol{u}=(I-\boldsymbol{K})^{-1}\,\boldsymbol{v}$ where here and onward $I$ denotes the $n\times n$ identity matrix.
Going back to equations~\eqref{eq:SPRT-NP-int-eqn}, let $\boldsymbol{K}_{i}$, $i=\{0,1\}$, denote the matrix approximation of the integral operator~\eqref{eq:int-op} induced by the kernel $K_{i}(z)$, $i=\{0,1\}$, given by~\eqref{eq:Ki-def}. Suppose also that the corresponding partition points are $\{x_j\}_{1\leqslant j\leqslant n}$, and introduce $\boldsymbol{1}\triangleq[1,1,\ldots,1]^{\top}$ and $\boldsymbol{F}_{i}\triangleq[F_{i}(a-x_1),F_{i}(a-x_2),\ldots,F_{i}(a-x_n)]^{\top}$ for $i=\{0,1\}$. Then, by linearization, the four equations~\eqref{eq:SPRT-NP-int-eqn} are reduced to $\boldsymbol{N}_{0}=\boldsymbol{1}+\boldsymbol{K}_{0}\,\boldsymbol{N}_{0}$, $\boldsymbol{P}_{0}=\boldsymbol{F}_0+\boldsymbol{K}_{0}\,\boldsymbol{P}_{0}$, $\boldsymbol{N}_{1}=\boldsymbol{1}+\boldsymbol{K}_{1}\,\boldsymbol{N}_{1}$, and $\boldsymbol{P}_{0}=\boldsymbol{F}_1+\boldsymbol{K}_{0}\,\boldsymbol{P}_{1}$. Here $\boldsymbol{N}_{i}$ and $\boldsymbol{P}_{i}$ are column-vectors comprised of approximate values of $N_{i}(x;a,b)$ and $P_{i}(x;a,b)$, respectively, evaluated at the partition nodes $\{x_j\}_{1\leqslant j\leqslant n}$. We are now in a position to make the following observation. While the system of linear equations for $\boldsymbol{N}_{0}$ and that for $\boldsymbol{P}_{0}$ have different nonhomogeneous terms, they both have the same matrix of coefficients $I-\boldsymbol{K}_0$. As a result, both systems can be (and should be) solved simultaneously by combining the nonhomogeneous terms into the $n\times 2$ matrix $[\boldsymbol{1},\boldsymbol{F}_0]$ and solving the system $(I-\boldsymbol{K}_0)\,[\boldsymbol{N}_0,\boldsymbol{P}_0]=[\boldsymbol{1},\boldsymbol{F}_0]$. Likewise, the system for $\boldsymbol{N}_{1}$ and that for $\boldsymbol{P}_{1}$ can be solved in exactly the same manner. There is no question that grouping the right-hand sides of any two or more different systems of linear equations with the same matrix of coefficients allows to cut down the overall number of operations needed to solve all of the systems. This a basic fact taught in any course on elementary linear algebra. The problem, however, is that of the four equations~\eqref{eq:SPRT-NP-int-eqn}, the top two (which correspond to the null hypothesis) and the bottom two (which correspond to the alternative hypothesis) appear to be unrelated, and therefore have to be solved {\em separately}. The main result of this paper is to prove otherwise. Specifically, it turns out that $K_1(z)$ and $K_0(z)$ {\em are} connected, and the connection is simple and allows one to show that the four equations~\eqref{eq:SPRT-NP-int-eqn} are all instances of the same single equation involving the same kernel and parameterized only by the nonhomogeneous term. As a result, all four equations~\eqref{eq:SPRT-NP-int-eqn} can be solved {\em simultaneously} by grouping the nonhomogeneous terms together, just as we described above for the generic equation~\eqref{eq:gen-int-eqn}. This would clearly lead to a reduction of the computational burden required to approximately recover $N_{i}(x;a,b)$ and $P_{i}(x;a,b)$ as $\boldsymbol{N}_{i}$ and $\boldsymbol{P}_{i}$, respectively. The specifics are discussed in the next section.
\section{The Main Result and Its Discussion}
\label{sec:main-results}
We begin with an observation that will be key to obtain a link between $K_0(z)$ and $K_1(z)$ given by~\eqref{eq:Ki-def}, and subsequently establish the main result of this paper. Let $P_i^{\LR}(t)\triangleq\mathbb{P}_i(\LR_1\leqslant t)$, $t\geqslant0$, $i=\{0,1\}$, denote the cdf of the LR under the hypothesis $H_i$, $i=\{0,1\}$, respectively. Since the LR is the Radon--Nikod\'{y}m derivative of the probability measure $\mathbb{P}_1$ with respect to the probability measure $\mathbb{P}_0$ (the two measures are assumed to be mutually absolutely continuous), one can deduce the following result.
\begin{lemma}\label{lem:change-of-measure-id1}
$dP_1^{\LR}(t)=t\,dP_0^{\LR}(t)$, $t\geqslant0$.
\end{lemma}
This result is nothing but Wald's~\cite{Wald:Book47} likelihood ratio identity (see also, e.g.,~\cite[p.~13]{Siegmund:Book85},~\cite[p.~4]{Woodroofe:Book82},~\cite{Lai:SA2004}, or~\cite[Theorem~2.3.3,~p.~32]{Tartakovsky+etal:Book2014}), and can be obtained from the following argument:
\begin{align*}
\begin{split}
dP_{1}^{\LR}(t)
&\triangleq d\mathbb{P}_{1}(\LR_{1}\leqslant t)\\
&=
d\mathbb{P}_{1}(X_1\leqslant \LR_{1}^{-1}(t))\\
&=
\LR_{1}(\LR_{1}^{-1}(t))\,d\mathbb{P}_0(X_{1}\leqslant \LR_{1}^{-1}(t))\\
&=
t\,d\mathbb{P}_{0}(\LR_{1}\leqslant t)\\
&=
t\,dP_{0}^{\LR}(t),
\end{split}
\end{align*}
whence $dP_1^{\LR}(t)=t\,dP_0^{\LR}(t)$, $t\geqslant0$, as needed; cf.~\cite{Polunchenko+etal:SA2014,Polunchenko+etal:ASMBI2014} and~\cite[Chapter~3]{Du:PhD-Thesis2015}.
As an immediate implication of Lemma~\ref{lem:change-of-measure-id1}, observe that since
\begin{align*}
\begin{split}
K_{i}(z)
&\triangleq
\frac{d}{dz}\mathbb{P}_{i}(\log\LR_1 \leqslant z)\\
&=
\frac{d}{dz}\mathbb{P}_{i}(\LR_1\leqslant e^z)\\
&=
\frac{d}{dz}P_{i}^{\LR}(e^z),\; i=\{0,1\},
\end{split}
\end{align*}
it follows that $K_{1}(z)=e^{z}\,K_{0}(z)$, $z\in\mathbb{R}$. Now, setting $z=y-x$, the following can be seen to hold true.
\begin{lemma}\label{lem:change-of-measure-id2}
$e^{-y}\,K_{1}(y-x)=e^{-x}\,K_{0}(y-x)$, $x,y\in\mathbb{R}$.
\end{lemma}
The foregoing lemma is the main result of this paper. It is an obvious extension of the results obtained previously in~\cite{Polunchenko+etal:SA2014,Polunchenko+etal:ASMBI2014} for the Generalized Shiryaev--Roberts procedure. As simple as it may seem, the established connection between $K_1(z)$ and $K_0(z)$ has far-reaching consequences. We shall now elaborate on this at greater length.
At the very least Lemma~\ref{lem:change-of-measure-id2} provides a ``shortcut'' to derive a formula for ${K}_1(z)$ from that for ${K}_{0}(z)$, or the other way around---whichever one of the two is found first. To illustrate this point, suppose that
\begin{align*}
f_0(x)
&=
\frac{1}{\sqrt{2\,\pi}}\,e^{-\tfrac{x^2}{2}}
\;\text{and}\;
f_1(x)
=
\frac{1}{\sqrt{2\,\pi}}\,e^{-\tfrac{(x-\theta)^2}{2}},
\end{align*}
where $x\in\mathbb{R}$ and $\theta\neq 0$, a known parameter (which is the ``off-target'' mean level). This basic Gaussian model is the standard ``testbed'' model widely used in the literature for demonstrational purposes. Under this model it is direct to see that the log-LR is of the form
\begin{align}\label{eq:LR-Gauss}
\log\LR_n
&\triangleq
\log\frac{f_1(X_n)}{f_0(X_n)}=
\theta\,X_n-\frac{\theta^2}{2},
\end{align}
whence
\begin{align*}
K_0(z)
&=
\frac{1}{\sqrt{2\,\pi\,\theta^2}}\,\exp\leqslantft\{-\frac{1}{2\theta^2}\leqslantft(z+\frac{\theta^2}{2}\right)^2\right\},\; z\in\mathbb{R},
\end{align*}
i.e., $K_0(z)$ is the pdf of a Gaussian distribution with mean $-\theta^2/2\,(<0)$ and variance $\theta^2\,(\neq 0)$. As a result, from $K_1(z)=e^{z}\,K_0(z)$ we obtain that
\begin{align*}
K_1(z)
&=
\frac{1}{\sqrt{2\,\pi\,\theta^2}}\,\exp\leqslantft\{-\frac{1}{2\theta^2}\leqslantft(z-\frac{\theta^2}{2}\right)^2\right\},\; z\in\mathbb{R},
\end{align*}
i.e., $K_1(z)$ is the pdf of a Gaussian distribution with mean $\theta^2/2\,(>0)$ and variance $\theta^2\,(\neq 0)$. This is exactly what $K_1(z)$ is supposed to be for the Gaussian model at hand. We stress that the formula for $K_1(z)$ was {\em not} obtained from~\eqref{eq:LR-Gauss}, i.e., from the log-LR formula: we used the log-LR formula~\eqref{eq:LR-Gauss} to recover $K_0(z)$ only, and then with $K_0(z)$ expressed explicitly, we exploited the identity $K_1(z)=e^{z}\,K_0(z)$ to find $K_1(z)$. When $f_i(x)$, $i=\{0,1\}$, are complicated, and the log-LR is not a simple function, obtaining both $K_i(z)$, $i=\{0,1\}$, in a closed-form {\em directly} from the log-LR formula may be rather ``calculusy''. It is in such cases that appealing instead to the identity $K_1(z)=e^{z}\,K_0(z)$ may prove especially advantageous, for the calculus involved is effectively half that required to get $K_1(z)$ directly from the log-LR formula.
More importantly, observe that in view of Lemma~\ref{lem:change-of-measure-id2}, the aforementioned four integral equations~\eqref{eq:SPRT-NP-int-eqn} on $N_i(x;a,b)$ and $P_i(x;a,b)$, $i=\{0,1\}$, can be rewritten as follows:
\begin{align*}
\begin{split}
N_0(x;a,b)
&=
1+\int_{a}^b K_0(y-x)\, N_0(y;a,b)\,dy,\\
e^x\,N_1(x;a,b)
&=
e^x+\int_{a}^b K_0(y-x)\,[e^{y}\,N_1(y;a,b)]\,dy,\\
P_0(x)
&=
F_0(a-x)+\int_{a}^b K_0(y-x)\,P_0(y;a,b)\,dy,\\
e^{x}\,P_1(x)
&=
e^{x}\,F_1(a-x)+\int_a^b K_0(y-x)\,[e^{y}\,P_1(y;a,b)]\,dy,
\end{split}
\end{align*}
so that the kernel $K_{1}(z)$ is eliminated entirely, and all of the equations turn out to involve only the kernel $K_0(z)$. It is now evident that the ASN and OC functions of the SPRT under the null and under the alternative are {\em all} governed by the same {\em one} integral equation but with different nonhomogeneous terms: the equation for $N_0(x;a,b)$ has 1 as its nonhomogeneous term, the equation for $P_0(x;a,b)$ has $F_0(a-x)$ as its nonhomogeneous term, the equation for $e^x\,N_1(x;a,b)$ has $e^x$ as its nonhomogeneous term, and the equation for $e^x\,P_1(x;a,b)$ has $e^{x}\,F_1(a-x)$ as its nonhomogeneous term. We note also that the latter two equations are to be solved not for $N_1(x;a,b)$ and $P_1(x;a,b)$, but for $e^x\, N_1(x;a,b)$ and $e^x\, P_1(x;a,b)$, respectively, and the exponential factor present in the obtained solutions is taken care of once $e^x\, N_1(x;a,b)$ and $e^x\, P_1(x;a,b)$ are found. Therefore, Lemma~\ref{lem:change-of-measure-id2} allows to find the ASNs and OCs of the SRPT under the null and under the alternative {\em simultaneously} in the manner that was explained at the end of Section~\ref{sec:preliminaries}, i.e., by simply grouping the nonhomogeneous terms into one matrix. Furthermore, via the relation~\eqref{eq:ARL-CS-ASN-OC-SPRT} the in- and out-of-control ARLs of the CUSUM chart can also be computed {\em concurrently}, thereby making a more efficient use of the computational resources available. This is the primary {\em practical} benefit of Lemma~\ref{lem:change-of-measure-id2}, i.e., the main result of this paper.
To provide yet another illustration of the practical benefits of Lemma~\ref{lem:change-of-measure-id2}, consider the problem of computing the entire distribution of the CUSUM Run Length under the hypothesis $H_i$, $i=\{0,1\}$. The respective integral equations and recurrences were obtained, e.g., by Ewan and Kemp~\cite{Ewan+Kemp:B1960} and then also by Woodall~\cite{Woodall:T1983}. See also, e.g., Waldmann~\cite{Waldmann:T1986}. Specifically, capitalizing on the fact that the CUSUM chart is a sequence of SPRTs, for each hypothesis $H_i$, $i=\{0,1\}$, Woodall~\cite{Woodall:T1983} expressed the distribution of the CUSUM Run Length through that of the Run Length of the SPRT. See~\cite[p.~296]{Woodall:T1983}. The distribution of the SPRT, in turn, is found using repetitive application of the linear integral operator
\begin{align}\label{eq:int-op-i}
\mathcal{K}_{i}\circ u
&\triangleq
\int_{a}^{b} K_{i}(y-x)\,u(y)\,dy,
\end{align}
where $i=\{0,1\}$. Therefore, from Lemma~\ref{lem:change-of-measure-id2} it is easy to see that the distribution of the SPRT Run Length under the hypothesis $H_0$ can be found {\em concurrently} with the distribution under the hypothesis $H_1$.
We would like to conclude this section with two remarks. First, note that the integral operator~\eqref{eq:int-op-i} was involved in all of the integral equations we considered. As a matter of fact, the integral operator~\eqref{eq:int-op-i} can be used as a universal index that summarizes the overall performance of the chart, whatever the specific metric be. By way of example, if one were interested in computing the higher-order $H_i$-moments of the CUSUM Run Length, i.e., the quantities $\mu_{i}^{(k)}\triangleq\EV_{i}[(\mathcal{C}_h^w)^k]$, $i=\{0,1\}$, $k=0,1,\ldots$, then the equivalence of the CUSUM chart to sequential application of the SPRT would again allow to express $\mu_{i}^{(k)}$ through the corresponding moments of the SPRT, and the latter moments would again satisfy integral equations similar to equations~\eqref{eq:SPRT-NP-int-eqn}. See, e.g.,~\cite{Ewan+Kemp:B1960}. Therefore, Lemma~\ref{lem:change-of-measure-id2} would again allow to lessen the computational cost, for it would allow to compute $\mu_{1}^{(k)}$ and $\mu_{0}^{(k)}$ {\em concurrently} for any fixed $k=0,1,\ldots$. Second, note that for Page's~\cite{Page:B54} symmetric two-sided CUSUM scheme, Lemma~\ref{lem:change-of-measure-id2} would allow for an even greater reduction of the computational complexity.
\section{Conclusion}
\label{sec:conclusion}
As part of the author's ongoing effort to bridge the gap between the theory and application of sequential analysis, this work sought to build on to the results obtained previously in~\cite{Polunchenko+etal:SA2014,Polunchenko+etal:ASMBI2014} and in~\cite[Chapter~3]{Du:PhD-Thesis2015}, and provide an example of an alternative, more {\em applicative} use of one of the central techniques of {\em theoretical} sequential analysis---Wald's~\cite{Wald:Book47} likelihood ratio identity. Specifically, by virtue of the latter we obtained a connection between a broad range of {\em in-control} characteristics of the CUSUM chart and their {\em out-of-control} counterparts. The obtained connection relates directly the integral equations on the {\em in-control} characteristics and the integral equations on the corresponding {\em out-of-control} characteristics. On a practical level, this relationship allows the {\em in-} and {\em out-of-control} characteristics to be recovered {\em simultaneously} as solutions of the respective integral equations, thereby improving the efficiency of any numerical method one may employ to compute the performance of the CUSUM chart as well as that of the SPRT.
\end{document} |
\begin{document}
\begin{abstract}
We prove an analogue for Stokes torsors of Deligne's skeleton conjecture and deduce from it the representability of the functor of relative Stokes torsors by an affine scheme of finite type over $\mathds{C}$. This provides, in characteristic 0, a local analogue of the existence of a coarse moduli for skeletons with bounded ramification, due to Deligne.
As an application, we use the geometry of this moduli to derive quite strong finiteness results for integrable systems of differential equations in several variables which did not have any analogue in one variable.
\end{abstract}
\title{Skeletons and moduli of Stokes torsors}
Consider the following linear differential equation $(E)$ with polynomial coefficients
$$
p_n\frac{d^n f}{ dz^n}+p_{n-1} \frac{d^{n-1} f}{ dz^{n-1}}+\dots + p_1\frac{d f}{dz}+p_0 f=0
$$
If $p_n(0)\neq 0$, Cauchy theorem asserts that a holomorphic solution to $(E)$ defined on a small disc around $0$ is equivalent to the values of its $n$ first derivatives at $0$. If $p_n(0)=0$, holomorphic solutions to $(E)$ on a small discs around $0$ may always be zero. Nonetheless, $(E)$ may have formal power series solutions. The \textit{Main asymptotic development theorem} \cite{SVDP},
due to Hukuhara and Turrittin asserts that for a direction $\theta$ emanating from 0, and for a formal power series solution $f$, there is a sector $\mathcal{S}_{\theta}$ containing $\theta$ such that $f$ can be "lifted" in a certain sense to a holomorphic solution $f_{\theta}$ of $(E)$ on $\mathcal{S}_{\theta}$. We say that \textit{$f_{\theta}$ is asymptotic to $f$ at $0$}. If $f_{\theta}$ is analytically continued around 0 into a solution $\tilde{f}_{\theta}$ of $(E)$ on the sector $\mathcal{S}_{\theta^{\prime}}$, where $\theta^{\prime}\neq \theta$, it may be that the asymptotic development of $\tilde{f}_{\theta}$ at $0$ is not $f$ any more. This is the \textit{Stokes phenomenon}.
As a general principle, the study of $(E)$ amounts to the study of its "formal type" and the study of how asymptotic developments of solutions jump via analytic continuation around $0$. To organize these informations, it is traditional to adopt a linear algebra point of view.
\\ \indent
The equation $(E)$ can be seen as a \textit{differential module}, i.e. a finite dimensional vector space $\mathcal{N}$ over the field $\mathds{C}\{z\}[z^{-1}]$ of convergent Laurent series, endowed with a $\mathds{C}$-linear endomorphism $\nabla : \mathcal{N}\longrightarrow \mathcal{N}$ satisfying the Leibniz rule.
In this language, solutions of $(E)$ correspond to elements of $\Ker \nabla$ (also called \textit{flat sections of $\nabla$}).
Furthermore, a differential equation with same "formal type" as $(E)$ corresponds to a differential module $\mathcal{M}$ with an isomorphism of \textit{formal} differential modules $\iso : \mathcal{M}_{\hat{0}}\longrightarrow \mathcal{N}_{\hat{0}}$. Since $\iso$ can be seen as a formal flat section of the differential module $\Hom(\mathcal{M}, \mathcal{N})$, the main asymptotic development theorem applies to it. The lifts of $\iso$ to sectors thus produce a cocycle $\gamma:=(\iso_\theta\iso_{\theta^{\prime}}^{-1})_{\theta,\theta^{\prime}\in S^1}$ with value into the sheaf of sectorial automorphisms of
$\mathcal{N}$ which are asymptotic to $\Id$ at $0$. This is the \textit{Stokes sheaf} of $\mathcal{N}$, denoted by $\St_{\mathcal{N}}(\mathds{C})$. A fundamental result of Malgrange \cite{MaSi} and Sibuya \cite{SI} implies that $(\mathcal{M}, \iso)$ is determined by the torsor under $\St_{\mathcal{N}}(\mathds{C})$ associated to the cocycle $\gamma$. Hence, \textit{Stokes torsors encode in an algebraic way analytic data and classifying differential equations amounts to studying Stokes torsors}. As a result, the study of Stokes torsors is meaningful.
\\ \indent
In higher dimension, the role played by differential modules is played by \textit{good meromorphic connections}. We will take such a connection $\mathcal{N}$ defined around $0\in \mathds{C}^n$ to be of the shape \begin{equation}\label{shape}
\mathcal{E}^{a_1} \otimes \mathcal{R}_{a_1}\oplus \dots \oplus \mathcal{E}^{a_d} \otimes \mathcal{R}_{a_d}
\end{equation}
where the $a_i$ are meromorphic functions with poles contained in a normal crossing divisor $D$, where $
\mathcal{E}^{a_i}$ stands for the rank one connection $(\mathcal{O}_{\mathds{C}^n, 0}(\ast D), d-da_i)$, and where the $\mathcal{R}_{a_i}$ are regular connections.
Note that from works of Kedlaya \cite{Kedlaya1}\cite{Kedlaya2} and Mochizuki \cite{Mochizuki2}\cite{Mochizuki1}, every meromorphic connection is (up to ramification) formally isomorphic at each point to a connection of the form \eqref{shape} at the cost of blowing-up enough the pole locus. If $(r_i, \theta_i)_{i=1, \dots, n}$ are the usual polar coordinates on $\mathds{C}^n$, the Stokes sheaf $\St_{\mathcal{N}}$ of $\mathcal{N}$ is a sheaf of complex unipotent algebraic groups over the torus $\mathds{T}:=(S^1)^n$ defined by $r_1=\cdots = r_n=0$.
\\ \indent
By a \textit{good $\mathcal{N}$-marked connection}, we mean the data $(\mathcal{M}, \nabla, \iso)$ of a good meromorphic connection $(\mathcal{M}, \nabla)$ around $0$ endowed with an isomorphism of formal connections at the origin $\iso: \mathcal{M}_{\hat{0}}\longrightarrow \mathcal{N}_{\hat{0}}$. As in dimension 1, Mochizuki \cite{MochStokes}\cite{Mochizuki1} showed that good $\mathcal{N}$-marked connections are determined by their associated Stokes torsor, so we consider them as elements in $H^{1}(\mathds{T}, \St_{\mathcal{N}}(\mathds{C}))$.\\ \indent
Since $\St_{\mathcal{N}}$ is a sheaf of complex algebraic groups, its sheaf of $R$-points $\St_{\mathcal{N}}(R)$ is a well-defined sheaf of groups on $\mathds{T}$ for any commutative $\mathds{C}$-algebra $R$. Consequently, one can consider the functor of relative Stokes torsors $R\longrightarrow H^1(\mathds{T}, \St_{\mathcal{N}}(R))$, denoted by $H^1(\mathds{T}, \St_{\mathcal{N}})$. Following a strategy designed by Deligne, Babbitt and Varadarajan \cite{BV} proved that $H^1(S^1, \St_{\mathcal{N}})$ is representable by an affine space. Hence in dimension 1, the set of torsors under $\St_{\mathcal{N}}$ has a structure of a complex algebraic variety. \\ \indent
The interest of this result is to \textit{provide a framework in which questions related to differential equations can be treated with the apparatus of algebraic geometry}.
This might look like a wish rather than a documented fact since the local theory of linear differential equations is fully understood in dimension $1$ by means of analysis. In dimension $\geq 2$ however, new phenomena appear and this geometric perspective seems relevant. As we will show, the representability of $H^1(\mathds{T}, \St_{\mathcal{N}})$ in any dimension implies for differential equations quite strong finiteness results which have no counterparts in dimension 1 and which seem out of reach with former technology. Thus, we prove the following
\begin{theorem}\label{bigtheorem}
The functor $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ is representable by an affine scheme of finite type over $\mathds{C}$.
\end{theorem}
Before explaining how the proof relates to Deligne's skeleton conjecture, let us describe two applications to finiteness results. \\ \indent
Suppose that $\mathcal{N}$ is \textit{very good}, that is, for functions $a_i$, $a_j$ appearing in \eqref{shape} with $a_i\neq a_j$, the difference $a_i- a_j$ has poles along all the components of the divisor $D$ along which $\mathcal{N}$ is localized. Let $V$ be a manifold containing $0$ and let us denote by $\mathcal{N}_{V}$ the restriction of the connection $\mathcal{N}$ to $V$. We prove the following
\begin{theorem}\label{restriction}
If $V$ is transverse to every irreducible component of $D$, there is only a finite number of equivalence classes of good $\mathcal{N}$-marked connections with given restriction to $V$. Furthermore, this number depends only on $\mathcal{N}$ and on $V$.
\end{theorem}
This theorem looks like a weak differential version of Lefschetz's theorem. A differential Lefschetz theorem would assert that for a generic choice of $V$, good $\mathcal{N}$-marked connections are determined by their restriction to $V$. It is a hope of the author that such a question is approachable by geometric means using
the morphism of schemes
\begin{equation}\label{res}
\res_V: H^{1}(\mathds{T}, \St_{\mathcal{N}})\longrightarrow H^{1}(\mathds{T}^{\prime}, \St_{\mathcal{N}_{V}})
\end{equation}
induced by the restriction to $V$. \\ \indent
To give flesh to this intuition, let us indicate how geometry enters the proof of Theorem \ref{restriction}. Since unramified morphisms of finite type have finite fibers, it is enough to show that $\mathcal{N}$-marked connections lie
in the unramified locus of $\res_V$, which is the locus where
the tangent map of $\res_V$ is injective. We show in \ref{tanandirr} a canonical identification
\begin{equation}\label{relation}
T_{(\mathcal{M}, \nabla, \iso)} H^1(\mathds{T}, \St_\mathcal{N})\simeq \mathcal{H}^1 (\Sol \End\mathcal{M})_0
\end{equation}
where the left-hand side denotes the tangent space of $H^1(\mathds{T}, \St_\mathcal{N})$ at $(\mathcal{M}, \nabla, \iso)$ and where $\mathcal{H}^1 \Sol$ denotes the first cohomology sheaf of the solution complex of a $\mathcal{D}$-module. Note that the left-hand side of \eqref{relation} is \textit{algebraic}, whereas the right-hand side is \textit{transcendental}. From that we deduce a similar interpretation to the kernel $\Ker T_{(\mathcal{M}, \nabla, \iso)} \res_V$ and prove its vanishing using a perversity theorem due to Mebkhout \cite{Mehbgro}. \\ \indent
Using an invariance theorem due to Sabbah \cite{SabRemar}, we further prove the following rigidity result:
\begin{theorem}\label{rigidity}
Suppose that $D$ has at least two components and that $\mathcal{N}$ is very general. Then there is only a finite number of equivalence classes of good $\mathcal{N}$-marked connections.
\end{theorem}
In this statement, very general means that $\mathcal{N}$ is very good and that the monodromy of each regular constituent contributing to $\mathcal{N}$ in \eqref{shape} has eigenvalues away from a denombrable union of strict Zariski closed subsets of an affine space. \\ \indent
Let us finally explain roughly the proof of Theorem \ref{bigtheorem}. The main idea is to import and prove a conjecture from the field of Galois representations. Let $X$ be a smooth variety over a finite field of characteristic $p>0$, and let $\ell\neq p$ be a prime number. To any $\ell$-adic local system $\mathcal{F}$
on $X$ up to semi-simplification, one can associate its
\textit{skeleton} $\sk \mathcal{F}$, that is the collection of restrictions of $\mathcal{F}$ to curves
drawn on $X$ endowed with compatibilities at intersection points. It has bounded ramification
at infinity in an appropriate sense, see \cite{HenKerz}. As a consequence
of Cebotarev's density theorem, the assignment $\mathcal{F}\longrightarrow \sk
\mathcal{F}$ is injective and Deligne conjectured that any skeleton with
bounded ramification comes from an $\ell$-adic local system. Building on the work of
Wiesend \cite{Wiesend}, this has been proved in the tame case by Drinfeld
\cite{DriDel} and for arbitrary ramification in the rank one case by Kerz and Saito
\cite{KerzSaito}. \\ \indent
Back to characteristic $0$, let $\mathscr{C}$ be the family of smooth curves $i: C \hookrightarrow \mathds{C}^n$ containing $0$. For $C\in \mathscr{C}$, passing to polar coordinates induces a map $(r, \theta)\longrightarrow i(r, \theta)$. Restricting to $r=0$ thus produces an embedding in $\mathds{T}$ of the circle $S^1_C$ of directions inside $C$ emanating from $0$.
We define for every $\mathds{C}$-algebra $R$ a \textit{Stokes skeleton relative to $R$} as a collection of Stokes torsors $(\mathcal{T}_C \in H^{1}(S^1_C, \St_{\mathcal{N}_C}(R)))_{C\in \mathscr{C}}$
endowed with compatibilities at points of $\mathds{T}$ where two circles $S^1_{C_1}$ and $S^1_{C_2}$ intersect, where $C_1, C_2\in \mathscr{C}$.
We observe that a naive version of Deligne's conjecture stating that every Stokes skeleton comes from a Stokes torsor is false, and introduce a
combinatorial condition \ref{admissible} satisfied by skeletons of Stokes torsors that we call \textit{admissibility}. We finally prove the following
\begin{theorem}\label{equcat}
Restriction induces a bijection between $H^{1}(\mathds{T}, \St_{\mathcal{N}}(R))$ and the set $\St_{\mathcal{N}}(R)\text{-}\Sk_{\mathscr{C}}$ of $\St_{\mathcal{N}}(R)$-admissible skeletons relative to $R$.
\end{theorem}
The Stokes skeleton functor $R\longrightarrow \St_{\mathcal{N}}(R)\text{-}\Sk_{\mathscr{C}}$ is easily seen to be representable by an affine scheme by Babbitt-Varadarajan's theorem. Hence, to prove Theorem \ref{bigtheorem}, the whole point is to show that the same is true when admissibility is imposed. To do this, we first use a general theorem \cite[6.3]{ArtinOnStack}, see also \cite[10.4]{LMB} and \cite[04S6]{SP} to obtain the representability of $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ by an algebraic space of finite type over $\mathds{C}$. Finally, for a
finite family $\mathscr{C}_f\subset \mathscr{C}$ of carefully chosen curves, we show that the formation of the $\mathscr{C}_f$-skeleton
$$
H^{1}(\mathds{T}, \St_{\mathcal{N}})\longrightarrow \St_{\mathcal{N}}\text{-}\Sk_{\mathscr{C}_f}
$$
is a closed immersion.
\section{Generalities on Stokes torsors}
\subsection{}\label{notation}
Let $D$ be a germ of normal crossing divisor at $0\in \mathds{C}^n$ and let $i: D\longrightarrow \mathds{C}^n$ be the canonical inclusion. Let $D_1, \dots, D_m$ be the irreducible components of $D$. Let $\mathcal{I}$ be a good set of irregular values with poles contained in $D$ in the sense of \cite{Mochizuki1}. For every $a\in \mathcal{I}$, set
$$
\mathcal{E}^{a}=(\mathcal{O}_{\mathds{C}^n, 0}(\ast D), d-da)
$$
We fix once for all a germ of unramified good meromorphic flat bundle of rank $r$ of the form
\begin{equation}\label{formalmodel}
(\mathcal{N}, \nabla_{\mathcal{N}}):=\bigoplus_{a\in \mathcal{I}}\mathcal{E}^{a} \otimes \mathcal{R}_a
\end{equation}
where $\mathcal{R}_a$ is a germ of regular meromorphic connection with poles along $D$. We also fix a basis $\mathbf{e}_a$ of $\mathcal{R}_a$.
We denote by $E_a$ the $\mathds{C}$-vector space generated by $\mathbf{e}_{a}$, $E:= \oplus_{a\in \mathcal{I}} E_a$, $i_a : E_a\longrightarrow E$ the canonical inclusion and $p_a : E \longrightarrow E_a$ the canonical projection.\\ \indent
Note that in dimension $>1$, an arbitrary meromorphic connection may not have a formal model of the shape \eqref{formalmodel}, but we know from works of Kedlaya \cite{Kedlaya1}\cite{Kedlaya2} and Mochizuki \cite{Mochizuki2}\cite{Mochizuki1} that such a model exists (up to ramification) at each point after enough suitable blow-ups.
\subsection{Recollection on asymptotic analysis} \label{rappelasan}
As references for asymptotic ana\-lysis and good connections, let us mention \cite{Maj}, \cite{Stokes} and \cite{Mochizuki1}.
For $i=1, \dots, m$, let $\tilde{X}_i\longrightarrow \mathds{C}^n$ be the real blow-up of $\mathds{C}^n$ along $D_i$ and let $p:\tilde{X}\longrightarrow \mathds{C}^n$ be the fiber product of the $\tilde{X}_i$. We have $\tilde{X}\simeq ([0, +\infty [ \times S^{1})^{m}\times \mathds{C}^{n-m}$ and $p$ reads
$$
((r_k,\theta_k )_{k},y)\longrightarrow ((r_k e^{i\theta_k} )_{k},y)
$$
In particular, we have an open immersion $j_{D}: \mathds{C}^n\setminus D\longrightarrow \tilde{X}$ and $\mathds{T}:=p^{-1}(0)$ is isomorphic to $(S^1)^m$.\\ \indent
Let $\mathcal{A}$ be the sheaf on $\mathds{T}$ of holomorphic functions on $\mathds{C}^n\setminus D$ admitting an asymptotic development along $D$. Let $\mathcal{A}^{<D}$ be the subsheaf of $\mathcal{A}$ of functions asymptotic to $0$ along $D$. These sheaves are endowed with a structure of $p^{-1}\mathcal{D}_{\mathds{C}^n,0}$-module and there is a canonical \textit{asymptotic development} morphism
$$
\xymatrix{
\AS_0: \mathcal{A} \ar[r] & \mathds{C}\llbracket z_1, \dots, z_n\rrbracket
}
$$
where $\mathds{C}\llbracket z_1, \dots z_n\rrbracket$ has to be thought of as the constant sheaf on $\mathds{T}$. \\ \indent
For a germ $(\mathcal{M}, \nabla)$ of flat meromorphic connection at $0$, the module $p^{-1}\mathcal{M}$ makes sense in a neighbourhood of $\mathds{T}$ in $\tilde{X}$. Thus,
$$
\tilde{\mathcal{M}}:=\mathcal{A}\otimes_{\mathcal{O}_{\mathds{C}^n, 0}} p^{-1}\mathcal{M}
$$
is a $p^{-1}\mathcal{D}_{\mathds{C}, 0}$-module on $\mathds{T}$. Let $\DR \tilde{\mathcal{M}}$ be the De Rham complex of $ \tilde{\mathcal{M}}$ and let $\DR^{<D} \mathcal{M}$ be the De Rham complex of $ \tilde{\mathcal{M}}$ with coefficients in $\mathcal{A}^{<D}$. \\ \indent
For $\xi \in \mathcal{H}^{0}\DR \mathcal{M}_{\hat{0}}$, we denote by $\mathcal{H}^{0}_{\xi}\DR \tilde{\mathcal{M}}$ the subsheaf of $\mathcal{H}^{0}\DR \tilde{\mathcal{M}}$ of sections $\tilde{\xi}$ which are asymptotic to $\xi$, that is
$$
(\AS_0\otimes \id_{ \mathcal{M}})(\tilde{\xi})=\xi
$$
We set $$\Isom_{\iso}( \mathcal{M}, \mathcal{N}):=\mathcal{H}^{0}_{\iso}\DR \MCHom( \tilde{\mathcal{M}}, \tilde{\mathcal{N}})
$$
Mochizuki's asymptotic development theorem \cite{Mochizuki1}\footnote{see also \cite{HienInv} for an account of the proof.} implies that the sheaf $\Isom_{\iso}(\mathcal{M}, \mathcal{N})$ is a torsor under $\St_{\mathcal{N}}(\mathds{C})$ defined below.
\subsection{Stokes hyperplane}
For $a, b \in \mathcal{I}$, the function
$$
F_{a,b}:= \Real(a-b)|z^{-\ord(a-b)}|
$$
induces a $C^{\infty}$ function on $\partial \tilde{X}$. We denote by $\mathcal{H}_{a,b}$ its vanishing locus on $\mathds{T}$. The
$\mathcal{H}_{a,b}$ are the \textit{Stokes hyperplanes of $\mathcal{I}$}. Concretely, $$a-b=f_{ab}z^{\ord(a-b)}$$ with $f_{ab}(0)\neq 0$ and $\ord(a-b)=-(\alpha_{ab}(1), \dots, \alpha_{ab}(m))$, where $\alpha_{ab}(k)\geq 0$ for $k=1, \dots, m$. Hence, $F_{a,b}$ induces
\begin{equation}\label{Fabinduit}
((r_k, \theta_k)_{1\leq k \leq m}, (z_k)_{m+1\leq k \leq n})\longrightarrow \Real f_{ab}(r_k e^{i \theta_k}, z_k) e^{-i\sum_{k=1}^{m}\alpha_{ab}(k) \theta_k}
\end{equation}
Set $f_{ab}(0)=r_{ab}e^{i\theta_{ab}}$ with $0\leq \theta_{ab}<2\pi$. The restriction of \eqref{Fabinduit} to $\mathds{T}$ is
\begin{equation}\label{FabinduitsurT}
(\theta_1, \dots, \theta_m)\longrightarrow \cos(\theta_{ab}-\sum_{k=1}^{m}\alpha_{ab}(k) \theta_k)
\end{equation}
From now on, we see $\mathds{T}$ as $\mathds{R}^m/2\pi \mathds{Z}^m$ and we denote by $\pi : \mathds{R}^m \longrightarrow \mathds{T}$ the canonical projection.
For $l\in \mathds{Z}$, let $\mathcal{H}_{a,b}(l)$ be the hyperplane of $\mathds{R}^{m}$ given by
$$\sum_{k=1}^{m}\alpha_{ab}(k) \theta_k= \theta_{ab}+\frac{\pi}{2}+ l\pi
$$
and define
$$
Z_{ab}=\displaystyle{\bigsqcup_{l\in \mathds{Z}} \mathcal{H}_{a,b}(l)}
$$
Then $\mathcal{H}_{a,b}=\pi(Z_{ab})$.\\ \indent
If $\mathcal{S}$ is a product of strict intervals, $\mathcal{S}$ is homeomorphic via $\mathds{R}^{m}\longrightarrow \mathds{T}$ to an open $U(\mathcal{S})\subset \mathds{R}^m$. Since $Z_{ab}=Z_{ab}+ 2\pi \mathds{Z}^m$ we have
$$
\mathcal{H}_{a,b}\cap \mathcal{S} = \pi(Z_{ab})\cap \mathcal{S}=\pi(Z_{ab}\cap U(\mathcal{S}))\simeq Z_{ab}\cap U(\mathcal{S})=\displaystyle{\bigsqcup_{l\in \mathds{Z}} \mathcal{H}_{a,b}(l)\cap U(\mathcal{S})}
$$
Hence, the connected components of $ \mathcal{H}_{a,b}\cap \mathcal{S}$ correspond via $U(\mathcal{S})\simeq \mathcal{S}$ to the non empty intersections between $U(\mathcal{S})$ and the $\mathcal{H}_{a,b}(l)$.
\subsection{The Stokes sheaf. Definition and local structure}\label{localstructure}
For $a, b \in \mathcal{I}$ and $\mathcal{S}$ any subset of $\mathds{T}$, we define following \cite{Mochizuki1} a partial order $<_\mathcal{S}$ on $\mathcal{I}$ as follows
\begin{equation}\label{order}
a<_\mathcal{S} b \text{ if and only if } F_{a,b}(x)<0 \text{ for all } x\in \mathcal{S}
\end{equation}
We use notations from \ref{notation}. For every $R\in \mathds{C}$-alg, we define $\St_{\mathcal{N}}(R)$ as the subsheaf of $R\otimes_{\mathds{C}} (j_{D\ast}\mathcal{H}^{0}\DR\End \mathcal{N})_{|\mathds{T}}$ of sections of the form $\Id+f$ where $p_ a f i_b = 0$ unless $a<_\mathcal{S} b$. \\ \indent
Suppose that $\mathcal{S}$ is contained in a product of strict open intervals. For $a\in \mathcal{I}$, the regular connection $\mathcal{R}_a$ admits in the basis $\mathbf{e}_a$ a fundamental matrix of flat sections $F_{a}$ on $\mathcal{S} $. Set $F:=\oplus_{a\in \mathcal{I}}F_a$ and $D:= \oplus_{a\in \mathcal{I}} a \Id$. For every $h\in (j_{D\ast}\mathcal{H}^{0}\DR\End \mathcal{N} )_{|\mathcal{S}}$, a standard computation shows that the derivatives of
$e^{-D} F^{-1} h F e^{D}$ are $0$. We thus have a well-defined isomorphism
\begin{equation}\label{inj}
\xymatrix{
(j_{D\ast}\mathcal{H}^{0}\DR\End \mathcal{N} )_{|\mathcal{S}}\ar[r]^-{\sim} & \underline{\End E}
}
\end{equation}
For every $\mathds{C}$-vector space $I$, we obtain a commutative diagram
\begin{equation}\label{tri}
\xymatrix{
\Gamma(\mathcal{S}, I\otimes_{\mathds{C}}j_{D\ast}\mathcal{H}^{0}\DR\End \mathcal{N})\ar[r]^-{\sim} & I\otimes_{\mathds{C}}\End E \\
I\otimes_{\mathds{C}}\Gamma(\mathcal{S}, j_{D\ast}\mathcal{H}^{0}\DR\End \mathcal{N}) \ar[ru]_-{\sim} \ar[u]
}
\end{equation}
Hence, the canonical morphism
\begin{equation}\label{sortleR}
I\otimes_{\mathds{C}}\Gamma(\mathcal{S}, j_{D\ast}\mathcal{H}^{0}\DR\End \mathcal{N}) \longrightarrow\Gamma(\mathcal{S}, I\otimes_{\mathds{C}}j_{D\ast}\mathcal{H}^{0}\DR\End \mathcal{N})
\end{equation}
is an isomorphism. Applying this to $I=R$, we see that \eqref{tri} identifies $\Gamma(\mathcal{S},\St_{\mathcal{N}}(R))$ with the space of $h\in R\otimes_{\mathds{C}}\End E$ such that $p_ a (h-\Id) i_b = 0$ unless $a<_\mathcal{S} b$. In particular, $\St_{\mathcal{N}}(R)$ is a sheaf of unipotent algebraic groups over $R$. \\ \indent
If $\mathcal{S}^{\prime}\subset \mathcal{S}$ are as above and if $R\longrightarrow S$ is a morphism of rings, the following diagram
$$
\xymatrix{
\Gamma(\mathcal{S} , \St_{\mathcal{N}}(R)) \ar[rr] \ar[ddd] \ar[dr] & & \Gamma(\mathcal{S}^{\prime}, \St_{\mathcal{N}}(R)) \ar[ddd] \ar[dl] \\
& R \otimes_{\mathds{C}} \End E \ar[d] & \\
& S \otimes_{\mathds{C}} \End E & \\
\Gamma(\mathcal{S} , \St_{\mathcal{N}}(S)) \ar[rr] \ar[ur] & & \Gamma(\mathcal{S}^{\prime}, \St_{\mathcal{N}}(S)) \ar[ul]
}
$$
commutes. Hence, horizontal arrows are injective. If $R\longrightarrow S$ is injective, vertical arrows are injective and we have further
\begin{equation}\label{uneegalite}
\Gamma(\mathcal{S} , \St_{\mathcal{N}}(R)) =\Gamma(\mathcal{S}^{\prime}, \St_{\mathcal{N}}(R)) \cap \Gamma(\mathcal{S} , \St_{\mathcal{N}}(S))
\end{equation}
\subsection{Restriction to curves}
Let $\iota : C\longrightarrow \mathds{C}^n$ be a germ of smooth curve passing through $0$ and non included in $D$. Let $\tilde{\iota}:\tilde{C}\longrightarrow \tilde{X}$ be the induced morphism at the level of the real blow-ups and. We still denote by $\tilde{\iota}:\partial\tilde{C}\longrightarrow \mathds{T}$ the induced morphism at the level of the boundaries. Note that $\tilde{\iota}$ is injective. We set $\iota^{\ast}\mathcal{I}:=\{a\circ \iota, a\in \mathcal{I}\}$. By goodness property of $\mathcal{I}$, restriction to $C$ induces a preserving order bijection $\mathcal{I}\longrightarrow \iota^{\ast}\mathcal{I}$, that is for every $x\in \partial\tilde{C}$ and $a, b\in \mathcal{I}$, we have $a<_{\tilde{\iota}(x)} b$ if and only if $a\circ \iota <_{x} b \circ \iota$. Hence, the canonical isomorphism
$$
\xymatrix{
\tilde{\iota}^{-1}(j_{D\ast}\mathcal{H}^{0}\DR\End \mathcal{N}) \ar[r]^-{\sim} & (j_{0\ast}\mathcal{H}^{0}\DR\End \mathcal{N}_{C} )_{|\partial \tilde{C}}
}
$$
deduced from Cauchy-Kowaleska theorem for flat connections induces for every $R\in \mathds{C}$-alg a canonical isomorphism
$$
\xymatrix{
\tilde{\iota}^{-1}\St_{\mathcal{N}}(R) \ar[r]^-{\sim} & \St_{\mathcal{N}_C}(R)
}
$$
compatible with \eqref{inj}.
\subsection{Preferred matricial representations in dimension 1}\label{prefered}
Let us now restrict to the dimension 1 case and let $d$ and $d^{\prime}$ be two consecutive Stokes line of $\mathcal{I}$. Let $a_1, \dots, a_k$ be the elements of $\mathcal{I}$ noted in increasing order for the total order $<_{]d, d^{\prime}[}$. In the basis $\mathbf{e}:=(\mathbf{e}_{a_1}, \dots, \mathbf{e}_{a_k})$, the morphism \eqref{inj} identifies $g\in \Gamma(]d, d^{\prime}[ , \St_{\mathcal{N}}(R))$ with the subgroup of $\GL_r(R)$ of upper-triangular matrices with only $1$ on the diagonal. Let $I$ be a strict open interval meeting $]d, d^{\prime}[$. For $i\in \llbracket 1, r \rrbracket$, let $j_i \in \llbracket 1, k \rrbracket$ such that $\mathbf{e}_i$ is an element of $\mathbf{e}_{a_{j_i}}$. Note that $j_i$ increases with $i$. We denote by $\Jump_{\mathcal{N}}(I)$ the set of $( i_1,i_2)$, $1 \leq i_1<i_2\leq r$, such that $j_{i_1}<j_{i_2}$ and $a_{j_{i_1}} \nless_I a_{j_{i_2}}$.
\subsection{Stokes torsors}
For $\mathcal{T} \in H^{1}(\mathds{T}, \St_{\mathcal{N}}(R))$ and a map of ring $\varphi: R\longrightarrow S$, we denote by $\mathcal{T}(S)$ the push-forward of $\mathcal{T}$ to $S$. Concretely, if $\mathcal{T}$ is given by a cocycle $(g_{ij})$, a cocycle for $\mathcal{T}(S)$ is $(\varphi(g_{ij}))$. There is a canonical morphism of sheaves $\mathcal{T}\longrightarrow \mathcal{T}(S)$ equivariant for the morphism of sheaves of groups $\St_{\mathcal{N}}(R)\longrightarrow \St_{\mathcal{N}}(S)$. If $t$ is a section of $\mathcal{T}$, we denote by $t(S)$ the associated section of $\mathcal{T}(S)$. \\ \indent
For $\mathcal{T} \in H^{1}(\mathds{T}, \St_{\mathcal{N}}(R))$, let
$T_{\mathcal{T}}H^{1}(\mathds{T}, \St_{\mathcal{N}})$
be the tangent space of $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ at $\mathcal{T}$. By definition, this is the set of $\mathcal{T}^{\prime} \in H^{1}(\mathds{T}, \St_{\mathcal{N}}(R[\epsilon]))$ such that
$\mathcal{T}^{\prime}(R)=\mathcal{T}$.
\subsection{Automorphisms of Stokes torsors}\label{auto}
In this subsection, we give a proof of the following
\begin{proposition}\label{autoId}
Stokes torsors have no non trivial isomorphisms.
\end{proposition}
\begin{proof}
Using restriction to curve, we are left to treat the one dimensional case. To simplify notations, we denote by $H^1$ the affine space representing the functor $H^{1}(S^{1}, \St_{\mathcal{N}})$ and by $\mathds{C}[H^1]$ its algebra of functions. Let $d_1, \dots , d_N$ be the Stokes lines of $\mathcal{N}$ indexed consecutively by $\mathds{Z}/N\mathds{Z}$. We denote by
$$\St_{\mathcal{N}, d_i}^{H^1}:=H^1\times_{\mathds{C}} \St_{\mathcal{N}, d_i}$$
the base change of the complex algebraic group $\St_{\mathcal{N}, d_i}$ to $H^1$. Let $\mathcal{T}^{\univ}_{\mathcal{N}}$ be the universal Stokes torsor given by Babbitt-Varadarajan representability theorem in dimension 1. For each $i\in \mathds{Z}/N\mathds{Z}$, let us choose a trivialisation of $\mathcal{T}^{\univ}_{\mathcal{N}}$ on the interval $]d_{i-1}, d_{i+1}[$ and let
$$g_{ii+1}^{\univ}\in \Gamma\left(]d_i , d_{i+1}[, \St_{\mathcal{N}}(\mathds{C}[H^1])\right)$$
be the associated cocycle. Let $G\longrightarrow H^1$ be the subgroup scheme of
$$
\prod_{i\in \mathds{Z}/N\mathds{Z}} \St_{\mathcal{N}, d_i}^{H^1}
$$
of $N$-uples $(h_1, \dots, h_N)$ satisfying
$$
h_i g_{ii+1}^{\univ}=g_{ii+1}^{\univ} h_{i+1}
$$
in $H^1\times_{\mathds{C}}\GL_r$. For $R\in \mathds{C}$-alg, $\mathcal{T}\in H^{1}(S^1, \St_{\mathcal{N}}(R))$ corresponds to a unique morphism of $\mathds{C}$-algebras $f: \mathds{C}[H^1]\longrightarrow R$, and a cocycle for $\mathcal{T}$ is given by applying $f$ to the $g_{ii+1}^{\univ}$. Hence, automorphisms of $\mathcal{T}$ are in bijection with $R$-points of $\Spec R \times_{H^1} G$. To prove \ref{autoId}, we are thus left to prove that $G$ is the trivial group scheme over $H^1$, that is, that the structural morphism of $G$ is an isomorphism.\\ \indent
As a complex algebraic group, $G$ is smooth. The affine scheme $H^1$ is smooth as well. So to prove that
$G\longrightarrow H^1$ is an isomorphism, it is enough to prove that it induces a bijection at the level of the underlying topological spaces. This can be checked above each point of $H^1$, that is to say after base change to a field $K$ of finite type over $\mathds{C}$. It is enough to show the bijectivity after base change to an algebraic closure $\overline{K}$ of $K$. Hence, we are left to prove Theorem \ref{autoId} for $R=\overline{K}$. By Lefschetz principle, we can suppose $R=\mathds{C}$. \\ \indent
Let $\mathcal{T}\in H^{1}(S^1, \St_{\mathcal{N}}(\mathds{C}))$ and let $(g_{ij})$ be a cocycle of $\mathcal{T}$ with respect to an open cover $(U_i)$. As already seen, an automorphism of $\mathcal{T}$ is equivalent to the data of sections $h_i\in \Gamma (U_i, \St_{\mathcal{N}}(\mathds{C}))$ satisfying
\begin{equation}\label{petiterelation}
h_i g_{ij}=g_{ij} h_j
\end{equation}
The cocycle $g$ defines an element of $H^{1}(S^1, \Id+ M_r( \mathcal{A}^{<0}))$. At the cost of refining the cover, Malgrange-Sibuya theorem \cite[I 4.2.1]{BV} asserts the existence of sections $x_i\in \Gamma(U_i, \GL_r(\mathcal{A}))$ such that $x_i x_j^{-1}=g_{ij}$ on $U_{ij}$. Since
$$
x_i^{-1} h_i x_i=x_j^{-1}(g_{ij}^{-1}h_i g_{ij}) x_j = x_j^{-1} h_j x_j
$$
the $x_i^{-1} h_i x_i$ glue into a global section of $\Id+ M_r( \mathcal{A}^{<0})$. Since $\mathcal{A}^{<0}$ has no non zero global section, we deduce $h_i =\Id$ for every $i$ and \ref{autoId} is proved.
\end{proof}
\subsection{$\mathcal{I}$-good open sets}\label{Igoodopen}
A $\mathcal{I}$-good open set at $x\in\mathds{T}$
is a product $\mathcal{S}$ of strict intervals containing $x$ and such that
$$
\left\{
\begin{array}{ll}
\mathcal{S}\cap \mathcal{H}_{a,b}=\emptyset & \text{ if } x\notin \mathcal{H}_{a,b}\\
\mathcal{S}\cap \mathcal{H}_{a,b} \text { is connected } & \text{ if } x\in \mathcal{H}_{a,b}
\end{array}
\right.
$$
Every $x\in\mathds{T}$ admits a fundamental system of neighbourhoods which are $\mathcal{I}$-good open sets.
\begin{lemme}\label{crittri}
For every $R\in \mathds{C}$-alg and every $\St_{\mathcal{N}}(R)$-torsors $\mathcal{T}$, the restriction of $\mathcal{T}$ to a $\mathcal{I}$-good open set $\mathcal{S}$ at $x\in \mathds{T}$ is trivial.
\end{lemme}
\begin{proof}
Let $\mathcal{H}_{a_1, b_1}, \dots, \mathcal{H}_{a_k, b_k}$ be the Stokes hyperplanes passing through $x$. For $I\subset \llbracket 1, k \rrbracket$ non empty, we set
$$
\mathcal{H}_{I}:=\bigcap_{i\in I}(\mathcal{H}_{a_i, b_i}\cap \mathcal{S})
$$
Finally, we set $\mathcal{H}_{\emptyset}=\mathcal{S}$. Let us choose $t\in \mathcal{T}_x$. We argue by decreasing recursion on $d$ that $t$ extends to
$$
\mathcal{H}(d):= \bigcup_{I\subset \llbracket 1, k \rrbracket, |I|=d}\mathcal{H}_{I}
$$
We know by $\mathcal{I}$-goodness that $\mathcal{H}(k)$ is homeomorphic to a $\mathds{R}$-vector space. Since the order \eqref{order} is constant on $\mathcal{H}(k)$, the sheaf of group $\St_{\mathcal{N}}(R)$ is constant on $\mathcal{H}(k)$, the section $t$ extends uniquely to $\mathcal{H}(k)$. We now suppose $d<k$ and assume that $t$ extends to $\mathcal{H}(d+1)$ into a section that we still denote by $t$. If we set
$$
\mathcal{H}_{I^+} :=\displaystyle{\bigcup_{i \in \llbracket 1, k \rrbracket \setminus I}} \mathcal{H}_{I\cup \{i\}} \text{ and } \mathcal{H}_{I}^- := \mathcal{H}_{I}\setminus \mathcal{H}_{I^+}
$$
We have set theoretically
$$
\mathcal{H}(d)=\mathcal{H}(d+1)\bigsqcup \bigsqcup_{\underset{|I|=d}{I\subset \llbracket 1, k \rrbracket }}\mathcal{H}_{I}^-
$$
Hence, we have to extend $t$ to each $\mathcal{H}_{I}^-$ in a compatible way. Let $I\subset \llbracket 1, k \rrbracket$ of cardinal $d$. By admissibility, the data of the $\mathcal{H}_{I\cup \{i\}}$, $i\in \llbracket 1, k \rrbracket\setminus I$ inside $\mathcal{H}_{I}$ is topologically equivalent to that of a finite number of hyperplanes in a $\mathds{R}$-vector space. Hence, a connected component
$F_I\in \pi_0(\mathcal{H}_{I}^-)$ is contractible and
$\mathcal{H}_{I^+}\subset \mathcal{H}(d+1)$
admits an open neighbourhood $U$ whose trace on $F_I$ is connected. Since the order \eqref{order} is constant on $F_I$, the sheaf of group $\St_{\mathcal{N}}(R)$ is constant on $F_I$. Hence, $t_{|U\cap F_I}$ extends uniquely to a section $t_{F_I}$ on $F_I$. If $F_I^{\prime}\in \pi_0(\mathcal{H}_{I}^-)$ is distinct from $F_I$,
$$
\overline{F_I}\cap \overline{F_{I}^{\prime}}\subset \mathcal{H}_{I^+}
$$
hence $t_{| \mathcal{H}_{I^+}} $ and the $t_{F_I}$ glue into a section $t_I$ of $\mathcal{T}$ on $\mathcal{H}_I$. For
$I^{\prime}\subset \llbracket 1, k \rrbracket$ distinct from $I^{\prime}$ and of cardinal $d$, and $F_{I^{\prime}}\in \pi_0(\mathcal{H}_{I^{\prime}}^-)$,
we have
$$
\overline{F_I}\cap \overline{F_{I^{\prime}}}\subset \mathcal{H}_{I}\cap \mathcal{H}_{I^{\prime}}\subset \mathcal{H}(d+1)
$$
Hence $t$ and the $t_{I}$ and glue into a section of $\mathcal{T}$ over $\mathcal{H}(d)$ and \ref{crittri} is proved.
\end{proof}
\section{Skeletons and Stokes torsors}\label{partskeleton}
Let $X$ be a smooth real manifold. Let $\mathscr{C}$ be a set of closed curves in $X$, let $\Sh_X$ be the category of sheaves on $X$, set
$$\Int \mathscr{C}:=\displaystyle{\bigcup_{C, C^{\prime}\in \mathscr{C}}} C \cap C^{\prime} $$
and for every $x\in \Int \mathscr{C}$, set
$\mathscr{C}(x):=\{C\in \mathscr{C} \text{ with } x\in C\}$.
\subsection{Definition}\label{defskeleton}
We define the category of $\mathscr{C}$-skeleton $\Sk_{X}(\mathscr{C})$ on $X$ as the category whose objects are systems $(\mathcal{F}_\mathscr{C}, \iota_\mathscr{C})$ where $\mathcal{F}_\mathscr{C}=(\mathcal{F}_C)_{C\in \mathscr{C}}$
is a family of sheaves on the curves $C\in \mathscr{C}$, and where $\iota_\mathscr{C}$ is a collection of identifications
$$
\xymatrix{\iota_{C, C^{\prime}}(x): \mathcal{F}_{C, x}\ar[r]^-{\sim} & \mathcal{F}_{C^{\prime}, x}}
$$
for $x\in \Int \mathscr{C}$ and $C, C^{\prime}\in \mathscr{C}(x)$, satisfying
\begin{align*}
\iota_{C, C}(x)& =\id \\
\iota_{ C, C^{\prime}}(x) & = \iota_{ C^{\prime}, C}^{-1}(x) \\
\iota_{ C, C^{\prime \prime}}(x) & = \iota_{ C^{\prime}, C^{\prime \prime}} \iota_{ C, C^{\prime }}(x)
\end{align*}
A morphism $(\mathcal{F}_\mathscr{C}, \iota_\mathscr{C})\longrightarrow (\mathcal{G}_\mathscr{C}, \kappa_\mathscr{C})$ in $\Sk_{X}(\mathscr{C})$ is a collection of morphisms of sheaves $f_C :\mathcal{F}_C\longrightarrow \mathcal{G}_C$ such that the following diagrams commute for every $x\in \Int \mathscr{C}$ and $C, C^{\prime}\in \mathscr{C}(x)$
$$
\xymatrix{
\mathcal{F}_{C, x}\ar[r]^-{\sim} \ar[d]_-{f_{C,x}} & \mathcal{F}_{C^{\prime}, x} \ar[d]^-{f_{C^{\prime},x}} \\
\mathcal{G}_{C, x}\ar[r]^-{\sim} & \mathcal{G}_{C^{\prime}, x}
}
$$
Restriction induces a functor $\sk_\mathscr{C} : \Sh_X \longrightarrow \Sk_{X}(\mathscr{C})$ called the \textit{$\mathscr{C}$-skeleton functor}.
\subsection{Coskeleton}
We now suppose that $\mathscr{C}$ covers $X$, that is every $x\in X$ belongs to at least one $C\in \mathscr{C}$. Take $(\mathcal{F}_\mathscr{C}, \iota_\mathscr{C})\in \Sk_{X}(\mathscr{C})$. The set
$$
E(\mathcal{F}_\mathscr{C}):=\displaystyle{\bigsqcup_{C\in \mathscr{C}, x\in C}}\mathcal{F}_{C,x}
$$
is endowed with the equivalence relation
$$
(C, x, s\in \mathcal{F}_{C,x})\sim (C^{\prime}, x, s^{\prime}\in \mathcal{F}_{C^{\prime},x}) \text{ if and only if } s^{\prime}=\iota_{C, C^{\prime}}(x)(s)
$$
Let $E(\mathcal{F}_\mathscr{C}, \iota_{\mathscr{C}})$ be the quotient of $E(\mathcal{F}_\mathscr{C})$ by this relation.
The surjection $E(\mathcal{F}_\mathscr{C})\longrightarrow X$ induces a surjection $p: E(\mathcal{F}_\mathscr{C}, \iota_{\mathscr{C}}) \longrightarrow X$. Let
$\cosk_{\mathscr{C}}(\mathcal{F}_\mathscr{C}, \iota_{\mathscr{C}})$ be the functor associating to every open $U$ in $X$ the set of sections $s$ of $p$ over $U$ such that for every $C\in \mathscr{C}$, there exists $s_C \in \Gamma(U\cap C, \mathcal{F}_C)$ satisfying for every $x\in U\cap C$
$$
s(x)=(C,x, s_C(x)) \text{ in } E(\mathcal{F}_\mathscr{C}, \iota_{\mathscr{C}})
$$
Equivalently, if $s(C,x)$ denotes the unique representative of
$s(x)$ associated to $(C,x)$, the above equation means $s(C,x)=s_C(x)$ in $\mathcal{F}_{C,x}$.\\ \indent
The functor $\cosk_{\mathscr{C}}(\mathcal{F}_\mathscr{C}, \iota_{\mathscr{C}})$ is trivially a presheaf on $X$ and one checks that it is sheaf. We thus have a well-defined functor $\cosk_{\mathscr{C}} : \Sk_{X}(\mathscr{C}) \longrightarrow \Sh_X$ called the \textit{$\mathscr{C}$-coskeleton functor}.
\begin{lemme}\label{adjoint}
The functor $\cosk_{\mathscr{C}}$ is right adjoint to $\sk_{\mathscr{C}}$.
\end{lemme}
\begin{proof}
Let $\mathcal{F}\in \Sh_X$ and let $(\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}})\in \Sk_{X}(\mathscr{C})$.
We have to define a natural bijection
$$
\xymatrix{
\Hom_{\Sk_{X}(\mathscr{C}) }(\sk_{\mathscr{C}}\mathcal{F}, (\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}}))\ar[r] &
\Hom_{\Sh_{X}}(\mathcal{F}, \cosk_{\mathscr{C}}(\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}}))
}
$$
A morphism $f:= (f_C :\mathcal{F}_{|C}\longrightarrow \mathcal{G}_C)_{C\in \mathscr{C}}$ in the left-hand side induces a well-defined map
$$
\xymatrix{
E(\sk_{\mathscr{C}}\mathcal{F}) \ar[r] & E(\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}})
}
$$
associating to the class of $(C, x, s\in \mathcal{F}_x)$ the class of $(C, x, f_{C,x}(s)\in \mathcal{G}_{C,x})$.
Let $U$ be an open in $X$. A section $s\in \Gamma(U, \mathcal{F})$ induces a section to $E(\sk_{\mathscr{C}}\mathcal{F}) \longrightarrow X$ over $U$, from which we deduce a section $\adj(f)(s)$ of $E(\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}})\longrightarrow X$ over $U$. For $C\in \mathscr{C}$, the section $\adj(f)(s)$ is induced on $U\cap C$ by $f_C(s_{|U\cap C})\in \Gamma(U\cap C, \mathcal{G}_C)$. Hence, $\adj(f)(s)\in \Gamma(U, \cosk_{\mathscr{C}}(\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}}))$. We have thus constructed a morphism of sheaves
$$
\adj(f) : \mathcal{F}\longrightarrow \cosk_{\mathscr{C}}(\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}})
$$
such that the following diagram commutes
$$
\xymatrix{
\mathcal{F}_{|C} \ar[rr]^-{\adj(f)_{|C}} \ar[rrd]_-{f_C} & & \cosk_{\mathscr{C}}(\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}})_{|C} \ar[d] \\
& & \mathcal{G}_C
}
$$
where the vertical morphism sends a germ of section at $x\in C$ to its unique representative in $\mathcal{G}_{C, x}$. Thus, $\adj : \Hom_{\Sk_{X}(\mathscr{C}) }(\sk_{\mathscr{C}}\mathcal{F}, (\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}}))\longrightarrow
\Hom_{\Sh_{X}}(\mathcal{F}, \cosk_{\mathscr{C}}(\mathcal{G}_\mathscr{C}, \iota_{\mathscr{C}}))$ is well-defined and injective.
It is a routine check to see that $\adj$ is surjective.
\end{proof}
\subsection{Torsor and skeleton}\label{torsorandske}
Let $\mathcal{G}$ be a sheaf of groups on $X$. The canonical morphism
\begin{equation}\label{caninjectG}
\mathcal{G}\longrightarrow \cosk_{\mathscr{C}} \sk_{\mathscr{C}} \mathcal{G}
\end{equation}
is injective and we suppose from now on that it is also surjective.
\begin{definition}
A $\mathcal{G}$-skeleton torsor is the data of an object $(\mathcal{F}_\mathscr{C}, \iota_\mathscr{C})\in \Sk_{X}(\mathscr{C})$ such that for every $C\in \mathscr{C}$, the sheaf $\mathcal{F}_C$ is a $\mathcal{G}_{|C}$-torsor such that for every $C, C^{\prime}\in \mathscr{C}$ and for every $x\in C \cap C^{\prime}$, the following diagram commutes
\begin{equation}\label{compatcarré}
\xymatrixcolsep{6pc}\xymatrix{
\mathcal{G}_{|C, x}\times \mathcal{F}_{C, x}\ar[d] \ar[r]^-{\iota_{C, C^{\prime}, \mathcal{G}}(x) \times\iota_{C, C^{\prime}}(x)} & \mathcal{G}_{|C^{\prime}, x}\times \mathcal{F}_{C^{\prime}, x} \ar[d] \\
\mathcal{F}_{C, x} \ar[r]_-{\iota_{C, C^{\prime}}(x)} & \mathcal{F}_{C^{\prime}, x}
}
\end{equation}
where $\iota_{C, C^{\prime}, \mathcal{G}}(x)$ is the composite of the canonical morphisms $\mathcal{G}_{|C, x}\overset{\sim}{\longrightarrow} \mathcal{G}_{ x}\overset{\sim}{\longleftarrow}\mathcal{G}_{|C^{\prime}, x}$. We denote by $\mathcal{G}$-$\Sk_{\mathscr{C}}$ the category of $\mathcal{G}$-skeleton torsors on $X$ with respect to $\mathscr{C}$.
\end{definition}
Let $(\mathcal{F}_\mathscr{C}, \iota_\mathscr{C})$ be a $\mathcal{G}$-skeleton torsors. The morphism \eqref{caninjectG} and the compatibilities \eqref{compatcarré} show that $\cosk_\mathscr{C}(\mathcal{F}_\mathscr{C}, \iota_\mathscr{C})$ is endowed
with an action of $\mathcal{G}$. Let $U$ be an open of $X$ and let $s, t\in \Gamma(U, \cosk_\mathscr{C}(\mathcal{F}_\mathscr{C}, \iota_\mathscr{C}))$. For every $C\in \mathscr{C}$ meeting $U$, the sections $s$ and $t$ are induced on $C$ by $s_C, t_C\in \Gamma(U\cap C, \mathcal{F}_C)$. Since $\mathcal{F}_C$ is a $\mathcal{G}_{|C}$-torsor, there exists a unique $g_C\in \Gamma(U\cap C,\mathcal{G})$ such that $t_C= g_C s_C$. From \eqref{compatcarré}, we see that the $(g_C)_{C\in \mathscr{C}}$ define a section of $\cosk_\mathscr{C}\mathcal{G}$ over $U$. Since \eqref{caninjectG} is supposed to be an isomorphism, we deduce $t=gs$ for a unique $g\in \Gamma(U,\mathcal{G})$. Hence, $\cosk_\mathscr{C}(\mathcal{F}_\mathscr{C}, \iota_\mathscr{C})$ is a \textit{pseudo-torsor}. It may not be a torsor in general.
\subsection{Stokes Torsors and skeletons}\label{STandSk}
For $\theta=(\theta_{1}, \dots, \theta_{m})\in [0, 2\pi [^m$ and $\nu=(\nu_1, \dots, \nu_m)\in (\mathds{N^{\ast}})^{m}$, we define $C_{\theta, \nu}$ as the curve of $\mathds{C}^n$ defined by
$$
t\longrightarrow (e^{i\theta_1} t^{\nu_1}, \dots, e^{i\theta_m} t^{\nu_m}, 0)
$$
It gives rise to a curve $\partial \tilde{C}_{\theta, \nu}\simeq S^1$ on $\mathds{T}$ explicitly given by
$$
\theta\longrightarrow (\theta_{1}+ \nu_1 \theta, \dots ,\theta_{m}+ \nu_m \theta)
$$
From this point on, we apply the previous formalism to
$$
\left\{
\begin{array}{ll}
X=\mathds{T} \\
\mathscr{C}=\{\partial \tilde{C}_{\theta, \nu},\theta\in [0, 2\pi [^m, \nu\in (\mathds{N^{\ast}})^{m} \} \\
\mathcal{G}=\St_{\mathcal{N}}(R)
\end{array}
\right.
$$
where $R\in \mathds{C}$-alg.
\begin{remarque}
Proposition \ref{autoId} shows that the category $\St_{\mathcal{N}}(R)\text{-}\Sk_{\mathscr{C}}$ is a setoïd, that is a groupoïd whose objects have exactly one automorphism. We still denote by $\St_{\mathcal{N}}(R)\text{-}\Sk_{\mathscr{C}}$ the set of isomorphism classes of objects in this category.
\end{remarque}
As explained in \ref{torsorandske}, the coskeleton of a $\St_{\mathcal{N}}(R)$-skeleton torsor is a $\St_{\mathcal{N}}(R)$-pseudo torsor due to the following
\begin{lemme}
The canonical morphism
\begin{equation}\label{canmorphism}
\St_{\mathcal{N}}(R)\longrightarrow \cosk_{\mathscr{C}} \sk_{\mathscr{C}} \St_{\mathcal{N}}(R)
\end{equation}
is an isomorphism.
\end{lemme}
\begin{proof}
It is enough to show surjectivity over an $\mathcal{I}$-good open set $\mathcal{S}$ for $x\in \mathds{T}$. As explained in \ref{localstructure}, a choice of fundamental matrix for the $\mathcal{R}_a, a\in \mathcal{I}$ on $\mathcal{S}$ induces a commutative diagram
$$
\xymatrix{
\Gamma(\mathcal{S},\St_{\mathcal{N}}(R) ) \ar[d] \ar[r] & \bigsqcup_{x\in \mathcal{S}} \St_{\mathcal{N}}(R)_x \ar[d]& \Gamma(\mathcal{S},\cosk_\mathscr{C} \sk_\mathscr{C}\St_{\mathcal{N}}(R) ) \ar[l] \ar[ld] \\
\GL_r(R) \ar[r] & \GL_r(R)^{\mathcal{S}}&
}
$$
with injective arrows and where the bottom arrow associates to $g\in \GL_r(R)$ the function constant to $g$ on $\mathds{T}$. Hence, we have to show that the function $\mathcal{S}\longrightarrow \GL_r(R)$ induced by a section of $\cosk_\mathscr{C} \sk_\mathscr{C}\St_{\mathcal{N}}(R)$ over $\mathcal{S}$ is constant. Since it is constant over every admissible curve and since two points in $\mathcal{S}$ can always been connected by intervals lying on admissible curves, we are done.
\end{proof}
Let $\mathcal{S}$ be an $\mathcal{I}$-good open set of $\mathds{T}$.
A $\mathscr{C}$-polygon of $\mathcal{S}$ is the image by the canonical surjection $\pi:\mathds{R}^{m}\longrightarrow \mathds{T}$ of a convex polygon $P\subset \mathds{R}^{m}$ with at least 3 edges such that $\pi(P)\subset \mathcal{S}$ and the image of an edge $E$ lies on a curve $C(E)\in \mathscr{C}$. We introduce the following
\begin{definition}\label{admissible}
A $\St_{\mathcal{N}}(R)$-skeleton torsor $(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} })\in \St_{\mathcal{N}}(R)\text{-}\Sk_{\mathscr{C}}$ is said to be \textit{admissible} if for every $\mathcal{I}$-good open set $\mathcal{S}$, every $\mathscr{C}$-polygon
$\pi : P\longrightarrow \mathcal{S}$ with edge $E_i=[x_i, x_{i+1}]$, $i\in \mathds{Z}/N\mathds{Z}$, every $t_i\in \Gamma(\pi(E_i), \mathcal{T}_{C(E_i)})$, $i=1, \dots, N-1$ such that
$$
t_i(\pi(x_{i+1}))=t_{i+1}(\pi(x_{i+1})) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
there exists a (necessarily unique) $t_N\in \Gamma(\pi(E_N), \mathcal{T}_{C(E_N)})$ satisfying
$$
t_N(\pi(x_N))=t_{N-1}(\pi(x_N)) \text{ and } t_N(\pi(x_1))=t_{1}(\pi(x_1)) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
\end{definition}
Again, the category $\St_{\mathcal{N}}(R)\text{-}\AdSk_{\mathscr{C}}$ is a setoïd. We still denote by $\St_{\mathcal{N}}(R)\text{-}\AdSk_{\mathscr{C}}$ the set of isomorphism classes of objects in this category.
\subsection{Proof of Theorem \ref{equcat}}
If we prove that the skeleton of a Stokes-torsor is admissible, and that the coskeleton of an admissible Stokes-skeleton is a torsor, we are done since the adjunction maps provided by \ref{adjoint} will automatically be isomorphisms. \\ \indent
Let $\mathcal{T}\in H^{1}(\mathds{T}, \St_{\mathcal{N}}(R))$. Let $\mathcal{S}$ be an $\mathcal{I}$-good open set, let $\pi : P\longrightarrow \mathcal{S}$ be a $\mathscr{C}$-polygon, and let $t$ be a section of $\mathcal{T}$ on $\pi(P\setminus ]x_N, x_1[)$. From \ref{crittri}, $\mathcal{T}$ admits a section $s$ on $\mathcal{S}$. Hence, there exists a section $g$ of $ \St_{\mathcal{N}}(R)$ over $\pi(P\setminus ]x_N, x_1[)$ such that $t=g s$ and we are left to prove that $g$ extends to $\pi(P)$. For $a,b\in \mathcal{I}$ with $ \mathcal{H}_{a,b}$ meeting $\mathcal{S}$, we have to show that
$$
a<_{\pi(P\setminus ]x_N, x_1[)}b \Longrightarrow a <_{\pi(]x_N, x_1[)} b
$$
By $\mathcal{I}$-goodness, $\mathcal{S}\setminus (\mathcal{S}\cap \mathcal{H}_{a,b})$ has only two connected components $C^{\pm}_{ab}$. They are convex. If $\pi(P\setminus ]x_N, x_1[) \subset C^{-}_{ab}$, so does the segment $\pi( [x_N, x_1])$ by convexity and we are done.\\ \indent
Let $(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} })\in \St_{\mathcal{N}}(R)\text{-}\AdSk_{\mathscr{C}}$ and let us prove that $\cosk_{\mathscr{C}}(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} })$ is a torsor under $\St_{\mathcal{N}}(R)$.
For $x\in \mathds{T}$, we have to prove that the germ of $\cosk_{\mathscr{C}}(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} })$ at $x$ is not empty. Let
$$
\mathcal{S}(x, \epsilon):=\prod_{i=1}^{m} ]x_i-\epsilon, x_i + \epsilon [
$$
be an $\mathcal{I}$-good open set at $x$, choose $s\in E(\mathcal{T}_{\mathscr{C}})$ above $x$.
Set
$$
\mathcal{S}(x, \epsilon)^- :=\prod_{i=1}^{m} ]x_i-\epsilon, x_i [\text{ and } \mathcal{S}(x, \epsilon)^+ :=\prod_{i=1}^{m} ]x_i, x_i + \epsilon [
$$
We are going to "transport" the germ $s$ in two steps to the open $\mathcal{S}(x, \epsilon/2)$. We first transport it to $\mathcal{S}(x, \epsilon)^+$ and $\mathcal{S}(x, \epsilon)^-$.\\ \indent
For $C\in \mathscr{C}(x)$, let $C(x)$ be the connected component of $C\cap \mathcal{S}(x, \epsilon)$ containing $x$. By admissibility, the set $\mathcal{H}_{a,b}\cap C(x)$ is either empty or reduced to $\{x\}$. Hence, the restriction of $\mathcal{T}_C$ to $C(x)$ contains at most one Stokes line, which is $\{x\}$ if there is one. Lemma \ref{crittri} thus shows that the unique representative of $s$ in $\mathcal{T}_{C, x}$ extends uniquely to a section $s_{C(x)}\in \Gamma(C(x), \mathcal{T}_C)$. Two distinct $C(x)$ and $C^{\prime}(x)$ meet only at $x$, so we have a well-defined section of $E(\mathcal{T}_{\mathscr{C}}, \iota_{\mathscr{C}})\longrightarrow \mathds{T}$ above
$$
\mathcal{S}(x, \epsilon)^+ \times \{x\}\times \mathcal{S}(x, \epsilon)^-
$$
that we still denote by $s$, noting $s(y)$ its value at a point $y$.
We now extend $s$ to $\mathcal{S}(x, \epsilon/2)$ using the admissibility condition. Let $y \in \mathcal{S}(x, \epsilon/2)$. If $\mathbf{1}$ denotes the $m$-uple $(1, \dots, 1)$, we can choose $y_{\pm}\in \partial \tilde{C}_{y,\mathbf{1}}\cap \mathcal{S}(x, \epsilon)^{\pm}$.
Admissibility applied to the triangle $y_-x y_+$ and the sections $s_{[y_- , x]}, s_{[x , y_+]}$ gives a unique $t_{ y_-, y_+}\in \Gamma([y_- , y_+], \mathcal{T}_{\partial\tilde{C}_{y,\mathbf{1}}})$ such that
$$
t_{ y_-, y_+}(y_-)=s(y_-) \text{ and } t_{ y_-, y_+}(y_+)=s(y_+) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
If $y_+^{\prime} \in \partial \tilde{C}_{y,\mathbf{1}}\cap \mathcal{S}(x, \epsilon)^{+}$ is another choice, the sections $t_{ y_-, y_+}$ and $t_{ y_-, y_+^{\prime}}$
coincide at $y_-$. Hence they coincide on $[y_{-}, y_+] \cap [y_{-}, y_+^{\prime}]$. Arguing similarly with $y_-$, we deduce that $t_{ y_-, y_+}$ does not depend on the choice of $y_{+}$ and $y_-$. We have thus constructed a section
$$
t_y\in \Gamma(\partial \tilde{C}_{y,\mathbf{1}}\cap \mathcal{S}(x, \epsilon), \mathcal{T}_{\partial\tilde{C}_{y,\mathbf{1}}})
$$
We now show that $y\longrightarrow t_y(y)$ defines a section of
$\cosk_{\mathscr{C}}(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} })$ over $\mathcal{S}(x, \epsilon/2)$.
Let $C\in \mathscr{C}$ and let $y_0\in C\cap \mathcal{S}(x, \epsilon/2)$. It is enough to show that $y\longrightarrow t_y(y)$ is induced by a section of $\mathcal{T}_C$ on a small enough interval of $C$ contained in $\mathcal{S}(x, \epsilon/2)$ and containing $y_0$. One can choose such a non trivial interval $[y_1, y_2]$ in a way that it admits a translate $[y_{1+}, y_{2+}]$ contained in $\mathcal{S}(x, \epsilon/2)^+$ with $y_i^+\in \partial\tilde{C}_{y_i,\mathbf{1}}$ for $i=1, 2$. The situation can be depicted as follows:
In particular, $[y_{1+}, y_{2+}]$ is an interval of a translate
$C^+$ of $C$. Since $\mathscr{C}$ is stable by
translation, we have $C^+ \in \mathscr{C}$. Admissibility
applied to the triangle $y_{1+}x y_{2+}$ and the sections
$s_{[y_{1+}, x]}, s_{[y_{2+}, x]}$ gives a unique $s_{y_{1+},
y_{2+}}\in \Gamma([y_{1+}, y_{2+}], \mathcal{T}_{C^+})$ such
that
$$
s_{y_{1+},
y_{2+}}(y_{1+})=s(y_{1+}) \text{ and } s_{y_{2+},
y_{2+}}(y_{2+})=s(y_{2+}) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
Note that $s_{y_{1+},
y_{2+}}(y)=s(y)$ for every $y\in [y_{1+},
y_{2+}]$. Indeed, we know by admissibility applied to $yxy_{2+}$ that there exists a unique section $s_{y,y_{2+}}\in \Gamma([y_1, y_{2+}], \mathcal{T}_{C^+})$ such that
$$
s_{y,
y_{2+}}(y)=s(y) \text{ and } s_{y,
y_{2+}}(y_{2+})=s(y_{2+}) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
Since $s_{y_{1+},
y_{2+}}$ and $s_{y, y_{2+}}$ coincide at $y_{2+}$, they are equal on $[y,y_{2+}]$, so
$$
s_{y_{1+},
y_{2+}}(y)=s_{y,
y_{2+}}(y)=s(y) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
By admissibility applied to the parallelogram $y_1 y_{1+}y_{2+}y_2$ and the sections $t_{y_1|[y_1, y_{1+}]}$, $s_{y_{1+}, y_{2+}}$ and $t_{y_2|[y_2, y_{2+}]}$, there exists a unique section $t\in \Gamma([y_1, y_2], \mathcal{T}_C)$ such that
$$
t(y_1)=t_{y_1}(y_1) \text{ and } t(y_2)=t_{y_2}(y_2) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
We are left to show that $t$ coincides with $y\longrightarrow t_y(y)$ on $[y_1, y_2]$. Let $y\in [y_1, y_2]$. The line $\partial\tilde{C}_{y,\mathbf{1}}$ and the segment $[y_{1+}, y_{2+}]$ meet at a point $y_+$. Admissibility applied to the parallelogram $yy_2y_{2+}y_+$ and the sections $t_{y|[y, y_+]}, t_{y_2|[y_2, y_{2+}]}, s_{y_{1+}, y_{2+}|[y_{+}, y_{2+}]}$ gives a section $t^{\prime}\in \Gamma([y, y_2], \mathcal{T}_C)$ such that
$$
t^{\prime}(y)=t_{y}(y) \text{ and } t^{\prime}(y_2)=t_{y_2}(y_2) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
Since $t^{\prime}$ and $t$ are two sections of $\mathcal{T}_{C}$ coinciding at $y_{2}$, they are equal on $[y, y_2]$. Hence
$$
t(y)=t^{\prime}(y)=t_{y}(y) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
From the construction of $y\longrightarrow t_y(y)$ out of $s$, we deduce that
\begin{equation}\label{is}
\xymatrix{
\Gamma(\mathcal{S}(x, \epsilon/2), \cosk_{\mathscr{C}}(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} }))\ar[r] & E(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} })_x
}
\end{equation}
is surjective. In particular, the left-hand side of \eqref{is} is not empty. Since the sheaf $\cosk_{\mathscr{C}}(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} })$ is a pseudo-torsor, \eqref{is} is also injective. Taking the colimit over $\epsilon$ gives an identification
$$
\xymatrix{
\cosk_{\mathscr{C}}(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} })_x\ar[r]^-{\sim} & E(\mathcal{T_{\mathscr{C}},\iota_{\mathscr{C}} })_x
}
$$
and Theorem \ref{equcat} is proved. \\ \indent
As a corollary of Theorem \ref{equcat}, we see that admissibility is stable under base change. This is not clear a priori if one considers only a subset of $\mathscr{C}$. Hence, the assignment $R\longrightarrow \St_{\mathcal{N}}(R)\text{-}\AdSk_{\mathscr{C}}$ is a well-defined functor.
As a direct consequence of \ref{crittri}, we have the following
\begin{corollaire}\label{trivialsurI}
Let $(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})\in \St_{\mathcal{N}}(R)\text{-}\AdSk_{\mathscr{C}}$. For every $C\in \mathscr{C}$ and every interval $I\subset C$ contained in an $\mathcal{I}$-good open set, the torsor $\mathcal{T}_C$ is trivial on $I$.
\end{corollaire}
\subsection{A monomorphism into an affine scheme}\label{mono}
One can find a finite family $\mathscr{C}(\mathcal{I})\subset \mathscr{C}$ for which there exists a finite family $\mathscr{P}$ of closed parallelotops covering $\mathds{T}$ such that for every $\mathcal{P}\in \mathscr{P}$, the following holds
\begin{enumerate}
\item $\mathcal{P}$ is contained in an open which is $\mathcal{I}$-good for its center $x_\mathcal{P}$.
\item every edge of $\mathcal{P}$ is contained in a curve of $\mathscr{C}(\mathcal{I})$.
\item for $\mathcal{P}_1, \mathcal{P}_2 \in \mathscr{P}$, there exists $x_{12}\in \mathcal{P}_1\cap \mathcal{P}_2$, $x_1 \in \mathcal{P}_1$ and $x_2\in \mathcal{P}_2$ such that $[x_{\mathcal{P}_i}, x_i]$ and $[x_i, x_{12}]$, $i=1, 2$ lie on curves of $\mathscr{C}(\mathcal{I})$.
\end{enumerate}
Condition $(3)$ will be used in section \ref{prooftheorem} only.
\begin{lemme}\label{famillespecial}
For every $R\in \mathds{C}$-alg, the map
\begin{equation}\label{injfamillespeciale}
\xymatrix{
H^{1}(\mathds{T}, \St_{\mathcal{N}}(R)) \ar[r] & \St_{\mathcal{N}}(R)\text{-}\Sk_{\mathscr{C}(\mathcal{I})}
}
\end{equation}
is injective.
\end{lemme}
\begin{proof}
We set
$$
\mathcal{C}(\mathcal{I}):=\bigcup_{C\in \mathscr{C}(\mathcal{I})} C
$$
Let $\mathcal{T}_1, \mathcal{T}_2\in H^{1}(\mathds{T}, \St_{\mathcal{N}}(R))$ such that $\sk_{\mathscr{C}(\mathcal{I})}(\mathcal{T}_1)=\sk_{\mathscr{C}(\mathcal{I})}(\mathcal{T}_2)$ and let
$$
f:\mathcal{T}_{1| \mathcal{C}(\mathcal{I})}\longrightarrow \mathcal{T}_{2|\mathcal{C}(\mathcal{I})}
$$
be the induced isomorphism. Let $\mathcal{P} \in \mathscr{P}$. From \ref{crittri}, we can choose sections $t_i \in \Gamma(\mathcal{P},\mathcal{T}_i )$. Let $g\in \Gamma(\mathcal{P}\cap \mathcal{C}(\mathcal{I}),\St_{\mathcal{N}}(R))$ such that $f(t_1)= g t_2$ on $\mathcal{P}\cap \mathcal{C}(\mathcal{I})$. If a Stokes hyperplan $\mathcal{H}_{ab}$ meets $\mathcal{P}$, then it meets an edge of $\mathcal{P}$, so it meets $\mathcal{P}\cap \mathscr{C}(\mathcal{I})$. Hence, $g$ extends uniquely to $\mathcal{P}$, so $f$ extends uniquely to an isomorphism $f_{\mathcal{P}}$ over $\mathcal{P}$.
Let $\mathcal{P}_1, \mathcal{P}_2\in \mathscr{P}$ with $\mathcal{P}_1 \cap \mathcal{P}_2\neq \emptyset$. The transition functions of $\St_{\mathcal{N}}(R)$ between connected sets are injective. Hence, $f_{\mathcal{P}_1}$ and $f_{\mathcal{P}_2}$ coincide on the convex $\mathcal{P}_1 \cap \mathcal{P}_2$ if they coincide at a point of $ \mathcal{P}_1 \cap \mathcal{P}_2$. Since $\mathcal{P}_1 \cap \mathcal{P}_2$ contains a point lying on the edge of $\mathcal{P}_1$ or $\mathcal{P}_2$, we are done. Hence, the $f_{\mathcal{P}}$ glue into a global isomorphism between $\mathcal{T}_{1}$ et $\mathcal{T}_{2}$.
\end{proof}
To justify the title of this subsection, we are left to prove the representability of $\St_{\mathcal{N}}\text{-}\Sk_{\mathscr{C}(\mathcal{I})}$ by an affine scheme.
\begin{lemme}\label{finitereprske}
For every finite set of curves $\mathscr{C}_f\subset \mathscr{C}$, the functor $\St_{\mathcal{N}}$-$\Sk_{\mathscr{C}_f}$ is representable by an affine scheme of finite type over $\mathds{C}$.
\end{lemme}
\begin{proof}
For $C\in \mathscr{C}_f$, let $\mathcal{T}^{\univ}_{\mathcal{N}_C}$ be the universal Stokes torsor for $\mathcal{N}_C$ given by Babbitt-Varadarajan representability theorem in dimension 1. For $x\in \Int \mathscr{C}_f$ and $C\in \mathscr{C}_f(x)$, choose $t_{x, C}\in \mathcal{T}^{\univ}_{\mathcal{N}_C,x}$. Let $R\in \mathds{C}$-alg. \\ \indent
For every $C\in \mathscr{C}_f$, let $\mathcal{T}_C$ be a $\St_{\mathcal{N}_C}(R)$-torsor and let $(\iota_{C, C^{\prime}}(x))$ be a system of compatibilities as in \eqref{compatcarré}. There exists an isomorphism $\iso_{\mathcal{T}_C}:\mathcal{T}^{\univ}_{\mathcal{N}_C}(R)\longrightarrow \mathcal{T}_C$ and we know from Theorem \ref{autoId} that it is unique.
For $x\in \Int \mathscr{C}_f$ and $C, C^{\prime}\in \mathscr{C}_f(x)$, define $g_{x}(\mathcal{T}_C, \mathcal{T}_{C^{\prime}})$ as the unique element of $\St_{\mathcal{N}, x}(R)$ satisfying
$$
\iota_{C, C^{\prime}}(x)(\iso_{\mathcal{T}_C}(t_{x, C}(R)))= g_{x}(\mathcal{T}_C, \mathcal{T}_{C^{\prime}}) \iso_{\mathcal{T}_{C^{\prime}}}(t_{x, C^{\prime}}(R))
$$
From Theorem \ref{autoId}, we have a well-defined injective morphism of functors
$$
\St_{\mathcal{N}}\text{-}\Sk_{\mathscr{C}_f}\longrightarrow \displaystyle{\prod_{C\in \mathscr{C}_f} H^{1}(S^{1}, \St_{\mathcal{N}_{C}})}\times \displaystyle{\prod_{x\in \Int \mathscr{C}_f} \St_{\mathcal{N}, x}^{\mathscr{C}_f(x)^{2}}}
$$
identifying $\St_{\mathcal{N}}$-$\Sk_{\mathscr{C}_f}$ with the subfunctor of
\begin{equation}\label{grosfoncteur}
\displaystyle{\prod_{C\in \mathscr{C}_f} H^{1}(S^{1}, \St_{\mathcal{N}_{C}})}\times \displaystyle{\prod_{x\in \Int \mathscr{C}_f} \St_{\mathcal{N}, x}^{\mathscr{C}_f(x)^{2}}}
\end{equation}
of families $(\mathcal{T}, g)$ satisfying for every $x \in \Int \mathscr{C}_f$ and every $C, C^{\prime}\in \mathscr{C}_f(x)$,
\begin{align*}
g_{x, C, C}& =\id \\
g_{x, C, C^{\prime}} & = g_{x, C^{\prime}, C}^{-1} \\
g_{x, C, C^{\prime \prime}} & = g_{x, C, C^{\prime}} g_{x, C^{\prime}, C^{\prime \prime}}
\end{align*}
Those conditions are algebraic, so they identify $\St_{\mathcal{N}}$-$\Sk_{\mathscr{C}_f}$ with a closed subscheme of \eqref{grosfoncteur}.
\end{proof}
Taking the limit over finite subsets of $\mathscr{C}$ gives the following
\begin{corollaire}\label{Stskrepresentable}
The functor $\St_{\mathcal{N}}$-$\Sk_{\mathscr{C}}$ is representable by an affine scheme.
\end{corollaire}
\section{Sheaf property and tangent spaces of $H^{1}(\mathds{T}, \St_{\mathcal{N}})$}\label{Artincond}
\subsection{A technical lemma}
In this subsection, we work in dimension 1. Let $f: R\longrightarrow S$ be a morphism of rings. Let $I=]a,b[$ be a strict interval of $S^1$, let $a< d_1 < \cdots < d_k < b$ be the Stokes lines of $\mathcal{N}$ contained in $I$. Set $d_0=a$, $d_{k+1}=b$ and $I_i:=]d_{i-1}, d_{i+1}[$ for $i=1,\dots, k$. The following lemma is obvious when $f$ is surjective. This is however the injective case that will be relevant to us.
\begin{lemme}\label{deRdansS}
For every $\mathcal{T}\in H^{1}(I, \St_{\mathcal{N}}(R))$ such that $\mathcal{T}(S)$ is trivial, there exists $t_i \in \Gamma(I_i, \mathcal{T})$ for every $i=1, \dots, k$ such that the $t_i(S)$ glue into a global section of $\mathcal{T}(S)$ on $I$.
\end{lemme}
\begin{proof}
We argue by recursion on the number of Stokes lines in $I$. The case $k=1$ is implied by \ref{crittri}. Suppose $k>1$ and let $t_i^{\prime} \in \Gamma(I_i, \mathcal{T})$ for $i=1, \dots, k-1$ as given by the recursion hypothesis applied to $J := ]a,d_k[$. From \ref{crittri}, we can choose $t_k^{\prime}\in \Gamma(I_k, \mathcal{T})$ and we want to modify the $t_i^{\prime}$, $i=1, \dots, k$ so that the conclusion of \ref{deRdansS} holds. \\ \indent
Since $\mathcal{T}(S)$ is trivial, we choose $t\in \Gamma(I, \mathcal{T}(S))$ and denote by $t^{\prime}\in \Gamma(J, \mathcal{T}(S))$ the gluing of the $t_i^{\prime}(S)$ for $ i=0,\dots, k-1$. We write $t^{\prime}=g t$ with $g\in \Gamma(J, \St_{\mathcal{N}}(S))$ and $t^{\prime}_k(S)=h t$ with $h\in \Gamma(I_{k}, \St_{\mathcal{N}}(S))$. In the matricial representation \ref{prefered} induced by $<_{]d_{k-1}, d_k[}$, the automorphisms $g$ and $h$ correspond to upper triangular matrices. We argue by recursion on $d$ that at the cost of modifying the $t^{\prime}_i$, we can always suppose that $g$ and $h$ have the same $j$-diagonal for $j=1, \dots , d$. For $d=1$, there is nothing to do. Suppose $d>1$ and write $g=\Id + M$ and $h=\Id + N$ where $M$ and $N$ are nilpotent matrices. On $I_{k-1}\cap I_k$ we have $t_{k-1}^{\prime}= \gamma t_{k}^{\prime}$ with $\gamma \in \Gamma( I_{k-1}\cap I_k , \St_{\mathcal{N}}(R))$. Hence, on $I_{k-1}\cap I_k$
$$
gt=t^{\prime}=\gamma(S)t_k^{\prime}(S)=\gamma(S)h t
$$
so we deduce
$$
gh^{-1} = \id+\displaystyle{\sum_{j=0}^{r}}(-1)^{j} (M-N)N^j =\gamma(S)
$$
We denote by $\Diag_i(A)$ the $i$-upper diagonal of a matrix $A$.
Since $\Diag_i(M)=\Diag_i(N)$ for $i=1, \dots, d$,
we have $D_{d+1}((M-N) N^j)=0$ unless $j=0$. Hence
$$
\left\{
\begin{array}{ll}
M_{i, i+d}-N_{i, i+d}=f(\gamma_{i, i+d})& \mbox{if } (i, i+d)\notin \Jump_{\mathcal{N}}(J)\cup \Jump_{\mathcal{N}}(I_k)\\
M_{i, i+d} = f(\gamma_{i, i+d})& \mbox{if } (i, i+d) \in \Jump_{\mathcal{N}}(I_k) \\
N_{i, i+d} = -f(\gamma_{i, i+d})& \mbox{if } (i, i+d) \in \Jump_{\mathcal{N}}(J)
\end{array}
\right.
$$
Let us define $A, B\in \GL_r(R)$ as follows
$$
\left\{
\begin{array}{ll}
A_{i, i+d}=- \gamma_{i, i+d} & \mbox{if } (i, i+d) \in \Jump_{\mathcal{N}}(I_k) \setminus ( \Jump_{\mathcal{N}}(J)\cap \Jump_{\mathcal{N}}(I_k))\\
A_{i,j}=0 & \mbox{otherwise}
\end{array}
\right.
$$
and
$$
\left\{
\begin{array}{ll}
B_{i, i+d}= \gamma_{i, i+d} & \mbox{if } (i, i+d) \notin \Jump_{\mathcal{N}}(I_k) \cup\Jump_{\mathcal{N}}(J)\\
B_{i, i+d}=\gamma_{i, i+d} & \mbox{if } (i,i+d)\in \Jump_{\mathcal{N}}(J) \setminus ( \Jump_{\mathcal{N}}(J)\cap \Jump_{\mathcal{N}}(I_k))\\
B_{i,j}=0 & \mbox{otherwise}
\end{array}
\right.
$$
Note that $\Id+A\in \Gamma(J, \St_{\mathcal{N}}(R))$ and $\Id+B\in \Gamma(I_k, \St_{\mathcal{N}}(R))$. For $i=1, \dots, d+1$, we have $\Diag_i(f(A)M)=\Diag_i(f(B)N)=0$. If $i<d+1$, we deduce by recursion hypothesis
\begin{align*}
\Diag_i((\Id+f(A))g) & =\Diag_i(\Id) + \Diag_i(M) \\
& =\Diag_i(\Id) + \Diag_i(N)\\
& = \Diag_i((\Id+f(B))h)
\end{align*}
Finally, we have by definition of $A$ and $B$
\begin{align*}
\Diag_{d+1}((\Id+f(A))g)& =\Diag_{d+1}(f(A)+M)\\
& =\Diag_{d+1}(f(B)+N)\\
&=\Diag_{d+1}((\Id+f(B))h)
\end{align*}
Replacing the $t_i^{\prime}$ by $(\Id+A) t_i^{\prime}$ for $i=1, \dots, k-1$ and $t_k^{\prime}$ by $(\Id+B)t_k^{\prime}$ changes $g$ into $(\Id+f(A))g$ and $h$ into $(\Id+f(B))h$. Keeping on with this process leads to a situation where $g=h$.
\end{proof}
We deduce the following
\begin{corollaire}\label{trivialiff}
If $R\longrightarrow S$ is an etale cover and if $I$ is an interval of $S^1$, then $\mathcal{T}\in H^{1}(I, \St_{\mathcal{N}}(R))$ is trivial iff $\mathcal{T}(S)$ is trivial.
\end{corollaire}
\begin{proof}
The only statement requiring a proof is the converse statement. By Babbitt-Varadarajan representability and descent \cite{SGA1}, the functor $H^{1}(S^{1}, \St_{\mathcal{N}})$ is a sheaf so \ref{trivialiff} is known for $I=S^{1}$. If $I$ is a strict interval, this is a consequence of \ref{deRdansS} since an etale cover is an injective morphism of rings.
\end{proof}
\subsection{Sheaf property}
The goal of this subsection is to prove the following
\begin{lemme}\label{sheaf}
The functor $\St_{\mathcal{N}}$-$\AdSk_{\mathscr{C}}$ is a sheaf for the etale topology on $\mathds{C}$-alg.
\end{lemme}
\begin{proof}
From \ref{Stskrepresentable} and descent \cite{SGA1}, the functor $\St_{\mathcal{N}}$-$\Sk_{\mathscr{C}}$ is a sheaf for the etale topology. In particular, the subfunctor $\St_{\mathcal{N}}$-$\AdSk_{\mathscr{C}}$ is a presheaf. To prove that it is a sheaf, we are left to show that for every etale cover $R\longrightarrow S$, a Stokes skeleton $(\mathcal{T}_{\mathscr{C}}, \iota_{\mathscr{C}})\in \St_{\mathcal{N}}\text{-}\Sk_{\mathscr{C}}(R)$ is admissible if $(\mathcal{T}_{\mathscr{C}}(S), \iota_{\mathscr{C}}(S))$ is admissible.\\ \indent
Let $\mathcal{S}$ be a $\mathcal{I}$-good open set, let
$\pi : P\longrightarrow \mathcal{S}$ be a $\mathscr{C}$-polygon with vertices $E_i=[x_i, x_{i+1}]$, $i\in \mathds{Z}/N\mathds{Z}$, let $t_i\in \Gamma(\pi(E_i), \mathcal{T}_{C(E_i)})$, $i=1, \dots, N-1$ such that
$$
t_i(\pi(x_{i+1}))=t_{i+1}(\pi(x_{i+1})) \text{ in } E(\mathcal{T}_\mathscr{C}, \iota_{\mathscr{C}})
$$
Let $t^{\prime}\in \Gamma([x_N,x_1], \mathcal{T}_{C(E_N)}(S))$ be the unique section satisfying
$$
t^{\prime}(\pi(x_N))=t_{N-1}(S)(\pi(x_N)) \text{ and } t^{\prime}(\pi(x_1))=t_{1}(S)(\pi(x_1)) \text{ in } E(\mathcal{T}_\mathscr{C}(S), \iota_{\mathscr{C}}(S))
$$
Since $\mathcal{T}_{C(E_N)}(S)$ is trivial on the strict closed interval $\pi([x_N,x_1])$, it is trivial on a strict open interval containing $\pi([x_N,x_1])$. We deduce from \ref{trivialiff} that $\mathcal{T}_{C(E_N)}$ is trivial on $\pi([x_N,x_1])$. Let $t\in \Gamma(\pi([x_N,x_1]), \mathcal{T}_{C(E_N)})$ and let $g\in \Gamma(\pi([x_N,x_1]), \St_{\mathcal{N}}(S))$ such that $t^{\prime}=g t(S)$. From \eqref{uneegalite} applied to $Z=\pi([x_N,x_1])$ and $Y=\pi(x_1)$, we get that $g$ is defined over $R$. The section $gt$ is the sought-after section.
\end{proof}
\subsection{Twisted Lie algebras, tangent space and obstruction theory}\label{twisted}
For $R\in \mathds{C}$-alg, we denote by $\Lie \St_{\mathcal{N}}(R)$ the sheaf of Lie algebras over $R$ on $\mathds{T}$ induced by $\St_{\mathcal{N}}(R)$. Concretely, $\Lie \St_{\mathcal{N}}(R)$ is the subsheaf of $R\otimes_{\mathds{C}} (j_{D\ast}\mathcal{H}^{0}\DR\End \mathcal{N})_{|\mathds{T}}$ of sections $f$ satisfying $p_a f i_b = 0$ unless $a<_\mathcal{S} b$. \\ \indent
Let $\underline{\mathcal{S}}=(\mathcal{S}_i)_{i\in K}$ be a cover of
$\mathds{T}$ by good open subsets. For $i_1, \dots, i_k\in K$, we set as usual $\mathcal{S}_{i_1 \dots i_k}:= \cap_{j} \mathcal{S}_{i_j}$. We define $L_i(R):=\Lie \St_{\mathcal{N}}(R)_{|\mathcal{S}_i}$. Let $\mathcal{T}\in H^{1}(\mathds{T}, \St_{\mathcal{N}}(R))$ and let $g=(g_{ij})$ be a cocycle representing $\mathcal{T}$. The identifications
\begin{eqnarray*}
L_i(R)_{|S_{ij}} & \overset{\sim}{\longrightarrow} & L_j(R)_{|S_{ij}} \\
M & \longrightarrow & g_{ij}^{-1}M g_{ij}
\end{eqnarray*}
allow to glue the $L_i(R)$ into a sheaf of $R$-Lie algebras over $\mathds{T}$ denoted by $\Lie \St_{\mathcal{N}}(R)^{\mathcal{T}}$ and depending only on $\mathcal{T}$ and not on $g$.
Let us examine the first cocycle conditions in the Cech complex of $\Lie \St_{\mathcal{N}}(R)^{\mathcal{T}}$. For $t=(t_i)_{i\in K}\in \check{C}^{0}(\underline{\mathcal{S}}, \Lie \St_{\mathcal{N}}(R)^{\mathcal{T}})$, let $d_{i}$ be the unique representative of $t_i$ in $\Gamma(\mathcal{S}_{i}, L_i(R))$. Then
\begin{align*}
(dt)_{ij}& =t_i- t_j \\
& = [d_i]-[d_j]\\
& =[d_i- g_{ij}d_j g_{ij}^{-1}]
\end{align*}
where $[$ $]$ denotes the class of an element of $\Gamma(\mathcal{S}_{i}, L_i(R))$ in $\Gamma(S_{ij},\Lie \St_{\mathcal{N}}(R)^{\mathcal{T}})$.
For $t=(t_{ij})\in \check{C}^{1}(\underline{\mathcal{S}}, \Lie \St_{\mathcal{N}}(R)^{\mathcal{T}})$, let $\alpha_{ij}$ be the unique representative of $t_{ij}$ in $\Gamma(\mathcal{S}_{ij}, L_i(R))$. The relation $t_{ij}=- t_{ji}$ translates into $\alpha_{ji}=-g^{-1}_{ij}\alpha_{ij}g_{ij}$. Thus
\begin{align*}
(dt)_{ijk} &=t_{jk}+t_{ki}+t_{ij}\\
&=[g_{ij}\alpha_{jk}g_{ij}^{-1}+g_{ik}\alpha_{ki}g_{ik}^{-1}+\alpha_{ij}]
\end{align*}
Let us now compute the tangent space of $H^{1}(\mathds{T},\St_{\mathcal{N}})$ at $\mathcal{T}$.
\begin{lemme}\label{lemmecalculTangent}
There is a canonical isomorphism of $R$-modules
\begin{equation}\label{calculTangent}
\xymatrix{
T_{\mathcal{T}}H^{1}(\mathds{T}, \St_{\mathcal{N}})\ar[r]^-{\sim} & \check{H}^{1}(\underline{\mathcal{S}}, \Lie \St_{\mathcal{N}}(R)^{\mathcal{T}})
}
\end{equation}
\end{lemme}
\begin{proof}
Let $\mathcal{T}_{\epsilon} \in T_{\mathcal{T}}H^{1}(\mathds{T}, \St_{\mathcal{N}})$. A cocycle of $\mathcal{T}_{\epsilon}$ associated to the good cover $\underline{\mathcal{S}}$ can be written $(g_{ij}+\epsilon_{ij})$ where the $\epsilon_{ij}$ take value in the line $\mathds{C}\epsilon$. The relation $g_{ji}+\epsilon_{ji}=(g_{ij}+\epsilon_{ij})^{-1}$ is equivalent to
$$
\epsilon_{ji}=-g_{ij}^{-1}\epsilon_{ij}g_{ij}^{-1}
$$
The cocycle condition is equivalent to
$$
(g_{ij}+\epsilon_{ij})(g_{jk}+\epsilon_{jk})(g_{ki}+\epsilon_{ki})=\Id
$$
which is equivalent to the following equality in $\Gamma(S_{ij}, \Lie \St_{\mathcal{N}}(R))$
$$
\epsilon_{ij}g_{jk}g_{ki} + g_{ij}\epsilon_{jk}g_{ki} + g_{ij}g_{jk}\epsilon_{ki}=0
$$
That is
\begin{equation}\label{relation1retour}
\epsilon_{ij}g_{ij}^{-1}+g_{ij}\epsilon_{jk}g_{ki} +g_{ki}^{-1}\epsilon_{ki}=0
\end{equation}
Let us set $\alpha_{ij}=\epsilon_{ij}g_{ij}^{-1}$ viewed as a section of $ L_i$ over $\mathcal{S}_{ij}$ and let $t_{ij}(\epsilon)$ be the class of $\alpha_{ij}$ in $\Lie \St_{\mathcal{N}}(R)^{\mathcal{T}}$. We have $t_{ii}(\epsilon)=[\alpha_{ii}]=0$ and
$$
t_{ji}(\epsilon)=[\alpha_{ji}]=[\epsilon_{ji}g_{ij}]=
-[g_{ij}^{-1}\epsilon_{ij}]=-[\alpha_{ij}]=-t_{ij}(\epsilon)
$$
The relations explicited in \ref{twisted} show that \eqref{relation1retour} is equivalent to
the $t_{ij}(\epsilon)$ defining a cocycle of $\Lie \St_{\mathcal{N}}(R)^{\mathcal{T}}$. Its class in
$\check{H}^{1}(\underline{\mathcal{S}}, \Lie \St_{\mathcal{N}}(R)^{\mathcal{T}})$
does not depend on the choice of a cocycle representing $\mathcal{T}_{\epsilon}$. Indeed, suppose that $(g_{ij}+\epsilon_{ij})_{ij}$ is cohomologous to $(g_{ij}+\nu_{ij})_{ij}$, that is
\begin{eqnarray*}
g_{ij}+\nu_{ij}=(c_i+d_i)(g_{ij}+\epsilon_{ij})(c_j+d_j)^{-1}
\end{eqnarray*}
where $(c_i+d_i)_{i\in K}\in \check{C}^{0}(\underline{\mathcal{S}}, \St_{\mathcal{N}}(R[\epsilon]))$. We obtain
\begin{equation}\label{unauto}
g_{ij}=c_i g_{ij} c_j^{-1}
\end{equation}
and
\begin{equation}\label{vasesimplifier}
\nu_{ij}=c_i \epsilon_{ij}c_j^{-1}+ d_i g_{ij} c_j^{-1}- c_i g_{ij} c_j^{-1} d_j c_{j}^{-1}
\end{equation}
Relation \eqref{unauto} implies that the $c_i$ define an automorphism of $\mathcal{T}$. From \ref{autoId}, we deduce that $c_i= \Id$ for every $i$. Hence, \eqref{vasesimplifier} gives
$$
\nu_{ij}= \epsilon_{ij}+ d_i g_{ij} - g_{ij} d_j
$$
Multiplying by $g_{ij}^{-1}$ and taking the class in $\Lie \St_{\mathcal{N}}(R)^{\mathcal{T}}$ gives
$$
t_{ij}(\nu)= t_{ij}(\epsilon)+ [d_i - g_{ij} d_j g_{ij}^{-1}]
$$
Hence, the morphism \eqref{calculTangent} is well-defined and is injective. One easily checks that it is surjective.
\end{proof}
One can show similarly (but we will not need it) that $\check{H}^{2}(\underline{\mathcal{S}}, \Lie \St_{\mathcal{N}}(R)^{\mathcal{T}})$ provides an obstruction theory for the functor
$H^{1}(\mathds{T},\St_{\mathcal{N}})$ at $\mathcal{T}$.
\section{Proof of Theorem \ref{bigtheorem}}\label{prooftheorem}
\subsection{Representability by an algebraic space}\label{repalgspace}
Let $\underline{\mathcal{S}}$ be a finite cover by $\mathcal{I}$-good open sets of $\mathds{T}$. Since the cocycle condition in
$$
\prod_{\mathcal{S}, \mathcal{S}^{\prime}\in \underline{\mathcal{S}}} \Gamma(\mathcal{S}\cap \mathcal{S}^{\prime}, \St_{\mathcal{N}})
$$
is algebraic, it defines a closed subscheme denoted by $Z( \underline{\mathcal{S}}, \St_{\mathcal{N}})$. From \ref{crittri}, the morphism of presheaves
$$
\xymatrix{
Z(\underline{\mathcal{S}}, \St_{\mathcal{N}}) \ar[r] & H^{1}(\mathds{T}, \St_{\mathcal{N}})
}
$$
is surjective. Hence $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ is the quotient in the category of presheaves of $Z(\underline{\mathcal{S}}, \St_{\mathcal{N}})$ by the algebraic group
$$
\prod_{\mathcal{S}\in \underline{\mathcal{S}}} \Gamma(\mathcal{S}, \St_{\mathcal{N}})
$$
From \ref{autoId}, this action is free. By Artin theorem \cite[6.3]{ArtinOnStack}, see also \cite[10.4]{LMB} and \cite[04S6]{SP}, we deduce that the sheaf associated to $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ is representable by an algebraic space of finite type over $\mathds{C}$. From lemma \ref{sheaf}, the functor $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ is a sheaf for the etale topology on $\mathds{C}$-alg, so it is isomorphic to its sheafification. Hence, $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ is representable by an algebraic space of finite type over $\mathds{C}$.
\subsection{A closed immersion in an affine scheme of finite type}\label{closedimmersionpreuve}
The morphism of algebraic spaces \eqref{injfamillespeciale} is a monomorphism of finite type. Hence, it is separated and quasi-finite \cite[Tag 0463]{SP}. Since a separated quasi-finite morphism is representable \cite[Tag 03XX]{SP}, we deduce that \eqref{injfamillespeciale} is representable. Since $\St_{\mathcal{N}}\text{-}\Sk_{\mathscr{C}(\mathcal{I})}$ is a scheme, we deduce that $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ is representable by a scheme of finite type over $\mathds{C}$. We still denote by $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ this scheme. We are left to show that \eqref{injfamillespeciale} is a closed immersion. From \cite[18.12.6]{EGA4-4}, closed immersions are the same thing as proper
monomorphisms. We are thus left to prove that \eqref{injfamillespeciale} is proper.
By the valuative criterion for properness \cite[7.3.8]{EGA2}, we have to prove that if
$R$ is a discrete valuation ring with field of fraction $K$, for
$(\mathcal{T}_{\mathscr{C}(\mathcal{I})}, \iota_{\mathscr{C}(\mathcal{I})})\in
\St_{\mathcal{N}}\text{-}\Sk_{\mathscr{C}(\mathcal{I})}(R)$ such that
$$
(\mathcal{T}_{\mathscr{C}(\mathcal{I})}(K), \iota_{\mathscr{C}(\mathcal{I})}(K))=\sk_{\mathscr{C}(\mathcal{I})}(\mathcal{T}_K)
$$
where $\mathcal{T}_K\in H^{1}(\mathds{T},
\St_{\mathcal{N}}(K))$, there exists a (necessarily unique) $\mathcal{T}\in H^{1}
(\mathds{T}, \St_{\mathcal{N}}(R))$ such that $\sk_{\mathscr{C}(\mathcal{I})}
(\mathcal{T})=(\mathcal{T}_{\mathscr{C}(\mathcal{I})}, \iota_{\mathscr{C}(\mathcal{I})})$ (which then
automatically implies $\mathcal{T}(K)=\mathcal{T}_K$). Since $R\longrightarrow K$ is injective, it is enough to prove that $\mathcal{T}_K$ is defined over $R$. \\ \indent
We make here an essential use of condition $(3)$ of \ref{mono}, so we depict it.
For $\mathcal{P}\in \mathscr{P}$, let us choose $t_{\mathcal{P}}\in \Gamma(\mathcal{P}, \mathcal{T}_K)$. By assumption, $x_\mathcal{P}$ belongs to a curve $C_{\mathcal{P}}\in \mathscr{C}(\mathcal{I})$. Let $s_{\mathcal{P}}\in \mathcal{T}_{C_{\mathcal{P}}, x_\mathcal{P}}$, and let us write
$$
t_{\mathcal{P}, x_\mathcal{P}}=g_\mathcal{P} s_{\mathcal{P}}(K) \text{ with } g_\mathcal{P} \in \St_{\mathcal{N}}(K)_{x_\mathcal{P}}
$$
Since $\mathcal{P}$ is contained in an $\mathcal{I}$-good open set for $x_\mathcal{P}$, the section $g_\mathcal{P}$ extends to $\mathcal{P}$. Replacing
$t_{\mathcal{P}}$ by $g_\mathcal{P}^{-1}t_{\mathcal{P}}$, we can thus suppose that
$t_{\mathcal{P}, x_\mathcal{P}}=s_{\mathcal{P}}(K)$.
Let $\mathcal{P}_1, \mathcal{P}_2\in \mathscr{P}$ such that $\mathcal{P}_1 \cap \mathcal{P}_2\neq \emptyset $. We have to show that the transition matrix between $t_{\mathcal{P}_1}$ and $t_{\mathcal{P}_2}$ take value in $R$.
For $i=1, 2$ we choose $C_i\in \mathscr{C}(\mathcal{I})$ containing $[x_{\mathcal{P}_i}, x_i]$ and $C_i^{\prime}\in \mathscr{C}(\mathcal{I})$ containing $[x_i, x_{12}]$. The torsors $\mathcal{T}_{C_i}(K)$ are trivial on $[x_{\mathcal{P}_i}, x_i]$. Since $R\longrightarrow K$ is injective, \ref{deRdansS}
ensures that the same is true for $\mathcal{T}_{C_i}$. Similarly, the torsors $\mathcal{T}_{C_i^{\prime}}$ are trivial on $[x_i, x_{12}]$. Let us choose $s_i \in \Gamma([x_{\mathcal{P}_i}, x_i], \mathcal{T}_{C_i})$ and $s_i^{\prime} \in \Gamma([x_i, x],\mathcal{T}_{C_i^{\prime}})$. We have
$$
t_{\mathcal{P}_i}= g_i s_i(K) \text{ in } \Gamma([x_{\mathcal{P}_i}, x_i], \mathcal{T}_K)
$$
where $g_i \in \Gamma([x_{\mathcal{P}_i}, x_i], \St_{\mathcal{N}}(K))$. Then
$$
t_{\mathcal{P}_i, x_{\mathcal{P}_i}}= g_{i, x_{\mathcal{P}_i}} s_{i, x_{\mathcal{P}_i}}(K)=s_{\mathcal{P}_i}(K)
$$
Since $s_{\mathcal{P}_i}$ and $s_i$ are both defined over $R$, so is $g_{i, x_{\mathcal{P}_i}}$. From \eqref{uneegalite}, we deduce that $g_i$ has coefficients in $R$. Hence, we have similarly on $[x_i, x_{12}]$
$$
t_{\mathcal{P}_i}=g_i^{\prime}s_i^{\prime}(K) \text{ with } g_i^{\prime}\in \Gamma([ x_i, x_{12}], \St_{\mathcal{N}}(R))
$$
Finally, $s_1^{\prime}$ and $s_2^{\prime}$ compare at $x_{12}$ in terms of a matrix $h$ with coefficient in $R$. If we write $t_{\mathcal{P}_2}= g_{12}t_{\mathcal{P}_1}$ on $\mathcal{P}_1 \cap \mathcal{P}_2$, we deduce
$g_{12}= g_2^{\prime}hg_1^{\prime-1}$, so $g_{12}$ has coefficients in $R$. This concludes the proof of Theorem \ref{bigtheorem}.
\section{The case where $\mathcal{I}$ is very good}
\subsection{Differential interpretration of $\St_{\mathcal{N}}$}\label{diffint}
From now on, we suppose that $\mathcal{I}$ is very good. This means that for every $a,b\in \mathcal{I}$ with $a \neq b$, the pole locus of $ a-b$ is exactly $D$. In particular, for every $R\in \mathds{C}$-alg,
$$ \St_{\mathcal{N}}(R)=\Id+ R\otimes_{\mathds{C}}\mathcal{H}^{0}\DR^{<D}\End \mathcal{N}
$$
\subsection{Tangent space and irregularity}\label{tanandirr}
Let $(\mathcal{M}, \nabla, \iso)$ be a good $\mathcal{N}$-marked connection. A choice of local trivialisations for $\mathcal{T}:=\Isom_{\iso}(\mathcal{M}, \mathcal{N})$ gives rise to an isomorphism of sheaves on $\mathds{T}$
$$
\xymatrix{
\mathcal{H}^0 \DR^{<D}\End \mathcal{M}\ar[r]^-{\sim} & \Lie \St_{\mathcal{N}}(\mathds{C})^{\mathcal{T}}
}
$$
Hence, for every $i \in \mathds{N}$, there are canonical identifications
\begin{align*}
H^{i}(\mathds{T}, \Lie \St_{\mathcal{N}}(\mathds{C})^{\mathcal{T}})&\simeq H^i( \mathds{T}, \mathcal{H}^0 \DR^{<D}\End \mathcal{M}) \\
& \simeq
H^i(\mathds{T}, \DR^{<D}\End \mathcal{M}) \\
& \simeq (\mathcal{H}^i\Irr^{\ast}_D\End \mathcal{M})_{0}
\end{align*}
The second identification comes from the fact \cite[Prop. 1]{HienInv} that $\DR^{<D}\End \mathcal{M}$ is concentrated in degree $0$. The third identification comes from \cite[2.2]{SabRemar}. We deduce the following
\begin{lemme}\label{tspace}
The tangent space of $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ at $\mathcal{T}:=\Isom_{\iso}(\mathcal{M}, \mathcal{N})$ identifies in a canonical way to $(\mathcal{H}^1\Irr^{\ast}_D\End \mathcal{M})_{0}$.
\end{lemme}
\begin{proof}
Since $\mathds{T}$ is paracompact, sheaf cohomology is computed by Cech cohomology. Moreover, $\mathcal{I}$-good opens form a basis of the topology of $\mathds{T}$.
For two such covers $\underline{\mathcal{U}}$ and $\underline{\mathcal{V}}$ with $\underline{\mathcal{V}}$ refining $\underline{\mathcal{U}}$, lemma \ref{lemmecalculTangent} provides a commutative diagram
$$
\xymatrix{
& T_{\mathcal{T}}H^{1}(\mathds{T}, \St_{\mathcal{N}})\ar[rd] \ar[ld] & \\
\check{H}^{1}(\underline{\mathcal{U}}, \Lie \St_{\mathcal{N}}(\mathds{C})^{\mathcal{T}}) \ar[rr]& & \check{H}^{1}(\underline{\mathcal{V}}, \Lie \St_{\mathcal{N}}(\mathds{C})^{\mathcal{T}})
}
$$
where the diagonal arrows are isomorphisms. Hence, the vertical arrow is an isomorphism and \ref{tspace} is proved.
\end{proof}
\subsection{Proof of Theorem \ref{restriction}}
Unramified morphisms of finite type are quasi-finite. Hence, it is enough to prove that a good $\mathcal{N}$-marked connection $(\mathcal{M}, \nabla, \iso)$ belongs to the unramified locus of $\res_V$, which is open. The cotangent sequence of $\res_V$ reads
$$
\xymatrix{
\res_V^{\ast} \Omega^{1}_{H^{1}(\mathds{T}^{\prime}, \St_{\mathcal{N}_V})} \ar[r] & \Omega^{1}_{H^{1}(\mathds{T}, \St_{\mathcal{N}})} \ar[r] & \Omega^{1}_{\res_V} \ar[r] & 0
}
$$
Taking the fiber at $\mathcal{T}:=\Isom_{\iso}(\mathcal{M}, \mathcal{N})$ preserves cokernel, so after dualizing, we obtain the following exact sequence
\begin{equation}\label{ptitexasequ}
\xymatrix{
0 \ar[r] & \Omega^{1}_{\res_V}(\mathcal{T})^{\vee} \ar[r] & T_{\mathcal{T}}H^{1}(\mathds{T}, \St_{\mathcal{N}}) \ar[r] & T_{ \res_V(\mathcal{T})}H^{1}(\mathds{T}^{\prime}, \St_{\mathcal{N}_V})
}
\end{equation}
By Nakayama lemma, we have to prove that $\Omega^{1}_{\res_V}(\mathcal{T} )$ vanishes. Let $i_V: V\longrightarrow \mathds{C}^n$ be the canonical inclusion.
Since $\mathcal{M}$ is localized at $0$, we have
$$
(\Irr^{\ast}_{D}\End \mathcal{M})_0 \simeq \Irr^{\ast}_{0}\End \mathcal{M}
$$
Applying $\Irr^{\ast}_{0}$ \cite[3.4-2]{Mehbsmf} to the local cohomology triangle \cite{Kalivre}
$$
\xymatrix{
i_{V+}i_{V}^+\End \mathcal{M}[\dim V-n]\ar[r]& \End \mathcal{M} \ar[r]& \End \mathcal{M}(\ast V)\ar[r]^-{+1}&
}
$$
gives a distinguished triangle
$$
\xymatrix{
(\Irr^{\ast}_{V}\End \mathcal{M})_0 \ar[r]& \Irr^{\ast}_{0}\End \mathcal{M}\ar[r]&
\Irr^{\ast}_{0}\End \mathcal{M}_{V}
\ar[r]^-{+1}&
}
$$
Hence, we obtain an exact sequence
\begin{equation}\label{ptitexasequ2}
\xymatrix{
0\ar[r]& (\mathcal{H}^1\Irr^{\ast}_{V}\End \mathcal{M})_0
\ar[r]& \mathcal{H}^1\Irr^{\ast}_{0}\End \mathcal{M}\ar[r]& \mathcal{H}^1\Irr^{\ast}_{0}\End \mathcal{M}_{V}
}
\end{equation}
From \ref{tspace}, the sequence \eqref{ptitexasequ} identifies canonically to \eqref{ptitexasequ2}. Hence, we are left to show the following
\begin{proposition}
For every $\mathcal{N}$-marked connection $\mathcal{M}$, we have $$(\mathcal{H}^1\Irr^{\ast}_{V}\mathcal{M})_0\simeq 0$$
\end{proposition}
\begin{proof}
Let $p: X\longrightarrow \mathds{C}^{n}$ be the blow-up at the origin. Let us denote by $E$ the exceptional divisor, $V^{\prime}$ (resp. $D^{\prime}$) the strict transform of $V$ (resp. $D$). Since $p$ is an isomorphism above $\mathds{C}^{n}\setminus D$, we know from \cite[3.6-4]{Mehbsmf} that $p_+ p^{+ }\mathcal{M} \longrightarrow \mathcal{M}$
is an isomorphism. Applying $\Irr^{\ast}_{V}$ and the compatibility of $\Irr^{\ast}$ with proper push-forward, we obtain
\begin{align*}
(\mathcal{H}^1\Irr^{\ast}_{V}\mathcal{M})_0 & \simeq H^{1}({E},(\Irr^{\ast}_{p^{-1}(V)}p^{+}\mathcal{M})_{|E})\\
& \simeq H^{0}({E},(\mathcal{H}^{1}\Irr^{\ast}_{E\cup V^{\prime}}p^{+}\mathcal{M})_{|E})
\end{align*}
Set $\Omega:= E\setminus (E\cap D^{\prime})$. Since $V$ is transverse to every irreducible components of $D$, the set $\Omega \cap V^{\prime}$ is not empty. Pick $d\in \Omega \cap V^{\prime}$. Since $E$ is included in the pole locus of $p^{+}\mathcal{M}$,
\begin{equation}\label{isoirr}
(\Irr^{\ast}_{E\cup V^{\prime}}p^{+}\mathcal{M})_d \simeq (\Irr^{\ast}_{V^{\prime}}p^{+}\mathcal{M})_d
\end{equation}
Moreover, the multiplicities of the components of $\Char p^{+}\mathcal{M}$
passing through $x\in \Omega$ only depend on the formalization of $p^{+}\mathcal{M}$ at $x$. Since $p^{+}\mathcal{M}$ has good formal decomposition along the smooth divisor $\Omega$, we deduce that above $\Omega$, $\Char \mathcal{M}$ is supported on the conormal bundle of $\Omega$. In particular, $V^{\prime}$ is non-characteristic for $p^{+}\mathcal{M}$ at $d$. Thus, \cite{TheseKashiwara} asserts that
the Cauchy-Kowaleska morphism
$$
\RHom(p^+\mathcal{M}, \mathcal{O}_X)_{|V^{\prime}}\longrightarrow \RHom((p^+\mathcal{M})_{V^{\prime}}, \mathcal{O}_{V^{\prime}})
$$
is an isomorphism in a neighbourhood of $d$. From \cite[V 2.2]{MT}, we deduce that the right-hand side of \eqref{isoirr} is zero. \\ \indent
Take $s\in H^{0}({E},(\mathcal{H}^{1}\Irr^{\ast}_{E\cup V^{\prime}}p^{+}\mathcal{M})_{|E})$. Since
$$
(\Irr^{\ast}_{E\cup V^{\prime}}p^{+}\mathcal{M})_{|E}\simeq \Irr^{\ast}_{E}(p^{+}\mathcal{M})(\ast V^{\prime})
$$
we know from \cite{Mehbgro} that the complex $(\Irr^{\ast}_{E\cup V^{\prime}}p^{+}\mathcal{M})_{|E}[1]$ is a perverse sheaf on $E$. So to prove $s=0$, we are left to prove that the support of $s$ is contained in a closed subset of dimension $<\dim E$ \cite[(10.3.3)]{KS}. Hence, it is enough to prove that $s$ vanishes on $\Omega$. From the discussion above, $s$ vanishes on a neighbourhood $U$ of $\Omega \cap V^{\prime}$ in $\Omega$. Since $\Omega$ is path-connected, a point in $\Omega \setminus (\Omega \cap V^{\prime})$ can be connected to a point in $U \setminus (U \cap V^{\prime})$ via a path in $\Omega \setminus (\Omega \cap V^{\prime})$. So we are left to see that $\mathcal{H}^{1}\Irr^{\ast}_{E\cup V^{\prime}}p^{+}\mathcal{M}$ is a local system on $\Omega \setminus (\Omega \cap V^{\prime})$. This is a consequence of the following
\begin{lemme}
Let $X$ be a smooth manifold and let $Z$ be a smooth divisor of $X$. Let $\mathcal{M}$ be a meromorphic connection on $X$ with poles along $Z$ admitting a good formal structure along $Z$. Then $\Irr^{\ast}_Z\mathcal{M}$ is a local system concentrated in degree 1.
\end{lemme}
We can see this as a particular case of Sabbah theorem \cite{SabRemar}. Let us give an elementary argument. Since $\mathcal{M}$ has good formal structure along $Z$, any smooth curve transverse to $Z$ is non-characteristic for $\mathcal{M}$. Hence, $\Irr^{\ast}_Z\mathcal{M}$ is concentrated in degree 1 and $x\rightarrow \dim \mathcal{H}^1(\Irr^{\ast}_Z\mathcal{M})_x$ is constant to the generic irregularity of $\mathcal{M}$ along $Z$. We know from \cite{Mehbgro} that $\Irr^{\ast}_Z\mathcal{M}[1]$ is perverse on $Z$. We conclude using the fact that a perverse sheaf on $Z$ with constant Euler-Poincaré characteristic function is a local system concentrated in degree 0.
\end{proof}
\subsection{Proof of Theorem \ref{rigidity}}
Let $(\mathcal{M}, \nabla, \iso)$ be a good $\mathcal{N}$-marked connection. Since $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ is of finite type, it is enough to prove that the tangent space of $H^{1}(\mathds{T}, \St_{\mathcal{N}})$ at $\Isom_{\iso}(\mathcal{M}, \mathcal{N})$ vanishes. From \ref{tanandirr}
and Sabbah invariance theorem \cite{SabRemar}, we are left to prove the vanishing of $(\mathcal{H}^1\Irr^{\ast}_D\End \mathcal{N})_{0}$ for a generic choice of the regular parts $\mathcal{R}_a$, $a\in \mathcal{I}$. Generically, the monodromy is semi-simple, so we are left to prove that for every $i \in \mathds{N}$,
\begin{equation}\label{vanishing}
(\mathcal{H}^i\Irr^{\ast}_D z^{\alpha} \mathcal{E}^{a})_0\simeq 0
\end{equation}
where $\alpha=(\alpha_1, \dots, \alpha_m)\in \mathds{C}^m$ is generic, $z^{\alpha}=z_1^{\alpha_1}\cdots z_m^{\alpha_m}$ and $a$ is a good meromorphic function with poles along $D$.
By a change of variable, we can suppose $a=1/z_1^{a_1}\cdots z_m^{a_m}$ where $a_i\in \mathds{N}^{\ast}$, $i=1, \dots, m$. We are thus left to prove the following
\begin{lemme}\label{quasidernierlemme}
Suppose that there exists $i, j\in \llbracket 1,m\rrbracket$ with $i\neq j$ such that
$$
\alpha_i a_i +\alpha_j a_j \notin \mathds{Z}
$$
Then, $(\Irr^{\ast}_D z^{\alpha}\mathcal{E}^{a})_{0}\simeq 0$.
\end{lemme}
In a first draft of this paper, a proof using perversity arguments was given. We give here a simpler and more natural argument, kindly communicated to us by C. Sabbah.
\begin{proof}
From \cite[2.2]{SabRemar} and \cite[Prop. 1]{HienInv}, we have
$$
(\Irr^{\ast}_D z^{\alpha}\mathcal{E}^{a})_{0}\simeq R\Gamma( \mathds{T}, \mathcal{H}^{0}\DR^{<D}z^{\alpha}\mathcal{E}^{a})
$$
Let $\rho : \mathds{T}\longrightarrow S^1$ be the morphism $(\theta_1, \dots, \theta_m)\longrightarrow \sum_{i=1}^m a_i \theta_i$. Then, $\mathcal{H}^{0}\DR^{<D}z^{\alpha}\mathcal{E}^{a}$ is the extension by $0$ of the restriction $L_{\alpha}$ of $\DR z^{\alpha}$ to $U:=\rho^{-1}(]\pi/2, 3\pi/2[)$. By Leray spectral sequence
$$
E_{2}^{pq}=H_c^p(]\pi/2, 3\pi/2[, R^q\rho_{\ast} L_{\alpha}) \Longrightarrow R\Gamma_c(U, L_{\alpha})
$$
and proper base change, we are left to prove
$$
R\Gamma(\rho^{-1}(\theta), L_{\alpha})\simeq 0
$$
for every $\theta\in S^{1}$. Note that $\rho^{-1}(\theta)\simeq (S^1)^{m-1}$. By homotopy, it is enough to treat the case $\theta=0$. Without loss of generality, we can suppose $\alpha_1 a_1 +\alpha_2 a_2 \notin \mathds{Z}$. Let $p: \rho^{-1}(0)\longrightarrow (S^1)^{m-2}$ be the restriction to $\rho^{-1}(0)$ of the projection on coordinates $\theta_3, \dots, \theta_n$. By the same argument as above, we are left to show that the restriction of $ L_{\alpha}$ to the circles $p^{-1}(\theta), \theta \in (S^1)^{m-2}$ has no cohomology. Since the monodromy of such a restriction is the multiplication by $e^{2\pi i (\alpha_1 a_1 +\alpha_2 a_2)}$, we are done.
\end{proof}
\end{document} |
\begin{document}
\title[Polynomial Berezin transform]
{Berezin transform in polynomial Bergman spaces}
\author[Ameur]
{Yacin Ameur}
\address{Yacin Ameur\\ Department of Mathematics\\
Uppsala University\\
\\
Box 480\\
751 06 Uppsala\\
Sweden}
Mathrm email{[email protected]}
\author[Hedenmalm]
{H\aa{}kan Hedenmalm}
\address{H{\aa}kan Hedenmalm\\ Department of Mathematics\\
The Royal Institute of Technology\\
S -- 100 44 Stockholm\\
SWEDEN}
Mathrm email{[email protected]}
\thetaanks{Research supported by the G\"oran Gustafsson Foundation. The third author is supported by N.S.F. Grant No. 0201893.}
\author[Makarov]
{Nikolai Makarov}
\address{Nikolai Makarov\\ Department of Mathematics\\ California Institute of
Technology\\ Pasadena\\ CA 91125\\ USA}
Mathrm email{[email protected]}
\begin{abstract} We consider fairly general weight functions
$Q:{Mathbb C}\to {Mathbb R}$, and let
$K_{m,n}$ denote the reproducing kernel for the space
${H}_{m,n}$ of analytic polynomials $u$ of degree at most $n-1$
of finite norm $\|u\|_{mQ}^2=\int_{Mathbb C}\babs{u(z)}^2Mathrm e^{-mQ(z)}{{Mathrm d} A}(z),$
${{Mathrm d} A}$ denoting suitably normalized area measure in ${Mathbb C}$. For a
continuous bounded function $f$ on ${Mathbb C}$, we consider its
(polynomial) Berezin transform
\begin{equation*}Mathfrak B_{m,n}f(z)=\int_{Mathbb C} f(\zeta){Mathrm d}
B^{\langle z\rangle}_{m,n}(\zeta) \qquad\text{where}\qquad {Mathrm d}
B^{\langle z\rangle}_{m,n}(\zeta)=\frac {\babs{K_{m,n}(z,\zeta)}^2}
{K_{m,n}(z,z)}Mathrm e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta).
Mathrm end{equation*}
Let ${Mathbb S}etX=\{{Mathbb D}elta Q>0\}.$ For a parameter $\tau>0$ we prove that
there exists a compact subset ${Mathbb S}etS_\tau$ of ${Mathbb C}$ such that
\begin{equation}\label{berconv}Mathfrak B_{m,n}f(z)\to f(z)\qquad \text{as}
\quad m\to\infty\quad \text{and}\quad n- m\tau\to 0,Mathrm end{equation}
for all continuous bounded $f$ if $z$ is in the interior of
${Mathbb S}etS_\tau\cap {Mathbb S}etX$. Equivalently, the measures $B_{m,n}^{\langle
z\rangle}$ converge to the Dirac measure at $z$. The set
${Mathbb S}etS_\tau$ is the coincidence set for an associated obstacle
problem.
We also prove that the convergence in Mathrm eqref{berconv} is
\textit{Gaussian} when $z$ is in the interior of
${Mathbb S}etS_\tau\cap{Mathbb S}etX$, in the sense that with $F(\zeta)=fL^p(\D,dA)ar
{Mathbb S}qrt{m{Mathbb D}elta Q(z)}(\zeta-z)\right )$, we have
\begin{equation*}
Mathfrak B_{m,n}F(z)\to \int_{Mathbb C} f(\zeta){Mathrm d} P(\zeta),
Mathrm end{equation*}
${Mathrm d} P(\zeta)=Mathrm e^{-\babs{\zeta}^2}{Mathrm d} A(\zeta)$ denoting the
standard Gaussian.
In the "model case'' $Q(z)=\babs{z}^2$, ${Mathbb S}etS_\tau$ is the closed
disk with centre $0$ and radius ${Mathbb S}qrt{\tau}$. We prove that if $z$
is fixed with $\babs{z}>{Mathbb S}qrt{\tau}$, the corresponding measures
$B_{m,n}^{\langle z\rangle}$ converge to harmonic measure for $z$
relative to the domain ${Mathbb C}^*{Mathbb S}etminus{Mathbb S}etS_\tau$, ${Mathbb C}^*$ denoting
the extended plane.
Our auxiliary results include $L^2$ estimates for the
${\overline{\partial}}$-equation ${\overline{\partial}} u=f$ when $f$ is a suitable test function
and the solution $u$ is restricted by a growth constraint near
$\infty$.
Our results have applications e.g. to the study of weighted
distributions of eigenvalues of random normal matrices. In the companion paper \cite{AHM} we consider such applications, e.g. a proof of Gaussian field convergence
for fluctuations of linear statistics of eigenvalues of random normal matrices from the ensemble associated with $Q$.
Mathrm end{abstract}
Maketitle
\addtolength{\textheight}{2.2cm}
{Mathbb S}ection{Introduction} \label{intro}
{Mathbb S}ubsection{General introduction to Berezin quantization.}
\footnote{In this version we have but slightly modified the formulation of Theorem 2.8 and made some minor corrections.}
In a version of quantum theory, a Bargmann-Fock type space of
polynomials plays the role of the quantized system, while the
corresponding weighted $L^2$ space is the classical analogue. It is
therefore natural to study the asymptotics of the quantized system
as we approach the semiclassical limit. A particularly useful object
is the {Mathrm em reproducing kernel} of the Bargmann-Fock type space. To
make matters more concrete, let $Mu$ be a finite positive Borel
measure on ${Mathbb C}$, and $L^2({Mathbb C};Mu)$ the usual $L^2$ space with inner
product
$$\big\langle f,g\big\rangle_{L^2({Mathbb C};Mu)}=\int_{Mathbb C} f(z)\overline{g(z)}
{Mathrm d}Mu(z).$$ The subspace of $L^2({Mathbb C};Mu)$ of entire functions is
the Bergman space $A^2({Mathbb C};Mu)$.
We assume that the support of $Mu$ has infinitely many points so
that any given polynomial corresponds to a unique element of
$L^2({Mathbb C};Mu)$, and write
$$J_Mu={Mathbb S}up\big\{j\in{Mathbb Z};\,
\int_{Mathbb C}\babs{z}^{2(j-1)}{Mathrm d}Mu(z)<\infty\big\}.$$ Since $Mu$ is
finite, we are ensured that $1\le J_Mu\le \infty$.
Let ${Mathcal P}_n$ be the set of analytic polynomials of degree at most
$n-1$, and write
\begin{equation*}A^2_{Mu,n}=L^2({Mathbb C};Mu)\cap
{Mathcal P}_n{Mathbb S}ubset L^2({Mathbb C};Mu).Mathrm end{equation*} Putting
$n'=Min\{n,J_Mu\}$,
it is thus seen that $A^2_{Mu,n}$ equals ${Mathcal P}_{n'}$ in the
sense of sets, with the norm inherited from $L^2({Mathbb C};Mu)$. Hence
$A^2_{Mu,n}=A^2_{Mu,n'}$ for all $n$, and the reader who so
desires can without loss of generality assume that $n=n'$ in the
following.
Let $e_1,\ldots, e_{n'}$ be an orthonormal basis for $A^2_{Mu,n}$.
The reproducing kernel $K_{Mu,n}$ for the space $A^{2}_{Mu,n}$ is
the function
\begin{equation*}
K_{Mu,n}(z,w)={Mathbb S}um_{j=1}^{n'}e_j(z)\overline{e_j(w)}.
Mathrm end{equation*}
Then $K_{Mu,n}$ is independent of the choice of
an orthonormal basis
for $A^2_{Mu,n}$, and it is characterized by the
properties that $zMapsto K_{Mu,n}(z,z_0)$ is in $A^2_{Mu,n}$ and
\begin{equation}\label{reprod}u(z_0)=
\big\langle u,K_{Mu,n}(\cdot,z_0)\big\rangle_{L^2({Mathbb C};Mu)} =\int_{Mathbb C}
u(z) \overline{K_{Mu,n}(z,z_0)} {Mathrm d} Mu(z), \qquad u\in
A^2_{Mu,n},\quad z_0\in{Mathbb C}.Mathrm end{equation}
For a given complex number $z_0$ we now consider the measure
\begin{equation}\label{bm}{Mathrm d}
B^{\langle z_0\rangle}_{Mu,n}(z)=\frac {\babs{K_{Mu,n}(z,z_0)}^2}
{K_{Mu,n}(z_0,z_0)}{Mathrm d}Mu(z),Mathrm end{equation} which we may call the
\textit{Berezin measure} associated with $Mu$, $n$ and $z_0$. The
reproducing property Mathrm eqref{reprod} applied to
$u=K_{Mu,n}(\cdot,z_0)$ implies that
$B_{Mu,n}^{\langle z_0\rangle}$ is a probability measure. In
classical physics,
$B^{\langle z_0\rangle}_{Mu,n}$ corresponds to a
point mass at $z_0$, so our interest focuses on how closely this
measure approximates the point mass.
{Mathbb S}ubsection{Weights.} By a \textit{weight} we mean a
measurable function $\phi:{Mathbb C}\to {Mathbb R}$ such that the measure
\begin{equation*}{Mathrm d}Mu_\phi(z)=Mathrm e^{-\phi(z)}{Mathrm d} A(z),Mathrm end{equation*}
is a finite measure on ${Mathbb C}$, where ${Mathrm d} A(z)={Mathrm d} x{Mathrm d} y/\pi$
with $z=x+\operatorname{Im}ag y$.
We write $L^2_\phi$ for the space $L^2({Mathbb C};Mu_\phi)$ and $A^2_\phi$
for $A^2({Mathbb C};Mu_\phi)$ and the norm of an element $u\in L^2_\phi$
will be denoted by
\begin{equation*}\|u\|_\phi^2=\int_{Mathbb C}\babs{u}^2Mathrm e^{-\phi}{{Mathrm d} A}.Mathrm end{equation*}
For a positive integer $n$, we frequently write
\begin{equation}\label{ospaces}L^2_{\phi,n}=\big\{u\in L^2_\phi;\,
u(z)={Mathcal O}L^p(\D,dA)ar\babs{z}^{n-1}\right )\quad \text{when}\quad
z\to\infty\big\}Mathrm end{equation} and
\begin{equation}\label{os2}A^2_{\phi,n}=L^2_{\phi,n}\cap
A^2_\phi=L^2_{\phi,n}\cap{Mathcal P}_n.Mathrm end{equation} Observe that
$L^2_{\phi,n}$ usually is non-closed in $L^2_\phi$, whereas the
finite-dimensional $A^2_{\phi,n}$ is closed in $L^2_\phi$.
{Mathbb S}ubsection{The weights considered.} \label{weigh} Let $Q:{Mathbb C}\to {Mathbb R}$ be a given
measurable function, which satisfies a growth condition of the form
\begin{equation}\label{gro}Q(z)\ge \rho\log\babs{z}^2,
\qquad |z|\ge C,
Mathrm end{equation}
for some positive numbers $\rho$ and $C$.
For a positive number $m$, we now consider weights $\phi_m$ of the
form $\phi_m=mQ$. By abuse of notation, we will in this context
sometimes refer to $Q$ as the weight. To simplify notation, we set
\begin{equation*}{H}_{m,n}=A^2_{mQ,n}=L^2_{mQ}\cap{Mathcal P}_nMathrm end{equation*}
and denote by $K_{m,n}$ the reproducing kernel for ${H}_{m,n}$.
For a given $z_0$, the corresponding $B_{m,n}^{\langle z_0\rangle}$
is defined accordingly,
cf. Mathrm eqref{bm}.
We think of $Q$ as being fixed while the parameters $m$ and $n$
vary, and also fix a number $\tau$ satisfying $0<\tau<\rho$.
To avoid bulky notation, it is customary to reduce the number of
parameters to one, by regarding $n=n(m)$ as a function of $m>0$.
We will
adopt this convention and study the behaviour of the measures
$B^{\langle z_0\rangle}_{m,n}$ when
$m\to\infty$ and $n-m\tau\to 0$.
Adding a real constant to $Q$ means that the inner product in
${H}_{m,n}$ is only chanced by a positive multiplicative
constant, and $K_{m,n}$ gets multiplied by the inverse of that
number. Hence the corresponding Berezin measures are unchanged, and
we can for example w.l.o.g. assume that $Q\ge 1$ on ${Mathbb C}$ when this
is convenient.
\begin{rem} Replacing $Q$ by $\tau^{-1}Q$ one can assume that
$\tau=1$. The most important case (e.g. in random matrix theory)
occurs when $\tau=1$ and $m=n$. However, we shall take on the
slightly more general approach (with three parameters $m$, $n$ and
$\tau$, instead of just $n$) in this note.
Mathrm end{rem}
{Mathbb S}ubsection{A word on notation.} For real $x$, we write $]x[$ for
the largest integer
which is strictly smaller than $x$. We write frequently
$${\partial}_z=\frac{{\partial}}{{\partial}
z}=\frac1{2}\bigg(\frac{{\partial}}{{\partial} x}-\operatorname{Im}ag\frac{{\partial}}{{\partial} y}\bigg),\qquad
{\overline{\partial}}_z=\frac{{\partial}}{{\partial} \bar z}=\frac1{2}\bigg(\frac{{\partial}}{{\partial}
x}+\operatorname{Im}ag\frac{{\partial}}{{\partial} y}\bigg),\qquad z=x+\operatorname{Im}ag y,$$ and use
${Mathbb D}elta={Mathbb D}elta_z$ to denote the normalized Laplacian
$${Mathbb D}elta_z={\partial}_z{\overline{\partial}}_z=\frac{1}{4}\bigg(
\frac{{\partial}^2}{{\partial} x^2}+\frac{{\partial}^2}{{\partial} y^2}\bigg).$$ We write
${Mathbb D}(z_0;r)$ for the open disk $\{z\in{Mathbb C};\,|z-z_0|<r\}$, and we
simplify the notation to ${Mathbb D}$ when $z_0=0$ and $r=1$. The boundary
of ${Mathbb D}(z_0,r)$ is denoted ${Mathbb T}(z_0,r)$. When $S$ is a subset of ${Mathbb C}$,
we write $S^\circ$ for the interior of $S$, and $\bar{S}$ for its
closure; the support of a continuous function $f$ is denoted by
${Mathbb S}upp f$. The symbol $A{Mathbb S}ubset B$ means that $A$ is a subset of
$B$, while $A\Subset B$ means that $A$ is a compact subset of $B$.
We write $\operatorname{dist\,}(A,B)$ for the Euclidean distance between $A$ and
$B$. The symbol \textit{a.e.} is short for "${Mathrm d} A$-almost
everywhere'', and \textit{p.m.} is short for "probability measure''.
Finally, we use the shorthand $L^2$ to denote the unweighted space
$L^2_0=L^2({Mathbb C};{{Mathrm d} A})$.
{Mathbb S}ubsection{Random normal matrices and the Coulomb gas.} This paper is a slight reworking of a preprint from arxiv.org which was written
in 2007 and early 2008. During the work, it became convenient to collect, in a separate paper, some estimates needed for the paper \cite{AHM}, which was written simultaneously. More precisely, we wanted to make clear a variant of some results on asymptotic expansions of (polynomial) Bergman kernels which were known in a several variables context (\cite{BBS}, \cite{B}), as well as some uniform estimates in the off-diagonal case.
The relevant results for our
applications in \cite{AHM} are primarily theorems \operatorname{Re}f{th3} and \operatorname{Re}f{flock}. A discussion containing related estimates which hold near the boundary will appear in our later publication \cite{AHM2}.
{Mathbb S}ection{Main results}
\label{sec-main}
{Mathbb S}ubsection{Berezin quantization.} Fix a
non-negative
weight $Q$ and two positive numbers $\tau$ and $\rho$,
$\tau<\rho$, such that the growth condition Mathrm eqref{gro} is
satisfied, and form the
measures
\begin{equation*}{Mathrm d} B^{\langle z_0\rangle}_{m,n}=
\berd^{\langle z_0\rangle}_{m,n}{{Mathrm d} A}\qquad \text{where}\qquad
\berd^{\langle z_0\rangle}_{m,n}(z)=\frac {\babs{K_{m,n}(z,z_0)}^2}
{K_{m,n}(z_0,z_0)}Mathrm e^{-mQ(z)}.Mathrm end{equation*} We shall refer to the
function $\berd_{m,n}^{\langle z_0\rangle}$ as the \textit{Berezin
kernel} associated with $m$, $n$ and $z_0$.
Let us now assume that $Q\in {Mathcal C}^2({Mathbb C})$.
In the first instance, we ask for which $z_0$ we have convergence
\begin{equation*}B^{\langle z_0\rangle}_{m,n}\to {\partial}elta_{z_0}\qquad
\text{as}\quad
m\to\infty \quad \text{and}\quad n-m\tau\to0,Mathrm end{equation*} in the
sense of measures. In terms of the \textit{Berezin transform}
\begin{equation*}Mathfrak B_{m,n} f(z_0)=\int_{Mathbb C} f(z){Mathrm d} B^{\langle
z_0\rangle}_{m,n}(z),
Mathrm end{equation*}
we are asking whether
\begin{equation}\label{q2}
Mathfrak B_{m,n} f(z_0)\to f(z_0) \qquad \text{for all}\quad
f\in{Mathcal C}_b({Mathbb C})\quad \text{as}\quad
m\to \infty \quad \text{and} \quad n-m\tau\to0.
Mathrm end{equation}
Let ${Mathbb S}etX$ be the set of points where $Q$ is strictly subharmonic,
$${Mathbb S}etX=\{{Mathbb D}elta Q>0\}.$$
We shall find that there exists a compact set ${Mathbb S}etS_\tau$ such that
Mathrm eqref{q2} holds for all $z_0$ in the interior of ${Mathbb S}etS_\tau\cap
{Mathbb S}etX$, while Mathrm eqref{q2} fails whenever $z_0\nuot\in {Mathbb S}etS_\tau$.
To define ${Mathbb S}etS_\tau$, we first need to introduce some notions from
weighted potential theory, cf. \cite{ST} and \cite{HM}. It is here
advantageous to slightly relax the regularity assumption on $Q$ and
assume that $Q$ is in the class ${Mathcal C}^{1,1}({Mathbb C})$ consisting of all
${Mathcal C}^1$-smooth functions with Lipschitzian gradients. We will
have frequent use of the simple fact that the distributional
Laplacian ${Mathbb D}elta F$ of a function $F\in {Mathcal C}^{1,1}({Mathbb C})$ makes
sense almost everywhere and is of class $L^\infty_{{\rm loc}}({Mathbb C})$.
Let ${\rm SH}_\tau$ denote the set of all subharmonic functions $f:{Mathbb C}\to
{Mathbb R}$ which satisfy a growth bound of the form
\begin{equation*}f(z)\le\tau\log\babs{z}^2+{Mathcal O}(1)\quad\text{as}
\quad z\to\infty.Mathrm end{equation*} The \textit{equilibrium potential}
corresponding to $Q$ and the parameter $\tau$ is defined by
\begin{equation*}\widehat{Q}_\tau(z)={Mathbb S}up\big\{f(z);\,\,f\in{\rm SH}_\tau
\quad\text{and}\quad f\le Q\quad\text{on}\quad
{Mathbb C}\big\}.Mathrm end{equation*} Clearly, $\widehat{Q}_\tau$ is subharmonic on
${Mathbb C}$. We now define ${Mathbb S}etS_\tau$ as the coincidence set
\begin{equation}\label{coincidence}{Mathbb S}etS_\tau=\{Q=\widehat{Q}_\tau\}.
Mathrm end{equation}
Evidently, ${Mathbb S}etS_\tau$ increases with $\tau$, and that $Q$ is
subharmonic on ${Mathbb S}etS_\tau^\circ$.
We shall need the following lemma.
\begin{lem}
\label{drspock} The function $\widehat{Q}_\tau$ is of class
${Mathcal C}^{1,1}({Mathbb C})$, and ${Mathbb S}etS_\tau$ is compact. Furthermore,
$\widehat{Q}_\tau$ is harmonic in ${Mathbb C}{Mathbb S}etminus {Mathbb S}etS_\tau$ and there
is a number $C$ such that $\widehat{Q}_\tau(z)\le
\tau\log_+\babs{z}^2+C$ for all $z\in{Mathbb C}.$
Mathrm end{lem}
\begin{proof} This is [\cite{HM}, Prop. 4.2, p. 10]
and [\cite{ST}, Th. I.4.7, p. 54].Mathrm end{proof}
\nuoindent It is easy to construct subharmonic minorants of critical
growth; for $C$ large enough, the function $zMapsto \tau\log_+L^p(\D,dA)ar
\babs{z}^2/C\right )$ is a minorant of $Q$ of class ${\rm SH}_\tau$. It
yields that
\begin{equation}\label{qtau}\widehat{Q}_\tau(z)=\tau\log\babs{z}^2+
{Mathcal O}(1)\quad\text{as}\quad z\to \infty.Mathrm end{equation}
\begin{prop}\label{prop1} Let $Q\in {Mathcal C}^{1,1}({Mathbb C})$ and $z_0\in{Mathbb C}$ an arbitrary point.
Then $B_{m,n}^{\langle z_0\rangle}(\Lambda) \to1$ as $m\to\infty$
and $n\le m\tau+ 1$ for every open neighbourhood $\Lambda$ of
${Mathbb S}etS_\tau$.
Mathrm end{prop}
\begin{proof} See Sect. \operatorname{Re}f{sec3}.Mathrm end{proof}
\nuoindent It follows from Prop. \operatorname{Re}f{prop1} that $B_{m,n}^{\langle
z_0\rangle}({Mathbb C}{Mathbb S}etminus\Lambda)\to 0$. Hence if $z_0\nuot\in
{Mathbb S}etS_\tau$, then Mathrm eqref{q2} fails in general, and
\begin{equation*}Mathfrak B_{m,n}f(z_0)\to 0\qquad \text{as}\quad
m\to\infty\quad \text{and} \quad n\le m\tau+1,Mathrm end{equation*} for
all $f\in{Mathcal C}_b({Mathbb C})$ such that ${Mathbb S}upp f\cap{Mathbb S}etS_\tau=Mathrm emptyset$.
The situation is entirely different when the point $z_0$ is in the
interior of ${Mathbb S}etS_\tau\cap{Mathbb S}etX$.
\begin{thm}\label{th1} Assume that $Q\in{Mathcal C}^2({Mathbb C})$ and let
$z_0\in {Mathbb S}etS_\tau^\circ\cap {Mathbb S}etX$. Then, for any
$f\in{Mathcal C}_b({Mathbb C})$, and any real number $M\ge 0$, the measures
$B^{\langle z_0\rangle}_{m,n}$ converge to ${\partial}elta_{z_0}$ as
$m\to\infty$ and $n\ge m\tau-M$.
Mathrm end{thm}
\begin{proof} See Sect. \operatorname{Re}f{p2}. See also Rem. \operatorname{Re}f{smick}. Mathrm end{proof}
{Mathbb S}ubsection{A more elaborate estimate for the Berezin kernel.}
Th. \operatorname{Re}f{th1} suggests that if $z_0$ is a given point in the
interior of ${Mathbb S}etS_\tau\cap {Mathbb S}etX$, $m$ is large, and $n\ge
m\tau-1$, then the density $\berd_{m,n}^{\langle z_0\rangle}(z)$
should attain its maximum for $z$ close to $z_0$. The following
theorem implies that this is the case, and gives a good control for
$\berd_{m,n}^{\langle z_0\rangle}$ in the critical case, when
$n-m\tau\to 0$.
\begin{thm} \label{th1.5} Assume that $Q\in{Mathcal C}^2({Mathbb C})$. Let $K$ be a compact
subset of ${Mathbb S}etS_\tau^\circ\cap{Mathbb S}etX$ and fix a non-negative number
$M$. Put
\begin{equation*}d_K=\operatorname{dist\,}(K,{Mathbb C}{Mathbb S}etminus({Mathbb S}etS_\tau\cap{Mathbb S}etX))\qquad \text{and}\qquad
a_K=\inf\{{Mathbb D}elta Q(z);\, \operatorname{dist\,}(z,K)\le d_K/2\}.Mathrm end{equation*} Then
there exists positive numbers $C$ and $Mathrm epsilon$ such that
\begin{equation*}
\berd^{\langle z_0\rangle}_{m,n}(z)\le C mMathrm e^{-Mathrm epsilon
{Mathbb S}qrt{m}Min\{d_K,\babs{z-z_0}\}}Mathrm e^{-m(Q(z)-\widehat{Q}_\tau(z))},\qquad
z_0\in K,\,\, z\in {Mathbb C},\,\, m\tau-M\le n\le m\tau+1.
Mathrm end{equation*}
Here $C$ depends only on $K$, $M$ and $\tau$, while $Mathrm epsilon$ only
depends on $M$, $a_K$ and $\tau$.
Mathrm end{thm}
\begin{proof} See Sect. \operatorname{Re}f{point}.
Mathrm end{proof}
A similar result was proved independently by Berman \cite{B3} after the completion of this note.
\begin{rem} \label{smick} For fixed $M$ and $\tau$, the number
$Mathrm epsilon$ can be chosen proportional to $a_K$. A related result
with much more precise information on the dependence of $C$ and
$Mathrm epsilon$ on the various parameters in question is given in Th.
\operatorname{Re}f{propn1} and the subsequent remark.
We also remark that our proof for Th. \operatorname{Re}f{th1.5} is very different
from that for Th. \operatorname{Re}f{th1}. Thus Th. \operatorname{Re}f{th1.5} gives an
alternative method for obtaining Th. \operatorname{Re}f{th1} in the critical case
when $n-m\tau\to 0$.
Mathrm end{rem}
{Mathbb S}ubsection{Gaussian convergence.} Fix a point $z_0\in {Mathbb S}etX$. It
will be convenient to introduce the \textit{normalized}
Berezin measure $\widehat{B}_{m,n}^{\langle z_0\rangle}$ by
\begin{equation}
{Mathrm d}\widehat B^{\langle z_0\rangle}_{m,n}=\widehat{\berd}_{m,n}^{\langle
z_0\rangle}{{Mathrm d} A}\quad \text{where}\quad \widehat{\berd}_{m,n}^{\langle
z_0\rangle}(z)=\frac 1 {m{Mathbb D}elta Q(z_0)} {\berd^{\langle
z_0\rangle}_{m,n}\bigg(z_0+\frac{z}{{Mathbb S}qrt{m{Mathbb D}elta Q(z_0)}}\bigg)}
.Mathrm end{equation} We denote the standard Gaussian p.m. on ${Mathbb C}$ by
\begin{equation*}
{Mathrm d} P(z)=Mathrm e^{-\babs{z}^2}{Mathrm d} A(z).
Mathrm end{equation*}
We have the following CLT, which gives much more precise information
than Th. \operatorname{Re}f{th1}. We will settle for stating it for ${Mathcal C}^\infty$ weights $Q$
which are \textit{real-analytic} in a neighbourhood of ${Mathbb S}etS_\tau$, but remark that, using
well-known methods, our proof can be extended to cover the case of,
say, ${Mathcal C}^\infty$-smooth weights. We will give further details on
possible generalizations later on.
\begin{thm}\label{th2} Assume that $Q$ is real-analytic in a neighbourhood of ${Mathbb S}etS_\tau$.
Fix a compact subset $K\Subset {Mathbb S}etS_\tau^\circ\cap{Mathbb S}etX$, a point
$z_0\in K$, and a number $M\ge 0$. Then we have
\begin{equation}\label{gc1}
\int_{Mathbb C}\babs{\widehat{\berd}_{m,n}^{\langle z_0\rangle}(z)
-Mathrm e^{-\babs{z}^2}}{{Mathrm d} A}(z)\to0\qquad\text{as}\quad
m\to\infty\quad \text{and}\quad n\ge m\tau-M,Mathrm end{equation} with
uniform convergence for $z_0\in K$. Equivalently, $\widehat
B^{\langle z_0\rangle}_{m,n}\to P$ as
$m\to\infty$ and $n\ge m\tau-M$.
Mathrm end{thm}
\begin{proof} See Sect. \operatorname{Re}f{p2}. Mathrm end{proof}
{Mathbb S}ubsection{The expansion formula.} Most of our results in this paper
rely on a suitable approximation for
$K_{m,n}(z,w)$ when $z,w$ are both close to some point
$z_0\in{Mathbb S}etX$, and $m$ and $n$ are large. We will here assume that
$Q$ be \textit{real-analytic} in a neighbourhood of ${Mathbb S}etS_\tau$.
An adequate and rather far-reaching approximation formula was
recently stated by Berman \cite{B}, p. 9, depending on methods from
\cite{BBS}; we will here only require a special case of his result.
We will need to introduce some definitions.
For subsets $S{Mathbb S}ubset{Mathbb C}$ we write
$${\overline{\text{\rm diag}}}(S)=\big\{(z,\bar z)\in{Mathbb C}^2;\,z\in S\big\}.$$
That $Q$ is real-analytic in a neighbourhood of $S$ means that there exists a unique
holomorphic function ${Mathbb Q}ext$ defined in a neighbourhood of ${\overline{\text{\rm diag}}}(S)$ in ${Mathbb C}^2$ such that
\begin{equation*}
{Mathbb Q}ext(z,\bar{z})=Q(z)\quad\text{for all}\quad z\in{Mathbb C}.
Mathrm end{equation*}
Explicitly, one has
\begin{equation}\label{psi2}{Mathbb Q}ext(z+h,\overline{z+k})={Mathbb S}um_{i,j=0}^\infty
[{\partial}^i{\overline{\partial}}^j Q](z)\frac {h^i \bar{k}^j} {i!j!}Mathrm end{equation} when
$h$ and $k$ are sufficiently close to zero. In particular,
$[{\partial}_1^i{\partial}_2^j{Mathbb Q}ext](z,\bar{z})={\partial}^i{\overline{\partial}}^j Q(z)$ for all $z\in{Mathbb C}$.
Here, ${\partial}_1$ and ${\partial}_2$ denote differentiation w.r.t. the first and
second coordinates.
\begin{defn}
\label{def1} Let ${b}_0$ and ${b}_1$ denote the functions
\begin{equation*}{b}_0={\partial}_1{\partial}_2{Mathbb Q}ext,\quad
{b}_1=\frac12{\partial}_1{\partial}_2 \log\big[{\partial}_1{\partial}_2 {Mathbb Q}ext\big]=
\frac{{\partial}_1^2{\partial}_2^2 {Mathbb Q}ext {\partial}_1{\partial}_2 {Mathbb Q}ext -{\partial}_1^2{\partial}_2 {Mathbb Q}ext{\partial}_1{\partial}_2^2
{Mathbb Q}ext} {2\big[{\partial}_1{\partial}_2 {Mathbb Q}ext\big]^2}.Mathrm end{equation*} The function
${b}_1$ is well-defined where there is no division by $0$, in
particular in a neighborhood of ${\overline{\text{\rm diag}}}({Mathbb S}etX)$. Along the
anti-diagonal, we have
\begin{equation*}
{b}_0(z,\bar{z})={Mathbb D}elta Q(z)\quad \text{and}\quad
{b}_1(z,\bar{z})=\frac 12{Mathbb D}elta\log{Mathbb D}elta Q(z),\quad z\in {Mathbb S}etX.
Mathrm end{equation*}
We note for later use that ${b}_0$ and ${b}_1$ are connected via
\begin{equation}\label{funrel}{b}_1(z,w)=\frac 1 2 \frac {\partial} {{\partial} w}L^p(\D,dA)ar
\frac 1{{b}_0(z,w)}\frac {\partial} {{\partial} z}{b}_0(z,w)\right ).Mathrm end{equation}
We define the \textit{first order approximating Bergman kernel}
$K_m^1(z,w)$ by
\begin{equation*}
K_m^1(z,w)=L^p(\D,dA)ar m{b}_0(z,\bar{w})+{b}_1(z,\bar{w})\right ) Mathrm e^{m
{Mathbb Q}ext(z,\bar{w})}.Mathrm end{equation*}
Mathrm end{defn}
We have the following theorem.
\begin{thm}\label{th3} Assume that $Q$ is real-analytic in ${Mathbb C}$.
Let $K$ be a compact subset of ${Mathbb S}etS_\tau^\circ\cap{Mathbb S}etX$. Fix a
point $z_0\in K$ and a number $M\ge 0$. Then there exists a number
$m_0$ depending only on $M$ and $\tau$ and
a positive number $Mathrm eps$ depending only on $K$ and $M$ such that
\begin{equation*}
\babs{K_{m,n}(z,w)-K_m^1(z,w)}Mathrm e^{-m(Q(z)+Q(w))/2}\le C m^{-1},
\qquad z_0\in K,\quad z,w\in {Mathbb D}(z_0;Mathrm eps),
Mathrm end{equation*}
for all $m\ge m_0$ and $n\ge m\tau-M$.
Here $C$ is an absolute constant (depending only on $Q$).
In particular, by restricting
to the diagonal, we get
\begin{equation}\label{ber}\babs{K_{m,n}(z,z)Mathrm e^{-mQ(z)}-
L^p(\D,dA)ar m{Mathbb D}elta Q(z)+\frac 1 2{Mathbb D}elta\log{Mathbb D}elta Q(z)\right )}\le
\frac{C}{m}, \qquad z\in K,
Mathrm end{equation}
when $m\ge m_0$ and $n\ge m\tau-M$.
Mathrm end{thm}
\begin{proof} See Sect. \operatorname{Re}f{proof}.
See also Remark \operatorname{Re}f{bel} below. Mathrm end{proof}
\begin{rem} \label{bel}
We want to stress that Th. \operatorname{Re}f{th3} has a long history; analogous
expansions are well-known for weighted Bergman kernels, see, for
instance, \cite{F}, \cite{dBMS}, \cite{J}, \cite{E}, \cite{BBS},
\cite{B}, and the references therein. Moreover, as we mentioned
above, Th. \operatorname{Re}f{th3} is a slight modification of a more general
result in several complex variables stated by Berman \cite{B}, Th.
3.8, which may be obtained by adapting the methods from \cite{BBS}.
In our proof, we make frequent use of ideas and techniques developed
in \cite{B}, \cite{B2}, \cite{BBS}, and in the book \cite{S}.
Mathrm end{rem}
\begin{rem} (Simple properties of $K_m^1$.)
(1) Since ${b}_0$, ${b}_1$ and ${Mathbb Q}ext$ are real on the
anti-diagonal, a suitable version of the reflection principle
implies that $K_m^1$ is Hermitian, that is, $\overline{
K_m^1(z,w)}=K_m^1(w,z)$.
(2) Using the Taylor series for ${Mathbb Q}ext$ at $(z,\bar{z})$ (see
Mathrm eqref{psi2}) and ditto for $Q$ and $z$, we get
\begin{equation}\label{bbs}
2\operatorname{Re} {Mathbb Q}ext(z,\bar{w})-Q(z)-Q(w)=-{Mathbb D}elta
Q(w)\babs{w-z}^2+R(z,w),Mathrm end{equation} where
$R(z,w)={Mathcal O}\big(\babs{w-z}^3\big)$ for $z$, $w$ in a sufficiently
small neighbourhood of $z_0$, and the ${Mathcal O}$ is uniform for $z_0$
in a fixed compact subset of ${Mathbb S}etX$. In case ${Mathbb D}elta Q(z_0)>0$, it
follows that \begin{equation}\label{uniq}2\operatorname{Re}
{Mathbb Q}ext(z,\bar{w})-Q(z)-Q(w)\le-{\partial}elta_0\babs{w-z}^2,\qquad z,w\in
{Mathbb D}(z_0;2Mathrm eps),Mathrm end{equation} with ${\partial}elta_0=\frac12{Mathbb D}elta Q(z_0)>0$,
provided
that the positive number $Mathrm eps$ is chosen small enough. More
generally, if we fix a compact subset $K\Subset {Mathbb S}etX$, and put
\begin{equation*}
{\partial}elta_0=\frac 1 2\inf_{\zeta\in K}\,\{{Mathbb D}elta
Q(\zeta)\},Mathrm end{equation*} we may find an $Mathrm eps>0$ such that
Mathrm eqref{uniq} holds for all $z_0\in K$. With a perhaps somewhat
smaller $Mathrm eps>0$, we may also ensure that
the functions ${b}_0$ and ${b}_1$ are bounded and holomorphic in
the set $\{(z,w);\, z,\bar{w}\in{Mathbb D}(z_0;2Mathrm eps),\, z_0\in K\}$.
We infer that
\begin{equation}\label{lead}\babs{K_m^1(z,w)}^2Mathrm e^{-m(Q(z)+Q(w))}
\le Cm^2Mathrm e^{-m{\partial}elta_0\babs{z-w}^2},\qquad z,w\in {Mathbb D}(z_0;2Mathrm eps),
\quad z_0\in K,
Mathrm end{equation}
with a number $C$ depending only on $K$ and $Mathrm eps$.
Mathrm end{rem}
{Mathbb S}ubsection{Extensions and possible generalizations.} We discuss
possible extensions of Th. \operatorname{Re}f{th3} (and also of Th. \operatorname{Re}f{th2}).
Again, these extensions
(at least of Th. \operatorname{Re}f{th3}) are essentially implied by a result
[\cite{B}, Th. 3.8], stated by R. Berman.
For a given positive integer $k$, one may define a \textit{$k$-th
order approximating Bergman kernel}
\begin{equation*}K_m^k(z,w)=L^p(\D,dA)ar m{b}_0(z,\bar w)+{b}_1(z,\bar
w)+\ldots+m^{-k+1}{b}_k(z,\bar w)\right )Mathrm e^{m{Mathbb Q}ext(z,\bar
w)},Mathrm end{equation*} for $z$ and $w$ in a neighbourhood of
${\overline{\text{\rm diag}}}({Mathbb S}etX)$ where ${b}_i$ are certain holomorphic functions
defined in a neighbourhood of ${\overline{\text{\rm diag}}}({Mathbb S}etX)$. It can be shown that
for $z_0\in{Mathbb S}etX$ there exists $Mathrm eps>0$ such that
\begin{equation*}\babs{K_{m,n}(z,w)-K_m^k(z,w)}^2Mathrm e^{-m(Q(z)+Q(w))}\le
Cm^{-2k},Mathrm end{equation*} whenever $z,w\in{Mathbb D}(z_0;Mathrm eps)$ and $n\ge
m\tau-1$. The coefficients ${b}_i$ can in principle be determined
from a recursion formula involving partial differential equations of
increasing order, compare with [Berman et al. \cite{BBS}, (2.15),
p.9], where a closely related formula is given. However, the
analysis required for calculating higher order coefficients
${b}_i$ for $i\ge 2$ seems to be quite involved, and the first
order approximation seems to be sufficient for many practical
purposes, cf. \cite{AHM}. We therefore prefer a more direct approach
here.
Th. \operatorname{Re}f{th3} can also be generalized in another direction -- one
may relax the assumption that $Q$ be real-analytic, and instead
assume that e.g. $Q\in{Mathcal C}^\infty({Mathbb C})$. In this case, the functions
${b}_i$ and ${Mathbb Q}ext$ will be almost-holomorphic at the
anti-diagonal (cf. e.g. \cite{dBMS} for a relevant discussion of
such functions). The modifications needed for proving Th. \operatorname{Re}f{th3}
in this more general case are based on standard arguments; they are
essentially as outlined in [\cite{BBS}, Subsect. 2.6 and p. 15]. We
leave the details to the interested reader. However, the reader
should note that, using this generalized version of Th. \operatorname{Re}f{th3},
one may easily extend our proof of Th. \operatorname{Re}f{th2}, to the case of
${Mathcal C}^\infty$-smooth weights.
{Mathbb S}ubsection{The Bargmann--Fock case and harmonic measure.} When
$z_0\in{Mathbb C}{Mathbb S}etminus({Mathbb S}etS_\tau^\circ\cap {Mathbb S}etX)$, Prop. \operatorname{Re}f{prop1}
yields that the measures $B^{\langle z_0\rangle}_{m,n}$ tend to
concentrate on ${Mathbb S}etS_\tau$ as
$m\to\infty$ and $n-m\tau\to 0$.
However our general results provide no further information regarding
the asymptotic distribution.
We now specialize to the Bargmann--Fock weight $Q(z)=\babs{z}^{2}$.
We then obviously have ${Mathbb S}etX={Mathbb C}$. It will be convenient to
introduce the functions (truncated exponentials)
\begin{equation*}E_k(z)={Mathbb S}um_{j=0}^k\frac {z^j} {j!}.Mathrm end{equation*}
It is then easy to check that $K_{m,n}(z,w)=mE_{n-1}(mz\bar{w}),$
${Mathbb S}etS_\tau=\overline{{Mathbb D}}(0;{Mathbb S}qrt{\tau}),$ and
$\widehat{Q}_\tau(z)=\tau+\tau\logL^p(\D,dA)ar\babs{z}^2/\tau\right )$ when
$\babs{z}^2\ge\tau.$ We infer that
\begin{equation}\label{bfock}{Mathrm d} B^{\langle z_0\rangle}_{m,n}(z)=
m\frac{\babs{E_{n-1}(mz\bar{z}_0)}^2} {E_{n-1}(m\babs{z_0}^2)}
Mathrm e^{-m\babs{z}^2}{Mathrm d} A(z).
Mathrm end{equation}
Let ${Mathbb C}^*={Mathbb C}\cup\{\infty\}$ denote the extended plane.
\begin{thm}\label{th5} Let $Q(z)=\babs{z}^2$ and $z_0\in{Mathbb C}{Mathbb S}etminus {Mathbb S}etS_\tau$. Then, as
$m\to\infty$ and $n/m\to\tau$, the measures $B^{\langle
z_0\rangle}_{m,n}$ converge to harmonic measure at $z_0$ with
respect to ${Mathbb C}^*{Mathbb S}etminus{Mathbb S}etS_\tau$.
Mathrm end{thm}
\begin{proof} See Sect. \operatorname{Re}f{new}. Mathrm end{proof}
\begin{rem} The above theorem is a special case of Theorem 8.7 in \cite{AHM}.
Mathrm end{rem}
{Mathbb S}ubsection{Weighted $L^2$ estimates for the ${\overline{\partial}}$-equation with a
growth constraint.} Our analysis depends in an essential way on
having a good control for the norm-minimal solution $u_*$ in the
space $L^2_{mQ,n}$ to the ${\overline{\partial}}$-equation ${\overline{\partial}} u=f$, where $f$ is
a given suitable test-function such that ${Mathbb S}upp f{Mathbb S}ubset
{Mathbb S}etS_\tau\cap{Mathbb S}etX$. We would not have a problem if $u_*$ were just
required to be of class $L^2_{mQ}$ (and not $L^2_{mQ,n}$), for then
an elementary version of the well-known $L^2$ estimates of
H\"ormander apply, see e.g. \cite{H} or \cite{H2}. The requirement
that $u_*$ be of restricted growth gives rise to some difficulties,
but we shall see in Sect. \operatorname{Re}f{l2estimates} (notably Th. \operatorname{Re}f{strb})
that the classical estimates can be adapted to our situation.
(Estimates very similar to ours were given independently by Berman in \cite{B3} after this note was completed.)
{Mathbb S}ection{Preparatory lemmas} \label{sec3}
{Mathbb S}ubsection{Preliminaries.} In this section, we prove Prop. \operatorname{Re}f{prop1}. At the
same time we will get the opportunity to display some elementary
inequalities which will be used in later sections.
For the
results obtained here, it is enough to suppose that $Q$ is
${Mathcal C}^{1,1}$-smooth (and satisfies the growth assumption
Mathrm eqref{gro}). Thus ${Mathbb D}elta Q$ makes sense almost everywhere and is
of class $L^\infty_{\rm loc}({Mathbb C})$.
{Mathbb S}ubsection{Subharmonicity estimates.} We start by giving two
simple lemmas which are based on the sub-mean property of
subharmonic functions.
\begin{lem} \label{berndt} Let $\phi$ be a weight of class
${Mathcal C}^{1,1}({Mathbb C})$ and put
$A=Mathrm esssup\{{Mathbb D}elta\phi(\zeta);\,\zeta\in{Mathbb D}\}.$ Then if $u$ is
bounded and holomorphic in ${Mathbb D}$, we have that
$$|u(0)|^2Mathrm e^{-\phi(0)}\le\int_{Mathbb D}|u(\zeta)|^2Mathrm e^{-\phi(\zeta)}Mathrm e^{A|\zeta|^2}{{Mathrm d} A}(\zeta)
\leMathrm e^A\int_{Mathbb D}|u(\zeta)|^2Mathrm e^{-\phi(\zeta)}{{Mathrm d} A}(\zeta).$$
Mathrm end{lem}
\begin{proof} Consider the function
$F(\zeta)=|u(\zeta)|^2Mathrm e^{-\phi(\zeta)+A|\zeta|^2},$ which satisfies
$${Mathbb D}elta \log F={Mathbb D}elta\log|u|^2-{Mathbb D}elta\phi+A\ge0\qquad \text{a.e.\, on}\quad {Mathbb D},$$
making $F$ logarithmically subharmonic. But then $F$ is subharmonic
itself, and so
$$F(0)\le\int_{Mathbb D} F(z){{Mathrm d} A}(z).$$
The assertion of the lemma is immediate from this.
Mathrm end{proof}
\begin{lem} \label{lemm2} Let $Q\in {Mathcal C}^{1,1}({Mathbb C})$ and fix ${\partial}elta>0$ and $z\in{Mathbb C}$ and put
$A=A_{z,{\partial}elta}=Mathrm esssup\{{Mathbb D}elta Q(\zeta);\, \zeta\in{Mathbb D}(z;{\partial}elta)\}.$
Also, let $u$ be holomorphic and bounded in ${Mathbb D}(z;{\partial}elta/{Mathbb S}qrt{m})$.
Then
\begin{equation*}\babs{u(z)}^2e^{-mQ(z)}\le
m{\partial}elta^{-2}{Mathrm e^{A{\partial}elta^2}}
\int_{{Mathbb D}(z;{\partial}elta/{Mathbb S}qrt{m})}\babs{u(\zeta)}^2e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta).
Mathrm end{equation*}
Mathrm end{lem}
\begin{proof} The assertion follows if we make the
change of variables $\zeta=z+{\partial}elta\xi/{Mathbb S}qrt{m}$ where $\zeta\in
{Mathbb D}(z;{\partial}elta/{Mathbb S}qrt{m})$ and $\xi\in{Mathbb D}$, and apply Lemma \operatorname{Re}f{berndt}
with the weight $\phi(\xi)=mQ(\zeta)$.
Mathrm end{proof}
\nuoindent We note the following consequence of the subharmonicity
estimates. We will frequently need it in later sections.
\begin{lem} \label{lemm3} Let $K$ be a compact subset of ${Mathbb C}$ and
${\partial}elta$ a given positive number. We put
\begin{equation*}K_{\partial}elta=\{z\in{Mathbb C};\, \operatorname{dist\,}(z,K)\le {\partial}elta\}\qquad \text{and}
\qquad A=Mathrm esssup\{{Mathbb D}elta Q(z);\, z\in K_{\partial}elta\}.Mathrm end{equation*}
Then, for all $m,n\ge 1$,
\begin{equation*}\babs{K_{m,n}(z,w)}^{2}Mathrm e^{-mQ(z)}\le m{\partial}elta^{-2}Mathrm e^{{\partial}elta^2
A}\int_{{Mathbb D}(z;{\partial}elta/{Mathbb S}qrt{m})}\babs{K_{m,n}(\zeta,w)}^2Mathrm e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta),\quad
z\in K,\quad w\in {Mathbb C}.Mathrm end{equation*}
Mathrm end{lem}
\begin{proof} Apply Lemma \operatorname{Re}f{lemm2} to $u(\zeta)=K_{m,n}(\zeta,w)$.
Mathrm end{proof}
{Mathbb S}ubsection{A weak maximum principle for weighted polynomials.}
Maximum principles for weighted polynomials have a long history, see
e.g. \cite{ST}, Chap. III. The following simple lemma will suffice
for our present purposes; it is a consequence of \cite{ST}, Th.
III.2.1. (Recall that ${Mathbb S}etS_\tau$ denotes the coincidence set
$\{Q=\widehat{Q}_\tau\}$, see Mathrm eqref{coincidence}.)
\begin{lem} \label{wmax} Suppose that a polynomial $u$ of degree at most
$n-1$ satisfies $\babs{u(z)}^2Mathrm e^{-mQ(z)}\le 1$ on
${Mathbb S}etS_{\tau(m,n)}$, where $\tau(m,n)=(n-1)/m<\rho$. Then
$\babs{u(z)}^2Mathrm e^{-m\widehat{Q}_{\tau(m,n)}(z)}\le 1$ on ${Mathbb C}$.
Mathrm end{lem}
\begin{proof} Put $q(z)=m^{-1}\log\babs{u(z)}^2$. The assumptions
on $u$ imply that $q\in{\rm SH}_{\tau(m,n)}$ and that $q\le Q$ on
${Mathbb S}etS_{\tau(m,n)}$. Hence $q\le\widehat{Q}_{\tau(m,n)}$ on ${Mathbb C}$, as
desired.
Mathrm end{proof}
\begin{lem} \label{lemm4} Let
$u\in{H}_{m,n}$, and suppose that $n\le m\tau+1$. Then
\begin{equation*}\babs{u(z)}^2\le mMathrm e^A\|u\|_{mQ}^2Mathrm e^{m\widehat{Q}_\tau(z)},
Mathrm end{equation*}
where $A$ denotes the essential supremum of ${Mathbb D}elta Q$ over the set
$\{z\in{Mathbb C};\, \operatorname{dist\,}(z,{Mathbb S}etS_\tau)\le 1\}$.
Mathrm end{lem}
\begin{proof} The assertion that $n\le m\tau+1$ is equivalent to that
$\tau(m,n)\le \tau$ where $\tau(m,n)=(n-1)/m$. Thus
${Mathbb S}etS_{\tau(m,n)}{Mathbb S}ubset{Mathbb S}etS_\tau$ and $\widehat{Q}_{\tau(m,n)}\le
\widehat{Q}_\tau$. An application of Lemma \operatorname{Re}f{lemm2} with ${\partial}elta=1$
now gives
\begin{equation*}\babs{u(z)}^2Mathrm e^{-mQ(z)}
\le mMathrm e^A\int_{{Mathbb D}(z;1)}\babs{u(\zeta)}^2
Mathrm e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta)\le
mMathrm e^A\int_{Mathbb C}\babs{u(\zeta)}^2Mathrm e^{-mQ(\zeta)} {{Mathrm d} A}(\zeta),\quad
z\in{Mathbb S}etS_{\tau}.
Mathrm end{equation*}
As a consequence, the same estimate holds on ${Mathbb S}etS_{\tau(m,n)}$.
We can thus apply Lemma \operatorname{Re}f{wmax}. It yields that
\begin{equation*}\babs{u(z)}^2\le mMathrm e^A\|u\|_{mQ}^2Mathrm e^{m\widehat{Q}_{\tau(m,n)}(z)},
\quad z\in{Mathbb C}.Mathrm end{equation*}
The desired assertion follows, since $\widehat{Q}_{\tau(m,n)}\le\widehat{Q}_\tau$.
Mathrm end{proof}
{Mathbb S}ubsection{The proof of Proposition \operatorname{Re}f{prop1}} Fix two points $z$ and
$z_0$ in ${Mathbb C}$. We apply Lemma \operatorname{Re}f{lemm4} to the polynomial
\begin{equation*}u(\zeta)=\frac {K_{m,n}(\zeta,z_0)}{{Mathbb S}qrt{K_{m,n}(z_0,z_0)}},
Mathrm end{equation*}
which is of class ${H}_{m,n}$ and satisfies $\|u\|_{mQ}=1$. It
yields that $\babs{u(z)}^2\le mMathrm e^AMathrm e^{\widehat{Q}_\tau(z)}$, or
\begin{equation}\label{cerd}
\berd_{m,n}^{\langle w\rangle}(z)=\babs{u(z)}^2Mathrm e^{-mQ(z)} \le
mMathrm e^{A}Mathrm e^{m(\widehat{Q}_\tau(z)-Q(z))},\quad z\in{Mathbb C},\, n\le
m\tau+1.Mathrm end{equation}
Now let $\Lambda$ be an open neighborhood of ${Mathbb S}etS_\tau$. Since
$Q>\widehat{Q}_\tau$ on ${Mathbb C}{Mathbb S}etminus {Mathbb S}etS_\tau$, the continuity of
the functions involved coupled with the growth conditions
Mathrm eqref{qtau} and Mathrm eqref{gro} yield that $Q-\widehat{Q}_\tau$ is
bounded below by a positive number on ${Mathbb C}{Mathbb S}etminus\Lambda$. It
follows that \begin{equation*}B_{m,n}^{\langle
z_0\rangle}({Mathbb C}{Mathbb S}etminus\Lambda)=
\int_{{Mathbb C}{Mathbb S}etminus\Lambda}\berd^{\langle z_0\rangle}_{m,n}{Mathrm d} A\to
0Mathrm end{equation*} as
$m\to\infty$ and $n\le m\tau+1$.
Since $B_{m,n}^{\langle z_0\rangle}$ is a p.m. on ${Mathbb C}$, the
assertion of Prop. \operatorname{Re}f{prop1} follows.
$\qed$
{Mathbb S}ubsection{An estimate for the one-point function.} Our proof of
Prop. \operatorname{Re}f{prop1} implies a useful estimate for the one-point
function $zMapsto K_{m,n}(z,z)Mathrm e^{-mQ(z)}$. The following result is
implicit in \cite{B}.
\begin{prop}\label{lemma1} Suppose that $n\le m\tau+1$.
Then there exists a number $C$ depending only on $\tau$ such that
\begin{equation}\label{berg1}K_{m,n}(z,z)Mathrm e^{-mQ(z)}\le Cm
Mathrm e^{-m(Q(z)-\widehat{Q}_\tau(z))},\quad z\in{Mathbb C},Mathrm end{equation}
\begin{equation}\label{berg2}\babs{K_{m,n}(z,w)}^2Mathrm e^{-m(Q(z)+Q(w))}\le
Cm^2Mathrm e^{-m(Q(z)-\widehat{Q}_\tau(z))}Mathrm e^{-m(Q(w)-\widehat{Q}_\tau(w))},\quad
z,w\in{Mathbb C}.
Mathrm end{equation}
In particular, $\babs{K_{m,n}(z,w)}^2Mathrm e^{-m(Q(z)+Q(w))}\le Cm^2$ for
all $z$ and $w$.
Mathrm end{prop}
\begin{proof} The
function $\berd_{m,n}^{\langle z_0\rangle}(z)$ in the diagonal case
$z=z_0$ reduces to
\begin{equation*}\berd_{m,n}^{\langle z\rangle}(z)=K_{m,n}(z,z)Mathrm e^{-mQ(z)}.
Mathrm end{equation*}
Thus the estimate Mathrm eqref{cerd} implies
\begin{equation*}K_{m,n}(z,z)Mathrm e^{-mQ(z)}\le mMathrm e^AMathrm e^{m(\widehat{Q}_\tau(z)-Q(z))},
Mathrm end{equation*}
which proves Mathrm eqref{berg1}. In order to get Mathrm eqref{berg2} it now
suffices to use the Cauchy--Schwarz inequality
$\babs{K_{m,n}(z,w)}^2\le K_{m,n}(z,z)K_{m,n}(w,w)$ and apply
Mathrm eqref{berg1} to the two factors in the right hand side.
Mathrm end{proof}
{Mathbb S}ubsection{Cut-off functions.} \label{cutoff} We will in the following
frequently use cut-off functions with various properties. For
convenience of the reader who may not be familiar with these
details, we collect the necessary facts here.
Given any numbers ${\partial}elta>0$, $r>0$ and $C>1$, there exists $\chi\in
\coity({Mathbb C})$ such that $\chi=1$ on ${Mathbb D}(0;{\partial}elta)$, $\chi=0$ outside
${Mathbb D}(0;{\partial}elta(1+r))$, $\chi\le 1$ and $|{\overline{\partial}}\chi|^2\le (C/r^2)\chi$
on ${Mathbb C}$. \footnote{Such a $\chi$ can be obtained by standard
regularization applied to the Lipschitzian function
which equals $(1-({\partial}elta^{-1}\babs{z}-1)/r)^2$ for $1\le\babs{z}\le
r$, and is otherwise locally constant on ${Mathbb C}$.} Moreover, with such
a choice for $\chi$, it follows that $|{\overline{\partial}}\chi|^2\le
(C/r^2){\partial}elta^{-2}\chi$ on ${Mathbb C}$. It is then easy to check that
$\|{\overline{\partial}}\chi\|_{L^2}^2\le C(1+2/r)$.
Later on, we will often use the values ${\partial}elta=3Mathrm eps/2$ and
${\partial}elta(1+r)=2Mathrm eps$, where $Mathrm eps>0$ is given. We may then arrange
that
$\|{\overline{\partial}}\chi\|_{L^2}\le 3$.
{Mathbb S}ection{Weighted estimates for the ${\overline{\partial}}$-equation with a
growth constraint}\label{l2estimates}
{Mathbb S}ubsection{General introduction; the Bergman projection and the
${\overline{\partial}}$-equation.} \label{init} Let $\phi$ be a weight on ${Mathbb C}$ (cf.
Subsect. \operatorname{Re}f{weigh}). We assume throughout that $\phi$ is of class
${Mathcal C}^{1,1}({Mathbb C})$ (so that ${Mathbb D}elta\phi\in L^\infty_{{\rm
loc}}({Mathbb C})$), and that $\int_{Mathbb C}Mathrm e^{-\phi}{Mathrm d} A<\infty$ (so that
$L^2_\phi$ contains the constant functions). Also fix a positive
integer $n$ and recall the definition of the "truncated'' spaces
$L^2_{\phi,n}$ and $A^2_{\phi,n}$ (see Mathrm eqref{ospaces} and
Mathrm eqref{os2}).
Let $K_{\phi,n}$ denote the
reproducing kernel for $A^2_{\phi,n}$, and let
${P}_{\phi,n}:L^2_\phi\to A^2_{\phi,n}$ be the orthogonal
projection,
\begin{equation}\label{ooproj}{P}_{\phi,n}u(w)=\int_{Mathbb C}
u(z)K_{\phi,n}(w,z)Mathrm e^{-\phi(z)}{{Mathrm d} A}(z),\quad u\in
L^2_{\phi}.Mathrm end{equation} We will in later sections frequently need
to estimate ${P}_{\phi,n}u(w)$, especially when $u$ is holomorphic
in a neighbourhood of $w$.
Now note that for $f$ in the class ${Mathcal C}_0({Mathbb C})$ (continuous
functions with compact support), the Cauchy transform $u={Mathfrak C}chy f$
given by
\begin{equation*}{Mathfrak C}chy f(w)=\int_{Mathbb C}\frac {f(z)} {w-z}
{{Mathrm d} A}(z),Mathrm end{equation*} satisfies the ${\overline{\partial}}$-equation
\begin{equation}\label{dstreck}{\overline{\partial}} u=f.Mathrm end{equation}
Moreover, $u={Mathfrak C}chy f$ is bounded, and is therefore of class
$L^2_{\phi,n}$ for any $n\ge 1$.
Thus the function
\begin{equation}\label{star}u_{*}(w)=u(w)-{P}_{\phi,n}u(w),Mathrm end{equation}
solves Mathrm eqref{dstreck} and is of class $L^2_{\phi,n}$. It is easy to
verify that $u_{*}$ defined by Mathrm eqref{star} is the unique
norm-minimal solution to Mathrm eqref{dstreck} in $L^2_{\phi,n}$ whenever
$u\in L^2_{\phi,n}$ is a solution to Mathrm eqref{dstreck}. Hence the
study of the orthogonal projection ${P}_{\phi,n}u$ is equivalent
to the study of the \textit{the $L^2_{\phi,n}$-minimal} solution
$u_{*}$ to Mathrm eqref{dstreck}.
It is useful to observe that $u_{*}$ is characterized amongst the
solutions of class $L^2_{\phi,n}$ to Mathrm eqref{dstreck} by the
condition $u_{*}\boldsymbol\Omegat A^2_{\phi,n}$, or
\begin{equation}\label{orthog}
\int_{Mathbb C} u_{*}(z)\overline{h(z)}Mathrm e^{-\phi(z)}{{Mathrm d} A}(z)=0,\quad\text{for
all}\quad h\in A^2_{\phi,n}.Mathrm end{equation}
Our
principal result in this section, Th. \operatorname{Re}f{strb}, states that a
variant of the elementary one-dimensional form of the $L^2$
estimates of H\"ormander are valid for $u_{*}$. The important
results for the further developments in this paper are however Th.
\operatorname{Re}f{boh} and Cor. \operatorname{Re}f{bh}. The reader may consider to glance at
those results and skip to the next section, at a first read.
{Mathbb S}ubsection{$L^2$ estimates.} The
$L^2$ estimates of H\"{o}rmander in the most elementary,
one-dimensional form applies only to weights $\phi$ which are
strictly subharmonic in the entire plane ${Mathbb C}$. The result states
that $u_0$, the $L^2_\phi$-minimal solution to Mathrm eqref{dstreck}
(where $f\in{Mathcal C}_0({Mathbb C})$) satisfies
\begin{equation}\label{hoe0}\int_{Mathbb C}\babs{u_0}^2Mathrm e^{-\phi}{{Mathrm d} A}\le\int_{Mathbb C}
\babs{f}^2\frac {Mathrm e^{-\phi}} {{Mathbb D}elta\phi}{{Mathrm d} A},Mathrm end{equation} provided
that $\phi$ is ${Mathcal C}^2$-smooth on ${Mathbb C}$. See \cite{H}, eq. (4.2.6),
p. 250 (this is essentially just Green's formula).
It is important to observe that the estimate Mathrm eqref{hoe0} remains
valid for strictly subharmonic weights $\phi$ in the larger class
${Mathcal C}^{1,1}({Mathbb C})$ (that $\phi$ is strictly subharmonic then means
that ${Mathbb D}elta\phi>0$ a.e. on ${Mathbb C}$). The proof in \cite{H}, Subsect.
4.2 goes through without changes in this more general case.
{Mathbb S}ubsection{Weighted $L^2_{\phi,n}$ estimates.} We have the following theorem, where
we consider two weights $\phi$ and $\widehat{\phi}$ with various
properties. The practically minded reader may, with little loss of
generality, think of $\phi=mQ$,
$\widehat{\phi}(z)=m\widehat{Q}_\tau(z)+Mathrm eps\log(1+\babs{z}^2)$, and
${Mathbb S}etSp={Mathbb S}etS_\tau$. This will essentially be the case in all our
later applications.
\begin{thm}\label{strb} Let ${Mathbb S}etSp$ be a compact subset of ${Mathbb C}$,
$\phi$, $\widehat\phi$ and ${Mathbb T}fun$ three real-valued functions
of class ${Mathcal C}^{1,1}({Mathbb C})$, and $n$ a positive integer. We assume
that:
\begin{enumerate}
\item[(1)] $L^2_{\widehat{\phi}}$ contains the constants and there are
non-negative numbers $\alpha$ and $\beta$ such that
\begin{equation}\label{put}\widehat{\phi}\le
\phi+\alpha\quad\text{on}\quad {Mathbb C}\qquad\text{and}\qquad\phi\le
\beta+\widehat{\phi}\quad \text{on}\quad {Mathbb S}etSp,Mathrm end{equation}
\item[(2)] $A^2_{\widehat{\phi}}{Mathbb S}ubset A^2_{\phi,n}$,
\item[(3)] The function $\widehat{\phi}+{Mathbb T}fun$ is strictly subharmonic on
${Mathbb C}$,
\item[(4)] ${Mathbb T}fun$ is locally constant on ${Mathbb C}{Mathbb S}etminus {Mathbb S}etSp$,
\item[(5)] There exists a number $\kappa$, $0<\kappa<1$, such that
\begin{equation}\label{kappa}
\frac{|{\overline{\partial}} {Mathbb T}fun|^2}{{Mathbb D}elta\widehat{\phi}+{Mathbb D}elta{Mathbb T}fun}\le
\frac{\kappa^2}{Mathrm e^{\alpha+\beta}}\quad \text{a.e. on}\quad
{Mathbb S}etSp.Mathrm end{equation}
Mathrm end{enumerate}
Let $f\in \coity({Mathbb C})$ be such that
\begin{equation*}{Mathbb S}upp f{Mathbb S}ubset {Mathbb S}etSp.Mathrm end{equation*} Then
$u_*$, the $L^2_{\phi,n}$-minimal solution to ${\overline{\partial}} u=f$ satisfies
\begin{equation}\label{zapp}\int_{Mathbb C}\babs{u_*}^2Mathrm e^{{Mathbb T}fun-\phi}{{Mathrm d} A}\le \frac
{Mathrm e^{\alpha+\beta}} {(1-\kappa)^2}\int_{Mathbb C}\babs{f}^2\frac
{Mathrm e^{{Mathbb T}fun-\phi}} {{Mathbb D}elta\widehat{\phi}+{Mathbb D}elta{Mathbb T}fun} {{Mathrm d} A}.Mathrm end{equation}
Mathrm end{thm}
\nuoindent Before we give the proof, we note a simple lemma, which
will be put to repeated use.
\begin{lem}\label{convo} Suppose that $f\in{Mathcal C}_0({Mathbb C})$. Then
Mathrm eqref{dstreck} has a solution $u$ in $L^2_{\widehat{\phi}}$. Moreover
$v_*$, the $L^2_{\widehat{\phi}}$-minimal solution to Mathrm eqref{dstreck} is
of class $L^2_{\phi,n}$.
Mathrm end{lem}
\begin{proof}
The assumptions imply that the
Cauchy transform ${Mathfrak C}chy f$ is in $L^2_{\widehat\phi}$. Thus the
$L^2_{\widehat\phi}$-minimal solution $v_*$ to Mathrm eqref{dstreck} exists,
necessarily of the form $v_*={Mathfrak C}chy f+g$ with some $g\in
A^2_{\widehat\phi}$. In view of (3), we know that $g\in A^2_{\phi,n}$.
The assertion is now immediate.
Mathrm end{proof}
\begin{proof}[Proof of Theorem \operatorname{Re}f{strb}]
Assume that the right hand side in Mathrm eqref{zapp} is finite. In view
of Mathrm eqref{orthog}, the condition that $u_{*}$ is
$L^2_{\phi,n}$-minimal may be expressed as
\begin{equation*}\int_{Mathbb C} u_{*}Mathrm e^{Mathbb T}fun \bar{h} Mathrm e^{-(\phi+{Mathbb T}fun)}{{Mathrm d} A}=0
\quad\text{for all}\quad h\in A^2_{\phi,n}.
Mathrm end{equation*}
The latter relation means that the function $w_{0}=u_{*}Mathrm e^{Mathbb T}fun$
minimizes the norm in $L^2_{\phi+{Mathbb T}fun}$ over elements of the
(non-closed) subspace
\begin{equation*}Mathrm e^{{Mathbb T}fun}\cdot L^2_{\phi,n}=\big\{w;\quad w=hMathrm e^{Mathbb T}fun,\quad\text{where}\quad h\in
L^2_{\phi,n}\big\}{Mathbb S}ubset L^2_{\phi+{Mathbb T}fun}
Mathrm end{equation*}
which solve the (modified) ${\overline{\partial}}$-equation
\begin{equation}\label{mdbar}{\overline{\partial}} w={\overline{\partial}}L^p(\D,dA)ar u_{*}Mathrm e^{Mathbb T}fun\right )=
fMathrm e^{Mathbb T}fun+u_{*}{\overline{\partial}}L^p(\D,dA)ar Mathrm e^{Mathbb T}fun\right ).
Mathrm end{equation}
Now, since ${Mathbb T}fun$ is bounded on ${Mathbb C}$ and locally constant outside
${Mathbb S}etSp$, we have
\begin{equation*}L^2_{\phi+{Mathbb T}fun,n}=Mathrm e^{Mathbb T}fun\cdot L^2_{\phi,n}=
L^2_{\phi,n} \qquad (\text{as sets}),Mathrm end{equation*} and we conclude
that $w_{0}$ is the norm-minimal solution to Mathrm eqref{mdbar} in
$L^2_{\phi+{Mathbb T}fun,n}$.
Let $v_*$ denote the $L^2_{\widehat{\phi}}$-minimal solution to ${\overline{\partial}}
u=f$ (see Lemma \operatorname{Re}f{convo}). We form the continuous function
$g={\overline{\partial}}L^p(\D,dA)ar(u_{*}-v_*)Mathrm e^{Mathbb T}fun\right )=(u_{*}-v_*){\overline{\partial}}L^p(\D,dA)ar
Mathrm e^{Mathbb T}fun\right ),$ whose support is contained in ${Mathbb S}etSp$, and consider
the ${\overline{\partial}}$-equation
\begin{equation}\label{go}{\overline{\partial}}\xi=g=(u_{*}-v_*){\overline{\partial}}L^p(\D,dA)ar Mathrm e^{Mathbb T}fun\right ).
Mathrm end{equation}
The assertion of Lemma \operatorname{Re}f{convo} remains valid if $\widehat\phi$ is
replaced by $\widehat{\phi}+{Mathbb T}fun$; it follows that Mathrm eqref{go} has a
solution $\xi\in L^2_{\widehat{\phi}+{Mathbb T}fun}$, and moreover, if $\xi_{0}$
denotes the norm-minimal solution to Mathrm eqref{go} in
$L^2_{\widehat{\phi}+{Mathbb T}fun}$, we have $\xi_{0} \in L^2_{\phi+{Mathbb T}fun,n}$.
We now continue by forming the function
\begin{equation*}w_1=v_*Mathrm e^{{Mathbb T}fun}+\xi_{0}.Mathrm end{equation*}
It is clear that $w_1\in L^2_{\phi+{Mathbb T}fun,n}$, and a calculation
yields that
\begin{equation}\label{nu2}{\overline{\partial}} w_1=
f Mathrm e^{{Mathbb T}fun}+v_*{\overline{\partial}}L^p(\D,dA)ar Mathrm e^{{Mathbb T}fun}\right ) +(u_{*}-v_*){\overline{\partial}}L^p(\D,dA)ar
Mathrm e^{{Mathbb T}fun}\right ) ={\overline{\partial}}L^p(\D,dA)ar u_{*}Mathrm e^{{Mathbb T}fun}\right )={\overline{\partial}} w_{0}.
Mathrm end{equation}
Since $w_{0}$
is norm-minimal in $L^2_{\phi+{Mathbb T}fun,n}$ over functions $w$ such that
${\overline{\partial}} w={\overline{\partial}} w_{0}$, we must have
\begin{equation}
\label{ojoj}\int_{Mathbb C}\babs{w_{0}}^2 e^{-(\phi+{Mathbb T}fun)}{{Mathrm d} A}\le
\int_{Mathbb C}\babs{w_1}^2Mathrm e^{-(\phi+{Mathbb T}fun)}{{Mathrm d} A}\le Mathrm e^{\alpha}\int_{Mathbb C}
\babs{w_1}^2 Mathrm e^{-(\widehat{\phi}+{Mathbb T}fun)}{{Mathrm d} A},
Mathrm end{equation}
where we have used the condition $\widehat{\phi}\le\phi+\alpha$ to deduce
the second inequality. Moreover, since $\xi_0$
is norm-minimal in $L^2_{\widehat{\phi}+{Mathbb T}fun}$ amongst
solutions to
Mathrm eqref{go}, we have
\begin{equation*}
\int_{Mathbb C} w_1\bar{h}Mathrm e^{-(\widehat{\phi}+{Mathbb T}fun)}{{Mathrm d} A}=\int_{Mathbb C}
v_*\bar{h}Mathrm e^{-\widehat{\phi}}{{Mathrm d} A}+ \int_{Mathbb C} \xi_0
\bar{h}Mathrm e^{-(\widehat{\phi}+{Mathbb T}fun)}{{Mathrm d} A}=0\quad
Mathrm end{equation*}
for all $h\in A^2_{\widehat{\phi}+{Mathbb T}fun}$, so the function $w_1$ is in
fact
the norm-minimal solution to a ${\overline{\partial}}$-equation in
$L^2_{\widehat{\phi}+{Mathbb T}fun}$. The ${\overline{\partial}}$-equation satisfied by $w_1$ is
(see Mathrm eqref{nu2})
\begin{equation*}{\overline{\partial}} w_1=fMathrm e^{Mathbb T}fun+u_{*}{\overline{\partial}}L^p(\D,dA)ar Mathrm e^{Mathbb T}fun\right ) =
f Mathrm e^{Mathbb T}fun +u_{*}Mathrm e^{Mathbb T}fun{\overline{\partial}} {Mathbb T}fun.
Mathrm end{equation*}
By the estimate Mathrm eqref{hoe0} applied to the weight
$\widehat\phi+{Mathbb T}fun$, we obtain that
\begin{equation}\label{inter}\int_{Mathbb C}\babs{w_1}^2
Mathrm e^{-(\widehat{\phi}+{Mathbb T}fun)}{{Mathrm d} A}\le \int_{{Mathbb C}} \babs{fMathrm e^{Mathbb T}fun
+u_{*}Mathrm e^{Mathbb T}fun{\overline{\partial}} {Mathbb T}fun}^2 \frac {Mathrm e^{-(\widehat{\phi}+{Mathbb T}fun)}}
{{Mathbb D}eltaL^p(\D,dA)ar \widehat{\phi}+{Mathbb T}fun\right )}{{Mathrm d} A}=\int_{{Mathbb C}} \babs{f
+u_{*}{\overline{\partial}}{Mathbb T}fun}^2 \frac {Mathrm e^{{Mathbb T}fun-\widehat{\phi}}}
{{Mathbb D}elta\widehat{\phi}+{Mathbb D}elta{Mathbb T}fun}{{Mathrm d} A}.
Mathrm end{equation}
Since $f$ and ${\overline{\partial}}{Mathbb T}fun$ are supported in ${Mathbb S}etSp$, and since
$Mathrm e^{-\widehat{\phi}}\le Mathrm e^{\beta}Mathrm e^{-\phi}$ there (see Mathrm eqref{put}),
it is seen that the right hand side in Mathrm eqref{inter} can be
estimated by
\begin{equation}\label{inter2}Mathrm e^{\beta}
\int_{Mathbb C}\babs{f+u_*{\overline{\partial}}{Mathbb T}fun}^2\frac {Mathrm e^{{Mathbb T}fun-\phi}}
{{Mathbb D}elta\widehat{\phi}+{Mathbb D}elta{Mathbb T}fun} {{Mathrm d} A}.Mathrm end{equation} For $t>0$ we now
use the inequality $\babs{a+b}^2\le (1+t)\babs{a}^2+
(1+t^{-1})\babs{b}^2$
and the condition Mathrm eqref{kappa} to conclude that the
integral in Mathrm eqref{inter2} is dominated by
\begin{equation}\label{inter3}\begin{split}(1+t)&\int_{Mathbb C}
\babs{f}^2\frac {Mathrm e^{{Mathbb T}fun-\phi}} {{Mathbb D}elta\widehat{\phi}+{Mathbb D}elta{Mathbb T}fun}
{{Mathrm d} A}+(1+t^{-1})\int_{Mathbb C}\babs{u_*}^2\frac {|{\overline{\partial}}{Mathbb T}fun|^2}
{{Mathbb D}elta\widehat{\phi}+{Mathbb D}elta{Mathbb T}fun}
Mathrm e^{{Mathbb T}fun-\phi}{{Mathrm d} A}\le\\
&\le (1+t)\int_{Mathbb C}\babs{f}^2\frac {Mathrm e^{{Mathbb T}fun-\phi}}
{{Mathbb D}elta\widehat{\phi}+{Mathbb D}elta{Mathbb T}fun}{{Mathrm d} A}+(1+t^{-1})\frac {\kappa^2}
{Mathrm e^{\alpha+\beta}}
\int_{Mathbb C}\babs{u_*}^2Mathrm e^{{Mathbb T}fun-\phi}{{Mathrm d} A}.\\
Mathrm end{split}
Mathrm end{equation}
Tracing back through Mathrm eqref{ojoj}, Mathrm eqref{inter}, Mathrm eqref{inter2} and
Mathrm eqref{inter3}, we get
\begin{equation*}\int_{Mathbb C}\babs{u_*}^2Mathrm e^{{Mathbb T}fun-\phi}{{Mathrm d} A}\le
Mathrm e^{\alpha+\beta}(1+t)\int_{Mathbb C}\babs{f}^2\frac {Mathrm e^{{Mathbb T}fun-\phi}}
{{Mathbb D}elta\widehat{\phi}+{Mathbb D}elta{Mathbb T}fun}{{Mathrm d} A}+
(1+t^{-1})\kappa^2\int_{Mathbb C}\babs{u_*}^2Mathrm e^{{Mathbb T}fun-\phi}{{Mathrm d} A},Mathrm end{equation*}
which we write as
\begin{equation*}
\big(1-(1+t^{-1})\kappa^2\big)\int_{Mathbb C}|u_{*}|^2Mathrm e^{{Mathbb T}fun-\phi}{{Mathrm d} A}
\leMathrm e^{\alpha+\beta}(1+t)\int_{{Mathbb C}}\babs{f}^2\frac{Mathrm e^{{Mathbb T}fun-\phi}}
{{Mathbb D}elta\widehat{\phi}+{Mathbb D}elta{Mathbb T}fun}{{Mathrm d} A}.
Mathrm end{equation*}
The desired estimate Mathrm eqref{zapp} now follows if we pick
$t=\kappa/(1-\kappa).$
Mathrm end{proof}
{Mathbb S}ubsection{Implementation scheme.} We now fix
$Q\in {Mathcal C}^{1,1}({Mathbb C})$ satisfying the growth assumption Mathrm eqref{gro}
with a fixed $\rho>0$. Adding a constant to $Q$ does not change the
problem and so we may assume that $Q\ge 1$ on ${Mathbb C}$. Let us put
\begin{equation}\label{sigmat}q_\tau={Mathbb S}up_{z\in{Mathbb S}etS_\tau}\{Q(z)\}.Mathrm end{equation}
We next fix a positive number $\tau<\rho$ and two positive numbers $A^p(\D)ar$
and ${Mathbf p}ar$ such that
\begin{equation}\label{sss}
{Mathbf p}ar\log(1+\babs{z}^2)\leA^p(\D)ar \widehat{Q}_\tau(z),\qquad
z\in{Mathbb C}.Mathrm end{equation}
This is possible since $\widehat{Q}_\tau\ge 1$ and
$\widehat{Q}_\tau(z)=\tau\log\babs{z}^2+{Mathcal O}(1)$ when $z\to\infty$ (see
Mathrm eqref{qtau}). In particular, it yields that ${Mathbf p}ar\le A^p(\D)ar\tau$.
We now define
\begin{equation}\label{pha}\phi_m=mQ\quad
\text{and}\quad
\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}(z)=(m-A^p(\D)ar)\widehat{Q}_\tau(z)+{Mathbf p}ar\log(1+\babs{z}^2).Mathrm end{equation}
Note that $\widehat\phi_{m,A^p(\D)ar,{Mathbf p}ar}$ is strictly subharmonic on ${Mathbb C}$
whenever $m\ge A^p(\D)ar$ with
\begin{equation}\label{bye}{Mathbb D}elta \widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}(z)=(m-A^p(\D)ar){Mathbb D}elta \widehat{Q}_\tau(z)+
{Mathbf p}arL^p(\D,dA)ar 1+\babs{z}^2\right )^{-2} \ge {Mathbf p}arL^p(\D,dA)ar
1+\babs{z}^2\right )^{-2},Mathrm end{equation} and that Mathrm eqref{sss} implies
(since $\widehat{Q}_\tau\le Q$)
\begin{equation*}\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}\le \phi_m\qquad
\text{on}\quad {Mathbb C}.Mathrm end{equation*} Furthermore, Mathrm eqref{qtau} implies
that
\begin{equation}\label{qq}\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}=L^p(\D,dA)ar(m-A^p(\D)ar)\tau+{Mathbf p}ar\right )
\log\babs{z}^2+{Mathcal O}(1), \qquad \text{as}\quad
z\to\infty.Mathrm end{equation} It yields that
\begin{equation*}\phi_m(z)-\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}(z)\ge
m(\rho-\tau)\log\babs{z}^2+{Mathcal O}(1),\qquad\text{as}\quad
z\to\infty.Mathrm end{equation*} Note also that
\begin{equation*}A^2_{\phi_m,n}=A^2_{mQ}\cap{Mathcal P}_n={H}_{m,n}.
Mathrm end{equation*}
We now check the conditions (1) and (2) of Th. \operatorname{Re}f{strb}. (Recall
that $]x[$ denotes the largest integer which is strictly smaller
than $x$.)
\begin{lem} \label{putty}
Condition {\rm (1)} holds for $\phi=\phi_m$,
$\widehat\phi=\widehat\phi_{m,A^p(\D)ar,{Mathbf p}ar}$, ${Mathbb S}etSp={Mathbb S}etS_\tau$, $\alpha=0$
and
$\beta=A^p(\D)ar q_\tau$, provided that $(m-A^p(\D)ar)\tau+{Mathbf p}ar>1$.
Condition {\rm (2)}, i.e. $A^2_{\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}}{Mathbb S}ubset
{H}_{m,n}$, holds if $n\ge ](m-A^p(\D)ar)\tau+{Mathbf p}ar[$. Thus
conditions {\rm (1)} and {\rm (2)} hold whenever
\begin{equation}\label{alag}n\ge](m-A^p(\D)ar)\tau+{Mathbf p}ar[>0.Mathrm end{equation}
Mathrm end{lem}
\begin{proof}
We have already shown that $\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}\le \phi_m$ on
${Mathbb C}$. Moreover $\phi_m\le\beta+\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}$ on
${Mathbb S}etS_\tau$, since $Q=\widehat{Q}_\tau$ there. The assertion that
$\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}\le \phi_m$ implies that
$A^2_{\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}}{Mathbb S}ubset A^2_{\phi_m}$, and it
remains only to prove that $A^2_{\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}}{Mathbb S}ubset
{Mathcal P}_n$. But this follows from Mathrm eqref{qq}, Mathrm eqref{alag}, and the
fact that
$\int_{{Mathbb C}}(1+|z|^{2})^{-r}{{Mathrm d} A}(z)<\infty$ if and only if $r>1$.
Mathrm end{proof}
We now apply Th. \operatorname{Re}f{strb}.
It will be convenient to define
\begin{equation}\label{ctdef}c_\tau=\inf_{z\in{Mathbb S}etS_\tau}
\big\{L^p(\D,dA)ar 1+\babs{z}^2\right )^{-2}\big\}.Mathrm end{equation}
\begin{thm} \label{boh} Let $Q\in{Mathcal C}^{1,1}({Mathbb C})$ and $Q\ge 1$ on ${Mathbb C}$. Fix two positive
numbers $A^p(\D)ar$ and ${Mathbf p}ar$ such that relation Mathrm eqref{sss} is
satisfied, and define the positive numbers $q_\tau$ and $c_\tau$ by
Mathrm eqref{sigmat} and Mathrm eqref{ctdef}, and let
$\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}$ be defined by Mathrm eqref{pha}.
Suppose there are real-valued functions ${Mathbb T}fun_m\in{Mathcal C}^{1,1}({Mathbb C})$
which are locally constant on ${Mathbb C}{Mathbb S}etminus{Mathbb S}etS_\tau$ such that
$\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}+{Mathbb T}fun_m$ is strictly subharmonic on ${Mathbb C}$
and that for some
positive number $\kappa<1$ we have
\begin{equation*}
\frac {|{\overline{\partial}} {Mathbb T}fun_m|^2}
{{Mathbb D}elta\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}+{Mathbb D}elta{Mathbb T}fun_m}\le\frac{\kappa^2}
{Mathrm e^{A^p(\D)ar q_\tau}} \qquad \text{a.e. on}\quad {Mathbb C}.
Mathrm end{equation*}
Suppose furthermore that $f\in\coity({Mathbb C})$ satisfies
\begin{equation*}{Mathbb S}upp f{Mathbb S}ubset
{Mathbb S}etS_\tau.Mathrm end{equation*} Then, if $n\ge ](m-A^p(\D)ar)\tau+{Mathbf p}ar[>0$,
we have that $u_*$, the $L^2_{mQ,n}$-minimal solution to ${\overline{\partial}}
u=f$, satisfies
\begin{equation*}\int_{Mathbb C}\babs{u_*}^2
Mathrm e^{{Mathbb T}fun_m-mQ}{{Mathrm d} A}\le \frac{Mathrm e^{A^p(\D)ar
q_\tau}}{(1-\kappa)^2}\int_{{Mathbb C}}\babs{f}^2 \frac {Mathrm e^{{Mathbb T}fun_m-mQ}}{
(m-A^p(\D)ar){Mathbb D}elta Q+{Mathbb D}elta{Mathbb T}fun_m+{Mathbf p}ar c_\tau} {{Mathrm d} A}.Mathrm end{equation*}
Mathrm end{thm}
\begin{proof} All
conditions in Th. \operatorname{Re}f{strb} are fulfilled
with $\phi=mQ$, $\widehat{\phi}=\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}$,
${Mathbb T}fun={Mathbb T}fun_m$, $\alpha=0$ and $\beta=A^p(\D)ar q_\tau$. In view of
Mathrm eqref{bye}, it yields that
\begin{equation*}\label{whop3}\begin{split}\int_{Mathbb C}\babs{u_*}^2Mathrm e^{{Mathbb T}fun_m-mQ}{{Mathrm d} A}&\le
\frac {Mathrm e^{A^p(\D)ar q_\tau}} {(1-\kappa)^2}\int_{{Mathbb C}}\babs{f}^2\frac
{Mathrm e^{{Mathbb T}fun_m-mQ}} {{Mathbb D}elta\widehat{\phi}_{m,A^p(\D)ar,{Mathbf p}ar}+
{Mathbb D}elta{Mathbb T}fun_m}{{Mathrm d} A}\le\\
&\le \frac {Mathrm e^{A^p(\D)ar q_\tau}}
{(1-\kappa)^2}\int_{{Mathbb C}}\babs{f}^2\frac {Mathrm e^{{Mathbb T}fun_m-mQ}}
{(m-A^p(\D)ar){Mathbb D}elta\widehat{Q}_\tau+{Mathbb D}elta{Mathbb T}fun_m+{Mathbf p}ar c_\tau}{{Mathrm d} A}.\\
Mathrm end{split}Mathrm end{equation*} The assertion is now immediate from Mathrm eqref{whop3},
since ${Mathbb D}elta Q={Mathbb D}elta \widehat{Q}_\tau$ a.e. on ${Mathbb S}upp f$.
Mathrm end{proof}
\nuoindent In Sect. \operatorname{Re}f{point}, we shall use the full force of Th.
\operatorname{Re}f{boh}. As for now, we just mention the following simple
consequence, which also can be proved easily by more elementary
means, cf. \cite{B}, p. 10.
\begin{cor}\label{bh} Let $Q\in {Mathcal C}^{1,1}({Mathbb C})$ and $Q\ge 1$ on
${Mathbb C}$. Further, let ${Mathbb S}etSp$ be a compact subset of ${Mathbb S}etS_\tau$. Put
\begin{equation*}a=Mathrm essinf\{{Mathbb D}elta Q(z);\,
z\in{Mathbb S}etSp\}.Mathrm end{equation*} Let $m_0=Max\{2A^p(\D)ar,(1+A^p(\D)ar)/\tau\}$
and assume that $f\in \coity({Mathbb C})$ satisfies \begin{equation*}{Mathbb S}upp
f{Mathbb S}ubset{Mathbb S}etSp,Mathrm end{equation*} and that $n\ge ](m-A^p(\D)ar)\tau+{Mathbf p}ar[$
and $m\ge m_0$. Then $u_*$, the $L^2_{mQ,n}$-minimal solution to
${\overline{\partial}} u=f$ satisfies
\begin{equation*}\|u_*\|_{mQ}^2\le
\frac {2Mathrm e^{A^p(\D)ar q_\tau}}
{am+{Mathbf p}ar c_\tau}\|f\|_{mQ}^2.
Mathrm end{equation*}
Mathrm end{cor}
\begin{proof}
Take ${Mathbb T}fun_m=0$ in Th. \operatorname{Re}f{boh} and observe that $m-A^p(\D)ar\ge m/2$
and also $](m-A^p(\D)ar)\tau+{Mathbf p}ar[>0$ whenever $m\ge m_0$. (Any other
$m_0$ having these properties would work as well, of course.)
Mathrm end{proof}
{Mathbb S}ection{Approximate Bergman projections}
{Mathbb S}ubsection{Preliminaries} In this section we state and prove a result (Th. \operatorname{Re}f{mlem} below), which we
will use to prove the asymptotic expansion in Th. \operatorname{Re}f{th3} in the
next section.
The result in question is a modified version of [Berman et al.
\cite{BBS}, Prop. 2.5, p. 9]. Our proof is elementary but rather
lengthy, and the reader may find it worthwhile to look at the result
and go to the next section at the first read.
{Mathbb S}ubsection{A convention}
It will be convenient to be able to define integrals $\int_S
\chi(w)Y(w){Mathrm d}Mu(w)$ where $S$ is a $Mu$-measurable subset of
${Mathbb C}$, $\chi$ a cut-off function, and $Y$ a complex-valued
$Mu$-measurable function which is well-defined on ${Mathbb S}upp\chi$, but
not necessarily on the entire domain $S$. By convention, we extend
the integrand $Y(w)\chi(w)$ to ${Mathbb C}$ by defining $\chi(w)Y(w)=0$
whenever $\chi(w)=0$, i.e., \textit{we define}
\begin{equation*}\int_S
\chi(w)Y(w){Mathrm d}Mu(w):=\int_{S\cap
{Mathbb S}upp\chi}\chi(w)Y(w){Mathrm d}Mu(w).Mathrm end{equation*}
{Mathbb S}ubsection{Statement of the result} Recall the definition of the set ${Mathbb S}etX=\{{Mathbb D}elta Q>0\}$,
as well as of the holomorphic extension ${Mathbb Q}ext$ of $Q$ from the
anti-diagonal, see Mathrm eqref{psi2}.
For the approximating kernel $K_m^1$, see Definition \operatorname{Re}f{def1}. By
${\partial} S$ we mean the positively oriented boundary of a compact set
$S$. We have the following theorem.
\begin{thm} \label{mlem} Let $Q$ be real-analytic in ${Mathbb C}$. Let $S$ be a compact
subset of ${Mathbb S}etX$ and fix a point $z_0\in S$. Then there exists
\begin{enumerate}
\item[(1)] a number $Mathrm eps>0$ small enough that $K_m^1(z,w)$ makes
sense and is Hermitian for all $z,w\in{Mathbb D}(z_0;2Mathrm eps)$ and all $z_0\in
S$,
\item[(2)] a function $\chi\in\coity({Mathbb C})$ with $\chi=1$ in
${Mathbb D}(z_0;3Mathrm eps/2)$, $\chi=0$ outside ${Mathbb D}(z_0;2Mathrm eps)$, and
$\|{\overline{\partial}}\chi\|_{L^2}\le 3,$
\item[(3)] a real-analytic function $\nuu_0(z,w)$ defined for
$z,w\in{Mathbb D}(z_0;2Mathrm eps)$,
Mathrm end{enumerate}
such that with
\begin{equation}\label{izm}I_{m}u(z)=\int_{S}
u(w)\chi(w)K_m^1(z,w)Mathrm e^{-mQ(w)}{{Mathrm d} A}(w),\qquad u\in A^2_{mQ},\quad
z\in{Mathbb D}(z_0;Mathrm eps)\cap S^\circ,
Mathrm end{equation} we have
\begin{equation*}\begin{split}
&\babs{I_{m}u(z)-u(z)-\frac 1 {2\pi\operatorname{Im}ag}\int_{{\partial}
S}u(w)\chi(w)Mathrm e^{m({Mathbb Q}ext(z,\bar{w})-Q(w))}L^p(\D,dA)ar \frac 1
{z-w}+\nuu_0(z,w)\right ){Mathrm d}
w} \le\frac{C}{m^{3/2}}Mathrm e^{mQ(z)/2}\|u\|_{mQ},\\
Mathrm end{split}Mathrm end{equation*} for all $u$ and $z$ as above, with a number $C$ which only depends on $Mathrm eps$ and
$z_0$. Moreover, the numbers $C$ and $Mathrm eps$ may be chosen
independently of $z_0$ for $z_0\in S.$
In particular, if $z_0\in S^\circ$ and if $Mathrm eps$ is small enough
that ${Mathbb D}(z_0;2Mathrm eps){Mathbb S}ubset S^\circ$, then $\chi=0$ on ${\partial} S$ and so
\begin{equation}\label{ching}\babs{I_{m}u(z)-u(z)}\le
Cm^{-3/2}Mathrm e^{mQ(z)/2}\|u\|_{mQ}, \qquad u\in A^2_{mQ},\quad
z\in{Mathbb D}(z_0;Mathrm eps).Mathrm end{equation}
Mathrm end{thm}
\begin{rem} The function $\nuu_0$ is constructed in the
proof, and it depends only on the function ${Mathbb Q}ext$ and its
derivatives. See Mathrm eqref{nudef} below.
In the case when $z_0$ is in $S^\circ$ and $Mathrm eps$ is small enough
that ${Mathbb D}(z_0;2Mathrm eps){Mathbb S}ubset S^\circ$, the conclusion Mathrm eqref{ching}
makes it seem reasonable to call the functional $uMapsto I_m
u(z)$ an \textit{approximate Bergman projection}, at least locally,
for $z$ in a neighbourhood of $z_0$.
Mathrm end{rem}
{Mathbb S}ubsection{A rough outline; previous work.} In the proof we shall
construct three operators $I_m^j$, $j=1,2,3$, such that $I_m
u(z)=I_m^1 u(z)+I_m^2 u(z)+I_m^3 u(z)$ for $z$ near $z_0$.
The constructions will be made so that
\begin{equation*}\begin{split}&I_m^1 u(z)\quad\text{is close to}\quad u(z)+\frac 1
{2\pi\operatorname{Im}ag}\int_{{\partial}
S}\frac{u(w)\chi(w)Mathrm e^{m({Mathbb Q}ext(z,\bar{w})-Q(w))}} {z-w}{Mathrm d}
w,\\
&I_m^2 u(z)\quad \text{is close to}\quad \frac 1
{2\pi\operatorname{Im}ag}\int_{{\partial}
S}u(w)\chi(w)\nuu_0(z,w)Mathrm e^{m({Mathbb Q}ext(z,\bar{w})-Q(w))}{Mathrm d}
w,\\
&\text{and}\quad I_m^3 u(z)\quad \text{is "negligible" for
large\,} m.\\
Mathrm end{split}Mathrm end{equation*} The operator $I_m^1$ is easy to construct
directly, whereas the definitions of the other operators $I_m^j$
for $j=2,3$ are somewhat more involved. We have found it convenient
to start by considering $I_m^1$.
Our approach is based on the paper \cite{BBS} by Berman et al., in
which a somewhat different situation is treated (notably, $Q$ is
assumed strictly subharmonic in the entire plane in \cite{BBS},
which means that a somewhat different inner product $|u|^2_{mQ}=\int
\babs{u}^2Mathrm e^{-mQ}{Mathbb D}elta Q{{Mathrm d} A}$ is available).
We have found the remarks of Berman in \cite{B}, p. 9 quite useful;
the construction in \cite{BBS} is \textit{local} and hence the
requirement that $Q$ be globally strictly subharmonic can, at least
in principle, be removed. We have found it motivated to give a
detailed proof for this statement.
Our approach is heavily inspired by that of the aformentioned
papers, but is frequently more elementary. The rest of this section
is devoted to the proof of Th. \operatorname{Re}f{mlem}.
{Mathbb S}ubsection{The first approximation.} Throughout the proof, we keep
a point $z_0\in S$ arbitrary but fixed, where $S$ is a given compact
subset of ${Mathbb S}etX$.
Also fix a function $u\in A^2_{mQ}$.
The idea is to construct a suitable complex phase function
$\phi_z(w)$, such that for fixed $z$ in $S^\circ$ close to $z_0$,
the main contribution in Mathrm eqref{izm} is of the form
\begin{equation}\label{grep}I_m^1 u(z)=\int_{S} \frac {u(w)\chi(w)}
{z-w}{\overline{\partial}}_wL^p(\D,dA)ar Mathrm e^{m\phi_z(w)(z-w)}\right ){{Mathrm d} A}(w).Mathrm end{equation}
To construct $\phi_z$, we first define, for points $z,w,$ and
$\bar{\zeta}$ sufficiently near each other,
\begin{equation}\label{bri}\thetaeta(z,w,\zeta)=\int_0^1
{\partial}_1{Mathbb Q}ext(w+t(z-w),\zeta){Mathrm d} t.
Mathrm end{equation}
Then $\thetaeta$ is holomorphic in a neighbourhood of the subset
$\{(z,z,\bar{z});z\in {Mathbb C}\}{Mathbb S}ubset {Mathbb C}^3$ and
\begin{equation*}\thetaeta(z,w,\zeta)(z-w)={Mathbb Q}ext(z,\zeta)-{Mathbb Q}ext(w,\zeta).
Mathrm end{equation*}
Let $Mathrm eps>0$ be sufficiently small that ${Mathbb D}(z_0;2Mathrm eps)\Subset {Mathbb S}etX$
and such that the functions ${Mathbb Q}ext(z,w)$, ${b}_0(z,w)$ and
${b}_1(z,w)$ make sense and are holomorphic on
$\{(z,w);z,\bar{w}\in {Mathbb D}(z_0;2Mathrm eps)\}$ (see Definition \operatorname{Re}f{def1}).
Writing
\begin{equation*}{\partial}elta_0={Mathbb D}elta Q(z_0)/2,Mathrm end{equation*}
we may then use Mathrm eqref{bbs} to
choose $Mathrm eps>0$ somewhat smaller if necessary so that
\begin{equation}\label{goodc}2\operatorname{Re}\{\thetaeta(z,w,\bar{w})(z-w)\}=2\operatorname{Re}
{Mathbb Q}ext(z,\bar{w})-2Q(w)\le
-{\partial}elta_0\babs{z-w}^2+Q(z)-Q(w),Mathrm end{equation} for all $z,w\in
{Mathbb D}(z_0;2Mathrm eps)$. Note that the same $Mathrm eps$ can be used for all $z_0$
in $K$.
Next, we fix a cut-off function $\chi\in\coity({Mathbb C})$ such that
$\chi=1$ in ${Mathbb D}(z_0;3Mathrm eps/2)$, $\chi=0$ outside ${Mathbb D}(z_0;2Mathrm eps)$,
$0\le \chi\le 1$ on ${Mathbb C}$ and
$\|{\overline{\partial}}\chi\|_{L^2}\le 3$, see Subsect. \operatorname{Re}f{cutoff}.
Take a point $z\in S^\circ\cap{Mathbb D}(z_0;Mathrm eps)$. We now put
\begin{equation*}\phi_z(w)=\thetaeta(z,w,\bar{w}),\qquad\text{which means that}
\qquad \phi_z(w)(z-w)={Mathbb Q}ext(z,\bar{w})-Q(w),Mathrm end{equation*} and
consider the corresponding integral $I_m^1 u(z)$ given by
Mathrm eqref{grep}.
We have the following lemma. The result should be compared with
\cite{BBS}, Prop. 2.1.
\begin{lem} \label{blem}There exist positive numbers $C_1$ and ${\partial}elta$ depending
only on
$z_0$ such that
\begin{equation}\label{goo1}\babs{I_{m}^1 u(z)-\frac 1 {2\pi\operatorname{Im}ag}\int_{{\partial} S}
\frac {u(w)\chi(w)Mathrm e^{m\phi_z(w)(z-w)}} {z-w}{Mathrm d} w-u(z)}\le
C_1Mathrm eps^{-1}Mathrm e^{m(Q(z)/2-{\partial}eltaMathrm eps^2)}\|u\|_{mQ},Mathrm end{equation} for all $u\in
A^2_{mQ}$ and all $z\in{Mathbb D}(z_0;Mathrm eps)\cap S^\circ$. The numbers $C_1$
and ${\partial}elta$ may also be chosen independently of the point $z_0\in
S$. (Indeed, one may take ${\partial}elta={Mathbb D}elta Q(z_0)Mathrm eps^2/16$ and
$C_1=6/Mathrm eps$.)
Mathrm end{lem}
\begin{proof} Keep $z\in{Mathbb D}(z_0;Mathrm eps)\cap S^\circ$ arbitrary but
fixed and take $u\in A^2_{mQ}$. Since
$\chi(z)=1$ and since $\babs{w-z}\geMathrm eps/2$ when ${\overline{\partial}}\chi(w)\nue
0$, Mathrm eqref{grep} and Cauchy's formula implies (keep in mind that
$z\in S^\circ$)
\begin{equation}\begin{split}\label{pr}u(z)&+\frac 1 {2\pi\operatorname{Im}ag}\int_{{\partial} S}
\frac {u(w)\chi(w)Mathrm e^{m\phi_z(w)(z-w)}} {z-w}{Mathrm d} w=\int_{S} \frac
{{\overline{\partial}}_wL^p(\D,dA)ar
u(w)\chi(w)Mathrm e^{m\phi_z(w)(z-w)}\right )} {z-w}{{Mathrm d} A}(w)=\\
&=I_m^1 u(z)+\int_{S}\frac {u(w){\overline{\partial}}\chi(w)} {z-w}
Mathrm e^{m\phi_z(w)(z-w)}{{Mathrm d} A}(w)=\\
&=I_m^1 u(z)+\int_{\{w\in S;\,\babs{w-z}\ge Mathrm eps/2\}}\frac
{u(w){\overline{\partial}}\chi(w)} {z-w}
Mathrm e^{m\phi_z(w)(z-w)}{{Mathrm d} A}(w).\\
Mathrm end{split}
Mathrm end{equation}
Applying the estimate Mathrm eqref{goodc} to the last integral in
Mathrm eqref{pr} gives
\begin{equation*}\begin{split}&\babs{
\int_{\{w\in S;\,\babs{w-z}\ge Mathrm eps/2\}}\frac {u(w){\overline{\partial}}\chi(w)}
{z-w} Mathrm e^{m\phi_z(w)(z-w)}{{Mathrm d} A}(w)}\le\\
&\qquad\qquad\qquad\le \int_{\{w\in{Mathbb C};\, \babs{w-z}\ge
{Mathrm eps/2}\}}\frac{\babs{ u(w){\overline{\partial}}\chi(w)}}
{\babs{z-w}}Mathrm e^{m(Q(z)/2-Q(w)/2-{\partial}elta_0\babs{z-w}^2/2)}{{Mathrm d} A}(w)\le\\
&\qquad\qquad\qquad\le
(2/Mathrm eps)Mathrm e^{m(Q(z)/2-{\partial}elta_0Mathrm eps^2/8)}\int_{Mathbb C}
\babs{u(w){\overline{\partial}}\chi(w)}Mathrm e^{-mQ(w)/2}{{Mathrm d} A}(w)\le\\
&\qquad\qquad\qquad\le
(2/Mathrm eps)Mathrm e^{m(Q(z)/2-{\partial}elta_0Mathrm eps^2/8)}\|u\|_{mQ}\|{\overline{\partial}}\chi\|_{L^2}
\le (6/Mathrm eps)Mathrm e^{m(Q(z)/2-{\partial}elta_0Mathrm eps^2/8)},\\
Mathrm end{split}
Mathrm end{equation*}
where we have used
that $\|{\overline{\partial}}\chi\|_{L^2}\le 3$ to get the last inequality.
Recalling that ${\partial}elta_0={Mathbb D}elta Q(z_0)/2$ and combining with
Mathrm eqref{pr}, we get an estimate
\begin{equation*}\babs{u(z)+\frac 1 {2\pi\operatorname{Im}ag}\int_{{\partial} S}
\frac {u(w)\chi(w)Mathrm e^{m\phi_z(w)(z-w)}} {z-w}{Mathrm d} w-I_m^1
u(z)}\le (6/Mathrm eps)Mathrm e^{m(Q(z)/2-{Mathbb D}elta
Q(z_0)Mathrm eps^2/16)}.Mathrm end{equation*} The proof is finished.
Mathrm end{proof}
{Mathbb S}ubsection{The change of variables.}
Keep $z_0\in S$ fixed and note that the definition of $\thetaeta$ (see
Mathrm eqref{bri}) implies that
\begin{equation}\label{d3}[{\partial}_3\thetaeta](z,z,\bar{z})=
[{\partial}_1{\partial}_2{Mathbb Q}ext](z,\bar{z})={Mathbb D}elta
Q(z),Mathrm end{equation} for all $z\in {Mathbb C}$. Since ${Mathbb D}elta Q(z_0)>0$,
Mathrm eqref{d3} implies that there exists a neighbourhood $U$ of
$(z,w,\zeta)=(z_0,z_0,\bar{z}_0)$ such that the function
\begin{equation*}F:U\to F(U)\quad:\quad
(z,w,\zeta)Mapsto (z,w,\thetaeta(z,w,\zeta))Mathrm end{equation*} defines a
biholomorphic change of coordinates. In particular, we may regard
$\zeta$ as a function of the parameters $z$, $w$ and $\thetaeta$ when
$(z,w,\zeta)$ is in $U$.
In view of Mathrm eqref{d3} we can choose $Mathrm eps>0$ such that
$[{\partial}_3\thetaeta](z,w,\zeta)\nue 0$ whenever
$z,w,\bar{\zeta}\in{Mathbb D}(z_0;2Mathrm eps)$. Since
$\phi_z(w)=\thetaeta(z,w,\bar{w})$, we have
\begin{equation}\label{basic}
[{\partial}_3\thetaeta](z,w,\bar{w})={\overline{\partial}}\phi_z(w),Mathrm end{equation} and we infer
that when $z$ is fixed in ${Mathbb D}(z_0;Mathrm eps)$ we have
\begin{equation*}{\overline{\partial}} \phi_z(w)\nue 0\quad \text{when}\quad w\in{Mathbb D}(z_0;2Mathrm eps),
Mathrm end{equation*}
Moreover, choosing $Mathrm eps>0$ somewhat smaller if necessary, we can
assume that $U=\{(z,w,\zeta);z,w,\bar{\zeta}\in{Mathbb D}(z_0;2Mathrm eps)\}$.
Again, note that the same $Mathrm eps$ can be used for all $z_0$ in $S$.
We shall in the following fix a number $Mathrm eps>0$ with the above
properties, in addition to the properties which we required earlier,
i.e., ${Mathbb D}(z_0;2Mathrm eps)\Subset {Mathbb S}etX$, the inequality Mathrm eqref{goodc}
holds for all $z,w\in{Mathbb D}(z_0;2Mathrm eps)$, and the functions ${Mathbb Q}ext$,
${b}_0$ and ${b}_1$ are holomorphic in the set
$\{(z,w);z,\bar{w}\in{Mathbb D}(z_0;2Mathrm eps)\}$.
{Mathbb S}ubsection{The functions ${Mathbb T}heta_0$ and $\Upsilon_0$ and a formula for
$I_m$.} We now use the correspondence $F$ to define a holomorphic
function $\widetilde{{Mathbb Q}ext}$ in ${Mathbb D}(z_0;2Mathrm eps)\times F(U)$ via
\begin{equation*}\widetilde{{Mathbb Q}ext}(z,\xi,w,\thetaeta)={Mathbb Q}ext(z,\zeta(\xi,w,\thetaeta)).Mathrm end{equation*}
We also put
\begin{equation}\label{d0}{Mathbb T}heta_0(z,w,\thetaeta)=[{\partial}_1{\partial}_4
\widetilde{{Mathbb Q}ext}](z,z,w,\thetaeta)={b}_0(z,\zeta)\cdot[{\partial}_\thetaeta\zeta](z,w,\thetaeta)=
{b}_0(z,\zeta)/[{\partial}_\zeta\thetaeta](z,w,\zeta),\quad
\zeta=\zeta(z,w,\thetaeta),Mathrm end{equation}
where we recall that
\begin{equation*}{b}_0={\partial}_1{\partial}_2{Mathbb Q}ext.Mathrm end{equation*}
The function ${Mathbb T}heta_0$ was put to use by Berman et al. in
[\cite{BBS}, p. 9], where it
is called ${Mathbb D}elta_0$.
Note that Mathrm eqref{bri} implies that
$\thetaeta(z,z,\zeta)={\partial}_1{Mathbb Q}ext(z,\zeta)$ for all $z$ and $\zeta$, so
that, $[{\partial}_\zeta\thetaeta](z,z,\zeta)={b}_0(z,\zeta)$. Hence
Mathrm eqref{d0} gives
\begin{equation}\label{d01}{Mathbb T}heta_0(z,z,\thetaeta)=
{b}_0(z,\zeta)/{b}_0(z,\zeta)=1.Mathrm end{equation} Thus putting
\begin{equation*}\Upsilon_0(z,w,\thetaeta)=-\int_0^1[{\partial}_2{Mathbb T}heta_0](z,z+t(w-z),\thetaeta){Mathrm d}
t,Mathrm end{equation*} we obtain the relation
\begin{equation}\label{upsrel}\Upsilon_0(z,w,\thetaeta)(z-w)={Mathbb T}heta_0(z,w,\thetaeta)-1.Mathrm end{equation}
We have the following lemma.
\begin{lem}\label{funs}
\begin{equation}\label{boss}I_{m} u(z)=\int_{S} \frac
{u(w)\chi(w)}
{z-w}L^p(\D,dA)ar 1+\frac {{b}_1(z,\bar{w})}
{m{b}_0(z,\bar{w})}\right ) {Mathbb T}heta_0(z,w,\phi_z(w)){\overline{\partial}}_wL^p(\D,dA)ar
Mathrm e^{m\phi_z(w)(z-w)}\right ) {{Mathrm d} A}(w).Mathrm end{equation}
Mathrm end{lem}
\begin{proof}
Combining Mathrm eqref{d0} with Mathrm eqref{basic} we get the identity
\begin{equation}\label{neq}
{Mathbb T}heta_0(z,w,\phi_z(w))={b}_0(z,\bar{w})/{\overline{\partial}}\phi_z(w).Mathrm end{equation}
Recalling that
$K_m^1(z,w)=(m{b}_0(z,\bar{w})+{b}_1(z,\bar{w}))Mathrm e^{m{Mathbb Q}ext(z,\bar{w})}$
and $\phi_z(w)=\thetaeta(z,w,\bar{w})$ we thus have (see Mathrm eqref{izm})
\begin{equation*}\begin{split}I_{m} u(z)&=\int_{S}
u(w)\chi(w)K_m^1(z,\bar{w})Mathrm e^{-mQ(w)}{{Mathrm d} A}(w)=\\
&=\int_{S}
u(w)\chi(w)(m{b}_0(z,\bar{w})+{b}_1(z,\bar{w}))Mathrm e^{m\phi_z(w)(z-w)}
{{Mathrm d} A}(w)=\\
&=\int_{S} u(w)\chi(w) (m{b}_0(z,\bar{w})+{b}_1(z,\bar{w}))\frac
{{\overline{\partial}}_w (Mathrm e^{m\phi_z(w)(z-w)})} {m(z-w){\overline{\partial}} \phi_z(w)}{{Mathrm d} A}(w)=\\
&=\int_{S} \frac {u(w)\chi(w)}
{z-w}L^p(\D,dA)ar 1+\frac {{b}_1(z,\bar{w})}
{m{b}_0(z,\bar{w})}\right ) \frac{{b}_0(z,\bar{w})} {{\overline{\partial}}
\phi_z(w)}
{\overline{\partial}}_wL^p(\D,dA)ar Mathrm e^{m\phi_z(w)(z-w)}\right ) {{Mathrm d} A}(w)=\\
&=\int_{S} \frac {u(w)\chi(w)}
{z-w}L^p(\D,dA)ar 1+\frac {{b}_1(z,\bar{w})}
{m{b}_0(z,\bar{w})}\right ) {Mathbb T}heta_0(z,w,\phi_z(w)){\overline{\partial}}_wL^p(\D,dA)ar
Mathrm e^{m\phi_z(w)(z-w)}\right ) {{Mathrm d} A}(w),
Mathrm end{split}Mathrm end{equation*}
where we have used Mathrm eqref{neq} to get the last identity.
Mathrm end{proof}
Comparing Mathrm eqref{boss} with the formula Mathrm eqref{grep}, we now see a
connection between $I_m u(z)$ and $I_m^1 u(z)$. We shall
exploit this relation within short.
{Mathbb S}ubsection{The differential equation.} We will now show that the
functions ${b}_0$, ${b}_1$ and ${Mathbb T}heta_0$ are connected via an
important differential equation.
\begin{lem} \label{comp} For all $z,\bar{\zeta}\in{Mathbb D}(z_0;2Mathrm eps)$, we have the
identity
\begin{equation}\label{b1}\frac {{b}_1(z,\zeta)}
{{b}_0(z,\zeta)}=-L^p(\D,dA)ar\frac {{\partial}^2} {{\partial} w{\partial}\thetaeta}
{Mathbb T}heta_0(z,w,\thetaeta)\right )\biggm|_{z=w,\, \thetaeta=\thetaeta(z,z,\zeta)}.
Mathrm end{equation}
Mathrm end{lem}
\begin{proof}
In view of Mathrm eqref{funrel}, it suffices to show that the holomorphic
function
\begin{equation}\label{B1}B_1(z,\zeta)=-{b}_0(z,\zeta)\cdot
[{\partial}_2{\partial}_3{Mathbb T}heta_0](z,z,\thetaeta(z,z,\zeta)))Mathrm end{equation} satisfies
$B_1(z,\zeta)=\frac 1 2 \frac {\partial} {{\partial} \zeta}L^p(\D,dA)ar \frac 1
{{b}_0(z,\zeta)}\frac {\partial} {{\partial} z}{b}_0(z,\zeta)\right )$ for all $z$
and $\bar{\zeta}$ in a neighbourhood of $z_0$.
To this end, we first note that a Taylor expansion yields that
\begin{equation*}{Mathbb Q}ext(w,\zeta)={Mathbb Q}ext(z,\zeta)+ (w-z)\frac
{{\partial} {Mathbb Q}ext} {{\partial} z}(z,\zeta)+\frac {(w-z)^2} 2 \frac {{\partial}^2{Mathbb Q}ext} {{\partial}
z^2}(z,\zeta)+{Mathcal O} ((w-z)^3),\quad \text{as } w\to
z,Mathrm end{equation*} i.e.,
\begin{equation*}\thetaeta(z,w,\zeta)=\frac {{Mathbb Q}ext(w,\zeta)-{Mathbb Q}ext(z,\zeta)}{w-z}=
\frac {{\partial} {Mathbb Q}ext} {{\partial} z}(z,\zeta)+\frac {w-z} 2 \frac {{\partial}^2 {Mathbb Q}ext}
{{\partial} z^2}(z,\zeta)+{Mathcal O}((w-z)^2).
Mathrm end{equation*}
Differentiating with respect to $\zeta$ and using that
${b}_0={\partial}_1{\partial}_2{Mathbb Q}ext$ yields
\begin{equation}\label{burt}\frac {{\partial}\thetaeta} {{\partial}\zeta}={b}_0(z,\zeta)+
\frac {w-z} 2 \frac {{\partial} {b}_0} {{\partial}
z}(z,\zeta)+{Mathcal O}((w-z)^2).Mathrm end{equation} Dividing both sides of
Mathrm eqref{burt} by ${b}_0={b}_0(z,\zeta)$ and using Mathrm eqref{d0}, we
get
\begin{equation*}\frac 1 {{Mathbb T}heta_0}=
\frac 1 {{b}_0}\frac {{\partial}\thetaeta}{{\partial} \zeta}=
1+\frac {w-z} {2 {b}_0}\frac {{\partial} {b}_0} {{\partial}
z}+{Mathcal O}((w-z)^2).Mathrm end{equation*} Inverting this relation we obtain
\begin{equation*}\label{doeq}{Mathbb T}heta_0={b}_0/\frac {{\partial}\thetaeta}{{\partial}\zeta}=
1-\frac {w-z} {2{b}_0}\frac {{\partial} {b}_0} {{\partial}
z}+{Mathcal O}((w-z)^2).Mathrm end{equation*} In view of Mathrm eqref{B1}, it yields
that
\begin{equation*}\begin{split}B_1(z,\zeta(z,z,\thetaeta))
&=-{b}_0(z,\zeta)\cdot \frac {{\partial}^2} {{\partial}\thetaeta{\partial} w}
{Mathbb T}heta_0(z,w,\thetaeta)\biggm|_{w=z}=\\
&=\frac {{b}_0(z,\zeta)} 2\frac {\partial} {{\partial}\thetaeta}L^p(\D,dA)ar \frac 1
{{b}_0(z,\zeta)}\frac {\partial} {{\partial} z}
{b}_0(z,\zeta)\right )\biggm|_{z=w}=\\
&=\frac {{b}_0(z,\zeta)} 2 \frac {\partial} {{\partial}\zeta}L^p(\D,dA)ar \frac 1
{{b}_0(z,\zeta)}\frac{\partial} {{\partial} z} {b}_0(z,\zeta)\right )\biggm|_{z=w}/
\frac
{{\partial}\thetaeta}{{\partial}\zeta}(z,z,\zeta)=\\
&=\frac 1 2 \frac {\partial} {{\partial}\zeta}L^p(\D,dA)ar \frac 1
{{b}_0(z,\zeta)}\frac{\partial} {{\partial} z} {b}_0(z,\zeta)\right )={b}_1(z,\zeta).\\
Mathrm end{split}
Mathrm end{equation*}
Here we have used Mathrm eqref{d0}, Mathrm eqref{d01} and Mathrm eqref{funrel} to
obtain the last equality.
Mathrm end{proof}
{Mathbb S}ubsection{The operator ${Mathbb D}op_m$.} For the further analysis, it
will be convenient to consider a differential operator ${Mathbb D}op_m$
defined for complex smooth functions $A$ of the parameters $z$, $w$
and $\thetaeta$ by
\begin{equation}\label{nabla}{Mathbb D}op_m A=\frac {{\partial} A} {{\partial}\thetaeta}+m(z-w)A=
Mathrm e^{-m\thetaeta(z-w)}\frac {\partial} {{\partial} \thetaeta}L^p(\D,dA)ar
Mathrm e^{m\thetaeta(z-w)}A(z,w,\thetaeta)\right ).Mathrm end{equation} Cf. \cite{BBS},
p. 5. It will be useful to note that, evaluating at a point where
$\thetaeta=\phi_z(w)$ we have
\begin{equation}\label{eval}{Mathbb D}op_m A\biggm|_{\thetaeta=\phi_z(w)}=\frac {Mathrm e^{-m\phi_z(w)(z-w)}}
{{\overline{\partial}} \phi_z(w)}{\overline{\partial}}_wL^p(\D,dA)ar
Mathrm e^{m\phi_z(w)(z-w)}A(z,w,\phi_z(w))\right ),Mathrm end{equation} as is
easily verified by carrying out the differentiation and comparing
with Mathrm eqref{nabla}. From this, we now derive an important identity,
\begin{equation}\begin{split}\label{impo}\int_{S} u(w)&\chi(w)[{Mathbb D}op_m
A](z,w,\phi_z(w)){\overline{\partial}}_wL^p(\D,dA)arMathrm e^{m\phi_z(w)(z-w)}\right ){{Mathrm d} A}(w)=\\
&=m\int_{S} u(w)\chi(w){\overline{\partial}}_wL^p(\D,dA)ar
Mathrm e^{m\phi_z(w)(z-w)}A(z,w,\phi_z(w))\right ){{Mathrm d} A}(w)=\\
&=-m\int_{S} u(w){\overline{\partial}}\chi(w)
Mathrm e^{m\phi_z(w)(z-w)}A(z,w,\phi_z(w)){{Mathrm d} A}(w)+\\
&\quad +\frac m {2\pi\operatorname{Im}ag}\int_{{\partial}
S}u(w)\chi(w)Mathrm e^{m\phi_z(w)(z-w)}A(z,w,\phi_z(w)){Mathrm d}
w=\\
&=-m\int_{\{w\in S;\,\babs{z-w}\ge Mathrm eps/2\}} u(w){\overline{\partial}}\chi(w)
Mathrm e^{m\phi_z(w)(z-w)}A(z,w,\phi_z(w)){{Mathrm d} A}(w)+\\
&\quad +\frac m {2\pi\operatorname{Im}ag}\int_{{\partial}
S}u(w)\chi(w)Mathrm e^{m\phi_z(w)(z-w)}A(z,w,\phi_z(w)){Mathrm d} w ,\qquad
z\in{Mathbb D}(z_0;Mathrm eps)
.\\
Mathrm end{split}
Mathrm end{equation}
Here we have used Mathrm eqref{eval} to deduce the first equality. The
second equality follows from Green's formula. To get the last
equality we have used that $\babs{z-w}\ge Mathrm eps/2$ when
${\overline{\partial}}\chi(w)\nue 0$.
We now have the
following lemma, which is based on [\cite{BBS}, Lemma 2.3 and the
discussion on p. 9].
\begin{lem} \label{tlem} There exist holomorphic functions
$A_m(z,w,\thetaeta)$ and $B(z,w,\thetaeta)$, uniformly bounded on
$F(U)$, such that
\begin{equation}\label{tobe}L^p(\D,dA)ar 1+
\frac{{b}_1(z,\zeta)}{m{b}_0(z,\zeta)}\right )
{Mathbb T}heta_0(z,w,\thetaeta)=1+m^{-1}{Mathbb D}op_m
A_m(z,w,\thetaeta)+m^{-2}B(z,w,\thetaeta),Mathrm end{equation} where it is
understood that $\thetaeta=\thetaeta(z,w,\zeta)$. Moreover, we have that
\begin{equation}\label{alim}\lim_{m\to\infty}A_m=\Upsilon_0,Mathrm end{equation}
with uniform convergence on $F(U)$.
Mathrm end{lem}
\begin{proof} Put
\begin{equation*}{Mathbb T}heta_1(z,w,\thetaeta)=\frac {{b}_1(z,\zeta)}
{{b}_0(z,\zeta)}{Mathbb T}heta_0(z,w,\thetaeta),Mathrm end{equation*} so that the
left hand side in Mathrm eqref{tobe} can be written
\begin{equation}\label{obet}
L^p(\D,dA)ar 1+\frac {{b}_1(z,\zeta)}
{m{b}_0(z,\zeta)}\right ){Mathbb T}heta_0(z,w,\thetaeta)=
{Mathbb T}heta_0(z,w,\thetaeta)+m^{-1}{Mathbb T}heta_1(z,w,\thetaeta).Mathrm end{equation} Next,
recall that ${Mathbb T}heta_0(z,w,\thetaeta)-1=(z-w)\Upsilon_0(z,w,\thetaeta),$
cf. Mathrm eqref{upsrel}. Further, from the relation Mathrm eqref{b1}, which
reads
\begin{equation*}\frac {{b}_1(z,\zeta)} {{b}_0(z,\zeta)}=-L^p(\D,dA)ar \frac
{{\partial}^2} {{\partial}\thetaeta{\partial} w}{Mathbb T}heta_0(z,w,\thetaeta)\right )\biggm|_{z=w},\quad
\zeta=\zeta(z,z,\thetaeta),Mathrm end{equation*} it follows that
\begin{equation}\begin{split}\label{1029}{Mathbb T}heta_1(z,z,\thetaeta)&=-L^p(\D,dA)ar
\frac {{\partial}^2} {{\partial}\thetaeta{\partial} w}
{Mathbb T}heta_0(z,w,\thetaeta)\right ) \biggm|_{z=w}\cdot {Mathbb T}heta_0(z,z,\thetaeta)
= -L^p(\D,dA)ar \frac {{\partial}^2} {{\partial}\thetaeta{\partial} w} {Mathbb T}heta_0(z,w,\thetaeta)\right ) \biggm|_{z=w},\\
Mathrm end{split}Mathrm end{equation}
where we have used that ${Mathbb T}heta_0(z,z,\thetaeta)=1$ (see Mathrm eqref{d01})
to get the last equality. Mathrm eqref{1029} allows us to write
\begin{equation*}{Mathbb T}heta_1(z,w,\thetaeta)+{\partial}_\thetaeta{\partial}_w{Mathbb T}heta_0(z,w,\thetaeta)=(z-w)\Upsilon_1(z,w,\thetaeta),Mathrm end{equation*}
where $\Upsilon_1$ is holomorphic. Now define
\begin{equation*}A_m=\Upsilon_0+m^{-1}(\Upsilon_1-({\partial}_\thetaeta{\partial}_w \Upsilon_0))\quad, \quad
B=-{\partial}_\thetaeta(\Upsilon_1-({\partial}_\thetaeta{\partial}_w)\Upsilon_0).Mathrm end{equation*}
Then $A_m$ and $B$ are holomorphic and $A_m\to\Upsilon_0$ as
$m\to\infty$.
We will see that straightforward calculations show that
\begin{equation}\label{dopf}{Mathbb D}op_m A_m=mL^p(\D,dA)ar {Mathbb T}heta_0-1\right )+{Mathbb T}heta_1+m^{-1}\frac {\partial}
{{\partial}\thetaeta}L^p(\D,dA)ar \frac {{Mathbb T}heta_1} {z-w}-\frac {{\partial}_\thetaeta {Mathbb T}heta_0}
{(z-w)^2}\right )=mL^p(\D,dA)ar
{Mathbb T}heta_0-1\right )+{Mathbb T}heta_1-m^{-1}B.Mathrm end{equation} Indeed, when $z\nue
w$ we have
\begin{equation*}\begin{split}A_m&=\frac {{Mathbb T}heta_0-1} {z-w}+m^{-1}L^p(\D,dA)ar \frac
{{Mathbb T}heta_1+{\partial}_\thetaeta{\partial}_w{Mathbb T}heta_0} {z-w}-{\partial}_wL^p(\D,dA)ar\frac
{{\partial}_\thetaeta{Mathbb T}heta_0} {z-w}\right )\right )=\\
&=\frac {{Mathbb T}heta_0-1} {z-w}+m^{-1}L^p(\D,dA)ar \frac {{Mathbb T}heta_1} {z-w}-\frac
{{\partial}_\thetaeta{Mathbb T}heta_0} {(z-w)^2}\right )\\
Mathrm end{split}Mathrm end{equation*}
so that
\begin{equation*}{\partial}_\thetaeta A_m=\frac {{\partial}_\thetaeta {Mathbb T}heta_0}
{z-w}+m^{-1}L^p(\D,dA)ar \frac {{\partial}_\thetaeta{Mathbb T}heta_1} {z-w}-\frac
{{\partial}_\thetaeta^2{Mathbb T}heta_0} {(z-w)^2}\right ),Mathrm end{equation*} and
\begin{equation*}m(z-w)A_m=mL^p(\D,dA)ar {Mathbb T}heta_0-1\right )+{Mathbb T}heta_1-\frac
{{\partial}_\thetaeta{Mathbb T}heta_0} {z-w},Mathrm end{equation*} which gives Mathrm eqref{dopf},
since ${Mathbb D}op_m A_m={\partial}_\thetaeta A_m+m(z-w)A_m$.
Dividing through by $m$ in the Mathrm eqref{dopf} now gives
\begin{equation*}{Mathbb T}heta_0+m^{-1}{Mathbb T}heta_1=1+m^{-1}{Mathbb D}op_m
A_m+m^{-2}B,Mathrm end{equation*}
and glancing at Mathrm eqref{obet} we obtain
Mathrm eqref{tobe}.
Mathrm end{proof}
{Mathbb S}ubsection{Decomposition of $I_m$.} In view of Mathrm eqref{boss} and Lemma
\operatorname{Re}f{tlem}, we can now write
\begin{equation}\begin{split}\label{recon}I_{m} u(z)&=\int_{S} \frac {u(w)\chi(w)}
{z-w}L^p(\D,dA)ar 1+\frac {{b}_1(z,\bar{w})}
{m{b}_0(z,\bar{w})}\right )
{Mathbb T}heta_0(z,w,\bar{w}){\overline{\partial}}_wL^p(\D,dA)ar Mathrm e^{m\phi_z(w)(z-w)}\right ) {{Mathrm d} A}(w)=\\
&=\int_{S} \frac {u(w)\chi(w)} {z-w}
{\overline{\partial}}_w(Mathrm e^{m\phi_z(w)(z-w)}){{Mathrm d} A}(w)+\\
&+\frac 1 m\int_{S} \frac {u(w)\chi(w)} {z-w}[{Mathbb D}op_m
A_m](z,w,\phi_z(w)){\overline{\partial}}_w(Mathrm e^{m\phi_z(w)(z-w)}){{Mathrm d} A}(w)+\\
&+\frac 1 {m^2}\int_{S} \frac
{u(w)\chi(w)} {z-w}B(z,w,\phi_z(w)){\overline{\partial}}_w(Mathrm e^{m\phi_z(w)(z-w)}){{Mathrm d} A}(w),\\
Mathrm end{split}Mathrm end{equation}
where $A_m$ and $B$ are uniformly bounded and holomorphic near
$(z_0,z_0,\phi_{z_0}(z_0))$.
We recognize the first integral in the right hand side of
Mathrm eqref{recon} as $I_{m}^1 u(z)$ (see Mathrm eqref{grep}). Let us denote
the others by
\begin{equation*}I_{m}^2 u(z)=\frac 1 m \int_{S} \frac {u(w)\chi(w)}
{z-w}{Mathbb D}op_m
A_m(z,w,\phi_z(w)){\overline{\partial}}_w(Mathrm e^{m\phi_z(w)(z-w)}){{Mathrm d} A}(w),Mathrm end{equation*}
and
\begin{equation}\label{iih}
I_{m}^3 u(z)=\frac 1 {m^2} \int_{S} \frac {u(w)\chi(w)}
{z-w}B(z,w,\phi_z(w)){\overline{\partial}}_w(Mathrm e^{m\phi_z(w)(z-w)}){{Mathrm d} A}(w).Mathrm end{equation}
{Mathbb S}ubsection{Estimates for $I_m^2$.} We start by estimating
$I_{m}^2 u(z)$ when $z\in{Mathbb D}(z_0;Mathrm eps)$, and note that
Mathrm eqref{impo} yields that
\begin{equation}\begin{split}\label{ffsp}I_{m}^2 u(z)=&-\int_{\{w\in S;\,\babs{z-w}\geMathrm eps/2\}} u(w){\overline{\partial}} \chi(w)
Mathrm e^{m\phi_z(w)(z-w)}A_m(z,w,\phi_z(w)){{Mathrm d} A}(w)+\\
&+\frac 1 {2\pi\operatorname{Im}ag}\int_{{\partial}
S}u(w)\chi(w)Mathrm e^{m\phi_z(w)(z-w)}A_m(z,w,\phi_z(w)){Mathrm d}
w.\\
Mathrm end{split}
Mathrm end{equation}
Let us now define
\begin{equation}\label{nudef}\nuu_0(z,w)=\Upsilon_0(z,w,\phi_z(w)),Mathrm end{equation}
so that $A_m(z,w,\phi_z(w))\to\nuu_0(z,w)$ with uniform convergence
for $z,w\in{Mathbb D}(z_0;2Mathrm eps)$ when $m\to\infty$, by Lemma \operatorname{Re}f{tlem}.
Also let $C'=C'(Mathrm eps)$ be an upper bound for
$\{\babs{A_m(z,w,\phi_z(w))};\, z,w\in{Mathbb D}(z_0;2Mathrm eps),\, m\ge 1\}$. We
now use Mathrm eqref{ffsp} and the estimate Mathrm eqref{goodc} to get
\begin{equation}\begin{split}\label{goo2}&\babs{I_{m}^2 u(z)-\frac 1 {2\pi\operatorname{Im}ag}\int_{{\partial} S}
u(w)\chi(w)Mathrm e^{m\phi_z(w)(z-w)}\nuu_0(z,w){Mathrm d} w}\le\\
&\qquad\le C'\int_{\{w\in S;\,\babs{w-z}\ge Mathrm eps/2\}}
\babs{u(w){\overline{\partial}}\chi(w)}Mathrm e^{m(Q(z)/2-Q(w)/2-{\partial}elta_0\babs{z-w}^2/2)}{{Mathrm d} A}(w)\le\\
&\qquad\le C'Mathrm e^{m(Q(z)/2-{\partial}elta_0Mathrm eps^2/8)}\int_{Mathbb C}
\babs{u(w){\overline{\partial}}\chi(w)}Mathrm e^{-Q(w)/2}{{Mathrm d} A}(w)\le\\
&\qquad\le
C'Mathrm e^{m(Q(z)/2-{\partial}elta_0Mathrm eps^2/8)}\|u\|_{mQ}\|{\overline{\partial}}\chi\|_{L^2}\le
C_2Mathrm e^{m(Q(z)/2-{\partial}eltaMathrm eps^2)}\|u\|_{mQ},\\
Mathrm end{split}
Mathrm end{equation}
where ${\partial}elta={\partial}elta_0/8={Mathbb D}elta Q(z_0)/16$, and
$C_2=3C'$.
{Mathbb S}ubsection{Estimates for $I_m^3$.} To estimate
$I_m^3 u(z)$, we note that Mathrm eqref{iih} implies
\begin{equation*}I_m^3 u(z)=\frac 1 m \int_{Mathbb C} u(w)\chi(w)B(z,w,\phi_z(w))
Mathrm e^{m\phi_z(w)(z-w)}{\overline{\partial}}\phi_z(w){{Mathrm d} A}(w).Mathrm end{equation*} It suffices
to estimate this integral using Mathrm eqref{goodc}. This gives
\begin{equation*}\babs{I^3_{m} u(z)}\le \frac 1 m \int_{Mathbb C}
\babs{u(w)\chi(w)B(z,w,\phi_z(w))}Mathrm e^{m(Q(z)/2-Q(w)/2)-m{\partial}elta_0\babs{z-w}^2/2}\babs{{\overline{\partial}}
\phi_z(w)}{{Mathrm d} A}(w).Mathrm end{equation*} Since the function
$\babs{B(z,w,\phi_z(w)){\overline{\partial}}\phi_z(w)\chi(w)}$ can be estimated by a number
$C_3$ independent of $z$ and $w$ for $z\in{Mathbb D}(z_0;Mathrm eps)$ and $w\in
{Mathbb D}(z_0;2Mathrm eps)$, it gives, in view of the Cauchy--Schwarz inequality
\begin{equation}\label{goo3}\begin{split}\babs{I^3_{m}u(z)}&\le
C_3m^{-1}Mathrm e^{mQ(z)/2}\int_{Mathbb C}\babs{u(w)\chi(w)}Mathrm e^{-mQ(w)/2}{{Mathrm d} A}(w)\le\\
&\le C_3m^{-1}Mathrm e^{mQ(z)/2}\|u\|_{mQ}L^p(\D,dA)ar\int_{Mathbb C} Mathrm e^{-m{\partial}elta_0\babs{z-w}^2}{{Mathrm d} A}(w)\right )^{1/2}\le
C_3{\partial}elta_0^{-1/2}m^{-3/2}Mathrm e^{mQ(z)/2}\|u\|_{mQ},\\
Mathrm end{split}Mathrm end{equation} where
$C_3=C_3(Mathrm eps)={Mathbb S}up\big\{\babs{B(z,w,\phi_z(w)){\overline{\partial}}\phi_z(w)\chi(w)};\,
z,w\in {Mathbb D}(z_0;2Mathrm eps)\big\}$.
{Mathbb S}ubsection{Conclusion of the proof of Theorem \operatorname{Re}f{mlem}} By
Mathrm eqref{recon} and the estimates Mathrm eqref{goo1}, Mathrm eqref{goo2} and
Mathrm eqref{goo3}, (using that $\phi_z(w)(z-w)={Mathbb Q}ext(z,\bar{w})-Q(w)$)
\begin{equation*}\begin{split}&\babs{I_{m} u(z)-u(z)-\frac 1
{2\pi\operatorname{Im}ag}\int_{{\partial} S}u(w)\chi(w)L^p(\D,dA)ar \frac 1
{z-w}+\nuu_0(z,w)\right )Mathrm e^{m({Mathbb Q}ext(z,\bar{w})-Q(w))}{Mathrm d}
w}\le\\
&\le \babs{I_{m}^1 u(z)-u(z)-\frac 1 {2\pi\operatorname{Im}ag}\int_{{\partial}
S}\frac{u(w)\chi(w)Mathrm e^{m\phi_z(w)(z-w)}}
{z-w}{Mathrm d} w}+\\
&\qquad+\babs{I_{m}^2u(z)-\frac 1 {2\pi\operatorname{Im}ag}\int_{{\partial}
S}u(w)\chi(w)\nuu_0(z,w)Mathrm e^{m\phi_z(w)(z-w)}{Mathrm d} w}+
\babs{I_{m}^3u(z)}\le\\
&\le C_1Mathrm eps^{-1}Mathrm e^{m(Q(z)/2-{\partial}eltaMathrm eps^2)}\|u\|_{mQ}+C_2Mathrm e^{m(Q(z)/2-{\partial}eltaMathrm eps^2)}
\|u\|_{mQ}+C_3{\partial}elta_0^{-1}m^{-3/2}Mathrm e^{mQ(z)/2}\|u\|_{mQ},\quad z\in{Mathbb D}(z_0;Mathrm eps)\cap S^\circ.\\
Mathrm end{split}
Mathrm end{equation*}
It thus suffices to choose a number $C$ large enough that
\begin{equation}\label{itsp}C_1Mathrm e^{-{\partial}elta m}+C_2Mathrm e^{-{\partial}elta m}
+C_3m^{-3/2}
\le Cm^{-3/2},\quad m\ge 1.Mathrm end{equation}
Now recall that the numbers $C_1,$ $C_2,$ and $C_3$ only depend on
$Mathrm eps$, where $Mathrm eps>0$ can be chosen independently of $z_0$ for
$z_0$ in the given compact subset $S$ of ${Mathbb S}etX$. Moreover, we have
that ${\partial}elta={Mathbb D}elta Q(z_0)Mathrm eps^2/16$, and ${Mathbb D}elta Q(z_0)$ is bounded
below by a positive number for $z_0\in S$. It follows that $C$ in
Mathrm eqref{itsp} can be chosen independently of $z_0\in S$, and the
proof is finished.
$\qed$
{Mathbb S}ection{The proof of Theorem \operatorname{Re}f{th3}}\label{proof}
{Mathbb S}ubsection{Preliminaries.}
In this section we prove Th. \operatorname{Re}f{th3}. Our approach which follows
[\cite{BBS}, Sect. 3] is obtained by assembling the information from
Lemma \operatorname{Re}f{lemm2}, Th. \operatorname{Re}f{mlem}, and Cor. \operatorname{Re}f{bh}. To facilitate
the presentation, we divide the proof into steps.
First recall that adding a constant to $Q$ means that $K_{m,n}$ is
only changed by a multiplicative constant, and hence we can (and
will) assume that $Q\ge 1$ on ${Mathbb C}$.
Fix a compact subset $K\Subset {Mathbb S}etS_\tau^\circ\cap{Mathbb S}etX$ and a
point $z_0\in {Mathbb S}etS_\tau^\circ\cap {Mathbb S}etX$, and let $Mathrm eps>0$ and
$\chi\in \coity({Mathbb C})$ be as in Th. \operatorname{Re}f{mlem}.
Choosing $Mathrm eps>0$ somewhat smaller if necessary, we may ensure that
$2Mathrm eps<\operatorname{dist\,}(K,{Mathbb C}{Mathbb S}etminus({Mathbb S}etS_\tau\cap{Mathbb S}etX))$ and that
\begin{equation}\label{leed}\babs{K_m^1(z,w)}^2Mathrm e^{-m(Q(z)+Q(w))}\le
Cm^2Mathrm e^{-{\partial}elta_0 m\babs{z-w}^2},\quad z,\bar{w}\in
{Mathbb D}(z_0;2Mathrm eps),\quad z_0\in K,Mathrm end{equation} for some positive
numbers $C$ and ${\partial}elta_0$ (see Mathrm eqref{lead}). We also introduce the
set
\begin{equation*}{Mathbb S}etSp=\{z\in{Mathbb C};\, \operatorname{dist\,}(z,K)\le
2Mathrm eps\}\Subset {Mathbb S}etS_\tau^\circ\cap{Mathbb S}etX.Mathrm end{equation*} Our goal is
to prove an estimate
\begin{equation*}\babs{K_{m,n}(z,w)-K_m^1(z,w)}Mathrm e^{-m(Q(z)+Q(w))/2}\le
Cm^{-1},\quad z,w\in{Mathbb D}(z_0;Mathrm eps),\quad n\ge m\tau-M,\quad m\ge m_0,
Mathrm end{equation*}
where $M\ge 0$ is given, and $C$ and $m_0$ are independent of the specific
choice of the point $z_0\in K$.
In the following, we let $C$ denote various (more or less) absolute
constants which can change meaning from time to time, even within
the same chain of inequalities.
Let $P_{m,n}:L^2_{mQ}\to {H}_{m,n}$ denote the orthogonal
projection, that is,
\begin{equation*}P_{m,n}u(z)=
\int_{Mathbb C} u(w)K_{m,n}(z,w)Mathrm e^{-mQ(w)} {{Mathrm d} A}(w).Mathrm end{equation*}
Throughout the proof we \textit{fix} two points $z,w\in{Mathbb D}(z_0;Mathrm eps)$
and introduce the functions
\begin{equation}\label{func1}u_z(\zeta)=K_{m,n}(\zeta,z)\quad\text{and}\quad
v_w(\zeta)=\begin{cases} \chi(\zeta)K_m^1(\zeta,w),\quad& \zeta\in
{Mathbb D}(z_0;2Mathrm eps),\cr 0, &\text{otherwise},\cr
Mathrm end{cases}
Mathrm end{equation}
as well as
\begin{equation}\label{dwdef}
d_w(\zeta)=v_w(\zeta)-P_{m,n}v_w(\zeta),\qquad
\zeta\in{Mathbb C}.Mathrm end{equation} We note that $v_w\in{Mathcal C}_0^\infty({Mathbb C})$
with ${Mathbb S}upp v_w{Mathbb S}ubset {Mathbb S}etSp\Subset{Mathbb S}etS_\tau^\circ\cap {Mathbb S}etX.$
\begin{lem} \label{lemma2} There exists a positive number $C$ independent
of $m\ge 1$, $n$, $z$ and $w$, such that
\begin{equation}\label{app1}
\big|K_{m,n}(z,w)-P_{m,n}v_w(z)\big|\le Cm^{-1}
Mathrm e^{m(Q(z)+Q(w))/2}.Mathrm end{equation} Moreover, $C$ can be chosen
independent of $z_0$ for $z_0\in K$.
Mathrm end{lem}
\begin{proof}
By Th. \operatorname{Re}f{mlem}, there is a number $C$ depending only on $Mathrm eps$
and $K$, such that
\begin{equation*}\babs{I_{m} u_z(w)-u_z(w)}
\le Cm^{-3/2}Mathrm e^{mQ(w)/2}\|u_z\|_{mQ},
Mathrm end{equation*}
where $I_{m}$ is given by Mathrm eqref{izm} with $S=\Sigma$. Next note
that the estimate Mathrm eqref{berg1} implies
\begin{equation*}\|u_{z}\|_{mQ}^2=K_{m,n}(z,z)\le
CmMathrm e^{mQ(z)},
Mathrm end{equation*}
with a number $C$ depending only on $\tau$. We conclude that
\begin{equation}\label{fest}\babs{I_{m} u_z(w)-K_{m,n}(w,z)}\le
Cm^{-1}Mathrm e^{m(Q(z)+Q(w))/2}.Mathrm end{equation} We next note that
\begin{equation}\label{ik1}\overline{I_{m} u_z(w)}=
\int_{Mathbb C}
\chi(\zeta)K_m^1(\zeta,w)K_{m,n}(z,\zeta)Mathrm e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta)
=P_{m,n}v_w(z),Mathrm end{equation} with $v_w$ given by Mathrm eqref{func1}.
Thus the statement Mathrm eqref{app1} is immediate from Mathrm eqref{ik1} and
Mathrm eqref{fest}.
Mathrm end{proof}
\begin{lem} \label{lemma3} Let two positive numbers $A^p(\D)ar$ and ${Mathbf p}ar$ be given
such that Mathrm eqref{sss} is fulfilled, and put
$m_0=Max\{(1+A^p(\D)ar)/\tau,2A^p(\D)ar\}$. Then for all $m,n$ such that $n\ge
](m-A^p(\D)ar)\tau+{Mathbf p}ar[$ and $m\ge m_0$, there exist positive numbers
${\partial}elta$ and $C$, independent of $z_0$, $m$, $n$, $z$, and $w$, such
that
\begin{equation}\label{sest}
\babs{d_w(z)}^2\le Cm^3Mathrm e^{-m{\partial}elta}Mathrm e^{m(Q(z)+Q(w))}.
Mathrm end{equation}
Mathrm end{lem}
\begin{proof} The function $d_w$ is the
$L^2_{mQ,n}$-minimal solution to the ${\overline{\partial}}$-equation ${\overline{\partial}}
d_w={\overline{\partial}} v_w$. Since ${Mathbb S}upp v_w{Mathbb S}ubset {Mathbb S}etS_\tau$, Cor. \operatorname{Re}f{bh}
yields that
\begin{equation}\label{dbest}
\|d_w\|_{mQ}^2\le C\|{\overline{\partial}} v_w\|_{mQ}^2,\qquad m\ge m_0,\quad n\ge
](m-A^p(\D)ar)\tau+{Mathbf p}ar[,Mathrm end{equation} with a number $C$ depending
only on $\tau$, $A^p(\D)ar$ and ${Mathbf p}ar$. But ${\overline{\partial}} v_w(\zeta)={\overline{\partial}}
\chi(\zeta)K_m^1(\zeta,w)$, whence the estimate Mathrm eqref{leed} shows
that there are numbers $C$ and ${\partial}elta_0>0$ such that
\begin{equation}\label{test}|{\overline{\partial}} v_w(\zeta)|^2
Mathrm e^{-m(Q(\zeta)+Q(w))}\le
Cm^2|{\overline{\partial}}\chi(\zeta)|^2Mathrm e^{-m{\partial}elta_0\babs{\zeta-w}^2},\qquad
\zeta\in{Mathbb C}.
Mathrm end{equation} Now, since $\babs{\zeta-w}\ge Mathrm eps/2$ whenever
${\overline{\partial}}\chi(\zeta)\nue 0$, we deduce from Mathrm eqref{test} that
\begin{equation}\label{quest}|{\overline{\partial}} v_w(\zeta)|^2Mathrm e^{-mQ(\zeta)}\le
Cm^2|{\overline{\partial}}\chi(\zeta)|^2Mathrm e^{m(Q(w)-{\partial}elta)},\quad \zeta
\in{Mathbb C},Mathrm end{equation} where ${\partial}elta={\partial}elta_0Mathrm eps^2/4$. Using
Mathrm eqref{dbest} and then integrating the inequality Mathrm eqref{quest} with
respect to ${{Mathrm d} A}(\zeta)$, we get
\begin{equation}\label{cest}\|d_w\|_{mQ}^2\le
C\|{\overline{\partial}} v_w\|_{mQ}^2 \le
Cm^2Mathrm e^{m(Q(w)-{\partial}elta)}\|{\overline{\partial}}\chi\|_{L^2}^2\le
Cm^2Mathrm e^{m(Q(w)-{\partial}elta)}.
Mathrm end{equation}
Next note that the function $d_w$ is holomorphic in the disk
${Mathbb D}(z_0;3Mathrm eps/2)$, so that Lemma \operatorname{Re}f{lemm2} gives
\begin{equation}\label{desto}
|d_w(z)|^2Mathrm e^{-mQ(z)}\le
Cm\int_{{Mathbb D}(z;Mathrm eps/{Mathbb S}qrt{m})}\babs{d_w(\zeta)}^2Mathrm e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta)\le
Cm \|d_w\|^2_{mQ},
Mathrm end{equation}
where $C$ only depends on $Mathrm eps$ and $\tau$. Combining Mathrm eqref{desto}
with Mathrm eqref{cest}, we end up with Mathrm eqref{sest}, and so we are done.
Mathrm end{proof}
{Mathbb S}ubsection{Conclusion of the proof of Theorem \operatorname{Re}f{th3}.} Since $\chi(z)=1$,
we have that $v_w(z)=K_m^1(z,w)$, whence by Mathrm eqref{dwdef}
\begin{equation*}\babs{K_{m,n}(z,w)-K_m^1(z,w)}=
\babs{K_{m,n}(z,w)-v_w(z)}
\le\babs{K_{m,n}(z,w)-P_{m,n}v_w(z)}+\babs{d_w(z)}.
Mathrm end{equation*}
In view of Lemmas \operatorname{Re}f{lemma2} and \operatorname{Re}f{lemma3}, the right hand side
can be estimated by
\begin{equation*}CL^p(\D,dA)ar m^{-1}+m^{3/2}Mathrm e^{-m{\partial}elta/2}\right )
Mathrm e^{m(Q(z)+Q(w))/2},Mathrm end{equation*} whenever
$n\ge](m\tau-A^p(\D)ar)+{Mathbf p}ar[$ and $m\ge m_0$. For $m\ge 1$, the latter
expression is dominated by $Cm^{-1}Mathrm e^{m(Q(z)+Q(w))/2}$. The proof
is finished.
$\qed$
{Mathbb S}ection{Berezin quantization and Gaussian convergence} \label{p2}
{Mathbb S}ubsection{Preliminaries.} In this section we use the
expansion formula for $K_{m,n}$ (Th. \operatorname{Re}f{th3}) to prove theorems
\operatorname{Re}f{th1} and \operatorname{Re}f{th2}. In the proofs, we set $\tau=1$ and $m=n$.
(The argument in the general case follows the same pattern.) It then
becomes natural to write $K_n$ for $K_{n,n}$ etc. We also fix a
compact subset $K\Subset {Mathbb S}etS_1^\circ\cap{Mathbb S}etX$, a point $z_0\in
K$, and a positive number $Mathrm eps$ with the properties listed in Th.
\operatorname{Re}f{th3}, and we put
\begin{equation*}{\partial}elta_n=Mathrm eps\log n/{Mathbb S}qrt{n}.Mathrm end{equation*}
{Mathbb S}ubsection{The proof of Theorem \operatorname{Re}f{th1}} It suffices to show that
\begin{equation}\label{firstest}B_{n}^{\langle
z_0\rangle}({Mathbb D}(z_0;{\partial}elta_n))\to 1\quad \text{as}\quad
n\to \infty.Mathrm end{equation} Indeed, since $B_{n}^{\langle
z_0\rangle}$ is a p.m., this implies Th. \operatorname{Re}f{th1}.
In order to prove Mathrm eqref{firstest}, we apply Th. \operatorname{Re}f{th3}, which
gives
\begin{equation*}K_{n}(z_0,z)Mathrm e^{-n(Q(z)+Q(z_0))/2}=
L^p(\D,dA)ar
n{b}_0(z_0,\bar{z})+{b}_1(z_0,\bar{z})\right )Mathrm e^{n(\operatorname{Re}{Mathbb Q}ext(z_0,\bar{z})-(Q(z)+Q(z_0))/2)}+
{Mathcal O}(n^{-1}),\quad z\in{Mathbb D}(z_0;Mathrm eps)Mathrm end{equation*} when
$n\to\infty$, where the ${Mathcal O}$ is uniform for $z_0\in K$. In view
of Mathrm eqref{bbs}, we have
\begin{equation}\label{2ndest}K_{n}(z_0,z)Mathrm e^{-n(Q(z)+Q(z_0))/2}=
L^p(\D,dA)ar n{b}_0(z_0,\bar{z})+{Mathcal O}(1)\right ) Mathrm e^{n(-{Mathbb D}elta
Q(z_0)/2+R(z_0,z))}+ {Mathcal O}(n^{-1}),\quad
z\in{Mathbb D}(z_0;Mathrm eps)Mathrm end{equation} with a function $R$ satisfying
$\babs{R(z,w)}\le C\babs{z-w}^3$ whenever $z,w\in {Mathbb D}(z_0;Mathrm eps)$ and
$z_0\in K$. Note that
\begin{equation}\label{mroh}n\babs{R(z_0,z)}\le Cn{\partial}elta_n^3\le C\log^3
n/{Mathbb S}qrt{n},\qquad \text{when}\quad z\in{Mathbb D}(z_0;{\partial}elta_n),\quad z_0\in
K.Mathrm end{equation} Introducing this estimate in Mathrm eqref{2ndest} gives
\begin{equation*}\babs{K_{n}(z_0,z)}^2Mathrm e^{-n(Q(z)+Q(z_0))}=
L^p(\D,dA)ar n^2\babs{b_0(z_0,\bar{z})}^2+{Mathcal O}(n)\right )Mathrm e^{-n{Mathbb D}elta
Q(z_0)\babs{z_0-z}^2+{Mathcal O}(\log^3 n/{Mathbb S}qrt{n})}+{Mathcal O}(n^{-2}),
Mathrm end{equation*}
for $z\in{Mathbb D}(z_0;{\partial}elta_n)$ as
$n\to\infty$, where the ${Mathcal O}$-terms are uniform for $z_0\in K$.
Furthermore, Mathrm eqref{ber} yields that
\begin{equation}\label{3dest}\frac {\babs{K_{n}(z_0,z)}^2}
{K_{n}(z_0,z_0)}Mathrm e^{-nQ(z)}=\frac {\babs{K_{n}(z_0,z)}^2} {n{Mathbb D}elta
Q(z_0)+{Mathcal O}(1)}Mathrm e^{-n(Q(z_0)+Q(z))},\quad
z\in{Mathbb D}(z_0;Mathrm eps),Mathrm end{equation} as
$n\to\infty$. Note that the left hand side in Mathrm eqref{3dest} is the
density $\berd_{n}^{\langle z_0\rangle}(z)$, so that Mathrm eqref{2ndest}
and Mathrm eqref{3dest} implies
\begin{equation}\label{4thest}\berd_{n}^{\langle z_0\rangle}(z)=\frac
{n^2\babs{{b}_0(z_0,\bar{z})}^2+{Mathcal O}(n)} {n{Mathbb D}elta
Q(z_0)+{Mathcal O}(1)}Mathrm e^{-n{Mathbb D}elta
Q(z_0)\babs{z-z_0}^2+{Mathcal O}(\log^3n/{Mathbb S}qrt{n})} +{Mathcal O}(n^{-2}),\quad
z\in {Mathbb D}(z_0;{\partial}elta_n).Mathrm end{equation} Integrating Mathrm eqref{4thest} with
respect to ${{Mathrm d} A}$ over ${Mathbb D}(z_0;{\partial}elta_n)$ and using the mean-value
theorem for integrals now gives that there are positive numbers
$v_{n,z_0}$ converging to $1$ (uniformly for $z_0\in K$), and also
complex numbers $b_{n,z_0}$ converging to
${b}_0(z_0,\bar{z}_0)={Mathbb D}elta Q(z_0)$ (uniformly for $z_0\in K$) as
$n\to\infty$, such that
\begin{equation*}\begin{split}B_{n}^{\langle z_0\rangle}({Mathbb D}(z_0;{\partial}elta_n))&=
v_{n,z_0}\frac {n^2\babs{b_{n,z_0}}^2+{Mathcal O}(n)} {n{Mathbb D}elta
Q(z_0)+{Mathcal O}(1)}\int_{{Mathbb D}(z_0;{\partial}elta_n)} Mathrm e^{-n{Mathbb D}elta
Q(z_0)\babs{z-z_0}^2}{{Mathrm d} A}(z)+{Mathcal O}(n^{-2})\int_{{Mathbb D}(z_0;{\partial}elta_n)}{{Mathrm d} A}(z)=\\
&=v_{n,z_0}\frac {n^2\babs{b_{n,z_0}}^2+{Mathcal O}(n)} {n^2{Mathbb D}elta
Q(z_0)^2+{Mathcal O}(n)}L^p(\D,dA)ar 1-Mathrm e^{-{Mathbb D}elta Q(z_0)Mathrm eps^2\log^2
n}\right )+{Mathcal O}(n^{-2}).\\
Mathrm end{split}
Mathrm end{equation*}
The expression in the right hand side converges to $1$ as
$n\to\infty$. This proves Mathrm eqref{firstest}, and Th. \operatorname{Re}f{th1}
follows.
$\qed$
{Mathbb S}ubsection{The proof of Theorem \operatorname{Re}f{th2}}It suffices to to show that
there are numbers $Mathrm eps_n$ converging to zero as $n\to\infty$ such
that
\begin{equation}\label{5thest}\int_{Mathbb C}\babs{\berd_{n}^{\langle
z_0\rangle}(z)-n{Mathbb D}elta Q(z_0)Mathrm e^{-n{Mathbb D}elta
Q(z_0)\babs{z-z_0}^2}}{{Mathrm d} A}(z)\le Mathrm eps_n,\qquad z_0\in
K.Mathrm end{equation} Indeed, Mathrm eqref{gc1} follows form Mathrm eqref{5thest}
after the change of variables $zMapsto z_0+z/{Mathbb S}qrt{n{Mathbb D}elta Q(z_0)}$
in the integral in Mathrm eqref{5thest}.
To prove Mathrm eqref{5thest}, we split the integral with respect to the
decomposition $\{\babs{z-z_0}<{\partial}elta _n\}\cup\{\babs{z-z_0}\ge
{\partial}elta_n\}$, and put
\begin{equation*}A_{n,z_0}=
\int_{\{z;\babs{z-z_0}<{\partial}elta_n\}}\babs{\berd_{n}^{\langle
z_0\rangle}(z)-n{Mathbb D}elta Q(z_0)Mathrm e^{-n{Mathbb D}elta
Q(z_0)\babs{z-z_0}^2}}{{Mathrm d} A}(z),Mathrm end{equation*}
\begin{equation*}B_{n,z_0}=
\int_{\{z;\babs{z-z_0}\ge {\partial}elta_n\}}\babs{\berd_{n}^{\langle
z_0\rangle}(z)-n{Mathbb D}elta Q(z_0)Mathrm e^{-n{Mathbb D}elta
Q(z_0)\babs{z-z_0}^2}}{{Mathrm d} A}(z).Mathrm end{equation*} Considering $A_{n,z_0}$
first, we get from Mathrm eqref{5thest} that
\begin{equation*}A_{n,z_0}=
\int_{{Mathbb D}(z_0;{\partial}elta_n)}\babs{Mathrm e^{nR(z_0,\bar{z})}\frac
{n^2\babs{{b}_0(z_0,\bar{z})}^2+{Mathcal O}(n)} {n{Mathbb D}elta
Q(z_0)+{Mathcal O}(1)}-n{Mathbb D}elta Q(z_0)}Mathrm e^{-n{Mathbb D}elta
Q(z_0)\babs{z-z_0}^2}{{Mathrm d} A}(z)+{Mathcal O}(n^{-2}),Mathrm end{equation*} as
$n\to\infty$. Next we put
\begin{equation*}s_{n,z_0}={Mathbb S}up_{z\in{Mathbb D}(z_0;{\partial}elta_n)}\bigg\{\babs{Mathrm e^{nR(z_0,\bar{z})}\frac
{n^2\babs{{b}_0(z_0,\bar{z})}^2+{Mathcal O}(n)} {n{Mathbb D}elta
Q(z_0)+{Mathcal O}(1)}-n{Mathbb D}elta Q(z_0)}\bigg\},Mathrm end{equation*} and observe
Mathrm eqref{mroh} implies that $s_{n,z_0}/n\to 0$, uniformly for $z_0\in
K$. It yields that
\begin{equation*}A_{n,z_0}\le s_{n,z_0}\int_{{Mathbb D}(z_0;{\partial}elta_n)}Mathrm e^{-n{Mathbb D}elta
Q(z_0)\babs{z-z_0}^2}{{Mathrm d} A}(z)+{Mathcal O}(n^{-2})\le
Cs_{n,z_0}/n,Mathrm end{equation*} which converges to $0$ as $n\to \infty$
uniformly for $z_0\in K$.
To estimate $B_{n,z_0}$ we simply observe that
\begin{equation}\label{bmest}B_{n,z_0}\le
\int_{\{z;\babs{z-z_0}\ge{\partial}elta_n\}}\berd_{n}^{\langle
z_0\rangle}(z){{Mathrm d} A}(z)+\int_{\{z;\babs{z-z_0}\ge{\partial}elta_n\}}n{Mathbb D}elta
Q(z_0)Mathrm e^{-n{Mathbb D}elta Q(z_0)\babs{z-z_0}^2}{{Mathrm d} A}(z).
Mathrm end{equation}
Since $B_{n}^{\langle z_0\rangle}$ is a p.m., the estimate
Mathrm eqref{firstest} yields that the first integral in the right hand
side of Mathrm eqref{bmest} converges to $0$ as $n\to\infty$. Moreover, a
simple calculation yields that the second integral in Mathrm eqref{bmest}
converges to $0$ when $n\to\infty$. We have shown that $B_{n,z_0}\to
0$ as $n\to \infty$ with uniform convergence for $z_0\in K$. The
proof is finished.
$\qed$
{Mathbb S}ection{Off-diagonal damping}\label{point}
{Mathbb S}ubsection{An estimate for $K_{m,n}$.} In this section we prove
Th. \operatorname{Re}f{th1.5}. We shall obtain that theorem from Th. \operatorname{Re}f{flock}
below, which is of independent interest, and has applications in
random matrix theory, see \cite{AHM}. Our analysis depends on the
following lemma. It will be convenient to define the set
\begin{equation*}{Mathbb S}etS_{\tau,1}=\{\zeta;\,
\operatorname{dist\,}(\zeta,{Mathbb S}etS_\tau)\le 1\}.Mathrm end{equation*}
\begin{lem}\label{propn1} Assume that $Q\in{Mathcal C}^2({Mathbb C})$.
Let $z_0\in{Mathbb S}etS_\tau\cap{Mathbb S}etX$ and let $M$ be given non-negative
numbers. Put
\begin{equation*}8d=\operatorname{dist\,}(z_0,{Mathbb C}{Mathbb S}etminus({Mathbb S}etS_\tau\cap{Mathbb S}etX))\qquad
a=\inf\{{Mathbb D}elta Q(\zeta);\, \zeta\in{Mathbb D}(z_0;6d)\}\qquad A={Mathbb S}up\{{Mathbb D}elta
Q(\zeta);\, \zeta\in{Mathbb S}etS_{\tau,1}\}.Mathrm end{equation*} There then
exists positive numbers $C$ and $Mathrm epsilon$ such that for all $m,n\ge
1$ with $m\tau-M\le n\le m\tau+1$, we have
\begin{equation}\label{bupp}
\babs{K_{m,n}(z_0,z)}^2Mathrm e^{-m(Q(z_0)+Q(z))}\le C
m^2Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{d,\babs{z_0-z}\}} ,\qquad z\in
{Mathbb S}etS_\tau.
Mathrm end{equation}
Here $C$ depends only on $M$, $a$, $A$, and $\tau$, while $Mathrm epsilon$
only depends on $a$, $\tau$ and $M$.
Mathrm end{lem}
\begin{rem} For fixed
$\tau$ and $M$, our method of proof gives that $Mathrm epsilon$ can be
chosen proportional to $a$ while $C$ can be chosen proportional to
$a^{-1}$. Indeed, our proof shows that if there is a positive number
$c$ depending only on $\tau$ and $M$ such that, with $a^\prime=Min\{a,1\}$,
\begin{equation*}\babs{K_{m,n}(z_0,z)}^2Mathrm e^{-m(Q(z_0)+Q(z))}\le
C\frac {m^3} {am+c}Mathrm e^{-Mathrm epsilon'a'{Mathbb S}qrt{m}Min\{d,\babs{z_0-z}\}}
,\qquad z\in {Mathbb S}etS_\tau,
Mathrm end{equation*}
with $C$ and $Mathrm epsilon'$ independent of $a$. This estimate can
easily be extended to all $z\in{Mathbb C}$ by adapting the proof of Th.
\operatorname{Re}f{flock} below.
Mathrm end{rem}
\nuoindent Before we turn to the proof of Lemma \operatorname{Re}f{propn1}, we use
it to prove the main result of this section.
\begin{thm}\label{flock}
In the situation of Lemma \operatorname{Re}f{propn1} we also have an estimate
\begin{equation}\label{brest}\babs{K_{m,n}(z_0,z)}^2Mathrm e^{-m(Q(z_0)+Q(z))}\le
Cm^2Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{d,\babs{z_0-z}\}}Mathrm e^{-m
(Q(z)-\widehat{Q}_\tau(z))},\qquad z\in{Mathbb C},Mathrm end{equation} valid for all
$z_0\in {Mathbb S}etS_\tau\cap X$, $m\ge 1$, and all $n$ with $m\tau-M\le
n\le m\tau+1$.
Mathrm end{thm}
\begin{proof} In view of Lemma \operatorname{Re}f{propn1}, and since $Q=\widehat{Q}_\tau$
on ${Mathbb S}etS_\tau$, it suffices to show
the estimate Mathrm eqref{brest} when $z\nuot\in {Mathbb S}etS_\tau$. To this end, we
consider the function
\begin{equation*}f(\zeta)=\log\babs{K_{m,n}(z_0,\zeta)}^2-m\widehat{Q}_\tau(\zeta).
Mathrm end{equation*}
Since $\widehat{Q}_\tau$ is harmonic on ${Mathbb C}{Mathbb S}etminus{Mathbb S}etS_\tau$, $f$ is
subharmonic there. Moreover, since $n-1\le m\tau$ and since
$K_{m,n}(\cdot,z_0)\in {H}_{m,n}$, we have a simple estimate
\begin{equation*}\log\babs{K_{m,n}(z_0,\zeta)}^2\le (n-1)\log\babs{\zeta}^2+{Mathcal O}(1)\le m\tau\log\babs{\zeta}^2+
{Mathcal O}(1) \qquad \text{when}\quad \zeta\to\infty,Mathrm end{equation*}
while the relation Mathrm eqref{qtau} says that $\widehat{Q}_\tau(\zeta)=
\tau\log\babs{\zeta}^2+{Mathcal O}(1)$ when $\zeta\to \infty$. Hence $f$
is bounded above. Furthermore, it is clear that $f$ is harmonic in a
punctured neighbourhood of $\infty$, which yields that $f$ has a
representation $f(\zeta)=h(\zeta)-c\log\babs{\zeta}$ for all large
enough $|\zeta|$, where $c$ is a non-negative number and $h$ is
harmonic at $\infty$ (see e.g. [\cite{ST}, Cor. 0.3.7, p. 12]). In
particular, $f$ extends to a subharmonic function on ${Mathbb C}^*{Mathbb S}etminus
{Mathbb S}etS_\tau$. Finally, since Lemma \operatorname{Re}f{propn1} yields that
\begin{equation*}f(\zeta)\le \log(Cm^2)+mQ(z_0)-Mathrm epsilon{Mathbb S}qrt{m}d,\qquad
\text{when}\quad \zeta\in{\partial}{Mathbb S}etS_\tau,Mathrm end{equation*} the maximum
principle shows that the same estimate holds for all $\zeta \in
{Mathbb C}^*{Mathbb S}etminus {Mathbb S}etS_\tau$. This means that
\begin{equation}\babs{K_{m,n}(z_0,z)}^2Mathrm e^{-m\widehat{Q}_\tau(z)}\le Cm^2
Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}d}Mathrm e^{mQ(z_0)},\qquad \text{when}\quad z\nuot\in {Mathbb S}etS_\tau,
Mathrm end{equation}
which implies Mathrm eqref{brest} when $z\nuot\in {Mathbb S}etS_\tau$.
Mathrm end{proof}
\nuoindent\bf Background. \rm Estimates related to those considered
in Lemma \operatorname{Re}f{propn1} are known, see e.g. Lindholm's article
\cite{L}, Prop. 9, pp. 404--407. We will follow the basic strategy
used in that paper in our proof below. Since we are considering a
different situation with polynomial Bergman kernels (instead of the
full Bergman kernel of $A^2_{mQ}$), and since we are not assuming
the weight $Q$ to be globally strictly subharmonic, nontrivial
modifications of the classical arguments are needed. Our main tool
for accomplishing this is provided by the weighted $L^2$ estimates
in Th. \operatorname{Re}f{boh}.
{Mathbb S}ubsection{The proof of Lemma \operatorname{Re}f{propn1}} We can and will
assume that $Q\ge 1$ on ${Mathbb C}$. Moreover, we will denote various
constants by the same letter $C$, which can change meaning during
the course of the calculations. When $C$ depends on one or several
parameters, we will always specify this. To simplify the proof we
will first reduce the problem by treating three simple cases.
{Mathbb S}ubsection{Case 1: $md^2\le 1$.} Since we have assumed that $n\le m\tau+1$,
the estimate Mathrm eqref{berg2} of Prop. \operatorname{Re}f{lemma1} applies. It gives
that
\begin{equation}\label{spuc}\babs{K_{m,n}(z_0,z)}^2Mathrm e^{-m(Q(z_0)+Q(z))}\le
Cm^2,\quad z\in{Mathbb C},Mathrm end{equation} with a number $C$ depending only on
$\tau$. The assertion Mathrm eqref{bupp} follows immediately in case
$d=0$. In the remaining case we have $d>0$ and $m\le d^{-2}$. Then,
for any $Mathrm epsilon>0$ and any $z\in{Mathbb C}$, we have that
$Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{d,\babs{z_0-z}\}}\ge Mathrm e^{-Mathrm epsilon
d^{-1}d}=Mathrm e^{-Mathrm epsilon},$ so that
\begin{equation}\label{buss}\babs{K_{m,n}(z_0,z)}^2Mathrm e^{-m(Q(z_0)+Q(z))}\le
CMathrm e^Mathrm epsilonMathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{d,\babs{z_0-z}\}}.Mathrm end{equation}
This gives the desired estimate Mathrm eqref{bupp} with $C$ replaced by
$CMathrm e^Mathrm epsilon$.
{Mathbb S}ubsection{Notation}\label{nota}
We now fix positive numbers $A^p(\D)ar$ and ${Mathbf p}ar$ such that the
relation Mathrm eqref{sss} is satisfied. We further choose $A^p(\D)ar$ and
${Mathbf p}ar$ so that $](m-A^p(\D)ar)\tau+{Mathbf p}ar[\le m\tau-M$ for all $m\ge 1$,
and let $m_0=Max\{(1+A^p(\D)ar)/\tau,4A^p(\D)ar\}$. Let $n$ be a positive
integer such that $n\le m\tau+1$. We also fix a point
$z\in{Mathbb S}etS_\tau$ and let $R$ be a number such that ${Mathbb S}etS_\tau{Mathbb S}ubset {Mathbb D}(0;R)$.
{Mathbb S}ubsection{Case 2: $m\le m_0$.}
Given any $Mathrm epsilon>0$, we have that
$Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{\babs{z_0-z},d\}}\ge
Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m_0}R},$ and so Mathrm eqref{spuc} implies that
\begin{equation*}\babs{K_{m,n}(z_0,z)}^2Mathrm e^{-m(Q(z_0)+Q(z))}\le
CMathrm e^{Mathrm epsilon{Mathbb S}qrt{m_0}R}Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{\babs{z_0-z},d\}}.Mathrm end{equation*}
Thus Mathrm eqref{bupp} holds with $C$ replaced by
$CMathrm e^{Mathrm epsilon{Mathbb S}qrt{m_0}R}$.
{Mathbb S}ubsection{Case 3: $\babs{z-z_0}\le 8/{Mathbb S}qrt{m}$.} In this case,
$Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{d,\babs{z_0-z}\}}\ge Mathrm e^{-8Mathrm epsilon}$, and thus
Mathrm eqref{spuc} implies
\begin{equation*}\babs{K_{m,n}(z_0,z)}^2Mathrm e^{-m(Q(z_0)+Q(z))}\le
CMathrm e^{8Mathrm epsilon}m^2Mathrm e^{-{Mathbb S}qrt{m}Min\{d,\babs{z_0-z}\}}.
Mathrm end{equation*}
We have shown Mathrm eqref{bupp} in the case when $\babs{z_0-z}\le
8/{Mathbb S}qrt{m}$ with $C$ replaced by $CMathrm e^{8Mathrm epsilon}$.
{Mathbb S}ubsection{Case 4: $m\ge m_0$, $md^2\ge 1$ and $\babs{z-z_0}\ge 8/{Mathbb S}qrt{m}$.}
In the sequel we fix any integer $n$ with $n\ge
](m-A^p(\D)ar)\tau+{Mathbf p}ar[$. Here $](m-A^p(\D)ar)\tau+{Mathbf p}ar[>0$ for all $m\ge
m_0$ by our choice of $m_0$ (see Subsect. \operatorname{Re}f{nota}).
It is important to note that the assumption $md^2\ge 1$ means that
$1/{Mathbb S}qrt{m}\le d$ so that
\begin{equation*}8/{Mathbb S}qrt{m}\le
8d=\operatorname{dist\,}(z_0,{Mathbb C}{Mathbb S}etminus({Mathbb S}etS_\tau\cap{Mathbb S}etX)).Mathrm end{equation*} Our
starting point is Lemma \operatorname{Re}f{lemm3}, which yields that
\begin{equation}\label{sim}\babs{K_{m,n}(z,z_0)}^2Mathrm e^{-mQ(z)}\le
Cm\int_{{Mathbb D}(z;1/{Mathbb S}qrt{m})}\babs{K_{m,n}(\zeta,z_0)}^2Mathrm e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta),
\quad z\in {Mathbb S}etS_\tau,
Mathrm end{equation}
where the number $C$ only depends on $A$. Define
\begin{equation*}Mathrm eps_0(\zeta)=Min\{\babs{z_0-\zeta}/2,4d\},\quad \zeta\in {Mathbb C}.Mathrm end{equation*}
Note that
\begin{equation}\label{siz}4/{Mathbb S}qrt{m}\le Mathrm eps_0(z)\le 4d.Mathrm end{equation}
Let $\chi_0$ be a smooth non-negative function such that
\begin{equation}\chi_0=0\quad \text{on}\quad {Mathbb D}(z_0;Mathrm eps_0(z)/2)\quad
\text{and}\quad \chi_0=1\quad\text{outside}\quad
{Mathbb D}(z_0;Mathrm eps_0(z)),Mathrm end{equation} and also $\babs{{\overline{\partial}}\chi_0}^2\le
(C/Mathrm eps_0(z)^2)\chi_0$ with $C$ an absolute constant ($C=5$ will
do). In view of Mathrm eqref{siz}, it yields that
\begin{equation*}\babs{{\overline{\partial}}\chi_0}^2\le
Cm\chi_0\quad \text{on}\quad {Mathbb C},Mathrm end{equation*} where $C=5/16$.
Notice that ${\overline{\partial}}\chi_0$ is supported in the annulus
\begin{equation}\label{ann}U_0=U_0(z_0,z)=\{\zeta;\, Mathrm eps_0(z)/2\le \babs{z_0-\zeta}\le
Mathrm eps_0(z)\}.Mathrm end{equation}
Since $\chi_0$ is non-negative on ${Mathbb C}$ and since $\chi_0=1$ on
${Mathbb D}(z;1/{Mathbb S}qrt{m})$, the estimate Mathrm eqref{sim} implies that
\begin{equation}\label{gops}\babs{K_{m,n}(z,z_0)}^2Mathrm e^{-mQ(z)}\le
Cm\int_{Mathbb C}\chi_0(\zeta)\babs{K_{m,n}(\zeta,z_0)}^2Mathrm e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta),
Mathrm end{equation}
where $C$ only depends on $A$. Let $H_{\chi_0,m,n}$ be the linear
space ${H}_{m,n}$ with inner product
\begin{equation*}\langle f,g\rangle_{\chi_0,mQ}=\int_{Mathbb C} f\bar g\chi_0Mathrm e^{-mQ}
{{Mathrm d} A}.Mathrm end{equation*} We rewrite integral in the the right hand side
in Mathrm eqref{gops} in the following way
\begin{multline}
\int_{Mathbb C}
\chi_0(\zeta)\babs{K_{m,n}(\zeta,z_0)}^2Mathrm e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta)
={Mathbb S}up\biggl\{\babs{\langle
u,K_{m,n}(\cdot,z_0)\rangle_{\chi_0,mQ}}^2;\,
u\in{H}_{m,n},\, \int_{Mathbb C} \babs{u}^2\chi_0Mathrm e^{-mQ}{{Mathrm d} A}\le 1\biggr\}=\\
={Mathbb S}up\biggl\{\babs{\bigl\langle\chi_0 u,
K_{m,n}(\cdot,z_0)\bigr\rangle_{mQ}}^2;\,u\in{H}_{m,n},
\,\int_{Mathbb C}\babs{u}^2\chi_0 Mathrm e^{-mQ}{{Mathrm d} A}\le 1\biggr\}=\\
={Mathbb S}up\biggl\{\big|P_{m,n}[\chi_0
u](z_0)\big|^2;\,u\in{H}_{m,n},\, \int_{Mathbb C}\babs{u}^2\chi_0
Mathrm e^{-mQ}{{Mathrm d} A}\le 1 \biggr\}, \label{basic-1}
Mathrm end{multline}
where $P_{m,n}$ is the orthogonal projection of $L^2_{mQ}$ onto
${H}_{m,n}$. Now fix $u\in{H}_{m,n}$ with
$\int_{Mathbb C}\babs{u}^2\chi_0Mathrm e^{-mQ}{{Mathrm d} A}\le 1$ and recall that
$P_{m,n}[\chi_0 u]=\chi_0u-u_*,$ where $u_*$ is the
$L^2_{mQ,n}$-minimal solution to ${\overline{\partial}} u_*={\overline{\partial}}(\chi_0
u)=u{\overline{\partial}}\chi_0$. In particular, $u_*$ is holomorphic in ${Mathbb D}(z_0;Mathrm eps_0(z)/2)$ and
$u_*=-P_{m,n}[\chi_0 u]$ there. See Subsect. \operatorname{Re}f{init}.
We intend to apply Th. \operatorname{Re}f{boh} with a suitable real-valued
function ${Mathbb T}fun_m$. We shall at first only specify ${Mathbb T}fun_m$ by
requiring certain properties of it. An explicit construction of
${Mathbb T}fun_m$ is then given at the end of the proof.
Medskip
\nuoindent \bf Condition 1. \rm We require that
\begin{equation}\label{rc1}{Mathbb T}fun_m=0\quad \text{on}\quad
{Mathbb D}(z_0;1/(2{Mathbb S}qrt{m})),Mathrm end{equation}
Medskip
\nuoindent \bf Condition 2. \rm There exists a number $Mathrm epsilon>0$
depending only on $a$, $A^p(\D)ar$ and $\tau$, such that
\begin{equation*}{Mathbb T}fun_m(\zeta)\le
-Mathrm epsilon{Mathbb S}qrt{m}Mathrm eps_0(z)/4\quad \text{when}\quad \babs{\zeta-z_0}\ge
Mathrm eps_0(z)/2,Mathrm end{equation*}
Medskip
\nuoindent \bf Condition 3. \rm The various conditions on ${Mathbb T}fun_m$
in Th. \operatorname{Re}f{boh} are satisfied.
More precisely,
(i) ${Mathbb T}fun_m$ is ${Mathcal C}^{1,1}$-smooth and
\begin{equation*}(m-A^p(\D)ar){Mathbb D}elta Q(\zeta)+{Mathbb D}elta {Mathbb T}fun_m(\zeta)\ge
ma/2,\quad \text{for a.e.}\quad \zeta\in
\overline{{Mathbb D}}(z_0;6d),Mathrm end{equation*}
(ii) ${Mathbb T}fun_m$ is constant in ${Mathbb C}{Mathbb S}etminus \overline{{Mathbb D}}(z_0;6d)$,
(iii) We have that
\begin{equation*}\frac {|{\overline{\partial}} {Mathbb T}fun_m|^2} {ma/2}\le \frac 1
{4Mathrm e^{A^p(\D)ar q_\tau}}\quad \text{on}\quad {Mathbb C},Mathrm end{equation*}
where $q_\tau={Mathbb S}up_{{Mathbb S}etS_\tau}\{Q(\zeta)\}$.
It is clear that (i), (ii) and (iii) imply the conditions on
${Mathbb T}fun_m$ in Th. \operatorname{Re}f{boh}, with $\kappa=1/2$.
We now turn to consequences of the conditions on ${Mathbb T}fun_m$, and
start with condition 1. Applying Lemma \operatorname{Re}f{lemm2}, we find that,
for any real function ${Mathbb T}fun_m$ satisfying Mathrm eqref{rc1}, we have
\begin{equation}\label{gops4}\babs{u_*(z_0)}^2Mathrm e^{-mQ(z_0)}\le
Cm\int_{{Mathbb D}(z_0;1/(2{Mathbb S}qrt{m}))}\babs{u_*(\zeta)}^2Mathrm e^{-mQ(\zeta)}{{Mathrm d} A}(\zeta)\le
Cm\int_{Mathbb C}
\babs{u_*(\zeta)}^2Mathrm e^{{Mathbb T}fun_m(\zeta)-mQ(\zeta)}{{Mathrm d} A}(\zeta),Mathrm end{equation}
where $C$ depends only on $A$.
By Condition 3, we may apply Th. \operatorname{Re}f{boh} to the integral in the
right hand side in Mathrm eqref{gops4}. It yields that
\begin{equation}\label{gops5}\int_{Mathbb C}\babs{u_*}^2
Mathrm e^{{Mathbb T}fun_m-mQ}{{Mathrm d} A}\le C\int_{U_0}\babs{u{\overline{\partial}}\chi_0}^2 \frac
{Mathrm e^{{Mathbb T}fun_m-mQ}}{ma/2+{Mathbf p}ar c_\tau} {{Mathrm d} A},
Mathrm end{equation}
with a number $C$ depending only on $\tau$ and $A^p(\D)ar$ (note that
$(1-\kappa)^{-2}=4$). Here
$c_\tau=\inf_{{Mathbb S}etS_\tau}\{(1+\babs{\zeta}^2)^{-2}\}
\ge (1+R^2)^{-2}$ and $U_0$ is the annulus defined
in Mathrm eqref{ann}.
But since $\babs{{\overline{\partial}}\chi_0}^2\le Cm\chi_0$, Mathrm eqref{gops5} implies
\begin{equation*}\int_{Mathbb C}\babs{u_*}^2
Mathrm e^{{Mathbb T}fun_m-mQ}{{Mathrm d} A}\le \frac {Cm}{am+{Mathbf p}ar
c_\tau}\int_{U_0}\babs{u}^2\chi_0Mathrm e^{{Mathbb T}fun_m-mQ}{{Mathrm d} A}.Mathrm end{equation*}
Here $C$ only depends on $A^p(\D)ar$ and $\tau$. We now use Condition 2,
which implies that ${Mathbb T}fun_m(\zeta)\le-Mathrm epsilon{Mathbb S}qrt{m}Mathrm eps_0(z)$
whenever $\zeta\in U_0$. It yields that we may continue to estimate
\begin{equation}\label{gops7}
\int_{U_0}\babs{u}^2\chi_0Mathrm e^{{Mathbb T}fun_m-mQ}{{Mathrm d} A}\le
Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Mathrm eps_0(z)/4}\int_{Mathbb C}
\babs{u}^2\chi_0Mathrm e^{-mQ}{{Mathrm d} A}\le
Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{\babs{z_0-z}/8,d\}},Mathrm end{equation} where
we have used that $\int_{Mathbb C} \babs{u}^2\chi_0Mathrm e^{-mQ}{{Mathrm d} A}\le 1$ in the
last step.
Tracing back through Mathrm eqref{gops}--Mathrm eqref{gops7}, we infer that
\begin{equation*}\babs{K_{m,n}(z_0,z)}^2Mathrm e^{-m(Q(z_0)+Q(z))}\le
C\frac {m^3} {am+{Mathbf p}ar
c_\tau}Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{\babs{z_0-z}/8,d\}}\le
Ca^{-1}m^2Mathrm e^{-Mathrm epsilon{Mathbb S}qrt{m}Min\{\babs{z_0-z}/8,d\}},Mathrm end{equation*}
where $C$ depends on $A^p(\D)ar$, $\tau$, and $A$. This proves Lemma
\operatorname{Re}f{propn1} (with $Mathrm epsilon/8$ instead of $Mathrm epsilon$ and $Ca^{-1}$
in place of $C$) under the hypotheses that a function ${Mathbb T}fun_m$
satisfying conditions 1,2, and 3 above exists. To finish the proof
we must verify the existence of such a ${Mathbb T}fun_m$.
{Mathbb S}ubsection{Construction of ${Mathbb T}fun_m$.} We now construct a
function ${Mathbb T}fun_m$ and a positive number $Mathrm epsilon$ which satisfy the conditions 1, 2, and 3 above.
We look for a radial function of the form
\begin{equation*}{Mathbb T}fun_m(\zeta)=-Mathrm epsilon{Mathbb S}qrt{m}S_m(\babs{\zeta-z_0}),Mathrm end{equation*}
where the number $Mathrm epsilon>0$ will be fixed later. We start by
giving an explicit construction of $S_m$, the proof of conditions 1
though 3 will then be accomplished without difficulty.
We recall that $1/{Mathbb S}qrt{m}\le d$ and start by specifying the
derivative $S_m^\prime$ to be the piecewise linear continuous
function on $[0,\infty)$ such that
\begin{equation*}S_m^\prime=0\quad \text{on} \quad [0,1/(2{Mathbb S}qrt{m})]\cup
[6d,\infty)\quad \text{and}\quad S_m^\prime=1\quad \text{on}\quad
[1/{Mathbb S}qrt{m},5d],Mathrm end{equation*} and $S_m^\prime$ is affine on each
of the intervals $[1/(2{Mathbb S}qrt{m}),1/{Mathbb S}qrt{m}]$ and $[5d,6d]$. The
distributional derivative of $S_m^\prime$ is then a linear
combination of characteristic functions,
\begin{equation*}S_m^{\prime\prime}=2{Mathbb S}qrt{m}{Mathbf 1}_{[1/(2{Mathbb S}qrt{m}),
1/{Mathbb S}qrt{m}]}-(1/d){Mathbf 1}_{[5d,6d]},Mathrm end{equation*}
so that (since $md^2\ge 1$)
\begin{equation*}\babs{S_m^{\prime\prime}}\le
Max\{2{Mathbb S}qrt{m},1/d\}=2{Mathbb S}qrt{m}.Mathrm end{equation*} We now define $S_m$
by requiring that $S_m(0)=0$. Since $S_m^\prime=0$ on
$[0,1/(2{Mathbb S}qrt{m})]$ it is then clear that $S_m=0$ on $[0,1/(2{Mathbb S}qrt{m})]$.
Moreover, when $2/{Mathbb S}qrt{m}\le t\le 5d$, we get
\begin{equation*}S_m(t)=\int_0^t S_m^\prime(s){Mathrm d} s\ge
t-1/{Mathbb S}qrt{m}\ge t/2,\quad t\in [2/{Mathbb S}qrt{m},5d],Mathrm end{equation*} since
$S_m^\prime=1$ on $[1/{Mathbb S}qrt{m},5d]$. When $t\ge 5d$, we plainly have
\begin{equation*}S_m(t)\ge 5d-1/{Mathbb S}qrt{m}\ge 4d.Mathrm end{equation*}
We conclude that
\begin{equation}\label{auto}S_m(t)\ge Min\{t/2,4d\},\quad t\ge
2/{Mathbb S}qrt{m}.Mathrm end{equation} In particular, denoting by $c_m$ the
constant value that $S_m$ assumes on $[6d,\infty)$, we have $c_m\ge
4d$. This finishes the construction of $S_m$, and the corresponding
function ${Mathbb T}fun_m$ is clearly of class ${Mathcal C}^{1,1}({Mathbb C})$.
We now verify the conditions 1 through 3 above. First, Condition 1
is clear, since $S_m(t)=0$ when $t\le 1/(2{Mathbb S}qrt{m})$. Also, part (ii)
of condition 3 is clear; since $S_m(t)=c_m$ is constant when $t\ge
6d$, we have that ${Mathbb T}fun_m(\zeta)=-Mathrm epsilon{Mathbb S}qrt{m}c_m$ is constant
when $\zeta\nuot\in \overline{{Mathbb D}}(0;6d)$.
Since $Mathrm eps_0(z)\ge 4/{Mathbb S}qrt{m}$, Mathrm eqref{auto} implies that
$\varrho_m(\zeta)=-Mathrm epsilon S_m(\babs{\zeta-z_0})\le-Mathrm epsilonMathrm eps_0(\zeta)$ when
$\babs{\zeta-z_0}\geMathrm eps_0(z)/2$. Since moreover $Mathrm eps_0(\zeta)\ge Mathrm eps_0(z)/4$ in this case, we get that condition 2 is satisfied.
There remains to check parts (i) and (iii) of condition 3, and to make precise what we mean by "$Mathrm epsilon$". To this
end, we need the following estimates
\begin{equation}\label{827}\babs{{\overline{\partial}}
{Mathbb T}fun_m(\zeta)}^2=Mathrm epsilon^2m\babs{S_m^\prime(\babs{\zeta-z_0})}^2/4\le
Mathrm epsilon^2 m/4,\qquad \zeta\in {Mathbb C},Mathrm end{equation} and
\begin{equation}\label{828}\babs{{Mathbb D}elta {Mathbb T}fun_m(\zeta)}\le \frac Mathrm epsilon 4{Mathbb S}qrt{m}L^p(\D,dA)ar
\babs{S_m^{\prime\prime}L^p(\D,dA)ar \babs{\zeta-z_0}\right )}+\frac
{S_m^\primeL^p(\D,dA)ar\babs{z_0-\zeta}\right )}
{\babs{z_0-\zeta}}\right )\le Mathrm epsilon m,\qquad \zeta\in
{Mathbb C},Mathrm end{equation} which follows immediately from the properties of
$S_m$ (since $\babs{z_0-\zeta}\ge 1/(2{Mathbb S}qrt{m})$ when
$S_m^\prime(\babs{z_0-\zeta})\nue 0$).
To verify (i), we use Mathrm eqref{828}. It yields that it suffices to
choose $Mathrm epsilon>0$ such that $(m-A^p(\D)ar)a-Mathrm epsilon m\ge ma/2$ for
$m\ge m_0$, i.e. $Mathrm epsilon\le (1/2-A^p(\D)ar/m)a.$ Since we have assumed
that $m_0\ge 4A^p(\D)ar$, it thus suffices to choose $Mathrm epsilon=a/4$.
We finally verify (iii). By Mathrm eqref{827}, it suffices to choose an
$Mathrm epsilon>0$ such that $(Mathrm epsilon^2/4)/(a/2)\le 1/(4Mathrm e^{A^p(\D)ar
q_\tau})$, i.e. $Mathrm epsilon^2 \le a/(2Mathrm e^{A^p(\D)ar q_\tau})$.
We have verified the existence of $Mathrm epsilon>0$, of the form
$Mathrm epsilon=Min\{{Mathbb S}qrt{a/(2Mathrm e^{A^p(\D)ar q_\tau})},a/2\}$. This shows
that the choice $Mathrm epsilon=cMin\{a,1\}$ works with a proportionality constant
$c$ which depends on $A^p(\D)ar$ and $q_\tau$. The proof is finished.
$\qed$
{Mathbb S}ubsection{The proof of Theorem \operatorname{Re}f{th1.5}} Let $K$ be a
compact subset of ${Mathbb S}etS_\tau^\circ\cap {Mathbb S}etX$, and pick $M\ge 0$.
By Th. \operatorname{Re}f{th3} we have that $K_{m,n}(z_0,z_0)Mathrm e^{-mQ(z_0)}=m{Mathbb D}elta
Q(z_0)+{Mathcal O}(1)$ as $m\to\infty$ and $n\ge m\tau-M$, where the
${Mathcal O}$ is uniform for $z_0\in K$. It yields that
\begin{equation}\label{abo}\berd_{m,n}^{\langle z_0\rangle}(z)=\frac
{\babs{K_{m,n}(z,z_0)}^2} {K_{m,n}(z_0,z_0)}Mathrm e^{-mQ(z)}=\frac
{\babs{K_{m,n}(z,z_0)}^2} {m{Mathbb D}elta
Q(z_0)+{Mathcal O}(1)}Mathrm e^{-m(Q(z)+Q(z_0))},\quad z\in{Mathbb C},Mathrm end{equation} as
$m\to \infty$ and $n\ge m\tau-M$. Since ${Mathbb D}elta Q$ is bounded below
by a positive number on $K$, the right hand side in Mathrm eqref{abo} can
be estimated by
\begin{equation*}Cm^{-1}\babs{K_{m,n}(z,z_0)}^2Mathrm e^{-m(Q(z)+Q(z_0))},\quad z\in {Mathbb C}Mathrm end{equation*}
where $C$ depends on the lower bound of ${Mathbb D}elta Q$ on $K$. The
assertion now follows from Th. \operatorname{Re}f{flock}.
$\qed$
{Mathbb S}ection{The Bargmann--Fock case and harmonic measure} \label{new}
{Mathbb S}ubsection{Preliminaries.} In this section we prove
Th. \operatorname{Re}f{th5}. We therefore put $Q(z)=\babs{z}^2$. Recall that in
this case ${Mathbb S}etS_\tau=\overline{{Mathbb D}}(0;{Mathbb S}qrt{\tau}),$ and (see
Mathrm eqref{bfock})
\begin{equation}
\label{bfock-2} {Mathrm d} B^{\langle {z_0}\rangle}_{m,n}(z)=
m\frac{|E_{n-1}(mz\bar{z}_0)|^2}{E_{n-1}(m\babs{{z_0}}^2)}Mathrm e^{-m|z|^2}{Mathrm d}
A(z)\qquad\text{where}\qquad E_k(z)={Mathbb S}um_{j=0}^k\frac {z^j} {j!}.
Mathrm end{equation}
{Mathbb S}ubsection{The action on polynomials.}
\begin{prop} Fix a complex number ${z_0}\nue 0$, a positive integer $d$ and let $n$ be an integer, $n\ge d+1$.
Then, for all analytic polynomials
$u$ of degree at most $d$, we have
\begin{equation}
\label{pv} \operatorname{p.v.}\int_{Mathbb C} u(z^{-1}){Mathrm d} B^{\langle {z_0}\rangle}_{m,n}
\to u(z_0^{-1}),\qquad\text{as}\quad m\to\infty,
Mathrm end{equation}
uniformly in $n$, $n\ge d+1$. \label{prop-pv}
Mathrm end{prop}
\begin{proof} It is sufficient to prove the statement for $u(z)=z^j$
with $j\le d$. The left hand side in Mathrm eqref{pv}
can then be written
\begin{equation*}\operatorname{p.v.} \int_{Mathbb C} z^{-j}{Mathrm d} B^{\langle {z_0}\rangle}_{m,n}
=\frac{m{b}_{m,n}^j({z_0})}{E_{n-1}(m\babs{{z_0}}^2)},Mathrm end{equation*}
where we have put
\begin{equation*}{b}_{m,n}^j({z_0})=\operatorname{p.v.} \int_{{Mathbb C}}
z^{-j}\bigg|{Mathbb S}um_{k=0}^{n-1}\frac {(m{z_0}\bar{z})^k}
{k!}\bigg|^2Mathrm e^{-m\babs{z}^2}{{Mathrm d} A}(z).
Mathrm end{equation*}
Expanding the square yields
\begin{equation*}b_{m,n}^j({z_0})={Mathbb S}um_{k,l=0}^{n-1}
\frac {m^{k+l}{z_0}^k\bar{z}_0^l}
{k!l!}\operatorname{p.v.}\int_{{Mathbb C}}\bar{z}^kz^{l-j}Mathrm e^{-m\babs{z}^2}{{Mathrm d} A}(z).
Mathrm end{equation*}
Clearly only those $k,l$ for which $k=l-j$ give a non-zero
contribution to the sum, and therefore,
\begin{equation*}b_{m,n}^j({z_0})={z_0}^{-j}{Mathbb S}um_{l=j}^{n-1}\frac
{m^{2l-j}\babs{{z_0}}^{2l}}
{(l-j)!l!}\int_{{Mathbb C}}\babs{z}^{2(l-j)}Mathrm e^{-m\babs{z}^2}{{Mathrm d} A}(z)=
\frac{1}{m{z_0}^{j}}{Mathbb S}um_{l=j}^{n-1}\frac {m^{l}\babs{{z_0}}^{2l}} {l!}.
Mathrm end{equation*}
It follows that
\begin{equation*}b_{m,n}^j({z_0})=\frac{1}{m{z_0}^{j}}{Mathbb S}um_{l=j}^{n-1}
\frac {(m\babs{{z_0}}^2)^l} {l!}=\frac{1}{m{z_0}^j}L^p(\D,dA)ar E_{n-1}(m\babs{{z_0}}^2)-
E_{j-1}(m\babs{{z_0}}^2)\right ),Mathrm end{equation*}
and so
\begin{equation*}\frac {mb_{m,n}^j({z_0})} {E_{n-1}(m\babs{{z_0}}^2)}=\frac1{{z_0}^j}L^p(\D,dA)ar
1-\frac{E_{j-1}(m\babs{{z_0}}^2)}{E_{n-1}(m\babs{{z_0}}^2)}\right ).
Mathrm end{equation*}
Finally, since $j\le d<n$,
$$\frac{E_{j-1}(m\babs{{z_0}}^2)}{E_{n-1}(m\babs{{z_0}}^2)}\le
\frac{E_{d-1}(m\babs{{z_0}}^2)}{E_{d}(m\babs{{z_0}}^2)}\to0\quad\text{as}\quad
m\to\infty.$$
Mathrm end{proof}
\begin{prop} \label{bulk}
Let $0<r<{Mathbb S}qrt{\tau}$, ${z_0}\in{Mathbb C}{Mathbb S}etminus\overline{{Mathbb D}}(0;{Mathbb S}qrt{\tau})$
and $u$ an analytic polynomial. Then
\begin{equation*}\operatorname{p.v.}\int_{{Mathbb D}(0;r)}u(z^{-1})
{Mathrm d} B^{\langle {z_0}\rangle}_{m,n}(z)\to0
\qquad\text{as}\quad
m\to\infty\quad \text{and}\quad n/m\to\tau.
Mathrm end{equation*}
Mathrm end{prop}
\begin{proof} Put, for $\nuu=0,1,2,\ldots$,
$$b_{m,n}^\nuu({z_0},r)=\operatorname{p.v.}\int_{{Mathbb D}(0;r)}z^{-\nuu}
{Mathrm d} B^{\langle {z_0}\rangle}_{m,n}.$$ A straightforward calculation
based on Mathrm eqref{bfock-2} leads to
\begin{equation}
\label{grr}
b_{m,n}^\nuu({z_0},r)=\frac 1
{z_0^\nuu E_{n-1}(m\babs{{z_0}}^2)}{Mathbb S}um_{j=\nuu}^{n-1}\frac {(m\babs{{z_0}}^2)^j}
{j!(j-\nuu)!}\int_0^{mr^2} s^{j-\nuu}Mathrm e^{-s}{Mathrm d} s.
Mathrm end{equation}
We suppose $n$ is greater than $\nuu$ by at least two units, so that we may
pick an integer $k$ with $\nuu<k<n$, and split the sum Mathrm eqref{grr} accordingly:
\begin{equation}
\label{grr-1}
b_{m,n}^\nuu({z_0},r)=\frac 1
{z_0^\nuu E_{n-1}(m\babs{{z_0}}^2)}{Mathcal B}(\D)igg\{{Mathbb S}um_{j=\nuu}^{k-1}\frac {(m\babs{{z_0}}^2)^j}
{j!(j-\nuu)!}\int_0^{mr^2} s^{j-\nuu}Mathrm e^{-s}{Mathrm d} s
+{Mathbb S}um_{j=k}^{n-1}\frac {(m\babs{{z_0}}^2)^j}
{j!(j-\nuu)!}\int_0^{mr^2} s^{j-\nuu}Mathrm e^{-s}{Mathrm d} s{Mathcal B}(\D)igg\}.
Mathrm end{equation}
We estimate the first term trivially as follows:
\begin{equation}
\label{grr-2} {Mathbb S}um_{j=\nuu}^{k-1}\frac {(m\babs{{z_0}}^2)^j}
{j!(j-\nuu)!}\int_0^{mr^2} s^{j-\nuu}Mathrm e^{-s}{Mathrm d} s\le
{Mathbb S}um_{j=\nuu}^{k-1}\frac {(m\babs{{z_0}}^2)^j}
{j!(j-\nuu)!}\int_0^{\infty} s^{j-\nuu}Mathrm e^{-s}{Mathrm d} s={Mathbb S}um_{j=0}^{k-1}
\frac {(m\babs{{z_0}}^2)^j}{j!}=E_{k-1}(m|{z_0}|^2).
Mathrm end{equation}
As for the second term, we use the fact that the function $sMapsto
s^{j-\nuu}Mathrm e^{-s}$ is increasing on the interval $[0,j-\nuu]$, to say
that \begin{equation*}\int_0^{mr^2} s^{j-\nuu}Mathrm e^{-s}{Mathrm d}
s\le(mr^2)^{j-\nuu+1}Mathrm e^{-mr^2},Mathrm end{equation*} provided that $j\ge
mr^2+\nuu$. It follows that if $k\ge mr^2+\nuu$, then
\begin{equation*}{Mathbb S}um_{j=k}^{n-1}\frac {(m\babs{{z_0}}^2)^j}
{j!(j-\nuu)!}\int_0^{mr^2} s^{j-\nuu}Mathrm e^{-s}{Mathrm d} s\le
(mr^2)^{1-\nuu}Mathrm e^{-mr^2}{Mathbb S}um_{j=k}^{n-1}\frac
{(mr\babs{{z_0}})^{2j}} {j!(j-\nuu)!}.Mathrm end{equation*} By Stirling's
formula, $j!\ge {Mathbb S}qrt{2\pi}j^{j+1/2}Mathrm e^{-j},$ so that
\begin{equation*}\frac {(mr\babs{{z_0}})^{2j}}
{j!}Mathrm e^{-mr^2}\le\frac{1}{{Mathbb S}qrt{2\pi j}}m^{j}|{z_0}|^{2j}
L^p(\D,dA)ar\frac{mr^2}{j}Mathrm e^{1-\frac{mr^2}{j}}\right )^j.Mathrm end{equation*}
Since the function $xMapsto xMathrm e^{1-x}$ is increasing on the
interval $[0,1]$, it yields that \begin{equation*}\frac
{(mr\babs{{z_0}})^{2j}} {j!}Mathrm e^{-mr^2}\le\frac{1}{{Mathbb S}qrt{2\pi j}}
m^{j}|{z_0}|^{2j} L^p(\D,dA)ar\frac{mr^2}{k}Mathrm e^{1-\frac{mr^2}{k}}\right )^{j},
\qquad mr^2+\nuu\le k\le j.Mathrm end{equation*} We write
\begin{equation}
\label{eq-ckm} c_{k,m}=\frac{mr^2}{k}Mathrm e^{1-\frac{mr^2}{k}}\le1,
\qquad mr^2+\nuu\le k,
Mathrm end{equation}
and conclude that
\begin{multline}
{Mathbb S}um_{j=k}^{n-1}\frac {(m\babs{{z_0}}^2)^j}
{j!(j-\nuu)!}\int_0^{mr^2} s^{j-\nuu}Mathrm e^{-s}{Mathrm d} s\le
(mr^2)^{1-\nuu}{Mathbb S}um_{j=k}^{n-1}
\frac{(m|{z_0}|^2c_{k,m})^j}{(j-\nuu)!{Mathbb S}qrt{2\pi j}}\\
\le \frac{(mr^2)^{1-\nuu}}{{Mathbb S}qrt{2\pi}}
(c_{k,m})^k{Mathbb S}um_{j=k}^{n-1}\frac{(m|{z_0}|^2)^j}{(j-\nuu)!}\le
\frac{mr^2}{{Mathbb S}qrt{2\pi}}{Mathcal B}(\D)ig(\frac{|{z_0}|}{r}{Mathcal B}(\D)ig)^{2\nuu}
(c_{k,m})^kE_{n-\nuu-1}(m|{z_0}|^2). \label{eq-101}
Mathrm end{multline}
Now, a combination Mathrm eqref{grr-2} and Mathrm eqref{eq-101} applied to Mathrm eqref{grr-1}
yields
\begin{multline}
\label{grr-1'} \babs{{z_0^{\nuu}}b_{m,n}^\nuu({z_0},r)}\le\frac
{E_{k-1}(m\babs{{z_0}}^2)}{E_{n-1}(m\babs{{z_0}}^2)}
+\frac{mr^2}{{Mathbb S}qrt{2\pi}}{Mathcal B}(\D)ig(\frac{|{z_0}|}{r}{Mathcal B}(\D)ig)^{2\nuu}(c_{k,m})^k
\frac{E_{n-\nuu-1}(m|{z_0}|^2)}{E_{n-1}(m|{z_0}|^2)}\\
\le\frac {E_{k-1}(m\babs{{z_0}}^2)}{E_{n-1}(m\babs{{z_0}}^2)}
+\frac{mr^2}{{Mathbb S}qrt{2\pi}}{Mathcal B}(\D)ig(\frac{|{z_0}|}{r}{Mathcal B}(\D)ig)^{2\nuu}(c_{k,m})^k.
Mathrm end{multline}
We would like to show that each of the terms on the right hand side of
Mathrm eqref{grr-1'} can be made small by choosing $k$ cleverly.
As for the first term, we appeal to a theorem of Szeg\"o,
\cite{Sz}, Hilfssatz 1, p. 54, which states that
\begin{equation*}E_l(lx)=\frac 1 {{Mathbb S}qrt{2\pi l}}
(Mathrm e x)^l \frac x{x-1}L^p(\D,dA)ar 1+Mathrm eps_l(x)\right )\qquad x>1,
Mathrm end{equation*}
where $Mathrm eps_l(x)\to 0$ uniformly on compact subintervals of
$(1,\infty)$ as $l\to\infty$. It follows that
\begin{equation*}
\frac{E_{k-1}(m\babs{{z_0}}^2)}{E_{n-1}(m\babs{{z_0}}^2)}=
{Mathbb S}qrt{\frac{n-1}{k-1}}\frac{m|{z_0}|^2-n+1}{m|{z_0}|^2-k+1}
{Mathcal B}(\D)ig(\frac{Mathrm e m|{z_0}|^2}{k-1}{Mathcal B}(\D)ig)^{k-1}{Mathcal B}(\D)ig(\frac{Mathrm e
m|{z_0}|^2}{n-1}{Mathcal B}(\D)ig)^{1-n}
\frac{1+Mathrm eps_{k-1}(\frac{m|{z_0}|^2}{k-1})}{1+Mathrm eps_{n-1}(\frac{m|{z_0}|^2}{n-1})}.
Mathrm end{equation*}
Finally, we decide to pick $k$ such that
$$k/m\to\beta$$
as $k,m\to\infty$, where $r^2<\beta<\tau$. We observe that with this
choice of $k$, the above epsilons tend to zero as $k,m,n\to\infty$.
The function
$$yMapsto(Mathrm e/y)^{y},\qquad 0<y\le1$$
is is strictly increasing, so that with
$$y_1=\frac{k-1}{m|{z_0}|^2}A^p(\D)prox\frac{\beta}{|{z_0}|^2},
\quad y_2=\frac{n-1}{m|{z_0}|^2}A^p(\D)prox\frac{\tau}{|{z_0}|^2},$$
we have
$$\frac{(Mathrm e/y_1)^{y_1}}{(Mathrm e/y_2)^{y_2}}\le\thetaeta<1,$$
where at least for large $k,m,n$, the number $\thetaeta$ may be taken
to be independent of $k,m,n$. It follows that
$${Mathcal B}(\D)ig(\frac{Mathrm e m|{z_0}|^2}{k-1}{Mathcal B}(\D)ig)^{k-1}{Mathcal B}(\D)ig(\frac{Mathrm e m|{z_0}|^2}{n-1}{Mathcal B}(\D)ig)^{1-n}
\le \thetaeta^{-m|{z_0}|^2},$$
so that
\begin{equation*}
\frac{E_{k-1}(m\babs{{z_0}}^2)}{E_{n-1}(m\babs{{z_0}}^2)}\le(1+o(1))
{Mathbb S}qrt{\frac{\tau}{\beta}}\frac{|{z_0}|^2-\tau}{|{z_0}|^2-\beta}
\thetaeta^{-m|{z_0}|^2}\to0
Mathrm end{equation*}
exponentially quickly as $k,m,n\to\infty$.
Finally, as for the second term, we observe that the numbers $c_{k,m}$
defined by Mathrm eqref{eq-ckm} have the property that
$$c_{k,m}\to \frac{r^2}{\beta}Mathrm e^{1-\frac{r^2}{\beta}}<1,$$
as $k,m,n\to\infty$ in the given fashion. In particular, the second
term converges exponentially quickly to $0$. The proof is complete.
Mathrm end{proof}
\begin{cor} \label{korp} Let ${z_0}\in{Mathbb C}{Mathbb S}etminus\overline{{Mathbb D}}(0;{Mathbb S}qrt{\tau})$, and
let $\omega$ be an open set in ${Mathbb C}$ which contains the circle
${Mathbb T}(0,{Mathbb S}qrt{\tau})$. Further, let $u$ be an analytic polynomial.
Then
\begin{equation*}\int_{\omega}u(z^{-1}){Mathrm d} B^{\langle
{z_0}\rangle}_{m,n}(z)\to u(z_0^{-1})\qquad \text{as}\quad
m\to\infty\quad \text{and}\quad n/m\to\tau.Mathrm end{equation*}
Mathrm end{cor}
\begin{proof} This follows from propositions \operatorname{Re}f{bulk}
and \operatorname{Re}f{prop1}.
Mathrm end{proof}
{Mathbb S}ubsection{The proof of Theorem \operatorname{Re}f{th5}.}
Let ${{Mathcal H}}_\tau$ be the class of continuous functions ${Mathbb C}^*\to{Mathbb C}$
which are harmonic on ${Mathbb C}^*{Mathbb S}etminus{Mathbb S}etS_\tau$. For a function
$f\in{{Mathcal C}}_b({Mathbb C})$ we write $\widetilde{f}$ for the unique function of
class ${{Mathcal H}}_\tau$ which coincides with $f$ on
${Mathbb D}(0,{Mathbb S}qrt{\tau})$. We must show that \begin{equation*}\int_{Mathbb C}
f(z){Mathrm d} B^{\langle {z_0}\rangle}_{m,n}(z)\to \widetilde
f({z_0})Mathrm end{equation*} as $m\to\infty$ and $n/m\to\tau$. See e.g.
\cite{GM}, p. 90.
Convolving with the F\'ejer kernel, we see that $f$ may be uniformly
approximated by functions which on a neighborhood $\omega$ of
${Mathbb T}(0,{Mathbb S}qrt{\tau})$ are of the form $u(z^{-1})$, with $u$ a harmonic
polynomial. We may therefore w.l.o.g. suppose that $f$ itself is of
this form, i.e. $f(z)=u(z^{-1})$ when $z\in \omega$.
Thus $f(z)=\widetilde{f}(z)=u(z^{-1})$ on
$\omega$. By Prop. \operatorname{Re}f{prop1}, \begin{equation*}\int_{Mathbb C}
(f(z)-\widetilde{f}(z)) {Mathrm d} B^{\langle {z_0}\rangle}_{m,n}(z)\to
0,Mathrm end{equation*} as $m\to\infty$ and $n/m\to\tau$. Moreover, Cor.
\operatorname{Re}f{korp} gives that
\begin{equation*}\int_{Mathbb C} \widetilde{f}(z){Mathrm d} B^{\langle
{z_0}\rangle}_{m,n}(z)=\int_{Mathbb C} u(z^{-1}){Mathrm d} B^{\langle
{z_0}\rangle}_{m,n}(z)\to u(z_0^{-1})=\widetilde{f}({z_0}),Mathrm end{equation*}
as $m\to\infty$ and $n/m\to\tau$.
$\qed$
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Mathrm end{document} |
\begin{document}
\title{Classicalization by phase space measurements}
\author{Marduk Bola{\~n}os}
\address{Fakult{\"a}t f{\"u}r Physik, University of Duisburg-Essen,
Lotharstr. 1, 47048 Duisburg, Germany}
\begin{abstract}
This article provides an accessible illustration of the measurement
approach to the study of the quantum-classical transition suitable
for beginning graduate students. As an example, we apply it to a
quantum system with a general quadratic Hamiltonian and obtain the
exact solution of the dynamics for an arbitrary measurement
strength.
\end{abstract}
\pacs{03.65.-w, 03.65.Ta, 03.65.Yz}
\noindent{\it Keywords\/}: quantum-classical transition, measurement
master equation, phase space methods
\submitto{\EJP}
\section{Introduction}
\label{sec:introduction}
The study of the quantum-classical transition is an active field of
research that has provided important insights into the foundations of
quantum theory and plays a prominent role in the development of
quantum technologies \cite{Schlosshauer}.
Some of its achievements have been obtained through the measurement
approach, which consists in using the framework of generalized
measurements \cite{Busch_et_al} to model the dynamics of an open
quantum system, i.e. a quantum system in interaction with a large
number of quantum degrees of freedom, which are collectively called
environment. This has been used, for example, to develop a
quantitative assessment of the macroscopic character of a
superposition state based on the experimental observation of quantum
effects \cite{Nimmrichter}.
The concepts and methods used in this field are usually not familiar
to beginning graduate students. However, some of them have been
discussed in a pedagogical way in recent years
\cite{Case,LovettNazir,Pearle,XuLiMaLi}. The main contribution of this
article is to present an accessible illustration of the measurement
approach applied to an analytically tractable model for the
classicalization of a quantum system described by a general quadratic
Hamiltonian. This should be useful for instructors with an interest in
introducing graduate students to current research topics in quantum
mechanics.
The manuscript is organized as follows. In section 2, the
Heisenberg--Dirac formulation of quantum mechanics is briefly
discussed, with a focus on physical dimensions and algebraic
considerations. These two aspects will play an important role
throughout the article. We remark that the use of algebraic methods in
quantum mechanics has lead to a deeper understanding of nature, while
also offering an elegant and powerful framework to study a wide
variety of systems \cite{Woit,ThyssenCeulemans}. Section 3 contains a
brief account of the phase space formulation of quantum mechanics,
which allows one to establish a connection with classical
mechanics. In section 4 generalized measurements are defined and it is
shown how to construct a dynamical equation for the state of a system
subject to such measurement. Section 5 contains an example of
measurement-induced classicalization. In particular, we consider an
imprecise, simultaneous measurement of two canonically conjugate
observables of a quantum harmonic oscillator, which leads to the decay
of all the coherences in a superposition state.
\section{Heisenberg--Dirac quantum mechanics}
\label{sec:Heisenberg_Dirac_QM}
In analogy to classical Hamiltonian mechanics, the observable
quantities of an elementary quantum system are described by
operator-valued functions of two self-adjoint operators
$\mathsf{Q} = \mathsf{Q}^\dagger, \mathsf{P} = \mathsf{P}^\dagger$
satisfying the canonical commutation relation
\begin{equation}
\label{eq:CCR_x_p}
[\mathsf{Q}, \mathsf{P}] = i\hbar \mathsf{I}.
\end{equation}
The operators $\mathsf{Q}$, $\mathsf{P}$, and $\mathsf{I}$ (identity)
are the canonical basis of the Heisenberg Lie algebra
$\mathfrak{h}(3)$ \cite{KlimovChumakov}. It is customary to choose
units in which $\hbar = 1$, but this makes it difficult to verify that
an equation has the correct dimensions. For economy of notation it is
preferable to use the dimensionless basis of $\mathfrak{h}(3)$ given
by \cite{Serafini}
\begin{equation}
\label{eq:CCR_ladder_basis}
\mathsf{a} = \frac{1}{\sqrt{2}}\left(\frac{\mathsf{Q}}{\lambda} + \frac{i}{\hbar}\mathsf{P}\lambda\right),\quad \mathsf{a}^\dagger = \frac{1}{\sqrt{2}}\left(\frac{\mathsf{Q}}{\lambda} - \frac{i}{\hbar}\mathsf{P}\lambda\right),\quad \mathsf{I},
\end{equation}
where $\lambda$ has the same dimensions as $\mathsf{Q}$, $\mathsf{P}$ has dimensions
of $\hbar/\lambda$ and $\mathsf{a}, \mathsf{a}^\dagger$ satisfy the commutation relation
\begin{equation}
\label{eq:CCR_boson}
[\mathsf{a}, \mathsf{a}^\dagger] = \mathsf{I}.
\end{equation}
Correspondingly, the observable quantities of the system can be
expressed as operator-valued functions of $\mathsf{a}$ and
$\mathsf{a}^\dagger$. For example, the operator
$\mathsf{N} = \mathsf{a}^\dagger \mathsf{a}$ together with the
operators $\mathsf{a}$, $\mathsf{a}^\dagger$, $\mathsf{I}$ generates
the Lie algebra $\mathfrak{h}(4)$. From the commutators
\begin{equation}
\label{eq:ladder_operators}
[\mathsf{a}, \mathsf{N}] = \mathsf{a}, \quad [\mathsf{a}^\dagger, \mathsf{N}] = -\mathsf{a}^\dagger
\end{equation}
follows \cite{KlimovChumakov} that $\mathsf{a}$ and
$\mathsf{a}^\dagger$ are ladder operators with respect to the
eigenvectors
$|n\rangle = (\mathsf{a}^{\dagger})^n|0\rangle/\sqrt{n!}$ of
$\mathsf{N}$:
\begin{equation}
\label{eq:ladder_operators_N_eigenstates}
\mathsf{a} \,|n\rangle = \sqrt{n}\,|n-1\rangle, \quad \mathsf{a}^\dagger |n\rangle = \sqrt{n+1}\,|n+1\rangle,
\end{equation}
and we require that $\mathsf{a}\, |0\rangle = 0$ (because the energy of
a quantum system must be bounded from below). These eigenvectors form a
complete and orthonormal ($\langle n|m \rangle = \delta_{nm}$) set
that spans a Hilbert space $\mathcal{H}$, in which one can represent
the state of a system described by $\mathfrak{h}(4)$, e.g. the
quantum harmonic oscillator.
An eigenvector $|\alpha\rangle$ of the operator $\mathsf{a}$ is
called a coherent state. The corresponding eigenvalue, $\alpha$, is a
complex number, since this operator is not self-adjoint. In the basis
$|n\rangle$, a coherent state is expressed as \cite{KlimovChumakov}
\begin{equation}
\label{eq:coherent_state_Fock_basis}
|\alpha\rangle = e^{-\frac12 |\alpha|^2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle.
\end{equation}
These states play a fundamental role in the study of the
quantum-classical transition \cite{Zurek_et_al}. This becomes
apparent with the use of phase space methods, which will be discussed
in the following section.
\section{Quantum mechanics in phase space}
\label{sec:qm_in_phase_space}
Quantum mechanics can also be formulated in terms of unitary
operators, i.e. operators that satisfy $\mathsf{O}\mathsf{O}^\dagger = \mathsf{O}^\dagger \mathsf{O} = \mathsf{I}$. In this
framework, the operators associated with $\mathsf{Q}$ and $\mathsf{P}$ are
\cite{Tarasov}
\begin{equation}
\label{eq:Weyl_operators}
\mathsf{U}(a) = e^{\frac{i}{\hbar}a\mathsf{Q}}, \quad \mathsf{V}(b) = e^{\frac{i}{\hbar}b\mathsf{P}}.
\end{equation}
Since the argument of the exponential function must be dimensionless,
$a$ has the dimensions of $\mathsf{P}$ and $b$ has the dimensions of
$\mathsf{Q}$. The above operators are a special case of the Weyl
operators (or displacement operators)
\begin{equation}
\label{eq:displacement_operator}
\mathsf{D}(a,b) = e^{\frac{i}{\hbar}(a\,\mathsf{Q} \,+\, b\,\mathsf{P})},
\end{equation}
which satisfy the canonical commutation relations in Weyl (or integral) form \cite{Tarasov}
\begin{equation}
\label{eq:CCR_Weyl}
\mathsf{D}(a_1, b_1)\,\mathsf{D}(a_2, b_2) = e^{-\frac{i}{2\hbar}[a_1b_2 - a_2b_1]}\mathsf{D}(a_1 + a_2, b_1 + b_2).
\end{equation}
In terms of the operators $\mathsf{a}$ and $\mathsf{a}^\dagger$, the displacement
operator is given by \cite{KlimovChumakov}
\begin{equation}
\label{eq:Displacement_operator_a_a_dagger}
\mathsf{D}(\alpha) = e^{\,\alpha\, \mathsf{a}^\dagger -\, \alpha^\ast \mathsf{a}} = e^{-\frac12|\alpha|^2} e^{\alpha\, \mathsf{a}^\dagger} e^{-\alpha^\ast \mathsf{a}}.
\end{equation}
The coherent states can also be defined as displaced ground states
$|\alpha\rangle := \mathsf{D}(\alpha) |0\rangle$, as can be seen by
comparing this expression with (\ref{eq:coherent_state_Fock_basis}).
The operators acting on a representation space (such as the Hilbert
space spanned by the vectors $|n\rangle$) belong to a Hilbert space
called Liouville space, in which the scalar product is given by
$\langle \mathsf{A}|\mathsf{B} \rangle =
\mathrm{Tr}(\mathsf{A}^\dagger \mathsf{B})$. The set of displacement
operators $\mathsf{D}(\alpha)$ forms a delta-orthogonal basis of this
space:
$\langle \mathsf{D}(\alpha)|\mathsf{D}(\beta) \rangle = \pi
\delta^{(2)}(\alpha - \beta)$.
In this basis, the density operator (or state operator) $\rho$ of a
quantum system is described by the Weyl characteristic function (also
called ambiguity function) \cite{BarnettRadmore}
\begin{equation}
\label{eq:def_characteristic_function}
\chi(\eta, \eta^\ast) = \mathrm{Tr}[\rho\, \mathsf{D}(\eta)],
\end{equation}
which will figure prominently in section 5. The following Fourier
transform of $\chi(\eta, \eta^\ast)$ yields the Wigner function \cite{BarnettRadmore}:
\begin{equation}
\label{eq:Wigner_function}
W(\xi, \xi^\ast) = \frac{1}{\pi^2} \int \mathrm{d}^2\eta\, \chi(\eta, \eta^\ast) \,e^{\,\xi\eta^\ast -\, \xi^\ast\eta}, \quad \mathrm{d}^2\eta = \mathrm{d}\mathrm{Re}(\eta)\,\mathrm{d}\mathrm{Im}(\eta),
\end{equation}
which can also be defined as
$W(\xi, \xi^\ast) = \mathrm{Tr}[\rho\, \mathsf{\Pi}(\xi)]$, where
\begin{equation}
\label{eq:wigner_operator}
\mathsf{\Pi(\xi)} = \frac{2}{\pi}\exp{\bigl[ i\pi (\mathsf{a}^{\dagger} - \xi^\ast)(\mathsf{a} - \xi) \bigr]}
\end{equation}
is the Wigner operator (or displaced parity operator)
\cite{BishopVourdas}, which is self-adjoint and therefore is an
observable, unlike the displacement operator. This implies that the
Wigner function is real, whereas the characteristic function is
complex. The geometrical interpretation of these functions is
discussed in \cite{OzorioAlmeida}. For the experimental determination
of the Wigner function we refer the reader to \cite{HarocheRaimond}.
We remark that it is customary to denote the operators $\mathsf{D}$
and $\mathsf{\Pi}$ with a single complex argument even though $\chi$
and $W$ are functions of two complex variables. This mapping of
operators to functions, also known as a mapping from $q$-numbers to
$c$-numbers, is very useful in the study of the quantum-classical
transition, as will be shown in section 5. For a thorough discussion
of mappings of this kind we refer the reader to
\cite{KlauderSudarshan,AgarwalWolf}.
For a coherent state, $W$ is Gaussian, which can easily be seen using
(\ref{eq:Displacement_operator_a_a_dagger})\textendash(\ref{eq:Wigner_function})
together with the fact that a coherent state is an eigenvector of
$\mathsf{a}$. However, for a so-called cat state
\begin{equation}
\label{eq:cat_state}
|\psi\rangle = \mathcal{N}^{1/2}(|\alpha\rangle + |-\alpha\rangle), \quad \mathcal{N}^{-1} = 2(1 + e^{-2|\alpha|^2}),
\end{equation}
$W$ takes negative values:
\begin{equation}
\label{eq:Wigner_function_cat_state}
W(\xi, \xi^\ast) = \frac{2\, \mathcal{N}}{\pi} \Bigl[ e^{-2|\xi - \alpha|^2} +\, e^{-2|\xi + \alpha|^2} +\,
2\, e^{-2|\xi|^2} \cos{(4\, \mathrm{Im}(\alpha^\ast \xi))} \Bigr].
\end{equation}
This is a signature of a non-classical state
\cite{KimNoz}. Moreover, it can be proven that the only
positive-definite Wigner functions describing pure states are
Gaussian \cite{SotoClaverie}. This is one reason why coherent states
are considered the most classical quantum states.
It can be shown \cite{KimNoz} that for a Hamiltonian with a general
quadratic potential
$\mathsf{V}(\mathsf{q}) = a\,\mathsf{q}^2 + b\,\mathsf{q} + c$, the
evolution of the Wigner function is given by the classical Liouville
equation for a probability distribution in phase space:
\begin{equation}
\label{eq:Liouville_equation}
\frac{\partial}{\partial t} W(q,p,t) = \frac{\partial H}{\partial q} \frac{\partial W}{\partial p} - \frac{\partial H}{\partial p} \frac{\partial W}{\partial q}.
\end{equation}
However, this does not represent classical behavior unless the Wigner
function is positive. For this reason, $W$ is called a
quasi-probability phase space distribution. Moreover, as a consequence
of Heisenberg's uncertainty principle, the Wigner function cannot have
a width smaller than the size of a Planck cell:
$\Delta q \Delta p \geq \hbar$ \cite{KimNoz}. Therefore, quantum
mechanics introduces a coarse-graining in phase space.
One reason why quasi-probability distributions are useful is that they
enable calculating quantum-mechanical expectation values similarly to
averages in classical statistical mechanics. In particular,
distributions belonging to the Cohen class \cite{Cohen}, which
includes the Wigner function, have the property that integrating them
with respect to one canonical variable yields the probability
distribution of the canonically conjugate variable.
In the literature, it is common to describe the transition to
classical behavior as ``taking the limit $\hbar \rightarrow
0$''. However, this characterization is misleading, since $\hbar$ is a
constant of nature and it cannot be made arbitrarily small. What is
meant by this statement is that one may form a dimensionless parameter
involving $\hbar$ and other physical quantities, such that when this
parameter is made arbitrarily small, a quantum equation reduces to a
classical one (e.g. the Liouville equation above). This shows again
the importance of dimensional considerations in quantum mechanics. We
remark that this limiting procedure does not lead to the vanishing of
negative regions in the Wigner function.
\section{Quantum measurements and the measurement master equation}
\label{sec:phase_space_measurements}
\subsection{Measurement in quantum mechanics}
\label{sec:measurement_quantum_mechanics}
In the axiomatization of quantum mechanics carried out by von Neumann
\cite{vonNeumann}, given an observable with a discrete spectrum
$\mathsf{O} = \sum_n \lambda_n | \lambda_n \rangle \langle \lambda_n
|$, the probability that a measurement of $\mathsf{O}$ yields the
result $\lambda_n$ is
\begin{equation}
\label{eq:probability_von_Neumann_measurement}
\mathrm{Prob} (\lambda_n) = \mathrm{Tr} [\rho\, | \lambda_n \rangle
\langle \lambda_n |],
\end{equation}
and the state of the system after the measurement is
$\rho = | \lambda_n \rangle \langle \lambda_n |$. If the measurement
result is not known, the system is described by the mixed state
\begin{equation}
\label{eq:state_after_von_Neumann_measurement}
\rho = \sum_n \mathrm{Prob} (\lambda_n) | \lambda_n \rangle \langle
\lambda_n |.
\end{equation}
Instead of associating a projector with each measurement result $n$,
in general one may associate a positive operator $\pi_n$ with
it. These operators form a positive-operator-valued measure (POVM)
\cite{Busch_et_al} and must be such that $\sum_n \pi_n =
\mathsf{I}$. Moreover, each operator $\pi_n$ may be decomposed in
terms of pairs ($\mathsf{A}_k, \mathsf{A}_k^{\dagger}$) of operators:
\begin{equation}
\label{eq:Kraus_decomposition_effect}
\pi_n = \sum_k \mathsf{A}_{n k}^{\dagger} \mathsf{A}_{n k}.
\end{equation}
In this framework, the probability that a measurement yields the
result $n$ is $\mathrm{Prob} (n) = \mathrm{Tr} [\rho\, \pi_n]$ and the
state of the system after the measurement is
$\rho = (\mathrm{Tr} [\rho\, \pi_n])^{-1} \sum_k \mathsf{A}_{n k}\,
\rho\, \mathsf{A}_{n k}^{\dagger}$. If the measurement result is not
known, the state of a system after performing a generalized
measurement is given by
\begin{equation}
\label{eq:mixed_state_generalized_measurement}
\rho = \sum_n \mathrm{Prob} (n) \frac{\sum_k \mathsf{A}_{n k}\, \rho\, \mathsf{A}_{n
k}^{\dagger}}{\mathrm{Tr} [\rho\, \pi_n]} = \sum_{n,k} \mathsf{A}_{n k}\, \rho\,
\mathsf{A}_{n k}^{\dagger}.
\end{equation}
We remark that this formalism is quite general and when the
measurement result can take any real or complex value the sums are
replaced by corresponding integrals.
\subsection{Measurement master equation}
\label{sec:measurement_master_equation}
The dynamics of the state operator of an open quantum system is
described under certain approximations by the
Lindblad--Gorini--Kossakowski--Sudarshan master equation:
\begin{equation}
\label{eq:Lindblad_equation}
\frac{\partial \rho}{\partial t} = -\frac{i}{\hbar}[\mathsf{H},\rho] + \frac12\sum_{j=1}^d\kappa_j\Bigl( 2\,\mathsf{R}_j\rho \mathsf{R}_j^\dagger - \mathsf{R}_j^\dagger \mathsf{R}_j\rho - \rho\, \mathsf{R}_j^\dagger \mathsf{R}_j \Bigr),
\end{equation}
where $\mathsf{H}$ is a self-adjoint operator with dimension of
energy, $\mathsf{R}_j$ are arbitrary dimensionless operators and $\kappa_j$ are
non-negative real numbers with dimension of frequency. Here we are
only interested in using this equation and refer the reader to
\cite{BreuerPetruccione} for an in depth discussion of its derivation
and the physical considerations behind it.
The change in the state of a system subject to a generalized
measurement can be modeled as a Poisson process with rate $\gamma$
\cite{Hanson}, as follows. We assume that in a short time interval
$\Delta t$ the probability that a measurement occurs is
$\gamma \Delta t$. If a measurement occurs, then the state at the time
$t + \Delta t$ will be given by
(\ref{eq:mixed_state_generalized_measurement}). Otherwise, the state
at this time results from the unitary evolution of the system, given by
the first term in the right-hand side of
(\ref{eq:Lindblad_equation}). To first order in $\Delta t$, the state
of a system subject to this stochastic process is described at time
$t + \Delta t$ by
\begin{equation}
\label{eq:Poisson_process_measurement}
\rho (t + \Delta t) = (1 - \gamma \Delta t) \rho (t) -
\frac{i}{\hbar} [\mathsf{H}, \rho (t)] \Delta t + \gamma \Delta t \sum_{n, k}
\mathsf{A}_{n k}\, \rho (t)\, \mathsf{A}_{n k}^{\dagger}.
\end{equation}
In the limit $\Delta t \rightarrow 0$ one obtains the measurement master equation \cite{Cresser_et_al}
\begin{equation}
\label{eq:measurement_master_equation}
\frac{\partial \rho}{\partial t} = - \frac{i}{\hbar} [\mathsf{H}, \rho] + \gamma
\Biggl[ \sum_{n, k} \mathsf{A}_{n k}\, \rho\, \mathsf{A}_{n k}^{\dagger} - \rho \Biggr],
\end{equation}
which can be shown to be of the type (\ref{eq:Lindblad_equation}).
\section{Classicalization of systems with a quadratic Hamiltonian}
\label{sec:classicalization_quadratic_Hamiltonian}
A general quadratic Hamiltonian in the basis $\mathsf{Q}$, $\mathsf{P}$ is of the form:
\begin{equation}
\label{eq:quadratic_Hamiltonian_Q_P}
\mathsf{H} = c_1\mathsf{Q}^2 + c_2\mathsf{P}^2 + c_3(\mathsf{Q}\mathsf{P} + \mathsf{P}\mathsf{Q}) + c_4\mathsf{Q} + c_5\mathsf{P}, \quad c_i \in \mathbb{R}.
\end{equation}
The corresponding expression in a dimensionless basis analogous to
(\ref{eq:CCR_ladder_basis}) is:
\begin{equation}
\label{eq:quadratic_Hamiltonian_a_a_dagger}
\mathsf{H} = z_1\mathsf{b}^\dagger \mathsf{b} + z_2\mathsf{b}^2 + z_2^\ast (\mathsf{b}^\dagger)^2 + z_3\mathsf{b} + z_3^\ast \mathsf{b}^\dagger, \quad z_1 \in \mathbb{R},\quad z_2,z_3 \in \mathbb{C}.
\end{equation}
Using a canonical transformation (see \ref{sec:diagonalization_quadratic_Hamiltonian}) one obtains from
(\ref{eq:quadratic_Hamiltonian_a_a_dagger}) the harmonic
oscillator-like Hamiltonian ($z_0 = (z_1^2 - 4|z_2|^2)^{1/2}$, $\phi = \mathrm{Arg}(z_2)$):
\begin{equation}
\label{eq:harmonic_oscillator_hamiltonian}
\mathsf{H} = z_0\, \mathsf{a}^\dagger \mathsf{a} + c\,\mathsf{I},
\end{equation}
with
\begin{equation*}
\label{eq:ladder_operator_diagonal_basis}
\mathsf{b} = \biggl[\frac12 \frac{z_1 + z_0}{z_0} \biggr]^{1/2} e^{i\phi/2}\, \mathsf{a} + \biggl[\frac12 \frac{z_1 - z_0}{z_0} \biggr]^{1/2} e^{i\phi/2} \, \mathsf{a}^\dagger + \frac{2 z_2 z_3^\ast - z_1 z_3}{z_0^2}\mathsf{I}
\end{equation*}
and
\begin{equation*}
\label{eq:ground_state_energy_diagonalized_Hamiltonian}
c = \frac12(z_0 - z_1) + \frac{1}{z_0^2}\left(z_2 (z_3^\ast)^2 + z_2^\ast z_3^2 - z_1 |z_3|^2\right).
\end{equation*}
We note that the constant $c$ in the Hamiltonian does not affect the
dynamics and, therefore, in this basis the system behaves like a
harmonic oscillator with ground-state energy zero. This is a
demonstration of the usefulness of algebraic methods in quantum
mechanics. In the following we will set $z_0 = \hbar\,\omega$.
\subsection{Simultaneous imprecise measurement of the quadratures of the harmonic oscillator}
\label{measurement_of_quadratures}
For the quantum harmonic oscillator
(\ref{eq:harmonic_oscillator_hamiltonian}) one can define the
dimensionless self-adjoint quadrature operators:
\begin{equation}
\label{eq:quadratures}
\mathsf{X} = \frac12 (\mathsf{a} + \mathsf{a}^{\dagger}), \quad \mathsf{Y} = \frac{1}{2i}(\mathsf{a} - \mathsf{a}^{\dagger}), \quad [\mathsf{X}, \mathsf{Y}] = \frac{i}{2}.
\end{equation}
For any state of the harmonic oscillator, the Heisenberg uncertainty
product is given by $\Delta \mathsf{X} \,\Delta \mathsf{Y} \geq 1 / 4$,
with
$\Delta \mathsf{O} = (\langle \mathsf{O}^2 \rangle - \langle
\mathsf{O} \rangle^2)^{1/2}$. States that satisfy the equality are
called minimum-uncertainty states. For example, coherent states have
this property: $ \Delta \mathsf{X} = \Delta \mathsf{Y} =
\frac{1}{2}$. This is another reason why coherent states are
considered the most classical quantum states. The eigenvalues of
$\mathsf{X}$ and $\mathsf{Y}$ are the real and imaginary parts of the
amplitude $\alpha = \alpha_x + i \alpha_y$. A measurement of $\alpha$
can be described using the formalism of generalized measurements of
section 4, as follows.
Given a positive operator with unit trace $\Psi$, one can define the
operators \cite{Walker}:
\begin{equation}
\label{eq:POVM_quadrature_measurement}
\pi_{\alpha} = \frac{1}{\pi} \mathsf{D} (\alpha) \Psi \mathsf{D}^{\dagger} (\alpha), \quad
\int \mathrm{d}^2\alpha\, \pi_{\alpha} = \mathsf{I}.
\end{equation}
The probability density of obtaining the result $\alpha$ is
$\mathrm{Prob}(\alpha) = \mathrm{Tr} [\pi_{\alpha}\,\rho]$, and its moments are given by
\begin{equation}
\label{eq:moments_distribution_alpha}
M_{i j} = \int \mathrm{d}^2\alpha\, \mathrm{Prob}(\alpha)\, \alpha_x^i \,\alpha_y^j , \quad \mathrm{d}^2\alpha = \mathrm{d} \alpha_x \, \mathrm{d} \alpha_y.
\end{equation}
The expectation values of $\alpha_x$ and $\alpha_y$ are given by
\begin{eqnarray}
\label{eq:expectation_values_quadratures}
\langle \alpha_x \rangle & = & M_{10} = \langle \mathsf{X} \rangle_\rho - \langle \mathsf{X} \rangle_\Psi,\nonumber\\
\langle \alpha_y \rangle & = & M_{01} = \langle \mathsf{Y} \rangle_\rho - \langle \mathsf{Y} \rangle_\Psi,
\end{eqnarray}
so in order for the measurement to be unbiased, the expectation values
with respect to $\Psi$ must be zero. The second-order moments give the
mean square uncertainty for the simultaneous measurement of $\alpha_x$
and $\alpha_y$:
\begin{eqnarray}
\label{eq:second_moments_quadratures}
\langle (\Delta \alpha_x)^2 \rangle & = & M_{20} - (M_{10}^{})^2 = \langle
\mathsf{X}^2 \rangle_{\rho} + \langle \mathsf{X}^2 \rangle_{\Psi},\nonumber\\
\langle (\Delta \alpha_y)^2 \rangle & = & M_{02} - (M_{01}^{})^2 = \langle
\mathsf{Y}^2 \rangle_{\rho} + \langle \mathsf{Y}^2 \rangle_{\Psi},
\end{eqnarray}
which shows that the measurement contributes (classical noise) to the
uncertainty (quantum noise) of the measured state. Therefore, it is an
imprecise, simultaneous measurement of two non-commuting
observables. In the following we will show that this interaction
between the harmonic oscillator and the system described by $\Psi$
leads to classicalization of the former.
\subsection{Measurement-induced classicalization}
\label{sec:measurement_induced_classicalization}
From the theory of generalized measurements in section
\ref{sec:phase_space_measurements}, the operators
(\ref{eq:POVM_quadrature_measurement}) can be written as:
\begin{equation}
\label{eq:Kraus_decomposition_pi}
\pi_\alpha = \mathsf{A}^\dagger_\alpha \mathsf{A}_\alpha,\quad \mathsf{A}_\alpha = \frac{1}{\sqrt{\pi}} \, \mathsf{D}(\alpha) \Psi^{1/2} \mathsf{D}^\dagger(\alpha).
\end{equation}
In turn, $\mathsf{A}_\alpha$ can be written in the basis of displacement operators:
\begin{equation}
\label{eq:effect_basis_displacement_ops}
\fl \mathsf{A}_\alpha = \int \mathrm{d}^2\beta\, \mathsf{D}(\beta) \mathrm{Tr}[\mathsf{D}^\dagger(\beta) \mathsf{A}_\alpha]
= \frac{1}{\sqrt{\pi}}\int \mathrm{d}^2\beta\, \mathsf{D}(\beta)\, e^{\,\beta^\ast \alpha \,-\, \alpha^\ast \beta} \chi^\ast_{\Psi^{1/2}}(\beta),
\end{equation}
where we used the cyclic property of the trace, the definition
(\ref{eq:def_characteristic_function}) and the identity
\begin{equation}
\label{eq:displacement_of_displacement_operator}
\mathsf{D}^\dagger(\xi) \mathsf{D}(\zeta) \mathsf{D}(\xi) = \mathsf{D}(\zeta) \exp{(\xi^\ast \zeta - \zeta^\ast \xi)}.
\end{equation}
In this representation,
\begin{equation}
\label{eq:twirling_superoperator}
\int \mathrm{d}^2\alpha\, \mathsf{A}_\alpha \,\rho\, \mathsf{A}_\alpha^\dagger =
\pi \int \mathrm{d}^2\gamma\, \mathsf{D}(\gamma)\, \rho\, \mathsf{D}^\dagger(\gamma)\, |\chi_{\Psi^{1/2}}(\gamma)|^2,
\end{equation}
where we used the identity
\begin{equation}
\label{eq:2d_Dirac_delta}
\delta^{(2)}(\beta\, -\, \gamma) = \frac{1}{\pi^2} \int_{-\infty}^{\infty} \mathrm{d} ^2\alpha \exp[\alpha\beta^* - \alpha^*\beta] \exp[\alpha^*\gamma - \alpha\gamma^*].
\end{equation}
Defining the even, positive-definite function
$g(|\alpha|) := \mathcal{N}\,|\chi_{\Psi^{1/2}}(\alpha)|^2$, where
$\mathcal{N}$ is a normalization constant, we finally arrive at the
master equation corresponding to the measurement described above:
\begin{equation}
\label{eq:master_equation_quadrature_measurement}
\frac{\partial \rho}{\partial t} = - \frac{i}{\hbar} [\mathsf{H}, \rho] + \gamma
\int \mathrm{d}^2\alpha\, g (| \alpha |) [\mathsf{D} (\alpha) \,\rho\, \mathsf{D}^{\dagger} (\alpha) -
\rho],
\end{equation}
where $\mathsf{H}$ is defined in
(\ref{eq:harmonic_oscillator_hamiltonian}). The integral term can be
interpreted as describing phase space ``kicks'' occurring with rate
$\gamma$ and with a $g$-distributed strength $\alpha$.
Taking the trace of (\ref{eq:master_equation_quadrature_measurement})
multiplied with $\mathsf{D}(\eta)$ yields the following mapping from
operators to functions:
\begin{eqnarray}
\label{eq:Bargmann_rep_superoperators}
\mathsf{a}^\dagger \mathsf{a} \,\rho \rightarrow \biggl( - \frac{\partial}{\partial \eta} \frac{\partial}{\partial \eta^\ast} + \eta^\ast \frac{\partial}{\partial \eta^\ast} \biggr) \chi_\rho(\eta, \eta^\ast, t),\nonumber\\
\rho \, \mathsf{a}^\dagger \mathsf{a} \rightarrow \biggl( - \frac{\partial}{\partial \eta} \frac{\partial}{\partial \eta^\ast} + \eta \frac{\partial}{\partial \eta} \biggr) \chi_\rho(\eta, \eta^\ast, t),\nonumber\\
\int \mathrm{d}^2\alpha\, g(|\alpha|)\,\mathsf{D}(\alpha) \,\rho\, \mathsf{D}^\dagger(\alpha) \rightarrow \chi_\rho(\eta, \eta^\ast, t) \,\chi_g(|\eta|),
\end{eqnarray}
and the corresponding dynamic equation for the characteristic function:
\begin{equation}
\label{eq:equation_char_fun_stationary_frame}
\biggl( \frac{\partial}{\partial t} + i\omega[\eta^\ast \partial_{\eta^\ast} - \eta \partial_\eta] + \gamma[1 - \chi_g(|\eta|)] \biggr)\chi_\rho(\eta, \eta^\ast, t) = 0.
\end{equation}
Introducing the function
$\tilde{\chi}_\rho(\eta, \eta^\ast, t) = \chi_\rho(\eta e^{i \omega t},
\eta^\ast e^{-i \omega t} t)$, the equality of the total
differentials yields:
\begin{equation}
\label{eq:time_derivative_tilde}
\frac{\partial \tilde{\chi}_\rho}{\partial t} = i\omega\eta \frac{\partial \chi_\rho}{\partial \eta} - i\omega\eta^\ast \frac{\partial \chi_\rho}{\partial \eta^\ast} + \frac{\partial \chi_\rho}{\partial t},
\end{equation}
from which one obtains the equation of motion for the characteristic
function of the state in a frame rotating with frequency $\omega$:
\begin{equation}
\label{eq:equation_characteristic_function}
\frac{\partial}{\partial t} \tilde{\chi}_{\rho} (\eta, \eta^\ast, t) = -\, \gamma [1 - \chi_g (| \eta |)] \,\tilde{\chi}_{\rho} (\eta,\eta^\ast, t),
\end{equation}
which in the non-rotating frame has the solution:
\begin{equation}
\label{eq:solution_characteristic_function}
\chi_{\rho} (\eta, \eta^\ast, t) = \chi_0 (\eta e^{i \omega t}, \eta^\ast e^{-i \omega t}) \exp \{ -\, \gamma t[1
- \chi_g (| \eta |)] \}.
\end{equation}
For economy of notation, in the following we omit the second argument
of $\chi_\rho$. We recognize the exponential term as the
characteristic function of a compound Poisson process with rate
$\gamma t$ and jump-size distribution $g$ \cite{Hanson}. Since
$\chi_g$ is the characteristic function of a probability
distribution, it satisfies $\chi_g(0)=1$ and in the limit
$|\eta| \rightarrow \infty$ it vanishes for all $t$:
$\chi_g(|\eta|) \rightarrow 0$. Therefore, at any time the
exponential term is an even function that asymptotically decays in
phase space towards the plane $f(\eta) = e^{-\gamma t}$, which in
turn decays with time towards zero.
\begin{figure}
\caption{Top: $|\chi(\eta,0)|$ for the cat state
(\ref{eq:Wigner_function_cat_state}
\label{fig:char_fun_cat_state_w_noise_wo_noise}
\end{figure}
Let us assume that $\chi_0$ corresponds to the cat state
(\ref{eq:Wigner_function_cat_state}). Since $\chi_\rho$ is a complex
function related to the Wigner function through a Fourier transform,
we can interpret $|\chi_\rho(\eta, t)|$ as a phase space Fourier
spectrum \cite{Chountasis_et_al}. For $t=0$, it is shown in the top
panel in figure \ref{fig:char_fun_cat_state_w_noise_wo_noise}. The
central, structured peak corresponds to the low frequencies of the
Wigner function, whereas the two outer peaks correspond to the high
frequencies, which are associated to the non-classical character of
the state.
A snapshot of the dynamics given by
(\ref{eq:equation_characteristic_function}) is shown in the bottom
panel for $\gamma t = 1$. For illustration purposes, the distribution
$g$ is assumed to be Gaussian. The noticeable suppression of the
outer peaks is due to the ``filtering'' effect of the measurement,
described by the decaying exponential term in
(\ref{eq:solution_characteristic_function}). Moreover, repeating the
above analysis for a coherent state reveals that it will be
broadened, as expected from (\ref{eq:second_moments_quadratures}).
Therefore, it is clear that the measurement drives a superposition
state of the harmonic oscillator towards a mixture of Gaussian
states, which corresponds to a classical phase space function.
It is interesting to consider the case of very frequent and very
small phase-space kicks, which amounts to assuming that
$\gamma \rightarrow \infty$ and that the second moments of the
distribution $g$ are much larger than the higher moments, but their
product with $\gamma$ remains finite. Performing a series expansion
of the displacement operator to second order in $\eta$:
\begin{equation}
\label{eq:series_displacement_op_second_order}
\mathsf{D}(\eta) \approx \mathsf{I} + \eta \mathsf{a}^\dagger - \eta^\ast \mathsf{a} + \frac12 (\eta \mathsf{a}^\dagger - \eta^\ast \mathsf{a})(\eta \mathsf{a}^\dagger - \eta^\ast \mathsf{a}),
\end{equation}
one obtains from (\ref{eq:master_equation_quadrature_measurement}) a
master equation describing diffusion in phase space \cite{Agarwal}:
\begin{equation}
\label{eq:master_equation_diffusion}
\fl \frac{\partial \rho}{\partial t} = -i\omega [\mathsf{a}^\dagger \mathsf{a},\rho] - \kappa(-2\,\mathsf{a}\, \rho\, \mathsf{a}^\dagger + \mathsf{a}^\dagger \mathsf{a} \,\rho + \rho\, \mathsf{a}^\dagger \mathsf{a} - 2\,\mathsf{a}^\dagger \rho\, \mathsf{a} + \mathsf{a}\, \mathsf{a}^\dagger \rho + \rho\, \mathsf{a}\, \mathsf{a}^\dagger).
\end{equation}
In fact, one can describe this dynamics in terms of the Brownian
motion of the state vector of the system in Hilbert space
\cite{JacobsSteck}. This and other master equations describing
Gaussian dynamics are thoroughly discussed in \cite{Serafini}. It is
interesting to note that under this kind of evolution, the Wigner
function becomes positive everywhere in a finite time
\cite{BrodierOzorio}, in contrast to the example discussed here.
\section{Conclusions}
\label{sec:conclusions}
The model presented in section 5 illustrates the measurement approach
to the classicalization of a quantum system by means of a master
equation that can be solved exactly for an arbitrary measurement
strength (``kick'' size). Throughout the article, we aimed at keeping
the presentation general and, at the same time, accessible. This
should enable newcomers to the field to apply the measurement approach
to classicalization to more complex systems.
\ack
The author acknowledges the kind support of a CONACYT-DAAD PhD
scholarship and is grateful to Klaus Hornberger for his suggestions to
improve the quality of the manuscript.
\appendix
\section{Diagonalization of a quadratic Hamiltonian}
\label{sec:diagonalization_quadratic_Hamiltonian}
Following \cite{Zelevinsky}, we consider the canonical transformation
\begin{equation}
\label{eq:bogoliubov_trafo}
\mathsf{b} = \mu \mathsf{a} + \nu \mathsf{a}^\dagger + \delta.
\end{equation}
From
$[\mathsf{a}, \mathsf{a}^\dagger] = [\mathsf{b}, \mathsf{b}^\dagger] =
\mathsf{I}$, follows that $|\mu|^2 - |\nu|^2 = 1$. Substituting in
(\ref{eq:quadratic_Hamiltonian_a_a_dagger}) and collecting terms, in
order to get a diagonal operator the following equations must be
satisfied:
\begin{eqnarray}
\label{eq:equations_diagonalization}
z_1 \nu^\ast \delta + z_1 \delta^\ast \mu + 2 z_2^\ast \mu \delta + 2 z_2 \nu^\ast \delta^\ast + z_3^\ast \mu + z_3 \nu^\ast = 0\\
z_1 \mu^\ast \delta + z_1 \delta^\ast \nu + 2 z_2^\ast \nu \delta + 2 z_2 \mu^\ast \delta^\ast + z_3^\ast \nu + z_3 \mu^\ast = 0\\
z_1 \nu^\ast \mu + z_2^\ast \mu^2 + z_2 (\nu^\ast)^2 = 0.
\end{eqnarray}
Subtracting (A.2) multiplied by $\nu$ from (A.3) multiplied by $\mu$,
yields $\delta = -(2 z_2 \delta^\ast + z_3)/z_1$. Substituting in
(A.2) we obtain $\delta^\ast = (2 z_2^\ast z_3 - z_1 z_3^\ast)/z_0^2$,
with $z_0 = (z_1^2 - 4|z_2|^2)^{1/2}$. From this expression we arrive
at the condition $z_1 > 2|z_2|$.
In order to find $\mu$ and $\nu$ from (A.4) we use the polar
representations
\begin{equation}
\label{eq:polar_reps}
z_2 = |z_2| e^{i \phi}, \quad \mu = U e^{i \phi_u}, \quad \nu = V e^{i \phi_v},
\end{equation}
and choose $\phi_u = \phi_v = \frac12 \phi$ in order to obtain an
equation with real variables. We now use the parametrizations
$U = \cosh(\Theta/2)$ and $V = \sinh(\Theta/2)$, and recall the
identities
\begin{equation}
\label{eq:hyperbolic_identities}
\cosh^2(x) + \sinh^2(x) = \cosh(2x),\quad 2\sinh(x)\cosh(x) = \sinh(2x).
\end{equation}
In terms of $\Theta$, (A.4) has the form
$(z_1 \sinh \Theta)/2 + |z_2| \cosh \Theta = 0$. Using
$2\,\mathrm{atanh}(x) = \log[(1+x)/(1-x)]$, we obtain
$\Theta = \log \sqrt{(z_1 - 2|z_2|)/(z_1 + 2|z_2|)}$. Now we can
calculate $\sinh \Theta = -2|z_2|/z_0$ and $\cosh \Theta =
z_1/z_0$. Recalling the identities
\begin{equation}
\label{eq:more_identities}
\cosh \frac{x}{2} = \sqrt{(\cosh x + 1)/2}, \quad \sinh \frac{x}{2} = \sqrt{(\cosh x - 1)/2},
\end{equation}
we finally arrive at
\begin{equation}
\label{eq:U_and_V}
U = \Bigl[ \frac12\, \frac{z_1 + z_0}{z_0} \Bigl]^{1/2}, \quad V = \Bigl[ \frac12\, \frac{z_1 - z_0}{z_0} \Bigl]^{1/2}.
\end{equation}
Substituting all of the above in
(\ref{eq:quadratic_Hamiltonian_a_a_dagger}), one obtains
(\ref{eq:harmonic_oscillator_hamiltonian}).
\end{document} |
\begin{document}
\title{Decompositions of Bernstein-Sato polynomials and slices}
\begin{abstract}
Let $G$ be a linearly reductive group acting on a vector space $V$, and $f$ a (semi-)invariant polynomial on $V$. In this paper we study systematically decompositions of the Bernstein-Sato polynomial of $f$ in parallel with some representation-theoretic properties of the action of $G$ on $V$. We provide a technique based on a multiplicity one property, that we use to compute the Bernstein-Sato polynomials of several classical invariants in an elementary fashion. Furthermore, we derive a \lq\lq slice method" which shows that the decomposition of $V$ as a representation of $G$ can induce a decomposition of the Bernstein-Sato polynomial of $f$ into a product of two Bernstein-Sato polynomials -- that of an ideal and that of a semi-invariant of smaller degree. Using the slice method, we compute Bernstein-Sato polynomials for a large class of semi-invariants of quivers.
\end{abstract}
\section*{Introduction}
\label{sec:intro}
The classification of irreducible prehomogeneous vector spaces was achieved in \cite{saki}. The computation of $b$-functions (i.e. Bernstein-Sato polynomials) of their semi-invariants has been completed using sophisticated methods such as microlocal calculus (for example, see \cite{kimu,skko}). Extensive calculations have been done also in the case of \emph{reducible} prehomogeneous vector spaces (for example, \cite{en,sasu, sugi, ukai}).
In the article \cite{sasu}, a criterion has been given for the decomposition of the $b$-functions on a prehomogeneous space $V$ in terms of decomposing $V$ into smaller representations. Using this, the $b$-functions for quivers of type $\mathbb{A}$ are computed in \cite{sugi}. In this paper, we provide a more general computational technique based on a multiplicity one property that gives similar decompositions of $b$-functions. This technique gives a more elementary approach for the computation of the (local and global) $b$-functions of some classical semi-invariants, such as the determinant, symmetric determinant, Pfaffian and others. Furthermore, we derive a slice method leading to a reduction process that decomposes the $b$-function of a semi-invariant of $V$ into the product of the Bernstein-Sato polynomial of an ideal and the $b$-function of a semi-invariant on a slice of $V$. Applying this process, we can compute the $b$-functions for some semi-invariants of quivers, including those of Dynkin type $\mathbb{A},\mathbb{D}$ and other tree quivers.
In \cite{en}, the author gives a method by \lq\lq reflections'' that allows the computation of $b$-functions for semi-invariants of any Dynkin quiver. For quivers, the slice method has the advantage of yielding faster results in most cases (when applicable). Also, the slice technique does not require extensive knowledge of representations of quivers. For best results in the case of quivers, the two methods can (and should) be combined. We note that in \cite{robin} the $b$-function for a semi-invariant of a special quiver (with a loop) is investigated using different tools.
In his thesis \cite[Chapter 4]{phd}, the author considers a slice method similar to the one in this paper, but which is rather cumbersome to use. The methods in this paper are major improvements of the slice method considered there.
We consider the following examples. Take $X=(x_{ij})$ an $n\times n$ generic matrix of variables, and $\partial X$ the matrix formed by the partial derivatives $\dfrac{\partial}{\partial x_{ij}}$. Its determinant is a differential operator. The classical Capelli identity implies (see \cite{howumed}):
$$\det \partial X \cdot \det X^{s+1} = (s+1)(s+2)\cdots(s+n)\det(X)^s.$$
Hence the Bernstein-Sato polynomial of the determinant is $b(s)=(s+1)(s+2)\cdots (s+n)$. In Section \ref{sec:slice} we explain how one can use the technique based on the multiplicity one property to derive this result in an elementary way.
A simple, yet non-trivial example of interest is the following semi-invariant, coming from the quiver $\mathbb{D}_4$:
$$\det\begin{pmatrix}
X & Y & 0 \\
0 & Y & Z
\end{pmatrix}.$$
Here $X,Y,Z$ are generic matrices of variables, with $X\in M_{\beta_4,\beta_1},Y\in M_{\beta_4,\beta_2},Z\in M_{\beta_4,\beta_3}$ and $\beta_1+\beta_2+\beta_3=2\beta_4$. We compute its $b$-function (together with many other quivers) in Section \ref{subsec:exquiv} based on the slice method developed in Section \ref{subsec:method}.
The paper is organized as follows. In Section \ref{sec:b-func}, we focus on generalities about Bernstein-Sato polynomials, mostly in the equivariant setting.
In Section \ref{sec:slice}, we start by describing a method based on a multiplicity one property. We use Theorem \ref{thm:mult1} in order to compute the $b$-functions of several classical semi-invariants in Section \ref{subsec:exam}. Then we derive a slice method in Section \ref{subsec:method}, where the main result is Theorem \ref{thm:mainapp}. We also give the analogous result for $b$-functions of several variables (Theorem \ref{thm:multi}).
In Section \ref{sec:quiv}, after introducing some background material on quivers, we apply the slice method (Theorem \ref{thm:mainapp}) to arrows of quivers (Theorem \ref{thm:bquiv}). This gives a practical reduction method for computing $b$-functions of many (determinantal) quiver semi-invariants. This includes those of quivers of type $\mathbb{A},\mathbb{D}$ and other tree quivers (see Theorems \ref{thm:tree}), \ref{thm:dee}. We work out several examples in Section \ref{subsec:exquiv} of $b$-functions of one variable and $b$-functions of several variables.
Besides yielding the roots of $b$-functions, the slice method provides other useful information as well. For example, it gives an algorithm for determining the locally semi-simple representation corresponding to a semi-invariant (see Proposition \ref{prop:locsemi}). Based on slices, we also give an easy algorithm for the explicit description of generic representations for Dynkin quivers of type $\mathbb{D}$, as described in Appendix \ref{app:decomp}.
\begin{notation}\label{not:notation}
As usual, $\mathbb{N}$ will denote the set of all non-negative integers and $\mathbb{C}$ the set of complex numbers. For $a,b,d\in \mathbb{N}, a\leq b$, we use the following notation in $\mathbb{C}[s]$:
$$[s]^d_{a,b}:=\prod_{i=a+1}^b \prod_{j=0}^{d-1} (ds+i+j).$$
In the case $d=1$, we sometimes write $[s]_{a,b}:=[s]_{a,b}^1$. Also, if $a=0$, we sometimes write $[s]^d_{b}:=[s]^d_{0,b}$. Hence $[s]^{d}_{a,b}[s]^d_{a}=[s]^d_b$.
Now fix an $l$-tuple $\underline{m}=(m_1,\dots,m_l)\in \mathbb{N}^l$. Then for any $l$-tuple $(d_1,\dots,d_l)$, we use the following notation in $\mathbb{C}[s_1,\dots,s_l]$:
$$[s]^{d_1,\dots,d_l}_{a,b}=\prod_{i=a+1}^b \prod_{j=0}^{d-1} (d_1s_1+\dots+d_ls_l+i+j),$$
where $d=m_1d_1+\dots + m_ld_l$.
\end{notation}
\section{Bernstein-Sato polynomials}
\label{sec:b-func}
\subsection{Definition} \label{subsec:bern}
First we define and briefly recall some basic properties about Bernstein-Sato polynomials. We will interchangeably call them also $b$-functions, especially in the contexts of Theorem \ref{thm:bfun} and Lemma \ref{lem:several} from Section \ref{subsec:bfun}. For details on Bernstein-Sato polynomials, we refer the reader to \cite{gyoja,kashi}.
Throughout this paper we work over the complex field $\mathbb{C}$. Let $V$ be an $n$-dimensional vector space. Denote by $D$ the algebra of differential operators on $V$ (i.e. the Weyl algebra in $n$ variables), and by $D_v$ the algebra of differential operators regular at $v\in V$ (i.e. the localization of $D$ at $v$).
Let $f\in \mathbb{C}[V]$ be a non-zero polynomial, and let $R$ be one of the rings $D$ or $D_v$. Then there exits (see \cite{kashi} a differential operator $P(s)\in R[s]:=R\otimes \mathbb{C}[s]$ and a non-zero polynomial $b(s)\in \mathbb{C}[s]$ such that
$$P(s)\cdot f^{s+1}(x) = b(s)\cdot f^s(x).$$
The functions $b(s)$ satisfying such a relation form an ideal of $\mathbb{C}[s]$, whose monic generator we denote by $b_f(s)$ or $b_{f,v}(s)$, if $R=D$ or $D_v$, respectively. We call $b_f$ the (global) \textit{Bernstein-Sato polynomial} of $f$, and $b_{f,v}$ the local Bernstein-Sato polynomial of $f$ at $v$.
By \cite{kashi}, all roots of $b_f(s)$ are negative rational numbers. Moreover, if $f$ is a homogeneous polynomial, then $b_{f,0}(s)= b_f(s)$ (see \cite[Lemma 2.5.3]{gyoja}).
Throughout we work mostly in equivariant settings as seen in the next section.
\subsection{$b$-functions of semi-invariants}\label{subsec:bfun}
Let $G$ be a (connected) reductive algebraic group, acting rationally on $V$. That is, we have a morphism of algebraic groups $\rho: G \to \operatorname{GL}(V)$. Then we have an action of $G$ on $\mathbb{C}[V]$ by $(g\cdot f)(v)=f(g^{-1}\cdot v)$ for all $v\in V$, where $g\in G$, $f\in \mathbb{C}[V]$. We call a polynomial $f\in \mathbb{C}[V]$ a semi-invariant, if there is a character $\sigma\in \operatorname{Hom}(G,\mathbb{C}^{\times})$ such that $g\cdot f = \sigma(g) f$, that is, $f(gv)=\sigma(g)^{-1}f(v)$. In this case we say the weight of $f$ is $\sigma$. In the literature such $f$ is sometimes also called a relative invariant polynomial. We form the ring of semi-invariants
$$\operatorname{SI}(G,V)=\bigoplus_{\sigma} \operatorname{SI}(G,V)_{\sigma}=\mathbb{C}[V]^{[G,G]},$$
where the sum runs over all characters $\sigma$ and the weight spaces are
$$\operatorname{SI}(G,V)_{\sigma}=\{f\in \mathbb{C}[V] | f \text{ is a semi-invariant of weight } \sigma\}.$$
The multiplicity of $\sigma$ is $\dim\operatorname{SI}(G,V)_{\sigma}$. Following \cite{en}, we make the following definition (which makes sense even when $G$ is not reductive):
\begin{definition}\label{def:multfree}
We say that $\sigma$ is \emph{multiplicity-free}, if the multiplicity of $\sigma^k$ is $1$, for any $k\in\mathbb{N}$.
\end{definition}
By a standard argument, one can give the following geometric characterization of the above property when $G$ is a connected reductive group: a semi-invariant $f\in \mathbb{C}[V]$ has multiplicity-free weight $\sigma$ if and only if there is a unique closed orbit $\mathcal{O}$ in the open affine neighborhood $f\neq 0$. In the spirit of \cite{shme}, for an element $x\in \mathcal{O}$ of this orbit we say that $x$ is the \textit{locally semi-simple} point of $f$.
Given a semi-invariant $f$ of weight $\sigma$, for the results of this paper regarding $b$-functions to hold (see Theorem \ref{thm:bfun}) it is enough to require the multiplicity of $\sigma^k $ to be $1$ for just $k=\deg f-1$.
Let $x=(x_1,\dots, x_n)$ be the coordinate system with respect to a basis of $V$.
We denote the dual variables (partial derivatives) by
\[\partial x=(\partial_1, \dots, \partial_n).\]
Let $V^*$ be the dual space of $V$, with is naturally a $\operatorname{GL}(V)$-module. For any $d\geq 0$, let $\mathbb{C}[V]_d$ (resp. $\mathbb{C}[V^*]_d$) be the subspaces of homogeneous polynomials of degree $d$ in $\mathbb{C}[V]$ (resp. $\mathbb{C}[V^*]$). We have the $\operatorname{GL}(V)$-equivariant pairing between $\mathbb{C}[V]_d$ and $\mathbb{C}[V^*]_d$ by
\begin{equation}\label{eq:pair}
\langle P,P^*\rangle = P^*(\partial x) \cdot P(x).
\end{equation}
This gives a $\operatorname{GL}(V)$-equivariant isomorphism $\mathbb{C}[V^*]_d \cong (\mathbb{C}[V]_d)^*$.
Let $f\in \mathbb{C}[V]$ be a semi-invariant of weight $\sigma$, and assume $\sigma$ is multiplicity-free. Then $f$ must be homogeneous (see \cite[Lemma 1.3]{gyoja}). Since $G$ is reductive, by the above pairing there is a dual semi-invariant $f^*\in \mathbb{C}[V^*]$ of weight $\sigma^{-1}$ of the same degree, canonical up to constant. In fact, we can choose a basis of $V$ such that the subset $\rho(G)\subset \operatorname{GL}(V)$ is stable under conjugate transpose, in which case $f^*$ can be obtained from $f$ by taking the complex conjugates of the coefficients -- see \cite{saki}.
The next result follows by \cite[Lemma 1.6,1.7]{gyoja} and \cite[Corollary 2.5.10]{gyoja}.
\begin{theorem}\label{thm:bfun}
Let $f\in \mathbb{C}[V]$ be a semi-invariant with multiplicity-free weight, and let $f^*\in\mathbb{C}[V^*]$ be the dual semi-invariant. We have
\begin{equation}\label{eq:good}
f^*(\partial x)\cdot f(x)^{s+1}=b(s)f(x)^s.
\end{equation}
where $b(s)$ is a polynomial equal to the Bernstein-Sato polynomial $b_f(s)$ up to a non-zero constant factor and $\deg b_f(s) = \deg f$.
\end{theorem}
We call $(G,V)$ a prehomogeneous vector space, if $V$ has a dense open orbit $\mathcal{O}$, i.e. $\overline{\mathcal{O}}=V$. By Rosenlicht's Theorem (see \cite{preh}), $(G,V)$ is prehomogeneous iff all weight multiplicities of the ring of semi-invariants are at most $1$. Moreover, the following holds (see \cite{saki}):
\begin{theorem}\label{thm:satokimura}
Assume $(G,V)$ is a prehomogeneous vector space, and let $Z(f_1),Z(f_2),\dots, Z(f_k)$ be the irreducible components of $V\backslash \mathcal{O}$ of codimension $1$, for some $f_1,f_2,\dots,f_k \in \mathbb{C}[V]$. Then $f_1,f_2\dots,f_k$ are algebraically independent semi-invariants and $\operatorname{SI}(G,V)=\mathbb{C}[f_1,f_2,\dots,f_k]$.
\end{theorem}
The semi-invariants $f_1,f_2\dots,f_k$ as above are called \textit{fundamental} semi-invariants. We mention that many of our examples in this paper are prehomogeneous vector spaces, but we also work with spaces that are not necessarily prehomogeneous but have semi-invariants of multiplicity-free weights (for example, Theorem \ref{thm:tree}).
We have the following notion of $b$-function of several variables (see \cite{sata}).Let $f_1,\dots f_l \in \mathbb{C}[V]$ be semi-invariants of weights $\sigma_1,\dots, \sigma_l$, respectively. Assume that the product $\sigma_1\cdots \sigma_l$ is a multiplicity-free weight in $\mathbb{C}[V]$. In this case we can take respective dual semi-invariants $f^*_1,\dots, f^*_l\in \mathbb{C}[V^*]$. Put $\underline{f}=(f_1,\dots, f_l)$ and $\underline{f}^*=(f_1^*,\dots , f_l^*)$. For a multi-variable $\underline{s}=(s_1,\dots ,s_l)$, we define $\underline{f}^{\underline{s}}=\displaystyle\prod_{i=1}^l f_i^{s_i}$, and $\underline{f}^{*\underline{s}}=\displaystyle\prod_{i=1}^l f_i^{*s_i}$.
\begin{lemma}\label{lem:several}
Using the notation above, if $\sigma_1\cdots \sigma_l$ is multiplicity-free, then for any $l$-tuple $\underline{m}=(m_1,\dots,m_l)\in \mathbb{N}^l$ there is a polynomial $b_{\underline{f},\underline{m}}(\underline{s})$ of $l$ variables such that
\begin{equation}
\underline{f}^{*\underline{m}}(\partial x)\cdot\underline{f}^{\underline{s}+\underline{m}}(x)=b_{\underline{f},\underline{m}}(\underline{s})\underline{f}^{\underline{m}}(x).
\end{equation}
\end{lemma}
If $\sigma_1\cdots \sigma_l$ is multiplicity-free, then all the individual weights $\sigma_i$ are multiplicity-free, and one can easily recover the $b$-function $b_{f_i}(s)$ of one variable from $b_{\underline{f},\underline{m}}(\underline{s})$. Again, if $(G,V)$ is prehomogeneous then any $\sigma_1\cdots \sigma_l$ is automatically multiplicity-free.
\subsection{Bernstein-Sato polynomials of ideals}\label{subsec:bideal}
Now we consider tuples of polynomials $\underline{f}=(f_1,\dots,f_r)$ with $f_i \in \mathbb{C}[V]$, from a different viewpoint. Following \cite[Definition 3.3]{bmax}, we introduce (note that in the case of $r=1$ we recover Definition \ref{def:multfree}):
\begin{definition}\label{def:tuple}
A tuple $\underline{f}= (f_1,\dots,f_r)$ in $\mathbb{C}[V]$ is said to be a \textit{multiplicity-free tuple} if
\begin{itemize}
\item[(a)] For every $k\in \mathbb{N}$, the polynomials
\[\underline{f}^{\underline{k}} = f_1^{k_1}\cdots f_r^{k_r},\mbox{ for } \underline{k}=(k_1,\dots,k_r)\in\mathbb{N}^r \mbox{ satisfying } k_1+\dots+k_r=k,\]
span an irreducible $G$-subrepresentation $M_{k}\subset \mathbb{C}[V]$.
\item[(b)] For every $k\in\mathbb{N}$, the multiplicity of the $G$-representation $M_{k}$ inside $\mathbb{C}[V]$ is equal to one.
\end{itemize}
\end{definition}
We note that given any multiplicity-free tuple $\underline{f}=(f_1,\dots,f_r)$, any \lq\lq power'' of the tuple $\underline{f}$ is also multiplicity-free. Here the $d$th power of the tuple $\underline{f}$ is a new tuple formed by all elements of the form
\[\underline{f}^{\underline{d}} = f_1^{d_1}\cdots f_r^{d_r},\mbox{ for } \underline{d}=(d_1,\dots,d_r)\in\mathbb{N}^r \mbox{ satisfying } d_1+\dots+d_r=d.\]
Now fix a multiplicity-free tuple $\underline{f} = (f_1,\dots,f_r)$, which WLOG we assume that is a basis of $M_1$. Since $G$ is reducitive and the multiplicity of $M_1$ is in $\mathbb{C}[V]$ is one, there is a dual representation $M_1^*$ in $\mathbb{C}[V^*]$ of multiplicity one. We take a basis $f_1^*,\dots,f_r^*$ that is $G$-dual (up to constant) to $f_1,\dots,f_r$ with respect to the pairing (\ref{eq:pair}). Then the element
\[D_{\underline f}=\displaystyle\sum_{i=1}^r f^*_i(\partial x) f_i(x)\]
is a $G$-invariant differential operator. Denote by $I$ the ideal generated by $f_1,\dots,f_r$ in $\mathbb{C}[V]$, and let $b_I(s)=b_{\underline{f}}(s)$ be the Bernstein-Sato polynomial of $I$ -- for the definition of Bernstein-Sato polynomials of ideals (or tuples), we refer the reader to \cite{bideal}. By \cite[Proposition 3.4]{bmax}, we have the following result.
\begin{proposition}\label{prop:pf}
Consider a multiplicity-free tuple $\underline{f}=(f_1,\dots,f_r)$. If we let $s=s_1+\dots+s_r$ then there exists a polynomial $P_f(s)\in\mathbb{C}[s]$ such that
\[D_{\underline{f}}\cdot \underline{f}^{\underline s} = P_{\underline f}(s)\cdot \underline{f}^{\underline s},\]
and the Bernstein-Sato polynomial $b_{\underline{f}}(s)$ divides $P_{\underline{f}}(s)$.
\end{proposition}
As in the case $r=1$ (by Theorem \ref{thm:bfun}), we conjecture that for multiplicity-free tuples $\underline{f}$ we always have equality $b_{\underline{f}}(s)=P_{\underline{f}}(s)$. In \cite{bmax} this has been shown to be the case when $I$ is the ideal generated by maximal minors or the ideal generated by sub-maximal Pfaffians. We can also consider powers of ideals $I^d$, for positive integers $d$, as follows.
Let $M_{m,n}$ be the space of $m\times n$ matrices with $m\leq n$. Let $X$ be the $m\times n$ generic matrix of indeterminates and denote by $I$ the ideal of $\mathbb{C}[V]$ generated by all the $n\times n$ minors of $X$.
\begin{theorem}\label{thm:powermax}
Let $I^d$ the power of the ideal $I$ generated by maximal minors for some $d\in \mathbb{N}$. Then the Bernstein-Sato polynomial of $I^d$ is
\[b_{I^d}(s) = \prod_{i=n-m+1}^n \prod_{j=0}^{d-1} \left(s+\dfrac{i+j}{d}\right).\]
\end{theorem}
\begin{proof}
Consider the tuple formed by all maximal minors, which is a multiplicity-free tuple (see \cite{bmax}) by the FFT (see \cite[XI. Section 1.2]{proc}). Denote by $\underline{f}$ the $d$th power of this tuple as explained above, so $\underline{f}$ is also multiplicity-free, hence Proposition \ref{prop:pf} applies. One can obtain $P_{\underline f}(s)$ in several ways. For example, we can apply either the method in the proof of \cite[Theorem 3.5]{bmax} using the Fourier transform, or observe that by Schur's Lemma the polynomial $P_{\underline f}(s)$ is the same as the one computed in \cite[Theorem 3.3]{sasu} -- see also proof of Lemma \ref{lem:firstfac}. Hence, up to a constant we have (see Notation \ref{not:notation})
\[P_{\underline f}(s)=[s]^d_{n-m,n}.\]
To see that $b_{\underline{f}}(s)=P_{\underline{f}}(s)$, we note that the proof for the case $d=1$ from \cite[Section 4]{bmax} carries over, \textit{mutatis mutandis}, for an arbitary $d\in\mathbb{N}$.
\end{proof}
\section{Slices and the multiplicity one property}
\label{sec:slice}
In this section, we develop several techniques for calculating $b$-functions. These are similar to the methods used in \cite{sasu,ukai,wach}. The slice method developed in Section \ref{subsec:method} will be used further in Section \ref{sec:quiv}.
\subsection{Slices}\label{subsec:slice}
Let $H$ be a connected affine algebraic group and $V$ a rational $H$-module. Let $f\in \mathbb{C}[V]$ be a non-zero $H$-semi-invariant of weight $\sigma$. Denote by $\mathfrak{h}$ the Lie algebra of $H$. Fix an element $v\in V$ and let $H_v$ be the stabilizer of $v$. The tangent space at $v$ to the orbit $\mathcal{O}=G\cdot v$ of $v$ is $T_v(\mathcal{O})=\mathfrak{h}\cdot v$, on which $H_v$ acts naturally. By a theorem of Mostow \cite{mostow}, we can write $H_v= L_v \ltimes U_v$, where $U_v$ is the unipotent radical of $H_v$ and $L_v \cong H_v/U_v$ is reductive. Let $W$ be an $L_v$-complement to $T_v(\mathcal{O})$ in $V$, so that we have an $L_v$-decomposition $V=T_v(\mathcal{O})\operatornamelus W$. We call $(L_v,W)$ the \textit{slice representation} at $v$.
Given a polynomial $f\in \mathbb{C}[V]$, we construct a polynomial $f_v \in \mathbb{C}[W]$ defined by $f_v(w):=f(v+w)$ for $w\in W$. This gives an algebra map from $\mathbb{C}[V]$ to $\mathbb{C}[W]$ given by $f \mapsto f_v$.
Now if $f\in \mathbb{C}[V]$ is a $H$-semi-invariant of weight $\sigma$, then $f_v\in\mathbb{C}[W]$ is a $L_v$-semi-invariant of weight $\sigma|_{L_v}$. Hence the map $f \mapsto f_v$ induces the maps
\begin{equation}\label{eq:slice}
\phi_v : \operatorname{SI}(H,V) \to \operatorname{SI}(L_v,W)\,\, , \,\, \phi_v^\sigma : \operatorname{SI}(H,V)_{\sigma} \to \operatorname{SI}(L_v,W)_{\sigma |_{L_v}}.
\end{equation}
As in \cite{ukai}, we consider the map
$$\mu: H\times W \to V,$$
$$\mu(h,w)=h(v+w).$$
Computing the differential at the identity of $H$, we see that $\mu$ is a smooth map. In particular, the algebra map $\mu^*: \mathbb{C}[V] \to \mathbb{C}[H]\otimes\mathbb{C}[W]$ is injective. The map separates variables for a semi-invariant $f$ of weight $\sigma$, for we have
\begin{equation}
\mu^*(f)=\sigma^{-1}\otimes f_v.
\label{eq:basic}
\end{equation}
By the above discussion we obtain the following lemma (see also \cite[p. 57]{ukai}):
\begin{lemma}\label{lem:basic}
The map $\phi_v^\sigma$ is injective. Moreover, if $f$ is a semi-invariant of $(H,V)$ then $b_{f,v} = b_{f_v,0}$, that is, the local $b$-functions of $f$ at $v$ and of $f_v$ at $0$ coincide. In particular, if $f_v$ is homogeneous then $b_{f_v}|b_f$.
\end{lemma}
\begin{remark}\label{rem:works}
We note that in some situations one can choose algebraic groups (with corresponding complements $W$) different from $L_v$ and still make the above considerations work.
\end{remark}
\subsection{Expansions and the multiplicity one property}\label{subsec:mult1}
We recall and generalize some considerations from \cite{sasu}. Let $G$ be a (connected) reductive group with a Borel subgroup $B$ that contains a maximal torus $T$. The irreducible rational $G$-modules are parameterized by dominant $T$-weights. Let $V$ an algebraic $G$-module, and fix $f\in \operatorname{SI}(G,V)_\sigma$ with $\sigma$ \textit{multiplicity-free} as in Definition \ref{def:multfree}. Then $f$ is homogeneous, say of degree $d>0$. Take any integer $k$ with $0< k < n$. We have a $G$-equivariant map
\[ \mathbb{C}[V]_k \otimes \mathbb{C}[V]_{d-k} \to \mathbb{C}[V]_d.\]
The polynomial $f$ lies in the image of this onto map. Decomposing $\mathbb{C}[V]_k$ (resp. $\mathbb{C}[V]_{d-k}$) into irreducible $G$-modules and using that the multiplicity of $\sigma$ in $\mathbb{C}[V]$ is one, we see that there exits an irreducible $G$-submodule $M_\lambda$ of $\mathbb{C}[V]_k$ (resp. $M_{\lambda^*\cdot \sigma}$ of $\mathbb{C}[V]_{d-k}$) such that $f$ is in the image of the multiplication map
\[M_\lambda \otimes M_{\lambda^*\cdot\sigma} \to \mathbb{C}[V]_d.\]
Here $M_\lambda$ (resp. $M_{\lambda^*\cdot\sigma}$) is an irreducible $G$-module of highest weight $\lambda$ (resp. $\lambda^* \cdot \sigma$) for some dominant weight $\lambda$, and $M_{\lambda^*\cdot \sigma}$ is $G$-isomorphic to the dual space of $M_\lambda$ tensored with the character $\sigma$. Take a basis $f_1^{(1)},\dots, f_p^{(1)}$ of $M_\lambda$, and take a $G$-dual basis $f_1^{(2)},\dots,f_p^{(2)}$ of $M_{\sigma\cdot\lambda^*}$. The above shows that we have an \textit{expansion} (up to non-zero constant)
\begin{equation}\label{eq:expand}
f(x) = \displaystyle\sum_{i=1}^p f_i^{(1)}(x)f_i^{(2)}(x).
\end{equation}
In order to determine $M_\lambda \subset \mathbb{C}[V]_k$ for some fixed $k$, we discuss the following typical examples.
\begin{example}\label{ex:subvar}
Let $f_1^{(1)},\dots, f_p^{(1)}$ be a basis of an irreducible submodule $M_\lambda$ of $\mathbb{C}[V]_k$. If $f$ lies in the ideal generated by $f_1^{(1)},\dots, f_p^{(1)}$ in $\mathbb{C}[V]$, then we have an expansion (\ref{eq:expand}) as above. Geometrically, if $f_1^{(1)},\dots, f_p^{(1)}$ generate the (reduced) defining ideal of a closed subset of the zero-set $Z(f)$, then we have an expansion (\ref{eq:expand}) as above.
\end{example}
\begin{example}\label{ex:main}
The case considered in \cite{sasu} is when $V$ is reducible, that is, there is a non-trivial $G$-decomposition $V=E\operatornamelus F$. Then $\mathbb{C}[V]=\mathbb{C}[E]\otimes \mathbb{C}[F]$, and we can choose $M_\lambda$ (resp. $M_{\lambda^*\cdot \sigma})$ to be a $G$-irreducible isotypic component $\mathbb{C}[E]_\lambda$ (resp. $\mathbb{C}[F]_{\lambda^*\cdot \sigma}$), for a unique dominant weight $\lambda$ (see \cite[Proposition 1.6]{sasu}). We remark that in \cite{sasu} the roles of $E$ and $F$ are interchanged.
\end{example}
Since $G$ is reductive, the constructions above can be obtained for $\mathbb{C}[V^*]$ as well (see also \cite{sasu}). Namely, let $f^*\in \mathbb{C}[V^*]_d$ be the dual semi-invariant of $f$, which then has multiplicity-free weight $\sigma^{-1}$. Under the assumptions above, there exists an irreducible $G$-submodule $N_{\lambda^*}$ of $\mathbb{C}[V^*]_k$ that is $G$-isomorphic to the dual of $M_{\lambda}$, and an irreducible $G$-submodule $N_{\lambda\cdot \sigma^{-1}}$ of $\mathbb{C}[V^*]_{d-k}$ that is $G$-isomorphic to the dual of $M_{\lambda^*\cdot \sigma}$, such that $f^*$ is in the image of the map
\[N_{\lambda^*} \otimes N_{\lambda \cdot \sigma^{-1}} \to \mathbb{C}[V]_d.\]
Then we have an expansion of the form
\[f(x^*) = \displaystyle\sum_{i=1}^p f_i^{*(1)}(x^*)f_i^{*(2)}(x^*),\]
for $x^* \in V^*$. Here we can take $f_1^{*(1)},\dots, f_p^{*(1)}$ (resp. $f_1^{*(2)},\dots,f_p^{*(2)}$) to be a basis of $N_{\lambda^*}$ (resp. $N_{\lambda\cdot \sigma^{-1}}$) that is $G$-dual to $f_1^{(1)},\dots, f_p^{(1)}$ (resp. $f_1^{(2)},\dots,f_p^{(2)}$) with respect to the pairing \ref{eq:pair}.
As in \cite{sasu}, we assume that the following \textit{multiplicity one property} is satisfied: $\mathbb{C}[V]_{\lambda\cdot \sigma^{d-k-1}}=M_\lambda\cdot f^{d-k-1}$, or equivalently:
\begin{equation}\label{eq:mult1}
\mbox{The multiplicity of the irreducible $G$-module of highest weight } \lambda \cdot \sigma^{d-k-1} \mbox{ in } \mathbb{C}[V] \mbox{ is }1.
\end{equation}
We obtain the following generalization of \cite[Theorem 1.12]{sasu} (the proof is analogous):
\begin{theorem}\label{thm:mult1}
Let $f$ be a semi-invariant with multiplicity-free weight, and take an expansion (\ref{eq:expand}) as above. Assume that the multiplicity one property (\ref{eq:mult1}) holds. Then the $b$-function of $f$ decomposes as $b_f(s)=b_1(s)\cdot b_2(s)$ with:
\begin{enumerate}
\item[(1)] $\,\,\left[\displaystyle\sum_{i=1}^p f^{*(1)}_i(\partial x) f^{(1)}_i (x)\right] \cdot f^{s}(x) = b_1(s) f^s(x),$
\item[(2)] $\,\,f^{*(2)}_i(\partial x) \cdot f^{s+1}(x) = b_2(s) f^{(1)}_i (x) f^s(x)\,\,$ (for any $i=1,\dots ,p$).
\end{enumerate}
\end{theorem}
\begin{remark}\label{rem:loc}
We note that if $v$ is any element in $V$ with $f_i^{(1)}(v)\neq 0$ (for some $i$) then equation (2) above is a candidate for giving the local $b$-function of $f$ at $v$. In other words, $b_{f,v}(s) | b_2(s)$. In fact, we will see that in some situations equality holds, and that $b_2(s)$ can be itself a $b$-function of a semi-invariant of lower degree -- see Sections \ref{subsec:exam}, \ref{subsec:method}.
\end{remark}
Now we discuss the $k=1$ case for Theorem \ref{thm:mult1} in more detail:
\begin{corollary}\label{cor:euler}
Assume $(G,V)$ is an irreducible prehomogeneous vector space and $f\in \mathbb{C}[V]$ a semi-invariant of weight $\sigma$. Let $n=\dim V$ and $d=\deg f>1$, and assume the multiplicity of the irreducible representation $V^*\otimes \sigma^{d-2}$ in $\mathbb{C}[V]$ is one. Then $-n/d$ is a root of $b_f(s)/(s+1)$.
\end{corollary}
\begin{proof}
The multiplicity one property (\ref{eq:mult1}) holds, where $k=1$ and $M_\lambda=V^* = \mathbb{C}[V]_1$. By Theorem \ref{thm:mult1}, we have a decomposition $b_f(s) = b_1(s) \cdot b_2(s)$. Clearly, $-1$ is a root of $b_2(s)$, and $b_1(s)$ satisfies
\[\left(\displaystyle\sum_{i=1}^n \partial_{i} x_{i}\right) \cdot f^s = b_{1}(s) \cdot f^s.\]
The operator on the LHS equals $E+n$, where $E$ denotes the usual Euler operator. Hence, we have $b_1(s)=ds+n$, proving our claim.
\end{proof}
We note that for all irreducible prehomogeneous spaces considered in \cite{kimu}, $-n/d$ is indeed a root of the $b$-function, suggesting that the multiplicity-one property holds frequently among these (see examples in the next section).
\subsection{Examples of irreducible prehomogeneous spaces}\label{subsec:exam}
As explained in \cite[Section 3.1]{sasu}, the decomposition technique as in Example \ref{ex:main} can be used to obtain in an elementary way the $b$-functions of some classical (semi-)invariants such as the determinant and the Pfaffian. Previous proofs rely on sophisticated methods such as Capelli's identity (see \cite{howumed,proc}) or microlocal calculus (see \cite{kimu}). However, for the calculation of the $b$-function of the symmetric determinant, the technique as in Example \ref{ex:main} is not sufficient. As it turns out, considering more general expansions as (\ref{eq:expand}) is adequate for this purpose. Furthermore, in combination with methods from Section \ref{subsec:slice}, we obtain all the local $b$-functions of these classical invariants as well. For illustration, we now work out the case of the symmetric determinant and several others that do not arise from reducible representations as in Example \ref{ex:main}. These suggest that many $b$-functions of semi-invariants of prehomogeneous vector spaces can be computed with this method. Further examples will be provided for semi-invariants of quivers (Section \ref{sec:quiv}).
For the standard notation that we use for the representations below, cf. \cite{saki}.
\begin{example} \label{ex:symdet} $(\operatorname{GL}(n), 2\Lambda_1)$, the symmetric determinant.
We can think of elements $M\in V=\operatorname{Sym}^2 \mathbb{C}^n$ as symmetric matrices $M=M^t$, on which the action of $G=\operatorname{GL}(n)$ is given by $g\cdot M = gMg^t$. The semi-invariant $f$ is given by $f(M)=\det(M)$ and has degree $n$. We note that $V$ is a multiplicity-free space (cf. \cite{howumed}), i.e. $\mathbb{C}[V]$ has $G$-irreducible isotypic components. In particular, $f$ has multiplicity-free weight $\sigma=\det^2$.
We have $n+1$ orbits $\mathcal{O}_0,\mathcal{O}_1,\dots,\mathcal{O}_n$ in $V$ under the action of $G$, where $\mathcal{O}_i$ denotes the set of symmetric matrices of rank $i$. Fix any integer $k$ with $0<k<n$. The defining ideal of $\overline{\mathcal{O}}_{k-1}$ is generated by the $k\times k$ minors $f_1^{(1)},\dots, f_p^{(1)}$ of the generic symmetric matrix $X$ of variables (for example, see \cite[Theorem 6.3.1]{jerzy}), and these form a basis for an irreducible $G$-submodule $M_\lambda$ of $\mathbb{C}[V]$, where $\lambda$ is given by the partition $(2^k,0,\dots,0)$. Since $V$ is a multiplicity-free space, the multiplicity one property (\ref{eq:mult1}) holds. We have $\overline{\mathcal{O}}_{k-1} \subset Z(f)= \overline{\mathcal{O}}_{n-1}$, so by Example \ref{ex:subvar} we have a (Laplace) expansion of the form (\ref{eq:expand}). By Theorem \ref{thm:mult1}, the $b$-function of $f$ decomposes as $b_n(s)=b_{k,1}(s)\cdot b_{k,2}(s)$, and for any $i=1,\dots,p$ we have the equation
\begin{equation}\label{eq:loc}
\left(\dfrac{1}{f^{(1)}_i (x)} \, f^{*(2)}_i(\partial x) \right) \cdot f^{s+1}(x) = b_{k,2}(s) f^s(x).
\end{equation}
We can choose $f_1^{(1)}$ (resp. $f_1^{*(2)}$) to be $k\times k$ (resp. $(n-k) \times (n-k)$) minor formed by the first $k$ (resp. last $n-k$) rows and columns. We consider the equation (\ref{eq:loc}) with $i=1$, and specialize at
\[X=\begin{bmatrix}
I_k & 0\\
0 & X_{n-k}
\end{bmatrix},\]
where $X_{n-k}$ is the generic $(n-k)\times (n-k)$ matrix of respective variables. This readily gives the equation for the $b$-function of the symmetric determinant of size $(n-k)\times (n-k)$, hence we obtain $b_{k,2}(s)=b_{n-k}(s)$, and we have the decomposition
\[b_n(s) = b_{k,1}(s)\cdot b_{n-k}(s).\]
To determine $b_n(s)$ (and a fortiori, all $b_{k,1}(s)$), we consider the case $k=1$. By Corollary \ref{cor:euler} we have $b_{1,1}(s)=ns + \frac{n(n-1)}{2}$, and we can write (up to a non-zero constant)
\[b_n(s)=\left(s+\frac{n+1}{2}\right) \cdot b_{n-1}(s) = (s+1)\left(s+\frac{3}{2}\right)\cdots \left(s+\frac{n+1}{2}\right).\]
Now we show that the equations (\ref{eq:loc}) give local $b$-functions at elements in $\mathcal{O}_k$. Clearly, if $v\in \mathcal{O}_k$ then there is an $i$ such that $f_i^{(1)}(v)\neq 0$, and (\ref{eq:loc}) shows that the local $b$-function $b_{f,v}(s)$ divides $b_{k,2}(s) = b_{n-k}(s) = (s+1)(s+3/2)\cdots (s+\frac{n-k+1}{2}).$ To see that equality holds, by equivariance we have $b_{f,v} = b_{f,gv}$, for any $g\in G$, which we can denote by $b_{f,\mathcal{O}_k}$. So it is enough to consider the element $v=\begin{bmatrix} I_k & 0 \\ 0 & 0 \end{bmatrix}$. If we take the slice at $v$ as in Section \ref{subsec:slice}, we get a decomposition $V=\mathfrak{g}v \operatornamelus W$, where we can identify $W$ with the space of $(n-k)\times (n-k)$ symmetric matrices. The induced semi-invariant $f_v$ is the symmetric determinant on $W$. By Lemma \ref{lem:basic}, we have $b_{f,\mathcal{O}_k}=b_{f,v}(s) = b_{f_v}(s)=b_{n-k}(s)$, hence obtaining the desired equality. We will exploit techniques with slices more systematically in the next section.
\end{example}
\begin{example}\label{ex:ortho} $(\operatorname{SO}(m)\times \operatorname{GL}(n), \Lambda_1 \otimes \Lambda_1)$, where $m>n$.
This example is also considered in \cite{skko} (although we require only $m>n$). Here $G=\operatorname{SO}(m)\times \operatorname{GL}(n)$, where $\operatorname{SO}(m)$ denotes the special orthogonal group. We think of $V$ as the space of $m\times n$ matrices with the action of $G$ defined by $(h,g) \cdot M = h\cdot M \cdot g^t$, where $h \in \operatorname{SO}(m), g\in \operatorname{GL}(n)$ and $M\in V$. We have a semi-invariant $f_{m,n}$ defined by $f_{m,n}(M) =\det(M^t \cdot M)$ of degree $2n$ and with weight $\sigma = 1 \otimes \det^2$. Since $G$ acts on $V$ with finitely many orbits (see \cite{skko}), $\sigma$ is multiplicity-free. The orthogonal invariants are generated by the entries of $X^t \cdot X$, where $X$ denotes an $m\times n$ generic matrix of variables (see \cite[XI. Section 2.1]{proc}). In fact, this induces a $\operatorname{GL}(n)$-equivariant algebra isomorphism (see \cite[XI. Section 5.2]{proc})
\[\mathbb{C}[V]^{SO(m)} \cong \mathbb{C}[\operatorname{Sym}^2 \mathbb{C}^n].\]
In particular, we have a (Laplace) expansion (\ref{eq:expand}) as in the previous example if we take $M_\lambda$ to be the span of all the $r\times r$ minors $f_1^{(1)},\dots,f_p^{(1)}$ of $X^t\cdot X$ for any $r$ with $1\leq r \leq n-1$, where $\lambda = 1 \otimes (2^r,0,\dots,0)$. Moreover, the above isomorphism shows that the multiplicity one property (\ref{eq:mult1}) holds. By Theorem \ref{thm:mult1} the $b$-function $b_{m,n}(s)$ of $f_{m,n}$ has a decomposition $b_{m,n}(s)= b'_{r,1}(s)\cdot b'_{r,2}(s)$, where $b'_{r,2}(s)$ satisfies the equation
\[\left(\dfrac{1}{f^{(1)}_1 (x)} \, f^{*(2)}_1(\partial x) \right) \cdot f_{m,n}^{s+1}(x) = b'_{r,2}(s) f_{m,n}^s(x).\]
Here $f^{(1)}_1$ (resp. $f^{*(2)}_1$) is the $r\times r$ (resp. $(n-r)\times (n-r)$) minor formed by the first $r$ (resp. last $n-r$) rows and colums of $X^t \cdot X$ (resp. that in dual variables). Specializing the equation above at
\[X=\begin{bmatrix}
I_r & X_r\\
0 & X_{n-r}
\end{bmatrix},\]
and simplifying $f_{m,n}$, we obtain precisely the equation for the $b$-function of the semi-invariant $f_{m-r,n-r}$ in the variables of $X_{n-r}$. Hence $b'_{r,2}(s)=b_{m-r,n-r}(s)$, and we have a decomposition
\[b_{m,n}(s) = b'_{r,1}(s) \cdot b_{m-r,n-r}(s).\]
To compute $b_{m,n}(s)$ (hence, a fortiori all $b'_{r,1}(s)$ as well), we choose $r=1$. In this case $f_1^{(1)},\dots,f_p^{(1)}$ are just the entries of $X^t\cdot X$ and $f_1^{*(1)},\dots,f_p^{*(1)}$ the respective dual elements. By Theorem \ref{thm:mult1}, $b'_{1,1}(s)$ is given by
\[\left[\displaystyle\sum_{i=1}^p f^{*(1)}_i(\partial x) f^{(1)}_i (x)\right] \cdot f_{m,n}^{s}(x) = b'_{1,1}(s) f_{m,n}^s(x).\]
Since this is involves only a 2nd-order differential operator, by a direct computation we obtain (up to constant) that $b'_{1,1}(s) = (s+ \frac{n+1}{2}) (s+ \frac{m}{2})$. Hence we get
\[b_{m,n}(s) = \left(s+ \frac{n+1}{2}\right)\left(s+ \frac{m}{2} \right) \cdot b_{m-1,n-1}(s) = \prod_{i=1}^n \left(s+ \frac{i+1}{2} \right) \left(s+ \frac{m-n+i}{2}\right).\]
\end{example}
\begin{example}\label{ex:symp} $(\operatorname{Sp}(2m) \times \operatorname{GL}(2n), \Lambda_1 \otimes \Lambda_1)$, where $m>n$.
This example appears also in \cite{kimu} (although we require only $m>n$). Again, we think of $V$ as the space of $2m\times 2n$ matrices. The semi-invariant is the Pfaffian of $f(M)=\operatorname{Pf}(M^t \cdot J \cdot M)$, where $J=\begin{bmatrix} 0 & -I_m \\ I_m & 0 \end{bmatrix}$. The argument is entirely analogous to the previous example, so we omit the details. For each $r$, we obtain a decomposition of the $b$-function of $f$ as $b_{m,n}(s) = b'_{r,1}(s) \cdot b_{m-r,n-r}(s)$. Putting $r=1$, we obtain
\[b_{m,n}(s) =(s+2n-1)(s+2m)\cdot b_{m-1,n-1}(s) = \prod_{i=1}^n \left(s+ 2i-1\right)\left(s+2(m-n+i) \right).\]
\end{example}
\begin{example}\label{ex:cubic} $(\operatorname{GL}(2), 3\Lambda_1)$, the space of binary cubics.
This example appears also in \cite{skko}. Here $V=\operatorname{Sym}^3 \mathbb{C}^2$ is the space of binary cubic forms with the natural action of $G=\operatorname{GL}(2)$. If we choose $w_0,w_1$ to be a basis of $\mathbb{C}^2$, then we choose the basis $\{w_0^3, 3w_0^2 w_1, 3w_0 w_1^2, w_1^3\}$ for $V$. Let $x=(x_0, x_1, x_2,x_3)$ the respective coordinate system. The semi-invariant $f\in \mathbb{C}[V]_4$ is the discriminant
\[ f = 3x_1^2x_2^2-4x_0x_2^3-4x_1^3x_3-x_0^2x_3^2+6x_0 x_1 x_2 x_3.\]
Since $V$ has only $4$ orbits under the action of $G$, the weight $\sigma = \det^6$ is multiplicity-free. For each $k$ with $0<k<4$, we describe the expansion (\ref{eq:expand}) and show that in each case the multiplicity one property (\ref{eq:mult1}) holds. To this end, we use the $G$-decomposition of $\mathbb{C}[V]$ described as rational function in \cite[Section 6.1]{series} (we follow the notation as in \cite[Lemma 2.1]{binary})
\begin{equation}\label{eq:binpol}
\operatorname{Sym}(\operatorname{Sym}^3 \mathbb{C}^2) = \dfrac{1+(6,3)}{(1-(3,0)(1-(4,2))(1-(6,6))},
\end{equation}
where irreducible $G$-modules correspond to pairs of integers $(a,b)$ with $a\geq b$.
When $k=1$, then we have a decomposition (\ref{eq:expand}) with $M_\lambda = \mathbb{C}[V]_1$ so that $\lambda=(3,0)$ and $\lambda^* \cdot \sigma = (0,-3) + (6,6) = (6,3)$. By (\ref{eq:binpol}) we see that the multiplicity of $\lambda \cdot \sigma^2 = (3,0)+(12,12) = (15,12)$ in $\mathbb{C}[V]$ is one. Hence (\ref{eq:mult1}) holds, and by Theorem \ref{thm:mult1} we have a decomposition for the $b$-function $b(s)$ of $f$ as $b(s)=b_{1,1}(s) \cdot b_{1,2}(s)$. By Corollary \ref{cor:euler}, we have (up to a constant) $b_{1,1}(s)=s+1$. Since the equation for $b_{1,2}(s)$ involves only a 3rd-order differential operator, one can obtain by a direct calculation that $b_{1,2}(s)=(s+1)(s+5/6)(s+7/6)$.
When $k=2$. we can take $\lambda=(4,2)$ and $\lambda^* \cdot \sigma = (4,2)$. We see from (\ref{eq:binpol}) that the multiplicity of $\lambda \cdot \sigma^1= (10,8)$ in $\mathbb{C}[V]$ is one. Hence (\ref{eq:mult1}) holds, and by Theorem \ref{thm:mult1} we have a decomposition $b(s) = b_{2,1}(s) \cdot b_{2,2}(s)$. We give more details for this case. A basis of $M_\lambda=M_{\lambda^*\cdot \sigma}$ (resp. basis of $N_{\lambda^*}=N_{\lambda \cdot \sigma^{-1}}$) is given by the $2\times 2$ minors of
\[\begin{bmatrix}
x_0 & x_1 & x_2\\
x_1 & x_2 & x_3
\end{bmatrix}, \mbox{ resp. }
\begin{bmatrix}
3\partial_0 & \partial_1 & \partial_2\\
\partial_1 & \partial_2 & 3\partial_3
\end{bmatrix}.\]
We choose the basis $\{f_i^{(1)}\}$ and its the dual (up to constant) basis $\{f_i^{*(1)}\}$ with respect to the pairing (\ref{eq:pair}) as follows:
\[\arraycolsep=10pt\begin{array}{ll}
f_1^{(1)} = x_0 x_2 - x_1^2, & f_1^{*(1)} = 6 \partial_0 \partial_2 - 2\partial_1^2, \\
f_2^{(1)} = x_1 x_3 - x_2^2, & f_2^{*(1)} = 6\partial_1 \partial_3 - 2 \partial_2^2, \\
f_3^{(1)} = x_0 x_3 - x_1 x_2, & f_3^{*(1)} = 9 \partial_0 \partial_3 - \partial_1\partial_2.
\end{array}\]
Next, it is easy to see that we can make the choice $f_1^{*(2)}= 3\partial_1 \partial_3 - \partial_2^2$. Now by a direct computation we obtain by Theorem \ref{thm:mult1} that (up to constant) $b_{2,1}(s)=(s+1)(s+5/6)$ and $b_{2,2}(s)=(s+1)(s+7/6)$.
When $k=3$, we have the same expansion for $f$ as with $k=1$, but with the roles of $\lambda$ and $\lambda^*\cdot \sigma$ interchanged. Namely, now $\lambda=(6,3)$ and $\lambda^*\cdot \sigma = (3,0)$. It is easy to see from (\ref{eq:binpol}) that the multiplicity of $\lambda=(6,3)$ in $\mathbb{C}[V]$ is one, hence (\ref{eq:mult1}) holds. Again, by Theorem \ref{thm:mult1} we have a decomposition $b(s)=b_{3,1}(s)\cdot b_{3,2}(s)$, and it is immediate that $b_{3,2}(s)=s+1$, hence $b_{3,1}=(s+1)(s+5/6)(s+7/6)$.
\end{example}
\begin{example} $(\operatorname{GL}(6), \bigwedge^3 \mathbb{C}^6)$
This example appears in \cite{skko} and is very similar to the one above, so we omit the details. There exists a semi-invariant $f$ of degree $4$. The $G$-decomposition of $\mathbb{C}[V]$ is described in \cite[Section 6]{series}. Using this, it is easy to see that the multiplicity one property (\ref{eq:mult1}) holds for all cases $k=1,2,3$, just as in the above example. Hence one can apply Theorem \ref{thm:mult1} here as well and obtain decompositions of the $b$-function of $f$.
\end{example}
\subsection{The slice method}\label{subsec:method}
In general, the multiplicity one property (\ref{eq:mult1}) is not easy to check directly. Several criteria are given in \cite[Section 2]{sasu}, but these are not sufficient for our purposes. Indeed, the authors in \cite{sasu} bring attention to the problem of finding a more satisfactory criterion for the multiplicity one property to hold. Although difficult to answer in general, using slices as in Section \ref{subsec:slice} we derive an efficient criterion that is relatively easy to use. We call this process the slice method.
For the standard theory of reductive groups that we use, we refer the reader to \cite{borel}. Assume $G$ is a connected reductive group, $T$ a maximal torus of $G$ and $B$ a Borel subgroup and $B^-$ an opposite Borel subgroup so that $B\cap B^- = T$. In this section, $V$ is a rational $G$-module with a $G$-decomposition $V=E\operatornamelus F$ as in Example \ref{ex:main}. We have an algebra isomorphism $\mathbb{C}[V]=\mathbb{C}[E]\otimes \mathbb{C}[F]$. As explained before, for $f\in \operatorname{SI}(G,V)_\sigma$ with $\sigma$ multiplicity-free, we can write
\begin{equation}\label{eq:decomp}
f(x,y) = \displaystyle\sum_{i=1}^p f_i^{(1)}(x)f_i^{(2)}(y),
\end{equation}
for $x\in E, y\in F$, where $f_1^{(1)},\dots, f_p^{(1)}$ is a basis for a $G$-irreducible isotypic component $\mathbb{C}[E]_\lambda$, and $f_1^{(2)},\dots,f_p^{(2)}$ is a $G$-dual basis for the irreducible $\mathbb{C}[F]_{\sigma \cdot \lambda^*}$, for some dominant weight $\lambda$.
We can assume WLOG that $f_1^{(1)},\dots, f_p^{(1)}$ is a $T$-weight basis of $\mathbb{C}[E]_\lambda$ and $f_1^{(1)}$ is the highest weight vector, that is, $f_1^{(1)}$ is a $B$-semi-invariant of weight $\lambda$. Then $f_1^{(2)},\dots,f_p^{(2)}$ is a $T$-weight basis of $\mathbb{C}[F]_{\sigma \cdot \lambda^*}$, and $f_1^{(2)}$ is a lowest weight vector, that is, a $B^-$-semi-invariant of weight $\lambda^{-1} \cdot \sigma$. For simplicity, put $f^{(1)}:=f_1^{(1)}$ and $f^{(2)}:=f_1^{(2)}$.
Let $f^*\in \mathbb{C}[V^*]$ be the dual of $f$, which is a semi-invariant of multiplicity-free weight $\sigma^{-1}$. We have an algebra isomorphism $\mathbb{C}[V^*]=\mathbb{C}[E^*]\otimes \mathbb{C}[F^*]$, and we can write
\[f(x^*,y^*) = \displaystyle\sum_{i=1}^p f_i^{*(1)}(x^*)f_i^{*(2)}(y^*),\]
for $x^*\in E^*,y^*\in F^*$. Here $f_1^{*(1)},\dots, f_p^{*(1)}$ (resp. $f_1^{*(2)},\dots,f_p^{*(2)}$) is the dual basis of $f_1^{(1)},\dots, f_p^{(1)}$ (resp. $f_1^{(2)},\dots,f_p^{(2)}$) with respect to (\ref{eq:pair}). In particular, $f^{*(2)}:=f^{*(2)}_1$ is a highest weight vector, that is, a $B$-semi-invariant of weight $\lambda \cdot \sigma^{-1}$.
Since $f^{(1)}\in \mathbb{C}[E]$ is a highest weight vector with dominant weight $\lambda$, the stabilizer of the line $\mathbb{C} \cdot f^{(1)}$ is a parabolic subgroup $P$ of $G$. Moreover, since $f^{(2)}$ is a lowest weight vector of weight $\sigma\cdot\lambda^{-1}$, the opposite parabolic subgroup $P^-$ is the stabilizer of the line $\mathbb{C} \cdot f^{(2)}$. We have $P\cap P^-=L$, where $L$ is the Levi subgroup of $P$, which is a connected reductive group.
We assume that we have an element $v\in E$ such that $f^{(1)}(v)=1$, and $f^{(1)}_i(v)=0$, for $i\neq1$. Additionally, we assume that $v$ has a dense $P$-orbit in $E$ (for example, when the action of $G$ on $E$ is multiplicity-free, i.e. $E$ has a dense $B$-orbit -- see \cite{howumed}). With notation from Section \ref{subsec:slice} (choosing $H=P$), we have a decomposition $V=\mathfrak{p}v \operatornamelus F$, and we consider the (slice) representation $(L,F)$ at $v$. Putting $x=v$ in (\ref{eq:decomp}) we get that $f_v = f^{(2)}$ is an $L$-semi-invariant on $F$ of weight $(\lambda^{-1} \cdot \sigma)|_{L}$ (restriction to $L$). If $L_v$ denotes the stabilizer of $v$, then we will see that $f_v$ is an $L_v$-semi-invariant of weight $\sigma|_{L_v}$.
\begin{theorem}\label{thm:maintheor}
Let $v\in E$ as above, assume that the weight $\sigma|_{L_v}$ in $\mathbb{C}[F]$ is multiplicity-free. Then the multiplicity one property (\ref{eq:mult1}) holds in $\mathbb{C}[V]$, and we have a decomposition $b_f(s) = b_1(s)\cdot b_2(s)$ as in Theorem \ref{thm:mult1} with $b_2(s)=b_{f_v}(s)=b_{f,v}(s)$.
\end{theorem}
\begin{proof}
First, we show that $\lambda | _{L_v} = 1$. The polynomial $f^{(1)}\in\mathbb{C}[E]$ is an $L_v$-semi-invariant of weight $\lambda | _{L_v}$. In particular, we have $ (l \cdot f^{(1)})(v) = \lambda(l) f^{(1)}(v) = \lambda(l)$, for any $l \in L_v$. On the other hand, we have $(l \cdot f^{(1)})(v)=f^{(1)}(l^{-1} v) = f^{(1)}(v) =1$, hence $\lambda | _{L_v} = 1$. This implies that $f_v$ is an $L_v$-semi-invariant of weight $\sigma|_{L_v}$.
Now we show that property (\ref{eq:mult1}) holds. In fact, we prove that the multiplicity in $\mathbb{C}[V]$ of the irreducible corresponding to $\lambda \cdot \sigma^k$ is one, for any $k\in\mathbb{N}$. As noted in Remark \ref{rem:works}, the considerations in Section \ref{subsec:slice} work for the slice representation $(L_v,F)$ with the group $L_v$ (although this group is defined in a different way than the one defined in that section). Using that $\lambda | _{L_v} = 1$, the map (\ref{eq:slice}) in this case is
\[\phi_v^{\lambda\cdot\sigma^k} : \operatorname{SI}(P,V)_{\lambda\cdot\sigma^k} \to \operatorname{SI}(L_v,F)_{\sigma^k |_{L_v}}.\]
Since the weight of $\sigma|_{L_v}$ is multiplicity-free, the space $\operatorname{SI}(L_v,F)_{\sigma^k |_{L_v}} = \mathbb{C} \cdot f_v^k$ is one-dimensional. By Lemma \ref{lem:basic}, $\phi_v^{\lambda\cdot\sigma^k}$ is injective, hence $\operatorname{SI}(P,V)_{\lambda\cdot\sigma^k}=\mathbb{C} \cdot (f^{(1)}f^k)$. This implies that the multiplicity one property (\ref{eq:mult1}) holds. By Theorem \ref{thm:mult1} (2) we have an equation
\begin{equation}\label{eq:eval}
f^{*(2)}(\partial y) \cdot f^{s+1}(x,y) = b_2(s) f^{(1)}(x) f^s(x,y).
\end{equation}
Since $\sigma|_{L_v}$ is a multiplicity-free weight and $\lambda | _{L_v} = 1$, the $L$-semi-invariant $f_v$ has multiplicity-free weight $(\lambda^{-1} \cdot \sigma)|_{L}$ for the reductive group $L$ (in particular, $f$ is homogeneous). Recall that $f^{*(2)}$ has highest weight $\lambda\cdot \sigma^{-1}$, hence it is an $L$-semi-invariant of weight $(\lambda\cdot \sigma^{-1})|_L$. This shows that (up to constant) $f^{*(2)}$ is the dual $L$-semi-invariant of $f_v$ on $F$, that is $f_v^* = f^{*(2)}$. Now specializing at $x=v$ in the equation (\ref{eq:eval}) we obtain
\[f^*_v (\partial y) \cdot f_v^{s+1} (y) = b_2(s) f^s_v(y).\]
By Theorem \ref{thm:bfun} this equation gives precisely the $b$-function of $f_v$, so $b_2(s)=b_{f_v}(s)$. To see that equation (\ref{eq:eval}) gives indeed the local $b$-function of $f$ at $v$, we use Lemma \ref{lem:basic} again and obtain $b_{f_v}(s)=b_{f,v}(s)$.
\end{proof}
In the case of multiplicity-free tuples as in Definition \ref{def:tuple}, we can say more about the first factor $b_1(s)$ in Theorem \ref{thm:maintheor}:
\begin{lemma}\label{lem:firstfac}
Assume additionally, that $\underline{f}=(f_1^{(1)},\dots, f_p^{(1)})$ from (\ref{eq:decomp}) is a multiplicity-free tuple. Then $b_1(s)=P_{\underline{f}}(s)$ , with $P_{\underline{f}}(s)$ as in Proposition \ref{prop:pf}. In particular, the Bernstein-Sato polynomial $b_{\underline{f}}(s)$ divides $b_1(s)$.
\end{lemma}
\begin{proof}
It is enought to show $b_1(s)=P_{\underline{f}}(s)$ for an arbitrary positive integer $s$. Denote by $M_s$ the irreducible $G$-module as in Definition \ref{def:tuple}. By Theorem \ref{thm:mult1}, $b_1(s)$ is given by the equation
\[\left[\displaystyle\sum_{i=1}^p f^{*(1)}_i(\partial x) f^{(1)}_i (x)\right] \cdot f^{s}(x,y) = b_1(s) f^s(x,y).\]
We can evaluate the equation at any point $y=w\in F$. Choose $w\in F$ such that the polynomial $f(x,w)\in \mathbb{C}[E]$ is not zero. By the expansion (\ref{eq:decomp}), we see that $f^s(x,w) \in M_s$, for any $w\in F$. By Schur's Lemma and Proposition \ref{prop:pf}, $D_{\underline{f}}$ acts on $M_s$ by the scalar $P_{\underline{f}}(s)$, which then coincides with $b_1(s)$ by the equation above.
\end{proof}
Now we formulate a result for the important case when the representation $(G,V)$ is of the form
\begin{equation}\label{eq:matrix}
(\operatorname{GL}(m)\times \operatorname{GL}(n) \times G'\,,\, \Lambda_1^{(*)} \otimes \Lambda_1^{(*)} \otimes 1 \operatornamelus 1 \otimes \rho\,,\, M_{m,n}\operatornamelus F)
\end{equation}
with $m\leq n$. This is the main case considered also in \cite{sasu} and \cite{sugi}, and we use the notation as in \cite[Section 2.1]{sasu}. Namely, here $G'$ is an arbitary connected reductive group, $ \rho$ is an arbitary rational representation of $\operatorname{GL}(n) \times G'$, and $\Lambda_1^{(*)}$ is either the standard representation of $\operatorname{GL}$ or its dual (for simplicity, we take WLOG the duals $\Lambda_1^{*}$). Many prehomogeneous vector spaces are of this form -- see Sections \ref{sec:quiv} and the classification in \cite{saki}.
We define
\[H=\operatorname{GL}(m) \times \operatorname{GL}(n-m) \times G'\]
to be the the reductive subgroup of $\operatorname{GL}(n) \times G' \subset G$ , with the factor $\operatorname{GL}(m) \times \operatorname{GL}(n-m)$ of $H$ embeds into $\operatorname{GL}(n)$ as
\[\operatorname{GL}(m) \times \operatorname{GL}(n-m) = \left\{\begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix} : \mbox{ where } A\in \operatorname{GL}(m), B\in \operatorname{GL}(n-m) \right\}\subset \operatorname{GL}(n).\]
Let $I \subset \mathbb{C}[M_{m,n}]$ denote the ideal generated by the maximal minors as introduced in Section \ref{subsec:bideal}. Choose $v=\left(\begin{bmatrix}I_m & 0_{n-m} \end{bmatrix},0\right) \in M_{m,n}\operatornamelus F$.
\begin{theorem} \label{thm:mainapp}
Consider the space $V=M_{m,n}\operatornamelus F$ as in (\ref{eq:matrix}) and let $f\in \mathbb{C}[V]$ be a $G$-semi-invariant of weight $\sigma=\det^d \otimes \det^e \otimes \sigma'$, where $d,e\in \mathbb{N}$ and $\sigma'$ is a character of $G'$. Assume that $\det^{e-d} \otimes \det^e \otimes \sigma'$ is a multiplicity-free character of $H$ in $\mathbb{C}[F]$. Then $\sigma$ is a multiplicity-free character of $G$ in $\mathbb{C}[V]$ and the multiplicity one property (\ref{eq:mult1}) holds. Moreover, the Bernstein-Sato polynomial of $f$ decomposes as $b_f(s)=b_1(s) \cdot b_2(s)$ where:
\begin{itemize}
\item[(1)] $b_1(s) =b_{I^{d}}(s) = [s]_{n-m,n}^{d}$ is the Bernstein-Sato polynomial of the ideal $I^{d}$;
\item[(2)] $b_2(s) = b_{f,v}(s) = b_{f_v}(s)$ is the Bernstein-Sato polynomial of the induced semi-invariant $f_v$ on the slice $(H,F)$, which is also equal to the local Bernstein-Sato polynomial of $f$ at $v$.
\end{itemize}
\end{theorem}
\begin{proof}
The stabilizer $G_v$ of $v$ is formed by all elements of the form
\[\left(A^{-1}, \begin{bmatrix} A & 0 \\ C & B \end{bmatrix}, g\right) \subset \operatorname{GL}(n) \times \operatorname{GL}(m)\times G'.\]
Let $L_v$ be the reductive subgroup of $G_v$ formed by the elements as above with $C=0$. Clearly, $L_v$ is isomorphic to $H$ (by forgetting the first factor).
We have $\mathfrak{g}v=M_{m,n}$. As in Section \ref{subsec:slice}, we consider the map from $\mathbb{C}[V]$ to $\mathbb{C}[F]$ given by $h \mapsto h_v$, where $h\in \mathbb{C}[V]$, and $h_v\in\mathbb{C}[F]$ is defined by $h_v(y)=h(v+y)$, for $y\in F$. Fix $k\in\mathbb{N}$ and assume $h\in \mathbb{C}[V]$ is a $G$-semi-invariant of weight $\sigma^k$. As seen in Section \ref{subsec:slice}, $h_v\in\mathbb{C}[F]$ is then an $L_v$-semi-invariant of weight $\sigma^k|_{L_v}=(\det^{e-d} \otimes \det^e \otimes \sigma')^k$. Since the first factor $\operatorname{GL}(n)$ of $G$ acts on $F$ trivially, in fact $h_v$ is also an $H$-semi-invariant of weight $(\det^{e-d} \otimes \det^e \otimes \sigma')^k$. This shows that we have a map as (\ref{eq:slice}):
\[\phi_v^{\sigma^k} : \operatorname{SI}(G,V)_{\sigma^k} \to \operatorname{SI}(H,F)_{(\det^{e-d} \otimes \det^e \otimes \sigma')^k}.\]
Since $\det^{e-d} \otimes \det^e \otimes \sigma'$ is multiplicity-free, $ \operatorname{SI}(H,F)_{(\det^{e-d} \otimes \det^e \otimes \sigma')^k}=\mathbb{C} \cdot f_v^k$ is one-dimensional. By Lemma \ref{lem:basic} (taking into account Remark \ref{rem:works}) the map $\phi_v^{\sigma^k}$ is injective, hence $\operatorname{SI}(G,V)_{\sigma^k}=\mathbb{C} \cdot f^k$ and $\sigma$ is multiplicity-free. In particular, we have an expansion of the form (\ref{eq:decomp}). By FFT (see \cite[XI. Section 1.2]{proc}), the elements $f^{(1)}_1,\dots, f^{(1)}_p$ are a basis of the irreducible $\operatorname{GL}(m)\times\operatorname{GL}(n)$-module $M_\lambda$ of $\mathbb{C}[M_{m,n}]$, where the dominant weight is $\lambda = \det^d \otimes (d^m, 0^{n-m}) \otimes 1$ (see also \cite[Section 2.1]{sasu}). We choose $f^{(1)}_1,\dots, f^{(1)}_p$ to be elements that are products of $d$ maximal minors. We can take $f_1^{(1)}$ to be the $d$th power of the maximal minor corresponding to the first $m$ columns, which, by a standard choice of a Borel subgroup $B$ of $\operatorname{GL}(m)\times\operatorname{GL}(n)$, is highest weight vector. Note that under this choice the $B$-orbit of $v$ is dense in $M_{m,n}$. Also, $f_1^{(1)}(v)=1$, while $f_i^{(1)}(v) = 0$, for $i\neq 1$, so $f^{(2)}_1 = f_v$ (see the considerations before Theorem \ref{thm:maintheor}). Let $P$ be the parabolic subgroup of $G$ corresponding to $\lambda$, i.e. the stabilizer of the line $\mathbb{C} \cdot f_1^{(1)}$, and let $L$ be the corresponding Levi subgroup. Then it is easy to see that the stabilizer of $v$ in $L$ is the same as the group $L_v$ constructed above. Since $\sigma'|_H$ is multiplicity-free, $\sigma|_{L_v}$ is multiplicity-free on $F$ as well.
We showed that all the assumptions in Theorem \ref{thm:maintheor} are satisfied. This, together with Lemma \ref{lem:firstfac} and Theorem \ref{thm:powermax}, yields the conclusion.
\end{proof}
The technique can be used to determine an explicit representative for the locally semi-simple point of $f$ (see Section \ref{subsec:bfun}).
\begin{proposition}\label{prop:locsemi}
Consider a semi-invariant $f\in \mathbb{C}[V]$ as in Theorem \ref{thm:mainapp} with $d\neq 0$. Let $w\in F$ be the locally semi-simple point of $f_v \in \operatorname{SI}(H,F)$. Then $v+w$ is the locally semi-simple point of $f\in \operatorname{SI}(G,V)$.
\end{proposition}
\begin{proof}
Take any $z \in V$ such that $f(z)\neq 0$. We want to show that $v+w \in \overline{Gz}$. Since $d\neq 0$, the orbit $G\cdot z$ has an element the form $v+w'$, where $w' \in F$. Since $f_v(w') = f(v+w') \neq 0$ and $w$ is the locally semi-simple point of $f_v$, we must have that $w \in \overline{H\cdot w'}=\overline{L_v \cdot w'}$. Since $L_v$ fixes $v$, we have $v+w \in \overline{L_v \cdot (v+w')}$. This shows that $v+w$ is in the closure of the $G$-orbit of $z$.
\end{proof}
As seen in the proof of Theorem \ref{thm:mainapp}, the stabilizer $G_v$ of $v$ decomposes as a semi-direct product $G_v = L_v \ltimes U$, where $U\cong \operatorname{Hom}_\mathbb{C}(\mathbb{C}^m, \mathbb{C}^{n-m})$ is a unipotent subgroup.
\begin{proposition}\label{prop:algiso}
Consider the space $V=M_{m,n} \operatornamelus F$ as in (\ref{eq:matrix}). Then the map $\phi_v$ from (\ref{eq:slice}) induces an isomorphism of algebras
\[ \phi_v : \operatorname{SI}(G,V) \cong \operatorname{SI}(H,F)^{U}.\]
\end{proposition}
\begin{proof}
By Lemma \ref{lem:basic}, $\phi_v$ is injective on the level of weight spaces. A $G$-semi-invariant of weight $\det^d \otimes \det^e \otimes \sigma'$ is mapped to an $H$-semi-invariant of weight $\det^{e-d} \otimes \det^e \otimes \sigma'$. Since $m<n$, this shows that different weight spaces are mapped to different weights spaces, so $\phi_v$ is injective.
Now we show that $\phi_v$ is surjective. Let $f' \in \operatorname{SI}(H,F)^U$ be an $H$-semi-invariant of weight $\det^{a} \otimes \det^b \otimes \sigma'$, for some $a,b\in \mathbb{Z}$ and character $\sigma'$ of $G'$. Consider the character $\sigma$ of $G$ defined as $\sigma = \det^{b-a} \otimes \det^b \otimes \sigma'$. Consider the function $F$ defined on the open set $G\cdot v \times F \subset V$ by
\[ F(g\cdot v, y) = \sigma(g) \cdot f'(g^{-1} \cdot y), \mbox{ for } g\in G, y\in F.\]
Using that $f'$ is $G_v$-semi-invariant, we see that $F$ is a well-defined semi-invariant of weight $\sigma$. Since $m<n$, the open set $G\cdot v \times F$ has codimension $\geq 2$ in $V$. Hence $F$ extends to a global semi-invariant, and $F_v = f'$.
\end{proof}
\begin{remark}
We note that the results above regarding $b$-functions hold for the case $m=n$ in (\ref{eq:matrix}) as well. Moreover, in this case there is an algebra isomorphism analogous to Proposition \ref{prop:algiso}
\[\phi_v: \operatorname{SI}(G,V) / (\det X - 1) \cong \operatorname{SI}(H,F),\]
where $X$ is the generic matrix of variables on $M_{n,n}$. For results in this direction obtained by slicing at elements other then our choice $v$, cf. \cite[Section 4]{phd}.
\end{remark}
We conclude the section by mentioning that most results for $b$-functions of one variable can be extended readily to the case of $b$-functions of \textit{several} variables as in Lemma \ref{lem:several}. We will mention only the extension of Theorem \ref{thm:mainapp} to this case, the proof of which is analogous, \textit{mutatis mutandis}.
\begin{theorem}\label{thm:multi}
Consider the space $V=M_{m,n}\operatornamelus F$ as in (\ref{eq:matrix}), and let $\underline{f}=(f_1,\dots, f_l)$ be $G$-semi-invariants in $\mathbb{C}[V]$ of weights $\sigma_1,\dots, \sigma_l$, respectively, where $\sigma_i = \det^{d_i} \otimes \det^{e_i} \otimes \sigma'_i$, for $i=1,\dots,l$, with $d_i,e_i\in \mathbb{N}$ and $\sigma'_i$ a character of $G'$. Assume the product $\prod_{i=1}^l \det^{e_i-d_i} \otimes \det^{e_i} \otimes \sigma'_i$ is a multiplicity-free character of $H$ in $\mathbb{C}[F]$. Then the product $\sigma_1 \cdots \sigma_l$ is a multiplicity-free character of $G$ in $\mathbb{C}[V]$. Moreover, the $b$-function of several variables decomposes as
\[b_{\underline{f},\underline{m}}(\underline{s}) = [s]_{n-m,n}^{d_1,\dots,d_l} \cdot b_{\underline{f}_v,\underline{m}}(\underline{s}),\]
for any tuple $\underline{m}=(m_1,\dots,m_l)\in \mathbb{N}^l$, where $b_{\underline{f}_v,\underline{m}}(\underline{s})$ is the $b$-function of several variables of the tuple $\underline{f}_v=(f_{1,v},\dots, f_{l,v})$ of induced semi-invariants on the slice $(H,F)$.
\end{theorem}
\section{Semi-invariants of quivers and the slice method}
\label{sec:quiv}
In this section we apply the methods the slice method from Section \ref{subsec:method} to semi-invariants of quivers.
\subsection{Background on quivers and their semi-invariants}\label{subsec:backquiv}
In this section we will introduce some basics of quivers and semi-invariants. For more background material, we refer the reader to \cite{elements,harwey}. We follow similar notation to that in \cite{en}.
A quiver $Q$ is an oriented graph, i.e. a pair $Q=(Q_0,Q_1)$ formed by the set of vertices $Q_0$ and the set of arrows $Q_1$. An arrow $a$ has a head $ha$, and tail $ta$, that are elements in $Q_0$:
\[\xymatrix{
ta \ar[r]^{a} & ha
}\]
We assume in throughout that $Q$ is a quiver \itshape without oriented cycles \normalfont.
A representation $V$ of $Q$ is a family of finite dimensional vector spaces $\{V(x)\,|\, x\in Q_0\}$ together with linear maps $\{V(a) : V(ta)\to V(ha)\, | \, a\in Q_1\}$. The dimension vector $\underline{d} (V)\in \mathbb{N}^{Q_0}$ of a representation $V$ is the tuple $\mathbb{D}im V:=(\dim V(x))_{x\in Q_0}$. A morphism $\phi:V\to W$ of two representations is a collection of linear maps $\phi = \{\phi(x) : V(x) \to W(x)\,| \,x\in Q_0\}$, with the property that for each $a\in Q_1$ we have $\phi(ha)V(a)=W(a)\phi(ta)$. Denote by $\operatorname{Hom}_Q(V,W)$ the vector space of morphisms of representations from $V$ to $W$. For two vectors $\alpha, \beta\in \mathbb{Z}^{Q_0}$, we define the Euler product
$$\langle \alpha, \beta \rangle = \displaystyle\sum_{x\in Q_0} \alpha_x \beta_x - \displaystyle\sum_{a\in Q_1} \alpha_{ta} \beta_{ha}.$$
Let $E$ denote the Euler matrix corresponding to the Euler product. Then $C=-E^{-1} \cdot E^t$ is the \textit{Coxeter transformation} of $Q$ (see \cite{elements}).
We define the vector space of representations with dimension vector $\alpha\in \mathbb{N}^{Q_0}$ by
$$\operatorname{Rep}(Q,\alpha):=\displaystyle\bigoplus_{a\in Q_1} \operatorname{Hom}(\mathbb{C}^{\alpha_{ta}},\mathbb{C}^{\alpha_{ha}}).$$
The group
$$\operatorname{GL}(\alpha):= \prod_{x\in Q_0} \operatorname{GL}(\alpha_x)$$
acts on $\operatorname{Rep}(Q,\alpha)$ in a natural way by changing basis at each vertex. Under this action, two representations lie in the same orbit if and only if they are isomorphic representations.
For any two representations $V$ and $W$, we have the following exact sequence:
\begin{equation}\label{eq:ringel}
\begin{array}{rlc}
0 \to \operatorname{Hom}_Q (V,W) \stackrel{i}{\longrightarrow} \displaystyle\bigoplus_{x \in Q_0}& \!\!\!\!\!\operatorname{Hom}(V(x),W(x)) & \\
& \stackrel{d^V_W}{\longrightarrow} \displaystyle\bigoplus_{a\in Q_1} \operatorname{Hom}(V(ta),W(ha)) \stackrel{p}{\longrightarrow} \operatorname{Ext}_Q(V,W)\to 0 &
\end{array}
\end{equation}
Here, the map $i$ is the inclusion, $d_W^V$ is given by
$$\{\phi(x)\}_{x\in Q_0} \mapsto \{\phi(ha)V(a) - W(a)\phi(ta)\}_{a\in Q_1}$$
and the map $p$ builds an extension of $V$ and $W$ by adding the maps $V(ta)\to W(ha)$ to the direct sum $V\operatornamelus W$. From the exact sequence (\ref{eq:ringel}) we have that
\[\langle \mathbb{D}im V,\mathbb{D}im W \rangle = \dim \operatorname{Hom}_Q(V,W) - \dim \operatorname{Ext}_Q(V,W).\]
The orbit $\mathcal{O}_V$ is dense in $\operatorname{Rep}(Q,\alpha)$ if and only if $\operatorname{Ext}_Q(V,V)=0$, in which case we say that $V$ is a \textit{generic} representation, and $\alpha$ a \textit{prehomogeneous dimension vector}.
Now we turn to semi-invariants of a quiver representation space $\operatorname{Rep}(Q,\beta)$. As in Section \ref{sec:b-func}, form the ring of semi-invariants $\operatorname{SI}(Q,\beta)\subset \mathbb{C}[\operatorname{Rep}(Q,\beta)]$ by
\[\operatorname{SI}(Q,\beta)=\bigoplus_\sigma \operatorname{SI}(Q,\beta)_\sigma.\]
Here $\sigma$ runs through all the characters of $\operatorname{GL}(\beta)$. Each character $\sigma$ of $\operatorname{GL}(\beta)$ is a product of determinants, that is, of the form
$\prod_{x\in Q_0} \operatorname{det}_x^{\sigma(x)},$ where $\operatorname{det}_x$ is the determinant function on $\operatorname{GL}(\beta_x)$. In this way, we will view a character $\sigma$ as a function $\sigma : Q_0 \to \mathbb{Z}$, or equivalently, as an element $\sigma\in \operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^{Q_0},\mathbb{Z})$. With this convention, we view characters as duals to dimension vectors, namely:
$$\sigma(\beta)=\displaystyle\sum_{x\in Q_0} \sigma(x)\beta_x.$$
We recall the definition of an important class of determinantal semi-invariants, first constructed by Schofield in \cite{scho}. Fix two dimension vectors $\alpha,\beta$, such that $\langle \alpha, \beta \rangle=0$. The latter condition says that for every $V\in \operatorname{Rep}(Q,\alpha)$ and $W\in \operatorname{Rep}(Q,\beta)$ the matrix of the map $d^V_W$ in (\ref{eq:ringel}) will be a square matrix. We define the semi-invariant $c$ of the action of $\operatorname{GL}(\alpha)\times\operatorname{GL}(\beta)$ on $\operatorname{Rep}(Q,\alpha)\times \operatorname{Rep}(Q,\beta)$ by $c(V,W):=\det d^V_W$. Note that we have
$$c(V,W)=0 \Leftrightarrow \operatorname{Hom}(V,W)\neq 0 \Leftrightarrow \operatorname{Ext}(V,W)\neq 0.$$
Next, for a fixed $V$, restricting $c$ to $\{V\} \times \operatorname{Rep}(Q,\beta)$ defines a semi-invariant $c^V\in \operatorname{SI}(Q,\beta)$. Similarly, for a fixed $W$, restricting $c$ to $\operatorname{Rep}(Q,\alpha)\times \{W\}$, we get a semi-invariant $c_W\in \operatorname{SI}(Q,\alpha)$. The weight of $c^V$ is $\langle \alpha, \cdot \rangle \in \operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}^{Q_0},\mathbb{Z})$, and the weight of $c_W$ is $-\langle \cdot, \beta \rangle$. The semi-invariants $c^V$ and $c_W$ are well-defined up to scalar, that is, if $V$ is isomorphic to $V'$, then $c^V$ and $c^{V'}$ are equal up to a scalar.
\begin{theorem}[{\cite{harwey,schovan}}]\label{thm:span}
For a fixed dimension vector $\beta$, the ring of semi-invariants $\operatorname{SI}(Q,\beta)$ is spanned by the semi-invariants $c^V$, with $\langle \mathbb{D}im V,\beta \rangle=0$. The analogous result holds for the semi-invariants $c_W$.
\end{theorem}
By \cite[Lemma 1]{harwey}, the algebra of semi-invariants $\operatorname{SI}(Q,\beta)$ is generated by semi-invariants $c^V$, with $\langle \mathbb{D}im V, \cdot \rangle=0$ and $V$ a Schur representation (that is, $\operatorname{End}_Q(V)=\mathbb{C}$). We call a prehomogeneous dimension vector $\alpha$ a \itshape real Schur root\normalfont, if the generic representation $V\in\operatorname{Rep}(Q,\alpha)$ is a Schur representation. Note that in this case we have $\langle \alpha, \alpha \rangle=1$. Examples of real Schur roots include the dimension vectors of preprojective and preinjective representations (see \cite{en}).
In the case $\beta$ is a prehomogeneous dimension vector, $\operatorname{SI}(Q,\beta)$ a polynomial ring generated by semi-invariants $c^{V_i}$, where $V_i$ are the simple objects in an appropriate perpendicular category (see \cite[Theorem 4.3]{scho}).
To find semi-invaraints with multiplicity-free weights on spaces $\operatorname{Rep}(Q,\beta)$ with $\beta$ not necessarily prehomogeneous, the following reciprocity result is useful:
\begin{theorem}[{\cite[Corollary 1]{harwey}}]
\label{thm:recip}
Let $\alpha$ and $\beta$ be two dimension vectors, with $\langle \alpha,\beta \rangle=0$. Then
$$\dim \operatorname{SI}(Q,\beta)_{\langle \alpha, \cdot \rangle} = \dim \operatorname{SI}(Q,\alpha)_{-\langle \cdot , \beta \rangle}.$$
\end{theorem}
In particular, if $f$ is a non-zero semi-invariant of weight $\langle \alpha , \cdot \rangle$, with $\alpha$ prehomogeneous, then $f=c^V \in \operatorname{SI}(Q,\beta)$ has multiplicity-free weight, where $V$ is the generic representation in $\operatorname{Rep}(Q,\alpha)$.
\begin{remark}\label{rem:fulton}
By the proof of the Generalized Fulton Conjecture (see \cite[Theorem 2.22]{combi}), in order to show that a character $\sigma$ is multiplicity-free in $\operatorname{SI}(Q,\beta)$, it is enough to show that $\dim \operatorname{SI}(Q,\beta)_\sigma = 1$ (i.e. one does not need to check this for higher powers of $\sigma$).
\end{remark}
One can write down the semi-invariants $c^V$ explicitly as determinants of suitable block matrices (see \cite[Remark 3.3]{en}).
\begin{example}
\label{eq:deefor}
Let $Q$ be the following $\mathbb{D}_4$ quiver:
\[\xymatrix{
& 2 \ar[d] & \\
1 \ar[r] & 4 & \ar[l] 3
}\]
Let $V$ be the indecomposable $V=\begin{matrix} & 1 & \\ 1 & 1 & 1\end{matrix}$. Then $\langle \alpha, \beta \rangle=0$ gives $\beta=(\beta_1,\beta_2,\beta_3,\beta_4)$ with $\beta_1+\beta_2+\beta_3=2\beta_4$. Let $X,Y,Z$ be generic matrices of variables, with $X\in M_{\beta_4,\beta_1},Y\in M_{\beta_4,\beta_2},Z\in M_{\beta_4,\beta_3}$. Then $c^V$ is the determinant of the following square matrix of variables:
$$\det\begin{pmatrix}
X & 0 & 0 & I_{\beta_4}\\
0 & Y & 0 & I_{\beta_4}\\
0 & 0 & Z & I_{\beta_4}
\end{pmatrix}=\det\begin{pmatrix}
X & Y & 0 \\
0 & Y & Z
\end{pmatrix}.$$
Also, $c^V\neq 0$ if and only if $\beta_i\leq \beta_4$, for $i=1,2,3$, and $c^V$ is irreducible if and only if all these inequalities are strict.
\end{example}
In general, slicing a quiver results in a more complicated quiver. However, in some cases we can view a semi-invariant of a quiver as a function on a simpler quiver.
\begin{lemma}[{\cite[Lemma 3.4]{en}}]\label{lem:simp}
Let $Q$ be a quiver without oriented cycles, $\beta$ a dimension vector and $f$ a semi-invariant on $\operatorname{Rep}(Q,\beta)$ of weight $\sigma=\langle \alpha , \cdot \rangle$. Then we can view $f$ as a semi-invariant on a new quiver with new weight according to the following simplification rules:
\begin{itemize}
\item[(a)] If $\alpha_1=0$, then we have (we put the values of $\alpha$ on top of $\beta$):
\[\vcenter{\vbox{\[email protected]@C+1.8pc{
& \stackrel{\alpha_{x_1}}{\beta_{x_1}} \dots \\
& \stackrel{\alpha_{x_2}}{\beta_{x_2}} \dots \\
\stackrel{0}{\beta_{1}} \ar[uur] \ar[ur] & \dots \\
& \stackrel{\alpha_{y_2}}{\beta_{y_2}} \ar[ul] \dots \\
& \stackrel{\alpha_{y_1}}{\beta_{y_1}} \ar[uul] \dots
}}}\hspace{0.4in}{\xymatrix@C+1.5pc{ \ar@{~>}[r] & }}\hspace{0.4in}
\vcenter{\vbox{\xymatrix@R-2pc@C+2pc{
& \stackrel{\alpha_{x_1}}{\beta_{x_1}} \dots \\
& \stackrel{\alpha_{x_2}}{\beta_{x_2}} \dots \\
& \hspace{0.2in} \dots \\
\stackrel{0}{\beta_{1}} & \stackrel{\alpha_{y_2}}{\beta_{y_2}} \ar[l] \dots \\
\stackrel{0}{\beta_{1}} & \stackrel{\alpha_{y_1}}{\beta_{y_1}} \ar[l] \dots
}}} \]
\item[(b)] Write $\sigma=-\langle \cdot, \alpha^* \rangle$. If $\alpha^*_1=0$, then the same simplification rule holds as in part (a) by replacing $\alpha$ with $\alpha^*$, with the arrows reversed.
\end{itemize}
\end{lemma}
If we write $\sigma=\langle \alpha, \cdot \rangle = -\langle \cdot, \alpha^* \rangle$, then $\alpha^* = C\cdot \alpha$, where $C$ denotes the Coxeter transformation. This transformation can understood as applying reflections to sinks successively once at each vertex of the quiver (see \cite{elements},\cite{en}). In particular, if vertex $1$ in part (b) of the above lemma is a sink, then $\alpha_1^*=0$ is equivalent to $\alpha_1 = \displaystyle\sum_{i} \alpha_{x_i}$, in which case one can simply delete vertex $1$.
\subsection{The slice method for quivers}\label{subsec:slicequiv}
Let $Q$ be a quiver without oriented cycles. We say that an that arrow $a\in Q_1$ is a \itshape 1-source \normalfont (resp. \itshape 1-sink\normalfont) if $ta$ (resp. $ha$) is not a vertex of any arrow other than $a$. We will slice at such arrows $a$ as in Section \ref{subsec:method}. The following is an immediate consequence of Theorem \ref{thm:mainapp}:
\begin{theorem}\label{thm:bquiv}
Let $Q$ be a quiver, $\beta$ be a dimension vector. Let $\vec{a}\in Q_1$ a 1-source or 1-sink arrow, number its vertices by $1,2$, and assume $\beta_1 \leq \beta_2$. The slice at the arrow $\vec{a}$ is a representation space $\operatorname{Rep}(Q_a,\beta_a)$ corresponding to the following quiver (where the orientation of $\vec{a}$ is arbitrary):
\[(Q,\beta): \hspace{0.2in} \vcenter{\vbox{\[email protected]{
& & \beta_{x_1} \dots\\
& & \beta_{x_2} \dots\\
\beta_{1} \ar@{-}[r]^a & \beta_{2} \ar[uur] \ar[ur] & \dots\\
& & \beta_{y_2} \ar[ul] \dots\\
& & \beta_{y_1} \ar[uul] \dots
}}}\hspace{0.1in}\xymatrix@C+1pc{ \ar@{~>}[r] & }
(Q_a,\beta_a): \hspace{0.2in} \vcenter{\vbox{\[email protected]{
& & \beta_{x_1} \dots \\
& & \beta_{x_2} \dots \\
\beta_{1} \ar@{-->}[uurr] \ar@{-->}[urr] & \beta_{2}-\beta_{1} \ar[uur] \ar[ur] &\dots \\
& & \beta_{y_2} \ar[ul] \ar@{-->}[ull] \dots\\
& & \beta_{y_1} \ar[uul] \ar@{-->}[uull] \dots
}}}\]
Let $f$ be a semi-invariant on $\operatorname{Rep}(Q,\beta)$ of weight $\sigma=\langle \alpha,\cdot \rangle$ and $f_a$ be the induced semi-invariant on $\operatorname{Rep}(Q_a,\beta_a)$ with induced weight $\sigma_a= \langle \alpha_a ,\cdot \rangle$. Under the natural correspondence of vertices between $Q$ and $Q_a$, $\sigma_a$ differs from $\sigma$ only at vertex $1$, with $\sigma_a(1)=\sigma(1)+\sigma(2)$. Moreover, if $\sigma_a$ is a multiplicity-free weight on $\operatorname{Rep}(Q_a,\beta_a)$, then $\sigma$ is multiplicity-free as well and we have
$$b_f(s)=b_{f_a}(s)\cdot [s]^{|\sigma_1|}_{\beta_2-\beta_1,\beta_2}.$$
\end{theorem}
\begin{remark}\label{rem:impo}
In examples, we prefer working with dimension vectors $\alpha$ rather than the weights $\sigma$. Since we know the weight $\sigma_a=\langle \alpha_a, \cdot \rangle$ on the slice, we implicitly also know the dimension vector $\alpha_a$. Let $P_i$ be the indecomposable projective module (see \cite{elements}) of $Q_a$ at vertex $i$ and $S_i$ the simple module of $Q_a$ at vertex $i$. The formulas are:
\begin{itemize}
\item[(a)] If $a$ is a 1-source, then $\alpha_a=\alpha+(\alpha_2-\alpha_1)\mathbb{D}im P_1-\alpha_1 \cdot \mathbb{D}im S_2,$
\item[(b)] If $a$ is a 1-sink, then $\alpha_a=\alpha+\alpha_1 \cdot \mathbb{D}im P_1-\alpha_1 \cdot \mathbb{D}im S_1.$
\end{itemize}
Moreover, in these cases we can see by direct computation that if $f=c^V$, then $f_a=c^{V'}$ is again a Schofield semi-invariant, where the representation $V'\in \operatorname{Rep}(Q_a,\alpha_a)$ can be written down explicitly. Since we will be working with generic Schur representations $V$, we will write only the corresponding dimension vectors (which are real Schur roots).
Writing $\sigma=\langle \alpha, \cdot \rangle = \langle \cdot, \alpha^* \rangle$, we can write down the dual formulas for the relation between $\alpha^*$ and $\alpha_a^*$ as well. They be deduced easily from the formulas above if we note that the dual semi-invariant $f^*$ on the opposite quiver $Q^*$ of $Q$ (i.e. reverse all arrows) has weight of $-\sigma = \langle \alpha^*, \cdot \rangle_*\,$, where $\langle \cdot, \cdot \rangle_*$ denotes the Euler product on $Q^*$.
\end{remark}
\begin{definition} \label{def:sliceable}
For a semi-invariant $f$ of a quiver $Q$, we say $f$ is \itshape sliceable \normalfont if, after slicing repeatedly at $1$-sinks and $1$-arrows as described in Theorem \ref{thm:bquiv} (with possible simplifications, as in Lemma \ref{lem:simp}), we can reach the empty quiver (equivalently, a non-zero constant function).
\end{definition}
In the case $f$ is sliceable, we can compute the $b$-function and the locally semi-simple representation (see Proposition \ref{prop:locsemi}) of $f$ using the slice method. The following proposition gives a clearer picture of sliceable irreducible semi-invariants:
\begin{proposition}\label{prop:neg}
Let $f=c^V\in \operatorname{SI}(Q,\beta)$ be an irreducible semi-invariant of weight $\langle \alpha, \cdot \rangle=-\langle \cdot, \alpha^* \rangle$ and assume $f$ depends on all arrows of $Q$. If $\alpha$ (resp. $\alpha^*$) is not a real Schur root, then $f$ is not sliceable.
Furthermore, take an arrow $\vec{a}$ that is a $1$-source or $1$-sink between $1$ and $2$ such that $\beta_1\leq \beta_2$, and assume $\alpha$ is a real Schur root. Let $\langle \alpha_a , \cdot \rangle$ be the weight of the induced semi-invariant $f_a$ on the slice $(Q_a,\beta_a)$, and let $\langle \alpha_a', \cdot \rangle$ be the weight on $(Q_a',\beta_a')$ after possible simplifications as in Lemma \ref{lem:simp}. Then the following are equivalent:
\begin{itemize}
\item[(a)] $\alpha_a$ is a real Schur root;
\item[(b)] $\alpha_a'$ is a real Schur root;
\item[(c)] $\vec{a}$ is a $1$-source with $\alpha_1=\alpha_2$ or $\alpha^*_1=0$, or $\vec{a}$ is a $1$-sink with $\alpha_1=0$ or $\alpha_1^*=\alpha_2^*$.
\end{itemize}
\end{proposition}
\begin{proof}
We will assume $a$ is a $1$-source (the case with $1$-sink is similar). Since $f$ depends on all arrows of $Q$ and is irreducible, we have by Theorem \ref{thm:bquiv} part a) that $\beta$ and $\beta_a$ are sincere dimension vectors. Due to the isomorphism $\operatorname{SI}(Q,\beta)\cong\mathbb{C}[\operatorname{Rep}(Q_a,\beta_a)]^{U\rtimes\operatorname{SL}(\beta_a)}$, we also have that $f_a=c^{V'}$ is irreducible. Since $\beta$ and $\beta_a$ are sincere, $V$ and $V'$ are Schur representations by \cite[Lemma 1]{harwey}.
Note that $\langle \alpha , \alpha \rangle = -\langle \alpha, \alpha^*\rangle = \langle \alpha^*,\alpha^* \rangle$. By a direct computation, one obtains the formula
$$\langle \alpha_a, \alpha_a \rangle_a = \langle \alpha , \alpha \rangle -(\alpha_2-\alpha_1)\alpha_1^*,$$
where $\langle \cdot, \cdot \rangle_a$ is the Euler form on $Q_a$. This implies that this value decreases by slicing (at least before simplifications), and it remains the same iff $\alpha_2=\alpha_1$ or $\alpha_1^*=0$. However, we can simplify according to Lemma \ref{lem:simp} precisely under these conditions, and we get a reduced quiver $Q_a'$ with $\alpha_a'$. But an easy computation yields that the value $\langle \alpha_a' , \alpha_a' \rangle_a'=\langle \alpha_a, \alpha_a \rangle_a$ still remains the same. Since $V$ (resp. $V'$) are Schur representations, $\alpha$ (resp. $\alpha_a$) is a real Schur root if and only if $\langle \alpha, \alpha \rangle=1$ (resp. $\langle \alpha_a , \alpha_a \rangle = 1$). Now assume $f$ is sliceable. Since $V$ is a Schur representation, we have $\langle \alpha , \alpha \rangle \leq 1$. Since this value can only decrease by slicing and the last value (when the function is constant) is trivially $1$, we must have that all values are $1$, and the encountered dimension vectors are all real Schur roots.
\end{proof}
Finally, we summarize the rules of slicing in the most common situation described in part (c) of the above theorem, combining Lemma \ref{lem:simp}, Theorem \ref{thm:bquiv} and Remark \ref{rem:impo}.
\begin{corollary}\label{cor:rulz}
Take $Q$ and $f$ a semi-invariant of weight $\sigma=\langle \alpha,\cdot\rangle$ as in Theorem \ref{thm:bquiv}. Slicing at the arrow $\vec{a}$ in the following cases, we obtain the slice $(Q_a,\beta_a)$ and induced semi-invariant $f_a$ with weight $\sigma_a=\langle \alpha_a, \cdot \rangle$:
\begin{itemize}
\item[(a)] If $\vec{a}$ is a 1-source with $\alpha_1=\alpha_2$, then
\[(Q_a,\stackrel{\alpha_a}{\beta_a}): \hspace{0.2in} \vcenter{\vbox{\xymatrix@R-2pc@C+0.5pc{
& & \stackrel{\alpha_{x_1}}{\beta_{x_1}} \dots \\
& & \stackrel{\alpha_{x_2}}{\beta_{x_2}} \dots \\
& \stackrel{\alpha_2}{\beta_1} \ar[uur] \ar[ur] &\dots \\
\stackrel{0}{\beta_2-\beta_1} & & \stackrel{\alpha_{y_2}}{\beta_{y_2}} \ar[ul] \ar@{-->}[ll] \dots\\
\stackrel{0}{\beta_2-\beta_1} & & \stackrel{\alpha_{y_1}}{\beta_{y_1}} \ar[uul] \ar@{-->}[ll] \dots
}}}\]
\item[(b)] If $\vec{a}$ is a 1-sink with $\alpha_1=0$, then
\[(Q_a,\stackrel{\alpha_a}{\beta_a}): \hspace{0.2in} \vcenter{\vbox{\xymatrix@R-2pc@C+0.5pc{
& & \stackrel{\alpha_{x_1}}{\beta_{x_1}} \dots \\
& & \stackrel{\alpha_{x_2}}{\beta_{x_2}} \dots \\
& \stackrel{\alpha_2}{\beta_{2}-\beta_1} \ar[uur] \ar[ur] &\dots \\
\stackrel{0}{\beta_1} & & \stackrel{\alpha_{y_2}}{\beta_{y_2}} \ar[ul] \ar@{-->}[ll] \dots\\
\stackrel{0}{\beta_1} & & \stackrel{\alpha_{y_1}}{\beta_{y_1}} \ar[uul] \ar@{-->}[ll] \dots
}}}\]
\end{itemize}
Moreover, writing $\sigma= -\langle \cdot, \alpha^* \rangle$, we have rules dual to the above by replacing $\alpha$ with $\alpha^*$, with all arrows reversed. Furthermore, in all these four cases $\alpha$ is a real Schur root if and only if $\alpha_a$ is a real Schur root, in which case
\[b_f(s) = b_{f_a}(s) \cdot [s]_{\beta_2-\beta_1,\beta_2}^{\alpha_2}.\]
\end{corollary}
\begin{remark} For a semi-invariant $f$ to be non-zero, some inequalities must be satisfied between the dimensions $\beta_x$, where $x\in Q_0$. The isomorphism $\operatorname{SI}(Q,\beta)\cong\operatorname{SI}(Q_a,\beta_a)^{U}$ from Proposition \ref{prop:algiso} gives inductively these inequalities, and they will be encoded in the negativity of the roots of the $b$-function. For simplicity, we will work with dimension vectors $\beta$ so that these inequalities are strict.
\end{remark}
\subsection{Some computations of $b$-functions for quivers}\label{subsec:exquiv}
We now show how to use Theorem \ref{thm:bquiv} and Corollary \ref{cor:rulz} in examples. We place the values of $\alpha$ or $\alpha^*$ on top of the values of the dimension vector $\beta$, where $\langle \alpha , \cdot \rangle = - \langle \cdot, \alpha^*\rangle$ is the weight of the semi-invariant. When $\alpha^*$ is used, we label its values by $*$ at each vertex. We use a dashed line for the arrow at which we are slicing. We indicate (below the curly arrow) the slicing rule used from Corollary \ref{cor:rulz} (or Remark \ref{rem:impo} or Lemma \ref{lem:simp}) and retain (above the curly arrow) the decomposition of the $b$-function as given by Corollary \ref{cor:rulz} (or Theorem \ref{thm:bfun}).
\begin{example}\label{thm:beefor} We compute the $b$-function of the semi-invariant from Example \ref{eq:deefor}. Recall $\beta_1+\beta_2+\beta_3=2\beta_4$.
\[\vcenter{\vbox{\xymatrix{
& \stackrel{1}{\beta_2} \ar[d] & \\
\stackrel{1}{\beta_1} \ar@{-->}[r] & \stackrel{1}{\beta_4} & \ar[l] \stackrel{1}{\beta_3}}
}}\stackrel[\ref{cor:rulz} (a)]{[s]_{\beta_4-\beta_1,\beta_4}}{\xymatrix{ \ar@{~>}[r] & }}
\vcenter{\vbox{\xymatrix{
\stackrel{1}{\beta_2} \ar[d] \ar@{-->}[r] & \stackrel{0}{\beta_4-\beta_1} & \\
\stackrel{1}{\beta_1} & \ar[l] \stackrel{1}{\beta_3} \ar[r] & \stackrel{0}{\beta_4-\beta_1}}
}}\stackrel[\ref{cor:rulz} (b)]{[s]_{\beta_1+\beta_2-\beta_4,\beta_2}}{\xymatrix@C+1pc{ \ar@{~>}[r] & }}\]
\[
\vcenter{\vbox{\xymatrix{
\stackrel{1}{\beta_4-\beta_3} \ar@{-->}[d] & & \\
\stackrel{1}{\beta_1} & \ar[l] \stackrel{1}{\beta_3} \ar[r] & \stackrel{0}{\beta_4-\beta_1}}
}}\stackrel[\ref{cor:rulz} (a)]{[s]_{\beta_1+\beta_3-\beta_4,\beta_1}}{\xymatrix@C+1pc{ \ar@{~>}[r] & }}
\vcenter{\vbox{\[email protected]{ & \stackrel{1}{\beta_4-\beta_3} & \\
\stackrel{0}{\beta_4-\beta_2} & \ar[l]\ar[u] \stackrel{1}{\beta_3} \ar[r] & \stackrel{0}{\beta_4-\beta_1}}
}}\stackrel[\ref{lem:simp} (b)]{}{\xymatrix{ \ar@{~>}[r] & }}\]
\[\vcenter{\vbox{\xymatrix{
\stackrel{0}{\beta_4-\beta_2} & \ar@{-->}[l] \stackrel{1}{\beta_3} \ar[r] & \stackrel{0}{\beta_4-\beta_1}}}}
\stackrel[\ref{cor:rulz} (b)]{[s]_{\beta_2+\beta_3-\beta_4,\beta_3}}{\xymatrix@C+1pc{ \ar@{~>}[r] & }}
\vcenter{\vbox{\xymatrix{
\ar@{-->}[r] \stackrel{1}{\beta_4-\beta_1} & \stackrel{0}{\beta_4-\beta_1}}
}}\stackrel{[s]_{\beta_4-\beta_1}}{\xymatrix{ \ar@{~>}[r] & }} \emptyset
\]
Hence the $b$-function is
$$b(s)=[s]_{\beta_4-\beta_1,\beta_4}\cdot[s]_{\beta_1+\beta_2-\beta_4,\beta_2}\cdot[s]_{\beta_1+\beta_3-\beta_4,\beta_1}\cdot[s]_{\beta_2+\beta_3-\beta_4,\beta_3}\cdot[s]_{\beta_4-\beta_1}=$$
$$=[s]_{\beta_4}\cdot[s]_{\beta_1+\beta_2-\beta_4,\beta_2}\cdot[s]_{\beta_2+\beta_3-\beta_4,\beta_3}\cdot[s]_{\beta_1+\beta_3-\beta_4,\beta_1}.$$
Using Proposition \ref{prop:locsemi} at each step, we get that the locally semi-simple representation is
$$A=V_1^{\beta_4-\beta_1}\operatornamelus V_2^{\beta_4-\beta_2} \operatornamelus V_3^{\beta_4-\beta_3},$$
where the indecomposables are $V_1= \begin{matrix} & 1 & \\ 0 & 1 & 1\end{matrix} $, $V_2 = \begin{matrix} & 0 & \\ 1 & 1 & 1\end{matrix}$ , $V_3 =\begin{matrix} & 1 & \\ 1 & 1 & 0\end{matrix}$. Note that this is also the generic representation in $\operatorname{Rep}(Q,\beta)$. This is due to the fact that $\operatorname{Rep}(Q,\beta)\backslash \mathcal{O}_A$ is the hypersurface defined by the semi-invariant.
\end{example}
Now we formulate a result for tree quivers, that is, for quivers whose underlying graphs have no cycles. This includes the $b$-functions of semi-invariants for type $\mathbb{A}$ quivers determined in \cite{sugi}.
\begin{theorem} \label{thm:tree}
Let $Q$ be a tree quiver, and $f$ a non-zero semi-invariant on $Q$ of weight $\langle \alpha,\cdot \rangle=-\langle \cdot, \alpha^* \rangle$. If $\alpha_x\leq 1$ for any $x\in Q_0$ (resp. $\alpha_x^*\leq 1$ for any $x\in Q_0$), then $f$ is sliceable, and the roots of $b_f(s)$ are negative integers.
\end{theorem}
\begin{proof}
By duality, it is enough to consider the case $\alpha_x\leq 1$ for all $x\in Q_0$. It is immediate that $\alpha$ is a prehomogeneous dimension vector, hence the weight $\langle \alpha, \cdot \rangle$ is multiplicity-free. As usual, we work with the support of $f$, that is, we can drop arrows if $f$ doesn't depend on its corresponding variables. Since $Q$ is a tree, we can take an arrow $\vec{a}\in Q_1$ that is a 1-source or 1-sink. We use the notation as in Theorem \ref{thm:bquiv}.
First, assume $\vec{a}$ is 1-source. If $f$ depends on $\vec{a}$, we must have $\alpha_1=1$ by Lemma \ref{lem:simp}. Let $A$ be the generic matrix of variables corresponding to $\vec{a}$. If $\alpha_2=0$, then by Lemma \ref{lem:simp} part a) we can disconnect the quiver, $A$ has to be a square matrix, and we can separate variables $f=f'\cdot \det A$, where $f'$ is a semi-invariant on the smaller quiver without the arrow $\vec{a}$. Hence we can assume $\alpha_2=1$.
Similarly, if $\vec{a}$ is a 1-sink, we can assume WLOG that $\alpha_1=0$ and $\alpha_2=1$.
In any case, we are in the situation of slicing at $\vec{a}$ as in Corollary \ref{cor:rulz}, and get a quiver $Q_a$ which is still a tree quiver, and the weight $\alpha_a$ of the induced semi-invariant $f_a$ on $Q_a$ still satisfies $(\alpha_a)_x\leq 1$, for any $x\in (Q_a)_0$. By Theorem \ref{thm:bquiv}, we get
$$b_f(s)=b_{f_a}(s)\cdot [s]_{\beta_2-\beta_1,\beta_2}.$$
Since the dimension of the representation space strictly decreases by slicing, this procedure is finite and stops when we arrive at a constant function.
\end{proof}
For some geometric implications of the result above about singularities of the zero sets of such semi-invariants, see \cite[Theorem 3.13]{en2}. We consider the next family of Dynkin quivers:
\begin{theorem}\label{thm:dee}
All fundamental semi-invariants of quivers of type $\mathbb{D}_n$ are sliceable.
\end{theorem}
\begin{proof}
Proceeding as in Theorem \ref{thm:tree} and using Corollary \ref{cor:rulz}, one we can reduce the proof to the case when $\alpha$ is the longest root. We illustrate the proof with the orientation of $\mathbb{D}_n$ chosen so that all arrows point to the joint vertex.
\[\vcenter{\vbox{\[email protected]@R-1pc{
& & & & \stackrel{1}{\beta_{n-1}} \ar[d] & \\
\stackrel{1}{\beta_1} \ar[r] & \stackrel{2}{\beta_2} \ar[r] & \stackrel{2}{\beta_3} \ar[r] & \dots \ar[r]& \stackrel{2}{\beta_{n-2}} & \ar[l] \stackrel{1}{\beta_n}
}}}\stackrel{\alpha\to \alpha^*}{\xymatrix{ \ar@{~>}[r] & }}
\vcenter{\vbox{\[email protected]@R-1pc{
& & & & \stackrel{1^*}{\beta_{n-1}} \ar[d] & \\
\stackrel{0^*}{\beta_1} \ar@{-->}[r] & \stackrel{1^*}{\beta_2} \ar[r] & \stackrel{2^*}{\beta_3} \ar[r] & \dots \ar[r]& \stackrel{2^*}{\beta_{n-2}} & \ar[l] \stackrel{1^*}{\beta_n}
}}}\stackrel[\ref{cor:rulz} (b)^*]{[s]_{\beta_2-\beta_1,\beta_2}}{\xymatrix{ \ar@{~>}[r] & }}\]
\[\vcenter{\vbox{\[email protected]@R-0.8pc{
& \stackrel{0^*}{\beta_1}\ar@{-->}[d] & & & \stackrel{1^*}{\beta_{n-1}} \ar[d] & \\
\stackrel{1^*}{\beta_2-\beta_1} \ar[r] & \stackrel{2^*}{\beta_3} \ar[r] & \stackrel{2^*}{\beta_4}\ar[r] & \dots \ar[r]& \stackrel{2^*}{\beta_{n-2}} & \ar[l] \stackrel{1^*}{\beta_n}
}}}\!\!\!\!\stackrel[\ref{cor:rulz} (b)^*]{[s]_{\beta_3-\beta_1,\beta_3}^2}{\xymatrix{ \ar@{~>}[r] & }}\!\!\!\!
\vcenter{\vbox{\[email protected]@R-0.8pc{
& & \stackrel{0^*}{\beta_1}\ar@{-->}[d] & & \stackrel{1^*}{\beta_{n-1}} \ar[d] & \\
\stackrel{1^*}{\beta_2-\beta_1} \ar[r] & \stackrel{2^*}{\beta_3-\beta_1} \ar[r] & \stackrel{2^*}{\beta_4}\ar[r] & \dots \ar[r]& \stackrel{2^*}{\beta_{n-2}} & \ar[l] \stackrel{1^*}{\beta_n}
}}}\]
\[\stackrel[\ref{cor:rulz} (b)^*]{[s]_{\beta_4-\beta_1,\beta_4}^2}{\xymatrix{ \ar@{~>}[r] & }} \dots \stackrel[\ref{cor:rulz} (b)^*]{[s]_{\beta_{n-3}-\beta_1,\beta_{n-3}}^2}{\xymatrix{ \ar@{~>}[r] & }}\!\!\!\!
\vcenter{\vbox{\[email protected]@R-1pc{
& & & \stackrel{0^*}{\beta_1}\ar@{-->}[dr] & \stackrel{1^*}{\beta_{n-1}} \ar[d] & \\
\stackrel{1^*}{\beta_2-\beta_1} \ar[r] & \stackrel{2^*}{\beta_3-\beta_1} \ar[r] & \dots \ar[r] &\stackrel{2^*}{\beta_{n-3}-\beta_{1}}\ar[r] & \stackrel{2^*}{\beta_{n-2}} & \ar[l] \stackrel{1^*}{\beta_n}
}}}\!\!\!\stackrel[\ref{cor:rulz} (b)^*]{[s]_{\beta_{n-2}-\beta_1,\beta_{n-2}}^2}{\xymatrix{ \ar@{~>}[r] & }}\]
\[\vcenter{\vbox{\[email protected]@R-1pc{
& & & \stackrel{1^*}{\beta_{n-1}} \ar[d] & \\
\stackrel{1^*}{\beta_2-\beta_1} \ar[r] & \stackrel{2^*}{\beta_3-\beta_1} \ar[r] & \dots \ar[r]& \stackrel{2^*}{\beta_{n-2}-\beta_1} & \ar[l] \stackrel{1^*}{\beta_n}}}}
\!\!\!\!\!\!\!\stackrel{\alpha^*\to \alpha}{\xymatrix{ \ar@{~>}[r] & }}\!\!\!\!\!\!\!
\vcenter{\vbox{\[email protected]@R-1pc{
& & & \stackrel{1}{\beta_{n-1}} \ar[d] & \\
\stackrel{1}{\beta_2-\beta_1} \ar[r] & \stackrel{1}{\beta_3-\beta_1} \ar[r] & \dots \ar[r] & \stackrel{1}{\beta_{n-2}-\beta_1} & \ar[l] \stackrel{1}{\beta_n}
}}}\]
At this stage we know that the latter quiver is sliceable, by Theorem \ref{thm:tree}. Continuing,
\[\stackrel[\ref{cor:rulz} (a)]{[s]_{\beta_3-\beta_2,\beta_3-\beta_1}}{\xymatrix@C+1pc{ \ar@{~>}[r] & }}
\vcenter{\vbox{\[email protected]{
& & & \stackrel{1}{\beta_{n-1}} \ar[d] & \\
\stackrel{1}{\beta_2-\beta_1} \ar@{-->}[r] & \stackrel{1}{\beta_4-\beta_1} \ar[r] & \dots \ar[r] & \stackrel{1}{\beta_{n-2}-\beta_1} & \ar[l] \stackrel{1}{\beta_n}
}}}\stackrel[\ref{cor:rulz} (a)]{[s]_{\beta_4-\beta_2,\beta_4-\beta_1}}{\xymatrix@C+1pc{ \ar@{~>}[r] & }} \dots \]
\[
\dots \stackrel[\ref{cor:rulz} (a)]{[s]_{\beta_{n-3}-\beta_2,\beta_{n-3}-\beta_1}}{\xymatrix@C+1pc{ \ar@{~>}[r] & }} \vcenter{\vbox{\xymatrix{
& \stackrel{1}{\beta_{n-1}} \ar[d] & \\
\stackrel{1}{\beta_2-\beta_1} \ar[r] & \stackrel{1}{\beta_{n-2}-\beta_1} & \ar[l] \stackrel{1}{\beta_n}
}}}\stackrel[\text{Example \ref{thm:beefor}}]{}{\xymatrix@C+3pc{ \ar@{~>}[r] & }}\emptyset\]
Hence the $b$-function is:
$$b(s)=[s]_{\beta_2-\beta_1,\beta_2}\prod_{i=3}^{n-2}\left([s]_{\beta_i-\beta_1,\beta_i}^2 [s]_{\beta_i-\beta_2,\beta_i-\beta_1}\right) \cdot $$
$$\cdot [s]_{\beta_{n-2}-\beta_{n-1}-\beta_1,\beta_{n-2}-\beta_1}[s]_{\beta_{n-2}-\beta_{n}-\beta_1,\beta_{n-2}-\beta_1}[s]_{\beta_{n-2}-\beta_1}.$$
Accordingly, the homogeneous inequalities that are necessary and sufficient for the semi-invariant to be non-zero are:
$$\beta_1\leq \beta_2 \leq \beta_i, i=3,\dots,n-2,$$
$$\beta_{n-1},\beta_n \leq \beta_{n-2}-\beta_1.$$
If these inequalities are strict, then the semi-invariant is irreducible by Proposition \ref{prop:algiso}. Also, one can write down the corresponding locally semi-simple representation explicitly using Proposition \ref{prop:locsemi} in each step.
\end{proof}
We give an example of a quiver of extended Dynkin type:
\begin{example} We take $\overline{\mathbb{D}}_4$ with the dimension vector $\beta$, with $2\beta_1+\beta_2+\beta_3+\beta_4=3\beta_5$, semi-invariant (unique up to constant) $f=c^V$, where $\mathbb{D}im V=\alpha=(2,1,1,1,2)$ is a real Schur root:
\[
\vcenter{\vbox{\[email protected]{
& \stackrel{1}{\beta_2} \ar[d] & \\
\stackrel{2}{\beta_1} \ar@{-->}[r] & \stackrel{2}{\beta_5} & \ar[l] \stackrel{1}{\beta_3}\\
& \stackrel{1}{\beta_4} \ar[u] &
}}}\stackrel[\ref{cor:rulz} (a)]{[s]^2_{\beta_5-\beta_1,\beta_5}}{\xymatrix@C+2pc{ \ar@{~>}[r] & }}
\vcenter{\vbox{\[email protected]{
\stackrel{1}{\beta_2} \ar[d] \ar@{-->}[r] & \stackrel{0}{\beta_5-\beta_1} & \\
\stackrel{2}{\beta_1} & \ar[l] \stackrel{1}{\beta_3} \ar@{-->}[r] & \stackrel{0}{\beta_5-\beta_1}\\
\stackrel{1}{\beta_4} \ar[u] \ar@{-->}[r] & \stackrel{0}{\beta_5-\beta_1} &
}}}\stackrel{}{\xymatrix{ \ar@{~>}[r] & }}\]
\[\stackrel[\ref{cor:rulz} (a)]{[s]_{\beta_1+\beta_2-\beta_5,\beta_2}\cdot[s]_{\beta_1+\beta_3-\beta_5,\beta_3}\cdot[s]_{\beta_1+\beta_4-\beta_5,\beta_4}}{\xymatrix@C+10pc{ \ar@{~>}[r] & }}
\vcenter{\vbox{\xymatrix@C-1pc{
& \stackrel{1}{\beta_1+\beta_3-\beta_5} \ar[d] & \\
\stackrel{1}{\beta_1+\beta_2-\beta_5} \ar[r] & \stackrel{2}{\beta_1} & \ar[l] \stackrel{1}{\beta_1+\beta_4-\beta_5}}
}}\!\!\!\stackrel{[s]_{\beta_1}}{\xymatrix{ \ar@{~>}[r] & }}
\emptyset
\]
In the last step we noticed the shortcut that the semi-invariant is just the square determinant of size $\beta_1$. So the $b$-function of $f$ is
$$b_f(s)= [s]^2_{\beta_5-\beta_1,\beta_5} \cdot [s]_{\beta_1+\beta_2-\beta_5,\beta_2}\cdot[s]_{\beta_1+\beta_3-\beta_5,\beta_3}\cdot[s]_{\beta_1+\beta_4-\beta_5,\beta_4}\cdot[s]_{\beta_1}.$$
\end{example}
In contrast with the method by reflections from \cite{en}, we find a Dynkin quiver with a semi-invariant that is not sliceable.
\begin{example}\label{thm:noot}
Take the following quiver of type $\mathbb{E}_6$ with semi-invariant of weight $\langle \alpha, \cdot \rangle = -\langle \cdot, \alpha^*\rangle$, with $\alpha$ being the longest root:
\[\vcenter{\vbox{\xymatrix{
& & 2\ar[d] & & \\
1 \ar[r]& 2\ar[r] & 3 & \ar[l] 2& \ar[l] 1
}}}\stackrel{\alpha \to \alpha^*}{\xymatrix{ \ar@{~>}[r] & }}
\vcenter{\vbox{\xymatrix{
& & 1\ar[d] & & \\
1 \ar[r]& 2\ar[r] & 3 & \ar[l] 2& \ar[l] 1
}}}
\]
There are no $1$-sources (resp. $1$-sinks) $a$ with $\alpha_{ta}=\alpha_{ha}$ or with $\alpha^*_{ta}=0$ (resp. $\alpha^*_{ta}=\alpha^*_{ha}$ or $\alpha_{ha}=0$). By Proposition \ref{prop:neg} the semi-invariant is not sliceable. However, in order to compute the $b$-function one can apply the method by reflections from \cite{en}.
\end{example}
\begin{example}\label{ex:symm} Symmetric quivers.
Examples \ref{ex:symdet},\ref{ex:ortho},\ref{ex:symp} are particular cases of semi-invariants of \textit{symmetric} quivers, see \cite{semisymm,symmetric}. In \cite[Proposition 4.1]{sasu}, the $b$-function of a semi-invariant of the equioriented symmetric quiver of type $\mathbb{A}$ is computed based on the multiplicity one property. Many more $b$-functions of semi-invariants of symmetric quivers can be computed using the techniques developed in Section \ref{sec:slice}. A more systematic study of these will be pursued in a subsequent paper.
\end{example}
We show in the next example how to apply Theorem \ref{thm:bquiv} together with Theorem \ref{thm:multi} to compute $b$-functions of \textit{several variables}. The main difference in the process is that we can make only simultaneous simplifications for the semi-invariants as in Lemma \ref{lem:simp} or Corollary \ref{cor:rulz}.
\begin{example}\label{thm:several}($b$-function of several variables) Take the following $\mathbb{D}_5$ quiver with non-zero semi-invariants $f_i=c^{V_i}$, for $i=1,2$, $\alpha^1=\mathbb{D}im V_1=(0,1,1,0,1)$, $\alpha^2=\mathbb{D}im V_2=(1,1,0,0,0)$ and $\beta_1+\beta_4=\beta_3$, $\beta_2=\beta_5$. We put the values of $\alpha^1$ and $\alpha^2$ on top of $\beta$:
\[\vcenter{\vbox{\xymatrix{
& & \stackrel{1,0}{\beta_3} \ar[d] & \\
\stackrel{0,1}{\beta_1} & \stackrel{1,1}{\beta_2} \ar@{-->}[l] \ar[r] & \stackrel{1,0}{\beta_5} \ar[r] & \stackrel{0,0}{\beta_4}
}}}\stackrel[\ref{rem:impo} (b)]{[s]_{\beta_2-\beta_1,\beta_2}^{1,0}}{\xymatrix{ \ar@{~>}[r] & }}
\vcenter{\vbox{\xymatrix{
& \stackrel{1,0}{\beta_3} \ar[d] & \\
\stackrel{1,1}{\beta_2-\beta_1} \ar@{-->}[r] & \stackrel{1,1}{\beta_5} \ar[r] & \stackrel{0,1}{\beta_4}\\
& \stackrel{0,1}{\beta_1} \ar[u] &
}}}\stackrel[\ref{cor:rulz} (a)]{[s]_{\beta_1,\beta_5}^{1,1}}{\xymatrix{\ar@{~>}[r] & }}
\]
\[\vcenter{\vbox{\[email protected]@C-0.5pc{
\stackrel{1,0}{\beta_3} \ar[d] \ar[r]& \stackrel{0,0}{\beta_1} \\
\stackrel{1,1}{\beta_2-\beta_1} \ar[r] & \stackrel{0,1}{\beta_4}\\
\stackrel{0,1}{\beta_1} \ar@{-->}[r] \ar[u] & \stackrel{0,0}{\beta_1}
}}}\stackrel[\ref{cor:rulz} (b)]{[s]^{0,1}_{\beta_1}}{\xymatrix{ \ar@{~>}[r] & }}
\vcenter{\vbox{\xymatrix{
\stackrel{1,0}{\beta_3} \ar[d] \ar@{-->}[r]& \stackrel{0,0}{\beta_1} \\
\stackrel{1,0}{\beta_2-\beta_1} \ar[r] & \stackrel{0,0}{\beta_4}
}}}\stackrel[\ref{cor:rulz} (b)]{[s]^{1,0}_{\beta_4,\beta_3}}{\xymatrix{ \ar@{~>}[r] & }}
\vcenter{\vbox{\xymatrix{
\stackrel{1,0}{\beta_4} \ar@{-->}[d] & \\
\stackrel{1,0}{\beta_2-\beta_1} \ar[r] & \stackrel{0,0}{\beta_4}
}}}\]
\[ \stackrel[\ref{cor:rulz} (a)]{[s]^{1,0}_{\beta_2-\beta_3,\beta_2-\beta_1}}{\xymatrix{ \ar@{~>}[r] & }}
\vcenter{\vbox{\xymatrix{
\stackrel{1,0}{\beta_4} \ar@{-->}[r] & \stackrel{1,0}{\beta_4}
}}}\stackrel{[s]^{1,0}_{\beta_4}}{\xymatrix{ \ar@{~>}[r] & }}\emptyset\]
Hence we have
$$b_{\underline{m}}(s_1,s_2)=[s]_{\beta_1,\beta_2}^{1,1}[s]^{0,1}_{\beta_1}[s]^{1,0}_{\beta_3}[s]^{1,0}_{\beta_2-\beta_3,\beta_2}.$$
\end{example}
It is not difficult to see that for the quivers of type $\mathbb{A}_n$ the slice method is sufficient to compute all the $b$-functions of several variables (as are the methods in \cite{en},\cite{sugi}). However, the slice method is not always sufficient to obtain directly the $b$-functions of several variables for type $\mathbb{D}_n$ quivers (although the method in \cite{en} is). Nevertheless, given the individual $b$-function (of one variable) of each semi-invariant (see Theorem \ref{thm:dee}), one can in principle apply the Structure Theorem of $b$-functions as in \cite{sugi} for this purpose -- for an example, see \cite[Example 4.3.13]{phd}. To proceed as in \cite{sugi}, one needs an explicit description for the locally semi-simple representation (and use \cite[Lemma 4.2.4]{phd}) and the generic representation. One can describe the locally semi-simple representation of each semi-invariant by Proposition \ref{prop:locsemi}, and the generic representation using the procedure we present in Appendix \ref{app:decomp}.
\appendix
\section{Generic decomposition for Dynkin quivers of type $\mathbb{D}$} \label{app:decomp}
Based on slices, we give an easy procedure for determining the generic decomposition for type $\mathbb{D}_n$ quivers.
Let $Q$ be a quiver, and $\alpha$ a prehomogeneous dimension vector. Following \cite{kac2}, we call a decomposition
$$\alpha=\alpha_1 \operatornamelus \alpha_2 \operatornamelus \dots \operatornamelus \alpha_t$$
the \itshape generic decomposition \normalfont (also called canonical decomposition), if the generic representation of dimension vector $\alpha$ decomposes into indecomposable representations of dimension vectors $\alpha_1,\alpha_2,\dots \alpha_t$. As already discussed in Section \ref{sec:quiv}, in this case $\alpha_i$ are real Schur roots, with $\operatorname{Ext}_Q(\alpha_i,\alpha_j)=0$ (that is, the corresponding generic representations have no self-extensions). Moreover, rewriting
$$\alpha=\alpha_1^{\operatornamelus r_1}\operatornamelus \alpha_2^{\operatornamelus r_2} \operatornamelus \dots \operatornamelus \alpha_t^{\operatornamelus r_t}.$$
with $\alpha_i$ distinct, we may assume, after a suitable rearrangement, that $\operatorname{Hom}_Q(\alpha_i,\alpha_j)=0$, for $i<j$ (again, this means that there are no morphisms between the corresponding generic representations). For more details , see \cite{combi,kac2}.
Though there exist algorithms to determine the generic decomposition for a dimension vector (e.g. see \cite{canon}), it is of interest to give clear-cut procedures that are easy to work out by hand. There is such a rule for quivers of type $\mathbb{A}_n$, and this is described in \cite[Proposition 3.1]{abeasis}. We illustrate this construction by the following example:
\[\xymatrix{
3 \ar[r] & 5 & 6 \ar[l] \ar[r] & 3 \ar[r] & 5
}\]
The generic decomposition is given by the following diagram (the connected horizontal components are the indecomposables):
\[\[email protected]@[email protected]{
& \bullet & \bullet\ar[l] & & \\
& \bullet & \bullet\ar[l] & & \bullet \\
\bullet\ar[r] & \bullet & \bullet\ar[l] & & \bullet\\
\bullet \ar[r] & \bullet & \bullet \ar[l]\ar[r] & \bullet \ar[r] & \bullet\\
\bullet \ar[r] & \bullet & \bullet \ar[l]\ar[r] & \bullet \ar[r] & \bullet\\
& & \bullet\ar[r] & \bullet \ar[r] & \bullet
}\]
Based on the $\mathbb{A}_n$ case, we extend the rule for quivers of type $\mathbb{D}_n$. Take a quiver with underlying graph $\mathbb{D}_n$ and the following labeling:
\[\xymatrix{
& n & & & \\
1 & 2 \ar@{-}[l]\ar[u]\ar@{-}[r] & 3 \ar@{-}[r] & \dots \ar@{-}[r] & n-1
}\]
Since the generic decomposition of a quiver and its opposite quiver coincide, we will fix without loss of generality the orientation of the arrow $2\to n$. We illustrate the procedure by examples first. Take the following $\mathbb{D}_n$ quiver with $n=6$ and $\alpha=(3,5,6,3,5,4)$:
\[\[email protected]{& 4 & & & \\
3 \ar[r] & 5 \ar[u] & 6 \ar[l] \ar[r] & 3 \ar[r] & 5
}\]
First, take the generic decomposition of the $\mathbb{A}_{n-1}$ quiver by dropping the $n$-th vertex. This was done in the example above. Then, the indecomposables of $\mathbb{A}_{n-1}$ that have $0$ dimension at vertex $2$ will also appear in the generic decomposition for $\mathbb{D}_n$. Hence we drop them, and we are left with the following diagram:
\[\[email protected]@[email protected]{
& \bullet & \bullet\ar[l] & & \\
& \bullet & \bullet\ar[l] & & \\
\bullet\ar[r]\ar@{-}[]+<-2em,0.8em>;[rrrr]+<2em,0.8em> & \bullet & \bullet\ar[l] & & \\
\bullet \ar[r] & \bullet & \bullet \ar[l]\ar[r] & \bullet \ar[r] & \bullet\\
\bullet \ar[r] & \bullet & \bullet \ar[l]\ar[r] & \bullet \ar[r] & \bullet
}\]
We separated by a horizontal line the two classes of indecomposables with dimension at vertex $1$ equal to $0$ or equal to $1$. We call the indecomposables under this line of the first class and over the line of the second class. Now we place $\alpha_n$ symbols $\circ$ on the left of the diagram starting from the horizontal line and moving downwards ($\circ$ represents the simple representation $S_n$). When we stop, we put another horizontal line to the bottom. Then we move the indecomposables of the second class starting from the top of the diagram and add their dimension vectors starting from the bottom horizontal line and stop if either:
\begin{itemize}
\item[(a)] We reach the top horizontal line, or
\item[(b)] We run out of indecomposables of the second class, or
\item[(c)] There exists a non-zero morphism from the indecomposable of the second class that we want to move to corresponding indecomposable of the first class.
\end{itemize}
In this example we stop due to part (b) and the diagram we get is:
\[\[email protected]@[email protected]{
\ar@{-}[]+<-2em,0.8em>;[rrrrr]+<2em,0.8em> \circ &\bullet\ar[r] & \bullet & \bullet\ar[l] & & \\
\circ &\bullet \ar[r] & \bullet & \bullet \ar[l]\ar[r] & \bullet \ar[r] & \bullet\\
\circ &\bullet \ar[r] & \bullet\bullet & \bullet\bullet\ar[l]\ar[r] & \bullet \ar[r] & \bullet\\
\ar@{-}[]+<-2em,-0.8em>;[rrrrr]+<2em,-0.8em> \circ & & \bullet & \bullet\ar[l] & &
}\]
Now we are ready to read off the generic decomposition. The indecomposables outside the horizontal lines will stay the same (there are none in this example). Finally, for each row between the two horizontal lines the dimension vector will have dimension $1$ at vertex $n$. Hence we get in this case
\begin{multline*}
(3,5,6,3,5,4)=(1,1,1,0,0,1)\operatornamelus(1,1,1,1,1,1)\operatornamelus(1,2,2,1,1,1)\operatornamelus(0,1,0,0,0,1)\operatornamelus\\
\operatornamelus (0,0,0,0,1,0)^{\operatornamelus 2} \operatornamelus (0,0,1,1,1,0).
\end{multline*}
We give another example:
\[\[email protected]{& 4 & & & \\
3 & 6 \ar[l] \ar[u] & 5 \ar[l] \ar[r] & 3
}\]
The generic decomposition for the $\mathbb{A}_4$ part is
\[\[email protected]@[email protected]{
\bullet & \ar[l] \bullet & \bullet\ar[l] & \\
\bullet & \ar[l] \bullet & \bullet\ar[l] & \\
\bullet & \ar[l] \bullet & \bullet\ar[l] \ar[r] & \bullet\\
\ar@{-}[]+<-2em,0.8em>;[rrr]+<2em,0.8em> & \bullet & \bullet\ar[l] \ar[r] & \bullet\\
& \bullet & \bullet\ar[l] \ar[r] & \bullet\\
& \bullet & & \\
}\]
Note that all indecomposables have dimension $1$ at the vertex $2$. The diagram joining the two classes of indecomposables is:
\[\hspace{-0.25in}\[email protected]@[email protected]{
& \bullet & \ar[l] \bullet & \bullet\ar[l] \ar[r] & \bullet\\
\ar@{-}[]+<-2em,0.8em>;[rrrr]+<2em,0.8em> \circ & &\bullet & \bullet\ar[l] \ar[r] & \bullet\\
\circ & & \bullet & \bullet\ar[l] \ar[r] & \bullet\\
\circ & \bullet & \ar[l] \bullet \bullet & \bullet\ar[l] & \\
\ar@{-}[]+<-2em,-0.8em>;[rrrr]+<2em,-0.8em> \circ & \bullet & \ar[l] \bullet & \bullet\ar[l] &
}\]
Here we stopped due to condition (c) since there is a non-zero map from the indecomposable $1\leftarrow 1 \leftarrow 1 \rightarrow 1$ to the corresponding indecomposable $0\leftarrow 1 \leftarrow 1 \rightarrow 1$. Hence the generic decomposition is
$$(3,6,5,3,4)=(1,1,1,1,0)\operatornamelus(0,1,1,1,1)^{\operatornamelus 2}\operatornamelus (1,2,1,0,1)\operatornamelus (1,1,1,0,1).$$
\begin{theorem} The algorithm described above gives the generic decomposition for $\mathbb{D}_n$ quivers.
\end{theorem}
\begin{proof}
We give a proof using slices. First, write the generic decomposition for a generic representation $R$ of the $\mathbb{A}_{n-1}$ quiver in the form
$$R=\bigoplus_{i=1}^m V_i^{p_i} \operatornamelus \bigoplus_{i=1}^n W_i^{q_i} \operatornamelus \bigoplus_i Z_i.$$
Here $V_i$ and $W_i$ are representations of the first and second class, respectively (separated by the horizontal line as in the examples) and $Z_i$ are the representations with dimension $0$ at vertex $2$. We assume that the order is chosen such that:
\begin{itemize}
\item[(a)] There is a map from $V_i$ to $V_j$ iff $j \leq i$;
\item[(b)] There is a map from $W_i$ to $V_j$ iff $j \leq i$;
\item[(c)] There are no maps from $V_i$ to $W_j$ for all $i,j$.
\end{itemize}
We note that this can be achieved immediately from the generic decomposition algorithm for $\mathbb{A}_{n-1}$ (after dropping the representations $Z_i$): $V_i$ are the representations below the horizontal line, ordered from top to bottom, and $W_i$ are the representations above the horizontal line, ordered from top to bottom. With this in mind, we take the slice as in Section \ref{subsec:slice}. Take a representation of the form $V=Z+R$ in $\operatorname{Rep}(\mathbb{D}_n,\alpha)$, with $Z\in \operatorname{Hom}(\mathbb{C}^{\alpha_2},\mathbb{C}^{\alpha_n})$. Then $V$ has a dense $\operatorname{GL}(\alpha)$-orbit if and only if $Z$ has a dense orbit in $\operatorname{Hom}(\mathbb{C}^{\alpha_2},\mathbb{C}^{\alpha_n})$ under the action of the stabilizer $G_R=\operatorname{GL}(\alpha_n)\times \operatorname{GL}(\underline{p})\times\operatorname{GL}(\underline{q}) \times U \times U'$, where $U=\prod_{j < i} \operatorname{Hom}(\mathbb{C}^{p_i},\mathbb{C}^{p_j}) \prod_{j < i} \operatorname{Hom}(\mathbb{C}^{q_i},\mathbb{C}^{q_j})$ and $U'=\prod_{i,j} \operatorname{Hom}(W_i,V_j)^{p_jq_i}$. It can be easily seen that forgetting about the action of $U'$, the following element already has a dense orbit in $\operatorname{Hom}(\mathbb{C}^{\alpha_2},\mathbb{C}^{\alpha_n})$:
$$Z=\kbordermatrix{
~ & V_1 & V_1 & \dots & V_m & \vrule & W_1 & W_1 & \dots & W_n\cr
& 1 & 0 & \dots & 0 &\vrule & & & &\cr
& 0 & 1 & \dots & 0 &\vrule & & &\cr
& & & \ddots & & \vrule & & & \iddots & \cr
& & & & & \vrule & 0 & 1 & \dots & 0\cr
& & & & & \vrule & 1 & 0 & \dots & 0
}.$$
Here there are $p_i$ (resp. $q_i$) columns corresponding to $V_i$ (resp. $W_i$), and we put the ones diagonally in the first (resp. second) block starting from the top left (resp. bottom left) until we reach the bottom or right (resp. top or right) edge of the block. The arrangement of ones corresponds to stopping under condition (a) or (b). Now using the action of $U'$, if two ones are in the same row corresponding to the columns of $V_i$ and $W_j$, and $\operatorname{Hom}_Q(W_j,V_i)\neq 0$, then we can cancel the $1$ in the column of $W_j$. This corresponds to stopping under condition (c).
\end{proof}
\begin{remark}
The article \cite{abeasisd} describes the generic decomposition for an equioriented quiver of type $\mathbb{D}$. The explicit description of generic representations for type $\mathbb{A}$ and $\mathbb{D}$ quivers is also pursued in the recent paper \cite{riedtrec}.
\end{remark}
\end{document} |
\begin{document}
\title{Strongly regular graphs satisfying the 4-vertex condition}
\footnotetext[1]{Retired}
\footnotetext[2]{Dept.~of Mathematics:~Analysis, Logic and Discrete
Math., Ghent University, Belgium.
E-mail: {\tt [email protected]}}
\begin{abstract}
We survey the area of strongly regular graphs satisfying
the 4-vertex condition and find several new families.
We describe a switching operation on collinearity graphs of polar spaces
that produces cospectral graphs. The obtained graphs satisfy
the 4-vertex condition if the original graph belongs to
a symplectic polar space.
\end{abstract}
\section{Introduction}
In this note we look at graphs with high combinatorial regularity,
where this regularity is not an obvious consequence of properties
of their group of automorphisms.
A graph $\Gamma$ is said to satisfy the {\em $t$-vertex condition} if,
for all triples $(T,x_0,y_0)$ consisting of a $t$-vertex graph $T$
together with two distinct distinguished vertices $x_0,y_0$ of $T$,
and all pairs of distinct vertices $x,y$ of $\Gamma$, the number of
isomorphic copies of $T$ in $\Gamma$, where the isomorphism maps
$x_0$ to $x$ and $y_0$ to $y$, does not depend on the choice of the pair
$x,y$ but only on whether $x,y$ are adjacent or nonadjacent.
This concept was introduced by Hestenes \& Higman \cite{HestenesHigman71}
(who refer to the unpublished Sims \cite{Sims-unpub})
in order to study rank 3 graphs.
Clearly, a rank 3 graph satisfies the $t$-vertex condition for all $t$.
If the graph $\Gamma$ satisfies the $t$-vertex condition, where $\Gamma$
has $v$ vertices and $3 \le t \le v$, then $\Gamma$ also satisfies the
$(t-1)$-vertex condition. A graph satisfies the 3-vertex condition
if and only if it is strongly regular (or complete or edgeless).
It satisfies the $v$-vertex condition if and only if it is rank 3.
Thus, we get a hierarchy of conditions of increasing
strength between strongly regular and rank 3.
The present paper will focus almost exclusively on the case $t=4$.
A simple criterion for the 4-vertex condition is given in
Proposition \ref{prop:sims}.
Previously not many graphs were known
that satisfy the 4-vertex condition without being rank 3.
Here we survey the known examples and give several new constructions.
One of our constructions proceeds by switching symplectic graphs
(see Section \ref{sec:switched_col}). As a consequence we find
\begin{Theorem} \label{thm:4vtxsrgs}
For $v \ge 4$ there are at least $\lfloor v^{1/6} \rfloor !$
strongly regular graphs of order at most $v$ satisfying
the $4$-vertex condition.
\end{Theorem}
It follows that among all non-isomorphic strongly regular graphs
of order at most $v$ that satisfy the $4$-vertex condition
the fraction that is determined by their spectrum goes to 0
when $v$ goes to infinity.
\section{The 4-vertex condition} \label{sec:def}
{
\footnotesize
A graph of order $v$ is called {\em strongly regular} with parameters
$(v, k, \lambda, \mu)$ if it is neither complete nor edgeless,
each vertex has degree $k$, any two adjacent vertices have exactly
$\lambda$ common neighbors, and any two non-adjacent vertices have exactly
$\mu$ common neighbors.
A graph with vertex set $V$ has {\em rank $r$} if
its automorphism group is transitive on $V$ and
has exactly $r$ orbits on $V \times V$.
Rank 3 graphs are strongly regular.
If $x$ is a vertex of the graph $\Gamma$, then the {\em local graph}
$\Gamma(x)$ of $\Gamma$ at $x$ is the induced subgraph in $\Gamma$
on the neighborhood of $x$. We say that $\Gamma$ is {\em locally} P
when all local graphs of $\Gamma$ have property P.
If $\Gamma$ is strongly regular, then its {\em 1st subconstituent}
(at a vertex $x$) is the local graph at $x$, while its {\em 2nd subconstituent}
(at $x$) is the induced subgraph on the non-neighborhood of $x$.
If $xy$ is an edge (resp. nonedge) in $\Gamma$, then the subgraph
induced on $\Gamma(x) \cap \Gamma(y)$ is called a $\lambda$-graph
(resp. $\mu$-graph).
See \cite{BrouwerVM21} for further information about
strongly regular graphs.
\par
}
Details on the parameters of graphs satisfying the 4-vertex condition
are given in \cite{HestenesHigman71}. In particular, we have the following
simple criterion for the 4-vertex condition:
\begin{Proposition} {\rm (Sims \cite{Sims-unpub})} \label{prop:sims}
A strongly regular graph $\Gamma$ with parameters
$(v,k,\allowbreak\lambda,\allowbreak\mu)$
satisfies the 4-vertex condition, with parameters $(\alpha,\beta)$,
if and only if the number of edges in $\Gamma(x) \cap \Gamma(y)$
is $\alpha$ (resp. $\beta$) whenever the vertices $x,y$ are adjacent
(resp. nonadjacent).
In this case, $k\big (\binom{\lambda}{2} - \alpha\big) = \beta(v-k-1)$.
\end{Proposition}
The equality here follows by counting 4-cliques minus an edge.
It immediately follows that the collinearity graph of a generalized quadrangle
(cf.~\cite{PayneThas84}) or partial quadrangle (cf.~\cite{Cameron74})
satisfies the 4-vertex condition (with $\alpha = \binom{\lambda}{2}$ and
$\beta = 0$). The same holds for a graph $\Gamma$ with $\lambda \le 1$.
If $\Gamma$ is locally strongly regular,
say with local parameters $(v',k',\lambda',\mu')$
(where clearly $v' = k$ and $k' = \lambda$),
then $\Gamma(x) \cap \Gamma(y)$ has valency $\lambda'$ (resp. $\mu'$)
when $x \sim y$ (resp. $x \not\sim y$)
so that $\Gamma$ satisfies the 4-vertex condition with
$\alpha = \lambda\lambda'/2$ and $\beta = \mu\mu'/2$.
\subsection{A few rank 4 examples}
Below we give a small table with the parameters of some edge-transitive
rank 4 graphs satisfying the 4-vertex condition.
Except for the example with group $HJ.2$ due to Reichard \cite{Reichard00},
these do not seem to have been noticed in print.
{
\footnotesize\noindent
\setlength{\tabcolsep}{5pt}
\begin{tabular}{@{\,}cccccccc@{~~~}l@{~~}l@{~~}l}
$v$ & $k$ & $\lambda$ & $\mu$ & $\lambda'$ & $\mu'$ & $\alpha$ & $\beta$ & group & name & ref \\
\hline
144 & 55 & 22 & 20 & - & 9 & 87 & 90 & ${\rm M}_{12} . 2$ \\
280 & 36 & 8 & 4 & - & 2 & 1 & 4 & ${\rm HJ} . 2$ & & \cite{Reichard00} \\
300 & 104 & 28 & 40 & - & 8 & 78 & 160 & ${\rm PGO}_5(5)$ & $NO_5^-(5)$ & \S\ref{sec:Oex} \\
325 & 144 & 68 & 60 & - & 30 & 1153 & 900 & ${\rm PGO}_5(5)$ & $NO_5^+(5)$ & \S\ref{sec:Oex} \\
512 & 196 & 60 & 84 & 14 & 20 & 420 & 840 & $2^9 . {\rm \Gamma L}_3(8)$
& dual hyperoval & \S\ref{sec:hyperovals} \\
729 & 112 & 1 & 20 & 0 & 0 & 0 & 0 & $3^6 . 2 . {\rm L}_3(4) . 2$ & Games graph & \cite{BrouwerVanLint84} \\
1120 & 729 & 468 & 486 & 297 & 306 & 69498 & 74358 & ${\rm PSp}_6(3) . 2$
& disj.~t.i.~planes & \S\ref{sec:Sp6} \\
1849 & 462 & 131 & 110 & - & - & 2980 & 1845 & $43^2 {:} (42 {\times} {\rm D}_{22})$
& power~diff.~set & \S\ref{sec:cyclo} \\
\end{tabular}\par}
The numbers $\lambda',\mu'$ give the valency of the $\lambda$- and $\mu$-graphs
in case these are regular (and then $\alpha = \lambda\lambda'/2$ and
$\beta = \mu\mu'/2$).
The examples on 144 and 729 vertices also satisfy the 5-vertex condition.
\subsection{Strongly regular graphs with strongly regular subconstituents}
\label{srgsubs}
As we saw, graphs that are locally strongly regular satisfy
the 4-vertex condition.
Sometimes it follows that also the 2nd subconstituents must be
strongly regular.
\begin{Lemma} \label{lem:paras_for_local_srg}
Suppose that a strongly regular graph with parameters
$(v,k,\lambda,\mu) = (4t^2, 2t^2-\varepsilon t,\allowbreak t^2-\varepsilon t, t^2-\varepsilon t)$
(where $\varepsilon = \pm1$)
has first subconstituents that are strongly regular with parameters
$(v',k',\lambda',\mu') =\big (2t^2-\varepsilon t, t^2-\varepsilon t,
\frac12 t(t - \varepsilon), t(\frac12 t - \varepsilon)\big)$.
Then its second subconstituents are strongly regular
with parameters $(v'',k'',\allowbreak \lambda'',\mu'') = \big(2t^2 + \varepsilon t - 1, t^2,
\frac12 t(t - \varepsilon), \frac12 t^2\big)$.
\end{Lemma}
{\footnotesize
More generally, the spectrum of the 2nd subconstituent at any vertex
of a strongly regular graph follows from that of the 1st subconstituent
---see \cite{CameronGoethalsSeidel78}, Theorem 5.1.\par
}
Call the three parameter sets in the above lemma $A(\varepsilon t)$, $B(\varepsilon t)$,
and $C(\varepsilon t)$, respectively. They occur again in \S\ref{binaryvs}.
The parameter sets $A(t)$ and $A(-t)$ are known as ({\em negative})
{\em Latin square parameters} ${\rm LS}_t(2t)$ (resp. ${\rm NL}_t(2t)$).
The complementary graphs have parameters ${\rm LS}_{t+1}(2t)$
(resp. ${\rm NL}_{t-1}(2t)$).
Cameron, Goethals \& Seidel \cite{CameronGoethalsSeidel78}
studied the situation of a primitive strongly regular graph such that,
for some vertex, both subconstituents are strongly regular, and found
that such a graph either has a vanishing Krein parameter
$q_{11}^1$ or $q_{22}^2$, or has Latin square or negative Latin square
parameters. They conjectured that every non-grid example of the latter
has parameters as in the above lemma or has a complement with
these parameters.
\section{Survey of the known examples and results} \label{sec:knownknowns}
\subsection{Complements}
A graph satisfies the $t$-vertex condition if and only if its complement does.
\subsection{Generalized quadrangles}\label{sec:GQ}
Higman \cite{Higman71} observed that the collinearity graphs of
generalized quadrangles satisfy the 4-vertex condition
(and there are many examples that are not rank~3, cf.~\cite{Kantor86}).
{
\footnotesize
More generally the 4-vertex condition holds for partial quadrangles.
For example, the Hill graph with parameters
$(v,k,\lambda,\mu) = (4096,234,2,14)$
(derived from the cap constructed in \cite{Hill73})
has a rank 10 group and satisfies the 4-vertex condition with
$\alpha=1$, $\beta=0$.
\par
}
Reichard \cite{Reichard15} showed that the collinearity graphs of
generalized quadrangles satisfy the 5-vertex condition, and that the
collinearity graphs of generalized quadrangles ${\rm GQ}(s,s^2)$
satisfy the 7-vertex condition.
{
\footnotesize
More generally the 5-vertex condition holds for partial quadrangles.
\par}
\subsection{Binary vector spaces with a quadratic form}\label{binaryvs}
The first non-rank-3 graph satisfying the 5-vertex condition was
constructed by A. V. Ivanov \cite{Ivanov89}: a strongly regular graph
$\Gamma_0$ whose subconstituents
$\Gamma_1, \Gamma_2$ satisfy the 4-vertex condition.
The parameters are as follows.
{
\footnotesize\noindent
\begin{tabular}{c|cccccccl}
& $v$ & $k$ & $\lambda$ & $\mu$ & $\alpha$ & $\beta$ & $|G|$ & remarks \\
\hline
$\Gamma_0$ & 256 & 120 & 56 & 56 & 784 & 672
& $2^{20} \cdot 3^2 \cdot 5 \cdot 7$
& rank 4: $1+120+120+15$ \\
$\Gamma_1$ & 120 & 56 & 28 & 24 & 216 & 144
& $2^{12} \cdot 3^2 \cdot 5 \cdot 7$
& rank 4: $1+56+56+7$ \\
$\Gamma_2$ & 135 & 64 & 28 & 32 & 168 & 192
& $2^{12} \cdot 3^2 \cdot 5 \cdot 7$
& intransitive: $120+15$
\end{tabular}\par}
In \cite{BrouwerIvanovKlin89} an infinite family of graphs $\Gamma^{(m)}$
($m \ge 1$) is constructed by taking as vertex set ${\mathbb F}ss{2}{2m}$,
where vectors are adjacent when the line joining them~meets
the hyperplane at infinity in a fixed hyperbolic quadric
minus a maximal t.i.~subspace.
The graphs $\Gamma^{(m)}$ have parameters $A(2^{m-1})$ (see \S\ref{srgsubs}).
They have a rank 4 group (for $m \ge 4$) and satisfy the 4-vertex condition.
The local graphs $\Delta^{(m)}$ are strongly regular with parameters
$B(2^{m-1})$.
They have a rank 4 group (for $m \ge 4$) and satisfy the 4-vertex condition.
By Lemma \ref{lem:paras_for_local_srg} also the 2nd subconstituents
${\rm E}^{(m)}$ are strongly regular, with parameters $C(2^{m-1})$.
We checked by computer that the graph $\Gamma^{(4)}$ is isomorphic to the
above $\Gamma_0$.
{
\footnotesize
In \cite{Reichard00} it is shown that the graphs $\Gamma^{(m)}$
satisfy the 5-vertex condition.
In \cite{PechPech19} it is shown that the graphs $\Gamma^{(m)}$
are triplewise 5-regular, a.k.a. (3,5)-regular, where
$(s,t)$-regularity is the analog of the $t$-vertex condition
where $s$ instead of two vertices are distinguished.
It follows that the 2nd subconstituents ${\rm E}^{(m)}$
of the graphs $\Gamma^{(m)}$ also satisfy the 4-vertex condition.
\par}
In \cite{Ivanov94}, two infinite families of graphs are constructed.
One is the above $\Gamma^{(m)}$. The second family has members
$\Sigma^{(m)}$ with vertex set ${\mathbb F}ss{2}{2m}$, where vectors are
adjacent when the line joining them hits the hyperplane at infinity
either in a fixed elliptic quadric minus a maximal t.i.~subspace $S$
or in $S^\perp\backslash S$. The graphs $\Sigma^{(m)}$ have
parameters $A(-2^{m-1})$, have rank 5 (for $m \ge 5$),
and satisfy the 4-vertex condition.
{
\footnotesize
Let $\Gamma(V,X)$ be the graph on a vector space $V$ where two vectors
are adjacent precisely when the joining line hits the subset $X$ of the
hyperplane $PV$ at infinity. Since $\Gamma(V,X)$ is strongly regular
if and only if $X$ is a 2-character set (\cite{Delsarte72a}),
that is, if and only if $|X \cap H|$ takes only two distinct values
when $H$ runs through the hyperplanes of $PV$,
the set $(Q \setminus S) \cup (S^\perp \setminus S)$
is a 2-character set when $Q$ is an elliptic quadric,
and $S$ a maximal t.i. subspace.
Let $V$ be a vector space over ${\mathbb F}_2$.
Then the local graph of $\Gamma(V,X)$ is the collinearity graph
of the partial linear space with point set $X$ and whose lines are the
projective lines (of size 3) contained in $X$.
\par
}
The local graphs ${\rm T}^{(m)}$ are strongly regular with parameters
$B(-2^{m-1})$. They are intransitive (for $m \geq 5$).
\mysqueeze{0.2pt}{It follows from Lemma \ref{lem:paras_for_local_srg} that
also the 2nd subconstituents $\Upsilon^{(m)}$
are~strongly} regular, with parameters $C(-2^{m-1})$.
There is a tower of graphs here:
If $\Upsilon$ is the 2nd subconstituent of $\Sigma^{(m)}$ at a vertex $x$,
and $s \in S$, then the local graph of $\Upsilon$ at its vertex $x+s$
is isomorphic to $\Sigma^{(m-1)}$. (For a proof, see Appendix A.)
{
\footnotesize
In \cite{Ivanov94} it is conjectured that the graphs $\Sigma^{(m)}$
satisfy the 5-vertex condition, and that the graphs ${\rm T}^{(m)}$
and $\Upsilon^{(m)}$ satisfy the 4-vertex condition.
The former was proved in \cite{Reichard00}.
The latter is proved in Appendix~A.
In \cite{PechPech19} it is announced that $\Sigma^{(m)}$ is even
$(3,5)$-regular, but we are not aware of a proof in print.
\par}
\subsection{Block graphs of Steiner triple systems}
Higman \cite{Higman71} investigated for which $v$-point Steiner triple systems
the block graph satisfies the 4-vertex condition.
He found that either the system is a projective space
${\rm PG}(m,2)$ or $v$ is one of
9, 13, 25. In \cite{Kaski-et-al12} the cases 13 and 25 are ruled out,
so that the only other example is the affine plane ${\rm AG}(2,3)$.
The examples are rank 3.
\subsection{Smallest example}
In \cite{Klin-et-al05} it is shown that the smallest non-rank-3
strongly regular graphs satisfying the 4-vertex condition have
$v = 36$ vertices. There are three examples. All have
$(v,k,\lambda,\mu) = (36,14,4,6)$ and $\alpha=0$, $\beta=4$.
\subsection{Cyclotomic examples} \label{sec:cyclo}
Given $(q,e,J)$, where $e \,|\, (q-1)/2$ and $J$ is a set of
nonnegative integers, and a fixed primitive element $\eta$ of ${\mathbb F}_q$,
consider the cyclotomic graph with vertex set ${\mathbb F}_q$, where two elements
are adjacent when their difference is in
$D = \{ \eta^{ie+j} \mid 0 \le i < (q-1)/e, ~~ j \in J \}$.
In some cases this yields a strongly regular graph that satisfies
the 4-vertex condition. We give a few examples.
The examples on $11^2$ and $23^2$ vertices are due to
Klin \& Pech \cite{KlinPech}.
{
\footnotesize
\begin{tabular}{cccccccc}
$q$ & $p^f$ & $e$ & $J$ & $\eta$ & $\alpha$ & $\beta$ & rk \\
\hline
1849 & $43^2$ & 4 & $\{0\}$ & any & 2980 & 1845 & 4 \\
146689 & $383^2$ & 4 & $\{0\}$ & any & 11353825 & 10662960 & 4 \\
121 & $11^2$ & 6 & $\{0,1,2\}$ & any & 200 & 206 & 5 \\
625 & $5^4$ & 6 & $\{0,1,2\}$ & any & 5913 & 6022 & 5 \\
5041 & $71^2$ & 6 & $\{0,1,2\}$ & any & 395641 & 396270 & 5 \\
529 & $23^2$ & 8 & $\{0,1,2,3\}$ & $\eta^2 = \eta+4$ & 4215 & 4300 & 5 \\
\end{tabular}\par}
In all cases $q = p^f$ where $p$ is semiprimitive mod $e$
(that is, $e \,|\, (p^i+1)$ for some $i$),
so that the parameters of the strongly regular graph can be found
in \cite[Thm.~7.3.2]{BrouwerVM21}.
\section{Graphs from hyperovals}\label{sec:hyperovals}
In \cite{HuangHuangLin09}, Huang, Huang \& Lin constructed
various families of graphs. The complement of one of them
can be described as follows (\cite{Brouwer16}).
For $q = 2^m$, take ${\mathbb F}ss{q}{3}$ as the vertex set of $\Gamma$.
Let $\pi$ be the plane at infinity of ${\mathbb F}ss{q}{3}$.
Let $H^*$ be a~dual hyperoval of $\pi$ (that is, a set of
$q+2$ lines, no three on a point).
The plane $\pi$ is partitioned into two parts,
$\frac12 (q+1)(q+2)$ points on two lines of $H^*$
and $\frac12 q(q-1)$ exterior points on no line of $H^*$.
Two vertices of $\Gamma$ are adjacent when the line joining them hits
$\pi$ in one of the exterior points.
Then $\Gamma$ is strongly regular and has parameters
$$
(v, k, \lambda, \mu) =
\big(q^3, \tfrac12 q(q-1)^2, \tfrac14 q(q-2)(q-3), \tfrac14 q(q-1)(q-2)\big).
$$
Its local graphs are strongly regular with parameters
$$
\big(\tfrac12 q(q-1)^2, \tfrac14 q(q-2)(q-3), \tfrac18 q(q^2-9q+22),
\tfrac18q(q-3)(q-4)\big).
$$
Hence, as noted in Section \ref{sec:def}, $\Gamma$ satisfies the
$4$-vertex condition.
If $m=3$, then $\Gamma$ has rank $4$.
\section{Disjoint t.i.~planes in symplectic 6-space}\label{sec:Sp6}
Let $V$ be a 6-dimensional vector space over ${\mathbb F}_q$,
provided with a nondegenerate symplectic form.
Let $\Gamma$ be the graph with as vertices the totally isotropic planes,
adjacent when disjoint.
\begin{Proposition}\label{disj-planes}
The graph $\Gamma$ is strongly regular, with parameters
$v = (q^3+1)(q^2+1)(q+1)$, $k = q^6$,
$\lambda = q^2(q^3-1)(q-1)$, $\mu = (q-1)q^5$.
If $q$ is even, then $\Gamma$ is rank $3$, otherwise rank $4$.
Its local graph $\Delta$ is strongly regular with parameters
$v' = k$, $k' = \lambda$, $\lambda' = \mu' - q^2(q-2)$
and $\mu' = q^2(q-1)(q^3-q^2-1)$.
It follows that $\Gamma$ satisfies the $4$-vertex condition.
\end{Proposition}
For convenience, we give the parameters of $\bar{\Delta}$,
the complement of $\Delta$:\\
$\bar{v} = q^6$, $\bar{k} = (q^2+1)(q^3-1)$,
$\bar{\lambda} = q^4+q^3-q^2-2$, $\bar{\mu} = q^4+q^2$.
{\footnotesize
\noindent{\bf Proof.}\quad
The dual polar graph $\Sigma$ belonging to ${\rm Sp}_6(q)$
is distance-regular of diameter 3 and has eigenvalue $-1$.
It follows that its distance-3 graph $\Gamma$ is strongly regular
(see \cite{BCN}, Prop.~4.2.17).
More generally, the distance 1-or-2 graph of the symplectic
dual polar space ${\rm Sp}_{2m}(q)$ is distance-regular
(cf.~\cite{BCN}, Prop.~9.4.10).
For $m=3$ it is the complement of $\Gamma$.
For any vertex $x$, the subgraph induced by $\Sigma$ on $\Sigma_3(x)$
is isomorphic to the symmetric bilinear forms graph on ${\mathbb F}ss{q}{3}$
(see \cite{BCN}, Prop.~9.5.10).
If $q$ is odd, then distance $j$ ($j=0,1,2,3$) in $\Sigma_3(x)$
corresponds to ${\rm rk}(f-g) = j$ in the symmetric bilinear forms graph
and hence to distance $\lfloor (j+1)/2 \rfloor$ in the quadratic forms
graph (see \cite{BCN}, \S9.6). It follows that $\Delta$ is the
complement of the quadratic forms graph, and has parameters as claimed.
If $q$ is even, then $\Gamma$ is rank 3 (by triality, it is
the complement of the $O_8^+(q)$ polar graph), and $\Delta$
is the complement of the rank 3 graph $\smash{VO_6^+(q)}$,
with parameters as claimed.
$\Box$
\par}
\section{Nonsingular points joined by a tangent}\label{sec:Oex}
Let $V$ be a vector space of dimension $2m+1$ over ${\mathbb F}_q$
with $q$ odd, and let $Q$ be a nondegenerate quadratic form on $V$.
We also use $Q$ as the symbol for the set of singular projective points.
The projective space $PV$ has $(q^{2m+1}-1)/(q-1)$ points,
$(q^{2m}-1)/(q-1)$ singular, and $q^{2m}$ nonsingular.
The nonsingular points come in two types:
there are $\frac12 q^m(q^m + \varepsilon)$ points of type $\varepsilon$
(where $\varepsilon = \pm 1$), with $\varepsilon=+1$ (resp. $-1$) for points $x$
for which $x^\perp$, the hyperplane of points orthogonal to $x$,
is hyperbolic (resp. elliptic).
Consider the graph $NO_{2m+1}^\varepsilon(q)$ that has as vertex set the set
of nonsingular points of type $\varepsilon$, where two points are adjacent
when the joining line is a tangent.
\begin{Proposition} {\rm (Wilbrink \cite{Wilbrink},
cf.~\cite{BrouwerVanLint84})}
Let $m \ge 2$. The graph $NO_{2m+1}^\varepsilon(q)$ is strongly regular
with parameters
$v = \frac12 q^m(q^m + \varepsilon)$,
$k = (q^{m-1}+\varepsilon)(q^m-\varepsilon)$,
$\lambda = 2(q^{2m-2}-1) + \varepsilon q^{m-1}(q-1)$,
$\mu = 2q^{m-1}(q^{m-1}+\varepsilon)$.
\end{Proposition}
For $m=1$, $\varepsilon=-1$ the graph is edgeless.
For $m=1$, $\varepsilon=1$ we have the triangular graph $T(q+1)$.
Wilbrink also handled the case of even $q$.
We give an explicit proof here;
for a different and more general proof see \cite{BannaiHaoSong90}.
{
\footnotesize
\noindent{\bf Proof.}\quad
The neighbors of a vertex $x$ lie on the tangents
joining $x$ with a singular point of $x^\perp$,
and $x^\perp$ has $(q^{m-1}+\varepsilon)(q^m-\varepsilon)/(q-1)$ singular points.
This gives the value of $k$.
A common neighbor $z$ of two adjacent vertices $x,y$
lies on the line $xy$ (and there are $q-2$ choices) or on
some other tangent $T$ on $x$.
In the latter case the plane $\langle x,y,z \rangle$ meets $Q$
in a conic or double line. If it is a conic, then $z$ is
uniquely determined on $T$ by the fact that $yz$ is the tangent
on $y$ other than $xy$. If it is a double line, then each
nonsingular point of $T \setminus \{x\}$ is suitable.
Let $p$ be the singular point on $xy$.
Then $\{p,x\}^\perp/\langle p \rangle$ is a nondegenerate
$(2m-2)$-space of type $\varepsilon$, and has
$a = (q^{m-2}+\varepsilon)(q^{m-1}-\varepsilon)/(q-1)$ singular points.
It follows that $xy$ is in $a$ planes that hit $Q$ in a double line,
and in $q^{2m-2}$ planes that hit $Q$ in a conic.
Consequently, $\lambda = q-2 + q^{2m-2} + (q-1)qa$,
as desired.
A common neighbor $z$ of two nonadjacent vertices $x,y$
determines a nondegenerate plane $\pi = \langle x,y,z \rangle$ in which
$xz$ and $yz$ are tangents, so that $x,y,z$ are exterior points.
Now $x,y$ are on two tangents each, and $\pi$ contains 4
common neighbors of $x,y$.
If $Q$ is a quadratic form on a $(2m+1)$-space, then a point $p$
is exterior if and only if $(-1)^m \det(Q)\,Q(p)$ is a nonzero square.
In order to have $p$ exterior in $\pi$ but a $\varepsilon$-point in $V$,
the $(2m-2)$-space $\pi^\perp$ must be an $\varepsilon$-subspace of the
$(2m-1)$-space $\{x,y\}^\perp$. Since there are
$b = \frac12 q^{m-1}(q^{m-1}+\varepsilon)$ such $\varepsilon$-subspaces,
we find $\mu = 4b$, as desired.
$\Box$
\par}
The automorphism group ${\rm P\Gamma{}O}_{2m+1}(q)$
of the graph contains ${\rm PGO}_{2m+1}(q)$.
The latter has $(q+3)/2$ orbits on pairs of vertices \cite{BannaiHaoSong90}.
Hence, the graph has rank $(q+3)/2$ if $q$ is prime.
For $m=2, \varepsilon = -1$, this is the collinearity graph of a semi-partial geometry
found by Metz. Its lines have size $s+1 = q$ and there are $t+1 = q^2+1$
lines on each point. Each point outside a line has either $0$ or
$\alpha = 2$ neighbors on the line.
See Debroey \cite{Debroey78}, voorbeeld 1.1.3d,
and Debroey-Thas \cite{DebroeyThas78}, example 1.4d,
and Hirschfeld-Thas \cite{HirschfeldThas80}, p.~268,
and Brouwer-van\,Lint \cite{BrouwerVanLint84}, \S7A,
and Brouwer-Van\,Maldeghem \S8.7, example (ix).
For $m=2, \varepsilon = +1$ this is the collinearity graph of a geometry
with $t+1 = (q+1)^2$ lines of size $s+1 = q$ on each point,
such that each point outside a line has 0, 2, or $q$ neighbors on the line
(\cite{BrouwerVanLint84}, \S7B).
We shall prove that these graphs satisfy the 4-vertex condition.
First a lemma.
\begin{Lemma}\label{lem:tri_adj_sd}
Let $S$ be a solid such that $Q\restrictedto{S}$ is nondegenerate.
Let $x, y, z$ be distinct nonsingular points of the same type $\varepsilon$
such that $\langle z, x \rangle$ and $\langle z, y \rangle$ are tangents
and $\langle x, y \rangle$ is nondegenerate.
Put $\pi = \langle x,y,z \rangle$. Then there are
either $0$ or $2$ nonsingular points $w \in S \setminus \pi$ of type $\varepsilon$
such that $\langle x, w \rangle$, $\langle y, w \rangle$,
and $\langle z, w \rangle$ are tangents.
For $x, y, z$ given, the number of $w$ only depends on the type of $S$.
It equals $2$ if and only if the nonzero number
$2(\frac{B(z,z)B(x,y)}{B(x,z)B(y,z)}-1) \det (Q\restrictedto{S})$
is a square.
\end{Lemma}
\noindent{\bf Proof.}\quad
Replace $x$ by $\frac{B(z,z)}{B(x,z)} x$ and
$y$ by $\frac{B(z,z)}{B(y,z)} y$.
Then $B(x,z) = B(z,z) = B(y,z)$.
Put $x_0 = x-z$, $y_0 = y-z$, $w_0 = w-z$, then
$B(x_0,z) = B(y_0,z) = B(w_0,z) = 0$.
Since the lines $\langle z, x \rangle$, $\langle z, y \rangle$, and
$\langle z, w \rangle$ are tangents, the points $x_0,y_0,z_0$ are singular,
that is, $Q(x_0) = Q(y_0) = Q(w_0) = 0$.
The line $\langle x, w \rangle$ is a tangent, so $Q(x+tw) = 0$
has a unique solution $t$. Now
\begin{align*}
Q(x+tw) &= Q(z + x_0 + t(z + w_0)) = Q((1+t)z + x_0 + t w_0)\\
&= (1+t)^2 Q(z) + Q(x_0 + t w_0) = (1+t)^2 Q(z) + tB(x_0, w_0).
\end{align*}
It follows that $(2+\frac{B(x_0, w_0)}{Q(z)})^2 = 4$,
that is $\frac{B(x_0, w_0)}{Q(z)} \in \{ 0, -4 \}$.
As $Q\restrictedto{S}$ is nondegenerate, $z^\perp \cap S$ is
a nondegenerate plane.
If $B(x_0, w_0) = 0$, then $\langle x_0, w_0 \rangle$ is a
totally singular line in this plane, impossible.
Hence, $B(x_0, w_0) = -4 Q(z)$. Similarly, $B(y_0, w_0) = -4 Q(z)$.
In the plane $z^\perp \cap S$, let $u$ be the point of intersection of
the tangents through the points $x_0$ and $y_0$ and write
$w_0 = ax_0 + by_0 + cu$. Then $B(x_0, u) = B(y_0, u) = 0$ and
$-4 Q(z) = B(x_0, w_0) = B(x_0, ax_0 + by_0 + cu) = bB(x_0, y_0)$.
Similarly, $-4 Q(z) = B(y_0, w_0) = aB(x_0, y_0)$,
so that $a = b = \frac{-4Q(z)}{B(x_0, y_0)}$, independent of $w$. Also,
\begin{align*}
0 &= Q(w_0) = Q(ax_0 + by_0 + cu) = abB(x_0, y_0) + c^2 Q(u)
= \frac{16Q(z)^2}{B(x_0, y_0)} + c^2 Q(u).
\end{align*}
If $-B(x_0, y_0)Q(u)$ is a square, then we have two solutions for $c$
(so also $w_0$ and, therefore, $w$) and otherwise none.
Since $u$ is an exterior point in the plane $\sigma = z^\perp \cap S$,
the number $-Q(u) \det Q\restrictedto{\sigma}$ is a square.
Also, $\det Q\restrictedto{S} = Q(z) \det Q\restrictedto{\sigma}$
and $B(x,y) = B(x_0,y_0) + B(z,z)$.
$\Box$
\begin{Proposition}
The graph $NO_{2m+1}^\varepsilon(q)$ satisfies the 4-vertex condition.
\end{Proposition}
\noindent{\bf Proof.}\quad
By Proposition \ref{prop:sims} it suffices to check for $x \ne y$ that
the number of edges in $\Gamma(x) \cap \Gamma(y)$ does not depend on
the choice of the points $x,y$, but only on whether $x,y$
are adjacent or not.
Since ${\rm Aut}~\Gamma$ is edge-transitive,
we only need to check $\Gamma(x) \cap \Gamma(y)$ for $x \not\sim y$.
Claim: this subgraph $\Gamma(x) \cap \Gamma(y)$
is regular of valency $4q^{2m-3} + 3\varepsilon q^{m-1} - 4\varepsilon q^{m-2} - 1$.
In other words, this is the value of $\mu$ in the local graph
(which is regular, but not strongly regular).
If $x \sim z \sim y$, $x \not\sim y$, then $\pi = \langle x,y,z \rangle$
is a nondegenerate plane in which the common neighbors of $x,y$
form a 4-cycle, so that $x,y,z$ have two common neighbors in $\pi$,
say $a$ and $b$.
The plane $\pi$ lies in $(q^{2m-3}-\varepsilon q^{m-2})/2$ solids of type
$O^-(4, q)$, equally many solids of type $O^+(4, q)$,
and $(q^{m-2}+\varepsilon)(q^{m-1}-\varepsilon)/(q-1)$ degenerate solids.
If $S$ is a degenerate solid through $\pi$ with apex $p$, we see that
$w \in S \setminus \pi$ is in $\Gamma(x) \cap \Gamma(y) \cap \Gamma(z)$
if and only if gets projected from $p$ onto an element of $\{ a, b, z \}$
in $\pi$. Hence,
$|\Gamma(x) \cap \Gamma(y) \cap \Gamma(z) \cap S \setminus \pi| = 3(q-1)$.
Hence, the total number of choices for $w$ equals $3(q^{m-2}+\varepsilon)(q^{m-1}-\varepsilon)$.
Now let $S$ be a nondegenerate solid on $\pi$,
and let $p = S \cap \pi^\perp$.
By Lemma \ref{lem:tri_adj_sd}, the number of $w$ in $S$ is 0 or 2,
depending on the determinant of $Q$ restricted to $S$.
Since $\pi^\perp$ contains equally many points $p$ with $Q(p)$
a square as with $Q(p)$ a non-square, the total number of choices
for $w$ equals the number of choices for $p$ which is $q^{2m-3}-\varepsilon q^{m-2}$.
So the induced subgraph on $\Gamma(x) \cap \Gamma(y)$ has valency
$2 + 3(q^{m-2}+\varepsilon)(q^{m-1}-\varepsilon) + (q^{2m-3}-\varepsilon q^{m-2})
= 4q^{2m-3} + 3\varepsilon q^{m-1} - 4\varepsilon q^{m-2} - 1$.
$\Box$
\section{Polar switching}\label{sec:switched_col}
{
\footnotesize
A {\em polar space} is a partial linear space such that for each line $L$
any point outside $L$ is collinear to either all or precisely one of the
points of $L$. A {\em singular subspace} is a line-closed set of points,
any two of which are collinear. The polar space is called
{\em nondegenerate} when no point is collinear to all points.
Finite nondegenerate polar spaces are the sets of totally isotropic
(t.i.) or totally singular (t.s.) points and lines in a vector space
over a finite field provided with a suitable symplectic, quadratic or
hermitian form. The {\em rank} of the polar space is the
(vector space) dimension of its maximal singular subspaces.
\par}
Let ${\bf P}$ be a nondegenerate polar space of rank $d \ge 3$
in a vector space $V$ over ${\mathbb F}_q$. Its collinearity graph
$\Gamma_0$ is strongly regular and satisfies the 4-vertex condition
(since it is rank 3).
We shall construct cospectral graphs that satisfy the 4-vertex condition
(but are not rank 3) by a switching construction.
Let $x^\perp$ be the set of points collinear with $x$ (including $x$ itself).
{
\footnotesize
Suppose $U$ is a maximal singular subspace of ${\bf P}$
(i.e., a maximal clique in $\Gamma_0$), and let $H_1,H_2$ be two
hyperplanes of $U$. We can redefine adjacency and make the points
$x$ with $x^\perp \cap U = H_1$ or $H_2$ adjacent to the points in
$H_2$ or $H_1$, respectively, and leave all other adjacencies unchanged.
This is an example of WQH-switching (Wang, Qiu \& Hu \cite{WangQiuHu19},
cf.~\cite{Ihringer19}) and yields a graph cospectral with $\Gamma_0$.
One can repeat this interchange of hyperplanes and get arbitrary
permutations of all hyperplanes. We generalize this, even allowing
different designs on $U$.
\par}
\subsection{Construction}
Let $P$ be the point set of ${\bf P}$, and let the subset
$U$ be (the set of points of) a totally isotropic $d$-space.
Let ${\bf D}$ be a symmetric design with the same parameters
as the symmetric design of points and hyperplanes of ${\rm PG}(d-1, q)$, so its
parameters are
2-$\big(\downstrut\frac{q^{d}-1}{q-1}, \frac{q^{d-1}-1}{q-1},
\frac{q^{d-2}-1}{q-1} \big)$.
Let $\varphi$ be a bijection from the set ${\mathcal H}$ of hyperplanes of $U$
to the blocks of ${\bf D}$. We assume that the points of $U$ are also
the points of ${\bf D}$.
Following ideas in \cite{Kantor94}
and \cite{DempwolffKantor08}
we define a graph $\Gamma_\varphi$ on the vertex set of $\Gamma_0$ as follows:
\begin{enumerate}
\item Vertices in $U$ are pairwise adjacent.
\item Distinct vertices $x,y \notin U$ are adjacent if $x \in y^\perp$.
\item Vertices $x \in U$, $y \notin U$ are adjacent if $x \in (y^\perp \cap U)^\varphi$.
\end{enumerate}
Clearly, $\Gamma_\varphi = \Gamma_0$ if we take the hyperplanes of $U$
for the blocks of ${\bf D}$ and $\varphi$ as the identity.
\begin{Theorem}
The graph $\Gamma_\varphi$ is strongly regular with
the same parameters as the classical graph $\Gamma_0$.
\end{Theorem}
{\footnotesize
\noindent{\bf Proof.}\quad
Let $x$ and $y$ be any two vertices. We show that
the number of common neighbors $z$ of $x,y$ in $\Gamma_\varphi$
does not depend on $\varphi$ (but depends on whether $x,y$ are equal,
adjacent or nonadjacent in $\Gamma_\varphi$).
If $x,y \in U$, then any $z \in U$ is a common neighbor.
The number of $z \in P \setminus U$ such
that $x,y \in (z^\perp \cap U)^\varphi$ does not depend on $\varphi$:
each hyperplane $H$ of $U$ such that $x, y \in H^\varphi$
contributes $| H^\perp \setminus U |$ such $z$.
Suppose that $x,y \notin U$. Then we are counting the $z$ in
$(x^\perp \cap U)^\varphi \cap (y^\perp \cap U)^\varphi$,
and also the $z$ in $(x^\perp \cap y^\perp) \setminus U$.
The numbers of such $z$ does not depend on $\varphi$.
The remainder of the proof concerns the case $x \in U$, $y \notin U$.
If $z \in U$ then the requirements are $z \ne x$ and
$z \in (y^\perp \cap U)^\varphi$. The number of such $z$
does not depend on $\varphi$.
So we need to count the $z \notin U$. First set $I := y^\perp \cap U$,
so $Y := \langle y, I \rangle$ is totally isotropic.
If $z \in Y$ then $I^\varphi = (z^\perp \cap U)^\varphi$,
and $x,z$ are adjacent if and only if $x,y$ are adjacent. The number of
such $z$ is independent of $\varphi$.
It remains to count the $z$ in $y^\perp \setminus Y$ such that
$x \in (z^\perp \cap U)^\varphi$; here $z^\perp \cap U \ne I$ as $z\notin Y$.
Let $H \neq I$ be a hyperplane of $U$ such that $x \in H^\varphi$.
The number of $H$ does not depend on $\varphi$
(note that $x \in I^\varphi$ if and only if $x,y$ are adjacent in $\Gamma_\varphi$).
We show that the number of $z$ in $y^\perp \setminus Y$
with $z^\perp \cap U = H$ does not depend on $\varphi$ or $H$.
Using bars to project $(H \cap I)^\perp$ into the nondegenerate
rank 2 polar space $(H \cap I)^\perp/(H \cap I)$, we see
totally isotropic lines $\bar U$ and $\bar Y$ meeting at a point $\bar I$,
and a nondegenerate 2-space $\langle \bar y , \bar H \rangle$;
the number of $\bar z$ in $\langle \bar y , \bar H \rangle^\perp\backslash \bar I$
does not depend on $\varphi$ or $H$, so neither does the number of required $z$.
$\Box$
\par
}
\subsection{Isomorphisms}
\subsubsection*{Emptying bijections $\varphi$}
Call a vertex $e \in U$ {\em emptying} for $\varphi$ if
$\bigcap \{ H \mid H \in {\mathcal H},~ e \in H^\varphi \} = \emptyset$.
Call $\varphi$ {\em emptying} if the subspace $U$ is spanned by
emptying vertices.
Call a vertex $f \in U$ {\em dually emptying} for $\varphi$ if
$\bigcap \{ H^\varphi \mid f \in H \in {\mathcal H} \} = \emptyset$.
Call $\varphi$ {\em dually emptying} if the subspace $U$ is spanned by
dually emptying vertices.
{\footnotesize
If $a$ is not emptying, then
$\bigcap \{ H \mid H \in {\mathcal H},~ a \in H^\varphi \} = \{b\}$ for some vertex $b$.
If $b$ is not dually emptying, then
$\bigcap \{ H^\varphi \mid b \in H \in {\mathcal H} \} = \{a\}$ for some vertex $a$.
This establishes a 1-1 correspondence between not emptying vertices $a$
and not dually emptying vertices $b$.
\par}
\begin{Proposition}\label{prop:all_is_empty}
If a permutation $\varphi$ of ${\mathcal H}$ is not dually emptying,
then it is in ${\rm P\Gamma{}L}(U)$.
\end{Proposition}
{\footnotesize
\noindent{\bf Proof.}\quad
Let $E$ denote the set of emptying vertices of $U$, and put
$A = U \setminus E$. Let $F$ denote the set of dually emptying
vertices of $U$, and put $B = U \setminus F$.
Let $\psi \colon B \to A$ be the 1-1 correspondence found above.
We show that if $L$ is a line in $U$ with $|L \cap B| \ge q$,
then $L \subseteq B$ and $L^\psi$ is a line.
Indeed, let $b,b' \in L \cap B$ and set $M = \langleb^\psi, b'^\psi\rangle$.
Then $L \subseteq H$ is equivalent to $M \subseteq H^\varphi$
so that $(L \cap B)^\psi = M \cap A$.
If all points of $L$ are in $B$ with a single exception $w$,
then all points of $M$ are in $A$ with a single exception $v$,
and all hyperplanes $H$ with $w \in H$ satisfy $v \in H^\varphi$
(since every line meets every hyperplane), and $v = w^\psi$,
that is, $w$ was no exception.
If $\varphi$ is not dually emptying, then there
exists a hyperplane $H$ such that $U \setminus H \subseteq B$.
By the above this implies $B = U$ and $\psi$ is in
${\rm P\Gamma{}L}(U)$ and induces $\varphi$ on the set ${\mathcal H}$.
$\Box$
\par}
\subsubsection*{Large cliques}
We use the presence of maximal cliques of various sizes to study
the structure of the graphs $\Gamma_\varphi$ when $\varphi$ is a permutation.
Abbreviate the size $\frac{q^i-1}{q-1}$ of an $i$-space with $m_i$,
so that maximal singular subspaces have size $m_d$.
Since $m_d$ is the Delsarte-Hoffman upper bound for the size of cliques
in $\Gamma_\varphi$, each vertex outside a clique of this size is
adjacent to precisely $m_{d-1}$ vertices inside,
cf. \cite[Proposition 1.1.7]{BrouwerVM21}.
\begin{Lemma}\label{maxcliques}
Let $d \ge 3$.
(i) If $M \ne U$ is a maximal singular subspace of ${\bf P}$,
then $C = (M \setminus U) \,\cup\,
\bigcap \{ H^\varphi \mid M \cap U \subseteq H \in {\mathcal H} \}$
is a maximal clique in $\Gamma_\varphi$ of size at least $q^{d-2}(q+1)$
(and $C \setminus U = M \setminus U$).
(ii) If $C \ne U$ is a maximal clique in $\Gamma_\varphi$ of size
at least $q^{d-2}(q+1)$, then $M = \langle C \setminus U \rangle$ is
a maximal singular subspace of ${\bf P}$.
If, moreover, $|C| = m_d$, then $M \setminus U = C \setminus U$.
\end{Lemma}
\noindent{\bf Proof.}\quad
(i)
Let $M$ be a maximal singular subspace other than $U$.
Then $C = (M \setminus U) \cup
\bigcap \{ H^\varphi \mid M \cap U \subseteq H \in {\mathcal H} \}$
is the largest clique in $\Gamma_\varphi$ containing $M \setminus U$.
(Indeed, the set of hyperplanes of $U$ of the form $m^\perp \cap U$
where $m \in M \setminus U$ equals the set of hyperplanes containing
$M \cap U$, so $C$ is a clique. No further point outside $U \cup C$ can be
adjacent to all of $C$, since $|M \setminus U| > m_{d-1}$.)
If $\dim M \cap U = d-1$, then $|C| = |M| = m_d$.
If $\dim M \cap U \le d-2$, then $|C| \ge |M \setminus U|
\ge m_d - m_{d-2} = q^{d-2}(q+1)$.
(ii)
Let $C \ne U$ be a maximal clique of size at least $q^{d-2}(q+1)$.
If $|C \setminus U| \le m_{d-1}$, then
$|C \cap U| \ge q^{d-2}(q+1) - m_{d-1} > m_{d-2}$.
The set $C \cap U$ is the intersection of sets $H^\varphi$,
each of size $m_{d-1}$, and any two distinct such sets meet in
$m_{d-2}$ points. It follows that no two different $H$ occur, that is,
$H = c^\perp \cap U$ is independent of the choice of $c \in C \setminus U$.
Now $C$ is contained in, and hence equals, $H^\varphi \cup (C \setminus U)$,
and $|C \setminus U| = m_d - m_{d-1} > m_{d-1}$, a contradiction.
If $S$ is a clique in $\Gamma_0$, then also $\langle S \rangle$ is a clique
in $\Gamma_0$. In particular, $\langle C \setminus U \rangle$ is
a singular subspace. It is maximal since $|\langle C \setminus U \rangle| > m_{d-1}$.
If $|C| = m_d$, then each vertex outside $C$ is adjacent to
precisely $m_{d-1}$ vertices inside. Hence no point
outside $C \cup U$ can be adjacent to all of $C \setminus U$.
$\Box$
\begin{Lemma}\label{lem:detU}
If the permutation $\varphi$ is dually emptying,
then $U$ is uniquely determined within the graph $\Gamma_\varphi$.
\end{Lemma}
\noindent{\bf Proof.}\quad
The subspace $U$ is a clique of size $m_d$ in $\Gamma_\varphi$,
with the two properties
(i) in the subgraph induced on its complement $P \setminus U$
all maximal cliques $N$ have size $m_d - m_i$
(where $m_i = | \langle N \rangle \cap U |$) for some $i$, $0 \le i \le d-1$,
and
(ii) the number of maximal cliques of size $m_d$ disjoint from $U$
equals the number of maximal singular subspaces disjoint from any
given one.
Let $E \ne U$ be a clique of $\Gamma_\varphi$ of size $m_d$
with the same two properties.
First we use (i) to see that $E \cap U$ must be a hyperplane in $U$.
Since $E$ is a maximal clique, and $\varphi$ is
a permutation, $E \cap U$ is an intersection of hyperplanes
and hence a subspace of $U$. By hypothesis, we can find a
dually emptying point $f$ of $U$ not in $E$.
If $g \in f^\perp \cap (E \setminus U)$
($g$ will exist unless $f^\perp \cap E = U \cap E$)
and $M$ is a maximal singular subspace containing $f$ and $g$,
and meeting $U$ in $\{f\}$, then $C = M \setminus \{ f \}$
is a maximal clique in $\Gamma_\varphi$ of size $m_d-1$.
And $N = C \setminus E$ is a maximal clique in $P \setminus E$
of size $m_d - m_i - 1$ in case $|M \cap E| = m_i$.
(Note that $C \setminus U = M \setminus U$.)
{\footnotesize
Why is N maximal? No point can be added since $|N| > m_{d-1}$,
unless $q=2$ and $|N|=|M \cap E|=m_{d-1}$. In that case,
no point outside U can be added since $\langleN\rangle=M$.
And no point inside $U$ can be added since $N$ determines all
hyperplanes on $f$, and $f$ is dually emptying.
\par}
Since $M \cap E \ne \emptyset$, we have $1 \le i \le d-1$,
and $m_d - m_i - 1$ is not of the form $m_d - m_h$, violating (i).
Therefore, $f^\perp \cap E = U \cap E$, so that
$H = \langleE \setminus U \rangle \cap U$ and $H^\varphi = E \cap U$ are hyperplanes.
Now we use (ii) to arrive at a contradiction.
We claim that if a maximal clique $F$ of size $m_d$ is disjoint from
$E$, then $\langle F \setminus U \rangle$ is disjoint from $\langle E \setminus U \rangle$.
Suppose not. Since $\langle E \setminus U \rangle \setminus U = E \setminus U$
and $\langle F \setminus U \rangle \setminus U = F \setminus U$ by Lemma \ref{maxcliques}(ii),
a common vertex must lie in $U$.
If $\langle F \setminus U \rangle$ meets $U$ in $m_e$ vertices with $e \geq 2$,
then $F$ meets $U$ in a subspace of dimension $e$, but that would meet
$H^\varphi$, impossible.
So, $\langle F \setminus U \rangle$ meets $U$ in a singleton $\{ f \}$
on the hyperplane $H$.
As $F$ has size $m_d$, $f$ is not dually emptying,
so $\bigcap \{ H^\varphi \mid f \in H \} = \{ f' \}$ for some point $f'$.
Now $f' \in E \cap F$, a contradiction. This shows our claim.
By the claim and Lemma \ref{maxcliques}, we have an injection from the set
of maximal cliques of size $m_d$ disjoint from $E$ into the set of
maximal singular subspaces disjoint from $\langle E \setminus U \rangle$.
Since $E$ satisfies (ii), both sets have the same size, so
the injection is also a surjection.
On the other hand, since $\varphi$ is dually emptying, there is a
dually emptying point $o$ in $U \setminus H$. This $o$ lies in
a maximal singular subspace $O$ disjoint from $\langle E \setminus U \rangle$,
and this $O$ is not in the image of the surjection. Contradiction.
$\Box$
\begin{Lemma}\label{lem:extending}
Let ${\bf P}$ be a nondegenerate polar space with point set $P$,
and $U$ a maximal totally isotropic subspace.
Let $h \colon P \setminus U \to P \setminus U$ be a bijection preserving
collinearity. Then $h$ can be uniquely extended to an automorphism
$h'$ of ${\bf P}$.
\end{Lemma}
{\footnotesize
\noindent{\bf Proof.}\quad
Indeed, we can extend $h$ as follows.
For $u \in U$, let $R$ be a maximal t.i.~subspace with $U \cap R = \{u\}$.
Then $R \setminus \{u\}$ is a subspace of ${\bf L}$ of size
$|U| - 1$ and is mapped by $h$ to a similar subspace $S$.
In ${\bf P}$ this subspace is contained in a unique maximal t.i.~subspace
$\langleS\rangle$ ($= S^\perp$) and we can define $h'(u) = v$ when
$\langleS\rangle \setminus S = \{v\}$.
This is well-defined: if $R'$ is a maximal t.i.~subspace
with $U \cap R' = \{u\}$ and $R$, $R'$ meet in codimension 1,
and $h$ maps $R' \setminus \{u\}$ to $S'$, then
$\langle S \cap S' \rangle = (S \cap S') \cup \{v\}$.
Since the graph on such subspaces $R$, adjacent when they meet
in codimension 1, is connected, $v$ is well-defined.
This preserves orthogonality: if $u \in x^\perp$, then there is
a maximal t.i.~subspace $R$ containing $u,x$ with $R \cap U = \{u\}$.
Now $h(u) = v$ lies in the t.i.~subspace $\langle h(R \setminus \{u\}) \rangle$
which also contains $h(x)$.
$\Box$
\par}
\begin{Proposition}\label{prop:isos_kantor}
\mysqueeze{0.378pt}{Let ${\bf P}$ be a nondegenerate polar space and $U$ a maximal t.i.~subspace.}
Let $\varphi$ and $\chi$ be permutations of ${\mathcal H}$ such that
$\Gamma_\varphi$ is isomorphic to $\Gamma_\chi$. Then $\varphi$ and $\chi$ are
in the same ${\rm P\Gamma{}L}(U)$-double coset in Sym$({\mathcal H})$.
\end{Proposition}
\noindent{\bf Proof.}\quad
If $\varphi \in {\rm P\Gamma{}L}(U)$, then $\Gamma_\varphi$ is isomorphic to
$\Gamma_0$ and its group of automorphisms is transitive on the set of
maximal singular subspaces.
If $\varphi \notin {\rm P\Gamma{}L}(U)$, then according to
Lemma \ref{lem:detU} and Proposition \ref{prop:all_is_empty}
the maximal singular subspace $U$ can be recognized
in $\Gamma_\varphi$, and hence $\Gamma_\varphi$ is not isomorphic to $\Gamma_0$.
Since by assumption $\Gamma_\varphi$ and $\Gamma_\chi$ are isomorphic,
either both or neither are isomorphic to $\Gamma_0$.
In the former case both $\varphi$ and $\chi$ are in ${\rm P\Gamma{}L}(U)$
and the claim holds. Assume in the following that $\varphi$ and $\chi$
are not in ${\rm P\Gamma{}L}(U)$.
We have the set $P$, the point set of ${\bf P}$,
with three structures defined on it.
The polar space structure ${\bf P}$, with relation $\perp$,
and the two graph structures $\Gamma_\varphi$ and $\Gamma_\chi$.
We translate what it means for $\Gamma_\varphi$ and $\Gamma_\chi$
to be isomorphic in terms of the polar space.
Let $g: \Gamma_\varphi \rightarrow \Gamma_\chi$ be an isomorphism.
By Lemma \ref{lem:detU}, it sends $U$ to itself.
The number of common neighbors of a triple of points in $U$
equals $\lambda-1$ for collinear triples and is smaller
for noncollinear triples. It follows that $g$ preserves
projective lines in $U$, and hence induces a permutation
$\bar{g}$ of ${\mathcal H}$ that is in ${\rm P\Gamma{}L}(U)$.
Let $h$ denote the restriction of $g$ to $P \setminus U$.
Then $h$ preserves collinearity (since we have
$\{x,y,z\}^\perp \cap (P \setminus U) = \{x,y\}^\perp \cap (P \setminus U)$
for a triple of pairwise adjacent points $x,y,z$ of $P \setminus U$
if and only if $x,y,z$ are collinear).
By Lemma \ref{lem:extending}, $h$ can be uniquely extended
to an automorphism $h'$ of ${\bf P}$.
Let $\bar{h}$ be the permutation of ${\mathcal H}$ induced by $h'$.
Then $\bar{h} \in {\rm P\Gamma{}L}(U)$.
For $x \in U$ and $y \notin U$, if $x$ and $y$ are adjacent in
$\Gamma_\varphi$, then $x^g$ and $y^g$ are adjacent in $\Gamma_\chi$.
This says that $x \in (y^\perp \cap U)^\varphi$ implies that
$x^g \in (y^{g\perp} \cap U)^\chi$: $g$ maps the points of any hyperplane
of $U$ to the points of another hyperplane.
Then $(y^\perp \cap U)^{\varphi g} = (y^{g\perp} \cap U)^\chi
= (y^{h\perp} \cap U)^\chi = (y^\perp \cap U)^{\bar{h}\chi}$,
so that $\varphi\bar{g} = \bar{h}\chi$.
$\Box$
\begin{Theorem}\label{thm:low_bnd}
Let $d \ge 3$. There are at least $q^{d-2}!$ pairwise
nonisomorphic strongly regular graphs having the same parameters
as the collinearity graph $\Gamma_0$ of the polar space ${\bf P}$.
\end{Theorem}
\noindent{\bf Proof.}\quad
Let $q = p^e$, where $p$ is prime.
Then $|{\rm P\Gamma{}L}(U)| < e q^{d^2}$.
In view of Proposition \ref{prop:isos_kantor}, we have obtained at least
$m_d!/|{\rm P\Gamma{}L}(U)|^2 > q^{d-2}!$
pairwise nonisomorphic strongly regular graphs unless $(d, q) = (3, 2)$.
For $(d, q) = (3, 2)$, we have four ${\rm P\Gamma{}L}(U)$-double cosets
in Sym$({\mathcal H})$.
$\Box$
Similar estimates would follow if one generalized Lemma \ref{lem:detU}
to show that $U$ is uniquely determined in ${\bf P}$ for arbitrary designs
${\bf D}$ (that is, for $\varphi$ that are not permutations).
The blocks of ${\bf D}$ are then found as
$\{ \Gamma_\varphi(x) \cap U \mid x \in P \setminus U \}$.
In \cite[Corollary 3.2]{Kantor94} it is shown that
for $d \geq 4$ there are at least $q^{d-2}!$
choices for ${\bf D}$.
Hence, one would obtain the same estimate as in Theorem \ref{thm:low_bnd}
for $d \geq 4$.
\subsection{Switched symplectic graphs with 4-vertex condition}
We show that in the symplectic case the graphs
$\Gamma_\varphi$ satisfy the 4-vertex condition.
Let ${\bf P}$ be ${\rm Sp}_{2d}(q)$, and
let $V$ be a $2d$-dimensional vector space over ${\mathbb F}_q$,
provided with a nondegenerate symplectic form.
{
\footnotesize
The parameters of $\Gamma_0$ are
$v=(q^{2d}-1)/(q-1)$,
$k=q(q^{2d-2}-1)/(q-1)$,
$v-k-1=q^{2d-1}$,
$\lambda=q^2(q^{2d-4}-1)/(q-1)+q-1$,
$\mu=(q^{2d-2}-1)/(q-1)$
and
$\binom{\lambda}{2} - \alpha = \frac12 q^{2d-1}(q^{2d-4}-1)/(q-1)$,
$\beta = \frac12 q(q^{2d-2}-1)(q^{2d-4}-1)/(q-1)^2$,
and those of $\Gamma_\varphi$ will turn out to be the same.
\par
}
\begin{Proposition}
The graph $\Gamma_\varphi$ satisfies the $4$-vertex condition.
\end{Proposition}
\noindent{\bf Proof.}\quad
Let $x,y$ be two vertices of $\Gamma_\varphi$.
We show that the number of edges in $\Gamma_\varphi(x) \cap \Gamma_\varphi(y)$
is independent of $\varphi$, and only depends on whether $x,y$ are adjacent
or nonadjacent.
Since $\Gamma_0$ satisfies the 4-vertex condition, $\Gamma_\varphi$ does too.
Count edges $ab$ in $\Gamma_\varphi(x) \cap \Gamma_\varphi(y)$.
The vertices $x,y,a,b$ are pairwise adjacent, except that
$x$ and $y$ need not be adjacent. We distinguish several cases depending on
which of $x,y,a,b$ are in $U$. Each of the separate counts will be
independent of $\varphi$.
If $x \notin U$ then let $X = x^\perp \cap U$.
If $y \notin U$ then let $Y = y^\perp \cap U$.
\paragraph*{Case $x,y,a,b \notin U$.}
In this case adjacencies and counts do not involve $\varphi$.
\paragraph*{Case $a,b \in U$.}
Here $a,b$ must be chosen distinct from $x,y$ in case
$x,y \in U$, or distinct from $x$ and in $Y^\varphi$ in case
$x \in U$, $y \notin U$ (and the count depends on whether $x \sim y$), or in
$X^\varphi \cap Y^\varphi$ in case $x,y \notin U$
(and the count depends on whether $X = Y$).
In all cases the count is independent of $\varphi$.
\paragraph*{Case $x,y,a \in U$, $b \notin U$.}
For each hyperplane $H$ such that $x,y \in H^\varphi$ we count
the $b \in H^\perp \setminus U$ and the $a \in H^\varphi$
distinct from $x,y$.
\paragraph*{Case $x,y \in U$, $a,b \notin U$.}
For any two hyperplanes $H,H'$ of $U$ with $x, y \in H^\varphi \cap H'^\varphi$
count adjacent $a,b$ with $a \in H^\perp \setminus U$
and $b \in H'^\perp \setminus U$.
(The counts will depend on whether $H = H'$, but not on $\varphi$.)
\paragraph*{Case $x,a \in U$, $y,b \notin U$.}
For each hyperplane $H$ with $x \in H^\varphi$, count
the $a \in H^\varphi \cap Y^\varphi$ distinct from $x$, and
$b \in H^\perp \setminus U$ adjacent to $y$.
(Here $H = Y$ occurs when $x \sim y$. The counts for $H \ne Y$
do not depend on $H$.)
\paragraph*{Case $x \in U$, $y,a,b \notin U$.}
For any two hyperplanes $H,H'$ with $x \in H^\varphi \cap H'^\varphi$, count
edges $ab$ with $a \in H^\perp$ and $b \in H'^\perp$
in $y^\perp \setminus (U \cup \{y\})$.
(Here $H = Y$ or $H' = Y$ occur when $x \sim y$.
The counts for $H,H' \ne Y$ do not depend on the hyperplanes chosen
but only on whether $H = Y$ or $H' = Y$ or $H = H'$.)
Finally the least trivial case.
\paragraph*{Case $a \in U$, $x,y,b \notin U$.}
Count $a,H,b$ with $a \in X^\varphi \cap Y^\varphi$ and $H$
a hyperplane of $U$ on $a$ and $b \in \langlex,y,H\rangle^\perp
\setminus (U \cup \{x,y\})$. The count for $a$ depends
on whether $X = Y$, that for $b$ depends on whether
$H = X$ or $H =Y$ or $H \supseteq X \cap Y$,
but does not otherwise depend on the choice of $H$.
Since all counts were independent of $\varphi$, this proves
our proposition.
$\Box$
By Theorem \ref{thm:low_bnd}, this shows that there
are many strongly regular graphs which satisfy the
4-vertex condition. But we still have to show the
simplified version of this statement given in
the introduction as Theorem \ref{thm:4vtxsrgs}.
\noindent{\bf Proof of Theorem \ref{thm:4vtxsrgs}.}\quad
Note that here $v$ refers to a nonnegative integer
as in Theorem \ref{thm:4vtxsrgs} and no longer is the
number of vertices in $\Gamma_\varphi$.
Apply Theorem \ref{thm:low_bnd} for $d=3$ to find at least
$q!$ strongly regular graphs satisfying the 4-vertex condition
on $\tilde{v}$ vertices, for $\tilde{v} = \smash{\frac{q^6-1}{q-1}}$.
Given $v$, there is a prime $q$ between $v^{1/6}$ and $2v^{1/6}$
by Bertrand's postulate. Now $\tilde{v} < 2q^5 < 64 v^{5/6} < v$
for $v > 2^{36}$.
Checking the prime powers $q$ for $7 \le q \le 64$
one sees that there is a $q$ with $\tilde{v} \le v \le q^6$
for $v \ge 19608$.
One easily verifies the assertion for $v < 19608$ using rank 3 graphs.
$\Box$
{
\footnotesize
Further graphs with the same parameters satisfy the 4-vertex condition.
Additional examples can be obtained by repeated WQH-switching,
see \S\ref{subsec:small} and \cite{Ihringer19}, and there are more examples
among the graphs constructed in \cite{Ihringer17}.
We have not tried (much) to determine precisely which
graphs in \cite{Ihringer17} do satisfy the 4-vertex condition. Similarly,
we do not know when WQH-switching preserves the 4-vertex
condition.
\par}
\subsection{Small examples}\label{subsec:small}
\subsubsection*{Examples on 63 vertices}
In \cite{Ihringer20} a large number of strongly regular graphs are found
by applying GM-switching to the ${\rm Sp}_6(2)$ polar graph.
Among these are 280 non-rank-3 strongly \mysqueeze{0.25pt}{regular graphs
with $(v,k,\lambda,\mu) = (63, 30, 13, 15)$ satisfying
the 4-vertex condition. All have $\alpha=30$ and $\beta=45$.
Three of these are among the $\Gamma_\varphi$ constructed above.}
We list for each occurring group size the number of examples found.
{
\footnotesize\noindent
\setlength{\tabcolsep}{4pt}
\begin{tabular}{c|cccccccccccccccc}
$|G|$ & 4 & 8 & 16 & 32 & 48 & 64 & 96 & 128 & 192 & 256 & 384 & 512 & 768 & 1344 & 1536 & 4608 \\
\hline
\# & 3 & 16 & 76 & 62 & 1 & 60 & 2 & 30 & 5 & 12 & 3 & 3 & 2 & 1 & 3 & 1
\end{tabular}
\par
}
None of these examples has a transitive group.
We list the orbit lengths in the seven cases with fewer than six orbits.
{
\footnotesize\noindent
\begin{tabular}{c|cccccccccccccccc}
$|G|$ & 768 & 768 & 1344 & 1536 & 1536 (twice) & 4608 \\
\hline
orbits & $3{+}12{+}48$ & $1{+}6{+}24{+}32$ & $7 {+} 56$ & $1{+}6{+}24{+}32$ & $3{+}4{+}8{+}48$ & $3{+}12{+}48$
\end{tabular}
\par
}
\subsubsection*{Permutations of hyperplanes}
Let ${\bf P}$ be ${\rm Sp}_{2d}(q)$, and let $\varphi$ be a permutation
of the set ${\mathcal H}$ of hyperplanes of $U$.
For $(d, q) = (3, 2)$, $(3, 3)$, $(4, 2)$, the number of double cosets
of ${\rm P\Gamma{}L}(d, q)$ in ${\rm Sym}({\mathcal H})$
is $4$, $252$, and $3374$, respectively, and these are the numbers
of non-isomorphic graphs $\Gamma_\varphi$. In each case, exactly one has rank $3$.
None of the others has a transitive group (since $U$ can be recognized).
The pointwise stabiliser of $U$ in ${\rm Aut}(\Gamma_0)$ has size
$N = q^{\binom{d+1}{2}} (q-1)$
and is always contained in ${\rm Aut}(\Gamma_\varphi)$.
Hence, $N$ divides $|{\rm Aut}(\Gamma_\varphi)|$.
{\em Case $(d, q) = (3, 3)$.}
Here $N = 1458$.
We list the group sizes for the 251 graphs $\Gamma_\varphi$ other than
$\Gamma_0$.
{
\footnotesize\noindent
\setlength{\tabcolsep}{4pt}
\begin{tabular}{c|cccccccccccccc}
$|G|/N$ & 1 & 2 & 3 & 4 & 6 & 8 & 12 & 16 & 18 & 24 & 39 & 54 & 72 & 144 \\
\hline
\# & 172 & 26 & 29 & 6 & 3 & 2 & 2 & 2 & 1 & 1 & 3 & 1 & 2 & 1
\end{tabular}
\par
}
We list the orbit lengths in the five cases with fewer than six orbits.
{
\footnotesize\noindent
\begin{tabular}{c|ccc}
$|G|/N$ & 39 (thrice) & 72 & 144 \\
\hline
orbits & $13{+}351$ & $1{+}12{+}108{+}243$ & $1{+}12{+}108{+}243$
\end{tabular}
\par}
{\em Case $(d, q) = (4, 2)$.}
Here $N = 1024$.
We list the group sizes for the 3373 graphs $\Gamma_\varphi$ other than
$\Gamma_0$.
{
\footnotesize\noindent
\setlength{\tabcolsep}{2.75pt}
\begin{tabular}{c|cccccccccccccccccccc}
$|G|/N$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 12 & 16 & 18 & 21 & 24 & 32 & 56 & 60 & 96 & 192 & 288 & 1344 \\
\hline
\# & 3148 & 85 & 40 & 24 & 4 & 10 & 6 & 26 & 1 & 4 & 1 & 2 & 11 & 2 & 2 & 1 & 2 & 2 & 1 & 1
\end{tabular}
\par
}
We list the orbit lengths in the eight cases with fewer than six orbits.
{
\footnotesize\noindent
\begin{tabular}{c|cccccc}
$|G|/N$ & 12 & 18 & 24 & 56 (twice) \\
\hline
orbits & $3{+}12{+}48{+}192$ & $6 {+} 9 {+} 96 {+} 144$ & $3 {+} 12 {+} 48 {+} 192$ & $1 {+} 14 {+} 112 {+} 128$
\end{tabular}
\noindent
\begin{tabular}{c|ccccc}
$|G|/N$ & 60 & 288 & 1344 \\
\hline
orbits & $15 {+} 240$ & $3{+}12{+}48{+}192$ & $7{+}8{+}16{+}224$
\end{tabular}
\par
}
\subsubsection*{Other polar spaces}
We made the same exhaustive investigation of all permutations $\varphi$
for the other choices of ${\bf P}$ in the cases
$(d, q) \in \{ (3, 2), (3, 3), (4, 2) \}$.
The only non-rank-3 examples satisfying the
$4$-vertex condition occur for ${\rm O}_7(3)$.
Here we obtain $252$ graphs in total, of which one is rank $3$,
and three more satisfy the $4$-vertex condition.
They all have two orbits (of sizes $13{+}351$) and
an automorphism group of size 56862.
All other graphs $\Gamma_\varphi$ obtained from $O_7(3)$ have
more than two orbits.
One might wonder whether
a graph $\Gamma_\varphi$ from ${\rm O}_{2d+1}(q)$ satisfies
the $4$-vertex condition if and only if it has
at most two orbits. And whether a non-rank-3 graph $\Gamma_\varphi$
can only satisfy the 4-vertex condition if ${\bf P}$
is ${\rm Sp}_{2d}(q)$ or ${\rm O}_{2d+1}(q)$.
\subsubsection*{Other designs}
There are four 2-$(15, 7, 3)$ designs ${\bf D}$
other than that of the hyperplanes of ${\rm PG}(3,2)$.
We investigated the case where $(d,q) = (4, 2)$ and ${\bf P}$ is ${\rm Sp}_2(8)$,
so that the resulting examples satisfy the 4-vertex condition.
We generated several hundred thousand graphs $\Gamma_\varphi$
for each of these designs.
None of these graphs occurs for two different designs.
We believe our enumeration to be complete.
\begin{tabular}{cccc}
$|{\rm Aut}({\bf D})|$ & point orbits & block orbits & \# $\Gamma_\varphi$ \\ \hline
576 & $3{+}12$ & $3{+}12$ & 113519 \\
168 & $7{+}8$ & $1{+}14$ & 340730 \\
168 & $1{+}14$ & $7{+}8$ & 328078 \\
96 & $1{+}6{+}8$ & $1{+}6{+}8$ & 677460
\end{tabular}
\begin{appendix}
\section*{Appendix A --- Details on Ivanov's graphs}
{\footnotesize
In Section \ref{binaryvs} we discussed the graphs
$\Gamma^{(m)}$ from \cite{BrouwerIvanovKlin89}
and $\Sigma^{(m)}$ from \cite{Ivanov94}.
Here we give some more detail on the latter.
For $m \ge 2$, consider $V = {\mathbb F}ss{2}{2m}$ provided with
the elliptic quadratic form $q(x) = x\subsupr{1}{2} + x\subsupr{2}{2} +
x_1x_2 + x_3x_4 + ... + x_{2m-1}x_{2m}$.
Identify the set of projective points (1-spaces) in $V$
with $V^* = V \setminus \{0\}$.
Let $Q = \{ x \in V^* \mid q(x) = 0 \}$ and let $S$ be the
maximal t.s.~subspace given by $S = \{ x \in V^* \mid
x_1 = x_2 = 0 ~{\rm and}~ x_{2i-1}=0 ~(2 \le i \le m) \}$.
Then $S^\perp = \{ x \in V^* \mid x_{2i-1}=0 \allowbreak ~(2 \le i \le m) \}$.
The graph $\Sigma^{(m)}$ has $V$ as vertex set, where two distinct vertices
$v,w$ are adjacent when $v-w \in (Q \cup S^\perp) \setminus S$.
Let ${\rm T}^{(m)}$ and $\Upsilon^{(m)}$ be the induced
subgraphs on the neighbors (nonneighbors) of the vertex 0.
Put $R = V^* \setminus (Q \cup S^\perp)$.
\noindent{\bf Proposition.}
(i) For $m \le 4$, the graphs $\Sigma^{(m)}$ are rank $3$,
and are isomorphic to the complement of $VO_{2m}^-(2)$.
(ii) For $m \ge 5$, the automorphism group of ${\rm T}^{(m)}$ has
two vertex orbits $S^\perp \setminus S$ and $Q \setminus S$,
of sizes $3 \cdot 2^{m-1}$ and $2^{2m-1}-2^m$, respectively.
For $2 \le m \le 4$, the group is rank $3$,
and the graph is the complement of $NO_{2m}^-(2)$.
(iii) For $m \ge 5$, the automorphism group of $\Upsilon^{(m)}$
has two vertex orbits $S$ and $R$ of sizes
$2^{m-1}-1$ and $2^{2m-1}-2^m$, respectively.
For $3 \le m \le 4$, the group is rank $3$,
and the graph is the complement of $O_{2m}^-(2)$.
(iv) The $\lambda$- and $\mu$-graphs in $\Upsilon^{(m)}$
and the $\mu$-graphs in ${\rm T}^{(m)}$ are all regular
of valency $2^{m-2}(2^{m-2}+1)$.
In particular, $\Upsilon^{(m)}$ satisfies the 4-vertex condition.
(v) The $\lambda$-graphs in ${\rm T}^{(m)}$ have vertices of valencies
in $0$, $2^{2m-4}-2^m$, $2^{2m-4}$, $2^{2m-3}-2^m$.
Edges not in a line contained in $Q$ have $\lambda$-graphs
with a single isolated vertex and $\lambda-1$ vertices of valency $2^{2m-4}$.
For edges in a line contained in $Q$ the $\lambda$-graphs have
a single vertex with valency $2^{2m-3}-2^m$, and
$2^{m-3}-1$ vertices with valency $2^{2m-4}-2^m$,
and the remaining $2^{2m-3}+2^{m-3}$ vertices have valency $2^{2m-4}$.
In particular, ${\rm T}^{(m)}$ satisfies the 4-vertex condition,
with $\alpha = 2^{2m-5}(2^{2m-3}+2^{m-2}-1)$ and
$\beta = \frac12 \mu \mu' = 2^{2m-4} (2^{m-2}+1)^2$.
(vi) The local graph of $\Upsilon^{(m)}$ at a vertex $s \in S$
is isomorphic to $\Sigma^{(m-1)}$.
\noindent \noindent{\bf Proof.}\quad
(i)--(iii) This is clear, and can also be found in \cite{Ivanov94}.
(iv)-(v) (the part about ${\rm T}^{(m)}$):
Let $(v,w) = q(v+w)-q(v)-q(w)$ be the symmetric bilinear form
belonging~to~$q$.
Let $X = (Q \cup S^\perp) \setminus S$. Then ${\rm T}^{(m)}$
is the graph with vertex set $X$, where two vertices $x,y$ are adjacent
when the projective line $\{x,y,x+y\}$ they span is contained in $X$.
If at least one of $x,y$ is in $S^\perp \setminus S$, then this is
equivalent to $(x,y)=1$. If both are in $Q \setminus S$, then this is
equivalent to ($(x,y)=0$ and $x+y \notin S$) or ($(x,y)=1$ and
$x+y \in S^\perp \setminus S$).
Let $x,y,z$ be pairwise adjacent vertices. The valency $c$ of $z$ in
the $\lambda$-graph $\lambda(x,y)$ is the number of common neighbors
of $x,y,z$. Distinguish several cases.
If $z = x+y$, then if $x,y,z \in Q$ we find
$c = | \{x,y\}^\perp \cap (Q \setminus S) | - 3 = 2^{2m-3}-2^m$.
If $z = x+y$ and at least one of $x,y,z$ lies in $S^\perp$, then $c = 0$.
Now let $z \ne x+y$.
The claims are true for $m \le 4$. Let $m \ge 5$ and use induction on $m$.
Choose coordinates so that $x,y,z$ have final coordinates $00$
and let $x',y',z'$ be these points without the final two coordinates.
If they have $c'$ common neighbors $w'$ in ${\rm T}^{(m-1)}$,
then we find $2c'$ common neighbors $w = (w',0,*)$.
Moreover (since $x,y,z$ are linearly independent), we find
$2^{2m-5}$ common neighbors $(w',1,q'(w'))$ in $Q$, where $w'$ runs through
all vectors with the desired inner products with $x',y',z'$.
Altogether $c = 2c'+2^{2m-5}$, as claimed.
For the $\mu$-graphs the argument is similar and simpler:
by the definition of adjacency three dependent vertices are pairwise adjacent,
so that the case $z = x+y$ does not occur here.
(iv) (the part about $\Upsilon^{(m)}$):
Let $Y = V^* \setminus X$. Then $\Upsilon^{(m)}$ is the graph
with vertex set $Y$, where two vertices $x,y$ are adjacent when
the projective line $\{x,y,x+y\}$ they span is not contained in $Y$.
The same argument as before yields the valencies of the
$\lambda$- and $\mu$-graphs.
(vi) Consider the graph $\Sigma^{(m)}$.
The nonneighbors $z$ of 0 that are neighbors of $s$ are the vertices
of the form $z=s+b$ with $z \in S \cup R$
and $b \in (Q \cup S^\perp) \setminus S$.
It follows that $s+z \in Q \setminus s^\perp$.
Let $s = (0\ldots 01)$, then $Q \setminus s^\perp$ can be identified with
$W = {\mathbb F}ss{2}{2m-2}$ via $w \to i(w)=(w,1,\bar{q}(w))$
for $w \in {\mathbb F}ss{2}{2m-2}$ and $\bar{q}(w)$ determined by $q(i(w))=0$.
The local graph of $\Upsilon$ at $s$ can be identified with
the graph with vertices $w$, where $w,w'$ are adjacent when the line joining
$i(w),i(w')$ has third point $(w+w',0,*) \in (Q \cup S^\perp) \setminus S$,
that is, the line joining $w,w'$ has third point $w''=w+w'$ satisfying
$w'' \notin T$ and $(\bar{q}(w'')=0$ or $w'' \in T^\perp)$ where
$T = \{ w \in W \mid w_1=w_2=w_3=w_5=...=w_{2m-3}=0 \}$.
But this is $\Sigma^{(m-1)}$.
$\Box$
\par}
\end{appendix}
\paragraph*{Acknowledgment} The second author is supported by a
postdoctoral fellowship of the Research Foundation -- Flanders (FWO).
\end{document} |
\begin{document}
\title{Lectures on the free period Lagrangian action functional}
\author{Alberto Abbondandolo}
\address{Ruhr Universit\"at Bochum,
Fakult\"at f\"ur Mathematik,
Geb\"aude NA 4/33,
D-44801 Bochum,
Germany}
\email{[email protected]}
\maketitle
\let\thefootnote\relax\footnote{The present work is part of the author's activities within CAST, a Research Network Program of the European Science Foundation.}
\centerline{\em To Kazimierz G\c{e}ba on the occasion of his 80th birthday}
\begin{abstract}
In this expository article we study the question of the existence of periodic orbits of prescribed energy for classical Hamiltonian systems on compact configuration spaces. We use a variational approach, by studying how the behavior of the free period Lagrangian action functional changes when the energy crosses certain values, known as the Ma\~n\'e critical values.
\end{abstract}
\tableofcontents
\section*{Introduction}
The main topic of this expository article is the question of the existence of periodic orbits of prescribed energy for classical Hamiltonian systems on compact configuration spaces.
More precisely, we consider a connected closed manifold $M$ and a smooth {\em Tonelli Lagrangian} $L$ on the tangent bundle $TM$ of $M$: The Tonelli assumption means that $L$ is fiberwise uniformly convex and superlinear. It is a very natural assumption: For instance, it guarantees that the Legendre transform is well defined and produces a diffeomorphism between the tangent and the cotangent bundle of $M$, it allows to prove that every pair of points in $M$ is connected by a curve $\gamma$ which minimizes the Lagrangian action
\[
\int_{t_0}^{t_1} L(\gamma(t),\gamma'(t)) \, dt
\]
and is a solution of the Euler-Lagrange equation associated to $L$, which is unique whenever the two points are sufficiently close to each other (see e.g. \cite{bgh98} or \cite{maz11}). Typical examples of Tonelli Lagrangians are {\em electromagnetic Lagrangians}, that is functions of the form
\[
L(x,v) = \frac{1}{2} |v|_x^2 + \theta(x)[v] - V(x), \qquad \forall (x,v)\in TM,
\]
where $|\cdot|_x$ denotes the norm associated to a Riemannian metric on $M$ (the kinetic energy), $\theta$ is a smooth one-form (the magnetic potential) and $V$ is a smooth function (the scalar potential) on $M$.
The Euler-Lagrange equations associated to $L$ induce a smooth flow on $TM$ which preserves the energy function $E:TM \rightarrow {\mathbb{R}}$,
\[
E(x,v) := d_v L(x,v)[v] - L(x,v), \qquad \forall (x,v)\in TM.
\]
Given a number $\kappa\in [\min E,+\infty)$, the problem under considerations is the existence of a periodic orbit on the energy level $E^{-1}(\kappa)$.
Such a problem has been studied by several authors, for several classes of Tonelli Lagrangians and energy ranges, and by several techniques. For instance, in the case of Lagrangians of the form
\[
L(x,v) = \frac{1}{2} |v|_x^2 - V(x),
\]
one can use the Maupertuis-Jacobi metric as in \cite{ben84} and reduce the problem to the existence of closed Riemannian geodesics either on $M$, if $\kappa$ is larger than the maximum of $V$, or on the domain $\{V\leq \kappa\}\subset M$, which is endowed with a metric which degenerates on the boundary $V^{-1}(\kappa)$, if $\kappa$ is smaller than the maximum of $V$. For a general Tonelli Lagrangian, the role of the maximum of $V$ is played by the {\em Ma\~{n}\'e critical values}. More precisely, important values of the energy are the numbers
\[
\min E \leq e_0(L) \leq c_u(L) \leq c_0(L),
\]
where $e_0(L)$ is the maximal critical value of $E$, $c_u(L)$ is minus the infimum of the mean Lagrangian action
\[
\frac{1}{T} \int_0^T L(\gamma(t)\, \gamma'(t))\, dt
\]
over all contractible closed curves $\gamma$, and $c_0(L)$ is minus the infimum of the mean Lagrangian action over all null-homologous closed curves. In the case of electromagnetic Lagrangians, $e_0(L)=c_u(L)=c_0(L)$ when the magnetic potential $\theta$ vanishes, but the first and the second values are in general distinct when $\theta$ does not vanish ($c_u(L)$ and $c_0(L)$ can be distinct only when the fundamental group of $M$ is sufficiently non-abelian). The importance of $e_0(L)$ is clear, since it marks a change in the topology of $E^{-1}(\kappa)$: If $\kappa>e_0(L)$, then $E^{-1}(\kappa)$ is diffeomorphic to the unit tangent bundle of $M$, if $\kappa<e_0(L)$ then the projection of $E^{-1}(\kappa)$ to $M$ is not surjective anymore. The {\em lowest} Ma\~{n}\'e critical value $c_u(L)$ affects directly the behavior of the
{\em free period Lagrangian action functional}
\[
{\mathbb{S}}_{\kappa}(\gamma) := \int_0^T \Bigl( L\bigl(\gamma(t),\gamma'(t)\bigr) + \kappa \Bigr) \, dt, \qquad \gamma: {\mathbb{R}}/T{\mathbb{Z}} \rightarrow M.
\]
The critical points of this functional, whose domain is a suitable space of closed curves $\gamma$ in $M$ of arbitrary period $T$, are exactly the closed orbits of energy $\kappa$. The functional ${\mathbb{S}}_{\kappa}$ is bounded from below on every connected component of the free loop space whenever $\kappa\geq c_u(L)$, and it is unbounded from below on every such connected component when $\kappa< c_u(L)$.
The {\em strict} Ma\~{n}\'e critical value $c_0(L)$ is not directly related to the topology of ${\mathbb{S}}_{\kappa}$, but it has dynamical and geometric significance: For $\kappa>c_0(L)$ the energy surface $E^{-1}(\kappa)$ is of {\em restricted contact type}, and the Euler-Lagrangian flow on it is conjugated, up to a time reparametrization, to a {\em Finsler geodesic flow} on $M$, whereas both facts are in general false for $\kappa\leq c_0(L)$. Furthermore, the Ma\~{n}\'e critical values are related to compactness properties of the functional ${\mathbb{S}}_{\kappa}$, such as the {\em Palais-Smale condition}.
By exploiting these facts, the free period action functional ${\mathbb{S}}_{\kappa}$ can be effectively used as a variational principle for our problem and allows to prove various results, which we summarize into the following theorem.
\begin{Thm*}
Let $L$ be a Tonelli Lagrangian on the tangent bundle of the closed manifold $M$.
\begin{enumerate}
\item If $\kappa>c_u(L)$ and $M$ is not simply connected, then the energy level $E^{-1}(\kappa)$ has a ${\mathbb{S}}_{\kappa}$-minimizing periodic orbit in each non-trivial homotopy class of the free loop space of $M$.
\item If $\kappa>c_u(L)$ and $M$ is simply connected, then the energy level $E^{-1}(\kappa)$ has a periodic orbit with positive action ${\mathbb{S}}_{\kappa}$.
\item For almost every $\kappa\in (\min E,c_u(L))$ the energy level $E^{-1}(\kappa)$ has a periodic orbit with positive action ${\mathbb{S}}_{\kappa}$.
\item If the energy level $E^{-1}(\kappa)$ is stable then $E^{-1}(\kappa)$ has a periodic orbit.
\end{enumerate}
\end{Thm*}
Notice that in (iii) only existence for {\em almost every} energy level in the interval $(\min E,c_u(L))$ (in the sense of Lebesgue measure) is stated: existence for {\em all} energy levels in this range is still unknown, although no counterexamples have been found so far. This issue is related to the fact that the Palais-Smale condition does not hold anymore below $c_u(L)$. The {\em stability condition} which is assumed in (iv) is a weaker form of the contact type condition.
The above theorem was first proved in this form by G.~Contreras \cite{con06} (assuming contact type instead of stable in (iv)), building on previous geometric ideas of I.~A.~Taimanov \cite{tai83,tai92b}. Contreras' long paper \cite{con06} contains many other beautiful results, such as the study of the invariant probability measures which one obtains as limits of Palais-Smale sequences which do not converge in the free loop space. This article is meant to be a gentle introduction, including some technical simplifications, to the part of \cite{con06} which concerns periodic orbits.
Unlike in a typical survey article, we are more concerned with detailed proofs, which we try to make accessible to a large audience including students, than with a systematic overview of the literature, for which we refer to the beautiful survey of Taimanov \cite{tai92b}, to \cite{cmp04} and to the already cited \cite{con06}. In particular, we start by proving well known abstract results, such as the mountain pass theorem, the general minimax principle, and the construction of the structure of infinite dimensional Hilbert manifold on the space of closed loops on $M$ of Sobolev class $W^{1,2}$ (Sections \ref{mpsec} and \ref{hmlsec}). In Section \ref{sec3} we introduce the already mentioned free period action functional ${\mathbb{S}}_{\kappa}$, which plays a fundamental role in this article and gives it its title: Unlike in \cite{con06}, we use it also to get existence of closed orbits for energies below $e_0(L)$. The Ma\~{n}\'e critical values which are relavant for this article are introduced in Section \ref{mcv}, together with some of their characterizations and with the discussion of two geometric properties of an energy level, namely the contact type and the stability condition. The analysis of Palais-Smale sequences is carried out in Section \ref{pss}, and in Section \ref{pohe} we prove statements (i) and (ii) of the above theorem. The topology of the free period action functional for $\kappa<c_u(L)$ is studied in Section \ref{tfpafle}, and in Section \ref{pole} we finally prove statements (iii) and (iv), using {\em Struwe's monotonicity argument}, together with a weaker version of (iii), using an alternative argument.
\partialaragraph{\bf Acknowledgments} This expository article is the outcome of two series of lectures that the author gave at two summer schools at the Korea Institute for Advanced Study of Seul in 2010 and at the Universit\'e de Neuch\^atel in 2011, respectively. I am grateful to Urs Frauenfelder and Felix Schlenk for organizing these two events, and to Gabriel Paternain, who was also a speaker at the first school, for many fruitful discussions. I would like to thank Jungsoo Kang, who participated to the first school, organized the material and typed a first version of the notes which eventually became this article. I would like to thank also Luca Asselle, who participated to the second school and suggested Lemma \ref{PST0} below, which allows to avoid extra technicalities and the detailed analysis of Palais-Smale sequences with infinitesimal periods which was present in the previous notes.
\numberwithin{equation}{section}
\section{The minimax principle}
\label{mpsec}
\partialaragraph{\bf The mountain pass theorem} Let $H$ be a real Hilbert space and let $f$ be a continuously differentiable real function on $H$. The symbol ${\mathbb{C}}rit f$ denotes the set of critical points of $f$.
We assume that a certain open sublevel $\{f<a\}$ is
not connected, say $\{f<a\}=A\cup B$, with $A$ and $B$ disjoint non-empty open sets. We may think of $A$ and $B$ as two valleys, and consider the set of paths going from one valley to the other one, that is the set
$$
{\bf G}amma:=\{\textrm{curves in $H$ with one end in $A$ and the other in $B$}\}.
$$
We can define the minimax value of $f$ on ${\bf G}amma$ as
$$
c:=\inf_{\gamma\in{\bf G}amma}\max_{x\in\gamma} f(x),
$$
and we notice that $a\leq c < +\infty$, because ${\bf G}amma$ is non empty and each of its elements intersects the set $H\setminus (A\cup B) = \{f\geq a\}$.
One would expect this mountain pass level $c$ to be a critical value of $f$.
The next simple example shows that this is not always the case.
\begin{Ex}
Consider the smooth function $f$ on ${\mathbb{R}}^2$ defined by
\[
f(x,y)=e^x-y^2.
\]
Then $\{f<0\}$ has two connected components, $c=0$, but $f$ has no critical points. The problem here is that the critical point is pushed to infinity: Indeed, $f(-n,0)=e^{-n}$ converges to the mountain pass level $c=0$ and $df(-n,0)=e^{-n} dx$ tends to zero.
\end{Ex}
This example suggests the following definition.
\begin{Def}
A sequence $(x_n)_{n\in{\mathbb{N}}}\subset H$ is called a {\em Palais-Smale sequence} at level $c$ ($(\mathrm{PS})_c$ for short) if
$$
\lim_{n\to\infty}f(x_n)=c\quad\mbox{and}\quad \lim_{n\to\infty}df(x_n)=0.
$$
The function $f$ is said to satisfy $(\mathrm{PS})_c$ if all $(\mathrm{PS})_c$ sequences are compact. It is said to satisfy $(\mathrm{PS})$ if it satisfies $(\mathrm{PS})_c$ for every $c\in {\mathbb{R}}$.
\end{Def}
Notice that limiting points of $(\mathrm{PS})_c$ sequences are critical points at level $c$. We can now state the celebrated mountain pass theorem of Ambrosetti and Rabinowitz \cite{ar73} in the following form:
\begin{Thm}[Mountain Pass Theorem]
\label{mp}
Let $f\in C^{1,1}(H)$ be such that $\{f<a\}$ is not connected and let $c$ be defined as above. Then $f$ admits a $(\mathrm{PS})_c$ sequence. In particular, if $f$ satisfies $(\mathrm{PS})_c$, then $c$ is a critical value.
\end{Thm}
Here $C^{1,1}$ denotes the set of functions whose differential is locally Lipschitz-continuous.
\begin{proof}
By contradiction, suppose that there exists $\epsilon>0$ such that $||df|| \geq\epsilon$ on the set $\{|f-c|\leq\epsilon\}$. We denote by $\nabla f$ the gradient of $f$ and we assume for sake of simplicity that the locally Lipschitz vector field $-\nabla f$ is positively complete, meaning that its flow $\partialhi$, that is the solution of
\[
\left\{
\begin{aligned}
&\frac{\partial}{\partial t}\partialhi_t (u)=-\nabla f\bigl(\partialhi_t(u)\bigr),\\[1ex]
&\partialhi_0 (u)=u,
\end{aligned}
\;\;\right.
\]
is defined for every $t\geq 0$ and every $u\in H$. This holds, for instance, if $\nabla f$ is globally Lipschitz (in this case the flow of $-\nabla f$ is defined on the whole ${\mathbb{R}}\times H$). See Remark \ref{noncomp} below for a hint on how to remove this extra assumption. Notice that
\begin{equation}
\label{decr}
\frac{d}{dt} f\bigl(\partialhi_t(u)\bigr) = df\bigl( \partialhi_t(u) \bigr) \bigl[-\nabla f\bigl(\partialhi_t(u)\bigr) \bigr] = - \bigl\|df\bigl(\partialhi_t(u)\bigr)\bigr\|^2,
\end{equation}
so the function $t\mapsto f(\partialhi_t(u))$ is decreasing. If $|f(\partialhi_t(u))-c|\leq\epsilon$ for all $t\in[0,T]$, we have
\[
2\epsilon \geq f(u)-f(\partialhi_T(u)) =-\int_0^T\frac{d}{dt}f(\partialhi_t(u))dt
=\int_0^T \bigl\|d f\bigl(\partialhi_t(u)\bigr)\bigr\|^2dt
\geq\epsilon^2T,
\]
from which we conclude that $T\leq 2/\epsilon$. Choose $\gamma\in{\bf G}amma$ such that $\max_\gamma f\leq c+\epsilon$ and set
$$
\tilde\gamma=\partialhi_T(\gamma), \qquad\textrm{for some } T>\frac{2}{\epsilon}.
$$
The fact that $f$ decreases along the orbits of $\partialhi$ implies that $\tilde\gamma$ belongs to ${\bf G}amma$. Since $f\leq c+\epsilon$ on $\gamma$, any $x\in\gamma$ satisfies either (i) $|f(x)-c|\leq\epsilon$ or (ii) $f(x)<c-\epsilon$. Let $x\in \gamma$.
If (i) holds, then $f(\partialhi_T(x))<c-\epsilon$ because $T>2/\epsilon$. If (ii) holds, then $f(\partialhi_T(x))<c-\epsilon$ because $f$ decreases along the orbits of $\partialhi$. Therefore we conclude that $\tilde\gamma\subset\{f< c-\epsilon\}$, which contradicts the definition of $c$.
\end{proof}
\begin{Rmk}
\label{noncomp}
If the vector field $-\nabla f$ is not positively complete, we can replace it by the complete one
$-\nabla f/\sqrt{||\nabla f||^2+1}$. The above proof goes through with minor adjustments.
\end{Rmk}
\begin{Rmk}
The mountain pass theorem holds also for $f\in C^{1,1}(\mathcal{M}M)$ where $(\mathcal{M}M,g)$ is a Hilbert manifold equipped with a complete Riemannian metric $g$.
In this case, $(x_n)_{n\in{\mathbb{N}}}\subset\mathcal{M}M$ is said to be a $(\mathrm{PS})_c$ sequence if $\lim_{n\to\infty}f(x_n)=c$ and $\lim_{n\to\infty}||df(x_n)||=0$, where $\|\cdot\|$ denotes the dual norm induced by $g$. Notice that the (PS) condition and the completeness of $g$ are somehow antagonist requirements: One may always achieve the completeness of an arbitrary Riemannian metric $g$ by multiplying it by a positive function which diverges at infinity (such an operation reduces the set of the Cauchy sequences), while the (PS) condition could be achieved by multiplying $g$ by a positive function which is infinitesimal at infinity (since the dual norm is multiplied by the inverse of this function, this operation reduces the set of the (PS) sequences).
\end{Rmk}
\begin{Rmk}
The mountain pass theorem holds also if $f$ is just continuously differentiable. In this case, its negative gradient vector field is just continuous and may not induce a continuous flow. In order to prove the above theorem, one needs to construct a locally Lipschitz pseudo-gradient vector field for $f$, see for instance \cite[Lemma 3.2]{str00}. The same construction allows to prove the mountain pass theorem for continuously differentiable functions on Banach spaces, or more generally on Banach manifolds.
\end{Rmk}
\begin{Rmk}
When dealing with functions on manifolds, it is sometimes useful to have a formulation of the mountain pass theorem which does not involve the choice of a metric. Here is such a formulation. Assume that $f$ is a continuously differentiable function on a Hilbert manifold $\mathcal{M}M$ and that $V$ is a positively complete locally Lipschitz vector field such that $df[V]<0$ on $\mathcal{M}M \setminus{\mathbb{C}}rit f$. Then the mountain pass theorem holds, provided that we define $(x_n)_{n\in{\mathbb{N}}}\subset \mathcal{M}M$ to be a $(\mathrm{PS})_c$ sequence if $f(x_n)$ tends to c and $df(x_n)[V(x_n)]$ is infinitesimal. Now the antagonism is between this form of the (PS) condition and the positive completeness of $V$.
\end{Rmk}
\partialaragraph{\bf The general minimax principle} In the proof of Theorem \ref{mp} we have not used the fact that ${\bf G}amma$ is a set of curves, but rather that ${\bf G}amma$ is positively invariant with respect to the negative gradient flow $\partialhi$ of $f$, meaning that $\partialhi_t(\gamma)\in{\bf G}amma$ for all $\gamma\in{\bf G}amma$ and $t\geq 0$. Here $\partialhi$ is either the flow of $-\nabla f$, when this vector field is positively complete, or the flow of some conformally equivalent positively complete vector field, such as
$-\nabla f/\sqrt{\|\nabla f\|^2+1}$, in the general case.
This simple observation leads to the following powerful generalization of the mountain pass theorem.
\begin{Thm}[\bf General Minimax Principle]
\label{thm:finite c induces PS}
Let $f$ be a $C^{1,1}$ function on the complete Riemannian Hilbert manifold $(\mathcal{M}M,g)$ and let ${\bf G}amma$ be a set of subsets of $\mathcal{M}M$ which is positively invariant with respect to the negative gradient flow of $f$. If the number
\[
c= \inf_{\gamma\in{\bf G}amma}\sup_\gamma f
\]
is finite, then $f$ admits a $(\mathrm{PS})_c$ sequence. In particular, if $f$ satisfies $(\mathrm{PS})_c$, then $c$ is a critical value.
\end{Thm}
The proof is a straightforward modification of the proof of Theorem \ref{mp}.
\begin{Ex}
Let $f\in C^{1,1}(H)$, where $H$ is a Hilbert space. If $\partiali_k(\{f<a\})\ne 0$ for some $k\geq 0$ and $f$ satisfies $(\mathrm{PS})$, then $f$ has a critical point. Indeed, we can consider the set
\[
\begin{split}
{\bf G}amma := \bigl\{ z(\overline{B}^{k+1}) \; \Big| \; & z: (\overline{B}^{k+1},\partialartial B^{k+1}) \rightarrow (H,\{f<a\}) \mbox{ continuous map such that }\\ & [z|_{\partialartial B^{k+1}}] \neq 0 \mbox{ in } \partiali_k(\{f<a\})\Bigr\},
\end{split}
\]
where $B^{k+1}$ denotes the unit open ball of dimension $k+1$. By applying Theorem \ref{thm:finite c induces PS} with such a ${\bf G}amma$ we get the existence of a critical point at level $c\geq a$. The case $k=0$ is precisely the Mountain Pass Theorem \ref{mp}.
\end{Ex}
\begin{Rmk}
\label{mini}
If ${\bf G}amma$ is the class of all one-point sets in $\mathcal{M}M$, then $c$ is the infimum of $f$. Therefore, the general minimax principle has as a particular case the following existing result for minimizers: Assume that $f\in C^{1,1}(\mathcal{M}M)$ is bounded from below, has complete sublevels and satisfies $(\mathrm{PS})_c$ at the level $c=\inf f$; then $f$ has a minimizer.
\end{Rmk}
\begin{Rmk}
\label{trunc}
It is sometimes useful to replace the negative gradient flow by a flow which fixes a certain sublevel of $f$. Let $\rho:{\mathbb{R}}\partialf{\mathbb{R}}^+$ be a smooth bounded function such that $\rho=0$ on $(-\infty,b]$ and $\rho>0$ on $(b,+\infty)$. Then we consider the vector field $V=-\rho(f)\cdot\nabla f$ (or $V= - \rho(f) \nabla f/\sqrt{\|\nabla f\|^2+1}$ in the non-positively complete case) and denote its flow by $\partialhi$. It is a negative gradient flow truncated below level $b$: The function $t\mapsto f(\partialhi_t(u))$ is constant if $u\in {\mathbb{C}}rit f \cup \{f\leq b\}$ and it is strictly decreasing otherwise.
If ${\bf G}amma$ is positively invariant with respect to this negative gradient flow truncated below level $b$ and the minimax value $c$ is strictly larger than $b$, then $f$ has a $(\mathrm{PS})_c$ sequence.
\end{Rmk}
\section{A Hilbert manifold of loops}
\label{hmlsec}
Let $(M,g)$ be a closed Riemannian manifold of dimension $n$ and consider the Sobolev space of loops
$$
W^{1,2}({\mathbb{T}},M):=\Big\{x:{\mathbb{T}}\partialf M\,\Big|\,x\textrm{ is absolutely continuous and } \int_{\mathbb{T}}|x'(s)|^2_{x(s)}ds<\infty\Big\},
$$
where ${\mathbb{T}}:={\mathbb{R}}/{\mathbb{Z}}$ and $|\cdot|_{\cdot}$ denotes the norm induced by $g$. This set of loops is clearly independent from the choice of the Riemannian metric $g$.
\partialaragraph{\bf The smooth structure of $\mathbf{W^{1,2}(T,M)}$} Let us recall the construction of the smooth Hilbert manifold structure on $W^{1,2}({\mathbb{T}},M)$.
Fix $x_0\in C^\infty({\mathbb{T}},M)$. Assume for simplicity that $x_0$ preserves the orientation, so that $x_0^*(TM)$ has a trivialization
$$
{\mathbb{P}}hi: {\mathbb{T}}\times{\mathbb{R}}^n\partialf x_0^*(TM).
$$
Let $B_r$ be the open ball of radius $r$ about $0$ in ${\mathbb{R}}^n$. Consider a smooth map
\[
\varphi:{\mathbb{T}}\times B_r \partialf M,
\]
such that $\varphi(t,0)=x_0(t)$ and $\varphi(t,\cdot)$ is a diffeomorphism onto an open subset in $M$, for every $t\in {\mathbb{T}}$. For instance, the map
\[
\varphi(t,\timesi)= \exp_{x_0(t)}\big({\mathbb{P}}hi(t,\timesi)\big),
\]
satisfies the above requirements if $r$ is small enough.
The map $\varphi$ induces the following parameterization:
\begin{equation}
\label{eq:chart of the loop space}
\varphi_*: W^{1,2}({\mathbb{T}},B_r)\partialf W^{1,2}({\mathbb{T}},M), \quad
\zeta\mapsto\varphi\big(\cdot,\zeta(\cdot)\big),
\end{equation}
where $W^{1,2}({\mathbb{T}},B_r)$ denotes the open subset of the Hilbert space $W^{1,2}({\mathbb{T}},{\mathbb{R}}^n)$ which consists of loops taking values into $B_r$. The collection of all these parameterizations, for every $x_0\in C^{\infty}({\mathbb{T}},M)$ and every $\varphi$ as above, defines a smooth atlas for $W^{1,2}({\mathbb{T}},M)$, which is then a smooth manifold modeled on the Hilbert space $W^{1,2}({\mathbb{T}},{\mathbb{R}}^n)$. Indeed, the smoothness of the transition maps is an immediate consequence of the chain rule.
It is worth noticing that the image of the parameterization $\varphi_*$ is $C^0$-open.
\begin{Rmk}
If $x_0$ is not orientation preserving, the natural model for the connected component of $W^{1,2}({\mathbb{T}},M)$ which contains $x_0$ is the space of $W^{1,2}$ sections of the vector bundle $x_0^*(TM)$. Alternatively, one can define a manifold structure on $W^{1,2}([0,1],M)$ without encountering topological problems, and then see $W^{1,2}({\mathbb{T}},M)$ as the inverse image of the diagonal of $M \times M$ by the smooth submersion
\[
W^{1,2}([0,1],M) \rightarrow M \times M, \qquad x \mapsto (x(0),x(1)).
\]
\end{Rmk}
The tangent space of $W^{1,2}({\mathbb{T}},M)$ at $x$ is naturally identified with the space of $W^{1,2}$ sections of $x^*(TM)$. Therefore, we can define a Riemannian metric on $W^{1,2}({\mathbb{T}},M)$ by setting
\begin{equation}
\label{metr}
\langle \timesi,\eta\rangle_x:=\int_{{\mathbb{T}}} \Bigl( g(\timesi,\eta)+g(\nabla_t\timesi,\nabla_t\eta) \Bigr) \,dt, \quad \forall \timesi,\eta\in T_x W^{1,2}({\mathbb{T}},M),
\end{equation}
where $\nabla_t$ denotes the Levi-Civita covariant derivative along $x$. The distance induced by this Riemannian metric is compatible with the topology of $W^{1,2}({\mathbb{T}},M)$.
The fact that $M$ is compact implies that this metric on $W^{1,2}({\mathbb{T}},M)$ is complete (more generally, this metric is complete whenever $g$ is complete).
The gradient of functionals on $W^{1,2}({\mathbb{T}},M)$ is the one which is associated to such a Riemannian metric.
\begin{Rmk}
If $\varphi$ is the restriction of a smooth map $B_{r'}\times {\mathbb{T}}\rightarrow M$ with the same properties, for some $r'>r$, then the parameterization $\varphi_*$ is bi-Lipschitz.
\end{Rmk}
See e.g. \cite{kli82} for more details on the Hilbert manifold structure of $W^{1,2}({\mathbb{T}},M)$.
\partialaragraph{\bf The homotopy type of $\mathbf{W^{1,2}(T,M)}$} The inclusions
$$
C^\infty({\mathbb{T}},M)\hookrightarrow W^{1,2}({\mathbb{T}},M)\hookrightarrow C({\mathbb{T}},M)
$$
are dense homotopy equivalences. These facts can be proved by embedding $M$ into a Euclidean space ${\mathbb{R}}^N$, by regularizing the loops $x:{\mathbb{T}}\rightarrow M\subset {\mathbb{R}}^N$ by convolution, and by projecting the regularized loop back to $M$ using the tubular neighborhood theorem. In particular, the connected components of $W^{1,2}({\mathbb{T}},M)$ are in one-to-one correspondence with the conjugacy classes of
$\partiali_1(M)$. See, e.g., \cite[Chapter 10]{lee03} for more details.
\section{The free period action functional}
\label{sec3}
\partialaragraph{\bf Tonelli Lagrangians} Let $M$ be a connected closed manifold.
A function $L\in C^\infty(TM)$ is called a {\em Tonelli Lagrangian} if:
\begin{enumerate}
\item $L$ is fiberwise uniformly convex, i.e.\ $d_{vv}L(x,v)>0$ for every $(x,v)\in TM$, where $d_{vv}L$ denotes the fiberwise second differential of $L$;
\item $L$ has superlinear growth on each fiber, i.e.
\[
\lim_{|v|\to +\infty}\frac{L(x,v)}{|v|_x}=+\infty.
\]
\end{enumerate}
The main example of Tonelli Lagrangians is given by the {\em electromagnetic Lagrangians}, that is functions of the form
\begin{equation}
\label{elmag}
L(x,v)=\frac{1}{2}|v|^2_x+\theta(x)[v]-V(x),
\end{equation}
where $|\cdot|_x$ denotes the norm associated to a Riemannian metric (the kinetic energy), $\theta$ is a smooth one-form (the magnetic potential) and $V$ is a smooth function (the scalar potential) on $M$. We shall omit the subscript $x$ in $|\cdot|_x$ when the point $x$ is clear from the context.
The Tonelli assumptions imply that the Euler-Lagrange equation, which in local coordinates can be written as
\begin{equation}
\label{EL}
\frac{d}{dt}\bigl( \partial_v L (\gamma(t),\gamma'(t)) \bigr)= \partial_x L (\gamma(t),\gamma'(t)),
\end{equation}
is well-posed and defines a smooth flow on $TM$. This flow preserves the energy
\[
E:TM\rightarrow {\mathbb{R}}, \quad
E(x,v):=d_v L(x,v)[v]-L(x,v),
\]
where $d_v$ denotes the fiberwise differential.
When $L$ has the form (\ref{elmag}), then
\begin{equation}
\label{ene}
E(x,v) = \frac{1}{2} |v|^2 + V(x).
\end{equation}
In general, the energy function of a Tonelli Lagrangian satisfies the following properties:
\begin{itemize}
\item[(i)] $E$ is fiberwise uniformly convex and superlinear.
\item[(ii)] For any $x\in M$, the restriction of $E$ to $T_x M$ achieves its minimum at $v=0$.
\item[(iii)] The point $(\bar x,0)$ is singular for the Euler-Lagrange flow if and only if $(\bar x,0)$ is a critical point of $E$.
\end{itemize}
We are interested in proving the existence of periodic orbits on a given energy level $E^{-1}(\kappa)$. Since such an energy level is compact, up to the modification of $L$ far away from it, we may assume that the Tonelli Lagrangian $L(x,v)$ is electromagnetic for $|v|$ large. In particular, we have the inequalities
\begin{eqnarray}
\label{boundsonL}
L(x,v) \geq L_0 |v|^2 - L_1, \qquad & \forall (x,v)\in TM,\\
\label{bd2L}
d^2_{vv} L(x,v)[u,u] \geq 2 L_0 |u|^2, \qquad &\forall (x,v)\in TM, \; u\in T_x M,
\end{eqnarray}
for some numbers $L_0>0$ and $L_1\in {\mathbb{R}}$. Moreover, $E$ has the form (\ref{ene}) for $|v|$ large.
\partialaragraph{\bf The free period action functional} We would like to study the Lagrangian action on the space of closed curves of arbitrary period. The latter space can be given a manifold structure by reparametrizing each curve on ${\mathbb{T}}$ and by keeping track of its period as a second variable:
Let $\gamma :{\mathbb{R}}/{T{\mathbb{Z}}}\partialf M$ be an absolutely continuous $T$-periodic curve and define $x:{\mathbb{T}} \rightarrow M$ as $x(s):=\gamma(sT)$. The closed curve $\gamma$ is identified with the pair $(x,T)$.
The action of $\gamma$ on the time interval $[0,T]$ is the number
\[
\int_0^TL\bigl(\gamma(t),\gamma'(t)\bigr)\, dt= T\int_{{\mathbb{T}}} L\bigl(x(s),x'(s)/{T}\bigr)\, ds.
\]
Fix a real number $\kappa$, the value of the energy for which we would like to find periodic solutions. Consider the {\em free period action functional} corresponding to the energy $\kappa$
\[
{\mathbb{S}}_{\kappa}(\gamma) =
{\mathbb{S}}_{\kappa} (x,T) := T\int_{{\mathbb{T}}} \Bigl( L\bigl(x(s),x'(s)/T\bigr) + \kappa \Bigr)\,ds = \int_0^T \Bigl( L\bigl( \gamma(t),\gamma'(t)\bigr)+\kappa\Bigr)\, dt.
\]
The fact that $L$ is electromagnetic outside a compact subset of $TM$ implies that ${\mathbb{S}}_{\kappa}(x,T)$ is well-defined when $x\in W^{1,2}({\mathbb{T}},M)$. Therefore, we obtain a functional
\[
{\mathbb{S}}_{\kappa} : W^{1,2}({\mathbb{T}},M) \times (0,+\infty) \rightarrow {\mathbb{R}}.
\]
The Hilbert manifold $W^{1,2}({\mathbb{T}},M) \times (0,+\infty)$ is denoted by $\mathcal{M}M$.
\begin{Lemma}(Regularity properties of ${\mathbb{S}}_{\kappa}$)
\begin{itemize}
\item[(i)] ${\mathbb{S}}_{\kappa}$ is in $C^{1,1}(\mathcal{M}M)$ and is twice Gateaux differentiable at every point.
\item[(ii)] ${\mathbb{S}}_{\kappa}$ is twice Fr\'ech\'{e}t differentiable at every point if and only if $L$ is electromagnetic on the whole $TM$. In this case, ${\mathbb{S}}_{\kappa}$ is actually smooth on $\mathcal{M}M$.
\end{itemize}
\end{Lemma}
See e.g.\ \cite{as09b} for a detailed proof.
If $d_x$ denotes the horizontal differential with respect to some horizontal-vertical splitting of $TTM$, the differential of ${\mathbb{S}}_{\kappa}$ with respect to the first variable at some $(x,T)\in \mathcal{M}M$ has the form
\begin{equation}
\label{diff1}
\begin{split}
d{\mathbb{S}}_{\kappa} (x,T) \bigl[(\timesi,0)\bigr] & = T \int_0^1 \Bigl( d_x L\bigl( x, x'/T \bigr) [ \timesi ] + d_v L\bigl( x, x'/T \bigr) \bigl[ \nabla_s \timesi/T \bigr] \Bigr)\, ds \\ & = \int_0^T \Bigl( d_x L \bigl(\gamma,\gamma'\bigr) [\zeta] + d_v L \bigl(\gamma,\gamma'\bigr) [\nabla_t \zeta] \Bigr)\, dt,
\end{split}\end{equation}
where $\timesi\in T_x W^{1,2}({\mathbb{T}},M)$, $\gamma(t)= x(t/T)$ and $\zeta(t):=\timesi(t/T)$.
Let $(x,T)$ be a critical point of ${\mathbb{S}}_{\kappa}$. The above formula and an integration by parts imply that $\gamma$ is a $T$-periodic solution of (\ref{EL}). Moreover
\begin{equation}
\label{eq:diff A w.r.t. T}
\begin{split}
\frac{\partialartial {\mathbb{S}}_{\kappa}}{\partialartial T} (x,T) &= \int_{{\mathbb{T}}} \Bigl( L\bigl( x(s),x'(s)/T\bigr) + \kappa + T \, d_v L\bigl( x(s),x'(s)/T\bigr) \bigl[-x'(s)/T^2\bigr] \Bigr)\, ds \\ &= \int_{{\mathbb{T}}} \Bigl( \kappa - E\bigl( x(s),x'(s)/T\bigr) \Bigr)\, ds = \frac{1}{T}\int_0^T \Bigl( \kappa - E\bigl( \gamma(t),\gamma'(t)\bigr) \Bigr) \, dt.
\end{split}
\end{equation}
Together with the fact that $E$ is constant along the orbits of the Euler-Lagrange flow, the above identity shows that the $T$-periodic orbit $\gamma$ belongs to the energy levek $E^{-1}(\kappa)$. We conclude that $(x,T)$ is a critical point of ${\mathbb{S}}_{\kappa}$ on $\mathcal{M}M$ if and only if $\gamma(t):= x(t/T)$ is a $T$-periodic orbit of energy $\kappa$ ($T$ is not necessarily the minimal period).
\partialaragraph{\bf Behavior of ${\mathbb{S}}_{\kappa}$ for $\mathbf{T\to 0}$} The Hilbert manifold $\mathcal{M}M=W^{1,2}({\mathbb{T}},M)\times (0,+\infty)$ is endowed with the product Riemannian structure of (\ref{metr}) and the Euclidean metric of $(0,+\infty) \subset {\mathbb{R}}$. As such, it is not complete, the non-converging Cauchy sequences being the sequences $(x_h,T_h)$ with $x_h \rightarrow x\in W^{1,2}({\mathbb{T}},M)$ and $T_h \rightarrow 0$. Therefore, we need to understand the behavior of ${\mathbb{S}}_{\kappa}$ on such sequences.
We decompose $\mathcal{M}M$ as $\mathcal{M}M = \mathcal{M}M^{\mathrm{contr}} \sqcup \mathcal{M}M^{\mathrm{noncontr}}$, where $\mathcal{M}M^{\mathrm{contr}}$ denotes the connected component consisting of contractible loops.
\begin{Lemma}
\label{Lem:3}
\begin{enumerate}
\item On $\mathcal{M}M^\mathrm{noncontr}$ the sublevels $\{{\mathbb{S}}_{\kappa}\leq c\}$ are complete. More precisely, if $(x_h,T_h)\in \mathcal{M}M^\mathrm{noncontr}$ and $T_h \rightarrow 0$, then ${\mathbb{S}}_{\kappa}(x_h,T_h)\rightarrow +\infty$.
\item If $(x_h,T_h)\in\mathcal{M}M^\mathrm{contr}$ and $T_h\to 0$, then $\liminf_h{\mathbb{S}}_{\kappa}(x_h,T_h)\geq0$.
\end{enumerate}
\end{Lemma}
\begin{proof}
By (\ref{boundsonL}), we have the chain of inequalities
\begin{equation}
\label{lowbdonA}
\begin{split}
{\mathbb{S}}_{\kappa} (x,T) &= T \int_{{\mathbb{T}}} \Bigl( L\bigl(x,x'/T\bigr) + \kappa \Big)\, ds \geq T \int_{{\mathbb{T}}} \Bigl( L_0 \frac{|x'|^2}{T^2} -L_1 + \kappa \Bigr)\, ds
\\ &= \frac{L_0}{T} \int_{{\mathbb{T}}} |x'|^2\, ds - (L_1-\kappa) T
\geq \frac{L_0}{T} \ell(x)^2 - (L_1 - \kappa)T,
\end{split} \end{equation}
where $\ell(x)$ denotes the length of the loop $x$. The length of the non-contractible loops in $M$ is bounded away from zero. Therefore, the estimate (\ref{lowbdonA}) implies statement (i). Statement (ii) is also an immediate consequence of (\ref{lowbdonA}).
\end{proof}
Since $\mathcal{M}$ is not complete, we cannot expect the vector field $-\nabla {\mathbb{S}}_{\kappa}$ to be positively complete. However, the only sources of non-completeness is the second component approaching zero. The next result says that this may happen only at level zero:
\begin{Lemma}\label{Lem:4}
Let $(x,T):[0,\sigma^*)\partialf\mathcal{M}M^\mathrm{contr}$, $0<\sigma^*<\infty$, be a flow line of $-\nabla {\mathbb{S}}_{\kappa}$ such that
\[
\liminf_{\sigma\to\sigma^*}T(\sigma)=0.
\]
Then
\[
\lim_{\sigma\to\sigma^*}{\mathbb{S}}_{\kappa} \big(x(\sigma),T(\sigma)\big)=0.
\]
\end{Lemma}
\begin{proof}
Since both $E$ and $L$ are quadratic in $v$ for $|v|$ large, we have the estimate
\[
E(x,v) \geq C_0 \, L(x,v) - C_1,
\]
for some $C_0>0$ and $C_1\in {\mathbb{R}}$. From (\ref{eq:diff A w.r.t. T}) we obtain the inequality
\begin{equation*}
\begin{split}
\frac{\partialartial {\mathbb{S}}_{\kappa}}{\partialartial T} ( x,T) &= \frac{1}{T} \int_0^T \bigl( \kappa - E(\gamma,\gamma') \bigr)\, dt
\leq \frac{1}{T} \int_0^T \bigl( \kappa - C_0 \,L(\gamma,\gamma') + C_1 \bigr)\, dt \\
&= \kappa + C_1 - \frac{C_0}{T} \int_0^T \bigl( L(\gamma,\gamma')+\kappa \bigr)\, dt + C_0 \kappa = (C_0+1) \kappa + C_1 - \frac{C_0}{T} {\mathbb{S}}_{\kappa} (x,T),
\end{split} \end{equation*}
which can be rewritten as
\begin{equation}
\label{boundsondAdt}
{\mathbb{S}}_{\kappa} (x,T) \leq \frac{T}{C_0} \Bigl( C - \frac{\partialartial {\mathbb{S}}_{\kappa}}{\partialartial T} (x,T) \Bigr),
\end{equation}
for a suitable constant $C$.
By the assumption, there exists an increasing sequence $(\sigma_h)$ which converges to $\sigma^*$ and satisfies $T'(\sigma_h)\leq 0$ and
$T(\sigma_h)\rightarrow 0$. Since $\sigma\mapsto (x(\sigma),T(\sigma))$ is a flow line of $-\nabla {\mathbb{S}}_{\kappa}$,
\[
0 \geq T'(\sigma_h) = - \frac{\partialartial {\mathbb{S}}_{\kappa}}{\partialartial T} \bigl(x(\sigma_h),T(\sigma_h) \bigr),
\]
and by (\ref{boundsondAdt}) we have
\[
{\mathbb{S}}_{\kappa} \bigl(x(\sigma_h),T(\sigma_h) \bigr) \leq \frac{T(\sigma_h)}{C_0} \Bigl( C - \frac{\partialartial {\mathbb{S}}_{\kappa}}{\partialartial T} \bigl(x(\sigma_h),T(\sigma_h) \bigr) \Bigr) \leq \frac{C}{C_0} T(\sigma_h).
\]
Since $T(\sigma_h)$ is infinitesimal, we obtain
\[
\limsup_{h\rightarrow \infty} {\mathbb{S}}_{k} \bigl(x(\sigma_h),T(\sigma_h) \bigr) \leq 0.
\]
Together with statement (ii) of Lemma \ref{Lem:3} and the monotonicity of the function $\sigma \longmapsto {\mathbb{S}}_{\kappa}(x(\sigma),T(\sigma))$,
this concludes the proof.
\end{proof}
\section{Ma\~n{\'e} critical values, contact type and stability conditions}
\label{mcv}
\partialaragraph{\bf The critical values} The following numbers should be interpreted as energy levels and mark important dynamical and geometric changes for the Euler-Lagrange flow induced by the Tonelli Lagrangian $L$:
\begin{equation*}
\begin{split}
c_0(L)&:=\inf\{\kappa\in{\mathbb{R}} \, | \,{\mathbb{S}}_{\kappa}(x,T)\geq 0 \; \forall (x,T)\in \mathcal{M}M \mbox{ with }x\mbox{ homologous to zero} \} \\
&= - \inf \Bigl\{ \frac{1}{T} \int_0^T L(\gamma(t),\gamma'(t)) \, dt \, \Big| \, \gamma\in C^{\infty}({\mathbb{R}}/T{\mathbb{Z}},M) \mbox{ homologous to zero, } T>0\Bigr\}, \\
c_u(L)&:=\inf\{\kappa\in{\mathbb{R}} \, | \,{\mathbb{S}}_{\kappa}(x,T)\geq 0 \; \forall (x,T)\in \mathcal{M}M \mbox{ with }x\mbox{ contractible} \}\\
&= - \inf \Bigl\{ \frac{1}{T} \int_0^T L(\gamma(t),\gamma'(t)) \, dt \, \Big| \, \gamma\in C^{\infty}({\mathbb{R}}/T{\mathbb{Z}},M) \mbox{ contractible, } T>0 \Bigr\}, \\
e_0(L)&:=\max_{x\in M} E(x,0) = \max \set{E(x,v)}{(x,v) \in {\mathbb{C}}rit E}.
\end{split} \end{equation*}
The number $c_0(L)$ is known as the {\em strict Ma\~n\'e critical value}, while $c_u(L)$ is the {\em lowest Ma\~{n}\'e critical value} (see \cite{man97}). When the fundamental group of $M$ is rich, there are other Ma\~n\'e critical values, which are associated to the different coverings to $M$, but the above ones are those which are more relevant for the question of existence of periodic orbits.
It is easy to see that
\[
\min E\leq e_0(L)\leq c_u(L)\leq c_0(L).
\]
When $L$ has the form \eqref{elmag}, $\min E$ is the minimum of the scalar potential $V$, while $e_0(L)$ is its maximum. When the magnetic potential $\theta$ vanishes, the identities $e_0(L)=c_u(L)=c_0(L)$ hold, but in general $e_0(L)$ is strictly lower than the other two numbers. The values $c_u(L)$ and $c_0(L)$ coincide when the fundamental group of $M$ is Abelian and, more generally, when it is ameanable (see \cite{fm07}).
The lowest Ma\~{n}\'e critical value $c_u(L)$ plays a decisive role in the geometry of the action functional ${\mathbb{S}}_{\kappa}$, as the next result shows:
\begin{Lemma}\label{Lem:5}
If $\kappa\geq c_u(L)$, then ${\mathbb{S}}_{\kappa}$ is bounded from below on every connected component of $\mathcal{M}M$. If $\kappa<c_u(L)$, then the functional ${\mathbb{S}}_{\kappa}$ is unbounded from below on each connected component of $\mathcal{M}M$.
\end{Lemma}
\begin{proof}
Choose $\gamma:{\mathbb{R}}/T{\mathbb{Z}}\rightarrow M$ in some connected component of the free loop space and let $\tilde\gamma:[0,T]\partialf\widetilde M$ be the its lift to the universal covering $\partiali:\widetilde M \rightarrow M$. We lift the metric of $M$ to $\widetilde{M}$ and notice that the fact of having fixed the connected component of the free loop space implies that the quantity $\mathrm{dist}\big(\tilde\gamma(T),\tilde\gamma(0)\big)$ is uniformly bounded. Therefore, there exists a path $\tilde{\alpha}:[0,1]\rightarrow \widetilde M$ which joins $\tilde{\gamma}(T)$ to $\tilde{\gamma}(0)$ and has uniformly bounded action
\[
\widetilde{{{\mathbb{S}}}}_{\kappa} (\tilde{\alpha}) := \int_0^1 \Bigl( \widetilde L\bigl(\tilde{\alpha}(t), \tilde{\alpha}'(t) \bigr) + \kappa \Bigr) \, dt \leq C,
\]
where $\widetilde{L}$ denotes the Lagrangian on $T\widetilde M$ which is obtained by lifting $L$. If $\alpha := \partiali\circ \tilde{\alpha}$, the juxtaposition $\gamma\# \alpha$ is a contractible loop in $M$. Since $\kappa\geq c_u(L)$, we have
\[
0 \leq {\mathbb{S}}_{\kappa} (\gamma \# \alpha) = {\mathbb{S}}_{\kappa} (\gamma) + {\mathbb{S}}_{\kappa} (\alpha) = {\mathbb{S}}_{\kappa} (\gamma) + \widetilde{{{\mathbb{S}}}}_{\kappa} (\tilde{\alpha})
\leq {\mathbb{S}}_{\kappa} (\gamma)+C,
\]
from which ${\mathbb{S}}_{\kappa}(\gamma)\geq -C$.
When $\kappa<c_u(L)$, the functional ${\mathbb{S}}_{\kappa}$ is unbounded from below on each connected component of $\mathcal{M}M$. In fact, if $\alpha$ is a contractible closed curve with ${\mathbb{S}}_{\kappa}(\alpha)<0$, we can modify any closed curve $\gamma$ within its free homotopy class and make it have arbitrarily low action ${\mathbb{S}}_{\kappa}$: Join $\gamma(0)$ to $\alpha(0)$ by some path, wind around $\alpha$ several times, come back to $\gamma(0)$ by the inverse path, and finally go once around $\gamma$.
\end{proof}
The strict Ma\~n\'e critical value $c_0(L)$ is not directly related to the geometry of ${\mathbb{S}}_{\kappa}$, but has the following important characterization (see \cite{fat97} and \cite{cipp98}):
\begin{equation}
\label{cara}
c_0(L) = \inf \left\{ \max_{x\in M} H(x,\alpha(x)) \, \Big| \, \alpha \mbox{ smooth closed one-form on } M\right\},
\end{equation}
where $H:T^*M \rightarrow {\mathbb{R}}$ is the Hamiltonian associated to the Lagrangian $L$ via Legendre duality:
\[
H(x,p) := \max_{v\in T_x M} \bigl( p[v] - L(x,v) \bigr).
\]
Then $H$ is a Tonelli Hamiltonian, meaning that it is fiberwise superlinear and uniformly convex (see the beginning of Section \ref{sec3}).
Let $X_H$ be the induced Hamiltonian vector field on $T^*M$, which is defined by the identity
\[
\omegaega(X_H(z),\zeta) = - dH(z)[\zeta], \quad \forall z\in T^*M, \; \zeta\in T_z T^*M,
\]
where $\omegaega=dp\wedge dx$ is the standard symplectic form on $T^*M$. The flow of $X_H$ preserves each level $H^{-1}(\kappa)$, where it is conjugated to the Euler-Lagrange flow of $L$ on $E^{-1}(\kappa)$ by the Legendre transform
\[
TM \rightarrow T^*M, \quad (x,v) \mapsto \bigl( x , d_v L(x,v) \bigr).
\]
Assume that $\kappa$ is a regular value of $H$, so that $\mathcal{S}igma:= H^{-1}(\kappa)$ is a hypersurface. Up to a time reparametrization, the dynamics on $\mathcal{S}igma$ is determined only by the geometry of $\mathcal{S}igma$ and not by the Hamiltonian of which $\mathcal{S}igma$ is an energy level: In fact the nowhere vanishing vector field $X_H|_{\mathcal{S}igma}$ belongs to the one-dimensional distribution
\[
\mathcal{L}_{\mathcal{S}igma}:=\ker \omegaega|_{\mathcal{S}igma},
\]
whose integral curves are hence the orbits of $X_H|_{\mathcal{S}igma}$.
The characterization (\ref{cara}) has a dynamical and a geometric consequence. We begin with the dynamical one:
\begin{thm}
\label{finsler}
If $\kappa>c_0(L)$, then the Euler-Lagrange flow on $E^{-1}(\kappa)$ is conjugated up to a time-reparametrization to the geodesic flow which is induced by a Finsler metric on $M$.
\end{thm}
\begin{proof}
Since $\kappa>c_0(L)$, there is a smooth closed one-form $\alpha$ whose image is contained in the sublevel $\{H<\kappa\}$. Since $\alpha$ is closed, the diffeomorphism of $T^*M$ defined by $(x,p) \mapsto (x,p+\alpha(x))$ is symplectic and conjugates the Hamiltonian flow of $H$ to that of $K(x,p):= H(x,p+\alpha(x))$. The energy level $K^{-1}(\kappa)$ is now the boundary
of a fiberwise uniformly convex bounded open set which contains the zero section of $T^*M$. Therefore, there exists a fiberwise convex and 2-homogeneous function $F:T^*M \rightarrow [0,+\infty)$ such that $F^{-1}(1)=K^{-1}(\kappa)$. Thus, the Hamiltonian flow of $F$ on $F^{-1}(1)=K^{-1}(\kappa)$ is related to that of $K$ - hence to that of $H$ on $H^{-1}(\kappa)$ - by a time reparametrization. But the Legendre dual of the fiberwise convex and 2-homogeneous Hamiltonian $F$ is the square of a Finsler structure on $M$. We conclude that the orbits of the Euler-Lagrange flow of $L$ of energy $\kappa$ are reparametrized Finsler geodesics.
\end{proof}
In order to derive the geometric consequence of (\ref{cara}), we need to recall some notions from symplectic topology.
\partialaragraph{\bf Contact type and stable energy hypersurfaces}
The energy level $\mathcal{S}igma$ is said to be of {\em contact type} if there is a one-form $\eta$ on $\mathcal{S}igma$ which is a primitive of $\omegaega|_{\mathcal{S}igma}$ and is such that $\eta$ does not vanish on $\mathcal{L}_{\mathcal{S}igma}$. Equivalently, there is a smooth vector field $Y$ in a neighborhood of $\mathcal{S}igma$ which is transverse to $\mathcal{S}igma$ and such that $L_Y \omegaega = \omegaega$ (the vector field $Y$ and the one-form $\eta$ are related by the identity $\mathrm{im}ath_Y \omegaega|_{\mathcal{S}igma} = \eta$). The energy level $\mathcal{S}igma$ is said to be of {\em restricted contact type} if the one-form $\eta$ extends to a primitive of $\omegaega$ on the whole $T^*M$.
If the surface $\mathcal{S}igma\subset T^*M$ bounds an open fiberwise convex set which contains the zero-section (or more generally an open set which is starshaped with respect to the zero section), then it is of restricted contact type: As $\eta$ one can take the Liouville form $p\, dq$. Therefore, arguing as in the first part of the proof of Theorem \ref{finsler}, we obtain the following geometric consequence of the characterization (\ref{cara}):
\begin{thm}
\label{contact}
If $\kappa>c_0(L)$, then $H^{-1}(\kappa)$ is of restricted contact type.
\end{thm}
If $c_u(L) \leq \kappa \leq c_0(L)$ and $M$ is not the 2-torus, $H^{-1}(\kappa)$ is not of contact type (see \cite[Proposition B.1]{con06}), and it is conjectured that the same is true for $e_0(L)<\kappa<c_u(L)$. If $\min E < \kappa < e_0(L)$, $H^{-1}(\kappa)$ might or might not be of contact type: For instance, if the one-form $\theta(x)[v]:=d_vL(x,0)[v]$ is closed, then every regular energy level is of contact type (see \cite[Proposition C.2]{con06}, in this case $e_0(L)=c_u(L)=c_0(L)$).
The contact condition has the following dynamical consequence: If $\mathcal{S}igma$ is a contact type compact hypersurface in a symplectic manifold $(W,\omegaega)$ (in our case, $W=T^*M$), then there is a diffeomorphism
\[
(-\epsilon, \epsilon) \times \mathcal{S}igma \rightarrow W, \qquad (r,x) \mapsto \partialsi_r(x),
\]
onto an open neighborhood of $\mathcal{S}igma$ such that $\partialsi_0$ is the identity on $\mathcal{S}igma$ and
\[
\partialsi_r : \mathcal{S}igma \rightarrow \mathcal{S}igma_r := \partialsi_r(\mathcal{S}igma)
\]
induces an isomorphism between the line bundles $\mathcal{L}_{\mathcal{S}igma}$ and $\mathcal{L}_{\mathcal{S}igma_r}$ (the hypersurface $\mathcal{S}igma_r$ is the image of $\mathcal{S}igma$ by the flow at time $r$ of the vector field $Y$ given by the contact condition, see \cite[page 122]{hz94}). Therefore, if the hypersurfaces $\mathcal{S}igma_r$ are level sets of a Hamiltonian $K$, the dynamics of $X_K$ on $\mathcal{S}igma_r$ is conjugate to the one on $\mathcal{S}igma_0=\mathcal{S}igma$ up to a time reparametrization.
Hypersurfaces with the above propery are called {\em stable} (see \cite[page 122]{hz94}). The stability condition is weaker than the contact condition, as the following characterization, which is due to K.\ Cieliebak and K.\ Mohnke \cite[Lemma 2.3]{cm05}, shows:
\begin{Prop}
Let $\mathcal{S}igma$ be a compact hypersurface in the symplectic manifold $(W,\omegaega)$. Then the following facts are equivalent:
\begin{enumerate}
\item $\mathcal{S}igma$ is stable;
\item there is a vector field $Y$ on a neighborhood of $\mathcal{S}igma$ which is transverse to $\mathcal{S}igma$ and satisfies $\mathcal{L}_{\mathcal{S}igma} \subset \ker ( L_Y \omegaega|_{\mathcal{S}igma})$;
\item there is a one-form $\eta$ on $\mathcal{S}igma$ such that $\mathcal{L}_{\mathcal{S}igma} \subset \ker d\eta$ and $\eta$ does not vanish on $\mathcal{L}_{\mathcal{S}igma}$.
\end{enumerate}
\end{Prop}
\begin{proof}
(i) ${\mathbb{R}}ightarrow$ (ii). By stability, a neighborhood of $\mathcal{S}igma$ can be identified with $(-\epsilon,\epsilon)\times \mathcal{S}igma$ in such a way that $\mathcal{L}_{\{r\}
\times \mathcal{S}igma}$ does not depend on $r$. Set $Y:= \partialartial/\partialartial r$ and denote by $\partialhi_t(r,x)=(r+t,x)$ its flow. Then $\ker ( \partialhi_t^* \omegaega|_{\{0\} \times \mathcal{S}igma} )$ does not depend on $t$ and differentiating in $t$ at $t=0$ we get
\[
\mathcal{L}_{\mathcal{S}igma} = \ker \omegaega|_{\mathcal{S}igma} \subset \ker ( L_Y \omegaega|_{\mathcal{S}igma}).
\]
(ii) ${\mathbb{R}}ightarrow$ (iii). If we set $\eta:= \mathrm{im}ath_Y \omegaega|_{\mathcal{S}igma}$, by Cartan's identity we have
\[
d\eta = d \mathrm{im}ath_Y \omegaega|_{\mathcal{S}igma} = (L_Y \omegaega - \mathrm{im}ath_Y d \omegaega)|_{\mathcal{S}igma} = L_Y \omegaega|_{\mathcal{S}igma},
\]
so $\mathcal{L}_{\mathcal{S}igma} \subset \ker(L_Y \omegaega|_{\mathcal{S}igma}) = \ker d\eta$. If $\timesi\neq 0$ is a vector in $\mathcal{L}_{\mathcal{S}igma}$, then
\[
\eta(\timesi) = \omegaega(Y,\timesi) \neq 0,
\]
because $Y\notin T\mathcal{S}igma$.
\noindent (iii) ${\mathbb{R}}ightarrow$ (i). Consider the closed two-form on $(-\epsilon,\epsilon) \times \mathcal{S}igma$
\[
\tilde{\omegaega} = \omegaega|_{\mathcal{S}igma} + d(r\eta) = \omegaega|_{\mathcal{S}igma} + rd\eta + dr\wedge \eta.
\]
If $\epsilon$ is small enough, the form $\omegaega|_{\mathcal{S}igma} + rd\eta$ is non-degenerate on $\ker \eta$ for every $r\in (-\epsilon,\epsilon)$, from which we deduce that $\tilde{\omegaega}$ is a symplectic form. Since $\tilde{\omegaega}|_{\{0\} \times \mathcal{S}igma}$ coincides with $\omegaega|_{\mathcal{S}igma}$, by the coisotropic neighborhood theorem (see \cite{got82}, or \cite[Exercise 3.36]{ms98} for the particular case of a hypersurface), a neighborhood of $\mathcal{S}igma$ in $W$ is symplectomorphic to $((-\epsilon,\epsilon) \times \mathcal{S}igma,\tilde{\omegaega})$, up to the choice of a smaller $\epsilon$. Since for $\timesi\in \mathcal{L}_{\mathcal{S}igma}(x)$ and $\zeta \in T_{(r,x)} (\{r\} \times \mathcal{S}igma) = (0) \times T_x \mathcal{S}igma$ there holds
\[
\tilde{\omegaega}(\timesi,\zeta) = \omegaega(\timesi,\zeta) + r d\eta (\timesi,\zeta) = 0,
\]
we deduce that $\ker (\tilde{\omegaega}|_{\{r\} \times \mathcal{S}igma}) = \mathcal{L}_{\mathcal{S}igma}$ does not depend on $r$. Therefore, $\{0\} \times \mathcal{S}igma$ is stable in $((-\epsilon,\epsilon) \times \mathcal{S}igma,\tilde{\omegaega})$ and hence $\mathcal{S}igma$ is stable in $(W,\omegaega)$.
\end{proof}
\begin{Rmk}
L.~Macarini and G.~P.~Paternain have constructed examples of Tonelli Lagrangians on the tangent bundle of $\mathbb{T}^n$ such that $H^{-1}(\kappa)$ is stable for $\kappa=c_u(L)=c_0(L)$, see \cite{mp10}.
\end{Rmk}
\section{Palais-Smale sequences}
\label{pss}
Palais-Smale sequences $(x_h,T_h)$ with $T_h\rightarrow 0$ are a possible source of non-compactness, but the next result shows that they occur only at level zero.
\begin{Lemma}
\label{PST0}
Let $(x_h,T_h)$ be a $(\mathrm{PS})_c$ sequence for ${\mathbb{S}}_{\kappa}$ with $T_h\to0$. Then $c=0$.
\end{Lemma}
\begin{proof}
Set $\gamma_h(t):= x_h(t/T_h)$.
Since $(x_h,T_h)$ is a $(\mathrm{PS})$ sequence, the identity (\ref{eq:diff A w.r.t. T}) shows that the sequence
\[
\alpha_h := \frac{1}{T_h} \int_0^{T_h} \bigl( E(\gamma_h,\gamma_h') - \kappa \bigr) \, dt = - \frac{\partialartial {\mathbb{S}}_{\kappa}}{\partialartial T} (x_h,T_h)
\]
is infinitesimal and, in particular, bounded. Since $L(x,v)$ is electromagnetic for $|v|$ large, $E(x,v)$ has the form (\ref{ene}) for $|v|$ large and hence satisfies the estimate
\[
E(x,v) \geq E_0 |v|^2 - E_1, \qquad \forall (x,v)\in TM,
\]
for suitable numbers $E_0>0$ and $E_1\in {\mathbb{R}}$. It follows that
\[
\alpha_h \geq \frac{1}{T_h} \int_0^{T_h} \bigl( E_0 |\gamma_h'|^2 - E_1 - \kappa \bigr)\, dt = \frac{E_0}{T_h} \int_0^{T_h} |\gamma_h'|^2\,dt - (E_1 + \kappa),
\]
and hence
\[
\int_0^{T_h} |\gamma_h'|^2\,dt \leq \frac{T_h}{E_0} ( \alpha_h + E_1 + \kappa) = O(T_h)
\]
for $h\rightarrow \infty$. From the lower bound (\ref{boundsonL}) and from the analogous upper bound
\[
L(x,v) \leq L_2 |v|^2 + L_3, \qquad \forall (x,v)\in TM,
\]
for suitable positive number $L_2,L_3$, we find
\[
{\mathbb{S}}_{\kappa}(x_h,T_h) = \int_0^{T_h} \bigl( L(\gamma_h,\gamma_h') + \kappa \bigr)\, dt \geq L_0 \int_0^{T_h} |\gamma_h'|^2\, dt + T_h (\kappa - L_1) = O(T_h)
\]
and
\[
{\mathbb{S}}_{\kappa}(x_h,T_h) = \int_0^{T_h} \bigl( L(\gamma_h,\gamma_h') + \kappa \bigr)\, dt \leq L_2 \int_0^{T_h} |\gamma_h'|^2\, dt + T_h (L_3 + \kappa) = O(T_h)
\]
for $h\rightarrow \infty$. Since $T_h\rightarrow 0$, we conclude that ${\mathbb{S}}_{\kappa}(x_h,T_h)$ is infinitesimal and hence $c=0$.
\end{proof}
\begin{Rmk}
By choosing a suitable metric on the Hilbert manifold $\mathcal{M}$, one can also obtain that for every Palais-Smale sequences $(x_h,T_h)$ with $T_h\rightarrow 0$ the sequence $(x_h)$ converges to an equilibrium solution of the Euler-Lagrange equations which has energy $\kappa$ (see \cite[Proposition 3.8]{con06}). In particular, such Palais-Smale sequences can exist only when $\kappa$ is a critical value of $E$.
\end{Rmk}
The next result says that Palais-Smale sequences $(x_h,T_h)$ with $(T_h)$ bounded and bounded away from zero are always compact.
\begin{Lemma}\label{Lem;2}
Let $(x_h,T_h)$ be a $(\mathrm{PS})_c$ sequence for ${\mathbb{S}}_{\kappa}$ with $0<T_*\leq T_h\leq T^*<\infty$. Then $(x_h,T_h)$ is compact in $\mathcal{M}M$.
\end{Lemma}
\begin{proof}
Up to a subsequence, we may assume that $(T_h)$ converges to some $T\in [T_*,T^*]$.
By (\ref{boundsonL}) we have
\begin{equation}
\label{isbd}
\begin{split}
c+o(1) = {\mathbb{S}}_{\kappa}(x_h,T_h) = T_h \int_0^1 \Bigr( L\bigl(x_h,x_h'/T_h \bigr)+\kappa \Bigr)\, ds \\ \geq T_h \int_0^1\Bigr( L_0 \frac{|x_h'|^2}{T_h^2}-(L_1-\kappa)\Bigr)\, ds \geq \frac{L_0}{T^*}\|x_h'\|_2^2-T^* |L_1-\kappa|,
\end{split} \end{equation}
where $\|\cdot\|_2$ denotes the $L^2$ norm with respect to the fixed Riemannian metric on $M$. Therefore, $\|x_h'\|_2$ is uniformly bounded and $(x_h)$ is 1/2-equi-H\"older-continuous:
\[
\mathrm{dist}\big(x_h(s'),x_h(s)\big)\leq\int_s^{s'}|x_h'(r)|\, dr \leq |s'-s|^{1/2} \|x_h'\|_2.
\]
By the Ascoli-Arzel\`a theorem, up to a subsequence $(x_h)$ converges uniformly to some $x\in C({\mathbb{T}},M)$. In particular, $(x_h)$ eventually belongs to the image of the parameterization $\varphi_*$ induced by a smooth map
\[
\varphi: {\mathbb{T}}\times B_r \rightarrow M.
\]
See \eqref{eq:chart of the loop space} and recall that the image of this parameterization is $C^0$-open.
Then $x_h=\varphi_*(\timesi_h)$, where $\timesi_h$ belongs to $W^{1,2}({\mathbb{T}},B_r)$ and is a $(\mathrm{PS})$ sequence for the functional
$$
\widetilde{{\mathbb{S}}}(\timesi,T)=T\int_{{\mathbb{T}}}\widetilde L\bigr(s,\timesi,\timesi'/T \bigr)\,ds,
$$
with respect to the standard Hilbert product on $W^{1,2}({\mathbb{T}},{\mathbb{R}}^n)$,
where the Lagrangian $\widetilde{L}\in C^{\infty}({\mathbb{T}}\times B_r \times {\mathbb{R}}^n)$ is obtained by pulling back $L+\kappa$ by $\varphi$. Moreover, $(\timesi_h)$ converges uniformly and, since $\|\timesi_h'\|_2$ is bounded, weakly in $W^{1,2}$ to some $\timesi$ in
$W^{1,2}({\mathbb{T}},B_r)$. We must prove that this convergence is actually strong in $W^{1,2}$.
Since $\widetilde{L}(s,x,v)$ is electromagnetic for $|v|$ large, we have the bounds
\begin{equation}
\label{bddL}
\bigl| d_x \widetilde L (s,x,v) \bigr| \leq C(1+|v|^2),\quad\bigl| d_v \widetilde L (s,x,v)\bigr| \leq C(1+|v|),
\end{equation}
for a suitable constant $C$. Since $(\timesi_h,T_h)$ is a $(\mathrm{PS})$ sequence with $(\timesi_h)$ bounded in $W^{1,2}$, we have by (\ref{diff1})
\begin{equation*}
\begin{split}
o(1) &= d\widetilde{{\mathbb{S}}}(\timesi_h,T_h)[(\timesi_h-\timesi,0)] \\ &= T_h \int_{{\mathbb{T}}} d_x \widetilde L \bigl(s,\timesi_h,\timesi_h'/T_h\bigr) [\timesi_h-\timesi]\, ds + T_h \int_{{\mathbb{T}}} d_v \widetilde{L} \bigl(s,\timesi_h,\timesi_h'/T_h\bigr) \bigl[ (\timesi_h'-\timesi')/T_h \bigr] \, ds.
\end{split} \end{equation*}
By the first bound in (\ref{bddL}) and the uniform convergence $\timesi_h\rightarrow \timesi$, the first integral is infinitesimal. Together with the fact that $(T_h)$ is bounded away from zero, we obtain
\begin{equation}
\label{uno}
\int_{{\mathbb{T}}} d_v \widetilde{L} \bigl(s,\timesi_h,\timesi_h'/T_h\bigr) \bigl[ (\timesi_h'-\timesi')/T_h \bigr] \, ds=o(1).
\end{equation}
From the fiberwise uniform convexity of $\widetilde{L}$, we have the bound
\[
d_{vv} \widetilde{L} (s,x,v) [u, u] \geq \delta |u|^2, \quad \forall (s,x,v)\in {\mathbb{T}}\times B_r \times {\mathbb{R}}^n, \; u\in {\mathbb{R}}^n,
\]
for a suitable positive number $\delta$. It follows that
\begin{equation*}
\begin{split}
d_v \widetilde L \left(s,\timesi_h,\frac{\timesi_h'}{T_h}\right) & \left[ \frac{\timesi_h'-\timesi'}{T_h} \right] - d_v\widetilde L \left(s,\timesi_h,\frac{\timesi'}{T_h} \right)\left[\frac{\timesi_h'-\timesi'}{T_h} \right] \\
&= \int_0^1 d_{vv} \widetilde L \left(s,\timesi_h,\frac{\timesi'}{T_h}+\sigma \frac{\timesi_h'-\timesi'}{T_h} \right) \left[ \frac{\timesi_h'-\timesi'}{T_h}, \frac{\timesi_h'-\timesi'}{T_h} \right] \, d\sigma \geq \frac{\delta}{T_h^2} |\timesi_h'- \timesi'|^2.
\end{split} \end{equation*}
By integrating this inequality over $s\in {\mathbb{T}}$ and by (\ref{uno}), we obtain
\[
o(1)- \int_{{\mathbb{T}}} d_v \widetilde L \bigl(s,\timesi_h,\timesi'/T_h)\bigl[ (\timesi_h'-\timesi')/T_h \bigr] \, ds \geq \frac{\delta}{T_h^2} \|\timesi_h' - \timesi'\|_2^2.
\]
By the second bound in (\ref{bddL}), the sequence
\[
d_v \widetilde L \bigl(s,\timesi_h,\timesi'/T_h\bigr)
\]
converges strongly in $L^2$. By the weak $L^2$ convergence to $0$ of $(\timesi_h'-\timesi)$, we deduce that the integral on the left-hand side of the above inequality is infinitesimal. We conclude that $(\timesi_h)$ converges to $\timesi$ strongly in $W^{1,2}$.
\end{proof}
In general, ${\mathbb{S}}_{\kappa}$ might have Palais-Smale sequences $(x_h,T_h)$ with $(T_h)$ unbounded. However, this does not occur when $\kappa$ is larger than the lowest Ma\~{n}\'e critical value $c_u(L)$:
\begin{Lemma}\label{Lem:6}
If $\kappa>c_u(L)$, then any $(\mathrm{PS})$ sequence $(x_h,T_h)$ in a given connected component of $\mathcal{M}M$ with $T_h\geq T_*>0$ is compact.
\end{Lemma}
\begin{proof}
By Lemma \ref{Lem;2}, it is enough to show that $(T_h)$ is bounded from above. Since
\[
{\mathbb{S}}_{\kappa} (x,T)={\mathbb{S}}_{c_u(L)} (x,T) + \big(\kappa -c_u(L)\big)T,
\]
the period
\[
T_h = \frac{1}{\kappa -c_u(L)} \bigr({\mathbb{S}}_{\kappa} (x_h,T_h) - {\mathbb{S}}_{c_u(L)} (x_h,T_h)\bigr)
\]
is bounded from above, because ${\mathbb{S}}_{\kappa}$ is bounded on the $(\mathrm{PS})$ sequence $(x_h,T_h)$ and ${\mathbb{S}}_{c_u(L)}(x_h,T_h)$ is bounded from below by Lemma \ref{Lem:5}.
\end{proof}
\begin{Rmk}
By choosing a suitable metric on $\mathcal{M}$, it is possible to characterize $c_u(L)$ as the infimum of all $\kappa_0$'s such that ${\mathbb{S}}_{\kappa}$ satisfies the Palais-Smale condition for every $\kappa\in [\kappa_0,+\infty)$. See \cite{cipp00}.
\end{Rmk}
\section{Periodic orbits with high energy}
\label{pohe}
The following result shows that the energy levels above $c_u(L)$ have always periodic orbits and proves statements (i) and (ii) of the theorem in the Introduction.
\begin{Thm}
\label{highthm}
Assume that $\kappa>c_u(L)$. Then the following facts hold.
\begin{enumerate}
\item If $M$ is not simply connected, then the energy level $E^{-1}(\kappa)$ has a $\kappa$-action-minimizing periodic orbit in each non-trivial homotopy class of the free loop space.
\item If $M$ is simply connected, then the energy level $E^{-1}(\kappa)$ has a periodic orbit with positive action ${\mathbb{S}}_{\kappa}$.
\end{enumerate}
\end{Thm}
\begin{proof}
(i) Assume that $M$ is not simply connected. Let $\alpha \in[{\mathbb{T}},M]$ be a non-trivial homotopy class and let
$\mathcal{M}M_{\alpha}$ be the connected component of $\mathcal{M}M^\mathrm{noncontr}$ corresponding to $\alpha$. By Lemma \ref{Lem:5}, the functional ${\mathbb{S}}_{\kappa}$ is bounded from below on $\mathcal{M}M_{\alpha}$. By Lemma \ref{Lem:3} (i), the sublevels
\[
\{ (x,T)\in \mathcal{M}M_{\alpha} \, | \, {\mathbb{S}}_{\kappa}(x,T)\leq c \}
\]
are complete. Let $(x_h,T_h)\subset \mathcal{M}M_{\alpha}$ be a (PS) sequence for ${\mathbb{S}}_{\kappa}$. By Lemma \ref{Lem:3} (i), $(T_h)$ is bounded away from zero, so Lemma \ref{Lem:6} implies that ${\mathbb{S}}_{\kappa}$ satisfies the (PS) condition on $\mathcal{M}M_{\alpha}$. By Remark \ref{mini}, we conclude that ${\mathbb{S}}_{\kappa}$ has a minimizer on $\mathcal{M}M_{\alpha}$, as we wished to prove.
(ii) Assume that $M$ is simply connected, so that $\mathcal{M}M=\mathcal{M}M^{\mathrm{contr}}$. In this case, ${\mathbb{S}}_{\kappa}$ is strictly positive everywhere, because $\kappa>c_u(L)$, but the infimum of ${\mathbb{S}}_{\kappa}$ is zero, as one readily checks by looking at sequences of the form $(x_0,T_h)$, with $x_0$ a constant loop and $T_h\rightarrow 0$. So the infimum is not achieved. We will find the periodic orbit by considering the same minimax class which Lusternik and Fet \cite{lf51} considered in their proof of the existence of a closed geodesic on a simply connected compact manifold.
Since the closed manifold $M$ is simply connected, there exists $l\geq 2$ such that $\partiali_l(M)\ne 0$ (a manifold all of whose homotopy groups vanish is contractible, but closed manifolds are never contractible, for instance because their $n$-dimensional homology group with ${\mathbb{Z}}_2$ coefficients does not vanish). We fix a non-zero homotopy class $\mathcal{G} \in\partiali_l(M)$. Thanks to the isomorphism
$\partiali_{l-1}(C({\mathbb{T}},M))\cong\partiali_l(M)$, we have an induced non-zero homotopy class
\[
\mathcal{H} \in [S^{l-1},C({\mathbb{T}},M)] \cong [S^{l-1},\mathcal{M}M],
\]
and we consider the minimax value
\[
c=\inf_{\substack{h:S^{l-1}\to \mathcal{M}M\\ h\in \mathcal{H}}} \max_{\timesi\in S^{l-1}} {\mathbb{S}}_{\kappa} (h(\timesi)).
\]
Let us show that $c>0$. Since $\mathcal{H}$ is non-trivial, there exists a positive number $a$ such that for every map $h=(x,T): S^{l-1} \rightarrow \mathcal{M}M$ belonging to the class $\mathcal{H}$ there holds
\[
\max_{\timesi \in S^{l-1}} \ell(x(\timesi)) \geq a,
\]
where $\ell(x(\timesi))$ denotes the length of the loop $x(\timesi)$ (see \cite[Theorem 2.1.8]{kli78}). If $(x,T)$ is an element of $\mathcal{M}M$ with $\ell(x)\geq a$, then (\ref{boundsonL}) implies
\begin{equation*}
\begin{split}
{\mathbb{S}}_{\kappa}(x,T) &= T\int_{{\mathbb{T}}} \Bigl(L\bigl(x,x'/T\bigr)+\kappa \Bigr) \,ds
\geq T \int_{{\mathbb{T}}} \Bigl( L_0\frac{|x'|^2}{T^2} -L_1+\kappa \Bigr)\, ds \\
& \geq \frac{L_0}{T}\ell(x)^2-T(L_1-\kappa)
\geq \frac{L_0}{T}a^2-T(L_1-\kappa).
\end{split} \end{equation*}
Since $a>0$, the above chain of inequalities implies that there exists $T_0>0$ such that for every $(x,T)\in \mathcal{M}M$ with $\ell(x)\geq a$ and ${\mathbb{S}}_{\kappa}(x,T)\leq c+1$, the period $T$ is at least $T_0$. Now let $h\in \mathcal{H}$ be such that the maximum of ${\mathbb{S}}_{\kappa}$ on $h(S^{l-1})$ is less than $c+1$. By the above considerations, there exists $(x,T)$ in $h(S^{l-1})$ with $T\geq T_0$, whence
\[
{\mathbb{S}}_{\kappa} (x,T)={\mathbb{S}}_{c_u(L)} (x,T)+ \bigl(\kappa-c_u(L)\bigr)T \geq \bigl(\kappa-c_u(L)\bigr) T_0.
\]
This shows that the minimax value $c$ is strictly positive.
Theorem \ref{thm:finite c induces PS}, together with Remark \ref{trunc} and Lemma \ref{Lem:4}, implies the existence of a $(\mathrm{PS})_c$ sequence $(x_h,T_h)$. Lemma \ref{PST0} guarantees that $(T_h)$ is bounded away from zero, so by Lemma \ref{Lem:6} the sequence $(x_h,T_h)$ has a limiting point in $\mathcal{M}M$, which gives us the required periodic orbit.
\end{proof}
\begin{Rmk}
If $M$ is not simply connected and $\kappa>c_u(L)$, the energy level $E^{-1}(\kappa)$ might have no contractible periodic orbits. Consider for instance the Lagrangian $L(x,v)=|v|^2/2$ on the torus ${\mathbb{T}}^n$, equipped with the flat metric. The corresponding Euler-Lagrange flow is the geodesic flow on $T{\mathbb{T}}^n$, $c_u(L)=0$, and all the non-constant closed geodesics on the flat torus are non-contractible.
\end{Rmk}
\begin{Rmk}
\label{veryhigh}
If $\kappa>c_0(L)$, the existence of a periodic orbit on $E^{-1}(\kappa)$ also follows from Theorem \ref{finsler}, because every Finsler metric on a closed manifold has a closed geodesic.
\end{Rmk}
\begin{Rmk}
Theorem \ref{finsler} implies that the multiplicity results for closed Finsler geodesics hold also for Hamiltonian orbits at energy levels $\kappa>c_0(L)$. Actually, most of these results remain true for $\kappa>c_u(L)$.
In fact, as the proof of Theorem \ref{highthm} suggests, the (PS) condition and the topology of the sublevels of the functional ${\mathbb{S}}_{\kappa}$ are analogous to the corresponding properties of the Finsler geodesic energy functional (with the notable exception of the zero level).
\end{Rmk}
\section{Topology of the free period action functional for low energies}
\label{tfpafle}
As we have seen, when $\kappa<c_u(L)$ the functional ${\mathbb{S}}_{\kappa}$ is unbounded from below on each connected component of $\mathcal{M}M$. The aim of this section is to show that, nevertheless, some sublevels of ${\mathbb{S}}_{\kappa}$ have a sufficiently rich topology.
We start by proving a simple lemma about the integral of a one form. The integral of a given one-form on a curve $x$ is clearly bounded by a constant times the length of $x$. When the support of the curve is contained in a ball of $M$, one may also take the square of the length in this bound, which is a better estimate for short curves. More precisely, we have the following:
\begin{Lemma}\label{Lemma:isoper}
Let $\theta$ be a smooth one-form on $M$ and let $U\subset M$ be an open set whose closure is diffeomorphic to a closed ball in ${\mathbb{R}}^n$. Then there exists a number ${\mathbb{T}}heta>0$ such that
\[
\bigg| \int_{\mathbb{T}} x^*(\theta) \bigg| \leq {\mathbb{T}}heta \cdot\ell(x)^2,
\]
for every closed curve $x:{\mathbb{T}}\rightarrow U$.
\end{Lemma}
\begin{proof}
Up to the change of the constant ${\mathbb{T}}heta$,
we may assume that $U=B_r$ is the ball of radius $r$ around the origin in ${\mathbb{R}}^n$, equipped with the Euclidean metric. Given the closed curve $x:{\mathbb{T}}\rightarrow B_r$, we consider the map
\[
X:[0,1]\times {\mathbb{T}} \rightarrow B_r, \quad X(s,t) = x(0) + s \bigl( x(t) - x(0) \bigr).
\]
Then $X(1,t)=x(t)$ and $X(0,t)=x(0)$, hence by Stokes theorem
\begin{equation*}
\begin{split}
\left| \int_{{\mathbb{T}}} x^*(\theta) \right| &= \bigg| \int_{[0,1] \times {\mathbb{T}}} X^* (d\theta) \bigg| = \left| \int_0^1 ds \int_{{\mathbb{T}}} d\theta\bigl( X(s,t) \bigr) \Bigl[ \frac{\partial X}{\partial s} , \frac{\partial X}{\partial t} \Bigr] \, dt \right| \\ &\leq \|d\theta\|_{\infty} \int_0^1 ds \int_{{\mathbb{T}}} \left| \frac{\partial X}{\partial s} \right| \, \left| \frac{\partial X}{\partial t} \right| \, dt
= \|d\theta\|_{\infty} \int_0^1 ds \int_{{\mathbb{T}}} \bigl| x(t) - x(0) \bigr| s \bigl| x'(t) \bigr|\, dt \\ &\leq \frac{1}{2} \|d\theta\|_{\infty}\, \ell(x) \int_0^1 ds \int_{{\mathbb{T}}} s \bigl| x'(t)\bigr|\, dt = \frac{1}{4} \|d\theta\|_{\infty}\, \ell(x)^2,
\end{split} \end{equation*}
as claimed.
\end{proof}
\partialaragraph{\bf The energy range $\mathbf{(e_0(L),c_u(L))}$} If $\kappa<c_u(L)$, there are contractible closed curves with negative action ${\mathbb{S}}_{\kappa}$. Since the space of contractible loops is connected, we can consider the following class of continuous paths in $\mathcal{M}M$:
\begin{equation}
\label{curve}
\mathcal{Z}_0 := \bigl\{ (x,T):[0,1] \rightarrow \mathcal{M}M\, \big| \, x(0) \mbox{ is a constant loop and } {\mathbb{S}}_{\kappa}(x(1),T(1)) <0 \bigr\}.
\end{equation}
Notice that if $x_0$ is a constant loop and $T>0$, then
\begin{equation}
\label{constants}
{\mathbb{S}}_{\kappa}(x_0,T) = T \bigl( L(x_0,0) + \kappa \bigr) = T \bigl( \kappa - E(x_0,0) \bigr).
\end{equation}
When $\kappa>e_0(L)=\max_{x\in M} E(x,0)$, the above quantity is strictly positive (and tends to zero for $T\rightarrow 0$). The next lemma shows that when $e_0(L) < \kappa < c_u(L)$, ${\mathbb{S}}_{\kappa}$ has a sort of mountain pass geometry:
\begin{Lemma}
\label{Lem:mountain pass}
Assume that $e_0(L) <\kappa < c_u(L)$. Then there exists $a>0$ such that for every $z\in \mathcal{Z}_0$ there holds
\[
\max_{\sigma\in[0,1]} {\mathbb{S}}_{\kappa} \bigl(z(\sigma)\bigr)\geq a.
\]
\end{Lemma}
\begin{proof}
Consider the smooth one-form on $M$,
\[
\theta(x) [v] := d_v L(x,0)[v].
\]
By taking a Taylor expansion and by using the bound (\ref{bd2L}), we get the estimate
\begin{equation}
\label{bbb}
L(x,v) = L(x,0)+d_vL(x,0)[v]+\frac{1}{2}d_{vv}L(x,s v)[v,v]
\geq -E(x,0)+\theta(x)[v]+ L_0 |v|^2,
\end{equation}
where $s\in [0,1]$. Let $\{U_1,\dots,U_N\}$ be a finite covering of $M$ consisting of open sets whose closures are diffeomorphic to closed Euclidean balls, and let ${\mathbb{T}}heta >0$ be such that the conclusion of Lemma
\ref{Lemma:isoper} holds for the one-form $\theta$, for each of the open sets $U_j$'s. Let $r_0$ be a Lebesgue number for this covering, meaning that every ball of radius $r_0$ is contained in one of the $U_j$'s.
We claim that if ${\mathbb{S}}_{\kappa}(x,T)<0$ then
\begin{equation}
\label{claim}
\ell(x)>\min\bigg\{r_0,\frac{\sqrt{L_0(\kappa-e_0(L))}}{{\mathbb{T}}heta}\bigg\}=:r_1.
\end{equation}
In fact, assuming that $\ell(x)\leq r_0$, we have that $x({\mathbb{T}})$ is contained in some $U_j$, for $1\leq j \leq N$. Set as usual $\gamma(t)=x(t/T)$.
By Lemma \ref{Lemma:isoper} and by (\ref{bbb}), we obtain the chain of inequalities
\begin{equation}
\label{stma}
\begin{split}
0 &>{\mathbb{S}}_{\kappa}(x,T) ={\mathbb{S}}_{\kappa}(\gamma) = \int_0^T \bigr(L(\gamma,\gamma')+\kappa \bigr)\,dt
\\ &\geq \int_0^T\bigr( -E(\gamma,0) + \theta(\gamma)[\gamma'] + L_0 |\gamma'|^2 + \kappa \bigr)\, dt
\\ &= \int_0^T \bigl( \kappa - E(\gamma,0)\bigr)\, dt + \int_{{\mathbb{R}}/T{\mathbb{Z}}} \gamma^*(\theta) + L_0 \int_0^T |\gamma'|^2\, dt \\
&\geq \bigl(\kappa-e_0(L)\bigr)T-{\mathbb{T}}heta\cdot \ell(\gamma)^2+\frac{L_0}{T}\ell(\gamma)^2.
\end{split} \end{equation}
Since we are assuming $\kappa>e_0(L)$, the above estimate implies that
$T> L_0/{\mathbb{T}}heta$ and that
\[
\ell(\gamma)^2>\frac{\bigl(\kappa-e_0(L)\bigr)T}{{\mathbb{T}}heta-L_0/T}>\frac{\bigl(\kappa-e_0(L)\bigr)T}{{\mathbb{T}}heta}>\frac{\bigl(\kappa-e_0(L)\bigr)L_0}{{\mathbb{T}}heta^2},
\]
which proves (\ref{claim}).
Fix some number $r$ in the open interval $(0,r_1)$.
Since $z=(x,T)\in \mathcal{Z}_0$, ${\mathbb{S}}_{\kappa}(x(1),T(1))$ is negative, so by (\ref{claim}) the length of $x(1)$ is larger than $r_1$. By continuity, using the fact that $x(0)$ is a constant loop, we get the existence of $\sigma\in(0,1)$ for which $\ell(x(\sigma))=r$. Then (\ref{stma}) implies
\[
{\mathbb{S}}_{\kappa}(x(\sigma),T(\sigma)) \geq \bigl(\kappa-e_0(L)\bigr) T +\Bigr(\frac{L_0}{T}-{\mathbb{T}}heta\Bigr)r^2.
\]
Minimization in $T$ yields
\[
{\mathbb{S}}_{\kappa}(x(\sigma),T(\sigma)) \geq
r \bigr(\sqrt{L_0(\kappa-e_0(L))}-{\mathbb{T}}heta r \bigr)=: a.
\]
The number $a$ is positive because $r<r_1$. This concludes the proof.
\end{proof}
\partialaragraph{\bf The energy range $\mathbf{(\min E,e_0(L))}$} When $\kappa<e_0(L)$, the identity (\ref{constants}) shows that ${\mathbb{S}}_{\kappa}$ takes negative values on some constant loops, and the conclusion of Lemma \ref{Lem:mountain pass} cannot hold. Instead than considering the class of paths which go from some constant loop to a loop of negative action, one has to consider the class of deformations of the space of constant loops - which is diffeomorphic to $M$ - into the space of loops with negative action. More precisely, we consider the set of continuous maps
\[
\mathcal{Z}_M = \bigl\{ (x,T): [0,1]\times M \rightarrow \mathcal{M}M \, \big| \, x(0,x_0) = x_0 \mbox{ and } {\mathbb{S}}_{\kappa}\bigl(x(1,x_0),T(1,x_0)\bigr)<0, \; \forall x_0\in M\bigr\}.
\]
\begin{Lemma}
\label{notempty}
If $\kappa<c_u(L)$, then the set $\mathcal{Z}_M$ is not empty.
\end{Lemma}
We just sketch the proof, referring to \cite{tai83} for more details (see also \cite{tai10}). The argument follows closely Bangert's technique of ``pulling one loop at a time'' (see \cite{ban80} and \cite{bk83}).
Let $M_0\subset M_1 \subset \dots \subset M_n = M$ be a CW-complex decomposition of $M$. Since $\kappa<c_u(L)$ and since the 0-skeleton $M_0$ is a finite set, it is easy to construct a continuous map
\[
z_0:[0,1]\times M \rightarrow \mathcal{M}M, \quad z_0(\sigma,x_0) = \bigl(y_0(\sigma,x_0),T_0(\sigma,x_0)\bigr),
\]
such that
\begin{enumerate}
\item $y_0(0,x_0) = x_0$ for every $x_0\in M$;
\item ${\mathbb{S}}_{\kappa}\circ z_0(1,x_0)<0$ for every $x_0\in M_0$.
\end{enumerate}
Given a positive integer $h$, we may iterate each loop $h$ times and obtain the map
\[
z_0^h: [0,1]\times M\rightarrow \mathcal{M}M, \quad z_0^h(\sigma,x_0) = \bigl(y_0^h(\sigma,x_0),hT_0(\sigma,x_0)\bigr),
\]
where
\[
y_0^h(\sigma,x_0)(s) := y_0(\sigma,x_0)(hs), \quad \forall (\sigma,x_0)\in [0,1]\times M, \; \forall s\in {\mathbb{T}}.
\]
Consider an edge $S$ in $M_1$ with end-points $x_0,x_1\in M_0$. The map $z_0^h(1,\cdot)$ maps the the end-points of $S$ into the $h$-th iterates $\alpha^h$ and $\beta^h$ of two loops $\alpha$ and $\beta$ with negative action ${\mathbb{S}}_{\kappa}$. By pulling one of the $h$ loops at a time from $\alpha^h$ to $\beta^h$, one can construct a new map from $S$ into $\mathcal{M}M$ with end-points $\alpha^h$ and $\beta^h$ and such that ${\mathbb{S}}_{\kappa}$ is negative on its image, provided that $h$ is large enough. By relying on the map $z_0^h$, this construction can be done globally, and one ends up with a continuous map
\[
z_1:[0,1]\times M\rightarrow \mathcal{M}M, \quad z_1(\sigma,x_0) = \bigl(y_1(\sigma,x_0),T_1(\sigma,x_0)\bigr),
\]
such that
\renewcommand{\roman{enumi}}{\roman{enumi}}
\renewcommand{(\theenumi)}{(\roman{enumi}')}
\begin{enumerate}
\item $y_1(0,x_0) = x_0$ for every $x_0\in M$;
\item ${\mathbb{S}}_{\kappa}\circ z_1(1,x_0)<0$ for every $x_0\in M_1$.
\end{enumerate}
Iterating this process, one can construct continuous maps
\[
z_k:[0,1]\times M\rightarrow \mathcal{M}M, \quad z_k(\sigma,x_0) = \bigl(y_k(\sigma,x_0),T_k(\sigma,x_0)\bigr),
\]
such that
\renewcommand{\roman{enumi}}{\roman{enumi}}
\renewcommand{(\theenumi)}{(\roman{enumi}'')}
\begin{enumerate}
\item $y_k(0,x_0) = x_0$ for every $x_0\in M$;
\item ${\mathbb{S}}_{\kappa}\circ z_k(1,x_0)<0$ for every $x_0\in M_k$.
\end{enumerate}
\renewcommand{\roman{enumi}}{\roman{enumi}}
\renewcommand{(\theenumi)}{(\roman{enumi})}
The map $z_n$ is an element of $\mathcal{Z}_M$. This concludes our sketch of the proof of Lemma \ref{notempty}.
The proof of the following result is analogous to the proof of Lemma \ref{Lem:mountain pass}.
\begin{Lemma}
\label{mtpass2}
Assume that $\min E <\kappa < c_u(L)$. Then there exists $a>0$ such that for every $z\in \mathcal{Z}_M$ there holds
\[
\max_{(\sigma,x_0) \in[0,1]\times M} {\mathbb{S}}_{\kappa} \bigl(z(\sigma,x_0)\bigr)\geq a.
\]
\end{Lemma}
\section{Periodic orbits with low energy}
\label{pole}
\partialaragraph{\bf The Struwe monotonicity argument} When $\kappa\leq c_u(L)$, the periods in a (PS) sequence need not be bounded anymore. Because of this fact, the question of the existence of periodic orbits for every energy $\kappa$ in the interval $[\min E,c_u(L)]$ is open, although no counterexamples are known.
The known system which is closer to being a counterexample is the horocycle flow on a closed surface $M$ with constant negative curvature (see e.g. \cite{man91, cmp04}): Such a flow has no periodic orbits (actually, every orbit is dense) and it is the restriction of a Hamiltonian flow to an energy surface at a Ma\~{n}\'e critical value, but the corresponding Lagrangian is well defined only on the (non compact) universal cover of $M$ (such a system belongs to the family of {\em non-exact magnetic flows}, whereas only {\em exact} magnetic flows can be described by a Tonelli Lagrangian on $TM$).
The following argument is a version of an argument of Struwe, which says that when dealing with a minimax value associated to a family of functionals depending on a real parameter in a suitable monotone way, there exist compact (PS) sequences for almost every value of the parameter. This argument has applications both to Hamiltonian periodic orbits and to semilinear elliptic equations
(see \cite{str90}, \cite[section II.9]{str00} and references therein).
Let us assume that $\min E<c_u(L)$, otherwise the interval of low energies collapses to a single level and there is nothing to prove.
Given $\kappa\in (\min E,c_u(L))$, let ${\bf G}amma$ be the set of the images of the maps either in $\mathcal{Z}_0$ or in $\mathcal{Z}_M$, which were introduced in the previous section: If $e_0(L)< \kappa < c_u(L)$ we may take $\mathcal{Z}_0$, while in general we should take $\mathcal{Z}_M$. Let $I$ be either the interval $(e_0(L),c_u(L))$ - if we are dealing with $\mathcal{Z}_0$ - or the interval $(\min E,c_u(L))$ - if we are dealing with $\mathcal{Z}_M$. For every $\kappa\in I$, consider the minimax value
\begin{equation}
\label{minimax}
c(\kappa) := \inf_{K \in {\bf G}amma} \max_{(x,T)\in K} {\mathbb{S}}_{\kappa} (x,T).
\end{equation}
By Lemmas \ref{Lem:mountain pass}, \ref{notempty}, and \ref{mtpass2}, $c(\kappa)$ is finite and positive, and since ${\mathbb{S}}_{\kappa}$ depends monotonically on $\kappa$, the function
\[
c:I \rightarrow (0,+\infty)
\]
is weakly increasing. By Lebesgue Theorem, the set of points of $I$ at which $c$ has a linear modulus of continuity, that is
\[
J := \bigl\{ \bar\kappa\in I\, \big| \, \exists \delta>0, \; \exists M>0 \mbox{ s.t. } |c(\kappa) - c(\bar{\kappa})| \leq M |\kappa - \bar{\kappa}| \mbox{ for every } \kappa \in I \mbox{ with }|\kappa - \bar\kappa| < \delta\bigr\},
\]
has full Lebesgue measure in $I$.
\begin{Lemma}
\label{strarg}
If $\bar \kappa\in J$, then ${\mathbb{S}}_{\bar\kappa}$ admits a bounded (PS) sequence at level $c(\bar\kappa)$, which consists of contractible loops.
\end{Lemma}
\begin{proof}
First recall that ${\bf G}amma$ is a class of subsets of $\mathcal{M}M^{\mathrm{contr}}$.
Let $(\kappa_h)\subset I$ be a strictly decreasing sequence which converges to $\bar\kappa$, and set $\epsilon_h:=\kappa_h-\bar \kappa\downarrow0$. We pick $K_h \in {\bf G}amma$ such that
\[
\max_{K_h}{\mathbb{S}}_{\kappa_h}\leq c(\kappa_h)+\epsilon_h.
\]
Let $z=(x,T)\in K_h$ be such that ${\mathbb{S}}_{\bar \kappa}(z)>c(\bar \kappa)-\epsilon_h$. Since $\bar\kappa$ belongs to $J$, we have
\[
T = \frac{{\mathbb{S}}_{\kappa_h}(z)-{\mathbb{S}}_{\bar \kappa}(z)}{\kappa_h-\bar\kappa} \leq \frac{c(\kappa_h)+\epsilon_h-c(\bar\kappa)+\epsilon_h}{\epsilon_h}\leq M+2.
\]
Moreover,
\[
{\mathbb{S}}_{\bar \kappa}(z) \leq {\mathbb{S}}_{\kappa_h}(z) \leq c(\kappa_h)+\epsilon_h \leq c(\bar \kappa)+(M+1)\epsilon_h.
\]
By the above considerations,
\[
K_h \subset A_h \cup\big\{{\mathbb{S}}_{\bar \kappa}\leq c(\bar \kappa)-\epsilon_h\big\},
\]
where
\[
A_h := \big\{(x,T)\,\big|\, T\leq M+2\mbox{ and }{\mathbb{S}}_{\bar \kappa} (x,T) \leq c(\bar \kappa)+(M+1)\epsilon_h\big\}.
\]
If $(x,T)$ belongs to $A_h$, we have the estimate
\[
{\mathbb{S}}_{\bar\kappa}(x,T) \geq \frac{L_0}{M+2}\|x'\|_2^2-(M+2)(L_1-\bar\kappa),
\]
(see (\ref{isbd})), which shows that $A_h$ is bounded in $\mathcal{M}M$, uniformly in $h$. Let $\partialhi$ be the flow of the vector field obtained by multiplying $-\nabla{\mathbb{S}}_{\bar \kappa}$ by a suitable non-negative function, whose role is to make the vector field bounded on $\mathcal{M}M$ and vanishing on the sublevel $\{{\mathbb{S}}_{\bar\kappa} \leq c(\bar\kappa)/4\}$, while keeping the uniform decrease condition
\begin{equation}
\label{ud}
\frac{d}{d\sigma} {\mathbb{S}}_{\bar\kappa} \bigl(\partialhi_{\sigma}(z)\bigr) \leq - \frac{1}{2} \min \bigl\{
\|d{\mathbb{S}}_{\bar\kappa} (\partialhi_{\sigma}(z)) \|^2,1 \bigr\}, \quad \mbox{if } {\mathbb{S}}_{\bar\kappa} (\partialhi_{\sigma}(z)) \geq c(\bar\kappa)/2.
\end{equation}
See (\ref{decr}) and Remarks \ref{noncomp}, \ref{trunc}. Then Lemma \ref{Lem:4} implies that $\partialhi$ is well-defined on $[0,+\infty[\times \mathcal{M}M$, and the class of sets ${\bf G}amma$ is positively invariant with respect to $\partialhi$. Since $\partialhi$ maps bounded sets into bounded sets, we have
\begin{equation}
\label{dove}
\partialhi([0,1]\times K_h) \subset B_h \cup \bigl\{{\mathbb{S}}_{\bar \kappa}\leq c(\bar \kappa)-\epsilon_h \bigr\},
\end{equation}
for some uniformly bounded set
\begin{equation}
\label{bdd}
B_h\subset \bigl\{{\mathbb{S}}_{\bar \kappa}\leq c(\bar \kappa)+(M+1)\epsilon_h \bigr\}.
\end{equation}
We claim that there exists a sequence $(z_h)\subset \mathcal{M}M^{\mathrm{contr}}$ with
\[
z_h \in B_h\cap \bigl\{{\mathbb{S}}_{\bar \kappa}\geq c(\bar \kappa)-\epsilon_h \bigr\},
\]
and $\| d{\mathbb{S}}_{\bar \kappa}(z_h)\|$ infinitesimal. Such a sequence is clearly a bounded (PS) sequence at level $c(\bar\kappa)$.
Assume, by contradiction, the above claim to be false. Then there exists $0<\delta<1$ which satisfies
\[
\|d{\mathbb{S}}_{\bar \kappa}\| \geq \delta \quad \mbox{on } B_h \cap \bigl\{{\mathbb{S}}_{\bar \kappa}\geq c(\bar \kappa)-\epsilon_h\bigr\},
\]
for every $h$ large enough. Together with (\ref{ud}), (\ref{dove}) and (\ref{bdd}), this implies that, for $h$ large enough, for any $z\in K_h$ such that
\[
\partialhi([0,1]\times \{z\}) \subset \bigl\{ {\mathbb{S}}_{\kappa} \geq c(\bar \kappa)-\epsilon_h\bigr\},
\]
there holds
\[
{\mathbb{S}}_{\bar \kappa} \big(\partialhi_1(z)\big) \leq {\mathbb{S}}_{\bar \kappa}(z)- \frac{1}{2} \delta^2 \leq c(\bar \kappa)+ (M+1) \epsilon_h- \frac{1}{2} \delta^2.
\]
It follows that
\[
\max_{\partialhi_1(K_h)} {\mathbb{S}}_{\bar \kappa} \leq c(\bar \kappa)-\epsilon_h,
\]
for $h$ large enough. Since $\partialhi_1(K_h)$ belongs to ${\bf G}amma$, this contradicts the definition of $c(\bar\kappa)$ and concludes the proof.
\end{proof}
\partialaragraph{\bf Existence of periodic orbits of low energy} We are finally ready to prove the following result, which is statement (iii) in the theorem of the Introduction:
\begin{Thm}
\label{ae}
For almost every $\kappa \in (\min E,c_u(L))$, there is a contractible periodic orbit $\gamma$ of energy $E(\gamma,\gamma')=\kappa$ and positive action ${\mathbb{S}}_{\kappa}(\gamma)=c(\kappa)$.
\end{Thm}
\begin{proof}
Let $\kappa$ be an element of the full measure set $J\subset I$.
By Lemma \ref{strarg}, ${\mathbb{S}}_{\kappa}$ admits a (PS)$_{c(\kappa)}$ sequence $(x_h,T_h)\subset \mathcal{M}M^{\mathrm{contr}}$ with $(T_h)$ bounded. By Lemma \ref{PST0}, $(T_h)$ is bounded away from zero, because $c(\kappa)>0$.
By Lemma \ref{Lem;2}, the sequence $(x_h,T_h)$ has a limiting point in $\mathcal{M}M^{\mathrm{contr}}$, which gives us the required contractible periodic orbit.
\end{proof}
\begin{Rmk}
The existence of a periodic orbit for almost energy level in $(\min E,e_0(L))$ can be proved also by an argument from symplectic topology. In fact, let $H:T^*M \rightarrow {\mathbb{R}}$ be the Hamiltonian which is Legendre dual to $L$.
The fact that $\kappa< e_0(L)$ implies that the restriction of the projection $T^*M\rightarrow M$ to $H^{-1}(\kappa)$ is not surjective. Therefore, one can build a Hamiltonian diffeomorphism of $T^*M$ which displaces $H^{-1}(\kappa)$ from itself (see \cite[Proposition 8.2]{con06}). Sets which are displaceable by a Hamiltonian diffeomorphism have finite $\partiali_1$-sensitive Hofer-Zehnder capacity (see \cite{sch06} and \cite{fs07}) and this fact implies the almost everywhere existence result for periodic orbits (see \cite{hz94}). See \cite[Corollary 8.3]{con06} for more details on such a proof.
\end{Rmk}
The next result shows that stable energy levels of Tonelli Hamiltonians posses periodic orbits, proving statement (iv) of the theorem in the Introduction. In particular, the same is true for contact type energy levels.
\begin{Cor}
Assume that $\kappa$ is a regular value of the Tonelli Hamiltonian $H\in C^{\infty}(T^*M)$ and that the hypersurface
$\mathcal{S}igma=H^{-1}(\kappa)$ is stable. Then $\mathcal{S}igma$ carries a periodic orbit.
\end{Cor}
\begin{proof}
By stability, we can find a diffeomorphism
\[
(-\epsilon, \epsilon) \times \mathcal{S}igma \rightarrow T^*M, \qquad (r,x) \mapsto \partialsi_r(x),
\]
onto an open neighborhood of $\mathcal{S}igma$ such that $\partialsi_0$ is the identity on $\mathcal{S}igma$ and
\[
\partialsi_r : \mathcal{S}igma \rightarrow \mathcal{S}igma_r := \partialsi_r(\mathcal{S}igma)
\]
induces an isomorphism between the line bundles $\mathcal{L}_{\mathcal{S}igma}$ and $\mathcal{L}_{\mathcal{S}igma_r}$. Up to the choice of a smaller $\epsilon$, we may assume that all the hypersurfaces $\mathcal{S}igma_r$ are levels of a uniformly convex function. Therefore, they are the level sets of a Tonelli Hamiltonian $K\in C^{\infty}(T^* M )$ (see \cite{mp10} for a detailed construction of $K$). Since the Legendre transform of $K$ is a Tonelli Lagrangian, Theorems \ref{highthm} and \ref{ae} imply that $K^{-1}(\kappa)$ has periodic orbits for almost every $\kappa$. In particular, $\mathcal{S}igma_r$ has periodic orbits for almost every $r$, but since the dynamics on $\mathcal{S}igma_r$ and on $\mathcal{S}igma$ are conjugated, the same is true for $\mathcal{S}igma$.
\end{proof}
\begin{Rmk}
The above proof shows the usefulness of having a theory which works with Tonelli Lagrangians, rather than just electromagnetic ones. In fact even if the stable hypersurface $\mathcal{S}igma$ is the level set of an electromagnetic Hamiltonian (that is, it is fiberwise an ellipsoid), the hypersurfaces $\mathcal{S}igma_r$ given by the stability assumption may be more general fiberwise uniformly convex hypersurfaces.
\end{Rmk}
\begin{Rmk}
If $E^{-1}(\kappa)$ is of contact type and $\partiali_* : H_1(E^{-1}(\kappa),{\mathbb{R}}) \rightarrow H_1(M,{\mathbb{R}})$ is injective, then ${\mathbb{S}}_{\kappa}$ satisfies the Palais-Smale condition (with a suitable choice of the metric of $\mathcal{M}$). See \cite[Proposition F]{con06}. Therefore, in this case the existence of a periodic orbit can be obtained also without using stability.
\end{Rmk}
\begin{Rmk}
It can be proved that when $M$ is a closed surface and $L$ is of the form (\ref{elmag}) with $V=0$ (that is, in the case of exact magnetic flows), there are periodic orbits on {\em every} energy level below $c_0(L)$ (see \cite{tai92b}, \cite{tai92c}, \cite{tai92} and \cite{cmp04}). In fact, the advantage of dealing with a surface is that when $\kappa<c_0(L)$ one can minimize ${\mathbb{S}}_{\kappa}$ on a suitable space of {\em embedded} closed curves. In the same setting, one can prove that for almost every energy level below $c_u(L)$ there are infinitely many periodic orbits, at least if all periodic orbits are assumed to be non-degenerate (see \cite{amp13}).
\end{Rmk}
\partialaragraph{\textbf{The two Lyapunov functions argument}} We conclude these notes by discussing an alternative argument to deal with the lack of (PS) which is exhibited by ${\mathbb{S}}_{\kappa}$ when $\kappa<c_u(L)$. It allows to prove that the set of energy levels $\kappa$ such that the Euler-Lagrange flow has a periodic orbit of energy $\kappa$ is {\em dense} in $(\min E,c_u(L))$, a weaker statement than Theorem \ref{ae}. However, it has some advantages, which are discussed in Remark \ref{adv} below. This argument is used, in a different context, in \cite{ama08}. Here we shall use it in order to prove the following weaker version of Theorem \ref{ae}:
\begin{Thm}
\label{dense}
Let $\min E<\bar \kappa<c_u(L)$ and assume that there are no contractible periodic orbits of energy $\bar\kappa$ and non-negative action ${\mathbb{S}}_{\bar\kappa}$.
Then there exists a strictly decreasing sequence $(\kappa_h)$ which converges to $\bar\kappa$ and is such that the Euler-Lagrange flow has a contractible periodic orbit $\gamma_h$ with energy $\kappa_h$ and period $T_h$, which satisfies ${\mathbb{S}}_{\kappa_h}(\gamma_h)/T_h \downarrow 0$.
\end{Thm}
\begin{proof}
We argue by contradiction and assume that there exists $\tilde\kappa>\bar\kappa$ and $\delta>0$ such that for any $\kappa \in[\bar \kappa,\tilde \kappa]$ all the periodic orbits $\gamma$ of
energy $\kappa$ and period $T$ satisfy either ${\mathbb{S}}_{\kappa}(\gamma)/T\geq\delta$ or ${\mathbb{S}}_{\kappa} (\gamma)\leq 0$. Fix real numbers $a>c(\bar \kappa)$ and $\kappa^*\in(\bar \kappa,\tilde \kappa]$. Assume that we can find $\lambda\in[0,1]$ and $(x,T)\in\mathcal{M}M$ such that
\[
\lambda \, d{\mathbb{S}}_{\bar \kappa}(x,T)+(1-\lambda)\, d{\mathbb{S}}_{\kappa^*}(x,T)=0, \quad 0<{\mathbb{S}}_{\bar \kappa}(x,T)\leq a.
\]
Then $(x,T)$ is a critical point of ${\mathbb{S}}_{\lambda\bar \kappa+(1-\lambda)\kappa^*}$, hence it is a $T$-periodic orbit with energy $\lambda\bar \kappa+(1-\lambda)\kappa^*$. By what we have assumed at the beginning, we have
\[
\delta \leq \frac{1}{T}{\mathbb{S}}_{\lambda\bar \kappa+(1-\lambda)\kappa^*}(x,T)
= \frac{1}{T} {\mathbb{S}}_{\bar \kappa}(x,T)+(1-\lambda)(\kappa^*-\bar \kappa)
\leq \frac{a}{T} + \kappa^*-\bar\kappa.
\]
Up to the choice of a smaller $\kappa^*>\bar\kappa$, we may assume that $\kappa^*-\bar \kappa\leq \delta/2$. Then the above estimate implies that
\[
T\leq \frac{2a}{\delta} =: T^*.
\]
With these choices of $\kappa^*$ and $T^*$, we can restate what we have proved so far as:
\begin{Lemma}
\label{oppo}
If $T>T^*$ and $0<{\mathbb{S}}_{\bar \kappa}(x,T)\leq a$, then the segment
\[
\mathrm{conv} \bigl\{ d{\mathbb{S}}_{\bar \kappa}(x,T), d{\mathbb{S}}_{\kappa^*}(x,T) \bigr\} \subset T_{(x,T)}^* \mathcal{M}M
\]
does not contain $0$.
\end{Lemma}
The above lemma allows us to construct a negative pseudo-gradient vector field for ${\mathbb{S}}_{\bar\kappa}$ which has all the good properties of $-\nabla {\mathbb{S}}_{\bar\kappa}$ and moreover has ${\mathbb{S}}_{\kappa^*}$ as a Lyapunov function on the open set
\[
A:= \bigl\{T>T^*\} \cap \{0< {\mathbb{S}}_{\bar\kappa} < a \bigr\}.
\]
In fact, the only obstruction to finding a vector field $W$ whose flow make both ${\mathbb{S}}_{\bar\kappa}$ and ${\mathbb{S}}_{\kappa^*}$ decrease in $A$, is that the differentials of ${\mathbb{S}}_{\bar\kappa}$ and ${\mathbb{S}}_{\kappa^*}$ point in opposite directions in some point of $A$, and this is precisely what is excluded by Lemma \ref{oppo}.
More precisely, one can prove the following:
\begin{Lemma}
\label{pg}
There exists a locally Lipschitz vector field $W$ on $\mathcal{M}M$ such that:
\begin{enumerate}
\item $d{\mathbb{S}}_{\bar\kappa}[W]<0$ on $\{{\mathbb{S}}_{\bar\kappa}>0\}$;
\item $W$ is forward complete and vanishes on $\{{\mathbb{S}}_{\bar\kappa}\leq 0\}$;
\item let $z_h=(x_h,T_h)$ be a sequence in $\mathcal{M}M^{\mathrm{contr}}$ such that
\[
0< \inf {\mathbb{S}}_{\bar{\kappa}}(z_h) \leq \sup {\mathbb{S}}_{\bar{\kappa}}(z_h) < +\infty, \quad \lim_{h\rightarrow \infty} d{\mathbb{S}}_{\bar\kappa}(z_h)[W(z_h)] = 0,
\]
and $(T_h)$ is bounded from above; then $(z_h)$ has a subsequence which converges in $\mathcal{M}M^{\mathrm{contr}}$;
\item $d{\mathbb{S}}_{\kappa^*}[W]<0$ on $A$.
\end{enumerate}
\end{Lemma}
In fact, one can choose $W$ to be given by the vector field
\[
\nabla{\mathbb{S}}_{\bar \kappa}+\chi \frac{\|\nabla{\mathbb{S}}_{\bar \kappa}\|}{\|\nabla{\mathbb{S}}_{\kappa^*}\|}\nabla{\mathbb{S}}_{\kappa^*}
\]
multiplied by a suitable non-positive function. Here $\chi$ is a suitable cut-off function. See \cite[Lemmas 5.1 and 5.4]{ama08} for a similar construction.
We can now prove Theorem \ref{dense}.
By the definition of $c(\bar\kappa)$, there is a set $K$ in ${\bf G}amma$ such that
\[
\max_{K} {\mathbb{S}}_{\bar \kappa} <a.
\]
By Lemma \ref{pg} (i) and (ii), for every $\sigma_0>0$ we have
\[
\inf_{\sigma\in \sigma_0} \Bigl| d{\mathbb{S}}_{\bar\kappa}\bigl(\partialhi_{\sigma}(z)\bigr)\bigl[W(\partialhi_{\sigma}(z))\bigr] \Bigr| \leq \frac{1}{\sigma_0} \int_0^{\sigma_0} \Bigl| d{\mathbb{S}}_{\bar\kappa}\bigl(\partialhi_{\sigma}(z)\bigr)\bigl[W(\partialhi_{\sigma}(z))\bigr]\Bigr| \, d\sigma = \frac{
{\mathbb{S}}_{\bar\kappa}(z) - {\mathbb{S}}_{\bar\kappa}(\partialhi_{\sigma_0}(z))}{\sigma_0},
\]
and, by the definition of $c(\bar\kappa)$,
\[
\max_{z\in K} {\mathbb{S}}_{\bar\kappa} \bigl( \partialhi_{\sigma_0}(z) \bigr) \geq c(\bar\kappa).
\]
By taking a limit for $\sigma_0\rightarrow +\infty$, thanks to Lemma \ref{pg} (ii), the above facts imply that $\partialhi_{{\mathbb{R}}^+}(K)\cap \{{\mathbb{S}}_{\bar\kappa}>0\}$ contains a sequence $z_h=(x_h,T_h)$ such that
\[
0< c(\bar\kappa) \leq \inf {\mathbb{S}}_{\bar{\kappa}}(z_h) \leq \sup {\mathbb{S}}_{\bar{\kappa}}(z_h) < a \quad \mbox{and} \quad \lim_{h\rightarrow \infty} d{\mathbb{S}}_{\bar\kappa}(z_h)[W(z_h)] = 0.
\]
It is enough to show that $(T_h)$ is bounded from above: Indeed, in this case Lemma \ref{pg} (iii) implies that $(z_h)$ has a limiting point, which is a critical point of ${\mathbb{S}}_{\bar\kappa}$ with positive action, contradicting the hypothesis of Theorem \ref{dense}.
The upper bound on $(T_h)$ is a consequence of the following claim: the period $T$ is bounded on the set $\partialhi_{{\mathbb{R}}^+}(K)\cap \{{\mathbb{S}}_{\bar\kappa}>0\}$.
In order to prove this claim, we first notice that
\begin{equation}
\label{hhh}
{\mathbb{S}}_{\bar \kappa}(x,T)\leq a, \;\; T\leq T^* \quad {\mathbb{R}}ightarrow \quad
{\mathbb{S}}_{\kappa^*}(x,T)\leq a+(\kappa^*-\bar \kappa)T^*=:b.
\end{equation}
Since $K$ is compact, we can find $d>b$ such that $K\subset \{{\mathbb{S}}_{\kappa^*} < d\}$. Let $\partialhi$ be the flow of the vector field $W$. We claim that
\begin{equation}
\label{cc}
\partialhi_{{\mathbb{R}}^+}(K) \cap \bigl\{{\mathbb{S}}_{\bar\kappa} > 0\bigr\} \subset \bigl\{{\mathbb{S}}_{\kappa^*} < d\bigr\}.
\end{equation}
In fact, let $z\in K$ and let $\sigma_0>0$ be the first instant such that ${\mathbb{S}}_{\kappa^*}(\partialhi_{\sigma_0}(z))=d$, while ${\mathbb{S}}_{\bar{\kappa}}(\partialhi_{\sigma_0}(z))> 0$. By Lemma \ref{pg} (i), ${\mathbb{S}}_{\bar\kappa}(\partialhi_{\sigma_0}(z)) \leq {\mathbb{S}}_{\bar\kappa}(z) < a$. By Lemma \ref{pg} (iv), the point $\partialhi_{\sigma_0}(z)$ cannot belong to $A$, so $\partialhi_{\sigma_0}(z)=(x,T)$ with $T\leq T^*$ and (\ref{hhh}) implies that ${\mathbb{S}}_{\kappa^*}(\partialhi_{\sigma_0}(z)) \leq b < d$. This contradiction proves (\ref{cc}).
If ${\mathbb{S}}_{\bar\kappa}(x,T)>0$ and ${\mathbb{S}}_{\kappa^*}(x,T)<d$, then
\[
d > {\mathbb{S}}_{\kappa^*}(x,T) = {\mathbb{S}}_{\bar\kappa}(x,T) + (\kappa^*-\bar\kappa) T > (\kappa^*-\bar\kappa) T.
\]
This shows that the period $T$ is bounded on the set
\[
\bigl\{{\mathbb{S}}_{\bar\kappa} > 0\bigr\} \cap \bigl\{{\mathbb{S}}_{\kappa^*} < d\bigr\},
\]
and by (\ref{cc}) it is bounded also on
\[
\partialhi_{{\mathbb{R}}^+}(K) \cap \bigl\{{\mathbb{S}}_{\bar\kappa}>0\},
\]
as claimed.
\end{proof}
\begin{Rmk}
\label{adv}
In the Struwe monotonicity argument, one gets the existence of bounded (PS) sequences at level $c(\bar{\kappa})$, but has no control on the (PS) sequences at other levels. Therefore, it is not clear whether the space of negative gradient flow lines for ${\mathbb{S}}_{\bar\kappa}$ which connect two given critical points - say with positive action - is bounded.
An advantage of the two Lyapunov functions argument, is that the latter fact is true for the flow lines of the vector field $W$ constructed in Lemma \ref{pg}: the second Lyapunov function ${\mathbb{S}}_{\kappa^*}$ allows to exclude the existence of flow lines which go arbitrarily far and come back. This fact would allow to develop some global critical point theory for ${\mathbb{S}}_{\bar\kappa}$, such as Morse theory or Lusternik-Schnirelmann theory. This is not useful here, because the a priori estimates which lead to the existence of the pseudo-gradient vector field $W$ come from a contradiction argument. However, it might be useful in situations where these a priori bounds have a different origin, such as for example in the case of tame energy levels (see \cite{cfp10} for the definition of tameness and for motivating examples).
\end{Rmk}
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\begin{document}
\begin{abstract}
We construct several new families of Fano varieties of K3 type. We give a geometrical explanation of the K3 structure and we link some of them to projective families of irreducible holomorphic symplectic manifolds.
\end{abstract}
\maketitle
\tableofcontents
\mathfrak{sl}ection{Introduction}
Fano varieties and Irreducible Holomorphic Symplectic manifolds (for short, IHS) are two of the most studied classes of varieties in algebraic geometry. They are very different in nature (for example, they have different Kodaira dimensions) and they are often studied using different tools. Indeed, Fano varieties are at the core of birational geometry, while IHS manifolds (sometimes called hyperk\"ahler when the context is more differential-geometric) can be considered as a higher dimensional analogue of K3 surfaces, with lattice theory as one of the most relevant operative tools.
One of the most important properties of Fano varieties is their \emph{boundedness}: indeed it is well known that in every dimension there exists a finite number of families of Fano varieties up to deformations. This still holds if we allow some mild singularities, see \cite{birkar} for an up--to--date survey. It is therefore natural to aim for a classification, but such a problem is currently out of reach. A complete answer is known when the dimension is up to three, see for example \cite{ip99} for the smooth case. In the singular case the classification is still an open problem, even in low dimension and with mild singularities (for example terminal).
From dimension four onwards, only partial results are known. In particular a known explicit bound in terms on the canonical volume is assumed to be hugely non-sharp (being a large number as $(n+2)^{(n+2)^{n2^{3n}}}$), already for $n=2$. The strategy for a partial classification usually is to consider special subclasses of Fano varieties, or fixing some other invariant, such as the \emph{index}. Recall that this the integer $\iota_X$ which is the maximal number for which the anticanonical class is divisible in the Picard group.
It is a classical result that whenever a Fano $X$ is smooth, the index satisfies $\iota_X \leq \ddim X+1$, with the equality attained only in the case of projective space. Prime Fano varieties, that is Fano varieties with Picard rank $\rho=1$, of index $\ddim X -2 \leq \iota_X \leq \ddim X+1$ are completely classified, as they are when $\iota_X \geq \frac{\ddim X+1}{2}$ and $\rho_X >1$. Again see \cite{ip99} for a complete list of results. Mukai's conjecture further bounds the Picard rank in terms of the index: namely the conjecture states that for a smooth Fano $\rho_X(\iota_X-1) \leq \ddim X.$ The general philosophy is therefore that high index Fano varieties are somewhat easier to classify than low index.\\
On the contrary the main problem in the study of IHS is the lack of examples.
Similarly to the case of Calabi-Yau manifolds, no result of boundedness is known in general for IHS manifolds (although there are some partial results if one fixes for example the Beauville-Bogomolov-Fujiki form). However, finding examples is definitely harder than in the Calabi-Yau case. The known deformation types include two series of examples for every even dimension found by Beauville for every even dimension (Hilbert scheme of points on a K3 surface and a similar construction, called \emph{generalised Kummer variety} on an abelian surface), and two sporadic examples in dimension 6 and 10, found by O'Grady. Even if we fix the deformation type and we look for \emph{polarised} families (in analogy with the K3 case) the situation does not improve much: very few examples of projective families are known. A survey of this story can be found for example in \cite{beauville}.\\
The interplay between special classes of Fano varieties and IHS manifold is not a new story: a main example is the one of a maximal family of IHS fourfolds (deformation equivalent to the Hilbert Scheme of tow points on a K3 surface) as the Fano variety of lines of a smooth cubic fourfold, due to Beauville and Donagi. We remark that this is not the unique IHS that can be linked to a cubic fourfold, as the recent constructions of Lehn-Lehn-Sorger-van Straten, \cite{llsvs} (an 8-fold of K3$^{[4]}$-type) and Laza-Sacc\`a-Voisin, \cite{lsv} (example of OG10 manifold) highlight. The cubic fourfold is not the only Fano to which we can associate polarised families of IHS: this is indeed a common feature of a special subclass of Fano varieties, called \emph{Fano varieties of K3 type} (FK3 for short) whose study is the central topic of this paper. We give here the key definitions.
\begin{definition} Let $X$ be a smooth, projective $n$-dimensional Fano variety and $j$ be a non-negative integer. The cohomology group $H^j(X, \mathbb{C}) \cong \bigoplus_{p+q=j} H^{p,q}(X)$ (with $j \geq k$) is said to be of $k$ Calabi-Yau type if \begin{itemize} \item $h^{\frac{k+j}{2},\frac{j-k}{2}}=1$; \item $h^{p,q}=0$, for all $p+q=j, \ p <\frac{k+j}{2}$.
\end{itemize}
$X$ is said to be of $k$ (pure) Calabi-Yau type (k--FCY or Fano of k-CY type for short) if there exists at least a positive $j$ such that $H^j(X, \mathbb{C})$ is of $k$ Calabi-Yau type. Similarly, $X$ is said to be of mixed $(k_1, \ldots, k_s)$ Calabi-Yau type if the cohomology of $X$ has different level CY structures in different weights.
\end{definition}
In the above definition, we consider all sub-Hodge structures, even those naturally arising using Lefschetz's Theorems (for us, a sixfold with a 2 Calabi Yau structure in $H^4$ and $H^4\cong H^6\cong H^8$ is a $(2,2,2)$ Calabi-Yau type). In the paper, we will say that a Fano variety is \emph{central} if all its cohomology groups have level 0.\\
A Fano variety of K3 type (FK3) is nothing but a 2-FCY. Fano varieties of CY type were first introduced and studied by Iliev and Manivel in \cite{ilievmanivel}. The authors focus on the case $k=3$, adding moreover an extra condition on the $H^1(T_X)$ (which we do not ask, since it would rule out already the cubic fourfold and many other interesting examples). They classify 3-FCY that can be obtained by slicing homogeneous spaces with linear and quadratic equations. We remark that our definition is purely Hodge-theoretical, but there are deep links with the concept of CY subcategories, see for example \cite{kuzicy}. In particular, constructing examples of Fano varieties of K3 and CY type might help in finding new playground for testing Kuznetsov's conjecture on rationality.\\
We are especially interested in the FK3 case, due to its deep relation with IHS manifolds. Indeed, a result of Kuznetsov and Markushevich in \cite{kuzm} shows that if $\mathfrak{M}$ is a moduli space of stable or simple sheaves on $X$, then any form in $H^{n-q-2}(X, \Omega^{n-q})$ defines a closed 2-form in $H^0(\mathfrak{M}^{\textrm{smooth}}, \Omega^2)$. This is indeed a good starting point in the hunt for examples of IHS. In particular, let us mention the IHS linked to the Debarre-Voisin twentyfold hypersurface, or to a Gushel--Mukai fourfold, or to a section of a product of $\mathbb{P}^3$, all examples of FK3 varieties, see \cite{debarrevoisin}, \cite{dk16}. \cite{ilievmanivel2}. \\
Although FK3 are definitely easier to hunt than IHS, there are not many known examples in the literature. For example, as complete intersections in (weighted) projective spaces one finds only the cubic fourfold, see \cite{ps18}. More examples are found if one allows terminal and $\mathbb{Q}$-factorial singularities, see \cite{frz19} but no new examples of IHS are produced anyway. In \cite{eg1} we conjectured that even taking complete intersection in Grassmannian one does not get any new example other than a complete intersection with four linear hypersurfaces in the Grassmannian $\mathbb{G}r(2,8)$ and the above mentioned examples. This paper deals with the construction of examples of FK3 as zero locus of general global section of homogeneous vector bundles in Grassmannians or products of such. This is motivated by the list of K\"uchle, see \cite{kuchle}, of index 1 Fano fourfolds obtained in such a way, where few more interesting FK3 are found. Therefore the aim of this paper is twofold:
\begin{aim*} \begin{enumerate}\item Construct new examples of Fano varieties of K3 type;
\item Construct examples of polarised families of IHS from our FK3.
\end{enumerate}
\end{aim*}
This paper is a first step of this project. One of its main aim is to show that there might be a lot of examples \emph{out there}. Even if obtaining a complete classification of all Fano of K3 type might be out of reach, a classification might be attainable if we restrict ourselves to some special subcases, for example Fano obtained as zero locus on Grassmannians and homogeneous varieties. The main problem here is that in general translating the (Hodge-theoretical) requirement of being of K3 type into algebraic conditions is not easy. Using some tools that we developed in \cite{eg1} we were anyway able to find some numerological condition useful to produce examples of FK3, see Numerology \ref{num}. Unfortunately the condition in \ref{num} are still too general for replicating a classification-type argument as the original one from K\"uchle. However, \ref{num} has the advantage of highlighting the connection between FK3 and \emph{central} Fano varieties, that is Fano such that $h^{p,q}\neq 0$ if and only if $p=q$. When this happens, we say that all the cohomology groups of $X$ has \emph{level} (lv) 0, see \ref{num}. Indeed a future problem we are interested in is the following, possibily up to restriction to some special subcases, with zero locus of sections of homogeneous vector bundles as a first step.
\begin{problem} Classify Fano varieties such that lv $(H^j(X,\mathbb{C}))=0$ for all $j$.
\end{problem}
\mathfrak{sl}ubsection{How we subdivide the examples}
We first write down the list of examples that we have found. Later on in the paper we will explain the numerology behind our list, and give a detailed geometrical description of our examples. Our purpose its twofold. Indeed to a Fano of K3-type we want to associate (whenever possible) both a K3 category and an IHS manifold. For the definition of K3 or CY (sub) category we follow \cite{kuzicy}. Before doing this, we need to prove first that the families of Fano that we consider are of K3 type. This is done usually with either Riemann-Roch type computations as for example in \ref{m7} or using our Griffiths ring-type construction as in Proposition \ref{s1}, or via a Borel-Bott-Weil computation, as in Proposition \ref{t129}.
In particular we divide our list into three distinct blocks. We say that a FK3 $X$ is of \emph{blow-up} type (\textbf{B}) if there exists a pair $(Y,S)$, with $S \mathfrak{sl}ubset Y$, $Y$ Fano, $S$ K3 surface such that $X \cong Bl_S Y$. Examples of this type are already included in K\"uchle list, \cite{kuchle}, called \emph{c7} and \emph{d3}. We say that a FK3 $X$ is of \emph{Mukai type} (\textbf{M}) if we can reduce systematically the study of its derived category to Mukai's classification of Fano threefolds. We say that a FK3 $X$ is \emph{sporadic} (\textbf{S}) if it does not fall in one of the two previous categories. We collect all our list of examples of FK3 in Table \ref{table}. \\
For FK3 of blow-up and Mukai type the question on the existence of a K3-subcategory admits always a positive answer. This is the content of Propositions \ref{lem:blowup}, \ref{rennemo} and Theorem \ref{cayley}. However the question of existence of an IHS linked to any FK3 is far from being answered. We give an example in Proposition \ref{gp2} . For the FK3 of sporadic type, we do not have any information a priori. For all of them the question on the existence of a K3-subcategory is open, and even we have to cook up ad-hoc methods even to show that they are of K3 type (in the Hodge theoretical sense). Here as well there is no easy answer from the IHS viewpoint. A new construction is given for example in Proposition \ref{o2}. Special attention must be placed upon example (\textbf{S6}) and (\textbf{S7}). Indeed they are cut by irreducible vector bundles which are not linear. We observe as well the appereance of mixed structures of $(2,3)$-CY type. The last part of the paper is devoted to the study of these varieties. The results about IHS are collected in Table \ref{table2}. We point out that we believe that to any of the example in Table \ref{table} we will eventually be able to construct an example of polarised IHS. We added in both our tables two examples found independently by Iliev and Manivel in \cite {ilievmanivel2}, while our work was still in the very early stage. These are the families \textbf{B1} and \textbf{S3}. Although they were already known we decided to include them anyway in our list, since they fit perfectly in our pattern.
We highlight now the main results and the structure of this paper.
\mathfrak{sl}ubsection{Results and Structure}
This paper is devoted to the construction of a meaningful bunch of examples of Fano varieties of K3 type. We mainly exploit our numerological condition in \ref{num}, coming from a similar analysis to the one we carried out in \cite{eg1}. Our main result can be summarised in
\begin{thm}There exists 23 examples of families of Fano varieties of K3 type obtained as zero locus of general global section of homogeneous vector bundles over Grassmannians or products of such. These Fano varieties have dimension $4 \leq n \leq 20$, Picard rank $1 \leq \rho_X \leq 3$ and index $\frac{n-1}{2}\leq \iota_X \leq \frac{n}{2}$.
\end{thm}
See Table \ref{table} for the list of this Fano varieties. We point out that since they have an index which is comparatively high with respect to the dimension (close to Wisniewski's bound), there could be hope for a classification.
For each of this Fano we first needed to prove that they are of K3 type. We either explain geometrically in a systematic way (whenever possible) the presence of K3 structure (both from a Hodge-theoretical and derived category viewpoint) or we give an ad-hoc description for the sporadic cases. We point out that new examples may and will be discovered and analysed in a series of future works.\\
Some of the Fano we analyse have new and interesting behaviours. We collect some of the results here.
\begin{thm} There exists prime Fano varieties with multiple CY structures (see Proposition \ref{3k3}) and with mixed Calabi-Yau (2,3) structure, (see Proposition \ref{23cy}).
\end{thm}
To the best of our knowledge, these are the first examples of known prime Fano varieties with this property. The prime hypothesis eliminates the possibility for these CY structure to come from a blow-up, a projective bundle or other related constructions. We link some of these Fano varieties to projective families of IHS manifolds. Unfortunately, up to now we have only found new ways of describing old examples, but we believe that a further extensive examination of our list could lead to new constructions. We collect our results here.
\begin{thm}We show that the Hilbert square on a K3 of genus 8 is isomorphic to the zero locus of a certain bundle on $\mathbb{G}r(4,6) \times \mathbb{G}r(2,6)$, see Proposition \ref{gp2}. We show that the Debarre-Voisin IHS 4-folds are isomorphic to the space of special rational fourfolds on varieties of type $\ThetaT(2,10)$, see Proposition \ref{t2} and to the compactification of the space of $(\mathbb{P}^1)^3$ on a linear section of $\Ml(3,8)$, see Theorem \ref{hk}.
\end{thm}
These results are collected in Table \ref{table2}. We spend a few words on the structure of this paper. In \textbf{Section 2} we explain how our numerological condition creates the list and we
explain some straightforward geometric tricks and a general strategy to attack these Fano varieties. In \textbf{Section 3} we perform a case--by--case analysis of the most interesting examples and we prove our main results. We finish with a bunch of \textbf{Appendices}, where we describe three related cases we encountered: some extra Fano varieties of 3CY type, a trio of infinite series of Calabi--Yau varieties and a Fano variety with a fake K3 structure.
\mathfrak{sl}ubsubsection*{Acknowledgements}
This paper was completed throughout the course of the past year and a half. The work was carried out mainly at Roma Tre and Bologna University, in several of its campus sites (although some of the latter were not officially recognised by its own administration). Many people gave useful comments and suggestions throughout the whole process. We mention in particular Atanas Iliev, for sharing with us some of the ideas that led to Theorem \ref{hk}, and Alexander Kuznetsov, for many suggestions, ideas shared and comments on an early draft of this manuscript. Many of the computations were carried out using a Macaulay2 code written by the first author together with Fabio Tanturri. We thank as well for various ideas, conversations and support Hamid Ahmadinezhad, Vladimiro Benedetti, Marcello Bernardara, Daniele Faenzi, Lorenzo Federico, Camilla Felisetti, Michal and Grzegorz Kapustka, Laurent Manivel, Luca Migliorini, Claudio Onorati, Miles Reid and J\o rgen Rennemo. EF was supported by MIUR-project FIRB 2012 "Moduli spaces and their applications" and by an EPSRC Doctoral Prize. GM was supported by ``Progetto di ricerca INdAM per giovani ricercatori: Pursuit of IHS''. Both authors are member of the INDAM-GNSAGA and received support from it.
\mathfrak{sl}ection{The quest for examples}
\textbf{Notation for the paper and for the tables}
With $\mathcal{R}$ and $\mathcal{Q}$ we denote (respectively) the rank $k$ tautological and the rank $n-k$ quotient bundle on the Grassmanian $\mathbb{G}r(k,n)$. We fix the convention that $\mathcal{O}_G(1) =\mathrm{det}(\mathcal{Q})=\mathrm{det}(\mathcal{R}^{\vee})$.
$\mathrm{S}_i\mathbb{G}r(k,n)$ denotes the $i$-th symplectic Grassmannian. The most relevant cases for us are for $k=1$ and $k=2$. For $i=1$ this variety is nothing but the usual symplectic Grassmannian (usually called Lagrangian when $2k=n$), for $i=2$ it is the \emph{bisymplectic Grassmannian}, which will be better defined and characterised later in the paper. If $k=1$ we will simply write $\SGr(k,n)$. $\overline{\SGr(3,6)}$ in the table will denote a linear section of $\SGr(3,6)$. $\OGr(k,n)$ denotes the orthogonal Grassmannian and $\mathbb{S}_{n} $ we denote one of the two connected components of $\OGr(n,2n)$ in its spinor embedding.
$\ThetaT(k,n)$ denotes the subvariety of $\mathbb{G}r(k,n)$ cut by the zero locus of a general three-form $\mathfrak{sl}igma \in \bigwedge^3 V_n^\vee$. According to $k$, $\ThetaT(k,n)$ can be represented as the zero locus of a general global section of a different vector bundle. As an example, if $k=3$, $\ThetaT(3,n)$ is nothing but a linear section of the Grassmannian $\mathbb{G}r(3,n)$, if $k=2$, $\ThetaT(2,n)$ is the congruence of lines given by the bundle $\mathcal{Q}^*(1)$ and if $k=4$ the bundle is of course $\bigwedge^3 \mathcal{R}^{\vee}$.\\
We use $X_1 \mathfrak{sl}ubset G$ to denote a linear section of the variety $G$ (and similarly for higher degree or multidegree). Whenever there might be ambiguity or we want to emphasize the choice of the linear subspace we might write $X_H$. Similarly, sometimes we will use the shorthand $X_{\mathcal{F}} \mathfrak{sl}ubset G$ to denote the zero locus of a general global section of the vector bundle $\mathcal{F}$ over $G$.\\
The notation $H^n_{\van}(X)$ (and similarly for the $(p,q)$ part) will denote the vanishing subspace of the cohomology group, see \cite[2.27]{voisin2} for a definition. \\
If $X$ and $Y$ are smooth projective variety we will use the shortand $D^b(X) \mathcal{H}ookrightarrow D^b(Y)$ to mean that one can construct a semiorthogonal decomposition for $D^b(Y)$ where $D^b(X)$ appears as one of the factors, up to a fully faithful functor.\\
The notation $S_g$ means a K3 surface of genus g. With $Q_k$ we indicate the $k$-dimensional quadric hypersurface.
\begin{table}[ht]
\centering
\begin{tabular}{@{} *9l @{}l @{}l @{}l @{}l @{}l @{}} \toprule
no. & \emph{$X \mathfrak{sl}ubset Y$}& $\ddim X$ & $\iota_X$ & $\rho_X$ & Comments \\ \midrule
B1 & $X_{(2,1,1)} \mathfrak{sl}ubset \mathbb{P}^3 \times \mathbb{P}^1 \times \mathbb{P}^1$ &4& 1 & 3& $X \cong Bl_{S_7} (\mathbb{P}^3 \times \mathbb{P}^1)$\\
B2 & $X_{(2,1)} \mathfrak{sl}ubset \mathbb{G}r(2,4) \times \mathbb{P}^1$& 4&1&2& $X \cong Bl_{S_5} \mathbb{G}r(2,4)$\\
M1 & $X_{(1,1,1)} \mathfrak{sl}ubset \mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3$& 8& 3&$ 3$ &$D^b(S_{3} )\mathcal{H}ookrightarrow D^b(X)$ \cite[Section 4]{ilievmanivel2} \\
M2 & $X_{(1,1,1)} \mathfrak{sl}ubset Q_3 \times \mathbb{P}^2 \times \mathbb{P}^2$ & 6 & 2& 3& $D^b(S_{4} )\mathcal{H}ookrightarrow D^b(X)$\\
M3 & $X_{(1,1)} \mathfrak{sl}ubset \mathbb{G}r(2,5)\times Q_5$& 10 &4& 2& $D^b(S_6) \mathcal{H}ookrightarrow D^b(X)$ \\
M4 & $X_{(1,1)} \mathfrak{sl}ubset \SGr(2,5) \times Q_4$&8&3& 2&$D^b(S_6) \mathcal{H}ookrightarrow D^b(X)$ \\
M5& $X_{(1,1)} \mathfrak{sl}ubset \Ml(2,5) \times Q_3$& 6&2&2& $D^b(S_6) \mathcal{H}ookrightarrow D^b(X)$\\
M6& $X_{(1,1)} \mathfrak{sl}ubset \mathbb{S}_{5} \times \mathbb{P}^7$& 16& 7& 2&$D^b(S_7) \mathcal{H}ookrightarrow D^b(X)$\\
M7 & $X_{(1,1)} \mathfrak{sl}ubset \mathbb{G}r(2,6) \times \mathbb{P}^5$& 12 & 5& 2 & $D^b(S_{8}) \mathcal{H}ookrightarrow D^b(X)$\\
M8& $X_{(1,1)} \mathfrak{sl}ubset \SGr(2,6) \times \mathbb{P}^4$& 10& 4& 2&$D^b(S_8) \mathcal{H}ookrightarrow D^b(X)$\\
M9& $X_{(1,1)} \mathfrak{sl}ubset \Ml(2,6) \times \mathbb{P}^3$& 8 &3& 2 & $D^b(S_8) \mathcal{H}ookrightarrow D^b(X)$\\
M10 & $X_{(1,1)} \mathfrak{sl}ubset \SGr(3,6) \times \mathbb{P}^3$&8&3& 2&$D^b(S_9) \mathcal{H}ookrightarrow D^b(X)$ \\
M11 & $X_{(1,1)} \mathfrak{sl}ubset \overline{\SGr(3,6)} \times \mathbb{P}^2$&6&2& 2&$D^b(S_9) \mathcal{H}ookrightarrow D^b(X)$ \\
M12& $X_{(1,1)} \mathfrak{sl}ubset \mathrm{G}_2 \times \mathbb{P}^2$& 6& 2& 2& $D^b(S_{10}) \mathcal{H}ookrightarrow D^b(X)$\\
M13& $X_{(1,1)}\mathfrak{sl}ubset \mathbb{G}r(2,8)\times \mathbb{P}^3$ & 14 & 1 & 2 & $D^b(S_3) \mathcal{H}ookrightarrow D^b(X)$\\
S1 & $X_{1^4} \mathfrak{sl}ubset \mathbb{G}r(2,8)$& 8 & 4& 1 & $D^b(S_3) \mathcal{H}ookrightarrow D^b(X)$ \\
S2& $X_1 \mathfrak{sl}ubset \OGr(3,8)$ &8&3& 2& $D^b(S_7) \mathcal{H}ookrightarrow D^b(X)$\\
S3& $X_1 \mathfrak{sl}ubset \SGr(3,9)$ &14&6& 1 & \cite[Section 5]{ilievmanivel2}\\
S4& $X_1 \mathfrak{sl}ubset \Ml (3,8)$ &8&3& 1 & \\
S5& $X_1 \mathfrak{sl}ubset \ThetaT(2,9)$ &6&2& 1 &\\
S6& $\ThetaT(2,10)$ &8&3& 1 &$ 3 \times $ K3 structure\\
S7& $X_1 \mathfrak{sl}ubset \ThetaT(2,10)$ &7&2& 1 & $ 2 \times $ K3 structure, $ 1 \times $ 3CY\\
S8& $X_{L} \mathfrak{sl}ubset \ThetaT(k,10)$ &&& 1 & invariants depending by $k$ and $L$\\
\bottomrule
\mathcal{H}line
\end{tabular}
\captionof{table}{Fano of K3 type with invariants} \label{table}
\end{table}
\begin{table}[ht]
\centering
\begin{tabular}{@{} *9l @{}l @{}l @{}l @{}l @{}l @{}} \toprule
no. & \emph{$X \mathfrak{sl}ubset Y$} & IHS $Z$& Comments\\ \midrule
M1 & $X_{(1,1,1)} \mathfrak{sl}ubset \mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3$ & \cite[Section 4]{ilievmanivel2} & $Z \cong Hilb^2 S_3$ \\
M7 & $X_{(1,1)} \mathfrak{sl}ubset \mathbb{G}r(2,6) \times \mathbb{P}^5$& Prop.\ref{gp2}& $Z \cong Hilb^2 S_8$\\
S2& $X_1 \mathfrak{sl}ubset \OGr(3,8)$ & Prop.\ref{o2}&$Z \cong S_7$\\
S3& $X_1 \mathfrak{sl}ubset \SGr(3,9)$ & \cite[Section 5]{ilievmanivel2}& $Z \cong Z_{DV}$\\
S4& $X_1 \mathfrak{sl}ubset \Ml (3,8)$ & Thm. \ref{hk} &$Z \cong Z_{DV}$\\
S6&$\ThetaT(2,10)$ & Prop. \ref{t2}&$Z \cong Z_{DV}$\\
\bottomrule
\mathcal{H}line
\end{tabular}
\captionof{table}{Projective families of IHS linked to FK3} \label{table2}
\end{table}
\mathfrak{sl}ubsection{What are we looking for?}
Many of the examples in the above table are obtained by chasing up the same numerology. Indeed from arguments similar to the one used in \cite{eg1} one can come up with a numerical criterion (cf. \cite{ilievmanivel} and \cite{kuzicy} for comparison and similar criteria). For a smooth projective variety we define the \emph{level} of $H^j(X, \mathbb{C})$ as the largest difference $|p-q|$ for which $H^{p,q}(X) \neq 0$, with $p+q=j$.
It is obvious that lv$(H^j(X, \mathbb{C})) \leq $ wt $(H^j(X, \mathbb{C})) \leq \ddim X$. For a Fano variety by Kodaira vanishing the first inequality is always strict. For example, if $X$ is a Fano of dimension $n$, then $\mathrm{lv} (H^n(X,\mathbb{C})) \leq \ddim X-2$. Moreover we say that a variety $X$ is \emph{central} if all of its $H^j$ have level zero, or equivalently if $h^{p,q}(X)=0$ for $p \neq q$.
\begin{criterion}\label{num} Let $Y$ be a smooth projective Fano variety of dimension $2t+1$ and index $\iota_Y$.
Assume that $t$ divides $\iota_Y $ and that lv$(H^{2t+1}(Y))\leq 1$. Then a generic $ X \in | -\frac{1}{t} K_Y |$ is a Fano variety of K3 type, with the K3-type structure located in degree $2t$.
\end{criterion}
The above criterion is not necessary. Notable exceptions are \textbf{(S1)} (where the divisibility relation does not hold) and \textbf{(S6)}, where there the decomposition in irreducibles of the bundle that cuts the variety has no linear factor (albeit the variety has the correct ratio between dimension and index), and moreover two K3 sub-Hodge structures are present, in degree 6 and 8.\\
To the best of our knowledge the above numerology admits no counterexample. However the cohomological vanishing required to potentially prove the statement are ad-hoc, and there seems to us no easy way to transform the above statement in a proper theorem. However it is a cheap and easy way to produce several candidates, which turn out to be all of the desired type. We do not feel confident enough to state it as conjecture, as it stands. There could be ways of turning it into a statement or a conjecture. For example we could ask for $Y$ to have a rectangular Lefschetz decomposition in the categorical sense. Or, whenever $Y$ itself is cut by a section of an homogeneous vector bundle $\mathcal{F}= \bigoplus \mathcal{F}_i$ on $\mathbb{G}r(k,n)$, we might ask that the slope $\mu (-\frac{1}{t} K_Y) > \mu(\mathcal{F}_i)$ for all $i$. However, for the purpose of the current paper, we prefer to leave it as is is, and we plan to formalise this statement in a future work.
\mathfrak{sl}ubsubsection{Some numerology (and how the list is created)}
The list of FK3 in the tables has no presumption of being complete. The main problem is the condition on the level of Hodge theory of the ambient variety, which is quite hard to control. The first case to investigate is the one of complete intersections in homogeneous varieties. We conjectured in \cite{eg1} that there are no more FK3 as complete intersection in $\mathbb{G}r(k,n)$ other than the well-known cubic fourfold, the Gushel-Mukai fourfold, the Debarre-Voisin twentyfold hypersurface and a codimension four linear section of Grassmannian $\mathbb{G}r(2,8)$. We have not been able to prove this conjecture yet, but no counterexample has been found either.\\
We tried as well hypersurfaces in other homogeneous varieties other than $\mathbb{G}r(k,n)$, for example using the list of Konno in \cite{konno2}, but none of them satisfied the above condition. For the complete intersections in homogeneous space, we do not have any reasonable conjecture. Atanas Iliev informed us that a FK3 variety can be obtained by taking a 6-codimensional linear section of the $E_6$ variety $\mathbb{O}\mathbb{P}^2$, but we have not pursued this direction yet.\\
Already in this paper we analyse some extra case that do not fit in our numerological pattern. This is for example the case of $\ThetaT(k,10)$ (and its linear section). However, since this the only reasonable systematic way to produce examples, we decided to write few lines to explain how the list was found and why it stops. To do this, we decided to use as key varieties $Y$ examples that automatically satisfied the Hodge theoretical condition in \ref{num}.
Let $G$ be one of the varieties below. Consider the positive integer $m$ such that $\omega_G\cong \mathcal{O}_G(-m)$ and $D=\ddim G$. The equations in \ref{num} become
\begin{equation}\label{condition}2t+1=D \textrm{ and } at=m.
\end{equation}
\mathfrak{sl}ubsubsection*{$\mathbb{G}r(k,k+l)$}
For the Grassmannian $\mathbb{G}r(k,k+l)$ the dimension is $D=lk$ and the index equals $k+l$. First notice that $D$ must be odd.
The equations are $2t+1=kl$ and $at=k+l$, some $a$. Substituting we get $\frac{a(kl-1)}{2}=k+l$ and thus $akl=a+2k+2l$. Since $a\geq 1$ we have $kl \leq a+2k+2l$. It is easy to see that there are no solutions if $k\geq 5$, and for obvious reasons the case $k=2,4$ are excluded. In the case case $k=3$ substituting we get $l=\frac{a+6}{3a-2}$. This implies $a< 3$ for the previous number to be an integer. The case $a=2$ gives an even dimensional Grassmannian, so we discard it. The case $a=1$ corresponds to $G=\mathbb{G}r(3,10)$.
The associated FK3 is the Debarre-Voisin variety.
\mathfrak{sl}ubsubsection*{$\SGr(k,k+l)$}
The symplectic Grassmannian $\SGr(k,k+l)$ has dimension $kl- {k \choose 2}$ and index equal to $l+1$. If we substitute this in the equation above and look for solutions we find as triple $(k,l,a)=(2,3,2),(3,6,1), (5,3,2), (10,6,1)$. However, if $\omega$ is a non-degenerate skew symmetric $(k+l) \times (k+l)$ matrix, there are no $k$-dimensional isotropic subspaces if $k>l$ and $k+l$ even. We can therefore discard the last two triples and we are left with $X_2 \mathfrak{sl}ubset \SGr(2,5)$ (Gushel-Mukai fourfold) and $X_1 \mathfrak{sl}ubset \SGr(3,9)$, already considered in
\cite{ilievmanivel2}.
\mathfrak{sl}ubsubsection*{$\Ml(k,k+l)$}
The bisiymplectic Grassmannian $\Ml(k,k+l)$ has dimension $kl-k(k+1)$ and index equal to $l-k+2$. If we substitute in the equation above and look for solutions we find as triple $(k,l,a)=(3,5,1), (5,5,1)$. The second one can be identified with a (multi)-linear section of $(\mathbb{P}^1)^5$, see \cite{kuznetsovpicard}, the first one, an 8-fold linear section of $\Ml(3,8)$ is new.\\
A similar computation can be done for the tri-symplectic Grassmannian $\mathrm{S}_3\mathbb{G}r(k,k+l)$. This is relevant since two K3 by Mukai (genus 6 and genus 12) can be considered as (respectively) quadratic and linear section of it. However, no more examples have been found.
\mathfrak{sl}ubsubsection*{$\OGr(k,k+l)$}
The orthogonal Grassmannian $\OGr(k,k+l)$ has dimension $kl-{k+1 \choose 2}$ and index $l-1$ (with respect to the Pl\"ucker line bundle $\mathcal{O}_G(1)$, albeit non-irreducible in the Picard group). The only admissible triple is $(3,5,1)$. This is a linear section of the orthogonal Grassmannian $\OGr(3,8)$.
\mathfrak{sl}ubsubsection*{$Z_{\mathcal{Q}^*(1)}$}
This variety is the zero locus of a general global section of the bundle $\mathcal{Q}^*(1)$ on $\mathbb{G}r(k, k+l)$. If $k=2$, it is $\ThetaT(k,k+l)$. It has dimension $l(k-1)$ and index $k+1$. There are two admissible triples, $(2,7,1), (6,3,1)$. However the second one can be identified with $X_1 \mathfrak{sl}ubset \SGr(3,9)$. The first one $X_1 \mathfrak{sl}ubset \ThetaT(2,9)$ is new. Notice that we find as well the generic K3 of genus 4 as $(2,3,3)$ since the zero locus of $Q^*(1)$ on $\mathbb{G}r(2,5)$ is a quadric threefold. There are as well some FK3 obtained by $\ThetaT(2,n)$. However, they do not fall in this pattern, and we will examine them separately.
\mathfrak{sl}ubsubsection*{\textit{Other varieties}}
We tried other bundles to produce varieties of K3 type, such as $\mathcal{R}^{\vee}(1)$ or the locus of $\Sym^2 \mathcal{R}^{\vee} \oplus \bigwedge^2 \mathcal{R}^{\vee}$ (the \emph{orthosymplectic Grassmannian}). Even without no guarantee on the weight of the Hodge structure, our attempt was motivated by some example in the list of K\"uchle, see \cite{kuznetsovpicard}. However, we found no more new example.
\mathfrak{sl}ubsubsection*{\textit{Products}}
Products of projective spaces do produce a handful more of examples. One can easily see that no more than 5 projectives can be involved, with the extremal case being $X_{(1^5)} \mathfrak{sl}ubset (\mathbb{P}^1)^5$. Other examples are $X_{(1,1,1)} \mathfrak{sl}ubset (\mathbb{P}^3)^3$, and $X_{(2,1,1)} \mathfrak{sl}ubset \mathbb{P}^3 \times \mathbb{P}^1 \times \mathbb{P}^1$. In the products of Grassmannians when $k>1$, no further example is found. Indeed the index of a product of Grassmannians has index the gcd$(k_i+l_i)$. Substituting in the equations, one first find that no more than two Grassmannians can be used, and only one of them can have $k>1$. The possible cases are $X_{(1,1)} \mathfrak{sl}ubset \mathbb{G}r(2,6) \times \mathbb{P}^5$ and $X_{(2,1)} \mathfrak{sl}ubset \mathbb{G}r(2,4) \times \mathbb{P}^1$. Identical computations yield all the remaining cases.
\mathfrak{sl}ubsection{Geometric tools and tricks}
\mathfrak{sl}ubsubsection{A blow-up lemma}
We state here a blow up lemma. Although it merely descends from definitions, it is worth to recall it. It is worth to point out that a similar lemma is used in \cite{kuznetsovpicard}.
\begin{lemma}\label{lem:blowup} Let $X=X_{(d, 1)} \mathfrak{sl}ubset Z \times \mathbb{P}^1$. Then $X \cong Bl_S Z,$ where $S$ is the intersections of $2$ divisors of degree $d$ on $ Z$.
\end{lemma}
\begin{proof}
Let $\mathbb{P}^1=\Proj(\mathbb{C}[y_0,y_1])$ and $V^{\vee} \cong \mathbb{C}[y_0,y_1]_1$. (that is, homogeneous forms of degree 1). Denote by $W^{\vee} \cong H^0(\mathcal{O}_{Z}(d)) $. $X$ is given by definition by a choice of $\lambda \in W^{\vee} \otimes V^{\vee}$, or equivalently by a map (that we will still denote by $\lambda$) $\lambda: V \longrightarrow W^{\vee}.$
This map gives a 2-dimensional subspace of $ W^{\vee}$, or equivalently a pencil of divisors in $ Z$. The base locus of this pencil coincides with the $S$ defined in the lemma. The (only) incidence equation for the blow up of $Z$ in $S$ is $y_0f_d+y_1g_d$ and this is of course the same equation defining $X$. This proof admits an obvious generalisation when $\rho(Z) >1$.
\end{proof}
\mathfrak{sl}ubsubsection{Higher codimension case and Cayley trick(s)}
The above blow-up lemma admits a higher-codimensional generalisation. Indeed, when $X$ is the zero locus of a $(1,1)$ divisor in $U \times \mathbb{P}^{r-1}$ (with the obvious generalisation if $\rho(U) >1$) then $X$ can be given either by an element of $ W^{\vee} \otimes V_r^{\vee}$ or as a map $$\lambda: V_r \longrightarrow W^{\vee}.$$ If $r>2$ we cannot identify $X$ with any birational modification of the pair $(U,S)$, where $S$ is the base locus of the above linear system. However $X$ and $S$ share a deep relation, known as the \emph{Cayley trick}. More precisely the result is the following
\begin{thm}[Thm. 2.10 in \cite{orlov}, Thm 2.4 in \cite{kimkim}] \label{cayley}
Let $q:E \rightarrow U$ be a vector bundle of rank $r\ge 2$ over a smooth projective variety $U$
and let $S=s^{-1}(0)\mathfrak{sl}ubset U$ denote the zero locus of a regular section $s \in H^0(U,E)$ such that $ \dim S = \dim U - \mathrm{rank}\, E$.
Let $X=w^{-1}(0) \mathfrak{sl}ubset \mathbb{P} E^\vee$ be the zero locus of the section $w\in H^0(\mathbb{P} E^\vee, \mathcal{O}_{\mathbb{P} E^\vee}(1))$
determined by $s$ under the natural isomorphism
$H^0(U,E)\cong H^0(\mathbb{P} E^\vee, \mathcal{O}_{\mathbb{P} E^\vee}(1))$.
Then we have the semiorthogonal decomposition
$$ D^b(X)= \langle q^*D^b(U), \cdots, q^*D^b(U) \otimes_{\mathcal{O}_X} {\mathcal{O}_X}(r-2), D^b(S) \rangle .$$
\end{thm}
When this happens, we will write $D^b(S) \mathcal{H}ookrightarrow D^b(X)$. There is as well an (older) analogue Hodge-theoretic statement, cf. Prop. 4.3 in \cite{konno}, stating that the vanishing cohomologies of $S$ and $X$ are isomorphic up to a shift. When the hypotheses of the above Theorem are verified, this therefore proves at once that $X$ is of K3-type.\\
The Cayley trick can be generalised in the following way, using the formalism of Homological projective duality.
\begin{proposition} \label{rennemo} Let $Y_1$ and $Y_2$ be a pair of varieties with Lefschetz decompositions and embedded in $\mathbb{P}(V)$. Let $Z_H$ be the intersection of $Y_1 \times Y_2$ with a general (1,1)-divisor $H$. Let $f_H$ be the map that $H$ naturally defines from $\mathbb{P}(V)$ to $\mathbb{P}(V^\vee)$. Let $X_H = Y_1 \cap f_H^{-1}(Y_2^\vee$), where $Y_2^\vee$ is the Homological Projective dual to $Y_2$. Then $D(X_H) \mathcal{H}ookrightarrow D(Z_H)$.
\end{proposition}
\begin{proof}
Let $D(Y_2)=\langle A_0,A_1(1),\dots A_{m}(m) \rangle$ be the given Lefschetz decomposition of $Y_2$.
The divisor $H$ parametrizes, for every point of $Y_1$, an hyperplane section of $Y_2$, hence it defines a map $f_H\,:Y_1\,\rightarrow\,\mathbb{P}(V^\vee)$. In this way, $Z_H$ is identified with the pullback through $f_H$ of the universal hyperplane section $\mathcal{Y}_2\mathfrak{sl}ubset Y_2\times \mathbb{P}(V^\vee)$. Now, by \cite[Lemma 3.3]{kuzi_sod2} we have
$$D(\mathcal{Y}_2)=\langle D(Y_2^\vee), A_1(1)\boxtimes D(\mathbb{P}(V^\vee)),\dots, A_m(m)\boxtimes D(\mathbb{P}(V^\vee)) \rangle.$$
By applying base change \cite[Thm 5.6]{kuzi_sod} to the diagram
$$\xymatrix{ Z_H \ar[d]^\iota \ar[r]
& \mathcal{Y}_2\ar[d]^{\pi_2} \\ Y_1\ar[r]^{f_H} & \mathbb{P}(V^\vee), } $$
we obtain:
$$D(Z_H)=\langle D(Y^\vee_2 \times_{\mathbb{P}(V^\vee)} Y_1), A_1(1)\boxtimes D(Y_1),\dots, A_m(m)\boxtimes D(Y_1)\rangle.$$
And the variety in the first factor here is precisely $X_H=Y_2^\vee \times_{\mathbb{P}(V^\vee)} Y_1=Y_1 \cap f_H^{-1} (Y_2)^\vee.$
\end{proof}
\mathfrak{sl}ection{Case-by-case analysis}
\mathfrak{sl}ubsection{Identifications}
Before analysing in details the examples in our list, we want to eliminate some varieties that are well-known examples in disguise. We recall some results of Kuznetsov, that we conveniently bundle together. Recall that the variety $\Ml(k,n)$ is the \emph{bisymplectic Grassmannian}. It can be thought either as the intersection of two symplectic Grassmannian $\SGr(k,n)$ inside $\mathbb{G}r(k,n)$ or as the zero locus over $\mathbb{G}r(k,n)$ of a general global section of the bundle $\bigwedge^2 \mathcal{R}^{\vee} \oplus \bigwedge^2 \mathcal{R}^{\vee}$. We will better describe this variety later in the paper.
\begin{thm}[Thm 3.1 and Cor. 3.5 in \cite{kuznetsovpicard}]\label{kuzzolo}The following hold:
\begin{itemize}
\item There is an isomorphism $\Ml(n,2n) \cong \prod (\mathbb{P}^1)^n$;
\item The variety $X_{(1,1,1,1,1)} \mathfrak{sl}ubset \prod (\mathbb{P}^1)^5$ is isomorphic to $W=Bl_S(\prod (\mathbb{P}^1)^4)$, where $S=S_{(1,1,1,1)^2}$ is a non-generic K3 surface of genus $g=13$, given as the intersection of two divisors of multidegree $(1,1,1,1)$.
\end{itemize}
\end{thm}
Some of the Fano of K3 type that we found in our search can be actually identified with the $W$ above. For this reason they are not included in our main table. More precisely we have
\begin{lemma} Let $W$ the Fano of K3 type in \cite{kuznetsovpicard} defined above. Then the following Fano of K3 type \begin{itemize}
\item $X_{(1,1,1,1,1)} \mathfrak{sl}ubset Q_2 \times Q_2 \times \mathbb{P}^1$;
\item $X_{(1,1,1,1,1)} \mathfrak{sl}ubset \Ml(4,8) \times \mathbb{P}^1;$
\item $X_{(1,1,1,1,1)} \mathfrak{sl}ubset \Ml(3,6) \times \Ml(2,4);$
\end{itemize}
are isomorphic to $W$.
\end{lemma}
\begin{proof}
The first case is obvious, since $Q_2 \cong \mathbb{P}^1 \times \mathbb{P}^1$. For the other two cases, by definition and Kuznetsov's result $\Ml(n,2n)$ coincides with $\prod (\mathbb{P}^1)^n$.
\end{proof}
There is one more identification between two numerological candidates.
\begin{lemma} $X_{(1,1,1)} \mathfrak{sl}ubset \mathbb{S}_3 \times \mathbb{P}^1 \times \mathbb{P}^1 \cong X_{(2,1,1)} \mathfrak{sl}ubset \mathbb{P}^3 \times \mathbb{P}^1 \times \mathbb{P}^1$.
\end{lemma}
\begin{proof} It follows from the well known identification $ \mathbb{S}_3 \cong \mathbb{P}^3$, see for example \cite{kuznetsovs}. The difference in the degree is explained since the line bundle giving the spinor embedding for $\mathbb{S}_3 $ is the square root of the Pl\"ucker one.
\end{proof}
\mathfrak{sl}ubsection{Blow-up and Mukai type}
To prove that each of the variety of type M and B are of K3 type one can use the Cayley trick statement, as in Theorem \ref{cayley}. Indeed the (stronger) derived category statement implies the Hodge theoretical one. Indeed this can be seen by writing down such a semiorthogonal decomposition as prescribed by \ref{cayley} and then taking Hochshild homology. Alternatively one can use Riemann-Roch and standard exact sequences to compute the relevant Hodge numbers. We did these calculations as sanity checks for all our examples, however we believe it is neither worth nor interesting to list all of them, since they are quite similar. Therefore we will include just one example, namely Proposition \ref{gq1}, where Theorem \ref{cayley} does not apply in a straightforward way. For the families B1 and B2, Lemma \ref{lem:blowup} settles the matter. \\
In terms of construction of polarised families of IHS, we investigate another construction of the Hilbert scheme of points on a genus 8 K3, see Proposition \ref{gp2}. We believe that each of the examples in our list of Fano could lead to similar constructions: this would be especially interesting, considering the lack of examples of polarised families of Hilbert schemes of points on K3 surfaces.
\mathfrak{sl}ubsection{M3: a (different) computation in intersection theory} The variety M3 is $X_{(1,1)} \mathfrak{sl}ubset \mathbb{G}r(2,5) \times Q_5$. It has dimension 10 and index 4. It is neither a blow up with a center in a K3 surface, nor we can apply the Cayley trick. However we can show that it is a Fano of K3 type using Proposition \ref{rennemo}. Indeed we have
\begin{lemma} Let $S_6$ be a K3 surface of genus 6 in the Mukai model and $X$ our M3 as defined in the table. Then $D^b(S_6) \mathcal{H}ookrightarrow D^b(X)$.
\end{lemma}
\begin{proof} It suffices to apply Proposition \ref{rennemo}, since the Grassmannian $\mathbb{G}r(2,5)$ (or even a quadric hypersurface) is projectively self-dual. The intersection of the Grassmannian $\mathbb{G}r(2,5)$ with a 5-dimensional quadric (or, equivalently, the intersection of $\mathbb{G}r(2,5)$ with a quadric and 3 hyperplanes in its Pl\"ucker embedding) is a K3 of genus 6 and degree 10 by Mukai's classification. To conclude one needs to argue that the orthogonal complement to the derived category of $D^b(S_6)$ in $D^b(X)$ is generated by an exceptional collection, and then taking Hochshild homology (which is additive on semi-orthogonal decompositions), together with the Hochshild-Konstant-Rosenberg isomorphism cf. \cite[Theorem 7.5, 8.3]{kuzhkr}.
\end{proof}
As an alternative methods we can show that M3 is of K3 type using a lengthy (but rather standard) play with long exact sequences and cohomological vanishings.
\begin{proposition}\label{gq1}Let $X=X_{(1,1)} \mathfrak{sl}ubset \mathbb{G}r(2,5) \times Q_5$. Then $X$ is of K3 type.
\end{proposition}
The proof of the above proposition can be split in two lemma. The first one is a Chern class computation, the second one is essentially an application of Bott's theorem.
\begin{lemma} \label{erchar}The topological Euler characteristic of $X$ is $e(X)=72$.
\end{lemma}
\begin{proof}This is a lengthy (but direct) exercise in intersection theory, and we will spare the details to the reader. Let us denote $Y=\mathbb{G}r(2,5) \times Q_5$. Denote by $\alpha_1=c_1(\mathcal{O}_Q(-1))$ and $\beta_1=c_1(\mathcal{O}_G(-1))$. Denote by $\beta_2=c_2(\mathcal{R})$. One has $H^4(\mathbb{G}r(2,5), \mathbb{Z}) = \langle \beta_1^2, \beta_2 \rangle$. One easily compute $c(Q)$, $c(G)$ and $c(Y)=c(G)c(Q)$. In particular by Gauss-Bonnet $c_{11}(Y)=-6 \alpha_1^5 \beta_1^6$ and $$e(Y)= \int_{Y} -6 \alpha_1^5 \beta_1^6=60.$$
We then use the normal sequence associated to $X$ $$ 0 \to T_X \to TY|_X \to \mathcal{O}_X(1,1) \to 0.$$
This implies $c(TY|_X)=c(X)(1-\alpha_1-\beta_1)$. We can compute recursively the Chern classes of $X$, with in particular $$c_{10}(X)=(9\alpha_1^5\beta_1\beta_2^2+9\alpha_1^4\beta_1^2\beta_2^2)|_X.$$ To compute the restriction we evaluate against the class of $X$, and we have $ c_{10}(X) \cdot X=18\alpha_1^5\beta_1^2\beta_2^2.$
Using the relation in $A(G)$ given by $2\beta_1^5=5\beta_1\beta_2^2$ we get $$c_{10}(X) \cdot X= \frac{2 \cdot 18}{5}\alpha_1^5 \beta_1^6=\frac{6}{5}e(Y)=72.$$
\end{proof}
\begin{lemma} For $0 \leq i \leq 3$ we have $h^{i, 10-i}(X)=0$. Moreover $h^{6,4}(X)=h^{4,6}(X)=1$.
\end{lemma}
\begin{proof} As before let us denote $Y=\mathbb{G}r(2,5) \times Q_5$, and with $\mathcal{L} \cong \mathcal{O}_Y(1,1)$ (and its restriction to $X$ as well). We use the following two exact sequences \begin{equation} \label{seq1}
0 \to \Omega_X^{k-1} \otimes \mathcal{L}^{\vee} \to \Omega^k_{Y|X} \to \Omega^k_X \to 0 \end{equation} and
\begin{equation}\label{seq2} 0 \to \Omega^k_Y \otimes \mathcal{L}^{\vee} \to \Omega^k_Y \to \Omega^k_{Y|X}\to 0,
\end{equation}
possibly twisting for some positive multiple of $\mathcal{L}^{\vee}$ when required. The computation is rather lengthy and technical, and we will skip most of the details. To find similar computations the reader can refer to \cite{eg1}. For the results on the cohomological vanishings for both $\mathbb{G}r(2,5)$ and $Q_5$ one can consult for example \cite{peternell}, \cite{snow}.\\
The first vanishing $h^{0,10}(X)$ is obvious. Let us show the first non-obvious one, that is $h^{1,9}(X)=0$. Consider the two sequences \ref{seq1} and \ref{seq2} above with $k=1$. Using the K\" unneth formula one easily see that the cohomology of $\mathbb{G}r(2,5) \times Q_5$ is of Lefschetz-type. Moreover from Kodaira vanishing and since $H^{10}(X, \mathcal{L}) \cong H^{0}(X, \mathcal{O}_X(-3,-3))=0$ one reduces to
$$ 0 \to H^9(\Omega^1_{Y|X}) \to H^9 (\Omega^1_X) \to 0 $$ and $$ 0 \to H^9(\Omega^1_{Y|X}) \to H^{10}(\Omega^1_Y \otimes \mathcal{L}^{\vee}) \to 0.$$
However, if we denote with $\pi_1$ (resp. $\pi_2$) the projection on $\mathbb{G}r(2,5)$ (resp. $Q_5$) we have $\Omega^1_Y \cong \pi_1^* \Omega^1_{\mathbb{G}r(2,5)} \oplus \pi_2^* \Omega^1_Q$, and from the K\"unneth formula for the box product and the well known vanishings for the twisted cohomologies of $\mathbb{G}r(2,5)$ and $Q_5$ we have $$H^{10}(\Omega^1_Y \otimes \mathcal{L}^{\vee})\cong H^9(\Omega^1_{Y|X}) \cong H^9 (\Omega^1_X)=0.$$
For $h^{2,8}(X)$ we use the sequences \ref{seq1} and \ref{seq2} with $k=2$ and $k=1$ twisted by $\mathcal{L}^{\vee}$. Indeed one has from \ref{seq1}
$$ 0 \to H^8(\Omega^2_{Y|_X}) \to H^8(\Omega^2_X) \to H^9(\Omega^1_X) \to H^8(\Omega^2_{Y|_X}) \to 0.$$ The two external terms can be checked to be 0 using \ref{seq2}, again together with the K\"unneth formula and the usual vanishings (using the decomposition for $\Omega^2_Y$). Using the twisted version of \ref{seq1} and \ref{seq2} we reduce to the isomorphism $H^8(\Omega^2_X) \cong H^{10}((\mathcal{L}_X^{\vee})^{\otimes 2})=0$. The same argument works as well for $h^{3,7}(X)=0$, where for $h^{4,6}(X)$ we get $$H^6(\Omega^4_X) \cong H^{10}((\mathcal{L}_X^{\vee})^{\otimes 4})\cong H^0(\mathcal{O}_X) \cong \mathbb{C}.$$
\end{proof}
The last Lemma is enough to prove that $X$ is of K3 type. In particular, when combined with Lemma \ref{erchar} we explicitely compute all the Hodge numbers. The following corollary is in fact proved bundling the two results above, together with Lefschetz theorem on hyperplane section and a direct application of the K\"unneth formula.
\begin{corollary} Suppose $p+q \neq 10$. The only non-zero Hodge numbers $h^{p,q}$ of $X$ are $$h^{0,0}=h^{10,10}=1, \ h^{1,1}=h^{9,9}=2, \ h^{2,2}=h^{8,8}=4, \ h^{3,3}=h^{7,7}=6, \ h^{4,4}=h^{6,6}=8.$$
For $p+q=n$ the only non-zero Hodge numbers are $$h^{6,4}=h^{4,6}=1, \ h^{5,5}=28,$$ with moreover the dimension of the vanishing cohomology subspace $h^{5,5}_{\van}=19$.
\end{corollary}
\mathfrak{sl}ubsection{M7: another construction of $S_8^{[2]}$}
The 12-fold $X_{M7}$ is given by the zero locus of a (1,1) section on $\mathbb{G}r(2,6) \times \mathbb{P}^5$. Let $S_8=\mathbb{G}r(2,6) \cap H_1 \cap \ldots \cap H_6$. Then $S_8$ is a general K3 surface of genus 8 in Mukai's model. From the Cayley trick argument one has that $D^b(S_8) \mathcal{H}ookrightarrow D^b(X_{M7})$. On the Hodge-theoretical level indeed we have:
\begin{lemma}\label{m7} Let $X_{M7}$ as above. Then $X_{M7}$ is of K3 type with $h^{6,6}=31$ and the vanishing subspace $h_{\van}^{6,6}=19$.
\end{lemma}
\begin{proof} Since $\mathbb{G}r(2,6) \times \mathbb{P}^5$ is a central variety, it is enough to compute the Euler characteristics $\chi(\Omega^i)$ for $i=5,6$. This can be done for example via Riemann-Roch or using Macaulay2.
\end{proof}
As expected, we can associate to $X_{M7}$ an IHS, which is linked to the genus 8 K3. To do this, let $Z$ be given by the zero locus of a general global section of the bundle $\bigwedge^2 \mathcal{R}^{\vee}_{4,6} \otimes \mathcal{R}^{\vee}_{2,6}$ on $\mathbb{G}r(4,6) \times \mathbb{G}r(2,6)$. We have the following proposition.
\begin{proposition} $Z$ is an IHS fourfold.
\end{proposition}
\begin{proof}Recall the formula for the first Chern class of a product $c_1(\bigwedge^2 \mathcal{R}^{\vee}_{4,6} \otimes \mathcal{R}^{\vee}_{2,6})= \mathrm{rk}(\mathcal{R}^{\vee}_{2,6})\cdot c_1(\bigwedge^2 \mathcal{R}^{\vee}_{4,6} )+\mathrm{rk}(\bigwedge^2 \mathcal{R}^{\vee}_{4,6})\cdot c_1( \mathcal{R}^{\vee}_{2,6}).$ By adjunction it follows that for a general section $Z$ is a smooth fourfold with $c_1=0$. We compute now its holomorphic Euler characteristic $\chi(\mathcal{O}_Z)$.
This can be done for example via a Riemann-Roch computation, since $$\chi(\mathcal{O}_Z) = \frac{c_2^2-c_4}{720}.$$ We will use a Macaulay2 code in order to speed up the calculation.
\begin{verbatim}
loadPackage "Schubert2"
k1=2, l1=4, k2=4, l2=2;
G26=flagBundle({k1,l1}, VariableNames=>{r1,q1});
(R1,Q1)=G26.Bundles;
V=abstractSheaf(G26, Rank=>6);
G46=flagBundle({k2,l2}, V, VariableNames=>{r2,q2});
(R2,Q2)=G46.Bundles;
p=G46.StructureMap;
R1G46=p^*(dual R1);
F=R1G46**exteriorPower_2 dual R2;
Z=sectionZeroLocus F;
chi(OO_Z);
\end{verbatim}
Running the previous code one verifies $\chi(\mathcal{O}_Z)=3$. in particular the statement follows by simply applying Beauville-Bogomolov decomposition theorem.
\end{proof}
The deformation type of $Z$ can be shown to be the expected one as follows.
\begin{proposition}\label{gp2} $Z$ is isomorphic to Hilb$^2(S^8)$.
\end{proposition}
\begin{proof} Let $h \in \bigwedge^2V_6^* \otimes V_6^*$ defining $X_{M_7}$. As above, we can consider $h$ as a morphism $$h: V_6 \to \bigwedge^2 V_6^*.$$ A point in Hilb$^2(S_8)$ is therefore given by a pair $(u_1, u_2)$, $u_i \in \bigwedge^2 V_6$ on both of which $h$ vanishes. Consider $W \mathfrak{sl}ubset \bigwedge^2 V_6$ spanned by $u_1, u_2$. Consider further the restricted morphism $\overline{h}^t: W \to V^\vee_6$. This has rank 2, and we can take $P= \mathrm{Im}(\overline{h}^t)$. By construction $h$ vanishes on the pair $(W,P) \in \mathbb{G}r(4,6) \times \mathbb{G}r(2,6)$, thus defining a point in $Z$. From this construction, it is clear that $W$ determines $P$. Moreover, the map we constructed inside $\mathbb{G}r(4,6)$ can be seen as the same map (after duality) which associates to Hilb$^2(S_8)$ a line in the pfaffian cubic fourfold, hence it is an isomorphism. \end{proof}
We point out the similarities between this contruction and \cite[Proposition B.6.3]{kps}. Here it is proved how the variety of lines (resp. conics) of a smooth cubic threefold (resp. a generic Fano threefold of genus 8) is isomorphic to a section of the bundle $\bigwedge^2 \mathcal{R}^{\vee}_{4,5} \otimes \mathcal{R}^{\vee}_{2,5}$ over $\mathbb{G}r(4,5) \times \mathbb{G}r(2,5)$. In turn, their proof can be modified to give an alternative proof of \ref{gp2}.
\mathfrak{sl}ubsection{Sporadic examples}
This subset of the list is the most interesting one. Indeed for these Fano we cannot produce a systematic method as in the \emph{Mukai} case. For each one of them already proving that they are of K3 type requires an ad-hoc strategy. Our most interesting results comes indeed from this section: indeed we reinterprete the Debarre-Voisin IHS fourfold as moduli space of relevant objects on a Fano of K3 type in two different ways, namely as in Theorem \ref{hk} and Proposition \ref{t2}. Moreover we produce the first examples of a Fano with multiple K3 structures, cf. Proposition \ref{3k3} and with a mixed $(2,3)$ CY structure, cf. Proposition \ref{23cy}. Moreover we do not limit ourselves to the computation of the Hodge numbers: we give indeed geometrical descriptions of many of the examples we consider, since we believe them to be a rich and beautiful sources of geometries.
\mathfrak{sl}ubsection{S1: four codimensional linear section of $\mathbb{G}r(2,8)$}
We already considered this example in our previous work \cite{eg1}, therefore we will not spend too much time on it. It is described in a surprisingly simple way as a codimensional 4 linear section of the Grassmannian $\mathbb{G}r(2,8)$.
\begin{proposition}\label{s1} Let $X_{1,1,1,1} \mathfrak{sl}ubset \mathbb{G}r(2,8)$ given by a generic section of $\mathcal{O}_G(1)^{\oplus 4} $. Then $X$ is an 8-fold of K3 type,with $h^{4,4}_{\van}(X)=19$.
\end{proposition}
We remark that there is another FK3 closely related to S1. This is $X_{(1,1)} \mathfrak{sl}ubset \mathbb{G}r(2,8) \times \mathbb{P}^3$. In our main table this is listed as M13. We chose this notation since, although there is no K3 in the Mukai model related, it shares many similarities with the other Fano in the \textbf{M} group. In particular one can apply directly Cayley trick to prove that this Fano is of K3 type.\\
As already remarked in our previous work the projective dual of $X_{1,1,1,1} \mathfrak{sl}ubset \mathbb{G}r(2,8)$ is quartic K3 surface $S_3 \mathfrak{sl}ubset \mathbb{P}^3$. An embedding of the derived category of the quartic K3 inside the derived category of the above linear section is proved in \cite{segalthomas}, Thm 2.8.\\
We already conjectured that this complete intersection in $\mathbb{G}r(k,n)$ should be the only FK3 obtained in this way. We repeat the precise formulation of this conjecture here.
\begin{conj} Let $X=X_{d_1, \ldots, d_c} \mathfrak{sl}ubset \mathbb{G}r(k,n)$ a Fano smooth complete intersection of even dimension. Then $X$ is not of K3-type unless $$(\lbrace d_i \rbrace, k,n)=(\lbrace 3 \rbrace,1,6),(\lbrace 2,1\rbrace,2,5), (\lbrace 1,1,1,1\rbrace,2,8), (\lbrace 1 \rbrace, 3,10).$$
\end{conj}
\mathfrak{sl}ubsection{S2: a K3 of genus 7 from $\OGr(3,8)$}
This sporadic example is a linear section $X= \OGr(3,8) \cap H$ of the orthogonal Grassmannian $\OGr(3,8)$. It is worth to spend few words on the ambient variety. In general the orthogonal Grassmannian $\OGr(n-1, 2n)$ behaves differently from $\OGr(k,2n)$, which for $k \neq n-1$ is a prime Fano variety. Indeed $\OGr(n-1, 2n)$ can be realised as a $\mathbb{P}^{n-1}$ bundle over (both of) $\mathbb{S}_{n}^i$, the latter denoting the two connected component of the maximal orthogonal Grassmannian $\OGr(n, 2n)$ in the spinor embedding. In particular the Picard group of $\OGr(n-1, 2n)$ has rank 2 with the Pl\"ucker line bundle $\mathcal{L} := \mathcal{O}_{\mathbb{S}^1} (1)\boxtimes \mathcal{O}_{\mathbb{S}^2}(1)$ is very ample. $\OGr(n-1, V_{2n})$ is non-degenerate in the Pl\"ucker embedding, and $$H^0(\OGr(n-1, 2n), \mathcal{L}) \cong \bigwedge^{n-1} V_{2n}^{\vee}.$$
With $X=X_1 \mathfrak{sl}ubset \OGr(3,8)$ in the introductory table we mean the zero locus of a generic global section of $\mathcal{L}$. Such $X$ is an 8-fold of index $\iota=3$. Since it is a linear section of a central variety, to compute its Hodge numbers it suffices to compute the Euler characteristics $\chi(\Omega^i_X)$, together with the knowledge of the cohomology of $\OGr(3,8)$. A full computation by the means of Borel-Bott-Weil theorem, can be found in the PhD thesis of the first author. We recall here the result.
\begin{lemma}[cf. \cite{thesis}, Proposition A.1.1] $X$ is a Fano 8-fold of K3 type with $h^{4,4}(X)=24$, and its vanishing subspace of rank 19.
\end{lemma}
We explain now a link between this 8-fold $X$ and a K3 of genus 7. Recall from the work of Mukai that a generic K3 of such genus can be obtained by cutting $\mathbb{S}_{10}$ with 8 hyperplanes. Here we use a different description of the aforementioned K3. Let $X \mathfrak{sl}ubset \OGr(3,8)$ defined by $V(\mathfrak{sl}igma)$, $\mathfrak{sl}igma \in H^0(\mathcal{L})$. Let $\mathbb{S}_8$ be (one of the two connected component of) the orthogonal Grassmannian $\OGr(4,8)$, denote with $\mathcal{R}$ the restriction of the tautological bundle. Since $\mathfrak{sl}igma$ can be seen as an element in $H^0(\mathbb{S}_8, \bigwedge^3\mathcal{R}^{\vee})$ we can denote by $S= V(\mathfrak{sl}igma) \mathfrak{sl}ubset \mathbb{S}_8$. It is easy to check that $S$ is a K3 of genus 7 (notice that $\mathbb{S}_8$ is nothing but a 6-dimensional quadric hypersurface in disguise, either using triality or checking dimension and invariants). Such $S$ is responsible for the interesting part of the derived category (and therefore the Hodge theory of $X$). Indeed we quote the following result of Ito-Miura-Okawa-Ueda. Denote $\pi$ the restriction of the projection $p$ from $X$ to (one of the two) $\mathbb{S}_8$.
\begin{lemma}[Lemma 2.1 in \cite{ito}] The morphism $\pi$ is a $\mathbb{P}^2$-bundle over $\mathbb{S}_8 \mathfrak{sl}mallsetminus S$ and a $\mathbb{P}^3$-bundle over $S$, locally trivial in the Zariski topology.
\end{lemma}
In turn we can use an adapted version of Orlov's blow-up formula to this case. This is indeed a generalisation of the Cayley trick. We borrow this result from the forthcoming \cite{nested}, where it will be shown in full details and generality. For this reason, the proof will be omitted here.\\
First, in the notation above, denote by $\iota: S \mathfrak{sl}ubset \mathbb{S}_8 $. The above Lemma is equivalent to the following commutative diagram
$$\xymatrix{
F \ar@{^{(}->}[r]^j \ar[d]_p & X \ar[d]^\pi \\
S \ar@{^{(}->}[r]^\iota & \mathbb{S}_8,}$$
with $F$ a smooth projective subvariety, $j:F \mathfrak{sl}ubset X$ of codimension $d=4+2-3=3$ and a locally free sheaf $\mathcal{F}$ of rank $4$ on $S$ such that $p:F \mathfrak{sl}imeq \mathbb{P}_S(\mathcal{F}) \to S$.
We denote by $\mathcal{O}_F(H)$ the relative ample bundle of $p$ and we assume that there is a line
bundle $\mathcal{O}_Y(H)$ such that $\mathcal{O}_Y(H)_{\vert F} \mathfrak{sl}imeq \mathcal{O}_F(H)$ and that there is a vector
bundle $\mathcal{E}$ of rank $d$ on $X$ such that $F$ is the zero locus of a general section
of $\pi^*\mathcal{E} \otimes \mathcal{O}_Y(-H)$.
We define the functors $\Phi_l:\Db(S) \to \Db(F)$ by the formula
$\Phi_l(A)= j_* (p^* A \otimes \mathcal{O}(lH))$.
\begin{proposition}\label{o2}
In the configuration above, $\Phi_l$ is fully faithful for any integer $l$, and there is a semiorthogonal
decomposition:
$$\Db(Y)=\mathfrak{sl}od{\Phi_{-1}\Db(Z),\pi^*\Db(X),\ldots,\pi^*\Db(X)\otimes \mathcal{O}_Y(2H)}$$
\end{proposition}
\mathfrak{sl}ubsection{S3: bisymplectic Grassmannian $\Ml(3,8)$ and Debarre-Voisin IHS} The variety $\Ml(k,n)$ is given by the vanishing of a global section of the bundle $\bigwedge^2 \mathcal{R}^{\vee} \oplus \bigwedge^2 \mathcal{R}^{\vee}$ on the Grassmannian $\mathbb{G}r(k,n)$. Equivalently, given a pencil $\lambda: \mathbb{C}^2 \to \bigwedge^2 V_n^{\vee}$ it parametrises k-dimensional subspaces isotropic for all skew-forms in the pencil. In \cite{kuznetsovpicard} this variety is studied by Kuznetsov when $k=n/2$ and by Benedetti in \cite{ben18} with a strong emphasis in the case $k=2$. Let us recall some key facts of the construction. Assume that $n=2m$ is of even dimension. To a general pencil $\lambda$ are canonically associated $m$ degenerate skew-forms $\lbrace \lambda_1, \ldots, \lambda_m \rbrace$, given by the intersection bewteen the line $L_{\lambda} \mathfrak{sl}ubset \mathbb{P}(\bigwedge^2 V^{\vee})$ and the (Pfaffian) discriminant hypersurface $D$, corresponding to degenerate skew-forms. Denote by $K_i$ the kernel of $\lambda_i$. The smoothness of $\Ml$ is equivalent to the $\lambda_i$ being distinct, and moreover we can decompose $V=K_1 \oplus \ldots \oplus K_m$ as a direct sum.\\
Kuznetsov gives as well the canonical form for the pencil, espressing the two generators (up to dividing by 2) as $$\omega_1= x_{1,2}+x_{3,4}+\ldots +x_{n-1,n}, \ \ \omega_2= a_1x_{1,2}+a_2x_{3,4}+\ldots +a_m x_{n-1,n},$$ with the $a_i$ pairwise distinct, and $x_{i,j}:= x_i \wedge x_j$. This way, we can identify $K_1:=\langle e_1, e_2 \rangle$, $K_2:=\langle e_3, e_4 \rangle$ and so on. When $m=k$ one has $\Ml(k, 2k) \cong \prod (\mathbb{P}^1)^k$, see the theorem already recalled in \ref{kuzzolo}. When $m\neq k$ however we do not have such a nice description as a product. For $k=2$ for example $\Ml(2,n)$ is an intersection of $\mathbb{G}r(2,n)$ with a linear subspace of codimension 2. \\
Let us now focus on the case $\Ml(3,8)$. We compute first the cohomology of a linear section of $\Ml(3,8)$.
\begin{proposition} \label{ah}A linear section $X_1=V(\mathfrak{sl}igma_1) \mathfrak{sl}ubset \Ml(3,8)$ is of K3 type.
\end{proposition}
\begin{proof}
The first thing to prove is that $\Ml(3,8)$ is a central variety. This can be done via a direct computation, for example using Borel-Bott-Weil theorem. There is however another (much easier) argument which is however more conceptual, and prove the similar statement for all $\Ml(k,n)$. Indeed in \cite[Proposition 2.10]{ben18} it is proved that there is a torus $T \cong (\mathbb{C}^*)^n$ acting on $\Ml(k,n)$ with the fixed locus constituted only by $2^k {n \choose k}$ points. This implies, thanks to \cite[Theorem 2]{sommese} that the $\Ml(k,n)$ is a central variety, with $2^k {n \choose k}$ being its topological Euler characteristic.\\
Lefschetz theorem on hyperplane section enables us to describe the cohomology of $Z$ except all the Hodge groups $h^{p,q}(Z)$ with $p+q=8$. We can determine these dimensions by computing the Euler characteristics of $\chi(\Omega_X^i)$. The latter can be computed via a direct but lengthy computation, and computer algebra systems as Macaulay2 can speed up everything. One has in particular \begin{align*}&\chi(\Omega^1_X)=\chi(\Omega^1_{\Ml(3,8)})=1\\
&\chi(\Omega^2_X)=\chi(\Omega^2_{\Ml(3,8)})=2\\
&\chi(\Omega^3_X)=\chi(\Omega^3_{\Ml(3,8)})+1=7\\
&\chi(\Omega^4_X)=26.
\end{align*}
\end{proof}
This gives as well all the Hodge numbers. We collect them in the next corollary for the reader's convenience.
\begin{corollary}The only non-zero Hodge numbers $h^{p,q}$ of $\Ml(3,8)$ are $$h^{0,0}=h^{9,9}=1, \ h^{1,1}=h^{8,8}=1, \ h^{2,2}=h^{7,7}=2, \ h^{3,3}=h^{6,6}=6, \ h^{4,4}=h^{5,5}=6.$$
\end{corollary}
\begin{corollary} \label{hodges2}Suppose $p+q \neq 8$. The only non-zero Hodge numbers $h^{p,q}$ of $X$ are $$h^{0,0}=h^{8,8}=1, \ h^{1,1}=h^{7,7}=1, \ h^{2,2}=h^{6,6}=2, \ h^{3,3}=h^{5,5}=6.$$
For $p+q=8$ the only non-zero Hodge numbers are $$h^{3,5}=h^{5,3}=1, \ h^{4,4}=26,$$ with moreover $h^{5,5}_{\van}=20$.
\end{corollary}
We want now to associate to our Fano of K3 type $X$ an IHS $Z$. To do this, at first notice that $\Ml(3,8)$ is degenerate in the Pl\"ucker embedding in $\mathbb{P}(\bigwedge^3 V_8)$. It lies indeed in $\mathbb{P}(U)$, where
$$U:= \mathcal{E}r (\varphi: \bigwedge^3 V_8 \mathfrak{sl}tackrel{(\lrcorner_1, \lrcorner_2)}{\longrightarrow} V_8 \oplus V_8),$$ where $\lrcorner_i$ denotes the contraction with the 2-skew form $\omega_i$. Equivalently, we have that $\Ml(3,8)$ is defined by a general $\mathfrak{sl}igma_1 \in U^{\vee}$.\\
Consider now the Grassmannian $\mathbb{G}r(6,8)$. Denote by $\bar{\lrcorner_i}$ the contraction with the restriction of the two form $\omega_i|_W$ to a 6-space $W$. For the generic $[W] \in \mathbb{G}r(6,8)$ the map $$\overline{\varphi}: \bigwedge^3 W \mathfrak{sl}tackrel{(\bar{\lrcorner_1}, \bar{\lrcorner_2})}{\longrightarrow} W \oplus W$$ remains surjective, since the rank of $\omega_1|_W$ and $\omega_2|_W$ is still maximal.
However, when $W$ is such that every element of the pencil restricted to such $W$ has rank 4, then the above map is not surjective anymore.
As a special example, one can take a subspace given by $x_1=x_3=0$. Then for example the vector $(e_5, d e_5)$, $d \neq 1$ is not in the image of $\overline{\varphi}$.
To identify in general the locus $D$ where $\overline{\varphi}$ is not surjective let us write (in the notation above) $V_8=K_1 \oplus K_2 \oplus K_3 \oplus K_4$. We can then describe $D$ as $$D:= \lbrace W_6 \mathfrak{sl}ubset V_8 \ | \ \ddim (W_6 \cap K_i) \geq 1, \ \forall i \rbrace.$$
$D$ is therefore isomorphic to a $\mathbb{G}r(2,4)$-bundle over $(\mathbb{P}^1)^4 \cong \Ml(4,8)$, and over it we have a cokernel sheaf $\mathcal{G}$ of rank 4 on its support, given by the Kernel of the rank 4 map $W \to W^* \oplus W^*$. Summing up, we have the following result
\begin{proposition} On $\mathbb{G}r(6,8)$ there is an exact sequence of sheaves
$$ 0 \to F \to \bigwedge^3 \mathcal{R} \to \mathcal{R} \oplus \mathcal{R} \to \mathcal{G} \to 0.$$
\end{proposition}
\begin{corollary} $F^\vee$ is a globally generated vector bundle of rank 8 and $H^0(F^{\vee})= U^{\vee}$.
\end{corollary}
\begin{proof}
Dually, there is a surjective morphism of sheaves $\bigwedge^3 \mathcal{R}^\vee \to F^\vee$, which is surjective on stalks. Hence, global sections of $F^\vee$ which are images of global sections of $\bigwedge^3 \mathcal{R}^\vee$ are sufficient to generate stalks, so that $F^\vee$ is globally generated.
\end{proof}
Moreover, since $\mathcal{G}$ is a torsion sheaf supported in codimension 4 we have the following corollary.
\begin{corollary} $c_1(F^{\vee})=8h.$
\end{corollary}
\begin{proposition} Let $Z \mathfrak{sl}ubset \mathbb{G}r(6,8)$ defined by the zero locus of a general global section of the vector bundle $F^{\vee}$. Then $Z$ is a fourfold with canonical class $\omega_Z \cong \mathcal{O}_Z$.
\end{proposition}
\begin{thm} \label{hk}Let $Z$ as above, and let $Z_{DV} \mathfrak{sl}ubset \mathbb{G}r(6,10)$ the Debarre-Voisin IHS. Then $Z$ is isomorphic to $Z_{DV}$. Moreover, $Z$ can be interpreted as (the compactification of) the space of $\Ml(3,6) \cong (\mathbb{P}^1)^3$ inside $X_1 \mathfrak{sl}ubset \Ml(3,8)$.
\end{thm}
\begin{proof}
With a non canonical choice of a two-space $\langle v,w\rangle = V_2\mathfrak{sl}ubset V_{10}$, the three form $\omega$ defining $Z_{DV}$ can be written as $\omega=\omega_8+v^\vee\wedge \mathfrak{sl}igma_1+w^\vee\wedge \mathfrak{sl}igma_2$, where $\omega_8$ is a three form on an eight dimensional vector space $V_8$ and $\mathfrak{sl}igma_i$ are two forms on the same space. The natural projection from $\mathbb{P}(V_{10})$ to $\mathbb{P}(V_8)$ induces a rational map from $\mathbb{G}r(6,10)$ to $\mathbb{G}r(6,8)$. For this map, there are three kinds of six-spaces:
\begin{itemize}
\item[Type 0] Six spaces which do not intersect the fixed two space $V_2$.
\item[Type 1] Six spaces meeting the fixed $V_2$ in a line $U_1$.
\item[Type 2] Six spaces containing the fixed $V_2$.
\end{itemize}
By a dimension count and the genericity assumption on $Z_{DV}$, spaces of type 2 do not occur inside $Z_{DV}$. Spaces of type 1 are given by the Schubert cycle $\mathfrak{sl}igma_{3,0^5}(V_2)$, and inside $Z_{DV}$ this is a curve of degree $132$, which is smooth since one can check that the Schubert cycle we use to obtain it is smooth as well. The blow up of $Z_{DV}$ along this curve maps into a subvariety of $\mathbb{G}r(6,8)$ given by six planes where the three form $\omega_8$ is given as the sum of $\mathfrak{sl}igma_1$ and $\mathfrak{sl}igma_2$ wedged with the dual of some vectors of the six space itself. This is precisely the variety $Z$ for the forms $\omega_8,\mathfrak{sl}igma_1,\mathfrak{sl}igma_2$.
The local picture in the exceptional divisor is given by sending a six plane $U_1\mathfrak{sl}ubset U_6$ to the set of all possible six planes in $V_8$ containing $U_6/U_1$, which is a $\mathbb{P}^2$.
The image $\pi(U_6)$ of a six space $U_6\in Z_{DV}$ contains three spaces parametrized by $X_1 \mathfrak{sl}ubset \Ml(3,8)$ where the form $\omega_8$ restricts to zero, hence also the two forms $\mathfrak{sl}igma_1,\mathfrak{sl}igma_2$ are zero. That is, a point of $Z$ parametrizes a copy $\Ml(3,6)\cong (\mathbb{P}^1)^3$ contained in $X_1$ as claimed above.
We proved that $Z$ has trivial canonical bundle and, if the rational map we defined above from $Z_{DV}$ has degree one, $Z$ and $Z_{DV}$ would be birational minimal models, hence the map given by the blow-up of $Z_{DV}$ along the curve composed with the projection would be a flop. But a flop is not defined in codimension at most two on an IHS fourfold, hence the map was already well defined and is an isomorphism. Let us prove that this map has indeed degree one: Let $V_6$ and $W_6$ be two points of $Z_{DV}$ with the same projection. Therefore, their basis differ only for multiples of v and w and, after a linear combination, we can suppose that at most two elements differ by these vectors. Let us treat first the case of a single vector: let $V_6=\langle v_1,v_2,v_3,v_4,v_5,v_6 \rangle$ and let $W_6=\langle v_1,v_2,v_3,v_4,v_5,v_6+av+bw \rangle$. As the choice of $V_6$ varies, the coefficients $a,b$ are not constant, hence we can suppose $a=1,b=0$ (which happens in codimension one). Thus on $W$ we have $\omega(v_6+v,x,y)=v\wedge \mathfrak{sl}igma_1(x,y)$. So, if the six space annihilates such a three form, it must be isotropic for $\mathfrak{sl}igma_1$, which is clearly impossible on a six space, unless the two form degenerates, which happens in codimension two.\\ On the other hand, if $W_6=\langle v_1,v_2,v_3,v_4,v_5+w,v_6+v \rangle$ we have $\omega(v_6+v,x,y)=v\wedge \mathfrak{sl}igma_1(x,y)$ and $\omega(v_5+w,x,y)=w\wedge \mathfrak{sl}igma_2(x,y)$. This implies that the residual four space is isotropic with respect to both forms, which is a codimension twelve condition on the six spaces themselves. Indeed, this is $\Ml(4,8)\cong (\mathbb{P}^1)^4$ inside $\mathbb{G}r(4,8)$. Hence, by the genericity assumption on $\omega$, this does not happen in our case.
\end{proof}
\mathfrak{sl}ubsection{S5: a section of a non-central variety} This sporadic Fano of K3 type is rather different from the others. It is a linear section of a certain 7-fold of index 3 that we call $\ThetaT(2,9)$, which is not even central, let alone homogeneous. This 7-fold is the zero locus of a general global section of the bundle $\mathcal{Q}^*(1)$ on the Grassmannian $\mathbb{G}r(2,9)$. By Borel-Bott-Weil we interpret $H^0(\mathbb{G}r(2,9), \mathcal{Q}^*(1)) \cong \bigwedge^3 V_9^{\vee},$ therefore $\ThetaT(2,9)$ (sometimes shortened as $\ThetaT$ in the following proofs) is given by the locus of two-spaces in a 9-dimensonal space which are annihilated by a 3-form. This 7-fold, which is indeed a \emph{congruence of lines} has been considered in the recent work (\cite{faenzi}, Ex. 4.14). As we said, the variety $\ThetaT(2,9)$ is not central, therefore we cannot apply any trick as in Proposition \ref{blowup} to compute the Hodge numbers of its linear section. Therefore we will need to go through a proper Borel-Bott-Weil computation.\\
We will start by stating the final result on the Hodge numbers.
\begin{proposition} \label{t129}The Hodge numbers of $\ThetaT(2,9)$ are
\begin{center}
{\mathfrak{sl}mall
\[\begin{matrix}
&&&&&&&&1 &&&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&0 &&1&&0&&&&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&0&&0 & & 2 & &2 && 0 && 0 &&&\\
&&0 && 0 && 0 &&2 &&0 &&0 && 0&&\\
&0 && 0 && 0 && 2&& 2 &&0 &&0 && 0&\\
&&0 && 0 && 0 &&2 &&0 &&0 && 0&&\\
&&&0&&0 & & 2 & &2 && 0 && 0 &&&\\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&&&0 &&1&&0&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&&&1 &&&&&&&&
\end{matrix}\]}
\end{center}
\end{proposition}
From the above diamond it immediately follows that holomorphic Euler characteristics for $\ThetaT$ are $\chi(\Omega^1_{\ThetaT})=-1, \ \chi(\Omega^2_{\ThetaT})=0, \ \chi(\Omega^3_{\ThetaT})=2$. These can be easily double-checked using Macaulay2. Moreover the topological Euler characteristic $e_{\textrm{top}}(\ThetaT)=0$ (cf. \cite{faenzi}, Ex. 4.14).
\begin{corollary} \label{cor29}Let $X= \ThetaT(2,9) \cap H$ be a linear section of $\ThetaT(2,9)$. This is a Fano of K3 type with Hodge diamond
\begin{center}
{\mathfrak{sl}mall
\[\begin{matrix}
&&&&&&&&1 &&&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&0 &&1&&0&&&&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&0&&0 & & 2 & &2 && 0 && 0 &&&\\
&&0 && 0 && 1 &&22 &&1 &&0 && 0&&\\
&&&0&&0 & & 2 & &2 && 0 && 0 &&&\\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&&&0 &&1&&0&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&&&1 &&&&&&&&
\end{matrix}\]}
\end{center}
The vanishing subspace is $h^{2,2}_{\van}(X)=20$. The holomorphic Euler characteristics for $X$ are $\chi(\Omega^1_{X})=-1, \ \chi(\Omega^2_{X})=1, \ \chi(\Omega^3_{\ThetaT})=-18$. Moreover the topological Euler characteristic $e_{\textrm{top}}(X)=24$.\end{corollary}
\begin{proof} The Hodge numbers for $X$ follows from those of $\ThetaT(2,9)$ together with the computations of $\chi(\Omega^i)$, which can be easily done a priori via Riemann-Roch and the help of computer algebra.
\end{proof}
\mathfrak{sl}ubsubsection{Borel-Bott-Weil computation for $\ThetaT(2,9)$}
Borel-Bott-Weil theorem is a powerful tool for computing cohomologies of vector bundles on homogeneous spaces. Together with some well-known sequences it is often sufficient to compute Hodge numbers for varieties cut by general global sections of homogeneous vector bundles. Although rather long and involved, the procedure is mostly algorithmic. We will include the general setup (skipping most details for the sake of readability) in order to give the reader a toolbox for further computations.
\mathfrak{sl}ubsubsection*{General BBW strategy}
Let $\mathbb{G}r(k,n)$ be the
Grassmannian of $k$-dimensional subspaces of $V_n$. Consider two dominant weights $\alpha
= (\alpha_1, \dots ,\alpha_{n-k})$ and $\beta = (\beta_1, \dots
,\beta_{k})$ for the Schur functors $\Sigma$ applied to $\mathcal{Q}$ and $\mathcal{R}$ and their concatenation $\gamma =
(\gamma_1,\dots,\gamma_n)$. Let $\partiallta$ the decreasing sequence $\partiallta = (n-1, \dots , 0)$ and
consider $\gamma + \partiallta$. Write $\mathfrak{sl}ort(\gamma + \partiallta)$ for the sequence obtained
by arranging the entries of $\gamma + \partiallta$ in non-increasing order, and define
$\tilde{\gamma} = \mathfrak{sl}ort(\gamma + \partiallta)- \partiallta$.
If $\gamma + \partiallta$ has repeated entries, then
$$H^i(\mathbb{G}r(k,n),\Sigma_{\alpha} \mathcal{Q} \otimes \Sigma_{\beta} \mathcal{R})=0$$ for all $i
\ge 0$. Otherwise,
writing $l$ for the \emph{number of disorders}, that is the number of pairs $(i, j)$ with $1 \le i < j \le n$
and $\gamma_i - i
< \gamma_j - j$ we have$$H^l(\mathbb{G}r(k,n),\Sigma_{\alpha} \mathcal{Q} \otimes
\Sigma_{\beta} \mathcal{R} ) = \Sigma_{\tilde{\gamma}} V$$
and
$H^i(\mathbb{G}r(k,n),\Sigma_{\alpha} \mathcal{Q}\otimes \Sigma_{\beta}\mathcal{R} )=0$ for $i \ne
l$.
Let now $Z \mathfrak{sl}ubset \mathbb{G}r(k,n)$ a variety which is the zero locus of a general section of a rank $r$ globally generated vector bundle $F^{\vee}$. We have the Koszul complex for $Z$, which is indeed a resolution
\begin{equation}\label{koszul} 0 \to \partialt(F) \to \bigwedge^{r-1} F \to \ldots \to F \to \mathcal{O}_G \to \mathcal{O}_Z \to 0. \end{equation}
If $H$ is another globally generated vector bundle on $\mathbb{G}r(k,n)$ we can tensor the above sequence by $H$: we have the spectral sequence
$$\mathbf{E}_1^{-q,p} =H^p(\mathbb{G}r(k,n), H \otimes \bigwedge^q F)\mathbb{R}ightarrow H^{p-q}(Z, H|_Z),$$
if moreover both $F$ and $H$ are homogeneous we can compute all terms on the left by BBW formula. We can now compute the Hodge numbers for our $X$. Notice that the $F$ in the Koszul complex above is the dual of bundle we start with. In this case it will be $\mathcal{Q}(-1)$.
\mathfrak{sl}ubsubsection*{The Hodge numbers $h^{1,i}(\ThetaT(2,9))$}
We apply the above formula together with the conormal sequence, which since $N^{\vee}_{\ThetaT/\mathbb{G}r} \cong F$ becomes
$$ 0 \to F|_{\ThetaT}\to \Omega^1_G|_{\ThetaT} \to \Omega^1_{\ThetaT} \to 0.$$
We can compute the cohomologies of the first two bundles using the above strategy. $F|_X$ turns out to be acyclic, whereas the only non-zero cohomology of $\Omega^1_G|_{\ThetaT}$ is $H^1(\Omega^1_G|_{\ThetaT}) \cong H^1(\Omega^1_G) \cong \mathbb{C}$. It follows that the Hodge numbers $h^{1,i}(\ThetaT)=0$, $i \neq 1$ and $h^{1,1}(\ThetaT)=1$.
\mathfrak{sl}ubsubsection*{The Hodge numbers $h^{2,i}(\ThetaT(2,9))$}
In order to compute these other Hodge numbers we need to rise the conormal sequence to the second exterior power, that is
$$ 0\to \Sym^2 F|_{\ThetaT} \to (F \otimes \Omega^1_G)|_{\ThetaT} \to \Omega^2_{G}|_{\ThetaT} \to \Omega^2_{\ThetaT} \to 0. $$
$\Sym^2 F \otimes \bigwedge^i F$ is acylic for $i\neq 7$. This can be checked using first the Littlewood-Richardson formula to determine the irreducible decomposition of each of these bundles, and then applying several iteration of the BBW formula.
For $i=7$ it is $\Sigma_{3,1^6}\mathcal{Q}\otimes \Sigma_{9,9}\mathcal{R}$ that has
$H^{12}(\Sym^2 F \otimes \bigwedge^7 F) \cong \mathbb{C}$ (and
therefore $H^5(\Sym^2 F|_{\ThetaT}) \cong \mathbb{C}$). The bundle $\Omega^1 \otimes F \otimes \bigwedge^i F$ is acylic for all i.
The bundle $\Omega^2 \otimes \bigwedge^i F$ is not acylic for $i=0$ (and $H^2(\Omega^2_G|_{\ThetaT}) \cong\mathbb{C}^2$) and for $i=3$.
Indeed in the case $i=3$ its decomposition in irreducibles contains the summand
$ \Sigma_{3,3,3,2,2,1,1}\mathcal{Q}\otimes \Sigma_{7,5} \mathcal{R}$.
This gives $H^6(\Omega^2 \otimes \bigwedge^3 F)= \mathbb{C}$.
Putting all these data together one obtains $H^2(\Omega^2_{\ThetaT})=H^3(\Omega^2_{\ThetaT})\cong \mathbb{C}^2$ with the other Hodge $h^{2,i}=0$.
\mathfrak{sl}ubsubsection*{The Hodge numbers $h^{3,i}(\ThetaT(2,9))$}
By Riemann-Roch one gets $\chi(\Omega^3_{\ThetaT})=2$. Thanks to the knowledge of $h^{i,3}(\ThetaT)$ for $i \neq 3,4$, this implies $ h^{3,3}(\ThetaT)=h^{4,3}(\ThetaT).$
We use the third power of the conormal sequence, namely
$$0 \to \Sym^3 F |_{\ThetaT} \to (\Omega^1 \otimes \Sym^2F)|_{\ThetaT} \to (\Omega^2 \otimes
F)|_{\ThetaT} \to \Omega^3_G|_{\ThetaT} \to \Omega^3_{\ThetaT} \to 0.$$
One strategy is to split the sequence above in three short one, namely
\begin{equation}0 \to \Sym^3 F |_{\ThetaT} \to (\Omega^1 \otimes \Sym^2F)|_{\ThetaT} \to J_2 \to 0 ,\end{equation}
\begin{equation} 0 \to J_2 \to (\Omega^2 \otimes F)|_{\ThetaT} \to J_1 \to 0 ,\end{equation}
\begin{equation} \label{finaleq} 0\to J_1 \to \Omega^3_G|_{\ThetaT} \to \Omega^3_{\ThetaT} \to 0. \end{equation}
The only cohomological contributions come from
\begin{enumerate}[(a)]
\item $H^{12}(Sym^ 3F \otimes \bigwedge^6 F)= \mathbb{C}^{81} \cong \mathcal{E}nd(V_9) \cong \mathfrak{gl}(V_9)$;
\item $H^{12}(Sym^3 F \otimes \bigwedge^7 F) = \mathbb{C}^{84} \cong \bigwedge^3 V_9$;
\item $H^{13}(\Omega^1 \otimes Sym^2F \otimes \bigwedge^7 F) = \mathbb{C} \cong H^6 ((\Omega^1 \otimes \Sym^2F)|_{\ThetaT})$;
\item $H^6(\Omega^2 \otimes F \otimes \bigwedge^2 F) = \mathbb{C}\cong H^4((\Omega^2 \otimes F)|_{\ThetaT})$;
\item $H^{10}(\Omega^2 \otimes F \otimes \bigwedge^5 F)= \mathbb{C} \cong H^5((\Omega^2 \otimes F)|_{\ThetaT})$;
\item $H^3(\Omega^3) = \mathbb{C}^2 \cong H^3(\Omega^3_G|_{\ThetaT})$;
\item $H^7(\Omega^3 \otimes \bigwedge^3 F)= \mathbb{C} \cong H^4(\Omega^3_G|_{\ThetaT} )$;
\item $H^{11}(\Omega^3 \otimes \bigwedge^6 F) = \mathbb{C} \cong H^5(\Omega^3_G|_{\ThetaT} )$.
\end{enumerate}
Except in the case of (a) and (b) one can compute immediately the cohomology of the restriction of the bundles to $\ThetaT$. The only non obvious case is given by the exact sequence
$$ 0 \to H^5 (\Sym^3 F |_{\ThetaT} ) \to \bigwedge^3 V \mathfrak{sl}tackrel{\phi_f}{\to} \mathcal{E}nd(V_9) \to H^6 (\Sym^3 F |_{\ThetaT}) \to 0.$$
The situation is analogous to (\cite{klm}, Appendix B). Indeed the dual of the map $\phi_f$ is the map $\varphi_f: \mathcal{E}nd(V_9) \to \bigwedge^3 V_9^{\vee}$ mapping $u \mapsto u(f)$, where $f$ is the defining section for $\ThetaT$ and $u$ is the Lie action. This is because one can do the same computation in family, use the $GL(V)$ equivariance to ensure that $\varphi_f$ depends linearly on $f$. Since up to a scalar there is a unique equivariant map from $\bigwedge^3 V^{\vee}$ to $\Hom(\mathcal{E}nd(V), \bigwedge^3 V^{\vee})$ we can conclude. Therefore for general $f$ the map $\varphi_f$ is injective (this can be verified for example using the general form for $f$ given in (\cite{faenzi}, 4.14) with sufficiently general coefficients and therefore $\phi_f$ is surjective as required.\\
If we plug in these cohomological informations in the long exact sequence associate to the sequence \ref{finaleq} we get several non-zero cohomology groups. In particular the final groups in this sequence are
$$ \ldots \to \mathbb{C} \mathfrak{sl}tackrel{\epsilon}{\to} H^4 (\Omega^3_{\ThetaT}) \mathfrak{sl}tackrel{\mu}{\to} \mathbb{C}^2 \mathfrak{sl}tackrel{\nu}{\to} \mathbb{C} \to 0$$
Therefore $h^{3,3}(\ThetaT)=h^{3,4}(\ThetaT)= \ddim (\mathcal{E}r \mu) + \ddim (\mathrm{Im} \ \mu)$ and by standard properties of long exact sequences $h^{3,3}(\ThetaT)=h^{3,4} \leq 2$. On the other hand by Hard Lefschetz $h^{3,3}(\ThetaT)=h^{3,4} \geq 2$. This concludes the proof of the theorem.
\mathfrak{sl}ubsubsection{Geometry of $\ThetaT(2,9)$ and $X$}
This rather atypical (for our setting) Hodge structure for $\ThetaT(2,9)$ has a geometrical explanation.\\
First consider a linear section $X_H \mathfrak{sl}ubset \mathbb{G}r(3,9)$. It is a Fano 17-fold of index 8. One can compute that its central Hodge structure has level 1, with the same numerology of a genus 2 curve. Consider the configuration in the diagram below.
The map $p: \mathrm{Fl}(2,3,9) \to \mathbb{G}r(3,9)$ is a $\mathbb{P}^2$ bundle, given by the choice of $V_2 \mathfrak{sl}ubset V_3$. It remains as well a $\mathbb{P}^2$ bundle if we restrict $p$ from $X_{p^* H} \to X_H$. The Hodge structure of $X_H \mathfrak{sl}ubset \mathbb{G}r(3,9)$ is therefore repeated three times in $X_{p^* H}$. Consider as well the projection $\phi$ from $\mathrm{Fl}(2,3,9) \cong \mathbb{P}_{\mathbb{G}r(2,9}(\mathcal{Q}(-1))$ to $\mathbb{G}r(2,9)$, that is a $\mathbb{P}^6$-bundle. Restricting $\phi$ to $X_{p^* H}$ this gives a $\mathbb{P}^5$ bundle generically on $\mathbb{G}r(2,9)$, that degenerates on a $\mathbb{P}^6$ on the zero locus $Z_H$ of a section of the dual of $\mathcal{Q}(-1)$, that is $\ThetaT(2,9)$.
\begin{equation} \label{diagrammoneswaggone}\xymatrix{
& F \ar[dl]^\phi & X_{p^* H} \ar[dr]^p \ar[dl]^ \phi \ar@{^{(}->}[r] & \mathrm{Fl}(2,3,9) \ar[dr]^p \\
Z_H \ar@{^{(}->}[r] & \mathbb{G}r(2,9) & & X_H \ar@{^{(}->}[r] & \mathbb{G}r(3,9)
}\end{equation}
One can prove that the Hodge structure of $\ThetaT(2,9)$ can be pushed down from $X_{p^* H}$, which in turn can be calculated from $X_H \mathfrak{sl}ubset \mathbb{G}r(3,9)$. This can be considered as an alternative (and a bit more geometrical) proof of Thm. \ref{t129}. The precise details of this construction and extension to the derived category case will appear in \cite{nested}. In particular a similar argument, albeit in a more complicated version, can be used to derive directly Corollary \ref{cor29} and geometrically explain the K3 structure.
We do not produce here a result interpreting some moduli space on $X$ as an IHS: however we expect a similar result to Proposition \ref{t2} to hold here as well.
\mathfrak{sl}ubsection{S6: $3 \times $K3 structure }
This sporadic Fano has some interesting features. First of all, unlike all our other examples, it is not a section of another Fano by the zero locus of a line bundle. Then it is a Fano of K3 type in two different ways. \\
The variety $\ThetaT(2,10)$ is the zero locus of a general global section of the bundle $\mathcal{Q}^*(1)$ on the Grassmannian $\mathbb{G}r(2,10)$. As in the previous case \textbf{S5} we have $$H^0(\mathbb{G}r(2,10), \mathcal{Q}^*(1)) \cong \bigwedge^3 V_{10}^{\vee},$$ therefore $\ThetaT(2,10)$ is given by the locus of two-spaces in a 10-dimensonal space which are annihilated by a 3-form. It is straightforward to check that $\ThetaT(2,10)$ is a Fano 8-fold of index $\iota=3$. We compute first its Hodge numbers
\begin{proposition} \label{3k3}The Hodge numbers of $\ThetaT(2,10)$ are
\begin{center}
{\mathfrak{sl}mall
\[\begin{matrix}
&&&&&&&&1 &&&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&0 &&1&&0&&&&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&0&&0 & & 0 & &0 && 0 && 0 &&&\\
&&0 && 0 && 1 &&22 &&1 &&0 && 0&&\\
&0 && 0 && 0 && 0&& 0 &&0 &&0 && 0&\\
0 && 0 && 0 && 1&& 23 &&1 &&0 && 0&&0\\
&0 && 0 && 0 && 0&& 0 &&0 &&0 && 0&\\
&&0 && 0 && 1 &&22 &&1 &&0 && 0&&\\
&&&0&&0 & & 0 & &0 && 0 && 0 &&&\\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&&&0 &&1&&0&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&&&1 &&&&&&&&
\end{matrix}\]}
\end{center}
\end{proposition}
As we can see from the above theorem, $\ThetaT(2,10)$ has a Hodge structure of K3 type both in $H^6$ (and therefore in $H^{10}$ by duality) and in $H^8$, making it a rather peculiar example. Indeed by Hard Lefschetz the K3 structure in $H^6$ immediately implies the presence of a K3 sub-structure in $H^{10}$. The surprising bit is that this is the whole of $H^8$, with the exception of a primitive cycle. The computation of the above Hodge numbers is done via a Borel-Bott-Weil computation, as in the previous section. Since these are rather long computations (and not really different from the previous case) we will just sketch it.
\begin{proof} Let $F$ be the dual of the bundle that cuts $\ThetaT$.The computations of the Hodge numbers until $h^{2,i}$ does not present any challenge. In the third exterior power of the conormal exact sequence
$$0 \to \Sym^3 F |_{\ThetaT} \to (\Omega^1 \otimes \Sym^2F)|_{\ThetaT} \to (\Omega^2 \otimes
F)|_{\ThetaT} \to \Omega^3_G|_{\ThetaT} \to \Omega^3_{\ThetaT} \to 0$$
we have that $\Omega^2 \otimes
F)|_{\ThetaT}$ is acylic, for $(\Omega^1 \otimes \Sym^2F)|_{\ThetaT}$ the unique cohomology group is $H^7((\Omega^1 \otimes \Sym^2F)|_{\ThetaT}) \cong \mathbb{C}$ and for the third cotangent we have $H^3( \Omega^3_G|_{\ThetaT}) \cong \mathbb{C}^2$. The only tricky part comes when considering $\Sym^3 F |_{\ThetaT}$. Indeed from the spectral sequence associated to the Koszul resolution for $\Sym^3 F |_{\ThetaT}$ one finds an exact sequence
$$ 0 \to H^{13}(K_7) \to H^{14}( \bigwedge^8 F \otimes \Sym^3 F) \to H^{14}( \bigwedge^7 F \otimes \Sym^3 F) \to H^{14}(K_7) \to 0 $$
where $K_7$ is the sheaf which we used to complete the sequence $0 \to \bigwedge^8 F \otimes \Sym^3 F \to \bigwedge^7 F \otimes \Sym^3 F$. The above sequence is equal to:
$$ 0 \to H^{13}(K_7) \to \bigwedge^3 V_{10} \to \mathcal{E}nd(V_{10}) \to H^{14}(K_7) \to 0 $$
As in the previous section case, one can argue that the middle map is surjective, and therefore chasing the sequence one gets that the unique cohomology group for $\Sym^3 F |_{\ThetaT}$ is $H^6( \Sym^3 F |_{\ThetaT}) \cong \mathbb{C}^{20}$. Collecting all these data together in the above long exact sequence we get $h^{3,3}(\ThetaT)=22$ and $h^{5,3}(\ThetaT)=1$. The missing number can be obtained from the computation of the Euler characteristic.
\end{proof}
This strange Hodge structure can be explained with a construction absolutely equivalent to the one of \eqref{diagrammoneswaggone}, with of course $\mathrm{Fl}(2,3,10)$ being the relevant Flag. In particular, one can repeat the construction of 3.9.2 and do the computations in $K_0(\textrm{Var})$ as an alternative way of computing Hodge numbers. Indeed this is the same Hodge structure coming from the Debarre-Voisin twentyfold $Y_1 \mathfrak{sl}ubset \mathbb{G}r(3,10)$. It is therefore not surprising that we can relate the IHS fourfold $Z_{DV} \mathfrak{sl}ubset \mathbb{G}r(6,10)$ to $\ThetaT(2,10)$. \\
Define first $Z_{\mathcal{O}(1)^4}$ to be the zero locus of four general linear sections in the Grassmannian $\mathbb{G}r(2,6)$. Moreover we denote by $\ThetaT_{,\omega}(2,10)$ a distinguished element of the family defined by a specified 3-form $\omega$.
\begin{proposition} \label{t2}
The Debarre-Voisin fourfold $F_{\omega}$ is birational to the moduli space (contained in the Hilbert scheme) of fourfolds $Z_{\mathcal{O}(1)^4}$ contained in the variety $\ThetaT_{,\omega}(2,10)$.
\end{proposition}
\begin{proof}
Let $W$ be a general point in the Debarre-Voisin fourfold given by a general three form $\omega$. Let us consider the subscheme of $\ThetaT_{,\omega}(2,10)$ given by all two spaces contained inside $W$. This does not coincide with the full Grassmannian $\mathbb{G}r(2,6)$, as the condition $\omega(W)=0$ does not imply $\omega \lrcorner \bigwedge^2 U=0$ for all $U\mathfrak{sl}ubset W$ two-spaces. Notice that this is not the case if one considers three spaces contained in $W$, that is the construction of the Debarre-Voisin IHS fourfold as a moduli space of $\mathbb{G}r(3,6)$ contained in the respective twentyfold.\\
On $\mathbb{G}r(k,10)$ for all $k$ we have a sequence $0\to \mathcal{R} \to V_{10}\otimes\mathcal{O}\to (V_{10}\otimes\mathcal{O})/\mathcal{R}\to 0$ which dually gives a sequence $0 \to \mathcal{R}^\perp \to V_{10}^\vee\otimes\mathcal{O}\to \mathcal{R}^\vee \to 0$. This gives a filtration of $\bigwedge^3 V_{10}^\vee\otimes\mathcal{O}$ with factors $\bigwedge^3 \mathcal{R}^\perp,$ $\bigwedge^2\mathcal{R}^\perp\otimes \mathcal{R}^\vee$, $\mathcal{R}^\perp\otimes \bigwedge^2\mathcal{R}^\vee$ and $\bigwedge^3\mathcal{R}^\vee$. The three-form $\omega$ is a section of the last factor $\bigwedge^3\mathcal{R}^\vee$ on $\mathbb{G}r(6,10)$. On the zero locus of such a section, this lift to a section of $\mathcal{R}^\perp\otimes\bigwedge^2\mathcal{R}^\vee$, which corresponds to a map
$V_{10}/W\rightarrow \bigwedge^2 W^\vee.$
The image of such a map is a four dimensional space $H_4$ of two forms on $W$, for every six space $W$ in the Debarre-Voisin twentyfold given by $\omega$.\\
Let $U\mathfrak{sl}ubset W$ be a point of $\ThetaT_{,\omega}(2,10)$. The space $U$ is isotropic for all two forms in $H_4$, indeed if this were not the case we would have a two form $\mathfrak{sl}igma\in H_4$ such that $\mathfrak{sl}igma_{|U}$ is non degenerate and, by how forms in $H_4$ are obtained, this would imply $\omega \lrcorner \bigwedge^2 U\neq 0$. On the contrary, in an appropriate basis, it is not difficult to show that $\omega \lrcorner \bigwedge^2 U =0$ is implied by $\mathfrak{sl}igma(U)=0$ for all $\mathfrak{sl}igma\in H_4$. Thus, the scheme of subspaces $U\mathfrak{sl}ubset W$ with fixed $W$ is parametrized by a fourfold $Z_{\mathcal{O}(1)^4} \mathfrak{sl}ubset \mathbb{G}r(2,W)$, which a Fano fourfold of index two, rational by \cite[Thm. 2.2.1]{fei}, with central cohomology $(h^{1,1}, h^{2,2})=(1,8)$. This gives a rational map between the Debarre-Voisin fourfold and the space of $Z_{\mathcal{O}(1)^4}$ contained in $\ThetaT(2,10)$ (and in a fixed $\mathbb{G}r(2,6)$). As by changing the point of the Debarre Voisin fourfold we change the ambient Grassmannian $\mathbb{G}r(2,6)$, it is clear that such a map is generically injective, hence birational.
\end{proof}
\mathfrak{sl}ubsection{S7: a mixed (2,3) CY structure}
A curious yet interesting thing happens when we take a linear section $X_H$ of the above $\ThetaT(2,10)$. Indeed by Lefschetz's hyperplane section theorem we know that the K3 structure of $\ThetaT(2,10)$ in $H^6$ and $H^8$ must transfer to its linear section: what is most interesting is that the $H^7$ presents as well a Calabi-Yau structure of level three. To the best of our knowledge, this is the first example of a prime variety that has 2 different examples of CY-structure, of course in different weights. The precise result is
\begin{proposition}\label{23cy} The Hodge numbers of a linear section $X_H \mathfrak{sl}ubset \ThetaT(2,10)$ are
\begin{center}
{\mathfrak{sl}mall
\[\begin{matrix}
&&&&&&&&1 &&&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&0 &&1&&0&&&&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&0&&0 & & 0 & &0 && 0 && 0 &&&\\
&&0 && 0 && 1 &&22 &&1 &&0 && 0&&\\
&0 && 0 && 1 && 44&& 44 &&1&&0 && 0&\\
&&0 && 0 && 1 &&22 &&1 &&0 && 0&&\\
&&&0&&0 & & 0 & &0 && 0 && 0 &&&\\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&&&0 &&1&&0&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&&&1 &&&&&&&&
\end{matrix}\]}
\end{center}
\end{proposition}
The above proposition can be proved with a Borel-Bott-Weil computation similar to the ones above. We will not add further details here in order to preserve the readability of the current paper. We will indeed give a sketch of a geometrical explanation of why such numbers appear.\\
Indeed as an expert reader might notice, the 3CY structure in our $X_H$ has the same dimension of the 3CY structure appearing in the $H^{23}$ of a linear section $X_1 \mathfrak{sl}ubset \mathbb{G}r(3,11)$. We will give now an explanation on why and how this 3CY structure gets projected from such varieties to our $X_H \mathfrak{sl}ubset \ThetaT(2,10)$. This will be only sketched, since the details (in a more general context) will appear in the forthcoming \cite{nested}. The first steps are the following lemmata.
\begin{lemma} A linear section $X_1 \mathfrak{sl}ubset \mathbb{G}r(3,11)$ is a Fano 23-fold of 3CY type. Indeed its non-zero Hodge numbers of weight 23 are $(h^{10,13},h^{11,12}, h^{12,11}, h^{13,10})=(1,44,44,1)$.
\end{lemma}
This lemma can easily be proved, for example using our results in \cite{eg1}. We notice that such a variety is of 3CY even in the (stronger) categorical sense, see \cite[4.5]{kuzicy}. The orthogonal complement to the Calabi-Yau category is generated by 150 exceptional objects. The following Lemma is less obvious
\begin{lemma}\label{symphodge} A linear section $Y_1 \mathfrak{sl}ubset \SGr(3,10)$ is a Fano 17-fold of 3CY type. Indeed its non-zero Hodge numbers of weight 17 are $(h^{7,10},h^{8,9}, h^{9,8}, h^{10,7})=(1,44,44,1)$.
\end{lemma}
It can be proved for example with similar calculations to Corollary \ref{hodges2}, since we already know that the symplectic Grassmannian $\SGr(k, n)$ is a central variety. However one can prove that the statement is more than merely numerological. Indeed one can show the existence of a fully faithful functor $\Phi: D^b(Y_1) \to D^b(X_1)$ and a semiorthogonal decomposition of $D^b(X_1)$ with $\Phi D^b(Y_1)$ as first component, together with a bunch of exceptional objects. This obviously proves the Hodge-theoretical statement as well. This in turn explains the 3CY structure in $X_H \mathfrak{sl}ubset \ThetaT(2,10)$. Indeed it is possible to write a diagram like the one for $\ThetaT(2,9)$ in \ref{diagrammoneswaggone}, appropriately modified; in particular we have to pass through the symplectic partial flag $\mathrm{SFL}(2,3,9)$. The construction is more involved, but it is enough to explain that this mixed (2,3) Calabi-Yau structure ultimately comes from an hyperplane section of (respectively) $\mathbb{G}r(3,10)$ and $\mathbb{G}r(3,11)$. An interesting problem is therefore to look for other examples of varieties with mixed CY structure that are not induced by these constructions tricks outlined in \cite{nested}.
\mathfrak{sl}ubsection{S8: other K3 structures as $X_L \mathfrak{sl}ubset \ThetaT(k,10)$}
A similar construction can be applied to $\ThetaT(4,10)$, $\ThetaT(5,10)$ and their linear sections. Indeed both of them will inherite a bunch of K3 type structure as in \ref{diagrammoneswaggone}. As an example, in the case of $\ThetaT(4,10)$ the diagram will be
\begin{equation}\xymatrix{
& F \ar[dl]^\phi & X_{\pi^* H} \ar[dr]^p \ar[dl]^ \phi \ar@{^{(}->}[r] & \mathrm{Fl}(3,4,10) \ar[dr]^p \\
Z_H \ar@{^{(}->}[r] & \mathbb{G}r(4,10) & & X_H \ar@{^{(}->}[r] & \mathbb{G}r(3,10)
}\end{equation}
The map $p$ is a $\mathbb{P}^6$ bundle, whereas $\phi$ is generically a $\mathbb{P}^2$ bundle specialising to a $\mathbb{P}^3$ bundle over $Z_H$. This suggests that $\ThetaT(4,10)$ should have 7xK3 type structure, and a Borel-Bott-Weil calculation confirms this. A similar construction, albeit more complicated can be performed as well for $\ThetaT(5,10)$, where the fibers of the map on the right hand side of the diagram are $\mathbb{G}r(2,7)$. Moreover on the left side of the diagram there are three type of fibers, corresponding to (generically) smooth hyperplane sections of $\mathbb{G}r(2,5)$, singular sections in codimension 3 and the whole of $\mathbb{G}r(2,5)$ in codimension 10. Of course the linear sections of both $\ThetaT(4,10)$ and $\ThetaT(5,10)$ inherits some structure of K3 type by Lefschetz theorem (depending of course by the dimension of the linear subspace). It is interesting to notice codimensional 1 and 2 linear section will be of mixed CY type, with an argument equivalent to the one of the previous section. Finally we remark that even $\ThetaT(6,10)$ and $\ThetaT(1,10)$ admits structures of K3 type: the first one is nothing but the IHS fourfold of Debarre-Voisin, while a the second one can be used to construct the Peskine variety in $\mathbb{P}^9$, although formally the latter is given by a degeneracy locus. In \cite{nested} we will indeed use this approach to compute the Hodge numbers of this special variety.
\mathfrak{sl}ection*{Appendix A: some extra Fano of 3-CY type}
The methods of this paper can be used to produce Fano of $k$- CY type for every $k$. Another interesting case is when the variety is 3CY. This has been already considered by Iliev and Manivel in \cite{ilievmanivel}. They classified the Fano of 3CY type that can be obtained as linear or quadratic section of homogeneous space, under the additional assumption that the $H^1(T_X)$ was to be isomorphic to one of the Hodge groups of $X$. Many more examples can be found using our method, especially if this condition is not assumed. We do not write the full list here, since we believe it would not fit well with rest of the paper. However it is worthy to point out that many of the examples can be produced as linear sections of symplectic and bisymplectic Grassmannian, with an explanation as in Lemma \ref{symphodge}. \\
Indeed such examples include $ X_1 \mathfrak{sl}ubset \SGr(3,10)$ and $X_1 \mathfrak{sl}ubset \SGr(4,9)$ in the symplectic Grassmannian and $X_1 \mathfrak{sl}ubset \Ml(3,9), X_1 \mathfrak{sl}ubset \Ml(4,9)$ and $X_2 \mathfrak{sl}ubset \Ml(2,6)$ for the bisymplectic. We point out that the Hodge structure of the linear section of $\SGr(3,10)$ and $\Ml(3,9)$ comes from an hyperplane section of $\mathbb{G}r(3,11)$ (which is as well of 3CY type) with an argument similar to Lemma \ref{symphodge} to be fully spelled out in \cite{nested}. A different but not dissimilar argument can be made for $X_2 \mathfrak{sl}ubset \Ml(2,6)$ and explain how this structure of 3CY comes from $X_2 \mathfrak{sl}ubset \mathbb{G}r(2,6)$. In the symplectic Grassmannian we find as well $X_2 \mathfrak{sl}ubset \SGr(4,7)$. In the Orthogonal Grassmannian we find the examples of linear sections of $\OGr(3,9), \OGr(4,9)$ and $\OGr^+(5,10)$. The latter is equivalent to a quadratic section of $\mathbb{S}_{10}$ in the spinor embedding (since the line bundle required is the square root of the Pl\"ucker one). This is already in the list of Iliev and Manivel, so we will not include it.\\
Another interesting example is a section $X_H \mathfrak{sl}ubset \mathrm{SO}(3,8)$ the \emph{ortho-symplectic} Grassmannian. The latter is given by the zero locus of $\bigwedge^2 \mathcal{R}^\vee \oplus \Sym^2 \mathcal{R}^\vee$ on $\mathbb{G}r(3,8)$. We use the notation $X_H$ to point out that, as in the case of Orthogonal Grassmannian $\OGr(3,8)$, $\mathrm{SO}(3,8)$ has Picard rank equal to 2. We checked that there are no other examples of Fano of CY3 type in the orthosymplectic Grassmannian. \\
The cohomology of the orthosymplectic Grassmannian can be computed using a torus action on it (as remarked also in \cite{benphd}, and then Lefschetz's theorem and Borel-Bott-Weil theorem allow us to compute the cohomology of its linear sections in many cases. First, notice that two general symmetric and skew symmetric forms $s,\lambda$ on a space of dimension $2n$ can be put in the following canonical form:
$$s =\mathfrak{sl}um^{2n} s_ix_i^2; \ \ \lambda =\mathfrak{sl}um^{n} x_{2i}\wedge x_{2i+1}.$$
In this way, the stabilizer of these forms contains as maximal torus the group $(\mathbb{C}^*)^n$. In a similar fashion to \cite[Prop 4.2.1]{benphd}, one can prove that this maximal torus has only isolated fixed points (more precisely, $2^k{n\choose k}$) and therefore the cohomology of the orthosymplectic grassmannian is concentrated in the $(p,p)$ part (and the characteristic of the cotangent sheaf and its exterior powers give us the desired cohomology).\\
From this, we obtain the following cohomology for $X_H$:
\begin{center}
{\mathfrak{sl}mall
\[\begin{matrix}
&&&&&&&&1 &&&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&0 &&2&&0&&&&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&0 &&0 && 7 &&0 && 0 &&&&\\
&&&0&&1 & & 45 & &45 && 1 && 0 &&&\\
&&&&0 &&0 && 7 &&0 && 0 &&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&&&0 &&2&&0&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&&&1 &&&&&&&&
\end{matrix}\]}
\end{center}
We point out that $X_H \mathfrak{sl}ubset \mathrm{SO}(3,8), \ X_2 \mathfrak{sl}ubset \SGr(4,7)$ and $X_2 \mathfrak{sl}ubset \Ml(2,6)$ are particularly interesting as Fano of 3CY type, since they are of dimension 5 (the minimal possible) and therefore relevant for testing a modified version of Kuznetsov's conjecture on rationality and derived categories. We collect in the next table the 3CY structure mentioned in the above discussion. We mention that in analogy with the K3 case, $\ThetaT(2,11)$ can be considered as well as an example of 3$\times$CY structure. Taking other $k$ and appropriate number of linear sections is possible as well to produce examples of \emph{mixed} $(3,j)$ CY structure. However we do not include them in the following table. Of course more examples could be found by considering products and such as in the FK3 case, but we decided to not consider them here in order to keep the length of this paper within an acceptable limit. We of course do not include the examples already considered in \cite{ilievmanivel}.
\begin{center}
\begin{tabular}{@{} *9l @{}l @{}l @{}l @{}l @{}l @{}} \toprule
Type &dim. &$\iota_X$& $h^{n-1/2,n+1/2}$ \\ \midrule
$X_1 \mathfrak{sl}ubset \OGr(3,9)$ & 11 &4& 49\\
$X_1 \mathfrak{sl}ubset \OGr(4,9)$ & 9 &3&70\\
$X_1 \mathfrak{sl}ubset \SGr(3,10)$ & 17 &7&44 \\
$X_1 \mathfrak{sl}ubset \SGr(4,9)$ & 13 &5&45 \\
$X_1 \mathfrak{sl}ubset \Ml(3,9)$ & 11 &4& 44\\
$X_1 \mathfrak{sl}ubset \Ml(4,9)$ & 7 &2&45 \\
$X_2 \mathfrak{sl}ubset \Ml(2,6)$ & 5 &2&67 \\
$X_H \mathfrak{sl}ubset \mathrm{SO}(3,8)$ & 5 &1& 45 \\
$X_2 \mathfrak{sl}ubset \SGr(4,7)$ & 5 &2& 72 \\
\bottomrule
\mathcal{H}line
\end{tabular}
\captionof{table}{3CY structure in $\OGr, \SGr, \Ml$ and $\mathrm{SO}$} \label{table3}
\end{center}
\mathfrak{sl}ection*{Appendix B: infinite CY series}
During our search we identified some interesting class of varieties. Although not directly related to the main story of this paper, they have some interesting features that is worth to underline. As an example we identified some interesting infinite families of varieties (of every dimension) with trivial canonical bundle obtained using the same bundles in different Grassmannians. We checked that these varieties are actually Calabi-Yau for low dimension (up to 6). We expect them to be always like this. \\
We describe now these series of varieties, according to the type of bundles involved.
\begin{align*} &A(k,l):= \mathcal{Q}(1) \oplus \bigwedge^2\mathcal{R}^{\vee} \textrm{ on } \mathbb{G}r(k, k+l);\\
&B(k,l):= \mathcal{Q}^{\vee}(1) \oplus \Sym^2 \mathcal{R}^{\vee}\textrm{ on } \mathbb{G}r(k, k+l);\\
&C(k, k+1):=\Sym^2 \mathcal{R}^{\vee}\oplus \bigwedge^2\mathcal{R}^{\vee}\oplus\mathcal{O}(1) \textrm{ on } \mathbb{G}r(k,2k+1).
\end{align*}
$A(k,l)$ has dimension $l(k-1)-{k \choose 2}$, $B(k,l)$ has dimension $l(k-1)-{k+1 \choose 2}$ and $C(k)$ has dimension $k-1$. Notice that $A(k,l)$ can naturally be seen as $Z_{\mathcal{Q}(1)} \mathfrak{sl}ubset \SGr(k,k+l)$, $B(k,l)$ as $Z_{\mathcal{Q}^{\vee}(1)} \mathfrak{sl}ubset \OGr(k, k+l$) and $C$ is a linear section of the ortho-symplectic Grassmannian. In particular, as in \cite{kuznetsovc5} in the case of bisymplectic Grassmannian, one can prove that \[ C(k,k+1) \cong X_{(2, \ldots, 2)} \mathfrak{sl}ubset (\mathbb{P}^1)^k.\]\\
When $k=2$, the zero locus of a general global section of $A(2,l)$ is indeed deformation of a complete intersection given by $(\mathcal{O}(1))^{l+3} $ on $\mathbb{G}r(2, l+3)$. Indeed first notice that on $\mathbb{G}r(2, l+3)$ we have $(\mathcal{O}(1))^{l+3} \cong \mathcal{Q}(1) \oplus \mathcal{R}(1)\cong \mathcal{Q}(1) \oplus \mathcal{R}^{\vee} $. Then notice that the zero locus of a general global section of $ \mathcal{Q}(1) \oplus \mathcal{R}^{\vee}$ on $\mathbb{G}r(2, l+3)$ is isomorphic to the zero locus of a general global section of $\mathcal{Q}(1) \oplus \mathcal{O}(1)$ on $\mathbb{G}r(2, l+2)$. This follows from the standard fact that $Z_{\mathcal{R}^{\vee}} \mathfrak{sl}ubset \mathbb{G}r(k, n) \cong \mathbb{G}r(k, n-1)$ and under the following isomorphism $\mathcal{Q}(1)_{k,n}$ projects to $\mathcal{Q}(1)_{k,n-1} \oplus \mathcal{O}(1)$ .\\
For dimension $d=2,3,4$ we refer to \cite{benedetti}, \cite{inoue2016complete}. For $d=5,6$ the Calabi-Yaus in the series $A$ and $B$ are the following. We do not include $B(5,5)$, since it can bee seen as a deformation of the double spinor variety studied by Manivel in \cite{manivelspinor}. The Hodge numbers are computed in Proposition 3.3, together with the fact that this family is locally complete. Since $C$ is indeed a well-known class of variety in disguise, we will not compute the Hodge numbers for the first values of the series. In the following list of invariants we do not include either trivially known Hodge numbers such as $h^{0, n}$. Moreover the number not listed are always $0$.
\begin{center}
\begin{tabular}{@{} *9l @{}l @{}l @{}l @{}l @{}l @{}} \toprule
dim. & Type &$h^{1,1}$& $h^{2,2}$&$h^{4,1}$&$h^{3,2}$\\ \midrule
5 & $A(2,6)$ & 1&2& 163 & 1784 \\
5 & $A(3,4)$ & 1&2& 148&1619 \\
5 & $B(4,5)$ & 1&2& 165&1806 \\
\bottomrule
\mathcal{H}line
\end{tabular}
\captionof{table}{First values of infinite series for fivefolds} \label{table5}
\end{center}
\begin{center}
\begin{tabular}{@{} *9l @{}l @{}l @{}l @{}l @{}l @{}} \toprule
dim. & Type &$h^{1,1}$& $h^{2,2}$&$h^{5,1}$&$h^{4,2}$&$h^{3,3}$\\ \midrule
6 & $A(2,7)$ & 1&2& 251&5202&14004 \\
6 & $A(4,4)$ & 1&1& 251&5181&13960\\
6 & $B(2,9)$ & 1&2& 120 &2254&6274\\
6 & $B(3,6)$ & 1&2& 125&2380&6596 \\
\bottomrule
\mathcal{H}line
\end{tabular}
\captionof{table}{First values of infinite series for sixfolds} \label{table4}
\end{center}
An interesting question, which however falls beyond the scope of this paper, is to investigate whether the varieties constructed in this way are generic in moduli, that is whether all of their deformations are embedded in the same Grassmannian. This can be done by a direct computation of $h^1(TG|_X)$ using Koszul complex and Borel-Bott-Weil theorem, however these calculations are quite demanding in each specific case, and a general argument is out of reach.
\mathfrak{sl}ection*{Appendix C: fake K3 structure}
The numerical condition in \eqref{condition} restricted most of our search to vector bundles in which one of the irreducible summand is linear. One can of course try to rearrange this condition in order to eliminate the constraint. Indeed this is geometrically meaningful, as for example $\ThetaT(2,10)$ shows (it is a zero locus of an indecomposable bundle that is non-linear, with slope $\mu=c_1(E)/r(E)=7/8$). It is possible, and we plan to do so, to fully investigate this case. \\
During a preliminary search we found this example, the zero locus $X_{\mathcal{R}^\vee(1)} \mathfrak{sl}ubset \mathbb{G}r(2,6)$. It is a sixfold of index 3, defined by a bundle of slope $\mu=3/2$, satisfying all our preliminary numerological condition. Although it is not of K3 type, it is rather curious, and we decided to add it anyway. Indeed its Hodge numbers are
\begin{center}
{\mathfrak{sl}mall
\[\begin{matrix}
&&&&&&&&1 &&&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&0 &&1&&0&&&&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&0&&0 & & 0 & &0 && 0 && 0 &&&\\
&&0 && 0 && 0 &&22 &&0 &&0 && 0&&\\
&&&0&&0 & & 0 & &0 && 0 && 0 &&&\\
&&&&0 &&0 && 2 &&0 && 0 &&&&\\
&&&&& 0 && 0 && 0 &&0&&& \\
&&&&&&0 &&1&&0&&&&&&\\
&&&&&&&0&&0&&&&&&&\\
&&&&&&&&1 &&&&&&&&
\end{matrix}\]}
\end{center}
The absence of the 1 in $h^{4,2}$ is explained by a Borel-Bott-Weil computation, since an inconvenient cancellation in the spectral sequence occurs. It is possible that some higher-dimensional analogue of this \textit{false positive} may occur, although we expect this to be quite an exception and not the general rule.
\end{document} |
\betagin{document}
\title
[On bodies floating in equilibrium in every orientation ]
{On bodies floating in equilibrium in every orientation}
\author[D. Ryabogin]{Dmitry Ryabogin}
\address{Department of Mathematics, Kent State University,
Kent, OH 44242, USA} \varepsilonmail{[email protected]}
\thanks{The author is supported in
part by Simons Collaboration Grant for Mathematicians program 638576,
by U.S.~National Science Foundation Grant
DMS-1600753 and by United States - Israel Binational Science Foundation (BSF)}
\keywords{Floating bodies, Ulam's problem, normal curvature}
\betagin{abstract}
Ulam's problem 19 from the Scottish Book asks: {\it is a solid of uniform density which floats in water in every position necessarily a sphere?} We obtain several results related to this problem.
\varepsilonnd{abstract}
\maketitle
\section{Introduction}
Let the density of water be $1$ and assume that a convex body $K\subset {\mathbb R^3}$ of uniform density ${\mathcal D}\in(0,1)$ is submerged into water. We say that $K$ floats in equilibrium in the direction $\xi$ orthogonal to the water surface if the line $\varepsilonll(\xi)$ connecting the center of mass of $K$ and the center of mass of the submerged part is parallel to $\xi$. We say that
$K$ floats in equilibrium in every orientation if $\varepsilonll(\xi)$ is parallel to $\xi$ for every $\xi$.
The following intriguing problem was proposed by Ulam C^{\infty}te[Problem 19]{U}: {\it If a convex body $K\subset {\mathbb R^3}$ made of material of uniform density ${\mathcal D}\in(0,1)$ floats in equilibrium in any orientation in water, must $K$ be spherical? }
Schneider C^{\infty}te{Sch1} and Falconer C^{\infty}te{Fa} showed that this is true, provided $K$ is centrally symmetric and ${\mathcal D}=\fracrac{1}{2}$. However, it has been recently proven in C^{\infty}te{R2} that there are non-centrally-symmetric convex bodies of density ${\mathcal D}=\fracrac{1}{2}$ that float in equilibrium in every orientation.
The ``two-dimensional version" of the problem is also very interesting. In this case, we consider floating logs of uniform cross-section, and seek for the ones that will float in every orientation with the axis horizontal. In other words, our cross-section $K$ is a convex set in ${\mathbb R^2}$ and the water surface is a line that cuts off a set of the given area from $K$.
If ${\mathcal D}=\fracrac{1}{2}$, Auerbach C^{\infty}te{A} has exhibited logs with non-circular cross-section, both convex and non-convex, whose boundaries are so-called Zindler curves C^{\infty}te{Zi}.
More recently, Bracho, Montejano and Oliveros C^{\infty}te{BMO} showed that
for densities ${\mathcal D}=\fracrac{1}{3}$, $\fracrac{1}{4}$, $\fracrac{1}{5}$ and $\fracrac{2}{5}$ the answer is affirmative, while Wegner
proved that for some other values of ${\mathcal D}\neq \fracrac{1}{2}$ the answer is negative, C^{\infty}te{Weg1}, C^{\infty}te{Weg2}; see also related results of V\'arkonyi C^{\infty}te{V1}, C^{\infty}te{V2}. Overall, the case of general ${\mathcal D}\in (0,1)$ is notably involved and widely open.
No results in ${\mathbb R^3}$ are known for densities ${\mathcal D}\in (0,1)$ different from $\fracrac{1}{2}$ and no counterexamples have been found so far.
In this paper we prove and recall several results which were used in the case of density $\fracrac{1}{2}$, C^{\infty}te{R2}, and which, we believe, would help to attack the problem for other densities.
We begin with
\begin{theorem}\lambdabel{Dpr}
Let $d\ge 3$, let $K\subset {\mathbb R^d}$ be a convex body and let $\deltalta\in (0,\textnormal{vol}_d(K))$.
If $K$ floats in equilibrium at the level $\deltalta$ in every orientation, then, for all hyperplanes $H$ that cut off the parts of volume $\deltalta$ from $K$, the cutting sections $K\cap H$ have equal moments of inertia with respect to all $(d-2)$-dimensional planes $\Pi\subset H$ passing through the center of mass of $K\cap H$ and these moments are independent of $H$ and $\Pi$.
Conversely, let $K$ have a $C^1$-smooth boundary and let the center of mass of $K$ coincide with the center of mass of the surface of centers, i.e., the locus of the centers of mass of all parts of volume $\deltalta$ that are cut off by the cutting hyperplanes $H$. If all cutting sections $K\cap H$ have equal moments of inertia with respect to all $(d-2)$-dimensional planes $\Pi\subset H$ passing through the center of mass of $K\cap H$ and these moments are independent of $H$ and $\Pi$, then $K$ floats in equilibrium at the level $\deltalta$ in every orientation.
\varepsilont
This Theorem \fracootnote{This result was also recently obtained in C^{\infty}te[Theorem 1.1]{FSWZ}, but the case $\deltalta= \fracrac{\textrm{vol}_d(K)}{2}$
is considered under the assumption that the Dupin floating body coincides with the B\'ar\'any-Larman-Sh\"utt-Werner floating body and it
is a single point.} gives an affirmative answer to a question mentioned in C^{\infty}te[page 20, line 14 from below]{CFG}: ``It seems that the floating body problem is just (V, I)".
An analogous Theorem for $d=2$ was obtained by Davidov C^{\infty}te{Da} and independently by Auerbach C^{\infty}te{A}, see Theorem
\ref{Dakr} and Remark \ref{nun} at the end of Section \ref{Au}.
\begin{corollary}\lambdabel{equi}
Let $d\ge 3$,
let a convex body $K$ have a $C^1$-smooth boundary and let
$\deltalta\in (0, \textnormal{vol}_d(K))$.
Assume also that the center of mass of $K$ coincides with the center of mass of the surface of centers.
If for every hyperplane $H$ that cuts off the part of volume $\deltalta$ from $K$
every cutting section $K\cap H$ is $(d+1)$-equichordal, i.e., if there exists a constant $c$ such that for every line $l\subset
K\cap H$ passing through the center of mass ${\mathcal C}(K\cap H)$ and having two points of intersection $\zetata_{\partialm}(l)$ with the boundary of $K$ one has
$$
\textnormal{dist}^{d+1}({\mathcal C}(K\cap H),\zetata_{+}(l))+\textnormal{dist}^{d+1}({\mathcal C}(K\cap H),\zetata_{-}(l))=c,
$$
then $K$ floats in equilibrium in every orientation.
\varepsilonc
Using the results in C^{\infty}te{R1} and C^{\infty}te{R2} one can show that
the converse is not true, provided $\deltalta=\fracrac{\textnormal{vol}_d(K)}{2}$, i.e., there exists a non-centrally-symmetric body of revolution $K$ that floats in equilibrium in every orientation, yet not every section $K\cap H$ by the hyperplane that cuts off the part of volume $\deltalta$ is $(d+1)$-equichordal. On the other hand, it was proved in C^{\infty}te{R1} that if $K$ is a body of revolution, then the condition
that $K\cap H$ is $(d+1)$-equichordal for every hyperplane $H$ that cuts off the part of volume $\deltalta$ from
$K$ yields that it is the Euclidean ball.
\begin{proof}rob\lambdabel{abdul1}
Is it possible to construct a convex body $K$ and find $\deltalta\in (0, \textrm{vol}_d(K))$, $\deltalta\neq \fracrac{\textnormal{vol}_d(K)}{2}$,
so that $K\cap H$ is $(d+1)$-equichordal for every hyperplane $H$ that cuts off the part of volume $\deltalta$ from $K$, but
$K$ is not an Euclidean ball?
\varepsilonprob
We refer the reader to C^{\infty}te[pgs. 9-11]{CFG}, C^{\infty}te[Chapter 6]{Ga} and references therein for the information about equichordal bodies.
We also have
\begin{corollary}\lambdabel{Al}
Let $d\ge 2$ and let a sequence $(\deltalta_n)_{n=1}^{\infty}$ of positive numbers be such that the Dupin floating body $K_{[{\deltalta_n}]}$ coincides with the floating body $K_{\deltalta_n}$ for all $n\in {\mathbb N}$ and $\deltalta_n\to 0$ as $n\to\infty$.
If $K$ floats in equilibrium in every orientation for all levels $\deltalta_n$, then $K$ is a Euclidean ball.
\varepsilonc
Using Theorem \ref{Dpr} and the results of Myroshnychenko and Saroglou C^{\infty}te{MRS}, one can also give a different proof \fracootnote{See C^{\infty}te[Theorem 1.2]{FSWZ} for a third proof of this statement.} of the aforementioned result of Schneider and Falconer obtained in C^{\infty}te{Sch1} and C^{\infty}te{Fa} via spherical harmonics.
\begin{theorem}\lambdabel{CS}
Let $d\ge 3$ and let $K\subset {\mathbb R^d}$ be a centrally-symmetric convex body.
If $K$ floats in equilibrium in every orientation at the level $\deltalta=\fracrac{\textnormal{vol}_d(K)}{2}$
then
$K$ is a Euclidean ball.
\varepsilont
Most of the results of this paper, as well as many other results on floating bodies, follow from the classical theorems of Dupin which, we believe, were missed by the mathematical community, C^{\infty}te[Chapter XXIV]{DVP}, C^{\infty}te[Hydrostatics, Part I]{Zh}). In Sections \ref{El} and \ref{DP} we formulate and prove these theorems in ${\mathbb R^d}$, $d\ge 3$ (see also C^{\infty}te[Appendices A and B]{R2}).
We refer the interested reader to C^{\infty}te[pgs. 90-93]{M}, C^{\infty}te[pgs. 19-20]{CFG}, C^{\infty}te[pgs. 376-377]{Ga}, C^{\infty}te[pgs. 560-563]{Sch2}, and C^{\infty}te{G}, for an exposition of known results related to Ulam's Problem 19; see also C^{\infty}te{O}, C^{\infty}te{Od}, C^{\infty}te{HSW}, C^{\infty}te{KO}, C^{\infty}te{Gr} and C^{\infty}te{Mo} for related results.
The {\it floating body problems} appear in several areas of mathematics and, among other things, are related to the Busemann-Petty problems in asymptotic geometric analysis C^{\infty}te{BP}, to problems in statistics C^{\infty}te{NSW}, and to
problems about polytopal approximation, C^{\infty}te{B}, C^{\infty}te{BL}, C^{\infty}te{S2},
C^{\infty}te{BLW}.
We also refer the reader to C^{\infty}te{MR}, C^{\infty}te{St}, C^{\infty}te{S1}, C^{\infty}te{SW1}, C^{\infty}te{SW2}, C^{\infty}te{W}, and references therein for other works on floating bodies.
The paper is structured as follows. In the next section we recall some well-known facts about floating bodies and formulate the Theorems of Dupin
in ${\mathbb R^d}$, $d\ge 3$. We prove these theorems in Section \ref{DP}. The proofs of Lemma \ref{tr}, Theorems \ref{Dpr} and \ref{CS}, and Corollaries \ref{equi} and \ref{Al} are given in Section \ref{Au}.
\section{Notation, basic definitions and Theorems of Dupin}\lambdabel{El}
\subsection{Notation and basic definitions}
A convex body $K\subset {\mathbb R^d}$, $d\ge 2$, is a convex compact set with a non-empty interior $\textrm{int} K$.
The boundary of $K$ is denoted by
$\partialartial K$. We say that $K$ is strictly convex
if $\partialartial K$ does not contain a segment.
We say that $K$ is origin-symmetric if $K=-K$ and centrally-symmetric if there exists $p\in{\mathbb R^d}$ such that $K-p=\{q-p:\,q\in K\}$ is origin-symmetric.
For $d\ge 2$ we denote by $S^{d-1}$ the unit sphere in ${\mathbb R^d}$ centered at the origin. Given $\xi\in S^{d-1}$ we denote by
$\xi^{\partialerp}=\{p\in{\mathbb R^d}:\, p\mathcal Dot \xi=0\}$ the subspace orthogonal to $\xi$, where $p\mathcal Dot\xi=p_1\xi_1+\dots +p_d\xi_d$ is a usual inner product in ${\mathbb R^d}$.
The symbol $``+"$ stands for the usual Minkowski (vector) addition, i.e., given two sets $D$ and $E$ in ${\mathbb R^d}$, $D+E=\{d+e:\,d\in D,\,e\in E \}$.
Let $W_j$ be a $j$-dimensional plane in ${\mathbb R^d}$, $1\le j\le d$.
The {\it center of mass} of a compact convex set $K\subset W_j$ with a non-empty relative interior will be denoted by ${\mathcal C}(K)$,
$$
{\mathcal C}(K)=\fracrac{1}{\textrm{vol}_j(K)}\intL^{\infty}mits_{K}xdx,
$$
where $\textrm{vol}_j(K)$ is the $j$-dimensional volume of $K$ in ${\mathbb R^j}$.
We say that a hyperplane $H$ is the supporting hyperplane of a convex body $K$ if $K\cap H\neq \varepsilonmptyset$, but $\textrm{int}\,K\cap H=\varepsilonmptyset$.
If $K\subset {\mathbb R^d}$ is a convex body containing a point $p$ in its interior, the {\it radial function} of $K$ with respect to $p$ in the direction $\theta\in S^{d-1}$ is defined as
$$
\rho_{K,\,p}(\theta)=\max\{\lambdambda>0:\,p+\lambdambda \theta\in K\}.
$$
In particular, if $p$ is the origin, we will use the notation $$\rho_{K}(\theta)=\max\{\lambdambda>0:\lambdambda \theta\in K\}.
$$
Let $m\in{\mathbb N}$. We say that a convex body $K$ is of class $C^m({\mathbb R^d})$ (or $K$ has a $C^m$-smooth boundary) if for every point $z$ on the boundary $\partialartial K$ of $K\subset {\mathbb R^d}$ there exists a neighborhood $U_z$ of $z$ in ${\mathbb R^d}$ such that $\partialartial K\cap U_z$ can be written as a graph of a function having
all continuous partial derivatives up to the $m$-th order.
The {\it Hausdorff distance} between two convex bodies $K$ and $L$ is defined as
$$
d(K,L)=\supL^{\infty}mits_{\{\theta\in S^{d-1}\}}|h_K(\theta)-h_L(\theta)|,
$$
where
$h_K$, $h_L$ are the {\it support functions} of bodies $K$, $L$, and for any $\theta\in S^{d-1}$,
$h_K(\theta)=\supL^{\infty}mits_{\{y\in K \}}\theta\mathcal Dot y$.
A symbol $\,\square\,$ denotes end of the proof.
We recall several well-known facts and definitions.
Let $d\ge 3$, let $K\subset {\mathbb R^d}$ be a convex body
and let $\deltalta\in (0, \textrm{vol}_d(K))$ be fixed. Given a direction $\xi\in S^{d-1}$ and $t=t(\xi)\in{\mathbb R}$, we call a hyperplane
\betagin{equation}\lambdabel{p2}
H(\xi)=H_t(\xi)=\{p\in{\mathbb R^{d}}:\,p\mathcal Dot\xi=t\},
\varepsilonnd{equation}
the {\it cutting hyperplane} of $K$ in the direction $\xi$, if it cuts out of $K$ the given volume $\deltalta$,
i.e., if
\betagin{equation}\lambdabel{fubu}
\textrm{vol}_d(K\cap H^-(\xi))=\deltalta,\quadquad H^-(\xi)=\{p\in{\mathbb R^{d}}:\,p\mathcal Dot\xi\le t(\xi)\},
\varepsilonnd{equation}
(see Figure \ref{width1}).
\betagin{figure}[h]
\mathcal Entering
\includegraphics[height=2.8in]{ImageH2.pdf}
\caption{A body $K$ and its submerged part $K\cap H^-(\xi)$}
\lambdabel{width1}
\varepsilonnd{figure}
Now we recall the notions of {\it floating in equilibrium} and the {\it surface of centers}, C^{\infty}te{DVP}, C^{\infty}te{Zh}.
\begin{definition}\lambdabel{efb}
Let $\xi\in S^{d-1}$ and let ${\mathcal C}(\xi)={\mathcal C}_{\deltalta}(\xi)$ be the center of mass of the submerged part $K\cap H^-(\xi)$ satisfying (\ref{fubu}).
A convex body $K$ {\it floats in equilibrium in the direction $\xi\in S^{d-1}$ at the level $\deltalta$} if (\ref{fubu}) holds and the line $l(\xi)$ connecting ${\mathcal C}(K)$ with ${\mathcal C}_{\deltalta}(\xi)$ is orthogonal to the ``free water surface" $H(\xi)$, i.e., the line $l(\xi)$ is ``vertical" $\textnormal{(}$parallel to $\xi$, see Figure \ref{width1}$\textnormal{)}$. We say that $K$ floats in equilibrium in every orientation at the level $\deltalta$ if $l(\xi)$ is parallel to $\xi$ for every $\xi\in S^{d-1}$.
\varepsilond
\begin{definition}\lambdabel{scn}
Let $K$ be a convex body, let $\xi\in S^{d-1}$ and let ${\mathcal C}(\xi)={\mathcal C}_{\deltalta}(\xi)$ be the center of mass of the submerged part $K\cap H^-(\xi)$ satisfying (\ref{fubu}). The geometric locus $\{{\mathcal C}_{\deltalta}(\xi):\,\xi\in S^{d-1}\}$ is called the {\it surface of centers} ${\mathcal S}={\mathcal S}_{\deltalta}$ or the {\it surface of buoyancy} $\textnormal{(}$see Figure \ref{width2}$\textnormal{)}$.
\varepsilond
One can show, see Theorem \ref{D1} below, that the surface of centers is a boundary of a strictly convex body.
\begin{remark}\lambdabel{ew}
It was recently proved in C^{\infty}te{HSW} that the surface of centers ${\mathcal S}$ is $C^{k+1}$-smooth, provided $K$ is of class $C^k$, $k\ge 0$.
In particular, if $K$ is an arbitrary convex body, then ${\mathcal S}$ is $C^1$-smooth.
\varepsilonr
The following result is well-known, see
C^{\infty}te[page 203]{G}, C^{\infty}te[Section 2.1]{V1} and
C^{\infty}te[Corollary 2.4]{HSW}.
In the next section we give a different proof.
\begin{lemma}\lambdabel{tr}
Let $d\ge 2$, let $K$ be a convex body and let $\deltalta\in (0, \textnormal{vol}_d(K))$.
If $K$
floats in equilibrium in every orientation at the level $\deltalta$, then the surface of centers ${\mathcal S}$ is a sphere. Conversely, if ${\mathcal S}$ is a sphere centered at ${\mathcal C}(K)$, then $K$ floats in equilibrium in every orientation.
\varepsilonl
It is known that the condition of ${\mathcal S}$ being centered at ${\mathcal C}(K)$ is satisfied for
$\deltalta=\fracrac{\textnormal{vol}(K)}{2}$ (${\mathcal C}(K)$ is an arithmetic average of ${\mathcal C}(K\cap H^+(\xi))$ and ${\mathcal C}(K\cap H^-(\xi))$ for every $\xi\in S^{d-1}$), and for any
$\deltalta\in (0, \textrm{vol}_d(K))$, provided $K$ is centrally-symmetric.
Now we recall the notion of a {\it floating body}.
A floating body $K_{[\deltalta]}$ of $K$ was introduced by C. Dupin in 1822, C^{\infty}te{D}.
\begin{definition}\lambdabel{dupa}
A non-empty convex set $K_{[\deltalta]}$ is the Dupin floating body of $K$ if each supporting plane of $K_{[\deltalta]}$ cuts off a set of volume $\deltalta\in (0, \textnormal{vol}_d(K))$ from $K$.
\varepsilond
We remark that $K_{[\deltalta]}$ does not necessarily exist for every convex $K$, see C^{\infty}te{L} or C^{\infty}te[Chapter 5]{NSW}, but
if $K$ has a sufficiently smooth boundary and $\deltalta>0$ is small enough, then $K_{[\deltalta]}$ exists, C^{\infty}te[Satz 2]{L}.
The notion of a {\it convex floating body} was introduced independently in C^{\infty}te{BL} and C^{\infty}te{SW1}.
\begin{definition}\lambdabel{blsw}
A body $K_{\deltalta}$ is called the convex
floating body of $K$, provided
$$
K_{\deltalta}=\bigcapL^{\infty}mits_{\{\xi\in S^{d-1}\}}H^+(\xi),\quadquad H^+(\xi)=\{p\in{\mathbb R^d}:\,p\mathcal Dot\xi\ge t(\xi)\}.
$$
\varepsilond
If $K_{[\deltalta]}$ exists, then $K_{[\deltalta]}=K_{\deltalta}$;
$K_{\deltalta}$ is allowed to be an empty set, C^{\infty}te{SW1}.
It was proved in C^{\infty}te[Theorem 3, page 334]{MR} that $K_{[\deltalta]}=K_{\deltalta}$ for any
$0<\deltalta\le\fracrac{\textrm{vol}_d(K_{\deltalta})}{2}$,
provided $K$ is centrally-symmetric.
It was also shown in C^{\infty}te{MR} that the boundary of $K_{\deltalta}$ is $C^2$-smooth, provided the boundary of $K$ is $C^1$-smooth and
for every $x$ on the boundary of $K$ there is a unique supporting hyperplane
of $K$ through $x$.
Let $K$ float in equilibrium in every orientation for some $\deltalta\in (0,\textrm{vol}_d(K))$, $\deltalta\neq \fracrac{\textrm{vol}_d(K)}{2}$. It is not clear if the additional condition
$K_{[{\deltalta}]}=K_{\deltalta}$
yields an affirmative answer to Ulam's Problem 19.
\subsection{Theorems of Dupin}
The solution of the problem of finding the directions in which the given convex body floats in equilibrium is contained in the following three results, proved by Dupin, (cf. C^{\infty}te[pgs. 658-660]{Zh} and C^{\infty}te{Da} for $d=2$, and C^{\infty}te[pgs. 287-288]{DVP} for $d=3$; see also C^{\infty}te{G}).
For convenience of the reader, in this section we formulate these theorems for all $d\ge 3$ and include sketches of the proofs in the next section.
\betagin{figure}[h]
\mathcal Entering
\includegraphics[height=1.5in]{image0.jpeg}
\caption{Floating body $K_{\deltalta}$ and surface of centers ${\mathcal S}$}
\lambdabel{width2}
\varepsilonnd{figure}
Let $\xi\in S^{d-1}$ and let ${\mathcal H}(\xi)$ be a tangent hyperplane to ${\mathcal S}$ at ${\mathcal C}(\xi)$ which is the center of mass of $K\cap H^-(\xi)$, see Remark \ref{ew}.
The First Theorem of Dupin reads as follows.
\begin{theorem}\lambdabel{D1}
Let $d\ge 2$, $K\subset {\mathbb R^d}$ be convex, and let $\deltalta\in (0,\textnormal{vol}_d(K))$. If $H(\xi)$, $\xi\in S^{d-1}$, is a cutting hyperplane, then ${\mathcal H}(\xi)$
is parallel to
$H(\xi)$. Moreover,
the bounded set $L({\mathcal S})$ with boundary ${\mathcal S}$ is a strictly convex body.
\varepsilont
The Second Theorem of Dupin is
\begin{theorem}\lambdabel{D2}
Let $d\ge 2$, $K\subset {\mathbb R^d}$ be convex, and let $\deltalta\in (0,\textnormal{vol}_d(K))$.
Assume that $H(\xi)$, $\xi\in S^{d-1}$, is a cutting hyperplane and
$\{H_n\}_{n=1}^{\infty}$, $H_n=H(\xi_n)$, is any sequence of cutting hyperplanes converging to $H(\xi)$ as $\xi_n\to\xi$ for $n\to\infty$
and such that the limit $L^{\infty}mL^{\infty}mits_{n\to\infty}H(\xi)\cap H(\xi_n)$ exists.
Then the $(d-2)$-dimensional plane $\Pi=L^{\infty}mL^{\infty}mits_{n\to\infty}H(\xi)\cap H(\xi_n)$ passes through the center of mass of $K\cap H(\xi)$.
\varepsilont
\betagin{figure}[h]
\mathcal Entering
\includegraphics[height=3in]{Meta.pdf}
\caption{The metacenter $M=l(\xi)\cap l(\varepsilonta)$ of $K$}
\lambdabel{meta}
\varepsilonnd{figure}
In order to formulate the third Theorem of Dupin in the case $d\ge 3$, we recall the notions of a {\it metacenter} C^{\infty}te[page 284]{DVP} and of a {\it moment of inertia} C^{\infty}te[page 553]{Zh}.
To define the metacenter {\it heuristically}, assume
that a body $K\subset {\mathbb R^3}$ is ``cylindrical".
In naval architecture, C^{\infty}te{Tu},
a ship floating originally at a horizontal waterline $H(\xi)\subset E$ is rotated through a small angle by an external force and then floats at waterline $H(\varepsilonta)\subset E$ (it is assumed that $H(\xi)$ and $H(\varepsilonta)$ intersect at the center of mass of $K$). Then the point $M=l(\xi)\cap l(\varepsilonta)$ is the metacenter, where
$l(\xi)$ is the line parallel to $\xi$ passing through
the old center of boyancy ${\mathcal C}(\xi)$ and
$l(\varepsilonta)$ is the line parallel to $\varepsilonta$ passing through
the new center of boyancy ${\mathcal C}(\varepsilonta)$, see Figure \ref{meta}.
Now we recall a rigorous definition, C^{\infty}te[pgs. 284, 285]{DVP}.
\begin{definition}\lambdabel{MGF}
Let ${\mathcal S}$ be the surface of centers and
let ${\mathcal C}$ be a point on ${\mathcal S}$ at which the normal curvatures exist.
Assume that ${\mathcal C}$ belongs to
some curve $\gammamma\subset{\mathcal S}$ with the tangent $\zetata$ at ${\mathcal C}$. Take ${\mathcal C}'\in\gammamma$ close to ${\mathcal C}$ and consider the normal lines $l_{\mathcal C}$, $l_{{\mathcal C}'}$, to
${\mathcal S}$ at ${\mathcal C}$ and ${\mathcal C}'$. If $\mu\mu'$ is a shortest distance between these lines, $\mu\in l_{\mathcal C}$, $\mu'\in l_{{\mathcal C}'}$, then the limiting position of the end $\mu$ of the segment
$[\mu,\mu']$, when ${\mathcal C}'$ tends to ${\mathcal C}$,
is the metacenter $M_{{\mathcal C}}(\zetata)$ related to ${\mathcal C}$ in the tangential direction $\zetata$.
\varepsilond
Let ${\mathcal S}$ be $C^2$-smooth. One can assume without loss of generality that
the tangent hyperplane ${\mathcal H}$ to ${\mathcal S}$ at ${\mathcal C}$ is horizontal, i.e., ${\mathcal H}$ is the $x_1\dots x_{d-1}$-hyperplane and that ${\mathcal C}$ is the origin. Then, choosing properly the directions of the axes in
${\mathcal H}$ one can assume that the equation of ${\mathcal S}$ in a small neighborhood of ${\mathcal C}$ is
\betagin{equation}\lambdabel{DVFrench1}
2x_d=k_1x_1^2+\dots +k_{d-1}x_{d-1}^2+o(x_1^2,\dots, x_{d-1}^2),
\varepsilonnd{equation}
where $k_j$, $j=1,\dots, d-1$, are some non-negative coefficients, $k_1\le k_2\le$ $\dots\le k_{d-1}$.
\begin{lemma}\lambdabel{DVPkr2}
The $x_d$-coordinate of $M_{{\mathcal C}}(\zetata)$ is
\betagin{equation}\lambdabel{DVPkr1}
{\mathcal C}\mu=\fracrac{k_1\zetata_1^2+\dots+k_{d-1}\zetata_{d-1}^2}{k_1^2\zetata_1^2+\dots+k_{d-1}^2\zetata_{d-1}^2}, \quadquad \textrm{where}\quaduad\zetata=(\zetata_1,\dots,\zetata_{d-1})\in S^{d-2}.
\varepsilonnd{equation}
\varepsilonl
This formula is proved in C^{\infty}te[page 285]{DVP} for $d=3$, the general case can be shown similarly. For convenience of the reader we prove (\ref{DVPkr1}) in Appendix.
\begin{remark}\lambdabel{Fkrut11}
We see that $\fracrac{1}{k_{d-1}} \le {\mathcal C}\mu\le \fracrac{1}{k_1} $ and that ${\mathcal C}\mu$ is equal to one of $\fracrac{1}{k_{j}}$,
$j=1,\dots,d-1$, provided $\zetata$ is one of the corresponding principal directions of ${\mathcal S}$ at ${\mathcal C}$.
\varepsilonr
We refer the reader to C^{\infty}te[pgs. 103-106]{Sch2} and C^{\infty}te[pgs. 82-89]{T} for the definition of the principal directions and the normal curvatures.
Alexandrov proved that if $M$ is a convex body and $G(\xi)$ is its supporting hyperplane, then the normal curvatures exist at $M\cap G(\xi)$ for almost every $\xi\in S^{d-1}$, C^{\infty}te{BF}, C^{\infty}te{Al}, C^{\infty}te{H}. Hence, for an arbitrary convex body the metacenter is defined for almost every $\xi\in S^{d-1}$.
Now we define the moment of inertia.
Let $d\ge 3$, let $\deltalta\in (0,\fracrac{\textrm{vol}_d(K)}{2})$, and let $\xi\in S^{d-1}$ be any direction. Consider a convex body $K$ and the hyperplane $H(\xi)$ defined by (\ref{p2}) such that (\ref{fubu}) holds.
Choose any $(d-2)$-dimensional plane $\Pi\subset H(\xi)$ passing through the center of mass ${\mathcal C}(K\cap H(\xi))$ and let
$\varepsilonta_1,\dots,\varepsilonta_{d-2}, \varepsilonta_{d-1}$ be an orthonormal basis of $\xi^{\partialerp}=\{p\in{\mathbb R^d}:\,p\mathcal Dot\xi=0 \}$ such that
\betagin{equation}\lambdabel{base}
\Pi={\mathcal C}(K\cap H(\xi))+\textrm{span}(\varepsilonta_1,\dots,\varepsilonta_{d-2}),\quaduad H(\xi)={\mathcal C}(K\cap H(\xi))+\xi^{\partialerp}.
\varepsilonnd{equation}
\begin{definition}\lambdabel{miab}
The moment of inertia $I_{K\cap H(\xi)}(\Pi)$ of $K\cap H(\xi)$ with respect to $\Pi$ is calculated by summing $\textnormal{dist}(\Pi, v)^2$ for every ``particle" $v$ in the set $K\cap H(\xi)$, where $\textnormal{dist}(\Pi,v)=\minL^{\infty}mits_{\{x\in \Pi\} }|v-x|$,
$\textnormal{(}$see Figure 4$\textnormal{)}$,
i.e.,
\betagin{equation}\lambdabel{moment}
I_{K\cap H(\xi)}(\Pi)=\intL^{\infty}mits_{K\cap H(\xi)}\textnormal{dist}(\Pi,v)^2dv=
\intL^{\infty}mits_{K\cap H(\xi)-{\mathcal C}(K\cap H(\xi))}(u\mathcal Dot\varepsilonta_{d-1})^2\,du.
\varepsilonnd{equation}
\varepsilond
\betagin{figure}[h]
\mathcal Entering
\includegraphics[height=2.8in]{ImageH1.pdf}
\caption{Two-dimensional body $K\cap H(\xi)$ with center of mass at the origin, and a line $\Pi$ parallel to $\varepsilonta_1$; we have $\textrm{dist}(\Pi,v)^2=|v|^2-(v\mathcal Dot \varepsilonta_1)^2=(v\mathcal Dot \varepsilonta_2)^2$.}
\lambdabel{width4}
\varepsilonnd{figure}
The Third Theorem of Dupin reads as follows (cf. C^{\infty}te{DVP}, page 288).
\begin{theorem}\lambdabel{D3}
Let $d\ge 3$, let $K\subset {\mathbb R^d}$ be a convex body and let $\deltalta\in (0,\textnormal{vol}_d(K))$.
If $H(\xi)$, $\xi\in S^{d-1}$, is a cutting hyperplane and ${\mathcal C}={\mathcal C}(\xi) \in {\mathcal S}$ is the corresponding center of mass at which the normal curvatures of ${\mathcal S}$ exist in all directions and if a sequence of cutting hyperplanes $\{H_n\}_{n=1}^{\infty}$, $H_n=H(\xi_n)$, converging to $H(\xi)$ as $n\to\infty$, is such that the limit
$L^{\infty}mL^{\infty}mits_{n\to\infty}H(\xi)\cap H(\xi_n)$
exists,
then for the corresponding sequence of the centers of mass $\{{\mathcal C}_n\}_{n=1}^{\infty}$, ${\mathcal C}_n={\mathcal C}(\xi_n)$, ${\mathcal C}=L^{\infty}mL^{\infty}mits_{n\to\infty}{\mathcal C}_n$,
one has
$$
{\mathcal R}_{{\mathcal C}(\xi)}(\zetata):=\textrm{dist}({\mathcal C}(\xi), M_{{\mathcal C}(\xi)}(\zetata))=\fracrac{1}{\deltalta} I_{K\cap H(\xi)}(\Pi),
$$
where $\zetata=L^{\infty}mL^{\infty}mits_{n\to\infty}\fracrac{{\mathcal C}{\mathcal C}_n}{|{\mathcal C}{\mathcal C}_n|}$ and $I_{K\cap H(\xi)}(\Pi)$ is the moment of inertia of
$K\cap H(\xi)$ with respect to the $(d-2)$-dimensional plane $\Pi=L^{\infty}mL^{\infty}mits_{n\to\infty}H(\xi)\cap H(\xi_n)$.
\varepsilont
If the reader does not want to deal with subtleties related to
the almost everywhere existence of tangent hyperplanes or normal curvatures
for general convex bodies, C^{\infty}te{BF}, C^{\infty}te{Al}, C^{\infty}te{H}, one can assume from now on that $K$ is $C^1$. In this case, ${\mathcal S}$ is $C^2$-smooth, C^{\infty}te{HSW}, and Theorem \ref{D3} holds for every $\xi\in S^{d-1}$.
The following theorem can be found in C^{\infty}te[page 23]{Da} and C^{\infty}te{A} in the case when $K$ has $C^1$-smooth boundary. It is the Third Theorem of Dupin for $d=2$.
\begin{theorem}\lambdabel{Dakr}
Let $K\subset {\mathbb R^2}$ be convex and let $\deltalta\in (0,\textnormal{area}(K))$. Then
$$
R(\xi)=\fracrac{\textnormal{length}^3(K\cap H(\xi))}{12\, \textnormal{area}(K\cap H^-(\xi))} \quadquad\textrm{for almost every}\quaduad \xi\in S^1,
$$
where $H(\xi)$ and $H^-(\xi)$ are defined by (\ref{p2}) and (\ref{fubu}), and
$R(\xi)$ is the radius of curvature of ${\mathcal S}$ at the point of tangency ${\mathcal S}\cap {\mathcal H}(\xi)$.
\varepsilont
\section{Proofs of Theorems of Dupin}\lambdabel{DP}
\subsection{Proof of Theorem \ref{D1}} Rotating and translating if necessary we can assume that $\xi$ is such that $H(\xi)$ is ``horizontal", i.e., $H(\xi)=e_d^{\partialerp}$. Let $\varepsilonta\in S^{d-1}$, $\varepsilonta\neq \xi$ and let ${\mathcal H}(\xi)$ be a hyperplane parallel to $H(\xi)$ and passing through ${\mathcal C}_{\deltalta}(\xi)$. We claim that ${\mathcal C}_{\deltalta}(\varepsilonta)$ is ``above"
${\mathcal H}(\xi)$, i.e.,
$x_d({\mathcal C}_{\deltalta}(\xi))<x_d({\mathcal C}_{\deltalta}(\varepsilonta))$.
Since $x_d>0$
$\fracorall x\in (K\cap H^-(\varepsilonta))\setminus (K\cap H^-(\xi))$ but $x_d\le 0$
$\fracorall x\in (K\cap H^-(\xi))\setminus (K\cap H^-(\varepsilonta)$, we have
$$
x_d({\mathcal C}_{\deltalta}(\xi))=\fracrac{1}{\deltalta}\Big( \intL^{\infty}mits_{(K\cap H^-(\xi))\setminus (K\cap H^-(\varepsilonta))}x_d dx+\intL^{\infty}mits_{K\cap H^-(\varepsilonta)\cap H^-(\xi)}x_d dx\Big)<
$$
$$
\fracrac{1}{\deltalta}\Big( \intL^{\infty}mits_{(K\cap H^-(\varepsilonta))\setminus (K\cap H^-(\xi))}x_d dx+\intL^{\infty}mits_{K\cap H^-(\varepsilonta)\cap H^-(\xi)}x_d dx\Big)=\,x_d({\mathcal C}({\mathcal C}_{\deltalta}(\varepsilonta))
$$
and the claim is proved.
Thus, for any $\xi\in S^{d-1}$ we have
${\mathcal S}\subset {\mathcal H}^+(\xi)$,
${\mathcal S}\cap {\mathcal H}(\xi)={\mathcal C}_{\deltalta}(\xi)$ and $\minL^{\infty}mits_{\{\xi\in S^{d-1}\}}|{\mathcal C}(K)-{\mathcal C}_{\deltalta}(\xi)|>0$. We conclude that
$L({\mathcal S})=\bigcapL^{\infty}mits_{\{\xi\in S^{d-1}\}}{\mathcal H}^+(\xi)$ is a strictly convex body.
$\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad \,\,\,\square $
\subsection{Proof of Theorem \ref{D2}}
Rotating and translating if necessary, assume that $H(\xi)$ is ``horizontal", i.e., $H(\xi)=e_d^{\partialerp}$. Take $n$ large enough and consider the $(d-2)$-dimensional plane $\Pi_n=H(\xi)\cap H(\xi_n)$. Introduce the ``moving" coordinates $(x_1,x_2,\dots, x_{d-1}, x_d)$ so that $\Pi_n$ is the
$(x_2,\dots, x_{d-1})$-plane.
Denote by $A\triangle B$ the symmetric difference of two sets $A$ and $B$, i.e.,
$A\triangle B=(A\setminus B)\cup (B\setminus A)$, and
let
$\Lambdambda_n= (K\cap H(\xi))\triangle P_{H(\xi)} (K\cap H(\xi_n))
$, where
$P_{H(\xi)}$ is the orthogonal projection onto $H(\xi)$. Then,
\betagin{equation}\lambdabel{kr1}
\varDelta V=\textnormal{vol}_d(K\cap H^-(\xi))-\textnormal{vol}_d(K\cap H^-(\xi_n))=
\varepsilonnd{equation}
$$
\intL^{\infty}mits_{K\cap H(\xi)}x_1\tan \varepsilon_n \,dx-
\intL^{\infty}mits_{\Lambdambda_n}\zetata_d \,dx=0,
$$
where $x_1=x_1(\xi,\xi_n)$ and
$\zetata_d=\zetata_d(\xi,\xi_n)$ is an error of $x_d=x_1\tan\varepsilon_n$ in $\Lambdambda_n$ which is obtained during the computation of $\varDelta V$ using the first integral above (see Figure \ref{Fig7}; observe that
$H(\xi)\cap H(\xi_n)\cap \textrm{int}K\neq\varepsilonmptyset$
(see C^{\infty}te[p. 116]{O} or C^{\infty}te[Appendix A]{R2})).
To see (\ref{kr1}), consider on $e_d^{\partialerp}$ an infinitesimally small element of the $(d-1)$-dimensional volume $dx$ as a base of an infinitesimally small prism ``between"
$H(\xi)$ and $H(\xi_n)$ of ``height" $\tan\varepsilon_n |x_1|$, where $\varepsilon_n$ is a small angle between
$H(\xi)$ and $H(\xi_n)$. The $d$-dimensional volume of the prism is $\tan\varepsilon_n|x_1|dx$. Summing up the volumes of the corresponding prisms we obtain (\ref{kr1}).
\betagin{figure}[ht]
\includegraphics[width=360pt]{Fig7.pdf}
\caption{The function $\zetata_d$.}
\lambdabel{Fig7}
\varepsilonnd{figure}
By (\ref{kr1}), we have
$$
x_1({\mathcal C}(K\cap H(\xi))=\fracrac{\intL^{\infty}mits_{K\cap H(\xi)}x_1 \,\,\,dx}{\textrm{vol}_{d-1}(K\cap H(\xi))}=\fracrac{\intL^{\infty}mits_{\Lambdambda_n}\zetata_d \,\,\,dx}{\textrm{vol}_{d-1}(K\cap H(\xi))\tan \varepsilon_n}.
$$
Since $\textrm{vol}_{d-1}(\Lambdambda_n)\to 0$ as $n\to\infty$ (see C^{\infty}te[p. 116]{O} or C^{\infty}te[Appendix A]{R2}), and since $|\zetata_d|\le D\tan\varepsilon_n$, where $D$ is the diameter of $K$,
we obtain
$$
|x_1({\mathcal C}(K\cap H(\theta)))|\le \fracrac{D\tan \varepsilon_n\,\,\textrm{vol}_{d-1}(\Lambdambda_n)}{\textrm{vol}_{d-1}(K\cap H(\xi))\tan \varepsilon_n}\to 0
$$
as $n\to \infty$.
We see that the $(d-2)$-dimensional plane
$H(\xi)\cap H(\xi_n)$ tends, as $n\to \infty$, to a limiting position $\Pi$ that passes through the center of mass of $K\cap H(\xi)$.
$\quadquad\quadquad\quadquad\quadquad \quadquad\quadquad\quadquad\quadquad\quadquad \quadquad\quadquad\quadquad\quadquad\, \square $
\betagin{figure}[h]
\mathcal Entering
\includegraphics[height=3.5in]{DVPbbb.pdf}
\caption{The normals ${\mathcal C}\mu$ and ${\mathcal C}_n\mu_n$ to the surface of centers}
\lambdabel{VPbeauty}
\varepsilonnd{figure}
\subsection{Proof of Theorem \ref{D3}}
As in the previous proofs, we assume that $H(\xi)=e_d^{\partialerp}$. We take $n$ large enough and put $\Pi_n=H(\xi)\cap H(\xi_n)$. As above we introduce the ``moving" coordinates $(x_1,x_2,\dots, x_{d-1}, x_d)$ so that the $(d-2)$-dimensional plane $\Pi_n$ is the
$(x_2,\dots, x_{d-1})$-plane. Denote by $v_{1,n}$ and $v_{2,n}$ the $d$-dimensional bodies with the $x_1$-coordinates having opposite
signs,
$$
v_{1,n}=(K\cap H^-(\xi_n))\setminus (K\cap H^-(\xi)),\quaduad v_{2,n}=(K\cap H^-(\xi))\setminus (K\cap H^-(\xi_n)),
$$
and let $y_{1,n}$, $z_{1,n}$ be the $x_1$-coordinates of ${\mathcal C}={\mathcal C}_{\deltalta}(\xi)$ and ${\mathcal C}_n={\mathcal C}_{\deltalta}(\xi_n)$, see Figure \ref{VPbeauty} (cf. Figure 59, page 289 from C^{\infty}te{DVP}). Then
$$
\deltalta y_{1,n}=\intL^{\infty}mits_{ K\cap H^-(\xi)}x_1dx,\quaduad \deltalta z_{1,n}=\intL^{\infty}mits_{ K\cap H^-(\xi_n)}x_1dx,
$$
and looking at the difference, we have
$$
\deltalta (y_{1,n}-z_{1,n})=\intL^{\infty}mits_{v_{1,n}\cup\, v_{2,n}}|x_1|dx.
$$
Repeating the argument from the proof of Theorem \ref{D2} showing that the volumes $\textrm{vol}_d(v_{1,n})=\textrm{vol}_d(v_{2,n})$
are (up to $o(\varepsilon_n)$) the sums of volumes $\varepsilon_n x_1dx$ of infinitesimal prisms, we obtain
\betagin{equation}\lambdabel{dvp33}
\deltalta (z_{1,n}-y_{1,n}) =\tan\varepsilon_n \intL^{\infty}mits_{K\cap H(\xi)}x_1^2d\sigmagma_{d-1}(x)+o(\varepsilon_n)=
\varepsilonnd{equation}
$$
\tan\varepsilon_n I_{K\cap H(\xi)}(\Pi_n)+o(\varepsilon_n).
$$
On the other hand, consider the normals ${\mathcal C}\mu$ and ${\mathcal C}_n\mu_n$ to ${\mathcal S}$ at the points ${\mathcal C}={\mathcal C}_{\deltalta}(\xi)$ and ${\mathcal C}_n={\mathcal C}_{\deltalta}(\xi_n)$. The angle $\varepsilon_n$ between these normals
is equal to the one between the hyperplanes
$H(\xi)$ and $H(\xi_n)$. At the same time this is the angle between the $x_d$-axis and ${\mathcal C}_n\mu_n$. By definition of the metacenter, the vector $\mu\mu_n$ is ``parallel" to $\Pi_n$, so $\mu$ and $\mu_n$ have the same $x_1$-coordinate;
it is the $x_1$-coordinate of the intersection of orthogonal projections of lines $\varepsilonll$, $\varepsilonll_n$, containing
${\mathcal C}\mu$, ${\mathcal C}_n\mu_n$, onto the $x_1x_d$-plane.
We conclude that $z_{1,n}-y_{1,n}$ is the projection of ${\mathcal C}_n\mu_n$ onto the $x_1$-axis, i.e.,
$z_{1,n}-y_{1,n}=\sigman\varepsilon_n |{\mathcal C}_n\mu_n|$.
Substituting this expression into (\ref{dvp33}) and passing to the limit as $n\to\infty$ we see that
$$
|{\mathcal C}\mu|=L^{\infty}mL^{\infty}mits_{n\to\infty}|{\mathcal C}_n\mu_n|=\fracrac{I_{K\cap H(\xi)}(\Pi)}{\deltalta},
$$
which is the desired conclusion. $\quadquad\quadquad\quadquad\quadquad \quadquad\quadquad\quadquad\quadquad\quadquad \square $
\section{Proofs of Lemma \ref{tr}, Theorems \ref{Fedja1}, \ref{CS}, and Corollaries \ref{Al}, \ref{equi}}\lambdabel{Au}
We start with the proof of Lemma \ref{tr} (cf. C^{\infty}te{Gr}, C^{\infty}te{Mo}, C^{\infty}te[page 203]{G} and C^{\infty}te[Corollary 2.4 and Proposition 2.2]{HSW}).
\begin{proof}
At first we prove the converse statement. Using the fact that all normals of the sphere intersect at its center and Theorem \ref{D1}, we see that for every $\xi\in S^{d-1}$, the lines $\varepsilonll(\xi)$ passing through ${\mathcal C}(K)$ and ${\mathcal C}_{\deltalta}(\xi)$ are orthogonal to $H(\xi)$.
Now we prove the {\it if} part.
Let $\xi\in S^{d-1}$ and let $\varepsilonll(\xi)$ be a line passing through ${\mathcal C}(K)$ and the center of mass ${\mathcal C}(\xi)$ of $K\cap H^-(\xi)$.
By
Theorem \ref{D1}, ${\mathcal H}(\xi)$ is parallel to $H(\xi)$.
Since $K$ floats in equilibrium in the direction $\xi$, the line $\varepsilonll(\xi)$ is orthogonal to $H(\xi)$. Since ${\mathcal H}(\xi)$ is parallel to $H(\xi)$, $\varepsilonll(\xi)$ is the normal line to
${\mathcal S}$ at ${\mathcal C}(\xi)$, and since the body floats in equilibrium in all directions $\xi\in S^{d-1}$,
we know that
the lines $\varepsilonll(\xi)$ passing through ${\mathcal C}(K)$
are the normal lines to ${\mathcal S}$ for every $\xi$;
we recall that ${\mathcal S}$ is $C^1$-smooth, C^{\infty}te{HSW}. Consider any two-dimensional plane $\Pi$ passing through ${\mathcal C}(K)$. Parametrizing the plane curve
$ {\mathcal S}\cap\Pi$ by the radius vector ${\mathbf r}$ going from ${\mathcal C} (K)$ to the corresponding
$ {\mathcal S}\cap l(\xi)$, we see that
${\mathbf r}$ is orthogonal to ${\mathbf r}'$, i.e.,
${\mathbf r}\mathcal Dot {\mathbf r}'=0$, $|{\mathbf r}|$ is constant,
and $ {\mathcal S}\cap \Pi$ is a circle.
Since $\Pi$ was chosen arbitrarily,
applying C^{\infty}te[Corollary 7.1.4, page 272]{Ga} to $L({\mathcal S})$ from Theorem \ref{D1}, we obtain that ${\mathcal S}$ is a sphere.
This gives the desired conclusion.
\varepsilonp
\subsection{Proof of Corollary \ref{Al}}
Let $\deltalta_n\to0$ and let ${\mathcal S}_n$ be the corresponding surfaces of centers, which are all spheres of the radii $r_n$, $r_n\to r$ as $n\to\infty$.
Since $d(K_{{\deltalta}_n}, K)\to 0$ as $n\to \infty$, and since $K_{{\deltalta}_n}\subset B_{r_n}^2(0)\subset K$, we have
$d(B_{r_n}^2(0), K)\to 0$ as $n\to \infty$. Hence, $K$ is the Euclidean ball $B^2_r(0)$.
$\quadquad\,\, \square$
\subsection{Proof of Theorem \ref{Dpr}}
It is a consequence of Lemma \ref{tr} and Theorems of Dupin.
It will be convenient to reformulate Theorem \ref{Dpr} in terms of the radial function.
Given a direction $\xi\in S^{d-1}$ and a hyperplane (\ref{p2}) for which (\ref{fubu}) holds,
we will use the notation $\rho_{K\cap H(\xi)}(w)$ for the radial function of the $(d-1)$-dimensional convex body $K\cap H(\xi)$ with respect to the center of mass ${\mathcal C}(K\cap H(\xi))$ in the direction $w\in S^{d-1}\cap \xi^{\partialerp}$, i.e., for
$$
\rho_{K\cap H(\xi),\,{\mathcal C}(K\cap H(\xi)) }(w)=\max\{\lambdambda>0:\,{\mathcal C}(K\cap H(\xi))+\lambdambda w\in (K\cap H(\xi))\}.
$$
\begin{theorem}\lambdabel{Fedja1}
Let $d\ge 3$, let $K$ be a convex body and let $\deltalta\in (0, \textnormal{vol}_d(K))$.
If $K$ floats in equilibrium at the level $\deltalta$ in every orientation, then $\fracorall \xi\in S^{d-1}$
the cutting sections $K\cap H(\xi)$ have equal principal moments, i.e.,
we have
\betagin{equation}\lambdabel{eq112}
\intL^{\infty}mits_{S^{d-1}\cap \xi^{\partialerp}}w_k^2\,\rho_{K\cap H(\xi)}^{d+1}(w)dw=(d+1)\deltalta {\mathcal R},\quaduad k=1,2,\dots, d-1,
\varepsilonnd{equation}
\betagin{equation}\lambdabel{eq112B}
\intL^{\infty}mits_{S^{d-1}\cap \xi^{\partialerp}}w_jw_k\,\rho_{K\cap H(\xi)}^{d+1}(w)dw=0,\quaduad 1\le k,j\le d-1, \quaduad j\neq k,
\varepsilonnd{equation}
where ${\mathcal R}$
is the radius of the spherical surface of centers ${\mathcal S}$.
Conversely, if ${\mathcal C}({\mathcal S})={\mathcal C}(K)$
and for every cutting hyperplane $H(\xi)$, $\xi\in S^{d-1}$, the cutting section $K\cap H(\xi)$ satisfies (\ref{fubu}), (\ref{eq112}) and (\ref{eq112B}) with some constant ${\mathcal R}$, then the body $K$ with $C^1$-smooth boundary floats in equilibrium in every orientation
at the level $\deltalta$.
\varepsilont
\begin{proof}
Let $d\ge 3$.
Fix any $\xi\in S^{d-1}$ and a cutting hyperplane $H(\xi)$.
Let $\Pi\subset H(\xi)$ be a $(d-2)$-dimensional plane passing through ${\mathcal C}(K\cap H(\xi))$,
let $\Pi_n\subset H(\xi)$ be a sequence of $(d-2)$-dimensional planes converging and parallel to $\Pi$ as $n\to\infty$, and let $H_n=H(\xi_n)$, $H_n\cap H(\xi)=\Pi_n$, be the corresponding cutting hyperplanes.
If ${\mathcal C}_n={\mathcal C}(\xi_n)$ are the centers of mass of $K\cap H_n^-$ converging to ${\mathcal C}={\mathcal C}(\xi)$ as $n\to\infty$, then,
by
Theorem \ref{D3},
for $\zetata=L^{\infty}mL^{\infty}mits_{n\to\infty}\fracrac{{\mathcal C}{\mathcal C}_n}{|{\mathcal C}{\mathcal C}_n|}$ we have
\betagin{equation}\lambdabel{huhuh}
{\mathcal R}_{\mathcal C(\xi)}(\zetata)\stackrel{\text{for a.e \,$\xi$}}{=}\fracrac{1}{\deltalta}I_{K\cap H(\xi)}(\Pi).
\varepsilonnd{equation}
By Lemma \ref{tr} the surface of centers ${\mathcal S}$ is a sphere of certain radius ${\mathcal R}$ centered at ${\mathcal C}(K)$. Since the radii of the normal curvatures of the sphere of radius ${\mathcal R}$ are equal to ${\mathcal R}$ at all points ${\mathcal C}\in{\mathcal S}$ in all directions and since $\Pi$ was chosen arbitrarily,
by Remark \ref{Fkrut11}, we see that the function in the right-hand side of (\ref{huhuh}) is constant for almost every $\xi\in S^{d-1}$ and for all $\Pi$.
Since the function
$(\xi,\Pi)\to I_{K\cap H(\xi)}(\Pi)$
is continuous,
the right-hand side of (\ref{huhuh}) is constant for every $\xi\in S^{d-1}$ and for all $\Pi$.
Hence, using (\ref{moment}) we obtain that for all $\xi\in S^{d-2}$ one has
\betagin{equation}\lambdabel{cognac}
\fracrac{1}{\deltalta}\intL^{\infty}mits_{K\cap H(\xi)-{\mathcal C}(K\cap H(\xi))}(v\mathcal Dot\varepsilonta_{d-1})^2\,dv={\mathcal R}\quadquad\fracorall \varepsilonta_{d-1}\in S^{d-1}\cap \xi^{\partialerp},
\varepsilonnd{equation}
where we recall that $\varepsilonta_1,\dots,\varepsilonta_{d-2}, \varepsilonta_{d-1}$ is the orthonormal basis of $\xi^{\partialerp}$ such that (\ref{base}) holds.
Passing to polar coordinates in $H(\xi)$ with respect to ${\mathcal C}(K\cap H(\xi))$, we have
\betagin{equation}\lambdabel{cognac22}
\intL^{\infty}mits_{K\cap H(\xi)-{\mathcal C}(K\cap H(\xi))}\!\!(v\mathcal Dot\varepsilonta_{d-1})^2dv=\intL^{\infty}mits_{S^{d-1}\cap \xi^{\partialerp}}\!\!dw\!\!\intL^{\infty}mits_0^{\rho_{K\cap H(\xi)}(w)}\!\!( r w\mathcal Dot \varepsilonta_{d-1})^2r^{d-2}dr=
\varepsilonnd{equation}
$$
\fracrac{1}{d+1}\intL^{\infty}mits_{S^{d-1}\cap \xi^{\partialerp}}(w\mathcal Dot\varepsilonta_{d-1})^2\rho^{d+1}_{K\cap H(\xi)}(w)dw,
\quadquad\fracorall \varepsilonta_{d-1}\in S^{d-1}\cap \xi^{\partialerp}.
$$
This identity and (\ref{cognac}) yield
\betagin{equation}\lambdabel{vodochka1}
\intL^{\infty}mits_{S^{d-1}\cap \xi^{\partialerp}}(w\mathcal Dot\varepsilonta_{d-1})^2\rho^{d+1}_{K\cap H(\xi)}(w)dw=(d+1)\,\deltalta \,{\mathcal R},
\varepsilonnd{equation}
where the right-hand side is independent of $\varepsilonta_{d-1}\in S^{d-1}\cap \xi^{\partialerp}$.
By choosing $\varepsilonta_{d-1}$ to be the standard coordinate vectors in $\xi^{\partialerp}$,
we obtain (\ref{eq112}). By taking $\varepsilonta_{d-1}=(0,\dots,\underbrace{\fracrac{\sqrt{2}}{2}}_j,0,\dots,0,\underbrace{\fracrac{\sqrt{2}}{2}}_k, 0,\dots, 0 )$ for different $1\le j,k\le d-1$, $j\neq k$, and using (\ref{eq112}) we obtain (\ref{eq112B}).
Since $\xi$ was arbitrary, the proof of the {\it if} part is complete.
Now we prove the converse statement. Our goal is to show that the surface of centers is a sphere.
We will show at first that
for almost every $\xi\in S^{d-1}$ the points
${\mathcal C}(\xi)={\mathcal S}\cap{\mathcal H}(\xi)$ are umbilical.
Let $\xi\in S^{d-1}$ be such that the normal curvatures at the corresponding point ${\mathcal C}(\xi)\in{\mathcal S}$ exist.
Assume that (\ref{eq112}) and (\ref{eq112B}) are true. We can also assume that $\Pi$ satisfies (\ref{base}).
Then, expanding the expression $(w\mathcal Dot \varepsilonta_{d-1})^2$ by writing $w$ in the basis $\varepsilonta_1$, $\dots$, $\varepsilonta_{d-1}$
and
using
the identities (\ref{cognac}) and (\ref{cognac22}), we see that (\ref{vodochka1}) holds with some constant ${\mathcal R}$ in
the right-hand side, i.e., it is independent of $\varepsilonta_{d-1}\in S^{d-1}\cap \xi^{\partialerp}$.
Hence, using (\ref{moment}), (\ref{cognac}) and (\ref{cognac22}), we see that the right-hand side of
(\ref{huhuh}) is independent of $\Pi$ and $\xi$.
Now let $\zetata$ be any unit principal direction in the hyperplane ${\mathcal H}(\xi)$ tangent to ${\mathcal S}$ at ${\mathcal C}(\xi)$, and let $\Pi$ be a two-dimensional subspace spanned by $\zetata$ and the normal to ${\mathcal S}$ at ${\mathcal C}(\xi)$. Consider
a sequence of unit directions $\zetata_n$
tangent to the two-dimensional curve ${\mathcal S}\cap\Pi$ at the corresponding points
${\mathcal C}(\xi_n)\in({\mathcal S}\cap\Pi)$
and such that $\zetata_n\to\zetata$, ${\mathcal C}(\xi_n)\to {\mathcal C}(\xi)$, as $n\to\infty$.
If $\{H(\xi_n)\}_{n=1}^{\infty}$
is
a sequence of cutting hyperplanes $H(\xi_n)$ converging to $H(\xi)$ as $n\to\infty$ with ${\mathcal C}(\xi_n)$ being the centers of mass of $K\cap H^-(\xi_n)$, applying
Theorem \ref{D3} and passing to a subsequence if necessary to ensure the existence of $L^{\infty}mL^{\infty}mits_{n\to\infty}H(\xi)\cap H(\xi_n)$,
we see that the radii of the principal normal curvatures of
${\mathcal S}$ at ${\mathcal C}(\xi)$ in the principal directions are the same and the value of the radii is independent of $\xi$ and $\zetata$
for almost every $\xi\in S^{d-1}$
and for every principal direction $\zetata$ parallel to ${\mathcal H}(\xi)$.
Thus, for almost every $\xi\in S^{d-1}$ the points
${\mathcal C}(\xi)$ are umbilical.
We claim that ${\mathcal S}$ is a sphere.
Indeed, recall that by Remark \ref{ew} the surface of centers is $C^2$. Hence, by continuity, all the points on ${\mathcal S}$ are umbilical. Using C^{\infty}te[Proposition 4, page 147]{DC} and C^{\infty}te[Corollary 7.1.4, page 272]{Ga} we conclude that
${\mathcal S}$ must be a $(d-1)$-dimensional sphere.
An application of Lemma \ref{tr} finishes the proof. \varepsilonp
\begin{remark}\lambdabel{nun}
In the planar case an analogous result is a consequence of Lemma \ref{tr} and Theorem
\ref{Dakr}.
\varepsilonr
\subsection{Proof of Corollary \ref{equi}}
The condition of the corollary reads as
\betagin{equation}\lambdabel{radhop1}
\fracorall \xi\in S^{d-1},\quadquad \rho^{d+1}_{K\cap H(\xi)}(w)+ \rho^{d+1}_{K\cap H(\xi)}(-w)=c\quadquad\fracorall w\in S^{d-1}\cap\xi^{\partialerp}.
\varepsilonnd{equation}
The result follows from the second part of Theorem \ref{Al} by writing $\rho_K^{d+1}$ as the sum of even and odd parts and
substituting the even part from (\ref{radhop1}) into (\ref{eq112}) and (\ref{eq112B}).$\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quaduad\quaduad\,\,\,\,
\square$
\subsection{Proof of Theorem \ref{CS}}
We recall that a measurable function $f:\, S^{d-1}\to {\mathbb R}$ is isotropic if the signed measure $fdx$ is isotropic,
i.e., its center of mass is at the
origin and the map
$$
S^{d-1}\ni y\quaduad \to\quaduad \intL^{\infty}mits_{S^{d-1}}(y\mathcal Dot w)^2f(w)dw
$$
is constant, C^{\infty}te{MP}. The following result was obtained in C^{\infty}te{MRS}.
\begin{theorem}\lambdabel{CrSa}
Let $f:\, S^{d-1}\to {\mathbb R}$ be a measurable, bounded a. e. and even function, $d\ge 3$. If
for almost every $\xi\in S^{d-1}$ the restriction $f|_{S^{d-1}\cap \xi^{\partialerp}}$ to
$S^{d-1}\cap \xi^{\partialerp}$
is isotropic $\textnormal{(}$i.e. the
restriction of $f$ to almost every equator is isotropic$\textnormal{)}$, then $f$ is almost everywhere equal to a
constant.
\varepsilont
By the origin-symmetry,
the centers of mass of all cutting sections are equal to the center of mass of $K$. Hence, we may apply
Theorem \ref{Fedja1} to see that there exists a constant $c$ such that all second moments of the central sections $K\cap\xi^{\partialerp}$ are equal to $c$ for all $ \xi\in S^{d-1}$. The result follows from Theorem \ref{CrSa} with $f=\rho_K^{d+1}$. $\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\quadquad\,\square$
\section{Appendix: proof of Lemma \ref{DVPkr2} from C^{\infty}te[page 285]{DVP}}
Let $M$ be a point on $C^2$-smooth ${\mathcal S}$ and let $\gammamma\subset {\mathcal S}$ be a curve passing through $M$. Let $M'\in\gammamma$ be a point infinitesimally close to $M$. Consider two normal lines $M\Gammamma$ and $M'N'$ to ${\mathcal S}$ at $M$ and $M'$ and let $\mu\mu'$ be the shortest distance between these normal lines.
We can assume that the tangent hyperplane to ${\mathcal S}$ at $M$ is $e_d^{\partialerp}$ and that its boundary is locally described by (\ref{DVFrench1}).
Now drop the terms of the orders higher than $2$.
We have $\fracrac{\partialartial x_d}{\partialartial x_j}=k_jx_j$ for $j=1,\dots,d-1$.
The normal line at $M'=M'(x_1,\dots,x_d)$ can be expressed in terms of the ``running" coordinates $(y_1,\dots,y_d)$ by equations
$y_j-x_j=k_jx_j(y_d-x_d)$, $j=1,\dots,d-1$.
The square of the distance between $(y_1,\dots, y_{d-1})$ and $M\Gammamma$ is
$$
\sumL^{\infty}mits_{j=1}^{d-1}y_j^2=\sumL^{\infty}mits_{j=1}^{d-1}(x_j-k_jx_j(y_d-x_d))^2.
$$
The ``ordinate" $y_d={\mathcal C}\mu$ of the metacenter gives the minimum of the above expression and annihilates its derivatives (at $x_d=0$). Hence,
$$
\sumL^{\infty}mits_{j=1}^{d-1}k_jx_j(x_j-k_jx_jy_d)=0,\quadquad\textrm{i.e.,}\quadquad {\mathcal C}\mu=\fracrac{\sumL^{\infty}mits_{j=1}^{d-1}k_jx_j^2}{\sumL^{\infty}mits_{j=1}^{d-1}k_j^2x_j^2}\,.
$$
If $MT$ is the unit tangent vector to $\gammamma$ at $M$, then, identifying $e_d^{\partialerp}$ with ${\mathbb R^{d-1}}$, writing $MT$ in spherical coordinates $\zetata=(\zetata_1,\dots,\zetata_{d-1})\in S^{d-2}$ and putting $(\zetata_1,\dots,\zetata_{d-1})=\fracrac{(x_1,\dots,x_{d-1})}{\sqrt{x_1^2+\dots+x_{d-1}^2}}$, we obtain (\ref{DVPkr1}).
{\bf Acknowledment}. The author is very thankful to Mariangel Alfonseca, Alexander Fish, Carsten Sch\"utt, Elisabeth Werner, Vlad Yaskin and Ning Zhang for very useful discussions. He is also very grateful to Daniel Hug and Christos
Saroglou for explaining several results and providing references, and to Peter V\'arkonyi for pointing out the possible necessity of an extra condition in Lemma \ref{tr} for non-symmetric convex bodies.
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\begin{document}
\title{\bf\Large $C^{1,\alpha}$
regularity for fully nonlinear elliptic equations with superlinear growth in the gradient}
\author[1]{Gabrielle Nornberg\footnote{Corresponding author. Email address: [email protected]}\footnote{The work was supported by Capes PROEX/PDSE grant 88881.134627/2016-01.}}
\affil[1]{Pontifícia Universidade Católica do Rio de Janeiro, Brazil}
\; \mathrm{d}ate{}
\maketitle
{\mathbb{S}mall\mathbb{N}oindent{\bf{Abstract.}} We extend the Caffarelli-Świech-Winter $C^{1,\alpha}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded coefficients. As an application, in addition to the usual $W^{2,p}$ results, we prove the existence of positive eigenvalues for proper operators with nonnegative unbounded weight, in particular for Pucci's operators with unbounded coefficients.
{\mathbb{S}mall\mathbb{N}oindent{\bf{Résumé.}} Dans cet article on étend les résultats de régularité $C^{1, \alpha}$ de Caffarelli-Świech-Winter aux solutions de $ L^p$ viscosité des équations complètement non-linéaires, uniformément elliptiques, sous forme non-divergence, avec croissance super-linéaire par rapport au gradient et coefficients non bornés. Dans le cadre d'une application, en plus des résultats habituels $W ^{2, p}$, on prouve l’existence de valeurs propres positives pour les opérateurs propres avec poids non borné non négatif, en particulier pour les opérateurs de Pucci à coefficients non bornés.
{\mathbb{S}mall\mathbb{N}oindent{\bf{Keywords.}} {Regularity, Estimates, Superlinear gradient growth, Nondivergence form.}
{\mathbb{S}mall\mathbb{N}oindent{\bf{MSC2010.}} {35J15, 35B65, 35J60, 35P30, 35P15.}
\mathbb{S}ection{Introduction}\label{Introduction}
The seminal work of Caffarelli \cite{Caf89} in 1989 brought an innovative approach of looking at Schauder type results via iterations from the differential quotients that are perturbations of solutions of the respective autonomous equations.
The techniques in \cite{Caf89}, which contains in particular $C^{1,\alpha}$ estimates for $L^p$-viscosity solutions of uniformly elliptic equations $F(x,D^2u)=f(x)$, allowed Świech \cite{Swiech} to extend them to more general operators $F(x,u,Du,D^2u)$ and later Winter \cite{Winter} to boundary and global bounds. However, everything that is available in the literature, to our knowledge, for $L^p$-viscosity solutions in the fully nonlinear framework, concerns only structures with either linear gradient growth or bounded coefficients, except for some particular cases of extremal equations with small coefficients, see \cite{KSexist2009}.
It is our goal here to obtain $C^{1,\alpha}$ regularity and estimates for general fully nonlinear uniformly elliptic equations, with at most quadratic growth in the gradient and unbounded coefficients.
The study of such quasilinear elliptic equations with quadratic dependence in the gradient had its beginning in the '80s, essentially with the works of Boccardo, Murat and Puel \cite{BMP2}, \cite{BMP1} and became a relevant research topic which still develops. This type of nonlinearity often appears in risk-sensitive stochastic problems, as well as in large deviations, control and game theory, mean-fields problems. Moreover, the class of equations in the form $Lu = g (x,u,D u)$, where $L$ is a second order general operator and $g$ has quadratic growth in the gradient, is invariant under smooth changes of the function $u$ and the variable $x$. Due to this fact, this class is usually referred as having \textit{natural} growth in the gradient.
Rather complete $C^\alpha$ regularity results for fully nonlinear uniformly elliptic equations with up to quadratic growth in the gradient were obtained in \cite{arma2010}, in the most general setting of unbounded coefficients, for $L^p$-viscosity solutions. Then, the question of $C^{1,\alpha}$ regularity for the same class arises naturally. In the present work we show, as can be expected, that $C^{1,\alpha}$ regularity and estimates are valid in this context. These $C^{1,\alpha}$ estimates are instrumental in the recent study of multiplicity for nonproper equations in \cite{multiplicidade}.
We note that Trudinger, independently from \cite{Caf89}, in \cite{T89} proved $C^{1,\alpha}$ regularity in a less general scenario than Świech and Winter, under a continuity hypothesis for $F$, dealing with $C$-viscosity solutions and approximations under supconvolutions. In that paper, it was stated that a priori estimates for solutions in $C^{1,\alpha}$ of superlinear equations could be derived from the arguments in \cite{T89} and \cite{T88}. However, the question of regularity is more complicated (for a discussion on differences between a priori bounds and regularity results we refer to \cite{SB}).
We also quote some other papers on $C^{1,\alpha}$ regularity, the classical works \cite{KrylovBook}, \cite{LU}, \cite{LU89} for linear equations, \cite{MilSilv} for Neumann boundary conditions, \cite{ST} for asymptotically convex operators, \cite{IS} (local) and \cite{BDC1beta} (global) for degenerate elliptic operators,
\cite{CKS} and \cite{Krylov} for parabolic equations possibly with VMO coefficients.
Furthermore, Wang \cite{W2} has made an important contribution to $C^{1,\alpha}$ regularity for the parabolic equation $u_t+F(x,D^2 u)=g(t,x,Du)$, where $|g(t,x,p)|\leq A\, |p|^2+g(t,x)$, for bounded coefficients, see lemma 1.6 in \cite{W2} (which uses theorem 4.19 in \cite{W1}). Sharp regularity results for general parabolic equations with linear gradient growth can be found in \cite{JE}, and very complete $C^{1,\alpha}$ estimates on the boundary for solutions in the so called $S^*$-class for equations with linear gradient growth and unbounded coefficients in \cite{DEWboundary}.
It is also essential to mention an important series of papers due to Koike and Świech \cite{KSweakharnack}, \cite{KSmp2004}, \cite{KSmpite2007}, \cite{KSexist2009}, in which they proved ABP and weak Harnack inequalities for $L^p$-viscosity solutions of equations with superlinear growth in the gradient, together with several theorems about existence, uniqueness and $W^{2,p}$ estimates for solutions of extremal equations involving Pucci's operators with unbounded coefficients, see in particular theorem 3.1 in \cite{KSexist2009}. Many of our arguments depend on the machinery in these works.
Next we list our hypotheses.
For $F(x,r,p,X)$ measurable in $x$, we consider the general structure condition
\begin{align} \label{SCmu}
\mathcal{M}_{\lambda, \Lambda}^- (X-Y)-b(x)|p-q|-\mu |p-q|(|p|+|q|)-d(x)\,\omega (|r-s|) \mathbb{N}onumber \\
\leq F(x,r,p,X) - F(x,s,q,Y) \tag*{$(SC)^\mu$} \\
\leq \mathcal{M}_{\lambda, \Lambda}^+ (X-Y)+b(x)|p-q|+\mu |p-q|(|p|+|q|)+d(x)\,\omega (|r-s|) \; \textrm{ for } x\in \Omega\mathbb{N}onumber
\end{align}
where $F(\cdot,0,0,0)\equiv 0$ and $0<\lambda \leq \Lambda$, $b\in L^p_+ (\Omega)$ for some $ p>n$, $d\in L^\infty_+ (\Omega)$, $\mu\geq 0$ and $\omega$ is a modulus of continuity (see section \ref{Preliminaries}).
In order to measure the oscillation of $F$ in the $x$ entry, we define, as in \cite{Caf89}, \cite{Winter},
\begin{align} \label{def beta}
\beta(x,x_0)=\beta_F (x,x_0):= \mathbb{S}up_{X\in \mathbf{S}^n\mathbb{S}etminus \{0\} } \frac{|F(x,0,0,X)-F(x_0,0,0,X)|}{\|X\|} .
\end{align}
Notice that $\beta$ is a bounded function by \ref{SCmu} and consider the usual hypothesis, as in \cite{Caf89}, \cite{Winter}: given $\theta>0$, there exists $r_0=r_0\, (\theta)>0$ such that
\begin{align}\label{Htheta} \tag{$H_{\theta}$}
\;\left( \frac{1}{r^n} \int_{B_r(x_0)\cap\Omega} \beta (x,x_0)^p \; \mathrm{d} x \right)^{\frac{1}{p}} \leq \theta \, ,\;\textrm{ for all } r\leq r_0 .
\end{align}
The following is our main result. To simplify its statement, here we assume $\omega (r)\leq \omega(1)r$ for all $r\geq 0$.
\begin{teo}
\label{C1,alpha regularity estimates geral}
Assume $F$ satisfies \ref{SCmu}, $f \in L^p (\Omega)$, where $p>n$, and $\Omega\mathbb{S}ubset {\mathbb{R}^n} $ is a bounded domain.
Let $u$ be an $L^p$-viscosity solution of
\begin{align}\label{F=f}
F(x,u,Du,D^2 u)=f(x) \quad \textrm{ in }\;\Omega
\end{align}
with $\| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} \leq C_0$.
Then, there exists $\alpha\in (0,1)$ and $\theta=\theta (\alpha)$, depending on $n,p,\lambda,\Lambda,\|b\|_{L^p(\Omega)}$, such that if \eqref{Htheta} holds for all $r\leq \min \{ r_0, \mathrm{dist} (x_0,\partial\Omega)\}$, for some $r_0>0$ and for all $x_0 \in \Omega$, this implies that $u\in C^{1,\alpha}_{\mathrm{loc}} (\Omega)$ and for any subdomain $\Omega^\prime \mathbb{S}ubset\mathbb{S}ubset \Omega$,
\begin{align}\label{estim local}
\|u\|_{C^{1,\alpha}(\overline{\Omega^\prime})} \leq C \,\{ \| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} \}
\end{align}
where $C$ depends only on $\,r_0,n,p,\lambda,\Lambda,\alpha, \mu, \| b \|_{L^p (\Omega)},\omega(1)\| d \|_{L^\infty (\Omega)},\mathrm{diam} (\Omega)$, $\mathrm{dist} (\Omega^\prime,\partial\Omega), C_0$.
If in addition, $\partial\Omega\in C^{1,1}$ and $u\in C(\overline{\Omega})\cap C^{1,\tau} (\partial \Omega )$ is such that and $\| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} + \|u\|_{C^{1,\tau} (\partial\Omega)} \leq C_1$, then there exists $\alpha\in (0,\tau)$ and $\theta =\theta (\alpha)$, depending on $n,p,\lambda , \Lambda , \|b\|_{L^p(\Omega)}$, so that if \eqref{Htheta} holds for some $r_0>0$ and for all $x_0 \in \overline{\Omega}$, this implies that $u\in C^{1,\alpha}(\overline{\Omega})$ and satisfies the estimate
\begin{align}\label{estim global}
\|u\|_{C^{1,\alpha}(\overline{\Omega})} \leq C\, \{ \| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} + \|u\|_{C^{1,\tau} (\partial\Omega)} \}
\end{align}
where $C$ depends on $\,r_0,n,p,\lambda,\Lambda,\alpha,\mu$,$\| b \|_{L^p (\Omega)},\omega(1)\| d \|_{L^\infty (\Omega)},\mathrm{diam} (\Omega),C_1$ and on the $C^{1,1}$ diffeomorphisms that describe the boundary.
If $\mu=0$, then the constant $C$ does not depend on $C_0$, $C_1$.
\end{teo}
We also consider, as in \cite{Swiech} and chapter 8 in \cite{CafCab}, a slightly different (smaller) version of $\beta$,
\begin{align} \label{def beta bar}
\bar{\beta}(x,x_0)=\bar{\beta}_F (x,x_0):= \mathbb{S}up_{X\in \mathbf{S}^n} \frac{|F(x,0,0,X)-F(x_0,0,0,X)|}{\|X\|+1} .
\end{align}
Consider the hypothesis ${(\overline{H})}_\theta$, which is \eqref{Htheta} with $\beta$ replaced by $\bar{\beta}$. This hypothesis is trivially satisfied if $F(x,0,0,X)$ is uniformly continuous in $x$.
\begin{rmk}
If $\omega$ is an arbitrary modulus, we still have regularity and estimates for bounded~$b$, with the same dependence on constants as before, by adding 1 on the right hand side of \eqref{estim local} and \eqref{estim global}.
In this case, we can replace \eqref{Htheta} by ${(\overline{H})}_\theta$ in Theorem \ref{C1,alpha regularity estimates geral}, see remark \ref{Remark qualquer modulo}.
Of course, explicit zero order unbounded terms that only depend on $u$ and $x$, can always be handled as being part of the right hand side $f(x)$.
\end{rmk}
\begin{rmk}
If $\mu = 0$ we can also obtain Theorem \ref{C1,alpha regularity estimates geral} in terms of ${(\overline{H})}_\theta$, see remark \ref{remark mu=0}.
\end{rmk}
The proof of Theorem \ref{C1,alpha regularity estimates geral} is based on Caffarelli's iteration method. Compared to \cite{Swiech}, \cite{Winter}, we use a simplified rescaling of variable which allows us to carry out the proof, without needing to use a twice differentiability property of viscosity solutions (whose validity is unknown for unbounded coefficients).
We also use ideas of Wang to deal with superlinear terms.
The structure of the paper is as follows. In section \ref{Preliminaries} we recall some known results which are used along the text. In section \ref{proof main th} we give a detailed proof of theorem \ref{C1,alpha regularity estimates geral}, splitting it into local and boundary parts. The final sections \ref{W2,p regularity} and \ref{First eigenvalue} are devoted to applications. Section \ref{W2,p regularity} deals with $W^{2,p}$ regularity -- see theorem \ref{W2,p quad} for the main regularity result; we also present a generalized Nagumo's lemma \ref{Nagumo}.
Section \ref{First eigenvalue} is related to existence of eigenvalues for general operators with a nonnegative unbounded weight, see theorem \ref{exist eig for F c geq 0} (these results play an important role in \cite{multiplicidade}).
Significant contributions on eigenvalues of continuous operators in nondivergence form in bounded domains include the fundamental work \cite{BNV} for linear operators; \cite{QB} for convex fully nonlinear operators; \cite{MJ} for nonlocal operators; \cite{BD}, \cite{BDru}, \cite{L83}, and the recent \cite{BDPR} for degenerate elliptic operators. Theorem~\ref{exist eig for F c geq 0} is a slight improvement to the general existence theory about nonconvex operators possessing first eigenvalues in \cite{Arms2009} (see also \cite{IY}), since we are not supposing that our nonlinearity is uniformly continuous in $x$.
If, in addition, we have $W^{2,p}$ regularity of solutions, we can extend theorem \ref{exist eig for F c geq 0} even further, allowing an unbounded first order coefficient. Eigenvalues for fully nonlinear operators with such coefficients have been previously studied, to our knowledge, only for radial operators and eigenfunctions, in \cite{II1} and \cite{II2}.
As a particular case of theorem \ref{exist eig for F c geq 0}, we obtain the existence of positive eigenvalues with a nonnegative unbounded weight for the extremal Pucci's operators with unbounded coefficients.
\begin{prop}\label{corolPucci}
Let $\Omega\mathbb{S}ubset {\mathbb{R}^n} $ a bounded $C^{1,1}$ domain, $b,\, c\in L^p_+(\Omega)$, $c\gneqq 0$, for $p>n$. Then, there exists $\varphi_1^\pm\in W^{2,p}(\Omega)$ such that, for $\lambda_1^{\pm}$ defined in section \ref{First eigenvalue}, we have $\lambda_1^{\pm}>0$ and
\begin{align}
\left\{
\begin{array}{rclcc}
\mathcal{M}^\pm_{\lambda,\Lambda} (D^2 \varphi_1^\pm )\pm b(x)|D \varphi_1^\pm |+\lambda_1^\pm c(x) \varphi_1^\pm &=& 0 &\mbox{in} & \Omega \\
\varphi_1^\pm &>& 0 &\mbox{in} &\Omega \\
\varphi_1^\pm &=& 0 &\mbox{on} &\partial\Omega.
\end{array}
\right.
\end{align}
\end{prop}
\textit{Acknowledgments.} I would like to thank my Ph.D. advisor, Professor Boyan Sirakov, for years of patient guidance, many valuable discussions and helpful suggestions.
I am also deeply indebted to Prof.\ Diego Moreira, whose valuable remarks and suggestions led to a substantial improvement of the paper, in particular by clarifying some arguments.
\mathbb{S}ection{Preliminaries}\label{Preliminaries}
We start detailing the hypothesis \ref{SCmu}. Notice that the condition over the highest order term $X$, for $p=q$ and $ r=s$, implies that $F$ is a uniformly elliptic operator.
In \ref{SCmu},
$$
\mathcal{M}^+_{\lambda,\Lambda}(X):=\mathbb{S}up_{\lambda I\leq A\leq \Lambda I} \mathrm{tr} (AX)\;\;\textrm{ and} \quad \mathcal{M}^-_{\lambda,\Lambda}(X):=\inf_{\lambda I\leq A\leq \Lambda I} \mathrm{tr} (AX)
$$
are the Pucci's extremal operators. See, for example, \cite{CafCab} and \cite{QB} for their properties.
By \textit{modulus} we mean a function $\omega :[0,+\infty] \rightarrow [0,+\infty]$ continuous at $0$ with $\omega (0)=0$. We may consider $\omega$ increasing and continuous, up to replacing it by a larger function. We can also suppose $\omega$ subadditive, from where $\omega (k)\leq (k+1)\,\omega (1)$ for all $k\geq 0$.
Moreover, we may assume that $d$ is bounded everywhere, up to defining it in a zero measure set.
We stress that our results are not restricted to bounded zero order terms, since unbounded ones which only depend on $x$ and $u$ can always be treated as being part of the right hand side.
Next we recall the definition of $L^p$-viscosity solution.
\begin{defin}\label{def Lp-viscosity sol}
Let $f\in L^p_{\textrm{loc}}(\Omega)$.
We say that $u\in C(\Omega)$ is an $L^p$-viscosity subsolution $($respectively, supersolution$)$ of \eqref{F=f} if whenever $\phi\in W^{2,p}_{\mathrm{loc}}(\Omega)$, $\varepsilon>0$ and $\mathcal{O}\mathbb{S}ubset\Omega$ open are such that
\begin{align*}
F(x,u(x),D\phi(x),D^2\phi (x))-f(x) \leq -\varepsilon\;\;
( F(x,u(x),D\phi(x),D^2\phi (x))-f(x) \geq \varepsilon )
\end{align*}
for a.e. $x\in\mathcal{O}$, then $u-\phi$ cannot have a local maximum $($minimum$)$ in $\mathcal{O}$.
\end{defin}
We can think about $L^p$-viscosity solutions for any $p>\frac{n}{2}$, since this restriction makes all test functions $\phi\in W^{2,p}_{\mathrm{loc}}(\Omega)$ continuous and having a second order Taylor expansion \cite{CCKS}.
We are going to deal mostly with the case $p>n$. In particular, for $\Omega$ bounded with $\partial\Omega\in C^{1,1}$, this implies that the continuous injection $W^{2,p}(\Omega)\mathbb{S}ubset C^1(\overline{\Omega})$ is compact, for all $n\geq 1$.
Further, when $p>n$ and $F$ possesses the quadratic structure \ref{SCmu}, the maximum or minimum in the definition \ref{def Lp-viscosity sol} can taken to be strict (see for example proposition 1 in \cite{KoikePerronRevisited}).
If $F$ and $f$ are continuous in $x$, we can use the more usual notion of \textit{$C$-viscosity} sub and supersolutions, as in \cite{user}.
Both definitions are equivalent when, moreover, $F$ satisfies \ref{SCmu} for bounded $b$, with $\mu,d\equiv 0$ and $p\geq n$, by proposition 2.9 in \cite{CCKS}; we will be using them interchangeably, in this case, throughout the text.
A \textit{strong} sub or subsolution belongs to $W^{2,p}_{\mathrm{loc}}(\Omega)$ and satisfies the inequality at almost every point. Such notions are related, up to quadratic growth, as shows the next proposition.
\begin{prop}\label{Lpiffstrong.quad}
Assume $F$ satisfies \ref{SCmu} with $b\in L^q_+(\Omega)$, $q\geq p\geq n$, $q>n$ and $f\in L^p(\Omega)$.
Then, $u\in W^{2,p}_{\mathrm{loc}}(\Omega)$ is a strong subsolution $($supersolution$)$ of $F=f$ in $\Omega$ if and only if it is an $L^p$-viscosity subsolution $($supersolution$)$ of it.
\end{prop}
See theorem 3.1 and proposition 9.1 in \cite{KSweakharnack} for a proof, even for more general conditions on $\mu$ and the exponents $p,q$.
A \textit{solution} is always both sub and supersolution of the equation.
The next proposition follows from theorem 4 in \cite{arma2010} in the case $p=n$.
For a version with more general exponents and coefficients, we refer to proposition 9.4 in \cite{KSweakharnack}.
\begin{prop} \label{Lpquad} {$($Stability$)$}
Let $F$, $F_k$ operators satisfying \ref{SCmu},\,$b\in L^q_+(\Omega), q\geq p\geq n, q>n$, $f, \, f_k\in L^p(\Omega)$. Let $u_k\in C(\Omega)$ be an $L^p$-viscosity subsolution $($supersolution$)$ of $F_k=f_k$ i.e.
$$
F_k(x,u_k,Du_k,D^2u_k)\geq f_k(x) \;\;\;\textrm{in} \;\;\;\Omega\quad (\leq)\quad\textrm{ for all }\; k\in \mathbb{N} .
$$
Suppose $u_k\rightarrow u$ in $L_{\mathrm{loc}}^\infty (\Omega)$ as $k\rightarrow \infty$ and for each ball $B\mathbb{S}ubset\mathbb{S}ubset \Omega$ and $\varphi\in W^{2,p}(B)$, setting
\begin{align*}
g_k(x):=F_k(x,u_k,D\varphi,D^2\varphi)-f_k(x) ; \;\;
g(x):=F(x,u,D\varphi,D^2\varphi)-f(x),
\end{align*}
we have $\| (g_k-g)^+\|_{L^p(B)}$ $(\| (g_k-g)^-\|_{L^p(B)}) \rightarrow 0$ as $k\rightarrow \infty$. Then $u$ is an $L^p$-viscosity subsolution $($supersolution$)$ of $F=f$ i.e.
$F(x,u,Du,D^2u)\geq f(x)$ \;$ (\leq)\;$ in $\Omega$.
If $F$ and $f$ are continuous in $x$, then it is enough that the above holds for every $\varphi\in C^2 (B)$, in which case $u$ is a $C$-viscosity subsolution $($supersolution$)$ of $F=f$ in $\Omega$.
\end{prop}
\begin{rmk} \label{Lpquadencaixados}
Proposition \ref{Lpquad} is valid if we have $f_k \in L^p(\Omega_k)$, $u_k\in C(\Omega_k)$, for an increasing sequence of domains $\Omega_k \mathbb{S}ubset \Omega_{k+1}$ such that $\Omega :=\bigcup_{k\in \mathbb{N}} \Omega_k\,$, see proposition 1.5 in \cite{Winter}.
\end{rmk}
Denote $\mathcal{L}^\pm[u]:=\mathcal{M}^\pm_{\lambda,\Lambda} (D^2u)\pm b(x)|Du|$, where $b\in L^p_+(\Omega)$, and $F[u]:=F(x,u,Du,D^2u)$.
We recall Alexandrov-Bakelman-Pucci type results with unbounded ingredients and quadratic growth, which will be referred simply by ABP.
\begin{prop} \label{wABPquad}
Let $\Omega$ bounded, $\mu\geq 0$, $b\in L^q_+ (\Omega)$ and $f\in L^p (\Omega)$, for $q\geq p \geq n$, $q>n$.
Then, there exist $\; \mathrm{d}elta=\; \mathrm{d}elta (n,p,\lambda, \Lambda, \mathrm{diam}(\Omega), \|b\|_{L^q(\Omega)})>0$ such that if
\begin{align*}
\quad\quad\quad\quad \mu \|f^-\|_{L^p(\Omega)} \, (\mathrm{diam}(\Omega))^{\frac{n}{p}} \leq \; \mathrm{d}elta \quad\quad\quad (\mu \|f^+\|_{L^p(\Omega)} \, (\mathrm{diam}(\Omega))^{\frac{n}{p}} \leq \; \mathrm{d}elta )
\end{align*}
then every $u\in C(\overline{\Omega})$ which is an $L^p$-viscosity subsolution $($supersolution$)$ of
\begin{align*}
\mathcal{L}^+[u]+\mu |D u|^2 \geq f(x)\;\; \mathrm{in}\;\; \Omega\cap\{u>0\} \quad
\left( \mathcal{L}^-[u]-\mu |D u|^2 \leq f(x)\;\; \mathrm{in}\;\; \Omega\cap\{u<0\} \,\right)
\end{align*}
satisfies, for a constant $C_A$ depending on $n,p,\lambda,\Lambda, \|b\|_{L^q(\Omega)},\mathrm{diam}(\Omega)$, the estimate
\begin{align*}
\max_{\overline{\Omega}} u \leq \max_{\partial \Omega} u +C_A \, \|f^-\|_{L^p(\Omega)} \quad \left( \min_{\overline{\Omega}} u \geq \min_{\partial \Omega} u -C_A \|f^+\|_{L^p(\Omega)} \right).
\end{align*}
Moreover, $C_A$ remains bounded if these quantities are bounded.
\end{prop}
As a matter of fact, ABP is valid under more general conditions, even for unbounded $\mu$. We refer to theorem 2.6 and lemma 9.3 in \cite{KSweakharnack}, and theorem 3.4 in \cite{Naka}, for a precise dependence on constants (see also \cite{KSmpite2007} and \cite{KSexist2009}).
For a simplified proof in the case where $\mu> 0$ is constant and $p>n$ (which is the only superlinear case that we need along the text) we also refer to \cite{tese}.
\begin{prop}{$(C^\beta$ Regularity\,$)$}
\label{Cbetaquad}
Assume $F$ satisfies \ref{SCmu} for $N=0$, $q=0$, $s=0$ and $b\in L^q_+(\Omega)$, for $q\geq p \geq n$, $q>n$. Let $u\in C(\Omega)$ be an $L^p$-viscosity solution of \eqref{F=f} with $f\in L^p(\Omega)$. Then there exists $\beta\in (0,1)$ depending on $n,p,\lambda, \Lambda$ and $ \|b\|_{L^q(\Omega)}$ such that $u\in C^{\beta}_{\textrm{loc}}(\Omega)$ and for any subdomain $\Omega^\prime \mathbb{S}ubset\mathbb{S}ubset\Omega$ we have
\begin{align*}
\|u\|_{C^\beta(\Omega^\prime)}\leq K_1 \,\{\|u\|_{L^\infty (\Omega)} + \|f\|_{L^p(\Omega)}+\|d\|_{L^p(\Omega)}\,\omega(\|u\|_{L^\infty (\Omega)}) \}
\end{align*}
where $K_1$ depends only on $n,p,\lambda,\Lambda,\mu, \|b\|_{L^q(\Omega)}, \|u\|_{L^\infty (\Omega^\prime)}, \mathrm{dist}(\Omega^\prime, \partial\Omega)$.
If, in addition, $u\in C (\overline{\Omega})\cap C^\tau (\partial\Omega)$ and $\Omega$ satisfies a uniform exterior cone condition with size $L$, then there exists $\beta_0= \beta_0 (n,p,\lambda, \Lambda, L,\|b\|_{L^q(\Omega)}) \in (0,1)$ and $\beta = \min (\beta_0, \frac{\tau}{2})$ such that
\begin{align*}
\|u\|_{C^\beta(\overline{\Omega})}\leq K_1 \,\{\|u\|_{L^\infty (\Omega)} + \|f\|_{L^p(\Omega)} + \|u\|_{C^\tau (\partial\Omega)} +\|d\|_{L^p(\Omega)}\,\omega(\|u\|_{L^\infty (\Omega)})\}
\end{align*}
where $K_1$ depends on $n,p,\lambda,\Lambda,\mu, L, \|b\|_{L^q(\Omega)},\omega (1)\|d\|_{L^p(\Omega)} ,\mathrm{diam} (\Omega),\|u\|_{L^\infty (\Omega)}$.
In both cases, $K_1$ remains bounded if these quantities are bounded.
The same result holds if, instead of a solution of \eqref{F=f}, $u$ is only an $L^p$-viscosity solution of the inequalities
$\mathcal{L}^-[u]-\mu |Du|^2\leq g(x)$ and $ \mathcal{L}^+[u]+\mu |Du|^2\geq -g(x)$ in $\Omega$.
If $\mu=0$, the final constant does not depend on a bound from above on $\|u\|_{L^\infty (\Omega)}$.
\end{prop}
\begin{proof} This is a direct consequence of the proof of theorem 2 in \cite{arma2010}, reading the $L^n$-viscosity sense there as $L^p$-viscosity one, changing $b\in L^p, \, d,f\in L^n$ there by $b\in L^q, \,d,f\in L^p$. The corresponding growth lemmas and exponents concerning $\rho$ must be replaced by $\rho^{1-\frac{n}{p}}$, which appear by using proposition \ref{wABPquad} (for $\mu=0$) instead of theorem 3 there.
The zero order term is handled as being part of the right hand side, since the whole proof is valid if we only have $u$ as an $L^p$-viscosity solution of inequalities $\mathcal{L}^+ [u]\geq - g(x)$ and $\mathcal{L}^-[u]\leq g(x)$ in the case $\mu=0$ (see the final remark in the end of the proof of theorem 2 in \cite{arma2010}).
\end{proof}
Next we recall some important results concerning the strong maximum principle and Hopf lemma. For a proof for $L^p$-viscosity solutions with unbounded coefficients, see \cite{B2016}, which in particular generalize the results for $C$-viscosity solutions in \cite{BardidaLio}.
We will refer to them simply by \textit{SMP} and \textit{Hopf} throughout the text.
\begin{teo}{$($SMP\,$)$}\label{SMP}
Let $\Omega$ be a $C^{1,1}$ domain and $u$ an $L^p$-viscosity solution of $\mathcal{L}^-[u]-du\leq 0$, $u\geq 0$ in $\Omega$, where $d\in L^p(\Omega)$. Then either $u>0$ in $\Omega$ or $u\equiv 0$ in $\Omega$.
\end{teo}
\begin{teo}{$($Hopf\,$)$}\label{Hopf}
Let $\Omega$ be a $C^{1,1}$ domain and $u$ an $L^p$-viscosity solution of $\mathcal{L}^-[u]-du\leq 0$, $u> 0$ in $\Omega$, where $d\in L^p(\Omega)$. If $u(x_0)=0$ for some $x_0\in\partial\Omega$, then $\partial_\mathbb{N}u u(x_0)>0$, where $\partial_\mathbb{N}u$ is the derivative in the direction of the interior unit normal.
\end{teo}
In \cite{B2016}, theorems \ref{SMP} and \ref{Hopf} are proved for $d\equiv 0$, but exactly the same proofs there work for any coercive operator. Moreover, since the function $u$ has a sign, they are also valid for nonproper operators, by splitting the positive and negative parts of $d$ and using $d^- u\geq 0$.
We finish the section recalling some results about pure second order operators $F(D^2 u)$, i.e. uniformly elliptic operators $F$ depending only on $X$ (so Lipschitz continuous in $X$) and satisfying $F(0)=0$. These operators will play the role of $F(0,0,0,X)$ in the approximation lemmas.
The next proposition is corollary 5.7 in \cite{CafCab}, which deals with $C^{1,\bar{\alpha}}$ interior regularity.
\begin{prop}
\label{C1,baralpha}
Let $u$ be a $C$-viscosity solution of $F(D^2 u)=0$ in $B_1$. Then $u\in C^{1,\bar{\alpha}} (\overline{B}_{1/2})$ for some universal $\bar{\alpha}\in (0,1)$ and there exists a constant $K_2$, depending on $n,\lambda$ and $\Lambda$, such that
$$\|u\|_{C^{1,\bar{\alpha}} (\overline{B}_{1/2})} \leq K_2 \, \|u\|_{L^\infty (B_1)} . $$
\end{prop}
We also need the following result about solvability of the Dirichlet problem for $F(D^2 u)$.
\begin{prop}
\label{ExisUnicF(D2u)}
Let $\Omega$ satisfies a uniform exterior cone condition, $\psi\in C(\partial\Omega)$. Then there exists a unique $C$-viscosity solution $u\in C(\overline{\Omega})$ of
\begin{align*}
\left\{
\begin{array}{rclcc}
F(D^2 u)&=& 0 &\mbox{in} &\Omega\\
u &=& \psi &\mbox{on} & \partial\Omega\, .
\end{array}
\right.
\end{align*}
\end{prop}
\begin{proof}
Uniqueness is corollary 5.4 in \cite{CafCab}. Let us recall how to obtain existence via Perron's Method, proposition II.1 in \cite{IL90} (see also \cite{Ishii89}). Surely, comparison principle holds for $F(D^2 u)$ by theorem 5.3 and corollary 3.7 in \cite{CafCab}. Further, we obtain a pair of strong sub and supersolutions $\underline{u},\, \overline{u} \in W^{2,p}_{\mathrm{loc}}(\Omega) \cap C(\overline{\Omega})$ of Pucci's equations $\mathcal{M}^+ (D^2 \overline{u}) \leq 0 \leq \mathcal{M}^- (D^2 \underline{u})$ in $\Omega$ with $\underline{u}= \overline{u}=\psi$ on $\partial\Omega$ by lemma 3.1 of \cite{CCKS}. They are $L^p$ (so $C$) viscosity
sub and supersolutions of $F(D^2 u)=0$.
\end{proof}
We use the following notation from \cite{MilSilv} and \cite{Winter},
$$B_r^\mathbb{N}u (x_0):=B_r(x_0)\cap \{x_n >-\mathbb{N}u\} ,\;\;\mathbb{T}^{\mathbb{N}u}_r(x_0) :=B_r (x_0)\cap \{ x_n =-\mathbb{N}u \},\textrm{ for } r>0 ,\,\mathbb{N}u >0,$$
simply $\mathbb{T}:=B_1 \cap\{x_n=0\}$ and $B_r^+:=B_r \cap \{ x_n >0 \}$, where $B_r=B_r(0)$.
\begin{prop}
\label{C1,baralphaglobal}
Let $u\in C(\overline{B_1^\mathbb{N}u })$ be a $C$-viscosity solution of
\begin{align*}
\left\{
\begin{array}{rclcc}
F(D^2 u)& =& 0 &\mbox{in} & B_1^\mathbb{N}u \\
u &=&\psi &\mbox{on} &\mathbb{T}_1^\mathbb{N}u
\end{array}
\right.
\end{align*}
such that $\psi\in C(\partial B_1^\mathbb{N}u)\cap C^{1,\tau}(\mathbb{T}_1^\mathbb{N}u)$ for some $\tau >0$. Then $u\in C^{1,\bar{\alpha}}( \overline{B_{1/2}^\mathbb{N}u})$, where $\bar{\alpha}=\min (\tau, \alpha_0)$ for a universal $\alpha_0$. Moreover, for a constant $K_3$, depending only on $n,\lambda,\Lambda$ and $\tau$, we have
\begin{align*}
\|u\|_{C^{1,\bar{\alpha}}( \overline{B_{1/2}^\mathbb{N}u})}\leq K_3 \, \{ \|u\|_{L^\infty (B_1^\mathbb{N}u)} + \|\psi\|_{C^{1,\tau} (\mathbb{T}_1^\mathbb{N}u )} \}.
\end{align*}
\end{prop}
For a proof of proposition \ref{C1,baralphaglobal} see proposition 2.2 in \cite{MilSilv}; see also remark 3.3 in \cite{Winter}.
\mathbb{S}ection{Proof of theorem \ref{C1,alpha regularity estimates geral}.}\label{proof main th}
\mathbb{S}ubsection{Local Regularity}\label{local regularity}
Fix a domain $\Omega^\prime\mathbb{S}ubset\mathbb{S}ubset\Omega$.
Consider $K_1$ and $\beta$ the pair given by the $C^\beta$ local superlinear estimate (proposition \ref{Cbetaquad}) for $\Omega^\prime $, related to the initial $n,p,\lambda, \Lambda,\mu,\|b\|_{L^p(\Omega)}$, $\mathrm{dist}(\Omega^\prime,\partial\Omega)$ and $C_0$ such that
$$\|u\|_{C^\beta(\Omega^\prime)}\leq K_1 \,\{\|u\|_{L^\infty (\Omega)} + \|f\|_{L^p(\Omega)}+\|d\|_{L^p(\Omega)}\,\omega(\|u\|_{L^\infty (\Omega)})\}.$$
Also, let $K_2$ (which we can suppose greater than 1) and $\bar{\alpha}$ be the constants of $C^{1,\bar{\alpha}}$ local estimate (proposition \ref{C1,baralpha}) associated to $n,\lambda,\Lambda$ in the ball $B_1(0)$.
By taking $K_1$ larger and $\beta$ smaller, we can suppose $K_1\geq \widetilde{K}_1$ and $\beta\leq \widetilde{\beta}$, where $\widetilde{K}_1,\widetilde{\beta}$ is the pair of $C^\beta$ local estimate in the ball $B_1$ (or $B_{1/2}$), with respect to an equation with given constants $n,p,\lambda,\Lambda$ and bounds for the coefficients $\mu\leq 1$, $\|b\|_{L^p(B_2)}\leq 1+2K_2|B_1|^{1/p}$ and $\omega(1)\|d\|_{L^p(B_2)}\leq 1$, for all solutions in the ball $B_2$ with $\|u\|_{L^\infty(B_2)}\leq 1$ (or for all solutions in the ball $B_1$ with bounds on the coefficients in $B_1$).
The first step is to approximate our equation with one which already has the corresponding regularity and estimates that we are interested in. Denote $\|\cdot\|_{ p}=\|\cdot\|_{ L^p (B_1)}$.
\begin{lem}
\label{AproxLem}
Assume $F$ satisfies \ref{SCmu} in $B_1$, $f \in L^p (B_1)$, where $p>n$. Let $\psi \in C^\tau (\partial B_1)$ with $\|\psi\|_{C^\tau (\partial B_1)}\leq K_0$.
Moreover, set $L(x)=A+B\cdot x$ in $B_1$, for $A\in \mathbb{R}$, $B\in {\mathbb{R}^n} $.
Then, for every $\varepsilon>0$, there exists $\; \mathrm{d}elta\in (0,1)$, $\; \mathrm{d}elta=\; \mathrm{d}elta (\varepsilon,n,p,\lambda,\Lambda,\tau,K_0)$, such that
\begin{align*}
\|{\bar{\beta}}_F(\cdot,0)\|_{ p}\, ,\;\|f\|_{ p}\, ,\; \mu(|B|^2 +|B|+1) \, ,\;
\|b\|_{p}(|B|+1) \, ,\;\omega (1) \|d\|_{ p}(|A|+|B|+1)\leq \; \mathrm{d}elta
\end{align*}
imply that any two $L^p$-viscosity solutions $v$ and $h$ of
\begin{align*}
\left\{
\begin{array}{rclcc}
F(x,v+L(x),Dv+B,D^2v)&=& f(x)& \mbox{in} & B_1 \\
v &=& \psi \; & \mbox{on} & \partial B_1
\end{array}
\right., \;
\left\{
\begin{array}{rclcc}
F(0,0,0,D^2h)&=& 0 & \mbox{in} & B_1 \\
h &=&\psi & \textrm{on} & \partial B_1
\end{array}
\right.
\end{align*}
respectively, with $\omega (1) \|d\|_{ p}\|v\|_{\infty}\leq \; \mathrm{d}elta$, satisfy $\|v-h\|_{L^\infty (B_1)}\leq \varepsilon$.
\end{lem}
\begin{proof}
In this proof denote $\alpha_n$ as the measure of the ball $B_1(0)$.
We are going to prove that for all $\varepsilon>0$, there exists a $\; \mathrm{d}elta\in (0,1)$ satisfying the above, with $\; \mathrm{d}elta\leq 2^{-\frac{n}{2p}}C_n^{-\frac{1}{2}}\,\widetilde{\; \mathrm{d}elta}^{1/2}$, where $\widetilde{\; \mathrm{d}elta}$ is the constant from proposition \ref{wABPquad} for $\tilde{b}=b+2|B|\mu$, $C_n=4+2\alpha_n^{{1}/{p}}$.
Assume the conclusion is not satisfied, then there exist some $\varepsilon_0>0$ and a sequence of operators $F_k$ satisfying $(SC)^{\mu_k}$ for $b_k,\, d_k\in L^p_+(B_1)$, $\mu_k\geq 0$, $\omega_k$ modulus, also $f_k\in L^p(B_1)$, $A_k\in \mathbb{R}$, $B_k\in {\mathbb{R}^n} $, $L_k(x)=A_k+B_k\cdot x$, and $\; \mathrm{d}elta_k\in(0,1)$ such that
$
\; \mathrm{d}elta_k \leq 2^{-\frac{n}{2p}}C_n^{-\frac{1}{2}} \,\widetilde{\; \mathrm{d}elta}_k^{1/2}$ for all $k\in\mathbb{N}$,
where $\widetilde{\; \mathrm{d}elta}_k$ is the number from ABP related to $\tilde{b}_k=b_k+2|B_k|\mu_k$, in addition to
\begin{align*}
\|{\bar{\beta}}_{F_k}(\cdot,0)\|_{ p},\; \|f_k\|_{p} ,\; \mu_k (|B_k|^2+|B_k|+1),\;
\|b_k\|_{ p}(|B_k|+1),\; \omega_{k}(1) \|d_k\|_{ p}(|A_k|+|B_k|+1)\leq \; \mathrm{d}elta_k
\end{align*}
with $\; \mathrm{d}elta_k\rightarrow 0$, and $v_k$, $h_k\in C(\overline{B}_1)$ $L^p$-viscosity solutions of
\begin{align*}
\left\{
\begin{array}{rclc}
F_k(x,v_k+L_k(x),Dv_k+B_k,D^2v_k)&=&f_k(x) & B_1 \\
v_k &=& \psi_k & \partial B_1
\end{array}
\right.
, \;
\left\{
\begin{array}{rclc}
F_k(0,0,0,D^2h_k)&=& 0 & B_1 \\
h_k &=& \psi_k & \partial B_1
\end{array}
\right.
\end{align*}
where $\|\psi_k\|_{C^\tau (\partial B_1)}\leq K_0$, $\omega_k (1) \|d_k\|_{ p}\|v_k\|_{\infty}\leq \; \mathrm{d}elta_k$, but
$\|v_k-h_k\|_{L^\infty (B_1)} >\varepsilon_0.$
We first claim that
\begin{align} \label{v,hbound}
\|v_k\|_{L^\infty (B_1)}\,, \;\|h_k\|_{L^\infty (B_1)}\leq C_0
\end{align}
for large $k$, where $C_0=C_0(n,p,\lambda,\Lambda, K_0)$.
Indeed, in the first place, since we have $\mathcal{M}^-(D^2h_k) \leq 0\leq \mathcal{M}^+(D^2h_k)$ in the viscosity sense, we obtain directly that
$
\|h_k\|_{L^\infty (B_1)}\leq \|\psi_k\|_{L^\infty (\partial B_1)}\leq K_0.
$
For $v_k$, we initially observe that
$$
2^{\frac{n}{p}} C_n\, \mu_k \, \; \mathrm{d}elta_k \leq 2^{\frac{n}{p}} C_n \,\; \mathrm{d}elta_k^2\leq \widetilde{\; \mathrm{d}elta}_k\, , \;\;\; \textrm{for all } k\in\mathbb{N}.
$$
Further, $v_k$ is an $L^p$-viscosity solution of
$$\mathcal{{L}}_k^+[v_k]+\mu_k|D v_k|^2 +d_k(x) \omega_k (|v_k+L_k|)+g_k \geq f_k \geq \mathcal{{L}}_k^-[v_k]-\mu_k |Dv_k|^2-d_k(x)\omega_k(|v_k+L_k|)-g_k.$$
Here $\mathcal{{L}}_k^\pm[w]=\mathcal{M}^\pm(D^2w)\pm \widetilde{b}_k|Dw|$, and $g_k(x)=b_k(x)|B_k|+\mu_k |B_k|^2$.
Then, applying ABP in its quadratic form in $B_1(0)$, with RHS $d_k \,\omega_k (|v_k+L_k|)+g_k+|f_k|$, we obtain that
\begin{align*}
\|v_k\|_{L^\infty(B_1)} &\leq\| v_k\|_{L^\infty(\partial B_1)} +C_A^k \, \{\|{f}_k\|_{p}+\|{g}_k\|_{p} +\|d_k\|_{p}\,\omega_k(1) (|A_k|+|B_k|+\|v_k\|_{L^\infty (B_1)}+1) \}.
\end{align*}
Since $\|\tilde{b}_k\|_{L^n(B_1)}\leq \alpha_n^{\frac{p-n}{np}}$ for large $k$, then the constant in ABP is uniformly bounded, say $C_A^k\leq C_A$. Using the assumptions and $C_A\,\omega_k(1)\|d_k\|_{p}\leq 1/2$ for large $k$ as in \cite{Winter}, we obtain that $\|v_k\|_{L^\infty(B_1)} \leq C_0 $, with $C_0=C_0(n,p,\lambda,\Lambda, K_0)$, proving the claim \eqref{v,hbound}.
Then, by the $C^\beta$ global estimate (proposition \ref{Cbetaquad}), there exists $\beta\in (0,1)$ such that
\begin{align*}
\|v_k\|_{C^\beta (\overline{B}_1)}\,, \;\|h_k\|_{C^\beta (\overline{B}_1)}\leq C ,
\end{align*}
where $\beta=\min {(\beta_0,\frac{\tau}{2})}$ for some $\beta_0=\beta_0 (n,p, \lambda, \Lambda)$, $C=C(n,p, \lambda, \Lambda, C_0)$. Here, $\beta$ and $C$ do not depend on $k$, since $\mu_k, \, \|\widetilde{b}_k\|_{L^p(B_1)}, \, \omega_k (1)\|d_k\|_{L^p(B_1)}, \, \|f_k\|_{L^p(B_1)}\leq 1$ for large $ k$.
Then, by the compact inclusion $C^\beta(\overline{B}_1) \mathbb{S}ubset C(\overline{B}_1)$ we have, up to subsequences, that
$$
v_k\longrightarrow v_\infty\,, \;\;\; h_k\longrightarrow h_\infty\;\;\; \textrm{in}\; C(\overline{B}_1)\;\;\;\; \textrm{as}\;{k\rightarrow\infty},
$$
for some $v_\infty, \; h_\infty \in C(\overline{B}_1)$ with $v_\infty = h_\infty = \psi_\infty$ on $\partial B_1$. Moreover, by Arzelà-Ascoli theorem, a subsequence of $F_k(0,0,0,X)$ converges uniformly on compact sets of $\mathbb{S}^n$ to some uniformly elliptic operator $F_\infty(X)$, since $ \mathcal{M}^-_{\lambda,\Lambda}(X-Y) \leq F_k(0,0,0,X)-F_k(0,0,0,Y) \leq \mathcal{M}^+_{\lambda,\Lambda}(X-Y)$.
We claim that both $v_\infty$ and $h_\infty$ are viscosity solutions of
\begin{align*}
\left\{
\begin{array}{rclcc}
F_\infty (D^2 u)&=&0 & \mbox{in} & B_1 \\
u&=&\psi_\infty & \mbox{on} & \partial B_1 \,.
\end{array}
\right.
\end{align*}
This implies that they are equal, by proposition \ref{ExisUnicF(D2u)},
which contradicts $\|v_\infty-h_\infty\|_{L^\infty (B_1)}\geq \varepsilon_0$.
The claim for $h_\infty$ follows by taking the uniform limits at the equation satisfied by $h_k$. On the other hand, for $v_\infty$ we apply stability (proposition \ref{Lpquad}) by noticing that, for $\varphi\in C^2(B_1)$,
\begin{align*}
F_k(x,v_k+L_k,D\varphi+B_k, D^2 \varphi)-f_k(x)-F_\infty (D^2 \varphi)= \{ F_k(x,v_k+L_k,D\varphi+B_k, D^2 \varphi)-\\ F_k(x,0,0, D^2 \varphi) \}
+\{ F_k(x,0,0, D^2 \varphi)- F_k(0,0,0, D^2 \varphi) \}
+ \{ F_k(0,0,0, D^2 \varphi) - F_\infty (D^2 \varphi) \} -f_k(x)
\end{align*}
and that each one of the addends in braces tends to zero in $L^p$ as $k\rightarrow\infty$. Indeed, the first one in modulus is less or equal than
$\mu_k (|D\varphi |^2+2B_k|D\varphi|+|B_k|^2)+ b_k(x)( |D\varphi|+|B_k|)+\omega_k (\|v_k\|_{\infty } +|A_k|+|B_k| ) \, d_k(x)$, so its $L^p$-norm is bounded by $\mu_k \alpha_n^{1/p} \|D\varphi\|_{\infty}^2 + \| b_k\|_{p} \|D\varphi\|_{\infty} + (C_0+1)\,\omega_{k}(1) \, \| d_k\|_{p} +C_n\; \mathrm{d}elta_k$; while the $L^p$-norm of the second and third are bounded by $\|{\bar{\beta}}_{F_k}(\cdot,0)\|_{p} (\|D^2 \varphi \|_{\infty} +1)$ and $\alpha_n^{{1}/{p}} \| F_k(0,0,0, D^2 \varphi) - F_\infty (D^2 \varphi) \|_{L^\infty (B_1)}$ respectively, what concludes the proof.
\end{proof}
\begin{proof}[\textit{Proof of Local Regularity Estimates in the set $\Omega^\prime$.}] The main difference from the case $\mu= 0$, in the present proof, consists of defining a slightly different scaling on the function, which allows us to have $\mu$ small in order to obtain the conditions of the approximation lemma \ref{AproxLem}. For this, we will bring forward an argument due to Wang \cite{W2}, that uses the $C^\beta$ regularity of $u$.
Set $W:=\| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)}+\|d\|_{L^p(\Omega)}\,\omega(\|u\|_{L^\infty (\Omega)})$, which is less or equal than $W_0$, a constant that depends on $C_0$ and $\omega (1)\|d\|_{L^p(\Omega)}$.
Suppose, to simplify the notation, that $0\in \Omega^\prime$ and set $s_0:=\min (r_0, \mathrm{dist}(0,\partial\Omega^\prime))$. Recall that this $r_0 = r_0 (\theta)$ is such that \eqref{Htheta} holds for all $r\leq \min \{ r_0, \mathrm{dist} (x_0,\partial\Omega)\}$, for all $x_0 \in \Omega$. We will see, in the sequel, how the choice of $\theta$ is done.
We start assigning some constants. Fix an $\alpha\in(0,\bar{\alpha})$ with $\alpha\leq\min (\beta,1- {n}/{p} )$. Then, choose $\gamma=\gamma(\alpha,\bar{\alpha},K_2)\in (0,\frac{1}{4}]$ such that
\begin{align} \label{gamma}
2^{2+\bar{\alpha}}K_2\, \gamma^{\bar{\alpha}}\leq \gamma^{\alpha}
\end{align}
and define
\begin{align} \label{epsilon}
\varepsilon=\varepsilon (\gamma):=K_2\, (2\gamma)^{1+\bar{\alpha}}.
\end{align}
This $\varepsilon$ provides a $\; \mathrm{d}elta=\; \mathrm{d}elta(\varepsilon)\in (0,1)$, the constant of the approximation lemma \ref{AproxLem} that, up to diminishing, can be supposed to satisfy
\begin{align} \label{delta}
(5+2K_2)\,\; \mathrm{d}elta \leq \gamma^{\alpha}.
\end{align}
Now let $\mathbb{S}igma=\mathbb{S}igma (s_0,n,p,\alpha , \bar{\alpha},\beta, \; \mathrm{d}elta,\mu ,\|b\|_{L^p (\Omega)},\omega(1)\|d\|_{L^\infty (\Omega)},K_1,K_2,C_0) \leq \frac{s_0}{2}$ such that
\begin{align} \label{sigma}
\mathbb{S}igma^{\min{({1-\frac{n}{p}},\beta)}} m \leq {\; \mathrm{d}elta} \,\{ {32K^2(K_2+K+1)|B_1|^{1/p}}\}^{-1}
\end{align}
where $m:=\max{ \{ 1, \|{b}\|_{L^p (\Omega)}, \omega(1)\|{d}\|_{L^\infty (\Omega)},\mu (1+2^\beta K_1) W_0 \} }$. Consider the constant $K (\gamma, \alpha, K_2)$ defined as $K={K_2}\,{{\gamma^{-\alpha}}(1-\gamma^\alpha)^{-1}} +{K_2}\,{{\gamma^{-1-\alpha}}(1-\gamma^{1+\alpha})^{-1}} $ which is greater than $K_2\geq 1$. Hence, in particular, $\overline{B}_{2\mathbb{S}igma} (0)\mathbb{S}ubset \Omega^\prime$ and we can define
\begin{align*}
N=N_\mathbb{S}igma (0):= \mathbb{S}igma W + \mathbb{S}up_{x\in B_2} |u(\mathbb{S}igma x) - u(0)|.
\end{align*}
By construction and $C^\beta$ local quadratic estimate, $N$ is uniformly bounded by
\begin{align}\label{N}
\mathbb{S}igma W \leq N \leq (\mathbb{S}igma + 2^\beta K_1 \mathbb{S}igma ^\beta)W
\leq (1+2^\beta K_1) W_0\,\mathbb{S}igma ^\beta .
\end{align}
\begin{claim}\label{claim C1,alpha local 1a mud.var.}
$\widetilde{u}(x):=\frac{1}{N} \{ u(\mathbb{S}igma x)-u(0) \}$
is an $L^p$-viscosity solution of $\widetilde{F}[\,\widetilde{u}\,]=\widetilde{f}(x)$ in $B_2$, where
$$\widetilde{F}(x,r,p,X):= \frac{\mathbb{S}igma^2}{N} F\left( \mathbb{S}igma x,Nr+u(0),\frac{N}{\mathbb{S}igma}p, \frac{N}{\mathbb{S}igma^2} X \right)-\frac{\mathbb{S}igma^2}{N} F\left( \mathbb{S}igma x,u(0),0, 0 \right)$$
and $\widetilde{f}:=\widetilde{f}_1+\widetilde{f}_2$ for
\begin{align*}
\widetilde{f}_1(x):={\mathbb{S}igma^2} f(\mathbb{S}igma x)/N, \quad
\widetilde{f}_2(x):= -
{\mathbb{S}igma^2} F\left( \mathbb{S}igma x,u(0),0, 0 \right)/N,
\end{align*}
with $\widetilde{F}$ satisfying $(\widetilde{SC})^{\widetilde{\mu}}$ for $\widetilde{b}(x):= \mathbb{S}igma b(\mathbb{S}igma x)$, $\widetilde{\mu}:=N\mu$,
$\widetilde{d}(x):= \mathbb{S}igma^2 d(\mathbb{S}igma x)$ and $\widetilde{\omega}(r):=\omega(Nr)/N $.
\end{claim}
\begin{proof}
Let $\varepsilon >0$ and $\widetilde{\varphi}\in W^{2,p}_{\textrm{loc}}(B_2)$ such that $\widetilde{u}-\widetilde{\varphi}$ has a minimum (maximum) at $x_0\in B_2$.
Define $\varphi (x):={N} \widetilde{\varphi} (x/\mathbb{S}igma)+u(0)$ in $B_{2\mathbb{S}igma} (0)$ and notice that $u-\varphi$ has a minimum (maximum) at $\mathbb{S}igma x_0 \in B_{2\mathbb{S}igma}$. Since $u$ is an $L^p$-viscosity solution on $B_{2\mathbb{S}igma}$, for this $\varepsilon >0$ there exists $r>0$ such that
\begin{align*}
F(\mathbb{S}igma x, u(\mathbb{S}igma x), D\varphi (\mathbb{S}igma x),D^2\varphi (\mathbb{S}igma x)) \leq (\geq)\, f(\mathbb{S}igma x)+(-)\,{N\varepsilon}/{\mathbb{S}igma^2} \quad \textrm{a.e. in }B_{r}(x_0),
\end{align*}
which is equivalent to
\begin{align*}
\frac{\mathbb{S}igma^2}{N}&F\left(\mathbb{S}igma x, N \widetilde{u}(x)+u(0),\frac{N}{\mathbb{S}igma}D \widetilde{\varphi}(x),\frac{N}{\mathbb{S}igma^2}D^2\widetilde{\varphi} (x)\right) \leq (\geq)\,\frac{\mathbb{S}igma^2}{N} f(\mathbb{S}igma x)+ (-)\,\varepsilon \quad \textrm{a.e. in }B_{r}(x_0).
\end{align*}
Adding $-{\mathbb{S}igma^2} F\left( \mathbb{S}igma x,u(0),0, 0 \right)/N$ in both sides, we have
\begin{align*}
\widetilde{F}(x, \widetilde{u}(x),D \widetilde{\varphi}(x),D^2\widetilde{\varphi}(x)) \leq (\geq)\, \widetilde{f}( x)+ (-)\,\varepsilon \quad \textrm{a.e. in }B_{r}(x_0).
\end{align*}
Furthermore, $\widetilde{F}(x,0,0,0)= 0\,$ for $x\in B_2$ and for all $r\in \mathbb{R}$, $p\in {\mathbb{R}^n} $, $X\in \mathbb{S}^n$, we have
\begin{align*}
\widetilde{F} &(x,r,p,X)-\widetilde{F}(x,s,q,Y)\\
&= \frac{\mathbb{S}igma^2}{N} \left\{ F\left( \mathbb{S}igma x, Nr+u(0),\frac{N}{\mathbb{S}igma}p,\frac{N}{\mathbb{S}igma^2} X\right)- F\left( \mathbb{S}igma x, Ns+u(0),\frac{N}{\mathbb{S}igma}q,\frac{N}{\mathbb{S}igma^2} Y\right) \right\} \\
&\leq \mathcal{M}^+_{\lambda,\Lambda} (X-Y)+\mathbb{S}igma b(\mathbb{S}igma x) \,|p-q|+N\mu |p-q|(|p|+|q|)+\mathbb{S}igma^2 d(\mathbb{S}igma x) \,{\omega(N|r-s|)}/{N}\\
&=\mathcal{M}^+_{\lambda,\Lambda}(X-Y)+\widetilde{b}(x)|p-q|+\widetilde{\mu} |p-q|(|p|+|q|) +\widetilde{d}(x)\,{\widetilde{\omega} }(|r-s|).
\end{align*}
The estimate from below in $(\widetilde{SC})^{\widetilde{\mu}}$ is analogous.
\qedhere{\textit{Claim \ref{claim C1,alpha local 1a mud.var.}.}}
\end{proof}
Notice that, with this definition and the choice of $\mathbb{S}igma$ in \eqref{sigma}, we have
\begin{itemize}
\item $\|\widetilde{u}\|_{L^\infty (B_2)} \leq 1$ since $N\geq \mathbb{S}up_{B_2} |u(\mathbb{S}igma x)-u(0)|$;
\item $\|\widetilde{f}_1\|_{L^p (B_2)}=\frac{\mathbb{S}igma^{2-\frac{n}{p}}}{N}\|{f}\|_{L^p (B_{2\mathbb{S}igma})}\leq \mathbb{S}igma^{1-\frac{n}{p}} \frac{\|{f}\|_{L^p (\Omega)}}{W} \leq \frac{\; \mathrm{d}elta}{16}$;
\item $\|\widetilde{f}_2\|_{L^p (B_2)}\leq \frac{\mathbb{S}igma^{2-\frac{n}{p}}}{N} \omega (|u(0)|) \,\|d\|_{L^p (B_{2\mathbb{S}igma})}$ $\leq \mathbb{S}igma^{1-\frac{n}{p}}\frac {\omega (\|u\|_\infty) \|d\|_{L^p (\Omega)} }{W} \leq \frac{\; \mathrm{d}elta}{16}$\,; thus $\|\widetilde{f}\|_{L^p (B_2)}\leq \frac{\; \mathrm{d}elta}{8}$;
\item $\widetilde{\mu}=N\mu\leq (1+2^\beta K_1)W_0 \, \mu \, \mathbb{S}igma^\beta \leq \frac{\; \mathrm{d}elta}{8K^2|B_1|^{1/p} }$;
\item $\|\widetilde{b}\|_{L^p (B_2)}= \mathbb{S}igma^{1-\frac{n}{p}} \|{b}\|_{L^p (B_{2\mathbb{S}igma})} \leq \frac{\; \mathrm{d}elta}{16K} $;
\item $\widetilde{\omega}(1) \|\widetilde{d}\|_{L^p (B_2)}={\mathbb{S}igma^{2-\frac{n}{p}}} \frac{\omega(N)}{N} \|{d}\|_{L^p (B_{2\mathbb{S}igma})}\leq \mathbb{S}igma^{2-\frac{n}{p}} \omega(1) \|{d}\|_{L^p (\Omega)} \leq \frac{\; \mathrm{d}elta}{32(K_2+K+1)}$ from the hypothesis $\omega(r)\leq \omega (1)r$ for all $r\geq 0$;
\item $\|{\bar{\beta}}_{\widetilde{F}}(\cdot,0)\|_{L^p (B_1)} \leq \; \mathrm{d}elta/4 $, by choosing $\theta=\; \mathrm{d}elta/8$. Indeed,
\end{itemize}
\begin{align}\label{mudandobetabar}
\bar{\beta}_{\widetilde{F}}(x,x_0)&\leq \frac{\mathbb{S}igma^2}{N}\mathbb{S}up_{X\in \mathbb{S}^n} \frac{| F(\mathbb{S}igma x,u(0),0,\frac{N}{\mathbb{S}igma^2}X)-F(\mathbb{S}igma x,0,0,\frac{N}{\mathbb{S}igma^2}X)|}{\|X\|+1} \mathbb{N}onumber\\
&\quad +\mathbb{S}up _{X\in \mathbb{S}^n} \frac{| F(\mathbb{S}igma x,0,0,\frac{N}{\mathbb{S}igma^2}X)-F(\mathbb{S}igma x_0,0,0,\frac{N}{\mathbb{S}igma^2}X)|}{\frac{N}{\mathbb{S}igma^2}(\|X\|+1)} \mathbb{N}onumber\\
&\quad +\frac{\mathbb{S}igma^2}{N}\mathbb{S}up_{X\in \mathbb{S}^n} \frac{| F(\mathbb{S}igma x_0,0,0,\frac{N}{\mathbb{S}igma^2}X)-F(\mathbb{S}igma x_0,u(0),0,\frac{N}{\mathbb{S}igma^2}X)|}{\|X\|+1} \mathbb{N}onumber\\
&\quad + \frac{\mathbb{S}igma^2}{N}\mathbb{S}up_{X\in \mathbb{S}^n} \frac{| F(\mathbb{S}igma x,u(0),0,0)|+|F(\mathbb{S}igma x_0,u(0),0,0)|}{\|X\|+1}\mathbb{N}onumber\\
&\leq \frac{2\mathbb{S}igma^2}{N} \{d(\mathbb{S}igma x)+d(\mathbb{S}igma x_0)\} \,\omega(|u(0)|)\mathbb{S}up_{X\in \mathbb{S}^n} ({\|X\|+1} )^{-1}+\beta_F(\mathbb{S}igma x,\mathbb{S}igma x_0)
\end{align}
and therefore,
\begin{align*}
\|\bar{\beta}_{\widetilde{F}}(\cdot,0)\|_{L^p (B_1)}&\leq 4 \mathbb{S}igma |B_1|^{\frac{
1}{p}}\, \frac{\omega (\|u\|_{L^\infty(\Omega)})\|d\|_{L^\infty(\Omega)}}{W} + \left( \frac{1}{\mathbb{S}igma^n} \int_{B_\mathbb{S}igma(0)} \beta_F(y,0)^p \mathrm{d}y \right)^{\frac{1}{p}}\\
&\leq \; \mathrm{d}elta/8 +\theta\,=\, \; \mathrm{d}elta/4.
\end{align*}
In particular $\widetilde{F}, \widetilde{u}, \widetilde{\mu}, \widetilde{b}, \widetilde{d}, \widetilde{\omega}, A=0, B=0$ satisfy lemma \ref{AproxLem} hypotheses. Further, if we show $\|\widetilde{u}\|_{C^{1,\alpha}(\overline{B}_1)} \leq C$, we will obtain
$
\| {u(\mathbb{S}igma x)-u(0)} \|_{C^{1,\alpha}(\overline{B}_1)} \leq CN \leq (1+2^\beta K_1)C W
$
by \eqref{N}, then
$$
\| {u} \|_{C^{1,\alpha}(\overline{B}_\mathbb{S}igma)}\leq C\, \{ \| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)}\} ,
$$
where the constant depends on $\mathbb{S}igma$; the local estimate following by a covering argument.
\begin{rmk}\label{Remark qualquer modulo} In the case we have an arbitrary modulus of continuity, we define
$
N= \mathbb{S}igma \max \{W,1\} \\ + \mathbb{S}up_{x\in B_2} |u(\mathbb{S}igma x) - u(0)|,
$
which by construction and $C^\beta$ local superlinear estimate,
$$
\mathbb{S}igma \leq N \leq (\mathbb{S}igma + 2^\beta K_1 \mathbb{S}igma ^\beta)\max\{W,1\}
\leq (1+2^\beta K_1) W_0\,\mathbb{S}igma ^\beta \leq 1.
$$
Then we have
$\widetilde{\omega}(1) \|\widetilde{d}\|_{L^p (B_2)}=\frac{\mathbb{S}igma^{2-\frac{n}{p}}}{N} \omega(N) \|{d}\|_{L^p (B_{2\mathbb{S}igma})}\leq \mathbb{S}igma^{1-\frac{n}{p}} \omega(1) \|{d}\|_{L^p (\Omega)} $.
Moreover, we can consider the smallness assumption in terms of $(\overline{H})_\theta$, with $\bar{\beta}$ instead of $\beta$ in \eqref{mudandobetabar}. In fact, in this case we use $N/{\mathbb{S}igma^2}\geq 1$. In the end, we obtain that the original function $u$ is such that $\|u\|_{C^{1,\alpha}(\overline{\Omega})}$ is bounded by $C\max\{W,1\}\leq C (W+1)$, in place of $CW$.
\end{rmk}
Notice that the only place we had to use the dependence on the bound $C_0$ is to measure the smallness of $\mu$. Thus, if $\mu=0$, the final constant does not depend on $W_0$, neither on $C_0$.
\begin{rmk}\label{remark mu=0}
Still for $\mu=0$, if we split our analysis in two cases $($as usual for linear growth in the gradient, see for instance \cite{Bbook}$\,)$, then we can obtain the conditions in terms of $(\overline{H}_\theta)$. Indeed, set $N:=W$. If $N\leq 1$, we just define $\widetilde{u}=u (\mathbb{S}igma x)$ and use that each of the addends in $W$ is less or equal than $1$, and
$\widetilde{\omega}(1) \|\widetilde{d}\|_{L^p (B_2)}={\mathbb{S}igma^{2-\frac{n}{p}}} {\omega(N)}\|{d}\|_{L^p (B_{2\mathbb{S}igma})}\leq \mathbb{S}igma^{2-\frac{n}{p}} \omega(1) \|{d}\|_{L^p (\Omega)}$; this yields the estimate
$\|u\|_{C^{1,\alpha}}\leq C\leq C (N+1)$.
If $N\geq 1$, $\widetilde{u}$ is as in claim \ref{claim C1,alpha local 1a mud.var.}, and using $N/{\mathbb{S}igma^2}\geq 1$ we can replace $\beta$ by $\bar{\beta}$ in \eqref{mudandobetabar}; for the estimate in $\widetilde{\omega}(1) \|\widetilde{d}\|_{L^p (B_2)}$ we only need $\omega(r)\leq \omega (1)r$ for $r\geq 1$.
\end{rmk}
With these rescalings in mind, we write $F,u,M, \mu,b,d,\omega$ as the shorthand notation for $\widetilde{F},\widetilde{u}, \widetilde{\mu},\widetilde{b},\widetilde{d},\widetilde{\omega}$.
Now we can proceed with Caffarelli's iterations as in \cite{Caf89}, \cite{CafCab}, \cite{Swiech}, which consists of finding a sequence of linear functions $l_k (x):=a_k +b_k \cdot x$ such that
$(i)_k \;\;\| u-l_k \|_{L^\infty (B_{r_k})} \leq r_k^{1+\alpha}$
$(ii)_k \;\;|a_k-a_{k-1}| \leq K_2 \,r_{k-1}^{1+\alpha}\;,\;\; |b_k - b_{k-1}|\leq K_2 \,r_{k-1}^\alpha$
$(iii)_k \;\;|(u-l_k)(r_k x)-(u-l_k)(r_k y)|\leq (1+3K_1) \, r_k^{1+\alpha} |x-y|^\beta \;\; \textrm{for all } x,y\in B_1 $\\
for $r_k=\gamma^k$ for some $\gamma\in(0,1)$, for all $ k\geq 0$, with the convention that $l_{-1}\equiv 0$.
Observe that this proves the result. Indeed, $b_k = b_0 + (b_1-b_0) + \ldots + (b_k-b_{k-1})$ converges to some $b$, since $\mathbb{S}um_{k=0}^\infty |b_{k} - b_{k-1}| \leq K_2 \mathbb{S}um_{k=0}^\infty (\gamma^{\alpha} )^{k-1} < \infty$; also
$|b_k - b| \leq \mathbb{S}um_{l=k}^\infty |b_{l+1} - b_l| \leq K_2 \mathbb{S}um_{l=k}^\infty \gamma^{\alpha l} = K_2 \frac{\gamma^{\alpha k}}{1-\gamma^\alpha}.$
Similarly, $|a_k - a|\leq K_2\frac{\gamma^{k(1+\alpha)}}{1-\gamma^{1+\alpha}}$ and $a_k$ converges to some $a$.
Next, for each $x\in B_1$, there exists $k\geq 0$ such that $r_{k+1} < |x|\le r_k$. Then, $|u(x) - a_k - b_k \cdot x|=|u(x)-l_k(x)| \leq r_k^{1+\alpha}$,
since $x\in \overline{B}_{r_k}$, thus
\begin{align*}
|u(x) - a - b\cdot x| &\leq |u(x) - a_k - b_k \cdot x| + |a_k - a| + |b_k - b| \,|x| \\
&\leq r_k^{1+\alpha} + K_2 \frac{r_k^{1+\alpha}}{1-\gamma^{1+\alpha}} + K_2 \frac{r_k^{\alpha}}{1-\gamma^\alpha} r_{k} \\
&= \left\{ 1+\frac{K_2 }{1-\gamma^{1+\alpha}} + \frac{K_2 }{1-\gamma^\alpha} \right\} \frac{1}{\gamma^{1+\alpha}} \, r_{k+1}^{1+\alpha} \leq C_\gamma\, |x|^{1+\alpha}.
\end{align*}
By definition of a differentiable function, $a=u(0)\, ,\; b=D u(0)$ and we will have obtained $| u(x) - u(0) - D u(0)\cdot x | \leq C|x|^{1+\alpha}$ and $|D u(0)| \leq C$.
Notice that there was nothing special in doing the initial argument around $0$, which we had supposed in the beginning of the proof, belonging to $\Omega^\prime$. Actually, by replacing it by any $x_0\in\Omega$ and setting the corresponding $s_0=\min \{r_0,\mathrm{dist}(x_0,\partial\Omega^\prime)\}$, we define $N=N_\mathbb{S}igma (x_0)$ by changing $0$ by $x_0$ in there. With this, we show that our initial function $u$ is differentiable at $x_0$ with
$$
\,| u(x)-u(x_0)-Du(x_0)\cdot(x-x_0) | \leq CW|x-x_0|^{1+\alpha},\quad |Du(x_0)| \leq CW
$$
which implies\footnote{This is just a property of functions. See, for example, a simple proof done by Sirakov in \cite{Bbook}, or \cite{tese}.} that $Du\in C^\alpha (\overline{B}_{\mathbb{S}igma})$ and $\|u\|_{C^{1,\alpha} (\overline{B}_{\mathbb{S}igma})} \leq CW$.
Thus, for the complete local estimate, we just take finitely many such points in order to cover $\Omega^\prime$.
We stress that $(i)_k$ and $(ii)_k$ are completely enough to imply the result, as above, while $(iii)_k$ is an auxiliary tool to get them.
So, let us prove $(i)_k-(iii)_k$ by induction on $k$.
For $k=0$ we set $a_0 = b_0 = 0$. Recall that $\beta$ and $K_1$ are the constants from the $C^\beta$ superlinear local estimate in $B_1$ such that
$
\|u\|_{C^\beta(\overline{B}_1)}\leq \widetilde{K}_1 (1+\; \mathrm{d}elta+1)\leq 3K_1,
$
which implies $(iii)_0$. Obviously $(i)_0$ and $(ii)_0$ are satisfied too.
Notice that
$
|b_k|\leq \mathbb{S}um_{l=0}^k |b_{l} - b_{l-1}| \leq \frac{K_2}{\gamma^\alpha} \mathbb{S}um_{k=0}^\infty \gamma^{\alpha k}=\frac{K_2}{{\gamma^\alpha}(1-\gamma^\alpha)}\leq K
$
and also, for all $x\in B_1$,
$
|l_k(x)|\leq |a_k|+|b_k|| x|\leq \frac{K_2}{\gamma^{(1+\alpha)}} \mathbb{S}um_{k=0}^\infty \gamma^{(1+\alpha) k} + \frac{K_2}{\gamma^\alpha}\mathbb{S}um_{k=0}^\infty \gamma^{\alpha k}= K.
$
As the induction step, we suppose the items $(i)_k-(iii)_k$ valid in order to construct $a_{k+1}$ and $b_{k+1}$ for which $(i)_{k+1}-(iii)_{k+1}$ hold. Define
$$
v(x)=v_k({x}) := \frac{(u-l_k)(r_k {x})}{r_k^{1+\alpha}}=\frac{u(r_k{x}) - a_k - b_k\cdot x r_k}{r_k^{1+\alpha}}\, , \;\textrm{ for all }\, {x}\in B_2 .
$$
Note that $(i)_k$ says precisely that $ |v({x})| \leq 1$ for all $x\in B_1.$ Further, from this and $(iii)_k$ we get
$$
\|v\|_{C^\beta (\overline{B}_1)}=\|v\|_{L^\infty ({B}_1)}+ \mathbb{S}up_{\mathbb{S}ubstack{ x,y \in B_1 \\ x \mathbb{N}eq y }} \frac{|v(x)-v(y)|}{|x-y|^\beta} \leq 2+3K_1=:K_0.
$$
\begin{claim} \label{claim C1,alpha local 2a mud.var.}
$v$ is an $L^p$-viscosity solution of
$F_k [v]=f_k (x) $ in $B_2$,
for $f_k:=f_k^1+f_k^2$ with
$
f_k^1(x):=r_k^{1-\alpha} f(r_k x); \;
f_k^2(x):=- r_k^{1-\alpha} F(r_k x,l_k(r_k x),b_k, 0)
$
and $F_k$ satisfying $(SC)_{F_k}^{\mu_{F_k}}$, where
\begin{align*}
F_k(x,s,p,X):=r_k^{1-\alpha} F(r_k x,r_k^{1+\alpha} s+l_k(r_kx), r_k^\alpha p+b_k, r_k^{\alpha -1}X)- r_k^{1-\alpha} F(r_k x,l_k(r_k x),b_k, 0),
\end{align*}
$b_{F_k}(x):=r_k b(r_k x)+2r_k \mu K$, $\mu_{F_k}:=r_k^{1+\alpha} \mu$, $d_{F_k}(x):=r_k^{2} d(r_k x)$ and $\omega_{F_k}(s):=r_k^{-1-\alpha} \omega(r_k^{1+\alpha}s)$.
\end{claim}
\begin{proof}
Let $\varepsilon >0$ and $\psi\in W^{2,p}_{\textrm{loc}}(B_2)$ such that $v-\psi$ has a minimum (maximum) at $x_0$.
Define $\varphi (x):=r_k^{1+\alpha}\psi (x/r_k)+l_k(x)$ for all $x\in B_{2r_k}$; then $u-\psi$ has a minimum (maximum) at $r_k x_0$. Since $u$ is an $L^p$-viscosity solution in $B_{2r_k}(0)$, there exists $r\in (0,2)$ such that
\begin{align*}
F(r_kx,u(r_kx),D\varphi (r_k x), D^2 \varphi (r_k x))\leq (\geq) \,f(r_kx) +(-)\, r_k^{\alpha-1} \varepsilon\;\; \textrm{ a.e. in }B_r(x_0).
\end{align*}
Using that $D\psi (x)=r_k^{-\alpha} \{D\varphi(r_k x)-b_k\}$ and $D^2\psi (x)=r_k^{1-\alpha}D^2\varphi (r_kx)$ a.e., we get
\begin{align*}
r_k^{1-\alpha}&F(r_kx,r_k^{1+\alpha}v(x)+l_k(r_kx),r_k^\alpha D\psi (x)+b_k, r_k^{\alpha-1}D^2 \psi (x))
\leq (\geq)\, r_k^{1-\alpha}f(r_kx) +(-)\, \varepsilon
\end{align*}
a.e. in $B_r(x_0)$. Adding $- r_k^{1-\alpha} F(r_k x,l_k(r_k x),b_k, 0)$ in both sides we obtain
$$
F_k(x,v(x),D\psi, D^2\psi)\leq (\geq )\, f_k(x)+(-) \,\varepsilon\quad\textrm{a.e. in }B_r(x_0).
$$
Moreover, $F_k$ satisfies $(SC)_{F_k}^{\mu_{F_k}}$, since $F_k(x,0,0,0)=0$ for $x\in B_2$ and
\begin{align*}
&F_k(x,r,p,X)-F_k(x,s,q,Y)= r_k^{1-\alpha}\{F(r_kx,r_k^{1+\alpha}r+l_k(r_kx),r_k^\alpha p+b_k,r_k^{\alpha-1}X)\\
&\qquad-F(r_kx, r_k^{1+\alpha}s+l_k(r_kx),r_k^\alpha q+b_k,r_k^{\alpha-1}Y)\} \\&
\leq \mathcal{M}^+_{\lambda,\Lambda} (X-Y)+ r_k b(r_kx)|p-q| +r_k\mu|p-q|\{r_k^\alpha (|p|+|q|)+b_k\} + r_k^{1-\alpha} d(r_kx) \omega(r_k^{1+\alpha}|r-s|)\\
&= \mathcal{M}^+_{\lambda,\Lambda}(X-Y)+ b_{F_k}(x)|p-q| +\mu_{F_k}|p-q|(|p|+|q|)+ d_{F_k}(x)\omega_{F_k}(|r-s|)
\end{align*}
and the left hand side is completely analogous.
\qedhere{\textit{Claim \ref{claim C1,alpha local 2a mud.var.}.} }
\end{proof}
Notice that $F_k,v,\mu_{F_k},b_{F_k},d_{F_k},\omega_{F_k}, A=0, B=0$ satisfy the hypotheses of lemma \ref{AproxLem}, since
\begin{align*}
&\|b_{F_k}\|_{L^p (B_1)}\leq r_k^{1-\frac{n}{p}} \|b\|_{L^p (B_{r_k})}+2\mu K |B_1|^{1/p} \leq \; \mathrm{d}elta;\;
\|f_k^1\|_{L^p(B_1)}\leq r_k^{1-\frac{n}{p}-\alpha} \|f\|_{L^p(B_{r_k})}\leq \frac{\; \mathrm{d}elta}{2}\, ;\\
&\|f_k^2\|_{L^p(B_1)}\leq r_k^{1-\frac{n}{p}-\alpha} \{\|b\|_{L^p(B_{r_k})}|b_k|+ (K+1) \omega (1) \|d\|_{L^p(B_{r_k})} \}+r_k^{1-\alpha}\mu |b_k|^2 |B_1|^{\frac{1}{p}}\leq \frac{\; \mathrm{d}elta}{2};\\
&\omega_{{F_k}}(1)\|d_{F_k}\|_{L^p(B_1)}= r_k^{1-\frac{n}{p}-\alpha} \omega (r_k^{1+\alpha})\|d\|_{L^p(B_{r_k})}\leq r_k^{1-\frac{n}{p}-\alpha} \omega (1)\|d\|_{L^p(B_1)} \leq \; \mathrm{d}elta ; \;\|v\|_\infty \leq 1;
\end{align*}
(recall tilde notations from claim \ref{claim C1,alpha local 1a mud.var.}), and up to defining $b$ in a zero measure set,
\begin{align*}
\bar{\beta}_{F_k}(x,x_0) &\leq r_k^{1-\alpha}\mathbb{S}up_{X\in \mathbb{S}^n} \frac{| F(r_k x,l_k(r_k x),b_k,r_k^{\alpha-1}X)-F(r_k x,0,0,r_k^{\alpha-1}X)|}{\|X\|+1} \\
&\qquad +\mathbb{S}up _{X\in \mathbb{S}^n} \frac{| F(r_k x,0,0,r_k^{\alpha-1} X)-F(r_k x_0,0,0,r_k^{\alpha-1} X)|}{r_k^{\alpha-1}(\|X\|+1)} \\
&\qquad +r_k^{1-\alpha}\mathbb{S}up_{X\in \mathbb{S}^n} \frac{| F(r_k x_0,0,0,r_k^{\alpha-1}X)-F(r_k x_0,l_k(r_k x_0),b_k,r_k^{\alpha-1}X)|}{\|X\|+1} \\
&\qquad + r_k^{1-\alpha}\mathbb{S}up_{X\in \mathbb{S}^n} \frac{| F(r_k x,l_k(r_k x),b_k,0)|+|F(r_k x_0,l_k(r_k x_0),b_k,0)|}{\|X\|+1}\\
&\leq {2r_k^{1-\alpha}}\{ (d(r_k x)+d(r_k x_0))\,\omega(\|l_k(r_kx)\|_{L^\infty(\Omega)})+ (\,b(r_kx)+b(r_kx_0)\,)|b_k|+ \mu |b_k|^2 \} \\
&\qquad \mathbb{S}up_{X\in \mathbb{S}^n} ({\|X\|+1} )^{-1}+ \bar{\beta}_F(r_k x,r_k x_0)
\end{align*}
since $r_k^{\alpha-1}\geq 1$; then if $b$ is bounded,
\begin{align*}
\|\bar{\beta}_{F_k}(\cdot,0)\|_{L^p (B_1)}&\leq 4 r_k^{1-\alpha} |B_1|^{\frac{1}{p}} (K+1) \,\omega(1)\|d\|_{L^\infty(B_{r_k})} + 4 Kr_k^{1-\alpha}|B_1|^{\frac{1}{p}}\|b\|_{L^\infty (B_{r_k})} +2 \mu K^2 |B_1|^{\frac{1}{p}}\\
& + \|\bar{\beta}_{F}(\cdot,0)\|_{L^p (B_{r_k})}\leq \; \mathrm{d}elta .
\end{align*}
In particular, this gives an alternative proof of $C^{1,\alpha}$ results in \cite{Swiech} in the case $\mu=0$.
\mathbb{S}mallskip
On the other hand, without boundedness assumption on $b$, we note that it also follows from the proof of claim \ref{claim C1,alpha local 2a mud.var.} that $v$ is an $L^p$-viscosity solution of
\begin{center}
$F_k(x,v+L_k (x), Dv+B_k, D^2 v) = f_k^1(x)$\, in $B_1$,
\end{center}
where $F_k$ is now defined as
$F_k(x,s,p,X)=r_k^{1-\alpha} F(r_kx,r_k^{1+\alpha}s,r_k^\alpha p,r_k^{\alpha-1} X)$, for $L_k (x)=A_k +B_k \cdot x$, $A_k=r_k^{-1-\alpha} a_k $ and $B_k=r_k^{-\alpha}b_k$, which satisfies $(SC)^{\mu_{F_k}}_{F_{k}}$ for $b_{F_k}(x)=r_k b(r_kx)$,
but $\mu_{F_k}$, $d_{F_k}$, $\omega_{F_k}$, $f_k^1$ remaining as in claim \ref{claim C1,alpha local 2a mud.var.}.
Observe that we trivially have
$\|\bar{\beta}_{F_k} (\cdot,0)\|_{L^p(B_1)}\leq \; \mathrm{d}elta$ for such $F_k$.
Furthermore,
$|B_k|\|b_{F_k}\|_{L^p (B_1)}$ $=r_k^{1-\alpha-\frac{n}{p}}\|b\|_{L^p(B_{r_k})} |b_k|\leq K\|b\|_{L^p(B_1)}\leq \; \mathrm{d}elta$;
$\mu_{F_k}|B_k| (|B_k|+1)=(r_k^{1-\alpha} |b_k|^2 +r_k |b_k|)\mu \leq K(K+1)\mu \leq \; \mathrm{d}elta$; and we finally get
$\omega_{F_k}(1)\|d_{F_k}\|_{L^p (B_1)}(|A_k|+|B_k|)\leq r_k^{1-\alpha - \frac{n}{p}} ( |a_k| + r_k|b_k|)\omega(1)\|d\|_{L^p(B_{r_k})}$ $\leq 2K\omega(1)\|d\|_{L^p(B_1)}\leq \; \mathrm{d}elta$ if $\omega (r)\leq \omega (1) r$ for $r\geq 0$.
Thus, such $F_k,v,\mu_{F_k},b_{F_k},d_{F_k}, \omega_{F_k}, A_k, B_k$ also satisfy lemma \ref{AproxLem} hypotheses if $\omega$ is a Lipschitz modulus.
In any case, let $h=h_k\in C(\overline{B}_1)$ be the $C$-viscosity solution of
\begin{align*}
\left\{
\begin{array}{rclcc}
F_k (0,0,0,D^2 h) &=& 0 &\mbox{in} & B_1 \\
h &=& v &\mbox{on} & \partial B_1\,.
\end{array}
\right.
\end{align*}
By ABP we have $\|h\|_{L^\infty (B_1)}\leq \|h\|_{L^\infty (\partial B_1)}\leq 1$ and by the $C^{1,\bar{\alpha}}$ local estimate (proposition \ref{C1,baralpha}), $\|h\|_{C^{1,\bar{\alpha}} (\overline{B}_{1/2})}\leq K_2\,\|h\|_{L^\infty (B_1)} \leq K_2$.
Hence, by lemma \ref{AproxLem} applied to $F_k,v,\mu_{F_k},b_{F_k}, d_{F_k},\omega_{F_k}$, $\psi:=v\mid_{\partial B_1},\tau:=\beta$, $K_0 $ and $h$ we get, for $\varepsilon$ given in \eqref{epsilon}, that
$
\|v-h\|_{L^\infty (B_1)}\leq \varepsilon.
$
\mathbb{S}mallskip
Define $\overline{l}(x)=\overline{l}_k(x):=h(0)+Dh(0)\cdot x$ in $B_1$, then,
\begin{align} \label{wlinfty local}
\|v-\overline{l}\|_{L^\infty (B_{2\gamma})}\leq \gamma^{1+\alpha}.
\end{align}
In fact, by the choice of $\gamma\leq \frac{1}{4}$ in \eqref{gamma}, we have for all $x\in B_{2\gamma}(0)$ that
\begin{align*}
|v(x)-\overline{l}(x)|&\leq |v(x)-h(x)| + |h(x) - h(0) - D h(0)\cdot x|\\
&\leq K_2 \,(2\gamma)^{1+\bar{\alpha}}+ K_2|x|^{1+\bar{\alpha}}\leq 2 K_2 \,(2\gamma)^{1+\bar{\alpha}}\leq \gamma^{1+\alpha}.
\end{align*}
However, inequality \eqref{wlinfty local} and the definition of $v$ imply
\begin{align*}
|u(r_k {x}) -l_k(r_kx) - r_k^{1+\alpha} h(0) - r_k^{1+\alpha} D h(0) \cdot x |\leq r_k^{1+\alpha} \gamma^{1+\alpha} = r_{k+1}^{1+\alpha}\, \; \textrm{ for all } x \in B_{2\gamma}\,,
\end{align*}
which is equivalent to
\begin{align*}
|u(y) -l_{k+1}(y)|\leq r_k^{1+\alpha} \gamma^{1+\alpha} = r_{k+1}^{1+\alpha}\, \; \textrm{ for all } y=r_kx \in B_{2\gamma r_k}= B_{2r_{k+1}}\, ,
\end{align*}
where $l_{k+1}(y):=l_k(y)+r_k^{1+\alpha} h(0) + r_k^{\alpha} D h(0) \cdot y \,$. Then, we define
$$
a_{k+1} := a_k + h(0)\, r_k^{1+\alpha} , \;\;\;
b_{k+1} := b_k + D h(0)\, r_k^{\alpha}
$$
obtaining $(i)_{k+1}$. Further, $|a_{k+1} - a_k|\leq K_2 \, r_k^{1+\alpha}$, $|b_{k+1} - b_k|\leq K_2 \, r_k^{\alpha}$, which is $(ii)_{k+1}$. To finish we observe that, in order to prove $(iii)_{k+1}$, it is enough to show
\begin{align}\label{iii}
\|v-\overline{l}\|_{C^\beta (\overline{B}_\gamma)}\leq (1+2K_1) \,\gamma^{1+\alpha-\beta}.
\end{align}
Indeed, if $x,y\in B_1$ and \eqref{iii} is true, then
\begin{align*}
&|(v-\overline{l})(\gamma x)-(v-\overline{l})(\gamma y)|\leq (1+2K_1)
\gamma^{1+\alpha-\beta} |\gamma x - \gamma y|^\beta \\
& \Leftrightarrow |(u-l_k)(\gamma r_{k}x)-(u-l_k)(\gamma r_{k}y)-r_k^{\alpha} Dh(0)\cdot (x-y)\gamma r_k| \leq (1+2K_1) \gamma^{1+\alpha} r_{k}^{1+\alpha} |x-y|^\beta \\
& \Leftrightarrow |(u-l_{k+1})(r_{k+1}x)-(u-l_{k+1})(r_{k+1}y)| \leq (1+2K_1) \, r_{k+1}^{1+\alpha} |x-y|^\beta.
\end{align*}
Now, we obtain \eqref{iii} applying the local quadratic $C^\beta$ estimate (proposition \ref{Cbetaquad}) to the function $w:=v-\overline{l}$, which is an $L^p$-viscosity solution in $B_2$ of the inequalities
\begin{align} \label{Cbetafinal}
\mathcal{L}_{k}^- [w]-\mu_{F_k}|Dw|^2\leq g_k(x) ,\;\;
\mathcal{L}_{k}^+ [w]+\mu_{F_k} |Dw|^2 \geq - g_k(x),
\end{align}
where $g_k:=g_k^1+g_k^2\,$, for $g_k^1(x):=|f_k(x)-F_k(x,\overline{l}(x),Dh(0),0)|$ and $g_k^2 (x):=d_{F_k}(x)\omega_{F_k}(|w|)$, with $\mathcal{L}^\pm_k[u]:=\mathcal{M}^\pm_{\lambda,\Lambda}(D^2 u)\pm (b_{F_k}+2K_2\,\mu_{F_k})|Du|$.
Surely, this finishes the proof of \eqref{iii}, since
$$
|g_k^1(x)|\leq |f_k (x)|+b_{F_k} (x) |Dh(0)|+\omega_{F_k} ( |\overline{l}(x) | )\, d_{F_k}(x)+ \mu_{F_k}|Dh(0)|^2,
$$
then using that $|\overline{l}(x)|\leq |h(0)|+|Dh(0)|\,|x|\leq \|h\|_{C^{1,\bar{\alpha}} (\overline{B}_{1/2})}\leq K_2$ for all $ x \in B_1$,
we have
\begin{align*}
\|\,g_k\|_{L^p(B_1)}& \leq \|f_k\|_{L^p(B_1)} +\|b_{F_k}\|_{L^p(B_1)} K_2 + (K_2+1)\,\omega_{{F_k}}(1) \|d_{F_k}\|_{L^p(B_1)} \\
&+\mu K_2^2 \, |B_1 |^{\frac{1}{p}}+(1+ \|w\|_{L^\infty ({B_1})})\,\omega_{F_k} (1)\|d_{F_k}\|_{L^p(B_1)} \leq (5+2K_2)\,\; \mathrm{d}elta \leq \gamma^{\alpha}
\end{align*}
from the definition of $\; \mathrm{d}elta$ in \eqref{delta}. Thus, using the estimate above and \eqref{wlinfty local} in the $C^\beta$ local estimate, properly scaled to the ball of radius $\gamma$, we obtain in particular that
\begin{align*}
[w]_{\beta,\overline{B_\gamma}} &\leq \gamma^{-\beta} \widetilde{K}_1 \,\{ \, \|w\|_{L^\infty ({B_{2\gamma}})} +\gamma^{2-\frac{n}{p}} \|g_k\|_{L^p ({B_{2\gamma}})} \,\} \\
&\leq \gamma^{-\beta} {K}_1 \,\{ \,\gamma^{1+\alpha} + \gamma^{2-\frac{n}{p}}\gamma^\alpha\,\}\leq 2 K_1 \,\gamma^{1+\alpha-\beta}
\end{align*}
and so
$
\|w\|_{C^\beta (\overline{B_\gamma})}=\|w\|_{L^\infty ({B_\gamma})} +
[w]_{\beta,\overline{B_\gamma}}
\leq \gamma^{1+\alpha} +2K_1 \,\gamma^{1+\alpha-\beta}
\leq (1+2K_1)\,\gamma^{1+\alpha-\beta}
$
as desired.
\end{proof}
\begin{rmk}\label{obs plafond C1,alpha}
By the proof above we see that, under $\mu, \|b\|_{L^p(\Omega)}, \omega(1) \|d\|_{L^\infty(\Omega)} \leq C_1$, both $\mathbb{S}igma$ and the final constant $C$ depends on $n,p,\lambda,\Lambda, \alpha,\beta, K_1,K_2,C_0$ and $C_1$.
This is very useful in applications, when we have, for example, a sequence of solutions $u_k$ with their respective coefficients uniformly bounded; with $\|u_k\|_{L^\infty}$ and the $L^p$ norm of the right hand side a priori bounded. Then we can uniformly bound the $C^{1,\alpha}$ norm of $u_k$.
\end{rmk}
\mathbb{S}ubsection{Boundary Regularity}\label{boundary regularity}
Since our equation is invariant under diffeomorphisms and $\partial\Omega\in C^{1,1}$, we only need to prove regularity and estimates for some half ball, say $B_1^+(0)$.
Indeed, near a boundary point we make a diffeomorphic change of independent variable, which takes a neighborhood of $\partial\Omega$ into $B_1^+$. This change only depends on the coefficients of the equation and the $C^{1,1}$ diffeomorphisms that describe the boundary, see details in \cite{Winter} (see also \cite{tese} for a version with superlinear growth).
Then, consider $K_1$ and $\beta$ the pair of $C^\beta$ global superlinear estimate (proposition \ref{Cbetaquad}) in $B_1^+$, related to the initial $n,p,\lambda, \Lambda,\mu$, $\|b\|_{L^p(\Omega)}, \tau$ and $C_1$, such that $$\|u\|_{C^\beta(\overline{B}_1^+)}\leq K_1 \,\{\|u\|_{L^\infty (B_1^+)} + \|f\|_{L^p(B_1^+)} + \|u\|_{C^\tau (\mathbb{T})} +\|d\|_{L^p(B_1^+)}\,\omega(\|u\|_{L^\infty (B_1^+)})\}.$$
As in \cite{Winter}, we start proving a boundary version of the approximation lemma in $B_1^\mathbb{N}u$.
For this set, let $K_3\geq 1$ and $\bar{\alpha}$ be the pair of $C^{1,\bar{\alpha}}$ boundary estimate (proposition \ref{C1,baralphaglobal}) associated to $n,\lambda,\Lambda$ and $\tau$, independently of $\mathbb{N}u>0$.
We can suppose that $K_1\geq \widetilde{K}_1$ and $\beta\leq \widetilde{\beta}$, where $\widetilde{K}_1,\widetilde{\beta}$ is the pair of $C^\beta$ global estimate for the set $B_1^\mathbb{N}u$ (or $B_{1/2}^\mathbb{N}u$), independently of $\mathbb{N}u>0$, with respect to an equation with given constants $n,p,\lambda,\Lambda$ and bounds for the coefficients $\mu\leq 1$, $\|b\|_{L^p(B_2^\mathbb{N}u)}\leq 1+2K_3\, (3+2C_n) |B_1|^{1/p}$ (for a constant $C_n$, depending only on $n$, from lemma 6.35 of \cite{GT} for $\epsilon=1/2$, that will appear in the sequel) and $\omega(1)\|d\|_{L^p(B_2^\mathbb{N}u)}\leq 1$, for any solution in $B_2^\mathbb{N}u$ satisfying $\|u\|_{L^\infty(B_2^\mathbb{N}u)}\leq 1$ and $\|\psi\|_{C^{1,\tau} (\mathbb{T}_2^\mathbb{N}u)}\leq 2$ (or for any solution in $B_1^\mathbb{N}u$ with coefficients in $B_1^\mathbb{N}u$).
Denote $ \|\cdot\|_{L^p_\mathbb{N}u} =\|\cdot \|_{ L^p (B_1^\mathbb{N}u)}$.
\begin{lem}\label{AproxLemBoundary}
Assume $F$ satisfies \ref{SCmu} in $B_1^\mathbb{N}u$ for some $\mathbb{N}u \in [0,1]$ and $f \in L^p (B_1^\mathbb{N}u)$, where $p>n$. Let $\psi \in C^\tau (\partial B_1^\mathbb{N}u)$ with $\|\psi\|_{C^\tau (\partial B_1^\mathbb{N}u)}\leq K_0$. Set $L(x)=A+B\cdot x$ in $B_1^\mathbb{N}u$, for $A\in \mathbb{R}$, $B\in {\mathbb{R}^n} $.
Then, for every $ \varepsilon>0$, there exists $\; \mathrm{d}elta\in (0,1)$, $\; \mathrm{d}elta=\; \mathrm{d}elta (\varepsilon,n,p,\lambda,\Lambda,\tau,K_0)$, such that if
\begin{align*}
\|{\bar{\beta}}_F(\cdot,0)\|_{ L^p_\mathbb{N}u}\, ,\;\|f\|_{L^p_\mathbb{N}u}\, ,\; \mu(|B|^2 +|B|+1) \, ,\;
\|b\|_{L^p_\mathbb{N}u}(|B|+1) \, ,\;\omega (1) \|d\|_{L^p_\mathbb{N}u}(|A|+|B|+1)\leq \; \mathrm{d}elta
\end{align*}
then any two $L^p$-viscosity solutions $v$ and $h$ of
\begin{align*}
\left\{
\begin{array}{rclcc}
F(x,v+L(x),Dv+B,D^2v)&=&f(x) & \mbox{in} & B_1^\mathbb{N}u \\
v &=& \psi & \mbox{on} & \partial B_1^\mathbb{N}u
\end{array}
\right.
,\;
\left\{
\begin{array}{rclcc}
F(0,0,0,D^2h)&=& 0 & \mbox{in} & B_1^\mathbb{N}u \\
h &=& \psi & \mbox{on} & \partial B_1^\mathbb{N}u
\end{array}
\right.
\end{align*}
respectively, with $\omega (1) \|d\|_{L^p_\mathbb{N}u}\|v\|_{L^\infty_\mathbb{N}u}\leq \; \mathrm{d}elta$, satisfy $\|v-h\|_{L^\infty (B_1^\mathbb{N}u)}\leq \varepsilon$.
\end{lem}
\begin{proof}
For $\varepsilon>0$, we will prove the existence of $\; \mathrm{d}elta\in (0,1)$ as above with $\; \mathrm{d}elta\leq 2^{-\frac{n}{2p}}C_n^{-\frac{1}{2}}\,\widetilde{\; \mathrm{d}elta}^{1/2}$, for $\widetilde{\; \mathrm{d}elta}$ as in lemma \ref{AproxLem}.
Suppose the contrary, then there exist $\varepsilon_0>0$ and sequences $\mathbb{N}u_k \in [0,1]$, $F_k$ satisfying $(SC)^{\mu_k}$ for $b_k,\, d_k\in L^p_+(B_1^{\mathbb{N}u_k} )$, $\mu_k\geq 0$, $\omega_k$ modulus; $f_k\in L^p(B_1^{\mathbb{N}u_k})$, $A_k\in \mathbb{R}$, $B_k\in {\mathbb{R}^n} $, $L_k(x)=A_k+B_k\cdot x$, and $\; \mathrm{d}elta_k\in(0,1)$ with $\; \mathrm{d}elta_k\leq 2^{-\frac{n}{2p}}C_n^{-\frac{1}{2}}\,\widetilde{\; \mathrm{d}elta_k}^{1/2}$ for $\widetilde{b}_k=b_k+2|B_k|\mu_k$, such that
\begin{align*}
\|\bar{\beta}_{F_k}(\cdot,0)\|_{ L^p_{\mathbb{N}u_k} },\; \|f_k\|_{ L^p_{\mathbb{N}u_k} } ,\; \mu_k (|B_k|^2 +|B_k|+1),\;
\|b_k\|_{ L^p_{\mathbb{N}u_k} } (|B_k|+1) ,\; \omega_k(1) \|d_k\|_{ L^p_{\mathbb{N}u_k} } (|A_k|+|B_k|+1)
\end{align*}
are less or equal than $\; \mathrm{d}elta_k $ with $ \; \mathrm{d}elta_k \rightarrow 0$, and $v_k$, $h_k\in C(\overline{B_1^{\mathbb{N}u_k}})$ are $L^p$-viscosity solutions of
\begin{align*}
\left\{
\begin{array}{rclc}
F_k(x,v_k+L_k(x),Dv_k+B_k,D^2v_k)&=& f_k(x) & B_1^{\mathbb{N}u_k} \\
v_k &=& \psi_k & \partial B_1^{\mathbb{N}u_k}
\end{array}
\right.
,\;
\left\{
\begin{array}{rclc}
F_k(0,0,0,D^2h_k)&=&0 & B_1^{\mathbb{N}u_k} \\
h_k &=& \psi_k & \partial B_1^{\mathbb{N}u_k}
\end{array}
\right.
\end{align*}
where $\|\psi_k\|_{C^\tau (\partial B_1^{\mathbb{N}u_k})}\leq K_0$, $\omega_k (1) \|d_k\|_{L^p_{\mathbb{N}u_k}}\|v_k\|_{L^\infty_{\mathbb{N}u_k}} \leq \; \mathrm{d}elta_k$, but $
\|v_k-h_k\|_{L^\infty (B_1^{\mathbb{N}u_k})} > \varepsilon_0.$
Analogously to the proof of lemma \ref{AproxLem}, ABP implies that
$\|v_k\|_{L^\infty (B_1^{\mathbb{N}u_k})}\,, \;\|h_k\|_{L^\infty (B_1^{\mathbb{N}u_k})}\leq C_0 $
for large $k$, where $C_0$ is a constant that depends only on $n,p,\lambda,\Lambda$ and $K_0$.
Notice that $B_1^{\mathbb{N}u_k}$ has the exterior cone property, then by the $C^\beta$ global quadratic estimate (proposition \ref{Cbetaquad}) we obtain $\beta\in (0,1)$ such that
\begin{align} \label{CbetaApBoundary}
\|v_k\|_{C^\beta (\overline{B_1^{\mathbb{N}u_k}})}\,, \;\|h_k\|_{C^\beta (\overline{B_1^{\mathbb{N}u_k}})}\leq C \, , \quad \textrm{for large }k,
\end{align}
where $\beta=\min {(\beta_0,{\tau}/{2})}$ for some $\beta_0=\beta_0 (n,p, \lambda, \Lambda)$ and $C=C(n,p, \lambda, \Lambda, C_0)$. Observe that $\beta$ and $C$ do not depend on $k$, since $\mu_k, \,\|\tilde{b}_k\|_{L^p(B_1^{\mathbb{N}u_k})},\, \omega_k(1) \,\|d_k\|_{L^p(B_1^{\mathbb{N}u_k})}, \,\|f_k\|_{L^p(B_1^{\mathbb{N}u_k})}\leq 1$ and $\mathrm{diam} (B_1^{\mathbb{N}u_k}) \leq 2$, for large $k $. Here we have different domains, what prevents us from directly using the compact inclusion $C^\beta$ into the set of continuous functions in order to produce convergent subsequences. But this is just a technicality, as in \cite{Winter}, by taking a subsequence of $\mathbb{N}u_k$ that converges to some $\mathbb{N}u_\infty \in [0,1]$, which we can suppose monotonous. Hence we consider two cases: $B_1^{\mathbb{N}u_\infty}\mathbb{S}ubset B_1^{\mathbb{N}u_k} \mathbb{S}ubset B_1^{\mathbb{N}u_{k+1}}\mathbb{S}ubset ...$ or $...\mathbb{S}ubset B_1^{\mathbb{N}u_{k+1}}\mathbb{S}ubset B_1^{\mathbb{N}u_k} \mathbb{S}ubset B_1^{\mathbb{N}u_\infty}$, for all $k \in \mathbb{N}$.
In the first one, we use the compact inclusion on $\overline{B_1^{\mathbb{N}u_\infty}}$. In the second, we make a trivial extension of our functions to the larger domain $\overline{B_1^{\mathbb{N}u_\infty}}$, i.e. by defining $\psi_k$ in $\widetilde{B}_k=B_1 \cap \{ - \mathbb{N}u_\infty \leq x_n \leq - \mathbb{N}u_k \}$ in such a way that $\|\psi_k\|_{C^\tau (\widetilde{B}_k)}\leq C_0$, from where we may suppose that \eqref{CbetaApBoundary} holds on $\overline{B_1^{\mathbb{N}u_\infty}}$ for the extended $v_k$ and $h_k$.
In both cases, we obtain convergent subsequences
$v_k\longrightarrow v_\infty$, $ h_k\longrightarrow h_\infty$ in $C(\overline{B_1^{\mathbb{N}u_\infty}})$ as ${k\rightarrow\infty}$,
for some continuous functions $v_\infty, \; h_\infty$ in $\overline{B_1^{\mathbb{N}u_\infty}}$, with $v_\infty = h_\infty = \psi_\infty$ on $\partial B_1^{\mathbb{N}u_\infty}$.
Finally, we claim that $v_\infty$ and $h_\infty$ are viscosity solutions of
\begin{align*}
\left\{
\begin{array}{rclcc}
F_\infty (D^2 u)&=&0 & \mbox{in} & B_1^{\mathbb{N}u_\infty} \\
u &=& \psi_\infty \; & \mbox{on} & \partial B_1^{\mathbb{N}u_\infty}
\end{array}
\right.
\end{align*}
and therefore equal by proposition \ref{ExisUnicF(D2u)}, which contradicts $\|v_\infty-h_\infty\|_{L^\infty (B_1^{\mathbb{N}u_\infty})}\geq \varepsilon_0$.
For $h_\infty$, it follows by taking the uniform limits on the inequalities satisfied by $h_k$. For $v_\infty$, we apply proposition \ref{Lpquad} together with observation \ref{Lpquadencaixados}, since we have, for each $\varphi\in C^2(D)$, where $D\mathbb{S}ubset B_1^{\mathbb{N}u_\infty}$, that
$F_k(x,v_k+L_k(x),D\varphi+B_k, D^2 \varphi)-f_k(x)-F_\infty (D^2 \varphi)\rightarrow 0$ as ${k\rightarrow\infty}$
in $L^p(D)$, analogously to the end of the proof of lemma \ref{AproxLem}.
\end{proof}
\begin{proof}[\textit{Proof of Boundary Regularity Estimates in the set $B_1^+$.}]
We proceed as in the local case, introducing the corresponding changes, in order to deal with the boundary. Our approach is similar to \cite{Winter}. Now we set
$W:=\| u \|_{L^{\infty} (B_1^+)} + \|f \|_{L^p (B_1^+ )} + \|u\|_{C^{1,\tau} (\mathbb{T} )} +\|d\|_{L^p(B_1^+)}\,\omega(\|u\|_{L^\infty (B_1^+)}) \leq W_0$ and $s_0:=\min (r_0, \frac{1}{2})$.
Fix $\alpha\in (0,\bar{\alpha})$ with $\alpha\leq\min (\beta,1- \frac{n}{p},\tau,\bar{\alpha} (1-\tau))$ and choose $\gamma=\gamma(n,\alpha,\bar{\alpha},K_3)\in (0,\frac{1}{4}]$ such that
$2^{2+\bar{\alpha}}K_4 \, \gamma^{\bar{\alpha}}\leq \gamma^{\alpha}$,
where $K_4=K_4 \,(K_3,n)\geq 1$ will be specified later. Thus, define
$\varepsilon=\varepsilon (\gamma)$ by $K_4 \, (2\gamma)^{1+\bar{\alpha}}.$
This $\varepsilon$ provides a $\; \mathrm{d}elta=\; \mathrm{d}elta(\varepsilon)\in (0,1)$, the constant of the approximation lemma~\ref{AproxLemBoundary} which, up to diminishing, can be supposed to satisfy
$(5+2K_4) \,\; \mathrm{d}elta \leq \gamma^\alpha.$
Next we chose $\mathbb{S}igma=\mathbb{S}igma (s_0,n,p,\alpha , \bar{\alpha},\beta, \; \mathrm{d}elta,\mu, \|b\|_{L^p(B_1^+)},\omega(1)\|d\|_{L^\infty (B_1^+)},K_1,K_3,C_0) \leq \frac{s_0}{2}$ such that
$$\mathbb{S}igma^{\min{({1-\frac{n}{p}},\beta)}} m \leq {\; \mathrm{d}elta} \,\{{32K^2 (K_4+K+1)|B_1|^{1/p}}\}^{-1}$$
where $m:=\max{ \{ 1, \|{b}\|_{L^p (B^+_1)}, \omega(1)\|{d}\|_{L^\infty (B^+_1)},\mu (1+2^\beta K_1) W_0 \} }$ and $K:={K_4}\,{{\gamma^{-\alpha}}(1-\gamma^\alpha)^{-1}} +{K_4}\,{{\gamma^{-1-\alpha}}(1-\gamma^{1+\alpha})^{-1}}\geq K_4\geq 1$.
Fix $z=(z^\prime,z_n) \in B_{1/2}^+ (0)$. We split our analysis in two cases, depending on the distance of the point $z$ to the bottom boundary: 1) $z_n < \frac{\mathbb{S}igma}{2}\; \Leftrightarrow\; \mathbb{N}u < \frac{1}{2}$ and 2) $z_n \geq \frac{\mathbb{S}igma}{2} \;\Leftrightarrow\; \mathbb{N}u \geq \frac{1}{2}$, for $\mathbb{N}u :=\frac{z_n}{\mathbb{S}igma}$.
Suppose the first one. In this case we will be proceeding as in \cite{Winter} by translating the problem to the set $B_{2}^\mathbb{N}u $, in order to use the approximation lemma in its boundary version \ref{AproxLemBoundary}. Notice that
$$
x \in B_2^\mathbb{N}u (0) \; \Leftrightarrow\; \mathbb{S}igma x +z \in {B_{2 \mathbb{S}igma}^+} (z) \mathbb{S}ubset B_1^+(0).
$$
\begin{figure}
\caption{ Illustration of the change of variable, from $B^+_{2\mathbb{S}
\label{Rotulo}
\end{figure}
Then we define
$
N=N_\mathbb{S}igma (z):= \mathbb{S}igma W + \mathbb{S}up_{x\in B_2^\mathbb{N}u (0)} |u(\mathbb{S}igma x +z) - u(z)|.
$
The $C^\beta$ quadratic estimate, this time the global one, restricted to the set $B_{2 \mathbb{S}igma}^+ (z)$, yields
\begin{align}\label{Nboundary}
\mathbb{S}igma W \leq N \leq (\mathbb{S}igma + 2^\beta K_1 \mathbb{S}igma ^\beta)W
\leq (1+2^\beta K_1) \mathbb{S}igma ^\beta W_0.
\end{align}
Next we set $\widetilde{u}(x)=\frac{1}{N}\{u(\mathbb{S}igma x +z)-u(z)\}$, which is, as in claim \ref{claim C1,alpha local 1a mud.var.}, an $L^p$-viscosity solution of
\begin{align*}
\left\{
\begin{array}{rclcc}
\widetilde{F}(x,\widetilde{u},D \widetilde{u},D^2 \widetilde{u}) &=& \widetilde{f} (x) &\mbox{in} & B_2^\mathbb{N}u \\
\widetilde{u} &=& \widetilde{\psi} &\textrm{on} & \mathbb{T}^{\mathbb{N}u}_2
\end{array}
\right.
\end{align*}
for
$$\widetilde{F}(x,r,p,X):= \frac{\mathbb{S}igma^2}{N} F\left( \mathbb{S}igma x +z,Nr+u(z),\frac{N}{\mathbb{S}igma}p, \frac{N}{\mathbb{S}igma^2} X \right)-\frac{\mathbb{S}igma^2}{N}
F(\mathbb{S}igma x+z, u(z),0,0),$$
$\widetilde{\psi}(x):=\frac{1}{N} \{ \psi (\mathbb{S}igma x +z)-u(z) \}$ and $\widetilde{f}:=\widetilde{f}_1+\widetilde{f}_2$ where
\begin{align*}
\widetilde{f}_1(x):={\mathbb{S}igma^2} f(\mathbb{S}igma x+z)/{N};\;\;\widetilde{f}_2(x):= -{\mathbb{S}igma^2}F(\mathbb{S}igma x+z, u(z),0,0)/{N},
\end{align*}
$\widetilde{F}$ satisfying $(\widetilde{SC})^{\widetilde{\mu}}$ for $\widetilde{b}(x)= \mathbb{S}igma b(\mathbb{S}igma x+z)$, $\widetilde{\mu}=N\mu$, $\widetilde{d}(x)= \mathbb{S}igma^2 d(\mathbb{S}igma x+z)$ and $\widetilde{\omega}(r)=\omega(Nr)/N$.
With this definition and the choice of $\mathbb{S}igma$ in \eqref{sigma}, we obtain $\|\widetilde{u}\|_{L^\infty (B_2^\mathbb{N}u )} \leq 1$, $\|\widetilde{f}\|_{L^p (B_2^\mathbb{N}u)}\leq \frac{\; \mathrm{d}elta}{8}$, $\widetilde{\mu} \leq \frac{\; \mathrm{d}elta}{8K^2 |B_1|^{1/p} }$,
$\|\widetilde{b}\|_{L^p (B_2^\mathbb{N}u)}\leq \frac{\; \mathrm{d}elta}{16K} $,
$\widetilde{\omega} (1) \|\widetilde{d}\|_{L^p (B_2^\mathbb{N}u)}\leq \frac{\; \mathrm{d}elta}{32(K_4+K+1)}$ and $\|\bar{\beta}_{\widetilde{F}}(0,\cdot)\|_{L^p (B_1^\mathbb{N}u)}\leq \; \mathrm{d}elta/4 $ by choosing $\theta=\; \mathrm{d}elta/8$, as in the local case.
Furthermore, we have $\|\widetilde{\psi} \|_{L^\infty (\mathbb{T}^{\mathbb{N}u}_2)} \leq\|\widetilde{u}\|_{L^\infty (B_2^\mathbb{N}u )} \leq 1$
and then $\|D \widetilde{\psi}\|_{C^\tau (\mathbb{T}^{\mathbb{N}u}_2)}$ is bounded by
\begin{align*}
\frac{\mathbb{S}igma}{N} \| D\psi \|_{L^\infty (B_{2\mathbb{S}igma} (z) \cap \mathbb{T})} + \frac{\mathbb{S}igma}{N} \mathbb{S}up_{\mathbb{S}ubstack{ x\mathbb{N}eq y \in \mathbb{T}_2^\mathbb{N}u }} \frac{|D\psi (\mathbb{S}igma x +z)-D\psi (\mathbb{S}igma y+z)|}{|\mathbb{S}igma x-\mathbb{S}igma y|^\tau} \mathbb{S}igma^\tau\leq \frac{ \|\psi \|_{C^{1,\tau} (\mathbb{T})}}{W}\leq 1
\end{align*}
since $N\geq \mathbb{S}igma W$. Therefore, we obtain $\|\widetilde{\psi} \|_{C^{1,\tau} (\mathbb{T}^{\mathbb{N}u}_2)}=\|\widetilde{\psi} \|_{L^\infty (\mathbb{T}^{\mathbb{N}u}_2)} +\|D \widetilde{\psi}(x)\|_{C^\tau (\mathbb{T}^{\mathbb{N}u}_2)} \leq 2$.
We can suppose, up to this rescaling, that $F,u, \mu,b,d,\omega$ satisfy the former hypotheses related to $\widetilde{F},\widetilde{u},\widetilde{\mu},\widetilde{b},\widetilde{d},\widetilde{\omega}$. Thus, we move to the construction of $l_k (x):=a_k +b_k \cdot x$ such that
$(i)_k \;\;\| u-l_k \|_{L^\infty (B_{r_k}^\mathbb{N}u)} \leq r_k^{1+\alpha}$
$(ii)_k \;\;|a_k-a_{k-1}| \leq K_4 \,r_{k-1}^{1+\alpha}\;,\;\; |b_k - b_{k-1}|\leq K_4\, r_{k-1}^\alpha$
$(iii)_k \;\;|(u-l_k)(r_k x)-(u-l_k)(r_k y)|\leq C_{1,4} \, r_k^{1+\alpha} |x-y|^\beta \;\; \textrm{ for all } x,y\in B_1^{\mathbb{N}u_k}$
\\
where $C_{1,4}=C_{1,4}\,(K_1,K_4)$ and $\mathbb{N}u_k := \frac{\mathbb{N}u}{r_k}$, $r_k=\gamma^k$ for some $\gamma\in(0,1)$, for all $k\geq 0$ ($l_{-1}\equiv 0$).
We emphasize that these iterations will prove that the function $u$ (which plays the role of $\widetilde{u}$) is differentiable at $0$ and provide
$| u(x) - u(0) - D u(0)\cdot x | \leq C|x|^{1+\alpha} $, $|D u(0)| \leq C$
for every $x\in B_1^\mathbb{N}u$.
In terms of our original function defined on $B_1^+$, it means that $u$ will be differentiable at $z$, for all $z$ with $z_n<\frac{\mathbb{S}igma}{2}$. On the other hand, the second case $z_n\geq \frac{\mathbb{S}igma}{2}$ is covered by the local part, section \ref{local regularity}, since in this situation we are far away from the bottom boundary. Consequently, boundary superlinear regularity and estimates on $B_1^+$ will follow by a covering argument.
For the proof of $(i)_k-(iii)_k$, we use induction on $k$. For $k=0$ we set $a_0 = b_0 = 0$. Recall that $\beta$ and $K_1$ are the constants from $C^\beta$ quadratic global estimate in the set $B_1^{\mathbb{N}u}$, then we have
$
\|u\|_{C^\beta(\overline{B_1^\mathbb{N}u})}\leq \widetilde{K}_1 (1+\; \mathrm{d}elta+2+1)\leq 5K_1
$
and so $(iii)_0$ for $\mathbb{N}u_0=\mathbb{N}u$, $(i)_0$ and $(ii)_0$ are valid.
Analogously to the the local case, we have
$|b_k|, \; \|l_k\|_{L^\infty (B_{r_k}^\mathbb{N}u)} \leq K .$
For the induction's step we suppose $(i)_k-(iii)_k$ and construct $a_{k+1}$, $b_{k+1}$ such that $(i)_{k+1}-(iii)_{k+1}$ are valid.
Define
$$
v(x)=v_k({x}) := \frac{(u-l_k)(r_k {x})}{r_k^{1+\alpha}}=\frac{u(r_k{x}) - a_k - b_k\cdot x r_k}{r_k^{1+\alpha}}\, , \;\textrm{ for all }\,{x}\in B_2^{\mathbb{N}u_k} .
$$
Since $r_k x\in B_{r_k}^{\mathbb{N}u} \Leftrightarrow x \in B_1^{\mathbb{N}u_k}$, $(i)_k$ says that $ |v| \leq 1$ in $B_1^{\mathbb{N}u_k}.$ From this and $(iii)_k$, we get
$$
\|v\|_{C^\beta (\overline{B_1^{\mathbb{N}u_k}})}=\|v\|_{L^\infty ({B}_1^{\mathbb{N}u_k})}+ \mathbb{S}up_{\mathbb{S}ubstack{ x,y \in B_1^{\mathbb{N}u_k} \\ x \mathbb{N}eq y }} \frac{|v(x)-v(y)|}{|x-y|^\beta} \leq 1+C_{1,4}=:K_0.
$$
Notice that, as in the local case, it follows from claim \ref{claim C1,alpha local 2a mud.var.} that $v$ is as an $L^p$-viscosity solution of
\begin{align*}
\left\{
\begin{array}{rclcc}
F_k(x,v+L_k (x), Dv+B_k, D^2 v) &=& f_k (x) &\mathrm{in} & B_2^{\mathbb{N}u_k} \\
v & =& \psi_k &\mathrm{on} & \mathbb{T}_2^{\mathbb{N}u_k}
\end{array}
\right.
\end{align*}
where
$f_k(x):=r_k^{1-\alpha} f(r_k x)$, $L_k(x)=A_k+B_k\cdot x$ in $B_2^{\mathbb{N}u_k}$ for $A_k=r_k^{-1-\alpha} a_k$, $B_k=r_k^{-\alpha} b_k$, and
\begin{align*}
F_k(x,s,p,X):=r_k^{1-\alpha} F(r_k x,r_k^{1+\alpha} s, r_k^\alpha p, r_k^{\alpha -1}X)
\end{align*}
satisfying $(SC)_{F_k}^{\mu_{F_k}}$ for $b_{F_k}(x)=r_k b(r_k x)$, $\mu_{F_k}=r_k^{1+\alpha}\mu$, $d_{F_k}(x)=r_k^2 d(r_k x)$, and $\omega_{F_k}(s)=r_k^{-1-\alpha}\omega(r_k^{1+\alpha} s)$, for the Lipschitz modulus $\omega$.
The above coefficients satisfy the hypotheses of lemma \ref{AproxLemBoundary}, since $\|b_{F_k}\|_{L^p (B_1^{\mathbb{N}u_k})}(|B_k|+1) \leq \; \mathrm{d}elta$, $\mu_{F_k}(|B_k|^2 +|B_k|+1)\leq \; \mathrm{d}elta$, $\omega_{F_k}(1)\|d_{F_k}\|_{L^p(B_1^{\mathbb{N}u_k})} (|A_k|+|B_k|+1)\leq \; \mathrm{d}elta$, $\|v\|_{\infty} \leq 1$, $\|f_k\|_{L^p(B_1^{\mathbb{N}u_k})}\leq \; \mathrm{d}elta$,
and $\|\bar{\beta}_{F_k}(\cdot,0)\|_{L^p(B_1^{\mathbb{N}u_k})}\leq \; \mathrm{d}elta$, see section \ref{local regularity}.
Let $h=h_k\in C(\overline{B_1^{\mathbb{N}u_k}})$ be the $C$-viscosity solution of
\begin{align*}
\left\{
\begin{array}{rclcc}
F_k (0,0,0,D^2 h)&=& 0 &\mbox{in} & B_1^{\mathbb{N}u_k} \\
h &=& v \quad &\mbox{on} & \partial B_1^{\mathbb{N}u_k}
\end{array}
\right.
\end{align*}
given by proposition \ref{ExisUnicF(D2u)}, since $B_1^{\mathbb{N}u_k}$ has the uniform exterior cone condition. From ABP we get $\|h\|_{L^\infty (B_1^{\mathbb{N}u_k})}\leq \|h\|_{L^\infty (\partial B_1^{\mathbb{N}u_k})}\leq 1$. Further, $h=v=\psi_k \in C^{1,\tau}(B_1\cap \{x_n=-\mathbb{N}u_k \})$ and we can find a uniform bound for the $C^{1,\tau}$ norm of $\psi_k$. Indeed, $\|\psi_k\|_{L^\infty (\mathbb{T}_1^{\mathbb{N}u_k} )}\leq \|v\|_{L^\infty (\overline{B_1^{\mathbb{N}u_k}} )} \leq 1$ and
$$
[D\psi_k]_{\tau,\mathbb{T}_1^{\mathbb{N}u_k} } =\mathbb{S}up_{\mathbb{S}ubstack{ x,y \in \mathbb{T}_1^{\mathbb{N}u_k} \\ x \mathbb{N}eq y }} \frac{|D\psi_k (x)-D\psi_k (y)|}{|x-y|^\tau} = \mathbb{S}up_{\mathbb{S}ubstack{ \tilde{x}, \tilde{y} \in \mathbb{T}_{r_k}^{\mathbb{N}u} \\ \tilde{x}=r_k x, \,\tilde{y}= r_k y }} \frac{|D\psi (\tilde{x})-D\psi (\tilde{y})|}{|\tilde{x}-\tilde{y}|^\tau} r_k^{\tau - \alpha} \leq 1
$$
since $\|D\psi\|_{C^{\tau} (\mathbb{T}_1^{\mathbb{N}u} )} \leq 1$ and $\alpha \leq \tau$. Moreover, using the global Holder interpolation in smooth domains, lemma 6.35 of \cite{GT}, for $\epsilon = \frac{1}{2}$, there exists a constant\footnote{The proof of lemma 6.35 in \cite{GT} is based on an interpolation inequality (6.89) for adimensional Holder norms (that does not depend on the domain); followed by a partition of unity that straightens the boundary (not necessary in our case $\mathbb{T}_1^{\mathbb{N}u_k}\mathbb{S}ubset \mathbb{R}^{n-1} $). Then we have an estimate independently on $k$.} $C_n$ such that
$$
\|\psi_k\|_{C^{1} (\mathbb{T}_1^{\mathbb{N}u_k} )} \leq C_n \, \|\psi_k\|_{C (\mathbb{T}_1^{\mathbb{N}u_k} )} +\frac{1}{2} \|\psi_k\|_{C^{1,\tau} (\mathbb{T}_1^{\mathbb{N}u_k} )}
$$
hence
$$
\|\psi_k\|_{C^{1,\tau} (\mathbb{T}_1^{\mathbb{N}u_k} )} =\|\psi_k\|_{C^{1} (\mathbb{T}_1^{\mathbb{N}u_k} )} + [D\psi_k]_{\tau,\mathbb{T}_1^{\mathbb{N}u_k}} \leq C_n +\frac{1}{2} \|\psi_k\|_{C^{1,\tau} (\mathbb{T}_1^{\mathbb{N}u_k} )} +1
$$
i.e. $ \|\psi_k\|_{C^{1,\tau} (\mathbb{T}_1^{\mathbb{N}u_k} )} \leq 2(C_n+1)$.
Thus, the $C^{1,\bar{\alpha}}$ global estimate (proposition \ref{C1,baralphaglobal}) yields
$$
\|h\|_{C^{1,\bar{\alpha}} (\overline{B_{1/2}^{\mathbb{N}u_k}})}\leq K_3 \,\{ \|h\|_{L^\infty (B_1^{\mathbb{N}u_k})} + \|\psi_k\|_{C^{1,\tau} (\mathbb{T}_1^{\mathbb{N}u_k} )} \} \leq K_3\, (3+2C_n)=: K_4.
$$
Now, the approximation boundary lemma \ref{AproxLem} applied to $F_k,v,h,\mathbb{N}u_k, \mu_{F_k},b_{F_k}, d_{F_k},\omega_{F_k}, A_k, B_k, \psi_k $, $\beta, K_0$ gives us that
$
\|v-h\|_{L^\infty (B_1^{\mathbb{N}u_k})}\leq \varepsilon.
$
Therefore, defining $\overline{l}(x)=\overline{l}_k(x):=h(0)+Dh(0)\cdot x$ in $B_{1}^{\mathbb{N}u_k}$, we have
\begin{align} \label{wlinftyboundary}
\|v-\overline{l}\|_{L^\infty (B_{2\gamma}^{\mathbb{N}u_k})}\leq \gamma^{1+\alpha}.
\end{align}
In fact, by the choice of $\gamma$ we have, for all $x\in B_{2\gamma}^{\mathbb{N}u_k}(0)$,
\begin{align*}
|v(x)-\overline{l}(x)|\leq |v(x)-h(x)| + |h(x) - h(0) - D h(0)\cdot x|\leq 2 K_4 \,(2\gamma)^{1+\bar{\alpha}}\leq \gamma^{1+\alpha}.
\end{align*}
Next, \eqref{wlinftyboundary} and the definition of $v$ imply
\begin{align*}
|u(r_k {x}) -l_k(r_kx) - r_k^{1+\alpha} h(0) - r_k^{1+\alpha} D h(0) \cdot x |\leq r_k^{1+\alpha} \gamma^{1+\alpha} = r_{k+1}^{1+\alpha}\, \; \textrm{ for all } x \in B_{2\gamma}^{\mathbb{N}u_k} \, ,
\end{align*}
which is equivalent to
\begin{align*}
|u(y) -l_{k+1}(y)|\leq r_k^{1+\alpha} \gamma^{1+\alpha} = r_{k+1}^{1+\alpha}\, \; \textrm{ for all } y=r_kx \in B_{2\gamma r_k}^{\mathbb{N}u}= B_{2r_{k+1}}^\mathbb{N}u\, ,
\end{align*}
where $l_{k+1}(y):=l_k(y)+r_k^{1+\alpha} h(0) + r_k^{\alpha} D h(0) \cdot y $. Then, we define
$a_{k+1} := a_k + h(0) r_k^{1+\alpha}$, $b_{k+1} := b_k + D h(0) r_k^{\alpha}$,
obtaining $(i)_{k+1}$. Also, $|a_{k+1} - a_k|\leq K_4 \,r_k^{1+\alpha}$, $|b_{k+1} - b_k|\leq K_4\, r_k^{\alpha}$, which is $(ii)_{k+1}$. As in the local case, to finish the proof of $(iii)_{k+1}$, it is enough to show that
$$\|v-\overline{l}\|_{C^\beta (\overline{B_\gamma^{\mathbb{N}u_k}})}\leq C_{1,4} \,\gamma^{1+\alpha-\beta}.$$
Let us see that this is obtained by applying the global superlinear $C^\beta$ estimate in proposition \ref{Cbetaquad} to the function $w:=v-\overline{l}$.
Analogously to the local case, $w$ is an $L^p$-viscosity solution in $B_2^{\mathbb{N}u_k}$ of \eqref{Cbetafinal} (see notations and coefficients there), in addition to $w=\psi_k - \overline{l}$ on $ \mathbb{T}_2^{\mathbb{N}u_k}$.
The definition of $\; \mathrm{d}elta$ gives us $\|g_k\|_{L^p(B_1^{\mathbb{N}u_k})}\leq (5+2K_4)\; \mathrm{d}elta \leq \gamma^{\alpha}$.
Further, using that $\psi_k=h$ on $\mathbb{T}_{2\gamma}^{\mathbb{N}u_k}$, we obtain $\|\psi_k-\overline{l}\|_{L^\infty (\mathbb{T}_{2\gamma}^{\mathbb{N}u_k})}\leq \gamma^{1+{\alpha}}.$
Now, since $\psi_k-\overline{l}\in C^1(\mathbb{T}_{2\gamma}^{\mathbb{N}u_k})$, it is a Lipschitz function with constant less or equal than $\|D\psi_k-D\overline{l}\,\|_{C (\mathbb{T}_{2\gamma}^{\mathbb{N}u_k})}\leq 2(C_n+1)+K_4\leq 2 K_4$ and thus
\begin{align*}
|(\psi_k-\overline{l})(x)-(\psi_k-\overline{l})(y)|&=|(\psi_k-\overline{l})(x)-(\psi_k-\overline{l})(y)|^\tau |(\psi_k-\overline{l})(x)-(\psi_k-\overline{l})(y)|^{1-\tau}\\
&\leq (2K_4)^\tau (2K_4)^{1-\tau} |x-y|^\tau \,\gamma^{(1+\bar{\alpha})(1-\tau)}= 2K_4\, |x-y|^\tau \,\gamma^{1-\tau+\bar{\alpha}(1-\tau)}.
\end{align*}
Then, the choice of $\alpha$ implies that
$
[\psi_k-\overline{l}]_{\tau,\mathbb{T}_{2\gamma}^{\mathbb{N}u_k}}\leq 4K_4\,\gamma^{1-\tau+\alpha}.
$
Hence, from this, \eqref{wlinftyboundary} and $C^\beta$ global estimate, properly scaled for the radius $\gamma$, we obtain
\begin{align*}
[w]_{\beta,\overline{B_\gamma^{\mathbb{N}u_k}}} &\leq \gamma^{-\beta} \widetilde{K}_1 \,\{ \, \|w\|_{L^\infty ({B_{2\gamma}^{\mathbb{N}u_k}})} +\gamma^{2-\frac{n}{p}} \|g_k\|_{L^p ({B_{2\gamma}^{\mathbb{N}u_k}})} +\|\psi_k-\overline{l}\|_{L^\infty (\mathbb{T}_{2\gamma}^{\mathbb{N}u_k}) }+\gamma^\tau [\psi_k-\overline{l}]_{\tau,\mathbb{T}_{2\gamma}^{\mathbb{N}u_k}} \,\} \\
&\leq \gamma^{-\beta} K_1 \,\{ \,2\gamma^{1+\alpha} + \gamma^{2-\frac{n}{p}}\gamma^\alpha+ 4K_4\,\gamma^{1+\alpha} \,\}\leq K_1 \,(3+4K_4)\,\gamma^{1+\alpha-\beta}
\end{align*}
and finally, for $C_{1,4}:=1+(3+4K_4)K_1=C_{1,4}\,(K_1,K_4)$, we conclude
\begin{align*}
\|w\|_{C^\beta (\overline{B_\gamma^{\mathbb{N}u_k}})}=\|w\|_{L^\infty ({B_\gamma^{\mathbb{N}u_k}})} +
[w]_{\beta,\overline{B_\gamma^{\mathbb{N}u_k}}}
\leq \gamma^{1+\alpha} +(3+4K_4)K_1 \,\gamma^{1+\alpha-\beta}
\leq C_{1,4}\,\gamma^{1+\alpha-\beta} .
\end{align*}
\end{proof}
Therefore, the complete proof of regularity and estimates in the global case is done by a covering argument over the domain $\Omega$, using local and boundary results.
\mathbb{S}ection{$W^{2,p}$ Results}\label{W2,p regularity}
The first application of the $C^{1,\alpha}$ theory is $W^{2,p}$ regularity for solutions of fully nonlinear equations with superlinear growth in the gradient, which are convex or concave in the variable $X$. This extends the results in \cite{Winter} to superlinear growth in the gradient in the case $p>n$.
In the next two sections we make the convention that $\omega$ is a Lipschitz modulus in the sense that $\omega (r)\leq \omega (1) r$, for all $r\geq 0$, unless otherwise specified.
\begin{teo} \label{W2,p quad}{$(W^{2,p}$ Regularity\,$)$}
Let $\Omega\mathbb{S}ubset {\mathbb{R}^n} $ be a bounded domain and $u\in C(\Omega)$ an $L^p$-viscosity solution of
\begin{align}\label{eqW2,p}
F(x,u,Du,D^2 u)+g(x,Du) =f(x) \quad \textrm{in}\;\;\;\Omega
\end{align}
where $f\in L^p(\Omega)$, $p>n$, $g$ is a measurable function in $x$ such that $g(x,0)=0$ and $|g(x,p)-g(x,q)|\leq \gamma|p-q|+ \mu |p-q|(|p|+|q|)$,
$F$ is convex or concave in $X$ satisfying $(SC)^{0}$, for $b,\, d \in L^\infty_+(\Omega)$ and $\omega$ a Lipschitz modulus. Also, suppose $\| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} \leq C_0$.
Then, there exists $\theta=\theta (n,p,\lambda,\Lambda,\|b\|_{L^ p(\Omega)})$ such that, if \eqref{Htheta} holds for all $r\leq \min \{ r_0, \mathrm{dist} (x_0,\partial\Omega)\}$, for some $r_0>0$ and for all $x_0 \in \Omega$, this implies that $u\in W^{2,p}_{\mathrm{loc}} (\Omega)$ and for every $\Omega^\prime \mathbb{S}ubset\mathbb{S}ubset \Omega$,
\begin{align*}
\|u\|_{W^{2,p}(\Omega^\prime)} \leq C \,\{ \| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} \}
\end{align*}
where $C$ depends on $\,r_0,n,p,\lambda,\Lambda, \mu,\| b \|_{L^p (\Omega)},\omega (1)\|d\|_{L^\infty(\Omega)}, \mathrm{dist} (\Omega^\prime,\partial\Omega),\mathrm{diam} (\Omega)$ and $C_0$.
If, moreover, $\partial\Omega\in C^{1,1}$, $u\in C(\overline{\Omega})$ and $u=\psi$ on $\partial\Omega$ for some $\psi \in W^{2,p} (\Omega )$ with $\| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} + \|\psi\|_{W^{2,p}(\Omega)} \leq C_1$ then, there exists $\theta =\theta (n,p,\lambda , \Lambda,\|b\|_{L^ p(\Omega)})$ such that, if \eqref{Htheta} holds for some $r_0>0$ and for all $x_0 \in \overline{\Omega}$, this implies that $u\in W^{2,p}(\Omega)$ and satisfies the estimate
\begin{align*}
\|u\|_{W^{2,p}(\Omega)} \leq C\, \{ \| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} + \|\psi\|_{W^{2,p} (\Omega)} \}
\end{align*}
where $C$ depends on $r_0,n,p,\lambda,\Lambda, \mu,\| b \|_{L^p (\Omega)},\omega (1)\|d\|_{L^\infty(\Omega)},\partial\Omega,\mathrm{diam} (\Omega)$ and $C_1$.
\end{teo}
\begin{proof}
We prove only the global case, since in the local one we just ignore the term with $\psi$, by considering it equal to zero in what follows.
Notice that $\psi\in W^{2,p}(\Omega)\mathbb{S}ubset C^{1,\tau}(\overline{\Omega})$ for some $\tau\in (0,1)$ with continuous inclusion, then $\| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} + \|\psi\|_{C^{1,\tau} (\partial\Omega)}\leq C_2$.
Thus, by $C^{1,\alpha}$ regularity theorem, we have that $\bar{f}(x):= f(x)-g(x,Du)\in L^p(\Omega)$ and also $$\|u\|_{C^{1,\alpha}(\overline{\Omega})}\leq C_3\, \{\| u \|_{L^{\infty} (\Omega)} + \|\bar{f} \|_{L^p (\Omega)} + \|\psi\|_{C^{1,\tau} (\partial\Omega)} \} .$$
\begin{claim}\label{claimW2,p}
$u$ is an $L^p$-viscosity solution of $F(x,u,Du,D^2u)=\bar{f}(x)$ in $\Omega$.
\end{claim}
\begin{proof}
Let us prove the subsolution case; for the supersolution it is analogous.
Assuming the contrary, there exists some $\phi\in W^{2,p}_{\mathrm{loc}}(\Omega)$, $x_0\in \Omega$ and $\varepsilon>0$ such that $u-\phi$ has a local maximum at $x_0$ and
$F(x,u,D \phi,D^2 \phi)-\bar{f}(x)\leq -\varepsilon$\; a.e. in $B_r(x_0)$.
In turn, by the definition of $u$ being an $L^p$-viscosity subsolution of \eqref{eqW2,p}, we have that
\begin{center}
$F(x,u,D \phi,D^2 \phi) +g(x,D\phi) \geq f(x)-\varepsilon /2$\; a.e. in $B_r(x_0)$
\end{center}
up to diminishing $r>0$. By subtracting the last two inequalities, we obtain that
\begin{align}\label{estW2,p absurdo}
-\{\gamma +\mu (|Du|+|D\phi|)\}\,|Du-D\phi|\leq g(x,Du)-g(x,D\varphi)\leq - \varepsilon /2 \, \textrm{ a.e. in } B_r(x_0).
\end{align}
Since $u-\phi \in C^1(B_r(x_0))$ has a local maximum at $x_0$, we have $D (u-\phi)(x_0)=0$ and, moreover, $|D(u-\phi)(x)|<\varepsilon \, \{\gamma+\mu (\|Du\|_{L^\infty (B_r(x_0))}+\|D\phi\|_{L^\infty (B_r(x_0)}) +1\}^{-1}/4\,$ for all $x\in B_r(x_0)$, possibly for a smaller $r$, which contradicts \eqref{estW2,p absurdo}.
\qedhere{\textit{Claim \ref{claimW2,p}.} }
\end{proof}
Thus by Winter's result, theorem 4.3 in \cite{Winter} (or Świech \cite{Swiech} in the local case), we have that $u\in W^{2,p}(\Omega)$ (respectively $u\in W^{2,p}_{\mathrm{loc}}(\Omega)$) and
\begin{align*}
&\|u\|_{W^{2,p}(\Omega)} \leq C\, \{ \| u \|_{L^{\infty} (\Omega)} + \|\bar{f} \|_{L^p (\Omega)} + \|\psi\|_{W^{2,p} (\Omega)} \}\\
&\leq C\,\{ \| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} +\mu \|u\|_{C^1(\overline{\Omega})}^2+\gamma \|u\|_{C^1(\overline{\Omega})} + \|\psi\|_{W^{2,p} (\Omega)} \}\\
&\leq C\,\{ \| u \|_{L^\infty(\Omega)} + \|f \|_{L^p(\Omega)} + \|\psi\|_{W^{2,p}(\Omega)} + (\mu\, C_2+\gamma) C_3\{\| u \|_{L^\infty (\Omega)} +\|f \|_{L^p(\Omega)} + \|\psi\|_{C^{1,\tau} (\partial\Omega)} \} \}
\end{align*}
which implies the estimate.
\end{proof}
In theorem \ref{W2,p quad}, the final constants only depend on the $L^p$-norm of the coefficient $b$, despite the boundedness hypothesis on it. The latter hypothesis is needed to conclude that solutions are twice differentiable a.e.
Observe that, in \cite{Winter} (see theorem 4.3 there), $W^{2,p}$ results consist of two parts: (i) introducing a new equation $F(x,0,0,D^2 u)=\tilde{f}(x)$ (via corollary 1.6 in \cite{Swiech}), in which $u$ remains a solution in the $L^p$-viscosity sense; (ii) obtaining $W^{2,p}$ estimates for solutions of $F(x,0,0,D^2u)=\widetilde{f}(x)$, which are independent of the zero and first order coefficients.
From the regularity and estimates related to $\mu =0$, we can give an alternative proof of proposition 2.4 in \cite{KSexist2009}, concerning existence and uniqueness for the Pucci's extremal operators with unbounded coefficients in the case $p>n$.
\begin{prop} \label{W2,p solv}{$($Solvability of the Dirichlet problem\,$)$}
Let $\Omega\mathbb{S}ubset {\mathbb{R}^n} $ be a bounded $C^{1,1}$ domain. Let $b,\, d\in L^p_+ (\Omega)$, $p>n$ and $\omega$ a Lipschitz modulus.
Let $f\in L^p(\Omega)$ and $\psi\in W^{2,p}(\Omega)$.
Then, there exists $u_\pm \in C(\overline{\Omega})$ which are the unique $L^p$-viscosity solutions of the problems
\begin{align*}
\left\{
\begin{array}{rllll}
\mathcal{M}^\pm_{\lambda,\Lambda} (D^2 u_\pm)\pm b(x)|Du_\pm|\pm d(x)w((\mp\, u_\pm)^+) &=& f(x)&\mbox{in}&\Omega\\
u_\pm &=& \psi&\mbox{on}&\partial\Omega\, .
\end{array}
\right.
\end{align*}
Moreover, $u_\pm \in W^{2,p}(\Omega)$ and satisfies the estimate
\begin{align*}
\|u_\pm\|_{W^{2,p}(\Omega)} \leq C\, \{ \| u_\pm \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} + \|\psi\|_{W^{2,p} (\Omega)} \}
\end{align*}
where $C$ depends only on $n,p,\lambda,\Lambda,\| b \|_{L^p (\Omega)},\omega (1)\,\| d \|_{L^p (\Omega)},\partial\Omega$ and $\mathrm{diam} (\Omega)$.
\end{prop}
\begin{proof}
It is enough to treat the upper extremal case.
Let $b_k, \, d_k\in L^\infty_+ (\Omega)$ be such that $b_k\rightarrow b$ and $d_k\rightarrow d$ in $L^p(\Omega)$.
Let $u_k \in W^{2,p}(\Omega)$ be the unique $L^p$-viscosity solution of
\begin{align*}
\left\{
\begin{array}{rllll}
\mathcal{M}^+_{\lambda,\Lambda} (D^2 u_k)+ b_k(x)|Du_k|+ d_k(x)\,\omega(u_k^-) &=& f(x)&\mbox{in}&\Omega\\
u_k &=& \psi&\mbox{on}&\partial\Omega
\end{array}
\right.
\end{align*}
given by theorem 4.6 of \cite{Winter}.
From the estimates in theorem \ref{W2,p quad}, with $d_k(x)\omega (u_k^-)$ as RHS,
\begin{align} \label{ukW2,p}
\|u_k\|_{W^{2,p}(\Omega)} \leq C_k \, \{ \| u_k \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} + \|\psi\|_{W^{2,p} (\Omega)} \},
\end{align}
where $C_k$ remains bounded, since $b_k$ and $d_k$ are bounded in $L^p(\Omega)$.
Now, by ABP we have that that
$\|u_k\|_{L^\infty (\Omega)}\leq \|\psi\|_{L^\infty (\partial\Omega)} + C\, \|f\|_{L^p(\Omega)}$.
From this and \eqref{ukW2,p} we get
$\|u_k\|_{W^{2,p}(\Omega)} \leq C$ and hence there exists $u\in C^{1}(\overline{\Omega})$ such that $u_k\rightarrow u$ in $C^{1}(\overline{\Omega})$.
Next, proposition \ref{Lpquad} implies that $u$ is an $L^p$-viscosity solution of
\begin{align}\label{uLpvis}
\left\{
\begin{array}{rllll}
\mathcal{M}^+_{\lambda,\Lambda} (D^2 u)+ b(x)|Du|+ d(x)\,\omega(u^-) &=& f(x)&\mbox{in}&\Omega\\
u&=& \psi&\mbox{on}&\partial\Omega\, .
\end{array}
\right.
\end{align}
Notice that $W^{2,p}(\Omega)$ is reflexive, so there exists $\tilde{u}\in W^{2,p}(\Omega)$ such that $u_k$ converges weakly to $\tilde{u}$. By uniqueness of the limit, $\tilde{u}=u$ a.e. in $\Omega$, and $u$ is a strong solution of \eqref{uLpvis}.
Finally, if there would exist another $L^p$-viscosity solution of \eqref{uLpvis}, say $v\in C(\overline{\Omega})$, then the function $w:=u-v$ satisfies $w=0$ on $\partial\Omega$ and it is an $L^p$-viscosity solution of
$\mathcal{L}^+[w]\geq 0$ in $\Omega\cap\{w>0\}$.
Indeed, since $u$ is strong, we can apply the definition of $v$ as an $L^p$-viscosity supersolution with $u$ as a test function; we also use that $u^- \leq v^- + (u-v)^-$, monotonicity and subadditivity of the modulus.
Then, by ABP we have that $w\leq 0$ in $\Omega$. Analogously, from the definition of subsolution of $v$, we obtain $w\geq 0$ in $\Omega$, and so $w\equiv 0$ in $\Omega$.
\end{proof}
The approximation procedure in the above proof cannot be used to extend theorem \ref{W2,p quad} for unbounded $b$ and $d$, since in this case we do not have uniqueness results to infer that the limiting function is the same as the one we had started with.
However, knowing a priori that the solution is strong, we can obtain $W^{2,p}$ a priori estimates in the general case, as a kind of generalization of Nagumo's lemma (for instance, lemma 5.10 in \cite{Troiani}).
\begin{lem} \label{Nagumo}{$($Generalized Nagumo's lemma\,$)$}
Let $\Omega\mathbb{S}ubset {\mathbb{R}^n} $ be a bounded $C^{1,1}$ domain. Let $F$ be a convex or concave operator in the $X$ entry, satisfying \ref{SCmu}, with $b,\, d\in L^p_+ (\Omega)$ for $p>n$ and $\omega$ an arbitrary modulus.
Suppose that there exists $\theta>0$ such that \eqref{Htheta} holds for some $r_0>0$ and for all $x_0 \in \overline{\Omega}$.
Let $f\in L^p(\Omega)$, $\psi\in W^{2,p}(\Omega)$ and let $u\in W^{2,p}(\Omega)$ be a strong solution of
\begin{align*}
\left\{
\begin{array}{rllll}
F(x,u,Du,D^2 u) &=& f(x)&\mbox{in}&\Omega\\
u&=& \psi &\mbox{on}&\partial\Omega
\end{array}
\right.
\end{align*}
such that $\| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} + \|\psi\|_{W^{2,p}(\Omega)} \leq C_1$.
Then we have
\begin{align}\label{est nagumo}
\|u\|_{W^{2,p}(\Omega)} \leq C\, \{ \| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} + \|\psi\|_{W^{2,p} (\Omega)} +\| d \|_{L^p (\Omega)}\,\omega(\| u \|_{L^{\infty} (\Omega)} ) \}
\end{align}
where $C$ depends on $r_0,n,p,\lambda,\Lambda,\mu,\| b \|_{L^p (\Omega)},C_1,\partial\Omega$ and $\mathrm{diam} (\Omega)$. The local case is analogous.
If $\mu = 0$, then the above constant $C$ does not depend on $C_1$.
\end{lem}
\begin{proof}
Note that, in particular, $u\in C^{1,\alpha}(\overline{\Omega})$ and satisfies $F(x,0,0,D^2 u)=g(x)$ a.e. in $\Omega$,
where $$g(x):=f(x)-F(x,u,Du,D^2u)+F(x,0,0,D^2u)\in L^p (\Omega),$$
since
$|F(x,u,Du,D^2u)-F(x,0,0,D^2u)|\leq b(x)|Du|+\mu |Du^2|+d(x)\,\omega(|u|) \in L^p(\Omega).$
Now, by theorem \ref{W2,p quad} (for $b,d,\mu,\gamma= 0$) and the proof there dealing with $C^{1,\alpha}$ estimates,
\begin{align*}
&\hspace{3cm}\|u\|_{W^{2,p}(\Omega)} \leq C\, \{ \| u \|_{L^{\infty} (\Omega)} + \|g \|_{L^p (\Omega)} + \|\psi\|_{W^{2,p} (\Omega)} \}\\
&\leq C\,\{ \| u \|_{L^{\infty} (\Omega)} + \|f \|_{L^p (\Omega)} +\mu \|u\|_{C^1(\overline{\Omega})}^2 + \|\psi\|_{W^{2,p} (\Omega)}+\|b\|_{L^p(\Omega)}\|u\|_{C^1(\overline{\Omega})}+\|d\|_{L^p(\Omega)}\,\omega(\| u \|_\infty ) \}
\end{align*}
from where \eqref{est nagumo} follows.
\end{proof}
\mathbb{S}ection{The weighted eigenvalue problem}\label{First eigenvalue}
We start recalling some notations.
A subset $K\mathbb{S}ubset E$ of a Banach space is an ordered cone if it is closed, convex, $\lambda K\mathbb{S}ubset K$ for all $\lambda\geq0$ and $K\cap (-K)=\{0\}$. This cone induces a partial order on $E$, for
$ u,v\in E$, given by $u\leq v \Leftrightarrow v-u\in K.$
We say that $K$ is solid if $\mathrm{int}K\mathbb{N}eq\emptyset$.
Further, a \textit{completely continuous} operator, defined in $E$, is continuous and takes bounded sets into precompact ones.
Following the construction of \cite{BR}, \cite{Quaas2004}, we have the following Krein-Rutman theorem for nonlinear operators -- see \cite{tese}.
\begin{teo}\label{KRquaas}{$($Generalized Krein-Rutman\,$)$}
Let $K\mathbb{S}ubset E$ be an ordered solid cone and let $T:K\rightarrow K$ be a completely continuous operator that is also
\begin{enumerate}[(i)]
\item positively 1-homogeneous, i.e. $T(\lambda u)=\lambda Tu$, for all $ \lambda\geq 0, u\in K$;
\item monotone increasing, i.e. for all $ u,v\in K$, $u\leq v$ we have $ Tu\leq Tv$;
\item strongly positive with respect to the cone, in the sense that $T(K\mathbb{S}etminus \{0\})\mathbb{S}ubset \mathrm{int}K$.
\end{enumerate}
Then $T$ has a positive eigenvalue $\alpha_1>0$ associated to a positive eigenfunction $w_1\in \mathrm{int}K$.\label{itemKpositive}
\end{teo}
Consider $\Omega\mathbb{S}ubset {\mathbb{R}^n} $ a bounded $C^{1,1}$ domain. The application of Krein-Rutman is very standard for positive weights \cite{BR}, \cite{HessKato}. Let us recall its use when we have a fully nonlinear operator with unbounded coefficients.
About structure, we suppose
\begin{align}\mathcal{M}_{\lambda, \Lambda}^- (X-Y)-b(x)|p-q|-d(x)\,\omega ((r-s)^+) \leq F(x,r,p,X) - F(x,s,q,Y) \label{SC}\tag{SC} \\ \leq \mathcal{M}_{\lambda, \Lambda}^+ (X-Y)+b(x)|p-q|+d(x)\,\omega ((s-r)^+)\;\; \textrm{ for } x\in \Omega \mathbb{N}onumber\end{align}
with $F(\cdot,0,0,0)\equiv 0$, where $0<\lambda \leq \Lambda$, $ b\in L^p_+ (\Omega)$, $ p>n$, $d\in L^\infty_+ (\Omega)$, $\omega$ a Lipschitz modulus.
Here, the condition over the zero order term in \eqref{SC} means that $F$ is proper, i.e.\ decreasing in~$r$.
Consider $E=C_0^1(\overline{\Omega})$ and the usual ordered solid cone $K=\{u\in E;\,u\geq 0 \textrm{ in }\Omega\}$ in $E$.
Let $c(x)\in L^p_+(\Omega)$ with $c>0$ in $\overline{\Omega}$, $p> n$.
As the operator on $K$, we take $T=-F^{-1}\circ c$ in the sense that $U=Tu$ iff $U$ is the unique $L^n$-viscosity solution of the Dirichlet problem
\begin{align}\label{Tu}\tag*{$(T_u)$}
\left\{
\begin{array}{rclcc}
F(x,U,DU,D^2U)&=& -c(x)u & \mbox{in} & \Omega\\
U &=& 0 &\mbox{on}& \partial\Omega
\end{array}
\right.
\end{align}
where $F$ satisfies the following hypotheses
\begin{align}\label{HF} \tag{$H$}
\left\{
\begin{array}{l}
\textrm{there exists } \theta>0 \textrm{ such that } (\overline{H})_\theta \textrm{ holds for for all } x_0\in \overline{\Omega}, \\
\eqref{SC} \textrm{ and } \eqref{ExistUnic T bem definido} \textrm{ hold},\;\,F(x,tr,tp,tX)=tF(x,r,p,X) \textrm{ for all } t\geq 0.
\end{array}
\right.
\end{align}
Here, hypothesis \eqref{ExistUnic T bem definido} means the solvability in $L^n$- sense with data in $L^p$, i.e. for any $f\in L^p(\Omega)$,
\begin{align}\label{ExistUnic T bem definido} \tag{$S$}
\textrm{there exists a unique } u\in C(\overline{\Omega})\; L^n\textrm{-viscosity solution of }
F[u]=f(x) \textrm{ in }\Omega; \;u=0 \textrm{ on }\partial\Omega.
\end{align}
Of course, Pucci's extremal operators
$$\mathcal{L}^\pm [u]:=\mathcal{M}^\pm (D^2 u)\pm b(x)|Du|\pm d(x)\omega(u^\mp), \;\;b,\,d\in L^p_+(\Omega),$$
where $\omega$ is a Lipschitz modulus, are particular examples of $F$ satisfying \eqref{HF}. Indeed, recall that proposition \ref{W2,p solv} provides a strong solution $u\in W^{2,p}(\Omega)\mathbb{S}ubset W^{2,n}(\Omega)$, which is an $L^n$-viscosity solution by proposition \ref{Lpiffstrong.quad}. Furthermore, since it is unique among $L^p$-viscosity solutions, it is also unique among $L^n$-viscosity ones. Here all the coefficients can be unbounded.
Observe that \eqref{ExistUnic T bem definido} and $(\overline{H})_\theta$ also holds when $F$ is a uniformly continuous operator in $x$ satisfying the growth conditions in \cite{Arms2009} (see also \cite{IY}), in this case concerning $C$-viscosity notions of solutions.
On the other hand, $(\overline{H})_\theta$, \eqref{SC} and \eqref{ExistUnic T bem definido} are completely enough to ensure existence, uniqueness and $C^{1,\alpha}$ global regularity and estimates for the problem \ref{Tu} from Theorem \ref{C1,alpha regularity estimates geral}, which in turn implies that the operator $T$ is well defined and completely continuous.
Furthermore, $T=-F^{-1}\circ c$ is strictly positive with respect to the cone, thanks to SMP and Hopf.
In general, without the strict positiveness of $c$ in $\Omega$ there is no guarantee on this property, i.e., under $c\geq 0$ and $c\mathbb{N}ot\equiv 0$ in $\Omega$, we only obtain that $T(K\mathbb{S}etminus\{0\})\mathbb{S}ubset K$.
Notice that $T=-F^{-1}\circ c\,$ has an eigenvalue $\alpha_1>0$ associated to the positive eigenfunction $\varphi_1$ if and only if $\varphi_1$ is an $L^n$-viscosity solution of
\begin{align*}
\left\{
\begin{array}{rclcc}
F[\varphi_1]+1/\alpha_1 \, c(x) \varphi_1 &=& 0 &\mbox{in} & \Omega \\
\varphi_1 &>& 0 &\mbox{in} &\Omega \\
\varphi_1 &=& 0 &\mbox{on} &\partial\Omega\, .
\end{array}
\right.
\end{align*}
For any $c\in L^p(\Omega)$ with $p> n$ and $F$ satisfying \eqref{HF}, we can define, as in \cite{BNV}, \cite{QB},
\begin{align*}
\lambda_1^\pm=\lambda_1^\pm\,(F(c),\Omega)&=\mathbb{S}up\left\{ \lambda>0; \; \Psi^\pm(F(c),\Omega,\lambda)\mathbb{N}eq \emptyset\right\}
\end{align*}
where
$\Psi^\pm (F(c),\Omega,\lambda):=\left\{ \psi\in C(\overline{\Omega}); \; \pm\psi>0 \textrm{ in }\Omega,\; \pm (F[\psi]+\lambda c(x)\psi )\leq 0 \textrm{ in }\Omega \right\}$;
with inequalities holding in the $L^n$-viscosity sense. Notice that, by definition,
$\lambda_1^\pm (G(c),\Omega)=\lambda_1^\mp (F(c),\Omega),$
where $G(x,r,p,X):=-F(x,-r,-p,-X)$.
With an approximation procedure by positive weights given by Krein-Rutman theorem as above, for $F$ satisfying \eqref{HF}, we obtain existence of eigenvalues with nonnegative weight.
\begin{teo}\label{exist eig for F c geq 0}
Let $\Omega\mathbb{S}ubset {\mathbb{R}^n} $ be a bounded $C^{1,1}$ domain, $c\in L^p(\Omega)$, $c\gneqq 0$ for $p>n$ and $F$ satisfying \eqref{HF} for $b,\, d\in L_+^\infty (\Omega)$. Then $F$ has two positive weighted eigenvalues $\alpha_1^\pm>0$ corresponding to normalized and signed eigenfunctions $\varphi_1^\pm\in C^{1,\alpha}(\overline{\Omega})$ that satisfies
\begin{align} \label{eq exist eigen F c lambda1+}
\left\{
\begin{array}{rclcc}
F[\varphi_1^\pm]+\alpha_1^\pm c(x) \varphi_1^\pm &=& 0 &\mbox{in} & \Omega \\
\pm \varphi_1^\pm &>& 0 &\mbox{in} &\Omega \\
\varphi_1^\pm &=& 0 &\mbox{on} &\partial\Omega
\end{array}
\right.
\end{align}
in the $L^p$-viscosity sense, with $\max_{\overline{\Omega}}\,(\pm \varphi_1^\pm) =1$.
If, moreover, the operator $F$ has $W^{2,p}$ regularity of solutions $($in the sense that every $u\in C(\overline{\Omega})$ which is an $L^p$-viscosity solution of $F[u]=f(x)\in L^p(\Omega)$, $u=0$ on $\partial\Omega$, satisfies $u\in W^{2,p}(\Omega))$, then $\alpha_1^\pm=\lambda^\pm_1$ and the conclusion is valid also for $b\in L^p_+(\Omega)$.
\end{teo}
Notice that we obtain positive eigenvalues because $F$ is proper. For general existence related to nonproper operators see the script in \cite{QB} for bounded coefficients.
We also stress that, without regularity assumptions on the domain, it is still possible to obtain the existence of an eigenpair, as in \cite{BNV} and \cite{QB}; in such cases the eigenfunction belongs to $C^{1,\alpha}_{\mathrm{loc}}(\Omega)\cap C(\overline{\Omega})$ by using $C^{1,\alpha}$ local regularity instead of the global one.
We start proving some auxiliary results which take into account the unboundedness of $c$.
\begin{prop}\label{th4.1 QB} Let $u,v\in C(\overline{\Omega})$ be $L^n$-viscosity solutions of
\begin{align}\label{eq th1.4 2case ineq}
\left\{
\begin{array}{rclcc}
F[u]+c(x)u &\geq & 0 & \mbox{in} &\Omega \\
u &<& 0& \mbox{in} & \Omega
\end{array}
\right. ,\quad
\left\{
\begin{array}{rcll}
F[v]+c(x)v &\leq & 0 &\mbox{in} \;\;\; \Omega \\
v &\geq & 0 & \mbox{on} \;\; \partial\Omega\\
v(x_0) &< & 0 &x_0 \in\Omega
\end{array}
\right.
\end{align}
with $F$ satisfying \eqref{HF}, $c\in L^p(\Omega)$, $p>n$. Suppose one, $u$ or $v$, is a strong solution. Then, $u=tv$ for some $t>0$.
The conclusion is the same if $F[u]+c(x)u\leq 0$, $F[v]+c(x)v\geq 0 $ in $\Omega$, with $u>0$ in $\Omega$, $v\leq 0$ on $\partial\Omega$ and $v(x_0)>0$ for some $x_0 \in\Omega$.
\end{prop}
For the proof of proposition \ref{th4.1 QB}, as in \cite{Arms2009}, \cite{BNV}, \cite{QB}, we need the following consequence of ABP, which is MP for small domains.
\begin{lem}\label{MP small domain}
Assume $F$ satisfies \eqref{SC} and $c\in L^p(\Omega)$, $p>n$. Then there exists $\varepsilon_0>0$, depending on $n,p,\lambda,\Lambda$, $\|b\|_{L^p(\Omega)}$, $\|c^+\|_{L^p(\Omega)}$ and $\mathrm{diam}(\Omega)$, such that if $|\Omega|\leq \varepsilon_0$ then any $u\in C(\overline{\Omega})$ which is an $L^n$-viscosity solution of
\begin{align}\label{MP small domain eq u}
\left\{
\begin{array}{rclcc}
F[u]+c(x)u &\geq & 0 &\mbox{in} &\Omega\\
u &\leq & 0 &\mbox{on} & \partial\Omega
\end{array}
\right.
\end{align}
satisfies $u\leq 0$ in $\Omega$. Analogously, any $v\in C(\overline{\Omega})$ that is an $L^n$-viscosity solution of $F[v]+c(x)v\leq 0$ in $\Omega$, with $v\geq 0$ on $\partial\Omega$, is such that $v\geq 0$ in $\Omega$ provided $|\Omega|\leq \varepsilon_0$.
\end{lem}
\begin{proof}
Assume $u$ satisfies \eqref{MP small domain eq u}. In order to obtain a contradiction, suppose that $\Omega^+:=\{u>0\}$ is not empty. By \eqref{SC}, we have that $u$ is an $L^n$-viscosity solution of
\begin{center}
$\mathcal{M}^+(D^2u)+b(x)|Du|\geq \mathcal{M}^+(D^2u)+b(x)|Du|-c^-(x)u\geq -c^+(x)u$ \; in $\Omega^+$.
\end{center}
Hence, ABP gives us that
$$\mathbb{S}up_{\Omega^+}u\leq C_1 \,\textrm{diam}(\Omega)\, \|c^+\|_{L^n(\Omega)}\mathbb{S}up_{\Omega^+}u \leq C_1 \,\textrm{diam}(\Omega)\, |\Omega|^{1-\frac{n}{p}}\|c^+\|_{L^p(\Omega)}\mathbb{S}up_{\Omega^+}u. $$
Then we choose $\epsilon_0>0$ such that $C_1 \,\textrm{diam}(\Omega)\, \varepsilon_0^{1-\frac{n}{p}}\|c^+\|_{L^p(\Omega)}\leq 1/2$ to produce a contradiction. If $v$ is a supersolution it is similar, by using ABP in the opposite direction.
\qedhere{\,\textit{Lemma \ref{MP small domain}.}}
\end{proof}
\begin{proof}[\textit{Proof of Proposition \ref{th4.1 QB}.}]
We are going to prove the first case, since the second is analogous.
Let $u,v$ be $L^n$-viscosity solutions of \eqref{eq th1.4 2case ineq}. Say both are strong, otherwise just use test functions for one of them and read all inequalities below in the $L^n$-viscosity sense.
Set $z_t:=tu-v$ for $t>0$. Then, using 1-homogeneity and \eqref{SC}, we have that $z_t$ is a solution of
\begin{align}\label{eq th 1.4 sem DF}
\mathcal{M}^+(D^2 z_t)&+b(x)|Dz_t|+d(x)\omega((-z_t)^+)+c(x)z_t \geq F[tu]-F[v]+c(x)z_t \mathbb{N}onumber\\
&=t\,\{F[u]+c(x)u\}-\{F[v]+c(x)v\}\geq 0 \;\;\textrm{in }\Omega\, .
\end{align}
Let $K$ be a compact subset of $\Omega$ such that $x_0\in K$ and MP lemma \ref{MP small domain} holds for $\Omega\mathbb{S}etminus K$. Further, let $t_0>0$ be large enough such that $z_{t_0}\leq 0$ in $K$. In fact, this $t_0$ can be taken as $\min_K v / \max_K u >0$, since $u<0$ in $K$ and $\min_K v\leq v(x_0)<0$.
Then, since $z_{t_0}\leq 0$ in $\partial\, (\Omega\mathbb{S}etminus K)\mathbb{S}ubset \partial\Omega\cup\partial K$, we obtain from lemma \ref{MP small domain} that $z_{t_0}\leq 0$ in $\Omega\mathbb{S}etminus K$ and so in $\Omega$.
Define $\tau:=\inf \{t>0;\;z_t\leq 0\;\textrm{ in }\Omega\}\geq t_0>0$.
Hence, using \eqref{SC}, we have that $z_\tau$ is a nonpositive solution of
$
\mathcal{L}^-[-z_\tau]+\{c(x)-d(x)\,\omega (1)\}(-z_\tau) \leq 0 \;\;\textrm{ in } \Omega
$
and so by SMP we have either $z_\tau\equiv 0$ or $z_\tau <0$ in $\Omega$. In the first case we are done. Suppose, then, $z_\tau <0$ in $\Omega$ in order to obtain a contradiction.
Next we choose some $\varepsilon >0$ such that $z_{\tau-\varepsilon}<0$ in $K$. Indeed, we can take, for example, $\varepsilon =\min \{-\min_K z_\tau / (2\|u\|_{L^\infty (K)}), \tau/2 \}$, which implies, as in \cite{Patrizi},
$$z_{\tau -\varepsilon}=z_\tau -\varepsilon u\leq \min_K z_\tau +\varepsilon\|u\|_{L^\infty (K)}<0 \textrm{ in }K.$$
In particular, $z_t$ satisfies \eqref{eq th 1.4 sem DF} for $t=\tau -\varepsilon >0$ . Thus, $z_{\tau-\varepsilon}\leq 0$ by MP in $\Omega \mathbb{S}etminus K$. By SMP, $z_{\tau-\varepsilon}<0$ in $\Omega$, which contradicts the definition of $\tau$ as an infimum.
\qedhere{\textit{Proposition \ref{th4.1 QB}. }}
\end{proof}
The next result was first introduced in \cite{BNV} and extended in \cite{QB} to nonlinear operators. When we add an unbounded weight $c$, all we need is its positiveness on a subset of positive measure in order to obtain a bound from above on $\lambda_1$.
\begin{lem} \label{boundedness eig QB}
Suppose \eqref{HF} with $b,\, d\in L_+^\infty (\Omega)$. If $c\geq \; \mathrm{d}elta>0$ a.e. in $B_R\mathbb{S}ubset\mathbb{S}ubset\Omega$, for $R\leq 1$, then
$$\lambda_1^\pm (F(c),\Omega)\leq\frac{C_0}{\; \mathrm{d}elta R^2}$$
for a positive constant $C_0$ that depends on $n,\lambda,\Lambda,R,\|b\|_{L^\infty(\Omega)} $ and $\omega (1)\|d\|_{L^\infty(\Omega)}$.
If, moreover, $F$ has no term of order zero (i.e. $d$ or $\omega$ is equal to zero), then $R$ can be any positive number. On the other hand, if $b\equiv 0$, then $C_0$ does not depend on $R$.
\end{lem}
\begin{proof}
Observe that $\lambda_1^\pm(F(c),\Omega) \leq \lambda_1^\pm (F(c),B_R)$ by definition.
Consider, as in \cite{BNV} and \cite{QB}, the radial function $\mathbb{S}igma (x):=-(R^2-|x|^2)^2<0$ in $B_R$.
Let us treat the case of $\lambda_1^-$, since for $\lambda_1^+$ it is just a question of looking at $-\mathbb{S}igma$.
Suppose, in order to obtain a contradiction, that there exists some $\lambda>\frac{C_0}{\; \mathrm{d}elta R^2}$ such that $\Psi^-(F(c),\Omega,\lambda)\mathbb{N}eq\emptyset$, i.e. let $\psi\in C(\overline{\Omega})$ be a negative $L^n$-viscosity solution of $F[\psi]+\lambda c(x)\psi\geq 0$ in $\Omega$; also of $F[\psi]+\frac{C_0}{\; \mathrm{d}elta R^2} c(x)\psi\geq 0$ in $B_R$.
\begin{claim} \label{claim sigma}
We have $F[\mathbb{S}igma]+ \frac{C_0}{\; \mathrm{d}elta R^2} c(x)\, \mathbb{S}igma \leq 0$ a.e. in $B_R$.
\end{claim}
\begin{proof}
Say, for example, $b(x)\leq\gamma$ and $d(x)\leq \eta$ a.e., then it holds (see \cite{BNV} or \cite{tese})
\begin{align*}
\frac{F[\mathbb{S}igma]}{\mathbb{S}igma}\geq \frac{8\lambda\, |x|^2}{(R^2-|x|^2)^2} -\frac{4n\Lambda}{R^2-|x|^2}-\frac{4\gamma R}{R^2-|x|^2}-\eta \,\omega (1) \;\textrm{ a.e. in } B_R\, .
\end{align*}
Hence, if we take
$\alpha=({n\Lambda +\gamma R})(2\lambda+n\Lambda+\gamma R)^{-1}\in (0,1)$,
we have two cases.
$(a)$ $|x|^2\geq \alpha R^2$: From construction,
${F[\mathbb{S}igma]}/{\mathbb{S}igma}\geq - \eta\, \omega (1)\geq-{\eta\, \omega (1)}\,c(x)/({\; \mathrm{d}elta R^2} ).$
$(b)$ $|x|^2\leq \alpha R^2$: In this case we just bound the first term by zero; the others are such that
${F[\mathbb{S}igma]}/{\mathbb{S}igma}\geq - {4(n\Lambda +\gamma R)}/({(1-\alpha)R^2}) - \eta\, \omega (1)\geq - {C_0}\,c(x)/({\; \mathrm{d}elta R^2}).$
\qedhere{\textit{Claim \ref{claim sigma}.}}
\end{proof}
Now we apply proposition \ref{th4.1 QB}, since $\mathbb{S}igma\in C^2(\overline{B}_R)$, obtaining that $\psi = t\mathbb{S}igma$, for some $t>0$. However, this is not possible, since $\psi <0$ on $\partial B_R\mathbb{S}ubset\Omega$ while $\mathbb{S}igma =0$ on $\partial B_R\,$.
\qedhere{\textit{Lemma \ref{boundedness eig QB}.} }
\end{proof}
Moving to the last statement in theorem \ref{exist eig for F c geq 0}, we first prove an eigenvalue bound that takes into account an unbounded $b$, when the weight is a continuous and positive function in $\overline{\Omega}$.
Note that, in this case, theorem \ref{KRquaas} gives us a pair $\alpha_1>0$ and $\varphi_1\in C^{1}(\overline{\Omega})$ such that
\begin{align} \label{eq exist eigen G 1}
\left\{
\begin{array}{rclcc}
G[\varphi_1]+\alpha_1 \,c(x) \varphi_1 &= &0 &\mbox{in} & \Omega \\
\varphi_1 &> &0 &\mbox{in} &\Omega \\
\varphi_1 &=& 0 &\mbox{on} &\partial\Omega
\end{array}
\right.
\end{align}
in the $L ^n$-viscosity sense, with $\max_{\overline{\Omega}}\,\varphi_1=1$ and $0<\alpha_1\leq \lambda_1^+(G(c),\Omega)=\lambda_1^-(F(c),\Omega).$
The following lemma is a delicate point in our construction of an eigenpair. It states that $\alpha_1$ in \eqref{eq exist eigen G 1} is bounded, and this does not seem to be a consequence of the usual methods for bounding a first eigenvalue, such as the one in lemma \ref{boundedness eig QB}. Instead, we use the classical blow-up method \cite{GS} of Gidas and Spruck.
\begin{lem}\label{limit unbounded b cont c}
Let $c\in C(\overline{\Omega})$, $c>0$ in $\overline{\Omega}$ and $G$ satisfying \eqref{HF} with $b\in L^p_+(\Omega)$, $d\in L_+^\infty (\Omega)$. Let $\alpha_1$ and $\varphi_1$ as in \eqref{eq exist eigen G 1}. Then $\alpha_1\leq C$, for $C=C(n,\lambda,\Lambda,\Omega, \|b\|_{L^p(\Omega)},\omega (1)\|d\|_{L^\infty(\Omega)})$.
\end{lem}
\begin{proof}
If the conclusion is not true, then exists a sequence $b_k\in L^\infty_+ (\Omega)$, with $\|b_k\|_{L^p(\Omega)}\leq C$, $\|b_k\|_{L^\infty (\Omega)}\rightarrow +\infty$ and the respective eigenvalue problem
\begin{align} \label{eq exist eigen Gk c epsilon}
\left\{
\begin{array}{rclcc}
G_k\,[\varphi_k]+\alpha_1^k \,c(x) \varphi_k &= &0 &\mbox{in} & \Omega \\
\varphi_k &> &0 &\mbox{in} &\Omega \\
\varphi_k &=& 0 &\mbox{on} &\partial\Omega
\end{array}
\right.
\end{align}
in the $L^n$-viscosity sense, with $\max_{\overline{\Omega}}\,\varphi_k=1$ for all $k\in\mathbb{N}$ and $\alpha_1^k\rightarrow +\infty$ as $k\rightarrow +\infty$, where $G_k$ is a fully nonlinear operator satisfying $(H)_k\,$, i.e. \eqref{HF} for $b_k$ and $d_k$. Say $d_k\leq \eta$ and $\max_{\overline{\Omega}}\varphi_k=\varphi_k(x_0^k)$ for $x_0^k\in\Omega$. Then, $x_0^k\rightarrow x_0\in \overline{\Omega}$ as $k\rightarrow +\infty$, up to a subsequence.
\textit{Case 1:} $x_0\in\Omega$. Let $2\rho=\mathrm{dist}(x_0,\partial\Omega)>0$ and notice that $x_0^k\in B_\rho (x_0)$ for all $k\geq k_0$.
Set $r_k={(\alpha_1^k)}^{-{1}/{2}}$ and define $\psi_k(x)=\varphi_k(x_0^k+r_k x)$. Thus, $\psi_k$ is an $L^n$ (so $L^p$) viscosity solution of
\begin{align*}
\widetilde{G}_k(x,\psi_k,D\psi_k,D^2\psi_k)+c_k (x)\psi_k(x)=0 \quad \textrm{ in }\; \widetilde{B}_k:=B_{\rho/{r_k}}(0)
\end{align*}
where ${c}_k (x):=c(x_0^k+r_k x)$, $\widetilde{G}_k(x,r,p,X):=r^2_k\, G_k(x_0^k+r_k x,r,p/{r_k},X/{r_k^2})$ satisfies $(\widetilde{H})_k$, i.e. \eqref{HF} for $\widetilde{b_k}$ and $\eta_k$, where $\widetilde{b}_k(x):=r_k \,b_k(x_0^k+r_k x) $ and ${\eta}_k=r_k^2 \,\eta$. Notice that $b_k$ and $\eta_k$ converge locally to zero in $L^p(\widetilde{B}_k)$ as $k\rightarrow +\infty$, since $p>n$.
Furthermore, $\mathbb{S}up_{\widetilde{B}_k}\psi_k=\psi_k(0)=1$ for all $k\in\mathbb{N}$ and $B_R(0)\mathbb{S}ubset\mathbb{S}ubset \widetilde{B}_k$ for large $k$, for any fixed $R>0$.
By theorem \ref{C1,alpha regularity estimates geral} we have that $\psi_k$ is locally in $C^{1,\alpha}$ and satisfies the estimate
\begin{align*}
\|\psi_k\|_{C^{1,\alpha}(\overline{B}_R(0))}\leq C_k \|\psi_k\|_{L^\infty (\widetilde{B_k})}\leq C,
\end{align*}
since $\psi_k$ attains its maximum at $0$ and $C_k$ depends only on the $L^p$-norm of the coefficients $b_k$ and $c_k$, which are uniformly bounded in there.
Hence, by compact inclusion we have that there exists $\psi\in C^1(\overline{B}_R(0))$ such that $\psi_k\rightarrow \psi$ as $k\rightarrow +\infty$, up to a subsequence. Doing the same for each ball $B_R(0)$, for every $R>0$, we obtain in particular that $\psi_k\rightarrow \psi$ in $L^{\infty}_{\mathrm{loc}}( {\mathbb{R}^n} )$, by using the uniqueness of the limit for $\psi_k$ in the smaller balls.
Using stability (proposition \ref{Lpquad} together with observation \ref{Lpquadencaixados}) and the continuity of $c$, we have that $\psi$ is an $L^p$-viscosity solution of $J(x,D^2\psi)+{c} (x_0)\psi=0$ in $ {\mathbb{R}^n} $ for some measurable operator $J$ still satisfying \eqref{HF} with coefficients of zero and first order term, $d$ and $b$, equal to zero.
Also, $\psi(0)=1$ and $\psi>0$ in $ {\mathbb{R}^n} $ by SMP.
This implies that $1\leq \lambda_1^+ (J({c(x_0)}),B_R)\leq \frac{C_0}{c(x_0)R^2}$ for all $R>0$, which gives a contradiction when we take $R\rightarrow +\infty$.
\textit{Case 2:} $x_0\in\partial\Omega$.
By passing to new coordinates, that come from the smoothness property of the domain $\partial\Omega\in C^{1,1}$, we can suppose that $\partial\Omega\mathbb{S}ubset \{x_n=0\}$ and $\Omega\mathbb{S}ubset\{ x_n>0\}$.
Set $\rho_k=\mathrm{dist}(x_0^k,\partial\Omega)=x_0^k\cdot e_n=x_{0,n}^k\,$, where $e_n=(0,\ldots,0,1)$, $x_{0}^k=(x_{0,1}^k,\ldots,x_{0,n}^k)$.
Analogously, consider $\psi_k (y)$ in $y\in B_{\rho_k/{r_k}}(0)$ and the respective equation $\widetilde{G}_k$ as in case 1.
Thus, we have for $x,y$ satisfying $r_k y=x-x_0^k\,$, that the set $\{x_n>0\}$ is equivalent to $A_k:=\{y_n=(x-x_0^k)\cdot e_n / {r_k}>-\rho_k/{r_k}\}$. Now we need to analyze the behavior of the set $A_k$ when we take the limit as $k\rightarrow +\infty$.
We first claim that $\rho_k/{r_k}$ is bounded below by a constant $C_1>0$, which means that $A_k$ does not converge to $\{y_n>0\}$.
This is an easy consequence of our $C^{1,\alpha}$ boundary regularity and estimates in a half ball, applied to $\psi_k$ and $\widetilde{G}_k$.
Indeed, since $\|D \psi_k\|_{L^\infty (B_r^+(0))}\leq C$, then $1=|\varphi (x_0^k)-\varphi(\bar{x}_0^k)|=|\psi_k(0,0)-\psi_k(0,-\rho_k/{r_k})|\leq C \rho_k/{r_k}$, with $\bar{x}_0^k=(x_{0,1}^k,\ldots,x_{0,n-1}^k,0)\in \partial\Omega$ and fixed $r>0$, from where we obtain the desired bound.
Next observe that we have two possibilities about the fraction $\rho_k/ {r_k}$, either it converges to $+\infty$ or it is uniformly bounded. In the first one, $A_k\rightarrow {\mathbb{R}^n} $ and we finish as in case 1. In the second, $A_k\rightarrow \{y_n> \varrho\}$, $\varrho\in (0,+\infty)$, by passing to a subsequence, and the proof carries on as in the case 1, since we have a smooth domain that contains a ball with radius $R=(2\,C_0/c(x_0)\,)^{1/2}$; this derives the final contradiction.
\end{proof}
\begin{lem}\label{limit final}
Let $c\in L^p(\Omega)$, $c\geq \; \mathrm{d}elta$ in $B_R$ for some $B_R\mathbb{S}ubset\mathbb{S}ubset\Omega$ and $F$ satisfying \eqref{HF}, then
$$\lambda_1^\pm (F(c),\Omega) \leq \frac{\lambda_1^\pm (F(1),B_R)}{\; \mathrm{d}elta}. $$
\end{lem}
\begin{proof}
Let us prove the $\lambda^+_1$ case; for $\lambda_1^-$ we use $G$ instead of $F$. We already know that both quantities are nonnegative, by the properness of the operator $F$. Hence, it is enough to verify that $\mathcal{A}\cap \{\lambda\geq 0\} \mathbb{S}ubset\mathcal{B}/\; \mathrm{d}elta \cap \{\lambda\geq 0\}$, where
$$
\lambda_1^+(F(c),\Omega)=\mathbb{S}up_\mathcal{A} \lambda = \mathbb{S}up_{\mathcal{A}\cap \{\lambda\geq 0\}} \lambda\; ,\;\; \lambda_1^+(F(1),B_R)=\mathbb{S}up_\mathcal{B} \lambda = \mathbb{S}up_{\mathcal{B}\cap \{\lambda\geq 0\}} \lambda
$$
as defined before. Let $\lambda\in \mathcal{A}\cap \{\lambda\geq 0\}$, then there exists $\psi\in C(\overline{\Omega})$ a nonnegative $L^n$-viscosity solution of $F[\psi]+c(x)\lambda\psi \leq 0$ in $\Omega$. Then, $\psi$ is also a nonnegative $L^n$-viscosity solution of $F[\psi]+\; \mathrm{d}elta\lambda\psi \leq 0$ in $B_R$\,, from where $\; \mathrm{d}elta\lambda\in \mathcal{B}$.
\end{proof}
\begin{proof}[\textit{Proof of Theorem \ref{exist eig for F c geq 0}.}]
First, from the fact that $c>0$ in a set of positive measure, there exists $\; \mathrm{d}elta>0$ such that $\{c\geq \; \mathrm{d}elta\}$ is a nontrivial set. In fact, if this was not true, i.e. if $|\{c\geq\; \mathrm{d}elta\}|=0$ for all $\; \mathrm{d}elta$, then $\{c>0\}=\bigcup_{\; \mathrm{d}elta>0} \{c\geq\; \mathrm{d}elta\}$ would have measure zero, as the union of such sets, contradicting the hypothesis.
Namely, then, $c\geq \; \mathrm{d}elta>0$ a.e. in some ball $B_R\mathbb{S}ubset\mathbb{S}ubset\Omega$.
Let us prove the $\lambda_1^-$ case, applying Krein-Rutman results to $G$; for $\lambda_1^+$ replace $G$ by $F$.
Let $\varepsilon\in (0,1)$ and define $c_\varepsilon:=c+\varepsilon >0$ in $\Omega$, for all $\varepsilon$.
From theorem \ref{KRquaas}, we obtain the existence of pairs $\alpha_1^\varepsilon>0$ and $\varphi_1^\varepsilon\in C^{1}(\overline{\Omega})$ such that
\begin{align} \label{eq exist eigen G c epsilon}
\left\{
\begin{array}{rclcc}
G[\varphi_1^\varepsilon]+\alpha_1^\varepsilon \,c_\varepsilon(x) \varphi_1^\varepsilon &= &0 &\mbox{in} & \Omega \\
\varphi_1^\varepsilon &> &0 &\mbox{in} &\Omega \\
\varphi_1^\varepsilon &=& 0 &\mbox{on} &\partial\Omega
\end{array}
\right.
\end{align}
with $\max_{\overline{\Omega}}\,\varphi_1^\varepsilon=1$ for all $\varepsilon\in (0,1)$. Then,
\begin{align}\label{cota lambda1 b,d limit}
0<\alpha_1^\varepsilon\leq \lambda_1^+(G(c_\varepsilon),\Omega)=\lambda_1^-(F(c_\varepsilon),\Omega)\leq \frac{C_0}{\; \mathrm{d}elta R^2}\,\;\textrm{ for all }\varepsilon\in (0,1).
\end{align}
Next, $\alpha_1^\varepsilon\rightarrow\alpha_1 \in [0,C_0/{\; \mathrm{d}elta R^2}]$ up to a subsequence.
Then, applying $C^{1,\alpha}$ global regularity and estimates (theorem \ref{C1,alpha regularity estimates geral}) in the case $\mu=0$ (recall again that $L^n$-viscosity solutions are $L^p$-viscosity for $p>n$), by considering $\alpha_1^\varepsilon \,c_\varepsilon (x)\,\varphi_1^\varepsilon \in L^p (\Omega)$ as the right hand side, we obtain
\begin{align*}
\|\varphi_1^\varepsilon\|_{C^{1,\alpha}(\overline{\Omega})} &\leq C\,\{\,\|\varphi_1^\varepsilon\|_{L^\infty(\Omega)} +\alpha_1^\varepsilon\, \|c_\varepsilon\|_{L^p(\Omega)} \,\|\varphi_1^\varepsilon \|_\infty +1\,\}
\leq C\, C_1 \,(\|c\|_{L^p(\Omega)} +1) \,\} \leq C.
\end{align*}
Hence the compact inclusion $C^{1,\alpha}(\overline{\Omega})\mathbb{S}ubset C^1(\overline{\Omega})$ yields $\varphi_1^\varepsilon \rightarrow \varphi_1\in C^1(\overline{\Omega})$, up to a subsequence. Of course this implies that $\max_{\overline{\Omega}}\,\varphi_1=1$, $\varphi_1\geq 0$ in $\Omega$ and $\varphi_1=0$ on $\partial\Omega$.
Since $c_\varepsilon\rightarrow c$ in $L^p(\Omega)$ as $\varepsilon\rightarrow 0$, by proposition \ref{Lpquad} we have that $\varphi_1$ is an $L^p$-viscosity solution of $G[\varphi_1]+\alpha_1 c(x)\varphi_1=0$ in $\Omega$, which allows us to apply $C^{1,\alpha}$ regularity again to obtain that $\varphi_1\in C^{1,\alpha}(\overline{\Omega})$.
Using now that $\varphi_1$ is an $L^p$-viscosity solution of
$\mathcal{L}^-[\varphi_1]-(d(x)\omega (1)-\alpha_1 c(x))\,\varphi_1 \leq 0$ in $ \Omega$, together with SMP, we have that $\varphi_1>0$ in $\Omega$, since $\max_{\overline{\Omega}}\,\varphi_1=1$.
Moreover, we must have $\alpha_1>0$, because the case $\alpha_1=0$ would imply that $\varphi_1$ is an $L ^p$-viscosity solution of $\mathcal{L}^+[\varphi_1] \geq 0$ in $\Omega\cap \{\varphi_1>0\}$ (since $F$ is proper, and so $G$) which, in turn, would give us $\varphi_1\leq 0$ in $\Omega$, by ABP. Thus, the existence property is completed.
In order to conclude that, under $W^{2,p}$ regularity assumptions over $F$, the $\alpha_1$ obtained is equal to $\lambda_1^-=\lambda_1^-(F(c),\Omega)$, related to $\varphi_1^-=\varphi_1^-(F(c),\Omega)=-\varphi_1<0$ in $\Omega$, we have to work a little bit more, as in proposition 4.7 in \cite{QB}.
We already have that $\alpha_1\leq\lambda_1^-$. Suppose by contradiction that $\alpha_1<\lambda_1^-$.
By definition of $\lambda_1^-$ as a supremum, we know that $\alpha_1$ cannot be an upper bound, i.e. there exists $\lambda>0$ such that $\Psi^-(F(c),\Omega,\lambda)\mathbb{N}eq\emptyset$ and $\alpha_1<\lambda\leq\lambda_1^-$. Then we obtain $\psi\in C(\overline{\Omega})$ such that
$F[\psi]+\lambda c(x)\psi\geq 0$ in $\Omega$
in the $L^n$-viscosity sense, with $\psi<0$ in $\Omega$.
Now, since $c\gneqq 0$, we have $c(x)(\lambda-\alpha_1)\gneqq 0$. Next, $\psi$ is an $L^n$-viscosity solution of
\begin{align}\label{contradction exist eig}
F[\psi]+\alpha_1 \,c(x)\psi\gneqq F[\psi]+\lambda\, c(x)\psi\geq 0\;\textrm{ in }\Omega\, .
\end{align}
Then, under $W^{2,p}$ regularity, we have that $\varphi_1^-\in W^{2,p}(\Omega)\mathbb{S}ubset W^{2,n}(\Omega)$ is a strong solution of
\begin{align*}
\left\{
\begin{array}{rclcc}
F[\varphi_1^-]+\alpha_1 \,c(x) \varphi_1^- &=& 0 &\mbox{in} & \Omega \\
\varphi_1^- &<& 0 &\mbox{in} &\Omega \\
\varphi_1^- &=& 0 &\mbox{on} &\partial\Omega\, .
\end{array}
\right.
\end{align*}
Applying proposition \ref{th4.1 QB} we obtain that $\psi=t\varphi^-_1$ for some $t>0$; but this contradicts the strict inequality in \eqref{contradction exist eig}. Thus, we must have $\alpha_1=\lambda_1^-$. The case of $\lambda_1^+$ is completely analogous, by reversing the inequalities.
From this last paragraph, under $W^{2,p}$ regularity of the solutions, the only possibility to $\alpha_1$ is to coincide with $\lambda_1$.
Therefore, by using lemmas \ref{limit unbounded b cont c} (with $c\equiv 1$) and \ref{limit final}, we obtain that
$\lambda_1^-(F(c_\varepsilon),\Omega)\leq C_1/\; \mathrm{d}elta$, for all $\varepsilon\in (0,1)$,
where $C_1$ depends on $n,\lambda,\Lambda,R$, $\|b\|_{L^p(\Omega)}$ and $\omega (1)\|d\|_{L^\infty(\Omega)}$.
Thus, we carry on this bound on $\lambda_1$, instead of \eqref{cota lambda1 b,d limit}, in the limiting procedure, in order to get the desired existence result for $b\in L^p(\Omega)$.
\qedhere{\textit{Theorem \ref{exist eig for F c geq 0}.}}
\end{proof}
\end{document} |
\begin{document}
\title{Undecidability of Multiplicative Subexponential Logic}
\begin{abstract}
Subexponential logic is a variant of linear logic with a family of
exponential connectives---called \cdoth{subexponentials}---that are
indexed and arranged in a pre-order.
Each subexponential has or lacks associated structural properties of
weakening and contraction.
We show that classical propositional multiplicative linear logic
extended with one unrestricted and two incomparable linear
subexponentials can encode the halting problem for two register
Minsky machines, and is hence undecidable.
\end{abstract}
\section{Introduction}
\label{sec:introduction}
The decision problem for classical propositional multiplicative
exponential linear logic (\MELL), consisting of formulas constructed
from propositional atoms using the connectives $\set{\TENS,
\ONE, \PAR, \BOT, \BANG, \QM}$, is perhaps the longest standing open
problem in linear logic.
\MELL is bounded below by the purely multiplicative fragment (\MLL),
which is decidable even in the presence of first-order quantification,
and above by \MELL with additive connectives (\MAELL), which is
undecidable even for the propositional fragment~\cite{lincoln90apal}.
This paper tries to make the undecidable upper bound a bit tighter by
considering the question of the decision problem for a family of
propositional multiplicative \cdoth{subexponential} logics
(\MSEL)~\cite{nigam09phd,nigam09ppdp-alt}, each of which consists of
formulas constructed from propositional atoms using the (potentially
infinite) set of connectives $\set{\TENS,\ONE,\PAR,\BOT} \cup \
\bigcup_{u \in \Upsigma} \set{\BANG^u, \QM^u}$, where $\Upsigma$ is a
pre-ordered set of subexponential \cdoth{labels}, called a
\cdoth{subexponential signature}, that is a parameter of the family of
logics.
In particular, we show that a particular \MSEL with a subexponential
signature consisting of exactly three labels can encode a two register
Minsky machine (\TWORM), which is Turing-equivalent.
This is the same strategy used in~\cite{lincoln90apal} to show the
undecidability of \MAELL, but the encoding in \MSEL is
different---simpler---for the branching instructions, and shows that
additive behaviour is not essential to implement branching.
We use the classical dialect of linear logic to show these results.
The intuitionistic dialect has the same decision problem because it is
possible to faithfully encode (\ie linearly simulate the sequent
proofs of) the classical dialect in the intuitionistic dialect without
changing the signature~\cite{local:chaudhuri10csl}.
This short note is organized as follows: in
section~\ref{sec:background} we sketch the one-sided sequent
formulation of \MSEL and recall the definition of a \TWORM.
In section~\ref{sec:encoding} we encode the transition system of a
\TWORM in a \MSEL with a particular signature.
In section~\ref{sec:adequacy} we argue that the encoding is
\cdoth{adequate}, \ie that the halting problem for a \TWORM is reduced
to the proof search problem for this \MSEL-encoding, by appealing to a
focused sequent calculus for \MSEL.
The final section~\ref{sec:perspectives} discusses some of the
ramifications of this result.
\section{Background}
\label{sec:background}
\subsection{Propositional Subexponential Logic}
\label{sec:subexponential-logic}
Let us quickly recall propositional subexponential logic (\SEL) and
its associated sequent calculus proof system.
This logic is sometimes called subexponential \cdoth{linear} logic
(\SELL), but since it is possible for the subexponentials to have
linear semantics it is redundant to include both adjectives.
Formulas of \SEL ($A, B, \dotsc$) are built from \cdoth{atomic
formulas} ($a, b, \dotsc$) according to the following grammar:
\begin{tikzpicture}[node distance=1ex]
\node [matrix of math nodes] (gr) {
A, B, \dotsc & ::= &
\node(a){a}; & \UpgammaOR & \node(tens){A \TENS B}; & \UpgammaOR & \ONE & \UpgammaOR &
\node(plus){A \PLUS B}; & \UpgammaOR & \ZERO &
\UpgammaOR & \node(bang){\BANG^u A}; \\
& \node[right] {\UpgammaOR} ; &
\node(a'){\NEG a}; & \UpgammaOR &
\node(par){A \PAR B}; & \UpgammaOR & \node(bot){\BOT}; & \UpgammaOR &
\node(with){A \UpomegaITH B}; & \UpgammaOR & \node(top){\TOP}; &
\UpgammaOR & \node(qm){\QM^u A}; \\
} ;
\begin{scope}[on background layer]
\fill[rounded corners,color=green!5!white]
($(a.north west)-(.2,0)$) rectangle ($(a'.south east)-(0,.5)$) ;
\node at ($(a'.south west)!.5!(a'.south east)-(0.05,.2)$){
\tiny\scshape atomic
} ;
\fill[rounded corners,color=blue!5!white]
(tens.north west) rectangle ($(bot.south east)-(0,.5)$) ;
\node at ($(par.south west)!.5!(bot.south east)-(0,.2)$) {
\tiny\scshape multiplicative
} ;
\fill [rounded corners,color=red!5!white]
(plus.north west) rectangle ($(top.south east)-(0,.5)$) ;
\node at ($(with.south west)!.5!(top.south east)-(0,.2)$) {
\tiny\scshape additive
} ;
\fill [rounded corners,color=orange!15!white]
(bang.north west) rectangle ($(qm.south east)+(1,-.5)$) ;
\node at ($(qm.south west)!.5!(qm.south east)+(.5,-.2)$) {
\tiny \scshape subexponential
} ;
\end{scope}
\end{tikzpicture}
\noindent
Each column in the grammar above is a De Morgan dual pair.
A \cdoth{positive formula} (depicted with $P$ or $Q$ when relevant) is
a formula belonging to the first line of the grammar, and a
\cdoth{negative formula} (depicted with $N$ or $M$) is a formula
belonging to the second line.
The \cdoth{labels} ($u, v, \dotsc$) on the subexponential connectives
$\BANG^u$ and $\QM^u$ belong to a \cdoth{subexponential signature}
defined below.
The additive fragment of this syntax is just used in this section for
illustration; we will not be using the additives in our encodings.
The fragment without the additives will be called \cdoth{multiplicative
subexponential logic} (\MSEL).
\begin{definition}
A \cdoth{subexponential signature} $\Upsigma$ is a structure $\str{\LOC,
U, \le}$ where:
\begin{itemize}
\item $\LOC$ is a countable set of \cdoth{labels};
\item $U \subseteq \LOC$, called the \cdoth{unbounded labels}; and
\item ${\le} \subseteq \LOC \times \LOC$ is a pre-order on $\LOC$---
\ie it is reflexive and transitive---and $\le$-upwardly closed
with respect to $U$, \ie for any $u,v\in \LOC$, if $u \in U$ and
$u \le v$, then $v \in U$. \defin
\end{itemize}
\end{definition}
\noindent
We will assume an ambient signature $\Upsigma$ unless we need to
disambiguate particular instances of \MSEL, in which case we will
use $\Upsigma$ in subscripts.
For instance, $\MSEL_\Upsigma$ is a particular instance of \MSEL for $\Upsigma$.
\begin{figure}
\caption{
Inference rules for a cut-free one-sided sequent calculus formulation of
\SEL.
Only the rules on the last line are sensitive to the signature. }
\label{fig:sel-rules}
\end{figure}
The true formulas of \MSEL are derived from a \cdoth{sequent calculus}
proof system consisting of sequents of the form $|- A_1, \dotsc, A_n$
(with $n > 0$) and abbreviated as $|- \Upgamma$.
The \cdoth{contexts} ($\Upgamma, \Updelta, \ldots$) are multi-sets of formulas of
\SEL, and $\Upgamma,\Updelta$ and $\Upgamma, A$ stand as usual for the multi-set union
of $\Upgamma$ with $\Updelta$ and $\set{A}$, respectively.
The inference rules for \SEL sequents are displayed in
figure~\ref{fig:sel-rules}.
Most of the rules are shared between \SEL and linear logic and will
not be elaborated upon here.
The differences are with the subexponentials, for which we use the
following definition.
\begin{definition}
For any $n \in N$ and lists $\vec u = [u_1, \dotsc, u_n]$ and $\vec
A = [A_1, \dotsc, A_n]$, we write $\QM^{\vec u} \vec A$ to stand for
the context $\QM^{u_1} A_1, \dotsc, \QM^{u_n} A_n$.
For $\vec v = [v_1, \dotsc, v_n]$, we write $u \le \vec v$ to mean
that $u \le v_1$, \ldots, and $u \le v_n$. \defin
\end{definition}
The rule for $\BANG$, sometimes called \cdoth{promotion}, has a side
condition that checks that the label of the principal formula is less
than the labels of all the other formulas in the context.
This rule cannot be used if there are non-$\QM$-formulas in the
context, nor if the labels of some of the $\QM$-formulas are strictly
smaller or incomparable with that of the principal $\BANG$-formula.
Both these properties will be used in the encoding in the next section.
The structural rules of weakening and contraction apply to those
principal $\QM$-formulas with unbounded labels.
\subsection{Two Register Minsky Machines}
\label{sec:two-register-minsky}
Like Turing machines, Minsky register machines have a finite state
diagram and transitions that can perform I/O on some unbounded storage
device, in this case a bank of registers that can store arbitrary
natural numbers.
We shall limit ourselves to machines with two registers (\TWORM) $\ka$
and $\kb$, which are sufficient to encode Turing machines.
\begin{definition}
A \TWORM is a structure $\str{Q, *, \CONF, \TRANS{}}$ where:
\begin{itemize}
\item $Q$ is a non-empty finite set of \cdoth{states};
\item $* \in Q$ is a distinguished \cdoth{halting state};
\item $\CONF$ is a set of \cdoth{configurations}, each of which is a
structure of the form $\str{q, v}$, with $q \in Q$ and $v :
\set{\ka, \kb} \to N$, that assigns values (natural numbers) to
the registers $\ka$ and $\kb$ in state $q$;
\item $\TRANS{} \subseteq \CONF \times I \times \CONF$ is a
deterministic labelled transition relation between configurations
where the label set $I = \set{\k{halt}, \k{incra}, \k{incrb},
\k{decra}, \k{decrb}, \k{isza}, \k{iszb}}$ (called the
\cdoth{instructions}).
\end{itemize}
By usual convention, we write $\TRANS{}$ infix with the instruction
atop the arrow.
We require that every element of $\TRANS$ fits one of the following
schemas, where in each case $q, r \in Q$ and $q \neq r$:
\begin{gather}
\begin{array}{rcl}
\str{q, v}
&\TRANS[\kl{halt}]&
\str{*, \set{\ka:0, \kb:0}}
\rlap{\hspace{1cm} $(\text{with }q \neq *)$}
\\
\str{q, \set{\ka:m, \kb:n}}
&\TRANS[\kl{incra}]&
\str{r, \set{\ka: m + 1, \kb:n}}
\\
\str{q, \set{\ka:m, \kb:n}}
&\TRANS[\kl{incrb}]&
\str{r, \set{\ka:m, \kb:n + 1}}
\\
\str{q, \set{\ka:m + 1, \kb:n}}
&\TRANS[\kl{decra}]&
\str{r, \set{\ka: m, \kb:n}}
\\
\str{q, \set{\ka:m, \kb:n + 1}}
&\TRANS[\kl{decrb}]&
\str{r, \set{\ka:m, \kb:n}}
\\
\str{q, \set{\ka:0, \kb:n}}
&\TRANS[\kl{isza}]&
\str{r, \set{\ka:0, \kb:n}}
\\
\str{q, \set{\ka:m, \kb:0}}
&\TRANS[\kl{iszb}]&
\str{r, \set{\ka:m, \kb:0}}
\end{array}
\label{eq:trans}
\end{gather}
For a \cdoth{trace} $\vec i = [i_1, \dotsc, i_n]$, we write
$\str{q_0,v_0} \TRANS[\vec i] \str{q_n, v_n}$ if $\str{q_0, v_0}
\TRANS[i_1] \dotsm \TRANS[i_n] \str{q_n, v_n}$.
The \TWORM \cdoth{halts from} an initial configuration $\str{q_0,
v_0}$ if there is a trace $\vec i$ such that $\str{q_0, v_0}
\TRANS[\vec i] \str{*, \set{\ka:0, \kb:0}}$.
(The configuration $\str{*, \set{\ka:0, \kb:0}}$ will be called the
\cdoth{halting configuration}.)
The \cdoth{halting problem} for a \TWORM is the decision problem of
whether the machine halts from an initial configuration. \defin
\end{definition}
The requirement that $\TRANS$ be deterministic amounts to: $\str{q, v}
\TRANS[i] \str{q_1, v_1}$ and $\str{q, v} \TRANS[j] \str{q_2, v_2}$
imply that $i = j$, $q_1 = q_2$, and $v_1 = v_2$.
Note that a trace that does not end with a halting configuration will
not be considered to be halting, even if there is no possible
successor configuration.
It is an easy exercise to transform a given \TWORM into one where
every configuration has a successor except for the halting
configuration.
\begin{theorem}[\cite{minsky61aom}] \label{thm:2rm-undec}
The halting problem for {\TWORM}s is recursively unsolvable. \qed
\end{theorem}
\section{The Encoding}
\label{sec:encoding}
For a given \TWORM, which we fix in this section, we will encode its
halting problem as the derivability of a particular \MSEL sequent that
encodes its labelled transition system and the initial configuration.
We will use the following subexponential signature in the rest of this
section.
\begin{definition}
Let $\TWOSI$ stand for the signature $\str{\set{\UNB, \ka, \kb},
\set{\UNB}, \le}$ where $\le$ is the reflexive-transitive closure
of $\le_0$ defined by $\ka \le_0 \UNB$ and $\kb \le_0 \UNB$.
\defin
\end{definition}
\begin{definition}[encoding configurations]
For $c = \str{q, v}$, we write $\ENC{c}$ for the following
$\MSEL_\TWOSI$ context:
\begin{gather*}
\underbrace{\QM^\ka \NEG \kra, \QM^\ka \NEG \kra, \dotsc,
\QM^\ka \NEG \kra}_{\text{length } =\ v(\ka)},
\underbrace{\QM^\kb \NEG \krb, \QM^\kb \NEG \krb, \dotsc,
\QM^\kb \NEG \krb}_{\text{length } =\ v(\kb)},
\NEG q
\tag*{\defin}
\end{gather*}
\end{definition}
\begin{definition}[encoding transitions]
The transitions~(\ref{eq:trans}) of the \TWORM are encoded as a
context $\Pi$ with:
\begin{itemize}
\item to represent $\str{q,v} \TRANS[\k{halt}] \str{*,
\set{\ka:0,\kb:0}}$, the elements: $q \TENS \NEG \kh, \kh \TENS
\BANG^\ka \kra \TENS \NEG \kh, \kh \TENS \BANG^\kb \krb \TENS \NEG
\kh, \kh \TENS \BANG^\infty \ONE$ (for some $\kh \notin Q$):
\item to represent $\str{q, \set{\ka:m, \kb:n}} \TRANS[\k{incra}]
\str{r, \set{\ka:m + 1, \kb:n}}$, the element $q \TENS (\NEG
r \PAR \QM^\ka \NEG \kra)$;
\item to represent $\str{q, \set{\ka:m, \kb:n}} \TRANS[\k{incrb}]
\str{r, \set{\ka:m, \kb:n + 1}}$, the element: $q \TENS (\NEG
r \PAR \QM^\kb \NEG \krb)$;
\item to represent $\str{q, \set{\ka:m + 1, \kb:n}}
\TRANS[\k{decra}] \str{r, \set{\ka:m, \kb:n}}$, the element: $q
\TENS \BANG^\ka \kra \TENS \NEG r$;
\item to represent $\str{q, \set{\ka:m, \kb:n + 1}}
\TRANS[\k{decrb}] \str{r, \set{\ka:m, \kb:n}}$, the element: $q
\TENS \BANG^\kb \krb \TENS \NEG r$;
\item to represent $\str{q, \set{\ka:0, \kb:n}} \TRANS[\k{isza}]
\str{r, \set{\ka:0, \kb:n}}$, the element: $q \TENS \BANG^\kb \NEG
r$; and
\item to represent $\str{q, \set{\ka:m, \kb:0}} \TRANS[\k{iszb}]
\str{r, \set{\ka:m, \kb:0}}$, the element: $q \TENS \BANG^\ka \NEG
r$.
\end{itemize}
Note that $\Pi$ contains a finite number of elements. \defin
\end{definition}
\begin{definition}[encoding the halting problem]
If $\Upgamma$ is $A_1, \dotsc, A_n$, then let $\QM^u \Upgamma$ stand for $\QM^u
A_1, \dotsc, \QM^u A_n$.
The encoding of the halting problem for the \TWORM from the initial
configuration $c_0 = \str{q_0, v_0}$ is the $\MSEL_\TWOSI$ sequent
$|- \QM^\infty \Pi, \ENC{c_0}$. \defin
\end{definition}
\begin{theorem} \label{thm:forward}
If the \TWORM halts from $c_0$, then $|-_\TWOSI \QM^\infty \Pi,
\ENC{c_0}$ is derivable.
\end{theorem}
\begin{proof}
We will show that if $c = \str{q_1, v_1} \TRANS[i] \str{q_2, v_2} =
d$ (for some $i$), then the following $\MSEL_\TWOSI$ rule is
derivable:
\begin{gather*}
\infer{
|- \QM^\infty \Pi, \ENC{c}
}{
|- \QM^\infty \Pi, \ENC{d}
}
\end{gather*}
This is largely immediate by inspection. Here are three
representative cases.
\begin{itemize}
\item The case of $i = \k{incra}$: it must be that $v_2(\ka) =
v_1(\ka) + 1$ and $v_2(\kb) = v_1(\kb)$, so $\ENC{d} =
\ENC{c}\setminus\set{\NEG q_1}, \NEG q_2, \QM^\ka \kra$.
Moreover, $q_1 \TENS (\NEG q_2 \PAR \QM^\ka \NEG \kra) \in \Pi$.
So:
\begin{gather*}
\infer[\rn{contr}, \QM]{
|- \QM^\infty \Pi, \ENC{c}
}{
\infer[\TENS]{
|- \QM^\infty \Pi, \ENC{c}, q_1 \TENS (\NEG q_2 \PAR \QM^\ka \NEG \kra)
}{
\infer[\rn{init}]{|- \NEG q_1, q_1}{} &
\infer[\PAR]{
|- \QM^\infty \Pi, \ENC{c}\setminus\set{\NEG q_1}, \NEG q_2 \PAR \QM^\ka \NEG \kra
}{
|- \QM^\infty \Pi, \ENC{c}\setminus\set{\NEG q_1}, \NEG q_2, \QM^\ka \NEG \kra
}}}
\end{gather*}
The cases for \k{incrb}, \k{decra}, and \k{decrb} are similar.
\item The case of $i = \k{isza}$: it must be that $v_2(\ka) =
v_1(\ka) = 0$ and $v_2(\kb) = v_1(\kb)$.
Hence, $\ENC{d} = \ENC{c}\setminus\set{\NEG q_1}, \NEG q_2$ and
$\QM^\ka \kra \notin \ENC{c} \cup \ENC{d}$.
Moreover, $q_1 \TENS \BANG^\kb \NEG q_2 \in \Pi$. So:
\begin{gather*}
\infer[\rn{contr}, \QM]{
|- \QM^\infty \Pi, \ENC{c}
}{
\infer[\TENS]{
|- \QM^\infty \Pi, \ENC{c},
q_1 \TENS \BANG^\kb \NEG q_2
}{
\infer[\rn{init}]{|- \NEG q_1, q_1}{}
&
\infer[\BANG]{
|- \QM^\infty \Pi, \ENC{c}\setminus\set{\NEG q_1},
\BANG^\kb \NEG q_2
}{
|- \QM^\infty \Pi, \ENC{c}\setminus\set{\NEG q_1},
\NEG q_2
}}}
\end{gather*}
The instance of $\BANG$ is justified because $\kb \le \infty$ and
$\kb \le \kb$, and there are no $\QM$-formulas labelled $\ka$ or
non-$\QM$ formulas in the sequent.
The case of \k{iszb} is similar.
\item The case of $i = \k{halt}$.
Here, we know that $q_1 \TENS \NEG \kh \in \Pi$, so:
\begin{gather*}
\infer[\rn{contr}, \QM]{
|- \QM^\infty \Pi, \ENC{c}
}{
\infer[\TENS]{
|- \QM^\infty \Pi, \ENC{c}, q_1 \TENS \NEG \kh
}{
\infer[\rn{init}]{|- \NEG q_1, q_1}{}
&
|- \QM^\infty \Pi, \ENC{c} \setminus\set{\NEG q_1}, \NEG \kh
}}
\end{gather*}
Now, as long as there are any occurrences of $\QM^\ka \kra$ or
$\QM^\ka \krb$ in $\ENC{c}$, we can apply one of the decrementing
rules $\kh \TENS \BANG^\ka \kra \TENS \NEG \kh$ or $\kh \TENS
\BANG^\kb \krb \TENS \NEG \kh \in \Pi$.
The general case looks something like this, where $\Updelta_\kra =
\set{\NEG \kra, \dotsc, \NEG \kra}$ and $\Updelta_\krb = \set{\NEG \krb,
\dotsc, \NEG \krb}$.
\begin{gather*}
\infer[\rn{contr},\QM]{
|- \QM^\infty \Pi,
\ENC{c} \setminus\set{\smash{\NEG q_1,
\QM^\ka \Updelta_\kra, \QM^\kb \Updelta_\krb, \QM^\ka \NEG \kra}},
\QM^\ka \NEG \kra,
\NEG \kh
}{
\infer[\TENS, \TENS]{
|- \QM^\infty \Pi,
\ENC{c} \setminus\set{\smash{\NEG q_1,
\QM^\ka \Updelta_\kra, \QM^\kb \Updelta_\krb, \QM^\ka \NEG \kra}},
\QM^\ka \NEG \kra,
\NEG \kh, \kh \TENS \BANG^\ka \kra \TENS \NEG \kh
}{
\infer[\rn{init}]{|- \kh, \NEG \kh}{}
&
\infer[\BANG]{
|- \QM^\ka \NEG \kra, \BANG^\ka \kra
}{
\infer[\QM]{
|- \QM^\ka \NEG \kra, \kra
}{
\infer[\rn{init}]{
|- \NEG \kra, \kra
}{
}}}
&
|- \QM^\infty \Pi,
\ENC{c} \setminus\set{\smash{\NEG q_1,
\QM^\ka \Updelta_\kra, \QM^\kb \Updelta_\krb, \QM^\ka \NEG \kra}},
\NEG \kh
}}
\end{gather*}
There is a symmetric case for contracting the $\kh \TENS \BANG^\kb
\krb \TENS \NEG \kh$.
Eventually, the right branch just becomes $|- \QM^\infty \Pi, \NEG
\kh$, at which point we have:
\begin{gather*}
\infer[\rn{contr}, \QM]{
|- \QM^\infty \Pi, \NEG \kh
}{
\infer[\TENS]{
|- \QM^\infty \Pi, \NEG \kh, \kh \TENS \BANG^\infty \ONE
}{
\infer[\rn{init}]{|- \kh, \NEG \kh}{}
&
\infer[\BANG]{
|- \QM^\infty \Pi, \BANG^\infty \ONE
}{
\infer*[\rn{weak}]{
|- \QM^\infty \Pi, \ONE
}{
\infer[\ONE]{|- \ONE}{}
}}
}}
\tag*{\qedhere}
\end{gather*}
\end{itemize}
\end{proof}
\section{Adequacy of the Encoding via Focusing}
\label{sec:adequacy}
By the contrapositive of theorem~\ref{thm:forward}, if the sequent
$|-_\TWOSI \QM^\infty \Pi, \ENC{c_0}$ is not derivable, then the
\TWORM does not halt from $c_0$.
This gives half of the reduction.
For the converse of theorem~\ref{thm:forward}, we need to show how to
recover a halting trace by searching for proofs of a $\MSEL_\TWOSI$
encoding of a halting problem.
The best way to do this is to build a focused proof which will have
the derived inference rules in the above proof as the only possible
\cdoth{synthetic} rules, in a sense made precise below.
We will begin by sketching the focused proof system for \SEL that is
sound and complete for the unfocused system of
figure~\ref{fig:sel-rules}, and then show how the synthetic rules for
the encoding are in bijection for all instructions (with a small
correction needed for \k{halt}).
\begin{figure}
\caption{
Inference rules for a focused sequent calculus formulation of
\SEL.}
\label{fig:fsel-rules}
\end{figure}
Focusing is a general technique to restrict the non-determinism in a
cut-free sequent proof system.
Though originally defined for classical linear logic
in~\cite{andreoli92jlc}, it is readily extended to many other
logics~\cite{chaudhuri08jar,liang09tcs,nigam09phd}.
This section sketches the basic design of a focused version of the
rules of figure~\ref{fig:sel-rules}, and omits most of the
meta-theoretic proofs of soundness and completeness, for which the
general proof techniques are by now well
known~\cite{chaudhuri08jar,miller07cslb,simmons14tocl}.
To keep things simple, we will define a focused calculus by adding to
the unfocused system a new kind of \cdoth{focused sequent}, $|- \Upomega,
\foc{A}$, where the formula $A$ is \cdoth{under focus}.
Contexts written with $\Upomega$, which we call \cdoth{neutral contexts}, can
contain only positive formulas, atoms, negated atoms, and
$\QM$-formulas.
The rules of the focused proof system for \SEL are depicted in
figure~\ref{fig:fsel-rules}.
Focused sequents are created---reading from conclusion upwards to
premises---from unfocused sequents with neutral contexts by means of
the rules \rn{decide}, \rn{ldecide}, or \rn{udecide}.
In a focused sequent, only the formula under focus can be principal,
and the focus persists on the immediate subformulas of this formula in
the premises, with the exception of the rule $\foc{\BANG}$.
In the base case, for \focrn{init}, the focused atom must find its
negation in the context, while all formulas in the context must be
$\QM$-formulas with unbounded labels.
When the focused formula is negative, the focus is released with the
\focrn{blur} rule, at which point any of the unfocused rules
$\set{\PAR,\BOT,\UpomegaITH,\TOP}$ of figure~\ref{fig:sel-rules} can be used
to decompose the formula and its descendants further.
Eventually, when there are no more negative descendants---\ie the
whole context has the form $\Upomega$---a new focused phase is launched
again and the cycle repeats.
Note that the structural rules \rn{contr} and \rn{weak} of the
unfocused calculus are removed in the focused system.
Instead, weakening is folded into \focrn{init}, $\foc{\BANG}$, and
$\foc{\ONE}$, and contraction is folded into $\foc{\TENS}$ and
\rn{udecide}.
The rules \rn{contr} and \rn{weak} remain admissible for either
sequent form in the focused calculus.
\begin{theorem} \label{thm:foc-compl}
The \SEL sequent $|- \Upgamma$ is provable in the unfocused system of
figure~\ref{fig:sel-rules} iff it is provable in the focused system
of figure~\ref{fig:fsel-rules}.
\end{theorem}
\begin{proof}[Sketch]
Straightforward adaptation of existing proofs of the soundness and
completeness of focusing, such
as~\cite{chaudhuri08jar,miller07cslb,simmons14tocl}.
An instance for \SEL can be found in~\cite[chapter 5]{nigam09phd}.
\end{proof}
\begin{theorem} \label{thm:backward}
The \TWORM halts from $c_0$ if $|-_\TWOSI \QM^\infty \Pi, \ENC{c_0}$
is derivable.
\end{theorem}
\begin{proof}
We will show instead that the \TWORM halts from $c_0$ if the sequent
$|-_\TWOSI \QM^\infty \Pi, \ENC{c_0}$ is derivable in the focused
calculus, and we will moreover extract the halting trace from such a
focused proof.
The required result will then follow immediately from
theorem~\ref{thm:foc-compl}, since any provable \SEL sequent has a
focused proof.
Let a focused proof of $|-_\TWOSI \QM^\infty \Pi, \ENC{c}$ (for $c =
\str{q, v}$) be given.
We proceed by induction on the lowermost instance of \rn{udecide} in
this proof.
Note that the $\MSEL_\TWOSI$ context $\QM^\infty \Pi, \ENC{c}$ is
neutral; moreover, all the elements of $\ENC{c}$ are either negated
atoms or $\QM$-prefixed negated atoms with bounded labels.
So, the only rules of the focusing system that apply to this sequent
are \rn{ldecide} or \rn{udecide}.
However, if we use \rn{ldecide}, then the premise becomes
unprovable, as there is no way to remove an occurrence of $\NEG
\kra$ or $\NEG \krb$ from a context that also contains $\NEG q$.
Thus, the only possible rule will be an instance of \rn{udecide},
with the focused formula in the premise being one of the $\Pi$.
First, consider the case where the focused formula does not contain
$\k{h}$, \ie it corresponds to one of the instructions in
$I\setminus\set{\k{halt}}$.
In each of these cases, the focused phase that immediately follows
is deterministic.
As a characteristic case, suppose the focused formula is $q \TENS
\BANG^\kb \NEG r$; then we have:
\begin{gather*}
\infer[\rn{udecide}]{
|- \QM^\infty \Pi, \ENC{c}
}{
\infer[\foc{\TENS}]{
|- \QM^\infty \Pi, \ENC{c},
\foc{\smash{q \TENS \BANG^\kb \NEG r}}
}{
\infer[\rn{init}]{|- \NEG q, q}{}
&
\infer[\foc{\BANG}]{
|- \QM^\infty \Pi, \ENC{c}\setminus\set{\NEG q},
\foc{\smash{\BANG^\kb \NEG r}}
}{
|- \QM^\infty \Pi, \ENC{c}\setminus\set{\NEG q},
\NEG r
}}}
\end{gather*}
The right premise is now itself neutral and an encoding of a
different configuration.
We can appeal to the inductive hypothesis to find a halting trace
for it, to which we can prepend the instruction \k{isza} to get the
halting trace from $c$.
A similar argument can be used for the other instructions in
$I\setminus\set{\k{halt}}$.
This leaves just the formulas involving $\k{h}$ for the lowermost
\rn{udecide}.
We cannot select any formula but $q \TENS \NEG \kh$ from $\Pi$, for
the derivation would immediately fail because $\kh \notin Q$ and
there is no $\NEG \kh$ in $\ENC{c}$ to use with \focrn{init}.
So, as the formula selected is $q \TENS \NEG \kh$, we have:
\begin{gather*}
\infer[\rn{udecide}]{
|- \QM^\infty \Pi, \ENC{c}
}{
\infer[\TENS]{
|- \QM^\infty \Pi, \ENC{c}, \foc{q \TENS \NEG \kh}
}{
\infer[\focrn{init}]{|- \NEG q, \foc{q}}{}
&
\infer[\focrn{blur}]{
|- \QM^\infty \Pi, \ENC{c} \setminus\set{\NEG q}, \foc{\NEG \kh}
}{
|- \QM^\infty \Pi, \ENC{c} \setminus\set{\NEG q}, \NEG \kh
}
}}
\end{gather*}
The context of the right premise is now neutral, so the only rule
that applies to it is \rn{udecide}.
A simple nested induction will show that sequents of this form $|-
\QM^\infty \Pi, \ENC{c} \setminus\set{\NEG q}, \NEG \kh$ are always
derivable in the focused calculus.
Therefore, the trace that corresponds to the configuration $c$ is
just the singleton \k{halt}.
\end{proof}
\begin{corollary}
The derivability of $\MSEL_\TWOSI$ sequents is recursively
unsolvable.
\end{corollary}
\begin{proof}
Directly from theorems~\ref{thm:2rm-undec}, \ref{thm:forward}, and
\ref{thm:backward}.
\end{proof}
\section{Conclusion and Perspectives}
\label{sec:perspectives}
We have given a fairly obvious encoding of a \TWORM in a suitable
instance of \MSEL containing a three element subexponential signature.
The encoding of the \TWORM halting problem is very similar to that
of~\cite{lincoln90apal} for \MAELL; the main difference is in the
encoding of the \k{isz} transitions where we can directly check for
emptiness of the relevant zone instead of making an additive copy of
the world and checking this property in the copy.
Additives are therefore not necessary for undecidability.
Yet, this conclusion is not entirely satisfactory.
If $\MSEL_\TWOSI$ can simulate Turing machines, then it can obviously
simulate a theorem prover that implements a complete search procedure
for \MAELL.
Thus, in an indirect fashion, this paper establishes that additive
behaviour can be encoded using subexponentials and multiplicatives
alone.
It would be interesting to build this encoding of additives more
directly as an embedding of \MAELL---or even just \MALL---in \MSEL.
This work leaves open the questions of decidability of an arbitrary
\MSEL with a two-element signature or a one-element signature; the
latter is equivalent to the decidability of \MELL itself.
We also conjecture that the decision problem for an arbitrary \MSEL
with no unbounded subexponentials is PSPACE-hard, because it is very
likely possible to polynomially and soundly encode a \MALL sequent in
such an \MSEL.
Finally, this undecidability result should be taken as a word of
caution for the increasingly popular uses of \SEL as a logical
framework for the encodings of other systems, such
as~\cite{nigam13concur,nigam11entcs}.
If one is to avoid encoding a decidable problem in terms of an
undecidable one, subexponentials must be used very carefully.
\noindent
\textbf{Historical note}: The undecidability result presented here is
from an unpublished paper from 2009, cited as the source of the result
in Nigam's Ph.D. thesis from the same year~\cite[p. 103]{nigam09phd}.
Nigam has also published an indirect proof in~\cite{nigam12lics},
using the same strategy and roughly the same encoding, but this
version also uses the additive unit $\TOP$ for halting states and is
therefore not strictly in \MSEL.
\end{document} |
\begin{equation}gin{document}
\title[Diffusive Hamilton-Jacobi equations]
{The profile of boundary gradient blow-up \\ for the diffusive Hamilton-Jacobi equation}
\subjclass[2010]{35K55, 35B40, 35B44, 82C24.}
\keywords{Diffusive Hamilton-Jacobi equations, KPZ model, gradient blow-up, isolated boundary singularities, blow-up profile, tangential profile, anisotropic singularity.}
\author[Porretta]{Alessio Porretta}
\address{Universit\`a di Roma Tor Vergata,
Dipartimento di Matematica,
Via della Ricerca Scientifica 1,
00133 Roma, Italia}
\email{[email protected]}
\author[Souplet]{Philippe Souplet}
\address{Universit\'e Paris 13, Sorbonne Paris Cit\'e,
CNRS UMR 7539, Laboratoire Analyse, G\'{e}om\'{e}trie et Applications,
93430 Villetaneuse, France}
\email{[email protected]}
\begin{equation}gin{abstract}
We consider the diffusive Hamilton-Jacobi equation
$$u_t-\Delta u=|\nabla u|^p,$$
with Dirichlet boundary conditions in two space dimensions,
which is a typical model-case in the theory of parabolic PDEs
and also arises in the KPZ model of growing interfaces.
For $p>2$, solutions may develop gradient singularities on the boundary in finite time,
and examples of single-point gradient blowup on the boundary are known,
but the space-profile in the tangential direction has remained a completely open problem.
In the parameter range $2<p\le 3$, for the case of a flat boundary and an isolated singularity at the origin,
we give an answer to this question, obtaining the precise final asymptotic profile, under the form
$$u_y(x,y,T) \sim d_p\Bigl[y+C|x|^{2(p-1)/(p-2)}\Bigr]^{-1/(p-1)},\quad\hbox{ as $(x,y)\to (0,0)$.}$$
Interestingly, this result displays a new phenomenon of strong anisotropy of the profile,
quite different from what is observed in other blowup problems for nonlinear parabolic equations,
with the exponents $1/(p-1)$ in the normal direction
$y$ and $2/(p-2)$ in the tangential direction $x$.
Furthermore, the tangential profile violates the (self-similar) scale invariance of the equation,
whereas the normal profile remains self-similar.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
\subsection{Background}
This article is devoted to the qualitative study of solutions of the diffusive Hamilton-Jacobi equation
\begin{equation}gin{equation}\label{VHJ}
u_t-\Delta u=|\nabla u|^p.
\end{equation}
Beside being the viscosity approximation of Hamilton-Jacobi type
equations from stochastic control theory \cite{L82}, equation (\ref{VHJ}) is involved
in certain physical models, for instance of ballistic deposition processes,
were it describes the evolution of the profile of a growing interface.
It is actually the deterministic version of the well-known
Kardar-Parisi-Zhang (KPZ) equation (see \cite{KPZ86} and cf. also Krug and Spohn \cite{KS88}).
In its stochastic version, it has undergone spectacular development recently with the work of M. Hairer~\cite{Hai13}.
Finally, equation (\ref{VHJ}) is a typical model-case in the theory of parabolic PDEs.
Indeed it is the simplest example of a parabolic equation with a nonlinearity depending on the first-order
spatial derivatives of $u$. As such, it is important to study its qualitative properties.
Equation (\ref{VHJ}) has been intensively studied in the past twenty years, and it is well known that two fundamentally different situations occur.
If the equation is considered in the whole space $\mathbb{R}^n$ (with, say, bounded $C^1$ initial conditions), then
all solutions exist globally in the classical sense and remain bounded in $C^1$; see e.g. \cite{AB98,BL99,BSW02,LS03,BKL04,GGK03,G05,S07,QS07},
where the large time behavior is investigated in detail).
At the opposite, if the equation is posed on a
domain with homogeneous Dirichlet boundary conditions, then for $p>2$ and suitably large initial data,
the local classical solution develops singularities in finite time.
These singularities are of gradient blowup type (GBU), the function~$u$~itself remaining bounded, and are located
on some part of the boundary; see~e.g.~\cite{ABG89,FL94,CG96,A96,S02,ARS04,HM04,BaD04,SV06,SZ06,QS07,GH08,LS}
and the references therein.
For the classical blowup problem associated with the nonlinear heat equation
\begin{equation}gin{equation}\label{NLH}
u_t-\Delta u=u^p,
\end{equation}
a considerably developed theory is available for the description of the asymptotic profile of the solution
near a finite time singularity (see \cite{QS07} and the references therein). In comparison, very little is known for equation (\ref{VHJ}).
In particular, in the case of an isolated boundary singularity, {\bf the final blowup profile of $\nabla u$
in the tangential direction is completely unknown.}\footnote{It is only known that
$|\nabla u(X,t)|\le C [{\rm dist}(X,\partial\Omega)]^{-1/(p-1)}$ (see \cite{ABG89, ARS04, SZ06, LS}),
which gives an upper estimate in the normal direction but provides no information on how the profile is damped away from the
point of singularity along the boundary.}
\subsection{Main result: final gradient blowup profile near an isolated boundary singularity}
The goal of this paper is to fill this gap by giving a substantial contribution to this question.
In the range of exponents $2<p\le 3$, we will give a sharp description of
the final blowup profile of $\nabla u$ near an isolated boundary singularity (in both normal and tangential directions).
Since the question is quite involved, we shall restrict ourselves to a rather simple setting,
but which captures the essence of the problem. Namely, we consider the two-dimensional case,
where the domain is assumed to coincide locally with a half-plane near the point of singularity.
To this end, for given $\rho>0$, we set
$$\omega_\rho=(-\rho,\rho)\times (0,\rho)\subset \mathbb{R}^2,\qquad \omega_\rho^+=\omega_\rho\cap \{x>0\}.$$
Next we fix some $L,T>0$ and put
\begin{equation}gin{equation}\label{DefOmega1}
\omega=\omega_L,\quad \omega'=\omega_{L/2},
\end{equation}
\begin{equation}gin{equation}\label{DefOmega2}
Q_T:=\omega\times(0,T),\quad\Gamma_T= (-L,L)\times\{0\}\times (0,T).
\end{equation}
\begin{equation}gin{defn}
Let $L,T>0$ and let $u\in C^{2,1}(\overline\omega\times(0,T))$ be a nonnegative classical solution of (\ref{VHJ}) in $Q_T$,
with $u=0$ on $\Gamma_T$. We say that $u$ has an {\bf isolated gradient blowup point at~$(0,0,T)$} if
\begin{equation}gin{equation}\label{GBU0}
\limsup_{(x,y,t)\to (0,0,T)} |\nabla u(x,y,t)|=\infty
\end{equation}
and
\begin{equation}gin{equation}\label{isolGBU}
\nabla u\ \hbox{ is bounded on $K\times (0,T)$ for any $K\subset\subset \overline\omega\setminus\{(0,0)\}$.}
\end{equation}
\end{defn}
If $u$ has an isolated gradient blowup point at $(0,0,T)$ then
we may define the {\bf final blowup profile} of $\nabla u$, given by
$$\nabla u(x,y,T):=\lim_{t\to T} \nabla u(x,y,t),
\quad\hbox{ for all $(x,y)\in \overline\omega'\setminus\{(0,0)\}.$}$$
Indeed the limit above exists and is finite due to (\ref{isolGBU}), as a consequence of standard parabolic estimates.
Our main result is the following.
\begin{equation}gin{thm}\label{thm:GBUprofile}
Assume
$$2<p\le 3.$$
Let $L,T>0$, let $u\in C^{1,2}(\overline\omega\times(0,T))$ be a nonnegative classical solution of (\ref{VHJ}) in $Q_T$,
with $u=0$ on $\Gamma_T$. Assume that $u$ has an isolated gradient blowup point at $(0,0,T)$ and that $u$ satisfies the monotonicity condition
\begin{equation}gin{equation}\label{MonotonicityHyp}
x{\hskip 0.4pt}u_x\le 0\quad\hbox{in $Q_T$.}
\end{equation}
Then there exist constants $C_1,C_2,C_3>0$, $\rho\in (0,L)$ (possibly depending on $u$) such that, for all $(x,y)\in ([-\rho,\rho]\times [0,\rho])\setminus\{(0,0)\}$,
the final blowup profile satisfies
\begin{equation}gin{equation}\label{mainEstimate}
d_p\Bigl[y+C_1 |x|^{2(p-1)/(p-2)}\Bigr]^{-\begin{equation}ta}-C_3 \le u_y(x,y,T)
\le d_p\Bigl[y+C_2 |x|^{2(p-1)/(p-2)}\Bigr]^{-\begin{equation}ta} +C_3
\end{equation}
where
$$ \begin{equation}ta=1/(p-1)\quad\hbox{and}\quad d_p=\begin{equation}ta^\begin{equation}ta.$$
In particular, the final profile of the normal derivative on the boundary satisfies
$$C_4|x|^{-2/(p-2)}\le u_y(x,0,T)
\le C_5 |x|^{-2/(p-2)},$$
for all $0<|x|\le \rho$ and some $C_4,C_5>0$.
Also, for some $C>0$, we have
$$u\le C,\quad |u_x|\le C,\quad\hbox{ for all $(x,y)\in \omega'$}.$$
\end{thm}
\hskip 1.5cm \includegraphics[width=13cm, height=7.5cm]{Shape}
\vskip -2mm
\centerline{\it Fig. 1: The shape of the final profile of $u$ near the origin.}
\subsection{Discussion and remarks}\hskip 3mm
(a) Interestingly, this result shows that the GBU profile is strongly {\bf anisotropic,}
i.e.~the exponents of the singularity profile in the normal and in the tangential directions are different,
respectively $1/(p-1)$ and $2/(p-2)$.
Moreover, whereas the exponent of the normal profile obeys the natural scaling of the equation,
the latter is violated by the tangential profile. Indeed, recall that equation (\ref{VHJ}) is invariant under the group of transformations
$$u \mapsto u_\lambda(x,y,t):=\lambda^m u(\lambda x,\lambda y, \lambda^2t)\quad\hbox{ with $m=(2-p)/(p-1)$, \quad for all $\lambda>0$},$$
whose gradient is given by $\nabla u_\lambda= \lambda^{1/(p-1)} \nabla u(\lambda x,\lambda y, \lambda^2t)$.
(b) As far as we know, no similar example of anisotropic, isolated blowup singularity is known in parabolic problems.
For the nonlinear heat equation~(\ref{NLH}), the stable blowup profile at an isolated blowup point is known to be
isotropic\footnote{this concerns blowup at an interior point -- actually only the whole space case is considered in these works; however no blowup can occur at a boundary point for equation (\ref{NLH}), at least in a convex domain}
(see \cite{Ve93H, MZ97, MZ98, FMZ} and the references therein),
with
$$u(X,T)\sim c(p)|X|^{-2/(p-1)}|\log |X||^{-1/(p-1)}\quad\hbox{as $X\to 0$}.$$
Here $X\in\mathbb{R}^n$ with $n\ge 2$ and $1<p<(n+2)/(n-2)$, and this profile occurs for instance for any symmetric, radially decreasing solution.
The case of the linear heat equation with nonlinear boundary conditions
\begin{equation}gin{equation}\label{NLBC}
\begin{equation}gin{cases}
u_t-\Delta u=0\quad\hbox{in $\Omega\times(0,T) $}, \\
\noalign{\vskip 1mm}
\displaystyle\frac{\partial u}{\partial\nu}=u^p\quad\hbox{on $\partial \Omega\times (0,T)$}\\
\end{cases}
\end{equation}
was studied in \cite{CF00, HY, Ha13, Ha15}.
Like for (\ref{VHJ}), this problem involves boundary singularities (however $u$ itself blows up).
It was recently found in \cite{Ha13, Ha15} that for $\Omega=\mathbb{R}^2_+=\{(x,y);\ y>0\}$ under assumption (\ref{MonotonicityHyp}),
the singularity profile satisfies
$$u(x,y,T)\sim
\begin{equation}gin{cases}
y^{-1/(p-1)} &\quad\hbox{for $y\to 0$ with $|x|=O(y)$}\\
\noalign{\vskip 1mm}
x^{-1/(p-1)}|\log x|^{-1/2(p-1)} &\quad\hbox{for $x\to 0$ and $y=0$.}
\end{cases}
$$
A similar result holds in dimension $n\ge 3$ if $1<p<n/(n-2)$. Note that this profile is only weakly anisotropic (by a logarithmic correction)
in comparison with (\ref{mainEstimate}).
On the other hand we also observe that, unlike in problems (\ref{NLH}) and (\ref{NLBC}),
the profile that we find for (\ref{VHJ}) is given by pure powers, without (e.g. logarithmic) corrections.
This situation seems typical of type II blow-up problems (see \cite{MM09} and cf. Remark~(c)).
(c) The exponent $2/(p-2)$ appears to be new in this problem. However, it is worth noting that, in some cases, the time rate of GBU involves
a related exponent.
Namely, for monotone in time solutions in 1 space dimension, we have \cite{GH08}:
\begin{equation}gin{equation}\label{TimeRate}
\|\nabla u(\cdot,t)\|_\infty\sim (T-t)^{-1/(p-2)}.
\end{equation}
However, the question of the time GBU rate is still open in 2 dimensions.
Note that the rate (\ref{TimeRate}) corresponds to a type II blow-up, in the sense that this rate is more singular than what the natural scaling of the equation would suggest
(see \cite{GH08,QS07} for details).
A possible heuristic explanation of the appearance of the number $2/(p-2)$ in this problem, based on ideas of quasi-stationary approximation, is
given in Section~6.
(d) It remains an open problem what is the actual tangential singularity exponent for $p>3$
-- see~Remark~\ref{remp3} for details.
Actually the lower estimate in (\ref{mainEstimate}) remains true for any $p>2$ (cf.~Theorem \ref{thm:GBUprofileLower}).
As for the upper estimate, for $p>3$, our method would allow to obtain
an estimate of the form in (\ref{mainEstimate}), with some power, greater than $2(p-1)/(p-2)$,
which could be explicitly computed in terms of $p$. However, due to the gap between the upper and
lower estimates in this case, we are unable to determine the exponent of the actual profile.
Therefore, and in order not to further increase the technicality of the article,
we have refrained from expanding on this.
On the other hand, it might be possible to extend our results to more general (nonflat) domains
and to higher dimensions, at the expense of further complication.
But since the main goal of this work is to present a new phenomenon, we have decided to leave this aside.
(e) Actually, the upper estimate in (\ref{mainEstimate}) is satisfied by $u_y(x,y,t)$ for all $t<T$
(this is a consequence of the proof, cf.~in particular formula (\ref{conclusionUpper})).
(f) By direct integration of (\ref{mainEstimate}) between $0$ and $y$, one easily obtains the corresponding estimate for the profile of the function $u$ itself
(whose shape is depicted in Fig.~1).
In the course of the proof of Theorem \ref{thm:GBUprofile}, we also establish additional estimates, of possible independent interest. In particular, we show that
for any $p>2$, there holds
$$|u_t|\le C,\qquad |u_x|\le C|x|,\qquad u_{xx}\ge -C$$
for $(x,y,t)$ close to $(0,0,T)$. Moreover, for $2<p\le 3$, we show that
$$-C\le u_{xx}(0,y,T) \le -c<0$$
for $y>0$ small (see Remark~\ref{rem51}). In particular, since $u_{xx}(x,0,T)=0$ for $x\ne 0$, we see that $u_{xx}(\cdot,T)$ is discontinuous near the origin.
\subsection{Existence of single-point gradient blow-up solutions}
In order to obtain solutions satisfying all the assumptions in Theorem \ref{thm:GBUprofile}
we now recall a result from \cite{LS} concerning the initial-boundary value problem
\begin{equation}gin{eqnarray}
&u_t-\Delta u=|\nabla u|^p,\ \ \ &x\in \Omega,\ t>0,\label{sys main 1}\\
&u(x,t)=0,\ &x\in \partial \Omega,\ t>0,\label{sys main 2}\\
&u(x,0)=u_0(x),\ &x\in \Omega.\label{sys main 3}
\end{eqnarray}
Here, it is assumed that
\begin{equation}gin{equation}\label{hypSPGBU}
\hbox{$\Omega\subset \mathbb{R}^2$ is a $C^{2+\alpha}$-smooth bounded domain,
$u_0\in C^1(\overline{\Omega})$ with $u_0\geq 0$ and ${u_0}_{|\partial \Omega}=0$.}
\end{equation}
It follows from \cite[Theorem 10, p.~206]{F64} that problem (\ref{sys main 1})-(\ref{sys main 3}) admits a unique maximal,
nonnegative classical solution $u\in C^{2,1}(\overline{\Omega}\times(0,T))\cap
C^{1,0}(\overline{\Omega}\times[0,T))$, where $T=T(u_0)$ is the
maximal existence time.
Also, by the maximum principle, we immediately have
$$
\|u(t)\|_\infty\leq \|u_0\|_\infty,\quad 0<t<T.
$$
On the other hand, by the Bernstein-type estimate in \cite{SZ06}, we know that
$$|\nabla u(t,X)|\le C[{\rm dist}(X,\partial\Omega)]^{-1/(p-1)},\quad \hbox{in $\Omega\times (0,T)$},$$
so that in particular GBU can take place only on $\partial\Omega$.
The following result was proved in \cite{LS}:
{\bf Theorem A.} {\it
Assume (\ref{hypSPGBU}) and
\begin{equation}gin{eqnarray}
&&\hskip -1cm\hbox{$\Omega$ and $u_0$ are symmetric with respect to the line $x=0$}, \label{def dom 1}\\
&&\hskip -1cm\hbox{$\Omega$ coincides locally near $0$ with the half-plane $\{y>0\}$ and is convex in the $x$-direction}, \label{def dom 2}\\
&&\hskip -1cm\hbox{$xu_{0,x}\leq 0$ in $\Omega$}. \label{hyp ID 2}
\end{eqnarray}
If $u_0$ is suitably concentrated near the origin (see Remark~\ref{concentrated} below), then the solution of (\ref{sys main 1})-(\ref{sys main 3}) satisfies
$$\hbox{$T=T(u_0)<\infty$ and $\nabla u$ blows up only at the origin}$$
(i.e. (\ref{GBU0}) is true and $\nabla u$ is bounded on $K\times (0,T)$ for any $K\subset\subset \overline\Omega\setminus\{(0,0)\}$).
}
Also we note that, as a consequence of the assumptions of Theorem~A,
we have $x{\hskip 0.4pt}u_x\leq 0$ in $\Omega\times (0,T)$.
\begin{equation}gin{rem}\label{concentrated}
As an example of data ``suitably concentrated near the origin'' for Theorem A, the following was given in \cite{LS}:
\begin{equation}gin{equation}\label{def ID}
u_0(x)=C\varepsilon^k\varphi\Bigr(\varepsilon^{-1}\sqrt{x^2+(y-\varepsilon)^2}\Bigl),
\end{equation}
where $k=(p-2)/(p-1)$, $C\geq C_0(p)>0$,
$\varepsilon>0$ is sufficiently small, and
$\varphi\in C^\infty([0,\infty))$ satisfies
\begin{equation}gin{equation}\label{def ID2}
\varphi'\leq 0,\qquad \varphi(s)=1,\ s\leq 1/4, \qquad\varphi(s)=0,\
s\geq 1/2.
\end{equation}
Note that these functions have support concentrated near the boundary point $(0,0)$
(and large derivatives for $\varepsilon$ small).\end{rem}
\subsection{Ideas of proof}
The proof of Theorem \ref{thm:GBUprofile} is long and technical, and it requires to combine many ingredients.
Let us briefly describe the main ideas.
To establish the lower estimate, we start with an estimation of the normal derivative on the boundary,
which is obtained in three steps (see Fig.~2 in Section~3): we start from the vertical line $\{x=0\}$, where the
precise final profile follows rather easily from ODE arguments. We then extend the lower estimate to the region
above the curve
$y=K{\hskip 0.1pt}x^{2/(1-\begin{equation}ta)}$, by using a lower bound of $u_{xx}$ along horizontal segments.
The extension to the region below the curve $y=K{\hskip 0.1pt}x^{2/(1-\begin{equation}ta)}$ is then achieved
by means of a boundary Harnack-type estimate in suitable boxes connecting this curve to the boundary $\{y=0\}$.
Once $u_y$ is estimated from below on the boundary, the full lower estimate of $u_y$ is
obtained by suitable integration along vertical lines, plus some horizontal averaging
made possible by an estimate of the mixed derivative $u_{xy}$.
As for the proof of the upper estimate, it combines two ingredients. The first one is an auxiliary function of the form
$$J(x,y,t)=u_x+kx\,(1+ y)\,y^{-(1-\begin{equation}ta)q}u^q,$$
with suitable parameters $k, q>0$,
which is shown to be nonpositive via the maximum principle.
The integration of the inequality $J\le 0$ along horizontal lines then yields a sharp upper estimate of H\"older type for $u$.
The second ingredient is a family of suitable regularizing barriers, which allow us to improve the H\"older estimate of $u$ to
a pointwise upper bound of $u_y$ on the boundary.
We note that rougher versions of both ingredients were used in \cite{LS},
in order to prove single-point GBU.{\footnote {\,The function $J$ in \cite{LS} was itself
motivated by a device from \cite{FML85}, where a function of the form $J(r,t)=r^{n-1}u_r+\varepsilon r^n u^q$
was introduced to study the blowup of radial solutions of equation (\ref{NLH}).}
However, these ideas need to be considerably refined in order to obtain the sharp tangential GBU profile.
In particular, the derivation of the parabolic inequality satisfied by $J$ requires a delicate analysis in terms of
the auxiliary quantities
$$\xi=y\frac{u_y}{u}\quad\hbox{ and }\quad\Theta=y(u_y)^{p-1}.$$
This latter step turns out to require the restriction $p\le 3$ and leaves open the question whether the upper estimate remains true for $p>3$ as well (see~Remark~\ref{remp3}).
The rest of the paper is devoted to the proof of Theorem \ref{thm:GBUprofile}.
Some preliminary properties, mostly based on the maximum principle, are given in Section~2.
The lower estimate is established in Section~3. In Section~4, we construct the regularizing barriers.
In Section~5, the analysis of the parabolic inequality for the function $J$ is carried out, and the proof of the upper estimate is then completed.
Finally, a possible heuristic explanation of the appearance of the number $2/(p-2)$ in this problem, based on ideas of quasi-stationary approximation, is
given in Section~6.
\section{Preliminary properties}
In the following propositions, we state a number of useful bounds and properties of the solution,
which will be used in the proof of the main result Theorem 1.1.
All the proofs will be given after the statements.
Here and in the rest of the paper, letters such as $C, C_1, c, \dots$ will denote possibly different positive constants,
whose dependence will be specified only when necessary.
We start with some simple bounds, which follow rather easily from the maximum principle.
Let us recall that $\omega,\omega', Q_T$ and $\Gamma_T$ are defined in \rife{DefOmega1}-\rife{DefOmega2}.
\begin{equation}gin{prop} \label{prop:maxprin}
Assume $p>2$, let $L,T>0$ and let $u\in C^{1,2}(\overline\omega\times(0,T))$ be a nonnegative classical solution of (\ref{VHJ}) in $Q_T$,
with $u=0$ on $\Gamma_T$. Assume that $u$ has an isolated gradient blowup point at~$(0,0,T)$.
(i) Then $u$ extends to a function
\begin{equation}gin{equation}\label{extension}
u\in C^{1,2}(\tilde Q), \qquad\hbox{with $\tilde Q:=\bigl(\overline\omega'\times[T/2,T]\bigr)\setminus\{(0,0,T)\}$}.
\end{equation}
(This extension will still be denoted by $u$ without risk of confusion.)
(ii) There exists a constant $C>0$
(possibly depending on the solution $u$), such that $u$ satisfies the following bounds in $\tilde Q$:
\begin{equation}gin{eqnarray}
&&|u_t|\le C \label{bound0}\\
&&u_y\ge -C \label{bound2}\\
&&u_{xx}\ge -C. \label{bound3}
\end{eqnarray}
If, moreover, $u$ satisfies (\ref{MonotonicityHyp}), then we have
\begin{equation}gin{equation}\label{bound1}
|u_x|\le C|x|
\end{equation}
in $\tilde Q$.
\end{prop}
We next show that the gradient blowup does occur in a pointwise sense: $u_y$ becomes uniformly large near the blow-up time and the origin.
\begin{equation}gin{prop}\label{prop:blowup}
Assume $p>2$, let $L,T>0$ and let $u\in C^{1,2}(\overline\omega\times(0,T))$ be a nonnegative classical solution of (\ref{VHJ}) in $Q_T$,
with $u=0$ on $\Gamma_T$. Assume that $u$ has an isolated gradient blowup point at~$(0,0,T)$
and that $u$ satisfies the monotonicity condition (\ref{MonotonicityHyp}).
Then we have
\begin{equation}gin{equation}\label{limuy}
\lim_{{t\to T \atop (x,y)\to (0,0)}}u_y(x,y,t)=+\infty.
\end{equation}
\end{prop}
As a consequence of Proposition~\ref{prop:blowup}, there exists $0<\rho_0<\min(L/2,T/2)$ such that
\begin{equation}gin{equation}\label{uylarge}
u_y>1\quad\hbox{ in $\bigl(\overline\omega_{\rho_0}\times [T-{\rho_0},T]\bigl)\setminus\{(0,0,T)\}$.}
\end{equation}
We now give upper bounds which essentially follow by integrating in the vertical direction.
\begin{equation}gin{prop} \label{prop:upper1d}
Under the assumptions of Proposition \ref{prop:blowup},
there exist a constant $C>0$ and $\rho\in(0,\rho_0)$
(possibly depending on $u$), such that the solution $u$ satisfies
\begin{equation}gin{equation}\label{bound5a}
u_y(x,y,t)\le\bigl[(u_y)^{1-p}(x,0,t)+(p-1)y\bigr]^{-\begin{equation}ta}+2C y \quad\hbox{ in $\omega_\rho\times [T-\rho,T)$.}
\end{equation}
In particular, we have
\begin{equation}gin{eqnarray}
&&u_y(x,y,t)\le d_p y^{-\begin{equation}ta}+2Cy \quad\hbox{ in $\omega_\rho\times [T-\rho,T)$} \label{bound5}\\
&&u(x,y,t)\le c_p y^{1-\begin{equation}ta}+Cy^2 \quad\hbox{ in $\omega_\rho\times [T-\rho,T)$,}\label{bound6}
\end{eqnarray}
where
\begin{equation}gin{equation}\label{defcpdp}
\begin{equation}ta=\frac{1}{p-1}, \qquad d_p=\begin{equation}ta^\begin{equation}ta,\qquad c_p= (1-\begin{equation}ta)^{-1}d_p.
\end{equation}
\end{prop}
Our next result shows that similar lower bounds are true at $ x=0$
(of course they cannot be true for $x\neq 0$ in view of the profile eventually found in (\ref{mainEstimate})).
\begin{equation}gin{prop} \label{prop:lower1d}
Under the assumptions of Proposition \ref{prop:blowup},
there exist a constant $C_1>0$ and $\rho\in(0,\rho_0)$
(possibly depending on $u$) such that
\begin{equation}gin{equation}\label{bound7}
u_y(0,y,t)\ge \bigl[(u_y)^{1-p}(0,0,t)+(p-1)y\bigr]^{-\begin{equation}ta}-C_1.
\quad 0<y<\rho,\ \ T-\rho\le t<T.
\end{equation}
Moreover, we have
\begin{equation}gin{equation}\label{bound8}
u_y(0,y,T)\ge d_p y^{-\begin{equation}ta}-C_1, \quad 0<y<\rho
\end{equation}
and
\begin{equation}gin{equation}\label{bound9}
u(0,y,T)\ge c_p y^{1-\begin{equation}ta}-C_1y, \quad 0<y<\rho.
\end{equation}
\end{prop}
The following relationship between second order derivates, whose proof is rather delicate,
will play an important role to establish the lower pointwise estimates in (\ref{mainEstimate}).
\begin{equation}gin{prop} \label{prop:uxy-uxx}
Under the assumptions of Proposition \ref{prop:blowup}, for any $\eta>0$, there exists a constant $C_\eta>0$
(possibly depending on $u$), such that the solution $u$ satisfies
\begin{equation}\label{bound10}
u_{xy}\le \eta\, u_{xx}+C_\eta \qquad\hbox{in $\overline\omega_L^+\times [T/2,T]$.}
\end{equation}
\end{prop}
We now turn to the proofs of the results that we have just stated.
\vskip2em
\noindent \textit{Proof of Proposition \ref{prop:maxprin}}.
$\bullet$ Property (\ref{extension}) is a consequence of standard parabolic estimates.
$\bullet$ Proof of (\ref{bound0})
and (\ref{bound2}). Set $z=u_t$ or $z=u_y$. Then $z\in C^{1,2}(\tilde Q)$ by parabolic regularity and it satisfies
\begin{equation}\label{eqdrift}
z_t-\Delta z=A\cdot\nabla z,
\end{equation}
where $A=p|\nabla u|^{p-2}\nabla u$.
Since $u_t=0$ on $\Gamma_T$, using (\ref{extension}), we see that the supremum of $|u_t|$ on the parabolic boundary of $\omega'\times[T/2,T)$ is finite.
Denoting this supremum by $C$, the maximum principle then guarantees
$$|u_t|\le C\quad\hbox{ in $\omega'\times[T/2,T)$,}$$
which implies (\ref{bound0}) in view of (\ref{extension}).
We can apply a similar reasoning to $u_y$. Since $u\ge 0$ and $u=0$ on $\Gamma_T$, we have $u_y\ge 0$ on $\Gamma_T$.
By (\ref{extension}), we see that the infimum of $u_y$ on the parabolic boundary of $\omega'\times[T/2,T)$ is finite.
Denoting this infimum by $-C$, the maximum principle then guarantees
$$u_y\ge -C\quad\hbox{ in $\omega'\times[T/2,T)$,}$$
which implies (\ref{bound2}) in view of (\ref{extension}).
$\bullet$ Proof of (\ref{bound3}). The function $Z:=u_{xx}\in C^{1,2}(\tilde Q)$ by parabolic regularity and it satisfies
\begin{equation}\label{equxx}
\begin{equation}gin{split}
Z_t-\Delta Z
&= p\bigl[|\nabla u|^{p-2}\nabla u\cdot\nabla u_{x}\bigr]_{x} \\
&= A\cdot\nabla Z+p|\nabla u|^{p-2}|\nabla u_x|^2+p(p-2)|\nabla u|^{p-4}(\nabla u\cdot\nabla u_x)^2 \\
&\ge A\cdot\nabla Z.
\end{split}
\end{equation}
Since $Z=0$ on $\Gamma_T$, using (\ref{extension}) we see
that the infimum of $Z$ on the parabolic boundary of $\omega'\times[T/2,T)$ is finite.
It then follows from the maximum principle that $u_{xx}\ge -C$ in $\omega'\times[T/2,T)$,
which implies (\ref{bound3}) in view of (\ref{extension}).
$\bullet$ Proof of (\ref{bound1}).
As a consequence of~(\ref{MonotonicityHyp}), we have
\begin{equation}gin{equation}\label{uxzero}
u_x(0,y,t)=0\qquad\hbox{for all $(y,t)\in \bigl([0,L/2]\times[T/2,T]\bigr)\setminus\{(0,T)\}$}.
\end{equation}Consequently, we get
$$u_x(x,y,t)=\int_0^x u_{xx}(t,s,y)\, ds\ge -Cx\qquad\hbox{in $\bigl(\omega'\times[T/2,T)\bigr)\cap\{x>0\}$,}$$
and a similar estimate for $x<0$.
This implies \rife{bound1}.
$\Box$
\vskip1em
In view of the proofs of Propositions \ref{prop:blowup}--\ref{prop:lower1d}, we prepare the following Lemma.
\begin{equation}gin{lem}\label{LemLower1}
Under the assumptions of Proposition \ref{prop:blowup}, we have
\begin{equation}gin{equation}\label{uyinfty}
\limsup_{t\to T}u_{y}(0,0,t)=+\infty
\end{equation}
and
\begin{equation}gin{equation}\label{bound8a}
u_y(0,y,T)\ge d_p y^{-\begin{equation}ta}-C_1, \quad 0<y<L/2.
\end{equation}
\end{lem}
\noindent \textit{Proof.}
As a consequence of (\ref{MonotonicityHyp}) and $u=0$ on $\Gamma_T$, we have
\begin{equation}gin{equation}\label{uyxneg}
xu_{yx}(x,0,t)\le 0\quad\hbox{ for $0<|x|\le L/2$ and $t\in [T/2,T]$,}
\end{equation}
hence
$$u_{y}(0,0,t)=\sup_{|x|\le L/2}u_{y}(x,0,t)\quad\hbox{ for $t\in [T/2,T)$.}$$
On the other hand, by (\ref{eqdrift}) and the maximum principle, for each $\tau\in (T/2,T)$, the maximum of $u_y$
on $Q'_\tau:=\omega'\times (T/2,\tau)$ is attained on the parabolic boundary $\partial_PQ'_\tau$ of $Q'_\tau$.
Moreover, by (\ref{isolGBU}), we have
$$M_0=\sup_{(x,y,t)\in \Gamma'} u_y<\infty,
\quad\hbox{ where $\Gamma':= \partial_PQ'_T\setminus\bigl([-L/2,L/2]\times\{0\}\times [T/2,T)\bigr)$.}$$
Therefore,
$$\sup_{Q'_\tau} u_y\le \max\Bigl(M_0,\sup_{t\in [T/2,\tau]} u_y(0,0,t)\Bigr).$$
By our assumption (\ref{GBU0}), the LHS goes to $\infty$ as $\tau\to T$
and (\ref{uyinfty}) follows.
Let us now prove (\ref{bound8a}).
By~(\ref{MonotonicityHyp}), we have
\begin{equation}gin{equation}\label{uxxzero}
u_{xx}(0,y,t)\le 0\qquad\hbox{for all $(y,t)\in (0,L/2]\times(T/2,T]$}.
\end{equation}
Also, we know from Proposition \ref{prop:maxprin}
that $u_y\ge -C$ hence $|u_y|\le u_y+2C$ and that $|u_t|\le C$ in $\omega'\times (T/2,T)$.
Set $C_1:=2C+C^{1/p}$.
For $(y,t)\in (0,L/2)\times (T/2,T)$, using (\ref{uxzero}) and (\ref{uxxzero}) it follows that
$$ -u_{yy}(0,y,t)=[u_{xx}+|\nabla u|^p-u_t](0,y,t)\le |u_y(0,y,t)|^p+C\le (u_y(0,y,t)+C_1)^p.$$
Observing that $u_y(0,y,t)+C_1>0$ and integrating in $y$, we obtain
\begin{equation}gin{equation}\label{inequyyp}
(u_y(0,y,t)+C_1)^{1-p}\le (u_y(0,0,t)+C_1)^{1-p}+(p-1)y.
\end{equation}
By (\ref{extension}) and (\ref{uyinfty}), we deduce that
$$(u_y(0,y,T)+C_1)^{1-p}\le \liminf_{t\to T}(u_y(0,0,t)+C_1)^{1-p}+(p-1)y=(p-1)y\quad\hbox{ for $y\in(0,L/2)$},$$
which yields (\ref{bound8a}).
$\Box$
\vskip1em
\noindent \textit{Proof of Proposition \ref{prop:blowup}.}
Assume by contradiction that there exist a constant $K>0$ and a sequence $(t_n, x_n, y_n)$ such that
$$
(x_n, y_n, t_n) \to (0,0, T)\quad\hbox{ and }\quad u_y(x_n,y_n, t_n) \leq K\,.
$$
By (\ref{bound0}) and (\ref{bound3}), we have
\begin{equation}gin{equation}\label{bounduyyp}
u_{yy}+|u_y|^p\le u_{yy}+|\nabla u|^p=u_t-u_{xx}\le 2C,
\end{equation}
hence in particular $u_{yy}\le 2C$.
Fix any $y\in (0,L/2)$. For $n$ large enough, we have $0<y_n<y$, hence
$$
u_y(x_n,y,t_n)\le u_y(x_n,y_n,t_n)+2C(y-y_n)\leq K + CL.
$$
Letting $n\to \infty$ and using (\ref{extension}), we get
$$
u_y(0,y,T) \leq K + CL.
$$
This is in contradiction with \rife{bound8a}.
$\Box$
\vskip1em
\noindent \textit{Proof of Proposition \ref{prop:upper1d}.}
Fix any $x\in (-L/2,L/2)$ and $t\in (T/2,T)$. By (\ref{uylarge}), there exists $\rho\in(0,\rho_0)$ such that
$$h(y) := u_y(x,y,t)-2Cy>0,\quad\hbox{ for $y\in (0,\rho)$}.$$
By (\ref{bounduyyp}), the function $h$ satisfies
$$h'+h^p =u_{yy}-2C+(u_y -2Cy)^p
\le u_{yy}-2C+(u_y)^p \le 0.$$
By integration, we obtain
$$h(y)\le\bigl[h^{1-p}(0)+(p-1)y\bigr]^{-\begin{equation}ta},\quad\hbox{ for $y\in (0,\rho)$},$$
hence (\ref{bound5a}) and in particular (\ref{bound5}). Property (\ref{bound6}) follows by further integration.
$\Box$
\vskip1em
\noindent \textit{Proof of Proposition \ref{prop:lower1d}}.
Estimate (\ref{bound7}) is an immediate consequence of (\ref{inequyyp}).
As for (\ref{bound8}), it was already proved in Lemma~\ref{LemLower1}. Finally, (\ref{bound9}) follows from (\ref{bound8}) by integration.
$\Box$
\vskip1em
We finally prove Proposition \ref{prop:uxy-uxx}.
\vskip1em
\noindent \textit{Proof of Proposition \ref{prop:uxy-uxx}}. In view of estimate \rife{bound3}, there is no loss of generality if we only consider $\eta \leq 1$.
First we recall from \rife{equxx} that $u_{xx}$ satisfies
$$
(u_{xx})_t-\Delta u_{xx}
- A\cdot\nabla u_{xx}\geq 0\,,\qquad A=p |\nabla u|^{p-2}\nabla u \,.
$$
We compute the same equation for $u_{xy}$, and we get
\begin{equation}\label{uxy1}
\begin{equation}gin{split}
(u_{xy})_t-\Delta u_{xy}
- A\cdot\nabla u_{xy} & = p|\nabla u|^{p-2}\Bigl[ \nabla u_y\cdot\nabla u_x+(p-2)\frac{(\nabla u\cdot\nabla u_x)(\nabla u\cdot\nabla u_y)}{|\nabla u|^2}\Bigr]
\\
& = p|\nabla u|^{p-4}\Bigl[ |\nabla u|^2\nabla u_y\cdot\nabla u_x+(p-2) (\nabla u\cdot\nabla u_x)(\nabla u\cdot\nabla u_y)\Bigr].
\end{split}
\end{equation}
Notice that this is justified close enough to the singularity, due to $u_y>0$ (cf.~(\ref{uyposQr}) below) and parabolic regularity.
Now, given $\eta \leq 1$, we consider the function
$$
z:= u_{xy}-\eta\, u_{xx}
$$
and the operator ${\mathcal L}(z)= z_t-\Delta z-A\cdot\nabla z$. On account of \rife{uxy1} we have
$$
{\mathcal L}(z)\leq F\qquad \hbox{in $ Q_T$}
$$
and
$$
F:= p|\nabla u|^{p-4}\Bigl[ |\nabla u|^2\nabla u_y\cdot\nabla u_x+(p-2) (\nabla u\cdot\nabla u_x)(\nabla u\cdot\nabla u_y)\Bigr]\,.
$$
We analyze now the sign of $F$ at large values of $z$. First of all, we develop each component of the scalar products and
we find
\begin{equation}\label{uxy2}
\begin{equation}gin{split}
\frac F{p|\nabla u|^{p-4}} & = u_{xy} \left\{ u_{xx}( (p-1)(u_x)^2 + (u_y)^2) + u_{yy} ( (u_x)^2 + (p-1)(u_y)^2)\right\}
\\
& \quad + (p-2) u_yu_xu_{yy}u_{xx} + (p-2) u_xu_yu_{xy}^2.
\end{split}
\end{equation}
Due to Proposition \ref{prop:blowup}, we may choose $r>0$ small so that $u_y$ is sufficiently large in $Q'_r:=(0,r)^2\times (T-r,T)$. In particular, we may assume that
\begin{equation}\label{uyposQr}
u_y >0\,,\qquad u_{yy}<0 \,,\qquad u_{xx}< - u_{yy}\qquad \hbox{in $Q'_r$,}
\end{equation}
the last two inequalities coming from the equation $u_{xx}+ u_{yy} = - |\nabla u|^p+ u_t $ together with the bounds \rife{bound0} and \rife{bound3}.
In addition, since $Q'_r\subset \omega_L^+\times (0,T)$, we have $u_x\leq 0$ and therefore the last term in \rife{uxy2} is nonpositive.
Dropping this term we get
\begin{equation}gin{align*}
\frac F{p|\nabla u|^{p-4}} & \leq u_{xy} \left\{ u_{xx}( (p-1)(u_x)^2 + (u_y)^2) + u_{yy} ( (u_x)^2 + (p-1)(u_y)^2)\right\}
\\
& \quad + (p-2) u_yu_xu_{yy}u_{xx} \,.
\end{align*}
Now, since $u_{xx}\geq -C$ (cf.~\rife{bound3}), at any point where $z\ge M_\eta:=\eta C$
we have $u_{xy} \geq M_\eta + \eta u_{xx}\geq 0$,
and since $u_{xx}\leq -u_{yy}$ we estimate
\begin{equation}\label{uxy3}\begin{equation}gin{split}
\frac F{p|\nabla u|^{p-4}} & \leq (p-2) \Bigl[ u_{xy} u_{yy} ( (u_y)^2 - (u_x)^2) + u_yu_xu_{yy}u_{xx} \Bigr]
\\
& = (p-2) u_{yy} \Bigl[ u_{xy} ( (u_y)^2 - (u_x)^2) + u_yu_x u_{xx} \Bigr]\,.
\end{split}
\end{equation}
We conclude by noticing that the right hand side of \rife{uxy3} is negative if $u_y$ is large enough. Indeed,
by Proposition \ref{prop:blowup} and (\ref{bound1}), we may chose $r$ so that
$$
u_y > \frac2\eta \|u_x\|_\infty \qquad \hbox{ in $Q'_r$.}
$$
Then, at any point of $Q'_r$ where $z\ge M_\eta$, we have
$$
u_{xy} \geq M_\eta + \eta u_{xx}
= \eta (u_{xx}+ C) > \frac{2\|u_x\|_\infty}{u_y} (u_{xx}+ C)> -\frac{2u_x}{u_y} u_{xx}\,,
$$
hence
$$
u_{xy} ( (u_y)^2 - (u_x)^2) + u_yu_x u_{xx} \geq \frac12 u_{xy} (u_y)^2 + u_yu_x u_{xx}
= u_y \Bigl( \frac12 u_{xy} u_y + u_x u_{xx}\Bigr) >0.
$$
So from \rife{uxy3} we get $F<0$ at any point of $Q'_r$ such that $z\ge \eta C$.
On the other hand, considering the parabolic boundary of $Q'_r$, we have $ z\leq \eta C$ at $\{x=0\}$ due to \rife{uxzero} and \rife{bound3}.
At $\{ y=0\}$, we have $u_{xy}-\eta u_{xx}= u_{xy}\leq 0$ by (\ref{uyxneg}). On the rest of the lateral boundary,
as well as at $t=T-r$, the function is bounded by some constant $C_\eta$.
By the maximum principle applied to ${\mathcal L}$, we deduce that $ z\leq \max(\eta C,C_\eta)$ in $Q'_r$.
Therefore, the bound \rife{bound10} is proved in $Q'_r$. In view of the regularity of $u$ outside of $(0,0,T)$, the bound can of course be extended to
$\omega_L^+ \times [T/2,T]$ up to an extra uniform constant.
$\Box$
\section{Proof of main result: the lower estimate}
In this section, we shall prove the following Theorem, which is valid for any $p>2$, and in particular implies the lower estimate in Theorem \ref{thm:GBUprofile}.
\begin{equation}gin{thm}\label{thm:GBUprofileLower}
Assume $p>2$, let $L,T>0$ and recall the notation in (\ref{DefOmega1})-(\ref{DefOmega2}).
Let $u\in C^{1,2}(\overline\omega\times(0,T))$ be a nonnegative classical solution of (\ref{VHJ}) in $Q_T$,
with $u=0$ on $\Gamma_T$. Assume that $u$ has an isolated gradient blowup point at $(0,0,T)$ and that $u$ satisfies the monotonicity condition
\begin{equation}gin{equation}\label{MonotonicityHyp2}
x{\hskip 0.4pt}u_x\le 0\quad\hbox{in $Q_T$.}
\end{equation}
Then there exist constants $C_1,C_2>0$, $\rho\in (0,L)$ (possibly depending on $u$) such that, for all $(x,y)\in \overline\omega_\rho\setminus\{(0,0)\}$,
the final blowup profile satisfies
\begin{equation}gin{equation}\label{mainEstimateLower}
u_y(x,y,T)\ge d_p\Bigl[y+C_1 |x|^{2(p-1)/(p-2)}\Bigr]^{-\begin{equation}ta}-C_2,
\end{equation}
where
$$ \begin{equation}ta=1/(p-1)\quad\hbox{and}\quad d_p=\begin{equation}ta^\begin{equation}ta.$$
In particular, the final profile of the normal derivative on the boundary satisfies
\begin{equation}gin{equation}\label{mainEstimateLowerBoundary}
u_y(x,0,T) \ge C_3 |x|^{-2/(p-2)},
\end{equation}
for all $0<|x|\le \rho$ and some $C_3>0$.
\end{thm}
In the rest of this section we denote the final profile at the blow-up time by
$$v:= u(\cdot,T)\in C^2(\overline\omega'\setminus\{(0,0)\}).$$
Theorem \ref{thm:GBUprofileLower} is proved in two steps. First, in Lemma~\ref{LemLower2}, we establish
the estimate of the normal derivative on the boundary
(i.e. (\ref{mainEstimateLowerBoundary})).
To do so, the idea is as follows (see fig.~2 below): we start from the vertical line $\{x=0\}$, where the
precise lower bound of the final profile $v$ is already known thanks to (\ref{bound9}).
We then extend the lower estimate of $v$ to the region $\Sigma_+$ above the curve
\begin{equation}gin{equation}\label{defParabolaSigma}
\Sigma_0=\bigl\{(x,y)\,: y=K{\hskip 0.1pt}x^{2/(1-\begin{equation}ta)}\bigr\},
\end{equation}
which plays an important role in our arguments.
This relies on the lower bound of $u_{xx}$ in Proposition~\ref{prop:maxprin}, used along horizontal segments.
This is not sufficient since the region $\Sigma_+$ does not touch the boundary $\{y=0\}$.
However, the extension to the region $\Sigma_-$ below the curve (\ref{defParabolaSigma}) can then be achieved
by using a Harnack-type estimate in suitable boxes connecting the curve $\Sigma_0$ to the boundary $\{y=0\}$,
in terms of the distance to the boundary.
Finally, once the normal derivative is estimated on the boundary, the full lower estimate of $u_y$ is
obtained (cf.~Lemma~\ref{LemLower3}) by suitable integration along vertical lines, plus some horizontal averaging
made possible by the estimate of the mixed derivative $u_{xy}$ given in Proposition \ref{prop:uxy-uxx}.
\begin{equation}gin{lem}\label{LemLower2}
Under the assumptions of Theorem \ref{thm:GBUprofileLower},
there exist constants $c_0>0$ and $\rho\in (0,L)$ such that we have
\begin{equation}\label{vy0}
v_y (x,0) \geq c_0{\hskip 1pt} x^{- 2/(p-2)},\quad \hbox{ for $0<x<\rho$.}
\end{equation}
\end{lem}
For the proof of Lemma \ref{LemLower2}, we shall use a well-known quantitative version of the Hopf Lemma (or
boundary Harnack inequality) \cite{BC},
which we state in a suitably scale invariant form.
\begin{equation}gin{lem} \label{LemLowerBC}
Let $D_1$ be a $C^2$ domain such that
$$ (-1, 1)\times (0,2) \subseteq D_1 \subseteq (-2, 2)\times (0,2).$$
For any $(x_0,y_0)\in \mathbb{R}^2$ and $\lambda>0$, we set
$$D_\lambda:= (x_0,y_0) + \lambda D_1$$
and
$$\delta_{D_\lambda}(x,y)={\rm dist}\bigl((x,y),\partial D_\lambda\bigr).$$
There exists $c_1>0$ depending only on $D_1$ such that for any $(x_0,y_0)\in \mathbb{R}^2$,
any $\lambda>0$ and all $f\in L^\infty(D_\lambda)$, $f\ge 0$,
the solution $z$ of
\begin{equation}\label{laplace}
\begin{equation}gin{cases}
-\Delta z=f & \hbox{in $D_\lambda$,}\\
z=0 & \hbox{on $\partial D_\lambda$}
\end{cases}
\end{equation}
satisfies
$$
\frac{z(x,y)}{\delta_{D_\lambda}(x,y)}\geq c_1\lambda^{-2} \int\!\!\!\!\int_{D_\lambda} f(x',y')\delta_{D_\lambda} (x',y') \,dx'dy',
\quad \hbox{ for all $(x,y)\in D_\lambda.$}
$$
\end{lem}
\vskip1em
\noindent \textit{Proof of Lemma \ref{LemLowerBC}.}
By translation invariance, we may assume $x_0=y_0=0$.
If $z$ solves (\ref{laplace}) in $D_\lambda$, then $Z(X,Y):=z(\lambda X,\lambda Y)$ solves
$$
\begin{equation}gin{cases}
-\Delta Z=f_\lambda(X,Y):=\lambda^2f(\lambda X,\lambda Y) & \hbox{in $D_1$,}\\
Z=0 & \hbox{on $\partial D_1$}.
\end{cases}
$$
The inequality for $\lambda =1$ is well known; see \cite{BC}.
Using the fact that
\begin{equation}\label{scalingLambda}
\delta_{D_1} (\lambda^{-1}x,\lambda^{-1}y)=\lambda^{-1}\delta_{D_\lambda} (x,y)\quad \hbox{ for all $(x,y)\in D_\lambda,$}
\end{equation}
and changing variables, it follows that
\begin{equation}gin{align*}
z(x,y)=Z(\lambda^{-1}x,\lambda^{-1}y)
& \geq c_1 \delta_{D_1} (\lambda^{-1}x,\lambda^{-1}y)\int\!\!\!\!\int_{D_1} f_\lambda(X',Y')\delta_{D_1} (X',Y') \,dX'dY'\\
& = c_1 \delta_{D_\lambda} (x,y)\int\!\!\!\!\int_{D_1} f(\lambda X',\lambda Y')\delta_{D_\lambda} (\lambda X',\lambda Y') \,dX'dY'\\
& = c_1 \lambda ^{-2}\delta_{D_\lambda} (x,y)\int\!\!\!\!\int_{D_\lambda} f(x',y')\delta_{D_\lambda} (x',y') \,dx'dy',
\end{align*}
which proves the lemma.
$\Box$
\vskip1em
\noindent \textit{Proof of Lemma \ref{LemLower2}.}
Starting from the lower estimate (\ref{bound9}) on $\{ x=0\}$, i.e.
\begin{equation}\label{estiminfv00}
v(0,y) \geq c_p\, y^{1-\begin{equation}ta }- Cy\quad \hbox{ for $0<y<\rho$,}
\end{equation}
the proof is done in three steps (cf. Fig.2 below).
{\bf Step 1.} {\it Lower estimate of $v$ in the region $\omega_\rho\cap\{y\ge K x^{2/(1-\begin{equation}ta)}\}$.} We claim that
there exist constants $K>0$ and $\rho\in (0,L)$ (depending on $v$) such that
\begin{equation}\label{estiminfv0}
v(x,y) \geq \frac{c_p}{2}\,y^{1-\begin{equation}ta } \quad \hbox{ for $(x,y)\in \omega_\rho\cap\{y\ge K x^{2/(1-\begin{equation}ta)}\}$.}
\end{equation}
Let $\rho$ be given by Proposition \ref{prop:upper1d}.
Using the lower estimate (\ref{estiminfv00}) on $\{ x=0\}$, the fact that $v_x(0,y)=0$ and $v_{xx}\geq -C$
(cf. (\ref{uxzero}) and (\ref{bound3})) and Taylor's expansion, we obtain
$$
v(x,y) \geq c_p\, y^{1-\begin{equation}ta }- Cy- C \,x^2 \quad \hbox{ for $(x,y)\in \omega_\rho$.}
$$
hence
$$
v(x,y) \geq \bigl(c_p-CK^{\begin{equation}ta-1}-Cy^\begin{equation}ta\bigr)\, y^{1-\begin{equation}ta } \quad \hbox{ for $(x,y)\in \omega_\rho\cap\{y> K x^{2/(1-\begin{equation}ta)}\}$.}
$$
The claim (\ref{estiminfv0}) follows by taking $\rho\le (c_p/4C)^{1/\begin{equation}ta}$ and $K\ge (c_p/4C)^{-1/(1-\begin{equation}ta)}$.
{\bf Step 2.} {\it Harnack-type estimate in suitable boxes near the boundary.}
We claim that there exist constants $c, \tilde c>0$ and $\rho\in (0,L)$ such that, for all $x\in (0,\rho/2)$ and all $\lambda\in (0,x/4)$,
\begin{equation}\label{estiminfv1}
\frac{v(x,y)}{y} \geq c\, \lambda^{1-2p}\, \left(\int_{x-\lambda}^x\int_0^{2\lambda} |v_y| \,dx'dy'\right)^{p} - \tilde c \, \lambda
\quad \hbox{ for $0<y<\lambda$.}
\end{equation}
By (\ref{bound3}) and (\ref{limuy}), reducing $\rho$ if necessary, we may assume that
$$-\Delta v \ge |\nabla v|^p-C\ge 0\quad \hbox{ in $\omega_\rho$.}$$
Let $D_1$ be a $C^2$ domain such that
$$ (-1, 1)\times (0,2) \subseteq D_1 \subseteq (-2, 2)\times (0,2).$$
For given $x\in (0,\rho/2)$ and $\lambda\in (0,x/4)$, we set
$$
D=D_{x,\lambda}:= (x,0) + \lambda D_1\subset (x/2,3x/2)\times (0,\rho)\subset \omega_\rho\,.
$$
Observe that $-\Delta v \geq f_{x,\lambda}:=|\nabla v|^p- C$ in $D$ with $f_{x,\lambda}\in L^\infty(D)$ and $f_{x,\lambda}\ge 0$.
Since $v\ge 0$, it follows from Lemma \ref{LemLowerBC}
and the maximum principle that, for some constant $c_1>0$,
\begin{equation}\label{estiminfv2}
\frac{v(x,y)}{\delta_{D}(x,y)} \geq c_1 \lambda^{-2} \int\!\!\!\!\int_{D} \left \{ |\nabla v|^p(x',y')- c \right\}\delta_{D} (x',y')\,dx'dy'.
\end{equation}
By H\"older's inequality, we have
$$
\int\!\!\!\!\int_{D} |v_y| \,dx'dy' \leq \left(\int\!\!\!\!\int_{D} |v_y|^p\delta_{D}(x',y')\,dx'dy'\right)^{\frac1p}
\left(\int\!\!\!\!\int_{D} \delta_{D}^{-\frac1{p-1}}(x',y') \,dx'dy'\right)^{\frac{p-1}{p}}.
$$
Using (\ref{scalingLambda}), we see that
$$\int\!\!\!\!\int_{D} \delta_{D}^{-\frac1{p-1}}(x',y') \,dx'dy'
=\lambda^2\int\!\!\!\!\int_{D_1} \delta_{D}^{-\frac1{p-1}}(\lambda X',\lambda Y') \,dX'dY',
=\lambda^{2-\frac1{p-1}}\int\!\!\!\!\int_{D_1} \delta_{D_1}^{-\frac1{p-1}}(X',Y') \,dX'dY',$$
where the integral on the RHS is finite due to $1/(p-1)<1$ (see e.g. \cite{S02}).
Therefore,
$$
\left(\int\!\!\!\!\int_{D} |v_y| \,dx'dy'\right)^p \leq C \, \lambda^{2p-3} \int\!\!\!\!\int_{D} |\nabla v|^p\delta_{D}(x',y')\, dx'dy'.
$$
Using also $\int\!\!\!\!\int_{D} \delta_{D}(x',y') \,dx'dy'=C\lambda^3$, we deduce
from (\ref{estiminfv2}) that
$$
\frac{v(x,y)}{\delta_{D}(x,y)} \geq c\, \lambda^{1-2p}\, \left(\int\!\!\!\!\int_{D} |v_y| \,dx'dy'\right)^{p} - \tilde c \, \lambda \,.
$$
Since $\delta_{D}(x,y)= y$ for $0<y<\lambda$, the claim (\ref{estiminfv1}) follows.
{\bf Step 3.} {\it Conclusion.}
Fix $x\in (0,\rho/2)$ (the case $x\in (-\rho/2,0)$ can be treated similarly).
We proceed to estimate from below the integral in (\ref{estiminfv1}).
To this end, we choose
$$\lambda = K{\hskip 0.1pt}x^{2/(1-\begin{equation}ta)},$$
where $K$ is from (\ref{estiminfv0}).
Note that this implies $\lambda\in (0,x/4)$, taking a smaller $\rho$ if necessary.
By (\ref{estiminfv0}), we have
$$
\int_{x-\lambda}^x\int_0^{2\lambda} |v_y|\, dx'dy'\ge \int_{x-\lambda}^x\int_0^{2\lambda} v_y\, dx'dy'= \int_{x-\lambda}^x v(x', 2\lambda) dx'
\geq \frac{c_p}{2}\lambda (2\lambda)^{1-\begin{equation}ta}=c \lambda^{2-\begin{equation}ta}.
$$
Combining this with (\ref{estiminfv1}) and using $p\begin{equation}ta=\begin{equation}ta+1$ we obtain
$$
\frac{v(x,y)}{y} \geq c\, \lambda^{-\begin{equation}ta} -\tilde c\,\lambda
\quad \hbox{ for $0<y<\lambda$,}
$$
which implies, reducing $\rho>0$ again if necessary,
\begin{equation}\label{estiminfv3}
\frac{v(x,y)}{y} \geq \frac{c}{2}\, x^{-2\begin{equation}ta/(1-\begin{equation}ta)}, \quad \hbox{ for $0<y<\lambda$.}
\end{equation}
Since $2\begin{equation}ta/(1-\begin{equation}ta)= 2/(p-2)$, letting $y\to 0$ we get \rife{vy0}.
$\Box$
\hskip 1.5cm \includegraphics[width=13cm, height=7.5cm]{PicturePS00}
\centerline{\it Fig. 2: The scheme of the proof of Lemma 3.2.}
\centerline{\it The marks A, B, C correspond to estimates
(\ref{estiminfv00}), (\ref{estiminfv0}) and (\ref{estiminfv3}), respectively}
The proof of Theorem \ref{thm:GBUprofileLower} is then completed by the following lemma.
\begin{equation}gin{lem}\label{LemLower3}
Under the assumptions of Theorem \ref{thm:GBUprofileLower},
for each $\varepsilon\in(0,1)$, there exists constants $C>0$ and $\rho\in (0,L/2)$ such that
\begin{equation}\label{pre-media1}
v_y(x,y) \geq \bigl[(v_y)^{1-p}((1+\varepsilon)x, 0) +(p-1)y\big]^{-\begin{equation}ta} - C\quad\hbox{ in $\overline\omega_\rho\setminus\{(0,0)\}$.}
\end{equation}
\end{lem}
\proof
{\bf Step 1.} Let $\rho_0$ be given by (\ref{uylarge}) and take $\rho=\min(\rho_0,1)\le L/2$. We first claim that there exists a constant $A>0$ such that
\begin{equation}\label{pre-media}
v_y(x,y) \ge \bigl[(v_y)^{1-p}(x,0)+(p-1)y\big]^{-\begin{equation}ta} - \int_0^y v_{xx}\,dy-A \quad\hbox{ in $\overline\omega_\rho\setminus\{(0,0)\}$.}
\end{equation}
From \rife{bound0}, \rife{bound1}, \rife{bound3} and (\ref{uylarge}), we know that
\begin{equation}\label{lower1}
|u_t(\cdot,T)|\le C,\quad |v_x|\le C,\quad v_y>0, \quad v_{xx}\ge -C \quad\hbox{ in $\overline\omega_\rho\setminus\{(0,0)\}$}
\end{equation}
for some constant $C\ge 1$.
Let $A=3C$ and set
$$
z(x,y):=v_y+\int_0^y v_{xx}\,dy+A.
$$
Observe that, for all $(x,y)\in \overline\omega_\rho\setminus\{(0,0)\}$
\begin{equation}\label{lower2}
-z_y=-\Delta v \leq |\nabla v|^p + C.
\end{equation}
Since $y\leq \rho\le 1$, using \rife{lower1}, we see that
\begin{equation}\label{lower3}
z = v_y+\Bigl(C+\int_0^y v_{xx}\,dy\Bigr)+2C\ge v_y+2C.
\end{equation}
Therefore, by \rife{lower2}--\rife{lower3}, we obtain
$$
-z_y
\le [(v_y)^2+C^2]^{p/2}+C
\le [v_y+C]^p+C^p
\le [v_y+2C]^p
\le z^p,
$$
hence
$$(z^{1-p})_y=-(p-1)z_yz^{-p}\le p-1\quad\hbox{ in $\overline\omega_\rho\setminus\{(0,0)\}$}.$$
After integration, it follows that
$$z^{1-p}(x,y)\le z^{1-p}(x,0)+(p-1)y
\le (v_y)^{1-p}(x,0)+(p-1)y \quad\hbox{ in $\overline\omega_\rho\setminus\{(0,0)\}$,}
$$
hence the claim.
{\bf Step 2.} We may assume $0<x<\rho/2$ without loss of generality.
To prove \rife{pre-media1}, we now take the average of inequality \rife{pre-media} and we get
$$
\frac1{\varepsilon x} \int_x^{(1+\varepsilon)x}v_y(s,y)ds \ge \frac1{\varepsilon x} \int_x^{(1+\varepsilon)x} \bigl[(v_y)^{1-p}(s,0)+(p-1)y\big]^{-\begin{equation}ta} ds- \frac1{\varepsilon x} \int_x^{(1+\varepsilon)x} \int_0^y v_{xx}\,dsdy-A\,.
$$
By \rife{uyxneg}, the function $x\mapsto v_y(x,0)$ is nonincreasing for $0<x<L$, so that we obtain
\begin{equation}\label{lower4}
\begin{equation}gin{split}
\frac1{\varepsilon x} \int_x^{(1+\varepsilon)x}v_y(s,y)ds &
\geq \bigl[(v_y)^{1-p}((1+\varepsilon)x, 0) +(p-1)y\big]^{-\begin{equation}ta} - \int_0^y \frac1{\varepsilon x} \int_x^{(1+\varepsilon)x} v_{xx}\,dsdy- A
\\ &
\geq \bigl[(v_y)^{1-p}((1+\varepsilon)x, 0) +(p-1)y\big]^{-\begin{equation}ta} - C \frac y \varepsilon- A,
\end{split}
\end{equation}
where we used \rife{bound1} in the last inequality.
Now we recall from Proposition \ref{prop:uxy-uxx} that, for some constant $C_1>0$ we have
$$
(v_y- v_x- C_1 x)_x \leq 0\qquad \hbox{in $\omega_L^+$.}
$$
Therefore, for some constant $C_2>0$, we get
\begin{equation}gin{align*}
v_y(x,y) & \geq \frac1{\varepsilon x} \int_x^{(1+\varepsilon)x}v_y(s,y)ds - \frac1{\varepsilon x} \int_x^{(1+\varepsilon)x}( v_x(s,y)+ C_1 s)ds
+ v_x(x,y) + C_1 x
\\
& \geq \frac1{\varepsilon x} \int_x^{(1+\varepsilon)x}v_y(s,y)ds - C_2 x\,,
\end{align*}
where again we also used \rife{bound1}. Jointly with inequality \rife{lower4}, we conclude that
$$
v_y(x,y) \geq \bigl[(v_y)^{1-p}((1+\varepsilon)x, 0) +(p-1)y\big]^{-\begin{equation}ta} - C_\varepsilon
$$
i.e., \rife{pre-media1}.
$\Box$
Finally, using Lemma \ref{LemLower3} with $\varepsilon=1$, combined with Lemma \ref{LemLower2}, immediately yields \rife{mainEstimateLower}.
The proof of Theorem \ref{thm:GBUprofileLower} is concluded.
\section{Regularizing barriers}
The following lemma shows that a suitable local H\"older bound of exponent $1-\begin{equation}ta$,
near a boundary point, actually guarantees a bound for the normal derivative at this point.
\begin{equation}gin{lem}\label{barrier}
Let $p>2$, $r,d\in (0,1)$, $d<L$, $t_0\in [0,T)$ and $x_0$ be such that $[x_0-r,x_0+r]\subset [-L/2,L/2]$.
Let
$$D=(x_0-r,x_0+r)\times(0,d).
$$
There exist constants $C_0=C_0(p)>0$ and $\eta_0=\eta_0(p,T)\in (0,1)$
with the following property. Let $\eta\in(0,\eta_0)$ and $u\in C^{2,1}(\overline\omega\times(0,T))$
be a nonnegative classical solution of (\ref{VHJ}) in $Q_T$, with $u=0$ on $\Gamma_T$. If
\begin{equation}gin{equation}\label{hypcontroluc0}
u(x,y,t)\leq c_py^{1-\begin{equation}ta}- \kappa \frac{ y^2}2\ \ \ {\rm in}\ \
\overline D \times[t_0,T),
\end{equation}
with $c_p=(1-\begin{equation}ta)^{-1}d_p$,
$\kappa=C_0 \eta^{1-\begin{equation}ta}(r^2+ T-t_0)$ and
\begin{equation}\label{toit}
u(x,d,t) \leq c_p \left[ (d+ \eta \, (t-t_0)^{\frac1{1-\begin{equation}ta}}\,r^{\frac2{1-\begin{equation}ta}})^{1-\begin{equation}ta}- \eta^{1-\begin{equation}ta}\, (t-t_0)\,r^2 \right] - \kappa \frac{d^2}2 \ \ {\rm in}\ \
[x_0-r,x_0+r]\times[t_0,T),
\,
\end{equation}
then
\begin{equation}gin{equation}\label{conclcontroluc0}
u_y(x_0,0,t)\le d_p \eta^{-\begin{equation}ta} \bigl( (t-t_0)\,r^2 \bigr)^{-\frac\begin{equation}ta{1-\begin{equation}ta}}\ \ \ {\rm for }\ t\in ( t_0,T).
\end{equation}
\end{lem}
\begin{equation}gin{rem}
Lemma \ref{barrier} is an improvement with respect to \cite[Lemma 2.2]{LS}, where a similar result was proved,
but required a small constant instead of $c_p$ in assumption \rife{hypcontroluc0}.
This improvement is crucial in order to obtain the exact power in the upper estimate
of the GBU profile in the next section.
We note that assumption \rife{hypcontroluc0} is essentially sharp, since $u=c_py^{1-p}$ is an exact solution of (\ref{VHJ})
with $u=0$ and $u_y=\infty$ at $y=0$.
\end{rem}
\begin{equation}gin{proof}
Let us define the comparison function
\begin{equation}gin{eqnarray}\label{defzphi}
z=z(x,y,t)=c_p\bigl[(y+\varphi(x,t))^{1-\begin{equation}ta}-\varphi^{1-\begin{equation}ta}(x,t)\bigr]-\kappa\frac{y^2}2\ \ \ {\rm in}\
\overline D\times[t_0,T),
\end{eqnarray}
with
$$\varphi(x,t)=\eta\, (t-t_0)^{\frac1{1-\begin{equation}ta}}\left(\frac{r^2-(x-x_0)^2}r\right)^{\frac2{1-\begin{equation}ta}},
$$
where $\eta>0$ and $\kappa\in (0,1)$.
Let us denote by $C$ possibly different constants only depending on $p$ (often through the value of $\begin{equation}ta$). We first notice that there exists $C>0$ such that
\begin{equation}\label{dervfi}
0\leq \varphi_t \leq C \eta^{1-\begin{equation}ta} r^2 \varphi^{\begin{equation}ta}\,,\qquad |\varphi_x|^2\leq C \eta^{ 1-\begin{equation}ta } (t-t_0) \varphi^{1+\begin{equation}ta}\,,\qquad |\varphi_{xx}|\leq C\eta^{1-\begin{equation}ta} (t-t_0)\, \varphi^{\begin{equation}ta}\,.
\end{equation}
Moreover, if $\kappa$ is sufficiently small (depending only on $p,T$), we have
\begin{equation}gin{align}\label{kappasmall}
y(y+\varphi)^{\begin{equation}ta } \kappa \, \le \kappa(1+T^{\frac{\begin{equation}ta}{1-\begin{equation}ta}}) < c_p(1-\begin{equation}ta) =d_p \,.
\end{align}
This implies
$$
|z_y|^2 = d_p^2 (y+\varphi)^{-2\begin{equation}ta} \left( 1-\frac{\kappa y(y+\varphi)^{\begin{equation}ta}} d_p \right)^2
\leq
d_p^2 (y+\varphi)^{-2\begin{equation}ta}
$$
and since
$$
|z_x|^2 = d_p^2 \left( (y+\varphi)^{-\begin{equation}ta}-\varphi^{-\begin{equation}ta}\right)^2 \varphi_x^2
\leq d_p^2 \varphi^{-2\begin{equation}ta} \varphi_x^2,
$$
we deduce
$$
|\nabla z|^p = (z_x^2+ z_y^2)^{\frac p2}\leq d_p^p\, (y+\varphi)^{-p\begin{equation}ta} \left[ 1+ (y+\varphi)^{2\begin{equation}ta}\varphi_x^2 \varphi^{-2\begin{equation}ta}\right]^{\frac p2}\,.
$$
Since, from \rife{dervfi}, we have $\varphi_x^2 \varphi^{-2\begin{equation}ta}\leq C\eta^{1-\begin{equation}ta} (t-t_0)\varphi^{1-\begin{equation}ta}$, by taking $\eta_0=\eta_0(p,T)$ sufficiently small, it follows that $(y+\varphi)^{2\begin{equation}ta}\varphi_x^2 \varphi^{-2\begin{equation}ta} \le C.$ Therefore we have
$$
|\nabla z|^p\leq d_p^p\, (y+\varphi)^{-p\begin{equation}ta} \left[ 1+ C (y+\varphi)^{2\begin{equation}ta}\varphi_x^2 \varphi^{-2\begin{equation}ta}\right],
$$
which implies
\begin{equation}gin{align*}
|\nabla z|^p & \leq d_p^p\, (y+\varphi)^{-p\begin{equation}ta} + C (y+\varphi)^{\begin{equation}ta-1}\varphi_x^2 \varphi^{-2\begin{equation}ta}
\\ &
\leq d_p^p\, (y+\varphi)^{-p\begin{equation}ta} + C\varphi_x^2 \varphi^{-\begin{equation}ta-1} \,.
\end{align*}
Thus, for $(x,t)\in D\times(t_0,T)$, we estimate
\begin{equation}gin{eqnarray*}
z_t-\Delta z- |\nabla z|^p & \geq & d_p\, [(y+\varphi)^{-\begin{equation}ta}-\varphi^{-\begin{equation}ta} ] (\varphi_t -\varphi_{xx}
)\\
\noalign{\vskip 1mm} &&
+ d_p\, \begin{equation}ta[(y+\varphi)^{-\begin{equation}ta-1}-\varphi^{-\begin{equation}ta-1} ] \varphi_x^2
\\
\noalign{\vskip 1mm} &&
+ d_p\,\begin{equation}ta(y+\varphi)^{-\begin{equation}ta-1} + \kappa
\\ \noalign{\vskip 1mm} &&
- d_p\, (y+\varphi)^{-p\begin{equation}ta} - C\varphi_x^2 \varphi^{-\begin{equation}ta-1}.
\end{eqnarray*}
Using
$$
d_p \begin{equation}ta(y+\varphi)^{-\begin{equation}ta-1} = d_p^p\, (y+\varphi)^{-p\begin{equation}ta}
$$
we deduce
$$
z_t-\Delta z- |\nabla z|^p \geq - C \varphi^{-\begin{equation}ta} (|\varphi_t| + |\varphi_{xx}|)
)
-C \varphi^{-\begin{equation}ta-1} \varphi_x^2 +\kappa
$$
and thanks to \rife{dervfi} we conclude
$$
z_t-\Delta z- |\nabla z|^p \geq \kappa- C (r^2+ T-t_0) \eta^{1-\begin{equation}ta}\,.
$$
In particular, we have
\begin{equation}gin{eqnarray}\label{ineq:Pv negative}
z_t-\Delta z\geq |\nabla z|^p\ \ \ {\rm in}\ D\times (t_0,T)
\end{eqnarray}
with the choice $\kappa= C (r^2+ T-t_0) \eta^{1-\begin{equation}ta}$
(which in turn guarantees (\ref{kappasmall}) for $\eta_0=\eta_0(p,T)$ small).
On $D\times \{ t_0\}$, as well as on
the lateral boundary part $\{x_0-r,x_0+r\}\times [0,d]\times (t_0,T)$, we have $\varphi=0$ and so
$$
z= c_p y^{1-\begin{equation}ta} - \kappa \frac{y^2}2 \geq u
$$
by \rife{hypcontroluc0}. Next, on the part $[x_0-r,x_0+r]\times\{0\}\times (t_0,T)$, we
have, for $t_0<t<T$,
$$
u(\cdot,\cdot,t)=z(\cdot,\cdot,t)=0.
$$
On the remaining part $[x_0-r,x_0+r]\times\{d\}\times (t_0,T)$, thanks to \rife{toit}
and to the fact that the expression in \rife{defzphi}
is a decreasing function of $\varphi$, we have
$$
u(x,d,t)\leq c_p \left[ (d+ \eta \, (t-t_0)^{\frac1{1-\begin{equation}ta}}\,r^{\frac2{1-\begin{equation}ta}})^{1-\begin{equation}ta}- \eta^{1-\begin{equation}ta}\, (t-t_0)\,r^2 \right] - \kappa \frac{d^2}2 \leq z(x,d,t)\,.
$$
By the comparison principle, we deduce
\begin{equation}gin{equation}\label{res:u leq v}
u\leq z \ \ \ {\rm in}\
\overline D\times[t_0,T).
\end{equation}
In particular, we have
$$u_y(x_0,0,t)\le z_y(x_0,0,t)$$
hence (\ref{conclcontroluc0}).
\end{proof}
\section{Proof of main result: the upper estimate}
We first establish the following
\begin{equation}gin{prop}\label{propJ} Let $p\in (2,3]$, $q>p-1$
and let $u$ be as in Theorem 1.1. There exist $k\in (0,1)$ and $x_1, y_1, \sigma>0$ such that we have
$$
u_x+kxy^{-q(1-\begin{equation}ta)}(1+ y)u^q\leq 0 \ \ \ {\rm in}\ (0,x_1)\times(0,y_1)\times ( T-\sigma,T).
$$
\end{prop}
\begin{equation}gin{rem}\label{rem51}
It follows from Proposition~\ref{propJ} and estimate (\ref{bound9}) that there exists a constant $c>0$ such that, for $y>0$ small,
$$u_{xx}(0,y,T)=\lim_{x\to 0}\frac{u_x(x,y,T)}{x}\le -k(1+ y)\Bigl(\frac{u(0,y,T)}{y^{1-\begin{equation}ta}}\Bigr)^q \le -c<0.$$
\end{rem}
\noindent \textit{Proof of Proposition \ref{propJ}.}
It is divided into several steps.
\textbf{Step 1. Preparations.}
We consider the auxiliary function
\begin{equation}gin{equation*}
J(x,y,t)=u_x+c(x)d(y)F(u)\ \ \ {\rm in}\ D\times(0,T),
\end{equation*}
where $D=(0,x_1)\times(0,y_1)$ and the smooth positive functions $c,d,F$ will be chosen below.
Our aim is to use the maximum principle to prove that,
for sufficiently small $x_1, y_1, \sigma>0$, there holds
\begin{equation}gin{equation}\label{res:J negative}
J\leq 0\ \ \ {\rm in}\ D\times[T -\sigma,T).
\end{equation}
This will be done in the subsequent steps.
\textbf{Step 2. Derivation of a parabolic inequality for $J$.}
The following basic computation was made in \cite{LS}.
For completeness and for the convenience of readers, we repeat it here.
We have
\begin{equation}gin{eqnarray*}
J_t&=&u_{xt}+cdF'u_t,\\
\Delta J&=&\Delta u_x+cdF'\Delta u+cdF''|\nabla u|^2
+2c'dF'u_x+2cd'F'u_y+(cd''+ c''d)F.
\end{eqnarray*}
Then we obtain
$$ J_t-\Delta J =(|\nabla u|^p)_x+cdF'|\nabla u|^p
-2c'dF'u_x-2cd'F'u_y-cdF''|\nabla u|^2-(cd''+ c''d) F.$$
Using $u_x=J-cdF$, we write
\begin{equation}gin{eqnarray*}
(|\nabla u|^p)_x&=&p|\nabla u|^{p-2}\nabla u\cdot\nabla u_x\\
&=&p|\nabla u|^{p-2}\nabla u\cdot\nabla J
-p|\nabla u|^{p-2}\nabla u\cdot\nabla (cdF)\\
&=&p|\nabla u|^{p-2}\nabla u\cdot\nabla J
-p|\nabla u|^{p-2}
(cdF'|\nabla u|^2+u_xc'dF+u_ycd'F),
\end{eqnarray*}
hence
\begin{equation}gin{eqnarray*}
(|\nabla u|^p)_x&=&p|\nabla u|^{p-2}\nabla u\cdot\nabla J
-pcdF'|\nabla u|^p \nonumber\\
&&-pc'dF|\nabla u|^{p-2}J+pcc'd^2F^2|\nabla u|^{p-2}
-pcd'F|\nabla u|^{p-2}u_y.
\end{eqnarray*}
We also have
\begin{equation}gin{equation*}
-2c'dF'u_x=-2c'dF'J+2cc'd^2FF'.
\end{equation*}
So we get
\begin{equation}gin{eqnarray}\label{eq:J}
J_t-\Delta J&=&aJ+b\cdot\nabla J\nonumber\\
&&-(p-1)cdF'|\nabla u|^p+pcc'd^2F^2|\nabla u|^{p-2}
-pcd'F|\nabla u|^{p-2}u_y
\nonumber\\
&&+2cc'd^2FF'-2cd'F'u_y-cdF''|\nabla u|^2-(cd''+ c''d) F,
\end{eqnarray}
where
\begin{equation}gin{equation}\label{def a b}
a=-pc'dF|\nabla u|^{p-2}-2c'dF' \ \ \ {\rm and}\ \ \
b=p|\nabla u|^{p-2}\nabla u.
\end{equation}
Let
\begin{equation}gin{equation*}
\mathcal{P}J=J_t-\Delta J-aJ-b\cdot\nabla J.
\end{equation*}
We can rewrite the above equality as follows
\begin{equation}gin{eqnarray}\label{ineq:J first estimate0}
\frac{\mathcal{P}J}{cdF}&
&=-(p-1){F'\over F}|\nabla u|^p-{F''\over F}|\nabla u|^2 -{c''\over c}-{d''\over d} \\
&&\quad -2{d'\over d}{F'\over F}u_y-p{d'\over d}|\nabla u|^{p-2}u_y
+pc'dF|\nabla u|^{p-2}+2c'dF'.\nonumber
\end{eqnarray}
\textbf{Step 3. Estimation of the RHS of (\ref{ineq:J first estimate0}).}
We now specialize the previous computation to the following choice:
\begin{equation}gin{eqnarray}
&&F(u)=u^q,\ \ \ \label{def F}\\
&&d(y)=y^{-\gamma}\varphi(y),\label{def d}\\
&&c(x)=kx,\label{def c}
\end{eqnarray}
where $k\in (0,1)$, $q>1$, $\gamma>0$ and $\varphi>0$ is a smooth function with $\varphi'\ge 0$.
Using
$$
d'(y)=-\gamma y^{-\gamma-1}\varphi(y)+y^{-\gamma}\varphi'(y),
$$
$$
d''(y)=\gamma(\gamma+1) y^{-\gamma-2}\varphi(y)-2\gamma y^{-\gamma-1}\varphi'(y)+y^{-\gamma}\varphi''(y),
$$
the equality \rife{ineq:J first estimate0} implies
\begin{equation}gin{eqnarray}\label{ineq:J first estimate}
\frac{y^2\mathcal{P}J}{cdF}&=
&-(p-1)q\frac{y^2|\nabla u|^p}u-q(q-1)\frac{y^2|\nabla u|^2}{u^2}-\gamma(\gamma+1) +2\gamma \frac{y\varphi'}{\varphi}-\frac{y^2\varphi''}{\varphi}\\
&&+2q\Bigl(\gamma-\frac{y\varphi'}{\varphi}\Bigr)\frac{yu_y}{u}+p\Bigl(\gamma-\frac{y\varphi'}{\varphi}\Bigr)y|\nabla u|^{p-2}u_y
+pky^{2-\gamma}\varphi u^q|\nabla u|^{p-2}+2kqu^{q-1}y^{2-\gamma}\varphi.\nonumber
\end{eqnarray}
Also, taking $\sigma,x_1,y_1\in(0,1)$ sufficiently small and setting
$Q=(0,x_1)\times (0,y_1)\times (T-\sigma,T)$, we have, by \rife{bound1}, \rife{uylarge}, \rife{bound5}, \rife {bound6},
$$
|u_x|\le Cx \qquad\hbox{in $Q$},
$$
as well as
\begin{equation}\label{u1d}
y\le u\le c_py^{1-\begin{equation}ta}+Cy^2,\quad 1\le u_y\le d_py^{-\begin{equation}ta}+Cy\qquad\hbox{in $Q$}.
\end{equation}
In particular we have, close enough to the singularity,
\begin{equation}gin{align*}
y|\nabla u|^{p-2}u_y
&=y(u_y)^{p-1}\Bigl[1+\Bigl(\frac{u_x}{u_y}\Bigr)^2\Bigr]^{(p-2)/2}\le y(u_y)^{p-1}
\Bigl[1+\frac{p-2}2\Bigl(\frac{u_x}{u_y}\Bigr)^2\Bigr]\\
&=y(u_y)^{p-1}+ \frac{p-2}2 y(u_y)^{p-3}(u_x)^2
\\
& \le y(u_y)^{p-1}+Cy^m\, x^2
\end{align*}
with $m=\min(1,2\begin{equation}ta)$. In particular, for $p\leq 3$ (i.e. $\begin{equation}ta\geq \frac12$), we have
$$
y|\nabla u|^{p-2}u_y\leq y(u_y)^{p-1}+ Cy\, x^2.
$$
Similarly, using \rife{u1d} we estimate
$$
pky^{2-\gamma}\varphi u^q|\nabla u|^{p-2}+2kqu^{q-1}y^{2-\gamma}\varphi\le kqC\varphi\,y^{(q-1)(1-\begin{equation}ta)+2-\gamma}
\leq q C\varphi\, y^{1+\begin{equation}ta +q(1-\begin{equation}ta)-\gamma}
$$
for any $k\in (0,1)$.
Consequently, we get from \rife{ineq:J first estimate}
\begin{equation}\label{pregam}
\begin{equation}gin{split}
{y^2{\mathcal P}J\over cdF}
&\le-(p-1)q\frac{y^2(u_y)^p}{u} -q(q-1)\frac{y^2(u_y)^2}{u^2}-\gamma(\gamma+1)\\
&\quad +2q\Bigl(\gamma-\frac{y\varphi'}{\varphi}\Bigr)\frac{yu_y}{u}+p\Bigl(\gamma-\frac{y\varphi'}{\varphi}\Bigr)y(u_y)^{p-1}\\
&\quad +2\gamma \frac{y\varphi'}{\varphi}-\frac{y^2\varphi''}{\varphi}
+ Cy(x^2+ q\varphi y^{\begin{equation}ta +q(1-\begin{equation}ta)-\gamma})
\qquad\hbox{ in $Q$}.
\end{split}
\end{equation}
We will conclude Step 3 through the following lemma.
\begin{equation}gin{lem}\label{pjneg}
Let $p\in (2,3]$, $q>p-1$, and take $\gamma=q (1-\begin{equation}ta)$ and $\varphi(y)=1+ y$ in \rife{def d}.
There exist $x_1, y_1, \sigma>0$ sufficiently small such that,
for any $k\in (0,1)$, we have
$${\mathcal P}J\le 0 \quad\hbox{in $Q=(0,x_1)\times(0,y_1)\times (T-\sigma,T)$.}
$$
\end{lem}
\noindent \textit{Proof of Lemma \ref{pjneg}.} To shorten notations, we set
$$
\xi=\xi(x,y,t)=\frac{yu_y}{u}\, \ge 0,\qquad \Theta=\Theta(x,y,t)= y(u_y)^{p-1}\, \ge 0, \qquad \psi=\psi(y)=\frac{y\varphi'}{\varphi} .
$$
Notice that $\psi(y)= \frac{ y}{1+ y}$ is small provided $y_1$ is sufficiently small.
We wish to show that the right-hand side of \rife{pregam} is nonpositive in $Q$. To this purpose we distinguish two cases according to whether $\xi\leq 1-\begin{equation}ta$ or not.
\noindent{\hskip 4mm}{\bf Case 1: $\xi\leq 1-\begin{equation}ta$.}
By Young's inequality, we have
\begin{equation}\label{add-rev1}
p\Bigl(\gamma-\psi(y)\Bigr)y(u_y)^{p-1} \leq (p-1) q \frac{y^2(u_y)^p}{u}+
\Bigl(\gamma-\psi(y)\Bigr)^{p} \frac{u^{p-1}}{q^{p-1}y^{p-2}}\,.
\end{equation}
Using \rife{u1d}, and $1-\begin{equation}ta= (p-2)/(p-1)$, we have
\begin{equation}gin{align*}
\Bigl(\gamma-\psi(y)\Bigr)^{p} \frac{u^{p-1}}{q^{p-1}y^{p-2}} & \leq \gamma^p\Bigl(1-\frac{\psi(y)}\gamma\Bigr)^{p} \frac{(c_p y^{1-\begin{equation}ta}+ C y^2)^{p-1}}{q^{p-1}y^{p-2}}
\\
& \quad \leq \gamma \Bigl(1-\frac{\psi(y)}\gamma\Bigr)^{p} \left( \frac{c_p\gamma}q\right)^{p-1} \left(1+ C y^{1+\begin{equation}ta}\right)^{p-1}.
\end{align*}
The precise value of $c_p$ in \rife{defcpdp} and the choice $\gamma=q (1-\begin{equation}ta)$ then imply
$$
\Bigl(\gamma-\psi(y)\Bigr)^{p} \frac{u^{p-1}}{q^{p-1}y^{p-2}} \leq
\gamma\begin{equation}ta \Bigl(1-\frac{\psi(y)}\gamma\Bigr)^{p} \left(1+ C y^{1+\begin{equation}ta}\right)^{p-1} \leq q(1-\begin{equation}ta)\begin{equation}ta- p \begin{equation}ta \psi(y) + C y^{1+\begin{equation}ta}
$$
for some $C$ (possibly depending on $q$).
Hence from \rife{add-rev1} we obtain
$$
p\Bigl(\gamma-\psi(y)\Bigr)y(u_y)^{p-1} \leq (p-1) q \frac{y^2(u_y)^p}{u}+
q(1-\begin{equation}ta)\begin{equation}ta- p \begin{equation}ta \psi(y) + C y^{1+\begin{equation}ta}\,.
$$
Therefore, using $\varphi''=0$, \rife{pregam} implies
\begin{equation}gin{align*}
{y^2{\mathcal P}J\over cdF}
&\le -q(q-1)\xi^2 +2q\Bigl(\gamma-\psi(y)\Bigr)\xi \\
&\quad - \gamma(\gamma+1)+ q(1-\begin{equation}ta)\begin{equation}ta- p \begin{equation}ta \psi(y) + 2\gamma \psi(y)+ Cy(x^2+ y^\begin{equation}ta).
\end{align*}
Now we remark that the function
$$
\xi\mapsto -q(q-1)\xi^2 +2q\Bigl(\gamma-\psi(y)\Bigr)\xi
$$
is increasing for $\xi \leq 1-\begin{equation}ta$ and $y$ sufficiently small, so we get
\begin{equation}gin{align*}
{y^2{\mathcal P}J\over cdF}
&\le -q(q-1)(1-\begin{equation}ta)^2+2q\Bigl(\gamma-\psi(y)\Bigr)(1-\begin{equation}ta) \\
&\quad - \gamma(\gamma+1)+ q(1-\begin{equation}ta)\begin{equation}ta- p \begin{equation}ta \psi(y) + 2\gamma \psi(y)+ Cy(x^2+ y^\begin{equation}ta)
\end{align*}
which implies, using $\gamma=(1-\begin{equation}ta)q$,
$$
{y^2{\mathcal P}J\over cdF} \leq - p \begin{equation}ta \psi(y) + Cy(x^2+ y^\begin{equation}ta)=- p \begin{equation}ta \frac{ y}{1+ y} + Cy(x^2+ y^\begin{equation}ta)\,.
$$
Therefore, we have ${\mathcal P}J\leq 0$ provided $x_1$, $y_1$ are taken sufficiently small.
\noindent{\hskip 4mm}{\bf Case 2: $\xi > 1-\begin{equation}ta$.}
With the above notations, and using $\varphi''=0$, \rife{pregam} can be written as
\begin{equation}\label{pregam2}
\begin{equation}gin{split}
{y^2{\mathcal P}J\over cdF}
&\le-(p-1)q\xi\,\Theta -q(q-1)\xi^2-\gamma(\gamma+1) \\
&\quad +2q\Bigl(\gamma-\psi(y)\Bigr)\xi+ p\bigl(\gamma-\psi(y)\bigr) \Theta \\
&\quad +2\gamma \psi(y)+C y(x^2+ y^\begin{equation}ta)
\end{split}
\end{equation}
hence
$$
{y^2{\mathcal P}J\over cdF} \leq H( y, \xi,\Theta) + Cy(x^2+ y^\begin{equation}ta),
$$
where
\begin{equation}\label{defH}
\begin{equation}gin{split}
H(y, \xi, \Theta) & =\Bigl\{p \bigl[q(1-\begin{equation}ta)-\psi(y)\bigr]- q(p-1) \xi\Bigr\} \Theta
\\
&\quad - q(q-1)\xi^2+2q \bigl[q(1-\begin{equation}ta)-\psi(y)\bigr]\xi \\
&\quad -q(1-\begin{equation}ta) \bigl(q(1-\begin{equation}ta)+1\bigr)+2q(1-\begin{equation}ta) \psi(y)\,.
\end{split}
\end{equation}
\noindent{\hskip 8mm}{\bf Subcase 2.1: } $p \bigl[q(1-\begin{equation}ta)-\psi(y)\bigr]- q(p-1) \xi \le 0$.
Using $\xi\ge 0$ and $0\le\psi(y)\le y_1$ in $Q$, we first observe that this implies $H\leq q\,f(\xi)$,
where
$$
f(\xi):= - (q-1)\xi^2 +2 q(1-\begin{equation}ta)\xi- (1-\begin{equation}ta) \bigl(q(1-\begin{equation}ta)+1\bigr)+2 (1-\begin{equation}ta)y_1. \\
$$
Computing the reduced discriminant of this trinomial we notice that
\begin{equation}gin{align*}
\Delta
&=q^2(1-\begin{equation}ta)^2-(q-1)(1-\begin{equation}ta) \bigl(q(1-\begin{equation}ta)+1\bigr)+2 (q-1)(1-\begin{equation}ta)y_1\\
&= (1-\begin{equation}ta)\bigl[1-q\begin{equation}ta+2 (q-1)y_1\bigr]<0
\end{align*}
provided $q>\frac 1\begin{equation}ta= p-1$
and $y_1$ is sufficiently small. Therefore $f(\xi) <0$ for every $\xi \, \ge 0$ and we have
$$
p \bigl[q(1-\begin{equation}ta)-\psi(y)\bigr]- q(p-1) \xi \le 0 \quad \Longrightarrow \quad H\leq q\,\max_{\xi\ge 0} f(\xi) <0\,.
$$
Hence ${\mathcal P}J<0$ provided $y_1$ is sufficiently small.
\noindent{\hskip 8mm}{\bf Subcase 2.2: } $p \bigl[q(1-\begin{equation}ta)-\psi(y)\bigr]- q(p-1) \xi > 0$.
Here, we use
\rife{u1d} to estimate
$$
\Theta \leq [c_p(1-\begin{equation}ta)]^{p-1} (1+ Cy^{1+\begin{equation}ta})^{p-1} = \begin{equation}ta (1+ Cy^{1+\begin{equation}ta})^{p-1}\le \begin{equation}ta + Cy^{1+\begin{equation}ta}.
$$
Since $p \bigl[q(1-\begin{equation}ta)-\psi(y)\bigr]- q(p-1) \xi >0$, it follows that
\begin{equation}gin{align*}
H & \leq \left\{p \bigl[q(1-\begin{equation}ta)-\psi(y)\bigr]- q(p-1) \xi\right\} \begin{equation}ta
\\
&\quad - q(q-1)\xi^2+2q \bigl(q(1-\begin{equation}ta)-\psi(y)\bigr)\xi\\
&\quad -q(1-\begin{equation}ta) \bigl(q(1-\begin{equation}ta)+1\bigr)+2q(1-\begin{equation}ta) \psi(y) + C y^{1+\begin{equation}ta}
\end{align*}
which yields, using $p\begin{equation}ta=1+\begin{equation}ta$,
\begin{equation}gin{align*}
H & \leq q \left\{ \xi^2-\xi+\begin{equation}ta(1-\begin{equation}ta) - q\bigl[\xi-(1-\begin{equation}ta)\bigr]^2-2\psi(y) \bigl[\xi-(1-\begin{equation}ta)\bigr] \right\}
\\
&\quad - (1+\begin{equation}ta)\psi(y) + C y^{1+\begin{equation}ta}.
\end{align*}
Now we observe that, for any $y\in [0,y_1]$, the function
$$
\phi(\xi):= \xi^2-\xi+\begin{equation}ta(1-\begin{equation}ta) - q[\xi-(1-\begin{equation}ta)]^2-2\psi(y) [\xi-(1-\begin{equation}ta)]
$$
is concave and we have
\begin{equation}\label{ineqphi1}
\phi'(1-\begin{equation}ta)= 1-2\begin{equation}ta-2\psi(y) \leq 0
\end{equation}
since $\begin{equation}ta\geq \frac12$ ($\iff p\leq 3$). Therefore, we have
\begin{equation}\label{ineqphi2}
\phi(\xi)\leq \phi(1-\begin{equation}ta)= 0 \quad\hbox{ for all $\xi\geq 1-\begin{equation}ta$}
\end{equation}
and we conclude that
$$
H \leq - (1+\begin{equation}ta)\psi(y) + C y^{1+\begin{equation}ta},
$$
hence
\begin{equation}gin{align*}
{y^2{\mathcal P}J\over cdF} & \leq - (1+\begin{equation}ta)\psi(y) + C y^{1+\begin{equation}ta} +Cy(x^2+ y^\begin{equation}ta)
\\
&
\leq - (1+\begin{equation}ta)\frac{ y}{1+ y} + Cy(x^2+ y^\begin{equation}ta) \leq 0
\end{align*}
provided $x_1$ and $y_1$ are taken sufficiently small.
The proof of Lemma \ref{pjneg} is complete.
\qed
\noindent \textit{Continuation of proof of Proposition \ref{propJ}.}
\textbf{Step 4. Initial and boundary conditions for $J$.}
First observe that, for each $T'<T$, we have
\begin{equation}gin{equation}\label{verif regul J}
u\leq Cy \ \ \ {\rm in}\ D\times [0,T']
\end{equation}
for some $C=C(T')>0$. Since $\gamma<q$, we have in particular
\begin{equation}gin{equation}\label{regul J}
J\in C(\overline D\times [0,T))\cap C^{2,1}(D\times(0,T)).
\end{equation}
Also in view of (\ref{verif regul J}) and $\gamma<q-1$, the coefficient $a(x,y,t)$ of the operator $\mathcal{P}$ (cf.~(\ref{def a b})) satisfies
\begin{equation}gin{equation}\label{bdd a b}
\hbox{$a$ is bounded in $D\times(0,T')$ for each $T'<T$.}
\end{equation}
Next, since $w=u_x$ satisfies
\begin{equation}gin{eqnarray}\label{eq:u x}
w_t-\Delta w=p|\nabla u|^{p-2}\nabla u\cdot\nabla w\ \ \ {\rm
in}\ Q_T,
\end{eqnarray}
and is nonnegative nontrivial in $ (0,L)^2\times(0,T)$,
by the maximum principle and after a time shift, we may assume that
\begin{equation}gin{equation}\label{res:u x negative}
u_x<0\ \ \ {\rm
in}\ (0,L)^2\times(0,T).
\end{equation}
Let now $x_1,y_1,\sigma$ be given by Lemma \ref{pjneg} and assume $\sigma<T/2$ without loss of generality.
By (\ref{extension}), (\ref{eq:u x}), (\ref{res:u x negative}), (\ref{sys
main 1}) and Hopf's Lemma, there exist constants $c_1,c_2>0$ such that
\begin{equation}gin{eqnarray*}
&&u_x\leq -c_1y\ \ \ {\rm on}\ \{x_1\}\times(0,y_1)\times(T/4,T),\\
&&u_x\leq -c_1x\ \ \ {\rm on}\
(0,x_1)\times\{y_1\}\times(T/4,T),\\
&&u\leq c_2y\ \ \ {\rm on}\ \{x_1\}\times(0,y_1)\times(0,T).
\end{eqnarray*}
With the above estimates, we check the function $J$ on the lateral boundary: if $y=y_1$, we have,
by~\rife{bound6},
\begin{equation}\label{cond:J boundary rectangle 3}
J(x,y_1,t)\leq \bigl[-c_1+ky_1^{-\gamma} (1+ y_1)(c_p y_1^{1-\begin{equation}ta}+ c y_1^2)^q \bigr]x\leq 0
\ \ \ {\rm on}\ (0,x_1)\times\{y_1\}\times(T/4,T),
\end{equation}
if $k$ is sufficient small. If $x=x_1$, we have
\begin{equation}gin{eqnarray}
J(x_1,y,t)
&\leq& -c_1y+kx_1c_2^qy^{q-\gamma} (1+ y) \notag \\
&\leq& \bigl[-c_1+kx_1c_2^qy_1^{q-\gamma-1} (1+ y_1)\bigr]y\leq 0
\ \ \ {\rm on}\ \{x_1\}\times(0,y_1)\times(T/4,T), \label{cond:J boundary rectangle 4}
\end{eqnarray}
if $k$ is sufficiently small, since $\gamma= q(1-\begin{equation}ta)<q-1$. Moreover, we clearly have
\begin{equation}gin{eqnarray}
&&J(x,0,t)=0\ \ \ {\rm on}\ (0,x_1)\times\{0\}\times(T/4,T),\label{cond:J boundary rectangle 1} \\
&&J(0,y,t)=0\ \ \ {\rm on}\ \{0\}\times(0,y_1)\times(T/4,T).\label{cond:J boundary rectangle 2}
\end{eqnarray}
Finally, we recall that there exists $c_3>0$ such that
\begin{equation}gin{equation}\label{claim corner}
u_x(x,y, T-\sigma)\leq -c_3xy\ \ \ {\rm in}\ D.
\end{equation}
(This is a parabolic version of ``Serrin's corner Lemma''; see \cite[p.512]{LS}).
Now (\ref{claim corner}) implies
\begin{equation}gin{eqnarray}
J(x,y, T-\sigma) &\leq& -c_3xy+kxc_2^qy^{q-\gamma} (1+ y) \notag \\
&\leq& \bigl[-c_3+kc_2^qy_1^{q-\gamma-1} (1+ y_1)\bigr]xy\leq 0\ \ \ {\rm in}\ D
\label{cond:J initial rectangle}
\end{eqnarray}
if $k$ is sufficient small, since $\gamma<q-1$.
Then (\ref{res:J negative}) follows from
Lemma \ref{pjneg}, (\ref{cond:J boundary rectangle 3})-(\ref{cond:J boundary rectangle 2}), (\ref{cond:J
initial rectangle}) and the maximum principle.
Note that the use of the maximum principle is justified in view of (\ref{regul J}) and (\ref{bdd a b})
(or, alternatively, of the fact that $a<0$).
The proof of Proposition \ref{propJ} is complete.
\qed
\vskip1em
By combining Proposition \ref{propJ} and Lemma \ref{barrier}, we shall now prove the upper estimate.
\noindent
\textit{Proof of Theorem 1.1: the upper estimate in (\ref{mainEstimate}).}
It suffices to prove it for $x>0$ sufficiently small (the case $x<0$ will follow by considering $u(-x,y,t)$).
By Proposition \ref{propJ}, we know now that, for some $k>0$, $q>p-1$ and $t_0\in (0,T)$, we have
$$
u_x+kxy^{-q(1-\begin{equation}ta)}(1+y)u^q\leq 0 \ \ \ {\rm in}\ Q=(0,x_1)\times(0,y_1)\times ( t_0,T).
$$
Integrating over $(0,x)$ and using \rife{bound6}, we obtain
\begin{equation}gin{equation}\label{ConclLemmeCore2}
\begin{equation}gin{split}
u^{1-q}(x,y,t) & \geq (q-1) k\frac {x^2}2 y^{-q(1-\begin{equation}ta)}+ u^{1-q}(0,y,t)
\\ & \geq (q-1) k\frac {x^2}2 y^{-q(1-\begin{equation}ta)} + [c_p y^{1-\begin{equation}ta}+ c y^2]^{1-q} \ \ \ {\rm in}\ Q.
\end{split}
\end{equation}
Starting with this estimate, we shall now apply the regularizing barrier lemma (Lemma \ref{barrier}).
Fix $x_0\in (0,x_1)$. Let $\eta\in(0,\eta_0)$, where $\eta_0$ is given by Lemma \ref{barrier}, and set
\begin{equation}gin{equation}\label{defrdD}
r=\displaystyle\frac {x_0}2, \quad d=\eta x_0,\quad D= (x_0-r, x_0+r)\times(0,d).
\end{equation}
Next we recall $\kappa$, given by Lemma \ref{barrier}:
\begin{equation}gin{equation}\label{defKappa}
\kappa=C_0(p) \,\eta^{1-\begin{equation}ta}(r^2+ T-t_0).
\end{equation}
We shall also use the notation $\tau=t-t_0$.
We claim now that there exists $\eta\in (0,\eta_0)$ such that, for any $x_0$ sufficiently small, we have
\begin{equation}gin{align}\label{compuz0}
\begin{equation}gin{cases}
&c_p y^{1-\begin{equation}ta} -\kappa \frac{y^2}2>0, \\
\noalign{\vskip 2mm}
&\bigl[c_p y^{1-\begin{equation}ta} -\kappa \frac{y^2}2\bigr]^{1-q}-\bigl[c_p y^{1-\begin{equation}ta}+ c y^2\bigr]^{1-q}
\leq (q-1) k\frac {x^2}2 y^{-q(1-\begin{equation}ta)}
\end{cases}
\qquad\hbox{ in $\overline D\times [t_0,T)$}
\end{align}
and
\begin{equation}gin{align}\label{compuz}
\begin{equation}gin{cases}
&c_p \bigl[(d+ \eta \tau^{\frac1{1-\begin{equation}ta}} r^{\frac2{1-\begin{equation}ta}})^{1-\begin{equation}ta}-\eta^{1-\begin{equation}ta} \tau r^2 \bigr]
-\kappa \frac{d^2}2>0, \\
\noalign{\vskip 2mm}
&\left\{c_p \bigl[(d+ \eta \tau^{\frac1{1-\begin{equation}ta}} r^{\frac2{1-\begin{equation}ta}})^{1-\begin{equation}ta}-\eta^{1-\begin{equation}ta}\tau r^2 \bigr]
-\kappa \frac{d^2}2\right\}^{1-q} \\
\noalign{\vskip 2mm}
&\qquad\qquad-\bigl[c_p d^{1-\begin{equation}ta}+ c d^2\bigr]^{1-q} \leq (q-1) k\frac {x^2}2 d^{-q(1-\begin{equation}ta)}
\end{cases}
\qquad\hbox{ in $[x_0-r, x_0+r]\times [t_0,T)$.}
\end{align}
\vskip0.5em
Assume for the moment that \rife{compuz0}-\rife{compuz} hold; together with \rife{ConclLemmeCore2}, this implies
$$
u^{1-q}(x,y,t) \geq \left\{c_p y^{1-\begin{equation}ta} -\kappa \frac{y^2}2\right\}^{1-q}
\quad\hbox{ in $\overline D\times [t_0,T)$}
$$
and
$$
u^{1-q}(x,d,t) \geq\left\{c_p [d+ \eta \tau^{\frac1{1-\begin{equation}ta}} r^{\frac2{1-\begin{equation}ta}})^{1-\begin{equation}ta}-\eta^{1-\begin{equation}ta} \tau r^2]-\kappa \frac{d^2}2\right\}^{1-q}
\quad\hbox{ in $[x_0-r, x_0+r]\times [t_0,T)$,}
$$
and so both \rife{hypcontroluc0} and \rife{toit} will hold.
We may then apply Lemma \ref{barrier} to deduce
\begin{equation}gin{equation}\label{conclusionUpper}
u_y(x_0,0,t) \le d_p \eta^{-\begin{equation}ta} \bigl( \tau r^2 \bigr)^{-\frac\begin{equation}ta{1-\begin{equation}ta}} \quad\hbox{ in $(t_0,T)$.}
\end{equation}
At time $t=T$ this gives
$$
u_y(x_0,0,T )\le C \eta^{-\begin{equation}ta}\, x_0^{-\frac{2\begin{equation}ta}{1-\begin{equation}ta}} = C \eta^{-\begin{equation}ta}\, x_0^{-\frac2{p-2}}
$$
which, jointly with (\ref{bound5a}), implies the upper estimate in (\ref{mainEstimate}).
\vskip0.5em
To conclude, we are thus left to prove \rife{compuz0}-\rife{compuz}. We note right away that the
first part of \rife{compuz0} is true whenever $d$ is small enough, a condition which holds as soon as $x_0$ is small enough (independently of $\eta$).
Similarly, observe that
if $x_0$ --~hence~$d$~-- is sufficiently small (independently of $\eta$), then
\begin{equation}gin{align*}
&\Bigl[c_p y^{1-\begin{equation}ta} -\kappa \frac{y^2}2\Bigr]^{1-q}-\bigl[c_p y^{1-\begin{equation}ta}+ c y^2\bigr]^{1-q} \\
&\qquad\qquad \leq (c_p y^{1-\begin{equation}ta})^{1-q}\left[\bigl(1 - \frac\kappa2 c_p^{-1} y^{1+\begin{equation}ta}\bigr)^{1-q}-\bigl(1 +c c_p^{-1} y^{1+\begin{equation}ta}\bigr)^{1-q}\right] \\
&\qquad\qquad \leq C y^{(1-\begin{equation}ta)(1-q)}y^{1+\begin{equation}ta}=C y^{2-q(1-\begin{equation}ta)} \leq C d^2 y^{-q(1-\begin{equation}ta)}=C \eta^2 x_0^2\, y^{-q(1-\begin{equation}ta)}
\end{align*}
in $D\times (t_0,T)$. Here and in the rest of the proof, $C$ denotes a generic constant independent of $x_0$ and $\eta$.
Consequently, \rife{compuz0} holds as soon as $\eta$ is sufficiently small.
In order to prove \rife{compuz}, setting $\zeta=\eta \tau^{\frac1{1-\begin{equation}ta}} r^{\frac2{1-\begin{equation}ta}}$, we write
\begin{equation}gin{align*}
c_p \bigl[(d+ \eta \tau^{\frac1{1-\begin{equation}ta}} r^{\frac2{1-\begin{equation}ta}})^{1-\begin{equation}ta}-\eta^{1-\begin{equation}ta}\tau r^2\bigr]-\kappa \frac{d^2}2
&= c_p d^{1-\begin{equation}ta} \left[\left( 1+ \frac{\zeta}d \right)^{1-\begin{equation}ta}-\left(\frac\zeta d\right)^{1-\begin{equation}ta}- \frac\kappa{2c_p} d^{1+\begin{equation}ta}\right] \\
&\ge c_p d^{1-\begin{equation}ta} \left[1-\left(\tau^{\frac1{1-\begin{equation}ta}}r^{\frac{1+\begin{equation}ta}{1-\begin{equation}ta}}\right)^{1-\begin{equation}ta}- \frac\kappa{2c_p} (\eta r)^{1+\begin{equation}ta}\right].
\end{align*}
By (\ref{defKappa}), (\ref{defrdD}), it follows that
\begin{equation}gin{align} \label{compare1}
c_p \bigl[(d+ \eta \tau^{\frac1{1-\begin{equation}ta}} r^{\frac2{1-\begin{equation}ta}})^{1-\begin{equation}ta}-\eta^{1-\begin{equation}ta}\tau r^2\bigr]-\kappa \frac{d^2}2
\ge c_p d^{1-\begin{equation}ta} \left[1-Cx_0^{1+\begin{equation}ta}\left(T+(x_0^2+T)\eta_0^2 \right) \right]>0,
\end{align}
provided
\begin{equation}gin{align} \label{smallx0}
Cx_0^{1+\begin{equation}ta}\left(T+(x_0^2+T)\eta_0^2 \right) < 1.
\end{align}
Note that (\ref{smallx0}) is true for $x_0$ sufficiently small, which guarantees the first part of \rife{compuz}.
Next notice that
the convexity inequality $ (a+b)^{1-q}\geq a^{1-q}+ (1-q)a^{-q}b$ implies
$$
[c_p d^{1-\begin{equation}ta}+ c d^2]^{1-q}\geq (c_p d^{1-\begin{equation}ta})^{1-q} - C d^{-q(1-\begin{equation}ta)+2}.
$$
Combining this with (\ref{compare1}) and (\ref{smallx0}), it follows that
\begin{equation}gin{align*} &
\left\{c_p \bigl[(d+ \eta \tau^{\frac1{1-\begin{equation}ta}} r^{\frac2{1-\begin{equation}ta}})^{1-\begin{equation}ta}-\eta^{1-\begin{equation}ta}\tau r^2\bigr]-\kappa \frac{d^2}2\right\}^{1-q}-\bigl[c_p d^{1-\begin{equation}ta}+ c d^2\bigr]^{1-q} \\
& \qquad \leq (c_p d^{1-\begin{equation}ta})^{1-q} \left\{ \left[1-Cx_0^{1+\begin{equation}ta}\left(T+(x_0^2+T)\eta_0^2 \right) \right]^{1-q} -1\right\} + C d^{-q(1-\begin{equation}ta)+2} \\
& \qquad \leq C d^{(1-\begin{equation}ta)(1-q)} x_0^{1+\begin{equation}ta}\left(T+(x_0^2+T)\eta_0^2 \right) + C d^{-q(1-\begin{equation}ta)+2} \\
& \qquad \leq C \eta^{1-\begin{equation}ta}\left(T+(x_0^2+T)\eta_0^2+ \eta_0^{1+\begin{equation}ta} \right) x_0^2\,d^{-q(1- \begin{equation}ta)}.
\end{align*}
Therefore, \rife{compuz} is satisfied as soon as $\eta$ is chosen sufficiently small. \qed
\section{A heuristic explanation of the singularity exponents through quasi-stationary approximation}
A possible heuristic explanation of the appearance of the number $2/(p-2)$ in the tangential singularity profile (\ref{mainEstimate}) can be obtained
using the idea of {\bf quasi-stationary approximation} along the family of 1D steady states.
Recall the following family of 1D steady states, given by the translates of the reference solution
$$ V(y)=c_p y^{1-\begin{equation}ta},$$
i.e.
$$V_a(y)=V(y+a)-V(a), \quad y\ge 0, \ a\ge 0.$$
These special solutions verify
$$-V_a''={V_a'}^p, \qquad V_a(0)=0, \qquad V_a'(0)=d_p a^{-\begin{equation}ta}.$$
The idea is then to look for an approximate solution obtained by modulating in $a$, or moving on the manifold of steady-states $(V_a)_{a\ge 0}$.
More precisely, we set $U=u_{approx}$ given by
\begin{equation}gin{equation}\label{defApproxSol}
U(x,y,t) = V(y+h(t,x))-V(h(t,x)),
\end{equation}
which amounts to parametrize the solution by $a=h(t,x)$.
In particular, we have $U_y(x,y,t) = V'(y+h(t,x)).$
The function $h(t,x)$ is positive for $t<T$ and must satisfy $h(T,0)=0$
so that $U_y(0,0,T)=\infty.$
Note that this Ansatz means in some sense that
$-u_{yy} \sim (u_y)^p$ and $u_t \sim u_{xx}$ near the singularity,
already giving a rough clue to the parabolic nature of the scaling
of the profiles in $t$ and $x$.
With the above Ansatz, one has an interpretation of the lower estimate of the tangential profile in (\ref{mainEstimate})
as being {\bf a consequence of the constraint $U_{xx} \ge -C$, which comes from the maximum principle} (cf.~Proposition \ref{prop:maxprin}).
Indeed, $U_{xx} \ge -C$ and $U_x(0,y,t)=0$ imply that
\begin{equation}gin{equation}\label{yrate0}
U_x \ge -Cx,\quad x>0.
\end{equation}
For $t<T$, restricting without loss of generality to $x>0$, we note that
$$U_x(x,y,t)=d_p\bigl[(y+h(t,x))^{-\begin{equation}ta}-h^{-\begin{equation}ta}(t,x)\bigr]h_x\ge -d_ph_xh^{-\begin{equation}ta}(t,x),$$
where we used $h_x>0$ due to $U_x<0$, and that
$$U_x(x,h(t,x),t)=-ch_xh^{-\begin{equation}ta}(t,x).$$
Consequently, (\ref{yrate0}) is equivalent to $h_xh^{-\begin{equation}ta}(t,x) \le Cx$. By integration, it follows that
$h^{1-\begin{equation}ta}(t,x)\le h^{1-\begin{equation}ta}(t,0)+Cx^2$. Letting $t\to T$, we get $h(T,x)\le Cx^{2/(1-\begin{equation}ta)}$, which leads to
\begin{equation}gin{equation}\label{yrate}
U_y(x,0,T)=V'(h(T,x))=d_p (h(T,x))^{-\begin{equation}ta}\ge Cx^{-2\begin{equation}ta/(1-\begin{equation}ta)}=Cx^{-2/(p-2)}.
\end{equation}
The fact that the upper estimate in (\ref{mainEstimate}) is exactly of this order means that
the constraint $U_{xx} \ge -C$ is satisfied in a minimal way by the parabolic flow.
The same analysis can be done with the
time rate as well and actually enables one to recover also the exponent $1/(p-2)$
of the time rate\footnote{this observation
doesn't seem to have been made in previous work on the time rate.}, namely
$$
\|\nabla u(t)\|_\infty\sim (T-t)^{-1/(p-2)}
$$
(the lower estimate is always true -- see \cite{CG96,GH08,QS07} -- whereas the upper estimate is only known for monotone increasing solutions in 1D;
see \cite{GH08} and cf. also \cite{QS07}).
This time the essential constraint is $|U_t| \le C$ (cf.~Proposition \ref{prop:maxprin}).
Indeed, one can easily see that
$|U_t| \le C$ is equivalent to $|(V(h))_t| \le C$.
Since $h(T,0)=0$,
we thus have $V(h(t,0))\le C(T-t)$,
i.e. $h(t,0)\le C(T-t)^{1/(1-\begin{equation}ta)}$, or
\begin{equation}gin{equation}\label{timerate}
U_y(0,0,t)
=V'(h(t,0))=d_p (h(t,0))^{-\begin{equation}ta} \ge C(T-t)^{-\begin{equation}ta/(1-\begin{equation}ta)}=C(T-t)^{-1/(p-2)}.
\end{equation}
Since the rates (\ref{yrate}) and (\ref{timerate}) violate the self-similar structure, or natural scaling, of the equation
(cf.~Remark (a) in Section 1.3),
so one can say that the maximum principle here wins against self-similarity.
\begin{equation}gin{rem}\label{remp3}
For $p>3$, the proof of the upper estimate in (\ref{mainEstimate})
fails at the level of inequalities \rife{ineqphi1}--\rife{ineqphi2}.
Actually, it can be seen along the lines of the proof of Lemma \ref{pjneg}
that ${\mathcal P}J>0$ in some regions near the singularity
(more precisely, where $yu_yu^{-1}\sim (1-\begin{equation}ta)_+$ and $yu_y^{p-1}\sim\begin{equation}ta$).
However this might be technical, and it is presently open whether or not
the actual behavior of $u$ changes for $p>3$.
As for the above heuristic argument, although it does not a priori make a difference between the ranges
$2<p\le 3$ or $p>3$, it is not clear if such an argument can suggest more than a lower estimate of the profile.
Indeed, we stress that the heuristic argument gives a justification of the lower estimate in (\ref{mainEstimate})
only in view of the one-sided estimate
\begin{equation}gin{equation}\label{onesided}
u_{xx}\ge -C.
\end{equation}
For $p\le 3$, our proof of the upper estimate also shows that the estimate $u_{xx}\ge -C$
is really optimal and that $u_{xx}$ is discontinuous near the singularity at $t=T$ (cf.~Remark~\ref{rem51}).
If one could show for $p>3$ that $u_{xx}$ remains continuous (i.e., has a zero limit)
near the singularity at $t=T$,
then the proof of Theorem \ref{thm:GBUprofileLower} (as well as the heuristic argument),
would imply that the final profile of $u_y$
is more singular than (\ref{mainEstimate}).
However, such property of $u_{xx}$ should be rather unstable and proving this might be quite delicate.
Indeed, for any $p>2$ and {\it any} $\alpha\ge (p-1)/(p-2)$,
a simple computation shows that the function
$$u(x,y,t)=c_p\bigl[(|x|^{2\alpha}+(T-t)^\alpha+y)^{1-\begin{equation}ta}-(|x|^{2\alpha}+(T-t)^\alpha)^{1-\begin{equation}ta}\bigr],$$
modeled after \rife{defApproxSol}, is a solution of
$$u_t-\Delta u=|\nabla u|^p+f$$
in $Q=(-1,1)\times (0,1)\times (0,T)$
with $u(x,0,t)=0$ and some $f\in L^\infty(Q)$. Moreover $u$ has an isolated
gradient blowup point at $(0,0,T)$,
$u$ satisfies \rife{onesided} and $|u_t|\le C$, but
$u_{xx}$ is continuous near $(0,0,T)$ if and only if $\alpha> (p-1)/(p-2)$.
\end{rem}
\noindent{\bf Acknowledgement.} \quad
Part of this work was done during a visit of A.~Porretta at the Laboratoire Analyse, G\'{e}om\'{e}trie et Applications of Universit\'e Paris 13.
A.~Porretta wishes to thank these institutions for the invitation and the very kind hospitality given on this occasion.
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\end{document} |
\begin{document}
\title{Automatic Generation of Probabilistic Programming from Time Series Data}
\author{Anh Tong \and Jaesik Choi\\
Ulsan National Institute of Science and Technology\\
Ulsan, 44919 Korea\\
\{anhth,jaesik\}@unist.ac.kr}
\nocopyright
\maketitle
\begin{abstract}
\begin{quote}
Probabilistic programming languages represent complex data with intermingled models in a few lines of code. Efficient inference algorithms in probabilistic programming languages make possible to build unified frameworks to compute interesting probabilities of various large, real-world problems. When the structure of model is given, constructing a probabilistic program is rather straightforward. Thus, main focus have been to learn the best model parameters and compute marginal probabilities. In this paper, we provide a new perspective to build expressive probabilistic program from continue time series data when the structure of model is not given.
The intuition behind of our method is to find a descriptive covariance structure of time series data in nonparametric Gaussian process regression. We report that such descriptive covariance structure efficiently derives a probabilistic programming description accurately.
\end{quote}
\end{abstract}
\section{Introduction}
Probabilistic programming has potential impacts on various works in artificial intelligence, machine learning, statistics, robotics (\cite{robotLBDM04}), vision (\cite{Kulkarni_2015_CVPR}), neuroscience (\cite{neuroLake1332}), and cognitive science (\cite{cogniFRT12}).
Many different approaches and probabilistic programming languages are introduced, for example, Church \citep{church}, Problog \citep{problog}, BUGS \citep{bugs}, \texttt{Stan} \citep{stan}. Although each probabilistic programming language has its own strength and domain, we choose \texttt{Stan} to present our work in this paper since it is suitable for modeling continuous signal.
The Automatic Bayesian Covariance Discovery (ABCD) system which is so-called automatic statistician system, is proposed \citep{lloyd2014automatic} with the aim of automating the process of statistical modeling. It focuses on regression problems. Specifically, it takes time series as input, searches and then produces a learned Gaussian process model which has interpretable properties (smoothness, periodicity, changepoints) and is summarized with a report. With the same purpose, \citep{1511.08343} proposed a relational approach to handle multiple data. However, such kinds of modeling usually require a body of work and efforts to make build a new model. One of the advantages of probabilistic programming which helps in this situation is the ease of creating generative models with several lines of code \citep{Kulkarni_2015_CVPR}. In order to facilitate the process of building new ABCD-based models, we propose a method generating \texttt{Stan} probabilistic programming from ABCD results. Time series are stored in the compact representation of encoded ABCD probabilistic programmings which potentially allow construct a complex model from heterogeneous models.
This paper is organized as follows. We briefly introduce the background of Gaussian Processes (GP) , the ABCD system and its relational version for multiple data. Then, we present our main contribution on how a probabilistic programming is generated from time series data; Next is the experiment section; Finally, we conclude our work.
\section{Background}
\subsection{Automatic Statistician System}
\input{./2.1-abcd.tex}
\subsection{Relational Automatic Statistic System}
\input{./2.2-rabcd.tex}
\subsection{Probabilistic Programming}
\input{./2.3-probabilistic-programming.tex}
\section{Automatic Generation of Probabilistic Programming}
\par Generating probabilistic programming from ABCD and/or relational ABCD is a crucial component in the system we aim to build (see Figure \ref{fig:model}). Prior to this component, one can choose either ABCD or relational ABCD based on whether the preference is for a single time series or for global information in multiple time series. Both ABCD and relational ABCD play as producers which output compositional kernels from data (Step 1 and 2). Step 4 and 5 are an example application. We will discuss what is inside Step 3 in this section.
\begin{figure*}
\caption{An overview of the system in which the relational ABCD framework (or ABCD framework) combines with probabilistic programming. Step 1 executes the input in the relational ABCD framework (ABCD framework); Step 2 retrieves the output as a kernel; Step 3 generates probabilistic programs; Step 4 executes a query into system; Step 5 automatically makes a report with respect to query. Step 1, 2 are procedures in the relational ABCD framework (ABCD framework). We address the step 3 in this paper.}
\label{fig:model}
\end{figure*}
\par From now on, we use the notation \texttt{StanABCD} to indicate the \texttt{Stan} probabilistic programs generated automatically from ABCD.
\subsection{Base kernels}
An ABCD's result is represented by a compositional structure which is a sum of products of base kernels. This summation is the outcome of simplifications: the multiplication of two SE kernel produces another SE with different parameter values. The product of WN and any stationary kernel including C, PER, WN, SE results a new WN kernel. Multiplying C with any kernel does not change the kernel but changes the scale parameter of that kernel \citep{lloyd2014automatic}. Hence, let $G$ be a set of all possible kernel expressions, written as
$$G = \{\sum k \prod_m \textrm{LIN}^{(m)} \sigma^{(n)}\}$$
where $\sigma$ is the sigmoid function, $k$ is in
$$K = \{\textrm{WN}, \textrm{C}, \prod_k\textrm{PER}^{(k)}, \textrm{SE}\prod_k\textrm{PER}^{(k)}\}$$
For example, a learned compositional kernel is described as
\begin{equation}
\label{eq:2}
\textrm{SE} \times \textrm{LIN}
\end{equation}
Here, the square exponential kernel and linear kernel are written respectively as
$$\textrm{SE}(x, x') = \sigma^2 \exp\left( - \frac{(x - x')^2}{2l^2} \right),$$
$$\textrm{LIN}(x, x') = \sigma^2(x - l)(x' - l).$$
The kernel (\ref{eq:2}) can be understood linguistically as 'a smooth function with linearly (LIN) increasing amplitude'.\\
Another real-world example is a chosen currency exchange data set (see Figure \ref{fig:indo}). The data set contains exchange value of Indonesian Rupiad from 2015-06-28 to 2015-12-30 acquired from Yahoo Finance \citep{yahoofinance}. Carried experiments on this data set, relational ABCD found the best compositional kernel which is shortly written as
\begin{equation}
\textrm{CW}(\textrm{SE} + \textrm{CW}(\textrm{WN} + \textrm{SE},\textrm{WN} ), \textrm{C} )
\label{eq:indo}
\end{equation}
This kernel is well-explained for several currency exchanges sharing common financial behaviors. A qualitative result shows that the changewindow (CW) kernel occurs around mid September 2015 which reflects big financial events (FED’s announcement about policy changes in interest rates, China’s foreign exchange reserves falls) \citep{1511.08343} . We take this compostional kernel as a typical example for demonstration purpose.
Given a data set and a learned kernel, we are interested in encoding them into a \texttt{Stan} program. In order to do that, we first prepare built-in base kernels in \texttt{Stan} version. A base kernel is written as a \texttt{Stan} function which takes data, and hyperparameters as input and returns a matrix. The matrix has elements reflecting how similar (correlated) the data points in data set are. Each base kernel have a specific number of hyperparameters itself. For the SE kernel case, it has two hyperparameters: a scale factor $\sigma$, and a lengthscale $l$; then we build a \texttt{Stan} code as showed in Figure~\ref{fig:se} (The detail implementations of other kernels are in the Appendix).
\begin{figure}
\caption{The implementation of SE kernel on \texttt{Stan}
\label{fig:se}
\end{figure}
\input{./code/kernel.tex}
Next, we will discuss how a \texttt{StanABCD} organize. \texttt{StanABCD}s share common conventional blocks (as in previous section) but the compositional kernel. We briefly describe what is required in each blocks.
\subsubsection{Data block}
In general, \texttt{StanABCD} contains training data points $X$ and test data points $X_{\star}$. Abiding by the \texttt{Stan} convention, \texttt{StanABCD} declares data as following:
\input{./code/data_block.tex}
Here, we provide the information of training data through \texttt{N1} (number of training data points) and vectors \texttt{x1}, \texttt{y1}. Similarly, test data is specified by the number of test data points \texttt{N2} and a vector \texttt{x2}.
For example, we analyze Indonesian Rupiah exchange data with the period from July 2015 to December 2015 as shown in Figure \ref{fig:indo}. This data set consists of 132 data points in which we take the first 120 data points as training data, and the next 12 data points as test data. We have \texttt{N1} = 120, \texttt{x1} be the days in training data set, \texttt{y1} be the exchange value, \texttt{N2} = 12, and \texttt{x2} be the days in test data set. We want to predict the exchange value on \texttt{x2}.\\
\begin{figure}
\caption{Indonesian Rupiad exchange data. Dot: raw data point. Line with shade: Gaussian Process prediction with 95\% confidence region. Vertical dash line separates the training data and test data.}
\label{fig:indo}
\end{figure}
\subsubsection{Parameter block and transformed parameters block}
These blocks consist of all necessary parameters to construct the model. With the purpose of sampling data on test points,
\texttt{StanABCD} has the parameter block containing a parameter \texttt{z} as an array with length \texttt{N2} equal to the number of test data points. \texttt{z} follows unit normal distribution to increase the sampling performance which is discussed later in the transformed parameters block. If we use \texttt{StanABCD} not only as a sampler, the parameter block should be customized by adding more parameters for our desired models.
What the transformed parameters block does is to make computation for the posterior distribution of GPs. Based on GP prior, the joint distribution of training output $\mathbf{y}$ and test output $\mathbf{y}_\star$ is represented as
\begin{equation*}
\begin{pmatrix}
\mathbf{y}\\
\textbf{y}_\star
\end{pmatrix}
\sim
\mathcal{N}\left(
\mathbf{0},
\begin{bmatrix}
K(X,X) & K(X,X_\star) \\
K(X_\star, X) & K(X_\star, X_\star)
\end{bmatrix}
\right).
\end{equation*}
Assume we already know the structure of compositional kernel $K(.,.)$ from ABCD.
From this compositional kernel structure, $K(X,X_\star)$ is a $n \times n_\star$ matrix evaluated at all pairs of training and test points, where $n = |X|$ is the number of training data points, $n_\star = |X_\star|$ is the number of test data points. Analogously, we compute $K(X,X)$, $K(X_\star, X)$, and $K(X_\star,X_\star)$. Note that $K(X,X_\star) = K(X_\star, X)^T$. Now, using the conditioning Gaussians, it follows that
\begin{equation}
\begin{split}
\mathbf{y}_\star | X_\star, X, \mathbf{y} & \sim \mathcal{N}(K(X, X_\star)K(X, X)^{-1}\mathbf{y}, \\
& K(X_\star, X_\star) - K(X, X_\star)K(X,X)^{-1}K(X,X_\star))
\end{split}
\label{eq:post}
\end{equation}
The posterior distribution is analytically tractable. This makes easier to represent on a \texttt{Stan} programming because it supports most of distributions. Here, we only need to specify a mean $\mu = K(X, X_\star)K(X, X)^{-1}\mathbf{y}$ and a covariance matrix $\Sigma = K(X_\star, X_\star) - K(X, X_\star)K(X,X)^{-1}K(X,X_\star)$, in order to declare a multivariate normal distribution in the next blocks.
Taking a consideration about the efficiency of implementation, the Cholesky decomposition (which is available as a built-in function in \texttt{Stan}) of $\Sigma$ is pre-computed. Let us denote the Cholesky decompositon of $\Sigma$ be $L$. \texttt{Stan} only perform its sampling method to produce a unit normal distribution
\begin{equation}
z \sim \mathcal{N}(\mathbf{0}, \mathbf{I})
\label{eq:unit_normal}
\end{equation}
Then, we transform $z$ into $\mu + Lz$ to obtain our target distribution
$$\mu + Lz \sim \mathcal{N}(\mu, \Sigma = LL^T).$$
Here is the sample code for this block:
\input{./code/trans.tex}
The above variables \texttt{Sigma}, \texttt{Omega}, \texttt{K} correspond respectively to the terms $K(X,X)$, $K(X_\star, X_\star), K(X, X_\star)$.
We still leave the question how to build a compositional kernel from base kernels. Basically, the compositional kernel is the sum of product of base kernels. To represent the compositional kernel in \texttt{Stan}, the summation is used as a matrix sum operation and the multiplication between base kernels is replaced by the Hadamard product. Here is an example
\input{./code/kernel.tex}
\subsubsection{Model block}
\texttt{Stan} allows us to quickly design a Bayesian hierarchical model. Utilizing the mean and covariance computed in the previous block helps us declare a normal distribution which plays a role as the first level in the multiple levels of hierarchical model. However, we set this aside and only illustrate the case that we sample the posterior distribution on test data points. From (\ref{eq:unit_normal}), we declare a Gaussian distribution $\mathcal{N}(\mathbf{0}, \mathbf{I})$ to serve the sampling purpose in the generated quantities block.
\input{./code/model.tex}
\subsubsection{Generated quantities block}
For purpose of generating sample extrapolation value, the normal distribution declared in the model block will be called one time.
\input{./code/generate.tex}
In order to get the sample of test output \texttt{y2}, the above \texttt{Stan} code performs the linear transformation on sample values generated from \texttt{z} ($\mathcal{N}(\mathbf{0}, \mathbf{I})$) as we explained in the transformed parameters block.
\section{Experiment}
\subsubsection{Data set} Beside the Indonesian Rupiah exchange data mentioned in previous section, we select a airline data set to perform experiments on. The data set describes monthly international airline passenger numbers for the period between January 1949 and December 1960 (\cite{Box}). The number of passengers was periodic with a typical period 1 year. The total number of passengers per year increased monotonically. ABCD captures this information well, and explain the data set by a compositional kernel: LIN + SE $\times$ PER $\times$ LIN + SE + WN $\times$ LIN.
\begin{figure}
\caption{A comparison of ABCD result and \texttt{StanABCD}
\label{fig:exp}
\end{figure}
\subsubsection{Sampling data}
We provide a complete sample \texttt{Stan} code in the appendix. We want to get extrapolation sample values on test points (from January 1961) of airline data set. During the experiment, we use Python to retrieve data set then pass to \texttt{Stan} compiler through \textbf{PyStan} \citep{pystan}. Figure \ref{fig:exp} and \ref{fig:exp_indo} show that \texttt{StanABCD} provides similar results as ABCD in the view of extrapolation performance. Our generating method guarantees a reliable way to perform one-to-one mapping from ABCD result into a probabilistic programming.
\begin{figure}
\caption{The extrapolation sampling (from vertical dash line) from \texttt{StanABCD}
\label{fig:exp_indo}
\end{figure}
\section{Conclusion}
We propose a beautiful blend between the automatic statistician and probabilistic programming. As the result, it opens a broad direction to explore on encoded \texttt{Stan} programs because of their potentiality to make further inference or perform statistical relational learning. On the other hand, \texttt{StanABCD} provides a promising way to accelerate the learning kernel in ABCD framework which requires an exhaustive search procedure. A database of \texttt{StanABCD}s is one of the possible solutions.
\section*{Acknowledgments}
This work is supported by Basic
Science Research Program through the National Research
Foundation of Korea (NRF) grant funded by the Ministry
of Science, ICT \& Future Planning (MSIP) (NRF-
2014R1A1A1002662) and the NRF grant funded by the MSIP
(NRF-2014M2A8A2074096).
\appendix
\section*{Appendix}
\subsection{Base kernels}
\subsubsection{White noise kernel}
A white noise kernel is written as
$$\textrm{WN}(x, x') = \sigma^2 \delta_{x,x'}$$
We implement this kernel in
\input{./kernel/wn.tex}
\subsubsection{Constant kernel}
A constant kernel is written as
$$\textrm{C}(x, x') = \sigma^2$$
\input{./kernel/const.tex}
\subsubsection{Linear kernel}
A linear kernel is written as
$$\textrm{LIN}(x, x') = \sigma^2(x - l)(x' - l)$$
\input{./kernel/lin.tex}
\subsubsection{Squared exponential kernel}
A squared exponential is defined as follows:
$$\textrm{SE}(x, x') = \sigma^2 \exp\left( - \frac{(x - x')^2}{2l^2} \right)$$
\input{./kernel/se.tex}
\subsubsection{Periodic kernel}
A periodic kernel is defined as
$$\textrm{PER}(x,x') = \sigma^2 \frac{\exp \left(\frac{\cos \frac{2\pi(x - x')}{p}}{l^2}\right) - I_0(\frac{1}{l^2})}{\exp(\frac{1}{l^2}) - I_0(\frac{1}{l^2})}, $$
where $I_0$ is the modified Bessel function of the first kind of order zero
\input{./kernel/per.tex}
\subsubsection{Changepoint operator}
A changepoint operator on kernels $k_1$ and $k_2$ is defined as
\begin{equation*}
\begin{split}
\textrm{CP}(k_1,k_2)(x,x') = & \sigma(x)k_1(x,x')\sigma(x') + \\
& (1- \sigma(x))k_2(x,x')(1 - \sigma(x'))
\end{split}
\end{equation*}
where $\sigma(x) = \frac{1}{2}(1 + \textrm{tanh}(\frac{l - x}{s}))$
\input{./kernel/changepoint.tex}
\onecolumn
\subsection{A sample code}
\input{./code/complete.tex}
\pagebreak
\end{document} |
\begin{document}
{\bf Comment on ``Weak value amplification is suboptimal for estimation and detection'' }
In a recent Letter, Ferrie and Combes \cite{FC} defined the practical tasks ``detect'' and ``estimate'' and concluded that
``Post-selection cannot aid in detect and estimate for any interaction parameter''. In particular, they argued that
``there is no sense in which WVA [Weak Value Amplification] provides an ``amplification'' for quantum metrology''.
At 1988 Aharonov, Albert and Vaidman \cite{AAV} discovered that a sufficiently weak coupling to any observable of a pre- and postselected quantum system is a coupling to the ``weak value'' of this observable and, since the weak value can be much larger than the eigenvalues of the observable, this method provides an effective amplification of the weak coupling. This amplification is the WVA discussed in the Letter of Ferrie and Combes. The WVA method has been implemented in several experiments in recent years. The spin Hall effect for light was first detected using this method \cite{HK}. A record precision of a mirror angle estimation was obtained using WVA in another experiment \cite{How}. So, definitely, the post-selection aided in detecting and estimating interaction parameters.
How can it then be that the ``statistically rigorous arguments'' of Ferrie and Combes contradict these experimental results? The explanation is that the assumptions in their statistical analysis are irrelevant for realistic experimental situations.
I found the main erroneous assumption which led Ferrie and Combes to their incorrect conclusions thank to my direct involvement in two weak measurement experiments \cite{Xi,Danan}. The limiting factor in these and other experiments is not the number of preselected quantum systems (photons) considered by Ferrie and Combes, but the number of detected, post-selected photons. The saturation of the detectors generally happens much before the power limitation of the laser source kicks in. Thus, the low probability of the postseletction, the main negative factor in experiments with large weak values, is not relevant. This then undermines the conclusions of Ferrie and Combes.
In their Letter, Ferrie and Combes quote other recent papers analyzing the limitations of the WVA method \cite{KnBr,TaYa,Zhu,KnGa}, which they improve and complement.
These limitations were obtained by using the same assumptions, but the authors of these works specify (some of them maybe not clearly enough) that their conclusions are conditioned on these assumptions. Zhu et al. \cite{Zhu} do it very precisely. They conclude: ``We have shown that weak measurements cannot effectively improve
the SNR [Signal to Noise Ratio] and the MS [measurement sensitivity] when the probability decrease due
to postselection needs to be considered; while for practical cases when the probability reduced by postselection need not
be considered, weak measurements can significantly improve both the SNR and the MS.''
This work has been supported in part by grant number 32/08 of the Binational Science Foundation and the Israel Science Foundation Grant No. 1125/10.
L. Vaidman\\
Raymond and Beverly Sackler School of Physics and Astronomy\\
Tel-Aviv University, Tel-Aviv 69978, Israel
\end{document} |
\begin{document}
\title{Partitioning transitive tournaments into isomorphic digraphs}
\begin{abstract}
In an earlier paper (see \cite{SaSi}) the first two authors have shown that
self-complementary
graphs can always be oriented in such a way that the union of the oriented
version and its isomorphically oriented complement gives a transitive
tournament. We investigate the possibilities of generalizing this theorem to
decompositions of the complete graph into three or more isomorphic graphs.
We find that a complete characterization of when an orientation with similar
properties is possible seems
elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and
also imply that decompositions of odd vertex complete graphs to Hamiltonian
cycles admit such an orientation. These conditions are further generalized and
some necessary conditions are given as well.
\par\noindent
{\em Keywords:} graph orientation, decomposition to isomorphic graphs, transitive tournament
\end{abstract}
\section{Introduction}
\message{Introduction}
Investigating the relationship between the Shannon capacity of graphs and the Sperner capacity of their oriented versions the first two authors proved the following theorem.
\begin{thm} \label{thm:selfc} {\rm (\cite{SaSi}, cf. also \cite{GyA})} Let $G$ be a graph isomorphic to its complement $F=\overline{G}$. Then $G$ and $F$ can be oriented so that they remain isomorphic as digraphs while the tournament formed by their union is the transitive tournament.
Moreover, the above can be done for any fixed isomorphism between $G$ and $F$. That is, for any such isomorphism $f$ one can find oriented versions $\vec G$ and $\vec F$ of $G$ and $F$, respectively, such that $f$ provides and isomorphism between $\vec G$ and $\vec F$ and the union of $\vec G$ and $\vec F$ is a transitive tournament.
\end{thm}
The goal of the present paper is to investigate the possibilities of
generalizing the above theorem to three or more graphs, that is, to the
situation when (the edge set of) the complete graph is partitioned into three
or more isomorphic graphs.
As already observed by G\"orlich, Kalinowski, Meszka, Pil\'sniak, and Wo\'zniak \cite{GKMPW} in this case it will not be true that the three graphs can always be oriented
in an isomorphic manner so that their union forms a transitive
tournament. Moreover, a complete characterization of when this is possible
seems to be elusive. In \cite{GKMPW, GKMPW2} the authors determine all digraphs with at most four edges that can decompose a transitive tournament. (For related results, see also \cite{GPW} and \cite{GP}.)
In our approach we fix the number of isomorphic graphs in the decompositions considered. We start with the case when this number is $3$. We will give some sufficient conditions when the isomorphic parts of a decomposition of $K_n$ can be isomorphically oriented to get a decomposition of the transitive tournament. This result gives a generalization of Theorem~\ref{thm:selfc}.
It is well-known that complete graphs on an odd number of vertices decompose
into Hamiltonian cycles. One can directly show how to obtain a decomposition of the transitive tournament of odd order into isomorphically oriented Hamiltonian cycles, but this also follows from our sufficient condition mentioned above.
First we extend our sufficient condition to a more general one, then we show with an example (found by computer) that this more general condition is still not necessary. A
complete characterization seems out of reach, but we are able to give some non-trivial necessary conditions.
As usual $K_n$ denotes the complete graph on $n$ vertices, while we denote the transitive tournament on $n$ vertices by $T_n$. The vertex set of $K_n$ and $T_n$ is assumed to be $[n]=\{0,1\dots,n-1\}$ and we consider these vertices as residue classes modulo $n$, that is equality between vertices will be understood modulo $n$. We denote the cyclic permutation of $[n]$ bringing $i$ to $i+1$ by $\sigma_n$.
\section{Small examples and problem formulation}\label{sect:smallex}
First we recall an example from \cite{GKMPW} of three isomorphic graphs partitioning the complete graph that cannot be isomorphically oriented so that their union is a transitive tournament even if the functions giving the isomorphism among them are not fixed.
Let $n=|V(G)|=4$ and the three isomorphic graphs be paths on $3$ vertices. One
easily partitions $K_4$ into three such graphs. It is also easy to see that
whatever way we orient these paths in an isomorphic manner, we cannot put them
together to obtain a transitive tournament on $4$ vertices. This is simply because from no orientation can we produce simultaneously a vertex of outdegree $0$ and a vertex of outdegree $3$. Note that this example is just a very special case of Theorem 5 from \cite{GKMPW}.
Let us assume that the edge set of the complete graph $K_n$ is partitioned into three isomorphic graphs $F$, $G$ and $H$. We can ask whether there are isomorphic orientations $\vec F$, $\vec G$ and $\vec H$ of the graphs $F$, $G$ and $H$, respectively, such that their union gives a transitive tournament. But we can be more specific and fix an isomorphism $\sigma$ from $F$ to $G$, an isomorphism $\rho$ from $G$ to $H$ and ask whether there are orientations $\vec F$, $\vec G$ and $\vec H$ of the graphs $F$, $G$ and $H$ whose union is a transitive tournament and such that $\sigma$ is an isomorphism between $\vec F$ and $\vec G$ and $\rho$ is an isomorphism between $\vec G$ and $\vec H$.
To illustrate the difference between these two questions let us consider the smallest possible example. The graph $K_3$ can be partitioned into three (isomorphic) single edge graphs: $F$, $G$ and $H$. Clearly, the three oriented edges of $T_3$ also form
isomorphic graphs. This answers the first question above for this specific partition affirmatively. If, however, we fix a cyclic permutation $\sigma=\rho$ that brings $F$ to $G$ and $G$ to $H$, then the answer to the second question is negative. Indeed, if $\vec F$ is either orientation of $F$, $\vec G=\sigma(\vec F)$ and $\vec H=\sigma^2(\vec F)$, then the union of these three directed graphs is a directed cycle and thus not transitive.
In this paper we will concentrate on the question with fixed permutations $\sigma$ and $\rho$. We will only consider the special case $\sigma=\rho$. Although this assumption is restrictive, it is
in complete analogy with the case of two self-complementary graphs, and we
believe that understanding this special case would largely improve our
knowledge about the situation.
\begin{defi}
Let $\sigma$ be a permutation of the vertex set of $K_n$. We call the partition of the edge set of $K_n$ into three graphs $F$, $G$ and $H$ such that the permutation $\sigma$ brings $F$ to $G$ and $G$ to $H$ a {\em $\sigma$-partition}. In this case $\sigma$ brings $H$ to $F$ and $\sigma^3$ is an automorphism of all three of these graphs. We call a transitive orientation $T$ of $K_n$ a {\em transitive $\sigma$-orientation} of this $\sigma$-partition if the subgraphs $\vec F$, $\vec G$ and $\vec H$ of $T$ that are the orientations of the graphs $F$, $G$ and $H$, respectively, satisfy $\sigma(\vec F)=\vec G$ and $\sigma(\vec G)=\vec H$. We say that $\sigma$ {\em reverses the orientation} of an edge $e$ in $T$ if $\sigma(e)$ is oriented in the other direction, that is, if $e$ goes from $a$ to $b$, and the edge of $T$ between $\sigma(a)$ and $\sigma(b)$ is oriented toward $\sigma(a)$. Observe that a transitive orientation $T$ of $K_n$ is a transitive $\sigma$-orientation of the $\sigma$-partition of $K_n$ to $F$, $G$ and $H$ if and only if $\sigma$ reverses no edges of $T$ that belong to $F$ or $G$.
\end{defi}
Just as it was the case with
self-complementary graphs, we may assume that the permutation $\sigma$ is cyclic as the case of general $\sigma$ reduces to the cyclic case. Indeed, let the cycle decomposition of the permutation of $\sigma$ be $\rho_1\rho_2\dots\rho_k$. Let $F$, $G$ and $H$ form a $\sigma$-partition $P$. The subgraphs $F_i$, $G_i$, $H_i$ induced by the domain of the cycle $\rho_i$ form a $\rho_i$-partition $P_i$ for all $i$. Clearly, if $P$ has a transitive $\sigma$-orientation, then it restricts to transitive $\rho_i$-orientations of $P_i$. On the other hand, if $P_i$ has a transitive $\rho_i$-orientation for all $i$, then $P$ has a transitive $\sigma$-orientation. To see this last statement simply keep the orientations of the edges in the transitive $\rho_i$-orientations and orient edges connecting vertices from distinct cycles $\rho_i$ and $\rho_j$ toward the higher indexed cycle. None of these latter type of edges is reversed by $\sigma$.
It is easy to see that a $\sigma$-partition exists if and only if $\sigma$ has at most one fixed point and the length of all non-trivial cycles of the cycle decomposition of $\sigma$ is divisible by $3$.
From now on we do make the assumption that $\sigma$ consists of a single cycle on $n>1$ vertices,
namely $\sigma=\sigma_n$, where $\sigma_n$ stands for the permutation on the set $[n]=\{0,1,2,\dots,n-1\}$ bringing $i$ to $i+1\bmod n$. The vertices of our graphs will therefore be the elements of $[n]$ and we consider them as the residue classes modulo $n$, that is, equalities about them are always understood modulo $n$. We assume $n$ is divisible by $3$ as otherwise there is no $\sigma_n$-partition.
We denote the graphs of the $\sigma_n$-partition by $F_0$, $F_1=\sigma_n(F_0)$ and $F_2=\sigma_n^2(F_0)$. The {\em label} $\ell(a,b)$ of an edge $\{a,b\}$ of $K_n$ is the index of the subgraph the edge belongs to, so the label of the edges of $F_i$ are $i$. As $\sigma_n$ brings $F_0$ to $F_1$ to $F_2$ and back to $F_0$ we must have $\ell(a+1,b+1)\equiv\ell(a,b)+1$ for all $a$ and $b$, where the congruence is modulo $3$ (and, as noted above, the vertices are understood modulo $n$). With the same convention we have the more general congruence for any edge $\{a,b\}$ and integer $i$:
\begin{equation}\label{shift}
\ell(a+i,b+i)\equiv\ell(a,b)+i\pmod3.
\end{equation}
\begin{defi}
The {\em defining sequence} of the $\sigma_n$-partition $\{F_0,F_1,F_2\}$ is $a_1,a_2,\dots,a_m$, where $m=\lfloor n/2\rfloor$ and $a_j=\ell(0,j)$. By the congruence above, this sequence determines all other labels and thus the entire $\sigma_n$-partition. On the other hand, it is easy to see that (as $n$ is divisible by $3$) every sequence of length $\lfloor n/2\rfloor$ over the alphabet $\{0,1,2\}$ is a defining sequence of a $\sigma_n$-partition. This is
analogous to the case of self-complementary graphs, cf. \cite{english, Ringel, Sachs}. By symmetry, we may and will often assume that $\ell(0,1)=0$, that is, the defining sequence starts with $a_1=0$. This can be achieved by shifting the $\sigma_n$-partition by $\sigma_n$ or $\sigma_n^2$.
\end{defi}
In the smallest $n=3$ case, there is just one $\sigma_3$-partition and we have already seen that it has no transitive $\sigma_3$-orientation. Let us look at the next case $n=6$ a bit closer. By the
foregoing, there are $3^2=9$ $\sigma_6$-partitions to consider according to the labeling of the edges $\{0,2\}$ and $\{0,3\}$.
The corresponding graphs $F_0$ are depicted in Figure~1. It turns out that transitive $\sigma_6$-orientations exist in
exactly four of the nine cases. (We have indicated such an orientation in Figure~1 whenever it exists.)
Notice that the $F_0$ is simply a path on the six vertices in four cases
but a transitive $\sigma_6$-orientation exists for only two of
them. (The truth of this statement will follow from the results of the next
section.) Thus, in spite of the isomorphism of these four graphs they behave
differently according to the different effect of permutation $\sigma_6$ on them.
\begin{figure}
\caption{Four of the nine possible $3$-partitions of $K_6$ can be oriented as
required. In the remaining five cases such orientations do not exist.}
\label{fig:kilenc}
\end{figure}
\section{The standard orientation}\label{sect:standard}
We want to decide whether a given $\sigma_n$-partition $P$ has a transitive
$\sigma_n$-orientation. For our notation including the labeling of edges and
the definition of the defining sequence see the previous section. We will
describe a transitive $\sigma_n$-orientation with an ordering
$\tau(1),\tau(2),\dots,\tau(n)$ on the vertices. We say that an orientation is
{\em consistent} with $\tau$ if all edges point towards the vertex that come
later in the order. Clearly, the transitive orientation of $K_n$ and the
ordering it is consistent with mutually determine each other. Recall that a
transitive orientation $T$ is a transitive $\sigma_n$-orientation of our
$\sigma_n$-partition $P$ if and only if the orientation of no label-$0$ or
label-$1$ edge is reversed by $\sigma_n$.
\begin{thm}\label{standard}
If $n=2m$ and the defining sequence $a_1\dots a_m\in\{0,1,2\}^m$ of a
$\sigma_n$-partition satisfies that for every $j\in\{1,\dots,m-1\}$ either
$a_{j+1}=a_j$ or $a_{j+1}\equiv a_j+1 \pmod3$, then there exists a transitive $\sigma_n$-orientation for this $\sigma_n$-partition.
\end{thm}
\proof
The proof is an extension of the argument given by Gy\'arf\'as \cite{GyA} for our Theorem~\ref{thm:selfc}. We give a linear order $\tau$ of the vertices $0,1,\dots,n-1$ and show that orienting the edges consistently with this order gives a transitive $\sigma_n$-orientation.
Let us first recall our assumption that $a_1=0$. This can be achieved by appropriately relabeling the vertices. The relabeling changes the defining sequence but does not affect the condition in the theorem. Now we define $\tau$. We set $\tau(1)=0$ and declare that $\tau$ will have the property, that for any $i$, the set of vertices $A_i:=\{\tau(1), \tau(2),\dots,\tau(i)\}$ forms a consecutive arc of the cycle formed by the vertices $0,1,\dots,n-1$, i.e., it is equal to $\{j_i+1,\dots,j_i+i\}$ for some $j_i\in \{n~-~i,n~-~i~+~1,\dots,n~-~1\}$. Recall that the names of the vertices are understood modulo $n$. Now
$\tau(i+1)$, that is the unique element of $A_{i+1}\setminus A_i$ is either
$j_i$ or $j_i+i+1$ for every $i$.
Thus $\tau$ is determined if we give a rule for deciding which of the two elements $j_i$ and $j_i+i+1$ should be taken as $\tau(i+1)$ if $i<n-1$. (No rule is needed for $i=n-1$ as then $j_i=j_i+i+1$ is the only vertex outside $A_i$.) This choice for $\tau(i+1)$ depends on the label of the edge $\{j_i,j_i+i+1\}$. If $\ell(j_i,j_i+i+1)=0$, then we set $\tau(i+1)=j_i$ making $j_{i+1}=j_i-1$. If $\ell(j_i,j_i+i+1)=2$, then we set $\tau(i+1)=j_i+i+1$ making $j_{i+1}=j_i$. We claim that the third possibility, namely $\ell(j_i,j_i+i+1)=1$ will not happen for any $1\le i<n-1$.
First we show this last statement by induction. Note that all congruences are modulo $3$. The base case is all right as
$\tau(1)=0$ and $\ell(n-1,1)\equiv a_2-1$, see Equation~(\ref{shift}). By the assumption on the defining sequence this is either $a_1-1\equiv2$ or $(a_1+1)-1=0$. Now assume the statement to be true for the edge $\{j_{i-1},j_{i-1}+i\}$, and we show that it is also true for $\{j_i,j_i+i+1\}$. By Equation(\ref{shift}), we have $\ell(j_{i-1},j_{i-1}+i)\equiv\ell(0,i)+j_{i-1}$ and $\ell(j_i,j_i+i+1)\equiv\ell(0,i+1)+j_i$. We have $j_i=j_{i-1}-1$ if $\tau(i)=j_{i-1}$, i.e., if $\ell(j_{i-1},j_{i-1}+i)=0$, while otherwise this label is $2$, so we have $j_i=j_{i-1}$. Therefore, we can formulate the congruence:
$$j_i\equiv j_{i-1}-\ell(j_{i-1},j_{i-1}+i)-1\equiv j_{i-1}-(\ell(0,i)+j_{i-1})-1=-\ell(0,i)-1.$$
We also have:
\begin{equation}\label{bar}
\ell(j_i,j_i+i+1)\equiv\ell(0,i+1)+j_i\equiv\ell(0,i+1)-\ell(0,i)-1.
\end{equation}
For $1\le i\le n/2-1$ we simply have $\ell(0,i+1)=a_{i+1}$ and $\ell(0,i)=a_i$, so
\begin{equation}\label{kicsi}
\ell(j_i,j_i+i+1)\equiv a_{i+1}-a_i-1,
\end{equation}
so by assumption it cannot be $1$. If $n/2\le i<n$ we use Equation~(\ref{shift}) again to see that $\ell(0,i)=\ell(i,0)\equiv\ell(0,n-i)+i=a_{n-i}+i$ and similarly, $\ell(0,i+1)\equiv a_{n-i-1}+i+1$. Therefore \begin{equation}\label{nagy}
\ell(j_i,j_i+i+1)\equiv a_{n-i-1}-a_{n-i},
\end{equation}
which cannot be $1$ either by the same assumption.
Note that we used the fact that $n$ is even. For $n$ odd and $i=(n-1)/2$ we would have $\ell(j_i,j_i+i+1)\equiv\ell(0,i+1)-\ell(0,i)-1\equiv(a_i-i)-a_i-1=-i-1\equiv1$.
\par\noindent
We need to show that the orientation consistent with the order $\tau$ is a transitive $\sigma_n$-orientation. As noted above, for this we have to show that $\sigma_n$ reverses the orientation only of edges of label $2$. Equivalently, if an edge $\{u,v\}$ is oriented from $u$ to $v$ and
it has label $1$ or $2$ then the edge $\{u-1,v-1\}$ is oriented from
$u-1$ to $v-1$. Assume $\{u,v\}$ is oriented from $u$ to $v$, that is, $\tau^{-1}(u)<\tau^{-1}(v)$. We distinguish cases according to the order of $u$ and $v$. Note that while in most formulas we consider the vertices as residue classes modulo $n$ (and thus equality really means congruence modulo $n$) in inequalities the vertices are treated as integers between $0$ and $n-1$.
In the simplest case we have $0<u<v$. In this case $u-1\in A_{\tau^{-1}(u)}$ while
$v\notin A_{\tau^{-1}(u)}$ and either $(v-1)\notin A_{\tau^{-1}(u)}$ or
$v-1=u$. In both cases we have $\tau^{-1}(u-1)<\tau^{-1}(v-1)$
implying that the edge $\{u-1,v-1\}$ is oriented from $u-1$ to $v-1$ as we
need.
If $u=0$ and $\{u-1,v-1\}$ is oriented toward $u-1$, then $\tau^{-1}(u-1)>\tau^{-1}(v-1)$ and therefore $\ell(u-1,v-1)=2$ implying $\ell(u,v)=0$ and therefore it does
not matter that $\{u-1,v-1\}$ is not oriented from $u-1$ to $v-1$.
Finally assume $v<u$. Note that $v>0$ as otherwise the edge $\{u,v\}$ could not be directed toward $v$. If $\{u-1,v-1\}$ is not oriented from $u-1$ to
$v-1$, then the arc $\{u,u+1,\dots,v-1\}$ is either $A_{\tau^{-1}(u)}$ or
$A_{\tau^{-1}(v-1)}$. In the former case our rule implies that $\ell(u,v)=0$,
in the latter case it implies $\ell(u-1,v-1)=2$ and thus again $\ell(u,v)=0$
in which case we have no problem.
\par\noindent
This proves that our rule gives a transitive $\sigma_n$-orientation and completes the
proof of the theorem.
${\cal B}ox$
\par\noindent
Note that the linear order obtained on the vertex set of $K_n$ by the
orientation in the proof above has some special properties. To formulate them
we introduce the following notions.
\begin{defi}
Let $\tau(1)\dots\tau(n)$ be an ordering of the numbers $0,1,\dots,n-1$. We say that $j\in\{0,1,\dots,n-1\}$
is a {\em local minimum} in this order if $j$ precedes both $j-1$ and $j+1$
(addition is meant modulo $n$), that is
$\tau^{-1}(j)<\tau^{-1}(j-1)$ and $\tau^{-1}(j)<\tau^{-1}(j+1)$.
Similarly, $j\in\{0,1,\dots,n-1\}$
is a {\em local maximum} if $j$ is preceded by both $j-1$ and $j+1$, that is
$\tau^{-1}(j)>\tau^{-1}(j-1)$ and $\tau^{-1}(j)>\tau^{-1}(j+1)$.
We call $\tau$ {\em bitonic} if it has a unique local minimum and a unique local maximum. We call a transitive $\sigma_n$-orientation of a $\sigma_n$-partition {\em standard} if it is consistent with a bitonic ordering of the vertices.
\end{defi}
\par\noindent
\begin{prop}\label{felez}
The transitive $\sigma_n$-orientations given in the proof of Theorem~\ref{standard} are standard. The unique local minimum of the corresponding ordering is at $0$, while the unique local maximum is at $n/2$.
\end{prop}
\proof
The fact that $\tau$ given in the proof is bitonic and thus the transitive $\sigma_n$-orientations are standard follows immediately from the construction. It is also clear that $\tau(1)=0$ is the local minimum and we need only prove that the local maximum (that is $\tau(n)$) is $n/2$.
Recall from the proof of Theorem~\ref{standard} that $A_i=\{\tau(1),\dots,\tau(i)\}$ is a consecutive interval in the cycle formed by the $i$ vertices from $j_i+1$ to $j_i+i$ for all $1\le i\le n$. The sequence $(A_i)_{i=1}^n$ starts at $A_1=\{0\}$ and we obtain $A_{i+1}$ from $A_i$ by extending $A_i$ with $\tau(i+1)$ at one end of this interval. The label $\ell(j_i,j_i+i+1)$ (which is never $1$) determines which end we place $\tau(i+1)$, namely if the label is $0$ we chose one end, while if it is $2$ we chose the other end. For $1\le i\le n/2-1$ we have $\ell(j_i,j_i+i+1)\equiv a_{i+1}-a_i-1$ by Equation~(\ref{kicsi}) and $\ell(j_{n-i-1},j_{n-i-1}+n-i)\equiv a_i-a_{i+1}$ by Equation~(\ref{nagy}). This means that $\ell(j_i,j_i+i+1)+\ell(j_{n-i-1},j_{n-i-1}+n-i)\equiv2$, so one of these labels must be $0$ and the other $2$ and thus
we extend the interval $A_i$ on one end to get $A_{i+1}$ while we extend $A_{n-i-1}$ on the other end to obtain $A_{n-i}$. This partitions the $n-2$ extension steps bringing $A_1$ to $A_{n-1}$ into $n/2-1$ pairs and shows that we use $n/2-1$ extensions of the interval at either side. Thus $n/2$ (the vertex in distance $n/2$ from $A_1=\{0\}$ in either direction) must be the only vertex outside $A_{n-1}$ and therefore we have $\tau(n)=n/2$ as stated.
${\cal B}ox$
The following proposition is a sort of converse of the previous one.
\begin{prop}\label{converse-standard}
If there exists a standard transitive $\sigma_n$-orientation for a $\sigma_n$-partition, then the conditions of Theorem~\ref{standard} are satisfied, namely the number $n$ of vertices is even and for the defining sequence $a_1\dots a_{n/2}$ of the $\sigma_n$-partition either $a_{j+1}=a_j$ or $a_{j+1}\equiv a_j+1 \pmod 3$ holds for each $1\le j\le n/2-1$. If there exists a standard transitive $\sigma_n$-orientation for a $\sigma_n$-partition, then it is unique up to shifts of the automorphism $\sigma_n^3$.
\end{prop}
\proof
Consider a standard transitive $\sigma_n$-orientation $T$ for a $\sigma_n$-partition. Without loss of generality we may assume that the defining sequence of the $\sigma_n$-partition starts with $a_1=0$. Consider the bitonic ordering $\tau$ of the vertices consistent with $T$. The first element $a=\tau(1)$ is clearly the local minimum. All edges are directed away from $a$, so $\sigma$ reverses the orientation of the edge $\{a-1,a\}$, as it brings it to $\{a,a+1\}$. Therefore we have $\ell(a,a+1)=0$ and thus $a\equiv0\pmod3$. We may and will assume $a=0$ as this can be achieved with a shift of a suitable power of the automorphism $\sigma_n^3$.
As $\tau$ is bitonic, the set $A_i=\{\tau(1),\dots,\tau(i)\}$ must be an interval $\{j_i+1,\dots,j_i+i\}$ along the cycle formed by the vertices. Clearly, $\tau(i+1)$ is either $j_i$ or $j_i+i+1$. We show the uniqueness of $\tau$ by observing that the value of $\tau(i+1)$ depends on the label of the edge $e=\{j_i,j_i+i+1\}$ exactly as in the construction in the proof of Theorem~\ref{standard}, namely $\tau(i+1)=j_i$ if this label is $0$ and $\tau(i+1)=j_i+i+1$ if the label is $2$ and the label of $e$ cannot be $1$. Indeed, if $\tau(i+1)=j_i+i+1$, then $\sigma_n$ reverses the orientation of the edge $e$, so its label is $2$, but if $\tau(i+1)=j_i$, then $\sigma_n^{-1}$ reverses the orientation of $e$, so its label must be $0$.
From the rule established above we can derive Equation~(\ref{bar}) just as in the proof of Theorem~\ref{standard}. In case $n$ is odd we can apply this formula to $i=(n-1)/2$ and using also Equation~(\ref{shift}) we obtain $\ell(j_i,j_i+i+1)\equiv\ell(0,i+1)-\ell(0,i)-1=(\ell(0,i)+i+1)-\ell(0,i)-1\equiv i\equiv1$ contradicting that we saw that the label of the edge $\{j_i,j_i+i+1\}$ cannot be $1$. This proves that $n$ is even.
From Equation~(\ref{bar}) we can deduce Equation~(\ref{kicsi}) for $1\le i\le n/2-1$. Using that the label of the edge $\{j_i,j_i+i+1\}$ cannot be $1$ this implies that $a_{i+1}=a_i$ or $a_{i+1}\equiv a_i+1$ finishing the proof of the proposition.
${\cal B}ox$
\subsection{Several parts}
In this subsection we extend the earlier results to partitions to more than three parts. The extensions are straightforward, the proofs carry over almost verbatim. We just list the results here and show how they apply to decompositions into Hamiltonian paths and cycles. Note that in the discussion below we also allow the case when the number of parts is $k=2$.
\begin{defi}
Let $k$ and $n$ be integers larger than $1$ and (as before) let $\sigma_n$ be the cyclic permutation on the set $[n]=\{0,1,\dots,n-1\}$ bringing $i$ to $i+1$. Recall that elements of $[n]$ are understood modulo $n$. Let us call a partition of the edge set of the complete graph on the vertex set $[n]$ into the subgraphs $F_0,\dots,F_{k-1}$ a {\em $\sigma_n$-$k$-partition} if $\sigma_n(F_d)=F_{d+1}$ for $0\le d<k-1$. This implies that $n$ is divisible by $k$, further if $k$ is even, then $n$ is divisible by $2k$. It also implies that $\sigma_n(F_{k-1})=F_0$ and that $\sigma_n^k$ is an automorphism of all the graphs $F_d$. We say that the {\em label} of an edge $e$ of $F_d$ is $\ell(e)=d$.
The defining sequence of this partition is the sequence $a_1,\ldots,a_{\lfloor n/2\rfloor}$, where $a_i=\ell(\{0,i\})$. The defining sequence uniquely determines the $\sigma_n$-$k$-partition, namely $F_d$ consists of the edges
$\{b,b+i\}$ for which $b\equiv d-a_i\pmod k$. This is indeed a $\sigma_n$-$k$-partition for any sequence $a_1,\dots,a_{\lfloor n/2\rfloor}$ over the letters $0,1,\dots,k-1$ if the divisibility conditions are satisfied by $n$ and $k$.
We call a transitive orientation $T$ of $K_n$ a {\em transitive $\sigma_n$-orientation} of a $\sigma_n$-$k$-partition if $\sigma_n(\vec{F_d})=\vec{F}_{d+1}$ for $0\le d<k-1$, where $\vec F_d$ is the orientation of $F_d$ obtained as a subgraph of $T$. A transitive orientation $T$ satisfies this condition if and only if $\sigma_n$ reverses the orientation of no edges of $F_d$ with $d\ne k-1$. We call a transitive $\sigma_n$-orientation {\em standard} if it is consistent with a bitonic ordering of the vertices.
\end{defi}
The common generalization of Theorems~\ref{thm:selfc} and \ref{standard} is as
follows.
\begin{thm}\label{genstandard}
There exists a standard orientation for a $\sigma_n$-$k$-partition if and only if $n$ is divisible by $2k$ and the defining sequence $a_1a_2\dots a_{n/2}$ of the partition satisfies that for every $1\le j\le n/2-1$ either $a_{j+1}=a_j$ or $a_{j+1}\equiv a_j+1\pmod k$. If a standard orientation exists, then it is unique up to shifts of $\sigma^k$ and the corresponding bitonic ordering of the vertices satisfies that the first and last elements in the ordering (the unique local minimum and maximum) differ by $n/2$.
\end{thm}
\par\noindent
\begin{remark}
It is straightforward to see that Theorem~\ref{genstandard} generalizes
Theorem~\ref{standard}. Note that Theorem~\ref{thm:selfc} is also implied by
Theorem~\ref{genstandard} since in case of two self-complementary graphs, that
is, when $k=2$, the condition $a_{j+1}\equiv a_j\ \ {\rm or}\ \ a_j+1\pmod k$
is automatically satisfied as $a_{j+1}$ cannot take any value other than $0$
or $1$.
\end{remark}
We formulate the following simple generalization of the trivial observation that the (only) $\sigma_3$-partition has no transitive $\sigma_3$-orientation.
\begin{prop}\label{trivi}
If there exists a transitive $\sigma_n$-orientation for a $\sigma_n$-$k$-partition, then $n\ge2k$ and $n\ne3k$. If $n=2k$, then any transitive $\sigma_n$-orientation of a $\sigma_n$-$k$-partition is standard.
\end{prop}
\proof
From the existence of this partition we know that $k$ divides $n$. Note that $\sigma_n$ reverses the orientation of an even number of the $n$ edges $\{i,i+1\}$ in any orientation $T$ of $K_n$. But if $k=n$, then only one of these edges has label $k-1$, so if $T$ is a transitive $\sigma_n$-orientation, then the orientation of none of these edges are reversed. But then they form a directed cycle, so $T$ is not transitive.
If $n=2k$ or $3k$, then two or three of these $n$ edges have label $k-1$, so exactly two of them have to be reversed by $\sigma_n$ to avoid the directed cycle. This makes the transitive $\sigma_n$-orientation standard. But we know from Theorem~\ref{genstandard} that if a $\sigma_n$-$k$-partition admits a standard orientation, then $n$ is divisible by $2k$, so $n=3k$ is not an option.
${\cal B}ox$
Complete graphs of odd order can be decomposed into Hamiltonian cycles. This
is a classical result as mentioned in Adrian Bondy's chapter \cite{Bondy} of
the {\em Handbook of Combinatorics}. (It is added there that ``one such
construction, due to a Monsieur Walecki, is described in the book by Lucas
(1891, pp. 161--164)'', cf. \cite{Lucas}.)
This result extends to the decomposition of odd order transitive tournaments to identically oriented Hamiltonian cycles. This decomposition can be found directly, but it can also be arrived at by applying our result to a certain decomposition of the non-oriented complete graph as shown below.
\begin{cor}\label{Hpaths}
If $n$ is even, then we can decompose $T_n$ into $\frac{n}{2}$ alternatingly
oriented Hamiltonian paths.
\end{cor}
\proof
Let $F_0$ be the path defined as
$0,1,n-1,2,n-2,\dots,i,n-i,(i+1),\dots,(\frac{n}{2}+1),\frac{n}{2}$. It is easy to see that the graphs
$F_d:=\sigma_n^d(F_0)$ for $d=0,1,\dots,n/2-1$
partition the edge set of $K_n$ into $\frac{n}{2}$ Hamiltonian paths. This is a $\sigma_n$-$n/2$-partition.
One can also readily check that $\{0,j\}\in E(F_{\lfloor j/2\rfloor})$ holds for every
$j\in\{1,\dots,n-1\}$. This means that the defining sequence of this
$\sigma$-$n/2$-partition is $a_1,\dots,a_{n/2}$ with $a_i=\lfloor
i/2\rfloor$. By Theorem~\ref{genstandard} there is a standard orientation for
this partition that is unique up to a shift with $\sigma_n^{n/2}$. It is not
hard to check that the standard orientation orients the Hamiltonian paths in
this partition alternatingly, that is, each vertex will be a source or a sink of each Hamiltionian path. This construction is illustrated for
$n=6$ by the oriented path in the very central picture of Figure~1.
${\cal B}ox$
\begin{cor}\label{Hcycles}
If $n$ is odd, then we can decompose $T_n$ into $\frac{n-1}{2}$ isomorphically
oriented Hamiltonian cycles.
\end{cor}
\proof
Consider the decomposition of $T_{n-1}$ given in Corollary~\ref{Hpaths} on
vertices labeled by $[n-1]$. Add the extra vertex $v$ and
connect it to the two endpoints of each of the Hamiltonian paths thus extending
them to Hamiltonian cycles. Orient all edges incident to $v$ away from $v$. These isomorphically oriented Hamiltonian cycles decompose the transitive tournament on $n$ vertices.
${\cal B}ox$.
\begin{remark}
The orientation of the Hamiltonian cycles in our decomposition in
Corollary~\ref{Hcycles} is such that all but one of the vertices is either a source or a sink. This kind of orientation of odd cycles is called
alternating in \cite{KPS}, where it is shown that these oriented versions of
odd cycles have maximal Sperner capacity. In the special case of $n=5$ this
orientation already appeared in \cite{GGKS}, where it was observed that its
Sperner capacity is $\sqrt{5}$, that is, it achieves the Shannon capacity of the
underlying undirected graph which is $C_5$, whose capacity was determined in
the celebrated paper by Lov\'asz \cite{LL}. This observation was the starting
point of our investigations in \cite{SaSi}.
In Section~\ref{sect:smallex} we explained how to find transitive $\sigma$-orientations for $\sigma$-partitions if the permutation $\sigma$ is not cyclic. First we solve the restriction of the problem for the domain of each cycle in the cycle decomposition of $\sigma$, then extend the obtained orientations by orienting the edges between distinct cycles consistently with a linear ordering of these cycles. We used the same strategy here for the two cycles of the permutation $\sigma$ that has $v$ as a fixed point and acts on $[n-1]$ as $\sigma_{n-1}$. The decomposition has $v$ as the domain of a trivial cycle and $[n-1]$ as the domain of $\sigma_{n-1}$. Had we defined these notions for arbitrary permutations (not just for cyclic ones), we could call the decomposition of the complete graph in Corollary~\ref{Hcycles} a $\sigma$-$(n-1)/2$-partition of $K_n$ and the orientations of the Hamiltonian cycles would form a transitive $\sigma$-orientation of this partition.
\end{remark}
\section{Non-standard orientations}
The results in the previous section may make one hope that the conditions of Theorem~\ref{standard} are not just sufficient but also necessary for a $\sigma_n$-partition to have transitive $\sigma_n$-orientations. This is not the case.
In this section we give some sufficient conditions that go beyond the ones in Theorems~\ref{standard} and \ref{genstandard} .
Our construction takes a transitive $\sigma_m$-orientation for a
$\sigma_m$-$k$-partition and uses that to get transitive
$\sigma_n$-orientations of related $\sigma_n$-$k$-partitions. Here $n$ is a multiple of $m$. Even for some standard $\sigma_m$-$k$-orientations, the resulting transitive $\sigma_n$-orientations are not always standard and in some cases the $\sigma_n$-$k$-partitions do not have standard orientations at all.
\begin{defi}\label{defi:blowup}
Let $k>1$ and $m$ be such that $\sigma_m$-$k$-partitions exist (that is, $m$ is a multiple of $k$ and if $k$ is even $m$ is also a multiple of $2k$). Let $n$ be a multiple of $m$. We call the $\sigma_n$-$k$-partition $P$ with defining sequence $a_1,\dots,a_{\lfloor n/2\rfloor}$ a {\em blow-up} of the $\sigma_m$-$k$-partition $Q$ with defining sequence $b_1,\ldots,b_{\lfloor m/2\rfloor}$ if $a_i=b_{(i\bmod m)}$ whenever $1\le i\le\lfloor n/2\rfloor$ and $i$ is not divisible by $m$. For this to make sense even if $i\bmod m>\lfloor m/2\rfloor$ we extend the sequence $b_i$ by setting $b_i=(b_{m-i}+i)\bmod k$ for $\lfloor m/2\rfloor<i<m$. This makes $b_i$ the label of the edge $\{0,i\}$ in the $\sigma_m$-$k$-partition $Q$ for any $i$. Note that we have no requirement for the value of $a_i$ if $i$ is divisible by $m$, so $P$ is not determined by $Q$ and $n$.
\end{defi}
Notice that if the $\sigma_m$-$k$-partition $Q$ has a standard orientation, then its defining sequence satisfies the requirements of Theorem~\ref{genstandard} and therefore the defining sequence $a_1,\dots,a_{n/2}$ of its blow-up $P$ also satisfies $a_{i+1}=a_i$ or $a_{i+1}\equiv a_i+1\pmod k$ whenever neither $i$
nor $i+1$ is divisible by $m$. However, by the free choice of the value $a_i$
whenever $i$ is divisible by $m$, this property need not hold for
the other indices, thus $P$ may violate the
conditions of Theorem~\ref{genstandard}. Nevertheless, as we will show, $P$ admits a transitive $\sigma_n$-orientation in this case. We call the transitive orientation constructed in the next theorem the {\em blow-up} of the transitive $\sigma_m$-orientation of $Q$.
\begin{thm}\label{thm:blowup}
If the $\sigma_n$-$k$-partition $P$ is a blow-up of the $\sigma_m$-$k$-partition $Q$ and $Q$ admits a transitive $\sigma_m$-orientation, then $P$ admits a transitive $\sigma_n$-orientation.
\end{thm}
\par\noindent
\proof
Let $T$ be the transitive $\sigma_m$-orientation of $Q$ we assumed to exist and let $\tau$ be the ordering of the vertex set $[m]$ that $T$ is consistent with. The theorem claims that $P$ has transitive $\sigma_n$-orientation $T'$. We construct $T'$ by finding the ordering $\tau'$ on the vertex set $[n]$ that $T'$ is consistent with. The only requirement we have to satisfy is that all edges of $T'$ whose orientation $\sigma_n$ reverses should have label $k-1$.
Let us set $d=n/m$ and for $i\in[m]$, let $H_i=\{jm+i\mid j\in[d]\}$. Each of these $m$ sets has $d$ elements and together they partition $[n]$. The ordering $\tau'$ starts with the elements of $H_{\tau(1)}$ followed by the elements of $H_{\tau(2)},\dots,H_{\tau(m-1)}$. The order within the sets $H_i$ will be specified later. We call an edge of $K_n$ an {\em outer edge} if it connects vertices from distinct sets $H_i$ and $H_j$, otherwise it is an {\em inner edge}. The orientation of the outer edges in $T'$ are not influenced by the order within the sets $H_i$. namely for $a\in H_i$ and $b\in H_j$ ($i\ne j$) the orientation of the edge $\{a,b\}$ in $T'$ is determined by the orientation of the of edge $\{i,j\}$ in $T$. This means that $\sigma_n$ reverses the orientation of the edge $\{a,b\}$ in $T'$ if and only if $\sigma_m$ reverses the orientation of the edge $\{i,j\}$ in $T$. Our definition of a blow-up ensures that the label of the edge $\{a,b\}$ for the partition $P$ is the same as the label of the edge $\{i,j\}$ for the partition $Q$. Therefore $\sigma_n$ reverses the orientation of outer edges in $T'$ only if their label is $k-1$.
To finish the proof we have to specify the ordering $\tau'$ within the sets $H_i$ in such a way that the same can be said about inner edges of $T'$: $\sigma_n$ reserves the orientation of them only if their label is $k-1$.
For $i\in[m]$ let $\tau_i(1),\tau_i(2),\dots,\tau_i(d)$ be an ordering of $[d]$ to be specified later and let us say that $\tau'$ orders the elements of $H_i$ in the following order: $\tau_i(1)m+i,\tau_i(2)m+i,\dots,\tau_i(d)m+i$.
Consider an inner edge $e=\{a,b\}$, with $a,b\in H_i$, $i\in[m]$. We have $a=Am+i$ and $b=Bm+i$ for some $A,B\in[d]$ and the orientation of $e$ in $T'$ is determined by the order of $A$ and $B$ in $\tau_i$. In case $i\ne m-1$ we have $\sigma_n(e)=\{a+1,b+1\}$, $a+1,b+1\in H_{i+1}$ and the orientation of $\sigma_n(e)$ is determined by the order of $A$ and $B$ in $\tau_{i+1}$. Thus, $\sigma_n$ reverses the orientation of $e$ in $T'$ if and only if $A$ and $B$ are in different order in $\tau_i$ and in $\tau_{i+1}$. We must make sure that this only happens if the label of $e$ is $k-1$.
The situation is a bit different for inner edges $e=\{a,b\}$ with $a,b\in H_{m-1}$. We still have $\sigma_n(e)=\{a+1,b+1\}$ (recall that these vertices are understood modulo $n$). But now $a+1,b+1\in H_0$ and $a+1=(A+1)m$, $b+1=(B+1)m$. Thus, $\sigma_n$ reverses the orientation of $e$ if and only if $A$ and $B$ are in different order in $\tau_{m-1}$, then $A+1$ and $B+1$ are in $\tau_0$. Here both $A+1$ and $B+1$ are understand modulo $d$. We will make sure that this only happens if the label of $e$ is $k-1$.
We will now specify the orderings $\tau_i$. As explained above, this finishes the proof if we can show the following two properties:
\begin{description}
\item{(a)} If for some $0\le i<m-1$ and $A\ne B\in[d]$ the order of $A$ and $B$ is different in $\tau_i$ and $\tau_{i+1}$, then we have $\ell(Am+i,Bm+i)=k-1$.
\item{(b)} If $A$ precedes $B$ in $\tau_{m-1}$ but $(B+1)\bmod d$ precedes $(A+1)\bmod d$ in $\tau_0$, then we have $\ell(Am+m-1,Bm+m-1)=k-1$.
\end{description}
We set $\tau_0$ to be the order: $0,1,d-1,2,d-2,\dots,\lceil d/2\rceil$. Note that the pair of elements $j$ and $d-j$ are consecutive for all $1\le j<d/2$. For $0\le i<k$ we obtain $\tau_{i+1}$ from $\tau_i$ by swapping the order of $j$ and $d-j$ for each $j$ satisfying $\ell(jm+i,(d-j)m+i)=k-1$. In this way we maintain that the $j$ and $d-j$ are consecutive in $\tau_i$ for all $0\le i\le k$ and $1\le j<d/2$. Also, with this rule condition (a) is satisfied for $0\le i<k$. For any $i$ and $j$ as above we have $\ell(jm+i,(d-j)m+i)\equiv\ell(jm,(d-j)m)+i\pmod k$. Therefore, for any such $j$, the label will be $k-1$ for exactly one of the indices $0\le i<k$ and thus $\tau_k$ will have all the pairs $(j,d-j)$ swapped. Namely, $\tau_k$ is the order $0,d-1,1,d-2,2,d-3,\ldots,\lfloor d/2\rfloor$.
Proposition~\ref{trivi} tells us that $m\ge2k$. Let us assume for now that $m>2k$. We will come back to the case $m=2k$ later. Observe that $j$ and $d-j-1$ are consecutive in $\tau_k$ for any $0\le j\le d/2-1$. For $k\le i<2k$ we obtain $\tau_{i+1}$ from $\tau_i$ by swapping the order of $j$ and $d-j-1$ for each $j$ satisfying $\ell(jm+i,(d-j-1)m+i)=k-1$. In this way we maintain that the vertices $j$ and $d-j-1$ are consecutive in $\tau_i$ for all $k\le i\le2k$ and $0\le j\le d/2-1$. Also, this rule makes condition (a) satisfied for $k\le i<2k$. Just as before, for any $j$ as above the label condition is satisfied for exactly one index $k\le i<2k$ and thus $\tau_{2k}$ will have all the pairs $(j,d-j-1)$ swapped. Namely, $\tau_{2k}$ is the order $d-1,0,d-2,1,d-3,2,\ldots\lceil d/2\rceil-1$.
We set $\tau_i=\tau_{2k}$ for $2k<i<m$. This makes condition (a) hold vacuously for $2k\le i\le m-1$ as $\tau_i=\tau_{i+1}$. Condition (b) is also satisfied vacuously, since $A$ precedes $B$ in $\tau_{m-1}=\tau_{2k}$ if and only if $(A+1)\bmod d$ precedes $(B+1)\bmod d$ in $\tau_0$. This is so because $\tau_0$ can be obtained from $\tau_{2k}$ by replacing each element $j$ by $(j+1)\bmod d$. This finishes the proof of the theorem in the case $m>2k$.
It remains to consider the case $m=2k$. We define the orders $\tau_i$ for $k<i\le2k$ the same way as above. We do not use $\tau_{2k}$ in the definition of $\tau'$, but we will use it in our argument below.
Condition (a) is satisfied as before. But now condition (b) is not vacuous. We still have that $A$ precedes $B$ in $\tau_{2k}$ if and only if $(A+1)\bmod d$ precedes $(B+1)\bmod d$ in $\tau_0$, so if $A$ precedes $B$ in $\tau_{2k-1}$ but $(B+1)\bmod d$ precedes $(A+1)\bmod d$ in $\tau_0$ as called for in condition (b), then $A$ and $B$ appear in different order in $\tau_{2k-1}$ and $\tau_{2k}$ and therefore $\ell(Am+m-1,Bm+m-1)=k-1$. This makes condition (b) satisfied and finishes the proof of the theorem.
${\cal B}ox$
To illustrate Theorem~\ref{thm:blowup} we show a $\sigma_n$-partition that admits a transitive $\sigma_n$-orientation but does not admit a standard one. We take $n=12$ and consider the $\sigma$-partition $P$ with defining sequence $000121$. Recall that this is a partition of the edge set of $K_{12}$ to three isomorphic subgraphs and it has no standard orientation by Proposition~\ref{converse-standard}. But it is a blow-up of the $\sigma_6$-partition $Q$ with defining sequence $000$. $Q$ has a transitive $\sigma_6$-orientation, even a standard one by Theorem~\ref{standard}. The first part of $Q$ and its orientation is depicted in the first illustration on Figure~1. By Theorem~\ref{thm:blowup} $P$ has a transitive $\sigma_{12}$-orientation. For example orienting the edges consistent with the following ordering of the vertices gives a transitive $\sigma_{12}$-orientation: $0,6,1,7,2,8,11,5,4,10,3,9$.
\section{Complications}
After seeing Theorems~\ref{standard} and \ref{thm:blowup} one might be curious
to know whether they describe all transitive $\sigma_n$-orientations of a
$\sigma_n$-partition, namely if all such orientations are blow-ups of standard
orientations. This is, however, not the case, moreover, there exist
$\sigma_n$-partitions that neither admit a standard orientation nor are
blow-ups of $\sigma_m$-partitions for some $m<n$, yet they do admit a
transitive $\sigma_n$-orientation. We found such examples by computer and do
not see a general pattern that would still suggest a complete characterization.
(We note that our examples in this section all concern the simplest possible case $k=3$ again.)
The $\sigma_{24}$-partition with the defining sequence
$$0,0,0,1,2,0,0,0,1,1,2,1$$
is such an example. The orientation consistent with the following order of the vertices is a transitive $\sigma_{24}$-orientation for this partition:
$$0,1,2,23,22,21,3,9,4,10,20,5,11,8,7,19,6,18,12,13,14,17,16,15.$$
(To check that this defines a transitive $\sigma_{24}$-orientation one has only to verify that the edges whose orientation $\sigma_{24}$ reverses have label $2$. For example, the edge $e=\{4,8\}$ is oriented towards $8$ as $8$ appears later in this sequence than $4$. But $\sigma_{24}(e)=\{5,9\}$ is oriented towards $5$ as $5$ appears later than $9$. So $\sigma_{24}$ reverses the orientation of $e$. Now $\ell(4,8)\equiv\ell(0,4)+4\pmod3$ and $\ell(0,4)$ is the fourth number of the defining sequence, namely $1$, so $\ell(4,8)=2$ as required.)
\section{Necessary conditions}
So far we have seen sufficient conditions for $\sigma_n$-partitions (or $\sigma_n$-$k$-partitions) to posses a transitive $\sigma_n$-orientation. While a complete characterization seems elusive it makes sense to look also for non-trivial necessary conditions.
Here we give a simple such condition.
\begin{thm}\label{prop:kkkz} Let $P$ be a $\sigma_n$-$k$-partition with defining sequence $a_1,\ldots,a_{\lfloor n/2\rfloor}$. Assume $a_1=0$ and let $i$ be an index with $1\le i<\lfloor n/2\rfloor$ such that $a_j\ne k-1$ for $1\le j\le i$. If $P$ has a transitive $\sigma_n$-orientation, then $a_{i+1}\le a_i+1$.
\end{thm}
\proof
Fix a transitive $\sigma_n$-orientation $T$ of $P$. When referring to the orientation of edges we speak about the orientation in $T$.
We call a vertex $m\in[n]$ a {\em leader} if it is divisible by $k$. Notice that if $m$ is a leader, then the label $\ell(m,m+j)$ is $a_j$ for all $1\le j\le\lfloor n/2\rfloor$. (Recall that vertices are always understood modulo $n$.) We call a leader $m$ an {\em in-leader} if the edge $\{m,m+1\}$ is oriented towards $m$, otherwise it is an {\em out-leader}.
We claim that if $m$ is an in-leader, then all the edges $\{m,m+j\}$ for $1\le j\le i+1$ are oriented toward $m$, while if $m$ is an out-leader, then all these edges are oriented away from $m$. By symmetry, it is enough to prove one of the statements. We prove the latter one by induction on $j$. The statement of the claim is assumed for $j=1$. So let $1\le j\le i$ and assume $\{m,m+j\}$ is oriented away from $m$. The label of this edge is $a_j\ne k-1$, so $\sigma_n$ does not reverse its orientation. Therefore the edge $\{m+1,m+j+1\}$ is oriented towards $m+j+1$. As $m$ is an out-vertex the edge $\{m,m+1\}$ is oriented toward $m+1$. To get a transitive orientation $\{m,m+j+1\}$ must therefore be oriented away from $m$ finishing the inductive proof.
Consider the cycle formed by the edges $\{m,m+1\}$ for all $m\in[n]$. If all leader vertices were in-leaders or all of them were out-leaders, then this cycle would be a directed cycle contradicting the transitivity of $T$. So we must have at least one in-leader and also at least one out-leader. We can therefore fix an out-leader $m$ such that the very next leader vertex, namely $m+k$ is an in-leader.
By the claim above (and since $m$ is an out-leader) the edge $\{m,m+i\}$ is oriented towards $m+i$. The label of this edge is $a_i$, so one can apply the permutation $\sigma_n$ to this edge $k-1-a_i$ times without it reversing the orientation. Thus, the edge $\{m+k-1-a_i,m+i+k-1-a_i\}$ is oriented toward $m+i+k-1-a_i$. Recall that $a_i\ne k-1$, so (as $m$ is an out-leader) the edge $\{m+k-2-a_i,m+k-1-a_i\}$ is oriented toward $m+k-1-a_i$. The transitivity of $T$ therefore implies that the edge $e=\{m+k-2-a_i,m+i+k-1-a_i\}$ is oriented toward $m+i+k-1-a_i$. We have $\sigma_n^{a_i+2}(e)=\{m+k,m+k+i+1\}$ and this edge is oriented toward the in-leader $m+k$ by the claim above. We see that $\sigma_n^{a_i+2}$ reverses the orientation of $e$, therefore the label $a_{i+1}$ of $\sigma_n^{a_i+2}(e)$ must be less than $a_i+2$.
${\cal B}ox$
\begin{defi}
Let $a_1,\ldots,a_{\lfloor n/2\rfloor}$ be the defining sequence of a $\sigma_n$-$k$-partition $P$. We say that the sequence {\em halts} at the index $i$ ($1\le i<\lfloor n/2\rfloor$) if $a_{i+1}=a_i$. We say that it {\em steps} at the index $i$ if $a_{i+1}\equiv a_i+1\pmod k$. We say that it {\em jumps} at the index $i$ if it neither halts nor steps there.
We call the $\sigma_n$-$k$-partition with defining sequence $b_1,\ldots,b_{\lfloor n/2\rfloor}$ the {\em dual} of $P$ if $b_i\equiv i-1-a_i\pmod k$ for all $i$.
\end{defi}
Note that if a the defining sequence of a $\sigma_n$-$k$-partition halts at an index $i$, then the defining sequence of its dual steps there and vice versa. The defining sequences of a $\sigma_n$-$k$-partition and its dual jump at the same indices.
We can rephrase Theorem~\ref{genstandard} as follows: A $\sigma_n$-$k$-partition has a standard orientation if and only if its defining sequence does not jump at all.
\begin{thm}\label{dual}
If a $\sigma_n$-$k$-partition admits a transitive $\sigma_n$-orientation, then so does its dual.
If a $\sigma_n$-$k$-partition $P$ admits a transitive $\sigma_n$-orientation and the defining sequence of $P$ jumps at an index $i$, then there is an index $j<i$ where it halts and at least $k-1$ distinct indices $j'<i$ where it steps, or the other way around: there is an index $j<i$ where it steps and at least $k-1$ distinct indices $j'<i$ where it halts. In particular, there is no jump at indices $i\le k$.
\end{thm}
\proof
Let $P$ be the $\sigma_n$-$k$-partition, with parts $F_0,\dots,F_{k-1}$. The dual $Q$ of $P$ can be obtained by first relabeling the vertices, namely switching the label $v$ and $n-v$ for $1\le v<k$, and then considering the parts in reverse order, namely as $F_{k-1},F_{k-2},\dots,F_0$. If a transitive orientation $T$ of $K_n$ is a transitive $\sigma_n$-orientation of $P$, then $T$ (after the relabeling) is also a transitive $\sigma_n$-orientation of $Q$. This proves the first claim in the theorem.
To verify the second claim we assume without loss of generality that the defining sequence $(a_j)$ of $P$ starts with $a_1=0$. Note that this makes the defining sequence $(b_j)$ of its dual $Q$ start with $b_1=0$. Assume that the first jump in $(a_j)$ is at the index $i$, so at indices $1\le j<i$ the sequence halts or steps.
By Theorem~\ref{prop:kkkz} we have $a_{i+1}\le a_i+1$ unless the sequence steps for at least $k-1$ such indices. Further, it has to step for at least one such index, as otherwise we would have $a_j=0$ for all $j\le i$ and $a_{i+1}\le a_i+1=1$ contradicting our assumption that the sequence jumps at $i$.
Note that $i$ is also the first index where the sequence $(b_j)$ jumps, so we similarly have that this sequence steps for at least one index $j<i$ and we have $b_{i+1}\le b_i+1$ unless it steps at $k-1$ or more such indices. To finish the proof it is enough to note that $(a_j)$ steps where $(b_j)$ halts and vice versa and if both step at fewer than $k-1$ indices $j<i$, then $a_i\le a_{i+1}\le a_i+1$ contradicting our assumption that the sequence $(a_j)$ jumps at $i$.
${\cal B}ox$
\noindent
We conclude the paper with two conjectures.
All $\sigma_n$-partitions for which we found a transitive $\sigma_n$-orientation were for even values of $n$. We tried to find such examples with odd $n$ but failed even with a computer. This suggests the following.
\begin{conj}
No $\sigma_n$-partition with $n$ odd has a transitive $\sigma_n$-orientation.
\end{conj}
Although we did not do any computer search for partitions with more than three parts, we still venture the following stronger conjecture:
\begin{conj}
No $\sigma_n$-$k$-partition with $k>1$ and $n$ odd has a transitive $\sigma_n$-orientation.
\end{conj}
\end{document} |
\betagin{document}
\widetildetle[Estimates for parabolic measures]{On the fine properties of parabolic measures\\ associated to strongly degenerate parabolic\\ operators of Kolmogorov type}
\address{Malte Litsg{\aa}rd \\mathcal Deltapartment of Mathematics, Uppsala University\\
S-751 06 Uppsala, Sweden}
\boldsymbol email{[email protected]}
\address{Kaj Nystr\"{o}m\\mathcal Deltapartment of Mathematics, Uppsala University\\
S-751 06 Uppsala, Sweden}
\boldsymbol email{[email protected]}
\thanks{K. N was partially supported by grant 2017-03805 from the Swedish research council (VR)}
\author{Malte Litsg{\aa}rd and Kaj Nystr{\"o}m}
\mathbb R^{nN}ketitle
\betagin{abstract}
\boldsymbol \nuoindentndent
We consider strongly degenerate parabolic operators of the form
\betagin{eqnarray*}
\mathcal L:=\boldsymbol \nuabla_X\cdot(A(X,Y,t)\boldsymbol \nuabla_X)+X\cdot\boldsymbol \nuabla_Y-\partialartial_t
\boldsymbol end{eqnarray*}
in unbounded domains
\betagin{eqnarray*}
\mathbb R^{nN}thcal Omegaega=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{nN}thbb R^{m-1}\widetildemes\mathbb R^{nN}thbb R\widetildemes\mathbb R^{nN}thbb R^{m-1}\widetildemes\mathbb R^{nN}thbb R\widetildemes\mathbb R^{nN}thbb R\mid x_m>\partialsii(x,y,t)\}.
\boldsymbol end{eqnarray*}
We assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix in $\mathbb R^{nN}thbb R^{m}$) and concerning $\partialsii$ and $\mathbb R^{nN}thcal Omegaega$ we assume that $\mathbb R^{nN}thcal Omegaega$ is what we call an (unbounded) Lipschitz domain: $\partialsii$ satisfies a
uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying
the operator $\mathcal L$. We prove, assuming in addition that $\partialsii$ is independent of the variable $y_m$, that $\partialsii$ satisfies an additional regularity condition formulated in terms of a Carleson measure, and additional conditions on $A$, that the associated parabolic measure is absolutely continuous with respect to a surface measure and that
the associated Radon-Nikodym derivative defines an $A_\infty$-weight with respect to the surface measure.\\
\boldsymbol \nuoindentndent
2000 {\boldsymbol em Mathematics Subject Classification.} 35K65, 35K70, 35H20, 35R03.
\boldsymbol \nuoindentndent
\boldsymbol \nuoindentndent
{\it Keywords and phrases: Kolmogorov equation, parabolic, ultraparabolic, hypoelliptic, operators in divergence form, Lipschitz domain, doubling measure, parabolic measure, Carleson measure, $A_\infty$, Lie group.}
\boldsymbol end{abstract}
\setcounter{equation}{0} \setcounter{theorem}{0}
\section{Background and motivation}
In this paper we are concerned with the fine properties of parabolic measures, defined with respect to appropriate domains $\mathbb R^{nN}thcal Omegaega$, and associated to the operator
\betagin{eqnarray}\lambdabel{e-kolm-nd}
\mathcal L=\mathcal L_A:=\boldsymbol \nuabla_X\cdot(A(X,Y,t)\boldsymbol \nuabla_X)+X\cdot\boldsymbol \nuabla_Y-\partialartial_t,
\boldsymbol end{eqnarray}
in $\mathbb R^{nN}thbb R^{N+1}$, $N=2m$, $m\geq 1$, equipped with coordinates $(X,Y,t):=(x_1,...,x_{m},y_1,...,y_{m},t)\in \mathbb R^{nN}thbb R^{m}\widetildemes\mathbb R^{nN}thbb R^{m}\widetildemes\mathbb R^{nN}thbb R$. We assume that $A=A(X,Y,t)=\{a_{i,j}(X,Y,t)\}_{i,j=1}^{m}$ is a real-valued $m\widetildemes m$-dimensional symmetric matrix
satisfying
\betagin{eqnarray}\lambdabel{eq2}
\kappa^{-1}|\xi|^2\leq \sum_{i,j=1}^{m}a_{i,j}(X,Y,t)\xi_i\xi_j,\quad \ \ |A(X,Y,t)\xi\cdot\mathbf Zata|\leq \kappa|\xi||\mathbf Zata|,
\boldsymbol end{eqnarray}
for some $\kappa\in [1,\infty)$, and for all $\xi,\mathbf Zata\in \mathbb R^{nN}thbb R^{m}$, $(X,Y,t)\in\mathbb R^{nN}thbb R^{N+1}$. We refer to $\kappa$ as the constant of $A$. Throughout the paper we will also assume that
\betagin{eqnarray}\lambdabel{eq2+}
a_{i,j}\in C^\infty(\mathbb R^{nN}thbb R^{N+1})
\boldsymbol end{eqnarray}
for all $i,j\in\{1,...,m\}$. While the assumption in \boldsymbol eqref{eq2+} will only be used in a qualitative fashion, the constants of our quantitative estimates will depend on $m$ and $\kappa$.
The starting point for our analysis is the recent results concerning the local regularity of weak solutions to the equation $\mathcal L u=0$ established in \cite{Ietal}. In \cite{Ietal} the authors extended the De Giorgi-Nash-Moser (DGNM) theory, which in its original form only considers elliptic
or parabolic equations in divergence form, to hypoelliptic equations with rough coefficients including the ones in \boldsymbol eqref{e-kolm-nd} assuming \boldsymbol eqref{eq2} and, implicitly, also \boldsymbol eqref{eq2+}. Their result is the correct scale- and translation-invariant estimates for local H{\"o}lder continuity and the Harnack inequality for weak solutions.
We recall that the prototype for the operators in \boldsymbol eqref{e-kolm-nd}, i.e. $A\boldsymbol equiv 1_m$ and the operator
$$\mathcal K:=\boldsymbol \nuabla_X\cdot \boldsymbol \nuabla_X+X\cdot\boldsymbol \nuabla_Y-\partialartial_t,$$
was originally introduced and studied by
Kolmogorov in a famous note published in 1934 in Annals of Mathematics, see \cite{K}. Kolmogorov noted that $\mathcal K$ is an example of a degenerate parabolic
operator having strong regularity properties and he proved that $\mathcal K$ has a fundamental solution which is smooth off its diagonal.
As a consequence,
\betagin{eqnarray}\lambdabel{uu3}
\mathcal K u = f \in C^\infty \quad \mathbb R^{nN}thbb Rightarrow \quad u \in C^\infty,
\boldsymbol end{eqnarray}
for every distributional solution of $\mathcal K u=f$. These days, using the terminology introduced by H{\"o}rmander, see \cite{Hm}, the property in \boldsymbol eqref{uu3} is stated
\betagin{eqnarray}\lambdabel{uu2}
\mbox{$\mathcal K$ is hypoelliptic}.
\boldsymbol end{eqnarray}
Naturally, for operators as in \boldsymbol eqref{e-kolm-nd}, assuming only measurable coefficients and \boldsymbol eqref{eq2}, the methods of Kolmogorov and H{\"o}rmander can not be directly applied to establish the DGNM theory and related estimates.
The results in \cite{Ietal} represent an important achievement which paves the way for developments concerning
operators as in \boldsymbol eqref{e-kolm-nd} in several fields of analysis and in the theory of PDEs. In this paper we contribute to the understanding of the fine properties of the Dirichlet problems for operators of the form stated in \boldsymbol eqref{e-kolm-nd}
in appropriate domains $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ and we note that in general there is a rich interplay between the operators considered, applications and geometry. Indeed, today the Kolmogorov operator, and the more general operators of Kolmogorov-Fokker-Planck type with variable coefficients considered in this paper, play central roles in many application in analysis, physics and finance and depending on the application different model cases for the local geometry of $\mathbb R^{nN}thcal Omegaega$ may be relevant:
\betagin{eqnarray}\lambdabel{dom-mod}
(i)&&\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{nN}thbb R^{N+1}\mid x_m>\partialsii_1(x,Y,t)\},\boldsymbol \nuotag\\
(ii)&&\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{nN}thbb R^{N+1}\mid y_m>\partialsii_2(X,y,t)\},\\
(iii)&&\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{nN}thbb R^{N+1}\mid t>\partialsii_3(X,Y)\}.\boldsymbol \nuotag
\boldsymbol end{eqnarray}
In particular, in finance and in the context of option pricing and associated free boundary problems, case $(i)$ is relevant. In kinetic theory it is relevant to restrict the particles to a container making case $(ii)$ relevant. Case $(iii)$ captures, as a special case, the initial value or Cauchy problem.
In this paper we consider solutions to $\mathcal L u=0$ in $\mathbb R^{nN}thcal Omegaega$ assuming \boldsymbol eqref{eq2} and \boldsymbol eqref{eq2+}. Concerning $\mathbb R^{nN}thcal Omegaega$ we restrict ourselves to case $(i)$ and unbounded domains $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ of the form
\betagin{eqnarray}\lambdabel{dom-}
\mathbb R^{nN}thcal Omegaega=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{nN}thbb R^{N+1}\mid x_m>\partialsii(x,y,t)\}.
\boldsymbol end{eqnarray}
We impose restrictions on $\partialsii$ of Lipschitz character and the importance of the additional assumption that $\partialsii$ is independent of $y_m$ will be explained.
Assuming that $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is a (unbounded) Lipschitz domain in the sense of
Definition \mathbb R^{nN}thbf{R}f{car} below, it follows that given $\varphi\in
C_0(\partialartial\mathbb R^{nN}thcal Omegaega)$, there exists
a unique weak solution $u=u_\varphi$, $u\in C(\bar \mathbb R^{nN}thcal Omegaega)$, to the Dirichlet problem
\betagin{equation} \lambdabel{e-bvpuu}
\betagin{cases}
\mathcal L u = 0 &\text{in} \ \mathbb R^{nN}thcal Omegaega, \\
u = \varphi & \text{on} \ \partialartial \mathbb R^{nN}thcal Omegaega.
\boldsymbol end{cases}
\boldsymbol end{equation}
Furthermore, there exists, for every $(Z, t):=(X,Y,t)\in \mathbb R^{nN}thcal Omegaega$, a unique probability
measure $\omegaega(Z,t,\cdot)$ on $\partialartial\mathbb R^{nN}thcal Omegaega$ such that
\betagin{eqnarray} \lambdabel{1.1xxuu}
u(Z,t)=\iint_{\partialartial\mathbb R^{nN}thcal Omegaega}\varphi(\widetildelde Z,\widetildelde t)\, \mathbb R^{nN}thrm{d} \omegaega(Z,t,\widetildelde Z,\widetildelde t).
\boldsymbol end{eqnarray}
The measure $\omegaega(Z,t,E)$ is referred to as the parabolic measure associated to $\mathcal L$ in $\mathbb R^{nN}thcal Omegaega$ and at $(Z, t)\in \mathbb R^{nN}thcal Omegaega$ and of $E\subset\partialartial\mathbb R^{nN}thcal Omegaega$. Properties of $\omegaega(Z,t,\cdot)$ govern the Dirichlet problem in \boldsymbol eqref{e-bvpuu}.
If $\mathbb R^{nN}thcal Omegaega=\mathbb R^{nN}thcal Omegaega_\partialsii\subset\mathbb R^{nN}thbb R^{N+1}$ is an unbounded ($y_m$-independent) Lipschitz domain we introduce the (physical) measure
$\sigmagma$ on $\partialartial\mathbb R^{nN}thcal Omegaega$ as
\betagin{eqnarray}\lambdabel{surfac+}d\sigmagma(X,Y,t):=\sqrt{1+|\boldsymbol \nuabla_{x}\partialsii(x,y,t)|^2}\, \mathbb R^{nN}thrm{d} x\, \mathbb R^{nN}thrm{d} Y\, \mathbb R^{nN}thrm{d} t,\ (X,Y,t)\in\partialartial\mathbb R^{nN}thcal Omegaega.
\boldsymbol end{eqnarray}
We will refer to $\sigmagma$ as the surface measure on $\partialartial\mathbb R^{nN}thcal Omegaega$.
Two fundamental questions concerning $\omegaega(Z,t,\cdot)$ can be stated as follows. Under what assumptions on $A$ and $\partialsii$, $\mathbb R^{nN}thcal Omegaega$ as in \boldsymbol eqref{dom-}, is it true that
\betagin{align} \lambdabel{problems}
(i)&\mbox{ $\omegaega(Z,t,\cdot)$ is a doubling measure ?}\boldsymbol \nuotag\\
(ii)&\mbox{ $\omegaega(Z,t,\cdot)$ satisfies scale-invariant absolute continuity estimates with respect}\\
&\mbox{ to the physical (surface) measure $\sigmagma$ on $\partialartial\mathbb R^{nN}thcal Omegaega$ ?}\boldsymbol \nuotag
\boldsymbol end{align}
In \cite{LN} we developed a potential theory for operators $\mathcal L$ as in \boldsymbol eqref{e-kolm-nd}, assuming only \boldsymbol eqref{eq2} and \boldsymbol eqref{eq2+}, in unbounded ($y_m$-independent) Lipschitz domains in the sense of Definition \mathbb R^{nN}thbf{R}f{car} below. As part of this theory we proved that $\omegaega(Z,t,\cdot)$ is a doubling measure, hence establishing \boldsymbol eqref{problems} $(i)$. The additional assumption that the function $\partialsii$ defining the domain is independent of $y_m$ was cruical in this part of \cite{LN}. In this paper we refine the result of \cite{LN} considerably by proving, under additional assumptions on $\partialsii$ ( i.e. on $\mathbb R^{nN}thcal Omegaega$) and the coefficients $A$, that $\omegaega(Z,t,\cdot)$ defines an $A_\infty$ weight with respect to the surface measure $\sigmagma$ in \boldsymbol eqref{surfac+} giving a quantitative answer to \boldsymbol eqref{problems} $(ii)$. In the prototype case $A\boldsymbol equiv 1_m$, i.e. in the case of the operator $\mathcal K$, the corresponding results were established in \cite{NP} and \cite{N1}, respectively, and this seems to be the only previous results of their kind for operators of Kolmogorov type.
To put the results of \cite{LN} and this paper into perspective it is relevant to outline the progress on the corresponding problems in the case of uniformly parabolic equations in $\mathbb R^{nN}thbb R^{m+1}$, i.e. in the case when all dependence on the variable $Y$ is removed in \boldsymbol eqref{e-kolm-nd} leaving us with the operator
\betagin{eqnarray}
\boldsymbol \nuabla_X\cdot(A(X,t)\boldsymbol \nuabla_X)-\partialartial_t.
\boldsymbol end{eqnarray}
In this setting the questions in \boldsymbol eqref{problems} have in recent times been discussed and resolved in a number of fundamental papers and we here highlight the main contributions to the field.
First, for uniformly parabolic equations with bounded measurable coefficients in Lipschitz type domains, scale and translation invariant boundary comparison principles,
boundary Harnack inequalities and doubling properties of associated parabolic measures were settled in a number of fundamental papers including \cite{FS}, \cite{FSY}, \cite{SY}, \cite{FGS} and \cite{N}. This type of results find their applications in many fields of analysis including the analysis of free boundary problems, see \cite{C1}, \cite{C2} and \cite{ACS} for instance.
Second, in \cite{LS}, \cite{LM}, \cite{HL}, \cite{H}, see also \cite{HL1}, the correct notion
of time-dependent Lipschitz type cylinders, correct from the perspective of parabolic measure, parabolic singular integral operators, parabolic
layer potentials, as well as from the perspective of the Dirichlet, Neumann and Regularity
problems with data in $L^p$ for the heat operator, was found. In particular, in \cite{LS}, \cite{LM} the mutual absolute continuity of the
parabolic measure with respect to surface measure, and the $A_\infty$-property, was studied/established and in \cite{HL} the authors
solved the Dirichlet, Neumann and Regularity
problems with data in $L^2$. For further related results concerning the fine
properties of parabolic measures we refer to the impressive and influential work \cite{HL2}. In \cite{HL2} the authors consider equations modeled on certain refined pull-backs of the heat operator to the parabolic upper half space $\mathbb R^{nN}thbb R^{m+1}_+=\{(x,x_m,t)\mid x_m>0\}$. These pull-back operators take the form
\betagin{eqnarray}\lambdabel{e-kolm-ndfl+a}
\boldsymbol \nuabla_X\cdot(A\boldsymbol \nuabla_Xu)+B\boldsymbol \nuabla_Xu-\partialartial_tu=0,
\boldsymbol end{eqnarray}
where the coefficient $B$ now gives rise to a singular drift term and the regularity of $A$ and $B$ are measured using certain Carleson measures. The singular drift term complicates matters considerably as there seem to be no positive answer to \boldsymbol eqref{problems} $(i)$ in this case. It should be mentioned that in
\cite{NR}, \cite{DPP}, parts of \cite{HL2} have been simplified.
Third, very recently there has been significant progress in the theory of boundary value problems for second order parabolic equations (and systems) of the form
\betagin{eqnarray}\lambdabel{eq1}
\boldsymbol \nuabla_X\cdot(A(x,t)\boldsymbol \nuabla_Xu)-\partialartial_tu=0,
\boldsymbol end{eqnarray}
in the parabolic upper half space $\mathbb R^{nN}thbb R_+^{m+1}$ with boundary determined by $x_m=0$, assuming only bounded, measurable, uniformly elliptic and complex coefficients. In~\cite{N2, CNS, N3}, the solvability for Dirichlet, Regularity and Neumann problems with data in $L^2$ were established for the class of parabolic equations \boldsymbol eqref{eq1} under the additional assumptions that the elliptic part is also independent of the time variable $t$ and that it has either constant (complex) coefficients, real symmetric coefficients, or small perturbations thereof. Focusing on parabolic measure, a particular consequence of Theorem 1.3 in~\cite{CNS} is the generalization of~\cite{FSa} to equations of the form \boldsymbol eqref{eq1} but with $A$ real, symmetric and time-independent. This analysis in \cite{N2, CNS, N3} was advanced further in~\cite{AEN}, where a first order strategy to study boundary value problems of parabolic systems with second order elliptic part in the upper half-space was developed. The outcome of~\cite{AEN} was the possibility to address arbitrary parabolic equations (and systems) as in \boldsymbol eqref{eq1} with coefficients depending also on time and on the transverse variable with additional transversal regularity. Finally, in \cite{AEN1} the authors consider parabolic equations as in \boldsymbol eqref{eq1}, assuming that the coefficients are real, bounded, measurable, uniformly elliptic, but not necessarily symmetric. They prove that the associated parabolic measure is absolutely continuous with respect to the surface measure on
$\mathbb R^{nN}thbb R^{m+1}$ (i.e. $\mathbb R^{nN}thrm{d} x\, \mathbb R^{nN}thrm{d} t$) in the sense defined by the Muckenhoupt class $A_\infty(\mathbb R^{nN}thrm{d} x\, \mathbb R^{nN}thrm{d} t)$.
In light of the above outline concerning the progress on uniformly parabolic equations, \cite {LN} and the main result of this paper, Theorem \mathbb R^{nN}thbf{R}f{Ainfty} stated below, represent important steps towards a corresponding theory concerning the Dirichlet problem for operators of Kolmogorov type with bounded and measurable coefficients in Lipschitz type domains adapted to the (non-Euclidean) group structure.
The rest of the paper is organized as follows. Section \mathbb R^{nN}thbf{R}f{sec2} is of preliminary nature. In Section \mathbb R^{nN}thbf{R}f{sec3} we state our main result, Theorem \mathbb R^{nN}thbf{R}f{Ainfty}, the proof of which we start in
Section \mathbb R^{nN}thbf{R}f{sec4}. In Section \mathbb R^{nN}thbf{R}f{sec4} we prove how Theorem \mathbb R^{nN}thbf{R}f{Ainfty} can be reduced to three lemmas: Lemmas \mathbb R^{nN}thbf{R}f{existcover}- \mathbb R^{nN}thbf{R}f{lemmacruc+}. We consider the proof of Lemma \mathbb R^{nN}thbf{R}f{lemmacruc+} a rather difficult part in the proof of Theorem \mathbb R^{nN}thbf{R}f{Ainfty} and in Section \mathbb R^{nN}thbf{R}f{sec4} we show that this lemma can be reduced to one key lemma: Lemma \mathbb R^{nN}thbf{R}f{Carleson}. Section \mathbb R^{nN}thbf{R}f{sec5} is devoted to the proof of Lemma \mathbb R^{nN}thbf{R}f{Carleson}. Finally, in Section \mathbb R^{nN}thbf{R}f{sec6} we prove Lemma \mathbb R^{nN}thbf{R}f{existcover} and Lemma \mathbb R^{nN}thbf{R}f{lemmacruc} by partially relying on a number of estimates for non-negative solutions recently established in \cite{LN}.
\setcounter{equation}{0} \setcounter{theorem}{0}
\section{Preliminaries}\lambdabel{sec2}
\subsection{Group law and metric} The natural family of dilations for $\mathcal L$, $(\, \mathbb R^{nN}thrm{d}eltalta_r)_{r>0}$, on $\mathbb R^{nN}thbb R^{N+1}$,
is defined by
\betagin{equation}\lambdabel{dil.alpha.i}
\, \mathbb R^{nN}thrm{d}eltalta_r (X,Y,t) =(r X, r^3 Y,r^2 t),
\boldsymbol end{equation}
for $(X,Y,t) \in \mathbb R^{nN}thbb R^{N +1}$, $r>0$. Our class of operators is closed under the group law
\betagin{equation}\lambdabel{e70}
(\widetildelde Z,\widetildelde t)\circ (Z,t)=(\widetildelde X,\widetildelde Y,\widetildelde t)\circ (X, Y,t)=(\widetildelde X+X,\widetildelde Y+Y-t\widetildelde X,\widetildelde t+t),
\boldsymbol end{equation}
where $(Z,t),\ (\widetildelde Z,\widetildelde t)\in \mathbb R^{nN}thbb R^{N+1}$. Note that
\betagin{equation}\lambdabel{e70+}
(Z,t)^{-1}=(X,Y,t)^{-1}=(-X,-Y-tX,-t),
\boldsymbol end{equation}
and hence
\betagin{equation}\lambdabel{e70++}
(\widetildelde Z,\widetildelde t)^{-1}\circ (Z,t)=(\widetildelde X,\widetildelde Y,\widetildelde t)^{-1}\circ (X,Y,t)=(X-\widetildelde X,Y-\widetildelde Y+(
t-\widetildelde t)\widetildelde X,t-\widetildelde t),
\boldsymbol end{equation}
whenever $(Z,t),\ (\widetildelde Z,\widetildelde t)\in \mathbb R^{nN}thbb R^{N+1}$. Given $(Z,t)=(X,Y,t)\in \mathbb R^{nN}thbb R^{N+1}$ we let \betagin{equation}\lambdabel{kolnormint}
\|(Z, t)\|= \|(X,Y, t)\|:=|(X,Y)|\!+|t|^{\frac{1}{2}},\ |(X,Y)|=\big|X\big|+\big|Y\big|^{1/3}.
\boldsymbol end{equation}
We recall the following pseudo-triangular
inequality: there exists a positive constant ${c}$ such that
\betagin{eqnarray}\lambdabel{e-ps.tr.in}
\|(Z,t)^{-1}\|\le {c} \| (Z,t) \|,\quad \|(Z,t)\circ (\widetildelde Z,\widetildelde t)\| \le {c} (\| (Z,t) \| + \| (\widetildelde Z,\widetildelde t)
\|),
\boldsymbol end{eqnarray}
whenever $(Z,t),(\widetildelde Z,\widetildelde t)\in \mathbb R^{nN}thbb R^{N+1}$. Using \boldsymbol eqref{e-ps.tr.in} it follows directly that
\betagin{equation} \lambdabel{e-triangularap}
\|(\widetildelde Z,\widetildelde t)^{-1}\circ (Z,t)\|\le c \, \|(Z,t)^{-1}\circ (\widetildelde Z,\widetildelde t)\|,
\boldsymbol end{equation}
whenever $(Z,t),(\widetildelde Z,\widetildelde t)\in \mathbb R^{nN}thbb R^{N+1}$. Let
\betagin{equation}\lambdabel{e-ps.distint}
d((Z,t),(\widetildelde Z,\widetildelde t)):=\frac 1 2\bigl( \|(\widetildelde Z,\widetildelde t)^{-1}\circ (Z,t)\|+\|(Z,t)^{-1}\circ (\widetildelde Z,\widetildelde t)\|).
\boldsymbol end{equation}
Using \boldsymbol eqref{e-triangularap} it follows that
\betagin{equation}\lambdabel{e-ps.dist}
\|(\widetildelde Z,\widetildelde t)^{-1}\circ (Z,t)\|\approx d((Z,t),(\widetildelde Z,\widetildelde t))\approx \|(Z,t)^{-1}\circ (\widetildelde Z,\widetildelde t)\|
\boldsymbol end{equation}
for all $(Z,t),(\widetildelde Z,\widetildelde t)\in \mathbb R^{nN}thbb R^{N+1}$ and with uniform constants. Again using \boldsymbol eqref{e-ps.tr.in} we also see that
\betagin{equation} \lambdabel{e-triangular}
d((Z,t),(\widetildelde Z,\widetildelde t))\le {c} \bigl(d((Z,t),(\hat Z,\hat t))+d((\hat Z,\hat
t),(\widetildelde Z,\widetildelde t))\bigr ),
\boldsymbol end{equation}
whenever $(Z,t),(\hat Z,\hat t),(\widetildelde Z,\widetildelde t)\in \mathbb R^{nN}thbb R^{N+1}$, and hence $d$ is a symmetric quasi-distance. Based on $d$ we introduce the balls
\betagin{equation}\lambdabel{e-BKint}
\mathbb R^{nN}thcal{B}_r(Z,t):= \{ (\widetildelde Z,\widetildelde t) \in\mathbb R^{nN}thbb R^{N+1} \mid d((\widetildelde Z,\widetildelde t),(Z,t)) <
r\},
\boldsymbol end{equation}
for $(Z,t)\in \mathbb R^{nN}thbb R^{N+1}$ and $r>0$. The measure of the ball $\mathbb R^{nN}thcal{B}_r(Z,t)$ is $|\mathbb R^{nN}thcal{B}_r(Z,t)|\approx r^{{\bf q}}$, independent of $(Z,t)$, and where
$${\bf q}:=4m+2.$$
Similarly, given $(z,t)=(x,y,t)\in \mathbb R^{nN}thbb R^{N-1}=\mathbb R^{nN}thbb R^{m-1}\widetildemes\mathbb R^{nN}thbb R^{m-1}\widetildemes\mathbb R^{nN}thbb R$ we let
\betagin{equation}\lambdabel{e-BKint+}
\mathbb R^{nN}thcal{B}_r(z,t):= \{ (\widetildelde z,\widetildelde t) \in\mathbb R^{nN}thbb R^{N-1} \mid d((\widetildelde x,0,\widetildelde y, 0,\widetildelde t),(x,0,y,0,t)) <
r\}.
\boldsymbol end{equation}
The measure of the ball $\mathbb R^{nN}thcal{B}_r(z,t)$ is $|\mathbb R^{nN}thcal{B}_r(z,t)|\approx r^{{\bf q}-4}$, independent of $(z,t)$. We will by $\mathbb R^{nN}thcal{B}_r(Z,t)$ always denote a ball in $\mathbb R^{nN}thbb R^{N+1}$, with capital $Z$, and by $\mathbb R^{nN}thcal{B}_r(z,t)$ we will always denote a ball in $\mathbb R^{nN}thbb R^{N-1}$, with lowercase $z$.
\subsection{Geometry} We consider domains of the form stated in \boldsymbol eqref{dom-} and we here outline the assumptions we impose on the defining function $\partialsii$. Let
$\mathcal P\in C_0^\infty(\mathbb R^{nN}thcal{B}_1(0,0))$, $\mathbb R^{nN}thcal{B}_1(0,0)\subset\mathbb R^{nN}thbb R^{N-1}$, be a standard approximation of the identity. Let
$$\mathcal P_\lambdambda(x,y,t)=\lambdambda^{-{(\bf q}-4)}\mathcal P(\lambdambda^{-1}x,\lambdambda^{-3}y,\lambdambda^{-2}t),$$ for $\lambdambda>0$. Given a function $f$ defined on
$\mathbb R^{nN}thbb R^{N-1}$ we let
\betagin{eqnarray}\lambdabel{eq1vi}
\mathcal P_{\lambdambda}f(x,y, t)&:=&\iint_{\mathbb R^{nN}thbb R^{N-1}}f(\bar x,\bar y,\bar t)\mathcal P_\lambdambda((\bar x,\bar y, \bar t)^{-1}\circ (x,y, t))\, \, \mathbb R^{nN}thrm{d}\bar x \, \mathbb R^{nN}thrm{d}\bar y \, \mathbb R^{nN}thrm{d}\bar t\boldsymbol \nuotag\\
&=&\iint_{\mathbb R^{nN}thbb R^{N-1}}f(\bar x,\bar y,\bar t)\mathcal P_\lambdambda(x-\bar x,y-\bar y+(t-\bar t)\bar x,t-\bar t)\, \, \mathbb R^{nN}thrm{d}\bar x \, \mathbb R^{nN}thrm{d}\bar y \, \mathbb R^{nN}thrm{d}\bar t.
\boldsymbol end{eqnarray}
$\mathcal P_{\lambdambda}f$ represents a regularization of $f$. Given $(\widetildelde z, \widetildelde t)\in \mathbb R^{nN}thbb R^{N-1}$, $\lambdambda>0$, we let $\gammamma_\partialsii(\widetildelde z, \widetildelde t,\lambdambda)$ denote the number
\betagin{eqnarray*}\lambdabel{eq1apa}
\biggl (\lambdambda^{-{({\bf q}-4)}}\iint_{\mathbb R^{nN}thcal{B}_\lambdambda(\widetildelde z, \widetildelde t)}\biggl |\frac {\partialsii(\bar x,\bar y,\bar t)-\partialsii(\widetildelde x,\widetildelde y,\widetildelde t)-\mathcal P_\lambdambda(\boldsymbol \nuabla_x\partialsii)(\widetildelde x,\widetildelde y,\widetildelde t)(\bar x-\widetildelde x)}{\lambdambda}\biggr |^2\, \, \mathbb R^{nN}thrm{d}\bar x\, \mathbb R^{nN}thrm{d}\bar y\, \mathbb R^{nN}thrm{d}\bar t\biggr )^{1/2}.
\boldsymbol end{eqnarray*}
\betagin{definition}\lambdabel{car} Assume that there exist constants $0<M_1,M_2<\infty$, such that
\betagin{eqnarray}\lambdabel{Lip-++a1}
|\partialsii(z,t)-\partialsii(\widetildelde z,\widetildelde t)|\leq M_1||(\widetildelde z,\widetildelde t)^{-1}\circ(z,t)||,
\boldsymbol end{eqnarray}
whenever $(z,t),\ (\widetildelde z,\widetildelde t)\in\mathbb R^{nN}thbb R^{N-1}$ and such that
\betagin{eqnarray}\lambdabel{Lip-++a2}
\sup_{(z,t)\in\mathbb R^{nN}thbb R^{N-1},\ r>0}\quad r^{-{({\bf q}-4)}}\int_0^r\iint_{\mathbb R^{nN}thcal{B}_\lambdambda(z,t)}\gammamma_\partialsii^2(\widetildelde z, \widetildelde t,\lambdambda)\, \frac {\, \mathbb R^{nN}thrm{d}\widetildelde z\, \mathbb R^{nN}thrm{d}\widetildelde t\, \mathbb R^{nN}thrm{d}\lambdambda}\lambdambda\leq M_2.
\boldsymbol end{eqnarray}
Let $\mathbb R^{nN}thcal Omegaega=\mathbb R^{nN}thcal Omegaega_\partialsii$ be defined as in \boldsymbol eqref{dom-}. We say that $\mathbb R^{nN}thcal Omegaega$, defined by a function $\partialsii$ satisfying \boldsymbol eqref{Lip-++a1}, is
a ($y_m$-independent) Lipschitz domain with constant $M_1$. We say that $\mathbb R^{nN}thcal Omegaega$, defined by a function $\partialsii$ satisfying \boldsymbol eqref{Lip-++a1} and \boldsymbol eqref{Lip-++a2}, is an admissible ($y_m$-independent) Lipschitz domain with constants $(M_1,M_2)$.
\boldsymbol end{definition}
Given $\rho>0$ and $\mathcal Lambdambda>0$ we introduce points of reference in $\mathbb R^{nN}thbb R^{m-1}\widetildemes\mathbb R^{nN}thbb R\widetildemes\mathbb R^{nN}thbb R^{m-1}\widetildemes\mathbb R^{nN}thbb
R\widetildemes\mathbb R^{nN}thbb R$,
\betagin{align}\lambdabel{pointsref2}
A_{\rho,\mathcal Lambdambda}^\partialm:= \left(0,\mathcal Lambdambda\rho,0,\mp\tfrac 2 3\mathcal Lambdambda\rho^3,\partialm\rho^2\right),\ A_{\rho,\mathcal Lambdambda}&:=\left(0,\mathcal Lambdambda\rho,0,0,0\right).
\boldsymbol end{align}
Given $(Z_0,t_0)\in\mathbb R^{nN}thbb R^{N+1}$ we let $$A_{\rho,\mathcal Lambdambda}^\partialm(Z_0,t_0):=(Z_0,t_0)\circ A_{\rho,\mathcal Lambdambda}^\partialm,\ A_{\rho,\mathcal Lambdambda}(Z_0,t_0):=(Z_0,t_0)\circ A_{\rho,\mathcal Lambdambda}.$$
\subsection{Dyadic grids, Whitney cubes and Carleson boxes}\lambdabel{dya} Assuming that $\mathbb R^{nN}thcal Omegaega=\mathbb R^{nN}thcal Omegaega_\partialsii\subset\mathbb R^{nN}thbb R^{N+1}$ is a Lipschitz domain, with constant $M_1$, in the sense of Definition \mathbb R^{nN}thbf{R}f{car}, we let
$$\mathcal Sigmagma:=\partialartial \mathbb R^{nN}thcal Omegaega=\{(x,x_{m},y,y_{m},t)\in\mathbb R^{nN}thbb R^{N+1} \mid x_m=\partialsii(x,y,t)\}.$$
Then $(\mathcal Sigmagma,d,d\sigmagma)$, where the symmetric quasi-distance $d$ was introduced in \boldsymbol eqref{e-ps.distint}, is a space of homogeneous type in the sense of \cite{CW} with homogeneous dimension ${\bf q}-1$. Furthermore, $(\mathbb R^{nN}thbb R^{N+1},d,dZdt)$ is also a space of homogeneous type in the sense of \cite{CW}, but with homogeneous dimension ${\bf q}$. By the results in \cite{Ch} there exists what we here will refer to as a dyadic grid on
$\mathcal Sigmagma$ having a number of important properties in relation to $d$. To formulate this we introduce, for any $(Z,t)=(X,Y,t)\in\mathcal Sigmagma$ and
$E\subset \mathcal Sigmagma$,
\betagin{equation}
{\rm dist} ((Z,t),E):=\inf \{ d((Z,t),(\widetildelde Z,\widetildelde t)) \mid (\widetildelde Z,\widetildelde t)\in E\},
\boldsymbol end{equation}
and we let
\betagin{equation}
\textnormal{diam}m(E):=\sup \{ d((Z,t),(\widetildelde Z,\widetildelde t)) \mid (Z,t),\ (\widetildelde Z,\widetildelde t)\in E\}.
\boldsymbol end{equation}
Using \cite{Ch} we can conclude that there exist
constants $ \alphapha>0,\, \betata>0$ and $c_*<\infty$, such that for each $k \in \mathbb R^{nN}thbb{Z}$
there exists a collection of Borel sets, $\mathbb R^{nN}thbb{D}_k$, which we will call cubes, such that
$$
\mathbb R^{nN}thbb{D}_k:=\{Q_{j}^k\subset\mathcal Sigmagma \mid j\in \mathbb R^{nN}thfrak{I}_k\},$$ where
$\mathbb R^{nN}thfrak{I}_k$ denotes some index set depending on $k$, satisfying
\betagin{eqnarray}\lambdabel{cubes}
(i)&&\mbox{$\mathcal Sigmagma={\rm B}p_{j}Q_{j}^k\,\,$ for each
$k\in{\mathbb R^{nN}thbb Z}$.}\boldsymbol \nuotag\\
(ii)&&\mbox{If $m\geq k$ then either $Q_{i}^{m}\subset Q_{j}^{k}$ or
$Q_{i}^{m}u_{\rm B}p Q_{j}^{k}=\boldsymbol emptyset$.}\boldsymbol \nuotag\\
(iii)&&\mbox{For each $(j,k)$ and each $m<k$, there is a unique
$i$ such that $Q_{j}^k\subset Q_{i}^m$.}\boldsymbol \nuotag\\
(iv)&&\mbox{$\textnormal{diam}m\big(Q_{j}^k\big)\leq c_* 2^{-k}$.}\boldsymbol \nuotag\\
(v)&&\mbox{Each $Q_{j}^k$ contains $\mathcal Sigmagmau_{\rm B}p \mathbb R^{nN}thcal{B}_{\alphapha2^{-k}}(Z^k_{j},t^k_{j})$ for some $(Z^k_{j},t^k_j)\in\mathcal Sigmagma$.}\boldsymbol \nuotag\\
(vi)&&\mbox{$\sigmagma(\{(Z,t)\in Q^k_j\mid{\rm dist}((Z,t),\mathcal Sigmagma\setminus Q^k_j)\leq \rho \,2^{-k}\big\})\leq
c_*\,\rho^\betata\,\sigmagma(Q^k_j),$}\boldsymbol \nuotag\\
&&\mbox{for all $k,j$ and for all $\rho\in (0,\alphapha)$.}
\boldsymbol end{eqnarray}
In the setting of a general space of homogeneous type, this result is due to Christ
\cite{Ch}, with the
dyadic parameter $1/2$ replaced by some constant $\, \mathbb R^{nN}thrm{d}eltalta \in (0,1)$. In fact, one may always take $\, \mathbb R^{nN}thrm{d}eltalta = 1/2$, see \cite[Proof of Proposition 2.12]{HMMM}. We shall denote by $\mathbb R^{nN}thbb{D}=\mathbb R^{nN}thbb{D}(\mathcal Sigmagma)$ the collection of all
$Q^k_j$, i.e. $$\mathbb R^{nN}thbb{D} := {\rm B}p_{k} \mathbb R^{nN}thbb{D}_k.$$
Note that \boldsymbol eqref{cubes} $(iv)$ and $(v)$ imply that for each cube $Q\in\mathbb R^{nN}thbb{D}_k$,
there is a point $(Z_Q,t_Q)=(X_Q,Y_Q,t_Q)\in \mathcal Sigmagma$, and a ball $\mathbb R^{nN}thcal{B}_{r}(Z_Q,t_Q)$ such that
$r\approx 2^{-k} \approx {\rm diam}(Q)$
and \betagin{equation}\lambdabel{cube-ball}
\mathcal Sigmagmau_{\rm B}p\mathbb R^{nN}thcal{B}_{r}(Z_Q,t_Q)\subset Q \subset \mathcal Sigmagmau_{\rm B}p \mathbb R^{nN}thcal{B}_{cr}(Z_Q,t_Q),\boldsymbol end{equation}
for some uniform constant $c$. We will denote the associated surface ball by
\betagin{equation}\lambdabel{cube-ball2}
\mathcal Deltalta_Q:= \mathcal Sigmagmau_{\rm B}p \mathbb R^{nN}thcal{B}_{r}(Z_Q,t_Q)\boldsymbol end{equation}
and we shall refer to the point $(Z_Q,t_Q)$ as the center of $Q$. Given a dyadic cube $Q\subset\mathcal Sigmagma$, we define its $\gammamma$ dilate by
\betagin{equation}\lambdabel{dilatecube}
\gammamma Q:= \mathcal Sigmagmau_{\rm B}p \mathbb R^{nN}thcal{B}_{\gammamma \textnormal{diam}m(Q)}(Z_Q,t_Q).
\boldsymbol end{equation}
For a dyadic cube $Q\in \mathbb R^{nN}thbb{D}_k$, we let $\boldsymbol ell(Q) = 2^{-k}$, and we shall refer to this quantity as the length
of $Q$. Clearly, $\boldsymbol ell(Q)\approx \textnormal{diam}m(Q).$ For a dyadic cube $Q \in \mathbb R^{nN}thbb{D}$, we let $k(Q)$ denote the dyadic generation
to which $Q$ belongs, i.e. we set $k = k(Q)$ if
$Q\in \mathbb R^{nN}thbb{D}_k$, thus, $\boldsymbol ell(Q) =2^{-k(Q)}$. For any $Q\in \mathbb R^{nN}thbb D(\mathcal Sigmagma)$, we set $\mathbb R^{nN}thbb D_Q:= \{Q'\in\mathbb R^{nN}thbb D \mid Q'\subset Q\}\,.$
Using that also $(\mathbb R^{nN}thbb R^{N+1},d,dZdt)$ is a space of homogeneous type we see that we can partition
$\mathbb R^{nN}thcal Omegaega$ into a collection
of (closed) dyadic Whitney cubes $\{I\}$, in the following denoted $\mathbb R^{nN}thcal{W}=\mathcal W(\mathbb R^{nN}thcal Omegaega)$, such that the cubes in $\mathbb R^{nN}thcal{W}$
form a covering of $\mathbb R^{nN}thcal Omegaega$ with non-overlapping interiors, and
\betagin{equation}\lambdabel{eqWh1} 4\, {\rm{diam}}\,(I)\leq \textnormal{dist}st(4 I,\mathcal Sigmagma) \leq \textnormal{dist}st(I,\mathcal Sigmagma) \leq 40 \, {\rm{diam}}\,(I)\boldsymbol end{equation}
and
\betagin{equation}\lambdabel{eqWh2}\textnormal{diam}m(I_1)\approx \textnormal{diam}m(I_2), \mbox{ whenever $I_1$ and $I_2$ touch.}
\boldsymbol end{equation}
Given $I\in \mathbb R^{nN}thcal{W}$ we let $\boldsymbol ell(I)$ denote its size. Given $Q\in \mathbb R^{nN}thbb D(\mathcal Sigmagma)$ we set
\betagin{equation}\lambdabel{eq2.1}
\mathcal W_Q:= \left\{I\in \mathcal W\mid \,100^{-1} \boldsymbol ell(Q)\leq \boldsymbol ell(I)
\leq 100\,\boldsymbol ell(Q),\, {\rm and}\, \textnormal{dist}st(I,Q)\leq 100\, \boldsymbol ell(Q)\right\}.
\boldsymbol end{equation}
We fix a small, positive parameter $\tau$, and given $I\in\mathcal W$,
we let
\betagin{equation}\lambdabel{eq2.3*}I^* =I^*(\tau) := (1+\tau)I
\boldsymbol end{equation}
denote the corresponding ``fattened" Whitney cube. Choosing $\tau$ small we see that the cubes $I^*$ will retain the usual properties of Whitney cubes;
in particular, that
$$\textnormal{diam}m(I) \approx \textnormal{diam}m(I^*) \approx \textnormal{dist}st(I^*,\mathcal Sigmagma) \approx \textnormal{dist}st(I,\mathcal Sigmagma)\,.$$
We then define a Whitney region
with respect to $Q$ by setting
\betagin{equation}\lambdabel{eq2.3}
U_Q:= \bigcup_{I\in \mathcal W_Q}I^*\,.
\boldsymbol end{equation}
Given $Q\in \mathbb R^{nN}thbb D(\mathcal Sigmagma)$ we let
\betagin{equation}\lambdabel{eq2.box-}
T_Q:={\rm int}\left( \bigcup_{Q'\in \mathbb R^{nN}thbb D_Q} U_{Q'}\right),
\boldsymbol end{equation}
denote the Carleson box
associated to $Q$. Furthermore, given $\gammamma\geq 1$ we let
\betagin{equation}\lambdabel{eq2.box}
T_{\gammamma Q}:={\rm int}\left( \bigcup_{Q':\ Q'u_{\rm B}p (\gammamma Q)\boldsymbol \nueq \boldsymbol emptyset} U_{Q'}\right),
\boldsymbol end{equation}
denote the Carleson set
associated to the $\gammamma$ dilate of $Q$. Finally, given $Q\in\mathbb R^{nN}thbb{D}$ and $\mathcal Lambdambda>0$, we let
\betagin{equation}\lambdabel{pointsref2apa}
\betagin{split}
A_{Q,\mathcal Lambdambda}^\partialm&:=(Z_Q,t_Q)\circ (0,\mathcal Lambdambda l(Q),0,\mp\frac 2 3\mathcal Lambdambda l(Q)^3,\partialm l(Q)^2),\\
A_{Q,\mathcal Lambdambda}&:=(Z_Q,t_Q)\circ (0,\mathcal Lambdambda l(Q),0,0,0).
\boldsymbol end{split}
\boldsymbol end{equation}
\subsection{Weak solutions} Consider $U_X\widetildemes U_Y\widetildemes J\subset\mathbb R^{nN}thbb R^{N+1}$ with $U_X\subset\mathbb R^{nN}thbb R^{m}$, $U_Y\subset\mathbb R^{nN}thbb R^{m}$ being bounded domains, i.e, open, connected and bounded sets, and $J=(a,b)$ with $-\infty<a<b<\infty$. Then
$u$ is said to be a weak solution to the equation
\betagin{eqnarray}\lambdabel{e-kolm-nd-}
\mathcal L u=\boldsymbol \nuabla_X\cdot(A(X,Y,t)\boldsymbol \nuabla_Xu)+X\cdot\boldsymbol \nuabla_Yu-\partialartial_tu=0,
\boldsymbol end{eqnarray}
in $U_X\widetildemes U_Y\widetildemes J\subset\mathbb R^{nN}thbb R^{N+1}$ if
\betagin{eqnarray}\lambdabel{weak1}
u\in L_{Y,t}^2(U_Y\widetildemes J,H_X^1(U_X)),
\boldsymbol end{eqnarray}
and
\betagin{eqnarray}\lambdabel{weak2}
-X\cdot\boldsymbol \nuabla_Yu+\partialartial_tu\in L_{Y,t}^2(U_Y\widetildemes J,H_X^{-1}(U_X)),
\boldsymbol end{eqnarray}
and if $\mathcal L u=0$ in the sense of distributions, i.e,
\betagin{eqnarray}\lambdabel{weak3}
\iiint_{}\ \bigl(A(X,Y,t)\boldsymbol \nuabla_Xu\cdot \boldsymbol \nuabla_X\partialhii+(X\cdot \boldsymbol \nuabla_Y\partialhii)u-u\partialartial_t\partialhii\bigr )\, \, \mathbb R^{nN}thrm{d} X \, \mathbb R^{nN}thrm{d} Y \, \mathbb R^{nN}thrm{d} t=0,
\boldsymbol end{eqnarray}
whenever $\partialhii\in C_0^\infty(U_X\widetildemes U_Y\widetildemes J)$.
We say that $u$ is a weak solution
to the equation $\mathcal L u=0$ in $\mathbb R^{nN}thcal Omegaega$ if $u$ is a weak solution to $\mathcal L u=0$ in $U_X\widetildemes U_Y\widetildemes J\subset\mathbb R^{nN}thbb R^{N+1}$, where $U_X\subset\mathbb R^{nN}thbb R^{m}$, $U_Y\subset\mathbb R^{nN}thbb R^{m}$ are bounded domains, and $J=(a,b)$ with $-\infty<a<b<\infty$, whenever $U_X\widetildemes U_Y\widetildemes J$ is compactly contained in $\mathbb R^{nN}thcal Omegaega$.
\setcounter{equation}{0} \setcounter{theorem}{0}
\section{Statement of the main result}\lambdabel{sec3}
Assume that $\mathbb R^{nN}thcal Omegaega=\mathbb R^{nN}thcal Omegaega_\partialsii\subset\mathbb R^{nN}thbb R^{N+1}$ is a Lipschitz domain, with constant $M_1$, in the sense of Definition \mathbb R^{nN}thbf{R}f{car}, and recall that $\mathbb R^{nN}thbb{D}$ is the set of dyadic cubes on $\partialartial\mathbb R^{nN}thcal Omegaega$. Given $Q\in\mathbb R^{nN}thbb{D}$, recall the definitions of $l(Q)$, $(Z_Q,t_Q)$, $\gammamma Q$, $T_Q$, $A_{Q,\mathcal Lambdambda}^\partialm$, introduced in Subsection \mathbb R^{nN}thbf{R}f{dya}.
Using this notation a version of one of the main results (namely, Theorem 3.6) proved in \cite{LN} can be stated as follows.
\betagin{theorem}\lambdabel{dub} Let $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is an unbounded ($y_m$-independent) Lipschitz domain with constant $M_1$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}. Assume that $A$ satisfies \boldsymbol eqref{eq2} with constant $\kappa$, \boldsymbol eqref{eq2+} and that
\betagin{eqnarray}\lambdabel{struct}
A(X,Y,t)=A(x,x_m,y,y_m,t)=A(x,x_m,y,t)
\boldsymbol end{eqnarray}
whenever $(x,x_{m},y,y_{m},t)\in\mathbb R^{nN}thbb R^{N+1}$, i.e. also $A$ is assumed to be independent of the variable $y_m$. Then there exist
$\mathcal Lambdambda=\mathcal Lambdambda(m,M_1)$, $1\leq \mathcal Lambdambda<\infty$, $c=c(m,\kappa,M_1)$, $1\leq c<\infty$, such that the following is true. Consider $Q_0\in\mathbb R^{nN}thbb{D}$ and let $\omegaega(\cdot):=\omegaega\bigl (A_{cQ_0,\mathcal Lambdambda}^+,\cdot\bigr )$. Then
\betagin{eqnarray*}
\omegaega\bigl (2Q\bigr )\leq
c\omegaega\bigl (Q\bigr )
\boldsymbol end{eqnarray*}
for all $Q\in\mathbb R^{nN}thbb{D}$ such that $4Q\subset Q_0$.
\boldsymbol end{theorem}
Given an unbounded ($y_m$-independent) Lipschitz domain $\mathbb R^{nN}thcal Omegaega=\mathbb R^{nN}thcal Omegaega_\partialsii\subset\mathbb R^{nN}thbb R^{N+1}$ we let $\, \mathbb R^{nN}thrm{d}eltalta=\, \mathbb R^{nN}thrm{d}eltalta(X,Y,t)$ denote the distance from $(X,Y,t)\in\mathbb R^{nN}thcal Omegaega$ to $\partialartial\mathbb R^{nN}thcal Omegaega$, i.e.
\betagin{equation}\lambdabel{deltadist}
\, \mathbb R^{nN}thrm{d}eltalta(X,Y,t)=\min\{d((X,Y,t),(\widetildelde X,\widetildelde Y,\widetildelde t)) \mid (\widetildelde X,\widetildelde Y,\widetildelde t)\in\partialartial\mathbb R^{nN}thcal Omegaega\}.
\boldsymbol end{equation}
Consider the following measures $\mu_1$ and $\mu_2$ defined on $\mathbb R^{nN}thcal Omegaega$:
\betagin{equation}\lambdabel{measure1}
\betagin{split}
\, \mathbb R^{nN}thrm{d}\mu_1(X,Y,t)&:=|\boldsymbol \nuabla_XA(X,Y,t)|^2\, \mathbb R^{nN}thrm{d}eltalta(X,Y,t)\ \, \mathbb R^{nN}thrm{d} X\, \mathbb R^{nN}thrm{d} Y\, \mathbb R^{nN}thrm{d} t,\\
\, \mathbb R^{nN}thrm{d}\mu_2(X,Y,t)&:=|(X\cdot\boldsymbol \nuabla_Y-\partialartial_t)A(X,Y,t)|^2\, \mathbb R^{nN}thrm{d}eltalta^3(X,Y,t)\ \, \mathbb R^{nN}thrm{d} X\, \mathbb R^{nN}thrm{d} Y\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{split}
\boldsymbol end{equation}
We say that $\mu_1$ and $\mu_2$ are Carleson measures on $\mathbb R^{nN}thcal Omegaega$ with constant $\mathcal Gammamma$ if
\betagin{equation}\lambdabel{measure2}
\betagin{split}
\sup_{Q\in\mathbb R^{nN}thbb{D}}\quad l(Q)^{-{({\bf q}-1)}}\iiint_{T_Q}\, \mathbb R^{nN}thrm{d}\mu_1(\widetildelde X,\widetildelde Y,\widetildelde t)&\leq\mathcal Gammamma,\\
\sup_{Q\in\mathbb R^{nN}thbb{D}}\quad l(Q)^{-{({\bf q}-1)}}\iiint_{T_Q}\, \mathbb R^{nN}thrm{d}\mu_2(\widetildelde X,\widetildelde Y,\widetildelde t)&\leq\mathcal Gammamma.
\boldsymbol end{split}
\boldsymbol end{equation}
The following is the main result proved in this paper.
\betagin{theorem}\lambdabel{Ainfty} Assume that $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is an (unbounded) admissible ($y_m$-independent) Lipschitz domain with constants $(M_1,M_2)$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}. Assume that $A$ satisfies \boldsymbol eqref{eq2} with constant $\kappa$, \boldsymbol eqref{eq2+} and \boldsymbol eqref{struct}, i.e. also $A$ is independent of $y_m$. Assume that the measures $\mu_1$ and $\mu_2$ defined in \boldsymbol eqref{measure1} are Carleson measures on $\mathbb R^{nN}thcal Omegaega$ with constant $\mathcal Gammamma$ in the sense of \boldsymbol eqref{measure2}. Then there exist
$\mathcal Lambdambda=\mathcal Lambdambda(m,M_1)$, $1\leq \mathcal Lambdambda<\infty$, $c=c(m,\kappa,M_1)$, $1\leq c<\infty$, $\widetildelde c=\widetildelde c(m,\kappa, M_1,M_2,\mathcal Gammamma)$, $1\leq \widetildelde c<\infty$,
$\boldsymbol eta=\boldsymbol eta(m,\kappa,M_1,M_2,\mathcal Gammamma)$, $0<\boldsymbol eta<1$, such that the following is true. Consider $Q_0\in\mathbb R^{nN}thbb{D}$ and let $\omegaega(\cdot):=\omegaega\bigl (A_{cQ_0,\mathcal Lambdambda}^+,\cdot\bigr )$. Then
\betagin{eqnarray*}
\quad\widetildelde c^{-1}\biggl (\frac{ \sigmagma ( E ) }{ \sigmagma(Q)}\biggr )^{1/\boldsymbol eta}\leq \frac {\omegaega\bigl (E\bigr )}{\omegaega\bigl ( Q\bigr )}\leq \widetildelde c\biggl (\frac{ \sigmagma ( E ) }{ \sigmagma(Q)}\biggr )^\boldsymbol eta
\boldsymbol end{eqnarray*}
whenever $E\subset Q$ for some $Q\in\mathbb R^{nN}thbb{D}$ such that $Q\subseteq Q_0$.
\boldsymbol end{theorem}
As mentioned before, in the prototype case $A\boldsymbol equiv 1_m$, i.e. in the case of the operator $\mathcal K$, Theorem \mathbb R^{nN}thbf{R}f{Ainfty} is proved in \cite{N} and this seems to be the only previous result of its kind for operators of Kolmogorov type.
\section{Proof of Theorem \mathbb R^{nN}thbf{R}f{Ainfty}: preliminary reductions}\lambdabel{sec4}
Using Lemma \mathbb R^{nN}thbf{R}f{lemmacruc-} and Lemma \mathbb R^{nN}thbf{R}f{T:doubling} below it follows that it suffices to prove Theorem \mathbb R^{nN}thbf{R}f{Ainfty} with $Q=Q_0$. In the following we let $Q_0\in \mathbb R^{nN}thbb{D}$ and we let $\omegaega(\cdot)$ be as in the statement of Theorem \mathbb R^{nN}thbf{R}f{Ainfty}. Our proof of Theorem \mathbb R^{nN}thbf{R}f{Ainfty} is based on ideas introduced in \cite{KKPT} in the context of elliptic measures and we will use the notion of good $\varepsilonsilon_0$ covers.
\betagin{definition}\lambdabel{deff1} Let $E\subset {Q_0}$ be given, let $\varepsilonsilon_0\in (0,1)$ and let $k$ be an integer. A good $\varepsilonsilon_0$ cover of $E$, of length $k$, is a collection
$\{\mathbb R^{nN}thcal{O}_l\}_{l=1}^k$ of nested (relatively) open subsets of ${Q_0}$, together with collections
$\mathcal F_l=\{\mathcal Deltalta_i^l\}_i\subset Q_0$, $\mathcal Deltalta_i^l\in \mathbb R^{nN}thbb D$, such that
\betagin{eqnarray}\lambdabel{cover1}
E\subset \mathbb R^{nN}thcal{O}_k\subset\mathbb R^{nN}thcal{O}_{k-1}\subset....\subset\mathbb R^{nN}thcal{O}_1\subset Q_0,
\boldsymbol end{eqnarray}
\betagin{eqnarray}\lambdabel{cover2}
\mathbb R^{nN}thcal{O}_l=\bigcup_{\mathcal F_l}\mathcal Deltalta_i^l,
\boldsymbol end{eqnarray}
and
\betagin{eqnarray}\lambdabel{cover3}\omegaega(\mathbb R^{nN}thcal{O}_lu_{\rm B}p \mathcal Deltalta_i^{l-1})\leq \varepsilonsilon_0\omegaega(\mathcal Deltalta_i^{l-1}),\mbox{ for all }\mathcal Deltalta_i^{l-1}\in\mathcal F_{l-1}.
\boldsymbol end{eqnarray}
\boldsymbol end{definition}
Using the notion of good $\varepsilonsilon_0$ covers we can reduce the proof of Theorem \mathbb R^{nN}thbf{R}f{Ainfty} to the proof of the following three lemmas.
\betagin{lemma}\lambdabel{existcover} Let $E\subset {Q_0}$ be given, consider $\varepsilonsilon_0\in (0,1)$ and let $k$ be a positive integer. There exist $\gammamma=\gammamma(m,\kappa,M_1)$, $0<\gammamma\ll 1$, and $\mathcal Upsilon=\mathcal Upsilon(m,\kappa,M_1)$, $1\ll\mathcal Upsilon$, such that if we let $\, \mathbb R^{nN}thrm{d}eltalta_0=\gammamma(\varepsilonsilon_0/\mathcal Upsilon)^k$, and if
$\omegaega(E)\leq\, \mathbb R^{nN}thrm{d}eltalta_0$, then
$E$ has a good $\varepsilonsilon_0$ cover of length $k$.
\boldsymbol end{lemma}
\betagin{lemma}\lambdabel{lemmacruc} Let $\mathcal Upsilon\gg 1$ be given and consider $\, \mathbb R^{nN}thrm{d}eltalta_0\in (0,1)$. Assume that $E\subset {Q}_0$ with $\omegaega(E)\leq\, \mathbb R^{nN}thrm{d}eltalta_0$. If
$\, \mathbb R^{nN}thrm{d}eltalta_0=\, \mathbb R^{nN}thrm{d}eltalta_0(m,\kappa,M_1,\mathcal Upsilon)$ is chosen sufficiently small, then there exists a Borel set $S\subset\partialartial\mathbb R^{nN}thcal Omegaega$, and a constant $c=c(m,\kappa,M_1)$, $1\leq c<\infty$, such that if we let $u(Z,t):=\omegaega(Z,t,S)$, then
$$\mathcal Upsilon^2\sigmagma(E)\leq c \iiint_{T_{cQ_0}}|\boldsymbol \nuabla_Xu|^2\, \mathbb R^{nN}thrm{d}eltalta\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.$$
Here
$\, \mathbb R^{nN}thrm{d}eltalta=\, \mathbb R^{nN}thrm{d}eltalta(Z,t)$ is as in \boldsymbol eqref{deltadist}, i.e. the distance from $(Z,t)\in \mathbb R^{nN}thcal Omegaega$ to $\mathcal Sigmagma$, and $T_{cQ_0}$ is the Carleson set associated to $cQ_0$ as defined in \boldsymbol eqref{eq2.box}.
\boldsymbol end{lemma}
\betagin{lemma}\lambdabel{lemmacruc+} Let $u(Z,t):=\omegaega(Z,t,S)$ and $c$ be as stated in Lemma \mathbb R^{nN}thbf{R}f{lemmacruc}.
Then there exists $\widetildelde c=\widetildelde c(m,\kappa, M_1,M_2,\mathcal Gammamma)$, $1\leq \widetildelde c<\infty$, such that
$$\iiint_{ T_{cQ_0}}|\boldsymbol \nuabla_Xu|^2\, \mathbb R^{nN}thrm{d}eltalta\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\leq \widetildelde c\sigmagma(Q_0).$$
\boldsymbol end{lemma}
The proofs of Lemmas \mathbb R^{nN}thbf{R}f{existcover}- \mathbb R^{nN}thbf{R}f{lemmacruc+} are given in the forthcoming sections of the paper. To prove Theorem \mathbb R^{nN}thbf{R}f{Ainfty} using these auxiliary lemmas, we note that first using Lemma \mathbb R^{nN}thbf{R}f{lemmacruc} and
Lemma \mathbb R^{nN}thbf{R}f{lemmacruc+} we can, for $\mathcal Upsilon\gg 1$ given, choose $\, \mathbb R^{nN}thrm{d}eltalta_0=\, \mathbb R^{nN}thrm{d}eltalta_0(m,M_1,\mathcal Upsilon)$, so that if
$E\subset {Q_0}$ with $\omegaega(E)\leq\, \mathbb R^{nN}thrm{d}eltalta_0$, then
\betagin{eqnarray}
\mathcal Upsilon^2\sigmagma(E)\leq \hat c\sigmagma(Q_0),
\boldsymbol end{eqnarray}
for some $\hat c=\hat c(m,\kappa, M_1,M_2,\mathcal Gammamma)$, $1\leq \hat c<\infty$. In particular, we can conclude that
there exists, for every $\varepsilon>0$, a positive $\, \mathbb R^{nN}thrm{d}eltalta_0=\, \mathbb R^{nN}thrm{d}eltalta_0(m,\kappa,M_1,M_2,\mathcal Gammamma,\varepsilon)$ such that
\betagin{eqnarray}\omegaega(E)\leq\, \mathbb R^{nN}thrm{d}eltalta_0\leq c\, \mathbb R^{nN}thrm{d}eltalta_0\omegaega({Q_0})\implies\sigmagma(E)\leq \varepsilon\sigmagma({Q_0}),
\boldsymbol end{eqnarray}
where we have also applied Lemma \mathbb R^{nN}thbf{R}f{bourg} stated below. Theorem \mathbb R^{nN}thbf{R}f{Ainfty} now follows from the doubling property of $\omegaega$, see Lemma \mathbb R^{nN}thbf{R}f{T:doubling}, and the classical result in \cite{CF}.
The rest of the paper is devoted to the proofs of Lemmas \mathbb R^{nN}thbf{R}f{existcover}- \mathbb R^{nN}thbf{R}f{lemmacruc+} and we consider the proof of Lemma \mathbb R^{nN}thbf{R}f{lemmacruc+} a rather difficult part in the proof of Theorem \mathbb R^{nN}thbf{R}f{Ainfty}. We here show how to reduce Lemma \mathbb R^{nN}thbf{R}f{lemmacruc+} to a core technical estimate. To prove Lemma \mathbb R^{nN}thbf{R}f{lemmacruc+} we can without loss of generality assume that
$(Z_{Q_0},t_{Q_0})=(0,0)$ and we let $\rho_0:=l(Q_0)$. Throughout the rest of the paper we let $\mathcal P$ denote a parabolic approximation of the identity: $\mathcal P\in C_0^\infty(\mathcal B_1(0,0))$, $\mathcal B_1(0,0)\subset\mathbb R^{nN}thbb R^{N-1}$, $\mathcal P\geq 0$ is real-valued, and $\iint \mathcal P\, \, \mathbb R^{nN}thrm{d} z \, \mathbb R^{nN}thrm{d} t=1$. We will assume, as we may
by imposing a product structure on $\mathcal P$, that $\mathcal P$ is even in the sense that
\betagin{eqnarray}\lambdabel{even}
\iint x_i\mathcal P(z,t)\, \, \mathbb R^{nN}thrm{d} z\, \mathbb R^{nN}thrm{d} t=\iint y_i\mathcal P(z,t)\, \, \mathbb R^{nN}thrm{d} z\, \mathbb R^{nN}thrm{d} t=\iint t\mathcal P(z,t)\, \, \mathbb R^{nN}thrm{d} z\, \mathbb R^{nN}thrm{d} t=0
\boldsymbol end{eqnarray}
for $i\in\{1,...,m-1\}$. We set $\mathcal P_\lambdambda(z,t)=\mathcal P_\lambdambda(x,y,t)=\lambdambda^{-{({\bf q}-4)}}\mathcal P(\lambdambda^{-1}x,\lambdambda^{-3}y,\lambdambda^{-2}t)$ whenever $\lambdambda>0$. Given $\mathcal P$ we let
$\mathcal P_\lambdambda$ define a convolution operator as introduced in \boldsymbol eqref{eq1vi}. To prove Lemma \mathbb R^{nN}thbf{R}f{lemmacruc+} we need to enable partial integration and we therefore use the mapping,
\betagin{eqnarray}\lambdabel{dom+ggaint}
U \owns (w,w_m,y,y_m,t) \mathbb R^{nN}psto (w,w_m+\mathcal P_{\gammamma w_m}\partialsii(w,y,t),y,y_m,t),
\boldsymbol end{eqnarray} where
\betagin{eqnarray}\lambdabel{dom+gint}
U&=&\{(W,Y,t)=(w, w_m,y,y_m, t)\in\mathbb R^{nN}thbb R^{m-1}\widetildemes\mathbb R^{nN}thbb R\widetildemes\mathbb R^{nN}thbb R^{m-1}\widetildemes\mathbb R^{nN}thbb R\widetildemes\mathbb R^{nN}thbb R \mid w_m>0\}.
\boldsymbol end{eqnarray}
We will need the following two lemmas proved in \cite{N1}. Lemma \mathbb R^{nN}thbf{R}f{carlemma-} and Lemma \mathbb R^{nN}thbf{R}f{carlemma} correspond to Lemma 2.1 and Lemma 2.2 in \cite{N1}, respectively.
\betagin{lemma}\lambdabel{carlemma-} Let $\partialsii$ be a function satisfying \boldsymbol eqref{Lip-++a1} for some constant $0<M_1<\infty$, let
$\gammamma\in (0,1)$ and let $\mathcal P_{\gammamma w_m}\partialsii$ be defined as above for $w_m>0$. Let
$\theta,\widetildelde\theta\geq 0$ be integers and let $(\partialhii_1,..,\partialhii_{m-1})$ and $(\widetildelde\partialhii_1,..,\widetildelde\partialhii_{m-1})$ denote multi-indices.
Let $\boldsymbol ell:=(\theta+|\partialhii|+3|\widetildelde\partialhii|+2\widetildelde\theta)$. Then
\betagin{eqnarray}\lambdabel{con1}
\biggl |\frac {\partialartial^{\theta+|\partialhii|+|\widetildelde\partialhii|}}{\partialartial w_m^{\theta}\partialartial w^{\partialhii}\partialartial y^{\widetildelde\partialhii}} \biggl ((w\cdot\boldsymbol \nuabla_y-\partialartial_t)^{\widetildelde \theta}(\mathcal P_{\gammamma w_m}\partialsii(w,y,t)) \biggr )\biggr |\leq c(m,l)\gammamma^{1-(l-\theta)}w_m^{1-l}M_1,
\boldsymbol end{eqnarray}
whenever $(W,Y,t)\in U$.
\boldsymbol end{lemma}
\betagin{lemma}\lambdabel{carlemma} Let $\partialsii$ be a function satisfying \boldsymbol eqref{Lip-++a1} and \boldsymbol eqref{Lip-++a2} for some constants $0<M_1,M_2<\infty$, let
$\gammamma\in (0,1)$ and let $\mathcal P_{\gammamma w_m}\partialsii$ be defined as above for $w_m>0$. Let
$\theta,\widetildelde\theta\geq 0$ be integers and let $(\partialhii_1,..,\partialhii_{m-1})$ and $(\widetildelde\partialhii_1,..,\widetildelde\partialhii_{m-1})$ denote multi-indices. Let $\boldsymbol ell:=(\theta+|\partialhii|+3|\widetildelde\partialhii|+2\widetildelde\theta)$. Let
\betagin{eqnarray*}
\, \mathbb R^{nN}thrm{d}\mu=\, \mathbb R^{nN}thrm{d}\mu(W,Y,t):=\biggl |\frac {\partialartial^{\theta+|\partialhii|+|\widetildelde\partialhii|}}{\partialartial w_m^{\theta}\partialartial w^{\partialhii}\partialartial y^{\widetildelde\partialhii}} \biggl (( w\cdot\boldsymbol \nuabla_y-\partialartial_t)^{\widetildelde \theta}(\mathcal P_{\gammamma w_m}\partialsii( w,y,t)) \biggr )\biggr |^2 w_m^{2l-3} \, \mathbb R^{nN}thrm{d} W\, \mathbb R^{nN}thrm{d} Y\, \mathbb R^{nN}thrm{d} t,
\boldsymbol end{eqnarray*}
be defined on $U$. Then
\betagin{eqnarray*}\lambdabel{con2}
\mu(Uu_{\rm B}p \mathbb R^{nN}thcal{B}_r)\leq c(m,l,M_1,M_2)\gammamma^{2-2(l-\theta)}r^{{\bf q}-1},
\boldsymbol end{eqnarray*}
for all balls $\mathbb R^{nN}thcal{B}_r=\mathbb R^{nN}thcal{B}_r(Z_0,t_0)\subset\mathbb R^{nN}thbb R^{N+1}$ centered on $\partialartial U$, $r>0$.
\boldsymbol end{lemma}
Using Lemma \mathbb R^{nN}thbf{R}f{carlemma-} we see that that there exists $\hat\gammamma=\hat\gammamma(m,M_1)\in (0,1)$ such that if $\gammamma\in (0,\hat\gammamma)$ then
\betagin{eqnarray}\lambdabel{1-1}\frac 1 2\leq 1+\frac {\partialartial}{\partialartial w_m}(\mathcal P_{\gammamma w_m}\partialsii)(w,y,t)\leq \frac 32,
\boldsymbol end{eqnarray}
whenever $(w,w_m,y,y_m,t)\in U$. This implies, in particular, that the map in \boldsymbol eqref{dom+ggaint} is one-to-one.
Defining $v$ as the pull-back of $u(Z,t):=\omegaega(Z,t,S)$ under the map in \boldsymbol eqref{dom+ggaint}, i.e.
\betagin{eqnarray}v(w,w_m,y,y_m,t):=u(w,w_m+\mathcal P_{\gammamma w_m}\partialsii(w,y,t),y,y_m,t)),
\boldsymbol end{eqnarray}
we see that to prove Lemma \mathbb R^{nN}thbf{R}f{lemmacruc+} it suffices to prove that
\betagin{eqnarray}\lambdabel{keyestalla}
I_\varepsilonsilon :=\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}|\boldsymbol \nuabla_{W}v|^2\mathcal Psii_\varepsilonsilon^2w_m\, \, \mathbb R^{nN}thrm{d} W\, \mathbb R^{nN}thrm{d} Y\, \mathbb R^{nN}thrm{d} t \leq c(m,\kappa, M_1, M_2,\mathcal Gammamma)\rho_0^{{\bf q}-1},
\boldsymbol end{eqnarray}
for $\varepsilonsilon>0$ small, where $$\mathbb R^{nN}thbb R^{N+1}_+=\mathbb R^{nN}thbb R^{m-1}\widetildemes\lbrace w_m>0 \rbrace\widetildemes \mathbb R^{nN}thbb R^{m}\widetildemes \mathbb R^{nN}thbb R,$$
and where $\mathcal Psii_\varepsilonsilon$ is a smooth cut-off function such that $\mathcal Psii_\varepsilonsilon\boldsymbol equiv 1$ on $$([-c,c]^m\widetildemes [-c,c]^m\widetildemes [-c,c])u_{\rm B}p \{w_m\geq 2\varepsilonsilon\},$$ and $\mathcal Psii_\varepsilonsilon\boldsymbol equiv 0$ on $$\bigl (([-2c,2c]^m\widetildemes [-2c,2c]^m\widetildemes [-2c,2c])\setminus ([-c,c]^m\widetildemes [-c,c]^m\widetildemes [-c,c])\bigr )u_{\rm B}p \{w_m<\varepsilonsilon\},$$ where $c=c(m,\kappa,M_1)\gg 1$.
Furthermore, the pull-back $v$ is a (weak) solution to
\betagin{eqnarray}\lambdabel{e-kolm-ndggha-int}
\widetildelde{\mathcal L} v=\boldsymbol \nuabla_{W}\cdot (\widetildelde A\boldsymbol \nuabla_{W} v)+\widetildelde B\cdot\boldsymbol \nuabla_{W} v+ D\cdot\boldsymbol \nuabla_{Y,t} v=0
\boldsymbol end{eqnarray}
in $U$ where the $\widetildelde A=(\widetildelde a_{i,j})$ and $\widetildelde B=(\widetildelde b_i)$ depend on $\mathcal L$ and the pull-back map in \boldsymbol eqref{dom+ggaint}. Here and in the following $\boldsymbol \nuabla_{Y,t}=(\boldsymbol \nuabla_Y,\partialartial_t)$ and $D$ is the vector valued function
\betagin{eqnarray}\lambdabel{e-kolm-ndggha-intD}
D:=(w,w_m+\mathcal P_{\gammamma w_m}\partialsii(w,y,t),-1).
\boldsymbol end{eqnarray}
Using that $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is an (unbounded) admissible ($y_m$-independent) Lipschitz domain with constants $(M_1,M_2)$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}, that $A$ satisfies \boldsymbol eqref{eq2} with constant $\kappa$, \boldsymbol eqref{struct}, and Lemma \mathbb R^{nN}thbf{R}f{carlemma-}, it follows that
$\widetildelde A$ and $\widetildelde B$ are measurable and locally bounded satisfying
\betagin{eqnarray}\lambdabel{eq2++}
\widetildelde\kappa^{-1}|\xi|^2\leq \sum_{i,j=1}^{m}\widetildelde a_{i,j}(W,Y,t)\xi_i\xi_j,\quad \ \ |\widetildelde A(W,Y,t)\xi\cdot\mathbf Zata|\leq \widetildelde\kappa|\xi||\mathbf Zata|,
\boldsymbol end{eqnarray}
for some $\widetildelde\kappa\in [1,\infty)$, and for all $\xi,\mathbf Zata\in \mathbb R^{nN}thbb R^{m}$, $(W,Y,t)\in\mathbb R^{nN}thbb R^{N+1}$, and
\betagin{eqnarray}\lambdabel{eq2++a}
w_m|\boldsymbol \nuabla_W \widetildelde A(W,Y,t)|+w_m|\widetildelde B(W,Y,t)|\leq c.
\boldsymbol end{eqnarray}
Here $\widetildelde \kappa$ and $c$ depends on $m$, $\kappa$ and $M_1$ only. In addition it is important to note that $\widetildelde A$ is symmetric. Furthermore, using Lemma \mathbb R^{nN}thbf{R}f{carlemma}, and that the measures $\mu_1$ and $\mu_2$ defined in \boldsymbol eqref{measure1} are Carleson measures on $\mathbb R^{nN}thcal Omegaega$ with constant $\mathcal Gammamma$, we see that if we introduce $\, \mathbb R^{nN}thrm{d}\widetildelde \mu_i=\, \mathbb R^{nN}thrm{d}\widetildelde\mu_i(W,Y,t)$, $i\in\{1,2,3\}$,
\betagin{equation}
\betagin{split}
\, \mathbb R^{nN}thrm{d}\widetildelde\mu_1&:= |\boldsymbol \nuabla_W\widetildelde A|^2w_m\, \, \mathbb R^{nN}thrm{d} W\, \mathbb R^{nN}thrm{d} Y\, \mathbb R^{nN}thrm{d} t,\\
\, \mathbb R^{nN}thrm{d}\widetildelde\mu_2&:=|\widetildelde B|^2w_m\, \, \mathbb R^{nN}thrm{d} W\, \mathbb R^{nN}thrm{d} Y\, \mathbb R^{nN}thrm{d} t,\\
\, \mathbb R^{nN}thrm{d}\widetildelde\mu_3&:=|D\cdot\boldsymbol \nuabla_{Y,t}\widetildelde A|^2w_m^3\, \, \mathbb R^{nN}thrm{d} W\, \mathbb R^{nN}thrm{d} Y\, \mathbb R^{nN}thrm{d} t,
\boldsymbol end{split}
\boldsymbol end{equation}
as measures on $U$, then
\betagin{eqnarray}\lambdabel{con2+}
\widetildelde\mu_i(Uu_{\rm B}p \mathbb R^{nN}thcal{B}_\rho(w_0,0,Y_0,t_0))\leq c(m,\kappa, M_1, M_2,\mathcal Gammamma)\rho^{{\bf q}-1},
\boldsymbol end{eqnarray}
whenever $(w_0,0,Y_0,t_0)\in\partialartial U$, $\rho>0$, and $\mathbb R^{nN}thcal{B}_\rho(w_0,0,Y_0,t_0)\subset\mathbb R^{nN}thbb R^{N+1}$, and for $i\in\{1,2,3\}$. In particular, all measures in $\{\widetildelde\mu_i\}$ define Carleson measures on $U$. Furthermore, we emphasize that by our assumptions
\betagin{eqnarray}\lambdabel{e-kolm-ndggha-lla+jja}
\mbox{$\widetildelde A$ and $\widetildelde B$ are independent of $y_m$}.
\boldsymbol end{eqnarray}
To prove \boldsymbol eqref{keyestalla} it suffices to prove the following lemma.
\betagin{lemma}\lambdabel{Carleson}
Let $\sigmagma\in (0,1)$ be a given degree of freedom. Then there exists a finite constant $c=c(m,\kappa, M_1, M_2,\mathcal Gammamma,\sigmagma)$, such that
\betagin{eqnarray*}
I_{\varepsilonsilon}\leq \sigmagma I_{\varepsilonsilon}+c\rho_0^{{\bf q}-1}.
\boldsymbol end{eqnarray*}
\boldsymbol end{lemma}
Note that by construction $I_{\varepsilonsilon}$ is finite. The proof of Lemma \mathbb R^{nN}thbf{R}f{Carleson} is given in the next section and from here on, and hence also in the proof, we will not indicate the dependence on $\varepsilonsilon$ and simply write $I$ for $I_\varepsilonsilon$ and we note -- and this is a consequence of the introduction of $\varepsilonsilon$ -- that no boundary terms will survive when we perform partial integration. In addition we will also here, with a slight abuse of notation, let $Z:=(W,Y)$ and $\, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t:=\, \mathbb R^{nN}thrm{d} W\, \mathbb R^{nN}thrm{d} Y\, \mathbb R^{nN}thrm{d} t$.
In the proof of Lemma \mathbb R^{nN}thbf{R}f{Carleson} we will also use the quantities
\betagin{eqnarray}\lambdabel{e-kolm-ndggha-lla+gg}
J&:=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}|D\cdot\boldsymbol \nuabla_{Y,t}v|^2\mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
K&:=&\sum_{i=1}^m\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ |\boldsymbol \nuabla_{W}(\partialartial_{w_i}v)|^2 \mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
L&:=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ |\boldsymbol \nuabla_{Y}v|^2 \mathcal Psii^6w_m^5\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\\
L_{i}&:=&
\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}|\partialartial_{y_i} v|^2\mathcal Psii^6w_m^5\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
M&:=&\sum_{i=1}^m \iiint_{\mathbb R^{nN}thbb R^{N+1}_+} |\boldsymbol \nuabla_W(\partialartial_{y_i}v)|^2\mathcal Psii^8w_m^7\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.\boldsymbol \nuotag
\boldsymbol end{eqnarray}
In the rather technical proof to follow, the crucial estimate in the proof of Lemma \mathbb R^{nN}thbf{R}f{Carleson} is stated in \boldsymbol eqref{sv} below and states that
\betagin{eqnarray*}
L_{m}\lesssim M^{1/2}J^{1/2}+I+J,
\boldsymbol end{eqnarray*}
where $\lesssim$ means that we can control the constants. This estimate uses, in a crucial way it seems, that $\partialsii$ and $A$, and hence $\widetildelde A$, do not depend on $y_m$. It seems that this additional degree of freedom is crucial for us to be able to complete the argument.
\section{Proof of Lemma \mathbb R^{nN}thbf{R}f{Carleson}}\lambdabel{sec5}
We will first prove that
\betagin{eqnarray}\lambdabel{auxest1}
I\leq c\rho_0^{{\bf q}-1}+\sigmagma I+\widetildelde\sigmagma J
\boldsymbol end{eqnarray}
where $\sigmagma,\widetildelde\sigmagma\in (0,1)$ are degrees of freedom and $c$ is a positive constant which, unless otherwise stated, only depends on $(m,\kappa, M_1, M_2,\mathcal Gammamma)$ and $\sigmagma, \widetildelde\sigmagma$. In general, in the following $c$ will denote a generic such constant, not necessarily the same at each instance. We often write $c_1\lesssim c_2$ and this means that $c_1/c_2$ is bounded by a constant depending only on $(m,\kappa, M_1, M_2,\mathcal Gammamma)$, $\sigmagma$ and $\widetildelde\sigmagma$.
To start the proof of \boldsymbol eqref{auxest1} we note, using ellipticity, that
\betagin{eqnarray*}\lambdabel{est4}
I\lesssim \sum_{i,j=1}^m I_{i,j},
\boldsymbol end{eqnarray*}
where
\betagin{align*}
I_{i,j}:=2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}\widetildelde a_{i,j}(\partialartial_{w_i}v)(\partialartial_{w_j}v)\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{align*}
Assume first that $i\boldsymbol \nueq m$. Then, integrating by parts in $I_{i,j}$ with respect to $w_i$ we see that
\betagin{eqnarray*}
I_{i,j}&=&-2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v\partialartial_{w_i}(\widetildelde a_{i,j}\partialartial_{w_j}v)\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\partialartial_{w_i}\widetildelde a_{m,m}^{-1}\widetildelde a_{i,j}v(\partialartial_{w_j}v)\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-4\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1} {\widetildelde a_{i,j}}v(\partialartial_{w_j}v)w_m\mathcal Psii\partialartial_{w_i}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{eqnarray*}
Similarly we see that
\betagin{eqnarray*}
I_{m,j}&=&-2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v\partialartial_{w_m}(\widetildelde a_{m,j}\partialartial_{w_j}v)\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\partialartial_{w_m}\widetildelde a_{m,m}^{-1}\widetildelde a_{m,j} v(\partialartial_{w_j}v)\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}}{\widetildelde a_{m,j}}v(\partialartial_{w_j}v)\mathcal Psii^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-4\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}}{\widetildelde a_{m,j}}v(\partialartial_{w_j}v)w_m\mathcal Psii\partialartial_{w_m}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{eqnarray*}
Put together
\betagin{eqnarray*}
I\leq I_1+I_2+I_3+I_4,
\boldsymbol end{eqnarray*}
where
\betagin{eqnarray*}
I_1&:=&-2\sum_{i,j}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v\partialartial_{w_i}(\widetildelde a_{i,j}\partialartial_{w_j}v)\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
I_2&:=&-2\sum_{i,j}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\partialartial_{w_i}\widetildelde a_{m,m}^{-1}\widetildelde a_{i,j}v(\partialartial_{w_j}v)\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
I_3&:=&-4\sum_{i,j}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1} {\widetildelde a_{i,j}}v(\partialartial_{w_j}v)w_m\mathcal Psii\partialartial_{w_i}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
I_4&:=&-2\sum_{j}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}}{\widetildelde a_{m,j}}v(\partialartial_{w_j}v)\mathcal Psii^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{eqnarray*}
We first analyze $I_1$. Using the equation, i.e. \boldsymbol eqref{e-kolm-ndggha-int}, we obtain
\betagin{eqnarray*}
I_1&=&2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v (D\cdot\boldsymbol \nuabla_{Y,t}v)\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+2\sum_{i}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}vb_i\partialartial_{w_i}v\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=:&I_{11}+I_{12},
\boldsymbol end{eqnarray*}
and \betagin{eqnarray*}
I_{12}&\leq&c\biggl (\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}v^2|\widetildelde B|^2\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr )^{1/2}
\biggl (\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}|\boldsymbol \nuabla_W v|^2\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr )^{1/2}\boldsymbol \nuotag\\
&\leq& c\rho_0^{{\bf q}-1}+\sigmagma I,
\boldsymbol end{eqnarray*}
by \boldsymbol eqref{con2+} applied to $\widetildelde\mu_2$. Furthermore, integrating by parts with respect to $w_m$ we see that
\betagin{eqnarray*}
I_{11}=I_{111}+I_{112}+I_{113}+I_{114},
\boldsymbol end{eqnarray*}
where
\betagin{eqnarray*}
I_{111}&=&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\partialartial_{w_m}\widetildelde a_{m,m}^{-1}v(D\cdot\boldsymbol \nuabla_{Y,t}v) \mathcal Psii^2w_m^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
I_{112}&=&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}\partialartial_{w_m}v(D\cdot\boldsymbol \nuabla_{Y,t}v) \mathcal Psii^2w_m^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
I_{113}&=&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v\partialartial_{w_m}(D\cdot\boldsymbol \nuabla_{Y,t}v) \mathcal Psii^2w_m^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
I_{114}&=&-2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v(D\cdot\boldsymbol \nuabla_{Y,t}v) w_m^2\mathcal Psii \partialartial_{w_m}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{eqnarray*}
Focusing on $I_{111}$ we see that
\betagin{eqnarray*}
I_{111}&=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\biggl (\frac {\partialartial_{w_m} \widetildelde a_{m,m}}{\widetildelde a_{m,m}^2}\biggr )v(D\cdot\boldsymbol \nuabla_{Y,t}v)\mathcal Psii^2w_m^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&\leq&c\biggl (\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}|\partialartial_{w_m}\widetildelde a_{m,m}|^2v^2 w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr)^{1/2}J^{1/2}\boldsymbol \nuotag\\
&\leq & c\rho_0^{{\bf q}-1}+\widetildelde\sigmagma J,
\boldsymbol end{eqnarray*}
by \boldsymbol eqref{con2+} applied to $\widetildelde\mu_1$. To continue we see that
\betagin{eqnarray*}
I_{113}&=&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v(D\cdot\boldsymbol \nuabla_{Y,t}\partialartial_{w_m}v) \mathcal Psii^2w_m^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
&&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v(1+\partialartial_{w_m}\mathcal P_{\gammamma w_m}\partialsii(w,y,t))(\partialartial_{y_m}v) \mathcal Psii^2w_m^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=:&I_{1131}+I_{1132}.
\boldsymbol end{eqnarray*}
To estimate $I_{1132}$ we write
\betagin{eqnarray*}
I_{1132}&=&-\frac 12\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}(1+\partialartial_{w_m}\mathcal P_{\gammamma w_m}\partialsii(w,y,t))(\partialartial_{y_m}v^2) \mathcal Psii^2w_m^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}(1+\partialartial_{w_m}\mathcal P_{\gammamma w_m}\partialsii(w,y,t)) v^2 w_m^2\mathcal Psii\partialartial_{y_m}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,
\boldsymbol end{eqnarray*}
where we have used that $\widetildelde a_{m,m}$ and $\partialsii$ are independent of $y_m$. In particular, $|I_{1132}|\leq c\rho_0^{{\bf q}-1}$. Focusing on
$I_{1131}$,
\betagin{eqnarray*}
I_{1131}&=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}((D\cdot\boldsymbol \nuabla_{Y,t})\widetildelde a_{m,m}^{-1})v(\partialartial_{w_m}v) \mathcal Psii^2w_m^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
&&+\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}(D\cdot\boldsymbol \nuabla_{Y,t}v)(\partialartial_{w_m}v) \mathcal Psii^2w_m^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v(\partialartial_{w_m}v) w_m^2\mathcal Psii(D\cdot\boldsymbol \nuabla_{Y,t})\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=:&I_{11311}+I_{11312}+I_{11313}.
\boldsymbol end{eqnarray*}
Again using \boldsymbol eqref{con2+} applied to $\widetildelde\mu_3$ and elementary estimates we see that
\betagin{eqnarray*}
|I_{11311}|+|I_{11313}|\leq c\rho_0^{{\bf q}-1}+\sigmagma I.
\boldsymbol end{eqnarray*}
Furthermore, \betagin{eqnarray*}
I_{11312}=-I_{112}.
\boldsymbol end{eqnarray*}
In particular, we have proved that
\betagin{eqnarray*}
I_1\leq c\rho_0^{{\bf q}-1}+\sigmagma I+\widetildelde\sigmagma J+|I_{114}|.
\boldsymbol end{eqnarray*}
To estimate $I_{114}$ we write
\betagin{eqnarray*}
I_{114}&=&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}(D\cdot\boldsymbol \nuabla_{Y,t}v^2)w_m^2\mathcal Psii \partialartial_{w_m}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}((D\cdot\boldsymbol \nuabla_{Y,t})\widetildelde a_{m,m}^{-1})v^2w_m^2\mathcal Psii \partialartial_{w_m}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v^2w_m^2\mathcal Psii((D\cdot\boldsymbol \nuabla_{Y,t}) \partialartial_{w_m}\mathcal Psii)\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\widetildelde a_{m,m}^{-1}v^2w_m^2 ((D\cdot\boldsymbol \nuabla_{Y,t})\mathcal Psii)\partialartial_{w_m}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=:&I_{1141}+I_{1142}+I_{1143}.
\boldsymbol end{eqnarray*}
Using \boldsymbol eqref{con2+} applied to $\widetildelde\mu_3$, and by now familiar arguments, we see that $|I_{114}|\leq c\rho_0^{{\bf q}-1}$. Put together we can conclude that
\betagin{eqnarray*}
I_1\leq c\rho_0^{{\bf q}-1}+\sigmagma I+\widetildelde\sigmagma J.
\boldsymbol end{eqnarray*}
It is straightforward to see that
\betagin{eqnarray*}
|I_2|+|I_3|\leq c\rho_0^{{\bf q}-1}+ \sigmagma I. \boldsymbol end{eqnarray*}
To estimate $I_4$ we write
\betagin{eqnarray*}
I_4=-2\sum_{j}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}}v(\partialartial_{w_j}v)\mathcal Psii^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.\boldsymbol \nuotag\\
\boldsymbol end{eqnarray*}
We first consider the term in the definition of $I_4$ which corresponds to $j=m$ and we note that
\betagin{eqnarray*}
\biggl|\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\partialartial_{w_m}(v^2)\mathcal Psii^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr |=2\biggl|\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}v^2\mathcal Psii\partialartial_{w_m}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr|\leq c\rho_0^{{\bf q}-1}.
\boldsymbol end{eqnarray*}
Next we consider the terms
in the definition of $I_4$ which corresponds to $j\boldsymbol \nueq m$. By integration by parts we see that
\betagin{equation*}
\betagin{split}
-2&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}}v(\partialartial_{w_j}v)\partialartial_{w_m}(w_m)\mathcal Psii^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\\
&= 2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\partialartial_{w_m}({\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}})v\partialartial_{w_j}v\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\\
&\quad +2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}}\partialartial_{w_m} v\partialartial_{w_j}v\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\\
&\quad +2\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}}v\partialartial_{w_mw_j}v\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&\quad +4\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}}v\partialartial_{w_j}vw_m\mathcal Psii\partialartial_{w_m}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{split}
\boldsymbol end{equation*}
Let
\betagin{eqnarray*}
I_{41}&:=&2\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}}\partialartial_{w_m}v\partialartial_{w_j}v\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
I_{42}&:=&2\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}}v\partialartial_{w_mw_j}v\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{eqnarray*}
By the above deductions, and using by now familiar arguments, we can conclude that \betagin{eqnarray*}
|I_4-I_{41}-I_{42}|\leq c\rho_0^{{\bf q}-1}+\sigmagma I.
\boldsymbol end{eqnarray*}
To estimate $I_{42}$ we use that $j\boldsymbol \nueq m$. Integrating by parts
\betagin{eqnarray*}
I_{42}&=&-2\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\partialartial_{w_j}({\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}})v\partialartial_{w_m}v\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-2\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}}\partialartial_{w_j}v\partialartial_{w_m}v\mathcal Psii^2w_m\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-4\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}{\widetildelde a_{m,m}^{-1}} {\widetildelde a_{m,j}}v\partialartial_{w_m}vw_m\mathcal Psii\partialartial_{w_j}\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=:&I_{421}+I_{422}+I_{423}. \boldsymbol end{eqnarray*}
Note that
\betagin{eqnarray*}
I_{422}=-I_{41},
\boldsymbol end{eqnarray*}
and that
\betagin{eqnarray*}
|I_{421}|+|I_{423}|\leq c\rho_0^{{\bf q}-1}+\sigmagma I,
\boldsymbol end{eqnarray*}
by familiar arguments. Summarizing we can conclude that
\betagin{eqnarray*}
I&\lesssim&|I_1|+|I_2|+|I_3|+|I_4|\leq c\rho_0^{{\bf q}-1}+\sigmagma I+\widetildelde\sigmagma J,
\boldsymbol end{eqnarray*}
where $\sigmagma$, $\widetildelde\sigmagma$, are degrees of freedom. This completes the proof of the estimate in \boldsymbol eqref{auxest1}.
The next step is to estimate $J$ in a similar fashion.
\subsection{Estimating the term $J$}
To estimate $J$ we write
\betagin{eqnarray*}\lambdabel{e-kolm-ndggha-lla+ggco}
J=-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}(D\cdot\boldsymbol \nuabla_{Y,t}v)
\bigl (\boldsymbol \nuabla_{W}\cdot (\widetildelde A\boldsymbol \nuabla_{W} v)+\widetildelde B\cdot\boldsymbol \nuabla_{W}v\bigr)\mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,
\boldsymbol end{eqnarray*}
and
\betagin{eqnarray}
J=J_{1}+J_{2}+J_{3}+J_{4},
\boldsymbol end{eqnarray}
where
\betagin{eqnarray*}
J_{1}&:=&-\sum_{j}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (D\cdot\boldsymbol \nuabla_{Y,t}v)\partialartial_{w_m}(\widetildelde a_{m,j}\partialartial_{w_j}v)
\mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
J_{2}&:=&-\sum_{i\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (D\cdot\boldsymbol \nuabla_{Y,t}v)\partialartial_{w_i}(\widetildelde a_{i,m}\partialartial_{w_m}v)
\mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
J_{3}&:=&-\sum_{i\boldsymbol \nueq m}\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (D\cdot\boldsymbol \nuabla_{Y,t}v)
\partialartial_{w_i}(\widetildelde a_{i,j}\partialartial_{w_j}v) \mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
J_{4}&:=&-\sum_{i}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (D\cdot\boldsymbol \nuabla_{Y,t}v){\widetildelde b_i}\partialartial_{w_i}v \mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{eqnarray*}
Using \boldsymbol eqref{eq2++a} we immediately see that
\betagin{eqnarray}\lambdabel{bound}
|J_{1}|+|J_{2}|+|J_{4}|\lesssim I^{1/2}J^{1/2}+\biggl (\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ |\boldsymbol \nuabla_W (\partialartial_{w_m}v)|^2 \mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr )^{1/2}J^{1/2}.
\boldsymbol end{eqnarray}
Focusing on $J_{3}$, and integrating by parts with respect to $w_i$, we see that
\betagin{eqnarray}
J_{3}&=&\sum_{i\boldsymbol \nueq m}\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ \partialartial_{w_i}(D\cdot\boldsymbol \nuabla_{Y,t}v)(\widetildelde a_{i,j}\partialartial_{w_j}v) \mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+4\sum_{i\boldsymbol \nueq m}\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (D\cdot\boldsymbol \nuabla_{Y,t}v)(\widetildelde a_{i,j}\partialartial_{w_j}v) \partialartial_{w_i}(\mathcal Psii)w_m^3\mathcal Psii^3 \, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\\
&=:&J_{31}+J_{32},\boldsymbol \nuotag
\boldsymbol end{eqnarray}
and that $|J_{32}|\leq cI^{1/2}J^{1/2}$. Furthermore,
\betagin{eqnarray*}
J_{31}&=&\sum_{i\boldsymbol \nueq m}\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (\partialartial_{y_i}v)(\widetildelde a_{i,j}\partialartial_{w_j}v) \mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+\sum_{i\boldsymbol \nueq m}\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (D\cdot\boldsymbol \nuabla_{Y,t}(\partialartial_{w_i}v))(\widetildelde a_{i,j}\partialartial_{w_j}v) \mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+\sum_{i\boldsymbol \nueq m}\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (\partialartial_{w_i}\mathcal P_{\gammamma w_m}\partialsii(w,y,t))(\partialartial_{y_m}v)(\widetildelde a_{i,j}\partialartial_{w_j}v) \mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=:&J_{311}+J_{312}+J_{313}.
\boldsymbol end{eqnarray*}
Then
\betagin{eqnarray}
|J_{311}|+|J_{313}|\lesssim \biggl (\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ |\boldsymbol \nuabla_{Y}v|^2 \mathcal Psii^6w_m^5\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr )^{1/2} I^{1/2}.
\boldsymbol end{eqnarray}
To estimate $J_{312}$ we lift
the vector field $D\cdot\boldsymbol \nuabla_{Y,t}$ through partial integration and
use the symmetry of the matrix $\{\widetildelde a_{i,j}\}$
to see that
\betagin{eqnarray*}
2J_{312}&=&-\sum_{i\boldsymbol \nueq m}\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (\partialartial_{w_i}v)(D\cdot\boldsymbol \nuabla_{Y,t}(\widetildelde a_{i,j}))\partialartial_{w_j}v \mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-4\sum_{i\boldsymbol \nueq m}\sum_{j\boldsymbol \nueq m}\iiint_{\mathbb R^{nN}thbb R^{N+1}_+}\ (\partialartial_{w_i}v)(\widetildelde a_{i,j}\partialartial_{w_j}v)
(D\cdot\boldsymbol \nuabla_{Y,t}(\mathcal Psii))w_m^3\mathcal Psii^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=:&J_{3121}+J_{3122}.
\boldsymbol end{eqnarray*}
Then, by familiar arguments,
\betagin{eqnarray}
|J_{3121}|+|J_{3122}|\leq cI.
\boldsymbol end{eqnarray}
Let $K$ and $L$ be as introduced in \boldsymbol eqref{e-kolm-ndggha-lla+gg}. Then, putting all estimates together we can conclude that
\betagin{equation}
\betagin{split}
J&\leq |J_{1}|+|J_{2}|+|J_{3}|+|J_{4}|\\
&\lesssim I+I^{1/2}J^{1/2}+J^{1/2}K^{1/2}+I^{1/2}L^{1/2}.
\boldsymbol end{split}
\boldsymbol end{equation}
Hence
\betagin{eqnarray}
J&\lesssim& I+K+I^{1/2}L^{1/2}.
\boldsymbol end{eqnarray}
To proceed we have to estimate $K$ and $L$.
\subsection{Estimating the term $K$} To start the argument for $K$ we introduce $\widetildelde v=\partialartial_{w_i} v$ and we use \boldsymbol eqref{e-kolm-ndggha-int} to conclude that
$\widetildelde v$ solves
\betagin{equation}\lambdabel{e-kolm-nd+a}
\betagin{split}
&\boldsymbol \nuabla_{W}\cdot (\widetildelde A\boldsymbol \nuabla_{W} \widetildelde v)+\widetildelde B\cdot\boldsymbol \nuabla_{W} \widetildelde v+(D\cdot\boldsymbol \nuabla_{Y,t})\widetildelde v\\
&=-\boldsymbol \nuabla_{W}\cdot (\partialartial_{w_i} \widetildelde A\boldsymbol \nuabla_{W} v)-\partialartial_{w_i} \widetildelde B\cdot\boldsymbol \nuabla_{W} v-\partialartial_{y_i}v-\partialartial_{w_i}\mathcal P_{\gammamma w_m}\partialsii(w,y,t)\partialartial_{y_m}v
\boldsymbol end{split}
\boldsymbol end{equation}
in $U$.
Multiplying the equation in \boldsymbol eqref{e-kolm-nd+a} with $\widetildelde v\mathcal Psii^4w_m^3$, integrating and using Cauchy-Schwarz we see that
\betagin{eqnarray}\lambdabel{Lintbparts}
K\lesssim I+|K_1|+|K_2|+|K_3|+|K_4|,
\boldsymbol end{eqnarray}
where
\betagin{eqnarray*}
K_1&:=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} \bigl ((D\cdot\boldsymbol \nuabla_{Y,t})\widetildelde v\bigr )\widetildelde v\mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
K_2&:=& \iiint_{\mathbb R^{nN}thbb R^{N+1}_+} \bigl (\boldsymbol \nuabla_{W}\cdot ((\partialartial_{w_i} \widetildelde A)\boldsymbol \nuabla_{W}v)\bigr )\widetildelde v\mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
K_3&:=& \iiint_{\mathbb R^{nN}thbb R^{N+1}_+} \bigl (\partialartial_{w_i}\widetildelde B\cdot \boldsymbol \nuabla_{W}v\bigr )\widetildelde v\mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,\boldsymbol \nuotag\\
K_4&:=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (\partialartial_{y_i}v+\partialartial_{w_i}\mathcal P_{\gammamma w_m}\partialsii(w,y,t)\partialartial_{y_m}v)\widetildelde v\mathcal Psii^4w_m^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{eqnarray*}
Using \boldsymbol eqref{eq2++a} we immediately see that
\betagin{eqnarray}
|K_2|+|K_3|+|K_4|\lesssim I+I^{1/2}K^{1/2}+I^{1/2}L^{1/2}.
\boldsymbol end{eqnarray}
Furthermore,
\betagin{equation}\lambdabel{acom}
\betagin{split}
2|K_1| &\leq 4\biggl |\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} \widetildelde v^2\bigl ((w,w_m+\mathcal P_{\gammamma w_m}\partialsii(w,y,t))
\cdot \boldsymbol \nuabla_Y-\partialartial_t\bigr)(\mathcal Psii)w_m^3\mathcal Psii^3\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr |\\
&\lesssim I,
\boldsymbol end{split}
\boldsymbol end{equation}
and we can conclude that
\betagin{eqnarray}
K\lesssim I+|K_1|+|K_2|+|K_3|+|K_4|\lesssim K\lesssim I+I^{1/2}K^{1/2}+I^{1/2}L^{1/2}.
\boldsymbol end{eqnarray}
Hence
\betagin{eqnarray}
K\lesssim I+I^{1/2}L^{1/2}.
\boldsymbol end{eqnarray}
\subsection{Estimating the term $L$ ($L_i$)}
Focusing on $L$ we write
\betagin{eqnarray}
L=\sum_{i=1}^m L_{i}
\boldsymbol end{eqnarray}
where $L_i$ is defined in \boldsymbol eqref{e-kolm-ndggha-lla+gg}.
Note that
\betagin{eqnarray}\lambdabel{dyivrel}
\partialartial_{y_i} v=-(D\cdot\boldsymbol \nuabla_{Y,t})(\partialartial_{w_i}v)+\partialartial_{w_i}(D\cdot\boldsymbol \nuabla_{Y,t})(v)-(\partialartial_{w_i}\mathcal P_{\gammamma w_m}\partialsii(w,y,t))\partialartial_{y_m}v.
\boldsymbol end{eqnarray}
Hence,
\betagin{eqnarray*}
L_{i}&=&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (\partialartial_{y_i} v)(D\cdot\boldsymbol \nuabla_{Y,t})(\partialartial_{w_i}v)w_m^5\mathcal Psii^6\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (\partialartial_{y_i} v)\partialartial_{w_i}(D\cdot\boldsymbol \nuabla_{Y,t})(v)w_m^5\mathcal Psii^6\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (\partialartial_{y_i} v)((\partialartial_{w_i}\mathcal P_{\gammamma w_m}\partialsii(w,y,t))\partialartial_{y_m}v)w_m^5\mathcal Psii^6\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=:&L_{i,1}+L_{i,2}+L_{i,3}.
\boldsymbol end{eqnarray*}
Using partial integration
we immediately see that
\betagin{eqnarray}
|L_{i,2}|\lesssim M^{1/2}J^{1/2}+L_{i}^{1/2}J^{1/2}
\boldsymbol end{eqnarray}
where also $M$ was defined in \boldsymbol eqref{e-kolm-ndggha-lla+gg}. Furthermore,
\betagin{eqnarray*}
L_{i,1}&=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (D\cdot\boldsymbol \nuabla_{Y,t})(\partialartial_{y_i} v)(\partialartial_{w_i}v)w_m^5\mathcal Psii^6\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+6\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (\partialartial_{y_i} v)(\partialartial_{w_i}v)w_m^5\mathcal Psii^5(D\cdot\boldsymbol \nuabla_{Y,t})\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&=&\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} \partialartial_{y_i} (D\cdot\boldsymbol \nuabla_{Y,t})(v)(\partialartial_{w_i}v)w_m^5\mathcal Psii^6\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (\partialartial_{y_i} \mathcal P_{\gammamma w_m}\partialsii(w,y,t))(\partialartial_{y_m} v)(\partialartial_{w_i}v)w_m^5\mathcal Psii^6\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+6\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (\partialartial_{y_i} v)(\partialartial_{w_i}v)w_m^5\mathcal Psii^5(D\cdot\boldsymbol \nuabla_{Y,t})\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{eqnarray*}
Integrating by parts we have
\betagin{eqnarray*}
L_{i,1} &=&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (D\cdot\boldsymbol \nuabla_{Y,t})(v)(\partialartial_{w_iy_i}v)w_m^5\mathcal Psii^6\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-6\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (D\cdot\boldsymbol \nuabla_{Y,t})(v)(\partialartial_{w_i}v)w_m^5\mathcal Psii^5\partialartial_{y_i} \mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&-\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (\partialartial_{y_i} \mathcal P_{\gammamma w_m}\partialsii(w,y,t))(\partialartial_{y_m} v)(\partialartial_{w_i}v)w_m^5\mathcal Psii^6\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\boldsymbol \nuotag\\
&&+6\iiint_{\mathbb R^{nN}thbb R^{N+1}_+} (\partialartial_{y_i} v)(\partialartial_{w_i}v)w_m^5\mathcal Psii^5(D\cdot\boldsymbol \nuabla_{Y,t})\mathcal Psii\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t.
\boldsymbol end{eqnarray*}
Hence we can first conclude that
\betagin{eqnarray}
|L_{i,1}|\lesssim J^{1/2}M^{1/2}+I^{1/2}J^{1/2}+I^{1/2}(L_{i}^{1/2}+L_{m}^{1/2})
\boldsymbol end{eqnarray}
and then by collecting the estimates
\betagin{eqnarray}\lambdabel{hata}
L_{i}\lesssim I^{1/2}J^{1/2}+I^{1/2}(L_{i}^{1/2}+L_{m}^{1/2})+M^{1/2}J^{1/2}+L_{i}^{1/2}J^{1/2}+|L_{i,3}|.
\boldsymbol end{eqnarray}
We now first consider the case $i=m$. Using \boldsymbol eqref{1-1} and the above we immediately see that
\betagin{eqnarray}\lambdabel{sv}
L_{m}\lesssim M^{1/2}J^{1/2}+I+J.
\boldsymbol end{eqnarray}
Consider now $i\boldsymbol \nueq m$. Then, using \boldsymbol eqref{hata} we have
\betagin{eqnarray}\lambdabel{hata+}
L_{i}\lesssim I^{1/2}J^{1/2}+I^{1/2}(L_{i}^{1/2}+L_{m}^{1/2})+M^{1/2}J^{1/2}+L_{i}^{1/2}J^{1/2}+L_{i}^{1/2}L_{m}^{1/2}.
\boldsymbol end{eqnarray}
Hence
\betagin{eqnarray}
L_{i}\lesssim I+I^{1/2}J^{1/2}+I^{1/2}L_{m}^{1/2}+M^{1/2}J^{1/2}+J+L_{m}
\boldsymbol end{eqnarray}
for $i\boldsymbol \nueq m$. In particular, using \boldsymbol eqref{sv}
\betagin{eqnarray}
L_{i}\lesssim I^{1/2}J^{1/2}+M^{1/2}J^{1/2}+I+J\mbox{ for all }i\in\{1,...,m\}.
\boldsymbol end{eqnarray}
Still the auxiliary term $M$ has to be estimated.
\subsection{Estimating the term $M$} To estimate $M$ we
introduce $\widetildelde v=\partialartial_{y_i} v$ and using the equation we see that
$\widetildelde v$ solves
\betagin{equation}\lambdabel{e-kolm-nd+auu}
\betagin{split}
&\boldsymbol \nuabla_{W}\cdot (\widetildelde A\boldsymbol \nuabla_{W} \widetildelde v)+\widetildelde B\cdot\boldsymbol \nuabla_{W} \widetildelde v+(D\cdot\boldsymbol \nuabla_{Y,t})\widetildelde v\\
&=-\boldsymbol \nuabla_{W}\cdot (\partialartial_{y_i}\widetildelde A\boldsymbol \nuabla_{W} v)-\partialartial_{y_i}\widetildelde B\cdot\boldsymbol \nuabla_{W} v+\partialartial_{y_i}\mathcal P_{\gammamma w_m}\partialsii(w,y,t)\partialartial_{y_m}v,
\boldsymbol end{split}
\boldsymbol end{equation}
in $U$. Multiplying this equation with $\widetildelde v w_m^7\mathcal Psii^8$ and arguing similarly as to the estimates in the case of the expression $K$ we derive
\betagin{eqnarray}\lambdabel{acomuu}
M\lesssim L+I^{1/2}L^{1/2}+M^{1/2}L^{1/2}+K^{1/2}L^{1/2}.
\boldsymbol end{eqnarray}
Hence,
\betagin{eqnarray}\lambdabel{acomuu+}
M\lesssim L+I^{1/2}L^{1/2}+K^{1/2}L^{1/2}.
\boldsymbol end{eqnarray}
\subsection{Completing the proof of Lemma \mathbb R^{nN}thbf{R}f{Carleson}}
We are now ready to complete the proof of Lemma \mathbb R^{nN}thbf{R}f{Carleson} by collecting all the terms and estimates developed above. To summarize we have proved that
\betagin{equation}\lambdabel{acomuu++}
\betagin{split}
I&\leq c\rho_0^{{\bf q}-1}+\sigmagma I+\widetildelde\sigmagma J,\\
J&\lesssim I+K+I^{1/2}L^{1/2},\\
K&\lesssim I+I^{1/2}L^{1/2},\\
L&\lesssim I^{1/2}J^{1/2}+M^{1/2}J^{1/2}+I+J,\\
M&\lesssim L+I^{1/2}L^{1/2}+K^{1/2}L^{1/2}.
\boldsymbol end{split}
\boldsymbol end{equation}
We again note that by construction of the test function $\mathcal Psii$ we can ensure that $I,\, \mathbb R^{nN}thrm{d}ots,M$ are finite. Using \boldsymbol eqref{acomuu++} we first see that
\betagin{eqnarray*}\lambdabel{acomuu++a}
J+K&\lesssim& I+\varepsilonsilon_1L,\boldsymbol \nuotag\\
L&\lesssim& I +J+\varepsilonsilon_2M,\boldsymbol \nuotag\\
M&\lesssim& L+I+\varepsilonsilon_3K,
\boldsymbol end{eqnarray*}
where $\varepsilonsilon_1,\varepsilonsilon_2$ and $\varepsilonsilon_3$ are positive degrees of freedom. Using the estimates for $L$ and $M$ we have
\betagin{eqnarray*}\lambdabel{acomuu++b}
L&\lesssim& I +J+\varepsilonsilon_4K.
\boldsymbol end{eqnarray*}
Hence
\betagin{eqnarray*}\lambdabel{acomuu++c}
J+K\leq c(I+c\varepsilonsilon_1( I +J+\varepsilonsilon_4K)),
\boldsymbol end{eqnarray*}
and we can conclude that
\betagin{eqnarray*}\lambdabel{acomuu++d}
J+K\lesssim I.
\boldsymbol end{eqnarray*}
In particular,
\betagin{eqnarray*}
I\leq c\rho_0^{{\bf q}-1}+\sigmagma I+\widetildelde \sigmagma I
\boldsymbol end{eqnarray*}
and the proof is complete.
\mathbb R^{nN}thbb Qd
\section{Proof of auxiliary lemmas}\lambdabel{sec6}
In this section we prove Lemma \mathbb R^{nN}thbf{R}f{existcover} and Lemma \mathbb R^{nN}thbf{R}f{lemmacruc} and in the proof we need a number of estimates for non-negative solutions recently established in \cite{LN}.
Lemma \mathbb R^{nN}thbf{R}f{lem4.7bol} and Lemma \mathbb R^{nN}thbf{R}f{lem4.5-Kyoto1} below are Lemma 4.15 and Lemma 5.2 in \cite{LN}, respectively.
\betagin{lemma}\lambdabel{lem4.7bol} Let $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is an unbounded ($y_m$-independent) Lipschitz domain with constant $M_1$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}. Then there exist
$\mathcal Lambdambda=\mathcal Lambdambda(m,M_1)$, $1\leq \mathcal Lambdambda<\infty$, $c=c(m,\kappa,M_1)$, $1\leq c<\infty$, and $\gammamma=\gammamma(m,\kappa,M_1)$, $0<\gammamma<\infty$, such that the following is true. Let $(Z_0,t_0)\in\partialartial\mathbb R^{nN}thcal Omegaega$ and $r>0$. Assume that $u$ is a non-negative (weak) solution to $\mathcal L u=0$ in $\mathbb R^{nN}thcal Omegaegau_{\rm B}p \mathbb R^{nN}thcal{B}_{2r}(Z_0,t_0)$ and consider $\rho$, $\widetildelde\rho$, $0<\widetildelde\rho\leq\rho<r/c$. Then
\betagin{equation}\lambdabel{coneset-lem39}
\betagin{split}
&u(A_{\widetildelde\rho,\mathcal Lambdambda}^+(Z_0,t_0))\leq c(\rho/\widetildelde\rho)^\gammamma u(A_{\rho,\mathcal Lambdambda}^+(Z_0,t_0)),\\
&u(A_{\widetildelde\rho,\mathcal Lambdambda}^-(Z_0,t_0))\geq c^{-1} (\widetildelde\rho/\rho)^\gammamma u(A_{\rho,\mathcal Lambdambda}^-(Z_0,t_0)).
\boldsymbol end{split}
\boldsymbol end{equation}
\boldsymbol end{lemma}
\betagin{lemma}\lambdabel{lem4.5-Kyoto1} Let $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is an unbounded ($y_m$-independent) Lipschitz domain with constant $M_1$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}.
Let $(Z_0,t_0)\in\partialartial\mathbb R^{nN}thcal Omegaega$ and $r>0$. Let $\theta\in (0,1)$ be given. Then there exists $c=c(m,\kappa,M_1,\theta)$,
$1 \leq c < \infty$, such that following holds. Assume that $u$ is a non-negative (weak) solution to $\mathcal L u=0$ in $\mathbb R^{nN}thcal Omegaegau_{\rm B}p \mathbb R^{nN}thcal{B}_{2r}(Z_0,t_0)$,
vanishing continuously on $\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p \mathbb R^{nN}thcal{B}_{2r}(Z_0,t_0)$. Then
\betagin{eqnarray}
\sup_{\mathbb R^{nN}thcal Omegaegau_{\rm B}p \mathbb R^{nN}thcal{B}_{r/c}(Z_0,t_0)}u\leq \theta\sup_{\mathbb R^{nN}thcal Omegaegau_{\rm B}p \mathbb R^{nN}thcal{B}_{2r}(Z_0,t_0)}u.
\boldsymbol end{eqnarray}
\boldsymbol end{lemma}
\betagin{remark}\lambdabel{remnot}
Let $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ be an unbounded ($y_m$-independent) Lipschitz domain with constant $M_1$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}. The constants $\mathcal Lambdambda=\mathcal Lambdambda(m,M_1)$, $1\leq \mathcal Lambdambda<\infty$, $c=c(m,\kappa,M_1)$, $1\leq c<\infty$, referred to in Lemma \mathbb R^{nN}thbf{R}f{lem4.7bol} are fixed in \cite{LN}. In the following we also let $\mathcal Lambdambda$ and $c$ be determined accordingly.
\boldsymbol end{remark}
\betagin{lemma}\lambdabel{bourg} Let $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is an unbounded ($y_m$-independent) Lipschitz domain with constant $M_1$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}. Let $\mathcal Lambdambda=\mathcal Lambdambda(m,M_1)$ be in accordance with Remark \mathbb R^{nN}thbf{R}f{remnot}. Let $(Z_0,t_0)\in\partialartial\mathbb R^{nN}thcal Omegaega$ and $r>0$. Then
\betagin{eqnarray*}
&&\omegaega( A_{r/c,\mathcal Lambdambda}^+(Z_0,t_0),\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p \mathbb R^{nN}thcal{B}_{r}(Z_0,t_0))\geq c^{-1}.
\boldsymbol end{eqnarray*}
\boldsymbol end{lemma}
\betagin{proof}
This follows immediately from Lemma \mathbb R^{nN}thbf{R}f{lem4.5-Kyoto1}.
\boldsymbol end{proof}
Lemmas \mathbb R^{nN}thbf{R}f{T:doubling}- \mathbb R^{nN}thbf{R}f{lemmacruc-} below are Theorem 3.6, Lemma 12.2 and Lemma 12.3 in \cite{LN}, respectively.
\betagin{lemma}\lambdabel{T:doubling} Let $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is an unbounded ($y_m$-independent) Lipschitz domain with constant $M_1$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}. Let $\mathcal Lambdambda=\mathcal Lambdambda(m,M_1)$ be in accordance with Remark \mathbb R^{nN}thbf{R}f{remnot}. Let $(Z_0,t_0)\in\partialartial\mathbb R^{nN}thcal Omegaega$ and $r>0$. Then there exists $c=c(m,\kappa,M_1)$, $1\leq c<\infty$, such that
\betagin{eqnarray*}
&&\omegaega( A_{r,\mathcal Lambdambda}^+(Z_0,t_0),\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p \mathbb R^{nN}thcal{B}_{2\widetildelde r}(\widetildelde Z_0,\widetildelde t_0))\leq c
\omegaega( A_{r,\mathcal Lambdambda}^+(Z_0,t_0),\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p \mathbb R^{nN}thcal{B}_{\widetildelde r}(\widetildelde Z_0,\widetildelde t_0))
\boldsymbol end{eqnarray*}
whenever $(\widetildelde Z_0,\widetildelde t_0)\in\partialartial\mathbb R^{nN}thcal Omegaega$, $\mathbb R^{nN}thcal{B}_{\widetildelde r}(\widetildelde Z_0,\widetildelde t_0)\subset \mathbb R^{nN}thcal{B}_{r/c}(Z_0,t_0)$.
\boldsymbol end{lemma}
\betagin{lemma}\lambdabel{lem4.5-Kyoto1ha}Let $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is an unbounded ($y_m$-independent) Lipschitz domain with constant $M_1$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}. Let $\mathcal Lambdambda=\mathcal Lambdambda(m,M_1)$ be in accordance with Remark \mathbb R^{nN}thbf{R}f{remnot}. Let $(Z_0,t_0)\in\partialartial\mathbb R^{nN}thcal Omegaega$ and $r>0$. Let $(\widetildelde Z_0,\widetildelde t_0)\in\partialartial\mathbb R^{nN}thcal Omegaega$ and $\widetildelde r>0$ be such that $\mathbb R^{nN}thcal{B}_{\widetildelde r}(\widetildelde Z_0,\widetildelde t_0)\subset \mathbb R^{nN}thcal{B}_{r}(Z_0,t_0)$. Then there exists $c=c(m,\kappa,M_1)$, $1\leq c<\infty$, such that
\betagin{eqnarray}\lambdabel{ad}
K(A_{c\widetildelde r,\mathcal Lambdambda}^+(\widetildelde Z_0,\widetildelde t_0),\bar Z,\bar t):=\lim_{\bar r\to 0}\frac{\omegaega(A_{c\widetildelde r,\mathcal Lambdambda}^+(\widetildelde Z_0,\widetildelde t_0),\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p\mathbb R^{nN}thcal{B}_{\bar r}(\bar Z,\bar t))}{\omegaega(A_{cr,\mathcal Lambdambda}^+(Z_0,t_0),\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p\mathbb R^{nN}thcal{B}_{\bar r}(\bar Z,\bar t))}
\boldsymbol end{eqnarray}
exists for $\omegaega(A_{cr,\mathcal Lambdambda}^+(Z_0,t_0),\cdot)$-a.e. $(\bar Z,\bar t)\in \partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p\mathbb R^{nN}thcal{B}_{\widetildelde r}(\widetildelde Z_0,\widetildelde t_0)$, and
\betagin{eqnarray}\lambdabel{ad1}
c^{-1}\leq {\omegaega(A_{cr,\mathcal Lambdambda}^+(Z_0,t_0), \partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p\mathbb R^{nN}thcal{B}_{\widetildelde r}(\widetildelde Z_0,\widetildelde t_0))}K(A_{c\widetildelde r,\mathcal Lambdambda}^+(\widetildelde Z_0,\widetildelde t_0),\bar Z,\bar t)\leq c
\boldsymbol end{eqnarray}
whenever $(\bar Z,\bar t)\in \partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p\mathbb R^{nN}thcal{B}_{\widetildelde r}(\widetildelde Z_0,\widetildelde t_0)$.
\boldsymbol end{lemma}
\betagin{lemma}\lambdabel{lemmacruc-} Let $\mathbb R^{nN}thcal Omegaega\subset\mathbb R^{nN}thbb R^{N+1}$ is an unbounded ($y_m$-independent) Lipschitz domain with constant $M_1$ in the sense of Definition \mathbb R^{nN}thbf{R}f{car}. Let $\mathcal Lambdambda=\mathcal Lambdambda(m,M_1)$ be in accordance with Remark \mathbb R^{nN}thbf{R}f{remnot}. Let $(Z_0,t_0)\in\partialartial\mathbb R^{nN}thcal Omegaega$ and $r>0$. Let $(\widetildelde Z_0,\widetildelde t_0)\in\partialartial\mathbb R^{nN}thcal Omegaega$ and $\widetildelde r>0$ be such that $\mathbb R^{nN}thcal{B}_{\widetildelde r}(\widetildelde Z_0,\widetildelde t_0)\subset \mathbb R^{nN}thcal{B}_{r}(Z_0,t_0)$. Then there exist $c=c(m,\kappa,M_1)$, $1\leq c<\infty$, and $\widetildelde c=\widetildelde c(m,\kappa,M_1)$, $1\leq \widetildelde c<\infty$, such that
\betagin{eqnarray*}
\quad \widetildelde c^{-1}\omegaega(A_{c\widetildelde r,\mathcal Lambdambda}^+(\widetildelde Z_0,\widetildelde t_0),E)\leq \frac {\omegaega(A_{cr,\mathcal Lambdambda}^+(Z_0,t_0),E)}{\omegaega(A_{cr,\mathcal Lambdambda}^+(Z_0,t_0),\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p \mathbb R^{nN}thcal{B}_{\widetildelde r}(\widetildelde Z_0,\widetildelde t_0))}\leq \widetildelde c \omegaega(A_{c\widetildelde r,\mathcal Lambdambda}^+(\widetildelde Z_0,\widetildelde t_0),E),
\boldsymbol end{eqnarray*}
whenever $E\subset \mathbb R^{nN}thcal{B}_{\widetildelde r}(\widetildelde Z_0,\widetildelde t_0)$.
\boldsymbol end{lemma}
\subsection{Proof of Lemma \mathbb R^{nN}thbf{R}f{existcover}} Let in the following $Q_0\in \mathbb R^{nN}thbb{D}$, $\varepsilonsilon_0\in (0,1)$, and let $\omegaega(\cdot)$ be as in the statement of Theorem \mathbb R^{nN}thbf{R}f{Ainfty}. Let $k\in \mathbb R^{nN}thbb Z_+$ be given. Let $\gammamma$, $0<\gammamma\ll 1$, and $\mathcal Upsilon$, $1\ll\mathcal Upsilon$, be degrees of freedom to be chosen depending only on $m$, $\kappa$ and $M_1$. Let $\, \mathbb R^{nN}thrm{d}eltalta_0=\gammamma(\varepsilonsilon_0/\mathcal Upsilon)^k$. Suppose that $\omegaega(E)\leq\, \mathbb R^{nN}thrm{d}eltalta_0$. Using that $\omegaega$ is a regular Borel measure, we see that there exists a (relatively) open subset of ${Q}_0$, containing $E$, which we denote by $\mathbb R^{nN}thcal{O}_{k+1}$, satisfying
$\omegaega(\mathbb R^{nN}thcal{O}_{k+1})\leq 2\omegaega(E)$. Using Lemma \mathbb R^{nN}thbf{R}f{bourg} and the Harnack inequality, see Lemma \mathbb R^{nN}thbf{R}f{lem4.7bol}, we see that there exists $c=c(m,\kappa,M_1)$, $1\leq c<\infty$, such that
\betagin{eqnarray}\lambdabel{yy1}
\omegaega(\mathbb R^{nN}thcal{O}_{k+1})\leq 2\, \mathbb R^{nN}thrm{d}eltalta_0\leq c\, \mathbb R^{nN}thrm{d}eltalta_0\omegaega({Q}_0)\leq \frac 1 2 \biggl (\frac {\varepsilonsilon_0}{\mathcal Upsilon}\biggr )^k\omegaega({Q}_0)
\boldsymbol end{eqnarray}
if we let $\gammamma:=1/(2c)$. Let $f\in L^1_{\mbox{loc}}(\mathcal Sigmagma,\, \mathbb R^{nN}thrm{d}\omegaega)$, and let
$$M_{\omegaega}(f)(Z,t):=\sup_{\{\mathbb R^{nN}thcal{B}_r(\widetildelde Z,\widetildelde t):\ (\widetildelde Z,\widetildelde t)\in\mathcal Sigmagma,\ (Z,t)\in\mathbb R^{nN}thcal{B}_r(\widetildelde Z,\widetildelde t)\}}\frac 1{\omegaega(\mathbb R^{nN}thcal{B}_r(\widetildelde Z,\widetildelde t))}\iint_{\mathbb R^{nN}thcal{B}_r(\widetildelde Z,\widetildelde t)} |f|\, \, \mathbb R^{nN}thrm{d}\omegaega,$$
denote the Hardy-Littlewood maximal function of $f$, with respect to $\omegaega$, and
where the supremum is taken over all balls $\mathbb R^{nN}thcal{B}_r(\widetildelde Z,\widetildelde t)$, $(\widetildelde Z,\widetildelde t)\in\mathcal Sigmagma$, containing $(Z,t)$. Set
\betagin{equation}\lambdabel{lem:coverexists_Okdef}
\mathbb R^{nN}thcal{O}_k:=\{(Z,t)\in {Q}_0\mid M_{\omegaega}(\chii_{\mathbb R^{nN}thcal{O}_{k+1}})\geq {\varepsilonsilon_0}/{\bar c}\},
\boldsymbol end{equation}
where $\chii_{\mathbb R^{nN}thcal{O}_{k+1}}$ denotes the indicator function for the set $\mathbb R^{nN}thcal{O}_{k+1}$, and where we let $\bar c=\bar c(m,\kappa,M_1)$, $1\leq \bar c<\infty$, denote the constant appearing in Lemma \mathbb R^{nN}thbf{R}f{T:doubling}. Then, by construction,
$\mathbb R^{nN}thcal{O}_{k+1}\subset\mathbb R^{nN}thcal{O}_k$, $\mathbb R^{nN}thcal{O}_k$ is relatively open in
${Q}_0$ and $\mathbb R^{nN}thcal{O}_k$ is properly contained in ${Q}_0$. As $\omegaega$ is doubling, see Lemma \mathbb R^{nN}thbf{R}f{T:doubling}, we have that $(2Q_0,d,\omegaega)$ is a space of homogeneous type and weak $L^1$ estimates for the Hardy-Littlewood maximal function apply. Hence
\betagin{eqnarray}\lambdabel{yy2}\omegaega(\mathbb R^{nN}thcal{O}_k)\leq \widetildelde c\frac {\bar c}{\varepsilonsilon_0}\omegaega(\mathbb R^{nN}thcal{O}_{k+1})\leq \frac 1 2 \biggl (\frac {\varepsilonsilon_0}{\bar c}\biggr )^{k-1}\omegaega({Q}_0),
\boldsymbol end{eqnarray}
if we let $\mathcal Upsilon:=\widetildelde c\bar c$ and where $\widetildelde c=\widetildelde c(m,\kappa,M_1)$, $1\leq \widetildelde c<\infty$. By definition and by the construction, see \boldsymbol eqref{cubes} $(i)$-$(iii)$, ${Q}_0$ can be dyadically subdivided, and we can select a collection
$\mathcal F_k=\{\mathcal Deltalta_i^k\}_i\subset {{Q}_0}$, comprised
of the cubes that are maximal with respect to containment in $\mathbb R^{nN}thcal{O}_k$, and thus $\mathbb R^{nN}thcal{O}_k := {\rm B}p_i \mathcal Deltalta^k_i$. The cubes in $\mathcal F_k$ are maximal in the sense that
\betagin{equation}\lambdabel{Fkmaximal}
\mathcal Deltalta_i^k \in \mathcal F_k \:\iff\:\mathcal Deltalta_i^k\subset\mathbb R^{nN}thcal{O}_k \:\text{and}\: Q \subset \mathcal Deltalta_i^k, \:\forall Q\in\mathbb R^{nN}thbb{D}_{Q_0}\:\text{such that}\:Q\subset\mathbb R^{nN}thcal{O}_k.
\boldsymbol end{equation}
Using \boldsymbol eqref{Fkmaximal}, \boldsymbol eqref{lem:coverexists_Okdef}, and Lemma \mathbb R^{nN}thbf{R}f{T:doubling}, we see that
\betagin{equation}\lambdabel{yy3}
\betagin{split}
\omegaega(\mathbb R^{nN}thcal{O}_{k+1}u_{\rm B}p\mathcal Deltalta_i^k) &\leq \omegaega(\mathbb R^{nN}thcal{O}_{k+1}u_{\rm B}p 2\mathcal Deltalta_i^k)\\
&\leq \omegaega(2\mathcal Deltalta_i^k)\frac{1}{\omegaega(2\mathcal Deltalta_i^k)}\iint_{2\mathcal Deltalta_i^k}\chii_{\mathbb R^{nN}thcal{O}_{k+1}}\, \mathbb R^{nN}thrm{d}\omegaega\\&\leq \varepsilonsilon_0\omegaega(\mathcal Deltalta_i^k),
\boldsymbol end{split}
\boldsymbol end{equation}
for all $\mathcal Deltalta_i^k\in\mathcal F_k$. We now iterate this argument, to construct $\mathbb R^{nN}thcal{O}_{j-1}$ from $\mathbb R^{nN}thcal{O}_j$, for $2\leq j\leq k$, just as we constructed $\mathbb R^{nN}thcal{O}_k$ from $\mathbb R^{nN}thcal{O}_{k+1}$. It is then a routine matter to verify that the sets $\mathbb R^{nN}thcal{O}_1$,...., $\mathbb R^{nN}thcal{O}_k$, form a good $\varepsilonsilon_0$ cover of $E$. We omit further details.
\mathbb R^{nN}thbb Qd
\subsection{Additional notation}
\betagin{remark}\lambdabel{gc} In the following we let $\mathcal Pi(Z,t)$ denote the projection of $(Z,t)\in\mathbb R^{nN}thbb R^{N+1}$ along $x_m$ onto $\partialartial\mathbb R^{nN}thcal Omegaega$. Furthermore, from now on we fix two small dyadic numbers $\boldsymbol eta_1=2^{-k_1}$ and $\boldsymbol eta_2=2^{-k_2}$ where $1\leq k_1\ll k_2$ are to be chosen depending at most on $m$, $\kappa$ and $M_1$. Given $Q\in \mathbb R^{nN}thbb D$, we let $A_{\boldsymbol eta_1 Q}^+:=A_{c\boldsymbol eta_1 l(Q),\mathcal Lambdambda}^+(Z_Q,t_Q)$, we consider the point
$A_{c\boldsymbol eta_1^2 l(Q),\mathcal Lambdambda}^-(Z_Q,t_Q)$ and we let $\widetildelde Q\in \mathbb R^{nN}thbb D$ be such that $l(\widetildelde Q)=\boldsymbol eta_1^2 l(Q)$ and such that $\widetildelde Q$ contains the point $\mathcal Pi(A_{c\boldsymbol eta_1^2 l(Q),\mathcal Lambdambda}^-(Z_Q,t_Q))$. We can and will choose $\boldsymbol eta_1$ so small that
$$\mathcal Pi(A_{\boldsymbol eta_1 Q}^+)\subset \frac 1 4 Q$$
and such that
$$\widetildelde Q\subset Q.$$
\boldsymbol end{remark}
\betagin{remark}\lambdabel{gc+} Given $Q\in \mathbb R^{nN}thbb D$ and $A_{\boldsymbol eta_1 Q}^+=A_{c\boldsymbol eta_1l(Q),\mathcal Lambdambda}^+(Z_Q,t_Q)$ as in Remark \mathbb R^{nN}thbf{R}f{gc}, consider the point $\mathcal Pi(A_{\boldsymbol eta_1 Q}^+)\in\partialartial\mathbb R^{nN}thcal Omegaega$. We let
$\hat Q\in \mathbb R^{nN}thbb D$ be such that $l(\hat Q)=\boldsymbol eta_2 l(Q)$ and such that $\hat Q$ contains the point $\mathcal Pi(A_{\boldsymbol eta_1 Q}^+)$. We let $d_Q:=|A_{\boldsymbol eta_1 Q}^+-\mathcal Pi(A_{\boldsymbol eta_1 Q}^+)|$. Furthermore, we let
$\bar Q\in \mathbb R^{nN}thbb D$ be such that $l(\bar Q)=\boldsymbol eta_2^2 l(\hat Q)$ and $\bar Q$ contains the point $(Z_{\hat Q},t_{\hat Q})$. Note that by construction, if we choose $\boldsymbol eta_1$ (and hence $\boldsymbol eta_2$) small enough,
$$\bar Q\subset\hat Q\subset Q.$$
Given
$\bar Q$ we let
\betagin{equation}\lambdabel{yy4-}
\betagin{split}
&S_{\bar Q}^+:=\{(x,\partialsii(x,y,t)+d_Q,y,y_m,t)\mid\ (x,\partialsii(x,y,t),y,y_m,t)\in \bar Q\},\\
&S_{\bar Q}^-:=\{(x,\partialsii(x,y,t)+l(\bar Q),y,y_m,t)\mid\ (x,\partialsii(x,y,t),y,y_m,t)\in \bar Q\}.
\boldsymbol end{split}
\boldsymbol end{equation}
Then $S_{\bar Q}^-$ and $S_{\bar Q}^+$ are two pieces of surfaces above $\partialartial\mathbb R^{nN}thcal Omegaega$ in the direction of $x_m$, $S_{\bar Q}^+$ is above $S_{\bar Q}^-$, and each point on $S_{\bar Q}^-$ can be connected to a point on $S_{\bar Q}^+$ by a straight line in the direction of $x_m$.
\boldsymbol end{remark}
\betagin{remark} Note that Remark \mathbb R^{nN}thbf{R}f{gc} and Remark \mathbb R^{nN}thbf{R}f{gc+} are generic constructions for dyadic cubes. Consider now the special case $\mathcal Deltalta := \mathcal Deltalta_i^l\in\mathcal F_l$, i.e. $\mathcal Deltalta$ is a cube arising in some good $\varepsilonsilon_0$ cover. We then set $\widetildelde \mathcal Deltalta_i^l:=\widetildelde \mathcal Deltalta$, where $\widetildelde \mathcal Deltalta$ is defined as in Remark \mathbb R^{nN}thbf{R}f{gc}, and we define
\betagin{eqnarray}\lambdabel{yy4}\widetildelde \mathbb R^{nN}thcal O_l:=\bigcup_{\mathcal Deltalta_i^l\in\mathcal F_l}\widetildelde \mathcal Deltalta_i^l.
\boldsymbol end{eqnarray}
Furthermore, let $E\subset {Q}_0$ and consider the set up of Lemma \mathbb R^{nN}thbf{R}f{existcover}. We note that for every $(Z_0,t_0)\in E$ we have $(Z_0,t_0)\in \mathbb R^{nN}thcal{O}_l$, for all $l=1,2,...,k$,
and that therefore there exists, for each $l$, a cube $\mathcal Deltalta_i^l=\mathcal Deltalta_i^l(Z_0,t_0)\in\mathbb R^{nN}thcal{F}_l$ containing $(Z_0,t_0)$.
\boldsymbol end{remark}
\subsection{Proof of Lemma \mathbb R^{nN}thbf{R}f{lemmacruc}} Let in the following $Q_0\in \mathbb R^{nN}thbb{D}$ and let $\omegaega(\cdot)$ be as in the statement of Theorem \mathbb R^{nN}thbf{R}f{Ainfty}. To prove Lemma \mathbb R^{nN}thbf{R}f{lemmacruc}, let $\varepsilonsilon_0>0$ be a degree of freedom to be specified below and depending only on $m,\kappa,M_1$, let
$\, \mathbb R^{nN}thrm{d}eltalta_0=\gammamma(\varepsilonsilon_0/\mathcal Upsilon)^k$ be as specified in Lemma \mathbb R^{nN}thbf{R}f{existcover} where $k$ is to be chosen depending only on $m,\kappa,M_1$ and $\mathcal Upsilon$. Consider $E\subset {Q_0}$ with $\omegaega(E)\leq\, \mathbb R^{nN}thrm{d}eltalta_0$. Using Lemma \mathbb R^{nN}thbf{R}f{existcover} we see that $E$ has a good $\varepsilonsilon_0$ cover of length $k$, $\{\mathbb R^{nN}thcal{O}_l\}_{l=1}^k$ with corresponding collections
$\mathcal F_l=\{\mathcal Deltalta_i^l\}_i\subset Q_0$. Let $\{\widetildelde{\mathbb R^{nN}thcal{O}}_l\}_{l=1}^k$ be defined as in \boldsymbol eqref{yy4}. Using this good $\varepsilonsilon_0$ cover of $E$ we let
$$F(Z,t):=\sum_{j=2}^k \chii_{\widetildelde\mathbb R^{nN}thcal O_{j-1}\setminus\mathbb R^{nN}thcal O_j}(Z,t),$$
where $\chii_{\widetildelde\mathbb R^{nN}thcal O_{j-1}\setminus\mathbb R^{nN}thcal O_j}$ is the indicator function for the set $\widetildelde\mathbb R^{nN}thcal O_{j-1}\setminus\mathbb R^{nN}thcal O_j$. Then $F$ equals the indicator function of some Borel set $S\subset\mathcal Sigmagma$ and we let $u(Z,t):=\omegaega(Z,t, S)$. Consider \mbox{$(Z_0,t_0)\in E$} and an index \mbox{$l\in \{1,...,k\}$}. Let in the following $$\mbox{$\mathcal Deltalta_i^l\in \mathcal F_l$ be a cube in the collection $\mathcal F_l$ which contains $(Z_0,t_0)$.}$$ Given $k_0\in\mathbb R^{nN}thbb Z_+$ and $A_{\boldsymbol eta\mathcal Deltalta_i^l}^+=A_{c\boldsymbol eta l(\mathcal Deltalta_i^l),\mathcal Lambdambda}^+(Z_{\mathcal Deltalta_i^l},t_{\mathcal Deltalta_i^l})$ we let
$$\mbox{$\hat\mathcal Deltalta_i^l$, $\bar\mathcal Deltalta_i^l$, $S_{\bar\mathcal Deltalta_i^l}^-$ and $S_{\bar\mathcal Deltalta_i^l}^+$}$$
be as defined as in Remark \mathbb R^{nN}thbf{R}f{gc+} relative to $\mathcal Deltalta_i^l$ and using $\boldsymbol eta_j:=2^{-k_j}$.
Hence, based on $(Z_0,t_0)\in E$ and an index $l\in \{1,...,k\}$, we have specified $\mathcal Deltalta_i^l$, $\hat \mathcal Deltalta_i^l$, $\bar \mathcal Deltalta_i^l$ and the surfaces
$S_{\bar\mathcal Deltalta_i^l}^-$ and $S_{\bar\mathcal Deltalta_i^l}^+$, and, by construction,
$$ \bar \mathcal Deltalta_i^l\subset\hat \mathcal Deltalta_i^l\subset \mathcal Deltalta_i^l,\ \widetildelde \mathcal Deltalta_i^l\subset \mathcal Deltalta_i^l.$$
We first intend to prove that there exists $\betata>0$, depending only on $m,\kappa,M_1$,
such that if $\varepsilonsilon_0$ and $\boldsymbol eta_j=2^{-k_j}$ are chosen sufficiently small, then
\betagin{eqnarray}\lambdabel{yy6}u(P+d_{\mathcal Deltalta_i^l}e_m)-u( P+l(\bar\mathcal Deltalta_i^l)e_m)\geq \betata,\ \forall P\in\{(x,\partialsii(x,y,t),y,y_m,t)\in \bar\mathcal Deltalta_i^l\},
\boldsymbol end{eqnarray}
where $e_m$ denotes the unit vector in $\mathbb R^{nN}thbb R^{N+1}$ which points in the direction of $x_m$. Given $P\in\{(x,\partialsii(x,y,t),y,y_m,t)\in \bar\mathcal Deltalta_i^l\}$ we let
$$P^+:=P+d_{\mathcal Deltalta_i^l}e_m,\ P^-:=P+l(\bar\mathcal Deltalta_i^l)e_m,$$
and we want to estimate $u(P^+)-u(P^-)$. To start the proof we note that, by construction, $\widetildelde \mathcal Deltalta_i^l\subset\mathcal Deltalta_i^l$ and by using Lemma \mathbb R^{nN}thbf{R}f{bourg} and Lemma \mathbb R^{nN}thbf{R}f{lem4.7bol} we see that
there exists $ c_{\boldsymbol eta_1}=c_{\boldsymbol eta_1}(m,\kappa,M_1,\boldsymbol eta_1)$, $1\leq c_{\boldsymbol eta_1}<\infty$, for $\boldsymbol eta_1$ small enough,
such that
\betagin{eqnarray}\lambdabel{boundbelow}\omegaega(P^+,\widetildelde \mathcal Deltalta_i^l)\geq c_{\boldsymbol eta_1}^{-1}.
\boldsymbol end{eqnarray}
To estimate $u( P^+)$ we first note that
\betagin{equation}
\betagin{split}
u(P^+)&\geq\iint_{\widetildelde \mathcal Deltalta_i^l}\chii_{\widetildelde{\mathbb R^{nN}thcal{O}}_{l}\setminus\mathbb R^{nN}thcal{O}_{l+1}}\, \, \mathbb R^{nN}thrm{d}\omegaega(P^+,\bar Z,\bar t)\\
&=\omegaega(P^+,\widetildelde \mathcal Deltalta_i^l)-\omegaega( P^+,\widetildelde\mathcal Deltalta_i^lu_{\rm B}p \mathbb R^{nN}thcal{O}_{l+1}),
\boldsymbol end{split}
\boldsymbol end{equation}
as all terms in the definition of $F$ are non-negative. Consider $(\bar Z,\bar t)\in \mathcal Deltalta_i^l$. Then, using Lemma \mathbb R^{nN}thbf{R}f{lem4.5-Kyoto1ha} we have that
\betagin{eqnarray}
K( P^+,\bar Z,\bar t):=\lim_{\rho\to 0}\frac{\omegaega( P^+,\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p\mathbb R^{nN}thcal{B}_\rho(\bar Z,\bar t))}{\omegaega(\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p\mathbb R^{nN}thcal{B}_\rho(\bar Z,\bar t))},
\boldsymbol end{eqnarray}
exists for $\omegaega$-a.e. $(\bar Z,\bar t)\in \mathcal Deltalta_i^l$, and
\betagin{eqnarray}
K(P^+,\bar Z,\bar t)\leq \frac c{\omegaega(\mathcal Deltalta_i^l)}\mbox{ whenever }(\bar Z,\bar t)\in \mathcal Deltalta_i^l.
\boldsymbol end{eqnarray}
In the last conclusion we have also used Lemma \mathbb R^{nN}thbf{R}f{T:doubling}. Using this estimate, and the fact that by construction $\widetildelde \mathcal Deltalta_i^l\subset\mathcal Deltalta_i^l$, we see that
\betagin{eqnarray}
\omegaega( P^+,\widetildelde\mathcal Deltalta_i^lu_{\rm B}p \mathbb R^{nN}thcal{O}_{l+1})\leq \frac {c_{\boldsymbol eta_1}}{\omegaega(\mathcal Deltalta_i^l)}\omegaega(\widetildelde\mathcal Deltalta_i^lu_{\rm B}p \mathbb R^{nN}thcal{O}_{l+1})\leq C_{\boldsymbol eta_1}\varepsilonsilon_0,
\boldsymbol end{eqnarray}
by the construction. In particular, using \boldsymbol eqref{boundbelow} we deduce
\betagin{eqnarray}\lambdabel{yy10}
u(P^+)\geq c_{\boldsymbol eta_1}^{-1}-C_{\boldsymbol eta_1}\varepsilonsilon_0.
\boldsymbol end{eqnarray}
To estimate $u( P^-)$ we write
\betagin{eqnarray*}
u( P^-)&=&\iint_{Q_0\setminus \hat\mathcal Deltalta_i^l}F(\bar Z,\bar t)\, \, \mathbb R^{nN}thrm{d}\omegaega( P^-,\bar Z,\bar t)+\iint_{\hat\mathcal Deltalta_i^l}F(\bar Z,\bar t)\, \, \mathbb R^{nN}thrm{d}\omegaega( P^-,\bar Z,\bar t)\boldsymbol \nuotag\\
&=:&I+{II}.
\boldsymbol end{eqnarray*}
Using Lemma \mathbb R^{nN}thbf{R}f{lem4.5-Kyoto1} and the definition of $ P^-$ we see that
\betagin{eqnarray}
|I|\leq \omegaega(P^-,Q_0\setminus \hat\mathcal Deltalta_i^l)\leq c\boldsymbol eta_2^\sigmagma,
\boldsymbol end{eqnarray}
for some $c=c(m,\kappa,M_1)$, $\sigmagma=\sigmagma(m,\kappa,M_1)\in (0,1)$.
We split ${II}$ as
\betagin{eqnarray}
{II}= {II}_1+{II}_2+{II}_3,
\boldsymbol end{eqnarray}
where
\betagin{eqnarray}
{II}_1&:=& \sum_{j=2}^l\iint_{\hat\mathcal Deltalta_i^l}1_{\widetildelde{\mathbb R^{nN}thcal{O}}_{j-1}\setminus\mathbb R^{nN}thcal{O}_j}\, \, \mathbb R^{nN}thrm{d}\omegaega( P^-,\bar Z,\bar t),\boldsymbol \nuotag\\
{II}_2&:=&\sum_{j=l+2}^k\iint_{\hat\mathcal Deltalta_i^l}1_{\widetildelde{\mathbb R^{nN}thcal{O}}_{j-1}\setminus\mathbb R^{nN}thcal{O}_j}\, \, \mathbb R^{nN}thrm{d}\omegaega( P^-,\bar Z,\bar t),\\
{II}_3&:=&\iint_{\hat\mathcal Deltalta_i^l}1_{\widetildelde{\mathbb R^{nN}thcal{O}}_{l}\setminus\mathbb R^{nN}thcal{O}_{l+1}}\, \, \mathbb R^{nN}thrm{d}\omegaega( P^-,\bar Z,\bar t).\boldsymbol \nuotag
\boldsymbol end{eqnarray}
Note that if $j\leq l$, then $\hat\mathcal Deltalta_i^l\subset\mathcal Deltalta_i^l\subset\mathbb R^{nN}thcal{O}_l\subset \mathbb R^{nN}thcal{O}_j$ and $(\widetildelde{\mathbb R^{nN}thcal{O}}_{j-1}\setminus\mathbb R^{nN}thcal{O}_j)u_{\rm B}p \mathcal Deltalta_i^l=\boldsymbol emptyset$. Hence $II_1 = 0$. Furthermore,
\betagin{equation}
\betagin{split}
|II_2|&\leq\sum_{j=l+2}^k\omegaega( P^-,(\widetildelde{\mathbb R^{nN}thcal{O}}_{j-1}\setminus\mathbb R^{nN}thcal{O}_j)u_{\rm B}p \mathcal Deltalta_i^l)\\
&\leq c_{\boldsymbol eta_2}\sum_{j=l+2}^k\omegaega( A_{\hat\mathcal Deltalta_i^l}^+(\mathcal Pi(A_{\boldsymbol eta\mathcal Deltalta_i^l}^+)),(\widetildelde{\mathbb R^{nN}thcal{O}}_{j-1}\setminus\mathbb R^{nN}thcal{O}_j)u_{\rm B}p \mathcal Deltalta_i^l),
\boldsymbol end{split}
\boldsymbol end{equation}
by the Harnack inequality, see Lemma \mathbb R^{nN}thbf{R}f{lem4.7bol}. Consider $(\bar Z,\bar t)\in (\widetildelde{\mathbb R^{nN}thcal{O}}_{j-1}\setminus\mathbb R^{nN}thcal{O}_j)u_{\rm B}p \mathcal Deltalta_i^l$. Then, again using Lemma \mathbb R^{nN}thbf{R}f{lem4.5-Kyoto1ha} we have that
\betagin{eqnarray}
K( A_{\hat\mathcal Deltalta_i^l}^+(\mathcal Pi(A_{\boldsymbol eta\mathcal Deltalta_i^l}^+)),\bar Z,\bar t):=\lim_{\rho\to 0}\frac{\omegaega( A_{\hat\mathcal Deltalta_i^l}^+(\mathcal Pi(A_{\boldsymbol eta\mathcal Deltalta_i^l}^+)),\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p\mathbb R^{nN}thcal{B}_\rho(\bar Z,\bar t))}{\omegaega(\partialartial\mathbb R^{nN}thcal Omegaegau_{\rm B}p\mathbb R^{nN}thcal{B}_\rho(\bar Z,\bar t))},
\boldsymbol end{eqnarray}
exists for $\omegaega$-a.e. $(\bar Z,\bar t)\in \mathcal Deltalta_i^l$, and
\betagin{eqnarray}
K(A_{\hat\mathcal Deltalta_i^l}^+(\mathcal Pi(A_{\boldsymbol eta\mathcal Deltalta_i^l}^+)),\bar Z,\bar t)\leq \frac c{\omegaega(\mathcal Deltalta_i^l)}\mbox{ whenever }(\bar Z,\bar t)\in \mathcal Deltalta_i^l.
\boldsymbol end{eqnarray}
In the last conclusion we have also used Lemma \mathbb R^{nN}thbf{R}f{T:doubling}. Using these facts, and using the definition of the good $\varepsilonsilon_0$ cover, we see that
\betagin{equation}\lambdabel{yy8}
\betagin{split}
|II_2|&\leq\frac {c_{\boldsymbol eta_2}}{\omegaega(\mathcal Deltalta_i^l)}\sum_{j=l+2}^k\omegaega((\widetildelde{\mathbb R^{nN}thcal{O}}_{j-1}\setminus\mathbb R^{nN}thcal{O}_j)u_{\rm B}p \mathcal Deltalta_i^l)\\
&\leq\frac {c_{\boldsymbol eta_2}}{\omegaega(\mathcal Deltalta_i^l)}\sum_{j=l+2}^k\omegaega(\mathbb R^{nN}thcal{O}_{j-1}u_{\rm B}p \mathcal Deltalta_i^l)\leq \frac {c_{\boldsymbol eta_2}}{ \omegaega(\mathcal Deltalta_i^l)}\sum_{j=l+2}^k\varepsilonsilon_0^{j-1-l}\omegaega(\mathcal Deltalta_i^l)\leq C_{\boldsymbol eta_2}\varepsilonsilon_0.
\boldsymbol end{split}
\boldsymbol end{equation}
To estimate the term $II_3$ we first observe that $\hat \mathcal Deltalta_i^lu_{\rm B}p \widetildelde{\mathbb R^{nN}thcal{O}}_{l}=\boldsymbol emptyset$ by the definition of $\widetildelde{\mathbb R^{nN}thcal{O}}_{l}$. Hence,
\betagin{eqnarray}
II_3&=&\omegaega( P^-,\hat\mathcal Deltalta_i^lu_{\rm B}p (\widetildelde{\mathbb R^{nN}thcal{O}}_{j-1}\setminus\mathbb R^{nN}thcal{O}_j))=0
\boldsymbol end{eqnarray}
and we can conclude that
\betagin{eqnarray}\lambdabel{yy11}
u( P^-)\leq c\boldsymbol eta_2^\sigmagma+C_{\boldsymbol eta_2}\varepsilonsilon_0.
\boldsymbol end{eqnarray}
Combining \boldsymbol eqref{yy10} and \boldsymbol eqref{yy11} we can conclude, in either case, that
\betagin{eqnarray}
u( P^+)-u( P^-)\geq c_{\boldsymbol eta_1}^{-1}-C_{\boldsymbol eta_1}\varepsilonsilon_0-c\boldsymbol eta_2^\sigmagma-C_{\boldsymbol eta_2}\varepsilonsilon_0.
\boldsymbol end{eqnarray}
We now first choose $\boldsymbol eta_1=\boldsymbol eta_1(m,\kappa,M_1)$ small. We then choose $\boldsymbol eta_2=\boldsymbol eta_2(m,\kappa,M_1)$ so that $c_{\boldsymbol eta_1}^{-1}=2c\boldsymbol eta_2^\sigmagma$. Having fixed $\boldsymbol eta_1$ and $\boldsymbol eta_2$ we
choose $\varepsilonsilon_0=\varepsilonsilon_0(m,\kappa,M_1)$ so that $c\boldsymbol eta_2^\sigmagma=2(C_{\boldsymbol eta_1}+C_{\boldsymbol eta_2})\varepsilonsilon_0$. By these choices we can conclude that there exists $0<\betata=\betata(m,\kappa,M_1)\ll 1$ such that
\betagin{eqnarray}\lambdabel{yy12}
u(P+d_{\mathcal Deltalta_i^l}e_m)-u( P+l(\bar\mathcal Deltalta_i^l)e_m)\geq \betata,\ \forall P\in\{(x,\partialsii(x,y,t),y,y_m,t)\in \bar\mathcal Deltalta_i^l\}.
\boldsymbol end{eqnarray}
In particular, fix $P\in\{(x,\partialsii(x,y,t),y,y_m,t)\in \bar\mathcal Deltalta_i^l\}$. Then \boldsymbol eqref{yy12} implies
\betagin{eqnarray}
\betata^2\leq cl(\mathcal Deltalta_i^l)\int_{P^-}^{P^+}|\partialartial_{x_m}u(x,x_m,y,y_m,t)|^2\, \, \mathbb R^{nN}thrm{d} x_m.
\boldsymbol end{eqnarray}
Integrating with respect to $P\in \bar\mathcal Deltalta_i^l$ we see that
\betagin{eqnarray}\lambdabel{yy14}
\betata^2\sigmagma(\bar\mathcal Deltalta_i^l)\leq cl(\mathcal Deltalta_i^l)\iiint_{R_{\bar\mathcal Deltalta_i^l}}|\boldsymbol \nuabla_Xu(Z,t)|^2\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t,
\boldsymbol end{eqnarray}
where $R_{\bar\mathcal Deltalta_i^l}$ is a naturally defined Whitney type region. Recall that $\sigmagma(\bar\mathcal Deltalta_i^l)\approx \sigmagma(\mathcal Deltalta_i^l)$. In particular, by an elementary connectivity/covering argument we see that
\betagin{eqnarray*}
\quad c^{-1}\betata^2\leq \iiint_{\widetildelde W_{\mathcal Deltalta_i^l}} |\boldsymbol \nuabla_Xu|^2\, \mathbb R^{nN}thrm{d}eltalta^{2-{\bf q}}\, dZdt,
\boldsymbol end{eqnarray*}
where $\widetildelde W_{\mathcal Deltalta_i^l}$ is a natural Whitney type region associated to $\mathcal Deltalta_i^l$, $\, \mathbb R^{nN}thrm{d}eltalta=\, \mathbb R^{nN}thrm{d}eltalta(Z,t)$ is the distance from $(Z,t)$ to $\mathcal Sigmagma$, and $c=c(m,M_1,\kappa)$, $1\leq c<\infty$. Consequently, for
$(Z_0,t_0)\in E$ fixed we find, by summing over all indices $i$, $l$, such that
$(Z_0,t_0)\in \mathcal Deltalta_i^l$, that
\betagin{eqnarray}
\quad\quad c^{-1}\betata^2k\leq\sum_{i,l: (Z_0,t_0)\in \mathcal Deltalta_i^l} \biggl (\iiint_{\widetildelde W_{\mathcal Deltalta_i^l}} |\boldsymbol \nuabla_Xu|^2\, \mathbb R^{nN}thrm{d}eltalta^{2-{\bf q}}\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr ).
\boldsymbol end{eqnarray}
The construction can be made so that the Whitney type regions $\{\widetildelde W_{\mathcal Deltalta_i^l}\}$ have bounded overlaps measured by a constant depending only on $m$, $M_1$, and such that $W_{\mathcal Deltalta_i^l}\subset T_{cQ_0}$ for some $c=c(m,M_1)$, $1\leq c<\infty$, where $T_{cQ_0}$ is defined in \boldsymbol eqref{eq2.box}. Hence, integrating with respect to $\, \mathbb R^{nN}thrm{d} \sigmagma$, we deduce that
\betagin{eqnarray}\lambdabel{yy13+}
\quad c^{-1}\betata^2k\sigmagma(E)\leq \biggl (\iiint_{ T_{cQ_0}} \bigl |\boldsymbol \nuabla_Xu|^2\, \mathbb R^{nN}thrm{d}eltalta\, \, \mathbb R^{nN}thrm{d} Z\, \mathbb R^{nN}thrm{d} t\biggr )
\boldsymbol end{eqnarray}
where, resolving the dependencies, $c=c(m,\kappa,M_1)$, $1\leq c<\infty$. Furthermore,
$$k\approx \frac {\log(\, \mathbb R^{nN}thrm{d}eltalta_0)}{\log (\varepsilonsilon_0)},$$
where $\boldsymbol eta$ and $\varepsilonsilon_0$ now have been fixed, and $\, \mathbb R^{nN}thrm{d}eltalta_0$ is at our disposal. Given $\mathcal Upsilon$
we obtain the conclusion of the lemma by specifying $\, \mathbb R^{nN}thrm{d}eltalta_0=\, \mathbb R^{nN}thrm{d}eltalta_0(m,\kappa,M_1,\mathcal Upsilon)$ sufficiently small. This completes the proof of Lemma \mathbb R^{nN}thbf{R}f{lemmacruc}.
\mathbb R^{nN}thbb Qd
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\begin{document}
\title[One distribution function on the Moran sets]{One distribution function on the Moran sets}
\author{Symon Serbenyuk}
\address{
45~Shchukina St. \\
Vinnytsia \\
21012 \\
Ukraine}
\email{[email protected]}
\subjclass[2010]{11K55, 11J72, 28A80, 26A09}
\keywords{
s-adic representation, Moran set, Hausdorff dimension, monotonic function, distribution function.}
\begin{abstract}
In the present article, topological, metric, and fractal properties of certain sets are investigated. These sets are images of sets whose elements have restrictions on using digits or combinations of digits in own s-adic representations, under the map $f$, that is a certain distribution function.
\end{abstract}
\maketitle
\section{Introduction}
Let us consider space $\mathbb R^n$. In \cite{Moran1946}, P. A. P. Moran introduced the following construction of sets and calculated the Hausdorff dimension of the limit set
\begin{equation}
\label{eq: Cantor-like set}
E=\bigcap^{\infty} _{n=1}{\bigcup_{i_1,\dots , i_n\in A_{0,p}}{\Delta_{i_1i_2\ldots i_n}}}.
\end{equation}
Here $p$ is a fixed positive integer, $A_{0,p}=\{1, 2, \dots , p\}$, and sets $\Delta_{i_1i_2\ldots i_n}$ are basic sets having the following properties:
\begin{itemize}
\item any set $\Delta_{i_1i_2\ldots i_n}$ is closed and disjoint;
\item for any $i\in A_{0,p}$ the condition $\Delta_{i_1i_2\ldots i_ni}\subset\Delta_{i_1i_2\ldots i_n}$ holds;
\item
$$
\lim_{n\to\infty}{d\left(\Delta_{i_1i_2\ldots i_n}\right)}=0, \text{where $d(\cdot)$ is the diameter of a set};
$$
\item each basic set is the closure of its interior;
\item at each level the basic sets do not overlap (their interiors are disjoint);
\item any basic set $\Delta_{i_1i_2\ldots i_ni}$ is geometrically similar to $\Delta_{i_1i_2\ldots i_n}$;
\item
$$
\frac{d\left(\Delta_{i_1i_2\ldots i_ni}\right)}{d\left(\Delta_{i_1i_2\ldots i_n}\right)}=\sigma_i,
$$
where $\sigma_i\in (0,1)$ for $i=\overline{1,p}$.
\end{itemize}
The Hausdorff dimension $\alpha_0$ of the set $E$ is the unique root of the following equation
$$
\sum^{p} _{i=1}{\sigma^{\alpha_0} _i}=1.
$$
It is easy to see that set \eqref{eq: Cantor-like set} is a Cantor-like set and a self-similar fractal. The set $E$ is called \emph{the Moran set}.
Much research has been devoted to Moran-like constructions and Cantor-like sets (for example, see \cite{{Falconer1997},{Falconer2004}, {Mandelbrot1977}, {PS1995}, DU2014, DU2014(2), HRW2000, PS1995, {S.Serbenyuk 2017}, {S. Serbenyuk fractals}} and references therein).
Fractal sets are widely applicated in computer design, algorithms of the compression to information, quantum mechanics, solid-state physics, analysis and categorizations of signals of various forms appearing in different areas (e.g. the analysis of exchange rate fluctuations in economics), etc. In addition, such sets are useful for checking of preserving the Hausdorff dimension by certain functions \cite{{S. Serbenyuk abstract1},{S.Serbenyuk 2017}}. However, for much classes of fractals the problem of the Hausdorff dimension calculation is difficult and the estimate of parameters on which the Hausdorff dimension of certain classes of fractal sets depends is left out of consideration.
Let $s>1$ be a fixed positive integer. Let us consider the s-adic representation of numbers from~$[0,1]$:
$$
x=\Delta^s _{\alpha_1\alpha_2...\alpha_n...}=\sum^{\infty} _{n=1}{\frac{\alpha_n}{s^n}},
$$
where $\alpha_n\in A=\{0,1,\dots, s-1\}$.
In addition, we say that the following representation
$$
x=\Delta^{-s }_{\alpha_1\alpha_2...\alpha_n...}=\sum^{\infty} _{n=1}{\frac{\alpha_n}{(-s)^n}},
$$
is the nega-s-adic representation of numbers from $\left[-\frac{s}{s+1}, \frac{1}{s+1}\right]$. Here $\alpha_n\in A$ as well.
Some articles (see \cite{ DU2014, DU2014(2), {S. Serbenyuk fractals},{S. Serbenyuk abstract 2}, {S. Serbenyuk abstract 3},{S. Serbenyuk abstract 5}, {Symon1}, {Symon2}, {S. Serbenyuk 2013}, {S. Serbenyuk 2017 fractals}} ) were devoted to sets whose elements have certain
restrictions on using combinations of digits in own s-adic representation. Let us consider the following results.
Suppose $s>2$ be a fixed positive integer.
Let us consider a class $\Upsilon_s$ of sets $\mathbb S_{(s,u)}$ represented in the form
\begin{equation*}
\mathbb S_{(s,u)}= \left\{x: x=\frac{u}{s-1} +\sum^{\infty} _{n=1} {\frac{\alpha_n - u}{s^{\alpha_1+\dots+\alpha_n}}}, (\alpha_n) \in L, \alpha_n \ne u, \alpha_n \ne 0 \right\},
\end{equation*}
where $u=\overline{0,s-1}$, the parameters $u$ and $s$ are fixed for the set $\mathbb S_{(s,u)}$. That is the class $\Upsilon_s$ contains the sets $\mathbb S_{(s,0)}, \mathbb S_{(s,1)},\dots,\mathbb S_{(s,s-1)}$. We say that $\Upsilon$ is a class of sets such that contains the classes $\Upsilon_3, \Upsilon_4,\dots ,\Upsilon_n,\dots$.
It is easy to see that the set $\mathbb S_{(s,u)}$ can be defined by the s-adic representation in the following form
\begin{equation*}
\mathbb S_{(s,u)}=\left\{x: x= \Delta^{s}_{{\underbrace{u\ldots u}_{\alpha_1-1}} \alpha_1{\underbrace{u\ldots u}_{\alpha_2 -1}}\alpha_2 ...{\underbrace{u\ldots u}_{ \alpha_n -1}}\alpha_n...}, (\alpha_n) \in L, \alpha_n \ne u, \alpha_n \ne 0 \right\},
\end{equation*}
\begin{theorem}[\cite{{Symon2}, {S. Serbenyuk 2017 fractals}, {S. Serbenyuk fractals}}]
\label{th: theorem1}
For an arbitrary $u \in A$ the set $\mathbb S_{(s,u)}$ is an uncountable, perfect, nowhere dense set of zero Lebesgue measure, and a self-similar fractal whose Hausdorff dimension $\alpha_0 (\mathbb S_{(s,u)})$ satisfies the following equation
$$
\sum _{p \ne u, p \in A_0} {\left(\frac{1}{s}\right)^{p \alpha_0}}=1.
$$
\end{theorem}
\begin{remark}
We note that the statement of the last-mentioned theorem is true for all sets $\mathbb S_{(s,0)}, \mathbb S_{(s,1)},\dots,\mathbb S_{(s,s-1)}$ (for fixed parameters $u=\overline{0,s-1}$ and any fixed $2<s\in\mathbb N$ ) without the sets $ S_{(3,1)}$ and $ S_{(3,2)}$.
\end{remark}
\begin{theorem}[\cite{{Symon2}, {S. Serbenyuk 2017 fractals}, {S. Serbenyuk 2013}, {S. Serbenyuk fractals}}]
\label{th: theorem2}
Let $E$ be a set, whose elements contain (in own s-adic or nega-s-adic representation) only
digits or combinations of digits from a certain fixed finite set $\{\sigma_1, \sigma_2,\dots,\sigma_m\}$ of s-adic digits or combinations of digits.
Then the Hausdorff dimension $\alpha_0$ of $E$ satisfies the following equation:
$$
N(\sigma^1 _m)\left(\frac{1}{s}\right)^{\alpha_0}+N(\sigma^2 _m)\left(\frac{1}{s}\right)^{2\alpha_0}+\dots+N(\sigma^{k} _m)\left(\frac{1}{s}\right)^{k\alpha_0}=1,
$$
where $N(\sigma^k_m)$ is a number of k-digit combinations $\sigma^k_m$ from the set $\{\sigma_1, \sigma_2,\dots,\sigma_m\}$,
$k \in \mathbb N$, and $N(\sigma^1 _m)+N(\sigma^2 _m)+\dots+ N(\sigma^{k} _m)=m$.
\end{theorem}
Now we will describe the main function of our investigation.
Let $\eta$ be a random variable, that defined by the s-adic representation
$$
\eta= \frac{\xi_1}{s}+\frac{\xi_2}{s^2}+\frac{\xi_3}{s^3}+\dots +\frac{\xi_{k}}{s^{k}}+\dots = \Delta^{s} _{\xi_1\xi_2...\xi_{k}...},
$$
where $\xi_k=\alpha_k$
and digits $\xi_k$ $(k=1,2,3, \dots )$ are random and taking the values $0,1,\dots ,s-1$ with positive probabilities ${p}_{0}, {p}_{1}, \dots , {p}_{s-1}$. That is $\xi_k$ are independent and $P\{\xi_k=i_k\}={p}_{i_k}$, $i_k \in A$.
From the definition of a distribution function and the following expressions
$$
\{\eta<x\}=\{\xi_1<\alpha_1(x)\}\cup\{\xi_1=\alpha_1(x),\xi_2<\alpha_2(x)\}\cup \ldots
$$
$$
\cup\{\xi_1=\alpha_1(x),\xi_2=\alpha_2(x),\dots ,\xi_{k-1}=\alpha_{k-1}(x), \xi_{k}<\alpha_{k}(x)\}\cup \dots,
$$
$$
P\{\xi_1=\alpha_1(x),\xi_2=\alpha_2(x),\dots ,\xi_{k-1}=\alpha_{k-1}(x), \xi_{k}<\alpha_{k}(x)\}
=\beta_{\alpha_{k}(x)}\prod^{k-1} _{j=1} {{p}_{\alpha_{j}(x)}},
$$
where
$$
\beta_{\alpha_k}=\begin{cases}
\sum^{\alpha_k(x)-1} _{i=0} {p_{i}(x)}&\text{whenever $\alpha_k(x)>0$}\\
0&\text{whenever $\alpha_k(x)=0$,}
\end{cases}
$$
it is easy to see that the following statement is true.
\begin{statement}
The distribution function ${f}_{\eta}$ of the random variable $\eta$ can be represented in the following form
$$
{f}_{\eta}(x)=\begin{cases}
0&\text{whenever $x< 0$}\\
\beta_{\alpha_1(x)}+\sum^{\infty} _{k=2} {\left({\beta}_{\alpha_k(x)} \prod^{k-1} _{j=1} {{p}_{\alpha_j(x)}}\right)}&\text{whenever $0 \le x<1$}\\
1&\text{whenever $x\ge 1$,}
\end{cases}
$$
where ${p}_{\alpha_{j(x)}}>0$.
\end{statement}
The function
$$
{f}(x)=\beta_{\alpha_1(x)}+\sum^{\infty} _{n=2} {\left({\beta}_{\alpha_n(x)}\prod^{n-1} _{j=1} {{p}_{\alpha_j(x)}}\right)},
$$
can be used as a representation of numbers from $[0,1]$. That is
$$
x=\Delta^{P} _{\alpha_1(x)\alpha_2(x)...\alpha_n(x)...}=\beta_{\alpha_1(x)}+\sum^{\infty} _{n=2} {\left({\beta}_{\alpha_n(x)}\prod^{n-1} _{j=1} {{p}_{\alpha_j(x)}}\right)},
$$
where $P=\{p_0,p_1,\dots , p_{s-1}\}$, $p_0+p_1+\dots+p_{s-1}=1$, and $p_i>0$ for all $i=\overline{0,s-1}$. The last-mentioned representation is \emph{the P-representation of numbers from $[0,1]$}.
In the present article, we will consider properties of images of the sets considered in Theorem~\ref{th: theorem1} and Theorem~\ref{th: theorem2} under the map $f$.
We begin with definitions.
Let $s$ be a fixed positive integer, $s> 2$.
Let $c_1, c_2,\dots ,c_m$ be an ordered tuple of integers such that $c_i\in\{0,1,\dots ,s-1\}$ for $i=\overline{1,m}$.
\begin{definition}
{\itshape A cylinder of rank $m$ with base $c_1c_2\ldots c_m$} is a set $\Delta^{P} _{c_1c_2\ldots c_m}$ formed by all numbers
of the segment $[0,1]$ with P-representations in which the first $m$ digits coincide with $c_1,c_2,\dots ,c_m$, respectively, i.e.,
$$
\Delta^{P} _{c_1c_2\ldots c_m}=\left\{x: x=\Delta^{P} _{\alpha_1\alpha_2\ldots\alpha_n\ldots}, \alpha_j=c_j, j=\overline{1,m}\right\}.
$$
\end{definition}
Cylinders $\Delta^{P} _{c_1c_2\ldots c_m}$ have the following properties:
\begin{enumerate}
\item any cylinder $\Delta^{P} _{c_1c_2\ldots c_m}$ is a closed interval;
\item
$$
\inf \Delta^{P} _{c_1c_2\ldots c_m}= \Delta^{P} _{c_1c_2\ldots c_m000...},
\sup \Delta^{P} _{c_1c_2\ldots c_m}= \Delta^{P} _{c_1c_2\ldots c_m[s-1][s-1][s-1]...};
$$
\item
$$
| \Delta^{P} _{c_1c_2\ldots c_m}|=p_{c_1}p_{c_2}\cdots p_{c_m};
$$
\item
$$
\Delta^{P} _{c_1c_2\ldots c_mc}\subset \Delta^{P} _{c_1c_2\ldots c_m};
$$
\item
$$
\Delta^{P} _{c_1c_2\ldots c_m}=\bigcup^{s-1} _{c=0} { \Delta^{P} _{c_1c_2\ldots c_mc}};
$$
\item
$$
\lim_{m \to \infty} { |\Delta^{P} _{c_1c_2\ldots c_m}|}=0;
$$
\item
$$
\frac{| \Delta^{P} _{c_1c_2\ldots c_mc_{m+1}}|}{| \Delta^{-D} _{c_1c_2\ldots c_m}|}=p_{c_{m+1}};
$$
\item
$$
\sup\Delta^{P} _{c_1c_2...c_mc}=\inf \Delta^{P} _{c_1c_2...c_m[c+1]},
$$
where $c \ne s-1$;
\item
$$
\bigcap^{\infty} _{m=1} {\Delta^{-D} _{c_1c_2\ldots c_m}}=x=\Delta^{-D} _{c_1c_2\ldots c_m\ldots}.
$$
\end{enumerate}
\begin{definition} A number $x \in[0,1]$ is called {\itshape P-rational} if
$$
x=\Delta^{P} _{\alpha_1\alpha_2\ldots\alpha_{n-1}\alpha_n000\ldots}
$$
or
$$
x=\Delta^{P} _{\alpha_1\alpha_2\ldots\alpha_{n-1}[\alpha_n-1][s-1][s-1][s-1]\ldots}.
$$
The other numbers in $[0,1]$ are called {\itshape P-irrational}.
\end{definition}
\section{The objects of research}
Let $2<s$ be a fixed positive integer, $A=\{0,1,\dots ,s-1\}$,
$A_0=A \setminus \{0\}=\{1,2,\dots , s -1\}$, and
$$
L \equiv (A_0)^{\infty}= (A_0) \times (A_0) \times (A_0)\times\dots
$$
be the space of one-sided sequences of elements of $ A_0$.
Let $P=\{p_0,p_1, \dots , p_{s-1}\}$ be a fixed set of positive numbers such that $p_0+p_1+\dots + p_{s-1}=1$.
Let us consider a class $\Gamma$ that contains classes $\Gamma_{P_s}$ of sets $\mathbb S_{(P_s,u)}$ represented in the form
\begin{equation}
\label{S(s,u)1}
\mathbb S_{(P_s,u)}\equiv\left\{x: x= \Delta^{P}_{{\underbrace{u...u}_{\alpha_1-1}} \alpha_1{\underbrace{u...u}_{\alpha_2 -1}}\alpha_2 ...{\underbrace{u...u}_{ \alpha_n -1}}\alpha_n...}, (\alpha_n) \in L, \alpha_n \ne u, \alpha_n \ne 0 \right\},
\end{equation}
where $u=\overline{0,s-1}$, the parameters $u$ and $s$ are fixed for the set $\mathbb S_{(P_s,u)}$. That is the class $\Gamma_{P_s}$ contains the sets $\mathbb S_{(P_s,0)}, \mathbb S_{(P_s,1)},\dots,\mathbb S_{(s,s-1)}$.
\begin{lemma}
An arbitrary set $\mathbb S_{(P_s,u)}$ is a uncountable set.
\end{lemma}
\begin{proof} Let us consider the mapping $g: \mathbb S_{(P_s,u)} \to S_u$. That is
$$
\forall (\alpha_n)\in L: x= \Delta^{P}_{{\underbrace{u...u}_{\alpha_1-1}} \alpha_1{\underbrace{u...u}_{\alpha_2 -1}}\alpha_2 ...{\underbrace{u...u}_{ \alpha_n -1}}\alpha_n...} \stackrel{g}{\longrightarrow}
\Delta^{s}_{\alpha_1\alpha_2 ...\alpha_n...}=y=g(x).
$$
It follows from the definition of an arbitrary set $S_u$ that s-adic-rational numbers of the form
$$
\Delta^{s} _{\alpha_1\alpha_2\ldots\alpha_{n-1}\alpha_n000\ldots}
$$
do not belong to $ S_u$ (since the condition $\alpha_n\notin\{0,u\}$ holds). Hence each element of $ S_u$ has the unique s-adic representation.
For any $x\in \mathbb S_{(P_s,u)}$ there exists $y=g(x)\in S_u$ and for any $y\in S_u$ there exists $x=g^{-1}(y)\in \mathbb S_{(P_s,u)}$. Since P-rational numbers do not belong to $\mathbb S_{(P_s,u)}$, we have that for arbitrary $x_1\ne x_2$ the inequality $f(x_1)\ne f(x_2)$ holds.
So from the uncountability of $ S_u$ follows the uncountability of the set $\mathbb S_{(P_s,u)}$.
\end{proof}
To investigate topological and metric properties of $\mathbb S_{(P_s,u)}$, we will study properties of cylinders.
Let $c_1, c_2,\dots , c_n$ be an ordered tuple of integers such that $c_i\in\{0,1,\dots ,s-1\}$ for $i=\overline{1,n}$.
\begin{definition}
{\itshape A cylinder of rank $n$ with base $c_1c_2\ldots c_n$} is a set $\Delta^{(P,u)} _{c_1c_2\ldots c_n}$ of the form:
$$
\Delta^{(P,u)} _{c_1c_2\ldots c_n}=\left\{x: x=\Delta^{P}_{{\underbrace{u...u}_{c_1-1}} c_1{\underbrace{u...u}_{c_2 -1}}c_2 ...{\underbrace{u...u}_{ c_n -1}}c_n{\underbrace{u...u}_{\alpha_{n+1}-1}}\alpha_{n+1}{\underbrace{u...u}_{\alpha_{n+2}-1}}\alpha_{n+2}...}, \alpha_j=c_j, j=\overline{1,n}\right\}.
$$
\end{definition}
By $(a_1a_2\ldots a_k)$ denote the period $a_1a_2\ldots a_k$ in the representation of a periodic number.
\begin{lemma} Cylinders $ \Delta^{(P,u)} _{c_1...c_n} $ have the following properties:
\label{lm: Lemma on cylinders}
\begin{enumerate}
\item
$$
\inf \Delta^{(P,u)} _{c_1...c_n}=\begin{cases}
\Delta^{P} _{{\underbrace{0...0}_{c_1-1}} c_1{\underbrace{0...0}_{c_2 -1}}c_2 ...{\underbrace{0...0}_{ c_n -1}}c_n({\underbrace{0...0}_{ s-2}}[s-1])} &\text{if $u=0$}\\
\Delta^{P} _{{\underbrace{1...1}_{c_1-1}} c_1{\underbrace{1...1}_{c_2 -1}}c_2 ...{\underbrace{1...1}_{ c_n -1}}c_n({\underbrace{1...1}_{ s-2}}[s-1])} &\text{if $u=1$}\\
$$\\
\Delta^{P} _{{\underbrace{u...u}_{c_1-1}} c_1{\underbrace{u...u}_{c_2 -1}}c_2 ...{\underbrace{u...u}_{ c_n -1}}c_n(1)}&\text{if $ u \in \{2,3,\dots ,s-1\}$,}
\end{cases}
$$
$$
\sup \Delta^{(P,u)} _{c_1...c_n...}=\begin{cases}
\Delta^{P} _{{\underbrace{[s-1]...[s-1]}_{c_1-1}} c_1 ...{\underbrace{[s-1]...[s-1]}_{ c_n -1}}c_n({\underbrace{[s-1]...[s-1]}_{ s-3}}[s-2])} &\text{if $u=s-1$}\\
\Delta^{P} _{{\underbrace{u...u}_{c_1-1}} c_1{\underbrace{u...u}_{c_2 -1}}c_2 ...{\underbrace{u...u}_{ c_n -1}}c_n({\underbrace{u...u}_{ u}}[u+1])} &\text{if $u\in\{1,\dots, s-2\}$}\\
$$\\
\Delta^{P} _{{\underbrace{0...0}_{c_1-1}} c_1{\underbrace{0...0}_{c_2 -1}}c_2 ...{\underbrace{0...0}_{ c_n -1}}c_n(1)}&\text{if $ u=0$.}
\end{cases}
$$
\item If $d(\cdot) $ is the diameter of a set, then
$$
d(\Delta^{(P,u)} _{c_1...c_n})=d(\mathbb S_{(P_s,u)})p^{c_1+c_2+\dots+c_n-n} _{u}\prod^{n} _{j=1}{p_{c_j}}.
$$
\item
$$
\frac{d(\Delta^{(P,u)} _{c_1...c_nc_{n+1}})}{d(\Delta^{(P,u)} _{c_1...c_n})}=p_{c_{n+1}}p^{c_{n+1}-1} _{u}.
$$
\item
$$
\Delta^{(P,u)} _{c_1c_2...c_n} =\bigcup^{s-1} _{i=1} { \Delta^{(P,u)} _{c_1c_2...c_ni}}~~~\forall c_n \in A_0,~~~n \in \mathbb N,~ i \ne u.
$$
\item The following relationships are satisfied:
\begin{enumerate}
\item if $ u\in \{0,1\}$, then
$$
\inf \Delta^{(P,u)} _{c_1...c_np}> \sup \Delta^{(P,u)} _{c_1...c_n[p+1]};
$$
\item if $ u \in \{2,3,\dots ,s-3\}$, then
$$
\begin{cases}
\sup \Delta^{(P,u)} _{c_1...c_np}< \inf \Delta^{(P,u)} _{c_1...c_n[p+1]}&\text{for all $p+1\le u$}\\
$$\\
\inf \Delta^{(P,u)} _{c_1...c_np}> \sup \Delta^{(P,u)} _{c_1...c_n[p+1]},&\text{for all $u<p$;}
\end{cases}
$$
\item if $ u \in \{s-2,s-1\}$, then
$$
\sup \Delta^{(P,u)} _{c_1...c_np}< \inf \Delta^{(P,u)} _{c_1...c_n[p+1]} ~~~(\text{in this case, the condition $p\ne s-1$ holds}).
$$
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof} \emph{The first property} follows from the equality
$$
x=\Delta^{P}_{{\underbrace{u...u}_{c_1-1}} c_1{\underbrace{u...u}_{c_2 -1}}c_2 ...{\underbrace{u...u}_{ c_n -1}}c_n{\underbrace{u...u}_{\alpha_{n+1}-1}}\alpha_{n+1}{\underbrace{u...u}_{\alpha_{n+2}-1}}\alpha_{n+2}...}
$$
$$
=\Delta^{P}_{{\underbrace{u...u}_{c_1-1}} c_1{\underbrace{u...u}_{c_2 -1}}c_2 ...{\underbrace{u...u}_{ c_n -1}}c_n(0)}+p^{c_1+\dots +c_n-n} _{u}\left(\prod^{n} _{k=1}{p_{c_k}}\right)\Delta^{P}_{{\underbrace{u...u}_{\alpha_{n+1}-1}}\alpha_{n+1}{\underbrace{u...u}_{\alpha_{n+2}-1}}\alpha_{n+2}...}
$$
and the definition of $ \mathbb S_{(P_s,u)}$.
It is easy to see that \emph{the second property} follows from the first property, \emph{the third property} is a corollary of the first and second properties, and \emph{Property 4} follows from the definition of the set.
Let us show that \emph{Property 5} is true. We now prove that the first inequality holds for $ u=1$. In fact,
$$
\inf \Delta^{(P,0)} _{c_1...c_np}- \sup \Delta^{(P,0)} _{c_1...c_n[p+1]}=
\Delta^{P} _{{\underbrace{0...0}_{c_1-1}} c_1{\underbrace{0...0}_{c_2 -1}}c_2 ...{\underbrace{0...0}_{ c_n -1}}c_n{\underbrace{0...0}_{p -1}}p({\underbrace{0...0}_{ s-2}}[s-1])}-\Delta^{P} _{{\underbrace{0...0}_{c_1-1}} c_1{\underbrace{0...0}_{c_2 -1}}c_2 ...{\underbrace{0...0}_{ c_n -1}}c_n{\underbrace{0...0}_{p}}[p+1](1)}
$$
$$
=\beta_pp^{c_1+...+c_n-n+p-1} _0\prod^{n} _{j=1}{p_{c_j}}+p_pp^{c_1+...+c_n-n+p-1} _0\left(\prod^{n} _{j=1}{p_{c_j}}\right)\inf{\mathbb S_{(P_s,0)}}
$$
$$
-\beta_{p+1}p^{c_1+...+c_n-n+p} _0\prod^{n} _{j=1}{p_{c_j}}-p_{p+1}p^{c_1+...+c_n-n+p} _0\left(\prod^{n} _{j=1}{p_{c_j}}\right)\sup{\mathbb S_{(P_s,0)}}
$$
$$
=p^{c_1+...+c_n-n+p} _0\left(\prod^{n} _{j=1}{p_{c_j}}\right)\left(\beta_pp^{-1} _0+p_pp^{-1} _0\inf{\mathbb S_{(P_s,0)}}-\beta_{p+1}-p_{p+1}\sup{\mathbb S_{(P_s,0)}}\right)
$$
$$
=p^{c_1+...+c_n-n+p-1} _0\left(\prod^{n} _{j=1}{p_{c_j}}\right)(p_0(1-p_0-p_p-p_{p+1}\sup{\mathbb S_{(P_s,0)}})+(1-p_0)(p_1+...+p_{p-1})+p_p\inf{\mathbb S_{(P_s,0)}})>0
$$
because
$$
1-p_0-p_p-p_{p+1}\sup{\mathbb S_{(P_s,0)}}=1-p_0-p_p-p_{p+1}\frac{p_0}{1-p_1}=
\frac{\sum_{i\notin\{0,1,p,p+1\}}p_i+p_{p+1}(1-p_0)+p_0p_1+p_1p_p}{1-p_1}>0.
$$
Also,
$$
\inf \Delta^{(P,1)} _{c_1...c_np}- \sup \Delta^{(P,1)} _{c_1...c_n[p+1]}=
\Delta^{P} _{{\underbrace{1...1}_{c_1-1}} c_1{\underbrace{0...0}_{c_2 -1}}c_2 ...{\underbrace{1...1}_{ c_n -1}}c_n{\underbrace{1...1}_{p -1}}p({\underbrace{1...1}_{ s-2}}[s-1])}-\Delta^{P} _{{\underbrace{1...1}_{c_1-1}} c_1{\underbrace{1...1}_{c_2 -1}}c_2 ...{\underbrace{1...1}_{ c_n -1}}c_n\underbrace{1...1}_{p}[p+1](12)}
$$
$$
=\beta_pp^{c_1+...+c_n+p-n-1} _1\prod^{n} _{j=1}{p_{c_j}}+p_pp^{c_1+...+c_n-n+p-1} _1\left(\prod^{n} _{j=1}{p_{c_j}}\right)\inf{\mathbb S_{(P_s,1)}}
$$
$$
-\beta_{p+1}p^{c_1+...+c_n+p-n} _1\prod^{n} _{j=1}{p_{c_j}}-p_{p+1}p^{c_1+...+c_n-n+p} _1\left(\prod^{n} _{j=1}{p_{c_j}}\right)\sup{\mathbb S_{(P_s,1)}}
$$
$$
=p^{c_1+...+c_n-n+p-1} _1\left(\prod^{n} _{j=1}{p_{c_j}}\right)\left(\beta_p+p_p\inf{\mathbb S_{(P_s,1)}}-\beta_{p+1}p_1-p_{p+1}p_1\sup{\mathbb S_{(P_s,1)}}\right)
$$
$$
=p^{c_1+...+c_n-n+p-1} _1\left(\prod^{n} _{j=1}{p_{c_j}}\right)(p_p\inf{\mathbb S_{(P_s,1)}}+p_1(1-p_1-p_p-p_{p+1}\sup{\mathbb S_{(P_s,1)}})+(1-p_1)(p_0+p_2+...+p_{p-1}))>0,
$$
since
$$
\sup{\mathbb S_{(P_s,1)}}=\Delta^P _{(12)}=\beta_1+\sum^{\infty} _{k=1}{\beta_1p^k _{1} p^k _2}+\sum^{\infty} _{k=1}{\beta_2p^k _{1} p^k _2}=\frac{p_0+p_0p_1+p^2 _1}{1-p_1p_2}>0
$$
and
$$
1=p_0+p_1+\dots+ p_{s-1}.
$$
Let us prove the system of inequalities. Consider the first inequality. For the case when $p+1\le u$ we get
$$
\inf \Delta^{(P,u)} _{c_1...c_n[p+1]}-\sup \Delta^{(P,u)} _{c_1...c_np}=\Delta^{P} _{{\underbrace{u...u}_{c_1-1}} c_1{\underbrace{u...u}_{c_2 -1}}c_2 ...{\underbrace{u...u}_{ c_n -1}}c_n{\underbrace{u...u}_{p}} [p+1](1)}-\Delta^{P} _{{\underbrace{u...u}_{c_1-1}} c_1{\underbrace{u...u}_{c_2 -1}}c_2 ...{\underbrace{u...u}_{ c_n -1}}c_n{\underbrace{u...u}_{p-1}} p({\underbrace{u...u}_{ u}}[u+1])}
$$
$$
=\beta_up^{c_1+...+c_n-n+p-1} _u\prod^{n} _{j=1}{p_{c_j}}+\beta_{p+1}p^{c_1+...+c_n-n+p} _u\prod^{n} _{j=1}{p_{c_j}}+p_{p+1}p^{c_1+...+c_n-n+p} _u\left(\prod^{n} _{j=1}{p_{c_j}}\right)\cdot\Delta^P _{(1)}
$$
$$
-\beta_pp^{c_1+...+c_n-n+p-1} _u\prod^{n} _{j=1}{p_{c_j}}-p_{p}p^{c_1+...+c_n-n+p-1} _u\left(\prod^{n} _{j=1}{p_{c_j}}\right)\cdot\Delta^P _{({\underbrace{u...u}_{ u}}[u+1])}
$$
$$
=p^{c_1+...+c_n-n+p-1} _u\left(\prod^{n} _{j=1}{p_{c_j}}\right)\left(\beta_u+\beta_{p+1}p_u+p_{p+1}p_u\Delta^P _{(1)}-\beta_p-p_p\Delta^P _{({\underbrace{u...u}_{ u}}[u+1])}\right)
$$
$$
=p^{c_1+...+c_n-n+p-1} _u\left(\prod^{n} _{j=1}{p_{c_j}}\right)\left(p_{p+1}p_u\Delta^P _{(1)}+(\beta_u-\beta_p)+p_up_p+\beta_pp_u-p_p\Delta^P _{({\underbrace{u...u}_{ u}}[u+1])}\right)>0
$$
since the conditions $p<u$, $\beta_u-\beta_p>0$, and $\beta_{p+1}=\beta_p+p_{p}$ hold.
Let us prove that the second inequality is true. Here $ p>u $, i.e., $p-u \ge 1$. Similarly,
$$
\inf \Delta^{(P,u)} _{c_1...c_np}-\sup \Delta^{(P,u)} _{c_1...c_n[p+1]}=\Delta^{P} _{{\underbrace{u...u}_{c_1-1}} c_1{\underbrace{u...u}_{c_2 -1}}c_2 ...{\underbrace{u...u}_{ c_n -1}}c_n{\underbrace{u...u}_{p-1}}p(1)}-\Delta^{P} _{{\underbrace{u...u}_{c_1-1}} c_1{\underbrace{u...u}_{c_2 -1}}c_2 ...{\underbrace{u...u}_{ c_n -1}}c_n{\underbrace{u...u}_{p}}[p+1]({\underbrace{u...u}_{ u}}[u+1])}
$$
$$
=\beta_pp^{c_1+...+c_n-n+p-1} _u\prod^{n} _{j=1}{p_{c_j}}+p_{p}p^{c_1+...+c_n-n+p-1} _u\left(\prod^{n} _{j=1}{p_{c_j}}\right)\cdot\Delta^P _{(1)}
$$
$$
-\beta_up^{c_1+...+c_n-n+p-1} _u\prod^{n} _{j=1}{p_{c_j}}-\beta_{p+1}p^{c_1+...+c_n-n+p} _u\prod^{n} _{j=1}{p_{c_j}}
-p_{p+1}p^{c_1+...+c_n-n+p} _u\left(\prod^{n} _{j=1}{p_{c_j}}\right)\cdot\Delta^P _{({\underbrace{u...u}_{ u}}[u+1])}
$$
$$
=p^{c_1+...+c_n-n+p-1} _u\left(\prod^{n} _{j=1}{p_{c_j}}\right)\left(\beta_p+p_{p}\Delta^P _{(1)}-\beta_u-\beta_{p+1}p_u-p_{p+1}p_u\Delta^P _{({\underbrace{u...u}_{ u}}[u+1])}\right)
$$
$$
=p^{c_1+...+c_n-n+p-1} _u\left(\prod^{n} _{j=1}{p_{c_j}}\right)\left(p_{p}\Delta^P _{(1)}
+(p_{u+1}+...+p_{p+1})+p_u(p_{p+1}+...+p_{s-1}-p_{p+1}\Delta^P _{({\underbrace{u...u}_{ u}}[u+1])})\right)>0
$$
since the conditions $p>u$, $\beta_p-\beta_u=p_u+p_{u+1}+...+p_{p-1}$, and $1-\beta_{p+1}=p_{p+1}+...+p_{s-1}$ hold.
Suppose that $u=s-2$. Then
$$
\inf\Delta^{(P,s-2)} _{c_1c_2...c_n[p+1]}-\sup\Delta^{(P,s-2)} _{c_1c_2...c_np}
$$
$$
=\Delta^{P} _{{\underbrace{[s-2]...[s-2]}_{c_1-1}} c_1{\underbrace{[s-2]...[s-2]}_{c_2 -1}}c_2 ...{\underbrace{[s-2]...[s-2]}_{ c_n -1}}c_n{\underbrace{[s-2]...[s-2]}_{p}} [p+1](1)}
$$
$$
- \Delta^{P} _{{\underbrace{[s-2]...[s-2]}_{c_1-1}} c_1{\underbrace{[s-2]...[s-2]}_{c_2 -1}}c_2 ...{\underbrace{[s-2]...[s-2]}_{ c_n -1}}c_n{\underbrace{[s-2]...[s-2]}_{p-1}} p({\underbrace{[s-2]...[s-2]}_{ s-2}}[s-1])}
$$
$$
=\beta_{s-2}p^{c_1+...+c_n-n+p-1} _{s-2}\prod^{n} _{j=1}{p_{c_j}}+\beta_{p+1}p^{c_1+...+c_n-n+p} _{s-2}\prod^{n} _{j=1}{p_{c_j}}
$$
$$
+p_{p+1}p^{c_1+...+c_n-n+p} _{s-2}\left(\prod^{n} _{j=1}{p_{c_j}}\right)\cdot\Delta^{P} _{(1)}-\beta_pp^{c_1+...+c_n-n+p-1} _{s-2}\prod^{n} _{j=1}{p_{c_j}}
$$
$$
-p_pp^{c_1+...+c_n-n+p-1} _{s-2}\left(\prod^{n} _{j=1}{p_{c_j}}\right)\cdot\Delta^P _{({\underbrace{[s-2]...[s-2]}_{ s-2}}[s-1])}
$$
$$
=p^{c_1+...+c_n-n+p-1} _{s-2}\left(\prod^{n} _{j=1}{p_{c_j}}\right)(\beta_{s-2}+\beta_{p+1}p_{s-2}+p_{s-2}p_{p+1}\Delta^{P} _{(1)}-\beta_{p}-p_p\Delta^P _{({\underbrace{[s-2]...[s-2]}_{ s-2}}[s-1])})
$$
$$
=p^{c_1+...+c_n-n+p-1} _{s-2}\left(\prod^{n} _{j=1}{q_{c_j}}\right)(p_p(1-\Delta^P _{({\underbrace{[s-2]...[s-2]}_{ s-2}}[s-1])})+(p_{p+1}+...+p_{s-3})+\beta_{p+1}p_{s-2}+p_{s-2}p_{p+1}\Delta^{P} _{(1)})>0
$$
since $\beta_{s-2}-\beta_{p}=p_p+p_{p+1}+\dots+p_{s-3}$. Here $p\ne s-1$.
Suppose that $u=s-1$. Then
$$
\inf\Delta^{(P,s-1)} _{c_1c_2...c_n[p+1]}-\sup\Delta^{(P,s-1)} _{c_1c_2...c_np}
$$
$$
=\Delta^{P} _{{\underbrace{[s-1]...[s-1]}_{c_1-1}} c_1{\underbrace{[s-1]...[s-1]}_{c_2 -1}}c_2 ...{\underbrace{[s-1]...[s-1]}_{ c_n -1}}c_n{\underbrace{[s-1]...[s-1]}_{p}} [p+1](1)}
$$
$$
-\Delta^{P} _{{\underbrace{[s-1]...[s-1]}_{c_1-1}} c_1 ...{\underbrace{[s-1]...[s-1]}_{ c_n -1}}c_n{\underbrace{[s-1]...[s-1]}_{p-1}}p({\underbrace{[s-1]...[s-1]}_{ s-3}}[s-2])}
$$
$$
=\beta_{s-1}p^{c_1+...+c_n-n+p-1} _{s-1}\prod^{n} _{j=1}{p_{c_j}}+\beta_{p+1}p^{c_1+...+c_n-n+p} _{s-1}\prod^{n} _{j=1}{p_{c_j}}
$$
$$
+p_{p+1}p^{c_1+...+c_n-n+p} _{s-1}\left(\prod^{n} _{j=1}{p_{c_j}}\right)\cdot\Delta^{P} _{(1)}-\beta_pp^{c_1+...+c_n-n+p-1} _{s-1}\prod^{n} _{j=1}{p_{c_j}}
$$
$$
-p_pp^{c_1+...+c_n-n+p-1} _{s-1}\left(\prod^{n} _{j=1}{p_{c_j}}\right)\cdot\Delta^P _{({\underbrace{[s-1]...[s-1]}_{ s-3}}[s-2])}
$$
$$
=p^{c_1+...+c_n-n+p-1} _{s-1}\left(\prod^{n} _{j=1}{p_{c_j}}\right)(\beta_{s-1}+\beta_{p+1}p_{s-1}+p_{s-1}p_{p+1}\Delta^{P} _{(1)}-\beta_{p}-p_p\Delta^P _{({\underbrace{[s-1]...[s-1]}_{ s-3}}[s-2])})>0.
$$
\end{proof}
\begin{theorem}
The set $\mathbb S_{(P_s,u)}$ is a perfect and nowhere dense set of zero Lebesgue measure.
\end{theorem}
\begin{proof}
We now prove that \emph{the set $\mathbb S_{(P_s,u)}$ is a nowhere dense set}. From the definition it follows that there exist cylinders $ \Delta^{(P,u)} _{c_1...c_n}$ of rank $n$ in an arbitrary subinterval of the segment $I=[\inf\mathbb S_{(P_s,u)},\sup\mathbb S_{(P_s,u)}]$. Since Property 5 from Lemma~\ref{lm: Lemma on cylinders} is true for these cylinders, we have that for any subinterval of $ I$ there exists a subinterval such that does not contain points from $\mathbb S_{(P_s,u)}$. So $\mathbb S_{(P_s,u)}$ is a nowhere dense set.
Let us show that \emph{$\mathbb S_{(P_s,u)}$ is a set of zero Lebesgue measure}. Suppose that $ I^{(P_s,u)} _{c_1c_2...c_n} $ is a closed interval whose endpoints coincide with endpoits of the cylinder $ \Delta^{(P,u)} _{c_1c_2...c_n}$,
$$
|I^{(P_s,u)} _{c_1c_2...c_n}|=d(\Delta^{(P,u)} _{c_1c_2...c_n})=d(\mathbb S_{(P_s,u)})p^{c_1+c_2+\dots+c_n-n} _{u}\prod^{n} _{j=1}{p_{c_j}},
$$
and
$$
\mathbb S_{(P_s,u)}= \bigcap^{\infty} _{k=1} E^{(P_s,u)} _k,
$$
where
$$
E^{(P_s,u)} _1=\bigcup_{c_1\in A_0\setminus\{u\}}{I^{(P_s,u)} _{c_1}},
$$
$$
E^{(P_s,u)} _2=\bigcup_{c_1,c_2\in A_0\setminus\{u\}}{I^{(P_s,u)} _{c_1c_2}},
$$
$$
\dots\dots\dots\dots\dots\dots\dots
$$
$$
E^{(P_s,u)} _k= \bigcup_{c_1,c_2,...,c_k\in A_0\setminus\{u\}}{I^{(P_s,u)} _{c_1c_2...c_k}},
$$
$$
\dots\dots\dots\dots\dots\dots\dots
$$
In addition, since $ E^{(P_s,u)} _{k+1} \subset E^{(P_s,u)} _k $, we have
$$
E^{(P_s,u)} _k= E^{(P_s,u)} _{k+1} \cup \bar E^{(P_s,u)} _{k+1}.
$$
Let $ I$ be an initial closed interval such that $ \lambda(I)=d_0 $ and $\mathbb [\inf \mathbb S_{(P_s,u)}, \sup\mathbb S_{(P_s,u)}]=I$, $\lambda(\cdot)$ be the Lebesgue measure of a set. Then
$$
\lambda(E^{(P_s,u)} _1)=\sum_{c_1\in A_0\setminus\{u\}}{|I^{(P_s,u)} _{c_1}|}=d(\mathbb S_{(P_s,u)})\sum_{c_1\in A_0\setminus\{u\}}{p^{c_1-1} _{u}}=\gamma_0.
$$
We get
$$
\lambda(\bar E^{(P_s,u)} _1)=d_0-\lambda(E^{(P_s,u)} _1)=d_0 - \gamma_0 d_0= d_0(1 - \gamma_0).
$$
Similarly,
$$
\lambda(\bar E^{(P_s,u)} _2)=\lambda(E^{(P_s,u)} _1)-\lambda(E^{(P_s,u)} _2)=\gamma_0d_0-\gamma^2 _0d_0=d_0\gamma_0(1-\gamma_0),
$$
$$
\lambda(\bar E^{(P_s,u)} _3)=\lambda(E^{(P_s,u)} _2)-\lambda(E^{(P_s,u)} _3)=\gamma^2 _0d_0-\gamma^3 _0d_0=(1-\gamma_0)\gamma^2 _0d_0,
$$
$$
\dots\dots\dots\dots\dots\dots\dots
$$
So,
$$
\lambda(\mathbb S_{(P_s,u)})=d_0-\sum^{n} _{k=1}{\lambda(\bar E^{(P_s,u)} _k)}=d_0-\sum^{n} _{k=1}{\gamma^{k-1} _0d_0(1-\gamma_0)}=d_0-\frac{d_0(1-\gamma_0)}{1-\gamma_0}=0.
$$
The set $\mathbb S_{(P_s,u)}$ is a set of zero Lebesgue measure.
Let us prove that \emph{$\mathbb S_{(P_s,u)}$ is a perfect set}. Since
$$
E^{(P_s,u)} _k= \bigcup_{c_1,c_2,...,c_k\in A_0\setminus\{u\}}{I^{(P_s,u)} _{c_1c_2...c_k}}
$$
is a closed set ($E^{(P_s,u)} _k$ is a union of segments), we see that
$$
\mathbb S_{(P_s,u)}= \bigcap^{\infty} _{k=1} E^{(P_s,u)} _k
$$
is a closed set.
Let $ x \in \mathbb S_{(P_s,u)} $, $ P$ be any interval that contains $ x $, and $ J_n $ be a segment of $ E^{(P_s,u)} _n $ that contains $ x $. Choose a number $ n $ such that $ J_n \subset P $. Suppose that $ x_n $ is the endpoint of $ J_n $ such that the condition
$ x_n \ne x $ holds. Hence $ x_n \in \mathbb S_{(P_s,u)} $ and $ x $ is a limit point of the set.
Since $\mathbb S_{(P_s,u)}$ is a closed set and does not contain isolated points, we obtain that $\mathbb S_{(P_s,u)}$ is a perfect set.
\end{proof}
\begin{theorem}
The set $\mathbb S_{(P_s,u)} $ is a self-similar fractal and the Hausdorff dimension $\alpha_0 (\mathbb S_{(P_s,u)})$ of the set satisfies the following equation
$$
\sum _{i\in A_0\setminus\{u\}} {\left(p_ip^{i-1} _u\right)^{\alpha_0}}=1.
$$
\end{theorem}
\begin{proof}
Since
$ \mathbb S_{(P_s,u)} \subset I$ and $ \mathbb S_{(P_s,u)}$ is a perfect set, we obtain that $\mathbb S_{(P_s,u)}$ is a compact set. In addition,
$$
\mathbb S_{(P_s,u)}=\bigcup_{i\in A_0\setminus\{u\}}{\left[I^{(P_s,u)} _i\cap \mathbb S_{(P_s,u)}\right]}
$$
and $\left[I^{(P_s,u)} _i\cap \mathbb S_{(P_s,u)}\right]\stackrel{p_ip^{i-1} _u}{\sim}\mathbb S_{(P_s,u)}$ for all $i\in A_0\setminus\{u\}$.
Since the set $\mathbb S_{(P_s,u)}$ is a compact self-similar set of space $ \mathbb R^1 $, we have that the self-similar dimension of this set is equal to the Hausdorff dimension of $\mathbb S_{(P_s,u)}$. So the set $\mathbb S_{(P_s,u)} $ is a self-similar fractal, and its Hausdorff dimension $\alpha_0$ satisfies the equation
$$
\sum _{i\in A_0\setminus\{u\}} {\left(p_ip^{i-1} _u\right)^{\alpha_0}}=1.
$$
\end{proof}
\begin{theorem}
Let $E$ be a set whose elements represented in terms of the P-representation by a finite number of fixed combinations $\tau_1, \tau_2,\dots,\tau_m$ of digits from the alphabet $A$. Then the Hausdorff dimension $\alpha_0$ of $E$ satisfies the following equation:
$$
\sum^{m} _{j=1}{\left(\prod^{s-1} _{i=0}{p^{N_i(\tau_j)} _i}\right)^{\alpha_0}}=1,
$$
where $N_i(\tau_k)$ ($k=\overline{1,m}$) is a number of the digit $i$ in $\tau_k$ from the set $\{\tau_1, \tau_2,\dots,\tau_m\}$.
\end{theorem}
\begin{proof} Let $\{\tau_1, \tau_2,\dots,\tau_m\}$ be a set of fixed combinations of digits from $A$ and the P-representation of any number from $E$ contains only such combinations of digits.
It is easy to see that there exist combinations $\tau', \tau''$ from the set $\Xi=\{\tau_1, \tau_2,\dots,\tau_m\}$ such that
$\Delta^P _{\tau^{'}\tau^{'}...}=\inf E$, $\Delta^P _{\tau^{''}\tau^{''}...}=\sup E$,
and
$$
d(E)=\sup E - \inf E=\Delta^P _{\tau^{''}\tau^{''}...}-\Delta^s _{\tau^{'}\tau^{'}...}.
$$
\emph{A cylinder $ \Delta^{(P,E)} _{\tau^{'} _1\tau^{'} _2\ldots\tau^{'} _n}$ of rank $n$ with base $\tau^{'} _1\tau^{'} _2\ldots\tau^{'} _n$} is a set formed by all numbers of $E$ with the P-representations in which the first $n$ combinations of digits are fixed and coincide with $\tau^{'} _1,\tau^{'} _2,\dots,\tau^{'} _n$ respectively ($\tau^{'} _j\in \Xi$ for all $j=\overline{1,n}$).
It is easy to see that
$$
d( \Delta^{(P,E)} _{\tau^{'} _1\tau^{'} _2...\tau^{'} _n})=d(E)\cdot p^{N_0(\tau^{'} _1\tau^{'} _2...\tau^{'} _n)} _0p^{N_1(\tau^{'} _1\tau^{'} _2...\tau^{'} _n)} _1\cdots p^{N_{s-1}(\tau^{'} _1\tau^{'} _2...\tau^{'} _n)} _{s-1},
$$
where ${N_i(\tau^{'} _1\tau^{'} _2...\tau^{'} _n)}$ is a number of the digit $i\in A$ in $\tau^{'} _1\tau^{'} _2...\tau^{'} _n$.
Since $E$ is a closed set, $ E \subset [\inf E, \sup E] $, and
$$
\frac{d\left( \Delta^{(P,E)} _{\tau^{'} _1\tau^{'} _2...\tau^{'} _n\tau^{'} _{n+1}}\right)}{d\left( \Delta^{(P,E)} _{\tau^{'} _1\tau^{'} _2...\tau^{'} _n}\right)}=\prod^{s-1} _{i=0}{p^{N_i(\tau^{'} _{n+1})} _i},
$$
$$
E=[I_{\tau_{1}} \cap E]\cup [I_{\tau_{2}} \cap E]\cup\ldots\cup[I_{\tau_m}\cap E],
$$
where
$I_{\tau_j}=[\inf \Delta^{(P,E)} _{\tau_j},\sup \Delta^{(P,E)} _{\tau_j}]$ and $j=1,2,\dots,m,$
we have
$$
{[I_{\tau_j} \cap E]} \stackrel{\omega_j}{\sim}E~\text{for all}~j=\overline{1,m},
$$
where
$$
\omega_j=\prod^{s-1} _{i=0}{p^{N_i(\tau_j)} _i}.
$$
This completes the proof.
\end{proof}
\end{document} |
\begin{document}
\tilde tle[M-Function Asymptotics and Borg-type Theorems]{
Weyl-Titchmarsh $M$-Function Asymptotics, Local Uniqueness
Results, Trace Formulas, and Borg-type
Theorems for Dirac Operators}
\author[Clark and Gesztesy]{Steve Clark and Fritz Gesztesy}
\address{Department of Mathematics and Statistics, University
of Missouri-Rolla, Rolla, MO 65409, USA}
\email{[email protected]}
\urladdr{http://www.umr.edu/\~{ }clark}
\address{Department of Mathematics,
University of
Missouri, Columbia, MO
65211, USA}
\email{[email protected]\newline
\text{\rm{ind}}ent{\it URL:}
http://www.math.missouri.edu/people/fgesztesy.html}
\dedicatory{Dedicated to F.~V.~Atkinson, one of the pioneers of this subject.}
\thanks{Supported in part by NSF grant INT-9810322.}
\subjclass{Primary 34B20, 34E05, 34L40; Secondary 34A55.}
\keywords{Weyl-Titchmarsh matrices, high-energy expansions, uniqueness
results, trace formulas, Borg theorems, Dirac operators.}
\begin{abstract}
We explicitly determine the high-energy asymptotics for
Weyl-Titchmarsh matrices associated with general
Dirac-type operators on half-lines and on ${\mathbb{R}}$. We
also prove new local uniqueness results for Dirac-type operators in terms
of exponentially small differences of Weyl-Titchmarsh matrices. As
concrete applications of the asymptotic high-energy expansion we derive
a trace formula for Dirac operators and use it to prove a Borg-type
theorem.
\end{abstract}
\maketitle
\section{Introduction}\label{s1}
While the high-energy asymptotics, $|z|\to\infty$, of scalar-valued
Weyl-Titchmarsh functions, $m_+(z,x_0)$, associated with general
half-line Dirac-type differential expressions of the form
\begin{equation}
J\frac{d}{dx}-B(x), \quad J=\begin{pmatrix} 0 &-1\\1 &0
\end{pmatrix}
\end{equation}
and $B$ a self-adjoint $2\tilde mes 2$ matrix with
real-valued coefficients, $B^{(n)}\in L^1([x_0,c])^{2\tilde mes 2}$ for some
$n\in{\mathbb{N}}_0(={\mathbb{N}}\cup\{0\})$ and all $c>x_0$, received some attention
over the past two decades as can be inferred, for instance, from
\cite{EHS83}, \cite{Ha85}, \cite{HKS89a}, \cite{HKS89b}, \cite{Mi91}
(and the literature therein), it may perhaps come as a surprise
that the corresponding matrix extension of this problem,
considering general matrix-valued differential expressions of the
type
\begin{equation}
J\frac{d}{dx}-B(x), \quad J=\begin{pmatrix} 0 &-I_m\\I_m &0\end{pmatrix}
\end{equation}
with $I_m$ the identity matrix in ${\mathbb{C}}^m$, $m\in{\mathbb{N}}$, and $B$ a
self-adjoint
$2m\tilde mes 2m$ matrix satisfying $B^{(n)}\in L^1([x_0,c])^{2m\tilde mes 2m}$
for some $n\in{\mathbb{N}}_0$ and all $c>x_0$, apparently, received no attention
at all. (It should be noted that this observation discounts papers in the
special scattering theoretic case concerned with short-range coefficients
$B^{(n)}\in L^1([x_0,\infty); (1+|x|)dx)^{2m\tilde mes 2m}$, where
iterations of Volterra-type integral equations yield the asymptotic
high-energy expansion of $M_+(z,x_0)$ as $|z|\to\infty$ to any order,
cf.~Lemma~\ref{l4.1}.) This is not because of a lack of interest in this
type of problem (we will discuss its relevance below), but simply since
it is a nontrivial one, which, in many of its aspects, must be regarded as
more difficult than the corresponding matrix-valued
Schr\"odinger operator case, which in turn, was only very recently
settled in \cite{CG99}. The results proven in this paper show that in
leading order (and independently of the self-adjoint boundary condition
chosen at $x_0$),
\begin{equation}\label{1.1}
M_+(z,x_0) \underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} iI_{m} +o(1),
\end{equation}
where $C_{\varepsilon}$ denotes the open sector in the open upper
complex half-plane ${\mathbb{C}}_+$ with vertex at zero, symmetry axis along the
positive imaginary axis, and opening angle $\varepsilon$, with
$0<\varepsilon <\pi/2$. We are interested in proving the
asymptotic expansion \eqref{1.1} and especially in its higher-order
analogs in powers of $1/z$, under optimal smoothness hypotheses on $B$.
Such results are then also derived for the $2m\tilde mes 2m$ analog $M(z,x)$
of $M_+(z,x)$ associated with Dirac-type operators on ${\mathbb{R}}$.
Our principal motivation in studying this problem stems from our
general interest in operator-valued Herglotz functions
(cf.~\cite{Ca76}, \cite{GKMT98}, \cite{GKM00}, \cite{GM99}, \cite{GM99a},
\cite{GMN98}, \cite{GMT98}, \cite{GT97},
\cite{Sh71}) and their possible applications in the areas of inverse
spectral theory and completely integrable systems. More precisely,
using higher-order asymptotic expansions of $M_+(z,x)$, one can prove
trace formulas for $B(x)$ and certain higher-order differential
polynomials in $B(x)$ (similar in spirit to an approach pioneered in
\cite{GS96} (see also \cite{GH97}, \cite{GHSZ95}) in connection with
Schr\"odinger operators). These trace formulas, in turn, then can be used
to prove various results in inverse spectral theory for matrix-valued
Dirac-type operators $D=J\frac{d}{dx}-B$ in $L^2({\mathbb{R}})^{2m}$. For instance,
using one of the principal results of this paper, Theorem~\ref{t4.6}, and
its straightforward application to the asymptotic high-energy
expansion of the diagonal Green's matrix
$G(z,x,x)=(D-z)^{-1}(x,x)$ of $D$, the following matrix-valued
analog of a classical uniqueness result of Borg \cite{Bo46} for
one-dimensional Schr\"odinger operators will be proven in
in the context of Dirac-type operators in Section~\ref{s6}.
\begin{theorem} \label{t1.1} Suppose that $B$ is of the normal form
$B(x)=\left(\begin{smallmatrix} B_{1,1}(x)
& B_{1,2}(x)\\B_{1,2}(x) & -B_{1,1}(x)\end{smallmatrix}\right)$, with
$B_{1,1}(x)$ and $B_{1,2}(x)$ self-adjoint for a.e.~$x\in{\mathbb{R}}$, and
assume that $D$ is reflectionless $($e.g., $B$ is periodic and D has
uniform spectral multiplicity $2m$$)$. In addition, suppose that $D$
has spectrum equal to ${\mathbb{R}}$. Then,
\begin{equation}
B(x)=0 \text{ for a.e.~$x\in{\mathbb{R}}$}. \label{1.2}
\end{equation}
\end{theorem}
For related results see, for instance, \cite{Am93}, \cite{AG96},
\cite{CJ87}, \cite{Ge91}, \cite{GSS91}, \cite{GJ84},
\cite{GG93}. Incidentally, the higher-order differential polynomials in
$B(x)$ just alluded to represent the Ablowitz-Kaup-Newell-Segur (AKNS) or
Zakharov-Shabat (ZS) invariants (i.e., densities associated with the
AKNS-ZS conservation laws) and hence provide a link to
infinite-dimensional completely integrable systems (cf., e.g.,
\cite{AK90}, \cite{Ch96}, \cite{Di91}, \cite{Du77}, \cite{Du83},
\cite{Ma78}, \cite{Ma88}, \cite{Sa88}, \cite{Sa99}, \cite{Sa94a},
\cite{Sa99a}, and the references therein), especially, hierarchies of
matrix-valued (i.e., nonabelian) nonlinear Schr\"odinger equations.
Although various aspects of inverse spectral theory for scalar
Schr\"odinger, Jacobi, and Dirac-type operators, and more
generally, for $2\tilde mes 2$ Hamiltonian systems, are
well-understood by now (cf.~the extensive list of references provided in
\cite{GKM00}), the corresponding theory for such operators and
Hamiltonian systems with $m\tilde mes m$ matrix-valued coefficients,
$m\in{\mathbb{N}}$, is still in its infancy. A particular inverse spectral
theory aspect we have in mind is that of determining isospectral sets
(manifolds) of such systems. It may, perhaps, come as a surprise that
determining the isospectral set of Hamiltonian systems with
matrix-valued periodic coefficients is a completely open problem. It
appears to be no exaggeration to claim that absolutely
nothing seems to be known about the corresponding isospectral sets of
periodic Dirac-type operators in the case $m\geq 2$.\ (More or less the
same ignorance applies to Schr\"odinger, Jacobi, and more generally, to
periodic $2m\tilde mes 2m$ Hamiltonian systems with $m\geq 2$.)
Theorem~\ref{t1.1} can be viewed as a first (and very modest) step
toward the construction of isospectral manifolds of certain classes of
matrix-valued potential coefficients $B$ for Dirac-type operators.
However, asymptotic high-energy expansions for Weyl-Titchmarsh matrices
on half-lines and on ${\mathbb{R}}$, their applications to trace formulas for
$B(x)$, and the derivation of Borg-type theorems for Dirac operators
are not the only topics under consideration in this paper. We also provide a
comprehensive and new treatment of local uniqueness theorems for $B$
in terms of exponentially close Weyl-Titchmarsh matrices. More
precisely, in Section \ref{s5} we will prove the following
result ($\|\cdot\|_{{\mathbb{C}}^{m\tilde mes m}}$ denotes a matrix norm on
${\mathbb{C}}^{m\tilde mes m}$).
\begin{theorem} \label{t1.2}
Fix $x_0\in{\mathbb{R}}$ and suppose that $B_j\in
L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$, posseses the normal form
given in Theorem~\ref{t1.1}
a.e.~on $(x_0,\infty)$, $j=1,2$. Denote by $M_{j,+}(z,x)$, $x\geq
x_0$, the unique Weyl-Titchmarsh matrices corresponding to the half-line
Dirac-type operators in $L^2([x_0,\infty))^{2m}$ associated with
$B_j$, $j=1,2$ $($fixing some self-adjoint boundary condition at $x_0$$)$.
Then,
\begin{equation}
\text{if for some $a>0$, }\, B_1(x)=B_2(x) \, \text{ for a.e. $x\in
(x_0,x_0+a)$,} \label{1.3}
\end{equation}
one obtains
\begin{equation}
\|M_{1,+}(z,x_0)-M_{2,+}(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes
m}}\underset{\substack{\abs{z}
\to\infty\\ z\in \rho_{+}}}{=}
O\bibitemg(e^{-2\text{\rm Im}(z)a}\bibitemg) \label{1.4}
\end{equation}
along any ray $\rho_+\subset{\mathbb{C}}_+$ with $0<\arg(z)<\pi$ $($and for all
self-adjoint boundary condition at $x_0$$)$. Conversely, if $m>1$, assume
in addition that $B_j\in L^\infty([x_0,x_0+a])^{2m\tilde mes 2m}$, $j=1,2$.
Moreover, suppose that for all $\varepsilon >0$,
\begin{equation}
\|M_{1,+}(z,x_0)-M_{2,+}(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes
m}}\underset{\substack{|z|\to\infty\\z\in \rho_{+,\ell}}}{=}
O\bibitemg(e^{-2\text{\rm Im}(z)(a-\varepsilon)}\bibitemg), \quad \ell=1,2, \label{1.5}
\end{equation}
along a ray $\rho_{+,1}\subset{\mathbb{C}}_+$ with $0<\arg(z)<\pi/2$ and along a
ray $\rho_{+,2}\subset{\mathbb{C}}_+$ with $\pi/2<\arg(z)<\pi$. Then
\begin{equation}
B_1(x)=B_2(x) \text{ for a.e. } x\in [x_0,x_0+a]. \label{1.6}
\end{equation}
\end{theorem}
\nonumberindent We also prove the analog of Theorem~\ref{t1.2} for the $2m\tilde mes 2m$
Weyl-Titchmarsh matrices $M_j(z,x)$ associated with Dirac-type
operators on ${\mathbb{R}}$ corresponding to $B_j$, $j=1,2$.
In the context of scalar Schr\"odinger operators, the analog of
Theorem~\ref{t1.2} was first proved by Simon \cite{Si98}. An alternative
proof, applicable to matrix-valued Schr\"odinger operators was presented
in \cite{GS99} (cf.~also \cite{GKM00}). More recently, yet another proof
was found by Bennewitz \cite{Be00} (following some ideas in
\cite{Bo52}). In fact, our proof of Theorem~\ref{t1.2} is based on that
of Bennewitz \cite{Be00} with additional modifications necessary to
accomodate Dirac-type operators. These results extend the
classical (global) uniqueness results due to Borg \cite{Bo52} and
Marchenko \cite{Ma50}, \cite{Ma52} which state
that half-line $m$-functions uniquely determine the corresponding
potential coefficient. The Dirac-type results such as Theorem~\ref{t1.2}
appear to be new, even in the special case $m=1$. Previous results in the
Dirac case focused on global uniqueness questions only. We refer to
Gasymov and Levitan \cite{GL66} in the case $m=1$ and to Lesch and Malamud
\cite{LM00} in the matrix case $m\in{\mathbb{N}}$.
Next, we briefly sketch the content of each section. Section~\ref{s2}
provides the necessary background results on
Dirac-type operators and recalls the basic notions of Weyl-Titchmarsh
theory for Hamiltonian systems on a half-line as well
as on ${\mathbb{R}}$, as developed in detail by Hinton and Shaw in a series of
papers \cite{HS81}--\cite{HS86} (see also \cite{At64}, \cite{Jo87},
\cite{HSH93}, \cite{HSH97}, \cite{JNO00}, \cite{KR74}, \cite{KS88},
\cite{Kr89a}, \cite{Kr89b}, \cite{LM00a}, \cite{Or76}, \cite{Sa94a}). In
fact, most of these references deal with more general singular
Hamiltonian systems and hence we specialize some of this material to the
Dirac-type operator case at hand. While our treatment of
Weyl-Titchmarsh theory in Section~\ref{s2} is somewhat
detailed, the results presented appear to
be of vital importance for our asymptotic expansions in Sections~\ref{s3}
and \ref{s4}. At any rate, we intended to present this
material as concisely as possible.
Section~\ref{s3} is devoted to a proof of the
leading-order for the asymptotic high-energy expansion
\eqref{1.1} of $M_+(z,x)$ for the Dirac case. We follow the strategy
developed in the context of matrix-valued Schr\"odinger operators in our
joint paper \cite{CG99} by appealing to the theory of Riccati equations.
By doing so, we follow the lead of Atkinson who highlighted the
importance of Riccati equations, in this regard, first in \cite{At81},
subsequently in \cite{At82}, \cite{At88a} and ultimately in the
unpublished manuscript \cite{At88} in which he obtains the leading order
for the asymptotic high-energy expansion of $M_+(z,x)$ for the
matrix-valued Schr\"odinger case.
Theorems \ref{t3.6} and \ref{t3.7} contain two characterizations
of the {\it Weyl disk} (cf.~Definition~\ref{dWD}). These
characterizations provide an answer in Remark~\ref{r3.3} to a point
raised in \cite{CG99} concerning the nature of the Weyl disk.
{}From these characterizations of the Weyl disk, we obtain a
realization of $M_+(z,x)$ as a differentiable function of $x$ which
satisfies a certain Riccati equation globally and whose imaginary part
is strictly positive. We observe, in Remark~\ref{r3.6a}, that the
totality of Weyl disks, $D_+(z,x)$ (cf.~Defintion~\ref{dLWD}), represents
the phase space for these solutions. Thus, the asymptotic expansion we
seek, represents the asymptotic high-energy behavior for certain
solutions of a given Riccati equation.
Section~\ref{s4} develops a systematic
higher-order high-energy asymptotic expansion of $M_+(z,x)$ as
$|z|\to\infty$, combining the leading-order asymptotic result in
Section~\ref{s3} with matrix-valued extensions of some methods based
again on an associated Riccati equation. More precisely, following a
technique in \cite{GS98} in the scalar Schr\"odinger operator context, we
show how to derive the general high-energy asymptotic expansion of
$M_+(z,x)$ as $|z|\to\infty$ by combining Atkinson's
leading-order term in \eqref{1.1} and the corresponding asymptotic
expansion of $M_+(z,x)$ in the special case where $B$ has
compact support. Section~\ref{s5} then contains our new local uniqueness
results for $B(x)$ in terms of exponentially small differences of
Weyl-Titchmarsh matrices as indicated in Theorem~\ref{t1.2}. Finally, in
Section~\ref{s6} we derive a new trace formula for
Dirac-type operators $D$ in $L^2({\mathbb{R}})^{2m}$, using
appropriate Herglotz representation results for the diagonal Green's
matrix $G(z,x,x)$ discussed in Section~\ref{s2}. Moreover, we derive the
Borg-type Theorem~\ref{t1.1} for Dirac operators and
close with an application to the case of periodic potentials
coefficients $B$.
\section{Weyl-Titchmarsh Matrices for Hamiltonian Systems}
\label{s2}
We now turn to the Weyl-Titchmarsh theory for Hamiltonian
systems as developed by Hinton and Shaw in a series
of papers devoted to the spectral theory of (singular) Hamiltonian systems
\cite{HS81}--\cite{HS86} (see also \cite{HSH93}, \cite{HSH97},
\cite{Kr89a}, \cite{Kr89b}, \cite{Sa92}, \cite{Sa94a}, \cite{Sa99},
\cite{Sa99a}). Throughout this paper all matrices will be considered over
the field of complex numbers ${\mathbb{C}}$. The basic assumptions throughout are
described in the following three hypotheses.
\begin{hypothesis} \label{h2.1}
Fix $m\in{\mathbb{N}}$ and define the $2m\tilde mes 2m$ matrix
\begin{subequations}\label{2.1}
\begin{equation}\label{2.1a}
J=\begin{pmatrix}0& -I_m \\ I_m & 0 \end{pmatrix},
\end{equation}
where $I_m$ denotes the identity matrix in ${\mathbb{C}}^{m\tilde mes m}$.
Suppose
\begin{equation}
A_{j,k}, B_{j,k} \in
L_{\text{\rm{loc}}}^1({\mathbb{R}})^{m\tilde mes m}, \quad j,k = 1,2
\end{equation}
and assume
\begin{align}
A(x)&=\begin{pmatrix}A_{1,1}(x)&A_{1,2}(x)
\\A_{2,1}(x) & A_{2,2}(x) \end{pmatrix}\ge 0, \label{2.1c}\\
\quad B(x)&=\begin{pmatrix}B_{1,1}(x)&B_{1,2}(x) \\B_{2,1}(x) &
B_{2,2}(x) \end{pmatrix}=B(x)^*,\label{2.1d}
\end{align}
\end{subequations}
for a.e.~$x\in {\mathbb{R}}$.
\end{hypothesis}
\nonumberindent
$L_{\text{\rm{loc}}}^1({\mathbb{R}})$ denotes the set of locally integrable
functions on ${\mathbb{R}}$. With $M\in{\mathbb{C}}^{m\tilde mes m}$, let
$M^t$ denote the transpose, let $M^*$ denote the adjoint or conjugate
transpose of the matrix $M$ and let $M\ge 0$ and $M\le 0$ denote
nonnegative and nonpositive matrices
$M$ (i.e., positive and negative semidefinite matrices). Moreover, let
$\text{\rm Im}(M)=(M-M^*)/(2i)$ and
$\text{\rm Re}(M)=(M+M^*)/2$ denote, respectively, the imaginary and real parts
of the matrix $M$.
Given Hypothesis~\ref{h2.1}, our Hamiltonian system is given by
\begin{subequations}\label{HS}
\begin{equation}\label{HSa}
J \varPsi'(z,x)=(zA(x)+B(x))\varPsi(z,x), \quad z\in{\mathbb{C}}
\end{equation}
for a.e. $x\in {\mathbb{R}}$, where $z$ plays the role of the spectral
parameter, and where
\begin{equation}\label{HSb}
\varPsi(z,x) = \begin{pmatrix}\psi_1(z,x)\\
\psi_2(z,x) \end{pmatrix}, \quad \psi_j(z,\cdot)\in
AC_{\text{\rm{loc}}}({\mathbb{R}})^{m\tilde mes r}, \,\, j=1,2.
\end{equation}
\end{subequations}
$AC_{\text{\rm{loc}}}({\mathbb{R}})$ denotes the set of locally absolutely
continuous functions on ${\mathbb{R}}$. The parameter $r$ in \eqref{HSb}
will be context dependent and range between $1\leq r \leq m$.
For our discussions of the Weyl-Titchmarsh theory for
the Hamiltonian system \eqref{HS}, we introduce the
definiteness assumption found in Atkinson~\cite{At64}.
\begin{hypothesis}\label{h2.2}
For all nontrivial solutions $\varPsi$ of \eqref{HSa} with $r=1$ in
\eqref{HSb}, we assume that
\begin{equation}\label{2.3}
\int_{a}^b dx \, \varPsi(z,x)^*A(x)\varPsi(z,x) > 0\; ,
\end{equation}
for every interval $(a,b)\subset {\mathbb{R}}$, $a<b$.
\end{hypothesis}
A principal example of such a system is the Dirac-type
system obtained when
\begin{equation}\label{DS}
A(x) = I_{2m},
\end{equation}
and the subject of the present paper; another example being
the matrix-valued Schr\"odinger system, obtained when
\begin{equation}\label{SS}
A(x)= \begin{pmatrix}I_m& 0 \\ 0 & 0 \end{pmatrix},
\qquad B(x) = \begin{pmatrix}-Q(x)& 0 \\ 0 & I_m \end{pmatrix},
\end{equation}
and the subject of \cite{CG99}. When \eqref{SS} holds, we
note that
\eqref{HSa} is equivalent to
\begin{align}
-\psi_1^{\prime\prime}(z,x)+Q(x)\psi_1(z,x)&
=z\psi_1(z,x), \label{2.6} \\
\psi_2(z,x)&=\psi_1^\prime(z,x) \label{2.7}
\end{align}
for a.e.~$x\in{\mathbb{R}}$. Hypothesis \ref{h2.2} clearly holds in both
examples.
Next, we introduce a set of matrices that will serve as
boundary data for separated boundary conditions.
\begin{hypothesis}\label{h2.3}
Let $\gamma = (\gamma_1\; \gamma_2)$ with $\gamma_j \in {\mathbb{C}}^{m\tilde mes m}$,
$j=1,2$. We assume that $\gamma$ satisfies the following
conditions,
\begin{subequations}\label{BD}
\begin{equation}\label{BDa}
\text{\rm{rank}} (\gamma) = m,
\end{equation}
and that either
\begin{equation}\label{BDc}
\text{\rm Im} (\gamma_{2}\gamma_{1}^*) \le 0,
\quad \text{or}
\quad \text{\rm Im} (\gamma_{2}\gamma_{1}^*) \ge 0,
\end{equation}
where $(2i)^{-1}\, \gamma J\gamma^*=\text{\rm Im} (\gamma_{2}\gamma_{1}^*)$.
Given the rank condition in \eqref{BDa},
we assume, without loss of generality in what follows, the
normalization
\begin{equation}\label{BDd}
\gamma\gamma^* = I_m.
\end{equation}
\end{subequations}
\end{hypothesis}
\begin{remark} \label{r2.4}
With $\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$, the conditions
\begin{equation}
\alpha\alpha^*=I_m, \quad \alpha J\alpha^*=0 \label{2.8e}
\end{equation}
imply that $\alpha$ satisfies Hypothesis~\ref{2.3}, and they explicitly
read
\begin{equation}
\alpha_1\alpha_1^* +\alpha_2\alpha_2^*=I_m, \quad \alpha_2\alpha_1^*
-\alpha_1\alpha_2^*=0. \label{2.8f}
\end{equation}
In fact, one also has
\begin{equation}
\alpha_1^*\alpha_1 +\alpha_2^*\alpha_2=I_m, \quad \alpha_2^*\alpha_1
-\alpha_1^*\alpha_2=0, \label{2.8g}
\end{equation}
as is clear from
\begin{equation}
\begin{pmatrix} \alpha_1 & \alpha_2\\ -\alpha_2 & \alpha_1 \end{pmatrix}
\begin{pmatrix} \alpha_1^* & -\alpha_2^*\\ \alpha_2^* & \alpha_1^*
\end{pmatrix}=I_{2m}=\begin{pmatrix} \alpha_1^* & -\alpha_2^*\\
\alpha_2^* & \alpha_1^* \end{pmatrix}\begin{pmatrix} \alpha_1 & \alpha_2\\
-\alpha_2 & \alpha_1 \end{pmatrix}, \label{2.8h}
\end{equation}
since any left inverse matrix is also a right inverse, and vice versa.
Moreover, from \eqref{2.8g} we obtain
\begin{equation}\label{2.8i}
\alpha^*\alpha J + J\alpha^*\alpha = J.
\end{equation}
\end{remark}
With $\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying \eqref{2.8e},
let $\Psi(z,x,x_0,\alpha)$ be a normalized fundamental system of
solutions of \eqref{HS} at some
$x_0\in{\mathbb{R}}$. That is, $\Psi(z,x,x_0,\alpha)$ satisfies \eqref{HS}
for a.e.\ $x\in{\mathbb{R}}$, and
\begin{subequations}\label{FS}
\begin{equation}\label{FSa}
\Psi(z,x_0,x_0,\alpha)=(\alpha^* \; J\alpha^*)=
\begin{pmatrix} \alpha_1^* & -\alpha_2^* \\
\alpha_2^* & \alpha_1^* \end{pmatrix}.
\end{equation}
We partition $\Psi(z,x,x_0,\alpha)$ as follows,
\begin{align}
\Psi(z,x,x_0,\alpha)&=(\Theta(z,x,x_0,\alpha)\;
\Phi(z,x,x_0,\alpha))\label{FSb}\\
&=\begin{pmatrix}\theta_1(z,x,x_0,\alpha) &
\phi_1(z,x,x_0,\alpha)\\
\theta_2(z,x,x_0,\alpha)& \phi_2(z,x,x_0,\alpha)
\end{pmatrix},\label{FSc}
\end{align}
\end{subequations}
where $\theta_j(z,x,x_0,\alpha)$ and $\phi_j(z,x,x_0,\alpha)$
for $j=1,2$ are $m\tilde mes m$ matrices, entire with
respect to $z\in{\mathbb{C}}$, and normalized according to
\eqref{FSa}.~One can now prove the following result.
\begin{lemma}\label{l2.4}
Let $ \Theta(z,x,x_0,\alpha)$ and $\Phi(z,x,x_0,\alpha)$
be defined in \eqref{FSb} with $\alpha$ and $\beta$ satisfying
Hypothesis~\ref{h2.3} and with $\text{\rm Im}(\alpha_2\alpha_1^*)=0$. Then, for
$c\ne x_0$, $\beta\Phi(z,c,x_0,\alpha)$ is singular if and
only if $z$ is an eigenvalue for the regular boundary value
problem given by \eqref{HSa} on $[x_0,c]$ if $c>x_0$ and on
$[c,x_0]$ if $c<x_0$ together with the separated boundary conditions
\begin{equation}\label{BC}
\alpha\varPsi(z,x_0)=0, \quad \beta\varPsi(z,c)=0,
\end{equation}
where $\varPsi (z,x)=(\psi_1(z,x)^t\; \psi_2(z,x)^t)^t$ with
$\psi_j(z,\cdot)\in AC([x_0,c])$ if $c>x_0$ and $\psi_j(z,\cdot)\in
AC([c,x_0])$ if $c<x_0$, $j=1,2$.
\end{lemma}
\nonumberindent Note that the regular boundary value problem described
in Lemma~\ref{l2.4} is self-adjoint when $\text{\rm Im}(\beta_2\beta_1^*)=0$.
In light of Lemma~\ref{l2.4}, it is possible to introduce, under appropriate
conditions, the $m\tilde mes m$ matrix-valued function,
$M(z,c,x_0,\alpha,\beta)$, as follows.
\begin{definition}\label{dMF}
Let $ \Theta(z,x,x_0,\alpha)$, and $\Phi(z,x,x_0,\alpha)$ be
defined in \eqref{FSb} with $\alpha$ and $\beta$ satisfying
Hypothesis~\ref{h2.3} and with $\text{\rm Im}(\alpha_2\alpha_1^*)=0$. For
$c\ne x_0$, and $\beta\Phi(z,c,x_0,\alpha)$ nonsingular let
\begin{equation}\label{MF}
M(z,c,x_0,\alpha,\beta) =
-[\beta\Phi(z,c,x_0,\alpha)]^{-1}[\beta\Theta(z,c,x_0,\alpha)].
\end{equation}
$M(z,c,x_0,\alpha,\beta)$ is said to be the
{\it Weyl-Titchmarsh $M$-function} for the regular boundary value
problem described in Lemma~\ref{l2.4}.
\end{definition}
The Weyl-Titchmarsh $M$-function is an
$m\tilde mes m$ matrix-valued function with meromorphic entries whose
poles correspond to eigenvalues for the regular boundary value
problem given by \eqref{HSa} and \eqref{BC}. Moreover, if $M\in
{\mathbb{C}}^{m\tilde mes m}$, and one defines
\begin{equation}\label{2.14}
U(z,x,x_0,\alpha)= \begin{pmatrix}
u_1(z,x,x_0,\alpha)\\u_2(z,x,x_0,\alpha) \end{pmatrix}=
\Psi(z,x,x_0,\alpha)\begin{pmatrix}I_m\\M\end{pmatrix},
\end{equation}
with $u_j(z,x,x_0,\alpha)\in {\mathbb{C}}^{m\tilde mes m}$, $j=1,2$, then
$U(z,x,x_0,\alpha)$ will satisfy the boundary condition
at $x=c$ in
\eqref{BC} whenever $M=M(z,c,x_0,\alpha,\beta)$.
Intimately connected with the matrices
introduced in Definition~\ref{dMF} is the set of
$m\tilde mes m$ complex matrices known as the Weyl disk. Several
characterization of this set have appeared in the literature
(see, e.g., \cite{At64}, \cite{At88a}, \cite{At88}, \cite{HSH93},
\cite{HS81}, \cite{Kr89a}, \cite{Or76}). We now mention two, and
will introduce two others in Section~\ref{s3} which we use
in the derivation of the asymptotic expansions that are the subject
of Sections \ref{s3} and \ref{s4}.
To describe this set, we first introduce the matrix-valued function
$E_c(M)$: With $c\ne x_0$, $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, and with
$U(z,c,x_0,\alpha)$ defined by \eqref{2.14} in terms of a matrix
$M\in{\mathbb{C}}^{m\tilde mes m}$, let
\begin{equation}\label{2.380}
E_c(M) = \sigma(x_0,c,z)U(z,c,x_0,\alpha)^*(iJ)U(z,c,x_0,\alpha),
\end{equation}
where
\begin{equation}
\sigma(s,t,z)=\fracrac{(s-t)\text{\rm Im} (z)}{|(s-t)\text{\rm Im} (z)|},\quad
\sigma(s,t)=\sigma(s,t,i),\quad
\sigma(z)= \sigma(1,0,z),
\end{equation}
with $s\ne t$, and $s,t\in{\mathbb{R}}$.
\begin{definition}\label{dWD}
Let the following be fixed: Real numbers $x_0$ and $c\ne x_0$,
an $m\tilde mes 2m$ matrix $\alpha$ satisfying \eqref{2.8e},
and $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$.
${\mathcal D}(z,c,x_0,\alpha)$ will denote the collection of all $M\in
{\mathbb{C}}^{m\tilde mes m}$ for which $E_c(M)\le 0$, where $E_c(M)$ is
defined in
\eqref{2.380}. ${\mathcal D}(z,c,x_0,\alpha)$ is said to be a
{\it Weyl disk}.
The set of $M\in {\mathbb{C}}^{m\tilde mes m}$ for which $E_c(M) = 0$ is
said to be
a {\it Weyl circle} (even when $m>1$).
\end{definition}
This definition leads to a presentation
that is a generalization of the description first given
by Weyl~\cite{We10}; a presentation which is geometric in nature,
involves the contractive matrices
$V\in{\mathbb{C}}^{m\tilde mes m}$, such that $VV^*\le I_m$, and provides the
justification for the geometric terms of circle and disk
(cf., e.g., \cite{HS81}, \cite{HSH93}, \cite{Kr89a}, \cite{Or76}).
The disk has also been characterized
in terms of matrices which statisfy Hypothesis~\ref{h2.3} and which
serve as boundary data for the regular boundary value problem
described in Lemma~\ref{l2.4} (cf., e.g., \cite{At88a}, \cite{At88}).
More precisely, one could have used the following alternative
definition. \\
\vspace*{-2mm}
\nonumberindent {\bf Definition 2.7A.}
${\mathcal D}(z,c,x_0,\alpha)$ denotes the collection of all $M\in
{\mathbb{C}}^{m\tilde mes m}$ obtained by the construction given in \eqref{MF}
where $c\ne x_0$, $z\in{\mathbb{C}}/{\mathbb{R}}$, where $\alpha$ and $\beta$ are
the $m\tilde mes m$ matrices defined in Hypothesis~\ref{h2.3} for which
$\sigma(c,x_0,z)\text{\rm Im}(\beta_2\beta_1^*)\ge 0$, and
$\text{\rm Im}{(\alpha_2\alpha_1^*)}=0$. \\
\vspace*{-2mm}
\nonumberindent However, in this paper we take
Definition~\ref{dWD} as our point of departure.
We note that the Weyl
circle corresponds to the regular boundary value problems in
Lemma~\ref{l2.4} with separated, self-adjoint boundary conditions. For
convenience of the reader, and to achieve a reasonable level of
completeness, we reproduce the corresponding short proof below.
\begin{lemma}[\cite{HS84}, \cite{HSH93}, \cite{Kr89a}]\label{l2.11}
Let $M\in{\mathbb{C}}^{m\tilde mes m}$, $c\ne x_0$, and $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$.
Then, $E_c(M)=0$ if and only if
there is a $\beta\in {\mathbb{C}}^{m\tilde mes 2m}$ satisfying
\eqref{2.8e} such that
\begin{equation}\label{2.24}
0=\beta U(z,c,x_0,\alpha),
\end{equation}
where $U(z,c)=U(z,c,x_0,\alpha)$ is defined in \eqref{2.14} in
terms of $M$. With $\beta$ so defined,
\begin{equation}\label{2.25}
M=-[\beta\Phi(z,c,x_0,\alpha)]^{-1}[\beta\Theta(z,c,x_0,\alpha)],
\end{equation}
that is, $M=M(z,c,x_0,\alpha,\beta)$. Moreover, $\beta$ may be chosen
to satisfy \eqref{BDd}, and hence Hypothesis~\ref{2.3}.
\end{lemma}
\begin{proof}
Let $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, and suppose
for a given $M\in{\mathbb{C}}^{m\tilde mes m}$ that there is
a $\beta\in{\mathbb{C}}^{m\tilde mes 2m}$ which satisfies \eqref{2.8e}
and such that \eqref{2.24} is satisfied. Given that
$\beta J \beta^* = 2i\text{\rm Im} (\beta_2\beta_1^*)=0$, and given
that $\text{\rm{rank}} (\beta)=\text{\rm{rank}}(J\beta^*)=m$, there is a
nonsingular $C\in{\mathbb{C}}^{m\tilde mes m}$ such that
$U(z,c) = J\beta^*C$. Hence,
$E_c(M)=i\sigma(c,x_0,z)C^*\beta J\beta^*C=0$.
Upon showing that $\beta\Phi(z,c)=\beta\Phi(z,c,x_0,\alpha)$
is nonsingular, \eqref{2.25}
will then follow from \eqref{2.24}. If $\beta\Phi(z,c)$ is singular,
then there are nonzero vectors
$v, w \in {\mathbb{C}}^{m}$ such that $\beta\Phi(z,c)v=0$, and such that
$\Phi(z,c)v = J\beta^*w$.
Let $\varPsi_j=\varPsi_j(z,x)$, $j=1,2$ denote solutions of
\eqref{HSa} with $z=z_j$, $j=1,2$. Then,
\begin{equation}\label{2.19}
(\varPsi_1^*J\varPsi_2)'=(z_2 - \begin{align}r{z}_1)\varPsi_1^*A\varPsi_2.
\end{equation}
Using \eqref{2.19}, and recalling that $\Phi(z,x)$ is defined in \eqref{FS}, we obtain
\begin{subequations}\label{2.230}
\begin{align}
2i\text{\rm Im}(z)\int_{x_0}^c dx\, v^*\Phi(z,x)^* A(x) \Phi(z,x)v&=v^*\Phi(z,c)^* J
\Phi(z,c)v \label{2.230a} \\
&=w^*\beta J \beta^*w =0.
\end{align}
\end{subequations}
Thus, by Hypothesis~\ref{h2.2}, $\text{\rm Im}(z)=0$. This contradicts the
assumption that $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$.
Conversely, if $E_c(M)=0$ for a given $M\in {\mathbb{C}}^{m\tilde mes m}$,
then for $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$ let
$\beta =(I_m\; M^*)\Psi(z,c,x_0,\alpha)^*J =
U(z,c,x_0,\alpha)^* J$. One observes that \eqref{2.24} is
satisfied and that $\text{\rm{rank}} (\beta) =m$. Moreover,
$0=\sigma(x_0,c,z)E_c(M)/2=\text{\rm Im}(\beta_2\beta_1^*)$.
If for this choice of $\beta$, \eqref{BDd} is not yet
satisfied, one introduces $\Delta = (\beta\beta^*)^{-1/2}\beta$ and
observes that $0=\Delta U(z,c,x_0,\alpha)$,
$\text{\rm Im}(\Delta_2\Delta_1^*)=
(\beta\beta^*)^{-1/2} \text{\rm Im}(\beta_2\beta_1^*)
(\beta\beta^*)^{-1/2}$, and that $\Delta$ satisfies all
requirements of \eqref{2.8e}.
\end{proof}
Next, we recall a fundamental property associated with matrices
in ${\mathcal D}(z,c,x_0,\alpha)$.
\begin{lemma} \label{l2.8}
If $M\in{\mathcal D}(z,c,x_0,\alpha)$, then
\begin{equation}\label{Hgz}
\sigma(c,x_0,z) \text{\rm Im} (M)>0.
\end{equation}
Moreover, whenever $\beta\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfies \eqref{2.8e},
\begin{equation}\label{2.270}
M(\begin{align}r z,c,x_0,\alpha,\beta)= M(z,c,x_0,\alpha,\beta)^*.
\end{equation}
\end{lemma}
\begin{proof}
Let $\varPsi_j=\varPsi_j(z,x)$, $j=1,2$ denote solutions of
\eqref{HSa} with $z=z_j$, $j=1,2$. Then $(\varPsi_1^*J\varPsi_2)'=(z_2
- \begin{align}r{z}_1)\varPsi_1^*A\varPsi_2$ as in \eqref{2.19}. This implies
\begin{align}
2i\text{\rm Im}(z)\int_{x_0}^c dx\, U(z,x)^* A(x) U(z,x) &=
U(z,x)^* J U(z,x)\bibitemg |_{x_0}^c \nonumber \\
&=2i\text{\rm Im}(M) + U(z,c)^* J U(z,c), \label{2.280}
\end{align}
with $U(z,x)=U(z,x,x_0,\alpha)$ defined in \eqref{2.14}. Moreover, by the
definition of $E_c(M)$ given in \eqref{2.380}, one obtains
\begin{align}\label{2.290}
&2\sigma(c,x_0,z)\text{\rm Im} (M)\\
&= -E_c(M) + 2\sigma(c,x_0)|\text{\rm Im}(z)|\int_{x_0}^c ds\, U(z,s)^*A(s)U(z,s).\nonumber
\end{align}
By Hypothesis~\ref{h2.2} and Definition~\ref{dWD}, one infers that
$\sigma(c,x_0,z) \text{\rm Im} (M)>0$. To prove \eqref{2.270}, let
$\Psi(z,x)=\Psi(z,x,x_0,\alpha)$, where
$\Psi$ is defined in \eqref{HS}. Then, by \eqref{2.19},
\begin{equation}\label{2.310}
\Psi(\begin{align}r{z},x)^*J\Psi(z,x)=J,
\end{equation}
which implies $J\Psi(z.x)(\Psi(\begin{align}r{z},x)J)^*=I_{2m}$
and hence
\begin{equation}\label{2.330}
\Psi(z,x)J\Psi(\begin{align}r{z},x)^*=J.
\end{equation}
Thus one concludes
\begin{equation}
\beta\Phi(z,c)(\beta\Theta(\begin{align}r{z},c))^*-
\beta\Theta(z,c)(\beta\Phi(\begin{align}r{z},c))^*=
\beta J\beta^*=0,
\end{equation}
from which \eqref{2.270} follows immediately by Lemma~\ref{l2.11}.
\end{proof}
For $c>x_0$, the function $M(z,c,x_0,\alpha,\beta)$,
defined by \eqref{MF}, and satisfying \eqref{Hgz}, is said to be
a matrix-valued {\it Herglotz} function of rank $m$. Hence, for
$\text{\rm Im}(\beta_2\beta_1^*)=0$, poles of
$M(z,c,x_0,\alpha,\beta)$, $c>x_0$, are at most of first order, are
real, and have nonpositive residues. Such functions admit a
representation of the form
\begin{align}\label{NP}
M(z,c,x_0,\alpha,\beta)=& \; C_1(c,x_0,\alpha,\beta) +
zC_2(c,x_0,\alpha,\beta)
\nonumber\\
&+\int_{-\infty}^\infty
d\Omega(\lambda,c,x_0,\alpha,\beta)\,\left(
\fracrac{1}{\lambda-z} -\fracrac{\lambda}{1+\lambda^2} \right), \quad c>x_0,
\end{align}
where $C_2(c,x_0,\alpha,\beta)\ge 0$ and $C_1(c,x_0,\alpha,\beta)$ are
self-adjoint $m\tilde mes m$ matrices, and where
$\Omega(\lambda,c,x_0,\alpha,\beta)$ is a nondecreasing $m\tilde mes m$
matrix-valued function such that
\begin{subequations}\label{Mrep}
\begin{align}
&\int_{-\infty}^{\infty}\|
d\Omega(\lambda,c,x_0,\alpha,\beta)\|_{{\mathbb{C}}^{m\tilde mes m}}\,
(1+\lambda^2)^{-1} < \infty, \label{NPa} \\
&\Omega((\lambda, \mu],c,x_0,\alpha,\beta)= \lim_{\delta\downarrow 0}
\lim_{\epsilon \downarrow 0}\fracrac{1}{\pi}\int_{\lambda
+ \delta}^{\mu
+ \delta }d\nu\, \sigma(c,x_0)\text{\rm Im}\left( M(\nu
+i\epsilon,c,x_0,\alpha,\beta)\right). \label{NPb}
\end{align}
\end{subequations}
In general, for self-adjoint boundary value problems,
$\Omega(\lambda,c,x_0,\alpha,\beta)$ is
piecewise constant with jump discontinuities precisely at the eigenvalues
of the boundary value problem, and that in the matrix-valued Schr\"odinger and
Dirac-type cases $C_2=0$ in \eqref{NP} (and later in \eqref{2.42} and
\eqref{2.64}). Analogous statements apply to
$-M(z,c,x_0,\alpha,\beta)$ if $c<x_0$. For such problems, we note in the
subsequent lemma that for fixed $\beta$, varying the boundary data
$\alpha$ produces Weyl-Titchmarsh matrices $M(z,c,x_0,\alpha,\beta)$
related to each other via linear fractional transformations (see also
\cite{GMT98}, \cite{GT97} for a general approach to such linear fractional
transformations).
\begin{lemma}\label{l2.9}
Suppose $\alpha, \beta, \gamma\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfy \eqref{2.8e}.
Let $M_{\alpha}=M(z,c,x_0,\alpha,\beta)$, and
$M_{\gamma}=M(z,c,x_0,\gamma,\beta)$. Then,
\begin{equation}
M_{\alpha}= [-\alpha J \gamma^* + \alpha\gamma^* M_{\gamma}]
[\alpha\gamma^* + \alpha J\gamma^*M_{\gamma}]^{-1}. \label{2.360}
\end{equation}
\end{lemma}
\begin{proof}
Let $U_{\alpha}(z,x)=U(z,x,x_0,\alpha)$ and $U_{\gamma}(z,x)=U(z,x,x_0,\gamma)$
be defined in \eqref{2.14} with $M=M_{\alpha}$ and $M=M_{\gamma}$
respectively. Then,
\begin{equation}
0=\beta U_{\alpha}(z,c)=\gamma U_{\gamma}(z,c).
\end{equation}
By the rank condition \eqref{BDa},
\begin{equation}
U_{\alpha}(z,c)= J\beta^*C_{\alpha}\, ,\qquad
U_{\gamma}(z,c)= J\beta^*C_{\gamma}
\end{equation}
for nonsingular $C_{\alpha}, \; C_{\gamma}\in {\mathbb{C}}^{m\tilde mes m}$.
Thus, by \eqref{FSa}, and by the uniqueness of solution of \eqref{HSa},
there is a nonsingular
$C\in {\mathbb{C}}^{m\tilde mes m}$ for which
\begin{equation}\label{2.410}
\begin{pmatrix} \alpha^*\hspace{-5pt}&J\alpha^*\end{pmatrix}
\begin{pmatrix} I_m \\ M_{\alpha}\end{pmatrix}=U_{\alpha}(z,x_0)=U_{\gamma}(z,x_0)C=
\begin{pmatrix} \gamma^*\hspace{-5pt}&J\gamma^*\end{pmatrix}
\begin{pmatrix} I_m \\ M_{\gamma}\end{pmatrix}C.
\end{equation}
By \eqref{2.8i},
\begin{equation}
\begin{pmatrix} \alpha^*\hspace{-5pt}&J\alpha^*\end{pmatrix}^{-1}=
\begin{pmatrix} \alpha \\ -\alpha J\end{pmatrix};
\end{equation}
and hence, by \eqref{2.410} we see that
\begin{subequations}
\begin{align}
I_m&=(\alpha \gamma^* + \alpha J \gamma^* M_{\gamma} )C\\
M_{\alpha}&= (-\alpha J\gamma^* + \alpha \gamma^* M_{\gamma} )C,
\end{align}
\end{subequations}
from which \eqref{2.360} immediately follows.
\end{proof}
\begin{remark}
{}From the proof of the previous lemma one infers, in general, that
\begin{equation}
U_{\gamma}(z,x) =
U_{\alpha}(z,x)(\alpha \gamma^* + \alpha J \gamma^* M_{\gamma} ).
\end{equation}
Moreover, if $\alpha_0 =(I_m\; 0)$ and $\gamma_0=(0\ I_m)$
one observes, in particular,
\begin{equation}
M(z,c,x_0,\alpha_0,\beta)=-M(z,c,x_0,\gamma_0,\beta)^{-1}.
\end{equation}
\end{remark}
We further note that the sets ${\mathcal D}(z,c,x_0,\alpha)$ are closed,
and convex, (cf., e.g., \cite{HS84}, \cite{HSH93},
\cite{Kr89a}, \cite{Or76}). Moreover, by \eqref{2.290} and Hypothesis~\ref{h2.2},
one concludes that $E_c(M)$ is strictly increasing. This fact together
with Lemma~\ref{l2.11} implies that, as a function of $c$, the sets
${\mathcal D}(z,c,x_0,\alpha)$ are strictly nesting in the sense that
\begin{equation}\label{2.28}
{\mathcal D}(z,c_2,x_0,\alpha)\subset {\mathcal D}(z,c_1,x_0,\alpha) \quad
\text{for}\quad x_0<c_1< c_2\quad \text{or}
\quad c_2< c_1<x_0.
\end{equation}
Hence, the intersection of this nested sequence, as $c\to \pm \infty$,
is nonempty, closed and convex. We say that this intersection is
a limiting set for the nested sequence.
\begin{definition}\label{dLWD}
Let ${\mathcal D}_\pm(z,x_0,\alpha)$ denote the closed, convex set in the space
of $m\tilde mes m$ matrices which is the limit, as
$c\to \pm\infty$, of the nested collection of sets
${\mathcal D}(z,c,x_0,\alpha)$ given in Definition~\ref{dWD}.
${\mathcal D}_\pm(z,x_0,\alpha)$ is said to be a limiting {\em disk}.
Elements of ${\mathcal D}_\pm(z,x_0,\alpha)$ are denoted by $M_\pm(z,x_0,\alpha)\in
{\mathbb{C}}^{m\tilde mes m}$.
\end{definition}
In light of the containment described in
\eqref{2.28}, for $c\ne x_0$ and $z\in {\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$,
\begin{equation}\label{2.32}
{\mathcal D}_\pm(z,x_0,\alpha)\subset {\mathcal D}(z,c,x_0,\alpha),
\end{equation}
with emphasis on strict containment of the disks in \eqref{2.32}.
Moreover, by \eqref{2.290},
\begin{equation}\label{2.320}
M\in {\mathcal D}_\pm(z,x_0,\alpha) \text{ precisely when }E_c(M)<0
\text{ for all } c \in(x_0, \pm\infty).
\end{equation}
The following Lemma appears to have gone unnoted in the literature.
\begin{lemma}\label{l2.12}
Let $M\in{\mathbb{C}}^{m\tilde mes m}$, $c\ne x_0$, and $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$.
Then, $E_c(M)<0$ if and only if
there is a $\beta\in {\mathbb{C}}^{m\tilde mes 2m}$ satisfying the condition
\begin{equation}\label{2.27a}
\sigma(c,x_0,z) \text{\rm Im}(\beta_2\beta_1^*)>0,
\end{equation}
and such that \eqref{2.24} holds
with $u_j(z,c)= u_j(z,c,x_0,\alpha)$, $j=1,2$, defined in
\eqref{2.14} in terms of $M$. With $\beta$ so defined,
\eqref{2.25} holds; that is, $M=M(z,c,x_0,\alpha,\beta)$. Moreover,
$\beta$ maybe chosen to satisfy \eqref{BDd}, and hence Hypothesis~\ref{h2.3}.
\end{lemma}
\begin{proof}
Let $z\in {\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, and for a
given $M\in {\mathbb{C}}^{m\tilde mes m}$ suppose that there is a
$\beta\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying
\eqref{2.27a} such that \eqref{2.24} holds. The matrices
$\beta_j$, $j=1,2$, are
invertible by \eqref{2.27a}, and by \eqref{2.24} it follows that
\begin{equation}\label{2.26}
U(z,c)=\begin{pmatrix}
-\beta_1^{-1}\beta_2\\I_m \end{pmatrix}
u_2(z,c).
\end{equation}
By \eqref{2.380} and \eqref{2.26}, one then concludes that
\begin{equation}\label{2.27}
E_c(M) = -2\sigma(c,x_0,z) u_2(z,c)^*\beta_1^{-1}
\text{\rm Im} (\beta_2\beta_1^*)
(\beta_1^*)^{-1} u_2(z,c),
\end{equation}
and hence that $E_c(M)<0$ whenever \eqref{2.27a} holds.
Upon showing that $\beta\Phi(z,c)$ is nonsingular, \eqref{2.25} will follow from
\eqref{2.24}. If $\beta\Phi(z,c)$ is singular, then there is a nonzero
vector $v\in {\mathbb{C}}^{m}$ such that $\beta\Phi(z,c)v=0$. By the
nonsingularity of $\beta_j$, $j=1,2$, $\phi_1(z,c)v =
-\beta_1^{-1}\beta_2\phi_2(z,c)v$, and as a result, \eqref{2.230a}
yields
\begin{align}
&2\sigma(c,x_0)|\text{\rm Im}(z)|\int_{x_0}^c dx \,v^*\Phi(z,x)^*A(x)\Phi(z,x)v
\nonumber\\
&= -2\sigma(c,x_0,z)v^*\phi_2(z,c)^*\beta_1^{-1}\text{\rm Im}(\beta_2\beta_1^*)
(\beta_1^*)^{-1}\phi_2(z,c)v,
\end{align}
and hence, a contradiction given \eqref{2.27a} (cf.~\eqref{2.3}).
Conversely, if $E_c(M)<0$ for a given $M\in{\mathbb{C}}^{m\tilde mes m}$, then for
$z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, $u_j(z,c)$, $j=1,2$, defined by
\eqref{2.14}, are nonsingular. Indeed, if either $u_1(z,c)$
or $u_2(z,c)$
are singular, then there is a $v\in {\mathbb{C}}^m$, $v\ne 0$,
such that $v^*E_c(M)v=0$, a contradiction. Next, let $\beta_1 =I_m$
and let $\beta_2=-u_1(z,c)u_2(z,c)^{-1}$.
Then, for these $\beta_j$, $j=1,2$, \eqref{2.24}
holds. Equation~\eqref{2.27} now implies that
$\sigma(c,x_0,z)\text{\rm Im}(\beta_2\beta_1^*)> 0$ for
$c\ne x_0$ and $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$.
For this choice, $\beta$ does not satisfy \eqref{BDd}. However, one
can normalize $\beta$ as described in the proof of
Lemma~\ref{l2.11}.
\end{proof}
Hence by Lemma~\ref{l2.12} and \eqref{2.320}, we see that if
$M\in{\mathcal D}_\pm(z,x_0,\alpha)$, then for some
$\beta\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying \eqref{2.27a}
\begin{equation}\label{2.33}
M_\pm(z,x_0,\alpha)=M(z,c,x_0,\alpha,\beta).
\end{equation}
\begin{remark}\label{r2.14}
To the reader of \cite{CG99}, our study of the high-energy asymptotics of
the Weyl-Titchmarsh $M$-function for matrix-Schr\"odinger operators,
we offer this cautionary note:
In \cite{CG99}, $D(z,c,x_0,\alpha)$ represents the set of matrices
characterized by Lemmas~\ref{l2.11} and \ref{l2.12}.
However, the homeomorphism that exists between the contractive matrices
$V\in{\mathbb{C}}^{m\tilde mes m},\ VV^*\le I_m$, and the Weyl disk, $D(z,c,x_0,\alpha)$,
(cf., \cite{HS84}, \cite{HSH93}, \cite{Kr89a}, \cite{Or76}) shows that
those $M\in{\mathbb{C}}^{m\tilde mes m}$ characterized in
Lemma~\ref{l2.11} correspond to the set of unitary matrices while those
characterized in Lemma~\ref{l2.12} correspond
to the contractive matrices for which $VV^*<I_m$. Hence, Lemma~\ref{l2.11}
characterizes part of the boundary while Lemma~\ref{l2.12} characterizes
the interior of the Weyl disk as it is defined in Defintion~\ref{dWD}.
As a result, the closure of the set consisting of those
$M\in{\mathbb{C}}^{m\tilde mes m}$ characterized by these two lemmas (i.e., those $M$
which correspond to $VV^*<I_m$, or to $VV^*=I_m$) is the Weyl disk.
Thus, for deriving high-energy asymptotics
for $M_\pm(z,x_0,\alpha)$, it is sufficient to consider the subset
of the Weyl disk consisting of those matrices, $M\in{\mathbb{C}}^{m\tilde mes m}$,
characterized in Lemma~\ref{l2.11} and Lemma~\ref{l2.12}.
This was the approach taken in \cite{CG99}.
\end{remark}
When ${\mathcal D}_\pm(z,x_0,\alpha)$ is a singleton matrix, the
system \eqref{HSa} is said to be in the {\it limit point} (l.p.) case at
$\pm\infty$. If ${\mathcal D}_\pm(z,x_0,\alpha)$ has nonempty interior, then
\eqref{HSa} is said to be in the {\it limit circle} (l.c.) case at
$\pm\infty$. Indeed, for the case $m=1$, the limit point case
corresponds to a point in ${\mathbb{C}}$, whereas the limit circle case
corresponds to ${\mathcal D}_\pm(z,x_0,\alpha)$ being a disk in ${\mathbb{C}}$.
These apparent geometric properties for the disk
correspond to analytic properties for the solutions of the
Hamiltonian system \eqref{HSa}. To recall this correspondence, we
introduce the following spaces in which we assume that
$ -\infty\le a< b \le \infty$,
\begin{subequations}\label{2.29}
\begin{align}
L_A^2((a,b))&=\bibitemgg\{\phi:(a,b)\to{\mathbb{C}}^{2m} \bibitemgg| \int_a^b dx\,
(\phi(x),A\phi(x))_{{\mathbb{C}}^{2m}}<\infty \bibitemgg\}, \label{2.29a}
\\
N(z,\infty)&=\{\phi\in L_A^2((c,\infty)) \mid J\phi^\prime
=(zA+B)\phi
\text{ a.e. on $(c,\infty)$} \}, \label{2.29b}
\\
N(z,-\infty)&=\{\phi\in L_A^2((-\infty,c)) \mid
J\phi^\prime=(zA+B)\phi
\text{ a.e. on $(-\infty,c)$} \}, \label{2.29c}
\end{align}
\end{subequations}
for some $c\in{\mathbb{R}}$ and $z\in{\mathbb{C}}$. (Here
$(\phi,\psi)_{{\mathbb{C}}^n}=\sum_{j=1}^n \overline\phi_j\psi_j$
denotes the standard scalar product in ${\mathbb{C}}^n$, abbreviating
$\chi\in{\mathbb{C}}^n$ by
$\chi=(\chi_1,\dots,\chi_n)^t$.) Both dimensions of the
spaces in \eqref{2.29b} and \eqref{2.29c},
$\dim_{\mathbb{C}}(N(z,\infty))$ and $\dim_{\mathbb{C}}(N(z,-\infty))$,
are constant for $z\in{\mathbb{C}}_\pm=\{\zeta\in{\mathbb{C}}
\mid \pm\text{\rm Im}(\zeta)> 0 \}$ (see, e.g., \cite{At64},
\cite{KR74}). One then observes that the Hamiltonian
system \eqref{HSa} is in the limit point case at
$\pm\infty$ whenever
\begin{equation}\label{2.30}
\dim_{\mathbb{C}}(N(z,\pm\infty))=m \text{ for all
$z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$}
\end{equation}
and in the limit circle case at $\pm\infty$ whenever
\begin{equation}\label{2.31}
\dim_{\mathbb{C}}(N(z,\pm\infty))=2m \text{ for all $z\in{\mathbb{C}}$.}
\end{equation}
Next we show that the Dirac-type systems considered in this paper
are always in the limit point case at $\pm\infty$. Results of this
type, under varying sets of assumptions on $B(x)$, are well-known
to experts in the field. For instance, in the case $m=1$ and with
$B_{1,2}(x)=B_{2,1}(x)$ this fact can be found in \cite{We71}. For
$B\in C({\mathbb{R}})^{2m\tilde mes 2m}$ and a more general constant matrix $A$,
this result is proven in \cite{LM00} (their proof, however,
extends to the current $B\in L^1_{\text{\rm{loc}}} ({\mathbb{R}})$ case). More generally,
multi-dimensional Dirac operators with $L^2_{\text{\rm{loc}}} ({\mathbb{R}}^n)$-type
coefficients (and additional conditions) can be found in \cite{LO82}. A
short proof in the case $m=1$ has recently been sent to us by Don Hinton
\cite{Hi99}. For convenience of the reader we present its elementary
generalization to $m\in{\mathbb{N}}$ below (see also \cite{Cl94} for a sketch of
such a proof). After completion of this paper we became aware of a recent
preprint by Lesch and Malamud \cite{LM00a} which provides a thorough
study of self-adjointness questions for more general Hamiltonian systems
than those studied in this paper.
\begin{lemma} \label{l2.15}
The limit point case holds for Dirac-type systems
{\rm (}i.e., for $A=I_{2m}$ in \eqref{HSa}{\rm )} at $\pm \infty$.
\end{lemma}
\begin{proof}
Let $\{y_\ell(z,x)\}_{\ell=1,\dots,k}$ and
$\{w_j(z,x)\}_{j=1,\dots,k^\prime}$ denote bases for
$N(z,\pm\infty)$
and $N(\overline z,\pm\infty)$, respectively. By Theorem~9.11.1 of
Atkinson \cite{At64}, one has $k,k^\prime\geq m$ for
$z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$. We now assume that $k>m$.
One observes that $\{y_1(z,c),\dots,y_k(z,c)\}$ and
$w_1(\overline z,c),\dots,w_{k^\prime}(\overline z,c)\}$ are linearly
independent in ${\mathbb{C}}^{2m+1}$, where $k+k^\prime\geq 2m+1$.
Consequently, there is some $s\in\{1,\dots,k\}$ and some
$r\in\{1,\dots,k^\prime\}$ such that
\begin{equation}
w_r(\overline z,c)^*Jy_s(z,c)\neq 0. \label{2.32a}
\end{equation}
By Lagrange's identity,
\begin{equation}
w_r(\overline z,x)^*Jy_s(z,x)=w_r(\overline z,c)^*Jy_s(z,c) \label{2.33a}
\end{equation}
is constant with respect to $x$. On the other hand, an application
of Cauchy's inequality shows that the left-hand side of \eqref{2.33a} is in
$L^1 ((c,\pm\infty))$. By \eqref{2.32a} one obtains a contradiction
and hence concludes that
\begin{equation}
\dim_{\mathbb{C}} (N(z,\pm\infty))=m. \label{2.34}
\end{equation}
The analogous argument then also yields
\begin{equation}
\dim_{\mathbb{C}} (N(\overline z,\pm\infty))=m \label{2.35}
\end{equation}
and hence the limit point property of Dirac-type systems with
$A(x)=I_{2m}$ in \eqref{HSa}.
\end{proof}
Returning to the general case \eqref{HSa}, in either the limit point or
limit circle cases, $M_\pm(z,x_0,\alpha)\in \partial
{\mathcal D}_{\pm}(z,x_0,\alpha)$ is said to be a {\em half-line
Weyl-Titchmarsh matrix}. Each such matrix is associated with the
construction of a self-adjoint operator acting on
$L_A^2([x_0,\pm \infty))\cap \text{\rm{AC}}([x_0,\pm\infty))^{2m}$ for the
Hamiltonian system \eqref{HSa}. However, for those intermediate cases
where
$m<\dim_{\mathbb{C}}(N(z,\pm\infty))<2m$, Hinton and Schneider have noted that
not every element of $\partial{\mathcal D}_{\pm}(z,x_0,\alpha)$ is a
half-line Weyl-Titchmarsh matrix, and have characterized those elements
of the boundary that are (cf.~\cite{HSH93}, \cite{HSH97}).
For convenience of the reader we summarize some of the
principal results on half-line Weyl-Titchmarsh matrices next.
\begin{theorem}
[\cite{AD56}, \cite{Ca76}, \cite{GT97}, \cite{HS81},
\cite{HS82}, \cite{HS86}, \cite{KS88}] \label{t2.3}
Suppose Hypotheses \ref{h2.1} and \\ \ref{h2.2}. Let
$z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$,
$x_0\in{\mathbb{R}}$, and denote by $\alpha, \gamma\in{\mathbb{C}}^{m\tilde mes 2m}$ matrices
satisfying \eqref{2.8e}. Then, \\
$(i)$ $\pm M_{\pm}(z,x_0,\alpha)$ is an $m\tilde mes m$
matrix-valued Herglotz
function of maximal rank.
In particular,
\begin{gather}
\text{\rm Im}(\pm M_{\pm}(z,x_0,\alpha)) > 0, \quad z\in{\mathbb{C}}_+, \\
M_{\pm}(\overline z,x_0,\alpha)=M_{\pm}(z,x_0,\alpha)^*, \label{2.38} \\
\text{\rm{rank}} (M_{\pm}(z,x_0,\alpha))=m, \\
\lim_{\varepsilon\downarrow 0} M_{\pm}(\lambda+
i\varepsilon,x_0,\alpha) \text{
exists for a.e.\
$\lambda\in{\mathbb{R}}$},\\
\begin{split}\label{2.41}
M_\pm(z,x_0,\alpha) &= [-\alpha J \gamma^* +
\alpha\gamma^* M_\pm(z,x_0,\gamma)]\tilde mes \\
&\quad \tilde mes[ \alpha\gamma^*
+ \alpha J \gamma^*M_\pm(z,x_0,\gamma)]^{-1}.
\end{split}
\end{gather}
Local singularities of $\pm M_{\pm}(z,x_0,\alpha)$ and
$\mp M_{\pm}(z,x_0,\alpha)^{-1}$ are necessarily real and at most of first
order in the sense that
\begin{align}
&\mp \lim_{\epsilon\downarrow0}
\left(i\epsilon\,
M_{\pm}(\lambda+i\epsilon,x_0,\alpha)\right) \geq 0, \quad \lambda\in{\mathbb{R}},
\label{2.24b} \\
& \pm \lim_{\epsilon\downarrow0}
\left(\frac{i\epsilon}{M_{\pm}(\lambda+i\epsilon,x_0,\alpha)}\right)
\geq 0, \quad \lambda\in{\mathbb{R}}. \label{2.24c}
\end{align}
$(ii)$ $\pm M_{\pm}(z,x_0,\alpha)$ admit the representations
\begin{align}
&\pm M_{\pm}(z,x_0,\alpha)=F_\pm(x_0,\alpha)+\int_{\mathbb{R}}
d\Omega_\pm(\lambda,x_0,\alpha) \,
\bibitemg((\lambda-z)^{-1}-\lambda(1+\lambda^2)^{-1}\bibitemg) \label{2.42} \\
&=\exp\bibitemgg(C_\pm(x_0,\alpha)+\int_{\mathbb{R}} d\lambda \, \Xi_{\pm}
(\lambda,x_0,\alpha)
\bibitemg((\lambda-z)^{-1}-\lambda(1+\lambda^2)^{-1}\bibitemg)
\bibitemgg), \label{2.43}
\end{align}
where
\begin{align}
F_\pm(x_0,\alpha)&=F_\pm(x_0,\alpha)^*, \quad \int_{\mathbb{R}}
\|d\Omega_\pm(\lambda,x_0,\alpha)\|_{{\mathbb{C}}^{m\tilde mes m}} \,
(1+\lambda^2)^{-1}<\infty,
\\
C_\pm(x_0,\alpha)&=C_\pm(x_0,\alpha)^*,
\quad 0\le\Xi_\pm(\,\cdot\,,x_0,\alpha)
\le I_m \, \text{ a.e.}
\end{align}
Moreover,
\begin{align}
\Omega_\pm((\lambda,\mu],x_0,\alpha)&
=\lim_{\delta\downarrow
0}\lim_{\varepsilon\downarrow 0}\frac1\pi
\int_{\lambda+\delta}^{\mu+\delta} d\nu \, \text{\rm Im}(\pm
M_\pm(\nu+i\varepsilon,x_0,\alpha)), \\
\Xi_\pm(\lambda,x_0,\alpha)&=\lim_{\varepsilon\downarrow 0}
\pi^{-1}\text{\rm Im}(\text{\rm ln}(\pm
M_\pm(\lambda+i\varepsilon,x_0,\alpha))) \text{ for a.e.\
$\lambda\in{\mathbb{R}}$}.
\end{align}
$(iii)$ Define the $2m\tilde mes m$ matrices
\begin{align}
U_\pm(z,x,x_0,\alpha)&=\begin{pmatrix}u_{\pm,1}(z,x,x_0,\alpha) \\
u_{\pm,2}(z,x,x_0,\alpha) \end{pmatrix}
=\Psi(z,x,x_0,\alpha)\begin{pmatrix} I_m \\
M_\pm(z,x_0,\alpha) \end{pmatrix} \nonumber \\
&=\begin{pmatrix}\theta_1(z,x,x_0,\alpha)
& \phi_1(z,x,x_0,\alpha)\\
\theta_2(z,x,x_0,\alpha)
& \phi_2(z,x,x_0,\alpha)\end{pmatrix}
\begin{pmatrix} I_m \\
M_\pm(z,x_0,\alpha) \end{pmatrix}, \label{2.52}
\end{align}
with $\theta_j(z,x,x_0,\alpha)$, and
$\phi_j(z,x,x_0,\alpha)$, $j=1,2$, defined by \eqref{FSc}. Then,
\begin{equation}
\text{\rm Im}(M_\pm(z,x_0,\alpha))=\text{\rm Im}(z) \int_{x_0}^{\pm\infty}ds\,
U_\pm(z,s,x_0,\alpha)^* A(s)
U_\pm(z,s,x_0,\alpha).
\end{equation}
\end{theorem}
In the Dirac-type context, where $A=I_{2m}$, the
$m$ columns of $U_\pm (z,\cdot,x_0,\alpha)$ span $N(z,\pm\infty)$.
Up to this point, we focused exclusively on Hamiltonian systems and
neglected the notion of a linear operator associated with \eqref{HS}. We
did this on purpose as the formalism presented thus far is widely
applicable and goes beyond the prime candidates such as Schr\"odinger and
Dirac-type systems. However, in the remainder of this section and for
the bulk of the material from Section~\ref{s3} on, we will focus on
the Dirac-type case. Thus, in addition to Hypotheses~\ref{h2.1}--\ref{h2.3},
which are assumed throughout this paper, we introduce the following
hypothesis taylored to these occasions.
\begin{hypothesis}\label{h2.4}
Assume Hypotheses~\ref{h2.1} and \ref{h2.3} as well as the
Dirac-type assumption \eqref{DS}.
\end{hypothesis}
Assuming the Dirac-type Hypothesis~\ref{h2.4}, we now describe
the associated Dirac-type operator $D$ on ${\mathbb{R}}$ by first introducing the
Green's matrix associated with \eqref{HS} and \eqref{DS}. Define the
$2m\tilde mes 2m$ matrix $G$ by
\begin{align}
G(z,x,x^\prime)=U_\mp(z,x,x_0,\alpha_0)[M_-(z,x_0,\alpha_0) &
-M_+(z,x_0,\alpha_0)]^{-1} U_\pm(\overline z,x^\prime,x_0,\alpha_0)^*,
\nonumber \\
& \alpha_0=(I_m\; 0), \quad x\lessgtr x^\prime,\,
z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}} \label{2.56}
\end{align}
Next, let $\phi\in L^2({\mathbb{R}})^{2m}$ and consider
\begin{equation}
J\psi^\prime(z,x)=(zI_{2m}+B(x))\psi(z,x)+ \phi(x),
\quad z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}} \label{2.58}
\end{equation}
for a.e.\ $x\in{\mathbb{R}}$. Then, as inferred from \cite{HS81},
\cite{HS83}, \eqref{2.58} has a
unique solution $\psi(z,\,\cdot\,)\in
L^2({\mathbb{R}})^{2m}\cap\text{\rm{AC}}_{\text{\rm{loc}}}({\mathbb{R}})^{2m}$ given by
\begin{equation}
\psi(z,x)=\int_{\mathbb{R}} dx^\prime\, G(z,x,x^\prime)
\phi(x^\prime). \label{2.59}
\end{equation}
The Dirac-type operator $D$ in $L^2({\mathbb{R}})^{2m}$ associated with the
Hamiltonian system \eqref{HS} and \eqref{DS} is then defined by
\begin{equation}
((D-z)^{-1}\psi)(x)= \int_{\mathbb{R}} dx^\prime\,
G(z,x,x^\prime)\psi(x^\prime), \quad \psi\in
L^2({\mathbb{R}})^{2m}, \; z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}. \label{2.60}
\end{equation}
Explicitly, one obtains
\begin{align}
D&=J \frac{d}{dx}-B, \label{2.61} \\
\text{\rm{dom}}(D)&=\{\phi\in L^2({\mathbb{R}})^{2m}\mid \phi
\in\text{\rm{AC}}_{\text{\rm{loc}}}({\mathbb{R}})^{2m}; \,(J\phi^\prime-B\phi)\in
L^2({\mathbb{R}})^{2m} \}, \nonumber
\end{align}
taking into account the limit point property of Dirac-type
systems as described in Lemma~\ref{l2.15}. Thus, $D$ is a
self-adjoint operator in $L^2({\mathbb{R}})^{2m}$.
As described in \cite{HS81}--\cite{HS86}, the $2m\tilde mes 2m$
Weyl-Titchmarsh matrix $M(z,x_0,\alpha_0)$ associated with $D$ is then
defined by
\begin{align}
M(z,x_0,\alpha_0)
&=\bibitemg(M_{j,j^\prime}(z,x_0,\alpha_0)\bibitemg)_{j,j^\prime=1,2} \nonumber \\
&=[G(z,x_0,x_0+0)+G(z,x_0,x_0-0)]/2,
\quad z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}. \label{2.62}
\end{align}
Actually, one can replace $\alpha_0=(I_m\; 0)$ by an arbitrary matrix
$\alpha=[\alpha_1\ \alpha_2]\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying
\eqref{2.8e} and hence introduces
\begin{subequations}\label{2.620}
\begin{align}
M(z,x_0,\alpha)
&=\bibitemg(M_{j,j^\prime}(z,x_0,\alpha)\bibitemg)_{j,j^\prime=1,2}, \quad
z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}, \label{2.62A} \\
M_{1,1}(z,x_0,\alpha)&=[M_-(z,x_0,\alpha)-M_+(z,x_0,\alpha)]^{-1},
\label{2.62B} \\
M_{1,2}(z,x_0,\alpha)&=2^{-1}
[M_-(z,x_0,\alpha)-M_+(z,x_0,\alpha)]^{-1}
[M_-(z,x_0,\alpha)+M_+(z,x_0,\alpha)], \nonumber \\
M_{2,1}(z,x_0,\alpha)&=2^{-1} [M_-(z,x_0,\alpha)+M_+(z,x_0,\alpha)]
[M_-(z,x_0,\alpha)-M_+(z,x_0,\alpha)]^{-1},\nonumber \\
M_{2,2}(z,x_0,\alpha)&=M_\pm(z,x_0,\alpha)
[M_-(z,x_0,\alpha)-M_+(z,x_0,\alpha)]^{-1}M_\mp(z,x_0,\alpha). \nonumber
\end{align}
\end{subequations}
\
The basic results on $M(z,x_0,\alpha)$ then read as follows.
\begin{theorem} [\cite{GT97}, \cite{HS81}, \cite{HS82},
\cite{HS86}, \cite{KS88}] \label{thm2.19}
Assume Hypothesis~\ref{h2.4} and suppose \, that $z\in{\mathbb{C}}
\begin{align}ckslash {\mathbb{R}}$, $x_0\in{\mathbb{R}}$, and that $\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$
satisfies \eqref{2.8e}. Then, \\
$(i)$ $M(z,x_0,\alpha)$ is a matrix-valued Herglotz function of rank
$2m$ with representation
\begin{align}
&M(z,x_0,\alpha)=F(x_0,\alpha)+\int_{\mathbb{R}} d\Omega(\lambda,x_0,\alpha)\,
\bibitemg((\lambda-z)^{-1}-\lambda(1+\lambda^2)^{-1}\bibitemg), \label{2.64} \\
&=\exp\bibitemgg(C(x_0,\alpha)+\int_{\mathbb{R}} d\lambda \, \Upsilon
(\lambda,x_0,\alpha)
\bibitemg((\lambda-z)^{-1}-\lambda(1+\lambda^2)^{-1}\bibitemg) \bibitemgg), \label{2.65}
\end{align}
where
\begin{align}
F(x_0,\alpha)&=F(x_0,\alpha)^*, \quad \int_{\mathbb{R}} \Vert
d\Omega(\lambda,x_0,\alpha)
\Vert_{{\mathbb{C}}^{2m\tilde mes 2m}} \,(1+\lambda^2)^{-1}<\infty, \label{2.66} \\
C(x_0,\alpha)&=C(x_0,\alpha)^*, \quad 0\le\Upsilon(\,\cdot\,,x_0,\alpha)
\le I_{2m} \, \text{ a.e.} \label{2.67}
\end{align}
Moreover,
\begin{align}
\Omega((\lambda,\mu],x_0,\alpha)&=\lim_{\delta\downarrow
0}\lim_{\varepsilon\downarrow 0}\frac1\pi
\int_{\lambda+\delta}^{\mu+\delta} d\nu \,
\text{\rm Im}(M(\nu+i\varepsilon,x_0,\alpha)), \label{2.68} \\
\Upsilon(\lambda,x_0,\alpha)&=\lim_{\varepsilon\downarrow 0}
\pi^{-1}\text{\rm Im}(\text{\rm ln}(M(\lambda+i\varepsilon,x_0,\alpha))) \text{ for a.e.\
$\lambda\in{\mathbb{R}}$}. \label{2.69}
\end{align}
$(ii)$ $z\in{\mathbb{C}}\begin{align}ckslash\text{\rm{spec}}(D)$ if and only if $M(z,x_0,\alpha)$ is
holomorphic near $z$.
\end{theorem}
Here $\text{\rm{spec}} (T)$ abbreviates the spectrum of a linear operator $T$.
Next, we explicitly discuss the elementary Dirac-type example where
$A=I_{2m}$ and $B=0$.
\begin{example}\label{e2.20}
Suppose $A=I_{2m}$, $B=0$ and let $\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfy
\eqref{2.8e}. Then,
\begin{align}
\Theta(z,x,x_0,\alpha)&=\begin{pmatrix}\theta_{1}(z,x,x_0,\alpha) \\
\theta_{2}(z,x,x_0,\alpha) \end{pmatrix}=\begin{pmatrix}
\alpha_1^*\cos(z(x-x_0))+\alpha_2^*\sin(z(x-x_0)) \\
\alpha_2^*\cos(z(x-x_0))-\alpha_1^*\sin(z(x-x_0)) \end{pmatrix}, \nonumber \\
& \hspace*{7.5cm} \quad z\in{\mathbb{C}}, \label{2.80} \\
\Phi(z,x,x_0,\alpha)&=\begin{pmatrix}\phi_{1}(z,x,x_0,\alpha) \\
\phi_{2}(z,x,x_0,\alpha) \end{pmatrix}=\begin{pmatrix}
-\alpha_2^*\cos(z(x-x_0))+\alpha_1^*\sin(z(x-x_0)) \\
\alpha_1^*\cos(z(x-x_0))+\alpha_2^*\sin(z(x-x_0)) \end{pmatrix}, \nonumber \\
& \hspace*{7.5cm} \quad z\in{\mathbb{C}}, \label{2.81} \\
U_\pm (z,x,x_0,\alpha)&=\begin{pmatrix} u_{\pm,1}(z,x,x_0,\alpha) \\
u_{\pm,2}(z,x,x_0,\alpha) \end{pmatrix}
=\begin{pmatrix} \alpha_1^* \mp i\alpha_2^* \\
\pm i(\alpha_1^* \mp i\alpha_2^*) \end{pmatrix}\exp(\pm iz(x-x_0)),
\nonumber \\
& \hspace*{7cm} \quad z\in{\mathbb{C}}_+, \label{2.82} \\
M_\pm (z,x,\alpha)&=\pm iI_m, \quad z\in{\mathbb{C}}_+. \label{2.83}
\end{align}
\end{example}
Compared to the case of Schr\"odinger operators, it is remarkable that
$M_\pm(z,x,\alpha)$ in \eqref{2.83} is, in fact, independent of $\alpha$.
Put differently, in Dirac-type situations, $M_\pm(z,x,\alpha)$ may contain
no information on the boundary condition indexed by
$\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$.
In Sections~\ref{s4} and \ref{s5} we will also refer to half-line Dirac
operators $D_+(\alpha)$ in $L^2([x_0,\infty))^{2m}$ associated with a
self-adjoint boundary condition at $x_0$ indexed by
$\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying \eqref{2.8e},
and hence briefly introduce
\begin{align}
D_+(\alpha)&=J \frac{d}{dx}-B, \label{2.84} \\
\text{\rm{dom}}(D_+(\alpha))&=\{\phi\in L^2([x_0,\infty))^{2m} \mid \phi
\in\text{\rm{AC}}([x_0,R])^{2m} \text{ for all $R>0$}; \nonumber \\
& \hspace*{2.4cm} \alpha\phi(x_0)=0; \, (J\phi^\prime-B\phi)\in
L^2([x_0,\infty))^{2m} \}, \nonumber
\end{align}
taking into account the limit point property of Dirac-type
systems at $+\infty$ as described in Lemma~\ref{l2.15}. Thus,
$D_+(\alpha)$ is a self-adjoint operator in $L^2([x_0,\infty))^{2m}$. In
complete analogy one introduces $D_-(\alpha)$ in $L^2((-\infty,
x_0])^{2m}$.
Next, we recall a few formulas in connection with Lagrange's identity
needed in the proof of Theorem~\ref{t4.10} assuming
$\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfies \eqref{2.8e}. Then,
explicitly, \eqref{2.310} and \eqref{2.330} read
\begin{align}
\theta_2(\begin{align}r z,x,x_0,\alpha)^*\theta_1(z,x,x_0,\alpha)-
\theta_1(\begin{align}r z,x,x_0,\alpha)^*\theta_2(z,x,x_0,\alpha)&=0, \label{2.72}
\\
\phi_2(\begin{align}r z,x,x_0,\alpha)^*\phi_1(z,x,x_0,\alpha)-
\phi_1(\begin{align}r z,x,x_0,\alpha)^*\phi_2(z,x,x_0,\alpha)&=0, \label{2.73} \\
\phi_2(\begin{align}r z,x,x_0,\alpha)^*\theta_1(z,x,x_0,\alpha)-
\phi_1(\begin{align}r z,x,x_0,\alpha)^*\theta_2(z,x,x_0,\alpha)&=I_m, \label{2.74}
\\
\theta_1(\begin{align}r z,x,x_0,\alpha)^*\phi_2(z,x,x_0,\alpha)-
\theta_2(\begin{align}r z,x,x_0,\alpha)^*\phi_1(z,x,x_0,\alpha)&=I_m, \label{2.75}
\end{align}
and
\begin{align}
\phi_1(z,x,x_0,\alpha)\theta_1(\begin{align}r z,x,x_0,\alpha)^*-
\theta_1(z,x,x_0,\alpha_0)\phi_1(\begin{align}r z,x,x_0,\alpha)^*&=0, \label{2.92}
\\
\phi_2(z,x,x_0,\alpha)\theta_2(\begin{align}r z,x,x_0,\alpha)^*-
\theta_2(z,x,x_0,\alpha)\phi_2(\begin{align}r z,x,x_0,\alpha)^*&=0, \label{2.93} \\
\phi_2(z,x,x_0,\alpha)\theta_1(\begin{align}r z,x,x_0,\alpha)^*-
\theta_2(z,x,x_0,\alpha)\phi_1(\begin{align}r z,x,x_0,\alpha)^*&=I_m, \label{2.94}
\\
\theta_1(z,x,x_0,\alpha)\phi_2(\begin{align}r z,x,x_0,\alpha)^*-
\phi_1(z,x,x_0,\alpha)\theta_2(\begin{align}r z,x,x_0,\alpha)^*&=I_m. \label{2.95}
\end{align}
Finally, we note the connection between $\Phi$ defined in \eqref{FSb},
for different boundary value data
$\alpha, \gamma\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying \eqref{2.8e}, namely
\begin{equation}\label{2.96}
\Phi(z,x,x_0,\gamma)=\Phi(z,x,x_0,\alpha)\alpha\gamma^* +
\Theta(z,x,x_0,\alpha)\alpha J \gamma^*.
\end{equation}
This connection formula follows by the uniqueness of solutions of
\eqref{HS} and by the identity given in \eqref{2.8i}. It
is needed in the proof of Theorem~\ref{t4.10}.
\section{The Leading Order Term in the Asymptotic \\ Expansion
of $M_\pm (z,x,\alpha)$} \label{s3}
Assuming Hypothesis~\ref{h2.4}, the principal result proven in this
section will be the following leading-order asymptotic result for
half-line Weyl-Titchmarsh matrices $M_\pm(z,x_0,\alpha_0)$ associated
with the Dirac-type operator \eqref{2.61},
\begin{equation}\label{3.1}
M_\pm(z,x_0,\alpha_0) \underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} \pm iI_{m} +o(1).
\end{equation}
Here $\alpha_0 =(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$, and
$C_{\varepsilon} \subset {\mathbb{C}}_+$ denotes the open sector with vertex at
zero, symmetry axis along the positive imaginary axis, and
opening angle $\varepsilon$, with $0<\varepsilon <\pi/2$.
This particular topic originates with the order result of
Hille~\cite{Hil63}
and the asymptotic formulas of Everitt~\cite{Ev72} and of Everitt and
Halvorsen~\cite{EH78}. By appealing to the theory of Riccati equations,
Atkinson in \cite{At81}, \cite{At82}, and \cite{At88a} obtains results
like those of Hille, Everitt, and Halvorsen, both for the Schr\"odinger
case as well as for the scalar-Dirac ($m=1$) case. Through a deeper
understanding of the role played by Riccati theory, Atkinson obtains
the first order asymptotic expansion of $M_+(z,x,\alpha_0)$ for the
matrix-valued Schr\"odinger case in an unpublished manuscript \cite
{At88}. Our strategy of proof for \eqref{3.1} is patterned after
Atkinson's approach which also appears in our recent work on the full
asymptotic expansion for $M_+(z,x,\alpha_0)$ in the matrix-valued
Schr\"odinger case \cite{CG99}.
We begin our discussion by noting two additional characterizations
for the Weyl disk, ${\mathcal D}(z,c,x_0,\alpha)$, for the general Hamiltonian system \eqref{HSa}.
\begin{lemma}\label{l3.3}
Assume Hypotheses~\ref{h2.1} and \ref{h2.2}.
Let $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, $c\ne x_0$, and define
$U(z,x,x_0,\alpha)$, in terms of $M\in{\mathbb{C}}^{m\tilde mes m}$ by
\eqref{2.14}. Then $M\in {\mathcal D}(z,c,x_0,\alpha)$ if and only if
\begin{equation}\label{3.3}
\sigma(c,x_0,z)\text{\rm Im} (u_1(z,x,x_0,\alpha)^*u_2(z,x,x_0,\alpha)) > 0,
\quad x \in[x_0,c),
\end{equation}
or equivalently, if and only if
\begin{equation}\label{3.4}
\sigma(c,x_0,z)\text{\rm Im} (u_2(z,x,x_0,\alpha)u_1(z,x,x_0,\alpha)^{-1}) > 0,
\quad x \in[x_0,c).
\end{equation}
Moreover, $M\in {\mathcal D}_{\pm}(z,x_0,\alpha)$ if and only if \eqref{3.3} and
\eqref{3.4} hold for $c=\pm\infty$.
\end{lemma}
\begin{proof}
Let $U(z,x)=U(z,x,x_0,\alpha)$, and let $u_j(z,x)=u_j(z,x,x_0,\alpha)$, $j=1,2$
with $x\in[x_0,c)$. By \eqref{2.280},
\begin{align}\label{3.6}
&2\sigma(c,x_0)|\text{\rm Im}(z)|\int_x^c ds\, U(z,s)^*A(s)U(z,s)\nonumber \\
&=\sigma(x_0,c,z)U(z,s)^*(iJ)U(z,s)\Big |_x^c.
\end{align}
By \eqref{2.380}, this yields
\begin{align}\label{3.5}
&2\sigma(c,x_0,z)\text{\rm Im} (u_1(z,x)^*u_2(z,x))\nonumber\\
&= -E_c(M) + 2\sigma(c,x_0)|\text{\rm Im}(z)|\int_x^c ds\, U(z,s)^*A(s)U(z,s).
\end{align}
The integral expression in \eqref{3.5} is strictly positive by
Hypothesis~\ref{h2.2}. This yields the equivalence of $-E_c(M)\ge 0$, and hence
of $M\in {\mathcal D}(z,c,x_0,\alpha)$, with the condition given in \eqref{3.3}.
The equivalence of \eqref{3.3} and \eqref{3.4} follows
from the observation that
\begin{equation}
\text{\rm Im} (u_2(z,x)u_1(z,x)^{-1}) = (u_1(z,x)^{-1})^*\text{\rm Im} (u_1(z,x)^*u_2(z,x))u_1(z,x)^{-1}.
\end{equation}
The analogous characterization of ${\mathcal D}_{\pm}(z,x_0,\alpha)$ now follows
from Definition~\ref{dLWD}.
\end{proof}
In Lemma~\ref{l3.3}, $u_j(z,c)$, $j=1,2$, are well-defined and
$E_c(M)=0$ precisely when $\sigma(c,x_0,z)\text{\rm Im} (u_1(z,c)^*u_2(z,c))= 0$.
A similar statement might not hold for \eqref{3.4} since
$u_1(z,c,x_0,\alpha)$ might be singular. In part, the
latter point motivates the next characterization of the disk.
\begin{lemma}\label{l3.4}
Assume Hypotheses~\ref{h2.1} and \ref{h2.2}.
Let $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, $c\ne x_0$, and define
$u_j(z,x)=u_j(z,x,x_0,\alpha)$, $j=1,2$, by
\eqref{2.14}. Then $M\in {\mathcal D}(z,c,x_0,\alpha)$ if and only if
\begin{equation}\label{3.8}
u_1(z,x) -i\sigma(c,x_0,z)u_2(z,x)
\end{equation}
is nonsingular for $x\in[x_0,c]$ and
\begin{equation}\label{3.9}
\begin{split}
\vartheta(z,x)=\vartheta(z,x,x_0,\alpha)&= [u_1(z,x)
+i\sigma(c,x_0,z)u_2(z,x)]\tilde mes\\
& \quad \tilde mes [u_1(z,x)
-i\sigma(c,x_0,z)u_2(z,x)]^{-1}
\end{split}
\end{equation}
satisfies
\begin{equation}\label{3.10}
I_m-\vartheta(z,x)^*\vartheta(z,x)>0,\quad
x\in[x_0,c),
\end{equation}
with nonnegativity holding at $x=c$.
Moreover, $M\in {\mathcal D}_{\pm}(z,x_0,\alpha)$ if and only if \eqref{3.9}
is well-defined on $[x_0,\pm\infty)$ and
\eqref{3.10} holds for $c=\pm\infty$.
\end{lemma}
\begin{proof}
Let $M\in{\mathcal D}(z,c,x_0,\alpha)$ and suppose that
$u_1(z,\xi)v=i\sigma(c,x_0,z)u_2(z,\xi)v$ for $\xi\in[x_0,c]$ and
$v\in{\mathbb{C}}^m$, $v\ne 0$. Then,
\begin{equation}
v^*\sigma(c,x_0,z)\text{\rm Im} (u_1(z,\xi)^*u_2(z,\xi))v =
-v^*u_1(z,\xi)^*u_1(z,\xi)v.
\end{equation}
By \eqref{3.3}, an immediate contradiction results if $\xi\ne c$. However,
if $\xi=c$, then either $v^*E_c(M)v>0$ or $u_j(z,c)v=0$, $j=1,2$. In either
case, a contradiction results since $E_c(M)\leq 0$ by Definition~\ref{dWD}
and $U=(u_1^t,u_2^t)^t$ satisfies the first-order system \eqref{HSa}.
Hence, $\vartheta(z,x)$ is well-defined on $[x_0,c]$. For $x\in [x_0,c)$
and $M\in {\mathcal D}(z,c,x_0,\alpha)$, \eqref{3.3} implies that
\begin{equation}\label{3.12}
2i\sigma(c,x_0,z)(u_1(z,x)^*u_2(z,x) - u_2(z,x)^*u_1(z,x) )< 0.
\end{equation}
This is equivalent to
\begin{equation}\label{3.13}
\begin{split}
&[u_1(z,x)^*-i\sigma(c,x_0,z)u_2(z,x)^*]
[u_1(z,x)+i\sigma(c,x_0,z)u_2(z,x)] \\
&<[u_1(z,x)^*+i\sigma(c,x_0,z)u_2(z,x)^*][u_1(z,x)
-i\sigma(c,x_0,z)u_2(z,x)]
\end{split}
\end{equation}
on $[x_0,c)$. Given the nonsingularity of
$u_1(z,x)-i\sigma(c,x_0,z)u_2(z,x)$ on
$[x_0,c]$, \eqref{3.13} implies \eqref{3.10}, with nonnegativity holding
at $x=c$. \\
Next, let $M\in{\mathbb{C}}^{m\tilde mes m}$, and suppose that
$\vartheta(z,x)$, defined by \eqref{3.9}, is well-defined on $[x_0,c]$,
and satisfies \eqref{3.10}. Then, on $[x_0,c)$,
\eqref{3.13} and consequently \eqref{3.12} follow, which implies
that \eqref{3.3} holds, and hence that $M\in{\mathcal D}(z,c,x_0,\alpha)$.
The analogous characterization of ${\mathcal D}_{\pm}(z,x_0,\alpha)$
follows from Definition~\ref{dLWD}.
\end{proof}
By Lemma~\ref{l3.3} one notes, for $z\in {\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, that $M\in
{\mathcal D}(z,c,x_0,\alpha)$ if and only if
\begin{equation}\label{3.14}
V(z,x,x_0,\alpha)= u_2(z,x,x_0,\alpha)u_1(z,x,x_0,\alpha)^{-1},
\quad x\in [x_0,c),
\end{equation}
is well-defined while satisfying
\begin{equation}\label{3.15}
\sigma(c,x_0,z)\text{\rm Im} (V(z,x,x_0,\alpha)) > 0, \quad x\in [x_0,c).
\end{equation}
In terms of $V(z,x,x_0,\alpha)$ and by \eqref{3.9}, one notes that
\begin{equation}\label{3.16}
\begin{split}
\vartheta(z,x,x_0,\alpha) &=
[I_m + i\sigma(c,x_0,z)V(z,x,x_0,\alpha)]\tilde mes\\
& \quad \tilde mes[I_m - i\sigma(c,x_0,z)V(z,x,x_0,\alpha)]^{-1},\quad
x\in[x_0,c),
\end{split}
\end{equation}
is a Cayley-type transformation of
$V(z,x,x_0,\alpha)$. In the scalar context, this
transformation corresponds to a conformal
mapping of the complex upper half-plane to the
unit disk.
Moreover, defined as it is, $V(z,x,x_0,\alpha)$
satisfies a Riccati differential equation
that is associated with the Hamiltonian system
\eqref{HSa} while $\vartheta(z,x,x_0,\alpha)$
satisfies a Riccati equation obtained by the
Cayley-type transformation \eqref{3.16} applied to
the differential equation satisfied by $V(z,x,x_0,\alpha)$.
For the Dirac-type case of \eqref{HSa}, one observes by a simple
calculation that $V(z,x,x_0,\alpha_0)$ is seen to satisfy a particular
initial value problem for a Riccati differential equation.
\begin{lemma} \label{l3.5}
Assume Hypotheses~\ref{h2.1}, \ref{h2.2}, and \ref{h2.4}.
Let $\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$, let $u_j(z,x)=u_j(z,x,x_0,\alpha_0)$,
$j=1,2$, be defined by \eqref{2.14}, and suppose that $V(z,x,x_0,\alpha_0)$
is well-defined by \eqref{3.14}. Then, $V(z,\cdot)=V(z,\cdot,x_0,\alpha_0)$ satisfies,
\begin{subequations}\label{3.17}
\begin{align}
&V'(z,x) +zV(z,x)^2 + V(z,x) B_{2,2}(x)V(z,x) + B_{1,2}(x)V(z,x)
+ V(z,x)B_{2,1}(x) \nonumber \\\
& + B_{1,1}(x) +zI_m =0, \label{3.17a}\\
&V (z,x_0)=M, \label{3.17b}
\end{align}
\end{subequations}
where $B_{j,k}\in {\mathbb{C}}^{m\tilde mes m}$, $j,k= 1,2$, are defined in
\eqref{2.1d}.
\end{lemma}
Hence, by Lemma~\ref{l3.3}, the associated relations \eqref{3.14} and
\eqref{3.15}, and the uniqueness of solutions for \eqref{3.17}, we
obtain the following result for the Dirac-type case.
\begin{theorem}\label{t3.6}
Assume Hypotheses~\ref{h2.1}, \ref{h2.2}, and \ref{h2.4}, and let
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$. Then, $M\in
{\mathcal D}(z,c,x_0,\alpha_0)$ if and only if the initial value problem
given by \eqref{3.17} has a solution, $V(z,\cdot)$, well-defined and
satisfying
\begin{equation}\label{3.18}
\sigma(c,x_0,z)\text{\rm Im} (V(z,x)) > 0,\quad x\in[x_0,c).
\end{equation}
Moreover, $M\in {\mathcal D}_{\pm}(z,x_0,\alpha_0)$
if and only if \eqref{3.18} holds for $c=\pm\infty$.
\end{theorem}
\begin{remark} \label{r3.6a}
An important consequence of Theorem~\ref{t3.6} and
the uniqueness of
solutions for \eqref{3.17} is that solution
trajectories for \eqref{3.17}, which satisfy
\eqref{3.18}, consist of elements of Weyl disks;
that is,
\begin{equation}\label{3.19}
V(z,x,x_0,\alpha_0)\in {\mathcal D}(z,c,x,\alpha_0),
\quad x\in [x_0,c).
\end{equation}
Given the characterization of ${\mathcal D}(z,c,x_0,\alpha_0)$ in
Defintion~2.7A, for each $x\in [x_0,c)$ there is
a $\beta\in{\mathbb{C}}^{m\tilde mes 2m}$ with
$\sigma(c,x_0,z)\text{\rm Im} (\beta_2\beta_1^*)\ge 0$, such that
\begin{equation}
V(z,x,x_0,\alpha_0)=M(z,c,x,\alpha_0,\beta).
\end{equation}
It is in this sense that we let $M(z,c,x,\alpha_0)$
denote our solution of the initial value problem \eqref{3.17}
that satisfies \eqref{3.18}. Analogously,
\begin{equation}
V(z,x,x_0,\alpha_0)\in {\mathcal D}_{\pm}(z,x,\alpha_0),\quad x\in
[x_0,\pm\infty),
\end{equation}
for trajectories of \eqref{3.17} that satisfy
\eqref{3.18} for $c=\pm\infty$.
Hence, in this sense, we let
$M_{\pm}(z,x,\alpha_0)$ denote those
solutions of \eqref{3.17} that satisfy
\eqref{3.18} for $c=\pm\infty$.
However, by Lemma~\ref{l2.15}, our Dirac system
is in the limit point case at $\pm\infty$. Each
${\mathcal D}_{\pm}(z,x,\alpha_0)$ consists of a
unique matrix, and thus
$M_{\pm}(z,x,\alpha_0)$ describes {\em
unique} trajectories for \eqref{3.17a}. This
contrasts with the matrix-valued Schr\"odinger case
considered in \cite{CG99} where there are as many
trajectories, each denoted by either
$M_{+}(z,x,\alpha_0)$ or
$M_{-}(z,x,\alpha_0)$, as there are
matrices in a given initial disk
${\mathcal D}_{\pm}(z,x_0,\alpha_0)$.
\end{remark}
Now for the Dirac-type case \eqref{DS}
with $\alpha_0=(I_m\; 0)\in {\mathbb{C}}^{m\tilde mes 2m}$, with
$\vartheta(z,x)=\vartheta(z,x,x_0,\alpha_0)$ defined in
\eqref{3.9} and \eqref{3.16}, and with $x\in[x_0,c)$,
one concludes that
\begin{equation}\label{3.22}
\vartheta(z,x)[u_1(z,x)-i\sigma(c,x_0,z)u_2(z,x)]=u_1(z,x)+
i\sigma(c,x_0,z)u_2(z,x),
\end{equation}
and hence that
\begin{subequations}\label{3.23}
\begin{align}
I_m + \vartheta(z,x)&=
2u_1(z,x)[u_1(z,x)-i\sigma(c,x_0,z)u_2(z,x)]^{-1},\\ I_m -
\vartheta(z,x)&= -2i\sigma(c,x_0,z)u_2(z,x)[u_1(z,x)-i
\sigma(c,x_0,z)u_2(z,x)]^{-1}.
\end{align}
\end{subequations}
Differentiating \eqref{3.22} one obtains
\begin{equation}
\begin{split}
\vartheta'(u_1-i\sigma u_2)&=(I_m -\vartheta)
(zu_2 + B_{2,1}u_1 + B_{2,2}u_2) \\
& \quad +i\sigma (I_m +\vartheta)(-zu_1 - B_{1,1}u_1 -B_{1,2}u_2).
\end{split}
\end{equation}
By \eqref{3.23} one concludes that $\vartheta(z,\cdot,x_0,\alpha_0)$
satisfies the initial value problem given by
\begin{subequations}\label{3.25}
\begin{align}
\vartheta'(z,x)&= \fracrac{1}{2} \begin{pmatrix}I_m +
\vartheta(z,x)^t\\ I_m - \vartheta(z,x)^t\end{pmatrix}^t\tilde mes \nonumber \\
& \quad \tilde mes \begin{pmatrix} -i\sigma(c,x_0,z)(zI_m +B_{1,1}(x)) &
B_{1,2}(x)\\B_{2,1}(x)& i\sigma(c,x_0,z)(zI_m +B_{2,2}(x))
\end{pmatrix}\tilde mes\nonumbertag \\
& \quad \tilde mes\begin{pmatrix}I_m +
\vartheta(z,x)\\ I_m - \vartheta(z,x)\end{pmatrix}, \label{3.25a} \\[5pt]
\vartheta(z,x_0)&=(I_m +i\sigma(c,x_0,z) M)
(I_m -i\sigma(c,x_0,z) M)^{-1},\label{3.25b}
\end{align}
\end{subequations}
where $B_{j,k}\in {\mathbb{C}}^{m\tilde mes m}$,
$j,k= 1,2$, satisfy Hypothesis~\ref{h2.1}.
By Lemma~\ref{l3.4} and the uniqueness of solutions for \eqref{3.25},
one obtains the following result in the Dirac-type case \eqref{DS}.
\begin{theorem}\label{t3.7}
Assume Hypothesis~\ref{h2.4}. Then
$M\in {\mathcal D}(z,c,x_0,\alpha_0)$ if and only if the initial value
problem given by \eqref{3.25} has a solution, $\vartheta(z,\cdot)$
which is well-defined on $[x_0,c]$ and satisfies
\begin{equation}\label{3.26}
I_m-\vartheta(z,x)^*\vartheta(z,x)> 0,\quad
x\in[x_0,c).
\end{equation}
Moreover, $M\in {\mathcal D}_{\pm}(z,x_0,\alpha_0)$ if and only if
\eqref{3.26} holds for $c=\pm\infty$.
\end{theorem}
Given the positivity present in \eqref{3.26}, we note the exact
correspondence which exists, by \eqref{3.16}, between solutions of
\eqref{3.17} that satisfy \eqref{3.18} and those solutions of
\eqref{3.25} that satisfy \eqref{3.26}. In particular, given Remark~\ref{r3.6a},
we rewrite \eqref{3.16} as
\begin{equation}\label{3.27}
\begin{split}
\vartheta(z,x,x_0,\alpha_0) &=
[I_m + i\sigma(c,x_0,z)M(z,c,x,\alpha_0)]\tilde mes\\
& \quad \tilde mes[I_m - i\sigma(c,x_0,z)M(z,c,x,\alpha_0)]^{-1},
\quad x\in[x_0,c),
\end{split}
\end{equation}
Moreover, our Dirac system is in the limit point case at $\pm\infty$.
Consequently, there are unique solutions
of \eqref{3.25}, $\vartheta_{\pm}(z,\cdot,x_0,\alpha_0)$,
$z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, which satisfy \eqref{3.26} for
$c=\pm\infty$, and which correspond to the unique solutions of
\eqref{3.17}, $M_{\pm}(z,x,\alpha_0)$, which satisfy \eqref{3.18}
for $c=\pm\infty$; specifically,
\begin{equation}\label{3.28}
\vartheta_{\pm}(z,x,x_0,\alpha_0)= [I_m \pm
i\sigma(z)M_{\pm}(z,x,\alpha_0)][I_m \mp
i\sigma(z)M_{\pm}(z,x,\alpha_0)]^{-1}.
\end{equation}
These relationships form the basis for the
analysis to follow. The asymptotic result \eqref{3.1} is obtained
by an analysis of the corresponding asymptotic behavior for all
solutions $\vartheta(z,\cdot,x_0,\alpha_0)$
described in \eqref{3.25}, these include among
them the particular solutions
$\vartheta_{\pm}(z,\cdot,x_0,\alpha_0)$. Thus
asymptotic behavior is deduced for all
corresponding solutions
$M(z,c,\cdot,\alpha_0)$ of \eqref{3.17} which
include among them the solutions
$M_{\pm}(z,\cdot,\alpha_0)$. The advantage of
this approach comes from the compactification
inherent in the Cayley-type transformation
\eqref{3.27}, and the resulting boundedness of the
solutions as a consequence of \eqref{3.26}.
We pause for a moment to address, in the following remark,
a point raised by us in \cite{CG99} for the matrix-valued
Schr\"odinger case described in \eqref{SS}.
\begin{remark}\label{r3.3}
With $u_j(z,x)=u_j(z,x,x_0,\alpha)$, $j=1,2$, defined in \eqref{2.14} for
the general Hamiltonian system \eqref{HSa}, an analog to Lemma~\ref{l3.4}
for the characterization of ${\mathcal D}(z,c,x_0,\alpha)$ is obtained by replacing
the expression in \eqref{3.8} with
\begin{equation}
u_1(z,x) -i|z|^{-1/2}\sigma(c,x_0,z)u_2(z,x),
\end{equation}
and by replacing the definition for $\vartheta(z,x)=\vartheta(z,x,x_0,\alpha)$
given in \eqref{3.9} with
\begin{equation}
\begin{split}
\vartheta(z,x) =& (u_1(z,x) +i|z|^{-1/2}\sigma(c,x_0,z)u_2(z,x))\tilde mes\\
& \tilde mes (u_1(z,x) -i|z|^{-1/2}\sigma(c,x_0,z)u_2(z,x))^{-1}.
\end{split}
\end{equation}
Specific to the matrix-valued Schr\"odinger case, we obtain analogs of
Lemma~\ref{l3.5}, Theorem~\ref{t3.6}, and Theorem~\ref{t3.7} by replacing
equation \eqref{3.17a} with
\begin{equation}
V'(z,x) + V(z,x)^2 - Q(x) +zI_m = 0
\end{equation}
and by replacing the equations in \eqref{3.25} with
\begin{subequations}\label{3.280}
\begin{align}
\vartheta'(z,x)&= \sigma(c,x_0,z)\fracrac{1}{2}
\begin{pmatrix}I_m +\vartheta(z,x)^t\\
I_m - \vartheta(z,x)^t
\end{pmatrix}^t
\begin{pmatrix} -i|z|^{-1/2}(zI_m - Q(x)) & 0 \\
0 & i|z|^{-1/2}I_m
\end{pmatrix}\tilde mes\nonumbertag \\
& \quad \tilde mes\begin{pmatrix}I_m + \vartheta(z,x)\\
I_m - \vartheta(z,x)\end{pmatrix},\label{3.280a} \\[5pt]
\vartheta(z,x_0)&=(I_m + i|z|^{-1/2}\sigma(c,x_0,z)M)
(I_m - i|z|^{-1/2}\sigma(c,x_0,z)M)^{-1}.\label{3.280b}
\end{align}
\end{subequations}
${\mathcal D}^{{\mathcal R}}(z,c,x_0,\alpha_0)$ was defined in \cite{CG99} to be the set
of those $M\in {\mathbb{C}}^{m\tilde mes m}$ for which the intial value problem given
by \eqref{3.280} has a solution, $\vartheta(z,x)$, which is well-defined
on $[x_0,c]$ and satisfies \eqref{3.26}. In \cite{CG99} we showed that
${\mathcal D}(z,c,x_0,\alpha_0)\subseteq{\mathcal D}^{{\mathcal R}}(z,c,x_0,\alpha_0)$.
This was sufficient for the subsequent analysis
in \cite{CG99}. However, as the analog of Theorem~\ref{t3.7} now shows,
one actually has equality of the two disks in \cite{CG99}, that is,
\begin{equation}
{\mathcal D}(z,c,x_0,\alpha_0)={\mathcal D}^{{\mathcal R}}(z,c,x_0,\alpha_0).
\end{equation}
\end{remark}
To obtain a proof of \eqref{3.1} for the Dirac-type case, we adapt an
approach due to Atkinson \cite{At88} for proving a result
analogous to \eqref{3.1} for the matrix-valued Schr\"odinger case
(cf., e.g., \cite[Theorem 3.1]{CG99}) In light of Remark~\ref{r3.2}, we
begin by restricting our attention to $z\in{\mathbb{C}}_+$, and as in the
previous discussion, take $\alpha_0=(I_m\; 0)\in
{\mathbb{C}}^{m\tilde mes 2m}$.
First we introduce two systems related to \eqref{3.25} by means of a
change of variables. Let
\begin{equation}\label{3.29}
\varphi(z,t)=\vartheta(z,x),\qquad
t= (x-x_0)|z|,\qquad x \in [ x_0 , c ).
\end{equation}
With this change, \eqref{3.25} becomes
\begin{subequations}\label{3.30}
\begin{align}\label{3.30a}
\varphi'(z,t)&= \fracrac{1}{2} |z|^{-1}\begin{pmatrix}I_m
+ \varphi(z,t)^t\\ I_m - \varphi(z,t)^t\end{pmatrix}^t
\begin{pmatrix}\mp i(zI_m +\widetilde B_{1,1}(t)) & \widetilde
B_{1,2}(t)\\[1mm] \widetilde B_{2,1}(t)& \pm i(zI_m +\widetilde B_{2,2}(t))
\end{pmatrix}\tilde mes \nonumbertag \\
& \quad \tilde mes\begin{pmatrix}I_m +
\varphi(z,t)\\ I_m - \varphi(z,t)\end{pmatrix}.
\end{align}
\nonumberindent With $M=M(z,c,x_0,\alpha_0)\in
{\mathcal D}(z,c,x_0,\alpha_0)$
\eqref{3.25b} becomes
\begin{equation}\label{3.30b}
\varphi(z,0)=(iI_m \mp M(z,c,x_0,\alpha_0))
(iI_m \pm M(z,c,x_0,\alpha_0))^{-1},
\end{equation}
\nonumberindent and \eqref{3.26} becomes
\begin{equation}\label{3.30c}
\varphi(z,t)^*\varphi(z,t)< I_m
\qquad t\in[ 0, (c-x_0)|z|),
\end{equation}
where in \eqref{3.30a},
\begin{equation}\label{3.30d}
\widetilde B_{j,k}(t)= B_{j,k}(x_0 +t|z|^{-1}), \quad j,k=1,2.
\end{equation}
\end{subequations}
In the complete system \eqref{3.30}, one now has a set of
conditions equivalent to system \eqref{3.25} and \eqref{3.26}.
We recall that $C_\varepsilon \subset {\mathbb{C}}_+$ represents
the open sector with vertex at zero, symmetry axis along the
positive imaginary axis, and opening angle $\varepsilon$,
with $0<\varepsilon <\pi/2$. Next, consider a sequence, $z_n \in
{\mathbb{C}}_{\varepsilon}$, $n\in{\mathbb{N}} $, such that $|z_n| \to
\infty$ as $n\to \infty$ and such that
\begin{equation}\label{3.31}
0< \varepsilon < \delta_n = \arg{(z_n)}
< \pi - \varepsilon.
\end{equation}
By choosing an appropriate subsequence, we may assume that
\begin{equation}\label{3.32}
\delta_n \to \delta \in [\varepsilon, \pi - \varepsilon].
\end{equation}
Let $\varphi (z_n ,t)$ denote a corresponding
sequence of functions that satisfy \eqref{3.30a}
and \eqref{3.30c}, with initial data, $\varphi
(z_n ,0)$, defined by \eqref{3.30b} for a sequence
of points $M(z_n,c,x_0,\alpha_0)$, where each
$M(z_n,c,x_0,\alpha_0)$ is chosen to be an element of the disk
${\mathcal D}(z_n,c,x_0,\alpha_0)$. Note that as $z_n\to
\infty$, the intervals described in \eqref{3.30c}
eventually cover all compact subintervals of $
{\mathbb{R}}_+$. Given the uniform boundedness of
$\varphi_n(t)=\varphi (z_n ,t)$ described in
\eqref{3.30c}, we assume, upon passing to an
appropriate subsequence still denoted by
$\varphi_n (0)$, that
\begin{equation}\label{3.33}
\varphi_n(0) = \varphi (z_n,0) \rightarrow
\varphi_\pm (\delta), \ \text{for}
\ \pm (c-x_0) > 0 \
\text{ as } n\rightarrow \infty ,
\end{equation}
and as a consequence, that
\begin{equation}\label{3.34}
{\varphi_\pm(\delta)}^* \varphi_\pm(\delta) \le I_m.
\end{equation}
With $\varphi_\pm(\delta)$ defined in \eqref{3.33} as
$|z_n|\to\infty$, we
consider limiting systems associated with \eqref{3.30}:
\begin{subequations}\label{3.35}
\begin{align}
\eta_\pm '(t)&= \fracrac{1}{2}
\begin{pmatrix}
I_m+ \eta_\pm (t)^t \\ I_m-\eta_\pm (t)^t
\end{pmatrix} ^t
\begin{pmatrix}
\mp ie^{i\delta}I_m & 0\\ 0 & \pm ie^{i\delta}I_m
\end{pmatrix}\begin{pmatrix}
I_m+\eta_\pm (t)\\ I_m- \eta_\pm (t)
\end{pmatrix}, \quad \pm t\ge 0, \label{3.35a} \\
\eta_\pm (0)&= \varphi_\pm(\delta). \label{3.35b}
\end{align}
\end{subequations}
\begin{theorem}\label{t3.8} Assume Hypothesis~\ref{h2.4}. Then
the solution $\eta_\pm$ of \eqref{3.35} satisfies
\begin{equation}\label{3.36}
\eta_\pm (t)^* \eta_\pm(t) \le I_m,\quad
t\in [0, \pm\infty).
\end{equation}
Moreover, the
solutions $\varphi_n =\varphi (z_n,\cdot)$
of \eqref{3.30} converge to $\eta_\pm$ uniformly on
$[0,\pm T]$ for every $T>0$, as $n\to \infty $.
\end{theorem}
\begin{proof}
In this proof, we consider only the case corresponding
to $t\ge 0$, that is, $\eta_+(0)=\varphi_+(\delta)$ in
\eqref{3.35b}. The other case follows in a similar manner.
For this reason, we let $\eta( t)= \eta_+( t)$ in the
remaining discussion. We also let $T\in {\mathbb{R}}_+$ be the
greatest value such that
\eqref{3.36} holds for $t\in [0, T] $ and show that
\eqref{3.36} must hold for some $[0, T'] $ with $ T' > T $,
thus proving $T=\infty.$\\
The solution of \eqref{3.35}, $\eta$,
presumed to be defined on $[0, T]$, can be continued
onto some $[0, T']$ with $T' > T$; $\eta$ then satisfies
\begin{equation}\label{3.37}
\eta (t)^* \eta (t) \le \kappa^2 I_m
\end{equation}
for $0\le t \le T'$ and for some $\kappa\ge 1$. \\
For brevity, let $\varphi'_n (t)= G_n(\varphi_n,t) $ denote
\eqref{3.30a} with
$z=z_n$, and let $\eta'(t)= H(\eta,t) $ denote \eqref{3.35a}
in the following. Integrating \eqref{3.35a} and
\eqref{3.30a}, one obtains
\begin{align}\label{3.38}
\varphi_n (t) -\eta (t) &= \varphi_n(0) -\varphi_0(\delta)
+ \int_0^t ds \{
G_n(\eta,s) - H(\eta,s)\} \nonumber \\
&\quad + \int_0^t ds \{ G_n(\varphi_n,s) - G_n(\eta,s)\}.
\end{align}
We note that
\begin{align}\label{3.39}
G_n(\eta ,s) - H(\eta ,s) &=
\fracrac{1}{2}i(e^{i\delta} - e^{i\delta_n})
(I_m +\eta(s) )^2
-\fracrac{1}{2}i(e^{i\delta} - e^{i\delta_n})
(I_m -\eta(s) )^2 +\nonumber\\
& \quad + \sum_{j,k =1}^2 F_{j,k}(z_n,s),
\end{align}
where,
\begin{subequations}\label{3.40}
\begin{align}
F_{1,1}(z_n,s)&=-\fracrac{1}{2}i|z_n|^{-1}
(I_m+\eta(s))\widetilde B_{1,1}(s)(I_m+\eta(s)),\\
F_{2,2}(z_n,s)&=\fracrac{1}{2}i|z_n|^{-1}
(I_m-\eta(s))\widetilde B_{2,2}(s)(I_m-\eta(s)),\\
F_{1,2}(z_n,s)&=\fracrac{1}{2}i|z_n|^{-1}
(I_m+\eta(s))\widetilde B_{1,2}(s)(I_m-\eta(s)),\\
F_{2,1}(z_n,s)&=\fracrac{1}{2}i|z_n|^{-1}
(I_m-\eta(s))\widetilde B_{2,1}(s)(I_m+\eta(s)).
\end{align}
\end{subequations}
Thus, for $t\in [0,T']$, \eqref{3.37} implies that
as $n\to\infty$
\begin{equation}
|e^{i\delta}-e^{i\delta_n}|\int_0^t
\|I_m \pm \eta (s) \|_{{\mathbb{C}}^{m\tilde mes m}}^2 ds= o (1),
\end{equation}
and together with \eqref{3.29} and \eqref{3.30d} that
\begin{equation}\label{3.42}
\int_0^t \| F_{j,k}(s) \|_{{\mathbb{C}}^{m\tilde mes m}}ds =
O\bibitemgg( \int_{x_0}^{x_0 + t|z_n|^{-1}} \| \widetilde
B_{j,k}(s) \|_{{\mathbb{C}}^{m\tilde mes m}} ds\bibitemgg) =o(1).
\end{equation}
(Here $\|\cdot\|_{{\mathbb{C}}^{m\tilde mes m}}$ denotes a norm on ${\mathbb{C}}^{m\tilde mes
m}$.) Hence, by \eqref{3.39}--\eqref{3.42}, one infers that for $t\in
[0,T']$ and as $n\to\infty$,
\begin{equation}\label{3.43}
\int_0^t \{ G_n(\eta,s) -H(\eta,s)\} ds = o(1).
\end{equation}
Next, one notes that
\begin{equation}\label{3.44}
G_n(\varphi_n,s) - G_n(\eta,s) = 2ie^{i\delta_n}
(\eta(s) - \varphi_n(s)) + \sum_{j,k =1}^2 K_{j,k}(z_n,s),
\end{equation}
where
\begin{subequations}
\begin{align}
K_{1,1}(z_n,s) &= \fracrac{-i}{2}|z_n|^{-1}
\{ (I_m +\varphi_n) B_{1,1}(s) (\varphi_n -\eta)
+ (\varphi_n
-\eta)B_{1,1}(s) (I_m +\eta) \},\\ K_{2,2}(z_n,s) &=
\fracrac{i}{2}|z_n|^{-1}\{ (I_m -\varphi_n) B_{2,2}(s) (\eta
-\varphi_n) + (\eta -\varphi_n)B_{2,2}(s) (I_m -\eta) \},\\
K_{1,2}(z_n,s) &= \fracrac{1}{2}|z_n|^{-1}\{(I_m
+\varphi_n)B_{1,2}(s)(\eta -\varphi_n) + (\varphi_n
-\eta)B_{1,2}(s)(I_m -\eta) \},\\ K_{2,1}(z_n,s) &=
\fracrac{1}{2}|z_n|^{-1}\{(I_m -\varphi_n) B_{2,1}(s) (\varphi_n
-\eta) + (\eta -\varphi_n)B_{2,1}(s) (I_m +\eta) \}.
\end{align}
\end{subequations}
By \eqref{3.34} and \eqref{3.37}, for $s\in [0,T']$,
\begin{equation}\label{3.46}
\| I_m \pm \varphi_n(s) \|_{{\mathbb{C}}^{m\tilde mes m}}\le 2,
\qquad \| I_m \pm \eta(s) \|_{{\mathbb{C}}^{m\tilde mes m}}\le \kappa +1,
\end{equation}
and hence by \eqref{3.44}--\eqref{3.46},
\begin{align}\label{3.47}
&\| G_n(\varphi_n,s) - G_n(\eta,s) \|_{{\mathbb{C}}^{m\tilde mes m}} \nonumber \\
&\le \|
\eta (s)- \varphi_n (s)\|_{{\mathbb{C}}^{m\tilde mes m}} \bibitemgg\{ 2 +
\fracrac{|z_n|^{-1}}{2}(3+\kappa)\sum_{j,k=1}^2\| \widetilde
B_{j,k}(s) \|_{{\mathbb{C}}^{m\tilde mes m}} \bibitemgg\}.
\end{align}
Of course, by \eqref{3.33} as $n\to \infty$,
\begin{equation}\label{3.48}
\phi_n(0) - \phi_+(\delta) = o(1).
\end{equation}
Thus, by \eqref{3.42}, \eqref{3.47} and \eqref{3.48},
one concludes for $t\in [0,T']$ and as $n\to \infty$, that
\begin{align}\label{3.49}
&\|\varphi_n(t)-\eta(t) \|_{{\mathbb{C}}^{m\tilde mes m}} \le o(1) \nonumber \\
&+ \int_0^t \|\varphi_n(s)-\eta(s) \|_{{\mathbb{C}}^{m\tilde mes m}} \bibitemgg\{ 2 +
\fracrac{|z_n|^{-1}}{2}(3+\kappa)\sum_{j,k=1}^2\| \widetilde
B_{j,k}(s) \|_{{\mathbb{C}}^{m\tilde mes m}} \bibitemgg\}ds.
\end{align}
Gronwall's inequality applied to \eqref{3.49}
together with a consideration of the effect of the variable
change \eqref{3.29}, as illustrated in \eqref{3.42}, yields
\begin{equation}\label{3.50}
\varphi_n(t)-\eta(t)\to 0 \,\text{ as } \, n\to \infty
\end{equation}
uniformly for $t\in [0,T']$. Thus by \eqref{3.30c},
the contradiction results that for all $t\in [0,T']$,
$\eta$ satisfies \eqref{3.36}.
\end{proof}
What solutions of \eqref{3.35} satisfy \eqref{3.36}?
\begin{lemma} Assume Hypothesis~\ref{h2.4}.
If $\eta_\pm$ is a solution of \eqref{3.35a}
which satisfies \eqref{3.36}, then
\begin{equation}\label{3.51}
0=\eta_\pm (t) ,\quad t\in [0, \pm\infty).
\end{equation}
\end{lemma}
\begin{proof}
We note that \eqref{3.35a} is equivalent to \eqref{3.30a}
with $\widetilde B=0$. By the variable change \eqref{3.29},
\eqref{3.35a} is also equivalent to \eqref{3.25a} with
$B=0$. Next, we recall the connection between the Riccati-type
equations \eqref{3.25a}, and \eqref{3.17a} by means of the
Cayley transformation \eqref{3.27}. Solution matrices of
\eqref{3.35a} which statisfy \eqref{3.36} at $t=0$ thus
correspond to solution matrices, $V(z,\cdot)$, of \eqref{3.17a} for
which $\text{\rm Im} (V(z,x_0))\ge 0$. Moreover, solutions of \eqref{3.17a}
for which for which $\text{\rm Im} (V(z,x_0))\ge 0$ are obtainable from
solutions of \eqref{HSa}, with $B=0$, by means of
\eqref{2.14} with $\text{\rm Im} (M)\ge 0$. Thus, by utilizing this
connection between explicit exponential solutions of
\eqref{HSa} with $B=0$ and solutions of the Riccati-type
equation \eqref{3.17a}, and by performing on the resulting
solution of \eqref{3.17a} the conformal mapping \eqref{3.27}
followed by the variable transformation \eqref{3.29}, one
obtains the following solution for \eqref{3.30a},
\begin{equation}\label{3.52}
\varphi (z,t)= (iI_m \mp M)(iI_m \pm M)^{-1}
\exp(\mp 2ite^{i\delta}),
\end{equation}
for $\pm t\ge 0$, $\text{\rm Im} (\pm M)\ge 0$, and $z\in {\mathbb{C}}_+$.
By hypothesis, $0<\delta<\pi$. Thus the exponential
term in \eqref{3.52} will result in
\begin{equation}
|| \varphi (z,t) ||_{{\mathbb{C}}^{m\tilde mes m}} >1 \ \text{ as } t\to \pm\infty
\end{equation}
unless
\begin{equation}
M = \pm i I_m,
\end{equation}
thus implying \eqref{3.51}.
\end{proof}
\nonumberindent One then obtains the following result.
\begin{corollary}\label{c3.10}
With $\phi_\pm(\delta) $ defined in \eqref{3.33},
$\eta_\pm(0)=\phi_\pm(\delta)=0$.
\end{corollary}
For $M(z_n,c,x_0,\alpha_0)\in{\mathcal D}(z_n,c,x_0,\alpha_0)$,
it follows by \eqref{3.30b}, \eqref{3.33}, and
Corollary~\ref{c3.10} that
\begin{equation}\label{3.55}
[iI_m \mp M(z_n,c,x_0,\alpha_0)][iI_m
\pm M(z_n,c,x_0,\alpha_0)]^{-1} = o(1), \qquad\pm (c-x_0)>0,
\end{equation}
as $n\to \infty$. Hence one infers,
for elements of ${\mathcal D}(z_n,c,x_0,\alpha_0)$, that
\begin{equation}\label{3.56}
M(z_n,c,x_0,\alpha_0)= \pm iI_m + o(1), \qquad \pm (c-x_0)>0,
\end{equation}
as $|z|\to \infty$ in $C_\varepsilon$. This proves \eqref{3.1}.
Actually, \eqref{3.56} is a statement for all elements of
${\mathcal D}(z,c,x_0,\alpha_0)$ including the particular element
$M_{\pm}(z,x_0,\alpha_0)$, for $\pm (c-x_0)>0$.
In \eqref{3.1} an asymptotic expansion is given that
is uniform with respect to $\arg(z)$ for $|z| \to \infty$ in
$C_\varepsilon$. We now vary the reference point, $x_0$, and
observe that the asymptotic expansion in \eqref{3.1} is also
uniform with respect to $x_0$ whenever $x_0$ is confined to a
compact subset of ${\mathbb{R}}$.
\begin{theorem} \label{t3.12} Assume Hypothesis~\ref{h2.4}.
Let $\alpha_0 =(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$, and denote by
$C_\varepsilon\subset {\mathbb{C}}_+$ the open sector with vertex at zero,
symmetry axis along the positve imaginary axis and opening
angle $\varepsilon$, with $0<\varepsilon<\pi/2$. Let
$M_\pm(z,x_0,\alpha_0)$ be the unique elements of the limit
disks ${\mathcal D}_\pm (z,x_0,\alpha_0)$ for the Dirac system given by
\eqref{HS} and \eqref{DS}. Then,
\begin{equation}\label{3.57}
M_\pm(z,x,\alpha_0) \underset{\substack{|z|
\to\infty\\z\in C_\varepsilon}}{=} \pm iI_m +o(1)
\end{equation}
uniformly with respect to $\arg(z)$, for $|z|
\to \infty$ in $C_\varepsilon$, and uniformly with respect to
$x$, as long as $x$ varies in compact subsets of $[x_0,\pm\infty)$.
\end{theorem}
\begin{proof}
We note that the system \eqref{3.35} is independent of
the reference point $x_0$. Next, we recall that $\delta$, defined in
\eqref{3.32} is determined by an apriori choice of the
sequence $z_n$, subject only to $z_n$ being in $C_\varepsilon$
(c.f.~\eqref{3.31}). Moreover, we note that $\varphi_\pm(\delta)$,
defined as a limit in \eqref{3.33}, described explicity in
Corollary~\ref{c3.10}, and which gives solutions of
\eqref{3.35} satisfying \eqref{3.36} for $t\in [0, \pm\infty)$,
is also independent of the reference point
$x_0$. Thus, had we chosen a different point of reference,
$x_0'\ne x_0$, at the start, the asymptotic analysis begun in
Theorem~\ref{t3.8} and continued through \eqref{3.55}, would
remain the same after the variable change in \eqref{3.29},
except for the integral expression in \eqref{3.42} in which
$x_0$ would be replaced by $x_0'$. However, given the local
integrability assumption on $B$ in Hypothesis~\ref{h2.1},
one concludes that this integral expression is uniformly continuous
with respect to $x_0$ whenever $x_0$ is confined to a compact
subset of ${\mathbb{R}}$. Thus \eqref{3.42}, and consequently \eqref{3.50}, are
uniform with respect to $t$ and with respect to $x_0$ whenever both are
confined to compact subsets of ${\mathbb{R}}$. Consequently, \eqref{3.55}
holds for elements ${\mathcal D}(z,c,x_0,\alpha_0)$, that this asymptotic
expansion is uniform with respect to $\arg (z)$ for $|z|\to \infty$ in
$C_\varepsilon$, and that it is uniform with respect to $x_0$ when $x_0$
is confined to compact subsets of ${\mathbb{R}}$.
\end{proof}
\begin{remark}\label{r3.2}
(i) In the special case $m=1$, the leading-order asymptotics \eqref{3.57}
was published by Everitt, Hinton, and Shaw \cite{EHS83} in 1983.
For asymptotic estimates of Weyl solutions in the case
$m=1$ we refer to \cite{Mi91}. \\
(ii) A comparison of \eqref{3.57} with
\eqref{2.41} then proves that the leading-order asymptotic behavior
\eqref{3.57} is in fact independent of the boundary condition at $x_0$
indexed by
$\alpha$, that is,
\begin{equation}\label{3.58}
M_\pm(z,x_0,\alpha) \underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} \pm iI_{m} +o(1)
\end{equation}
for any $\alpha$ satisfying the conditions stated in
\eqref{2.8e}. In the scalar case $m=1$ this fact had been
noticed in \cite{EHS83}. This boundary condition independence of
the leading-order asymptotic behavior of $M_\pm(z,x_0,\alpha)$ is in
sharp contrast to the case of matrix-valued Schr\"odinger operators
(see, e.g., \cite{CG99}). Moreover, regarding the conclusion of
Theorem~\ref{t3.12}, no generality is lost by assuming that
$C_\varepsilon \subset {\mathbb{C}}_+$ because of \eqref{2.38}.
\end{remark}
\section{Higher Order Terms in the Asymptotic Expansion of
$M_\pm(z,x,\alpha)$} \label{s4}
In this section we shall prove one of our principal
results of this paper, the asymptotic high-energy expansion of
$M_+(z,x,\alpha_0)$ to arbitrarily high orders in sectors of the type
$C_\varepsilon\subset{\mathbb{C}}_+$ as defined in Theorem~\ref{t3.12}.
Throughout this section we choose $z\in{\mathbb{C}}_+$. We also recall the
following notion: $x\in [a,b)$ (resp.,
$x\in (a,b]$) is called a right (resp., left) Lebesgue point of an
element $q\in L^1 ((a,b))$, $a<b$ if$\int_0^\varepsilon dx^\prime \,
|q(x+x^\prime)-q(x)|=o (\varepsilon)$ (resp., $\int_0^\varepsilon
dx^\prime \, |q(x-x^\prime)-q(x)|=o (\varepsilon)$) as
$\varepsilon\downarrow 0$. Similarly, $x\in (a,b)$ is called a Lebesgue
point of $q\in L^1 ((a,b))$ if $\int_{-\varepsilon}^\varepsilon dx^\prime
\, |q(x+x^\prime)-q(x)|=o (\varepsilon)$ as $\varepsilon\downarrow 0$.
The set of all such points is then denoted the right (resp., left)
Lebesgue set of $q$ on $[a,b]$ in the former case and simply the Lebesgue
set of $q$ on $[a,b]$ in the latter case. The
analogous notions are applied to $2m\tilde mes 2m$ matrices $B\in L^1
((a,b))^{2m\tilde mes 2m}$ by simultaneously considering all $4m^2$ entries of
$B$. The right (resp., left) Lebesgue set of $B$ on $[a,b]$ is then
simply the intersection of the right (resp., left) Lebesgue sets of
$B_{j,k}$ for all $1\leq j,k\leq 2m$, and similarly for the Lebesgue
set of $B$, etc.
Finally, we need one more ingredient, recently proven by Rybkin
\cite[Lemma~3]{Ry99} using appropriate maximal functions. Let $q\in L^1
((x_0,\infty))$, $\text{\rm{supp}}(q)\subseteq [x_0,x_0+R]$ for some $R>0$, and
suppose $x\in [x_0,x_0+R]$ is a right Lebesgue point of $q$. Then
\begin{equation}
\int_x^{x_0+R} dx^\prime \, q(x^\prime)\exp(2iz(x^\prime -x))
\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=}-\frac{q(x)}{2iz} + o\bibitemg(|z|^{-1}\bibitemg). \label{4.-2}
\end{equation}
An alternative proof of \eqref{4.-2} follows from
\cite[Theorem~I.13]{Ti86}, which implies
\begin{equation}
\lim_{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}z^{-1} \int_x^{x_0+R} dx^\prime \,
|q(x^\prime) -q(x)|\exp(2iz(x^\prime -x)) = 0 \label{4.-1}
\end{equation}
for any right Lebesgue point $x$ of $q$.
We start with the simpler case where $B$ has compact support contained in
some interval $[x_0,y_0]$. Below in \eqref{4.0} and in
analogous formulas in this section, $\|\cdot\|_{{\mathbb{C}}^{\ell\tilde mes \ell}}$
denotes a norm in ${\mathbb{C}}^{\ell\tilde mes \ell}$.
\begin{lemma}\label{l4.1}
Fix $x_0, y_0\in{\mathbb{R}}$ with $y_0>x_0$ and let $x\geq x_0$. Suppose
$A=I_{2m}$, $B\in L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$,
$B=B^*$ a.e.~on $(x_0,\infty)$. In addition, assume that $B$ has compact
support contained in $[x_0,y_0]$, that
$B^{(N-1)}\in L^1([x_0,y_0])^{2m\tilde mes 2m}$ for some $N\in{\mathbb{N}}$, that $x$
is a right Lebesgue point of $B^{(N-1)}$, and that
\begin{align}
&\underset{y\in[x_0,y_0]}{\text{\rm{ess}}sup} \, \bibitemgg\|\int_y^{y_0}
dx'\,B^{(N-1)}(x')\exp(2iz(x'-y))
+\frac{1}{2iz}B^{(N-1)}(y)\bibitemgg\|_{{\mathbb{C}}^{2m\tilde mes 2m}} \nonumber \\
& \underset{\substack{\abs{z}
\to\infty\\ z\in C_\varepsilon}}{=}o\bibitemg(|z|^{-1}\bibitemg). \label{4.0}
\end{align}
If $N=1$, suppose in addition $B_{k,k'}B_{\ell,\ell'}\in
L^1([x_0,y_0])^{m\tilde mes m}$ for all $k,k',\ell,\ell'\in\{1,2\}$. Let
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$ and
denote by $M_+(z,x,\alpha_0)$, $x\geq x_0$, the unique
Weyl-Titchmarsh matrix associated with the half-line
Dirac-type operator $D_+(\alpha_0)$ in \eqref{2.84}. Then, as
$\abs{z}\to\infty$ in $C_\varepsilon$, $M_+(z,x,\alpha_0)$ has
an asymptotic expansion of the form
\begin{equation}
M_+(z,x,\alpha_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=}
i I_m +\sum_{k=1}^N m_{+,k}(x,\alpha_0)z^{-k}+
o\bibitemg(|z|^{-N}\bibitemg), \quad N\in{\mathbb{N}}. \label{4.1}
\end{equation}
The expansion \eqref{4.1} is uniform with respect to $\arg\,(z)$ for $|z|
\to \infty$ in
$C_\varepsilon$ and uniform in $x$ as long as $x$ varies in compact
subintervals of $[x_0,\infty)$ intersected with the right Lebesgue set
of $B^{(N-1)}$. The expansion coefficients $m_{+,k}(x,\alpha_0)$ can be
recursively computed from
\begin{align}
m_{+,1}(x,\alpha_0)&=-\frac{1}{2} \bibitemg(
B_{1,2}(x)+B_{2,1}(x)\bibitemg) +\frac{i}{2} \bibitemg(
B_{1,1}(x)-B_{2,2}(x)\bibitemg), \nonumber \\
m_{+,k+1}(x,\alpha_0)&=\frac{i}2\bibitemgg(m_{+,k}^\prime(x,\alpha_0)+
\sum_{\ell=1}^{k} m_{+,\ell}(x,\alpha_0)
m_{+,k+1-\ell}(x,\alpha_0) \nonumber \\
& \qquad \quad +\sum_{\ell=0}^{k} m_{+,\ell}(x,\alpha_0)
B_{2,2}(x) m_{+,k-\ell}(x,\alpha_0) \label{4.2} \\
& \qquad \quad + B_{1,2}(x) m_{+,k}(x,\alpha_0)
+ m_{+,k}(x,\alpha_0) B_{2,1}(x)\bibitemgg), \nonumber \\
& \hspace*{5.7cm} 1 \leq k\leq N-1. \nonumber
\end{align}
\end{lemma}
\begin{proof}
In the following let $z\in{\mathbb{C}}_+$, and $x\geq x_0$.
The existence of an expansion of the type \eqref{4.1} is shown as
follows. First one considers a matrix Volterra integral equation of the
type
\begin{equation}
\widetilde U_+(z,x,\alpha_0)=\begin{pmatrix}I_m\\
iI_m\end{pmatrix}\exp(iz(x-x_0)) +\int_x^\infty dx'\, K(z,x,x') J
B(x')\widetilde U_+(z,x',\alpha_0), \label{4.3}
\end{equation}
where
\begin{equation}
\widetilde U_+(z,x,\alpha_0)=\begin{pmatrix} \widetilde
u_{+,1}(z,x,\alpha_0)\\
\widetilde u_{+,2}(z,x,\alpha_0)\end{pmatrix}\in
L^2([x_0,\infty))^{2m\tilde mes m},
\label{4.3A}
\end{equation}
and $K$ abbreviates the $2m\tilde mes 2m$ Volterra Green's kernel
\begin{equation}
K(z,x,x')=\begin{pmatrix} \cos(z(x-x'))I_m &\sin(z(x-x'))I_m \\
-\sin(z(x-x'))I_m &\cos(z(x-x'))I_m \end{pmatrix}. \label{4.3B}
\end{equation}
Clearly, $\widetilde U_+(z,\cdot,\alpha_0)$ solves the
Dirac-type system \eqref{HS} and \eqref{DS}. In addition, it satisfies
$\widetilde U_+(z,\cdot,\alpha_0)\in L^2 ([x_0,\infty))^{2m\tilde mes 2m}$.
Thus, up to normalization, $\widetilde U_+(z,\cdot,\alpha_0)$ represents
the Weyl solution associated with $B$ on the half-line
$[x_0,\infty)$. Next, introducing
\begin{equation}
\widetilde V_+(z,x,\alpha_0)=\begin{pmatrix} \widetilde
v_{+,1}(z,x,\alpha_0)\\
\widetilde v_{+,2}(z,x,\alpha_0) \end{pmatrix} =\widetilde
U_+(z,x,\alpha_0)\exp(-iz(x-x_0)), \label{4.3a}
\end{equation}
one rewrites \eqref{4.3} in the form
\begin{equation}
\widetilde V_+(z,x,\alpha_0)=\begin{pmatrix}I_m\\ iI_m\end{pmatrix}
+\int_x^{y_0} dx'\, \widetilde K(z,x,x') J B(x')\widetilde
V_+(z,x',\alpha_0), \label{4.3b}
\end{equation}
where
\begin{equation}
\widetilde K(z,x,x')=\frac{1}{2}\begin{pmatrix} (1+\exp(2iz(x'-x)))I_m
&-i(1-\exp(2iz(x'-x)))I_m \\[1mm]
i(1-\exp(2iz(x'-x)))I_m &(1+\exp(2iz(x'-x)))I_m \end{pmatrix}. \label{4.3C}
\end{equation}
Thus, one infers,
\begin{equation}
M_+(z,x,\alpha_0)=\widetilde u_{+,2}(z,x,\alpha_0)\widetilde
u_{+,1}(z,x,\alpha_0)^{-1} =\widetilde v_{+,2}(z,x,\alpha_0)\widetilde
v_{+,1}(z,x,\alpha_0)^{-1}.
\label{4.4}
\end{equation}
Introducing
\begin{equation}
R=\begin{pmatrix} C_1 & -iC_2 \\ iC_1 & C_2
\end{pmatrix}, \quad S=\begin{pmatrix} D_1 & iD_2 \\
-iD_1 & D_2 \end{pmatrix}, \label{4.4a}
\end{equation}
where
\begin{align}
C_1&=-B_{1,2}^* -iB_{1,1}, \quad C_2=
B_{1,2}-iB_{2,2}, \label{4.4b} \\
D_1&=-B_{1,2}^*+iB_{1,1}, \quad D_2=
B_{1,2}+iB_{2,2}, \label{4.4c}
\end{align}
\eqref{4.3b} results in
\begin{align}
\widetilde V_+(z,x,\alpha_0)&=\begin{pmatrix}I_m\\ iI_m\end{pmatrix}
+\int_x^{y_0} dx'\, \bibitemg(R(x')+S(x')\exp(2iz(x'-x))
\bibitemg) \widetilde V_+(z,x',\alpha_0) \label{4.4d} \\
&=\bibitemgg(I_{2m}+\sum_{k=1}^\infty 2^{-k}\int_x^{y_0} dx_1 \,
\bibitemg(R(x_1)+S(x_1)e^{2iz(x_1-x)} \bibitemg) \tilde mes \nonumber \\
& \hspace*{2.5cm} \tilde mes \int_{x_1}^{y_0} dx_2
\, \bibitemg(R(x_2)+S(x_2)e^{2iz(x_2-x_1)} \bibitemg)\dots \label{4.4e} \\
& \hspace*{2.3cm} \dots \int_{x_{k-1}}^{y_0} dx_k
\, \bibitemg(R(x_k)+S(x_k)e^{2iz(x_k-x_{k-1})} \bibitemg)
\bibitemgg)\begin{pmatrix}I_m\\ iI_m\end{pmatrix}. \nonumber
\end{align}
This yields
\begin{equation}
\|\widetilde v_{+,j}(z,x,\alpha_0) \| \leq C_j, \quad z\in{\mathbb{C}}_+, \;
\text{\rm Im}(z)> 0, \; x\geq x_0, \; j=1,2 \label{4.4ca}
\end{equation}
for some $C_j>0$, $j=1,2$, depending on $\|B\|_1$. Integrating
by parts in \eqref{4.4e}, repeatedly applying \eqref{4.-2} and \eqref{4.0}
to $q(x)=(S(x))_{j,k}$ for all $1\leq j,k\leq 2m$ then results in the
existence of an asymptotic expansion for $\widetilde V_+(z,x,\alpha_0)$
of the type
\begin{equation}
\widetilde V_+(z,x,\alpha_0)=\begin{pmatrix} \widetilde v_{+,1}(z,x,\alpha_0)\\
\widetilde v_{+,2}(z,x,\alpha_0) \end{pmatrix}=\sum_{k=0}^{N} \widetilde
V_{+,k}(x,\alpha_0)\, z^{-k} +o\bibitemg(|z|^{-N}\bibitemg). \label{4.3f}
\end{equation}
Inserting the expansions for $\widetilde v_{+,2}(z,x,\alpha_0)$ and
$\widetilde v_{+,1}(z,x,\alpha_0)^{-1}$ into \eqref{4.4} (using a geometric
series expansion for $\widetilde v_{+,1}(z,x,\alpha_0)^{-1}$) then yields the
existence of an expansion of the type \eqref{4.1} for
$M_+(z,x,\alpha_0)$. The actual expansion coefficients and the
associated recursion relation \eqref{4.2} then follow upon inserting
expansion \eqref{4.1} into the Riccati-type equation \eqref{3.17a}. The
stated uniformity assertions concerning the asymptotic expansion
\eqref{4.1} then follow from iterating the system of Volterra integral
integral equations
\eqref{4.3b}.
\end{proof}
\begin{remark} \label{r4.2}
The analogous solution $\widetilde U_-(z,\cdot,\alpha_0)$ of the Dirac-type
operator \eqref{2.61} on the interval $(-\infty,x_0]$ satisfies
\begin{align}
\widetilde U_-(z,x,\alpha_0)&=\begin{pmatrix}I_m\\
-iI_m\end{pmatrix}\exp(-iz(x-x_0)) \nonumber \\
& \quad -\int_{-\infty}^x dx'\, K(z,x,x') J
\widetilde B(x')\widetilde U_-(z,x',\alpha_0), \label{4.14A}
\end{align}
with integral kernel $K$ given by \eqref{4.3B}. (Again $\widetilde U_-$
coincides with the Weyl solution $U_-$ up to normalization.) A closer look
at the system of Volterra integral equations
\eqref{4.3},
\eqref{4.4d},
\eqref{4.4e}, and similarly in connection with \eqref{4.14A}, then
reveals that $\widetilde U_\pm(z,\cdot,\alpha_0)$ have the asymptotic behavior
\begin{equation}
\widetilde U_\pm (z,x,\alpha_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} \left(\sum_{k=0}^N \begin{pmatrix} \widetilde
v_{\pm,k,1}(x,\alpha_0) \\ \widetilde v_{\pm,k,2}(x,\alpha_0)
\end{pmatrix}z^{-k} +o\bibitemg(|z|^{-N}\bibitemg)\right)\exp(\pm iz(x-x_0)),
\label{4.14B}
\end{equation}
with leading asymptotics determined as follows.
\begin{align}
\begin{split} \label{4.14C}
\widetilde v_{\pm,0,1}(x,\alpha_0)&=I_m+\widetilde w_{\pm,0,1}(x,\alpha_0), \\
\widetilde v_{\pm,0,2}(x,\alpha_0)&=\pm i\bibitemg(I_m+\widetilde
w_{\pm,0,1}(x,\alpha_0)\bibitemg),
\end{split}
\end{align}
where $\widetilde w_{\pm,0,1}(x,\alpha_0)$ satisfies
\begin{equation}
\widetilde w_{\pm,0,1}'(x,\alpha_0)=\frac{1}{2}\bibitemg[\widetilde B_{2,1}(x) -\widetilde
B_{1,2}(x) \pm i\widetilde B_{2,2}(x) \pm i\widetilde B_{1,1}(x)\bibitemg]\bibitemg(I_m+\widetilde
w_{\pm,0,1}(x,\alpha_0)\bibitemg), \label{4.14D}
\end{equation}
and
\begin{equation}
\lim_{x\to\pm\infty} \widetilde w_{\pm,0,1}(x,\alpha_0)=0
\label{4.14F}
\end{equation}
(in fact, $\widetilde v_{\pm,0,1}(\cdot,\alpha_0)=I_m$, $\widetilde
v_{\pm,0,2}(\cdot,\alpha_0)=\pm iI_m$, and $\widetilde
v_{\pm,k,j}(\cdot,\alpha_0)=0$, $j=1,2$, $1\leq k\leq N$ outside the
support of $\widetilde B$). In particular,
\begin{equation}
\widetilde w_{\pm,0,1}(x,\alpha_0)=0 \label{4.14G}
\end{equation}
and hence
\begin{equation}
\widetilde U_\pm (z,x,\alpha_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} \left(\begin{pmatrix} I_m \\ \pm iI_m
\end{pmatrix} +o(1)\right)\exp(\pm iz(x-x_0)), \label{4.14H}
\end{equation}
if and only if $\widetilde B$ is in the normal form
\begin{equation}
\widetilde B(x)=\begin{pmatrix} \widetilde B_{1,1}(x) & \widetilde B_{1,2}(x)\\
\widetilde B_{1,2}(x) &-\widetilde B_{1,1}(x)\end{pmatrix}, \quad
\widetilde B_{1,1}^*(x)=\widetilde B_{1,1}(x),
\; \widetilde B_{1,2}^*(x)=\widetilde B_{1,2}(x) \text{ a.e.} \label{4.14I}
\end{equation}
\end{remark}
For more details we refer to Lemma~\ref{l4.9}.
Next we recall an elementary result on finite-dimensional
evolution equations essentially taken from \cite{MPS90} (cf.~also
\cite[Lemma~4.2]{CG99}).
\begin{lemma} \mbox{\rm (\cite{MPS90}.)}
\label{l4.2} Let $\Gamma_j\in L^1_{\text{\rm{loc}}}({\mathbb{R}})^{m\tilde mes m}$, $j=1,2$. Then
any
$m\tilde mes m$ matrix-valued solution $X$ of
\begin{equation}
X'(x)=\Gamma_1(x)X(x)+X(x)\Gamma_2(x) \text{ for ~a.e. } x\in{\mathbb{R}},
\label{4.5}
\end{equation}
is of the type
\begin{equation}
X(x)=Y(x)CZ(x), \label{4.6}
\end{equation}
where $C$ is a constant $m\tilde mes m$ matrix and $Y$ is a
fundamental system of solutions of
\begin{equation}
\Psi'(x)=\Gamma_1(x)\Psi(x) \label{4.7}
\end{equation}
and $Z$ is a fundamental system of solutions of
\begin{equation}
\Phi'(x)=\Phi(x)\Gamma_2(x). \label{4.8}
\end{equation}
\end{lemma}
The next result provides the proper extension of Lemma~4.3 in
\cite{CG99} in the context of matrix-valued Schr\"odinger operators
(which in turn extended Proposition~2.1 in the scalar context in
\cite{GS98} to the matrix-valued case) to the Dirac-type case under
consideration.
\begin{lemma} \label{l4.3}
Fix $x_0,y_0\in{\mathbb{R}}$ with $y_0>x_0$. Suppose
$A_j=I_{2m}$, $B_j\in L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$,
$B_j=B_j^*$ a.e.~on $[x_0,\infty)$, $j=1,2$, and $B_1=B_2$ a.e.~on
$[x_0,y_0]$. Let $\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$ and denote by
$M_{j,+}(z,x,\alpha_0)$, $x\geq x_0$, the unique Weyl-Titchmarsh matrix
corresponding to the half-line Dirac operators $D_{+,j}(\alpha_0)$,
$j=1,2$, in \eqref{2.84}. Then,
\begin{align}
&[M_{1,+}'(z,x,\alpha_0)-M_{2,+}'(z,x,\alpha_0)] \nonumber \\
&=-(z/2)[M_{1,+}(z,x,\alpha_0)+M_{2,+}(z,x,\alpha_0)]
[M_{1,+}(z,x,\alpha_0)-M_{2,+}(z,x,\alpha_0)] \nonumber \\
& \quad -(z/2)[M_{1,+}(z,x,\alpha_0)-M_{2,+}(z,x,\alpha_0)]
[M_{1,+}(z,x,\alpha_0) +M_{2,+}(z,x,\alpha_0)] \nonumber \\
& \quad
-[M_{1,+}(z,x,\alpha_0)+M_{2,+}(z,x,\alpha_0)]B_{2,2}(x)
[M_{1,+}(z,x,\alpha_0)-M_{2,+}(z,x,\alpha_0)]/2 \nonumber \\
& \quad
-[M_{1,+}(z,x,\alpha_0)-M_{2,+}(z,x,\alpha_0)]B_{2,2}(x)
[M_{1,+}(z,x,\alpha_0)+M_{2,+}(z,x,\alpha_0)]/2 \nonumber \\
& \quad -B_{1,2}(x)[M_{1,+}(z,x,\alpha_0)-M_{2,+}(z,x,\alpha_0)]
\nonumber \\
& \quad -[M_{1,+}(z,x,\alpha_0)-M_{2,+}(z,x,\alpha_0)]B_{2,1}(x)
\; \text{ for~a.e. $x\in [x_0,y_0]$,} \label{4.14}
\end{align}
where we denoted $B_1=B_2=\left(\begin{smallmatrix}B_{1,1}
&B_{1,2}\\ B_{2,1} &B_{2,2} \end{smallmatrix}\right)$ a.e.~on $(x_0,y_0)$.
\end{lemma}
\begin{proof}
This is obvious from \eqref{3.17a}.
\end{proof}
\begin{lemma} \label{l4.4}
Fix $x_0,y_0\in{\mathbb{R}}$ with $y_0>x_0$. Suppose
$A_j=I_{2m}$, $B_j\in L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all
$R>0$, and $B_j=B_j^*$ a.e.~on $[x_0,\infty)$, $j=1,2$. Let
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$ and denote by
$M_{j,+}(z,x,\alpha_0)$, $x\geq x_0$, the unique Weyl-Titchmarsh matrix
corresponding to the half-line Dirac operators $D_{+,j}(\alpha_0)$,
$j=1,2$, in \eqref{2.84}. Define
\begin{align}
\Gamma_1(z,x)&=-(z/2)[M_{1,+}(z,x,\alpha_0)
+M_{2,+}(z,x,\alpha_0)] \nonumber \\
& \quad -(1/2)[M_{1,+}(z,x,\alpha_0)
+M_{2,+}(z,x,\alpha_0)]B_{2,2}(x)-B_{1,2}(x), \label{4.15} \\
\Gamma_2(z,x)&=-(z/2)[M_{1,+}(z,x,\alpha_0)
+M_{2,+}(z,x,\alpha_0)] \nonumber \\
& \quad -(1/2)B_{2,2}(x)[M_{1,+}(z,x,\alpha_0)
+M_{2,+}(z,x,\alpha_0)]-B_{2,1}(x), \label{4.15a}
\end{align}
for a.e.~$x\in [x_0,y_0]$. In addition, assume $Y_+(z,\cdot)$
and $Z_+(z,\cdot)$ to be fundamental matrix solutions of
\begin{equation}
\Psi'(z,x)=\Gamma_1(z,x)\Psi(z,x) \text{ and }
\Phi'(z,x)=\Phi(z,x)\Gamma_2(z,x) \label{4.16}
\end{equation}
on $[x_0,y_0]$, respectively, with
\begin{equation}
Y_+(z,y_0)=I_m, \quad Z_+(z,y_0)=I_m. \label{4.17}
\end{equation}
Then, as $|z|\to\infty$, $z\in C_\varepsilon$,
\begin{equation}
\|Y_+(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes m}}, \|Z_+(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes m}}
\leq \exp(-\text{\rm Im}(z)(y_0-x_0)(1+o (1))). \label{4.18}
\end{equation}
\end{lemma}
\begin{proof}
Define $\widetilde\Gamma_j(z,x)$, $j=1,2$, by
\begin{equation}
\widetilde\Gamma_j (z,x)=\Gamma_j (z,x)+izI_m, \quad j=1,2, \label{4.19}
\end{equation}
then
\begin{equation}
\int_{x_0}^{y_0} dx\, \|\widetilde\Gamma_j (z,x)\|_{{\mathbb{C}}^{m\tilde mes m}}
\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} o (z), \quad j=1,2 \label{4.20}
\end{equation}
due to the uniform nature of the asymptotic expansion
\eqref{3.57} for $x$ varying in compact intervals. Next,
introduce
\begin{equation}
E_+(z,x,y_0)=I_m\exp(iz(y_0-x)), \quad x\leq y_0,
\label{4.21}
\end{equation}
then
\begin{align}
Y_+(z,x)&=E_+(z,x,y_0)-\int_x^{y_0}dx'\,E_+(z,x,x')
\widetilde\Gamma_1(z,x')Y_+(z,x'), \label{4.22} \\
Z_+(z,x)&=E_+(z,x,y_0)-\int_x^{y_0}dx'\,Z_+(z,x')
\widetilde\Gamma_2(z,x')E_+(z,x,x'). \label{4.23}
\end{align}
Using
\begin{equation}
\|E_+(z,x_0,y_0)\|_{{\mathbb{C}}^{m\tilde mes m}}\leq\exp(-\text{\rm Im}(z)(y_0-x_0)),
\label{4.24}
\end{equation}
a standard Volterra-type iteration argument in
\eqref{4.22}, \eqref{4.23} then yields
\begin{align}
\|Y_+(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes m}}& \leq
\exp\left(-\text{\rm Im}(z)(y_0-x_0)+\int_{x_0}^{y_0}dx
\,\|\widetilde\Gamma_1(z,x)\|\right), \label{4.25} \\
\|Z_+(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes m}}& \leq
\exp\left(-\text{\rm Im}(z)(y_0-x_0)+\int_{x_0}^{y_0}dx
\,\|\widetilde\Gamma_2(z,x)\|\right), \label{4.25a}
\end{align}
and hence \eqref{4.18}.
\end{proof}
\begin{theorem} \label{t4.5}
Fix $x_0,y_0\in{\mathbb{R}}$ with $y_0>x_0$. Suppose
$A_j=I_{2m}$, $B_j\in L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$,
$B_j=B_j^*$ a.e.~on $[x_0,\infty)$, $j=1,2$, and $B_1=B_2$ a.e.~on
$[x_0,y_0]$. Let
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$ and denote by
$M_{j,+}(z,x,\alpha_0)$, $x\geq x_0$, the unique Weyl-Titchmarsh matrix
corresponding to the half-line Dirac operators $D_{+,j}(\alpha_0)$,
$j=1,2$, in \eqref{2.84}. Then, as $\abs{z}\to\infty$ in $C_\varepsilon$,
\begin{equation}
\|M_{1,+}(z,x_0,\alpha_0)-M_{2,+}(z,x_0,\alpha_0)\|_{{\mathbb{C}}^{m\tilde mes m}}
\leq C\exp(-2\text{\rm Im}(z)(y_0-x_0)(1+o (1))) \label{4.26}
\end{equation}
for some constant $C>0$.
\end{theorem}
\begin{proof}
Define for $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, $x\in [x_0,y_0]$,
\begin{equation}
X_+(z,x)=M_{1,+}(z,x,\alpha_0)-M_{2,+}(z,x,\alpha_0), \label{4.26a}
\end{equation}
and for $z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$ and a.e.~$x\in [x_0,y_0]$,
\begin{align}
\Gamma_1(z,x)&=-(z/2)[M_{1,+}(z,x_0,\alpha_0)+M_{2,+}(z,x_0,\alpha_0)]
\nonumber \\
& \quad -(1/2)[M_{1,+}(z,x_0,\alpha_0)+M_{2,+}(z,x_0,\alpha_0)]B_{2,2}(x)
-B_{1,2}(x), \label{4.26b} \\
\Gamma_2(z,x)&=-(z/2)[M_{1,+}(z,x_0,\alpha_0)+M_{2,+}(z,x_0,\alpha_0)]
\nonumber \\
& \quad -(1/2)B_{2,2}(x)[M_{1,+}(z,x_0,\alpha_0)+M_{2,+}(z,x_0,\alpha_0)]
-B_{2,1}(x). \label{4.26c}
\end{align}
By Lemma~\ref{l4.3},
\begin{equation}
X_+'=\Gamma_1X_++X_+\Gamma_2
\end{equation}
and hence by Lemma~\ref{l4.2},
\begin{equation}
X_+(z,x)=Y_+(z,x)X_+(z,x_1)Z_+(z,x), \label{4.30}
\end{equation}
where $Y_+(z,x)$ and $Z_+(z,x)$ are fundamental solution matrices of
\begin{equation}
\Psi'(z,x)=\Gamma_1(z,x)\Psi(z,x) \text{ and }
\Phi'(z,x)=\Phi(z,x)\Gamma_2(z,x),
\end{equation}
respectively, with
\begin{equation}
Y_+(z,y_0)=I_m, \quad Z_+(z,y_0)=I_m.
\end{equation}
By Lemma~\ref{l4.4},
\begin{equation}
\|Y_+(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes m}}, \|Z_+(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes m}}
\leq \exp(-\text{\rm Im}(z)(y_0-x_0))(1+o (1))) \label{4.33}
\end{equation}
as $|z|\to\infty$, $z\in C_\varepsilon$. Thus, as
$|z|\to\infty$, $z\in C_\varepsilon$,
\begin{align}
\|X_+(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes m}}&\leq
\|X_+(z,y_0)\|_{{\mathbb{C}}^{m\tilde mes m}}\,\|Y_+(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes
m}}\,\|Z_+(z,x_0)\|_{{\mathbb{C}}^{m\tilde mes m}} \nonumber \\
&\leq C\exp(-2\text{\rm Im}(z)(y_0-x_0)(1+o(1)))
\label{4.34}
\end{align}
for some constant $C>0$ by \eqref{3.57}, \eqref{4.30}, and
\eqref{4.33}.
\end{proof}
Given these preparations we can now drop the compact
support assumption on $B$ in Lemma~\ref{l4.1} and hence arrive at one
of the principal results of this paper.
\begin{theorem} \label{t4.6}
Fix $x_0,y_0\in{\mathbb{R}}$ with $y_0>x_0$ and suppose $A=I_{2m}$, $B\in
L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$, and $B=B^*$ a.e.~on
$(x_0,\infty)$. In addition, assume that for some $N\in{\mathbb{N}}$,
$B^{(N-1)}\in L^1([x_0,c])^{2m\tilde mes 2m}$ for all $c>x_0$, that $x_0$ is
a right Lebesgue point of $B^{(N-1)}$, and that
\begin{align}
&\underset{y\in [x_0,y_0]}{\text{\rm{ess}}sup} \, \bibitemgg\|\int_y^{y_0}
dx'\,B^{(N-1)}(x')\exp(2iz(x'-y))
+\frac{1}{2iz}B^{(N-1)}(y)\bibitemgg\|_{{\mathbb{C}}^{2m\tilde mes 2m}} \nonumber \\
&\underset{\substack{\abs{z}
\to\infty\\ z\in C_\varepsilon}}{=}o\bibitemg(|z|^{-1}\bibitemg). \label{4.34a}
\end{align}
If $N=1$, suppose in addition
$B_{k,k'}B_{\ell,\ell'}\in L^1([x_0,y_0])^{m\tilde mes m}$ for all
$k,k',\ell,\ell'\in\{1,2\}$. Let $\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes
2m}$ and denote by $M_+(z,x_0,\alpha_0)$ the unique element of the
limit disk ${\mathcal D}_+ (z,x_0,\alpha_0)$ for the half-line Dirac operator
$D_+(\alpha_0)$ in \eqref{2.84}. Then, as
$\abs{z}\to\infty$ in $C_\varepsilon$, $M_+(z,x_0,\alpha_0)$ has an
asymptotic expansion of the form
\begin{equation}
M_+(z,x_0,\alpha_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} i I_m +\sum_{k=1}^N
m_{+,k}(x_0,\alpha_0)z^{-k}+ o\bibitemg(|z|^{-N}\bibitemg), \quad N\in{\mathbb{N}}.
\label{4.35}
\end{equation}
The expansion \eqref{4.35} is uniform with respect to $\arg\,(z)$
for $|z|\to \infty$ in $C_\varepsilon$. The expansion coefficients
$m_{+,k}(x_0,\alpha_0)$ can be recursively computed from \eqref{4.2}.
\end{theorem}
\begin{proof}
Define
\begin{equation}
\widetilde B(x)=\begin{cases} B(x) &\text{ for }
x\in [x_0,y_0], \; x_0<y_0 \\
0 &\text{ otherwise} \label{4.36} \end{cases}
\end{equation}
and apply Theorem~\ref{t4.5} with $B_1=B$, $B_2=\widetilde
B$. Then (in obvious notation)
\begin{equation}
\|M_+(z,x_0,\alpha_0)-\widetilde M_+(z,x_0,\alpha_0)\|_{{\mathbb{C}}^{m\tilde mes
m}}\leq C \exp(-2\text{\rm Im}(z)(y_0-x_0)(1+o(1))) \label{4.36a}
\end{equation}
as $|z|\to\infty$, $z\in C_\varepsilon$, and hence the
asymptotic expansion \eqref{4.1} for $\widetilde M_+(z,x_0,\alpha_0)$
in Lemma~\ref{l4.1} coincides with that of $M_+(z,x_0,\alpha_0)$.
\end{proof}
In analogy to Theorem~\ref{t3.12}, the asymptotic expansion
\eqref{4.35} extends to one for $M_+(z,x,\alpha_0)$ valid uniformly
with respect to $x$ as long as $x$ varies in compact subintervals
of $[x_0,\infty)$ intersected with the right Lebesgue set of
$B^{(N-1)}$.
\begin{theorem} \label{t4.7}
Fix $x_0\in{\mathbb{R}}$ and let $x\geq x_0$. Suppose $A=I_{2m}$, $B\in
L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$, and $B=B^*$ a.e.~on
$(x_0,\infty)$. In addition, assume that for some $N\in{\mathbb{N}}$,
$B^{(N-1)}\in L^1([x_0,c))^{2m\tilde mes 2m}$ for all $c>x_0$, that $x$ is
a right Lebesgue point of $B^{(N-1)}$, and that for all $R>0$,
\begin{align}
&\underset{y\in [x_0,x_0+R]}{\text{\rm{ess}}sup} \,
\bibitemgg\|\int_y^{x_0+R} dx'\,B^{(N-1)}(x')\exp(2iz(x'-y))
+\frac{1}{2iz}B^{(N-1)}(y)\bibitemgg\|_{{\mathbb{C}}^{2m\tilde mes 2m}} \nonumber \\
& \underset{\substack{\abs{z}
\to\infty\\ z\in C_\varepsilon}}{=}o\bibitemg(|z|^{-1}\bibitemg). \label{4.36b}
\end{align}
If $N=1$, suppose in addition
$B_{k,k'}B_{\ell,\ell'}\in L^1([x_0,x_0+R])^{m\tilde mes m}$ for all $R>0$
and all $k,k',\ell,\ell'\in\{1,2\}$. Let
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$ and denote by
$M_+(z,x,\alpha_0)$, $x\geq x_0$, the unique element of the limit disk
${\mathcal D}_+(z,x,\alpha_0)$ for the half-line Dirac operator
$D_+(\alpha_0)$ in \eqref{2.84}. Then, as $\abs{z}\to\infty$ in
$C_\varepsilon$, $M_+(z,x,\alpha_0)$ has an asymptotic expansion of the
form
\begin{equation}
M_+(z,x,\alpha_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} i I_m +\sum_{k=1}^N
m_{+,k}(x,\alpha_0)z^{-k}+ o\bibitemg(|z|^{-N}\bibitemg), \quad N\in{\mathbb{N}}. \label{4.37}
\end{equation}
The expansion \eqref{4.37} is uniform with respect to $\arg\,(z)$ for
$|z|\to \infty$ in $C_\varepsilon$ and uniform in $x$ as long as $x$
varies in compact subsets of ${\mathbb{R}}$ intersected with the right Lebesgue
set of $B^{(N-1)}$. The expansion coefficients $m_{+,k}(x,\alpha_0)$ can
be recursively computed from \eqref{4.2}.
\end{theorem}
\begin{proof}
To see that uniformity holds for this expansion, first recall the role of
Theorem~\ref{t3.12} in providing uniformity in the asymptotic expression
\eqref{4.20} which then leads to \eqref{4.18} holding uniformly with
respect to $x_0$ varying within compact subsets of ${\mathbb{R}}$ and with respect
to $\arg\,(z)$ for $|z|\to \infty$ in $C_\varepsilon$. This in turn leads
to a similar uniformity holding for \eqref{4.26} which is the key to
\eqref{4.35} holding with respect to $x_0$ varying within compact subsets
of ${\mathbb{R}}$ and with respect to $\arg\,(z)$ for $|z|\to \infty$ in
$C_\varepsilon$.
\end{proof}
\begin{remark} \label{r4.9}
For simplicity, we focused thus far on the expansion of
$M_+(z,x_0,\alpha_0)$ as $|z|\to\infty$. Of course,
Theorem~\ref{t4.7} holds also for $M_-(z,x_0,\alpha_0)$ replacing the
hypotheses concerning right Lebesgue points by those of left
Lebesgue points, etc. For convenience we just state the corresponding
expansion and associated nonlinear recursion formula which covers both
cases.
\begin{equation}
M_\pm (z,x,\alpha_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} \sum_{k=0}^N
m_{\pm,k}(x,\alpha_0)z^{-k}+ o\bibitemg(|z|^{-N}\bibitemg), \quad N\in{\mathbb{N}}.
\label{4.100}
\end{equation}
\begin{align}
m_{\pm,0}(x,\alpha_0)&=\pm iI_m, \nonumber \\
m_{\pm,1}(x,\alpha_0)&=-\frac{1}{2} \bibitemg(
B_{1,2}(x)+B_{2,1}(x)\bibitemg) \pm \frac{i}{2} \bibitemg(
B_{1,1}(x)-B_{2,2}(x)\bibitemg), \nonumber \\
m_{\pm,k+1}(x,\alpha_0)&=\pm\frac{i}2\bibitemgg(m_{\pm,k}^\prime(x,\alpha_0)+
\sum_{\ell=1}^{k}m_{\pm,\ell}(x,\alpha_0)
m_{\pm,k+1-\ell}(x,\alpha_0) \nonumber \\
& \qquad \quad +\sum_{\ell=0}^{k}m_{\pm,\ell}(x,\alpha_0)B_{2,2}(x)
m_{\pm,k-\ell}(x,\alpha_0) \label{4.101} \\
& \qquad \quad +B_{1,2}(x)m_{\pm,k}(x,\alpha_0)
+m_{\pm,k}(x,\alpha_0)B_{2,1}(x)\bibitemgg), \nonumber \\
& \hspace*{5.8cm} 1\leq k\leq N-1. \nonumber
\end{align}
\end{remark}
Combining Theorem~\ref{t4.7} and \eqref{2.620} then yields the
analogous asymptotic expansion for $M(z,x,\alpha_0)$.
\begin{theorem} \label{t4.10a}
Assume Hypothesis~\ref{h2.1} with $A=I_{2m}$, and let
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$. Fix $x_0\in{\mathbb{R}}$ and let
$x\in{\mathbb{R}}$. Suppose that for some $N\in{\mathbb{N}}$,
$B^{(N-1)}\in L^1_{\text{\rm{loc}}}({\mathbb{R}})^{2m\tilde mes 2m}$, that
$x$ is a right and a left Lebesgue point of $B^{(N-1)}$, and that for
all $R>0$,
\begin{align}
& \quad \underset{y\in [x_0,x_0+R]}{\text{\rm{ess}}sup} \,
\bibitemgg\|\int_y^{x_0+R} dx'\,B^{(N-1)}(x')\exp(2iz(x'-y))
+\frac{1}{2iz}B^{(N-1)}(y)\bibitemgg\|_{{\mathbb{C}}^{2m\tilde mes 2m}} \nonumber \\
&+ \underset{y\in [x_0-R,x_0]}{\text{\rm{ess}}sup} \,
\bibitemgg\|\int_{x_0-R}^{y} dx'\,B^{(N-1)}(x')\exp(2iz(x'-y))
-\frac{1}{2iz}B^{(N-1)}(y)\bibitemgg\|_{{\mathbb{C}}^{2m\tilde mes 2m}} \nonumber \\
& \underset{\substack{\abs{z}
\to\infty\\ z\in C_\varepsilon}}{=}o\bibitemg(|z|^{-1}\bibitemg). \label{4.102}
\end{align}
If $N=1$, assume in addition
$B_{k,k'}B_{\ell,\ell'}\in L^1_{\text{\rm{loc}}}({\mathbb{R}})^{m\tilde mes m}$
for all $k,k',\ell,\ell'\in\{1,2\}$. Let $M(z,x,\alpha_0)$ be defined as
in \eqref{2.62} $($see also
\eqref{2.620}$)$. Then, as $\abs{z}\to\infty$ in
$C_\varepsilon$, $M(z,x,\alpha_0)$ has an asymptotic expansion of the
form
\begin{equation}
M(z,x,\alpha_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} (i/2) I_{2m} +\sum_{k=1}^N
M_{k}(x,\alpha_0)z^{-k}+ o\bibitemg(|z|^{-N}\bibitemg), \quad N\in{\mathbb{N}}, \label{4.103}
\end{equation}
where
\begin{align}
M_1(x,\alpha_0)&=-\frac{i}8 \begin{pmatrix}B_{1,1}(x+0)-B_{2,2}(x+0) &
B_{1,2}(x+0)+B_{2,1}(x+0) \\
B_{1,2}(x+0)+B_{2,1}(x+0)& B_{2,2}(x+0)-B_{1,1}(x+0)\end{pmatrix} \nonumber \\
& \quad -\frac{i}8 \begin{pmatrix}B_{1,1}(x-0)-B_{2,2}(x-0) &
B_{1,2}(x-0)+B_{2,1}(x-0) \\
B_{1,2}(x-0)+B_{2,1}(x-0)& B_{2,2}(x-0)-B_{1,1}(x-0)\end{pmatrix},
\text{ etc.} \label{4.104}
\end{align}
The expansion \eqref{4.103} is uniform with respect to $\arg\,(z)$ for
$|z|\to \infty$ in $C_\varepsilon$ and uniform in $x$ as long as $x$
varies in compact subsets of ${\mathbb{R}}$ intersected with the right and
left Lebesgue set of $B^{(N-1)}$. \\
\nonumberindent If one merely assumes Hypothesis~\ref{h2.1} with $A=I_{2m}$,
$\alpha_0=(I_m\; 0)$, and $B\in L^1_{\text{\rm{loc}}}({\mathbb{R}})^{2m\tilde mes 2m}$, then
\begin{equation}
M(z,x,\alpha_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} (i/2)I_{2m}+o(1). \label{4.105}
\end{equation}
Again the asymptotic expansion \eqref{4.105} is uniform with respect to
$\arg\,(z)$ for $|z|\to \infty$ in $C_\varepsilon$ and uniform in
$x\in{\mathbb{R}}$ as long as $x$ varies in compact intervals.
\end{theorem}
\nonumberindent The higher-order coefficients in \eqref{4.103} can be derived
upon inserting \eqref{4.100} into \eqref{3.17a}, taking into account
\eqref{2.620}.
Theorems~\ref{t4.6} and \ref{t4.7} (with $N\in{\mathbb{N}}$) are new even in the
scalar case $m=1$ with respect to the regularity assumptions on $B$. For
previous results in the case $m=1$ under stronger hypotheses on $B$ we
refer to \cite{EHS83}, \cite{Ha85}, \cite{HKS89a}, \cite{HKS89b},
\cite{Mi91}. In particular, \cite{Ha85}, \cite{HKS89a}, and \cite{HKS89b}
derived alternative high-energy expansions for the Weyl-Titchmarsh
$m$-function in the case $m=1$.
Throughout this section we fixed $\alpha$ to be $\alpha_0=(I_m\; 0)$. The
case of general $\alpha\in{\mathbb{C}}^{2m\tilde mes m}$ satisfying \eqref{2.8e} then
follows from \eqref{2.41}.
\section{A Local Uniqueness Result} \label{s5}
In this section we assume that $B$ is in the normal form given in
Theorem~\ref{t1.1},
\begin{equation}\label{4.115}
B(x)=\begin{pmatrix} B_{1,1}(x)
& B_{1,2}(x)\\B_{1,2}(x) & -B_{1,1}(x) \end{pmatrix},
\end{equation}
with $B_{1,1}$ and $B_{1,2}$ self-adjoint a.e.
We prove fundamental new local uniqueness results for $B$ in terms of
exponentially small differences of Weyl-Titchmarsh
matrices $M_+(z,x,\alpha)$ and $M(z,x,\alpha)$. These results, in turn, yield
new global ramifications. We start with an auxiliary result concerning
asymptotic expansions.
\begin{lemma} \label{l4.9} Suppose $\alpha=(\alpha_1 \;
\alpha_2)\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfies \eqref{2.8e}, fix
$x_0,y_0\in{\mathbb{R}}$ with
$y_0>x_0$, and let
$x\geq x_0$. Assume $A=I_{2m}$, $B\in L^1([x_0,\infty))^{2m\tilde mes 2m}$,
$\text{\rm{supp}}(B)\subseteq [x_0,y_0]$, with $B$ in the normal form given in
\eqref{4.115} a.e.~on
$(x_0,y_0)$. Then, the following asymptotic expansions hold for
$\Theta(z,x,x_0,\alpha)$, $\Phi(z,x,x_0,\alpha)$, and
$U_{+}(z,x,x_0,\alpha)$ associated with \eqref{HSa},
\begin{align}
\Theta(z,x,x_0,\alpha)&\underset{\substack{\abs{z}
\to\infty\\ z\in {\mathbb{C}}_+}}{=}\frac{1}{2}\begin{pmatrix}\alpha_1^*+i\alpha_2^*\\
-i(\alpha_1^*+i\alpha_2^*)\end{pmatrix}\exp(-iz(x-x_0))\bibitemg(1+o(1)\bibitemg),
\quad x>x_0, \label{4.48} \\
\Phi(z,x,x_0,\alpha)&\underset{\substack{\abs{z}
\to\infty\\ z\in {\mathbb{C}}_+}}{=}\frac{i}{2}\begin{pmatrix}-\alpha_2^*+i\alpha_1^*\\
-i(-\alpha_2^*+i\alpha_1^*)\end{pmatrix}\exp(-iz(x-x_0))\bibitemg(1+o(1)\bibitemg),
\quad x>x_0, \label{4.54} \\
U_{+}(z,x,x_0,\alpha)&\underset{\substack{\abs{z}
\to\infty\\ z\in {\mathbb{C}}_+}}{=}\begin{pmatrix}\alpha_1^*-i\alpha_2^*\\
i(\alpha_1^*-i\alpha_2^*)\end{pmatrix}\exp(iz(x-x_0))\bibitemg(1+o(1)\bibitemg),
\quad x\geq x_0. \label{4.60}
\end{align}
Next, we introduce the abbreviation
\begin{equation}
C=-B_{1,2}-iB_{1,1}, \quad C^*=-B_{1,2}+iB_{1,1}, \label{4.38f}
\end{equation}
and suppose in addition that
\begin{equation}
\underset{y\in [x_0,y_0]}{\text{\rm{ess}}sup} \, \bibitemgg\|\int_y^{y_0}
dx'\,B(x')\exp(2iz(x'-y)) +\frac{1}{2iz}B(y)\bibitemgg\|_{{\mathbb{C}}^{2m\tilde mes 2m}}
\underset{\substack{\abs{z}
\to\infty\\ z\in\rho_+}}{=}o\bibitemg(|z|^{-1}\bibitemg), \label{4.39a}
\end{equation}
along a ray $\rho_+\subset{\mathbb{C}}_+$, and that
\begin{equation}
B_{1,1}^2,\ B_{1,2}^2,\ B_{1,1}B_{1,2},\ B_{1,2}B_{1,1} \in
L^1([x_0,y_0])^{m\tilde mes m}. \label{4.39f}
\end{equation}
Then,
\begin{align}
&\Theta(z,x,x_0,\alpha)\underset{\substack{\abs{z}
\to\infty\\ z\in
\rho_+}}{=}\bibitemgg(\frac{1}{2}\begin{pmatrix}\alpha_1^*+i\alpha_2^*\\
-i(\alpha_1^*+i\alpha_2^*)
\end{pmatrix}-\frac{i}{4z}\begin{pmatrix}C(x_0)^*(\alpha_1^*-i\alpha_2^*)\\
-iC(x_0)^*(\alpha_1^*-i\alpha_2^*)\end{pmatrix} \nonumber \\
& -\frac{i}{4z}\begin{pmatrix}C(x)(\alpha_1^*+i\alpha_2^*)\\
iC(x)(\alpha_1^*+i\alpha_2^*)\end{pmatrix} \nonumber \\
& +\frac{i}{4z}\int_{x_0}^x dx' \,
\begin{pmatrix}C(x')^*C(x')(\alpha_1^*+i\alpha_2^*)\\
-iC(x')^*C(x')(\alpha_1^*+i\alpha_2^*)\end{pmatrix}\bibitemgg)
e^{-iz(x-x_0)}\bibitemg(1+o\bibitemg(|z|^{-1}\bibitemg)\bibitemg), \nonumber \\
& \hspace*{9.5cm} x>x_0, \label{4.49} \\
&\Phi(z,x,x_0,\alpha)\underset{\substack{\abs{z}
\to\infty\\ z\in \rho_+}}{=}\bibitemgg(\frac{i}{2}\begin{pmatrix}
-\alpha_2^*+i\alpha_1^*\\
-i(-\alpha_2^*+i\alpha_1^*)\end{pmatrix}-\frac{1}{4z}
\begin{pmatrix}C(x_0)^*(-\alpha_2^*-i\alpha_1^*)\\
-iC(x_0)^*(-\alpha_2^*-i\alpha_1^*)\end{pmatrix} \nonumber \\
&+\frac{1}{4z}\begin{pmatrix}C(x)(-\alpha_2^*+i\alpha_1^*)\\
iC(x)(-\alpha_2^*+i\alpha_1^*)\end{pmatrix}
\nonumber \\
& -\frac{1}{4z}\int_{x_0}^x dx' \,
\begin{pmatrix}C(x')^*C(x')(-\alpha_2^*+i\alpha_1^*)\\
-iC(x')^*C(x')(-\alpha_2^*+i\alpha_1^*)\end{pmatrix}\bibitemgg)
e^{-iz(x-x_0)}\bibitemg(1+o\bibitemg(|z|^{-1}\bibitemg)\bibitemg), \nonumber \\
& \hspace*{9.8cm} x>x_0, \label{4.55}
\end{align}
whenever $x_0$ is a right Lebesgue point of $B$ and $x$ is a left
Lebesgue point of $B$, and
\begin{align}
&U_{+}(z,x,x_0,\alpha)\underset{\substack{\abs{z}
\to\infty\\ z\in \rho_+}}{=}\bibitemgg(\begin{pmatrix} \alpha_1^*-i\alpha_2^*\\
i(\alpha_1^*-i\alpha_2^*)\end{pmatrix}+\frac{i}{2z}
\begin{pmatrix}(C(x)^*-C(x_0)^*)(\alpha_1^*-i\alpha_2^*)\\
-i(C(x)^*+C(x_0)^*)(\alpha_1^*-i\alpha_2^*)\end{pmatrix} \nonumber \\
& -\frac{i}{2z}\int_{x_0}^{x} dx' \,
\begin{pmatrix}C(x')C(x')^*(\alpha_1^*-i\alpha_2^*)\\
iC(x')C(x')^*(\alpha_1^*-i\alpha_2^*) \end{pmatrix}\bibitemgg)
e^{iz(x-x_0)}\bibitemg(1+o\bibitemg(|z|^{-1}\bibitemg)\bibitemg), \quad
x\geq x_0, \label{4.61}
\end{align}
whenever $x$ is a right Lebesgue point of $B$.
\end{lemma}
\begin{proof}
Since $x_0$ and $\alpha$ are fixed throughout this proof, we will
temporarily suppress these variables whenever possible to simplify
notations. Introducing
\begin{equation}
\widehat \Theta(z,x)= 2\Theta(z,x)\exp(iz(x-x_0)), \label{4.41}
\end{equation}
the Volterra integral equation for $\Theta$ (cf. \eqref{4.3B}),
\begin{align}
\Theta(z,x)&=\begin{pmatrix}\alpha_1^*\cos(z(x-x_0))
+\alpha_2^*\sin(z(x-x_0))\\
\alpha_2^*\cos(z(x-x_0))-\alpha_1^*\sin(z(x-x_0))\end{pmatrix} \nonumber \\
& \quad -\int_{x_0}^x dx'\, K(z,x,x') J B(x')\Theta(z,x'), \label{4.42}
\end{align}
can be rewritten in terms of that of $\widehat \Theta$ in the form
\begin{align}
\widehat\Theta(z,x)&=\begin{pmatrix}\alpha_1^*+i\alpha_2^*\\
-i(\alpha_1^*+i\alpha_2^*)\end{pmatrix}
+\begin{pmatrix}\alpha_1^*-i\alpha_2^*\\
i(\alpha_1^*-i\alpha_2^*)\end{pmatrix}\exp(2iz(x-x_0)) \nonumber \\
& \quad -\frac{1}{2}\int_{x_0}^x dx'
\bibitemg(R(x')\exp(2iz(x-x'))+S(x')\bibitemg)\widehat\Theta(z,x'), \label{4.43}
\end{align}
where we abbreviated
\begin{equation}
R=\begin{pmatrix} C&iC\\ iC &-C \end{pmatrix}, \quad
S=\begin{pmatrix} C^* & -iC^*\\ -iC^* & -C^*\end{pmatrix}. \label{4.44}
\end{equation}
Using the elementary algebraic facts
\begin{equation}
R\begin{pmatrix}a\\ ia\end{pmatrix}=0, \quad
R\begin{pmatrix}b\\ -ib\end{pmatrix}
=2\begin{pmatrix} Cb\\ iCb\end{pmatrix}, \quad
S\begin{pmatrix} a\\ ia\end{pmatrix}
=2\begin{pmatrix} C^* a\\ -iC^* a\end{pmatrix}, \quad
S\begin{pmatrix}b\\ -ib\end{pmatrix}=0 \label{4.46}
\end{equation}
for any $a,b \in {\mathbb{C}}^{m\tilde mes m}$, iterating \eqref{4.43} yields
\begin{align}
&\widehat\Theta(z,x)=\begin{pmatrix}\alpha_1^*+i\alpha_2^*\\
-i(\alpha_1^*+i\alpha_2^*)\end{pmatrix}+
\begin{pmatrix}\alpha_1^*-i\alpha_2^*\\
i(\alpha_1^*+i\alpha_2^*)\end{pmatrix}e^{2iz(x-x_0)} \nonumber \\
& \quad +\sum_{m=1}^\infty (-2)^{-m}\int_{x_0}^x d\xi_1 \,
\bibitemg(R(\xi_1)e^{2iz(x-\xi_1)}+S(\xi_1)\bibitemg)\tilde mes \nonumber \\
& \quad \tilde mes \int_{x_0}^{\xi_1} d\xi_2 \,
\bibitemg(R(\xi_2)e^{2iz(\xi_1-\xi_2)}+S(\xi_2)\bibitemg)\dots \label{4.47} \\
& \quad \;\, \dots \int_{x_0}^{\xi_{m-2}}
d\xi_{m-1} \,
\bibitemg(R(\xi_{m-1})e^{2iz(\xi_{m-2}-\xi_{m-1})}+S(\xi_{m-1})\bibitemg)
\tilde mes \nonumber \\
& \quad \tilde mes \int_{x_0}^{\xi_{m-1}} d\xi_{m} \,
\bibitemgg(R(\xi_{m})\begin{pmatrix}\alpha_1^*-i\alpha_2^*\\
-i(\alpha_1^*-i\alpha_2^*)\end{pmatrix}e^{2iz(\xi_{m-1}-\xi_{m})}
\nonumber \\
& \hspace*{2.8cm} +S(\xi_{m}) \begin{pmatrix}\alpha_1^*+i\alpha_2^*\\
i(\alpha_1^*+i\alpha_2^*)\end{pmatrix}e^{2iz(\xi_m-x_0)}\bibitemgg). \nonumber
\end{align}
Applying the Riemann-Lebesgue lemma to \eqref{4.47} then proves
\eqref{4.48} assuming $B\in L^1([x_0,\infty))^{2m\tilde mes 2m}$,
only. Assuming also \eqref{4.39a} and \eqref{4.39f} one can compute the
next term in the asymptotic expansion \eqref{4.48} and then obtains
\eqref{4.49} using \eqref{4.47} and the finite-interval variant of
\eqref{4.-2}, whenever $x_0$ is a right Lebesgue point of $B$ and $x$ is a
left Lebesgue point of $B$. \\
Exactly the same arguments apply to $\Phi$. Introducing
\begin{equation}
\widehat \Phi(z,x)= 2\Phi(z,x)\exp(iz(x-x_0)), \label{4.50}
\end{equation}
the Volterra integral equation for $\Phi$,
\begin{align}
\Phi(z,x)&=\begin{pmatrix}-\alpha_2^*\cos(z(x-x_0))
+\alpha_1^*\sin(iz(x-x_0))\\
\alpha_1^*\cos(iz(x-x_0))+\alpha_2^*\sin(z(x-x_0))\end{pmatrix} \nonumber \\
& \quad -\int_{x_0}^x dx'\, K(z,x,x') J B(x')\Phi_j(z,x'), \label{4.51}
\end{align}
can be rewritten in terms of that of $\widehat \Phi$ in the form
\begin{align}
\widehat\Phi(z,x)&=i\begin{pmatrix}-\alpha_2^*+i\alpha_1^*\\
-i(-\alpha_2^*+i\alpha_1^*)\end{pmatrix}
-i\begin{pmatrix}-\alpha_2^*-i\alpha_1^*\\
i(-\alpha_2^*-i\alpha_1^*)\end{pmatrix}\exp(2iz(x-x_0)) \nonumber \\
& \quad -\frac{1}{2}\int_{x_0}^x
\bibitemg(R(x')\exp(2iz(x-x'))+S(x')\bibitemg)\widehat\Phi(z,x'). \label{4.52}
\end{align}
Iterating \eqref{4.52}, taking into account \eqref{4.46}, yields
\begin{align}
&\widehat\Phi(z,x)=i\begin{pmatrix}-\alpha_2^*+i\alpha_1^*\\
-i(-\alpha_2^*+i\alpha_1^*)\end{pmatrix}
-i\begin{pmatrix}-\alpha_2^*-i\alpha_1^*\\
i(-\alpha_2^*-i\alpha_1^*)\end{pmatrix}e^{2iz(x-x_0)} \nonumber \\
& \quad +\sum_{m=1}^\infty (-2)^{-m}\int_{x_0}^x d\xi_1 \,
\bibitemg(R(\xi_1)e^{2iz(x-\xi_1)}+S(\xi_1)\bibitemg)\tilde mes \nonumber \\
& \quad \tilde mes \int_{x_0}^{\xi_1} d\xi_2 \,
\bibitemg(R(\xi_2)e^{2iz(\xi_1-\xi_2)}+S(\xi_2)\bibitemg)\dots \label{4.53} \\
& \quad \;\, \dots \int_{x_0}^{\xi_{m-2}}
d\xi_{m-1} \,
\bibitemg(R(\xi_{m-1})e^{2iz(\xi_{m-2}-\xi_{m-1})}+S(\xi_{m-1})\bibitemg)
\tilde mes \nonumber \\
& \quad \tilde mes \int_{x_0}^{\xi_{m-1}} d\xi_{m} \,
\bibitemgg(iR(\xi_{m})\begin{pmatrix}-\alpha_2^*+i\alpha_1^*\\
-i(-\alpha_2^*+i\alpha_1^*)\end{pmatrix}
e^{2iz(\xi_{m-1}-\xi_{m})} \nonumber \\
& \hspace*{2.8cm} -iS(\xi_{m})
\begin{pmatrix}-\alpha_2^*-i\alpha_1^*\\
i(-\alpha_2^*-i\alpha_1^*)\end{pmatrix}e^{2iz(\xi_m-x_0)}\bibitemgg). \nonumber
\end{align}
Applying the Riemann-Lebesgue lemma to \eqref{4.53} the proves
\eqref{4.54} assuming $B\in L^1([x_0,\infty))^{2m\tilde mes 2m}$,
only. Assuming also \eqref{4.39a} and \eqref{4.39f} one can compute the
next term in the asymptotic expansion \eqref{4.54} and then obtains
\eqref{4.55} using \eqref{4.53} and the finite-interval variant of
\eqref{4.-2}, whenever $x_0$ is a right Lebesgue point of $B$ and $x$ is
a left Lebesgue point of $B$. \\
Finally, we turn to $U_{+}(z,x)$. Introducing
\begin{equation}
{\widetilde V}_{+}(z,x)= \widetilde U_{+}(z,x)\exp(-iz(x-x_0)), \label{4.56}
\end{equation}
the Volterra integral equation for $\widetilde U_{+}$,
\begin{equation}
\widetilde U_{+}(z,x)=\begin{pmatrix} \alpha_1^*-i\alpha_2^*\\
i(\alpha_1^*-i\alpha_2^*)\end{pmatrix}\exp(iz(x-x_0)) +\int_x^\infty
dx'\, K(z,x,x') J B(x')\widetilde U_{+}(z,x'), \label{4.57}
\end{equation}
can be rewritten in terms of that of ${\widetilde V}_{+}$ in the form
\begin{equation}
{\widetilde V}_{+}(z,x)=\begin{pmatrix} \alpha_1^*-i\alpha_2^*\\
i(\alpha_1^*-i\alpha_2^*)\end{pmatrix}
+\frac{1}{2}\int_{x}^{y_0} dx'\,
\bibitemg(R(x')+S(x')\exp(2iz(x'-x))\bibitemg){\widetilde V}_{+}(z,x'). \label{4.58}
\end{equation}
Iterating \eqref{4.58}, taking into account \eqref{4.46}, yields
\begin{align}
{\widetilde V}_{+}(z,x)&=\begin{pmatrix} \alpha_1^*-i\alpha_2^*\\
i(\alpha_1^*-i\alpha_2^*)\end{pmatrix}+\sum_{k=1}^\infty
2^{-2k}\int_{x}^{y_0} d\xi_1 \, R(\xi_1) \int_{\xi_1}^{y_0} d\xi_2 \,
S(\xi_2)e^{2iz(\xi_2-\xi_1)}\tilde mes
\nonumber \\
& \quad \;\, \tilde mes \int_{\xi_2}^{y_0} d\xi_3 \,
R(\xi_3) \dots \int_{\xi_{2k-2}}^{y_0} d\xi_{2k-1} \, R(\xi_{2k-1})
\tilde mes \nonumber \\
& \quad \tilde mes \int_{\xi_{2k-1}}^{y_0} d\xi_{2k} \,
S(\xi_{m})\begin{pmatrix} \alpha_1^*-i\alpha_2^*\\
i(\alpha_1^*-i\alpha_2^*)\end{pmatrix}e^{iz(\xi_{2k}-\xi_{2k-1})} \nonumber \\
& \quad +\sum_{\ell=0}^\infty 2^{-2\ell+1}
\int_x^{y_0} d\xi_1 \, S(\xi_1)e^{2iz(\xi_1-x)} \int_{\xi_1}^{y_0} d\xi_2
\, R(\xi_2)\tilde mes \nonumber \\
& \quad \;\, \tilde mes \int_{\xi_2}^{y_0} d\xi_3 \,
S(\xi_3)e^{2iz(\xi_3-\xi_2)} \dots \int_{\xi_{2\ell-1}}^{y_0}
d\xi_{2\ell} \, R(\xi_{2\ell}) \tilde mes \nonumber \\
& \quad \tilde mes \int_{\xi_{2\ell}}^{y_0} d\xi_{2\ell+1} \,
S(\xi_{2\ell+1})\begin{pmatrix} \alpha_1^*-i\alpha_2^*\\
i(\alpha_1^*-i\alpha_2^*)\end{pmatrix}e^{iz(\xi_{2\ell+1}-\xi_{2\ell})}.
\label{4.59}
\end{align}
Next, we take into account the different normalizations of $U_+$ and
$\widetilde U_+$. Using $U_+(z,x_0)=[I_m \; M_+(z,x_0)^t]^t$ (cf., \eqref{2.52}
and $\Psi(z,x_0,x_0,\alpha_0)=I_{2m}$), one readily verifies the
relationship
\begin{equation}
u_{+,1}(z,x)={\widetilde u}_{+,1}(z,x){\widetilde u}_{+,1}(z,x_0)^{-1},
\quad
u_{+,2}(z,x)={\widetilde u}_{+,2}(z,x){\widetilde u}_{+,1}(z,x_0)^{-1}.
\label{4.59a}
\end{equation}
Thus, applying the Riemann-Lebesgue lemma to \eqref{4.59} then proves
\eqref{4.60} (in agreement with \eqref{4.14H}), assuming $B\in
L^1([x_0,\infty))^{2m\tilde mes 2m}$, only. Assuming also
\eqref{4.39a} and \eqref{4.39f} one can compute the next term in the
asymptotic expansion \eqref{4.60} and then obtains \eqref{4.61} using
\eqref{4.59} and \eqref{4.-2}, whenever $x$ is a right Lebesgue point of
$B$.
\end{proof}
In the special case $m=1$ (and for $\alpha=(1\; 0)$), the expansion
\eqref{4.61} was stated in \cite{Gr92}.
Next, we note an elementary result concerning the boundary data
independence of exponentially close Weyl-Titchmarsh matrices.
\begin{lemma} \label{l4.9a}
Fix $x_0\in{\mathbb{R}}$ and suppose $A_j=I_{2m}$, $B_j=B_j^*\in
L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$. Denote by
$M_{+,j}(z,x,\alpha)$, $x\geq x_0$, the unique Weyl-Titchmarsh matrices
corresponding to the half-line Dirac-type operators $D_{+,j}(\alpha)$,
$j=1,2$, in \eqref{2.84}. Fix an $\hat\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$
satisfying
\eqref{2.8e} and assume that for all $\varepsilon >0$,
\begin{equation}
\|M_{+,1}(z,x_0,\hat\alpha)-M_{+,2}(z,x_0,\hat\alpha)\|_{{\mathbb{C}}^{m\tilde mes
m}}\underset{\substack{|z|\to\infty\\z\in \rho_{+}}}{=}
O\bibitemg(e^{-2\text{\rm Im}(z)(a-\varepsilon)}\bibitemg) \label{4.24a}
\end{equation}
along some ray $\rho_{+}\subset{\mathbb{C}}_+$. Then, for all
$\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying \eqref{2.8e} and for all
$\varepsilon >0$,
\begin{equation}
\|M_{+,1}(z,x_0,\alpha)-M_{+,2}(z,x_0,\alpha)\|_{{\mathbb{C}}^{m\tilde mes
m}}\underset{\substack{|z|\to\infty\\z\in \rho_{+}}}{=}
O\bibitemg(e^{-2\text{\rm Im}(z)(a-\varepsilon)}\bibitemg) \label{4.25aa}
\end{equation}
along the ray $\rho_{+}$.
\end{lemma}
\begin{proof}
Using \eqref{2.38} and \eqref{2.41} one estimates
\begin{align}
&\|M_{+,1}(z,x_0,\alpha)-M_{+,2}(z,x_0,\alpha)\|_{{\mathbb{C}}^{m\tilde mes m}} \nonumber \\
&=\|M_{+,1}(\begin{align}r z,x_0,\alpha)^* -M_{+,2}(z,x_0,\alpha)\|_{{\mathbb{C}}^{m\tilde mes
m}} \nonumber \\
&\leq \|[\hat\alpha\alpha^* -M_{+,1}(\begin{align}r z,x_0,\hat\alpha)^*\hat\alpha
J\alpha^*]^{-1}\|_{{\mathbb{C}}^{m\tilde mes m}} \tilde mes \nonumber \\
& \quad \tilde mes
\|M_{+,1}(z,x_0,\hat\alpha)-M_{+,2}(z,x_0,\hat\alpha)\|_{{\mathbb{C}}^{m\tilde mes
m}}\tilde mes \nonumber \\ & \quad \tilde mes \|[\alpha\hat\alpha^* +\alpha J\hat\alpha^*
M_{+,2}(z,x_0,\hat\alpha)]^{-1}\|_{{\mathbb{C}}^{m\tilde mes m}},
\label{4.26aa}
\end{align}
since by \eqref{2.8i}
\begin{equation}
\hat\alpha J\alpha^*\alpha\hat\alpha^*+\hat\alpha\alpha^*\alpha
J\hat\alpha^*=0, \quad \hat\alpha\alpha^*\alpha\hat\alpha^*- \hat\alpha
J\alpha^*\alpha J\hat\alpha^*=I_m. \label{4.27a}
\end{equation}
Moreover, since
\begin{equation}
[\hat\alpha\alpha^*-i\hat\alpha J\alpha^*][\hat\alpha\alpha^*
-i\hat\alpha J\alpha^*]^*=I_m, \label{5.29a}
\end{equation}
by \eqref{4.27a}, one infers \eqref{4.25aa} from
\eqref{4.26aa} and $M_{+,j}(z,x_0,\alpha)=iI_m +o(1)$ as $|z|\to\infty$,
$z\in{\mathbb{C}}_+$, $j=1,2$ (cf.~\eqref{3.1}).
\end{proof}
Our principal new local uniqueness result for Dirac-type operators in
terms of Weyl-Titchmarsh matrices then reads as follows.
\begin{theorem} \label{t4.10}
Fix $x_0\in{\mathbb{R}}$ and suppose $A_j=I_{2m}$, $B_j\in
L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$. Suppose also that $B_j$
is in the normal form given in \eqref{4.115} a.e.~on $(x_0,\infty)$,
$j=1,2$. Denote by $M_{j,+}(z,x,\alpha)$, $x\geq
x_0$, the unique Weyl-Titchmarsh matrices corresponding to the half-line
Dirac-type operators $D_{+,j}(\alpha)$, $j=1,2$, in \eqref{2.84}. Then,
\begin{equation}
\text{if for some $a>0$, }\, B_1(x)=B_2(x) \, \text{ for a.e. $x\in
(x_0,x_0+a)$,} \label{4.38AA}
\end{equation}
one obtains
\begin{equation}
\|M_{1,+}(z,x_0,\alpha)-M_{2,+}(z,x_0,\alpha)\|_{{\mathbb{C}}^{m\tilde mes
m}}\underset{\substack{\abs{z}
\to\infty\\ z\in \rho_{+}}}{=}
O\bibitemg(e^{-2\text{\rm Im}(z)a}\bibitemg) \label{4.38}
\end{equation}
along any ray $\rho_+\subset{\mathbb{C}}_+$ with $0<\arg(z)<\pi$ and for all
$\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying \eqref{2.8e}. Conversely,
fix an $\hat\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying
\eqref{2.8e} and if $m>1$, assume in addition that
$B_j\in L^\infty([x_0,x_0+a])^{2m\tilde mes 2m}$, $j=1,2$. Moreover, suppose
that for all $\varepsilon >0$,
\begin{equation}
\|M_{1,+}(z,x_0,\hat\alpha_1)-M_{2,+}(z,x_0,
\hat\alpha_1)\|_{{\mathbb{C}}^{m\tilde mes
m}}\underset{\substack{|z|\to\infty\\z\in \rho_{+,\ell}}}{=}
O\bibitemg(e^{-2\text{\rm Im}(z)(a-\varepsilon)}\bibitemg), \quad \ell=1,2, \label{4.39}
\end{equation}
along a ray $\rho_{+,1}\subset{\mathbb{C}}_+$ with $0<\arg(z)<\pi/2$ and along a
ray $\rho_{+,2}\subset{\mathbb{C}}_+$ with $\pi/2<\arg(z)<\pi$. Then
\begin{equation}
B_1(x)=B_2(x) \text{ for a.e. } x\in [x_0,x_0+a]. \label{4.40}
\end{equation}
\end{theorem}
\begin{proof}
Since \eqref{4.38} follows from Theorem~\ref{t4.5} and Lemma~\ref{l4.9a},
it suffices to focus on the proof of \eqref{4.40}. Moreover, applying
Theorem~\ref{t4.5}, we may without loss of generality assume for the rest
of the proof that
\begin{equation}
\text{\rm{supp}}(B_j)\subseteq [x_0,x_0+a], \quad j=1,2. \label{4.40a}
\end{equation}
In the following, we will adapt the principal ingredients of a recent
proof of the local Borg-Marchenko uniqueness theorem for scalar
Schr\"odinger operators (i.e., for $m=1$) by Bennewitz \cite{Be00}, to the
current Dirac-type situation. First we recall that by
Lemma~\ref{l4.9a}, \eqref{4.39} holds along the rays $\rho_{+,j}$,
$j=1,2$ for all $\alpha=(\alpha_1\; \alpha_2)\in{\mathbb{C}}^{m\tilde mes 2m}$
satisfying \eqref{2.8e}. To simplify notations in the following
we will again suppress $x_0$ and $\alpha$ whenever possible and hence
abbreviate, $\Theta(z,x,x_0,\alpha)$, $\Phi(z,x,x_0,\alpha)$, and
$U_{j,+}(z,x,x_0,\alpha)$ by $\Theta(z,x)$, $\Phi(z,x)$, and
$U_{j,+}(z,x)$, respectively. Next, denoting in obvious notation by
\begin{align}
&\Theta_j(z,x)=\begin{pmatrix}\theta_{j,1}(z,x)\\
\theta_{j,2}(z,x)\end{pmatrix}, \quad
\Phi_j(z,x)=\begin{pmatrix}\phi_{j,1}(z,x)\\
\phi_{j,2}(z,x)\end{pmatrix}, \quad
U_{j,+}(z,x)=\begin{pmatrix} u_{j,+,1}(z,x)\\
u_{j,+,2}(z,x)\end{pmatrix}, \nonumber \\
& \hspace*{8.5cm} j=1,2, \; x\geq x_0, \label{4.62}
\end{align}
the solutions associated with $B_j$, $j=1,2$, which are defined in
\eqref{FSb} and \eqref{2.14}, we introduce
\begin{equation}
g_{j,k}(z,x)=\phi_{j,k}(z,x) u_{j,+,k}(\begin{align}r z,x)^*, \quad j,k
\in\{1,2\}, \; x\geq x_0. \label{4.63}
\end{equation}
Using the asymptotic expansions \eqref{4.48}--\eqref{4.60} for
$\Theta_j(z,x)$, $\Phi_j(z,x)$, and $U_{j,+}(z,x)$, and the analogous
ones for $\Theta_j(\begin{align}r z,x)^*$, $\Phi_j(\begin{align}r z,x)^*$, and $U_{j,+}(\begin{align}r
z,x)^*$, one verifies for each fixed $x>x_0$,
\begin{equation}
g_{j,k}(z,x) \underset{\substack{\abs{z}
\to\infty\\ z\in {\mathbb{C}}_+}}{=}(i/2)I_m + o(1), \quad j,k \in\{1,2\},
\label{4.65}
\end{equation}
assuming $B_j\in L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$, $j=1,2$,
only. Next, using the fact that for each fixed $x>x_0$,
\begin{align}
\phi_{1,k}(z,x)^{-1}\phi_{2,k}(z,x)&\underset{\substack{\abs{z}
\to\infty\\ z\in {\mathbb{C}}_+}}{=}I_m + o(1), \quad k=1,2, \label{4.65a} \\
(u_{1,+,k}(\begin{align}r z,x)^*)^{-1} u_{2,+,k}(\begin{align}r
z,x)^*&\underset{\substack{\abs{z}
\to\infty\\ z\in {\mathbb{C}}_+}}{=}I_m + o(1), \quad k=1,2, \label{4.66}
\end{align}
by \eqref{4.54}, \eqref{4.60}, one concludes
\begin{align}
&\phi_{1,k}(z,x) u_{2,+,j}(\begin{align}r z,x)^*-u_{1,+,k}(z,x)
\phi_{2,k}(\begin{align}r z,x)^* \nonumber \\
&=\phi_{1,k}(z,x) \theta_{2,k}(\begin{align}r z,x)^*-\theta_{1,k}(z,x)
\phi_{2,k}(\begin{align}r z,x)^* \nonumber \\
& \quad +\phi_{1,k}(z,x) \bibitemg(M_{2,+}(z)-M_{1,+}(z) \bibitemg)
\phi_{2,k}(\begin{align}r z,x)^* \underset{\substack{\abs{z}\to\infty\\ z\in
{\mathbb{C}}_+}}{=}o(1), \label{4.67}
\end{align}
using \eqref{4.65}, \eqref{4.66}, and
$M_{2,+}(\begin{align}r z)^*=M_{2,+}(z)$. Combining hypothesis \eqref{4.39} and
\eqref{4.54}, one infers
\begin{equation}
\bibitemg\|\phi_{1,k}(z,x) \bibitemg(M_{2,+}(z)-M_{1,+}(z) \bibitemg)
\phi_{2,k}(\begin{align}r z,x)^*\bibitemg\| \underset{\substack{\abs{z}\to\infty\\ z\in
\rho_{+,\ell}}}{=}o(1), \ \ x\in (x_0,x_0+a) \label{4.68}
\end{equation}
along the rays $\rho_{+,\ell}$, $\ell=1,2$. Thus, \eqref{4.67} implies
\begin{equation}
\bibitemg\|\phi_{1,k}(z,x) \theta_{2,k}(\begin{align}r z,x)^*-\theta_{1,k}(z,x)
\phi_{2,k}(\begin{align}r z,x)^*\bibitemg\| \underset{\substack{\abs{z}\to\infty\\ z\in
\rho_{+,\ell}}}{=}o(1), \ \ x\in (x_0,x_0+a) \label{4.69}
\end{equation}
along the rays $\rho_{+,\ell}$, $\ell=1,2$. The analogous estimate
\eqref{4.69} holds along the complex conjugate rays $\begin{align}r \rho_{+,\ell}$,
$\ell=1,2$, in the lower complex half-plane ${\mathbb{C}}_-$. To simplify
notations we denote the open sector generated by $\rho_{+,1}$ and its
complex conjugate $\begin{align}r \rho_{+,1}$ by ${\mathcal S}_1$, the open sector
generated by the $\rho_{+,2}$ and its complex conjugate $\begin{align}r \rho_{+,2}$
by
${\mathcal S}_2$, the remaining sector in ${\mathbb{C}}_+$ is denoted by ${\mathcal S}_3$, and
its complex conjugate sector in ${\mathbb{C}}_-$ is denoted by ${\mathcal S}_4$. Thus,
one obtains a partition of ${\mathbb{C}}$ into
\begin{equation}
{\mathbb{C}}=\bibitemgcup_{\ell=1}^4 \overline {{\mathcal S}_\ell}, \label{4.70}
\end{equation}
where each sector ${\mathcal S}_\ell$, $1\leq \ell\leq 4$, has opening angle
strictly less than $\pi$. Since (each matrix element of) the expression
under the norm in \eqref{4.69} is entire and of order less or equal to
one, one can apply the Phragm\'en-Lindel\"of principle (cf., e.g.,
\cite[No.~322, p.~166--167, 379]{PS72}) to each sector ${\mathcal S}_\ell$,
$1\leq \ell\leq 4$, and obtains that each matrix element under the norm in
\eqref{4.69} is uniformly bounded in each sector and hence on all of
${\mathbb{C}}$. By Liouville's theorem, these matrix elements are all equal to
certain constants. By the right-hand side of \eqref{4.69}, these
constants all vanish. Thus, we proved
\begin{equation}
\phi_{1,k}(z,x) \theta_{2,k}(\begin{align}r z,x)^*=\theta_{1,k}(z,x)
\phi_{2,k}(\begin{align}r z,x)^* \, \text{ for all $x\in (x_0,x_0+a)$}
\label{4.71}
\end{equation}
and hence
\begin{equation}
\phi_{1,k}(z,x)^{-1}\theta_{1,k}(z,x)=\theta_{2,k}(\begin{align}r z,x)^*
(\phi_{2,k}(\begin{align}r z,x)^*)^{-1} \, \text{ for all $x\in (x_0,x_0+a)$.}
\label{4.72}
\end{equation}
Differentiating $\phi_{j,k}(z,x)^{-1}\theta_{j,k}(z,x)$, $j,k=1,2$, with
respect to $x$ yields
\begin{align}
&\bibitemg(\phi_{j,1}(z,x)^{-1}\theta_{j,1}(z,x)\bibitemg)' \nonumber \\
&=\phi_{j,1}(z,x)^{-1}((B_j)_{1,1}(x)-z)(\phi_{j,2}(z,x)
\phi_{j,1}(z,x)^{-1}
\theta_{j,1}(z,x)-\theta_{j,2}(z,x)), \label{4.73} \\
&\bibitemg(\phi_{j,2}(z,x)^{-1}\theta_{j,2}(z,x)\bibitemg)' \nonumber \\
&=\phi_{j,2}(z,x)^{-1}((B_j)_{1,1}(x)+z)(\phi_{j,1}(z,x)
\phi_{j,2}(z,x)^{-1}\theta_{j,2}(z,x)-\theta_{j,1}(z,x)). \label{4.74}
\end{align}
Multiplying \eqref{4.73} by $\phi_{j,1}(\begin{align}r z,x)^*(\phi_{j,1}(\begin{align}r
z,x)^*)^{-1}$ and using \eqref{2.92}, \eqref{2.94}, and similarly,
multiplying
\eqref{4.74} by $\phi_{j,2}(\begin{align}r z,x)^*(\phi_{j,2}(\begin{align}r z,x)^*)^{-1}$
and using \eqref{2.93}, \eqref{2.95} then yields
\begin{align}
\bibitemg(\phi_{j,1}(z,x)^{-1}\theta_{j,1}(z,x)\bibitemg)' &=
\phi_{j,1}(z,x)^{-1}((B_j)_{1,1}(x)-z)(\phi_{j,1}(\begin{align}r z,x)^*)^{-1},
\label{4.75}
\\
\bibitemg(\phi_{j,2}(z,x)^{-1}\theta_{j,2}(z,x)\bibitemg)' &=
\phi_{j,2}(z,x)^{-1}((B_j)_{1,1}(x)+z)(\phi_{j,2}(\begin{align}r z,x)^*)^{-1}.
\label{4.76}
\end{align}
In exactly the same way one derives
\begin{align}
&\bibitemg(\theta_{j,1}(\begin{align}r z,x)^*(\phi_{j,1}(\begin{align}r z,x)^*)^{-1}\bibitemg)' \nonumber \\
&=
(\theta_{j,1}(\begin{align}r z,x)^*(\phi_{j,1}(\begin{align}r z,x)^*)^{-1}\phi_{j,2}(\begin{align}r
z,x)^*-\theta_2(\begin{align}r z,x)^*)((B_j)_{1,1}(x)-z)(\phi_{j,1}(\begin{align}r
z,x)^*)^{-1} \nonumber \\
&= \phi_{j,1}(z,x)^{-1}((B_j)_{1,1}(x)-z)(\phi_{j,1}(\begin{align}r
z,x)^*)^{-1}, \label{4.77}
\\ &\bibitemg(\theta_{j,2}(\begin{align}r z,x)^*(\phi_{j,2}(\begin{align}r z,x)^*)^{-1}\bibitemg)' \nonumber
\\ &=(\theta_{j,2}(\begin{align}r z,x)^*(\phi_{j,2}(\begin{align}r z,x)^*)^{-1}\phi_{j,1}(\begin{align}r
z,x)^*-\theta_{j,1}(\begin{align}r z,x)^*)((B_j)_{1,1}(x)+z)(\phi_{j,2}(\begin{align}r
z,x)^*)^{-1} \nonumber \\
&=\phi_{j,2}(z,x)^{-1}((B_j)_{1,1}(x)+z)(\phi_{j,2}(\begin{align}r z,x)^*)^{-1},
\label{4.78}
\end{align}
using \eqref{2.72}--\eqref{2.75}. Thus, \eqref{4.72} implies
\begin{align}
\phi_{1,1}(\begin{align}r z,x)^*((B_1)_{1,1}(x)-z)^{-1}\phi_{1,1}(z,x)&=
\phi_{2,1}(\begin{align}r z,x)^*((B_2)_{1,1}(x)-z)^{-1}\phi_{2,1}(z,x), \label{4.79} \\
\phi_{1,2}(\begin{align}r z,x)^*((B_1)_{1,1}(x)+z)^{-1}\phi_{1,2}(z,x)&=
\phi_{2,2}(\begin{align}r z,x)^*((B_2)_{1,1}(x)+z)^{-1}\phi_{2,2}(z,x), \label{4.80} \\
\theta_{1,1}(\begin{align}r z,x)^*((B_1)_{1,1}(x)-z)^{-1}\theta_{1,1}(z,x)&=
\theta_{2,1}(\begin{align}r z,x)^*((B_2)_{1,1}(x)-z)^{-1}\theta_{2,1}(z,x),
\label{4.81}
\\
\theta_{1,2}(\begin{align}r z,x)^*((B_1)_{1,1}(x)+z)^{-1}\theta_{1,2}(z,x)&=
\theta_{2,2}(\begin{align}r z,x)^*((B_2)_{1,1}(x)+z)^{-1}\theta_{2,2}(z,x)
\label{4.82}
\end{align}
for a.e.~$x\in (x_0,x_0+a)$. Thus far we only used $B_j\in
L^1([x_0,x_0+R])^{2m\tilde mes 2m}$ for all $R>0$, $j=1,2$ and \eqref{4.40a}.
In the special case $m=1$, each of the equations
\eqref{4.79}--\eqref{4.82} allows for the completion of the proof of \eqref{4.40}.
Indeed, using the fact that
\begin{equation}
\overline{ \phi_{j,k}(\begin{align}r z,x)}=\phi_{j,k}(z,x), \quad
\overline{ \theta_{j,k}(\begin{align}r z,x)}=\theta_{j,k}(z,x), \quad j,k\in\{1,2\},
\label{4.83}
\end{equation}
and taking for instance
\eqref{4.79}, one infers for a.e.~$x\in (x_0,x_0+a)$, that
\begin{equation}
\frac{\phi_{1,1}(z,x)^2}{\phi_{2,1}(z,x)^2}=\frac{(B_1)_{1,1}(x)-z}
{(B_2)_{1,1}(x)-z}. \label{4.84}
\end{equation}
Since all zeros (and poles) of the left-hand side of \eqref{4.84} have
even multiplicity, while all zeros (and poles) of the right-hand side of
\eqref{4.83} are simple, one concludes, assuming only that $B_j\in
L^1([x_0,x_0+R])^{2\tilde mes 2}$ for all $R>0$, $j=1,2$, that
\begin{equation} \label{4.84a}
(B_1)_{1,1}(x)=(B_2)_{1,1}(x) \, \text{ for a.e.~$x\in (x_0,x_0+a)$.}
\end{equation}
Thus for the case $m=1$, we see by \eqref{4.79},
and \eqref{4.80}, \eqref{4.83}, and \eqref{4.84a}, for a.e.~$x\in (x_0,x_0+a)$, that
\begin{equation}\label{4.84b}
\phi_{1,k}^2(z,x) = \phi_{2,k}^2(z,x), \ \ k=1,2.
\end{equation}
Now, \eqref{2.75}, \eqref{4.72}, and \eqref{4.83} show, for a.e.~$x\in (x_0,x_0+a)$,
that
\begin{align}\label{4.84c}
\phi_{1,1}(z,x)\phi_{1,2}(z,x)&= \fracrac{\phi_{1,1}(z,x)}{\theta_{1,1}(z,x)}-
\fracrac{\phi_{1,2}(z,x)}{\theta_{1,2}(z,x)} \nonumber \\
&=\fracrac{\phi_{2,1}(z,x)}{\theta_{2,1}(z,x)}-
\fracrac{\phi_{2,2}(z,x)}{\theta_{2,2}(z,x)}=\phi_{2,1}(z,x)\phi_{2,2}(z,x).
\end{align}
By \eqref{HSa} we see that
\begin{equation}\label{4.84d}
(\phi_{j,1}^2(z,x))' = 2(z-(B_j(x))_{1,1} )\phi_{j,1}(z,x)\phi_{j,2}(z,x)
+ (B_j(x))_{1,2} \phi_{j,1}^2(z,x), \quad j=1,2.
\end{equation}
Thus, by \eqref{4.84a}, \eqref{4.84b}, and \eqref{4.84c},
\begin{equation}\label{4.84e}
(B_1(x))_{1,2} = (B_2(x))_{1,2} \,\text{ for a.e.~$x\in (x_0,x_0+a)$.}
\end{equation}
Together, \eqref{4.84a} and \eqref{4.84e} imply \eqref{4.40} in the
special case $m=1$.
Unfortunately, the case $m>1$ appears to be quite a bit more
involved. To deal with this case we first note that taking determinants
in \eqref{4.79} yields
\begin{equation}
\frac{\det(\phi_{1,1}(\begin{align}r z,x,x_0,\alpha)^*)\det(\phi_{1,1}(z,x,x_0,\alpha))}
{\det(\phi_{2,1}(\begin{align}r z,x,x_0,\alpha)^*)\det(\phi_{2,1}(z,x,x_0,\alpha)}=
\frac{\det((B_1)_{1,1}(x)-zI_m))}{\det((B_2)_{1,1}(x)-zI_m))} \label{4.85}
\end{equation}
for a.e.~$x\in (x_0,x_0+a)$.
Next, we intend to prove that
\begin{equation}
\det((B_1)_{1,1}(x)-zI_m))=\det((B_2)_{1,1}(x)-zI_m)) \,
\text{ for a.e.~$x\in (x_0,x_0+a)$.} \label{4.86}
\end{equation}
Given the fact that $(B_j)_{1,1}(x)$, $j=1,2$, is self-adjoint, showing
\eqref{4.86} is equivalent to showing that $B_1(x)$ and $B_2(x)$ are
unitarily equivalent for a.e.~$x\in (x_0,x_0+a)$. Arguing by
contradiction, we assume that at least one pair of eigenvalues of
$B_1(x)$ and $B_2(x)$ differs. Thus, fixing $x_1\in (x_0,x_0+a)$, let
$\lambda(x_1)$ be an eigenvalue of
$B_1(x_1)$ but not of $B_2(x_1)$. Then
\eqref{4.85} implies, for all $\alpha\in {\mathbb{C}}^{m\tilde mes 2m}$ satisfying
\eqref{2.8e}, that
\begin{equation}
\det(\phi_{1,1}(\lambda(x_1),x_1,x_0,\alpha))=0. \label{4.87}
\end{equation}
Next, for $\lambda\in{\mathbb{R}}$ and $x>x_0$ define
\begin{align}\label{4.88}
N(\lambda,x,\alpha)=&\theta_{1,2}(\lambda,x,x_0,\alpha)\theta_{1,2}
(\lambda,x,x_0,\alpha)^* \nonumber \\
&+\phi_{1,2}(\lambda,x,x_0,\alpha)\phi_{1,2}(\lambda,x,x_0,\alpha)^*.
\end{align}
Then, $N(\lambda,x,\alpha)$ is strictly positive definite,
\begin{equation}
N(\lambda,x,\alpha)>0. \label{4.89}
\end{equation}
Indeed, suppose $Nf=0$ for some $f\in{\mathbb{C}}^m$, then
\begin{equation}
\theta_{1,2}(\lambda)\theta_{1,2}(\lambda)^*f+
\phi_{1,2}(\lambda)\phi_{1,2}(\lambda)^*f=0 \label{4.90}
\end{equation}
implies
\begin{equation}
\theta_{1,2}(\lambda)\theta_{1,2}(\lambda)^*f=0,
\; \phi_{1,2}(\lambda)\phi_{1,2}(\lambda)^*f=0 \label{4.91}
\end{equation}
and hence
\begin{equation}
\theta_{1,2}(\lambda)^*f=0, \; \phi_{1,2}(\lambda)^*f=0. \label{4.92}
\end{equation}
Thus,
\begin{equation}
f=(\theta_{1,1}(\lambda)\phi_{1,2}(\lambda)^*-
\phi_{1,1}(\lambda)\theta_{1,2}(\lambda)^*)f=0 \label{4.93}
\end{equation}
by \eqref{2.95}, and hence $f=0$ proves \eqref{4.89}. Introducing
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$ and $\gamma=(\gamma_1 \;
\gamma_2)\in{\mathbb{C}}^{m\tilde mes 2m}$ defined by
\begin{align}
\gamma_1 &=[\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)
\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^* \label{4.94} \\
& \quad +\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)
\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^*]^{-1/2}
\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0), \nonumber \\
\gamma_2 &=[\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)
\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^* \label{4.95} \\
& \quad +\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)
\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^*]^{-1/2}
\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0), \nonumber
\end{align}
one verifies $\gamma\gamma^*=I_m$ (by \eqref{4.94} and \eqref{4.95}) and
$\gamma J\gamma^*=0$ (by \eqref{2.93}). Thus, $\gamma$ satisfies
\eqref{2.8e}. Next, since
\begin{equation}
\phi_{1,1}(\lambda(x_1),x_1,x_0,\gamma)=
\phi_{1,1}(\lambda(x_1),x_1,x_0,\alpha_0)\gamma_1^*-
\theta_{1,1}(\lambda(x_1),x_1,x_0,\alpha_0)\gamma_2^* \label{4.96}
\end{equation}
as a special case of \eqref{2.96}, one derives
\begin{align}
\phi_{1,1}(\lambda(x_1),x_1,x_0,\gamma)&=
[\phi_{1,1}(\lambda(x_1),x_1,x_0,\alpha_0)
\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^* \nonumber \\
& \quad -\theta_{1,1}(\lambda(x_1),x_1,x_0,\alpha_0)
\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^*]\tilde mes \nonumber \\
& \quad \tilde mes [\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)
\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^* \nonumber \\
& \quad +\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)
\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^*]^{-1/2} \nonumber \\
& =-[\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)
\theta_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^* \label{4.97} \\
& \quad +\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)
\phi_{1,2}(\lambda(x_1),x_1,x_0,\alpha_0)^*]^{-1/2}<0. \nonumber
\end{align}
using \eqref{2.95}. This contradiction to \eqref{4.87} proves
\eqref{4.86}. Hence for $\lambda\in {\mathbb{R}}$ and for a.e.~$x\in (x_0,x_0+a)$
\begin{equation}
\abs{\det(\phi_{1,1}(\lambda,x,x_0,\alpha))}=
\abs{\det(\phi_{2,1}(\lambda,x,x_0,\alpha))},
\label{4.98}
\end{equation}
by \eqref{4.85}. Equation \eqref{4.98} implies that for
a.e.~$x_1\in(x_0,x_0+a)$, the family of Dirac
operators $D_+(\alpha,\alpha_0)$ in $L^2([x_0,x_1])^{2m}$, defined by
\begin{align}
D_+(\alpha,\alpha_0)&=J \frac{d}{dx}-B, \label{4.99} \\
\text{\rm{dom}}(D_+(\alpha,\alpha_0))&=\{\phi\in L^2([x_0,x_1])^{2m} \mid \phi
\in\text{\rm{AC}}([x_0,x_1])^{2m}; \nonumber \\
& \qquad \alpha\phi(x_0)=0, \, \alpha_0\phi(x_1)=0; \,
(J\phi^\prime-B\phi)\in L^2([x_0,x_1])^{2m} \}, \nonumber
\end{align}
with $\alpha_0 = (I_m\;0)$, have identical spectra for all boundary data
$\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying \eqref{2.8e}. Hence,
assuming $B_j\in L^\infty([x_0,x_0+a])^{2m\tilde mes 2m}$, $j=1,2$, one can
apply Theorem~2.3 of Malamud \cite{Ma99} and obtains \eqref{4.40}.
\end{proof}
We should note that Malamud's Theorem~2.3 in \cite{Ma99} only requires
the equality of $m^2+1$ spectra (associated with linearly independent
boundary data indexed by $\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$) in order to
conclude \eqref{4.40}.
There is no particular significance of the rays $\rho_\ell$,
$\ell=1,2$, in Theorem~\ref{t4.10}. Any non-selfintersecting Jordan
arc that tends to infinity in the sectors $\varepsilon\leq\arg(z)\leq
(\pi/2)-\varepsilon$ and $(\pi/2)+\varepsilon\leq\arg(z)\leq
\pi-\varepsilon$ for some $0<\varepsilon<\pi/4$ will do.
\begin{remark} \label{r4.10}
We were not able to prove \eqref{4.40} directly from
\eqref{4.79}--\eqref{4.82}, without resorting to the arguments involving
\eqref{4.98} and \eqref{4.99}. To conclude the proof according to the
Borg-type Theorem~2.3 of Malamud \cite{Ma99} (cf.~also Theorem~4 in
\cite{Ma99a}), requires the introduction of the extra hypothesis
$B_j\in L^\infty([x_0,x_0+a])^{2m\tilde mes 2m}$, $j=1,2$ in
the matrix context $m>1$, since the construction of transformation
operators for Dirac-type systems, to date, uses such an additional
hypothesis on $B$. This extra hypothesis is clearly superfluous in the
case $m=1$. Obviously, one conjectures that this extra hypothesis on $B_j$
should also be redundant in Theorem~\ref{t4.10}, but this appears
to require nontrivial future efforts. In this context it might be
interesting to note that the higher-order expansions
\eqref{4.49}--\eqref{4.61} do not determine $B$ uniquely. An explicit
analysis shows that while they do determine $B_{1,2}$, they only determine
$B_{1,1}^2$, not $B_{1,1}$ itself. So that approach does not aide in
proving \eqref{4.40} (besides, it would require the additional hypotheses
\eqref{4.39a} on $B$).
\end{remark}
The corresponding local uniqueness result in terms of $M(z,x_0,\alpha)$
then reads as follows.
\begin{theorem} \label{t4.11}
Fix $x_0\in{\mathbb{R}}$ and suppose $A_j=I_{2m}$, $B_j\in
L^1_{\text{\rm{loc}}}({\mathbb{R}})^{2m\tilde mes 2m}$, and $B_j=B_j^*$
a.e.~on ${\mathbb{R}}$, $j=1,2$. Suppose also that $B_j$
is in the normal form given in \eqref{4.115} a.e.~on $(x_0,\infty)$,
$j=1,2$. Denote by $M_{j}(z,x_0,\alpha)$, the unique
Weyl-Titchmarsh matrices \eqref{2.62} corresponding to the Dirac-type
operators $D_{j}$, $j=1,2$, in \eqref{2.61}. Then,
\begin{equation}
\text{if for some $a>0$, }\, B_1(x)=B_2(x) \, \text{ for a.e. $x\in
(x_0-a,x_0+a)$,} \label{4.110}
\end{equation}
one obtains
\begin{equation}
\|M_{1}(z,x_0,\alpha)-M_{2}(z,x_0,\alpha)\|_{{\mathbb{C}}^{2m\tilde mes
2m}}\underset{\substack{\abs{z} \to\infty\\ z\in \rho_{+}}}{=}
O\bibitemg(e^{-2\text{\rm Im}(z)a}\bibitemg) \label{4.111}
\end{equation}
along any ray $\rho_+\subset{\mathbb{C}}_+$ with $0<\arg(z)<\pi$ and for all
$\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying \eqref{2.8e}. Conversely,
fix a $\hat\alpha\in{\mathbb{C}}^{m\tilde mes 2m}$ satisfying
\eqref{2.8e} and if $m>1$, assume in addition that
$B_j\in L^\infty([x_0-a,x_0+a])^{2m\tilde mes 2m}$, $j=1,2$. Moreover, suppose
that for all $\varepsilon >0$,
\begin{equation}
\|M_{1}(z,x_0,\hat\alpha_1)-M_{2}(z,x_0,\hat\alpha_1)\|_{{\mathbb{C}}^{2m\tilde mes
2m}}\underset{\substack{|z|\to\infty\\z\in \rho_{+,\ell}}}{=}
O\bibitemg(e^{-2\text{\rm Im}(z)(a-\varepsilon)}\bibitemg), \quad \ell=1,2, \label{4.112}
\end{equation}
along a ray $\rho_{+,1}\subset{\mathbb{C}}_+$ with $0<\arg(z)<\pi/2$ and along a
ray $\rho_{+,2}\subset{\mathbb{C}}_+$ with $\pi/2<\arg(z)<\pi$. Then
\begin{equation}
B_1(x)=B_2(x) \text{ for a.e. } x\in [x_0-a,x_0+a]. \label{4.113}
\end{equation}
\end{theorem}
\begin{proof}
\eqref{4.111} is proved by combining \eqref{2.620}, and
\eqref{4.38AA}, \eqref{4.38}, and \eqref{4.113} then follows by combining
\eqref{2.620}, and \eqref{4.39}, \eqref{4.40}, taking into
account the asymptotic expansions
\begin{equation}
M_\pm (z,x_0)\underset{|z|\to\infty}{=}\pm iI_m + o(1) \label{4.114}
\end{equation}
along any ray with $\varepsilon<\arg(z)<\pi-\varepsilon$ in the case
of Dirac-type operators (cf.~\eqref{3.1}).
\end{proof}
\begin{remark} \label{r4.12}
Theorem~\ref{t4.10} and Theorem~\ref{t4.11} yield new global
uniqueness theorems for half-line and full-line Dirac-type operators,
extending the classical Borg-Marchenko-type results. Indeed,
if \eqref{4.39} (resp., \eqref{4.112}) holds for all $a>0$, then
\eqref{4.40} (resp. \eqref{4.113}) holds for a.e.~$x\in[x_0,\infty)$
(resp., for a.e.~$x\in{\mathbb{R}}$).
\end{remark}
In the case of scalar Schr\"odinger operators, the analog of
Theorem~\ref{t4.10} is due to Simon \cite{Si98}. An alternative proof,
applicable to matrix-valued Schr\"odinger operators was presented in
\cite{GS99} (cf.~also \cite{GKM00}). More recently, yet another proof was
found by Bennewitz \cite{Be00} (following some ideas in \cite{Bo52}).
These results extend the classical (global) uniqueness results due to
Borg \cite{Bo52} and Marchenko \cite{Ma50}, \cite{Ma52} (cf. also
\cite{Be00a}), which state that half-line $m$-functions uniquely
determine the corresponding potential coefficient. The Dirac-type results
presented in this section (especially, all local considerations) appear
to be new, even in the special case $m=1$. Previous results in the Dirac
case focused on global uniqueness questions only. We refer to Gasymov and
Levitan \cite{GL66} in the case $m=1$ and to Lesch and Malamud
\cite{LM00} in the matrix case $m\in{\mathbb{N}}$. Most recently, Alexander
Sakhnovich kindly informed us that his integral representation of the
Weyl-Titchmarsh matrix in \cite{Sa88a} can be used to derive asymptotic
expansions for the Weyl-Titchmarsh matrix and its associated
matrix-valued spectral function, and also yields a result analogous to
Theorem~\ref{t4.10}\,(i) for a certain class of canonical systems.
Moreover, in the case of skew-adjoint Dirac-type systems, similar results
are discussed in \cite{Sa90} and applied to the nonlinear Schr\"odinger
equation on a half-axis.
Although not directly used in this paper, it should be pointed out that
inverse monodromy problems for canonical systems received a lot of
attention (some of it very recently). The interested reader is referred
to \cite{AD97}, \cite{AD00}, \cite{AD00a}, \cite{Ma95}, \cite{Ma99},
\cite{Ma99a}, \cite{Sa94}, \cite{Sa99a} and the extensive literature
cited therein. Moreover, inverse spectral theory associated with
canonical systems is discussed in \cite{MST01}, \cite{Sa90}, \cite{Sa97b},
\cite{Sa01}, \cite{Sa94}, \cite{Sa94a}, \cite{Sa99}, \cite{Sa99a} (see
also the extensive literature cited in \cite{GKM00}).
\section{Trace Formulas and Borg-Type Theorems}\label{s6}
In our final section we derive a trace formula for $B$ and then discuss
its application to Borg-type uniqueness theorems for Dirac-type
operators.
\begin{theorem} \label{t5.1}
Assume Hypothesis~\ref{h2.1} with $A=I_{2m}$, and let
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$. Fix $x_0\in{\mathbb{R}}$ and suppose
that for all $R>0$,
\begin{align}
& \quad \underset{y\in [x_0,x_0+R]}{\text{\rm{ess}}sup} \,
\bibitemgg\|\int_y^{x_0+R} dx'\,B(x')\exp(2iz(x'-y))
+\frac{1}{2iz}B(y)\bibitemgg\|_{{\mathbb{C}}^{2m\tilde mes 2m}} \nonumber \\
& + \underset{y\in [x_0-R,x_0]}{\text{\rm{ess}}sup} \,
\bibitemgg\|\int_{x_0-R}^{y} dx'\,B(x')\exp(2iz(x'-y))
-\frac{1}{2iz}B(y)\bibitemgg\|_{{\mathbb{C}}^{2m\tilde mes 2m}} \nonumber \\
& \underset{\substack{\abs{z}
\to\infty\\ z\in \rho_+}}{=}o\bibitemg(|z|^{-1}\bibitemg) \label{5.0}
\end{align}
along a ray $\rho_+\subset{\mathbb{C}}_+$. In addition, assume
$B_{k,k'}B_{\ell,\ell'}\in
L^1_{\text{\rm{loc}}}({\mathbb{R}})^{m\tilde mes m}$ for all
$k,k',\ell,\ell'\in\{1,2\}$. Then, with $\Upsilon(\lambda,x,\alpha_0)$
defined in \eqref{2.69},
\begin{align}
& \begin{pmatrix}B_{1,1}(x)-B_{2,2}(x) &
B_{1,2}(x)+B_{2,1}(x) \\
B_{1,2}(x)+B_{2,1}(x)& B_{2,2}(x)-B_{1,1}(x)\end{pmatrix} \nonumber \\
&=\lim_{\substack{|z|\to \infty\\z\in\rho_+}} 2 \int_{\mathbb{R}}
d\lambda \, z^2(\lambda-z)^{-2}\Upsilon(\lambda,x,\alpha_0)
\, \text{ for a.e. $x\in{\mathbb{R}}$}. \label{5.1}
\end{align}
\end{theorem}
\begin{proof}
By \eqref{2.64},
\begin{equation}
\frac{d}{dz}\text{\rm ln}(M(z,x,\alpha_0)) =\int_{\mathbb{R}} d\lambda\,
(\lambda-z)^{-2}\Upsilon(\lambda,x,\alpha_0). \label{5.2}
\end{equation}
Next, suppose that $x\in{\mathbb{R}}$ is a left and right Lebesgue point of $B$.
By \eqref{4.103}, \eqref{4.104} one obtains
\begin{align}
&\frac{d}{dz}\text{\rm ln}(M(z,x,\alpha_0)) \nonumber \\
&\underset{\substack{|z|\to \infty\\z\in\rho_+}}{=}\frac{1}{4}
\begin{pmatrix}B_{1,1}(x+0)-B_{2,2}(x+0) & B_{1,2}(x+0)+B_{2,1}(x+0) \\
B_{1,2}(x+0)+B_{2,1}(x+0)&
B_{2,2}(x+0)-B_{1,1}(x+0)\end{pmatrix}z^{-2} \label{5.3} \\
& \quad \;\,\, +\frac{1}{4}
\begin{pmatrix}B_{1,1}(x-0)-B_{2,2}(x-0) & B_{1,2}(x-0)+B_{2,1}(x-0) \\
B_{1,2}(x-0)+B_{2,1}(x-0)&
B_{2,2}(x-0)-B_{1,1}(x-0)\end{pmatrix}z^{-2} +o\bibitemg(z^{-2}\bibitemg) \nonumber
\end{align}
and hence
\begin{align}
& \quad \, \frac{1}{2}\begin{pmatrix}B_{1,1}(x+0)-B_{2,2}(x+0) &
B_{1,2}(x+0)+B_{2,1}(x+0) \\
B_{1,2}(x+0)+B_{2,1}(x+0)& B_{2,2}(x+0)-B_{1,1}(x+0)\end{pmatrix} \nonumber \\
&+\frac{1}{2}\begin{pmatrix}B_{1,1}(x-0)-B_{2,2}(x-0) &
B_{1,2}(x-0)+B_{2,1}(x-0) \\
B_{1,2}(x-0)+B_{2,1}(x-0)& B_{2,2}(x-0)-B_{1,1}(x-0)\end{pmatrix} \nonumber \\
&=\lim_{\substack{|z|\to \infty\\z\in\rho_+}} 2 \int_{\mathbb{R}}
d\lambda \, z^2(\lambda-z)^{-2}\Upsilon(\lambda,x,\alpha_0). \label{5.2a}
\end{align}
Since a.e. $x\in{\mathbb{R}}$ is a Lebesgue point of $B$, one concludes
\eqref{5.1}.
\end{proof}
In the case $m=1$, a trace formula for Dirac-type operators, using Krein
spectral shift functions and exponential representations of Herglotz
functions, was discussed in \cite{Ti95}. This circle of ideas was first
introduced in connection with trace formulas of Schr\"odinger operators
in \cite{GS96} (see also \cite{GHS95}, \cite{GHSZ95}, \cite{Ry99},
\cite{Ry99a} in the scalar case $m=1$. The corresponding case of trace
formulas for matrix-valued Schr\"odinger operators was introduced in
\cite{GH97} (see also \cite{CGHL00}).
Analogous trace formulas can be drived for all higher-order coefficients
$M_k(x,\alpha_0)$ in \eqref{4.103} (see, e.g., \cite{GHSZ95} in connection
with scalar Schr\"odinger operators).
A comparison of the trace formula (3.20) in
\cite{CGHL00} for Schr\"{o}dinger operators with its Dirac-type
counterpart \eqref{5.1} reveals characteristic differences. While in the
Schr\"{o}dinger case the trace formula directly involves the potential
coefficient $Q(x)$, $M_1(x,\alpha_0)$ differs markedly from a constant
multiple of $B(x)$, and consequently, the Dirac-type trace formula
\eqref{5.1} does not directly involve $B(x)$ but certain linear
combinations of $B_{j,k}(x)$. This is related to the fact that
$M(z,x_0,\alpha_0)$ (or equivalently, $\Upsilon(\lambda,x_0,\alpha_0)$),
in general, does not uniquely determine $B$ a.e. In fact, there exists a
typical ambiguity concerning the coefficients of $D$ related to unitary
gauge-transformations of $D$. In the case $m=1$ this ambiguity is
well-known and discussed, e.g., in \cite{GL66}, \cite[Sect.~I.10]{LS75},
\cite[Ch.~7]{LS91}. These gauge transformations leave the spectrum of
$D$ invariant and suggest that we focus our attention on certain normal
forms of $D$ in connection with inverse spectral problems for Dirac-type
operators.
\begin{lemma} \label{l5.2}
Assume Hypothesis~\ref{h2.4}. Then $D=J\frac{d}{dx}-B$ is unitarily
equivalent to $\widetilde D$, where $\widetilde D$ in $L^2({\mathbb{R}})^{2m}$ is of the
normal form
\begin{equation}
\widetilde D=J\frac{d}{dx}-\widetilde B= \begin{pmatrix} -\widetilde B_{1,1} & -I_m
\frac{d}{dx}- \widetilde B_{1,2} \\[1mm]
I_m \frac{d}{dx}-\widetilde B_{1,2}& \widetilde B_{1,1} \end{pmatrix}. \label{5.4}
\end{equation}
Here $\widetilde B=\widetilde B^*$ a.e. and
\begin{align}
\widetilde B_{1,1}&=-(1/2)\text{\rm Im}\bibitemg(U_{1,1}^{-1}[(B_{1,2}+B_{2,1})
-i(B_{1,1}-B_{2,2})] U_{2,2}\bibitemg)= \widetilde B_{1,1}^*, \label{5.5} \\
\widetilde B_{1,2}&=(1/2)\text{\rm Re}\bibitemg(U_{1,1}^{-1}[(B_{1,2}+B_{2,1})
-i(B_{1,1}-B_{2,2})] U_{2,2}\bibitemg)=\widetilde B_{1,2}^*, \label{5.6}
\end{align}
with $U_{j,j}\in{\mathbb{C}}^{m\tilde mes m}$, $j=1,2$, satisfying the
first-order system of ordinary differential equations
\begin{align}
iU_{j,j}^\prime(x)&=-(1/2)\bibitemg((-1)^j(B_{1,1}(x)
+B_{2,2}(x))+i(B_{1,2}(x)-B_{2,1}(x)) \bibitemg)U_{j,j}(x), \nonumber \\
&\hspace*{6cm} \text{for a.e. $x\in{\mathbb{R}}$}, \quad j=1,2. \label{5.7}
\end{align}
\end{lemma}
\begin{proof}
We start with the unitary transformation $V$ in $L^2({\mathbb{R}})^{2m}$ defined
by
\begin{equation}
V=\frac1{\sqrt{2}}\begin{pmatrix} i I_m & I_m \\ I_m & i I_m \end{pmatrix},
\quad V^{-1}=\frac1{\sqrt{2}}\begin{pmatrix} -i I_m & I_m \\ I_m & -i I_m
\end{pmatrix}, \label{5.8}
\end{equation}
which maps $D$ to $D_1$, where
\begin{align}
D_1&=V^{-1}DV= i \begin{pmatrix} I_m\frac{d}{dx} & 0 \\
0 & -I_m\frac{d}{dx} \end{pmatrix} \nonumber \\
& \quad -\frac12 \begin{pmatrix}B_{1,1}+B_{2,2}-i(B_{1,2}-B_{2,1}) &
B_{1,2}+B_{2,1}-i(B_{1,1}-B_{2,2}) \\ B_{1,2}+B_{2,1}+i(B_{1,1}-B_{2,2}) &
B_{1,1}+B_{2,2}+i(B_{1,2}-B_{2,1})\end{pmatrix}. \label{5.9}
\end{align}
Next, we introduce the unitary operator $U$ in $L^2({\mathbb{R}})^{2m}$ defined
by
\begin{equation}
U=\begin{pmatrix}U_{1,1} & 0 \\ 0 & U_{2,2}\end{pmatrix}, \label{5.10}
\end{equation}
where the unitary $m\tilde mes m$ matrices $U_{j,j}\in{\mathbb{C}}^{m\tilde mes m}$ are
solutions of the first-order system \eqref{5.7}. Since by hypothesis
$B_{j,k}\in L^1_{\text{\rm{loc}}}({\mathbb{R}})^{m\tilde mes m}$, $1\leq j,k \leq 2$, the
solutions of equation \eqref{5.7} are well-defined and
$U_{j,j}\in\text{\rm{AC}}_{\text{\rm{loc}}}({\mathbb{R}})^{m\tilde mes m}$, $j=1,2$. One computes
\begin{equation}
\widehat D=U^{-1} D_1 U= \begin{pmatrix}i I_m \frac{d}{dx} & -\widehat B_{1,2}
\\[1mm] -\widehat B_{1,2}^* & -i I_m \frac{d}{dx}\end{pmatrix}, \label{5.11}
\end{equation}
where $\widehat B_{1,2}\in L^1_{\text{\rm{loc}}}({\mathbb{R}})^{m\tilde mes m}$ and
\begin{equation}
\widehat B_{1,2}(x)=(1/2) U^{-1}_{1,1}(x)\bibitemg(B_{1,2}(x)+B_{2,1}(x)
-i(B_{1,1}(x)-B_{2,2}(x)\bibitemg)U_{2,2}(x). \label{5.12}
\end{equation}
Finally, defining $\widetilde D=V\widehat D V^{-1}$, one arrives at
\eqref{5.4}--\eqref{5.6}.
\end{proof}
Thus, unitary invariants of $D$ (such as the spectrum, $\text{\rm{spec}} (D)$, of $D$
and its multiplicity) cannot determine $B$ in general but at best a
potential matrix of the type (normal form) $\widetilde B$ in \eqref{5.4}. A
further restriction on the solvability of inverse spectral
problems for Dirac-type operators is mentioned in the following result.
\begin{lemma} \label{l5.3}
Assume Hypotheses~\ref{h2.4} and let
$\omega=\omega^*\in{\mathbb{C}}^{m\tilde mes m}$ be a constant self-adjoint $m\tilde mes
m$ matrix. Then $D=J\frac{d}{dx}-B$ is unitarily equivalent to $\widetilde
D_\omega$ in $L^2({\mathbb{R}})^{2m}$, where
\begin{equation}
\widetilde D_\omega=J\frac{d}{dx}-\widetilde B_\omega=\begin{pmatrix} -\widetilde
B_{\omega,1,1} & -I_m\frac{d}{dx}-\widetilde B_{\omega,1,2} \\[1mm]
I_m\frac{d}{dx}-\widetilde B_{\omega,1,2} & \widetilde B_{\omega,1,1}\end{pmatrix},
\label{5.13}
\end{equation}
with
\begin{align}
\widetilde B_{\omega,1,1}&=-(1/2)\text{\rm Im}\bibitemg(e^{i\omega}U_{1,1}^{-1}[(B_{1,2}
+B_{2,1})-i(B_{1,1}-B_{2,2})]
U_{2,2}e^{i\omega}\bibitemg)=\widetilde B_{\omega,1,1}^*, \nonumber \\
\widetilde B_{\omega,1,2}&=(1/2)\text{\rm Re}\bibitemg(e^{i\omega}U_{1,1}^{-1}[(B_{1,2}
+B_{2,1})-i(B_{1,1}-B_{2,2})] U_{2,2}e^{i\omega}\bibitemg)=\widetilde
B_{\omega,1,2}^*, \label{5.14}
\end{align}
and with $U_{j,j}$, $j=1,2$, satisfying the first-order system
\eqref{5.7}.
\end{lemma}
\begin{proof}
Define
\begin{equation}
U_\omega=\begin{pmatrix}e^{i\omega}& 0 \\
0 & e^{-i\omega} \end{pmatrix}. \label{5.15}
\end{equation}
Using the notation employed in the proof of Lemma~\ref{l5.2} one verifies
that
\begin{equation}
\widetilde D_\omega=V U_\omega (VU)^{-1}DVU(VU_\omega)^{-1}. \label{5.16}
\end{equation}
\end{proof}
In particular, choosing $\omega=(\pi/2)I_m$ effects the sign change
$\widetilde B\to -\widetilde B$, with $\widetilde B$ given by \eqref{5.5}, \eqref{5.6}.
For detailed discussions of various normal forms for Dirac-type operators
we refer to \cite{GL66}, \cite{HJKS91}, \cite[Ch.~9]{LS75},
\cite[Ch.~7]{LS91} in the case $m=1$ and to \cite{Ga68}, \cite{LM00},
\cite{Ma99}, \cite[p.~193--195]{Ma86}, \cite{Ma65} in the general
matrix-valued case. Perhaps it should be noted that if $D$ is in its
normal form $\widetilde D$ as in \eqref{5.4}, ${\widetilde D}^2$ turns into a
$2m\tilde mes 2m$ matrix-valued Schr\"odinger operator under appropriate
regularity assumptions on $\widetilde B$. Details on this fact and the relation
between the $M$-matrices of $\widetilde D$ and ${\widetilde D}^2$ can be found in
Section~3 of \cite{GKM00}.
Next, we turn to Borg-type theorems, one of the principal topics of this
paper. In 1946 Borg \cite{Bo46} proved, among a variety of other inverse
spectral theorems, the following result.
\begin{theorem}[\cite{Bo46}] \label{t5.4}
Assume $q\in L^2_{\text{\rm{loc}}} ({\mathbb{R}})$ to be real-valued and periodic and let
\begin{equation}
h=-d^2/dx^2+q \label{5.16a}
\end{equation}
be the associated self-adjoint Schr\"odinger operator in
$L^2({\mathbb{R}})$. Moreover, suppose that $\text{\rm{spec}} (h)=[e_0,\infty)$ for some
$e_0\in{\mathbb{R}}$. Then
\begin{equation}
q(x)=e_0 \, \text{ for a.e. $x\in{\mathbb{R}}$}. \label{5.17}
\end{equation}
\end{theorem}
The analog of Theorem~\ref{t5.4} for Dirac-type operators (in the case
$m=1$) was proven by Giacheti and Johnson \cite{GJ84} in 1984 (see also
\cite{Ge89}, \cite{Ge91}, \cite{GSS91} in the special case where $p$ is
constant and \cite{GG93} in the case where $p,q\in L^2 ({\mathbb{R}})$ are
real-valued and periodic).
\begin{theorem}[\cite{GJ84}] \label{t5.4a}
Assume $p,q\in L^\infty ({\mathbb{R}})$ to be real-valued and periodic and let
\begin{equation}
d=\begin{pmatrix}-p &-\frac{d}{dx}-q\\ \frac{d}{dx}-q & p\end{pmatrix}
\label{5.17a}
\end{equation}
be the associated self-adjoint Dirac-type operator in
$L^2({\mathbb{R}})^2$. Moreover, suppose that $\text{\rm{spec}} (d)={\mathbb{R}}$. Then
\begin{equation}
p(x)=q(x)=0 \, \text{ for a.e. $x\in{\mathbb{R}}$}. \label{5.17b}
\end{equation}
\end{theorem}
Traditionally, uniqueness results such as Theorems~\ref{t5.4} and
\ref{t5.4a} are called Borg-type theorems. (However, this terminology is
not uniquely adopted and hence a bit unfortunate. Indeed, inverse spectral
results on finite intervals recovering the potential coefficient(s) from
several spectra, were also pioneered by Borg in his celebrated paper
\cite{Bo46}, and hence are also coined Borg-type theorems in the
literature, see, e.g., \cite{Ma94}, \cite{Ma99}, \cite{Ma99a}.)
A quick and natural proof of Theorem~\ref{t5.4}, based on a trace formula
for $q$, was presented in \cite{CGHL00}. This strategy of proof was then
applied to the case of matrix-valued Schr\"odinger operators and the
corresponding matrix-valued analog of Theorem~\ref{t5.4} was also proved
in \cite{CGHL00} along these lines. A closer examination of the proof of
Theorem~\ref{t5.4} shows that periodicity of $q$ is not the crucial element
in the proof of the
uniqueness result \eqref{5.17}. The key ingredient (besides
$\text{\rm{spec}} (h)=[e_0,\infty)$) is clearly
the fact that for all $x\in{\mathbb{R}}$,
\begin{equation}
\xi(\lambda,x)=1/2 \, \text{ for a.e. } \lambda\in\text{\rm{spec}}_{\text{\rm{ess}}}(h)
\label{5.18}
\end{equation}
($\text{\rm{spec}}_{\text{\rm{ess}}}(\,\cdot\,)$ the essential spectrum), where $\xi(\cdot,x)$ is
defined by
\begin{equation}
\xi(\lambda,x)=\lim_{\varepsilon\to
0}\pi^{-1}\text{\rm Im}(\text{\rm ln}(g(\lambda+i\varepsilon,x))) \, \text{ for a.e.
$\lambda\in{\mathbb{R}}$}, \label{5.19}
\end{equation}
and $g(z,x)$ denotes Green's function (i.e., the integral kernel of the
resolvent) of $h$ on the diagonal,
\begin{equation}
g(z,x)=(h-z)^{-1}(x,x). \label{5.20}
\end{equation}
Completely analogous considerations apply to the Dirac-type case.
Real-valued periodic potentials are known to satisfy \eqref{5.18} but so
are certain classes of real-valued quasi-periodic and almost-periodic
potentials $q$ (see, e.g., \cite{CJ87}, \cite{Cr89}, \cite{DS83},
\cite{Jo82}, \cite{JM82}, \cite{Ko84}, \cite{Ko87a},
\cite{KK88}, \cite{KS88}, \cite{SY95}). In particular, the class of
real-valued algebro-geometric finite-gap potentials $q$ (a subclass of
the set of real-valued quasi-periodic potentials) is a prime example
satisfying
\eqref{5.18} without necessarily being periodic. Traditionally,
potentials $q$ satisfying \eqref{5.18} are called \textit{reflectionless}
(see \cite{Cr89}, \cite{DS83}, \cite{KK88}, \cite{SY95}).
Again the analogous notions apply to the Dirac-type case (cf., e.g.,
\cite{CJ87}, \cite{GJ84}, \cite{Jo87}).
Taking this circle of ideas as the point of departure for our derivation
of Borg-type results for Dirac-type operators, we now use the
reflectionless situation described in \eqref{5.18}, actually, its proper
analog for Dirac-type systems, as the model for the subsequent definition.
\begin{definition}\label{d5.5}
Assume Hypothesis~\ref{h2.1} with $A=I_{2m}$, and let
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$. Then $B$ is called
{\it reflectionless} if for all $x\in{\mathbb{R}}$,
\begin{equation}
\Upsilon(\lambda,x,\alpha_0)= (1/2) I_{2m} \, \text{ for a.e.\
$\lambda\in\text{\rm{spec}}_{\text{\rm{ess}}}(D)$}. \label{5.21}
\end{equation}
\end{definition}
Since hardly any confusion can arise, we will also call the Dirac-type
operator $D$ reflectionless if \eqref{5.21} is satisfied.
Given Definition~\ref{d5.5}, we turn to a Borg-type uniqueness theorem
and formulate the analog of Theorem~\ref{t5.4} for (reflectionless)
Dirac-type operators.
\begin{theorem}\label{t5.6}
Assume Hypothesis \ref{h2.1} with $A=I_{2m}$, and let
$\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{m\tilde mes 2m}$. If for all $x\in{\mathbb{R}}$,
$\Upsilon(\lambda,x,\alpha_0)=C$ is a constant $2m\tilde mes 2m$ matrix for
a.e.\ $\lambda\in{\mathbb{R}}$, especially, if $B$ is reflectionless and
$\text{\rm{spec}}(D)={\mathbb{R}}$, then
\begin{equation}
B_{1,1}(x)=B_{2,2}(x), \quad B_{1,2}(x)=-B_{2,1}(x) \, \text{ for a.e.
$x\in{\mathbb{R}}$}. \label{5.22}
\end{equation}
In particular, if $D$ is assumed to be in its normal form \eqref{5.4},
that is, of the type $\widetilde D = J\frac{d}{dx} - \widetilde B$, then
\begin{equation}
\widetilde B (x)=0 \text{ for a.e. $x\in{\mathbb{R}}$}. \label{5.23}
\end{equation}
\end{theorem}
\begin{proof}
The fact that $\int_{\mathbb{R}} d\lambda\, (\lambda-z)^{-2}=0$ for all
$z\in{\mathbb{C}}\begin{align}ckslash{\mathbb{R}}$, that a.e.~$x\in{\mathbb{R}}$ is a Lebesgue point of
$B$, and the trace formula \eqref{5.1}, imply \eqref{5.22}. Together with
Lemma \ref{l5.2} this yields \eqref{5.23}.
\end{proof}
The analog of Theorem~\ref{t5.6} for matrix-valued Schr\"odinger
operators was recently proved in \cite{CGHL00}.
In the remainder of the section we will show that the
case of periodic $B$ is covered by Theorem~\ref{t5.6} under appropriate
uniform multiplicity assumptions on $\text{\rm{spec}}(D)$. In order to handle Floquet
theoretic aspects of periodic Dirac-type operators $D$, we adopt the
following assumptions until the end of this section.
\begin{hypothesis} \label{h5.7}
In addition to Hypothesis~\ref{h2.1} assume $A=I_{2m}$ and suppose that
$B$ is periodic, that is, there is an $\omega>0$ such that
$B(x+\omega)=B(x)$ for a.e.~$x\in{\mathbb{R}}$.
\end{hypothesis}
The following result has been proven in \cite[Theorem~4.6]{CGHL00}.
\begin{theorem} [\cite{CGHL00}, Theorem~4.6] \label{t5.8}
Assume Hypothesis~\ref{h5.7} and let $\alpha_0=(I_m\; 0)\in{\mathbb{C}}^{2m\tilde mes
m}$. If $D$ has uniform spectral multiplicity $2m$, then for all
$x\in{\mathbb{R}}$ and all $\lambda\in\text{\rm{spec}}(D)^o$,
\begin{equation}
M_+(\lambda+i0,x,\alpha_0)=M_-(\lambda+i0,x,\alpha_0)^*
=M_-(\lambda-i0,x,\alpha_0).
\label{5.24}
\end{equation}
In particular, $M_-(z,x,\alpha_0)$ is the analytic continuation of
$M_+(z,x,\alpha_0)$ {\rm (}and vice versa{\rm )} through $\text{\rm{spec}}(D)^o$.
\end{theorem}
\nonumberindent Here $A^o$ denotes the open interior of a set $A\subseteq{\mathbb{R}}$.
Strictly speaking, Theorem~4.6 in \cite{CGHL00} was proved for
matrix-valued Schr\"odinger operators. But the proof extends line by line
to the corresponding Dirac-type situation and was predominantly
formulated in terms of Hamiltonian systems notation (rather than
Schr\"odinger operator specifics) in order to be applicable to the
present context. In particular, the spectrum, $\text{\rm{spec}}(H)$, of the
Schr\"odinger operator $H$ should be replaced by that of $D$, the point
spectrum, $\text{\rm{spec}}_p(H^D_{x_0})$, of the Dirichlet Schr\"odinger operator
$H^D_{x_0}$ with a Dirichlet boundary condition at the point $x_0$ should
simply be replaced by the set $\{\lambda\in{\mathbb{R}}\,|\,
\det(\phi_1(\lambda,x_0+\omega,x_0,\alpha_0))=0\}$, etc.
\begin{theorem} \label{t5.9}
Suppose Hypothesis~\ref{h5.7} and let $\alpha_0=(I_m\; 0)
\in{\mathbb{C}}^{2m\tilde mes m}$. If $D$ has uniform spectral multiplicity
$2m$, then $D$ is reflectionless and for all $x\in{\mathbb{R}}$ and all
$\lambda\in\text{\rm{spec}}(D)^o$,
\begin{equation}
\Upsilon(\lambda,x,\alpha_0)= (1/2) I_{2m}. \label{5.30}
\end{equation}
\end{theorem}
\begin{proof}
This is clear from \eqref{2.620} and \eqref{5.24}, which imply
\begin{equation}
M(\lambda+i0,x,\alpha_0)=-M(\lambda+i0,x,\alpha_0)^*. \label{5.27}
\end{equation}
\end{proof}
Theorems~\ref{t5.8} and \ref{t5.9} extend to more general
situations (not necessarily periodic ones) as is clear from the
corresponding results in \cite{CJ87}, \cite{GJ84}, \cite{GKT96},
\cite{Ko84}, \cite{Ko87a}, \cite{KK88}, \cite{SY95} in
the scalar case $m=1$ (replacing the phrase ``for all
$\lambda\in\text{\rm{spec}}(D)^o$'' by ``for~a.e.~$\lambda\in\text{\rm{spec}}(D)^o$'', etc.).
For the corresponding matrix-valued Schr\"odinger operator case we refer
to \cite{KS88}.
\begin{corollary} \label{c5.10}
Assume Hypothesis~\ref{h5.7}. If $D$ has uniform spectral multiplicity
$2m$ and $\text{\rm{spec}}(D)={\mathbb{R}}$, then
\begin{equation}
B_{1,1}(x)=B_{2,2}(x), \quad B_{1,2}(x)=-B_{2,1}(x) \, \text{ for a.e.
$x\in{\mathbb{R}}$}. \label{5.31}
\end{equation}
In particular, if $D$ is assumed to be in its normal form $\widetilde D =
J\frac{d}{dx} - \widetilde B$, with $\widetilde B$ given by \eqref{5.4}, then
\begin{equation}
\widetilde B (x)=0 \, \text{ for a.e. $x\in{\mathbb{R}}$}. \label{5.32}
\end{equation}
\end{corollary}
\begin{remark} \label{r5.11}
The assumption of uniform (maximal) spectral multiplicity $2m$ in
Corollary~\ref{c5.10} is an essential one. Otherwise, one can easily
construct nonconstant potentials $B$ such that the associated operator
$D$ has overlapping band spectra and hence spectrum the whole real line.
Also self-adjointness of $B$ is crucial for Corollary~\ref{c5.10} to hold
(cf.~the corresponding discussion in Remark~4.2 of \cite{CGHL00} in the
context of Schr\"odinger operators).
\end{remark}
The analog of Corollary~\ref{c5.10} for periodic matrix-valued
Schr\"odinger operators was first proved by Depres \cite{De95} and
recently rederived using such a trace formula approach in \cite{CGHL00}.
We note that all results presented in this paper also apply to
matrix-valued finite-difference Hamiltonian systems. We refer the
reader to \cite{CGR01} in this direction.
Finally, Borg-type uniqueness theorems for Hamiltonian systems are just a
beginning. There is a natural extension of Borg's Theorem~\ref{t5.4} to
self-adjoint periodic Schr\"{o}dinger, respectively, Dirac-type
operators with one gap, respectively, two gaps in their spectrum. In the
case of (scalar) Schr\"odinger operators, such an extension is due to
Hochstadt \cite{Ho65} and the resulting potential $q$ becomes twice the
elliptic Weierstrass function. In the case of Dirac-type operators (with
$m=1$ and vanishing diagonal coefficients in $B$) such an extension
involving elliptic functions can be found in \cite{Ge89}, \cite{Ge91},
\cite{GSS91} (see also \cite{GW98}). Extensions to matrix-valued versions
(i.e., for
$m\geq 2$) are currently under active investigations.
\vspace*{3mm}
\nonumberindent {\bf Acknowledgements.}
We would like to thank Suzanne Collier, Helge Holden, Konstantin Makarov,
Fedor Rofe-Beketov, Alexei Rybkin, Lev Sakhnovich, and Barry Simon for
helpful discussions and many hints regarding the literature, and
especially, Don Hinton, Boris Levitan, Mark Malamud, and Alexander
Sakhnovich for repeated correspondence on various parts of the material
in this paper. \\
S.~C. would like to thank the Mathematics Department of the
University of Missouri-Columbia for the great hospitality extended to
him during his 2000/2001 sabbatical when this work was completed.
\end{document} |
\begin{document}
\title[Multilinear pseudo-differential operators]
{Boundedness of multilinear pseudo-differential operators with $S_{0,0}$ class symbols on Besov spaces}
\author[N. Shida]{Naoto Shida}
\date{\today}
\address[N. Shida]
{Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,
Aichi, 464-8602, Japan}
\email[N. Shida]{[email protected]}
\keywords{Multilinear pseudo-differential operators, multilinear H\"ormander symbol classes, Besov space}
\thanks{This work was supported by Grant-in-Aid for JSPS KAKENHI Fellows,
Grant Numbers 23KJ1053.}
\subjclass[2020]{35S05, 42B15, 42B35}
\begin{abstract}
We consider multilinear pseudo-differential operators with symbols in the multilinear H\"ormander class $S_{0, 0}$.
The aim of this paper is to discuss the boundedness of these operators in the settings of Besov spaces.
\end{abstract}
\maketitle
\section{Introduction}
For a bounded function $\sigma = \sigma(x, \xi_1, \dots, \xi_N)$ on $(\mathbb{R}^n)^{N+1}$, the ($N$-fold) multilinear pseudo-differential operator is defined by
\[
T_\sigma(f_1, \dots, f_N)(x)
=
\frac{1}{(2\pi)^{Nn}}
\int_{(\mathbb{R}^n)^N}
e^{i x \cdot (\xi_1 + \dots + \xi_N)}
\sigma(x, \xi_1, \dots, \xi_N)
\prod_{j=1}^N
\widehat{f}_j(\xi_j)
\,
d\xi_1
\dots
d\xi_N,
\]
where $x \in \mathbb{R}^n$, $f_j \in \mathcal{S}(\mathbb{R}^n)$, $j=1, \dots, N$, and $\widehat{f}_j$ denotes the Fourier transform of $f_j$.
For $m \in \mathbb{R}$,
the symbol class $S^m_{0, 0}(\mathbb{R}^n, N)$ denotes the set of all $\sigma = \sigma(x, \xi_1, \dots, \xi_N) \in C^{\infty}((\mathbb{R}^n)^{N+1})$ satisfying
\[
|
\partial^\alpha_x
\partial^{\beta_1}_{\xi_1}
\dots
\partial^{\beta_N}_{\xi_N}
\sigma(x, \xi_1, \dots, \xi_N)
|
\le
C_{\alpha, \beta_1, \dots, \beta_N}
(1+|\xi_1|+ \dots + |\xi_N|)^{m}
\]
for all multi-indices $\alpha, \beta_1, \dots, \beta_N \in \mathbb{N}^n_0 = \{0, 1, 2, \dots\}^n$.
The subject of this paper is to study the boundedness of multilinear pseudo-differential operators.
We will use the following notations.
Let $X_1, \dots, X_N$, and $Y$ be function spaces on $\mathbb{R}^n$ equipped with quasi-norms
$\|\cdot\|_{X_j}$ and $\|\cdot\|_{Y}$, respectively.
We say that $T_{\sigma}$ is bounded from $X_1 \times \dots \times X_N$ to $Y$
if there exists a positive constant $C$ such that the inequality
\begin{equation}\langlebel{bdd-dfn}
\|T_{\sigma}(f_1, \dots, f_N)\|_{Y}
\le
C
\prod_{j=1}^N
\|f_j\|_{X_j}
\end{equation}
holds for all $f_j \in \mathcal{S} \cap X_j$, $j=1, \dots, N$.
The smallest constant $C$ of \eqref{bdd-dfn} is defined by
$\|T_{\sigma}\|_{X_1 \times \dots \times X_N \to Y}$.
If $T_{\sigma}$ is bounded from $X_1 \times \dots \times X_N$ to $Y$
for all $\sigma \in S^m_{0,0}(\mathbb{R}^n, N)$, then we write
\[
\mathop{\mathrm{Op}}(S^m_{0,0}(\mathbb{R}^n, N)) \subset B(X_1 \times \dots \times X_N \to Y).
\]
In the linear case, the class $S^m_{0,0}(\mathbb{R}^n, 1)$
is well known as the H\"ormander symbol class of $S_{0,0}$,
and the boundedness of linear pseudo-differential operators with symbols in this class
was well studied.
More precisely, the following is widely known
(for the definition of the function spaces
$h^p$ and $bmo$,
see Section \ref{preliminaries}).
\begin{thmA}[\cite{CV, CM-Asterisque, Miyachi-MathNachr, PS, KMT-JFA}]
Let
$0< p \le \infty$.
Then the boundedness
\[
\mathop{\mathrm{Op}}(
S^{m}_{0, 0}
)
\subset
B(h^{p} \to h^{p}).
\]
holds if and only if
\[
m
\le
-n\left|\frac{1}{p} - \frac{1}{2} \right|,
\]
where $h^{p}$
should be replaced by $bmo$ if $p = \infty$ .
\end{thmA}
In Theorem A,
the ``if'' part was proved by
Calder\'on--Vaillancourt \cite{CV} for the case $p=2$,
Coifman--Meyer \cite{CM-Asterisque} for the case $1 < p < \infty$,
and
Miyachi \cite{Miyachi-MathNachr} and P\"aiv\"arinta--Somersalo \cite{PS} for the case $0 < p \le \infty$.
The proof of ``only if'' part can be found in \cite{KMT-JFA}.
As a generalization of Theorem A,
the boundedness of linear pseudo-differential operators of $S_{0,0}$
was studied in the settings of Besov spaces $B^s_{p, q}$
(for the definition of Besov spaces, see Section \ref{preliminaries}).
\begin{thmB}[\cite{Marschall, Sugimoto, Park}]
Let $0< p \le \infty$, $0< q \le \infty$, and $s, t \in \mathbb{R}$.
Let
\[
m
=
-n\left|\frac{1}{p} - \frac{1}{2} \right|
+s-t.
\]
Then
\[
\mathop{\mathrm{Op}}(S^{m}_{0, 0})
\subset
B(B^s_{p, q} \to B^{t}_{p, q}).
\]
\end{thmB}
The above result was given by
Marschall \cite{Marschall} for the case $p=q=\infty$,
and by Sugimoto \cite{Sugimoto} for the case $1 \le p, q \le \infty$.
Recently, Theorem B was proved by Park \cite{Park}.
In the multilinear settings,
the class $S^m_{0,0}(\mathbb{R}^n, N)$ was first studied by B\'enyi-Torres \cite{BT-MRL}
for the case $N=2$, that is, the bilinear case.
The authors proved that,
for $1 \le p, p_1, p_2 < \infty$ satisfying $1/p = 1/p_1 + 1/p_2$,
there exist $x$-independent symbols in $S^0_{0,0}(\mathbb{R}^n, 2)$ such that
the corresponding bilinear operators are not bounded from $L^{p_1} \times L^{p_2}$ to $L^p$.
In particular, they pointed out that the class $S^0_{0,0}(\mathbb{R}^n, 2)$ does not assure
the $L^2 \times L^2 \to L^1$ boundedness in contrast to the Calder\'on-Vaillancourt theorem for linear pseudo-differential operators.
Then, the number $m$ which assures the $L^{p_1} \times \dots \times L^{p_N} \to L^p$ boundedness of these operators
was investigated by
B\'enyi-Bernicot-Maldonado-Naibo-Torres \cite{BBMNT}
and
Michalowski-Rule-Staubach \cite{MRS},
and after that,
a complete description of $m$ was given by Miyachi-Tomita \cite{MT-IUMJ} for $N = 2$
(for the case $1/p = 1/p_1+1/p_2$)
and Kato-Miyachi-Tomita\cite{KMT-JFA} for $N \ge 2$.
\begin{thmC}[\cite{MT-IUMJ, KMT-JFA}]
Let $N \ge 2$, $0< p, p_1, \dots, p_N \le \infty$, $1/p \le 1/p_1 + \dots + 1/p_N$
and $m \in \mathbb{R}$.
Then the boundedness
\[
\mathop{\mathrm{Op}}(S^m_{0, 0}(\mathbb{R}^n, N))
\subset
B(h^{p_1} \times \dots \times h^{p_N} \to h^p)
\]
holds if and only if
\[
m
\le
\min
\left\{
\frac{n}{p},
\frac{n}{2}
\right\}
-
\sum_{j=1}^N
\max
\left\{
\frac{n}{p_j},
\frac{n}{2}
\right\}.
\]
where $h^{p_j}$ can be replaced by $bmo$ if $p_j= \infty$ for some $j=1, \dots, N$.
\end{thmC}
In the scale of Besov spaces, some partial boundedness results for mulitilinear pseudo-differential operators with symbols in $S^m_{0,0}(\mathbb{R}^n, N)$ were given
for the bilinear case.
In Hamada-Shida-Tomita \cite{HST}, it was proved that
all bilinear pseudo-differential operators with symbols in $S^{-n/2}_{0,0}(\mathbb{R}^n, 2)$
is bounded from $L^2 \times L^2$ to $B^0_{p, q}$ if and only if $1 \le p \le 2$ and $1 \le q \le \infty$.
Since $B^0_{1, 1} \hookrightarrow h^1$, this result improves the $L^2 \times L^2 \to h^1$ boundedness
given by \cite{MT-IUMJ}.
In \cite{Shida-PAMS}, the following is given.
\begin{thmD}[\cite{Shida-PAMS}]
Let $1 \le p \le 2 \le p_1, p_2 \le \infty$ be such that $1/p \le 1/p_1 + 1/p_2$,
and let $0 < q_1, q_2, q \le \infty$ be such that $1/q \le 1/q_1 + 1/q_2$,
and let $s_1, s_2, s \in \mathbb{R}$ be such that $s_1+s_2=s$.
If $s_1$, $s_2$ and $s$ satisfy
\begin{equation} \langlebel{condi-PAMS}
s_1 < \frac{n}{2},
\quad
s_2 < \frac{n}{2},
\quad
\text{and}
\quad
s > - \frac{n}{2},
\end{equation}
then
\begin{align*}
\mathop{\mathrm{Op}}(S^{-n/2}_{0, 0}(\mathbb{R}^n, 2))
\subset
B(B^{s_1}_{p_1, q_1} \times B^{s_2}_{p_2, q_2} \to B^{s}_{p, q}).
\end{align*}
\end{thmD}
\noindent
In \cite{Shida-PAMS},
the sharpness of the condition \eqref{condi-PAMS} is also considered.
The purpose of this paper is
to extend the partial results on the bilinear case stated in Theorem D to the multilinear case in the full range $0< p, p_1, \dots, p_N \le \infty$.
Our main result reads as follows.
\begin{thm}\langlebel{main1}
Let $N \ge 2$, $0< p, p_1, \dots, p_N \le \infty$, $1/p \le 1/p_1 + \dots + 1/p_N$,
$0 < q, q_1, \dots, q_N \le \infty$, $1/q \le 1/q_1 + \dots + 1/q_N$,
and $s, s_1, \dots, s_N \in \mathbb{R}$.
Let
\begin{equation}\langlebel{criticalorder}
m
=
\min
\left\{
\frac{n}{p},
\frac{n}{2}
\right\}
-
\sum_{j=1}^N
\max
\left\{
\frac{n}{p_j},
\frac{n}{2}
\right\}
+
\sum_{j=1}^N
s_j
-s.
\end{equation}
If $s_1$, \dots, $s_N$ and $s$ satisfy
\begin{align}\langlebel{sassum}
\begin{split}
&s_j
<
\max \left\{ \frac{n}{p_j}, \frac{n}{2} \right\},
\quad
j=1, \dots, N,
\quad
\text{and}
\quad
s
>
-\max \left\{ \frac{n}{p^{\prime}}, \frac{n}{2} \right\},
\end{split}
\end{align}
then
\begin{equation}\langlebel{boundedness_1}
\mathop{\mathrm{Op}}(S^{m}_{0,0}(\mathbb{R}^n, N))
\subset
B(B^{s_1}_{p_1, q_1} \times \dots \times B^{s_N}_{p_N, q_N} \to B^s_{p, q}).
\end{equation}
\end{thm}
The condition \eqref{sassum} is sharp in the following sense.
\begin{thm}\langlebel{thmnec}
Let $0 < p, p_1, \dots, p_N \le \infty$, $0< q, q_1, \dots, q_N \le \infty$ and $s, s_1, \dots, s_N \in \mathbb{R}$.
If the boundedness \eqref{boundedness_1} holds with $m$ given in \eqref{criticalorder}, then
$s_j \le \max \{n/p_j, n/2\}$, $j= 1, \dots, N$,
and
$s \ge -\max \{n/p^{\prime}, n/2\}$.
\end{thm}
We shall explain some connection between our main results and previous results.
Firstly, Theorem \ref{main1} yields that,
for $0< p, p_1, \dots, p_N \le \infty$, $1/p \le 1/p_1 + \dots + 1/p_N$ and
$0< q, q_1, \dots, q_N \le \infty$, $1/q \le 1/q_1 + \dots +1/q_N$,
the boundedness
\begin{align}\langlebel{bdd-main1-part}
\mathop{\mathrm{Op}}(S^{m}_{0,0}(\mathbb{R}^n, N))
\subset
B(B^{0}_{p_1, q_1}
\times \dots \times
B^{0}_{p_N, q_N}
\to
B^{0}_{p, q})
\end{align}
holds with
\begin{align*}
m
=
\min\left\{\frac{n}{p}, \frac{n}{2}\right\}
-
\sum_{j=1}^N
\max\left\{\frac{n}{p_j}, \frac{n}{2}\right\}.
\end{align*}
If $0< p < \infty$ and $0 < p_1, \dots, p_N \le \infty$ satisfy
\begin{align}\langlebel{addcondforexpo}
\frac{1}{\min\{p, 2\}}
\le
\sum_{j=1}^N
\frac{1}{\max\{p_j, 2\}},
\end{align}
then the boundedness \eqref{bdd-main1-part} improves the boundedness result in Theorem C.
In fact, for $0< p < \infty$ and $0< p_1, \dots, p_N \le \infty$ satisfying \eqref{addcondforexpo}, we can choose $q_j = \max\{p_j, 2\}$, $j=1, \dots, N$, and $q = \min\{p, 2\}$ in \eqref{bdd-main1-part}, and hence we obtain
\[
\mathop{\mathrm{Op}}(S^{m}_{0,0}(\mathbb{R}^n, N))
\subset
B(B^{0}_{p_1, \max\{p_1, 2\}}
\times \dots \times
B^{0}_{p_N, \max\{p_N, 2\}}
\to
B^{0}_{p, \min\{p, 2\}}).
\]
Since we have the embedding relations
$B^{0}_{r, \min\{r, 2\}} \hookrightarrow h^r \hookrightarrow B^{0}_{r, \max\{r, 2\}}$
for $0 < r < \infty$,
and
$bmo \hookrightarrow B^{0}_{\infty, \infty}$,
this boundedness gives an improvement of the corresponding $h^{p_1} \times \dots \times h^{p_N} \to h^p$ boundedness given in Theorem C.
For the case $p= \infty$,
if $0< p_1, \dots, p_N \le \infty$ satisfy
\[
1
\le
\sum_{j=1}^N
\frac{1}{\max\{p_j, 2\}},
\]
then we can take $q_j = \max\{p_j, 2\}$, $j=1, \dots, N$ and $q =1$,
and consequently we obtain
\begin{align*}
&\mathop{\mathrm{Op}}(S^m_{0,0}(\mathbb{R}^n, N))
\subset
B(B^0_{p_1, \max\{p_1, 2\}} \times \dots \times B^0_{p_N, \max\{p_N, 2\}} \to B^0_{\infty, 1}).
\end{align*}
This improves the corresponding boundedness results given in Theorem C
since $h^r \hookrightarrow B^{0}_{r, \max\{r, 2\}}$, $0< r < \infty$, $bmo \hookrightarrow B^0_{\infty, \infty}$ and
$B^0_{\infty, 1}\hookrightarrow L^{\infty}$.
Secondly, the condition \eqref{sassum} is peculiar to the multilinear case.
In fact, we can take any $s$ and $t$ in Theorem B, however, in the multilinear settings, Theorem \ref{thmnec} says that the conditions \eqref{sassum} are (almost) necessary
to assure the boundedness on Besov spaces.
We also notice that the condition \eqref{sassum} can be found in the author's paper \cite{Shida-Sobolev}.
In \cite{Shida-Sobolev}, it is proved that bilinear pseudo-differential operators with symbols in $S^m_{0, 0}(\mathbb{R}^n, 2)$ with the critical $m$ are bounded on Sobolev spaces under the assumption \eqref{sassum} with $N=2$.
The organization of this paper is as follows.
In Section 2,
we give some notations and recall the definitions of some function spaces and embedding relations between them.
In Section 3, we give the proof of Theorem \ref{main1}.
In Section 4, we prove Theorem \ref{thmnec}.
\section{Preliminaries}\langlebel{preliminaries}
For two nonnegative quantities $A$ and $B$,
the notation $A \lesssim B$ means that
$A \le CB$ for some unspecified constant $C>0$,
and $A \approx B$ means that
$A \lesssim B$ and $B \lesssim A$.
For $0 < p \le \infty$,
$p^{\prime}$ is the conjugate exponent of $p$,
that is, $p^{\prime}$ is defined by
$1/p+1/p^{\prime}=1$ if $1 < p \le \infty$
and
$p^{\prime} = \infty$ if $0 < p \le 1$.
For a finite set $\Lambda$,
$|\Lambda|$ denotes the number of the elements of $\Lambda$.
Let $\mathcal{S}(\mathbb{R}^n)$ and $\mathcal{S}'(\mathbb{R}^n)$ be the Schwartz space of
rapidly decreasing smooth functions on $\mathbb{R}^n$ and its dual,
the space of tempered distributions, respectively.
We define the Fourier transform $\mathcal{F} f$
and the inverse Fourier transform $\mathcal{F}^{-1}f$
of $f \in \mathcal{S}(\mathbb{R}^n)$ by
\[
\mathcal{F} f(\xi)
=\widehat{f}(\xi)
=\int_{\mathbb{R}^n}e^{-i \xi \cdot x} f(x)\, dx
\quad \text{and} \quad
\mathcal{F}^{-1}f(x)
=\frac{1}{(2\pi)^n}
\int_{\mathbb{R}^n}e^{i x \cdot \xi} f(\xi)\, d\xi.
\]
For $m \in L^{\infty}(\mathbb{R}^n)$,
the Fourier multiplier operator $m(D)$ is defined by
$m(D)f=\mathcal{F}^{-1}[m\widehat{f}]$ for $f \in \mathcal{S}(\mathbb{R}^n)$.
For a countable set $J$, the sequence space $\ell^q(J)$, $0 < q \le \infty$, is defined to be the set
of all complex sequences $a = \{a_j\}_{j \in J}$ such that
\begin{align*}
\|a\|_{\ell^q(J)}
=
\begin{cases}
\left(\sum_{j \in J} |a_j|^q \right)^{1/q}
&\text{if $0 < q < \infty$},
\\
\sup_{j \in J} |a_j|
&\text{if $q = \infty$}
\end{cases}
\end{align*}
is finite. For $a = \{a_j\}_{j \in J}$, we will use the notation $\|a_j\|_{\ell^q_j(J)}$ instead of $\|a\|_{\ell^q(J)}$
when we indicate the variable explicitly.
Let $\phi \in \mathcal{S}(\mathbb{R}^n)$ be such that $\int_{\mathbb{R}^n} \phi(x)\, dx \neq 0$.
For $0< p \le \infty$, the local Hardy space $h^p = h^p(\mathbb{R}^n)$ consists of all $f \in \mathcal{S}^\prime(\mathbb{R}^n)$ such that
\begin{equation*}
\|f\|_{h^p}
=
\left\|
\sup_{0< t < 1}
|\phi_t * f|
\right\|_{L^p}
<
\infty,
\end{equation*}
where $\phi_t(x) = t^{-n} \phi(t^{-1} x)$.
It is known that the definition of $h^p$ is independent of the choice of the function $\phi$
up to equivalence of quasi-norms.
It is also known
that $h^p = L^p$ for $1 < p \le \infty$
and $h^1 \hookrightarrow L^1$.
The space $bmo = bmo(\mathbb{R}^n)$ consists of all locally integrable functions $f$ on $\mathbb{R}^n$
such that
\begin{equation*}
\|f\|_{bmo}
=
\sup_{|Q| \le 1}
\frac{1}{|Q|}
\int_{Q}
|f(x) -f_Q|
\, dx
+
\sup_{|Q| \ge 1}
\frac{1}{|Q|}
\int_{Q}
|f(x)|
\, dx
< \infty,
\end{equation*}
where $Q$ ranges over all cubes in $\mathbb{R}^n$.
It is known that $L^\infty \subset bmo$.
It is also known that the dual space of $h^1$ coincides with $bmo$.
We recall the definition of Besov spaces.
Let $\psi_k \in \mathcal{S}(\mathbb{R}^n), k \ge 0,$ be such that
\begin{align} \langlebel{partition}
\begin{split}
&\mathop{\mathrm{supp}} \psi_0 \subset \{\xi \in \mathbb{R}^n : |\xi| \le 2\},
\quad
\mathop{\mathrm{supp}} \psi_k \subset \{\xi \in \mathbb{R}^n : 2^{k-1} \le |\xi| \le 2^{k+1}\},
\quad
k \ge 1,
\\
&
\|\partialrtial^\alpha \psi_k\|_{L^\infty}
\le
C_\alpha 2^{-k |\alpha|},
\quad
\alpha \in \mathbb{N}^n_0,\ k \ge 0,
\\
&
\sum_{k = 0}^\infty
\psi_{k}(\xi)
=
1,
\quad
\xi \in \mathbb{R}^n.
\end{split}
\end{align}
The Besov space
$B^s_{p, q} = B^s_{p, q}(\mathbb{R}^n)$,
$0< p, q \le \infty$,
$s \in \mathbb{R}$,
is defined to be the set of all
$f \in \mathcal{S}^\prime(\mathbb{R}^n)$
such that
\begin{align*}
\|f\|_{B^s_{p, q}}
=
\left\|
2^{k s}
\left\|
\psi_k(D)f(x)
\right\|_{L^p_x(\mathbb{R}^n)}
\right\|_{\ell^q_k(\mathbb{N}_0)}
<
\infty.
\end{align*}
It is known that the definition of Besov spaces
is independent of the choice of $\psi_k$, $k=0, 1, 2, \dots$, up to the equivalence of quasi-norms.
If $1\le p, q < \infty$, then
the dual space of $B^s_{p, q}$ coincides with
$B^{-s}_{p^\prime, q^\prime}$.
The following embedding relations are well known;
\begin{align}
&B^s_{p, q_1} \hookrightarrow B^s_{p, q_2},
\langlebel{Bq1Bq2}
\quad
\text{if}
\quad
q_1 \le q_2,
\\
& B^0_{p, \min\{p, 2\}}
\hookrightarrow
h^p
\hookrightarrow
B^0_{p, \max\{p, 2\}},
\quad
\text{if}
\quad
0<p<\infty,
\langlebel{BhB}
\\
&
B^{0}_{\infty, 1}
\hookrightarrow
L^{\infty}
\hookrightarrow
B^{0}_{\infty, \infty},
\langlebel{BLBinfty}
\\
&
bmo
\hookrightarrow
B^{0}_{\infty, \infty}.
\notag
\end{align}
As a consequence of \eqref{Bq1Bq2}, \eqref{BhB} and \eqref{BLBinfty}, we have $h^p \hookrightarrow B^0_{p, \infty}$,
$0 < p \le \infty$,
which means
\begin{equation} \langlebel{embd-hpB0pinfty}
\sup_{k \in \mathbb{N}_0}
\|\psi_k(D)f\|_{L^p}
\lesssim
\|f\|_{h^p}.
\end{equation}
For more basic properties about Besov spaces,
see, e.g., Triebel \cite{Triebel-ToFS}.
It is known that
the $L^p$-norm in the definition of $B^s_{p, q}$-norm
can be replaced by the $h^p$-norm.
More precisely, the following proposition was given by Qui \cite{Qui}.
\begin{prop}[\cite{Qui}]\langlebel{propQui}
Let $0< p, q \le \infty$ and $s \in \mathbb{R}$. Then,
\begin{equation}\langlebel{Bhpequiv}
\|f\|_{B^s_{p, q}}
\approx
\left\|
2^{k s}
\left\|
\psi_k(D)f(x)
\right\|_{h^p_x(\mathbb{R}^n)}
\right\|_{\ell^q_{k}(\mathbb{N}_0)}.
\end{equation}
\end{prop}
We end this section by recalling the definition and some properties of the Wiener amalgam space.
Let
$\kappa \in \mathcal{S}(\mathbb{R}^n)$ be such that $\mathop{\mathrm{supp}} \kappa$ is compact and
\begin{align} \langlebel{part-Wiener}
\left|
\sum_{\mu \in \mathbb{Z}^n}
\kappa(\xi - \mu)
\right|
\ge 1,
\quad
\xi \in \mathbb{R}^n.
\end{align}
The Wiener amalgam space $W^{p, q}_s =W^{p, q}_s(\mathbb{R}^n)$, $0< p, q \le \infty$, $s \in\mathbb{R}$,
consists of all $f \in \mathcal{S}^\prime(\mathbb{R}^n)$ such that
\[
\|f\|_{W^{p, q}_s}
=
\left\|
\left\|
\langle \mu \rangle^{s}
\mathcal{B}ox_{\mu}f(x)
\right\|_{\ell^q_{\mu}(\mathbb{Z}^n)}
\right\|_{L^p_x(\mathbb{R}^n)}
<
\infty,
\]
where $\mathcal{B}ox_{\mu}f= \kappa(D-\mu)f = \mathcal{F}^{-1}[\kappa(\cdot-\mu) \widehat{f}]$.
We simply write $W^{p, q} = W^{p, q}_{0}$.
The space $W^{p, q}_{s}$ does not depend on the choice of $\kappa$ up to equivalence of quasi-norms.
For the definition of the Wiener amalgam space,
see Triebel \cite{Triebel-ZAA}.
The embedding relations between Lebesgue, local Hardy spaces
and Wiener amalgam spaces are well investigated as follows.
\begin{lem}[\cite{CKS, GWYZ}] \langlebel{embd}
Let $0< p, p_1, p_2, q_1, q_2 \le \infty$. Then,
\begin{align}
&W^{p_1, q_1}_s \hookrightarrow W^{p_2, q_2}_s
\quad
\text{if}
\quad
p_1 \le p_2,
\quad
\text{and}
\quad
q_1 \le q_2;
\langlebel{monotone}
\\
&W^{p, 2}_{\alpha(p)} \hookrightarrow h^p
\quad
\text{if}
\quad
0< p< \infty,
\quad
\text{where}
\quad
\alpha(p)
=
(n/2)
-
\min
\{n/2, n/p\};
\langlebel{Wh}
\\
&h^p \hookrightarrow W^{p, 2}_{\beta(p)}
\quad
\text{if}
\quad
0< p< \infty,
\quad
\text{where}
\quad
\beta(p)
=
(n/2)
-
\max
\{n/2, n/p\};
\langlebel{hW}
\\
&W^{\infty, 1} \hookrightarrow L^\infty.
\langlebel{WLinfty}
\end{align}
\end{lem}
The embedding relations \eqref{Wh} and \eqref{hW} are given
by Cunanan-Kobayashi-Sugimoto \cite{CKS} for the case $1< p< \infty$
and Guo-Wu-Yang-Zhao\cite{GWYZ} for the $0< p \le 1$.
The embedding \eqref{WLinfty} is given by \eqref{WLinfty}
The proof of \eqref{monotone} can be found in Kato-Miyachi-Tomita \cite{KMT-JFA}.
We will use these embedding relations in the proof of Theorem \ref{main1}.
The idea of using the Wiener amalgam spaces comes from the recent works of T. Kato, A. Miyachi and N. Tomita (see \cite{KMT-JPDOA, KMT-JMSJ, KMT-JFA}).
In the rest of this section, we shall show the following proposition.
\begin{prop} \langlebel{Keyprop}
Let $N \ge 2$, $0 < p_0, p_1, \dots, p_N \le \infty$, $1/p_0 = 1/p_1 + \dots + 1/p_N$, and $M_0 \in \mathbb{N}_0$.
Let $R, R_1, \dots, R_N \ge 1$.
Let $\Lambda, \Lambda_1, \dots, \Lambda_N$ be subsets of $\mathbb{Z}^n$ satisfying
\begin{align*}
\Lambda = \{\nu \in \mathbb{Z}^n : |\nu| \lesssim R\},
\quad
\Lambda_j = \{\nu \in \mathbb{Z}^n : |\nu| \approx R_j\},
\quad
j=1, \dots, N.
\end{align*}
Suppose that $R_1 = \max_{1 \le j \le N} R_j$ and that $R_2= \max_{2 \le j \le N} R_j$.
Then the estimate
\begin{align*}
&
\left\|
\left\|
\sum_{\substack{ \mathbb{N}u \in \Lambda_1 \times \dots \times \Lambda_N
\\
\nu_1 + \dots + \nu_N = \tau}
}
\prod_{j=1}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^2_\tau(\Lambda)}
\right\|_{L^{p_0}}
\lesssim
\min \{R_2^{n/2}, R^{n/2}\}
\prod_{j=3}^N
R_j^{n/2}
\prod_{j=1}^N
R_j^{-\beta(p_j)}
\|f_j\|_{h^{p_j}}
\end{align*}
holds with the implicit constant independent of $R_1, \dots, R_N$ and $R$.
\end{prop}
\begin{proof}
Notice that $|\Lambda| \lesssim R^n$ and $|\Lambda_j| \approx R_j^n$, $j=1, \dots, N$.
First,
we have by Young's inequality and H\"older's inequality
\begin{align*}
\left\|
\sum_{\substack{
\mathbb{N}u \in \Lambda_1 \times \dots \times \Lambda_N
\\
\nu_1 + \dots + \nu_N = \tau
}}
\prod_{j=1}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^2_\tau(\Lambda)}
&\le
\left\|
\sum_{\substack{
\mathbb{N}u \in \Lambda_1 \times \dots \times \Lambda_N
\\
\nu_1 + \dots + \nu_N = \tau
}}
\prod_{j=1}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^2_\tau(\mathbb{Z}^n)}
\\
&\le
\left\|
\mathcal{B}ox_{\nu_1} f_1
\right\|_{\ell^2_{\nu_1}(\Lambda_1)}
\prod_{j=2}^N
\left\|
\mathcal{B}ox_{\nu_j} f_j
\right\|_{\ell^1_{\nu_j}(\Lambda_j)}
\\
&\lesssim
\left\|
\mathcal{B}ox_{\nu_1} f_1
\right\|_{\ell^2_{\nu_1}(\Lambda_1)}
\prod_{j=2}^N
R_j^{n/2}
\left\|
\mathcal{B}ox_{\nu_j} f_j
\right\|_{\ell^2_{\nu_j}(\Lambda_j)}.
\end{align*}
Hence, this estimate and H\"older's inequality yield that
\begin{align} \langlebel{est-R2}
\left\|
\left\|
\sum_{\substack{
\mathbb{N}u \in \Lambda_1 \times \dots \times \Lambda_N
\\
\nu_1 + \dots + \nu_N = \tau
}}
\prod_{j=1}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^2_\tau(\Lambda)}
\right\|_{L^{p_0}}
\lesssim
\prod_{j = 2}^N
R_j^{n/2}
\prod_{j=1}^N
\left\|
\left\|
\mathcal{B}ox_{\nu_j} f_j
\right\|_{\ell^2_{\nu_j}(\Lambda_j)}
\right\|_{L^{p_j}}.
\end{align}
On the other hand,
it follows from H\"older's inequality that
\begin{align*}
\sum_{\substack{
\mathbb{N}u \in \Lambda_1 \times \dots \times \Lambda_N
\\
\nu_1 + \dots + \nu_N = \tau
}}
\prod_{j=1}^N
|\mathcal{B}ox_{\nu_j} f_j|
&=
\sum_{\nu_1 \in \Lambda_1}
|\mathcal{B}ox_{\nu_1} f_1|
\sum_{
\substack{
(\nu_2, \dots, \nu_N) \in \Lambda_2 \times \dots \times \Lambda_N
\\
\nu_2 + \dots + \nu_N = \tau -\nu_1}
}
\prod_{j=2}^N
|\mathcal{B}ox_{\nu_j} f_j|
\\
&\le
\left\|
\mathcal{B}ox_{\nu_1} f_1
\right\|_{\ell^2_{\nu_1}(\Lambda_1)}
\left\|
\sum_{
\substack{
(\nu_2, \dots, \nu_N) \in \Lambda_2 \times \dots \times \Lambda_N
\\
\nu_2 + \dots + \nu_N = \tau - \nu_1}
}
\prod_{j=2}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^2_{\nu_1}(\mathbb{Z}^n)}
\\
&=
\left\|
\mathcal{B}ox_{\nu_1} f_1
\right\|_{\ell^2_{\nu_1}(\Lambda_1)}
\left\|
\sum_{
\substack{
(\nu_2, \dots, \nu_N) \in \Lambda_2 \times \dots \times \Lambda_N
\\
\nu_2 + \dots + \nu_N = \mu}
}
\prod_{j=2}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^2_{\mu}(\mathbb{Z}^n)}.
\end{align*}
By Young's inequality and H\"older's inequality, we have
\begin{equation*}
\left\|
\sum_{
\substack{
(\nu_2, \dots, \nu_N) \in \Lambda_2 \times \dots \times \Lambda_N
\\
\nu_2 + \dots + \nu_N = \mu}
}
\prod_{j=2}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^2_{\mu}(\mathbb{Z}^n)}
\lesssim
\prod_{j=3}^N
R_j^{n/2}
\prod_{j=2}^N
\left\|
\mathcal{B}ox_{\nu_j} f_j
\right\|_{\ell^2_{\nu_j}(\Lambda_j)}.
\end{equation*}
Hence we obtain by H\"older's inequality
\begin{align} \langlebel{est-R}
\left\|
\left\|
\sum_{\substack{
\mathbb{N}u \in \Lambda_1 \times \dots \times \Lambda_N
\\
\nu_1 + \dots + \nu_N = \tau
}}
\prod_{j=1}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^2_\tau(\Lambda)}
\right\|_{L^{p_0}}
&\lesssim
R^{n/2}
\prod_{j=3}^N
R_j^{n/2}
\prod_{j=1}^N
\left\|
\left\|
\mathcal{B}ox_{\nu_j} f_j
\right\|_{\ell^2_{\nu_j}(\Lambda_j)}
\right\|_{L^{p_j}}.
\end{align}
Therefore, combining \eqref{est-R2} and \eqref{est-R}, we obtain
\[
\left\|
\left\|
\sum_{\substack{
\mathbb{N}u \in \Lambda_1 \times \dots \times \Lambda_N
\\
\nu_1 + \dots + \nu_N = \tau
}}
\prod_{j=1}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^2_\tau(\Lambda)}
\right\|_{L^{p_0}}
\lesssim
\min \{R_2^{n/2}, R^{n/2}\}
\prod_{j=3}^N
R_j^{n/2}
\prod_{j=1}^N
\left\|
\left\|
\mathcal{B}ox_{\nu_j} f_j
\right\|_{\ell^2_{\nu_j}(\Lambda_j)}
\right\|_{L^{p_j}}.
\]
Since $\langle \nu_j \rangle \approx R_j$ if $\nu_j \in \Lambda_{j}$, it follows from the embedding $h^{p_j} \hookrightarrow W^{p_j, 2}_{\beta(p_j)}$ that
\begin{align*}
\left\|
\left\|
\mathcal{B}ox_{\nu_j} f_j
\right\|_{\ell^2_{\nu_j}(\Lambda_j)}
\right\|_{L^{p_j}}
&\lesssim
R_j^{-\beta(p_j)}
\left\|
\left\|
\langlengle \nu_j \ranglengle^{\beta(p_j)}
\mathcal{B}ox_{\nu_j} f_j
\right\|_{\ell^2_{\nu_j}(\mathbb{Z}^n)}
\right\|_{L^{p_j}}
\\
&
\lesssim
R_j^{-\beta(p_j)}
\|f_j\|_{h^{p_j}},
\quad
j=1, \dots, N.
\end{align*}
The proof is complete.
\end{proof}
\begin{rem} \langlebel{Keyrem}
By the Cauchy-Schwartz inequality and Proposition \ref{Keyprop}, we also have
\begin{align*}
\left\|
\left\|
\sum_{\substack{ \mathbb{N}u \in \Lambda_1 \times \dots \times \Lambda_N
\\
\nu_1 + \dots + \nu_N = \tau}
}
\prod_{j=1}^N
|\mathcal{B}ox_{\nu_j} f_j|
\right\|_{\ell^1_\tau(\Lambda)}
\right\|_{L^{p_0}}
\lesssim
R^{n/2}
\min \{R_2^{n/2}, R^{n/2}\}
\prod_{j=3}^N
R_j^{n/2}
\prod_{j=1}^N
R_j^{-\beta(p_j)}
\|f_j\|_{h^{p_j}}.
\end{align*}
We also use this estimate in the proof of Theorem \ref{main1}.
\end{rem}
\section{Proof of Theorem \ref{main1}}
In this section, we shall prove Theorem \ref{main1}.
Let $0< p, p_j, q, q_j \le \infty$ and $s, s_j \in \mathbb{R}$, $j=1, \dots, N$, be the same as in Theorem \ref{main1}.
Throughout this section,
we always assume that $\sigma \in S^{m}_{0,0}(\mathbb{R}^n, N)$ with $m$ given by \eqref{criticalorder}.
We use the notation $\boldsymbol{\xi} = (\xi_1, \dots, \xi_N) \in (\mathbb{R}^n)^N$.
The following method using the Fourier series expansion
goes back at least to Coifman-Meyer \cite{CM-Asterisque, CM-AIF}.
Let $\varphi, \widetilde{\varphi} \in \mathcal{S}(\mathbb{R}^n)$ be such that
\begin{align*}
&\mathop{\mathrm{supp}} \varphi \subset [-1, 1]^n,
\quad
\sum_{\nu \in \mathbb{Z}^n} \varphi(\xi-\nu) = 1,
\quad
\xi \in \mathbb{R}^n,
\\
&\mathop{\mathrm{supp}} \widetilde{\varphi} \subset [-3, 3]^n,
\quad
0 \le \widetilde{\varphi} \le 1,
\quad
\widetilde{\varphi} = 1
\quad
\text{on}
\quad
[-1, 1]^n.
\end{align*}
We remark that $\varphi$ and $\widetilde{\varphi}$
satisfy \eqref{part-Wiener}.
We decompose the symbol $\sigma = \sigma(x, \boldsymbol{\xi})$ as
\begin{align*}
\sigma(x, \boldsymbol{\xi})
&=
\sum_{\mathbb{N}u = (\nu_1, \dots, \nu_N) \in (\mathbb{Z}^n)^N}
\sigma(x, \boldsymbol{\xi})
\prod_{j=1}^N
\varphi(\xi_j-\nu_j)
=
\sum_{\mathbb{N}u \in (\mathbb{Z}^n)^N}
\sigma_{\mathbb{N}u}(x, \boldsymbol{\xi}),
\end{align*}
where $\sigma_{\mathbb{N}u}(x, \boldsymbol{\xi}) = \sigma(x, \boldsymbol{\xi}) \prod_{j=1}^N \varphi(\xi_j-\nu_j)$.
We define
\[
S_{\mathbb{N}u}(x, \boldsymbol{\xi})
=
\sum_{\boldsymbol{\ell} \in (\mathbb{Z}^n)^N}
\sigma_{\mathbb{N}u} (x, \boldsymbol{\xi}-2\pi \boldsymbol{\ell}).
\]
Since $S_{\mathbb{N}u}(x, \boldsymbol{\xi}) = \sigma_{\mathbb{N}u}(x, \boldsymbol{\xi})$ if $\boldsymbol{\xi} \in \mathbb{N}u + [-3, 3]^{Nn}$, we have
\[
\sigma_{\mathbb{N}u}(x, \boldsymbol{\xi})
=
S_{\mathbb{N}u}(x, \boldsymbol{\xi})
\prod_{j=1}^N
\widetilde{\varphi}(\xi_j-\nu_j)
\]
Furthermore, since $S_{\mathbb{N}u}$ is a $2\pi \mathbb{Z}^{Nn}$-periodic function with respect to the $\boldsymbol{\xi}$-variable,
the Fourier series expansion yields that
\[
\sigma_{\mathbb{N}u}(x, \boldsymbol{\xi})
=
\sum_{\boldsymbol{\mu} = (\mu_1, \dots, \mu_N) \in (\mathbb{Z}^n)^N}
P_{\mathbb{N}u, \boldsymbol{\mu}}(x)
\prod_{j=1}^N
e^{i \mu_j \cdot \xi_j}
\widetilde{\varphi}(\xi_j-\nu_j),
\]
where
\begin{align*}
P_{\mathbb{N}u, \boldsymbol{\mu}}(x)
=
\frac{1}{(2\pi)^{Nn}}
\int_{\mathbb{N}u + [-\pi, \pi]^{Nn}}
e^{-i \boldsymbol{\mu} \cdot \boldsymbol{y}}
\sigma_{\mathbb{N}u}(x, \boldsymbol{y})
\,
d\boldsymbol{y}.
\end{align*}
It follows from integration by parts that
\begin{align*}
&P_{\mathbb{N}u, \boldsymbol{\mu}}(x)
=
\langle \boldsymbol{\mu} \rangle^{-2M}
Q_{\mathbb{N}u, \boldsymbol{\mu}}(x),
\end{align*}
where
\begin{align*}
&Q_{\mathbb{N}u, \boldsymbol{\mu}}(x)
=
\frac{1}{(2\pi)^{Nn}}
\int_{\mathbb{N}u + [-\pi, \pi]^{Nn}}
e^{-i \boldsymbol{\mu} \cdot \boldsymbol{y}}
(I-\Delta_{\boldsymbol{y}})^{M}
\sigma_{\mathbb{N}u}(x, \boldsymbol{y})
\,
d\boldsymbol{y}.
\end{align*}
We remark that, for $\alpha \in \mathbb{N}^n_0$,
\begin{equation}\langlebel{estPnu}
|\partial^\alpha_x Q_{\mathbb{N}u, \boldsymbol{\mu}}(x)|
\lesssim
\langle \mathbb{N}u \rangle^{m}
\end{equation}
holds for all $\mathbb{N}u, \boldsymbol{\mu} \in (\mathbb{Z}^n)^N$,
since $\sigma_{\mathbb{N}u}$ satisfies
\[
|
\partial_x^{\alpha}
\partial_{\boldsymbol{\xi}}^{\boldsymbol{\beta}}
\sigma_{\mathbb{N}u}(x, \boldsymbol{\xi})
|
\le
C_{\alpha, \boldsymbol{\beta}}
\langle \mathbb{N}u \rangle^m,
\quad
\alpha \in \mathbb{N}_0^n,\ \ \boldsymbol{\beta} \in (\mathbb{N}_0^n)^N.
\]
Thus we can write $T_{\sigma}$ as
\[
T_{\sigma}(f_1, \dots, f_N)(x)
=
\sum_{\boldsymbol{\mu} \in (\mathbb{Z}^n)^N}
\langle \boldsymbol{\mu} \rangle^{-2M}
\sum_{\mathbb{N}u \in (\mathbb{Z}^n)^N}
Q_{\mathbb{N}u, \boldsymbol{\mu}}(x)
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j}f_j(x+ \mu_j)
\]
Choosing the number $M$ as large as $2M \min \{1, p, q\}>Nn$, we obtain
\[
\left\|
T_{\sigma}(f_1, \dots, f_N)
\right\|_{B^{s}_{p, q}}
\lesssim
\sup_{\boldsymbol{\mu} \in (\mathbb{Z}^n)^N}
\left\|
\sum_{\mathbb{N}u \in (\mathbb{Z}^n)^N}
Q_{\mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j}f_j(\cdot + \mu_j)
\right\|_{B^s_{p, q}}.
\]
Let $\psi_{\ell_j} \in \mathcal{S}(\mathbb{R}^n),\ \ell_j \in \mathbb{N}_0$, $j=0, 1, \dots, N$, be the same partition of unities as in the definition of Besov spaces.
We further decompose the sum on the right hand side above as follows;
\begin{align*}
&\sum_{\mathbb{N}u \in (\mathbb{Z}^n)^N}
Q_{\mathbb{N}u, \boldsymbol{\mu}}(x)
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j}f_j(x+ \mu_j)
\\
&=
\sum_{\boldsymbol{\ell} = (\ell_0, \ell_1, \dots, \ell_N) \in (\mathbb{N}_0)^{N+1}}
\sum_{\mathbb{N}u \in (\mathbb{Z}^n)^N}
\psi_{\ell_0}(D)Q_{\mathbb{N}u, \boldsymbol{\mu}}(x)
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j}
\psi_{\ell_j}(D)f_j(x+ \mu_j)
\\
&=
\sum_{\boldsymbol{\ell} \in (\mathbb{N}_0)^{N+1}}
\sum_{\mathbb{N}u \in (\mathbb{Z}^n)^N}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}(x)
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}(x),
\end{align*}
where we set $Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}} = \psi_{\ell_0}(D)Q_{\mathbb{N}u, \boldsymbol{\mu}}$ and $F^{j}_{\ell_j, \mu_j} = \psi_{\ell_j}(D)f_j(\cdot + \mu_j)$, $j=1, \dots, N$.
Now, we divide the sum with respect to the variable $\boldsymbol{\ell}$ into the following $N$ parts:
\begin{align*}
&
\Lambda_1
=
\left\{\boldsymbol{\ell} \in (\mathbb{N}_0)^{N+1}
:
\ell_j \le \ell_1, \
j=2, \dots, N \right\},
\\
&
\Lambda_k
=
\left\{\boldsymbol{\ell} \in (\mathbb{N}_0)^{N+1}
:
\begin{array}{l}
\ell_j < \ell_k, \ j= 1, \dots, k-1,
\\
\ell_j \le \ell_k, \ j = k+1, \dots, N
\end{array}
\right\},
\quad
k=2, \dots, N-1,
\\
&
\Lambda_{N} = \left\{\boldsymbol{\ell} \in (\mathbb{N}_0)^{N+1} : \ell_j < \ell_N, \ j=1, \dots, N-1 \right\}.
\end{align*}
By symmetry, it is sufficient to deal with the sum concerning with $\Lambda_1$.
Furthermore, we divide the set $\Lambda_1$ into the following three parts;
\begin{align*}
\Lambda_1
=
&\big\{
\boldsymbol{\ell} \in \Lambda_1
:
\ell_0 \ge \ell_1-3
\big\}
\\
&\cup
\big\{
\boldsymbol{\ell} \in \Lambda_1
:
\ell_0 \le \ell_1-4,
\quad
\max \{\ell_2, \dots, \ell_N\} \le \ell_1-N-2
\big\}
\\
&\cup
\big\{
\boldsymbol{\ell} \in \Lambda_1
:
\ell_0 \le \ell_1-4,
\quad
\max\{ \ell_2, \dots, \ell_N \}
\ge
\ell_1-N-1
\big\}.
\end{align*}
By symmetry, it is sufficient to consider the case $\max\{\ell_2, \dots, \ell_N\} = \ell_2$.
In particular we may assume that $\ell_j \le \ell_2$, $j=3, \dots, N$.
Summarizing the above observations,
it is sufficient to prove that the estimates
\begin{equation}\langlebel{GOAL!!!!}
S_i :=
\left\|
\sum_{\boldsymbol{\ell} \in D_{i}}
\sum_{\mathbb{N}u \in (\mathbb{Z}^n)^N}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^j_{\ell_j, \mu_j}
\right\|_{B^{s}_{p, q}}
\lesssim
\prod_{j=1}^N
\|f_{j}\|_{B^{s_j}_{p_j, q_j}},
\quad
i=1, 2, 3,
\end{equation}
hold with the implicit constant independent of $\boldsymbol{\mu}$, where
\begin{align*}
&
D_{1}
=
\big\{
\boldsymbol{\ell} \in (\mathbb{N}_0)^{N+1}
:
\ell_0 \ge \ell_1-3,
\quad
\ell_j \le \ell_1, \
j=2, \dots, N
\big\},
\\
&
D_{2}
=
\left\{
\boldsymbol{\ell} \in (\mathbb{N}_0)^{N+1}
:
\ell_0 \le \ell_1-4,
\quad
\ell_2 \le \ell_1-N-2,
\quad
\ell_j \le \ell_2 \le \ell_1, \
j=3, \dots, N
\right\},
\\
&
D_{3}
=
\left\{
\boldsymbol{\ell} \in (\mathbb{N}_0)^{N+1}
:
\ell_0 \le \ell_1-4,
\quad
\ell_2 \ge \ell_1-N-1,
\quad
\ell_j \le \ell_2 \le \ell_1,\
j=3, \dots, N
\right\}.
\end{align*}
\begin{lem}\langlebel{EST-x}
Let
$m \in \mathbb{R}$
and $L \in \mathbb{N}_0$.
If $\sigma \in S^{m}_{0,0}(\mathbb{R}^n, N)$, then
\[
\|Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}\|_{L^\infty}
\lesssim
2^{-\ell_0L}
\langle \mathbb{N}u \rangle^{m}
\]
holds for all $\mathbb{N}u, \boldsymbol{\mu} \in (\mathbb{Z}^n)^N$ and $\ell_0 \in \mathbb{N}_0$.
\end{lem}
\begin{proof}
We first consider the case $\ell_0 \ge 1$.
Since $\mathcal{F}^{-1}\psi_{\ell_0}$ has the moment condition, Taylor's expansion gives that
\begin{align*}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}(x)
&=
\int_{\mathbb{R}^n}
\mathcal{F}^{-1}\psi_{\ell_0}(y) Q_{\mathbb{N}u, \boldsymbol{\mu}}(x-y)
\,
dy
\\
&=
\int_{\mathbb{R}^n}
\mathcal{F}^{-1}\psi_{\ell_0}(y)
\left(
Q_{\mathbb{N}u, \boldsymbol{\mu}}(x-y)
-
\sum_{|\alpha| \le L-1}
\frac{(-y)^\alpha}{\alpha !}
\partial^\alpha Q_{\mathbb{N}u, \boldsymbol{\mu}}(x)
\right)
\,
dy
\\
&=
\int_{\mathbb{R}^n}
\mathcal{F}^{-1}\psi_{\ell_0}(y)
\left(
L
\sum_{|\alpha| = L}
\frac{(-y)^\alpha}{\alpha !}
\int_0^1
(1-t)^{L-1}
[\partial^\alpha Q_{\mathbb{N}u, \boldsymbol{\mu}}](x-ty)
\,
dt
\right)
\,
dy
\end{align*}
Hence, it follows from \eqref{estPnu} that
\begin{align*}
|Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}(x)|
&\lesssim
\int_{\mathbb{R}^n}
|\mathcal{F}^{-1}\psi_{\ell_0}(y)|
\left(
\sum_{|\alpha| = L}
|(-y)^\alpha|
\int_0^1
|\partial^\alpha Q_{\mathbb{N}u, \boldsymbol{\mu}}(x-ty)|
\,
dt
\right)
\,
dy
\\
&\lesssim
\langle \mathbb{N}u \rangle^{m}
\int_{\mathbb{R}^n}
\frac{2^{\ell_0 n}|y|^L}{(1+2^{\ell_0}|y|)^{L+n+\epsilon}}
\,
dy
\\
&\lesssim
2^{-\ell_0L}
\langle \mathbb{N}u \rangle^{m}.
\end{align*}
Here, we used the estimate $|\mathcal{F}^{-1} \psi_{\ell_0}(x)| \lesssim 2^{\ell_0n}(1+2^{\ell_0}|x|)^{-(L+n+\epsilon)}$ in the second inequality.
We obtain the same estimate for $\ell_0=0$ without using the moment condition. The proof is complete.
\end{proof}
Now, we shall prove the estimate \eqref{GOAL!!!!}.
In what follows, we use the notation $f_{j, k} = \psi_k(D)f_j$ for $j = 1, \dots, N$ and $k \in \mathbb{N}_0$.
Since
\begin{align}
&\mathop{\mathrm{supp}} \mathcal{F} [Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}] \subset \{|\xi| \le 2^{\ell_0+1}\},
\langlebel{suppP}
\\
&\mathop{\mathrm{supp}} \mathcal{F} [\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}] \subset \nu_j + [-3, 3]^n, \quad j=1, \dots, N,
\langlebel{suppF}
\end{align}
we have
\begin{equation} \langlebel{suppunif}
\mathop{\mathrm{supp}} \mathcal{F}
\left[
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right]
\subset
\nu_1 + \dots + \nu_N
+
\left[
-2^{\ell_0+d}, 2^{\ell_0+d}
\right]^n
\end{equation}
for some $d = d_N > 0$ depending on $N$.
Thus we have
\begin{align} \langlebel{diag-rest}
\begin{split}
&\psi_k(D)
\left[
\sum_{\boldsymbol{\ell} \in D_{i}}
\sum_{\mathbb{N}u \in (\mathbb{Z}^n)^N}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^j_{\ell_j, \mu_j}
\right]
\\
&=
\psi_k(D)
\left[
\sum_{\boldsymbol{\ell} \in D_i}
\sum_{\mathbb{N}u : \nu_1 + \dots + \nu_N \in \Lambda_{k, \ell_0}}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right]
\end{split}
\end{align}
with
\begin{align*}
\Lambda_{k, \ell_0}
=
\{
\nu \in \mathbb{Z}^n
\,
:
\,
\mathop{\mathrm{supp}} \psi_k
\cap
(\nu + [-2^{\ell_0+d}, 2^{\ell_0+d}]^{n})
\neq
\emptyset
\}.
\end{align*}
We remark that $|\nu| \lesssim 2^{\ell_0+k}$ if $ \nu \in \Lambda_{k, \ell_0}$,
and consequently $|\Lambda_{k, \ell_0}| \lesssim 2^{(\ell_0+k)n}$.
We set
\begin{align*}
R_{\boldsymbol{\ell}, k}
=
R_{\boldsymbol{\ell}, k, \boldsymbol{\mu}}
=
\sum_{\mathbb{N}u : \nu_1 + \dots + \nu_N \in \Lambda_{k, \ell_0}}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}.
\end{align*}
For $M_0 \in \mathbb{R}$, we now prove that
the following estimate holds for all $\boldsymbol{\ell} = (\ell_0, \ell_1, \dots, \ell_N) \in (\mathbb{N}_0)^{N+1}$ and $k \in \mathbb{N}_0$:
\begin{align}\langlebel{Hulk}
&\left\|
R_{\boldsymbol{\ell}, k}
\right\|_{h^p}
\lesssim
2^{-\ell_0 M_0}
2^{k \alpha(p)}
2^{\ell_1m}
2^{\min\{\ell_2, k \}n/2}
\prod_{j=3}^{N}
2^{\ell_j n/2}
\prod_{j=1}^N
2^{-\ell_j\beta(p_j)}
\|f_{j, \ell_j}\|_{h^{p_j}}.
\end{align}
Here the implicit constant does not depend on $\boldsymbol{\mu}$.
Firstly, we prove that the estimate \eqref{Hulk} holds with $0< p< \infty$.
By the embedding $W^{p_0, 2}_{\alpha(p)} \hookrightarrow W^{p, 2}_{\alpha(p)} \hookrightarrow h^p$, we have
\begin{align*}
\|R_{\boldsymbol{\ell}, k}\|_{h^p}
&\lesssim
\|R_{\boldsymbol{\ell}, k}\|_{W^{p_0, 2}_{\alpha(p)}}
=
\left\|
\left\|
\langlengle \tau \ranglengle^{\alpha(p)}
\mathcal{B}ox_{\tau}
R_{\boldsymbol{\ell}, k}
\right\|_{\ell^2_{\tau}(\mathbb{Z}^n)}
\right\|_{L^{p_0}}.
\end{align*}
Recalling that \eqref{suppP}, \eqref{suppF} and that the function $\phi$ has compact support (see the definition of $\mathcal{B}ox_{\tau}$),
we write
\begin{align*}
&\|R_{\boldsymbol{\ell}, k}\|_{h^{p}}
\lesssim
\left\|
\left\|
\langle \tau \rangle^{\alpha(p)}
\mathcal{B}ox_{\tau}
\left[
\sum_{\substack{\mathbb{N}u : \nu_1 + \dots + \nu_N \in \Lambda_{k, \ell_0}
\\
|\nu_1 + \dots + \nu_N - \tau| \lesssim 2^{\ell_0}
}
}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right]
\right\|_{\ell^2_\tau(\mathbb{Z}^n)}
\right\|_{L^{p_0}}
\\
&\lesssim
2^{\ell_0 n/p_0}
\sum_{|\langlembda| \lesssim 2^{\ell_0}}
\left\|
\left\|
\langlengle \tau-\langlembda \ranglengle^{\alpha(p)}
\mathcal{B}ox_{\tau-\langlembda}
\left[
\sum_{\mathbb{N}u : \nu_1 + \dots + \nu_N = \tau}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right]
\right\|_{\ell^2_\tau(\Lambda_{k, \ell_0})}
\right\|_{L^{p_0}}
\\
&\lesssim
2^{\ell_0 (\alpha(p) + n/p_0)}
2^{k \alpha(p)}
\sum_{|\langlembda| \lesssim 2^{\ell_0}}
\left\|
\left\|
\mathcal{B}ox_{\tau-\langlembda}
R_\tau
\right\|_{\ell^2_k(\Lambda_{k, \ell_0})}
\right\|_{L^{p_0}},
\end{align*}
where
we set
\begin{equation} \langlebel{Nanjakore}
R_\tau
=
\sum_{\mathbb{N}u : \nu_1 + \dots + \nu_N = \tau}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}.
\end{equation}
and used the inequality $\langle \tau -\langlembda \rangle \lesssim 2^{\ell_0+k}$ for
$|\langlembda| \lesssim 2^{\ell_0}$ and $\tau \in \Lambda_{k, \ell_0}$.
Now, by recalling \eqref{suppunif},
we have $\mathop{\mathrm{supp}} \mathcal{F}[\mathcal{B}ox_{k-\langlembda}R_\tau] \subset \{|\zeta - \tau| \lesssim 2^{\ell_0}\}$.
Hence Nikol'skij's inequality (see, e.g, \cite[Remark 1.3.2/1]{Triebel-ToFS}) gives that
\begin{align} \langlebel{pointwiseNikolskij}
|\mathcal{B}ox_{k-\langlembda} R_\tau(x)|
\le
\|\Phi(\cdot)R_\tau(x-\cdot)\|_{L^1}
\lesssim
2^{\ell_0 n (1/r_0 -1)}
\|\Phi(\cdot)R_\tau(x-\cdot)\|_{L^{r_0}},
\end{align}
where $\Phi = \mathcal{F}^{-1} \phi$ and $r_0 = \min \{1, p_0\}$.
By Minkowski's inequality, we have
\begin{align*}
\left\|
\left\|
\mathcal{B}ox_{k-\langlembda}
R_\tau(x)
\right\|_{\ell^2_\tau(\Lambda_{k, \ell_0})}
\right\|_{L^{p_0}_x}
&\lesssim
2^{\ell_0 n (1/r_0-1)}
\left\|
\left\|
\left\|
\Phi(y)R_\tau(x-y)
\right\|_{L^{r_0}_y}
\right\|_{\ell^2_\tau(\Lambda_{k, \ell_0})}
\right\|_{L^{p_0}_x}
\\
&\lesssim
2^{\ell_0 n (1/r_0-1)}
\left\|
\left\|
R_\tau(x)
\right\|_{\ell^2_\tau(\Lambda_{k, \ell_0})}
\right\|_{L^{p_0}_x}.
\end{align*}
Hence, by applying Lemma \ref{EST-x} with $L$ replaced by
$M_0+\alpha(p) + n/p_0 +n/r_0 +n/2$,
we obtain
\begin{align*}
\|R_{\boldsymbol{\ell}, k}\|_{h^{p}}
&\lesssim
2^{-\ell_0(M_0+n/2)}
2^{k \alpha(p)}
\left\|
\left\|
\sum_{\mathbb{N}u : \nu_1 + \dots + \nu_N = \tau}
\langle \mathbb{N}u \rangle^{m}
\prod_{j=1}^N
\left|
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right|
\right\|_{\ell^2_\tau(\Lambda_{k, \ell_0})}
\right\|_{L^{p_0}}.
\end{align*}
If $\mathop{\mathrm{supp}} \varphi(\cdot-\nu_j) \cap \mathop{\mathrm{supp}} \psi_{\ell_j} = \emptyset$, then $\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j} = 0$.
Hence the sum over the $\nu_j$-variable can be restricted to
\[
\Lambda_{\ell_j}
=
\{\nu_j \in \mathbb{Z}^n
:
\mathop{\mathrm{supp}} \varphi(\cdot-\nu_j)
\cap
\mathop{\mathrm{supp}} \psi_{\ell_j}
\neq
\emptyset
\},
\quad
j=1, \dots, N.
\]
Notice that $|\nu_j| \approx 2^{\ell_j}$ if $\nu_j \in \Lambda_{\ell_j}$.
Furthermore, since $\langle \mathbb{N}u \rangle \approx 2^{\ell_1}$ for all $\mathbb{N}u = (\nu_1, \dots, \nu_N) \in \Lambda_{\ell_1} \times \dots \times \Lambda_{\ell_N}$
if $\ell_1 \ge \ell_j$, $j=2, \dots, N$,
we have
\begin{align*}
\sum_{\mathbb{N}u : \nu_1 + \dots + \nu_N = \tau}
\langle \mathbb{N}u \rangle^{m}
\prod_{j=1}^N
\left|
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right|
\lesssim
2^{\ell_1 m}
\sum_{\substack{\mathbb{N}u \in \Lambda_{\ell_1} \times \dots \times \Lambda_{\ell_N}
\\
\nu_1 + \dots + \nu_N = \tau}}
\prod_{j=1}^N
\left|
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right|,
\end{align*}
Recalling that $|\nu_j| \approx 2^{\ell_j}$ if $\nu_j \in \Lambda_{\ell_j}$
and $|\tau| \lesssim 2^{\ell_0 + k}$ if $\tau \in \Lambda_{k, \ell_0}$,
we have by Proposition \ref{Keyprop}
\begin{align*}
\left\|
\left\|
\sum_{\substack{\mathbb{N}u \in \Lambda_{\ell_1} \times \dots \times \Lambda_{\ell_N}
\\
\nu_1 + \dots + \nu_N = \tau}}
\prod_{j=1}^N
\left|
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right|
\right\|_{\ell^2_\tau(\Lambda_{k, \ell_0})}
\right\|_{L^{p_0}}
&\lesssim
2^{\min\{\ell_2, \ell_0+k\}n/2}
\prod_{j=3}^N
2^{\ell_jn/2}
\prod_{j=1}^N
2^{-\ell_j\beta(p_j)}
\|F^{j}_{\ell_j, \mu_j}\|_{h^{p_j}}
\\
&\le
2^{\ell_0 n/2}
2^{\min\{\ell_2, k \}n/2}
\prod_{j=3}^N
2^{\ell_jn/2}
\prod_{j=1}^N
2^{-\ell_j\beta(p_j)}
\left\|f_{j, \ell_j}\right\|_{h^{p_j}},
\end{align*}
where we used the fact that the $h^{p_j}$-norms are translation invariant in the last inequality.
Collecting the above estimates, we obtain \eqref{Hulk} with $0 < p < \infty$.
Next we shall show that the estimate \eqref{Hulk} holds with $p=\infty$.
Notice that $\alpha(\infty) = n/2$.
By the embedding relation $ W^{p_0, 1} \hookrightarrow W^{\infty, 1} \hookrightarrow L^{\infty}$,
we have
\[
\left\|
R_{\boldsymbol{\ell}, k}
\right\|_{L^\infty}
\lesssim
\left\|
R_{\boldsymbol{\ell}, k}
\right\|_{W^{p_0, 1}}
=
\left\|
\left\|
\mathcal{B}ox_\tau
R_{\boldsymbol{\ell}, k}
\right\|_{\ell^1_\tau}
\right\|_{L^{p_0}}.
\]
By the same argument as in the case $0< p < \infty$, it holds that
\begin{align*}
\left\|
\left\|
\mathcal{B}ox_\tau
R_{\boldsymbol{\ell}, k}
\right\|_{\ell^1_\tau}
\right\|_{L^{p_0}}
&=
\left\|
\left\|
\mathcal{B}ox_\tau
\left[
\sum_{\substack{\mathbb{N}u : \nu_1 + \dots + \nu_N \in \Lambda_{k, \ell_0}
\\
|\nu_1 + \dots + \nu_N - \tau| \lesssim 2^{\ell_0}
}}
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\left|
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right|
\right]
\right\|_{\ell^1_\tau}
\right\|_{L^{p_0}}
\\
&\lesssim
2^{\ell_0 n/p_0}
\sum_{|\langlembda| \lesssim 2^{\ell_0}}
\left\|
\left\|
\mathcal{B}ox_{\tau-\langlembda}
R_\tau
\right\|_{\ell^1_\tau(\Lambda)}
\right\|_{L^{p_0}}
\end{align*}
with $R_\tau$ given by \eqref{Nanjakore}.
Furthermore, it follows from \eqref{pointwiseNikolskij} that
\begin{align*}
\left\|
\left\|
\mathcal{B}ox_{\tau-\langlembda}
R_\tau
\right\|_{\ell^1_\tau(\Lambda)}
\right\|_{L^{p_0}}
\lesssim
2^{\ell_0 n (1/r_0-1)}
\left\|
\left\|
R_\tau
\right\|_{\ell^1_\tau(\Lambda)}
\right\|_{L^{p_0}}.
\end{align*}
Combining these estimates with Lemma \ref{EST-x} with $L$ replaced by
$M_0+ n/p_0 +n/r_0 +n$, we obtain
\begin{align*}
\left\|
R_{\boldsymbol{\ell}, k}
\right\|_{L^\infty}
&\lesssim
2^{-\ell_0(M_0+n)}
\left\|
\left\|
\sum_{\mathbb{N}u : \nu_1 + \dots + \nu_N = \tau}
\langle \mathbb{N}u \rangle^{m}
\prod_{j=1}^N
\left|
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right|
\right\|_{\ell^1_\tau(\Lambda)}
\right\|_{L^{p_0}}
\\
&\lesssim
2^{-\ell_0(M_0+n)}
2^{\ell_1 m}
\left\|
\left\|
\sum_{\substack{\mathbb{N}u \in \Lambda_{\ell_1} \times \dots \times \Lambda_{\ell_N}
\\
\nu_1 + \dots + \nu_N = \tau}}
\prod_{j=1}^N
\left|
\mathcal{B}ox_{\nu_j} F^{j}_{\ell_j, \mu_j}
\right|
\right\|_{\ell^1_\tau(\Lambda)}
\right\|_{L^{p_0}}.
\end{align*}
Hence, it follows from the estimate in Remark \ref{Keyrem} that
the right hand side just above can be estimated by
\begin{align*}
&
2^{-\ell_0(M_0+n)}
2^{\ell_1 m}
\times
2^{(\ell_0+j)n/2}
2^{\min \{\ell_2, \ell_0+j\} n/2}
\times
2^{-\ell_1\beta(p_1)}
\|f_{1, \ell_1}\|_{h^{p_1}}
\times
2^{-\ell_2\beta(p_2)}
\|f_{2, \ell_2}\|_{h^{p_2}}
\\
&\le
2^{-\ell_0 M_0}
2^{jn/2}
2^{\ell_1 m}
2^{\min \{\ell_2, j\}n/2}
\times
2^{-\ell_1\beta(p_1)}
\|f_{1, \ell_1}\|_{h^{p_1}}
\times
2^{-\ell_2\beta(p_2)}
\|f_{2, \ell_2}\|_{h^{p_2}}.
\end{align*}
The proof of \eqref{Hulk} is complete.
Now, we shall return to the proof the estimate \eqref{GOAL!!!!}.
By the embedding relation \eqref{Bq1Bq2},
it is sufficient to prove \eqref{GOAL!!!!} with $0 < q, q_1, \dots, q_N \le \infty$
satisfying $1/q = \sum_{j=1}^N 1/q_j$.
We set $r=\min\{1, p, q\}$.
\noindent
\textbf{Estimate for $S_1$ :}
If $\ell_0 \ge \ell_1-3$ and $\ell_1 \ge \ell_j$, $j=2, \dots, N$, then we have
\begin{equation*}
\mathop{\mathrm{supp}}
\mathcal{F}
\left[
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^j_{\ell_j, \mu_j}
\right]
\subset
\big\{
|\zeta| \le 2^{\ell_0+a_N}
\big\}
\end{equation*}
with some positive integer $a_N$ depending only on $N$.
Hence
\[
\psi_k(D) R_{\boldsymbol{\ell}, k} = 0
\quad
\text{if}
\quad
\ell_0 \le k-1-a_N,
\]
and consequently,
\begin{align} \langlebel{psildel}
\left\|
\psi_k(D) \sum_{\boldsymbol{\ell} \in D_1} R_{\boldsymbol{\ell}, k}
\right\|_{L^{p}}^r
\le
\sum_{\substack{\boldsymbol{\ell} \in D_1 \\ \ell_0 \ge k-a_N}}
\left\|
\psi_k(D)
R_{\boldsymbol{\ell}, k}
\right\|_{L^{p}}^r
\lesssim
\sum_{
\substack{
\boldsymbol{\ell} \in D_1
\\
\ell_0 \ge k-a_N
}}
\left\|
R_{\boldsymbol{\ell}, k}
\right\|_{h^p}^r,
\end{align}
where we used the estimate \eqref{embd-hpB0pinfty} in $\lesssim$.
Then, by \eqref{Hulk}, the right hand side above is estimated by
\begin{align*}
&
\sum_{
\substack{
\boldsymbol{\ell} \in D_1
\\
\ell_0 \ge k-a_N
}}
\mathcal{B}ig(
2^{-\ell_0 M_0}
2^{k \alpha(p)}
2^{\ell_1 m}
\prod_{j=2}^N
2^{\ell_j n/2}
\prod_{j=1}^N
2^{-\ell_j\beta(p_j)}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}
\mathcal{B}ig)^{r}
=
2^{k \alpha(p) r} U_k,
\end{align*}
where
\[
U_k
=
\sum_{\ell_0 : \ell_0 \ge k - a_N}
2^{-\ell_0 M_0 r}
\sum_{\ell_1 : \ell_1 \le \ell_0+3}
2^{\ell_1 (m - \beta(p_1)) r}
\|f_{1, \ell_1}\|_{h^{p_1}}^{r}
\prod_{j=2}^N
\sum_{\ell_j : \ell_j \le \ell_1}
2^{\ell_j\theta(p_j) r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}
\]
and
\begin{align}\langlebel{MsMarvel}
\theta(p_j)
=
(n/2)
- \beta(p_j)
=
\max\{n/p_j, n/2\}.
\end{align}
Notice that we have $\left\| f_{j, \ell_j} \right\|_{h^{p_j}} = \|\psi_{\ell_j}(D)f_j\|_{h^{p_j}} \lesssim 2^{-\ell_j s_j} \|f_j\|_{B^{s_j}_{p_j, q_j}}$
by \eqref{Bhpequiv} and \eqref{Bq1Bq2}.
Hence we have
\begin{align*}
U_k
&\lesssim
\left(
\sum_{\ell_0 : \ell_0 \ge k - a_N}
2^{-\ell_0 M_0 r}
\sum_{\ell_1 : \ell_1 \le \ell_0+3}
2^{\ell_1(m -\beta(p_1)-s_1) r}
\prod_{j=2}^N
\sum_{\ell_j : \ell_j \le \ell_1}
2^{\ell_j(\theta(p_j) - s_j)r}
\right)
\prod_{j=1}^N
\|f_{j}\|_{B^{s_j}_{p_j, q_j}}^r
\\
&\lesssim
\left(
\sum_{\ell_0 : \ell_0 \ge k-a_N}
2^{-\ell_0(M_0-C) r}
\right)
\prod_{j=1}^N
\|f_{j}\|_{B^{s_j}_{p_j, q_j}}^r.
\end{align*}
Here $C > 0$ is the sufficiently large number satisfying
\[
\sum_{\ell_1 : \ell_1 \le \ell_0+3}
2^{\ell_1(m -\beta(p_1)-s_1) r}
\prod_{j=2}^N
\sum_{\ell_j : \ell_j \le \ell_1}
2^{\ell_j(\theta(p_j) - s_j)r}
\le
2^{\ell_0 C r}.
\]
Choosing $M_0$ sufficiently large, we obtain
\begin{align*}
S_1
&\lesssim
\left\|
2^{k s}
\left\|
\psi_k(D)
\sum_{\boldsymbol{\ell} \in D_1}
R_{\boldsymbol{\ell}, k}
\right\|_{L^p_{}}
\right\|_{\ell^q_k
}
\\
&\lesssim
\left\|
2^{k (s +\alpha(p))}
\left(
\sum_{\ell_0 : \ell_0 \ge k-a_N}
2^{-\ell_0(M_0-C) r}
\right)^{1/r}
\right\|_{\ell^q_k}
\prod_{j=1}^N
\|f_{j}\|_{B^{s_j}_{p_j, q_j}}
\lesssim
\prod_{j=1}^N
\|f_{j}\|_{B^{s_j}_{p_j, q_j}}.
\end{align*}
The estimate for $S_1$ is complete.
\noindent
\textbf{Estimate for $S_2$ :}
If $\ell_0 \le \ell_1-4$ and $\ell_j \le \ell_1-N-2$ for all $j =2, \dots, N$, then
\begin{equation*}
\mathop{\mathrm{supp}}
\mathcal{F}
\left[
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^j_{\ell_j, \mu_j}
\right]
\subset
\big\{
2^{\ell_1-b_N} \le |\zeta| \le 2^{\ell_1+b_N}
\big\}.
\end{equation*}
with some positive integer $b_N > 0$ depending only on $N$.
Thus we see that
\[
\psi_k(D) R_{\boldsymbol{\ell}, k} = 0
\quad
\text{if}
\quad
|\ell_1 - k| \ge b_N + 1.
\]
Hence it follows from the same argument as in \eqref{psildel} that
\begin{align*}
\left\|
\psi_k(D) \sum_{\boldsymbol{\ell} \in D_2} R_{\boldsymbol{\ell}, k}
\right\|_{L^{p}}^r
&\lesssim
\sum_{
\substack{
\boldsymbol{\ell} \in D_{2}
\\
|\ell_1-k| \le b_N
}
}
\left\|
R_{\boldsymbol{\ell}, k}
\right\|_{h^{p}}^r.
\end{align*}
By using \eqref{Hulk} and taking the sum over $\ell_0$, we have
\begin{align*}
\sum_{
\substack{
\boldsymbol{\ell} \in D_{2}
\\
|\ell_1-k| \le b_N
}
}
\left\|
R_{\boldsymbol{\ell}, k}
\right\|_{h^{p}}^r
&\lesssim
2^{k \alpha(p) r}
V_k,
\end{align*}
where
\begin{align*}
V_k
&=
\sum_{\ell_1 : |\ell_1-k| \le b_N}
2^{\ell_1(m - \beta(p_1)) r}
\|f_{1, \ell_1}\|_{h^{p_1}}^{r}
\prod_{j=2}^N
\sum_{\ell_j : \ell_j \le \ell_1}
2^{\ell_j\theta(p_j) r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}
\\
&=
\sum_{\ell : |\ell|\le b_N}
2^{(k+\ell)(m-\beta(p_1))r}
\|f_{1, k+\ell}\|_{h^{p_1}}^{r}
\prod_{j=2}^N
\sum_{\ell_j : \ell_j \le k+\ell}
2^{\ell_j \theta(p_j) r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}.
\end{align*}
Thus, we have
\begin{align*}
S_2
&\lesssim
\left\|
2^{k s}
\left\|
\psi_k(D) \sum_{\boldsymbol{\ell} \in D_2} R_{\boldsymbol{\ell}, k}
\right\|_{L^{p}_{}}
\right\|_{\ell^q_k}
\\
&\lesssim
\sum_{\ell : |\ell|\le b_N}
\left\|
2^{k (s +\alpha(p))}
2^{(k+\ell)(m-\beta(p_1))}
\|f_{1, k+\ell}\|_{h^{p_1}}
\prod_{j=2}^N
\left(
\sum_{\ell_j : \ell_j \le k+\ell}
2^{\ell_j \theta(p_j) r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}
\right)^{1/r}
\right\|_{\ell^q_k}
\\
&\le
\sum_{\ell : |\ell|\le b_N}
2^{-\ell (s +\alpha(p))}
\left\|
2^{k(m-\beta(p_1) +s +\alpha(p))}
\|f_{1, k}\|_{h^{p_1}}
\prod_{j=2}^N
\left(
\sum_{\ell_j : \ell_j \le k}
2^{\ell_j \theta(p_j) r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}
\right)^{1/r}
\right\|_{\ell^q_k}
\\
&\lesssim
\left\|
2^{k(m-\beta(p_1) +s +\alpha(p))}
\|f_{1, k}\|_{h^{p_1}}
\prod_{j=2}^N
\left(
\sum_{\ell_j : \ell_j \le k}
2^{\ell_j \theta(p_j) r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}
\right)^{1/r}
\right\|_{\ell^q_k}
\end{align*}
where we change a variable with respect to $\ell$ in the third inequality.
Since $\alpha(p) - n/2 =-\min\{n/p, n/2\},$
we have
$m -\beta(p_1) + s + \alpha(p) = s_1 + \sum_{j=2}^N (s_j -\theta(p_j))$.
Hence we obtain
\begin{align*}
S_2
&\lesssim
\left\|
2^{k s_1}
\|f_{1, k}\|_{h^{p_1}}
\prod_{j=2}^N
\left(
\sum_{\ell_j : \ell_j \le k}
2^{(k-\ell_j)(s_j - \theta(p_j)) r}
2^{\ell_j s_j r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}
\right)^{1/r}
\right\|_{\ell^q_k}
\\
&\le
\left\|
2^{k s_1}
\|f_{1, k}\|_{h^{p_1}_{}}
\right\|_{\ell^{q_1}_k}
\prod_{j=2}^N
\left\|
\left(
\sum_{\ell_j : \ell_j \le k}
2^{(k-\ell_j) (s_j -\theta(p_j)) r}
2^{\ell_j s_j r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}_{}}^{r}
\right)^{1/r}
\right\|_{\ell^{q_j}_k}
\\
&\approx
\|f_{1}\|_{B^{s_1}_{p_1, q_1}}
\prod_{j=2}^N
\left\|
\sum_{\ell_j : \ell_j \le k}
2^{(k-\ell_j) (s_j -\theta(p_j)) r}
2^{\ell_j s_j r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}
\right\|_{\ell^{q_j/r}_k}^{1/r},
\end{align*}
where we used \eqref{Bhpequiv} in the last inequality.
Since $q_j \ge q \ge r$ and $s_j -\theta(p_j) <0$, $j=2, \dots, N$, it follows from Young's inequality that
\begin{align*}
\left\|
\sum_{\ell_j : \ell_j \le k}
2^{(k-\ell_j) (s_j -\theta(p_j)) r}
2^{\ell_j s_j r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}
\right\|_{\ell^{q_j/r}_k}^{1/r}
&\le
\left\|
2^{k (s_j -\theta(p_j))r}
\right\|_{\ell^1_k}^{1/r}
\left\|
2^{k s_j r}
\left\|f_{j, k}\right\|_{h^{p_j}_{}}^r
\right\|_{\ell^{q_j/r}_k}^{1/r}
\\
&\lesssim
\left\|
2^{k s_j}
\left\|f_{j, k}\right\|_{h^{p_j}_{}}
\right\|_{\ell^{q_j}_k}
\approx
\|f_{j}\|_{B^{s_j}_{p_j, q_j}},
\quad
j=2, \dots, N,
\end{align*}
where we used Proposition \ref{propQui} in the last inequality.
Combining these estimates, we obtain the desired estimate.
\noindent
\textbf{Estimate for $S_3$ :}
Since if $\ell_0 \le \ell_1-4$ and $\ell_1 \ge \ell_j$, $j =2, \dots, N$, then
\begin{equation*}
\mathop{\mathrm{supp}}
\mathcal{F}
\left[
Q_{\ell_0, \mathbb{N}u, \boldsymbol{\mu}}
\prod_{j=1}^N
\mathcal{B}ox_{\nu_j} F^j_{\ell_j, \mu_j}
\right]
\subset
\big\{
|\zeta| \le 2^{\ell_1 + c_N}
\big\}
\end{equation*}
with some positive integer $c_N$ depending only on $N$.
Hence
\[
\psi_k(D) R_{\boldsymbol{\ell}, k} = 0
\quad
\text{if}
\quad
\ell_1 \le k-1-c_N,
\]
and, by the same argument as above,
\begin{align*}
\left\|
\psi_k(D) \sum_{\boldsymbol{\ell} \in D_3} R_{\boldsymbol{\ell}, k}
\right\|_{L^{p}}^r
&\lesssim
\sum_{
\substack{
\boldsymbol{\ell} \in D_{3}
\\
\ell_1 \ge k-c_N
}
}
\left\|
R_{\boldsymbol{\ell}, k}
\right\|_{h^{p}}^r.
\end{align*}
Since
$
\alpha(p)
+
(n/2)
=
\max\{n/p^{\prime}, n/2\}
=
\theta(p^{\prime}),
$
it follows from \eqref{Hulk} and \eqref{MsMarvel} that
\begin{align*}
&\left\|
\psi_k(D) \sum_{\boldsymbol{\ell} \in D_3} R_{\boldsymbol{\ell}, k}
\right\|_{L^{p}}^r
\lesssim
2^{k \theta(p^{\prime}) r}
W_k,
\end{align*}
where
\begin{align*}
W_k=
&\sum_{\ell_1 : \ell_1 \ge k-c_N}
2^{\ell_1(m-\beta(p_1)) r}
\|f_{1, \ell_1}\|_{h^{p_1}}^{r}
\\
&\qquad\qquad\qquad
\times
\sum_{\ell_2 : \ell_1-N-1 \le \ell_2 \le \ell_1}
2^{-\ell_2\beta(p_2) r}
\|f_{2, \ell_2}\|_{h^{p_2}}^{r}
\prod_{j=3}^N
\sum_{\ell_j : \ell_j \le \ell_1}
2^{\ell_j\theta(p_j) r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}.
\end{align*}
By a change of variable with respect to $\ell_2$, we have
\begin{align*}
W_k
=
W_{k, 0} + W_{k, 1} + \dots + W_{k, N+1}
=
\sum_{i=0}^{N+1}
W_{k, i}
\end{align*}
with
\begin{align*}
W_{k, i}
=
\sum_{\ell_1 : \ell_1 \ge k-c_N}
2^{\ell_1 (m-\beta(p_1)) r}
2^{-(\ell_1 -i) \beta(p_2) r}
\|f_{1, \ell_1}\|_{h^{p_1}}^{r}
\|f_{2, \ell_1 -i}\|_{h^{p_2}}^{r}
\prod_{j=3}^N
\sum_{\ell_j : \ell_j \le \ell_1}
2^{\ell_j \theta(p_j) r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}}^{r}.
\end{align*}
Then we have
\begin{align*}
S_3
\lesssim
\left\|
2^{k s}
\left\|
\psi_k(D)
\sum_{\boldsymbol{\ell} \in D_3} R_{\boldsymbol{\ell}, k}
\right\|_{L^p_{}}
\right\|_{\ell^q_k}
\lesssim
\sum_{i=0}^{N+1}
\left\|
2^{k (s + \theta(p^{\prime}))}
W^{1/r}_{k, i}
\right\|_{\ell^q_k}.
\end{align*}
It is sufficient to prove the estimate for $W_{k,0}$,
The same argument below works for the other terms.
By using the notation $\theta(p_j)$ given in \eqref{MsMarvel}, we can write
$m -\beta(p_1) -\beta(p_2) = -s -\theta(p^{\prime})
+s_1+s_2
+\sum_{j=3}^N (s_j - \theta(p_j) )
$.
Hence
\begin{align*}
W_{k, 0}
&=
\sum_{\ell_1 : \ell_1 \ge k-c_N}
2^{-\ell_1(s+\theta(p^{\prime}))r}
\prod_{j=1}^2
2^{\ell_1 s_j r} \|f_{j, \ell_1}\|_{h^{p_j}}^r
\prod_{j=3}^N
\sum_{\ell_j : \ell_j \le \ell_1}
2^{(\ell_1-\ell_j)(s_j - \theta(p_j)) r}
2^{\ell_j s_j r}
\|f_{j, \ell_j}\|_{h^{p_j}}^{r}
\\
&=
\sum_{\ell_1 : \ell_1 \ge k-c_N}
2^{-\ell_1(s+\theta(p^{\prime}))r}
\widetilde{W}_{\ell_1},
\end{align*}
where
\begin{align*}
\widetilde{W}_{\ell_1}
=
\prod_{j=1}^2
2^{\ell_1 s_j r} \|f_{j, \ell_1}\|_{h^{p_j}}^r
\prod_{j=3}^N
\sum_{\ell_j : \ell_j \le \ell_1}
2^{(\ell_1-\ell_j)(s_j - \theta(p_j)) r}
2^{\ell_j s_j r}
\|f_{j, \ell_j}\|_{h^{p_j}}^{r}.
\end{align*}
Since $q \ge r$, it follows from Young's inequality that
\begin{align*}
\left\|
2^{k (s +\theta(p^{\prime}))}
W_{k, 0}^{1/r}
\right\|_{\ell^q_k}
&=
\left\|
\sum_{\ell_1 : \ell_1 \ge k-c_N}
2^{-(\ell_1-k)(s+\theta(p^{\prime}))r}
\widetilde{W}_{\ell_1}
\right\|_{\ell^{q/r}_k}^{1/r}
\\
&\lesssim
\left\|
\sum_{\ell_1 \in \mathbb{N}_0}
2^{-|\ell_1-k|(s+\theta(p^{\prime}))r}
\widetilde{W}_{\ell_1}
\right\|_{\ell^{q/r}_k}^{1/r}
\\
&\le
\left\|
2^{-k(s+\theta(p^{\prime}))r}
\right\|_{\ell^1_k}^{1/r}
\|
\widetilde{W}_{k}
\|_{\ell^{q/r}_k}^{1/r}
\lesssim
\|
\widetilde{W}_{k}
\|_{\ell^{q/r}_k}^{1/r},
\end{align*}
where we used the assumption $s+ \theta(p^{\prime}) > 0$ in the last inequality.
Furthermore,
by H\"older's inequality and the same argumet as in the estimate for $S_2$, we obtain
\begin{align*}
\|
\widetilde{W}_{k}
\|_{\ell^{q/r}_k}^{1/r}
&\lesssim
\prod_{j=1}^2
\left\|
2^{k s_j r} \left\|f_{j, k}\right\|_{h^{p_j}_{}}^r
\right\|_{\ell^{q_j/r}_k}^{1/r}
\prod_{j=3}^N
\left\|
\sum_{\ell_j : \ell_j \le k}
2^{(k-\ell_j)(s_j - \theta(p_j)) r}
2^{\ell_j s_j r}
\left\| f_{j, \ell_j} \right\|_{h^{p_j}_{}}^{r}
\right\|_{\ell^{q_j/r}_k}^{1/r}
\\
&\lesssim
\prod_{j=1}^N
\|f_j\|_{B^{s_j}_{p_j, q_j}}.
\end{align*}
Thus we obtain the estimate \eqref{GOAL!!!!}. The proof of Theorem \ref{main1} is complete.
\section{Proof of Theorem \ref{thmnec}}
In this section, we shall show the sharpness of conditions of $s_1, \dots, s_N$ and $s$.
In particular, we shall give the proof of Theorem \ref{thmnec}.
We now assume that the boundedness
\begin{equation} \langlebel{nec-bdd}
\mathop{\mathrm{Op}}(S^m_{0,0}(\mathbb{R}^n, N))
\subset
B(B^{s_1}_{p_1, q_1} \times \dots \times B^{s_N}_{p_N, q_N} \to B^s_{p, q})
\end{equation}
holds with
\begin{equation}\langlebel{mcrit}
m = \min \left\{ \frac{n}{p}, \frac{n}{2} \right\}
-
\sum_{j=1}^N \max\left\{ \frac{n}{p_j}, \frac{n}{2} \right\}
+
\sum_{j=1}^N s_j -s.
\end{equation}
By the closed graph theorem, the assumption \eqref{nec-bdd} implies that there exists
$M \in \mathbb{N}$ such that
\begin{align}\langlebel{operator-norm}
\begin{split}
&\|T_{\sigma}\|_{B^{s_1}_{p_1, q_1} \times \dots \times B^{s_N}_{p_N, q_N} \to B^{s}_{p, q}}
\\
&\quad
\lesssim
\max_{|\alpha|, |\beta_1|, \dots, |\beta_N| \le M}
\|
\langle (\xi_1, \dots, \xi_N)\rangle^{-m}
\partial_x^{\alpha}\partial_{\xi_1}^{\beta_1} \dots \partial_{\xi_N}^{\beta_N}
\sigma(x, \xi_1, \dots, \xi_N)
\|_{L^\infty_{x, \xi_1, \dots, \xi_N}}
\end{split}
\end{align}
for all $\sigma \in S^{m}_{0, 0}(\mathbb{R}^n, N)$.
For the argument using the closed graph theorem, see \cite[Lemma 2.6]{BBMNT}.
We recall the following fact given
by Wainger \cite{Wainger} and Miyachi-Tomita \cite{MT-IUMJ}.
\begin{lem}[\cite{Wainger, MT-IUMJ}]\langlebel{lemWainger}
Let $0< a < 1$, $0< b < n$, $1 \le p \le \infty$ and $\varphi \in \mathcal{S}(\mathbb{R}^n)$.
For $\epsilon > 0$, we set
\[
f_{a, b, \epsilon}(x)
=
\sum_{\nu \in \mathbb{Z}^n \setminus \{0\}}
e^{-\epsilon|\nu|}
|\nu|^{-b} e^{i|\nu|^a} e^{i\nu \cdot x} \varphi(x).
\]
If $b > (1-a)(n/2-n/p) +n/2$, then $\sup_{\epsilon >0}\|f_{a, b, \epsilon}\|_{L^p} < \infty$.
\end{lem}
In this section, we will use the following partition of unity $\psi_\ell$, $\ell= 0, 1, 2, \dots$,
in the definition of Besov spaces;
\begin{align*}
&\mathop{\mathrm{supp}} \psi_0 \subset \{|\xi| \le 2^{3/4}\},
\quad
\psi_0 = 1 \quad \text{on} \quad \{|\xi| \le 2^{1/4}\}
\\
&\mathop{\mathrm{supp}} \psi_\ell \subset \{2^{\ell-3/4} \le |\xi| \le 2^{\ell+3/4}\},
\quad
\psi_\ell= 1 \quad \text{on} \quad \{2^{\ell -1/4} \le |\xi| \le 2^{\ell + 1/4}\},
\quad
\ell \ge 1,
\\
&\|\partial^{\alpha}\psi_\ell\|_{L^\infty}
\le
C_{\alpha} 2^{-\ell |\alpha|},
\quad \alpha \in \mathbb{N}_0^n, \xi \in \mathbb{R}^n,
\\
&\sum_{\ell = 0}^\infty \psi_{\ell}(\xi) = 1,
\quad \xi \in \mathbb{R}^n.
\end{align*}
We take functions $\widetilde{\psi}_\ell \in \mathcal{S}(\mathbb{R}^n)$, $\ell = 0, 1, 2, \dots$, such that
\begin{align*}
&\mathop{\mathrm{supp}} \widetilde{\psi}_0
\subset
\{|\xi| \le 2^{1/4}\}
\quad
\text{and}
\quad
\widetilde{\psi}_0
=
1
\quad
\text{on}
\quad
\{|\xi| \le 2^{1/8}\},
\\
&\mathop{\mathrm{supp}} \widetilde{\psi}_\ell \subset \{2^{\ell-1/4} \le |\xi| \le 2^{\ell+1/4}\},
\quad
\widetilde{\psi}_\ell = 1 \quad \text{on} \quad \{2^{\ell -1/8} \le |\xi| \le 2^{\ell + 1/8}\},
\quad
\ell \ge 1,
\\
&\sup_{\ell \in \mathbb{N}_0}\|\mathcal{F}^{-1} \widetilde{\psi}_{\ell}\|_{L^1} < \infty.
\end{align*}
We also use the functions $\varphi, \widetilde{\varphi} \in \mathcal{S}(\mathbb{R}^n)$ satisfying the following;
\begin{align*}
&\mathop{\mathrm{supp}} \varphi \subset [-1/4, 1/4]^n,
\quad
|\mathcal{F}^{-1}\varphi(x)| \ge 1 \quad \text{on} \quad [-1, 1]^n,
\\
&
\mathop{\mathrm{supp}} \widetilde{\varphi} \subset [-1/2, 1/2]^n,
\quad
\widetilde{\varphi} = 1 \quad \text{on} \quad [-1/4, 1/4]^n.
\end{align*}
\begin{lem}\langlebel{lemNecessity1}
Let $1< p_1, \dots, p_N < \infty$, $0< p< \infty$, $0 < q, q_1, \dots, q_N < \infty$ and $s, s_1, \dots, s_N \in \mathbb{R}$.
If the boundedness \eqref{nec-bdd} holds with $m$ given in \eqref{mcrit}, then
$s \ge -\max\{n/p^{\prime}, n/2\}$.
\end{lem}
\begin{proof}
Let $\{c_{\boldsymbol{\mu}}\}_{\boldsymbol{\mu} \in (\mathbb{Z}^n)^{N}}$ be a sequence of complex numbers satisfying $\sup_{\boldsymbol{\mu}} |c_{\boldsymbol{\mu}}| \le 1$.
We take sufficiently large number $L > 0$.
For sufficiently large $\ell \in \mathbb{N}$ satisfying $\ell > L$,
we set
\begin{align*}
&\sigma_{\ell}(\xi_1, \dots, \xi_N)
=
\sum_{\boldsymbol{\mu} = (\mu_1,\dots, \mu_N) \in D_\ell}
c_{\boldsymbol{\mu}} \langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=1}^N
\varphi(\xi_j - \mu_j),
\\
&
\widehat{f_{1, \ell}}(\xi_1)
=
\widetilde{\psi}_{\ell}(\xi_1) \widehat{f_{a_1, b_1, \epsilon}}(\xi_1),
\quad
\widehat{f_{2, \ell}}(\xi_2)
=
\widetilde{\psi}_{\ell}(\xi_2) \widehat{f_{a_2, b_2, \epsilon}}(\xi_2),
\\
&
\widehat{f_{j, \ell}}(\xi_j)
=
\widetilde{\psi}_{\ell- L}(\xi_j) \widehat{f_{a_j, b_j, \epsilon}}(\xi_j),
\quad
j=3, \dots, N.
\end{align*}
where
\begin{align*}
&D_{\ell}
=
\left\{
\boldsymbol{\mu} = (\mu_1, \dots, \mu_N) \in (\mathbb{Z}^n)^N
:
\begin{array}{l}
\mu_1+ \mu_2 + \dots + \mu_N = 0,
\quad
2^{\ell -\delta} \le |\mu_2| \le 2^{\ell + \delta},
\\
2^{\ell -L-\delta} \le |\mu_j| \le 2^{\ell -L + \delta},
\quad
j=3, \dots, N
\end{array}
\right\}
\end{align*}
with sufficiently small $\delta >0$ and
\[
0< a_j < 1,
\quad
b_j
=
(1-a_j)
\left(
n/2-n/p_j
\right)
+
n/2
+
\epsilon_j,
\quad
\epsilon_j > 0,
\quad
j=1, \dots, N.
\]
We choose the number $L$ sufficiently large enough to satisfy
\begin{equation}\langlebel{setassum1}
\boldsymbol{\mu} \in D_{\ell}
\implies
2^{\ell -2 \delta}
\le
|\mu_2 + \dots + \mu_N|
\le
2^{\ell +2\delta}.
\end{equation}
Then, since $(1+|\xi_1|+\dots +|\xi_N|) \approx (1+|\mu_1|+\dots +|\mu_N|)$
if $(\xi_1, \dots, \xi_N) \in \mathop{\mathrm{supp}} \sigma_\ell$ and since $\sup_{\boldsymbol{\mu}} |c_{\boldsymbol{\mu}}| \le 1$,
we have $\sigma_{\ell} \in S^m_{0,0}(\mathbb{R}^n, N)$.
Furthermore, since
$\psi_{k} \widetilde{\psi}_{\ell} = \widetilde{\psi}_{\ell}$
if $k = \ell$
and
$0$
otherwise,
Lemma \ref{lemWainger} and Young's inequality yield that
\begin{align}\langlebel{f1}
\begin{split}
\|f_{j, \ell}\|_{B^{s_j}_{p_j, q_j}}
&=2^{\ell s_j}\|\widetilde{\psi}_{\ell}(D)f_{a_j, b_j, \epsilon}\|_{L^{p_j}}
\\
&\le
2^{\ell s_j}\|\mathcal{F}^{-1}\widetilde{\psi}_{\ell}\|_{L^1} \|f_{a_j, b_j, \epsilon}\|_{L^{p_j}}
\lesssim
2^{\ell s_j},
\quad
j=1, 2,
\end{split}
\end{align}
and similarly,
\begin{align}\langlebel{f3N}
\|f_{j, \ell}\|_{B^{s_j}_{p_j, q_j}}
\lesssim
2^{\ell s_j},
\quad
j=3, \dots, N.
\end{align}
Here the implicit constants are independent of $\ell$ and $\epsilon$.
Now, since $\ell$ is sufficiently large and $\delta$ is small, we have
\begin{align*}
&
\mathop{\mathrm{supp}} \varphi (\cdot - \mu_j)
\subset
\{2^{\ell-1/8} \le |\xi_j| \le 2^{\ell + 1/8}\},
\quad
j=1, 2,
\\
&
\mathop{\mathrm{supp}} \varphi (\cdot - \mu_j)
\subset
\{2^{\ell-L-1/8} \le |\xi_j| \le 2^{\ell-L + 1/8}\},
\quad
j=3, \dots, N,
\end{align*}
for $\boldsymbol{\mu} = (\mu_1, \dots, \mu_N) \in D_{\ell}$,
and consequently we have
\begin{align*}
&
\varphi(\xi_j - \mu_j)
\widetilde{\psi}_{\ell}(\xi_j)
=
\varphi(\xi_j - \mu_j),
\quad
j=1,2,
\\
&
\varphi(\xi_j - \mu_j)
\widetilde{\psi}_{\ell-L}(\xi_j)
=
\varphi(\xi_j - \mu_j),
\quad
j=3, \dots, N.
\end{align*}
Hence, we obtain
\begin{align*}
T_{\sigma_\ell}(f_{1, \ell}, \dots, f_{N, \ell})(x)
&=
\sum_{\boldsymbol{\mu} \in D_{\ell}}
c_{\boldsymbol{\mu}} \langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=1}^N
e^{-\epsilon|\mu_j|}
|\mu_j|^{-b_j}
e^{i|\mu_j|^{a_j}}
\mathcal{F}^{-1}[\varphi(\cdot - \mu_j)](x)
\\
&=
\{\Phi(x)\}^N
\sum_{\boldsymbol{\mu} \in D_{\ell}}
c_{\boldsymbol{\mu}} \langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=1}^N
e^{-\epsilon|\mu_j|}
|\mu_j|^{-b_j}
e^{i|\mu_j|^{a_j}}
\end{align*}
with $\Phi = \mathcal{F}^{-1} \varphi$,
where we used the fact that $\mu_1 + \dots + \mu_N = 0$ for $\boldsymbol{\mu} = (\mu_1, \dots, \mu_N) \in D_{\ell}$.
Thus, if we choose $c_{\boldsymbol{\mu}} = \prod_{j=1}^N e^{-i|\mu_j|^{a_j}}$, then
\[
T_{\sigma_\ell}(f_{1, \ell}, \dots, f_{N, \ell})(x)
=
\{\Phi(x)\}^N
\sum_{\boldsymbol{\mu} \in D_{\ell}}
\langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=1}^N
e^{-\epsilon|\mu_j|}
|\mu_j|^{-b_j}.
\]
Hence we obtain
\begin{equation}\langlebel{normest33}
\|T_{\sigma_\ell}(f_{1, \ell}, \dots, f_{N, \ell})\|_{B^s_{p, q}}
\approx
\sum_{\boldsymbol{\mu} \in D_{\ell}}
\langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=1}^N
e^{-\epsilon|\mu_j|}
|\mu_j|^{-b_j}.
\end{equation}
Combining \eqref{nec-bdd}, \eqref{f1}, \eqref{f3N}, and \eqref{normest33}, we obtain
\begin{align*}
\sum_{\boldsymbol{\mu} \in D_{\ell}}
\langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=1}^N
e^{-\epsilon|\mu_j|}
|\mu_j|^{-b_j}
\lesssim
2^{\ell \sum_{j=1}^N s_j}
\end{align*}
where the implicit constant does not depend on $\epsilon$.
Hence, taking the limit $\epsilon \to 0$, we have
\begin{align*}
\sum_{\boldsymbol{\mu} \in D_{\ell}}
\langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=1}^N
|\mu_j|^{-b_j}
\lesssim
2^{\ell \sum_{j=1}^N s_j}.
\end{align*}
By \eqref{setassum1}, the left hand side above can be written as
\begin{align*}
&
\sum_{\boldsymbol{\mu} \in D_{\ell}}
\langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=1}^N
|\mu_j|^{-b_j}
\\
&=
\sum_{\substack{
2^{\ell-\delta} \le |\mu_2| \le 2^{\ell+\delta}
\\
2^{\ell-L-\delta} \le |\mu_j| \le 2^{\ell-L+\delta},
\
j=3, \dots, N
}}
\langle (\mu_2 + \dots + \mu_N, \mu_2, \dots, \mu_N) \rangle^m
|\mu_2+ \dots + \mu_N|^{-b_1}
\prod_{j=2}^N
|\mu_j|^{-b_j}
\\
&\approx
2^{\ell(m-\sum_{j=1}^N b_j)}
\\
&\qquad\qquad
\times
\mathrm{card}
\left\{
(\mu_2, \dots, \mu_N) \in (\mathbb{Z}^n)^{N-1}
:
\begin{array}{l}
2^{\ell-\delta} \le |\mu_2| \le 2^{\ell+\delta},
\\
2^{\ell-L-\delta} \le |\mu_j| \le 2^{\ell-L+\delta},
\
j=3, \dots, N
\end{array}
\right\}
\\
&\approx
2^{\ell(m-\sum_{j=1}^N b_j+(N-1)n)}.
\end{align*}
Thus we obtain
\begin{equation*}
2^{\ell(m-\sum_{j=1}^N b_j+(N-1)n)}
\lesssim
2^{\ell \sum_{j=1}^N s_j}
\end{equation*}
with the implicit constant independent of $\ell$.
Since $\ell$ is arbitrarily large, we obtain
\begin{align*}
\sum_{j=1}^N
s_j
&\ge
m-\sum_{j=1}^N b_j +(N-1)n
=
m
-\sum_{j=1}^N
\left\{
(1-a_j)\left(\frac{n}{2}-\frac{n}{p_j}\right) +\frac{n}{2}
+ \epsilon_j
\right\} +(N-1)n.
\end{align*}
Taking the limits as $a_j \to 0$ if $1 < p_j \le 2$, $a_j \to 1$ if $ 2 < p_j < \infty$, and $\epsilon_j \to 0$, we conclude that
\begin{align*}
\sum_{j=1}^N
s_j
\ge
m
-
n
+
\sum_{j =1}^N
\max
\left\{
\frac{n}{p_j}, \frac{n}{2}
\right\}
=
-n
+\min
\left\{
\frac{n}{p}, \frac{n}{2}
\right\}
+
\sum_{j=1}^N
s_j
-
s
,
\end{align*}
which means
$s \ge -\max \{n/p^{\prime}, n/2 \}$.
The proof is complete.
\end{proof}
\begin{lem}\langlebel{lemNecessity2}
Let $1< p_1, \dots, p_N < \infty$, $0< p< \infty$, $1 < q, q_1, \dots, q_N < \infty$ and $s, s_1, \dots, s_N \in \mathbb{R}$.
If the boundedness \eqref{nec-bdd} holds with $m$ given in \eqref{mcrit}, then $s_j \le \max \{n/p_j, n/2\}$, $j=1, \dots, N$.
\end{lem}
\begin{proof}
It is sufficient to prove that $s_1 \le \max\{n/p_1, n/2\}$ by symmetry.
\textbf{Case I : $2 < p < \infty$.}
It is well known that
the multilinear pseudo-differential operator $T_{\sigma}$ with $\sigma \in S^m_{0, 0}(\mathbb{R}^n, N)$ is bounded from
$B^{s_1}_{p_1, q_1}
\times
B^{s_2}_{p_2, q_2}
\times
\dots
\times B^{s_N}_{p_N, q_N}$ to $B^{s}_{p, q}$, then
$T_{\sigma^{*1}}$ is bounded from
$B^{-s}_{p^\prime, q^\prime} \times \dots \times B^{s_N}_{p_N, q_N} \times $
to
$B^{-s_1}_{p_1^\prime, q_1^\prime}$.
Here $\sigma^{*1}$ is a multilinear symbol defined by
\[
\int_{\mathbb{R}^n}
T_{\sigma}(f_1, f_2, \dots, f_N)(x) g(x)
\,
dx
=
\int_{\mathbb{R}^n}
T_{\sigma^{*1}}(g, f_2, \dots, f_N)(x) f_1(x)
\,
dx.
\]
Since $\sigma \in S^m_{0, 0}(\mathbb{R}^n, N)$ if and only if $\sigma^{*1} \in S^{m}_{0,0}(\mathbb{R}^n, N)$,
the boundedness \eqref{nec-bdd} holds if and only if
\[
\mathop{\mathrm{Op}}(S^m_{0, 0}(\mathbb{R}^n, N))
\subset
B(B^{-s}_{p^\prime, q^\prime} \times \dots \times B^{s_N}_{p_N, q_N} \to B^{-s_1}_{p_1^\prime, q_1^\prime}).
\]
Thus it follows from Lemma \ref{lemNecessity1} that
$-s_1 \ge -\max\{n/(p_1^\prime)^{\prime}, n/2 \}$,
that is,
$s_1 \le \max\{n/p_1, n/2\}$.
\textbf{Case I\hspace{-0.1em}I : $0< p \le 2$.}
Let $L \in \mathbb{N}_0$ be a sufficently large.
For $\ell \in \mathbb{N}$ satisfying $\ell > L$, we set
\begin{align*}
&\sigma_{\ell}(\xi_1, \dots, \xi_N)
=
\varphi(\xi_1)
\left(
\sum_{\boldsymbol{\mu} = (\mu_2, \dots, \mu_N) \in D_{\ell}}
c_{\boldsymbol{\mu}} \langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=2}^{N}
\varphi(\xi_j-\mu_j)
\right)
,
\\
&
\widehat{f}_1(\xi_1) = \widetilde{\varphi}(\xi_1),
\quad
\widehat{f_{2, \ell}}(\xi_2)
=
\widetilde{\psi}_{\ell}(\xi_2) \widehat{f_{a_2, b_2, \epsilon}}(\xi_{2}),
\\
&\widehat{f_{j, \ell}}(\xi_j)
=
\widetilde{\psi}_{\ell-L}(\xi_j) \widehat{f_{a_j, b_j, \epsilon}}(\xi_j),
\quad
j=3, \dots, N,
\end{align*}
where $\{c_{\boldsymbol{\mu}}\}_{\boldsymbol{\mu}}$ be a sequence of complex numbers such that $\sup_{\boldsymbol{\mu}} |c_{\boldsymbol{\mu}}| \le 1$, and
\begin{align*}
D_{\ell}
=
\left\{
\boldsymbol{\mu}
=
(\mu_2, \dots, \mu_{N})
\in (\mathbb{Z}^n)^{N-1}
:
\begin{array}{l}
2^{\ell - \delta}
\le
|\mu_2 + \dots + \mu_{N}|
\le
2^{\ell + \delta},
\\
2^{\ell - L - \delta}
\le
|\mu_j|
\le
2^{\ell - L + \delta},
\quad
j=3, \dots, N
\end{array}
\right\}.
\end{align*}
with sufficiently small $\delta > 0$.
We take the number $L$ sufficiently large so that
\begin{align}\langlebel{murange2}
\boldsymbol{\mu}
\in
D_{\ell}
\implies
2^{\ell - 2\delta}
\le
|\mu_{2}|
\le
2^{\ell + 2\delta}.
\end{align}
We see that $\sigma_{\ell} \in S^m_{0,0}(\mathbb{R}^n, N)$ by the same argument as in the proof of Lemma \ref{lemNecessity1}, and hence \eqref{operator-norm} yields that
\begin{equation}\langlebel{operator-norm-2}
\|T_{\sigma_\ell}\|_{B^{s_1}_{p_1, q_1} \times \dots \times B^{s_N}_{p_N, q_N} \to B^{s}_{p, q}} \lesssim
1
\end{equation}
with the implicit constant independent of $c_{\boldsymbol{\mu}}$ and $\ell$.
We also have
\begin{align}\langlebel{f2}
\|f_1\|_{B^{s_1}_{p_1,q_1}} \lesssim 1,
\quad
\|f_{j, \ell}\|_{B^{s_j}_{p_j, q_j}} \lesssim 2^{\ell s_j}, \quad j=2, \dots, N.
\end{align}
Moreover, since $\varphi\widetilde{\varphi} = \varphi$ and
\begin{align*}
&\varphi(\xi_2 - \mu_2) \widetilde{\psi}_{\ell}(\xi_2) = \varphi(\xi_2 - \mu_2)
\\
&\varphi(\xi_j-\mu_j)\widetilde{\psi}_{\ell-L}(\xi_j) = \varphi(\xi_j-\mu_j), \quad j=3, \dots, N,
\end{align*}
if $\boldsymbol{\mu} \in D_{\ell}$, then we have
\begin{align*}
T_{\sigma_{\ell}}(f_{1}, f_{2, \ell} \dots, f_{N, \ell})(x)
&=
\{\Phi(x)\}^N
\sum_{\boldsymbol{\mu} \in D_{\ell}}
c_{\boldsymbol{\mu}} \langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=2}^{N}
e^{-\epsilon |\mu_j|}
|\mu_j|^{-b_j}
e^{i|\mu_j|^{a_j}}
e^{i \mu_j \cdot x}
\end{align*}
Let $\{r_{\mu}(\omega)\}_{\mu \in \mathbb{Z}^n}$, $\omega \in [0, 1]^n$, be a sequence of the Rademacher functions
enumerated in such a way that their index set is $\mathbb{Z}^n$
(for the definition of the Rademacher function, see, e.g., \cite[Appendix C]{Grafakos-Classical}).
If we choose $\{c_{\boldsymbol{\mu}}\}_{\boldsymbol{\mu}}$ as
\[
c_{\boldsymbol{\mu}} = r_{\mu_2+ \dots + \mu_{N}}(\omega) \prod_{j=2}^{N} e^{-i |\mu_j|^{a_j}},
\]
then
\begin{align*}
T_{\sigma_{\ell}}(f_{1}, f_{2, \ell} \dots, f_{N, \ell})(x)
&=
\{\Phi(x)\}^N
\sum_{\boldsymbol{\mu} \in D_{\ell}}
r_{\mu_2 + \dots + \mu_N}(\omega)
\langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=2}^{N}
e^{-\epsilon |\mu_j|}
|\mu_j|^{-b_j}
e^{i \mu_j \cdot x}
\\
&=
\{\Phi(x)\}^N
\sum_{\nu : 2^{\ell-\delta} \le |\nu| \le 2^{\ell + \delta}}
r_{\nu}(\omega)
e^{i \nu \cdot x}
\sum_{\substack{\boldsymbol{\mu} \in D_{\ell} \\ \mu_2 + \dots + \mu_{N} = \nu}}
\langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=2}^{N}
e^{-\epsilon |\mu_j|}
|\mu_j|^{-b_j}.
\end{align*}
Since $\ell$ is sufficiently large, we have
\begin{align*}
\mathop{\mathrm{supp}} \mathcal{F}[T_{\sigma_{\ell}}(f_1, f_{2, \ell}, \dots, f_{N, \ell})]
&
\subset
\bigcup_{\nu : 2^{\ell-\delta} \le |\nu| \le 2^{\ell + \delta}}
\mathop{\mathrm{supp}}[\varphi * \dots * \varphi](\cdot - \nu)
\subset
\{
2^{\ell-1/4}
\le
|\zeta|
\le
2^{\ell+1/4}
\},
\end{align*}
and consequently, we obtain
$\|T_{\sigma_{\ell}}(f_1, f_{2, \ell}, \dots, f_{N, \ell})\|_{B^{s}_{p,q}}= 2^{\ell s}\|T_{\sigma_{\ell}}(f_1, f_{2, \ell}, \dots, f_{N, \ell})\|_{L^p}$.
By the assumption $|\Phi| \ge 1$ on $[-1, 1]^n$,
\begin{align*}
\int_{[0, 1]^n}
\|T_{\sigma_{\ell}}(f_1, f_{2, \ell}, \dots, f_{N, \ell})\|_{B^{s}_{p,q}}^p
\,
d\omega
&=
2^{\ell s p}
\int_{[0, 1]^n}
\|T_{\sigma_{\ell}}(f_1, f_{2, \ell}, \dots, f_{N, \ell})\|_{L^p}^p
\,
d\omega
\\
&\gtrsim
2^{\ell s p}
\int_{[-1, 1]^n}
\left(
\int_{[0, 1]^n}
\left|
\sum_{\nu : 2^{\ell-\delta} \le |\nu| \le 2^{\ell + \delta}}
r_{\nu}(\omega)
e^{i \nu \cdot x}
d_{\nu, \epsilon}
\right|^p
\,
d\omega
\right)
dx,
\end{align*}
where $d_{\nu, \epsilon}$ is defined by
\[
d_{\nu, \epsilon}
=
\sum_{\substack{\boldsymbol{\mu} \in D_{\ell} \\ \mu_2 + \dots + \mu_{N} =\nu} }
\langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=2}^{N}
e^{-\epsilon |\mu_j|}
|\mu_j|^{-b_j}.
\]
Hence, by Khintchine's inequality (see, e.g., \cite[Appendix C]{Grafakos-Classical}),
we have
\begin{align*}
\int_{[-1, 1]^n}
\left(
\int_{[0, 1]^n}
\left|
\sum_{\nu : 2^{\ell-\delta} \le |\nu| \le 2^{\ell + \delta}}
r_{\nu}(\omega)
e^{i \nu \cdot x}
d_{\nu, \epsilon}
\right|^p
\,
d\omega
\right)
dx
&\approx
\int_{[-1, 1]^n}
\left(
\sum_{\nu : 2^{\ell-\delta} \le |\nu| \le 2^{\ell + \delta}}
|d_{\nu, \epsilon}|^{2}
\right)^{p/2}
dx
\\
&\approx
\left(
\sum_{\nu : 2^{\ell-\delta} \le |\nu| \le 2^{\ell + \delta}}
|d_{\nu, \epsilon}|^{2}
\right)^{p/2}
\end{align*}
Combining
\eqref{nec-bdd},
\eqref{operator-norm-2}
and
\eqref{f2},
we obtain
\begin{align*}
\left(
\sum_{\nu : 2^{\ell-\delta} \le |\nu| \le 2^{\ell + \delta}}
|d_{\nu, \epsilon}|^{2}
\right)^{1/2}
&\lesssim
2^{-\ell s}
\times
\left(
\int_{[0, 1]^n}
\|T_{\sigma_{\ell}}(f_1, f_{2, \ell}, \dots, f_{N, \ell})\|_{B^{s}_{p,q}}^p
\,
d\omega
\right)^{1/p}
\\
&\lesssim
2^{-\ell s}
\times
\|f_1\|_{B^{s_1}_{p_1, q_1}}
\prod_{j=2}^N
\|f_{j, \ell}\|_{B^{s_j}_{p_j, q_j}}
\\
&\lesssim
2^{\ell (\sum_{j=2}^{N} s_j -s)}
\end{align*}
Since the sums over $\nu$ and $\boldsymbol{\mu}$ are finite sums,
we have by taking the limit $\epsilon \to 0$
\begin{equation} \langlebel{estfordnu}
\left(
\sum_{\nu : 2^{\ell-\delta} \le |\nu| \le 2^{\ell + \delta}}
|d_{\nu}|^{2}
\right)^{1/2}
\lesssim
2^{\ell (\sum_{j=2}^{N} s_j -s)}
\end{equation}
with
\[
d_{\nu}
=
\sum_{\substack{\boldsymbol{\mu} \in D_{\ell} \\ \mu_2 + \dots + \mu_{N} =\nu} }
\langle \boldsymbol{\mu} \rangle^{m}
\prod_{j=2}^{N}
|\mu_j|^{-b_j}.
\]
Recalling \eqref{murange2},
for $\nu \in \mathbb{Z}^n$ satisfying $\nu : 2^{\ell-\delta} \le |\nu| \le 2^{\ell + \delta}$, we have
\begin{align*}
d_{\nu}
&\approx
2^{\ell (m-\sum_{j=2}^{N} b_j)}
\\
&\qquad
\times
\mathrm{card}
\left\{
(\mu_3, \dots, \mu_N) \in (\mathbb{Z}^n)^{N-2} :
2^{\ell - L - \delta}
\le
|\mu_j|
\le
2^{\ell - L + \delta},
\
j=3, \dots, N
\right\}
\\
&\approx
2^{\ell (m-\sum_{j=2}^{N} b_j-(N-2)n)}.
\end{align*}
Combining the above estimates, we obtain
\[
2^{\ell(m -\sum_{j=2}^{N} b_j +(N-2)n + n/2)}
\lesssim
2^{\ell (\sum_{j=2}^{N} s_j -s)}
\]
with the implicit constant independent of $\ell$.
Since $\ell$ is sufficiently large, we obtain
\begin{align*}
\sum_{j=2}^{N}
s_j
-s
&\ge
m
-\sum_{j=2}^{N} b_j
+(N-2)n + \frac{n}{2}
\\
&=
m
-
\left\{
\sum_{j=2}^{N}
(1-a_j)
\left(
\frac{n}{2}-\frac{n}{p_j}
\right)
+\frac{n}{2}
+
\epsilon_j
\right\}
+(N-2)n + \frac{n}{2}
\end{align*}
If we take limits as $a_j \to 0$ if $1 < p_j \le 2$,
$a_j \to 1$ if $2 < p_j < \infty$ and $\epsilon_j \to 0$,
then we obtain
\begin{align*}
\sum_{j=2}^{N}
s_j
-s
\ge
m
+
\sum_{j=2}^{N}
\max\left\{ \frac{n}{p_j}, \frac{n}{2}\right\}
- \frac{n}{2}
=
-\max\left\{\frac{n}{p_1}, \frac{n}{2}\right\}
+
\sum_{j=1}^N
s_j
-s,
\end{align*}
which means
$s_1 \le \max\{n/p_1, n/2\}$.
The proof is complete.
\end{proof}
In the rest of this section, we prove Theorem \ref{thmnec}.
The basic ideas go back to \cite{KMT-JFA} and \cite{Shida-Sobolev}.
\begin{proof}[Proof of Theorem \ref{thmnec}]
Let $0 < p, p_1, \dots, p_N \le \infty$, $0 < q, q_1, \dots, q_N \le \infty$ and $s, s_1 \dots, s_N \in \mathbb{R}$.
We assume that the boundedness
\begin{align*}
&
\mathop{\mathrm{Op}}(S^m_{0, 0}(\mathbb{R}^n, N))
\subset
B(B^{s_1}_{p_1, q_1} \times \dots \times B^{s_N}_{p_N, q_N} \to B^{s}_{p, q})
\end{align*}
holds with
\begin{align*}
m = \min \left\{ \frac{n}{p}, \frac{n}{2} \right\}
-
\sum_{j=1}^N \max\left\{ \frac{n}{p_j}, \frac{n}{2} \right\}
+
\sum_{j=1}^N s_j -s.
\end{align*}
It is already proved in Theorem \ref{main1} that the boundedness
\[
\mathop{\mathrm{Op}}(S^{-(N-1)(n/2-t)}_{0, 0}(\mathbb{R}^n, N))
\subset
B(B^{t}_{2, 2} \times \dots \times B^{t}_{2, 2} \to B^{t}_{2, 2})
\]
holds for any $t \in \mathbb{R}$ satisfying $-n/2 < t < n/2$.
Then, by complex interpolation, we have
\[
\mathop{\mathrm{Op}}(S^{\widetilde{m}}_{0, 0}(\mathbb{R}^n, N))
\subset
B(B^{\widetilde{s}_1}_{\widetilde{p}_1, \widetilde{q}_1}
\times \dots \times
B^{\widetilde{s}_N}_{\widetilde{p}_N, \widetilde{q}_N} \to B^{\widetilde{s}}_{\widetilde{p}, \widetilde{q}}),
\]
where $0< \theta < 1$,
\begin{align*}
&1/\widetilde{p}_j
=
(1-\theta)/2
+
\theta/p_j,
\quad
1/\widetilde{q}_j
=
(1-\theta)/2
+
\theta/q_j,
\quad
\widetilde{s_j}
=
(1-\theta)t + \theta s_j ,
\quad
j=1, \dots, N,
\\
&1/\widetilde{p}
=
(1-\theta)/2
+
\theta/p,
\quad
1/\widetilde{q}
=
(1-\theta)/2
+
\theta/q,
\quad
\widetilde{s}
=
(1-\theta)t + \theta s,
\end{align*}
and
\begin{align*}
\widetilde{m}
&=
- (N-1)\left(\frac{n}{2} - t\right) \times (1-\theta)
+
\left(
\min
\left\{
\frac{n}{p}, \frac{n}{2}
\right\}
-
\sum_{j=1}^N
\max
\left\{
\frac{n}{p_j}, \frac{n}{2}
\right\}
+\sum_{j=1}^N s_j -s
\right)
\times
\theta
\\
&=
\min
\left\{
\frac{n}{\widetilde{p}}, \frac{n}{2}
\right\}
-
\sum_{j=1}^N
\max
\left\{
\frac{n}{\widetilde{p}_j}, \frac{n}{2}
\right\}
+
\sum_{j=1}^N
\widetilde{s}_j
-\widetilde{s}.
\end{align*}
Since $1 < \widetilde{p}_j, \widetilde{q}_j, \widetilde{q} < \infty$ for sufficiently small $\theta$, it follows from
Lemma \ref{lemNecessity2} that
\begin{align*}
&\widetilde{s}_j
\le
\max
\left\{
\frac{n}{\widetilde{p}_j},
\frac{n}{2}
\right\},
\quad
j=1, \dots, N,
\end{align*}
which means that
\begin{align*}
&s_j
\le
\max
\left\{
\frac{n}{p_j},
\frac{n}{2}
\right\}
+
\frac{1-\theta}{\theta} \left(\frac{n}{2} - t \right),
\quad
j=1, \dots, N,
\end{align*}
Taking the limit $t \to n/2$,
we obtain the desired conclusion.
On the other hand, since $1 < \widetilde{p}_j < \infty$ if $\theta$ is sufficiently small, Lemma \ref{lemNecessity1} yields that
\begin{align*}
&\widetilde{s}
\ge
-
\max
\left\{
\frac{n}{{\widetilde{p}}^{\prime}},
\frac{n}{2}
\right\}.
\end{align*}
We remark that $1< \widetilde{p} < \infty$ for sufficiently small $\theta$,
and hence we can write the above inequality as
\[
s
\ge
-
\max
\left\{
\frac{n}{p^{\prime}},
\frac{n}{2}
\right\}
-
\frac{1-\theta}{\theta}
\left(
t
+
\frac{n}{2}
\right).
\]
We conclude that $s \ge -\max\{n/p^\prime, n/2\}$ by taking the limit $t \to -n/2$.
The proof of Theorem \ref{thmnec} is complete.
\end{proof}
\end{document} |
\begin{document}
\title[strongly solid II$_1$ factors with an exotic MASA]{strongly solid II$_1$ factors \\ with an exotic MASA}
\begin{abstract}
Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid $\rm{II}_1$ factor $M$ containing an ``exotic'' maximal abelian subalgebra $A$: as an $A$,$A$-bimodule, $L^2(M)$ is neither coarse nor discrete. Thus we show that there exist $\rm{II}_1$ factors with such property but without Cartan subalgebras. It also follows from Voiculescu's free entropy results that $M$ is not an interpolated free group factor, yet it is strongly solid and has both the Haagerup property and the complete metric approximation property.
\end{abstract}
\author[C. Houdayer]{Cyril Houdayer}
\address{CNRS ENS Lyon \\
UMPA UMR 5669 \\
69364 Lyon cedex 7 \\
France}
\email{[email protected]}
\author[D. Shlyakhtenko]{Dimitri Shlyakhtenko*}
\address{UCLA\\
Department of Mathematics\\
520 Portola Plaza\\
LA\\
CA 90095}
\email{[email protected]}
\thanks{* Research partially supported by NSF grant DMS-0555680}
\subjclass[2000]{46L10; 46L54}
\keywords{Free group factors; Deformation/rigidity; Intertwining techniques; Free probability}
\maketitle
\section{Introduction}
In their breakthrough paper \cite{ozawapopa}, Ozawa and Popa showed that the free group factors $L(\mathbf{F}_n)$ are {\em strongly solid}, i.e. the normalizer $\mathscr{N}_{L(\mathbf{F}_n)}(P)=\{u\in \mathscr{U}(L(\mathbf{F}_n)): uPu^*=P\}$ of any diffuse amenable subalgebra $P\subset L(\mathbf{F}_n)$ generates an amenable von Neumann algebra, thus AFD by Connes' result \cite{connes76}. This strengthened two well-known indecomposability results for free group factors: Voiculescu's celebrated result in \cite{voiculescu96}, showing that $L(\mathbf{F}_n)$ has no Cartan subalgebra, which in fact exhibited the first examples of factors with no Cartan decomposition; and Ozawa's result in \cite{ozawa2003}, showing that the commutant in $L(\mathbf{F}_n)$ of any diffuse subalgebra must be amenable ($L(\mathbf{F}_n)$ are {\it solid}). Furthermore in \cite{ozawapopaII}, Ozawa and Popa showed that for any lattice $\Gamma$ in $\operatorname{op}eratorname{SL}(2, \mathbf{R})$ or $\operatorname{op}eratorname{SL}(2, \mathbf{C})$, the group von Neumann algebra $L(\Gamma)$ is strongly solid as well.
In this paper, we use a combination of Popa's deformation and intertwining techniques \cite{{popasup}, {popamal1}, {popa2001}} and the techniques of Ozawa and Popa \cite{ozawapopa, ozawapopaII} to give another example of a strongly solid $\rm{II_1}$ factor not isomorphic to an amplification of a free group factor, i.e. to an {\em interpolated} free group factor \cite{{dykema94}, {radulescu1994}} (the first example of this kind was constructed by the first-named author in \cite{houdayer7}, answering an open question of Popa \cite{popa07}).
Our example is rather canonical: it is the crossed product of a free group factor $L(\mathbf{F}_\infty)$ by $\mathbf{Z}$, acting by a free Bogoljubov transformation obtained via Voiculescu's free Gaussian functor (cf. \cite{voiculescu92}). Roughly speaking, recall \cite{voiculescu92} that to any separable real Hilbert space $H_\mathbf{R}$, one can associate a finite von Neumann algebra $\Gamma(H_\mathbf{R})''$ which is precisely isomorphic to the free group factor $L(\mathbf{F}_{\dim H_\mathbf{R}})$. To any orthogonal representation $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ of $\mathbf{Z}$ on $H_\mathbf{R}$ corresponds a trace-preserving action $\sigma^\pi : \mathbf{Z} \curvearrowright \Gamma(H_\mathbf{R})''$, called the {\em Bogoljubov action} associated with the orthogonal representation $\pi$. Alternatively, our algebra can be viewed as a free Krieger algebra in the terminology of \cite{shlya99}, constructed from an abelian subalgebra and a certain completely positive map (related to the spectral measure of the $\mathbf{Z}$-action). It is in this way rather similar to a core of a free Araki-Woods factor \cite{shlya98,shlya97}. Along these lines, our main results are the following.
\begin{TheoremA}
Let $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ be an orthogonal representation on the real Hilbert space $H_\mathbf{R}$ such that the spectral measure of $\pi$ has no atoms. Denote by $M = L(\mathbf{F}_\infty) \rtimes_{\sigma^\pi} \mathbf{Z}$ the crossed product under the Bogoljubov action. Then for any maximal abelian subalgebra $A \subset M$, the normalizer $\mathscr{N}_M(A)$ generates an amenable von Neumann algebra.
\end{TheoremA}
In particular, the ${\rm II_1}$ factor $M = L(\mathbf{F}_\infty) \rtimes_{\sigma^\pi} \mathbf{Z}$ has no Cartan subalgebras. Under additional assumptions on the orthogonal representation $\pi$, we can obtain a stronger result.
\begin{TheoremB}
Let $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ be a {\em mixing} orthogonal representation on the real Hilbert space $H_\mathbf{R}$. Then $M = L(\mathbf{F}_\infty) \rtimes_{\sigma^\pi} \mathbf{Z}$ is a non-amenable strongly solid ${\rm II_1}$ factor, i.e. for any $P \subset M$ diffuse amenable subalgebra, $\mathscr{N}_M(P)''$ is an amenable von Neumann algebra.
\end{TheoremB}
Note that in both cases, $M$ has the Haagerup property and the complete metric approximation property, i.e. $\Lambda_{\operatorname{cb}}(M) = 1$.
The proof of Theorems A and B, following a {\textquotedblleft deformation/rigidity\textquotedblright} strategy, is a combination of the ideas and techniques in \cite{{houdayer7}, {ozawapopa}, {ozawapopaII}, {popamal1}}. We will use the {\textquotedblleft free malleable deformation\textquotedblright} by automorphisms $(\alpha_t, \beta)$ defined on $\Gamma(H_\mathbf{R})'' \ast \Gamma(H_\mathbf{R})'' = \Gamma(H_\mathbf{R} \operatorname{op}lus H_\mathbf{R})''$. This deformation naturally arises as the {\textquotedblleft second quantization\textquotedblright} of the rotations/reflection defined on $H_\mathbf{R} \operatorname{op}lus H_\mathbf{R}$ that commute with the $\mathbf{Z}$-representation $\pi \operatorname{op}lus \pi$.
The proof of Theorem B then consists in two parts. Let $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ be a mixing orthogonal representation and denote by $M = L(\mathbf{F}_\infty) \rtimes_{\sigma^\pi} \mathbf{Z}$ the corresponding crossed product ${\rm II_1}$ factor. First, we show that given any amenable subalgebra $P \subset M$ such that $P$ does not embed into $L(\mathbf{Z})$ inside $M$, the normalizer $\mathscr{N}_M(P)$ generates an amenable von Neumann algebra (see Theorem \ref{step}). For this, we will exploit the facts that the deformation $(\alpha_t)$ does not converge uniformly on the unit ball $(P)_1$ and that $P \subset M$ is {\it weakly compact}, and use the technology from \cite{{ozawapopa}, {ozawapopaII}}. So if $P \subset M$ is diffuse, amenable such that $\mathscr{N}_M(P)''$ is not amenable, $P$ must embed into $L(\mathbf{Z})$ inside $M$. Exploiting Popa's intertwining techniques and the fact that the $\mathbf{Z}$-action $\sigma^\pi$ is mixing, we prove that $\mathscr{N}_M(P)''$ is {\textquotedblleft captured\textquotedblright} in $L(\mathbf{Z})$ and finally get a contradiction.
In proving that free group factors $L(\mathbf{F}_n)$ have no Cartan subalgebras \cite{voiculescu96}, Voiculescu
proved that they actually have a formally stronger property: for any MASA (maximal abelian subalgebra) $A\subset N=L(\mathbf{F}_n)$, $L^2(N)$ (when viewed as an $A$,$A$-bimodule) contains a sub-bimodule of $L^2(A) \otimes L^2(A)$.
In more classical language, for every MASA $L^\infty[0,1]\operatorname{co}ng A\subset N$, every vector $\xi \in L^2(N)$ gives rise to a measure $\psi=\psi_\xi$ on $[0,1]^2$ determined by $$\int f(x) g(y) d\psi (x,y) = \langle f Jg^*J \xi , \xi\rangle,\qquad f,g\in A.$$ Voiculescu proved that, for any such $A\subset N\operatorname{co}ng L(\mathbf{F}_n)$, there exists a nonzero vector $\xi$ for which $\psi$ is Lebesgue absolutely continuous. Any $N$ with this property cannot of course have Cartan subalgebras, since if $A$ is a Cartan subalgebra, the measure $\psi$ will have to be ``$r$-discrete'' (i.e., $\psi (B) = \int \nu_t(B) dt$ for some family of discrete measures $\nu_t$).
This raised the obvious question: if $N$ has no Cartan subalgebras, must it be that for any diffuse MASA $A\subset N$, the $A$,$A$-bimodule $L^2(N)$ contains a sub-bimodule of $L^2(A) \otimes L^2(A)$?
We answer this question in the negative. Our examples $M = L(\mathbf{F}_\infty) \rtimes \mathbf{Z}$, while strongly solid (or having no Cartan subalgebras), have an ``exotic'' MASA $A = L(\mathbf{Z})$, so that $L^2(M)$, when viewed as an $A$,$A$-bimodule, contains neither coarse nor $r$-discrete sub-bimodules. In other words, for all $\xi\neq 0$, $\psi_\xi$ is neither $r$-discrete nor Lebesgue absolutely continuous. In particular, combined with Voiculescu's results, this property shows that our examples $M$ are not interpolated free group factors. Thus we prove:
\begin{CorollaryA}
Let $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ be an orthogonal representation on the real Hilbert space $H_\mathbf{R}$ such that the spectral measure of $\bigoplus_{n \geq 1} \pi^{\otimes n}$ is singular w.r.t. the Lebesgue measure and has no atoms. Then the non-amenable ${\rm II_1}$ factor $M = L(\mathbf{F}_\infty) \rtimes_{\sigma^\pi} \mathbf{Z}$ has no Cartan subalgebra and is not isomorphic to any interpolated free group factor $L(\mathbf{F}_t)$, $1 < t \leq +\infty$.
\end{CorollaryA}
Assuming that the representation $\pi$ is mixing, we can obtain (see Theorem \ref{singular-ssolid}) new examples of strongly solid ${\rm II_1}$ factors not isomorphic to interpolated free group factors (see \cite{{houdayer7}, {popa07}}).
\begin{CorollaryB}
Let $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ be a mixing orthogonal representation on the real Hilbert space $H_\mathbf{R}$ such that the spectral measure of $\bigoplus_{n \geq 1} \pi^{\otimes n}$ is singular w.r.t. the Lebesgue measure. Then the non-amenable ${\rm II_1}$ factor $M = L(\mathbf{F}_\infty) \rtimes_{\sigma^\pi} \mathbf{Z}$ is strongly solid and is not isomorphic to any interpolated free group factor $L(\mathbf{F}_t)$, $1 < t \leq +\infty$.
\end{CorollaryB}
In Section \ref{examples}, we will present examples of orthogonal representations $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ which satisfy the assumptions of Corollaries A and B. After recalling the necessary background in Section \ref{preliminaries}, Theorems A and B are proven in Section \ref{keyresult}.
{\bf Acknowledgements.} Part of this work was done while the first-named author was at University of California, Los Angeles. The first-named author is very grateful to the warm hospitality and the stimulating atmosphere at UCLA. He finally thanks Stefaan Vaes for fruitful discussions regarding this work during his visit at University of Leuven.
\section{Preliminaries}\label{preliminaries}
\subsection{Popa's intertwining techniques}
We first recall some notation. Let $P \subset M$ be an inclusion of finite von Neumann algebras. The {\it normalizer of} $P$ {\it inside} $M$ is defined as
\begin{equation*}
\mathscr{N}_M(P) := \left\{ u \in \mathscr{U}(M) : \operatorname{op}eratorname{Ad}(u) P = P \right\},
\end{equation*}
where $\operatorname{op}eratorname{Ad}(u) = u \cdot u^*$. The inclusion $P \subset M$ is said to be {\it regular} if $\mathscr{N}_M(P)'' = M$. The {\it quasi-normalizer of} $P$ {\it inside} $M$ is defined as
\begin{equation*}
\mathscr{QN}_M(P) := \left\{ a \in M : \exists b_1, \dots, b_n \in M, aP \subset \sum_i Pb_i, Pa \subset \sum_i b_iP \right\}.
\end{equation*}
The inclusion $P \subset M$ is said to be {\it quasi-regular} if $\mathscr{QN}_M(P)'' = M$. Moreover,
\begin{equation*}
P' \cap M \subset \mathscr{N}_M(P)'' \subset \mathscr{QN}_M(P)''.
\end{equation*}
Let $A, B$ be finite von Neumann algebras. An $A, B$-{\it bimodule} $H$ is a complex (separable) Hilbert space $H$ together with two {\it commuting} normal $\ast$-representations $\pi_A : A \to \mathbf{B}(H)$, $\pi_B : B^{\operatorname{op}} \to \mathbf{B}(H)$. We shall intuitively write $a \xi b = \pi_A(x)\pi_B(y^{\operatorname{op}}) \xi$, $\forall x \in A, \forall y \in B, \forall \xi \in H$. We say that $H_B$ is {\it finitely generated} as a right $B$-module if $H_B$ is of the form $p L^2(B)^{\operatorname{op}lus n}$ for some projection $p \in \mathbf{M}_n(\mathbf{C}) \otimes B$.
In \cite{{popamal1}, {popa2001}}, Popa introduced a powerful tool to prove the unitary conjugacy of two von Neumann subalgebras of a tracial von Neumann algebra $(M, \tau)$. We will make intensively use of this technique. If $A, B \subset (M, \tau)$ are (possibly non-unital) von Neumann subalgebras, denote by $1_A$ (resp. $1_B$) the unit of $A$ (resp. $B$).
\begin{theo}[Popa, \cite{{popamal1}, {popa2001}}]\label{intertwining1}
Let $(M, \tau)$ be a finite von Neumann algebra. Let $A, B \subset M$ be possibly non-unital von Neumann subalgebras. The following are equivalent:
\begin{enumerate}
\item There exist $n \geq 1$, a possibly non-unital $\ast$-homomorphism $\psi : A \to \mathbf{M}_n(\mathbf{C}) \otimes B$ and a non-zero partial isometry $v \in \mathbf{M}_{1, n}(\mathbf{C}) \otimes 1_AM1_B$ such that $x v = v \psi(x)$, for any $x \in A$.
\item The bimodule $\vphantom{}_AL^2(1_AM1_B)_B$ contains a non-zero sub-bimodule $\vphantom{}_AH_B$ which is finitely generated as a right $B$-module.
\item There is no sequence of unitaries $(u_k)$ in $A$ such that
\begin{equation*}
\lim_{k \to \infty} \|E_B(a^* u_k b)\|_2 = 0, \forall a, b \in 1_A M 1_B.
\end{equation*}
\end{enumerate}
\end{theo}
If one of the previous equivalent conditions is satisfied, we shall say that $A$ {\it embeds into} $B$ {\it inside} $M$ and denote $A \preceq_M B$. For simplicity, we shall write $M^n := \mathbf{M}_n(\mathbf{C}) \otimes M$.
\subsection{The complete metric approximation property}
\begin{df}[Haagerup, \cite{haagerup79}]
A finite von Neumann algebra $(M, \tau)$ is said to have the {\it complete metric approximation property} (c.m.a.p.) if there exists a net $\Phi_n : M \to M$ of ($\tau$-preserving) normal finite rank completely bounded maps such that
\begin{enumerate}
\item $\lim_n \|\Phi_n(x) - x\|_2 = 0$, $\forall x \in M$;
\item $\lim_n \|\Phi_n\|_{\operatorname{cb}} = 1$.
\end{enumerate}
\end{df}
It follows from Theorem $4.9$ in \cite{anan95} that if $G$ is a countable amenable group and $Q$ is a finite von Neumann algebra with the c.m.a.p., then for any action $G \curvearrowright (Q, \tau)$, the crossed product $Q \rtimes G$ has the c.m.a.p. as well. The notation $\bar{\otimes}$ will be used for the {\it spatial} tensor product.
\begin{df}[Ozawa \& Popa, \cite{ozawapopa}]
Let $\Gamma$ be a discrete group, let $(P, \tau)$ be a finite von Neumann algebra and let $\sigma : \Gamma \curvearrowright P$ be a $\tau$-preserving action. The action is said to be {\it weakly compact} if there exists a net $(\eta_n)$ of unit vectors in $L^2(P \bar{\otimes} \bar{P})_+$ such that
\begin{enumerate}
\item $\lim_n \|\eta_n - (v \otimes \bar{v})\eta_n\|_2 = 0$, $\forall v \in \mathscr{U}(P)$;
\item $\lim_n \|\eta_n - (\sigma_g \otimes \bar{\sigma}_g)\eta_n\|_2 = 0$, $\forall g \in \Gamma$;
\item $\langle (a \otimes 1)\eta_n, \eta_n \rangle = \tau(a) = \langle \eta_n, (1 \otimes \bar{a}) \eta_n \rangle$, $\forall a \in M, \forall n$.
\end{enumerate}
These conditions force $P$ to be amenable. A von Neumann algebra $P \subset M$ is said to be {\it weakly compact inside} $M$ if the action by conjugation $\mathscr{N}_M(P) \curvearrowright P$ is weakly compact.
\end{df}
\begin{theo}[Ozawa \& Popa, \cite{ozawapopa}]\label{weakcompact}
Let $M$ be a finite von Neumann algebra with the complete metric approximation property. Let $P \subset M$ be an amenable von Neumann subalgebra. Then $P$ is weakly compact inside $M$.
\end{theo}
\subsection{Voiculescu's free Gaussian functor \cite{DVV:free,voiculescu92}}
Let $H_\mathbf{R}$ be a real separable Hilbert space. Let $H = H_\mathbf{R} \otimes_\mathbf{R} \mathbf{C}$ be the corresponding complexified Hilbert space. The \emph{full Fock space} of $H$ is defined by
\begin{equation*}
\mathscr{F}(H) =\mathbf{C}\Omega \operatorname{op}lus \bigoplus_{n = 1}^{\infty} H^{\otimes n}.
\end{equation*}
The unit vector $\Omega$ is called the \emph{vacuum vector}. For any $\xi \in H$, we have the \emph{left creation operator}
\begin{equation*}
\ell(\xi) : \mathscr{F}(H) \to \mathscr{F}(H) : \left\{
{\begin{array}{l} \ell(\xi)\Omega = \xi, \\ \ell(\xi)(\xi_1 \otimes \cdots \otimes \xi_n) = \xi \otimes \xi_1 \otimes \cdots \otimes \xi_n.
\end{array}} \right.
\end{equation*}
For any $\xi \in H$, we denote by $s(\xi)$ the real part of $\ell(\xi)$ given by
\begin{equation*}
s(\xi) = \frac{\ell(\xi) + \ell(\xi)^*}{2}.
\end{equation*}
The crucial result of Voiculescu \cite{voiculescu92} is that the distribution of the operator $s(\xi)$ w.r.t. the vacuum vector state $\langle \cdot \Omega, \Omega\rangle$ is the semicircular law supported on the interval $[-\|\xi\|, \|\xi\|]$, and for any subset $\Xi\subset H_\mathbf{R}$ of pairwise orthogonal vectors, the family $\{s(\xi):\xi\in \Xi\}$ is freely independent. Set
\begin{equation*}
\Gamma(H_\mathbf{R})'' = \{s(\xi) : \xi \in H_\mathbf{R}\}''.
\end{equation*}
The vector state $\tau = \langle \cdot \Omega, \Omega\rangle$ is a faithful normal trace on $\Gamma(H_\mathbf{R})''$, and
\begin{equation*}
\Gamma(H_\mathbf{R})'' \operatorname{co}ng L(\mathbf{F}_{\dim H_\mathbf{R}}).
\end{equation*}
Since $\Gamma(H_\mathbf{R})''$ is a free group factor, $\Gamma(H_\mathbf{R})''$ has the Haagerup property and the c.m.a.p. \cite{haagerup79}.
\begin{rem}[\cite{speicher:noncrossing, voiculescu92}]\label{value}
Explicitely the value of $\tau$ on a word in $s(\xi_\iota)$ is given by
\begin{equation}\label{formula}
\tau(s(\xi_1) \cdots s(\xi_n)) = 2^{-n}\sum_
{(\{\beta_i, \gamma_i\})
\in \mathbf{N}C(n),
\beta_i < \gamma_i}
\prod_{k = 1}^{n/2}\langle \xi_{\beta_k}, \xi_{\gamma_k}\rangle.
\end{equation}
for $n$ even and is zero otherwise. Here $\mathbf{N}C(2p)$ stands for all the non-crossing pairings of the set $\{1, \dots, 2p\}$, i.e. pairings for which whenever $a < b < c < d$, and $a, c$ are in the same class, then $b, d$ are not in the same class. The total number of such pairings is given by the $p$-th Catalan number
\begin{equation*}
C_p = \frac{1}{p + 1}\begin{pmatrix}
2p \\
p
\end{pmatrix}.
\end{equation*}
\end{rem}
Let $G$ be a countable group together with an orthogonal representation $\pi : G \to \mathscr{O}(H_\mathbf{R})$. We shall still denote by $\pi : G \to \mathscr{U}(H)$ the corresponding unitary representation on the complexified Hilbert space $H = H_\mathbf{R} \otimes_\mathbf{R} \mathbf{C}$. The {\it free Bogoljubov shift} $\sigma^\pi : G \curvearrowright (\Gamma(H_\mathbf{R})'', \tau)$ associated with the representation $\pi$ is defined by
\begin{equation*}
\sigma_g^\pi = \operatorname{op}eratorname{Ad}(\mathscr{F}(\pi_g)), \forall g \in G,
\end{equation*}
where $\mathscr{F}(\pi_g) = \bigoplus_{n \geq 0} \pi_g^{\otimes n} \in \mathscr{U}(\mathscr{F}(H))$.
\begin{nota}
For a countable group $G$ together with an orthogonal representation $\pi : G \to \mathscr{O}(H_\mathbf{R})$, we shall denote by
\begin{equation*}
\Gamma(H_\mathbf{R}, G, \pi)'' = \Gamma(H_\mathbf{R})'' \rtimes_{\sigma^\pi} G.
\end{equation*}
\end{nota}
\begin{exam}
If $(\pi, H) = (\lambda_G, \ell^2(G))$ is the left regular representation of $G$, it is easy to see that the action $\sigma^{\lambda_G} : G \curvearrowright \Gamma(\ell^2(G))''$ is the {\it free} Bernoulli shift and in that case $\Gamma(\ell^2(G), G, \lambda_G)'' \operatorname{co}ng L(\mathbf{Z}) \ast L(G)$.
\end{exam}
For any $n \geq 0$, denote by $K_\pi^{(n)} = H^{\otimes n} \otimes \ell^2(G)$ with the $L(G),L(G)$-bimodule structure given by:
\begin{eqnarray*}
u_g \cdot (\xi_1 \otimes \cdots \otimes \xi_n \otimes \delta_h) & = & \pi_g \xi_1 \otimes \cdots \otimes \pi_g \xi_n \otimes \delta_{gh} \\
(\xi_1 \otimes \cdots \otimes \xi_n \otimes \delta_h) \cdot u_g & = & \xi_1 \otimes \cdots \otimes \xi_n \otimes \delta_{hg}.
\end{eqnarray*}
It is then straightforward to check that as $L(G), L(G)$-bimodules, we have the following isomorphism
\begin{equation*}
L^2(\Gamma(H_\mathbf{R}, G, \pi)'') \operatorname{co}ng \bigoplus_{n \geq 0} K_\pi^{(n)}.
\end{equation*}
Recall that $\pi$ is said to be {\it mixing} if
\begin{equation*}
\lim_{g \to \infty} \langle \pi_g \xi, \eta\rangle = 0, \forall \xi, \eta \in H.
\end{equation*}
The following proposition is an easy consequence of Remark \ref{value} and Kaplansky density theorem.
\begin{prop}
Let $G$ be a countable group together with an orthogonal representation $\pi : G \to \mathscr{O}(H_\mathbf{R})$. The following are equivalent:
\begin{enumerate}
\item The representation $\pi : G \to \mathscr{O}(H_\mathbf{R})$ is mixing.
\item The $\tau$-preserving action $\sigma^\pi : G \curvearrowright \Gamma(H_\mathbf{R})''$ is mixing, i.e.
\begin{equation*}
\lim_{g \to \infty} \tau(\sigma_g^\pi(x)y) = 0, \forall x, y \in \Gamma(H_\mathbf{R})'' \ominus \mathbf{C}.
\end{equation*}
\end{enumerate}
\end{prop}
\section{Proof of Theorems A and B}\label{keyresult}
\subsection{The free malleable deformation on $\Gamma(H_\mathbf{R}, G, \pi)''$}
Let $G$ be a countable group together with an orthogonal representation $\pi : G \to \mathscr{O}(H_\mathbf{R})$. Set
\begin{itemize}
\item $M = \Gamma(H_\mathbf{R}, G, \pi)''$.
\item $\widetilde{M} = \Gamma(H_\mathbf{R} \operatorname{op}lus H_\mathbf{R}, G, \pi \operatorname{op}lus \pi)''$.
\end{itemize}
Thus, we can regard $\widetilde{M}$ as the amalgamated free product
\begin{equation*}
\widetilde{M} = M \ast_{L(G)} M,
\end{equation*}
where we view $M \subset \widetilde{M}$ under the identification with the left copy. Consider the following orthogonal transformations on $H_\mathbf{R} \operatorname{op}lus H_\mathbf{R}$:
\begin{eqnarray*}
U_t & = &
\begin{pmatrix}
\operatorname{co}s(\frac{\pi}{2} t) & -\sin(\frac{\pi}{2} t) \\
\sin(\frac{\pi}{2} t) & \operatorname{co}s(\frac{\pi}{2} t)
\end{pmatrix}, \forall t \in \mathbf{R}, \\
V & = &
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}.
\end{eqnarray*}
Define the associated deformation $(\alpha_t, \beta)$ on $\Gamma(H_\mathbf{R} \operatorname{op}lus H_\mathbf{R})''$ by
\begin{equation*}
\alpha_t = \operatorname{op}eratorname{Ad}(\mathscr{F}(U_t)), \; \beta = \operatorname{op}eratorname{Ad}(\mathscr{F}(V)).
\end{equation*}
Since $U_t, V$ commute with $\pi \operatorname{op}lus \pi$, it follows that $\alpha_t, \beta$ commute with the diagonal action $\sigma^\pi \ast \sigma^\pi$. We can then extend the deformation $(\alpha_t, \beta)$ to $\widetilde{M}$ by ${\alpha_t}_{|L(G)} = \beta_{|L(G)} = \operatorname{op}eratorname{Id}$. Moreover it is easy to check that the deformation $(\alpha_t, \beta)$ is {\it malleable} in the sense of Popa:
\begin{prop}
The deformation $(\alpha_t, \beta)$ satisfies:
\begin{enumerate}
\item $\lim_{t \to 0} \|x - \alpha_t(x)\|_2 = 0$, $\forall x \in \widetilde{M}$.
\item $\beta^2 = \operatorname{op}eratorname{Id}$, $\alpha_t \beta = \beta \alpha_{-t}$, $\forall t \in \mathbf{R}$.
\item $\alpha_1(x \ast_{L(G)} 1) = 1 \ast_{L(G)} x$, $\forall x \in M$.
\end{enumerate}
\end{prop}
We recall at last that the s-malleable deformation $(\alpha_t, \beta)$ automatically features a certain {\it transversality} property.
\begin{prop}[Popa, \cite{popasup}]\label{transversality}
We keep the same notation as before. We have the following:
\begin{equation}\label{trans}
\|x - \alpha_{2t}(x)\|_2 \leq 2 \|\alpha_t(x) - (E_{M} \circ \alpha_t)(x)\|_2, \; \forall x \in M, \forall t > 0.
\end{equation}
\end{prop}
The following result of the first-named author about intertwining subalgebras inside the von Neumann algebras $\Gamma(H_\mathbf{R}, G, \pi)''$ (see Theorems $5.2$ in \cite{houdayer3} and $3.4$ in \cite{houdayer6}) will be a crucial tool in the next subsection.
\begin{theo}[\cite{{houdayer3}, {houdayer6}}]\label{intertwining}
Let $G$ be a countable group. Let $\pi : G \to \mathscr{O}(H_\mathbf{R})$ be any orthogonal representation. Set $M = \Gamma(H_\mathbf{R}, G, \pi)''$. Let $p \in M$ be a non-zero projection. Let $P \subset pMp$ be a von Neumann subalgebra such that the deformation $(\alpha_t)$ converges uniformly on the unit ball $(P)_1$. Then $P \preceq_M L(G)$.
\end{theo}
\subsection{The key result}
Let $M, N, P$ be finite von Neumann algebras. For any $M, N$-bimodules $H, K$, denote by $\pi_H$ (resp. $\pi_K$) the associated $\ast$-representation of the binormal tensor product $M \otimes_{\operatorname{bin}} N^{\operatorname{op}}$ on $H$ (resp. on $K$). We refer to \cite{EL} for the definition of $\otimes_{\operatorname{bin}}$. We say that $H$ is {\em weakly contained} in $K$ and denote it by $H \prec K$ if the representation $\pi_H$ is weakly contained in the representation $\pi_K$, that is if $\ker(\pi_H) \supset \ker(\pi_K)$. Let $H, K$ be $M, N$-bimodules. The following are true:
\begin{enumerate}
\item Assume that $H \prec K$. Then, for any $N$-$P$ bimodule $L$, we have $H \otimes_N L \prec K \otimes_N L$, as $M, P$-bimodules. Exactly in the same way, for any $P, M$-bimodule $L$, we have $L \otimes_M H \prec L \otimes_M K$, as $P, N$-bimodules (see Lemma $1.7$ in \cite{anan95}).
\item A von Neumann algebra $B$ is amenable iff $L^2(B) \prec L^2(B) \otimes L^2(B)$, as $B$-$B$ bimodules.
\end{enumerate}
Let $B, M, N$ be von Neumann algebras such that $B$ is amenable. Let $H$ be any $M, B$-bimodule and let $K$ be any $B, N$-bimodule. Then, as $M, N$-bimodules, we have $H \otimes_B K \prec H \otimes K$ (straightforward consequence of $(1)$ and $(2)$).
\begin{lem}\label{weakcontainment}
Let $G$ be an amenable group together with an orthogonal representation $\pi : G \to \mathscr{O}(H_\mathbf{R})$. Let $M = \Gamma(H_\mathbf{R}, G, \pi)''$. The $M, M$-bimodule $\mathscr{H} = L^2(\widetilde{M}) \ominus L^2(M)$ is weakly contained in the coarse bimodule $L^2(M) \otimes L^2(M)$. In particular, the left $M$-action on $\mathscr{H}$ extends to a u.c.p. map $\Psi : \mathbf{B}(L^2(M)) \to \mathbf{B}(\mathscr{H})$ whose range commutes with the right $M$-action.
\end{lem}
\begin{proof}
Set $B = L(G)$ which is amenable by assumption. By definition of the amalgamated free product $\widetilde{M} = M \ast_{L(G)} M$ (see \cite{voiculescu92}), we have as $M, M$-bimodules
\begin{equation*}
L^2(\widetilde{M}) \ominus L^2(M) \operatorname{co}ng \bigoplus_{n \geq 1} \mathscr{H}_n,
\end{equation*}
where
\begin{equation*}
\mathscr{H}_n = L^2(M) \otimes_B \mathop{\overbrace{(L^2(M) \ominus L^2(B)) \otimes_B \cdots \otimes_B (L^2(M) \ominus L^2(B))}}^{2n - 1} \otimes_B L^2(M).
\end{equation*}
Since $B = L(G)$ is amenable, the identity bimodule $L^2(B)$ is weakly contained in the coarse bimodule $L^2(B) \otimes L^2(B)$. From the standard properties of composition and weak containment of bimodules (see Lemma $1.7$ in \cite{anan95}), it follows that as $M, M$-bimodules
\begin{equation*}
\mathscr{H}_n \prec L^2(M) \otimes \mathop{\overbrace{(L^2(M) \ominus L^2(B)) \otimes \cdots \otimes (L^2(M) \ominus L^2(B))}}^{2n - 1} \otimes L^2(M).
\end{equation*}
Consequently, we obtain as $M, M$-bimodules
\begin{equation*}
\mathscr{H} = L^2(\widetilde{M}) \ominus L^2(M) \prec L^2(M) \otimes L^2(M).
\end{equation*}
Now the rest of the proof is the same as the one of Lemma $5.1$ in \cite{ozawapopaII}. The binormal representation $\mu$ of $M \odot M^{\operatorname{op}}$ on $\mathscr{H}$ is continuous w.r.t. the minimal tensor product. Hence $\mu$ extends to a u.c.p. map $\tilde{\mu}$ from $\mathbf{B}(L^2(M)) \bar{\otimes} M^{\operatorname{op}}$ to $\mathbf{B}(\mathscr{H})$. Define $\Psi(x) = \tilde{\mu}(x \otimes 1)$, $\forall x \in \mathbf{B}(L^2(M))$. Since $M^{\operatorname{op}}$ is in the multiplicative domain of $\tilde{\mu}$, it follows that the range of $\Psi$ commutes with the right $M$-action.
\end{proof}
The next theorem, which is the key result of this section in order to prove Theorems A and B, can be viewed as an analog of Theorems $4.9$ in \cite{ozawapopa}, B in \cite{ozawapopaII} and $3.3$ in \cite{houdayer7}.
\begin{theo}\label{step}
Let $G$ be an amenable group together with an orthogonal representation $\pi : G \to \mathscr{O}(H_\mathbf{R})$. Let $M = \Gamma(H_\mathbf{R}, G, \pi)''$. Let $P \subset M$ be an amenable subalgebra such that $P \npreceq_M L(G)$. Then $\mathscr{N}_M(P)''$ is amenable.
\end{theo}
\begin{proof}
The proof is conceptually similar to the one of Theorem 4.9 in \cite{ozawapopa} under weaker assumptions: the malleable deformation $(\alpha_t)$ defined on $M = \Gamma(H_\mathbf{R}, G, \pi)''$ is not assumed to be {\textquotedblleft compact over $L(G)$\textquotedblright} and the bimodule $L^2(\widetilde{M}) \ominus L^2(M)$ is merely weakly contained in the coarse bimodule $L^2(M) \otimes L^2(M)$. To overcome these technical difficulties, we will use ideas from the proof of Theorem B in \cite{ozawapopaII}. Note that the symbol {\textquotedblleft Lim\textquotedblright} will be used for a state on $\ell^\infty(\mathbf{N})$, or more generally on $\ell^\infty(I)$ with $I$ directed, which extends the ordinary limit.
Let $G$ be an amenable group and let $\pi : G \to \mathscr{O}(H_\mathbf{R})$ be an orthogonal representation. Let $M = \Gamma(H_\mathbf{R}, G, \pi)''$. Let $P \subset M$ be an amenable von Neumann subalgebra such that $P \npreceq_M L(G)$. Since $M$ has the c.m.a.p., $P$ is weakly compact inside $M$. Then there exists a net $(\eta_n)$ of vectors in $L^2(P \bar{\otimes} \bar{P})_+$ such that
\begin{enumerate}
\item $\lim_n \|\eta_n - (v \otimes \bar{v})\eta_n\|_2 = 0$, $\forall v \in \mathscr{U}(P)$;
\item $\lim_n \|\eta_n - \operatorname{op}eratorname{Ad}(u \otimes \bar{u})\eta_n\|_2 = 0$, $\forall u \in \mathscr{N}_M(P)$;
\item $\langle (a \otimes 1)\eta_n, \eta_n\rangle = \tau(a) = \langle \eta_n, (1 \otimes \bar{a}) \eta_n \rangle$, $\forall a \in M, \forall n$.
\end{enumerate}
We consider $\eta_n \in L^2(M \bar{\otimes} \bar{M})_+$, and note that $(J \otimes \bar{J}) \eta_n = \eta_n$, where $J$ denotes the canonical anti-unitary on $L^2(M)$. We shall simply denote $\mathscr{N}_M(P)$ by $\mathscr{G}$.
Let $z \in \mathscr{Z}(\mathscr{G}' \cap M)$ be a non-zero projection. Since $P \npreceq_M L(G)$ and $z \in P' \cap M$, it follows that $Pz \npreceq_M L(G)$. Theorem \ref{intertwining} then yields that the deformation $(\alpha_t)$ does not converge uniformly on $(Pz)_1$. Since any selfadjoint element $x \in (Pz)_1$ can be written
\begin{equation*}
x = \frac12 \|x\|_\infty (u + u^*)
\end{equation*}
where $u \in \mathscr{U}(Pz)$, it follows that $(\alpha_t)$ does not converge uniformly on $\mathscr{U}(Pz)$ either. Combining this with the inequality $(\ref{trans})$ in Proposition \ref{transversality}, we get that there exist $0 < c < 1$, a sequence of positive reals $(t_k)$ and a sequence of unitaries $(u_k)$ in $\mathscr{U}(P)$ such that $\lim_{k} t_k = 0$ and $\| \alpha_{t_k}(u_k z) - (E_M \circ \alpha_{t_k})(u_k z) \|_2 \geq c \|z\|_2$, $\forall k \in \mathbf{N}$. Since $\|\alpha_{t_k}(u_k z)\|_2 = \|z\|_2$, by Pythagora's theorem, we obtain
\begin{equation}\label{key}
\|(E_M \circ \alpha_{t_k})(u_k z)\|_2 \leq \sqrt{1 - c^2} \|z\|_2, \forall k \in \mathbf{N}.
\end{equation}
Set $\delta = \frac{1 - \sqrt{1 - c^2}}{6} \|z\|_2$. Choose and fix $k_0 \in \mathbf{N}$ such that
\begin{equation}\label{delta}
\|\alpha_{t_k}(z) - z\|_2 \leq \delta, \forall k \geq k_0.
\end{equation}
Define for any $n$ and any $k \geq k_0$,
\begin{eqnarray*}
\eta_n^k & = & (\alpha_{t_k} \otimes 1)(\eta_n) \in L^2(\widetilde{M}) \otimes L^2(\bar{M}) \\
\xi_n^k & = & (e_M\alpha_{t_k} \otimes 1)(\eta_n) \in L^2(M) \otimes L^2(\bar{M}) \\
\zeta_n^k & = & (e_M^\perp\alpha_{t_k} \otimes 1)(\eta_n) \in (L^2(\widetilde{M}) \ominus L^2(M)) \otimes L^2(\bar{M}).
\end{eqnarray*}
We observe that
\begin{equation}\label{norm2}
\|(x \otimes 1) \eta_n^k\|_2^2 = \tau(E_M(\alpha_{t_k}^{-1}(x^*x))) = \|x\|_2^2, \forall x \in \widetilde{M}.
\end{equation}
As in the proof of Theorem $4.9$ in \cite{ozawapopa}, noticing that $L^2(\widetilde{M}) \otimes L^2(\bar{M})$ is an $M \bar{\otimes} \bar{M}$-module and since $\eta_n^k = \xi_n^k + \zeta_n^k$, Equation \eqref{norm2} gives that for any $u \in \mathscr{G}$, and for any $k \geq k_0$,
\begin{eqnarray}\label{crucial}
\mathop{\operatorname{Lim}}_n \|[u \otimes \bar{u}, \zeta^k_n]\|_2
& \leq & \mathop{\operatorname{Lim}}_n \|[u \otimes \bar{u}, \eta_n^k]\|_2 \\ \nonumber
& \leq & \mathop{\operatorname{Lim}}_n \|(\alpha_{t_k} \otimes 1)([u \otimes \bar{u}, \eta_n])\|_2 + 2 \|u - \alpha_{t_k}(u) \|_2 \\
& = & 2 \|u - \alpha_{t_k}(u)\|_2. \nonumber
\end{eqnarray}
Moreover, for any $x \in M$,
\begin{eqnarray*}
\| (x \otimes 1) \zeta^k_n \|_2 & = & \|(x \otimes 1) (e_M^\perp \otimes 1) \eta_n^k\|_2 \\
& = & \|(e_M^\perp \otimes 1) (x \otimes 1) \eta_n^k\|_2 \\
& \leq & \| (x \otimes 1) \eta_n^k\|_2 = \|x\|_2.
\end{eqnarray*}
\begin{claim}\label{claim1}
For any $k \geq k_0$,
\begin{equation}\label{crucial2}
\mathop{\operatorname{Lim}}_n \|(z \otimes 1) \zeta_n^k\|_2 \geq \delta.
\end{equation}
\end{claim}
\begin{proof}[Proof of Claim $\ref{claim1}$]
We prove the claim by contradiction. Exactly as in the proof of Theorem 4.9 in \cite{ozawapopa}, noticing that $e_M z = z e_M$ (since $z \in M$) and $z u_k = u_k z$ (since $z \in \mathscr{Z}(\mathscr{G}' \cap M)$), and using $(\ref{delta})$ we have
\begin{eqnarray*}
&& \mathop{\operatorname{Lim}}_n \|(z \otimes 1)\eta_n^k - (e_M \alpha_{t_k}(u_k) z \otimes \bar{u}_k)\xi_n^k\|_2 \\
& \leq & \mathop{\operatorname{Lim}}_n \|(z \otimes 1)\eta_n^k - (e_M \alpha_{t_k}(u_k) z \otimes \bar{u}_k)\eta_n^k\|_2 + \mathop{\operatorname{Lim}}_n \|(z \otimes 1) \zeta_n^k\|_2 \\
& \leq & \mathop{\operatorname{Lim}}_n \|(z \otimes 1)\eta_n^k - (e_M z \alpha_{t_k}(u_k) \otimes \bar{u}_k)\eta_n^k\|_2 + \|[\alpha_{t_k}(u_k), z]\|_2 + \delta \\
& \leq & \mathop{\operatorname{Lim}}_n\|(z \otimes 1)\zeta_n^k\|_2 + \mathop{\operatorname{Lim}}_n \|\eta_n^k - (\alpha_{t_k}(u_k) \otimes \bar{u_k})\eta_n^k\|_2 \\
& & + 2\|z - \alpha_{t_k}(z)\|_2 + \delta \\
& \leq & \mathop{\operatorname{Lim}}_n \|(\alpha_{t_k} \otimes 1)(\eta_n - (u_k \otimes \bar{u}_k)\eta_n)\|_2 + 4\delta = 4 \delta.
\end{eqnarray*}
Thus, we would get
\begin{eqnarray*}
\|(E_M \circ \alpha_{t_k})(u_k z)\|_2 & \geq & \|(E_M \circ \alpha_{t_k})(u_k)z\|_2 - \|z - \alpha_{t_k}(z)\|_2 \\
& \geq & \mathop{\operatorname{Lim}}_n \|((E_M \circ \alpha_{t_k})(u_k)z \otimes \bar{u}_k)\eta_n^k\|_2 - \delta \\
& \geq & \mathop{\operatorname{Lim}}_n \|(e_M \otimes 1) ((E_M \circ \alpha_{t_k})(u_k) z \otimes \bar{u}_k)\eta_n^k\|_2 - \delta \\
& = & \mathop{\operatorname{Lim}}_n \|(e_M \alpha_{t_k}(u_k) z \otimes \bar{u}_k) \xi_n^k\|_2 - \delta \\
& \geq & \mathop{\operatorname{Lim}}_n \|(z \otimes 1)\eta_n^k\|_2 - 5\delta \\
& = & \|z\|_2 - 5\delta > \sqrt{1 - c^2} \|z\|_2,
\end{eqnarray*}
which is a contradiction according to $(\ref{key})$.
\end{proof}
We now use the techniques of the proof of Theorem B in \cite{ozawapopaII}. Define a state $\varphi^{z, k}$ on $\mathbf{B}(\mathscr{H}) \cap \rho(M^{\operatorname{op}})'$, where $\rho(M^{\operatorname{op}})$ is the right $M$-action on $\mathscr{H}$, by
\begin{equation*}
\varphi^{z, k}(x) = \mathop{\operatorname{Lim}}_n \frac{1}{\|\zeta_n^{z, k}\|_2^2} \langle (x \otimes 1) \zeta_n^{z, k}, \zeta_n^{z, k}\rangle,
\end{equation*}
where $\zeta_n^{z, k} = (z \otimes 1) \zeta_n^k$. Note that
\begin{equation*}
\varphi^{z, k}(x) = \varphi^{z, k}(z x) = \varphi^{z, k}(x z), \forall x \in \mathbf{B}(\mathscr{H}) \cap \rho(M^{\operatorname{op}})'.
\end{equation*}
\begin{claim}\label{claim2}
Let $a \in \mathscr{G}''$. Then one has
\begin{equation*}
\mathop{\operatorname{Lim}}_k | \varphi^{z, k} (a x - x a) | = 0,
\end{equation*}
uniformly for $x \in \mathbf{B}(\mathscr{H}) \cap \rho(M^{\operatorname{op}})'$ with $\|x\|_\infty \leq 1$.
\end{claim}
\begin{proof}[Proof of Claim $\ref{claim2}$]
For $u \in \mathscr{G}$, since $z \in \mathscr{Z}(\mathscr{G}' \cap M)$, one has
\begin{eqnarray*}
\mathop{\operatorname{Lim}}_n \|\zeta_n^{z, k} - (u \otimes \bar{u}) \zeta_n^{z, k} (u \otimes \bar{u})^*\|_2 & \leq & \mathop{\operatorname{Lim}}_n \|\zeta_n^{k} - (u \otimes \bar{u}) \zeta_n^{k} (u \otimes \bar{u})^*\|_2 \\
& \leq & 2 \|u - \alpha_{t_k}(u)\|_2.
\end{eqnarray*}
For every $x \in \mathbf{B}(\mathscr{H}) \cap \rho(M^{\operatorname{op}})'$, one has
\begin{equation*}
\varphi^{z, k}(u^* x u) = \mathop{\operatorname{Lim}}_n \frac{1}{\|\zeta_n^{z, k}\|_2^2} \langle(x \otimes 1)(u \otimes \bar{u}) \zeta_n^{z, k} (u \otimes \bar{u})^*, (u \otimes \bar{u}) \zeta_n^{z, k} (u \otimes \bar{u})^*\rangle,
\end{equation*}
so that with $(\ref{crucial}) - (\ref{crucial2})$,
\begin{equation*}
|\varphi^{z, k}(u^* x u) - \varphi^{z, k}(x)| \leq \frac{4}{\delta^2} \|x\|_\infty \|u - \alpha_{t_k}(u)\|_2.
\end{equation*}
This implies that
\begin{equation*}
\mathop{\operatorname{Lim}}_k |\varphi^{z, k}(a x - x a)| = 0,
\end{equation*}
for each $a \in \mbox{span }\mathscr{G}$ and uniformly for $x \in \mathbf{B}(\mathscr{H}) \cap \rho(M^{\operatorname{op}})'$ with $\|x\|_\infty \leq 1$. However, for any $a \in M$,
\begin{eqnarray*}
|\varphi^{z, k}(x a)| & = & \mathop{\operatorname{Lim}}_n \frac{1}{\|\zeta_n^{z, k}\|_2^2} |\langle (x \otimes 1)(a \otimes 1) \zeta_n^{z, k}, \zeta_n^{z, k}\rangle| \\
& \leq & \frac{1}{\delta^2} \|x\|_\infty \|z a\|_2 \\
& \leq & \frac{1}{\delta^2} \|x\|_\infty \| a\|_2,
\end{eqnarray*}
and likewise for $|\varphi^{z, k}(a x)|$. An application of Kaplansky density theorem does the job.
\end{proof}
To prove at last that $\mathscr{G}''$ is amenable, we will use (as in Theorem B in \cite{ozawapopaII}) Connes' criterion for finite amenable von Neumann algebras (see Theorem $5.1$ in \cite{connes76} for the type ${\rm II_1}$ case and Lemma 2.2 in \cite{haagerup83} for the general case). For any non-zero projection $z \in \mathscr{Z}(\mathscr{G}' \cap M)$ and any finite subset $F \subset \mathscr{U}(\mathscr{G}'')$, we need to show
\begin{equation*}
\| \sum_{u \in F} uz \otimes \overline{uz} \|_{M \bar{\otimes} \bar{M}} = |F|.
\end{equation*}
Let $z \in \mathscr{Z}(\mathscr{G}' \cap M)$ be a non-zero projection and let $F \subset \mathscr{U}(\mathscr{G}'')$ be a finite subset. Since the $M, M$-bimodule $\mathscr{H}$ is weakly contained in the coarse bimodule $L^2(M) \otimes L^2(M)$, let $\Psi : \mathbf{B}(L^2(M)) \to \mathbf{B}(\mathscr{H}) \cap \rho(M^{\operatorname{op}})'$ be the u.c.p. map which extends the left $M$-action on $\mathscr{H}$ (see Lemma \ref{weakcontainment}). Note that $M$ is contained in the multiplicative domain of $\Psi$. Define $\psi^{z, k} = \varphi^{z, k} \circ \Psi$ a state on $\mathbf{B}(L^2(M))$. Let $u \in \mathscr{G}''$. By Claim \ref{claim2}, one has
\begin{eqnarray*}
\mathop{\operatorname{Lim}}_k |\psi^{z, k}((uz)^* x (uz) - x)| & = & \mathop{\operatorname{Lim}}_k |\varphi^{z, k}( \Psi((uz)^* x (uz)) - \Psi(x))| \\
& = & \mathop{\operatorname{Lim}}_k |\varphi^{z, k}( (uz)^* \Psi(x) (uz) - \Psi(x))| \\
& = & \mathop{\operatorname{Lim}}_k |\varphi^{z, k}( u^* \Psi(x) u - \Psi(x))| = 0,
\end{eqnarray*}
uniformly for $x \in \mathbf{B}(L^2(M))$ with $\|x\|_\infty \leq 1$. By a standard recipe of the theory together with the Hahn-Banach separation theorem, we can find a net $(\mu^{z, k})$ of positive norm-one elements in $S_1(L^2(M))$ (trace-class operators on $L^2(M)$) such that
\begin{equation*}
\lim_k \|\mu^{z, k} - \operatorname{op}eratorname{Ad}(uz)\mu^{z,k}\|_1= 0, \forall u \in \mathscr{U}(\mathscr{G}'').
\end{equation*}
Since the above is satisfied in particular for $u = 1$ and since $F \subset \mathscr{\mathscr{G}''}$ is finite, replacing $\mu^{z, k}$ by $z \mu^{z, k} z/\|z \mu^{z, k} z\|_1$ we may assume that $\mu^{z, k} \in S_1(L^2(M))$ satisfies $\mu^{z, k} \geq 0$, $z\mu^{z, k}z = \mu^{z, k}$, $\|\mu^{z, k}\|_1 = 1$ and
\begin{equation*}
\lim_k \|\mu^{z, k} - \operatorname{op}eratorname{Ad}(uz)\mu^{z,k}\|_1= 0, \forall u \in F.
\end{equation*}
Define now $\nu^{z, k} = (\mu^{z, k})^{1/2} \in S_2(L^2(M))$ (Hilbert-Schmidt operators on $L^2(M)$). The net $(\nu^{z, k})$ satisfies $z\nu^{z, k}z = \nu^{z, k}$, $\|\nu^{z, k}\|_2 = 1$ and
\begin{equation*}
\lim_k \|\nu^{z, k} - \operatorname{op}eratorname{Ad}(uz)\nu^{z,k}\|_2= 0, \forall u \in F.
\end{equation*}
by Powers-St\o rmer inequality. With the identification
\begin{equation*}
S_2(L^2(M)) = L^2(M) \otimes L^2(\bar{M})
\end{equation*}
as $M, M$-bimodules it follows that the $\ast$-representations of $M$ and $\bar{M}$ given by the left and right $M$-actions induce the spatial tensor norm. Thus,
\begin{eqnarray*}
|F| & = & \| \sum_{u \in F} \nu^{z, k}\|_2 \\
& \leq & \lim_k \| \sum_{u \in F} (uz)\nu^{z, k}(uz)^*\|_2 + \lim_k \| \sum_{u \in F} \nu^{z, k} - (uz) \nu^{z, k} (uz)^*\|_2 \\
& \leq & \|\sum_{u \in F} uz \otimes \overline{uz}\|_{M \bar{\otimes} \bar{M}}.
\end{eqnarray*}
Since the other inequality is trivial, the proof is complete.
\end{proof}
\subsection{Proof of Theorem A}
We refer to Section \ref{examples} for the necessary background on spectral measures of unitary representations. Let's begin with a few easy observations first. Assume that $(N, \tau)$ is a finite von Neumann algebra with no amenable direct summand, i.e. $Nz$ is not amenable, $\forall z \in \mathscr{Z}(N)$, $z \neq 0$. Then for any non-zero projection $q \in N$, $qNq$ is non-amenable. Moreover, if $N$ has no amenable direct summand and $N \subset N_1$ is a unital inclusion of finite von Neumann algebras, then $N_1$ has no amenable direct summand either.
\begin{lem}\label{diffuse}
Let $G$ be a countable group together with an action $G \curvearrowright (N, \tau)$ on a finite von Neumann algebra. Write $M = N \rtimes G$ for the crossed product. Let $B \subset N$ be a diffuse subalgebra. Then $B \npreceq_M L(G)$.
\end{lem}
\begin{proof}
We denote by $(v_g)$ the canonical unitaries which generate $L(G) \subset N \rtimes G = M$. Let $B \subset N$ be a diffuse subalgebra. Let $(u_n)$ be a sequence of unitaries in $B$ such that $u_n \to 0$ weakly, as $n \to \infty$. Let $I, J \subset G$ be finite subsets and
\begin{eqnarray*}
x & = & \sum_{g \in I} x_g v_g \\
y & = & \sum_{h \in J} y_h v_h,
\end{eqnarray*}
where $x_g, y_h \in N$. Then we have
\begin{equation*}
E_{L(G)}(x^* u_n y) = \sum_{(g, h) \in I \times J} \tau(x_g^* u_n y_h) v_g^* v_h.
\end{equation*}
In particular,
\begin{equation*}
\|E_{L(G)}(x^* u_n y)\|_2 \leq \sum_{(g, h) \in I \times J} |\tau(x^*_g u_n y_h)|.
\end{equation*}
Since $u_n \to 0$ weakly, as $n \to \infty$, we get $\lim_n \|E_{L(G)}(x^* u_n y)\|_2 = 0$. Finally, using Kaplansky density theorem, we obtain
\begin{equation*}
\lim_n \|E_{L(G)}(x^* u_n y)\|_2 = 0, \forall x, y \in M.
\end{equation*}
By $(3)$ of Theorem \ref{intertwining1}, it follows that $B \npreceq_M L(G)$.
\end{proof}
\begin{theo}[Theorem A]\label{nocartan}
Let $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ be an orthogonal representation such that the spectral measure of $\pi$ has no atoms. Then $M = \Gamma(H_\mathbf{R}, \mathbf{Z}, \pi)''$ is a non-amenable ${\rm II_1}$ factor and for any maximal abelian subalgebra $A \subset M$, $\mathscr{N}_M(A)''$ is an amenable von Neumann algebra.
\end{theo}
\begin{proof}
Since the spectral measure of $\pi : \mathbf{Z} \to \mathscr{U}(H)$ has no atoms, it follows that $\pi$ has no eigenvectors. So the representation $\mathscr{F}(\pi) : \mathbf{Z} \to \mathscr{U}(\mathscr{F}(H))$ has no eigenvectors either. Thus, the corresponding free Bogoljubov action $\sigma^\pi : \mathbf{Z} \curvearrowright \Gamma(H_\mathbf{R})''$ is necessarily outer (see Theorem~\ref{thm:bogoOuter}) and then $M = \Gamma(H_\mathbf{R}, \mathbf{Z}, \pi)''$ is a ${\rm II_1}$ factor. Moreover, $L(\mathbf{Z})$ is clearly a MASA in $M$.
We prove the result by contradiction. Assume that $A \subset M= \Gamma(H_\mathbf{R}, \mathbf{Z}, \pi)''$ is a MASA such that $\mathscr{N}_M(A)''$ is not amenable. Write $1 - z \in \mathscr{Z}(\mathscr{N}_M(A)'')$ for the maximal projection such that $\mathscr{N}_M(A)''(1 - z)$ is amenable. Then $z \neq 0$ and $\mathscr{N}_M(A)''z$ has no amenable direct summand. Notice that $z \in A' \cap M = A$ and
\begin{equation*}
\mathscr{N}_M(A)''z = \mathscr{N}_{zMz}(Az)'',
\end{equation*}
by Lemma 3.5 in \cite{popamal1}. Moreover $Az \subset zMz$ is a MASA.
Since the action $\sigma^\pi : \mathbf{Z} \curvearrowright \Gamma(H_\mathbf{R})''$ is outer, it follows that $\Gamma(H_\mathbf{R})' \cap M = \mathbf{C}$. Thanks to Theorem 3.3 in \cite{masapopa}, we can find a diffuse abelian subalgebra $B \subset \Gamma(H_\mathbf{R})''$ which is a MASA in $M$. Since $M$ is a ${\rm II_1}$ factor and $B$ is diffuse, there exist a projection $p \in B$ and a unitary $u \in \mathscr{U}(M)$ such that $p = u z u^*$. Define $\tilde{A} = u Az u^*$. Then $\tilde{A} \subset pMp$ is a MASA and $\mathscr{N}_{pMp}(\tilde{A})''$ has no amenable direct summand. Let $C = \tilde{A} \operatorname{op}lus B(1 - p) \subset M$. Note that $C \subset M$ is still a MASA. Since $\mathscr{N}_M(C)''$ is not amenable and $C \subset M$ is weakly compact, Theorem \ref{step} yields $C \preceq_M L(\mathbf{Z})$. Since $L(\mathbf{Z})$ is a MASA, if we apply Theorem A.1 of \cite{popa2001}, we obtain $v \in M$ a nonzero partial isometry such that $v^*v \in C' \cap M = C$, $q = vv^* \in L(\mathbf{Z})$ and $v C v^* \subset L(\mathbf{Z})q$. Since $C \subset M$ is also a MASA, we get $v C v^* = L(\mathbf{Z})q$. Note that $v p v^* \neq 0$, because otherwise we would have $v B v^* = L(\mathbf{Z})q$ and this would imply that $B \preceq_M L(\mathbf{Z})$, a contradiction according to Lemma \ref{diffuse}. Thus, with $q' = v p v^*$ we obtain $v \tilde{A} v^* = L(\mathbf{Z})q'$. Consequently $\mathscr{N}_{q' M q'}(L(\mathbf{Z}) q')''$ is not amenable.
However, as $L(\mathbf{Z}), L(\mathbf{Z})$-bimodules we have the following isomorphism
\begin{equation*}
L^2(M) \operatorname{co}ng \bigoplus_{n \geq 0} K^{(n)}_{\pi},
\end{equation*}
where $K^{(n)}_{\pi} = H^{\otimes n} \otimes \ell^2(\mathbf{Z})$ (see Section \ref{preliminaries}). Since the spectral measure of $\pi$ has no atoms, it follows that $L(\mathbf{Z}) \subset M$ is a singular MASA, i.e. $\mathscr{N}_M(L(\mathbf{Z}))'' = L(\mathbf{Z})$, and {\em a fortiori} $\mathscr{N}_{q' M q'}(L(\mathbf{Z}) q')'' = L(\mathbf{Z}) q'$ (by Lemma 3.5 in \cite{popamal1}). We have reached a contradiction.
\end{proof}
\subsection{Proof of Theorem B}
\begin{theo}[Theorem B]\label{stronglysolid}
Let $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ be a mixing orthogonal representation. Then the non-amenable ${\rm II_1}$ factor $M = \Gamma(H_\mathbf{R}, \mathbf{Z}, \pi)''$ is strongly solid.
\end{theo}
\begin{proof}
Since the representation $\pi : \mathbf{Z} \to \mathscr{O}(H_\mathbf{R})$ is mixing, it has no eigenvectors. So the representation $\mathscr{F}(\pi) : \mathbf{Z} \to \mathscr{U}(\mathscr{F}(H))$ has no eigenvectors either. Thus, the free Bogoljubov action $\sigma^\pi : \mathbf{Z} \curvearrowright \Gamma(H_\mathbf{R})''$ is necessarily outer (see Theorem~\ref{thm:bogoOuter}) and then $M = \Gamma(H_\mathbf{R}, \mathbf{Z}, \pi)''$ is a ${\rm II_1}$ factor.
Let $P \subset M$ be a diffuse amenable von Neumann subalgebra. By contradiction assume that $ \mathscr{N}_M(P)''$ is not amenable. Write $1 - z \in \mathscr{Z}(\mathscr{N}_M(P)'')$ for the maximal projection such that $\mathscr{N}_M(P)''(1 - z)$ is amenable. Then $z \neq 0$ and $\mathscr{N}_M(P)''z$ has no amenable direct summand. Notice that
\begin{equation*}
\mathscr{N}_M(P)''z \subset \mathscr{N}_{zMz}(Pz)''.
\end{equation*}
Since this is a unital inclusion (with unit $z$), $\mathscr{N}_{zMz}(Pz)''$ has no amenable direct summand either. Let $A \subset \Gamma(H_\mathbf{R})''$ be a diffuse abelian subalgebra. Since $M$ is a ${\rm II_1}$ factor and $A$ is diffuse, there exist a projection $q \in A$ and a unitary $u \in \mathscr{U}(M)$ such that $q = u z u^*$. Define $Q = u Pz u^*$. Then $Q \subset qMq$ is diffuse, amenable and $\mathscr{N}_{qMq}(Q)''$ has no amenable direct summand. Let $B = Q \operatorname{op}lus A(1 - q) \subset M$. Note that $B \subset M$ is a unital diffuse amenable subalgebra. Since $\mathscr{N}_M(B)''$ is not amenable and $B \subset M$ is weakly compact, Theorem \ref{step} yields $B \preceq_M L(\mathbf{Z})$.
Thus, there exists $n \geq 1$, a non-zero partial isometry $v \in \mathbf{M}_{1, n}(\mathbf{C}) \otimes M$ and a (possibly non-unital) $\ast$-homomorphism $\psi : B \to L(\mathbf{Z})^n$ such that $x v = v \psi(x)$, $\forall x \in B$. Observe that $q v \neq 0$, because otherwise we would have $vv^* \leq 1 - q$ and $x v = v \psi(x)$, $\forall x \in A(1 - q)$. This would mean that $A(1 - q) \preceq_M L(\mathbf{Z})$ and so $A \preceq_M L(\mathbf{Z})$, which is a contradiction according to Lemma \ref{diffuse}. Write $q v = w |q v|$ for the polar decomposition of $qv$. It follows that $w \in \mathbf{M}_{1, n}(\mathbf{C}) \otimes M$ is a non-zero partial isometry such that $x w = w \psi(x)$, $\forall x \in Q$. This means exactly that $Q \preceq_M L(\mathbf{Z})$. Note that $ww^* \in Q' \cap qMq \subset \mathscr{N}_{qMq}(Q)''$ and $w^*w \in \psi(Q)' \cap \psi(q) M^n \psi(q)$.
Since the $\tau$-preserving action $\mathbf{Z} \curvearrowright \Gamma(H_\mathbf{R})''$ is mixing by assumption and $\psi(Q) \subset \psi(q)L(\mathbf{Z})^n\psi(q)$ is diffuse, it follows from Theorem $3.1$ in \cite{popamal1} (see also Theorem D.4 in \cite{vaesbern}) that $w^*w \in \psi(q)L(\mathbf{Z})^n\psi(q)$, so that we may assume $w^*w = \psi(q)$. Note that $w^* Q w = \psi(Q)$. Moreover since $\psi(Q)$ is diffuse, Theorem 3.1 in \cite{popamal1} yields that the quasi-normalizer of $\psi(Q)$ inside $\psi(q)M^n\psi(q)$ is contained in $\psi(q) L(\mathbf{Z})^n \psi(q)$. In particular, we get
\begin{equation*}
\operatorname{op}eratorname{Ad}(w^*)(ww^*\mathscr{N}_{qMq}(Q)'' ww^*) \subset \psi(q) L(\mathbf{Z})^n \psi(q).
\end{equation*}
Note that $\operatorname{op}eratorname{Ad}(w^*) : ww^* M ww^* \to w^*w M^n w^*w$ is a $\ast$-isomorphism. Since $\psi(q)L(\mathbf{Z})^n \psi(q)$ is amenable and $ww^*\mathscr{N}_{qMq}(Q)'' ww^*$ is non-amenable, we finally get a contradiction, which finishes the proof.
\end{proof}
The above theorem is still true for any amenable group $G$ (instead of $\mathbf{Z}$), and any mixing orthogonal representation $\pi : G \to \mathscr{O}(H_\mathbf{R})$ such that the corresponding Bogoljubov action $\sigma^\pi : G \curvearrowright \Gamma(H_\mathbf{R})''$ is properly outer, i.e. $\sigma^\pi_g$ is outer, for any $g \neq e$.
\section{New examples of strongly solid ${\rm II_1}$ factors}\label{examples}
\subsection{Spectral measures and unitary representations} Let $H$ be a separable complex Hilbert space. Let $G$ be a locally compact second countable (l.c.s.c.) abelian group together with $\pi : G \to \mathscr{U}(H)$ a $\ast$-strongly continuous unitary representation. Denote by $\widehat{G}$ the dual of $G$. It follows that $C^*(G) \operatorname{co}ng C_0(\widehat{G})$ and $\pi$ gives rise to a $\ast$-representation $\sigma : C_0(\widehat{G}) \to \mathbf{B}(H)$ such that $\sigma(f_g) = \pi(g)$, for every $g \in G$, where $f_g(\chi) = \chi(g)$, $\forall \chi \in \widehat{G}$.
Recall that for any unit vector $\xi \in H$, there exists a unique probability measure on $\mu_\xi$ on $\widehat{G}$ such that
\begin{equation*}
\int_{\widehat{G}} f \, d\mu_\xi = \langle \sigma(f) \xi, \xi\rangle.
\end{equation*}
Note that the formula makes sense for every bounded Borel function $f$ on $\widehat{G}$.
\begin{df}
Let $G$ be a l.c.s.c. abelian group together with $\pi : G \to \mathscr{U}(H)$ a $\ast$-strongly continuous unitary representation. The {\it spectral measure} $\mathscr{C}_\pi$ of the unitary representation $\pi$ is defined as the measure class on $\widehat{G}$ generated by all the probability measures $\mu_\xi$, for $\xi \in H$, $\|\xi\| = 1$.
\end{df}
Recall that the {\em support} of a measure is the (closed) subset of all points for which every neighborhood has positive measure. The spectral measure $\mathscr{C}_\pi$ is said to be {\it singular} if for all the probability measures $\mu$ in $\mathscr{C}_\pi$, the support of $\mu$ has $0$ Haar measure. From now on, we will only consider the cases when $G = \mathbf{Z}$ or $\mathbf{R}$.
We identify the Pontryagin dual of $\mathbf{R}$ with $\mathbf{R}$ by the pairing $\mathbf{R} \times \mathbf{R} \ni (x, y) \mapsto e^{2\pi i x y}$. Define
\begin{eqnarray*}
p: \mathbf{R} & \to & \mathbf{T} = \mathbf{R}/\mathbf{Z} \\
x & \mapsto & x + \mathbf{Z}
\end{eqnarray*}
the canonical projection. For $\mu$ a probability measure on $\mathbf{R}$, the push-forward measure of $\mu$ on $\mathbf{T}$ is defined by $(p_\ast\mu)(A) = \mu(p^{-1}(A))=\mu(A+\mathbf{Z})$, $\forall A \subset \mathbf{T}$ Borel subset. The convolution product is denoted by $\ast$. We shall write
\begin{equation*}
\mu^{\ast k} = \mu \ast \cdots \ast \mu
\end{equation*}
for the $k$-fold convolution product.
\begin{lem}\label{pushforward}
Let $\mu$ be a probability measure on $\mathbf{R}$. Write $\nu = p_\ast \mu$.
\begin{enumerate}
\item If $\mu$ is singular, then $\nu$ is singular.
\item For any $k \geq 1$, $(p_\ast \mu)^{\ast k}$ and $p_\ast(\mu^{\ast k})$ are absolutely continuous to each other.
\end{enumerate}
\end{lem}
\begin{proof}
Denote by $\lambda$ the Lebesgue measure on $\mathbf{R}$. We may identify $(\mathbf{T}, \operatorname{op}eratorname{Haar})$ with $([0, 1], \lambda)$ as probability spaces. We use the notation $\mu_1 \sim \mu_2$ for two measures absolutely continuous to each other.
$(1)$ Assume that $\mu$ is singular. Write $K$ for the support of $\mu$ and $K_n = K \cap [n, n+ 1[$. Clearly, $\operatorname{op}eratorname{supp}(\nu) \subset p(K)$. We have
\begin{eqnarray*}
\operatorname{op}eratorname{Haar}(p(K)) & \leq & \sum_{n \in \mathbf{Z}} \operatorname{op}eratorname{Haar}(p(K_n)) \\
& = & \sum_{n \in \mathbf{Z}} \lambda(K_n) = 0.
\end{eqnarray*}
Thus $\operatorname{op}eratorname{Haar}(\operatorname{op}eratorname{supp}(\nu)) = 0$ and $\nu$ is singular.
$(2)$ Under the previous identification, we have for any $A \subset \mathbf{T}$ Borel subset
\begin{eqnarray*}
\nu(B) & = & \mu(B + \mathbf{Z}) \\
& = &\sum_{n \in \mathbf{Z}} (\mu \ast \delta_n)(B).
\end{eqnarray*}
Thus for any $k \geq 1$, we have
\begin{eqnarray*}
\nu^{\ast k} & = & \left( \sum_{n \in \mathbf{Z}} \mu \ast \delta_n \right)^{\ast k} \\
& \sim & \sum_{n \in \mathbf{Z}} \left( \sum \mu^{\ast k} \ast \delta_n \right) \\
& \sim & \sum_{n \in \mathbf{Z}} \mu^{\ast k} \ast \delta_n.
\end{eqnarray*}
Consequently $(p_\ast \mu)^{\ast k} \sim p_\ast(\mu^{\ast k})$.
\end{proof}
\subsection{Examples of strongly solid ${\rm II_1}$ factors}
Erd\"os showed in \cite{erdos} that the symmetric probability measure $\mu_\theta$ on $\mathbf{R}$, with $\theta = 5/2$, obtained as the weak limit of
\begin{equation*}
\left( \frac12 \delta_{-\theta^{-1}} + \frac12 \delta_{\theta^{-1}} \right) \ast \cdots \ast \left( \frac12 \delta_{-\theta^{-n}} + \frac12 \delta_{\theta^{-n}} \right)
\end{equation*}
is singular w.r.t. the Lebesgue measure $\lambda$ and has a Fourier Transform
\begin{equation*}
\widetilde{\mu}_\theta(t) = \prod_{n \geq 1} \operatorname{co}s\left(\frac{t}{\theta^n}\right)
\end{equation*}
which vanishes at infinity, i.e. $\widetilde{\mu}(t) \to 0$, as $|t| \to \infty$.
\begin{exam}\label{singularmeasure}
Modifying the measure $\mu_\theta$, Antoniou \& Shkarin (see Theorem $2.5, {\rm v}$ in \cite{antoniou}) constructed an example of a symmetric probability measure $\mu$ on $\mathbf{R}$ such that:
\begin{enumerate}
\item The Fourier Transform of $\mu$ vanishes at infinity, i.e. $\widetilde{\mu}(t) \to 0$, as $|t| \to \infty$.
\item For any $n \geq 1$, the $n$-fold convolution product $\mu^{\ast n}$ is singular w.r.t. the Lebesgue measure $\lambda$.
\end{enumerate}
\end{exam}
Let $\mu$ be a symmetric probability measure on $\mathbf{R}$ as in Example \ref{singularmeasure} and consider $\nu = p_\ast \mu$ the push-forward measure on the torus $\mathbf{T}$. Since $\mu(X) = \mu(-X)$, for any Borel set $X \subset \mathbf{R}$, it follows that $\nu(A) = \nu(\overline{A})$, for any Borel set $A \subset \mathbf{T}$, where $\overline{A} = \{\bar{z} : z \in A\}$.
Let $\pi^\nu : \mathbf{Z} \to \mathscr{U}(L^2(\mathbf{T}, \nu))$ be the unitary representation defined by $(\pi^\nu_n f)(z) = z^n f(z)$, $\forall f \in L^2(\mathbf{T}, \nu)$, $\forall n \in \mathbf{Z}$. Note that moreover
\begin{equation*}
H_\mathbf{R}^\nu = \left\{ f \in L^2(\mathbf{T}, \nu) : \overline{f(z)} = f(\bar{z}), \forall z \in \mathbf{T} \right\}
\end{equation*}
is a real subspace of $L^2(\mathbf{T}, \nu)$ invariant under $\pi^\nu$. Indeed, for all $f, g \in H^\nu_\mathbf{R}$,
\begin{eqnarray*}
\langle f, g \rangle & = & \int_{\mathbf{T}} f(z) \overline{g(z)} \, d\nu(z) \\
& = & \int_{\mathbf{T}} \overline{f(\bar{z})} g(\bar{z}) \, d\nu(z) \\
& = & \int_{\mathbf{T}} \overline{f(\bar{z})} g(\bar{z}) \, d\nu(\bar{z}) \\
& = & \int_{\mathbf{T}} \overline{f(z)} g(z) \, d\nu(z) \\
& = & \overline{\langle f, g \rangle}.
\end{eqnarray*}
By assumption and using Lemma \ref{pushforward}, it follows that:
\begin{enumerate}
\item The unitary representation $\pi^\nu : \mathbf{Z} \to \mathscr{U}(L^2(\mathbf{T}, \nu))$ is mixing.
\item The spectral measure of $\bigoplus_{n \geq 1} (\pi^\nu)^{\otimes n}$ is singular.
\end{enumerate}
Consider now the non-amenable ${\rm II_1}$ factor $M = \Gamma(H_\mathbf{R}^\nu, \mathbf{Z}, \pi^\nu)''$. Let $A = L(\mathbf{Z})$. Since $\pi^\nu$ is mixing, $A$ is maximal abelian in $M$ and {\em singular}, i.e. $\mathscr{N}_M(A)'' = A$. Since the spectral measure of the unitary representation $\bigoplus_{n \geq 1} (\pi^\nu)^{\otimes n}$ is singular and because of the $A, A$-bimodule isomorphism
\begin{equation*}
L^2(M) \operatorname{co}ng \bigoplus_{n \geq 0} K^{(n)}_{\pi^\nu},
\end{equation*}
where $K^{(n)}_{\pi^\nu} = L^2(\mathbf{T}, \nu)^{\otimes n} \otimes \ell^2(\mathbf{Z})$ (see Section \ref{preliminaries}), it follows that the $A, A$-bimodule $L^2(M)$ is disjoint form the coarse bimodule $L^2(A) \otimes L^2(A)$. Combining Voiculescu's result (see Corollary 7.6 in \cite{voiculescu96}) and the second-named author's result (see Proposition $9.2$ in \cite{shlya99}), it follows that the non-amenable ${\rm II_1}$ factor $M$ is not isomorphic to any interpolated free group factor $L(\mathbf{F}_t)$, $1 < t \leq \infty$. Moreover, our Theorem \ref{stronglysolid} yields that $M$ is strongly solid, hence has no Cartan subalgebra.
\begin{theo}[Corollary B]\label{singular-ssolid}
The ${\rm II_1}$ factor $M = \Gamma(H_\mathbf{R}^\nu, \mathbf{Z}, \pi^\nu)''$ is strongly solid, hence has no Cartan subalgebra. Nevertheless, for the maximal abelian subalgebra $A = L(\mathbf{Z})$, the $A, A$-bimodule $L^2(M)$ is disjoint from the coarse bimodule $L^2(A) \otimes L^2(A)$. Thus, $M$ is never isomorphic to an interpolated free group factor.
\end{theo}
\begin{rem}
For $\theta = 3$, $\mu_\theta$ is the Cantor-Lebesgue measure on the ternary Cantor set. If we set $\nu = p_\ast \mu_\theta$, we get that for any $n \geq 1$, the $n$-fold convolution product $\nu^{\ast n}$ is singular w.r.t. the Lebesgue measure $\lambda$. In that case, the ${\rm II_1}$ factor $M = \Gamma(H_\mathbf{R}^\nu, \mathbf{Z}, \pi^\nu)''$ has no Cartan subalgebras and is not isomorphic to any interpolated free group factor (Corollary A).
\end{rem}
\subsection{Bimodule decompositions over MASAs.}
Recall that if $\mu$ is a probability measure on $[0,1]\times [0,1]$ so that its push-forwards by the projection maps onto the two copies of $[0,1]$ are Lebesgue absolutely continuous, then $L^2([0,1]\times [0,1],\mu)$ can be regarded as an $L^\infty[0,1]$, $L^\infty[0,1]$-bimodule via the action
\begin{multline*}
(f_1 \cdot \xi \cdot f_2)(x,y) = f_1 (x) \xi(x,y) f_2 (y),
\\ x,y\in [0,1],\quad f_j\in L^\infty[0,1], \quad \xi\in L^\infty ([0,1]\times[0,1],\mu).\end{multline*}
For a von Neumann algebra $M$, consider the collection $\mathscr{C}(M)$ of measure classes $[\mu]$ on $[0,1]\times [0,1]$ with the property that there exists a MASA $L^\infty [0,1]\operatorname{co}ng A\subset M$ so that $L^2(M)$, when regarded as an $A,A$-bimodule, contains a copy of $L^2([0,1]^2,\mu)$.
Also let $\mathscr{D}(M)$ be the collection of all measure classes $[\mu]$ so that for {\em every} MASA $L^\infty[0,1]\operatorname{co}ng A\subset M$, $L^2(M)$ contains a sub-bimodule of $L^2([0,1]^2,\mu)$. Clearly, $\mathscr{C}\supset \mathscr{D}$.
Then (as is well known) $M$ has a Cartan subalgebra if and only if $\mathscr{C}(M)$ contains an $r$-discrete measure class (i.e., a measure class $[\mu]$ for which $\mu(B) = \int \mu_t (B) dt$ and $\mu_t$ are a.e. discrete).
Voiculescu in \cite{voiculescu96} proved that $\mathscr{D}(L(\mathbf{F}_n)) \ni \{\textrm{Lebesgue Measure}\}$.
It thus remained open whether every II$_1$ factor $N$ must either contain a Cartan subalgebra, or satisfy that $\mathscr{D}(N) \ni \{\textrm{Lebesgue Measure}\}$. Our main example $M= \Gamma(H_\mathbf{R}^\nu, \mathbf{Z}, \pi^\nu)''$ answers this question in the negative, as $\mathscr{D}(M)$ does not contain Lebesgue measure and yet $M$ has no Cartan subalgebra.
\section{Outerness of free Bogoljubov actions}
Although we do not need the following result in the rest of the paper, we record the following observation, which is well-known to the experts and is most likely folklore (although we could not find a precise reference).
\begin{theo}\label{thm:bogoOuter} Let $G$ be a countable group, and let $\pi: G \to \mathscr{O}(H_\mathbf{R})$ be a $\ast$-strongly continuous orthogonal
representation of $G$ on a real Hilbert space $H_\mathbf{R}$. Then $\sigma_g^\pi$ is inner iff $\pi_g=1$.
In particular, if $\pi_g \neq 1$ for any $g\neq e$, the Bogoljubov action $\sigma^\pi$ of $G$ on $\Gamma(H_\mathbf{R})''$ is outer.
\end{theo}
\begin{proof}
Let $g$ be an element of $G$ so that $\pi_g \neq 1$,
and let $\alpha = \sigma_g^\pi$ acting on $M=\Gamma(H_\mathbf{R})''$. Let $T=\pi_g$. We may assume without loss of generality that $H_\mathbf{R}$ has dimension at least $2$, so that $M$ is a factor (otherwise, $M$ is abelian, and any non-trivial $T$ gives rise to an outer transformation).
Suppose for a contradiction that $\alpha = \operatorname{op}eratorname{Ad}(u)$ for some unitary $u\in M$. Then for any $x\in M$,
$$
\alpha(x) = u x u^*
$$ and so $\alpha (u) = u$.
Let $H = H_\mathbf{R} \otimes_\mathbf{R} \mathbf{C}$ be the complexification of $H_\mathbf{R}$. We continue to denote the complexification of $T$ by the same letter.
Let $H^a \subset H$ be the closed linear span of eigenvectors of $T$, $H^a_\mathbf{R} = H^a \cap H_\mathbf{R}$ be its real part. Then $N=\Gamma(H^a_\mathbf{R})''\subset \Gamma(H_\mathbf{R})''=M$. Moreover, it is clear from the Fock space decomposition of $L^2(M)$ that any eigenvectors for $\alpha$ must lie in $L^2(N)$, so $u\in N$. Thus we may, without loss of generality, assume that $N=M$ and that eigenvectors of $T$ densely span $H$.
Thus we may assume that
\begin{equation*}
H_\mathbf{R} = \mathbf{R}^n \operatorname{op}lus \bigoplus_{k\in J} H^k_\mathbf{R},
\end{equation*}
where $n\in \{0,1,\dots,+\infty\}$,
each $H^k_\mathbf{R}\operatorname{co}ng \mathbf{R}^2$ and $T$ acts trivially on $\mathbf{R}^n$ and acts on $H^k_\mathbf{R}$ by a rotation of period $2\pi /\log \lambda_k$. If we denote by $h_k, g_k$ an orthonormal basis for $H^k_\mathbf{R}$ and we set $c_k = s(h_k) + i s(g_k)\in M$, then $M\operatorname{co}ng L(\mathbf{F}_n) * W^*(c_k : k\in J)$, and $\alpha = \operatorname{op}eratorname{id} * \beta$ where
$\beta (c_j ) = \exp(2\pi i \lambda_j) c_j$. Let $c_j = u_j b_j$ be the polar decomposition of $c_j$; thus $\beta(u_j)=\exp(2\pi i \lambda_j) u_j$ and $\beta(b_j)=b_j$.
By \cite{DVV:circular}, $b_j$ and $u_j$ are freely independent and $W^*(b_k:k\in J) \operatorname{co}ng W^*(u_k : k\in J) \operatorname{co}ng L(\mathbf{F}_{2|J|})$. It follows that $M\operatorname{co}ng L(\mathbf{F}_n) * W^*(b_k : k\in J) * W^*(u_k : k\in J) \operatorname{co}ng L(\mathbf{F}_{n+|J|}) * L(\mathbf{F}_{|J|}) = N * P$ in such a way that $\alpha$ corresponds to the action $\operatorname{op}eratorname{id} * \gamma$ where $\gamma : P\to P = W^*(u_k : k\in J)$ is given by $\gamma(u_k)=\exp(2\pi i \lambda_k) u_k$.
Since by assumption $T$ is non-trivial, $|J|\geq 1$ and also $|J| + n \geq 1$. Thus if $\alpha (x) = uxu^*$ for all $x\in M$, then $u$ must commute with $N\subset N * P\operatorname{co}ng M$. But $N'\cap M = N'\cap N = \mathscr{Z}(N)$ (e.g. because as an $N$,$N$-bimodule, $L^2(M) = L^2(N) \operatorname{op}lus (\textrm{a multiple of coarse $N$,$N$-bimodule})$), so $u\in \mathscr{Z}(N)$. But then $u P u^* =\alpha(P)\subset P$, which is easily seen to be impossible by using the free product decomposition of $L^2(M)$ in terms of $L^2(N)$ and $L^2(P)$, unless $u=\tau(u)$. But this is impossible, since $\alpha(s(h)) = s(Th)$ is a non-trivial automorphism.
\end{proof}
\section{Free Krieger algebras}
Let $\nu$ be a probability measure on the torus $\mathbf{T}$. Note that $\nu$ gives rise to unital completely positive map $\eta : A \to A$, ($A = L^\infty(\mathbf{T})$), determined by
$$\eta(f)(x) = \int f(x-y) d\nu(y) = (f * \nu)(x), \forall f \in C(\mathbf{T}).$$
It is not hard to see that the von Neumann algebra $M=\Gamma(H_\mathbf{R}^\nu, \mathbf{Z},\pi^\nu)'' \operatorname{co}ng \Phi (A,\eta)$ in the notation of \cite{shlya99}, i.e., it is an example of a von Neumann algebra generated by an $A$-valued semicircular system with covariance $\eta$ (these were called ``free Krieger algebras'' in \cite{shlya99}, following the analogy between the operation $A\mapsto \Phi(A,\eta)$ and the crossed product operation $A\mapsto A\rtimes_\sigma \mathbf{Z}$).
As we have seen, $M$ has both the c.m.a.p. and the Haagerup property, and thus for this specific choice of $\eta$, $\Phi(A,\eta)$ has these properties.
We point out that in general (even for abelian $A$), $\Phi(A,\eta)$ may fail to have the Haagerup property for other choices of the completely positive maps $\eta$. It is an interesting question to determine exactly when $\Phi(A,\eta)$ has this property (and/or c.m.a.p.) as a condition on the completely-positive map $\eta : A\to A$, $A\operatorname{co}ng L^\infty[0,1]$. It is likely that the techniques of the present paper would then apply to give solidity of $\Phi(A,\eta)$.
\begin{prop}
There exists a choice of $\eta : A\to A$, $A\operatorname{co}ng L^\infty [0,1]$, so that $\Phi(A,\eta)$ does not have the Haagerup property and is not weakly amenable, i.e. $\Lambda_{\operatorname{cb}}(\Phi(A,\eta)) = \infty$.
\end{prop}
\begin{proof}
Let $\alpha$ be an action of a free group $\mathbf{F}_2$ on $A\operatorname{co}ng L^\infty[0,1]$ so that $M=A\rtimes_\alpha \mathbf{F}_2$ does not have the Haagerup property and is not weakly amenable (one could take, for example, an action measure equivalent to the action of $\operatorname{op}eratorname{SL}(2, \mathbf{Z})$ on $A=L(\mathbf{Z}^2)$; the crossed product in this case has relative property (T) and does not have the Haagerup property \cite{popa2001}. Moreover it is not weakly amenable, i.e. $\Lambda_{\operatorname{cb}}(M) = \infty$ (see \cite{dorofaeff}). Denote the two automorphisms of $A$ corresponding to the actions of the two generators of $\mathbf{F}_2$ by $\alpha_1$, $\alpha_2$, and let $\eta_j = \alpha_j + \alpha_j^{-1}$, $\eta = \eta_1+\eta_2$.
Let $\sigma$ be the free shift action of $\mathbf{Z}$ on $\mathbf{F}_\infty$. Then by \cite{shlya99},
\begin{multline}
\Phi(A,\eta) \operatorname{co}ng \Phi(A,\eta_1) *_A \Phi(A,\eta_2) \operatorname{co}ng \\
\left((A \bar{\otimes} L(\mathbf{F}_\infty)) \rtimes_{\alpha_1 \otimes \sigma} \mathbf{Z}\right)
*_A
\left((A \bar{\otimes} L(\mathbf{F}_\infty)) \rtimes_{\alpha_2\otimes \sigma} \mathbf{Z}\right) \operatorname{co}ng \\
(A \bar{\otimes} [L(\mathbf{F}_\infty)*L(\mathbf{F}_\infty)])\rtimes_{\alpha \otimes \sigma*\sigma}\mathbf{F}_2.
\end{multline}
Thus $\Phi(A,\eta)$ contains $M$ as a subalgebra. Since the Haagerup property and the weak amenability are inherited by subalgebras, it follows that $\Phi(A,\eta)$ cannot have the Haagerup property and is not weakly amenable.
\end{proof}
\end{document} |
\begin{document}
\pagestyle{plain}
\title{
Khovanov-Lipshitz-Sarkar homotopy type for links in thickened surfaces and those in $S^3$ with new modulis}
\author{Louis H. Kauffman
, Igor Mikhailovich Nikonov,
and
Eiji Ogasa}
\maketitle
\begin{abstract}
We define a family of Khovanov-Lipshitz-Sarkar stable homotopy types for the homotopical Khovanov homology of links in thickened surfaces indexed by moduli space systems. This family includes the Khovanov-Lipshitz-Sarkar stable homotopy type for the homotopical Khovanov homology of links in higher genus surfaces (see the content of the paper for the definition). The question whether different choices of moduli spaces lead to the same stable homotopy type is open.
\end{abstract}
\tableofcontents
\bigbreak
\section{Introduction}\label{intro}
\noindent
In this paper, a surface means a closed oriented surface unless otherwise stated.
Of course, a surface may or may not be the sphere.
We discuss links in thickened surfaces.
If $\mathcal L$ is a link in a thickened surface,
then a link diagram $L$ which represents $\mathcal L$ lies in the surface.
Since our theory has a special behaviour at genus one, in this paper a higher genus surface means a surface with genus greater than one unless otherwise stated.
In the previous paper~\cite{KauffmanNikonovOgasa},
the authors discussed the higher genus case.
In the present paper, we mainly discuss the torus case.
\unskip, \ignorespaces
Let $\mathcal K$ be a link in the thickened torus.
Let $K$ be a link diagram in the torus which represents $\mathcal K$.
Call a poset associated with a decorated Kauffman state, a {\it dposet}.
See \cite{KauffmanNikonovOgasa, LSk} for
decorated Kauffman states, or decorated resolution configurations.
Dposets are defined for all pairs of enhanced Kauffman states.
We discuss the following two cases, which will be introduced in \S\ref{mod}.
\unskip, \ignorespaces
The first case will be called Case (C) in \S\ref{subsect:case_C}.
We choose the right pair or the left one for the ladybug Kauffman state
(see \cite{KauffmanNikonovOgasa, LSk}).
We determine a degree 1 homology class $\lambda$ of $T^2$.
After that,
we define a cubic moduli for any dposet of $K$,
and construct
{\it Khovanov-Lipshitz-Sarkar stable homotopy type}
for the homotopical Khovanov chain complex (\cite{MN}) of $K$.
We define
{\it Khovanov-Lipshitz-Sarkar stable homotopy type}
for $\mathcal K$ to be that for $K$.
Make
the set of all Khovanov-Lipshitz-Sarkar stable homotopy types for all $\lambda$ and for a fixed choice of the right and the left. It
is a link type invariant.
There are infinitely many $\lambda$ but there are finite numbers of stable homotopy types.
\unskip, \ignorespaces
Recall in \cite{KauffmanNikonovOgasa} that in the higher genus case,
we give only one stable homotopy type for any link diagram
after we choose the right pair or the left one,
and therefore for any link type.
\unskip, \ignorespaces
The second case will be called Case (D) in \S\ref{subsect:case_D}.
We construct a different set of stable homotopy types for $K$.
We choose the right pair or the left one for the ladybug Kauffman state.
After that, we made a way to give
a set of moduli spaces for all dposets.
We do not fix a degree 1 homology class.
It is important that a moduli is not cubic.
This is a surprisingly new feature.
We choose a moduli for a dposet and construct a CW complex.
We construct no less than one CW complex for a link diagram.
The set of such all Khovanov-Lipshitz-Sarkar stable homotopy types
gives a link type invariant.
\unskip, \ignorespaces
We prove that
the set of our Khovanov-Lipshitz-Sarkar stable homotopy types
is stronger than
the homotopical Khovanov homology of $\mathcal K$ in both cases.
It is a meaningful Khovanov stable homotopy type of links in
a 3-manifold other than the 3-sphere.
In the previous paper \cite{KauffmanNikonovOgasa}
we introduced a Khovanov-Lipshitz-Sarkar stable homotopy type
for links in the thickened higher genus surface.
In this present paper we discuss
the case of links in the thickened torus.
\unskip, \ignorespaces
\begin{mth}\label{main}
$(1)$
We define
Khovanov-Lipshitz-Sarkar stable homotopy type
for
`links in the thickened torus, a degree 1 homology class $\lambda$, and
a choice of the right and the left pair'.
In this case, all modulis which we use are cube modulis.
\smallbreak\noindent $(2)$
We define
Khovanov-Lipshitz-Sarkar stable homotopy type for `links in the thickened torus,
and a choice of the right and the left pair', in a different method from $(1)$.
In this case, there is a case that uses a non-cubic moduli.
\smallbreak\noindent $(3)$
Each of the invariants (the stable homotopy type) in $(1)$ and that in $(2)$
gives an invariant stronger than
the homotopical Khovanov homology
as invariants of links in the thickened torus.
We use the second Steenrod square to prove it.
\end{mth}
\bigbreak
We give non-cubic modulis for Khovanov chain complex of classical links in $S^3$, which are not used in Lipshitz and Sarkar's construction in~\cite{LSk}. We show such a new moduli in Section~\ref{subsect:mutlivalued_moduli_system}. We use these modulis and can give a set of stable homotopy types to a link in $S^3$ as we did in Case (D) of this paper.
Although we use different modulis from Lipshitz and Sarkar's construction in~\cite{LSk}, it is an open question whether our stable homotopy types are different from that in Lipshitz and Sarkar's construction of~\cite{LSk}.
\subsection{Homotopical Khovanov homology}
Let $F$ be a closed oriented surface.
\begin{definition}\label{def:resolution_configuration}
A {\it resolution configuration} $D$ on the surface $F$ is a pair $(Z(D),A(D))$,
where $Z(D)$ is a set of pairwise-disjoint embedded circles in $F$,
and $A(D)$ is an ordered collection of disjoint arcs embedded in $F$,
with $A(D)\cap Z(D)=\partial A(D)$.
The number of arcs in $A(D)$ is the {\it index} of the resolution configuration $D$,
denoted by $\mathrm{ind}(D)$.
A {\it labeled resolution configuration} is a pair $(D, x)$ of a resolution configuration
$D$ and a labeling $x$ of each element of $Z(D)$ by either $x_+$ or $x_-$.
\end{definition}
\begin{example}
Consider a link $\mathcal L$ in the thickening $F\times[-1,1]$ of $F$. Let $L\subset F$ be a diagram of the link $\mathcal L$.
Assume that the diagram $L$ has $n$ crossings ordered somehow.
For any vector $v\in\{0,1\}^n$ one can define the {\it associated resolution configuration} $D_L(v)$
obtained by taking the resolution of the diagram $L$ corresponding to $v$
(that is, taking the 0-resolution at the $i$-th crossing
if $v_i =0$, and the 1-resolution otherwise)
and then placing arcs corresponding to each of the crossings labeled
by 0's in $v$ (that is, at the $i$-th crossing if $v_i=0$), see Fig.~\ref{fig:resolution}.
The index of the associated configuration is $\mathrm{ind}(D_L(v))=n-|v|$.
Let $\Lambda(L)$ be the set of all labeling with $x_+$ and $x_-$ of the associated resolution configurations of the link diagram $L$. The elements of this set are called {\em enhanced Kauffman states} or {\em Khovanov basis elements}.
\begin{figure}
\caption{\bf The 0- and 1-resolutions}
\label{fig:resolution}
\end{figure}
\end{example}
The set $\Lambda(L)$ of Khovanov basis elements has several grading on it.
Let $n$ (respectively, $n_+$, $n_-$) be the number of crossings (respectively, positive crossings, negative crossings) of $L$.
For a labeled resolution configurations $(D_L(u), x)\in\Lambda(L)$, its
{\em homological grading} is
\begin{equation}\label{eq:homological_grading}
\mathrm{gr}_h(D_{L(u)}, x) = -n_- + |u|,
\end{equation}
and the {\em quantum grading} is
\begin{equation}\label{eq:quantum_grading}
\mathrm{gr}_q(D_L(u), x) = n_+ - 2n_- + |u|+
\sharp\{Z\in Z(D_L(u)) | x(Z) = x_+\}
-\sharp\{Z\in Z(D_L(u)) | x(Z) = x_-\}.
\end{equation}
Let us consider the set $\mathfrak L = [S^1; F]$
of all the homotopy classes of free oriented loops in $F$.
Let $\bigcirc\in \mathfrak L$ be the homotopy class of contractible loops.
For any closed curve $\gamma$,
one can consider the curve $-\gamma$ obtained from $\gamma$
by the orientation change.
Let $\mathfrak H$ be the quotient group of the free abelian group
with generator set $\mathfrak L$ modulo the relations
$\bigcirc= 0$ and $[\gamma]=[-\gamma]$ for all free loops $\gamma$.
Define the {\em homotopical grading} of the Khovanov basis element $(D_L(u),x)$ as follows
\begin{equation}\label{eq:homotopical_grading}
\mathrm{gr}_{\mathfrak H}(D_L(u), x)=
\sum_{Z\in Z(D_L(u))}
\deg x(Z)\cdot[Z] \in\mathfrak H,
\end{equation}
\noindent
where $\deg(x_\pm)=\pm 1$.
\begin{definition}\label{def:surgery}
Given a resolution configuration $D$ and a subset $A'\subseteq A(D)$
there is a new resolution configuration $s_{A'}(D)$, the
{\em surgery of $D$ along $A'$}, obtained as follows.
The circles
$Z(s_{A'}(D))$ of $s_{A'}(D)$ are obtained by performing embedded
surgery along the arcs in $A'$; in other words, $Z(s_{A'}(D))$ is
obtained by deleting a neighborhood of $ A'$ from $Z(D)$ and
then connecting the endpoints of the result using parallel
translates of $A'$.
The arcs of $s_{A'}(D)$ are the arcs of $D$ not
in $A'$, i.e., $A(s_{A'}(D))=A(D)- A'$.
Let $s(D)=s_{A(D)}(D)$ denote the maximal surgery on $D$.
\end{definition}
\begin{definition}\label{2.10}
There is a partial order $\prec$ on labeled resolution configurations defined as
follows.
We declare that $(E, y)\prec(D, x)$ if:
\begin{enumerate}
\item
$D$ is obtained from $E$ by surgering along a single arc of $A(E)$
\item
The labelings $x$ and $y$ induce the same labeling on $D\cap E = E\cap D$.
\item
$\mathrm{gr}_q(E, y)=\mathrm{gr}_q(D, x)$, $\mathrm{gr}_{\mathfrak H}(E, y)=\mathrm{gr}_{\mathfrak H}(D, x)$.
\end{enumerate}
The possible cases of the order are drawn in Fig.~\ref{resolC} and~\ref{resolNC}.
\begin{figure}
\caption{{\bf
The partial order of labeled resolution configurations with contractible circles
}
\label{resolC}
\end{figure}
\begin{figure}
\caption{{\bf
The partial order of labeled resolution configurations with non-contractible circles.
Non-contractible circles are marked with (H).
}
\label{resolNC}
\end{figure}
Now, we close the order $\prec$ by transitivity.
\end{definition}
\begin{definition}\label{korekos}
Given an oriented link diagram $L$ with $n$ crossings and an ordering of the crossings in $L$,
the {\em Khovanov chain} complex $KC(L)$ is defined as the $\mathbb{Z}$-module freely generated
by labeled resolution configurations of the form $(D_L(u), x)$ for $u\in\{0, 1\}^n$.
Thus, the set of all labeled resolution configurations of $L$ is
a basis of $KC(L)$.
The {\em Khovanov differential} preserves the quantum grading and the homotopical grading,
increases the homological grading by 1, and is defined as
\begin{equation}\label{bibun}
{\displaystyle
\delta(D_L(v),y)
=
\sum_{(D_L(u),x)\succ (D_L(v),y)\colon |u|=|v|+1}
(-1)^{s_0(\mathcal C_{u,v})
}}(D_L(u),x),
\end{equation}
\noindent
where for $u= (\epsilon_1,..., \epsilon_{i-1}, 1, \epsilon_{i+1}, . . . , \epsilon_n)$ and
$v=(\epsilon_1,..., \epsilon_{i-1}, 0,\epsilon_{i+1}, . . . , \epsilon_n)$,
one defines $s_0(C_{u,v}) = \epsilon_1+\cdot\cdot\cdot+ \epsilon_{i-1}$.
The homology $KH(L)$ of the complex $(KC(L),\delta)$ are the {\em Khovanov homology} of the link $L$.
\end{definition}
\begin{definition}\label{def:decorated_resolution_configuration}
A {\em decorated resolution configuration} is a triple $(D, x, y)$
where $D$ is a resolution configuration and
$x$ (respectively, $y$) is a labeling of each component of $Z(s(D))$
(respectively, $Z(D)$) by an element of $\{x_+, x_-\}$. The labeled resolution configuration $i=(D,y)$ is the {\em initial configuration} of the decorated resolution configuration, and the labeled resolution configuration $f=(s(D),x)$ is the {\em final configuration}.
Associated to a decorated resolution configuration $(D, x, y)$ is the poset $P(D, x, y)$
consisting of all labeled resolution configurations $(E, z)$ with $(D, y)\prec(E, z)\prec(s(D), x)$.
We call $P(D, x, y)$ the poset for $(D, x, y)$.
For any resolution configuration $D'=s_A(D)$, $A\subset A(D)$, we define its {\em multiplicity} as the number of its labelings which belong to $P(D,x,y)$:
$$
\mu_{(D, x, y)}(D')=\sharp\{z\,|\, (D, y)\prec(D', z)\prec(s(D), x)\}.
$$
The {\em multiplicity} $\mu(D, x, y)$ of a decorated resolution configuration $(D, x, y)$ is the maximum of the multiplicities $\mu_{(D, x, y)}(D')$:
$$
\mu(D, x, y)=\max_{A\subset A(D)}\mu_{(D, x, y)}(s_A(D)).
$$
\end{definition}
\begin{definition}\label{def:core_configuration}
The {\it core} $c(D)$ of a resolution configuration $D$ is the resolution configuration
obtained from $D$ by deleting all the circles in $Z(D)$ that are disjoint from all the arcs in $A(D)$.
A resolution configuration $D$ is called {\em basic} if $D = c(D)$, that is, if every circle in $Z(D)$ intersects an arc in $A(D)$.
In the same way one can define the core $c(D, x)$ of a labeled resolution configuration $(D,x)$, and basic labeled resolution configurations.
The core of a decorated resolution configuration $(D,x,y)$ is the decorated configuration
$$c(D,x,y)=(c(D),x\mid_{s(c(D))},y\mid_{c(D)}).$$
A decorated resolution configuration is basic if it coincides with its core.
\end{definition}
\begin{remark}
Given two comparable labeled resolution configurations $\alpha=(D,y)\prec (D',x)=\beta$, $D'=s_A(D)$, $A\subset A(D)$, one can assign a basic decorated resolution configuration ${\mathcal D}(\alpha,\beta)$ to it. Consider two resolution configurations $\bar D=(Z(D),A)$, $\bar D'=s(\bar D)=(Z(D'),\emptyset)$. Then the decorated configuration ${\mathcal D}(\alpha,\beta)$ is defined as the core of $(\bar D, x, y)$.
If $(D,y)\not\prec (D',x)$ we say that the corresponding decorated resolution configuration is empty.
\end{remark}
\begin{remark}
(Basic) decorated resolution configurations form a partially ordered set by inclusion relation: $(D',x',y')\subset (D,x,y)$ if
$(D',y')$ and $(s(D'),x')$ belong the poset $P(D,x,y)$.
\end{remark}
\subsection{Khovanov homotopy type}
Let us remind the construction of Khovanov homotopy type by R. Lipshitz and S. Sarkar~\cite{LSk}.
\begin{definition}\label{def:manifold_with_corners}
A {\em $k$-dimensional manifold with corners} is a topological space $X$ which is locally homeomorphic to an open subset of $\mathbb R^k_+=(\mathbb R_+)^k$ where $\mathbb R_+=[0,\infty)$.
For $x\in X$, let $c(x)$ be the number of zero coordinates of the corresponding point in $\mathbb R^k_+$. The set $\{x\in X\,|\, c(x)=i\}$ is the {\em codimension-$i$ boundary} of $X$.
A {\em connected facet} of $X$ is the closure of a connected component of the codimension-$1$ boundary of $X$. A {\em facet} is a union of disjoint connected facets.
\end{definition}
\begin{definition}\label{def:n_manifold}
A manifold with corners $X$ is called a {\em manifold with facets} if every point $x\in X$ belongs to exactly $c(x)$ connected facets. An $\langle n\rangle$-manifold is a manifold with facets $X$ along with an ordered $n$-tuple $(\partial_1X,\dots,\partial_n X)$ of facets of $X$ such that
\begin{itemize}
\item $\bigcup_{i=1}^n \partial_i X=\partial X$;
\item for all distinct $i,j$ the intersection $\partial_i X\cap \partial_j X$ is a facet of both $\partial_i X$ and $\partial_j X$.
\end{itemize}
\end{definition}
For any $A\subset \{1,\dots,n\}$ denote $X(A)=\bigcap_{i\in A}\partial_i X$.
\begin{definition}\label{def:neat_embedding}
Given a $(n+1)$-tuple ${\mathbf d}=(d_0,\dots,d_n)\in\mathbb N^{n+1}$, let
$$
\mathbb E^{\mathbf d}_n=\mathbb R^{d_0}\times\mathbb R_+\times\mathbb R^{d_1}\times\mathbb R_+\times\dots\times\mathbb R_+\times\mathbb R^{d_n}.
$$
$\mathbb E^{\mathbf d}_n$ is a $\langle n\rangle$-manifold with
$$
\partial_i(\mathbb E^{\mathbf d}_n)=\mathbb R^{d_0}\times\dots\times\mathbb R^{d_{i-1}}\times\{0\}\times\dots\times\mathbb R^{d_n}.
$$
A {\em neat immersion} of an $\langle n\rangle$-manifold is a smooth immersion
$\iota\colon X\looparrowright\mathbb E_n^{\mathbf d}$ for some $\mathbf d$ such that:
\begin{enumerate}
\item $\iota^{-1}(\partial_i(\mathbb E^{\mathbf d}_n))=\partial_i X$ for all $i$,\unskip, \ignorespaces
\item for any $A\subset B\subset\{1,\dots,n\}$ the sets $\iota(X(A))$ and $\mathbb E^{\mathbf d}_n(B)$ are transversal.
\end{enumerate}
A {\em neat embeding} is a neat immersion that is also an embedding.
\end{definition}
\begin{definition}\label{def:flow_category}
A {\em flow category} is a pair $(\mathscr C, \mathrm{gr})$ where $\mathscr C$ is a category with finitely many objects $Ob(\mathscr C)$ and $\mathrm{gr}\colon Ob(\mathscr C)\to\mathbb{Z}$ is a function, satisfying the following conditions:
\begin{enumerate}
\item $\mathrm{Hom}(x,x)={id}$ for all $x\in Ob(\mathscr C)$, and for distinct $x,y\in Ob(\mathscr C)$, $\mathrm{Hom}(x,y)$ is a compact $(\mathrm{gr}(x)-\mathrm{gr}(y)-1)$-dimensional $\langle \mathrm{gr}(x)-\mathrm{gr}(y)-1\rangle$-manifold;
\item for distinct $x,y,z\in Ob(\mathscr C)$ with $\mathrm{gr}(z)-\mathrm{gr}(y)=m$ the composition map
$$
\circ\colon \mathrm{Hom}(z,y)\times\mathrm{Hom}(x,z)\to\mathrm{Hom}(x,y)
$$
is an embedding into $\partial_m\mathrm{Hom}(x,y)$. Furthermore,
$$
\circ^{-1}(\partial_i\mathrm{Hom}(x,y))=\left\{\begin{array}{cl}
\partial_i\mathrm{Hom}(z,y)\times\mathrm{Hom}(x,z) & \mbox{for } i<m,\unskip, \ignorespaces
\mathrm{Hom}(z,y)\times\partial_{i-m}\mathrm{Hom}(x,z) & \mbox{for } i>m.\end{array}\right.
$$
\item for distinct $x,y\in Ob(\mathscr C)$ the composition induces a diffeomorphism
$$
\partial_i\mathrm{Hom}(x,y)\cong\bigsqcup_{z\in Ob(\mathscr C)\colon \mathrm{gr}(z)=\mathrm{gr}(y)+i}\mathrm{Hom}(z,y)\times\mathrm{Hom}(x,z).
$$
\end{enumerate}
\end{definition}
For any objects $x,y$ in a flow category define the {\em moduli space} from $x$ to $y$ to be
$$
\mathcal{M}(x,y)=\left\{\begin{array}{cl}
\emptyset & \mbox{if } x=y,\unskip, \ignorespaces
\mathrm{Hom}(x,y) & \mbox{otherwise}.\end{array}\right.
$$
Let ${\mathbf d}=(\dots, d_{-1},d_0,d_1,\dots)$ be a sequence of natural numbers. For any $a<b$ denote $\mathbb E_{\mathbf d}[a:b]=\mathbb E_{b-a-1}^{d_a,\dots,d_{b-1}}$.
\begin{definition}
A neat immersion (embedding) of a flow category $\mathscr C$ is a collection of neat immersions (embeddings) $\iota_{x,y}\colon \mathcal{M}(x,y)\looparrowright
\mathbb E_{\mathbf d}[\mathrm{gr}(y):\mathrm{gr}(x)]$ such that
\begin{enumerate}
\item for all $i,j$ the map
$$\iota_{i,j}=\sqcup_{x,y}\iota_{x,y}\colon
\bigsqcup_{x,y\in Ob(\mathscr C)\colon \mathrm{gr}(x)=i,\mathrm{gr}(y)=j}\mathcal{M}(x,y)\to \mathbb E_{\mathbf d}[j:i]
$$
is a neat immersion (embedding);
\item for all objects $x,y,z$ and all points $p\in\mathcal{M}(x,z)$, $q\in\mathcal{M}(z,y)$
$$
\iota_{x,y}(q\circ p)=(\iota_{z,y}(q),0,\iota_{x,z}(p)).
$$
\end{enumerate}
\end{definition}
\begin{definition}\label{def:framed_flow_category}
Let $\iota$ be a neat immersion of a flow category $\mathscr C$. For objects $x,y$, let $\nu_{x,y}$ be the normal bundle on the moduli space $\mathcal{M}(x,y)$, induced by the immersion $\iota_{x,y}$. A {\em coherent framing} $\phi$ of the normal bundle is a framing for $\nu_{x,y}$ for all objects $x,y$ such that the product framing $\nu_{z,y}\times\nu_{x,z}$ equals to the pullback $\circ^*(\nu_{x,y})$ for all $x,y,z$.
A flow category with a fixed coherent framing of the normal bundle to some neat immersion is called a {\em framed flow category}.
\end{definition}
For a framed flow category there is an associated cochain complex $C^*(\mathscr C)$. The chain space of the complex is the free abelian group generated by the objects of the category: $C^*(\mathscr C)=\mathbb{Z}[Ob(\mathscr C)]$; the differential is given by the formula
$$
\delta y=\sum_{x\in Ob(\mathscr C)\colon \mathrm{gr}(x)=\mathrm{gr}(y)+1} \left(\sum_{f\in\mathcal{M}(x,y)}\phi(f)\right)x.
$$
The moduli space in the formula is a compact zero-dimensional manifold, i.e. a finite set, and the framing $\phi$ is given by signs of the elements of that set.
To a framed flow category one can associate a based CW complex in the following way.
\begin{definition}\label{def:flow_category_realization}
Let $\mathscr C$ be a framed flow categoty with a neat embedding $\iota$ into $\mathbb E_{\mathbf d}$ and a framing $\phi$.
Let $B=\min_{x\in Ob(\mathscr C)} \mathrm{gr}(x)$ and $B=\max_{x\in Ob(\mathscr C)} \mathrm{gr}(x)$.. Using framing, extend the embedding $\iota_{x,y}$ for some small $\epsilon>0$ to an embedding
$$
\tilde\iota_{x,y}\colon \mathcal{M}(x,y)\times [-\epsilon,\epsilon]^{d_{\mathrm{gr}(y)}+\cdots+d_{\mathrm{gr}(x)-1}}\to \mathbb E_{\mathbf d}[\mathrm{gr}(y):\mathrm{gr}(x)]
$$
where $\mathbb E_{\mathbf d}[\mathrm{gr}(y):\mathrm{gr}(x)]=
\mathbb R^{d_{\mathrm{gr}(y)}}\times\mathbb R_+\times\dots\times\mathbb R_+\times\mathbb R^{d_{\mathrm{gr}(x)-1}}.$
Choose $R$ sufficiently large so that for all $x,y$
$$\tilde\iota_{x,y}\colon \mathcal{M}(x,y)\times [-\epsilon,\epsilon]^{d_{\mathrm{gr}(y)}+\cdots+d_{\mathrm{gr}(x)-1}}$$
lies in $[-R,R]^{d_{\mathrm{gr}(y)}}\times[0,R]\times\dots\times[0,R]\times[-R,R]^{d_{\mathrm{gr}(x)-1}}$.
To any object $x$ assign the cell
\begin{multline*}
C(x)=[0,R]\times[-R,R]^{d_{B}}\times\dots\times[0,R]\times[-R,R]^{d_{\mathrm{gr}(x)-1}}\times\{0\}\times\unskip, \ignorespaces
[-\epsilon,\epsilon]^{d_{\mathrm{gr}(x)}}\times\dots\times\{0\}\times[-R,R]^{d_{A-1}}.
\end{multline*}
For any other object $y$ such that $\mathrm{gr}(y)<\mathrm{gr}(x)$ one identifies $C(y)\times\mathcal{M}(x,y)$ with the subset
\begin{multline*}
C_y(x)=[0,R]\times[-R,R]^{d_{B}}\times\dots\times[0,R]\times[-R,R]^{d_{\mathrm{gr}(y)-1}}\times\{0\}\times\unskip, \ignorespaces
\iota_{x,y}(\mathcal{M}(x,y))\times\{0\}\times[-\epsilon,\epsilon]^{d_{\mathrm{gr}(x)}}\times\dots\times\{0\}\times[-\epsilon,\epsilon]^{d_A-1}.
\end{multline*}
Define the attaching map for $C(x)$ as the map which is the projection $C_y(x)\cong C(y)\times\mathcal{M}(x,y)$ to $C(y)$ on $C_y(x)$ and the map to the basepoint on the $\partial C(x)\setminus \bigcup_y C_y(x)$.
These gluing maps define a CW complex $|\mathscr C|$ which is the {\em Cohen--Jones--Segal realization} of the framed flow category $\mathscr C$.
\end{definition}
\begin{theorem}[\cite{LSk}]
The CW complex $|\mathscr C|$ is well defined and its cellular cochain complex is isomorphic to the associated cochain complex $C^*(\mathscr C)$.
\end{theorem}
\begin{example}[Cube flow category]
Let $X=[0,1]^n$ be the $n$-dimensional cube and $f_n(x_1,\dots,x_n)=f(x_1)+\cdots+f(x_n)$, where $f(x)=3x^2-2x^3$, be a Morse function on it.
Define the {\em $n$-dimensional cube flow category} $\mathscr C_C(n)$ as the Morse flow category of the function $f_n$. This means that the objects of $\mathscr C_C(n)$ are the critical points of $f_n$, i.e. the vertices $\{0,1\}^n$ of the cube $X$.
Denote the object $(0,\dots,0)$ by $\bar 0$, and the object $(1,\dots, 1)$ by $\bar 1$.
The grading function is defined as $\mathrm{gr}(u)=|u|=\sum_{i=1}^n u_i$, $u=(u_1,\dots,u_n)\in\{0,1\}^n$. The moduli space $\mathcal{M}(x,y)$ consists of the lines of the gradient flow which starts at $x$ and ends at $y$. One can identify the moduli space $\mathcal{M}(x,y)$ with the permutahedron of dimension $\mathrm{gr}(x)-\mathrm{gr}(y)-1$.
The cube flow category can be framed (by induction on the moduli spaces dimension).
\end{example}
\begin{example}[Khovanov flow category]
Let $\mathcal L$ in $S^3$ be a link and $L$ in $S^2$ be its diagram.
The {\it Khovanov flow category} $\mathscr C_K(L)$ has one object
for each
Khovanov basis element.
That is, an object of $\mathscr C_K(L)$ is a labeled resolution configuration of
the form $\mathbf{x}=(D_L(u), x)$ with $u\in\{0, 1\}^n$.
The grading on the objects is
the homological grading gr$_h$;
the quantum grading gr$_q$ is an additional grading on the objects.
We need the orientation of $L$ in order to define these
gradings, but the rest of the construction of $\mathscr C_K(L)$ is independent of the orientation.
Consider objects $\mathbf{x}=(D_L(u), x)$ and
$\mathbf{y}=(D_L(v), y)$ of $\mathscr C_K(L)$.
The space $\mathcal M_{\mathscr C_K(L)}(\mathbf{x},\mathbf{y})$
is defined to be empty unless $y\prec x$
with respect to the partial order from Definition \ref{2.10}.
So,
assume that $y\prec x$.
Let $x|$ denote the restriction of $x$ to
$s(D_L(v)-D_L(u))=D_L(u)-D_L(v)$
and
let $y|$ denote the restriction of $y$ to $D_L(v)-D_L(u)$.
Therefore, $(D_L(v)-D_L(u), x|, y|)$ is
a basic decorated resolution configuration.
In \cite[\S5 and \S6]{LSk} Lipshitz and Sarkar associate to each index $n$ basic decorated resolution configuration
$(D, x, y)$ an $(n-1)$-dimensional $\left<n-1\right>$-manifold $\mathcal M(D, x, y)$
together with a $\mu(D,x,y)$-fold trivial covering
$$\mathcal F :\mathcal M(D, x, y) \to\mathcal M_{\mathscr C(n)}(\overline1, \overline0).$$
Use it, and define
$$\mathcal{M}_{\mathscr C_K(L)}(\mathbf{x},\mathbf{y})=\mathcal{M}(D_L(v)-D_L(u),x|, y|).$$
The framing of the cube flow category can be lifted to the Khovanov category.
\end{example}
\begin{definition}
The Cohen--Jones--Segal realization ${\mathcal X}(L)=|\mathscr C_K(L)|$ of the Khovanov framed flow category
is called the {\em Khovanov-Lipshitz-Sarkar stable homotopy type}
of the link diagram $L$.
\end{definition}
\begin{theorem}[\cite{LSk}]\label{thm:stable_khovanov_homology}
The Khovanov-Lipshitz-Sarkar stable homotopy type ${\mathcal X}(L)$ is a link invariant.
\end{theorem}
\section{Moduli systems}
Let $F$ be a closed oriented surface. We consider links in the thickening of the surface $F$.
In order to define Khovanov homotopy type of such links we need fix a set of moduli spaces for the decorated resolution configurations of the link. This observation leads to the following definition, cf.~\cite[Section 5.1]{LSk}.
\begin{definition}\label{def:moduli_system}
A {\em branched covering moduli system} on the surface $F$ is a family of correspondences between the basic decorated resolution configurations of index $n$ in the surface $F$ and $(n-1)$-dimensional $\langle n-1\rangle$-manifolds:
$$\mathcal{M}\colon (D,x,y) \mapsto \mathcal{M}(D, x, y)$$
together with $\langle n-1\rangle$-maps
$$
\mathcal F_{(D, x, y)}\colon\mathcal{M}(D, x, y)\to\mathcal{M}_{\mathcal{C}_C(n)}(\bar 1,\bar 0)
$$
These correspondences must obey the following conditions:
\begin{enumerate}
\item the moduli space $\mathcal{M}(D, x, y)$ of a basic decorated resolution configuration $(D, x, y)$ depends only on the isotopy class of the configuration;
\item $\mathcal{M}(D, x, y)=\emptyset$ if $P(D,x,y)=\emptyset$;
\item for any $(E,z)\in P(D,x,y)$ there are embeddings
$$ \circ\colon \mathcal{M}(D\setminus E, z|, y|)\times \mathcal{M}(E\setminus s(D), x|, z|)\to \mathcal{M}(D, x, y);$$
\item the faces of $\mathcal{M}(D, x, y)$ are determined by
$$
\partial_i\mathcal{M}(D, x, y)=
\coprod_{(E,z)\in P(D,x,y),\ ind(D\setminus E)=i} \circ(\mathcal{M}(D\setminus E, z|, y|)\times \mathcal{M}(E\setminus s(D), x|, z|));
$$
\item the composition is compatible with the maps $\mathcal F$: for any $E=D_D(v)$
\begin{equation*}\label{eq:moduli_space_equivarity}
\timesymatrix{
\mathcal{M}(D\setminus E,\gen{z}|,\gen{y}|)\times
\mathcal{M}(E\setminus s(D),\gen{x}|,\gen{z}|)
\ar[r]^-\circ\ar[d]_{{\mathcal F}\times{\mathcal F}} &
\mathcal{M}(D,\gen{x},\gen{y})\ar[dd]^{{\mathcal F}}\unskip, \ignorespaces
\mathcal{M}_{\mathcal{C}FlowCat(n-m)}(\overline{1},\overline{0})\times
\mathcal{M}_{\mathcal{C}FlowCat(m)}(\overline{1},\overline{0})
\ar[d]_{}&
\unskip, \ignorespaces
\mathcal{M}_{\mathcal{C}FlowCat(n)}({v},\overline{0}) \times
\mathcal{M}_{\mathcal{C}FlowCat(n)}(\overline{1},{v})\ar[r]^-\circ &
\mathcal{M}_{\mathcal{C}FlowCat(n)}(\overline{1},\overline{0}).
}
\end{equation*}
\item the map $\mathcal F_{(D, x, y)}$ is a $\mu(D,x,y)$-fold branched covering.
\end{enumerate}
\end{definition}
Note that we can define the covering by pointing out the branching set $P\subset \mathcal{M}_{\mathcal{C}_C(n)}(\bar 1,\bar 0)$ of codimension 2.
\begin{definition}\label{def:cases_C_and_D}
The moduli system $\{\mathcal{M}(D, x, y)\}$ is {\em cubical} or {\em of type C} if all the covering maps over the cubic moduli spaces are trivial. Otherwise, the moduli system is called {\em dodecagonal} or {\em of type D}.
\end{definition}
\begin{theorem}\label{thm:moduli_system_existence}
\begin{enumerate}
\item For any closed oriented surface $F$ there exists a covering moduli system of type $C$.
\item For any closed oriented surface $F$ there exists a covering moduli system of type $D$.
\end{enumerate}
\end{theorem}
We present the corresponding moduli systems in Sections~\ref{subsect:case_C} and~\ref{subsect:case_D}
Given a (branched covering) moduli system $\mathcal{M}=\{\mathcal{M}(D, x, y)\}$, let $\mathscr C_\mathcal{M}(L)$ be the Khovanov flow category whose objects are labeled resolution configurations and moduli spaces are
$$\mathcal{M}((D_L(u),x),(D_L(v),y))=\mathcal{M}(D_L(v)-D_L(u),x|, y|).$$
\begin{proposition}
There is a structure of framed flow category on $\mathscr C_\mathcal{M}(L)$.
\end{proposition}
\begin{proof}
Let $\iota_C$ be a neat embedding of the cube flow category $\mathcal{C}_C$ with a coherent framing $\phi_C$. Then $\iota_0=\iota_C\circ\mathcal F$ is a neat map into some $\mathbb E_{\mathbf d}[a:b]$. Although the map $\iota_0$ is not a neat immersion in general, the normal bundle to the image $\iota_0(\mathscr C_\mathcal{M}(L))=\iota_C(\mathcal{C}_C)$ is well-defined and framed (see Fig.~\ref{fig:normal_frame_lift}).
\begin{figure}
\caption{{\bf Lifting of a normal framing to a branch cover
}
\label{fig:normal_frame_lift}
\end{figure}
By Lemma 3.16 of~\cite{LSk} there exists a neat embedding $\iota_1$ of the flow category $\mathscr C_\mathcal{M}(L)$ into $\mathbb E_{\mathbf d'}[a:b]$.
By Lemma 3.17 of~\cite{LSk} there exists a family of neat maps $\tilde\iota_t$ connecting the neat maps $\iota_0[\mathbf{d}+\mathbf{d'}+1]$ and $\iota_1[\mathbf{d}+\mathbf{d'}+1]$ such that for any $t>0$ the map $\tilde\iota_t$ is a neat embedding. This family of maps admits an explicit formula
\begin{gather*}
\tilde\iota_t\colon \mathscr C_\mathcal{M}(L) \to \mathbb E_{\mathbf{d}+\mathbf{d'}+1}[a:b]=\mathbb R^{d_a+d'_a+1}\times\mathbb R_+\times\mathbb R^{d_{a+1}+d'_{a+1}+1}\times\mathbb R_+\times\dots\times\mathbb R_+\times\mathbb R^{d_{b-1}+d'_{b-1}+1},\unskip, \ignorespaces
\tilde\iota_t=((1-t)\iota_0^{a}+t\iota_1^a,f(t)\iota_1^a,f(t)\bar\iota_1^{a+1},(1-t)\bar\iota_0^{a+1}+t\bar\iota_1^{a+1}, (1-t)\iota_0^{a+1}+t\iota_1^{a+1},\dots,
(1-t)\iota_0^{b-1}+t\iota_1^{b-1},f(t)\iota_1^{b-1},0)
\end{gather*}
where $f(t)=e^{-\frac 1{t(1-t)}}$ and
$$
\iota_0=(\iota_0^a,\bar\iota_0^{a+1},\iota_0^{a+1},\dots,\bar\iota_0^{b-1},\iota_0^{b-1})\colon \mathscr C_\mathcal{M}(L) \to \mathbb R^{d_a}\times\mathbb R_+\times\mathbb R^{d_{a+1}}\times\dots\times\mathbb R_+\times\mathbb R^{d_{b-1}}.
$$
By Lemma~3.19 of~\cite{LSk} we can extend the coherent framing $\phi$ for the map $\tilde\iota_0=\iota_0[\mathbf{d'}+\mathbf{d''}]$ to a family of coherent framings $\phi_t$ for the maps $\tilde\iota_t$. In particular, the neat embedding $\tilde\iota_1=\iota_1[\mathbf{d}+\mathbf{d''}]$ admits the coherent framing $\phi_1$.
\end{proof}
Define the Khovanov homotopy type ${\mathcal X}_\mathcal{M} (L)$ associated with the moduli system $\mathcal{M}$ as the realization of the framed flow category $\mathscr C_\mathcal{M}(L)$.
\begin{theorem}\label{thm:moduli_system_invariance}
Khovanov homotopy type ${\mathcal X}_\mathcal{M} (L)$ is a link invariant.
\end{theorem}
\begin{proof}
We can use the reasoning of~\cite[Propositions 6.2, 6.3, 6.4]{LSk} without any changes.
Indeed, let the diagram $L'$ differ from $L$ by an increasing first Reidemeister move, Fig.~\ref{fig:reidemeister1}.
\begin{figure}
\caption{{\bf First Reidemeister move
}
\label{fig:reidemeister1}
\end{figure}
The Khovanov complex of the diagram $L'$ contains a contractible subcomplex $C_1$, Fig.~\ref{fig:reidemeister1_complex} left. The quotient complex $C_2$ (Fig.~\ref{fig:reidemeister1_complex} right) can be identified with the Khovanov complex of $L$. On the level of flow category this means that the Khovanov flow category $\mathscr C_K(L)$ contains a closed subcategory $\mathscr C_1$ which corresponds to $C_1$ and a subcategory $\mathscr C_2$ that corresponds to $C_2$. The subcategory $\mathscr C_2$ is isomorphic to the category $\mathscr C_K(L)$. The geometric realization of $\mathscr C_1$ is contractible, hence, ${\mathcal X}_\mathcal{M} (L')=|\mathscr C_K(L')|=|\mathscr C_1|\vee |\mathscr C_2|=|\mathscr C_2|=|\mathscr C_K(L)|={\mathcal X}_\mathcal{M} (L)$.
\begin{figure}
\caption{{\bf Parts of Khovanov complex of the diagram $L'$
}
\label{fig:reidemeister1_complex}
\end{figure}
Analogously, one establishes the invariance under the second and third Reidemeister moves.
\end{proof}
\subsection{Multivalued moduli systems}
\begin{definition}
A {\em multivalued moduli system} is a correspondence between basic decorated resolution configurations of index $n$ and sets of $(n-1)$-dimensional $\langle n-1\rangle$-manifolds ${\mathfrak M}(D, x, y)$. We reformulate the boundary condition as follows: for any $\mathcal{M} \in {\frak M}(D, x, y)$
$$
\partial_i\mathcal{M}=\coprod_{(E,z)\in P(D,x,y),\ ind(D\setminus E)=i} \circ(\mathcal{M}_{D\setminus E}\times \mathcal{M}_{E\setminus s(D)}),
$$
for some $\mathcal{M}_{D\setminus E}\in{\mathfrak M}(D\setminus E, z|, y|)$, $\mathcal{M}_{E\setminus s(D)}\in{\mathfrak M}(E\setminus s(D), x|, z|)$.
A multivalued moduli system is {\em extendable} if for any decorated resolution configuration $(D',z,w)$ and any choice of moduli spaces $M(D,x,y)\in {\mathfrak M}(D, x, y)$ for all the proper decorated resolution sub-configurations $(D,x,y)\subsetneq(D',z,w)$ which satisfies the conditions (2)-(6) of Definition~\ref{def:moduli_system}, there exists $M(D',z,w)\in {\mathfrak M}(D', z, w)$ such that
\[
\partial_i M(D',z,w)=\coprod_{(E,x)\in P(D',z,w),\ ind(D'\setminus E)=i} \circ(M{(D'\setminus E,x,w)}\times M{(E\setminus s(D'),z,x)}),
\].
\end{definition}
In other words, any compatible choice of moduli spaces of the proper decorated resolution sub-configurations can be extended to a compatible choice of a moduli space for the whole decorated resolution configuration.
\begin{theorem}\label{thm:multivalued_moduli_system}
There exists an extendable multi-valued moduli system.
\end{theorem}
Note that since any branched covering moduli system is an example of a multivalued moduli system with moduli sets ${\mathfrak M}(D, x, y)$ consisting at most of one moduli space, Theorem~\ref{thm:moduli_system_existence} implies Theorem~\ref{thm:multivalued_moduli_system}. In particular, the moduli systems of types (C) and (D) constructed in Sections~\ref{subsect:case_C} and~\ref{subsect:case_D} are multivalued moduli systems. In Section~\ref{subsect:mutlivalued_moduli_system} we present a multivalued moduli systems which assigns several spaces to decorated resolution configurations.
Using a multi-valued moduli system, one can construct a framed flow category by choosing moduli spaces $\mathcal{M}_{D}\in{\mathfrak M}(D, x, y)$ for each decorated resolution configuration. The geometric realization of this flow category is a topological space which can be thought of as a value of Khovanov homotopy type. By considering all possible choices we get a {\em multivalued Khovanov homotopy type} ${\mathfrak X}_{\mathfrak M}(L)$ which is a set of CW complexes considered up to stable homotopy.
\begin{theorem}
Let $\mathfrak M$ be an extendable multivalued moduli system. Then the multivalued Khovanov homotopy type ${\mathfrak X}_{\mathfrak M}(L)$ is a link invariant.
\end{theorem}
\begin{proof}
We need to prove that the set ${\mathfrak X}_{\mathfrak M}(L)$ is invariant under the Reidemeister moves. Let $L$ and $L'$ be two diagrams connected by a Reidemeister move. Take a homotopy type $\mathcal X\in {\mathfrak X}_{\mathfrak M}(L)$. We must find a homotopy type $\mathcal X'\in {\mathfrak X}_{\mathfrak M}(L')$ which is stable homotopic to $\mathcal X$. If the move is an increasing first or second Reidemeister move then the moduli spaces of a Khovanov flow category $\mathscr C_K(L)$ which produces the CW complex $\mathcal X$ can be extended to some moduli spaces for the decorated resolution configurations in a Khovanov flow category $\mathscr C_K(L')$. We can take these moduli spaces and construct a Khovanov homotopy type $\mathcal X'$. Then the flow category $\mathscr C_K(L)$ is a subcategory in the flow category $\mathscr C_K(L')$, and $\mathcal X$ is embedded in $\mathcal X'$ (or some factor-space of $\mathcal X'$) This map induces isomorphism in cohomology. Hence, by Whitehead theorem $\mathcal X$ homotopic to $\mathcal X'$.
For the third Reidemeister move, we consider its braid-like version (see Fig.~\ref{fig:braid}). The left diagram $L$ can be obtained by smoothing the right diagram $L'$. Hence, the resolution cube for $L$ is a subcube in the resolution cube for $L'$. Then we can use the reasonings above to construct a map between the stable homotopy types or the diagrams which will be a homotopy equivalence by Whitehead theorem.
\begin{figure}
\caption{{\bf The braid-like Reidemeister move $III$}
\label{fig:braid}
\end{figure}
Thus, for any homotopy type $\mathcal X\in {\mathfrak X}_{\mathfrak M}(L)$ we can extend the corresponding framed flow category to a framed flow category of $L'$ and get a homotopy type $\mathcal X'\in {\mathfrak X}_{\mathfrak M}(L')$ such that $\mathcal X\simeq \mathcal X'$. On the other hand, for a homotopy type $\mathcal X'\in {\mathfrak X}_{\mathfrak M}(L')$ the restriction of its framed flow category to the resolution configurations of the link $L$ gives a homotopy type $\mathcal X\in {\mathfrak X}_{\mathfrak M}(L)$ such that $\mathcal X\simeq \mathcal X'$.
\end{proof}
\section{Decorated resolution configurations}\label{sect:decorated}
Let $F$ be a closed oriented surface. Let us describe decorated resolution configurations of link diagrams in the surface $F$ in more details.
Let $\mathcal D = (D,x,y)$ be a decorated resolution configuration in $F$. It is a trivalent graph $F$ consisting of cycles and arcs between the cycles.
Recall that $P(D,x,y)$ is the partially ordered set consisting of the labeled resolution configurations between $(D,y)$ and $(s(D),x)$.
Let $A=A(D)$ be the set of arcs. Assuming the arcs are ordered, there is a map $\pi\colon P(D,x,y)\to C_n$, where $n=|A|$ is the index of $D$ and $C_n\simeq\{0,1\}^n$ is the vertex set of $n$-dimensional cube. If $(s_{A'}(D),z)\in P(D,x,y)$ where $A'\subset A=\{a_i\}_{i=1,\dots,n}$ then one sets $\pi(s_{A'}(D),z)=\chi_{A}=(\epsilon_1,\dots,\epsilon_n)$ where $\epsilon_i=1$ if $a_i\in A'$ and $\epsilon_i=0$ if not.
Recall that the multiplicity of the decorated resolution configuration is the number $\mu(D,x,y)=\max_{v\in C_n} |\pi^{-1}(v)|$.
Let $\mathcal D_i = (D_i, x|_{D_i}, y|_{D_i})$, $i=1,...,l$ be the connected components of the decorated resolution configuration $(D,x,y)$.
\begin{proposition}
$\mu(\mathcal D)=\mu(\mathcal D_1)\times\cdots\times\mu(\mathcal D_l)$.
\end{proposition}
\begin{proof}
Indeed, for each component $\mathcal D_i$ we can take a vector $v_i$ with the maximal preimage $\pi^{-1}|_{\mathcal D_i}(v_i)$. Then the concatenation $v$ of the vectors $v_1,\dots,v_l$ gives the maximal preimage of the map $\pi$, and $|\pi^{-1}(v)|=\prod_{i=1}^l \left|\pi^{-1}|_{\mathcal D_i}(v_i)\right|$.
\end{proof}
Below we assume that the decorate resolution configuration $D$ is connected.
Let $(D,y)$ be a labeled resolution configuration. Denote the number of circles in $D$ by $\gamma(D)$, and let $|y|$ be the sum of labels of the circles. Then $-\gamma(D)\le |y|\le\gamma(D)$. The quantum degree of the enchanced state $(D,y)$ is equal to $|\pi(D)|+|y|=|y|$. Let $A'\subset A(D)$ and $(s_{A'}(D),z)\succ (D,y)$. Since the quantum degrees of comparable configurations coincide, $|y|=|\pi(s_{A'}(D))|+|z|=|A'|+|z|$. Hence, $-\gamma(s_{A'}(D))\le |z|=|y|-|A'|$ and $\gamma(s_{A'}(D))\ge |A'|-|y|$.
In particular, if $\gamma(D)=1$ and $|y|=-1$ then $\gamma(s_{A'}(D))\ge |A'|+1$, hence, $\gamma(s_{A'}(D))=|A'|+1$.
If $\gamma(D)=1$ and $|y|=1$ then $\gamma(s_{A'}(D))=|A'|\pm 1$.
\begin{proposition}
Let $a\in A(D)$ be a leaf or coleaf (see Fig.~\ref{fig:leaf_coleaf}). Then there exists a unique labeled resolution configuration $(s_a(D),z)$ such that $(D,y)\prec(s_a(D),z)\prec(s(D),x)$, and
$$P(D,x,y)=P(s_a(D),z,y)\times\{0,1\}.$$
\begin{figure}
\caption{{\bf Leaf (left) and coleaf (right) in a resolution configuration
}
\label{fig:leaf_coleaf}
\end{figure}
\end{proposition}
\begin{proof}
If $a$ is a leaf then $(s_a(D),z)$ is uniquely determined. If $a$ is a coleaf then $s_a(D)$ splits into two components $(D_1,z_1)$ and $(D_2,z_2)$. Then $s(D)=s(D_1)\sqcup s(D_2)$. The ambiguity of $z_1, z_2$ concerns the labels of the circles incident to $a$. But one can restore the labels using the quantum grading of the component: $|\pi(D_i)|+|z_i|=|\pi(s(D_i))|+|x|_{s(D_i)}|$, so $|z_i|=|\pi(s(D_i))|+|x|_{s(D_i)}|-|\pi(D_i)|$. If one knows the labels of all the circles in $(D_i,z_i)$ except one and the sum of all labels $|z_i|$, then the last label is uniquely determined.
Let $P_0=\{(s_{A}(D),u)\in P(D,x,y)\,|\, a\not\in A\}$ and $P_1=\{(s_{A}(D),u)\in P(D,x,y)\,|\, a\in A\}$. Using the reasonings above, we can prove that the surgery $s_a$ establishes an bijection between $P_0$ and $P_1=P(s_a(D),z,y)$. This bijection is compatible with the order in $P_0$ and $P_1$.
\end{proof}
Let $(D,x,y)$ be a nonempty decorated resolution configuration with one circle. Then $D$ is a chord diagram. Let $M=(m_{ab})_{a,b\in A(D)}$ be the $\mathbb{Z}_2$-valued interlacement matrix: $m_{ab}=lk(a,b)$, that is $m_{ab}=1$ if the arcs $a$ and $b$ are linked, and $m_{ab}=0$ if they are not. Then $M$ is a skew-symmetric matrix.
For any labeled resolution $(s_{A'}(D),z)\in P(D,x,y)$ the number of circles $\gamma(s_{A'}(D))$ is given by the curcuit-nullity formula
\begin{proposition}
$\gamma(s_{A'}(D))=corank M|_{A'}+1$.
\end{proposition}
On the other hand, the quantum degree gives the equality $|z|+|A'|=|y|$, so $\gamma(s_{A'}(D))\ge |A'|-|y|$.
If $|y|=-1$ then $\gamma(s_{A'}(D))=|A'|+1$ and all labels in $z$ are equal to $x_-$. Thus, $\mu(D,x,y)=1$.
If $|y|=1$ then $\gamma(s_{A'}(D))\ge |A'|-1$. Hence,
$$rank M|_{A'}=|A'|-corank M|_{A'}=|A'|-\gamma(s_{A'}(D))+1\le 2.$$
In particular, $rank M\le 2$. Since, $M$ is skew-symmetric then $rank M$ is even.
If $rank M=0$ then $M=0$ and all the arcs are coleafs. Thus, $\mu(D,x,y)=1$.
If $rank M=2$ then there are two independent vectors $a,b\in\mathbb{Z}_2^n$ such that any row of the matrix is equal to $0,a,b$ or $a+b$. This means the arcs of the chord diagram split into four subsets: three sets of parallel chords and coleaves, see Fig.~\ref{fig:one_circle_state}. Then $A(D)=A_a\sqcup A_b\sqcup A_{a+b}\sqcup A_0$.
\begin{figure}
\caption{{\bf Combinatorial structure of a non empty decorated resolution configuration with one circle}
\label{fig:one_circle_state}
\end{figure}
\begin{proposition}\label{prop:decorated_resolution_1}
Let $(D,x,y)$ be a non empty decorated resolution configuration with one circle such that its interlacement matrix has $rank M = 2$. Then
\begin{enumerate}
\item the circle of the configuration $D$ is contractible
\item the arcs which belong to one subset $A_a, A_b, A_{a+b}$ are homotopical in $F$.
\end{enumerate}
\end{proposition}
\begin{proof}
1. Assume $D$ is not contractible. Take arcs $a\in A_a$, $b\in A_b$. Let $D'=s_{\{a,b\}}(D)$. Then $D$ has label $x_+$ and $D'$ has label $x_-$, hence, $\mathrm{gr}_{\mathfrak H}(D,x_+)=[D]$ and $\mathrm{gr}_{\mathfrak H}(D',x_-)=-[D']$. But $[D]\ne -[D']$.
2. Let $a_1,a_2\in A_a$ be two non homotopical arcs, and $b\in A_b$. Then the resolution $s_{\{a_1,a_2, b\}}(D)$ consists of two circles of homotopy type $[a_1b^\pm a_2^{-1}b^\mp]$ and $[a_1^{-1}a_2]$ with labels $x_-,x_-$. Then the homotopical grading of this labeled resolution can not be zero. But $\mathrm{gr}_{\mathfrak H}(D,x_+)=0$ because $D$ is contractible.
\end{proof}
Let $(D,x,y)$ be a non empty decorated resolution configuration with one circle such that its interlacement matrix has $rank M = 2$. The complement to the circle of the resolution configuration consists of two components, one of which is contractible. We will call an arc {\em inner} if it lies in the contractible component, and {\em outer} otherwise.
\begin{proposition}\label{prop:decorated_resolution_2}
Let $(D,x,y)$ be a non empty decorated resolution configuration with one circle such that its interlacement matrix has $rank M = 2$. Then
\begin{enumerate}
\item the arcs from one of the subsets $A_a, A_b, A_{a+b}$ are either all inner or all outer
\item let the subsets $A_a, A_b$ consist of outer arcs. Then the surface $F$ is the torus, and the homology type of the arcs in $A_{a+b}$ (when $A_{a+b}\ne\emptyset$) is the sum of homology classes of an arc in $A_a$ and an arc $A_b$.
\item let the subset $A_a$ consist of inner arcs. Then $A_b$ consists of outer arcs and $A_{a+b}=\emptyset$.
\end{enumerate}
\end{proposition}
\begin{proof}
1. Assume that $a_1,a_2\in A_a$ and $b\in A_b$. Since $a_i$, $i=1,2$, and $b$ are interlaced and do not intersect, then these chords lie in different components to the circle of $D$. Hence, the chords $a_1$ and $a_2$ lie in one component, i.e. they are both internal or both external. Thus, the chords from the subset $A_a$ are all internal or all external. The same statements holds for the subsets $A_b$ and $A_{a+b}$.
2. Let $a\in A_a$ and $b\in A_b$ be outer chords. The intersection index of the loops $a$ and $b$ is equal to $1$, hence, $a$ and $b$ present independent homology classes. The resolution configuration $s_{\{a,b\}}(D)$ consists of one circle with the label $x_-$. This circle must be contractible due to the homotopical grading. Hence, the surface $F$ can be obtained by gluing two disc along the arcs $a$ and $b$. Thus, $F$ is the torus. Let $c\in A_{a+b}$. Since $c$ is disjoint from $a$ and $b$ and $F$ is the torus, the homology class corresponding to $c$ is equal to $a\pm b$, i.e. it is the sum of the homology classes of $a$ and $b$ (up to signs).
2. Let $a\in A_a$ be inner and $b\in A_b$ and $c\in A_{a+b}$. Then $b$ and $c$ are outer chords. The intersection index of the loops $b$ and $c$ is equal to $1$, hence, $b$ and $c$ present independent homology classes. The resolution configuration $s_{\{a,b,c\}}(D)$ consists of two circles with the labels $x_-$. Then the homotopical grading of this labeled resolution is equal $-2[bc^{-1}]\ne 0$ but the initial resolution configuration has homotopical grading $0$. Thus, the decorated resolution configuration must be empty.
\end{proof}
Note that the necessary conditions of Propositions~\ref{prop:decorated_resolution_1},~\ref{prop:decorated_resolution_2} are also sufficient for the decorated resolution configuration to be non empty.
\begin{proposition}\label{prop:1circle_multiplicity}
1. The multiplicity of a resolution configuration $s_{A'}(D)$ is equal to $2$ if $A'$ contains chords from exactly one of subsets $A_a, A_b, A_{a+b}$, and is equal to $1$ otherwise.
2. Let $(D',z,w)\subset (D,x,y)$ be a decorated resolution configuration with an initial configuration $(s_{A'}(D),w)$ and final configuration $(s(A''),z)$, where $D'=s_{A'}(D)$ and $A'\subset A''\subset A(D)$. Then $\mu(D',z,w)=2$ if and only if $A'\subset A_0$ and $A''$ intersects at least two of the subsets $A_a$, $A_b$, $A_{a+b}$.
\end{proposition}
\begin{proof}
1) Let $A'\subset A(D)$. If $A'$ does not intersect with $A_a, A_b, A_{a+b}$ then it contains only coleaf arcs. The resolution configuration $s_{A'}(D)$ consists of circles among which only one contains a pair of linked arcs. Then the label of this circle in $s_{A'}(D)$ must be $x_+$ (otherwise the surgery by the pair of linked arcs gives zero) whereas the labels of the other circles must be $x_-$. Thus, the labels of the configuration $s_{A'}(D)$ are defined uniquely, and $\mu(s_{A'}(D))=1$.
If $A'$ contains a pair arcs from different subsets $A_a, A_b, A_{a+b}$ then $rank M|_{A'}=2$. This means that the labels of all the circles in $s_{A'}(D)$ are $x_-$. Thus, $\mu(s_{A'}(D))=1$.
If $A'$ contains chords from exactly one of subsets $A_a, A_b, A_{a+b}$ then the resolution configuration looks like in Fig.~\ref{fig:resolution_mult2}. There are two circles connected with an arc in $s_{A'}(D)$. Hence, the label of one of these circle must be $x_+$. The labels of the other circles are $x_-$. Hence, the multiplicity is $2$.
\begin{figure}
\caption{{\bf Labelings of a resolution configuration of multiplicity $2$}
\label{fig:resolution_mult2}
\end{figure}
2) The condition of the second statement means the decorated configuration $(D',z,w)$ contains a resolution configuration of multiplicity $2$. Thus, it follows from the first statement of the proposition.
\end{proof}
Let us now consider resolution configurations with several circles in the initial state.
Let $(D,x,y)$ be a connected nonempty decorated resolution configuration with $k\ge 1$ circles in the initial state. Then there are $k-1$ arcs $\noindentat A=\{c_1,\dots,c_{k-1}\}\in A(D)$ such that the resolution $D'=s_{\noindentat A}(D)$ consists of one circle (Fig.~\ref{fig:decorated_many_circles}). In the diagram $D'$ we can draw the arc adjoint to the arcs of $\noindentat A$. With some abuse of notation, we denote the set of the adjoint arcs by $\noindentat A$. Then $D=s_{\noindentat A}(D')\cup\noindentat A$.
\begin{figure}
\caption{Decorated resolution configurations $D$ and $D'$. The set $\noindentat A$ consists of green arcs}
\label{fig:decorated_many_circles}
\end{figure}
Let $M$ be the interlacement matrix of the chords $A(D')\cup \noindentat A\simeq(A(D)\setminus\noindentat A)\cup\noindentat A=A(D)$ on the circle $D'$. Then for any $A'\subset A(D)$ the number circles in the resolution configuration is equal to $corank M|_{A'\bigtriangleup\noindentat A}+1$ where $A'\bigtriangleup\noindentat A$ is the symmetric difference of the sets $A'$ and $\noindentat A$.
\begin{proposition}\label{prop:kcircle_multiplicity}
Let $(D,x,y)$ be a connected nonempty decorated resolution configuration such that $rank M|_{A(D)\setminus\noindentat A}=2$ and $A'\subset A(D)$. Let $M_{A'}$ be the submatrix of the interlacement matrix $M$ whose rows correspond to the subset $A'\setminus\noindentat A$ and columns correspond to the subset $A(D)\setminus(A'\cap \noindentat A)$, and $M^0_{A'}$ be the submatrix with rows from $A'\setminus\noindentat A$ and columns from $\noindentat A\setminus A'$. Then the resolution configuration $s_{A'}(D)$ has in $(D,x,y)$ multiplicity $2$ if and only if $rank M_{A'}-rank M^0_{A'}=1$, and multiplicity $1$ otherwise.
\end{proposition}
\begin{proof}
The matrix $M_{A'}$ includes $M_{A'}^0$, so $rk M_{A'}-rk M^0_{A'}\ge 0$.
Let $\tilde D=s_{\noindentat A\cap A'}(D)$ be the resolution configuration obtained from $D$ by surgery along arcs in $\noindentat A$. The resolution configuration $\tilde D$ has $|\noindentat A\setminus A'|+1$ circles with labels $x_+$.
Assume that $rk M^0_{A'}=0$. This means that the resolution configuration $\tilde D$ consists of $|\noindentat A\setminus A'|+1$ distinct components.
The chords from different components are not interlaced, i.e. the coefficient correspondent to them in the matrix $M_{A'}$ is zero. Indeed, let $c$ and $c'$ belong to different components. Then the resolution configuration $s_{\{c,c'\}}(\tilde D)$ has $|\noindentat A\setminus A'|+3$ circles, and the resolution configuration $s_{\{c,c'\}\cup (\noindentat A\setminus A')}(\tilde D)$ has $|\noindentat A\setminus A'|+3-|\noindentat A\setminus A'|=3$ circles. But the number of circles is equal to $corank M(\{c,c'\})+1$. Hence, $corank M(\{c,c'\})=2$ and $M(\{c,c'\})=0$, so $m_{cc'}=0$.
Among the components of $\tilde D$ only one can include interlaced chords. Indeed, if we have pairs $c_1,c_1'$ and $c_2,c_2'$ of interlaced chords from different component then the surgery $s_{\{c_1,c_1',c_2,c_2'\}\cup (\noindentat A\setminus A')}(\tilde D)$ would give a resolution configuration with one circle and a label of degree $-2$, but $\deg(x_-)=-1$ is the minimal grading. Thus, the label is zero and the decorated resolution configuration $(D,x,y)$ is empty.
Thus, we have only one component with interlaced chord. For the matrix $M(A')$ this means that its nonzero elements can correspond only to the distinguished component. Thus, $rk M_{A'}$ is equal to the rank of the submatrix which corresponds to the component with interlaced chords. Then the statement of the proposition follows from Proposition~\ref{prop:1circle_multiplicity}.
If $rk M^0_{A'}>0$ then there are chords in $\tilde D$ that connect different circles. Then we can chose subsets $A''\subset A'$ and $\noindentat A''\subset \noindentat A\setminus A'$ such that $|A''|=|\noindentat A''|=rank M^0_{A'}$ and surgery along the set $\noindentat A\setminus (A'\cup \noindentat A'')\cup A''$ transforms the resolution configuration $\tilde D$ to a configuration with one circle. Let $\noindentat A_1=(\noindentat A\setminus\noindentat A'')\cup A''$ and $M'$ be the interlacement matrix on the circle $D_1=s_{\noindentat A_1}(D)$. Then one can check that $corank M|_{A'\bigtriangleup\noindentat A}=corank M'|_{A'\bigtriangleup\noindentat A_1}$, $rank M_{A'}-rank M^0_{A'}=rank M'_{A'}-rank (M')^0_{A'}$ and $rank (M')^0_{A'}=0$. Thus, the statement of the proposition reduces to the case considered above.
\end{proof}
\begin{example}
Consider the resolution configuration in Fig.~\ref{fig:decorated_many_circles}. With labels $x_+$ on each circle, it defines an initial labeled resolution configuration of a decorated resolution configuration. One can choose the set $\noindentat A=\{a_1,a_2\}$ to merge the circles of the configuration. The interlacement matrix then is equal to
$$
M=\left(\begin{array}{cccccc}
0 & 0 & 1 & 0 & 0 & 0\unskip, \ignorespaces
0 & 0 & 1 & 0 & 0 & 0\unskip, \ignorespaces
1 & 1 & 0 & 0 & 0 & 0\unskip, \ignorespaces
0 & 0 & 0 & 0 & 1 & 0\unskip, \ignorespaces
0 & 0 & 0 & 1 & 0 & 0\unskip, \ignorespaces
0 & 0 & 0 & 0 & 0 & 0
\end{array}\right).
$$
Consider the surgery along the subset $A'=\{a_1,a_3,a_5,a_6\}$. The matrix $M_{A'}$ is equal
$$
M_{A'}=\left(\begin{array}{c|ccc}
1 & 0 & 0 & 0 \unskip, \ignorespaces
0 & 0 & 1 & 0 \unskip, \ignorespaces
0 & 0 & 0 & 0
\end{array}\right)
$$
where the left part is the submatrix $M^0_{A'}$. Then $rank M_{A'}=2$ and $rank M^0_{A'}=1$. Since $rank M_{A'}-rank M^0_{A'}=1$, the multiplicity of the resolution configuration $s_{A'}(D)$ is equal to $2$. The labelings of the resolution configuration $s_{A'}(D)$ are shown in Fig.~\ref{fig:decorated_many_circles_surgery}.
\begin{figure}
\caption{Resolution configuration $s_{\{a_1,a_3,a_5,a_6\}
\label{fig:decorated_many_circles_surgery}
\end{figure}
\end{example}
Let us enumerate the isotopy classes of connected decorated resolution configuration of multiplicity $>1$.
\subsection{Decorated resolution configurations of index $1$}
All nonempty decorated resolution configurations consists of a comparable pair of labeled resolution configurations and have multiplicity $1$.
\subsection{Decorated resolution configurations of index $2$}
According to Propositions~\ref{prop:1circle_multiplicity} and~\ref{prop:kcircle_multiplicity} the multiplicity of a decorated resolution configuration $(D, x, y)$ with two arcs is equal to $1$ except three cases (see Fig.~\ref{fig:ladybug_cases}).
If $F=S^2$ then there is a unique up to isomorphism decorated resolution configuration $L_0$ of multiplicity $2$. When $F\ne T^2$, the decorated resolution configurations of multiplicity $2$ are the ladybug resolution configuration of type $L_\alpha$ where $\alpha$ is a homotopy class of a nontrivial simple loop in $F$. In the torus case $F=T^2$, there is a two-parameter series of decorated resolution configurations of multiplicity $2$. The parameters are two homology classes $\alpha$ and $\beta$ in $H_1(T^2)$.
\begin{figure}
\caption{Ladybug configurations of type $L_0$ (left) and $L_\alpha$ (middle), and a quasi-ladybug configuration of type $Q_{\alpha,\beta}
\label{fig:ladybug_cases}
\end{figure}
Let us consider these configuration more attentively.
\subsubsection{\bf The ladybug configuration for link diagrams in $S^2$}\label{tamago}
\noindent
We review the ladybug configuration for link diagrams in $S^2$,
which is introduced in \cite[section 5.4]{LSk}.
Lipshitz and Sarkar introduced it in the case of link diagrams in $S^2$.
We cite the definition of it,
that of the right pair, and that of the left pair associated with it
from \cite[section 5.4.2]{LSk}. \unskip, \ignorespaces
\begin{definition}\label{teten} {\bf (\cite[Definition 5.6]{LSk}).}
An index 2 basic resolution configuration $D$ in $S^2$
is said to be a ladybug configuration
if the following conditions are satisfied (See Figure \ref{tento}.).
$\bullet$ $Z(D)$ consists of a single circle, which we will abbreviate as $Z$;
$\bullet$ The endpoints of the two arcs in $A(D)$, say $A_1$ and $A_2$,
alternate around $Z$
\noindentskip3mm (that is, $\partial A_1$ and $\partial A_2$ are linked in $Z$).\unskip, \ignorespaces
\end{definition}
\begin{definition}\label{rl} {\bf (\cite[section 5.4.2]{LSk}).}
Let $D$ be as above.
Let $Z$ denote the unique circle in $Z(D)$.
The surgery $s_{A_1}(D)$ (respectively, $s_{A_2}(D)$) consists of two circles;
denote these $Z_{1,1}$ and $Z_{1,2}$ (respectively, $Z_{2,1}$ and $Z_{2,2}$);
that is, $Z(s_{A_i}(D)) = \{Z_{i,1}, Z_{i,2}\}$.
As an intermediate step, we distinguish two of the four arcs in
$Z - (\partial A_1\cup \partial A_2)$.
Assume that the point $\infty\in S^2$ is not in $D$,
and view $D$ as lying in the plane $S^2-\{\infty\}\cong\mathbb R^2$.
Then one of $A_1$ or $A_2$ lies outside $Z$ (in the plane)
while the other lies inside $Z$.
Let $A_i$ be the inside arc and $A_o$ the outside arc.
The circle $Z$ inherits an orientation from the disk it bounds in $\mathbb R^2$.
With respect to this orientation, each component of
$Z - (\partial A_1\cup\partial A_2)$
either runs from the outside arc $A_o$ to an inside arc $A_i$ or vice-versa.
The {\it right pair} is the pair of components of $Z-(\partial A_1\cup\partial A_2)$
which run from the outside arc $A_o$ to the inside arc $A_i$.
The other pair of components is the {\it left pair}. See \cite[Figure 5.1]{LSk}.
\end{definition}
We explain why the ladybug configuration is important, below.
\begin{proposition}\label{4}
Let ${\bf x}$ $($respectively, ${\bf y})$ be a labelled resolution configuration in $S^2$
of
homological grading $n$ $($respectively, $n+2).$
Then the cardinality of the set
\noindentskip3mm $\{p| p$ is a labelled resolution configuration.
${\bf x}\prec p, p\prec{\bf y},$ $p\neq{\bf x}$, $p\neq{\bf y}\}$
\noindent
is 0, 2, or 4, where $\prec$ represents the partial order defined in
\cite[Definition 2.10]{LSk}.
\end{proposition}
Let $D$ be the ladybug configuration in $S^2$.
Since each of $D$ and $s(D)$ has only one circle,
we can let $x_+$ or $x_-$ denote a labeling on it.
Give $D$ (respectively, $s(D)$) a labeling $x_+$ (respectively, $x_-$).
We call the resultant labeled resolution configuration
$(D,x_+)$ (respectively, $(s(D),x_-)$).
We obtain a decorated resolution configuration $(D,x_-,x_+)$
as drawn in Figure \ref{uenp}.
\begin{fact}\label{tenten}
The case of 4 in Proposition \ref{4} occurs in the above case $(D,x_-,x_+)$.
\end{fact}
Fact \ref{tenten} is also explained in \cite[section 5.4]{LSk}.
\begin{figure}
\caption{{\bf The ladybug configuration
}
\label{tento}
\end{figure}
\begin{figure}
\caption{
{\bf The poset for
the decorated resolution configuration associated with
a ladybug configuration in $S^2$
}
\label{uenp}
\end{figure}
\bigbreak
\subsection{Ladybug
and quasi-ladybug configurations for link diagrams in surfaces}\label{hiyoko}
\bigbreak
\begin{definition}({\bf\cite{KauffmanNikonovOgasa}})\label{LQ}
Let $D$ be a resolution configuration
which is made of one circle and two m-arcs (multiplication arc). \unskip, \ignorespaces
Stand at a point in the circle where you see an arc to your right.
Go ahead along the circle. Go around one time.
Assume that you encounter the following pattern:
In the order of travel you next touch the other arc.
Then you touch the first arc.
Then you touch the other arc again.
Finally, you come back to the point at the beginning. \unskip, \ignorespaces
Since both arcs are m-arcs,
both satisfy the following property:
At both endpoints of each arc,
you see the arc in the same side -- either on
the right hand side and on the left hand side.\unskip, \ignorespaces
If you see the arcs
both in the right hand side
and in the left hand side
(respectively, only in the right
hand side)
while you go around one time,
we call $D$ a {\it ladybug configuration}
(respectively,
{\it quasi-ladybug configuration}).
If $F$ is the 2-sphere,
our definition of ladybug configurations
is the same as
that
in in \S\ref{tamago}.
\bigbreak
Let $D$ be
a ladybug (respectively, quasi-ladybug) configuration.
Then $Z(D)$ have only one circle and $A(D)$ have only two arcs.
Make $s(D)$.
Give $D$ (respectively, $s(D)$) a labeling $x$ (respectively, $y$).
We call the decorated resolution configuration $(D,y,x)$
a {\it decorated resolution configuration associated with
the ladybug $($respectively, quasi-ladybug$)$ configuration $D$}.
\end{definition}
Note that $(D,y,x)$ may be empty
as explained below.
Since each of $D$ and $s(D)$ has only one circle,
we can let $x_+$ or $x_-$ denote $x$ (respectively, $y$).
See Figure \ref{quasiT2}.
The partial order defined in
\cite[Definition 2.10]{LSk} is defined in the case of link diagrams in $S^2$.
The authors \cite{KauffmanNikonovOgasa} generalized it to the case of links in thickened surfaces.
\begin{figure}
\caption{{\bf
The poset for
a decorated resolution configuration $(D,x_-,x_+)$ associated with
a quasi-ladybug configuration on $T^2$:
We envelope $T^2$ along two circles as usual,
and draw six labeled resolution configurations.
Here, we have $[\timesi;a]\cdot[a;\eta]=[\timesi;b]\cdot[b;\eta]=-[\timesi;c]\cdot[c;\eta]=-[\timesi;d]\cdot[d;\eta].$
}
\label{quasiT2}
\end{figure}
\begin{proposition}{\bf(\cite{KauffmanNikonovOgasa})}\label{daijida}
\noindent$(1)$
Let $D$ be a quasi-ladybug configuration in a surface $F$.
Assume that the only one circle in $D$ is contractible.
Let $F$ be the torus.
Then there is a non-vacuous decorated resolution configuration
$(D, x_-,x_+)$ associated with $D$.
\smallbreak
\noindent$(2)$
Let $D$ be a quasi-ladybug configuration in a surface $F$.
Assume that the only one circle in $D$ is contractible.
Let $(D,y,x)$ be a decorated resolution configuration associated with $D$.
Assume that the genus of $F$ is greater than one.
Then $(D,y,x)$ is empty
for arbitrary $x$ and $y$.
\smallbreak
\noindent$(3)$
Let $D$ be a ladybug $($respectively, quasi-ladybug$)$ configuration in a surface $F$.
Let $(D,y,x)$ be a decorated resolution configuration associated with $D$.
Assume that the only one circle in $D$ is non-contractible.
Then $(D,y,x)$ is empty
for arbitrary $x$ and $y$.
\smallbreak
\noindent$(4)$
Let $F$ be an arbitrary surface.
There is a ladybug configuration $D$ in $F$
such that
a decorated resolution configuration $(D,y,x)$ associated with $D$
is non-empty.
\end{proposition}
\subsection{Decorated resolution configurations of index $3$}
All resolution configurations with three arcs are made from
graphs in Figure \ref{fig:cases3}.
\begin{figure}
\caption{{\bf
Connected graphs of the resolution configurations of index 3:
The segments denote arcs. We do not use dotted segments here.
}
\label{fig:cases3}
\end{figure}
Most of the configurations contain a leaf or a coleaf. Hence, they can be reduced to decorated configurations of smaller index.
Using Propositions~\ref{prop:1circle_multiplicity} and~\ref{prop:kcircle_multiplicity} we can enumerate the diagrams without leaves and coleaves, see Fig.~\ref{fig:cases}.
\begin{figure}
\caption{{\bf Initial configurations of decorated resolutions of index $3$ with one circle and without leaves and coleaves. The labels of the circles are $x_+$.}
\label{fig:cases}
\end{figure}
The diagrams (1)--(2'') are local, i.e. can be drawn in a disk of the surface. Note that the diagrams (1) and (1') ((2), (2') and (2'')) are isotopic if one considers them as diagrams in the sphere $S^2$. Then diagrams (3)--(6) are parameterized by the homotopy class $\alpha$ of a nontrivial simple loop in the sphere. The diagrams (7)--(8) contain a pair of interlaced outer chords (quasi-ladybug configuration) and can occur only when $F=T^2$ is the torus.
\section{Moduli spaces}\label{mod}
In the subsequent sections we consider concrete examples of branched covering moduli systems.
\subsection{Case (C) for links in surfaces}\label{subsect:case_C}
We starts with the cubic case, i.e. with moduli systems which cover trivially the moduli spaces of the cubic flow category.
We define the moduli spaces $\mathcal{M}(D, x, y)$ by induction on the index $n$ of decorated resolution configuration $(D, x, y)$.
Case $n=1$. We set $\mathcal{M}(D, x, y)\simeq \mathcal{M}_{\mathcal{C}_C(1)}(\bar 1, \bar 0)$ to be one point.
Case $n=2$. The moduli space $\mathcal{M}_{\mathcal{C}_C(2)}(\bar 1, \bar 0)$ can be identified with the segment $I=[0,1]$.
In all cases except the ladybug and quasi-ladybug configurations (see Fig.~\ref{fig:ladybug_cases}) there are four labeled resolution configurations in $(D, x, y)$ that correspond to the vertices of a square, i.e. the objects of $\mathcal{C}_C(2)$. Thus, we can identify the moduli space $\mathcal{M}(D, x, y)$ with $\mathcal{M}_{\mathcal{C}_C(2)}(\bar 1, \bar 0)=I$.
In a (quasi)-ladybug configuration the boundary $\partial\mathcal{M}(D, x, y)$ consists of four points $a,b,c,d$ which correspond to the paths in the diagram of the decorated resolution configuration, see Fig.~\ref{fig:ladybug_right} and~\ref{fig:quasi-ladybug_pairs}. By induction, the points $a$ and $b$ project to one end of the segment $I$, and $c$ and $d$ project to the other end. We must extend this projection to a $2$-fold covering over $I$. There are two ways to do this, and we must choose one of them.
\begin{figure}
\caption{Right pairs (right column) and left pairs (left column) in ladybug resolution configurations}
\label{fig:ladybug_right_left_pairs}
\end{figure}
For a ladybug configuration we can use left-right pair convention from the paper~\cite{LSk}. The ends of the arcs splits the cycle of the decorated resolution configuration into four segments. For the {\em right pairs}, we take the segments that start in the endpoint of the arc which goes to the right, and end in the endpoint of the arc which goes to the left, see Fig.~\ref{fig:ladybug_right_left_pairs}. The we pair the labeled resolution configurations which have the same labels on the distinguished segments of the cycle: $a$ with $d$, $b$ with $c$. The moduli space $\mathcal{M}(D, x, y)$ is the disjoint union of segments $ad$ and $bc$, see Fig.~\ref{fig:ladybug_moduli_space}.
Analogously, the moduli space for the left pairs can be defined.
\begin{figure}
\caption{Decorated diagram in a ladybug case. The blue segments are the right pairs in the cycle.}
\label{fig:ladybug_right}
\end{figure}
\begin{figure}
\caption{The moduli space in (quasi-)ladybug case}
\label{fig:ladybug_moduli_space}
\end{figure}
In the quasi-ladybug case, the cycle must be contractible (otherwise the decorated resolution configuration will be empty due to the homotopical grading). Then the orientation of the torus induces a canonical orientation of the cycle.
Any arc with the ends on the cycle determines a homology class in $H_1(T^2,\mathbb{Z})$ (we take the class of the loop that the arc becomes after contraction the cycle to a point).
Let us define an analogue of right pairs in the quasi-ladybug case. Fix a prime element $\lambda\in H_1(T^2,\mathbb{Z})$. It defines a simple curve in the torus. Choose a class $\mu$ such that $\lambda\cdot\mu=1$. Then $\lambda, \mu$ is a basis of $H_1(T^2,\mathbb{Z})$.
Any arc $a$ with ends on the cycle determines a homology class in $H_1(T^2,\mathbb{Z})$. Then $a=p\lambda+q\mu$, $p,q\in\mathbb{Z}$. We assign the number $-\frac pq$ to the arc $a$ (if $a=\pm\lambda$ we assign $-\infty$ to $a$). This numbering defines an order on the arcs of the decorated resolution configuration. Informally speaking, we number the arcs moving counterclockwise on the cycle, starting from an endpoint of the longitude on the cycle, see Fig.~\ref{fig:quasi-ladybug_order_pairs}.
In a quasi-ladybug decorated resolution configuration we take the segments of the cycle which start at the arc with the bigger number and end at the arc the smaller number, see Fig.~\ref{fig:quasi-ladybug_order_pairs}.
\begin{figure}
\caption{The $\lambda$-pair.}
\label{fig:quasi-ladybug_order_pairs}
\end{figure}
We shall call this pair of segments of the cycle the {\em $\lambda$-pair}. The other two segments form the {\em $\bar\lambda$-pair}, see Fig.~\ref{fig:quasi-ladybug_lambda_pairs}.
\begin{figure}
\caption{The $\lambda$-pairs (left) and the $\bar\lambda$-pairs (right).}
\label{fig:quasi-ladybug_lambda_pairs}
\end{figure}
Note that the $\lambda$-pair does not depend on the orientation of the cycle in the case when the arcs differ from $\lambda$ (as elements in $H_1(T^2,\mathbb{Z})$). The segments which form the $\lambda$-pair can be thought of as the segments of the cycle (with ends at the arcs) which intersect the longitude $\lambda$. But the orientation matters when one of the arcs is homologous to $\lambda$, see Fig.~\ref{fig:quasi-ladybug_lambda_pairs}.
\begin{figure}
\caption{The $\lambda$ and $\bar\lambda$-pairs when one of the arcs is homologous to $\lambda$.}
\label{fig:quasi-ladybug_order_pairs0}
\end{figure}
We pair the labeled resolution configurations which have the same labels on the distinguished segments of the cycle ($a$ with $d$, $b$ with $c$), see Fig.~\ref{fig:quasi-ladybug_pairs}. Thus, we get the moduli space $\mathcal{M}(D, x, y)$ to be equal the disjoint union of segments $ad$ and $bc$, see Fig.~\ref{fig:ladybug_moduli_space}.
\begin{figure}
\caption{Decorated diagram in the quasi-ladybug case. The red segments are the $\lambda$-pairs.}
\label{fig:quasi-ladybug_pairs}
\end{figure}
Thus, in order to define the moduli spaces of index $2$ we need to fix choices $\pi(0),\pi(\alpha)\in\{l,r\}$ of left/right pairs for ladybug configurations $L_0$ and $L_\alpha$, and choices $\pi(\alpha,\beta)=\pi(\alpha,\beta)\in\{\lambda,\bar\lambda\}$ between $\lambda$ and $\bar\lambda$-pairs for quasi-ladybug configurations $Q_{\alpha,\beta}$. This set of choices $\pi$ is called a {\em pairing}.
Let $\pi_{r,\lambda}$ denote the pairing such that $\pi_{r,\lambda}(0)=r$, $\pi_{r,\lambda}(\alpha)=r$ for all $\alpha$, and $\pi_{r,\lambda}(\alpha,\beta)=\lambda$ for all $\alpha,\beta$. Analogously, one defines the pairings $\pi_{l,\lambda}$, $\pi_{r,\bar\lambda}$, $\pi_{l,\bar\lambda}$. We will call these pairings {\em regular}.
From now on, we fix some pairing $\pi$ (regular or not) and define the moduli spaces $\mathcal{M}(D, x, y)$ of index $2$ according to it.
Case $n=3$. The moduli space $\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1, \bar 0)$ is a hexagon.
Let $(D, x, y)$ be a decorated resolution configuration of index $3$. The boundary $\partial\mathcal{M}(D, x, y)$ is defined by induction, and we need to extend it to a moduli space $\mathcal{M}(D, x, y)$ and a covering $\mathcal{M}(D, x, y)\to \mathcal{M}_{\mathcal{C}_C(3)}(\bar 1,\bar 0)$.
If the decorated resolution configuration does not include a (quasi)-ladybug configuration, there is a bijection between the labeled resolutions of $(D, x, y)$ and the vertices of the $3$-cube, and we set $\mathcal{M}(D, x, y)=\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1,\bar 0)$.
If the decorated resolution configuration includes only a ladybug of type $L_0$, we are in the classical situation that was treated in~\cite{LSk}. The boundary $\partial\mathcal{M}(D, x, y)$ is (the boundary of) two hexagons which project naturally to $\partial\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1, \bar 0)$. We extend this projection to a trivial $2$-fold covering $\mathcal{M}(D, x, y)\to \mathcal{M}_{\mathcal{C}_C(3)}(\bar 1,\bar 0)$.
If the decorated resolution configuration includes a ladybug of type $L_\alpha$, then it can not contain configurations of type $L_0$ or $Q_{\alpha,\beta}$ (otherwise the decorated resolution configuration is empty because of homotopical rading). Then we have the following initial labeled resolution configurations, see Fig.~\ref{fig:cases} (3)--(6). In all cases the homotopical grading does not interferes the poset structure of the decorated resolution configuration. Hence, we can treat the decorated configuration as if it were planar. Thus, the boundary $\partial\mathcal{M}(D, x, y)$ forms two hexagons as in the classical ladybug case, and the moduli space $\mathcal{M}(D, x, y)$ is defined as a trivial $2$-fold covering space over $\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1,\bar 0)$.
Let us consider the case when the decorated resolution configuration includes a quasi-ladybug configuration,
see Fig.~\ref{fig:cases} (7)--(10).
In the diagrams
(7), (9), a quasi-ladybug configuration of type $L_{\alpha,\beta}$ appears twice in the decorated resolution configuration. In these two cases the homotopical grading does not impose additional restrictions to the poset structure of the decorated resolution configuration. Then we can consider the horizontal (if $\pi(\alpha,\beta)=\lambda$) or vertical (if $\pi(\alpha,\beta)=\bar\lambda$) arcs as inner and work with the decorated configuration as with one including ladybug configuration of type $L_\beta$ (or $L_\alpha$). Thus, the moduli space $\mathcal{M}(D, x, y)$ is a trivial $2$-fold covering over the hexagon $\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1,\bar 0)$.
\begin{figure}
\caption{Initial labeled resolution configuration for the decorated configuration $DQ(\alpha,\beta,\gamma)$}
\label{fig:dodeca_quasiladybug}
\end{figure}
Let us consider the diagram in Fig.~\ref{fig:cases} (8). Denote the homology classes of the arcs by $\alpha$, $\beta$, $\gamma$, see Fig.~\ref{fig:dodeca_quasiladybug}. Then the decorated resolution configuration includes three quasi-ladybug configurations of type $Q_{\alpha,\beta}$, $Q_{\beta,\gamma}$ and $Q_{\alpha,\gamma}$, see Fig.~\ref{fig:case_8l}.
\begin{figure}
\caption{The decorated diagram $DQ(\alpha,\beta,\gamma)$ of index $3$}
\label{fig:case_8l}
\end{figure}
The boundary $\partial\mathcal{M}(D, x, y)$ consists of $12$ vertices which corresponds to paths from the initial to the final labeled resolution configuration in the diagram of the decorated configuration in Fig.~\ref{fig:case_8l}. Any path is determined by the intermediate labeled configuration, for example, $1a$, $4c$ etc. An edge of $\partial\mathcal{M}(D, x, y)$ corresponds to switching between two paths with a common edge, see Fig.~\ref{case_8_moduli_space1}.
There are six fixed edges $1a-1b$, $2a-2b$, $3a-3c$, $4a-4c$, $5b-5c$, $6b-6c$. The other six edges of $\partial\mathcal{M}(D, x, y)$ depend on the chosen pairing $\pi$.
\begin{figure}
\caption{Pairings in $\partial\mathcal{M}
\label{case_8_moduli_space1}
\end{figure}
Let us first consider the case when all pairs are $\lambda$-pairs: $\pi(\alpha,\beta)=\pi(\beta,\gamma)=\pi(\alpha,\gamma)=\lambda$.
Using the fixed longitude $\lambda$, we determine the $\lambda$-pairs for any pair of arcs, see Fig.~\ref{fig:case_8_order_pairs}.
\begin{figure}
\caption{Segments of the $\lambda$-pairs.}
\label{fig:case_8_order_pairs}
\end{figure}
We use the $\lambda$-pairs to find the moduli space of the quasi-ladybug faces. The moduli space of the face containing the labeled resolution configuration $a$ consists of segments $1a-3a$ and $2a-4a$, the other two faces give the segments $1b-5b$, $2b-6b$, $3c-5c$ and $4c-6c$. Thus, the boundary $\partial\mathcal{M}(D, x, y)$ forms two hexagons, see Fig.~\ref{fig:case_8_order_modulus}. Since the space $\partial\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1,\bar 0)$ is a hexagon, the covering map on the boundary is trivial.
\begin{figure}
\caption{The boundary $\partial\mathcal{M}
\label{fig:case_8_order_modulus}
\end{figure}
\begin{figure}
\caption{Initial labeled resolution configuration for the decorated configuration $DQ'(\alpha,\beta,\gamma)$. The classes $\alpha,\beta,\gamma$ are the ones of loops that appear after contracting the two cycles to points and deleting one of the three arcs.}
\label{fig:dodeca_quasiladybug1}
\end{figure}
Let us consider the diagram in Fig.~\ref{fig:dodeca_quasiladybug1}. This decorated resolution configuration $DQ'(\alpha,\beta,\gamma)$ (see Fig.~\ref{fig:case_8_dual_lambda}) is dual to the one considered above. Thus, it has the isomorphic moduli space $\mathcal{M}(D, x, y)$: two hexagons if the number of $\bar\lambda$-pairings among $\pi(\alpha,\beta)$, $\pi(\beta,\gamma)$, $\pi(\alpha,\gamma)$ is even, and a dodecagon branched over the hexagon $\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1, \bar 0)$ if the number $\bar\lambda$-pairings is odd.
\begin{figure}
\caption{The decorated diagram $DQ'(\alpha,\beta,\gamma)$ of index $3$}
\label{fig:case_8_dual_lambda}
\end{figure}
Thus, we see that the regular pairings $\pi_{r,\lambda}$, $\pi_{l,\lambda}$ induce trivial coverings of the moduli spaces $\mathcal{M}(D, x, y)\to\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1, \bar 0)$ of index three. Since there is no obstruction to extension of trivial covering in higher dimensions, the moduli spaces of regular $\lambda$-pairings are trivial coverings over the cubic moduli spaces.
Thus, we have proved Theorem~\ref{thm:moduli_system_existence} for the cubic case.
\subsection{Case (D) for links in surfaces}\label{subsect:case_D}
Now let us pass to the case when moduli spaces of the moduli system cover the moduli spaces of the cube flow category with branching points.
\begin{proof}[Proof of Theorem~\ref{thm:moduli_system_existence}]
We construct the moduli spaces $\mathcal{M}(D, x, y)$ by induction on the index of the decorated resolution configurations. We modify moduli spaces constructed for the case (C) in the previous section.
For indices $n=1,2$ we use the moduli spaces of the cubic case. These are points ($n=1$) and one or two segments ($n=2$).
Case $n=3$. The cubic moduli spaces are one or two hexagons. For two hexagons we add a handle connecting the hexagons and make the moduli space a cylinder (Fig.~\ref{fig:branched_moduli3}). This space corresponds to a branched covering over the hexagon with two branch points (Fig.~\ref{fig:branched_moduli3_cover}).
\begin{figure}
\caption{Branched moduli space.}
\label{fig:branched_moduli3}
\end{figure}
\begin{figure}
\caption{Branching set in the moduli space of the cube flow category.}
\label{fig:branched_moduli3_cover}
\end{figure}
Case $n\ge 4$. Assume we have construct moduli spaces for the decorated resolution configurations of index $\le n$. Let $(D, x, y)$ be a decorated resolution configurations of index $n$. By induction the boundary $\partial \mathcal{M}(D, x, y)$ of the moduli space is determined as a covering. This means that we have a branching submanifold of codimension 2 in the sphere $\partial\mathcal{M}_{\mathcal{C}_C(n)}(\bar 1, \bar 0)\simeq S^{n-2}$ such that the branched covering space is equal to
$$\partial \mathcal{M}(D, x, y)=\coprod_{(E,z)\in P(D,x,y)} \circ(\mathcal{M}(D\setminus E, z|, y|)\times \mathcal{M}(E\setminus s(D), x|, z|)).$$
The branching set $P\subset \partial \mathcal{M}_{\mathcal{C}_C(n)}(\bar 1, \bar 0)=S^{n-2}$ is an oriented closed $(n-4)$-dimensional manifold. Then there is a $(n-3)$-dimensional oriented manifold $Q\subset S^{n-2}$ with boundary such that $\partial Q = P$~\cite[p. 50, Theorem 3]{Kirby}.
Push the manifold $Q$ inside $B^{n-1}$ so that the deformed manifold $Q'$ have the boundary $P=\partial Q'=Q'\cap S^{n-2}$ and $Q'$ intersect the sphere $S^{n-2}$ transversely. Then we define the moduli space $\mathcal{M}(D,x,y)$ as the $2$-fold branched covering over $D^{n-1}$ with the branching set $Q'$.
\end{proof}
\begin{remark}
The moduli spaces constructed above differ from the moduli spaces of the case (C) by framed cobordisms. Then Pontryagin--Thom construction yields the same (up to homotopy) attaching maps in the cell complex for the Khovanov homotopy type~\cite{Stong}. Thus, the Khovanov homotopy type for the moduli system will be the same as the Khovanov homotopy type of the case (C).
\end{remark}
Let us construct another moduli system for links in the torus, which has some minimality properties:
\begin{itemize}
\item for index $3$ the branching set in the hexagon $\mathcal{M}_{\mathcal{C}_C(3)}$ consists of zero or one point;
\item for $n\ge 4$ the branching set in $\mathcal{M}_{\mathcal{C}_C(n)}$ has no components $B$ such that $B\cap\partial\mathcal{M}_{\mathcal{C}_C(n)}=\emptyset$.
\end{itemize}
We construct the moduli spaces $\mathcal{M}(D, x, y)$ by induction on the index of the decorated resolution configurations.
Case $n=1$. We set $\mathcal{M}(D, x, y)\simeq \mathcal{M}_{\mathcal{C}_C(1)}(\bar 1, \bar 0)$ to be one point.
Case $n=2$. The moduli space $\mathcal{M}_{\mathcal{C}_C(2)}(\bar 1, \bar 0)$ can be identified with the segment $I=[0,1]$.
In all cases except the ladybug and quasi-ladybug configurations (see Section~\ref{hiyoko}) the multiplicity of the decorated configuration is equal to $1$, hence, there are four labeled resolution configurations in $(D, x, y)$ that correspond to the vertices of a square, i.e. the objects of $\mathcal{C}_C(2)$. Thus, we can identify the moduli space $\mathcal{M}(D, x, y)$ with $\mathcal{M}_{\mathcal{C}_C(2)}(\bar 1, \bar 0)=I$.
For ladybug and quasi-ladybug configurations (case $\mu(D,x,y)=2$) we have four points in the boundary of $\partial \mathcal{M}(D, x, y)$ which can be connected with two segments. Thus, for those configurations $\mathcal{M}(D, x, y)$ is a disjoint union two segments.
In order to define the moduli spaces of index $2$ we fix choices $\pi(0),\pi(\alpha)\in\{l,r\}$ of left/right pairs for ladybug configurations $L_0$ and $L_\alpha$, and choices $\pi(\alpha,\beta)=\pi(\alpha,\beta)\in\{\lambda,\bar\lambda\}$ between $\lambda$ and $\bar\lambda$-pairs for quasi-ladybug configurations $Q_{\alpha,\beta}$. For the (D) case we can take one of the regular pairing $\pi_{r,\bar\lambda}$ or $\pi_{l,\bar\lambda}$ (see the previous section).
Case $n=3$. The moduli space $\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1, \bar 0)$ is a hexagon.
Let $(D, x, y)$ be a decorated resolution configuration of index $3$. The boundary $\partial\mathcal{M}(D, x, y)$ is defined by induction, and we extend it to a moduli space $\mathcal{M}(D, x, y)$ and a covering $\mathcal{M}(D, x, y)\to \mathcal{M}_{\mathcal{C}_C(3)}(\bar 1,\bar 0)$.
Let us show that the regular pairing $\pi_{r,\bar\lambda}$ gives branched moduli spaces. Consider the decorated resolution diagram $DQ(\alpha,\beta,\gamma)$ (Fig.~\ref{fig:case_8l}).
If we use $\bar\lambda$-pairing: $\pi(\alpha,\beta)=\pi(\beta,\gamma)=\pi(\alpha,\gamma)=\bar\lambda$, the boundary $\partial\mathcal{M}(D, x, y)$ of the moduli space is a dodecagon, see Fig.~\ref{fig:case_8_order_modulus1}. Then the moduli space $\mathcal{M}(D, x, y)$ should be the interior of the dodecagon. The map $\mathcal{M}(D, x, y)\to\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1, \bar 0)$ is a $2$-fold branched covering with one branch point in the center of the dodecagon.
\begin{figure}
\caption{The boundary $\partial\mathcal{M}
\label{fig:case_8_order_modulus1}
\end{figure}
For the decorated resolution configuration $DQ'(\alpha,\beta,\gamma)$ in Fig.~\ref{fig:dodeca_quasiladybug1} the $\bar\lambda$-pairing also gives a dodecagon moduli space.
Note that moduli spaces that covers the cubic moduli spaces with branching points can appear only for links in the torus. For this reason below we focus on the case when $F=T^2$.
Case $n=4$. The moduli space $\mathcal{M}_{\mathcal{C}_C(3)}(\bar 1, \bar 0)$ is a truncated octahedron.
Let $(D, x, y)$ be a decorated resolution configuration of index $4$. If it does not have a subconfiguration of index $3$ whose moduli space is a dodecagon, then the boundary $\partial\mathcal{M}(D, x, y)$ covers trivially the boundary $\partial\mathcal{M}_{\mathcal{C}_C(4)}(\bar 1,\bar 0)$. Hence, we define $\mathcal{M}(D, x, y)$ as a trivial cover over
$\mathcal{M}_{\mathcal{C}_C(4)}(\bar 1,\bar 0)$ which is $1$ fold when $(D, x, y)$ does not include (quasi)ladybug confgurations, $2$-fold when $(D, x, y)$ includes a (quasi)ladybug confguration, and $4$-fold when $(D, x, y)$ includes two disjoint (quasi)ladybug configurations.
Now, let $(D, x, y)$ includes a subconfiguration of index $3$ with a dodecagon moduli space.
At first, consider the case when this subconfiguration is $DQ(\alpha,\beta,\gamma)$. Then there exists a labeled resolution configuration $(\noindentat D,z)\in P(D,x,y)$ which contains the labeled resolution configuration of Fig.~\ref{fig:dodeca_quasiladybug}.
Let $A$ be the set of arcs in the decorated resolution configuration $(D, x, y)$. Then there exists a subset $A_-\subset A$ such that $\noindentat D=s_{A_-}D$. Let $A_+=A\setminus(A_-\cup\{\alpha, \beta,\gamma\})$. Since $A$ consists of four arcs, then $A\setminus\{\alpha, \beta,\gamma\}=\{\delta\}$.
Assume that $A_-=\{\delta\}$. We can draw $\delta$ as an arc in the resolution configuration $\noindentat D=s_\delta(D)$. That arc can not be an outer non-contractible arc. Otherwise, the diagram $D=s_\delta^{-1}(D)$ would have two nontrivial cycles with label $x_+$, hence the decorated resolution configuration must be empty because of homotopical grading. Then $\delta$ is an inner arc or an outer coleaf. In the latter case we can draw $\delta$ as an inner coleaf.
Assume that $A_+=\{\delta\}$. Then $\delta$ can not be an inner arc which interlaces with arc $\alpha$, $\beta$ or $\gamma$. Otherwise, we would have an empty decorated resolution configuration like in Fig.~\ref{case_9} by the homotopical grading. Thus, an inner $\delta$ is a trivial arc, so it can be drawn outside the cycle. Then $\delta$ is either outer nontrivial or outer trivial arc.
\begin{figure}
\caption{{\bf An empty decorated resolution configuration}
\label{case_9}
\end{figure}
Thus, we have several possibilities to draw the fourth arc in the resolution configuration, see Fig.~\ref{fig:dodeca_quasiladybug_index4}.
\begin{figure}
\caption{Possible layouts of the fourth arc in the diagram $\noindentat D$. The red arcs correspond to the cases when $\delta\in A_+$ and the blue arcs are cases for $\delta\in A_-$}
\label{fig:dodeca_quasiladybug_index4}
\end{figure}
The corresponding decorated resolution are given in Fig.~\ref{fig:dodeca_quasiladybug_index4_diagrams}.
\begin{figure}
\caption{Decorated resolution configuration of index $4$ which includes the decorated resolution configuration $DQ(\alpha,\beta,\gamma)$.}
\label{fig:dodeca_quasiladybug_index4_diagrams}
\end{figure}
Now, let the decorated resolution configuration $(D, x, y)$ include the subconfiguration is $DQ'(\alpha,\beta,\gamma)$. Then there exists a labeled resolution configuration $(\noindentat D,z)\in P(D,x,y)$ which contains the labeled resolution configuration of Fig.~\ref{fig:dodeca_quasiladybug1}. Let $a$ be one of the arcs of $DQ'(\alpha,\beta,\gamma)\subset\noindentat D$, and $\noindentat D'=s_a(\noindentat D)$.
Then $\noindentat D'$ has two nontrivial outer and one inner arc (which is $a$ after the surgery), see Fig.~\ref{fig:dodeca_quasiladybug1_index4}.
We should add a fourth arc $\delta$ to those three arc. As before, we have the following restrictions:
\begin{itemize}
\item if $\delta\in A_-$ and outer then $\delta$ must be homologically trivial
\item if $\delta\in A_-$ then it is not interlaced with $a$
\item if $\delta \in A_+$ and inner then it is not interlaced with $\alpha$, $\beta$ or $\gamma$
\end{itemize}
This we have the following possible cases.
\begin{figure}
\caption{Possible layouts of the fourth arc in the diagram $\noindentat D'$. The solid inner arc is the arc $a$. The red arcs correspond to the cases when $\delta\in A_+$ and the blue arcs are cases for $\delta\in A_-$}
\label{fig:dodeca_quasiladybug1_index4}
\end{figure}
The corresponding decorated resolution configurations are presented in Fig.~\ref{fig:dodeca_quasiladybug1_index4_diagrams}.
\begin{figure}
\caption{Decorated resolution configuration of index $4$ which includes the decorated resolution configuration $DQ'(\alpha,\beta,\gamma)$. Same numbers correspond to isomorphic resolution configurations}
\label{fig:dodeca_quasiladybug1_index4_diagrams}
\end{figure}
Now let us describe the moduli space structure for the cases 1--10. Let $\alpha$ be the homology class of the horizontal side and $\gamma$ be the vertical side of the square which represents the torus in Fig.~\ref{fig:dodeca_quasiladybug_index4_diagrams},~\ref{fig:dodeca_quasiladybug1_index4_diagrams}. Then $\beta=\alpha+\gamma$.
The boundary $\partial\mathcal{M}(D, x, y)$ covers the surface of the moduli space $\partial\mathcal{M}_{\mathcal{C}_C(n)}(\bar 1,\bar 0)$. The number of $2$-faces which can have branched points (of types $DQ$ and $DQ'$) for the cases 1--10 are given in the table.
\begin{center}
\begin{tabular}{|c|c|}
\noindentline
Case & Faces of types $DQ$ and $DQ'$ \unskip, \ignorespaces
\noindentline
1 & $2DQ(\alpha,\beta,\gamma)$ \unskip, \ignorespaces
2 & $2DQ(\alpha,\beta,\gamma)$ \unskip, \ignorespaces
3 & $2DQ(\alpha,\beta,\gamma)$ \unskip, \ignorespaces
4 & $2DQ(\alpha,\beta,\gamma)+2DQ(\alpha,\bar\beta,\gamma)+2DQ'(\alpha,\beta,\gamma)+2DQ'(\alpha,\bar\beta,\gamma)$ \unskip, \ignorespaces
5 & $DQ(\alpha,\beta,\gamma)+DQ'(\alpha,\beta,\gamma)$ \unskip, \ignorespaces
6 & $2DQ(\alpha,\beta,\gamma)$ \unskip, \ignorespaces
7 & $2DQ'(\alpha,\beta,\gamma)$ \unskip, \ignorespaces
8 & $2DQ'(\alpha,\beta,\gamma)$ \unskip, \ignorespaces
9 & $2DQ'(\alpha,\beta,\gamma)$ \unskip, \ignorespaces
10 & $2DQ'(\alpha,\beta,\gamma)$ \unskip, \ignorespaces
\noindentline
\end{tabular}
\end{center}
Here $\bar\beta=\gamma-\alpha$.
Thus, in cases 1--3,5--10, if we have $D$-case for the triple $(\alpha,\beta,\gamma)$ (i.e. the number of $\bar\lambda$ among the values $\pi(\alpha,\beta)$, $\pi(\alpha,\gamma)$, $\pi(\beta,\gamma)$ is odd) then the moduli space has two branched points on the boundary. The branched covering can be extended to the interior by choosing the branched set to be the segment that connects the branched points on the boundary, see Fig.~\ref{fig:branching_order4_2bpoints}.
\begin{figure}
\caption{The branched set in the cases 1,3,7,9 (left), 2,6,8,10 (middle) and 5 (right)}
\label{fig:branching_order4_2bpoints}
\end{figure}
The case 4 is more complicated. If there is one dodecagon triple among $(\alpha,\beta,\gamma)$ and $(\alpha,\bar\beta,\gamma)$ then there are 4 branched points on the boundary, see Fig.~\ref{fig:branching_order4_4bpoints} right. If both the triples $(\alpha,\beta,\gamma)$ and $(\alpha,\bar\beta,\gamma)$ are in $D$-case then the number of branched points is $8$, Fig.~\ref{fig:branching_order4_4bpoints} left. We can connect the branching points with two or four nonintersecting curves. Note that in this case there is no canonical way to extend the branched set from boundary to the whole moduli space.
\begin{figure}
\caption{Boundary branched points in the case 4}
\label{fig:branching_order4_4bpoints}
\end{figure}
Case $n\ge 5$. Assume we have construct moduli spaces for the decorated resolution configurations of index $\le n$. Let $(D, x, y)$ be a decorated resolution configurations of index $n$. By induction the boundary $\partial \mathcal{M}(D, x, y)$ of the moduli space is determined as a covering. This means that we have a branching submanifold of codimension 2 in the sphere $\partial\mathcal{M}_{\mathcal{C}_C(n)}(\bar 1, \bar 0)\simeq S^{n-2}$ such that the branched covering space is equal to
$$\partial \mathcal{M}(D, x, y)=\coprod_{(E,z)\in P(D,x,y)} \circ(\mathcal{M}(D\setminus E, z|, y|)\times \mathcal{M}(E\setminus s(D), x|, z|)).$$
The branching set $P\subset \partial \mathcal{M}_{\mathcal{C}_C(n)}(\bar 1, \bar 0)=S^{n-2}$ is an oriented closed $(n-4)$-dimensional manifold. Then there is a $(n-3)$-dimensional oriented manifold $Q\subset S^{n-2}$ with boundary such that $\partial Q = P$~\cite[p. 50, Theorem 3]{Kirby}.
Push the manifold $Q$ inside $B^{n-1}$ so that the deformed manifold $Q'$ have the boundary $P=\partial Q'=Q'\cap S^{n-2}$ and $Q'$ intersect the sphere $S^{n-2}$ transversely. Then we define the moduli space $\mathcal{M}(D,x,y)$ as the $2$-fold branched covering over $D^{n-1}$ with the branching set $Q'$.
\begin{remark}
Note that a ``minimal'' branching moduli systems such as considered above for links in the torus, present trivial covering (i.e. case (C)) moduli system for surfaces of other genus. Indeed, in this case quasi-ladybug configurations can not appear and for ladybug configurations any choice of left/right pairs lead to $2$-dimensional moduli spaces without branch point. Then branching sets can not appear in higher dimensions because by minimality they must intersect with the border.
\end{remark}
\subsection{Multivalued moduli systems for links in $S^3$}\label{subsect:mutlivalued_moduli_system}
As we have seen in the previous paragraph, for links in a surface except the torus, choice of a fixed moduli space for each decorated resolution leads (under some minimality condition) to a moduli system of type (C). We can overcome this situation by considering multivalued moduli system. If we allow to choose of left/right pairs for each ladybug configuration independently then we can get moduli spaces which cover the moduli spaces of the cubic flow category with some branch point.
Consider the following example (Fig.~\ref{fig:case_1}). The decorated resolution configuration has index $3$ and corresponds to a 2-dimensional moduli space. The moduli space has 12 vertices which correspond to the paths from the initial to the final state in the diagram in Fig.~\ref{fig:case_1}. The edges of the moduli space correspond to switching between the paths by changing one of the intermediate states (Fig.~\ref{fig:case_1_modulus}). The decorated resolution configuration includes two ladybug configurations, so we have two choices between the left (red in the figure) and the right (blue) pairs. If we choose the left pairs for one ladybug and the right pair for the other, we get a dodecagon as the moduli space. The dodecagon covers the hexagon (the moduli space of the cubic flow category) twofold with one branch point.
\begin{figure}
\caption{{\bf A decorated resolution configuration with two ladybug subconfigurations}
\label{fig:case_1}
\end{figure}
\begin{figure}
\caption{{\bf Boundary of the moduli space of the decorated resolution configuration}
\label{fig:case_1_modulus}
\end{figure}
Multivalued moduli systems generalize this example.
We construct a multivalued moduli system $\mathfrak M$ for links in a closed oriented surface $F$ by induction on the index of the decorated resolution configurations.
Case $n=1$. Let $\mathcal{M}(D, x, y)\simeq \mathcal{M}_{\mathcal{C}_C(1)}(\bar 1, \bar 0)$ be the moduli space consisting of one point. We set ${\mathfrak M}(D, x, y)=\{\mathcal{M}(D, x, y)\}$.
Case $n=2$. The moduli space $\mathcal{M}_{\mathcal{C}_C(2)}(\bar 1, \bar 0)$ can be identified with the segment $I=[0,1]$.
In all cases except the ladybug and quasi-ladybug configurations the moduli space $\mathcal{M}(D, x, y)$ is identified with $\mathcal{M}_{\mathcal{C}_C(2)}(\bar 1, \bar 0)=I$. Then we set ${\mathfrak M}(D, x, y)=\{I\}$.
For ladybug and quasi-ladybug configurations (case $\mu(D,x,y)=2$) we have four points in the boundary of $\partial \mathcal{M}(D, x, y)$ which can be connected with two segments. Thus, for those configurations we have two possible moduli spaces $\mathcal{M}'(D, x, y)$ and $\mathcal{M}''(D, x, y)$ which are disjoint unions of two segments. Then we take ${\mathfrak M}(D, x, y)=\{\mathcal{M}'(D, x, y),\mathcal{M}''(D, x, y)\}$.
Case $n>2$. Let $(D, x, y)$ be a decorated resolution configurations of index $n$. By definition the boundary $\partial \mathcal{M}(D, x, y)$ must be equal to
$$\partial \mathcal{M}(D, x, y)=\coprod_{(E,z)\in P(D,x,y)} \circ(\mathcal{M}(D\setminus E, z|, y|)\times \mathcal{M}(E\setminus s(D), x|, z|)).$$
We choose all possible moduli spaces $\mathcal{M}(D\setminus E, z|, y|)\in {\mathfrak M}(D\setminus E, z|, y|)$, $\mathcal{M}(E\setminus s(D), x|, z|)\in {\mathfrak M}(E\setminus s(D), x|, z|)$ which are compatible on the boundary. After the boundary $\partial \mathcal{M}(D, x, y)$ is defined, we extend the moduli space $\mathcal{M}(D, x, y)$ as in the previous section. Then we define ${\mathfrak M}(D, x, y)$ as the set consisting of all possible moduli space $\mathcal{M}(D, x, y)$ constructed by this scheme.
Note that the constructed moduli spaces $\mathcal{M}(D, x, y)\in {\mathfrak M}(D, x, y)$ will be branch coverings over the cubic moduli spaces.
\begin{example}
For the knot in Fig.~\ref{fig:case_1_knot} corresponding to the decorated resolution configuration in Fig.~\ref{fig:case_1}, we have $4$ possible variants to define moduli of index $2$. Two of these variants lead to the moduli space of index $3$ homeomorphic to two hexagons, and the other two variants yield a dodecagon. The decorated resolution configurations which are not sub-configurations of the one in Fig.~\ref{fig:case_1} have multiplicity one and correspond to moduli spaces of the cube flow category. Thus, we have two potentially different Khovanov homotopy types $\mathcal X_h,\mathcal X_d\in{\mathfrak X}_{\mathfrak M}(L)$ which are constructed on the hexagonal or dodecagonal moduli spaces correspondingly. But the knot is trivial, hence, its Khovanov homology is. Thus, the homotopy types coincide and ${\mathfrak X}_{\mathfrak M}(L)=\{S^{-1}\vee S^1\}$. This fact follows also from the invariance of the multi-valued homotopy type ${\mathfrak X}_{\mathfrak M}(L)$.
\begin{figure}
\caption{{\bf A diagram of the unknot}
\label{fig:case_1_knot}
\end{figure}
\end{example}
We hope to present a knot example with a nontrivial multi-valued Khovanov homotopy type in a subsequent paper.
\section{The second Steenrod square operator}\label{sq}
\subsection{The first Steenrod square operator $Sq^1$ }\label{sqq1}
\noindent
In \cite{Steenrod,SE}
the Steenrod square $Sq^* (*\in\mathbb{Z})$ is defined.
Let $X$ and $X'$ be compact CW complexes.
Let $\{C_i\}_{i\in\mathbb{Z}}$ be a chain complex. Assume that $\{C_i\}_{i\in\mathbb{Z}}$ is associated with both a CW decomposition on $X$
and a CW decomposition on $X'$.
It is well-known that $Sq^1(X)=Sq^1(X')$
(see e.g. \cite[Introduction]{LSs})
and that $Sq^2(X)$ and $Sq^2(X')$ are different in general
(see e.g. \cite{Seed}).
Therefore, $Sq^1$ is not informative as a link invariant. Let us pass to $Sq^2$.
\bigbreak
\subsection{The second Steenrod square operator $Sq^2$ }\label{sqq}
\noindent
We review the definition of the second Steenrod square~\cite{Steenrod, SE}.
\begin{definition}\label{secondsq}
Let $K_m$ be the Eilenberg--MacLane space $K(\mathbb{Z}_2,m)$ for any natural number $m>1$.
By definition, $K_m$ is connected and
$\pi_i(K_m)\cong\mathbb{Z}_2$ (respectively, $0$)
if $i=m$ (respectively, $i\neq m$ and $ i\ge 1$).
It is known that $H^{m+2}(K_m;\mathbb{Z}_2)\cong\mathbb{Z}_2$. Denote the generator of
$H^{m+2}(K;\mathbb{Z}_2)\cong\mathbb{Z}_2$ by $\timesi$.
Let $X$ be a CW complex and $[X,K]$ be the set of all homotopy classes of continuous maps $X\to K$.
Then $[X,K_m]=H^m(X;\mathbb{Z}_2)$.
For an arbitrary element $x\in H^m(X;\mathbb{Z}_2)$, take a continuous map $f_x:X\to K$ which corresponds to the class $x$.
Define the {\em second Steenrod square} $Sq^2(x)$ of $x$ to be $f^\ast_x(\timesi)\in H^{m+2}(X;\mathbb{Z}_2)$.
\end{definition}
This definition is reviewed and explained very well in \cite[section 3.1]{LSs}.
We review an important property of the second Steenrod square operator $Sq^2$, below.
\begin{proposition}\label{sq2}{\rm\bf(\cite[section 12]{Steenrod}.)}
Let $Y$ be any compact CW complex.
Let $Y^{(*)}$ be the $*$-skeleton of $Y (*\in\mathbb{Z})$.
Then the second Steenrod square \unskip, \ignorespaces
$Sq^2(Y):H^{m}(Y;\mathbb{Z}_2)\to H^{m+2}(Y;\mathbb{Z}_2)$
is determined by the homotopy type of $Y^{(m+2)}/Y^{(m-1)}$.
\end{proposition}
This proposition is reviewed and explained very well in \cite[section 3.1]{LSs}.
\subsection{The second Steenrod square $\mathcal Sq^2$
for links in the thickened torus in both cases (C) and (D)
}\label{vitaC}
\noindent
Let $\mathcal L$ be a link in the thickened torus.
We constructed stable homotopy types for $\mathcal L$.
Of course each of stable homotopy types has the second Steenrod square.
In the case (C), make a
Khovanov-Liphitzs-Sarkar stable homotopy type
for a triple of $\mathcal L$, a degree 1 homology class $\lambda\in H_1(T^2;\mathbb{Z}_2)$
and the right-left choice.
Its second Steenrod square gives an invariant of
a triple of $\mathcal L$, a degree 1 homology class $\lambda\in H_1(T^2;\mathbb{Z}_2)$,
and henceforce,
gives an invariant of $\mathcal L$.
Note. There are infinitely many choice of degree 1 homology classes.
However,
when we are given two link diagrams
and we compare the two,
we only have to calculate the second
Steenrod square in a finite cases.
In Case (D), take the set of
Khovanov-Liphitzs-Sarkar stable homotopy types
for a pair of $\mathcal L$ and the right-left choice.
The set of their second Steenrod squares
is
an invariant of a pair of $\mathcal L$ and the right-left choice,
and therefore,
gives an invariant of $\mathcal L$.
Note. Our invariant, the set of Steenrod squares,
is calculable.
\unskip, \ignorespaces
In the case of links in $S^3$,
in \cite{LSs} Lipshitz and Sarkar showed a way to calculate $\mathcal Sq^2$
by using classical link diagrams.
Seed calculated the second Steenrod square for links in $S^3$ by making a computer program of the method in \cite{LSs}. He found the following explicit pair.
\begin{theorem}\label{Seedrei} {\bf (\cite{Seed})}
There are links $\mathcal J$ and $\mathcal J'$ in $S^3$ such that
the Khovanov homologies are the same, but such that the second Steenrod squares are different.
Therefore
there are links $\mathcal J$ and $\mathcal J'$ in $S^3$ such that
the Khovanov homologies are the same, but the Khovanov stable homotopy types are different.
\end{theorem}
It is very natural to ask the following question. Are there a pair of links in the thickened torus such that the homotopical Khovanov homologies are the same, but such that the second Steenrod squares are different?
Note that all links in $B^3$ are regarded as links in the thickened torus if we regard $B^3$ is embedded in the thickened torus. Therefore,
by
Theorem~\ref{thm:moduli_system_existence}, Theorem~\ref{thm:moduli_system_invariance}
and Theorem \ref{Seedrei},
there are links $\mathcal J$ and $\mathcal J'$ in the thickened torus such that
the homotopical Khovanov homologies are the same (they coincide with the ordinary Khovanov homology), but such that the set of the second Steenrod squares are different.\label{proof:homotopy_type_stronger_homology} \unskip, \ignorespaces
Here, it is very natural to ask the following question. Are there a pair of links in the thickened torus which are not embedded in $B^3$ such that the homotopical Khovanov homologies are the same, but such that the second Steenrod squares are different?
The answer is a main result.
See the following proof.
\bigbreak
\noindent{\bf Proof of Main Theorem \ref{main}.}
Let $L$ be a link in the thickened torus. Consider the moduli system $\mathcal M$ on the torus constructed in Section~\ref{subsect:case_C}. Then ${\mathcal X}_\mathcal{M} (L)$ is a Khovanov homotopy type with cubic moduli spaces which proves Main Theorem \ref{main}.(1).
Then consider the moduli system $\mathcal M'$ on the torus constructed in Section~\ref{subsect:case_D}. The complex ${\mathcal X}_{\mathcal{M}'} (L)$ is a Khovanov homotopy type with non-cubic moduli spaces which proves Main Theorem \ref{main}.(2).
An easy example which proves Main Theorem \ref{main}.(3) is given above in the page~\pageref{proof:homotopy_type_stronger_homology}.
We show a little more complicated example below, and give an alternative proof of Main Theorem \ref{main}.(3).
Let $C$ be a circle in $T^2$ which represents
a
nontrivial
element of
$H_1(F;\mathbb{Z})$.
Regard $A$ as a knot in $F\times[-1,1]$.
Take
$\mathcal J$ and $\mathcal J'$
in a 3-ball $B$ embedded in $F\times[-1,1]$,
which are written in Theorem \ref{Seedrei}.
Assume that $C\cap B=\emptyset$.
Make a disjoint 2-component link
which is made from $C$ and
$\mathcal {J}$ (respectively, $\mathcal {J'}$).
By Theorem \ref{Seedrei},
these two links have different Steenrod squares and the same Khovanov homology.
See \cite[\S10.2]{LSk} for Khovanov-Lipshitz-Sarkar stable homotopy type
of disjoint links.
In this case, we do not have a quasi-ladybug configuration.
The right pair and the left one of ladybug situations
give the same Steenrod second square
by explicit calculus which uses that about the classical link diagram $K$.
(Note that the Steenrod square is only one element in this case.)
Replace $C$ with a link whose diagram is drawn in
Figure~\ref{fig:case_8l}.
This example gives a case which we use a dodecagon moduli.
\qed\unskip, \ignorespaces
The above example is just the beginning of many possible applications of the result in this paper.
Further applications require deeper computations of the virtual Khovanov homology
and will be the subject of a subsequent paper.\unskip, \ignorespaces
\noindent
Louis H. Kauffman
\noindent
Department of Mathematics, Statistics and Computer Science
\noindent
University of Illinois at Chicago
\noindent
851 South Morgan Street
\noindent
Chicago, Illinois 60607-7045
\noindent
USA
\noindent
and
\noindent
Department of Mechanics and Mathematics
\noindent
Novosibirsk State University
\noindent
Novosibirsk
\noindent
Russia
\noindent
[email protected]
\unskip, \ignorespaces
\noindent Igor Mikhailovich Nikonov
\noindent Department of Mechanics and Mathematics
\noindent Lomonosov Moscow State University
\noindent Leninskiye Gory, GSP-1
\noindent Moscow, 119991
\noindent Russia
\noindent [email protected]
\unskip, \ignorespaces
\noindent
Eiji Ogasa
\noindent
Meijigakuin University, Computer Science
\noindent
Yokohama, Kanagawa, 244-8539
\noindent
Japan
\noindent
[email protected]
\noindent
[email protected]
\end{document} |
\begin{equation}gin{document}
\title{{\bf\Large Hurst estimation of scale invariant processes with drift and stationary increments}
\begin{equation}gin{abstract}
The characteristic feature of the discrete scale invariant (DSI) processes is the invariance of their finite dimensional distributions by dilation for certain scaling factor.
DSI process with piecewise linear
drift and stationary increments inside prescribed scale intervals is introduced and studied.
To identify the structure of the process, first we determine the scale intervals, their linear drifts and eliminate them.
Then a new method for the estimation of the Hurst
parameter of such DSI processes is presented and applied to some period of the Dow Jones indices.
This method is based on fixed number equally spaced samples inside successive scale intervals. We also present some efficient
method for estimating Hurst parameter of self-similar processes with stationary increments. We compare the performance of this method with the celebrated FA, DFA and DMA on the simulated data of fractional Brownian motion.\\ \\
{\it Mathematics Subject Classification MSC 2010:} 62L12; 60G22; 60G18.\\ \\
{\it keywords:} Discrete scale invariance; Hurst estimation; Fractional Brownian motion; Scale parameter.
\end{abstract}
\title{{\bf\Large Hurst estimation of scale invariant processes with drift and stationary increments}
\section{Introduction}
Scale invariance or self-similarity has been discovered, analyzed and exploited in many frameworks, such as natural images \cite{rud}, fluctuations of stock market \cite{con}, \cite{fei} and traffic modeling in broadband networks \cite{abr}, \cite{rao}.
These processes are invariant in distribution under suitable scaling of time and space. Discrete scale invariant (DSI) processes are invariant by dilation for certain preferred scaling factors
or the observable obeys scale invariance for specific choices of scale \cite{bor}, \cite{sor}, \cite{zho}.
In practice, the main object is detecting the scale invariant property and estimating the Hurst index {\color{blue}$H$} and scale parameter {\color{blue}$\lambda$} of such processes \cite{b-1}.
Estimation methods depend on several factors, e.g, the estimation technique, sample size, time scale, level shifts, correlation and data structure.
Among all estimation methods for scale invariant and long-memory processes,
the rescaled adjusted range or $R/S$ statistic and semi-variogram are frequently used, see Beran \cite {b2}. Balasis et al. used the $R/S$ statistic in \cite{bal1} and the wavelet spectral analysis in \cite{bal2} to estimate the Hurst exponents for self-similar time series
originated from space physics applications.
Recently, Wang \cite{w1} presented a moving average method to estimate the Hurst exponent.
Vidacs and Virtamo obtained maximum likelihood estimator of the Hurst parameter based on some geometric sampling of the
fractional Brownian motion (fBm) traffic \cite{v1}.
Some estimation methods of Hurst index are based on variance-time, see \cite{c1}, \cite{c2}. In the econophysics, there are some celebrated methods for the estimation of Hurst parameters called
fluctuation analysis (FA)
\cite{p0}, detrended fluctuation analysis (DFA)
\cite{p00} and detrending moving average (DMA) which is described in \cite{a0}.
Regression analysis is a form of predictive modeling technique which allows to detect the trend of time series. We apply some piece-wise linear regression to detect drift to the DSI processes.
Especially we determine such piece-wise linear drift by applying regression lines to the plots of the corresponding scale intervals of DSI processes.\\
In real data scale invariant behavior often occures with some linear drift. To dtermine the structure of such processes one need to eliminate the drift first, and then estimate the Hurst parameter.
Brownian motion with drift is an example of scale invariant processes with drift which has lots of applications in mathematical finance and stock price modelings l \cite{ros}.
Usually the DSI behavior of the processes are characterized by detecting some regular behavior of the process inside successive scale intervals which can be identified by fitting homologous parabolas.
In this paper we present some flexible sampling scheme which provide some fixed number equally spaced sample points in each scale interval.
This sampling scheme provide a bases for our estimation method of Hurst parameter.
Then a DSI process with drift is introduced where by evaluating successive scale intervals, the evaluation and elimination of the linear piece-wise drift is studied.
Then a new innovative method for the estimation of Hurst parameter of DSI processes, having stationary increments inside scale intervals, is developed.
Finally we present some heuristic estimator of the Hurst parameter of self-similar processes with stationary increments (Hsssi).
The performance of this estimator are examined by using simulated data.
We show that our method is more efficient than the FA, DFA and DMA methods.
We compare our method using simulated samples of fractional Brownian motion (fBm) with drift and different Hurst parameters and show that the mean square error (MSE) of our method is much less than the compared methods.\\
The paper is structured as follows. In section 2, we present our flexible discrete sampling scheme and provide intuition on some basic notions of the scale invariance and DSI processes in discrete parameter space.
Section 3 is devoted to introducing DSI processes with drift, piece-wise linear drift and its elimination. We also present our estimation methods for scale and Hurst parameter of DSI processes while the increments are stationary inside scale intervals, and apply
our estimation methods to some part of Dow Jones indices in section 3.
A heuristic method for estimating the Hurst parameter of self-similar process with stationary increments is developed and its performance is compared with the celebrated methods FA, DFA and DMA for simulated data of fBm in section 4.
Conclusions are presented in section 5.
\section{Method of Flexible Discrete Sampling}
For our estimation method some appropriate sampling scheme is required. Current authors \cite{m11} considered geometric sampling at points $\alpha^k$, $k=1, 2, \ldots$ of DSI processes $\{X(t), t\in {\Bbb R^+}\}$ with scale $\lambda>1$, where $\alpha$ is determined by $\lambda=\alpha^T$ , and $T\in {\Bbb N}$ is some predefined number of observations in the scale interval $[\lambda^n, \lambda^{n+1})$, $n=0, 1,2,\cdots$.
Here we present some basic definitions and flexible sampling for DSI processes, see \cite{m3}.
A process $\{X(k),k\in {\check{T}}\}$ is called discrete time self similar (or scale invariant) process with parameter space $\check{T}$,
where $\check{T}$ is any subset of countable distinct points of positive real numbers, if $\{X(k_2)\}\stackrel{d}{=}(\frac{k_2}{k_1})^H\{X(k_1)\}$ for any $k_1, k_2 \in \check{T}$, where $\stackrel{d}{=}$ denotes equality of finite dimensional distributions.
The process is called discrete time DSI process with scale {\color{blue}$\lambda>0$} and parameter space $\check{T}$, if for any $k_1, k_2=\lambda k_1 \in \check{T}$, the above equation holds in distribution. So
sampling of $\{X(t), t\in {\Bbb R^+}\}$ at points $\alpha^{nT+k}, n\in {\Bbb Z}$, $k=0, 1, \ldots, T-1$, we have a discrete time scale invariant process with parameter space $\check{T}=\{\alpha^{nT+k}, n\in {\Bbb Z}\}$.
\\
A random process $\{X(k),k\in \check{T}\}$ is called self-similar ( scale invariant) in the wide sense with Hurst index $H>0$ and parameter space $\check{T}$, if for all $k, k_1\in \check{T}$ and all {\color{blue}$\lambda>0$}, where $\lambda k, \lambda k_1\in \check{T}$ we have that $E[X^2(k)]<\infty$, $E[X(\lambda k)]=\lambda^HE[X(k)]$ and $E[X(\lambda k)X(\lambda k_1)]=\lambda^{2H}E[X(k)X(k_1)]$. If these properties hold for some $\lambda=\lambda_0>0 $ then the process is called wide sense DSI with parameter space $\check{T}$, see \cite{m11}.\\
Following Modarresi and Rezakhah \cite{m3} we consider discrete flexible sampling of DSI process with scale $\lambda>1$ by
choosing arbitrary sample points of in the first scale interval as
as $1\leqslant s_{1}< s_{2}< \ldots< s_{q}< \lambda$ and sampling in the scale interval $I_j=[\lambda^j, \lambda^{j+1})$, $j\in{\Bbb N}$, at points $\lambda^js_i$, $i=1, \ldots, q$. So by recalling sample points with $t_j=\lambda^{[\frac{j-1}{q}]}s_{j-[\frac{j-1}{q}]q}$ our sample space are being $\check{T}=\{t_j, j\in {\Bbb W}\}$.
\section{DSI processes with drift}
For the real data, DSI behavior often occurs in short periods. The Dst time series \cite{b-1} and stock market indices \cite{b0}, \cite{m3}, \cite{m4} and \cite{m114} are some examples of such situations. The change of drift is another feature that specially occurs by the
changes in growth of stock markets. So their simultaneous effect can not be ignored.
There are many examples in modeling the behavior of stock prices by Brownian motion with drift \cite{hul}.
Besides the regression modeling approaches, Brownian motion with linear drift has drawn much attention in prognostics \cite{wang}. Here we present the definition of Brownian motion with linear drift.
\begin{equation}gin{definition}
The process $\{B_{\mu}(t), t\geqslant 0\}$ is a Brownian motion with drift coefficient $\mu$ and variance parameter $\sigma^2$ if $B_{\mu}(0)=0$, the process $\{B_{\mu}(t), t\geqslant 0\}$ has stationary and independent increments and $B_{\mu}(t)$ is normally distributed with mean $\mu t$ and variance $\sigma^2t$. Equivalently $B_{\mu}(t)=\sigma B(t)+\mu t$ is Brownian motion with drift, where{\color{blue} the standard Brownian motion $B(t)$ is $H=1/2$ self-similar}.
\end{definition}
Now we present the definition of DSI processes with piece-wise linear drift, which we apply in subsection 3.3 to model
some part of the Dow Jones indices .
\begin{equation}gin{definition}
A process $\{X(t), t\in{\Bbb R}\}$ is DSI process with drift if it satisfies the relation $X(t)=Y(t)+g(t)$, $t\in{\Bbb R}$ where $Y(t)$ is a DSI process and $g(t)$ is a drift function. We call the process $\{X(t)\}$ , DSI with piece-wise linear drift if the drift consist of different line in successive scale intervals of the DSI process as $g(t)=\sum_{k=1}^M (\alpha_k+\begin{equation}ta_k t) I_{B_k}(t)$, where $\alpha_k$ and $\begin{equation}ta_k$ are real numbers and $B_k$ is the $k$-th scale interval for $k=1,\cdots M$.
\end{definition}
Following the idea of
the Brownian motion with drift, described in \cite{ros}, and simple Brownian motion as a DSI process, described in \cite{m11}, we present an example, which we call simple Brownian motion with drift as a flexible pattern for modeling more comprehensive processes with DSI behavior.
\noindent{\bf Example:}
Following Modarresi et al. \cite{m4} we call a process $X(t)$ a simple Brownian motion with drift $g(t)$, the Hurst index $H>0$ and scale $\lambda>1$ if
$$X(t)=\sum_{n=1}^{M}\lambda^{n(H-\frac{1}{2})}I_{[\lambda^{n-1}, \lambda^{n})}(t)B(t)+g(t)$$
where $B(\cdot)$ is {\color{blue}the standard} Brownian motion, $I(\cdot)$ an indicator function.
The expected value of the process is
$$E(X(t))=\lambda^{n(H-\frac{1}{2})}E(B(t))+g(t)=g(t).$$
Also for $s\leqslant t$, the covariance function of the process is determined as
$$\mathrm{Cov}\big(X(t), X(s)\big)=\lambda^{(n+m)(H-\frac{1}{2})}\mathrm{Cov}\big(B(t)+g(t),
B(s)+g(s)\big)=\lambda^{(n+m)(H-\frac{1}{2})}s. $$
This by the fact that $\mathrm{Cov}\big(B(t), B(s)\big)=\min\{t,s\}$.
One can easily verify that
$\{ X(t) \}$ is a DSI process. As in the real world the DSI behavior just appears locally in different processes, one need to detect the DSI period first.
The rest of this section can be described as follows. In this section we introduce DSI processes with drift and give an example which has Markov property.
In subsection 3.1 we study the detection and elimination of the piece-wise linear drift to the DSI processes.
In subsection 3.2 we introduce some innovative method for estimating the Hurst parameters of DSI processes with stationary increments. In subsection 3.3 we apply our estimation method for estimating the Hurst parameter of real data as some part Dow Jones indices.
\subsection{Elimination of the drift}
We consider the DSI process with piece-wise linear drift as $X(t)=Y(t)+g(t)$ in some duration of time, say $[0, C]$, where $\{ Y(t), t\in [0,C]\} $ is a DSI process and the drift $g(t)$ is assumed to be
piece-wise linear function of time $t$ as $g(t)=\sum_{i=1}^k (\alpha_i+\begin{equation}ta_i)I_{B_i}(t) $ where $B_1, B_2, \cdots , B_n$ is some partition of this duration.
For decomposition of $X(t)$ as above and detecting the DSI behavior of $Y(t)$ one need to estimate such piece-wise linear drift and eliminate it from the main process $X(t)$ first. For this, we need to detect the corresponding scale intervals of the main process by fitting some parabola as has been applied in \cite{b0} and \cite{m3}. Then we fit separate regression lines to the samples of successive scale intervals of main process $X(t)$. These regression lines are considered as piece-wise linear drift to the process $\{ X(t), t \in [0,C] \}$.
Then we eliminate the drift by subtracting $g(t)$ from $X(t)$ to obtain the DSI process $Y(t)$. Eliminating this drift Then one can estimate the Hurst parameter of the corresponding DSI process $Y(t)$ by the following method.\\
Using detected scale intervals of $X(t)$ for the corresponding DSI process $Y(t)$, we consider some fixed number of equally spaced samples in each scale interval,
say $q$. By this and the assumption that the process have stationary increment property inside each scale interval, we conclude that the increments arises by this sampling method inside each scale interval are identically distributed.
Now we consider the following procedure for the estimation of the parameters of the DSI process $Y(t)$.
\subsection{Estimation procedure }
Let $Y=\{Y(t), t\geqslant 0\}$ be a DSI process with stationary increment property inside each scale interval. This cause the increments of equally spaced samples inside each scale interval to be identically distributed. Our
estimation method is presented by the following steps.\\ \\
1- The time interval that we study the DSI process $\{ Y(t) \}$ is considered as $[0,C]$.\\ \\
2- Following the methods of Bartolozzi et al. \cite{b0} and Modarresi et al. \cite{m3} and \cite{m4}, we evaluate scale intervals $I_i=(a_i, a_{i+1}]$ by fitting appropriate parabola to the samples of the period $[0,C]$.
So the scale of the process can be estimated by
$$\hat{\lambda}=\frac{1}{M}\sum_{i=1}^{M}\frac{a_{i+1}-a_i}{a_i-a_{i-1}}$$
where $M$ is the number of the scale intervals.\\ \\
3- We consider $q$ equally spaced sample points in each scale interval so that the sample points of $k$-th scale interval are determined as $t_{(k-1)q+i}=a_{k-1}+(i-1)d_k$, where $d_k=\frac{a_{k+1}-a_k}{q}$, $i=1, \ldots, q$ and $k= 1, \ldots, M$. So $q$ is the number of observations in each scale interval. Thus our parameter space is $\hat{\tau}=\{t_{(k-1)q+i}, i=1, \ldots, q, k= 1, \ldots, M\}$ and $\{Y(t), t\in \hat{\tau}\}$ is a DSI process with parameter space $\hat{\tau}$.\\ \\
4- Now we consider $\{U(t), t\in \hat{\tau}\}$ as the increment process, where $U(t_i)=Y(t_{i+1})-Y(t_{i})$,
and $S_{k}^2=\frac{1}{q}\sum_{i=1}^{q}(U(t_{(k-1)q+i})-\bar{U_k})^2$,
where $\bar{U_k}=\frac{1}{q}\sum_{i=1}^{q} U_{(k-1)q+i}$, the sample variance of increments in the $k$-th scale interval, $k=1, 2, \ldots, M$.\\ \\
5- By the scale invariant property of the process $\{Y(t), t\in \hat{\tau}\}$ we have that $U(t_{(k-1)q+i})\stackrel{d}{=}\hat{\lambda}^H U(t_{(k-2)q+i})$ for $i=1, \ldots, q$, so $\sigma_k^2=\hat{\lambda}^{2H}\sigma_{k-1}^2$, where $\sigma_k^2=\mbox{Var}\big(U(t_{(k-1)q+i})\big)$. Estimating $\sigma^2_k$ by $S^2_k$, one can evaluate the estimation of the Hurst parameter by $\hat{\lambda}^{2\hat{H}}=\frac{S^2_k}{S^2_{k-1}}$.
Denoting $\mu=\hat{\lambda}^{2H}$, we have $M-1$ estimate for $\mu$ as $\hat{\mu}_k=\frac{S^2_{k}}{S^2_{k-1}}$, $k=2, \ldots, M$ and the final estimation of $\mu$ is evaluated as the mean of these $\hat{\mu}_k$.\\ \\
This estimation method of the Hurst parameter is based on the first order of the increments. We also consider the second order difference of the increments inside scale intervals and evaluate the sample variance
of corresponding to $k$-th scale interval by $S_{k,2}^2=\frac{1}{q-1}\sum_{i=1}^{q-1}(Z(t_{(k-1)(q-1)+i})-\bar{Z_k})^2,$
$Z(t_i)=U(t_{i+1})-U(t_i)$ and $\bar{Z_k}=\frac{1}{q-1}\sum_{i=1}^{q-1} Z_{(k-1)(q-1)+i}$, $i=1, \ldots, q-1$.
Also we have that
${\sigma^2_{k,2}}=\hat{\lambda}^{2H}{\sigma_{k-1,2}}^2$, where $\sigma^2_{k,2}= \mbox{Var}\big(Z(t_{(k-1)(q-1)+i})\big) $.
Estimating $\sigma_{k,2}^2$ by $S^2_{k,2}$, one can estimate the Hurst parameter $H_2$ by
$\hat{\mu}_{k,2}=\hat{\lambda}^{2\hat{H}_2}=\frac{S^2_{k,2}}{S^2_{k-1,2 }}$.
Denoting ${{\mu}}_{k,2}={\hat{\lambda}}_{k}^{2{H_2}}$, $k=2, \ldots, M$, we have the final estimation of $\mu$ as mean of these $\hat{\mu}_{k,2}$.
\subsection{Real data analysis}
As an example of DSI process with drift we consider daily indices of Dow Jones from 25/10/2001 till 28/5/2014 and try to estimate the relevant parameters.
As these indices are no available on Saturdays, Sundays and holidays, the available indices for this duration are 3168 days, which are plotted in Figure 1 (left).
For the duration of the study 6/3/2009 till 14/11/2012 corresponding to the sample points $1800-2600$ we have evaluated a drift line as $g(t)=\hat{a}t+\hat{b}$. So the existence of the drift is clear by the fitted drift line which is shown in Figure 1. The DSI samples which has been evaluated by differencing the data set from this drift line is plotted in the bottom panel of the Figure 1 (right) which shows the DSI behavior.
The fitted red lines reveals the scale intervals of DSI variation with drift for the period 6/3/2009 till 14/11/2012. The end points of these scale intervals are $a_1=1854, a_2=2186, a_3=2466, a_4=2671, a_5= 2785$. The scale parameter is estimated as mean of the ratio of length of successive scale intervals, so the time dependent scale parameters are estimated as $\hat{\lambda}_1 =1.1857, \hat{\lambda}_2 =1.3659, \hat{\lambda}_3 =1.7982$ and their mean as $\hat{\lambda}= 1.4499$.
\begin{equation}gin{figure}[h!]
\begin{equation}gin{center}\label{daj}
\begin{equation}gin{tabular}{ c c }
\hspace{-.6in}\includegraphics[width=.9\textwidth, height=40mm]{unte.eps}\hspace{-.25in}
& \\
\hspace{-.55in}
\includegraphics[width=0.4\textwidth, height=40mm]{driftOne.eps}\hspace{-.05in}\hspace{-.1in}
\includegraphics[width=0.4\textwidth, height=40mm]{uneee.eps}\hspace{-.25in}
\\
\end{tabular}
\caption{\footnotesize The above figure shows the plot of Dow Jones indices that the DSI behavior is justified from 6/3/2009 till 14/11/2012. Red lines are borders of corresponding scale intervals. The left below figure shows the fitted drift line to the whole duration of DSI behavior, where the slop of the drift line is $5.2$ of share index per unit day. By the right below figure different drifts lines are fitted to the plots of successive scale intervals, where the slop of successive drift lines from the left are $9.7,\; 8.7, \; 10.6$ and $6.6$ respectively. }
\label{DOWJFig}
\end{center}
\end{figure}
\noindent Now we consider some fixed number of equally spaced samples in each two consecutive interval. Then we estimate their corresponding Hurst parameter as the logarithm of the ratio of sample variance of increments in such successive scale intervals.
This is done after eliminating the drifts by different drift lines which have been fitted to the samples of scale intervals separately. The slop of successive drift lines from the left are $9.7,\; 8.7, \; 10.6$ and $6.6$ share index unit per day, respectively.
Therefore $\log (\hat{\lambda}^{2H_i})=\log (S^2_i/S^2_{i-1})$ where $S^2_i$ is the sample variance of increments inside $i$-th scale interval after eliminating the corresponding drift. Dividing this value by $2\log\hat{\lambda}$ we estimate the time dependent Hurst parameters $H_{i}$ for the variation of $i$-th scale interval with respect to the previous one. A new method for the time dependent Hurst parameter estimation which is more accurate estimation was studied \cite{m114}.
This evaluation leads to the estimation of time dependent Hurst parameters as $H_1=0.5711, H_2=0.1375, H_3=0.8134$ with mean $0.5073$.\\
We should remind that considering such drift lines has two advantages. First it causes to model the mean changes of the process by such regression lines and the second advantage is that it reveals the true DSI behavior of the process after eliminating such drifts. For more clarification we represents here the estimated Hurst parameters without fitting such drift lines as
$H'_1= 0.7641 , \; H'_2=0.3105$ and $H'_3=1.9404 $ for the Hurst parameters of second, third and fourth scale intervals with respect to the previous scale intervals. Also when we fit just one drift line to the whole duration of DSI behavior and eliminating such drift from the data the corresponding Hurst parameters are estimated as $H''_1= 0.9409 , \; H''_2=1.5464$ and $H''_3=0.0567$. Comparing the variations of these estimates with the ones that were estimated by eliminating different drifts for successive scale intervals, reveals that those estimates are more close to each other and are more promising since all estimates are in the range $(0, 1)$. Also the bottom panel of Figure 1 (right) shows such drift provides much better estimation of share indices.
\\
\section{Comparison of Estimation methods for Hurst parameter of HSSSI process}
In this section we develop the method presented in \cite{R-M3} for the estimation of Hurst parameter of the scale invariant processes with stationary increments.
We consider the case that the scale invariant processes could have linear drift.
Let $\{X_{t}, t=1,2,\ldots, N\}$ be equally spaced samples of such a process.
{\color{blue} For this, first we eliminate the drift.}
We estimate this drift by the regression line $\hat{X}_t=\hat{a}+\hat{b}t$ by evaluating $\hat{a}$ and $\hat{b}$.
Then we eliminate the drift from the process as $ Z_t=X_t-\hat{X_t}$.
Now we are to estimate the Hurst parameter from this new process $Z_t$. Following the method of sampling of Rezakhah et al. \cite{R-M3}, we consider
sub-samples at points $\{Z_{ik}, i=1,2,\ldots, [N/k]\}$ as the $k$-th sub-sample for some fixed $k\in\mathbb{N}$.
Choosing $k$ depends on the sample size we take $k\in \{1,...,K^* \}$. We consider $K^*=\min \{20, N/30$\}.
For every $k\in \{1,...,K^*\}$ we consider two kind of sampled process $\{Z_{i}\}$ and $\{Z_{ik}\}$, where $i=1,2, \ldots, [N/k]$, and evaluate
first and second order sample variances for $r=1,2$
\begin{equation}gin{equation}
S^2_{r,k,2}=\frac{1}{[\frac{N}{k}]-r}\sum_{i=1}^{[\frac{N}{k}]-r}(Y_{r,i k}-\bar{Y}) ^2,\hspace{.5in} S^2_{r,k,1}=\frac{1}{[\frac{N}{k}]-r}\sum_{i=1}^{[\frac{N}{k}]-r}(Y_{r,i }-\bar{Y})^2 \label{Svar}
\end{equation}
where $Y_{1,j}=Z_{j+1}-Z_j$ and $Y_{2,j}=Z_{j+2}-2Z_{j+1}+Z_j$ correspondingly.
One can easily verify that $Y_{r,ik}\stackrel{d}{=}k^{H'_k}Y_{r,i}$ for $r=1,2$. So $\sigma^2_{r,k,2}=k^{2H'}\sigma^2_{r,k,1}$ ,
where $\sigma^2_{r,k}= \mbox {Var } (Y_{r,ik})$ and $\sigma^2_{r,1}= \mbox{Var}(Y_{r,i})$. Thus $S^2_{r,k,2}$ and $S^2_{r,k,1}$ are corresponding sample variances and estimates of $\sigma^2_{r,k}$ and $\sigma^2_{r,1}$ respectively. So
for different values of $k$ we evaluate $\hat{H'}_k$ by the relation.
\begin{equation}gin{equation}
\frac{S^2_{r,k,2}}{S^2_{r,k,1}}=k^{2\hat{H'_k}}\label{eq:1},
\end{equation}
and estimate $H'$ as the mean of such different $\hat{H'_k}$'s by
\begin{equation}gin{equation}
\hat{H'}= \frac{1}{2(K^*-1)}\sum_{k=2}^{K^*}\log\big(\frac{S^2_{r,k,2}}{S^2_{r,k,1}}\big)/\log(k).\label{Hes}\\
\end{equation}
\noindent Now we compare the performance of the introduced method for estimation of Hurst parameter with FA, DFA and DMA methods.
First we simulate 10000 samples from fBm with different Hurst parameters as $H=0.1, 0.2, \ldots, 0.9$. Then we estimate the Hurst parameters by different methods, FA, DFA, DMA and our two methods first difference (diff 1) and second difference (diff 2) ones with 500 repetitions. The MSE of the methods are plotted in Figure 2. As it is shown in the figure, the diff 2 method always have much better performance than the previous methods. The diff 1 method is the best for the Hurst parameters between $0.1$ and $0.5$, but for Hurst between $0.6$ and $0.9$ the diff 2 method is the best.
\input{epsf}
\epsfxsize=4.3in
\epsfysize=2.3in
\begin{equation}gin{figure}
\centerline{$\hspace{-.7in}$\epsffile{CU.eps}
}
\caption{\footnotesize
Mean square error in estimation of Hurst index of using 10000 samples of fBm with 500 repetitions.
}
\end{figure}
\section{Conclusions}
In this paper some heuristic method for the estimation of scale and Hurst parameter of discrete scale invariant processes with piece-wise linear drift.
As many DSI processes are accompanied by some piece-wise linear drift, the method of this paper in estimating and eliminating of such piecewise drifts, motivates
further research in this way and provide a platform to extend the applications
of DSI processes.
So one would expect to have a better understanding of the processes involving DSI behavior.
As an example, for the presented part of Dow Jones indices it is shown that the DSI behavior is accompany with piece-wise linear drift. We also presented new method to improve estimation of the Hurst parameter of DSI processes.
Comparing the presented method for the estimation of the Hurst parameter of Hsssi processes with the celebrated methods FA, DFA and DMA shows
its superior by producing less mean squared errors.
This paper could initiate further research and application in compare to the existing methods by applying our method in estimating and eliminating different piece-wise drifts.
Our method has the privilege to present a better estimation of the Hurst parameter of DSI processes by eliminating such additional structure for the variations.
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\begin{document}
\title{Super-Sensitive Ancilla-Based Adaptive Quantum Phase Estimation}
\author{Walker Larson}
\author{Bahaa Saleh}
\email{[email protected]}
\affiliation{
CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, Florida 32816, USA}
\date{\today}
\begin{abstract}
The super-sensitivity attained in quantum phase estimation is known to be compromised in the presence of decoherence. This is particularly patent at blind spots -- phase values at which sensitivity is totally lost. One remedy is to use a precisely known reference phase to shift the operation point of the sensor to a less vulnerable phase value. We present here an alternative approach based on combining the probe with an ancillary degree of freedom containing adjustable parameters to create an entangled quantum state of higher dimension. We validate this concept by simulating a configuration of a Mach-Zehnder interferometer with a two-photon probe and a polarization ancilla of adjustable parameters, entangled at a polarizing beam splitter. At the interferometer output, the photons are measured after an adjustable unitary transformation in the polarization subspace. Through calculation of the Fisher information and simulation of an adaptive estimation procedure, we show that optimizing the adjustable polarization parameters using an adaptive measurement process provides globally super-sensitive unbiased phase estimates for a range of decoherence levels, without prior information or a reference phase.
\end{abstract}
\maketitle
\section{\label{sec:into}Introduction}
It is well-known that bounds on the sensitivity of optical measurements based on coherent-state probes can be surpassed by use of non-classical light \cite{Giovannetti2011,Simon2016,Demkowicz-Dobrzanski2012,Holland1993,Braunstein1995,Banaszek2009}.
As a canonical example, estimates of the optical phase are bounded by the shot-noise or classical limit (CL) in classical sensing strategies and by the Heisenberg-limit (HL) \cite{Lee2002,Giovannetti2004,Giovannetti2006} in non-classical sensing strategies. As dictated by the Cram\'{e}r-Rao bound \cite{Helstrom1969,Paris2009}, the variance of estimates employing an average of $N$ photons can scale at best as $\frac{1}{\sqrt{N}}$ for classical probes, while the variance of estimates employing exactly $N$ photons can scale at best as $\frac{1}{N}$; an estimate that achieves a variance between these ranges is commonly referred to as super-sensitive.
The super-sensitivity attained in quantum phase estimation is compromised in the presence of any finite source of quantum state imperfection or decoherence \cite{Okamoto2008,Shaji2007,Koodynski2013,Dowling2008}, making it rather challenging to reach the HL goal. Even worse, for certain values of phase, which we refer to as blind spots, the measurement fails to provide \textit{any} sensitivity \cite{Israel2014} \cite{Matthews2016}. Traditional adaptive phase estimation overcomes this issue by employing a reference phase and iteratively moving the operation point of the interferometer to the range for which it is most sensitive. To our knowledge, in every demonstration of the use of two-photon interferometry, a reference phase has been required to observe super-sensitivity.
Lately, much work has been done to investigate how ancillary photons or degrees of freedom (DoFs) can be used to aid quantum estimation strategies against such deleterious effects \cite{Huang2016,Demkowicz-Dobrzanski2014,Dur2014}. More specifically, recent work \cite{Jachura2016} has shown that in the presence of partial two-photon spectral distinguishability, an effect that degrades two-photon interference while leaving single-photon interference unhindered, it is possible to employ an ancillary DoF to fortify super-sensitive two-photon states against the total loss of sensitivity at the blind spots. By coupling an ancillary DoF (ancilla) to the probe DoF it was possible to theoretically model and experimentally measure sensitivity above the CL using coincidence measurements at a blind spot.
In the present work, we consider ancilla-based phase estimation with a two-photon quantum state impaired by decoherence described by the 'depolarizing-channel' model, which is one of the most general models of decoherence in two-photon system. This effect degrades both two-photon and single-photon interference. We use a configuration of a Mach-Zehnder interferometer with a
two-photon probe and a polarization ancilla of adjustable parameters, entangled at a polarizing beam splitter. At the interferometer output, the photons are measured after an adjustable unitary
transformation in the polarization subspace. Through calculation of the Fisher information we show that fortification through an ancillary DoF protects the quantum advantage afforded to two-photon measurements for a range the depolarization probabilities (decoherence levels). Within this range, it is possible to use the ancillary DoF, rather than a reference phase, which must be placed within the interferometer itself, to retain the sensitivity of the interferometer.
We also show that adaptive phase estimation can be performed in this paradigm by tuning the polarization (ancilla) of the input two-photon states and the two-photon polarization measurements that are made at the output of the system, rather than tuning the optical system itself. Previous experimental and theoretical treatments of this topic have only considered the case where precise prior information of the phase exists, and this is, to our knowledge, the first theoretical work that considers the entire adaptive process. Our simulations suggest that just as in the case of using a reference phase, adaptive tuning of the ancillary degree of freedom provides unbiased estimators that are super-sensitive for a range of decoherence probabilities. The techniques developed here can therefore play a critical role in phase estimation tasks where introduction of a reference phase is not feasible.
\section{ Effect of Decoherence on Phase Sensitivity}
\subsection{Two-photon probe in a pure state}
The phase $\phi$ introduced by transmission through an optical element is typically measured by placing the element in one arm of a MZI and using an optical probe at the input ports together with an appropriate measurement at the output ports. An unbiased phase estimate $\tilde{\phi}$ based on a measurement outcome $M$ has a statistical variance satisfying the Cram\'er-Rao bound, ${\mathrm{Var}}(\tilde{\phi}) \geq \frac{1}{F^(\phi)}$, where $F(\phi)= p( M |\phi) \left[ \pd{}{\phi} \ln p( M |\phi) \right]^2$ is the Fisher information and $p( M |\phi)$ is the conditional probability distribution of measuring $M$ given $\phi$. This variance, which defines the sensitivity of the measurement, clearly depends on the choice of the probe and the measurement.
Here, we limit ourselves to two-photon optical probe states and two-photon measurements --- either through coincidence between single-photon detection, photon-number resolving detection, or both. When the quantum state is a pure state with one photon in each of the interferometer input ports, it turns out that measuring two-photon coincidence and double counts at the output ports of the interferometer is in fact the optimal measurement, with $F(\phi)=4$, $\forall$ $\phi$, achieving the HL that corresponds to the highest sensitivity achievable using two photons. For comparison, a classical optical probe in a coherent state with an average of two photons obtains a maximum value of $F(\phi)=2$. In this case, the quantum two-photon probe offers a factor of 2 advantage in the variance of estimates over a classical probe with the same mean number of photons \cite{Kacprowicz2010,Ono2013}.
\subsection{Two-photon probe with decoherence}
One would expect that the phase sensitivity achievable with an optical probe in a two-photon state subjected to decoherence would deteriorate gradually as the strength of decoherence increases. It turns out that the effect of decoherence also depends significantly on the actual value of the phase $\phi$. To verify this, we subject the otherwise pure input state to the 'depolarization' channel of decoherence. This is a model that randomly replaces the input state describing a single DoF of a photon (here the interferometer path mode) with a mixed state. The process occurs with probability $p$. The channel-sum representation of the operation can be used to model depolarization acting on a multi-degree-of-freedom or multi-photon states (See appendix).
To investigate how this affects the sensitivity of our system, we calculated the Fisher information $F(\phi;p)$ using the depolarized input state. Towards this calculation, a pure input state describing the interferometer-path-modes (probe) of a photon pair was represented as superpositions of vectors of the form $\ket{P_1} \otimes \ket{P_2}$, where $P$ denotes the binary probe DoF, and $1,2$ refer to the first and second photon. To create the optimal probe state \cite{C.K.Hong1987},
\begin{equation}
\ket{\psi_0}=\tfrac{1}{\sqrt{2}} \left[\ket{u}\ket{l}+\ket{u}\ket{l}\right],
\end{equation}
where $\ket{u}$ and $\ket{l}$ correspond to the upper and lower input ports of the MZI, respectively, was used. The density matrix $\rho_0=\ket{\psi_0}\bra{\psi_0}$ of this pure state is altered by decoherence, becoming a mixed state
\begin{equation}
\rho(p)= \mathcal{E}_p( \rho_0 ),
\end{equation}
where $\mathcal{E}_p$ represents the depolarization operation, which is a function of the probability $p$ (see appendix). This state is then evolved unitarily by the interferometer, encoding information about the phase difference $\phi$ into the state, leaving the output state
\begin{equation}
\rho(\phi;p)= U(\phi) \rho(p) U^{\dagger} (\phi).
\end{equation}
The probability of measuring a coincidence count is then
\begin{equation}
P_c(\phi;p)=\textrm{Tr}[\rho(\phi;p)\Pi_c],
\end{equation}
and the probability of measuring a double count is
\begin{equation}
P_d(\phi;p)=\textrm{Tr}[\rho(\phi;p)\Pi_d],
\end{equation}
where
\begin{equation}
\Pi_c=\ket{u}\ket{l}\bra{u}\bra{l}+\ket{l}\ket{u}\bra{l}\bra{u},
\end{equation}
and
\begin{equation}
\Pi_d=\ket{u}\ket{u}\bra{u}\bra{u}+\ket{l}\ket{l}\bra{l}\bra{l},
\end{equation}
are the operators describing coincidence and double counts, respectively.
From these probabilities, we find
\begin{equation}
F(\phi;p)=\frac{8 (p-1)^4 \sin^2(2 \phi)}{1-(p-1)^4 \cos(4 \phi)-(p-2) p ((p-2) p+2))},
\end{equation} which is plotted in Fig \ref{DepolMZI}.
When the depolarization operation contaminates the input state with any non-zero probability $p$, there is a drastic change in $F(\phi;p)$, and we see the emergence of blind spots --- phases for which the sensitivity afforded by both two-photon measurements drops sharply to zero. Specifically, for $\phi \in \lbrace \phi_{BS} \rbrace= \left \lbrace 0,\frac{\pi}{2},\pi \right \rbrace$, we find that $F(\phi;p)=0$. This behavior is plotted in Figure \ref{DepolMZI}. With this drastic change, it is clear that the once-optimal interferometer will now be completely insensitive to phase values in the neighborhood of any of these blind spots. At these spots, finding an unbiased estimator of $\phi$ becomes impossible as $F(\phi;p)$ approaches zero.
\begin{figure}
\caption{Fisher information as a function of the phase $\phi$ for two cases. (a) When a MZI is fed with the pure two-photon input state, the sum of the Fisher information provided by coincidence (blue) and double-count (red) measurements leads to the total (purple) $F(\phi;0)$ that can be gained from two-photon measurements at the output of an ideal Mach-Zehnder interferometer. (b) As in (a), but with non-zero decoherence probability $p$ (shown for $p=0.005$). Blind spots appear at $\phi=0,\frac{\pi}
\label{DepolMZI}
\end{figure}
\subsection{Reference Phase}
As a simple remedy to the loss of sensitivity at a blind spot, one could add a precisely known, tunable reference phase $\phi_r$ to a path of the interferometer \cite{Ono2013}. To show this, we calculate the Fisher information $F(\phi-\phi_r;p)$ that results from measuring the probabilities of coincidence and double counts when a reference phase is used. The probability of measuring coincidence or double counts are given by $P_c(\phi-\phi_r;p)=\textrm{Tr}[\rho(\phi-\phi_r;p)\Pi_c]$ or $ P_d(\phi-\phi_r;p)=\textrm{Tr}[\rho(\phi-\phi_r;p)\Pi_d]$, respectively, and give a sensitivity described by $F(\phi -\phi_r;p)$.
The optimal sensitivity afforded to this strategy is then given by maximizing $F(\phi -\phi_r;p)$
over $\phi_r$, for which the optimal operating point will be found at $\phi-\phi_r=\frac{\pi}{4}$. This optimization provides
\begin{equation}
F_{\phi_r}(p)=4 (1-p)^4,
\end{equation}
and shows that this strategy retains super-sensitivity for values of $p<0.1591$. For reference, this corresponds to an interference visibility of $\frac{2}{3}$ \cite{Rarity1990} if the depolarization operation is the only source of imperfect visibility.
While the introduction of a phase reference conveniently obviates the repercussions of the phase dependence on sensitivity, we are left with a question: could we perform some other modification to our system that does not require physical changes inside the interferometer itself? To answer this question affirmatively, we introduce a second degree of freedom, an ancilla, which we implement by means of the polarization of the photon pair.
\section{Ancilla Fortification}
\begin{figure}
\caption{A path-polarization two-photon entangled state is created by transmitting photon-pairs generated by a source S in the state $\ket{H}
\label{AncillaD}
\end{figure}
In a configuration using polarization of the photon pair used as an ancillary DoF, the input state begins as a pair of photons with crossed polarizations in a single spatial mode. The pure input state describing the interferometer-path-modes (probe) and polarization (ancillary) DoFs of a photon pair is represented as linear combinations of vectors of the product form $ \ket{P_1}\ket{A_1} \otimes \ket{P_2}\ket{A_2}$, where $P,A$ denote the binary probe and ancillary DoFs, respectively, and $1,2$ refer to the first and second photon, respectively. The added DoF creates a large set of possible states; while the ones we will proceed to use are not the most general forms of states in this set, we found them to provide the best sensitivity out of all the combinations that we have studied.
We start with an input state with a horizontally polarized photon in the same spatial mode as a vertically polarized photon is subjected to the transformation dictated by transmission through a half-wave plate (HWP) with an optic axis at an angle $\alpha_1$. Spatial-mode-polarization correlations are then created by use of a polarizing beam splitter (PBS)that enacts the transformation $\ket{H} \rightarrow \ket{l}\ket{H} $ and $\ket{V} \rightarrow \ket{u}\ket{V} $. Correlations are further tuned by placing a second HWP with an optic axis at an angle $\alpha_2$ in the upper arm, after the PBS, as shown in Fig \ref{AncillaD}. The states created by this process are states that, in general, are not optimal when $p=0$. This may not be surprising; for a large number of estimation tasks, the optimal quantum probe states are rarely optimal once decoherence is introduced to the system \cite{Demkowicz-Dobrzanski2009,Dorner2009,MacCone2009,Escher2011}. The input state after preparation is given by
\begin{multline}
\ket{\psi_{in}}^{(A)}=\cos^2 \alpha_1 \ket{l}\ket{H}\ket{l}\ket{H} -\sin^2 \alpha_1 \ket{u}\ket{\alpha_2}\ket{u}\ket{\alpha_2}\\
+\cos \alpha_1 \sin \alpha_1 \left( \ket{l}\ket{H}\ket{u}\ket{\alpha_2}- \ket{u}\ket{\alpha_2}\ket{l}\ket{H}\right),
\end{multline}
where the polarization state $\ket{ \alpha_2}=\cos \alpha_2 \ket{H} -\sin \alpha_2 \ket{V}$.
This state is then acted on by the depolarizing channel and transformed by transmission through the interferometer. A pair of HWPs and quarter-wave plates (QWP) are placed into each of the output arms of the interferometer. The HWPs have optics axes at angles $\beta_1$ (upper arm) and $\beta_2$ (lower arm), while the QWPs have optic axes at angles $2 \beta_1$ (upper arm) and $2 \beta_2$ (lower arm). Two-photon measurements are then made at the output ports of a PBS placed after each pair of wave plates. As a result, the output state before the final pair of of PBSs, $\rho^{(a)} (\phi; p, \Theta)$, is completely characterized by the parameter set $\Theta=\left \lbrace \alpha_1, \alpha_2, \beta_1, \beta_2\right \rbrace$.
Now, the calculation of the optimal Fisher information for a given value of $\phi$, $F_{\textrm{opt}}^{(a)}(\phi;p)$, becomes an optimization over the parameter set $\Theta$ as opposed to a reference phase $\phi_r$. The probabilities of measuring the two photon state in output path modes $\left\lbrace k_1, k_2\right\rbrace$ and polarization modes $\lbrace s_1,s_2 \rbrace$ needed to calculate $F_{\textrm{opt}}^{(a)}(\phi; p)$ are given by
\begin{equation}
P_{k_1,s_1,k_2,s_2}=\textrm{Tr} [ \rho^{(a)} (\phi; p, \Theta) \Pi_{k_1,s_1,k_2,s_2}],
\end{equation}
where
\begin{equation}
\Pi_{k_1,s_1,k_2,s_2}=\ket{k_1}\ket{s_1}\ket{k_2}\ket{s_2}\bra{k_1}\bra{s_1}\bra{k_2}\bra{s_2}.
\end{equation}
For each value of $\phi$, an exhaustive search over the sensitivity that results from a given set $\Theta$ determines the values that provide the optimal sensitivity $F_{\textrm{opt}}^{(a)}(\phi;p)=\underset{\left\lbrace \Theta \right\rbrace}{\textrm{max}} \left\lbrace F^{(a)}(\phi_;p,\Theta) \right\rbrace$. The sensitivity that results from this optimization is plotted in Figure 3 for $p=0.05$. While operation at the blind spots may not be as sensitive as the operation that could be attained using a reference phase, it is clear that it makes possible the quantum super-sensitive advantage in a sensing regime where no sensitivity was possible prior. We find that super-sensitivity is globally attainable for all phases for decoherence probabilities less than $p=0.072$.
\begin{figure}
\caption{Fisher information is plotted for a two-photon interferometer with (purple) and without (orange) the employment of an ancillary degree of freedom. Ancilla fortification allows for measurements that retain sensitivity at blind spots $0$,$\frac{\pi}
\label{RefVAnc}
\end{figure}
\section{Adaptive Phase Estimation}
To show that this methodology can find application in the general setting of quantum phase estimation, we have conducted simulations demonstrating how the ancilla-aided strategy provides a full platform for super-sensitive adaptive phase estimation. Simulations of adaptive phase estimation were conducted by supplementing simulations of maximum likelihood estimation with feedback based on measurement results. Traditional adaptive phase estimations uses feedback from measurements to update the value of the reference phase $\phi_r$, setting it to the value that maximizes the sensitivity of the optical system, assuming that the true value of $\phi$ is equal to the most recent estimate $\tilde{\phi}$ \cite{Pope2004,Hou2016,Lerch2014,Okamoto2012,Higgins2007}. Likewise, when using ancilla fortified states without a reference phase, feedback updates the parameter set $\Theta$, setting the parameters equal to the set that maximize the sensitivity of the system for the assumed value of $\phi=\tilde{\phi}$.
Specifically, performing an $n^{th}$ two-photon measurement provides a result $M_n$ that corresponds to the measurement operator element $\Pi_n \in \Pi_{k_1,k_2,s_1,s_2}$. The likelihood of the result is $\mathcal{L}_n (\phi)=\textrm{Tr}\left[\rho(\phi) \Pi_n\right] \mathcal{L}_{n-1} (\phi)$, from which an $n^{th}$ estimate of $\phi$, $\tilde{\phi}_n=\textrm{argmax}_{\phi}\textrm{ }\mathcal{L}_n (\phi)$, is made. With the estimate $\tilde{\phi}_n$, the settings of the parameters $\Theta$ that maximize $F^{(a)}(\tilde{\phi}_n;p,\Theta)$ are updated \cite{Chapeau-Blondeau2016}. To ensure that final estimate $\tilde{\phi}_N$ after $N$ measurements is unbiased (i.e. $\left \langle \tilde{\phi}_N\right \rangle=\phi$), the initial estimate $\tilde{\phi_0}$ is chosen at random. Finally, to avoid degeneracy in the final likelihood function (which would lead to equally likely estimates spaced some period apart), a final, non-optimal series of measurements must be made. By experimentation, we have found that setting $\Theta$ to the values that are optimal at $\phi=0$, then $\phi=\frac{\pi}{2}$, for the last $3 \%$ of adaptive iterations is sufficient in avoiding this discrepancy.
For our testing of this procedure for each phase $\phi$, we set $p=0.01$ and performed $S=5000$ simulations, each detecting adaptively a total of $N=1500$ two-photon states. From these simulated trials, $S$ final estimates $\tilde{\phi}_N$ were collected, and statistics were calculated on the ensemble of these estimates. The mean and variance of estimates as a function of the true value of $\phi$ for each simulation is plotted in Figures \ref{Mean} and \ref{Variance}. For the given sample size, we find that the strategy is unbiased, and the functional dependence of the variance on $\phi$ generally follows the trend predicted by the calculation of the Fisher information. Most importantly, for this probability of decoherence, our simulations show estimates that are super-sensitive for all values of $\phi$.
\begin{figure}
\caption{Mean of final phase estimates after $S=5000$ simulations of $N=1500$ adaptive two-photon state detections (Blue) are plotted for $p=0.01$ with the 1:1 correspondence expected for unbiased estimators (Orange). }
\label{Mean}
\end{figure}
\begin{figure}
\caption{Variance of final phase estimates after $S=5000$ simulations of $N=1500$ adaptive two-photon state detections (Blue) for $p=0.01$ follows the functional form of the Cram\'er-Rao Bound as dictated by the Fisher information (Orange). For each value of $\phi$, a variance below the classical estimation limit is observed.}
\label{Variance}
\end{figure}
\section{Conclusion}
The accuracy of a metrological measurement is limited by inherent noise in the probe. While quantum optical probes offer a global advantage over classical probes, they can be more vulnerable to minute imperfections or contamination by weak extraneous noise. A case in point is the interferometric measurement of phase by use of a two-photon probe. Although this quantum probe offers a sensitivity advantage (super-sensitivity) under ideal conditions, if the probe is subjected to weak decoherence, the sensitivity will be globally reduced and -- surprisingly -- lost altogether at certain phase values (blind spots). While the quantum advantage may be partially regained by shifting the phase to a less vulnerable value, the insertion of a precisely known reference phase into the interferometer may not be feasible.
We have investigated an alternative approach based on supplementing the path DoF of the probe with an ancillary DoF (polarization), and creating an entangled quantum state in a Hilbert space of twice the dimensionality. We have demonstrated that the diversity added into the probe can help avoid the blind-spot predicament. This of course requires tweaking polarization parameters of unitary transformations at both the input and output ports of the interferometer before detecting the outgoing two photons. These transformations tailor the input state and the corresponding output detection strategy for optimal estimation. In this paper, the number of these adjustable parameters was limited to four, two at the input of the interferometer and two at the output. Since values of the parameters that maximize the sensitivity depend on the unknown phase itself, the optimization must be conducted adaptively. Based on extensive simulation of the adaptive process, we conclude that for a range of decoherence strengths, super-sensitivity is indeed obtained for any phase.
While the example investigated in this paper uses path and polarization as the principal and ancillary DoFs, the reverse is possible and other DoFs, may also be used, as long as implementation of the prerequisite unitary transformations are practical. Likewise, we expect this methodology to find application in the larger of optical parameter estimation. In the more general task of estimating a unitary transformation, it is expected that blind spots will appear in the estimation of any of the parameters that encode the transformation: in this case, a binary DoF offering a secondary channel for enhanced estimation may be the only way to overcome these parameter blind spots.
\section{Appendix}
Decoherence of a quantum state is described by a transformation that converts the density operator $\rho$ into a density operator
\begin{equation}
\mathcal{E}(\rho)=\sum_{i} E_i \rho E_i^{\dagger},
\end{equation}
where $E_i$ are appropriate operation elements of this operator-sum representation.
In the case of the depolarizing channel acting on a binary quantum state, the operation elements are given by
\begin{equation}
\begin{aligned}
E_1=\sqrt{1-\frac{3p}{4}}\mathcal{I}, \qquad
E_2=\sqrt{\frac{p}{4}}\sigma_x, \\
E_3=\sqrt{\frac{p}{4}}\sigma_y, \qquad
E_4=\sqrt{\frac{p}{4}}\sigma_z,
\end{aligned}
\end{equation}
where $\hat{\sigma_i}$ are the two-dimensional Pauli matrices, and $\mathcal{I}$ is the two-dimensional identity matrix \cite{Nielsen2011}.
For composite systems, the decoherence operation can apply to individual subsystems. For example if $\rho=\rho^A \otimes \rho^B$, decoherence acting on subsystem $A$ alone is implemented by the operation elements
\begin{equation}
E_i^{AB}=E_i^A \otimes \mathcal{I}^B,
\end{equation}
where superscripts $A$ and $B$ denote transformations acting on respective systems $A$ and $B$.
With this definition, decoherence can act independently on any combination of degrees of freedom (DoF) of either photon.
In this paper, we consider a two-photon state with two DoFs, path and polarization, with polarization playing the role of the ancilla. Decoherence acts equally on both photons in the path DoF, leaving the polarization DoF unaltered. The operation is
\begin{equation}
\mathcal{E}_p(\rho)= \mathcal{E}^{(1)}_p( \mathcal{E}^{(2)}_p( \rho ))=\mathcal{E}^{(2)}_p( \mathcal{E}^{(1)}_p( \rho )),
\end{equation}
where $\mathcal{E}^{(i)}_p$ denotes the decoherence channel acting on the interferometer-path degree of freedom of the $i^{th}$ photon with probability $p$.
\begin{thebibliography}{36}
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\end{document} |
\begin{document}
\mathfrak{m}aketitle
\begin{abstract}
Let $k$ be an arbitrary field.
In this note, we show that if a sequence of relatively prime positive integers ${\bf a}=(a_1,a_2,a_3,a_4)$ defines a Gorenstein non complete intersection monomial curve ${\mathfrak{m}athcal C}({\bf a})$ in
${\mathfrak{m}athbb A}_k^4$, then there exist two vectors ${\bf u}$ and ${\bf v}$ such that
${\mathfrak{m}athcal C}({\bf a}+t{\bf u})$ and ${\mathfrak{m}athcal C}({\bf a}+t{\bf v})$ are also Gorenstein non complete intersection affine monomial curves
for almost all $t\geq 0$.
\\
{\sc Keywords:} affine monomial curve, numerical semigroup, Gorenstein curve.
\\
{\sc Mathematical subject classification:} 13C40, 14H45, 13D02, 20M25.
\end{abstract}
Let ${\bf a} = (a_1, \ldots a_n)$ be a sequence of positive integers and $k$ be an arbitrary field.
If $\phi: k[x_1, \dots, x_n ]\to k[t]$ is the ring homomorphism defined by
$\phi(x_i) = t^{a_i}$, then $I({\bf a}):= \ker \phi$ is a prime ideal of height $n-1$ in $R:=k[x_1, \ldots, x_n]$
which is a weighted homogeneous binomial ideal with the weighting $\deg x_i:= a_i$ on $R$. It is the
defining ideal of the affine monomial curve ${\mathfrak{m}athcal C}({\bf a})\subset {\mathfrak{m}athbb A}_k^n$ parametrically defined by ${\bf a}$
whose coordinate ring is $S({\bf a }):= {\rm Im }\phi = k[t^{a_1}, \ldots, t^{a_n}]\simeq R/I({\bf a})$. As $S({\bf a})$ is isomorphic to $S(d{\bf a})$ for all integer $d\geq 1$,
we will assume without loss of generality that $a_1, \ldots, a_n$ are relatively prime. Observe that $S({\bf a})$ is also the semigroup ring of
the numerical semigroup $\langle a_1, \ldots a_n\rangle\subset{\mathfrak{m}athbb N}$ generated by $a_1, \ldots ,a_n$.
\mathfrak{m}edskip
As observed in \cite{De} where Delorme characterizes sequences ${\bf a}$ such that $S({\bf a })$ is a complete intersection,
this fact does not depend on the field $k$ by \cite[Corollary~1.13]{He}.
On the other hand, it is well-known that $S({\bf a })$ is Gorenstein if and only if the numerical semigroup $\langle a_1, \ldots a_n\rangle\subset{\mathfrak{m}athbb N}$
is symmetric, which does not depend either on the field $k$. We will thus
say that ${\bf a}$ is a complete intersection (respectively Gorenstein) if the semigroup ring $S({\bf a})$ is a complete intersection (respectively Gorenstein).
In \cite{JS}, it is shown that if ${\bf a}$ is a complete intersection
with $a_1>>0$, then ${\bf a} +t(a_n-a_1)(1,\dots ,1)$ is also a complete intersection for all $t$. In this note, we will use the criterion for Gorenstein
monomial curves in ${\mathfrak{m}athbb A}_k^4$ due to Bresinsky in \cite{Br} to construct a class of Gorenstein
monomial curves in ${\mathfrak{m}athbb A}_k^4$.
\mathfrak{m}edskip
First observe that for each $i$, $1\le i\le n$, there exists a multiple of $a_i$ that belongs to the numerical semigroup generated by the rest of the elements in the sequence and denote by $r_i>0$ the smallest positive integer such that $r_ia_i \in \langle a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_n\rangle$. So we have that
\begin{equation}\label{principalrelations}
\forall i,\ 1\leq i\leq n,\ r_ia_i = \sum _{j\mathfrak{n}eq i} r_{ij}a_j,\ r_{ij} \ge 0,\ r_i >0\,.
\end{equation}
\begin{definition}{\rm
The $n\times n$ matrix $D(\bf {a}):=(r_{ij})$ where $r_{ii}:= -r_i$ is called a
{\it principal matrix} associated to $\bf {a}$.
}\end{definition}
\begin{lemma}
$D(\bf {a})$ has rank $n-1$.
\end{lemma}
\begin{proof}
Since the system $D({\bf a})X=0$ has solution $X={\bf a}^T$ by (\ref{principalrelations}), $D(\bf {a})$ has rank $\le n-1$. If ${\bf b}^T$ is another solution, then the ideal defining the curve $I({\bf b})$ contains $$f_i = x_i^{r_i} -\prod_{j\mathfrak{n}eq i}x_j^{r_{ij}}$$
for all $1\le i\le n$. But $I({\bf a})= \sqrt{(f_1, \ldots, f_n)}$ thus $I({\bf a})\subseteq I({\bf b})$ because $I({\bf b})$ is prime.
Since both $I({\bf a})$ and $I({\bf b})$ are primes of the same height $n-1$, they must be equal: $I({\bf a})= I({\bf b})$.
Since $x_i^{a_j}-x_j^{a_i}\in I({\bf a})$ for all $i$ and $j$, one has that
$a_ib_j = a_jb_i$ for all $i$ and $j$ and hence the $2\times n$ matrix whose rows are ${\bf a}$ and ${\bf b}$ is of rank 1, i.e.,
$a_i = cb_i$ for some $c$. So, rank of $D({\bf a}) = n-1$.
\end{proof}
\begin{remark}{\rm
Observe that $D({\bf a})$ is not uniquely defined. Although the diagonal entries $-r_i$ are uniquely determined, there is not a unique choice for $r_{ij}$ in general.
We have the ``map" $D: {\mathfrak{m}athbb N}^{[n]} \to T_n$ from the set ${\mathfrak{m}athbb N}^{[n]}$ of sequences of $n$ relatively prime positive integers to the subset $T_n$ of $n\times n$ matrices of rank $n-1$ with negative integers on the diagonal and non negative integers outside the diagonal.
Note that we can recover $\bf {a}$ from $D(\bf {a})$ by factoring out the greatest common divisor of the $n$ maximal minors
of the $n-1\times n-1$ submatrix of $D(\bf {a})$ obtained by removing the first row. In other words, call $D^{-1}:T_n \to N^{[n]}$
the operation that, for $M\in T_n$, takes the first column of $\operatorname{adj} (M)$ and then factors out the g.c.d. to get an element in $N^{[n]}$. Then,
$D^{-1}(D({\bf a}))={\bf a}$ for all ${\bf a}\in N^{[n]}$.
Now given a matrix $M\in T_n$, $D(D^{-1}(M)) \mathfrak{n}eq M$ in general as the following example shows: if
$M=\left[
\begin{matrix}
-4&0&1&1\\
1&-5&4&0\\
0&4&-5&1\\
3&1&0&-2\\
\end{matrix}
\right]$ then $D^{-1}(M)=(7,11,12,16)$ and $D(D^{-1}(M) \mathfrak{n}eq M$ (it is easy to check for example that $r_2=3<5$).
}\end{remark}
\mathfrak{m}edskip
We now focus on the case of Gorenstein monomial curves in ${\mathfrak{m}athbb A}_k^4$ so assume that $n=4$.
If ${\bf a}$ is Gorenstein but is not a complete intersection,
by the characterization in \cite[Theorems~3 and 5]{Br}, there is a principal matrix $D({\bf a})$ that has the following form:
\begin{equation}\label{embdim4GorMat}
\left[
\begin{matrix}
-c_1&0& d_{13} &d_{14}\\
d_{21}&-c_2&0&d_{24}\\
d_{31}&d_{32}& -c_3&0\\
0&d_{42}&d_{43}&-c_4\\
\end{matrix}
\right]
\end{equation}
with $c_i\ge 2$ and $d_{ij}>0$ for all $1\le i,j\le 4$, the columns summing to zero and all the columns of the adjoint being relatively prime.
The first column of the adjoint of this matrix is $-{\bf a}^T$ and Bresinsky's characterization also says that
the first column (after removing the signs) of the adjoint of a principal matrix $D({\bf a})$ of this form defines a Gorenstein curve
provided the entries of this column are relatively prime.
\mathfrak{m}edskip
The following is a slight strengthening of this criterion.
\begin{theorem}\label{criterion}
Let $A$ be a $4\times 4$ matrix of the form
$$
A=
\left[
\begin{matrix}
-c_1&0& d_{13} &d_{14}\\
d_{21}&-c_2&0&d_{24}\\
d_{31}&d_{32}& -c_3&0\\
0&d_{42}&d_{43}&-c_4\\
\end{matrix}
\right]
$$
with $c_i\ge 2$ and $d_{ij}>0$ for all $1\le i,j\le 4$, and all the columns summing to zero. Then the first column of the adjoint of $A$
{\rm(}after removing the signs{\rm)}
defines a monomial curve provided these entries are relatively prime.
\end{theorem}
\begin{proof}
Consider such a matrix $A$ and let $a_1, a_2, a_3, a_4$ be the entries in the first column of the adjoint of $A$ (after
removing the signs). Since we are assuming that they are relatively prime, there exist integers $\lambda_1,\ldots,\lambda_4$ such that
$\lambda_1 a_1+\cdots+\lambda_4 a_4=1$.
It suffices to show that the four relations in the rows of $A$ are principal relations.
We will show this for the first row and the other rows are similar.
Suppose that $b_{11} a_1 = b_{12}a_2+b_{13}a_3+b_{14}a_4$ is a relation with $b_{11}\geq 2$ and $b_{12}, b_{13}, b_{14}\geq 0$
and let's show that $b_{11}\geq c_1$.
Since the system
$
\left[
\begin{matrix}
-b_{11}&b_{12}& b_{13} &b_{14}\\
d_{21}&-c_2&0&d_{24}\\
d_{31}&d_{32}& -c_3&0\\
0&d_{42}&d_{43}&-c_4\\
\end{matrix}
\right]Y = 0$ has a nontrivial solution, namely $Y = (a_1, a_2, a_3, a_4)^T$,
we see that it has determinant zero.
So there exist $x_i$ such that
\begin{equation}\label{detzero}
(1,x_2, x_3, x_4)\left[
\begin{matrix}
-b_{11}&b_{12}& b_{13} &b_{14}\\
d_{21}&-c_2&0&d_{24}\\
d_{31}&d_{32}& -c_3&0\\
0&d_{42}&d_{43}&-c_4\\
\end{matrix}
\right]= 0\,.
\end{equation}
Consider the matrix $T_4 =\left[
\begin{matrix}
-b_{11}&b_{12}& b_{13} &b_{14}\\
d_{21}&-c_2&0&d_{24}\\
d_{31}&d_{32}& -c_3&0\\
\lambda_1&\lambda_2 &\lambda_3 &\lambda_4\\
\end{matrix}
\right]$.
If the determinant of $T_4$ is $-t_4$, then the last column of its adjoint is $-t_4 (a_1, a_2, a_3, a_4)^T$.
This is because $T_4 (a_1, a_2, a_3, a_4)^T = (0,0,0,1)^T$.
Hence, looking at the element in the last row and last column of the adjoint of $T_4$, one gets using (\ref{detzero}) that
$$- t_4 a_4= \left| \begin{matrix}
-b_{11}&b_{12}& b_{13} \\
d_{21}&-c_2&0 \\
d_{31}&d_{32}& -c_3 \\
\end{matrix}
\right| = \left| \begin{matrix}
0&-x_4d_{42}&-x_4 d_{43} \\
d_{21}&-c_2&0 \\
d_{31}&d_{32}& -c_3 \\
\end{matrix}
\right| = -x_4a_4\,.$$
Hence $t_4 = x_4$, and since $t_4$ is an integer, so is $x_4$.
Now, looking at the element in the last column and first row of the adjoint of $T_4$, one has
$$t_4a_1 = \left| \begin{matrix}
b_{12}&b_{13}& b_{14} \\
-c_2&0&d_{24} \\
d_{32}& -c_3&0 \\
\end{matrix}
\right| = b_{12}c_3d_{24}+ b_{13}d_{32}d_{24}+b_{14}c_2c_3>0\,.$$
So, $x_4 = t_4$ is now a positive integer.
Consider the matrix $T_2 = \left[
\begin{matrix}
-b_{11}&b_{12}& b_{13} &b_{14}\\
d_{31}&d_{32}&-c_3&0 \\
0&d_{42}&d_{43}& -c_4\\
\lambda_1&\lambda_2 &\lambda_3 &\lambda_4\\
\end{matrix}
\right]$
which determinant is denoted by $-t_2$. By similar calculations, we see that $x_2=t_2$ is an integer and,
focusing on the element in the last column and third row of the adjoint of $T_2$, one gets that
$$
t_2a_3 = \left| \begin{matrix}
-b_{11}&b_{12}& b_{14} \\
d_{31}&d_{32}&0 \\
0& d_{42}&-c_4 \\
\end{matrix}
\right| = b_{11}c_4d_{32}+ b_{12}d_{31}c_{4}+b_{14}d_{31}d_{42}>0
$$
so $x_2=t_2$ is also a positive integer.
Similarly, using the matrix $T_3= \left[
\begin{matrix}
-b_{11}&b_{12}& b_{13} &b_{14}\\
d_{21}&-c_2&0&d_{24}\\
0&d_{42}&d_{43}& -c_4\\
\lambda_1&\lambda_2 &\lambda_3 &\lambda_4\\
\end{matrix}
\right]$ of determinant $-t_3$, one gets that
$x_3 = -t_3$ and hence $x_3$ is an integer.
However, by calculating the entry in the last column and second row of the adjoint of $T_3$, one gets that
$$(-t_3)a_2 = \left| \begin{matrix}
- b_{11}&b_{13}& b_{14} \\
d_{21}&0&d_{24} \\
0& d_{43}&-c_4 \\
\end{matrix}
\right| = b_{11}d_{43}d_{24}+b_{13}d_{21}c_4+b_{14}d_{21}d_{43}>0,$$
and hence, $x_3$ is again a positive integer.
So, $b_{11} = x_2d_{21}+x_3d_{31}\ge d_{21}+d_{31} = c_1$ as desired.
Since we can make any of the $c_i$'s the first row, by rearranging the $a_i$'s suitably, this proves that all of the rows are principal relations and this is a principal matrix. Hence ${\bf a}=(a_1, a_2, a_3, a_4)$ is Gorenstein by Bresinsky's criterion.
\end{proof}
Denote now the sequence of positive integers by ${\bf a}=(a,a+x,a+y,a+z)$ for some $x,y,z>0$. In other words,
we assume that the first integer in the sequence is the smallest but after that we do not assume any ascending order.
Recall that we have assumed, without loss of generality, that ${\rm gcd}(a,x,y,z)=1$.
The following result gives two families of Gorenstein monomial curves in ${\mathfrak{m}athbb A}_k^4$ by translation from a given Gorenstein curve.
\begin{theorem}\label{main}
Given any Gorenstein non complete intersection monomial curve ${\mathfrak{m}athcal C}({\bf a})$ in ${\mathfrak{m}athbb A}_k^4$, there exist two vectors ${\bf u}$ and ${\bf v}$ in ${\mathfrak{m}athbb N}^4$ such that for all $t\ge 0$, ${\mathfrak{m}athcal C}({\bf a} +t{\bf u})$ and ${\mathfrak{m}athcal C}({\bf a}+t{\bf v})$ are also Gorenstein non complete intersection monomial curves whenever the entries of the corresponding sequence {\rm(}${\bf a} +t{\bf u}$ for the first family, ${\bf a} +t{\bf v}$ for the second{\rm)} are
relatively prime.
\end{theorem}
\begin{proof}
Let $D({\bf a})$ be the principal matrix of ${\bf a}$ given in (\ref{embdim4GorMat}).
Then let ${\bf u}$ be the vector of $3 \times 3 $ minors of the $3\times 4$ matrix
with
$U = \left[
\begin {matrix}
d_{21}&-c_2&0&d_{24}\\
1&0 & -1&0\\
0&d_{42}&d_{43}&-c_4\\
\end{matrix}\right]
$
so that $U {\bf u}^T = (0,0,0)^T$.
Focusing on the second row, we see that $u_1 = u_3$. Similarly, let ${\bf v}$ be the third adjugate of the $3\times 4$ matrix
$V = \left[
\begin{matrix}
-c_1&0& d_{13} &d_{14}\\
0&-1&0&1\\
d_{31}&d_{32}& -c_3&0\\
\end{matrix}
\right]
$ so that $V {\bf v}^T = (0,0,0)^T$.
We will check now that, as long as their entries are relatively primes, the sequences
${\bf a} +t{\bf u}$ and ${\bf a}+t{\bf v}$ respectively have principal matrices
$$
A_t = \left[
\begin{matrix}
-c_1-t&0& d_{13} +t&d_{14}\\
d_{21}&-c_2&0&d_{24}\\
d_{31}+t&d_{32}& -c_3-t&0\\
0&d_{42}&d_{43}&-c_4\\
\end{matrix}
\right]
\quad\hbox{and}\quad
B_t = \left[
\begin{matrix}
-c_1&0& d_{13} &d_{14}\\
d_{21}&-c_2-t&0&d_{24}+t\\
d_{31}&d_{32}& -c_3&0\\
0&d_{42}+t&d_{43}&-c_4-t\\
\end{matrix}
\right]$$
and hence define Gorenstein curves.
It is a straightforward calculation to check that
the rows of these matrices are the relations of ${\bf a} +t{\bf u}$ and ${\bf a}+t{\bf v}$ respectively, i.e.,
$A_t\times ({\bf a} +t{\bf u})^T=(0,0,0,0)^T$ and $B_t\times ({\bf a} +t{\bf v})^T=(0,0,0,0)^T$ and it suffices to check it for $t=1$.
Consider the vector $A_1 \times ({\bf a} + {\bf u})^T$.
If we add the first row of $D({\bf a})$ to $U$ to make a square matrix $U'$ then the determinant of $U'$, expanding by its third row, is $a-(a+y)=-y$.
On the other hand, the adjoint of $U'$ has ${\bf u}$ as the first column
and ${\bf a}$ as the third column,
thereby the first row of $U'$ multiplied by ${\bf u}$ equals $-y$
and the third row of $U'$ multiplied by ${\bf a}$ also equals $-y$.
Since the first row of $A_1$ is the first row of $U'$ minus the third row of $U'$, we see that the first entry
of the vector $A_1 \times ({\bf a} + {\bf u})^T$ is zero, and a similar argument works to show that the third entry
of this vector is also zero.
Moreover, the second row of $A_1$ coincides with the second row of $D({\bf a})$ so one gets 0 multiplying by ${\bf a}$, and since it is also the
first row of $U$, one also gets 0 multiplying by ${\bf u}$ and hence the second entry in the vector $A_1\times ({\bf a}+{\bf u})^T$ is zero. The same argument
works for the fourth entry and the proof for ${\bf v}$ is similar.
Since the differences between the matrix $D({\bf a})$ and $A_t$ are all in the first and third rows, and $U$ comes from the second and fourth row, we have shown that if $A_1$ is a relation matrix for ${\bf a}+{\bf u}$ , $A_{0} = D({\bf a})$ is a relation matrix for ${\bf a}$.
Moreover, note that the changes in $D({\bf a})$ to get $A_t$ or $B_t$ did not alter the column sums and hence the columns still add up to zero.
Since $A_1$ has the form, with zeros above the diagonal, with zero in the last column first row, all the non diagonal entries non negative and has all the columns sum to zero, it is principal provided ${\bf a}+{\bf u}$ is relatively prime by Theorem \ref{criterion}.
Thus, if the cofactors are relatively prime, they form Gorenstein curves.
\end{proof}
Note that since we have assumed that the first entry in ${\bf a}$ is the smallest, the first principal relation can not be homogeneous (w.r.t. the usual grading
on $R$). Let us see what happens when 2 of the other 3 principal relations are homogeneous.
\begin {corollary}\label{corMultConj}
Let ${\bf a} = (a, a+x, a+y, a+z)$ be Gorenstein and not a complete intersection.
Suppose that both the second and the fourth rows of the matrix $D({\bf a})$ in (\ref{embdim4GorMat}) have their entries summing to zero.
Then, $x<z<y$ and ${\bf a} + t\alpha y(1,1,1,1)$ is Gorenstein for all $t\ge 0$, where $\alpha$ is a positive integer determined by $(a,a+x,a+y,a+z)$.
\end{corollary}
\begin {proof}
Since the entries of the second row sum to 0, i.e., $c_2=d_{21}+d_{24}$, we get from $c_2(a+x)=d_{21}a+d_{24}(a+z)$ that
\begin{equation}\label{row2sumto0}
c_2x=d_{24}z,
\end{equation}
i.e., $(d_{21}+d_{24})x=d_{24}z$, and hence $x<z$.
Similarly, the sum of the entries of the fourth row being 0 implies that
\begin{equation}\label{row4sumto0}
d_{42}x+d_{43}y=c_4z,
\end{equation}
i.e., $d_{42}x+d_{43}y=(d_{42}+d_{43})z$, and hence
$d_{43}(y-z) = d_{42}(z-x)$ and one has that $y>z$.
Moreover, the hypothesis in the corollary, we compute that the vector ${\bf u}$ in the proof of the theorem~\ref{main} is ${\bf u} = b(1,1,1,1)$, where $b = d_{21}c_4+d_{24}d_{43}$.
Now set $d:={\rm gcd}(x,z)$. Simplifying (\ref{row2sumto0}) by $d$, one gets that $c_2x/d=d_{24}z/d$ with ${\rm gcd}(x/d,z/d)=1$ so
$c_2=qz/d$ and $d_{24}=qx/d$ for some integer $q$.
Now simplifying (\ref{row4sumto0}) by $d$ also, one gets that $d$ divides $d_{43}y$ and $d_{43}y/d=c_4z/d-d_{42}x/d$
and hence $qd_{43}y/d=c_2c_4-d_{24}d_{42}=c_4d_{21}+d_{24}(c_4-d_{42})=c_4d_{21}+d_{24}d_{43}=b$.
Multiply $b$ by the smallest strictly positive integer $\beta$ so that $\beta b= \alpha y$ for some integer $\alpha$. Note that $\beta\leq d$ and if $t=1$, then $\alpha=qd_{43}$.
Then,
${\bf a} + t\alpha y(1, 1,1,1)$ are always Gorenstein for if they have a common factor, then it will be a common factor of
$a+ t\alpha y$ and $a+y+t\alpha y$ which necessarily is common factor of both $a$ and $y$.
But then, it will be a common factor of $x$ and $z$, i.e., of all entries in ${\bf a}$ which are relatively prime.
\end{proof}
\begin{remark}{\rm
The previous result shows that for the Gorenstein curves satisfying the hypothesis in Corollary~\ref{corMultConj}, the periodicity conjecture
holds with period $\alpha y$.
}\end{remark}
\begin{example}{\rm
The sequence ${\bf a}=(11,17,25,19)$ is Gorenstein and not a complete intersection and its principal matrix,
$\displaystyle{
D({\bf a})=
\left[
\begin{matrix}
-4&0&1&1\\
1&-4&0&3\\
3&1&-2&0\\
0&3&1&-4\\
\end{matrix}
\right]
}$, satisfies the conditions in Corollary~\ref{corMultConj}. Since $b=7$ and ${\rm gcd}(x,z) = 2$, we see that $2b$ is the smallest integral multiple of $y= 14$. Thus, adding any positive multiple of $14$ to all the entries of ${\bf a}$ will provide
a new Gorenstein sequence which is not a complete intersection.
Observe that adding $b=7$ to all entries in ${\bf a}$ they have a common factor of $2$ and it does not
result in a Gorenstein sequence.
}\end{example}
\begin{remark}{\rm
When two rows of the principal matrix different from the second and the fourth ones both have entries summing up to zero, one does not expect to see
Gorenstein curves that are not complete intersections if we take $(a+t, a+x+t, a+y+t, a+z+t)$ for $t$ large. For example,
the sequence ${\bf a}=(43,67,49,83)$ is Gorenstein and not a complete intersection and its principal matrix,
$\displaystyle{
D({\bf a})=
\left[
\begin{matrix}
-5&0&1&2\\
2&-5&0&3\\
3&1&-4&0\\
0&4&3&-5\\
\end{matrix}
\right]
}$, satisfies that both its second and third rows have entries summing up to zero.
Adding $t(1,1,1,1)$ to ${\bf a}$ will provide a Gorenstein sequence which is not a complete intersection for $t=15$, $49$ and $83$ but
does not seem to result in a Gorenstein sequence which is not a complete intersection for larger values of $t$.
}\end{remark}
We will end this note giving a precise description of a minimal graded free resolution of $S({\bf a})$ as an $R$-module
when ${\bf a}$ is Gorenstein and not a complete intersection. Assume that ${\bf a}=(a,a+x,a+y,a+z)$ is Gorenstein but not a complete intersection
and let $D({\bf a})$ be the principal matrix of ${\bf a}$ given in (\ref{embdim4GorMat}).
Since all Gorenstein grade 3 ideals in $k[x_1,\ldots,x_n]$ must be the ideal of $n$ order pfaffians of an $(n+1)\times (n+1)$ skew symmetric matrix,
so must the ideal $I({\bf a})$.
The ideal $I ({\bf a})$ described in \cite{Br} is indeed the ideal of minors of the skew symmetric matrix
$$
\phi({\bf a}) = \left[
\begin{matrix}
0&0& x_2^{d_{32}} &x_3^{d_{43}}&x_4^{d_{24}}\\
0&0&x_1^{d_{21}}&x_4^{d_{14}}&x_2^{d_{42}}\\
-x_2^{d_{32}}&-x_1^{d_{21}}&0&0&x_3^{d_{13}}\\
-x_3^{d_{43}}&-x_4^{d_{14}}&0&0&x_1^{d_{31}}\\
-x_4^{d_{24}}&-x_2^{d_{42}}&-x_3^{d_{13}}&-x_1^{d_{31}}&0\\
\end{matrix}
\right]\,.
$$
The graded resolution of $S({\bf a})$ is
$$
0 \rightarrow R(-(ac_1+(a+z)c_4+(a+x)d_{32})) \stackrel{\delta_3}{\rightarrow} R^5 \stackrel{\phi}{\rightarrow} R^{5}\stackrel{\delta_1} \rightarrow R \rightarrow S({\bf a}) \rightarrow 0
$$
where $\phi = \phi ({\bf a})$ and $\delta _1= (\delta _3) ^T =\delta({\bf a})$ for
$$\delta({\bf a}) =
[x_1^{c_1}-x_3^{d_{13}}x_4^{d_{14}},
x_3^{c_3}-x_1^{d_{31}}x_2^{d_{32}},
x_4^{c_4}-x_2^{d_{42}}x_3^{d_{43}},
x_2^{c_2}-x_1^{d_{21}}x_4^{d_{24}}, x_1^{d_{21}}x_3^{d_{43}}-x_2^{d_{32}}x_4^{d_{14}}]\,.$$
Observe that the socle degree is ${\bf a} . [c_1, d_{32}, 0, c_4]-3$ where $.$ is the dot product of the vectors.
\mathfrak{m}edskip
\begin{remark}{\rm
By following the proof of the theorem \ref {main}, we see that when
one translates ${\bf a}$ through ${\bf u}$, we get that
$$
\phi ({\bf a}+t {\bf u})= \left[
\begin{matrix}
0&0& x_2^{d_{32}} &x_3^{d_{43}}&x_4^{d_{24}}\\
0&0&x_1^{d_{21}}&x_4^{d_{14}}&x_2^{d_{42}}\\
-x_2^{d_{32}}&-x_1^{d_{21}}&0&0&x_3^{d_{13}+t}\\
-x_3^{d_{43}}&-x_4^{d_{14}}&0&0&x_1^{d_{31}+t}\\
-x_4^{d_{24}}&-x_2^{d_{42}}&-x_3^{d_{13}+t}&-x_1^{d_{31}+t}&0\\
\end{matrix}
\right]
$$
and the socle degree is increased by
$t^2u_1+t(u_1c_1+u_2d_{32}+u_4c_4+a)$ to get
$ t^2u_1 +t a + [{\bf a}+ t {\bf u}]. [c_1, d_{32}, 0, c_4]-3$.
If one translates through $\bf {v}$, we get
$$
\phi ({\bf a}+t{\bf v})= \left[
\begin{matrix}
0&0& x_2^{d_{32}} &x_3^{d_{43}}&x_4^{d_{24}+t}\\
0&0&x_1^{d_{21}}&x_4^{d_{14}}&x_2^{d_{42}+t}\\
-x_2^{d_{32}}&-x_1^{d_{21}}&0&0&x_3^{d_{13}}\\
-x_3^{d_{43}}&-x_4^{d_{14}}&0&0&x_1^{d_{31}}\\
-x_4^{d_{24}+t}&-x_2^{d_{42}+t}&-x_3^{d_{13}}&-x_1^{d_{31}}&0\\
\end{matrix}
\right]
$$ and the socle degree increases by
$t^2v_4+t(v_1c_1+v_2d_{32}+v_4c_4+a+z)$
to get $ t^2v_4 +t (a+z) + [{\bf a}+ t{\bf v}]. [c_1, d_{32}, 0, c_4]-3$.
}
\end{remark}
\end{document} |
\begin{document}
\begin{abstract}
Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities
along an irreducible reduced complete curve $C$
with a contraction $f: (X, C)\to (Z, o)$
such that $C=f^{-1}(o)_{\operatorname{red}}$ and $-K_X$
is ample.
This paper continues our study of such germs containing a point
of type \type{(IIA)} started in \cite{Mori-Prokhorov-IIA-1}.
\end{abstract}
\title{Threefold extremal contractions \ of type ype{(IIA)}
\section{Introduction}
Let $(X,C)$ be a germ of a threefold with terminal singularities
along a reduced complete curve. We say that $(X,C)$
is an \textit{extremal curve germ} if
there is a contraction $f: (X,C)\to (Z,o)$ such that
$C=f^{-1}(o)_{\operatorname{red}}$ and $-K_X$
is $f$-ample.
Furthermore, $f$ is called \textit{flipping} if
its exceptional locus coincides with $C$ and
\textit{divisorial}
if its exceptional locus is two-dimensional.
If $f$ is not birational, then $\dim Z=2$ and
$(X,C)$ is said to be
a \textit{$\mathbb{Q}$-conic bundle germ} \cite{Mori-Prokhorov-2008}.
In this paper we consider only extremal curve germs with \textit{irreducible}
central fiber $C$.
All the possibilities for the local behavior of $(X,C)$ are classified into types
\type{(IA)}, \type{(IC)}, \type{(IIA)}, \type{(IIB)},
\type{(IA^\vee)}, \type{(II^\vee)}, \type{(ID^\vee)}, \type{(IE^\vee)},
and \type{(III)},
whose definitions we
refer the reader to \cite{Mori-1988} and \cite{Mori-Prokhorov-2008}.
In this paper we complete the classification of extremal curve germs
containing points of type \type{(IIA)} started in \cite{Mori-Prokhorov-IIA-1}.
As in \cite{Kollar-Mori-1992}, \cite{Mori-Prokhorov-IA}, and
\cite{Mori-Prokhorov-IC-IIB}
the classification is done in terms of
a general hyperplane section,
that is, a general divisor $H$ of $|\mathscr{O}_X|_C$,
the linear subsystem of $|\mathscr{O}_X|$ consisting of sections containing $C$.
The case where $H$ is normal was treated in \cite{Mori-Prokhorov-IIA-1}.
In this paper we consider the case of non-normal $H$.
Our main result is the following.
\begin{theorem}\label{main}
Let $(X,C)$ be an extremal curve germ
and let $f: (X, C)\to (Z,o)$ be the corresponding
contraction.
Assume that $(X,C)$ it has a point $P$ of type \type{(IIA)}.
Furthermore, assume that the general member $H\in |\mathscr{O}_X|_C$ is not normal.
Then the following are the only possibilities
for the dual graph of $(H,C)$, and all the possibilities do occur.
\begin{enumerate}
\renewcommand\labelenumi{{\rm (\arabic{section}.\arabic{subsection}.\arabic{enumi})}\refstepcounter{equation}}
\renewcommand\theenumi{(\arabic{section}.\arabic{subsection}.\arabic{enumi})}
\item
\label{main-theorem-divisorial}
$f$ is divisorial\footnote{This case was erroneously
omitted in \cite[Th. 3.6 and Cor. 3.8]{Tziolas2005}.}, $f(H)\operatorname{n}i o$ is of type \type{D_{5}},
\begin{equation*}
\xy
\xymatrix@R=7pt@C=17pt{
&\circ\ar@{-}[d]
\\
\underset {} \bullet \ar@{-}[r]
&\underset 3\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
\\
&\circ\ar@{-}[u]
}
\endxy
\end{equation*}
\item
\label{main-theorem-conic-bundle}
$f$ is a $\mathbb{Q}$-conic bundle over a smooth surface,
\begin{equation*}
\vcenter{
\xy
\xymatrix@R=7pt@C=11pt{
&\circ\ar@{-}[r]&\overset {3}\circ\ar@{-}[d]\ar@{-}[r]&\circ
\\
\bullet \ar@{-}[r] &\underset {}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}
\endxy}
\end{equation*}
\end{enumerate}
In both cases
$X$ can have at most one extra point of type \type{(III)}.
\end{theorem}
\begin{remark}
If $(X,C)$ is an extremal curve germ of type \type{(IIA)}, then
according to \cite[Corollary 2.6]{Mori-Prokhorov-IIA-1}
the general member $H\in |\mathscr{O}_X|_C$ is not normal if and only if $\operatorname{H}^0(\operatorname{gr}_C^1\mathscr{O})=0$.
\end{remark}
Note that the description of a member $H\in |\mathscr{O}_X|_C$ is just
a part of our results. We also describe the infinitesimal structure
of the corresponding extremal curve germs.
Refer to \eqref{equation-possibilities-lP=3+III-b} and \ref{scorollary-9-6-8}
for the case \ref{main-theorem-divisorial} and
to \eqref{equation-lP=4-gr-2-C-O} and \ref{case-conic-bundle}
for the case \ref{main-theorem-conic-bundle}.
We also provide many examples (see \ref{example-divisorial-lP=3}, \ref{example-divisorial-lP=7},
\ref{example-conic-bundle-lP=4+III}, \ref{example-conic-bundle-lP=8}).
The proof of the main theorem splits into cases
according to the invariant $\ell(P)$ which, in our case, can take values $\ell(P)\in\{ 3,\, 4,\, 7,\, 8\}$
(see \ref{equation-iP} and Proposition \ref{proposition-cases-lP}).
Cases of odd and even $\ell(P)$
will be considered in Sections \ref{section-lP=3-III} and
\ref{section-lP=4}, respectively.
\section{Preliminaries}\label{section-Preliminaries}
\begin{setup}
\label{Set-up}
Let $(X,C)$ be an extremal
curve germ and let $f: (X, C)\to (Z,o)$ be the corresponding
contraction.
The ideal sheaf of $C$ in $X$ we denote by $I_C$ or simply by $I$.
Assume that $(X,C)$ has a point $P$ of type \type{(IIA)}.
Then by \cite[6.7, 9.4]{Mori-1988} and
\cite[8.6, 9.1, 10.7]{Mori-Prokhorov-2008} $P$ is the only non-Gorenstein point of $X$
and $(X,C)$ has at most one Gorenstein singular point $R$ \cite[6.2]{Mori-1988},
\cite[9.3]{Mori-Prokhorov-2008}.
If $\operatorname{H}^0(\operatorname{gr}_C^1\mathscr{O})=0$, then $(X,C)$ is not flipping
\cite[ch. 7]{Kollar-Mori-1992}.
\end{setup}
\begin{scase}\label{sde}
Thus, in the case $\operatorname{H}^0(\operatorname{gr}_C^1\mathscr{O})=0$, we have two possibilities:
\begin{itemize}
\item
$f$ is a $\mathbb{Q}$-conic bundle and $(Z,o)$ is smooth \cite[Th. 1.2]{Mori-Prokhorov-2008};
\item
$f$ is a divisorial contraction and $(Z,o)$ is a cDV point
(or smooth) \cite[Th. 3.1]{Mori-Prokhorov-IA}.
\end{itemize}
\end{scase}
\begin{case}\label{equation-iP}
Everywhere in this paper $(X,P)$
denotes a terminal singularity $(X,P)$ of type \type{cAx/4} and
$(X^\sharp, P^\sharp)\to (X,P)$
denotes its index-one cover.
Let
\begin{equation*}
\ell(P):=\operatorname{len}_P I^{\sharp (2)}/I^{\sharp 2},
\end{equation*}
where $I^\sharp$ is the ideal defining $C^\sharp$ in $X^\sharp$.
Recall (see \cite[(2.16)]{Mori-1988}) that in our case
\begin{equation*}
i_P(1)=\lfloor(\ell(P)+6)/4\rfloor.
\end{equation*}
\end{case}
\begin{case}
\label{equation-IIA-point}
According to \cite[A.3]{Mori-1988} we can express the \type{(IIA)} point as
\begin{equation}
\label{equation-XC}
\begin{split}
(X, P)&=
\{\alpha=0\}/{\boldsymbol{\mu}}_4(1, 1, 3, 2)\subset\mathbb{C}^4_{y_1,\dots, y_4}/{\boldsymbol{\mu}}_4(1, 1, 3, 2),
\\
C&=\{y_1\text{-axis}\}/{\boldsymbol{\mu}}_4,
\end{split}
\end{equation}
where $\alpha=\alpha(y_1,\dots, y_4)$ is a semi-invariant such that
\begin{equation}\label{equation-alpha}
\operatorname{wt}\alpha\equiv 2\mod 4,\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad \alpha\equiv y_1^{\ell(P)}y_j\mod (y_2, y_3, y_4)^2,
\end{equation}
where $j= 2$ (resp. $3$, $4$) if $\ell(P)\equiv 1$ (resp. $3$, $0$) $\mod 4$
\cite[(2.16)]{Mori-1988} and $(I^\sharp)^{(2)}=(y_j)+(I^\sharp)^{2}$.
Moreover, $y_2^2,\, y_3^2\in \alpha$ (because $(X,P)$ is of type
\type{cAx/4}).
\end{case}
\begin{case}\label{ge}
Recall that in our case
the general member $D\in |-K_X|$ does not contain $C$
\cite[Th. 7.3]{Mori-1988},
\cite[Prop. 1.3.7]{Mori-Prokhorov-2008}.
Hence $D\cap C=\{P\}$, $D\simeq f(D)$, and $D$ has at $P$ a singularity of type \type{D_{2n+1}}
\cite[6.4B]{Reid-YPG1987}.
In the coordinates $y_1,\dots,y_4$, the divisor $D$ is given by
\begin{equation*}
D=\{y_1= \xi\}/{\boldsymbol{\mu}}_4,\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad \xi\in (y_2,\, y_3,\, y_4).
\end{equation*}
\end{case}
\begin{scase}\label{sde1}
Let $H$ be a general member of $|\mathscr{O}_X|_C$ through $C$ and let $\beta\in \operatorname{H}^0(I_C)$ be a
non-zero section defining $H$.
Let $H_Z=f(H)$ and let $\psi: H^{\operatorname{n}}\to H$ be the normalization. The composition map
$H^{\operatorname{n}}\to H_Z$ has connected fibers. Moreover,
it is a rational curve fibration if $\dim Z=2$
and it is a birational contraction to a point $(H_Z, o)$
which is either smooth or Du Val point of type \type{A} or \type{D}
if $f$ is divisorial (see \ref{ge}).
In both cases $H^{\operatorname{n}}$ has only rational singularities.
\end{scase}
For convenience of the reader we formulate the following lemma which follows from the
standard exact sequence
\begin{equation*}
0\xrightarrow{\hspace*{20pt}} I^{(n+1)} \xrightarrow{\hspace*{20pt}} I^{(n)} \xrightarrow{\hspace*{20pt}} \operatorname{gr}_C^n\mathscr{O}\xrightarrow{\hspace*{20pt}} 0.
\end{equation*}
\begin{lemma}\label{lemma-grC}
Let $(X,C)$ be an extremal curve germ.
Then the following assertions hold.
\begin{enumerate}
\item \label{lemma-grC-1}
If $\operatorname{H}^1(\operatorname{gr}_C^n\mathscr{O})=0$ and the
map $\operatorname{H}^0(I^{(n)})\to \operatorname{H}^0(\operatorname{gr}_C^n\mathscr{O})$ is surjective, then
$\operatorname{H}^1(I^{(n+1)})\simeq \operatorname{H}^1(I^{(n)})$.
\item \label{lemma-grC-2}
If for all $i<n$ one has $\operatorname{H}^1(\operatorname{gr}_C^i\mathscr{O})=0$ and the
map $\operatorname{H}^0(I^{(i)})\to \operatorname{H}^0(\operatorname{gr}_C^i\mathscr{O})$ is surjective, then
$\operatorname{H}^1(I^{(n)})\simeq \operatorname{H}^1(\operatorname{gr}_C^n\mathscr{O})=0$.
\item \label{lemma-grC-3}
In particular, $\operatorname{H}^1(I)= \operatorname{H}^1(\operatorname{gr}_C^1\mathscr{O})=0$ and if
$\operatorname{H}^0(\operatorname{gr}_C^1\mathscr{O})=0$, then
$\operatorname{H}^1(I^{(2)})= \operatorname{H}^1(\operatorname{gr}_C^2\mathscr{O})=0$.
\end{enumerate}
\end{lemma}
The following auxiliary result can be proved by induction on $n$.
\begin{proposition}\label{proposition-lP=4-XC}
Let $(X,P)\subset \mathbb{C}^4_{x_1,\dots,x_4}$ be a hypersurface containing
$C:=\{\text{$x_1$-axis}\}$ with defining equation $h\in \mathbb{C}\{x_1,\dots,x_4\}$ such that
\begin{equation*}
h=x_1^mx_4+h_2(x_2,x_3)+h_3(x_1,\dots,x_4),
\end{equation*}
where $h_2$ is a quadratic form in $x_2$ and $x_3$, $h_3\in (x_2,x_3,x_4)^3$, and $m\ge 1$.
Let $I=(x_2,x_3,x_4)$ be the ideal of $C$. Let
\begin{equation*}
\operatorname{gr}_C^{\bullet}:= \bigoplus_{n\ge 0} \operatorname{gr}_C^n\mathscr{O}
\end{equation*}
be the graded $\mathscr{O}_C$-algebra with the degree $n$ part $\operatorname{gr}_C^n\mathscr{O}$.
Then the following assertions hold.
\begin{enumerate}
\item \label{proposition-lP=4-XC-1}
If $h_2=0$, then
\begin{equation*}
\operatorname{gr}_C^2\mathscr{O}= S^2\operatorname{gr}_C^1\mathscr{O}.
\end{equation*}
\item \label{proposition-lP=4-XC-2}
If $h_2\operatorname{n}eq 0$, then
\begin{equation*}
\operatorname{gr}_C^{\bullet}\mathscr{O}\simeq\mathscr{O}_C[x_2,x_3,x_4]/(x_1^mx_4+h_2),
\end{equation*}
where $x_2,x_3,x_4$ have degree $1$, $1$, $2$, respectively.
\item \label{proposition-lP=4-XC-4}
If $x_3^2\in h_2$, then
\begin{equation*}
\operatorname{gr}_C^{\bullet}\mathscr{O} =\mathscr{O}_C[x_2,x_4]\oplus x_3\mathscr{O}_C[x_2,x_4].
\end{equation*}
\item \label{proposition-lP=4-XC-5}
If $h_2=x_2x_3$, then
\begin{equation*}
\operatorname{gr}_C^{\bullet}\mathscr{O} =\mathscr{O}_C[x_4]\oplus x_2\mathscr{O}_C[x_2,x_4]\oplus x_3\mathscr{O}_C[x_3,x_4].
\end{equation*}
\end{enumerate}
\end{proposition}
\begin{proposition}\label{proposition-cases-lP}
Assume that $\operatorname{H}^0(\operatorname{gr}_C^1\mathscr{O})=0$. Then
\begin{equation}
\label{equation-gr1CO}
\operatorname{gr}_C^1\mathscr{O}\simeq \mathscr{O}(-1)\oplus \mathscr{O}(-1)
\end{equation}
\textup(as an abstract sheaf\textup) and one of the following possibilities holds:
\begin{enumerate}
\item
$\operatorname{Sing}(X)=\{P\}$, $i_P(1)=3$, and $\ell(P)=7$ or $8$,
\item
$\operatorname{Sing}(X)=\{P,\, R\}$, where $R$ is a type \type{(III)} point,
$i_P(1)=2$, $i_R(1)=1$, and $\ell(P)=3$ or $4$.
\end{enumerate}
\end{proposition}
\begin{proof}
Write $\operatorname{gr}_C^1\mathscr{O}\simeq \mathscr{O}(a_1)\oplus \mathscr{O}(a_2)$
for some $a_1$, $a_2$.
Since $\operatorname{H}^0(\operatorname{gr}_C^1\mathscr{O})=0$, we have $a_1, a_2<0$.
On the other hand, $\operatorname{H}^1(\operatorname{gr}_C^1\mathscr{O})=0$ (see Lemma \ref{lemma-grC}\ref{lemma-grC-3}).
Hence, $a_1=a_2=-1$. Recall that $\ell(P)\operatorname{n}ot\equiv 2\mod 4$.
Consider the case where $P$ is the only singular point of $X$.
Then $i_P(1)=3$ by
\cite[(2.3.2)]{Mori-1988} and \cite[(3.1.2), (4.4.3)]{Mori-Prokhorov-2008}.
According to \cite[2.16]{Mori-1988} we have $7\le \ell(P)\le 9$.
Assume that $\ell(P)=9$. Then using a deformation of the form
$\alpha_t=\alpha+t y_1y_2$ (see \eqref{equation-alpha}), we get a germ $(X_t,C_t)$
having a point $P_t$ of type \type{(IIA)} with $\ell(P_t)=1$
and two type \type{(III)} points. This is impossible by \cite[7.4.1]{Kollar-Mori-1992} and
\cite[9.1]{Mori-Prokhorov-2008}.
Suppose $\operatorname{Sing}(X)\operatorname{n}eq \{P\}$.
Then by \cite[6.7]{Mori-1988} and \cite[8.6, 9.1]{Mori-Prokhorov-2008}
we have $\operatorname{Sing}(X)=\{P,\, R\}$, where $R$ is a type \type{(III)} point.
If $i_R(1)>1$, then by using deformation at $R$ we obtain
an extremal curve germ with one point of type \type{(IIA)}
and at least two points of type \type{(III)}.
This is impossible again by \cite[6.7]{Mori-1988} and \cite[9.1]{Mori-Prokhorov-2008}.
Therefore, $i_R(1)=1$ and so $i_P(1)=2$.
By \cite[2.16]{Mori-1988} we have $3\le \ell(P)\le 5$
Assume that $\ell(P)=5$.
Using a deformation of the form
$\alpha_t=\alpha+t y_1y_2$, we obtain a germ $(X_t,C_t)$
having a point $P_t$ with $\ell(P_t)=1$
and two type \type{(III)} points. This is impossible by \cite[7.4.1]{Kollar-Mori-1992} and
\cite[9.1]{Mori-Prokhorov-2008}.
\end{proof}
\begin{slemma}
\label{lemma-lP=3+IIIa}
If $\operatorname{H}^0(\operatorname{gr}_C^1\mathscr{O})=0$, then
\begin{equation*}
\operatorname{gr}_C^2\mathscr{O} \simeq \mathscr{O}(a_1)\oplus\mathscr{O}(a_2)\oplus\mathscr{O}(a_3),
\end{equation*}
\textup(as an abstract sheaf\textup) with $a_i\ge -1$ and $\max \{a_1,\, a_2,\, a_3\}\ge 0$.
\end{slemma}
\begin{proof}
If $\operatorname{H}^0(\operatorname{gr}_C^1\mathscr{O})=0$, then the general member $H\in |\mathscr{O}_X|_C$ is singular along $C$.
According to \cite[Lemma 3.1.1]{Mori-Prokhorov-IIA-1}
there exists a section $\beta\in \operatorname{H}^0(I)$ containing $y_4^2$ and $y_2y_3$
at $P^\sharp$. Therefore, $\beta\in \operatorname{H}^0(I^{(2)})$ and the image
$\bar\beta$
of $\beta$ in $\operatorname{H}^0(\operatorname{gr}_C^2\mathscr{O})$ is non-zero.
In particular, $\operatorname{H}^0(\operatorname{gr}_C^2\mathscr{O})\operatorname{n}eq 0$. By Lemma \ref{lemma-grC}\ref{lemma-grC-3}
we have $\operatorname{H}^1(\operatorname{gr}_C^2\mathscr{O})=0$ and the assertion follows.
\end{proof}
\section{Cases $\ell(P)=3$ and $7$}
\label{section-lP=3-III}
In this section we assume that $\ell(P)\in \{3,\, 7\}$.
It will be shown that Computation \ref{computation-lP=3a-III} is applicable here
and the possibility \ref{main-theorem-divisorial} occurs.
\begin{case}\label{notation-lP=3+III}
By Proposition \ref{proposition-cases-lP} in the case $\ell(P)=3$ the variety
$X$ has a type \type{(III)} point $R$ with $i_R(1)=1$ and
$X$ is smooth outside $P$ in the case $\ell(P)=7$.
According to \ref{equation-IIA-point} the equation of $X$ at $P$ has the form
\begin{equation}\label{equation-alpha-lP=3-and-7}
\alpha=y_1^{\ell(P)}y_3+y_2^2+y_3^2+\delta y_4^{2k+1}+c y_1^2y_4^2+\epsilon y_1y_3y_4+\xi y_1^3y_2y_4
+\cdots=0.
\end{equation}
Thus
\begin{equation}\label{equation-alpha-lP=3-and-7-mod}
\alpha\equiv y_1^{\ell(P)}y_3+y_2^2
\mod (y_2y_4,\, y_4^2)+ I^{(3)},\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad y_3\in I^{(2)}.
\end{equation}
\end{case}
\begin{scase}
In the case $\ell(P)=3$
by \cite[Lemma 2.16]{Mori-1988}, since $i_R(1)=1$,
the equation of $X$ at $R$ has the form
\begin{equation}\label{equation-beta-lP=3+III-1}
\gamma=z_1z_3+\gamma_2(z_2,z_4)+\gamma_3(z_1,\dots,z_4),
\end{equation}
where $\gamma_2$ is a quadratic form, $\gamma_3\in (z_2,z_4)^3+(z_2,z_4)z_3+(z_3)^2$,
and $C$ is the $z_1$-axis.
\end{scase}
\begin{scase}
According to \eqref{equation-gr1CO}, since
$y_4$ and $y_2$ form an $\ell$-free $\ell$-basis of $\operatorname{gr}^1_C\mathscr{O}$ at $P$, we have the following $\ell$-isomorphism
\begin{equation}
\label{equation-(7.4.1.1)-lP=3+III}
\vcenter{
\xymatrix@R=6pt@C=-3pt{
\operatorname{gr}_C^1\mathscr{O}= &(-1+3P^\sharp)\ar@{=}[d]&\mathbin{\tilde\oplus}& (-1+2P^\sharp).\ar@{=}[d]
\\
& \mathscr{A} && \mathscr{B}
}}
\end{equation}
We choose the coordinates
$y_1,\dots, y_4$ at $P$ keeping $y_1$ and $y_3$ the same
so that
$y_2$ is an $\ell$-basis of $\mathscr{A}$
and
$y_4$ is an $\ell$-basis of $\mathscr{B}$.
\end{scase}
\begin{sremark}\label{remarkProposition-lP=3+IIIa}
By \eqref{equation-alpha-lP=3-and-7-mod} the semi-invariants
$y_4^2$, $y_2y_4$, $y_3$ form an $\ell$-basis of $\operatorname{gr}_C^2\mathscr{O}$.
\end{sremark}
\begin{lemma}\label{lemma-possibilities-lP=3+III}
{}For $\operatorname{gr}_C^2\mathscr{O}$, one of the following possibilities holds
\begin{numcases}{\operatorname{gr}_C^2\mathscr{O}=}
(a)\mathbin{\tilde\oplus} (-1+P^\sharp)^{\mathbin{\tilde\oplus} 2},\quad a=0,\ 1
\label{equation-possibilities-lP=3+III-a}
\\
(P^\sharp)\mathbin{\tilde\oplus} (0)\mathbin{\tilde\oplus} (-1+P^\sharp),
\label{equation-possibilities-lP=3+III-b}
\\
\mathscr{V}\mathbin{\tilde\oplus} (-1),
\label{equation-possibilities-lP=3+III-c}
\end{numcases}
where $\mathscr{V}$ is some $\ell$-sheaf.
\end{lemma}
\begin{proof}
Consider the natural map
\begin{equation}
\label{equation-varphi}
\varphi: \tilde S^2 \operatorname{gr}_C^1\mathscr{O}=
\mathscr{A}^{\mathbin{\tilde\otimes} 2}
\mathbin{\tilde\oplus} (\mathscr{A}\mathbin{\tilde\otimes}\mathscr{B})\mathbin{\tilde\oplus}
\mathscr{B}^{\mathbin{\tilde\otimes} 2}
\xrightarrow{\hspace*{25pt}}\operatorname{gr}_C^2\mathscr{O},
\end{equation}
where
\begin{equation*}
\mathscr{A}^{\mathbin{\tilde\otimes} 2}= (-1+2P^\sharp),\quad
\mathscr{A}\mathbin{\tilde\otimes}\mathscr{B}= (-1+P^\sharp),\quad
\mathscr{B}^{\mathbin{\tilde\otimes} 2}= (-1).
\end{equation*}
$\ell$-bases of these $\ell$-sheaves at $P$ are
$y_2^2$,\,
$y_2y_4$,\,
$y_4^2$, and respectively.
By Remark \ref{remarkProposition-lP=3+IIIa}
we see that
an $\ell$-basis of $\operatorname{gr}_C^2\mathscr{O}$ can be taken as
$y_4^2,\, y_2y_4,\, y_3$.
According to \eqref{equation-alpha-lP=3-and-7-mod}
we have $y_1^2y_2^2\equiv (\operatorname{unit})\cdot y_1^{\ell(P)+1}\cdot y_1y_3$.
Hence,
\begin{equation}
\label{equation-lP=3-7-Coker-P}
\operatorname{coker}_P \varphi=\mathscr{O}_C \cdot y_1y_3/\mathscr{O}_C\cdot y_1^2y_2^2=
\mathscr{O}_C/\bigl(y_1^{\ell(P)+1}\bigr)\cdot y_1y_3
\end{equation}
and $\operatorname{coker}_P \varphi^{\sharp}= \mathscr{O}_{C^{\sharp}}/\bigl(y_1^{\ell(P)}\bigr)\cdot y_3$.
In particular,
$\operatorname{len} _P\operatorname{coker}_P \varphi=(\ell(P)+1)/2$.
If $\ell(P)=3$, then
\begin{equation*}
\operatorname{coker}_R \varphi =
\begin{cases}
\mathscr{O}_C/(z_1)\cdot z_3 & \text{if $\gamma_2\operatorname{n}eq 0$,}
\\
0 & \text{if $\gamma_2= 0$,}
\end{cases}
\end{equation*}
In particular,
$\operatorname{len}_R \operatorname{coker}_R \varphi\le 1$.
By Lemma \ref{lemma-lP=3+IIIa} one of the following holds
\begin{equation*}
\operatorname{gr}_C^2\mathscr{O}\simeq
\mathscr{O}(-1)^{\oplus 2}\oplus\mathscr{O}(1),\quad
\mathscr{O}(-1)^{\oplus 2}\oplus\mathscr{O},\quad\text{or}\quad
\mathscr{O}^{\oplus 2}\oplus\mathscr{O}(-1).
\end{equation*}
By Remark \ref{remarkProposition-lP=3+IIIa} we get the only possibilities listed in
Lemma \xref{lemma-possibilities-lP=3+III}.
\end{proof}
\begin{lemma}\label{treating-equation-possibilities-lP=3+III-c}
The case \eqref{equation-possibilities-lP=3+III-c} does not occur.
\end{lemma}
\begin{proof}
Indeed, from the exact sequence
\begin{equation*}
0\xrightarrow{\hspace*{20pt}} \operatorname{gr}_C^1\omega\xrightarrow{\hspace*{20pt}} \omega /F^{2}\omega
\xrightarrow{\hspace*{20pt}} \omega/ F^1 \omega\xrightarrow{\hspace*{20pt}} 0,
\end{equation*}
we obtain $\chi(\omega /F^{2}\omega)=0$. Then we apply
\cite[Lemma 3.7(ii)]{Mori-Prokhorov-IIA-1} with ${\mathscr{K}}=I^{(2)}$.
\end{proof}
\begin{lemma}
\label{lemma-equation-possibilities-lP=3+IIIa}
The case \eqref{equation-possibilities-lP=3+III-a} does not occur.
\end{lemma}
\begin{proof}
The deformation of the form
\begin{equation}
\label{equation-lP=3-7-deformations}
\alpha'=\alpha+\delta' y_4^{3}+\epsilon' y_1y_3y_4
\end{equation}
does not change the case division of
Lemma \xref{lemma-possibilities-lP=3+III} because $y_4^3,\, y_1y_3y_4\in I^{(3)}$.
Since it suffices to disprove a small deformation of $X$, we may assume that in \eqref{equation-alpha-lP=3-and-7}
the coefficients $\delta$ and $\epsilon$ are general and $k=1$.
Let us analyze the map $\varphi$ (see \eqref{equation-varphi}) in our case.
Since the map $\mathscr{A}^{\mathbin{\tilde\otimes} 2}\to (-1+P^\sharp)$ is zero (by the degree consideration),
the image of
$\mathscr{A}^{\mathbin{\tilde\otimes} 2}=(-1+2P^\sharp) \hookrightarrow \operatorname{gr}_C^2\mathscr{O}$
must be contained in the first summand $(a)\subset \operatorname{gr}_C^2\mathscr{O}$. Since $(-1+P^\sharp)^{\mathbin{\tilde\oplus} 2}$ has
no global sections, $\beta$ must be a global section of $(a)$.
The map $\varphi$ is given by the following matrix:
\[
\arraycolsep=1.4pt
\begin{blockarray}{cc@{\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad}ccc}
&&\scriptstyle{(-1+2P^\sharp)}&\scriptstyle{(-1+P^\sharp)}&\scriptstyle{(-1)}
\\[7pt]
\begin{block}{rc@{\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad}(ccc)}
\scriptstyle{(a)}&\scriptstyle{v_1}&y_1^2h(y_1^4)&\star&\star\star
\\
\scriptstyle{(-1+P^\sharp)}&\scriptstyle{v_2}&0&b_1&b_3y_1
\\
\scriptstyle{(-1+P^\sharp)}&\scriptstyle{v_3}&0&b_2&b_4y_1
\\
\end{block}
\end{blockarray}
\]
where $b_1,\dots, b_4$ are constants
and $h$ is a polynomial of degree $\le a$. Since the matrix is non-degenerate, $(b_1b_4-b_2b_3)h\operatorname{n}eq 0$.
Applying elementary transformations of rows
and switching the second and the third rows
(which correspond to automorphisms of
$\operatorname{gr}_C^2\mathscr{O}$), one can reduce the matrix to the form
\begin{equation}
\label{equation-lP=3-7-matrix}
\begin{pmatrix}
y_1^2h(y_1^4)&0&b_5
\\
0&1&0
\\
0&0&y_1
\end{pmatrix}
\end{equation}
where $b_5$ is a constant.
If $b_5=0$, then
\[
(\operatorname{coker}_{P} \varphi)^\sharp \simeq
\mathscr{O}_C^\sharp /(y_1)\oplus \mathscr{O}_C^\sharp /(y_1^2h).
\]
This contradicts \eqref{equation-lP=3-7-Coker-P}. Hence, we may assume that $b_5=1$.
From the matrix \eqref{equation-lP=3-7-matrix} we see
\begin{eqnarray*}
y_2^2&=& y_1^2hv_1,
\\
y_4^2&=& v_1+y_1v_3.
\end{eqnarray*}
Eliminating $v_1$ we obtain the following relations in $\operatorname{gr}_C^2\mathscr{O}$:
\begin{equation}
\label{equation-lP=3-7-vv}
v_1=y_4^2-y_1v_3,\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad
y_1^3h v_3+y_2^2-y_1^2hy_4^2=0.
\end{equation}
The last one must a multiple of $\alpha$.
\begin{scase}
\label{scase-lP=3-7-new-treatment}
If $h$ is a unit, then
comparing with \eqref {equation-alpha-lP=3-and-7} we see that $\ell(P)=3$,
$c=h(0)\operatorname{n}eq 0$ and $v_3\operatorname{n}i y_3$.
If $h$ is linear, then $\ell(P)=7$,
$c=h(0)= 0$ and again $v_3\operatorname{n}i y_3$.
Since $\beta$ is a section of $(a)\subset \operatorname{gr}_C^2\mathscr{O}$, it must be proportional
to $v_1$. Therefore, $y_1y_3\in \beta$. Moreover, \eqref{equation-lP=3-7-vv}
shows that in the case $\ell(P)=3$ the term $y_1y_3$ appears in $\beta$ with coefficient $1/c$.
Note that the coefficients of $y_1^2y_4^2\in \alpha$ and $y_4^2,\, y_1y_3\in \beta$
are preserved under deformations \eqref{equation-lP=3-7-deformations}.
So we may assume that the condition $\epsilon c\operatorname{n}eq \delta$
of \ref{computation-lP=3+III-part2} is satisfied.
Thus in the case $\ell(P)=3$ we may apply Computation \ref{computation-lP=3+III-part2}.
In the case $\ell(P)=7$ we also may apply \ref{computation-lP=3+III-part2}
to $\alpha^o=\beta=0$, where $\alpha^o$ is a linear combination of
$\alpha$ and $y_1^2\beta$ (and so $y_1^2y_4^2\in \alpha^o$).
Then in both cases we obtain a contradiction by Lemma \ref{slemma-lP=3+III-generalityH} below.
\end{scase}
\begin{slemma}\label{lemma-computation-lP=3+III-part2}
Assume that $\Delta(H,C)$ at $P$ is as in
\eqref{graphs-computation-lP=3+III-part2}.
Then the contraction $f$ is birational and
$\Delta(H,C)$
has one of the following forms:
\begin{equation*}
\xy
\xymatrix@R=1pt@C=10pt{
\mathrm{a)}&&\hbox to 5pt{\hss {$\scriptstyle{C}$ $\overset{3}{\scriptstyle \odot}$}}\ar@{-}[d]&\circ\ar@{-}[d]
\\
&\underset C\bullet\ar@{-}[r] &\circ\ar@{-}[r] &\underset{3}\circ\ar@{-}[r]&\circ \ar@{-}[r]&\circ
}
\endxy
\hspace{17pt}
\xy
\xymatrix"M"@C=10pt@R=1pt{
\mathrm {b_n)}&&&\circ\ar@{-}[d]\ar@{-}[r]& \cdots\ar@{-}[r]&\circ
\\
&\underset {C}{\bullet}\ar@{-}[r] &\circ\ar@{-}[r]
&\underset{3}{\circ}\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}
\POS"M1,4"."M1,6"!C*\frm{^\}},+U*++!D\txt{$\scriptstyle{n\ge 1}$}
\endxy
\end{equation*}
\begin{equation*}
\xy
\xymatrix@R1pt@C13pt{
\mathrm {c)}&\overset{4}{\diamond}\ar@{-}[d]&&\circ\ar@{-}[d]
\\
&\underset{C}\bullet\ar@{-}[r]&\circ\ar@{-}[r] &\underset{3}\circ\ar@{-}[r] &\circ\ar@{-}[r] &\circ
}
\endxy
\end{equation*}
where $\bullet$, as usual, corresponds to a component of the proper transform of $C$
that is a $(-1)$-curve,
$\scriptstyle \odot$ corresponds to a component that is not a $(-1)$-curve,
and $\diamond$ corresponds to an exceptional divisor over a point on $C\setminus \{P\}$.
\end{slemma}
\begin{proof}
Let $H^{\operatorname{n}}\to \tilde H$ be the normalization, let
$\hat H\to H^{\operatorname{n}}$ be the minimal resolution,
and let $\hat C\subset \hat H$ be
the proper transform of $C$.
Assume that $\hat C$ has two components $\hat C_1$ and $\hat C_2$
(the case \eqref{graphs-computation-lP=3+III-part2}a)).
Then $\Delta(H,C)$ has the form
\begin{equation*}
\xymatrix@R1pt {
\ovalh{\phantom{PP}$\Gamma_2$\phantom{PP}}
&&\ar@{-}[ll]\scriptstyle{\hat C_2}\ar@{-}[d]&
\ovalh{\phantom{P}$\Gamma$\phantom{P}}
\\
\ovalh{\phantom{PP}$\Gamma_1$\phantom{PP}}\ar@{-}[r]
&\scriptstyle{\hat C_1}\ar@{-}[r]&
\circ\ar@{-}[r]&\underset3\circ\ar@{-}[r]\ar@{-}[u]&\circ\ar@{-}[r]&\circ
}
\end{equation*}
where subgraphs $\Gamma_1$ and $\Gamma_2$ correspond to singularities
of $H^{\operatorname{n}}$ outside $P$ and $\Gamma$ is a Du Val
subgraph corresponding to
$O'\in \tilde H$ (see \ref{claim-new-11-3}).
Since the whole configuration $\Delta(H,C)$ is contractible to
a Du Val point
or corresponds to a fiber of a rational curve fibration (see \ref{sde1}), it contains a $(-1)$-curve.
Thus we may assume by symmetry that $\hat C_1^2=-1$. Then contracting $\hat C_1$ we obtain
\begin{equation*}
\xymatrix@R1pt {
\ovalh{\phantom{PP}$\Gamma_2$\phantom{PP}}&&\ar@{-}[ll]\scriptstyle{\hat C_2}\ar@{-}[d]
&\ovalh{\phantom{P}$\Gamma$\phantom{P}}
\\
\ovalh{\phantom{PP}$\Gamma_1'$\phantom{PP}}\ar@{-}[rr]&&\bullet\ar@{-}[r]&
\underset3\circ\ar@{-}[r]\ar@{-}[u]&\circ\ar@{-}[r]&\circ
}
\end{equation*}
Then $\Gamma_1'$ must be empty. Contracting the black vertex we obtain
\begin{equation*}
\xymatrix@R1pt {
\ovalh{\phantom{PP}$\Gamma_2$\phantom{PP}}&&\ar@{-}[ll]\scriptstyle{\hat C_2'}\ar@{-}@/_3pt/[dr]
&\ovalh{\phantom{P}$\Gamma$\phantom{P}}
\\
&&&\circ\ar@{-}[r]\ar@{-}[u]&\circ\ar@{-}[r]&\circ
}
\end{equation*}
Recall that $\Gamma\operatorname{n}eq \varnothing$.
It is easy to see that configuration $\Delta(H,C)$ does not
correspond to a fiber of a rational curve fibration. Hence $f$ is birational.
Since $y_4^3\in \alpha$, the general member $D\in |-K_X|$ is of type \type{D_5}
(see \ref{ge}). Hence $f(H)$ is either of type \type{D_5} or ``better''.
This implies that
$\Gamma_2=\varnothing$, $\hat C_2'^2=-1$ and so $\hat C_2^2=-2$. Moreover, $\Gamma$ consists of a single vertex.
Thus we obtain the case a).
The cases where $\hat C$ is irreducible is treated in a similar way.
\end{proof}
\begin{slemma}\label{slemma-lP=3+III-generalityH}
Assume that $(H,C)$ is of type
\type{a)}, \type{b_n)} or \type{c)}
of Lemma \xref{lemma-computation-lP=3+III-part2}.
Then the chosen element $H$
is not general in $|\mathscr{O}_X|_C$.
\end{slemma}
\begin{proof}
Take a divisor $\Theta$ on the minimal resolution
whose coefficients for \type{a)} and \type{b_n)} are as follows:
\begin{equation*}
\xy
\xymatrix@R=0pt@C=10pt{
\mathrm{a)}&\overset{1}{\scriptstyle \odot}\ar@{-}[d]&\overset{1}\circ\ar@{-}[d]&&\overset{1}\vartriangle\ar@{-}[d]
\\
\underset 3\bullet\ar@{-}[r] &\underset 3\circ\ar@{-}[r] &
\underset{2}\circ\ar@{-}[r]&\underset 2\circ \ar@{-}[r]&\underset 2\circ&\underset 1\vartriangle\ar@{-}[l]
}
\endxy
\hspace{25pt}
\xy
\xymatrix@C=17pt@R=1pt{
\mathrm {b_n)}&&\overset{1}\circ\ar@{-}[d]\ar@{-}[r]& \cdots\ar@{-}[r]&\overset{1}\circ&\overset{1}\vartriangle\ar@{-}[l]
\\
\underset {1}{\bullet}\ar@{-}[r] &\underset{1}\circ\ar@{-}[r]
&\underset{1}{\circ}\ar@{-}[r]&\underset{1}\circ\ar@{-}[r]&\underset{1}\circ&\underset{1}\vartriangle\ar@{-}[l]
}
\endxy
\end{equation*}
where $\vartriangle$ corresponds to
an arbitrary smooth analytic curve meeting the corresponding component
transversely.
It is easy to verify that $\Theta$ is numerically trivial,
so $\Theta$ is the pull-back of a Cartier divisor $\Theta_Z$ on $H_Z$.
Clearly, $\Theta_Z$ extends to a Cartier divisor $G_Z$ on $Z$.
Let $G:=f^*G_Z$. Then $\Theta$ is the pull-back of $G|_H$.
In the case \type{a)} the normalization of $H$ at a general point of $C$
is locally reducible: $H=H_1+H_2$.
The diagram \type{a)} shows that for $H_i\cap G$ is a reduced divisor
for some $i\in \{1,\, 2\}$.
Hence, $G\in |\mathscr{O}_X|_C$ is normal which contradicts our assumptions.
In the case \type{b_n)} and \type{c)}
the normalization of $H$ is a bijection by
Corollary \xref{scorollary-lP=3+III-sing}.
In the case \type{b_n)}
it is easy to see that
the multiplicity of the intersection $H\cap G$ at a general point of $C$
is $\le 2$. This shows that the divisor $G\in |\mathscr{O}_X|_C$ is normal, a contradiction.
Similar arguments show that in the case \type{c)} the multiplicity of the intersection
$H\cap G$ at a general point of $C$
equals $4$.
By Corollary \ref{scorollary-lP=3+III-sing}
$H$ has a cuspidal singularity at a general point of $C$.
Let $D\subset X$ be a disk that intersects $C$ transversely at a general point.
Then the curves $H|_D$ and $G|_D$ are cuspidal.
Since $H|_D \cdot G|_D=H\cdot G\cdot D=4$, these cusps are in general position,
that is, the quadratic parts of the corresponding equations are not proportional.
But then the general member of the pencil generated by $H|_D$ and $G|_D$
has an ordinary double point at the origin.
Hence the chosen element $H\in |\mathscr{O}_X|_C$ is not general, a contradiction.
\end{proof}
Thus the case \eqref{equation-possibilities-lP=3+III-a} does not occur.
Lemma \ref{lemma-equation-possibilities-lP=3+IIIa} is proved.
\end{proof}
\begin{case}{\bf Case \eqref{equation-possibilities-lP=3+III-b}.}
\label{Subcase-equation-possibilities-lP=3+III-c}
We will show that Computation \ref{computation-lP=3a-III} is applicable in this case and
the possibility \ref{main-theorem-divisorial} occurs.
We have
\begin{equation*}
\vcenter{
\xymatrix@R=6pt@C=-3pt{
\operatorname{gr}_C^2\mathscr{O}= &(P^\sharp)\ar@{=}[d]&\mathbin{\tilde\oplus}& (0)\ar@{=}[d]&\mathbin{\tilde\oplus}& (-1+P^\sharp).\ar@{=}[d]
\\
& \mathscr{D} && {\mathscr{E}} && \mathscr{G}
}}
\end{equation*}
We apply the arguments similar to that used in
the proof of Lemma \ref{lemma-equation-possibilities-lP=3+IIIa}.
In our case the map $\varphi$ is given by the following matrix:
\[
\begin{blockarray}{cc@{\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad}ccc}
&&\scriptstyle{(-1+2P^\sharp)}&\scriptstyle{(-1+P^\sharp)}&\scriptstyle{(-1)}
\\[7pt]
\begin{block}{rc@{\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad}(ccc)}
\scriptstyle{(P^\sharp)}&\scriptstyle{w_1} &b_1y_1^3&h(y_1^4)&\star
\\
\scriptstyle{(0)} &\scriptstyle{w_2} &b_2y_1^2&b_3y_1^3&\star\star
\\
\scriptstyle{(-1+P^\sharp)}& \scriptstyle{w_3} &0&b_4&b_5y_1
\\
\end{block}
\end{blockarray}
\]
where $b_1,\dots,b_5$ are constants, $h$ is a polynomial of degree $\le 1$, and $\star$
is divisible by $y_1$.
Consider the map
\begin{equation}
\label{equation-lP=3-7-definition-pi}
\pi: (-1+P^\sharp)=\mathscr{A}\mathbin{\tilde\otimes}\mathscr{B} \xrightarrow{\hspace*{10pt}}
\operatorname{gr}^2_C\mathscr{O}
\xrightarrow{\makebox[20pt]{ $\scriptstyle\operatorname{pr}$}}
\mathscr{G}=(-1+P^\sharp),
\end{equation}
which is uniquely determined by $\mathscr{A}$ and $\mathscr{B}$.
We may regard $\pi$ as the multiplication by $b_4$.
\begin{slemma}
$b_4\operatorname{n}eq 0$.
\end{slemma}
\begin{proof}
Assume that $b_4=0$. Since the matrix is non-degenerate, $b_5\operatorname{n}eq0$.
Applying elementary transformations of rows, as in the proof of
Lemma \ref{lemma-equation-possibilities-lP=3+IIIa}, one can reduce the matrix to the form
\begin{equation}
\label{equation-lP=3-7-matrix-2}
\begin{pmatrix}
b_1y_1^3&h(y_1^4)&0
\\
b_2y_1^2&b_3y_1^3&b_6
\\
0&0&y_1
\end{pmatrix}
\end{equation}
where $b_6$ is a constant.
If $b_6=0$, then
\[
(\operatorname{coker}_{P} \varphi)^\sharp \simeq
\mathscr{O}_C^\sharp /(y_1)\oplus \text{(non-zero $\mathscr{O}_C^\sharp$-module)}.
\]
This contradicts \eqref{equation-lP=3-7-Coker-P}. Hence, we may assume that $b_6=1$.
Assume that $b_2=0$.
Applying elementary row transformations
we can reduce \eqref{equation-lP=3-7-matrix-2} to the form
\[
\begin{pmatrix}
y_1^3&h(y_1^4)&0
\\
0&b_3y_1^3&1
\\
0&0&y_1
\end{pmatrix}
\]
which gives us
\begin{eqnarray*}
y_2^2&=& y_1^3 w_1,
\\
y_2y_4&=& h(y_1^4) w_1+b_3y_1^3 w_2,
\\
y_4^2&=& w_2+y_1w_3.
\end{eqnarray*}
If $h(0)=0$, then one can see that
$(\operatorname{coker}_{P} \varphi)^\sharp$ cannot be a cyclic $\mathscr{O}_C^\sharp$-module.
Thus, $h$ is a unit and we can eliminate $w_1$ and $w_2$:
\begin{eqnarray*}
y_2^2&=& \textstyle{\frac 1h y_1^3y_2y_4-\frac {b_3}h y_1^6 y_4^2+\frac {b_3}h y_1^7w_3},
\\
w_1&=& \textstyle{\frac 1h y_2y_4-\frac {b_3}h y_1^3 y_4^2+\frac {b_3}h y_1^4w_3},
\\
w_2&=&y_4^2-y_1w_3.
\end{eqnarray*}
Comparing the first equation with \eqref{equation-alpha-lP=3-and-7}
we see that $\ell(P)=7$ and $w_3\operatorname{n}i y_3$.
Then from the second one we see $w_1\operatorname{n}ot\operatorname{n}i y_3$.
Clearly, $\beta$ is a linear combination of $y_1w_1$ and $w_2$
(with constant coefficients). Hence, $\beta\operatorname{n}i y_1y_3$.
As in the proof of Lemma \ref{lemma-equation-possibilities-lP=3+IIIa}
a deformation of the form \eqref{equation-lP=3-7-deformations} is trivial
modulo $I^{(3)}$ and so it
preserves case division \ref{lemma-possibilities-lP=3+III}, as well as,
the vanishing of $b_4$.
Then we can argue as in \ref{scase-lP=3-7-new-treatment} and get a contradiction.
Hence $b_2\operatorname{n}eq 0$. Then we may assume that $b_1=0$ and $b_2=1$. The relations in
$(\operatorname{coker}_{P} \varphi)^\sharp$ are $y_1^2w_2=w_2+y_1w_3=0$, $hw_1+b_3y_1^3w_2=0$.
Eliminating $w_2$ one can see
\[
(\operatorname{coker}_{P} \varphi)^\sharp\simeq
\mathscr{O}_C^\sharp /(y_1^3)\oplus \mathscr{O}_C^\sharp /(h).
\]
By \eqref{equation-lP=3-7-Coker-P}
we have $h(0)\operatorname{n}eq 0$ and $\ell(P)=3$.
From the matrix \eqref{equation-lP=3-7-matrix-2} we see
\begin{eqnarray*}
y_4^2&=& w_2+y_1w_3,
\\
y_2^2&=& y_1^2w_2,
\\
y_2y_4&=& h(y_1^4)w_1+ b_3y_1w_2.
\end{eqnarray*}
Eliminating $w_2$ we obtain the following relations in $\operatorname{gr}_C^2\mathscr{O}$:
\begin{equation}
\label{equation-lP=3-7-vv-2}
w_2=y_4^2-y_1w_3,\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad
y_2^2- y_1^2y_4^2+y_1^3w_3=0.
\end{equation}
The last must be congruent to $ \alpha\mod I^{(3)}$.
Comparing with \eqref {equation-alpha-lP=3-and-7} we see that $w_3=\frac 1c y_3$ in $\operatorname{gr}_C^2\mathscr{O}$.
Since $\beta$ is a section of $(0)\subset \operatorname{gr}_C^2\mathscr{O}$, it must be proportional
to $w_2$. Therefore, $y_1y_3\in \beta$. Moreover, \eqref{equation-lP=3-7-vv-2}
shows that $y_1y_3$ appears in $\beta$ with coefficient $1/c$.
Now we apply Computation \ref{computation-lP=3+III-part2},
Lemma \ref{lemma-computation-lP=3+III-part2},
and Lemma \ref{slemma-lP=3+III-generalityH}
and get a contradiction.
\end{proof}
\begin{scase}
\label{treating-6-5-5}
From now on we assume that $b_4\operatorname{n}eq 0$.
In other words, the map $\pi$
is non-zero.
The induced map
\[
\mathscr{B}^{\mathbin{\tilde\otimes} 2}=(-1)\longrightarrow \mathscr{G}=(-1+P^\sharp)
\]
can be regarded as the multiplication by $sy_1$ for some $s$.
For $\mu\in \mathbb{C}$, take a subsheaf $\mathscr{B}'\subset \mathscr{A}\mathbin{\tilde\oplus}\mathscr{B}$ so that
$y_4':=y_4+\mu y_1y_2$ is an $\ell$-basis of $\mathscr{B}'$.
Clearly, $\operatorname{gr}^1_C\mathscr{O} =\mathscr{A}\mathbin{\tilde\oplus}\mathscr{B}'$.
Regard $y_1$ as a map $\mathscr{B}\to \mathscr{A}$.
Then $(\mu y_1,1)(\mathscr{B})\subset \mathscr{A}\mathbin{\tilde\oplus} \mathscr{B}$ and
we have the following diagram
\begin{equation*}
\xymatrix@R10pt{
((\mu y_1,1)(\mathscr{B}))^{\mathbin{\tilde\otimes} 2}\ar@{=}[d]\ar@{^{(}->}[r]& \tilde S^2\operatorname{gr}_C^1\mathscr{O}\ar[r]^-{\operatorname{pr}}&\mathscr{G}
\\
(\mu^2 y_1^2,2\mu y_1,1)(\mathscr{B}^{\mathbin{\tilde\otimes} 2})\ar@/_15pt/[urr]_-{\cdot(2\mu y_1b_4+sy_1)}
}
\end{equation*}
Set $\mu:=-s/(2b_4)$. With this choice of $\mu$, the map
${\mathscr{B}'}^{\mathbin{\tilde\otimes} 2}\to \mathscr{G}$ is zero.
Thus
$\mathscr{A}^{\mathbin{\tilde\otimes} 2}\mathbin{\tilde\oplus}\mathscr{B}'^{\mathbin{\tilde\otimes} 2}\subset \mathscr{D} \mathbin{\tilde\oplus}{\mathscr{E}}$.
Let ${\mathscr{K}}$ be the ideal such that $I^{(2)}\supset {\mathscr{K}}\supset I^{(3)}$
and ${\mathscr{K}}/ I^{(3)}= \mathscr{D}\mathbin{\tilde\oplus} {\mathscr{E}}$.
Since $\mathscr{A}^{\mathbin{\tilde\otimes} 2}\to \mathscr{G}$ is zero, perturbing $\mathscr{B}$
with $\mu$ has no effect on $\pi: \mathscr{A} \mathbin{\tilde\otimes} \mathscr{B} \to \mathscr{G}$, and we use the
same notation $\pi: \mathscr{A} \mathbin{\tilde\otimes} \mathscr{B}' \to \mathscr{G}$.
\end{scase}
\begin{slemma}\label{lemma-lP=3-7-ci}
$I{\mathscr{K}}=I^{(3)}$ outside $P$ and
$I^{\sharp}{\mathscr{K}}^{\sharp}=(I^{(3)})^{\sharp}$ at $P$.
\end{slemma}
\begin{proof}
Consider the following digram with $\ell$-exact rows and injective vertical arrows:
\begin{equation}\label{big-diagram}
\vcenter{
\xy
\xymatrix@R=14pt@C=20pt{
0\ar[r]
&\mathscr{A}^{\mathbin{\tilde\otimes} 2}\mathbin{\tilde\oplus} \mathscr{B}'^{\mathbin{\tilde\otimes} 2}\ar[r]\ar@{^{(}->}[d]^{\upsilon}
&\tilde S^2\operatorname{gr}^1_C\mathscr{O}\ar[r]\ar@{^{(}->}[d]^{\varphi}&
\mathscr{A}\mathbin{\tilde\otimes} \mathscr{B}'\ar[r]\ar[d]_{{\simeq}}^{b_4}& 0
\\
0\ar[r]&
\mathscr{D}\mathbin{\tilde\oplus} {\mathscr{E}}\ar[r]
&\operatorname{gr}_C^2\mathscr{O}\ar[r]&
\mathscr{G}\ar[r] &0
}
\endxy
}
\end{equation}
At a point $Q\in C$ which is a smooth point of $X$,
we can choose coordinates $u_1,u_2,u_3$
for $(X,Q)$
so that $Q$ is the origin, $C$ is the $u_1$-axis,
and $u_2$ (resp. $u_3$) generates $\mathscr{A}$ (resp. $\mathscr{B}'$)
at $Q$. Then from \eqref{big-diagram} we see
\begin{equation*}
I^{(3)}=I^3=(u_2,u_3)^3, \mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad
{\mathscr{K}}=(u_2^2,u_3^2)+(u_2,u_3)^3,
\end{equation*}
from which follows $I^{(3)}={\mathscr{K}} I$.
At $P$, again from \eqref{big-diagram} we have
\begin{equation*}
\operatorname{coker}_{P^\sharp}\upsilon^\sharp \simeq \operatorname{coker}_{P^\sharp} \varphi^\sharp
\simeq \left(\mathscr{O}_{C^\sharp}/ (y_1^3)\right) y_3.
\end{equation*}
Thus, $(\mathscr{D}\mathbin{\tilde\oplus} {\mathscr{E}})^\sharp$ is generated by $y_3$ and $\varrho$, where
$\varrho:=y_2^2$ or $y_4'^2$. Therefore,
\begin{eqnarray*}
y_2^2,\ y_4'^2\in {\mathscr{K}}^\sharp&=&(y_3,\varrho)+(y_2, y_4)^3,
\\
{\mathscr{K}}^\sharp I^\sharp &=& y_3 I^\sharp+(y_2, y_4)^3.
\end{eqnarray*}
Whence,
\begin{equation*}
\mathscr{O}_{C^\sharp}\cdot y_3 \oplus \mathscr{O}_{C^\sharp}\cdot \varrho
\twoheadrightarrow {\mathscr{K}}^\sharp /{\mathscr{K}}^\sharp I^\sharp.
\end{equation*}
Since
\begin{equation*}
{\mathscr{K}}^\sharp /{\mathscr{K}}^\sharp I^\sharp \twoheadrightarrow
{\mathscr{K}}^\sharp/ {I^{(3)}}^\sharp
\simeq \mathscr{O}_{C^\sharp}\oplus \mathscr{O}_{C^\sharp},
\end{equation*}
the arrow above is an isomorphism and
$I^{\sharp}{\mathscr{K}}^{\sharp}=(I^{(3)})^{\sharp}$ at $P^{\sharp}$.
If $\ell(P)=3$, then at $R$, changing coordinates $z_1,\dots, z_4$ keeping $z_1$ and $z_3$
the same, we may assume that $z_2$ and $z_4$
are bases at $R$ of $\mathscr{A}$ and $\mathscr{B}'$, respectively.
Then in view of \eqref{big-diagram} and $\operatorname{coker}_R \varphi =\mathbb{C}_R$, we see that
$\mathscr{D}\mathbin{\tilde\oplus} {\mathscr{E}}$ is generated by $z_3$ and $z_i^2$ for some $i=2,\, 4$.
Therefore,
\begin{eqnarray*}
z_2^2,\, z_4^2\in {\mathscr{K}} &=&(z_3,\, z_i^2)= (z_2,\, z_4)^3,
\\
{\mathscr{K}} I &=& z_3 I+(z_2, y_4)^3.
\end{eqnarray*}
Whence,
\begin{equation*}
\mathscr{O}_{C} \cdot z_3 \oplus \mathscr{O}_{C}\cdot z_i^2
\twoheadrightarrow {\mathscr{K}} /{\mathscr{K}} I.
\end{equation*}
Since
\begin{equation*}
{\mathscr{K}} /{\mathscr{K}} I \twoheadrightarrow {\mathscr{K}}/ I^{(3)}
\simeq \mathscr{O}_C\oplus \mathscr{O}_C,
\end{equation*}
we have $I{\mathscr{K}}=I^{(3)}$ at $R$.
This proves Lemma \ref{lemma-lP=3-7-ci}.
\end{proof}
\begin{scorollary}\label{corollary-lP=3-7-ci}.
${\mathscr{K}}\mathbin{\tilde\otimes} \mathscr{O}_C\simeq (P^\sharp)\mathbin{\tilde\oplus} (0)$ and so ${\mathscr{K}}$ is an l.c.i.
ideal of codimension $2$ outside $P$ and ${\mathscr{K}}^\sharp$ is l.c.i. at $P^\sharp$.
\end{scorollary}
\begin{scase}
Thus,
\begin{eqnarray*}
{\mathscr{K}}/ ({\mathscr{K}}\mathbin{\tilde\otimes} I)&=& (P^\sharp)\mathbin{\tilde\oplus} (0),
\\
(\omega_X\mathbin{\tilde\otimes} {\mathscr{K}})/ (\omega_X\mathbin{\tilde\otimes}{\mathscr{K}}\mathbin{\tilde\otimes} I)&=& (0)\mathbin{\tilde\oplus} (-P^\sharp).
\end{eqnarray*}
Our goal is to extend a non-zero section $\bar \xi $ of
$(0)\subset \omega_X\mathbin{\tilde\otimes} {\mathscr{K}}/ \omega_X\mathbin{\tilde\otimes}{\mathscr{K}}\mathbin{\tilde\otimes} I$
to a section $\xi \in \operatorname{H}^0(\omega_X\mathbin{\tilde\otimes} {\mathscr{K}})$.
By the Formal Function Theorem
\begin{equation*}
\lim_{\longleftarrow}
\operatorname{H}^0\left(\frac{\omega_X\mathbin{\tilde\otimes} {\mathscr{K}}}{\omega_X\mathbin{\tilde\otimes} {\mathscr{K}}^{(n)}} \right)
\simeq
\lim_{\longleftarrow}\ \frac{
f_*(\omega_X\mathbin{\tilde\otimes} {\mathscr{K}})}{ {\mathfrak{m}}^n_{o,Z}f_*(\omega_X\mathbin{\tilde\otimes} {\mathscr{K}})}.
\end{equation*}
Thus, for lifting $\bar \xi$, it is sufficient
to show that the map
\begin{equation*}
\Phi_n: \operatorname{H}^0(\omega_X\mathbin{\tilde\otimes} {\mathscr{K}}/ \omega_X\mathbin{\tilde\otimes}{\mathscr{K}}^{(n)})
\xrightarrow{\hspace*{20pt}}
\operatorname{H}^0(\omega_X\mathbin{\tilde\otimes} {\mathscr{K}}/ \omega_X\mathbin{\tilde\otimes}{\mathscr{K}}\mathbin{\tilde\otimes} I)
\end{equation*}
is surjective for all $n>0$, or equivalently $\Phi_2$ and
\begin{equation*}
\Psi_n:
\operatorname{H}^0(\omega_X\mathbin{\tilde\otimes} {\mathscr{K}}/ \omega_X\mathbin{\tilde\otimes}{\mathscr{K}}^{(n)})
\xrightarrow{\hspace*{20pt}}
\operatorname{H}^0(\omega_X\mathbin{\tilde\otimes} {\mathscr{K}}/ \omega_X\mathbin{\tilde\otimes}{\mathscr{K}}^{(n-1)})
\end{equation*}
are surjective for all $n>0$.
We have
\begin{equation*}
0 \to
\omega_X\mathbin{\tilde\otimes} \left(
\frac{{\mathscr{K}}^{(n-1)}}{{\mathscr{K}}^{(n)}}\right)
\xrightarrow{\hspace*{20pt}}
\frac{\omega_X\mathbin{\tilde\otimes} {\mathscr{K}}}{ \omega_X\mathbin{\tilde\otimes}{\mathscr{K}}^{(n)}}
\xrightarrow{\makebox[35pt]{ $\scriptstyle\psi_n$}}
\frac{\omega_X\mathbin{\tilde\otimes} {\mathscr{K}}}{ \omega_X\mathbin{\tilde\otimes}{\mathscr{K}}^{(n-1)}}
\to 0.
\end{equation*}
Note that the sheaves
$\omega_X\mathbin{\tilde\otimes}(\operatorname{im}({\mathscr{K}}\mathbin{\tilde\otimes} I \to {\mathscr{K}}))/{\mathscr{K}}^{(2)})$ and
\begin{equation*}
\omega_X\mathbin{\tilde\otimes} {\mathscr{K}}^{(n-1)} /\omega_X\mathbin{\tilde\otimes}{\mathscr{K}}^{(n)}
\simeq \tilde S^{n-1}\left(\omega_X\mathbin{\tilde\otimes}{\mathscr{K}}/\omega_X\mathbin{\tilde\otimes}{\mathscr{K}}^{(2)}\right)
\end{equation*}
have filtrations with successive subquotients
\begin{equation*}
(-P^\sharp)\mathbin{\tilde\otimes} \tilde S^{n-1}\left((-P^\sharp)\mathbin{\tilde\oplus} (0)\right)
\mathbin{\tilde\otimes}
\begin{cases}
(0)
\\
(-1+2P^\sharp)
\\
(-1+3P^\sharp)
\\
(-1+P^\sharp)
\end{cases}
\end{equation*}
which are all $\ge (-1)$ and hence have vanishing $\operatorname{H}^1$.
Thus $\Psi_n=\operatorname{H}^0(\psi_n)$ and $\Phi_2$ are onto and so is
$\Phi_n=\Phi_2\circ \Psi_3\circ\cdots \circ \Psi_n$.
\end{scase}
\begin{scase}\label{sde-iP=3+III-sections}
Thus a non-zero section $\bar \xi $ of
$(0)\subset \omega_X\mathbin{\tilde\otimes} {\mathscr{K}}/ \omega_X\mathbin{\tilde\otimes}{\mathscr{K}}\mathbin{\tilde\otimes} I$
induces
a section $\xi \in \operatorname{H}^0(\omega_X\mathbin{\tilde\otimes} {\mathscr{K}})$ which in turns
induces a generator of $(P^\sharp)$. Let $G:=\{\xi =0\}$.
Then $G\supset 4C$ and $\mathscr{O}_H{\mathscr{K}}=\mathscr{O}_H(-G)$.
Hence, ${\mathscr{K}}$ is generated by $\xi$ and $\beta$:
\begin{scorollary}\label{scorollary-9-6-8}
The ideal ${\mathscr{K}}$ is a global complete intersection.
More precisely, ${\mathscr{K}}=(\beta,\xi)$.
\end{scorollary}
Moreover, $\xi$
can be locally written as
$\xi=y_3+(\text{higher degree terms})$. Thus we may assume that there exists
a global section of $\mathscr{O}_X$ which is locally written as
$y_1y_3$, i.e. $y_1y_3\in \beta$.
On the other hand by \ref{ge} the general member $D\in |-K_X|$
is given by $y_1+\xi'=0$ for some $\xi'\in (y_2,y_3,y_4)$.
Then replacing $\beta$ with a linear combination of $\beta$ and $(y_1+\xi')\xi$ we may assume that
$y_1y_3$ appears in $\beta$ with arbitrary
coefficient $\lambda$ and $y_4^2$ appears in $\beta$ with coefficient $1$.
In particular, there is a specific section $\beta^\circ$
which does not contain $y_1y_3$ (and contains $y_4^2$).
Then $H$ can be given by the equations $\alpha^\circ=\beta=0$,
where $\alpha^\circ:= \alpha+ y_1^2\beta^\circ$ contains $y_1^2y_4^2$.
\end{scase}
Now applying Computation \ref{computation-lP=3a-III} with $l=3$ or $7$,
we obtain the diagram \ref{main-theorem-divisorial}. The following examples show that
this case does occur.
\end{case}
\begin{example}\label{example-divisorial-lP=3}
Let $Z \subset {\mathbb{C}}^5_{z_1,\ldots,z_5}$ be
defined by
\begin{eqnarray*}
0&=& z_2^2+z_3+z_4z_5^k-z_1^3,\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad k\ge 1,\\
0&=& z_1^2z_2^2+z_4^2-z_3z_5.
\end{eqnarray*}
Then $(Z,0)$ is a
threefold singularity of type \type{cD_{5}}.
Let $B \subset Z$ be the $z_5$-axis and
let $f : X \to Z$ be the weighted $(1,1,4,2,0)$-blowup.
The origin
of the $z_3$-chart
is a type \type{(IIA)} point $P$ with $\ell(P)=3$:
\begin{equation*}
\{-y_1^3y_3+y_2^2+y_3^2+y_4(y_1^2y_2^2+y_4^2)^k=0\}/{\boldsymbol{\mu}}_{4}(1,1,3,2),
\end{equation*}
where $(C,P)$ is the $y_1$-axis.
In the $z_1$-chart we have type \type{(III)} a point.
\end{example}
\begin{example}
\label{example-divisorial-lP=7}
As in \ref{example-divisorial-lP=3}, let $Z \subset \mathbb{C}^5_{z_1,\ldots,z_5}$ be
defined by
\begin{eqnarray*}
0 &=& z_2^2+z_1^2z_5+ z_3+z_4z_5^k, \mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad k\ge 1,
\\
0 &=& z_3z_5+z_1^5+z_4^2.
\end{eqnarray*}
Then the point $(Z,0)$ is of type \type{cD_{5}}.
Let $B \subset Z$ be the $z_5$-axis and
let $f : X \to Z$ be the weighted $(1,1,4,2,0)$-blowup.
In the $z_1$-chart $X$ is smooth and
the origin of the $z_3$-chart
is a \type{(IIA)} point $P$ with $\ell(P)=7$:
\begin{equation*}
\{-y_1^7y_3+y_2^2+y_3^2-y_1^2y_4^2+y_4(y_1^5y_3+y_4^2)^k=0\}/{\boldsymbol{\mu}}_{4}(1,1,3,2),
\end{equation*}
where $(C,P)$ is the $y_1$-axis.
\end{example}
\section{Cases $\ell(P)=4$ and $8$}\label{section-lP=4}
In this section we assume that $\ell(P)\in \{4,\, 8\}$.
We will show that Computation \ref{computation-lP=4+III} is applicable here
and the possibility \ref{main-theorem-conic-bundle} occurs.
\begin{case}
According to \ref{equation-IIA-point} we may write
\begin{equation}\label{equation-lP=4-alpha}
\alpha=y_1^{\ell(P)}y_4+y_2^2+y_3^2+\delta y_4^3+c y_1^2y_4^2+\epsilon y_1y_3y_4+
\zeta y_1^2y_2y_3+\cdots,
\end{equation}
with $\delta,\, c,\, \epsilon,\, \zeta\in \mathbb{C}\{y_1^4\}$.
It is easy to see that $y_4\in I^{\sharp (2)}$. Hence,
\begin{equation}\label{equation-y14y4}
-y_1^{\ell(P)}y_4\equiv y_2^2+y_3^2+
\zeta y_1^2y_2y_3 \mod I^{\sharp (3)}.
\end{equation}
By Proposition \ref{proposition-cases-lP} in the case $\ell(P)=4$ the variety
$X$ has a type \type{(III)} point $R$ with $i_R(1)=1$ and
$X$ is smooth outside $P$ in the case $\ell(P)=8$.
\end{case}
\begin{case}
Taking Proposition
\ref{proposition-lP=4-XC} into account
for any $n\ge 1$ we can write
\begin{equation*}
(\operatorname{gr}_C^n \mathscr{O})^\sharp =\bigoplus_{\substack{a+b+2c= n\\ b=0,\ 1}} \mathscr{O}_{C^\sharp}\cdot y_2^ay_3^by_4^c,
\end{equation*}
where $a,\, b,\, c\ge 0$, and
\begin{equation}
\label{equation-lP=4-gr1O}
\vcenter{
\xymatrix@R=6pt@C=-3pt{
\operatorname{gr}_C^1\mathscr{O}= &(-1+3P^\sharp)\ar@{=}[d]&\mathbin{\tilde\oplus}& (-1+P^\sharp),\ar@{=}[d]
\\
& \mathscr{A} && \mathscr{B}
}}
\end{equation}
where $y_2$ (resp. $y_3$) is an $\ell$-basis of
$\mathscr{A}$ (resp. $\mathscr{B}$) at $P$.
\end{case}
\begin{case}
In the case $\ell(P)=4$
by \cite[Lemma 2.16]{Mori-1988}, since $i_R(1)=1$,
the equation of $X$ at $R$ can be written as follows
\begin{equation}\label{equation-lP=4-gamma}
\gamma(z)=z_1z_4+q_2(z_2, z_3)+q_3(z_1, \dots,z_4),\quad q_3\in (z_2,z_3,z_4)^3,
\end{equation}
where $C$ is the $z_1$-axis and
$q_2\in \mathbb{C}\cdot z_2^2+\mathbb{C}\cdot z_2z_3+\mathbb{C}\cdot z_3^2$. Hence, $z_4\in I^{(2)}$.
\end{case}
\begin{case}
Consider the map $\varphi: \tilde S^2\operatorname{gr}_C^1\mathscr{O} \hookrightarrow \operatorname{gr}_C^2\mathscr{O}$.
Clearly, it is an isomorphism outside $\{P,\, R\}$ (resp. $\{P\}$)
in the case $\ell(P)=4$ (resp. $\ell(P)=8$).
The equality \eqref{equation-lP=4-gr1O} implies
\begin{eqnarray*}
\tilde S^2\operatorname{gr}_C^1\mathscr{O}&=&
(-1+2P^\sharp)\mathbin{\tilde\oplus} (-1)\mathbin{\tilde\oplus} (-2+2P^\sharp),
\\
\deg \operatorname{gr}_C^2\mathscr{O} &=& -4+\operatorname{len} \operatorname{coker} \varphi\ge -2.
\end{eqnarray*}
Furthermore,
\begin{equation}
\label{equation-lP=4-cokerP-cokerP}
\operatorname{coker}_P \varphi =\mathbb{C}_{(\ell(P)/4)P}\cdot \overline{(y_1^2y_4)}.
\end{equation}
Hence, in the case $\ell(P)=4$,
$\operatorname{coker}_R \varphi\operatorname{n}eq 0$. Taking Proposition \ref{proposition-lP=4-XC}\ref{proposition-lP=4-XC-1}
into account in this case we obtain
$q_2\operatorname{n}eq 0$ (see \eqref{equation-lP=4-gamma}) and
\begin{equation}
\label{equation-lP=4-cokerR-cokerR}
\operatorname{coker}_R \varphi =\mathbb{C}_R\cdot \bar z_4\simeq \mathbb{C}.
\end{equation}
Thus in both cases $\ell(P)=4$ and $\ell(P)=8$ we have
$\deg \operatorname{gr}_C^2\mathscr{O}=-2$.
By Lemma \ref{lemma-lP=3+IIIa}
\begin{equation}
\label{equation-lP=4-gr-2-C-O-1}
\operatorname{gr}_C^2\mathscr{O} \simeq \mathscr{O}\oplus \mathscr{O}(-1)^{\oplus 2}.
\end{equation}
Furthermore, $\operatorname{gr}_C^2\mathscr{O}$ has an $\ell$-basis $y_2y_3$, $y_2^2$, $y_4$ at $P^\sharp$.
Thus,
\begin{equation}
\label{equation-lP=4-gr-2-C-O}
\operatorname{gr}_C^2\mathscr{O}=
(0)\mathbin{\tilde\oplus}(-1+2P^\sharp)\mathbin{\tilde\oplus} (-1+2P^\sharp),
\end{equation}
since $\operatorname{H}^1 (\operatorname{gr}_C^2\omega)=0$ (cf. Lemma \ref{treating-equation-possibilities-lP=3+III-c}).
\end{case}
\begin{case}
According to \eqref {equation-lP=4-cokerP-cokerP} and \eqref {equation-lP=4-cokerR-cokerR}
\begin{equation}
\label{equation-lP=4-quotient}
\operatorname{gr}_C^2\mathscr{O}/\tilde S^2\operatorname{gr}_C^1\mathscr{O}\simeq
\begin{cases}
\mathbb{C}_P\oplus\mathbb{C}_R& \text{in the case $\ell(P)=4$,}\\
\mathbb{C}_{2P}& \text{in the case $\ell(P)=8$.}
\end{cases}
\end{equation}
Let $\mathbb{F}F$ be the sheaf
with an $\ell$-structure defined by the conditions:
\begin{eqnarray*}
&&\tilde S^2\operatorname{gr}_C^1\mathscr{O} \subset \mathbb{F}F\subset \operatorname{gr}_C^2\mathscr{O},
\\
&&\operatorname{gr}_C^2\mathscr{O}/\mathbb{F}F=\mathbb{C}_P,
\\
&&\operatorname{gr}_C^2\mathscr{O}^\sharp/\mathbb{F}F^\sharp=\mathscr{O}^\sharp/(y_1^4)\cdot y_4^2.
\end{eqnarray*}
{}From \eqref{equation-lP=4-gr-2-C-O-1} one can see that
there are two possibilities:
\begin{numcases}{\mathbb{F}F\simeq}
\mathscr{O}(-1)^{\oplus 3},\label{lP=4-F-case-1}
\\
\mathscr{O}\oplus\mathscr{O}(-1)\oplus \mathscr{O}(-2).\label{lP=4-F-case-2}
\end{numcases}
\end{case}
\begin{case}{\bf Case \eqref{lP=4-F-case-2}.}
Since $\mathbb{F}F\subset \operatorname{gr}_C^2\mathscr{O}$, by \eqref {equation-lP=4-gr-2-C-O}
\begin{equation*}
\mathbb{F}F=(0)\mathbin{\tilde\oplus} (-1+2P^\sharp) \mathbin{\tilde\oplus} (-2+2P^\sharp).
\end{equation*}
Now we treat the cases $\ell(P)=4$ and $\ell(P)=8$ separately.
\end{case}
\begin{slemma}\label{lemma-lP=4-case-does-not-occur}
The case \eqref{lP=4-F-case-2} with $\ell(P)=4$ does not occur.
\end{slemma}
\begin{proof}
Consider the embedding
\begin{equation*}
z_1\cdot(0)\subset \mathscr{O}_C(-R)\cdot\mathbb{F}F \subset \tilde S^2\operatorname{gr}_C^1\mathscr{O}
= (-1+2P^\sharp) \mathbin{\tilde\oplus}(-1)\mathbin{\tilde\oplus} (-2+2P^\sharp).
\end{equation*}
Clearly, the image in the third summand is zero and the projection to the second summand is
multiplication by a constant. Moreover, if this constant is zero, then the image of $z_1\cdot(0)$
is contained in $(-1+2P^\sharp)$.
In other words, the summand $(0)\subset \mathbb{F}F\subset \operatorname{gr}_C^2\mathscr{O}$
is contained in $(2P^\sharp)$ which is impossible by \eqref{equation-lP=4-gr-2-C-O}.
By changing $\ell$-splitting as follows
\begin{equation*}
z_3 \longmapsto z_3+(\operatorname{const} ) z_2,\quad y_3 \longmapsto y_3+(\operatorname{const}) y_1^2y_2,
\end{equation*}
one can assume that $q_2\in \mathbb{C}^*\cdot z_2z_3$ and so $(0)\operatorname{n}i \overline{z_1z_4}=\overline{z_2z_3}$.
Furthermore,
$\mathbb{F}F\supset (0)=\mathscr{O}_C\cdot z_4$ at $R$ by changing coordinates as $z_4 \mapsto z_4+\cdots$.
Since $\mathbb{F}F\subset \operatorname{gr}_C^2\mathscr{O}$, $(0)$ is sent isomorphically to
$(0)\subset \operatorname{gr}_C^2\mathscr{O}$.
We have the inclusion $\operatorname{gr}_C^2\mathscr{O}\supset \mathscr{O}_C\cdot \bar\beta=\mathscr{A}\mathbin{\tilde\otimes} \mathscr{B}(R)$
(see \eqref{equation-lP=4-gr1O}). Hence, $\bar\beta=\operatorname{n}u y_2y_3$ at $P^\sharp$, where $\operatorname{n}u$ is a unit.
\begin{sclaim}
$\bar\beta \operatorname{gr}_C^1\mathscr{O}$ is an $\ell$-subbundle
of $\operatorname{gr}_C^3\mathscr{O}$ and and the natural map
$\mathscr{A}^{\mathbin{\tilde\otimes} 3}\to \operatorname{gr}_C^3\mathscr{O}/\bar\beta \operatorname{gr}_C^1\mathscr{O}$
induces the following $\ell$-exact sequence
\begin{equation}
\label{iP=4-equation-exact-sequence-l}
\vcenter{
\xymatrix@R=6pt@C=20pt{
0\ar[r]&\mathscr{A}^{\mathbin{\tilde\otimes} 3}(4P^\sharp) \ar[r]\ar@{=}[d]&\operatorname{gr}_C^3\mathscr{O}/\bar\beta \operatorname{gr}_C^1\mathscr{O}
\ar[r]& \mathscr{B}^{\mathbin{\tilde\otimes} 3}(4P^\sharp)\ar@{=}[d] \ar[r]& 0,
\\
&(P^\sharp)&&(-2+3P^\sharp)
} }
\end{equation}
where $y_2y_4$ \textup(resp. $y_3y_4$\textup) is an $\ell$-basis of $\mathscr{A}^{\mathbin{\tilde\otimes} 3}(4P^\sharp)$
\textup(resp. $\mathscr{B}^{\mathbin{\tilde\otimes} 3}(4P^\sharp)$\textup).
\end{sclaim}
\begin{proof}
To check the assertion at $R$ we apply Proposition \ref{proposition-lP=4-XC}\ref{proposition-lP=4-XC-5}
with $m=1$ and $\bar\beta=z_4$, and note that $\operatorname{gr}_C^3\mathscr{O}/\bar\beta \operatorname{gr}_C^1\mathscr{O}=\mathscr{O}_Cz_2^3\oplus \mathscr{O}_C z_3^3$.
At $P^\sharp$, we note that $\bar\beta=\operatorname{n}u y_2y_3$ and
use Proposition \ref{proposition-lP=4-XC}\ref{proposition-lP=4-XC-4} with $h=\alpha$
to show that $\operatorname{gr}_C^3\mathscr{O}$ has $\ell$-basis $y_2^3$, $y_2^2y_3$, $y_2y_4$, $y_3y_4$.
By \eqref{equation-y14y4}
\begin{equation*}
y_1^{4}y_4+y_2^2+y_3^2+\zeta y_1^2\bar\beta=0.
\end{equation*}
Then $\operatorname{gr}_C^3\mathscr{O}/\bar\beta\operatorname{gr}_C^1\mathscr{O}$ has an $\ell$-free $\ell$-basis
$y_2y_4$, $y_3y_4$ because
$y_3^2y_2\equiv -y_2^3-y_1^4y_2y_4$, and we have $y_2^3\equiv -y_1^4y_2y_4\mod (\bar\beta)$
and $y_2y_4\equiv -y_2^3/y_1^4 \mod (\bar\beta)$. This shows the exactness because
$y_3^3\equiv -y_1^4y_3y_4\mod (\beta)$.
\end{proof}
To complete the proof of Lemma \ref{lemma-lP=4-case-does-not-occur} we note that
the sequence \eqref{iP=4-equation-exact-sequence-l}
implies that $\operatorname{H}^1(\operatorname{gr}_C^3\mathscr{O}/\bar\beta \operatorname{gr}_C^1\mathscr{O} )\operatorname{n}eq 0$.
This contradicts Lemma \ref{lemma-grC}.
Thus the case \eqref{lP=4-F-case-2} with $\ell(P)=4$ does not occur.
\end{proof}
\begin{slemma}\label{lemma-lP=8-case-does-not-occur}
The case \eqref{lP=4-F-case-2} with $\ell(P)=8$ does not occur.
\end{slemma}
\begin{proof}
We have $0\operatorname{n}eq \bar\beta \in \operatorname{H}^0((0))\subset \operatorname{H}^0(\mathbb{F}F)$.
Since $\bar\beta \operatorname{n}otin \operatorname{H}^0(\tilde S^2\operatorname{gr}_C^1\mathscr{O})$ and
$\mathbb{F}F/\tilde S^2\operatorname{gr}_C^1\mathscr{O}=\mathbb{C}\cdot \overline{y_1^6y_4}$, we have
\begin{equation}
\label{equation-lP=4-barbeta}
\bar\beta=(\cdots )y_2^2+(\cdots )y_2y_3+(\operatorname{unit})y_1^6y_4.
\end{equation}
{}From the following relation
\begin{equation*}
\bar\beta\cdot (-1) \subset \mathbb{F}F(-4P^\sharp)\subset \tilde S^2\operatorname{gr}_C^1\mathscr{O}
= (-1+2P^\sharp) \mathbin{\tilde\oplus}(-1)\mathbin{\tilde\oplus} (-2+2P^\sharp)
\end{equation*}
we see that the image of $y_1^4\bar\beta$ in the
third summand is zero and the projection to the second summand is multiplication by a
constant. Moreover, if this constant is zero, then the image of $y_1^4\cdot(0)$
is contained in $(-1+2P^\sharp)$.
In other words, the summand $(0)\subset \mathbb{F}F\subset \operatorname{gr}_C^1\mathscr{O}$
is contained in $(2P^\sharp)$ which is impossible by \eqref{equation-lP=4-gr-2-C-O}.
Therefore,
\begin{equation*}
y_1^4\bar\beta =(\cdots)y_2^2+(\operatorname{unit}) y_2y_3.
\end{equation*}
Then \eqref{equation-lP=4-barbeta} implies
\begin{equation*}
y_1^{10}y_4\equiv (\cdots)y_2^2+(\operatorname{unit}) y_2y_3 \mod I^{(3)}.
\end{equation*}
On the other hand, $y_4$, $y_2^2$, $y_2y_3$ form an $\ell$-basis of
$\operatorname{gr}_C^2\mathscr{O}$, a contradiction.
This proves Lemma \ref{lemma-lP=8-case-does-not-occur}.
\end{proof}
\begin{case}{\bf Case \eqref{lP=4-F-case-1}.}\label{case-conic-bundle}
If the coefficient of $y_1^2y_4$ in $\bar\beta$ is zero,
then $\bar\beta \in \operatorname{H}^0(\mathbb{F}F)$. But
in our case $\operatorname{H}^0(\mathbb{F}F)=0$ which gives us a contradiction.
Thus for a general choice of $\beta\in \operatorname{H}^0(\mathscr{O}_X)$
at $P$ we can write $\bar\beta=\operatorname{n}u y_2y_3+\eta y_1^2y_4+\cdots$ and so
\begin{equation*}
\beta=\theta y_4^2 +\operatorname{n}u y_2y_3+\eta y_1^2y_4+\cdots,
\end{equation*}
where $\theta,\, \operatorname{n}u,\, \eta$ are units.
This means that $y_1^2y_4\in \beta$.
Since $\operatorname{h}^0(\operatorname{gr}_C^2\mathscr{O})=1$, the ratio of
the coefficients $\operatorname{n}u$ and $\eta$ is fixed.
On the other hand, the ratio of
the coefficients of $\operatorname{n}u$ and $\theta$
is general \cite[Lemma 3.1.1]{Mori-Prokhorov-IIA-1}.
Hence the ratio of coefficients $\theta$ and $\eta$
can be chosen general.
Then we apply Computation \ref{computation-lP=4+III}.
One can see that the graph \eqref{graph-diagram-non-normal-lP=4+III} corresponds
to a conic bundle.
We obtain the diagram \ref{main-theorem-conic-bundle}.
Examples \ref{example-conic-bundle-lP=4+III}
and \ref{example-conic-bundle-lP=8} below show that both possibilities $\ell(P)=4$ and $8$ do occur.
\end{case}
\begin{example}\label{example-conic-bundle-lP=4+III}
Let $X$ be the the hypersurface of weighted degree $10$ in
the weighted projective space $\mathbb{P}(1,1,3,2,4)_{x_1,x_2, x_3, x_4, w}$
given by the equation
\begin{equation*}w\phi_6 -x_1^6\phi_4=0,\quad \text{}\quad
\begin{array}{lll}
\phi_6&:=&x_1^4x_4+x_3^2+x_2^2w+\delta x_4^3,
\\
\phi_4&:=&x_4^2+\operatorname{n}u x_2x_3+\eta x_1^2x_4+\mu x_1^3x_2
\end{array}
\end{equation*}
(for simplicity we assume that the coefficients $\delta$, $\operatorname{n}u$, $\eta$ are
general).
Regard $X$ as a small analytic neighborhood of $C$.
In the affine chart $U_w:=\{w\operatorname{n}eq 0\}\simeq \mathbb{C}^4/{\boldsymbol{\mu}}_{4}(1,1,3,2)$
the variety $X$ is given by
\begin{equation*}
\phi_6(y_1,y_2,y_3,y_4, 1) - y_1^6\phi_4(y_1,y_2,y_3,y_4, 1)=0
\end{equation*}
and $C$ is the $y_1$-axis.
Clearly, it has the form \eqref{equation-lP=4-alpha}.
So, the origin $P\in (X,C)$ is a type \type{(IIA)} point with $\ell(P)=4$.
In the affine chart $U_1:=\{x_1\operatorname{n}eq 0\}\simeq \mathbb{C}^4$
the variety $X$ is defined by
\begin{equation*}
w\phi_6(1,z_2,z_3,z_4, w) - \phi_4(1,z_2,z_3,z_4, w)=0.
\end{equation*}
If $\mu\operatorname{n}eq 0$, then $X$ is smooth outside $P$, i.e. $(X,C)$ is as
in the case \cite[(1.1.4)]{Mori-Prokhorov-IIA-1}. If $\mu=0$, then
$(X,C)$ has a type \type{(III)} point at $(0,0,0,\eta)$.
Consider the surface $H=\{\phi_6=\phi_4=0\}\subset X$.
Let $\psi : H^{\operatorname{n}}\to H$ be the normalization (we put $H^{\operatorname{n}}=H$ if $H$ is normal)
and let $C^{\operatorname{n}}:=\psi^{-1}(C)$. Near $P$ the surface
$H$ has the form \cite[9.3]{Mori-Prokhorov-IIA-1} (resp.
\ref{computation-lP=4+III}) if
$\mu \operatorname{n}eq 0$ (resp. $\mu =0$). In particular, the singularities of $H^{\operatorname{n}}$ are rational.
Note that $H$ is a fiber of the fibration $\pi: X\to D$ over a small disk around the origin
given by the rational function $ \phi_4/w =\phi_6/x_1^6$
which is regular in a neighborhood of $C$.
By the adjunction formula $\mathscr{O}_{X}(K_X)=\mathscr{O}_{X}(-1)$.
Hence,
\begin{equation*}
-K_H\cdot C=-K_X\cdot C=\mathscr{O}_{\mathbb{P}}(1)\cdot C=\textstyle \frac14.
\end{equation*}
\begin{sclaim}
\begin{enumerate}[leftmargin=20pt]
\item
If $\mu \operatorname{n}eq 0$, then $H$ is smooth outside $P$.
\item
Assume that $\mu = 0$.
Let $P_1\in C$ be the point $\{4\eta^2w=\operatorname{n}u^2 x_1^4\}$.
Then $H$ is singular along $C$, the curve
$C^{\operatorname{n}}$ is irreducible and rational, and $\psi_C:= C^{\operatorname{n}}\to C$ is a double cover
branched over $\{P,\, P_1\}$.
Moreover, $\psi^{-1}(P)$ is the only singular point of $H^{\operatorname{n}}$.
\end{enumerate}
\end{sclaim}
\begin{proof}
Direct computations show that
$P_1\in H$ is a pinch point \textup(see \xref{definition-pinch-point}\textup) and any $Q\in C\setminus \{P,\, P_1\}$
is a double normal crossing point of $H$.
\end{proof}
\begin{sclaim}
\label{claim-conic-bundle-C-Q-Cartier}
If $\mu=0$ \textup(resp. $\mu\operatorname{n}eq 0$\textup), then
$4C^{\operatorname{n}}$ \textup(resp. $8C^{\operatorname{n}}$\textup) is a Cartier divisor on $H^{\operatorname{n}}$. Moreover,
$(C^{\operatorname{n}})^2=0$.
\end{sclaim}
\begin{proof}
We consider only the case where $H$ is not normal, i.e. $\mu=0$.
The case $\mu\operatorname{n}eq 0$ is easier and left to the reader.
Let $V\subset \mathbb{P}(1,1,3,2,4)$ be the weighted hypersurface given by $x_4=0$
and let $M:= H\cap V$.
We have $M=\{x_3^2+w x_2^2=x_2x_3=x_4=0\}$.
Let $\Gamma$ be the line $\{x_3=x_4=w=0\}$
and let $\Gamma^{\operatorname{n}}$ be its preimage on $H^{\operatorname{n}}$.
Then
$\psi^* 2M= 4 C^{\operatorname{n}}+ 2\Gamma^{\operatorname{n}}$.
Since $2M$ is Cartier near $C$ and $\Gamma^{\operatorname{n}}$ is contained in the smooth locus of $H^{\operatorname{n}}$,
the divisor $4 C^{\operatorname{n}}$ is Cartier on $H^{\operatorname{n}}$. Further,
by the projection formula
\begin{equation*}
\psi^* 2M\cdot C^{\operatorname{n}} =4 V\cdot C =2.
\end{equation*}
Since $\Gamma$ is smooth and $C^{\operatorname{n}}\to C$ is \'etale over the point $\Gamma\cap C$,
the curves $\Gamma^{\operatorname{n}}$ and $C^{\operatorname{n}}$ meet each other transversely
at one point which is a smooth point of $H^{\operatorname{n}}$.
Hence, $\Gamma^{\operatorname{n}}\cdot C^{\operatorname{n}}=1$ and so
\begin{equation*}
\label{equation-conic-bundle-H-c2}
4 (C^{\operatorname{n}})^2= \psi^* 2M \cdot C^{\operatorname{n}} - 2\Gamma^{\operatorname{n}}\cdot C^{\operatorname{n}}= 2-2=0.\qedhere
\end{equation*}
\end{proof}
\begin{sclaim}\label{claim-conic-bundle-surface-H}
There exists a rational curve fibration $f_H: H\to B$, where $B\subset \mathbb{C}$
is a small disk around the origin, such that $C=f_H^{-1}(0)_{\operatorname{red}}$.
\end{sclaim}
\begin{proof}
Using the explicit description of the minimal resolution
(see \cite[9.3]{Mori-Prokhorov-IIA-1}, \eqref{graph-diagram-non-normal-lP=4+III}) and Claim \ref{claim-conic-bundle-C-Q-Cartier},
one can see that the contraction exists on $H^{\operatorname{n}}$.
Then, clearly, it descends to $H$.
\end{proof}
\begin{sclaim}
One has $\operatorname{H}^1(\hat X,\mathscr{O}_{\hat X})=0$,
where $\hat X$ denotes the completion of $X$ along $C$.
\end{sclaim}
\begin{proof}
Consider the case $\mu=0$ (the case $\mu\operatorname{n}eq 0$ is similar and easier).
By Claim \ref{claim-conic-bundle-surface-H} \ $4C^{\operatorname{n}}=\operatorname{div} (\varphi)$
for some regular function $\varphi\in \operatorname{H}^0(\mathscr{O}_{H^{\operatorname{n}}})$. Since $\varphi|_{C^{\operatorname{n}}}=0$,
this function descends to $H$ and defines a Cartier divisor $\mathbb{C}C$ on $H$ such that
$\psi^* \mathbb{C}C=4C^{\operatorname{n}}$. Consider the standard injection
$\theta: \mathscr{O}_H\to \psi_* \mathscr{O}_{H^{\operatorname{n}}}$. Then there is the following commutative diagram
\begin{equation*}
\label{equation-conic-bundle-diagram}
\begin{xy}
\xymatrix@C=39pt{
& I_C\ar@{^{(}->}[d] & I_{C^{\operatorname{n}}}\ar@{^{(}->}[d]
\\
0\ar[r]&\mathscr{O}_H\ar@{->>}[d] \ar[r]^{\theta}& \psi_* \mathscr{O}_{H^{\operatorname{n}}}\ar@{->>}[d]\ar[r] &\operatorname{coker}(\theta)\ar[d]^{\simeq}\ar[r]&0
\\
0\ar[r]&\mathscr{O}_C \ar[r]^{\theta}& \psi_* \mathscr{O}_{C^{\operatorname{n}}}\ar[r] &\psi_* \mathscr{O}_{C^{\operatorname{n}}}^{\langle\iota=-1\rangle}\ar[r]&0
}
\end{xy}
\end{equation*}
where $\mathscr{O}_{C^{\operatorname{n}}}^{\langle\iota=-1\rangle}$ is the anti-invariant part with respect to the Galois
involution $\iota: C^{\operatorname{n}}\to C^{\operatorname{n}}$.
Since the last row in this diagram splits
and $\operatorname{H}^1(\mathscr{O}_{C^{\operatorname{n}}})=0$, we have $\operatorname{H}^1(\operatorname{coker}(\theta))=0$.
Using the snake lemma we see that the multiplication by $\varphi$ induces the following diagram
\begin{equation*}
\begin{xy}
\xymatrix@C=39pt@C=33pt{
&0\ar[r]&\mathscr{O}_H\ar@{^{(}->}[d]^{\cdot \varphi} \ar[r]^{\theta}& \psi_* \mathscr{O}_{H^{\operatorname{n}}}\ar@{^{(}->}[d]^{\cdot \varphi}
\ar[r] &\operatorname{coker}(\theta)\ar[d]^{\cdot \varphi=0}\ar[r]&0
\\
& 0\ar[r]&\mathscr{O}_H\ar@{->>}[d] \ar[r]^{\theta}& \psi_* \mathscr{O}_{H^{\operatorname{n}}}\ar@{->>}[d]\ar[r] &\operatorname{coker}(\theta)\ar[d]^{\simeq}\ar[r]&0
\\
0\ar[r]& \operatorname{coker}(\theta)\ar[r]&\mathscr{O}_{\mathbb{C}C}\ar[r]& \psi_* \mathscr{O}_{4C^{\operatorname{n}}}\ar[r] &\operatorname{coker}(\theta)\ar[r]&0
}
\end{xy}
\end{equation*}
Since $\operatorname{H}^1(\operatorname{coker}(\theta))=0$, from the last row we see
$\operatorname{H}^1(\mathscr{O}_{\mathbb{C}C})\simeq \operatorname{H}^1(\mathscr{O}_{4C^{\operatorname{n}}})$.
On the other hand, $4C^{\operatorname{n}}$ is a fiber of a rational curve fibration.
Hence, $\operatorname{H}^1(\mathscr{O}_{\mathbb{C}C})\simeq \operatorname{H}^1(\mathscr{O}_{4C^{\operatorname{n}}})=0$.
Similar arguments show that $\operatorname{H}^1(\mathscr{O}_{m\mathbb{C}C})=0$ for any $m>0$.
Then by the Formal Function Theorem $\operatorname{H}^1(\hat H, \mathscr{O}_{\hat H})=0$,
where $\hat H$ is the completion of $H$ along $C$.
Applying the Formal Function Theorem again we obtain $\operatorname{H}^1(\hat X, \mathscr{O}_{\hat X})=0$.
\end{proof}
\begin{sclaim}
The contraction $f_H: H\to B$ extends to a contraction $\hat f: \hat X\to \hat Z$.
\end{sclaim}
\begin{proof}
Since $\operatorname{H}^1(\mathscr{O}_{\hat X})=0$, from the exact sequence
\begin{equation*}
0 \xrightarrow{\hspace*{20pt}} \mathscr{O}_X \xrightarrow{\hspace*{20pt}} \mathscr{O}_X (H) \xrightarrow{\hspace*{20pt}} \mathscr{O}_H (H)\xrightarrow{\hspace*{20pt}} 0
\end{equation*}
we see that the map $\operatorname{H}^0(\mathscr{O}_{\hat X} (\hat H))\to \operatorname{H}^0(\mathscr{O}_{\hat H} (\hat H))$
is surjective.
Hence there exists a divisor $\hat H_1\in |\mathscr{O}_{\hat X}|_{\hat C}$ such that $\hat H_1|_{\hat H}=\hat \mathbb{C}C$. Then the divisors
$\hat H$ and $\hat H_1$ define a contraction $\hat f: \hat X\to \hat Z$.
\end{proof}
\begin{sclaim}\label{claim-contraction-exists}
There exists a contraction $f:X\to Z$ that approximates $\hat f: \hat X\to \hat Z$.
\end{sclaim}
\begin{proof}
Let $F$ be the scheme fiber of $f_H: H\to B$ over the origin.
The above arguments shows that the deformations of $F$ are unobstructed.
Therefore the corresponding component of the Douady space is smooth and two-dimensional.
This allow us to produce a contraction $f: X\to Z$.
\end{proof}
\end{example}
\begin{example}
\label{example-conic-bundle-lP=8}
Similar to Example \ref{example-conic-bundle-lP=4+III}, let
$X\subset \mathbb{P}(1,1,3,2,4)$ be a small analytic neighborhood of $C=
\{\text{$(x_1,w)$-line}\}$ given by the equation
$x_1^6\phi_4-w \phi_6 =0$, where
\begin{eqnarray*}
\phi_6&:=&x_3^2+x_2^2w+\delta x_4^3+cx_1^2x_4^2,
\\
\phi_4&:=&x_4^2+\operatorname{n}u x_2x_3+\eta x_1^2x_4.
\end{eqnarray*}
It is easy to check that $P:=(0:0:0:0:1)$ is the only
singular point of $X$ on $C$ and it is a type \type{(IIA)} point with $\ell(P)=8$.
The rational function $\phi_4/w=\phi_6/x_1^6$ near $C$ defines a fibration
whose central fiber $H$ is given by $\phi_4=\phi_6 =0$.
Existence of a contraction
$f: X\to Z$ can be shown similar to Claim \ref{claim-contraction-exists}.
Near $P$ the surface $H$ has the following form which can be reduced to \ref{computation-lP=4+III}:
\begin{equation*}
-c\eta y_1^4y_4+ y_3^2+y_2^2+\delta y_4^3-c\operatorname{n}u y_1^2 y_2y_3=\phi_4=0.
\end{equation*}
\end{example}
\begin{subexample-remark}
\label{example-conic-bundle-normal-H}
In a similar way we can construct an example of a $\mathbb{Q}$-conic bundle with $\ell(P)=5$
and normal $H$
\cite[(1.1.4)]{Mori-Prokhorov-IIA-1}.
Consider $X\subset \mathbb{P}(1,1,3,2,4)$ given by $w\phi_6-x_1^6\phi_4=0$, where
\begin{eqnarray*}
\phi_6&:=&x_1^5 x_2+x_2^2w+x_3^2+\delta x_4^3+cx_1 ^2 x_4^2
\end{eqnarray*}
and $\phi_4$ is as in \ref{example-conic-bundle-lP=4+III}.
In the affine chart $U_w\simeq \mathbb{C}^4/{\boldsymbol{\mu}}_{4}(1,1,3,2)$
the origin $P\in (X,C)$ is a type \type{(IIA)} point with and $\ell(P)=5$.
It is easy to see that $X$ is smooth outside $P$.
The rational function $\phi_4/w=\phi_6/x_1^6$ defines
a fibration on $X$ near $C$ with central fiber $H=\{\phi_4=\phi_6=0\}$.
\end{subexample-remark}
\section{Appendix}
In this section we collect computations of resolutions of
(non-normal) surface singularities appearing as general members
$H\in |\mathscr{O}_X|$. The techniques is very similar to that used in
\cite[\S 9]{Mori-Prokhorov-IIA-1}
\begin{assumption}
\label{notation-blowup-1}
Let $W:= \mathbb{C}^4_{y_1,\dots,y_4}/{\boldsymbol{\mu}}_4(1,1,3,2)$ and let $\sigma$ be the weight $\frac14(1,1,3,2)$.
Let $P\in X$ be a three-dimensional terminal singularity of type \type{cAx/4}
given in $W$ there by the equation $\alpha=0$ with
\begin{equation}\label{equation-alpha-computations}
\alpha=y_1^ly_j+y_2^2+y_3^2+\delta y_4^{2k+1}+c y_1^2y_4^2+\epsilon y_1y_3y_4
+y_2\alpha'+\alpha'',
\end{equation}
where $j=3$ or $4$,\ $l\in \mathbb{Z}_{>0}$,\
$c, \epsilon\in \mathbb{C}$, \ $\delta\in \mathbb{C}^*$, \
$\alpha'\in (y_2,\, y_3,\, y_4)$,\
$\alpha''\in (y_2,\, y_3,\, y_4)^2$,\
$\sigma\mbox{-}\ord (\alpha')= 5/4$,\
$\sigma\mbox{-}\ord (\alpha'')> 3/2$,\
$k\ge 1$, and $2k+1$ is the smallest exponent of $y_4$
appearing in $\alpha$.
We usually assume that all the summands in \eqref{equation-alpha-computations}
have no common terms.
\end{assumption}
\begin{sconstruction}\label{construction-w-blowup-X}
Consider the weighted $\sigma$-blowup $\Phi: \tilde W\to W$. Let
$\tilde X$ be the proper transform of $X$ on $\tilde W$ and
$\Pi\subset \tilde W$ be the $\Phi$-exceptional divisor. Then $\Pi\simeq \mathbb{P}(1,1,3,2)$ and
$\mathscr{O}_{\Pi}(\Pi)\simeq \mathscr{O}_{\mathbb{P}}(-4)$.
Put
\begin{equation}\label{equation-Lambda-computation-2}
\begin{aligned}
O&:=(1:0:0:0),\quad Q:=(0:0:1:0)\in \Pi,
\\
\Lambda&:=\{y_2=\alpha_{\sigma=6/4}=0\}\subset \Pi.
\end{aligned}
\end{equation}
Let $\tilde X\subset \tilde W$
be the proper transform of $X$.
\end{sconstruction}
\begin{sclaim}\label{claim-4-construction-blowup}
$\operatorname{Sing}(\tilde X)$ consists of the curve $\Lambda$, the point $Q$, and the point
$Q_1:=(0:0:0:1)$ \textup($Q_1\operatorname{n}otin \Lambda$ only if $k=1$\textup).
\end{sclaim}
\begin{sclaim}\label{claim-singularities-Lambda}
$\tilde X$ has singularity of
type \type{cA_1} at a general point of $\Lambda$.
\end{sclaim}
\begin{proof}
Let $D\in |-K_X|$ be a general member
and let $F$ be a general hyperplane section of $X$ passing through $0$.
We may assume that $D$ is given by
$y_1+y_2+\cdots $ (see \ref{ge}) and
$F$ is given by $y_1y_3+\cdots=0$.
It is easy to compute
\begin{equation*}
\Phi^*\left( K_X+D+\textstyle\frac 12 F\right)=
K_{\tilde X}+\tilde D+\textstyle\frac 12 \tilde F+E\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}} 0,
\end{equation*}
where $E=\left(\Pi|_{\tilde X}\right)_{\operatorname{red}}=\{y_2=0\}\subset \Pi$, so $E\simeq \mathbb{P}(1,3,2)$
with natural coordinates $y_1$, $y_3$, $y_4$.
By the adjunction formula \cite[Th. 16.5]{Utah}
\begin{equation}\label{equation-adjunction}
\textstyle
\left.
\left(K_{\tilde X}+\tilde D+\frac 12 \tilde F+E\right)\right|_E=K_{E}+\tilde D|_E+\frac 12 \tilde F|_E+\operatorname{Diff}_E(0)\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}} 0,
\end{equation}
where $\operatorname{Diff}_E(0)$ is an effective divisor supported on $\Lambda$.
Let $G:=\{y_1=0\}\subset E$. Then $G$ is a generator of $\operatorname{Cl}(E)\simeq \mathbb{Z}$.
It is easy to see that $\tilde D|_E\sim G$, $\tilde F|_E\sim 4G$, and
$\Lambda\sim 6G$.
By \eqref{equation-adjunction}
we have
$\operatorname{Diff}_E(0) \mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}} 3 G$, i.e. $\operatorname{Diff}_E(0)=\frac 12 \Lambda$.
By the inversion of adjunction $K_{\tilde X}+E$ is plt
at a general point of $\Lambda$ \cite[Th. 17.6]{Utah}.
Then by \cite[Th. 16.6]{Utah} the variety $\tilde X$ has singularity of
type \type{cA_1} at a general point of $\Lambda$.
\end{proof}
\begin{assumption}\label{notation-blowup}
In the notation of \xref{notation-blowup-1}
consider a non-normal
surface singularity $H\operatorname{n}i 0$ given in $W$ by
two $\sigma$-semi-invariant equations $\alpha=\beta=0$.
We assume that the following conditions are satisfied
\begin{itemize}[leftmargin=20pt]
\item
$H$ is singular along $C:=\{\text{$y_1$-axis}\}/{\boldsymbol{\mu}}_4$ and smooth outside $C$,
\item
$\alpha$ satisfies the assumptions of \xref{notation-blowup-1},
\item
$\operatorname{wt} \beta\equiv 0\mod 4$,
\item
$y_4^2$ appears in $\beta$ with coefficient $1$,
\item
$y_2y_3$ appears in $\beta$ with coefficient $\operatorname{n}u$ which can be taken general,
\item
the normalization of $H$ has only rational singularities
and, for any resolution, the total transform of $C$ has only normal crossings.
\end{itemize}
\begin{scase}\label{computations-notation}
We can write the equations of $H$ in the following form
\begin{eqnarray*}
\alpha&=&y_1^ly_j+y_2^2+y_3^2+\delta y_4^{2k+1}+c y_1^2y_4^2+\epsilon y_1y_3y_4
+y_2\alpha'+\alpha'',
\\
\beta&= &y_4^2+\operatorname{n}u y_2y_3+\lambda y_1y_3+\eta y_1^2y_4+y_2\beta'+\beta'',
\end{eqnarray*}
where $\alpha$ is as in \ref{notation-blowup-1},\quad
$\eta, \operatorname{n}u, \lambda\in \mathbb{C}$, \
$\beta',\, \beta''\in (y_2,\, y_3,\, y_4)$,\
$\sigma\mbox{-}\ord (\beta')= 3/4$,\ and
$\sigma\mbox{-}\ord (\beta'')> 1$.
We usually assume that all the summands in $\beta$
have no common terms. Then
$\beta'\in (y_1y_4,\, y_1y_2,\, y_2^2, y_2y_4)$.
\end{scase}
\end{assumption}
\begin{sremark}\label{remark-computation-normality}
Since $H$ is singular along $C$, we have $y_1^sy_r\operatorname{n}otin \beta$ for any $r\operatorname{n}eq j$ and any $s$.
Hence $\lambda\eta=0$. Moreover, if $\lambda\operatorname{n}eq 0$, then $j=3$
and if $\eta\operatorname{n}eq 0$, then $j=4$.
We also may assume that $\beta''\in (y_2,\, y_3,\, y_4)^2$.
\end{sremark}
\begin{sconstruction}\label{notation-computations--sing}
As in \xref{construction-w-blowup-X} consider the weighted
$\sigma$-blowup $\Phi: \tilde W\to W$.
Let $\tilde H\subset \tilde W$
(resp. $\tilde C\subset \tilde W$) be the proper transform of $H$
(resp. $C$).
Clearly, $\tilde C\cap \Pi=\{ O\}$.
Denote (scheme-theoretically)
\begin{equation*}
\Xi:=\tilde H\cap \Pi = \{y_2^2=\beta_{\sigma=1}=0\} \subset \Pi.
\end{equation*}
The surface $\tilde H$ is smooth outside $\tilde C\cup \operatorname{Supp}(\Xi)$ and the set $\tilde C\cup \operatorname{Supp}(\Xi)$
is covered by two affine charts in $\tilde W$
\begin{equation*}
U_1=\{y_1\operatorname{n}eq 0\}\simeq \mathbb{C}^4,\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad U_3=\{y_3\operatorname{n}eq 0\}\simeq \mathbb{C}^4/{\boldsymbol{\mu}}_3(1,1,2,2).
\end{equation*}
Let $\varphi: \hat H\overset{\tau}{\longrightarrow} \tilde H^{\operatorname{n}}\overset{\tilde \psi}{\longrightarrow} \tilde H$ be
the composition of the normalization and the minimal resolution
and let $\hat \Xi_i\subset \hat H$ be
the proper transform of $\Xi_i$.
Let $\tilde C^{\operatorname{n}}=\tilde \psi^{-1}(\tilde C)_{\operatorname{red}}$ and
let $\hat C\subset \hat H$ be the proper transform of $\tilde C^{\operatorname{n}}$.
\end{sconstruction}
\begin{sclaim}[{\cite[9.1.4]{Mori-Prokhorov-IIA-1}}]\label{claim-1-construction-blowup}
Any irreducible component $\Xi_i$ of $\Xi$ is a smooth rational curve
passing through $Q$.
Moreover, $\Xi=2\Xi_1$ \textup(resp. $\Xi=2\Xi_1+2\Xi_2$,\ $\Xi=4\Xi_1$\textup)
if and only if $\lambda\operatorname{n}eq 0$ \textup(resp. $\lambda=0$ and $\eta\operatorname{n}eq 0$,
$(\lambda, \eta)= (0,0)$\textup).
\end{sclaim}
\begin{sclaim}[{\cite[9.1.5]{Mori-Prokhorov-IIA-1}}]
\label{claim-2-construction-blowup}
The point $Q\in \tilde H$ is Du Val of type \type{A_2}.
In particular, $\tilde H$ is normal outside $\tilde C$.
\end{sclaim}
\begin{sclaim}\label{claim-3-construction-blowup-a}
If at least one of the constants $\lambda$ or $\eta$ is non-zero, then
the singular locus of $\tilde H$ coincides with
$\bigl(\operatorname{Supp}(\Xi)\cap \Lambda\bigr)\cup \{Q\}\cup \tilde C$.
\end{sclaim}
\begin{proof}
Direct computations.
\end{proof}
\begin{sremark}
Let $\psi: H^{\operatorname{n}}\to H$ be the normalization
and let $C^{\operatorname{n}}:=\psi^{-1}(C)_{\operatorname{red}}$.
Since
$H$ has double singularities at a general point of $C$,
the map $\psi_C: C^{\operatorname{n}}\to C$ is either birational or a double cover.
In particular, $C^{\operatorname{n}}$ has at most two components.
\end{sremark}
\begin{sdefinition}\label{definition-pinch-point}
A surface singularity $0\in S$ is called a \emph{pinch point} if it is analytically isomorphic to
\begin{equation*}
0\in \{z_2^2+z_1z_3^2=0\}\subset \mathbb{C}^3.
\end{equation*}
\end{sdefinition}
\begin{sremark}
The singular locus of a surface $S$ near a pinch point $0$ is a smooth curve $C$,
the normalization $\psi: S^{\operatorname{n}}\to S$ of $S$ is smooth, and $\psi_C: \psi^{-1}(C)\to C$
is a double cover ramified over $0$.
\end{sremark}
\begin{sclaim}\label{claim-construction-blowup-Du-Val}
The singularities of
$\tilde H^{\operatorname{n}}$ are Du Val outside the preimage of $\tilde C$.
If moreover $\beta$ contains either $y_1y_3$ or $y_1^2y_4$,
then the singularities of $\tilde H^{\operatorname{n}}$ are Du Val everywhere.
\end{sclaim}
\begin{proof}
By Claim \ref{claim-2-construction-blowup}
\ $\tilde H$ has a Du Val singularity at $Q$.
Note that near $O$ the surface $\tilde H$ is a hypersurface
singularity of the form $x_2^2=\phi(x_1,x_3)$, where $\tilde C$ is the $x_1$-axis.
The normalization $\tilde \psi: \tilde H^{\operatorname{n}}\to \tilde H$ can be obtained as a sequence of
successive blowups over $\tilde C$.
In particular, $\tilde H^{\operatorname{n}}$ has only hypersurface singularities.
Finally we note that a two-dimensional
rational Gorenstein singularity must be Du Val.
\end{proof}
\begin{sclaim}{\cite[9.1.9]{Mori-Prokhorov-IIA-1}}
\label{claim-5-equation-notation-blowup}
$K_{\tilde H}=\Phi^*K_H-\frac 34 \Xi$.
\end{sclaim}
\begin{sclaim}\label{claim-computation-Xi-n}
Assume that the singularities of
$\tilde H^{\operatorname{n}}$ are Du Val
\textup(cf. Claim \xref{claim-construction-blowup-Du-Val}\textup).
Write $K_{\tilde H^{\operatorname{n}}}=\tilde \psi^* K_{\tilde H}-\Upsilon$, where
$\Upsilon$ is the effective divisor defined by the conductor ideal.
\begin{itemize}
\item
If $\Xi=2\Xi_1$, then $\hat \Xi_1^2=-4+\tau^*\Upsilon \cdot \hat \Xi_1$.
\item
If $\Xi=2\Xi_1+2\Xi_2$, then $\hat \Xi_i^2=-3+\tau^*\Upsilon \cdot\hat \Xi_i$.
\end{itemize}
\end{sclaim}
\begin{proof}
Consider, for example, the first case $\Xi=2\Xi_1$.
As in \cite[Claim 9.1.10]{Mori-Prokhorov-IIA-1},
$K_{\tilde H}\cdot \Xi_1=2$.
Since $\tilde H$ has only Du Val singularities, we have
\begin{equation*}
K_{\hat H}=\varphi^*K_{\tilde H} - \tau^*\Upsilon,\mathbin{\sim_{\scriptscriptstyle{\mathbb{Q}}}}uad
K_{\hat H}\cdot \hat \Xi_1= K_{\tilde H}\cdot \Xi_1
-\hat \Xi_1\cdot\tau^*\Upsilon.
\end{equation*}
Therefore, $\hat \Xi_1^2=-2- K_{\hat H}\cdot \hat \Xi_1=
-4+\hat \Xi_1\cdot\tau^*\Upsilon$.
\end{proof}
\begin{computation}\label{computation-lP=3+III-part2}
In the notation of \xref{notation-blowup}, let
\begin{eqnarray*}
\alpha&=&y_1^3y_3+y_2^2+y_3^2+\delta y_4^3+c y_1^2y_4^2+\epsilon y_1y_3y_4
+y_2\alpha'+\alpha'',
\\
\beta&= &y_4^2+\operatorname{n}u y_2y_3+\textstyle{\frac 1c} y_1y_3+y_2\beta'+\beta'',
\end{eqnarray*}
where $c$, $\operatorname{n}u$, $\delta$, $\epsilon$ are constants such that
$c\operatorname{n}eq 0$ and $\epsilon c\operatorname{n}eq \delta$.
We assume that the hypothesis of \xref{computations-notation}
are satisfied.
Then the graph $\Delta(H,C)$ has one of
the following forms:
\begin{equation}\label{graphs-computation-lP=3+III-part2}
\vcenter{
\xy
\xymatrix@R=5pt@C=10pt{
\mathrm{a)}&\overset{C}\bullet\ar@{-}[d]&\ovalh{\phantom{P}}\ar@{-}[d]
\\
\underset{C}\bullet\ar@{-}[r] &\circ\ar@{-}[r] &\underset{3}\circ\ar@{-}[r]&\circ \ar@{-}[r]&\circ
}
\endxy
}
\hspace{30pt}
\vcenter{
\xy
\xymatrix@R=5pt@C=13pt{
\mathrm{b)}&&\ovalh{\phantom{P}}\ar@{-}[d]
\\
\underset{C}{\bullet}\ar@{-}[r] &\circ\ar@{-}[r] &\underset{3}\circ\ar@{-}[r]&\circ \ar@{-}[r]&\circ
}
\endxy
}
\end{equation}
where $\ovalh{\phantom{P}}$ is a non-empty
connected Du Val subgraph.
In the second case the normalization of $H$ is a bijection.
\end{computation}
\begin{proof}
We use the notation of \xref{notation-blowup}.
By Remark \ref{remark-computation-normality}, \ $y_1^jy_2\operatorname{n}otin \beta$ for any $j$.
By \ref{claim-1-construction-blowup} we have
$\Xi=2\Xi_1$, where $\Xi_1:=\{y_2=y_4^2+\frac 1c y_1y_3=0\}$.
The first equation modulo the second one can be rewritten in the form
\begin{equation*}
\alpha=y_2^2+y_3^2+\delta y_4^3+\epsilon y_1y_3y_4
+y_2\alpha'+\alpha''.
\end{equation*}
\begin{sclaim}
The point $O\in \tilde H$ is analytically
isomorphic to a hypersurface singularity of the form
\begin{equation*}
\{y_2^2+y_1y_4^3+ \theta y_1^ry_4^2=0\}\subset \mathbb{C}^3,
\end{equation*}
where
again $\tilde C$ is the $y_1$-axis, $\theta\in \mathbb{C}$, and
$r\ge 2$.
\end{sclaim}
\begin{proof}
In the affine chart $U_1$
the equations of $\tilde H$ have the following form
\begin{eqnarray*}
\alpha_{U_1}&=&y_2^2+y_1y_3^2+\delta y_1y_4^3+\epsilon y_1y_3y_4
+y_1y_2\alpha_\bullet+y_1^2\alpha_{\blacktriangle},
\\
\beta_{U_1}&=&y_4^2+\operatorname{n}u y_2y_3+\textstyle{\frac 1c} y_3+y_2\beta_\bullet+y_1\beta_{\blacktriangle},
\end{eqnarray*}
where $\alpha_\bullet\in (y_2,y_3,y_4)$,
$\alpha_{\blacktriangle}$, $\beta_{\blacktriangle}\in (y_2,y_3,y_4)^2$, $\beta_{\bullet}\in (y_2,y_4)$.
{}From the second equation we obtain
\begin{equation*}
y_3= -cu (y_4^2+y_2\beta_\circ+y_1\beta_{\scriptscriptstyle \triangle}),
\end{equation*}
where $\beta_\circ\in (y_2, y_4)$,
$\beta_{\scriptscriptstyle \triangle}\in (y_2, y_4)^2$, and $u$ is a unit such that $u(0)=1$.
Consider the ideal
\begin{equation*}
\mathfrak I:=\left(y_1^2y_4^2,\, y_2^3,\, y_1y_2^2,\,
y_1y_2y_4,\, y_1y_4^4\right).
\end{equation*}
Then we can eliminate $y_3$ in the first equation modulo $\mathfrak I$:
\begin{equation*}
\alpha_{U_1}\equiv y_2^2+(\delta -c\epsilon u) y_1y_4^3 \mod \mathfrak I.
\end{equation*}
Thus, for some $v_i\in \mathbb{C}\{y_1,y_2,y_4\}$, we can write
\begin{equation*}
\alpha_{U_1}=y_2^2+(\operatorname{unit}) y_1y_4^3+v_1y_1^2y_4^2+v_2y_2^3+v_3y_1y_2^2+v_4y_1y_2y_4+v_5y_1y_4^4.
\end{equation*}
Clearly, the last equation
is analytically equivalent to the desired form.
\end{proof}
\begin{scorollary}\label{scorollary-lP=3+III-sing}
Let $\tilde \psi: \tilde H^{\operatorname{n}}\to \tilde H$ be the blowup of $\tilde C$.
Then $\tilde H^{\operatorname{n}}$ coincides with the normalization and
has exactly one singular point which is of type \type{A_1}.
Moreover,
if $r=2$ and $\theta\operatorname{n}eq 0$, then
the preimage $\tilde C^{\operatorname{n}}:=\tilde \psi^{-1}(\tilde C)_{\operatorname{red}}$ has two components
and $\tilde C^{\operatorname{n}}\to \tilde C$ is a double cover.
If $\theta=0$, then $\tilde C^{\operatorname{n}}$ is irreducible and $\tilde C^{\operatorname{n}}\to \tilde C$
is a bijection (near $O$).
If $r>2$ and $\theta\operatorname{n}eq 0$, then the total
transform of $\tilde C^{\operatorname{n}}$ on the minimal resolution is not a normal crossing
divisor.
\end{scorollary}
\begin{sclaim}\label{claim-new-11-3}
The intersection $\Xi_1\cap \operatorname{Sing}(\tilde H)$ consists of three points:
$O$, $Q$, and the point $O'\in \Xi_1\cap \Lambda\setminus\{O\}=
\{(0:0: -(\delta -c\epsilon)c : \delta -c\epsilon)\}$.
\end{sclaim}
Now to finish the proof of \ref{computation-lP=3+III-part2}
we notice that by Claim \ref{claim-computation-Xi-n} we have $\hat \Xi_1^2=-3$ \
because $\tau^*\Upsilon$ meets $\hat \Xi_1$
transversely.
This completes the proof of \ref{computation-lP=3+III-part2}.
\end{proof}
\begin{computation}\label{computation-lP=3a-III}
In the notation of \xref{notation-blowup}, let
\begin{eqnarray*}
\alpha&=&y_1^ly_3+y_2^2+y_3^2+\delta y_4^{2k+1}+c y_1^2y_4^2+\epsilon y_1y_3y_4
+y_2\alpha'+\alpha'',
\\
\beta&=&y_4^2+\operatorname{n}u y_2y_3+\lambda y_1y_3+y_2\beta'+\beta'',
\end{eqnarray*}
where
$l\equiv 3\mod 4$,\
\ $k\ge 1$.
We assume that the hypothesis of \xref{computations-notation}
are satisfied, $\lambda$ is general with respect to $\delta$ and $c$, and
if $l>3$, then $c\operatorname{n}eq 0$.
Then the preimage of $C$ on the normalization is
irreducible and the graph $\Delta(H,C)$ has
the following form:
\begin{equation}\label{graph-diagram-non-normal}
\vcenter{\hbox{
\xy
\xymatrix@R=10pt@C=19pt{
&\circ\ar@{-}[d]
\\
\underset C \bullet \ar@{-}[r]
&\underset 3\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
\\
&\circ\ar@{-}[u]
}
\endxy
}
}
\end{equation}
\end{computation}
\begin{proof}
We use the notation of \xref{notation-blowup}.
By Remark \ref{remark-computation-normality}, \ $y_1^jy_2\operatorname{n}otin \beta$ for any $j$.
We also may assume that $\alpha''$ does not contain any terms of the form $y_4^r$.
By \ref{claim-1-construction-blowup} we have
$\Xi=2\Xi_1$, where $\Xi_1:=\{y_2=y_4^2+\lambda y_1y_3=0\}$.
Since $\lambda\operatorname{n}eq 0$, by Claim \ref{claim-3-construction-blowup-a} the set
$\operatorname{Sing}(\tilde H)$ is contained in
$\tilde C\cup \{Q\}\cup \Lambda$.
\begin{sclaim}
The intersection $\tilde H\cap \Lambda$ consists of $O$ and two more
distinct points $P_1$ and $P_2$.
Moreover, $\tilde H$ meets $\Lambda$ transversely at $P_1$ and $P_2$
and has singularities of type \type{A_1} at these points.
\end{sclaim}
\begin{proof}
Consider the hypersurface $V\subset W$ defined by $\beta=0$.
Let $\tilde V\subset \tilde W$ be its proper transform.
So, $\tilde H=\tilde X\cap\tilde V$.
We have $(\tilde V|_\Pi\cdot \Lambda)_{\Pi}=4$ and the local intersection number at $O$
equals $2$. Since the base locus of the linear system
on $\Pi$ generated by $\tilde V|_{\Pi}$ meets $\Lambda$ only at $O$, the last assertion follows by
Bertini's theorem and Claim \ref{claim-singularities-Lambda}.
\end{proof}
\begin{sclaim}
$\tilde H\operatorname{n}i O$ is a pinch point.
\end{sclaim}
\begin{proof}
In the affine chart $U_1$
the equations of $\tilde H$ have the form
\begin{equation*}
\begin{aligned}
0&=y_1^{(l+1)/4}y_3+y_2^2+y_1 (y_3^2+\delta y_1^{k-1}y_4^{2k+1}+c y_4^2+\epsilon y_3y_4
+y_2\alpha_{\bullet}+y_1\alpha_{\blacktriangle}),
\\
0&=y_4^2+\operatorname{n}u y_2y_3+\lambda y_3+y_2\beta_\bullet+y_1\beta_\blacktriangle,
\end{aligned}
\end{equation*}
where $\beta_\bullet\in (y_2,\, y_3,\, y_4)$,
$\beta_\blacktriangle\in (y_2,\, y_3,\, y_4)^2$.
{}From the second equation we have
\begin{equation*}
y_3= u(y_4^2+y_2\beta_\circ+y_1\beta_{\scriptscriptstyle \triangle}),
\end{equation*}
where $u$ is a unit such that $u(0)=-1/\lambda$ and $\beta_\circ,\, \beta_{\scriptscriptstyle \triangle}\in (y_2,\, y_4)$.
Eliminating $y_3$ we obtain
\begin{multline*}
uy_1^{(l+1)/4} (y_4^2+y_2\beta_\circ+y_1\beta_{\scriptscriptstyle \triangle})+y_2^2+
u^2y_1(y_4^2+y_2\beta_\circ+y_1\beta_{\scriptscriptstyle \triangle})^2+
\\
\delta y_1^ky_4^{2k+1}+c y_1y_4^2+\epsilon u y_1y_4(y_4^2+y_2\beta_\circ+y_1\beta_{\scriptscriptstyle \triangle})
+y_1y_2\alpha_{\bullet}+y_1^2\alpha_{\blacktriangle}=0,
\end{multline*}
{}From this we see that the equation of $\tilde H$ at $O$ can be written in the form
$y_2^2+y_1y_4^2+\cdots=0$,
i.e. $\tilde H\operatorname{n}i O$ is a pinch point.
\end{proof}
Now to finish the proof of \ref{computation-lP=3a-III}
we notice that by Claim \ref{claim-computation-Xi-n} we have $\hat \Xi_1^2=-3$ \
because $\tau^*\Upsilon$ is reduced and meets $\hat \Xi_1$
transversely.
\end{proof}
\begin{computation}\label{computation-lP=4+III}
In the notation of \xref{notation-blowup}, let
\begin{eqnarray*}
\alpha&=&y_1^{4l}y_4+y_2^2+y_3^2+\delta y_4^{2k+1}+c y_1^2y_4^2+\epsilon y_1y_3y_4+y_2\alpha'+\alpha'',
\\
\beta&=&y_4^2+\operatorname{n}u y_2y_3+\eta y_1^2y_4+ y_2\beta'+\beta'',
\end{eqnarray*}
where $l,\, k\ge 1$,\ $c,\, \epsilon \in \mathbb{C}$,
\ $\delta,\, \eta\in \mathbb{C}^*$, and $\eta$ is general with respect to $\alpha$.
We assume that the hypothesis of \xref{computations-notation}
are satisfied.
Then the graph $\Delta(H,C)$ has one of the following forms:
\begin{equation}\label{graph-diagram-non-normal-lP=4+III}
\vcenter{
\xy
\xymatrix@R=7pt@C=11pt{
&\circ\ar@{-}[r]&\overset {3}\circ\ar@{-}[d]\ar@{-}[r]&\circ
\\
\bullet \ar@{-}[r] &\underset {}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}
\endxy}
\end{equation}
\end{computation}
\begin{proof}
We use the notation of \xref{notation-blowup}.
In our case $\Xi=2\Xi_1+2\Xi_2$, where $\Xi_1:=\{y_2=y_4=0\}$, \
$\Xi_2:=\{y_2=\eta y_1^2+ y_4=0\}$, and
$\Xi_1\cap \Xi_2=\{Q\}$.
\begin{sclaim}\label{claim-construction-blowup-intersection-Lambda}
\begin{enumerate}[leftmargin=20pt]
\item
$\operatorname{Sing}(\tilde H)\cap \Xi_1=\{O,\, Q\}$.
\item
$\operatorname{Sing}(\tilde H)\cap \Xi_2=\Xi_2\cap\Lambda \cup \{Q\}$.
\end{enumerate}
\end{sclaim}
\begin{proof}
By Claim \ref{claim-3-construction-blowup-a} we have
$\operatorname{Sing}(\tilde H)\subset \Lambda\cup \tilde C\cup \{Q\}$.
On the other hand, $Q\operatorname{n}otin \Lambda$ and $\Xi_1\cap \Lambda=\{O\}$.
\end{proof}
\begin{sclaim}
$O\in \tilde H$ is a pinch point.
\end{sclaim}
\begin{proof}
In the affine chart $U_1$
the equations of $\tilde H$ have the form
\begin{eqnarray*}
\alpha_{U_1}&=&y_2^2+y_1(y_1^{l-1}y_4+y_3^2+\delta y_4^{2k+1}+c y_4^2+\epsilon y_3y_4+y_2\alpha_\bullet
+y_1\alpha_\blacktriangle),
\\
\beta_{U_1}&=&y_4^2+\operatorname{n}u y_2y_3+\eta y_4+ y_2\beta_\bullet+y_1\beta_\blacktriangle,
\end{eqnarray*}
where
$\alpha_\blacktriangle\in (y_2,\, y_3,\, y_4)^2$,
$\beta_\bullet\in (y_2,\, y_4)$,
$\alpha_\bullet\in (y_2,\, y_3,\, y_4)$,
and $\beta_\blacktriangle\in (y_2,\, y_3,\, y_4)^2$ by Remark \ref{remark-computation-normality}.
{}From $\beta_{U_1}$ we have
\begin{equation*}
y_4= u y_2y_3+ y_2\beta_1+y_1\beta_2,\quad \beta_1\in (y_2),\ \beta_2\in (y_2,y_3)^2,
\ u=\operatorname{unit}.
\end{equation*}
Then we can eliminate $y_4$ from $\alpha_{U_1}$:
\begin{equation*}
y_2^2+y_1y_3^2+
\gamma_1y_1y_2+\gamma_2y_1 +\gamma_3 y_1^2=0,
\end{equation*}
where $\gamma_1\in (y_2,y_3)$, $\gamma_2\in (y_3)^4$, $\gamma_3\in (y_3)^2$.
By completing the square we can put the equation of $\tilde H$ at $O$ to the following form
\begin{equation*}
y_2^2+(\operatorname{unit} )\cdot y_1y_3^2=0. \qedhere
\end{equation*}
\end{proof}
Recall that by Claim \ref{claim-construction-blowup-Du-Val} the surface
$\tilde H^{\operatorname{n}}$ has only Du Val singularities.
As in \cite[9.1.6]{Mori-Prokhorov-IIA-1}
we see that the pair $(\tilde H, \Xi_1+\Xi_2)$
is not lc at $Q$ and lc outside $Q$ and $\tilde C$.
Thus the dual graph $\Delta(H, C)$ has the form
\begin{equation}\label{graph-diagram-non-normal-lP=4+IIIa}
\vcenter{
\xy
\xymatrix@R=1pt@C=23pt{
&&{\ovalv{\phantom{P}$\scriptstyle P$\phantom{P}}}\ar@{-}[r]&\overset {\Xi_2}\circ \ar@{-}[d]
\\
\underset C \bullet\ar@{-}[rr] &
&\underset {\Xi_1}\circ\ar@{-}[r]&\circ\ar@{-}[r]&\circ
}
\endxy
}
\end{equation}
where \ovalh{$\scriptstyle P$}
is a Du Val subgraph which is not empty (but possibly disconnected).
By Claim \ref{claim-computation-Xi-n} we have $\hat \Xi_2^2=-3$ and $\hat \Xi_1^2= -2$.
Further,
\begin{equation*}
\Xi_2\cdot (\Xi_1+\Xi_2)=\textstyle\frac 12 \Xi_2\cdot \Pi= -\frac 23,\quad \Xi_1\cdot \Xi_2=\frac 23,
\quad \Xi_2^2=-\frac 43.
\end{equation*}
Then as in the proof of \cite[Lemma 3.8]{Mori-Prokhorov-IIA-1} we have
$\deg \operatorname{Diff}_{\Xi_2}(0)=5/3$. There are two possibilities:
$\operatorname{Diff}_{\Xi_2}(0)=\frac 23 Q+\frac 12 P_1+ \frac 12 P_2$ and
$\operatorname{Diff}_{\Xi_2}(0)=\frac 23 Q+P_1$.
Hence the singularities of $\tilde H$ on $\Xi_2\setminus \{Q\}$ are
either two points which are of type \type {A_1}
or one point which is of type \type{D_n} or \type{A_3}.
The second possibility occurs only for some specific
choice of $\eta$ (when two intersection points $\Lambda\cap \Xi_2$ coincide).
We obtain \eqref{graph-diagram-non-normal-lP=4+III}.
\end{proof}
\par
\operatorname{n}oindent
{\bf Acknowledgments.}
The paper was written during the second author's visits to RIMS, Kyoto University.
The author is very grateful to the institute for
the invitation, support, and hospitality.
\def\mathbb#1{\mathbf#1} \def\bblapr{April}
\end{document} |
\begin{document}
\title{Coupling for features of random walks}
\begin{abstract}
We use coupling to study the time taken until the distribution of a statistic on a Markov chain is close to its stationary distribution. Coupling is a common technique used to obtain upper bounds on mixing times of Markov chains, and we explore how this technique may be used to obtain bounds on the mixing of a statistic instead.
\end{abstract}
\section{Introduction}
\label{sec:intro}
We are interested in the following general problem.
\begin{Problem}
If $\mathcal{M}$ is a Markov chain, and $f$ is a function defined on the states of $\mathcal{M}$, how long must $\mathcal{M}$ be run to guarantee that the distribution of $f$ is close to what it would be on the stationary distribution of $\mathcal{M}$?
\end{Problem}
Much is known about various schemes for shuffling a deck of cards (\cite{dovetail}, \cite{randomtranspositions}), and how many shuffles are necessary before the deck is `random'. In some circumstances it might not be necessary that the entire deck be random, but just some part of it. For example, perhaps a certain game of poker only uses the top $17$ cards of the deck. In playing this game, only the identity and order of these top $17$ cards are important, not the order of the entire deck. It might be expected that to randomise the cards in these positions, fewer shuffles are required than are necessary to randomise the entire deck. This is one instance of the problem --- given a shuffling scheme, how many iterations are required to randomise the top $17$ cards of the deck?
The same question can be asked for other choices of $f$ --- how long until the four bridge hands dealt in blocks from this deck are random? That is, the sets of cards in positions 1-13, 14-26, 27-39 and 40-52, but not their exact locations within these blocks. What if the dealing is done to each player in turn rather than in blocks of $13$ consecutive cards? How many shuffles are necessary to randomise the location of the ace of spades, or the identity of the card immediately following the ace of spades, or the distance between the aces of spades and hearts?
Problems of this sort have been considered previously. \cite{ADS} studies the mixing time of a deck of cards where certain sets of cards are identified, for instance where the suits of cards do not matter, or all face cards are equivalent. That paper gives an explicit formula for the separation distance after $t$ steps in such a setting. \cite{better} discusses this same problem as well as the dual problem of asking about the hands of cards dealt from a shuffled deck, ignoring the order in which those cards were dealt. This may be seen as identifying sets of positions rather than sets of cards. \cite{better} also presents intriguing computational data showing that the number of riffle shuffles required for this latter problem changes depending on the dealing method used, that is, that identifying different sets of positions produces different results. In various contexts, the values of the function $f$ might be referred to as `statistics' or `features' of the Markov chain.
Both \cite{JCthesis} and \cite{ADS} show that the mixing time for the position of a single card under riffle shuffles is $\log_2(n) + c$, \cite{JCthesis} by calculating the eigenvalues of the walk and also via a coupling argument, and \cite{ADS} by explicit calculations. \cite{repeatedcards1} simulates the required number of shuffles for the games of bridge and blackjack. The introduction of \cite{ADS} has additional references and background.
Previous work on these problems has mostly involved explicit calculations and formulas, which are then analysed with calculus. An exception is \cite{rabinovich2016function}, which develops the use of eigenfunctions, where the statistic in question is expanded in an eigenbasis of the original chain. Appendix B of \cite{ADS} also contains a computation in this style.
The contributions of this paper and the sequel \cite{GWsstfeatures} are technical --- we describe how coupling (and in the sequel, strong stationary times) may be adapted to give upper bounds on the mixing of a function on a Markov chain. These appear to be the first general approaches which use more probabilistic methods.
In general, the answers to these problems will depend on the function $f$. There are choices for $f$ where the distribution of $f$ is correct after one step, and there are choices where the distribution of $f$ is not correct until the whole chain is near its stationary distribution. Instances of each behaviour are shown in Section \ref{sec:examples}. We will also examine what known couplings (those already used for upper bounds on mixing times) have to say about various choices of $f$.
In some cases the statistic $f$ may form a Markov chain in its own right, as a quotient chain of $\mathcal{M}$. For example, when shuffling a deck of cards, the location of the ace of spades at time $(t+1)$ depends only on its location at time $t$. In contrast, the knowledge of which cards are in the top half of the deck at time $t$ is usually not enough information to determine which cards will be in the top half of the deck at time $(t+1)$. We do not require that $f$ be a quotient chain of $\mathcal{M}$. When this does occur, the analysis will not use this fact, because the goal is to demonstrate techniques that are applicable in more general settings. For this reason, some of the simpler examples may work more nicely than expected.
In this paper, upper bounds on mixing times will come from coupling arguments, so mixing times will be according to total variation distance.
\begin{Definition}
Let $\mathcal{M}$ be an aperiodic and irreducible Markov chain and $f$ be a function on the state space of $\mathcal{M}$. The \defn{stationary distribution of $f$} is the distribution of $f$ on the stationary distribution of $\mathcal{M}$.
\end{Definition}
\begin{Example}
\label{ex:statlist}
Our first examples are shuffling schemes on a deck of $n$ cards. That is, they will be random walks on the group $S_n$.
Here are some statistics of interest. For riffle shuffles, statistics involving the locations of certain cards or cards in particular locations have been analysed in \cite{ADS} and \cite{repeatedcards1} in more detail than in this paper. The focus here is on developing probabilistic techniques for these problems.
\begin{itemize}
\item The value of the top, second-to-top, bottom, or $k$th card.
\item The values and order of the top $k$ cards, or of the cards in a particular set of positions.
\item The set of cards in a particular set of positions, ignoring their relative order. For example, the sets of cards in the top quarter of the deck, the next quarter, the next quarter, and the bottom quarter, as might be relevant if one were to deal cards in blocks. Alternatively, the sets of cards in positions congruent to $i$ modulo four, for each $i$, as if cards were to be dealt one at a time.
\item The location of a particular card or set of cards.
\item The parity of the permutation.
\end{itemize}
\end{Example}
Some answers to questions like these are the following, which are proven in Section \ref{sec:examples}.
\begin{Proposition}
Using the random-to-top shuffle on a deck of $n$ cards, it takes $n\log(n)$ steps to get the entire deck close to random (via a standard coupon-collector argument), but only $n\log(\frac{n}{n-16})$ steps to get the top 17 cards close to random, or $n\log(\frac{3}{2})$ steps to get the top third of the deck close to random.
\end{Proposition}
\begin{Proposition}
Using inverse GSR riffle shuffles on a deck of $n$ cards, it takes $\frac{3}{2}\log_2(n)$ steps to get the entire deck close to random, but only $\log_2(n)$ steps to get any of the following statistics close to random: The identity of the top card, the location of the ace of spades, the set of cards in the top quarter of the deck, or the sets of cards in each quarter of the deck.
\end{Proposition}
\begin{Example}
\label{ex:hypercube}
Consider the random walk on the $n$--dimensional hypercube where at each step there is a $\frac{1}{2}$ chance to move to a random neighbour and a $\frac{1}{2}$ chance to remain still. This can also be considered as a random walk on $n$--bit binary strings, where at each step a random bit is chosen and replaced with $0$ or $1$ with equal probability.
How long does it take until statistics such as the following are close to their stationary distributions?
\begin{itemize}
\item The value of the first bit
\item The number of `1' bits
\item The location of the first `1'.
\end{itemize}
\end{Example}
Section \ref{sec:hypercube} shows that the value of the first bit mixes after $n$ steps, and that the location of the first 1 mixes after $O(n)$ steps. This should be compared to the mixing time of the walk, which is $\frac{1}{2}n\log(n)$.
\section{Coupling for features of random walks}
\label{sec:coupling}
This section gives results relating coupling to the convergence of a statistic on a Markov chain. These will be used in Section \ref{sec:examples} to give examples of bounds on the convergence of the statistics mentioned in Example \ref{ex:statlist} for some simple shuffling techniques --- random-to-top shuffles, inverse riffle shuffles, and random transpositions.
For the sake of comparison, we will first give a proof that couplings give bounds on the mixing time of the whole Markov chain. This is a classical result (Theorem 5.2 of \cite{LPW}), but the perspective on the problem will be useful for the material which follows.
\begin{Proposition}
\label{prop:coupling1}
Let $C$ be a coupling on two instances of a Markov chain $\mathcal{M}$, $p$ be between 0 and 1, and $t$ be a positive integer. Let $X_t$ and $Y_t$ be the states of the two instances of $\mathcal{M}$ after taking $t$ steps according to $C$. If for any initial states $x_0$ and $y_0$, there is at least a probability $p$ that $X_t = Y_t$, then for any initial state $x_0$ the distribution of $X_t$ is within $(1-p)$ of the stationary distribution of $\mathcal{M}$ in total variation distance.
\end{Proposition}
\begin{proof}
A condition of this result is that for any initial states $x_0$ and $y_0$, there is at least a probability $p$ that $X_t = Y_t$. From Lemma \ref{lem:linearcombination}, this is also true if $X_0$ and $Y_0$ are allowed to be distributions. This result is used with $X_0$ being an arbitrary fixed state and $Y_0$ being the stationary distribution $\pi $.
Consider $X_t$ and $Y_t = \pi M^t = \pi$. To show that these distributions overlap in most of their area, let $\mathcal{P}_1$ be the set of paths of length $t$ starting at $X_0$, and let $\mathcal{P}_2$ be the set of paths of length $t$ starting at $\pi $, all paths being weighted by their probabilities. The goal is to pair proportion $p$ of the paths from $\mathcal{P}_1$ with paths from $\mathcal{P}_2$ which end at the same place. Because $X_t$ is the distribution of endpoints of paths in $\mathcal{P}_1$ and $Y_t$ is the distribution of endpoints of paths in $\mathcal{P}_2$, this will guarantee that these two distributions overlap in $p$ of their area.
This pairing is given by the coupling $C$. Start with two copies of $\mathcal{M}$, one in the state $X_0$ and the other in the stationary distribution $\pi $, and evolve them according to the coupling $C$. Pair the paths taken by the two chains. This is a pairing between paths from $\mathcal{P}_1$ and paths from $\mathcal{P}_2$ because $C$ is a coupling, so the behaviour in either chain is what it would be in isolation. There is at least probability $p$ that the two chains end in the same state, so at least $p$ of the paths in $\mathcal{P}_1$ are paired with a path from $\mathcal{P}_2$ which ends at the same state. Therefore $X_t$ and $Y_t = \pi$ overlap in at least $p$ of their area, as required.
\end{proof}
\begin{Lemma}
\label{lem:linearcombination}
Using the notation of Proposition \ref{prop:coupling1}, assume that for any two initial states $x_0$ and $y_0$, there is at least a probability $p$ that $X_t = Y_t$ when the two chains evolve according to the coupling $C$. Then if the two chains are started in arbitrary distributions $X_0$ and $Y_0$ rather than fixed states, there is still at least probability $p$ that $X_t = Y_t$.
\end{Lemma}
\begin{proof}
Conditioned on any pair of initial states $x_0$ and $y_0$, the probability that $X_t = Y_t$ is at least $p$. Averaging these probabilities over the distributions $X_0$ and $Y_0$ gives the required result.
\end{proof}
We now adapt Proposition \ref{prop:coupling1} to apply to features of a Markov chain.
\begin{Proposition}
\label{prop:coupling2}
As in Proposition \ref{prop:coupling1}, let $C$ be a coupling on two copies of a Markov chain $\mathcal{M}$, $p$ be between $0$ and $1$, $t$ a positive integer, and $f$ a function on the state space $\Omega$. If for any initial states $x_0$ and $y_0$ there is at least a probability $p$ that $f(X_t) = f(Y_t)$ when the chains $X$ and $Y$ evolve according to the coupling $C$, then for any initial state $x_0$ the distribution $f(X^t)$ is within $(1-p)$ of $f(\pi )$, the stationary distribution of $f$, in total variation distance.
\end{Proposition}
\begin{proof}
This proof is very similar to the proof of Proposition \ref{prop:coupling1}. As with that proof, start by using Lemma \ref{lem:linearcombinationf} to show that for any initial distributions $X_0$ and $Y_0$, there is at least a probability $p$ that $f(X_t) = f(Y_t)$.
Let $X_0$ be any initial distribution, and $Y_0$ be the stationary distribution $\pi$. Consider $f(X_t)$ and $f(Y_t) = f(\pi M^t) = f(\pi)$. As in the proof of Proposition \ref{prop:coupling1}, let $\mathcal{P}_1$ be the set of paths of length $t$ starting at $X_0$, and let $\mathcal{P}_2$ be the set of paths of length $t$ starting at $\pi $, weighted by their probabilities. The goal is to pair proportion $p$ of the paths from $\mathcal{P}_1$ with paths from $\mathcal{P}_2$ which end, not necessarily at the same state, but at a state with the same value of $f$. This guarantees that the two distributions $f(X_t)$ and $f(Y_t) = f(\pi)$ overlap in $p$ of their area.
As in the proof of Proposition \ref{prop:coupling1}, the pairing is given by the coupling $C$. Start with two copies of $\mathcal{M}$, one in the distribution $X_0$ and the other in the stationary distribution $\pi $, and evolve them according to $C$. Pair the paths taken by the two chains. This is a pairing between paths in $\mathcal{P}_1$ and $\mathcal{P}_2$, because $C$ is a coupling. There is at least probability $p$ that the two chains end in states with matching values of $f$, so at least $p$ of the paths in $\mathcal{P}_1$ are paired with a path from $\mathcal{P}_2$ which ends at a state where $f$ takes the same value. Therefore $f(X_t)$ and $f(\pi)$ overlap in at least $p$ of their area, as required.
\end{proof}
\begin{Lemma}
\label{lem:linearcombinationf}
Using the notation of Proposition \ref{prop:coupling2}, assume that for any two initial states $x_0$ and $y_0$, there is at least a probability $p$ that $f(X_t) = f(Y_t)$ when the two chains evolve according to the coupling $C$. Then if the two chains are started in arbitrary distributions $X_0$ and $Y_0$ rather than fixed states, there is still at least probability $p$ that $f(X_t) = f(Y_t)$.
\end{Lemma}
\begin{proof}
The proof is the same as that of Lemma \ref{lem:linearcombination}.
\end{proof}
It is not obvious that the bound on the mixing time given by Proposition \ref{prop:coupling2} decreases with $t$. Lemma \ref{lem:decreasing} will show that this is the case.
\begin{Remark}
\label{rem:stationarycontinue}
When constructing a coupling to be used with Proposition \ref{prop:coupling1}, it is possible to decree that if the two chains are in the same state, then they will move in the same way, guaranteeing that they will continue to agree with one another. When constructing a coupling for Proposition \ref{prop:coupling2}, this is still possible --- that is, two chains in the same state will continue to agree --- but the same cannot be done for values of $f$. It may be that two chains presently have the same value of $f$, but cannot be coupled so that after one step they have the same value of $f$. Example \ref{ex:cantcouplef} gives an example of this behaviour. This observation is the same as noticing that the values of $f$ may not form a quotient Markov chain of $\mathcal{M}$.
\end{Remark}
\begin{Example}
\label{ex:cantcouplef}
Consider the random-to-top walk, and let $f$ be the label of the second-to-top card of the deck. If two decks are currently in the states $x_0 = (1,2,3,4,5,\dots,n)$ and $y_0 = (3,2,1,4,5,\dots,n)$, then $f(x_0) = f(y_0) = 2$. However, the distributions of $f$ after one step, $f(x_1)$ and $f(y_1)$ are wildly different for the two chains, as shown in the following table.
\begin{center}
\begin{tabular}{ccc}
$a$ & $P(f(x_1) = a)$ & $P(f(y_1) = a)$ \\
\rule{0pt}{3ex}1 & $\frac{n-1}{n}$ & 0 \\
\rule{0pt}{3ex}2 & $\frac{1}{n}$ & $\frac{1}{n}$ \\
\rule{0pt}{3ex}3 & 0 & $\frac{n-1}{n}$ \\
\end{tabular}
\end{center}
\end{Example}
\begin{Remark}
\label{rem:exp}
When analysing the convergence of a Markov chain, it suffices to find the time at which the total variation distance from stationarity falls below, say, a quarter, because it then decays exponentially (for instance, see Section 4.5 of \cite{LPW}). The analogous result for the convergence of a statistic is false, as shown in Example \ref{ex:convergencetable}. It is still the case that the total variation distance of the distribution of the statistic from stationarity eventually falls off exponentially, just no longer that the speed of this decay is controlled by the time taken for the distance to fall below a quarter.
\end{Remark}
\begin{Example}
\label{ex:convergencetable}
Consider the following variation on the random-to-top walk on the permutations of a deck of $n$ cards. At each step choose a card uniformly at random, and move it to the top of the deck. The bottom card of the deck is stuck to the table, and attempts to move it only succeed with probability $\frac{1}{100}$, otherwise the order of the deck remains unchanged. Let $f$ be the label of the top card, and the original top and bottom cards be 1 and $n$.
After one step of the chain, the distribution of $f$ is
\begin{center}
\begin{tabular}{cc}
$a$ & $P(f(x) = a)$\\
\rule{0pt}{3ex}1 & $\frac{1.99}{n}$\\
\rule{0pt}{3ex}$2 \leq a \leq n-1$ & $\frac{1}{n}$\\
\rule{0pt}{3ex}$n$ & $\frac{0.01}{n}$\\
\end{tabular}
\end{center}
The total variation distance between the distribution of $f$ after just one step and the uniform distribution is less than $\frac{1}{n}$. After 50 steps, it is more likely than not that the bottom card has not moved, so the total variation distance between the distribution of $f(X_{50})$ and the uniform distribution is at least $\frac{1}{2n}$.
\end{Example}
\begin{Remark}
\label{rem:ptpairs}
As a consequence of Remark \ref{rem:exp}, when using a coupling to examine convergence of a statistic, the relevant information is not just how long it takes for the total variation distance to drop below one quarter (or any other fixed small number), as it might be while using a coupling for the mixing time, but rather is a sequence of data points of the form ``for any starting point, after time $t_i$, the total variation distance of the statistic is less than $p_i$''.
\end{Remark}
\begin{Lemma}
\label{lem:decreasing}
Let $\mathcal{M}$ be a Markov chain, $f$ a function on the state space of $\mathcal{M}$, and $d(t)$ be the maximum distance (either total variation or separation) of $f$ from uniform after $t$ steps of $\mathcal{M}$, over all possible starting configurations. Then $d$ is a nonincreasing function of $t$.
\end{Lemma}
\begin{proof}
For any starting configuration, the distribution of $f$ after $t+1$ steps of $\mathcal{M}$ is a distribution of $f$ after $t$ steps of $\mathcal{M}$, starting from $t=1$. The definition of $d(t)$ is the maximum distance over all initial states, including this one, so the distance after $t+1$ steps is at most $d(t)$. Therefore $d(t+1) \leq d(t)$.
\end{proof}
Unlike the convergence of actual Markov chains (Lemmas 3.7 and 4.5 of \cite{ADstrong}), in this setting the total variation distance from stationarity is not submultiplicative. The distance will be submultiplicative eventually, but not at all times. This should be understood as the distance sometimes being small earlier than expected due to factors which do not control the long-term rate of convergence. Example \ref{ex:convergencetable} illustrates this behaviour.
To this end, it will be convenient to work with the coupling time. The definition of a coupling time is modified in the natural way to allow for couplings of statistics on a Markov chain.
\begin{Definition}
\label{def:couplingtime2}
Let $\mathcal{M}$ be a Markov chain and $C$ be a coupling on $\mathcal{M}$. The \emph{coupling time} is the (random) time until the two copies of $\mathcal{M}$ are in the same state, or the time until they have matching values of $f$, depending on the aim of the coupling in question.
\end{Definition}
\section{Examples}
\label{sec:examples}
One way to apply Proposition \ref{prop:coupling2} is to consider a coupling that has been successful in obtaining a bound for the mixing time via Proposition \ref{prop:coupling1}, and check what it says about a function $f$ of interest. This section details what some well-known couplings say about various statistics on the respective state spaces.
Keep in mind that these are only upper bounds on the time taken for a statistic to mix --- some statistics may well mix faster than shown by this particular coupling. These examples are intended to show how Proposition \ref{prop:coupling2} may be applied to reduce a mixing time problem to a question regarding a coupling time. The goal is not to analyse these coupling times in detail, so most examples will not have detailed bounds on $t_{\text{mix}}(\epsilon)$ for each $\epsilon$, but rather will describe the process in question and give some idea of how long it takes. In many examples, Chebyshev's inequality will give good upper bounds on the time taken.
\subsection{Random-to-top shuffles}
\label{sec:rtt}
The random-to-top shuffle consists of choosing a random card at each step, and moving it to the top of the deck. A coupling for two copies of this process is to choose cards with the same label in each deck --- for instance, moving both copies of the ace of spades to the top of their respective decks, regardless of their prior positions. The time taken for two copies of this process to couple is the coupon collector time $n\log(n)$.
Consider what this coupling has to say about each of the following statistics on $S_n$.
\begin{enumerate}
\item The top card of each chain is the same after a single step, and this continues to be true after any number of steps. Therefore this statistic is exactly uniformly distributed after one step.
\item The second-to-top cards match as soon as two different labels have been chosen, and this continues to be true after this point. With $\mathcal{G}(p)$ denoting a geometric distribution, the coupling time is $$T \stackrel{d}{=} \mathcal{G}(1) + \mathcal{G}(\frac{n-1}{n}).$$ The expected time is $1 + \frac{n}{n-1}$. The probability that $T > 2$ is $\frac{1}{n}$, so the second-to-top card is within $\frac{1}{n}$ of uniform after two steps, by Proposition \ref{prop:coupling2}. This corresponds to the probability that the same card is chosen twice, so the original top card is more likely to be in the second position than other cards.
Likewise, the probability that $T > 3$ is $\frac{1}{n^2}$, so the second-to-top card is within $\frac{1}{n^2}$ of uniform after three steps.
\item The location of the card labelled $1$ is the same in each deck as long as that label has been chosen. This coupling time is $$T \stackrel{d}{=} \mathcal{G}(\frac{1}{n}).$$ The expected time until this happens is $n$ steps, and the variance is $n-1$.
\item The locations of the cards labelled $1$ and $2$ match in the two decks as long as both of those labels have been chosen. This coupling time is $$T \stackrel{d}{=} \mathcal{G}(\frac{1}{n}) + \mathcal{G}(\frac{2}{n}).$$The expected time until this event is $\frac{n}{2} + n$, and the variance is $\frac{n}{(n-1)^2}$. Note that this time is the sum of the two worst terms of the coupon collector problem, in contrast to matching the top two cards of the deck, which was the sum of the two best terms. This generalises to attempting to match the locations of $k$ fixed cards.
\end{enumerate}
These examples appear to behave quite differently. When the value of the top card is coupled, it may be that the two chains both have the $1$ at the top, or the $2$, or the $k$, for any $k$. In contrast, when the location of the $1$ is coupled, it is always at the top of the deck. This latter might seem disconcerting --- a result about the location of the $1$ mixing is proven by coupling two instances of the chain, but the coupling always happens in a certain position.
This issue is reconciled by recalling the definition of total variation distance. Upper bounds on the total variation distance between two distributions do not guarantee that they have some chance of agreeing in any possible value, just that there is a certain chance that they agree at some value(s). After a single random-to-top step, there is a $\frac{1}{n}$ chance that the location of the $1$ matches, so the distribution of this statistic overlaps with the uniform distribution in at least $\frac{1}{n}$ of their area. This is true --- both distributions have at least a $\frac{1}{n}$ chance that the $1$ is in the top position.
Likewise, after two random-to-top steps, there is a $\frac{2n-1}{n^2}$ chance that the location of the $1$ matches, so as previously, the distribution of the statistic after two steps overlaps with the uniform distribution in at least $\frac{2n-1}{n^2}$ of their area. Again this is true, because both distributions have at least a $\frac{n}{n^2}$ chance that the $1$ is in the top position and at least a $\frac{n-1}{n^2}$ chance that the $1$ is in the second position.
Perhaps, then, it should be surprising that the identity of the top card is equally likely to take any value when it is matched. Indeed, that this happens means that the coupling time is also a strong stationary time, and so can be used to obtain bounds in separation distance rather than total variation distance. This will be discussed further in \cite{GWsstfeatures}.
Continuing with a more detailed example,
\begin{enumerate}[resume]
\item The top $k$ cards match as long as $k$ different labels have been chosen. The coupling time is $$T \stackrel{d}{=} \mathcal{G}(1) + \mathcal{G}(\frac{n-1}{n}) + \cdots + \mathcal{G}(\frac{n-k+1}{n}).$$Let $T$ be the time taken for this to occur. The expected value is
\begin{align}
\label{eq:couponfirstkex}
E(T) &= \frac{n}{n} + \frac{n}{n-1} + \dots + \frac{n}{n-k+1}\\
& \leq \int_{0}^k \frac{n}{n-x} dx \nonumber \\
& = n \int_{0}^k \frac{1}{n-x} dx \nonumber \\
& = n\left[-\log(n-x)\right]_0^k \nonumber \\
& = n\log \frac{n}{n-k} \nonumber \\
\end{align}
The variance is
\begin{equation}
\label{eq:couponfirstkvar}
\Var(T) = 0 + \frac{n}{(n-1)^2} + \frac{2n}{(n-2)^2} + \dots + \frac{(k-1)n}{(n-k+1)^2}
\end{equation}
For instance, for a game of poker in which only the top 17 cards of a 52--card deck are to be used, it might be demanded that the distribution of the identities and order of the top 17 cards of the deck were within 0.01 of uniform (Of course, this is not a reasonable shuffling scheme for a real deck of cards).
In this instance, $n=52$ and $k=17$, so $E(T) \leq 20.6$ and $\Var(T) \leq 4.3$. By Chebyshev's inequality, there is at most a 0.01 chance that $T$ is more than $E(T) + 10\sqrt{\Var(T)} \approx 41$. Therefore for this purpose, 41 random-to-top moves suffice, compared to the approximately $52\log 52 \approx 205$ required to get the state of the entire deck just to within 0.25 of uniform,
\item The sets of cards in each quarter of the deck match once the top three quarters of the decks match. The coupling time is $$T \stackrel{d}{=} \mathcal{G}(1) + \mathcal{G}(\frac{n-1}{n}) + \cdots + \mathcal{G}(\frac{\frac{n}{4}+1}{n}).$$
\item The sets of cards in positions congruent to each $i$ modulo $4$ are not guaranteed to match until the entire decks are in the same configuration.
\item The parity of the permutations are not guaranteed to match until the entire decks are in the same order. This is a terrible bound, indicating only that the coupling was unsuited to this statistic. See Remark \ref{rem:badparity} for a better one.
\item The identity of the card immediately above the card labelled by $1$ matches as long as $1$ has been chosen. (The possible values of this statistic are $2$ to $n$, as well as a special value corresponding to the $1$ being on the top of the deck. If instead the definition of `previous card' were to wrap around, with the card above the top card being the bottom card, then this example would behave quite like the next).
\end{enumerate}
The coupling times so far considered in this section have been sums of independent geometric random variables. This will not always be the case. The reason it happens in these examples is that in each of them, the coupling is attempting to make matches. It either creates a match or does not, and the probability of creating a match depends only on the number of matches presently existing. It is also important that in all of these examples, matches are never destroyed. This is why it was necessary to check not only that the statistic matched at a certain time, but also that this would continue to be true after additional steps.
That is, in such examples, the coupling time is the hitting time of a relatively simple Markov chain. For example, to match the top four cards of the deck, the coupling time \[T \stackrel{d}{=} \mathcal{G}(\frac{4}{n})+\mathcal{G}(\frac{3}{n})+\mathcal{G}(\frac{2}{n})+\mathcal{G}(\frac{1}{n})\] is a hitting time for the Markov chain shown in Figure \ref{fig:topfourhitting}
In the next example, thinking of a coupling time as a hitting time enables the analysis of a more complicated Markov chain, where matches may be destroyed.
\begin{figure}\label{fig:topfourhitting}
\end{figure}
Some more statistics:
\begin{enumerate}[resume]
\item The identities of the cards immediately below the $1$ match as long as the $1$ has been chosen, and if fewer than $n-1$ distinct labels have been chosen, the $1$ must have been chosen more recently than at least one of the other chosen labels (equivalently, the $1$ should not be on the bottom of the block of matching cards at the top of the deck). (As in the previous example, the possible values of this statistic are $2$ to $n$ and a special value corresponding to the $1$ being on the bottom of the deck).
Notice that unlike the other statistics considered so far, it is possible that the coupling creates matches in this statistic and then breaks them again.
Unlike previous examples, this coupling time is not a sum of independent geometric random variables. To see why this is and how it may be analysed, consider running the coupling. The information needed to decide whether or not two copies of the chain have coupled is as follows
\begin{itemize}
\item How many distinct cards have been chosen
\item Whether or not the $1$ has been chosen
\item If the $1$ has been chosen, how many cards have been chosen and were last chosen before the last time the $1$ was chosen?
\end{itemize}
This information forms a quotient Markov chain, and understanding the behaviour of this chain suffices to understand the coupling time. Let $(k)$ denote the state where $k$ cards have been chosen, not including the $1$, and $(k,l)$ denote the state where $k$ cards have been chosen, including the $1$, and where $l$ of those cards were last chosen before the $1$ was. Equivalently, $l$ is the number of cards below the $1$ in the block of matching cards at the top of each deck. Figure \ref{fig:afteronehitting} illustrates this chain for $n=4$.
The goal is to understand the coupling time of the original chain. That is, after a certain number of steps, what is the probability that in each deck, the cards following the $1$'s are the same? If the quotient Markov chain is in state $(k,l)$ with $l > 0$ or $k = n$, then the two chains have coupled, so it suffices to understand the probability that after time $t$, the quotient chain is in such a state. When $n=4$, this is understanding the probability that if the Markov chain illustrated in Figure \ref{fig:afteronehitting} is started in the state $(0)$ then after $t$ steps it is at one of the blue states.
\begin{figure}\label{fig:afteronehitting}
\end{figure}
The following gives a sample bound on the coupling time when $n=52$. After $200$ steps, there is a $98\%$ chance that the $1$ has been chosen. As in Equations \ref{eq:couponfirstkex} and \ref{eq:couponfirstkvar}, after $200$ steps there is at least a 99\% chance that at least $42$ different cards have been chosen, so there is at least a $97\%$ chance that at least $42$ different cards have been chosen, including the $1$. In at most $\frac{1}{42}$ of paths leading to such outcomes, each other chosen card has been chosen after the $1$ last was. Thus with probability $\frac{41}{42}\cdot 0.97 \approx 95\%$, the cards immediately following the $1$ match after $200$ steps.
Two choices were made in this calculation --- the number of steps, but also to demand that at least $42$ distinct cards had been chosen. Changing these choices would produce slightly different bounds.
\item The identities of the $k$ cards immediately after the card labelled by $1$ match as long as $1$ has been chosen, and if fewer than $n-1$ distinct labels have been chosen, $1$ must have been chosen more recently than at least $k$ of the other chosen labels (equivalently, $1$ should not be in the bottom $k$ cards of the block of matching cards at the top of the deck). This statistic takes values of ordered $k$--tuples, or smaller ordered tuples when the $1$ is close to the bottom of the deck.
\item The relative order of $1$ and $2$ matches as long as either label has been chosen. The coupling time is $$T \stackrel{d}{=} \mathcal{G}(\frac{2}{n}).$$
\item The relative order of $1$, $2$, ..., $k$ matches as long as all but one of these labels have been chosen. The coupling time is $$T \stackrel{d}{=} \mathcal{G}(\frac{k}{n}) + \mathcal{G}(\frac{k-1}{n}) + \cdots + \mathcal{G}(\frac{2}{n}).$$
\item The number of cards between $1$ and $2$ matches as long as both of these labels have been chosen. The coupling time is $$T \stackrel{d}{=} \mathcal{G}(\frac{2}{n}) + \mathcal{G}(\frac{1}{n}).$$ This is no better a bound than given for the stronger condition that the actual positions of cards $1$ and $2$ should match. It is unclear whether the weaker statistic mixes faster. That the bound is the same may be a weakness of the method, or it may be that a better coupling could be constructed.
\end{enumerate}
Some of these statistics are quotient chains of the random-to-top shuffle, and some are not, as follows:
These statistics are quotient chains. The identity of the top card, the identity and order of the top $k$ cards, the locations of any given set of cards, the parity of the permutation, and the relative order of a subset of cards.
These statistics do not form quotient chains:
\begin{itemize}
\item Identity of the second-to-top card. (Although this information is a subset of that contained in the identity and order of the top two cards, and that is a quotient chain)
\item Identity and order of the cards in any set $A$ of positions unless $A$ is a contiguous block at the top of the deck or $|A| = n-1$. (If the position $k$ is in $A$ but position $k-1$ is not, then if the card from position $k$ is moved to the top, it must be possible to deduce which card is now in position $k$ from only the information of which cards were in the positions in $A$. If besides $k-1$ there was another position not in $A$, those cards could be swapped without changing the available information, showing that this information is insufficient. If $A$ has size $n-1$ then the card in position $k-1$ is the only remaining card.)
\item The sets of cards in each quarter of the deck, either in blocks or interleaved.
\item The identities of the $k$ cards after a specific card.
\item Relative positions of a subset of cards. As commented above, the coupling does not treat this statistic any more specifically than just attempting to couple the exactly positions of those cards, and that is a quotient chain.
\end{itemize}
\begin{Remark}
\label{rem:badparity}
Notice that when $n$ is even, the parity of the permutation of the deck actually mixes perfectly in a single step, because exactly half of the moves correspond to multiplying by an odd permutation (and when $n$ is odd, it gets to within $\frac{1}{2n}$). However, using Proposition \ref{prop:coupling1} with the standard coupling of just choosing matching cards in each deck gives an upper bound of $n \log n$ steps for the mixing of this quantity, which is not at all good. This shows that the coupling used for the convergence of the chain need not be the best to use for the convergence of a statistic --- for permutation parity, for instance, there is a much better coupling which multiplies by either permutations of the same parity or of opposite parities, so that the resulting permutations are of the same parity. This gives that the parity mixes in one step.
\end{Remark}
In general, it is unclear whether upper bounds are bad because this shuffling technique is just not a good one for the statistic under consideration, or because the coupling was poorly chosen. For example, it is true that the random-to-top shuffle mixes the top card of the deck after a single step and takes many steps to mix the bottom card (at least $\frac{3}{4}n$ steps to get within $\frac{1}{4}$ of uniform, because it is impossible to get the original top card into the bottom quarter of the deck in fewer steps.), so some random walks are more suited to some statistics than to others. On the other hand, the previous example regarding permutation parity shows that a coupling may be ill-suited to a particular statistic, even if it gives a good bound for the convergence of the chain itself.
\subsection{Inverse riffle shuffles}
\label{sec:inverserifflestats}
As with the previous section, Proposition \ref{prop:coupling2} gives upper bounds on the mixing times of some statistics on $S_n$ under inverse riffle shuffles, which are modelled by assigning independent bits to each card and then sorting by those bits, breaking ties by the original order of the cards. Multiple steps of this process may be seen as assigning several bits, and sorting by the resulting base--$2$ string.
Two instances of the inverse riffle shuffle may be coupled by, for each label, assigning that card either a $0$ or a $1$ independently with probability $\frac{1}{2}$, and making the same choice in each deck for the cards of that label. The two processes will agree when each pair of cards have been assigned different labels by at least one step.
It will be necessary to consider the strings assigned to the various cards. A subset of cards is considered to have distinct labels if each card in the subset has a different label, and unique labels if those labels are also not repeated among the remainder of the cards.
\begin{Remark}
\label{rem:backwardsstrings}
These strings are growing right-to-left --- that is, least significant digit first. Taking a fixed number of steps of this chain, it will sometimes be convenient to consider the last step first, so that the most extreme changes in position are dealt with first, and the order is gradually refined with less and less impactful moves.
\end{Remark}
The coupling times in this section will not be sums of independent geometric random variables as they were in the previous section --- heuristically, there seemed to be something one-dimensional about most of the examples for the the random-to-top chain, where progress was only made in one direction, and it was possible to check how long it would take for each step, until the chains had coupled. Inverse riffle shuffles, on the other hand, change the positions of most of the cards at the same time.
Analysis of coupling times using this coupling will require the treatment of an associated family of combinatorial problems regarding the strings assigned to the cards. Which cards have unique strings? Which positions contain cards with unique strings? The first results will be in answer to these questions, and these will be used to analyse some statistics on $S_n$.
\begin{Proposition}
\label{prop:matchcard}
For any card, the expected number of cards with the same string as this card after $t$ steps is $\frac{n-1}{2^t}$.
\end{Proposition}
\begin{proof}
The probability that two uniformly random binary strings of length $t$ are equal is $\frac{1}{2^t}$, and there are $n-1$ other cards.
\end{proof}
This result may be generalised:
\begin{Proposition}
\label{prop:matchcard2}
Let $A$ be a set of $N$ pairs of labels. After $t$ steps, the expected number of these pairs of cards which have the same string is $\frac{N}{2^t}$
\end{Proposition}
\begin{proof}
The proof is the same as that of Proposition \ref{prop:matchcard}.
\end{proof}
Perhaps surprisingly, the behaviour of the number of strings matching the card in a certain position (rather than the card of a certain value) behaves differently. This is because the position depends on the sorting, which depends on the assigned strings.
Exact calculations for this statistic are not included, but the following argument gives a heuristic for the scaling.
\begin{Proposition}
\label{prop:matchposition}
For some fixed position $i$, let $A_t$ be the expected number of cards with the same string as the card in position $i$ after $t$ steps. Let $q$ be any real number greater than $\frac{1}{2}$. Then there is a constant $c$ depending on $q$ so that $A_t < cq^t(n-1)$.
\end{Proposition}
\begin{proof}
For this proposition, consider the most significant bit to be assigned first, as in Remark \ref{rem:backwardsstrings}.
This proposition is subtly different from the previous one, demonstrated with the following example. Consider a deck of four cards, and examine the number of cards with the same string as the card second from the top, whichever card this may be. Before any digits have been assigned, this is four. Now, when the first digit of each string is assigned, there are probabilities of 1, 4, 6, 4 and 1 sixteenths that 0,1,2,3 or 4 ones are assigned, respectively. This results in the number of cards sharing a string with the second card being 4,3,2,3 or 4, respectively --- this is different from the number of cards matching a specific card, which would be distributed binomially.
This differs from the behaviour of Proposition \ref{prop:matchcard} because the identity of the card in any given position depends on the assigned strings. Fortunately, the impact of this change is not particularly large, as will now be shown.
Assume that there are $k$ cards with strings matching the string assigned to the card in the $m$th position. After one step, these cards have been split binomially. The worst case (for there to be as many matches as possible) is for the $k$th position to be in the larger of the two blocks. This happens if it was central in the initial block of matching cards --- that is, the $k$ cards with the same string were in positions $m - \frac{k-1}{2}$ to $m + \frac{k-1}{2}$.
So the number of cards with the same string as the card in any given position decays faster than the following process:
\begin{itemize}
\item Start with $n$
\item Repeatedly replace the current number $k$ with $k - r$, where $r$ is obtained by splitting $k$ into two binomial pieces and choosing the smaller.
\end{itemize}
This process iteratively replaces $k$ by $\frac{k}{2} + O(\sqrt{k})$, using Chebyshev's inequality on the binomial distribution. For large enough $k$, this decreases faster than replacing $k$ with $qk$. Let the constant $c$ be the difference for values of $k$ smaller than this.
\end{proof}
It seems likely that the bound of Proposition \ref{prop:matchposition} could be improved to $A_t \leq c\frac{n-1}{2^t}$.
\begin{Example}
\label{ex:invrifflefeatures}
Consider the following statistics on $S_n$, to be studied via the inverse riffle shuffle. The point of these examples is that Proposition \ref{prop:coupling2} has reduced a mixing time problem to analysis of a coupling time. Estimates of the coupling times are given for some of the examples, but detailed analysis is not the goal of this section. Those examples involving the positions of certain cards or identities of cards in certain locations are known results --- see \cite{ADS} and \cite{repeatedcards1}, where these problems are analysed in greater detail.
\begin{enumerate}
\item The top card matches once the lexicographically first string is distinct from all others (equivalently, from the second).
\begin{Proposition}
\label{prop:topcard}
The mixing time $t_{\text{mix}}(\epsilon)$ for the identity of the top card is at most $\log_2(n-1)-\log_2(\epsilon)$.
\end{Proposition}
\begin{proof}
If $m(t)$ is the number of strings equal to the first after $t$ steps, then $$\E{m(t+1)-1} < \frac{1}{2}(m(t)-1).$$ The quantity $m(0)$ is $n-1$, so after $\log_2(n-1)-\log_2(\epsilon)$ steps, there is at least a $1 - \epsilon$ chance that $m(t)$ is zero and so the top cards match. (This is better than would be given by Proposition \ref{prop:matchposition} because the first position is always at the top of its block, so is in the smaller piece exactly half the time)
\end{proof}
\item The second-to-top card matches once the second string (and hence also the first) is distinct from all others.
\begin{Proposition}
The mixing time $t_{\text{mix}}(\epsilon)$ for the identity of the second card is at most $\log_2(n-1)+1-\log_2(\epsilon)$.
\end{Proposition}
\begin{proof}
Proposition \ref{prop:matchposition} suggests that this should take $\log_2(n) + c$ steps. This can be improved because the second position is near the top of the deck, so may only be in the smaller piece exactly once more than average.
\end{proof}
\item The $k$th-from-top card matches once the $k$th string is distinct from all others (equivalently, from the $(k-1)$th and $(k+1)$th). Proposition \ref{prop:matchposition} suggests that this takes $\log_2(n)+c$ steps.
\item The set of the top $k$ cards matches once the $k$th and $(k+1)$th strings are distinct. Again, Proposition \ref{prop:matchposition} suggests that this takes $\log_2(n)+c$ steps.
\item The identity and order of the top $k$ cards match once the top $(k+1)$ strings are all different.
\begin{Proposition}
The mixing time $t_{\text{mix}}(\epsilon)$ for the identity and order of the top $k$ cards is $\log_2(n) + \log_2(k) - \log_2(\epsilon)$.
\end{Proposition}
\begin{proof}
After about $\log_2(\frac{n}{k})$ steps, there is a block of cards at the top of size slightly larger than $k$ with strings distinct from all others. Then by Proposition \ref{prop:matchcard2} with $A$ being the set of all pairs of those cards, after approximately another $2\log_2(k)+c$ steps, the probability that these cards will all have distinct strings is greater than $1-\frac{1}{2^c}$. So $\log_2(n) + \log_2(k) - \log_2(\epsilon)$ steps are enough.
\end{proof}
\item The location of the $1$ matches once the string assigned to that card is distinct from all others.
\begin{Proposition}
\label{prop:locof1}
The mixing time $t_{\text{mix}}(\epsilon)$ for the location of the $1$ satisfies $$t_{\text{mix}}(\epsilon) \leq \log_2(n-1)-\log_2(\epsilon).$$
\end{Proposition}
\begin{proof}
Use Proposition \ref{prop:matchcard2} with $A$ being the set of pairs including $1$.
\end{proof}
\item The locations of $k$ specific cards match once the strings assigned to each are distinct from all others.
\begin{Proposition}
\label{prop:locofk}
The mixing time $t_{\text{mix}}(\epsilon)$ for the locations of any $k$ specific cards is at most $\log_2(n) + \log_2(k)-\log_2(\epsilon)$.
\end{Proposition}
\begin{proof}
Use Proposition \ref{prop:matchcard2} with $A$ being the set of pairs including any of these cards. This gives that the time taken until the expected number of matches is below $\frac{1}{4}$ is at most $$\log_2(nk - \binom{k+1}{2}) -\log_2(\epsilon) \leq \log_2(n) + \log_2(k) -\log_2(\epsilon).$$
\end{proof}
\item The bridge hands in blocks match once the $(13a)$th and $(13a+1)$th strings are different for $a = 1,2$ and $3$. Proposition \ref{prop:matchposition} suggests that this takes about $\log_2(n)+O(1)$ steps.
\item The bridge hands distributed mod $4$ match once the entire deck matches.
\item The parity of the permutation matches once the entire deck matches. As before, this is an awful bound.
\item The card after the $1$ matches once both the string assigned to the $1$ and the next string are distinct from all others.
\item The relative order of the $1$ and $2$ match as soon as they are assigned different strings. \begin{Proposition} \label{prop:rel2}The mixing time $t_{\text{mix}}(\epsilon)$ of the relative order of the $1$ and the $2$ is $-\log_2(\epsilon)$\end{Proposition} \begin{proof}The relative order of the $1$ and $2$ matches once they are assigned different strings. This takes $\mathcal{G}(\frac{1}{2})$ steps.\end{proof}
\item The relative order of the $1$ through $k$ match as soon as they are all assigned different strings.\begin{Proposition} \label{prop:relk}The mixing time $t_{\text{mix}}(\epsilon)$ of the relative order of the $1$ through $k$ is $2\log_2(k)-\log_2(\epsilon)$\end{Proposition} \begin{proof}The relative order of these cards match once they are assigned different strings. Proposition \ref{prop:matchcard2} gives the result.\end{proof}
\end{enumerate}
\end{Example}
Each of these statistics regarding the inverse riffle shuffle process may be translated to the forwards riffle shuffle process. Typically, this interchanges the roles of card positions and card labels, as this is the difference between left-multiplication and right-multiplication in the symmtetric group. We do not give details of these translations here, but they may be found in Section 7.2.3 of \cite{GWThesis}.
\subsection{Random walk on the hypercube}
\label{sec:hypercube}
\begin{Example}
\label{ex:hypercubecoupling}
Consider the lazy nearest-neighbour walk on the hypercube described in Example \ref{ex:hypercube}. Given two instances of this walk in arbitrary initial states, they may be coupled as follows, as in \cite{aldous1983}:
\begin{Coupling}
\label{cou:hypercube}
At each step, choose a position $i$ and a random bit $x$, either $0$ or $1$. In each chain, change the value of the bit in position $i$ to $x$.
\end{Coupling}
The two chains will be in the same state once every position has been chosen at least once. The time taken until this happens is an instance of the coupon collector problem --- approximately $n\log(n)$ steps are required.
\end{Example}
Consider what this coupling says about some statistics on the hypercube using Proposition \ref{prop:coupling2}.
\begin{enumerate}
\item The value of the first bit (or the $k$th bit) matches once that bit is chosen, which takes on average $n$ steps.
\item The number of `1's matches once every bit has been chosen.
\item The position of the first `1' matches if there is some $k$ so that the first $k$ bits have all been chosen and for at least one of them, the last time it was chosen, it was set to 1.
Hence once the first $k$ bits have all been chosen, there is a probability of at least $(1-2^{-k})$ that the position of the first `1' matches. For example, if to find a time by which there is at least a $\frac{15}{16}$ chance that the position of the first `1' matches, consider the time taken until the first five (not four) bits have been chosen, which, by coupon collector theory, has expectation less than $\frac{7}{3}n$ and standard deviation less than about $\frac{4}{3}n$.
So after time $t = \frac{7}{3}n + 6\cdot\frac{4}{3}n = \frac{31}{3}n$, Chebyshev's inequality says that there is at least a $\frac{35}{36}$ chance that the first five bits have all been chosen. There is a $\frac{31}{32}$ chance that they were set to something other than all zeros, so the chance that the first `1' matches after time $\frac{31}{3}n$ is at least $1 - \frac{1}{32} - \frac{1}{36} > \frac{15}{16}$.
This statistic is another example of one where the coupling can create a match and then destroy it, as opposed to `nicer' statistics, where matches, once created, endure forever.
\end{enumerate}
\subsection{Random transpositions on \texorpdfstring{$S_n$}{the symmetric group}}
Consider the shuffling scheme on a deck of $n$ cards where at each step, two cards are chosen uniformly at random and interchanged. Choosing the same card twice is allowed, and in this case the order of the deck is left unchanged. Equivalently, this is the random walk on $S_n$ generated by the set of all $\binom{n}{2}$ transpositions, along with $n$ copies of the identity.
It will be more convenient to describe the moves slightly differently. Define $$a_{i,j} = \text{``swap the card with label $i$ with the card in position $j$''}.$$ The random transposition walk is equivalently described by choosing $i$ and $j$ uniformly between $1$ and $n$ and then applying $a_{i,j}$.
Two copies of this walk may be coupled, following \cite{AF}
\begin{Coupling}
\label{cou:randomtranspositions}
\begin{itemize}
\item Choose $i$ and $j$ uniformly, $1 \leq i,j \leq n$.
\item In each chain, apply $a_{i,j}$
\end{itemize}
\end{Coupling}
To analyse this coupling, define a `match' to be a card which is in the same position in both decks. Observe that the number of matches does not decrease, and increases whenever neither the cards of label $i$ nor the cards in position $j$ presently match. According to this coupling, it takes approximately $n^2$ steps to couple the two chains. This shuffle actually mixes in $\f12n\log(n)$ steps (see \cite{randomtranspositions}), but no Markovian coupling can give this bound (consider two decks whose orders differ by a single transposition, and note that there's only a $\frac{2}{n^2}$ chance that they move to the same state, however they are coupled). See \cite{bormashenko2011coupling} for an amazing (non-Markovian) coupling for the random transposition walk, and a description of related problems.
It will soon be convenient to have some slight variants on this coupling. The previous coupling has the property that if the cards labelled by $k$ match in the two decks, then this match cannot be destroyed by choosing $i=k$, but can be by choosing $j=k$, in which case the match is replaced by the two cards labelled by $i$ matching instead. The analysis of some statistics will be easier if the coupling is edited so that matches are never destroyed.
This does not represent any great change in what's going on --- there is a possibility that the cards labelled by $k$ match, and then this match is broken and replaced by the cards $i$ matching. This is counterbalanced by some other paths where a different pair of cards matches, but that match is broken and replaced by the cards $k$ matching.
To that end, here is a second coupling for this walk.
\begin{Coupling}
\label{cou:randomtranspositionslabels}
Define $$a_{i,j} = \text{``swap the card with label $i$ with the card in position $j$''}.$$ and $$b_{i,j} = \text{``swap the card in position $i$ with the card in position $j$''}.$$
\begin{itemize}
\item Choose $i$ and $j$ uniformly, $1 \leq i,j \leq n$.
\item If the cards in position $j$ do not match, then apply $a_{i,j}$ in both chains.
\item If the cards in position $j$ did match, then instead apply $b_{i,j}$ in both chains.
\end{itemize}
\end{Coupling}
To see that this coupling restricts to the original random walk on both instances of the chain, observe that for any fixed $j$, $$\{a_{i,j}\}_{1\leq i\leq n} = \{b_{i,j}\}_{1\leq i\leq n}.$$ Because the decision as to whether to apply $a_{i,j}$ or $b_{i,j}$ depended only on the value of $j$, the coupling does restrict to the random transposition walk on both instances of the chain.
The analysis of this new coupling is exactly the same as the old --- the number of matches never decreases, and increases by one whenever neither the cards of label $i$ nor the cards in position $j$ currently match. However, it has the property that individual matches are never destroyed, while the previous coupling would destroy matches and replace them by others.
This modification ensured that once the cards labelled by $k$ matched, they would continue to match, albeit possibly in different positions. Alternatively, it could have been defined so that once there was a match in position $k$, there would continue to be a match in that position, although potentially of cards of a different value.
To do this, here is a third coupling.
\begin{Coupling}
\label{cou:randomtranspositionspositions}
Define $$a_{i,j} = \text{``swap the card with label $i$ with the card in position $j$''}.$$ and $$c_{i,j} = \text{``swap the card with label $i$ with the card with label $j$''}.$$
\begin{itemize}
\item Choose $i$ and $j$ uniformly, $1 \leq i,j \leq n$.
\item If the cards of value $i$ do not match, then apply $a_{i,j}$ in both chains.
\item If the cards of value $i$ did match, then instead apply $c_{i,j}$ in both chains.
\end{itemize}
\end{Coupling}
As in the previous case, to see that this coupling restricts to the original random walk on both instances of the chain, note that for any fixed $i$, $$\{a_{i,j}\}_{1\leq j\leq n} = \{c_{i,j}\}_{1\leq j\leq n},$$ and the decision as to whether to apply $a_{i,j}$ or $c_{i,j}$ depended only on the value of $i$.
The analysis of this coupling is the same as the others. The number of matches never decreases. Matches will stay in the same position, but may change in value.
It is also possible to make only part of this variation:
\begin{Coupling}
\label{cou:randomtranspositionsposition1}
Define $$a_{i,j} = \text{``swap the card with label $i$ with the card in position $j$''}.$$ and $$c_{i,j} = \text{``swap the card with label $i$ with the card with label $j$''}.$$
\begin{itemize}
\item Choose $i$ and $j$ uniformly, $1 \leq i,j \leq n$.
\item If the cards of value $i$ match and are in position 1, then apply $c_{i,j}$ in both chains.
\item Otherwise apply $a_{i,j}$ in both chains.
\end{itemize}
\end{Coupling}
This coupling has the property that the number of matches never decreases, and that once there is a match in position $1$, there will always be a match in position $1$.
These couplings may be used to examine the convergence of some statistics. Appendix B of \cite{ADS} computes similar results for the mixing of the position of a single card and the positions of half of the cards, or equivalently, the card in a certain position and the values of the cards in a certain half of the positions.
\begin{enumerate}
\item The top card.
\begin{Proposition}
\label{prop:rtcardtop}
The mixing time $t_{\text{mix}}(\epsilon)$ for the top card is at most $T$, defined by $\mathcal{P}r(\mathcal{G}(\frac{1}{n}) > T) \leq \epsilon$.
\end{Proposition}
\begin{proof}
Consider Coupling \ref{cou:randomtranspositionsposition1}. At each step, if the cards in position 1 do not match, there is a $\frac{1}{n}$ chance of this happening, by choosing $j=1$ and any $i$. Once the cards in position 1 do match, this will remain true, though the matching values may change. Hence the coupling time is $\mathcal{G}(\frac{1}{n})$. This completes the proof.
\end{proof}
Coupling \ref{cou:randomtranspositionsposition1} was used for this purpose, because Couplings \ref{cou:randomtranspositions} and \ref{cou:randomtranspositionslabels} do not preserve matches in position $1$, while Coupling \ref{cou:randomtranspositionspositions} will attempt to preserve matches in other positions, which can increase the time taken to create a match in position $1$.
\item The mixing time for the $k$th card is the same as that of the top card. If Coupling \ref{cou:randomtranspositionsposition1} is changed to preserve matches in position $k$ rather than position $1$, then this is exactly the same as the previous example. That is, it takes $\mathcal{G}(\frac{1}{n})$ steps.
\item The top two cards.
\begin{Proposition}
The mixing time $t_{\text{mix}}(\epsilon)$ for the top two cards is at most $T$, defined by $\mathcal{P}r(\mathcal{G}(\frac{1}{n})+\mathcal{G}(\frac{n-1}{n^2}) > T) \leq \epsilon$.
\end{Proposition}
\begin{proof}For this statistic, vary Coupling \ref{cou:randomtranspositionsposition1} to preserve matches in either of the top two positions. Then while there are no matches in positions $1$ or $2$, each step has a chance of $\frac{2}{n}$ to create one, by choosing $j=1$ or $j=2$, and any $i$. Once there is a match in either of these positions, each step has a chance of $\frac{1}{n}\frac{n-1}{n}$ of creating a match in the other position --- by choosing $j$ to be the other of $\{1,2\}$ and $i$ to be anything but the value involved in the existing match. Therefore the coupling time is $\mathcal{G}(\frac{1}{n})+\mathcal{G}(\frac{n-1}{n^2})$.\end{proof}
\item Any two cards. As was the case for attempting to match the card in a single position, the previous argument did not rely on the positions chosen, so the time until there are matches in any two positions is the same.
\item The cards in any $k$ positions. \begin{Proposition}
\label{prop:anykcards}
The coupling time for the cards in any $k$ positions to match is \[T = \mathcal{G}(\frac{k}{n})+\mathcal{G}(\frac{(k-1)(n-1)}{n^2})+\dots+\mathcal{G}(\frac{n-k+1}{n^2}).\]
\end{Proposition}
\begin{proof}
Use a variation of Coupling \ref{cou:randomtranspositionsposition1} which preserves matches in the relevant positions.
\end{proof}
For example, to match the cards in any $\frac{3n}{4}$ positions takes on average time
\begin{align*}
& \frac{n}{\frac{3n}{4}} + \frac{n}{\frac{3n}{4}-1}\frac{n}{n-1} + \dots + n\frac{n}{n+1-\frac{3n}{4}} \\
&\leq 4n(1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{\frac{3n}{4}}) \\
&\approx 4n\log(\frac{3n}{4}) \\
&\approx 4n\log(n)
\end{align*}
The interesting point here is not Proposition \ref{prop:anykcards} itself --- this bound is still larger than the true value of the mixing time of order of the entire deck, obtained by other means. Rather, it is that a coupling which gives a mixing time of the chain too large by a factor of $n$ can give the correct order of the mixing time of a fairly large portion of the deck.
\item The position of the card labelled by $k$. \begin{Proposition}
The mixing time $t_{\text{mix}}(\epsilon)$ for the location of the $1$ is at most $T$, defined by $\mathcal{P}r(\mathcal{G}(\frac{1}{n}) > T) \leq \epsilon$.
\end{Proposition}
\begin{proof} In the same way that Coupling \ref{cou:randomtranspositionspositions} was modified to create Coupling \ref{cou:randomtranspositionsposition1}, Coupling \ref{cou:randomtranspositionslabels} may be modified to preserve matches only when the matching label is $1$. Then the proof is the same as that of Proposition \ref{prop:rtcardtop}, using positions rather than values. \end{proof}
\item The time for the positions of any $k$ cards to match is the same as in Proposition \ref{prop:anykcards}, but again working with positions rather than values.
\end{enumerate}
\subsection{Glauber dynamics for graph colourings}
Aldous and Fill in \cite{AF} present a coupling for a random walk on graph colourings. Consider a graph $G$ with $n$ vertices and maximal degree $r$, and a set of $c$ colours. A \emph{graph colouring} is an assignment of a colour to each vertex of the graph so that no two vertices of the same colour are connected by an edge.
In \cite{AF}, a coupling is used to show that if $c > 4r$ then the mixing time is bounded above by approximately $\frac{cn}{c-4r}\log(n)$. If the statistic of interest is the set of vertices of any given colour, then this coupling may be modified to show that this statistic mixes in approximately $\frac{cn}{c-3r}\log(n)$ steps. When $c$ is close to $4r$, this is significantly smaller.
This is another situation where these results are interesting only in contrast to one another --- techniques other than coupling give better bounds.
\section{Further work}
We have discussed how coupling and strong stationary times may be used to give bounds for the convergence of statistics of a Markov chain. It would be useful to be able to go in the reverse direction.
\begin{Question}
Given bounds on the convergence of a suitably large collection of statistics on a Markov chain, is it possible to obtain bounds on the convergence of the Markov chain itself?
\end{Question}
The examples of Section 7 of \cite{GWmutations} may be seen as examples where this is possible --- Propositions 30 and 31 of that section may be understood as making rigorous the heuristic that a deck of cards is mixed once each card is in a random position, which in those examples takes $n^3$ steps for any card, and a factor of $\log(n)$ because each card individually must have achieved this.
Of course, care is needed here. Repeatedly applying powers of a single $n$--cycle will randomise the position of each card, but will certainly not result in a shuffled deck, because the positions of each card will be perfectly correlated with one another. A physical example of this is cutting a deck and placing the bottom portion on top. Regardless of how many and which cuts are made, the order of the deck is preserved up to cycling.
For some of the statistics considered in the present paper, a coupling immediately gave a good bound. For others, like the parity of the permutation, the bound was terrible, and a different argument was necessary.
\begin{Question}
Given a coupling or strong stationary time that gives good bounds for the convergence of a Markov chain, is it possible to predict for which statistics it will give good or bad bounds? How can better couplings or strong stationary times be designed for some statistics?
\end{Question}
\end{document} |
\begin{document}
\title
[Location of concentrated vortices]{Location of concentrated vortices in planar steady Euler flows}
\author{Guodong Wang, Bijun Zuo}
\address{Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, P.R. China}
\email{[email protected]}
\address{College of Mathematical Sciences, Harbin Engineering University, Harbin {\rm150001}, PR China}
\email{[email protected]}
\begin{abstract}
In this paper, we study two-dimensional steady incompressible Euler flows in which the vorticity is sharply concentrated in a finite number of regions of small diameter in a bounded domain. Mathematical analysis of such flows is an interesting and physically important research topic in fluid mechanics.
The main purpose of this paper is to prove that in such flows the locations of these concentrated blobs of vorticity must be in the vicinity of some critical point of the Kirchhoff-Routh function, which is determined by the geometry of the domain. The vorticity is assumed to be only in $L^{4/3},$ which is the optimal regularity for weak solutions to make sense. As a by-product, we prove a nonexistence result for concentrated multiple vortex flows in convex domains.
\end{abstract}
\maketitle
\section{Introduction}
Let $D\subset\mathbb R^2$ be a simply-connected bounded domain with smooth boundary $\partial D$. Consider in $D$ an ideal fluid in steady state, the motion of which is described by the famous Euler equations
\begin{equation}\lambdabel{euler}
\begin{cases}
(\mathbf v\cdot\nabla)\mathbf v=-\nabla P&\mathbf x=(x_1,x_2)\in D,\\
\nabla\cdot\mathbf v=0&\mathbf x\in D,\\
\mathbf v\cdot\mathbf n =g&\mathbf x\in\partial D,
\end{cases}
\end{equation}
where $\mathbf v=(v_1,v_2)$ is the velocity field, $P$ is a scalar function that represents the pressure, $\mathbf n$ is the unit outward normal on $\partial D,$ and $g$ is a given function satisfying the following compatibility condition
\begin{equation}\lambdabel{g}
\int_{\partial D}gdS=0.
\end{equation}
Here we assume that the fluid is of unit density. The first two equations in \eqref{euler} are the momentum conservation and mass conservation respectively, and the boundary condition in \eqref{euler} means that the rate of mass flow across the boundary per unit area is $g$. In particular, if $g\equiv0,$ then there is no matter flow through the boundary.
The scalar vorticity $\omega$, defined as the signed magnitude of curl$\mathbf v,$ that is,
\[\omega=\partial_{x_1} v_2-\partial_{x_2}v_1,\]
is one of the fundamental physical quantities and plays an important role in the study of two-dimensional flows.
Below we reformulate the Euler equations \eqref{euler} as a single equation of $\omega$, which is much easier to handle mathematically.
First we show that $\mathbf v$ can be recovered from $\omega$. In fact, since $\mathbf v$ is divergence-free and $D$ is simply-connected, we can apply the Green's theorem to show that there is a scalar function $\psi,$ called the \emph{stream function}, such that
\begin{equation}\lambdabel{psi}
\mathbf v=(\partial_{x_2}\psi,-\partial_{x_1}\psi).
\end{equation}
For convenience, throughout this paper we will use the symbol
$\mathbf b^\perp$ to denote the clockwise rotation through $\pi/2$ of any planar vector $\mathbf b=(b_1,b_2)$, that is, $\mathbf b^\perp=(b_2,-b_1)$, and $\nabla^\perp f$ to denote $(\nabla f)^\perp$ for any scalar function $f$, that is, $\nabla^\perp f=(\partial_{x_2}f,-\partial_{x_1}f)$. Thus \eqref{psi} can also be written as
\begin{equation}\lambdabel{psi2}
\mathbf v=\nabla^\perp \psi.
\end{equation}
It is easy to check that $\psi$ and $\omega$ satisfy
\begin{equation}\lambdabel{poisson}
\begin{cases}
-\Delta\psi=\omega&\text{in }D,\\
\nabla^\perp\psi\cdot\mathbf n=g&\text{on }\partial D.
\end{cases}
\end{equation}
To deal with the boundary condition in \eqref{poisson}, we consider the following elliptic problem
\begin{equation}\lambdabel{q}
\begin{cases}
-\Delta \psi_0=0&\text{in }D,\\
\nabla^\perp \psi_0\cdot\mathbf n=g&\text{on }\partial D.
\end{cases}
\end{equation}
To solve \eqref{q}, we first solve the following Laplace equation with standard Neumann boundary condition
\begin{equation*}
\begin{cases}
-\Delta \psi_1=0&\text{in }D,\\
\frac{\partial \psi_1}{\partial\mathbf n}=g&\text{on }\partial D,
\end{cases}
\end{equation*}
then the harmonic conjugate of $\psi_1$ solves \eqref{q}. Note that by the maximum principle the solution to \eqref{q} is unique up to a constant.
Now it is easy to see that $\psi-\psi_0$ satisfies
\begin{equation}\lambdabel{gw}
\begin{cases}
-\Delta(\psi-\psi_0)=\omega&\text{in }D,\\
\nabla^\perp (\psi-\psi_0)\cdot\mathbf n=0&\text{on }\partial D.
\end{cases}
\end{equation}
The boundary condition in \eqref{gw} implies that $\psi-\psi_0$ is a constant on $\partial D$ (recall that $D$ is simply-connected). Without loss of generality by adding a suitable constant we assume that $\psi-\psi_0=0$ on $\partial D,$ thus $\psi-\psi_0$ can be expressed in terms of the Green's operator as follows
\begin{equation}\lambdabel{exp}
\psi-\psi_0=\mathcal G\omega:=\int_DG(\cdot,\mathbf y)\omega(\mathbf y)d\mathbf y,
\end{equation}
where $G(\cdot,\cdot)$ is the Green's function for $-\Delta$ in $D$ with zero boundary condition. Combining \eqref{psi2} and \eqref{exp}, we have recovered $\psi$ from $\omega$ in the following
\begin{equation}\lambdabel{bs}
\mathbf v=\nabla^\perp(\mathcal G\omega+\psi_0),
\end{equation}
which is usually called the Biot-Savart law in fluid mechanics.
On the other hand, taking the curl on both sides of the momentum equation in \eqref{euler} we get
\begin{equation}\lambdabel{ve1}
\mathbf v\cdot\nabla \omega=0.
\end{equation}
From \eqref{bs} and \eqref{ve1}, the Euler equations \eqref{euler} are reduced to a single equation of $\omega$
\begin{equation}\lambdabel{ve}
\nabla^\perp(\mathcal G\omega+\psi_0)\cdot\nabla \omega=0\quad\text{ in }D,
\end{equation}
which is usually called the \emph{vorticity equation}.
\begin{remark}
When $D$ is multiply-connected, the above discussion is still valid. The only difference is that one needs to replace the usual Green's function $G$ by the hydrodynamic Green's function (see \cite{Flu}, Definition 15.1), which does not cause any essential difficulty for the problem discussed in this paper.
\end{remark}
In the rest of this paper, we will be focused on the study of \eqref{ve}. Note that once we have obtained a solution $\omega$ to \eqref{ve}, we immediately get a solution to \eqref{euler} with
\[\mathbf v=\nabla^\perp(\mathcal G\omega+\psi_0),\quad P(\mathbf x)=\int_{L_{\mathbf x_0,\mathbf x}}\omega(\mathbf y)\mathbf v^\perp(\mathbf y)\cdot d\mathbf y-\frac{1}{2}|\mathbf v(x)|^2,\]
where $\mathbf x_0$ is a fixed point in $D$ and $L_{\mathbf x_0,\mathbf x}$ is any $C^1$ curve joining $\mathbf x_0$ and $\mathbf x$ (one can easily check that the above line integral is well defined by using Green's theorem and the fact that $\omega$ is a solution).
Since in many physical problems the vorticity is of low regularity, not even continuous, it is necessary to define the notion of weak solutions to \eqref{ve}. In the rest of this paper, we regard $\psi_0$ as a given function.
\begin{definition}\lambdabel{wsve}
Let $\omega\in L^{4/3}(D)$. If for any $\phi\in C_c^\infty(D)$ it holds that
\begin{equation}\lambdabel{int}
\int_D\omega\nabla^\perp(\mathcal G\omega+\psi_0)\cdot\nabla\phi d\mathbf x=0,\end{equation}
then $\omega$ is called a weak solution to the vorticity equation \eqref{ve}.
\end{definition}
The above definition is reasonable from the fact that one can multiply any test function
$\phi$ on both sides of \eqref{ve} and integrate by parts formally to get \eqref{int}.
\begin{remark}
Since $\psi_0$ is harmonic (thus smooth) and $\phi$ has compact support in $D$, we see that the integral $\int_D\omega \nabla^\perp \psi_0\cdot\nabla\phi d\mathbf x$ in \eqref{int} makes sense. Note that throughout this paper we do not impose any condition on the boundary value of $\psi_0$.
\end{remark}
\begin{remark}
By the Calderon-Zygmund inequality and Sobolev inequality, $\omega\in L^{4/3}(D)$ is the optimal regularity for the integral $\int_D\omega\nabla^\perp\mathcal G\omega\cdot\nabla\phi d\mathbf x$ in \eqref{int} to be well-defined.
\end{remark}
In the literature, there has been extensive study on the existence of weak solutions to \eqref{ve}. See \cite{B1,B2,CLW,CPY1,CPY2,CW1,CWZ2,CWZu,EM,LP,LYY,SV,T,W,WZ} for example. The solutions obtained in these papers have one common feature, that is, the vorticity is the function of the stream function ``locally". In this regard, Cao and Wang \cite{CW2} proved a general criterion for an $L^{4/3}$ function to be a weak solution.
\begin{theorem}[Cao--Wang, \cite{CW2}]\lambdabel{cwthm}
Let $k$ be a positive integer and $\psi_0\in C^1(\bar D)$.
Suppose that $\omega\in L^{4/3}(D)$ satisfies
\begin{equation}\lambdabel{fo}
\omega=\sum_{i=1}^k\omega_i, \,\,\min_{1\leq i< j\leq k}\{\text{dist}(\text{supp}\omega_i,\text{supp}\omega_j)\}>0,\,\,\omega_i=f^i(\mathcal G\omega+\psi_0), \text{a.e. in (supp}\omega_i)_\delta,
\end{equation}
where $\delta$ is a positive number,
\[\text{(supp}\omega_i)_\delta:=\{\mathbf x\in D\mid \text{dist}(\mathbf x,\text{supp}\omega_i)<\delta\},\]
and each $f^i$ is either monotone from $\mathbb R$ to $\mathbb R\cup\{\pm\infty\}$ or Lipschitz from $\mathbb R$ to $\mathbb R$.
Then $\omega$ is a weak solution to the vorticity equation \eqref{ve}.
\end{theorem}
Some examples of such flows are as follows. When $f_i$ in Theorem \ref{cwthm} is a Heaviside function, the solutions are called vortex patches, and related existence results can be found in \cite{CPY1,CW1,CWZu,T,WZ}. When $f_i$ is a power function, related papers are \cite{CLW,CPY2,LP,LYY,SV}. In \cite{B1,B2,EM}, the authors obtained some steady vortex flows by maximizing or minimizing the kinetic energy of the fluid on the rearrangement class of some given function. The solutions obtained in \cite{B1,B2,EM} still have the form \eqref{fo}, where each $f_i$ is a monotone function, but the precise expression of $f_i$ is unknown. Recently Cao, Wang and Zhan \cite{CWZ2,W} modified Turkington's method \cite{T} and proved the existence of a large class of solutions of the form \eqref{fo}, where each $f_i$ is a given function with few restrictions.
Among the flows mentioned above, some are of particular interest and attract more attention, that is, flows in which the vorticity is sharply concentrated in a finite number of small regions and vanishes elsewhere, just like a finite sum of Dirac measures. Mathematically, the vorticity in such flows has the form
\begin{equation}\lambdabel{cv}
\omega_\varepsilon=\sum_{i=1}^k\omega_{\varepsilon,i},\quad
{\rm supp}(\omega_{\varepsilon,i})\subset B_{o(1)}(\bar x_i),\quad\int_D\omega_{\varepsilon,i} d\mathbf x=\kappa_i+o(1),\quad i=1,\cdot\cdot\cdot,k,
\end{equation}
where $\varepsilon$ is a small positive parameter, $k$ is a positive integer, $\bar x_i\in D$, $\kappa_i$ is a fixed non-zero real number, $i=1,\cdot\cdot\cdot,k$, and $o(1)\to0$ as $\varepsilon \to0^+$. Papers concerning the existence of such solutions include \cite{CLW,CPY1,CPY2,CW1,CWZ2,CWZu,SV,T,W}. Note that all the flows constructed in these papers have bounded vorticity.
Euler Flows with vorticity of the form \eqref{cv} is closely related to a very famous Hamiltonian system in $\mathbb R^2$, the point vortex model (see \cite{L}), which describes the evolution of a finite number of point vortices with their locations being the canonical variables. The point vortex model is only an approximate model, and its precise connection with the 2D Euler equations with concentrated vorticity in the evolutionary case is a tough and unsolved problem. For a detailed discussion, we refer the interested readers to \cite{MP1,MP2,MP3,MPa,T2}. According to the point vortex model, the locations of concentrated blobs of vorticity in steady Euler flows are not arbitrary, but \emph{should} be near a critical point of the following Kirchhoff-Routh function
\begin{equation}\lambdabel{krf}
W(\mathbf x_1,\cdot\cdot\cdot,\mathbf x_k)=-\sum_{1\leq i<j\leq k}\kappa_i\kappa_jG(\mathbf x_i,\mathbf x_j)+\frac{1}{2}\sum_{i=1}^k\kappa_i^2H(\mathbf x_i)+\sum_{i=1}^k\kappa_i\psi_0(\mathbf x_i),
\end{equation}
where
\[(\mathbf x_1,\cdot\cdot\cdot,\mathbf x_k)\in \underbrace{D\times\cdot\cdot\cdot\times D}_{k\text { times}}\setminus \{(\mathbf x_1,\cdot\cdot\cdot,\mathbf x_k)\mid \mathbf x_i\in D, \mathbf x_i=\mathbf x_j \text{ for some }i\neq j\}\]
and $H(\mathbf x)=h(\mathbf x,\mathbf x)$ with $h$ being the regular part of Green's function, that is,
\[h(\mathbf x,\mathbf y):=-\frac{1}{2\pi}\ln|\mathbf x-\mathbf y|-G(\mathbf x,\mathbf y),\quad \mathbf x,\mathbf y\in D.\]
However, to our knowledge there is no complete and rigorous proof on this issue in the literature, although the solutions of the form \eqref{cv} constructed in \cite{CLW,CPY1,CPY2,CW1,CWZ2,EM,SV,T} are all based on the hypothesis that $(\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k)$ is a critical point of $ W$.
The aim of paper is prove that such a hypothesis is necessary.
This paper is organized as follows. In Section 2, we state our main results (Theorems \ref{mthm} and \ref{none}) and give some comments. In Sections 3 and 4 we provide the proofs of them.
\section{Main results}
In this section, we present our two main results.
The first result is about the necessary condition about the locations of concentrated vortices.
\begin{theorem}\lambdabel{mthm}
Let $k$ be a positive integer, $\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k\in D$ be $k$ different points and $\kappa_1,\cdot\cdot\cdot,\kappa_k$ be $k$ non-zero real numbers.
Assume that there exists a sequence of weak solutions $\{\omega_n\}_{n=1}^{+\infty}$ to the vorticity equation \eqref{ve}, satisfying
$\omega_n=\sum_{i=1}^k\omega_{n,i}$ with $\omega_{n,i}\in L^{4/3}(D)$ and
\[{\rm supp}(\omega_{n,i})\subset B_{o(1)}(\bar {\mathbf x}_i),\quad
\int_D\omega_{n,i} d\mathbf x=\kappa_i+o(1),\quad i=1,\cdot\cdot\cdot,k,\]
where $o(1)\to0$ as $n\to+\infty$.
Then $(\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k)$ must be a critical point of $W$ defined by \eqref{krf}.
\end{theorem}
Here we compare Theorem \ref{mthm} with two related results in \cite{CGPY} and \cite{CM}.
In \cite{CGPY}, Cao, Guo, Peng and Yan studied planar Euler flows with vorticity of the following patch form
\begin{equation}\lambdabel{ceq}
\omega^\lambdambda=\sum_{i=1}^k\omega^\lambdambda_i,\quad
\omega^\lambdambda_i=\lambdambda\chi_{\{\mathbf x\in D\mid \mathcal G\omega^\lambdambda(\mathbf x)>\mu_i^\lambdambda\}\cap B_{\delta}(\bar {\mathbf x}_i)},\quad \int_D\omega_i^\lambdambda d\mathbf x=\kappa_i, \quad i=1,\cdot\cdot\cdot,k,
\end{equation}
where $\lambdambda$ is a large positive parameter, $\chi$ denotes the characteristic function, each $\mu^\lambdambda_i$ is a real number depending on $\lambdambda$ and each $\kappa_i$ is a given non-zero number.
They proved that if supp$\omega^\lambdambda_i$ ``shrinks" to $\bar {\mathbf x}_i$ as $\lambdambda\to+\infty$, then $\bar {\mathbf x}_1\cdot\cdot\cdot,\bar {\mathbf x}_k$ must necessarily constitute a critical point of $W$ (see Theorem 1.1 in \cite{CGPY} for the precise statement) . Compared with their result, we consider more general flows and only impose very weak regularity on the vorticity in Theorem \ref{mthm}. Moreover, as we will see in the next section, the proof we provide is shorter and more elementary. The other relevant work is \cite{CM}. In \cite{CM}, Caprini and Marchioro studied the evolution of a finite number of blobs of vorticity in $\mathbb R^2$ and proved the finite-time localization property (see Theorem 1.2 in \cite{CM} for the precise statement). In their result, each $\omega_{n,i}$ is required additionally to have a definite sign and satisfy the growth condition
\begin{equation}\lambdabel{gwth}
\|\omega_{n,i}\|_{L^\infty}\leq M(\text{diam(supp}\omega_{n,i}))^{-\delta},
\end{equation}
where $M$ and $\delta$ are both fixed positive numbers. As a consequence of their result, Theorem \ref{mthm} holds true if the additional growth condition \eqref{gwth} is satisfied (although they only considered the whole plane case, similar result for a bounded domain can also be proved without any difficulty). In this sense, our result can be regarded as a strengthened version of Caprini and Marchioro's result in the steady case.
\begin{remark}
In Theorem 1.1 in \cite{CGPY}, for vorticity of the form \eqref{ceq}, $\bar{\mathbf x}_i\in D$ and $\bar {\mathbf x}_i\neq \bar {\mathbf x}_j$ for $ i\neq j$ are not assumptions but can be proved as conclusions. However, in the very general setting of this paper, these two conclusions may be false. For example, we can regard
a single blob of vorticity as two artificially, thus they may concentrate on the same point. Also, Cao, Wang and Zuo \cite{CWZu} constructed a pair steady vortex patches with opposite rotation directions in the unit disk (Theorem 5.1, \cite{CWZu}), and it can be checked that as the ratio of the circulations of the two patches goes to infinity, the patch with smaller circulation will approach the boundary of the disk.
\end{remark}
Our second result is about the nonexistence of concentrated multiple vortex flows in convex domains, which can be seen as a by-product of Theorem \ref{mthm}.
\begin{theorem}\lambdabel{none}
Let $\delta_0>0$ be fixed, $D$ be a smooth convex domain, $k\geq 2$ be a positive integer, $ \kappa_1,\cdot\cdot\cdot,\kappa_k$ be $k$ positive numbers and $f_1,\cdot\cdot\cdot,f_k$ be $k$ real functions satisfying
\[\lim_{t\to0^+}f_i(t)=0,\quad i=1\cdot\cdot\cdot,k.\]
If $\psi_0\equiv0,$ then there exists $\varepsilon_0>0$, such that for any
$\varepsilon\in(0,\varepsilon_0),$ there is no weak solution $\omega_\varepsilon$ to the vorticity equation \eqref{ve} satisfying
\begin{itemize}
\item[(1)] $\omega_\varepsilon=\sum_{i=1}^k\omega_{\varepsilon,i},$ $\omega_{\varepsilon,i}\in L^{4/3}(D), i=1\cdot\cdot\cdot,k;$
\item[(2)] $\text{dist(supp}\omega_{\varepsilon,i},\text{supp}\omega_{\varepsilon,j}) >\delta_0 \,\, \forall\,1\leq i<j\leq k$ and
$\text{dist(supp}\omega_{\varepsilon,i},\partial D)>\delta_0\,\, \forall\,1\leq i\leq k;$
\item[(3)] diam(supp$\omega_{\varepsilon,i}$)<$\varepsilon,\,\,i=1,\cdot\cdot\cdot,k.$
\item[(4)] $\int_D\omega_{\varepsilon,i}d\mathbf x=\kappa_i+f_i(\varepsilon),\,\,i=1,\cdot\cdot\cdot,k.$
\end{itemize}
\end{theorem}
\section{Proof of Theorem \ref{mthm}}
First we need the following lemma.
\begin{lemma}\lambdabel{lem}
Let $\omega\in L^{4/3}(\mathbb R^2)$ with compact support. Define
\begin{equation*}
f(\mathbf x)=\int_{\mathbb R^2}\ln|\mathbf x-\mathbf y|\omega(\mathbf y)d\mathbf y.
\end{equation*}
Then $f\in W^{2,4/3}_{\rm loc}(\mathbb R^2)$ and the distributional partial derivatives of $f$ can be expressed as
\begin{equation}\lambdabel{deri}
\partial_{x_i} f(\mathbf x)=\int_{\mathbb R^2}\frac{x_i-y_i}{|\mathbf x-\mathbf y|^2}\omega(\mathbf y)d\mathbf y\quad \text{a.e. }\,\mathbf x\in\mathbb R^2, \,\,i=1,2.
\end{equation}
\end{lemma}
\begin{proof}
By the Calderon-Zygmund estimate we have $f\in W^{2,4/3}_{\rm loc}(\mathbb R^2)$. The expression \eqref{deri} follows from Theorem 6.21 on page 157, \cite{LL}.
\end{proof}
Now we are ready to prove Theorem \ref{mthm}. The key point of the proof is to use the anti-symmetry of the singular part of the Biot-Savart kernel.
\begin{proof}[Proof of Theorem \ref{mthm}]
Fix $l\in\{1,\cdot\cdot\cdot,k\}$. It is sufficient to show that
\[\nabla_{{\mathbf x}_l}W(\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k)=\mathbf 0.\]
Let $r_0$ be a small positive number such that
\[r_0<\text{dist}(\bar{\mathbf x}_i,\partial D)\quad \forall\,1\leq i\leq k,\quad r_0<\frac{1}{2}\text{dist}(\bar{\mathbf x}_i,\bar{\mathbf x}_j)\quad \forall\,1\leq i<j\leq k.\]
Choose $\phi(\mathbf x)=\rho(\mathbf x)\mathbf b\cdot \mathbf x$ in Definition \ref{wsve}, where $\mathbf b$ is a constant planar vector and $\rho$ satisfies
\[\rho\in C_c^\infty( D),\,\,\rho\equiv 1\text{ in } B_{r_0}(\bar {\mathbf x}_l),\,\,\rho\equiv 0\text{ in } B_{r_0}(\bar {\mathbf x}_i)\,\,\forall\,i\neq l.\]
Existence of such $\rho$ can be easily obtained by mollifying a suitable patch function.
Then we have
\[\int_D\omega_n\nabla^\perp\left(\mathcal G\omega_n+\psi_0\right)\cdot\nabla\phi d\mathbf x
=0.\]
Denote
\[A_n=\int_D\omega_n\nabla^\perp\mathcal G\omega_n\cdot\nabla\phi d\mathbf x,\quad B_n=\int_D\omega_n\nabla^\perp\psi_0\cdot\nabla\phi d\mathbf x.\]
Then \begin{equation}\lambdabel{ab}
A_n+B_n=0,\quad n=1,2,\cdot\cdot\cdot.
\end{equation}
Below we analyze $A_n$ and $B_n$ separately.
For $A_n$, we have
\begin{align*}
A_n=&\int_D\omega_n(\mathbf x)\nabla^\perp_{\mathbf x}\int_D\left(-\frac{1}{2\pi}\ln|\mathbf x-\mathbf y|-h(\mathbf x,\mathbf y)\right)\omega_n(\mathbf y)d\mathbf y\cdot\nabla\phi d\mathbf x\\
=&-\frac{1}{2\pi}\int_D\omega_n(\mathbf x)\int_D\frac{(\mathbf x-\mathbf y)^\perp}{|\mathbf x-\mathbf y|^2}\omega_n(\mathbf y)d\mathbf y\cdot\nabla\phi d\mathbf x-\int_D\omega_n\int_D\nabla^\perp_{\mathbf x}h(\mathbf x,\mathbf y)\omega_n(\mathbf x)(\mathbf y)d\mathbf y\cdot\nabla\phi d\mathbf x.
\end{align*}
Here we used Lemma \ref{lem} and the facts that $h\in C^\infty(D\times D)$ and $\omega_n$ has compact support in $D$.
Since $\omega_n\in L^{4/3}(D)$, by the Hardy-Littlewood-Sobolev inequality (see Theorem 0.3.2 in \cite{SO}) we have
\[\int_D\frac{|\omega_n(\mathbf y)|}{|\mathbf x-\mathbf y|}d\mathbf y\in L^4(D).\]
Now we can apply Fubini's theorem (see \cite{ru}, page 164) to obtain
\[\frac{(\mathbf x-\mathbf y)^\perp}{|\mathbf x-\mathbf y|^2}\cdot\nabla\phi\omega_n(\mathbf x)\omega_n(\mathbf y)\in L^1(D\times D)\]
and
\[\int_D\omega_n(\mathbf x)\int_D\frac{(\mathbf x-\mathbf y)^\perp}{|\mathbf x-\mathbf y|^2}\omega_n(\mathbf y)d\mathbf y\cdot\nabla\phi d\mathbf x=\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp\cdot\nabla\phi}{|\mathbf x-\mathbf y|^2}\omega_n(\mathbf x)\omega_n(\mathbf y) d\mathbf xd\mathbf y.\]
Thus we have obtained
\begin{align*}
A_n=-\frac{1}{2\pi}\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp\cdot\nabla\phi}{|\mathbf x-\mathbf y|^2}\omega_n(\mathbf x)\omega_n(\mathbf y) d\mathbf xd\mathbf y-\int_D\int_D\nabla^\perp_{\mathbf x}h(\mathbf x,\mathbf y)\cdot\nabla\phi\omega_n(\mathbf x) \omega_n(\mathbf y)d\mathbf xd\mathbf y.
\end{align*}
Substituting $\phi(\mathbf x)=\rho(\mathbf x)\mathbf b\cdot\mathbf x$ in $A_n$, for sufficiently large $n$ we have
\begin{align*}
A_n&=-\frac{1}{2\pi}\int_D\int_{D}\frac{(\mathbf x-\mathbf y)^\perp\cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_n(\mathbf y) d\mathbf xd\mathbf y-\int_D\int_D\nabla^\perp_{\mathbf x}h(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x) \omega_n(\mathbf y)d\mathbf xd\mathbf y\\
=&-\frac{1}{2\pi}\sum_{j=1}^k\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y
-\sum_{j=1}^k\int_D\int_D\nabla_{\mathbf x}^\perp h(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x)\omega_{n,j}(y)d\mathbf xd\mathbf y\\
=&-\frac{1}{2\pi}\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y
-\frac{1}{2\pi}\sum_{j=1,j\neq l}^k\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y\\
&-\sum_{j=1}^k\int_D\int_D\nabla_{\mathbf x}^\perp h(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y\\
=&-\frac{1}{2\pi}\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y
+\sum_{j=1,j\neq l}^k\int_D\int_D\nabla_{\mathbf x}^\perp G(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y\\
&-\int_D\int_D\nabla_{\mathbf x}^\perp h(\mathbf x,\mathbf y)\cdot\mathbf b\omega_{n,l}(\mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y\\
:=&C_n+D_n,
\end{align*}
where
\[C_n=-\frac{1}{2\pi}\int_D\int_D\frac{(\mathbf x-\mathbf y)^\perp \cdot\mathbf b}{|\mathbf x-\mathbf y|^2}\omega_{n,l}(\mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y,\]
\[D_n=\left(\sum_{j=1,j\neq l}^k\int_D\int_D\nabla_{\mathbf x}^\perp G(\mathbf x,\mathbf y)\omega_{n,l}(\mathbf x)\omega_{n,j}(\mathbf y)d\mathbf xd\mathbf y
-\int_D\int_D\nabla_{\mathbf x}^\perp h(\mathbf x,\mathbf y)\omega_{n,l}(\mathbf \mathbf x)\omega_{n,l}(\mathbf y)d\mathbf xd\mathbf y\right)\cdot\mathbf b.\]
By the anti-symmetric property of the integrand in $C_n$, we see that
\[C_n=0 \quad \text{for sufficiently large $n$}.\]
For $D_n,$ it is clear that
\[\lim_{n\to+\infty}D_n=\left(\sum_{j=1,j\neq l}^k \kappa_l\kappa_j\nabla_{\mathbf x}^\perp G(\bar {\mathbf x}_l,\bar {\mathbf x}_j)
-\kappa_l^2\nabla_{\mathbf x}^\perp h(\bar {\mathbf x}_l,\bar {\mathbf x}_l)\right)\cdot\mathbf b.\]
To conclude, we have obtained
\begin{equation}\lambdabel{a}
\lim_{n\to+\infty}A_n=\left(\sum_{j=1,j\neq l}^k \kappa_l\kappa_j\nabla_x^\perp G(\bar {\mathbf x}_l,\bar {\mathbf x}_j)
-\kappa_l^2\nabla_{\mathbf x}^\perp h(\bar {\mathbf x}_l,\bar {\mathbf x}_l)\right)\cdot\mathbf b.
\end{equation}
For $B_n,$ it is also clear that
\begin{equation}\lambdabel{b}
\lim_{n\to+\infty}B_n=\kappa_l\nabla^\perp\psi_0(\bar{\mathbf x}_l)\cdot\mathbf b.
\end{equation}
Combining \eqref{ab}, \eqref{a} and \eqref{b} we immediately get
\[\left(\sum_{j=1,j\neq l}^k \kappa_l\kappa_j\nabla_{\mathbf x}^\perp G(\bar {\mathbf x}_l,\bar {\mathbf x}_j)
-\kappa_l^2\nabla_{\mathbf x}^\perp h(\bar {\mathbf x}_l,\bar {\mathbf x}_l)+\kappa_l\nabla^\perp\psi_0(\bar{\mathbf x}_l)\right)\cdot \mathbf b= 0\]
for sufficiently large $n$.
Since $\mathbf b$ can be any constant vector, we deduce that
\[\sum_{j=1,j\neq l}^k \kappa_l\kappa_j\nabla_{\mathbf x}^\perp G(\bar {\mathbf x}_l,\bar {\mathbf x}_j)
-\kappa_l^2\nabla_{\mathbf x}^\perp h(\bar {\mathbf x}_l,\bar {\mathbf x}_l)+\kappa_l\nabla^\perp\psi_0(\bar{\mathbf x}_l)= \mathbf 0,\]
which is exactly
\[\nabla_{{\mathbf x}_l}W(\bar {\mathbf x}_1,\cdot\cdot\cdot,\bar {\mathbf x}_k)=\mathbf 0.\]
\end{proof}
\section{Proof of Theorem \ref{none}}
In this section we give the proof of Theorem \ref{none}. To begin with, we need an important property of the Kirchhoff-Routh function in a convex domain proved by Grossi and Takahashi. We only state the following simple version of their result which is enough for our use.
\begin{theorem}[Grossi--Takahashi, Theorem 3.2, \cite{GT}]\lambdabel{gtt}
Let $D$ be a smooth convex domain, $k\geq 2$ be a positive integer and $\kappa_1,\cdot\cdot\cdot,\kappa_k$ be $k$ positive numbers. If $\psi_0\equiv0,$ then the Kirchhoff-Routh function $W$ defined by \eqref{krf} has no critical point in
\[\underbrace{D\times\cdot\cdot\cdot\times D}_{k\text { times}}\setminus \{(\mathbf x_1,\cdot\cdot\cdot,\mathbf x_k)\mid \mathbf x_i\in D, \mathbf x_i=\mathbf x_j \text{ for some }i\neq j\}.\]
\end{theorem}
\begin{proof}[Proof of Theorem \ref{none}]
Suppose, by contradiction, that there exist a sequence of positive numbers $\{\varepsilon_{n}\}_{n=1}^{+\infty}$, $\varepsilon_n\to0^+$ as $n\to+\infty$, and a sequence of weak solutions $\{\omega_n\}_{n=1}^{+\infty}$ to the vorticity equation \eqref{ve} such that
\begin{itemize}
\item[(i)] $\omega_n=\sum_{i=1}^k\omega_{n,i},$ $\omega_{n,i}\in L^{4/3}(D), i=1\cdot\cdot\cdot,k;$
\item[(ii)] $\text{dist(supp}\omega_{n,i},\text{supp}\omega_{n,j}) >\delta_0 \,\,\,\forall\,1\leq i<j\leq k$ and
$\text{dist(supp}\omega_{n,i},\partial D)>\delta_0\,\,\, \forall\,1\leq i\leq k;$
\item[(iii)] diam(supp$\omega_{n,i}$)<$\varepsilon_n,\,\,i=1,\cdot\cdot\cdot,k.$
\item[(iv)] $\int_D\omega_{n,i}d\mathbf x=\kappa_i+f_i(\varepsilon_n),\,\,i=1,\cdot\cdot\cdot,k.$
\end{itemize}
Define
\[{\mathbf x}_{n,i}=\left(\int_{D}\omega_{n,i}d\mathbf x\right)^{-1}\int_{D}\mathbf x\omega_{n,i}d\mathbf x,\quad i=1\cdot\cdot\cdot,k.\]
By (ii) and (iii) we see that
\[\text{dist}(\mathbf x_{n,i},\mathbf x_{n,j}) \geq\frac{\delta_0}{2} \,\,\, \forall\,1\leq i<j\leq k,\quad
\text{dist}(\mathbf x_{n,i},\partial D)\geq\frac{\delta_0}{2}\,\,\, \forall\,1\leq i\leq k\]
if $n$ is large enough.
Thus we can choose a subsequence $\{\mathbf x_{n_m,i}\}$ such that $\mathbf x_{n_m,i}\to\bar {\mathbf x}_i$ as $m\to+\infty, i=1,\cdot\cdot\cdot,k,$ where $\bar{\mathbf x}_1,\cdot\cdot\cdot,\bar{\mathbf x}_k$ satisfy
\[\text{dist}(\bar{\mathbf x}_{i},\bar {\mathbf x}_{j}) \geq\frac{\delta_0}{2} \,\,\, \forall\,1\leq i<j\leq k,\quad
\text{dist}(\bar{\mathbf x}_{i},\partial D)\geq\frac{\delta_0}{2}\,\,\, \forall\,1\leq i\leq k.\]
Now we can see that the sequence of solutions$\{\omega_{n_m}\}$ satisfies the assumptions in Theorem \ref{mthm}, and therefore $(\bar{\mathbf x}_1,\cdot\cdot\cdot,\bar{\mathbf x}_k)$ must be a critical point of $W$ (with $\psi_0\equiv 0$). This is a contradiction to Theorem \ref{gtt}.
\end{proof}
{\bf Acknowledgements:}
{G. Wang was supported by National Natural Science Foundation of China (12001135, 12071098) and China Postdoctoral Science Foundation (2019M661261).}
\phantom{s}
\thispagestyle{empty}
\end{document} |
\begin{equation}gin{document}
\title{Restricted Partition Functions as \\
Bernoulli and Euler Polynomials of Higher Order}
\author{Boris Y. Rubinstein${}^{\dag}$ and Leonid G. Fel${}^{\ddag}$\\
\\
${}^{\dag}$Department of Mathematics, University of California, Davis,
\\One Shields Dr., Davis, CA 95616, U.S.A. \\
and \\
${}^{\ddag}$Department of Civil and Environmental Engineering,\\
Technion, Haifa 32000, Israel}
\date{\today}
\maketitle
\begin{equation}gin{abstract}
Explicit expressions for restricted partition
function $W(s,{\bf d}^m)$ and its quasiperiodic components
$W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive
integers ${\bf d}^m = \{d_1, d_2, \ldots, d_m\}$ are derived.
The formulas are represented in a form of a finite sum over
Bernoulli and Euler polynomials of higher order with periodic coefficients.
A novel recursive relation for the Sylvester waves is established.
Application to counting algebraically independent homogeneous
polynomial invariants of the finite groups is discussed.
\end{abstract}
\section{Introduction}
\label{intro}
The problem of partitions of positive integers has long history started from
the work of Euler who laid a foundation of the theory of partitions
\cite{GAndrews}, introducing the idea of generating functions.
Many prominent mathematicians contributed to the development of the theory
using the Euler idea.
J.J. Sylvester provided a new insight and made a
remarkable progress in this field. He found \cite{Sylv1,Sylv2} the
procedure enabling to determine a {\it restricted} partition functions,
and described symmetry properties of such functions. The restricted
partition function $W(s,{\bf d}^m) \equiv W(s,\{d_1,d_2,\ldots,d_m\})$ is a
number of partitions of $s$ into positive integers $\{d_1,d_2,\ldots,d_m\}$,
each not greater than $s$. The generating function for $W(s,{\bf d}^m)$
has a form
\begin{equation}
F(t,{\bf d}^m)=\prod_{i=1}^m\frac{1}{1-t^{d_{i}}}
=\sum_{s=0}^{\infty} W(s,{\bf d}^m)\;t^s\;,
\label{genfunc}
\end{equation}
where $W(s,{\bf d}^m)$ satisfies the basic recursive relation
\begin{equation}
W(s,{\bf d}^m) - W(s-d_m,{\bf d}^m) = W(s,{\bf d}^{m-1})\;.
\label{SW_recursion}
\end{equation}
Sylvester also proved the statement about splitting of the partition
function into periodic and non-periodic parts and showed that the
restricted partition function may be presented as a sum of "waves", which
we call the {\em Sylvester waves}
\begin{equation}
W(s,{\bf d}^m) = \sum_{j=1} W_j(s,{\bf d}^m)\;,
\label{SylvWavesExpand}
\end{equation}
where summation runs over all distinct factors in the set ${\bf d}^m$.
The wave $W_j(s,{\bf d}^m)$ is a quasipolynomial in $s$
closely related to prime roots $\rho_j$ of unit.
Namely, Sylvester showed in \cite{Sylv2} that the wave
$W_j(s,{\bf d}^m)$ is a coefficient of
${t}^{-1}$ in the series expansion in ascending powers of $t$ of
\begin{equation}
F_j(s,t)=\sum_{\rho_j} \frac{\rho_j^{-s} e^{st}}{\prod_{k=1}^{m}
\left(1-\rho_j^{d_k} e^{-d_k t}\right)}\;.
\label{generatorWj}
\end{equation}
The summation is made over all prime roots of unit
$\rho_j=\exp(2\pi i n/j)$ for $n$ relatively prime to $j$
(including unity) and smaller than $j$.
This result is
just a recipe for calculation of the partition function and it
does not provide explicit formula.
Using the Sylvester recipe we find an explicit formula for the
Sylvester wave $W_j(s,{\bf d}^m)$ in a form of finite sum of the Bernoulli
polynomials of higher
order \cite{bat53,NorlundMemo} multiplied by a periodic function of integer
period $j$. The periodic factor is expressed through the generalized
Euler polynomials of higher order \cite{Carlitz1960}.
A special symbolic technique is developed in the theory of polynomials of higher
order, which significantly simplifies computations performed with these
polynomials. A short description of this technique required
for better understanding of this paper is given in Appendix
\ref{appendix1}.
\section{Sylvester wave $W_1(s,{\bf d}^m)$ and Bernoulli polynomials \\
of higher order}
\label{1}
Consider a polynomial part of the partition function
corresponding to the wave $W_1(s,{\bf d}^m)$. It may be found as a residue of
the generator
\begin{equation}
F_1(s,t) = \frac{e^{st}}{\prod_{i=1}^m (1-e^{-d_i t})}\;.
\label{generatorW1}
\end{equation}
Recalling the generating function for the Bernoulli polynomials of higher
order \cite{bat53}:
\begin{equation}
\frac{e^{st} t^m \prod_{i=1}^m d_i}{\prod_{i=1}^m (e^{d_it}-1)} =
\sum_{n=0}^{\infty} B^{(m)}_n(s|{\bf d}^m)
\frac{t^{n}}{n!}\;,
\label{genfuncBernoulli0}
\end{equation}
and a transformation rule
$$
B^{(m)}_n(s|-{\bf d}^m) = B^{(m)}_n(s+\sum_{i=1}^m d_i|{\bf d}^m)\;,
$$
we obtain the relation
\begin{equation}
\frac{e^{st}}{\prod_{i=1}^m (1-e^{-d_it})} =
\frac{1}{\pi_m} \sum_{n=0}^{\infty} B^{(m)}_n(s+s_m|{\bf d}^m)
\frac{t^{n-m}}{n!}\;,
\label{genfuncBernoulli}
\end{equation}
where
$$
s_m = \sum_{i=1}^m d_i, \ \ \pi_m = \prod_{i=1}^m d_i\;.
$$
It is immediately seen from (\ref{genfuncBernoulli}) that the coefficient
of $1/t$ in (\ref{generatorW1}) is given by the term with $n=m-1$
\begin{equation}gin{equation}
W_1(s,{\bf d}^m) =
\frac{1}{(m-1)!\;\pi_m}
B_{m-1}^{(m)}(s + s_m | {\bf d}^m)\;.
\label{W_1}
\end{equation}
The polynomial part also admits a symbolic form frequently used in theory of
higher order polynomials
\begin{equation}gin{equation}
W_1(s,{\bf d}^m) = \frac{1}{(m-1)!\;\pi_m}
\left(s+s_m + \sum_{i=1}^m d_i \;{}^i\! B\right)^{m-1}\;,
\label{W_1symb}
\end{equation}
where after expansion powers $r_i$ of ${}^i\! B$ are converted into orders
of the Bernoulli numbers
\begin{equation}
{}^i \! B^{r_i} \Rightarrow B_{r_i}\;.
\label{replacement_rule}
\end{equation}
It is easy to recognize in (\ref{W_1}) the explicit
formula reported recently in
\cite{Beck}, which was obtained by a straightforward computation of the complex
residue of the generator (\ref{generatorW1}).
Note that basic recursive relation for the Bernoulli polynomials
\cite{NorlundMemo}
\begin{equation}
B_{n}^{(m)}(s + d_m | {\bf d}^m) -
B_{n}^{(m)}(s | {\bf d}^m) =
n d_m B_{n-1}^{(m-1)}(s | {\bf d}^{m-1})
\label{Bernoulli_recursion}
\end{equation}
naturally leads to the basic recursive relation for the polynomial part of
the partition function:
\begin{equation}
W_1(s,{\bf d}^m) - W_1(s-d_m,{\bf d}^m) =
W_1(s,{\bf d}^{m-1})\;,
\label{SW1_recursion}
\end{equation}
which coincides with (\ref{SW_recursion}). This indicates that the
Bernoulli polynomials of higher order represent a natural
basis for expansion of the partition function and its waves.
\section{Sylvester wave $W_2(s,{\bf d}^m)$ and Euler numbers \\
of higher order}
\label{2}
In order to compute the Sylvester wave with period $j>1$ we note, that
the summand in the expression (\ref{generatorWj}) can be rewritten as
a product
\begin{equation}
F_j(s,t) = \sum_{\rho_j}
\frac{e^{st}}{\prod_{i=1}^{\omega_j} (1-e^{-d_it})} \times
\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i}e^{-d_i t})}\;,
\label{generator_product}
\end{equation}
where the elements in ${\bf d}^m$ are sorted in a way that $j$ is a divisor
for first $\omega_j$ elements (we say that $j$ has weight
$\omega_j$), and the rest elements in the set are not
divisible by $j$.
Then a 2-periodic
Sylvester wave $W_2(s,{\bf d}^m)$ is a residue of the generator
\begin{equation}
F_2(s,t) =
\frac{e^{st}}{\prod_{i=1}^{\omega_2} (1-e^{-d_it})} \times
\frac{(-1)^{s}}{\prod_{i=\omega_2+1}^m (1+e^{-d_i t})}\;,
\label{generatorW2}
\end{equation}
where first $\omega_2$ integers $d_i$ are even, and the summation is
omitted being trivially restricted to the only value $\rho_2=-1$.
Recalling the generating function for the Euler polynomials of higher order
\cite{bat53, NorlundMemo} and corresponding recursive relation
\begin{equation}gin{eqnarray}
\frac{2^m e^{st}}{\prod_{i=1}^m (e^{d_i t}+1)} =
\sum_{n=0}^{\infty} E_n^{(m)}(s | {\bf d}^m) \frac{t^n}{n!}\;,
\label{Euler_GF} \\
E_n^{(m)}(s+d_m | {\bf d}^m)+E_n^{(m)}(s | {\bf d}^m)=
2E_n^{(m-1)}(s | {\bf d}^{m-1})\;,\nonumber
\end{eqnarray}
we may rewrite (\ref{generatorW2}) as double infinite sum
\begin{equation}
\frac{(-1)^{s}}{2^{m-\omega_2} \pi_{\omega_2}}
\sum_{n=0}^{\infty} B^{(\omega_2)}_n(s+s_{\omega_2}|{\bf d}^{\omega_2})
\frac{t^{n-\omega_2}}{n!}
\sum_{l=0}^{\infty} E_l^{(m-\omega_2)}(s_m-s_{\omega_2} | {\bf d}^{m-\omega_2})
\frac{t^l}{l!}\;.
\label{g2sum}
\end{equation}
The coefficient of $1/t$ in the above series is found for
$n+l=\omega_2-1$, so that we end up with a finite sum:
\begin{equation}
W_2(s,{\bf d}^m) =
\frac{(-1)^{s}}{(\omega_2-1)!\; 2^{m-\omega_2} \pi_{\omega_2}}
\sum_{n=0}^{\omega_2-1} \binom{\omega_2-1}{n}
B^{(\omega_2)}_n(s+s_{\omega_2}|{\bf d}^{\omega_2})
E_{\omega_2-1-n}^{(m-\omega_2)}(s_m-s_{\omega_2} | {\bf d}^{m-\omega_2}).
\label{W2}
\end{equation}
This expression may be rewritten as a symbolic power similar to
(\ref{W_1symb}):
\begin{equation}
W_2(s,{\bf d}^m) =
\frac{(-1)^{s}}{(\omega_2-1)! \; 2^{m-\omega_2} \pi_{\omega_2}}
\left(
s+s_m + \sum_{i=1}^{\omega_2} d_i \;{}^i\! B +
\sum_{i=\omega_2+1}^{m} d_i \;{}^i\! E(0)
\right)^{\omega_2-1},
\label{W2symb}
\end{equation}
where the rule for the Euler polynomials at zero $E_n(0)$ similar to
(\ref{replacement_rule}) is applied.
It is easy to rewrite formula (\ref{W2symb}) in a form
\begin{equation}
W_2(s,{\bf d}^m) =
\frac{(-1)^{s}}{(\omega_2-1)!\; 2^{m-\omega_2} \pi_{\omega_2}}
\sum_{n=0}^{\omega_2-1} \binom{\omega_2-1}{n}
B^{(\omega_2)}_n(s+s_{m}|{\bf d}^{\omega_2})
E_{\omega_2-1-n}^{(m-\omega_2)}(0|{\bf d}^{m-\omega_2}),
\label{W2last}
\end{equation}
where $E_{n}^{(m)}(0|{\bf d}^{m})$ denote the Euler polynomials
of higher orders computed at zero as follows:
\begin{equation}
E_{n}^{(m)}(0|{\bf d}^{m}) =
\left[
\sum_{i=1}^{m} d_i \;{}^i\! E(0)
\right]^{n}.
\label{Enumbers}
\end{equation}
The formula (\ref{W2last}) shows that the wave $W_2(s,{\bf d}^{m})$ can be
written as an expansion
over the Bernoulli polynomials of higher order with constant coefficients,
multiplied by a 2-periodic function $(-1)^s$.
\section{Sylvester waves $W_j(s,{\bf d}^m) \ (j>2)$ and Euler \\
polynomials of higher order}
\label{j}
Frobenius \cite{Frobenius} studied in great detail the polynomials
$H_n(s,\rho)$ satisfying the generating function
\begin{equation}
\frac{(1-\rho) e^{st}}{e^t-\rho} = \sum_{n=0}^{\infty} H_n(s,\rho)
\frac{t^n}{n!}, \ \ (\rho \ne 1),
\label{EulerRegGFnew}
\end{equation}
which reduces to definition of the Euler polynomials at fixed value of the
parameter $\rho$
$$
E_n(s) = H_n(s,-1).
$$
The polynomials $H_n(\rho) \equiv H_n(0,\rho)$ satisfy the
symbolic recursion ($H_0(\rho)=1$)
\begin{equation}
\rho H_n(\rho) = (H(\rho)+1)^n, \ \ \ n>0.
\label{EulerSymbolic}
\end{equation}
The generalization of (\ref{Euler_GF}) is straightforward
\begin{equation}
\frac{ e^{st} \prod_{i=1}^m (1-\rho^{d_i})}
{\prod_{i=1}^m (e^{d_i t}-\rho^{d_i})} =
\sum_{n=0}^{\infty} H_n^{(m)}(s, \rho | {\bf d}^m) \frac{t^n}{n!},
\ \ (\rho^{d_i} \ne 1),
\label{Euler_GFnew}
\end{equation}
where the corresponding recursive relation for $H_n^{(m)}(s, \rho | {\bf
d}^m)$ has the form
\begin{equation}gin{equation}
H_n^{(m)}(s+d_m,\rho | {\bf d}^m)-\rho^{d_m}H_n^{(m)}(s,\rho | {\bf d}^m)=
\left(1-\rho^{d_m}\right)H_n^{(m-1)}(s,\rho | {\bf d}^{m-1})\;.
\label{Euler_GFnewrecur}
\end{equation}
The {\em generalized Euler polynomials of higher order}
$H_n^{(m)}(s, \rho | {\bf d}^m)$
introduced by L. Carlitz in \cite{Carlitz1960}
can be defined through the symbolic
formula
\begin{equation}
H_n^{(m)}(s, \rho | {\bf d}^m) =
\left(s + \sum_{i=1}^{m} d_i \;{}^i\! H(\rho^{d_i}) \right)^n,
\label{EulerNewSymbolic}
\end{equation}
where $H_n(\rho)$ computed from the
relation
$$
\frac{1-\rho}{e^t-\rho} = \sum_{n=0}^{\infty} H_n(\rho)
\frac{t^n}{n!},
$$
or using the recursion (\ref{EulerSymbolic}). Using the polynomials
$H_n^{(m)}(s, \rho | {\bf d}^m)$ we can compute Sylvester wave of
arbitrary period.
Consider a $j$-periodic Sylvester wave $W_j(s,{\bf d}^m)$,
and rewrite the summand in (\ref{generator_product}) as double infinite sum
\begin{equation}
\frac{\rho_j^{-s}}{\pi_{\omega_j} \; \prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
\sum_{n=0}^{\infty} B^{(\omega_j)}_n(s+s_{\omega_j}|{\bf d}^{\omega_j})
\frac{t^{n-\omega_j}}{n!}
\sum_{l=0}^{\infty} H_l^{(m-\omega_j)}(s_m-s_{\omega_j}, \rho_j | {\bf
d}^{m-\omega_j})
\frac{t^l}{l!}.
\label{gjsum}
\end{equation}
The coefficient of $1/t$ in the above series is found for
$n+l=\omega_j-1$, so that we have a finite sum:
\begin{equation}a
W_j(s,{\bf d}^m) & = &
\frac{1}{(\omega_j-1)! \; \pi_{\omega_j}}
\sum_{\rho_j}
\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
\times \nonumber \\
&&\sum_{n=0}^{\omega_j-1} \binom{\omega_j-1}{n}
B^{(\omega_j)}_n(s+s_{\omega_j}|{\bf d}^{\omega_j})
H_{\omega_j-1-n}^{(m-\omega_j)}(s_m-s_{\omega_j}, \rho_j | {\bf
d}^{m-\omega_j})\;.
\label{Wj}
\end{equation}a
This expression may be rewritten as a symbolic power similar to
(\ref{W2symb}):
\begin{equation}
W_j(s,{\bf d}^m) =
\frac{1}{(\omega_j-1)! \; \pi_{\omega_j}}
\sum_{\rho_j}
\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
\left(
s+s_m + \sum_{i=1}^{\omega_j} d_i \;{}^i\! B + \!\!\!
\sum_{i=\omega_j+1}^{m} \!\!\! d_i \;{}^i\! H(\rho_j^{d_i})
\right)^{\omega_j-1} \!\!\!\!\!\!\!\!\;,
\label{Wjsymb}
\end{equation}
which is equal to
\begin{equation}a
W_j(s,{\bf d}^m) & = &
\frac{1}{(\omega_j-1)! \; \pi_{\omega_j}}
\sum_{n=0}^{\omega_j-1} \binom{\omega_j-1}{n}
B^{(\omega_j)}_n(s+s_{m}|{\bf d}^{\omega_j})
\times \nonumber \\
&& \sum_{\rho_j}
\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
H_{\omega_j-1-n}^{(m-\omega_j)}[\rho_j |{\bf d}^{m-\omega_j}]\;,
\label{WjBern}
\end{equation}a
where
\begin{equation}
H_{n}^{(m)}[\rho |{\bf d}^{m}] =
H_{n}^{(m)}(0,\rho |{\bf d}^{m}) =
\left[\sum_{i=1}^{m} d_i \;{}^i\! H(\rho^{d_i})\right]^n\;,
\label{Zrhonumbers}
\end{equation}
are generalized Euler numbers of higher order
and it is assumed that
$$
H_{0}^{(0)}[\rho | \emptyset] = 1, \
H_{n}^{(0)}[\rho | \emptyset] = 0\;, \ n>0\;.
$$
It should be underlined that the presentation of the Sylvester wave as a
finite sum of the Bernoulli polynomials with periodic coefficients
(\ref{WjBern}) is not unique. The symbolic formula (\ref{Wjsymb}) can be
cast into a sum of the generalized Euler polynomials
\begin{equation}a
W_j(s,{\bf d}^m) & = &
\frac{1}{(\omega_j-1)! \; \pi_{\omega_j}}
\sum_{n=0}^{\omega_j-1} \binom{\omega_j-1}{n}
B^{(\omega_j)}_n[{\bf d}^{\omega_j}]
\times \nonumber \\
&& \sum_{\rho_j}
\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
H_{\omega_j-1-n}^{(m-\omega_j)}(s+s_m, \rho_j |{\bf d}^{m-\omega_j})\;,
\label{WjEuler}
\end{equation}a
where
$$
B^{(m)}_n[{\bf d}^{m}] = B^{(m)}_n(0|{\bf d}^{m})
$$
are the Bernoulli numbers of higher order.
Substitution of the expression (\ref{WjBern}) into the expansion
(\ref{SylvWavesExpand}) immediately produces the partition function
$W(s,{\bf d}^{m})$ as finite sum of the Bernoulli
polynomials of higher order multiplied by periodic functions with
period equal to the least common multiple of the elements in ${\bf d}^m$
\begin{equation}a
W(s,{\bf d}^m) & = & \sum_j
\frac{1}{(\omega_j-1)! \; \pi_{\omega_j}}
\sum_{n=0}^{\omega_j-1} \binom{\omega_j-1}{n}
B^{(\omega_j)}_n(s+s_{m}|{\bf d}^{\omega_j})
\times \nonumber \\
&& \sum_{\rho_j}
\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
H_{\omega_j-1-n}^{(m-\omega_j)}[\rho_j | {\bf d}^{m-\omega_j}]\;.
\label{WBern}
\end{equation}a
The partition function $W(s,{\bf d}^m)$ has several interesting properties.
Analysis of the generating function (\ref{genfunc}) shows that the partition
function is a homogeneous function of zero order with respect to all
its arguments, i.e.,
\begin{equation}
W(k s,k {\bf d}^m) = W(s,{\bf d}^m).
\label{homogeneity}
\end{equation}
This property appears very useful for computation of the partition
function in case when the elements $d_i$ have a common factor $k$, then
\begin{equation}
W(s, k {\bf d}^m) = W\left(\frac{s}{k},{\bf d}^m\right).
\label{factor}
\end{equation}
The case of $m$ identical elements ${\bf p}^m=\{p,\ldots,p\}$ appears to be the
simplest and is reduced to the known formula for Catalan partitions
\cite{catal838}: {\it the Diophantine equation $x_1+x_2+\;.\;.\;.\;+x_m=s$ has
${s+m-1\choose s}$ sets of non-negative solutions.}
Using (\ref{factor}) for $s$ divisible by $p$ we arrive at
$$
W(s,{\bf p}^m) = W\left(\frac{s}{p},{\bf 1}^m\right) =
W_1\left(\frac{s}{p},{\bf 1}^m\right) =
\frac{B_{m-1}^{(m)}(s/p + m | {\bf 1}^m)}{(m-1)!}.
$$
The straightforward computation shows that
$$
B_{m-1}^{(m)}(s + m | {\bf 1}^m) = \prod_{k=1}^{m-1} (s+k) =
\frac{(s+m-1)!}{s!},
$$
so that
\begin{equation}
W(s,{\bf p}^m) =
\left\{ \begin{equation}gin{array}{ll}
\prod_{k=1}^{m-1} \left(1+\frac{s}{kp} \right), & s=0 \pmod p,\\
0 \;, & s \ne 0 \pmod p.
\end{array}\right.
\label{identical_d}
\end{equation}
In the end of this Section we consider a special case of the tuple
$\{p_1,p_2,\ldots p_m\}$ of primes $p_j$ which leads to essential
simplification of the formula (\ref{WBern}).
The first Sylvester wave $W_1$ is given by (\ref{W_1}) while
all higher waves arising
are purely periodic
\begin{equation}gin{equation}
W_{p_{i}}(s;\{p_1,p_2,\ldots,p_m\})=
\frac{1}{p_i}\sum_{k=1}^{p_{i}-1}\frac{\rho_{p_{i}}^{-ks}}
{\prod_{j\neq i}^m\left(1-\rho_{p_{i}}^{kp_{j}}\right)}\;.
\label{prim2}
\end{equation}
The further simplification $m=2,\,s=ap_1p_2$ makes it possible to
verify the partition identity
\begin{equation}gin{eqnarray}
W(a p_1p_2,\{p_1,p_2\})=a+1\;,
\label{aa1}
\end{eqnarray}
which follows from the recursion relation (\ref{SW_recursion}) for
the restricted partition function and its definition
\begin{equation}gin{eqnarray}
&&W(ap_1p_2,\{p_1,p_2\}) - W(ap_1p_2-p_1,\{p_1,p_2\}) =
W(ap_1p_2,\{p_2\})\;,\nonumber\\
&&W(ap_1p_2,\{p_2\})=1\;,\;\;\;W(ap_1p_2-p_1,\{p_1,p_2\})=
W((a-l)p_1p_2+(lp_2-1)p_1,\{p_1,p_2\})=a\;,\nonumber
\end{eqnarray}
where $a$ solutions of the Diophantine equation
$p_1X+p_2Y=(a-l)p_1p_2+(lp_2-1)p_1$ correspond to $l=1,\ldots,a$.
The relation (\ref{aa1}) has an important geometrical interpretation, namely,
a line $p_1X+p_2Y=ap_1p_2$ in the $XY$ plane
passes exactly through $a+1$ points with non-negative integer
coordinates.
The verification of (\ref{aa1}) is straightforward
(see Appendix B for details):
\begin{equation}gin{eqnarray}
&&W_1(ap_1p_2,\{p_1,p_2\})=a+\frac{1}{2}\left(\frac{1}{p_1}+
\frac{1}{p_2}\right)\;,\nonumber\\
&&W_{p_1}(ap_1p_2,\{p_1,p_2\})=\frac{1}{2}-\frac{1}{2p_1}\;,\;\;\;
W_{p_2}(ap_1p_2,\{p_1,p_2\})=\frac{1}{2}-\frac{1}{2p_2}\;,
\label{sylvp1p2}
\end{eqnarray}
which produces the required result.
A generalization of (\ref{aa1}) is possible using the explicit form of
the partition function
\begin{equation}
W(s,\{p_1,p_2\}) =
\frac{1}{p_1p_2}\left(s+\frac{p_1+p_2}{2}\right)+
\frac{1}{p_1} \sum_{\rho_{p_1}} \frac{\rho_{p_1}^{-s}}{1-\rho_{p_1}^{p_2}} +
\frac{1}{p_2} \sum_{\rho_{p_2}} \frac{\rho_{p_2}^{-s}}{1-\rho_{p_2}^{p_1}}.
\label{2primes}
\end{equation}
Setting here $s=ap_1p_2+b, \, 0 \le b < p_1p_2$ and
noting that the value of two last terms in (\ref{2primes})
don't depend on the integer $a$, one can easily see that
\begin{equation}
W(ap_1p_2+b,\{p_1,p_2\}) = a + W(b,\{p_1,p_2\}),
\label{reduct}
\end{equation}
which reduces the procedure to computation of the first $p_1p_2$
values of $W(s,\{p_1,p_2\})$. Recalling that $W(0,\{p_1,p_2\}) = 1$ we
immediately recover (\ref{aa1}) as a particular case of (\ref{reduct}).
\section{Recursive Relation for Sylvester Waves}
\label{recurs}
In this Section we prove that the recursive relation similar to
(\ref{SW_recursion}) holds not only for the entire partition function
$W(s,{\bf d}^m)$ and its polynomial part $W_1(s,{\bf d}^m)$ but also
for each Sylvester wave
\begin{equation}
W_j(s,{\bf d}^m) - W_j(s-d_m,{\bf d}^m) =
W_j(s,{\bf d}^{m-1})\;.
\label{SWj_recursion}
\end{equation}
When $j$ is not a divisor of $d_m$, the weight $\omega_j$ doesn't change
in transition from ${\bf d}^{m-1}$ to ${\bf d}^{m}$. Denoting for brevity
$$
A(s) = s+s_{m-1} + \sum_{i=1}^{\omega_j}d_i \;{}^i\! B + \!\!\!
\sum_{i=\omega_j+1}^{m-1} \!\! d_i \;{}^i\! H(\rho_j^{d_i})\;, \ \
B_{\omega_j} = \frac{1}{(\omega_j-1)! \; \pi_{\omega_j}}\;,
$$
we have
\begin{equation}a
W_j(s,{\bf d}^m) & = &
B_{\omega_j}
\sum_{\rho_j}
\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
\left(
A(s) + d_m[1 + H(\rho_j^{d_m})]
\right)^{\omega_j-1} \nonumber \\
&=&
B_{\omega_j}
\sum_{\rho_j}
\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
\sum_{l=0}^{\omega_j-1}
\binom{\omega_j-1}{l} A^{\omega_j-1-l}(s) d_m^l
[1 + H(\rho_j^{d_m})]^l\;. \nonumber
\end{equation}a
Now using (\ref{EulerSymbolic}) we have
\begin{equation}a
W_j(s,{\bf d}^m) & = &
B_{\omega_j}
\sum_{\rho_j}
\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
\left\{A^{\omega_j-1}(s) + \rho_j^{d_m} \sum_{l=1}^{\omega_j-1}
\binom{\omega_j-1}{l} A^{\omega_j-1-l}(s) d_m^l H_l(\rho_j^{d_m})
\right\} \nonumber \\
&=&
B_{\omega_j}
\sum_{\rho_j}\frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
\left\{(1-\rho_j^{d_m})A^{\omega_j-1}(s) + \rho_j^{d_m}\left(
A(s) + d_m H(\rho_j^{d_m})\right)^{\omega_j-1}\right\} \nonumber \\
&=&
B_{\omega_j}
\sum_{\rho_j}
\frac{\rho_j^{-(s-d_m)}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
\left(
A(s) + d_m H(\rho_j^{d_m})
\right)^{\omega_j-1}
+
B_{\omega_j}
\sum_{\rho_j}
\frac{\rho_j^{-s}A^{\omega_j-1}(s)}
{\prod_{i=\omega_j+1}^{m-1} (1-\rho_j^{d_i})}
\nonumber \\
&=&
W_j(s-d_m,{\bf d}^m) +
W_j(s,{\bf d}^{m-1})\;.
\end{equation}a
In case of $j$ being divisor of $d_m$ the weight of $j$
for the set ${\bf d}^{m-1}$
is equal to $\omega_j-1$, and we have
\begin{equation}
W_j(s,{\bf d}^{m-1}) =
\frac{(\omega_j-1) d_m}{(\omega_j-1)! \; \pi_{\omega_j}}
\sum_{\rho_j}
\frac{\rho_j^{-s}}{\prod_{i=\omega_j}^{m-1} (1-\rho_j^{d_i})}
\left(
s+s_{m-1} + \sum_{i=1}^{\omega_j-1} d_i \;{}^i\! B + \!\!\!
\sum_{i=\omega_j}^{m-1} \!\!\! d_i \;{}^i\! H(\rho_j^{d_i})
\right)^{\omega_j-2} \!\!\!\!\!\!\!\!\;.
\label{Wm-1}
\end{equation}
Denoting
$$
A(s) = s+s_{m-1} + \sum_{i=1}^{\omega_j-1} d_i \;{}^i\! B + \!\!
\sum_{i=\omega_j}^{m-1} \! d_i \;{}^i\! H(\rho_j^{d_i}), \ \ \
D(s,\rho_j) = \frac{\rho_j^{-s}}{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}\;,
$$
and using the symbolic formula for the Bernoulli numbers
\cite{NorlundMemo}
$$
(B+1)^n = B^n = B_n \ \ (n \ne 1)\;,
$$
we obtain
\begin{equation}a
W_j(s,{\bf d}^m) & = &
B_{\omega_j} \sum_{\rho_j}
D(s,\rho_j)
[A(s) + d_m(B+1)]^{\omega_j-1} \nonumber \\
&=&
B_{\omega_j}
\sum_{\rho_j}
D(s,\rho_j)
\sum_{l=0}^{\omega_j-1}
\binom{\omega_j-1}{l} A^{\omega_j-1-l}(s) d_m^l
(B+1)^l \\
&=&
B_{\omega_j}
\sum_{\rho_j}
D(s,\rho_j)
[A(s) + d_m B]^{\omega_j-1}
+
B_{\omega_j} d_m (\omega_j-1)
\sum_{\rho_j}
D(s,\rho_j) A^{\omega_j-2}(s)
\nonumber \\
&=&
W_j(s-d_m,{\bf d}^m) +W_j(s,{\bf d}^{m-1})\;, \nonumber
\end{equation}a
which completes the proof.
\section{Partition function $W\left(s,\{\overline{m}\}\right)$ for a set
of natural \\numbers \label{Sm}}
Sylvester waves for a set of consecutive natural numbers
$\{1,2,\dots,m\}=\{\overline{m}\}$ was under special consideration in
\cite{Rama}. An importance of this case based on its relation to the
invariants of symmetric group $S_m$ (see next Section) and, second,
$W(s,\{\overline{m}\})$ form a natural basis to utilize the
partition functions for every subsets of $\{1,2,\dots,m\}$. This case is
also important due to the famous Rademacher formula \cite{Radem37} for
{\it unrestricted partition function} $W(s,\{\overline{s}\})$, but
the latter already belongs to the analytical number theory.
The representation for $W(s,\{\overline{m}\})$ in terms of higher
Bernoulli polynomials comes when we put into (\ref{WBern})
\begin{equation}gin{equation}
\omega_j=\left[\frac{m}{j}\right]\;,\;\;\;\;
\pi_{\omega_j}=\omega_j!\;j^{\omega_j}\;,\;\;\;\;
s_{\omega_j}=\frac{\omega_j(\omega_j+1)}{2}\;,
\label{symmet}
\end{equation}
where $[x]$ denotes integer part of $x$. The partition function in this
case reads
\begin{equation}a
W(s,\{\overline{m}\}) & = & \sum_{j=1}^m
\frac{j^{-\omega_j}}{(\omega_j-1)!\; \omega_j!}
\sum_{n=0}^{\omega_j-1} \binom{\omega_j-1}{n}
B^{(\omega_j)}_n\left(s+\frac{m(m+1)}{2}|{\bf d}^{\omega_j}\right)
\times \nonumber \\
&& \sum_{\rho_j} \frac{\rho_j^{-s}}
{\prod_{i=\omega_j+1}^m (1-\rho_j^{d_i})}
H_{\omega_j-1-n}^{(m-\omega_j)}[\rho_j|{\bf d}^{m-\omega_j}]\;,
\label{WSmWODenom}
\end{equation}a
where ${\bf d}^{\omega_j} = j\{\overline{\omega}\}$, so that for each $j$
we have elements
divisible by $j$ at first $\omega_j$ positions. The expression for the
Sylvester wave of the maximal period $m$ looks particularly simple
\begin{equation}
W_m(s,\{\overline{m}\}) = \frac{1}{m^2} \sum_{\rho_m}
\rho_m^{-s}\;.
\label{WmSm}
\end{equation}
The straightforward calculations show that the expression (\ref{WSmWODenom})
produces exactly the same formulas for $m=1,2,\ldots,12$ which were obtained
in \cite{Rama}.
\begin{equation}gin{figure}[t]
\psfig{figure=./grW21two.eps,width=6in}
\caption{Plots of the partition function $W(s,\{\overline{21}\})$ ({\it
black curve}) and its first Sylvester wave $W_1(s,\{\overline{21}\})$
({\it white curve}) showing that the polynomial part provides an
important information about the partition function behavior.}
\label{W21approx}
\end{figure}
It needs to be noted that typically the argument $s$ in all formulas
derived above is assumed to have integer values, but it is obvious that all
results can be extended to real values of $s$, though such extension is not
unique.
Continuous values of the argument provide a convenient way to analyze the
behavior of the partition function and its waves. In this work we choose
the natural extension scheme based on the trigonometric functions
$$
\rho_j^s = e^{2 \pi i n s/j} = \cos \frac{2 \pi n s}{j} +
i \sin \frac{2 \pi n s}{j}.
$$
We finish this Section with a brief discussion of a phenomenon better
observed in graphics of $W(s,\{\overline{m}\})$
with large $m$ rather from the explicit expressions (see formulas (52) and
Figures of restricted partition functions in \cite{Rama}).
In the range $[-\frac{m(m+1)}{2},0]$ where $W(s,\{\overline{m}\})$ has
all its zeroes, one can easily assume an existence
of a function $\widetilde{W}(s,\{\overline{m}\})$ which envelopes
$W(s,\{\overline{m}\})$ or approximates it in some sense. The
decomposition of $W(s,\{\overline{m}\})$ into the Sylvester waves shows
that this role may be assigned to the wave $W_1(s,\{\overline{m}\})$. The
Figures \ref{W21approx}, \ref{W21diff} show that $W_1(s,\{\overline{21}\})$
serves as a good approximant for $W(s,\{\overline{21}\})$ in this range
as well as for large $s$.
\begin{equation}gin{figure}[t]
\psfig{figure=./WS21diff0.eps,width=6in}
\caption{Plot of the normalized difference
$[W(s,\{\overline{21}\})/W_1(s,\{\overline{21}\})-1]$
showing that the polynomial
part $W_1(s,\{\overline{21}\})$ at large values of the argument $s$ gives
a very accurate approximation to the partition function
$W(s,\{\overline{21}\})$.}
\label{W21diff}
\end{figure}
\section{Application to invariants of finite groups}
\label{finitegroup}
The restricted partition function $W(s,{\bf d}^m)$ has a strong
relationship to the invariants of finite reflection groups $G$
acting on the vector space $V$ over the field of complex numbers. If $M^G(t)$
is a Molien function of the finite group, $d_{r}$ and $m$ are degrees
and a number of the basic homogeneous invariants respectively, then its
series expansion
in $t$ gives a number $P(s,G)$ of
algebraically independent invariants of the degree $s$. The set of natural
numbers $\{\overline{m}\}$ corresponds to the symmetric group
$S_m:\;W(s,\{\overline{m}\})=P(s,S_m)$. The list of $P(s,G)$ for all
indecomposable reflections groups $G$ acting over the field of real
numbers and known as {\it Coxeter groups} is presented in \cite{Rama}. It
is easy to extend these formulas over indecomposable pseudoreflections
groups acting over the field of complex numbers using the list of 37
groups given by Shepard and Todd \cite{Shepar54}. In this Section we
extend the results of Section \ref{j} to all finite groups.
First, we recall an algebraic setup of the problem.
The fundamental problem of the invariant theory consists in determination of an
algebra ${\sf R}^G$ of invariants. Its solution is given by the
Noether theorem \cite{Benson93}: ${\sf R}^G$ is generated by a
polynomial $\vartheta_k(x_j)$ as an algebra due to action of finite group
$G\subset GL(V^q)$ on the $q$-dimensional vector space $V^q(x_j)$
over the field of complex numbers by not more than ${|G|+q\choose q}$
homogeneous invariants, of degrees not exceeding the order $|G|$ of group
\begin{equation}
k\leq {|G|+q\choose q}\;,\;\;\;j\leq \dim V^q=q\;,\;\;\;
\deg \vartheta_k(x_j)\leq |G|\;.
\label{molien0}
\end{equation}
To enumerate the invariants explicitly, it is convenient to classify
them by their degrees (as polynomials). A classical theorem of
Molien \cite{Benson93} gives an explicit expression for a number $P(s,G)$
of all homogeneous invariants of degree $s$
\begin{equation}
M^G(t)=\frac{1}{|G|}\sum_{l=1}^{|G|}
\frac{\widetilde{\chi}({\widehat g}_l)}
{\det({\hat I} -t\; {\widehat g}_l)}=
\sum_{s=0}^{\infty} P(s,G) t^s\;,\;\;\;\;P(0,G)=1\;,
\label{molien1}
\end{equation}
where ${\widehat g}_l$ are non--singular $(n\times n)$--permutation
matrices with entries, which form the regular representation of $G$,
${\widehat I}$ is the identity matrix and $\widetilde{\chi}$ is the
complex conjugate to character $\chi$. The further progress is due to
Hilbert and his {\it syzygy theorem} \cite{Benson93}. For our purpose
it is important that $M^G(t)$ is a rational polynomial
\begin{equation}
M^G(t)=\frac{N^G(t)}{\prod_{l=1}^n \left(1-t^{d_l}\right)}\;,\;\;\;\;
N^G(t)=\sum_{k=0}Q(k,G)\;t^k.
\label{molien2a}
\end{equation}
The formula (\ref{molien2a}) is very convenient to express the function
$P(s,G)$ through the Sylvester waves $W(s,{\bf d}^m)$. Recalling the
definition (\ref{genfunc}) of the generating function $F(t,{\bf d}^m)$
consider a general term $t^k F(t,{\bf d}^m)$ of the Molien function
(\ref{molien2a})
\begin{equation}a
t^k F(t,{\bf d}^m) = \sum_{s=0}^{\infty} W(s,{\bf d}^m) t^{s+k} =
\sum_{s=k}^{\infty} W(s-k,{\bf d}^m) t^{s},
\label{gen_term_Molien}
\end{equation}a
so that the corresponding partition function is $W(s-k,{\bf d}^m)$, which
implies that the number $P(s,G)$ of all homogeneous invariants of degree
$s$ for the finite group $G$ can be expressed through the simple relation
\begin{equation}
P(s,G)=\sum_{k=0}^{s}Q(k,G) W(s-k,{\bf d}^m)\;.
\label{fin2}
\end{equation}
We consider several instructive examples for which the explicit
expression of the Molien function $M^G(t)$ and the corresponding number of
homogeneous invariants $P(s,G)$ are given.
1. Alternating group ${\sf A}_n$ generated by its natural $n$--dimensional
representation, $|{\sf A}_n|=n!/2$.
\begin{equation}a
M_{{\sf A}_n}(t) & = & \left[1+t^{\binom{n}{2}}\right]
\prod_{k=1}^n\frac{1}{1-t^k}\;. \nonumber \\
P(s,{\sf A}_n) & = &
W(s,\{\overline{n}\}) + W\left(s-\frac{n(n-1)}{2},\{\overline{n}\}\right).
\label{altern}
\end{equation}a
The group ${\sf A}_n$ is acting on Euclidean vector space ${\mathbb R}^n$.
2. Group ${\sf G}_2$ generated by matrix {\footnotesize $
\left(\begin{equation}gin{array}{cc}
\rho_{n} & 0 \\
0 & \rho_{n}^{-1}\end{array}\right)$}, where $\rho_{n}=e^{2\pi i/n}$ is
a primitive $n$--th root of unity, $|{\sf G}_2|=n$.
\begin{equation}a
M_{{\sf G}_2}(t)&=&
\frac{1+t^n}{(1-t^2)(1-t^n)}\;. \nonumber \\
P(s,{\sf G}_2) & = &
W(s,\{2,n\}) + W(s-n,\{2,n\}).
\label{rotation}
\end{equation}a
${\sf G}_2$ is isomorphic as an abstract group to the cyclic group ${\sf
Z}_n$ acting on Euclidean vector space ${\mathbb R}^2$.
3. Group ${\sf G}_3$ generated by the matrices {\footnotesize
$\left(\begin{equation}gin{array}{cc}
\rho_{n} & 0 \\
0 & \rho_{n}^{-1}\end{array}\right)$} and {\footnotesize
$\left(\begin{equation}gin{array}{cc}
0 & 1 \\
1 & 0\end{array}\right)$}, $|{\sf G}_3|=2n$.
\begin{equation}a
M_{{\sf G}_3}(t)=\frac{1}{(1-t^2)(1-t^n)}\;,\;\;\;
P(s,{\sf G}_3)=W(s,\{2,n\})
\label{dihedr}
\end{equation}a
${\sf G}_3$ is isomorphic as an abstract group to the dihedral group ${\sf
I}_n$ acting on Euclidean vector space ${\mathbb R}^2$.
4. Group ${\sf G}_4$ generated by $(n\times n)$--diagonal matrix
{\sf diag}$(-1,-1,\dots,-1)$, $|{\sf G}_4|=2$.
\begin{equation}a
M_{{\sf G}_4}(t)&=&
\frac{1}{(1-t^2)^n}\sum_{k=0}^{\left[\frac{n}{2}\right]}
\binom{n}{2k}t^{2k}\;. \nonumber \\
P(s,{\sf G}_4) & = &
\left\{ \begin{equation}gin{array}{ll}
\sum_{k=0}^{\left[\frac{n}{2}\right]} \binom{n}{2k}
W(s-2k,{\bf 2}^n) = W(s,{\bf 1}^n),
& s=0 \pmod 2,\\
0, & s \ne 0 \pmod 2.
\end{array}\right.
\label{groupG}
\end{equation}a
${\sf G}_4$ is isomorphic as an abstract group to the cyclic group ${\sf
Z}_2$ acting on Euclidean vector space ${\mathbb R}^n$.
It is easy to see that both groups ${\sf G}_2$ and ${\sf G}_4$
acting on ${\mathbb R}^2$ give rise to the same Molien function and
corresponding number of invariants
\begin{equation}a
M_{{\sf Z}_2}(t)=\frac{1+t^2}{(1-t^2)^2}\;,\;\;\;
P(s,{\sf Z}_2)=\left\{ \begin{equation}gin{array}{ll}
W(s,{\bf 1}^2),
& s=0 \pmod 2,\\
0, & s \ne 0 \pmod 2.
\end{array}\right.
\label{groupPR}
\end{equation}a
5. Group ${\sf Q}_{4n}$ generated by the matrices {\footnotesize
$\left(\begin{equation}gin{array}{cc}
\rho_{2n} & 0 \\
0 & \rho_{2n}^{-1}\end{array}\right)$} and {\footnotesize
$\left(\begin{equation}gin{array}{cc}
0 & i \\
i & 0\end{array}\right)$}, $|{\sf Q}_{4n}|=4n$.
\begin{equation}a
M_{{\sf Q}_{4n}}(t) &= &
\frac{1+t^{2n+2}}{(1-t^4)(1-t^{2n})}\;,
\label{groupQ4n} \\
P(s,{\sf Q}_{4n})&=&\left\{ \begin{equation}gin{array}{ll}
W(\frac{s}{2},\{2,n\}) + W(\frac{s}{2}-n-1,\{2,n\}),
& s=0 \pmod 2,\\
0, & s \ne 0 \pmod 2. \nonumber
\end{array}\right.
\end{equation}a
In the case of quaternion group ${\sf Q}_8$ formula (\ref{groupQ4n}) is
reduced to
\begin{equation}a
M_{{\sf Q}_8}(t) &= & \frac{1+t^6}{(1-t^4)^2}\;,\;\;\;
P(s,{\sf Q}_8) =\left\{ \begin{equation}gin{array}{ll}
W(s,{\bf 1}^2)/2,
& s=0 \pmod 4,\\
0, & s \ne 0 \pmod 4.
\end{array}\right.
\label{groupQ8}
\end{equation}a
More sophisticated examples of the finite groups one can find in Appendices A, B
of the book \cite{Benson93}.
\section{Conclusion}
1. The explicit expression for restricted partition function $W(s,{\bf
d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ ({\em Sylvester
waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, \ldots,
d_m\}$ is derived. The formulas are represented as a finite sum over
Bernoulli and Euler polynomials of higher order with periodic
coefficients.
\noindent
2. Every Sylvester wave $W_j(s,{\bf d}^m)$ satisfies the same
recursive relation as the whole partition function $W(s,{\bf d}^m)$.
\noindent
3. The application of restricted partition function to the problem of
counting all algebraically independent invariants of the degree $s$
which arise due to action of finite group $G$ on the vector space $V$ over
the field of complex numbers is discussed.
\section*{Appendices}
\appendix
\renewcommand{\thesection\arabic{equation}}{\thesection\arabic{equation}}
\section{Symbolic Notation \label{appendix1}}
\setcounter{equation}{0}
The symbolic technique for manipulating sums with binomial coefficients by
expanding
polynomials and then replacing powers by subscripts was developed in
nineteenth century by Blissard.
It has been known as symbolic notation and the classical umbral
calculus \cite{Roman1978}. This notation can be used \cite{Gessel} to
prove
interesting formulas not easily proved by other methods.
An example of this notation is also found in \cite{bat53} in
section devoted to the Bernoulli polynomials $B_k(x)$.
The well-known formulas
$$
B_n(x+y) = \sum_{k=0}^{n} {n\choose k} B_k(x) y^{n-k}, \ \
B_n(x) = \sum_{k=0}^{n} {n\choose k} B_k x^{n-k},
$$
are written
symbolically as
$$
B_n(x+y) = (B(x)+y)^n, \ \ B_n(x) = (B+x)^n.
$$
After the expansion the exponents of $B(x)$ and $B$ are converted into the
orders of the Bernoulli polynomial and the Bernoulli number, respectively:
\begin{equation}
[B(x)]^k \Rightarrow B_k(x), \ \ \ B^k \Rightarrow B_k.
\label{conv_rule}
\end{equation}
We use this notation in its extended version suggested in
\cite{NorlundMemo} in order to make derivation more clear and
intelligible.
N\"orlund introduced the Bernoulli polynomials of higher order defined
through
the recursion
\begin{equation}
B_{n}^{(m)}(x|{\bf d}^m) =
\sum_{k=0}^n \binom{n}{k} d^k B_k(0) B_{n-k}^{(m-1)}(x|{\bf d}^{m-1}),
\label{Bern_poly_HO_def}
\end{equation}
starting from $B_{n}^{(1)}(x|d_1) = d_1^n B_n(\frac{x}{d_1})$.
In symbolic notation it takes form
$$
B_{n}^{(m)}(x) =
\left(
d_m B(0) + B^{(m-1)}(x)
\right)^n,
$$
and recursively reduces to more symmetric form
\begin{equation}
B_{n}^{(m)}(x|{\bf d}^m) =
\left(
x + d_1 \;{}^1\! B(0) +
d_2 \;{}^2\! B(0) + \ldots + d_m \;{}^m\! B(0)
\right)^n =
\left(
x + \sum_{i=1}^m d_i \;{}^i\! B(0)
\right)^n,
\label{Bern_poly_HO_symm}
\end{equation}
where each $[{}^i \! B(0)]^k$ is converted into $B_k(0)$.
\label{appendix2}
\section{Partition function for two primes}
\setcounter{equation}{0}
The polynomial part is computed according to (\ref{W_1})
\begin{equation}
W_1(ap_1p_2,\{p_1,p_2\}) =
\frac{1}{p_1p_2}
B_{1}^{(2)}(ap_1p_2 + p_1+p_2 |
\{p_1,p_2\})=a+\frac{1}{2}\left(\frac{1}{p_1}+
\frac{1}{p_2}\right)\;.
\label{w1p1p2}
\end{equation}
Two other waves read
\begin{equation}
W_{p_1}(ap_1p_2,\{p_1,p_2\})=\frac{1}{p_1} \sum_{r=1}^{p_1-1}
\frac{1}{1-\rho_{p_1}^{r}}\;,\;\;\;
W_{p_2}(ap_1p_2,\{p_1,p_2\})=\frac{1}{p_2} \sum_{r=1}^{p_2-1}
\frac{1}{1-\rho_{p_2}^{r}}\;.
\label{w12}
\end{equation}
where we use trivial identity $\rho_{p_1}^{ap_1p_2}=\rho_{p_2}^{ap_1p_2}=1$.
Computation of the sums in (\ref{w12})
we start with the identity (see \cite{Vandiver1942})
\begin{equation}
\prod_{r=0}^{m-1} (x-\rho_m^r) = x^m-1,
\label{identity1}
\end{equation}
and differentiation it with respect to $x$, and division by $x^m-1$
\begin{equation}
\sum_{r=0}^{m-1} \frac{1}{x-\rho_m^r} = \frac{mx^{m-1}}{x^m-1}.
\label{identity1diff1}
\end{equation}
Subtracting $1/(x-1)$ from both sides of (\ref{identity1diff1}) and taking a
limit at $x \rightarrow 1$ we obtain
\begin{equation}
\sum_{r=1}^{m-1} \frac{1}{1-\rho_m^r} = \frac{m-1}{2}.
\label{form2}
\end{equation}
Using this result we have for the periodic waves in (\ref{w12})
\begin{equation}
W_{p_1}(ap_1p_2,\{p_1,p_2\})=\frac{p_1-1}{2p_1}\;,\;\;\;
W_{p_2}(ap_1p_2,\{p_1,p_2\})=\frac{p_2-1}{2p_2}\;.
\label{w12f}
\end{equation}
\section*{Acknowledgment}
We thank I. M. Gessel for information about Ref. \cite{Gessel}.
The research was supported in part (LGF) by the Gileadi Fellowship
program of the Ministry of Absorption of the State of Israel.
\begin{equation}gin{thebibliography}{99}
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G. E. Andrews, {\it The Theory of Partitions}, \\Encyclopedia of Mathematics and
its Applications, V.2, Addison--Wesley, 1976.
\bibitem{bat53}
H. Bateman and A. Erdel\'yi, {\it Higher Transcendental Functions}, V.1,
\\ McGraw-Hill Book Co., NY, 1953.
\bibitem{Beck}
M. Beck, I. M. Gessel and T. Komatsu, {\it The Polynomial Part of a Restriction
Partition Function Related to the Frobenius Problem}, \\
The Electronic Journal of Combinatorics, {\bf 8}, N7 (2001), 1-5.
\bibitem{Benson93}
D. J. Benson, {\it Polynomial Invariants of Finite Groups},\\
Cambridge Univ. Press, Cambridge, 1993.
\bibitem{Carlitz1960}
L. Carlitz, {\it Eulerian Numbers and Polynomials of Higher Order},\\
Duke Mathematical Journal, {\bf 27} (1960), 401-423.
\bibitem{catal838}
L. E. Dickson, {\it History of the Theory of Numbers},\\
Chelsea Pub. Co., NY, {\bf 2}, 1952, p.114.
\bibitem{NorlundMemo}
N. E. N\"orlund, {\it M\'emoire sur les Polynomes de Bernoulli},\\
Acta Mathematica, {\bf 43} (1922), 121-196.
\bibitem{Rama}
L. G. Fel and B. Y. Rubinstein, {\it Sylvester Waves in the Coxeter Groups}, \\
Ramanujan Journal, {\bf 6} (2002), 307-329.
\bibitem{Frobenius}
F. G. Frobenius, {\it \"Uber die Bernoullischen Zahlen und die Eulerischen
Polynome},\\ Sitzungsberichte der K\"oniglich
Preu\ss ischen Akademie der Wissenschaften zu Berlin (1910), 809-847.
\bibitem{Gessel}
I. M. Gessel, {\it Applications of the Classical Umbral Calculus}, \\
to appear in Algebra Universalis.
\bibitem{Radem37}
H. Rademacher, {\it On the Partition Function p(n)},\\
Proc. London Math. Soc. {\bf 43} (1937), p. 241-254.
\bibitem{Roman1978}
S. Roman and G.-C. Rota, {\it The Umbral Calculus}, \\
Adv. Math. {\bf 27} (1978), 95-188.
\bibitem{Shepar54}
G. C. Shepard and J. A. Todd, {\it Finite Unitary Reflection
Groups}, \\ Canad. J. Math., {\bf 6} (1954), 274-304.
\bibitem{stanl79}
R. P. Stanley, {\it Invariants of Finite Groups and
Their Applications to Combinatorics}, \\
Bulletin of Amer. Math. Soc., {\bf 1} (1979), 475-511.
\bibitem{Sylv1}
J. J. Sylvester, {\it On the Partition of Numbers}, \\
Quarterly Journal of Mathematics {\bf 1} (1857), 141-152.
\bibitem{Sylv2}
J. J. Sylvester, {\it On Subinvariants, i.e. Semi-invariants to Binary Quantics
of an
Unlimited Order. With an Excursus on Rational Fractions and Partitions},\\
American Journal of Mathematics {\bf 5} (1882), 79-136.
\bibitem{Vandiver1942}
H.S. Vandiver, {\it An Arithmetical Theory of the Bernoulli Numbers}, \\
Trans. Amer. Math. Soc., {\bf 51} (1942), 502-531.
\end{thebibliography}
\end{document}
\pagestyle{empty}
Author Affiliations:
Boris Y. Rubinstein,\\
Department of Mathematics\\
University of California, Davis\\
One Shields Dr. \\
Davis, CA 95616, U.S.A.
\vskip0.5cm
Leonid G. Fel\\
Department of Civil and Environmental Engineering,\\
Technion \\
Haifa 32000, Israel \\
\vskip1cm
The research was supported in part (LGF) by the Gileadi Fellowship
program of the Ministry of Absorption of the State of Israel.
\vskip1cm
2000 Mathematics Subject Classification:\\
Primary -- 11P81; Secondary -- 11B68, 11B37.
\vskip1cm
Key words: restricted partitions, Bernoulli polynomials of higher order,
Euler polynomials of higher order, recursive relation.
Contact author
\\
\\
Dr. Boris Rubinstein\\
Dept. of Mathematics\\
University of California at Davis,\\
One Shield Drive\\
Davis, CA 95616\\
USA\\
\\
tel.: (530)-400-6910\\
fax: (530)-752-6635\\
e-mail: [email protected]
\centerline{\bf Figure Captions}
Fig.1. Plots of the partition function $W(s,\{\overline{21}\})$ ({\it
black curve}) and its first Sylvester wave $W_1(s,\{\overline{21}\})$
({\it white curve}) showing that the polynomial part provides an
important information about the partition function behavior.
\vskip1cm
Fig.2. Plot of the normalized difference
$[W(s,\{\overline{21}\})/W_1(s,\{\overline{21}\})-1]$
showing that the polynomial
part $W_1(s,\{\overline{21}\})$ at large values of the argument $s$ gives
a very accurate approximation to the partition function
$W(s,\{\overline{21}\})$.
\begin{equation}gin{figure}[h]
\psfig{figure=./grW21two.eps,width=6in}
\end{figure}
\vskip10cm
\begin{equation}gin{figure}[b]
\psfig{figure=./WS21diff0.eps,width=6in}
\end{figure}
\end{document} |
\begin{document}
\title{On the ruin time distribution for a Sparre Andersen process with exponential claim sizes}
\author{Konstantin A Borovkov\footnote{Research supported by the ARC Centre
of Excellence for Mathematics and Statistics of Complex Systems.} \ and David C M
Dickson}
\date{}
\maketitle
\begin{abstract}
We derive a closed-form (infinite series) representation for the distribution of the
ruin time for the Sparre Andersen model with exponentially distributed claims. This
extends a recent result of Dickson et al.~\cite{DiHuZh05} for such processes with Erlang
inter-claim times. We illustrate our result in the cases of gamma and mixed exponential
inter-claim time distributions.
\end{abstract}
\noindent{\em Keywords}: Sparre Andersen model; time of ruin; exponential claims.
\noindent{\em 2000 Mathematics Subject Classification}: Primary 91B30; 60K10, 60G51.
\section{Introduction}
In the Sparre Anderson model, the (continuous-time) surplus process $\{U(t)\}_{t\ge 0}$
has the form
\[
U (t) = u + ct - \sum_{j\le N(t)} X_j,
\]
where $u\ge0$ is the initial surplus, $c>0$ is the premium rate, and $\{N(t)\}_{t\ge 0}$ is a
delayed renewal process generated by a sequence of inter-claim times $\{T_j\}_{j\ge 0}$:
\[
N(t) = \inf\{ j \ge 0: \, T_0 +\cdots + T_j \ge t\},
\]
and $\{X_j\}_{j\ge 1}$ is the sequence of claim sizes (so that a claim of size $X_1$ is
made at time $T_0$, etc). We assume that the random variables from the above sequences
are jointly independent, with $\{T_j\}_{j\ge 1}$ and $\{X_j\}_{j\ge 1}$ being
i.i.d.\ sequences. The goal of the present note is to derive an explicit formula for the
distribution of the ruin time
\[
\tau = \inf\{ t>0: \, U(t) <0\}
\]
in the special case when the $X_j$'s follow the exponential distribution.
When claims occur according to a Poisson process and the claim size distribution is exponential, a solution for the distribution of the ruin time $\tau$
has been known for many years. See, for example, \cite{Asm00}, \cite{DreWil03} and \cite{Seal78} for different solutions to this problem.
In the recent paper \cite{DiHuZh05}, the authors
used analytical techniques to obtain an explicit formula for the density of $\tau$ in
the case when the $T_j$'s have an Erlang distribution. In the present note, we present
an alternative probabilistic method, which enables one to derive such an explicit
formula in the more general case when the $T_j$'s follow an arbitrary distribution.
\section{The main result}
We assume that the claim sizes $X_j$ follow the exponential distribution with
parameter~$\lambda>0$:
\begin{equation}
\mathbb{P} (X_j > x)=e^{-\lambda x },\qquad x\ge 0,
\label{ExpX}
\end{equation}
while the positive random variables $T_0$ and $T_1 (\deq T_j$, $j>1)$ have densities
$f_0(t)$ and $f(t)$, respectively. By $g*h$ we denote the convolution of the functions
$g,h$ defined on $(0,\infty)$:
\[
(g*h) (t) = \int_0^t g(t-v) h(v) dv,
\]
and by $g^{*n}= g^{*(n-1)} *g,$ $n\ge 2$, the $n$-fold convolution of $g$ with itself.
\begin{theorem}
Under the above assumptions, the ruin time $\tau$ has a (defective) density $p_\tau (t)$ given
by
\begin{multline}
p_\tau (t) = e^{-\lambda (u+ct)} \biggl\{ f_0(t)
\\
+
\sum_{n=1}^\infty \frac{\lambda^n (u+ct)^{n-1}}{n!} \bigl[ u (f^{*n} * f_0 )(t)
+ c(f^{*n} * f_1 ) (t)\bigr]\biggr\},
\label{Main}
\end{multline}
where $f_1(t) = tf_0(t).$
\end{theorem}
\mathbb{P}oof{} The idea of the proof is similar to the one used in~\cite{Bo85}: first we will
translate our problem into the problem of the crossing of a linear boundary by the pure jump process
$U^0 (t) =U(t)-ct$ and then swap the roles of the time and space coordinates. Then we
notice that the generalised inverse of the function $U^0(t)$ is nothing else but the
trajectory of a compound Poisson process. Eventually, the original problem proves to be
equivalent to finding the distribution of the hitting time of a level by a skip-free
L\'evy process, of which the solution is well-known and is given by Kendall's identity (see
e.g. \S\,12, Theorem~1 in~\cite{Bo72}, or \cite{BoBu01}).
(i)~We will assume in parts (i)-(ii) of the proof that $T_0\equiv v=\text{const}$ (which
is equivalent to conditioning on $T_0$, but is more convenient from a notation
viewpoint).
As we have just said, it is easily seen that, for the pure jump process
\[
U^0 (t) =U(t)-ct \equiv u - \sum_{j\le N(t)} X_j,
\]
one has
\[
\tau = \inf\{ t>0: \, U^0 (t) - (-ct) <0\}.
\]
Next we `translate' the origin to the point $(v,u)$ and swap the roles of coordinates by
introducing the new `time' $s=u-x$ and `space' $y=t-v$ (where $t$ and $x$ respectively represent the original time and space). In the new system of
coordinates, the trajectory of our process $\{U^0 (t)\}$ is again a pure step function,
which starts at zero at `time' $s=0$ and has jumps of sizes $T_1, T_2,T_3,\dots,$ at
`times' $X_1, X_1+ X_2, X_1+ X_2 +X_3,\dots.$ Due to our assumption~\eqref{ExpX}, this
will be a trajectory of the compound Poisson process
\[
Z^0 (s) = \sum_{k\le M(s)} T_k,
\]
where
\[
M(s) = \inf\{ k\ge 1:\, X_1 +\cdots +X_k > s\} -1
\]
is a Poisson process with rate $\lambda$. The distribution of
the r.v.\ $Z^0 (s)$ with $s>0$ has an atom $e^{-\lambda s}$ at zero and a density on $(0,\infty)$ given by
\begin{equation}
p_{Z^0 (s)} (y)= e^{-\lambda s}\sum_{n=1}^\infty \frac{(\lambda s)^n}{n!} f^{*n}(y),
\qquad y>0.
\label{DenZ0}
\end{equation}
To a crossing of the (lower) linear boundary $x=-ct$ by the process $\{U^0 (t)\}$ at time
$\tau$ (this necessarily is a jump epoch) there corresponds a (continuous) crossing of the
(again lower linear) boundary
\[
y = s/c - (v+u/c), \qquad s>0,
\]
by the process $\{Z^0 (s)\}$ at `time' $\sigma = u+c\tau$, so that
\begin{equation}
\tau = (\sigma -u)/c.
\label{TauSigma}
\end{equation}
Finally, we notice that $\sigma$ is the crossing time of the (lower) level $-(v+u/c)$
by the process $Z (s)= Z^0 (s) -s/c,$ which is clearly a skip-free in the negative
direction L\'evy process.
Figure 1 illustrates the translation of the original problem. The original surplus process starts at level $u=1$, and ruin occurs at the fifth claim.
Rotating the figure anti-clockwise through 90 degrees we see the corresponding path of the pure jump process $\{Z^0(s)\}$.
\begin{figure}
\caption{Original and translated processes}
\label{Fig1}
\end{figure}
(ii)~Therefore, provided that $Z(s)$ has a density $p_{Z(s)} (y)$ at the point
$y=-(v+u/c),$ the crossing `time' $\sigma$ also has a density $p_\sigma (s)$ at the
point $s$, which is given by Kendall's identity (see e.g. \S\,12, Theorem~1
in~\cite{Bo72}, or \cite{BoBu01}):
\[
p_\sigma (s) = \frac{v+u/c}{s} p_{Z(s)} (-(v+u/c)).
\]
This together with \eqref{DenZ0} implies that, for $s> u+cv,$ the r.v. $\sigma$ has the
density
\[
p_\sigma (s) = \frac{v+u/c}{s}\, e^{-\lambda s}
\sum_{n=1}^\infty \frac{(\lambda s)^n}{n!} f^{*n} \bigl((s-u)/c -v\bigr).
\]
Therefore, it follows now from~\eqref{TauSigma} that, given $T_0 =v$, for $t>v$ the stopping time
$\tau$ has a conditional density given by
\begin{equation}
p_\tau (t|v) = c p_\sigma (u+ct)
= \frac{u + cv}{u+ct}\, e^{-\lambda (u+ct)}
\sum_{n=1}^\infty \frac{(\lambda (u+ct))^n}{n!} f^{*n} (t -v ).
\label{DenTauV}
\end{equation}
(iii) To obtain the density of $\tau$ in the general case, we observe that $\tau
\ge T_0$ always and so, using \eqref{ExpX},
\begin{multline*}
\mathbb{P} (\tau \le t) = \mathbb{P} (T_0 = \tau \le t) + \mathbb{P} (T_0 < \tau \le t)
\\
= \int_0^t \mathbb{P} (u+cv -X_1 <0) f_0 (v) dv
+ \int_0^t \mathbb{P} (v <\tau \le t|\, T_0 =v) f_0 (v) dv
\\
= \int_0^t e^{-\lambda (u+cv)} f_0 (v) dv
+ \int_0^t \biggl[ \int_v^t p_\tau (r|v) dr\biggr] f_0 (v) dv.
\end{multline*}
Differentiating both sides and substituting the representation for $ p_\tau (t|v)$ from
\eqref{DenTauV} yields the density of $\tau$:
\begin{multline*}
p_\tau (t) = e^{-\lambda (u+ct)} f_0 (t)
+ \int_0^t p_\tau (t|v) f_0 (v) dv
\\
= e^{-\lambda (u+ct)} \biggl[ f_0 (t) + \frac{1}{u+ct}
\sum_{n=1}^\infty \frac{(\lambda (u+ct))^n}{n!} \int_0^t (u + cv) f^{*n} (t -v ) f_0 (v) dv
\biggr]
\end{multline*}
(the change of the order of integration/summation is justified as the integrand is a
non-negative function). As the last expression is equivalent to the RHS
of~\eqref{Main}, the theorem is proved. \varepsilonroof
\section{Examples}
\subsection {Gamma inter-claim times}
Let us first consider the situation where claims occur according to an ordinary renewal process, so that
each $T_j$, $j=0,1,2,...$ has density function
\[
f(t)=f_{0}(t)=\frac{\beta ^{n}t^{n-1}e^{-\beta t}}{\Gamma (n)},
\]
where $n>0$ and $\beta >0$. It is well known that
\[
f^{\ast (m+1)}(t)=\frac{\beta ^{n(m+1)}t^{n(m+1)-1}e^{-\beta t}}{\Gamma
(n(m+1))}
\]
and it is straightforward to show that
\[
f^{\ast m}\ast f_{1}(t)=\frac{n}{\beta }\frac{\beta
^{n(m+1)+1}t^{n(m+1)}e^{-\beta t}}{\Gamma (n(m+1)+1)}.
\]
Then formula \eqref{Main} gives
\begin{eqnarray*}
p_{\tau }(t)
&=&(\beta t)^{n-1}\frac{u\beta e^{-\lambda (u+ct)-\beta t}}{u+ct}
\sum_{m=0}^{\infty }\frac{\lambda ^{m}(u+ct)^{m}}{m!}\frac{(\beta t) ^{nm}}{
\Gamma (n(m+1))} \\
&&+ (\beta t)^{n}\frac{c n e^{-\lambda (u+ct)-\beta t}}{u+ct
}\sum_{m=0}^{\infty }\frac{\lambda ^{m}(u+ct)^{m}}{m!}\frac{(\beta t)^{nm}
}{\Gamma (n(m+1)+1)}.
\end{eqnarray*}
In the special case when $n$ is a positive integer, we can compute this as
\begin{multline}
p_{\tau }(t)= \frac{\beta e^{-\lambda (u+ct)-\beta t}}{u+ct}\frac{(\beta t)^{n-1}}{
\Gamma (n)} \left( u\ _{0}F_{n}\left( 1,1+\frac{1}{n},...,1+\frac{n-1}{n};\frac{
\lambda (u+ct)(\beta t)^{n}}{n^{n}}\right) \right.
\\
+\left.
ct \ _{0}F_{n}\left( 1+\frac{1}{n},1+\frac{2}{n},...,1+\frac{n}{n};
\frac{\lambda (u+ct)(\beta t)^{n}}{n^{n}}\right) \right), \label{orderl2}
\end{multline}
where
\begin{equation*}
{}_{p}F_{q}(B_1,B_2,...,B_p,C_{1},C_{2},\ldots C_{q};Z)=\sum_{m=0}^{\infty }\frac{(B_1)_m (B_2)_m...(B_p)_m}{
(C_{1})_{m}(C_{2})_{m}...(C_{q})_{m}}\frac{Z^{m}}{m!}
\end{equation*}
is the generalised hypergeometric function (and $(a)_n =\Gamma(a+n)/\Gamma(a)$ is Pochhammer's symbol).
Formula \eqref{orderl2} follows from the identity
\[
\frac {\Gamma(n+1)} {\Gamma((n(m+1)+1)}=\frac{1}{n^{nm}}\mathbb{P}od_{k=0}^{n-1}
\frac {\Gamma(1+\frac{k+1}{n})}{\Gamma(m+1+\frac{k+1}{n})} ,
\]
which can be derived by applying the multiplication formula of Gauss as
described in~\cite{DiHuZh05}.
Formula \eqref{orderl2} is in a different form to the formula for $p_\tau (t)$ derived in~\cite{DiHuZh05}. A comparison of these
two formulae for $p_\tau (t)$ yields the identity
\begin{eqnarray*}
&&_{0}F_{n}\left( 1,1+\frac{1}{n},...,1+\frac{n-1}{n};\frac{
\lambda (u+ct)(\beta t)^{n}}{n^{n}}\right) \\
&-& _{0}F_{n}\left( 1+\frac{1}{n},1+\frac{2}{n},...,1+\frac{n}{n};\frac{
\lambda (u+ct)(\beta t)^{n}}{n^{n}}\right) \\
&=& \lambda (u+ct)(\beta t)^{n} \frac{n!}{(2n)!} \ _{0}F_{n}\left( 2+\frac{1}{n},2+\frac{2}{n},...,2+\frac{n}{n};\frac{
\lambda (u+ct)(\beta t)^{n}}{n^{n}}\right).
\end{eqnarray*}
In the special case $n=1$, by writing $z=\sqrt{4\lambda\beta t (u+ct)}$ this identity reduces to the well-known result (e.g.~\cite{AbSt65})
\[
I_0 (z)-\frac{2}{z} I_1 (z)=I_2(z),
\]
where $I_v$ is the modified Bessel function of order $v$.
Next, let us consider the special case when $n=2$, and let us further assume that claims occur according to
a stationary renewal process, so that the distribution of $T_0$ is the equilibrium distribution of
$T_1$. Then we find that
\[
f_{0}(t)=\frac{\beta }{2}e^{-\beta t}\left( 1+\beta t\right) =\frac{1}{2}
(\beta e^{-\beta t}+\beta ^{2}te^{-\beta t}),
\]
giving
\[
f^{\ast m}\ast f_{0}(t)=\frac{1}{2}\left( \frac{\beta ^{2m+1}t^{2m}e^{-\beta
t}}{\Gamma (2m+1)}+\frac{\beta ^{2m+2}t^{2m+1}e^{-\beta t}}{\Gamma (2m+2)}
\right) .
\]
Further,
\[
f_{1}(t)=tf_{0}(t)=\frac{1}{2}(\beta te^{-\beta t}+\beta ^{2}t^{2}e^{-\beta
t}),
\]
giving
\[
f^{\ast m}\ast f_{1}(t)=\frac{1}{2}\frac{\beta ^{2m+1}t^{2m+1}e^{-\beta t}}{
\Gamma (2m+2)}+\frac{\beta ^{2m+2}t^{2m+2}e^{-\beta t}}{\Gamma (2m+3)}.
\]
Then formula \eqref{Main} gives
\begin{eqnarray*}
p_{\tau }(t) &=&e^{-\lambda (u+ct)}\left( f_{0}(t)+\frac{u}{u+ct}
\sum_{m=1}^{\infty }\frac{\lambda ^{m}(u+ct)^{m}}{m!}\frac{1}{2}\left( \frac{
\beta ^{2m+1}t^{2m}e^{-\beta t}}{\Gamma (2m+1)}+\frac{\beta
^{2m+2}t^{2m+1}e^{-\beta t}}{\Gamma (2m+2)}\right) \right. \\
&&\hspace{0.75in}\left. +\frac{c}{u+ct}\sum_{m=1}^{\infty }\frac{\lambda
^{m}(u+ct)^{m}}{m!}\left( \frac{1}{2}\frac{\beta ^{2m+1}t^{2m+1}e^{-\beta t}
}{\Gamma (2m+2)}+\frac{\beta ^{2m+2}t^{2m+2}e^{-\beta t}}{\Gamma (2m+3)}
\right) \right) ,
\end{eqnarray*}
and we can incorporate $f_{0}(t)$ into the sums so that both start at $\ m=0$. For computational
purposes we can write this in terms of
generalised hypergeometric functions as
\begin{multline}
p_{\tau }(t) =\frac{\beta e^{-\lambda (u+ct)-\beta t}}{2(u+ct)}\left( u\
_{0}F_{2}\left( \frac{1}{2},1;\frac{\lambda (u+ct)(\beta t)^{2}}{4}\right)
\right. \\
\left. +t(\beta u+c)\ _{0}F_{2}\left( 1,\frac{3}{2};\frac{\lambda
(u+ct)(\beta t)^{2}}{4}\right) +c\beta t^{2}\ _{0}F_{2}\left( \frac{3}{2},2;
\frac{\lambda (u+ct)(\beta t)^{2}}{4}\right) \right). \label{staterl2}
\end{multline}
Table \ref{table1} shows some values of finite time ruin probabilities when $\lambda=1$, $\beta=2$ and $c=1.1$. We
use the notation $\psi (u,t)$ to denote the probability of ruin by time $t$
from initial surplus $u$ when the density of $\tau $ is given by formula \eqref{orderl2}
with $n=2$, and $\psi _{e}(u,t)$ denotes the corresponding probability when
the density of $\tau $ is given by formula \eqref{staterl2}. These values have been found
by integrating the density functions numerically using Mathematica. We can
observe from this table that for each combination of $u$ and $t$, the finite
time ruin probability is greater when the distribution of $T_{0}$ is the
equilibrium distribution of $T_{1}$. This arises because both the mean and variance of $T_0$ are
smaller than the corresponding values for $T_1$.
\begin{table}[h] \centering
\begin{tabular}{ccccccc}
$t$ & $\psi (0,t)$ & $\psi _{e}(0,t)$ & $\psi (10,t)$ & $\psi _{e}(10,t)$ & $
\psi (20,t)$ & $\psi _{e}(20,t)$ \\ \hline
$20$ & 0.7973 & 0.8463 & 0.0457 & 0.0509 & 0.0009 & 0.0010 \\
$40$ & 0.8332 & 0.8735 & 0.1008 & 0.1082 & 0.0060 & 0.0066 \\
$60$ & 0.8481 & 0.8848 & 0.1387 & 0.1469 & 0.0138 & 0.0148 \\
$80$ & 0.8564 & 0.8912 & 0.1651 & 0.1737 & 0.0218 & 0.0232 \\
$100$ & 0.8618 & 0.8952 & 0.1842 & 0.1930 & 0.0292 & 0.0309 \\ \hline
\end{tabular}
\caption{Finite time ruin probabilities.
\label{table1}}
\end{table}
\subsection{Mixed exponential inter-claim times}
Let us now consider the situation when the distribution of each $T_{j}$, $
j=0,1,2,...,$ is mixed exponential with density function
\[
f(t)=f_0(t)=p\alpha e^{-\alpha t}+q\beta e^{-\beta t},
\]
where $0<p<1$, $q=1-p$, and $\beta >\alpha >0$. Following ideas in \cite{WilWoo07}, it is shown in \cite{Dic07} that the $m$-fold convolution of $f$ with
itself as can be written as
\[
f^{\ast m}(t)=\sum_{j=0}^{\infty }\gamma _{m,j}\ e(m+j,\beta ;t),
\]
where $e(m,\beta ;t)$ denotes the Erlang($m$) density with scale parameter $
\beta $ and
\[
\gamma _{m,j}=q^{m}(1-\alpha /\beta )^{j}\sum_{r=0}^{m}\binom{m}{r}\frac{
(r)_{j}}{j!}\left( \frac{\alpha p}{\beta q}\right) ^{r}.
\]
We can find a similar type of expression for $f^{\ast m}\ast f_{1}(t)$ by
using Laplace transforms. For a function $w$, let
\[
\tilde{w}(s)=\int_{0}^{\infty }e^{-st}w(t)dt.
\]
Then
\[
\tilde{f}(s)=\frac{p\alpha }{\alpha +s}+\frac{q\beta }{\beta +s}
\]
and
\[
\tilde{f}_{1}(s)=\frac{p\alpha }{(\alpha +s)^{2}}+\frac{q\beta }{(\beta
+s)^{2}},
\]
leading to
\begin{eqnarray*}
\left[ \tilde{f}(s)\right] ^{m}\ \tilde{f}_{1}(s) &=&\frac{p}{\alpha }
\sum_{r=0}^{m}\binom{m}{r}p^{r}q^{m-r}\left( \frac{\alpha }{\alpha +s}
\right) ^{r+2}\left( \frac{\beta }{\beta +s}\right) ^{m-r} \\
&&+\frac{q}{\beta }\sum_{r=0}^{m}\binom{m}{r}p^{r}q^{m-r}\left( \frac{\alpha
}{\alpha +s}\right) ^{r}\left( \frac{\beta }{\beta +s}\right) ^{m-r+2}.
\end{eqnarray*}
Hence
\begin{eqnarray*}
f^{\ast m}\ast f_{1}(t) &=&\frac{p}{\alpha }\sum_{r=0}^{m}\binom{m}{r}
p^{r}q^{m-r}\int_{0}^{t}\frac{\alpha ^{r+2}y^{r+1}e^{-\alpha y}}{\Gamma (r+2)
}\frac{\beta ^{m-r}(t-y)^{m-r-1}e^{-\beta (t-y)}}{\Gamma (m-r)}dy \\
&&+\frac{q}{\beta }\sum_{r=0}^{m}\binom{m}{r}p^{r}q^{m-r}\int_{0}^{t}\frac{
\alpha ^{r}y^{r-1}e^{-\alpha y}}{\Gamma (r)}\frac{\beta
^{m-r+2}(t-y)^{m-r+1}e^{-\beta (t-y)}}{\Gamma (m-r+2)}dy \\
&=&p\alpha \sum_{r=0}^{m}\binom{m}{r}\left( \alpha p\right) ^{r}\left( \beta
q\right) ^{m-r}\frac{e^{-\beta t}t^{m+1}}{\Gamma (m+2)}\
_{1}F_{1}(r+2,m+2,(\beta -\alpha )t) \\
&&+q\beta \sum_{r=0}^{m}\binom{m}{r}\left( \alpha p\right) ^{r}\left( \beta
q\right) ^{m-r}\frac{e^{-\beta t}t^{m+1}}{\Gamma (m+2)}\
_{1}F_{1}(r,m+2,(\beta -\alpha )t).
\end{eqnarray*}
If we now replace the $ _1 F_1$ functions by their series representations, we find after a small amount of manipulation, that
\[
f^{\ast m}\ast f_{1}(t) = \sum_{i=0}^{\infty }\eta _{i,m}e(m+i+2,\beta ;t),
\]
where
\[
\eta _{i,m}=\frac{q^{m}}{\beta ^{2}}\left( 1-\alpha /\beta \right)
^{i}\sum_{r=0}^{m}\binom{m}{r}\frac{(r)_{i}}{i!}\left( \frac{\alpha p}{
\beta q}\right) ^{r} \left( (r+2)(r+1)\alpha p + \beta q \right).
\]
Thus, we have formulae for all the ingredients in formula \eqref{Main}.
Figure 2 shows three plots of the density of $\tau$ when $u=10$, $c=1.1$ and the parameters of the mixed
exponential distribution are as in Table \ref{table2}. In Figure 2, Case A is illustrated by the dotted line,
Case B by the solid line, and Case C by the bold line. We observe that the ordering of these three plots leads
to highest finite time ruin probabilities for Case A and lowest for Case C, consistent with the ordering of the
three values of $\mathbb{V}\,[T_{0}]$.
\begin{table}[h] \centering
\begin{tabular}{cccccc}
Case & $\alpha $ & $\beta $ & $p$ & $\mathbb{E}\,[T_{0}]$ & $\mathbb{V}\,[T_{0}]$ \\ \hline
A & 2/5 & 2 & 1/4 & 1 & 5/2 \\
B & 1/2 & 2 & 1/3 & 1 & 2 \\
C & 3/5 & 2 & 3/7 & 1 & 5/3\\ \hline
\end{tabular}
\caption{Parameters of mixed exponential distributions.
\label{table2}}
\end{table}
\begin{figure}
\caption{Densities of time to ruin for mixed exponential inter-claim times.}
\label{Fig2}
\end{figure}
\noindent Konstantin A Borovkov\newline
Department of Mathematics and Statistics\newline
The University of Melbourne\newline
Victoria 3010\newline
Australia\newline
[email protected]\newline
\noindent David C M Dickson\newline
Centre for Actuarial Studies\newline
Department of Economics\newline
The University of Melbourne\newline
Victoria 3010\newline
Australia\newline
[email protected]
\end{document} |
\begin{document}
\title[The structure of finitely generated modules]{On the structure of finitely generated modules over quotients of Cohen-Macaulay local rings}
\author[N.T. Cuong]{Nguyen Tu Cuong}
\address{Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307
Hanoi, Vietnam}
\email{[email protected]}
\author[P.H. Quy]{Pham Hung Quy}
\address{Department of Mathematics FPT University, Hanoi, Vietnam}
\email{[email protected]}
\thanks{2010 {\em Mathematics Subject Classification\/}: 13H10, 13D45, 13H15.\\
This work is partially supported by funds of Vietnam National Foundation for Science
and Technology Development (NAFOSTED)}
\keywords{Cohen-Macaulay module; local cohomology; system of parameters; unmixed component; Cohen-Macaulay deviated sequence; extended degree; unmixed degree.}
\begin{abstract}
Let $(R, \frak m)$ be a homomorphic image of a Cohen-Macaulay local ring and $M$ a finitely generated $R$-module. We use the splitting of local cohomology to shed a new light on the structure of non-Cohen-Macaulay modules. Namely, we show that every finitely generated $R$-module $M$ is associated by a sequence of invariant modules. This modules sequence expresses the deviation of $M$ with the Cohen-Macaulay property. Our result generalizes the unmixed theorem of Cohen-Macaulayness for any finitely generated $R$-module. As an application we construct a new extended degree in sense of Vasconcelos.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
Throughout this paper, let $(R, \frak m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. Let $x_1, \ldots, x_d$ be a system of parameters of $M$.\\
\operatorname{no}indent {\bf Standard setting.} We always assume that $R$ is a homomorphic image of a Cohen-Macaulay local ring.\\
Cohen-Macaulay rings and modules are the central objects of commutative algebra. The unmixed theorem says that $M$ is Cohen-Macaulay if and only if for every $i < d$ all associated prime ideals of $M/(x_1, \ldots, x_i)M$ have the same height $i$ (or dimension $d-i$), that is, $M/(x_1, \ldots, x_i)M$ is an unmixed module for all $i < d$ and for every system of parameters $x_1, \ldots, x_d$. Suppose $\cap_{\frak p \in \mathrm{Ass}M}N(\frak p) = 0$ is a reduced primary decomposition of the zero submodule of $M$, then the {\it unmixed component} of $M$ is defined by
$$U_M(0) = \bigcap_{\frak p \in \mathrm{Ass}M, \dim R/\frak p = d}N(\frak p).$$
Then $U_M(0)$ is just the largest submodule of $M$ of dimension strickly less than $d$. The following is the unmixed component version of unmixed theorem.\\
\operatorname{no}indent {\bf The unmixed theorem.} {\it A finitely generated $R$-module $M$ is Cohen-Macaulay if and only if for some (and hence for all) system of parameters $x_1, \ldots, x_d$ of $M$ all unmixed components
$$U_M(0), U_{M/x_1M}(0), \ldots, U_{M/(x_1, \ldots, x_{d-1})M}(0)$$
are vanished.}\\
The unmixed theorem can be expressed in another form as follows. A finitely generated $R$-module $M$ is Cohen-Macaulay if and only if every system of parameters $x_1, \ldots, x_d$ of $M$ is an $M$-regular sequence. Recall that $x_1, \ldots, x_d$ is an $M$-regular sequence if for all $i \le d$ all relations
$$x_1 a_1 + \operatorname{cd}ots + x_i a_i = 0$$
are trivial, that is, $a_i \in (x_1, \ldots, x_{i-1})M$ for all $i \le d$. In general we have $a_i \in (x_1, \ldots, x_{i-1})M:x_i$, so $x_1, \ldots, x_d$ is an $M$-regular sequence if the sub-quotient module
$$\frac{(x_1, \ldots, x_{i-1})M:x_i}{(x_1, \ldots, x_{i-1})M} = 0$$
for all $i = 1, \ldots, d$. Since
$$((x_1, \ldots, x_{i-1})M:x_i)/(x_1, \ldots, x_{i-1})M = (0:x_i)_{M/(x_1,\ldots ,x_{i-1})M}$$
is a submodule of $M/(x_1, \ldots, x_{i-1})M$ of dimension less than or equal to $d-i = \dim M/(x_1, \ldots, x_{i-1})M -1$, we have
$$((x_1, \ldots, x_{i-1})M:x_i)/(x_1, \ldots, x_{i-1})M \subseteq U_{M/(x_1, \ldots, x_{i-1})M}(0)$$
for all $i < d$. Set
$$\frak b(M) = \bigcap_{\operatorname{un}derline{x}, i \le d} \mathrm{Ann} \frac{(x_1, \ldots, x_{i-1})M:x_i}{(x_1, \ldots, x_{i-1})M},$$
where $\operatorname{un}derline{x} = x_1, \ldots, x_d$ runs over all systems of parameters of $M$. It is clear that the ideal $\frak b(M)$ kills all non-trivial relations of systems of parameters of $M$.\\
The Cohen-Macaulayness of $M$ can be characterized by local cohomology: $M$ is Cohen-Macaulay if and only if the local cohomology $H^i_{\frak m}(M) = 0$ for all $i<d = \dim M$. Thus if $M$ is not Cohen-Macaulay, then $H^i_{\frak m}(M) \neq 0$ for some $i < d$. Notice that $H^i_{\frak m}(M)$ is always Artinian but it is rarely Noetherian. So $H^i_{\frak m}(M)$ may not be annihilated by $\frak m$-primary ideals. The ideals $\frak a_i(M) = \mathrm{Ann}H^i_{\frak m}(M)$, $i = 0 , \ldots, d$, play important role in many areas in commutative algebra such as the homological conjectures, the tight closure theory, ect. Set $\frak a(M) = \frak
a_0(M) \ldots \frak a_{d-1}(M)$. Schenzel proved the following inclusions \cite[Satz 2.4.5]{Sch82}
$$\mathfrak{a}(M) \subseteq \mathfrak{b}(M) \subseteq \mathfrak{a}_0(M) \cap \operatorname{cd}ots \cap \mathfrak{a}_{d-1}(M).$$
Notice that our ring is always a homomorphic image of a Cohen-Macaulay local ring. This condition gives us a critical fact that $\dim M/ \frak a(M)< \dim M$ for all finitely generated $R$-modules. Therefore we can choose a parameter element $x$ contained in $\frak a(M)$ (and hence in $\frak b(M)$). Furthermore, we have a special system of parameters satisfying that
$$x_d \in \frak a(M), x_{d-1} \in \frak a(M/x_dM), \ldots, x_1 \in \frak a(M/(x_2, \ldots, x_d)M).$$
Such a system of parameters is called a {\it $p$-standard system of parameters} \cite{C95}. The $p$-standard systems of parameters play a key igredient in Kawasaki's proof for the Macaulayfication problem \cite{K00}. By \cite[Theorem 1.2]{CC17} $R$ is a homomorphism of a Cohen-Macaulay local ring if and only if every finitely generated $R$-module admits a $p$-standard system of parameters.\\
In this paper, we will use a kind of $p$-standard system of parameters to study the splitting of local cohomology modules. As mentioned above we know that $0:x \subseteq U_M(0)$ for every parameter element $x$ of $M$. Moreover, if $x \in \frak b(M)$ then we have $0:x = U_M(0)$, so we get the following short exact sequence
$$0 \to M/U_M(0) \overset{x}{\to} M \to M/xM \to 0.$$
Furthermore if $x \in \frak b(M)^2$ then the above short exact sequence deduces the short exact sequence of local cohomology for any ideal $I$ (see Lemma \ref{B3.2.3})
$$0 \rightarrow H^i_I(M) \rightarrow H^i_I(M/xM) \rightarrow
H^{i+1}_I(M/U_M(0)) \rightarrow 0$$
for all $i < d - \dim R/I - 1$. Using \cite{CQ11} we can study the splitting of of these local cohomology exact sequences. Namely, the following is the first main result of this paper.
\begin{theorem}\label{T1.1} Let $I$ be an ideal of $R$ and $x$ a parameter element of $M$ contained in $\frak b(M)^3$. Then for all $i < d - \dim R/I - 1$ we have
$$H^i_I(M/xM) \cong H^i_I(M) \oplus H^{i+1}_I(M/U_M(0)).$$
\end{theorem}
In the case $I = \frak m$ we have the following consequence.
\begin{corollary}\label{C1.2} Let $x $ be a parameter element of $M$ contained in $\frak b(M)^3$. Then
$$H^i_{\mathfrak{m}}(M/xM) \cong H^i_{\mathfrak{m}}(M) \oplus H^{i+1}_{\mathfrak{m}}(M/U_M(0))$$
for all $i<d-1$, and
$$0:_{H^{d-1}_{\mathfrak{m}}(M/xM)}\mathfrak{b}(M) \cong H^{d-1}_{\mathfrak{m}}(M) \oplus
0:_{H^{d}_{\mathfrak{m}}(M)}\mathfrak{b}(M).$$
\end{corollary}
These splitting results lead a new kind of system of parameters $x_1, \ldots, x_d$ satisfying that
$$x_d \in \frak b(M)^3, x_{d-1} \in \frak b(M/x_dM)^3, \ldots, x_1 \in \frak b(M/(x_2, \ldots, x_d)M)^3.$$
We call such a system of parameters a {\it $C$-system of parameters} of $M$. Similar to $p$-standard system of parameters, every finitely generated $R$-module admits $C$-systems of parameters if and only if $R$ is a quotient of a Cohen-Macaulay local ring. It should be noted that the right hand sides of the above isomorphisms do not depend of the choice of $C$-parameter element $x \in \frak b(M)^3$. Thus the local cohomology modules $H^i_I(M/xM)$, $i < d - \dim R/I-1$, are invariants (up to an isomorphism). As consequences, we can expect several invariant properties of quotient modules $M/(x_i, \ldots, x_d)M$ regarding $C$-systems of parameters. For example, by using the fact $U_M(0) = H^0_{\frak b(M)}(M)$, as the second main result of this paper, we generalize the unmixed theorem for any finitely generated $R$-module.
\begin{theorem}\label{T1.3}
Let $M$ be a finitely generated $R$-module of dimension $d$ and $\operatorname{un}derline{x} = x_1, \ldots, x_d$ a $C$-system of parameters of $M$. Then the unmixed component $U_{M/(x_{i+1},
\ldots,x_d)M}(0)$ is independent of the choice of
$\operatorname{un}derline{x}$ for all $1 \leq i \leq d$ (up to an isomorphism).
\end{theorem}
The above theorem assigns to any finitely generated $R$-module $M$ of dimension $d$ a sequence of modules $U_0(M), \ldots, U_{d-1}(M)$, which satisfies that
$U_i(M) \cong U_{M/(x_{i+2}, \ldots,x_d)M}(0)$ for every $C$-system of parameters $x_1, \ldots, x_d$ of $M$. Notice that $M$ is Cohen-Macaulay if and only if $U_i(M) = 0$ for all $i = 0, \ldots, d-1$ by the unmixed theorem. This modules sequence gives information about the distance between $M$ and the Cohen-Macaulayness. We call $U_0(M), \ldots, U_{d-1}(M)$ the {\it Cohen-Macaulay deviated sequence} of $M$. The name of Cohen-Macaulay deviated sequence comes from the notion of {\it Cohen-Macaulay deviation} of Vasconcelos in his theory of extended degrees.\\
Let $I$ be an $\frak m$-primary ideal. We denote by $\mathrm{deg}(I, M)$ the ordinary multiplicity of $M$ with respect to $I$, and call the {\it degree} of $M$ with respect to $I$. The degree, $\mathrm{deg}(I, M)$, is a basic invariant that measures the complexity of $M$ with respect to $I$. Vasconcelos et al. \cite{DGV98, V98-1, V98-2}
introduced the notion of {\it extended degree} in order to capture the size of a module along with some of the complexity of its
structure. It is a numerical function on the category of finitely generated modules
over local or graded rings which generalizes the ordinary degree. Let $\mathcal{M}(R)$ be the category of finitely generated
$R$-modules. An {\it extended degree} on $\mathcal{M}(R)$
with respect to $I$ is a numerical function
$$ \mathrm{Deg}(I, \bullet) : \mathcal{M}(R) \to \mathbb{R} $$
satisfying the following conditions
\begin{enumerate}[{(i)}]
\item $\mathrm{Deg} (I, M) = \mathrm{Deg}(I, \overline{ M}) + \ell(H^0_{\frak
m}(M))$, where $\overline{M} = M/H^0_{\frak m}(M)$.
\item (Bertini's rule) $\mathrm{Deg}(I, M) \geq \mathrm{Deg}(I, M/xM)$ for every generic element $x \in I\setminus \frak mI$ of $M$.
\item If $M$ is Cohen-Macaulay then $\mathrm{Deg}(I, M) =
\mathrm{deg}(I, M)$.
\end{enumerate}
The difference $\mathrm{Deg} (I, M) - \mathrm{deg} (I, M)$ is called the {\it Cohen-Macaulay deviation} of $M$ with respect to $I$. The prototype of an extended degree is the {\it homological degree}, $\mathrm{hdeg}(I, M)$, was introduced and studied by Vasconselos in \cite{V98-1} (see Definition \ref{D3.3.4}). Until nowadays, the homological degree is the unique extended degree that we can describe in an explicit formula. Using the Cohen-Macaulay deviated sequence we introduce a new degree of $M$, which we call the {\it unmixed degree} of $M$ with respect to
$I$, and denote by $\mathrm{udeg}(I, M)$. We define
$$\mathrm{udeg}(I, M) = \mathrm{deg}(I, M) + \sum_{i=0}^{d-1}\delta_{i, \dim U_i(M)}\mathrm{deg}(I, U_i(M)),$$
where $\delta_{i, \dim U_i(M)}$ is Kronecker's symbol. The unmixed degree is a natural generalization of the ordinary degree as well as the {\it arithmetic degree} (for the definition of arithmetic degree, $\mathrm{adeg}(I,M)$, we refer to Definition \ref{adeg}). We prove the last main result of this paper as follows.
\begin{theorem}\label{T1.4}
The unmixed degree $\mathrm{udeg}(I, \bullet)$ is an extended degree on the category of finitely generated $R$-modules $\mathcal{M}(R)$.
\end{theorem}
Let us talk about the structure of this paper.
In the next section we collect useful results about the annihilator of local cohomologogy, the unmixed component and some special systems of parameters. We also mention the method of \cite{CQ11} to study the splitting of local cohomology.
Section 3 is devoted the splitting of local cohomology Theorem \ref{T1.1} and Corollary \ref{C1.2} (see Theorem \ref{D3.2.4} and Corollary \ref{H3.2.5}). Then we introduce the notion of $C$-system of parameters, that plays a key role in this paper.
Theorem \ref{T1.3} is proved in Section 4. We also prove the invariance of local cohomology of quotient modules regarding $C$-systems of parameters (cf. Theorem \ref{D3.2.7}). As an application of the Cohen-Macaulay deviated sequence $U_0(M), \ldots, U_{d-1}(M)$, we compute the length function $\ell(M/(x_1^{n_1}, \ldots, x_d^{n_d})M)$ when $x_1, \ldots, x_d$ is a $C$-system of parameters (cf. Proposition \ref{M3.2.13}). Other applications for sequentially Cohen-Macaulay modules and the Serre condition $(S_2)$ are also given.
The unmixed degree will be introduced in Section 5. Theorem \ref{T1.4} follows from Proposition \ref{M3.3.9}, Theorems \ref{D3.3.8} and \ref{D3.3.17}. The most difficulty is to prove the Bertini rule of unmixed degree. For that we show that for certain {\it superficial element} $x$ of $M$ with respect to $I$ we have $\mathrm{udeg}(I, M/xM) \le \mathrm{udeg}(I, M)$. We also compare the unmixed degree with the ordinary degree, the arithmetic degree and the homological degree.
\section{Preliminaries}
We start with the notion of annihilator of local cohomology which will be used frecequently in this paper.
\begin{notation} \rm Let $(R, \frak m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d>0$.
\begin{enumerate}[{(i)}]
\item For all $i < d$ we set $\frak a_i(M) =
\mathrm{Ann}H^{i}_\mathfrak{m}(M)$, and set $\frak a(M) = \frak
a_0(M) \ldots \frak a_{d-1}(M)$.
\item Put $\mathfrak{b}(M) = \bigcap_{\operatorname{un}derline{x};i=1}^d
\mathrm{Ann}(0:x_i)_{M/(x_1,\ldots ,x_{i-1})M}$ where $\operatorname{un}derline{x} =
x_1, \ldots, x_d$ runs over all systems of parameters of $M$.
\end{enumerate}
\end{notation}
\begin{remark}\label{C3.1.2}\rm
\begin{enumerate}[{(i)}]
\item Schenzel \cite[Satz 2.4.5]{Sch82} proved that
$$\mathfrak{a}(M) \subseteq \mathfrak{b}(M) \subseteq \mathfrak{a}_0(M) \cap \operatorname{cd}ots \cap \mathfrak{a}_{d-1}(M).$$
\item If $R$ is a homomorphic image of a Cohen-Macaulay local ring, then $\dim R/\mathfrak{a}_i(M) \leq
i$ for all $i< d$ \cite[Theorem 1.2]{CC17}. Furthermore, $\dim R/\mathfrak{a}_i(M) = i$ if and only if there exists $\frak p
\in \mathrm{Ass}M$ such that $\dim R/\frak p = i$ (see \cite[Theorem
8.1.1]{BH98}).
\item If $R$ is a homomorphic image of a Cohen-Macaulay local ring, then Faltings' annihilator theorem claims that $\frak p \in
\mathrm{supp}(M)$ and $\frak p \operatorname{no}tin V(\frak a(M))$ if and only if
$M_{\frak p}$ is Cohen-Macaulay and $\dim M_{\frak p} + \dim R/\frak
p = d$ (see \cite[9.6.6]{BS98}, \cite{CC17}).
\item The condition that $R$ is a homomorphic image of a Cohen-Macaulay local ring can not be removed in (ii) and (iii) by Nagata's example \cite[Example 2, pp. 203$-$205]{N62}.
\end{enumerate}
\end{remark}
Since we always assume that $(R, \frak m)$ is a homomorphic image of a Cohen-Macaulay local ring, Remark \ref{C3.1.2} (ii)
ensures that $\dim R/\frak a(M) < d$. Therefore we can choose a parameter element $x \in \frak a(M)$. Following \cite{C95} such a parameter element is called {\it $p$-standard}.
\begin{definition}\rm A system of parameters $x_1,\ldots,x_d$ of $M$ is called {\it $p$-standard} if $x_d \in \frak
a(M)$ and $x_i \in \frak a(M/(x_{i+1},\ldots,x_d)M)$ for all $i =
d-1,\ldots,1$.
\end{definition}
We recall a property of $p$-standard system of parameters which will be used in the sequel. Let
$\operatorname{un}derline{x} = x_1,\ldots,x_d$ be a system of parameters of $M$. Let $\operatorname{un}derline{n} = (n_1,\ldots,n_d)$ be a
$d$-tuple of positive integers and $\operatorname{un}derline{x}^{\operatorname{un}derline{n}} =
x_1^{n_1},\ldots,x_d^{n_d}$. We consider the difference
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) = \ell(M/(\operatorname{un}derline{x}^{\operatorname{un}derline{n}})M) -
e(\operatorname{un}derline{x}^{\operatorname{un}derline{n}};M)$$ as function in $\operatorname{un}derline{n}$,
where $e(\operatorname{un}derline{x};M)$ is the Serre multiplicity of $M$ with
respect to the sequence $\operatorname{un}derline{x}$. Although
$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n})$ may be not a polynomial for
$n_1,\ldots,n_d$ large enough, it is bounded above by polynomials.
Moreover, the first author in \cite{C91} proved that the least degree of
all polynomials in $\operatorname{un}derline{n}$ bounding above
$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n})$ is independent of the choice of
$\operatorname{un}derline{x}$, and it is denoted by $p(M)$. The invariant $p(M)$
is called the {\it polynomial type} of $M$.
If $(R, \frak m)$ is a homomorphic image of a Cohen-Macaulay local ring, then $p(M) = \dim R/\frak a(M)$ (see \cite{C92}). In addition, if
$\operatorname{un}derline{x} = x_1,\ldots,x_d$ is $p$-standard then
we have the following.
\begin{proposition}[\cite{C95}, Theorem 2.6 (ii)]\label{M3.1.4}
Let $x_1,\ldots,x_d$ be a $p$-standard system of parameters of $M$. Then for all $n_1,\ldots,n_d>0$ we have
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) =
\sum_{i=0}^{p(M)} n_1\ldots n_i e_i,$$ where
$e_i = e(x_1,\ldots,x_i; 0:_{M/(x_{i+2},\ldots,x_d)M}x_{i+1})$ and $e_0 =
\ell(0:_{M/(x_{2},\ldots,x_d)M}x_{1})$.
\end{proposition}
Recently, Cuong and the first author introduced the notion of {\it $dd$-sequence} which is a special case of the notion of {\it $d$-sequences} of Huneke.
\begin{definition}[\cite{Hu82,GY86}]\rm
A sequence of elements $\operatorname{un}derline{x} = x_1,\ldots,x_s$ is called a {\it
$d$-sequence} of $M$ if $(x_1,\ldots,x_{i-1})M:x_j =
(x_1,\ldots,x_{i-1})M:x_ix_j$ for all $i \leq j \leq s$.
A sequence $\operatorname{un}derline{x} = x_1,\ldots,x_s$ is called a {\it
strong $d$-sequence} if $\operatorname{un}derline{x}^{\operatorname{un}derline{n}} =
x_1^{n_1},\ldots,x_s^{n_s}$ is a $d$-sequence for all $\operatorname{un}derline{n} =
(n_1,\ldots,n_s) \in \mathbb{N}^s$.
\end{definition}
For important properties of $d$-sequence we refer to
\cite{Hu82,Tr83}.
\begin{definition}[\cite{CC07-1}]\rm
A sequence of elements $\operatorname{un}derline{x} = x_1,\ldots,x_s$ is call a {\it
$dd$-sequence} of $M$ if $\operatorname{un}derline{x}$ is a strong $d$-sequence of
$M$ and the following conditions are satisfied
\begin{enumerate}[{(i)}]
\item $s=1$ or,
\item $s>1$ and $\operatorname{un}derline{x}' = x_1,\ldots,x_{s-1}$ is a $dd$-sequence of $M/x_s^n$ for all $n \geq 1$.
\end{enumerate}
\end{definition}
The following is a characterization of $dd$-sequence in terms of $I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n})$ (\cite[Theorem 1.2]{CC07-1}).
\begin{proposition}\label{M3.1.7}
A system of parameters $\operatorname{un}derline{x} = x_1,\ldots,x_d$ of $M$ is a $dd$-sequence if and only if for all $n_1,\ldots,n_d>0$ we have
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) =
\sum_{i=0}^{p(M)} n_1\ldots n_i e_i,$$ where
$e_i = e(x_1,\ldots,x_i; 0:_{M/(x_{i+2},\ldots,x_d)M}x_{i+1})$ and $e_0 =
\ell(0:_{M/(x_{2},\ldots,x_d)M}x_{1})$.
\end{proposition}
\begin{remark}\rm \label{R dd seq}
\begin{enumerate}[{(i)}]
\item By Propositions \ref{M3.1.4} and \ref{M3.1.7}, if a system of parameter $x_1,\ldots,x_d$ of $M$ is $p$-standard, then it is a $dd$-sequence. Conversely, if $x_1,\ldots,x_d$ is a $dd$-sequence then $x_1^{n_1},\ldots,x_d^{n_d}$ with $n_i \geq i, i = 1, \ldots,d$,
is $p$-standard (see \cite[Section
3]{CC07-1}).
\item An $R$-module $M$ admits a $p$-standard (or $dd$-sequence) system of parameters if and only if $R/\mathrm{Ann}M$ is a homomorphic image of a
Cohen-Macaulay local ring \cite[Theorem 1.2]{CC17}.
\end{enumerate}
\end{remark}
We next recall the notion of {\it unmixed component} of $M$ and its relations with the ideal $\frak b(M)$.
\begin{definition}\label{Dn3.2.1} \rm The largest submodule of $M$ of dimension less than $d$ is called the {\it unmixed component} of $M$, and denoted by $U_M(0)$.
\end{definition}
\begin{remark}\label{C3.2.2} \rm
\begin{enumerate}[{(i)}]
\item If $\cap_{\frak p \in \mathrm{Ass}M}N(\frak p) = 0$ is a reduced primary decomposition of the zero submodule of $M$, then
$U_M(0) = \cap_{\dim R/\frak p \in \mathrm{Assh}M}N(\frak p)$, where $\mathrm{Assh}M = \{\frak p \in \mathrm{Ass}M \mid \dim R/\frak p = d\}$.
\item Since $\dim U_M(0) < d$, there exists a parameter element $x$
of $M$ contained in $\mathrm{Ann}\, U_M(0)$. Therefore $U_M(0) \subseteq 0:x$.
But $x$ is a parameter element, so $\dim (0:x) < d$. Hence
$U_M(0) = 0:x$. Following the definition of $\frak b(M)$ we have $\frak b(M)
\subseteq \mathrm{Ann}U_M(0)$. Thus if $x \in \frak b(M)$ is a parameter element of $M$ then $U_M(0) = 0:x$. We also have $U_M(0) \cong H^0_{\frak b(M)}(M)$.
\item By (ii) we have $\cap_{x} \mathrm{Ann}(0:_Mx) =
\mathrm{Ann}U_M(0)$, where $x$ runs over all parameter elements of $M$. Therefore
\begin{eqnarray*}
\mathfrak{b}(M) &=& \bigcap_{\operatorname{un}derline{x};i=1}^d
\mathrm{Ann}\,(0:x_i)_{M/(x_1,\ldots,x_{i-1})M}\\
&=& \bigcap_{\operatorname{un}derline{x};i=1}^d
\mathrm{Ann}\,U_{M/(x_1,\ldots,x_{i-1})M}(0),
\end{eqnarray*}
where $\operatorname{un}derline{x} = x_1,\ldots,x_d$ runs over all systems of parameters of $M$.
\end{enumerate}
\end{remark}
Problem of the splitting of local cohomology is started in \cite{CQ11}. For convenience we recall some results of \cite{CQ11} (with slight generalizations).
Suppose we are given an integer $t$, an ideal $\frak
a$ of $R$ and a submodule $U$ of $M$. Set $\overline{M} =M/U$. We
say that an element $x \in \mathfrak{a}$ satisfies the condition
$(\operatorname{sh}arp)$ if $0:_Mx = U$ and the short exact sequence
$$0 \longrightarrowngrightarrow \overline{M} \overset{x}{\longrightarrowngrightarrow} M \longrightarrowngrightarrow M/xM \longrightarrowngrightarrow 0$$
induces short exact sequences
$$0 \longrightarrowngrightarrow H^{i}_\mathfrak{a}(M) \longrightarrowngrightarrow H^{i}_\mathfrak{a}(M/xM)
\longrightarrowngrightarrow H^{i+1}_\mathfrak{a}(\overline{M}) \longrightarrowngrightarrow
0$$ for all $i<t-1$. When this is the case, we consider the above
exact sequence as an extension of $H^{i+1}_\mathfrak{a}(\overline{M})$ by $H^{i}_\mathfrak{a}(M)$, therefore as an
element of $\mathrm{Ext}^1_R(H^{i+1}_\mathfrak{a}(\overline{M}),
H^{i}_\mathfrak{a}(M))$ (see \cite[Chapter 3]{Mac75}). We denote this
element by $E_x^i$. Especially, if $H^{t}_\mathfrak{a}(\overline{M})
\cong H^{t}_\mathfrak{a}(M)$, then we have the short exact sequence
$$0 \longrightarrowngrightarrow H^{t-1}_\mathfrak{a}(M) \longrightarrowngrightarrow H^{t-1}_\mathfrak{a}(M/xM)
\longrightarrowngrightarrow 0:_{H^{t}_\mathfrak{a}(\overline{M})}x \longrightarrowngrightarrow 0.$$
Let $\frak b$ be an ideal such that $x\in \frak b$. We denote by $F^{t-1}_x$ the element of
$\mathrm{Ext}^1_R(0:_{H^{t}_\mathfrak{a}(\overline{M})}\mathfrak{b},
0:_{H^{t-1}_\mathfrak{a}(M)}\mathfrak{b})$ which represented by the following short exact sequence
$$0 \longrightarrowngrightarrow 0:_{H^{t-1}_\mathfrak{a}(M)}\mathfrak{b} \longrightarrowngrightarrow
0:_{H^{t-1}_\mathfrak{a}(M/xM)}\mathfrak{b}
\longrightarrowngrightarrow 0:_{H^{t}_\mathfrak{a}(\overline{M})}\mathfrak{b} \longrightarrowngrightarrow 0$$
provided the exact sequence is determined by applying the $\mathrm{Hom}(R/\frak b, \bullet)$ functor.
It should be noted here that an extension of $R$-module $A$ by an $R$-module $C$ is split if it is the zero-element of
$\mathrm{Ext}^1_R(C, A)$. The two next theorems can be proven by the same method as used in \cite[Theorem 2.2]{CQ11}
\begin{theorem}\label{T2.13}
Let $t$ be a positive integer and $U$ a submodule of $M$. Let
$\overline{M} = M/U$. Suppose $x$ and $y$ are elements satisfying the condition $(\operatorname{sh}arp)$ and $0:_M (x+y)=U$. Then
\begin{enumerate}[{(i)}]\rm
\item {\it $x+y$ also satisfies the condition $(\operatorname{sh}arp)$ and $E_{x+y}^i =
E_x^i + E_y^i$ for all $i<t-1$.}
\item {\it If
$H^{t}_\mathfrak{a}(\overline{M}) \cong H^{t}_\mathfrak{a}(M)$ and
$F^{t-1}_x, F^{t-1}_{y}$ are determined, then $F^{t-1}_{x+y}$ is determined and $F^{t-1}_{x+y} = F^{t-1}_x + F^{t-1}_{y}$.}
\end{enumerate}
\end{theorem}
\begin{theorem}\label{T2.14} Let $t$ be a positive integer and $U$ a submodule of $M$. Let
$\overline{M} = M/U$. Suppose $x$ and $y$ are elements such that
$x$ satisfies the condition $(\operatorname{sh}arp)$ and $0:_M xy=U$. Then
\begin{enumerate}[{(i)}]\rm
\item {\it $xy$ satisfies the condition $(\operatorname{sh}arp)$ and $E_{xy}^i = yE_x^i$ for all $i<t-1$. Suppose that $H^{t}_\mathfrak{a}(\overline{M})
\cong H^{t}_\mathfrak{a}(M)$ and $F^{t-1}_x$ is determined.
Then $F^{t-1}_{xy}$ is determined and $F^{t-1}_{xy}=yF^{t-1}_x$.}
\item {\it Suppose that $H^{t}_\mathfrak{a}(\overline{M}) \cong
H^{t}_\mathfrak{a}(M)$ and $yH^{i}_\mathfrak{a}(M)=0$ for all $i<t$.
Then $E_{xy}^i =0$ for all $i<t-1$. Moreover, $F^{t-1}_{xy}$ is determined and $F^{t-1}_{xy} = 0$.}
\end{enumerate}
\end{theorem}
The following is a prime avoidance theorem for a product of ideals.
\begin{lemma}[\cite{CQ11} Lemma 3.1] \label{L2.15}
Let $(R, \mathfrak{m})$ be a Noetherian local ring, $\mathfrak{a}$,
$\mathfrak{b}$ ideals and $\mathfrak{p}_1, \ldots, \mathfrak{p}_n$
prime ideals such that $\mathfrak{ab} \nsubseteq \mathfrak{p}_j$ for
all $j \leq n$. Let $x \in \mathfrak{ab}$ with $x \operatorname{no}tin
\mathfrak{p}_j$ for all $j \leq n$. There are elements $a_1,
\ldots, a_r \in \mathfrak{a}$ and $ b_1, \ldots, b_r \in \mathfrak{b}$ such that
$x=a_1b_1+ \operatorname{cd}ots + a_rb_r$, and that $a_ib_i \operatorname{no}tin \mathfrak{p}_j$
and $a_1b_1+ \operatorname{cd}ots +a_ib_i \operatorname{no}tin \mathfrak{p}_j$ for all $i \leq
r$ and all $j \leq n$.
\end{lemma}
\begin{corollary}\label{C2.16}
Let $(R, \mathfrak{m})$ be a Noetherian local ring, $M$ a finitely generated $R$-module of dimension $d>0$, $\frak a$ and $\frak b$ two ideals such that $\dim R/\frak a \frak b<d$.
Let $x \in \frak a \frak b$ be a parameter element of $M$. There exist parameter elements $a_1,
\ldots, a_r \in \frak a$ and $b_1, \ldots, b_r \in \mathfrak{b}$ of $M$ such that
$x=a_1b_1+ \operatorname{cd}ots + a_rb_r$, and that $a_1b_1+ \operatorname{cd}ots +a_ib_i $ is a parameter element for all $i \leq
r$.
\end{corollary}
\begin{proof} Note that an element $x$ is a parameter element of $M$ if and only if $x \operatorname{no}tin \frak p$ for all $\frak{p} \in \mathrm{Assh}M$. The assertion now follows from Lemma \ref{L2.15}.
\end{proof}
\section{The splitting of local cohomology}
In this section we prove splitting theorems for local cohomology in local rings. These results lead a new kind of systems of parameters. We need the following key ingredient about the annihilator of local cohomology supported at an arbitrary ideal that is of independent interest.
\begin{proposition}\label{M3.1.11}
Let $M$ be a finitely generated $R$-module of dimension $d$ and $I$ an ideal of $R$. We have $\mathfrak{b}(M) H^i_{I}(M)=0$
for all $i < d - \dim R/I$.
\end{proposition}
To prove the above result we use the following isomorphism of Nagel and
Schenzel (see \cite[Proposition 3.4]{NS94}). Recall that a sequence $x_1,\ldots,x_t$ of elements contained in $I$ is an
{\it $I$-filter regular sequence} of $M$ if
$$\mathrm{Supp}\,
((x_1,\ldots,x_{i-1})M:x_i)/(x_1,\ldots,x_{i-1})M \subseteq V(I)$$
for all $i = 1,\ldots,t$, where $V(I)$ denotes the set of prime
ideals containing $I$. This condition is equivalent to that $x_i \operatorname{no}tin \frak{p}$ for all $\frak{p} \in \mathrm{Ass}_R M/(x_1, \ldots, x_{i-1})M \setminus V(I)$ for all $i = 1, \ldots, t$. Moreover we can choose an $I$-filter regular sequence on $M$ of any length by the prime avoidance theorem.
\begin{lemma}[Nagel-Schenzel's isomorphism]\label{B3.1.12}
Let $I$ be an ideal of $R$ and $x_1, \ldots, x_t$ an $I$-filter regular
sequence of $M$. For each $j \leq t$ we have
$$
H^j_{I}(M)\cong
\begin{cases}
H^j_{(x_1,\ldots,x_t)}(M) \quad \quad \quad\,\, \text{with}\,\, j<t\\
H^{j-t}_I(H^t_{(x_1,\ldots,x_t)}(M))\,\, \text{with}\,\, j\geq t.\\
\end{cases}
$$
\end{lemma}
\begin{proof}[Proof of Proposition \ref{M3.1.11}] Set $ t = d - \dim R/I$. Suppose $t<d$, by the prime avoidance theorem we can choose an element $x_1 \in I$ such that $x_1 \operatorname{no}tin \frak p$ for all $\frak p \in \mathrm{Assh}\, M \cup (\mathrm{Ass}\, M \setminus V(I) )$. Thus $x_1$ is a parameter element of $M$ that is also an $I$-filter regular element. We continue this progress to obtain a part of a system of parameters $x_1, \ldots, x_t$ of $M$ that is also an $I$-filter regular on $M$. By Lemma \ref{B3.1.12} for
$i<t$, we have
\begin{eqnarray*}
H^i_{I}(M) &\cong& H^{0}_I(H^i_{(x_1,\ldots,x_i)}(M))\\
&\cong& H^{0}_I(\lim_{\longrightarrowngrightarrow}M/{(x_1^n,\ldots,x_i^n)}M)\\
&\cong& \lim_{\longrightarrowngrightarrow} H^{0}_I(M/{(x_1^n,\ldots,x_i^n)}M)\\
&\cong& \lim_{\longrightarrowngrightarrow}
\frac{(x_1^n,\ldots,x_i^n)M : I^\infty}{(x_1^n,\ldots,x_i^n)M}\\
&\cong& \lim_{\longrightarrowngrightarrow}
\frac{(x_1^n,\ldots,x_i^n)M :
x_{i+1}^\infty}{(x_1^n,\ldots,x_i^n)M},
\end{eqnarray*}
where $(x_1^n,\ldots,x_i^n)M : I^\infty = \cup_{k \ge 1} (x_1^n,\ldots,x_i^n)M : I^k$. Since $x_1, \ldots, x_t$ is a part of a system of parameters of $M$ and $(x_1^n,\ldots,x_i^n)M : x_{i+1}^\infty = (x_1^n,\ldots,x_i^n)M :
x_{i+1}^k$ for some $k$, we have
$$\frak b(M) \frac{(x_1^n,\ldots,x_i^n)M :
x_{i+1}^\infty}{(x_1^n,\ldots,x_i^n)M} = 0$$
for all $n \ge 1$ by the definition of $\frak b(M)$. Hence $\mathfrak{b}(M) H^i_{I}(M)=0$ for all
$i < d - \dim R/I$. The proof is complete.
\end{proof}
\begin{lemma}\label{B3.2.3}
Let $I$ be an ideal of $R$ and $x, y \in \mathfrak{b}(M)$ parameter elements of $M$. Let $U_M(0)$ be the unmixed component of $M$. Put $\overline{M} = M/U_M(0)$ and $t = d
-\dim R/I$. Then for all $i<t-1$ we have the following short exact sequence
$$0 \rightarrow H^i_I(M) \rightarrow H^i_I(M/xyM) \rightarrow
H^{i+1}_I(\overline{M}) \rightarrow 0.$$ Furthermore, if $H^{t}_I(M)
\cong H^{t}_I(\overline{M})$ then we have the short exact sequence
$$0 \rightarrow H^{t-1}_I(M) \rightarrow H^{t-1}_I(M/xyM) \rightarrow
0:_{H^{t}_I(M)}xy \rightarrow 0.$$
\end{lemma}
\begin{proof}
By Remark \ref{C3.2.2} (ii) we have $U_M(0) = 0:_Mx = 0:_Mxy$. Therefore the following diagram commutes
\[\divide\dgARROWLENGTH by 2
\begin{diagram}
\operatorname{no}de{0}\arrow{e}\operatorname{no}de{\overline{M}}
\arrow{e,t}{x}\arrow{s,l}{\mathrm{id}}\operatorname{no}de{M}\arrow{e}\arrow{s,l}{y}
\operatorname{no}de{M/xM}\arrow{s}\arrow{e} \operatorname{no}de{0}\\
\operatorname{no}de{0}\arrow{e}\operatorname{no}de{\overline{M}}
\arrow{e,t}{xy}\operatorname{no}de{M}\arrow{e}\operatorname{no}de{M/xyM}\arrow{e} \operatorname{no}de{0.}
\end{diagram}
\]
Applying the functor $H^{i}_I(\bullet)$ to the above diagram we obtain the following commutative diagram for all $i < t-1$
\[\divide\dgARROWLENGTH by 2
\begin{diagram}
\operatorname{no}de{\operatorname{cd}ots}\arrow{e}\operatorname{no}de{H^{i}_I(\overline{M})}
\arrow{e,t}{\psi^i}\arrow{s,l}{\mathrm{id}}\operatorname{no}de{H^{i}_I(M)}\arrow{e}\arrow{s,l}
{y}
\operatorname{no}de{H^{i}_I(M/xM)}\arrow{s}\arrow{e} \operatorname{no}de{\operatorname{cd}ots}\\
\operatorname{no}de{\operatorname{cd}ots}\arrow{e}\operatorname{no}de{H^{i}_I(\overline{M})}
\arrow{e,t}{\varphi^i}\operatorname{no}de{H^{i}_I(M)}\arrow{e}\operatorname{no}de{H^{i}_I(M/xyM)}\arrow{e}
\operatorname{no}de{\operatorname{cd}ots,}
\end{diagram}
\]
where $\psi^i$ and $\varphi^i$ are derived from homomorphisms $\overline{M }\overset{x}{\to}M$ and $\overline{M}
\overset{xy}{\to}M$, respectively. By Proposition \ref{M3.1.11},
$yH^{i}_I(M)=0$ for all $i \leq t-1$, so $\varphi^i =0$ for all $i
\leq t-1$. Thus we have the short exact sequences
$$0 \rightarrow H^i_I(M) \rightarrow H^i_I(M/xyM) \rightarrow
H^{i+1}_I(\overline{M}) \rightarrow 0$$ for all $i < t-1$. Thus we have the exact sequence
$$0 \rightarrow H^{t-1}_I(M) \rightarrow H^{t-1}_I(M/xyM) \rightarrow
H^{t}_I(\overline{M}) \overset{xy}{\rightarrow} H^t_I(M).$$
Moreover,
if $H^{t}_I(M) \cong H^{t}_I(\overline{M})$ then we have the following short exact sequence
$$0 \rightarrow H^{t-1}_I(M) \rightarrow H^{t-1}_I(M/xyM) \rightarrow
0:_{H^{t}_I(M)}xy \rightarrow 0.$$
The proof is complete.
\end{proof}
Let $xy$ be a parameter element of $M$ such that $x, y \in \frak b(M)$. Lemma
\ref{B3.2.3} says that
$xy$ satisfies the condition $(\operatorname{sh}arp)$ mentioned in Section 2 with $t = d - \dim R/I$ and $U = U_M(0)$.
Let $x \in \mathfrak{b}(M)^2$ be a parameter element of $M$, for all $i
< t-1$, we denote by $E^i_x$ the element in
$\mathrm{Ext}(H^{i+1}_I(\overline{M}), H^i_I(M))$ represented by the following short exact sequence provided it is determined
$$0 \rightarrow H^i_I(M) \rightarrow H^i_I(M/xM) \rightarrow
H^{i+1}_I(\overline{M}) \rightarrow 0.$$ In the case $i=t-1$ and
assume that $H^{t}_I(M) \cong H^{t}_I(\overline{M})$, we have the short exact sequence
$$0 \rightarrow H^{t-1}_I(M) \rightarrow H^{t-1}_I(M/xM) \rightarrow
0:_{H^{t}_I(M)}x \rightarrow 0.$$
Suppose we obtain the following short exact sequence by applying the $\mathrm{Hom}(R/ \frak b(M), \bullet)$ to above short exact sequence
$$0 \rightarrow H^{t-1}_I(M) \rightarrow 0:_{H^{t-1}_I(M/xM)}\mathfrak{b}(M)
\rightarrow
0:_{H^{t}_I(M)}\mathfrak{b}(M) \rightarrow 0.$$
Then we denote by $F^{t-1}_{x}$ the element of $\mathrm{Ext}(0:_{H^{t}_I(M)}\mathfrak{b}(M), H^{t-1}_I(M))$
represented by the above short exact sequence. The main result of this section as follows.
\begin{theorem}\label{D3.2.4} Let $M$ be a finitely generated $R$-module of dimension $d$, $I$ an ideal of $R$ and
$x$ a parameter element of $M$. Let $U_M(0)$ be the unmixed component of $M$ and set $\overline{M} = M/U_M(0)$. Let
$t = d -\dim R/I$. Then
\begin{enumerate}[{(i)}]\rm
\item {\it If $x \in \mathfrak{b}(M)^2$ then $E^i_x$ is determined for all $i<t-1$.}
\item {\it If $x \in \mathfrak{b}(M)^3$ then $E^i_x = 0$ for all $i<t-1$. Moreover, if $H^{t}_I(M) \cong H^{t}_I(\overline{M})$
then $F^{t-1}_{x} = 0$.}
\end{enumerate}
\end{theorem}
\begin{proof} (i) Notice that $\frak b(M) \nsubseteq \frak p$ for all $\frak p \in \mathrm{Assh}M$. By Corollary \ref{C2.16} there exist parameter elements $a_1,
\ldots, a_r, b_1, \ldots, b_r \in \mathfrak{b}(M)$ of $M$ such that
$x=a_1b_1+ \operatorname{cd}ots + a_rb_r$, and $a_1b_1+ \operatorname{cd}ots +a_jb_j $ are parameter elements for all $j \leq
r$. By Lemma \ref{B3.2.3} $E^i_{a_kb_k}$ is determined for all $i < t-1$ and for all $1 \leq k \leq r$. By Theorem \ref{T2.13} we have
$$E^i_x = E^i_{a_1b_1} + \operatorname{cd}ots +
E^i_{a_rb_r}$$
is determined for all $i < t-1$.\\
(ii) Similarly, we choose parameter elements $a_1,
\ldots, a_r \in \frak b(M)^2$ and $b_1, \ldots, b_r \in \mathfrak{b}(M)$ of $M$ such that
$x=a_1b_1+ \operatorname{cd}ots + a_rb_r$, and $a_1b_1+ \operatorname{cd}ots +a_jb_j $ are parameter elements for all $j \leq
r$. By Theorem \ref{T2.14} (ii) we have $E^i_{a_kb_k} = 0$ for all $i < t-1$ and for all $1 \leq k \leq r$. So $E^i_x = 0$ for all $i<t-1$. For the last assertion, by the same method, it is sufficient to show that $F^{t-1}_{ab} = 0$ for all parameter elements $a \in \frak b(M)^2$ and $b \in \frak b(M)$ provided $H^t_I(M) \cong H^t_I(\overline{M})$. Indeed, since $E^i_a$ and $E^i_{ab}$ are determined for all $i<t-1$, the commutative diagram
\[\divide\dgARROWLENGTH by 2
\begin{diagram}
\operatorname{no}de{0}\arrow{e}\operatorname{no}de{\overline{M}}
\arrow{e,t}{a}\arrow{s,l}{\mathrm{id}}\operatorname{no}de{M}\arrow{e}\arrow{s,l}{b}
\operatorname{no}de{M/aM}\arrow{s}\arrow{e} \operatorname{no}de{0}\\
\operatorname{no}de{0}\arrow{e}\operatorname{no}de{\overline{M}}
\arrow{e,t}{ab}\operatorname{no}de{M}\arrow{e}\operatorname{no}de{M/abM}\arrow{e} \operatorname{no}de{0.}
\end{diagram}
\]
deduces the following diagram
\[\divide\dgARROWLENGTH by 2
\begin{diagram}
\operatorname{no}de{0}\arrow{e}\operatorname{no}de{H^{t-1}_I(M)}
\arrow{e,t}{i}\arrow{s,l}{b}\operatorname{no}de{ H^{t-1}_I(M/aM)}\arrow{e}\arrow{s,l}{\beta}
\operatorname{no}de{0:_{H^{t}_I(M)}a}\arrow{s,l}{\alphapha}\arrow{e} \operatorname{no}de{0}\\
\operatorname{no}de{0}\arrow{e}\operatorname{no}de{H^{t-1}_I(M)}
\arrow{e,t}{\delta}\operatorname{no}de{ H^{t-1}_I(M/abM)}\arrow{e,t}{\pi}\operatorname{no}de{0:_{H^{t}_I(M)}ab}\arrow{e} \operatorname{no}de{0,}
\end{diagram}
\]
where $\alphapha: 0:_{H^{t}_I(M)}a \to 0:_{H^{t}_I(M)}ab$ is injective. By Proposition \ref{M3.1.11} $b H^{t-1}_I(M) = 0$, so $\beta \circ i = 0$. Thus we have a homomorphism $\epsilon: 0:_{H^{t}_I(M)}a \to H^{t-1}_I(M/abM)$ which makes the following diagram
\[\divide\dgARROWLENGTH by 2
\begin{diagram}
\operatorname{no}de{0}\arrow{e}\operatorname{no}de{H^{t-1}_I(M)}
\arrow{e,t}{i}\arrow{s,l}{b}\operatorname{no}de{ H^{t-1}_I(M/aM)}\arrow{e}\arrow{s,l}{\beta}
\operatorname{no}de{0:_{H^{t}_I(M)}a}\arrow{s,l}{\alphapha}\arrow{sw,t}{\epsilon}\arrow{e} \operatorname{no}de{0}\\
\operatorname{no}de{0}\arrow{e}\operatorname{no}de{H^{t-1}_I(M)}
\arrow{e,t}{\delta}\operatorname{no}de{ H^{t-1}_I(M/abM)}\arrow{e,t}{\pi}\operatorname{no}de{0:_{H^{t}_I(M)}ab}\arrow{e} \operatorname{no}de{0,}
\end{diagram}
\]
By applying the $\mathrm{Hom}_R(R/\frak b(M), \bullet)$ to the above diagram we have the following diagram
\[\divide\dgARROWLENGTH by 2
\begin{diagram}
\operatorname{no}de{}\operatorname{no}de{} \operatorname{no}de{}
\operatorname{no}de{0:_{H^{t}_I(M)} \frak b(M)}\arrow{sw,t}{\epsilon}\arrow{s,l}{\mathrm{id}}\\
\operatorname{no}de{0}\arrow{e}\operatorname{no}de{H^{t-1}_I(M)}
\arrow{e}\operatorname{no}de{ 0:_{H^{t-1}_I(M/abM)} \frak b(M)}\arrow{e,t}{\pi}\operatorname{no}de{0:_{H^{t}_I(M)}\frak b(M),}
\end{diagram}
\]
where the row is an exact sequence and the vertical map is an identification. Since $\pi \circ \epsilon = \mathrm{id}$, the homomorphism $\pi$ is split. Thus $F^{t-1}_{ab} = 0$. The proof is complete.
\end{proof}
In the case $I = \frak m$, the following is a generalization of \cite[Corollary 4.1]{CQ11} and \cite[Proposition 3.4]{Q12}.
\begin{corollary}\label{H3.2.5} Let $x \in \mathfrak{b}(M)^3$ be a parameter element of $M$. Let $U_M(0)$ be the unmixed component of $M$ and set $\overline{M} = M/U_M(0)$.
Then
$$H^i_{\mathfrak{m}}(M/xM) \cong H^i_{\mathfrak{m}}(M) \oplus H^{i+1}_{\mathfrak{m}}(\overline{M})$$
for all $i<d-1$, and
$$0:_{H^{d-1}_{\mathfrak{m}}(M/xM)}\mathfrak{b}(M) \cong H^{d-1}_{\mathfrak{m}}(M) \oplus
0:_{H^{d}_{\mathfrak{m}}(M)}\mathfrak{b}(M).$$
\end{corollary}
By the above splitting theorems it is natural to consider the following system of parameters.
\begin{definition}[\cite{MQ16}, Definition 2.15] \rm A parameter element $x\in \frak b(M)^3$ is called a {\it $C$-parameter element} of $M$. A system of parameters $x_1, ..., x_d$ is called a {\it $C$-system of parameters} of $M$ if $x_d \in \mathfrak b(M)^3$ and $x_i \in \mathfrak b(M/(x_{i+1}, ..., x_d)M)^3$ for all $i = d-1, ..., 1$. A sequence of elements $x_i, \ldots, x_d$ is called {\it a part of a $C$-system of parameters} if we can expand it to a $C$-system of parameters $x_1, \ldots, x_d$.
\end{definition}
It is envident that $C$-systems of parameters are closely related with $p$-standard systems of parameters. Lemmas below will be very useful in the sequel.
\begin{lemma} \label{B3.1.9}
Let $x$ be a parameter element of $M$. Then $\frak
b(M) \subseteq \frak b(M/xM)$.
\end{lemma}
\begin{proof}
It follows from the definition of $\frak b(M)$.
\end{proof}
\begin{lemma}\label{B3.1.10} Let $x_1, \ldots, x_d$ be a $C$-system of parameters of $M$. Then, for all $j \leq d$ we have $x_1, \ldots, x_{j-1},x_{j+1},\ldots,x_d$
is a $C$-system of parameters $M/x_jM$.
\end{lemma}
\begin{proof}
The case $j=d$ is clear. For $j \neq d$ by Lemma
\ref{B3.1.9} we have $\frak b(M) \subseteq \frak b(M/x_jM)$. Therefore $x_d$ is a $C$-parameter element of $M/x_jM$. Notice that
$x_1, \ldots,x_{d-1}$ is a $C$-system of parameters of $M/x_dM$. The claim follows from the induction on $d$.
\end{proof}
\section{The Cohen-Macaulay deviated sequences}
In this section we use the splitting theorem \ref{D3.2.4} to shed a new light on the structure of non-Cohen-Macaulay modules. Let $M$ be a finitely generated $R$-module of dimension $d$. The unmixed characterization of Cohen-Macaulay modules says that $M$ is Cohen-Macaulay if and only if for some (and hence for all) system of parameters
$x_1,\ldots,x_d$ we have $U_{M/(x_{i+1},\ldots,x_d)M}(0) = 0$ for all $1 \leq i
\leq d$. If $M$ is a generalized Cohen-Macaulay module and $\frak
m^{n_0}H^i_{\mathfrak{m}}(M) = 0$ for all $i<d$ and for some positive integer $n_0$, then by \cite[Corollary 4.2]{CQ11}
we have
$$U_{M/(x_{i+1},\ldots,x_d)M}(0) = H^0_{\mathfrak{m}}(M/(x_{i+1},\ldots,x_d)M)
\cong \bigoplus_{j=0}^{d-i} H^j_{\mathfrak{m}}(M)^{\binom{d-i}{j}}.$$
for any system of parameters $x_1,\ldots,x_d \in \frak m^{2n_0}$. Thus $U_{M/(x_{i+1},\ldots,x_d)M}(0)$ is independent of the choice of system of parameters $x_1,\ldots,x_d$ contained in
$\frak m^{2n_0}$ for all $1 \leq i \leq d$ (up to an isomorphism). The main aim this section is to generalize this fact for any finitely generated $R$-module. Concretely, we will show that for all $1 \leq i \leq d$ the modules $U_{M/(x_{i+1},\ldots,x_d)M}(0)$
is independent (up to an isomorphism) of the choice of a $C$-system of parameters $x_1,
\ldots, x_d$. We start with the following result about the invariance of local cohomology of quotient modules regarding $C$-systems of parameters.
\begin{theorem} \label{D3.2.7}
Let $\operatorname{un}derline{x} = x_1, \ldots, x_d$ be a $C$-system of parameters of $M$. Then the local cohomology module $H^j_{\mathfrak{m}}(M/(x_{i+1}, \ldots,x_d)M)$
is independent of the choice of $\operatorname{un}derline{x}$ for all $j
< i < d$ (up to an isomorphism).
\end{theorem}
\begin{proof}
We set $M_i = M/(x_{i+1}, \ldots,x_d)M$ for all $i < d$. We consider another $C$-system of parameters $\operatorname{un}derline{y} = y_1, \ldots, y_d$ of $M$, and put $M_i'
= M/(y_{i+1},\ldots,y_d)M$ for all $i < d$. We proceed by induction on $d$ that $H^j_{\mathfrak{m}}(M_i) \cong
H^j_{\mathfrak{m}}(M_i')$
for all $j < i < d$. The assertion is trivial if $d=1$. For $d>1$ and $i=d-1$ since $x_d$ and $y_d$ are $C$-parameter elements, Corollary \ref{H3.2.5} implies that
$$H^j_{\mathfrak{m}}(M_{d-1}) \cong H^j_{\mathfrak{m}}(M) \oplus
H^{j+1}_{\mathfrak{m}}(M/U_M(0)) \cong
H^j_{\mathfrak{m}}(M_{d-1}')$$ for all $j<d-1$. Suppose $i< d-1$. Since
$\dim R/\frak b(M_{i+1}) < \dim M_{i+1} = i+1$ and $\dim R/\frak b(M_{i+1}') < \dim M'_{i+1} = i+1$
we can choose a $C$-parameter element
$z$ of both $M_{i+1}$ and $M_{i+1}'$. By the inductive hypothesis we have
$$H^j_{\mathfrak{m}}(M_{i}) = H^j_{\mathfrak{m}}(M_{i+1}/x_{i+1}M_{i+1})
\cong H^j_{\mathfrak{m}}(M/(z, x_{i+2},\ldots,x_d)M), \quad \quad \quad \quad (1)$$
and
$$H^j_{\mathfrak{m}}(M_{i}') = H^j_{\mathfrak{m}}(M_{i+1}'/y_{i+1}M_{i+1}') \cong
H^j_{\mathfrak{m}}(M/(z, y_{i+2},\ldots,y_d)M)\quad \quad \quad \quad (2)$$ for all $j<i$. Notice that $z, x_{i+2},\ldots,x_d$ and
$z, y_{i+2},\ldots,y_d$ are parts of $C$-systems of parameters of $M$. By Lemma \ref{B3.1.10} we have $x_{i+2},\ldots,x_d$ and
$y_{i+2},\ldots,y_d$ are parts of $C$-systems of parameters of $M/zM$. Applying the inductive hypothesis for $M/zM$ we have
$$H^j_{\mathfrak{m}}(M/(z, x_{i+2},\ldots,x_d)M) \cong H^j_{\mathfrak{m}}(M/(z, y_{i+2},\ldots,y_d)M) \quad \quad \quad \quad (3)$$
for all $j<i$. The assertion follows from the isomorphisms $(1)$, $(2)$ and $(3)$.
\end{proof}
\begin{corollary}\label{C invar ann}
Let $\operatorname{un}derline{x} = x_1, \ldots, x_d$ be a $C$-system of parameters of $M$. Then the ideals $\frak a(M/(x_{i+1}, \ldots,x_d)M)$ and $\sqrt{\frak a(M/(x_{i+1}, \ldots,x_d)M)} = \sqrt{\frak b(M/(x_{i+1}, \ldots,x_d)M)}$ are independent of the choice of $\operatorname{un}derline{x}$ for all $i < d$.
\end{corollary}
We need the following result.
\begin{lemma}\label{B3.2.8}
Let $x $ be a $C$-parameter element of $M$. Then
$U_{M/xM}(0)$ is independent of the choice of $x$ (up to an isomorphism).
\end{lemma}
\begin{proof}
By Corollary \ref{C invar ann} we have the ideal
$$\mathfrak{b}' = \sqrt{\mathfrak{a}(M/xM)} = \sqrt{\mathfrak{b}(M/xM)}$$
is independent of the choice of $C$-parameter element $x$. By Remark
\ref{C3.2.2} (ii) we have $U_{M/xM}(0) \cong
H^0_{\mathfrak{b}'}(M/xM)$. Since $\dim R/\mathfrak{b}' \leq \dim M/xM
-1 = d-2$, Theorem \ref{D3.2.4} (ii) implies that
$$H^0_{\mathfrak{b}'}(M/xM) \cong H^0_{\mathfrak{b}'}(M) \oplus H^1_{\mathfrak{b}'}(M/U_M(0)),$$
and the right hand side does not depend on $x$. Thus the unmixed component $U_{M/xM}(0)$ is independent of the choice of $C$-parameter element $x$
(up to an isomorphism).
\end{proof}
Using Lemma \ref{B3.2.8} and by the same method as used in the proof of Theorem
\ref{D3.2.7} we obtain the main result of this section as follows.
\begin{theorem}\label{D3.2.9}
Let $M$ be a finitely generated $R$-module of dimension $d$ and $\operatorname{un}derline{x} = x_1, \ldots, x_d$ a $C$-system of parameters of $M$. Then for all $1 \leq i \leq d$, the unmixed component $U_{M/(x_{i+1},
\ldots,x_d)M}(0)$ is independent of the choice of
$\operatorname{un}derline{x}$ (up to an isomorphism).
\end{theorem}
\begin{definition} \rm
For all $0 \leq i \leq d-1$ we denote by $U_i(M)$ the module satisfying that $U_i(M) \cong U_{M/(x_{i+2}, \ldots,x_d)M}(0)$
for all $C$-systems of parameters $x_1, \ldots, x_d$ of $M$.
Notice that $\dim U_i(M) \leq i$ for all $0 \leq i \leq d-1$, and $U_{d-1}(M) \cong U_M(0)$. We call the modules sequence $U_0(M), \ldots, U_{d-1}(M)$ the {\it Cohen-Macaulay deviated sequence} of $M$. Notice that the Cohen-Macaulay deviated sequence of $M$ vanishes if and only if $M$ is Cohen-Macaulay.
\end{definition}
We next use the Cohen-Macaulay deviated sequence to prove some properties of $C$-systems of parameters.
\begin{corollary}\label{H3.2.11}
Let $\operatorname{un}derline{x} = x_i, \ldots, x_d, i > 1$, be a part of a $C$-system of parameters of $M$. Then $\mathfrak{b}(M/(x_i, \ldots,x_d)M) =
\mathfrak{b}(M/(x_i^{n_i}, \ldots,x_d^{n_d})M)$ for all $n_j \geq 1$
and all $i \leq j \leq d$.
\end{corollary}
\begin{proof}
For $i = d$, notice that $\operatorname{un}derline{y} = y_1,\ldots,y_{d-1}$ is a system of parameters of $M/x_dM$ if and only if it is also a system of parameters of
$M/x_d^{n_d}M$ for all $n_d \geq 1$. By Lemma \ref{B3.1.9} we have $x_d$
and hence $x_d^{n_d}$ are contained in $ \frak b(M/(y_1,\ldots,y_{j-1})M)^3$ for all $1 \leq j \leq d-1$. So Theorem \ref{D3.2.9} claims that
$$U_{M/(y_1,\ldots,y_{j-1},x_d)M}(0) \cong
U_{M/(y_1,\ldots,y_{j-1},x_d^{n_d})M}(0)$$
for all $1 \leq j \leq d-1$.
By Remark \ref{C3.2.2} (iii) we have
\begin{eqnarray*}
\mathfrak{b}(M/x_dM) &=& \bigcap_{\operatorname{un}derline{y}; j=1}^{d-1}
\mathrm{Ann}\,U_{M/(y_1,\ldots,y_{j-1},x_d)M}(0)\\
&=& \bigcap_{\operatorname{un}derline{y}; j=1}^{d-1}
\mathrm{Ann}\,U_{M/(y_1,\ldots,y_{j-1},x_d^{n_d})M}(0)\\
&=&\mathfrak{b}(M/x_d^{n_d}M),
\end{eqnarray*}
where $\operatorname{un}derline{y} = y_1,\ldots,y_{d-1}$ runs over all systems of parameters of $M/x_dM$.\\
We now proceed by induction on $d$. The case $d=2$
follows from the above fact since $i = 2$. Suppose $d \geq 3$ and
$i<d$. Applying the inductive hypothesis for $M/(x_{i+1},\ldots,x_d)M$ we have
$$\frak
b(M/(x_i,x_{i+1},\ldots,x_d)M) = \frak
b(M/(x_i^{n_i},x_{i+1},\ldots,x_d)M)$$ for all $n_i \geq 1$. By Lemma \ref{B3.1.10} we have $x_{i+1},\ldots,x_d$ is a part of a $C$-system of parameters of $M/x_i^{n_1}M$. By using the inductive hypothesis for $M/x_i^{n_i}M$ we obtain
$$\frak b(M/(x_i^{n_i},x_{i+1},\ldots,x_d)M) =
\frak b(M/(x_i^{n_i},x_{i+1}^{n_{i+1}}\ldots,x_d^{n_d})M)$$ for all
$n_{i+1},\ldots,n_{d} \geq 1$. The proof is complete.
\end{proof}
\begin{corollary} \label{H3.2.12}
Let $\operatorname{un}derline{x} = x_1, \ldots, x_d$ be a $C$-system of parameters of $M$. Then for all $d$-tuples of positive integers $\operatorname{un}derline{n} =(n_1,\ldots,n_d)$ we have
$x_{1}^{n_1},
\ldots,x_d^{n_d}$ is also a $C$-system of parameters.
\end{corollary}
\begin{proof} The assertion follows immediately from Corollary \ref{H3.2.11} and the definition of $C$-system of parameters.
\end{proof}
\operatorname{no}indent {\bf An application to $dd$-sequences.} We use the Cohen-Macaulay deviated sequence to compute the function $I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n})$.
\begin{proposition} \label{M3.2.13}
Let $\operatorname{un}derline{x} = x_1, \ldots, x_d$ be a $C$-system of parameters of $M$. Let $U_i(M)$, $0 \leq i \leq d-1$, be the Cohen-Macaulay deviated sequence of $M$. Then the difference
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) =
\ell(M/(x_1^{n_1},\ldots,x_d^{n_d})M) - n_1\ldots n_d e(x_1,\ldots,x_d;M)$$ is a
polynomial in $\operatorname{un}derline{n} = n_1,\ldots,n_d$. More precisely
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) = \sum_{i=0}^{p(M)}n_1\ldots n_i e(x_1,\ldots,x_i;U_i(M))$$
for all $n_i \geq 1$, where $p(M)$ is the polynomial type of $M$. In particular,
$\operatorname{un}derline{x} = x_1, \ldots, x_d$ is a $dd$-sequence system of parameters.
\end{proposition}
\begin{proof}
For all $d$-tuples of positive integers $\operatorname{un}derline{n} = (n_1,\ldots,n_d)$ by
Corollary \ref{H3.2.12} we have $x_{1}^{n_1},
\ldots,x_d^{n_d}$ is a $C$-system of parameters. By Theorem \ref{D3.2.9} and Remark \ref{C3.2.2} (ii) we have
$${(x_{i+2}^{n_{i+2}},\ldots,x_d^{n_{d}})M:_M x_{i+1}^{n_{i+1}}}/{(x_{i+2}^{n_{i+2}},\ldots,x_d^{n_{d}})M} \cong U_i(M)$$
for all $0 \leq i \leq d-1$.
By the Auslander-Buchsbaum formula (cf. \cite[Corollary 4.3]{AB58}) we have
\begin{eqnarray*}
I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) &=&
\sum_{i=0}^{d-1}e(x_{1}^{n_{1}},
\ldots,x_i^{n_i};{(x_{i+2}^{n_{i+2}},\ldots,x_d^{n_{d}})M:_M x_{i+1}^{n_{i+1}}}/{(x_{i+2}^{n_{i+2}},\ldots,x_d^{n_{d}})M})\\
&=& \sum_{i=0}^{d-1}e(x_{1}^{n_{1}},
\ldots,x_i^{n_i};U_i(M))\\
&=& \sum_{i=0}^{d-1}n_1\ldots n_i e(x_1,\ldots,x_i;U_i(M))
\end{eqnarray*}
is a polynomial in $n_1,\ldots,n_d$. By Remark \ref{C3.2.2}
(iii) we have $\mathrm{Ann}U_i(M) \supseteq \frak b(M)$ for all $i
\leq d-1$. Thus $\dim U_i \leq p(M)$ for all $i \leq d-1$ since $\dim
R/\frak b(M) = \dim R/\frak a(M) = p(M)$. Therefore $e(x_1,\ldots,x_i;U_i(M)) = 0$ for all $p(M) < i \leq d-1$. Hence
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) = \sum_{i=0}^{p(M)}n_1\ldots n_i e(x_1,\ldots,x_i;U_i(M)).$$
The last assertion follows from Proposition \ref{M3.1.7}. The proof is complete.
\end{proof}
The following is in some sense a generalization of Proposition \ref{M3.1.7} (see also \cite[Theorem 3.7]{CN}).
\begin{corollary}
Let $\operatorname{un}derline{x} = x_1, \ldots, x_d$ be a $dd$-sequence system of parameters of $M$. Let $U_i(M)$, $0 \leq i \leq d-1$, be the Cohen-Macaulay deviated sequence of $M$. Then the difference
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) = \sum_{i=0}^{p(M)}n_1\ldots n_i e(x_1,\ldots,x_i;U_i(M))$$
for all $n_i \geq 1$, where $p(M)$ is the polynomial type of $M$.
\end{corollary}
\begin{proof} Notice that if $\operatorname{un}derline{x} = x_1, \ldots, x_d$ is a $dd$-sequence system of parameters of $M$, then $\operatorname{un}derline{x}^k = x_1^k, \ldots, x_d^k$ is a $C$-system of parameters for some $k \ge 1$ (see Remark \ref{R dd seq}). So we have
$$I_{M,\operatorname{un}derline{x}^k}(\operatorname{un}derline{n}) = \sum_{i=0}^{p(M)}k^in_1\ldots n_i e(x_1,\ldots,x_i;U_i(M))$$
for all $n_i \geq 1$. By Proposition \ref{M3.1.7} we have
$$I_{M,\operatorname{un}derline{x}}(kn_1, \ldots, kn_d) =
\sum_{i=0}^{p(M)} k^in_1\ldots n_i e(x_1,\ldots,x_i; 0:_{M/(x_{i+2},\ldots,x_d)M}x_{i+1})$$
for all $n_i \geq 1$. However it is clear that $I_{M,\operatorname{un}derline{x}^k}(\operatorname{un}derline{n}) = I_{M,\operatorname{un}derline{x}}(kn_1, \ldots, kn_d)$. By the above equality we have
$$e(x_1,\ldots,x_i;U_i(M)) = e(x_1,\ldots,x_i; 0:_{M/(x_{i+2},\ldots,x_d)M}x_{i+1})$$
for all $i \le p(M)$. Therefore
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) = \sum_{i=0}^{p(M)}n_1\ldots n_i e(x_1,\ldots,x_i;U_i(M))$$
for all $n_i \geq 1$ by Proposition \ref{M3.1.7} again. The proof is complete.
\end{proof}
\operatorname{no}indent{\bf Sequentially Cohen-Macaulay modules.} We give an application of the Cohen-Macaulay deviate sequence to characterize {\it sequentially Cohen-Macaulay} modules. This notion firstly introduced by Stanley in the graded rings \cite{St96}, and for modules over local rings by Schenzel in \cite{Sch98}, and by Nhan and the first author in \cite{CN03}.
\begin{remark}[\cite{CC07-2}] \rm
\begin{enumerate}[{(i)}]
\item The filtration of submodules $\mathcal{D}: D_0 \subset D_1 \subset \operatorname{cd}ots \subset D_t =M$ of $M$ is called the {\it the dimension filtration} if $D_i = U_{D_{i+1}}(0)$ for all $i \leq t-1$.
\item We call $M$ is a {\it sequentially Cohen-Macaulay} module if $D_{i+1}/D_i$ is Cohen-Macaulay for all $i \leq t-1$.
\item A system of parameters $\operatorname{un}derline{x} =
x_1,\ldots,x_d$ of $M$ is called {\it good} if $D_i \cap (x_{d_i+1},\ldots,x_d)M = 0$ for
$i= 0, 1, \ldots, t-1$, where $d_i = \dim D_i$ for all $i \leq t$. Notice that every $dd$-sequence system of parameters is good.
\end{enumerate}
\end{remark}
\begin{remark}\label{Q3.2.14} \rm Let $M$ be a finitely generated $R$-module of dimension $d$ with the dimension filtration
$$\mathcal{D}: D_0 \subset D_1 \subset \operatorname{cd}ots \subset D_t =M,$$
with $d_i = \dim D_i $ for all $i \leq t$. Let $\operatorname{un}derline{x} = x_1,
\ldots, x_d$ be a $C$-system of parameters of $M$. For each
$i<t$ and $d_i \leq j \leq d-1$ we have
$$D_i \cap (x_{j+2},\ldots,x_d)M = 0.$$
Therefore we can identify $D_i$ with a submodule of
$M/(x_{j+2},\ldots,x_d)M$. Moreover, since $\dim D_i = d_i < j+1 = \dim
M/(x_{j+2},\ldots,x_d)M$, $D_i$ is isomorphism to a submodule of
$U_j(M)$ for all $d_i \leq j \leq d-1$. So without of any confusion we write $D_i \subseteq U_j(M)$ for all $d_i \leq j \leq d-1$.
\end{remark}
The following is a characterization of sequentially Cohen-Macaulay modules.
\begin{proposition}\label{M3.2.15} Let $M$ be a finitely generated $R$-module of dimension $d$ with the dimension filtration
$$\mathcal{D}: D_0 \subset D_1 \subset \operatorname{cd}ots \subset D_t =M,$$
with $d_i = \dim D_i $ for all $i \leq t$. Let $U_i(M)$, $0 \leq i \leq d-1$, be the Cohen-Macaulay deviated sequence of $M$. The following statements are equivalent
\begin{enumerate}[{(i)}]\rm
\item {\it $M$ is a sequentially Cohen-Macaulay modules.}
\item {\it $D_i = U_j(M)$ for all $i<t$ and for all $d_i \leq j <d_{i+1}$.}
\end{enumerate}
\end{proposition}
\begin{proof} $\mathrm{(i)}\Rightarrow \mathrm{(ii)}$ Let $\operatorname{un}derline{x} = x_1,
\ldots, x_d$ be a $C$-system of parameters of $M$. By Proposition \ref{M3.2.13} it is a $dd$-sequence. By \cite[Lemma 6.4]{CC07-1}, $M/(x_{j+2},\ldots,x_d)M$ is a sequentially Cohen-Macaulay module with the dimension filtration \\
$$ D_0 \cong \frac{D_0+(x_{j+2},\ldots,x_d)M}{(x_{j+2},\ldots,x_d)M} \subset \operatorname{cd}ots \subset D_i\cong \frac{D_i+(x_{j+2},\ldots,x_d)M}{(x_{j+2},\ldots,x_d)M} \subset M/(x_{j+2},\ldots,x_d)M$$
for all $i<t$ and for all $d_i \leq j <d_{i+1}$. Thus $D_i = U_j(M)$ for all $i<t$ and for all $d_i \leq j <d_{i+1}$.\\
$\mathrm{(ii) }\Rightarrow \mathrm{(i)}$ Let $\operatorname{un}derline{x} = x_1,
\ldots, x_d$ be a $C$-system of parameters of $M$. By Proposition
\ref{M3.2.13} we have
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) = \sum_{j=0}^{d-1}n_1\ldots n_j e(x_1,\ldots,x_j;U_j(M))$$
for all $n_1,..,n_d \geq 1$.
Since $D_i = U_j(M)$ for all $i<t$ and for all $d_i \leq j <d_{i+1}$ we have
$e(x_1,\ldots,x_j;U_j(M)) = 0$ for all $i<t$ and for all $d_i < j <d_{i+1}$. Therefore
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) = \sum_{i=0}^{t-1}n_1\ldots n_{d_i} e(x_1,\ldots,x_{d_i};D_i)$$
for all $n_1,..,n_d \geq 1$. Hence $M$ is a sequentially Cohen-Macaulay module by \cite[Theorem 4.2]{CC07-2}. The proof is complete.
\end{proof}
\operatorname{no}indent{ \bf Relation with the Serre condition ($S_2$).} For each $R$-module $M$ we have a set of invariant modules $U_i(M)$, $0 \leq i \leq d-1$, as Theorem \ref{D3.2.9}. Therefore we have a special set of prime ideals, $\cup_{i=0}^{d-1}\mathrm{Ass}\, U_i(M)$, attached to $M$. If $\frak p \in
\mathrm{Ass}\, M$ and $\dim R/\frak p < d$, then $\frak p \in
\mathrm{Ass}\, U_M(0) = \mathrm{Ass}\, U_{d-1}(M)$. In the following we consider the relation between $\mathrm{Ass}\, U_{d-2}(M)$ and {\it the Serre condition
$(S_2)$}.
\begin{definition}\rm For all $n \geq 1$, we say that $M$ satisfies {\it the Serre condition
$(S_n)$} at the prime ideal $\frak p \in \mathrm{Supp}(M)$ if
$$\mathrm{depth}M_\frak p \geq \min \{\dim M_{\frak p}, n\}.$$
Moreover, $M$ has property $(S_n)$ if it satisfies the Serre condition $(S_n)$ at all $\frak p \in
\mathrm{Supp}(M)$.
\end{definition}
It is obvious that $R$ satisfies the condition $(S_1)$ if and only if
$\mathrm{Ass}\, R = \mathrm{minAss}R$. Furthermore, if $R$ satisfies the condition
$(S_2)$ and $R$ is {\it cartenary} (this condition is always true if $R$
is a homomorphic image of a Cohen-Macaulay ring), then $\mathrm{Ass}\, R =
\mathrm{Assh}R$ (see \cite[Corollary
2.24]{Sch98-1}). Conversely, Goto and Nakamura \cite[Lemma
3.2]{GN01} proved that if $\mathrm{Ass}\, R \subseteq
\mathrm{Assh}R \cup \{\frak m\}$, then the set
$$\mathcal{F}(R) = \{\mathfrak{p}\in \mathrm{Spec}(R)\,|\, \dim R_{\mathfrak{p}}>
1=\mathrm{depth}R_{\mathfrak{p}},\, \mathfrak{p} \neq \mathfrak{m}
\}$$ is finite, i.e. $R$ does not satisfy the Serre condition $(S_2)$ at only finitely many prime ideals. The set $\mathcal{F}(R)$ can be described as follows.
\begin{proposition}\label{M3.2.17}
Suppose that $\mathrm{Ass}\, M \subseteq \mathrm{Assh}M
\cup \{\frak m\}$. Set
$$\mathcal{F}(M) = \{\mathfrak{p}\in \mathrm{Supp}(M)\,|\, \dim M_{\mathfrak{p}}>
1=\mathrm{depth}M_{\mathfrak{p}},\, \mathfrak{p} \neq \mathfrak{m}
\}.$$ Then $\mathcal{F}(M) = \mathrm{Ass}\, U_{d-2}(M) \setminus
\{\frak m\}$.
\end{proposition}
\begin{proof} Let $x$ be a $C$-parameter element of $M$. For all $\frak p \in \mathrm{Ass}\, U_{d-2}(M) \setminus \{\frak
m\}$ we have $\frak p \in \mathrm{Ass}\, M/xM$ and $\dim R/\frak p \leq
d-2$. Hence $\dim M_{\mathfrak{p}}>
1=\mathrm{depth}M_{\mathfrak{p}}$. So $\mathrm{Ass}\, U_{d-2}(M)
\setminus \{\frak m\} \subseteq \mathcal{F}(M)$.\\
Conversely, let $\mathfrak{p}\in \mathcal{F}(M)$. Since
$\mathrm{depth}M_{\mathfrak{p}} = 1$, for every parameter element $z
\in \mathfrak{p}$ we have $\mathfrak{p}\in \mathrm{Ass}\, M/zM$. Therefore
$\mathfrak{p}\in \mathrm{Ass}\, M/(xz)M$. Notice that $xz$ is a $C$-parameter element of $M$ and $\dim R/\frak p \leq
d-2$, so $\frak p \in \mathrm{Ass}\, U_{M/(xz)M}(0) \cong
\mathrm{Ass}\, U_{d-2}(M)$. The proof is complete.
\end{proof}
\begin{remark}\label{C3.2.18}\rm Let $M$ be a finitely generated $R$-module.
\begin{enumerate}[{(i)}]
\item Suppose that $\mathrm{Ass}\, M \subseteq \mathrm{Assh}M \cup
\{\frak m\}$ and $\mathcal{F}(M)$ as the previous proposition. Let $x$ be a parameter element of $M$ such that $x
\operatorname{no}tin \frak p$ for all $\frak p \in \mathcal{F}(M)$. Then $M$
satisfies the Serre condition $(S_2)$ at all prime ideals $\frak p \in \mathrm{supp}M$ containing $x$ and $\frak p
\neq \frak m$. So $M/xM$ satisfies the Serre condition $(S_1)$ at all $\frak p \in \mathrm{Supp}(M/xM)$ and $\frak p
\neq \frak m$. Hence
$$\mathrm{Ass}\, (M/xM) \subseteq \mathrm{minAss}(M/xM) \cup \{\frak m\} = \mathrm{Assh}(M/xM) \cup \{\frak m\}.$$
\item Set $\overline{M} = M/U_M(0)$. Let $x \in \frak b(M)^3 \cap \frak
b(\overline{M})^3$ be a parameter element of $M$ and hence of $\overline{M}$. Put $\frak b' = \frak b(M/xM)$,
$\frak b'' = \frak b(\overline{M}/x\overline{M})$ and $\frak b =
\frak b' \cap \frak b''$. We have $\dim R/\frak b \leq d-2$. By Remark \ref{C3.2.2} (i) we have
$U_{d-2}(M) \cong H^0_{\frak b'}(M/xM) \subseteq H^0_{\frak
b}(M/xM)$. However $\dim H^0_{\frak b}(M/xM) < d-1$, so
$U_{d-2}(M) \cong H^0_{\frak b}(M/xM)$. Similarly, we have
$U_{d-2}(\overline{M}) \cong H^0_{\frak
b}(\overline{M}/x\overline{M})$. By the proof of Lemma \ref{B3.2.8} we have
$$U_{d-2}(M) \cong H^0_{\frak
b}(M) \oplus H^1_{\frak b}(\overline{M}/x\overline{M})$$ and
$$U_{d-2}(\overline{M}) \cong H^0_{\frak
b}(\overline{M}) \oplus H^1_{\frak b}(\overline{M}/x\overline{M}) =
H^1_{\frak b}(\overline{M}/x\overline{M}).$$
Therefore
$U_{d-2}(\overline{M})$ is isomorphism to a direct summand of $U_{d-2}(M)$.
\end{enumerate}
\end{remark}
The following plays an important role in the next section.
\begin{proposition}\label{M3.2.19} Let $M$ be a finitely generated $R$-module of dimension $d \geq
2$. Let $x$ be a parameter element of $M$ such that $x \operatorname{no}tin \frak p$ for all $\frak p \in \big( \mathrm{Ass}\, U_{M}(0) \cup \mathrm{Ass}\, U_{d-2}(M) \big)
\setminus \{\frak m\}$. Then we have the following short exact sequence
$$0 \to U_M(0)/xU_M(0) \to U_{M/xM}(0) \to H^0_{\frak m}(\overline{M}/x\overline{M}) \to 0,$$
where $\overline{M} = M/U_M(0)$.
\end{proposition}
\begin{proof}
Since $U_M(0) \cap xM = x(U_M(0) :_Mx) = xU_M(0)$, we have the following short exact sequence
$$0 \to U_M(0)/xU_M(0) \overset{\varphi}{\to} M/xM \to \overline{M}/x\overline{M} \to 0.$$
If $\dim U_M(0) = 0$ then $\dim U_M(0)/xU_M(0) < d-1$. If $\dim
U_M(0) > 0$ then $x$ is a parameter element of both $M$ and $U_M(0)$ so
$\dim U_M(0)/xU_M(0) = \dim U_M(0) - 1 < d-1$. Notice that $\mathrm{Im}(\varphi) = (U_M(0) + xM)/xM$. Thus we always have
$(U_M(0) + xM)/xM$ is a submodule of $M/xM$ of dimension less than $d-1$. Hence $\mathrm{Im}(\varphi) = (U_M(0) + xM)/xM \subseteq U_{M/xM}(0)$. So we have the short exact sequence
$$0 \to U_M(0)/xU_M(0) \to U_{M/xM}(0) \to U_{\overline{M}/x\overline{M}}(0) \to 0.$$
On the other hand $x \operatorname{no}tin \frak p$ for all $\frak p \in
\mathrm{Ass}\, U_{d-2}(M) \setminus \{\frak m\}$. So $x \operatorname{no}tin \frak p$
for all $\frak p \in \mathrm{Ass}\, U_{d-2}(\overline{M}) \setminus
\{\frak m\}$ by Remark \ref{C3.2.18} (ii). By Remark \ref{C3.2.18} (i) we have
$$\mathrm{Ass}\, (\overline{M}/x\overline{M}) \subseteq
\mathrm{Assh}(\overline{M}/x\overline{M}) \cup \{\frak m\}.$$ Therefore
$U_{\overline{M}/x\overline{M}}(0) = H^0_{\frak
m}(\overline{M}/x\overline{M})$. Thus we obtain the short exact sequence
$$0 \to U_M(0)/xU_M(0) \to U_{M/xM}(0) \to H^0_{\frak m}(\overline{M}/x\overline{M}) \to 0.$$
The proof is complete.
\end{proof}
\section{The unmixed degree}
In this section let $I$ be an $\frak m$-primary ideal and $M$ a finitely generated
$R$-module of dimension $d
> 0$. Let $U_i(M)$, $0 \leq i \leq d-1$, be the Cohen-Macaulay deviated sequence of $M$. The purpose of this section is to construct a new degree for
$M$ in terms of $U_i(M)$. Firstly, recalling that
the length function $\ell(M/I^nM)$ becomes a polynomial of degree $d$ when $n \gg 0$ and
$$\ell(M/I^{n+1}M) = \sum_{i=0}^d(-1)^i e_i(I,M) \binom{n+d-i}{d-i}.$$
The coefficients $e_i(I, M)$, $i = 0,\ldots,d$ are called
the Hilbert coefficients of $M$ with respect to $I$. Especially, the leading coefficient
$e_0(I, M)$ is called {\it the Hilbert-Samuel multiplicity} of $M$ with respect to $I$. If $I=\frak m$, the multiplicity is written by $e_0(M)$ for simply. In the present paper we denote by $\mathrm{deg}(I, M)$ (resp.
$\mathrm{deg}(M)$) the multiplicity $e_0(I, M)$ (resp. $e_0(M)$) and call {\it the degree} of $M$ with respect to $I$ (resp. the degree of $M$). The following associativity formula for degree says that $\mathrm{deg}(I, M)$ depends only on the associated prime ideals of the highest dimension
(see \cite[Corollary 4.7.8]{BH98})
$$\mathrm{deg}(I, M) = \sum_{\frak p \in \mathrm{Assh}M}\ell_{R_{\frak p}}(M_{\frak p}) \mathrm{deg}(I, R/\frak p). $$
Notice that if $\frak p \in \mathrm{minAss}M$, then $M_{\frak p}$ has finite length and
$M_{\frak p} = H^0_{\frak p R_{\frak p}}(M_{\frak p})$. So we have
$$\mathrm{deg}(I, M) = \sum_{\frak p \in \mathrm{Assh}M}
\ell_{R_{\frak p}}(H^0_{\frak p R_{\frak p}}(M_{\frak p}))
\mathrm{deg}(I, R/\frak p). $$ We next recall some other degrees of $M$ related to $\mathrm{deg}(I, M)$
(see \cite{V98-2}).
\begin{definition}\label{adeg}\rm
The {\it arithmetric degree} of $M$ with respect to $I$, denoted by $\mathrm{adeg}(I, M)$, is the integer
$$\mathrm{adeg}(I, M) =
\sum_{\frak p \in \mathrm{Ass}\, M}\ell_{R_{\frak p}}(H^0_{\frak p
R_{\frak p}}(M_{\frak p})) \mathrm{deg}(I, R/\frak p). $$
\end{definition}
\begin{remark} \label{C3.3.2} \rm
\begin{enumerate}[{(i)}]
\item Let $\mathcal{D} : D_0 \subseteq D_1 \subseteq \operatorname{cd}ots
\subseteq D_t = M$ be the dimension filtration of $M$ we have $\mathrm{adeg}(I, M)
= \sum_{i=0}^t \mathrm{deg}(I, D_i)$. So $\mathrm{adeg}(I, M) \geq \mathrm{deg}(I,
M)$ and the equation occurs if and only if $U_M(0) = 0$.
\item Moreover, if $(R, \frak m)$ is a homomorphic image of a Gorenstein local ring $(S, \frak n)$ of dimension
$n$. Then $\mathrm{adeg}(I, M)$ can be determined without the knowledge of the primary decomposition as follows
$$\mathrm{adeg}(I, M) = \sum_{i} \mathrm{deg}(I, \mathrm{Ext}^{i}_S(\mathrm{Ext}^{i}_S(M, S), S)).$$
\end{enumerate}
\end{remark}
Vasconcelos et al. \cite{DGV98, V98-1, V98-2} introduced the notion of {\it extended degree of graded modules} in order to capture the size of a module along with some of the complexity of its
structure. The prototype of an extended degree is the {\it homological degree} was introduced and studied by Vasconselos in \cite{V98-1} (see also \cite{V98-2}). The extended degree for local rings was considered by Rossi, Trung and Valla in \cite{RTV03}.
This notion is associated by an $\frak m$-primary ideal $I$ in \cite{L05}.
\begin{definition}\label{D3.3.3}\rm Let $\mathcal{M}(R)$ be the category of finitely generated
$R$-modules. An {\it extended degree} on $\mathcal{M}(R)$
with respect to $I$ is a numerical function
$$ \mathrm{Deg}(I, \bullet) : \mathcal{M}(R) \to \mathbb{R} $$
satisfying the following conditions
\begin{enumerate}[{(i)}]
\item $\mathrm{Deg} (I, M) = \mathrm{Deg}(I, \overline{ M}) + \ell(H^0_{\frak
m}(M))$, where $\overline{M} = M/H^0_{\frak m}(M)$;
\item (Bertini's rule) $\mathrm{Deg}(I, M) \geq \mathrm{Deg}(I, M/xM)$ for every generic element $x \in I\setminus \frak mI$ of $M$;
\item If $M$ is Cohen-Macaulay then $\mathrm{Deg}(I, M) =
\mathrm{deg}(I, M)$.
\end{enumerate}
\end{definition}
The homological degree is a typical extended degree that is defined as follows.
\begin{definition}[\cite{V98-1}] \label{D3.3.4} \rm Supppose that $(R, \frak m)$ be a homomorphic image of a Gorenstein local ring
$(S, \frak n)$ of dimension $n$, and $M$ a finitely generated $R$-module of dimension $d$. Then the {\it homological degree}, $\mathrm{hdeg}(I, M)$, of $M$ with respect to $I$ is defined by the following recursive formula
$$\mathrm{hdeg}(I, M) = \mathrm{deg}(I, M) + \sum_{i=n-d+1}^n \binom{d-1}{i-n+d-1} \mathrm{hdeg}(I, \mathrm{Ext}^i_S(M, S)).$$
\end{definition}
\begin{remark} \label{C3.3.5} \rm
\begin{enumerate}[{(i)}]
\item The Definition \ref{D3.3.4} is recursive on dimension since $\dim \mathrm{Ext}^i_S(M, S) \le n-i <
d$ for all $i = n-d+1,\ldots, n$.
\item $\mathrm{hdeg}(I,\bullet)$ is an extended degree on $\mathcal{M}(R)$, and $\mathrm{hdeg}(I,M) =
\mathrm{deg}(I, M)$ if and only if $M$ is Cohen-Macaulay.
\item If $M$ is a generalized Cohen-Macaulay module, then $\ell(\mathrm{Ext}^{n-i}_S(M,
S)) = \ell(H^i_{\frak m}(M))$ for all $i = 0, \ldots, d-1$ by the local duality theorem. We have
$$\mathrm{hdeg}(I, M) = \mathrm{deg}(I, M) + \sum_{i=0}^{d-1} \binom{d-1}{i} \ell(H^i_{\frak m}(M)).$$
\item (\cite[Proposition 3.5]{V98-2}) If $\dim M = \dim S =2$ then
$$\mathrm{hdeg}(I, M) = \mathrm{adeg}(I, M) + \ell(\mathrm{Ext}^2_S(\mathrm{Ext}^1_S(M, S),S)).$$
\end{enumerate}
\end{remark}
Until nowadays, the homological degree is the uniquely explicit extended degree. The purpose of this section is to introduce an other extended degree on $\mathcal{M}(R)$ in terms of the Cohen-Macaulay deviated sequence $U_i(M)$, $i = 0,\ldots,d-1$. Notice that $\dim U_i(M) \leq i$ for all $0 \leq
i \leq d-1$.
\begin{definition}\rm Let $M$ be a finitely generated $R$-module of dimension
$d$ and $U_i(M)$, $0 \leq i \leq d-1$, the Cohen-Macaulay deviated sequence of $M$. We define the {\it unmixed degree} of $M$ with respect to
$I$, $\mathrm{udeg}(I, M)$, as follows
$$\mathrm{udeg}(I, M) = \mathrm{deg}(I, M) + \sum_{i=0}^{d-1}\delta_{i, \dim U_i(M)}\mathrm{deg}(I, U_i(M)),$$
where $\delta_{i, \dim U_i(M)}$ is the Kronecker symbol.
\end{definition}
It is worth noting that in the above definition and Proposition \ref{M3.2.13} we consider the subsequence of modules of the Cohen-Macaulay deviated sequence consisting $U_i(M)$ with $\dim U_i(M) = i$. We call this subsequence the {\it reduced Cohen-Macaulay deviated sequence} of $M$. In the rest of this paper, we shall prove that the unmixed degree is an extended degree. The first condition of Definition \ref{D3.3.3} follows from the following.
\begin{proposition}\label{M3.3.9}
Let $N$ be a submodule of finite length of $M$. Then
$$\mathrm{udeg}(I,M) = \mathrm{udeg}(I,M/N) + \ell(N).$$
\end{proposition}
\begin{proof}
Let $x_1,\ldots,x_d$ be a $C$-system of parameters of both $M$ and $M/N$. By Proposition \ref{M3.2.13}
$x_1,\ldots,x_d$ is a $dd$-sequence of $M$. So $H^0_{\frak m}(M) \cap
(x_1,\ldots,x_d)M = 0$. For all $0 \leq j \leq d-1$, we have the short exact sequence
$$0 \to N \to M/(x_{j+2},\ldots,x_d)M \to M/(N+(x_{j+2},\ldots,x_d)M) \to 0.$$
Therefore $U_j(M/N) \cong U_j(M)/N$ for all $0 \leq j \leq d-1$. Thus
$$\delta_{j, \dim U_j(M/N)} \mathrm{deg}(I,U_j(M/N)) =
\delta_{j, \dim U_j(M)} \mathrm{deg}(I,U_j(M))$$
for all $1 \leq j \leq d-1$ and
$$\delta_{0, \dim U_0(M/N)} \mathrm{deg}(I,U_0(M/N)) =
\delta_{0, \dim U_0(M)} \mathrm{deg}(I,U_0(M)) - \ell(N).$$ The claim is now obvious.
\end{proof}
The next result shows that $\mathrm{udeg}(M)$ agrees with $\mathrm{hdeg}(M)$ for generalized Cohen-Macaulay modules.
\begin{proposition} \label{M3.3.10} Let $M$ be a generalized Cohen-Macaulay $R$-module of dimension $d$. Then
$$\mathrm{udeg}(I,M) = \mathrm{deg}(I,M) + \sum_{j=0}^{d-1} \binom{d-1}{j}\ell(H^j_\frak{m}(M)).$$
\end{proposition}
\begin{proof}
Let $x_1,\ldots,x_d$ be a $C$-system of parameters of $M$. By Corollary \ref{H3.2.5} (see also \cite[Corollary 4.2]{CQ11}) we have
$$U_i(M) \cong H^0_{\mathfrak{m}}(M/(x_{i+2},\ldots,x_d)M) \cong
\bigoplus_{j=0}^{d-i-1} H^j_{\mathfrak{m}}(M)^{\binom{d-i-1}{j}}$$
for all $0 \leq i \leq d-1$. So $\dim U_i(M) = 0$ for all $i \le d-1$. Therefore $\delta_{i, \dim U_i(M)}\mathrm{deg}(I,U_i(M))
= 0$ for all $1 \leq i \leq d-1$ and
$$\delta_{0, \dim U_0(M)}\mathrm{deg}(I,U_0(M)) = \sum_{j=0}^{d-1}
\binom{d-1}{j}\ell(H^j_\frak{m}(M)).$$
The proof is complete.
\end{proof}
We next compute the unmixed degree when $\dim M$ is small.
\begin{proposition}\label{M3.3.11} The following statements hold true.
\begin{enumerate}[{(i)}]\rm
\item {\it If $d = 1$ then $\mathrm{udeg}(I,M) = \mathrm{adeg}(I,M)$.}
\item {\it If $d = 2$ then $\mathrm{udeg}(I,M) = \mathrm{adeg}(I,M) + \ell(H^1_{\frak m}(M/U_M(0)))$.}
\end{enumerate}
\end{proposition}
\begin{proof} (i) It is clear.\\
(ii) We consider the following two cases.\\
The case $\dim U_M(0) = 0$, we have $M$ is a generalized
Cohen-Macaulay modules. Therefore by Proposition \ref{M3.3.10} we have
\begin{eqnarray*}
\mathrm{udeg}(I,M) &=& \mathrm{deg}(I,M) + \ell(H^0_{\frak m}(M)) +
\ell(H^1_{\frak m}(M))\\
&=& \mathrm{adeg}(I,M) + \ell(H^1_{\frak m}(M/H^0_{\frak m}(M))).
\end{eqnarray*}
The case $\dim U_M(0) = 1$. Consider the dimension filtration $H^0_{\frak m}(M) \subset U_M(0) \subset M$ of $M$. By Remark \ref{C3.3.2} (i) we have
$$\mathrm{adeg}(I,M) =
\mathrm{deg}(I,M) + \mathrm{deg}(I,U_M(0)) + \ell(H^0_{\frak
m}(M)).$$ On the other hand $U_1(M) \cong U_M(0)$ so
$\delta_{1, \dim U_1(M)}\mathrm{deg}(I,U_1(M)) = \mathrm{deg}(I,U_M(0)) $. Let
$x_2$ be a $C$-parameter element of $M$. By Corollary \ref{H3.2.5} we have
$$U_0(M) \cong H^0_{\frak m}(M/x_2M) \cong H^0_{\frak m}(M) \oplus H^1_{\frak m}(M/U_M(0)).$$
Thus $\delta_{0, \dim U_0(M)}\mathrm{deg}(I,U_0(M)) = \ell (H^0_{\frak m}(M)) + \ell (H^1_{\frak m}(M/U_M(0)))$. Therefore we also have
$$\mathrm{udeg}(I,M) = \mathrm{adeg}(I,M) + \ell(H^1_{\frak m}(M/U_M(0))).$$
The proof is complete.
\end{proof}
\begin{corollary} Suppose $(R, \frak m)$ is a homomorphic image of a
Gorenstein local ring and $\dim M = 2$. Then $\mathrm{udeg}(I,M) =
\mathrm{hdeg}(I,M) $.
\end{corollary}
\begin{proof} Without loss of generality we may assume that $(R, \frak
m)$ is a Gorenstein local ring of dimension two. If $U_M(0) = H^0_{\frak
m}(M)$ we have $M$ is generalized Cohen-Macaulay, the claim follows from Proposition \ref{M3.3.10} and Remark \ref{C3.3.5} (iii). Suppose $\dim
U_M(0) = 1$, by Proposition \ref{M3.3.11} and Remark \ref{C3.3.5} (iv)
we need only to show that
$$\ell (H^1_{\frak m}(M/U_M(0))) =
\ell(\mathrm{Ext}^2_R(\mathrm{Ext}^1_R(M, R),R)).$$
Since
$\mathrm{Ass}\, M/U_M(0) = \{\frak p \mid \frak p\in \mathrm{Ass}\, M,
\dim R/\frak p = 2 \}$ we have
$\mathrm{Ext}^1_R(M/U_M(0),R)$ is a module of finite length, and
$\ell(\mathrm{Ext}^1_R(M/U_M(0),R)) = \ell(H^1_{\frak m}(M/U_M(0)))$
by local duality theorem. By local duality theorem again we have
$\ell(\mathrm{Ext}^2_R(\mathrm{Ext}^1_R(M, R),R)) = \ell
(H^0_{\frak m}(\mathrm{Ext}^1_R(M, R)))$. So it is enough to prove that
$$\ell(\mathrm{Ext}^1_R(M/U_M(0),R)) = \ell (H^0_{\frak
m}(\mathrm{Ext}^1_R(M, R))).$$
Indeed, consider the short exact sequence
$$0 \to U_M(0) \to M \to M/U_M(0) \to 0.$$
Since $\dim U_M(0) = 1$ and $\mathrm{depth}M/U_M(0)>0$ we have $\mathrm{Hom}_R(U_M(0),R) = \mathrm{Ext}^2_R(M/U_M(0),R) = 0$. So we have the following short exact sequence
$$0 \to \mathrm{Ext}^1_R(M/U_M(0),R) \to \mathrm{Ext}^1_R(M,R) \to \mathrm{Ext}^1_R(U_M(0),R) \to 0.$$
By \cite[Lemma 1.9]{Sch98-1} (v) we have $\mathrm{Ext}^1_R(U_M(0),R)$ is $(S_2)$ and hence it is a Cohen-Macaulay module of dimension one. Thus $H^0_{\frak
m}(\mathrm{Ext}^1_R(U_M(0),R)) = 0$. Therefore
$$\mathrm{Ext}^1_R(M/U_M(0),R)) = H^0_{\frak
m}(\mathrm{Ext}^1_R(M/U_M(0),R))) \cong H^0_{\frak
m}(\mathrm{Ext}^1_R(M, R)).$$ The proof is complete.
\end{proof}
In the following we prove the third condition of Definition \ref{D3.3.3}. Moreover we also give a characterization of sequentially Cohen-Macaulay modules in terms of unmixed degrees.
\begin{theorem}\label{D3.3.8} Let $M$ be a finitely generated $R$-module of dimension
$d$. We have
$$\mathrm{deg}(I, M) \leq \mathrm{adeg}(I, M) \leq \mathrm{udeg}(I, M).$$
Furthermore
\begin{enumerate}[{(i)}]\rm
\item {\it $\mathrm{deg}(I, M) = \mathrm{udeg}(I, M)$ if and only if $M$ is a Cohen-Macaulay module.}
\item {\it $\mathrm{adeg}(I, M) = \mathrm{udeg}(I, M)$ if and only if $M$ is a sequentially
Cohen-Macaulay module.}
\end{enumerate}
\end{theorem}
\begin{proof} The first inequality is clear. Let
$$\mathcal{D}: D_0 \subset D_1 \subset \operatorname{cd}ots \subset D_t =M$$
be the dimension filtration of $M$ with $d_i = \dim D_i $ for all $i \leq t$. Recalling that
$$\mathrm{adeg}(I,M) = \mathrm{deg}(I,M) + \sum_{i=0}^{t-1}\mathrm{deg}(I,D_i).$$
For all $i<t$ by Remark \ref{Q3.2.14} we have $D_i \subseteq
U_{d_i}(M)$. So $\dim U_{d_i}(M) = d_i $ and then
$$\mathrm{deg}(I,D_i) \leq
\mathrm{deg}(I,U_{d_i}(M)) = \delta_{d_i, \dim U_{d_i}(M)}\mathrm{deg}(I,
U_{d_i}(M)).$$
Thus $\mathrm{adeg}(I,M) \leq \mathrm{udeg}(I,M)$. \\
We have (i) follows from (ii), so it is enough to prove (ii). If $M$ is sequentially Cohen-Macaulay, then by Proposition
\ref{M3.2.15} we have $\mathrm{adeg}(I,M) = \mathrm{udeg}(I,M)$.\\
Conversely, suppose
$\mathrm{adeg}(I,M) = \mathrm{udeg}(I,M)$. We have
$$\mathrm{deg}(I,D_i) = \mathrm{deg}(I,U_{d_i}(M)) \quad \quad \quad (\star)$$
for all $i<t$, and
$$\delta_{j, \dim U_j(M)}\mathrm{deg}(I,U_j(M)) = 0 \quad \quad \quad (\star \star)$$ for all $i< t$ and $d_i < j
< d_{i+1}$. Let $\operatorname{un}derline{x} = x_1, \ldots, x_d$ be a $C$-system of parameters of $M$. By $(\star)$ and the associative formula we have
$$e(x_1, \ldots, x_{d_i}; D_i) = \mathrm{deg}((\operatorname{un}derline{x}), D_i) = \mathrm{deg}((\operatorname{un}derline{x}), U_{d_i}(M)) = e(x_1, \ldots, x_{d_i}; U_{d_i}(M))$$
for all $i < t$. By $(\star \star)$ we have $\dim U_j(M)<j$ for all for all $d_i < j
< d_{i+1}$ and $i< t$, so
$$e(x_1, \ldots, x_j; U_j(M)) = 0$$
for all $d_i < j
< d_{i+1}$ and $i< t$. By Proposition \ref{M3.2.13} we have
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) = \sum_{j=0}^{d-1}n_1\ldots n_j e(x_1,\ldots,x_j;U_j(M)).$$
for all $n_1, \ldots, n_d \ge 1$. Thus we have
$$I_{M,\operatorname{un}derline{x}}(\operatorname{un}derline{n}) = \sum_{i=0}^{t-1}n_1\ldots n_{d_i} e(x_1,\ldots,x_{d_i};D_i).$$
for all $n_1, \ldots, n_d \ge 1$. Hence $M$ is a sequentially Cohen-Macaulay module by \cite[Theorem 4.2]{CC07-2}.
The proof is complete.
\end{proof}
In order to prove the Bertini rule of Definition \ref{D3.3.3}, we will show that the unmixed degree has good behavior by passing to the quotient modules regarding certain {\it superficial} elements.
\begin{definition}\rm An element $x \in I \setminus \frak
mI$ is called a {\it superficial} element of $M$ with respect to $I$ if
there exists a positive integer $c$ such that
$$(I^{n+1}M:x) \cap I^cM = I^nM$$
for all $n \geq c$.
\end{definition}
\begin{remark}\label{C3.3.14}\rm
\begin{enumerate}[{(i)}]
\item Let $G_I(R) = \oplus_{n \geq 0} I^n/I^{n+1}$ be the associated graded ring of $R$ with respect to $I$ and $G_I(M)=
\oplus_{n \geq 0} I^nM/I^{n+1}M$ the graded $G_I(R)$-module. Set
$(G_I(R))_+ = \oplus_{n \geq 1} I^n/I^{n+1}$. Then $x$ is a superficial element of $M$ with respect to $I$ if and only if the {\it initial}
$x^*$ of $x$ in $G_I(R)$ is a $(G_I(R))_+$-filter regular element of
$G_I(M)$ i.e. $\ell (0:_{G_I(M)}x^*) < \infty$ (notice that in our context $I$ is $\frak m$-primary). Moreover, if $x$ is a superficial element, then it is an $I$-filter regular element of $M$.
\item A superficial element of $M$
with respect to $I$ always exist if the residue field $R/\frak m$ is infinite, a hypothesis which never cause us any problem because we can replace $R$ by the local ring
$R[X]_{\frak mR[X]}$, where $X$ is an indeterminate. In the sequel we assume that the residue field is infinite.
\item (cf. \cite[22.6]{N62}) Let $x$ be a superficial element of $M$
with respect to $I$. For $n \gg 0$ we have $I^{n+1}M:_Mx = 0:_Mx + I^n
M$ so
$$\ell(M/(I^{n+1}+(x))M) = \ell(M/I^{n+1}M) - \ell(M/I^nM) + \ell(0:_Mx)$$
for all $n \gg 0$.
\item Let $x$ be a superficial element of $M$
with respect to $I$. By (iii) we have $\mathrm{deg}(I,M/xM) =
\mathrm{deg}(I,M) $ if $d \geq 2$, and $\ell(M/xM) =
\mathrm{deg}(I,M/xM) = \mathrm{deg}(I,M) + \ell(0:_Mx)$ if $d=1$.
\end{enumerate}
\end{remark}
We need some lemmas before proving the Bertini rule of unmixed degrees.
\begin{lemma}\label{B3.3.15} Let $M$ be a finitely generated $R$-module of dimension $d \geq
2$. Let $x$ be a parameter element of $M$ such that $x$ is a superficial element of
$U_M(0)$ with respect to $I$ and $x \operatorname{no}tin \frak p$ for all
$\frak p \in \mathrm{Ass}\, U_{d-2}(M) \setminus \{\frak m\}$. Then
$$\delta_{d-2, \dim U_{M/xM}(0)}\mathrm{deg}(I,U_{M/xM}(0)) = \delta_{d-1, \dim U_M(0)}\mathrm{deg}(I,U_{M}(0))$$
if $d \geq 3$, and
$$\delta_{0, \dim U_{M/xM}(0)}\mathrm{deg}(I,U_{M/xM}(0)) = \delta_{1, \dim U_M(0)}\mathrm{deg}(I,U_{M}(0)) +
\ell(0:_{H^0_{\frak m}(M)}x) + \ell(0:_{H^1_{\frak
m}(M/U_{M}(0))}x)$$ if $d=2$.
\end{lemma}
\begin{proof} Put $\overline{M} = M/U_M(0)$, by Proposition
\ref{M3.2.19} we have the short exact sequence
$$0 \to U_M(0)/xU_M(0) \to U_{M/xM}(0) \to H^0_{\frak m}(\overline{M}/x\overline{M}) \to 0.$$
The case $d \geq 3$. If $\dim U_M(0) < d-1$ then $\dim
U_M(0)/xU_M(0) < d-2$. Therefore $\dim U_{M/xM}(0) < d-2$. Hence
$$\delta_{d-2, \dim U_{M/xM}(0)}\mathrm{deg}(I,U_{M/xM}(0)) = 0 =
\delta_{d-1, \dim U_M(0)}\mathrm{deg}(I,U_{M}(0)).$$
If $\dim U_M(0) = d-1$ we have
$\dim U_{M/xM}(0) = d-2 > 0$. So $\mathrm{deg}(I,U_{M/xM}(0)) =
\mathrm{deg}(I,U_M(0)/xU_M(0))$. By Remark \ref{C3.3.14} (iv) we have $\mathrm{deg}(I,U_{M}(0)) =
\mathrm{deg}(I,U_M(0)/xU_M(0))$. Thus we also have
$$\delta_{d-2, \dim U_{M/xM}(0)}\mathrm{deg}(I,U_{M/xM}(0)) =\delta_{d-1, \dim U_M(0)}\mathrm{deg}(I,U_{M}(0)).$$
The case $d=2$, we have $U_{M/xM}(0)$ has finite length.
Therefore
$$\delta_{0, \dim U_{M/xM}(0)}\mathrm{deg}(I,U_{M/xM}(0)) = \ell(U_{M/xM}(0)) =
\ell(U_M(0)/xU_M(0)) + \ell(H^0_{\frak
m}(\overline{M}/x\overline{M})).$$ If $\dim U_M(0) = 1$, by Remark
\ref{C3.3.14} (iv) we have
$$\ell(U_M(0)/xU_M(0)) = \mathrm{deg}(I,U_{M}(0))
+ \ell(0:_{U_M(0)}x) = \delta_{1, \dim U_M(0)}\mathrm{deg}(I,U_{M}(0)) +
\ell(0:_{H^0_{\frak m}(M)}x).$$ If $\dim U_M(0) = 0$ then
we have $U_{M}(0) = H^0_{\frak
m}(M)$ and hence $\delta_{1, \dim U_M(0)}\mathrm{deg}(I,U_{M}(0)) = 0 $. Moreover one can check that $\ell(H^0_{\frak m}(M)/xH^0_{\frak m}(M)) = \ell(0:_{H^0_{\frak m}(M)}x)$. Thus we always have
$$\ell(U_M(0)/xU_M(0)) = \delta_{1, \dim U_M(0)}\mathrm{deg}(I,U_{M}(0)) +
\ell(0:_{H^0_{\frak m}(M)}x).$$ On the other hand the short exact sequence
$$0 \to \overline{M} \overset{x}{\to} \overline{M} \to \overline{M}/x\overline{M} \to 0$$
induces the exact sequence of local cohomology modules
$$0 \to H^0_{\frak m}(\overline{M}/x\overline{M}) \to H^1_{\frak m}(\overline{M}) \overset{x}{\to}
H^1_{\frak m}(\overline{M}).$$
Therefore $\ell(H^0_{\frak
m}(\overline{M}/x\overline{M})) = \ell(0:_{H^1_{\frak
m}(\overline{M})}x)$. Hence
$$\delta_{0, \dim U_{M/xM}(0)}\mathrm{deg}(I,U_{M/xM}(0)) = \delta_{1, \dim U_M(0)}\mathrm{deg}(I,U_{M}(0)) +
\ell(0:_{H^0_{\frak m}(M)}x) + \ell(0:_{H^1_{\frak
m}(M/U_{M}(0))}x).$$
The proof is complete.
\end{proof}
We need one more technical lemma.
\begin{lemma}\label{B3.3.16} Let $M$ be a finitely generated $R$-module of dimension $d \geq 2$. Let $x$ be a parameter element of $M$ such that
$x \operatorname{no}tin \frak p$ for all $\frak p \in
\mathrm{Ass}\, U_M(0) \setminus \{\frak m\}$. Then we can choose a $C$-parameter element $x_d$ of $M$ such that $x$ is a parameter element of $M/x_dM$.
\end{lemma}
\begin{proof}
If $\dim U_M(0) < d-1$ then $\dim
R/\frak b(M) \leq d-2$ by Remark \ref{C3.1.2} (ii). Therefore we can choose a $C$-parameter element
$x_d$ such that $x$ and $x_d$
is a part of a system of parameters of $M$ by the prime avoidance theorem. Hence $x$ is a parameter element of
$M/x_dM$.\\
We now assume that $\dim U_M(0) = d-1$. Set $\overline{M} = M/U_M(0)$. The short exact sequence
$$0 \to U_M(0) \to M \to \overline{M} \to 0.$$
induces the exact sequence of local cohomology modules
$$ \operatorname{cd}ots \to H^i_{\frak m}(U_M(0)) \to H^i_{\frak m}(M) \to H^i_{\frak m}(\overline{M}) \to \operatorname{cd}ots.$$
Hence $\frak a_i(M) = \mathrm{Ann}\, H^i_{\frak m}(M) \supseteq
\mathrm{Ann}\, U_M(0) \, \frak a_i(\overline{M})$ for all $i \geq 0$. So
$$\sqrt{\frak b(M)} = \sqrt{\frak a(M)} \supseteq \sqrt{\mathrm{Ann}\, U_M(0)\,\frak a(\overline{M})} =
\sqrt{\mathrm{Ann}\, U_M(0)\,\frak b(\overline{M})}.$$
We claim that $\frak b(M) \nsubseteq \frak q$ for all $\frak q \in
\mathrm{Assh}\, M/xM$. Indeed, by Remark \ref{C3.1.2} (ii) we have $\dim R/\frak
b(\overline{M}) \leq d-2$. Therefore $\frak b(\overline{M}) \nsubseteq \frak q$.
Suppose $\mathrm{Ann}\, U_M(0) \subseteq \frak q$. Then $\frak q \in
\mathrm{Assh}\, U_M(0)$ since $\dim U_M(0) = \dim R/\frak q = d-1$. It contrasts to
our assumption that $x \operatorname{no}tin \frak p$ for all $\frak p \in
\mathrm{Ass}\, U_M(0) \setminus \{\frak m\}$. So
$\mathrm{Ann}\, U_M(0) \nsubseteq \frak q$, and hence $\frak b(M)
\nsubseteq \frak q$ for all $\frak q \in \mathrm{Assh}\, M/xM$. Thus there exists
$x_d \in \frak b(M)^3$ such that $x_d$ is a parameter element of $M/xM$ by the prime avoidance theorem. Such an element $x_d$ satisfies the requirements. The proof is complete.
\end{proof}
We are now ready to prove that the unmixed degrees satisfy the Bertini rule of extended degrees.
\begin{theorem}\label{D3.3.17} Let $M$ be a finitely generated $R$-module of dimension $d$.
Let $x$ be a superficial element of $M$ and of all $U_i(M)$, $1 \leq i
\leq d-1$, with respect to $I$. Then
$$\mathrm{udeg}(I,M) \ge \mathrm{udeg}(I,M/xM).$$
\end{theorem}
\begin{proof} Notice that since $x$ is a superficial element of $U_i(M)$, $1 \leq i \leq d-1$, with respect to $I$ we have $x \operatorname{no}tin \frak p$ for all $\frak p \in
\mathrm{Ass}\, U_i(M) \setminus \{\frak m\}$, $1 \leq i \leq
d-1$ by Remark \ref{C3.3.14} (i). The case $d=1$ is clear since $\mathrm{udeg}(I,M) =
\mathrm{deg}(I,M) + \ell(H^0_{\frak m}(M))$ and
$\mathrm{udeg}(I,M/xM) = \ell(M/xM) = \mathrm{deg}(I,M) +
\ell(0:_Mx)$. Suppose $d \geq 2$, by Lemma \ref{B3.3.16} we can choose a part of a $C$-system of parameters
$x_2,\ldots,x_d$ of $M$ such that $x, x_2,\ldots,x_d$
is also a system of parameters of $M$. By Lemma \ref{B3.1.9} we have $x_2, \ldots, x_d$ is a $C$-system of parameters of $M/xM$. Therefore, we have
\begin{eqnarray*}
\mathrm{udeg}(I,M) &=& \mathrm{deg}(I,M) +
\sum_{i=0}^{d-1}\delta_{i, \dim U_i(M)}\mathrm{deg}(I,U_i(M))\\
&=& \mathrm{deg}(I,M) +
\sum_{j=2}^{d+1}\delta_{j-2, \dim U_{M/(x_j,\ldots,x_d)M}(0)}\mathrm{deg}(I,U_{M/(x_j,\ldots,x_d)M}(0)),
\end{eqnarray*}
and
\begin{eqnarray*}
\mathrm{udeg}(I,M/xM) &=& \mathrm{deg}(I,M/xM) +
\sum_{i=0}^{d-2}\delta_{i, \dim U_i(M/xM)}\mathrm{deg}(I,U_i(M/xM))\\
&=& \mathrm{deg}(I,M/xM) +
\sum_{j=3}^{d+1}\delta_{j-3, \dim U_{M/(x,x_j,\ldots,x_d)M}(0)}\mathrm{deg}(I,U_{M/(x,x_j,\ldots,x_d)M}(0)).
\end{eqnarray*}
Since $x$ is a superficial element of $M$ with respect to $I$ we have
$\mathrm{deg}(I,M/xM) = \mathrm{deg}(I,M)$. For $j>3$ we have
$\dim M/(x_j,\ldots,x_d)M = j-1 \geq 3$. By Lemma \ref{B3.3.15} we obtain
$$\delta_{j-2, \dim U_{M/(x_j,\ldots,x_d)M}(0)}\mathrm{deg}(I,U_{M/(x_j,\ldots,x_d)M}(0)) = \delta_{j-3, \dim U_{M/(x,x_j,\ldots,x_d)M}(0)}\mathrm{deg}(I,U_{M/(x,x_j,\ldots,x_d)M}(0))$$
for all $3<j \leq d+1$. For $j=3$, set $M' = M/(x_3,\ldots,x_d)M $ we have $\dim M'
= 2$. By Lemma \ref{B3.3.15} we have
$$\delta_{0, \dim U_{M'/xM'}(0)}\mathrm{deg}(I,U_{M'/xM'}(0)) = \delta_{1, \dim U_{M'}(0)}\mathrm{deg}(I,U_{M'}(0)) +
\ell(0:_{H^0_{\frak m}(M')}x) + \ell(0:_{H^1_{\frak
m}(M'/U_{M'}(0))}x).$$
By Corollary \ref{H3.2.5} we have
$$U_0(M') = H^0_{\frak m}(M'/x_2M') \cong H^0_{\frak m}(M') \oplus H^1_{\frak m}(M'/U_{M'}(0)).$$
So
\begin{eqnarray*}
\delta_{0, \dim U_0(M')}\mathrm{deg}(I,U_0(M')) &=& \ell(H^0_{\frak m}(M')) + \ell(H^1_{\frak m}(M'/U_{M'}(0)))\\
&\ge& \ell(0:_{H^0_{\frak m}(M')}x) + \ell(0:_{H^1_{\frak
m}(M'/U_{M'}(0))}x).
\end{eqnarray*}
Therefore
$$\delta_{0, \dim U_{M'/xM'}(0)}\mathrm{deg}(I,U_{M'/xM'}(0)) \le \delta_{1, \dim U_{M'}(0)}\mathrm{deg}(I,U_{M'}(0)) +\delta_{0, \dim U_0(M')}\mathrm{deg}(I,U_0(M')).$$
More precisely, we have
$$\delta_{0, \dim U_{M/(x,x_3,\ldots,x_d)M}(0)}\mathrm{deg}(I,U_{M/(x,x_3,\ldots,x_d)M}(0)) \le
\sum_{j=2}^{3}\delta_{j-2, \dim U_{M/(x_j,\ldots,x_d)M}(0)}\mathrm{deg}(I,U_{M/(x_j,\ldots,x_d)M}(0)).$$
In conlusion,
$\mathrm{udeg}(I,M) \ge \mathrm{udeg}(I,M/xM)$. The proof is complete.
\end{proof}
\begin{remark}\rm By the prime avoidance theorem we always can choose $x$ satisfying the condition of Theorem \ref{D3.3.17}. Furthermore, according to the above proof we have $\mathrm{udeg}(I,M/xM) = \mathrm{udeg}(I,M)$ provided $x$ annihilates
$H^0_{\frak m}(M')$ and $H^1_{\frak
m}(M'/U_{M'}(0))$, where $M' = M/(x_3,\ldots,x_d)M$. This is the case if $xU_0(M) = 0$ by Corollary \ref{H3.2.5}.
\end{remark}
By Proposition \ref{M3.3.9}, Theorems \ref{D3.3.8} and \ref{D3.3.17} we have the main result of this section.
\begin{theorem} For every $\frak m$-primary ideal $I$, the unmixed degree $\mathrm{udeg}(I, \bullet)$ is an extended degree on the category of finitely generated $R$-modules $\mathcal{M}(R)$.
\end{theorem}
We next compare the unmixed degree and the homological degree for sequentially Cohen-Macaulay modules.
\begin{remark}\rm
Suppose $(R, \frak m)$ be a homomorphic image of a Gorenstein local ring $S$ of dimension $n$, and $M$ a sequentially Cohen-Macaulay $R$-module. It is easy to see that
$\mathrm{Ext}^i_S(M, S)$ is either a Cohen-Macaulay module or zero module for all $i$. By Theorem
\ref{D3.3.8} we have
$$\mathrm{udeg}(I,M) = \mathrm{adeg}(I,M) = \mathrm{deg}(I,M) + \sum_{i=0}^{d-1} \mathrm{deg}(\mathrm{Ext}^{n-i}_S(M, S))$$
for the last equation see \cite[Theorem 3.11]{NR06}.
Furthermore by \cite[Theorem 3.5]{NR06} we have
$$\mathrm{hdeg}(I,M) = \mathrm{deg}(I,M) + \sum_{i=0}^{d-1} \binom{d-1}{i}\mathrm{deg}(\mathrm{Ext}^{n-i}_S(M, S)).$$
Therefore $\mathrm{udeg}(I,M) \leq \mathrm{hdeg}(I,M)$. The equation occurs if and only if
$\mathrm{Ext}^{n-i}_S(M, S) = 0$ for all $1 \leq i \leq d-2$. In this case the dimension filtration of $M$ is either $H^0_{\frak m}(M) \subseteq M$ or $H^0_{\frak m}(M) \subseteq U_M(0) \subseteq M$ with dim $U_M(0) = d-1$.
\end{remark}
We close this paper with some examples and an open question.
\begin{example}\rm Let $R = k[[X_1,\ldots,X_4]]/(X_1^2, X_1X_2,
X_1X_3)$ where $k$ is a field and $X_i, 1 \leq i \leq 4,$ are indeterminates. We denote by $x_i$ the image of $X_i$ in $R$. We have $R$ is a sequentially
Cohen-Macaulay ring of dimension $3$ with the dimension filtration $\mathcal{D}: 0
\subseteq (x_1) \subseteq R$. We have
$$\mathrm{deg}(R) = 1 < \mathrm{adeg}(R) = \mathrm{udeg}(R) = 2 < \mathrm{hdeg}(R) = 3.$$
\end{example}
\begin{example}\rm
Let $R = k[[X_1,\ldots,X_7]]/(X_1,X_2, X_3) \cap (X_4,X_5,X_6)$ where $k$
is a field and $X_i, 1 \leq i \leq 7,$ are indeterminates. It is easy to see that $\mathrm{deg}(R) = \mathrm{adeg}(R) = 2$. Moreover, we can compute that $\mathrm{hdeg}(R) = 5$ and $\mathrm{udeg}(R) = 4$.
\end{example}
\begin{question}\rm Is it true that $\mathrm{udeg}(I, M) \leq
\mathrm{hdeg}(I, M)$ for all finitely generated $R$-modules $M$ and all $\frak m$-primary ideals $I$?
\end{question}
\end{document} |
\begin{document}
\baselineskip 16pt
\title{On one generalization of finite nilpotent groups}
\author{Zhang Chi \thanks{Research of the first author is supported by
China Scholarship Council and NNSF of
China(11771409)}\\
{\small Department of Mathematics, University of Science and
Technology of China,}\\ {\small Hefei 230026, P. R. China}\\
{\small E-mail:
[email protected]}\\ \\
{ Alexander N. Skiba}\\
{\small Department of Mathematics and Technologies of Programming,
Francisk Skorina Gomel State University,}\\
{\small Gomel 246019, Belarus}\\
{\small E-mail: [email protected]}}
\date{}
\maketitle
\date{}
\maketitle
\begin{abstract} Let $\sigma =\{\sigma_{i} | i\in I\}$ be a
partition of the set $\Bbb{P}$ of all primes and $G$ a finite group.
A chief factor $H/K$ of $G$ is
said to be \emph{$\sigma$-central} if the semidirect
product $(H/K)\rtimes (G/C_{G}(H/K))$ is
a $\sigma_{i}$-group for some $i=i(H/K)$. $G$ is called \emph{$\sigma$-nilpotent}
if every chief factor of $G$ is $\sigma$-central. We say that $G$ is
\emph{semi-${\sigma}$-nilpotent} (respectively \emph{weakly semi-${\sigma}$-nilpotent}) if
the normalizer $N_{G}(A)$ of every non-normal (respectively every non-subnormal)
$\sigma$-nilpotent subgroup $A$ of $G$ is $\sigma$-nilpotent.
In this paper we determine the structure of finite
semi-${\sigma}$-nilpotent and weakly semi-${\sigma}$-nilpotent groups.
\end{abstract}
\footnotetext{Keywords: finite group,
${\sigma}$-soluble group, ${\sigma}$-nilpotent group, semi-${\sigma}$-nilpotent group,
weakly semi-${\sigma}$-nilpotent group.}
\footnotetext{Mathematics Subject Classification (2010): 20D10,
20D15, 20D30}
\let\thefootnote\thefootnoteorig
\section{Introduction}
Throughout this paper, all groups are finite and $G$ always denotes
a finite group. Moreover,
$\mathbb{P}$ is the set of all primes,
$\pi \subseteq \Bbb{P}$ and $\pi' = \Bbb{P} \setminus \pi$. If $n$ is an integer,
the symbol $\pi (n)$
denotes the
set of all primes dividing $n$; as usual, $\pi (G)=\pi (|G|)$, the set of all
primes dividing the order of $G$.
In what follows, $\sigma =\{\sigma_{i} | i\in I\}$ is some
partition of $\Bbb{P}$, that is,
$\Bbb{P}=\bigcup_{i\in I} \sigma_{i}$ and $\sigma_{i}\cap
\sigma_{j}= \emptyset $ for all $i\ne j$.
By the analogy with the notation $\pi (n)$, we write $\sigma (n)$ to denote
the set $\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne
\emptyset \}$; $\sigma (G)=\sigma (|G|)$. A group
is said to be \emph{$\sigma$-primary}
\cite{1} if it is a $\sigma _{i}$-group for some $i$.
A chief factor $H/K$ of $G$ is
said to be \emph{$\sigma$-central} (in $G$) \cite{1} if the semidirect
product $(H/K)\rtimes (G/C_{G}(H/K))$ is
$\sigma$-primary. The normal subgroup $E$ of $G$ is called
\emph{$\sigma$-hypercentral} in $G$ if either $E=1$ or every chief factor of $G$ below $E$ is
$\sigma$-central.
Recall also that $G$ is called \emph{$\sigma$-nilpotent} \cite{1}
if every chief factor of $G$ is $\sigma$-central.
An arbitrary group $G$ has two canonical $\sigma$-nilpotent subgroups of particular
importance in this context. The first of these is the
\emph{$\sigma$-Fitting subgroup } $F_{\sigma}(G)$, that is, the product
of all normal $\sigma$-nilpotent subgroups of $G$. The other useful
subgroup is the \emph{$\sigma$-hypercentre $Z_{\sigma}(G)$ of $G$}, that
is, the product of all $\sigma$-hypercentral subgroups of $G$.
Note that in the classical case,
when $\sigma = \sigma ^{1}=\{\{2\}, \{3\}, \ldots \}$ (we use here the notation in
\cite{alg12}), $F_{\sigma}(G)=F(G)$ is the Fitting subgroup
and $Z_{\sigma}(G)=Z_{\infty}(G)$ is the hypercentre of $G$.
In fact, the $\sigma$-nilpotent groups are exactly the groups $G$ which can be written in the
form $G=G_{1} \times \cdots \times G_{t}$ for some $\sigma$-primary groups
$G_{1}, \ldots , G_{t}$ \cite{1}, and such groups have proved to be very useful in
the formation theory (see, in particular, the papers \cite{19,
20} and the books \cite[Ch. IV]{bookShem}, \cite[Ch. 6]{15}).
In the recent years, the $\sigma$-nilpotent groups have found new and to
some extent unexpected
applications in the theories of permutable and generalized subnormal
subgroups (see, in particular, \cite{1, alg12}, \cite{3}--\cite{6} and the survey \cite{comm}).
In view of the results in the paper \cite{belon}, the $\sigma$-nilpotent groups can
be characterized as the groups in which
the normalizer of any $\sigma$-nilpotent subgroup is $\sigma$-nilpotent.
Groups in which normalizers of all non-normal $\sigma$-nilpotent subgroups are
$\sigma$-nilpotent may be
non-$\sigma$-nilpotent (see Example 1.3 below), and in the case when $\sigma = \sigma ^{1}$
such groups have been described in \cite[Ch. 4, Section 7]{We} (see also \cite{Sah}).
In this paper, we determine the structure of such groups $G$ for the case arbitrary
$\sigma$.
{\bf Definition 1.1.} We say that $G$ is
(i) \emph{semi-${\sigma}$-nilpotent} if
the normalizer of every non-normal $\sigma$-nilpotent subgroup
of $G$ is $\sigma$-nilpotent;
(ii) \emph{weakly semi-${\sigma}$-nilpotent} if
the normalizer of every non-subnormal $\sigma$-nilpotent subgroup
of $G$ is $\sigma$-nilpotent;
(iii) \emph{weakly semi-nilpotent} if $G$ is weakly
semi-${\sigma}^{1}$-nilpotent.
{\bf Remark 1.2.} (i) Every ${\sigma}$-nilpotent group is
semi-${\sigma}$-nilpotent, and every semi-${\sigma}$-nilpotent group is weakly
semi-${\sigma}$-nilpotent.
(ii) The semi-${\sigma}^{1}$-nilpotent groups
are exactly the \emph{semi-nilpotent groups} studied in
\cite[Ch. 4, Section 7]{We} (see also \cite{Sah}).
(iii) We show that $G$ is (weakly) semi-${\sigma}$-nilpotent if and only if
the normalizer of every non-normal (respectively non-subnormal) $\sigma$-primary subgroup
of $G$ is $\sigma$-nilpotent. Since every $\sigma$-primary group is
$\sigma$-nilpotent, it is enough to show that
if the normalizer of every non-normal (respectively non-subnormal) $\sigma$-primary
subgroup $A$
of $G$ is $\sigma$-nilpotent, then $G$ is ${\sigma}$-semi-nilpotent (respectively weakly
semi-${\sigma}$-nilpotent).
First note that $A\ne 1$ and $A=A_{1} \times \cdots \times A_{n}$,
where $\{A_{1}, \ldots , A_{n}\}$ is a complete Hall $\sigma$-set of
$A$. The subgroups $A_{i}$ are
characteristic in $A$, so $N_{G}(A)=N_{G}(A_{1})
\cap \cdots \cap N_{G}(A_{n})$, where either $N_{G}(A_{n})=G$ or $N_{G}(A_{n})$ is
$\sigma$-nilpotent. Since $A$ is non-normal (respectively non-subnormal) in $G$,
there is $i$ such that $N_{G}(A_{n})$ is
$\sigma$-nilpotent. Therefore $N_{G}(A)$ is
$\sigma$-nilpotent by Lemma 2.2(i) below. Hence $G$ is semi-${\sigma}$-nilpotent
(respectively weakly semi-${\sigma}$-nilpotent).
{\bf Example 1.3.} Let $p > q > r > t > 2$ be primes, where $q$ divides $p-1$ and
$t$ divides $r-1$,
and let $\sigma =\{\{p\}, \{q\}, \{p, q\}'\}$. Let
$R$
be the quaternion group of order 8, $A$ a group of order $p$, and let
$B=C_{p}\rtimes C_{q}$ be
a non-nilpotent group of order $pq$ and $C$
a non-nilpotent group of order $rt$. Then $B\times R$ is a non-$\sigma$-nilpotent
semi-${\sigma}$-nilpotent group and $B \times C$ is not semi-${\sigma}$-nilpotent.
Now let $G=A\times (Q\rtimes R)$, where $Q$ is a simple ${\mathbb F}_{q}R$-module which is faithful
for $R$. Then for every subgroup $V$ of $R$
we have $N_{G}(V)=A\times R$, so
$G$ is weakly
semi-${\sigma}$-nilpotent. On the other hand, $QV$ is supersoluble for
every subgroup $V$ of $R$ of order 2 and so for some subgroup $L$ of $Q$ with
$1 < L < Q$ we have $V \leq N_{G}(L)$ and $[L, V]\ne 1$. Hence
$G$ is not semi-${\sigma}$-nilpotent.
Recall that $G^{{\mathfrak{N}}_{\sigma}}$
is the \emph{$\sigma$-nilpotent residual of $G$}, that is, the
intersection of all normal subgroups $N$ of $G$ with $\sigma$-nilpotent
quotient $G/N$.
Our goal here is to determine the structure of weakly
semi-${\sigma}$-nilpotent and
semi-${\sigma}$-nilpotent groups. In fact, the following concept is an
important tool to achieve such a goal.
{\bf Definition 1.4.} Let $H$ be a
${\sigma}$-nilpotent subgroup of $G$. Then we say that $H$ is
\emph{$\sigma$-Carter subgroup} of $G$ if $H$ is an \emph{${\mathfrak{N}}_{\sigma}$-covering
subgroup of $G$} \cite[p. 101]{15}, that is, $U^{{\mathfrak{N}}_{\sigma}}H=U$
for every subgroup $U$ of $G$ containing $H$.
Note that in Example 1.3, the subgroup $C_{q}C$
is a $\sigma$-Carter subgroup of the group $B \times C$.
It is clear also that a group $H$ of a soluble group $G$ is a Carter subgroup of
$G$ if and only if it is a $\sigma ^{1}$-Carter subgroup of $G$.
A \emph{complete set of Sylow subgroups of $G$}
contains exactly one Sylow $p$-subgroup for each prime $p$ dividing $|G|$.
In general, we say that a set ${\cal H}$ of subgroups of $G$ is a
\emph{complete Hall $\sigma $-set} of $G$ \cite{2, comm} if
every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$
for some $i$ and ${\cal H}$ contains exactly one Hall
$\sigma _{i}$-subgroup of $G$ for every $\sigma _{i}\in \sigma (G)$.
Our first result is the following
{\bf Theorem A.} {\sl If $G$ is weakly semi-${\sigma}$-nilpotent, then:}
(i) {\sl $G$ has a complete Hall $\sigma$-set $\{H_{1}, \ldots , H_{t}\}$
such that for some $1\leq r \leq t$ the subgroups
$H_{1}, \ldots , H_{r}$ are normal in $G$, $H_{i}$ is not normal in $G$ for all $i > r$,
and $$\langle H_{r+1},
\ldots , H_{t} \rangle =H_{r+1}\times \cdots \times H_{t}.$$}
(ii) {\sl If $G$ is not ${\sigma}$-nilpotent, then $N_{G}(H_{i})$
is a $\sigma$-Carter subgroup of $G$ for all $i > r$.}
(iii) {\sl $F_{\sigma}(G)$ is a maximal
${\sigma}$-nilpotent subgroup of $G$ and
$F_{\sigma}(G)=F_{0\sigma }(G)Z_{\sigma}(G)$, where
$F_{0\sigma }(G)=H_{1} \cdots H_{r}$.}
(iv) {\sl $V_{G}= Z_{\sigma}(G)$
for every maximal ${\sigma}$-nilpotent subgroup $V$ of $G$ such that $G=F_{\sigma}(G)V$.
}
(v) {\sl $G/F(G)$ is $\sigma$-nilpotent. }
On the basis of Theorem A we prove also the following
{\bf Theorem B.} {\sl Suppose that $G$ is semi-${\sigma}$-nilpotent,
and let $\{H_{1}, \ldots , H_{t}\}$ be a complete Hall $\sigma$-set of $G$, where
$H_{1}, \ldots , H_{r}$ are normal in $G$ and $H_{i}$ is not normal in
$G$ for all $i > r$. Suppose also that non-normal Sylow subgroups of
any Schmidt subgroup $A\leq H_{i}$
have prime order for all $i > r$.
Then:}
(i) {\sl $G/F_{\sigma}(G)$ is abelian. }
(ii) {\sl If
$U$ is any maximal $\sigma$-nilpotent non-normal subgroup of $G$, then
$U$ is a $\sigma$-Carter subgroup of $G$ and $U_{G}= Z_{\sigma}(G)$.}
(iii) {\sl If the subgroups $H_{1}, \ldots , H_{r}$ are nilpotent, then $G/F_{\sigma}(G)$
is cyclic. }
(iv) {\sl Every quotient and every subgroup of $G$ are semi-${\sigma}$-nilpotent}.
Now we consider some of corollaries of Theorems A and B in the three classical cases.
First of all note that in the case
when $\sigma = \sigma ^{1}$, Theorems A and B not only cover the main results in
\cite[Ch. 5 Section 7]{We} but they also give the alternative proofs of them.
Moreover, in this case we get from the theorems the following
results.
{\bf Corollary 1.4.} {\sl If $G$ is weakly semi-nilpotent, then:}
(i) {\sl $G$ has a complete set of Sylow subgroups $\{P_{1}, \ldots , P_{t}\}$ such
that for some $1\leq r \leq t$ the subgroups
$P_{1}, \ldots , P_{r}$ are normal in $G$, $P_{i}$ is not normal in $G$ for all $i > r$,
and $\langle P_{r+1},
\ldots , P_{t} \rangle =P_{r+1}\times \cdots \times P_{t}.$}
(ii) {\sl $F(G)$ is a maximal
nilpotent subgroup of $G$ and
$F(G)=F_{0\sigma }(G)Z_{\infty}(G)$, where
$F_{0\sigma }(G)=P_{1} \cdots P_{r}$.}
(iii) {\sl If $G$ is not nilpotent, then $N_{G}(P_{i})$
is a Carter subgroup of $G$ for all $ i > r$.}
{\bf Corollary 1.5} (See Theorem 7.6 in \cite[Ch. 4]{We}). {\sl If $G$ is
semi-nilpotent and $F_{0}(G)$ denotes the product of its normal
Sylow subgroups, then
$G/F_{0}(G)$ is nilpotent. }
{\bf Corollary 1.6} (See Theorem 7.8 in \cite[Ch. 4]{We}). {\sl If $G$ is
semi-nilpotent, then: }
(a) {\sl $F(G)$ is a maximal nilpotent subgroup of $G$.}
(b) {\sl If $U$ is a maximal nilpotent subgroup of $G$ and $U$ is not normal in $G$, then
$U_{G}=Z_{\infty}(G)$.}
{\bf Corollary 1.7} (See Theorem 7.10 in \cite[Ch. 4]{We}).
{\sl The class of all semi-nilpotent groups is closed under taking subgroups
and homomorphic images.}
In the other classical case when $\sigma =\sigma ^{\pi}=\{\pi,
\pi'\}$,
$G$ is $\sigma ^{\pi}$-nilpotent
if and only if $G$ is \emph{$\pi$-decomposable}, that is,
$G=O_{\pi}(G)\times O_{\pi'}(G)$.
Thus $G$ is semi-${\sigma}^{\pi}$-nilpotent if and only if the
normalizer of every $\pi$-decomposable non-normal subgroup of $G$ is
$\pi$-decomposable; $G$ is weakly semi-${\sigma}^{\pi}$-nilpotent if and only if the
normalizer of every $\pi$-decomposable non-subnormal subgroup of $G$ is
$\pi$-decomposable.
Therefore in this case we get from Theorems A and B the following results.
{\bf Corollary 1.8.} {\sl Suppose that $G$ is not $\pi$-decomposable. If the
normalizer of every $\pi$-decomposable non-subnormal subgroup of $G$ is
$\pi$-decomposable, then:}
(i) {\sl $G$ has a Hall $\pi$-subgroup $H_{1}$ and a Hall
$\pi'$-subgroup $H_{2}$,
and exactly one of these subgroups, $H_{1}$ say, is normal in $G$.}
(ii) {\sl $G/F(G)$ is $\pi$-decomposable. }
(iii) {\sl $N_{G}(H_{2})$ is an
$\mathfrak{F}$-covering subgroup of $G$, where $\mathfrak{F}$ is the class
of all $\pi$-decomposable groups.}
(iv) {\sl $O_{\pi}(G)\times O_{\pi'}(G)=H_{1}\times O_{\pi'}(G)$ is a maximal
$\pi$-decomposable subgroup of $G$ and every element of $G$
induces a $\pi'$-automorphism on every chief factor of $G$
below $O_{\pi'}(G)$.}
{\bf Corollary 1.9.} {\sl Suppose that $G$ is not $\pi'$-closed and the
normalizer of every $\pi$-decomposable non-normal subgroup of $G$ is
$\pi$-decomposable. Then $G=H_{1}\rtimes H_{2}$, where $H_{1}$ is a Hall
$\pi$-subgroup and $H_{2}$ is a Hall
$\pi'$-subgroup of $G$. Moreover, if non-normal Sylow subgroups of
any Schmidt subgroup $A\leq H_{2}$
have prime order, then:}
(i) {\sl $G/O_{\pi}(G)\times O_{\pi'}(G)$ is abelian. }
(ii) {\sl Every
maximal $\pi$-decomposable non-normal subgroup of $G$ is an
$\mathfrak{F}$-covering subgroup of $G$, where $\mathfrak{F}$ is the class
of all $\pi$-decomposable groups.}
(iii) {\sl If $H_{1}$ is nilpotent, then
$G/O_{\pi}(G)\times O_{\pi'}(G) $ is cyclic. }
In fact, in the theory of $\pi$-soluble groups
($\pi= \{p_{1}, \ldots , p_{n}\}$) we deal with the partition
$\sigma =\sigma ^{1\pi }=\{\{p_{1}\}, \ldots , \{p_{n}\}, \pi'\}$.
Moreover, $G$ is $\sigma ^{1\pi }$-nilpotent
if and only if $G$ is \emph{$\pi$-special} \cite{Cun2},
that is, $G=O_{p_{1}}(G)\times \cdots \times
O_{p_{n}}(G)\times O_{\pi'}(G)$.
Thus $G$ is semi-${\sigma}^{1\pi}$-nilpotent if and only if the
normalizer of every $\pi$-special non-normal subgroup of $G$ is
$\pi$-special; $G$ is weakly semi-${\sigma}^{1\pi}$-nilpotent if and only if the
normalizer of every $\pi$-special non-subnormal subgroup of $G$ is
$\pi$-special.
Therefore in this case we get from Theorems A and B the following results.
{\bf Corollary 1.10.} {\sl Let $P_{i}$ be a
Sylow $p_{i}$-subgroup of $G$ for all $p\in \pi= \{p_{1}, \ldots , p_{n}\}$.
If the
normalizer of every $\pi$-special non-subnormal subgroup of $G$ is
$\pi$-special, then: }
(i) {\sl $G$ has a Hall $\pi'$-subgroup $H$
and at least one of subgroups $P_{1}, \ldots , P_{n}, H$ is normal in $G$.}
(ii) {\sl $O_{p_{1}}(G)\times \cdots \times
O_{p_{n}}(G)\times O_{\pi'}(G)$ is a maximal
$\pi$-special subgroup of $G$.}
(iii) {\sl $G/F(G)$ is $\pi$-special. }
{\bf Corollary 1.11.} {\sl Suppose that the
normalizer of every $\pi$-special non-normal subgroup of $G$ is
$\pi$-special. If non-normal Sylow subgroups of
any Schmidt $\pi'$-subgroup of $G$
have prime order, then:}
(i) {\sl $G/(O_{p_{1}}(G)\times \cdots \times
O_{p_{n}}(G)\times O_{\pi'}(G))$ is abelian. }
(ii) {\sl Every
maximal $\pi$-special non-normal subgroup of $G$ is an
$\mathfrak{F}$-covering subgroup of $G$, where $\mathfrak{F}$ is the class
of all $\pi$-special groups.}
(iii) {\sl If every normal in $G$ subgroup $A\in \{P_{1}, \ldots , P_{n}, H\}$ is nilpotent,
then $G/(O_{p_{1}}(G)\times \cdots \times
O_{p_{n}}(G)\times O_{\pi'}(G))$ is cyclic. }
\section{Preliminaries}
Recall that $G$ is said to be: a \emph{$D_{\pi}$-group} if $G$ possesses a Hall
$\pi$-subgroup $E$ and every $\pi$-subgroup of $G$ is contained in some
conjugate of $E$; a \emph{$\sigma$-full group
of Sylow type} \cite{1} if every subgroup $E$ of $G$ is a $D_{\sigma _{i}}$-group for every
$\sigma _{i}\in \sigma (E)$; \emph{$\sigma$-soluble} \cite{1} if
every chief factor of $G$ is $\sigma$-primary.
{\bf Lemma 2.1 } (See Theorem A and B in \cite{2}).
{\sl If $G$ is $\sigma$-soluble, then $G$ is a
$\sigma$-full group of Sylow type and, for every $i$, $G$ has
a Hall $\sigma _{i}'$-subgroup and every two
Hall $\sigma _{i}'$-subgroups of $G$ are conjugate. }
A subgroup $A$ of $G$ is said to be \emph{${\sigma}$-subnormal}
in $G$ \cite{1}
if there is a subgroup chain $A=A_{0} \leq A_{1} \leq \cdots \leq
A_{n}=G$ such that either $A_{i-1}\trianglelefteq A_{i}$ or
$A_{i}/(A_{i-1})_{A_{i}}$ is $\sigma$-primary
for all $i=1, \ldots , n$. Note that a subgroup $A$ of $G$ is subnormal
in $G$ if and only if $A$ is ${\sigma}^{1}$-subnormal in $G$ (where
${\sigma}^{1}=\{\{2\}, \{3\}, \ldots \}$).
{\bf Lemma 2.2. } (i) {\sl
The class of all $\sigma$-nilpotent groups ${\mathfrak{N}}_{\sigma}$ is closed under taking direct
products, homomorphic images and subgroups. Moreover, if $H$ is a normal
subgroup of $G$ and $H/H\cap \Phi (G)$ is $\sigma$-nilpotent, then
$H$ is $\sigma$-nilpotent } (See Lemma 2.5 in \cite{2}).
(ii) {\sl $G$ is $\sigma $-nilpotent if and
only if every subgroup of $G$ is ${\sigma}$-subnormal in $G$ } (See
\cite[Proposition 3.4]{6}).
(iii) {\sl $G$ is $\sigma $-nilpotent if and only if
$G=G_{1}\times \cdots \times G_{n}$ for some $\sigma$-primary groups $G_{1}, \ldots ,
G_{n}$ } (See
\cite[Proposition 3.4]{6}).
{\bf Lemma 2.3} (See Lemma 2.6 in \cite{1}). {\sl Let $A$, $K$ and $N$ be subgroups of $G$.
Suppose that $A$
is $\sigma$-subnormal in $G$ and $N$ is normal in $G$. }
(1) {\sl If $N\leq K$ and $K/N$ is $\sigma$-subnormal in $G/N$, then $K$
is $\sigma$-subnormal in $G$}.
(2) {\sl $A\cap K$ is $\sigma$-subnormal in $K$}.
(3) {\sl If $A$ is $\sigma $-nilpotent, then $A\leq F_{\sigma}(G)$.}
(4) {\sl $AN/N$ is $\sigma$-subnormal in $G/N$}.
(5) {\sl If $A$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $i$, then $A$ is normal in $G$.}
In view of Proposition 2.2.8 in \cite{15},
we get from Lemma 2.2 the following
{\bf Lemma 2.4.} {\sl If $N$ is a normal subgroup of $G$, then
$(G/N)^{{\frak{N}}_{\sigma}}=G^{{\frak{N}}_{\sigma}}N/N.$ }
{\bf Lemma 2.5.} {\sl If $G$ is ${\sigma}$-soluble and, for some
$i$ and some Hall $\sigma _{i}$-subgroup $H$ of $G$, $N_{G}(H)$
is ${\sigma}$-nilpotent, then $N_{G}(H)$ is a $\sigma$-Carter subgroup of $G$. }
{\bf Proof.} Let $N=N_{G}(H)$ and $N\leq U\leq G$.
Suppose that $U^{{\mathfrak{N}}_{\sigma}}N\ne U$ and let $M$ be a maximal
subgroup of $U$ such that $U^{{\mathfrak{N}}_{\sigma}}N\leq M$. Then $M$
is $\sigma$-subnormal in $U$ by Lemmas 2.2(i, ii) and 2.3(1), so $U/M_{U}$ is a
$\sigma _{j}$-group for some $j$ since $U$ is clearly ${\sigma}$-soluble.
Therefore $|U:M|$ is a $\sigma _{j}$-number,
so $j\ne i$ and hence $H\leq M_{U}$. But then $U=M_{U}N_{U}(H)\leq M < U$ by Lemma 2.1 and
the Frattini argument. This contradiction completes the proof of the
lemma.
It is clear that if $A$ is $\sigma$-Carter subgroup of $G$, then $A$ is
a
$\sigma$-Carter subgroup in every subgroup of $G$ containing $A$.
Moreover, in view of Proposition 2.3.14 in \cite{15}, the following useful facts are true.
{\bf Lemma 2.6.} {\sl Let $H$ and $R$ be subgroups of $G$,
where $R$ is normal in $G$. }
(i) {\sl If
$H$ is a $\sigma$-Carter subgroup of $G$, then $HR/R$ is
a $\sigma$-Carter subgroup of $G/R$. }
(ii) {\sl If $U/R$ is
a $\sigma$-Carter subgroup of $G/R$ and $H$ is
a $\sigma$-Carter subgroup of $U$, then
$H$ is a $\sigma$-Carter subgroup of $G$. }
{\bf Lemma 2.7.} {\sl Suppose that $G$
possesses a $\sigma$-Carter subgroup. If $G$ is ${\sigma}$-soluble, then any two
of its
$\sigma$-Carter subgroups are conjugate. }
{\bf Proof.} Assume that this lemma is false and let $G$ be a
counterexample of minimal order. Then $|\sigma (G)| > 1$.
Let $A$ and
$B$ be $\sigma$-Carter subgroups of $G$, and let
$R$ be a minimal normal subgroup of $G$. Then $AR/R$ and
$BR/R$ are $\sigma$-Carter subgroups of $G/R$ by Lemma 2.6(i).
Therefore for some $x\in G$ we have $AR/R=B^{x}R/R$ by the choice of $G$.
If $AR\ne G$, then $A$ and $B^{x}$ are conjugate in $AR$ by the choice of
$G$ and so $A$ and $B$ are conjugate.
Now assume that $AR=G=B^{x}R=BR$. If $R\leq A$, then $A=G$ is
$\sigma$-nilpotent and so $A=B$. Therefore we can assume that
$A_{G}=1=B_{G}$.
Since $G$ is ${\sigma}$-soluble, $R$ is a ${\sigma}_{i}$-group for
some $i$. Let $H$ be a Hall ${\sigma}_{i}'$-subgroup of $A$.
Since $|\sigma (G)| > 1$, it follows that $H\ne 1$ and so
$N=N_{G}(H)\ne 1$. Since $A$ and
$B$ be $\sigma$-Carter subgroups of $G$, both these subgroups
are $\sigma$-nilpotent. Hence $A\leq N$ and, for some $x\in G$, $B^{x}\leq N$ by Lemma 2.1.
But then the choice of $G$ implies that $A$ and $B^{x}$ are conjugate in $N$.
So we again get that $A$ and $B$ are conjugate. The lemma is proved.
If $G \not \in {\mathfrak{N}}_{\sigma}$ but
every proper subgroup of $G$ belongs to ${\mathfrak{N}}_{\sigma}$, then $G$ is called
an \emph{${\mathfrak{N}}_{\sigma}$-critical
} or a \emph{minimal non-$\sigma$-nilpotent} group. If $G$ is an
${\mathfrak{N}}_{{\sigma}^{1}}$-critical group, that is, $G$ is not nilpotent but
every proper subgroup of $G$ is nilpotent, then $G$ is said to be a \emph{Schmidt
group}.
{\bf Lemma 2.8} (See \cite[Ch. V, Theorem 26.1]{bookShem}). {\sl If
$G$ is a
Schmidt group, then $G=P\rtimes Q$, where $P=G^{\frak{N}}=G'$
is a Sylow $p$-subgroup of $G$ and $Q=\langle x \rangle $ is a cyclic
Sylow $q$-subgroup of $G$ with $\langle x^{q} \rangle \leq Z(G)\cap \Phi (G)$. Hence $Q^{G}=G$. }
{\bf Lemma 2.9.} {\sl If
$G$ is an ${\frak{N}}_{\sigma}$-critical
group, then $G$ is a Schmidt group.}
{\bf Proof.} For some $i$, $G$ is an ${\frak{N}}_{\sigma _{0}}$-critical
group, where $\sigma _{0}=\{\sigma_{i}, \sigma_{i}'\}$. Hence $G$ is a Schmidt group by
\cite{belon}.
{\bf Lemma 2.10. } {\sl Let $Z=Z_{\sigma}(G)$.
Let $A$, $B$ and $N$ be subgroups of $G$, where
$N$ is normal in $G$.}
(i) {\sl $Z$ is ${\sigma}$-hypercentral in $G$. }
(ii) {\sl If $ N\leq Z$, then $Z/N= Z_{\sigma}(G/N)$.}
(iii) {\sl $Z_{\sigma}(B)\cap A\leq Z_{\sigma}(B\cap A)$. }
(iv) {\sl If $A$ is $\sigma$-nilpotent, then $ZA$ is also $\sigma$-nilpotent. Hence
$Z$ is contained in each maximal $\sigma$-nilpotent subgroup of $G$.}
(v) {\sl If $G/Z$ is $\sigma$-nilpotent, then $G$ is also $\sigma$-nilpotent.}
{\bf Proof. } (i) It is enough to consider the case when
$Z=A_{1}A_{2}$, where
$A_{1}$ and $A_{2}$ are normal ${\sigma}$-hypercentral subgroups of
$G$. Moreover, in view of the Jordan-H\"{o}lder theorem for the chief series,
it is enough to show that if $A_{1}\leq K < H \leq A_{1}A_{2}$, then $H/K$
is $\sigma$-central. But in this case we
have $H=A_{1}(H\cap A_{2})$, where $H\cap A_{2}\nleq K$ and so
from
the $G$-isomorphism $(H\cap A_{2})/(K\cap A_{2})\simeq (H\cap
A_{2})K/K=H/K$ we get that $C_{G}(H/K)=C_{G}((H\cap A_{2})/(K\cap A_{2}))$
and hence $H/K$ is $\sigma$-central in $G$.
(ii) This assertion is a corollary of Part (i) and
the Jordan-H\"{o}lder theorem for the chief series.
(iii) First assume that $B=G$, and let $1= Z_{0} < Z_{1}
< \cdots < Z_{t} = Z$ be a chief
series of $G$ below $Z$ and $C_{i}=
C_{G}(Z_{i}/Z_{i-1})$. Now consider the series
$$1= Z_{0}\cap A \leq Z_{1}\cap A
\leq \cdots \leq Z_{t} \cap A= Z\cap A.$$
We can assume without loss of generality that this series is a chief
series of $A$ below $Z\cap A$.
Let $i\in \{1, \ldots , t \}$. Then, by Part (i),
$Z_{i}/Z_{i-1} $ is $\sigma$-central in $G$,
$(Z_{i}/Z_{i-1})\rtimes (G/C_{i})$ is a $\sigma _{k}$-group say.
Hence $(Z_{i}\cap A)/(Z_{i-1}\cap A)$ is a $\sigma _{k}$-group. On the
other hand,
$A/A\cap C_{i}\simeq C_{i}A/C_{i}$ is a $\sigma _{k}$-group and $$A\cap C_{i}\leq
C_{A}((Z_{i}\cap A)/(Z_{i-1}\cap A)).$$ Thus $(Z_{i}\cap
A)/(Z_{i-1}\cap A)$
is $\sigma$-central in $A$. Therefore, in view of the Jordan-H\"{o}lder theorem
for the chief series, we have $Z\cap A\leq Z_{\sigma}(A)$.
Now assume that $B$ is any subgroup of $G$. Then,
in view of the preceding paragraph, we have
$$ Z_{\sigma}(B) \cap A = Z_{\sigma}(B) \cap
(B\cap A)\leq Z_{\sigma}(B\cap A).$$
(iv) Since $A$ is $\sigma$-nilpotent, $ZA/Z\simeq A/A\cap Z$ is
$\sigma$-nilpotent by Lemma 2.2(i). On the other hand, $Z\leq Z_{\sigma}(ZA)$
by Part (iii). Hence $ZA$ is $\sigma$-nilpotent by Part (i).
(v) This assertion follows from Part (i).
The lemma is proved.
The following lemma is a corollary of Lemmas 2.2(i) and 2.10(v).
{\bf Lemma 2.11.} {\sl $F_{\sigma}(G)/\Phi (G)=F_{\sigma}(G/\Phi(G))$ and
$F_{\sigma}(G)/Z_{\sigma}(G)=F_{\sigma}(G/Z_{\sigma}(G))$.}
\section{Proofs of the main results}
{\bf Proof of Theorem A.} Assume that this theorem is false and let $G$ be a
counterexample of minimal order. Then $G$ is not $\sigma$-nilpotent.
(1) {\sl Every proper subgroup $E$ of $G$ is weakly
semi-${\sigma}$-nilpotent. Hence the conclusion of the theorem holds for $E$.}
Let $V$ be a non-subnormal $\sigma$-nilpotent subgroup
of $E$.
Then $V$ is not subnormal
in $
G$ by Lemma 2.3(2), so $N_{G}(V)$ is
$\sigma$-nilpotent by
hypothesis. Hence $N_{E}(V)=N_{G}(V)\cap E$ is $\sigma$-nilpotent by Lemma 2.2(i).
(2) {\sl Every proper quotient $G/N$ of $G$ (that is, $N\ne 1$)
is weakly semi-${\sigma}$-nilpotent. Hence the conclusion of the theorem holds for
$G/N$. }
In view of Remark 1.2(iii) and the choice of $G$, it is enough to show that if $U/N$ is any
non-subnormal
$\sigma$-primary
subgroup of $G/N$, then $N_{G/N}(U/N)$ is $\sigma$-nilpotent.
We can
assume without loss of generality that $N$ is a minimal normal subgroup of $G$.
Since $U/N$ is not
subnormal
in $G/N$, $U/N < G/N$ and $U$ is not
subnormal in $G$. Hence $U$ is a proper
subgroup of $G$, which implies that $U$ is $\sigma$-soluble by Claim (1).
Hence $N$ is a $\sigma _{i}$-group for some $i$.
If $U/N$ is
a $\sigma _{i}$-group, then $U$ is $\sigma$-primary and so $N_{G}(U)$ is
$\sigma$-nilpotent.
Hence $N_{G/N}(U/N)=N_{G}(U)/N$ is $\sigma$-nilpotent
by Lemma 2.2(i). Now suppose that
$U/N$ is
a $\sigma _{j}$-group for some $j\ne i$. Then $N$ has a complement $V$ in
$U$ by the Schur-Zassenhaus theorem. Moreover, from the Feit-Thompson
theorem it follows that at least one of the groups $N$ or $U/N$ is soluble
and so every two complements to $N$ in $U$ are conjugate in $U$. Therefore
$N_{G}(U)=N_{G}(NV)=NN_{G}(V)$. Since $U=NV$ is not
subnormal in $G$, $V$ is not
subnormal in $G$ by Lemma 2.3(1, 4)
and so $N_{G}(V)$ is $\sigma$-nilpotent. Hence
$N_{G/N}(U/N)=N_{G}(U)/N$ is $\sigma$-nilpotent.
(3) {\sl If $A$ is an ${\mathfrak{N}}_{\sigma}$-critical
subgroup of $G$, then $A=P\rtimes Q$, where $P=A^{\frak{N}}=A'$
is a Sylow $p$-subgroup of $A$ and $Q$ is a
Sylow $q$-subgroup of $A$ for some different primes $p$ and $q$.
Moreover, $P$ is subnormal in $G$ and so $P\leq O_{p}(G)$. }
The first assertion of the claim directly follows from
Lemmas 2.8 and 2.9. Since $A$ is not
$\sigma$-nilpotent, $P$ is subnormal in $G$ by hypothesis. Therefore
$ P\leq O_{p}(G)$ by Lemma 2.3(3).
(4) {\sl $G$ is $\sigma$-soluble.}
Suppose that this is false. Then $G$ is a non-abelian simple group since every
proper section of $G$ is $\sigma$-soluble by Claims (1) and (2). Moreover,
$G$ is not $\sigma$-nilpotent and so it has an ${\mathfrak{N}}_{\sigma}$-critical
subgroup $A$. Claim (3) implies that for some Sylow subgroup $P$ of $A$
we have $1 < P\leq O_{p}(G) < G$.
This contradiction shows that we have (4).
(5) {\sl Statements (i) and (ii) hold for $G$.}
Since $G$ is $\sigma$-soluble by Claim (4),
it is a $\sigma$-full group of Sylow type by Lemma 2.1. In particular, $G$
possesses a complete Hall $\sigma$-set $\{H_{1}, \ldots , H_{t}\}$. Then
there is an index $k$ such that $H_{k}$ is not subnormal in $G$
by Lemma 2.3(5)
since $G$ is not $\sigma$-nilpotent. Then $N_{G}(H_{k})$
is $\sigma$-nilpotent by hypothesis, so
$N_{G}(H_{i})$ is a $\sigma$-Carter subgroup of $G$ by Lemma 2.5
for all $i > r$.
If for some $j\ne k$ the subgroup $H_{j}$ is not subnormal in $G$, then
$N_{G}(H_{j})$ is also a $\sigma$-Carter subgroup of $G$. But then
$N_{G}(H_{k})$ and $N_{G}(H_{j})$ are conjugate in $G$ by Lemma 2.7.
Hence for some $x\in G$ we have $H_{k}^{x}\leq N_{G}(H_{j})$. Therefore,
since $G$ is not $\sigma$-nilpotent,
there is a complete Hall $\sigma$-set $\{L_{1}, \ldots , L_{t}\}$ of $G$
such that for some
$1\leq r < t$ the subgroups $L _{1}, \ldots , L_{r}$ are normal in $G$,
$L_{i}$ is not normal in $G$ for all $i > r$,
and
$\langle L_{r+1},
\ldots , L_{t} \rangle =L_{r+1}\times \cdots \times L_{t}$.
(6) {\sl Every subgroup $V$
of $G$ containing $ F_{\sigma}(G)$ is $\sigma$-subnormal in $G$, so
$F_{\sigma}(V)=F_{\sigma}(G)$.}
From Claim (5) it follows that $H_{1}, \ldots , H_{r} \leq F_{\sigma}(G)$ and
$$G/F_{\sigma}(G)=F_{\sigma}(G)(H_{r+1}\times \cdots \times
H_{t})/F_{\sigma}(G)\simeq (H_{r+1}\times \cdots \times
H_{t})/((H_{r+1}\times \cdots \times
H_{t}) \cap F_{\sigma}(G))$$ is ${\sigma}$-nilpotent. Hence every
subgroup of $G/F_{\sigma}(G)$ is ${\sigma}$-subnormal in
$G/F_{\sigma}(G)$ by Lemma 2.2(ii). Therefore $V$ is
${\sigma}$-subnormal in $G$ by Lemma 2.3(1), so
$F_{\sigma}(V)\leq F_{\sigma}(G)\leq F_{\sigma}(V)$ by Lemma 2.3(3). Hence
we have (6).
(7) {\sl Statement (iii) holds for $G$.}
First note that $F_{\sigma}(G)$ is a maximal
${\sigma}$-nilpotent subgroup of $G$ by Claim (6).
In fact, $F_{\sigma}(G)=F_{0\sigma }(G)\times O_{\sigma _{i_{1}}}(G)\times \cdots \times
O_{\sigma _{i_{m}}}(G)$ for some $i_{1}, \ldots , i_{m} \subseteq \{r+1,
\ldots , t\}$.
Moreover, in view of Claim (5), we get clearly that $G/C_{G}(O_{\sigma _{i_{k}}}(G))$ is a
$\sigma _{i_{k}}$-group and so $O_{\sigma _{i_{k}}}(G)\leq Z_{\sigma}(G)$.
Hence
$F_{\sigma}(G)=F_{0\sigma }(G)Z_{\sigma}(G)$.
(8) {\sl Statement (iv) holds for $G$.}
First we show that $U_{G}\leq Z_{\sigma}(G)$ for every ${\sigma}$-nilpotent subgroup $U$
of $G$ such that $G=F_{\sigma}(G)U$. Suppose that this is false. Then $U_{G}\ne 1$. Let $R$ be a minimal
normal subgroup of $G$ contained in $U$ and $C=C_{G}(R)$.
Then $$G/R=(F_{\sigma}(G)R/R)(U/R)=F_{\sigma}(G/R)(U/R),$$ so $$U_{G}/R=(U/R)_{G/R}\leq
Z_{\sigma}(G/R)$$ by Claim (2). Since $G$ is $\sigma$-soluble, $R$
is a $\sigma _{i}$-group for some $i$. Moreover, from $G=F_{\sigma}(G)U$
and Lemma 2.1 we get that for some Hall $\sigma _{i}'$-subgroups $E$, $V$ and
$W$ of $G$, of $F_{\sigma}(G)$ and of $U$, respectively, we have $E=VW$.
But $R\leq F_{\sigma}(G)\cap U$, where $F_{\sigma}(G)$ and $U$ are
$\sigma$-nilpotent. Therefore $E\leq C$, so $R/1$ is $\sigma$-central in
$G$. Hence $R\leq Z_{\sigma}(G)$ and so $Z_{\sigma}(G/R)=Z_{\sigma}(G)/R$ by Lemma 2.10(ii).
But then $U_{G}\leq Z_{\sigma}(G)$. Finally, if $V$ is any
maximal ${\sigma}$-nilpotent subgroup of $G$ with $G=F_{\sigma}(G)V$, then $Z_{\sigma}(G)\leq V$ by
Lemma 2.11(iv) and so $V_{G}= Z_{\sigma}(G)$.
(9) {\sl Statement (v) holds for $G$.}
In view of Lemma 2.2(i), it is enough to show that $D=G^{{\mathfrak{N}}_{\sigma}}$ is
nilpotent. Assume that this is false. Then $D\ne 1$, and for
any minimal normal subgroup $R$ of $G$ we have that
$(G/R)^{{\mathfrak{N}}_{\sigma}}=RD/R \simeq D/D\cap R $ is nilpotent by
Claim (2) and Lemmas
2.2(i) and 2.4. Moreover, Lemma 2.2(i) implies
that $R$ is a unique minimal normal subgroup of $G$, $R\leq D$ and $R\nleq
\Phi (G)$.
Since $G$ is not ${\sigma}$-nilpotent, Claim (3) and \cite[Ch. A, 15.6]{DH} imply that
$R=C_{G}(R)=O_{p}(G)=F(G)$
for some prime $p$.
Then $R < D$ and $G=R\rtimes M$, where $M$ is not $\sigma$-nilpotent,
and so
$M$ has an ${\mathfrak{N}}_{\sigma}$-critical
subgroup $A$. Claim (3) implies that for some prime $q$ dividing $|A|$ and for a Sylow
$q$-subgroup $Q$ of $A$ we have $1 < Q\leq F(G)\cap M=R\cap M=1$.
This contradiction completes the proof of (9).
From Claims (5), (7), (8) and (9) it follows that the conclusion of the theorem is
true for $G$, contrary to the choice of $G$. The theorem is proved.
{\bf Proof of Theorem B.}
Assume that this theorem is false and let $G$ be a
counterexample of minimal order. Then $G$ is not ${\sigma}$-nilpotent. Nevertheless, $G$ is
${\sigma}$-soluble by Theorem A.
Let $F_{0\sigma }(G)=H_{1} \cdots H_{r}$ and
$ E=H_{r+1} \cdots H_{t}$. Then $E$ is $\sigma$-nilpotent by Theorem A(ii).
(1) {\sl Every proper subgroup $E$ of $G$ is
semi-${\sigma}$-nilpotent.
Hence Statements (i) and (ii) hold for $E$} (See Claim (1) in the proof of Theorem A).
(2) {\sl The hypothesis holds for every proper quotient $G/N$ of $G$.
Hence Statements (i), (ii) and (iv) hold for
$G/N$. }
It is not difficult to show that $G/N$ is semi-${\sigma}$-nilpotent (see
Claim (2) in the proof of Theorem A).
Now let $U/N$ be any Schmidt $\sigma _{i}$-subgroup of $G/N$ such that
$U/N\leq W/N$ for some non-normal in $G/N$ Hall $\sigma _{i}$-subgroup $W/N$ of $G/N$.
In view of Lemma 2.1, we can assume without loss of generality that
$W/N=H_{i}N/N$.
Let $L$ be
any minimal supplement to $N$ in $U$. Then $L\cap N\leq \Phi (L)$ and, by Lemma 2.8,
$U/N=LN/N\simeq L/L\cap N$ is a $\sigma _{i}$-group and $L/L\cap N
=(P/L\cap N)\rtimes (Q/L\cap N)$, where
$P/L\cap N=(L/L\cap N)^{\frak{N}}=(L/L\cap N)'$
is a Sylow $p$-subgroup of $L/L\cap N$ and $Q/L\cap N=\langle x \rangle $ is a cyclic
Sylow $q$-subgroup of $L/L\cap N$ with
$V/L\cap N=\langle x^{q} \rangle =\Phi (Q/L\cap N)\leq \Phi
(L/L\cap N)\cap Z(L/L\cap N)$
and $p, q \in \sigma _{i}$. Suppose that
$|Q/L\cap N| > q$. Then $L\cap N < V$.
In view of Lemma 2.2(i), a Sylow $p$-subgroup of $L$ is normal in $L$.
Hence, in view of Lemma
2.8, for any Schmidt subgroup $A$ of $L$ we have
$A =A_{p}\rtimes A_{q}$, where $A_{p}$ is a Sylow $p$-subgroup of $A$,
$ A_{q}$ is a Sylow $q$-subgroup of $A$ and $(A_{q})^{A}=A$.
We can assume without loss of
generality that $A_{q}(L\cap N)/(L\cap N) \leq Q/L\cap N$.
Then $A_{q}(L\cap N)/(L\cap N)
\nleq V/L\cap N$ since $V\leq \Phi (L)$.
It follows that $A_{q}\nleq N$. Since $W/N=H_{i}N/N$ is not normal in
$G/N$, $H_{i}$ is not normal in $G$. But for some $x\in G$ we have $A^{x}\leq
H_{i}$, so $|A_{q}^{x}|=|A_{q}|=q$ by hypothesis.
Note that $|Q/V|=q$ since $Q/L\cap N$
is cyclic and $V/L\cap N=\Phi (Q/L\cap N)$. Hence
$$(V/L\cap N)(A_{q}(L\cap N)/(L\cap N))=(V/L\cap N)\times
(A_{q}(L\cap N)/(L\cap N))=Q/(L\cap N) ,$$
which implies that $Q/(L\cap N)$ is not cyclic. This contradiction shows that $|Q/L\cap N|=q$, so
for a Sylow $q$-subgroup
$S$ of $U/N$ we have $|S|=q$. Therefore the hypothesis holds for $G/N$.
Hence we have (2) by the choice of $G$
(3) {\sl If $A$ is an ${\mathfrak{N}}_{\sigma}$-critical
subgroup of $G$, then $A=P\rtimes Q$, where $P=A^{\frak{N}}=A'$
is a Sylow $p$-subgroup of $A$ and $Q$ is a
Sylow $q$-subgroup of $A$ for some different primes $p$ and $q$. Moreover, the
subgroup $P$ is normal in $G$. Hence $G$ has an abelian minimal normal subgroup $R$}
(See Claim (3) in the proof of Theorem A).
(4) {\sl Statement (i) holds for $G$.}
In view of Lemma
2.2(i), it is enough to show that $G'$ is ${\sigma}$-nilpotent. Suppose
that this is false.
(a) {\sl $R=C_{G}(R)=O_{p}(G)=F(G)\nleq \Phi (G)$ for some prime $p$ and $|R| > p$}.
From Claim (2) it follows that for every minimal normal subgroup $N$ of $G$,
$(G/N)'=G'N/N\simeq G'/G'\cap N$
is $\sigma$-nilpotent. If $R\ne N$, it follows that $G'/((G'\cap N)\cap
(G'\cap R))=G'/1$ is $\sigma$-nilpotent by Lemma 2.2(i). Therefore $R$ is a unique
minimal normal subgroup of $G$, $R\leq D$ and $R\nleq \Phi (G)$ by Lemma
2.2(i). Hence
$R=C_{G}(R)=O_{p}(G)=F(G)$ by Theorem 15.6 in \cite[Ch. A]{DH}, so
$|R| > p$ since otherwise
$G/R=G/C_{G}(R)$ is cyclic, which implies that $G'=R$ is ${\sigma}$-nilpotent.
(b) {\sl
$F_{\sigma}(V)=F_{\sigma}(G)$ for every subgroup $V$
of $G$ containing $ F_{\sigma}(G)$} (See Claim (6) in the proof of Theorem A).
(c) {\sl $G=H_{1} \rtimes H_{2}$, where $R\leq H_{1}=F_{\sigma}(G)$ and $H_{2}$ is
a minimal non-abelian group.}
From Theorem A and Claim (a) it follows that $r=1$ and $R\leq H_{1}=F_{\sigma}(G)$.
Now let
$W=F_{\sigma}(G)V$, where $V$ is a maximal subgroup of $E$. Then
$F_{\sigma}(G)=F_{\sigma}(W)$ by Claim (b), so $W/F_{\sigma}(W)=W/F_{\sigma}(G)\simeq V$ is
abelian by Claim (1). Therefore $E$ is not abelian but
every proper subgroup of $E$ is abelian, so $E=H_{2}$ since $E$ is
$\sigma$-nilpotent. Hence we have (c).
(d) {\sl $H_{1}=R$ is a Sylow $p$-subgroup of $G$ and every subgroup $H\ne 1$ of
$H_{2}$ acts irreducibly on $R$. Hence every proper subgroup $H$ of $H_{2}$ is cyclic.}
Suppose that $|\pi (H_{1})| > 1$. There is a Sylow $p$-subgroup $P$
of $H_{1}$ such that $H_{2}\leq N_{G}(P)$ by Claim (c) and the Frattini argument.
Let $K=PH_{2}$. Then $K < G$ and $P=H_{1}\cap K$ is normal in $K$, so
$R\leq P=F_{\sigma}(K)$ since $C_{G}(R)=R$ by Claim (a).
Then $K/F_{\sigma}(K)=K/P\simeq H_{2}$ is abelian by Claim (1), a contradiction.
Hence $H_{1}$ is a normal Sylow $p$-subgroup of $G$. Hence
$H_{1}\leq F(G)\leq C_{G}(R)=R$ by \cite[Ch. A, 13.8(b)]{DH}, so $H_{1}=R$.
Now let $S=RH$. By
the Maschke theorem, $R=R_{1}\times \cdots \times R_{n}$, where $R_{i}$
is a minimal normal subgroup of $S$ for all $i$. Then $R=C_{S}(R)=C_{S}(R_{1}) \cap
\cdots \cap C_{S}(R_{n})$. Hence, for some $i$,
the subgroup $R_{i}H$ is not $\sigma$-nilpotent and so it has
an ${\mathfrak{N}}_{\sigma}$-critical
subgroup $A$ such that $1 < A'$ is normal in $G$ by Claim (3). But then
$R\leq A$. Therefore $i=1$, so we have (d) since $H$ is abelian by Claim (c).
(e) {\sl $H_{2}$ is not nilpotent. Hence $ |\pi (H_{2})| > 1$.}
Suppose that $H_{2}=Q\times H$ is nilpotent, where $Q\ne 1$ is a Sylow $q$-subgroup of $H_{2}$.
If $H\ne 1$, then $Q$ and $H$ are proper subgroups of $H_{2}$ and so the groups $Q$,
$H$ and $H_{2}$ are abelian by Claim (c). Therefore $H_{2}=Q$ is a $q$-group. Then,
since every maximal subgroup of $H_{2}$ is cyclic by Claim (d), $q=2$ by
\cite[Ch. 5, Theorems 4.3, 4.4]{Gor}. Therefore $|R|=p$, contrary to
Claim (a). Hence we have (e).
(f) {\sl $H_{2}=A\rtimes B$, where $A=C_{H_{2}}(A)$ is
a group of prime order $q\ne p$ and $B=\langle a \rangle$ is
a group of order $r$ for some prime $r\not \in \{p, q\}$. }
From Claims
(d) and (e) it follows that $H_{2}$ is a Schmidt group with cyclic
Sylow subgroups. Therefore Claim (f) follows from the hypothesis and Lemma 2.8.
{\sl Final contradiction for (4). } Suppose that for some $x=yz\in RA$,
where $y\in R$ and $z\in A$,
we have $xa=ax$. Then $x\in N_{G}(B)$, so $R\cap \langle x \rangle =1$
since $B$ acts irreducible on $R$ by Claim (d).
Hence $\langle x \rangle $ is a $q$-group and $V=\langle x \rangle B$ is abelian group such that
$B\cap R=1$. Hence from the isomorphism $G/R\simeq H_{2}$ we get that $x=1$.
Therefore $a$ induces a fixed-point-free automorphism on $RA$ and hence $RA$
is nilpotent by the
Thompson theorem \cite[Ch. 10, Theorem 2.1]{Gor}. But then $A\leq
C_{G}(R)=R$. This contradiction completes the proof of (4).
(5) {\sl Statement (ii) holds for $G$.}
Suppose that this is false.
By Lemma 2.10(iv),
$Z_{\sigma}(G)\leq U$. On the other,
$U/Z_{\sigma}(G)$ is a
maximal $\sigma$-nilpotent non-normal subgroup of $G/Z_{\sigma}(G)$ by
Lemma 2.10(v). Hence
in the case $Z_{\sigma}(G)\ne 1$ Claim (2) implies that
$U/Z_{\sigma}(G)$ is a $\sigma$-Carter subgroup $G/Z_{\sigma}(G)$, so
$U$ is a $\sigma$-Carter subgroup of $G$ by Lemma 2.6(ii). Hence $Z_{\sigma}(G)=1$, so
Theorem A(iii) implies that $F_{\sigma}(G)=
F_{0\sigma}(G)=H_{1}\cdots H_{r}$. Hence $E\simeq G/F_{0\sigma}(G)$ is abelian by Claim (4).
Let $V=F_{\sigma}(G)U $.
If $V=G$, then for some $x$ we have $H_{r+1}^{x}\leq U$ by Lemma 2.1. Hence
$U\leq N_{G}(H_{r+1}^{x})$ and so $U= N_{G}(H_{r+1}^{x})$ is a
$\sigma$-Carter subgroup of $G$ by Theorem A(ii).
Therefore
$V=F_{\sigma}(G)U $ is a normal proper subgroup of $G$.
Let $x\in G$. If the subgroup $U^{x}$ is normal in $V$, then $U^{x}$ is
subnormal in $G$ and so $U^{x}, U\leq F_{\sigma}(G) $ by Lemma 2.3(3), which implies that
$U=F_{\sigma}(G) $ is normal in $G$ since $F_{\sigma}(G) $ and $U$ are
maximal $\sigma$-nilpotent subgroups of $G$ by Theorem A(iii). This contradiction shows
that $U^{x}$ and $U$ are non-normal maximal $\sigma$-nilpotent subgroups
of $V$. Since $ V < G$, Claim (1) implies that $U^{x}$ and $U$ are
$\sigma$-Carter subgroups of $V$.
Since $V$ is $\sigma$-soluble, $U$ and $U^{x}$ are conjugate in $V$
by Lemma 2.7. Therefore $G=VN_{G}(U)$ by the Frattini argument. Since $U$
is a maximal
$\sigma$-nilpotent non-normal subgroup of $G$, $U=N_{G}(U)$. Hence
$G=VU=(F_{\sigma}(G)U)U =F_{\sigma}(G)U < G$. This contradiction completes
the proof of the fact that every maximal $\sigma$-nilpotent non-normal subgroup $U$
of $G$ is a $\sigma$-Carter subgroup of $G$. But then
$G=F_{\sigma}(G)U$ since $G/F_{\sigma}(G)$ is $\sigma$-nilpotent by Claim (4) and so
$U_{G}= Z_{\sigma}(G)$ by Theorem A(iv). Hence we have (5).
(6) {\sl If $F_{0\sigma} (G)\leq F(G)$, then $G/F_{\sigma}(G)$ is cyclic.}
Assume that this is false.
(i) {\sl $\Phi (F_{0\sigma}(G))=1$. Hence $F_{0\sigma} (G)$ is the direct product of some
minimal normal subgroups $R_{1}, \ldots , R_{k}$ of $G$.}
Suppose that $\Phi (F_{0\sigma} (G))\ne 1$ and let $N$ be a minimal normal
subgroup of $G$ contained in
$\Phi (F_{0\sigma} (G))\leq \Phi (G)$. Then $N$ is a $p$-group for some prime $p$.
We show that the hypothesis holds for $G/N$. First note that $G/N$ is
semi-${\sigma}$-nilpotent by Claim (2). Now let $V/N$ be a normal Hall
$\sigma _{i}$-subgroup of $G/N$ for some $\sigma _{i}\in \sigma (G/N)$.
If $p\in \sigma _{i}$, then $V$ is normal Hall $\sigma _{i}$-subgroup of $G$, so
$V\leq F(G)$ by hypothesis
and hence $V/N\leq F(G)N/N\leq F(G/N)$. Now assume that $p\not \in \sigma _{i}$
and let $W$ be a
Hall $\sigma _{i}$-subgroup of $V$. Then $W$ is a
Hall $\sigma _{i}$-subgroup of $G$. Moreover, every two
Hall $\sigma _{i}$-subgroups of $V$ are conjugate in $V$ by Lemma 2.1, so
$G=VN_{G}(W)=NWN_{G}(W)=NN_{G}(W)=N_{G}(W)$ by the Frattini argument.
Therefore $W\leq F(G)$, so $V/N=WN/N\leq F(G/N)$.
Hence $F_{0\sigma}(G/N)\leq F(G/N)$, so the hypothesis holds for $G/N$.
The choice of $G$ and Lemma 2.11 imply that $(G/N)/F_{\sigma}(G/N) =
(G/N)/(F_{\sigma}(G)/N)\simeq G/F_{\sigma}(G)$
is cyclic, a contradiction. Hence $\Phi (F_{0\sigma}(G))=1$, so we have (i) by
\cite[Ch. A, Theorem 10.6(c)]{DH}.
(ii) {\sl $Z_{\sigma}(G)=1$. Hence $F_{0\sigma} (G)=F_{\sigma} (G)=F(G)$.}
Since $Z_{\sigma}(G/Z_{\sigma}(G))=1$ by Lemma 2.10(ii), Lemma 2.11 and Theorem A(iii)
imply that
$$
F_{0\sigma}(G/Z_{\sigma}(G))=F_{\sigma}(G/Z_{\sigma}(G))
=F_{\sigma}(G)/Z_{\sigma}(G)=F_{0\sigma}
(G)Z_{\sigma}(G)/Z_{\sigma}(G), $$ where
$F_{0\sigma} (G)\leq F(G)$ and so
$F_{0\sigma}(G/Z_{\sigma}(G))\leq F(G/Z_{\sigma}(G))$. Therefore the
hypothesis holds for $G/Z_{\sigma}(G)$ and hence, in the case when $Z_{\sigma}(G)\ne 1$,
$G/F_{\sigma}(G)\simeq
(G/Z_{\sigma}(G))/F_{\sigma}(G/Z_{\sigma}(G))$ is cyclic by the choice of $G$.
Hence we have (ii).
{\sl Final contradiction for (6).}
Since $E\simeq G/F(G)$ is abelian by Claims (4) and (ii) and $G$ is not nilpotent, there
is an index $i$ such that $V=R_{i}\rtimes E$ is not
nilpotent. Then $C_{R_{i}}(E)\ne R_{i}$. By the Maschke theorem, $R_{i}=
L_{1}\times \cdots \times L_{m}$
for some minimal normal subgroups $L_{1}, \ldots , L_{m}$ of $V$. Then, since
$C_{R_{i}}(E)\ne R_{i}$, for some $j$ we have $L_{j}\rtimes E\ne L_{j}\times E$. Hence
$L_{j}E$ contains a Schmidt subgroup $A_{p}\rtimes A_{q}$ such that $A_{p}=R_{i}$, so $m=1$.
But then $E$ acts irreducible
on $R_{i}$ and hence $G/F(G)\simeq E$ is cyclic. This
contradiction completes the proof of (6).
From Claims (1), (2), (4), (5) and (6) it follows that the conclusion of the theorem is
true for $G$, contrary to the choice of $G$. The theorem is proved.
\end{document} |
\tilde begin{document}
\tilde begin{abstract}
The free-boundary compressible 1-D Euler equations with moving {\it physical
vacuum} boundary are a system of hyperbolic conservation laws which are
both {\it characteristic} and {\it degenerate}. The {\it physical vacuum singularity}
(or rate-of-degeneracy) requires the sound speed $c= \gamma \tilde\rhoho^{ \gamma -1}$ to scale as the square-root of the distance to the
vacuum boundary, and has attracted a great deal of attention in recent years \check{I}te{IMA2009}. We establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev
spaces) all the way to the moving boundary. The proof is founded on a
new higher-order Hardy-type inequality in conjunction with an approximation of the Euler equations
consisting of a particular degenerate parabolic regularization. Our regular solutions can be viewed as
{\it degenerate viscosity solutions}.
\tilde\etand{abstract}
\maketitle
{\starmall
\tableofcontents}
\starection{Introduction}
\label{sec_introduction}
\starubsection{Compressible Euler equations and the physical vacuum boundary}
This paper is concerned with the evolving vacuum state in a compressible gas flow.
For $0 \le t \le T$,
the evolution of a one-dimensional compressible gas moving inside of a dynamic vacuum boundary is modeled by the one-phase compressible Euler equations:
\tilde begin{subequations}
\label{ceuler}
\tilde begin{alignat}{2}
\tilde\rhoho\left[u_t+ uu_x\tilde\rhoight] + p(\tilde\rhoho)_x &=0 &&\text{in} \ \ I (t) \,, \label{ceuler.a}\\
\tilde\rhoho_t+ (\tilde\rhoho u)_x&=0
&&\text{in} \ \ I (t) \,, \label{ceuler.b}\\
p &= 0 \ \ &&\text{on} \ \ \Gamma(t) \,, \label{ceuler.c}\\
\mathcal{V} (\Gamma(t))& = u &&\ \ \label{ceuler.d}\\
(\tilde\rhoho,u) &= (\tilde\rhoho_0,u_0 ) \ \ &&\text{on} \ \ I(0) \,, \label{ceuler.e}\\
I (0) &= I =\{0<x<1\} \,. && \label{ceuler.f}
\tilde\etand{alignat}
\tilde\etand{subequations}
The open, bounded interval
$I (t) \starubset \mathbb{R} $ denotes the changing domain occupied by the gas, $\Gamma(t):= \tilde partial I (t)$ denotes
the moving vacuum boundary, $ \mathcal{V} (\Gamma(t))$ denotes the
velocity of $\Gamma(t)$.
The scalar field $u $ denotes the Eulerian velocity
field, $p$ denotes the pressure function, and $\tilde\rhoho$ denotes the density of the gas.
The equation of state $p(\tilde\rhoho)$ is given by
\tilde begin{equation}\label{eos}
p(x,t)= C_\gamma\, \tilde\rhoho(x,t)^\gamma\ \ \text{ for } \ \ \gamma> 1 ,
\tilde\etand{equation}
where $C_\gamma $ is the adiabatic constant which we set to one, and
$$
\tilde\rhoho>0 \ \text{ in } \ I (t) \ \ \ \text{ and } \ \ \ \tilde\rhoho=0 \ \text{ on } \Gamma(t) \,.
$$
Equation (\tilde\rhoef{ceuler.a}) is the conservation of momentum; (\tilde\rhoef{ceuler.b}) is the conservation of mass; the boundary
condition (\tilde\rhoef{ceuler.c}) states that the pressure (and hence density) must vanish along the vacuum boundary; (\tilde\rhoef{ceuler.d}) states that the
vacuum boundary is moving with the fluid velocity, and (\tilde\rhoef{ceuler.e})-(\tilde\rhoef{ceuler.f}) are the
initial conditions for the density, velocity, and domain. Using the equation of state (\tilde\rhoef{eos}),
(\tilde\rhoef{ceuler.a}) is written as
\tilde begin{alignat}{2}
\tilde\rhoho [u_t+ uu_x]+ (\tilde\rhoho^{\gamma})_x &=0 \ \ \ &&\text{in} \ \ I (t) \,. \tag{ \tilde\rhoef{ceuler.a}'}
\tilde\etand{alignat}
With the sound speed given by $c^2(x,t)= \gamma \tilde\rhoho^{\gamma-1}(x,t)$, and with $c_0 = c(x,0)$, the condition
\tilde begin{equation}\label{phys_vac}
0 < \left|\frac{n-1}{n}rac{ \tilde partial c_0^2}{ \tilde partial x}\tilde\rhoight| < \infty \text{ on } \Gamma
\tilde\etand{equation}
defines a {\it physical vacuum} boundary (or singularity) (see \check{I}te{Liu1996}, \check{I}te{LiYa1997}, \check{I}te{LiYa2000},
\check{I}te{XuYa2005}). Since $ \tilde\rhoho_0 >0$ in $ I$, (\tilde\rhoef{phys_vac}) implies that for some positive constant $C$ and $x\in I$ near the vacuum boundary $\Gamma$,
\tilde begin{equation}\label{degen}
\tilde\rhoho_0^{\gamma-1}(x) \ge C \text{dist}(x, \Gamma) \,.
\tilde\etand{equation}
Equivalently, the physical vacuum condition (\tilde\rhoef{degen}) implies that for some $ \alpha >0$,
\tilde begin{equation}\label{degen1}
\left|\frac{n-1}{n}rac{ \tilde partial \tilde\rhoho_0^{\gamma-1}}{\tilde partial x}(x)\tilde\rhoight|\ge 1 \text{ for any $x$ satisfying $d(x,\tilde partial I)\le\alpha$} \,,
\tilde\etand{equation}
and for a constant $C_ \alpha $, depending on $ \alpha $,
\tilde begin{equation}\label{degen2}
\tilde\rhoho_0^{\gamma-1}(x)\ge C_\alpha>0 \text{ for any $x$ such that $d(x,\tilde partial I)\ge\alpha$} \,.
\tilde\etand{equation}
\tilde vspace{.05 in}
Because of condition (\tilde\rhoef{degen}), the compressible Euler system (\tilde\rhoef{ceuler}) is a {\it degenerate} and {\it characteristic} hyperbolic
system to which standard methods of symmetric hyperbolic conservation laws cannot be applied in standard Sobolev spaces. In
\check{I}te{CoLiSh2009}, we established a priori estimates for the multi-D compressible Euler equations with physical vacuum boundary.
The main result of this paper is the construction of unique solutions in the 1-D case, which are smooth all the way to the moving vacuum boundary on
a (short) time-interval $[0,T]$, where $T$ depends on the initial data.
We combine
the methodology of our a priori estimates \check{I}te{CoLiSh2009}, with a particular degenerate parabolic regularization of Euler, which
follows our methodology in \check{I}te{CoSh2006, CoSh2007}, as well as a new higher-order Hardy-type inequality which permits the
construction of solutions to this degenerate parabolic regularization. As we describe below in Section \tilde\rhoef{history}, our solutions
can be thought of as {\it degenerate viscosity solutions}. The multi-D existence theory is treated in \check{I}te{CoSh2009}.
\starubsection{Fixing the domain and the Lagrangian variables on the reference interval $ I$} We transform the system (\tilde\rhoef{ceuler}) into
Lagrangian variables.
We let $\tilde\etata(x,t)$ denote the ``position'' of the gas particle $x$ at time $t$. Thus,
\tilde begin{equation}
\nonumberonumber
\tilde begin{array}{c}
\tilde partial_t \tilde\etata = u \check{I}rc \tilde\etata $ for $ t>0 $ and $
\tilde\etata(x,0)=x
\tilde\etand{array}
\tilde\etand{equation}
where $\check{I}rc $ denotes composition so that
$[u \check{I}rc \tilde\etata] (x,t):= u(\tilde\etata(x,t),t)\,.$
We set
\tilde begin{align*}
v &= u \check{I}rc \tilde\etata \text{ (Lagrangian velocity)}, \\
f&= \tilde\rhoho \check{I}rc \tilde\etata \text{ (Lagrangian density)}.
\tilde\etand{align*}
The Lagrangian version of equations (\tilde\rhoef{ceuler.a})-(\tilde\rhoef{ceuler.b}) can be written on
the fixed reference domain $ I$ as
\tilde begin{subequations}
\label{ceuler00}
\tilde begin{alignat}{2}
f v_t + ( f^ \gamma)_x &=0 \ \ && \text{ in } I \times (0,T] \,, \label{ceuler00.a} \\
f _t + f v_x/\tilde\etata_x &=0 \ \ && \text{ in } I \times (0,T] \,,\label{ceuler00.b} \\
f &=0 \ \ && \text{ in } I \times (0,T] \,,\label{ceuler00.c} \\
(f,v,\tilde\etata) &=(\tilde\rhoho_0, u_0, e) \ \ \ \ && \text{ in } I \times \{t=0\} \,, \label{ceuler00.d}
\tilde\etand{alignat}
\tilde\etand{subequations}
where $e(x)=x$ denotes the identity map on $I $.
It follows from solving the equation (\tilde\rhoef{ceuler00.b}) that
\tilde begin{equation}\label{J}
f =\tilde\rhoho \check{I}rc \tilde\etata = \tilde\rhoho_0/\tilde\etata_x,
\tilde\etand{equation}
so that the initial density function $\tilde\rhoho_0$ can be viewed as a parameter in the Euler equations. Let $\Gamma:= \tilde partial I $ denote
the initial vacuum boundary; we write the compressible Euler equations (\tilde\rhoef{ceuler00}) as
\tilde begin{subequations}
\label{ceuler0}
\tilde begin{alignat}{2}
\tilde\rhoho_0 v_t + (\tilde\rhoho_0^ \gamma /\tilde\etata_x^\gamma)_x &=0 \ \ && \text{ in } I \times (0,T] \,, \label{ceuler0.a} \\
(\tilde\etata, v) &=( e, u_0) \ \ \ \ && \text{ in } I \times \{t=0\} \,, \label{ceuler0.b} \\
\tilde\rhoho_0^{\gamma-1}& = 0 \ \ &&\text{ on } \Gamma \,, \label{ceuler0.c}
\tilde\etand{alignat}
\tilde\etand{subequations}
with $ \tilde\rhoho _0^{ \gamma -1}(x) \ge C Iperatorname{dist}( x, \Gamma ) $ for $x \in I$ near $\Gamma$.
\starubsection{Setting $\gamma=2$}We will begin our analysis of (\tilde\rhoef{ceuler0}) by considering the
case that $\gamma=2$, in which case, we seek solutions $\tilde\etata$ to the following system:
\tilde begin{subequations}
\label{ce0}
\tilde begin{alignat}{2}
\tilde\rhoho_0 v_t + (\tilde\rhoho_0^2/\tilde\etata_x^2)_x&=0 &&\text{in} \ \ I \times (0,T]
\,, \label{ce0.a}\\
(\tilde\etata,v)&= (e,u_0 ) \ \ \ &&\text{on} \ \ I \times \{t=0\} \,, \label{ce0.b}\\
\tilde\rhoho_0& = 0 \ \ &&\text{ on } \Gamma \,, \label{ce0.c}
\tilde\etand{alignat}
\tilde\etand{subequations}
with $ \tilde\rhoho _0(x) \ge C Iperatorname{dist}( x, \Gamma ) $ for $x \in I$ near $\Gamma$.
The equation
(\tilde\rhoef{ce0.a}) is equivalent to
\tilde begin{equation}
v_t + 2 \tilde\etata_x ^{-1} (\tilde\rhoho_0 \tilde\etata_x^{-1} )_x=0 \label{ce_vor} \,,
\tilde\etand{equation}
and (\tilde\rhoef{ce_vor}) can be written as
\tilde begin{equation}
v_t +\tilde\rhoho_0 ( \tilde\etata_x^{-2})_x + 2(\tilde\rhoho_0)_x \tilde\etata_x^{-2} =0 \label{ce_elliptic} \,.
\tilde\etand{equation}
Because of the degeneracy caused by $\tilde\rhoho_0 =0$ on $\Gamma$, all three equivalent forms of the compressible
Euler equations are crucially used in our analysis. The equation (\tilde\rhoef{ce0.a}) is used for energy estimates, while (\tilde\rhoef{ce_vor})
and (\tilde\rhoef{ce_elliptic}) are used for additional elliptic-type estimates that rely on our higher-order Hardy-type inequality.
\starubsection{The reference domain $ I$}\label{subsec_domain}
As we have already noted above,
the initial domain $ I \starubset \mathbb{R} $ at time $t=0$ is given by
$$
I = (0,1)\,,
$$
and the initial boundary points are given by $\Gamma = \tilde partial I = \{0,1\}$.
\starubsection{The higher-order energy function for the case $\gamma=2$}
We will consider the following higher-order energy function:
\tilde begin{align}
E(t,v) & =\starum_{s=0}^4 \|\tilde partial_t^sv(t,\cdot)\|^2_{H^{2-\frac{n-1}{n}rac{s}{2}}(I)} + \starum_{s=0}^2 \|\tilde\rhoho_0 \tilde partial_t^{2s} v(t,\cdot)\|^2_{H^{3-s}(I)} \nonumber \\
& \tilde qquad \tilde qquad +\|\starqrt{\tilde\rhoho_0}\tilde partial_t \tilde partial_x^{2} v(t,\cdot)\|^2_{L^2(I)} +\|\starqrt{\tilde\rhoho_0}\tilde partial_t^{3}\tilde partial_x v(t,\cdot)\|^2_{L^2(I)}\,.
\label{E}
\tilde\etand{align}
We define the polynomial function $M_0$ by
\tilde begin{equation}\label{M0}
M_0 = P(E(0,v)) \,.
\tilde\etand{equation}
\starubsection{The Main Result}
\tilde begin{theorem} [Existence and uniqueness for the case $\gamma=2$] \label{theorem_main}
Given initial data $(u_0, \tilde\rhoho_0)$ such that $M_0< \infty $ and the physical vacuum condition (\tilde\rhoef{degen}) holds for $\tilde\rhoho_0$, there exists a solution to (\tilde\rhoef{ce0}) (and hence (\tilde\rhoef{ceuler})) on $[0,T]$ for $T>0$ taken
sufficiently small, such that
$$
\starup_{t \in [0,T]} E(t) \le 2M_0 \,.
$$
Moreover if the initial data satisfies
\tilde begin{equation}\label{uniquedata}
\starum_{s=0}^3 \|\tilde partial_t^sv(0,\cdot)\|^2_{{3-s}} + \starum_{s=0}^3 \|\tilde\rhoho_0 \tilde partial_t^{2s} v(0,\cdot)\|^2_{{4-s}} < \infty \,,
\tilde\etand{equation}
then the solution is unique.
\tilde\etand{theorem}
\tilde begin{remark} The case of arbitrary $\gamma >1$ is treated in Theorem \tilde\rhoef{thm_main2} below.
\tilde\etand{remark}
\tilde begin{remark} Given the regularity provided by the energy function (\tilde\rhoef{E}), we see that the Lagrangian flow map
$\tilde\etata \in C([0,T], H^2(I))$. In our estimates for the multi-D problem \check{I}te{CoLiSh2009}, we showed that $\tilde\etata$ gains
regularity with respect to the velocity field $v$, and this fact is essential to control the geometry of the evolving free-surface.
This improved regularity for $\tilde\etata$ also holds in the 1-D setting, but it is not necessary for our estimates, as no
geometry is involved.
\tilde\etand{remark}
\tilde begin{remark}
Given $u_0$ and $\tilde\rhoho_0$, and using the fact that $\tilde\etata(x,0)=x$, the quantity $v_t|_{t=0}$ is computed using (\tilde\rhoef{ce0.a}):
$$
v_t|_{{t=0}} =- \left.\left({\frac{n-1}{n}rac{1}{\tilde\rhoho_0}} (\tilde\rhoho_0^2/\tilde\etata_x^2)_x \tilde\rhoight)\tilde\rhoight|_{t=0} = - 2\frac{n-1}{n}rac{\tilde partial \tilde\rhoho_0}{\tilde partial x} \,.
$$
Similarly, for all $k \in \mathbb{N} $,
$$
\tilde partial_t^k v|_{{t=0}} =- \left. \left( {\frac{n-1}{n}rac{1}{\tilde\rhoho_0}} (\tilde\rhoho_0^2/\tilde\etata_x^2)_x \tilde\rhoight)\tilde\rhoight|_{t=0} \,,
$$
so that each $\tilde partial_t^k v|_{t=0}$ is a function of space-derivates of $u_0$ and $\tilde\rhoho_0$.
\tilde\etand{remark}
\tilde begin{remark}Notice that our functional framework provides solutions which have optimal Sobolev regularity all the way
to the boundary. Hence, in the physical case that $c \starim \starqrt{\text{dist}(\tilde partial I)}$, no singular behavior occurs near
the vacuum boundary, even though both families of characteristics cross, and in particular, meet tangentially to $\Gamma(t)$ at a point.
\tilde\etand{remark}
\tilde begin{remark} Because of the degeneracy of the density function $\tilde\rhoho_0$ at the initial boundary $ \tilde partial I$, no compatibility
conditions are required of the initial data.
\tilde\etand{remark}
\starubsection{History of prior results for the compressible Euler equations with vacuum boundary}\label{history}
Some of the early developments in the theory of vacuum states for compressible gas dynamics can be found in \check{I}te{LiSm1980} and
\check{I}te{Lin1987}.
We are aware of only a handful of previous theorems pertaining to the existence of solutions to the compressible and {\it undamped} Euler equations with moving vacuum boundary. Makino
\check{I}te{M1986} considered compactly supported initial data, and treated the compressible Euler equations for a gas as being set on $\mathbb{R}^3 \times (0,T]$. With
his methodology, it is not possible to track the location of the vacuum boundary (nor is it necessary); nevertheless, an existence theory was developed in
this context, by a variable change that permitted the standard theory of symmetric hyperbolic systems to be employed. Unfortunately, the constraints on the data are too severe to allow for the evolution of the physical vacuum boundary.
In \check{I}te{Li2005b}, Lindblad proved existence and uniqueness for the 3-D compressible Euler equations modeling a {\it liquid} rather than a gas.
For a compressible liquid, the density $\tilde\rhoho\ge \lambda>0$ is assumed to be a strictly positive constant on the moving vacuum boundary $\Gamma(t)$ and is
thus uniformly bounded below by a positive constant. As such, the compressible liquid provides a uniformly hyperbolic, but characteristic, system. Lindblad used Lagrangian variables combined with Nash-Moser iteration to construct solutions. More recently, Trakhinin \check{I}te{Tr2008}
provided an alternative proof for the existence of a compressible liquid, employing a solution strategy based on symmetric hyperbolic systems
combined with Nash-Moser iteration.
In the presence of damping, and with mild singularity, some existence results of smooth
solutions are available, based on the adaptation of the theory of symmetric hyperbolic systems.
In \check{I}te{LiYa1997}, a local existence theory was developed for the case that $c^ \alpha $ (with $0< \alpha \le 1$) is smooth across $\Gamma$,
using methods that are not applicable to the local existence theory
for the physical vacuum boundary. An existence theory for the small perturbation of a
planar wave was developed in \check{I}te{XuYa2005}. See also \check{I}te{LiYa2000} and \check{I}te{Yang2006}, for other features of the vacuum state problem.
The only existence theory for the physical vacuum boundary condition that we know of can be found in the recent paper by Jang and Masmoudi \check{I}te{JaMa2009} for the 1-D compressible gas, wherein weighted Sobolev norms are used for the energy
estimates. From these weighted norms, the regularity of the solutions cannot be directly determined. Letting $d$ denote the distance
function to the boundary $\tilde partial I$, and letting $\| \cdot \|_0 $ denote
the $L^2(I)$-norm, an example of the type
of bound that is proved for the velocity field in \check{I}te{JaMa2009} is the following:
\tilde begin{align}
&\| d\, v\|_0^2 + \|d\, v_x\|_0^2 + \|d\, v_{xx} + 2 v_x\|_0^2 + \|d\, v_{xxx} + 2 v_{xx} - 2 d^{-1}\, v_x\|_0^2 \nonumber \\
&\tilde qquad\tilde qquad \tilde qquad \tilde qquad \tilde qquad\tilde qquad \tilde qquad + \|d\, v_{xxxx} + 4 v_{xxx} - 4 d^{-1}\, v_{xx}\|_0^2 < \infty \,.
\label{jama}
\tilde\etand{align}
The problem with inferring the regularity of $v$ from this bound can already be seen at the level of an $H^1(I)$ estimate. In particular,
the bound on the norm $\|d\, v_{xx} + 2 v_x\|_0^2$ only implies a bound on $\|d\, v_{xx}\|_0^2$ and $\|v_x\|_0^2$ if the integration by
parts on the cross-term,
$$
4\int_I d\, v_{xx} \, v_x = -2\int_I d_x \, |v_x|^2 \,,
$$
can be justified, which in turn requires having better regularity for $v_x$ than the a priori bounds provide. Any methodology which
seeks regularity in (unweighted) Sobolev spaces for solutions must contend with this type of issue. We overcome this difficulty by
constructing (sufficiently) smooth solutions to a degenerate parabolic regularization, and thus the sort of integration-by-parts difficulty
just described is overcome. One can view our solutions as {\it degenerate viscosity solutions}. The key to their construction is
our higher-order Hardy-type inequality that we provide below.
\starection{Notation and Weighted Spaces}\label{notation}
\starubsection{Differentiation and norms in the open interval $I$}
Throughout the paper the symbol $D$ will be used to
denote $\frac{n-1}{n}rac{\tilde partial}{\tilde partial x}$.
For integers $k\ge 0$,
we define the Sobolev space $H^k(I)$ to
be the completion of $C^\infty(I)$
in the norm
$$\|u\|_k := \left( \starum_{a\le k}\int_\Omega \left| D^a u(x)
\tilde\rhoight|^2 dx\tilde\rhoight)^{1/2}\,.$$
For real numbers $s\ge 0$, the Sobolev spaces $H^s(I)$ and the norms $\| \cdot \|_s$ are defined by interpolation.
We use $H^1_0(I)$ to denote the subspace of $H^1(I)$ consisting of those functions $u(x)$ that vanish at $x=0$ and $x=1$.
\starubsection{The embedding of a weighted Sobolev space} Using $d$ to denote the distance function to
the boundary $\Gamma$, and letting $p=1$ or $2$,
the weighted Sobolev space $H^1_{d^p}(I)$, with norm given by
$\int_I d(x)^p (|F(x)|^2+| DF (x)|^2 )\, dx$ for any $F \in H^1_{d^p}(I)$,
satisfies the following embedding:
$$H^1_{d^p}(I) \hookrightarrow H^{1 - \frac{n-1}{n}rac{p}{2}}(I)\,,$$
so that there is a constant $C>0$ depending only on $I$ and $p$ such that
\tilde begin{equation}\label{w-embed}
\|F\|_{1-p/2} ^2 \le C \int_I d(x)^p \tilde bigl( |F(x)|^2 + \left| DF(x) \tilde\rhoight|^2\tilde bigr) \, dx\,.
\tilde\etand{equation}
See, for example, Section 8.8 in Kufner \check{I}te{Kufner1985}.
\starection{A higher-order Hardy-type inequality}
We will make fundamental use of the following generalization of the well-known Hardy inequality to higher-order derivatives:
\tilde begin{lemma}[Higher-order Hardy-type inequality]\label{Hardy} Let $s\ge 1$ be a given integer, and suppose that
\tilde begin{equation}\nonumberonumber
u\in H^s( I)\cap H^1_0( I)\,.
\tilde\etand{equation}
Then if $d$ denotes the distance fuction to $\tilde partial I$, we have that $\displaystyle\frac{n-1}{n}rac{u}{d}\in H^{s-1}( I)$ with
\tilde begin{equation}
\label{Hardys}
\displaystyle\left\|\frac{n-1}{n}rac{u}{d}\tilde\rhoight\|_{s-1}\le C \|u\|_s.
\tilde\etand{equation}
\tilde\etand{lemma}
\tilde begin{proof}
We use an induction argument.
The case $s=1$ is of course the classical Hardy inequality. Let us now assume that the inequality (\tilde\rhoef{Hardys}) holds for a given $s\ge 1$, and suppose that
$$u\in H^{s+1}( I)\cap H^1_0( I)\,.$$
Using $D$ to denote $\frac{n-1}{n}rac{\tilde partial}{\tilde partial x}$,
a simple computation shows that for $m \in \mathbb{N} $,
\tilde begin{equation}
\label{Hardy1}
D^m(\frac{n-1}{n}rac{u}{d})=\frac{n-1}{n}rac{f}{d^{m+1}},
\tilde\etand{equation}
with $$f=\starum_{k=0}^m C_m^k D^{m-k}u\ k! (-1)^k d^{m-k}$$
for a constant $C_m^k$ depending on $k$ and $m$.
From the regularity of $u$, we see that $f\in H^1_0( I)$. Next, with $D= \frac{n-1}{n}rac{\tilde partial}{\tilde partial x}$, we obtain the identity
\tilde begin{align}
Df&=\starum_{k=0}^s C_s^k D^{s+1-k}u\ k! (-1)^k d^{s-k}+\starum_{k=0}^{s-1} C_s^k D^{s-k}u\ k! (-1)^k d^{s-k-1}(s-k)\nonumber\\
&=D^{s+1}u\ s! (-1)^s d^s +\starum_{k=1}^s C_s^k D^{s+1-k}u\ k! (-1)^k d^{s-k}\nonumber\\
& \tilde qquad\tilde qquad\tilde qquad +\starum_{k=0}^{s-1} C_s^{k+1} D^{s-k}u\ (k+1)! (-1)^k d^{s-k-1}\nonumber\\
&=D^{s+1}u\ s! (-1)^s d^s \,. \label{cs1}
\tilde\etand{align}
Since $f\in H^1_0( I)$, we deduce from (\tilde\rhoef{cs1}) that for any $x\in (0,\frac{n-1}{n}rac{1}{2}]$,
\tilde begin{equation*}
f(x)=(-1)^s s!\ \int_0^x D^{s+1}u(y)\ y^s dy,
\tilde\etand{equation*}
which by substitution in (\tilde\rhoef{Hardy1}) yields the identity
\tilde begin{equation*}
\displaystyle D^s(\frac{n-1}{n}rac{u}{d})(x)=\frac{n-1}{n}rac{(-1)^s s!\ \int_0^x D^{s+1}u(y)\ y^s dy}{x^{s+1}},
\tilde\etand{equation*}
which by a simple majoration provides the bound
\tilde begin{equation*}
\displaystyle \tilde bigl|D^s(\frac{n-1}{n}rac{u}{d})(x)\tilde bigr|\le s!\ \frac{n-1}{n}rac{\tilde psi_1(x)\int_0^x |D^{s+1}u(y)|\ dy}{d(x)},
\tilde\etand{equation*}
where $\tilde psi_1$ is the piecewise affine function equal to $1$ on $[0,\frac{n-1}{n}rac{1}{2}]$ and to $0$ on $[\frac{n-1}{n}rac{3}{4},1]$. Next, for any $x\in [\frac{n-1}{n}rac{1}{2},1)$, we obtain similarly that
\tilde begin{equation*}
\displaystyle \tilde bigl|D^s(\frac{n-1}{n}rac{u}{d})(x)\tilde bigr|\le s!\ \frac{n-1}{n}rac{\tilde psi_2(x)\int_x^1 |D^{s+1}u(y)|\ dy}{d(x)},
\tilde\etand{equation*}
where $\tilde psi_2$ is the piecewise affine function equal to $0$ on $[0,\frac{n-1}{n}rac{1}{4}]$ and to $1$ on $[\frac{n-1}{n}rac{1}{2},1]$.
so that for any $x\in I$:
\tilde begin{equation}
\label{Hardy2}
\displaystyle \tilde bigl|D^s(\frac{n-1}{n}rac{u}{d})(x)\tilde bigr|\le s!\ \frac{n-1}{n}rac{\tilde psi_1(x)\int_0^x |D^{s+1}u(y)|\ dy+\tilde psi_2(x)\int_x^1 |D^{s+1}u(y)|\ dy}{d(x)}.
\tilde\etand{equation}
Now, with $g=\tilde psi_1(x)\int_0^x |D^{s+1}u(y)|\ dy+\tilde psi_2(x)\int_x^1 |D^{s+1}u(y)|\ dy$, we notice that $g\in H_1^0( I)$, with $$\|g\|_1\le C\|D^{s+1}u\|_0.$$ Therefore, by the classical Hardy inequality, we infer from (\tilde\rhoef{Hardy2}) that
\tilde begin{equation}
\label{Hardy3}
\tilde bigl\|D^s(\frac{n-1}{n}rac{u}{d})\tilde bigr\|_0\le C \|g\|_1\le C \|D^{s+1}u\|_0.
\tilde\etand{equation}
Since we assumed in our induction process that our generalized Hardy inequality is true at order $s$, we then have that
$$\tilde bigl\|\frac{n-1}{n}rac{u}{d}\tilde bigr\|_{s-1}\le C \|u\|_s,$$
which, together with (\tilde\rhoef{Hardy3}), implies that
$$\tilde bigl\|\frac{n-1}{n}rac{u}{d}\tilde bigr\|_{s}\le C \|u\|_{s+1},$$
and thus establishes the property at order $s+1$, and concludes the proof.
\tilde\etand{proof}
In order to obtain estimates independent of a regularization parameter $\kappa$ defined in Section \tilde\rhoef{statement}, we will also need the following Lemma, whose proof can be found in Lemma 1, Section 6 of \check{I}te{CoSh2006}:
\tilde begin{lemma}\label{kelliptic}
Let $\kappa>0$ and $g\in L^\infty(0,T;H^s(I)))$ be given, and let $f\in H^1(0,T;H^s(I))$ be such that $$f+\kappa f_t=g\ \ \ \text{in}\ (0,T)\times I.$$ Then, $$\|f\|_{L^\infty(0,T;H^s(I))}\le C\, \max\{\|f(0)\|_s,\|g\|_{L^\infty(0,T;H^s(I))}\}.$$
\tilde\etand{lemma}
\starection{A degenerate parabolic approximation of the compressible Euler equations in vacuum}
\label{statement}
For $ \kappa >0$, we consider the following nonlinear degenerate parabolic approximation of the compressible Euler
system (\tilde\rhoef{ce0}):
\tilde begin{subequations}
\label{approximate}
\tilde begin{alignat}{2}
\tilde\rhoho_0 v_t + (\tilde\rhoho_0^2/\tilde\etata_x^2)_x&= \kappa [\tilde\rhoho_0^2 v_x]_x &&\text{in} \ \ I \times (0,T]
\,, \label{approximate.a}\\
(\tilde\etata,v)&= (e,u_0 ) \ \ \ &&\text{on} \ \ I \times \{t=0\} \,, \label{approximate.b}\\
\tilde\rhoho_0& = 0 \ \ &&\text{ on } \Gamma \,, \label{approximate.c}
\tilde\etand{alignat}
\tilde\etand{subequations}
with $ \tilde\rhoho _0(x) \ge C Iperatorname{dist}( x, \Gamma ) $ for $x \in I$ near $\Gamma$.
We will first obtain the existence of a solution to (\tilde\rhoef{approximate}) on a short time interval $[0,T_ \kappa ]$ (with $T_ \kappa $ depending a priori on $\kappa$). We will then perform energy estimates on this solution that will show that the time of existence, in fact, does not depend on $\kappa$, and moreover that our a priori estimates for this sequence of solutions is also independently of
$\kappa$.
The existence of a solution to the compressible Euler equations (\tilde\rhoef{ce0}) is obtained as the weak limit as $ \kappa \to 0$ of the sequence of solutions to (\tilde\rhoef{approximate}).
\starection{Solving the parabolic $\kappa$-problem (\tilde\rhoef{approximate}) by a fixed-point method}
For notational convenience, we will write
$$ \tilde\etata ' = \frac{n-1}{n}rac{\tilde partial \tilde\etata}{ \tilde partial x}$$
and
similarly for other functions.
\starubsection{Assumption on initial data}
Given $u_0$ and $\tilde\rhoho_0$, and using the fact that $\tilde\etata(x,0)=x$, the quantity $v_t|_{t=0}$ for the degenerate parabolic $ \kappa $-problem is computed using (\tilde\rhoef{approximate.a}):
$$
v_t|_{{t=0}} = \left.\left(\frac{n-1}{n}rac{ \kappa }{\tilde\rhoho_0} [\tilde\rhoho_0^2 v']' - {\frac{n-1}{n}rac{1}{\tilde\rhoho_0}} (\tilde\rhoho_0^2/\tilde\etata'^2)' \tilde\rhoight)\tilde\rhoight|_{t=0} =
\left(\frac{n-1}{n}rac{ \kappa }{\tilde\rhoho_0} [\tilde\rhoho_0^2 u_0']' - 2 \tilde\rhoho_0' \tilde\rhoight) \,.
$$
Similarly, for all $k \in \mathbb{N} $,
$$
\tilde partial_t^k v|_{{t=0}} = \left. \frac{n-1}{n}rac{\tilde partial^k}{\tilde partial t^k}\left(\frac{n-1}{n}rac{ \kappa }{\tilde\rhoho_0} [\tilde\rhoho_0^2 v']' - {\frac{n-1}{n}rac{1}{\tilde\rhoho_0}} (\tilde\rhoho_0^2/\tilde\etata'^2)' \tilde\rhoight)\tilde\rhoight|_{t=0} \,.
$$
These formulae make it clear that each $\tilde partial_t^k v|_{t=0}$ is a function of space-derivates of $u_0$ and $\tilde\rhoho_0$.
For the purposes of constructing solutions to the degenerate parabolic $ \kappa $-problem (\tilde\rhoef{approximate}), it is convenient to
smooth the initial data. By standard convolution methods, we assume that $u_0 \in C^ \infty (I)$ and that $\tilde\rhoho_0 \in C^ \infty (I)$
and satisfies (\tilde\rhoef{degen1}) and (\tilde\rhoef{degen2}).
\starubsection{Functional framework for the fixed-point scheme and some notational conventions}
For $T>0$, we shall denote by $\mathcal{X}_T$ the following Hilbert space:
\tilde begin{align}
\mathcal{X}_T=\{&v\in L^2(0,T;H^2(I))|\ \tilde partial_t^4 v\in L^2(0,T;H^1(I)), \ \tilde\rhoho_0 \tilde partial_t^4 v\in L^2(0,T;H^2(I)) \\
& \ \ \tilde partial_t^3 v\in L^2(0,T;H^2(I)), \ \tilde\rhoho_0 \tilde partial_t^3 v\in L^2(0,T;H^3(I)) \}\,,
\tilde\etand{align}
endowed with its natural Hilbert norm:
\tilde begin{align*}
\|v\|_{\mathcal{X}_T}^2 & = \| v\|_{L^2(0,T;H^2(I))}^2 + \| \tilde partial_t^4 v\| _{L^2(0,T;H^1(I))}^2 + \| \tilde\rhoho_0 \tilde partial_t^4 v\| _{L^2(0,T;H^2(I))}^2 \\
& \tilde qquad
+ \| \tilde partial_t^3 v\| _{L^2(0,T;H^2(I))}^2 + \| \tilde\rhoho_0 \tilde partial_t^3 v\| _{L^2(0,T;H^3(I))}^2 \,.
\tilde\etand{align*}
For $M>0$ given sufficiently large, we define the following closed, bounded, convex subset of $\mathcal{X}_T$:
\tilde begin{align}\label{ctm}
\mathcal{C}_T(M)=\{ v\in \mathcal{X}_T \ : \ \|v\|_{\mathcal{X}_T}\le M\},
\tilde\etand{align}
which is indeed non empty if $M$ is large enough. Henceforth, we assume that $T>0$ is given such that independently
of the choice of $v \in \mathcal{C} _T(M)$,
$$
\tilde\etata(x,t) =x + \int_0^t v(x,s)ds
$$
is injective for $t \in [0,T]$, and that ${\frac{n-1}{n}rac{1}{2}} \le \tilde\etata'(x,t) \le {\frac{n-1}{n}rac{3}{2}} $ for $t\in [0,T]$ and $x \in Iverline I$. This can
be achieved by taking $T>0$ sufficiently small: with $e(x)=x$, notice that
$$
\| \tilde\etata'( \cdot ,t) -e\|_1 = \| \int_0^t v'(\cdot ,s) ds\|_1 \le \starqrt{T}M \,.
$$
The space $ \mathcal{X} _T$ will be appropriate for our fixed-point methodology to prove existence of a solution to our $\kappa$-regularized parabolic problem (\tilde\rhoef{approximate}).
Finally, we define the polynomial function $ \mathcal{N} _0$ of norms of the initial data as follows:
\tilde begin{equation}\label{N0}
\mathcal{N} _0 = P( \| \starqrt{\tilde\rhoho_0} \tilde partial_t^5 v'(0)\|_0, \| \tilde partial_t^4 v(0)\|_1, \| \tilde partial_t^3 v(0)\|_2 , \| \tilde partial_t^2 v(0)\|_2 ,
\| \tilde partial_t v(0)\|_2, \| u_0\|_2) \,.
\tilde\etand{equation}
\starubsection{A theorem for the existence and uniqueness of solutions to the parabolic $\kappa $-problem}
We will make use of the Tychonoff fixed-point Theorem in our fixed-point procedure
(see, for example, \check{I}te{Deimling1985}). Recall that this states that for a reflexive separable
Banach space $\mathcal{X}_T$, and $\mathcal{C}_T(M) \starubset \mathcal{X}_T$ a closed, convex, bounded subset, if $F : \mathcal{C}_T(M) \to \mathcal{C}_T(M)$ is
weakly sequentially continuous into $\mathcal{X}_T$, then $F$ has at least one fixed point.
\tilde begin{theorem}[Solutions to the parabolic $ \kappa $-problem]\label{thm_ksoln}
Given our smooth data, for $T $ taken sufficiently small, there exists a unique solution $v \in \mathcal{X}_T$ to the degenerate
parabolic $ \kappa $-problem (\tilde\rhoef{approximate}).
\tilde\etand{theorem}
\starubsection{Linearizing the degenerate parabolic $ \kappa $-problem}
Given $\tilde bar v \in \mathcal{C}_T(M)$, and defining $\tilde bar \tilde\etata (x,t) =x + \int_0^t \tilde bar v(x,\tau)d\tau$, we consider the linear equation for $v$:
\tilde begin{equation}
\label{linear1}
\tilde\rhoho_0 v_t+ \left[\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde bar\tilde\etata'^2}\tilde\rhoight]'-\kappa [\tilde\rhoho_0^2 v']'=0 \,.
\tilde\etand{equation}
We will prove the following:
\tilde begin{enumerate}
\item $v$ is a unique solution to (\tilde\rhoef{linear1});
\item $v \in \mathcal{C}_T(M)$ for $T$ taken sufficiently small;
\item the map $\tilde bar v \mapsto v: \mathcal{C}_T(M) \to \mathcal{C}_T(M)$, and is sequentially weakly continuous in $\mathcal{X}_T$.
\tilde\etand{enumerate}
The solution to our parabolic $ \kappa $-problem (\tilde\rhoef{approximate}) will then be obtained as a fixed-point of the map $\tilde bar v \mapsto v$
in $\mathcal{X}_T$ via the Tychonoff fixed-point Theorem.
In order to use our higher-order Hardy-type inequality, Lemma \tilde\rhoef{Hardy}, it will be convenient to introduce the new variable
$$X=\tilde\rhoho_0 v'$$
which then belongs to $H_0^1( I)$. By a simple computation, we see that (\tilde\rhoef{linear1}) is equivalent to
\tilde begin{equation*}
v_t'+ \left[\frac{n-1}{n}rac{2}{\tilde bar\tilde\etata'}\Bigl(\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde bar\tilde\etata'}\Bigr)'\tilde\rhoight]'-\kappa \left[\frac{n-1}{n}rac{1}{\tilde\rhoho_0}\Bigl[\tilde\rhoho_0^2 v'\Bigr]'\tilde\rhoight]'=0,
\tilde\etand{equation*}
and hence that
\tilde begin{subequations}
\label{div}
\tilde begin{alignat}{2}
\frac{n-1}{n}rac{X_t}{\tilde\rhoho_0}-\kappa \left[\frac{n-1}{n}rac{1}{\tilde\rhoho_0} (\tilde\rhoho_0\ X)'\tilde\rhoight]'&=-\left[\frac{n-1}{n}rac{2}{\tilde bar\tilde\etata'}\Bigl(\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde bar\tilde\etata'}\Bigr)'\tilde\rhoight]' && \ \text{ in }\ [0,T]\times I,\\
X&=0 && \ \text{ on }\ \ [0,T]\times\tilde partial I,\\
X|_{t=0}&=\tilde\rhoho_0 u_0' && \ \text{ on } \ I.
\tilde\etand{alignat}
\tilde\etand{subequations}
We shall therefore solve the degenerate linear parabolic problem (\tilde\rhoef{div}) with Dirichlet boundary conditions, which (as we will prove) will surprising admit a solution with arbitrarily high space regularity (depending on the regularity of the right-hand side and the initial data, of course), and not just an $H_0^1(I)$-type weak solution. After we obtain the solution $X$, we will then easily find our solution $v$ to (\tilde\rhoef{linear1}).
In order to construct our fixed-point, we will need to obtain estimates for $X$ (and hence $v$)
with high space regularity; in particular, we will need to study the fifth time-differentiated problem. For this purpose, it is convenient to define
the new variable
\tilde begin{equation}\label{defineY}
Y=\tilde partial_t^5 X =\tilde\rhoho_0 \tilde partial_t^5 v'\,,
\tilde\etand{equation}
and consider the following equation for $Y$:
\tilde begin{subequations}
\label{divt}
\tilde begin{alignat}{2}
\frac{n-1}{n}rac{Y_{t}}{\tilde\rhoho_0}-\kappa \left[\frac{n-1}{n}rac{1}{\tilde\rhoho_0} (\tilde\rhoho_0\ Y)'\tilde\rhoight]'&=-\tilde partial_t^5\left[\frac{n-1}{n}rac{2}{\tilde bar\tilde\etata'}\Bigl(\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde bar\tilde\etata'}\Bigr)'\tilde\rhoight]' && \ \hbox{in}\ [0,T]\times I, \label{divt.a}\\
Y&=0&& \ \hbox{on}\ [0,T]\times\tilde partial I, \label{divt.b} \\
Y|_{t=0}&=Y_{\text{init}} && \ \hbox{in}\ I \,, \label{divt.c}
\tilde\etand{alignat}
\tilde\etand{subequations}
where $Y_{\text{init}} = \tilde\rhoho_0 \tilde partial_t^5 v'|_{t=0}$.
\starubsection{Existence of a weak solution to the linear problem (\tilde\rhoef{divt}) by a Galerkin scheme}
Let $(e_n)_{n\in\mathbb N}$ denote a Hilbert basis of $H_0^1( I)$, with each $e_n$ being of class $H^2( I)$. Such a choice of basis is indeed possible as we can take for instance the eigenfunctions of the Laplace operator on $I$ with vanishing Dirichlet boundary conditions. We then define the Galerkin approximation at order $n\ge 1$ of (\tilde\rhoef{div}) as being under the form $Y_n=\starum_{i=0}^n \lambda_i^n(t) e_i$ such that:
\tilde begin{subequations}
\label{divn}
\tilde begin{align}
\frac{n-1}{n}orall k\in\{0,...,k\},\ \left(\frac{n-1}{n}rac{{Y_n}_t}{\tilde\rhoho_0} + \kappa (\tilde\rhoho_0 Y_n)' \,, \frac{n-1}{n}rac{ e_k' }{\tilde\rhoho_0} \tilde\rhoight)_{L^2( I)} &=\left(\tilde partial_t^5\tilde bigr[\frac{n-1}{n}rac{2}{\tilde bar\tilde\etata'}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde bar\tilde\etata'}\tilde bigr]'\tilde bigl] \ ,\ e_k'\tilde\rhoight)_{L^2( I)}\ \hbox{in}\ [0,T],\\
\lambda_i^n(0)&=(Y_{\text{init}},e_i)_{L^2( I)}.
\tilde\etand{align}
\tilde\etand{subequations}
Since each $e_i$ is in $H^2( I)\cap H_0^1( I)$, we have by our high-order Hardy-type inequality (\tilde\rhoef{Hardy}) that
$$\frac{n-1}{n}rac{e_i}{\tilde\rhoho_0}\in H^1( I) \,;$$
therefore, each integral written in (\tilde\rhoef{divn}) is well-defined.
Furthermore, as the $e_i$ are linearly independent, so are the $\frac{n-1}{n}rac{e_i}{\starqrt{\tilde\rhoho_0}}$ and therefore the determinant of the matrix
$$\Bigl(\tilde bigl(\frac{n-1}{n}rac{e_i}{\starqrt{\tilde\rhoho_0}},\frac{n-1}{n}rac{e_j}{\starqrt{\tilde\rhoho_0}}\tilde bigr)_{L^2( I)}\Bigr)_{(i,j)\in\mathbb N_n=\{1,...,n\}}$$
is nonzero. This implies that our finite-dimensional Galerkin approximation (\tilde\rhoef{divn}) is a well-defined first-order differential system of order $n+1$, which therefore has a solution on a time interval $[0,T_n]$, where $T_n$ a priori depends on the dimension $n$ of
the Galerkin approximation.
In order to prove that $T_n=T$, with $T$ independent of $n$, we notice that since $Y_n$ is a linear combination of the $e_i$ ($i\in \mathbb N_n$), we have that
\tilde begin{equation}
\label{g1}
\left(\frac{n-1}{n}rac{{Y_n}_t}{\tilde\rhoho_0}-\kappa [\frac{n-1}{n}rac{1}{\tilde\rhoho_0} (\tilde\rhoho_0\ Y_n)']',\ Y_n\tilde\rhoight)_{L^2( I)} =\left(\tilde partial_t^5\tilde bigr[\frac{n-1}{n}rac{2}{\tilde bar\tilde\etata'}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde bar\tilde\etata'}\tilde bigr]'\tilde bigl]\ ,\ Y_n'\tilde\rhoight)_{L^2( I)}.
\tilde\etand{equation}
Since ${Y_n}\in H_0^1( I)$ and $[\frac{n-1}{n}rac{1}{\tilde\rhoho_0} (\tilde\rhoho_0\ Y_n)']'\in H^1( I)$, integration by parts yields
\tilde begin{align}
- \int_I [\frac{n-1}{n}rac{1}{\tilde\rhoho_0} (\tilde\rhoho_0\ Y_n)']'\ Y_n=\int_I [\frac{n-1}{n}rac{1}{\tilde\rhoho_0} (\tilde\rhoho_0\ Y_n)']Y_n'
=\int_I {Y_n'}^2+ \int_I \tilde\rhoho_0'\frac{n-1}{n}rac{Y_n}{\tilde\rhoho_0} Y_n'.
\label{g2}
\tilde\etand{align}
Next, using our higher-order Hardy-type inequality, we see that $\frac{n-1}{n}rac{Y_n}{\tilde\rhoho_0}\in H^1( I)$, and thus
$$
\int_I \tilde\rhoho_0'\frac{n-1}{n}rac{Y_n}{\tilde\rhoho_0} Y_n' =-\int_I \tilde\rhoho_0'\frac{n-1}{n}rac{1}{\tilde\rhoho_0} Y_n' Y_n+\int_I \frac{n-1}{n}rac{\tilde\rhoho_0'^2}{\tilde\rhoho_0^2} {Y_n^2}-\int_I \tilde\rhoho_0''\frac{n-1}{n}rac{Y_n^2}{\tilde\rhoho_0},
$$
which implies that
$$
\int_I \tilde\rhoho_0'\frac{n-1}{n}rac{Y_n}{\tilde\rhoho_0} Y_n' =\frac{n-1}{n}rac{1}{2} \int_I \frac{n-1}{n}rac{\tilde\rhoho_0'^2}{\tilde\rhoho_0^2} {Y_n^2}-\frac{n-1}{n}rac{1}{2} \int_I \tilde\rhoho_0''\frac{n-1}{n}rac{Y_n^2}{\tilde\rhoho_0}.
$$
Substitution of this identity into (\tilde\rhoef{g1}) and (\tilde\rhoef{g2}) yields
\tilde begin{equation*}
\frac{n-1}{n}rac{1}{2} \tilde bigl[\frac{n-1}{n}rac{d}{dt}\int_I\frac{n-1}{n}rac{Y_n^2}{\tilde\rhoho_0}- \kappa \int_I \tilde\rhoho_0''\frac{n-1}{n}rac{Y_n^2}{\tilde\rhoho_0}\tilde bigr] +\kappa\int_I Y_n'^2+\frac{n-1}{n}rac{1}{2}\kappa\int_I \frac{n-1}{n}rac{\tilde\rhoho_0'^2}{\tilde\rhoho_0^2} Y_n^2=
-\int_I \tilde partial_t^5\tilde bigl[\frac{n-1}{n}rac{2}{\tilde bar\tilde\etata'}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde bar\tilde\etata'}\tilde bigr]'\tilde bigr] Y_n'\ ,
\tilde\etand{equation*}
which shows that (since our given $\tilde bar v\in \mathcal{C}_T(M)$):
\tilde begin{equation*}
\frac{n-1}{n}rac{d}{dt}\int_I\frac{n-1}{n}rac{Y_n^2}{\tilde\rhoho_0}- \kappa \|\tilde\rhoho_0''\|_{L^\infty} \int_I \frac{n-1}{n}rac{Y_n^2}{\tilde\rhoho_0}+{\kappa}\int_I Y_n'^2+\kappa\int_I \tilde\rhoho_0'^2\frac{n-1}{n}rac{Y_n^2}{\tilde\rhoho_0^2}\le C_\kappa,
\tilde\etand{equation*}
for a constant $C_ \kappa $ depending on $ \kappa $.
Consequently, $T_n=T$ with $T$ independent of $n$, and
\tilde begin{equation}
\starup_{[0,T]} \int_I \frac{n-1}{n}rac{Y_n^2}{\tilde\rhoho_0} +{\kappa}\int_0^T\int_I Y_n'^2\le C_\kappa T + C \mathcal{N} _0 \,,
\tilde\etand{equation}
where $ \mathcal{N} _0$ is defined in (\tilde\rhoef{N0}).
Thus, there exists a subsequence of $(Y_n)$ which converges weakly to some $Y\in L^2(0,T;H_0^1( I))$, which satisfies
\tilde begin{equation}
\label{g3}
\starup_{[0,T]}\int_I \frac{n-1}{n}rac{Y^2}{\tilde\rhoho_0} +{\kappa}\int_0^T\int_I Y'^2\le C_\kappa T + C \mathcal{N} _0 \,.
\tilde\etand{equation}
With (\tilde\rhoef{defineY}), we see that
\tilde begin{equation}\label{pt5v}
\|\tilde\rhoho_0 \tilde partial_t^5v' \|_{L^2(0,T; H^1(I))} \le C_\kappa T + C \mathcal{N} _0 \,.
\tilde\etand{equation}
Furthermore, it can also be shown using standard arguments that $Y$ is a solution of (\tilde\rhoef{divt}) (where (\tilde\rhoef{divt.a}) is satisfied almost everywhere in $[0,T]\times I$ and holds in a variational sense for all test functions in $L^2(0,T;H_0^1( I))$), and that
\tilde begin{equation}\nonumberonumber
\frac{n-1}{n}rac{Y_t}{\tilde\rhoho_0} \in L^2(0,T; H ^{-1} (I)) \,.
\tilde\etand{equation}
Now, with the functions
$$
X_i = \tilde\rhoho_0\left.\frac{n-1}{n}rac{ \tilde partial^i v'}{ \tilde partial t^i} \tilde\rhoight|_{t=0} \text{ for } i=0,1,2,3,4\,,
$$
we define
\tilde begin{equation}
\label{Z}
Z(t,x)= \int_0^t Y(\cdot,x) = \tilde\rhoho_0(x) \int_0^t \tilde partial_t^5 v' ( \cdot, x ) = \tilde\rhoho_0(x) \tilde partial_t^4 v'( t, x) - X_4(x) \,,
\tilde\etand{equation}
and
\tilde begin{equation}
\label{X}
X(t,x)=\starum_{i=0}^4 X_i t^i+\int_0^t\int_0^{t_4}\int_0^{t_3}\int_0^{t_2} Z(t_1,x) dt_1dt_2dt_3dt_4 \,.
\tilde\etand{equation}
We then see that $X\in C^0([0,T];H_0^1( I))$ is a solution of (\tilde\rhoef{div}), with $\tilde partial_t^5X=Y$. In order to obtain a fixed-point for the map
$\tilde bar v \mapsto v$, we need to establish better space regularity for $Z$, and hence $X$ and $v$.
\starubsection{Improved space regularity for $Z$}
We introduce the variable $\check v$ defined by
\tilde begin{equation*}
\check v(t,x)=\int_0^x \frac{n-1}{n}rac{X(t,\cdot)}{\tilde\rhoho_0(\cdot)} \,,
\tilde\etand{equation*}
so that $\check v$ vanishes at $x=0$, and will us to employ the Poincar\'{e} inequality with this variable.
It is easy to see that
\tilde begin{equation*}
X=\tilde\rhoho_0 \check v',
\tilde\etand{equation*}
and
\tilde begin{equation}\label{Z2}
Z=\tilde\rhoho_0 \tilde partial_t^4\check v'.
\tilde\etand{equation}
Thanks to the standard Hardy inequality, we thus have that
\tilde begin{equation*}
\|\tilde partial_t^4\check v'\|_0\le C \|Z\|_1,
\tilde\etand{equation*}
and hence by Poincar\'e's inequality,
\tilde begin{equation}
\label{g4}
\|\tilde partial_t^4\check v(t,\cdot)\|_1\le C\|Z(t,\cdot)\|_1.
\tilde\etand{equation}
With
$$
F_0= \frac{n-1}{n}rac{Y_{\text{init}}}{\tilde\rhoho_0} + \left. \tilde partial_t^4\tilde bigr[\frac{n-1}{n}rac{2}{\tilde bar\tilde\etata'}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde bar\tilde\etata'}\tilde bigr]'\tilde bigl]' \tilde\rhoight|_{t=0}\,,
$$
our starting point is the equation
\tilde begin{equation*}
\frac{n-1}{n}rac{Y}{\tilde\rhoho_0}-\kappa [\frac{n-1}{n}rac{1}{\tilde\rhoho_0} (\tilde\rhoho_0\ Z)']' =-\tilde partial_t^4\tilde bigr[\frac{n-1}{n}rac{2}{\tilde bar\tilde\etata'}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde bar\tilde\etata'}\tilde bigr]'\tilde bigl]' +
F_0
\ \hbox{in}\ [0,T]\times I,
\tilde\etand{equation*}
which follows from our definition of $Z$ given in (\tilde\rhoef{Z}) and time-integration of (\tilde\rhoef{divt.a}).
From this equation, we infer that
\tilde begin{equation*}
\kappa \tilde bigl\|[\frac{n-1}{n}rac{1}{\tilde\rhoho_0}(\tilde\rhoho_0\ Z)']'\tilde bigr\|_0\le \tilde bigl\|\tilde partial_t^4\tilde bigl[\frac{n-1}{n}rac{2}{\tilde bar\tilde\etata'}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde bar\tilde\etata'}\tilde bigr]'\tilde bigl]'\tilde bigr\|_0
+\tilde bigl\|\frac{n-1}{n}rac{Y}{\tilde\rhoho_0}\tilde bigr\|_0 + \|F_0\|_0 \,.
\tilde\etand{equation*}
By the standard Hardy inequality and the fact that $\tilde bar v\in \mathcal{C}_T(M)$, we obtain the estimate
\tilde begin{equation*}
\kappa \tilde bigl\|[\frac{n-1}{n}rac{1}{\tilde\rhoho_0}(\tilde\rhoho_0\ Z)']'\tilde bigr\|_0\le C_M \starqrt{T} +
C\|{Y}\|_1 + \mathcal{N} _0 \,,
\tilde\etand{equation*}
where $C_M$ is a constant that depends on $M$. In particular, using (\tilde\rhoef{Z2}), we see that
$$
\frac{n-1}{n}rac{1}{\tilde\rhoho_0}(\tilde\rhoho_0\ Z)' = \tilde\rhoho_0 \tilde partial_t^4\check v''+2 \tilde\rhoho_0'\tilde partial_t^4\check v'
$$
so that
\tilde begin{equation*}
\kappa \tilde bigl\|\tilde\rhoho_0 \tilde partial_t^4\check v'''+3 \tilde\rhoho_0'\tilde partial_t^4\check v'' + 2 \tilde\rhoho_0'' \tilde partial_t^4 \check v'\tilde bigr\|_0
\le C_M \starqrt{T} + C\|{Y}\|_1 + \mathcal{N} _0 \,,
\tilde\etand{equation*}
which implies that
\tilde begin{subequations}
\tilde begin{align}
\kappa \tilde bigl\|(\tilde\rhoho_0 \tilde partial_t^4\check v)'''\tilde bigr\|_0 & \le C_M \starqrt{T} +
C\|{Y}\|_1 + \mathcal{N} _0+ \kappa \|\tilde\rhoho_0''' \tilde partial_t^4 \check v\|_0 + \kappa \|3\tilde\rhoho_0''\tilde partial_t^4 \check v'\|_0\nonumber\\
& \le C_M \starqrt{T} +
C\|{Y}\|_1+ \mathcal{N} _0+ \kappa( \|\tilde\rhoho_0'''\|_{L^\infty}+ 3 \|\tilde\rhoho_0''\|_{L^\infty} )\|Z\|_1\nonumber \,,
\tilde\etand{align}
\tilde\etand{subequations}
where we have used (\tilde\rhoef{g4}) in the second inequality above. Having established in (\tilde\rhoef{g4}) that $(\tilde\rhoho_0\tilde partial_t^4\check v)\in H_0^1( I)$, elliptic regularity shows that
\tilde begin{equation}
\label{g5}
\kappa \|\tilde\rhoho_0 \tilde partial_t^4\check v\|_3 \le C_M \starqrt{T} +
C\|{Y}\|_1 + \mathcal{N} _0 + \kappa( \|\tilde\rhoho_0'''\|_{L^\infty}+ 3 \|\tilde\rhoho_0''\|_{L^\infty} )\|Z\|_1\,.
\tilde\etand{equation}
Now, thanks to our high-order Hardy-type inequality, we infer from from (\tilde\rhoef{g5}) that
\tilde begin{equation}
\label{g6}
\kappa \| \tilde partial_t^4\check v\|_2 \le C_M \starqrt{T} +
C\|Y\|_1 + \mathcal{N} _0 + \kappa( \|\tilde\rhoho_0'''\|_{L^\infty}+ 3 \|\tilde\rhoho_0''\|_{L^\infty} )\|Z\|_1\,.
\tilde\etand{equation}
Next we see that (\tilde\rhoef{g5}) implies that
\tilde begin{equation*}
\kappa \|\tilde\rhoho_0 \tilde partial_t^4\check v'+\tilde\rhoho' \tilde partial_t^4\check v\|_2 \le C_M \starqrt{T} +
C\|Y\|_1+ \mathcal{N} _0 + \kappa( \|\tilde\rhoho_0'''\|_{L^\infty}+ 3 \|\tilde\rhoho_0''\|_{L^\infty} )\|Z\|_1\,,
\tilde\etand{equation*}
which thanks to (\tilde\rhoef{g6}) and (\tilde\rhoef{Z2}) implies that
\tilde begin{equation}
\label{g7}
\kappa \|Z\|_2 \le C_M \starqrt{T} +
C\|Y\|_1 + \mathcal{N} _0+ \kappa( \|\tilde\rhoho_0'''\|_{L^\infty}+ 3 \|\tilde\rhoho_0''\|_{L^\infty} )\|Z\|_1\,.
\tilde\etand{equation}
\starubsection{Definition of $v$ and the existence of a fixed-point}
We are now in a position to define $v$ in the following fashion: let us first define on $[0,T]$
\tilde begin{equation*}
f(t)=u_0(0)-\int_0^t \frac{n-1}{n}rac{1}{\tilde\rhoho_0}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde bar\tilde\etata'^2}\tilde bigr]'(\cdot,0)+\kappa\int_0^t\frac{n-1}{n}rac{1}{\tilde\rhoho_0} [\tilde\rhoho_0 X]'(\cdot,0),
\tilde\etand{equation*}
which is well-defined thanks to (\tilde\rhoef{g7}) and (\tilde\rhoef{g3}). We next define
\tilde begin{equation}
v(t,x)=f(t)+\check v(t,x).
\tilde\etand{equation}
We then notice that from (\tilde\rhoef{div}), we immediately have that
\tilde begin{equation*}
v_t'+ \tilde bigl[\frac{n-1}{n}rac{1}{\tilde\rhoho_0} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde bar\tilde\etata'^2}\tilde bigr]'\tilde bigr]'-\kappa \tilde bigl[\frac{n-1}{n}rac{1}{\tilde\rhoho_0} [\tilde\rhoho_0^2 v']'\tilde bigr]'=0,
\tilde\etand{equation*}
from which we infer that in $[0,T]\times I$
\tilde begin{equation*}
v_t+ \frac{n-1}{n}rac{1}{\tilde\rhoho_0} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde bar\tilde\etata'^2}\tilde bigr]'-\kappa \frac{n-1}{n}rac{1}{\tilde\rhoho_0} [\tilde\rhoho_0^2 v']'=g(t),
\tilde\etand{equation*}
for some function $g$ depending only on $t$. By taking the trace of this equation on the left end-point $x=0$, we see that
\tilde begin{equation*}
v_t(t,0)+ \frac{n-1}{n}rac{1}{\tilde\rhoho_0} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde bar\tilde\etata'^2}\tilde bigr]'(t,0)-\kappa \frac{n-1}{n}rac{1}{\tilde\rhoho_0} [\tilde\rhoho_0^2 v']'(t,0)=g(t),
\tilde\etand{equation*}
which together with the identity
$$v_t(t,0)=f_t(t)=-\frac{n-1}{n}rac{1}{\tilde\rhoho_0} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde bar\tilde\etata'^2}\tilde bigr]'(t,0)+\kappa \frac{n-1}{n}rac{1}{\tilde\rhoho_0} [\tilde\rhoho_0^2 v']'(t,0)$$
shows that $g(t)=0$. Therefore, $v$ is a solution of (\tilde\rhoef{linear1}), and also satisfies by construction $v(0,\cdot)=u_0(\cdot)$.
We can now establish the existence of a fixed-point for the mapping $\tilde bar v\tilde\rhoightarrow v$ in $\mathcal{C}_T(M)$, with $T$ taken sufficiently small and depending a priori on $\kappa$.
We first notice that, thanks to the estimates (\tilde\rhoef{g7}) and (\tilde\rhoef{g3}), we have the inequality
\tilde begin{equation*}
\|\tilde partial_t^4 f(t)\|_{L^2(0,T)}\le \mathcal{N} _0+\starqrt{T} C_M,
\tilde\etand{equation*}
which together with (\tilde\rhoef{g6}) and (\tilde\rhoef{g3}) provides the estimate
\tilde begin{equation}
\label{g8}
\|\tilde partial_t^4 v\|_{L^2(0,T;H^2( I))}\le \mathcal{N} _0+C_\kappa \starqrt{T} C_M \,.
\tilde\etand{equation}
Then, (\tilde\rhoef{g5}) implies that
\tilde begin{equation}
\label{g8b}
\|\tilde\rhoho_0\tilde partial_t^4 v\|_{L^2(0,T;H^3( I))}+
\|\tilde partial_t^4 v\|_{L^2(0,T;H^2( I))}\le \mathcal{N} _0+C_\kappa \starqrt{T} C_M \,,
\tilde\etand{equation}
and combining this with (\tilde\rhoef{pt5v}) shows that
\tilde begin{equation}
\label{g8c}
\|\tilde partial_t^5 v\|_{L^2(0,T;H^1( I))}+
\|\tilde\rhoho_0 \tilde partial_t^5 v'\|_{L^2(0,T;H^1( I))}\le \mathcal{N} _0+C_\kappa \starqrt{T} C_M \,,
\tilde\etand{equation}
In turn, (\tilde\rhoef{g8b}) shows that for
\tilde begin{equation}
\label{g9}
T\le \frac{n-1}{n}rac{\mathcal{N} _0^2}{C_\kappa C_M},
\tilde\etand{equation}
$v\in \mathcal{C}_T(M)$.
Moreover, it is clear that there is only one solution $v\in L^2(0,T;H^2( I))$ of (\tilde\rhoef{linear1}) with $v(0)=u_0$ (where this initial
condition is well-defined as $\|v_t\|_{L^2(0,T;H^1( I))}\le \mathcal{N} _0 +C_\kappa \starqrt{T} C_M$),
since if we denote by $w$ another solution with the same regularity, the difference $\displaystyleelta v=v-w$ satisfies $\displaystyleelta v(0,\cdot)=0$ with $\tilde\rhoho_0\displaystyleelta v_t-\kappa [\tilde\rhoho_0^2\displaystyleelta v']'=0$ which implies
$$\frac{n-1}{n}rac{1}{2}\frac{n-1}{n}rac{d}{dt}\int_I \tilde\rhoho_0 \displaystyleelta v^2 +\kappa \int_I \tilde\rhoho_0^2 \displaystyleelta v^2=0,$$
which with $\displaystyleelta v(0,\cdot)=0$ implies $\displaystyleelta v=0$. So the mapping $\tilde bar v\tilde\rhoightarrow v$ is well defined, and thanks to (\tilde\rhoef{g8}) is a mapping from $\mathcal{C}_T(M)$ into itself for $T=T_\kappa$ satisfying \tilde\rhoef{g9}). As it is furthermore clear that it is weakly continuous in the $L^2(0,T_\kappa;H^2( I))$ norm, the Tychonoff fixed-point theorem \check{I}te{Deimling1985} provides us with the existence of a fixed-point to this mapping. Such a fixed-point, which we denote by $v_\kappa$, is a solution of the nonlinear degenerate parabolic
$\kappa$-problem (\tilde\rhoef{approximate}), with initial condition $v_\kappa(0,\cdot)=u_0(\cdot)$. It should be clear that the fixed-point
$v_ \kappa $ also satisfies (\tilde\rhoef{pt5v}) and (\tilde\rhoef{g8b}) so that
\tilde begin{align}
&\|\tilde\rhoho_0 \tilde partial_t^4 v_ \kappa \|_{L^2(0,T;H^3( I))} +
\|\tilde partial_t^4 v_ \kappa \|_{L^2(0,T;H^2( I))} \nonumber \\
& \tilde qquad +
\|\tilde\rhoho_0 \tilde partial_t^5 v'_ \kappa \|_{L^2(0,T;H^1( I))} +
\|\tilde partial_t^5 v_ \kappa \|_{L^2(0,T;H^1( I))}
\le M \,,
\label{g8d}
\tilde\etand{align}
In the next section, we establish $\kappa $-independent estimates for $v_\kappa$ in $L^2(0,T_\kappa;H^2( I))$ (which are indeed
possible because our parabolic approximate $ \kappa $-problem respects the stucture of the original compressible Euler
equations (\tilde\rhoef{ce0})), from
which we infer a short time-interval of existence $[0,T]$, with $T$ independent of $\kappa$. These $ \kappa $-independent estimates will
allows us to pass to the weak limit of the sequence $v_ \kappa $ as $ \kappa \to 0$ to obtain the solution to (\tilde\rhoef{ce0}).
\starection{Asymptotic estimates for $v_\kappa$ which are independent of $\kappa$.}
\starubsection{The higher-order energy function appropriate for the asymptotic estimates as $ \kappa \to 0$.}
Our objective in this section is to show that the higher-order energy function $E$ defined in (\tilde\rhoef{E}) satisfies the inequality
\tilde begin{equation}\label{poly}
\starup_{t \in [0,T]} E(t) \le M_0 + C\,T\, P( \starup_{t \in [0,T]} E(t)) \,,
\tilde\etand{equation}
where $P$ denotes a polynomial function, and
for $T>0$ taken sufficiently small, with $M_0$ defined in (\tilde\rhoef{M0}). The norms in $E$ are for solutions $v_ \kappa $ to
our degenerate parabolic $ \kappa $-problem (\tilde\rhoef{approximate}). According to Theorem \tilde\rhoef{thm_ksoln},
$v_\kappa \in X_{T_ \kappa }$ with the additional bound $ \|\tilde partial_t^4 v_ \kappa \|_{L^2(0,T_ \kappa ;H^2( I))} < \infty $ provided by (\tilde\rhoef{g8}).
As such, the energy function $E$ is continuous with respect to $t$, and the inequality (\tilde\rhoef{poly}) would thus establish a time
interval of existence and bound which are both independent of $ \kappa $.
For the sake of notational convenience, we shall denote $v_\kappa$ by $\tilde v$.
\starubsection{A $\kappa$-independent energy estimate on the fifth time-differentiated problem}
Our starting point shall be the fifth time-differentiated problem of (\tilde\rhoef{approximate}) for which we have,
by naturally using $\tilde partial_t^5 \tilde v\in L^2(0,T_\kappa;H^1( I))$ as a test function, the following identity:
\tilde begin{equation}
\label{a1}
\underbrace{\frac{n-1}{n}rac{1}{2} \frac{n-1}{n}rac{d}{dt}\int_I{\tilde\rhoho_0}|\tilde partial_t^5\tilde v|^2}_{\mathcal{I} _1} \
- \ \underbrace{\int_I \tilde partial_t^5\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^2}\tilde bigr] \tilde partial_t^5\tilde v'}_{ \mathcal{I} _2} \
+ \ \underbrace{\kappa\int_I \tilde\rhoho_0^2(\tilde partial_t^5\tilde v')^2=0 }_{ \mathcal{I} _3}\,.
\tilde\etand{equation}
In order to form the exact time-derivative in term $ \mathcal{I} _1$, we rely on the fact that solutions we constructed to
(\tilde\rhoef{approximate}) satisfy
$ \tilde partial_t^6 v \in L^2(0,T_ \kappa ; L^2 (I))$, which follows from the relation
\tilde begin{equation*}
\tilde partial_t^6 \tilde v = \tilde partial_t^5\left[\frac{n-1}{n}rac{2}{\tilde\tilde\etata'}\Bigl(\frac{n-1}{n}rac{\tilde\rhoho_0}{\tilde\tilde\etata'}\Bigr)'\tilde\rhoight]+ \frac{n-1}{n}rac{\kappa}{\tilde\rhoho_0}\Bigl[\tilde\rhoho_0^2 \tilde partial_t^5 v'\Bigr]' \,,
\tilde\etand{equation*}
and the estimate (\tilde\rhoef{g8d}).
Upon integration in time, both the terms $ \mathcal{I} _1$ and $ \mathcal{I} _3$ provide sign-definite energy contributions, so we
focus our attention on the nonlinear estimates required of the term $ \mathcal{I} _2$.
We see that
\tilde begin{align*}
- \mathcal{I} _2&=2\int_I \tilde partial_t^4\tilde v'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^3}\tilde bigr] \tilde partial_t^5\tilde v'+\starum_{a=1}^4 c_a \int_I \tilde partial_t^{5-a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde partial_t^{a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde\rhoho_0^2\tilde partial_t^5\tilde v'\nonumber\\
&=\frac{n-1}{n}rac{d}{dt}\int_I (\tilde partial_t^4\tilde v')^2\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^3}+3\int_I (\tilde partial_t^4\tilde v')^2\tilde v'\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^4}+\starum_{a=1}^4 c_a \int_I \tilde partial_t^{5-a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde partial_t^{a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde\rhoho_0^2\tilde partial_t^5\tilde v' \,.
\tilde\etand{align*}
Hence integrating (\tilde\rhoef{a1}) from $0$ to $t\in[0,T_\kappa]$, we find that
\tilde begin{align}
\label{a2}
\frac{n-1}{n}rac{1}{2} \int_I{\tilde\rhoho_0}\tilde partial_t^5\tilde v^2 (t) &+ \int_I (\tilde partial_t^4\tilde v')^2\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^3} (t) +\kappa\int_0^t\int_I \tilde\rhoho^2(\tilde partial_t^5\tilde v')^2\nonumber\\
&=\frac{n-1}{n}rac{1}{2} \int_I{\tilde\rhoho_0}\tilde partial_t^5\tilde v^2 (0) + \int_I (\tilde partial_t^4\tilde v')^2\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^3} (0)-3\int_0^t\int_I (\tilde partial_t^4\tilde v')^2\tilde v'\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^4}\nonumber\\
&\ \ \ -\starum_{a=1}^4 c_a \int_0^t\int_I \tilde partial_t^{5-a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde partial_t^{a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde\rhoho_0^2\tilde partial_t^5\tilde v'\ .
\tilde\etand{align}
We next show that all of the error terms, comprising the right-hand side of (\tilde\rhoef{a2}) can be bounded by $C t P( \starup_{[0,t]} E)$.
For the first spacetime integral appearing on the right-hand side of (\tilde\rhoef{a2}), it is clear that
\tilde begin{equation}
\label{a3}
-3\int_0^t\int_I (\tilde partial_t^4\tilde v')^2\tilde v'\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^4} \le C t \ P(\starup_{[0,t]} E).
\tilde\etand{equation}
We now study the last integrals on the right-hand side of (\tilde\rhoef{a2}). Using integration-by-parts in time, we have that
\tilde begin{equation}
\label{a4}
\int_0^t\int_I \tilde partial_t^{5-a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde partial_t^{a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde\rhoho_0^2\tilde partial_t^5\tilde v'=-\int_0^t\int_I [\tilde partial_t^{5-a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde partial_t^{a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}]_t\tilde\rhoho_0^2\tilde partial_t^4\tilde v'+\left. \int_I \tilde partial_t^{5-a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde partial_t^{a}\frac{n-1}{n}rac{1}{\tilde\tilde\etata'}\tilde\rhoho_0^2\tilde partial_t^4\tilde v'\tilde\rhoight|_0^t.
\tilde\etand{equation}
We first consider the spacetime integral on the right-hand side of (\tilde\rhoef{a4}). As the sum is taken for $a=1$ to $4$, we then see that it will be written under the form of the sum of spacetime integrals of the following type:
\tilde begin{align*}
I_1&=\int_0^t\int_I \tilde\rhoho_0 \tilde partial_t^4\tilde v' R(\tilde \tilde\etata) \tilde\rhoho_0\tilde partial_t^4\tilde v',\\
I_2&=\int_0^t\int_I \tilde\rhoho_0 \tilde partial_t^3\tilde v' v' \tilde partial_t \tilde v' R(\tilde \tilde\etata) \tilde\rhoho_0\tilde partial_t^4\tilde v',\\
I_3&=\int_0^t\int_I \tilde\rhoho_0 \tilde partial_t^2\tilde v' v' \tilde partial_t^2 \tilde v' R(\tilde \tilde\etata) \tilde\rhoho_0\tilde partial_t^4\tilde v',\\
I_4&=\int_0^t\int_I \tilde\rhoho_0 \tilde partial_t^2\tilde v' v' R(\tilde \tilde\etata) (\tilde partial_t\tilde v')^2\tilde\rhoho_0\tilde partial_t^4\tilde v'\ ,
\tilde\etand{align*}
Where $R(\tilde\tilde\etata)$ denotes a rational function of $\tilde\tilde\etata$. We first immediately see that
\tilde begin{equation}
\label{a5}
|I_1|\le C t \ P(\starup_{[0,t]} E).
\tilde\etand{equation}
Next, we have that
\tilde begin{align}
\label{a6}
|I_2| &\le C \int_0^t\int_I \|\tilde\rhoho_0 \tilde partial_t^3\tilde v'\|_{L^4} \|v'\|_{L^\infty} \|\tilde partial_t \tilde v'\|_{L^4} \|R(\tilde \tilde\etata)\|_{L^\infty} \|\tilde\rhoho_0\tilde partial_t^4\tilde v'\|_0\nonumber\\
&\le C \int_0^t\int_I \|\tilde\rhoho_0 \tilde partial_t^3\tilde v'\|_{H^{\frac{n-1}{n}rac{1}{2}}} \|v'\|_{L^\infty} \|\tilde partial_t \tilde v'\|_{H^{\frac{n-1}{n}rac{1}{2}}} \|R(\tilde \tilde\etata)\|_{L^\infty} \|\tilde\rhoho_0\tilde partial_t^4\tilde v'\|_0\nonumber\\
& \le C t \ P(\starup_{[0,t]} E).
\tilde\etand{align}
Similarly,
\tilde begin{align}
\label{a6b}
|I_3| &\le C \int_0^t \|\tilde\rhoho_0 \tilde partial_t^2\tilde v'\|_{L^\infty} \|v'\|_{L^\infty} \|\tilde partial_t^2 \tilde v'\|_0\| R(\tilde \tilde\etata)\|_{L^\infty} \|\tilde\rhoho_0\tilde partial_t^4\tilde v'\|_0\nonumber\\
&\le C t \ P(\starup_{[0,t]} E),
\tilde\etand{align}
and
\tilde begin{align}
\label{a7}
|I_4| &\le C \int_0^t \|\tilde\rhoho_0 \tilde partial_t^2\tilde v'\|_{L^\infty} \|v'\|_{L^\infty} \|\tilde partial_t \tilde v'\|^2_{L^4}\| R(\tilde \tilde\etata)\|_{L^\infty} \|\tilde\rhoho_0\tilde partial_t^4\tilde v'\|_0\nonumber\\
& \le C t \ P(\starup_{[0,t]} E),
\tilde\etand{align}
where we have used the fact that in 1-D, $\|\cdot\|_{L^\infty}\le C \|\cdot\|_{H^1}$ and $\|\cdot\|_{L^4}\le C \|\cdot\|_{H^{\frac{n-1}{n}rac{1}{2}}}$.
Therefore, estimates (\tilde\rhoef{a2})--(\tilde\rhoef{a6}) provide us with
\tilde begin{equation*}
\frac{n-1}{n}rac{1}{2} \int_I{\tilde\rhoho_0}\tilde partial_t^5\tilde v^2 (t) + \int_I (\tilde partial_t^4\tilde v')^2\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^3} (t) +\kappa\int_0^t\int_I \tilde\rhoho_0^2(\tilde partial_t^5\tilde v')^2
\le M_0+ C t \ P(\starup_{[0,t]} E),
\tilde\etand{equation*}
and thus, employing the fundamental theorem of calculus,
\tilde begin{align}
\label{a8}
&\frac{n-1}{n}rac{1}{2} \int_I{\tilde\rhoho_0}\tilde partial_t^5\tilde v^2 (t) + \int_I (\tilde\rhoho_0 \tilde partial_t^4\tilde v')^2 (t) +\kappa\int_0^t\int_I \tilde\rhoho_0^2(\tilde partial_t^5\tilde v')^2\nonumber\\
&\tilde qquad\tilde qquad\tilde qquad \tilde qquad \le M_0+ C t \ P(\starup_{[0,t]} E)+ 3\int_I (\tilde partial_t^4\tilde v')^2(t){\tilde\rhoho_0^2}\int_0^t \frac{n-1}{n}rac{\tilde v'}{\tilde\tilde\etata'^4}\nonumber\\
&\tilde qquad\tilde qquad\tilde qquad \tilde qquad \le M_0+ C t \ P(\starup_{[0,t]} E).
\tilde\etand{align}
\starubsection{Elliptic and Hardy-type estimates for $\tilde partial_t^2 v(t)$} Having obtained the energy estimate (\tilde\rhoef{a8}) for the
fifth time-differentiated problem, we can begin our bootstrapping
argument. We now consider the third time-differentiated version of
(\tilde\rhoef{approximate.a}),
\tilde begin{equation*}
\tilde bigl[\tilde partial_t^3\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^2}\tilde bigr]'-\kappa [\tilde\rhoho_0^2 \tilde partial_t^3 \tilde v']'=-\tilde\rhoho_0 \tilde partial_t^4 \tilde v,
\tilde\etand{equation*}
which can be written as
\tilde begin{equation*}
-2\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'}{\tilde\tilde\etata'^3}\tilde bigr]'-\kappa [\tilde\rhoho_0^2 \tilde partial_t^3 \tilde v']'=-\tilde\rhoho_0\tilde partial_t^4 \tilde v+c_1\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]' +c_2 \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde v'^3}{\tilde\tilde\etata'^5}\tilde bigr]',
\tilde\etand{equation*}
and finally rewritten as the following identity:
\tilde begin{align}
-2\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'}\tilde bigr]'-\kappa [\tilde\rhoho_0^2 \tilde partial_t^3 \tilde v']'=&-\tilde\rhoho_0\tilde partial_t^4 \tilde v+c_1\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]' +c_2 \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde v'^3}{\tilde\tilde\etata'^5}\tilde bigr]' \nonumber\\
&-2\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'}\tilde bigr]'(1-\frac{n-1}{n}rac{1}{\tilde\tilde\etata'^3})-6\tilde\rhoho_0^2\tilde partial_t^2\tilde v'\frac{n-1}{n}rac{\tilde\tilde\etata''}{\tilde\tilde\etata'^4}\ . \label{cs3}
\tilde\etand{align}
Here, $c_1$ and $c_2$ are constants whose exact value is not important.
Therefore, using Lemma \tilde\rhoef{kelliptic} together with the fundamental theorem of calculus for the fourth term on the right-hand side
of (\tilde\rhoef{cs3}), we obtain that for any $t\in [0,T_\kappa]$,
\tilde begin{align}
\label{a9}
\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{2}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'}\tilde bigr]'\tilde bigr\|_0 \le & \starup_{[0,t]}\|\tilde partial_t^4 \tilde v\|_0 +\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{c_1}{\tilde\rhoho_0}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]'\tilde bigr\|_0 +\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{c_2}{\tilde\rhoho_0} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde v'^3}{\tilde\tilde\etata'^5}\tilde bigr]'\tilde bigr\|_0\nonumber\\
&+\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{2}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'}\tilde bigr]'\|_0\|3\int_0^\cdot\frac{n-1}{n}rac{\tilde v'}{\tilde\tilde\etata'^4}\|_{L^\infty}+6\starup_{[0,t]} \tilde bigl\|\tilde\rhoho_0\tilde partial_t^2\tilde v'\frac{n-1}{n}rac{\tilde\tilde\etata''}{\tilde\tilde\etata'^4}\tilde bigr\|_0\ .
\tilde\etand{align}
We next estimate each term on the right hand side of (\tilde\rhoef{a9}). For the first term, we will use our estimate (\tilde\rhoef{a8}) from which we infer for each $t\in [0,T_\kappa]$:
\tilde begin{equation*}
\int_I \tilde\rhoho_0^2 [\tilde partial_t^4\tilde v^2+\tilde partial_t^4\tilde v'^2] (t) \le M_0+ C t \ P(\starup_{[0,t]} E).
\tilde\etand{equation*}
Note that the first term on the left-hand side of (\tilde\rhoef{a10}) comes from the first term of (\tilde\rhoef{a8}), together with the fact that $\displaystyle\tilde partial_t^4 v(t,x)=v_4(x)+\int_0^t \tilde partial_t^5 v(\cdot,x).$ Therefore, the Sobolev weighted embedding estimate (\tilde\rhoef{w-embed}) provides us with
the following estimate:
\tilde begin{equation}
\label{a10}
\int_I \tilde partial_t^4\tilde v^2 (t) \le M_0+ C t \ P(\starup_{[0,t]} E).
\tilde\etand{equation}
The remaining terms will be estimated by simply using the definition of the energy function $E$. For the second term on the right-hand side of (\tilde\rhoef{a9}), we have that
\tilde begin{align*}
\tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]'\tilde bigr\|_0
& \le \|(\tilde\rhoho_0 v_t')'\|_0 \tilde bigl\|\frac{n-1}{n}rac{\tilde v'}{\tilde\tilde\etata'^4}\tilde bigr\|_{L^\infty}+\tilde bigl\|\tilde v_t'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]'\tilde bigr\|_0\\
& \le C \|(\tilde\rhoho_0 \tilde v_t')'\|_0 \|\tilde v'\|_{\frac{n-1}{n}rac{3}{4}}+\tilde bigl\|\tilde v_t'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]\tilde bigr\|_0+\tilde bigl\|\tilde v_t'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0\ \tilde v''}{\tilde\tilde\etata'^4}\tilde bigr]\tilde bigr\|_0+4\tilde bigl\|\tilde v_t'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0\ \tilde v'\tilde\tilde\etata''}{\tilde\tilde\etata'^5}\tilde bigr]\tilde bigr\|_0\\
& \le C \|(\tilde\rhoho_0 v_1')'+\int_0^\cdot (\tilde\rhoho_0 v_{tt}')'\|_0 \|\tilde v'\|_1^{\frac{n-1}{n}rac{1}{2}} \|v_1'+\int_0^\cdot \tilde v_t'\|_{\frac{n-1}{n}rac{1}{2}}^{\frac{n-1}{n}rac{1}{2}}+C\|v_1'+\int_0^\cdot \tilde v_{tt}'\|_0 \|\tilde v'\|_{\frac{n-1}{n}rac{3}{4}}\\
& \tilde qquad \ \ +C \|\tilde v_t'\|_0 \|\tilde\rhoho_0 v''\|_{\frac{n-1}{n}rac{3}{4}} + C \|\tilde v'\|_{\frac{n-1}{n}rac{3}{4}}\|\int_0^\cdot \tilde v''\|_0 \|\tilde\rhoho_0 v_1'+\int_0^\cdot\tilde\rhoho_0 \tilde v_{tt}'\|_1\\
& \le \|(\tilde\rhoho_0 v_1')'+\int_0^\cdot (\tilde\rhoho_0 v_{tt}')'\|_0 \|\tilde v'\|_1^{\frac{n-1}{n}rac{1}{2}} \|v_1' +\int_0^\cdot \tilde v_t'\|_{\frac{n-1}{n}rac{1}{2}}^{\frac{n-1}{n}rac{1}{2}}\\
& \tilde qquad \ \ +C\|v_1'+\int_0^\cdot \tilde v_{tt}'\|_0 \|\tilde v'\|_1^{\frac{n-1}{n}rac{1}{2}} \|v_1'+\int_0^\cdot \tilde v_t'\|_{\frac{n-1}{n}rac{1}{2}}^{\frac{n-1}{n}rac{1}{2}} \\
& \tilde qquad \ \ +C \|\tilde v_t'\|_0 \|\tilde\rhoho_0 v''\|_1^{\frac{n-1}{n}rac{3}{4}} \|\tilde\rhoho_0 v_0''+\int_0^\cdot \tilde\rhoho_0 v_t''\|_0^{\frac{n-1}{n}rac{1}{4}},
\tilde\etand{align*}
where we have used the fact that $\|\cdot\|_{L^\infty}\le C \|\cdot\|_{\frac{n-1}{n}rac{3}{4}}$.
Thanks to the definition of $E$, the previous inequality provides us, for any $t\in [0,T_\kappa]$, with
\tilde begin{equation}
\label{a11}
\starup_{[0,t]}\tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]'\tilde bigr\|_0\le C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
For the third term on the right-hand side of (\tilde\rhoef{a9}), we have similarly that
\tilde begin{align*}
\tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde v'^3}{\tilde\tilde\etata'^5}\tilde bigr]'\tilde bigr\|_0
&\le \|(\tilde\rhoho_0 v')'\|_{L^\infty} \tilde bigl\|\frac{n-1}{n}rac{\tilde v'^2}{\tilde\tilde\etata'^5}\tilde bigr\|_{L^2}+\tilde bigl\|\tilde v'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0\ \tilde v'^2}{\tilde\tilde\etata'^5}\tilde bigr]'\tilde bigr\|_0\\
& \le C \|(\tilde\rhoho_0 \tilde v')'\|_{\frac{n-1}{n}rac{3}{4}} \|\tilde v'\|^2_{\frac{n-1}{n}rac{1}{2}}+\tilde bigl\|\tilde v'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0'\ \tilde v'^2}{\tilde\tilde\etata'^5}\tilde bigr]\tilde bigr\|_0+2\tilde bigl\|\tilde v'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0\ \tilde v''\tilde v'}{\tilde\tilde\etata'^5}\tilde bigr]\tilde bigr\|_0+5\tilde bigl\|\tilde v'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0\ \tilde v'^2\tilde\tilde\etata''}{\tilde\tilde\etata'^6}\tilde bigr]\tilde bigr\|_0\\
& \le C \|(\tilde\rhoho_0 v_0')'+\int_0^\cdot (\tilde\rhoho_0 v_{t}')'\|_0^{\frac{n-1}{n}rac{1}{4}} \|(\tilde\rhoho_0 \tilde v')'\|_1^{\frac{n-1}{n}rac{3}{4}} \|\tilde v_0'+\int_0^\cdot \tilde v'_t\|_{\frac{n-1}{n}rac{1}{2}}^2
+C \|\tilde v_0'+\int_0^\cdot \tilde v'_t\|_{\frac{n-1}{n}rac{1}{2}}^3\\
&\tilde qquad \ \ +C \|\tilde v'\|_{\frac{n-1}{n}rac{1}{2}}^2 \|\tilde\rhoho_0 v''\|_{L^4} + C \|\tilde v'\|_{\frac{n-1}{n}rac{1}{2}}^3\|\tilde\rhoho_0 \tilde\tilde\etata''\|_{L^4}\\
& \le C \|(\tilde\rhoho_0 v_0')'+\int_0^\cdot (\tilde\rhoho_0 v_{t}')'\|_0^{\frac{n-1}{n}rac{1}{4}} \|(\tilde\rhoho_0 \tilde v')'\|_1^{\frac{n-1}{n}rac{3}{4}} \|\tilde v_0'+\int_0^\cdot \tilde v'_t\|_{\frac{n-1}{n}rac{1}{2}}^2
+C \|\tilde v_0'+\int_0^\cdot \tilde v'_t\|_{\frac{n-1}{n}rac{1}{2}}^3\\
&\tilde qquad \ \ +C \|\tilde v_0'+\int_0^\cdot \tilde v_t'\|_{\frac{n-1}{n}rac{1}{2}}^2 \|\tilde\rhoho_0 v_0''+\int_0^\cdot \tilde\rhoho_0 v_t''\|_0^{\frac{n-1}{n}rac{1}{2}}\|\tilde\rhoho_0 \tilde v''\|_1^{\frac{n-1}{n}rac{1}{2}} + C \|\tilde v_0'+\int_0^\cdot \tilde v_t'\|_{\frac{n-1}{n}rac{1}{2}}^3 \|\int_0^\cdot \tilde\rhoho_0 v''\|_1,
\tilde\etand{align*}
where we have used the fact that $\|\cdot\|_{L^p}\le C_p \|\cdot\|_{\frac{n-1}{n}rac{1}{2}}$, for all $1<p<\infty$.
Again, using the definition of $E$, the previous inequality provides us for any $t\in [0,T_\kappa]$ with
\tilde begin{equation}
\label{a12}
\starup_{[0,t]}\tilde bigl\| \frac{n-1}{n}rac{1}{\tilde\rhoho_0} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde v'^3}{\tilde\tilde\etata'^5}\tilde bigr]' \tilde bigr\|_0\le C \starup_{[0,t]} E^{\frac{n-1}{n}rac{1}{2}}\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
For the fourth term of the right-hand side of (\tilde\rhoef{a9}), we see that
\tilde begin{align}
\label{a13}
\tilde bigl\|\frac{n-1}{n}rac{2}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'}\tilde bigr]'\|_0\|3\int_0^\cdot\frac{n-1}{n}rac{\tilde v'}{\tilde\tilde\etata'^4}\|_{L^\infty} (t) & \le C [\ \|\tilde\rhoho_0 \tilde partial_t^2 v''\|_0+\|\tilde partial_t v'\|_0\ ] t \starup_{[0,t]} \|\tilde v\|_2\nonumber\\
&\le C t P(\starup_{[0,t]} E)\ .
\tilde\etand{align}
Similarly, the fifth term of the right-hand side of (\tilde\rhoef{a9}) yields the following estimate:
\tilde begin{align}
\label{a14}
\tilde bigl\|\tilde\rhoho_0\tilde partial_t^2\tilde v'\frac{n-1}{n}rac{\tilde\tilde\etata''}{\tilde\tilde\etata'^4}\tilde bigr\|_0 (t) &\le C \|\tilde\rhoho_0\tilde partial_t^2\tilde v'\|_{L^\infty}\|\tilde\tilde\etata''\|_0\nonumber\\
&\le C \|\tilde\rhoho_0\tilde partial_t^2\tilde v'\|_1 \|\int_0^\cdot\tilde v''\|_0\nonumber\\
& \le C t P(\starup_{[0,t]} E)\ .
\tilde\etand{align}
Combining the estimates (\tilde\rhoef{a11})--(\tilde\rhoef{a14}), we obtain the inequality
\tilde begin{equation}
\label{a15}
\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{2}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'}\tilde bigr]'\tilde bigr\|_0 \le C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
At this stage, we remind the reader, that the solution $\tilde v$ to our parabolic $ \kappa $-problem is in $X_{T_ \kappa }$, so that for any $t\in [0,T_\kappa]$, $\tilde partial_t^2 \tilde v\in H^2(I)$. Notice that
$$
\frac{n-1}{n}rac{1}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'}\tilde bigr]' = \tilde\rhoho_0 \tilde partial_t^2 \tilde v'' + 2 \tilde\rhoho_0'\tilde partial_t^2\tilde v' \,,
$$
so (\tilde\rhoef{a15}) is equivalent to
\tilde begin{equation}\label{a15b}
\starup_{[0,t]} \tilde bigl\| \tilde\rhoho_0 \tilde partial_t^2 \tilde v'' + 2 \tilde\rhoho_0'\tilde partial_t^2\tilde v' \tilde bigr\|_0 \le C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
From this inequality, we would like to conclude that
both $\| \tilde partial_t^2 \tilde v '\|_0$ and $\| \tilde\rhoho_0 \tilde partial_t^2 \tilde v'' \|_0$ are bounded by the right-hand side of (\tilde\rhoef{a15b}); the regularity
provided by solutions of the $ \kappa $-problem allow us to arrive at this conclusion.
By expanding the left-hand side of (\tilde\rhoef{a15b}), we see that
\tilde begin{align}
\label{a16}
\starup_{[0,t]} \tilde bigl\| \tilde\rhoho_0 \tilde partial_t^2 \tilde v'' + 2\tilde\rhoho_0' \tilde partial_t^2\tilde v' \tilde bigr\|_0^2&=\|\tilde\rhoho_0\tilde partial_t^2\tilde v''\|_0^2+4\|\tilde\rhoho_0'\tilde partial_t^2\tilde v'\|_0^2+4\int_I \tilde\rhoho_0 \tilde partial_t^2 \tilde v'' \ \tilde\rhoho_0'\tilde partial_t^2\tilde v' \,.
\tilde\etand{align}
Given the regularity of $\tilde partial_t^2 \tilde v$ provide by our parabolic $ \kappa$-problem, we notice that the cross-term in (\tilde\rhoef{a16}) is an exact derivative,
$$
4\int_I \tilde\rhoho_0 \tilde partial_t^2 \tilde v'' \ \tilde\rhoho_0'\tilde partial_t^2\tilde v' = 2 \int_I \tilde\rhoho_0\tilde\rhoho_0' \frac{n-1}{n}rac{\tilde partial}{\tilde partial x} |\tilde partial_t^2 \tilde v'|^2 \,,
$$
so that by integrating-by-parts, we find that
$$
4\int_I \tilde\rhoho_0 \tilde partial_t^2 \tilde v'' \ \tilde\rhoho_0'\tilde partial_t^2\tilde v' = -2 \|\tilde\rhoho_0'\tilde partial_t^2\tilde v'\|_0^2 - \int_I \tilde\rhoho_0 \tilde partial_t^2
\tilde v' \ \tilde\rhoho_0'' \tilde partial_t^2 \tilde v' \,,
$$
and hence (\tilde\rhoef{a16}) becomes
\tilde begin{align}
\label{a16b}
\starup_{[0,t]} \tilde bigl\| \tilde\rhoho_0 \tilde partial_t^2 \tilde v'' + 2\tilde\rhoho_0' \tilde partial_t^2\tilde v' \tilde bigr\|_0^2&=\|\tilde\rhoho_0\tilde partial_t^2\tilde v''\|_0^2+2\|\tilde\rhoho_0'\tilde partial_t^2\tilde v'\|_0^2 - \int_I \tilde\rhoho_0 \tilde partial_t^2
\tilde v' \ \tilde\rhoho_0'' \tilde partial_t^2 \tilde v' \,.
\tilde\etand{align}
Since the energy function $E$ contains $\tilde\rhoho_0 \tilde partial_t^2 \tilde v(t) \in H^2(I)$ and $\tilde partial_t^2\tilde v(t) \in H^1(I)$, the
fundamental theorem of calculus shows that
$$
\int_I \tilde\rhoho_0 \tilde partial_t^2\tilde v' \ \tilde\rhoho_0'' \tilde partial_t^2 \tilde v' \le
C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)) \,.
$$
Combining this inequality with (\tilde\rhoef{a16b}) and (\tilde\rhoef{a15}), yields
\tilde begin{equation*}
\starup_{[0,t]} [ \|\tilde\rhoho_0\tilde partial_t^2\tilde v''\|_0+\|\tilde\rhoho_0'\tilde partial_t^2\tilde v'\|_0 ] \le C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)),
\tilde\etand{equation*}
and thus
\tilde begin{equation*}
\starup_{[0,t]} [ \|\tilde\rhoho_0\tilde partial_t^2\tilde v''\|_0+\|\tilde\rhoho_0'\tilde partial_t^2\tilde v'\|_0 + \|\tilde\rhoho_0\tilde partial_t^2\tilde v'\|_0 ] \le M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)),
\tilde\etand{equation*}
and hence with the physical vacuum conditions on $\tilde\rhoho_0$ given by (\tilde\rhoef{degen1}) and (\tilde\rhoef{degen2}), we have that
\tilde begin{equation*}
\starup_{[0,t]} [ \|\tilde\rhoho_0\tilde partial_t^2\tilde v''\|_0+\|\tilde partial_t^2\tilde v'\|_0 ] \le M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)),
\tilde\etand{equation*}
which, together with (\tilde\rhoef{a10}), provide us with the estimate
\tilde begin{equation}
\label{a17}
\starup_{[0,t]} [ \|\tilde\rhoho_0\tilde partial_t^2\tilde v''\|_0+\|\tilde partial_t^2\tilde v\|_1 ] \le M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
\starubsection{Elliptic and Hardy-type estimates for $v(t)$.} Having obtained the estimates for $\tilde partial_t^2 \tilde v(t)$ in (\tilde\rhoef{a17}), we can
next obtain our estimates for $\tilde v(t)$.
To do so, we consider the first time-differentiated version of (\tilde\rhoef{approximate.a}), which yields the equation
\tilde begin{equation*}
-2\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde v'}{\tilde\tilde\etata'^3}\tilde bigr]'-\kappa [\tilde\rhoho_0^2 \tilde partial_t \tilde v']'=-\tilde\rhoho_0\tilde partial_t^2 \tilde v,
\tilde\etand{equation*}
which we rewrite as the following identity:
\tilde begin{align}
-\frac{n-1}{n}rac{2}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2\tilde v'}\tilde bigr]'-\frac{n-1}{n}rac{\kappa}{\tilde\rhoho_0} [\tilde\rhoho_0^2 \tilde partial_t \tilde v']'=&-\tilde partial_t^2 \tilde v-\frac{n-1}{n}rac{2}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2 \tilde v'}\tilde bigr]'(1-\frac{n-1}{n}rac{1}{\tilde\tilde\etata'^3})\ . \label{cs11}
\tilde\etand{align}
Using Lemma \tilde\rhoef{kelliptic}, we see that for any $t\in [0,T_\kappa]$,
\tilde begin{align}
\label{a18}
\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2 \tilde v'}\tilde bigr]'\tilde bigr\|_1 \le & C\starup_{[0,t]}\| \tilde partial_t^2 \tilde v\|_1 +C\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2 \tilde v'}\tilde bigr]'\|_1\|\int_0^\cdot\frac{n-1}{n}rac{\tilde v'}{\tilde\tilde\etata'^4}\|_{L^\infty} +C \|\frac{n-1}{n}rac{1}{\tilde\rhoho_0}[\tilde\rhoho_0^2 \tilde v']'.\frac{n-1}{n}rac{\tilde\tilde\etata''}{\tilde\tilde\etata'^4}\|_0
\tilde\etand{align}
We next estimate each term on the right hand side of (\tilde\rhoef{a18}). The bound for the first term on the right-hand side of
(\tilde\rhoef{a18}) is provided by (\tilde\rhoef{a17}).
The second term of the right-hand side of (\tilde\rhoef{a18}) is estimated as follows:
\tilde begin{align}
\label{a20}
\tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2 \tilde v'}\tilde bigr]'\|_1\|\int_0^\cdot\frac{n-1}{n}rac{\tilde v'}{\tilde\tilde\etata'^4}\|_{L^\infty} (t) &\le C [ \| \tilde\rhoho_0 \tilde v'''\|_0+\|v\|_2]\ t \starup_{[0,t]} \|\tilde v\|_2\nonumber\\
&\le C t P(\starup_{[0,t]} E)\ .
\tilde\etand{align}
For the third term on the right-hand side of (\tilde\rhoef{a18}),
\tilde begin{align}
\label{a21}
\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0}[\tilde\rhoho_0^2 \tilde v']'.[\frac{n-1}{n}rac{\tilde\tilde\etata''}{\tilde\tilde\etata'^4}]\|_1 (t)
& \le C \|\frac{n-1}{n}rac{1}{\tilde\rhoho_0}[\tilde\rhoho_0^2 \tilde v']'\|_{L^\infty} \|\tilde\tilde\etata''\|_0\nonumber\\
&\le C [ \| \tilde\rhoho_0 \tilde v'''\|_0+\|v\|_2] \|\int_0^t \tilde v''\|\nonumber\\
& \le C t P(\starup_{[0,t]} E)\ .
\tilde\etand{align}
Combining these estimates provides the inequality
\tilde begin{equation*}
\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0}\tilde bigl[{\tilde\rhoho_0^2\tilde v'}\tilde bigr]'\tilde bigr\|_1 \le C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)),
\tilde\etand{equation*}
which leads us immediately to:
\tilde begin{equation}
\label{a22}
\starup_{[0,t]} \| \tilde\rhoho_0\tilde v'''+3\tilde\rhoho_0'v''\|_0 \le C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)) \,.
\tilde\etand{equation}
Now, since for any $t\in [0,T_\kappa]$, the solution $\tilde v$ to our parabolic $ \kappa $-problem is in $H^3(I)$, we infer that $\tilde\rhoho_0 v'''\in L^2(I)$. We can then apply the same integration-by-parts argument as in \check{I}te{CoLiSh2009} to find that
\tilde begin{align}
\label{a23}
\|\tilde\rhoho_0\tilde v'''+3\tilde\rhoho_0'v''\|^2_0&=\|\tilde\rhoho_0\tilde v'''\|_0^2+9\|{\tilde\rhoho_0'}\tilde v''\|_0^2+3\int_I \tilde\rhoho_0\tilde\rhoho_0' [|\tilde v''|^2]'\nonumber\\
&=\|\tilde\rhoho_0\tilde v'''\|_0^2+9\|\tilde\rhoho_0'\tilde v''\|_0^2-3\int_I [\tilde\rhoho_0\tilde\rhoho_0''+\tilde\rhoho_0'^2] |\tilde v''|^2\nonumber\\
&=\|\tilde\rhoho_0\tilde v'''\|_0^2+6\|\tilde\rhoho_0'\tilde v''\|_0^2-3\int_I \tilde\rhoho_0\tilde\rhoho_0'' |\tilde v''|^2
\tilde\etand{align}
Combined with (\tilde\rhoef{a22}), this yields:
\tilde begin{align*}
\starup_{[0,t]} [ \|\tilde\rhoho_0\tilde v'''\|_0+\|\tilde\rhoho_0'\tilde v''\|_0 ] \le & C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E))\nonumber\\
&+M_0+C\|\int_0^t \starqrt{\tilde\rhoho_0}\tilde v_t''\|_0,
\tilde\etand{align*}
and thus
\tilde begin{equation*}
\starup_{[0,t]} [ \|\tilde\rhoho_0\tilde v'''\|_0+\|\tilde\rhoho_0'\tilde v''\|_0 + \|\tilde\rhoho_0\tilde v''\|_0 ] \le M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0,
\tilde\etand{equation*}
With (\tilde\rhoef{degen1}) and (\tilde\rhoef{degen2}), it follows that
\tilde begin{equation*}
\starup_{[0,t]} [ \|\tilde\rhoho_0\tilde v'''\|_0+\|\tilde v''\|_0 ] \le M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)),
\tilde\etand{equation*}
and hence
\tilde begin{equation}
\label{a24}
\starup_{[0,t]} [ \|\tilde\rhoho_0\tilde v'''\|_0+\|\tilde v\|_2 ] \le M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
\starubsection{Elliptic and Hardy-type estimates for $\tilde partial_t^3v(t)$ and $\tilde partial_t v(t)$}
We consider the fourth time-differentiated version of (\tilde\rhoef{approximate.a}):
$$
\tilde bigl[\tilde partial_t^4\frac{n-1}{n}rac{\tilde\rhoho_0^2}{\tilde\tilde\etata'^2}\tilde bigr]'-\kappa [\tilde\rhoho_0^2 \tilde partial_t^4 \tilde v']'=-\tilde\rhoho_0 \tilde partial_t^5 \tilde v \,,
$$
which can be rewritten as
\tilde begin{equation*}
-2\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t^3 \tilde v'}{\tilde\tilde\etata'^3}\tilde bigr]'-\kappa [\tilde\rhoho_0^2 \tilde partial_t^4 \tilde v']'=-\tilde\rhoho_0\tilde partial_t^5 \tilde v+c_1\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]' +c_2 \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}} \tilde partial_t\tilde v'^2}{\tilde\tilde\etata'^5}\tilde bigr]',
\tilde\etand{equation*}
for some constants $c_1$ and $c_2$. By employing the fundamental theorem of calculus and dividing by $\tilde\rhoho_0^ {\frac{n-1}{n}rac{1}{2}} $,
we obtain the equation
\tilde begin{align*}
-\frac{n-1}{n}rac{2}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^3 \tilde v'}\tilde bigr]'-\frac{n-1}{n}rac{\kappa}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}} [\tilde\rhoho_0^2 \tilde partial_t^4 \tilde v']'=&-\starqrt{\tilde\rhoho_0}\tilde partial_t^5 \tilde v+\frac{n-1}{n}rac{c_1}{{\tilde\rhoho_0}^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]' +\frac{n-1}{n}rac{c_2}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde partial_t\tilde v'^2}{\tilde\tilde\etata'^5}\tilde bigr]'\\
&-\frac{n-1}{n}rac{2}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^3 \tilde v'}\tilde bigr]'(1-\frac{n-1}{n}rac{1}{\tilde\tilde\etata'^3})-6\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v'\frac{n-1}{n}rac{\tilde\tilde\etata''}{\tilde\tilde\etata'^4}\ .
\tilde\etand{align*}
For any $t\in [0,T_\kappa]$, Lemma \tilde\rhoef{kelliptic} provides the $ \kappa $-independent estimate
\tilde begin{align}
\label{a25}
\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{2}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^3 \tilde v'}\tilde bigr]'\tilde bigr\|_0 \le & \starup_{[0,t]}\|\starqrt{\tilde\rhoho_0}\tilde partial_t^5 \tilde v\|_0 +\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{c_1}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]'\tilde bigr\|_0 +\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{c_2}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde partial_t\tilde v'^2}{\tilde\tilde\etata'^5}\tilde bigr]'\tilde bigr\|_0\nonumber\\
&+\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{2}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^3 \tilde v'}\tilde bigr]'\|_0\|3\int_0^\cdot\frac{n-1}{n}rac{\tilde v'}{\tilde\tilde\etata'^4}\|_{L^\infty}+6\starup_{[0,t]} \tilde bigl\|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^2\tilde v'\frac{n-1}{n}rac{\tilde\tilde\etata''}{\tilde\tilde\etata'^4}\tilde bigr\|_0\ .
\tilde\etand{align}
We estimate each term on the right-hand side of (\tilde\rhoef{a25}). The first term on the right-hand side is bounded by
$ M_0+ C t \ P(\starup_{[0,t]} E)$ thanks to (\tilde\rhoef{a8}).
For the second term on the right-hand side of (\tilde\rhoef{a25}), we have that
\tilde begin{align*}
\tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]'\tilde bigr\|_0 &\le \|\starqrt{\tilde\rhoho_0}(\tilde\rhoho_0 \tilde partial_t^2\tilde v')'\|_0 \tilde bigl\|\frac{n-1}{n}rac{\tilde v'}{\tilde\tilde\etata'^4}\tilde bigr\|_{L^\infty}+\tilde bigl\|\starqrt{\tilde\rhoho_0}\tilde partial_t^2\tilde v'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]'\tilde bigr\|_0\\
&\le C \|\starqrt{\tilde\rhoho_0}(\tilde\rhoho_0 \tilde partial_t^2\tilde v')'\|_0 \|\tilde v'\|_{\frac{n-1}{n}rac{3}{4}}+\tilde bigl\|\starqrt{\tilde\rhoho_0}\tilde partial_t^2\tilde v'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]\tilde bigr\|_0+\tilde bigl\|\starqrt{\tilde\rhoho_0}\tilde partial_t^2\tilde v'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0\ \tilde v''}{\tilde\tilde\etata'^4}\tilde bigr]\tilde bigr\|_0\nonumber\\
&\ \ +4\tilde bigl\|\starqrt{\tilde\rhoho_0}\tilde partial_t^2\tilde v'\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0\ \tilde v'\tilde\tilde\etata''}{\tilde\tilde\etata'^5}\tilde bigr]\tilde bigr\|_0\\
& \le C \|\starqrt{\tilde\rhoho_0}(\tilde\rhoho_0 \tilde v_2')'+\int_0^\cdot \starqrt{\tilde\rhoho_0}(\tilde\rhoho_0 \tilde partial_t^3\tilde v')'\|_0 \|\tilde v'\|_1^{\frac{n-1}{n}rac{1}{2}} \|\tilde v_1'+\int_0^\cdot \tilde v_t'\|_{\frac{n-1}{n}rac{1}{2}}^{\frac{n-1}{n}rac{1}{2}}\nonumber\\
&\ \ +C\|\starqrt{\tilde\rhoho_0}\tilde v_2'+\int_0^\cdot \starqrt{\tilde\rhoho_0}\tilde partial_t^3\tilde v'\|_0 \|\tilde v'\|_{\frac{n-1}{n}rac{3}{4}}\\
&\ \ +C \|\starqrt{\tilde\rhoho_0}\tilde partial_t^2\tilde v'\|_0 \|\tilde\rhoho_0 v''\|_{\frac{n-1}{n}rac{3}{4}} + C \|\tilde v'\|_{\frac{n-1}{n}rac{3}{4}}\|\int_0^\cdot \tilde v''\|_0 \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}} \tilde v_2'+\int_0^\cdot\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}} \tilde partial_t^3\tilde v'\|_1\\
& \le \|\starqrt{\tilde\rhoho_0}(\tilde\rhoho_0 \tilde v_2')'+\int_0^\cdot \starqrt{\tilde\rhoho_0}(\tilde\rhoho_0 \tilde v_{tt}')'\|_0 \|\tilde v'\|_1^{\frac{n-1}{n}rac{1}{2}} \|\tilde v_1' +\int_0^\cdot \tilde v_t'\|_{\frac{n-1}{n}rac{1}{2}}^{\frac{n-1}{n}rac{1}{2}}\\
&\ \ +C\|\starqrt{\tilde\rhoho_0}\tilde v_2'+\int_0^\cdot \starqrt{\tilde\rhoho_0}\tilde v_{tt}'\|_0 \|\tilde v'\|_1^{\frac{n-1}{n}rac{1}{2}} \|\tilde v_1'+\int_0^\cdot \tilde v_t'\|_{\frac{n-1}{n}rac{1}{2}}^{\frac{n-1}{n}rac{1}{2}} \\
&\ \ +C \|\starqrt{\tilde\rhoho_0}\tilde partial_t^2\tilde v'\|_0 \|\tilde\rhoho_0 v''\|_1^{\frac{n-1}{n}rac{3}{4}} \|\tilde\rhoho_0 v_0''+\int_0^\cdot \tilde\rhoho_0 v_t''\|_0^{\frac{n-1}{n}rac{1}{4}}\nonumber\\
&\ \ +C \|\tilde v'\|_1^{\frac{n-1}{n}rac{1}{2}} \|\tilde v_1'+\int_0^\cdot \tilde v_t'\|_{\frac{n-1}{n}rac{1}{2}}^{\frac{n-1}{n}rac{1}{2}} \|\int_0^\cdot \tilde v''\|_0 \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}} \tilde v_2'+\int_0^\cdot\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}} \tilde partial_t^3\tilde v'\|_1 \,,
\tilde\etand{align*}
where we have again used the fact that $\|\cdot\|_{L^\infty}\le C \|\cdot\|_{\frac{n-1}{n}rac{3}{4}}$.
Thanks to the definition of $E$, the previous inequality shows that for any $t\in [0,T_\kappa]$,
\tilde begin{equation}
\label{a27}
\starup_{[0,t]}\tilde bigl\|\frac{n-1}{n}rac{1}{\starqrt{\tilde\rhoho_0}}\tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2\tilde partial_t^2 \tilde v'\ \tilde v'}{\tilde\tilde\etata'^4}\tilde bigr]'\tilde bigr\|_0\le C \starup_{[0,t]} E^{\frac{n-1}{n}rac{3}{4}}\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
For the third term on the right-hand side of (\tilde\rhoef{a25}), we have similarly that
\tilde begin{align}
\label{a28}
\tilde bigl\|\frac{n-1}{n}rac{1}{\starqrt{\tilde\rhoho_0}} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde partial_t\tilde v'^2}{\tilde\tilde\etata'^5}\tilde bigr]'\tilde bigr\|_0 (t) & \le 2\|(\tilde\rhoho_0 \tilde partial_t\tilde v')'\|_{0} \tilde bigl\|\starqrt{\tilde\rhoho_0}\frac{n-1}{n}rac{\tilde partial_t\tilde v'}{\tilde\tilde\etata'^5}\tilde bigr\|_{L^\infty}+5\tilde bigl\|\starqrt{\tilde\rhoho_0}\frac{n-1}{n}rac{\tilde partial_t\tilde v'^2}{\tilde\tilde\etata'^6}\|_{L^1}\|\tilde\rhoho_0 \tilde\tilde\etata''\|_{L^\infty}\nonumber\\
& \le C \|(\tilde\rhoho_0\tilde v_1')'+\int_0^t (\tilde\rhoho_0\tilde partial_{tt}\tilde v')'\|_0\|\starqrt{\tilde\rhoho_0}\tilde partial_t\tilde v'\|_{\frac{n-1}{n}rac{3}{4}}+C \|\tilde partial_t\tilde v'\|_0^2\|\int_0^t (\tilde\rhoho_0\tilde v'')'\|_0\nonumber\\
& \le C \|(\tilde\rhoho_0\tilde v_1')'+\int_0^t (\tilde\rhoho_0\tilde partial_{tt}\tilde v')'\|_0\|\starqrt{\tilde\rhoho_0} \tilde partial_t\tilde v'\|_{0}^{1-\alpha}\|(\starqrt{\tilde\rhoho_0}\tilde partial_t\tilde v')'\|_{L^{2-a}}^{\alpha}\nonumber\\
&\ \ +C \|\tilde partial_t\tilde v'\|_0^2\|\int_0^t (\tilde\rhoho_0\tilde v'')'\|_0\nonumber\\
& \le C \|(\tilde\rhoho_0\tilde v_1')'+\int_0^t (\tilde\rhoho_0\tilde partial_{tt}\tilde v')'\|_0\|\tilde v'_1+\int_0^t\tilde partial_{tt}\tilde v'\|_{0}^{1-\alpha}\|(\starqrt{\tilde\rhoho_0}\tilde partial_t\tilde v')'\|_{L^{2-a}}^{\alpha}\nonumber\\
&\ \ +C \|\tilde partial_t\tilde v'\|_0^2\|\int_0^t (\tilde\rhoho_0\tilde v'')'\|_0 \,,
\tilde\etand{align}
where $0<a<\frac{n-1}{n}rac{1}{2}$ is given and $0<\alpha=\frac{n-1}{n}rac{3-3a}{4+3a}<1$.
The only term on the right-hand side of (\tilde\rhoef{a28}) which is not directly contained in the definition of $E$ is $\|(\starqrt{\tilde\rhoho_0}\tilde partial_t\tilde v')'\|_{L^{2-a}}^\alpha$. To this end, we notice that
\tilde begin{align}
\label{a29}
\|(\starqrt{\tilde\rhoho_0}\tilde partial_t\tilde v')'\|_{L^{2-a}} &\le \|\frac{n-1}{n}rac{\tilde partial_t\tilde v'}{2\starqrt{\tilde\rhoho_0}}\|_{L^{2-a}}+\|\starqrt{\tilde\rhoho_0} \tilde v_{tt}''\|_0\nonumber\\
&\le \|\frac{n-1}{n}rac{1}{2\starqrt{\tilde\rhoho_0}}\|_{L^{2-\frac{n-1}{n}rac{a}{2}}}\|\tilde partial_t\tilde v'\|_{\frac{n-1}{n}rac{1}{2}}+\|\starqrt{\tilde\rhoho_0} \tilde v_{tt}''\|_0 \,,
\tilde\etand{align}
where we have used the fact that $\|\cdot\|_{L^p}\le C_p \|\cdot\|_{\frac{n-1}{n}rac{1}{2}}$, for all $1<p<\infty$.
Thanks to the definition of $E$, the previous inequality and (\tilde\rhoef{a28}) provides us for any $t\in [0,T_\kappa]$ with
\tilde begin{equation}
\label{a30}
\starup_{[0,t]}\tilde bigl\| \frac{n-1}{n}rac{1}{\tilde\rhoho_0} \tilde bigl[\frac{n-1}{n}rac{\tilde\rhoho_0^2 \tilde v'^3}{\tilde\tilde\etata'^5}\tilde bigr]' \tilde bigr\|_0\le C \starup_{[0,t]} E^\alpha\ (M_0+t P(\starup_{[0,t]}E)),
\tilde\etand{equation}
where we recall again that $0<\alpha=\frac{n-1}{n}rac{3-3a}{4+3a}<1$.
The fourth term on the right-hand side of (\tilde\rhoef{a25}) is easily treated:
\tilde begin{align}
\label{a31}
\tilde bigl\|\frac{n-1}{n}rac{1}{\starqrt{\tilde\rhoho_0}}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^3 \tilde v'}\tilde bigr]'\|_0\|\int_0^t\frac{n-1}{n}rac{\tilde v'}{\tilde\tilde\etata'^4}\|_{L^\infty} (t) & \le C [\ \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}} \tilde partial_t^3 v''\|_0+\|\starqrt{\tilde\rhoho_0}\tilde partial_t^3 v'\|_0\ ] t \starup_{[0,t]} \|\tilde v\|_2\nonumber\\
& \le C t P(\starup_{[0,t]} E)\ .
\tilde\etand{align}
Similarly, the fifth term on the right-hand side of (\tilde\rhoef{a25}) is estimated as follows:
\tilde begin{align}
\label{a32}
\tilde bigl\|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v'\frac{n-1}{n}rac{\tilde\tilde\etata''}{\tilde\tilde\etata'^4}\tilde bigr\|_0 (t) & \le C \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v'\|_{L^\infty}\|\tilde\tilde\etata''\|_0\nonumber\\
& \le C [\ \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}} \tilde partial_t^3 v''\|_0+\|\starqrt{\tilde\rhoho_0}\tilde partial_t^3 v'\|_0\ ] \tilde bigl\|\int_0^t\tilde v''\tilde bigr\|_0\nonumber\\
& \le C t P(\starup_{[0,t]} E)\ .
\tilde\etand{align}
Combining the estimates (\tilde\rhoef{a25})--(\tilde\rhoef{a32}), we can infer that
\tilde begin{equation}
\label{a33}
\starup_{[0,t]} \tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^3 \tilde v'}\tilde bigr]'\tilde bigr\|_0 \le C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^\alpha\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
Now, since for any $t\in [0,T_\kappa]$, solutions to our parabolic $ \kappa $-problem have the regularity $\tilde partial_t^2v\in H^2(I)$,
we integrate-by-parts:
\tilde begin{align}
\label{a34}
\tilde bigl\|\frac{n-1}{n}rac{1}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^3 \tilde v'}\tilde bigr]'\tilde bigr\|^2_0&=\|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v''\|_0^2+4\|{\tilde\rhoho_0}^{\frac{n-1}{n}rac{1}{2}}\tilde\rhoho_0'\tilde partial_t^3\tilde v'\|_0^2+2\int_I \tilde\rhoho_0'\tilde\rhoho_0^2 [|\tilde partial_t^3\tilde v'|^2]'\nonumber\\
&=\|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v''\|_0^2+4\|\tilde\rhoho_0'\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}\tilde partial_t^3\tilde v'\|_0^2-4\int_I \tilde\rhoho_0'^2\tilde\rhoho_0 |\tilde partial_t^3\tilde v'|^2-2\int_I \tilde\rhoho_0''\tilde\rhoho_0^2 |\tilde partial_t^3\tilde v'|^2\nonumber\\
&=\|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v''\|_0^2-2 \int_I \tilde\rhoho_0''\tilde\rhoho_0^2 |\tilde partial_t^3\tilde v'|^2.
\tilde\etand{align}
Combined with (\tilde\rhoef{a33}), and the fact that $\tilde\rhoho_0\tilde partial_t^3\tilde v'=\tilde\rhoho_0\tilde v_3+\int_0^cdot \tilde\rhoho_0\tilde partial_t^4\tilde v'$ for the second term on the right-hand side of (\tilde\rhoef{a34}), we find that
\tilde begin{equation}
\label{a35}
\starup_{[0,t]} \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v''\|_0 \le C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^\alpha\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
Now, since $\frac{n-1}{n}rac{1}{\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}}\tilde bigl[{\tilde\rhoho_0^2\tilde partial_t^3 \tilde v'}\tilde bigr]'=\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v''+2 {\tilde\rhoho_0}^{\frac{n-1}{n}rac{1}{2}}\tilde\rhoho_0'\tilde partial_t^3\tilde v'$, the estimates (\tilde\rhoef{a33}) and (\tilde\rhoef{a35}) also imply that
\tilde begin{equation}
\label{a36}
\starup_{[0,t]} \|{\tilde\rhoho_0}^{\frac{n-1}{n}rac{1}{2}}\tilde\rhoho_0'\tilde partial_t^3\tilde v'\|_0 \le C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^\alpha\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
Therefore,
\tilde begin{align*}
\starup_{[0,t]} [ \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v''\|_0+\|\tilde\rhoho_0'\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}\tilde partial_t^3\tilde v'\|_0 +\|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v'\|_0] \le &M_0+C t P(\starup_{[0,t]} E) \\
&+ C \starup_{[0,t]} E^\alpha\ (M_0+t P(\starup_{[0,t]}E)),
\tilde\etand{align*}
so that with (\tilde\rhoef{degen1}) and (\tilde\rhoef{degen2}),
\tilde begin{equation*}
\starup_{[0,t]} [ \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v''\|_0+\|\tilde\rhoho_0^{\frac{n-1}{n}rac{1}{2}}\tilde partial_t^3\tilde v'\|_0 ] \le M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^\alpha\ (M_0+t P(\starup_{[0,t]}E)) \,.
\tilde\etand{equation*}
Together with (\tilde\rhoef{a10}) and the weighted embedding estimate (\tilde\rhoef{w-embed}), the above inequality shows that
\tilde begin{equation}
\label{a37}
\starup_{[0,t]} [ \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t^3\tilde v''\|_0+\|\tilde partial_t^3\tilde v\|_{\frac{n-1}{n}rac{1}{2}} ] \le M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^\alpha\ (M_0+t P(\starup_{[0,t]}E)).
\tilde\etand{equation}
By studying the second time-differentiated version of (\tilde\rhoef{approximate.a}) in the same manner, we find that
\tilde begin{equation}
\label{a38}
\starup_{[0,t]} [ \|\tilde\rhoho_0^{\frac{n-1}{n}rac{3}{2}}\tilde partial_t\tilde v'''\|_0+\|\tilde partial_t\tilde v\|_{\frac{n-1}{n}rac{3}{2}} ] \le M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^\alpha\ (M_0+t P(\starup_{[0,t]}E)) \,.
\tilde\etand{equation}
\starection{Proof of Theorem \tilde\rhoef{theorem_main}}
\starubsection{Time of existence and bounds independent of $ \tilde\etapsilon $ and existence of solutions to (\tilde\rhoef{ce0})}\label{subsec_finish1}
Summing the inequalities (\tilde\rhoef{a8}), (\tilde\rhoef{a17}), (\tilde\rhoef{a24}), (\tilde\rhoef{a37}), (\tilde\rhoef{a38}), we find that
$$
\starup_{t \in [0,T]} E(t) \le
M_0+C t P(\starup_{[0,t]} E) + C \starup_{[0,t]} E^\alpha\ (M_0+t P(\starup_{[0,t]}E)) \,.
$$
As $ \alpha < 1$, by employing Young's inequality and readjusting the constants, we obtain
$$
\starup_{t \in [0,T]} E(t) \le
M_0 + C \, T\, P({\starup_{t\in[0,T]}} E(t)) \,.
$$
Just as in Section 9 of \check{I}te{CoSh2006}, this provides us with
a time of existence $T_1$ independent of $\kappa$ and an
estimate on $(0,T_1)$ independent of $\kappa$ of the type:
\tilde begin{equation}
\starup_{t \in [0,T_1]} E(t) \le 2 M_0 \,. \label{eq420}
\tilde\etand{equation}
In particular, our sequence of solutions $(v_ \kappa )$ satisfy the $ \kappa $-independent bound (\tilde\rhoef{eq420}) on
the $ \kappa $-independent time-interval $(0,T_1)$.
\starubsection{The limit as $ \kappa \tilde\rhoightarrow 0$}
By the $\kappa$-independent estimate (\tilde\rhoef{eq420}), there exists a subsequence of
$\{v_ \kappa \}$ which converges weakly to $v$ in $L^2(0,T; H^2(I))$. With $\tilde\etata(t,x) = x + \int_0^t v(s,x)ds$,
by standard compactness arguments, we see that a further subsequence of $v_ \kappa $ and $\tilde\etata'_ \kappa$
uniformly converges to $v$ and $\tilde\etata'$, respectively, which shows that $v$ is the solution to (\tilde\rhoef{ce0}) and
$v(0,x) = u_0(x)$.
\starubsection{Uniqueness of solutions to the compressible Euler equations (\tilde\rhoef{ce0})} For uniqueness, we require the initial
data to have one space-derivative better regularity than for existence. Given the assumption (\tilde\rhoef{uniquedata}) on the data $(u_0,\tilde\rhoho_0)$,
repeating our argument for existence, we can produce a solution $v$ on $[0,T_1]$ which satisfies the estimate
$$
\starum_{s=0}^3\left[ \|\tilde partial_t^{2s}v(t,\cdot)\|^2_{H^{3-s}(I)} + \|\tilde\rhoho_0 \tilde partial_t^{2s} v(t,\cdot)\|^2_{H^{4-s}(I)} \tilde\rhoight] < \infty \,,
$$
and has the flow $\tilde\etata(t,x)=x + \int_0^t v(s,x) ds$.
For the sake of contradiction, let us assume that $w$ is also a solution on $[0,T_1]$ with initial data $(u_0, \tilde\rhoho_0)$, satisfying the same estimate, with flow
$\tilde psi(t,x)=x + \int_0^t w(s,x)ds$.
We define
$$
\displaystyleelta v = v -w \,,
$$
in which case we have the following equation for $ \displaystyleelta v$:
\tilde begin{subequations}
\label{cunique}
\tilde begin{alignat}{2}
\tilde\rhoho_0 \displaystyleelta v_t + (\tilde\rhoho_0^2[{\tilde\etata' }^{-2} - {\tilde psi'} ^{-2} ])'&=0 &&\text{in} \ \ I \times (0,T_1]
\,, \label{cunique.a}\\
\displaystyleelta v&= 0 \ \ \ &&\text{on} \ \ I \times \{t=0\} \,, \label{cunique.b}\\
\tilde\rhoho_0& = 0 \ \ &&\text{ on } \tilde partial I \,. \label{cunique.c}
\tilde\etand{alignat}
\tilde\etand{subequations}
By considering the fifth time-differentiated version of (\tilde\rhoef{cunique.a}) and taking the $L^2(I)$ inner-product with $ \tilde partial_t \displaystyleelta v$, we obtain the analogue of (\tilde\rhoef{a8}) (with $ \kappa =0$) for
$ \displaystyleelta v$. The additional error terms which arise are easily controlled
by the fact that both $v$ and $w$ have one space-derivative better regularity than the energy function $E$. This produces a good
bound for $ \tilde partial_t^4 \displaystyleelta v \in L^ \infty (0,T_1; L^2(I))$. By repeating the elliptic and Hardy-type estimates for $\tilde partial_t^2 \displaystyleelta v \in
L^ \infty (0,T_1; H^1(I))$ and $\tilde partial \displaystyleelta v \in
L^ \infty (0,T_1; H^2(I))$, and using (\tilde\rhoef{cunique.b}), we obtain the inequality
\tilde begin{align*}
& \starup_{t \in [0,T_1]} (\|\tilde partial_t^4 \displaystyleelta v(t)\|_0^2 + \|\tilde partial_t^2 \displaystyleelta v(t)\|_1^2 + \| \displaystyleelta v(t)\|_2^2 ) \\
& \tilde qquad\tilde qquad\tilde qquad
\le C\, T_1 \,
P( \starup_{t \in [0,T_1]} (\|\tilde partial_t^4 \displaystyleelta v(t)\|_0^2 + \|\tilde partial_t^2 \displaystyleelta v(t)\|_1^2 + \| \displaystyleelta v(t)\|_2^2 )) \,,
\tilde\etand{align*}
which shows that $ \displaystyleelta v=0$.
\starubsection{Optimal regularity for initial data} We smoothed our initial data $(u_0,\tilde\rhoho_0)$ in order to construct solutions to
our degenerate parabolic $ \kappa $-problem (\tilde\rhoef{approximate}). Having obtained solutions which depend only on $E(0,v)$,
a standard density argument shows that the initial data needs only to satisfy $M_0 < \infty $.
\starection{The case $\gamma \nonumbereq 2$}
In this section, we describe the modifications to the energy function and the methodology for the case that $\gamma \nonumbereq 2$.
We denote by $a_0$ the integer satisfying the inequality
$$
1 < 1+ {\frac{n-1}{n}rac{1}{\gamma -1}} -a_0 \le 2 \,.
$$
Letting
$$
d(x) = \text{dist}(x, \tilde partial I) \,,
$$
We consider the following higher-order energy function:
\tilde begin{align*}
E_\gamma(t,v) & = \starum_{s=0}^4 \| v(t, \cdot )\|^2_{2 - {\frac{n-1}{n}rac{s}{2}} } + \starum_{s=0}^2 \| d \, \tilde partial_t^{2s} v(t, \cdot )\|^2_{3-s}
+ \| \starqrt{d} \, \tilde partial_t\tilde partial_x^{2} v(t, \cdot )\|^2_{0} + \| \starqrt{d} \, \tilde partial_t^3 \tilde partial_x v(t, \cdot )\|^2_{0} \\
& \tilde qquad + \starum_{a=0}^{a_0} \| \starqrt{d}^{1+ {\frac{n-1}{n}rac{1}{\gamma-1}} - a} \tilde partial_t^{4+a_0-a} v'( t, \cdot )\|_0^2 \,,
\tilde\etand{align*}
and define the polynomial function $M_0^\gamma = P( E_ \gamma (0, v))$.
Notice the last sum in $E_ \gamma $ appears whenever $ \gamma < 2$, and the number of time-differentiated problems increases
as $\gamma \to 1$.
Using the same procedure as we have detailed for the case that $\gamma =2$, we have the following
\tilde begin{theorem}[Existence and uniqueness for any $\gamma >1$]\label{thm_main2}
Given initial data $(u_0, \tilde\rhoho_0)$ such that $M^\gamma_0< \infty $ and the physical vacuum condition (\tilde\rhoef{degen}) holds for $\tilde\rhoho_0$, there exists a solution to (\tilde\rhoef{ceuler0}) (and hence (\tilde\rhoef{ceuler})) on $[0,T_\gamma]$ for $T_\gamma>0$ taken
sufficiently small, such that
$$
\starup_{t \in [0,T]} E(t) \le 2M^\gamma_0 \,.
$$
Moreover if the initial data satisfies
$$
\starum_{s=0}^3 \|\tilde partial_t^sv(0,\cdot)\|^2_{H^{3-s}(I)} + \starum_{s=0}^3 \|d\, \tilde partial_t^{2s} v(0,\cdot)\|^2_{H^{4-s}(I)}
+ \starum_{a=0}^{a_0} \| \starqrt{d}^{1+ {\frac{n-1}{n}rac{1}{\gamma-1}} - a} \tilde partial_t^{6+a_0-a} v'( 0, \cdot )\|_0^2< \infty \,,
$$
then the solution is unique.
\tilde\etand{theorem}
\tilde vspace{.1 in}
\nonumberoindent
{\tilde bf Acknowledgments.}
SS was supported by the National Science Foundation under
grant DMS-0701056.
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\tilde\etand{document} |
\begin{document}
\author{Ting He}
\affil{School of Finance, Capital University of Economics and Business, Beijing, China}
\title{\Large \bf Nonparametric Predictive Inference for Asian options}
\date{[email protected]}
\maketitle
\centerline{\bf ABSTRACT}
\noindent Asian option, as one of the path-dependent exotic options, is widely traded in the energy market, either for speculation or hedging. However, it is hard to price, especially the one with the arithmetic average price. The traditional trading procedure is either too restrictive by assuming the distribution of the underlying asset or less rigorous by using the approximation. It is attractive to infer the Asian option price with few assumptions of the underlying asset distribution and adopt to the historical data with a nonparametric method. In this paper, we present a novel approach to price the Asian option from an imprecise statistical aspect. Nonparametric Predictive Inference (NPI) is applied to infer the average value of the future underlying asset price, which attempts to make the prediction reflecting more uncertainty because of the limited information. A rational pairwise trading criterion is also proposed in this paper for the Asian options comparison, as a risk measure. The NPI method for the Asian option is illustrated in several examples by using the simulation techniques or the empirical data from the energy market.
\noindent Key words:
Asian Option; Imprecise Probability; Nonparametric Predictive Inference; Uncertainty
\section{Introduction}
\doublespacing
Asian options, as one kind of the exotic options, are strongly path-dependent and widely traded in the commodity and foreign exchange market \citep{Kl01}. The main advantages of the Asian option are that its usage of avoiding the risk of market manipulation of the underlying instrument at maturity, and it holds a cheaper price compared to European or American options. The Asian option payoff is contingent on the average value of the underlying asset price, either arithmetic or geometric. For the Asian option settled on the basis of the geometric average price, there are closed formulae by the Black-Scholes model under the assumption that the underlying asset price is the lognormal distributed, so the geometric average price also follows the lognormal distribution with different mean and variance. Although the geometric Asian options are easily priced they are rarely used in practice \citep{Mi98}. While the Asian option with the arithmetic average price is very hard to be evaluated since the density function of the arithmetic average price is unknown\citep{Ve12}.
Many scholars try to develop and improve the method for the Asian option with the arithmetic average price. One study direction is by assuming a lognormal diffusion process of the underlying asset price and approximating the density function of the arithmetic average price. The moment matching is used to do the approximation of the option payoff presented by \citet{Tu91}; \citet{Le92}. \citet{Cu94} approximates the payoff of the option by conditioning on the geometric mean price. Another method is to use the numerical method to obtain the solution of the PDE of the Asian option. The problem of this method is that when the explicit finite difference method is used in PDE of a path-dependent option pricing, it is numerically unstable. The implicit method is stable referring to the Asian option pricing, but it only provides the result for some specific volatility structure. \citet{Ve01} improves the instability problem by presenting a numerical one dimensional PDE for the Asian option pricing which is stable under the finite difference method. Monte Carlo simulation \citep{Bo77} as a very effective way to price the path-dependent option that has been developed for the Asian option pricing \citep{Hu09}. \citet{Ke90} present a Monte Carlo strategy of pricing the option with the arithmetic average price with the variance reduction elements. \citet{Ba13} study the Asian option pricing problem by presenting a joint Monte Carlo-Fourier transform sampling scheme under the CGMY process. The concern of Monte Carlo simulation of the option pricing is to estimate an accurate result is very time-consuming. Another popular method is the discrete lattice method. \citet{Hu93} propose the first tree pricing model for Asian options, which has some drawbacks of the approximation precision and the convergence to the continuous value. \citet{Kl01} and \citet{Da08} improve Hull and White's tree model considering the representative average prices to limit the approximation error. \citet{Li14} present a binomial approach for the Asian option pricing leading to the upper and lower bounds of the approximation result reducing the interpolation error. The study discussed above is based on the assumption that the future pattern of the Asian option is well known. When the information of the future market is limited, imprecise probability allows us to predict the Asian option price with the observed information.
Imprecise probability as a generalization of classical probability theory enables various less restrictive representations of uncertainty \citep{Au14}. Nonparametric predictive inference (NPI) is one of the statistical inference methods for imprecise probability, which is a frequentist statistics framework with strong consistency properties \citep{Au04}; \citep{Co11}. The NPI method provides the approach to calculate the upper and lower probabilities of the interested event aiming to do the prediction by making few assumptions in addition to observed data. One property of the NPI method is when multiple future observations are predicted, the observations are interdependent, meaning after one prediction, this predicted value is added to the observed data together forecasting the next future observations. Therefore, the NPI method reflects more uncertainty by increasing the variability if the multiple future observations are assumed to be conditionally independent. The NPI method has been applied to the finance area, predicting future stock returns when little further information is available and providing a way of the pairwise comparison between stock returns \citep{Ba17}. \citet{Co18} presented a new approach to quantify the credit risk by using the NPI method based on the ROC analysis. The implements of the NPI method for the vanilla option pricing well perform when there are less certain information of the underlying asset \citep{He19};\citep{He20}.
In this paper, we present the NPI method for Asian option pricing, which attempts to evaluate the Asian option price based on the historical data and offering a rational pairwise trading scenario. Some relevant background about the NPI method is summarized in Section 2. The Asian option pricing model based on the NPI method is proposed in Section 3 along with a rational trading criterion of the comparison of two Asian options. In Section 4, we illustrate the NPI method by using the simulation as well as the empirical examples of the energy market. Some conclusions and extensions are written in Section 5.
\section{Preliminaries}
Nonparametric Predictive Inference (NPI) is an inferential framework based on the assumption $A_{(n)}$ \citep{Hi68}, which directly provides probabilities for future observations by using few model assumptions and observed values of related random quantities. $A_{(n)}$ assumption makes sure that the future observation is equally likely to fall in the interval of a real value line created by $n$ observed random quantities, which keeps the consistency of the exchangeability. Based on the $A_{(n)}$ assumption, NPI offers the upper and lower probabilities for one or multiple future random quantities when $n$ observed random quantities are available, which follows De Finetti's fundamental theorem of probability \citep{De74}. NPI is a frequentist statistical method which has strong consistency properties \citep{Au04}.
NPI for $m$ multiple future random quantities is concerned in this paper, which is based on $A_{(n+m-1)}$. The NPI method predicts the future observation based on historical data and keeps updating the data, which means the prediction of $m$ future data is identical to the prediction of one future data. After the first one is predicted, the prediction is used as the historical information to forecast the next observation. NPI assumes that each future data is equally likely in the interval $I_j$, where $j=1,2,...,n+1$, created by $n$ observed data, which also means that all possible orderings of $n$ observed data and $m$ future data are equally likely. Totally, ${n+m \choose m}$ possible orderings can be derived, so for each ordering the probability is equal to ${n+m \choose m}^{-1}$.
To inference the arithmetic average price of the underlying asset based on the NPI method, we first forcast the return of the underlying asset. Baker \citep{Ba17} predicts the stock future return with few information by applying the NPI method, but the predicted return is under the frame of simple interest. In this paper, we follows the same idea and extend it to compound interest. Through the prediction of the underlying asset return, the imprecise arithmeric price of the underlying asset can inferred and utilized in the Asian option pricing procedure.
\section{NPI for Asian options}
In this section, an Asian option pricing method is presented based on the NPI method, which reflects the uncertainty not only from the stochastic environment but also from the limited prior information. And a trading criterion by comparing the Asian options contingent on two different product is shown as a risk measure.
\subsection{Prediction stock returns }
Define $S_t$ is the underlying asset price at time $t$. By assuming there are $n$ historical underlying asset price $S_t=s_t, t=1,2,\ldots,n$ available in the market, where the time intervals between these historical data are identical to each other. Then the continuous compounding rates of return of the underlying asset price $r_t$ is,
$$r_t=ln\left(\frac{S_t}{S_{t+1}}\right), \ t=1,2,\ldots,n$$
To predict future return of underlying asset price based on the NPI method, the exchangeability is assumed in our model meaning the order of the underlying asset return is irrelevant. After we calculate the compounding return, we rank these values from the lowest value to the highest value, $r_{(1)}, \ldots, r$ $_{(n)}$. Then on this real value line created by $r_{(1)}, \ldots, r_{(n)}$, there are $n+1$ intervals. To avoid the influences of $\infty$ and $-\infty$, we need to find the lowest and the highest returns, $r_{(0)}$ and $r_{(n+1)}$, which can be the extreme returns in a long-term historical period or the extreme values referring to the user's preference. On the basis of the historical information, we assume that the future data randomly falls in any interval on this real value line. From the assumption of multiple future data prediction through the NPI method, totally there are ${n+m \choose m}$ orderings of $m$ future compounding returns, which are equally likely. Investor can infer the future returns by counting the orderings fitted in one's investment criterion.
As the aim of this paper is to study the Asian option with arithmetic average and do the prediction, the aggregate compounding return is concerned. The general formula of the aggregate compounding return is
$$\hat{R_i}=\frac{\sum_{t=n+1}^{n+i}R_t}{i}$$
This presents the aggregate compounding return for $i$ future cumulative time, where $i=1,\ldots,m$. For example, when $i$ is equal to one, $\hat{R_1}$ represents the aggregate return during the first time step, and when $i=2$, $\hat{R_1}$ represents the aggregate return during the first and second time steps and so on. By applying this formula to the NPI framework, the upper bound $\hat{R_i^u}$ and the lower bound $\hat{R_i^l}$ of the aggregate compounding return for $i$ future period can be calculated. The fundamental idea is that for a specific ordering, $R_t$ randomly falls in the interval $I_j$, $j=1,\ldots,n+1$, which defines the upper bound $R_t^u$ of $R_t$ equal to $r_{(j)}$ and the lower bound $R_t^l$ equal to $r_{(j-1)}$. By putting the upper and lower bounds of $R_t$ into the aggregate calculation, the upper and lower bounds of $\hat{R_i}$ can be obtained.
\subsection{ Asian option expected prices based on the NPI method}
As we mentioned, the Asian option's payoff depends on two type of average value, the arithmetic average price and the geometric average price. In this paper, the Asian option with arithmetic average price is in consideration. According to the different type of the strike price , there are two types of Asian options, with the fixed strike price and the float price. The Asian option with fixed strike price is discussed in this paper. Therefore, if a $m$ period Asian option with fixed strike price $K$ is priced, the general pricing formula is,
\begin{equation}
\label{eq:AsianF}
V_0=B(0,m)[S_\mu^m-K]^+
\end{equation}
where $V_0$ is the initial expected price of this Asian option, $S_\mu^m$ is the arithmetic average price of the underlying asset during $m$ period, and $B(0,m)$ is the discount factor during $m$ period. From Equation (\ref{eq:AsianF}), we can conclude that the payoff of this type Asian option is the positive value of the subtraction between the average underlying asset price and the predetermined strike price.
To calculate the arithmetic average price of the underlying asset during $m$ period, the aggregate compounding returns for every $i\in(1,\ldots,n)$ period are needed.
\begin{equation}
\label{eq:AthS}
S_\mu^m=\frac{1}{m}\sum_{k=0}^mS_0e^{i\hat{R_i}}
\end{equation}
where $S_0$ is the initial underlying asset price and $R_0$ is set to be zero. By the definition of the Asian option, the exact value of the future underlying asset is less important. Rather than the explicit value of each time step $S_t$, the average behavior of the underlying asset is considered, where the aggregate return is the appropriate value to represent the asset behavior during a period. Thus, based on the NPI method, we do not concern about the exact value of $S_{1}, \ldots, S_{m}$. Instead, the upper and lower bounds of aggregate compounding returns for every $i\in(1,\ldots,n)$ time-steps are calculated. Putting the bounds of the compounding returns in Equation (\ref{eq:AthS}), we get the upper and lower bounds of the arithmetic average underlying asset price. For $m$ future time steps, the minimum average underlying asset price is,
\begin{equation}\label{eq:2.6}
\underline{S_\mu^m}=\frac{1}{m}\sum_{k=0}^mS_0e^{i\hat{R_i^l}}
\end{equation}
The maximum average underlying asset is,
\begin{equation}\label{eq:2.7}
\overline{S_\mu^m}=\frac{1}{m}\sum_{k=0}^mS_0e^{i\hat{R_i^u}}
\end{equation}
Thus, according to the definition of the Asian option payoff, we can calculate the upper and lower expected option price based on the NPI method, which is called the minimum selling price and the maximum buying price according to the trading intention.
\textbf{The minimum selling price for the call option}
\begin{equation}\label{eq:NPI1}
\overline{V_0}=B(0,m)[\overline{S_\mu^m}-K]^+=B(0,m)\left[ \frac{1}{m}\sum_{k=0}^mS_0e^{i\hat{R_i^l}}-K\right] ^+
\end{equation}
\textbf{The maximum buying price for the call option}
\begin{equation}\label{eq:NPI2}
\underline{V_0}=B(0,m)[\underline{S_\mu^m}-K]^+=B(0,m)\left[ \frac{1}{m}\sum_{k=0}^mS_0e^{i\hat{R_i^u}}-K\right] ^+
\end{equation}
\textbf{The minimum selling price for the put option}
\begin{equation}\label{eq:NPI3}
\overline{V_0}=B(0,m)[K-\overline{S_\mu^m}]^+=B(0,m)\left[ K-\frac{1}{m}\sum_{k=0}^mS_0e^{i\hat{R_i^u}}\right] ^+
\end{equation}
\textbf{The maximum buying price for the put option}
\begin{equation}\label{eq:NPI4}
\underline{V_0}=B(0,m)[K-\underline{S_\mu^m}]^+=B(0,m)\left[ K-\frac{1}{m}\sum_{k=0}^mS_0e^{i\hat{R_i^l}}\right] ^+
\end{equation}
The upper and lower bounds of the Asian option indicate the buying and selling thresholds of the investor. The investor who trades according to the result from the NPI method would not like to be in the game when the quoted price is in the interval of the minimum selling price and the maximum buying price. However, if the quoted price is higher than the minimum selling price, the investor prefers to sell the option. Or the longing position is triggered when the maximum buying price is greater than the quoted price. The advantage of this method is we do not assume any distribution of the underlying asset distribution. The prediction is based on the information from the historical data. Different from calculating the average price of historical data directly, this method considers the randomness of the stock price and its outcome is an interval, which avoids the error of the historical data bias and reflects more uncertainty of the underlying asset.
\subsection{Trading criteria of the Asian options contingent on two underlying asset}\label{s:2.2}
Other than pricing the Asian option by using the NPI method, this method also offers a way to make a reasonable decision in the Asian option trade. The NPI method provides the upper and lower probability that the investor can get a positive profit in this investment. Suppose there is a sequence of historical data with an amount of $n$, which is continuous, consistent and exchangeable. Same as what we did for the option pricing procedure, we calculate the historical aggregate compounding returns and rank them from the lowest one to the highest one $r_{(1)},r_{(2)}...,r_{(n)}$. And to avoid the influence from the infinity values, we set the new historical sequence started with $r_{(0)}$, and end with $ r_{(n+1)}$, which these two values can be determined by using the minimum historical price and the maximum historical price in a long-term time. By having the aggregate compounding returns, we can calculate the average price of the underlying asset $S_\mu^m$. Next, the NPI lower and upper probabilities of the positive payoff are derived for the Asian option involving the average stock price $S_\mu^m$ and the strike Price $K$. The investor can use these probabilities to compare the Asian options and set their trading criteria. The formulae are listed below.
\textbf{The upper and lower probability of a positive payoff}
\begin{equation}
\overline{P}(\text{Payoff}>0)=
\begin{cases}
\frac{1}{{n+m\choose m}}\sum_O \mathbbm{1} \lbrace \overline{S_\mu^m}>K\rbrace & \text{call option}\\
\frac{1}{{n+m\choose m}}\sum_O \mathbbm{1} \lbrace \underline{S_\mu^m}<K\rbrace & \text{put option}
\end{cases}
\end{equation}
\begin{equation}
\underline{P}(\text{Payoff}>0)=
\begin{cases}
\frac{1}{{n+m\choose m}}\sum_O\mathbbm{1}\lbrace \underline{S_\mu^m}>K\rbrace & \text{call option}\\
\frac{1}{{n+m\choose m}}\sum_O \mathbbm{1} \lbrace \overline{S_\mu^m}<K\rbrace & \text{put option}
\end{cases}
\end{equation}
where$\overline{S_\mu^m}$, $\underline{S_\mu^m}$ are calculated by Equations (\ref{eq:2.6}) and (\ref{eq:2.7}). $\sum_O$ is the summation over all the$\frac{1}{{n+m\choose m}}$possible orderings of the m future returns within the $n+1$ intervals, and $\mathbbm{1}\lbrace A \rbrace $ is an indicator function which is equal to 1 if $A$ is true or 0 otherwise.
This interval probabilities can help an investor to choose the better underlying asset in the Asian option investment as either a speculator or a hedger. As in the commodity market, especially in the crude oil market. there are a variety of underlying assets correlated to each other, so it is hard to choose which underlying asset is a better investment. By the NPI method, an investor is offered an indicator that can be referred according to the investor's risk aversion and character, speculator or hedger. Suppose there are two underlying assets $A$ and $B$ that have similar price values and trends. A speculator is a risk-taker whose purpose of an investment is to seek the opportunities to earn some profit. If the speculator would like to invest in either of these two assets, then the indicator below suggests the speculator invest in asset $A$, the lower probability of a positive payoff for asset $A$, $\underline{P}(\text{Payoff}_A)$, is greater than the lower probability of a positive payoff for asset $B$, $\underline{P}(\text{Payoff}_B)$, or the upper probability of a positive payoff for asset $A$, $\overline{P}(\text{Payoff}_A)$, is greater than the upper probability of a positive payoff for asset $B$, $\overline{P}(\text{Payoff}_B)$. For a hedger, the purpose involving in the option trading is to hedge the risk in the trade of the underlying asset, so the hedger has a high level of risk aversion. An absolute strength of asset $A$ needs to be revealed to instruct this hedger's action. Thus, when $\underline{P}(\text{Payoff}_A)>\overline{P}(\text{Payoff}_B)$, the investment in asset $A$ is appealing to the hedger.
\section{Illustrated examples}
Several examples are discussed in this section to illustrate the NPI method for the Asian option. We first study the performance of the NPI method for Asian option pricing by the simulation techniques. Then a performance study of the energy market is developed to assess the empirical value of this method.
\subsection{The simulation study}
As acknowledged, the Geometric Brownian Motion (GBM) is widely used to model the stock price behavior. Therefore, to start the illustration, an example based on the GBM is presented in this section. By utilizing the R program, we first simulate 100 paths of stock prices following the GBM with the return equal to 0.02 and the volatility equal to 0.02 as well. The simulated stock paths are displayed in Figure \ref{fg:GBM}. In each path, the initial price is 50, and the program simulated the stock price movement for 110 time steps. In our example, the time period 0 to 100, is assumed as the historical time period calling the corresponding data the historical data, while assuming the time period 100 to 110, to be the predictive time period calling the corresponding data the future data. The idea is using the NPI method for the Asian call option pricing formulae, Equations (\ref{eq:NPI1}) and (\ref{eq:NPI2}) to forecast the option price, and using the future data to calculate the option price based on payoff definition as the benchmark. In the following example, we predict the price of an at-the-money (ATM) call option where the strike price equals to the initial price 50.
\begin{figure}
\caption{Simulated stock price paths}
\label{fg:GBM}
\end{figure}
\begin{figure}
\caption{Asian option prices predicted by the NPI method with the GBM option price as the benchmark}
\label{fg:AS_NPI_GBM}
\end{figure}
Figure \ref{fg:AS_NPI_GBM} discloses that the NPI method can provide an interval that includes the benchmark price in most of the cases. In these cases, the NPI method's prediction includes the benchmark value, but this does not mean that NPI method can predict the price accurately. If the result from the NPI method is an interval with a large value gap, the benchmark price could be in the interval for sure. Some further discussions of the accuracy based on the NPI method are illustrated in the next paragraph. Figure \ref{fg:AS_NPI_GBM} also reveals that the fluctuations of NPI option prices have a similar pattern to that of the GMB option prices. In these 100 paths, for the GBM option price with a higher value, the maximum buying and minimum selling prices corresponding to this path also have a higher value, and the same conclude can be derived from the figure for the path with a lower GBM option price as well. However, it is also clear that the boundary prices from the NPI method are less fluctuated than the GBM prices. It is easy to understand from the perspective of deviation. As the GBM prices are predicted based on future data, it has a greater deviation from the initial price of the simulation than that of the historical data, while the NPI method derives the option prices from the historical data. So the patterns of NPI option boundary prices are more stable than the pattern of the GBM prices. Therefore, there are some extreme values manifested in the GBM price. But the corresponding extreme values in the NPI option boundary prices are less significant.
To investigate the influence of the volatility on the prediction outcome from the NPI method, we define three factors; coverage percentage, accuracy and precision. Coverage percentage estimates the percentage of NPI outcomes including the benchmark value in the interval. Accuracy is defined as the expectation of absolute difference between the median value of the NPI interval and the benchmark value, $E[|\text{median}_{\text{NPI}}-\text{benchmark}_{\text{GBM}}|]$, which reflects the deviation between the NPI outcomes and the benchmark. Precision is to calculate the mean value of the interval length from the NPI method. Including the precision in our study is because if the precision is very large, the result of the coverage percentage is supposed to be better than the case when the precision is very small. Herein, we study the influence of the varying volatilities on these three factors in order to estimate the performance of the NPI prediction result for the same ATM option as that in the last example. In this study, the volatility is in the range from 5\% to 10\% to simulate the daily volatility in the market. Three factors are monitored to assess the NPI results.
\begin{longtable}[tp]{ |p{1.8cm}||p{1.8cm}|p{1.8cm}||p{1.8cm}|p{1.8cm}|}
\hline
&\multicolumn{2}{c||}{Precision is large [3,4]}&\multicolumn{2}{c|}{Precision is small[1,1.5]}\\
\hline
volatility&percentage&accuracy&percentage&accuracy\\
\hline
0.5\%&1&0.5856104&0.9653&1.391795\\
\hline
1\%&0.9999&0.6747436&0.9625&1.401927\\
\hline
1.5\%&0.9963&0.8498949&0.9576&1.421381\\
\hline
2\% &0.9828 & 1.047153 & 0.9597 &1.501727 \\
\hline
2.5\%&0.951 &1.262822 &0.9345 &1.609755\\
\hline
3\% &0.9208 & 1.425547 &0.8975 & 1.72399\\
\hline
3.5\% &0.8797 &1.590992 &0.8744 &1.866814\\
\hline
4\% &0.8481 &1.752489 &0.8289&1.969898\\
\hline
4.5\%&0.8271 &1.937935 & 0.8001 & 2.082382 \\
\hline
5\% &0.7876 &2.064087 &0.7875&2.188028 \\
\hline
5.5\%&0.7638 &2.174651 & 0.7576&2.358233 \\
\hline
6\% &0.7387 & 2.301093 & 0.7376 &2.47124\\
\hline
6.5\%&0.7169 &2.459455 &0.7137 & 2.554012 \\
\hline
7\% &0.6921 &2.543524 &0.6897&2.7054 \\
\hline
7.5\% &0.6832 &2.706887 & 0.6722&2.819131 \\
\hline
8\% &0.6808 &2.821586 & 0.6625 &2.980532 \\
\hline
8.5\% &0.6514 &2.913285 &0.6504 &3.059432 \\
\hline
9\% &0.6403 &3.012923 &0.64 &3.16712 \\
\hline
9.5\% &0.6364 &3.167188 &0.6242 &3.296574\\
\hline
10\%&0.6153&3.283862&0.6147&3.356551\\
\hline
\caption{The study of volatility influence}\label{ta:3factors}
\end{longtable}
Table \ref{ta:3factors} displays the outcomes of three factors with the varying volatility in two simulations. When we calculate the precision with different volatilities, we find that as the volatility increases, the precision decreases. This result does not mean that high volatility has a positive effect on the precision. The reason why high volatility causes a small precision is when the underlying asset is more volatile, there are more times of the simulated paths with a zero payoff either from NPI method or from the GBM model. As the average precision value of all simulated paths is calculated as the estimator, the average result is getting smaller as the more zeros appearing in the simulation. Therefore, the value of precision with varying volatility is less instructive in this study. Based on this, we categorize the outcomes in two parts according to two sizes of precision, the large precision with the value from 3 to 4 and the small precision with the value from 1 to 1.5. No matter in the simulation with large or small precision, the result indicates that the percentage of the NPI interval including the benchmark value gets lower, and the NPI results are less accurate along with the increasing volatility. If the results are compared horizontally, it is not difficult to conclude that the NPI prediction presents a better result with a larger precision than that with a smaller precision. Thus, inputting a larger precision is a safer choice. From the finance perspective, it illustrates that a conservative investor who uses NPI method can do the prediction with a larger precision to behave safely, but at the same time he may miss a lot of trading chances in the market. In the simulation with a large precision, when the volatility is lower than 4.5\%, the NPI method can offer a good prediction with the percentage greater than 80\%, and accuracy less than 2. To get a better result with a percentage greater than 90\% and accuracy less than 1.5, the volatility needs to be restricted within 3\%. In the simulation with small precision, the NPI's result is good when the volatility is also lower than 4.5\%, and the corresponding accuracy is less than 2.08 worse than the one with large precision. But if a better result is required, the volatility should be lower than 2.5\% in order to make the percentage greater than 90\%, then the accuracy under these circumstances is less than 1.61. Overall, we can draw the conclusion that the NPI method performance is better with an option based on an underlying asset at a lower volatility less than 3\% daily. An Asian option is normally used in commodity and foreign exchange markets where the underlying asset is less volatile than the equity in the stock market. This allows the NPI method to provide a relatively good result for the Asian option pricing in these markets. To support the statement, an empirical example of the Asian option in the crude oil market is investigated.
\subsection{The empirical study of the energy market}
\begin{figure}
\caption{NPI predictions with a ten year historical data }
\label{fg:WTILarge}
\end{figure}
The crude oil commodity market is considered in this example. The set of data is the New York Mercantile Exchange (NYMEX) daily closed price of the WTI crude oil normally used as a benchmark in the oil pricing. The Asian option price is the Chicago Mercantile Exchange (CME group) the WTI average price call option started on 23/10/2019 and expired on 29/11/2019 with the strike price 54(\$). By the end of the trading time on 23/10/2019, the settlement price of this call option is 2.83(\$), which is a reference price provided by the CME group. The NPI method, an imprecise statistical framework based on the historical data, controls the precision of the prediction by managing the historical data size. By large historical data, the degree of prior information dispersion is more significant than a small historical data leading to a less precise interval result. In the following example, we forecast the average price option based on ten years of historical data, from 23/10/2009 to 23/10/2019, and the plotted result is shown in Figure \ref{fg:WTILarge}. We first calculate the daily volatility based on the historical data, which equals to 2.1\%. According to our volatility study by simulation, the NPI prediction is supposed to provide some valuable results under this volatility. After 10000 trails, we get the expected NPI maximum buying price is equal to 2.6152 and the expected NPI minimum selling price is equal to 3.29124, which are shown as the two horizontal lines in orange and green in Figure \ref{fg:WTILarge}. The dots in Figure \ref{fg:WTILarge} the NPI results in each trail. To make the graph clear and well recognized, only 1000 trails results are plotted in the figure. Figure \ref{fg:WTILarge} indicates that the NPI method provides a relatively good result that includes the real price in its interval.
\begin{figure}
\caption{NPI predictions with a one year historical data}
\label{fg:WTISmall}
\end{figure}
Next, we improve the precision of the NPI prediction by limiting the historical data size to one year, from 23/10/2018 to 23/10/2019. From Figure \ref{fg:WTISmall}, the trial outcomes are more concentrated leading to a more precise result with a smaller interval from the NPI method. The maximum buying price is 2.447181, and the minimum selling price is 2.538937. The daily volatility during this historical period is 2.4\%. Although the NPI result is more precise, the interval deviates distinctly from the real market price 2.83 comparing to the result from a large historical data. This means we scarify the accuracy in order to gain a more precise result. But this is not a good deal since the investor referring to the NPI result would lose money when he trades WTI in the market.
\begin{figure}
\caption{WTI crude oil price from 23/10/2018 to 23/10/2019 }
\label{fg:WTIPrice}
\end{figure}
The examples above are a rigid investigation of the NPI performance by controlling the size of historical data. To study the historical data more clearly, we display the WTI crude oil price in this recent one year in Figure \ref{fg:WTIPrice}. It is obvious that there is a deep drop that started in October 2018 ended in December 2018, which is the worst performance in nearly three years. The price is down to 44.48 on 27/12/2018 the lowest closing price since January 2016. There exist multiple reasons causing this drop, global oversupply keeping the investors away, investors with less confidence of economic recovery in the next year and the longest US government shutdown on 22 December 2018. Through the comprehensive consideration, the data from 23/04/2019 to 23/10/2019 has a better reference value to do the prediction. But it is an arbitrary decision to cut off the data of early half year crudely. What we do here is to adjust the sampling procedure making it focus more on the latter half years' data than the earlier one. To achieve this, we use the maximum and minimum one year historical prices as the boundary values, but the main sampling data is the historical data from 23/04/2019 to 23/10/2019. By doing this, the pricing procedure not only considers the probabilities of the unexpected event but also places emphasis on the historical information in a relatively stable market environment. The adjusted result is plotted in Figure \ref{fg:WTISmallAd}. It is obvious that after adjustment, the accuracy of the NPI result gets better, the maximum buying price at 2.757054 and the minimum selling price 2.936124. This interval covers the real market price which is a better investment indicator than the NPI result without adjustment. In addition, the precision of the NPI result is nearly as same as that without adjustment. This example manifests that the NPI method performs better combining with the assessment of historical data.
\begin{figure}
\caption{NPI predictions after adjustment}
\label{fg:WTISmallAd}
\end{figure}
\begin{table}
\centering
\begin{tabular}{c||cccc}
\hline
\\[0.3ex]
Trading Date & $\overline{P}_{\text{WTI}}$ & $\underline{P}_{\text{WTI}}$ & $\overline{P}_{\text{Brent}}$ & $\underline{P}_{\text{Brent}}$\\
\hline
2019-11-22& 0.15 & 0.13 & 0.04 & 0.03 \\
2019-11-23 & 0.88 & 0.87 & 0.27 & 0.25 \\
2019-11-24& 0.80 & 0.79 & 0.17 & 0.15 \\
2019-11-25 & 0.70 & 0.69 & 0.04 & 0.03 \\
2019-11-26 & 0.70 & 0.69 & 0.04 & 0.02 \\
2019-11-27 & 0.71 & 0.69 & 0.03 & 0.02 \\
2019-11-28 & 0.71 & 0.70 & 0.03 & 0.01 \\
2019-11-29 & 0.72 & 0.71 & 0.04 & 0.03 \\
2019-11-30& 0.97 & 0.96 & 0.41 & 0.40 \\
\hline
\end{tabular}
\caption{NPI probabilities for WTI vs Brent with $K=0.95*S_0$}\label{ta:WTIB}
\end{table}
The NPI method as discussed in Section \ref{s:2.2} can be used as the market director for an investor. As acknowledged, in the crude oil market, WTI from the American and Brent from the North Sea are two benchmark prices of the crude oil market that are both sweet and normally track one another. Their prices trend and pattern are similar to each other making the investor hard to compare these two values directly from the market price. According to the NPI method illustrated in Section \ref{s:2.2}, an investor can get an indicator of the trading action according to the investor's risk aversion. In the following example, we assume the investor wants to buy an Asian call option on either WTI or Brent that the average underlying asset price during the option life period is not less than 95\% of its spot price meaning the strike price $K$ is equal to 95\% of the spot price $S_0$. The call option's trading day is from 2019-11-22 to 2019-11-30, and the expire day is 2019-11-30, so the option period is from 9 days to 0 days. Then the upper and lower probabilities of both WTI and Brent average prices greater than strike price are calculated getting the results displayed in Table \ref{ta:WTIB}. From Table \ref{ta:WTIB}, it is obvious that the lower probabilities of the WTI price are greater than the upper probability of the Brent price. WTI definitely has a higher possibility to earn a positive payoff in the call option market. For the investor either as a speculator or a hedger, it is optimal to invest in WTI. The result also plotted in Figure \ref{fg:WTIB} showing that the probability pattern of the WTI price is similar to that of the Brent price but with greater values. Also, from the figure, we can tell the best time to get in the market, which is 2019-11-23 in this example, since the NPI probabilities of Nov 23rd are the greatest value among these dates except the one of Nov 30th. The WTI and Brent oil price returns from 2019-11-22 to 2019-11-30 are also calculated to assess our prediction. The WTI return equals to 4.102\% higher than the Brent return, 1.935\%, which confirms that the trading strategy based on NPI is profitable.
\begin{figure}
\caption{NPI probabilities for WTI and Brent}
\label{fg:WTIB}
\end{figure}
\begin{figure}
\caption{NPI probabilities for WTI and the heating oil}
\label{fg:WTIH}
\end{figure}
\begin{table}
\centering
\begin{tabular}{c||cccc}
\hline
\\[0.3ex]
Trading Date & $\overline{P}_{\text{WTI}}$ & $\underline{P}_{\text{WTI}}$ & $\overline{P}_{\text{HO}}$ & $\underline{P}_{\text{HO}}$\\
\hline
2019-11-22 & 0.15 & 0.13 & 0.34 & 0.33 \\
2019-11-23 & 0.88 & 0.87 & 0.92 & 0.92 \\
2019-11-24 & 0.80 & 0.79 & 0.91 & 0.90 \\
2019-11-25 & 0.70 & 0.69 & 0.69 & 0.68 \\
2019-11-26 & 0.70 & 0.69 & 0.70 & 0.69 \\
2019-11-27 & 0.71 & 0.69 & 0.71 & 0.70 \\
2019-11-28 & 0.71 & 0.70 & 0.83 & 0.82 \\
2019-11-29& 0.72 & 0.71 & 0.92 & 0.92 \\
2019-11-30& 0.97 & 0.96 & 0.99 & 0.99 \\
\hline
\end{tabular}
\caption{NPI probabilities for WTI vs Heating Oil (HO) with $K=0.95*S_0$}\label{ta:WTIH}
\end{table}
To end this study, we calculate the upper and the lower probabilities of the WTI price comparing to another oil product, the heating oil. The heating oil price is related to the WTI price, because it is a low viscosity, liquid petroleum product made from the WTI crude oil. So the price of the heating oil in the United States is depending on the supply of the WTI crude oil. Here we also pick the WTI and heating oil price data from the CME group during 2019-11-22 to 2019-11-30. The event of interest is the average price $S_\mu$ ended by Nov 30th is greater than the 95\% of the spot price $S_0$. From the perspective of the Asian option, we are interested in the probability of a call option with $K=0.95*S_0$ end up with a positive payoff. We plot the NPI upper and lower probabilities in Figure \ref{fg:WTIH}. The decision of the option selection is harder to make in this comparison group, because there are intersections and overlapping of the NPI probabilities between these two products. Unlike the result of WTI versus Brent that WTI always dominates, in this example, there are overlapping and intersections in Figure \ref{fg:WTIH}. The different underlying asset is picked according to the trading date. To specify the underlying asset selection based on the trading data, the exact value of NPI upper and lower probabilities are listed in Table \ref{ta:WTIH}. Between 2019-11-22 to 2019-11-24, the lower probability of $S_\mu >K$ for the heating oil dominates the upper probability of $S_\mu>K$ for WTI. During this period, a speculator or a hedger is better to invest in the heating oil. On Nov 25th, the upper and lower probabilities of WTI are greater than the corresponding value of the heating oil. So on this day, a speculator is supposed to choose the call option based on the WTI oil price, but a hedger would wait. On Nov 26th, the NPI probability intervals of WTI and the heating oil are overlapped with each other, so there is no indication which underlying asset is better. The next day's upper probabilities of these two oil price are still the same value, while the lower probability of WI is less than the lower probability of the heating oil. Thus, on Nov 27th, a speculator is better to get in the game of the call option based on the heating oil, and a hedger still waits for the sign of a more determined trading indicator. This indicator appears on Nov 28th and lasts until Nov 30th, the lower probability of the heating advantages over the upper probability of the WTI, leading to the trade for both a speculator or a hedger in the Asian call option contingent on the heating oil.
\section{Concluding Remarks}
This paper presents a novel approach to evaluate the Asian option with the arithmetic price from the imprecise probability aspect through the NPI method, which forecasts the option price on the basis of the historical data with few assumptions. This approach provides an interval of prices as the result, which not only contains the uncertainty from the probability perspective but also the uncertainty from limited prior information. This property makes it more advanced than the traditional method for the Asian option, especially the one in the energy market because of the less liquidity of the Asian option in the energy market. The NPI method also gives a risk measure by comparing two energy products inspiring the investor with a trading criterion. We study the performance of the NPI method first by the simulation using the GBM prediction as the benchmark. Three factors, precision, coverage percentage and accuracy are defined and investigated to help us assess the performance. It turns out the NPI forecasting has more reference value for the less volatile product. Then we predict the WTI crude oil price based on the NPI method comparing to the real market price. With a long period of historical data, the NPI forecasting interval contains the real market price, but the precision of the result is not entirely satisfactory. To get a more precise interval result, narrowing the size of the historical data is going to scarify the accuracy of the result. After the investigation, we found that using the extreme value of the historical data to control the precision and considering the historical event of adjusting the sampling period of the historical data can offer a better outcome. We also illustrate the risk measure, the NPI trading criterion, by two examples, the trade of WTI and Brent and the trade of WTI and the heating oil. An investor is guided according to their risk aversions by using this criterion.
In order to get a better result from the NPI method, there are several aspects we need to consider. The time period of the predictive data should be considered discreetly since we assume the exchangeability of all data including the historical data and the future data in Section 2. If the prediction period is too long, it challenges the reasonableness of the exchangeability assumption, because some significant fluctuation may happen in the market. To inference these fluctuations, a large historical data is needed to infer the situation, which as we discussed in the last paragraph, this will reduce the level of accuracy of the result. The extreme value of the historical data, $r_{(0)}$ and $r_{n+1}$, is also an important consideration that will affect the NPI prediction as we illustrated in the example. These two values play a very important role in balancing the precision of the result and the inference ability of the historical data for the significant fluctuations.
To avoid the effect from the unexpected historical event or seasonal effect, the sampling historical data $s_{(1)},...,s_{n}$ has been picked in the time period with a more stable market. However, dealing with these effects may be important, the adaption of the NPI method for the data with the seasonal effect is a meaningful topic for future study. In this paper, we have explained that the NPI method is more suitable to predict the price of a less volatile product. How to solve the prediction problem of a market with high volatility from the imprecise probability perspective is also appealing for future study.
\section*{Data avaliability statement}
The raw data that support the study in this paper is obtained from the CME websites and the S\& P Capital IQ database.
\end{document} |
\begin{document}
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\title{Geometry of symmetric determinantal loci}
\author{Shinobu Hosono and Hiromichi Takagi}
\begin{abstract}
We study algebro-geometric properties of determinantal loci of
$(n+1)\times(n+1)$ symmetric matrices and also their double covers for even
ranks. Their singularities, Fano indices and birational geometries are
studied in general. The double covers of symmetric determinantal loci
of rank four are studied with special interest by noting their relation to
the Hilbert schemes of conics on Grassmannians.
\end{abstract}
\maketitle
\section{{\bf Introduction}}
Throughout this paper,
we work over $\mathbb{C}$, the complex number field,
and we fix a vector space $V$ of dimension $n+1$.
We define $\textsc{S}_r\subset \mathbb{P}({\ft S}^2 V^*)$ to be the locus
of quadrics in $\mathbb{P}(V)$ of rank at most $r$.
Taking a basis of $V$, $\textsc{S}_r$ is defined by $(r+1)\times (r+1)$ minors
of the generic $(n+1)\times (n+1)$ symmetric matrix.
We call $\textsc{S}_r$ the {\it symmetric determinantal locus of rank at most $r$}.
For example, $\textsc{S}_1=v_2(\mathbb{P}(V^*))$ with
$v_2(\mathbb{P}(V^*))$ being the second Veronese variety of $\mathbb{P}(V^*)$
and $\textsc{S}_{n+1}=\mathbb{P}({\ft S}^2 V^*)$.
There is a natural stratification of $\mathbb{P}({\ft S}^2 V^*)$ by $\textsc{S}_r$:
\[
v_2(\mathbb{P}(V^*))=\textsc{S}_1\subset \textsc{S}_2\subset \cdots \subset \textsc{S}_n\subset
\textsc{S}_{n+1}=\mathbb{P}({\ft S}^2 V^*).
\]
We call a point of $\textsc{S}_r\setminus \textsc{S}_{r-1}$ {\it a rank $r$ point}.
Similarly we define the symmetric determinantal locus $\textsc{S}_r^*$ in
the dual projective space $\mathbb{P}(\ft{S}^2V)$. It is a well-known fact that the
stratification of $\mathbb{P}(\ft{S}^2V^*)$ by $\textsc{S}_r$ and that of $\mathbb{P}(\ft{S}^2V)$
by $\textsc{S}_r^*$ are reversed under the projective duality.
Recently, classical projective duality is highlighted in the study of derived categories of coherent sheaves on projective varieties, where the duality is called {\it homological projective duality} (HPD) due to Kuznetsov \cite{HPD1}. HPD is a powerful framework to describe the derived category of a projective variety with its dual variety, and has been worked out in several interesting examples such as Pfaffian varieties (i.e., determinantal loci of anti-symmetric matrices) \cite{HPD2} and the second Veronese variety $\textsc{S}^*_1$ \cite{Quad}. Interestingly, it is often the case that we have interesting pairs of Calabi-Yau manifolds associated to HPDs
\cite{BC,HPD2}.
In a series of papers \cite{HoTa1}--\cite{HoTa3},
we have studied the case $\textsc{S}_2^*$ and $\textsc{S}_4$ for $n=4$ in detail, where
a pair of smooth Calabi-Yau threefolds $X$ and $Y$ appears, respectively, as a
linear section of $\textsc{S}^*_2$ and the double cover of the orthogonal linear
section of $\textsc{S}_4$ branched along the set of rank 3 points.
It has been shown in \cite{HoTa3} that these $X$ and $Y$ are derived-equivalent, indicating that $\textsc{S}_2^*$ and the double cover $\textsc{T}_4$ of $\textsc{S}_4$ (called {\it double quintic symmetroids}) are HPD to each other.
Also, for $n=3$, we have established in \cite{ReyeEnr} the relations between the derived categories of a $2$-dimensional linear section $X$ of $\textsc{S}^*_2$ and the double cover $Y$ of the orthogonal linear section of $\textsc{S}_4$ branched along the set of rank 2 or 3 points after the inspiring works \cite{Lines} and \cite{IK}. In the latter case of $n=3$, $X$ is known as an Enriques surface of Reye congruence, while $Y$ is known as an Artin-Mumford double solid.
The aim of the present paper is to put an algebro-geometric ground for our work \cite{HoTa3}. Indeed this is an extended version of the first part of \cite{Arxiv}. In a companion paper \cite{DerSym}, we will study homological properties of $\textsc{S}_2^*$ and $\textsc{T}_4$ for the cases $n=3,4$ based on the results of this paper.
In this paper, we are concerned with the birational geometry of $\textsc{S}_r$ for general $n$ from the viewpoint of minimal model theory. In particular, for even $r$, we present a precise description of the double covers $\textsc{T}_r$ of $\textsc{S}_r$ branched along $\textsc{S}_{r-1}$. If $r\leq n$, we show that $\textsc{S}_r$ and $\textsc{T}_r$ are $\mathbb{Q}$-factorial $\frac{(2n+3-r)r-2}{2}$-dimensional Fano varieties with Picard number one and Fano index $\frac{r(n+1)}{2}$ with only canonical singularities
in Subsection \ref{subsection:Spr}.
As an interesting application of these general results, we will consider {\it orthogonal} linear sections of $\textsc{S}^*_{n+2-r}$ and $\textsc{T}_r$, which entail a pair of Calabi-Yau varieties of the same dimensions. These Calabi-Yau varieties naturally generalize those studied in \cite{HoTa3, Arxiv, DerSym} for $n=4$, and indicates that HPD holds for $\textsc{S}^*_{n+2-r}$ and $\textsc{T}_r$ (see Subsection \ref{subsection:Pl}).
Below is the summary of the birational geometry of the double covering $T_4$ of $S_4$ for genreal $n$ which we establish in this paper.
Note that a general point of $\textsc{S}_4$ corresponds to a quadric of rank four in $\mathbb{P}(V)$. It
has two connected $\mathbb{P}^1$-families of $(n-2)$-planes which we identify with the respective conics in $\mathrm{G}(n-1,V)$. The double cover $\textsc{T}_4$ will be defined as the space which parametrizes the connected families of $(n-2)$-planes in quadrics, and will be described by making precise connection to the Hibert scheme of conics in $\mathrm{G}(n-1,V)$. In Section \ref{section:BirY}, we show the following:
\begin{thm}
Set ${\mathscr Y}:=\textsc{T}_4$ and denote by ${\mathscr Y}_0$
the Hilbert scheme of conics in $\mathrm{G}(n-1,V)$.
Then there is a commutative diagram of birational maps as follow\,$:$
\[
\xymatrix{ & & {\mathscr Y}_0\ar[d]\\
{\mathscr Y}_3\ar[dr]\ar@{-->}[rr]^{\text{\tiny{\text{$($anti-$)$flip}}}} & & \widetilde{{\mathscr Y}}\ar[dl]\ar[dr]^{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}} & \\
& \overline{{\mathscr Y}}' & &\;\;\;\;{\mathscr Y}:=\textsc{T}_4,}
\]
where
\begin{itemize}
\item ${\mathscr Y}_3:=\mathrm{G}(3,\wedge^2 \mathfrak{Q})$
with the universal quotient bundle ${\mathfrak Q}$ of $\mathrm{G}(n-3,V)$,
\item $\overline{{\mathscr Y}}'$ is the normalization
of the subvariety $\overline{{\mathscr Y}}$ of $\mathrm{G}(3,\wedge^{n-1} V)$
parametrizing $3$-planes annihilated by at least $n-3$ linearly independent vectors in $V$ by the wedge product {$\text{\rm
(Propositions~\ref{prop:barY-1-2-3}, \ref{lem:appendixB-UU-solve})}$},
\item
${\mathscr Y}_3\to \overline{{\mathscr Y}}'$
is a small contraction
with non-trivial fibers being copies of $\mathbb{P}^{n-3}$
{$\text{\rm (Proposition~\ref{pro:barYsing})}$},
\item
${\mathscr Y}_3\dashrightarrow \widetilde{{\mathscr Y}}$
is the $($anti-$)$\,flip for the small contraction
${\mathscr Y}_3\to \overline{{\mathscr Y}}'$
{$\text{\rm (Section \ref{subsection:BlowUp})}$,}
\item
$\widetilde{{\mathscr Y}}\to \overline{{\mathscr Y}}'$
is a small contraction
with non-trivial fibers being copies of $\mathbb{P}^5$
{$\text{\rm (Proposition~\ref{prop:tildeY})}$},
\item
$\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}\colon \widetilde{{\mathscr Y}}\to {\mathscr Y}$ is an extremal
divisorial contraction
{$\text{\rm (Proposition~\ref{prop:gendescr}(2))}$},
\item
${\mathscr Y}_0\to \widetilde{{\mathscr Y}}$ is the blow-up along a smooth subvariety
{$\text{\rm (Section \ref{subsection:BlowUp})}$}.
\end{itemize}
\end{thm}
In the course of the proof, we give an explicit construction of the Hilbert scheme ${\mathscr Y}_0$ of conics in $\mathrm{G}(n-1,V)$ in Subsection \ref{subsection:Hilb}.
In Section \ref{section:FY}, the contraction $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}\colon \widetilde{{\mathscr Y}}\to {\mathscr Y}$ is studied in detail.
Let $F_{\widetilde{{\mathscr Y}}}$ be $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}$-exceptional divisor
and $G_{{\mathscr Y}}$ be its image in ${\mathscr Y}$.
We determine the biregular structure of $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$
by introducing a natural double cover of $F_{\widetilde{{\mathscr Y}}}$.
Flattening of the morphism $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$ is constructed in Section \ref{section:FY}. Despite its technical nature, the flat morphism plays crucial roles for our caluculations of the cohomologies of ${\mathscr Y}$ in \cite{DerSym}.
\begin{ackn}
This paper is supported in part by Grant-in
Aid Scientific Research (S 24224001, B 23340010 S.H.) and Grant-in
Aid for Young Scientists (B 20740005, H.T.). They also thank Nicolas
Addington and Sergey Galkin for useful communications.
\end{ackn}
\noindent
\noindent\textbf{\textcolor{black}{Notation: }}
We will denote by $V_i$ an $i$-dimensional vector subspace of $V$.
\global\long\def\mathrm{Hom}ega{H_{\mathbb{P}(\Omega(1))}}
\global\long\defH_{\mP(\Omega(1)^{\wedge2})}{H_{\mathbb{P}(\Omega(1)^{\wedge2})}}
\global\long\defL_{\mP(\Omega(1)^{\wedge2})}{L_{\mathbb{P}(\Omega(1)^{\wedge2})}}
\global\long\defH_{\mathrm{G}(\Omega(1)^{\wedge2})}{H_{\mathrm{G}(\Omega(1)^{\wedge2})}}
\global\long\defH_{\mathrm{G}(\wedge^{2} T(-1))}{H_{\mathrm{G}(\wedge^{2} T(-1))}}
\global\long\defL_{\mP(T(-1))}{L_{\mathbb{P}(T(-1))}}
\global\long\defL_{\mP(T(-1)^{\wedge2})}{L_{\mathbb{P}(T(-1)^{\wedge2})}}
\global\long\defH_{\mP(T(-1))}{H_{\mathbb{P}(T(-1))}}
\global\long\defH_{\mP(T(-1)^{\wedge2})}{H_{\mathbb{P}(T(-1)^{\wedge2})}}
\section{{\bf Basics for symmetric determinantal loci $\textsc{S}_r$}}
As introduced in the preceding section, we denote by $\textsc{S}_r\subset \mathbb{P}({\ft S}^2 V^*)$ the locus of quadrics in $\mathbb{P}(V)$ of rank at most $r$.
\subsection{Springer type resolution $\widetilde{\textsc{S}}_r$ of $\textsc{S}_r$}
\label{subsection:Spr}
Let $\mathfrak{Q}$ be
the universal quotient bundle of rank $r$ on $\mathrm{G}(n+1-r, V)$ and
define the following projective bundle over $\mathrm{G}(n+1-r, V)$:
\begin{equation} \label{eq:deftSr}
\widetilde{\textsc{S}}_r:=\mathbb{P}(\ft{S}^{2}\mathfrak{Q}^{*})\to \mathrm{G}(n+1-r,V).
\end{equation}
When $r=n+1$, we consider this as the projective bundle over a point
\[
\widetilde{\textsc{S}}_{n+1}=\mathbb{P}({\ft S}^2 V^*)\to {\rm pt}
\]
{with $\widetilde{\textsc{S}}_{n+1}=\textsc{S}_{n+1}$.}
Considering the (dual of the) universal exact sequence,
we see that there is a canonical injection $\mathfrak{Q}^{*}\hookrightarrow V^{*}\otimes\mathcal{O}$,
which entails the injection $\ft{S}^{2}\mathfrak{Q}^{*}\hookrightarrow\ft{S}^{2}V^{*}\otimes\mathcal{O}.$
With this injection, composed with the natural surjection $\mathbb{P}(\ft{S}^{2}V^{*}\otimes\mathcal{O})\to\mathbb{P}(\ft{S}^{2}V^{*})$,
we have a morphism\begin{equation}
\widetilde{\textsc{S}}_r=
\mathbb{P}(\ft{S}^{2} \mathfrak{Q}^{*})\to\mathbb{P}(\ft{S}^{2}V^{*}).\label{eq:UUtoS2V}\end{equation}
By construction,
the pull-back of $\mathcal{O}_{\mathbb{P}({\ft S}^2 V^*)}(1)$ to $\widetilde{\textsc{S}}_r$ is
the tautological divisor $\mathcal{O}_{\mathbb{P}({\ft S}^2 {\mathfrak Q}\,^*)}(1)$,
which we denote by $M_{\widetilde{\textsc{S}}_r}$.
\begin{prop}
\label{prop:Spr}~
\noindent{\rm (1)} The image of the morphism $($\ref{eq:UUtoS2V}$)$
coincides with $\textsc{S}_r$.
The induced morphism $\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}\colon \widetilde{\textsc{S}}_r\to \textsc{S}_r$
is a resolution of $\textsc{S}_r$.
\noindent{\rm (2)} $\widetilde{\textsc{S}}_r=\{([V_{n+1-r}],[Q])\mid V_{n+1-r}\subset
\mathrm{Sing}\, Q
\}\subset\mathrm{G}(n+1-r,V)\times\mathbb{P}(\ft{S}^{2}V^{*})$,
where $Q$ is {a quadric} in $\mathbb{P}(V)$.\end{prop}
\begin{proof}
(1) Since the fiber of $\mathfrak{Q}^{*}$ over a point $[V_{n+1-r}]\in\mathrm{G}(n+1-r,V)$
is $(V/V_{n+1-r})^{*}$, the fiber of the projective bundle $\widetilde{\textsc{S}}_r\to\mathrm{G}(n+1-r,V)$
over $[V_{n+1-r}]$ is $\mathbb{P}(\ft{S}^{2}(V/V_{n+1-r})^{*})$,
which parameterizes quadrics in $\mathbb{P}(V/V_{n+1-r})\simeq \mathbb{P}^{r-1}$.
The morphism $\mathbb{P}({\ft S}^2 {\mathfrak Q}^*)\to \mathbb{P}({\ft S}^2 V^*)$
sends $\mathbb{P}(\ft{S}^{2}(V/V_{n+1-r})^{*})$ into $\mathbb{P}(\ft{S}^{2}V^{*})$.
Then the image is identified with quadrics in $\mathbb{P}(V)$ which are
singular at $[V_{n+1-r}]$, or equivalently, symmetric matrices whose
kernels contain $[V_{n+1-r}]$.
Therefore the image is $\textsc{S}_r$.
The morphism $\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}\colon \widetilde{\textsc{S}}_r\to \textsc{S}_r$ is
one to one over the locus of matrices of rank $r$ in $\textsc{S}_r$, since
a symmetric matrix of rank $r$ with the kernel $V_{n+1-r}$
{determines} uniquely the corresponding
quadric in $\mathbb{P}(V/V_{n+1-r})$. Hence $\widetilde{\textsc{S}}_r$ is birational
to $\textsc{S}_r$ under {$\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}$.}
Finally, $\widetilde{\textsc{S}}_r$ is smooth since it is a projective bundle, and hence
$\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}$ is a resolution of $\textsc{S}_r$.
The assertion (2) easily follows from the proof of (1).
\end{proof}
Using the Springer type resolution $\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}$,
we can derive several properties of $\textsc{S}_r$.
\noindent $\bullet$ {\bf Dimension.}
Since $\widetilde{\textsc{S}}_r$ is a $\mathbb{P}^{\binom{r+1}{2}-1}$-bundle over $\mathrm{G}(n+1-r,V)$, it holds
\begin{equation}
\label{eq:dimSr}
\dim \textsc{S}_r=\dim \widetilde{\textsc{S}}_r=
\frac{(r+1)r}{2}-1+r(n+1-r).
\end{equation}
\noindent $\bullet$ {\bf Canonical divisor.}
Since $\widetilde{\textsc{S}}_r=\mathbb{P}({\ft S}^2 \mathfrak{Q}^*)$ and
$\det {\ft S}^2 \mathfrak{Q}\simeq \mathcal{O}_{\mathrm{G}(n+1-r,V)}(r+1)$,
we have
\begin{equation}
\label{eq:adjSr1}
K_{\widetilde{\textsc{S}}_r}=-\binom{r+1}{2} M_{\widetilde{\textsc{S}}_r}
-(n-r)L_{\widetilde{\textsc{S}}_r},
\end{equation}
where
$M_{\widetilde{\textsc{S}}_r}$ is the tautological divisor of
$\mathbb{P}({\ft S}^2 \mathfrak{Q}^*)$ and
$L_{\widetilde{\textsc{S}}_r}$ is the pull-back of
$\mathcal{O}_{\mathrm{G}(n+1-r,V)}(1)$.
In the sequel in this subsection, we assume that $r\leq n$.
\noindent $\bullet$ {\bf Exceptional divisor.}
By Proposition \ref{prop:Spr} (2) and $\rho(\widetilde{\textsc{S}}_r/\textsc{S}_r)=1$,
the exceptional locus $E_r$ of $\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}$ is a prime divisor
and the induced map $E_r\to \textsc{S}_{r-1}$ is a $\mathbb{P}^{n+1-r}$-bundle over
$\textsc{S}_{r-1}\setminus \textsc{S}_{r-2}$.
We have
\begin{equation}
\label{eq:Er}
E_r=rM_{\widetilde{\textsc{S}}_r}-2L_{\widetilde{\textsc{S}}_r}.
\end{equation}
Indeed,
note that we may write
$E_r=aM_{\widetilde{\textsc{S}}_r}-bL_{\widetilde{\textsc{S}}_r}$ with some integers $a$ and $b$ since $M_{\widetilde{\textsc{S}}_r}$ and $L_{\widetilde{\textsc{S}}_r}$ generate
$\mathrm{Pic}\, \widetilde{\textsc{S}}_r$.
Let $\mathbb{P}\simeq \mathbb{P}^{n+1-r}$ be the fiber of
$E_r\to \textsc{S}_{r-1}$ over a point of
$\textsc{S}_{r-1}\setminus \textsc{S}_{r-2}$.
Then, by (\ref{eq:adjSr1}) and
$M_{\widetilde{\textsc{S}}_r}|_{\mathbb{P}}=0$,
we have $K_{\widetilde{\textsc{S}}_r}|_{\mathbb{P}}=\mathcal{O}_{\mathbb{P}}(-(n-r))$.
Therefore, using $K_{\mathbb{P}}=
K_{E_r}|_{\mathbb{P}}=(K_{\widetilde{\textsc{S}}_r}+E_r)|_{\mathbb{P}}$,
we obtain
$E_r|_{\mathbb{P}}=\mathcal{O}_{\mathbb{P}}(-2)$.
Thus $b=2$.
We have $a=r$
since the restriction of $E_r$ to a fiber
$\mathbb{P}({\ft S}^2 (V/V_{n+1-r})^*)$
of $\widetilde{\textsc{S}}_r\to \mathrm{G}(n+1-r,V)$
is the locus of singular quadrics in
$\mathbb{P}(V/V_{n+1-r})$, and it is a degree $r$ hypersurface in
$\mathbb{P}({\ft S}^2 (V/V_{n+1-r})^*)$.
\noindent $\bullet$ {\bf Generic Singularity.}
By $E_r|_{\mathbb{P}}=\mathcal{O}_{\mathbb{P}}(-2)$,
we see that
\begin{equation}
\label{equation:SingSr}
\text{${\textsc{S}}_r$ has $\frac 12 (1^{n+2-r})$-singularities along $S_{r-1}\setminus S_{r-2}$},
\end{equation}
hence $\mathrm{Sing}\, \textsc{S}_r=\textsc{S}_{r-1}$.
\noindent $\bullet$ {\bf Discrepancy and Fano index.}
The two equalities (\ref{eq:adjSr1}) and (\ref{eq:Er}) give the following
presentation of
$K_{\widetilde{\textsc{S}}_r}$:
\begin{equation}
\label{eq:adjSr2}
K_{\widetilde{\textsc{S}}_r}{=}_{{\mathbb{Q}}} -\frac{r(n+1)}{2} M_{\widetilde{\textsc{S}}_r}
+\frac{n-r}{2} E_r.
\end{equation}
The pushforward of (\ref{eq:adjSr2}) immediately gives
\begin{equation}
\label{eq:Fanoindex}
K_{{\textsc{S}}_r}=_{\mathbb{Q}} -\frac{r(n+1)}{2} M_{{\textsc{S}}_r}.
\end{equation}
Combining (\ref{eq:adjSr2}) and (\ref{eq:Fanoindex}),
we obtain
\[
K_{\widetilde{\textsc{S}}_r}{=}_{{\mathbb{Q}}} \text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}^*K_{{\textsc{S}}_r}
+\frac{n-r}{2} E_r.
\]
In particular,
$\textsc{S}_r$ has only terminal singularities if $n>r$,
and canonical singularities if $n=r$.
$\textsc{S}_r$ is $\mathbb{Q}$-factorial since $\widetilde{\textsc{S}}_r$ is smooth and
$\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}_r}}$ is a divisorial contraction.
\noindent $\bullet$
{\bf Gorenstein index.}
$K_{\textsc{S}_r}$ is Cartier in case $n-r$ is even.
In case $n-r$ is odd, $2K_{\textsc{S}_r}$ is Cartier while $K_{\textsc{S}_r}$ is not.
Indeed, when $n-r$ is even,
the integral divisor $K_{\widetilde{\textsc{S}}_r}-\frac{n-r}{2} E_r$ is the pull-back of a Cartier
divisor on $\textsc{S}_r$
by the Kawamata-Shokurov base point free theorem.
Then, in this case, the formulas (\ref{eq:adjSr2})
and (\ref{eq:Fanoindex})
mean linear equivalences.
In particular, $K_{\textsc{S}_r}$ is Cartier.
In case $n-r$ is odd, we see the assertion by a similar argument
and (\ref{equation:SingSr}).
\subsection{Double cover $\textsc{T}_r$ of $\textsc{S}_r$ with even $r$}
Throughout in this subsection, we suppose $r$ is even.
When $r$ is even, due to the fact that a quadric of even rank contains
two connected families of maximal linear subspaces in it, the determinantal
locus $\textsc{S}_r$ has a natural double cover. We describe below the double cover by
formulating Springer type morphism.
Note that
any quadric of rank at most $r$ contains
$(n-\frac r2)$-planes.
We will introduce the variety $\textsc{U}_r$ which parameterizes pairs
$([\Pi],[Q])$ of quadrics $Q$ of rank at most $r$
and $(n-\frac r2)$-planes $\mathbb{P}(\Pi)$
such that $\mathbb{P}(\Pi)\subset Q$. To parametrize $(n-\frac r2)$-planes
in $\mathbb{P}(V)$, consider the Grassmannian $\mathrm{G}(n-\frac r2 +1,V)$.
Let \begin{equation}
0\to{\text{\eu W}}_{\frac r2}^{*}\to V^{*}\otimes\mathcal{O}_{\mathrm{G}(n-\frac r2+1,V)}\to\text{\eu U}_{n-\frac r2+1}^{*}\to0\label{eq:Q*S}\end{equation}
be the dual of the universal exact sequence on $\mathrm{G}(n-\frac r2+1,V)$,
where $\text{\eu W}_{\frac r2}$ is the universal quotient bundle of rank $\frac r2$
and $\text{\eu U}_{n-\frac r2+1}$ is the universal subbundle of rank $n-\frac r2+1$.
For brevity, we often omit the subscripts writing them by $\text{\eu U}$ and $\text{\eu W}$.
For an $(n-\frac r2)$-plane $\mathbb{P}(\Pi)\subset\mathbb{P}(V)$, there exists a natural
surjection $\ft{S}^{2}V^{*}\to\ft{S}^{2}H^{0}(\mathbb{P}(\Pi),\mathcal{O}_{\mathbb{P}(\Pi)}(1))$
such that the projectivization of the kernel consists of the quadrics
containing $\mathbb{P}(\Pi)$. By relativizing this surjection over $\mathrm{G}(n-\frac r2+1,V)$,
we obtain the following surjection: $\ft{S}^{2}V^{*}\otimes\mathcal{O}_{\mathrm{G}(n-\frac r2-1,V)}\to\ft{S}^{2}\text{\eu U}^{*}.$
Let $\mathcal{E}^{*}$ be the kernel of this surjection, and consider the
following exact sequence: \begin{equation}
0\to\mathcal{E}^{*}\to\ft{S}^{2}V^{*}\otimes\mathcal{O}_{\mathrm{G}(n-\frac r2+1,V)}\to\ft{S}^{2}\text{\eu U}^{*}\to0.\label{eq:sE0}\end{equation}
Now we set $\textsc{U}_r:=\mathbb{P}(\mathcal{E}^{*})$ and denote by $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\textsc{U}_r}$ the projection
$\textsc{U}_r\to\mathrm{G}(n-\frac r2+1,V)$. By (\ref{eq:sE0}), $\textsc{U}_r$
is contained in $\mathrm{G}(n-\frac r2+1,V)\times\mathbb{P}(\ft{S}^{2}V^{*})$.
Since the fiber of $\mathcal{E}^{*}$ over $[\Pi]$ parameterizes quadrics
in $\mathbb{P}(V)$ containing $\mathbb{P}(\Pi)$, we have \[
\textsc{U}_r=\{([\Pi],[Q])\mid\mathbb{P}(\Pi)\subset Q\}\subset\mathrm{G}(n-\frac r2+1,V)\times\mathbb{P}(\ft{S}^{2}V^{*}). \]
Note that $Q$ in $([\Pi],[Q])\in\textsc{U}_r$ is a quadric of rank at most $r$
since quadrics contain $(n-\frac r2)$-planes only when their ranks are
at most $r$.
Hence the
symmetric determinantal locus $\textsc{S}_r$ is the image of the natural projection $\textsc{U}_r\to\mathbb{P}(\ft{S}^{2}V^{*})$.
Now we let \[
\xymatrix{ & \textsc{U}_r\;\ar[r]^{\;\;\text{\large{\mbox{$\pi\hskip-2pt$}}}\,_{\textsc{U}_r}\;\;} & \;\textsc{T}_r\;\ar[r]^{\;\;\text{\large{\mbox{$\rho\hskip-2pt$}}}\,\,_{\textsc{T}_r}\;\;} & \;\textsc{S}_r}
\]
be the Stein factorization of $\textsc{U}_r\to\textsc{S}_r$. By (\ref{eq:sE0}),
the tautological divisor of $\mathbb{P}(\mathcal{E}^{*})\to\mathrm{G}(n-\frac r2+1,V)$
is nothing but the pull-back of a hyperplane section of $\textsc{S}_r$. We
set \[
M_{\textsc{U}_r}:=\text{\large{\mbox{$\pi\hskip-2pt$}}}\,_{\textsc{U}_r}^{\;*}\circ\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\textsc{T}_r}^{\;*}\mathcal{O}_{\textsc{S}_r}(1).\]
We denote by $\textsc{U}_{r[Q]}$ the fiber of $\textsc{U}_r\to\textsc{S}_r$ over a point
$[Q]\in\textsc{S}_r$.
\begin{prop}
\label{Z_Q}
For a quadric $Q$ of rank $r$, the fiber ${\textsc{U}}_{r[Q]}$
is the orthogonal Grassmannian $\mathrm{OG}(\frac r2,r)$ \textcolor{black}{which
consists of two connected components. }
\end{prop}
\begin{proof} Quadric $Q$ of even rank $r$ induces a non-degenerate
symmetric bilinear form $q$ on the quotient $V/V_{n+1-r}$, where $V_{n+1-r}$
is the $(n+1-r)$-dimensional vector space such that $[V_{n+1-r}]$ is the vertex
of $Q$. Then $(n-\frac r2)$-planes on $Q$ naturally correspond to
the maximal isotropic subspaces in $V/V_{n-r+1}$ with respect to $q$,
which are parameterized by the orthogonal Grassmannian $\mathrm{OG}(\frac r2,r)$.
\end{proof}
\begin{prop}
\label{cla:double} The finite morphism $\textsc{T}_r\to\textsc{S}_r$ is of degree two
and is branched along $\textsc{S}_{r-1}$.
\end{prop}
\begin{proof} By Proposition \ref{Z_Q}, the degree of $\textsc{T}_r\to\textsc{S}_r$
is two since $\textsc{U}_{r[Q]}$ has two connected components for a quadric
$Q$ of rank $r$. If a quadric $Q$ has rank at most $r-1$, the family of $(n-\frac r2)$-planes in $Q$ is connected.
Hence we have the assertion. \end{proof}
By this proposition, we see that $\textsc{T}_r$ parameterizes connected
families of $(n-\frac r2)$-planes in quadrics of rank at most $r$
in $\mathbb{P}(V)$
(cf. Fig.1).
\begin{defn}
We call $\textsc{T}_r$ the {\it double symmetric determinantal locus} of rank at most $r$.
We call a point of $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\textsc{T}_r}^{-1}(\textsc{S}_i\setminus \textsc{S}_{i-1})$
{\it a rank $i$ point} for $1\leq i\leq r$.
\end{defn}
$\textsc{T}_r$ inherits good properties from $\textsc{S}_r$ as follows:
\begin{prop}
\label{cla:ZY}
\begin{enumerate}[$(1)$]
\item
The Picard number of $\textsc{U}_r$ is two and $\text{\large{\mbox{$\pi\hskip-2pt$}}}\,_{\textsc{U}_r}\colon\textsc{U}_r\to\textsc{T}_r$
is a Mori fiber space. In particular, $\textsc{T}_r$ is $\mathbb{Q}$-factorial
and has Picard number one.
\item $\textsc{T}_r$ has only Gorenstein canonical singularities
and $\mathrm{Sing}\, \textsc{T}_r$ is contained in
the inverse image of $\textsc{S}_{r-2}$.
In particular, $\dim \mathrm{Sing}\, \textsc{T}_r$ is smaller than
$\dim \mathrm{Sing}\, \textsc{S}_r$ in case $r\leq n$.
\item
$\textsc{T}_r$ is a Fano variety with
\begin{equation}
\label{eq:FanoindexTr}
K_{{\textsc{T}}_r}= -\frac{r(n+1)}{2} M_{{\textsc{T}}_r},
\end{equation}
where $M_{{\textsc{T}}_r}$ is the pull-back of $\mathcal{O}_{\textsc{S}_r}(1)$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) The Picard number of $\textsc{U}_r$ is two
since $\textsc{U}_r$ is a projective bundle over
$\mathrm{G}(n-\frac r2 +1,V)$. Therefore
the Picard number of $\textsc{T}_r$ is one since the relative Picard number
of $\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\textsc{U}_r}\colon\textsc{U}_r\to\textsc{T}_r$ is one.
$\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\textsc{U}_r}$ is a Mori fiber
space since a general fiber of $\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\textsc{U}_r}$
is a Fano variety by Proposition \ref{Z_Q}.
$\textsc{T}_r$ is $\mathbb{Q}$-factorial by \cite[Lemma 5-1-5]{KMM}.
\noindent(2)
To show the claim (2), we will construct the following commutative diagram:
\begin{equation}
\label{eq:STcomm}
\begin{matrix}
\xymatrix{\widetilde{\textsc{U}}_r\ar[r]^{\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\widetilde{\textsc{U}}_r}}\ar[d]_{\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{U}}_r}} & \widetilde{\textsc{T}}_r\ar[r]^{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_r}}\ar[d]_{\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{T}}_r}} & \widetilde{\textsc{S}}_r\ar[d]_{\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}}\\
\textsc{U}_r\ar[r]_{\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\textsc{U}_r}} & \textsc{T}_r\ar[r]_{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\textsc{T}_r}} & \textsc{S}_r.}
\end{matrix}
\end{equation}
\noindent $\bullet$
$\widetilde{\textsc{U}}_r$ is defined in $\mathrm{G}(\frac r2,{\mathfrak Q})\times_{\mathrm{G}(n+1-r,V)}\mathbb{P}(\ft{S}^{2}{\mathfrak Q}^{*})$, in a similar way to $\textsc{U}_r$, by
\[
\widetilde{\textsc{U}}_r:=\{([\Pi],[Q];[V_{n+1-r}])\mid\mathbb{P}(\Pi)\subset Q\subset \mathbb{P}(V/V_{n+1-r})\}.
\]
\noindent $\bullet$ Then the projection to the second factor yields a morphism
$\widetilde{\textsc{U}}_r\to \widetilde{\textsc{S}}_r$ and
the morphism
$\mathrm{G}(\frac r2,{\mathfrak Q})\times_{\mathrm{G}(n+1-r,V)}\mathbb{P}(\ft{S}^{2}{\mathfrak Q}^{*})
\to
\mathrm{G}(n+1-\frac r2,V)\times \mathbb{P}(\ft{S}^{2}V^{*})$
induces a morphism
$\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{U}}_r}\colon \widetilde{\textsc{U}}_r\to \textsc{U}_r$.
It is easy to see that $\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{U}}_r}$ is a birational morphism.
\noindent $\bullet$ Let
\[
\xymatrix{ & \widetilde{\textsc{U}}_r\;\ar[r]^{\;\;\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\widetilde{\textsc{U}}_r}\;\;} & \;\widetilde{\textsc{T}}_r\;\ar[r]^{\;\;\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{\textsc{T}}_r}\;\;} & \;\widetilde{\textsc{S}}_r}
\]
be the Stein factorization of $\widetilde{\textsc{U}}_r\to\widetilde{\textsc{S}}_r$.
By the definition of Stein factorization,
we have
${\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\widetilde{\textsc{U}}_r}}_*\mathcal{O}_{\widetilde{\textsc{U}}_r}=\mathcal{O}_{\widetilde{\textsc{T}}_r}$
and
${\text{\large{\mbox{$\pi\hskip-2pt$}}}_{{\textsc{U}}_r}}_*\mathcal{O}_{{\textsc{U}}_r}=\mathcal{O}_{{\textsc{T}}_r}$.
Therefore, by
\begin{equation}
\label{eq:long}
{\text{\large{\mbox{$p\hskip-1pt$}}}_{\widetilde{\textsc{S}}_r}}_*{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_r}}_*\mathcal{O}_{\widetilde{\textsc{T}}_r}=
{\text{\large{\mbox{$p\hskip-1pt$}}}_{\widetilde{\textsc{S}}_r}}_*{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_r}}_*{\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\widetilde{\textsc{U}}_r}}_*\mathcal{O}_{\widetilde{\textsc{U}}_r}=
{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\textsc{T}}_r}}_*{\text{\large{\mbox{$\pi\hskip-2pt$}}}_{{\textsc{U}}_r}}_*{\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{U}}_r}}_*\mathcal{O}_{\widetilde{\textsc{U}}_r}={\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\textsc{T}}_r}}_*\mathcal{O}_{{\textsc{T}}_r},
\end{equation}
we see that the Stein factorization of $\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{S}_r}\circ
\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{T}_r}$ is $\widetilde{\textsc{T}}_r\to \textsc{T}_r\to \textsc{S}_r$.
We denote by $\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{T}}_r}\colon \widetilde{\textsc{T}}_r\to \textsc{T}_r$ the induced morphism.
Now we have completed the diagram (\ref{eq:STcomm}).
Similarly to the proof of Proposition \ref{cla:double},
we see that the branch locus of $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{T}_r}\colon \widetilde{\textsc{T}}_r\to \widetilde{\textsc{S}}_r$ is $\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}$-exceptional divisor $E_r$.
Since $\widetilde{\textsc{U}}_r\to \widetilde{\textsc{T}}_r$ is a Mori fiber space,
$\widetilde{\textsc{T}}_r$ has only rational singularities by \cite{Fuj} and
in particular is Cohen-Macaulay.
Therefore $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_r}$ is flat.
First we treat the case where $r=n+1$.
Then $\textsc{T}_{n+1}$ is Gorenstein since it is the double cover of $\textsc{S}_{n+1}=\mathbb{P}({\ft S}^2 V^*)$ branched along the divisor $\textsc{S}_{n}$.
Thus $\textsc{T}_{n+1}$ has only canonical singularities by \cite{Fuj}.
$\mathrm{Sing}\, \textsc{T}_{n+1}$ is contained in the inverse image of $\mathrm{Sing}\, \textsc{S}_n=\textsc{S}_{n-1}$. Now we have verified the assertion (2) in case $r=n+1$.
{Let us} assume that $r\leq n$.
Then, by (\ref{eq:Er}),
it holds that
\begin{equation}
\label{eq:redEr}
\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}_r}}^*(\frac r2 M_{\widetilde{\textsc{S}}_r}-L_{\widetilde{\textsc{S}}_r})\sim (\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}_r}}^*E_r)_{\mathrm{red}}
\end{equation}
and
${\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_r}}_*\mathcal{O}_{{\widetilde{\textsc{T}}_r}}
=\mathcal{O}_{{\widetilde{\textsc{S}}_r}}\oplus \mathcal{O}_{{\widetilde{\textsc{S}}_r}}(-\frac r2 M_{\widetilde{\textsc{S}}_r}+L_{\widetilde{\textsc{S}}_r}).$
By (\ref{eq:long}),
we see that
${\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\textsc{T}}_r}}_*\mathcal{O}_{{\textsc{T}}_r}
=\mathcal{O}_{{\textsc{S}}_r}\oplus \mathcal{O}_{{\textsc{S}}_r}(-\frac r2 M_{{\textsc{S}}_r}+L_{{\textsc{S}}_r})$
with $L_{{\textsc{S}}_r}:=\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\widetilde{\textsc{S}}_r *} L_{\widetilde{\textsc{S}}_r}$
and
\begin{equation}
\label{eq:specT}
{\textsc{T}}_r=\mathrm{Spec}\,_{{\textsc{S}}_r}
\big(\mathcal{O}_{{\textsc{S}}_r}\oplus \mathcal{O}_{{\textsc{S}}_r}(-\frac r2 M_{{\textsc{S}}_r}+L_{{\textsc{S}}_r})\big).
\end{equation}
Pushing (\ref{eq:redEr}) forward by $p_{\widetilde{\textsc{T}}_r}$,
we obtain
\begin{equation}
\label{eq:triv}
\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\textsc{T}_r}}^*(\frac r2 M_{{\textsc{S}}_r}-L_{{\textsc{S}}_r})\sim 0.
\end{equation}
In particular,
$\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\textsc{T}}_r}^* L_{{\textsc{S}}_r}$ is Cartier
since so is $M_{{\textsc{S}}_r}$.
Therefore $K_{\textsc{T}_r}$ is Cartier by (\ref{eq:adjSr1})
and the formula $K_{\textsc{T}_r}=\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\textsc{T}_r}^*K_{\textsc{S}_r}$.
Namely, $\textsc{T}_r$ is Gorenstein.
To show that $\textsc{T}_r$ has only canonical singularities,
let $f\colon \widetilde{\textsc{R}}_r\to \widetilde{\textsc{T}}_r$ be a resolution.
Then, by the ramification formula, we have $K_{\widetilde{\textsc{R}}_r}\geq
f^*\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_r}^* K_{\widetilde{\textsc{S}}_r}$.
Since $\textsc{S}_r$ has only canonical singularities, we have
$K_{\widetilde{\textsc{S}}_r}\geq \text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}^* K_{{\textsc{S}}_r}$.
Therefore
\[
K_{\widetilde{\textsc{R}}_r}\geq
f^*\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_r}^* K_{\widetilde{\textsc{S}}_r}\geq
f^*\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_r}^* \text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_r}^*K_{{\textsc{S}}_r}=
f^* \text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{T}}_r}^*\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\textsc{T}}_r}^*K_{{\textsc{S}}_r}
=f^* \text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{T}}_r}^*K_{{\textsc{T}}_r}.
\]
This means that $\textsc{T}_r$ has only canonical singularities.
By (\ref{equation:SingSr}) and (\ref{eq:specT}),
we see that
$\textsc{T}_r$ is smooth at the inverse image of
a rank $r-1$ point $s \in \textsc{S}_r$
since $L_{\textsc{S}_r}$ generates the divisor class group at $s$
and then $(\ref{eq:specT})$ coincides with punctured universal cover
near $s$.
(3)
If $r=n+1$, then
the canonical divisor of $\textsc{T}_r$ is given by \[
-\binom{n+2}{2} M_{\textsc{T}_r}+\frac {n+1}2 M_{\textsc{T}_r}=
-\frac{(n+1)^2}{2} M_{\textsc{T}_r}
\]
since the degree of the branch locus $\textsc{S}_n$ is $n+1$.
If $r\leq n$, then
the assertion follows from $K_{\textsc{T}_r}=\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\textsc{T}_r}^*K_{\textsc{S}_r}$,
(\ref{eq:adjSr1}) and (\ref{eq:triv}).
\end{proof}
\begin{rem}
It is useful to consider that
$\widetilde{\textsc{T}}_r\to\widetilde{\textsc{S}}_r$ as in the diagram (\ref{eq:STcomm})
is the family over $\mathrm{G}(n+1-r,V)$ of the double cover
$\textsc{T}_r\to \textsc{S}_r$ for $r$-dimensional vector spaces $V/V_{n+1-r}$ with
$[V_{n+1-r}]\in \mathrm{G}(n+1-r,V)$.
\end{rem}
\subsection{Dual situations and orthogonal linear sections}
To consider
projective duality for the symmetric determinantal loci
in $\mathbb{P}({\ft S}^2 V^*)$,
the symmetric determinantal loci in
$\mathbb{P}({\ft S}^2 V)$ naturally appear.
{Recall that we denote by $\textsc{S}^*_r$ the symmetric determinantal
locus of rank at most $r$ in $\mathbb{P}({\ft S}^2 V)$. Similarly to $\textsc{S}_r$,
$\textsc{S}^*_1$ is the second Veronese variety $v_2(\mathbb{P}(V))$ and
$\textsc{S}^*_r$ is the $r$-secant variety of $\textsc{S}^*_1$. Corresponding to our
definitions $\textsc{U}_r, \textsc{T}_r$ and $\widetilde{\textsc{S}_r}$ for $\textsc{S}_r$ in
$\mathbb{P}(\ft{S}^2V^*)$, we have similar definitions $\textsc{U}_r^*, \textsc{T}_r^*$ and
$\widetilde{\textsc{S}_r^*}$ for $\textsc{S}_r^*$ in
$\mathbb{P}(\ft{S}^2V)$.}
For a linear subspace $L_{k+1}\subset {\ft S}^2 V^*$ of dimension $k+1$,
we say that $\textsc{S}_r\cap \mathbb{P}(L_{k+1})$ is a {\it linear section} of
$\textsc{S}_r$ if $\textsc{S}_r\cap \mathbb{P}(L_{k+1})$ is of codimension
$\dim {\ft S}^2 V^*-(k+1)$ in $\textsc{S}_r$. {Linear sections of $\textsc{S}^*_r$ is
defined for linear subspaces in ${\ft S}^2 V$ in a
similar way. }
Let $L_{k+1}^{\perp} \subset {\ft S}^2 V$ be the linear subspace
orthogonal to $L_{k+1}$ with respect to the dual pairing.
For a triple $(\textsc{S}_r,\textsc{S}_s^*,L_{k+1})$, we say
that linear sections
$\textsc{S}_r\cap \mathbb{P}(L_{k+1})$ and $\textsc{S}^*_s\cap \mathbb{P}(L_{k+1}^{\perp})$
are mutually {\it orthogonal}.
By slight abuse of terminology, we also call the pull-back
of a linear section of $\textsc{S}_r$ by the double cover $\textsc{T}_r\to \textsc{S}_r$
{\it a linear section of $\textsc{T}_r$}.
\section{{\bf Pairs of Calabi-Yau sections and plausible duality}}
In this paper, we adopt the following definition of Calabi-Yau variety
and also Calabi-Yau manifold.
\begin{defn}
A normal projective variety $X$ is called \textit{a Calabi-Yau variety}
if $X$ has only Gorenstein canonical singularities, and its canonical
divisor is trivial and $h^{i}(\mathcal{O}_{X})=0$ for $0<i<\dim X$.
If $X$ is smooth, then $X$ is called \textit{a Calabi-Yau manifold}.
A smooth Calabi-Yau threefold is abbreviated as a Calabi-Yau threefold.
\end{defn}
\subsection{Calabi-Yau linear section of $\textsc{S}_r$}
\begin{prop}
\label{prop:CYSr}
Assume that
$n-r$ is even and $r<n+1$.
Then a general linear section $\textsc{S}_r^{\textsc{CY}}$
of codimension $\frac{r(n+1)}{2}$
is a Calabi-Yau variety
of dimension $\frac{r(n+2-r)}{2}-1$
with only terminal $($resp.~canonical\,$)$ singularities if $r<n$ $($resp.~$r=n)$.
Moreover, a general $\textsc{S}_r^{\textsc{CY}}$ is smooth
if and only if
$r\leq 2$.
\end{prop}
\begin{proof}
$\textsc{S}_r^{\textsc{CY}}$
has trivial canonical divisor
by (\ref{eq:Fanoindex}) since $K_{\textsc{S}_r}$ is Cartier
in case $n-r$ is even.
Since $\textsc{S}_r$ has only terminal (resp.~canonical) singularities
in case $r<n$ (resp.~$r=n$) and
is a Fano variety as we saw in
the subsection \ref{subsection:Spr},
it holds that $h^i(\mathcal{O}_{\textsc{S}_r})=0$ for any $i>0$
and $h^i(\mathcal{O}_{\textsc{S}_r}(-jM_{\textsc{S}_r}))=0$ for any $i<\dim \textsc{S}_r$ and
$j>0$
by the Kodaira-Kawamata-Viehweg vanishing theorem.
Therefore we have
$h^i(\mathcal{O}_{\textsc{S}_r^{\textsc{CY}}})=0$ for any $0<i<\dim \textsc{S}_r^{\textsc{CY}}$
by the Koszul complex.
By a version
of the Bertini theorem (cf.~\cite[Prop.~0.8]{And}),
a general
$\textsc{S}_r^{\textsc{CY}}$ has only
terminal (resp.~canonical) singularities
in case $r<n$ (resp.~$r=n$).
Therefore a general
$\textsc{S}_r^{\textsc{CY}}$ is a Calabi-Yau variety.
Since $r<n+1$,
$\mathrm{Sing}\, \textsc{S}_r=\textsc{S}_{r-1}$.
Thus the second assertion is equivalent to
that
$\dim \textsc{S}_{r-1}=\frac{r(r-1)}{2}-1+(r-1)(n+2-r)<\frac{r(n+1)}{2}$ holds
if and only if $r\leq 2$.
A proof of this claim is elementary.
\end{proof}
\begin{rem}
\label{rem:CYSr}
In case $n-r$ is odd,
we can show the following by the same argument as
in the proof of Proposition \ref{prop:CYSr}:
Linear sections of $\textsc{S}_r$ of codimension $\frac {r(n+1)}{2}$
does not have trivial canonical divisors but
bi-canonical divisors are trivial.
Except this, the same properties as $\textsc{S}_r^{\textsc{CY}}$
hold for them.
\end{rem}
By {the above} proposition,
we observe that
\begin{equation}
\label{eq:ob}
\dim
\textsc{S}_r^{\textsc{CY}}=
\dim \textsc{S}_{n+2-r}^{\textsc{CY}}=
\dim \textsc{S}_{n+2-r}^{* \textsc{CY}}.
\end{equation}
This indicates certain duality between
$\textsc{S}_r$ and $\textsc{S}_{n+2-r}^*$.
We will discuss this duality in Subsection \ref{subsection:Pl}.
If $r=1$,
then $\textsc{S}_1$ is isomorphic to
the second Veronese variety $v_2(\mathbb{P}(V))$.
Therefore its linear sections are
complete intersections of quadrics in $\mathbb{P}(V)$.
In the next subsection,
we adopt the dual setting and consider $\textsc{S}^*_2$ and
its linear sections $\textsc{S}_2^{*\textsc{CY}}$
in detail.
\subsection{Rank two case and Calabi-Yau manifold $X$ of a Reye congruence}
\label{subsection:ranktwo}
{Consider the determinant locus $\textsc{S}_2^*$ in $\mathbb{P}(\ft{S}^2V)$ and also
$\textsc{U}_2^*,\textsc{T}_2^*,\widetilde{\textsc{S}_2^*}$ defined in the same way as $\textsc{U}_2,\textsc{T}_2,
\widetilde{\textsc{S}_2}$ for $\textsc{S}_2$ in $\mathbb{P}(\ft{S}^2V^*)$. Note that $\textsc{U}^*_2\simeq \textsc{T}^*_2$ holds in this case.}
Let us {write} the exact sequence (\ref{eq:sE0}) for $\textsc{S}^*_2$
{by} noting that $\mathrm{G}(n,V^*)=\mathbb{P}(V)$ and $\text{\eu U}=\Omega^1_{\mathbb{P}(V)}$:
\begin{equation}
\label{eq:Er=2}
0\to\mathcal{E}^{*}\to\ft{S}^{2}V\otimes\mathcal{O}_{\mathbb{P}(V)}\to\ft{S}^{2}T_{\mathbb{P}(V)}(-1)\to0.
\end{equation}
\begin{prop}
\label{prop:sEtriv}
$\mathcal{E}\simeq V^*\otimes \mathcal{O}_{\mathbb{P}(V)}(1)$.
\end{prop}
\begin{proof}
Taking fibers of (\ref{eq:Er=2}) at a point $[V_1]\in \mathbb{P}(V)$,
we obtain the exact sequence
$0\to V\otimes V_1\to {\ft S}^2 V\to {\ft S}^2 (V/V_1)\to 0$.
Therefore the fiber of $\mathcal{E}^*$ at $[V_1]$ is $V\otimes V_1$,
which show the claim.
\end{proof}
Therefore it holds that
\[
\textsc{T}_2^*\simeq \textsc{U}^*_2:=\mathbb{P}(\mathcal{E}^*)\simeq \mathbb{P}(V)\times \mathbb{P}(V).
\]
Moreover, by the proof of Proposition \ref{prop:sEtriv},
we see that
the map $\textsc{T}^*_2\to \mathbb{P}({\ft S}^2 V)$
is given by
$\mathbb{P}(V)\times \mathbb{P}(V)\ni ([{\bf v}],[{\bf w}]) \mapsto
[{\bf v}\otimes {\bf w}+{\bf w}\otimes {\bf v}]\in \mathbb{P}({\ft S}^2 V)$.
Therefore $\textsc{S}^*_2$, which is the image of this map, is nothing but
the symmetric product
${\ft S}^2 \mathbb{P}(V)$.
In \cite{Arxiv}, we show that,
by identifying ${\ft S}^2 \mathbb{P}(V)$ with the Chow variety of
degree two $0$-cycles in $\mathbb{P}(V)$ (cf.~\cite{GKZ}),
$\widetilde{\textsc{S}^*_2}$ is isomorphic to the Hilbert scheme of
length two subschemes in $\mathbb{P}(V)$, and
the Springer resolution $\widetilde{\textsc{S}^*_2}\to \textsc{S}^*_2$ coincides with
the Hilbert-Chow morphism.
{For brevity of notation, we fix the following definitions
in what follows:}
\[
{\mathscr X}:=\textsc{S}^*_2
\;\text{ and }\;
X:=\text{a codimension $n+1$ linear section of } \textsc{S}_2^*.
\]
In \cite{Ol} (see also \cite{Arxiv}),
a general $X$ is called a
\textit{Reye congruence} since it is isomorphic to
a $(n-1)$-dimensional subvariety of $\mathrm{G}(2,V)$.
By Proposition \ref{prop:CYSr} and Remark \ref{rem:CYSr},
Reye congruence $X$ is a Calabi-Yau variety when
$n$ is even; when $n$ is odd, $X$ has similar properties except that
$2K_X\sim 0$. In particular, when $n=3$, $X$ is an
Enriques surface (see \cite{Co}).
The proof of the following proposition
is standard, so we omit it here (cf.~\cite{Arxiv}).
\begin{prop}
For a general $X$, it holds that
\[
\pi_{1}(X)\simeq\mathbb{Z}_{2},\;\;\mathrm{Pic}\, X\simeq\mathbb{Z}\oplus\mathbb{Z}_{2},\]
where the free part of $\mathrm{Pic}\, X$ is generated by the class $D$ of
a hyperplane section of ${\mathscr X}$ restricted to $X$.
\end{prop}
When $n=4$, $X$ is a Calabi-Yau threefold with the following invariants
\cite[Proposition 2.1]{HoTa1}: \[
\deg(X)=35,\;\; c_{2}.D=50,\;\; h^{2,1}(X)=26,\; h^{1,1}(X)=1,\]
where $c_{2}$ is the second Chern class of $X$.
\subsection{Calabi-Yau linear section of $\textsc{T}_r$}
In this subsection, we assume that $r$ is even.
\begin{prop}
\label{prop:CYTr}
A general linear section $\textsc{T}_r^{\textsc{CY}}$
of codimension $\frac{r(n+1)}{2}$
is a Calabi-Yau variety
{of dimension $\frac{r(n+2-r)}{2}-1$
with only canonical singularities.}
Moreover, a general $\textsc{T}_r^{\textsc{CY}}$ is smooth
if
$r\leq 4$.
\end{prop}
\begin{proof}
By (\ref{eq:FanoindexTr}),
$\textsc{T}_r^{\textsc{CY}}$
has trivial canonical divisor.
Since $\textsc{T}_r$ is a Fano variety with only canonical singularities
by Proposition \ref{cla:ZY},
We can show that
$h^i(\mathcal{O}_{\textsc{T}_r^{\textsc{CY}}})=0$ for any $0<i<\dim \textsc{T}_r^{\textsc{CY}}$,
and a general
$\textsc{T}_r^{\textsc{CY}}$ has only canonical singularities
{in} the same way as in the proof of Proposition \ref{prop:CYSr}.
Therefore a general
$\textsc{T}_r^{\textsc{CY}}$ is a Calabi-Yau variety.
Since
$\mathrm{Sing}\, \textsc{T}_r$ is contained in the inverse image of $\textsc{S}_{r-2}$
by Proposition \ref{cla:ZY} (2),
the second assertion follows once we show
that
$\dim \textsc{S}_{r-2}=\frac{(r-1)(r-2)}{2}-1+(r-2)(n+3-r)<\frac{r(n+1)}{2}$
{holds} if and only if
$r\leq 4$.
A proof of the latter is elementary.
\end{proof}
We have already studied $\textsc{T}_2^{\textsc{CY}}$ in the subsection
\ref{subsection:ranktwo}.
We deal with $\textsc{T}_4^{\textsc{CY}}$ in detail in the subsection \ref{subsection:CY3Y}.
\subsection{Rank four case and Calabi-Yau manifold $Y$}
\label{subsection:CY3Y}
For brevity of notation, we introduce the following definitions:
\[
{\mathscr H}:=\textsc{S}_4,\quad {\mathscr U}:=\widetilde{\textsc{S}}_4,\quad {\mathscr Y}:=\textsc{T}_4,
\quad {\mathscr Z}:=\textsc{U}_4,
\]
{while retaining} the notation $\textsc{S}_1,\textsc{S}_2,\textsc{S}_3\subset {\mathscr H}$.
{We denote by ${\mathscr Z}_{[Q]}$ the fiber of the morphism ${\mathscr Z}\to {\mathscr H}$ over
a point $[Q]$. Recall that $\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\textsc{U}_4}=\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\mathscr Z}:{\mathscr Z} \to {\mathscr Y}$ is defined
by the Stein factorization ${\mathscr Z}\to{\mathscr Y}\to{\mathscr H}$ of ${\mathscr Z}\to{\mathscr H}$.}
\def\resizebox{11cm}{!}{\includegraphics{Quad.eps}}{\resizebox{11cm}{!}{\includegraphics{Quad.eps}}}
\def\resizebox{11cm}{!}{\includegraphics{Quad.eps}}Display{
\begin{xy}
(0,0)*{\resizebox{11cm}{!}{\includegraphics{Quad.eps}}},
(-45,12)*{\mathbb{P}(V_{n-3})},
(-16,12)*{\mathbb{P}(V_{n-2})},
( 11,12)*{\mathbb{P}(V_{n-1})},
( 40,12)*{\mathbb{P}(V_n)},
(-45,-12)*{\mathbb{P}^1 \sqcup \mathbb{P}^1},
(-16,-12)*{\mathbb{P}^1},
( 11,-12)*{\mathbb{P}^{n-1}\sqcup_{1pt}\mathbb{P}^{n-1}},
( 40,-12)*{2\,\mathbb{P}^{n-1}} \end{xy} }
\[
\resizebox{11cm}{!}{\includegraphics{Quad.eps}}Display\]
\begin{fcaption}
\item \textbf{Fig.1. Quadrics $Q$ of rank at most four in $\mathbb{P}(V)$ and families of $(n-2)$-planes
therein.} The singular loci of $Q$ are written by $\mathbb{P}(V_{k})$ with
$k=n+1-{\rm {rk}\, Q}$. Also the parameter spaces of the planes in
each $Q$ are shown ($\mathbb{P}^{n-1}\sqcup_{1pt}\mathbb{P}^{n-1}$ represents the
union of $\mathbb{P}^{n-1}$'s intersecting at one point). See also Fig.2 in
the subsection \ref{sub:tildeY-Y}. \end{fcaption}
\begin{prop}
\label{cla:double2} If $\mathrm{rank}\, Q=4$, then ${\mathscr Z}_{[Q]}$ is a disjoint union of two smooth rational curves, {each of which}
is identified
with a conic in $\mathrm{G}(n-1,V)$. If $\mathrm{rank}\, Q=3$, then ${\mathscr Z}_{[Q]}$
is a smooth rational curve, which is also identified with a conic
in $\mathrm{G}(n-1,V)$. If $\mathrm{rank}\, Q=2$, then ${\mathscr Z}_{[Q]}$ is the
union of two $\mathbb{P}^{n-1}$'s intersecting at one point. If $\mathrm{rank}\, Q=1$,
then ${\mathscr Z}_{[Q]}$ is a $($non-reduced\,$)$ $\mathbb{P}^{n-1}$.
In particular, $\text{\large{\mbox{$\pi\hskip-2pt$}}}_{{\mathscr Z}}\colon {\mathscr Z}\to {\mathscr Y}$ is generically a conic bundle.
\end{prop}
\begin{proof} If $\mathrm{rank}\, Q=4$, the fiber ${\mathscr Z}_{[Q]}$ consists of
two disconnected components, and is isomorphic to the orthogonal Grassmannian
$\mathrm{OG}(2,4)$ by Proposition \ref{Z_Q}. To be more explicit, let $\mathbb{P}(V_{n-3})\subset\mathbb{P}(V)$
be the vertex of $Q$. Then the quadric $Q$ is the cone over $\mathbb{P}^{1}\times\mathbb{P}^{1}$
with the vertex $\mathbb{P}(V_{n-3})$. There are two distinct $\mathbb{P}^{1}$-families
of lines in $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Each of the families can be understood
as the corresponding conic in $\mathrm{G}(2,V/V_{n-3})$, which gives
one of the connected components of $\mathrm{OG}(2,4)$. Under the natural
map $\mathrm{G}(2,V/V_{n-3})\rightarrow\mathrm{G}(n-1,V)$, we have two
$\mathbb{P}^{1}$- families of $2$-planes in $Q$ parameterized by the conics
in $\mathrm{G}(n-1,V)$.
If $\mathrm{rank}\, Q=3$, the vertex of the quadric $Q$ is a $\mathbb{P}(V_{n-2})\subset\mathbb{P}(V)$.
The quadric $Q$ is the cone over a conic with the vertex $\mathbb{P}(V_{n-2})$.
The conic is contained in $\mathbb{P}(V/V_{n-2})=\mathrm{G}(1,V/V_{n-2})$, and
can be identified with a conic in $\mathrm{G}(n-1,V)$ under the natural
map $\mathrm{G}(1,V/V_{n-2})\rightarrow\mathrm{G}(n-1,V)$.
If $\mathrm{rank}\, Q=2$, then the quadric $Q$ has a vertex $\mathbb{P}(V_{n-1})\subset\mathbb{P}(V)$
and is the union of two $(n-1)$-planes intersecting along the $(n-2)$-plane
$\mathbb{P}(V_{n-1})$. Hence ${\mathscr Z}_{[Q]}\subset\mathrm{G}(n-1,V)$ is given by
the union of the corresponding $\mathbb{P}^{n-1}$'s, i.e., $\mathrm{G}(n-1,n)$'s
in $\mathrm{G}(n-1,V)$, which intersect at one point $\mathbb{P}(V_{n-1})$.
If $\mathrm{rank}\, Q=1$, then $Q$ is a double $(n-1)$-plane. Thus ${\mathscr Z}_{[Q]}$
is a (non-reduced) $\mathbb{P}^{n-1}\cong\mathrm{G}(n-1,n)$. \end{proof}
We write by $G_{{\mathscr Y}}^{1}$ (resp.~$G_{{\mathscr Y}}^{2}$, $G_{{\mathscr Y}}$) the inverse
image under $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\mathscr Y}}$ of $\textsc{S}_1$
(resp.~$\textsc{S}_2\setminus \textsc{S}_1$, $\textsc{S}_2$). We note that $G_{{\mathscr Y}}\simeq \textsc{S}_1\simeq \ft{S}^{2}\mathbb{P}(V^{*})$ and $G_{{\mathscr Y}}^{1}\simeq \textsc{S}_2 \simeq v_{2}(\mathbb{P}(V^{*}))$
since
$\textsc{S}_2$ is contained in the branch locus of $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\mathscr Y}}$. Using these,
we summarize our construction above in the following diagram:
\def\xyHYZG{ \begin{matrix}
\begin{xy}
(35,0)*+{{\mathscr Z}}="Z",
(65,0)*+{\mathrm{G}(n-1,V)}="G",
(35,-13)*+{{\mathscr Y}}="Y",
(35,-26)*+{\ {\mathscr H},}="H",
(18,-13)*+{G_{\mathscr Y}^1\;\;\subset \;\;\;G_{\mathscr Y} \;\subset\;\;},
(15,-20)*+{\vcorr{-90}{$\simeq$} \;\qquad\;\; \vcorr{-90}{$\simeq$}},
(15,-26)*+{v_2(\mathbb{P}(V^*))\subset \ft{S}^2\mathbb{P}(V^*) \;\subset\ \ \;},
\ar^{\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\mathscr Z}} "Z";"Y"
\ar^{\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\mathscr Y}} "Y";"H"
\ar_{\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\mathscr Z}\;\;}^{\;_{\text{proj.~bundle}}\;\;\;\;\;\quad} "Z";"G"
\end{xy} \end{matrix} }
\begin{equation}
\xyHYZG\label{eq:Z}\end{equation}
where $\text{\large{\mbox{$\pi\hskip-2pt$}}}_{{\mathscr Z}}$ is a $\mathbb{P}^{1}$-fibration over ${\mathscr Y}\setminus G_{{\mathscr Y}}$
by Proposition \ref{cla:double2}. In Section \ref{section:BirY},
we will construct a nice desingularization $\widetilde{{\mathscr Y}}$ of
${\mathscr Y}$. Also, in Sections \ref{section:BirY} and \ref{section:FY},
we will study the geometry of $\widetilde{{\mathscr Y}}\to{\mathscr Y}$
along the loci $G_{{\mathscr Y}}$ and $G_{{\mathscr Y}}^{1}$ in full detail.
Now {consider the linear section of ${\mathscr Y}=T_4$ and we set}
\[
Y:=\textsc{T}_4^{\textsc{CY}}.
\]
By Proposition \ref{prop:CYTr},
a general $Y$ is a Calabi-Yau manifold of dimension $2n-5$.
{
By using the fibration $\L\pi_{{\mathscr Z}}\colon {\mathscr Z}\to {\mathscr Y}$,
it is possible to compute several invariants of $Y$.
Computations have been done
for the case $n=4$ in \cite[Prop.3.11 and Prop.3.12]{HoTa1}, \cite{Arxiv},
which we summarize below:}
\begin{prop}
\label{prop:Y} A general $Y$ is a simply connected smooth Calabi-Yau
$3$-fold such that $\mathrm{Pic}\, Y=\mathbb{Z}[M]$, ${M}^{3}=10$, $c_{2}(Y).{M}=40$
and $e(Y)=-50$. In particular, $h^{1,1}(Y)=1$ and $h^{1,2}(Y)=26$.
\end{prop}
{ It should be noted here that the Spec construction
(\ref{eq:specT}) of $\textsc{T}_4={\mathscr Y}$ generalizes the covering
constructed in \cite[eq.(3.4)]{HoTa1} for $n=4$.}
\hspace{5pt}
In the following two subsections, we discuss
two plausible dualities between
$\textsc{S}^*_a$ and $\textsc{T}_b$ for certain pairs of $a$ and $b$.
\subsection{Linear duality and beyond}
The exact sequence (\ref{eq:sE0}) means that
the fibers of ${\ft S}^2 \text{\eu U}$ and $\mathcal{E}^*$ over a point of
$\mathrm{G}(n+1-\frac r2,V)$
are the orthogonal spaces to each other when we consider them
as subspaces in ${\ft S}^2 V$ and ${\ft S}^2 V^*$, respectively.
The pair ${\ft S}^2 \text{\eu U}$ and $\mathcal{E}^*$ is an example of
{\it orthogonal bundles}.
In \cite[\S 8]{HPD1}, Kuznetsov has established
the homological projective duality between
a projective bundle $\mathbb{P}(\mathcal{V})$ over a smooth base $S$
and its orthogonal bundle $\mathbb{P}(\mathcal{V}^{\perp})$ for a
globally generated vector bundle $\mathcal{V}$ on $S$.
He has called this duality {\it linear duality} in \cite{ICM}.
{Due to this general result, we know that} $\mathbb{P}({\ft S}^2 \text{\eu U})$
and $\mathbb{P}(\mathcal{E}^*)$ are homological projective dual. Note that $\mathbb{P}({\ft S}^2 \text{\eu U})=\widetilde{\textsc{S}^*}_{n+1-\frac r2}$ and $\mathbb{P}(\mathcal{E}^*)=\textsc{U}_r$.
Mutually orthogonal linear sections $X$ and $Z$
of $\mathbb{P}({\ft S}^2 \text{\eu U})$ and $\mathbb{P}(\mathcal{E}^*)$
of codimensions $\mathrm{rank}\, {\ft S}^2 \text{\eu U}$ and $\mathrm{rank}\, \mathcal{E}^*$ respectively
{have the equal dimensions,}
$\dim \mathrm{G}(n+1-\frac r2,V)-1=\frac r2 (n+1-\frac r2)-1$,
and are derived equivalent by \cite[\S 8]{HPD1}.
Let $Y$ be the double cover of the image of $Z$ on $\mathbb{P}({\ft S}^2 V^*)$.
{The derived equivalence between $X$ and $Z$}
indicates that {there is some} relationship between
non-commutative resolutions of $\mathcal{D}^b(X)$ and $\mathcal{D}^b(Y)$.
{Indeed, in \cite{ReyeEnr},
we have shown that this is the case when $n=3$ and $r=4$.}
Note that in this case,
a general $X$ is a so-called Enriques-Fano threefold and
a general $Y$ is a del Pezzo surface of degree two [ibid.].
{
In this case (of $n=3$ and $r=4$),
we can also investigate the derived categories of mutually orthogonal
linear sections of $\textsc{S}^*_2$ and $\textsc{T}_4$ for a triple $(\textsc{S}_4,\textsc{S}_2^*,L_4)$,
which define, respectively, an Enriques surface of Reye congruence and
Artin Mumford double solid. In \cite{DerSym}, we have found natural
Lefschetz collections, which indicates that certain non-commutative
resolutions of $\textsc{S}^*_2$ and $\textsc{T}_4$ are homological projective dual
to each other. One may suspect that, with finding
suitable Lefschetz collections, non-commutative resolutions of
$\textsc{S}^*_{n+1-\frac r2}$ and $\textsc{T}_r$ are homologically projective dual
to each other in general.}
\subsection{Plausible duality}
\label{subsection:Pl}
Assume that $r$ is even. Then $n-(n+2-r)$ is also even.
Therefore we obtain mutually orthogonal
Calabi-Yau linear sections $\textsc{S}^{* \textsc{CY}}_{n+2-r}$ and $\textsc{T}^{\textsc{CY}}_r$ by Propositions \ref{prop:CYSr} and \ref{prop:CYTr}.
We suspect an equivalence of
the derived categories of
certain non-commutative resolutions of
orthogonal linear sections
$\textsc{S}^{* \textsc{CY}}_{n+2-r}$ and $\textsc{T}^{\textsc{CY}}_r$
rather than
$\textsc{S}^{* \textsc{CY}}_{n+2-r}$ and $\textsc{S}^{\textsc{CY}}_r$.
More generally, we speculate
that
certain non-commutative resolutions of
$\textsc{S}^*_{n+2-r}$ and $\textsc{T}_r$ with suitable Lefschetz collections {for each}
are homologically projective dual.
In fact, this is established in case $r=n+1$ \cite{Quad}
({called} Veronese-Clifford duality).
Note that in case $n=r=4$,
both {$\textsc{S}^{* \textsc{CY}}_{2}=X$ and $\textsc{T}^{\textsc{CY}}_4=Y$}
are smooth, and hence they are of considerable interest.
In \cite{DerSym}, we {have} constructed
(dual) Lefschetz collections in the derived categories
of $\widetilde{\textsc{S}}^*_{2}$ and $\textsc{T}_4$, and
{have proved} the derived equivalence between
$\textsc{S}^{* \textsc{CY}}_{2}$ and $\textsc{T}^{\textsc{CY}}_4$ in \cite{HoTa3}
using the properties of these collections.
{Having these applications in mind, in the rest of this paper,
we study the birational geometry of ${\mathscr Y}=\textsc{T}_4$ for general $n$.
Since we will be concentrated on the case $r=4$, we will extensively
use the notation introduced in the beginning of
the subsection \ref{subsection:CY3Y}. }
\section{{\bf Birational geometry of ${\mathscr Y}$}}
\label{section:BirY}
Proposition \ref{cla:double2} indicates {a correspondence}
between points in ${\mathscr Y}$ and conics in $\mathrm{G}(n-1,V)$.
In this section, we explicitly construct a birational map between ${\mathscr Y}$
and the Hilbert scheme ${\mathscr Y}_0$
of conics in $\mathrm{G}(n-1,V)$.
\subsection{Conics and planes in $\mathrm{G}(n-1,V)$\label{sub:Conics-and-planes}}
Let $q$ be a conic in $\mathrm{G}(n-1,V)$ and $\mathbb{P}_q$ the plane spanned by $q$.
Noting that $\mathrm{G}(n-1, V)$ is the intersection of the
Pl\"ucker quadrics in $\mathbb{P}(\wedge^{n-1} V)$,
we see that either
$\mathbb{P}_q\subset \mathrm{G}(n-1,V)$ or
$\mathrm{G}(n-1,V)\cap \mathbb{P}_q=q$ holds for $\mathbb{P}_q$.
When $\mathbb{P}_q\subset \mathrm{G}(n-1,V)$,
we note that there are exactly two types of planes contained in
$\mathrm{G}(n-1,V)\subset \mathbb{P}(\wedge^{n-1} V)$:\begin{equation}
\begin{alignedat}{2}{\rm P}_{V_{n-2}}:= & \{[\Pi]\in\mathrm{G}(n-1,V)\mid V_{n-2}\subset\Pi\}\cong\mathbb{P}^{2} & (\rho\text{-plane}),\\
{\rm P}_{V_{n-3}V_{n}}:= & \{[\Pi]\in\mathrm{G}(n-1,V)\mid V_{n-3}\subset\Pi\subset V_{n}\}\cong\mathbb{P}^{2} & (\sigma\text{-plane})\end{alignedat}
\label{eq:xxxxx}\end{equation}
with some $V_{n-2}\subset V$ and $V_{n-3}\subset V_{n}\subset V$, respectively.
As displayed above, we call these planes \textit{$\rho$-plane} and
\textit{$\sigma$-plane}, respectively.
It is easy to deduce the following proposition:
\begin{prop}
\label{prop:barPrho-barPsigma} In $\mathrm{G}(3,\wedge^{n-1}V)$,
the set of $\rho$-planes $\overline{{\mathscr P}}_{\rho}$
and
the set of $\sigma$-planes $\overline{{\mathscr P}}_{\sigma}$
are given by
\[
\begin{aligned}
\overline{{\mathscr P}}_{\rho} & = \left\{ \big[\left(V/V_{n-2}\right)\wedge\left(\wedge^{n-2}V_{n-2}\right)\big]\mid[V_{n-2}]\in\mathrm{G}(n-2,V)\right\}\\
\overline{{\mathscr P}}_{\sigma} & = \left\{ \;\big[\wedge^{2}\left(V_{n}/V_{n-3}\right)\wedge (\wedge^{n-3}V_{n-3})\big]\;\mid[V_{n-3}\subset V_{n}]\in \mathrm{F}(n-3,n,V)\right\},
\end{aligned}
\]
where $\overline{{\mathscr P}}_{\rho}\simeq \mathrm{G}(n-2,V)$ and $\overline{{\mathscr P}}_{\sigma}
\simeq \mathrm{F}(n-3,n,V)$.
\end{prop}
Let us make the following definition:
\begin{defn}
We call a conic $q$ in $\mathrm{G}(n-1,V)$ a {\it $\tau$-conic} if
$\mathbb{P}_{q}\cap \mathrm{G}(n-1,V)=q$.
A conic $q$ is called a {\it $\rho$-conic} and
{\it $\sigma$-conic}
if the plane $\mathbb{P}_{q}$ is contained in $\mathrm{G}(n-1,V)$, {and in that case}
$\mathbb{P}_q$ is {called} a $\rho$-plane and $\sigma$-plane,
{respectively.}
\end{defn}
{Let us denote by $[Q_y]$ the image of $y \in {\mathscr Y}$ under ${\mathscr Y}\to{\mathscr H}$.
By slight abuse of terminology, we say $y$ is a rank $k$ point if
$\mathrm{rank}\, Q_y=k$. By Proposition \ref{cla:double2}, the fiber of ${\mathscr Z}\to {\mathscr Y}$
over a rank $3$ or $4$ point $y$ is a conic, which we denote it by $q_y$.}
\begin{prop}
\label{lem:qy}$(1)$ If $\mathrm{rank}\, Q_{y}=4$, then $q_{y}$ is a $\tau$-conic. $(2)$
If $\mathrm{rank}\, Q_{y}=3$, then
the plane $\mathbb{P}_{q_{y}}$ is a $\rho$-plane, hence $q_{y}$ is a $\rho$-conic.
\end{prop}
\begin{proof}
(1) If $q_y$ is a $\rho$-conic,
then $(n-2)$-planes in $Q_y$ parameterized by $q_y$ must contain a $\mathbb{P}(V_{n-2})$ in common
but this can not be the case.
If $q_y$ is a $\sigma$-conic,
then $(n-2)$-planes in $Q_y$ parameterized by $q_y$
must be contained in one $\mathbb{P}(V_{n})$
but this also can not be the case.
Hence $q_{y}$ is a $\tau$-conic. The claim (2) is clear since the planes
parametrized by $q_{y}$ contain the vertex $\mathbb{P}(V_{n-2})$ of $Q_{y}$
in common.
\end{proof}
\begin{example}
\label{ex:conics}
(Smooth Conics) Taking a basis ${\bf e}_{1},\dots,{\bf e}_{n+1}$
of $V$, consider the subspaces $V_{n-3}=\langle{\bf e}_{4},\dots, {\bf e}_{n}\rangle$,
$V_{n}=\langle{\bf e}_{1},\dots,{\bf e}_{n}\rangle$
and $V_{n-2}=\langle{\bf e}_{4},\dots, {\bf e}_{n+1}\rangle$. An
example of $\tau$-conic may be given\[
q_{\tau}=\left\{ [s{\bf e}_{1}+t{\bf e}_{2},s{\bf e}_{3}+t{\bf e}_{4},{\bf e}_{5},\dots, {\bf e}_{n+1}]\mid[s,t]\in\mathbb{P}^{1}\right\} .\]
Similarly, as a $\mathbb{P}^{1}$-family of planes in the $\rho$-plane ${\rm P}_{V_{n-2}}$
and $\tau$-plane ${\rm P}_{V_{n-3}V_{n}}$, respectively, we have the
following examples: \[
q_{\rho}=\left\{ [s^{2}{\bf e}_{1}+st{\bf e}_{2}+t^{2}{\bf e}_{3},{\bf e}_{4},\dots,{\bf e}_{n+1}]\right\} ,\; q_{\sigma}=\left\{ [s{\bf e}_{1}+t{\bf e}_{2},s{\bf e}_{2}+t{\bf e}_{3},{\bf e}_{4},\dots, {\bf e}_n]\right\} ,\]
where $[s,t]\in\mathbb{P}^{1}$ parameterizes each conic $q$.
$\square$
\end{example}
\begin{example}
\label{ex:ranktwo}
(Rank two conics)
Since a line in $\mathrm{G}(n-1,V)$ takes the form $l_{V_{n-2}V_{n}}=\left\{ [\Pi]\mid V_{n-2}\subset\Pi\subset V_{n}\right\} $
with some $V_{n-2}\subset V_{n}\subset V$, reducible conics $q$ have
the following form: \begin{equation}
q=l_{V_{n-2}V_{n}}\cup l_{V_{n-2}'V_{n}'}
\label{eq:conic-q-rk2}\end{equation}
with
$\bullet$ $\dim(V_{n-2}\cap V_{n-2}')\geq
n-3$,
$\bullet$ $V_{n-2},V_{n-2}'\subset V_{n}\cap V_{n}'$, and
$\bullet$ $V_{n-2}\not =V'_{n-2}$ or $V_n\not =V'_n$.
\noindent
These conics will be described in detail in the section \ref{section:FY}.
\end{example}
{Descriptions of rank one conics may be found in Appendix \ref{app:aU}.}
\subsection{Hilbert scheme ${\mathscr Y}_0$ of conics on $\mathrm{G}(n-1,V)$}
\label{subsection:Hilb}
Consider a point $[U]\in \mathrm{G}(3,\wedge^{n-1}V)$.
To describe conics in $\mathrm{G}(n-1,V)\subset \mathbb{P}(3,\wedge^{n-1}V)$,
it suffices to find a {condition} for a plane $\mathbb{P}(U)$
to be contained in $\mathrm{G}(n-1,V)$ or cut out a conic from
$\mathrm{G}(n-1,V)$.
For this, we introduce
the composite $\varphi$ of the following maps:
\begin{equation}
\varphi\colon \ft{S}^{2}(\wedge^{n-1}V)\simeq\ft{S}^{2}(\wedge^{2}V^{*})
\overset{\psi}{\to}\wedge^{4}V^{*},\label{eq:phiU}
\end{equation}
where the first map is {induced by}
the duality $\wedge^{n-1}V\simeq\wedge^{2}V^{*}$
coming from the wedge product pairing
$\wedge^{n-1}V\times\wedge^{2}V\to\wedge^{n+1}V$,
and $\psi$ is {induced by} the wedge product.
Note that the zero locus of $\psi$ is nothing but $\mathrm{G}(2,V^*)$
{since we obtain the Pl\"ucker quadrics defining $\mathrm{G}(2,V^*)$
by writing $\psi$ with coordinates.}
Moreover, the duality $\wedge^{n-1}V\simeq\wedge^{2}V^{*}$
induces an isomorphism $\mathrm{G}(n-1,V)\simeq \mathrm{G}(2,V^*)$.
Therefore $\mathrm{G}(n-1,V)$ is the zero locus of $\varphi$.
Now we consider the restriction of $\varphi$ to
a $3$-plane $U\subset \wedge^{n-1} V$:
\[
\varphi_{U}:=\varphi\vert_{\ft{S}^{2}U}\colon \ft{S}^{2}U\to \wedge^4 V^*.
\]
Let $U'$ be the $3$-plane of $\wedge^2 V^*$ corresponding to $U$
and denote by $\psi_{U'}$ the restriction of $\psi$ to $U'$.
Since $\mathrm{G}(2,V^*)$ is
the zero locus of $\psi$,
$\mathbb{P}(U')\subset \mathrm{G}(2,V^*)$ iff $\psi_{U'}=0$.
Similarly,
$\mathbb{P}(U')\cap \mathrm{G}(2,V^*)$ is a conic iff
the restrictions of the Pl\"ucker quadrics on $\mathbb{P}({\ft S}^2 {U'}^*)$ form
a point, i.e., one-dimensional subspace of ${\ft S}^2 {U'}^*$,
which is equivalent to the condition $\mathrm{rank}\, \psi_{U'}=1$.
Translating this, we immediately obtain the following
descriptions on the {intersection} $\mathbb{P}(U)\cap \mathrm{G}(n-1,V)$:
\begin{prop}
\label{pro:conic-varphiU}
For a $3$-plane $U\subset \wedge^{n-1}V$,
$\mathbb{P}(U)\cap \mathrm{G}(n-1,V)$ contains a conic
iff $\mathrm{rank}\, \phi_U\leq 1$.
Moreover, the following {properties} hold$\,:$
\begin{myitem2} \item[$(1)$]
$\left\{ [U]\in\mathrm{G}(3,\wedge^{n-1} V)\mid
\varphi_{U}=0\right\} =\overline{{\mathscr P}}_{\rho}\sqcup\overline{{\mathscr P}}_{\sigma}$.
\item[$(2)$] If $\mathrm{rank}\, \varphi_{U}=1$, then
$\mathbb{P}(U)\cap \mathrm{G}(n-1,V)$ is a conic which is the zero
locus of $\varphi_U$.
\end{myitem2}\end{prop}
Motivated from the above descriptions of conics, we {define}
the following scheme with reduced structure:
\begin{equation}
{\mathscr Y}_{0}:=\left\{ ([U],[c_{U}])\mid[U]\in\mathrm{G}(n-1,V),
[c_{U}]\in\mathbb{P}(\ft{S}^{2}U^{*})\text{ s.t. } (c_{U})_0\subset (\varphi_{U})_0
\right\},\label{eq:resolutionY0}\end{equation}
where
$(c_{U})_0$ and $(\varphi_{U})_0$ {represents} the zero locus in $\mathbb{P}(U)$ of
$c_U$ and $\varphi_{U}$, respectively.
\begin{thm}
\label{prop:Y0Hilb}
${\mathscr Y}_0$ is smooth and isomorphic to the Hilbert
scheme of conics on $\mathrm{G}(n-1,V)$.
\end{thm}
\begin{proof}
By definition, ${\mathscr Y}_0$ obviously parameterizes conics
in $\mathrm{G}(n-1,V)$ in one to one way.
Moreover, there is a family
in $\mathbb{P}(\wedge^{n-1} V)\times {\mathscr Y}_0$
of corresponding conics $(c_U)_0$ at each point $([U], [c_U])\in {\mathscr Y}_0$.
Therefore, by the universal property of the Hilbert scheme,
there is a unique map from ${\mathscr Y}_0$
to the Hilbert scheme {$\mathrm{Hilb}^{\mathrm{co}}\mathrm{G}(n-1,V)$} of
conics in $\mathrm{G}(n-1,V)$.
{Since the smoothness of the Hilbert scheme
is known in \cite{IM} and \cite{CHK},
we have ${\mathscr Y}_0\simeq \mathrm{Hilb}^{\mathrm{co}}\mathrm{G}(n-1,V)$.}
\end{proof}
{
Let us consider the natural projection ${\mathscr Y}_0\to \mathrm{G}(n-1,V)$ and denote
by $\overline{{\mathscr Y}}$ its image with the reduced structure.
Let $\nu\colon \overline{{\mathscr Y}}'\to \overline{{\mathscr Y}}$ be
the normalization (one should be able to show that $\overline{{\mathscr Y}}$
is normal in general extending the explicit
description given in \cite{Arxiv} for $n=4$).
The following descriptions of $\overline{{\mathscr Y}}$
and related properties are easy to derive:
}
\begin{prop} \label{prop:barY-1-2-3}
{$(1)$ We have
\[
\overline{{\mathscr Y}}=
\left\{ [U]\in\mathrm{G}(3,\wedge^{n-1}V)\mid\mathrm{rank}\, \varphi_U\leq 1\right\}.
\]
$(2)$ ${\mathscr Y}_0\to \overline{{\mathscr Y}}'$ is isomorphism
outside $\nu^{-1}\overline{{\mathscr P}}_{\rho}$ and $\nu^{-1}\overline{{\mathscr P}}_{\sigma}$.}
\noindent
$(3)$ Let $G_{\rho}$ and $F_{\sigma}$ be
the exceptional set over $\nu^{-1}\overline{{\mathscr P}}_{\rho}$ and $\nu^{-1}\overline{{\mathscr P}}_{\sigma}$, respectively.
Then $G_{\rho}\to \nu^{-1}\overline{{\mathscr P}}_{\rho}$ and $F_{\sigma}\to \nu^{-1}\overline{{\mathscr P}}_{\sigma}$ are $\mathbb{P}^5$-bundles whose fiber parameterizes $\rho$- or $\sigma$-conics in a fixed $\rho$- or $\sigma$-plane respectively.
\end{prop}
\subsection{Small resolution ${\mathscr Y}_{3}\to \overline{{\mathscr Y}}'$}
{We find a small resolution
${\mathscr Y}_3\to \overline{{\mathscr Y}}'$ by
translating the condition $\mathrm{rank}\, \varphi_U\leq 1$ into an equivalent
form.}
For each $v\in V$, {let us} define a linear map
$E_{v}:\wedge^{n-1}V\to\wedge^{n}V$
by $u\mapsto v\wedge u$. Consider the restriction $E_{v}\vert_{U}$
to $U\subset\wedge^{n-1}V$ and introduce
\[
a_{U}=\left\{ v\in V\mid E_{v}\vert_{U}=0\right\},
\]
which is nothing but the annihilator of $U$.
Note that $\dim U=3$ implies $\dim a_U\leq n-2$.
We prove the following proposition
in Appendix \ref{sec:Appendix-B}.
\begin{prop}
\label{lem:appendixB-UU-solve}
For $[U]\in\mathrm{G}(3,\wedge^{n-1}V)$,
$\dim a_{U}\geq n-3 \Longleftrightarrow \mathrm{rank}\,\varphi_{U}\leq 1$.
\end{prop}
By this proposition, {it is immediate to see} that
\[
\overline{{\mathscr Y}}=\left\{ [U]\in\mathrm{G}(3,\wedge^{n-1}V)
\mid\dim a_{U}\geq n-3\right\}.
\]
{Below we define} a Springer type resolution
${\mathscr Y}_3\to \overline{{\mathscr Y}}'$,
which turns out to be a small resolution.
\begin{defn}
\label{def:Y3} {For $n\geq 3$,} we define
\[
{\mathscr Y}_{3}=\left\{ ([U],[V_{n-3}])\mid V_{n-3}\subset a_{U}\right\}
\subset\mathrm{G}(3,\wedge^{n-1}V)\times\mathrm{G}(n-3,V),
\]
{where $\mathrm{G}(n-3,V)$ should be understood as one point when $n=3$.}
Obviously,
the image of the projection of ${\mathscr Y}_{3}$ to the
first factor
coincides with $\overline{{\mathscr Y}}$.
\end{defn}
Since $E_{v}\vert_{U}=0\,(\forall v\in V_{n-3})$ implies that $U$ is
the $\mathbb{C}$-span of non-vanishing vectors of
the form $\bar{u}_{i}\wedge v_1\wedge \dots \wedge v_{n-3}\,(i=1,2,3)$
with $\bar{u}_{i}\in\wedge^{2}(V/V_{n-3})$ and $v_1,\dots, v_{n-3}$
{being a} basis of $V_{n-3}$, the fiber of the natural
projection ${\mathscr Y}_{3}\to\mathrm{G}(n-3,V)$ over $[V_{n-3}]\in\mathrm{G}(n-3,V)$
can be identified with $\mathrm{G}(3,\wedge^{2}(V/V_{n-3}))$. Hence
we see that \[
{\mathscr Y}_{3}=\mathrm{G}(3,\wedge^2 \mathfrak{Q}),\]
and in particular ${\mathscr Y}_{3}$ is smooth.
\begin{prop}
\label{pro:barYsing}
The morphism $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{{\mathscr Y}_{3}}:{\mathscr Y}_{3}\to\overline{{\mathscr Y}}'$
{is isomorphic over $\overline{{\mathscr Y}}'\setminus\nu^{-1}\overline{{\mathscr P}}_{\rho}$ and is a small resolution with $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{{\mathscr Y}_{3}}^{\,-1}(x)\simeq \mathbb{P}^{n-3}$ for
each $x \in \nu^{-1}\overline{{\mathscr P}}_{\rho}$.}
In particular, $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{{\mathscr Y}_{3}}$ is an isomorphism if $n=3$, and
$\nu^{-1}\overline{{\mathscr P}}_{\rho}=\mathrm{Sing}\, \overline{{\mathscr Y}}'$ if $n\geq 4$.
\end{prop}
\begin{proof} {It is easy to see that the} fiber
of ${\mathscr Y}_{3}\to\overline{{\mathscr Y}}'$ over
each point of $\nu^{-1}\overline{{\mathscr P}}_{\rho}$
is $\mathrm{G}(n-3,n-2)\simeq\mathbb{P}^{n-3}$, and ${\mathscr Y}_{3}\to\overline{{\mathscr Y}}'$
is bijective over $\overline{{\mathscr Y}}'\setminus\nu^{-1}\overline{{\mathscr P}}_{\rho}$.
\end{proof}
\begin{rem}
\label{rem:Y3=Ybar}
In case $n=3$, we have
${\mathscr Y}_3=\overline{{\mathscr Y}}'=\overline{{\mathscr Y}}=\mathrm{G}(3,\wedge^2 V)$.
\end{rem}
\subsection{Small resolution $\widetilde{{\mathscr Y}}\to\overline{{\mathscr Y}}'$
via the Hilbert scheme ${\mathscr Y}_0$ \label{subsection:BlowUp}}
We construct another small resolution
$\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{{\mathscr Y}}}\colon
\widetilde{{\mathscr Y}}\to \overline{{\mathscr Y}}'$ for $n\geq 4$,
which is the (anti-)flip of ${\mathscr Y}_3\to \overline{{\mathscr Y}}'$.
{We give} $\widetilde{{\mathscr Y}}$ from ${\mathscr Y}_{0}$ by contracting
the exceptional set (divisor) over {$\nu^{-1}\overline{{\mathscr P}}_{\sigma}$.}
Let
$R_{\rho}$ (resp.~$R_{\sigma}$) be the extremal ray spanned by
lines in fibers of $G_{\rho}\to \nu^{-1}\overline{{\mathscr P}}_{\rho}$
(resp.~$F_{\sigma}\to \nu^{-1}\overline{{\mathscr P}}_{\sigma}$).
We show that $R_{\rho}\not =R_{\sigma}$.
Indeed, note that $F_{\sigma}$ is a prime divisor and
$G_{\rho}\cap F_{\sigma}=\emptyset$. Therefore, $F_{\sigma}\cdot R_{\rho}=0$
and $F_{\sigma}\cdot R_{\sigma}<0$ and hence $R_{\rho}\not =R_{\sigma}$.
Since $\overline{{\mathscr Y}}'$ is smooth along $\overline{{\mathscr P}}_{\sigma}$
by Proposition \ref{pro:barYsing},
the discrepancy of $F_{\sigma}$ is positive and then $R_{\sigma}$ is
$K_{{\mathscr Y}_0}$-negative. Therefore there exists a unique extremal
contraction ${\mathscr Y}_0\to\widetilde{{\mathscr Y}}$ over $\overline{{\mathscr Y}}'$
associated to $R_{\sigma}$, which is nothing but the contraction
of $F_{\sigma}$. We denote by $G_{\sigma}$ the image of $F_{\sigma}$.
{
The following proposition follows from the above construction
of $\widetilde{{\mathscr Y}}$:}
\begin{prop}
$\widetilde{{\mathscr Y}}$ parameteirizes $\tau$- and $\rho$-conics, and
$\sigma$-planes.
\end{prop}
{We retain the notation $G_{\rho}$ to represent the locus in
$\widetilde{{\mathscr Y}}$ parameterizing $\rho$-conics and denote by
$\mathcal{Q}_{\rho}$ the universal quotient bundle on $\mathrm{G}(n-2,V)$.}
\begin{prop} \label{prop:Grho}
$G_{\rho}$ is isomorphic to $\mathbb{P}({\ft S}^2 \mathcal{Q}_{\rho}^*)$.
{It} is also isomorphic to $\widetilde{\textsc{S}}_3$.
\end{prop}
\begin{proof}
The first claim {is clear} since
$\mathbb{P}(\mathcal{Q}_{\rho})\to \overline{{\mathscr P}}_{\rho}\simeq \mathrm{G}(n-2,V)$ is the
family of {$\rho$-planes.}
The second one follows {from the definition of the resolution
$\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{S}}_3}:\widetilde{\textsc{S}}_3\to\textsc{S}_3$
(see Proposition~\ref{prop:Spr}).}
\end{proof}
\begin{prop}
\label{prop:tildeY}
$\text{\large{\mbox{$p\hskip-1pt$}}}_{\widetilde{{\mathscr Y}}}\colon \widetilde{{\mathscr Y}}\to \overline{{\mathscr Y}}'$ is
a small resolution for $n\geq 4$,
and is the blow-up along $\nu^{-1}\overline{{\mathscr P}}_{\rho}$ for $n=3$.
Non-trivial fibers of $\text{\large{\mbox{$p\hskip-1pt$}}}_{\widetilde{{\mathscr Y}}}$ are copies of $\mathbb{P}^5$.
\end{prop}
\begin{proof}
$\widetilde{{\mathscr Y}}$ is smooth
since ${\mathscr Y}_0$ is smooth by Theorem \ref{prop:Y0Hilb}
and $\overline{{\mathscr Y}}'$ is smooth along $\nu^{-1}\overline{{\mathscr P}}_{\sigma}$
by Proposition \ref{pro:barYsing}.
Note that $G_{\rho}$ is the $\text{\large{\mbox{$p\hskip-1pt$}}}_{\widetilde{{\mathscr Y}}}$-exceptional locus
since the restriction of $\text{\large{\mbox{$p\hskip-1pt$}}}_{\widetilde{{\mathscr Y}}}|_{G_{\rho}}$
is a $\mathbb{P}^5$-bundle over $\nu^{-1}\overline{{\mathscr P}}_{\rho}\simeq
\overline{{\mathscr P}}_{\rho}$.
If $n\geq 4$, then $G_{\rho}$ is not a divisor by dimension count.
In case $n=3$, $G_{\rho}$ is a prime divisor.
Since $\overline{{\mathscr Y}}'$ is smooth by
Proposition \ref{pro:barYsing},
and $G_{\rho}\to \nu^{-1}\overline{{\mathscr P}}_{\sigma}$ is a $\mathbb{P}^5$-bundle,
we see that $K_{\widetilde{{\mathscr Y}}}=
\text{\large{\mbox{$p\hskip-1pt$}}}_{\widetilde{{\mathscr Y}}}^*K_{\overline{{\mathscr Y}}'}+5G_{\rho}$.
Let
$\text{\large{\mbox{$p\hskip-1pt$}}}{\,'}_{\widetilde{{\mathscr Y}}}\colon \widetilde{{\mathscr Y}}'\to \overline{{\mathscr Y}}'$
be the blow-up along $\nu^{-1}\overline{{\mathscr P}}_{\rho}$
and $G'_{\rho}$ the $\text{\large{\mbox{$p\hskip-1pt$}}}{\,}'_{\widetilde{{\mathscr Y}}}$-exceptional divisor.
Then we have
$K_{\widetilde{{\mathscr Y}}'}=
{\text{\large{\mbox{$p\hskip-1pt$}}}{\,'}}_{\widetilde{{\mathscr Y}}}^*K_{\overline{{\mathscr Y}}'}+5G'_{\rho}$.
It is well-known that there is only one valuation of $k(\overline{{\mathscr Y}}')$
associated to exceptional divisors with center
{$\nu^{-1}\overline{{\mathscr P}}_{\rho}$} and discrepancy $5$.
Therefore, $\widetilde{{\mathscr Y}}$ and $\widetilde{{\mathscr Y}}'$ are isomorphic in
codimension one. Moreover, since
$-K_{\widetilde{{\mathscr Y}}}$ and $-K_{\widetilde{{\mathscr Y}}'}$ are relatively ample
over $\overline{{\mathscr Y}}'$,
$\widetilde{{\mathscr Y}}$ and $\widetilde{{\mathscr Y}}'$
{must be} isomorphic by \cite[Lemma 5.5]{Tk}.
\end{proof}
\subsection{\label{sub:tildeYtoH}Rational map ${\mathscr Y}_3\dashrightarrow {\mathscr H}$ via double spin decomposition}
\label{subsection:spin}
Consider a point $({[U]},[V_{n-3}])
\in{\mathscr Y}_{3}=\mathrm{G}(3,\wedge^2 \mathfrak{Q})$
with ${[U]}\in\mathrm{G}(3,\wedge^{2}(V/V_{n-3}))$. To describe $\wedge^{3}{U}$,
we use the following irreducible decomposition as $sl(V/V_{n-3})$-modules
(see \cite[$\S 19.1$]{FH} for example):
\begin{equation}
\begin{aligned}
&\wedge^{3}(\wedge^{2}(V/V_{n-3})) = \\
&\ft{S}^{2}(V/V_{n-3})\otimes\det(V/V_{n-3})\oplus
\ft{S}^{2}(V/V_{n-3})^{*}\otimes\det(V/V_{n-3})^{\otimes2}.
\end{aligned}
\label{eq:spin}
\end{equation}
We will call this {}``double spin'' decomposition since the
{symmetric powers} in the r.h.s. are identified with
$V_{2\lambda_{s}}$ and $V_{2\lambda_{\bar{s}}}$
as $so(\wedge^{2}V/V_{n-3})(\simeq sl(V/V_{n-3}))$-modules, where $\lambda_{s}$
and $\lambda_{\bar{s}}$ represent the spinor and conjugate spinor
weights, respectively (see {[}\textit{loc. cit.}{]}).
{Considering this decomposition fiberwise in the projective
bundle $\mathbb{P}(\wedge^3(\wedge^2\mathfrak{Q}))$ over $\mathrm{G}(n-3,V)$,
we have} the following sequence of
(rational) maps:
\begin{equation}
\begin{aligned}
{\mathscr Y}_{3}\hookrightarrow
& \mathbb{P}(\ft{S}^{2}\mathfrak{Q}\otimes\mathcal{O}_{\mathrm{G}(n-3, V)}(-1)
\oplus\ft{S}^{2}\mathfrak{Q}^{*}) \\
&
\qquad\qquad
\dashrightarrow
{\mathscr U}=\mathbb{P}(\ft{S}^{2}\mathfrak{Q}^{*})\hookrightarrow\mathbb{P}(\ft{S}^{2}V^{*}\otimes\mathcal{O}_{\mathrm{G}(n-3, V)}),\end{aligned}
\label{eq:phi-Y-H-as-bundle-hom}
\end{equation}
where the rational map in the middle is the projection to the second
factor and the {last} inclusion comes from the surjection $V\otimes\mathcal{O}_{\mathrm{G}(n-3, V)}\to \mathfrak{Q}\to0$.
We further consider the natural projection $\mathbb{P}(\ft{S}^{2}V^{*}\otimes\mathcal{O}_{\mathrm{G}(n-3, V)})\to\mathbb{P}(\ft{S}^{2}V^{*})$.
Then the image of the composite is contained in the locus ${\mathscr H}$ of
the quadrics of rank $\leq 4$, and {hence} we have a rational map
\[
\phi\colon {\mathscr Y}_{3}\dashrightarrow{\mathscr H}\,{(:=\textsc{S}_4).}
\]
To obtain a morphism,
we consider the inverse {images}
${\mathscr P}_{\rho},{\mathscr P}_{\sigma}$ of {$\nu^{-1}\overline{{\mathscr P}}_{\rho}$
and $\nu^{-1}\overline{{\mathscr P}}_{\sigma}$, respectively, under
the resolution ${\mathscr Y}_{3}\to\overline{{\mathscr Y}}'$.}
Then it is clear from the definitions that \begin{equation}
{\mathscr P}_{\rho}\simeq \mathrm{F}(n-3,n-2;V)\simeq\mathbb{P}(\mathfrak{Q}),\;\;{\mathscr P}_{\sigma}\simeq \mathrm{F}(n-3,n;V)\simeq\mathbb{P}(\mathfrak{Q}^{*}).\label{eq:Prho-Ptau-by-Tangent}\end{equation}
\begin{prop}
\label{prop:Prt}
Under the embedding {${\mathscr Y}_{3}\subset\mathbb{P}(\ft{S}^{2}\mathfrak{Q}
\otimes\mathcal{O}_{\mathrm{G}(n-3,V)}(-1)\oplus\ft{S}^{2}\mathfrak{Q}^{*})$,}
${\mathscr P}_{\rho}$ and ${\mathscr P}_{\sigma}$ are identified with \[
{\mathscr P}_{\rho}=v_{2}(\mathbb{P}(\mathfrak{Q})),\;\;{\mathscr P}_{\sigma}=v_{2}(\mathbb{P}(\mathfrak{Q}^{*})).\]
Moreover,
${\mathscr P}_{\rho}={\mathscr Y}_3\cap \mathbb{P}(\ft{S}^{2}\mathfrak{Q}\otimes\mathcal{O}_{\mathrm{G}(n-3, V)}(-1))$ scheme-theoretically.
\end{prop}
\begin{proof} {The claims follows from the decomposition
(\ref{eq:spin}) and its explicit description given in
Proposition \ref{prop:B1}, (\ref{eq:Ivw}).}
\end{proof}
\begin{defn}\label{def:Y2}
We define $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{{\mathscr Y}_2}\colon{\mathscr Y}_2\to {\mathscr Y}_3$ to be the
blow-up along ${\mathscr P}_{\rho}$, {and denote}
by $F_{\rho}$ its exceptional divisor.
\end{defn}
{
Clearly there is a morphism ${\mathscr Y}_2\to\mathrm{G}(n-3,V)$ as well as
${\mathscr Y}_3\to\mathrm{G}(n-3,V)$.}
\subsection{The case $n=3$ ($\dim V=4$)}
\label{subsection:dimV=4}
{
When $n=3$, projective bundles over $\mathrm{G}(n-3,V)$ reduce to the corresponding projective spaces, and considerable simplifications may be observed, for example,
in
\[
{\mathscr Y}_3=\overline{{\mathscr Y}}'=\overline{{\mathscr Y}}=\mathrm{G}(3,\wedge^2V) \text{ and }
{\mathscr P}_\rho=v_2(\mathbb{P}(V))\subset \mathbb{P}(\ft{S}^2V).
\]
Also in this case, we have ${\mathscr Y}_2
\simeq\widetilde{{\mathscr Y}}$ by Propositions \ref{pro:barYsing} and
\ref{prop:tildeY}. Then the birational morphism
$\phi\colon {\mathscr Y}_{3}\dashrightarrow{\mathscr H}(=\mathbb{P}(\ft{S}^2V^*))$ lifts to
a morphism $\widetilde{\phi}\colon {\mathscr Y}_{2}\to{\mathscr H}$ by the last assertion in Proposition \ref{prop:Prt}.}
{In this subsection, we study the case $n=3 \,(\dim V=4)$ (where
${\mathscr H}=\mathbb{P}(\ft{S}^2V^*)$).
The results below will be used to study the case of
$n\geq 4 \,(\dim V\geq 5)$ (where ${\mathscr H}=\textsc{S}_4\subset \mathbb{P}(\ft{S}^2V^*)$) in
the next subsection. Also these will be used extensively in~\cite{ReyeEnr}.}
\begin{prop}
\label{prop:n=3fib}
\begin{enumerate}[$(1)$]
\item
The Stein factorization of $\widetilde{\phi}\colon \widetilde{{\mathscr Y}}\to
{\mathscr H}$ factors through the double cover $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\mathscr Y}}\colon {\mathscr Y}\to {\mathscr H}$.
\item Let $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}\colon \widetilde{{\mathscr Y}}\to {\mathscr Y}$ be the
induced morphism. Then $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}$ is birational
and a $K_{\widetilde{{\mathscr Y}}}$-negative extremal divisorial contraction.
\item
Let $F_{\widetilde{{\mathscr Y}}}$ be the $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}$-exceptional divisor. Then the image of $F_{\widetilde{{\mathscr Y}}}$ by $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}$
coincides with $G_{{\mathscr Y}}$, and $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$
is a $\mathbb{P}^1\times \mathbb{P}^1$-fibration outside $G_{{\mathscr Y}}^1$.
\item
It holds that
\[
K_{\widetilde{{\mathscr Y}}}=\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}^{\;*}K_{{\mathscr Y}}+F_{\widetilde{{\mathscr Y}}}.
\]
In particular, ${\mathscr Y}$ has only terminal singularities with
$\mathrm{Sing}\,{\mathscr Y}=G_{{\mathscr Y}}$.
\item
Let $w=(w_{kl})$ be the $4\times4$ matrix representing $[Q]\in\mathbb{P}(\ft{S}^{2}V^{*})$.
Then the fiber of $\tilde{\phi}$ is described according to the rank of $w$ as follow\,$:$
\begin{enumerate}[$(\rm a)$]
\item When $\mathrm{rank}\, w=4$, $\tilde{\phi}^{-1}([Q])$ consists of two points.
\item When $\mathrm{rank}\, w=3$, $\tilde{\phi}^{-1}([Q])$
consists of one point.
\item When $\mathrm{rank}\, w=2$,
$\tilde{\phi}^{-1}([Q])\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}.$
\item When $\mathrm{rank}\, w=1$,
$\tilde{\phi}^{-1}([Q])\simeq\mathbb{P}(1^{3},2).$
The vertex of $\tilde{\phi}^{-1}([Q])$
corresponds to the $\sigma$-plane ${\rm P}_{V_3}$,
where $Q=2\mathbb{P}(V_3)$, and $\tilde{\phi}^{-1}([Q])\cap F_{\rho}\simeq \mathbb{P}^2$
which is a hyperplane section of $\mathbb{P}(1^{3},2)\subset \mathbb{P}^6$.
\end{enumerate}
\end{enumerate}
\end{prop}
\begin{proof}
Let $\widetilde{{\mathscr Y}}\to {\mathscr Y}'\to {\mathscr H}$ be the Stein factorization of
$\widetilde{\phi}$. We denote by $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}$
and $F_{\widetilde{{\mathscr Y}}}$,
the induced morphism $\widetilde{{\mathscr Y}}\to {\mathscr Y}'$
and the $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}$-exceptional locus respectively
(this notation will be compatible
with {(2) and (3)} after showing that
the induced morphism ${\mathscr Y}'\to {\mathscr H}$ coincides with
the double cover $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{{\mathscr Y}}\colon{\mathscr Y}\to {\mathscr H}$).
Let us start with showing
that $\widetilde{\phi}(F_{\rho})=\textsc{S}_3$.
Let $Q$ be a rank three quadric $Q$ in $\mathbb{P}(V)$.
Then, from (I.3) in Appendix \ref{sec:Appendix-A},
$[Q]$ cannot be {in} the image of $\phi$.
Therefore the locus $\textsc{S}_3$ is contained in $\widetilde{\phi}(F_{\rho})$. Since $F_{\rho}$ and $\textsc{S}_3$ are prime divisors in $\widetilde{{\mathscr Y}}$ and ${\mathscr H}$ respectively,
it holds that $\widetilde{\phi}(F_{\rho})=\textsc{S}_3$.
\noindent {\bf Proof of (5) (a).}
Let $Q$ be a rank four quadric $Q$ in $\mathbb{P}(V)$,
{i.e., $[Q]\in\textsc{S}_4\setminus\textsc{S}_3$.}
From (I.2) in Appendix$\,$\ref{sec:Appendix-A},
$\phi^{-1}([Q])$ consists of two points $[v,w]$
satisfying $v.w=\pm\sqrt{\det w}\,\mathrm{id}_{4}$.
{Since} $\widetilde{\phi}(F_{\rho})=\textsc{S}_3$,
$\widetilde{\phi}^{-1}([Q])$ also consists of two points.
{We know now that} ${\mathscr Y}'\to {\mathscr H}$ is a finite morphism
of degree two, and its branch locus is contained in $\textsc{S}_3$.
\noindent {\bf Proof of a weaker assertion than (5) (c).}
Let $Q$ be a rank two quadric $Q$ in $\mathbb{P}(V)$ and $w$ an associated
symmetric matrix. We show that $\tilde{\phi}^{-1}([Q])$ contains a $\mathbb{P}^1\times \mathbb{P}^1$.
Changing the coordinate of $V$ suitably, we may assume that
$[w]$ is given in the form $w_{0}=\left(\begin{smallmatrix}\begin{smallmatrix}0 & 1\\
1 & 0\end{smallmatrix} & O_{2}\\
O_{2} & O_{2}\end{smallmatrix}\right)$ with $O_{2}$ being the $2\times2$ zero matrix. Then by the properties
(I.4) and (I.2), we obtain $v=\left(\begin{smallmatrix}O_{2} & O_{2}\\
O_{2} & \begin{smallmatrix}v_{11} & v_{12}\\
v_{12} & v_{22}\end{smallmatrix}\end{smallmatrix}\right)$ . Now substituting $[v,w]=[v,tw_{0}]\,(t\not=0)$ into the equation
in the first line of (\ref{eq:Ivw}), we have \[
v_{11}v_{22}-v_{12}^{2}+t^{2}=0\;\;(t\not=0).\]
The closure $S$ of this locus in {${\mathscr Y}_3=\mathrm{G}(3,\wedge^2V)$} is
isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$.
{Note that the restriction of the blow-up} $\widetilde{\mathscr Y} \to {\mathscr Y}_3$
over $S\subset {\mathscr Y}_3$ is the blow-up along the locus $t=0$.
Hence the strict
transform $S'$ of $S$ in $\widetilde{{\mathscr Y}}$
is also isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$.
Note that $S'$ is contained in the fiber {of the restriction over $S$.}
\noindent {\bf Proof of a similar statement to (2) for
$\widetilde{{\mathscr Y}}\to {\mathscr Y}'$.}
Since $\rho(\widetilde{{\mathscr Y}})=2$, we have
$\rho(\widetilde{{\mathscr Y}}/{\mathscr Y}')\leq 1$.
Moreover, since the fiber over a rank two point is at least $2$-dimensional
and $\dim \textsc{S}_2=6$, $F_{\widetilde{{\mathscr Y}}}$ is a prime divisor.
We see that
the contraction $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}$ is $K_{\widetilde{{\mathscr Y}}}$-negative by computing the intersection number between
$K_{\widetilde{{\mathscr Y}}}$ and a ruling of $S'$.
Thus ${\mathscr Y}'$ has only terminal singularities.
\noindent {\bf Proof of (1).}
${\mathscr Y}'$ is Cohen-Macaulay since it is terminal
and hence
${\mathscr Y}'\to {\mathscr H}$ is flat. Then its branch locus is
empty or a divisor but the former case cannot occur since ${\mathscr H}=\mathbb{P}({\ft S}^2 V^*)$ is simply connected. Therefore the branch locus of ${\mathscr Y}'\to {\mathscr H}$ coincides with $\textsc{S}_3$. Now we see that ${\mathscr Y}'\simeq {\mathscr Y}$ since
both ${\mathscr Y}'\to {\mathscr H}$ and ${\mathscr Y}\to {\mathscr H}$ are both flat,
finite of degree two and
are branched along $\textsc{S}_3$.
\noindent {\bf Proof of (5) (b).}
Since $\textsc{S}_3$ is the branch locus, $F_{\rho}\to \textsc{S}_3$ is birational.
Therefore we see that the fiber over a rank three points consists of one point
{as claimed.}
We have shown (1), (2), the {first} half of (3) and (5) (b).
The {second half} of (3) will follow from (5) (c).
We will show two resolutions $F_{\rho}\to \textsc{S}_3$ and
$\widetilde{\textsc{S}}_3\to \textsc{S}_3$ coincides with each other.
First we note that $\rho(F_{\rho})=\rho(\widetilde{\textsc{S}}_3)=2$ and then
$\rho(F_{\rho}/\textsc{S}_3)=\rho(\widetilde{\textsc{S}}_3/\textsc{S}_3)=1$.
Since $\textsc{S}_3$ is $\mathbb{Q}$-factorial, $F_{\rho}\to \textsc{S}_3$ is a divisorial contraction.
Let $G_1$ and $G_2$ be the exceptional divisors of
$F_{\rho}\to \textsc{S}_3$ and $\widetilde{\textsc{S}}_3\to \textsc{S}_3$ respectively.
Since $\widetilde{\textsc{S}}_3\to \textsc{S}_3$ is a crepant resolution,
the valuation of $G_2$ in $k(\textsc{S}_3)$ is a unique crepant valuation.
If the discrepancy of $G_1$ is positive, then we see that
any exceptional valuation in $k(\textsc{S}_3)$ must have positive discrepancy by computing the discrepancies of
exceptional divisors over $F_{\rho}$, {which is}
a contradiction to the existence of $G_2$.
Therefore $F_{\rho}\to \textsc{S}_3$ is crepant, and moreover
the valuations of $G_1$ and $G_2$ are the same by the uniqueness of
the crepant valuation. In particular,
$F_{\rho}\to \textsc{S}_3$ and $\widetilde{\textsc{S}}_3\to \textsc{S}_3$ are isomorphic in
codimension one.
Note that $-G_1$ and $-G_2$ are relatively ample over $\textsc{S}_3$.
Let $p\colon {\bf G}amma\to F_{\rho}$ and $q\colon {\bf G}amma\to \textsc{S}_3$
be a common resolution of $F_{\rho}$ and $\textsc{S}_3$.
Thus, by the standard argument using the negativity lemma,
we see that $p^*(-G_1)=q^*(-G_2)$.
This implies that
two resolutions $F_{\rho}\to \textsc{S}_3$ and
$\widetilde{\textsc{S}}_3\to \textsc{S}_3$ coincides with each other.
\noindent {\bf Proof of (5) (c).}
As we see above,
the fiber over a rank two point $[Q]$
contains at least $S'\simeq \mathbb{P}^1\times \mathbb{P}^1$.
The fiber of $F_{\rho}\to \textsc{S}_3$ over
$[Q]$ is isomorphic to $\mathbb{P}^1$ by the description of the fibers of
$\widetilde{\textsc{S}}_3\to \textsc{S}_3$.
Thus the fiber $\tilde{\phi}^{-1}([Q])$
coincides with $S'$.
\noindent {\bf Proof of (4).}
We obtain the {claimed formula}
by computing the intersection number between
$K_{\widetilde{{\mathscr Y}}}$ and a ruling of $S'$.
\noindent {\bf Proof of (5) (d).}
Let $Q$ be a rank one quadric in $\mathbb{P}(V)$
and $w$ an associated symmetric matrix.
Then $w$ can be written as $(a_{k}a_{l})$ with some
$a\in V^{*}$. Then from (I.5) in Appendix, we see that
$\mathrm{rank}\, v\leq 1$. Writing $v_{ij}=x_{i}x_{j}$ with $x\in V$
and also solving (\ref{eq:Ivw}) we obtain
\begin{equation}
\label{eq:WPSeq}
\phi^{-1}([Q])=\left\{ [x_{i}x_{j},ta_{k}a_{l}]\mid a.x=0,t\not=0\right\} .
\end{equation}
The closure of this locus in ${\mathscr Y}_3$ is isomorphic to
the cone over $v_{2}(\mathbb{P}^{2})\simeq\mathbb{P}^{2}$
from the vertex $[0,a_{k}a_{l}]\in\mathbb{P}(\ft{S}^{2} V\oplus\ft{S}^{2} V^{*})$,
which is isomorphic to $\mathbb{P}(1^{3},2)$.
Then we have the former assertion (5) (d) by a similar argument in case $\mathrm{rank}\, w=2$. The latter assertion is clear from the above description.
\end{proof}
\begin{rem}
\label{rem:WPS}
It is convenient to give a coordinate-free description of $\widetilde{\phi}^{-1}([Q])$ in case $\mathrm{rank}\, Q=1$.
{Instead of $\widetilde{\phi}^{-1}([Q])$,
we may} describe {its isomorphic}
image $\Phi \subset {\mathscr Y}_3$ {under
$\widetilde{\mathscr Y}={\mathscr Y}_2\to{\mathscr Y}_3$}.
Note that $\Phi$ is the closure {in ${\mathscr Y}_3$} of $\phi^{-1}([Q])$
and its equation is given by
(\ref{eq:WPSeq}).
Let $Q=2\mathbb{P}(V_3)$ as in Proposition \ref{prop:n=3fib} (5) (d).
The vertex of $\widetilde{\phi}^{-1}([Q])$ corresponds to
the $\sigma$-plane ${\rm P}_{V_3}=\{\mathbb{C}^2\subset V_3\}$.
By the equation (\ref{eq:WPSeq}),
points $[{\rm P}_{V_1}]$ which correspond to $\rho$-planes and are contained in $\Phi$ satisfy $V_1\subset V_3$.
Since $\Phi$ is the cone over the Veronese surface $v_2(\mathbb{P}(V_3))$,
it is swept out by lines joining $[{\rm P}_{V_3}]$ and $[{\rm P}_{V_1}]$ such that
$V_1\subset V_3$.
\end{rem}
\begin{prop}
\label{prop:conicquad}
For a $\tau$- or $\rho$-conic $q$,
$\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}([q])$ is the point corresponding to
the quadric generated by lines which $q$ parameterizes.
For a $\sigma$-plane $\rm{P}_{V_3}$,
$\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}([\rm{P}])$ is the point corresponding to
the rank one quadric $2\mathbb{P}(V_3)$.
In particular,
the exceptional locus $F_{\widetilde{{\mathscr Y}}}$ consists of the points
corresponding to $\tau$- or $\rho$-conics of rank at most two
or $\sigma$-planes,
and the image of $F_{\widetilde{{\mathscr Y}}}$ coincides with
$G_{{\mathscr Y}}$.
\end{prop}
\begin{proof}
We {have described} $\tau$-conics and $\sigma$-planes in
Examples \ref{ex:conics} and \ref{ex:ranktwo}
and Appendix \ref{sec:Appendix-B}.
The {assertions} for $\tau$-conics and $\sigma$-planes {follow from}
their descriptions and
direct computations
based on the results in Appendix \ref{sec:Appendix-A}.
{For $\rho$-conics, the assertion follows from the isomorphism}
$F_{\rho}\simeq \widetilde{\textsc{S}}_3$
as in the proof of Proposition \ref{prop:n=3fib}.
\end{proof}
\subsection{Divisorial contraction $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}\colon \widetilde{{\mathscr Y}}\to{\mathscr Y}$ for {$n\geq4$ ($\dim V\geq 5$)} \label{sub:tildeY-Y}}
{ Recall that we have the morphisms
\[
{\mathscr Y}_3\to \mathrm{G}(n-3,V),\;\; {\mathscr Y}_2\to\mathrm{G}(n-3,V) \;\text{ and } \;
{\mathscr U} \to \mathrm{G}(n-3,V)
\]
from Definition \ref{def:Y2} and (\ref{eq:deftSr})
with ${\mathscr U}:=\widetilde{\textsc{S}}_4.$
In this subsection, we consider the relative setting over $\mathrm{G}(n-3,V)$ for
$n\geq4$. Thus, for example, the geometry of ${\mathscr Y}_2$ is
considered as the family of the blow-ups
of $\mathrm{G}(3,\wedge^2(V/V_{n-3}))$ along
${\mathscr P}_\rho\vert_{[V_{n-3}]}=v_2(\mathbb{P}(V/V_{n-3}))$ for $[V_{n-2}]\in \mathrm{G}(n-3,V)$.
The results in the preceding subsection apply to each member of
the family with the 4-dimensional vector space $V/V_{n-3}$. }
{
\begin{lem}
There exists a morphism ${\mathscr Y}_2\to {\mathscr U}$ defined over $\mathrm{G}(n-3,V)$.
\end{lem}
}
{
\begin{proof}
Denote by ${\mathscr Y}_2\vert_{[V_{n-3}]}, {\mathscr Y}_3\vert_{[V_{n-3}]}, {\mathscr U}\vert_{[V_{n-3}]}$
the restrictions to the fibers over $[V_{n-3}]\in \mathrm{G}(n-3,V)$.
Then ${\mathscr Y}_2\vert_{[V_{n-3}]}$ is the blow-up of ${\mathscr Y}_3\vert_{[V_{n-3}]}=
\mathrm{G}(3,\wedge^2(V/V_{n-3}))$, as
described above, and ${\mathscr U}\vert_{[V_{n-3}]}=\mathbb{P}(\ft{S}^2(V/V_{n-3})^*)$. The claimed morphism is the one described in Proposition \ref{prop:n=3fib} (1).
\end{proof}
}
\begin{prop}
\label{prop:gendescr}
\begin{enumerate}[$(1)$]
\item There exists an extremal divisorial contraction
$\text{\large{\mbox{$\tilde\rho\hskip-2pt$}}}\,_{{\mathscr Y}_2}\colon {\mathscr Y}_2\to \widetilde{{\mathscr Y}}$ which is the blow-up
along $G_{\rho}$ with the exceptional divisor $F_{\rho}$.
Any fiber of $F_{\rho}\to G_{\rho}$ is a copy of $\mathbb{P}^{n-3}$
and is mapped to a fiber of ${\mathscr Y}_3\to \overline{{\mathscr Y}}'$ isomorphically.
\item There exists an extremal divisorial contraction
$\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}\colon \widetilde{{\mathscr Y}}\to {\mathscr Y}$.
\end{enumerate}
\end{prop}
\begin{proof}
We {reproduce} here a part of the diagram (\ref{eq:STcomm}):
\begin{equation}
\label{eq:STcommrev}
\begin{matrix}
\xymatrix{{\mathscr Y}_{{\mathscr U}}=
\widetilde{\textsc{T}}_4\ar[r]^{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_4}}\ar[d]_{\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{T}}_4}}
& {\mathscr U}=\widetilde{\textsc{S}}_4\ar[d]_{\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\widetilde{\textsc{S}}_4}}\\
{\mathscr Y}=\textsc{T}_4\ar[r]_{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\textsc{T}_4}} & {\mathscr H}=\textsc{S}_4.}
\end{matrix}
\end{equation}
By construction, we see
that $\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_4}\colon {\mathscr Y}_{{\mathscr U}}\to {\mathscr U}$ is
the family over $\mathrm{G}(n-3,V)$
of the double covers $\textsc{T}_4\to \textsc{S}_4$ for
$4$-dimensional vector spaces $V/V_{n-3}$.
{Consider the Stein factorization of the morphism ${\mathscr Y}_2\to{\mathscr U}$.
By the uniqueness of finite double cover, it is given by ${\mathscr Y}_2\to{\mathscr Y}_{\mathscr U}
\to{\mathscr U}$. Then the induced morphism
${\mathscr Y}_2\to {\mathscr Y}_{{\mathscr U}}$
is the family over $\mathrm{G}(n-3,V)$
of the divisorial contraction described in Proposition \ref{prop:n=3fib} (2)
(for $4$-dimensional vector spaces $V/V_{n-3}$).}
In particular, a birational morphism ${\mathscr Y}_2\to {\mathscr Y}$ is induced.
By Proposition~\ref{pro:barYsing} and the definition of ${\mathscr Y}_2$,
a birational morphism ${\mathscr Y}_2\to \overline{{\mathscr Y}}'$ is also induced.
Therefore we obtain a map ${\mathscr Y}_2\to
\overline{{\mathscr Y}}'\times {\mathscr Y}$.
Let $\widetilde{{\mathscr Y}}'$ be the normalization of the image of this map.
We will show that ${\mathscr Y}_2\to \widetilde{{\mathscr Y}}'$ is non-trivial.
Let $Q$ be a quadric in $\mathbb{P}(V)$ of rank $3$ and $\mathbb{P}(V_{n-2})$
its singular locus.
By Proposition \ref{prop:conicquad},
the fiber ${\bf G}amma$ of ${\mathscr Y}_2\to {\mathscr Y}$ over $[Q]$
is isomorphic to $\mathrm{G}(n-3,V_{n-2})$.
By Proposition~\ref{pro:barYsing} and the definition of ${\mathscr Y}_2$,
${\bf G}amma$ is also contracted by ${\mathscr Y}_2\to \overline{{\mathscr Y}}'$.
Therefore ${\mathscr Y}_2\to \widetilde{{\mathscr Y}}'$ is non-trivial.
{$\widetilde{{\mathscr Y}}'$ can not be isomorphic to $\overline{{\mathscr Y}}'$ nor
${\mathscr Y}$ since $\rho(\overline{{\mathscr Y}}')=\rho({\mathscr Y})=1$ and
$\overline{{\mathscr Y}}'\not \simeq {\mathscr Y}$.}
Therefore {
$\widetilde{{\mathscr Y}}'\to \overline{{\mathscr Y}}'$} is
a small birational morphism.
By the uniqueness of the flip (cf.~\cite{KM}),
we see that $\widetilde{{\mathscr Y}}'\simeq \widetilde{{\mathscr Y}}\ \text{or}\ {\mathscr Y}_3$.
There does not exist, however,
a contraction ${\mathscr Y}_3\to {\mathscr Y}$ since
{$\rho({\mathscr Y}_3)=2$ and there are }
two non-trivial contractions ${\mathscr Y}_3\to \mathrm{G}(n-3,V)$ and
${\mathscr Y}_3\to \overline{{\mathscr Y}}'$. {Therefore we must have}
$\widetilde{{\mathscr Y}}'\simeq \widetilde{{\mathscr Y}}$.
Now extending (\ref{eq:STcommrev}), we have
\begin{equation}
\begin{matrix}
\xymatrix{{\mathscr Y}_2\ar[rr]^{/\mathrm{G}(n-3,V)}\ar[d] & & {\mathscr Y}_{{\mathscr U}}=\widetilde{\textsc{T}}_4\ar[rr]^{/\mathrm{G}(n-3,V)}_{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\textsc{T}}_4}}\ar[d]_{\text{\large{\mbox{$p\hskip-1pt$}}}\,_{\widetilde{\textsc{T}}_4}} & & {\mathscr U}=\widetilde{\textsc{S}}_4\ar[d]_{\text{\large{\mbox{$\pi\hskip-2pt$}}}_{\widetilde{\textsc{S}}_4}}\\
\widetilde{{\mathscr Y}}\ar[rr]_{\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}}& & {\mathscr Y}=\textsc{T}_4\ar[rr]_{\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\textsc{T}_4}} & & {\mathscr H}=\textsc{S}_4.}
\end{matrix}
\end{equation}
Note that ${\mathscr Y}_2\to {\mathscr Y}_{{\mathscr U}}$ and ${\mathscr Y}_{{\mathscr U}}\to {\mathscr Y}$
are divisorial contractions.
Moreover, ${\mathscr Y}_2\to \widetilde{{\mathscr Y}}$ is also a divisorial contraction
contracting $F_{\rho}$ to $G_{\rho}$.
Therefore $\widetilde{{\mathscr Y}}\to {\mathscr Y}$ is a divisorial contraction,
and moreover its exceptional divisor $F_{\widetilde{{\mathscr Y}}}$ is
the image of the exceptional divisor of
${\mathscr Y}_2\to {\mathscr Y}_{{\mathscr U}}$.
Finally {we show that
${\mathscr Y}_2\to \widetilde{{\mathscr Y}}$ is the blow-up of $G_{\rho}$.
This morphism is given by forgetting the markings by
$[V_{n-3}]$ in $\mathrm{G}(n-3,V)$.
But, since $G_{\rho}\simeq \mathbb{P}({\ft S}^2 \mathcal{Q}_{\rho}^*)$
(see Proposition~\ref{prop:Grho}),
the markings by $[V_{n-2}]$ in $\mathrm{G}(n-2,V)$ are retained.}
Therefore the fiber of ${\mathscr Y}_2\to \widetilde{{\mathscr Y}}$
over a point $(q, [V_{n-2}])$ in $\mathbb{P}({\ft S}^2 \mathcal{Q}_{\rho}^*)$
is isomorphic to $\mathrm{G}(n-3, V_{n-2})\simeq \mathbb{P}^{n-3}$.
We may conclude that
${\mathscr Y}_2\to \widetilde{{\mathscr Y}}$ is the blow-up of $G_{\rho}$
by the same argument as in the proof of Proposition \ref{prop:tildeY}.
\end{proof}
\begin{rem}
\label{rem:blup}
In a similar way to the proof of Proposition \ref{prop:gendescr} (1),
we can show that ${\mathscr Y}_0\to \widetilde{{\mathscr Y}}$ is the blow-up along $G_{\sigma}$.
\end{rem}
By Propositions \ref{prop:conicquad} and \ref{prop:gendescr},
we have the following:
\begin{prop}
\label{prop:FYdisc}
For a $\tau$- or $\rho$-conic $q$,
$\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}([q])$ is the point corresponding to
the quadric generated by {$\mathbb{P}(V_{n-1})$'s} which $q$ parameterizes.
For a $\sigma$-plane {$\rm{P}_{V_{n-3}V_{n}}$,
$\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}([\rm{P}_{V_{n-3}V_{n}}])$} is the point corresponding to
the rank one quadric $2\mathbb{P}(V_{n})$.
In particular,
the exceptional locus $F_{\widetilde{{\mathscr Y}}}$ consists of the points
corresponding to $\tau$- or $\rho$-conics of rank at most two
or $\sigma$-planes,
and the image of $F_{\widetilde{{\mathscr Y}}}$ coincides with
$G_{{\mathscr Y}}$.
\end{prop}
\def\resizebox{10cm}{!}{\includegraphics{HilbCoYs.eps}}{\resizebox{10cm}{!}{\includegraphics{HilbCoYs.eps}}}
\def\vcorr{90}{$\in$}{\vcorr{90}{$\in$}}
\def\xyFigCoYs{
\begin{xy}
(0,0)*{\resizebox{10cm}{!}{\includegraphics{HilbCoYs.eps}}},
(-49,13)*{\widetilde{\mathscr Y}},
(-49,-20)*{{\mathscr H}},
(-32,-28)*{{\mathscr H}},
(-10,-28)*{\textsc{S}_3},
( 14,-26.5)*{\textsc{S}_2=G_{\mathscr Y}},
(33,-26)*{\textsc{S}_1=G_{\mathscr Y}^1},
(-15,20)*{\tau\text{-conics}},
(-15,17)*{\,_{(\text{rk} \tau =3)}},
(-7,-3)*{\rho\text{-conic}},
(-7,-6)*{\,_{(\text{rk} \rho =3)}},
( 13, 18)*{\tau\text{-conics}},
( 13, 15)*{\,_{(\text{rk} \tau =2)}},
( 13,-1)*{\rho\text{-conics}},
( 13,-4)*{\,_{(\text{rk} \rho =2)}},
(38, 22)*{\tilde{\phi}^{-1}([Q])},
(38,16)*{\text{\vcorr{90}{$\in$}}},
(38,-1)*{\rho\text{-conics}},
(38,-4)*{\,_{(\text{rk} \rho=1)}},
( 56, 18)*{\text{double lines}},
( 52, 14)*{\text{and}},
( 53, 10)*{\sigma\text{-planes}},
(0,0)*{} \end{xy} }
\vbox{
\[
\xyFigCoYs\]
\begin{fcaption}
\item \textbf{Fig.2. The fibers of $\tilde{\phi}=
\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\textsc{T}_4}\circ\text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{\mathscr Y}}\colon \widetilde{{\mathscr Y}}\to{\mathscr H}$
when $n=4$.}
\end{fcaption} }
\section{{\bf Geometry of $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$ and flattening}}
\label{section:FY}
In this section, we determine the structure of $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$ and construct its flattening.
\subsection{Birational model $F^{(1)}/\mathbb{Z}_2$ of $F_{\widetilde{{\mathscr Y}}}$}
From the description of the conics of rank two in Example \ref{ex:ranktwo}
and Proposition \ref{prop:FYdisc}, we introduce the following $\mathbb{Z}_{2}$-subvariety $F^{(1)}$
of $\mathrm{F}(n-2,n,V)^{\times 2}$ to study the exceptional locus $F_{\widetilde{{\mathscr Y}}} \subset \widetilde{{\mathscr Y}}$:
\begin{equation}
F^{(1)}:=\left\{ ([V_{n-2}],[V_{n-2}'];[V_{n}],[V_{n}'])\,\bigg\vert\;\begin{matrix}V_{n-2},V_{n-2}'\subset V_{n}\cap V_{n}'\\
\dim (V_{n-2}\cap V_{n-2}')\geq n-3\end{matrix}\right\},\label{eq:FsY}
\end{equation}
where $\mathbb{Z}_{2}$ acts by the simultaneous exchanges $V_{n-2}\leftrightarrow V_{n-2}'$
and $V_{n}\leftrightarrow V_{n}'$.
We set
\[
\widehat{G}:=\mathbb{P}(V^{*})\times\mathbb{P}(V^{*}),\, \Delta_{G}:=
\text{the diagonal of $\widehat{G}$},
\]
and note that the natural
projection
$F^{(1)}\to\widehat{G}$
is a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration outside $\Delta_G$.
Let $\oF{(1)}$ be the following open subset of $F^{(1)}$:
\begin{equation}
\oF{(1)}:=\left\{ ([V_{n-2}],[V_{n-2}'];[V_{n}],[V_{n}'])\,\bigg\vert\;
V_{n}\not =V_{n}'
\right\} \subset F^{(1)}.\end{equation}
\begin{prop}
\label{prop:begin}
The natural map
$\oF{(1)}/\mathbb{Z}_2\to (\widehat{G}\setminus \Delta_{G})/\mathbb{Z}_2$
is isomorphic to
$F_{\widetilde{{\mathscr Y}}}\setminus \text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})
\to G_{{\mathscr Y}}\setminus G^1_{{\mathscr Y}}$.
In particular,
$F_{\widetilde{{\mathscr Y}}}\setminus \text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})
\to G_{{\mathscr Y}}\setminus G^1_{{\mathscr Y}}$
is a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration.
\end{prop}
\begin{proof}
First note that
$\widehat{G}/\mathbb{Z}_2\simeq G_{{\mathscr Y}}$,
$\Delta_G/\mathbb{Z}_2\simeq G^1_{{\mathscr Y}}$ and hence
$(\widehat{G}\setminus \Delta_{G})/\mathbb{Z}_2\simeq
G_{{\mathscr Y}}\setminus G^1_{{\mathscr Y}}$.
Let us note that
$\oF{(1)}/\mathbb{Z}_2$ parameterizes line pairs in $\mathrm{G}(n-1,n+1)$
which are reducible conics of rank two and not on $\sigma$-planes (see
Example \ref{ex:ranktwo} for explicit descriptions).
Therefore we have
the unique injective morphism $\oF{(1)}/\mathbb{Z}_2\to {\mathscr Y}_0$ which is induced by
the universality of the Hilbert scheme ${\mathscr Y}_0$.
By Proposition \ref{prop:FYdisc},
the image of $\oF{(1)}/\mathbb{Z}_2$ coincides with $F_{\widetilde{{\mathscr Y}}}\setminus \text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})$,
and the map $\oF{(1)}/\mathbb{Z}_2\to F_{\widetilde{{\mathscr Y}}}\setminus \text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})$ induces the following commutative diagram:
\[
\xymatrix{\oF{(1)}/\mathbb{Z}_2\ar[r]\ar[d] & F_{\widetilde{{\mathscr Y}}}\setminus \text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})\ar[d]\\
(\widehat{G}\setminus \Delta_{G})/\mathbb{Z}_2\ar[r]^{\simeq} & {G_{{\mathscr Y}}\setminus G^1_{{\mathscr Y}}}.}
\]
Note that
$F_{\widetilde{{\mathscr Y}}}\setminus \text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})$
is normal. Indeed, $F_{\widetilde{{\mathscr Y}}}$ satisfies the $S_2$ condition
since it is a divisor on a smooth variety.
It also satisfies the $R_1$ condition since, by considering the $\mathrm{SL}\,(V)$-action, its singular locus is at most the locus of $\rho$-conics of rank two which is codimension $n-2\geq 2$ in $F_{\widetilde{{\mathscr Y}}}$ if $n\geq 4$ (resp.~it is smooth
if $n=3$ by Proposition \ref{prop:n=3fib} (5)). Hence
$F_{\widetilde{{\mathscr Y}}}\setminus \text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})$
is normal.
Therefore the bijective morphism
$\oF{(1)}/\mathbb{Z}_2\to F_{\widetilde{{\mathscr Y}}}\setminus \text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})$ is an isomorphism by the Zariski main theorem.
Finally,
the natural map $\oF{(1)}\to \widehat{G}$ is obviously a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration, and then so is
$\oF{(1)}/\mathbb{Z}_2\to (\widehat{G}\setminus \Delta_{G})/\mathbb{Z}_2$
since the $\mathbb{Z}_2$-action interchanges the fibers over
$(x,y)$ and $(y,x)$ in $\widehat{G}\setminus \Delta_G$.
\end{proof}
The following corollary will be used in the companion paper \cite{DerSym}.
\begin{cor}
\label{cor:Steinn=any}
It holds that
\begin{equation}
\label{eq:adjn=n}
K_{\widetilde{{\mathscr Y}}}=\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}^{\;*}K_{{\mathscr Y}}+(n-2)F_{\widetilde{{\mathscr Y}}}.
\end{equation}
\end{cor}
\begin{proof}
Let $a$ be the discrepancy of $F_{\widetilde{{\mathscr Y}}}$. We show $a=n-2$.
Let ${\bf G}amma\simeq \mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$ be a fiber of $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$ outside the diagonal of $G_{{\mathscr Y}}$ and
$l$ a line in a ruling of ${\bf G}amma\simeq \mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$.
Since $K_{{\bf G}amma}\cdot l=-(n-1)$ and $K_{{\bf G}amma}=K_{F_{\widetilde{{\mathscr Y}}}}|_{{\bf G}amma}=(a+1)F_{\widetilde{{\mathscr Y}}}|_{{\bf G}amma}$, we have
$(a+1)F_{\widetilde{{\mathscr Y}}}\cdot l=-(n-1)$.
Therefore we have only to show $F_{\widetilde{{\mathscr Y}}}\cdot l=-1$.
For this we take $l$ so that $l\cap G_{\rho}\not =\emptyset$.
Now we consider the diagram (\ref{eq:STcommrev}).
Since ${\bf G}amma\cap G_{\rho}$ is the diagonal by Proposition \ref{prop:begin},
the strict transform $l'$ is a ruling of a fiber $\simeq \mathbb{P}^1\times \mathbb{P}^1$
of ${\mathscr Y}_2\to {\mathscr Y}_{{\mathscr U}}$.
Therefore $F'_{\widetilde{{\mathscr Y}}}\cdot l'=-1$ where
$F'_{\widetilde{{\mathscr Y}}}$ is the strict transform of $F_{\widetilde{{\mathscr Y}}}$.
Since $G_{\rho}\not \subset F_{\widetilde{{\mathscr Y}}}$,
we have $F_{\widetilde{{\mathscr Y}}}\cdot l=F'_{\widetilde{{\mathscr Y}}}\cdot l'=-1$
as desired.
\end{proof}
By Proposition \ref{prop:begin},
we have a birational map
$F^{(1)}/\mathbb{Z}_2\dashrightarrow F_{\widetilde{{\mathscr Y}}}$
extending the isomorphism
$\oF{(1)}/\mathbb{Z}_2\simeq
F_{\widetilde{{\mathscr Y}}}\setminus \text{\large{\mbox{$\rho\hskip-2pt$}}}\,_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})$.
In the sequel of this section,
we will give an explicit description of this birational map
using the minimal model theory,
which leads to a precise description of $F_{\widetilde{{\mathscr Y}}}$.
We summarize our description in the following diagram:
\begin{equation}
\begin{matrix}\xymatrix{ & {F}^{(3)}\ar[dl]\ar[dr]\\
{F}^{(2)}\ar[ddr]_{\;_{\text{\ensuremath{\mathbb{P}^{n-2}\times\mathbb{P}^{n-2}}-fib.}}}\ar@{-->}[rr]^{\;_{\text{(anti-)flip}}}\ar[dr] & & {F}^{(4)}\ar[d]^{\;_{\text{div. cont.}}}\ar[dl]\\
& {F}^{(1)}\ar[dr] & \widehat{F}\ar[d]\ar[r]_{\;_{\text{\ensuremath{\mathbb{Z}_{2}}-quot.}}} & F_{\widetilde{{\mathscr Y}}}\ar[d]^{\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}|_{F_{\widetilde{{\mathscr Y}}}}}\\
& \widehat{G}'\ar[r]_{_{\text{{diag.blow up}}}} & \widehat{G}\ar[r]^{\;_{\text{\ensuremath{\mathbb{Z}_{2}}-quot.}}} & {G}_{{{\mathscr Y}}}.}
\end{matrix}\label{eq:house}\end{equation}
\subsection{Small resolution and flip}
First we determine the singularities of $F^{(1)}$.
\begin{prop}
\label{pro:F2-proof-appendix-1} $F^{(1)}$ is singular along the
diagonal set
\begin{equation}
\Delta_{{F}^{(1)}}:=\{([V_{n-2}],[V_{n-2}];[V_{n}],[V_{n}])\mid V_{n-2}\subset V_{n}\}\simeq\mathrm{F}(n-2,n,V)\subset{F}^{(1)}.\label{eq:DelF}\end{equation}
The singularity at each point on $\Delta_{F^{(1)}}$ is isomorphic
to the cone over the Segre variety $\mathbb{P}^{1}\times\mathbb{P}^{n-2}$. \end{prop}
\begin{proof}
Recall that $F^{(1)}$ is a subvariety of
$\mathrm{F}(n-2,n,V)^{\times 2}$ and consider the first projection
${F}^{(1)} \to \mathrm{F}(n-2,n,V)$.
Let ${\bf G}amma$ be a fiber of this projection over a point
$([V_{n-2}];[V_n])\in \mathrm{F}(n-2,n,V)$.
We consider ${\bf G}amma$ as a subvariety of $\mathrm{F}(n-2,n,V)$
parameterizing $V'_{n-2}\subset V'_n$
such that $V'_{n-2}\subset V_n$, $V_{n-2}\subset V'_n$ and $\dim (V_{n-2}\cap V'_{n-2})\geq n-3$.
To describe ${\bf G}amma$, we choose a basis $\{{\bf e}_1,\dots,{\bf e}_{n+1}\}$
of $V$ so that
$V_{n-2}=\langle {\bf e}_1,\dots, {\bf e}_{n-2}\rangle$ and
$V_n=\langle {\bf e}_1,\dots, {\bf e}_n\rangle$.
An $(n-2)$-dimensional subspace $V'_{n-2}$ of $V_n$
with $\dim (V_{n-2}\cap V'_{n-2})\geq n-3$
is spanned by $n-3$ vectors in $V_{n-2}$ and a vector
$b_1{\bf e}_1+\dots+b_n{\bf e}_n$ in $V_n$.
We arrange these vectors into an $(n-2)\times n$ matrix as
\begin{equation}
\label{eq:V'n-2}
\begin{pmatrix}
A & \bf{0} & \bf{0}\\
b_1\dots b_{n-2}& b_{n-1} & b_n
\end{pmatrix},
\end{equation}
where the row vectors of $A$ represents the $n-3$ vectors in $V_{n-2}$.
We denote by $q_{ij}$ the Pl\"ucker coordinate of
$V'_{n-2}$
given by the $(n-2)\times (n-2)$ minors of (\ref{eq:V'n-2}) with the
$i$- and $j$-th columns omitted.
Denote by $x_1, \dots,x_{n+1}$, and $y_1,\dots,y_{n+1}$
the homogeneous coordinates of $\mathbb{P}(V)$ and $\mathbb{P}(V^*)$, respectively,
associated to the basis $\{{\bf e}_1,\dots,{\bf e}_{n+1}\}$
and its dual basis.
An $n$-dimensional subspace $V'_n$ of $V$
containing $V_{n-2}$ is of the form
$\{c_{n-1} x_{n-1}+c_n x_n+c_{n+1} x_{n+1}=0\}$,
where we consider $(0,\dots,0, c_{n-1},c_n, c_{n+1})$ as
the coordinates of $[V'_n]$ in $V^*$.
Therefore $V'_n$ contains $V'_{n-2}$ if and only if
$c_{n-1} b_{n-1}+c_n b_n=0$.
From the above considerations,
we can deduce that
\[
{\bf G}amma=
\left\{(q_{ij};y_1,\dots, y_{n+1}) \; \bigg| \; \begin{array}{l}
q_{ij}=0\, \text{for $1\leq i,j \leq n-2$}, \\
\mathrm{rank}\, \begin{pmatrix}
q_{1\,n} & q_{2\,n} & ... & q_{n-2\,n} & -y_n \\
q_{1\,n-1} & q_{2\,n-1}& ... & q_{n-2\,n-1} & y_{n-1}
\end{pmatrix}\leq 1 \end{array} \right\}.
\]
From this, it is easy to see the assertion.
\end{proof}
The cone over $\mathbb{P}^1\times \mathbb{P}^{n-2}$ has exactly two small resolutions;
one of which has a $\mathbb{P}^1$ as the exceptional set and
another has a $\mathbb{P}^{n-2}$ as the exceptional set.
Corresponding to these, we have two small resolutions of $F^{(1)}$.
One of them is given by the following variety $F^{(2)}$:
\[
\begin{aligned}
F^{(2)} & :=\mathrm{F}(n-2,n-1,n,V)\times_{\mathrm{G}(n-1,V)} \mathrm{F}(n-2,n-1,n,V) \\
&=
\left\{ ([V_{n-2}],[V_{n-2}'];[V_{n-1}];[V_{n}],[V_{n}'])\,\bigg\vert\; V_{n-2},V_{n-2}'\subset V_{n-1}\subset V_{n}\cap V_{n}'\right\}.\end{aligned}
\]
We set
\[
\begin{aligned}
\hat{G}' &:=
\mathrm{F}(n-1,n,V)\times_{\mathrm{G}(n-1,V)} \mathrm{F}(n-1,n,V) \\
&=\left\{ ([V_{n-1}];[V_{n}],[V_{n}'])\mid V_{n-1}\subset V_{n}\cap V_{n}'\right\}.
\end{aligned}
\]
$F^{(2)}$ has a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration $F^{(2)}\to\hat{G}'$.
We note that there is a morphism $\widehat{G}'\to\widehat{G}=\mathbb{P}(V^{*})\times\mathbb{P}(V^{*})$
defined by $([V_{n-1}];[V_{n}],[V_{n}'])\mapsto([V_{n}],[V_{n}'])$,
which is nothing but the blow-up of $\widehat{G}$ along the diagonal
$\Delta_G$.
\begin{prop}
\label{pro:F2-proof-appendix-2}\noindent $(1)$ $F^{(2)}$ is smooth.
The natural projection $F^{(2)}\to F^{(1)}$ is a small resolution
with every non-trivial fiber $\gamma$
being isomorphic to $\mathbb{P}^{1}$.
\noindent $(2)$
The normal bundle $\mathcal{N}_{\gamma/F^{(2)}}$ of a non-trivial fiber $\gamma$ of $F^{(2)}\to F^{(1)}$
is isomorphic to
$\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus n-1}\oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus 3n-4}$.
\noindent $(3)$ There is another small resolution $F^{(4)}\to F^{(1)}$,
whose non-trivial fiber is isomorphic to $\mathbb{P}^{n-2}$. $F^{(2)}$ and
$F^{(4)}$ fit into the following diagram$:$\def\Fdiagram{
\begin{xy}
(0,0)*+{F^{(3)}}="Fiii",
(-13,-10)*+{F^{(2)}}="Fii",
(13,-10)*+{F^{(4)}}="Fiv",
(0,-20)*+{F^{(1)},}="Fi",
\ar_p "Fiii";"Fii",
\ar "Fiii";"Fiv",
\ar "Fii";"Fi",
\ar "Fiv";"Fi",
\end{xy}}\begin{equation}
\begin{matrix}{\Fdiagram}\end{matrix}\label{eq:F-diagram}\end{equation}
where $p\colon F^{(3)}\to F^{(2)}$ is the blow-up along the exceptional locus
of $F^{(2)}\to F^{(1)}$, and $F^{(3)}\to F^{(4)}$ is the contraction
of the exceptional divisor of the blow-up $F^{(3)}\to F^{(2)}$ in
another direction.\end{prop}
\begin{proof}
(1) $F^{(2)}$ is smooth since it has a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration
over a smooth variety $\widehat{G}'$.
We show that ${F}^{(2)}\to{F}^{(1)}$
is a small resolution. For a point
\[
([V_{n-2}],[V_{n-2}'];[V_{n-1}];[V_{n}],[V_{n}'])\in F^{(2)},
\]
$V_{n-1}=V_{n-2}+V_{n-2}'$ holds when $V_{n-2}\not=V_{n-2}'$, and also $V_{n-1}=V_{n}\cap V_{n}'$
when $V_{n}\not=V_{n}'$. Hence the morphism ${F}^{(2)}\to{F}^{(1)}$
is isomorphic outside the diagonal set $\Delta_{{F}^{(1)}}$.
The fiber of ${F}^{(2)}\to{F}^{(1)}$ over a point $([V_{n-2}],[V_{n-2}];[V_{n}],[V_{n}])\in\Delta_{{F}^{(1)}}$
is \[
\{([V_{n-2}],[V_{n-2}];[V_{n-1}];[V_{n}],[V_{n}])\mid[V_{n-1}]\in\mathrm{G}(n-1,V),V_{n-2}\subset V_{n-1}\subset V_{n}\}\simeq\mathbb{P}^{1}.\]
We calculate the dimension of the exceptional set of ${F}^{(2)}\to{F}^{(1)}$
as $\dim\Delta_{{F}^{(1)}}+1=3n-3$. Hence ${F}^{(2)}\to{F}^{(1)}$ is
small since $\dim F^{(1)}=4n-4$.
(2) The two small resolutions of $F^{(1)}$ locally coincide with
those of the cone over $\mathbb{P}^1\times \mathbb{P}^{n-3}$.
Therefore the description of the normal bundle of $\gamma$ follows
by that of a non-trivial fiber of the small resolutions of
the cone over $\mathbb{P}^1\times \mathbb{P}^{n-3}$.
(3) Let $D$ be the $p$-exceptional divisor.
Then any fiber of $D$ is $\mathbb{P}^1\times \mathbb{P}^{n-2}$ by
Proposition \ref{pro:F2-proof-appendix-1}.
Let $\gamma\simeq \mathbb{P}^1$ be a fiber of $F^{(2)}\to F^{(1)}$.
Since $K_{F^{(2)}}\cdot \gamma=n-3$ by (2), we see that $p^*K_{F^{(2)}}+(n-3)D$ is nef
and $(p^*K_{F^{(2)}}+(n-3)D)-K_{F^{(3)}}=-D$ is nef and big over $F^{(1)}$,
$p^*K_{F^{(2)}}+(n-3)D$ is semi-ample over $F^{(1)}$
by the Kawamata-Shokurov base point free theorem. Since $p^*K_{F^{(2)}}+D$ is
numerically trivial for any fiber $\gamma'$ of
$\mathbb{P}^1\times \mathbb{P}^{n-2}\to \mathbb{P}^{n-2}$,
the birational morphism $F^{(3)}\to F^{(4)}$ over $F^{(1)}$ defined by a sufficiently high multiple of
$p^*K_{F^{(2)}}+(n-3)D$
contracts $\gamma'$.
Since $-K_{F^{(3)}}\cdot \gamma'=1$ by (3), $F^{(4)}$ is smooth
and $F^{(3)}\to F^{(4)}$ is the blow-up along the image of $D$
(cf.~the proof of Proposition \ref{prop:tildeY} in case $n=3$).
\end{proof}
\subsection{Divisorial contraction}
Let $D^{(2)}$ be the inverse image in $F^{(2)}$ of the diagonal $\Delta_{G}$
of $\widehat{G}$, namely,
\[
D^{(2)}:=\mathrm{F}(n-2,n-1,n,V)\times_{\mathrm{F}(n-1,n,V)} \mathrm{F}(n-2,n-1,n,V).
\]
We denote by $D^{(1)}$ the image on $F^{(1)}$ of
$D^{(2)}$.
It is easy to verify the following properties:
\begin{lem}
\begin{enumerate}[$(1)$]
\item
$D^{(2)}$ is a prime divisor of $F^{(2)}$.
\item
The projection $D^{(2)}\to \mathrm{F}(n-1,n,V)$ is a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration.
\item All the non-trivial fibers of
$F^{(2)}\to F^{(1)}$ are contained in $D^{(2)}$,
namely, they coincide with the fibers of
$D^{(2)}\to D^{(1)}$.
Therefore
$D^{(2)}\to D^{(1)}$ is birational with
any non-trivial fiber being a copy of $\mathbb{P}^{1}$.
\end{enumerate}
\end{lem}
Now we set
\begin{equation}
\label{eqn:FsY''}
\begin{aligned}
{D}^{(4)} & :=
\mathrm{F}(n-3,n-2,n,V)\times_{\mathrm{F}(n-3,n,V)} \mathrm{F}(n-3,n-2,n,V) \\
&=
\left\{
([V_{n-3}];[V_{n-2}],[V'_{n-2}];[V_n],[V_n]) \, \big| \,
\begin{array}{l}
V_{n-3}\subset V_{n-2}\cap V'_{n-2}, \\
V_{n-2}, V'_{n-2} \subset V_n
\end{array}
\right\}.
\end{aligned}
\end{equation}
Then we can deduce easily the following commutative diagram:
\begin{equation}
\label{eq:smallhouse}
\begin{matrix}
\xymatrix{{D}^{(2)}\ar[d]_{\;_{\text{$\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fib.}}}
\ar@{-->}[rr]\ar[dr] & &
{D}^{(4)}\ar[d]^{\;_{\text{$\mathbb{P}^2\times \mathbb{P}^2$-fib.}}}\ar[dl]& & \\
\mathrm{F}(n-1,n,V)\ar[dr] & {D}^{(1)}\ar[d] &
\mathrm{F}(n-3,n,V)\ar[dl]\\
& \Delta_{G}, &}
\end{matrix}
\end{equation}
where
$D^{(4)}\to D^{(1)}$ is birational with
any non-trivial fiber being a copy of $\mathbb{P}^{n-3}$.
\begin{lem}
\label{cla:D_4}
${D}^{(4)}$
is the strict transform on
${F}^{(4)}$ of
${D}^{(2)}$, and
the diagram
$(\ref{eq:smallhouse})$
follows from the restriction of
$(\ref{eq:F-diagram})$.
\end{lem}
\begin{proof}
In a similar way to the case of $F^{(1)}$,
we may show that $D^{(1)}$
is singular along
$\Delta_{{F}^{(1)}}$, and
the singularity at each point on $\Delta_{F^{(1)}}$ is isomorphic
to the cone over the Segre variety $\mathbb{P}^{1}\times\mathbb{P}^{n-3}$ if $n\geq 4$
($D^{(1)}$ is smooth if $n=3$).
Moreover,
by restricting (\ref{eq:F-diagram}) to $D^{(1)}$ and its strict transforms,
we have a similar diagram for $D^{(1)}$.
In particular, the restriction of
(\ref{eq:F-diagram}) gives two small resolutions of $D^{(1)}$ if $n\geq 4$
(for $n=3$, the restriction of $F^{(2)}\to F^{(1)}$ is
the blow-up along $\Delta_{{F}^{(1)}}$, and
the restriction of $F^{(4)}\to F^{(1)}$ is
an isomorphism).
Let us define
\begin{equation}
\label{eqn:FsY'''}
\begin{aligned}
{D}^{(3)} & :=
\mathrm{F}(n\text{--}3,n\text{--}2,n\text{--}1,n,V)\times_{\mathrm{F}(n\text{--}3,n\text{--}1,n,V)}
\mathrm{F}(n\text{--}3,n\text{--}2,n\text{--}1,n,V)\\
&=\left\{
([V_{n-3}];[V_{n-2}],[V'_{n-2}];[V_{n-1}];[V_n],[V_n]) \,\big|\,
\begin{array}{l}
V_{n-3}\subset V_{n-2}, \\
V'_{n-2} \subset V_{n-1} \subset V_n.
\end{array}
\right\} \\
\end{aligned}
\end{equation}
Then $D^{(1)},\dots, D^{(4)}$ fit into the following diagram
with the natural projections:
\def\Fdiagram'{
\begin{xy}
(0,0)*+{D^{(3)}}="Fiii",
(-13,-10)*+{D^{(2)}}="Fii",
(13,-10)*+{D^{(4)}}="Fiv",
(0,-20)*+{D^{(1)}.}="Fi",
\ar "Fiii";"Fii",
\ar "Fiii";"Fiv",
\ar "Fii";"Fi",
\ar "Fiv";"Fi",
\end{xy}}\begin{equation}
\begin{matrix}{\Fdiagram'}\end{matrix}\label{eq:F-diagram'}\end{equation}
By construction,
it is easy to see that $D^{(2)}\to D^{(1)}$ and $D^{(4)}\to D^{(1)}$ are
two small resolutions of $D^{(1)}$ if $n\geq 4$ (for $n=3$,
$D^{(2)}\to D^{(1)}$ is the blow-up along
$\Delta_{{F}^{(1)}}$ and $D^{(4)}\to D^{(1)}$ is an isomorphism).
Therefore the diagram
(\ref{eq:F-diagram'}) coincides with the restriction of
(\ref{eq:F-diagram}) considered above, and hence the assertions follow.
\end{proof}
\begin{prop}
\label{pro:Div-cont-Flat}
There exists a divisorial contraction
${F}^{(4)}\to \widehat{F}$ over
$\widehat{G}$ which
contracts the strict transform ${D}^{(4)}$
of ${D}^{(1)}$ to
the locus isomorphic to the flag variety $\mathrm{F}(n-3,n,V)$.
The discrepancy of
${D}^{(4)}$ is two.
\end{prop}
\begin{proof}
Let $\Delta'_{\mathbb{P}}$ be the inverse image
in $\widehat{G}'$ of $\Delta_{G}$.
Note that $\Delta'_{\mathbb{P}}\simeq \mathrm{F}(n-1,n,V)$.
Let ${\bf G}amma$ be a fiber of the $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration
${D}^{(2)}\to
\Delta'_{\mathbb{P}}$.
Then ${\bf G}amma$ intersects the flipping locus of $F^{(2)}\dashrightarrow F^{(4)}$
along the diagonal transversally.
Take a line $r\subset \mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$
which is contained in a fiber of the second projection ${\bf G}amma\to \mathbb{P}^{n-2}$
and intersects the flipping locus.
$r$ is of the form with some fixed $V_{n-3}\subset V'_{n-2}\subset V_{n-1}\subset V_n$ and moving $V_{n-2}$ as follows:
\[
r:=\{([V_{n-2}], [V'_{n-2}];[V_{n-1}];[V_n],[V_n])\mid V_{n-3}\subset V_{n-2}\subset V_{n-1}\}.
\]
Then its strict transform $r'$ on
${D}^{(4)}$
is contracted by the morphism
${D}^{(4)}\to \mathrm{F}(n-3,n,V)$.
Since ${F}^{(2)}\to
\widehat{G}'$ is a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration and
${D}^{(2)}$ is the pull-back of
$\Delta'_{\mathbb{P}}$,
we see that $K_{{F}^{(2)}}\cdot r=-(n-1)$
and
${D}^{(2)}\cdot r=0$.
By the standard calculations of the changes of the intersection numbers
by the flip,
we have
$K_{{F}^{(4)}}\cdot r'=-(n-1)+(n-3)=-2$
and
${D}^{(4)}\cdot r'=0-1=-1$.
These equalities of the intersection numbers still hold
for any line in a ruling of a fiber of
${D}^{(4)}\to \mathrm{F}(n-3,n,V)$.
We show $-K_{F^{(4)}}+2D^{(4)}$ is relatively nef
over $\widehat{G}$. Let $\gamma$ be a curve on
$F^{(4)}$ which is contracted to a point $t$ on $\widehat{G}$.
If $t\not \in \Delta_{G}$,
then $(-K_{F^{(4)}}+2D^{(4)})\cdot \gamma>0$ since
$D^{(4)}\cap \gamma=\emptyset$ and $F^{(4)}\to \widehat{G}$
is a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$ fibration outside $\Delta_{G}$.
If $t\in \Delta_{G}$ and $\gamma$ is an exceptional curve of $F^{(4)}\to F^{(1)}$, then
$(-K_{F^{(4)}}+2D^{(4)})\cdot \gamma>0$
since $-K_{F^{(4)}}\cdot \gamma>0$ and $D^{(4)}\cdot \gamma>0$.
In the remaining cases, $t\in \Delta_{G}$ and $\gamma\subset D^{(4)}$.
Therefore we have only to consider the relative nefness of
$(-K_{F^{(4)}}+2D^{(4)})|_{D^{(4)}}$ over $\Delta_{G}$. Now we take as $\gamma$
any line in a ruling of a fiber of
${D}^{(4)}\to \mathrm{F}(n-3,n,V)$. As we see in the first paragraph,
$(-K_{F^{(4)}}+2D^{(4)})\cdot \gamma=0$.
Therefore $(-K_{F^{(4)}}+2D^{(4)})|_{D^{(4)}}$ is the pull-back of
some divisor $D_F$ on $\mathrm{F}(n-3,n,V)$. It suffices to show
$D_F$ is relatively nef over $\Delta_{G}$, which is true
since an exceptional curve of $D^{(4)}\to D^{(1)}$
is positive for $(-K_{F^{(4)}}+2D^{(4)})|_{D^{(4)}}$ as above and
is mapped to a curve on a fiber of
$\mathrm{F}(n-3,n,V)\to \Delta_{G}$.
Therefore
$-K_{F^{(4)}}+2D^{(4)}$ is relatively nef
over $\widehat{G}$.
Moreover, by this argument, we see that
$(-K_{F^{(4)}}+2D^{(4)})^{\perp}\cap \overline{\mathrm{NE}}(F^{(4)}/\widehat{G})$
is generated by the numerical class of the curves on fibers of
$D^{(4)}\to \mathrm{F}(n-3,n,V)$. In particular,
$(-K_{F^{(4)}}+2D^{(4)})^{\perp}\cap \overline{\mathrm{NE}}(F^{(4)}/\widehat{G})\subset (K_{F^{(4)}})^{<0}$. Therefore, by Mori theory,
there exists a contraction associated to
this extremal face, which is nothing but the divisorial contraction
contracting $D^{(4)}$ to $\mathrm{F}(n-3,n,V)$
such that
$-K_{F^{(4)}}+2D^{(4)}$ is the pull-back of $-K_{\widehat{F}}$.
Thus
the discrepancy of
${D}^{(4)}$ is two.
\end{proof}
\subsection{$\mathbb{Z}_2$-quotient}
All the above constructions are $\mathbb{Z}_2$-equivariant,
hence we can take $\mathbb{Z}_2$-quotient $\widehat{F}/\mathbb{Z}_2$.
Comparing the morphisms
$a\colon F_{\widetilde{{\mathscr Y}}}\to G_{{{\mathscr Y}}}$
and
$b\colon \widehat{F}/\mathbb{Z}_2
\to G_{{{\mathscr Y}}}$, we obtain
\begin{prop}
\label{prop:coincides}
$\widehat{F}/\mathbb{Z}_2\simeq F_{\widetilde{{\mathscr Y}}}$ over $G_{{\mathscr Y}}$.
\end{prop}
\begin{lem}
\label{lem:dim}
The fiber of $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$
at any point of $G^1_{{\mathscr Y}}$
is of dimension at most $3n-6$.
In particular, codimension of the inverse image in
$F_{\widetilde{{\mathscr Y}}}$ of $G^1_{{\mathscr Y}}$ is at least two.
\end{lem}
\begin{proof}
We consider the diagram (\ref{eq:STcommrev}).
By Proposition \ref{prop:n=3fib} (5),
the fiber of ${\mathscr Y}_2\to {\mathscr Y}_{{\mathscr U}}$ over
a rank one point in a fiber of ${\mathscr Y}_{{\mathscr U}}\to \mathrm{G}(n-3,V)$
is isomorphic to $\mathbb{P}(1^3,2)$.
The fiber of ${\mathscr Y}_{{\mathscr U}}\to {\mathscr Y}$ over a rank one point is
isomorphic to that of ${\mathscr U}\to {\mathscr H}$ over a rank one point $[2V_n]\in \textsc{S}_1$,
and hence is a copy of $\mathrm{G}(n-3,V_n)$.
Therefore,
the fiber of $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$
at any point of $G^1_{{\mathscr Y}}$
is of dimension at most $3+3(n-3)=3n-6$.
\end{proof}
\begin{proof}[{\bf Proof of Proposition $\ref{prop:coincides}$}]
Note that the morphisms $a$ and $b$ are isomorphic outside $G^1_{{\mathscr Y}}$
by Proposition \ref{prop:begin}.
Therefore,
by \cite[Lem.~5.5]{Tk} for example,
it suffices to check the following properties:
\begin{enumerate}
\item
The inverse images of $G^1_{{\mathscr Y}}$
by the morphisms $a$ and $b$ are of codimension at least two.
\item
Both $F_{\widetilde{{\mathscr Y}}}$ and
$\widehat{F}/\mathbb{Z}_2$
are normal.
\item
$-K_{F_{\widetilde{{\mathscr Y}}}}$ and
$-K_{\widehat{F}/\mathbb{Z}_2}$
are $\mathbb{Q}$-Cartier.
\item
$-K_{F_{\widetilde{{\mathscr Y}}}}$ is $a$-ample and
$-K_{\widehat{F}/\mathbb{Z}_2}$ is $b$-ample.
\end{enumerate}
We show these in order.
\noindent (1) The inverse image of $G^1_{{\mathscr Y}}$ by the morphism
$a$
has codimension at least two
in $F_{\widetilde{{\mathscr Y}}}$ by Lemma \ref{lem:dim}
and
the inverse image of $G^1_{{\mathscr Y}}$ by the morphism
$b$
has codimension two
in $\widehat{F}/\mathbb{Z}_2$ by the construction of
$\widehat{F}/\mathbb{Z}_2$.
\noindent
(2) The variety $F_{\widetilde{{\mathscr Y}}}$ is normal.
Indeed, it satisfies the $S_2$ condition
since it is a Cartier divisor on
a smooth variety. It satisfies the $R_1$ condition
since it is a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration
outside the locus of codimension at least two by Proposition
\ref{prop:begin} and Lemma \ref{lem:dim}.
We see that the variety $\widehat{F}/\mathbb{Z}_2$ is normal
by its explicit construction as above.
\noindent
(3), (4) The divisor $-K_{F_{\widetilde{{\mathscr Y}}}}$ is $\mathbb{Q}$-Cartier
since ${F_{\widetilde{{\mathscr Y}}}}$ is a divisor on
the smooth variety $\widetilde{{\mathscr Y}}$.
We see that $-K_{F_{\widetilde{{\mathscr Y}}}}$ is $a$-ample
since the relative Picard number $\rho(\widetilde{{\mathscr Y}}/{\mathscr Y})$ is one
and $a$ is generically a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration.
Arguments for the morphism $b$ are similar. Let us first show that
$-K_{\widehat{F}/\mathbb{Z}_2}$ is
$\mathbb{Q}$-Cartier.
Indeed, by Lemma \ref{pro:Div-cont-Flat},
the discrepancy of ${D}^{(4)}$ is two.
Then, by the Kawamata-Shokurov base point free theorem,
$-K_{{F}^{(4)}}-
2{D}^{(4)}$ is the pull-back of
a Cartier divisor on
$\widehat{F}$,
which turns out to be the anti-canonical divisor
$-K_{\widehat{F}}$.
Thus
$-K_{\widehat{F}/\mathbb{Z}_2}$ is
$\mathbb{Q}$-Cartier.
To show
$-K_{\widehat{F}/\mathbb{Z}_2}$ is $b$-ample,
it suffices to see
the relative Picard number $\rho((\widehat{F}/\mathbb{Z}_2)/G_{{{\mathscr Y}}})$ is one because $b$ is generically a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration.
We compute $\rho((\widehat{F}/\mathbb{Z}_2)/G_{{{\mathscr Y}}})$
using the above construction.
The relative Picard number $\rho({F}^{(2)}/\widehat{G}')$ is
two since ${F}^{(2)}
\to \widehat{G}'$ is
a $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$-fibration
and it is easy to see that
it is the composite of two $\mathbb{P}^{n-2}$-fibrations.
Moreover we have
$\rho^{\mathbb{Z}_2}({F}^{(2)}/
\widehat{G}')=1$
since
rulings
in two directions
of a fiber $\mathbb{P}^{n-2}\times \mathbb{P}^{n-2}$ of
${F}^{(2)}
\to \widehat{G}'$ are exchanged
by the $\mathbb{Z}_2$-action.
Therefore
$\rho^{\mathbb{Z}_2}({F}^{(2)})=3$ since
$\rho^{\mathbb{Z}_2}(\widehat{G}')=2$.
It holds that
$\rho^{\mathbb{Z}_2}({F}^{(4)})=3$ since
the flip preserves the Picard number
and the flip is $\mathbb{Z}_2$-equivariant.
Since a divisorial contraction drops
the Picard number at least by one,
we have
$\rho^{\mathbb{Z}_2}(\widehat{F})\leq 2$.
Now we see that
$\rho((\widehat{F}/\mathbb{Z}_2)/G_{{{\mathscr Y}}})$ is one
since
$\rho(G_{{{\mathscr Y}}})=1$
and
the morphism $\widehat{F}/\mathbb{Z}_2\to G_{{{\mathscr Y}}}$
is non-trivial.
Therefore we conclude $-K_{\widehat{F}/\mathbb{Z}_2}$ is $b$-ample.
\end{proof}
\subsection{Flattening $F^{(3)}\to\widehat{G}'$ \label{sub:Flatenning-F-G}}
We describe the fibers of $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$
in the diagram (\ref{eq:house}).
\begin{prop}
\label{lem:inverse-rk1}
There is a birational morphism $\mathbb{P}(\mathcal{O}_{\mathrm{G}(n-2,V_{n})}\oplus\mathcal{U}_{\mathrm{G}(n-2,V_n)}^{*}(1))\to\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}^{-1}([V_n])$
which contracts the divisor $\mathbb{P}(\mathcal{U}_{V_{n}}^{*}(1))$
to $\mathrm{G}(n-3,V_n)$, where $\mathcal{U}_{\mathrm{G}(n-2,V_n)}$ is the universal subbundle
of the Grassmannian $\mathrm{G}(n-2,V_{n})$.
\end{prop}
\begin{proof}
Since the fiber under consideration
is contained in the branched locus of
$\widehat{F} \to {F}_{\widetilde{{\mathscr Y}}}$,
we have only to describe
the fiber ${\bf G}amma$ of $\widehat{F}\to
\widehat{G}$ over $[V_n]$,
where we consider $[V_n]$ is a point of the diagonal
of $\widehat{G}$.
Let ${\bf G}amma'$ be the restriction over $[V_n]$ of the exceptional locus
of ${F}^{(4)}\to
{F}^{(1)}$.
Then the fiber ${\bf G}amma$ is nothing but the image of ${\bf G}amma'$
under the divisorial contraction $F^{(4)}\to \hat F$.
Since the fiber of
$\Delta_{{F}^{(1)}}\to \widehat{G}$
over $[V_n]$ is $\mathrm{G}(n-2,V_n)$,
the variety ${\bf G}amma'$ is a $\mathbb{P}^{n-2}$-bundle over $\mathrm{G}(n-2,V_n)$.
By the definition of ${D}^{(4)}$,
we see that ${D}^{(4)}|_{{\bf G}amma'}=\mathrm{F}(n-3,n-2,V_n)$,
which is isomorphic to $\mathbb{P}(\mathcal{U}_{\mathrm{G}(n-2,V_n)}^*(-1))$.
Therefore we may write ${\bf G}amma'=\mathbb{P}(\mathcal{A}^*)$,
where $\mathcal{A}$ is the locally free sheaf of rank $n-2$ on
$\mathrm{G}(n-2,V_n)$ with a surjection $\mathcal{A}\to \mathcal{U}_{\mathrm{G}(n-2,V_n)}(1)$.
Now we show the kernel of $\mathcal{A}\to \mathcal{U}_{\mathrm{G}(n-2,V_n)}(1)$ is
$\mathcal{O}_{\mathrm{G}(n-2,V_n)}(2)$.
Note that the image of $\mathrm{F}(n-3,n-2,V_n)$
by the divisorial contraction
${{F}^{(4)}}\to
{\widehat{F}}$ is $\mathrm{G}(n-3, V_n)$.
Therefore, since the discrepancy of ${D}^{(4)}$
for ${F}^{(4)}\to
\widehat{F}$ is two,
and $\mathcal{O}_{\mathbb{P}(\mathcal{U}_{\mathrm{G}(n-2,V_n)}^*(-1))}(1)$ is the pull-back of $\mathcal{O}_{\mathrm{G}(n-3, V_n)}(1)$,
we see that ${D}^{(4)}|_{{\bf G}amma'}=
H_{{\mathbb{P}(\mathcal{A}^*)}}-2L$,
where $L$ is the pull-back of $\mathcal{O}_{\mathrm{G}(n-2,V_n)}(1)$.
Thus
the kernel of $\mathcal{A}\to \mathcal{U}_{\mathrm{G}(n-2,V_n)}(1)$ is $\mathcal{O}_{\mathrm{G}(n-2,V_n)}(2)$.
Since the exact sequence
$0\to \mathcal{O}_{\mathrm{G}(n-2,V_n)}(2)\to \mathcal{A}\to \mathcal{U}_{\mathrm{G}(n-2,V_n)}(1)\to 0$ splits,
we have $\mathcal{A}^*\simeq \mathcal{O}_{\mathrm{G}(n-2,V_n)}(-2)\oplus \mathcal{U}_{\mathrm{G}(n-2,V_n)}^*(-1)
\simeq (\mathcal{O}_{\mathrm{G}(n-2,V_n)}\oplus \mathcal{U}^*_{\mathrm{G}(n-2,V_n)}(1))\otimes \mathcal{O}_{\mathrm{G}(n-2,V_n)}(-2)$.
\end{proof}
We have obtained the following diagram:
\begin{equation}
\begin{matrix}\xymatrix{F^{(3)}\;\ar[r]\ar[d] & \; F^{(4)}\;\ar[d]\ar[r]^{\;_{\text{{\rm div. cont.}}}} & \;\widehat{F}\;\ar[d]\ar[r]^{\;_{\mathbb{Z}_{2}\text{{\rm -quot.}}}} & \; F_{\widetilde{{\mathscr Y}}}\ar[d]\\
\widehat{G}'\;\ar[r] & \;\widehat{G}\;\ar@{=}[r] & \;\widehat{G}\;\ar[r]^{\;_{\mathbb{Z}_{2}\text{{\rm -quot.}}}} & \; G_{{\mathscr Y}}}.
\end{matrix}\label{eq:flat-diagram}
\end{equation}
We show that $F^{(3)}\to\widehat{G}'$
gives a flattening of the fibration $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$.
\begin{prop}
\label{pro:Components-A-B}
$F^{(3)}\to\widehat{G}'$ is flat.
More precisely,
the fiber $Fib^{(3)}(V_{n-1},V_{n},V_{n}')$
of $F^{(3)}\to\widehat{G}'$ over a point $([V_{n-1}];[V_{n}],[V_{n}'])$
have the following descriptions\,$:$
\begin{myitem2}\item[$(1)$] $Fib^{(3)}(V_{n-1},V_{n},V_{n}')\simeq\mathbb{P}(V_{n-1}^{*})\times\mathbb{P}(V_{n-1}^{*})$
if $V_{n}\not=V_{n}'$.
\item[$(2)$] $Fib^{(3)}(V_{n-1},V_{n},V_{n})$
consists of two irreducible components $A$ and $B$ with \[
A=\mathbb{P}(\mathcal{O}_{\mathrm{G}(n-2,V_{n})}\oplus\mathcal{U}_{\mathrm{G}(n-2,V_{n})}^{*}(1))\big\vert_{\mathrm{G}(n-2,V_{n-1})},\; B=Bl_{\Delta}\mathbb{P}(V_{n-1}^{*})\times\mathbb{P}(V_{n-1}^{*}),\]
where $A$ is the restriction of the projective bundle as in Lemma
$\ref{lem:inverse-rk1}$ over $\mathrm{G}(n-2,V_{n-1})\subset\mathrm{G}(n-2,V_{n})$.
\item[$(3)$] The intersection $E_{AB}:=A\cap B$ is given by $E_{AB}=\mathbb{P}(\mathcal{U}_{\mathrm{G}(n-2,V_{n})}^{*}(1))\big\vert_{\mathrm{G}(n-2,V_{n-1})}\simeq\mathbb{P}(T_{\mathbb{P}(V_{n-1}^{*})})$
in $A$. Also, $E_{AB}$ in $B$ is the exceptional divisor of $Bl_{\Delta}\mathbb{P}(V_{n-1}^{*})\times\mathbb{P}(V_{n-1}^{*})$.
\end{myitem2}\end{prop}
\begin{proof}
Part (1) follows from the construction of
${F}^{(2)}\to \widehat{G}'$.
We show Part (2).
The fiber of ${F}^{(2)}\to \widehat{G}'$
over a point $([V_{n-1}];[V_n],[V_n])$ is $\mathbb{P}(V_{n-1}^*)\times \mathbb{P}(V_{n-1}^*)$.
The intersection of the fiber $\mathbb{P}(V_{n-1}^*)\times \mathbb{P}(V_{n-1}^*)$ with
the exceptional locus of ${F}^{(2)}\to {F}^{(1)}$ is
\[
\{
([V_{n-2}],[V_{n-2}];[V_{n-1}];[V_n],[V_n])\mid
V_{n-2} \subset V_{n-1}
\}
\simeq \mathbb{P}^{n-2},
\]
which is nothing but the diagonal of
$\mathbb{P}(V_{n-1}^*)\times \mathbb{P}(V_{n-1}^*)$. Therefore we have $B$ as
an irreducible component of the fiber
of ${F}^{(3)}\to \widehat{G}'$
over the point $([V_{n-1}];[V_n],[V_n])$.
Another component $A$ is a $\mathbb{P}^{n-2}$-bundle over
the diagonal of $\mathbb{P}(V_{n-1}^*)\times \mathbb{P}(V_{n-1}^*)$
since the exceptional divisor of ${F}^{(3)}\to
{F}^{(2)}$ is a $\mathbb{P}^{n-2}$-bundle
over the exceptional locus of ${F}^{(2)}\to
{F}^{(1)}$.
Since the image on ${F}^{(1)}$
of the diagonal $\Delta_{V_{n-1}}$
of $\mathbb{P}(V_{n-1}^*)\times \mathbb{P}(V_{n-1}^*)$ is equal to
$\mathrm{G}(n-2,V_{n-1})=\mathbb{P}(V_{n-1}^*)$ in $\mathrm{G}(n-2,V_n)$,
the image of $A$ in ${F}^{(4)}$
is the restriction of $\mathbb{P}(\mathcal{O}_{\mathrm{G}(n-2,V_n)}\oplus \mathcal{U}^*_{\mathrm{G}(n-2,V_n)}(1))$
over $\mathrm{G}(n-2,V_{n-1})$.
Therefore we obtain the description of $A$ as in the statement
since $\mathcal{U}^*_{\mathrm{G}(n-2,V_n)}|_{\mathbb{P}(V_{n-1}^*)}\simeq T_{\mathbb{P}(V_{n-1}^*)}(-1)$
and $\mathcal{N}_{\Delta_{V_{n-1}}}\cong T_{\mathbb{P}(V_{n-1}^*)}$ for the normal
bundle $\mathcal{N}_{\Delta_{V_{n-1}}}$ of the diagonal $\Delta_{V_{n-1}}$.
It is easy to see the assertion about $A\cap B$.
\end{proof}
\begin{rem}
In \cite[Thm.~3.7]{CHK}, the authors studied
the relationship between the Hilbert scheme ${\mathscr Y}_0$ of conics in
$\mathrm{G}(n-1,V)$ and the stable map compactification of the space of smooth conics in $\mathrm{G}(n-1, V)$, which we denote by ${\mathscr Y}_{\rm{st}}$.
We interpret this by our study of the birational geometry of ${\mathscr Y}_0$.
By Remark \ref{rem:blup},
${\mathscr Y}_0\to \widetilde{{\mathscr Y}}$ is the blow-up along $G_{\sigma}$.
By the blow-up ${\mathscr Y}_0\to \widetilde{{\mathscr Y}}$,
the fiber $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}^{-1}([V_n])$ becomes
the $\mathbb{P}^{n-2}$-bundle $\mathbb{P}(\mathcal{O}_{\mathrm{G}(n-2,V_{n})}\oplus\mathcal{U}_{\mathrm{G}(n-2,V_{n})}^{*}(1))\to \mathrm{G}(n-2,V_n)$
as in Proposition \ref{lem:inverse-rk1}.
Therefore the strict transform ${\bf G}amma$ of $\text{\large{\mbox{$\rho\hskip-2pt$}}}_{\widetilde{{\mathscr Y}}}^{-1}(G^1_{{\mathscr Y}})$
is a $\mathbb{P}^{n-2}$-bundle to $\mathrm{F}(n-2,n,V)$, where we note that
$\mathrm{F}(n-2,n,V)$ is isomorphic to the Hilbert scheme of lines in $\mathrm{G}(n-1,V)$.
Let $\widetilde{{\mathscr Y}}_0\to {\mathscr Y}_0$ be the blow-up along ${\bf G}amma$.
Then ${\mathscr Y}_{\rm{st}}$ is obtained by contracting the exceptional divisor
over ${\bf G}amma$ to a $\mathbb{P}^2$-bundle over $\mathrm{F}(n-2,n,V)$.
\end{rem}
\subsection{The component $A$ of the fiber $Fib^{(3)}(V_{n-1},V_{n},V_{n})$ }
Let us fix $V_{n-1}$ and $V_{n}$ such that $V_{n-1}\subset V_n$
and consider the exceptional set $A$
in the fiber
\[
\text{$Fib^{(3)}(V_{n-1},V_{n},V_{n})\simeq A\cup B$ over $([V_{n-1}];[V_{n}],[V_{n}])\in\widehat{G}'$.}
\] Since $A$ is $\mathbb{Z}_{2}$-invariant,
this determines the corresponding set $A_{\widetilde{{\mathscr Y}}}$ in
the fiber $F_{\widetilde{{\mathscr Y}}}\to G_{{\mathscr Y}}$ over $[V_{n}]$.
We note that $A\simeq \mathbb{P}(\mathcal{O}_{\mathrm{G}(n-2,V_{n-1})}\oplus\mathcal{U}_{\mathrm{G}(n-2,V_{n-1})}^{*}(1))\simeq\mathbb{P}(\mathcal{O}_{\mathbb{P}(V_{n-1}^{*})}\oplus T_{\mathbb{P}(V_{n-1}^{*})})$
by Proposition \ref{lem:inverse-rk1}.
\begin{prop}
\label{pro:Prop-C-A}
Define $A_{{\mathscr Y}_{2}}$ to be the strict transform
of $A_{\widetilde{{\mathscr Y}}}\subset\widetilde{{\mathscr Y}}$ under ${\mathscr Y}_{2}\to\widetilde{{\mathscr Y}}$,
and $A_{{\mathscr Y}_{3}}$ by the image of $A_{{\mathscr Y}_{2}}$ under the morphism
${\mathscr Y}_{2}\to{\mathscr Y}_{3}$.
\begin{enumerate}[$(1)$]
\item The morphism $A\to A_{\widetilde{{\mathscr Y}}}$
contracts the divisor $E_{AB}=\mathbb{P}(\mathcal{U}_{\mathrm{G}(n-2,V_{n-1})}^{*}(1))$ to
$\mathrm{G}(n-3,V_{n-1})$.
\item
The image $\mathrm{G}(n-3,V_{n-1})$ of $E_{AB}$ on $A_{\widetilde{{\mathscr Y}}}$
is the locus of $\sigma$-planes.
The locus $s_A$ of $\rho$-conics in $A$ is a section
of $A\to \mathrm{G}(n-2,V_{n-1})$
corresponding to an injection
$\mathcal{O}_{\mathbb{P}(V_{n-1}^{*})}\to \mathcal{O}_{\mathbb{P}(V_{n-1}^{*})}\oplus T_{\mathbb{P}(V_{n-1}^{*})}$.
\item $A_{{\mathscr Y}_{2}}\to A_{\widetilde{{\mathscr Y}}}$ is the blow-up
along the image $\tilde{s}_{A}$ in $A_{\widetilde{{\mathscr Y}}}$ of the
section $s_{A}$.
\item Let $\widehat{A}:=Bl_{s_{A}} A$ be the blow-up $\hat{A}$
of $A$ along the section $s_{A}$. There exists a natural morphism
$\widehat{A}\to A_{{\mathscr Y}_{2}}$, which is the blow-up of $A_{{\mathscr Y}_{2}}$
along the singular locus of $A_{{\mathscr Y}_{2}}$.
\item $A_{{\mathscr Y}_{3}}\simeq A_{{\mathscr Y}_{2}}$ and $\text{\large{\mbox{$\pi\hskip-2pt$}}}_{A_{3}}\colon A_{{\mathscr Y}_{3}}\to\mathrm{G}(n-3,V_{n-1})$
is a quadric cone fibration,
where $\text{\large{\mbox{$\pi\hskip-2pt$}}}_{A_{3}}:=\text{\large{\mbox{$\pi\hskip-2pt$}}}_{{\mathscr Y}_{3}}\vert_{A_{{\mathscr Y}_{3}}}$.
\def\AAAA{
\begin{xy}
(0,0)*+{\widehat{A}}="Ah",
(-15,-10)*+{A_{{\mathscr Y}_2}}="Aii",
(15,-10)*+{A}="A",
(-30,-10)*+{A_{{\mathscr Y}_3}}="Aiii",
(-30,-25)*+{\mathrm{G}(n-3,V_{n-1})}="Pv",
(0,-20)*+{A_{\widetilde{{\mathscr Y}}}}="At",
(15,-25)*+{\mathbb{P}(V_{n-1}^*)}="Pvs",
\ar "Ah";"A",
\ar "Ah";"Aii",
\ar "A";"At",
\ar "A";"Pvs",
\ar "Aii";"At",
\ar_{\simeq} "Aii";"Aiii",
\ar_{\text{\large{\mbox{$\pi\hskip-2pt$}}}_{A_{3}}} "Aiii";"Pv",
\end{xy} }\[
\begin{matrix}{\AAAA}\end{matrix}\]
\end{enumerate}\end{prop}
\begin{proof}
(1) follow from Proposition \ref{lem:inverse-rk1}.
(4) is clear and (3) follows once we show (2)
since ${\mathscr Y}_2\to \widetilde{{\mathscr Y}}$ is
the blow-up along $G_{\rho}$ by Proposition \ref{prop:gendescr} (1) and
$\widetilde{s}_A=G_{\rho}\cap A_{\widetilde{{\mathscr Y}}}$.
To show (2) and (5),
as in the discussion of the subsections \ref{subsection:dimV=4}
and \ref{sub:tildeY-Y}, we first consider the case where $\dim V=4$ and
then use the results to the general cases.
In case $\dim V=4$,
$A_{{\mathscr Y}_2}=A_{\widetilde{{\mathscr Y}}}$ is isomorphic to $\mathbb{P}(1^2,2)$ by
Proposition \ref{pro:Components-A-B}.
Moreover,
by Proposition \ref{prop:n=3fib} (5) (d),
the vertex corresponds to a $\sigma$-plane and
$A_{\widetilde{{\mathscr Y}}}\cap G_{\rho}$ is a $\mathbb{P}^1$ which is the image of a section of
$A\simeq \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(2))$
associated to an injection $\mathcal{O}_{\mathbb{P}^1}\to\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(2)$.
Therefore, we also have $A_{{\mathscr Y}_3}\simeq A_{{\mathscr Y}_2}\simeq \mathbb{P}(1^2,2)$.
Now we have finished the proof in case $\dim V=4$.
We turn to the general cases.
First we immediately obtain (5) by the results in case $n=4$
since ${\mathscr Y}_3\to \mathrm{G}(n-3,V)$ is the family of ${\mathscr Y}_3=\mathrm{G}(3,\wedge^2 (V/V_{n-3}))$ for
$4$-dimensional spaces $V/V_{n-3}$.
By comparing the singularities between $A_{{\mathscr Y}_2}$ and $A_{\widetilde{{\mathscr Y}}}$, we see that the image of $E_{AB}$ is the locus of $\sigma$-planes.
Then the locus $s_A$ of $\rho$-conics in $A$ is disjoint from $E_{AB}$.
Since $s_A$ is a section of $A\to \mathrm{G}(n-2,V_{n-1})$,
$s_A$ corresponds to an injection
$\mathcal{O}_{\mathbb{P}(V_{n-1}^{*})}\to \mathcal{O}_{\mathbb{P}(V_{n-1}^{*})}\oplus T_{\mathbb{P}(V_{n-1}^{*})}$.
Finally we show ${\mathscr P}_{\rho}\cap A_{{\mathscr Y}_{3}}\simeq\mathbb{P}({\mathfrak Q}_{V_{n-1}})$.
Note that ${\mathscr P}_{\rho}\cap A_{{\mathscr Y}_{3}}$ is isomorphic to the exceptional
divisor $G$ of $\widehat{A}\to A$, which we determine now. Let $\mathcal{I}_{s_{A}}$
be the ideal sheaf of the section $s_{A}$ in $A$. Note that $\mathcal{O}_{\mathbb{P}(\mathcal{O}_{\mathbb{P}(V_{n-1}^{*})}\oplus T_{\mathbb{P}(V_{n-1}^{*})})}(1)|_{s_{A}}=\mathcal{O}_{s_{A}}$.
Tensoring $0\to\mathcal{I}_{s_{A}}\to\mathcal{O}_{A}\to\mathcal{O}_{s_{A}}\to0$ with $\mathcal{O}_{\mathbb{P}(\mathcal{O}_{\mathbb{P}(V_{n-1}^{*})}\oplus T_{\mathbb{P}(V_{n-1}^{*})})}(1)$
and pushing forward to $\mathbb{P}(V_{n-1}^{*})$, we see that $\mathcal{I}_{s_{A}}/\mathcal{I}_{s_{A}}^{2}\simeq\Omega_{\mathbb{P}(V_{n-1}^{*})}$.
Therefore $G$ is isomorphic to $\mathbb{P}(T_{\mathbb{P}(V_{n-1}^{*})})$. Since
$\mathbb{P}(T_{\mathbb{P}(V_{n-1}^{*})})$ is isomorphic to the incident variety $\{([V_{n-3}],[V_{n-2}])\mid V_{n-3}\subset V_{n-2}\}\subset\mathbb{P}(V_{n-1})\times\mathbb{P}(V_{n-1}^{*})$,
it follows that $\mathbb{P}(T_{\mathbb{P}(V_{n-1}^{*})})$ is isomorphic to $\mathbb{P}(T_{\mathbb{P}(V_{n-1})}(-1))$.
\end{proof}
\begin{rem}
\label{rem:freedescr}
Based on Remark \ref{rem:WPS} and Proposition \ref{pro:Prop-C-A},
we can obtain the following description of $A_{{\mathscr Y}_3}\to \mathrm{G}(n-3,V_{n-1})$,
which follows by noting the fiber of ${\mathscr Y}_3\to \mathrm{G}(n-3,V)$ over
$[V_{n-3}]$ is isomorphic to $\mathrm{G}(3,\wedge^2 (V/V_{n-3}))$:
Take a point $[V_{n-3}]\in \mathrm{G}(n-3,V_{n-1})$ and
let ${\bf G}amma$ be the fiber of $A_{{\mathscr Y}_3}\to \mathrm{G}(n-3,V_{n-1})$ over $[V_{n-3}]$.
The vertex of the quadric cone ${\bf G}amma$ corresponds to
the $\sigma$-plane ${\rm P}_{V_n/V_{n-3}}=\{\mathbb{C}^2\subset V_n/V_{n-3}\}$,
where we denote by ${\rm P}_{V_n/V_{n-3}}$ the $\sigma$-plane in
$\mathrm{G}(3,\wedge^2 (V/V_{n-3}))$ corresponding to the $\sigma$-plane
${\rm P}_{V_{n-3}V_n}$.
Points $[{\rm P}_{V_{n-2}/V_{n-3}}]$ which correspond to $\rho$-planes and are contained in ${\bf G}amma$ satisfy $V_{n-3}\subset V_{n-2}$,
where we follows the same convention for $\rho$-planes as for $\sigma$-planes.
Since ${\bf G}amma$ is the cone over the Veronese curve $v_2(\mathbb{P}(V_{n-1}/V_{n-3}))$,
it is swept out by lines joining $[{\rm P}_{V_n/V_{n-3}}]$ and $[{\rm P}_{V_{n-2}/V_{n-3}}]$ such that
$V_{n-3}\subset V_{n-2}\subset V_{n-1}$.
By this description, we see that
${\mathscr P}_{\rho}\cap A_{{\mathscr Y}_{3}}\simeq\mathbb{P}({\mathfrak Q}_{V_{n-1}})\simeq \mathrm{F}(n-3,n-2,V_{n-1}),$
where ${\mathfrak Q}_{V_{n-1}}$ is the universal quotient bundle on $\mathrm{G}(n-3,V_{n-1})$.
\end{rem}
\mathrm{ap}pendix
\section{\label{sec:Appendix-B}{\bf Proof of Proposition \ref{lem:appendixB-UU-solve}}}
\label{app:aU}
\begin{proof}[Proof of Proposition $\ref{lem:appendixB-UU-solve}$]
If $\dim a_{U}\geq n-3$, it is easy to see $\mathrm{rank}\,\varphi_{U}\leq1$
by writing down $U$ using a basis of $a_U$.
This shows one direction of (1).
We show the converse direction of (1).
If $\varphi_U=0$, then $\mathbb{P}(U)$ is a plane contained in $\mathrm{G}(n-1,V)$, and hence
is a $\rho$- or $\sigma$-plane. Therefore, we see that $\dim a_U\geq n-3$ holds
by (\ref{eq:xxxxx}).
Now we assume that $\mathrm{rank}\, \varphi_U=1$. Then $q:={\mathscr Y}_3\cap \mathbb{P}(U)$ is the $\tau$-conic
which is the zero locus of $\varphi_U$.
We will argue depending on the rank of the $\tau$-conic $q$.
Assume that $\mathrm{rank}\, q=3$.
Note that the dual of the universal subbundle $\text{\eu U}^{*}$ on $\mathrm{G}(n-1,V)$ restricts
as $\text{\eu U}^{*}|_{q}\simeq\mathcal{O}(1)_{\mathbb{P}^{1}}^{\oplus 2}\oplus\mathcal{O}_{\mathbb{P}^{1}}^{\oplus n-3}$,
or $\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}}^{\oplus n-2}$ since $\text{\eu U}^{*}$
is generated by its global sections and $\deg\text{\eu U}^{*}|_{q}=\deg \mathcal{O}_{\mathrm{G}(n-1,V)}(1)|_q=2$ since $q$ is a conic.
Let $Q$ be the image of $\mathbb{P}({\text{\eu U}}|_{q})$ under the natural map $\varphi_{\text{\eu U}}\colon \mathbb{P}(\text{\eu U})\to \mathbb{P}(V)$.
Then there are two possibilities; (i) the degree of $\mathbb{P}({\text{\eu U}}|_{q})\to Q$
is two and $Q$ is a $(n-1)$-plane, i.e., a quadric of rank 1, or (ii)
the degree of $\mathbb{P}({\text{\eu U}}|_{q})\to Q$ is one and $Q$ is a quadric
of rank $4$ or $3$ depending on
$\text{\eu U}^{*}|_{q}\simeq\mathcal{O}(1)_{\mathbb{P}^{1}}^{\oplus 2}\oplus\mathcal{O}_{\mathbb{P}^{1}}^{\oplus n-3}$,
or $\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}}^{\oplus n-2}$ respectively.
The case (i) is
excluded since
if $Q$ were a $(n-1)$-plane $\mathbb{P}(V_{n})$, then $q\subset\{[U]\in\mathrm{G}(n-1,V)\mid U\subset V_{n}\}$
and $q$ would be a $\sigma$-conic
by definition, a contradiction.
The case (ii) with
$\text{\eu U}^{*}|_{q}\simeq\mathcal{O}_{\mathbb{P}^{1}}(2)\oplus\mathcal{O}_{\mathbb{P}^{1}}^{\oplus n-2}$
also is excluded since if this happened, then $q$ would be a $\rho$-conic.
Therefore we have the case (ii) with
$\text{\eu U}^{*}|_{q}\simeq\mathcal{O}(1)_{\mathbb{P}^{1}}^{\oplus 2}\oplus\mathcal{O}_{\mathbb{P}^{1}}^{\oplus n-3}$.
Then we see that $q$ is a connected family of $(n-1)$-planes
in the rank four quadric $Q$.
Since all the rank four quadrics are $\mathrm{SL}\,(V)$-equivalent,
we see that any rank three conic $q$ is also $\mathrm{SL}\,(V)$-equivalent.
Therefore we may assume that $q$ is of the form
as in Example \ref{ex:conics}. Then it is easy to see that
$a_{U}=\langle {\bf e}_4,\dots, {\bf e}_n\rangle$ and hence $\dim a_U=n-3$.
Assume that $q$ is of rank two. Then $q$ is of the form as in
Example \ref{ex:ranktwo}. Since $q$ is a $\tau$-conic,
$V_{n-2}\not =V'_{n-2}$ and $V_n\not =V'_n$.
Then it is easy to see that $a_U=V_{n-2}\cap V'_{n-2}$ and hence
$\dim a_U=n-3$.
Finally we assume that
$q$ is of rank one. Then the support of $q$ is a line $l$ and
$l$ is of the form as in Example \ref{ex:ranktwo}.
Let ${\bf e}_1,\dots,{\bf e}_{n-2}$ be a basis of $V_{n-2}$ and
${\bf e}_1,\dots,{\bf e}_n$ be a basis of $V_n$.
Then $l$ is spanned by ${\bf e}_1\wedge \dots\wedge {\bf e}_{n-2}\wedge {\bf e}_{n-1}$ and ${\bf e}_1\wedge \dots\wedge {\bf e}_{n-2}\wedge {\bf e}_{n}$.
Now we pass from $\wedge^{n-1} V$ to $\wedge^2 V^*$
and let $U'$ and $l'$ the $3$-plane in $\wedge^2 V^*$ and
the line in $\mathbb{P}(\wedge^2 V^*)$.
Then $l'$ is spanned by ${\bf v}_1:={\bf e}^*_n\wedge {\bf e}^*_{n+1}$ and
${\bf v}_2:={\bf e}^*_{n-1}\wedge {\bf e}^*_{n+1}$.
Let ${\bf w}:=\sum_{i<j} a_{ij} {\bf e}^*_i\wedge {\bf e}^*_j$ be a vector such that ${\bf v}_1, {\bf v}_2, {\bf w}$ span $U'$.
Then $\mathrm{G}(2,V^*)\cap \mathbb{P}(U')$ is a rank one conic.
Solving the equation
\[
(\lambda_1 {\bf v}_1+\lambda_2 {\bf v}_2+\mu{\bf w})\wedge
(\lambda_1 {\bf v}_1+\lambda_2 {\bf v}_2+\mu{\bf w})=0,
\]
we obtain the equation of $\mathrm{G}(2,V^*)\cap \mathbb{P}(U')$.
Thus
$\mathrm{G}(2,V^*)\cap \mathbb{P}(U')$ is a rank one conic iff
${\bf v}_1\wedge {\bf w}={\bf v}_2\wedge {\bf w}=0$.
Therefore
we have ${\bf w}=a_{n-1n}{\bf e}^*_{n-1}\wedge {\bf e}^*_n+
(\sum_{i\leq n-2} a_{in+1} {\bf e}^*_i)\wedge {\bf e}^*_{n+1}$.
Taking these back to $\wedge^{n-1} V$, we see that
$U$ is spanned by
${\bf e}_1\wedge \dots\wedge {\bf e}_{n-2}\wedge {\bf e}_{n-1}$ and ${\bf e}_1\wedge \dots\wedge {\bf e}_{n-2}\wedge {\bf e}_{n}$
and
${\bf w}=a_{n-1n}{\bf e}_1\wedge\dots \wedge {\bf e}_{n-2}+
\sum_{i\leq n-2} a_{in+1} {\bf e}_1\wedge \dots \wedge \check{\bf e}_i\wedge\dots
{\bf e}_{n}$, where $\check{\bf e}_i$ means that ${\bf e}_i$ is removed.
Therefore it is easy to see that
$a_U$ is spanned by vectors $\sum b_i {\bf e}_i$ with
$b_{n-1}=b_n=b_{n+1}=0$ and
$\sum (-1)^{n-i} a_{in+1} b_i=0$.
Therefore $\dim a_U\geq n-3$.
\end{proof}
\section{\label{sec:Appendix-A}{\bf The {}``double spin'' coordinates of $\mathrm{G}(3,6)$}}
In this appendix, we set $V_{4}=\mathbb{C}^{4}$ with the standard
basis. We can write the irreducible decomposition (\ref{eq:spin})
as \[
\wedge^{3}(\wedge^{2}V_{4})=\Sigma^{(3,1,1,1)}V_{4}\,\oplus\,\Sigma^{(2,2,2,0)}V_{4}\simeq\mathsf{S}^{2}V_{4}\,\oplus\,\mathsf{S}^{2}V_{4}^{*},\]
where $\Sigma^{\beta}$ is the Schur functor. We define the projective
space $\mathbb{P}(\wedge^{3}(\wedge^{2}V_{4}))=\mathbb{P}(\mathsf{S}^{2}V_{4}\,\oplus\,\mathsf{S}^{2}V_{4}^{*})$.
The homogeneous coordinate of $\mathbb{P}(\mathsf{S}^{2}V_{4}\,\oplus\,\mathsf{S}^{2}V_{4}^{*})$
is naturally introduced by $[v_{ij},w_{kl}]$, where $v_{ij}$ and
$w_{kl}$ are entries of $4\times4$ symmetric matrices. Let $\mathcal{I}=\left\{ \{i,j\}\mid1\leq i<j\leq4\right\} $
the index set to write the standard basis of $\wedge^{2}V_{4}$, then
the homogeneous coordinate of $\mathbb{P}(\wedge^{3}(\wedge^{2}V_{4}))$
is naturally given by the $[p_{IJK}]$ where $p_{IJK}$ is totally
anti-symmetric for the indices $I,J,K\in\mathcal{I}.$ These two coordinates
are related by the above irreducible decomposition. Focusing on the
different symmetry properties of the Schur functors, it is rather
straightforward to decompose $p_{IJK}$ into the two components. When
we use the signature function defined by ${\bf e}_{i_{1}}\wedge{\bf e}_{i_{2}}\wedge{\bf e}_{i_{3}}\wedge{\bf e}_{i_{4}}=\epsilon^{i_{1}i_{2}i_{3}i_{4}}{\bf e}_{1}\wedge{\bf e}_{2}\wedge{\bf e}_{3}\wedge{\bf e}_{4}$
for a basis ${{\bf e}_{1},..,{\bf e}_{4}}$ of $V_{4}$, they are
given by \begin{equation}
v_{ij}=\frac{1}{6}\sum_{k,l,m,n}\epsilon^{klmn}p_{[ik][jl][mn]},\quad w_{kl}=\frac{1}{6}\sum_{a,b,c}\sum_{m,n,q}\epsilon^{kabc}\epsilon^{lmnq}p_{[am][bn][cq]},\label{eq:vw-general-fromula}\end{equation}
where the square brackets in $p_{[ij][kl][mn]}$ represents the anti-symmetric
extensions of the indices, i.e., $p_{[ij][J][K]}=p_{\{ij\}[J][K]}$
for $i<j$ while $p_{[ij][J][K]}=-p_{\{ji\}[J][K]}$ for $i\geq j$.
For convenience, we write them in the following (symmetric) matrices:
\begin{equation}
\begin{aligned}v=(v_{ij})=\left(\begin{matrix}2p_{{\bf 124}} & p_{{\bf 134}}+p_{{\bf 125}} & p_{{\bf 234}}+p_{{\bf 126}} & p_{{\bf 146}}-p_{{\bf 245}}\\
& 2p_{{\bf 135}} & p_{{\bf 235}}+p_{{\bf 136}} & p_{{\bf 156}}-p_{{\bf 345}}\\
& & 2p_{{\bf 236}} & p_{{\bf 256}}-p_{{\bf 346}}\\
& & & 2p_{{\bf 456}}\end{matrix}\right),\\
w=(w_{kl})=\left(\begin{matrix}2p_{{\bf 356}} & -p_{{\bf 346}}-p_{{\bf 256}} & p_{{\bf 345}}+p_{{\bf 156}} & p_{{\bf 235}}-p_{{\bf 136}}\\
& 2p_{{\bf 246}} & -p_{{\bf 245}}-p_{{\bf 146}} & p_{{\bf 126}}-p_{{\bf 234}}\\
& & 2p_{{\bf 145}} & p_{{\bf 134}}-p_{{\bf 125}}\\
& & & 2p_{{\bf 123}}\end{matrix}\right),\end{aligned}
\label{eq:vw-plucker}\end{equation}
where we ordered the index set $\mathcal{I}$ as $\{{\bf 1},{\bf 2},...,{\bf 6}\}=\{\{1,2\},\{1,3\},\{2,3\},\{1,4\},$
$\{2,4\},$ $\{3,4\}\}$. Inverting the relations (\ref{eq:vw-plucker}),
we can write the Pl\"ucker relations among $p_{IJK}$ in terms of
the entries of $v$ and $w$. After some algebra, we find:
\begin{prop}
\label{prop:B1}
The Pl\"ucker ideal $I_{G}$ of
$\mathrm{G}(3,6)\subset\mathbb{P}(\wedge^{3}(\wedge^{2}V_{4}))$ is
generated by \begin{equation}
\begin{aligned}|v_{IJ}|-\epsilon_{I\check{I}}\epsilon_{J\check{J}}|w_{\check{I}\check{J}}|\qquad(I,J\in\mathcal{I}),\qquad\qquad\\
(v.w)_{ij},\;\;(v.w)_{ii}-(v.w)_{jj}\;\;(i\not=j,1\leq i,j\leq4),\end{aligned}
\label{eq:Ivw}\end{equation}
where $\check{I}$ represents the complement of $I$, i.e., $x\in\mathcal{I}$
such that $x\cup I=\{1,2,3,4\}$ and similarly for $\check{J}$. $|v_{IJ}|$
and $|w_{IJ}|$ represent the $2\times2$ minors of $v$ and $w$,
respectively, with the rows and columns specified by $I$ and $J$.
$\epsilon_{I\check{I}}$ is the signature of the permutation of the
'ordered' union $I\cup\check{I}$. $(v.w)_{ij}$ is the $ij$-entry
of the matrix multiplication $v.w$.
\end{prop}
For all $[v,w]\in V(I_{G})\simeq\mathrm{G}(3,6)$, we show the following
relations (I.1)-(I.5):
\noindent \textbf{(I.1)} $\det\, v=\det\, w$.
By the Laplace expansion of the determinant of $4\times4$ matrix
$v$, we have $\det\, v=\sum_{J\in\mathcal{I}}\epsilon_{J\check{J}}|v_{IJ}||v_{\check{I}\check{J}}|$.
Then, using the first relations of (\ref{eq:Ivw}), we obtain the
equality.
\noindent \textbf{(I.2)} $v.w=\pm\sqrt{\det\, w}\, id_{4}$, where
$id_{4}$ is the $4\times4$ identity matrix.
Note that the second line of (\ref{eq:Ivw}) may be written in a matrix
form $v.w=d\, id_{4}$ with $d=(v.w)_{11}=\cdots=(v.w)_{44}$. Then,
by {(I.1)}, we have $\det v\cdot w=(\det\, w)^{2}=d^{4}$ and hence
$d^{4}-(\det\, w)^{2}=(d^{2}-\det\, w)(d^{2}+\det\, w)=0$. We consider
a special case; $v=a\, id_{4}$, $w=a\, id_{4}$. Then $d=(v.w)_{11}=a^{2}$.
Therefore $d^{2}=a^{4}=\det\, w$ must holds for all since $V(I_{G})\simeq\mathrm{G}(3,6)$
is irreducible. Hence $d=\pm\sqrt{\det\, w}$ as claimed.
\noindent \textbf{(I.3)} ${\rm rk}\, w\not=3$ and also ${\rm rk}\, v\not=3$.
Assume ${\rm rk}\, w=3$, then from (I.2) we have $v.w=0$, which
implies ${\rm rk}\, v\leq1$. However, this contradicts the first
relations of (\ref{eq:Ivw}). Hence ${\rm rk}\, w\not=3$. By symmetry,
we also have ${\rm rk}\, v\not=3$.
\noindent \textbf{(I.4)} ${\rm rk}\, w=2\Leftrightarrow{\rm rk}\, v=2$.
When ${\rm rk}\, w=2$, we see ${\rm rk}\, v\geq2$ by the first relations
of (\ref{eq:Ivw}). From (I.1) and (I.3), we must have ${\rm rk}\, v=2$.
The converse follows in the same way.
\noindent \textbf{(I.5)} ${\rm rk}\, w\leq1\Leftrightarrow{\rm rk}\, v\leq1$.
This is immediate from the the first relations of (\ref{eq:Ivw}).
$\;$
\noindent {\footnotesize Department of Mathematics, Gakushuin University,
Toshima-ku,Tokyo 171-8588,$\,$Japan }{\footnotesize \par}
\noindent {\footnotesize e-mail address: [email protected]}
\noindent {\footnotesize Graduate School of Mathematical Sciences,
University of Tokyo, Meguro-ku,Tokyo 153-8914,$\,$Japan }{\footnotesize \par}
\noindent {\footnotesize e-mail address: [email protected]}
\end{document} |
\begin{document}
\title[Global Lipschitz stability for polygonal inclusions]{Global Lipschitz stability estimates for polygonal conductivity inclusions from boundary measurements }
\author[E.~Beretta et al.]{Elena~Beretta}
\address{Dipartimento di Matematica ``Brioschi'',
Politecnico di Milano \& New York University Abu Dhabi}
\email{[email protected]}
\author[]{Elisa~Francini}
\address{Dipartimento di Matematica e Informatica ``U. Dini'',
Universit\`{a} di Firenze}
\email{[email protected]}
\keywords{polygonal inclusions, conductivity equation, stability, inverse problems}
\subjclass[2010]{35R30, 35J25}
\begin{abstract}
We derive Lipschitz stability estimates for the Hausdorff distance of polygonal conductivity inclusions in terms of the Dirichlet-to-Neumann map.
\end{abstract}
\maketitle
\section{Introduction}
In this paper we establish Lipschitz stability estimates for a certain class of discontinuous conductivities $\gamma$ in terms of the Dirichlet-to-Neumann map.\\
More precisely, we consider the following boundary value problem
\begin{equation}\label{conductivity}
\left\{\begin{array}{rcl}
\textrm{ div }((1+(k-1)\chi_{\mathcal{P}})\nabla u) & = & 0\mbox{ in }\Omega\subset\mathbb{R}^2, \\
u & = & \phi \mbox{ on }\partial\Omega,
\end{array}
\right.
\end{equation}
where $\phi\in H^{1/2}\left(\partial\Omega\right)$, $\mathcal{P}$ is a polygonal inclusion strictly contained in a planar, bounded domain $\Omega$ and $k\neq 1$ is a given, positive constant. \\Our goal is to determine the polygon $\mathcal{P}$ from the knowledge of the Dirichlet-to-Neumann map
\begin{equation*}
\Lambda_{\gamma}: H^{1/2}\left(\partial\Omega\right)\to H^{-1/2}\left(\partial\Omega\right)
\end{equation*}
with
\begin{equation*}
\Lambda_{\gamma}(f):=\gamma{\frac{\partial u}{\partial \nu}}\in H^{-1/2}\left(\partial\Omega\right).
\end{equation*}
This class of conductivity inclusions is quite common in applications, like for example in geophysics exploration, where the medium (the earth) under inspection contains heterogeneities in the form of rough bounded subregions (for example subsurface salt bodies) with different conductivity properties \cite{ZK}.
Moreover, polygonal inclusions represent a class in which Lipschitz stable reconstruction from boundary data can be expected \cite{BdHFV}.
In fact, it is well known that the determination of an arbitrary (smooth) conductivity inclusion from the Dirichlet-to-Neumann map is exponentially ill-posed \cite{DiCR}. On the other hand, restricting the class of admissible inclusions to a compact subset of a finite dimensional space regularizes the inverse problem and allows to establish Lipschitz stability estimates and stable reconstructions (see \cite{BV},\cite{BMPS}, \cite{AS}, \cite{H}).
In order to show our main result we follow a similar approach as the one in \cite{BdHFV} and take advantage of a recent result obtained by the authors in \cite{BFV17} where they prove Fr\'echet differentiability of the Dirichlet-to-Neumann map with respect to affine movements of vertices of polygons and where they establish an explicit representation formula for the derivative.\\
We would like to mention that our result relies on the knowledge of infinitely many measurements though one expects that finitely many measurements should be enough to determine a polygonal inclusion. In fact, in \cite{BFS} the authors show that if the inclusion is a convex polyhedron, then one suitably assigned current at the boundary of the domain $\Omega$ and the corresponding measured boundary potential are enough to uniquely determine the inclusion (see also \cite{S} for the unique determination of an arbitrary polygon from two appropriately chosen pairs of boundary currents and potentials and also \cite{KY} where a convex polygon is uniquely determined in the case of variable conductivities). Unfortunately in the aforementioned papers, the choice of the current fields is quite special and the proof of uniqueness is not constructive. In fact, to our knowledge, no stability result for polygons from few boundary measurements has been derived except for the local stability result obtained in \cite{BFI}. On the other hand, in several applications, like the geophysical one, many measurements are at disposal justifying the use of the full Dirichlet-to-Neumann map, \cite{BCFLM}.\\
The paper is organized as follows: in Section 2 we state our main assumptions and the main stability result. Section 3 is devoted to the proof of our main result and finally, Section 4 is devoted to concluding remarks about the results and possible extensions.
\section{Assumptions and main result}
Let $\Omega\subset{\mathbb{R}}^2$ be a bounded open set with $diam(\Omega)\leq L$.
We denote either by $x=(x_1,x_2)$ and by $P$ a point in ${\mathbb{R}}^2$.
We assume that $\partial\Omega$ is of Lipschitz class with constants $r_0$ and $K_0>1$ that means that for every point $P$ in $\partial\Omega$ there exists a coordinate system such that $P=0$ and
\[\Omega\cap \left([-r_0,r_0]\times[-K_0r_0,K_0r_0]\right)=\left\{(x_1,x_2)\,:\, x_1\in[-r_0,r_0], x_2>\phi(x_1)\right\}\]
for a Lipschitz continuous function $\phi$ with Lipschitz norm smaller than $K_0$.
We denote by $dist(\cdot,\cdot)$ the euclidian distance between points or subsets in ${\mathbb{R}}^2$. Later on we will also define the Haussdorff distance $d_H(\cdot,\cdot)$.
Let ${\mathcal{A}}$ the set of closed, simply connected, simple polygons $\mathcal{P}\subset \Omega$ such that:
\begin{equation}\label{lati}
\mathcal{P}\mbox{ has at most }N_0\mbox{ sides each one with length greater than }d_0;
\end{equation}
\begin{equation}\label{lip}\partial\mathcal{P}\mbox{ is of Lipschitz class with constants }r_0\mbox{ and }K_0,\end{equation}
there exists a constant $\beta_0\in (0,\pi/2]$ such that the angle $\beta$ in each vertex of $\mathcal{P}$ satisfies the conditions
\begin{equation}\label{angoli}
\beta_0\leq\beta\leq 2\pi-\beta_0\mbox{ and } |\beta-\pi|\geq\beta_0,
\end{equation}
and
\begin{equation}\label{distanza}
dist(\mathcal{P},\partial\Omega)\geq d_0.
\end{equation}
Notice that we do not assume convexity of the polygon.
Let us consider the problem
\begin{equation*}
\left\{\begin{array}{rcl}
\text{\normalfont div}(\gamma\nabla u) & = & 0\mbox{ in }\Omega, \\
u&=&\phi\mbox{ on }\partial\Omega,\\
\end{array}
\right.
\end{equation*}
where $\phi\in H^{1/2}(\partial\Omega)$ and
\begin{equation}\label{gamma}
\gamma=1+(k-1)\chi_{\mathcal{P}},
\end{equation}
for a given $k>0$, $k\neq 1$ and for $\mathcal{P}\in{\mathcal{A}}$.
The constants $k$, $r_0$, $K_0$, $L$, $d_0$, $N_0$ and $\beta_0$ will be referred to as the \textit{a priori data}.\\ In the sequel we will introduce a number of constants depending only on the \textit{a priori data} that we will always denote by $C$. The values of these constants might differ from one line to the other.
Let us consider the Dirichlet to Neumann map
\[\begin{array}{rcl}\Lambda_\gamma: H^{1/2}(\partial\Omega)&\to& H^{-1/2}(\partial\Omega)\\
\phi&\to&\gamma{\frac{\partial u}{\partial n}}_{|_{\partial\Omega}},\end{array}\]
whose norm in the space of linear operators $\mathcal{L}(H^{1/2}(\partial\Omega), H^{-1/2}(\partial\Omega))$ is defined by
\[\|\Lambda_\gamma\|_*=\sup\left\{\|\Lambda_\gamma\phi\|_{H^{-1/2}(\partial\Omega)}/\|\phi\|_{H^{1/2}(\partial\Omega)}\,:\,\phi\neq 0\right\}. \]
\begin{teo}\label{mainteo}
Let $\mathcal{P}^0,\mathcal{P}^1\in{\mathcal{A}}$ and let
\[\gamma_0=1+(k-1)\chi_{\mathcal{P}^0}\mbox{ and }\gamma_1=1+(k-1)\chi_{\mathcal{P}^1}.\]
There exist $\varepsilon_0$ and $C$ depending only on the a priori data such that, if
\[\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*\leq \varepsilon_0,\]
then $\mathcal{P}^0$ and $\mathcal{P}^1$ have the same number $N$ of vertices $\left\{P_j^0\right\}_{j=1}^N$ and $\left\{P_j^1\right\}_{j=1}^N$ respectively. Moreover,
\begin{equation}\label{stab1}
d_{H}\left( \partial \mathcal{P}^{0},\partial \mathcal{P}^{1}\right)\leq C
\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*\quad \mbox{ for every }j=1,\ldots,N.
\end{equation}
\end{teo}
\vskip 8truemm
\begin{rem}
Observe that our stability estimate is a global one.
In fact, if $\left \Vert \Lambda _{\gamma_0}-\Lambda _{\gamma_1}\right \Vert
_{\ast }>\varepsilon _{0}$, since the
following trivial inequality holds
\[
d_{H}\left( \partial \mathcal{P}^{0},\partial \mathcal{P}^{1}\right) \leq 2L,
\]
we have trivially
\begin{equation}
d_{H}\left( \partial \mathcal{P}^{0},\partial \mathcal{P}^{1}\right) \leq
2L\leq 2L\frac{\left \Vert \Lambda _{0}-\Lambda _{1}\right \Vert _{\ast }}{
\varepsilon _{0}} \label{stab2}
\end{equation}
Therefore, in \textit{any case, }by\textit{\ (\ref{stab1}), (\ref{stab2})} we obtain
the global estimate
\[
d_{H}\left( \partial \mathcal{P}^{0},\partial \mathcal{P}^{1}\right) \leq
\left( C+\frac{2L}{\varepsilon _{0}}\right) \left \Vert \Lambda _{0}-\Lambda
_{1}\right \Vert _{\ast }.
\]
\end{rem}
\section{Proof of the main result}
The proof of Theorem \ref{mainteo} follows partially the strategy used in \cite{BdHFV} in the case of the Helmholtz equation.
The first step of the proof is a rough stability estimate for $\|\gamma_0-\gamma_1\|_{L^2(\Omega)}$ which is stated in Section \ref{srozza} and which follows from a result by Clop, Faraco and Ruiz \cite{CFR}. Then, in section \ref{sgeo}, we show a rough stability estimate for the Hausdorff distance of the polygons. We also show that if $\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*$ is small enough, then the two polygons have the same number of vertices and that the distance from vertices of $\mathcal{P}^0$ and vertices of $\mathcal{P}^1$ is small. For this reason it is possible to define a coefficient $\gamma_t$ that goes smoothly from $\gamma_0$ to $\gamma_1$ and the corresponding Dirichlet to Neumann map.
We prove that the Dirichlet to Neumann map is differentiable (section \ref{sFdiff}), its derivative is continuous (section \ref{sderivcont}) and bounded from below (section \ref{sboundbelow}). These results finally give the Lipschitz stability estimate of Theorem \ref{mainteo}.
\subsection{A logarithmic stability estimate}\label{srozza}
As in \cite{BdHFV}, we can show that, thanks to Lemma 2.2 in \cite{MP} there exists a constant $\Gamma_0$, depending only on the a priori data, such that, for $i=0,1$,
\begin{equation}\label{hs}
\|\gamma_i\|_{H^s(\Omega)}\leq \Gamma_0\quad\forall s\in(0,1/2).
\end{equation}
Due to this regularity of the coefficients, we can apply Theorem 1.1 in \cite{CFR} and obtain the following logarithmic stability estimate:
\begin{prop}\label{strozza} There exist $\alpha<1/2$ and $C>1$, depending only on the a priori data, such that
\begin{equation}\label{stimarozzal2}\|\gamma_1-\gamma_0\|_{L^2(\Omega)}\leq C\left|\log\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*\right|^{-\alpha^2/C},\end{equation}
if $\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*<1/2$.
\end{prop}
\subsection{A logarithmic stability estimate on distance of vertices}\label{sgeo}
In this section we want to show that, due to the assumptions on polygons in ${\mathcal{A}}$, estimate \eqref{stimarozzal2} yields an estimate on the Hausdorff distance $d_H(\partial\mathcal{P}^0,\partial\mathcal{P}^1)$ and, as a consequence, on the distance of the vertices of the polygons.
It is immediate to get from \eqref{stimarozzal2} that
\begin{equation}\label{diffsimm}
\left|\mathcal{P}^0\Delta\mathcal{P}^1\right|\leq \frac{C}{|k-1|}\left|\log\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*\right|^{-\alpha^2/C}\end{equation}
Now,
we show that \eqref{diffsimm} implies an estimate on the Hausdorff distance of the boundaries of the polygons.
Let us recall the definition of the Hausdorff distance between two sets $A$ and $B$:
\[d_H(A,B)=max\{\sup_{x\in A}\inf_{y\in B}dist(x,y),\sup_{y\in B}\inf_{x\in A}dist(y,x)\}\]
The following result holds:
\begin{lm}\label{dadiffahaus}
Given two polygons $\mathcal{P}^0$ and $\mathcal{P}^1$ in ${\mathcal{A}}$, we have
\[d_H(\partial\mathcal{P}^0,\partial\mathcal{P}^1)\leq C\sqrt{\left|\mathcal{P}^0\Delta\mathcal{P}^1\right|}\]
where $C$ depends only on the a priori data.
\end{lm}
\begin{proof}
Let $d=d_H(\partial\mathcal{P}^0,\partial\mathcal{P}^1)$. Assume $d>0$ (otherwise the thesis is trivial) and let $x_0\in\partial \mathcal{P}^0$ such that $dist(x_0,\partial\mathcal{P}^1)=d$. Then,
\[B_d(x_0)\subset {\mathbb{R}}^2\setminus\partial \mathcal{P}^1.\]
There are two possibilities:\\
(i) $B_d(x_0)\subset {\mathbb{R}}^2\setminus\mathcal{P}^1$ or \\
(ii) $B_d(x_0)\subset \mathcal{P}^1$.
In case (i),
$B_d(x_0)\cap \mathcal{P}^0\subset \mathcal{P}^0\setminus\mathcal{P}^1$.
The definition of ${\mathcal{A}}$ implies that, if $d\leq d_0$, there is a constant $C>1$ depending only on the a priori data such that
\[\left|B_d(x_0)\cap \mathcal{P}^0\right|\geq \frac{d^2}{C^2}.\]
If $d\geq d_0$ we trivially have
\[\left|B_d(x_0)\cap \mathcal{P}^0\right|\geq \left|B_{d_0}(x_0)\cap \mathcal{P}^0\right|\geq \frac{d_0^2}{C^2},\]
hence, in any case, for
\[f(d)=\left\{\begin{array}{rl}d^2/C^2&\mbox{ if }d<d_0\\
d_0^2/C^2&\mbox{ if }d\geq d_0\end{array}\right.\]
we have
\[f(d)\leq \left|B_d(x_0)\cap \mathcal{P}^0\right|\leq \left|\mathcal{P}^0\Delta\mathcal{P}^1\right|
.\]
Now, if $\left|\mathcal{P}^0\Delta\mathcal{P}^1\right|<\frac{d_0^2}{C^2}$, then $f(d)=\frac{d^2}{C^2}\leq \left|\mathcal{P}^0\Delta\mathcal{P}^1\right|$ gives $d\leq C\sqrt{\left|\mathcal{P}^0\Delta\mathcal{P}^1\right|}$.
On the other hand, if $\left|\mathcal{P}^0\Delta\mathcal{P}^1\right|\geq \frac{d_0^2}{C^2}$ we have
\[\frac{d^2}{C^2}\leq \frac{L^2}{C^2}\leq \frac{L^2}{C^2}\frac{\left|\mathcal{P}^0\Delta\mathcal{P}^1\right|}{d_0^2/C^2}\]
that gives $d\leq \frac{LC}{d_0}\sqrt{\left|\mathcal{P}^0\Delta\mathcal{P}^1\right|}$.
In case (ii), $B_d(x_0)\subset \mathcal{P}^1$, hence
\[B_d(x_0)\setminus \mathcal{P}^0\subset \mathcal{P}^1\setminus\mathcal{P}^0\subset\mathcal{P}^1\Delta\mathcal{P}^0.\]
Proceeding as above we have
\[f(d)\leq \left|B_d(x_0)\setminus \mathcal{P}^0\right|\leq \left|\mathcal{P}^0\Delta\mathcal{P}^1\right|\]
and the same conclusion follows.
\end{proof}
\begin{prop}\label{propvertici}
Given the set of polygons ${\mathcal{A}}$ there exist $\delta_0$ and $C$ depending only on the a priori data such that, if for some $\mathcal{P}^0$, $\mathcal{P}^1\in {\mathcal{A}}$ we have
\[d_H(\partial\mathcal{P}^0,\partial\mathcal{P}^1)\leq \delta_0,\]
then
$\mathcal{P}^0$ and $\mathcal{P}^1$ have the same number $N$ of vertices $\{P^0_i\}_{i=1}^N$ and $\{P^1_i\}_{i=1}^N$, respectively, that can be ordered in such a way that
\[dist(P^0_i,P^1_i)\leq Cd_H(\partial\mathcal{P}^0,\partial\mathcal{P}^1) \mbox{ for every }i=1,\ldots,N.\]
\end{prop}
\begin{proof}
Let us denote by
\[\delta=d_H(\partial\mathcal{P}^0,\partial\mathcal{P}^1).\]
Assume $\mathcal{P}^0$ has $N$ vertices and that $\mathcal{P}^1$ has $M$ vertices. We now will show that for any vertex $P^0_i\in \partial\mathcal{P}^0$ there exists a vertex $P^1_j\in\partial\mathcal{P}^1$ such that
$dist(P^0_i,P^1_j)<C\delta$. By assumption \eqref{lati} this implies that $N\leq M$. Interchanging the role of $\mathcal{P}^0$ and $\mathcal{P}^1$ we get that $M\leq N$ which implies that $M=N$.
Let $P$ be one of the vertices in $\partial\mathcal{P}^0$ and let us consider the side $l^\prime$ of $\partial\mathcal{P}^1$ that is close to $P$. Let us set the coordinate system with origin in the midpoint of $l^\prime$ and let $(\pm l/2,0)$ be the endpoint of $l^\prime$.
By definition of the Hausdorff distance, $P\in \mathcal{U}_\delta=\left\{x\in{\mathbb{R}}^2\,:\,dist(x,l^\prime)\leq\delta\right\}$.
Now we want to show that, due to the assumptions on ${\mathcal{A}}$, for sufficiently small $\delta$ there is a constant $C$ such that the distance between $P$ and one of the endpoints of $l^\prime$ is smaller than $C\delta$. The reason is that if $P$ is too far from the endpoints, assumption \eqref{angoli} on $\mathcal{P}^0$ cannot be true.
Let us choose $\delta$ small enough to have:
\begin{equation}\label{cond1}
\delta<K_0 r_0
\end{equation}
(this guarantees that the $\delta$-neighborhood of each side of $\mathcal{P}^1$ does not intersect the $\delta$-neighborhood of a non adjacent side), and
\begin{equation}\label{cond2}
\delta<\frac{d_0\sin\beta_0}{16}.
\end{equation}
Notice that, by assumption \eqref{angoli} and by \eqref{cond1}, the rectangle
\[R=\left[-\frac{l}{2}+\frac{2\delta}{\sin\beta_0},\frac{l}{2}-\frac{2\delta}{\sin\beta_0}\right]\times[-\delta,\delta]\]
does not intersect the $\delta$-neighborhood of any other side of $\mathcal{P}^1$.
Let us now show that $P$ cannot be contained in a slightly smaller rectangle
\[R^\prime=\left[-\frac{l}{2}+\lambda,\frac{l}{2}-\lambda\right]\times[-\delta,\delta],\]
where $\lambda=\frac{6\delta}{\sin\beta_0}$.
Let us assume by contradiction that $P\in R^\prime$ and consider the two sides of $\partial\mathcal{P}^0$ with an endpoint at $P$. These sides have length greater than $d_0$, hence they intersect $\partial B_{\lambda/2}(P)$ in two points $Q_1$ and $Q_2$ in $R$ (because $\lambda/2<\lambda-\frac{2\delta}{\sin\beta_0}$).
Since $\lambda/2>2\delta$ the intersection $\partial B_{\lambda/2}(P)\cap R$ is the union of two disjoint arcs. We estimate the angle of $\mathcal{P}^0$ at $P$ in the two alternative cases:\\
(i) $Q_1$ and $Q_2$ are on the same arc or\\
(ii) $Q_1$ and $Q_2$ are on different arcs.
In case (i), the angle at $P$ is smaller than $\arcsin\left(\frac{4\delta}{\lambda}\right)$
(the angle is smaller than $\arcsin\left(\frac{2(\delta-b)}{\lambda}\right)+\arcsin\left(\frac{2(\delta+b)}{\lambda}\right)$, where $b$ is the $y$-coordinate of $P$, that is maximum for $b=\pm\delta$).
In order for \eqref{angoli} to be true we should have
\[\arcsin\left(\frac{4\delta}{\lambda}\right)=\arcsin\left(\frac{2}{3}\sin\beta_0\right)\leq \beta_0\]
that is not possible for $\beta_0\in (0,\pi/2)$.
In case (ii), the angle differs from $\pi$ at most by $\arcsin\left(\frac{4\delta}{\lambda}\right)$, which is again too small for \eqref{angoli} to be true.
Since neither of cases (1) and (2) can be true, it is not possibile that $P\in R^\prime$, hence, $P\in \mathcal{U}_\delta\setminus R^\prime$ which implies that there is one of the endpoints of $l^\prime$, let us call it $P^\prime$ such that
\[dist(P,P^\prime)\leq \delta \sqrt{1+\frac{16}{\sin^2\beta_0}}.\]
\end{proof}
\begin{prop}\label{strozzavert}
Under the same assumptions of Theorem \ref{mainteo}, there exist positive constants $\varepsilon_0$, $\alpha$ and $C>1$, depending only on the a priori data, such that, if
\[\varepsilon:=\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*<\varepsilon_0,\]
then $\mathcal{P}^0$ and $\mathcal{P}^1$ have the same number $N$ of vertices $\left\{P_j^0\right\}_{j=1}^N$ and $\left\{P_j^1\right\}_{j=1}^N$ respectively. Moreover, the vertices can be order so that
\begin{equation}\label{stimarozzavertici}
dist\left(P_j^0,P_j^1\right)\leq \Omegaega(\varepsilon) \mbox{ for every }j=1,\ldots,N,
\end{equation}
where $\Omegaega(\varepsilon)=C \left|\log \varepsilon\right|^{-\alpha^2/C}$.
\end{prop}
\begin{proof}It follows by the combination of Proposition \ref{strozza}, Lemma \ref{dadiffahaus} and Proposition \ref{propvertici}.\end{proof}
\subsection{Definition and differentiability of the function $F$}\label{sFdiff}
Let us denote by $\{P^j_i\}_{i=1}^N$ the vertices of polygon $\mathcal{P}^j$ for $j=0,1$ numbered in such a way that
$dist(P^0_i,P^1_i)\leq \Omegaega(\varepsilon)\mbox{ for }i=1,\ldots,N$,
for $\Omegaega(\varepsilon)$ as in Proposition \ref{strozzavert} and the segment $P^i_jP^i_{j+1}$ is a side of $\mathcal{P}^i$ for $i=0,1$ and $j=1,\ldots,N$.
Let us consider a deformation from $\mathcal{P}^0$ to $\mathcal{P}^1$: for $t\in[0,1]$ let
\[P_i^t=P^0_i+tv_i,\mbox{ where }v_i=P^1_i-P^0_i,\mbox{ for }i=1,\ldots,N\]
and denote by $\mathcal{P}^t$ the polygon with vertices $P^t_j$ and sides $P^t_jP^t_{j+1}$.
Let $\gamma_t=1+(k-1)\chi_{\mathcal{P}^t}$ and let $\Lambda_{\gamma_t}$ be the corresponding DtoN map.
As we proved in \cite[Corollary 4.5]{BFV17} the DtoN map $\Lambda_{\gamma_t}$ is differentiable with respect to $t$.
The function
\[F(t,\phi,\psi)=<\Lambda_{\gamma_t}(\phi), \psi>,\]
for $\phi,\psi\in H^{1/2}(\partial\Omega)$, is a differentiable function from $[0,1]$ to ${\mathbb{R}}$ and we can write explicitly its derivative.
Let $u_t,v_t\in H^1(\Omega)$ be the solutions to
\[
\left\{\begin{array}{rcl}
\text{\normalfont div}(\gamma_t\nabla u_t) & = & 0\mbox{ in }\Omega, \\
u_t&=&\phi\mbox{ on }\partial\Omega,\\
\end{array}
\right.
\mbox{ and }
\left\{\begin{array}{rcl}
\text{\normalfont div}(\gamma_t\nabla v_t) & = & 0\mbox{ in }\Omega, \\
v_t&=&\psi\mbox{ on }\partial\Omega,\\
\end{array}
\right.
\]
and denote by $u_t^e$ and $v_t^e$ their the restrictions to $\Omega\setminus\mathcal{P}^t$ (and by $u_t^i$ and $v_t^i$ their restrictions to $\mathcal{P}^t$).
Let us fix an orthonormal system $(\tau_t, n_t)$ in such a way that $n_t$ represents almost everywhere the outward unit
normal to $\partial \mathcal{P}_t$ and the tangent unit vector $\tau_t$ is oriented counterclockwise. Denote
by $M_t$ a $2\times 2$ symmetric matrix valued function defined on $\partial\mathcal{P}_t$ with eigenvalues $1$ and $1/k$
and corresponding eigenvectors $\tau_t$ and $n_t$.
Let $\Phi_t^v$ be a map defined on $\partial\mathcal{P}_t$, affine on each side of the polygon and such that
\[\Phi_t^v(P_i^t)=v_i\mbox{ for }i=1,\ldots,N.\]
Then, it was proved in \cite[Corollary 2.2]{BFV17} that, for all $t\in[0,1]$,
\[\frac{d}{dt}F(t,\phi,\psi)=(k-1)\int_{\partial \mathcal{P}^t}M_t\nabla u_t^e\nabla v_t^e (\Phi_t^v\cdot n_t). \]
\subsection{Continuity at zero of the derivative of $F$}\label{sderivcont}
\begin{lm}\label{lcontder}
There exist constants $C$ and $\beta$, depending only on the a priori data, such that
\begin{equation}\label{contder}\left|\frac{d}{dt}F(t,\phi,\psi)-{\frac{d}{dt}F(t,\phi,\psi)}_{|_{t=0}} \right|\leq C\|\phi\|_{H^{1/2}(\partial\Omega)}\|\psi\|_{H^{1/2}(\partial\Omega)}|v|^{1+\beta}t^\beta.\end{equation}
\end{lm}
\begin{proof}
This result corresponds to Lemma 4.4 in \cite{BFV17}. The dependence on $|v|$ is obtained by refining estimate (3.5) in \cite[Proposition 3.4]{BFV17} to get
\[\|u_t-u_0\|_{H^1(\Omega)}\leq C\|\phi\|_{H^{1/2}(\partial\Omega)}\left|\mathcal{P}^t\Delta\mathcal{P}^0\right|^\theta\leq C_1\|\phi\|_{H^{1/2}(\partial\Omega)}|v|^\theta t^\theta,\]
and by noticing that
\[\left|\Phi_t^v\right|\leq C|v|.\]
\end{proof}
\subsection{Bound from below for the derivative of $F$}\label{sboundbelow}
In this section we want to obtain a bound from below for the derivative of $F$ at $t=0$.
\begin{prop}\label{p3.3}
There exist a constant $m_1>0$, depending only on the a priori data, and a pair of functions $\tilde{\phi}$ and $\tilde{\psi}$ in $H^{1/2}(\partial\Omega)$ such that
\begin{equation}\label{tesibasso}\left|{\frac{d}{dt}F(t,\tilde{\phi},\tilde{\psi})}_{|_{t=0}}\right|\geq m_1 |v| \|\tilde{\phi}\|_{H^{1/2}(\partial\Omega)}\|\tilde{\psi}\|_{H^{1/2}(\partial\Omega)}. \end{equation}
\end{prop}
\begin{proof}
Let us first normalize the length of vector $v$ and introduce
\[H(\phi,\psi)=\int_{\partial\mathcal{P}_0} M_o\nabla u_0^e\nabla v_0^e\tilde{\Phi}_0^{v}\cdot n_0,\]
where
\[\tilde{\Phi}_0^{v}=\Phi_0^{v/|v|}.\]
By linearity, we have that ${\frac{d}{dt}F(t,\phi,\psi)}_{|_{t=0}}=|v|H(\phi,\psi)$.
Let $m_0=\|H\|_*=\sup\left\{\frac{H(\phi,\psi)}{\|\phi\|_{H^{1/2}(\partial\Omega)}\|\psi\|_{H^{1/2}(\partial\Omega)}}\,:\,\phi,\psi\neq 0\right\}$ be the operator norm of $H$, so that
\begin{equation}\label{normaH}\left|H(\phi,\psi)\right|\leq m_0\|\phi\|_{H^{1/2}(\partial\Omega)}\|\psi\|_{H^{1/2}(\partial\Omega)}\mbox{ for every }\phi,\psi\in H^{1/2}(\partial\Omega).\end{equation}
Let $\Sigma$ be an open non empty subset of $\partial\Omega$ and let us extend $\Omega$ to a open domain $\Omega_0=\Omega\cup D_0$ that
has Lipschitz boundary with constants $r_0/3$ and $K_0$ and such that $\Sigma$ is contained in $\Omega_0$ (see \cite{AV} for a detailed construction). Let us extend $\gamma_0$ by $1$ in $D_0$ (and still denote it by $\gamma_0$).
We denote by $G_0(x,y)$ the Green function corresponding to the operator $\text{\normalfont div}(\gamma_0 \nabla\cdot)$ and to the domain $\Omega_0$.
The Green function $G_0(x,y)$ behaves like the fundamental solution of the Laplace equation $\Gamma(x,y)$ for points that are far from the polygon. For points close to the sides of the polygon but far from its vertices, the asymptotic behaviour of the Green function has been described in
\cite[Theorem 4.2]{AV} or \cite[Proposition 3.4]{BF11}: Let $y_r=Q+rn(y_0)$, where $Q$ is a point on $\partial\mathcal{P}^0$ whose distance from the vertices of the polygons is greater than $r_0/4$ and $n(y_0)$ is the unit outer normal to $\partial\mathcal{P}^0$. Then, for small $r$,
\begin{equation}\label{stimagreen}
\left\|G_0(\cdot,y_r)-\frac{2}{k+1}\Gamma(\cdot,y_r)\right\|_{H^1(\Omega_0)}\leq C,
\end{equation}
where $C$ depends only on the a priori data.
Let us take $u_0=G_0(\cdot,y)$ and $v_0=G_0(\cdot,z)$ for $y,z\in K$, where $K$ is a compact subset of $D_0$ such that $dist(K,\partial\Omega)\geq r_0/3$ and $K$ contains a ball of radius $r_0/3$. The functions $u_0$ and $v_0$ are both solutions to the equation $\text{\normalfont div}(\gamma_0\nabla\cdot)=0$ in $\Omega$.
Define the function
\[S_0(y,z)=\int_{\partial\mathcal{P}_0}M_0\nabla G_0(\cdot,y)\nabla G_0(\cdot,z)(\tilde{\Phi}_0^v\cdot n_0)\]
that, for fixed $z$, solves $\text{\normalfont div}(\gamma_0 \nabla S_0(\cdot,z))=0$ in $\Omega\setminus\mathcal{P}^0$ and, for fixed $y$ it solves $\text{\normalfont div}(\gamma_0 \nabla S_0(y,\cdot))=0$ in $\Omega\setminus\mathcal{P}^0$.
For $y,z\in K$, $S_0(y,z)=H(u_0,v_0)$, hence, by \eqref{normaH}
\begin{equation}\label{12.1}
|S_0(y,z)|\leq \frac{C_0m_0}{r_0^2}\mbox{ for }y,z\in K,
\end{equation}
where $C_0$ depend on the a priori data.
Moreover, by \eqref{stimagreen}, there exist $\rho_0$ and $E$ depending only on the a priori data such that
\begin{equation}
\label{13.1}
|S_0(y,z)|\leq E(d_yd_z)^{-1/2} \mbox{ for every } y,z\in \Omega\setminus\left(\mathcal{P}^0 \cup_{i=1}^NB_{\rho_0}(P_i^0)\right),
\end{equation}
where $d_y=dist(y,\mathcal{P}^0)$.
Since $S_0$ is small for $y,z\in K$ (see \eqref{12.1} and consider $m_0$ small), bounded for $y,z\in \Omega\setminus\mathcal{P}^0$ far from the vertices of the polygon, and since it is harmonic in $\Omega\setminus\mathcal{P}^0$, we can use a three balls inequality on a chain of balls in order to get a smallness estimate close to the sides of the polygon.
To be more specific, let $l_i$ be a side of $\mathcal{P}^0$ with endpoints $P^0_i$ and $P^0_{i+1}$. Let $Q^0_i$ be the midpoint of $l_i$ and let $y_r=Q^0_i+rn_i$ where $n_i$ is the unit outer normal to $\partial\mathcal{P}^0$ at $Q^0_i$ and $r\in(0,K_0r_0)$.
\begin{lm}\label{small}
There exist constants $C>1$, $\beta$, and $r_1<r_0/C$ depending only on the a priori data, such that, for $r<r_1$
\begin{equation}
\label{1.3.1}
\left|S_0(y_r,y_r)\right|\leq C \left(\frac{\varepsilon_0}{\varepsilon_0+E}\right)^{\beta\tau^2_r}(\varepsilon_0+E)r^{-1},
\end{equation}
where $\varepsilon_0=m_0C_0r_0^{-2}$ and $\tau_r=\frac{1}{\log(1-r/r_1)}$.
\end{lm}
\begin{proof}For the proof of Lemma \ref{small} see \cite[Proposition 4.3]{BF11} where the estimate of $\tau_r$ is slightly more accurate.\end{proof}
Now, we want to estimate $\left|S_0(y_r,y_r)\right|$ from below. In order to accomplish this, let us take $\rho=\min\{d_0/4,r_0/4\}$ and write
\begin{align}\label{6.1}
\left|\frac{S_0(y_r,y_r)}{k-1}\right|\geq
&\left|\int_{\partial\mathcal{P}_0\cap B_\rho(Q_i^0)}M_0\nabla G_0(\cdot,y_r)\nabla G_0(\cdot,y_r)(\tilde{\Phi}_0^v\cdot n_0)\right|\\&-\left|\int_{\partial\mathcal{P}_0\setminus B_\rho(Q_i^0)}M_0\nabla G_0(\cdot,y_r)\nabla G_0(\cdot,y_r)(\tilde{\Phi}_0^v\cdot n_0)\right|
\\&:=I_1-I_2.\end{align}
The behaviour of the Green function (see \cite{AV}) gives immediately that, for $r<\rho/2$,
\begin{equation}\label{6.2}
I_2\leq C_1,
\end{equation}
for some $C_1$ depending only on the a priori data.
In order to estimate $I_1$, we add and subtract $\Gamma(\cdot,y_r)$ to $G_0(\cdot,y_r)$, then by Young inequality, \eqref{stimagreen}, and by the properties of $M_0$, we get
\begin{equation}\label{I1}
I_1\geq C_2\left|\int_{\partial\mathcal{P}_0\cap B_\rho(Q_i^0)}\left|\nabla \Gamma(\cdot,y_r)\right|^2(\tilde{\Phi}_0^v\cdot n_0^i)\right|-C_3,
\end{equation}
where $C_2$ and $C_3$ depend only on the a priori data.
By definition of $\tilde{\Phi}_0^v$ we have
\[\left|\tilde{\Phi}_0^v(x)-\tilde{\Phi}_0^v(Q_i^0)\right|\leq C_4|x-Q^0_i|,\]
so, by adding and subtracting $\Phi^v_0(Q_i^0)$ into the integral of \eqref{I1}, we can write
\begin{align*}
\left|\int_{\partial\mathcal{P}_0\cap B_\rho(Q_i^0)}\left|\nabla \Gamma(\cdot,y_r)\right|^2(\tilde{\Phi}_0^v\cdot n_0^i)\right|\geq
& \,\overline{\alpha} \int_{\partial\mathcal{P}_0\cap B_\rho(Q_i^0)}\left|\nabla \Gamma(\cdot,y_r)\right|^2\\&-
C_4\int_{\partial\mathcal{P}_0\setminus B_\rho(Q_i^0)}\left|\nabla \Gamma(\cdot,y_r)\right|^2|x-Q^0_i|,
\end{align*}
where $\overline{\alpha}=|\tilde{\Phi}_0^v(Q_i^0)\cdot n_0^i|$.
By straightforward calculations one can see that
\begin{equation}\label{8.2}
\int_{\partial\mathcal{P}_0\cap B_\rho(Q_i^0)}\left|\nabla \Gamma(\cdot,y_r)\right|^2\geq \frac{C_5}{r}
\end{equation}
and
\begin{equation}\label{9.1}
\int_{\partial\mathcal{P}_0\setminus B_\rho(Q_i^0)}\left|\nabla \Gamma(\cdot,y_r)\right|^2|x-Q^0_i| \leq C_6\left|\log (\rho/r)\right|.
\end{equation}
By putting together \eqref{6.1}, \eqref{6.2}, \eqref{8.2} and \eqref{9.1}, we get
\begin{equation}\label{9.1star}
\left|S_0(y_r,y_r)\right|\geq \frac{C_6 \overline{\alpha}}{r}-C_7\left|\log (\rho/r)\right|-C_8.\end{equation}
By comparing \eqref{1.3.1} and \eqref{9.1star} we get
\begin{equation}\label{9.2}
C_6\overline{\alpha}\leq C \left(\frac{\varepsilon_0}{\varepsilon_0+E}\right)^{\beta\tau^2_r}(\varepsilon_0+E) +C_7r|\log(\rho/r)|+C_8r.
\end{equation}
By an easy calculation one can see that $\beta\tau_r^2\geq r^2/C_9$, hence
\begin{equation}\label{9.3}
C_6\overline{\alpha}\leq C \left(\frac{\varepsilon_0}{\varepsilon_0+E}\right)^{r^2/C_9}(\varepsilon_0+E) +C_{10}\sqrt{r}.
\end{equation}
By choosing $r=\left|\log\left(\frac{\varepsilon_0}{\varepsilon_0+E}\right)\right|^{-1/4}$ and recalling that $\varepsilon_0=C_0m_0r_0^{-2}$ we have
\[|\tilde{\Phi}_0^v(Q_i^0)\cdot n_0^i|=\overline{\alpha}\leq \Omegaega_0(m_0),\]
where $\Omegaega_0(t)$ is an increasing concave function such that $\lim_{t\to 0^+}\Omegaega_0(t)=0$.
This estimate can also be obtained for $\tilde{\Phi}_0^v(y)\cdot n_0^i$ for every $y\in B_{\rho}(Q_i^0)\cap l_i$. Since $\tilde{\Phi}_0^v$ is linear on the bounded side $l_i$,
\[|\tilde{\Phi}_0^v(y)\cdot n_0^i|\leq\Omegaega_0(m_0)\quad\quad \mbox{for every }y\in l_i,\]
and, in particular
\begin{equation}\label{vec1}\left|\frac{v_i}{|v|}\cdot n_0^i\right|=|\tilde{\Phi}_0^v(P_i)\cdot n_0^i|\leq\Omegaega_0(m_0)\end{equation}
Repeating the same argument on the adjacent side, $l_{i+1}$, containing $P_i$ we obtain in particular that
\begin{equation}\label{vec2}
\left|\frac{v_i}{|v|}\cdot n_0^{i+1}\right|=|\tilde{\Phi}_0^v(P_i)\cdot n_0^{i+1}|\leq\Omegaega_0(m_0)\end{equation}
Then, there exists a constant $C>0$ depending on the a priori constants only such that
\[\left|\frac{v_i}{|v|}\right|\leq C\Omegaega_0(m_0)\]
and since one can apply the same procedure on each side of the polygon we have
\[\left|\frac{v_i}{|v|}\right|\leq C\Omegaega_0(m_0)\mbox{ for }i=1,\ldots,N\]
that yields
\[1\leq NC\Omegaega_0(m_0)\Rightarrow m_0\geq \Omegaega_0^{-1}(1/CN).\]
By definition of the operator norm of $H$, there exist $\tilde{\phi}$ and $\tilde{\psi}$ in $H^{1/2}(\partial\Omega)$ such that
\[|H(\tilde{\phi},\tilde{\psi})|\geq \frac{m_0}{2}\|\tilde{\phi}\|_{H^{1/2}(\partial\Omega)}\|\tilde{\psi}\|_{H^{1/2}(\partial\Omega)}\]
and \eqref{tesibasso} is true for $m_1=\frac{\Omegaega_0^{-1}(1/CN)}{2}$.
\end{proof}
\begin{rem}
Note that the lower bound for the derivative of $F$ in Proposition \ref{p3.3} holds for functions $\tilde{\phi}$ and $\tilde{\psi}$ with compact support on an open portion of $\partial\Omega$.
\end{rem}
\subsection{Lipschitz stability estimate}
In this section we conclude the proof of Theorem \ref{mainteo}.
Let $\tilde{\phi}$ and $\tilde{\psi}$ the functions the satisfy \eqref{tesibasso} in Proposition \ref{p3.3}.
By \eqref{tesibasso} and by \eqref{contder} we have
\begin{eqnarray*}\label{fine}
\left|<\left(\Lambda_{\gamma_1}-\Lambda_{\gamma_0}\right)(\tilde{\phi}),\tilde{\psi}>\right|\!\!&=\!\!&\left|F(1,\tilde{\phi},\tilde{\psi})-F(0,\tilde{\phi},\tilde{\psi})\right|=\left|\int_0^1\frac{d}{dt}F(t,\tilde{\phi},\tilde{\psi})dt\right|\\
\!\! &\geq\!\!& \frac{d}{dt}F(t,\tilde{\phi},\tilde{\psi})_{|_{t=0}} \!-\!\int_0^1\!\left| \frac{d}{dt}F(t,\tilde{\phi},\tilde{\psi})-\frac{d}{dt}F(t,\tilde{\phi},\tilde{\psi})_{|_{t=0}}\right|dt\\
&\geq&\left(m_1-C|v|^\beta\right)|v|\|\tilde{\phi}\|_{H^{1/2}(\partial\Omega)}\|\tilde{\psi}\|_{H^{1/2}(\partial\Omega)},
\end{eqnarray*}
that implies
\begin{equation}\label{fine2}
\varepsilon=\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*\geq \left(m_1-C|v|^\beta\right)|v|.
\end{equation}
From \eqref{stimarozzavertici}, since
$|v|\leq N\max_jdist(P^0_j,P^1_j)$ it follows that there exists $\varepsilon_0>0$ such that, if
\[\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*\leq\varepsilon_0,\] then
\[\left(m_1-C|v|^\beta\right)\geq m_1/2\]
and
\[|v|\leq \frac{2}{m_1}\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*.\]
Finally, since
\[d_{H}\left( \partial \mathcal{P}^{0},\partial \mathcal{P}^{1}\right)\leq C| v|\]
the claim follows.
$\square$
Finally, as a byproduct of Theorem \ref{mainteo} and of Proposition \ref{propvertici} we have the following
\begin{coro}
Let $\mathcal{P}^0,\mathcal{P}^1\in{\mathcal{A}}$ and let
\[\gamma_0=1+(k-1)\chi_{\mathcal{P}^0}\mbox{ and }\gamma_1=1+(k-1)\chi_{\mathcal{P}^1}.\]
There exist $\varepsilon_0$ and $C$ depending only on the a priori data such that, if
\[\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*\leq \varepsilon_0,\]
then $\mathcal{P}^0$ and $\mathcal{P}^1$ have the same number $N$ of vertices $\left\{P_j^0\right\}_{j=1}^N$ and $\left\{P_j^1\right\}_{j=1}^N$ respectively. Moreover, the vertices can be ordered so that
\begin{equation}
dist\left(P_j^0,P_j^1\right)\leq C
\|\Lambda_{\gamma_0}-\Lambda_{\gamma_1}\|_*\quad \mbox{ for every }j=1,\ldots,N.
\end{equation}
\end{coro}
\section{Final remarks and extensions}
We have derived Lipschitz stability estimates for polygonal conductivity inclusions in terms of the Dirichlet-to-Neumann map using differentiability properties of the Dirichlet-to-Neuman map. \\
The result extends also to the case where finitely many conductivity polygonal inclusions are contained in the domain $\Omega$ assuming that they are at controlled distance one from the other and from the boundary of $\Omega$.\\
We expect that the same result holds also when having at disposal local data. In fact, as we observed at the end of Proposition 3.6, the lower bound for the derivative of $F$ is obtained using solutions with compact support in a open subset of $\partial\Omega$ and a rough stability estimate of the Hausdorff distance of polygons in terms of the local Dirichlet-to-Neumann map could be easily derived following the ideas contained in \cite{MR}.\\
Finally, it is relevant for the geophysical application we have in mind to extend the results of stability and reconstruction to the 3-D setting possibly considering an inhomogeneous and/or anisotropic medium. This case is not at all straightforward since differentiability properties of the Dirichlet-to-Neumann map in this case are not known.
\textbf{Acknowledgment}
The paper was partially supported by GNAMPA - INdAM.
\end{document} |
\begin{document}
\title{ Analytical approach for treating unitary quantum systems with initial mixed states }
\author{Faisal A. A. El-Orany }
\email{[email protected], Tel:006-166206010, Fax:
0060386579404}\affiliation{ Department of Mathematics and Computer
Science, Faculty of Science, Suez Canal University, Ismailia,
Egypt; } \affiliation{ Cyberspace Security Laboratory, MIMOS
Berhad, Technology Park Malaysia, 57000 Kuala Lumpur, Malaysia}
\begin{abstract}
The mixed states are important in quantum optics since they
frequently appear in the decoherence problems. When one of the
components of the system is prepared in the mixed state and the
evolution operator of this system is not available, one cannot
deduce the density matrix.
We present analytical approach to accurately solve this problem.
The approach can be applied on the condition that
the Schr\"{o}dinger's equation of the system is solvable with
any arbitrary initial state.
In deriving the solution we exploit the fact that any mixed state
can be expressed in terms of a phase state.
The approach is illustrated by
deriving the density matrix of a single-mode heat
environment interacting asymmetrically with two qubits. Our
results are in good agreement with the available results in the
literature. This approach opens new perspectives for treating
complicated systems and may impact other applications in the
quantum theory.
\end{abstract}
\pacs{ 03.65.Ud, 03.67.-a,
42.50.Dv} \maketitle
{\bf Key words:} mixed state, thermal state, density matrix,
Schrodinger equation, entanglement, two-qubit
\section{Introduction}
In quantum theory, the state of the system has two
categories; namely pure state and mixed state. A pure state
implies perfect knowledge of the system. For the mixed case, we
do not have enough information to specify the state of the system
completely and hence cannot form its wavefunction. In this
situation, we can only describe the system via the density matrix
$\hat{\rho}_f$. Mixed state frequently appears in the decoherence
problems, e.g., \cite{bernet}. We can obtain the mixed state,
which is associated with any state, by considering its phase to be
totally randomized. The best example of the mixed state is the
thermal field, which represents an electromagnetic radiation
emitted by a source at temperature $T$. The examination of a
quantum beam with a thermal noise is an important topic from both
theoretical and practical points of view, e.g. \cite{thermal}.
In the quantum information theory, a considerable
attention has been paid to the entanglement of the bipartite and
the multipartite systems in which one of the subsystems exists
initially in the thermal equilibrium \cite{amesen,lee1,lee2}. The
conclusion of all these studies is that it is possible for the
thermal field, which is a highly chaotic field, to induce
entanglement between qubits. The previous studies have been
limited to the systems whose exact form of the evolution operators
is obtainable \cite{amesen,lee2}. For the other systems the
solution is difficult or even impossible. For some few systems of
the latter, one has to use the numerical methods to solve the
master equation of the system \cite{lee1}, but the results cannot
be totally trusted. In this paper we develop, for the first time,
a simple analytical approach solving this problem exactly. This
approach can provide
the dynamical density matrix of the unitary quantum system
whose one of its components is initially in the mixed state such
as the thermal field. It works only when the Schr\"{o}dinger's
equation of the system for any arbitrary initial state is
solvable. Furthermore, the approach can be applied as an
alternative technique for finding the solution of the systems
whose evolution operators exist but they are very complicated. For
instance, the evolution of the mixed field with the multi-level
atoms, e.g. three-level atom and four-level atom, etc.
In section II we describe the approach in details. In section
III we give an application for the approach by solving specific
problem. The example, which we have considered, is that the
problem of the two atoms in the cavity interact asymmetrically
with the thermal field. The reason behind this choice is that
this system is a subject of the current research in relation to
entanglement \cite{amesen,lee1,lee2}. Thus, we can easily validate
the approach by comparing our results with the available results
in the literature.
\section{Description of the approach}
In this section, we describe how one can deduce
the density matrix of the unitary quantum system
whose one of its components is initially prepared in the thermal
state. Before going into details, let us briefly state some
properties of the thermal field. The thermal field has a diagonal
expansion
in terms of Fock states as \cite{bernet}:
\begin{equation}
\hat{\rho}_f=\sum\limits_{n=0}^{\infty}p(n)|n\rangle\langle
n|=\frac{1}{\pi}\int\limits_{0}^{\pi}|z(\phi)\rangle\langle
z(\phi)| d\phi, \label{1.2}
\end{equation}
where $p(n)$ is the photon-number distribution of the thermal
field and $|z(\phi)\rangle$ is the associated phase state having
the form:
\begin{equation}
|z(\phi)\rangle=\sum\limits_{n=0}^{\infty}\sqrt{p(n)}\exp(in\phi)|n\rangle
=\sum\limits_{n=0}^{\infty}C_n(\phi)|n\rangle,
\label{1.3}
\end{equation}
where
\begin{equation}
C_n(\phi)=\sqrt{p(n)}\exp(in\phi),\quad
p(n)=\frac{\bar{n}^n}{(1+\bar{n})^{n+1}}
\label{1.4}
\end{equation}
and $\bar{n}$ is the mean-photon number of the thermal field
having the form $\bar{n}=(e^{\hbar w/k_b T}-1)^{-1}$ and $k_b$ is
the Boltzmann's constant. It is obvious that $\bar{n}$ increases
by increasing the temperature $T$.
Now we are in a position to explain the approach.
Assume that we have a bipartite system whose Hamiltonian is
$\hat{H}_{db}$, where $d$ and $b$ stand for the field and the other party,
respectively. These two components are initially prepared in the
states
$|\psi_d(0)\rangle$ and $|\psi_b(0)\rangle$.
The Schr\"{o}dinger's equation of this system is solvable and
hence we can obtain the wavefunction as $|\psi(t)\rangle$. From
the basic concepts of the quantum mechanics we have:
\begin{equation}\label{ficn1}
|\psi(t,0)\rangle=\hat{U}(t,0)|\psi_d(0)\rangle\bigotimes|\psi_b(0)\rangle,
\end{equation}
where $\hat{U}(t,0)$ is the evolution operator of the system
regardless if we can deduce its explicit form or not. Under these
assumptions we can solve this system accurately if the party $d$
is initially prepared in the thermal field $\hat{\rho}_f$ instead
of $|\psi_d(0)\rangle$. In this case the initial state of the
system reads:
\begin{equation}
\hat{\rho}(0)=\hat{\rho}_f\bigotimes |\psi_b(0)\rangle
\langle\psi_b(0)|=
\frac{1}{\pi}\int\limits_{0}^{\pi}|z(\phi)\rangle\langle
z(\phi)|\bigotimes|\psi_b(0)\rangle \langle\psi_b(0)| d\phi.
\label{ficn2}
\end{equation}
Under the evolution operator $\hat{U}(t,0)$, the system in the
initial state (\ref{ficn2}) evolves as:
\begin{equation}
\hat{\rho}(t)=\hat{U}(t,0)\hat{\rho}(0)\hat{U}^{\dagger}(t,0)
=\frac{1}{\pi}\int\limits_{0}^{\pi}|\psi_z(t,\phi)\rangle\langle\psi_z
(t,\phi)| d\phi, \label{ficn3}
\end{equation}
where
$|\psi_z(t,\phi)\rangle=\hat{U}(t,0)|z(\phi)\rangle\bigotimes|\psi_b(0)\rangle$.
Our target is to deduce the wavefunction
$|\psi_z(t,\phi)\rangle$. If the explicit form of $\hat{U}(t,0)$
is available, the solution is straightforward. If it is not, we
can solve the Schr\"{o}dinger's equation for the initial condition
$|z(\phi)\rangle$ and $|\psi_b(0)\rangle$. We have to emphasize
if the system is solvable for the arbitrary state
$|\psi_d(0)\rangle$, it will be automatically solvable for the
state $|z(\phi)\rangle$. This is based on the fact that any state
of the field is just a linear combination of the Fock states
weighted by a specific distribution. As soon as we derive
$|\psi_z(t,\phi)\rangle$ we substitute it into (\ref{ficn3}) and
carry out the integration over the phase $\phi$
to obtain the requested density matrix. From the above description, the theme of the approach
we transform the problem of obtaining the dynamical density
matrix to that of finding the wavefunction of the system when the
field is initially prepared in the state $|z(\phi)\rangle$.
In spite of the simplicity of the approach, it is efficient and
able to provide the exact solutions for some non-analytically
solvable problems so far. In the following section we show this
fact by developing the analytical solution for one of those
problems, which has been already numerically treated.
\section{Example: two-qubit problem}
In this section, we apply the approach described in the preceding
section to obtain the density matrix of the two-qubit problem.
Precisely, we deduce the dynamical density matrix of the system of
the single-mode thermal field interacting simultaneously with the
two two-level atoms in the cavity in the non-identical fashion.
The numerical technique has been exploited for solving this system
\cite{lee1}, however, here we present the analytical solution. It
is worthwhile mentioning that the two-identical-atom version of
this system has been already demonstrated, e.g.,
\cite{lee1,lee2,nf}. For the latter the dynamical density matrix
of the system has been derived by the Kraus representation
\cite{kraus}. That cannot be applied to the non-identical-atom
system. Also, we verify the validity of the approach by
comparing some of our results to those in the literature in
relation to entanglement.
We proceed, under the rotating wave approximation the
Hamiltonian describing the system takes the form
\cite{lee1,lee2,nf}:
\begin{eqnarray}
\begin{array}{lr}
\frac{\hat{H}}{\hbar}=\hat{H}_0+\hat{H}_I,\\
\\
\hat{H}_0= \omega\hat{a}^{\dagger}\hat{a}+
\frac{\omega_a}{2}(\hat{\sigma}_1^{z}+\hat{\sigma}_2^{z}),\quad
\hat{H}_I=\sum\limits_{j=1}^2 \lambda_j (\hat{a}\hat{\sigma}_j^{+}
+ \hat{a}^{\dagger }\hat{\sigma}_j^{-}),
\label{6}
\end{array}
\end{eqnarray}
where $\hat{H}_0$ and $\hat{H}_I$ are the free and interaction
parts of the Hamiltonian, $\hat{\sigma}_j^{\pm}$ and
$\hat{\sigma}_j^{z}$ are the Pauli spin operators of the $j$th
atom; $\hat{a}\quad (\hat{a}^{\dagger})$ is the annihilation
(creation) operator denoting the cavity mode, $\omega$ and
$\omega_a$ are the frequencies of the cavity mode and the atomic
systems (we consider that the two atoms have the same frequency)
and $\lambda_j$ is the coupling constant between the $j$th atom
and the field. Based on the relation between $\lambda_1$ and
$\lambda_2$ we have two cases; namely asymmetric case
($\lambda_1\neq \lambda_2$) and symmetric case ($\lambda_1=
\lambda_2$). We should stress that the asymmetric case is closer
to experiment than the symmetric one \cite{duan}. Finally,
throughout the investigation we assume that $\omega_a=\omega$.
Now, our aim is to deduce the explicit form of the density matrix
$\hat{\rho}(t)$ of the Hamiltonian (\ref{6}) when the field is
initially prepared in the thermal field (\ref{1.2}) and the atoms
are initially
in the following mixed
state:
\begin{equation}
\hat{\rho}_a=\sin^2 \theta \cos^2 \vartheta|e_1,e_2\rangle\langle
e_2,e_1|+\sin^2 \theta \sin^2 \vartheta |g_1,g_2\rangle\langle
g_2,g_1| + \cos^2 \theta |e_1,g_2\rangle\langle g_2,e_1|,
\label{1.1}
\end{equation}
where $|e_j\rangle$ and $|g_j\rangle $ stand for the excited and
the ground atomic states of the $j$th atom, respectively. The
variables $\theta$ and $\vartheta$ are phases, which can be
specified to give different types of the initial atomic states.
The subscript $a$ denotes the atomic system. We should stress that
the suggested approach is applicable for any initial atomic state
not only (\ref{1.1}).
Noteworthily, for $\lambda_1\neq \lambda_2$ we cannot derive
the explicit form of the evolution operator of the system
$\hat{U}(t,0)$, however, we can solve the Schr\"{o}dinger's
equation \cite{faisala}. The latter is the main demand to apply
the approach.
The whole initial state of the system can be easily written in the form (\ref{ficn2}).
Under the evolution operator $\hat{U}(t,0)$, this initial state
evolves as:
\begin{eqnarray}
\begin{array}{lr}
\hat{\rho}(t)=\hat{U}(t,0)\hat{\rho}(0)\hat{U}^{\dagger}(t,0)\\
\\
=\frac{\sin^2 \theta \cos^2 \vartheta}{\pi}\int\limits_{0}^{\pi}
|\Psi_1(t,\phi)\rangle\langle \Psi_1(t,\phi)|d\phi +\frac{\sin^2
\theta \sin^2 \vartheta}{\pi}\int\limits_{0}^{\pi}
|\Psi_2(t,\phi)\rangle\langle \Psi_2(t,\phi)|d\phi\\
\\
+\frac{\cos^2\theta}{\pi}\int\limits_{0}^{\pi}
|\Psi_3(t,\phi)\rangle\langle \Psi_3(t,\phi)|d\phi,
\label{1.7}
\end{array}
\end{eqnarray}
where
\begin{eqnarray}
\begin{array}{lr}
|\Psi_j(t,\phi)\rangle=\hat{U}(t,0)|\Psi_j(0,\phi)\rangle, \quad j=1,2,3,\\
\\
|\Psi_1(0,\phi)\rangle=|e_1,e_2,z(\phi)\rangle, \quad
|\Psi_2(0,\phi)\rangle=|g_1,g_2,z(\phi)\rangle, \quad
|\Psi_3(0,\phi)\rangle=|e_1,g_2,z(\phi)\rangle.
\label{ficn5}
\end{array}
\end{eqnarray}
Our main concern is to derive
the wave functions $|\Psi_j(t,\phi)\rangle$. These can be easily obtained by solving the Schr\"{o}dinger's equation
three times for
the given initial conditions as \cite{faisala}:
\begin{widetext}
\begin{eqnarray}
\begin{array}{lr}
\mid \Psi_j (t,\phi)\rangle =\sum\limits_{n=0}^{\infty
}C_n(\phi)\left[ X_{1}^{(j)}(t,n)\mid e_{1},e_{2},n\rangle
+X_{2}^{(j)}(t,n)\mid
e_{1},g_{2},n+1\rangle \right. \\
\\
+\left. X_{3}^{(j)}(t,n)\mid g_{1},e_{2},n+1\rangle
+X_{4}^{(j)}(t,n)\mid g_{1},g_{2,}n+2\rangle \right], \quad
j=1,2,3 \label{new3}
\end{array}
\end{eqnarray}
\end{widetext}
where
\begin{widetext}
\begin{eqnarray}
\begin{array}{lr}
X_{1}^{(1)}(t,n)=\mu^-_n\frac{\cos(t\Omega^+_n)}{2\Lambda_n}
+\mu^+_n\frac{\cos(t\Omega^-_n)}{2\Lambda_n},
\\
X_{2}^{(1)}(t,n)=\frac{-i\lambda_2\sqrt{n+1}}{2\Lambda_n}\{
[4\lambda_1^2(n+2)
-\mu^-_n]\frac{\sin(t\Omega^+_n)}{\Omega^+_n}
-[4\lambda_1^2(n+2)
-\mu^+_n]\frac{\sin(t\Omega^-_n)}{\Omega^-_n}\},
\\
X_{3}^{(1)}(t,n)=\frac{-i\lambda_1\sqrt{n+1}}{2\Lambda_n}\{
[4\lambda_2^2(n+2)
-\mu^-_n]\frac{\sin(t\Omega^+_n)}{\Omega^+_n}
-[4\lambda_2^2(n+2)
-\mu^+_n]\frac{\sin(t\Omega^-_n)}{\Omega^-_n}\},
\\
X_{4}^{(1)}(t,n)=\frac{2\lambda_1\lambda_2\sqrt{(n+2)(n+1)}}{\Lambda_n}[\cos(t\Omega^+_n)-
\cos(t\Omega^-_n)] ,\\
X_{1}^{(2)}(t,n)=\frac{2\lambda_1\lambda_2\sqrt{n(n-1)}}{\Lambda_{n-2}}\left[
\cos(t\Omega^+_{n-2})-\cos(t\Omega^-_{n-2})\right],
\\
X_{2}^{(2)}(t,n)=\frac{-i\lambda_1\sqrt{n}}{2\Lambda_{n-2}}\left\{
[4\lambda_2^2(n-1)
+\mu^+_{n-2}]\frac{\sin(t\Omega^+_{n-2})}{\Omega^+_{n-2}}
-[4\lambda_2^2(n-1)
+\mu^-_{n-2}]\frac{\sin(t\Omega^-_{n-2})}{\Omega^-_{n-2}}\right\},
\\
X_{3}^{(2)}(t,n)=\frac{-i\lambda_2\sqrt{n}}{2\Lambda_{n-2}}\left\{
[4\lambda_1^2(n-1)
+\mu^+_{n-2}]\frac{\sin(t\Omega^+_{n-2})}{\Omega^+_{n-2}}
-[4\lambda_1^2(n-1)
+\mu^-_{n-2}]\frac{\sin(t\Omega^-_{n-2})}{\Omega^-_{n-2}}\right\},
\\
X_{4}^{(2)}(t,n)=\frac{\mu^+_{n-2}}{2\Lambda_{n-2}}\cos(t\Omega^+_{n-2})-
\frac{\mu^-_{n-2}}{2\Lambda_{n-2}}\cos(t\Omega^-_{n-2}),\\
X_{1}^{(3)}(t,n)=\frac{i\lambda_2\sqrt{n}}{2\Lambda_{n-1}}\left\{[\mu^-_{n-1}
-4\lambda_1^2(n+1)]
\frac{\sin(t\Omega^+_{n-1})}{\Omega^+_{n-1}}
+[4\lambda_1^2(n+1)-\mu^+_{n-1}
]
\frac{\sin(t\Omega^-_{n-1})}{\Omega^-_{n-1}}\right\},
\\
X_{2}^{(3)}(t,n)=\frac{1}{2\Lambda_{n-1}}\left\{
[\lambda_1^2-\lambda_2^2+\Lambda_{n-1}]\cos(t\Omega^+_{n-1}) +
[\lambda_2^2-\lambda_1^2+\Lambda_{n-1}]\cos(t\Omega^-_{n-1})\right\},
\\
X_{3}^{(3)}(t,n)=\frac{\lambda_1\lambda_2(2n+1)}{\Lambda_{n-1}}[\cos(t\Omega^+_{n-1})
- \cos(t\Omega^-_{n-1})],
\\
X_{4}^{(3)}(t,n)=\frac{i\lambda_1\sqrt{n+1}}{2\Lambda_{n-1}}\left\{[\mu^-_{n-1}+4\lambda_2^2n]
\frac{\sin(t\Omega^-_{n-1})}{\Omega^-_{n-1}}-[\mu^+_{n-1}+4\lambda_2^2n]
\frac{\sin(t\Omega^+_{n-1})}{\Omega^+_{n-1}}\right\}
\label{new4}
\end{array}
\end{eqnarray}
\end{widetext}
and
\begin{widetext}
\begin{eqnarray}\label{insert1}
\begin{array}{lr}
\Lambda_n=\sqrt{(\lambda_1^2+\lambda_2^2)^2(2n+3)^2-4(\lambda_1^2-\lambda_2^2)^2(n+2)(n+1)},
\quad \mu^{\pm}_n=\lambda_1^2+\lambda_2^2\pm \Lambda_{n},
\\
\\
\Omega^\pm_n=\frac{1}{\sqrt{2}}\sqrt{(\lambda_1^2+\lambda_2^2)(2n+3)\pm
\Lambda_n}.
\end{array}
\end{eqnarray}
\end{widetext}
\begin{figure}
\caption{
Atom-atom entanglement induced by the interaction with a thermal
field having $\bar{n}
\label{fig1}
\end{figure}
The dynamical density matrix of the whole system can be obtained
by substituting (\ref{new3})--(\ref{insert1}) into (\ref{1.7})
and then carrying out the integration with respect to $\phi$. This
treatment provides the exact solution of the problem. Moreover,
it is easier and better than solving the Liouville equation
numerically \cite{lee1}. To evaluate the density matrix of the two
qubits, i.e. $\hat{\rho}_a(t)$, one has to trace over the field
variables the density matrix of the system as:
\begin{equation}\label{new5}
\hat{\rho}_a(t)=B_{e_1e_2}|e_1,e_2\rangle \langle e_1,e_2|+B_{g_1g_2}|g_1,g_2\rangle \langle
g_1,g_2|+(B_{e_1g_2}|e_1,g_2\rangle+B_{g_1e_2}|g_1,e_2\rangle)(\langle e_1,g_2|B^*_{e_1g_2}+\langle
g_1,e_2|B^*_{g_1e_2}),
\end{equation}
where
\begin{widetext}
\begin{eqnarray}\label{insert2}
\begin{array}{lr}
B_{e_1e_2}=\sum\limits_{n=0}^{\infty }P(n)\left[
|X_{1}^{(1)}(t,n)|^2\sin^2\theta\cos^2\vartheta+|X_{1}^{(2)}(t,n)|^2\sin^2\theta\sin^2\vartheta+
|X_{1}^{(3)}(t,n)|^2\cos^2\theta\right],
\\
B_{g_1g_2}=\sum\limits_{n=0}^{\infty }P(n)\left[
|X_{4}^{(1)}(t,n)|^2\sin^2\theta\cos^2\vartheta+|X_{4}^{(2)}(t,n)|^2\sin^2\theta\sin^2\vartheta+
|X_{4}^{(3)}(t,n)|^2\cos^2\theta\right],
\\
B_{e_1g_2}=\sum\limits_{n=0}^{\infty }P(n)\left[
X_{2}^{(1)}(t,n)X_{3}^{(1)*}(t,n)\sin^2\theta\cos^2\vartheta\right.\\
+\left.
X_{2}^{(2)}(t,n)X_{3}^{(2)*}(t,n)\sin^2\theta\sin^2\vartheta+
X_{2}^{(3)}(t,n)X_{3}^{(3)*}(t,n)\cos^2\theta\right],\quad
B_{g_1e_2}=B_{e_1g_2}^*.
\end{array}
\end{eqnarray}
\end{widetext}
For the best of our knowledge this is the first time the explicit
form of the density matrix of this system to be presented. This
reflects the power of the approach. The density matrix
(\ref{new5}) tends to that of the symmetric case \cite{lee2},
which was obtained by the Kraus representation, by simply setting
$\lambda_1=\lambda_2=\lambda$.
To verify the validity of the approach we comment on the
entanglement of the two qubits controlled by (\ref{new5}).
In this regard, we use the negative values of the partial
transposition \cite{lee}, which is frequently used for such
systems \cite{amesen,lee2}, and is defined as:
\begin{equation}\label{masu}
\xi=-2\sum_j \gamma_j^-,
\end{equation}
where $\gamma_j^-$ are the negative eigenvalues of the partial
transposition of $\hat{\rho}_a(t)$. The entanglement measure
ranges between $\xi=0$ for separable qubits and $\xi=1$ for
maximally entangled qubits.
For the density matrix (\ref{new5}) there is only
one eigenvalue of those of the partial transposition matrix, which
can approach negative values, having the form:
\begin{equation}\label{negativ}
2
\gamma_j^-=B_{e_1e_2}+B_{g_1g_2}-\sqrt{(B_{e_1e_2}-B_{g_1g_2})^2+4|B_{g_1e_2}|^2}.
\end{equation}
By means of (\ref{new4})--(\ref{negativ})
we have obtained the results of
\cite{lee2} for the same values of the interaction
parameters. Now it is therefore reasonable to make a comparison
with the results of the general case, which has been numerically
presented
in \cite{lee1}. In that article the analysis has been
confined to such forms of
the coupling constants $\lambda_1=1+\gamma$ and
$\lambda_2=1-\gamma$, where $\gamma$ (with $0\leq \gamma\leq 1$)
is the relative difference between the two atomic couplings. This
means that the strength of the interaction of one of the
bipartites (atom-field) is increasing, while the other is
decreasing simultaneously. As an example, in Figs. 1 we plot the quantity
$\xi$ for the same values of the interaction parameters as those
of Figs. 2 in \cite{lee1}, which were given for the concurrence.
It is obvious that our figures and those in \cite{lee1} are
identical even though they represent different measures. This
clearly confirm the validity of the approach. Obviously, the
analytical treatments are in general better than the numerical
ones as they can provide us
some analytical facts about the system.
For instance, the condition of involving the expression
(\ref{negativ}) negative values is:
\begin{equation}\label{final11}
\Upsilon=B_{e_1e_2}B_{g_1g_2}-|B_{g_1e_2}|^2<0.
\end{equation}
From (\ref{new4})--(\ref{insert2}), when $n=0$ and the two atoms
are in $|e_1,g_2\rangle$, the inequality (\ref{final11}) can be
simplified as:
\begin{equation}\label{final1}
\Upsilon=-\frac{\lambda_1\lambda_2}{(\lambda_1^2+\lambda_2^2)^2}
[\cos(t\Omega^+_{-1})-1][\lambda_1^2\cos(t\Omega^+_{-1})+\lambda_2^2]<0.
\end{equation}
It is evident that the expression (\ref{final1}) gives
$\Upsilon=\sin^2(t\Omega^+_{-1})/4$ for the symmetric case. This
means that the symmetric case (with $\bar{n}=0$) cannot generate
any entanglement between the two atoms, which are initially in
$|e_1,g_2\rangle$. Nevertheless, the asymmetric case can generate
entanglement
for certain values of the coupling constants, in particular,
for those satisfying the inequality
$\cos(t\Omega^+_{-1})<-\lambda_2^2/\lambda_1^2$. Furthermore, for
the atoms, which are initially in $|g_1,g_2\rangle$ and $n=0$,
one can easily prove that $X_j^{(3)}(t,0)=0$. Thus, the condition
(\ref{final11}) is not fulfilled, i.e. entanglement cannot be
established in this case.
It is worth prompting that we have aimed by the above
discussion to justify the validity of the approach not to repeat
the study of the entanglement of the two qubits. So that we have
selected few cases for the sake of comparison only. Nevertheless,
from the treatment we have performed, which was not presented here
for the sake of brevity, and the other treatments given in
\cite{amesen,lee1,lee2} we can conclude that a highly chaotic
state in an infinite-dimensional Hilbert space can entangle two
qubits depending on the type of their interaction with the field
as well as the initial conditions of the system. The amounts of
entanglement generated by the asymmetric case are much greater
than those generated by the symmetric one.
In conclusion, we have developed, for the first time, a simple
analytical approach for solving unitary system when the initial
field, as one of its components, is in the mixed state, e.g.
thermal light. The approach is applicable only when the
Schr\"{o}dinger's equation of the system is solvable for
any initial arbitrary state. We have verified the validity of the approach
for some selective cases for the entanglement of two
qubits. The approach enables us to check the results, which
have been numerically
obtained earlier \cite{lee1}.
As a final note, we believe that
our approach represents a powerful tool to solve
such type of problems and
may impact other applications in the quantum theory.
\section*{ Acknowledgment}
The author would like to thank Professors Z. Ficek and M. S. Abdalla
for the critical reading of the manuscript.
\end{document} |
\begin{document}
\global\long\def\mathbf{N}{\mathbf{N}}
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\global\long\def\mathbf{R}{\mathbf{R}}
\global\long\def{\color{blue}\mathcal{B}}{{\color{blue}\mathcal{B}}}
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\global\long\def\mathcal{C}{\mathcal{C}}
\global\long\def\mathcal{F}{\mathcal{F}}
\global\long\def\textnormal{\ensuremath{\frownie}}{\textnormal{\ensuremath{\frownie}}}
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\global\long\def\Padded#1{\padded{#1}}
\makeatletter
\global\long\def\labelifpresent#1{\ifpresent{#1}{}[\nonumber\@gobbletwo]}
\makeatother
\title{Ramsey numbers of Boolean lattices}
\author{D\'aniel Gr\'osz\thanks{Department of Mathematics, University of Pisa, Pisa. e-mail: \protect\href{mailto:[email protected]}{[email protected]}}
\and Abhishek Methuku\thanks{School of Mathematics, University of Birmingham, Birmingham. e-mail: \protect\href{mailto:[email protected]}{[email protected]}}
\and Casey Tompkins\thanks{Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon. e-mail: \protect\href{mailto:[email protected]}{[email protected]}}}
\maketitle
\begin{abstract}
\noindent \setlength\parskip
amount The \emph{poset Ramsey number} $R(\mathcal{Q}_{m},\mathcal{Q}_{n})$ is the smallest integer $N$
such that any blue-red coloring of the elements of the Boolean lattice $\mathcal{Q}_{N}$ has a blue induced copy of~$\mathcal{Q}_{m}$ or
a red induced copy of~$\mathcal{Q}_{n}$. The \emph{weak poset Ramsey number} $R_{w}(\mathcal{Q}_{m},\mathcal{Q}_{n})$ is defined analogously, with
weak copies instead of induced copies. It is easy to see that $R(\mathcal{Q}_{m},\mathcal{Q}_{n})\ge R_{w}(\mathcal{Q}_{m},\mathcal{Q}_{n})$.
\noindent Axenovich and Walzer~\cite{AxenovichWalzer} showed that $n+2\le R(\mathcal{Q}_{2},\mathcal{Q}_{n})\le2n+2$. Recently, Lu and Thompson~\cite{LuThompson}
improved the upper bound to $\frac{5}{3}n+2$. In this paper, we solve this problem asymptotically by showing that $R(\mathcal{Q}_{2},\mathcal{Q}_{n})=n+O(n/\log n)$.
\noindent In the diagonal case, Cox and Stolee~\cite{CoxStolee} proved $R_{w}(\mathcal{Q}_{n},\mathcal{Q}_{n})\ge2n+1$ using a probabilistic
construction. In the induced case, Bohman and Peng~\cite{BohmanPeng} showed $R(\mathcal{Q}_{n},\mathcal{Q}_{n})\ge2n+1$ using an explicit
construction. Improving these results, we show that $R_{w}(\mathcal{Q}_{m},\mathcal{Q}_{n})\ge n+m+1$ for all $m\ge2$ and large~$n$ by
giving an explicit construction; in particular, we prove that $R_{w}(\mathcal{Q}_{2},\mathcal{Q}_{n})=n+3$.
\end{abstract}
\section{Introduction}
\partitle{Background and definitions.} The classical Ramsey theorem asserts that for any $m$ and $n$, there is an integer~$N$
such that every blue-red edge coloring of the complete graph on $N$ vertices contains a blue clique on $m$ vertices or
a red clique on $n$ vertices. Determining the smallest such integer $N$, known as the Ramsey number is a central problem
in combinatorics. More generally, for any two graphs $G$ and $H$, the Ramsey number is the smallest integer $N$ such that
every blue-red edge coloring of the complete graph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. Several
natural variations of these problems such as multicolor Ramsey numbers, and hypergraph Ramsey numbers are major subjects
of ongoing research. For further examples, we refer the reader to the surveys ~\cite{ConlonFoxSudakov,MubayiSuk}.
In this paper, we will study poset Ramsey numbers. A \emph{partially ordered set} (or a \emph{poset} for short) is a set
with an accompanying relation $\le$ which is transitive, reflexive, and antisymmetric. A \emph{Boolean lattice} of dimension~$n$,
denoted by $\mathcal{Q}_{n}$, is the power set of $[n]\coloneqq\{1,2,\ldots,n\}$ equipped with the inclusion relation. If $(P,\le)$
and $(Q,\le')$ are posets, then an injection $f:P\to Q$ is \emph{order-preserving} if $f(x)\le'f(y)$ whenever $x\le y$;
we say that $f(P)$ is a \emph{weak copy} of~$P$ in~$Q$ and that $P$ is a \emph{weak subposet} of~$Q$. An injection
$f:P\to Q$ is an \emph{order-embedding} if $f(x)\le'f(y)$ if and only if $x\le y$; we say that $f(P)$ is an \emph{induced
copy} of $P$ in $Q$ and that $P$ is an \emph{induced subposet} of~$Q$.
For posets $P_{1}$ and $P_{2}$, the \emph{(induced) poset Ramsey number} $R(P_{1},P_{2})$ is defined to be the smallest
integer $N$ such that every blue-red coloring of the elements of the Boolean lattice $\mathcal{Q}_{N}$ contains an induced copy
of $P_{1}$ whose elements are blue or an induced copy of~$P_{2}$ whose elements are red. Similarly, the \emph{weak poset
Ramsey number} $R_{w}(P_{1},P_{2})$ is defined to be the smallest integer $N$ such that every blue-red coloring of the
elements of the Boolean lattice $\mathcal{Q}_{N}$ contains a weak copy of~$P_{1}$ whose elements are blue or a weak copy of~$P_{2}$
whose elements are red. (For convenience, we will call a copy of poset $P$ all of whose elements are blue is called a blue
copy of $P$, and a copy of poset~$P$ all of whose elements are red is called a red copy of~$P$.) It is easy to see that
$R(P_{1},P_{2})\ge R_{w}(P_{1},P_{2})$. The focus of this paper is the natural problem when $P_{1}$ and $P_{2}$ are Boolean
lattices $\mathcal{Q}_{m}$ and $\mathcal{Q}_{n}$ for $m,n\in\mathbf{N}$. Recently, variants of this problem, such as rainbow poset Ramsey
numbers have been studied in~\cite{Chang.etal,Chenetal,CoxStolee}.
\partitle{Induced poset Ramsey numbers.} For the diagonal poset Ramsey number $R(\mathcal{Q}_{n},\mathcal{Q}_{n}),$ Axenovich and Walzer~\cite{AxenovichWalzer}
showed that $2n\le R(\mathcal{Q}_{n},\mathcal{Q}_{n})\le n^{2}+2n$. Walzer~\cite{WalzerThesis} improved the upper bound to $R(\mathcal{Q}_{n},\mathcal{Q}_{n})\le n^{2}+1$.
Recently, Lu and Thompson~\cite{LuThompson} further improved it to $R(\mathcal{Q}_{n},\mathcal{Q}_{n})\le n^{2}-n+2$. On the other hand,
Cox and Stolee~\cite{CoxStolee} showed that for $n\ge13$, $R_{w}(\mathcal{Q}_{n},\mathcal{Q}_{n})\ge2n+1$, which implies that $R(\mathcal{Q}_{n},\mathcal{Q}_{n})\ge2n+1$.
More generally, Axenovich and Walzer~\cite{AxenovichWalzer} showed that $n+m\le R(\mathcal{Q}_{m},\mathcal{Q}_{n})\le mn+n+m$ for any integers
$n,m\ge1$. Lu and Thompson~\cite{LuThompson} improved this bound by showing that $R(\mathcal{Q}_{m},\mathcal{Q}_{n})\le(m-2+\frac{9m\text{\textminus}9}{(2m\text{\textminus}3)(m+1)})n+m+3$
for all $n\ge m\ge4$. See \cite{AxenovichWalzer,CoxStolee,LuThompson,WalzerThesis} for several other interesting results.
For the off-diagonal poset Ramsey number $R(\mathcal{Q}_{2},\mathcal{Q}_{n})$, Axenovich and Walzer~\cite{AxenovichWalzer} showed that $n+2\le R(\mathcal{Q}_{2},\mathcal{Q}_{n})\le2n+2$.
Recently, Lu and Thompson~\cite{LuThompson} improved the upper bound by proving that $R(\mathcal{Q}_{2},\mathcal{Q}_{n})\le\frac{5}{3}n+2$.
In this paper, we determine $R(\mathcal{Q}_{2},\mathcal{Q}_{n})$ asymptotically by proving the following theorem.
\begin{thm}
\label{thm:Upperbound}For every $c>2$, there exists an integer $n_{0}$ such that for all $n\ge n_{0}$, we have
\[
R(\mathcal{Q}_{2},\mathcal{Q}_{n})\le n+c\frac{n}{\log_{2}n}.
\]
\end{thm}
Combining \cref{thm:Upperbound} with the lower bound $R(\mathcal{Q}_{2},\mathcal{Q}_{n})\ge n+2$, we obtain that $R(\mathcal{Q}_{2},\mathcal{Q}_{n})$
is asymptotically equal to~$n$. We prove \cref{thm:Upperbound} in \cref{sec:upper}. In fact, it follows from
our proof of \cref{thm:Upperbound} that for all $n\ge2$, we have $R(\mathcal{Q}_{2},\mathcal{Q}_{n})\le n+6.14\frac{n}{\log_{2}n}$.
\partitle{Weak poset Ramsey numbers.} A chain of length~$k$ is a poset of~$k$ distinct, pairwise comparable elements
and is denoted by~$C_{k}$. Cox and Stolee~\cite{CoxStolee} showed that $R_{w}(C_{k},\mathcal{Q}_{n})=n+k-1$; since $\mathcal{Q}_{m}$~is
a weak subposet of~$C_{2^{m}}$, this implies that $R_{w}(\mathcal{Q}_{m},\mathcal{Q}_{n})\le n+2^{m}-1$. The lower bound $R_{w}(\mathcal{Q}_{m},\mathcal{Q}_{n})\ge m+n$
is obtained by a simple ``layered'' coloring of~$\mathcal{Q}_{m+n-1}$ considered by Axenovich and Walzer~\cite{AxenovichWalzer},
which is described as follows. The collection of all subsets of~$[N]$ of a given size~$k$ is called a \emph{layer}. A
coloring of~$\mathcal{Q}_{N}$ is \emph{layered} if for every layer, all sets on that layer have the same color. A layered coloring
of~$\mathcal{Q}_{m+n-1}$ with $m$~blue layers and $n$~red layers does not contain a (weak) blue copy of~$\mathcal{Q}_{m}$ or a (weak)
red copy of~$\mathcal{Q}_{n}$. Therefore, $R_{w}(\mathcal{Q}_{m},\mathcal{Q}_{n})\ge m+n$ (which implies $R(\mathcal{Q}_{m},\mathcal{Q}_{n})\ge m+n$). Despite the
work of several researchers, so far this lower bound on $R_{w}(\mathcal{Q}_{m},\mathcal{Q}_{n})$ has not been improved except in the diagonal
case: Cox and Stolee~\cite{CoxStolee} showed that $R_{w}(\mathcal{Q}_{n},\mathcal{Q}_{n})\ge2n+1$ for $n\ge13$ using a probabilistic construction.
Recently, in the induced case, Bohman and Peng \cite{BohmanPeng} gave an explicit construction showing the bound $R(\mathcal{Q}_{n},\mathcal{Q}_{n})\ge2n+1$.
Note that these constructions showing $R(\mathcal{Q}_{n},\mathcal{Q}_{n})\ge2n+1$ cannot be layered.
We give an explicit construction which yields a lower bound on $R_{w}(\mathcal{Q}_{m},\mathcal{Q}_{n})$ for all~$m$ and $n\ge68$, thereby
generalizing the results of Bohman and Peng to the weak poset case, and additionally extending their results and those of
Cox and Stolee to the off-diagonal case.
\begin{thm}
\label{thm:Lowerbound}For any $m\ge2$ and $n\ge68$, we have \textup{
\[
R_{w}(\mathcal{Q}_{m},\mathcal{Q}_{n})\ge m+n+1.
\]
}
\end{thm}
Note that \cref{thm:Lowerbound} shows that $R_{w}(\mathcal{Q}_{2},\mathcal{Q}_{n})=n+3$ since $R_{w}(\mathcal{Q}_{2},\mathcal{Q}_{n})\le n+2^{2}-1=n+3$
by the upper bound mentioned earlier.
We prove \cref{thm:Lowerbound} in \cref{subsec:ours-general}. The construction and the proof of \cref{thm:Lowerbound}
are simpler if we restrict ourselves to the case of $m=2$ and consider induced subposets rather than weak subposets . Therefore,
in order to illustrate the main ideas of our construction, we present a short proof showing the special case $R(\mathcal{Q}_{2},\mathcal{Q}_{n})\ge n+3$
(for $n\ge18$) in \cref{subsec:ours-induced}. We also give a probabilistic construction for $m\ge3$ and $n$ sufficiently
large in \cref{subsec:cox stolee} by generalizing a construction of Cox and Stolee~\cite{CoxStolee}.
\section{\label{sec:upper}Upper bound: Proof of \texorpdfstring{\cref{thm:Upperbound}}{Theorem~\ref{thm:Upperbound}}}
Let $k=\left\lfloor c\frac{n}{\log_{2}n}\right\rfloor $. Assume that ${\color{blue}\mathcal{B}},{\color{red}\mathcal{R}}\subset\mathcal{Q}_{n+k}$ such that ${\color{blue}\mathcal{B}}\sqcup{\color{red}\mathcal{R}}=\mathcal{Q}_{n+k}$,
and further assume that $\mathcal{Q}_{2}$~is not an induced subposet of ${\color{blue}\mathcal{B}}$, and $\mathcal{Q}_{n}$~is not an induced subposet of~${\color{red}\mathcal{R}}$.
Before continuing with the proof of \cref{thm:Upperbound}, let us provide an outline of the proof.
\emph{Outline of the proof.} We attempt to define an order-embedding $\varphi$ from $\mathcal{Q}_{n}$ into~${\color{red}\mathcal{R}}$ recursively, starting
with~$\emptyset$, in such a way that the image of each set only depends on the images of its proper subsets. For every
$A\subseteq[n]$, $\varphi(A)$ will be a superset of~$A$, possibly containing some additional elements from $[n+k]\setminus[n]$.
If $\emptyset\in{\color{red}\mathcal{R}}$, then we set $\varphi(\emptyset)=\emptyset$. More generally, in order for $\varphi$ to be order-preserving,
for any set $A\in\mathcal{Q}_{n}$, $\varphi(A)$ must be a superset of the images of all proper subsets of~$A$; as long as the
minimal set that is a superset of $A$ and also has this property is in~${\color{red}\mathcal{R}}$, we set it as $\varphi(A)$. If instead this
minimal set is in~${\color{blue}\mathcal{B}}$, then we proceed to add elements of $[n+k]\setminus[n]$ to it, in an order determined by some arbitrary
permutation~$\pi$ of $[n+k]\setminus[n]$, until we obtain a set that is in ${\color{red}\mathcal{R}}$. Throughout this recursive procedure,
in addition to the injection~$\varphi$, we construct a function~$\alpha$ where $\alpha(A)$ records the number of elements
of $[n+k]\setminus[n]$ we need to include in~$\varphi(A)$ as a result of hitting sets in~${\color{blue}\mathcal{B}}$ while attempting to embed
$A$ (and its subsets, during previous steps of the recursion); and another function~$f$, where $f(A)$ records an actual
chain of length~$\alpha(A)$, consisting of sets in~${\color{blue}\mathcal{B}}$ that we have encountered while trying to embed $A$ and its subsets.
For any fixed permutation~$\pi$ of $[n+k]\setminus[n]$, the above embedding procedure can only fail if, at some point,
as we try to define $\varphi(A)$ for some $A\in\mathcal{Q}_{n}$, we hit a set in~${\color{blue}\mathcal{B}}$, but we have already ``used up'' all $k$
elements of $[n+k]\setminus[n]$, so there are no elements left to add. In this event, we obtain a chain of length $k+1$,
contained in~${\color{blue}\mathcal{B}}$. As $\mathcal{Q}_{n}$~is not an induced subposet of ${\color{red}\mathcal{R}}$, the procedure must fail for all $k!$ permutations~$\pi$
of $[n+k]\setminus[n]$. This way, we can obtain a chain of length $k+1$ inside~${\color{blue}\mathcal{B}}$, corresponding to each of these permutations.
We show that these $k!$ chains must all be distinct. We then show that the existence of $k!$ distinct chains of length
$k+1$ inside ${\color{blue}\mathcal{B}}$ implies that $\mathcal{Q}_{2}$ is an induced subposet of ${\color{blue}\mathcal{B}}$, a contradiction.
Now we continue with the proof of \cref{thm:Upperbound}.
At the core of the proof is \cref{lem:functions}. We will use the following notation: for a chain of sets $\mathcal{C}$
in~$\mathcal{\mathcal{Q}}_{n+k}$ of length~$l$, we denote its sets by $\left(q_{0},q_{1},\ldots,q_{l-1}\right)$ where $q_{0}\subseteq q_{1}\subseteq\ldots\subseteq q_{l-1}$.
\begin{claim}
\label{lem:functions}Let $\pi:[n+k]\setminus[n]\rightarrow[n+k]\setminus[n]$ be a permutation. There exist $\varphi:\mathcal{Q}_{n}\rightarrow{\color{red}\mathcal{R}}\cup\{\textnormal{\ensuremath{\frownie}}\}$
(where $\textnormal{\ensuremath{\frownie}}$~is an arbitrary element, distinct from the members of~${\color{red}\mathcal{R}}$, and used solely to indicate failure to produce
an induced map into~${\color{red}\mathcal{R}}$), $\alpha:\mathcal{Q}_{n}\rightarrow\{0,1,\ldots,k,k+1\}$ and $f:\mathcal{Q}_{n}\rightarrow\mathcal{C}^{\le k+1}({\color{blue}\mathcal{B}})$,
where $\mathcal{C}^{\le k+1}({\color{blue}\mathcal{B}})$ is the family of all chains of length at most $k+1$ in~${\color{blue}\mathcal{B}}$, with the following properties:\gdef\labelwidthi{\widthof{\textbf{\textup{L0. }}}}
\gdef\labeli{\textbf{\textup{P\arabic{enumi}. }}}
\gdef\propertyref#1{P#1}
\gdef\refi{\propertyref{\arabic{enumi}}}
\begin{enumerate}[label=\labeli, ref=\refi, labelsep=0em, leftmargin=0em, labelwidth=\labelwidthi, itemindent=\labelwidth, align=left]
\item \label{enu:phi containment}If $B,A\in\mathcal{Q}_{n}$ and $\varphi(B),\varphi(A)\in{\color{red}\mathcal{R}}$, then $B\subsetneqq A\Longleftrightarrow\varphi(B)\subsetneqq\varphi(A)$.
(This implies that if $\textnormal{\ensuremath{\frownie}}\notin\operatorname{Im}\varphi$, then $\mathcal{Q}_{n}$~is an induced subposet of~${\color{red}\mathcal{R}}$.)
\item \label{enu:alpha order}If $B\subseteq A\in\mathcal{Q}_{n}$, then $\alpha(B)\le\alpha(A)$.
\item \label{enu:alpha phi}If $\alpha(A)=k+1$, then $\varphi(A)=\textnormal{\ensuremath{\frownie}}$. Otherwise $\varphi(A)\cap[n]=A$, and $\varphi(A)=A\cup\{\pi(n+\nobreak1),\pi(n+2),\ldots,\pi(n+\alpha(A))\}$.
\item \label{enu:f chain}For every $A\in\mathcal{Q}_{n}$, $f(A)=\left(f(A)_{0},f(A)_{1},\ldots,f(A)_{\alpha(A)-1}\right)$ is a chain
in~${\color{blue}\mathcal{B}}$ of length $\alpha(A)$ with the property that $f(A)_{i}\setminus[n]=\{\pi(n+1),\pi(n+2),\ldots,\pi(n+i)\}$.
\item \label{enu:f subset}If $A\in\mathcal{Q}_{n}$ such that $1\le\alpha(A)\le k$, then $f(A)_{\alpha(A)-1}\subseteq\varphi(A)$. (In
fact this implies that $f(A)_{\alpha(A)-1}\subsetneqq\varphi(A)$, since the elements of~$f(A)$ are in~${\color{blue}\mathcal{B}}$, while $\varphi(A)$
is in~${\color{red}\mathcal{R}}$. We do not use this observation.)
\end{enumerate}
\gdef\lastproperty{\propertyref{\arabic{enumi}}}
\end{claim}
\begin{proof}
We construct the functions $\varphi$, $\alpha$ and~$f$ recursively, and simultaneously prove the above properties by
induction: we set the values of these functions on a set $A\in\mathcal{Q}_{n}$ in such a way that they only depend on the values
of the functions on proper subsets of~$A$. (This includes the case of $A=\emptyset$ where no proper subsets exist, which
we do not treat in a special way for most of the proof. One can also consider the proof as a recursion and induction on the
size of the set $A$.) Let us fix an $A\in\mathcal{Q}_{n}$. Now we will define the values $\varphi(A)$, $\alpha(A)$ and $f(A)$,
and then prove that \propertyref{1} to \lastproperty{} hold for this set~$A$ under the assumption that they hold for
every proper subset of~$A$.
If there exists a $B\subsetneqq A$ such that $\varphi(B)=\textnormal{\ensuremath{\frownie}}$, then we pick such a set~$B$ arbitrarily, and set $\varphi(A)=\textnormal{\ensuremath{\frownie}}$,
$\alpha(A)=k+1$ and $f(A)=f(B)$. Otherwise let
\begin{align*}
\beta & =\min\bigl\{ i\in\{0,1,\ldots,k\}:\left(\forall B\subsetneqq A:\alpha(B)\le i\right)\bigr\}\\
& =\begin{cases}
{\displaystyle \max_{{\scriptscriptstyle B\subsetneqq A}}\alpha(B)} & \text{if } A\ne\emptyset,\\
0 & \text{if } A=\emptyset,
\end{cases}
\end{align*}
and let
\[
C=A\cup\{\pi(n+1),\pi(n+2),\ldots,\pi(n+\beta)\}=A\cup\left(\bigcup_{B\subsetneqq A}\varphi(B)\right)
\]
(note that $\{\pi(n+1),\pi(n+2),\ldots,\pi(n+\beta)\}=\emptyset$ if $\beta=0$). We get the last equality by applying \ref{enu:alpha phi}
to the proper subsets of~$A$. We want $\varphi(A)$ to be a superset of $C$. If $C\in{\color{red}\mathcal{R}}$, we set $\varphi(A)=C$. If
$C\in{\color{blue}\mathcal{B}}$, we keep adding $\pi(n+\beta+1),\pi(n+\beta+2),\ldots$ to it, until the set is not in~${\color{blue}\mathcal{B}}$, if possible. That
is, let
\[
\alpha(A)=\begin{cases}
\min\left\{ \begin{gathered}i\in\{\beta,\beta+1,\ldots,k\}:\\
C\cup\{\pi(n+\beta+1),\pi(n+\beta+2),\ldots,\pi(n+i)\}\in{\color{red}\mathcal{R}}
\end{gathered}
\right\} & \text{if such \ensuremath{i} exists,}\\
k+1 & \text{otherwise}.
\end{cases}
\]
Then let
\[
\varphi(A)=\begin{cases}
C\cup\{\pi(n+\beta+1),\pi(n+\beta+2),\ldots,\pi(n+\alpha(A))\} & \text{if }\alpha(A)\le k,\\
\textnormal{\ensuremath{\frownie}} & \text{if }\alpha(A)=k+1.
\end{cases}
\]
Note that in the first case,
\begin{gather*}
C\cup\{\pi(n+\beta+1),\pi(n+\beta+2),\ldots,\pi(n+\alpha(A))\}\\
=A\cup\{\pi(n+1),\pi(n+2),\ldots,\pi(n+\alpha(A))\}.
\end{gather*}
Furthermore if $A=\emptyset$, set $f(A)=()$, an empty chain. Otherwise pick a set $B\subsetneqq A$ such that $\alpha(B)=\beta$.
We set $f(A)$ to be a chain of length $\alpha(A)$ in~${\color{blue}\mathcal{B}}$:
\[
f(A)_{i}=\begin{cases}
f(B)_{i} & \text{if }0\le i<\beta,\\
A\cup\{\pi(n+1),\pi(n+2),\ldots,\pi(n+i)\} & \text{if }\beta\le i<\alpha(A).
\end{cases}
\]
Note that these definitions of $\varphi(A)$, $\alpha(A)$ and~$f(A)$ only depend on the values of these functions for
proper subsets of~$A$, so our recursive definitions make sense. It is easy to check that the definitions of~$\varphi$
and~$\alpha$ satisfy \ref{enu:alpha order}~and~\ref{enu:alpha phi}, which together imply \ref{enu:phi containment}.
\ref{enu:f chain}~and~\ref{enu:f subset} are also trivially satisfied when $\alpha(A)=0$. If $\alpha(A)=\beta=k+1$,
we have defined $f(A)=f(B)$ for some $B\subsetneqq A$ such that $\varphi(B)=\textnormal{\ensuremath{\frownie}}$; then \ref{enu:f chain} follows because
it holds for~$B$ by induction, and \ref{enu:f subset} is trivial.
Now we prove \ref{enu:f chain} and \ref{enu:f subset} when $\alpha(A)>0$ and $\beta\le k$. In the case where $\alpha(A)=\beta$
(equivalently if $C\in{\color{red}\mathcal{R}}$, $\varphi(A)=C$ and $\alpha(A)=\alpha(B)$), then $f(A)=f(B)$ is a chain satisfying \ref{enu:f chain}
by induction. Since \ref{enu:f subset} holds for~$B$ by induction, we get that $f(A)_{\alpha(A)-1}=f(B)_{\alpha(B)-1}\subset\varphi(B)\subset\varphi(A)$,
so \ref{enu:f subset} is satisfied for~$A$ as well. If $\alpha(A)>\beta$, then \ref{enu:f subset} follows from the definitions
of $f(A)$ and~$\varphi(A)$. Furthermore, $A\cup\{\pi(n+1),\pi(n+2),\ldots,\allowbreak\pi(n+i)\}\in{\color{blue}\mathcal{B}}$ for $\beta\le i<\alpha(A)$
because $\alpha(A)$~was chosen as the smallest~$i\ge\beta$ such that $A\cup\{\pi(n+1),\pi(n+2),\ldots,\pi(n+i)\}\in{\color{red}\mathcal{R}}$;
this is enough to show \ref{enu:f chain} if~$\beta=0$. Finally, if $\alpha(A)>\beta>0$, then $f(A)$~is obtained by
concatenating the chains $f(B)$ and $\bigl(A\cup\{\pi(n+1),\pi(n+2),\ldots,\pi(n+i)\}\bigr)_{\beta\le i<\alpha(A)}$. By
induction $f(B)$~is a chain satisfying the conditions of \ref{enu:f chain}, and $B$ satisfies \ref{enu:f subset}, so
$f(B)_{\beta-1}\subset\varphi(B)$. Using that \ref{enu:alpha phi} holds for~$B$ by induction, we also have $\varphi(B)=B\cup\{\pi(n+1),\pi(n+2),\ldots,\pi(n+\alpha(B))\}\subsetneqq A\cup\{\pi(n+1),\allowbreak\pi(n+2),\ldots,\pi(n+\beta)\}$
(recall that $B\subsetneqq A$ and $\beta=\alpha(B)$). Thus $f(A)$~is indeed a chain satisfying \ref{enu:f chain}. The
proof of the properties \propertyref{1} to \lastproperty{} of $\varphi$, $\alpha$ and~$f$ is now complete.
\end{proof}
For an arbitrary permutation $\pi:[n+k]\setminus[n]\rightarrow[n+k]\setminus[n]$, let $\varphi^{\pi}$, $\alpha^{\pi}$
and $f^{\pi}$ be the maps given by \cref{lem:functions}. If $\operatorname{Im}\varphi^{\pi}\subseteq{\color{red}\mathcal{R}}$, then $\varphi^{\pi}$~shows
that $\mathcal{Q}_{n}$~is an induced subposet of~${\color{red}\mathcal{R}}$ by \ref{enu:phi containment}. Assume that this is not the case. Then, for
some $A\in\mathcal{Q}_{n}$, $\varphi^{\pi}(A)=\textnormal{\ensuremath{\frownie}}$, $\alpha^{\pi}(A)=k+1$ by~\ref{enu:alpha phi}, and $f^{\pi}(A)$~is a chain
of length $k+1$ in~${\color{blue}\mathcal{B}}$ by~\ref{enu:f chain}. By \ref{enu:f chain}, we have $\pi(n+i)=\left(f^{\pi}(A)_{i}\setminus f^{\pi}(A)_{i-1}\right)\setminus[n]$
when $1\le i\le\alpha^{\pi}(A)-1$, so if $\alpha^{\pi}(A)=k+1$, then one can recover the permutation $\pi$ from the chain~$f^{\pi}(A)$.
Under our assumption that $\mathcal{Q}_{n}$~is not an induced subposet of~${\color{red}\mathcal{R}}$, we get a distinct chain $f^{\pi}$ of length $k+1$
in~${\color{blue}\mathcal{B}}$ for each of the $k!$ permutations~$\pi$ of $[n+k]\setminus[n]$, with the property that
\[
\forall\,0\le i\le k:f_{i}^{\pi}\setminus[n]=\{\pi(n+1),\pi(n+2),\ldots,\pi(n+i)\}.
\]
We claim that the map $\pi\mapsto\left(f_{0}^{\pi},f_{k}^{\pi}\right)$ is injective. Let $\pi_{1}$ and~$\pi_{2}$ be two
different permutations of $[n+k]\setminus[n]$. Let $i=\min_{j\in\{0,\ldots,k\}}\pi_{1}(n+j)\neq\pi_{2}(n+j)$. Then $\pi_{1}(i)\in f_{i}^{\pi_{1}}$,
$\pi_{1}(i)\notin f_{i}^{\pi_{2}}$,$\pi_{2}(i)\in f_{i}^{\pi_{2}}$ and $\pi_{2}(i)\notin f_{i}^{\pi_{1}}$, so $f_{i}^{\pi_{1}}$
and $f_{i}^{\pi_{2}}$ are unrelated. So if $f_{0}^{\pi_{1}}=f_{0}^{\pi_{2}}$ and $f_{k}^{\pi_{1}}=f_{k}^{\pi_{2}}$, then
${\color{blue}\mathcal{B}}$~would contain an induced copy of $\mathcal{Q}_{2}$, a contradiction.
Since the map $\pi\mapsto\left(f_{0}^{\pi},f_{k}^{\pi}\right)$ is injective,
\[
k!\le\left(2^{n+k}\right)^{2}=2^{2(n+k)}.
\]
Approximating the left-hand side:
\[
k!>\left(\frac{k}{e}\right)^{k}=2^{k(\log_{2}k-\log_{2}e)}\text{, so}
\]
\begin{equation}
k(\log_{2}k-\log_{2}e)<2(n+k).\label{eq:2(n+k)}
\end{equation}
Since $k=\left\lfloor c\frac{n}{\log_{2}n}\right\rfloor $,
\begin{equation}
k\log_{2}k>\left(c\frac{n}{\log_{2}n}-1\right)\left(\log_{2}c+\log_{2}n-\log_{2}\log_{2}n-1\right)=cn(1-o(1)).\label{eq:cn}
\end{equation}
Since $c>2$, \eqref{eq:cn} contradicts \eqref{eq:2(n+k)} for sufficiently large~$n$. This completes the proof of \cref{thm:Upperbound}.
\begin{remark*}
It follows the above proof that for all $n\ge2$, we have $R(\mathcal{Q}_{2},\mathcal{Q}_{n})\le\nobreak n+\nobreak6.14\frac{n}{\log_{2}n}$.
Here we give a sketch of the calculations.
For $c=6.14$, we have $k=\nobreak\bigl\lfloor6.14\frac{n}{\log_{2}n}\bigr\rfloor>5.611\frac{n}{\log_{2}n}$ for every integer
$n\ge2$, and therefore we have $k\log_{2}k>5.611\frac{n}{\log_{2}n}\allowbreak\bigl(\log_{2}n\bigl(1-\frac{\log_{2}\log_{2}n}{\log_{2}n}\bigr)+\log_{2}5.611\bigr)\ge2.977n+13.96\frac{n}{\log_{2}n}$.
Using \eqref{eq:2(n+k)}, it can be shown that $0.8797k\log_{2}k\le k(\log_{2}k-\log_{2}e)\overset{{\scriptscriptstyle \eqref{eq:2(n+k)}}}{<}2n+12.28\frac{n}{\log_{2}n}$
for every $n\ge2$, contradicting the lower bound on $k\log_{2}k$ shown earlier.
\end{remark*}
\section{\label{sec:lower}Lower bounds}
\subsection{\label{subsec:ours-induced}An explicit construction showing $R(\protect\mathcal{Q}_{2},\protect\mathcal{Q}_{n})\ge n+3$}
In this subsection, we prove a special case of \cref{thm:Lowerbound} to illustrate the basic ideas of the construction.
The fully general proof of \cref{thm:Lowerbound}, presented in \cref{subsec:ours-general}, is significantly more
involved (primarily due the fact that it is more difficult to deduce properties of a weak map $\mathcal{Q}_{n}\rightarrow\mathcal{Q}_{n+m}$).
\begin{thm}
\label{prop:m_is_2-1}For $n\ge18$, there exist ${\color{blue}\mathcal{B}},{\color{red}\mathcal{R}}\subset\mathcal{Q}_{n+2}$ such that ${\color{blue}\mathcal{B}}\sqcup{\color{red}\mathcal{R}}=\mathcal{Q}_{n+2}$, $\mathcal{Q}_{2}$~is
not an induced subposet of~${\color{blue}\mathcal{B}}$, and $\mathcal{Q}_{n}$~is not an induced subposet of~${\color{red}\mathcal{R}}$.
\end{thm}
Let $k=\left\lfloor \frac{n}{2}\right\rfloor $. Let ${\color{blue}\mathcal{B}}\supset\binom{[n+2]}{k}\cup\binom{[n+2]}{k+3}$, with some sets of
size $k+1$ which we will add later. Assume for a contradiction that $\mathcal{Q}_{n}$~is an induced subposet of~${\color{red}\mathcal{R}}$. Let $\varphi:\mathcal{Q}_{n}\rightarrow{\color{red}\mathcal{R}}$
be an injection such that $\varphi(A)\subseteq\varphi(B)$ if and only if $A\subseteq B$.
For any maximal chain $\emptyset\subsetneqq A_{1}\subsetneqq\ldots\subsetneqq A_{n-1}\subsetneqq[n]$, the sets in its image
satisfy $\varphi(\emptyset)\subsetneqq\varphi(A_{1})\subsetneqq\ldots\subsetneqq\varphi(A_{n-1})\subsetneqq\varphi([n])$,
and none of the sets in the image are of size $k$ or $k+3$. So for every $A\subseteq[n]$,
\begin{equation}
\left|\varphi(A)\right|=\begin{cases}
\left|A\right| & \text{if }\left|A\right|\le k-1,\\
\left|A\right|+1 & \text{if } k\le\left|A\right|\le k+1,\\
\left|A\right|+2 & \text{if } k+2\le\left|A\right|,
\end{cases}\label{eq:levels-1-1}
\end{equation}
thus the image of every singleton is a singleton (and the image of the complement of every singleton is the complement of
a singleton).
For $a\in[n]$, let $\tilde{\varphi}(a)$ denote the unique element of $\varphi(\{a\})$. The map $\tilde{\varphi}:[n]\rightarrow[n+2]$
is an injection. Note that, for a set $A\subseteq[n]$, $\tilde{\varphi}[A]$ denotes the image of $A$ under $\tilde{\varphi}$,
and for a set $B\subseteq[n+m]$, $\tilde{\varphi}^{-1}[B]$ denotes the preimage of $B$ under $\tilde{\varphi}$.
Let $X=\left\{ \tilde{\varphi}(a):a\in[n]\right\} $ and $Y=[n+2]\setminus X=\{y,z\}$. We have $\left|X\right|=n$ and $\left|Y\right|=2$.
We claim that for every $A\subseteq[n]$, $\varphi(A)\cap X=\tilde{\varphi}[A]$. Indeed, for every $b\in X$, there is an
$a\in[n]$ such that $\tilde{\varphi}(a)=b$, and we have $b=\tilde{\varphi}(a)\in\tilde{\varphi}[A]\Longleftrightarrow a\in A\Longleftrightarrow\{a\}\subseteq A\Longleftrightarrow\{\tilde{\varphi}(a)\}=\varphi(\{a\})\subseteq\varphi(A)\Longleftrightarrow b=\tilde{\varphi}(a)\in\varphi(A)$.
From~\eqref{eq:levels-1-1}, $\varphi(A)$ contains neither $y$ nor $z$ if $\left|A\right|\le k-1$, exactly one of them
if $k\le\left|A\right|\le k+1$, and both if $k+2\le\left|A\right|$.
\begin{claim}
\label{claim:y-1-1}For every set $A\in\binom{[n]}{k}$, $\varphi(A)$ contains the same element of $Y$.
\end{claim}
\begin{proof}
Assume for a contradiction that some sets of the form $\varphi(A)$ contain $y$, and others contain $z$. Since the Johnson
graph -- whose vertices are $\binom{[n]}{k}$, and whose edges connect sets with symmetric difference $2$ -- is connected,
there would be two sets $A,B\in\binom{[n]}{k}$ with a symmetric difference of size $2$ such that $y\in\varphi(A)$ and
$z\in\varphi(B)$. Then $\left|A\cup B\right|=k+1$, and $\varphi(A\cup B)$ would contain both $y$ and $z$ as it would
have to be a superset of both $\varphi(A)$ and $\varphi(B)$, contradicting that it contains exactly one of $y$ and $z$.
\end{proof}
Now we specify which sets of $\binom{[n+2]}{k+1}$ are added to~${\color{blue}\mathcal{B}}$ in addition to $\binom{[n+2]}{k}\cup\binom{[n+2]}{k+3}$.
Our goal is to add these sets in such a way that for every map $\varphi:\mathcal{Q}_{n}\to\mathcal{Q}_{n+2}$, assuming that $\operatorname{Im}\varphi\subseteq{\color{red}\mathcal{R}}$,
and $\varphi$ is an order-embedding, the above observations lead to a contradiction. (The map~$\varphi$, and the variables
dependent on it such as $X$, $Y$, $y$ and $z$, are not fixed now; we have to set ${\color{blue}\mathcal{B}}$ in such a way that the existence
of any order-embedding $\varphi:\mathcal{Q}_{n}\to{\color{red}\mathcal{R}}$ leads to a contradiction.) For every distinct~$y,z\in[n+2]$, pick a set $C_{y,z}\in\binom{[n+2]}{k+1}$
such that $y\in C_{y,z}$ but $z\notin C_{y,z}$, and $C_{y,z}$ is at a symmetric difference of size at least $4$ from
every previously chosen set $C_{y',z'}$, and add $C_{y,z}$ to ${\color{blue}\mathcal{B}}$. We can do this by greedily picking sets~$C_{y,z}$
one-by-one: At each step, we have picked at most $(n+2)(n+1)-1$ sets so far, and each previously picked set~$C_{y',z'}$
blocks at most $1+k(n-k)$ choices (because there are at most that many sets containing $y$ but not $z$ with a symmetric
difference of size at most~2 from~$C_{y',z'}$). In total, there are $\binom{n}{k}$ sets $C\in\binom{[n+2]}{k+1}$ that
satisfy $y\in C$ and $z\notin C$. For $n\ge18$ and $k=\left\lfloor \frac{n}{2}\right\rfloor $, we have
\[
\binom{n}{k}>\bigl((n+2)(n+1)-1\bigr)\bigl(1+k(n-k)\bigr),
\]
so we can always choose a set $C_{y,z}$ which satisfies the required conditions.
After adding such sets $C_{y,z}$ for every distinct $y,z\in[n+2]$, the resulting family ${\color{blue}\mathcal{B}}$ will not contain a copy of
$\mathcal{Q}_{2}$. Indeed, a copy of $\mathcal{Q}_{2}$ in~${\color{blue}\mathcal{B}}$ would have to consist of a set of size~$k$, a set of size $k+3$, and
two sets of size $k+1$; but the latter two sets would need to have a symmetric difference of size~$2$.
Now assume for a contradiction that ${\color{red}\mathcal{R}}=2^{[n+2]}\setminus{\color{blue}\mathcal{B}}$ contains an induced copy of~$\mathcal{Q}_{n}$. Consider an arbitrary
injection $\varphi:\mathcal{Q}_{n}\rightarrow{\color{red}\mathcal{R}}$, and define $\tilde{\varphi}$, $X$ and $Y=\{y,z\}$ as before, and apply \cref{claim:y-1-1}.
We can assume without loss of generality that for every $A\in\binom{[n]}{k}$, we have $y\in\varphi(A)$, and thus $\varphi(A)=\tilde{\varphi}[A]\cup\{y\}$.
There is a set $C_{y,z}\in\binom{[n+2]}{k+1}\cap{\color{blue}\mathcal{B}}$ such that $y\in C_{y,z}$, but $z\notin C_{y,z}$. We have $\tilde{\varphi}^{-1}[C_{y,z}]\in{[n] \choose k}$,
and
\[
\varphi\left(\tilde{\varphi}^{-1}[C_{y,z}]\right)=\left\{ \tilde{\varphi}(a):a\in[n],\tilde{\varphi}(a)\in C_{y,z}\right\} \cup\{y\}=C_{y,z}\in{\color{blue}\mathcal{B}},
\]
contradicting that the image of~$\varphi$ is in~${\color{red}\mathcal{R}}$.
\subsection{\label{subsec:ours-general}An explicit construction showing \textmd{\normalsize{}$R_{w}(\protect\mathcal{Q}_{m},\protect\mathcal{Q}_{n})\ge m+n+1$}}
\cref{thm:Lowerbound} will be an immediate consequence of the following \MakeLowercase{\crtcrefnamebylabel{prop:noninduced-lower-general}}.
\begin{thm}
\label{prop:noninduced-lower-general}Let $n,m\in\mathbf{N}$ such that $m\ge2$ and $n\ge\sqrt{32m+260}+18$. There exist
${\color{blue}\mathcal{B}},{\color{red}\mathcal{R}}\subset\mathcal{Q}_{n+m}$ such that ${\color{blue}\mathcal{B}}\sqcup{\color{red}\mathcal{R}}=\mathcal{Q}_{n+m}$, $\mathcal{Q}_{m}$~is not a weak subposet of~${\color{blue}\mathcal{B}}$, and $\mathcal{Q}_{n}$~is
not a weak subposet of~${\color{red}\mathcal{R}}$.
\end{thm}
To see that \cref{thm:Lowerbound} follows from \cref{prop:noninduced-lower-general}, notice that we can assume
without loss of generality that $n\ge m$, so for $n\ge68$, we have $n\ge\sqrt{32n+260}+18$.
For values of $m$ less than~$67$, the threshold for~$n$ in the hypothesis of~\cref{prop:noninduced-lower-general}
is smaller than $68$: for instance, it holds for $m=2$ and $n\ge36$. Also note that in the proof of \cref{lem:modp code},
we use Bertrand's postulate to obtain a prime $N\le p<2(N-1)$. By finding the smallest prime greater than or equal to $N$,
one may be able to relax the requirement on $k$ in \cref{lem:modp code}, and thereby extend \cref{prop:noninduced-lower-general}
to somewhat smaller values of $n$, for a given $m$.
In the proof of \cref{prop:noninduced-lower-general}, we will use \cref{lem:modp code}, which will follow from
\cref{lem:modp subset}.
\begin{lem}[Olson \cite{olson}]
\label{lem:modp subset}Let $p$ be a prime, and let $A\subseteq[p]$ such that $\left|A\right|\ge\sqrt{4p-3}$. Then for
every $a\in\mathbf{Z}$, there is a subset $B\subseteq A$ such that
\[
\sum B\equiv a\pmod p.
\]
\end{lem}
\begin{lem}
\label{lem:modp code}Let $N,k\in\mathbf{N}$ such that $N>3$ and $k\ge\sqrt{8N-15}$. Then there is a constant-weight
code $\mathcal{C}\subset\binom{[N]}{k+1}$ such that the symmetric difference between any two sets is of size at least $4$, and the
following holds:
\begin{numberedstatement}
Let $n,m\in\mathbf{N}$ such that $n+m=N$ and $k\le n-\sqrt{8N-15}$. For every $Y\in\binom{[N]}{m}$ and $y\in Y$, there
is a set $C\in\binom{[N]\setminus Y}{k}$ such that $C\cup\{y\}\in\mathcal{C}$.\label{eq:modp code statement}
\end{numberedstatement}
\end{lem}
\begin{proof}
By Bertrand's postulate
\begin{comment}
There are much better estimates for sufficiently large $N$, see \href{https://en.wikipedia.org/wiki/Bertrand\%27s_postulate\#Better_results}{https://en.wikipedia.org/wiki/Bertrand\%27s\_postulate\#Better\_results}
\end{comment}
, there is a prime~$p$ such that $N\le p<2(N-1)$. Let $d\in[p]$ be a fixed constant, and let
\[
\mathcal{C}=\left\{ S\in\binom{[N]}{k+1}:\sum S\equiv d\pmod p\right\} .
\]
Let $Y\in\binom{[N]}{m}$ and $y\in Y$. We have to find a set~$C$ that satisfies the condition in the statement. Let $l=\left\lceil \sqrt{8N-15}\right\rceil $.
It follows from the conditions of the lemma that $n\ge2l$. Let $[N]\setminus Y=\{x_{1},\ldots,x_{n}\}$ such that $x_{1}<x_{2}<\ldots<x_{n}$.
For $i=1,\ldots,l$, let $a_{i}=x_{i}$ and $b_{i}=x_{n-i+1}$. The numbers $b_{i}-a_{i}$ are in $[p]$, and they are different
because $b_{1}-a_{1}>b_{2}-a_{2}>\ldots>b_{l}-a_{l}$. Let $a=\sum_{i=1}^{l}a_{i}$. Let $E$ be a subset of $\{x_{l+1},\ldots,x_{n-l}\}$
with $k-l$ elements, and let $e=\sum E$. (It follows from the conditions that $0\le k-l\le n-2l$.)
Since $l\ge\sqrt{8N-15}\ge\sqrt{4p-3}$, by \cref{lem:modp subset}, there is a subset of $\{b_{1}-a_{1},\ldots,b_{l}-a_{l}\}$
such that its sum is congruent with $d-y-e-a$. That is, there is a set $I\subseteq[l]$ such that
\[
\sum_{i\in I}(b_{i}-a_{i})\equiv d-y-e-a\pmod p.
\]
Let
\[
C=E\cup\left\{ \Padded{\begin{cases}
a_{i} & \text{if } i\notin I\\
b_{i} & \text{if } i\in I
\end{cases}:i\in[l]}\right\} .
\]
We have
\[
\sum\left(C\cup\{y\}\right)=\sum E+\sum_{i=1}^{l}a_{i}+\sum_{i\in I}(b_{i}-a_{i})+y\equiv d\pmod p,
\]
so $C\in\mathcal{C}$.
\end{proof}
Let $k$ be an integer between $\sqrt{8(n+m)-15}$ and $n-1-\sqrt{8(n+m)-15}$ inclusive. (The conditions of the proposition
imply that $\sqrt{8(n+m)-15}+1\le n-1-\sqrt{8(n+m)-15}$, therefore such an integer $k$ exists.) Let ${\color{blue}\mathcal{B}}\supset\binom{[n+m]}{k}\cup\binom{[n+m]}{k+3}\cup\binom{[n+m]}{k+4}\cup\ldots\cup\binom{[n+m]}{k+m+1}\cup\mathcal{C}$,
where $\mathcal{C}$~is given by \cref{lem:modp code}, using $n+m$ in the place of~$N$. First we show that the family ${\color{blue}\mathcal{B}}$
does not contain a~$\mathcal{Q}_{m}$. Indeed, any two sets in~${\color{blue}\mathcal{B}}$ of size $k+1$ have a symmetric difference of size at least~$4$.
A copy of~$\mathcal{Q}_{m}$ in~${\color{blue}\mathcal{B}}$ would consist of a set of size~$k$, some sets of size $k+3,\ldots,k+m+1$ corresponding
to the sets of size~$2$~to~$m$ of the~$\mathcal{Q}_{m}$, and $m$~sets of size $k+1$ corresponding to the singletons of~$\mathcal{Q}_{m}$.
The latter $m$~sets would need to have a symmetric difference of size~$2$.
Assume for a contradiction that $\mathcal{Q}_{n}$~is a subposet of~${\color{red}\mathcal{R}}$. Let $\varphi:\mathcal{Q}_{n}\rightarrow{\color{red}\mathcal{R}}$ be an injection that
preserves relations.
For any maximal chain $\emptyset\subsetneqq A_{1}\subsetneqq\ldots\subsetneqq A_{n-1}\subsetneqq[n]$, we have $\varphi(\emptyset)\subsetneqq\varphi(A_{1})\subsetneqq\ldots\subsetneqq\varphi(A_{n-1})\subsetneqq\varphi([n])$,
and none of the sets in the image are of size $k$ or $k+3,k+4,\ldots,k+m+1$. So for every $A\subseteq[n]$,
\begin{equation}
\left|\varphi(A)\right|=\begin{cases}
\left|A\right| & \text{if }\left|A\right|\le k-1,\\
\left|A\right|+1 & \text{if } k\le\left|A\right|\le k+1,\\
\left|A\right|+m & \text{if } k+2\le\left|A\right|,
\end{cases}\label{eq:levels}
\end{equation}
thus the image of every singleton is a singleton, and the image of the complement of every singleton is the complement of
a singleton).
For $a\in[n]$, let $\varphi_{1}(a)$ denote the unique element of $\varphi(\{a\})$, and let $\varphi_{2}(a)$ denote the
unique element of $[n+m]\setminus\varphi\left([n]\setminus\{a\}\right)$. Note that, for a set $A\subseteq[n]$, $\varphi_{i}[A]$
denotes the image of $A$ under $\varphi_{i}$, and for a set $B\subseteq[n+m]$, $\varphi_{i}^{-1}[B]$ denotes the preimage
of $B$ under $\varphi_{i}$.
The maps $\varphi_{1}$ and $\varphi_{2}$ are injections. Furthermore, for any distinct $a,b\in[n]$, it holds that $\{a\}\subseteq[n]\setminus\{b\}$,
so $\{\varphi_{1}(a)\}=\varphi(\{a\})\subseteq\varphi\left([n]\setminus\{b\}\right)=[n+m]\setminus\{\varphi_{2}(b)\}$, so
$\varphi_{1}(a)\ne\varphi_{2}(b)$. Now take the sets $\left\{ \varphi_{1}(a),\varphi_{2}(a)\right\} \subseteq[n+m]$ for
each $a\in[n]$; these sets have 1~or~2 elements, depending on whether $\varphi_{1}(a)=\varphi_{2}(a)$. Based on the observations
in this paragraph, if $a\ne b$, we have
\[
\left\{ \varphi_{1}(a),\varphi_{2}(a)\right\} \cap\left\{ \varphi_{1}(b),\varphi_{2}(b)\right\} =\emptyset.
\]
Since $\bigcup_{a\in[n]}\{\varphi_{1}(a),\varphi_{2}(a)\}\subseteq[n+m]$, we have $\sum_{a\in[n]}\left|\{\varphi_{1}(a),\varphi_{2}(a)\}\right|\le n+m$,
so the number of these sets which have 2~elements is at most~$m$; in other words,
\[
\left|\left\{ a\in[n]:\varphi_{1}(a)\ne\varphi_{2}(a)\right\} \right|\le m.
\]
Let
\begin{align*}
D & =\left\{ a\in[n]:\varphi_{1}(a)=\varphi_{2}(a)\right\} ,\\
E & =[n]\setminus D,\\
X_{12} & =\varphi_{1}[D]=\varphi_{2}[D],\\
X_{1} & =\varphi_{1}[E],\\
X_{2} & =\varphi_{2}[E],\\
X_{\emptyset} & =[n+m]\setminus(X_{12}\cup X_{1}\cup X_{2}).
\end{align*}
Then
\begin{align}
[n+m] & =X_{12}\sqcup X_{1}\sqcup X_{2}\sqcup X_{\emptyset},\label{eq:disjoint}\\
\operatorname{Im}\varphi_{1} & =X_{12}\cup X_{1},\\
\operatorname{Im}\varphi_{2} & =X_{12}\cup X_{2},\nonumber \\
\left|X_{12}\right| & =\left|D\right|\ge n-m,\labelifpresent{grey}\label{eq:D ge n-m}\\
\left|X_{1}\right| & =\left|X_{2}\right|=\left|E\right|\le m,\nonumber \\
\left|X_{12}\right|+\left|X_{1}\right| & =\left|D\right|+\left|E\right|=n,\nonumber \\
\left|E\right|+\left|X_{\emptyset}\right| & =(n+m)-\left(\left|X_{12}\right|+\left|X_{1}\right|\right)=m.\label{eq:e-x0}
\end{align}
We have that, for every $A\subseteq[n]$,
\begin{equation}
\forall a\in A:\varphi_{1}(a)\in\varphi(A)\label{eq:a in A}
\end{equation}
because $a\in A{\color{red}\mathcal{R}}ightarrow\{a\}\subseteq A{\color{red}\mathcal{R}}ightarrow\{\varphi_{1}(a)\}=\varphi(\{a\})\subseteq\varphi(A){\color{red}\mathcal{R}}ightarrow\varphi_{1}(a)\in\varphi(A)$.
(Equivalently, $\varphi_{1}[A]\subseteq\varphi(A)$.) Symmetrically,
\begin{equation}
\forall a\in[n]\setminus A:\varphi_{2}(a)\notin\varphi(A)\label{eq:a notin A}
\end{equation}
because $a\in[n]\setminus A{\color{red}\mathcal{R}}ightarrow A\subseteq[n]\setminus\{a\}{\color{red}\mathcal{R}}ightarrow\varphi(A)\subseteq\varphi\left([n]\setminus\{a\}\right)=[n+m]\setminus\{\varphi_{2}(a)\}{\color{red}\mathcal{R}}ightarrow\varphi_{2}(a)\notin\varphi(A)$.
For an $A\subseteq[n]$, let
\[
F(A)=\left\{ \Padded{\begin{cases}
\varphi_{2}(a) & \text{if } a\in A\\
\varphi_{1}(a) & \text{if } a\notin A
\end{cases}:a\in E}\right\} \cup X_{\emptyset}.
\]
By \eqref{eq:disjoint}, \eqref{eq:a in A} and \eqref{eq:a notin A}, we have
\begin{equation}
\begin{aligned}\varphi(A)\cap\left([n+m]\setminus F(A)\right) & =\varphi(A)\cap\left(X_{12}\cup\left\{ \Padded{\begin{cases}
\varphi_{1}(a) & \text{if } a\in A\\
\varphi_{2}(a) & \text{if } a\notin A
\end{cases}:a\in E}\right\} \right)\\
& =\varphi(A)\cap\left\{ \Padded{\begin{cases}
\varphi_{1}(a) & \text{if } a\in A\\
\varphi_{2}(a) & \text{if } a\notin A
\end{cases}:a\in[n]}\right\} \overset{\eqref{eq:a in A},\eqref{eq:a notin A}}{=}\varphi_{1}[A],
\end{aligned}
\label{eq:a-phi}
\end{equation}
and therefore
\begin{equation}
\left|\varphi(A)\cap\left([n+m]\setminus F(A)\right)\right|=\left|\varphi_{1}[A]\right|=\left|A\right|.\label{eq:a-phi-size}
\end{equation}
Note that $\left|F(A)\right|\overset{\eqref{eq:e-x0}}{=}m$. The elements of $F(A)$ are the only elements of $[n+m]$ such
that \eqref{eq:a-phi} does not determine whether they are elements of $\varphi(A)$. In particular,
\begin{equation}
F(A)\subseteq[n+m]\setminus\varphi_{1}[A].\label{eq:F}
\end{equation}
From \eqref{eq:levels} and \eqref{eq:a-phi-size}, $\varphi(A)$ contains no element of $F(A)$ if $\left|A\right|\le k-1$,
exactly one if $k\le\left|A\right|\le k+1$, and all elements of $F(A)$ if $k+2\le\left|A\right|$. For $A\in\binom{[n]}{k}\cup\binom{[n]}{k+1}$,
let $f(A)$ be the single element of $\varphi(A)\cap F(A)$.
\begin{claim}
\label{claim:y}One of the following holds:
\gdef\labelwidthi{\widthof{\textbf{\textup{A0. }}}}
\gdef\labeli{\textbf{\textup{A\arabic{enumi}. }}}
\gdef\propertyref#1{A#1}
\gdef\refi{\propertyref{\arabic{enumi}}}
\begin{enumerate}[label=\labeli, ref=\refi, labelsep=0em, leftmargin=0em, labelwidth=\labelwidthi, itemindent=\labelwidth, align=left]
\item \label{enu:X0}There is a $y\in X_{\emptyset}\cup X_{2}$ such that, for every $A\in\binom{[n]}{k}$, we have $f(A)=y$.
(In fact in this case $y\in X_{\emptyset}$, since for a $y\in X_{2}$ and $A\in\binom{[n]\setminus\{\varphi_{2}^{-1}(y)\}}{k}$
we would have $y\notin F(A)$. We do not use this.)
\item \label{enu:X1}There is a $y\in X_{1}$ such that, for every $A\in\binom{[n]\setminus\{\varphi_{1}^{-1}(y)\}}{k}$, we have
$f(A)=y$. (Note that when $\varphi_{1}^{-1}(y)\notin A$, $y\in F(A)$ holds by the definition of $F(A)$.)
\end{enumerate}
\end{claim}
First we show that \cref{prop:noninduced-lower-general} follows from this claim. We use the constant-weight code $\mathcal{C}\subset{\color{blue}\mathcal{B}}$
given by \cref{lem:modp code}. If \ref{enu:X0} holds in \cref{claim:y}, then we use the statement~\eqref{eq:modp code statement}
in \cref{lem:modp code} with the same $n$~and~$m$ as in \eqref{prop:noninduced-lower-general}, $X_{2}\cup X_{\emptyset}$
in the place of~$Y$, and $y$ as given by \cref{claim:y}. There is a set $C\in\binom{X_{12}\cup X_{1}}{k}$ such
that $C\cup\{y\}\in\mathcal{C}\subset{\color{blue}\mathcal{B}}$. Then $\varphi_{1}^{-1}[C]\in\mathcal{Q}_{n}$, and $\varphi\left(\varphi_{1}^{-1}[C]\right)=C\cup\{y\}\in{\color{blue}\mathcal{B}}$,
contradicting that the image of~$\varphi$ is in~${\color{red}\mathcal{R}}$. If \ref{enu:X1} holds in \cref{claim:y}, then we use the
statement~\eqref{eq:modp code statement} with $n-1$ in the place of $n$, $m+1$ in the place of $m$, $X_{2}\cup X_{\emptyset}\cup\{y\}$
in the place of $Y$, and $y$ as given by \cref{claim:y}. There is a set $C\in\binom{(X_{12}\cup X_{1})\setminus\{y\}}{k}$
such that $C\cup\{y\}\in\mathcal{C}\subset{\color{blue}\mathcal{B}}$. Then $\varphi_{1}^{-1}[C]\in\binom{[n]\setminus\{\varphi_{1}^{-1}(y)\}}{k}\subset\mathcal{Q}_{n}$,
and $\varphi\left(\varphi_{1}^{-1}[C]\right)=C\cup\{y\}\in{\color{blue}\mathcal{B}}$, contradicting that the image of~$\varphi$ is in~${\color{red}\mathcal{R}}$.
To prove \cref{claim:y}, we need the following.
\begin{claim}
\label{obs:neighbors}If $A,B\in\binom{[n]}{k}$ with a symmetric difference of size~$2$, and $f(A)\ne f(B)$, then at
least one of the following holds:
\begin{itemize}
\item $f(A)=\varphi_{1}(b)$ where $\{b\}=B\setminus A$. (This implies $b\in E$ and $f(A)\in X_{1}$.)
\item $f(B)=\varphi_{1}(a)$ where $\{a\}=A\setminus B$. (This implies $a\in E$ and $f(B)\in X_{1}$.)
\end{itemize}
\begin{proof}
Indeed, $\left|A\cup B\right|=k+1$, so
\begin{equation}
\varphi(A\cup B)=\varphi_{1}[A\cup B]\cup\{f(A\cup B)\}=\varphi_{1}[A\cap B]\cup\{\varphi_{1}(a),\varphi_{1}(b),f(A\cup B)\}.\label{eq:a-b-phi}
\end{equation}
Furthermore,
\begin{align}
\varphi(A\cup B) & \supset\varphi(A)=\varphi_{1}[A]\cup\{f(A)\}=\varphi_{1}[A\cap B]\cup\{\varphi_{1}(a),f(A)\}\text{ and }final\label{eq:phi-a}\\
\varphi(A\cup B) & \supset\varphi(B)=\varphi_{1}[B]\cup\{f(B)\}=\varphi_{1}[A\cap B]\cup\{\varphi_{1}(b),f(B)\}.\label{eq:phi-b}
\end{align}
By \eqref{eq:a-b-phi}, \eqref{eq:phi-a} and~\eqref{eq:phi-b}, we have {\thickmuskip=5mu plus 3mu minus 3mu \medmuskip=2mu $\left|\{\varphi_{1}(a),f(A),\varphi_{1}(b),f(B)\}\right|\le\left|\{\varphi_{1}(a),\varphi_{1}(b),f(A\cup B)\}\right|=\nobreak3$},
therefore the elements on the left-hand side of the inequality are not distinct. We know that $\varphi_{1}$ is an injection,
and by~\eqref{eq:F}, $f(S)\notin\varphi_{1}[S]$ for any $S$ of size $k$ or $k+1$. We have assumed $f(A)\ne f(B)$.
It follows that $f(A)=\varphi_{1}(b)$ or $f(B)=\varphi_{1}(a)$. This completes the proof of \cref{obs:neighbors}.
\end{proof}
\end{claim}
\present{grey}We first prove \cref{claim:y} under the condition $m\le n-\sqrt{8(n+m)-15}-1$, as the proof is simpler
than the proof for arbitrary $m$\emph{.} (For large $n$, this condition holds whenever the ratio of $m$ and $n$ is not
very close to 1.)
\begin{boldproof}[Proof of \cref{claim:y} when \thinmuskip=2mu \medmuskip=3mu \thickmuskip=4mu $m\le n-\sqrt{8(n+m)-15}-1$]
At the beginning of the proof of \cref{prop:noninduced-lower-general}, we chose an arbitrary $k$ between {\medmuskip=3mu plus 3mu minus 3mu $\sqrt{8(n+m)-15}$
and $n-1-\sqrt{8(n+m)-15}$}. Now we will assume that $k\le n-m$; this is satisfied by choosing e.g. $k=\sqrt{8(n+m)-15}$.
Then, by~\eqref{eq:D ge n-m}, there exists an $B\in\binom{[n]}{k}$ such that $B\subseteq D$. We show that \cref{claim:y}
holds with $y=f(B)$. In fact, since $B\cap E=\emptyset$, we have $f(B)\in F(B)=X_{\emptyset}$, and we show that $f(A)=f(B)$
for every $A\in\binom{[n]}{k}$.
Take an $A\in\binom{[n]}{k}$. We can get from $B$ to $A$ by replacing one element at a time, in such a way that we never
add an element that is not an element of $A$, and we never remove an element of $A$ (whether it is also an element of $B$,
or we have added it). In particular, we never remove an element of $E$. That is, there is a sequence $B=B_{0},B_{1},\ldots,B_{l}=A$
such that $\left|B_{i}\triangle B_{i+1}\right|=2$ and $b_{i}^{\leftarrow}\in D$ where $\{b_{i}^{\leftarrow}\}=B_{i}\setminus B_{i+1}$.
Let $b_{i+1}^{\rightarrow}\in[n]$ such that $\{b_{i+1}^{\rightarrow}\}=B_{i+1}\setminus B_{i}$.
We show by induction that $f(B_{i})=f(B)$ for every $i=0,\ldots,l$. Assume that $f(B)=f(B_{i})\ne f(B_{i+1})$. By \cref{obs:neighbors},
either $f(B_{i})=\varphi_{1}(b_{i+1}^{\rightarrow})$ or $f(B_{i+1})=\varphi_{1}(b_{i}^{\leftarrow})$. The former implies
$f(B_{i})=f(B)\in X_{1}$, contradicting that it is in $X_{\emptyset}$. The latter implies $b_{i}^{\leftarrow}\in E$, contradicting
that it is in $D$.
\end{boldproof}
\begin{boldproof}[Proof of \cref{claim:y}]
The general form of \cref{claim:y} will be a consequence of the following lemma.
\begin{lem}
\label{lem:y inductive}Let $n,l\in\mathbf{N}$ such that $n\ge5$ and $1\le l\le n-3$, and let $X$ and $Y$ be disjoint
sets such that $\left|X\right|=n$. Let $g:\binom{X}{l}\rightarrow X\cup Y$ be a function such that for every $A\in\binom{X}{l}$,
$g(A)\notin A$; and for every $A,B\in\binom{X}{l}$ with a symmetric difference of size~$2$, where $g(A)\ne g(B)$, at
least one of $\{g(A)\}=B\setminus A$ and $\{g(B)\}=A\setminus B$ holds. Then there is a $y\in X\cup Y$ such that $g(A)=y$
for every $A\in\binom{X\setminus\{y\}}{l}$.
\end{lem}
\begin{subproof}
We prove \cref{lem:y inductive} by induction on~$l$.
If $l=1$, the sets are singletons, and every symmetric difference is of size~$2$. We define a graph on~$X$: we connect
two elements $a$ and~$b$ if $g(\{a\})\ne g(\{b\})$. This graph is the complement of a graph whose components are complete
graphs (with the components defined by the values of $a\mapsto g(\{a\})$). For every $a,b\in X$ such that $ab$ is an edge,
we have $g(\{a\})=b$ or $g(\{b\})=a$. Direct the graph such that we have the directed edge $(a,b)$ when $g(\{a\})=b$
(we may direct some edges in both directions).
The out-degree of every vertex is at most~1. Thus the number of edges is at most~$n$. By our assumptions $n\ge4$; the
only graphs with these properties on at least~5 vertices (ignoring the directions of the edges) are the empty graph and
a star on $n$ vertices. If it is an empty graph, then $g(\{a\})$ is the same for every $a\in X$ (and it is necessarily
in~$Y$); the statement of \cref{lem:y inductive} holds with $y=g(\{a\})$. If the graph is a star on $n$ vertices,
let $a$ be the center. Since at most one edge is directed outward from~$a$, all but at most one edge is directed towards~$a$.
That is, $g(\{b\})=a$ for all but at most one $b\in X\setminus\{a\}$. Since the leaves of the star are not connected, $g$~has
the same values on them as singletons, so in fact $g(\{b\})=a$ for every $b\in X\setminus\{a\}$, and the lemma holds with
$y=a$.
Now let $l\ge2$. If $g(A)$ is the same for every $A\in\binom{X}{l}$, the lemma holds with that value as~$y$. Assume
that $g(A)$ is not the same for every $A\in\binom{X}{l}$. Since the Johnson graph is connected, there are sets with a symmetric
difference of size~$2$ with different~$g$; consequently there is an~$a\in X$ such that there are sets containing~$a$
with different~$g$.
We use the induction hypothesis with $\tilde{l}=l-1$, $\tilde{n}=n-1$, $\tilde{X}=X\setminus\{a\}$, $\tilde{g}\bigl(\tilde{A}\bigr)=g\bigl(\{a\}\cup\tilde{A}\bigr)$
for $\tilde{A}\in\binom{\tilde{X}}{l-1}$, and $Y$~unchanged. Note that since $g\bigl(\{a\}\cup\tilde{A}\bigr)\notin\{a\}\cup\tilde{A}$,
in fact $\tilde{g}\bigl(\tilde{A}\bigr)\in\tilde{X}\cup Y$ and $\tilde{g}\bigl(\tilde{A}\bigr)\notin\tilde{A}$, so the
conditions of the induction hypothesis hold. So there is a $b\in\tilde{X}\cup Y$ such that $g\bigl(\tilde{A}\bigr)=b$ for
every $\tilde{A}\in\binom{\tilde{X}\setminus\{b\}}{l}$; equivalently, $b\in(X\cup Y)\setminus\{a\}$ such that $g(A)=b$
for every $A\in\binom{X}{l}$ that contains $a$ but not~$b$. If $b$ were in~$Y$, then $g(A)$ would be the same for
every $A\in\binom{X}{l}$ that contains $a$, contradicting our assumption. So $b\in X$.
\begin{figure}
\caption{\label{fig:g(au)=00003Db}
\caption{\label{fig:bullets}
\label{fig:g(au)=00003Db}
\label{fig:bullets}
\end{figure}
(See \cref{fig:g(au)=00003Db}.)
Take a $C\in\binom{X\setminus\{a,b\}}{l}$. We show that $g(C)\in\{a,b\}$. Take an arbitrary $c\in C$. $(C\setminus\{c\})\cup\{a\}$
contains $a$ but not $b$, so $g\bigl((C\setminus\{c\})\cup\{a\}\bigr)=b$. Since $\left|C\triangle\bigl((C\setminus\{c\})\cup\{a\}\bigr)\right|=2$,
either $g(C)=g\bigl((C\setminus\{c\})\cup\{a\}\bigr)=b$, or $g(C)=a$, or $g\bigl((C\setminus\{c\})\cup\{a\}\bigr)=c$ ---
but the last option is false.
Now we show that $g(C)$ is the same for every $C\in\binom{X\setminus\{a,b\}}{l}$. (See \cref{fig:bullets}.) If this
is not the case, there are $C,D\in\binom{X\setminus\{a,b\}}{l}$ such that $\left|C\triangle D\right|=2$, and $g(C)=a$
but $g(D)=b$. But this implies that $C\setminus D=\{b\}$ or $D\setminus C=\{a\}$, which is impossible because $a,b\notin C,D$.
We already know that $g(A)=b$ for every $A\in\binom{X}{l}$ that contains $a$ but not~$b$. If $g(C)=b$ for every $C\in\binom{X\setminus\{a,b\}}{l}$,
then the lemma holds with $y=b$ (see \cref{fig:y=00003Db}). So assume instead that $g(C)=a$ for every $C\in\binom{X\setminus\{a,b\}}{l}$
(\cref{fig:assumption}).
\begin{figure}
\caption{\label{fig:y=00003Db}
\caption{\label{fig:assumption}
\label{fig:y=00003Db}
\label{fig:assumption}
\end{figure}
Let $B\in\binom{X}{l}$ such that it contains $b$ but not~$a$. We show that $g(B)=a$. Take two different, arbitrary elements
$c,d\in X\setminus(B\cup\{a\})$.
\begin{figure}
\caption{\label{fig:g(B)=00003Da}
\caption{\label{fig:y=00003Da}
\label{fig:g(B)=00003Da}
\label{fig:y=00003Da}
\end{figure}
(There are at least two such elements because $l\le n-3$. See \cref{fig:g(B)=00003Da}.) Since $\left|B\triangle\bigl((B\setminus\{b\})\cup\{c\}\bigr)\right|=2$,
either $g(B)=g\bigl((B\setminus\{b\})\cup\{c\}\bigr)=a$, or $g(B)=c$, or $g\bigl((B\setminus\{b\})\cup\{c\}\bigr)=b$ ---
but the last option is false. So if $g(B)\ne a$, then $g(B)=c$. By the same reasoning applied with $d$ in the place of
$c$, if $g(B)\ne a$, then $g(B)=d$, a contradiction. So $g(B)=a$ for every $B\in\binom{X}{l}$ that contains $b$ but
not~$a$. Since we already know that $g(C)=a$ for every $C\in\binom{X\setminus\{a,b\}}{l}$, this implies that the lemma
holds with $y=a$ (see \cref{fig:y=00003Da}). This completes the proof of \cref{lem:y inductive}.
\end{subproof}
Using \cref{lem:y inductive}, we show \cref{claim:y}. Let $l=k$, $X=X_{12}\cup X_{1}$, $Y=X_{\emptyset}\cup X_{2}$,
and for a $B\in\binom{X_{12}\cup X_{1}}{k}$, let $g(B)=f\left(\varphi_{1}^{-1}[B]\right)$. (Since $\varphi_{1}$~is an
injection and its image is $X_{12}\cup X_{1}$, we have $\varphi_{1}^{-1}[B]\in\binom{[n]}{k}=\operatorname{Dom} f$.) The conditions of
\cref{lem:y inductive} hold by \cref{obs:neighbors}. By \cref{lem:y inductive}, there is a $y\in[n+m]$
such that $f\left(\varphi_{1}^{-1}[B]\right)=g(B)=y$ for every $B\in\binom{(X_{12}\cup X_{1})\setminus\{y\}}{k}$ (where
$(X_{12}\cup X_{1})\setminus\{y\}$ may coincide with $X_{12}\cup X_{1}$).
If $y\in X_{\emptyset}\cup X_{2}$, then for every $A\in\binom{[n]}{k}$, we have $\varphi_{1}[A]\in\binom{(X_{12}\cup X_{1})\setminus\{y\}}{k}=\binom{X_{12}\cup X_{1}}{k}$,
and $f(A)=f\left(\varphi_{1}^{-1}\left[\varphi_{1}[A]\right]\right)=y$, so \ref{enu:X0} holds in \cref{claim:y}.
If $y\in X_{12}\cup X_{1}$, then for every $A\in\binom{[n]\setminus\{\varphi_{1}^{-1}(y)\}}{k}$, we have $\varphi_{1}[A]\in\binom{(X_{12}\cup X_{1})\setminus\{y\}}{k}$,
and $f(A)=f\left(\varphi_{1}^{-1}\left[\varphi_{1}[A]\right]\right)=y$. Since $f(A)\in F(A)\subseteq X_{1}\cup X_{2}\cup X_{\emptyset}$,
we also have $y\in X_{1}$, so \ref{enu:X1} holds in \cref{claim:y}.
\end{boldproof}
\subsection{\label{subsec:cox stolee}A probabilistic construction showing $R_{w}(\protect\mathcal{Q}_{m},\protect\mathcal{Q}_{n})\ge m+n+1$ when $m\ge3$}
\begin{thm}
\label{prop:CS-generalized}If $n,m\in\mathbf{N}$, $n$~is sufficiently large, and $m\ge3$, then there exist ${\color{blue}\mathcal{B}},{\color{red}\mathcal{R}}\subset\mathcal{Q}_{n+m}$
such that ${\color{blue}\mathcal{B}}\sqcup{\color{red}\mathcal{R}}=\mathcal{Q}_{n+m}$, $\mathcal{Q}_{m}$~is not a weak subposet of~${\color{blue}\mathcal{B}}$, and $\mathcal{Q}_{n}$~is not a weak subposet of~${\color{red}\mathcal{R}}$.
\end{thm}
In most of this subsection, we prove \cref{prop:CS-generalized}. The core of the random construction will be in \cref{lem:random}.
In the proof of \cref{lem:random} we will use the asymmetric version of the Lov\'asz Local Lemma.
\begin{lem}[Asymmetric Lov\'asz Local Lemma]
\label{lem:asymlocal}Let $\mathcal{A}$ be a collection of events. For $A\in\mathcal{A}$, let $\Gamma(A)$~be the set of those events in~$\mathcal{A}$,
other than $A$ itself, that are not independent of~$A$. If there is a function $x:\mathcal{A}\to[0,1)$ such that for every $A\in\mathcal{A}$,
we have
\begin{equation}
P(A)\le x(A)\prod_{B\in\Gamma(A)}(1-x(B)),\label{eq:asymlocal-ineq}
\end{equation}
then there is a non-zero probability that none of the events occur.
\end{lem}
\begin{claim}
\label{lem:random}If $n,m\in\mathbf{N}$, $n$~is sufficiently large, and $3\le m\le n$, then there is a family of sets
$\mathcal{F}\subset\binom{[n+m]}{m}$ such that
\gdef\labelwidthi{\widthof{(ii)}}
\gdef\labeli{\textup{(\roman{enumi})}}
\gdef\refi{(\textup{\roman{enumi})}}
\begin{enumerate}[label=\labeli, ref=\refi, labelwidth=\labelwidthi, itemindent=\labelwidth, align=left]
\item \label{enu:supersets}for each $S\in\binom{[n+m]}{m-1}$, $\mathcal{F}$~contains at least~$2$ supersets of~$S$, and
\item \label{enu:subsets}for each $T\in\binom{[n+m]}{m+1}$, $\mathcal{F}$~contains at most $m-1$ subsets of~$T$.
\end{enumerate}
\end{claim}
\begin{proof}
Let $p=\left(4(m+1)\left(n^{2}-1\right)e\right)^{-1/m}$. Let $\mathcal{F}$ be a collection of sets given by taking each set $F\in\binom{[n+m]}{m}$
independently at random with probability~$p$.
\begin{comment}
Notice that the family $\mathcal{F}$ satisfies the conditions of the lemma if and only if the family $\left\{ F\in\binom{[n+m]}{n}:[n+m]\setminus F\notin\mathcal{F}\right\} $
satisfies the conditions with $n$ and $m$ swapped.
\end{comment}
{} For any $S\in\binom{[n+m]}{m-1}$, let $A_{S}$ be the event in which $\mathcal{F}$ contains at most~$1$ superset of~$S$, and
for any $T\in\binom{[n+m]}{m+1}$, let $B_{T}$ be the event in which $\mathcal{F}$ contains at least~$m$ subsets of~$T$. We
have
\begin{align*}
P(A_{S}) & =(n+1)(1-p)^{n}p+(1-p)^{n+1}\text{ and }final\\
P(B_{T}) & =(m+1)p^{m}(1-p)+p^{m+1}.
\end{align*}
A given event $A_{S}$ is independent of an event of the form $B_{T}$ unless there is a set $F\in\binom{[n+m]}{m}$ such
that $S\subset F\subset T$, i.e., if $S\subset T$. There are $\frac{(n+1)n}{2}$ such events $B_{T}$. $A_{S}$~is not
independent of another event $A_{S'}$ if there is a $F\in\binom{[n+m]}{m}$ such that $S,S'\subset F$, i.e., if the symmetric
difference of $S$ and $S'$ is of size~$2$. There are $(m-1)(n+1)$ such events $A_{S'}$. By symmetry, a given event
$B_{T}$ is independent of all but $\frac{(m+1)m}{2}$ events~$A_{S}$, and is independent of all but $(n-1)(m+1)$ other
events of the form $B_{T'}$.
We want to use \cref{lem:asymlocal} to prove that there is a non-zero probability that none of the events $A_{S}$
and $B_{T}$ occur, and thus $\mathcal{F}$ fulfills the conditions of \cref{lem:random}. Define a function
\[
x:\left\{ A_{S}:S\in\binom{[n+m]}{m-1}\right\} \cup\left\{ B_{T}:T\in\binom{[n+m]}{m+1}\right\} \to[0,1)\text{ by}
\]
\[
x(E)=\begin{cases}
y\coloneqq\frac{1}{4(m-1)(n+1)} & \text{if } E=A_{S}\text{ for some }S\in\binom{[n+m]}{m-1},\\
z\coloneqq\frac{1}{4(n-1)(n+1)} & \text{if } E=B_{T}\text{ for some }T\in\binom{[n+m]}{m+1}.
\end{cases}
\]
For an event~$E$ in the domain of~$x$, let $\Gamma(E)$ be the set of other events that are not independent of~$E$.
We will use the bounds
\begin{equation}
e^{-x}\ge1-x\ge e^{-2x},\label{eq:expbound}
\end{equation}
which hold when $0\le x\le\frac{1}{2}.$ For any set $T\in\binom{[n+m]}{m+1}$, we have
\begin{gather*}
x(B_{T})\prod_{E'\in\Gamma(B_{T})}(1-x(E'))=z\overbrace{(1-y)^{(m+1)m/2}}^{E'=A_{S}}\overbrace{(1-z)^{(n-1)(m+1)}}^{E'=B_{T'}}\\
\overset{\eqref{eq:expbound}}{\ge}ze^{-2(y(m+1)m/2+z(n-1)(m+1))}>\frac{1}{4(n-1)(n+1)e}\\
=(m+1)p^{m}>(m+1)p^{m}(1-p)+p^{m+1}=P(B_{T}).
\end{gather*}
For any set~$S\in\binom{[n+m]}{m-1}$, we have
\begin{gather*}
x(A_{S})\prod_{E'\in\Gamma(A_{S})}(1-x(E'))=y\overbrace{(1-z)^{(n+1)n/2}}^{E'=B_{T}}\overbrace{(1-y)^{(m-1)(n+1)}}^{E'=A_{S'}}\\
\overset{\eqref{eq:expbound}}{\ge}ye^{-2(z(n+1)n/2+y(m-1)(n+1))}>\frac{1}{4(m-1)(n+1)e}\ge\frac{1}{4(n-1)(n+1)e}\text{,\quad and}
\end{gather*}
\begin{equation}
\begin{gathered}P(A_{S})=(n+1)(1-p)^{n}p+(1-p)^{n+1}<\bigl((n+1)p+1\bigr)(1-p)^{n}\\
\overset{\eqref{eq:expbound}}{\le}\bigl((n+1)p+1\bigr)\cdot e^{-pn}\\
<\bigl((n+1)\left(4(m+1)\left(n^{2}-1\right)e\right)^{-1/m}+1\bigr)\cdot e^{-\bigl(4(m+1)e\bigr)^{-1/m}\cdot n^{1-2/m}}.
\end{gathered}
\label{eq:PAS}
\end{equation}
On the right-hand side of \eqref{eq:PAS}, $\bigl((n+1)\left(4(m+1)\left(n^{2}-1\right)e\right)^{-1/m}+1\bigr)$ is increasing
in $m$ and $e^{-\bigl(4(m+1)e\bigr)^{-1/m}\cdot n^{1-2/m}}$ is decreasing for $m\ge3$. So, by replacing $m$ with $n$
in the first factor, and $m$ with $3$ in the second factor, we have
\begin{gather*}
P(A_{S})\le\bigl((n+1)\left(4(n+1)\left(n^{2}-1\right)e\right)^{-1/n}+1\bigr)\cdot e^{-\bigl(16e\bigr)^{-1/3}\cdot n^{1/3}}\\
\le\frac{1}{4(n-1)(n+1)e}<x(A_{S})\prod_{E'\in\Gamma(A_{S})}(1-x(E'))
\end{gather*}
when $n$~is sufficiently large. Therefore the function $x$ satisfies the inequality \eqref{eq:asymlocal-ineq} required
by the asymmetric Lov\'asz Local Lemma, so $\mathcal{F}$ has the desired properties.
\end{proof}
Now we are ready to prove \cref{prop:CS-generalized} using the family of sets constructed in \cref{lem:random}.
We may assume without loss of generality that $m\le n$. Let $\mathcal{F}\subset\binom{[n+m]}{m}$ be the family of sets given by
\cref{lem:random}. Let ${\color{blue}\mathcal{B}}=\binom{[n+m]}{0}\cup\binom{[n+m]}{1}\cup\ldots\cup\binom{[n+m]}{m-2}\cup\mathcal{F}\cup\binom{[n+m]}{m+1}$,
and let ${\color{red}\mathcal{R}}=\mathcal{Q}_{n+m}\setminus{\color{blue}\mathcal{B}}$.
Assume that ${\color{blue}\mathcal{B}}$ contains a weak copy of~$\mathcal{Q}_{m}$ provided by the injection $\varphi:\mathcal{Q}_{m}\to{\color{blue}\mathcal{B}}$. Note that ${\color{blue}\mathcal{B}}$ has
height $m+1$ as a poset. Therefore, for $A\in\mathcal{Q}_{m}$, $\left|\varphi(A)\right|=m$ if $\left|A\right|=m-1$, and $\left|\varphi(A)\right|=m+1$
if $A=[m]$. The $m$ sets of size $m-1$ in~$\mathcal{Q}_{m}$ are mapped to subsets of $\varphi([m])$ in $\mathcal{F}=\binom{[n+m]}{m}\cap{\color{blue}\mathcal{B}}$.
But, by \ref{enu:subsets} in \cref{lem:random}, only at most $m-1$ subsets of $\varphi([m])$ are in $\mathcal{F}$, a contradiction.
Similarly, assume that ${\color{red}\mathcal{R}}$ contains a weak copy of~$\mathcal{Q}_{n}$ provided by the injection $\varphi:\mathcal{Q}_{n}\to{\color{red}\mathcal{R}}$. Note that
${\color{red}\mathcal{R}}$ has height $n+1$. Therefore, for $A\in\mathcal{Q}_{n}$, $\left|\varphi(A)\right|=m-1$ if $A=\emptyset$, $\left|\varphi(A)\right|=m$
if $\left|A\right|=1$, and $\left|\varphi(A)\right|=\left|A\right|+m$ if $\left|A\right|\in\{2,3,\ldots,n\}$. The $n$
singletons of~$\mathcal{Q}_{n}$ are mapped to supersets of $\varphi(\emptyset)$ in $\binom{[n+m]}{m}\cap{\color{red}\mathcal{R}}=\binom{[n+m]}{m}\setminus\mathcal{F}$.
But, by \ref{enu:supersets} in \cref{lem:random}, at least~$2$ supersets of $\varphi(\text{\ensuremath{\emptyset}})$
are in~$\mathcal{F}$, so at most~$n-1$ are in $\binom{[n+m]}{m}\setminus\mathcal{F}$, a contradiction.
\begin{remark*}
The above proof of \cref{prop:CS-generalized} cannot be easily made to work for $m=2$. More precisely, the following
\MakeLowercase{\crtcrefnamebylabel{claim:cox-stolee-fails}} holds.
\end{remark*}
\begin{claim}
\label{claim:cox-stolee-fails}The conclusion of \cref{lem:random} does not hold for $m=2$, and in fact there is no
$\mathcal{F}\subset\binom{[n+2]}{2}$ such that, for ${\color{blue}\mathcal{B}}=\{\emptyset\}\cup\mathcal{F}\cup\binom{[n+2]}{3}$ and ${\color{red}\mathcal{R}}=\mathcal{Q}_{n+2}\setminus{\color{blue}\mathcal{B}}$,
$\mathcal{Q}_{2}$~is not a subposet of~${\color{blue}\mathcal{B}}$, and $\mathcal{Q}_{n}$~is not a subposet of~${\color{red}\mathcal{R}}$.
\end{claim}
\begin{proof}
A family of sets $\mathcal{F}\subset\binom{[n+2]}{2}$ that satisfies the condition \ref{enu:supersets} in \cref{lem:random}
contains, for any $S\in\binom{[n+2]}{1}$, a pair of sets $A,B$ such that $S\subset A,B$. Note that $A$ and $B$ have
a symmetric difference of size~$2$. Then $A,B\subset A\cup B\in\binom{[n+2]}{3}$, which contradicts the condition \ref{enu:subsets}
in \cref{lem:random}. So the two conditions of \cref{lem:random} cannot be satisfied at the same time by a family
of sets $\mathcal{F}\subset\binom{[n+2]}{2}$.
It is easy to check that the conditions \ref{enu:supersets}~and~\ref{enu:subsets} on~$\mathcal{F}$ in \cref{lem:random}
are not only sufficient, but also necessary for the above coloring to satisfy the conditions of \cref{prop:CS-generalized},
that is, to have no $\mathcal{Q}_{2}$ as a subposet of ${\color{blue}\mathcal{B}}=\{\emptyset\}\cup\mathcal{F}\cup\binom{[n+2]}{3}$, and no $\mathcal{Q}_{n}$ as a subposet
of ${\color{red}\mathcal{R}}=\mathcal{Q}_{n+2}\setminus{\color{blue}\mathcal{B}}$.
\end{proof}
\section*{Acknowledgments\phantomsection\addcontentsline{toc}{section}{Acknowledgements}}
The second and third authors were supported by the grant IBS-R029-C1. The research of the second author was also partially
supported by the EPSRC, grant no. EP/S00100X/1 (A. Methuku).
\end{document} |
\begin{document}
\title{Regularizing Bayesian Predictive Regressions \thanks{We appreciate helpful comments from conference participants at 26th Annual Meeting of the Midwest Econometrics Group, Washington University in St. Louis, 2017 Vienna-Copenhagen Conference on Financial Econometrics, 2017 NBER-NSF Seminar on Bayesian Inference in Econometrics and Statistics, and 2017 NBER-NSF Time Series Conference. We thank Jianeng Xu for his excellent research assistance.}}
\author{Guanhao Feng\thanks{
Address: 83 Tat Chee Avenue, Kowloon Tong, Hong Kong. E-mail address:
\texttt{[email protected]}.} \\
\textit{College of Business}\\
\textit{City University of Hong Kong}
\and Nicholas G. Polson\thanks{
Address: 5807 S Woodlawn Avenue, Chicago, IL 60637, USA. E-mail address:
\texttt{[email protected]}.} \\
\textit{Booth School of Business}\\
\textit{University of Chicago}}
\date{This Version: September 12, 2017}
\maketitle
\begin{abstract}
\noindent Regularizing Bayesian predictive regressions provides a framework for prior sensitivity analysis via the regularization path. We jointly regularize both expectations and variance-covariance matrices using a pair of shrinkage priors. Our methodology applies directly to vector autoregressions (VAR) and seemingly unrelated regressions (SUR). By exploiting a duality between penalties and priors, we reinterpret two classic macro-finance studies: equity premium predictability and macro forecastability of bond risk premia. We find that there exist plausible prior specifications for predictability for excess S\&P 500 returns using predictors book-to-market ratios, CAY (consumption, wealth, income ratio), and T-bill rates. We evaluate our forecasts using a market-timing strategy and show how ours outperforms buy-and-hold. We also predict multiple bond excess returns involving a high-dimensional set of macroeconomic fundamentals with a regularized SUR model. We find the predictions from latent factor models such as PCA to be sensitive to prior specifications. Finally, we conclude with directions for future research.
\end{abstract}
\begin{flushleft}
Key words:
Bayesian predictive regression; prior sensitivity analysis; maximum-a-posteriori; equity-premium predictability; bond risk premia; predictor selection.
\end{flushleft}
\section{Introduction}
The Bayesian paradigm in finance and economics requires prior distribution motivated by economic theory, whereas regularization requires a penalty to trade off by optimizing out-of-sample predictive performance. A duality exists between Bayesian methods and statistical regularization which leads to a framework for prior sensitivity analysis. To illustrate our method, we examine the predictability of the equity premium and bond risk premia predictability using macro factors.
Sensitivity analysis from a Bayesian perspective is typically computationally intensive simulation. However, in economics and finance, performing prior sensitivity analysis across a wide range of prior hyper-parameters at a low computational cost is essential. A significant contribution of our paper is the use of a fast and scalable convex optimization algorithm to perform prior sensitivity analysis. The data-driven method for choosing the tuning parameter, in particular, leads to an alternative interpretation of prior hyper-parameters. For example, we provide a fast sparse covariance matrix approach as an alternative to full Bayesian inverse Wishart simulation.
Bayesian methods have become increasingly popular as a solution to the over-parameterization in VAR (vector autoregression) and SUR (seemingly unrelated regression) systems. An SUR model consists of a set of regressions that may seem unrelated but have correlated error terms. A VAR(p) system is an SUR model where each equation uses the same regressors. To learn about interaction effects among multiple shocks, we employ a Lasso-tilted inverse Wishart prior that forces sparsity in the variance-covariance matrix estimation. Our methodology jointly regularizes expected values and variance-covariance matrices in VAR and SUR systems in a computationally efficient way. Particular attention is paid to the sensitivity over a wide range of hyper-parameters.
Our method reinterprets Bayesian studies of equity-premium predictability of \cite{Kandel1996} and \cite{Barberis2000}. We find a plausible prior specification for predictability in S\&P 500. In a second study, we provide a sensitivity analysis for the bond risk premia prediction using macro factors, see \cite{ludvigson2009macro, ludvigson2010factor}. Our regularized SUR prediction, with a multi-response perspective, uses supervised learning to explore the common macro factors in excess returns for bonds of multiple maturities. Our fast prediction sensitivity check is an alternative to a typical MCMC sampling approach.
Rather than marginalizing out prior hyper-parameters in a fully Bayesian analysis, we calibrate hyper-parameters using an economically meaningful predictive cross-validation measure given a univariate target. We develop a proximal algorithm using majorize-minimize operations (see \cite{polson2015proximal} and \cite{Bien2011}) to provide a simple alternative to MCMC simulation for high-dimensional VAR and SUR systems. In addition to computation speed, both empirical studies show that our proposed method is very competitive in terms of predictor selection, as well as prediction accuracy.
The rest of the paper is outlined as follows. Section \ref{sec11} provides the intuition behind the connection between prior sensitivity analysis and model regularization, and Section \ref{sec12} adds a literature review. Section \ref{sec2} presents the regularized system for Bayesian VAR and SUR, as well as a discussion of the Bayesian MAP estimator. Section \ref{sec3} develops a proximal algorithm for both auto-regressive coefficients and variance-covariance matrices within a VAR setting. Section \ref{sec41} illustrates our method by revisiting the equity-premium analysis for return predictability using many economic fundamental predictors from \cite{Welch2008}. Section \ref{sec42} revisits a well-known study about bond prediction using a high-dimension of macro factors from \cite{ludvigson2009macro,ludvigson2010factor} and provides insights beyond those provided by the dynamic factor analysis. Finally, Section \ref{sec5} concludes with directions for future research.
\subsection{Prior Sensitivity and Regularization \label{sec11}}
To fix notation, define $B$ as an auto-regressive coefficient in a VAR(1) model, and $\Sigma$ as a variance-covariance matrix for multiple shocks. Statistical regularization requires a researcher to specify a measure of goodness of fit, denoted by $l(B, \Sigma)$, as well as a penalty function that achieves a parsimonious model, denoted by $ \phi(B, \Sigma)$. Probabilistically, $l(B, \Sigma)$ and $\phi(B, \Sigma)$ correspond to the negative logarithms of the likelihood and a prior distribution. Regularization leads to an optimization problem of the form.
\[ \underset{B, \Sigma \in \Re^d}{\min} l(B, \Sigma) + \phi(B, \Sigma). \]
For example, a regularized regression minimizes a least-squares objective (Gaussian likelihood) plus a penalty such as an $L^2$-norm (Ridge) Gaussian probability model or $L^1$-norm (Lasso) double exponential probability model.
A probabilistic approach, on the other hand, leads to a Bayesian hierarchical model
\[ p(y \mid B, \Sigma) \propto \exp\{-l(B, \Sigma)\}, \quad p(B, \Sigma) \propto \exp\{ -\phi(B, \Sigma) \}.\]
The solution to the minimization problem corresponds to maximizing the posterior density,
\[ p( B, \Sigma \mid y) \propto p(y \mid B, \Sigma)\times p(B, \Sigma) = \exp\{- l(B, \Sigma) - \phi(B, \Sigma)\} \]
\[ (\hat{B}, \hat{\Sigma}) = \arg\!\max_{B, \Sigma} \; p( B, \Sigma \mid y), \] where $ p(B, \Sigma \mid y)$ denotes the posterior distribution. Here, $(\hat{B}, \hat{\Sigma})$ is simply the posterior mode.
Under a decomposable penalty, $\phi(B, \Sigma) = \lambda\phi_1(B) + \gamma\phi_2(\Sigma) $, the $\lambda$ and $\gamma$ hyper-parameters of a prior distribution are tuning parameters in a regularization problem. Whereas a Bayesian study requires a prior distribution and is sensitive to its hyper-parameters $(\lambda, \gamma)$, a regularization problem uses $(\lambda, \gamma)$ to control the bias-variance tradeoff of the model complexity. Consequently, the regularized estimates $(\hat{B}, \hat{\Sigma})_{(\lambda,\gamma)}$ provide a regularization path, which can be interpreted as prior sensitivity analysis for the MAP estimator. Therefore, the pair of $\lambda$ and $\gamma$ is the key to connect the interpretation of prior sensitivity analysis and model regularization.
\subsection{Connection with Empirical Macro-Finance \label{sec12}}
\cite{Campbell1989} and \cite{Fama1988} provide early evidence of time-series predictability of stock returns and show that market returns can be predicted using lagged dividend yields. However, \cite{Welch2008} examine 14 predictor variables and find little forecasting power in univariate forecasting regressions. \cite{Cochrane2008} uses a VAR system of returns and dividend growth to explore their joint stochastic dynamics and defend the return predictability. \cite{Campbell2008} find an economically significant out-of-sample forecasting power after imposing economically reasonable parameter restrictions. Therefore, we build on the empirical finance literature by adopting a regularized Bayesian predictive framework to exploit the joint dynamics for the univariate predictor and provide a forecast sensitivity analysis.
Early studies in time-varying bond risk premia include \cite{fama1987information} and \cite{campbell1991yield} regarding forecasting yield changes with yield spreads. The excess returns on U.S. government bonds can be predictable by the cross-section of yield spreads or forward rates. \cite{cochrane2005bond} later propose a return-forecasting factor predicts excess returns on one- to five-year maturity bonds with R$^2$ up to 44\%. From a large number of macro series, \cite{ludvigson2009macro} use of principal components analysis to estimate common factors regarding bond prediction. \cite{ludvigson2010factor} use prior information to organize the macro series into 8 subgroups and estimate a dynamic factor model for each subgroup using a Bayesian estimation.
\cite{giannone2015prior} analyze the hyper-parameter uncertainty for the density forecasts in a VAR model. With predictor selection in a large VAR to reduce estimation uncertainty, they use informative priors to shrink the richly parameterized unrestricted model and reduce prediction errors. Whereas \cite{carriero2012forecasting} study a large BVAR and optimally select the amount of shrinkage by maximizing the marginal likelihood, our approach relies on a predictive data-driven selection. \cite{banbura2010large} also apply cross-validation to estimate one single hyper-parameter in a large VAR, but their approach performs neither variable selection nor variance-covariance regularization.
\cite{stock2002forecasting, Stock2002} find significant improvements for macro and financial predictions using common factors estimated from large data sets. \cite{jurado2015measuring} attempt to reduce the dimension of macro series to quantify the time-varying macroeconomic uncertainty. For the recent development in high-dimensional time-series models, \cite{chan2016large} propose a Bayesian approach for inference in VARMA, and \cite{Nicholson2015} introduce regularization to reduce the parameter space of VAR and VARX (VAR with exogenous variables) models. \cite{Zantedeschi2011} also develop a Bayesian procedure for macro-finance forecasting.
Given the extensive list of potential predictors for both market risk premium and bond risk premium, we consider the role of shrinkage priors to examine the model uncertainty about the existence and strength of predictors. \cite{Park2008} suggest using the posterior mode interpretation of Lasso regularization. \cite{xu2015bayesian} propose the posterior median estimator for the Bayesian group lasso and uses spike and slab priors for group variable selection. In a hierarchical Bayesian framework, \cite{yuan2005efficient} show the empirical Bayes estimator that is closely related to the Lasso estimator for variable selection.
\section{Regularizing Bayesian Predictive Regressions \label{sec2}}
To illustrate Bayesian SUR system, we discuss high-dimensional VAR regularization which is popular in the empirical macro-finance literature. We demonstrate our SUR regularization in the bond prediction in Section \ref{sec42}.
\subsection{Bayesian Seemingly Unrelated Regressions}
\cite{Zellner1962, zellner1963estimators} initially proposed SUR to estimate a system of stacked regression equations, which include cross-equation parameter restrictions and correlated error terms. The model is also referred to as a ``generalized multivariate regression model" and therefore can be solved in a generalized least squares approach. Specifically, we have a matrix formulation for a general SUR model of the form
\begin{equation}
\label{eqn:SUR}
Z = XB + E, \quad \mbox{where } E \sim \mathcal{N}(0, \Sigma \otimes I).
\end{equation}
Here the target Z is an $n\times m$ matrix. $\mathcal{N}(0, \Sigma \otimes I)$ denotes the multivariate normal distribution and $\otimes$ is the Kronecker product. $\Sigma$ is an $m\times m$ matrix. Throughout, we use the following stacked variables $Z^\intercal = \bmat{Z_1^\intercal, \cdots, Z_m^\intercal}$, $X = \mbox{diag}\{X_1, \cdots, X_m\}$, $B^\intercal = \bmat{\beta_1^\intercal, \cdots, \beta_m^\intercal}$, and $E^\intercal = \bmat{\epsilon_1^\intercal, \cdots, \epsilon_m^\intercal}$.
The regressor $X_i$ for each individual regression is a $T \times p$ matrix and $\beta_i$ is a $p \times 1$ vector. The likelihood function is
\begin{eqnarray*}
L(Z, X \mid B, \Sigma) &=& (2\pi)^{-nm/2} \det(\Sigma)^{-n/2} \exp\left(-\frac{1}{2}(Z - XB)^\intercal (\Sigma^{-1} \otimes I) (Z - XB) \right) \\
&=& (2\pi)^{-nm/2} \det(\Sigma)^{-n/2} \exp\left(-\frac{1}{2}\mbox{tr}\{S_B\Sigma^{-1}\}\right),
\end{eqnarray*}
where the (i, j)th element in $S_B$ equals to $(Z_i - X_i\beta_i)^\intercal (Z_j- X_j\beta_j)$. Given a non-informative Jeffreys' invariant prior, the joint posterior density function is
\[ p(B, \Sigma \mid Z, X) \propto \det(\Sigma)^{-(n+m+1)/2}\exp\left(-\frac{1}{2}\mbox{tr}\{S_B\Sigma^{-1}\}\right) \]
\cite{Zellner1971} introduces a Bayesian approach to calculating posterior densities for parameters to estimate the SUR model. In the recent advancement, \cite{Zellner2010} propose a direct Monte Carlo (DMC) approach to calculate Bayesian estimation and prediction results using diffuse or informative priors. \cite{rothman2010sparse} propose a penalized likelihood method with simultaneous estimation of the regression coefficients as well as the covariance structure. \cite{chen2012sparse} apply a group-lasso type penalty in predicting multiple response variables from the same set of predictor variables.
\subsection{Vector Auto-Regressions}
The class of VAR models are a popular forecasting tool for empirical financial and macroeconomic time-series analysis that can capture complex dynamic interrelationships among variables. \cite{Kandel1996} adapt a VAR formulation to investigate the predictability of the equity premium and build portfolios using the predictive return distribution. Our predictive cross-validation in Section \ref{sec33} follows their framework of univariate variable forecasting in a multivariate time-series model.
In general, a VAR(p) system jointly explores the stochastic dynamics of both stock market returns, denoted by $y_t$, and economic predictive variables, $x_t$, for $1\leq t \leq T$, where
\begin{equation*}
\bmat{y_t \\ x_t} = \alpha + \beta_1\bmat{y_{t-1} \\ x_{t-1}} + \cdots + \beta_p\bmat{y_{t-p} \\ x_{t-p}} + \epsilon_t, \quad \mbox{ where } \epsilon_t \sim \mathcal{N}(0, \Sigma).
\end{equation*}
Here $y_t$ is a continuously compounded excess market return, and $x_t$ is a vector of $K$ economic predictive variables, which typically include dividend yield, earning-price ratio, book-to-market ratio, and so on. For multi-step prediction, we can estimate the VAR parameters $(\alpha, \beta, \Sigma)$ and iterate the model forward with the parameters fixed at their estimated values.
To illustrate the model, consider the simplest case of a VAR(1) system. Given that most predictors proposed in the literature are lagged one period, we rewrite a VAR(1) model with a demeaned vector given by $Z_t = (y_t, x_t^\intercal)^\intercal - E_T[(y_t, x_t^\intercal)^\intercal]$ and the autoregressive structure:
\begin{equation} \label{VAR1}
Z_t = B Z_{t-1} + \epsilon_t
\end{equation}
In our empirical study of Section \ref{sec41}, the goal is to study the joint predictability of proposed predictors to stock returns and see the trade-off between model specification and forecasting power. To compute the posterior distribution, it is convenient to reformulate the VAR(1) model into the matrix formulation of SUR in (\ref{eqn:SUR}), and we write
$z = \mbox{Vec}(Z) = (L^\intercal \otimes I_{p})B+ \mbox{Vec}(E)$, where Vec($\cdot$) is the column stack operator, and we stack variables and parameters $Z = (Z_1, \cdots, Z_{T})$, $L = (Z_0, Z_1, \cdots, Z_{T-1})$, $E = (\epsilon_1, \epsilon_2, \cdots, \epsilon_{T})$ and $B = \mbox{Vec}(\beta)$.
\subsection{Bayesian MAP Estimator}
We now turn to the problem of computing a regularized Bayesian MAP (Maximum-a-Posteriori) estimation. With a joint penalty, denoted by $\phi(B, \Sigma)$, for the parameters $\beta$ and the variance-covariance matrix $\Sigma$, a MAP estimator corresponds to the mode of the posterior distribution. The usual Bayesian estimator is the posterior expectation that minimizes the quadratic loss in Bayesian decision studies, whereas the MAP estimator minimizes the 0-1 loss. When the prior density is flat, the posterior mode turns out to be the maximum likelihood estimator.
This problem is equivalent to solving a penalized likelihood with regularization where $\phi(B, \Sigma)$ is considered a penalty function to control the favorable bias-variance trade-off. \cite{polson2015mixtures} describe such a duality between specifying a regularization penalty and a prior distribution. The prior hyper-parameter $\lambda$ can be viewed as the amount of regularization. We demonstrate that using some shrinkage priors performs variable selection, whereas $\lambda$ is used to control the model size and assess the bias-variance trade-off.
\cite{Kandel1996} study the equity-premium puzzle under a Gaussian prior for B, which is equivalent to a shrinkage $L^2$ penalty, where $B \sim \mathcal{N}(0, \tau^2 I) \mbox{ and } p(B) \propto \exp\Big(-\frac{B^\intercal B}{2\tau^2}\Big)$. With a Jeffrey's prior, we set $p(\Sigma) = 1$ and then can obtain the $L^2$ regularized MAP posterior mode estimate to compare the Bayesian posterior mean estimate of Kandel and Stambaugh by finding
\[ B^\star = \arg\!\min_{B} l(B \mid \Sigma) + \lambda\lVert B\rVert_2^2. \]
These penalized model comparison criteria is shown to correspond to a hierarchical Bayes model selection procedure under a particular class of priors by \cite{george2000calibration}. The process of selecting predictor variables can then be simplified using Laplace shrinkage priors, where $B_i \iid \mbox{Laplace}(0, \lambda) \mbox{ with } \phi(B) \propto \exp\{-\lambda\sum_{i=1}^P\lvert B_i \rvert \}$. An $L^1$ penalty performs variable selection through the sparsity of B and the optimization problem $ B^\star = \arg\!\min_{B}\Big\{ l(B \mid \Sigma) + \lambda\lVert B\rVert_1\Big\}$. Such a shrinkage prior is a solution to the unstable inference and inaccurate out-of-sample forecasts in large (Bayesian) VARs with dense parameterization.
\subsection{Prior Distribution of B and $\Sigma$}
For regularizing large variance-covariance matrices, \cite{Das2010} also present a generalized multivariate gamma distribution and discuss the MAP covariance estimation. \cite{cai2011adaptive} consider estimation of sparse covariance matrices and propose an adaptive thresholding procedure that outperforms the universal thresholding estimators. \cite{fan2013large} introduce the principal orthogonal complement thresholding method to deal with a sparse error covariance matrix in an approximate factor model.
When estimating a covariance matrix, $\Sigma$, two common prior distributions are Jeffreys' prior and the inverse Wishart. We now wish to model sparsity in the elements of the correlation matrix, denoted by $\rho$, as a sparse structure. The standard prior distribution for a variance-covariance matrix is the family of inverse Wishart probability densities, $p_{IW}(\Sigma)$, given by
\[ p_{IW}( \Sigma \mid v, S_0 ) \propto \det(\Sigma)^{\nu+p/2}\exp \left\{ - \frac{1}{2} \mbox{tr}( S_0 \Sigma^{-1} ) \right\}. \]
\noindent Here $ (S_0,\nu)$ are prior hyper-parameters. There are a number of approaches for specifying these parameters. For example, Jeffrey's prior corresponds to the special case $p(\Sigma) \equiv det(\Sigma)^{p/2}$. Zeros in a covariance matrix correspond to marginal independencies between variables and are desirable for impulse-response functions in VAR analysis. We will achieve this estimation by imposing an $L^1$ regularization penalty.
One of the main advantages of using an $L^1$-penalty is that the MAP estimator, $\Sigma^\star$, has zeros forced with its solution. Modeling $L^1$-sparsity in $\Sigma$ corresponds to adding an appropriate scale matrix and hyper-parameters for degrees of freedom to get a so-called Lasso-tilted inverse Wishart density, denoted by $p_{LIW}(\Sigma)$, where
\begin{equation}
p_{LIW}(\Sigma \mid B, Z) \propto \det(\Sigma)^{v+p/2}\exp\{-\mbox{tr}(S_0\Sigma^{-1})\}\exp\{-\gamma \Vert \Sigma \Vert_1\}.
\end{equation}
Jointly regularizing $(B, \Sigma)$ in a VAR system leads to a posterior density with $S_B = \frac{1}{T} \sum_{t=1}^T \epsilon_t \epsilon_t^\intercal$,
\[ p_{LIW}(\Sigma \mid B, Z, Y) \propto \det(\Sigma)^{v+(T+p)/2}\exp\{-\mbox{tr}((S_B+S_0)\Sigma^{-1})\}\exp\{-\gamma \Vert \Sigma \Vert_1\}. \]
\cite{Bien2011} provide convex optimization with an $L^1$-penalty can be applied to a variance-covariance matrix by penalizing the entries of the covariance matrix. Specifically, they show how to minimize a penalized negative log likelihood
\begin{equation}
\label{BienOptim}
\Sigma^\star = \arg\!\min_{\Sigma}\Big\{ \log(\det(\Sigma)) + \mbox{tr}(S \Sigma^{-1}) + \gamma \Vert \Sigma \Vert_1 \Big\},
\end{equation}
where the $L^1$ norm definition for a matrix is $\lVert \Sigma \rVert_1 = \lVert \mbox{vec}(\Sigma) \rVert_1 = \sum_{i,j}\lvert \sigma_{i,j}\rvert$. \cite{Bien2011} use a ``Majorize-Minimization" (MM) approach to solve the minimization problem and provide a \texttt{R} package [\texttt{spcov}]. We show how to adapt their optimization solution into our joint regularization problem in Section \ref{sec32}.
\section{Regularizing VARs \label{sec3}}
The sparse Lasso-VAR and shrinkage Ridge-VAR are identical in model reformulation and penalized estimation via coordinate descent algorithms. Both of them can be reformulated to an elastic net regularization, which linearly combines the $L^1$ and $L^2$ penalties of the Lasso and Ridge methods. Regularization produces a convenient tool to calculate posterior modes and graphically display prior sensitivity analysis via a regularization path. The model setup below is based on Ridge-VAR, and empirical results of Ridge-VAR are provided in Section \ref{sec41} and \ref{sec42}.
A high-dimensional Bayesian VAR is computationally convenient and interpretable by shrinkage regularization. However, \cite{Song2011} show the negative consequences of directly applying Lasso-type penalties for time series without considering the temporal dependence. Then \cite{Davis2015} use a maximum likelihood approach and propose a two-stage approach for fitting sparse VAR. Their procedure is based on the property that, for a given variance-covariance matrix, the penalized likelihood can be recast into a penalized regression.
The idea is the same as stacking equations to solve these simultaneous systems, such as VAR and SUR. The likelihood estimation of Lasso-VAR models is not straightforward, because the likelihood function involves the unknown parameter $\Sigma$ and estimation of the variance-covariance matrix $\Sigma$ is usually difficult because of the curse of dimensionality. We have adapted their likelihood reformulation into our regularized VAR framework.
\subsection{Regularization of B and $\Sigma$}
Given a sequence of $T$ multivariate normal $p$-dimensional random vectors $\{\epsilon_t\}$, and then the negative log likelihood using $B$ and $\Sigma$ is
\[ l(B, \Sigma) = \frac{T\times p}{2}\log 2\pi + \frac{T}{2}\log\{\det(\Sigma)\} + \frac{T}{2}\mbox{tr}(\Sigma^{-1}S_B). \]
\noindent where $S_B = \frac{1}{T}\sum_1^{T}\epsilon_t\epsilon_{t}^\intercal$ and $\epsilon_t = Z_t - B Z_{t-1}$.
Given our model specification and stack variable notations in (\ref{VAR1}), the conditional negative log-likelihood $l(B \mid \Sigma)$ of the VAR(1) model is given by
\[ l(B \mid \Sigma) \propto T\log\{det(\Sigma)\} + [z - (L^\intercal \otimes I_{p})B]^\intercal(I_T \otimes \Sigma^{-1})[z - (L^\intercal\otimes I_{p})B]. \]
\noindent The negative log-likelihood of $\Sigma$ conditional on B is given by
\[ l(\Sigma \mid B) \propto \log\{\det\Sigma\} + \mbox{tr}(\Sigma^{-1}S_B). \]
To regularize B, we add the penalty function on $ l(B \mid \Sigma)$ and calculate the estimate $B^\star = \arg\!\min_{B} \Big\{ l(B \mid \Sigma) + \lambda\lVert B \rVert_1 \Big\}$. Maximizing such a penalized Gaussian likelihood is equivalent to minimizing a penalized least-squares errors. Therefore, the optimization problem can be efficiently computed using the \texttt{R} package [\texttt{glmnet}]. Similarly, for the Ridge regularization for a Gaussian prior specification, we can solve $B^\star = \arg\!\min_{B} \Big\{l(B \mid \Sigma) + \lambda\lVert B \rVert_2\Big\}$.
Similarly, we add a scaled $L^1$-penalty function on $ l(\Sigma \mid B)$ for the joint regularization in ($B, \Sigma$ ). But we need to solve the following optimization problem, where $\Sigma \succ 0$ means positive definiteness for covariance matrix,
\begin{equation}
\arg\!\max_{B, \Sigma} p(B, \Sigma \mid y) = \arg\!\min_{\Sigma \succ 0} \{l(\Sigma \mid B) + \psi(B) + \gamma\lVert P \odot \Sigma \rVert_1\},
\end{equation}
\noindent where $P \odot \Sigma_\epsilon$ is defined as element-to element multiplication and P is a $p\times p$ matrix of all 1's with 0's on the diagonal to ensure the positive-definiteness. When using an infinite penalty $\gamma$, the solution is a diagonal variance-covariance matrix with all zero covariance pairs. The optimization algorithm is allowed to regularized variance-covariance matrix, including the level of variance in the diagonal, but we only regularize the correlation matrix in the empirical analysis.
\subsection{Proximal Algorithm \label{sec32}}
The goal is to calculate the mode of the posterior distribution, $\arg\!\max_{B, \Sigma} p(B, \Sigma \mid y)$. Although the objective function in (\ref{BienOptim}) is not convex, it is the sum of a convex function and a concave function. We can see $tr(S\Sigma^{-1}) + \gamma\lVert \Sigma \rVert_1$ is convex in $\Sigma$, but $\log(\det(\Sigma))$ is concave. Therefore, we adapt the MM algorithm suggested in \cite{Bien2011} to solve this optimization for $(B^\star, \Sigma^\star)$. In short, the MM method is a prescription for constructing interactive optimization algorithms that exploit the convexity of a function to find the minima.
We now combine these two convex programming problems to construct a proximal algorithm that jointly regularizes B and $\Sigma$. We have a composite objective of the negative log-likelihood function, regularization of B and regularization of $\Sigma$. Our joint objective depends on two global regularization parameters $(\lambda, \gamma) >0$ and is specified by
\begin{equation}
(B^\star, \Sigma^\star) = \arg\!\min_{\Sigma \succ 0, B} \Big\{ Q(B,\Sigma) = l(B,\Sigma) + \lambda\lVert B \rVert_1 + \gamma\lVert P \odot \Sigma \rVert_1 \Big\}
\end{equation}
\noindent by iterating the optimal parameters of B and $\Sigma$ from the individual optimizations.
\[ B^{(k+1)} = \arg\!\min_{B} Q(B \mid \Sigma^{(k)}) = l(B,\Sigma) + \lambda\lVert B \rVert_1 \]
\[ \Sigma^{(k+1)} = \arg\!\min_{\Sigma \succ 0} Q(\Sigma \mid B^{(k+1)}) = l(B,\Sigma) + \gamma\lVert P \odot \Sigma \rVert_1 \]
Hence, we have constructed a sequence $\{B^{(k+1)}, \Sigma^{(k+1)} \}$ that converges to the Bayesian MAP estimator $(B^\star, \Sigma^\star)$. As we vary the regularization parameters $(\lambda, \gamma)$, we trace out a full regularization solution path, thereby providing our sensitivity diagnostics. \cite{polson2015proximal} provides a discussion of convergence properties of proximal-point algorithms and shows how gains in efficiency can be achieved with Nesterov's acceleration.
One advantage of our joint regularization is that it is computationally efficient because we divide the composite objective and conquer each optimization sequentially. In the empirical analysis, we need no more than ten iterations for a sufficient convergence. The empirical results with a full prior sensitivity analysis are especially useful in empirical finance and macroeconomics forecasting. For example, we can learn about variable selection via increasing sparsity in B, and how target-predictor impulse-responses vary through to increasing the sparse correlation $\rho$ of $\Sigma$.
\subsection{Predictive Cross-Validation \label{sec33}}
Given the duality between the pair of tuning parameters $(\lambda, \gamma)$ with the hyper-parameters of a prior distribution of $(\beta, \Sigma)$, we can use predictive cross-validation to help select optimal prior hyper-parameters rather than directly marginalizing them out. The goal of out-of-sample prediction is to show regularization paths of prediction error as a function of tuning parameters. From the perspective of predictive ability, one can search the optimal pair of $(\lambda, \gamma)$ to achieve the best out-of-sample performance in a hold-out sample of data.
If our goal is the one-step-ahead prediction of the equity premium, we can apply a predictive cross-validation to calculate the model-prediction error in history. Cross-validation is an intuitively data-driven resampling method to assess the model out-of-sample performance and is extremely useful for model selection. We can then estimate a sequence of VAR models by using a rolling window of data and obtain the one-step-ahead prediction error. \cite{Song2011} also suggest choosing tuning parameters via a data-driven rolling scheme method to optimize the forecasting performance. For a robustness check, we have also implemented AIC and BIC for tuning-parameter selections. For the market-timing strategy in Section \ref{sec413}, the AIC selection demonstrates strong predictability in the multiple-predictor model.
Our equity-premium examples in Section \ref{sec41} show 63 years of quarterly data, and hence we pick the window size as 20 years or 80 quarters. Eighty observations is still an adequate number for estimating a regular 10-dimensional VAR(1) model. We have 172 overlapping rolling schemes in the sample, which is the number of VAR model estimation. The model-comparison criteria are the sample mean of the prediction errors in S\&P 500 excess returns. Other variables in the VAR system are viewed as ``instruments" in the forecast. Therefore, we can find a way to determine the level of regularization with the best out-of-sample prediction performance.
For multi-step-ahead forecasts, we can change the objective function to minimize the multi-step-ahead prediction errors. In this scenario, our proposed data-driven selection method is more ``informative" than any prior specification that is only related to one-step-ahead forecasts. Our VAR formulation could be applied to a multi-dimension forecast by using the rest of variables as ``predictive instruments" in the time series system. In the excess bond return prediction example of Section \ref{sec42}, we only show the sensitivity analysis, but we can use the same predictive cross-validation to determine the best tuning parameters. The advantage for the supervised learning in a multiple-response model is to specify the common tuning parameter.
In short, predictive cross-validation provides a powerful connection between a regularizing predictive regression and a fully Bayesian approach. Traditionally, we specify the prior hyper-parameters and calculate out-of-sample predictions, but this approach is optimal only under some specific priors. Our regularization approach demonstrates exactly how different prior hyper-parameters affect predictive power through regularization plots. Specifying one data-driven prior in the Bayesian predictive system is possible. We suggest one can view a regularization approach as a quick precursor to a more detailed full Bayesian analysis.
\section{Regularizing Macro-Finance Predictions}
\subsection{Equity-Premium Prediction \label{sec41}}
To illustrate our methodology, we revisit the equity-premium predictors surveyed in \cite{Welch2008}, who argue all these predictors lack out-of-sample forecastability. We examine the robustness of their predictability in both single-predictor and multiple-predictor models. For Bayesian VAR studies, \cite{Kandel1996} provide a framework of Bayesian predictive regressions using multiple predictors and a zero-mean prior that implies no time-series predictability and market efficiency. \cite{Barberis2000} studies a single-predictor model using the same conventional non-informative prior. Moreover, we examine the sensitivity of the evidence on predictability through a market-timing strategy as one out-of-sample study: whether investors can exploit the predictability and earn profits more than a buy-and-hold strategy in the market index.
\subsubsection{Single-Predictor Model}
In our analysis, the excess market return is the quarterly return on the S\&P 500 index minus the short-term risk-free rate. Table \ref{Tab0} has a description of the 14 economic fundamental variables proposed by academics. Our analysis uses quarterly data for excess returns and the dividend-price ratio of the S\&P 500 index from 1952 to 2015. We use a Gaussian prior for the AR coefficient and the Lasso-tilted inverse Wishart prior for the variance-covariance matrix. The optimal amount of regularization on B and $\Sigma$ is calibrated to the least predictive MSE for one-step-ahead excess returns.
\cite{Barberis2000} uses a ``non-informative" prior that is equivalent to the case of $\lambda=1$ for B and $\gamma=0$ for $\Sigma$ in our regularization. Table \ref{Tab1} shows the estimation comparison between Barberis's Bayesian posterior mean approach and our corresponding MAP mode approach. We can compare Barberis's coefficient B and our $\beta_{[:, 2]}$, which are coefficients of (D/P)$_{t-1}$ on return$_t$ and (D/P)$_{t}$. First, the dividend-price ratio is a random walk in a full Bayes estimation and must underperform the regularized estimator. Second, the large difference in $\Sigma_{[2, 2]}$ is due to this random-walk specification, and our only focus is the excess-return variable. Finally, The amount of optimal regularization suggests the predictability of the dividend-price ratio should be nonzero but slightly lower.
Figure \ref{Fig1} plots summaries for the single-predictor model from 1952 Quarter 1 to 2015 Quarter 4. To isolate the regularization effect, we regularize the AR coefficients for the top two plots and the variance-covariance matrix for the bottom two plots, respectively, in Figure \ref{Fig1}. The top-left plot shows how the predicted values vary over a wide range of tuning parameters. We see sequentially incorporating the belief of predictability dramatically changes the predicted value of S\&P 500 excess returns for 2015 Quarter 4. The shrinkage priors allow us to observe the signal strength sequentially through regularization. The minimum values of prediction regularization are in the middle and are different in Ridge and Lasso shrinkages. The left end corresponds to a least regularized prediction (VAR), and the right end corresponds to a most regularized prediction (Average).
Figure \ref{Fig1} bottom-left panel shows the regularization path of correlations, and the bottom-right panel shows the regularization path of one-step-ahead orthogonalized impulse response from the dividend-price ratio to S\&P 500 excess returns. When we increase the shrinkage amount $\gamma$, we can see the shock correlation shrinks to zero, but the orthogonal impulse response hardly changes (see the y-axis values). The advantage of using an orthogonal impulse response is the consideration of the shock relationship. But we find regularizing $\Sigma$ has negligible effects on the cross-impulse-response function from the dividend-price ratio to excess returns.
\subsubsection{Multiple-Predictor Mode}
Many previous discoveries of economic predictors for the equity premium are in the univariate forecasting model. \cite{Welch2008} perform their influential study with individual predictors instead of a combination of all predictors, and we revisit their proposed predictors at the lens of our high-dimensional regularized VAR. However, \cite{Rapach2010} find combinations of different model forecasts outperform the historical average on a consistent basis over time. Variable selection is an additional insight that requires spike-and-slab priors (see \cite{ishwaran2005spike}) within the Bayesian MCMC. Using the shrinkage regularization, we can learn about the prior influence at the regularization-parameter cutoffs where variable selection occurs.
We explore the equity-premium predictability with many economic variables in a joint dynamics from 1952 Quarter 1 to 2015 Quarter 4. Figure \ref{Fig2} plots the estimation summaries of the multiple-predictor model and illustrates the power of regularization for high-dimensional data. Similarly, we regularize the AR coefficients for the top two plots and the variance-covariance matrix for the bottom two plots, respectively. The top-left plot shows how the predicted values vary over a wide range of tuning parameters. We see sequentially incorporating the belief of predictability lowers the predicted value of S\&P 500 returns for 2015 Quarter 4 from positive to negative. The top-right plot shows the changes in predictor existence and the strength for the predictor excess stock returns. The property of sparsity allows us to observe the signal strength sequentially through regularization. We find the top three predictors are the book-to-market ratio, CAY (consumption, wealth, income ratio), and T-bill rates.
The bottom two plots show a regularization path for regularizing the variance-covariance matrix without regularization on the AR coefficients. The bottom-left panel shows correlations between predictors and returns for the covariance matrix. The top three predictors that have the most correlated shock with returns are the book-to-market ratio, earning-price ratio, and long-term rate of return. The bottom-right panel plots the one-step-ahead orthogonalized impulse response from predictors to returns. The top three predictors that have largest impulse response to returns are dividend yield, dividend payout ratio, and earnings-price ratio. Also, given that all impulse-response plots are flat, the uncorrelated-shock assumption does not affect the study of the impulse response.
\subsubsection{Market-Timing Strategy \label{sec413}}
\cite{Samuelson1969} and \cite{Merton1969} find an investor with power utility wants to hold a fixed portfolio of stocks and bonds when facing i.i.d. Market returns. If the excess market return is predictable, one can perform a market-timing strategy to change his fixed portfolio split. Therefore, implementing a market-timing strategy can test the out-of-sample performance of our regularization strategies over the buy-and-hold strategy. We also provide three strategies to check the robustness: the most regularized forecast (historical moving average), the optimally regularized one-step-ahead forecast, and the least regularized VAR(1) forecast.
Admittedly, all return forecasts should be considered pseudo-out-of-sample in our analysis, because we recursively re-estimate the model and then perform prediction. Previous studies about market-timing strategies do not consider the impact of taxes but might consider a fixed transaction cost. We consider neither of the tax impact and transaction cost in our analysis because we only update the portfolio quarterly and limit the portfolio change to no more than 50\%. For example, I have a portfolio with value \$1. The maximum change we can make in this quarter is either sell \$0.5 S\&P 500 and buy \$1 risk-free rate or sell \$0.5 risk-free rate and buy \$1 S\&P 500. Also, we restrict short selling in this simple exercise.
In the market-timing exercise, we look for the one-step-ahead market-return forecast and update the mean-variance optimal portfolio between the stock and risk-free rate every quarter. To avoid look-ahead bias, we fix the optimal regularization tuning parameter that is estimated using the training data from 1970 Quarter 1 to 1989 Quarter 4. Therefore, setting the optimal regularization amounts for ($\gamma$, $\lambda$) is one way to perform out-of-sample testing. For every quarter from 1990 to 2015 in the testing data, we use a moving training period of the previous 20 years (80 quarters) to re-estimate the regularized model and obtain the one-quarter ahead forecast for equity premium. Therefore, parameters are then re-estimated for each forecasting model, and the market-timing strategy is performed recursively.
With a maximum 50\% monthly portfolio change, we update the stock-bond split to the mean-variance optimal level quarterly based on the model prediction and sample variances. We do not model the portfolio variance but calculate the sample variance using the same training period (80 quarters). The maximum 50\% change might be an unrealistic assumption, but our purpose is to adapt a smooth loss function without considering transaction cost to compare the performances of three approaches. We also compare all three strategies with the buy-and-hold strategy, the fixed-split portfolio, with the same starting stock and bond shares.
The top two panels of Figure \ref{Fig3a} are the one-quarter-ahead returns from both single- and multiple-predictor models using predictive cross-validation. The least regularized prediction corresponds to the VAR prediction, and the most regularized prediction corresponds to the 80-quarter moving average. Though the range of return forecasts are mostly nonzero from 1990 to 2015, we find the forecasts are not robust to the regularization specification. We find one interesting pattern--that, in the multiple-predictor plot, the one-step-ahead optimal forecast has a high co-movement with the VAR forecast during the periods of the internet bubble and 2008 financial crisis. In other periods, the one-step-ahead optimal forecast almost overlaps with the moving average forecast.
The bottom two panels of Figure \ref{Fig3a} plot the wealth evolution of one dollar invested in a market-timing strategy using three procedures. The lines are the cumulative returns for the quarterly updated mean-variance efficient portfolios using both single- and multiple-predictor models. All procedures are updated using a rolling window of 80 quarters' observations. We also plot the performance of the buy-and-hold portfolio for comparison. We find substantial evidence of return predictability that all three strategies outperform the buy-and-hold portfolio.
Table \ref{Tab2} lists the portfolio return distribution and finds the portfolio performances are sensitive to the regularization specification. The portfolio return distributions match the bias-variance intuition that the least regularized forecasts (VAR) tend to have a large range and possibly a smaller bias, whereas the most regularized forecast (moving average) has the lowest standard error. In the period from 1990 to 2015, the moving-average strategy has the best performance in this period, followed by the one-step-ahead optimal strategy.
For a robustness check, we also repeat the same exercise in Figure \ref{Fig3b} using a modified AIC as the model-selection criterion, which is not necessarily optimal for the one-step-ahead forecast. The penalized model size is the number of predictors selected. We see similar time-series patterns of all three one-step-ahead forecasts as well as the non-sensitivity. We also see all three strategies outperform the fixed-split portfolio buy-and-hold strategy. However, we find a strong performance of the optimal strategy using multiple predictors. In the recent period after the 2008 financial crisis, this strategy's strong outperformance supports the evidence of using some predictors such as book-to-market ratio, CAY, and T-bill rates.
\subsection{Bond Premia Prediction \label{sec42}}
Since the failure of the expectations hypothesis, the literature focus on forecasting the variation in one-year expected excess returns for bonds of multiple maturities. \cite{ludvigson2009macro} build an empirical linkage between cyclical fluctuations in excess bond returns and macroeconomic fundamentals in a dynamic factor model. They use a small number of principal components instead of observed macroeconomic predictors in the predictive regressions and find these latent factors associated with real economic activity have significant predictive power beyond financial predictors.
Our exercise investigates the sensitivity of their findings with special attention paid to the joint prediction sensitivity. The factors that explain most of the variation on the right-hand side need not be the same as the factors most important for predicting the left-hand side. For the same robustness check, \cite{ludvigson2010factor} estimate a dynamic factor model for each of the eight subgroups using a Bayesian procedure, which can also be quickly implemented using our procedure.
Let $rx_{t+1}$ denote the continuously compounded (log) excess return on an n-year discount bond in period t + 1. We study the log 1-year holding period returns for two- to five- bond over the log yield of one-year bond from January 1964 to December 2003. Below is the SUR system for the joint prediction. For $rx_{t+1}^\intercal = \bmat{rx_{t+1,1}^\intercal, \cdots, rx_{t+1,4}^\intercal} $,
\[ rx_{t+1} = X_tB + E_{t+1}, \]
where $X_{t} = \mbox{diag}\{X_{t,1}, \cdots, X_{t,4}\}$, $B^\intercal = \bmat{\beta_1^\intercal, \cdots, \beta_4^\intercal}$, and $E_{t+1} = \bmat{\epsilon_{t+1,1}^\intercal, \cdots, \epsilon_{t+1,4}^\intercal} \sim \mathcal{N}(0, \Sigma \otimes I)$.
The SUR implementation is the same as the VAR system and we stack equations to form a univariate linear model. The difference is, in addition to the standard Lasso and Ridge regularization on $B$, we can apply a group regularization in the perspective of \cite{yuan2006model}. This group regularization can be implemented as a multivariate gaussian model in the package [\texttt{glmnet}]. An attractive property is the group factor selection from the group Lasso regularization, then we can see the common macro predictors in excess returns for bonds of multiple maturities.
\[ B^\star = \arg\!\min_{B} l(B \mid \Sigma) + \lambda \sum^p_{k=1} \sum^4_{j=1}\left|\beta_{j,k}\right|. \]
We have two models for one-month ahead prediction comparison. The first model includes the Cochrane-Piazzesi factor, a linear combination of five forward spreads, plus eight principal components from \cite{ludvigson2009macro} as well as two single factors constructed as a linear combination of five and six estimated factors. The first model includes the Cochrane-Piazzesi factor and a panel of 131 monthly macroeconomic time series. We show the regularization path for SUR forecasts in Figure \ref{Fig4} and the group regularized SUR forecasts in Figure \ref{Fig5}. In Figure \ref{Fig6}, we show the regularization path for predictor loadings in the first 11-predictor model.
We can see the significant variation for predicted value for the Dec. 2003. In both Figure \ref{Fig4} and \ref{Fig5}, we can see the sensitive change for prediction over a wide range of tuning parameters. Though the 11-predictor model has a smoother prediction over the 132-predictor model, the predicted values for the same month are entirely different for all four bonds in both SUR and group regularized SUR models. The latent factor extraction does not seem a robust approach concerning different priors, and useful macro information could be lost due to the ad-hoc decision in dimension reduction (or change of the shrinkage prior hyper-parameter).
Furthermore, in the bottom two plots, we can see it is more efficient to study the prediction in an SUR framework because the correlations of the shocks among four bonds are high with a low penalty. After hundreds of predictors, the cross-sectional correlations among bonds are still high. With the Cochrane-Piazzesi factor and macroeconomic series, there are still considerable co-movement in the regularization path for predicted values and shock correlations.
However, in the SUR prediction, the predictive factors have different strengths and existence for various bonds. The top three predictors stand out include f2 (2nd PC), CP (Cochrane-Piazzesi), and F6 (the single factor constructed as a linear combination of six estimated factors in \cite{ludvigson2009macro}). An attractive feature for the group regularized SUR prediction is the common factor selection in the cross-sectional of bond excess returns. The regularization path is similar to every bond with the top two predictors f2 and F6. Therefore, we can see the supervised learning is robust for a different penalty or prior distribution specification for bond prediction.
\section{Discussion \label{sec5}}
By exploiting the fundamental duality between regularization penalties and prior distributions, we provide a MAP approach that jointly regularizes both expected values and variance-covariance matrices for the high-dimensional VAR and SUR systems. Also, we use an iterative proximal algorithm that solves two convex subproblems on the predictor coefficients and the variance-covariance matrix for maximizing the posterior mode or penalized likelihood. Moreover, for the curse of dimensionality about the variance-covariance matrix, we introduce a Lasso-tilted inverse Wishart prior for regularization.
Our regularization approach offers several computational and empirical advantages, which include MAP's computational convenience over traditional MCMC procedures, the regularization-path plots for prior sensitivity analysis to forecasting power, and the possibility of building a high-dimension VAR for the model uncertainty and feedback effects. Furthermore, a regularization penalty using shrinkage priors, such as double-exponential distribution, provides many new empirical insights of variable selection in the Bayesian predictive analysis.
In the equity-premium prediction example, we demonstrate the change in out-of-sample prediction performance and orthogonal impulse response due to the change of specification in shrinkage priors. We find the risk premium forecasts are sensitive to the regularization penalty or the prior hyper-parameters in the Bayesian language. We also find significant predictability of excess S\&P 500 returns only using book-to-market ratio, CAY, and T-bill rates when implementing the market-timing strategies.
The trade-off is we obtain only a posterior-mode prediction instead of the full posterior distribution. Consequently, one caveat is we do not fully account for parameter uncertainty as in \cite{Kandel1996} and \cite{Barberis2000}. Although our results are quantitatively similar to their full Bayesian analysis, our regularized estimation can be computationally faster than MCMC procedures. Moreover, we can easily model the parameter sparsity in our Lasso specification, and we provide full prior sensitivity analysis rather having to specify the prior hyper-parameters.
Our approach builds on previously underexploited relationships between prior hyper-parameter selection and out-of-sample prediction power. In empirical finance and macroeconomics, most predictors lack statistical and even economic significance. For example, \cite{feng2017taming} provide a post-selection inference method to tame the zoo of factors in the cross-sectional asset pricing literature. For these problems involving large variable selection, our regularization procedure provides a lens to link the model uncertainty and prediction power. One possible direction is to apply the SUR system in the cross-sectional returns to extract common factors.
When implementing the elastic net package for variable selection, statisticians usually standardized the variables into the same scale. Otherwise, predictors with higher variance have smaller coefficients and therefore less penalization. It is equivalent to set a vector of heterogeneous hyper-parameter $\{\lambda_p\}$ of the ``informative" prior distribution for every predictor p. In our empirical examples, we follow the economic literature instead and do not standardize the predictors to have a direct comparison with the current empirical findings. Researchers use our regularization method should be aware of this variable selection issue.
In the bond prediction study, we find a significant information loss for the ad-hoc decision in dimension reduction technique. We find the predictions from their latent factor models to be sensitive to prior specifications. However, the shrinkage prior for predictor selection is robust for the penalty or prior distribution specification. The SUR system is another underexploited model in the empirical finance literature to explore the cross-sectional signals. There are some directions possible for future research, including the time-varying specification for the existence and strength of common predictors in the Bayesian regularization framework.
\begin{table}[]
\begin{center}
\caption{Predictor Description. \label{Tab0}}
\resizebox{\textwidth}{!}{
\begin{tabular}{@{}ll@{}}
\toprule
Predictor & Description \\ \midrule
Dividend Yield & Difference between the log of dividends and the log of lagged prices. \\
Earning Price Ratio & Difference between the log of earnings and the log of prices. \\
Book to Market Ratio & Ratio of book value to market value for the Dow Jones Industrial Average. \\
Dividend Payout Ratio & Difference between the log of dividends and the log of earnings. \\
T-bill rates & 3- Month Treasury Bill \\
Long Term Rate of Return & Long term yield on government bonds. \\
Default Return Spread & Difference between long-term corporate and government bond returns. \\
Investment to Capital Ratio & Ratio of aggregate investment to aggregate capital for the whole economy. \\
CAY & Consumption, wealth, income ratio \\ \bottomrule
\end{tabular}
}
\end{center}
\small
The predictor variables used in Section \ref{sec41} are defined above. Full details of variable definitions and sources are given in \cite{Welch2008}. Our quarterly data is from 1952 to 2015.
\end{table}
\begin{table}
\begin{center}
\caption{Comparison between Bayesian Analysis and Regularization.\label{Tab1}}
\begin{tabular}{cccccc}
\hline
& \multicolumn{2}{c}{Barberis} & & \multicolumn{2}{c}{Regularization} \\ \hline
\multirow{6}{*}{1952-2015} & a & B & & \multicolumn{2}{c}{$\beta$} \\ \cline{2-3}\cline{5-6}
& 6.764e-02 & 1.656e-2 & & 6.498e-02 & 1.263e-02 \\
& -6.167e-14 & 1.000 & & -5.590e-02 & 9.591e-01 \\ \cline{2-3}\cline{5-6}
& \multicolumn{2}{c}{$\Sigma$} & & \multicolumn{2}{c}{$\Sigma$} \\ \cline{2-3}\cline{5-6}
& 6.191e-03 & -1.700e-18 & & 6.191e-03 & 1.173e-04 \\
& & 1.019e-27 & & & 6.158e-03 \\ \hline
\multirow{6}{*}{1986-2015} & a & B & & \multicolumn{2}{c}{$\beta$} \\ \cline{2-3}\cline{5-6}
& 1.417e-01 & 3.357e-02 & & 4.145e-03 & 2.857e-02 \\
& 1.293e-13 & 1.000 & & 1.089e-02 & 9.1567e-01 \\ \cline{2-3}\cline{5-6}
& \multicolumn{2}{c}{$\Sigma$} & & \multicolumn{2}{c}{$\Sigma$} \\ \cline{2-3}\cline{5-6}
& 6.362e-03 & -2.197e-18 & & 6.397e-03 & 3.002e-05 \\
& & 5.110e-28 & & & 6.524e-03 \\ \cline{2-6}
\end{tabular}
\end{center}
\small This table shows the posterior estimates from the fully Bayesian analysis from \cite{Barberis2000} and our corresponding Ridge regularization MAP estimates. The first comparison is between B and $\beta_{[:, 2]}$, coefficients of D/P$_{t-1}$ on ret$_t$ and D/P$_{t}$. The second comparison is between the variance-covariance matrix.
\end{table}
\begin{table}
\begin{center}
\caption{Comparison for Model Prediction Distribution \label{Tab2}}
\begin{tabular}{@{}cccccccc@{}}
\toprule
& CV.S & AIC.S & VAR.S & CV.M & AIC.M & VAR.M & Average \\ \midrule
Min. & 0.05\% & -1.40\% & -2.35\% & -21.84\% & -21.65\% & -61.19\% & 0.05\% \\
1st Qu. & 0.78\% & 0.75\% & 0.21\% & 0.60\% & 0.20\% & -0.65\% & 0.80\% \\
Median & 1.21\% & 1.32\% & 1.06\% & 1.13\% & 1.26\% & 1.19\% & 1.24\% \\
Mean & 1.35\% & 1.24\% & 1.16\% & 1.33\% & 1.27\% & 0.76\% & 1.14\% \\
3rd Qu. & 1.80\% & 1.76\% & 2.23\% & 1.82\% & 2.81\% & 3.10\% & 1.49\% \\
Max. & 4.71\% & 2.63\% & 4.71\% & 10.35\% & 8.70\% & 12.72\% & 2.04\% \\
Sd. & 0.85\% & 0.74\% & 1.53\% & 2.86\% & 3.54\% & 7.22\% & 0.50\% \\
Sharpe Ratio & 1.58 & 1.67 & 0.76 & 0.47 & 0.36 & 0.11 & 2.26 \\ \bottomrule
\end{tabular}
\end{center}
\small This table shows the empirical distribution of model predictions for quarterly risk premium from 1990 to 2015 used in the market-timing analysis. All strategies (single or multiple predictors) are updated using data from a rolling window of 80 quarters. The single predictor approach uses dividend-price ratio. The regularization criteria include one-step ahead Predictive Cross-Validation and AIC selection. VAR is the regular VAR(1) estimation, and the average is the 80-quarter moving average.
\end{table}
\begin{figure}
\caption{Regularizing Single-Predictor Model.\label{Fig1}
\label{Fig1}
\end{figure}
\begin{figure}
\caption{Regularizing Multiple-Predictor Model.\label{Fig2}
\label{Fig2}
\end{figure}
\begin{figure}
\caption{Market Timing Strategy (Predictive Cross-Validation) \label{Fig3a}
\label{Fig3a}
\end{figure}
\begin{figure}
\caption{Market Timing Strategy (AIC Selection) \label{Fig3b}
\label{Fig3b}
\end{figure}
\begin{figure}
\caption{Regularized Seemingly Unrelated Regressions Forecasts \label{Fig4}
\label{Fig4}
\end{figure}
\begin{figure}
\caption{Group Regularized SUR Forecasts \label{Fig5}
\label{Fig5}
\end{figure}
\begin{figure}
\caption{Macro Factor Selection \label{Fig6}
\label{Fig6}
\end{figure}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{The simple harmonic urn}
\runtitle{The simple harmonic urn}
\begin{aug}
\author[A]{\fnms{Edward} \snm{Crane}\ead[label=e1]{[email protected]}},
\author[A]{\fnms{Nicholas} \snm{Georgiou}\ead[label=e2]{[email protected]}},
\author[B]{\fnms{Stanislav} \snm{Volkov}\corref{}\ead[label=e3]{[email protected]}},
\author[C]{\fnms{Andrew~R.} \snm{Wade}\ead[label=e4]{[email protected]}}
and
\author[A]{\fnms{Robert J.} \snm{Waters}\ead[label=e5]{[email protected]}}
\runauthor{E. Crane et al.}
\affiliation{University of Bristol, University of Bristol, University
of Bristol, University~of~Strathclyde and University of Bristol}
\address[A]{E. Crane\\
N. Georgiou\\
R. J. Waters\\
Heilbronn Institute for Mathematical Research\\
Department of Mathematics\\
University of Bristol\\
Bristol BS8~1TW\\
United Kingdom\\
\printead{e1}\\
\hphantom{E-mail: }\printead*{e2}\\
\hphantom{E-mail: }\printead*{e5}}
\address[B]{S. Volkov\\
Department of Mathematics\\
University of Bristol\\
Bristol BS8~1TW\\
United Kingdom\\
\printead{e3}\\}
\address[C]{A. R. Wade\\
Department of Mathematics and Statistics\\
University of Strathclyde\\
Glasgow G1 1XH\\
United Kingdom\\
\printead{e4}}
\end{aug}
{e}ceived{\smonth{11} \syear{2009}}
{e}vised{\smonth{8} \syear{2010}}
\begin{abstract}
We study a generalized P\'olya urn model with two types of ball.
If the drawn ball is red, it is replaced together with a black
ball, but if the drawn ball is black it is replaced and a red ball
is thrown out of the urn. When only black balls remain, the
roles of the colors are swapped and the process restarts. We
prove that the resulting Markov chain is transient but that if we
throw out a ball every time the colors swap, the process is
recurrent. We show that the embedded process obtained by observing
the number of balls in the urn at the swapping times has a scaling
limit that is essentially the square of a Bessel diffusion. We
consider an oriented percolation model naturally associated with
the urn process, and obtain detailed information about its
structure, showing that the open subgraph is an infinite tree with
a single end. We also study a natural continuous-time embedding of
the urn process that demonstrates the relation to the simple
harmonic oscillator; in this setting, our transience result
addresses an open problem in the recurrence theory of
two-dimensional linear birth and death processes due to Kesten and
Hutton. We obtain results on the area swept out by the process. We
make use of connections between the urn process and birth--death
processes, a uniform renewal process, the Eulerian numbers, and
Lamperti's problem on processes with asymptotically small drifts;
we prove some new results on some of these classical objects that
may be of independent interest. For instance, we give sharp new
asymptotics for the first two moments of the counting function of
the uniform renewal process. Finally, we discuss some related
models of independent interest, including a ``Poisson earthquakes''
Markov chain on the homeomorphisms of the plane.
\end{abstract}
\begin{keyword}[class=AMS]
\kwd[Primary ]{60J10}
\kwd[; secondary ]{60J25}
\kwd{60K05}
\kwd{60K35}.
\end{keyword}
\begin{keyword}
\kwd{Urn model}
\kwd{recurrence classification}
\kwd{oriented percolation}
\kwd{uniform renewal process}
\kwd{two-dimensional linear birth and death process}
\kwd{Bessel process}
\kwd{coupling}
\kwd{Eulerian numbers}.
\end{keyword}
\end{frontmatter}
\section{Introduction}
\lambdaabel{secIntro}
Urn models have a venerable history in probability theory, with
classical contributions
having been made by the Bernoullis and Laplace, among others. The
modern view of many urn
models is as prototypical reinforced stochastic processes. Classical
urn schemes
were often employed as ``thought experiments'' in which to frame
statistical questions; as stochastic
processes, urn models have wide-ranging applications in economics, the
physical sciences,
and statistics. There is a large literature on urn
models and their applications---see, for example, the monographs
\cite{JK,Mahmoud} and the surveys \cite{BK,Review}---and some important contributions have been made in the last few
years: see, for example, \cite{Ja,Fl}.
A generalized P\'olya\ urn with 2 types of ball, or 2 colors, is a
discrete-time Markov chain $(X_n,Y_n)_{n
\in{\mathbb{Z}}_+}$ on ${\mathbb{Z}}_+^2$, where ${\mathbb{Z}}_+ :=\{0,1,2,\lambdadots\}$. The
possible transitions of the chain are specified
by a $2 \times2$ \textit{reinforcement matrix} $A = (a_{ij})_{i,j=1}^2$
and the transition probabilities depend on the current state:
\begin{eqnarray}\lambdaabel{eqtrprob}
{\mathbb{P}}\bigl((X_{n+1},Y_{n+1})=(X_n+a_{11},Y_n+a_{12}) \bigr) &=& \frac
{X_n}{X_n+Y_n}, \nonumber\\[-8pt]\\[-8pt]
{\mathbb{P}}\bigl((X_{n+1},Y_{n+1})=(X_n+a_{21},Y_n+a_{22}) \bigr) &=& \frac
{Y_n}{X_n+Y_n}.\nonumber
\end{eqnarray}
This process can be viewed as an urn which at time $n$ contains $X_n$
red balls and $Y_n$ black balls. At each
stage, a ball is drawn from the urn at random, and then returned
together with $a_{i1}$ red balls and
$a_{i2}$ black balls, where $i=1$ if the chosen ball is red and $i=2$
if it is black.
A fundamental problem is to study the long-term behavior of
$(X_n,Y_n)$, defined by ({e}f{eqtrprob}),
or some function
thereof, such as the fraction of red balls~$X_n/\allowbreak(X_n+Y_n)$. In many cases,
coarse asymptotics for such
quantities are gover\-ned
by the eigenvalues of the reinforcement matrix $A$ (see, e.g., \cite{AK} or \cite{AN}, Section V.9).
However, there are some interesting special cases (see, e.g.,~\cite{PV1999}), and analysis of
finer behavior is in several cases still an open problem.
A large body of asymptotic theory is known under
various conditions on~$A$ and its eigenvalues.
Often it is assumed
that all $a_{ij}\ge0$, for example, $A=$
$ \bigl[
{{1 \atop 0} \enskip {0 \atop 1}}
\bigr]$ specifies the \textit{standard P\'olya\ urn},
while $A= \bigl[
{{a\atop b }\enskip {b \atop a}}
\bigr]$ with $a, b
>0$ specifies a \textit{Friedman urn}.
In general, the entries $a_{ij}$ may be negative, meaning that balls
can be
thrown away as well as added, but nevertheless in the literature \emph
{tenability} is usually imposed. This
is the condition that regardless of the stochastic path taken by the process,
it is never required to remove a ball of a color not currently present
in the urn.
For example, the \textit{Ehrenfest
urn},
which models the diffusion of a gas between two chambers of a box,
is tenable despite its reinforcement matrix $ \bigl[
{{ -1 \atop 1 }\enskip{ 1 \atop -1}}
\bigr]$ having some negative entries.
Departing from tenability, the \textit{OK Corral model} is the 2-color
urn with reinforcement matrix
$ \bigl[
{{0 \atop -1}\enskip{ -1 \atop 0}}
\bigr]$. This\vspace*{1pt} model for destructive competition was studied by Williams
and McIlroy \cite{WM} and Kingman \cite{K99} (and
earlier as a~stochastic version of Lanchester's combat model; see,
e.g., \cite{watson}
and references therein). Kingman and Volkov \cite{KV} showed that the OK
Corral model can be viewed as a time-reversed Friedman urn with $a=0$
and $b=1$.
In this paper, we will study the 2-color urn model with reinforcement matrix
\begin{equation}
\lambdaabel{eqharmm}
A = \lambdaeft[
\matrix {0 & 1 \cr -1 & 0
}
\right].
\end{equation}
To reiterate the urn model, at each time period
we draw a ball at random from the urn; if it is red, we replace it
and add an additional black ball, if it is black we replace it and
throw out a red ball.
The eigenvalues of $A$ are
$\pm i$, corresponding to the ordinary differential equation
$\dot{{\bf v}}=A{\bf v}$,
which governs the phase diagram of the simple harmonic oscillator. This
explains the name \textit{simple harmonic urn}.
Na\"{\i}vely, one might hope
that the behavior of the Markov chain is closely related to the paths
in the phase diagram.
We will see that it is, but that the exact behavior is somewhat more subtle.
\section{Exact formulation of the model and main results}
\lambdaabel{secResults}
\subsection{The simple harmonic urn process}
The definition of the process~given by the transition probabilities
({e}f{eqtrprob}) and
the matrix ({e}f{eqharmm}) only makes sense for $X_n,Y_n\ge0$;
however, it is easy to see that
almost surely (a.s.) $X_n < 0$ eventually.
Therefore, we reformulate the process
$(X_n,Y_n)$ rigorously as follows.
For $z_0 \in{\mathbb{N}}:= \{1,2,\lambdadots\}$ take $(X_0,Y_0) = (z_0,0)$;
we start on the positive $x$-axis for convenience but
the choice of initial state does not affect any of our asymptotic
results.
For $n \in{\mathbb{Z}}_+$, given $(X_n,Y_n) = (x,y) \in{\mathbb{Z}}^2 \setminus\{
(0,0)\}$,
we define the transition law of the process by
\begin{equation}
\lambdaabel{tranprobs}
(X_{n+1}, Y_{n+1} )=
\cases{
\bigl(x,y+\operatorname{sgn}(x)\bigr), &\quad with probability $\displaystyle
\frac{|x|}{|x|+|y|}$,\cr
\bigl(x- \operatorname{sgn}(y),y\bigr), &\quad with probability $\displaystyle \frac{|y|}{|x|+|y|}$,
}
\end{equation}
where $\operatorname{sgn}(x)=-1,0,1$ if $x<0$, $x=0$, $x>0$, respectively.
The process~$(X_n,\allowbreak Y_n)_{n \in{\mathbb{Z}}_+}$
is an irreducible Markov chain with state-space
${\mathbb{Z}}^2 \setminus\{ (0,0) \}$.
See Figure~{e}f{Fig2} for some simulated trajectories of the simple harmonic urn
process.
\begin{figure}
\caption{Two sample trajectories of the simple harmonic urn process,
starting at $(50,0)$ and running for about $600$ steps
(\textup{left}
\end{figure}
Let $\nu_0:=0$, and recursively define stopping times
\[
\nu_k := \min\{n>\nu_{k-1}\dvtx X_n Y_n =0\}\qquad (k \in{\mathbb{N}}),
\]
where throughout the paper we adopt the usual convention $\min
\varnothing\,{:=}\,\infty$.~Thus, $(\nu_k)_{k \in{\mathbb{N}}}$ is the sequence
of times at which the process visits one of the axes.
It is easy to see that every $\nu_k$ is almost surely finite.
Moreover, by
construction, the process $(X_{\nu_k},Y_{\nu_k})_{k \in{\mathbb{N}}}$ visits
in cyclic (anticlockwise)
order
the half-lines \mbox{$\{y>0\}$,} $\{x<0\}$, $\{y<0\}$, $\{x>0\}$.
It is natural
(and fruitful) to consider the embedded process $(Z_k)_{k \in{\mathbb{Z}}_+}$ obtained
by taking
$Z_0 := z_0$ and $Z_k := | X_{\nu_k} | + | Y_{\nu_k}|$ ($k \in{\mathbb{N}}$).
If $(X_n,Y_n)$ is viewed
as a random walk on ${\mathbb{Z}}^2$, the process $Z_k$
is the embedded process of the
distances from $0$ at the instances of hitting the axes.
To interpret the process $(X_n,Y_n)$ as the urn model
described in Section {e}f{secIntro},
we need a slight modification to the description there.
Starting with $z_0$ red balls, we run the process as described in
Section {e}f{secIntro},
so the process traverses the first quadrant via an up/left path
until the red balls run out (i.e., we first hit the half-line $\{ y >0\}
$). Now we interchange the
roles of the red and black balls, and we still use $y$ to count the
black balls, but we switch to using $-x$ to count the number of red balls.
Now the process traverses the second quadrant via a left/down path
until the
black balls run out, and so on. In the urn model, $Z_k$ is the number
of balls remaining in the urn
when the urn becomes monochromatic for the $k$th time ($k \in{\mathbb{N}}$).
The strong Markov property and the transition law of $(X_n,Y_n)$
imply that~$Z_k$ is an irreducible Markov chain on ${\mathbb{N}}$.
Since our two Markov chains just described are irreducible, there
is the usual recurrence/transience dichotomy, in that either
the process is recurrent, meaning that with probability $1$ it returns
infinitely
often to any finite subset of the state space, or it is transient,
meaning that with probability $1$ it eventually escapes to infinity.
Our main question is whether the process $Z_k$
is recurrent or transient. It is easy to see that, by the nature of the
embedding, this also
determines whether the urn model $(X_n, Y_n)$ is recurrent or transient.
\begin{thm}
\lambdaabel{th:main1}
$\!\!\!\!\!$The process $Z_k$ is transient; hence so is the process~$(X_n, Y_n)$.
\end{thm}
Exploiting a connection between the increments of the
process $Z_k$ and a~renewal process whose inter-arrival times
are uniform on $(0,1)$ will enable us to prove the following basic\vadjust{\goodbreak}
result.
\eject
\begin{thm}
\lambdaabel{th:zinc}
Let $n \in{\mathbb{N}}$. Then
\begin{equation}
\lambdaabel{zinc}
{\mathbb{E}}[ Z_{k+1} \mid Z_k = n ]
= n + \tfrac{2}{3} + O ( {e}^{\alpha_1 n} )
\end{equation}
as $n \to\infty$, where $\alpha_1 + \beta_1 i = -(2.088843\dots) +
(7.461489\dots)i$
is a root of $\lambdaambda- 1 + {e}^{-\lambdaambda} = 0$.
\end{thm}
The error term in ({e}f{zinc}) is sharp, and we obtain it from
new (sharp) asymptotics for the uniform renewal process:
see Lemma {e}f{lem: asymptotic} and Corollary {e}f{uniform},
which improve on known results. To prove Theorem {e}f{th:main1},
we need more than Theorem {e}f{th:zinc}: we need to know about
the second moments of the increments of $Z_k$, amongst other things; see
Section {e}f{S:zinc}. In fact, we prove Theorem {e}f{th:main1}
using martingale arguments applied to $h(Z_k)$ for a well-chosen
function $h$; the analysis of the function $h(Z_k)$ rests
on a recurrence relation satisfied by the transition probabilities
of $Z_k$, which are related to the Eulerian numbers (see Section {e}f
{secRubin}).
\subsection{The leaky simple harmonic urn}
\lambdaabel{S:leaky}
In fact the transience demonstrated in Theorem {e}f{th:main1} is
rather delicate, as one can see by simulating the process. To
illustrate this, we consider a slight modification of the process,
which we call the \textit{leaky} simple harmonic urn. Suppose that
each time the roles of the colors are reversed, the addition of
the next ball of the new color causes one ball of the other
color to leak out of the urn; subsequently the usual simple
harmonic urn transition law applies. If the total number of balls
in the urn ever falls to one, then this modified rule causes the
urn to become monochromatic at the next step, and again it
contains only one ball. Thus, there will only be one ball in total
at all subsequent times, although it will alternate in color. We
will see that the system almost surely does reach this steady
state, and we obtain almost sharp tail bounds on the time that it
takes to do so. The leaky simple harmonic urn arises naturally in
the context of a percolation model associated to the simple
harmonic urn process, defined in Section {e}f{S: percolation}
below.
As we did for the simple harmonic urn, we will represent the leaky urn
by a Markov chain $(X'_n, Y'_n)$.
For this version of the model,
it turns out to be more convenient to start just above the axis;
we take $(X'_0,Y'_0)=(z_0,1)$, where $z_0 \in{\mathbb{N}}$.
The distribution of $(X'_{n+1},Y'_{n+1})$ depends only on $(X'_n, Y'_n)
= (x,y)$.
If $x y \neq0$, the transition law is the same as that of the simple harmonic
urn process. The difference is when $x = 0$ or $y=0$; then the transition
law is
\begin{eqnarray*}
( X'_{n+1} , Y'_{n+1} ) & = &\bigl( - \operatorname{sgn}(y) , y - \operatorname{sgn}(y)
\bigr) \qquad (x = 0), \\
( X'_{n+1} , Y'_{n+1} ) & = &\bigl( x - \operatorname{sgn}(x) , \operatorname{sgn}(x) \bigr)\qquad\hspace*{10pt} (y = 0) .
\end{eqnarray*}
Now $(X'_n,Y'_n)$ is a reducible Markov chain whose state-space
has two communicating classes, the closed class $\mathcal{C} = \{ (x,y)
\in{\mathbb{Z}}^2 \dvtx | x | + | y| = 1\}$
and the class $\{ (x,y) \in{\mathbb{Z}}^2 \dvtx |x| + |y| \geq2 \}$; if the
process enters the closed
class $\mathcal{C}$ it remains there for ever, cycling round the origin.
Let $\tau$ be the hitting time of the set $\mathcal{C}$, that is
\[
\tau:= \inf\{ n \in\mathbb{Z} \dvtx |X'_n| + |Y'_n| = 1 \} .
\]
\begin{thm}
\lambdaabel{th:leaky} For the leaky urn, ${\mathbb{P}}( \tau< \infty) =1$.
Moreover, for any $\varepsilon>0$, ${\mathbb{E}}[ \tau^{1-\varepsilon}] < \infty$ but ${\mathbb{E}}
[ \tau^{1+\varepsilon}] = \infty$.
\end{thm}
In contrast, Theorem {e}f{th:main1} implies that the analogue of
$\tau$ for the ordinary urn process has ${\mathbb{P}}(\tau= \infty) >0$ if
$z_0 \geq2$.
\subsection{The noisy simple harmonic urn}
\lambdaabel{S:noisy}
In view of Theorems {e}f{th:main1} and {e}f{th:leaky}, it is
natural to ask about the properties of the hitting time $\tau$ if
at the time when the balls of one color run out we only discard a
ball of the other color with some probability $p \in(0,1)$. For
which $p$ is $\tau$ a.s. finite? (Answer: for $p \geq1/3$; see
Corollary {e}f{noisyclassification} below.)
We consider the following natural generalization of the model
specified by~({e}f{tranprobs}) in order to probe more
precisely the recurrence/transience transition. We call this
generalization the \textit{noisy} simple harmonic urn process.
In a~sense that we will describe,
this model includes the leaky urn and also the
intermittent leaky urn mentioned at the start of this section.
The basic idea is to throw out (or add)
a random number of balls at each
time we are at an axis, generalizing the idea of the leaky urn.
It is more convenient here to work with irreducible Markov chains,
so we introduce a ``barrier'' for our process. We now describe the model
precisely.
Let $\kappa,\kappa_1,\kappa_2,\lambdadots$ be a sequence of
i.i.d. ${\mathbb{Z}}$-valued
random variables such that
\begin{equation}
\lambdaabel{kappabound}
{\mathbb{E}}\bigl[ {e}^{\lambdaambda|\kappa|} \bigr] < \infty
\end{equation}
for some $\lambdaambda>0$, so in particular ${\mathbb{E}}[\kappa]$ is finite.
We now define the Markov chain $({\tilde X}_n,{\tilde Y}_n)_{n \in{\mathbb{Z}}_+}$ for the
noisy urn process. As for the leaky urn,
we start one step above the axis:
let $z_0 \in{\mathbb{N}}$, and take $({\tilde X}_0,{\tilde Y}_0) = (z_0,1)$.
For $n \in{\mathbb{Z}}_+$, given $({\tilde X}_n,{\tilde Y}_n) = (x,y) \in{\mathbb{Z}}^2 \setminus\{
(0,0)\}$,
we define the transition law as follows. If $xy \neq0$, then
\[
({\tilde X}_{n+1}, {\tilde Y}_{n+1} )=
\cases{
\bigl(x,y+\operatorname{sgn}(x)\bigr), &\quad with probability $\displaystyle
\frac{|x|}{|x|+|y|}$,\cr
\bigl(x- \operatorname{sgn}(y),y\bigr), &\quad with probability $\displaystyle \frac{|y|}{|x|+|y|}$,
}
\]
while if $x = 0$ or $y=0$ we have
\begin{eqnarray*}
({\tilde X}_{n+1} , {\tilde Y}_{n+1} ) & = & \bigl( - \operatorname{sgn}(y), \operatorname{sgn}(y) \max(1, |y| -
\kappa_n) \bigr)\qquad (x=0), \\
({\tilde X}_{n+1} , {\tilde Y}_{n+1} ) & = & \bigl(\operatorname{sgn}(x) \max(1, |x| - \kappa_n),
\operatorname{sgn}(x) \bigr)\qquad \hspace*{10pt} (y=0).
\end{eqnarray*}
In other words, the transition law is the same
as ({e}f{tranprobs}) except when
the process is on an axis at time $n$, in which case
instead of just moving one step away in the anticlockwise
perpendicular direction
it also moves an additional distance $\kappa_n$
parallel to the axis toward the origin
(stopping distance 1 away if it would otherwise
reach the next axis or overshoot).
Then $({\tilde X}_n, {\tilde Y}_n)_{n \in{\mathbb{Z}}_+}$ is an irreducible Markov
chain on ${\mathbb{Z}}^2 \setminus\{(0,0)\}$. The case where
${\mathbb{P}}( \kappa= 0 ) = 1$ corresponds
to the original process $(X_n, Y_n)$ starting one unit later in time.
A fundamental random variable is the first passage time
to within distance~1 of the origin:
\[
\tau:= \min\{ n \in{\mathbb{Z}}_+ \dvtx | \tilde X_n | + | \tilde Y_n | = 1 \}=
\min\{ n \in{\mathbb{Z}}_+ \dvtx ({\tilde X}_n, {\tilde Y}_n) \in\mathcal{C} \} .
\]
Define a sequence of stopping times $\tilde\nu_k$
by setting $\tilde\nu_0 := -1$ and for $k \in{\mathbb{N}}$,
\[
\tilde\nu_k := \min\{ n > \tilde\nu_{k-1} \dvtx {\tilde X}_n {\tilde Y}_n = 0 \} .
\]
As an analogue of $Z_k$, set ${\tilde Z}_0 := z_0$ and
for $k \in{\mathbb{N}}$ define
\[
{\tilde Z}_k := \max\{ | {\tilde X}_{1+\tilde\nu_k }| , | {\tilde Y}_{1+\tilde\nu_k
} | \} =
| {\tilde X}_{1+\tilde\nu_k }| + | {\tilde Y}_{1+\tilde\nu_k } | -1 ;
\]
then $({\tilde Z}_k)_{k \in{\mathbb{Z}}_+}$ is an irreducible Markov chain
on ${\mathbb{N}}$.
Define the return-time to
the state $1$ by
\begin{equation}
\lambdaabel{returntime}
\tau_q := \min\{ k \in{\mathbb{N}}\dvtx {\tilde Z}_k = 1 \},
\end{equation}
where the subscript $q$ signifies the fact that a time unit is one
traversal of a~quadrant here. By our embedding, $\tau= \tilde\nu
_{\tau_q}$.
Note that in the case ${\mathbb{P}}( \kappa=0)=1$, $({\tilde Z}_k)_{k \in{\mathbb{Z}}_+}$ has
the same distribution as the
original $(Z_k)_{ k \in{\mathbb{Z}}_+}$. The noisy urn with ${\mathbb{P}}(\kappa=1)=1$
coincides with the leaky urn
described in Section {e}f{S:leaky} up until the time $\tau$ (at which
point the leaky urn becomes
trapped in $\mathcal{C}$). Similarly, the embedded process ${\tilde Z}_k$ with
${\mathbb{P}}(\kappa=1)=1$ coincides
with the process of distances from the origin of the leaky urn at the
times that it visits the axes,
up until time $\tau_q$ (at which point the leaky urn remains at
distance $1$ forever). Thus, in the
${\mathbb{P}}(\kappa=1)=1$ cases of all the results that follow in this
section, $\tau$ and $\tau_q$ can be
taken to be defined in terms of the leaky urn $(X'_n,Y'_n)$.
The next result thus includes Theorem {e}f{th:main1}
and the first part of Theorem~{e}f{th:leaky}
as special cases.
\begin{thm}
\lambdaabel{thm:mixed}
Suppose that $\kappa$ satisfies ({e}f{kappabound}). Then
the process ${\tilde Z}_k$ is:
\begin{itemize}[(iii)]
\item[(i)] transient if ${\mathbb{E}}[\kappa] < 1/3$;
\item[(ii)] null-recurrent if $1/3 \lambdaeq{\mathbb{E}}[\kappa] \lambdaeq2/3$;
\item[(iii)] positive-recurrent if ${\mathbb{E}}[\kappa] > 2/3$.
\end{itemize}
\end{thm}
Of course, part (i) means that ${\mathbb{P}}(\tau_q < \infty)<1$,
part (ii) that ${\mathbb{P}}(\tau_q < \infty)=1$ but ${\mathbb{E}}[\tau_q] = \infty$,
and part (iii) that ${\mathbb{E}}[ \tau_q ]< \infty$. We can in fact obtain
more information
about the tails of $\tau_q$.
\begin{thm}
\lambdaabel{thm:moments}
Suppose that $\kappa$ satisfies ({e}f{kappabound})
and ${\mathbb{E}}[\kappa] \geq1/3$. Then
${\mathbb{E}}[ \tau_q^p ] < \infty$ for $p < 3 {\mathbb{E}}[\kappa] - 1$ and
${\mathbb{E}}[ \tau_q^p ] = \infty$ for $p > 3 {\mathbb{E}}[\kappa] - 1$.
\end{thm}
It should be possible, with some extra work, to show that ${\mathbb{E}}[
\tau_q^p ] = \infty$ when $p = 3 {\mathbb{E}}[\kappa] - 1$, using the
sharper results of \cite{ai} in place of those from \cite{aim}
that we use below in the proof of Theorem {e}f{thm:moments}.
In the recurrent case, it is of interest to obtain more detailed
results on the tail of~$\tau$
(note that there is a change of time between $\tau$ and $\tau_q$).
We obtain the following upper and lower bounds, which are close to sharp.
\begin{thm}
\lambdaabel{thm:moments2}
Suppose that $\kappa$ satisfies ({e}f{kappabound}) and ${\mathbb{E}}[\kappa]
\geq1/3$.\vspace*{1pt}
Then
${\mathbb{E}}[ \tau^p] < \infty$ for $p < \frac{3 {\mathbb{E}}[ \kappa] -1}{2}$ and
${\mathbb{E}}[ \tau^p] = \infty$ for $p > \frac{3 {\mathbb{E}}[ \kappa] -1}{2}$.
\end{thm}
Theorems {e}f{thm:mixed} and {e}f{thm:moments2} have
an immediate corollary for the noisy urn process $(\tilde X_n, \tilde Y_n)$.
\begin{corollary}
\lambdaabel{noisyclassification} Suppose that $\kappa$ satisfies
({e}f{kappabound}). The noisy simple harmonic urn process $(\tilde
X_n, \tilde Y_n)$ is recurrent if ${\mathbb{E}}[\kappa] \geq1/3$ and
transient if ${\mathbb{E}}[ \kappa] < 1/3$. Moreover, the process is
null-recurrent if $1/3 \lambdaeq{\mathbb{E}}[\kappa] < 1$ and positive-recurrent
if ${\mathbb{E}}[ \kappa] > 1$.
\end{corollary}
This result is close to sharp but leaves open the question of whether
the process
is null- or positive-recurrent when ${\mathbb{E}}[ \kappa]=1$ (we suspect the former).
We also study the distributional limiting behavior of
${\tilde Z}_k$ in the appropriate scaling regime when
${\mathbb{E}}[\kappa] < 2/3$. Again the case ${\mathbb{P}}(\kappa= 0) =1$ reduces
to the original~$Z_k$.
\begin{thm}
\lambdaabel{thm:difflimit}
Suppose that $\kappa$ satisfies ({e}f{kappabound})
and that ${\mathbb{E}}[\kappa] < 2/3$.
Let $(D_t)_{t \in[0,1]}$ be a diffusion process taking values in ${\mathbb{R}}
_+ := [0,\infty)$
with $D_0 = 0$ and
infinitesimal mean $\mu(x)$ and variance $\sigma^2(x)$ given for $x
\in{\mathbb{R}}_+$
by
\[
\mu(x) = \tfrac{2}{3} -{\mathbb{E}}[\kappa],\qquad \sigma^2 (x) = \tfrac{2}{3} x.
\]
Then as $k \to\infty$,
\[
( k^{-1} {\tilde Z}_{ k t} )_{t \in[0,1]} \to(D_t)_{t \in[0,1]} ,
\]
where the convergence is in the sense of finite-dimensional distributions.
Up to multiplication by a scalar,
$D_t$ is the square of a Bessel process with parameter $4 - 6 {\mathbb{E}}
[\kappa] > 0$.
\end{thm}
Since a Bessel process with parameter $\gamma\in{\mathbb{N}}$
has the same law as the norm of a $\gamma$-dimensional Brownian motion,
Theorem {e}f{thm:difflimit} says, for example,
that if ${\mathbb{E}}[\kappa] =0$ (e.g., for the original urn process)
the scaling limit of ${\tilde Z}_t$ is a scalar multiple of the norm-square
of $4$-dimensional Brownian motion,
while if ${\mathbb{E}}[ \kappa] = 1/2$
the scaling limit is a scalar multiple of the square
of one-dimensional Brownian motion.
To finish this section, consider the
\textit{area} swept out by the path of the noisy simple harmonic
urn on its first excursion (i.e., up to time $\tau$).
Additional motivation for studying this
random quantity is provided by the percolation model of Section~{e}f{S: percolation}. Formally, for $n \in{\mathbb{N}}$ let $T_n$ be the area
of the triangle with vertices $(0,0)$, $(\tilde X_{n-1}, \tilde Y_{n-1})$,
and $(\tilde X_{n}, \tilde Y_{n})$, and define $A := \sum_{n=1}^{\tau
} T_n$.
\begin{thm}
\lambdaabel{thm:area}
Suppose that $\kappa$ satisfies ({e}f{kappabound}).
\begin{itemize}[(ii)]
\item[(i)] Suppose that ${\mathbb{E}}[\kappa]< 1/3$. Then ${\mathbb{P}}( A = \infty) > 0$.
\item[(ii)] Suppose that ${\mathbb{E}}[\kappa]\ge1/3$. Then ${\mathbb{E}}[ A^p ] <
\infty$ for $p < \frac{3 {\mathbb{E}}[\kappa] -1}{3}$.
\end{itemize}
\end{thm}
In particular, part (ii) gives us information about the leaky urn
model, which corresponds to the case where ${\mathbb{P}}(\kappa=1)=1$, at
least up until the hitting time of the closed cycle; we can still
make sense of the area swept out by the leaky urn up to this
hitting time. We then have ${\mathbb{E}}[A^p] < \infty$ for $p<2/3$, a
result of significance for the percolation model of the next
section. We suspect that the bounds in Theorem~{e}f{thm:area}(ii) are
tight. We do not prove this but have the following result in the
case ${\mathbb{P}}(\kappa=1)=1$.
\begin{thm}
\lambdaabel{thm:areamean}
Suppose ${\mathbb{P}}( \kappa=1) =1$ (or equivalently take the leaky urn). Then
${\mathbb{E}}[ A] = \infty$.
\end{thm}
\subsection{A percolation model}\lambdaabel{S: percolation}
Associated to the simple harmonic urn is a~per\-colation model
which we describe in this section. The percolation model, as well as
being of
interest in its own right, couples many different instances
of the simple harmonic urn, and exhibits naturally an instance
of the leaky version of the urn in terms of the planar dual percolation
model. Our results
on the simple harmonic urn will enable us to establish
some interesting properties of the percolation model.
The simple harmonic urn can be viewed as a spatially inhomogeneous
random walk
on a directed graph whose vertices are ${\mathbb{Z}}^2 \setminus\{ (0,0)\}$; we
make this statement
more precise shortly.
In this section, we will view
the simple harmonic urn process not as a random path through a
predetermined directed graph
but as a deterministic path through a random directed graph.
To do this, it is helpful to consider a slightly larger state-space
which keeps track
of the number of times that the urn's path has wound around the origin.
We construct
this state-space as the vertex set of a graph $G$ that is embedded in
the Riemann surface
$\mathcal{R}$ of the complex logarithm, which is the universal cover of
$\mathbb{R}^2 \setminus\{(0,0)\}$.
To construct $G$, we take the usual square-grid lattice and delete the
vertex at the origin to obtain a graph on the vertex set $\mathbb
{Z}^2\setminus\{(0,0)\}$.
Make this into a directed graph by orienting each edge in the direction
of increasing argument;
the paths of the simple harmonic urn only
ever traverse edges in this direction. Leave undirected those edges
along any of the coordinate axes;
the paths of the simple harmonic urn never traverse these edges.
Finally, we let $G$ be the lift of this graph to the covering surfa\-ce~$\mathcal{R}$.
\begin{figure}
\caption{Simulated realizations of the simple harmonic urn
percolation model: on a single sheet of $\mathcal{R}
\end{figure}
We will interpret a path of the simple harmonic urn as the unique
oriented path from some starting vertex through a random subgraph $H$
of $G$.
For each vertex $v$ of $G$, the graph $H$ has precisely one of the
out-edges from $v$ that are in $G$. If the projection of $v$ to
$\mathbb{Z}^2$ is $(x,y)$,
then the graph $H$ contains the edge from $v$ that projects onto the
edge from $(x,y)$ to $(x-\operatorname{sgn}(y), y)$ with probability
$|y|/(|x|+|y|)$, and otherwise it contains
the edge from $v$ that projects onto the edge from $(x,y)$ to $(x, y +
\operatorname{sgn}(x))$. These choices are to be made independently for all vertices
$v$ of $G$.
In particular, $H$ does not have any edges that project onto either of
the coordinate axes.
The random directed graph $H$ is an oriented percolation model that
encodes a coupling of many different paths of the simple harmonic urn.
To make this precise, let $v_0$ be any vertex of $G$.
Then there is a unique oriented path $v_0, v_1, v_2, \dots$ through~$H$.
That is, $(v_i,v_{i+1})$ is an edge of $H$ for each $i \ge0$.
Let the projection of~$v_i$ from $\mathcal{R}$ to $\mathbb{R}^2$ be
the point $(X_i, Y_i)$.
Then the sequence $(X_i, Y_i)_{i=0}^{\infty}$ is a~sample of the
simple harmonic urn process.
If $w_0$ is another vertex of $G$, with unique oriented path $w_0, w_1,
w_2, \lambdadots,$ then its projection to~$\mathbb{Z}^2$ is also a sample
path of the simple harmonic urn process, but we will show (see Theorem
{e}f{thm:perc} below)
that with probability one the two paths eventually couple, which is to
say that there exist random finite $m \ge0$ and $n \ge0$ such that
for all $i \ge0$ we have $v_{i+m} = w_{i+n}$. Thus, the percolation
model encodes\vadjust{\goodbreak} many coalescing copies of the simple
harmonic urn process. Next, we show that it also encodes many copies of
the leaky urn of Section~{e}f{S:leaky}.
We construct another random graph $H'$ that is the dual percolation
model to $H$. We begin with the planar dual of the square-grid lattice,
which is another square-grid lattice with vertices at the points
$(m+1/2,n+1/2)$, $m,n \in\mathbb{Z}$. We orient all the edges in the
direction of decreasing argument, and lift to the covering surface
$\mathcal{R}$ to obtain the dual graph $G'$. Now let $H'$ be the directed
subgraph of $G'$ that consists of all those edges of $G'$ that do not
cross an edge of $H$. It turns out that $H'$ can be viewed as an
oriented percolation model that encodes a coupling of many different
paths of the leaky simple harmonic urn.
To explain this, we define a mapping ${\mathbb{P}}hi$ from the vertices of $G'$
to $\mathbb{Z}^2$. Let $(x,y)$ be the coordinates of the projection of
$v \in G'$ to the shifted square lattice $\mathbb{Z}^2 + (1/2,1/2)$. Then
\[
{\mathbb{P}}hi(v) = \bigl( x + \tfrac{1}{2} \operatorname{sgn} y, -\bigl(y - \tfrac{1}{2} \operatorname{sgn} x\bigr) \bigr) .
\]
Thus, we project from $\mathcal{R}$ to $\mathbb{R}^2$, move to the
nearest lattice point in the clockwise direction, and then reflect in
the $x$-axis.
If $v_0$ is any vertex of $H'$, there is a unique oriented path $v_0,
v_1, v_2, \dots$ through $H'$, this time winding clockwise. Take $v_0
= (z_0 - 1/2, 1/2)$.
A little thought shows that the sequence $(X'_i, Y'_i) = {\mathbb{P}}hi(v_i )$
has the distribution of the leaky simple harmonic urn process. This is
because the choice of edge in $H'$ from $v$ is determined by the choice
of edge in $H$ from the nearest point of $G$ in the clockwise direction.
The map ${\mathbb{P}}hi$ is not quite a graph homomorphism onto the square
lattice because of its behavior at the axes; for example, it
sends $(3\frac{1}{2},\frac{1}{2})$
and $(3\frac{1}{2}, -\frac{1}{2})$ to $(4,0)$ and $(3, 1)$, respectively.
The decrease of $1$ in the $x$-coordinate corresponds to the leaked
ball in the leaky urn model. If some $v_i$ has projection $(x_i, y_i)$
with $|x_i| + |y_i| = 1$,
then the same is true of all subsequent vertices in the path. This
corresponds to the closed class $\mathcal{C}$.
From results on our urn processes, we will deduce the following quite
subtle properties of the percolation model $H$.
Let $I(v)$ denote the number of vertices in the \textit{in-graph} of the
vertex $v$ in $H$, which is the subgraph of $H$ induced by all
vertices from which it is possible to reach $v$ by following an
oriented path.\vspace*{-3pt}
\begin{thm}
\lambdaabel{thm:perc} Almost surely, the random oriented graph $H$ is,
ignoring orientations, an infinite tree with a single
semi-infinite end in the out direction. In particular, for any
$v$, $I(v) < \infty$ a.s. and moreover ${\mathbb{E}}[ I(v)^p ] < \infty$
for any $p<2/3$; however, ${\mathbb{E}}[ I(v) ] =\infty$.
The dual graph $H'$ is also an infinite tree a.s., with a single
semi-infinite end in the out direction. It has a doubly-infinite
oriented path and the in-graph of any vertex not on this path is finite a.s.
\end{thm}
\subsection{A continuous-time fast embedding of the simple harmonic urn}
\lambdaabel{S: fast}
$\!\!\!$There is a\vadjust{\goodbreak} natural continuous-time embedding of the simple harmonic urn process.
Let $(A(t),B(t))_{t \in{\mathbb{R}}_+}$ be a $\mathbb{Z}^2$-valued
continuous-time Markov
chain with $A(0) = a_0$, $B(0) = b_0$, and transition rates
\begin{eqnarray*}
{\mathbb{P}}\bigl(A(t+ {d} t) = A(t) - \operatorname{sgn}(B(t))\bigr) & =& | B(t) | \,{d} t , \\
{\mathbb{P}}\bigl(B(t+ {d} t) = B(t) + \operatorname{sgn}(A(t))\bigr) & =& | A(t) | \,{d} t .
\end{eqnarray*}
Given that $(A(t),B(t)) = (a,b)$, the wait until the next jump after
time $t$ is
an exponential random variable with mean $1/(|a|+|b|)$.
The next jump is a~change in the first coordinate with probability
$|b|/(|a|+|b|)$, so the process considered at its sequence
of jump times does indeed follow the law of the simple harmonic urn.
Note that the process does not explode in finite time since the jump rate
at $(a,b)$ is $|a|+|b|$, and $|X_n| + |Y_n| = O(n)$ (as jumps are of
size $1$),
so $\sum_n ( |X_n| + |Y_n| )^{-1} = \infty$ a.s.
The process $(A(t),B(t))$ is an example of a \textit{two-dimensional
linear birth and death process}. The recurrence classification of
such processes defined on ${\mathbb{Z}}_+^2$ was studied by
Kesten \cite{kesten} and Hutton \cite{hutton}. Our case (which has
$B_{1,1}+B_{2,2}=0$ in their notation) was not covered by the
results in \cite{kesten,hutton}; Hutton
remarks (\cite{hutton}, page 638), that ``we do not yet know whether
this case is recurrent or transient.'' In the ${\mathbb{Z}}_+^2$ setting
of \cite{kesten,hutton}, the boundaries of the quadrant would
become absorbing in our case. The model on ${\mathbb{Z}}^2$ considered here
thus seems a natural setting in which to pose the
recurrence/transience question left open by \cite{kesten,hutton}.
Our Theorem {e}f{th:main1} implies that $(A(t), B(t))$ is in fact
transient.
We call $(A(t),B(t))$ the \textit{fast} embedding of the urn since
typically many jumps occur in unit time
(the process jumps faster the farther away from the origin it is).
There is another continuous-time
embedding of the urn model that is also very useful in its analysis,
the \textit{slow}
embedding described in Section {e}f{secRubin} below.
The mean of the process $(A(t),B(t))$ precisely follows the simple
harmonic oscillation suggested by the name
of the model.
This fact is most neatly expressed in the complex plane $\mathbb{C}$.
Recall that a~complex martingale
is a~complex-valued stochastic process whose real and imaginary parts
are both martingales.
\begin{lemma}
\lambdaabel{complexmartingale}
The process $(M_t)_{t \in{\mathbb{R}}_+}$ defined by
\[
M_t := {e}^{-it} \bigl( A(t) + i B(t) \bigr)
\]
is a complex martingale. In particular, for $t > t_0$ and $z \in
\mathbb{C}$,
\[
{\mathbb{E}}[ A(t) + iB(t) \mid A(t_0) + i B(t_0) = z ] = z {e}^{i(t-t_0)} .
\]
\end{lemma}
As can be seen directly from the definition,
the continuous-time
Markov chain $(A(t),B(t))$ admits a constant
invariant measure;
this fact is closely related to the ``simple harmonic
flea circus'' that we describe in Section {e}f{S: stationary model}.
Returning to the dynamics of the process,
what is the expected time taken to traverse a quadrant in the fast
continuous-time embedding? Define
$\tau_f := \inf\{ t \in{\mathbb{R}}_+ \dvtx A(t) = 0 \}$.
We use the notation ${\mathbb{P}}_n ( \cdot)$ for ${\mathbb{P}}( \cdot\mid A(0) =
n,\allowbreak
B(0) = 0)$,
and similarly for ${\mathbb{E}}_n$.
Numerical calculations strongly suggest the following:
\begin{conjecture}
\lambdaabel{conj:fasttime}
Let $n \in{\mathbb{N}}$. With $\alpha_1 \approx-2.0888$ as in Theorem {e}f
{th:zinc} above, as $n \to\infty$,
\[
{\mathbb{E}}_n [ \tau_f ] = \pi/2 + O\bigl({e}^{\alpha_1 n}/\sqrt{n}\,\bigr) .
\]
\end{conjecture}
We present a possible approach to the
resolution of Conjecture {e}f{conj:fasttime} in Section~{e}f
{S:polys}; it turns out that ${\mathbb{E}}_n [\tau_f]$
can be expressed as a rational polynomial of degree $n$ evaluated at
${e}$.
The best result that we have been able to prove along the lines of
Conjecture {e}f{conj:fasttime}
is the following,
which shows not only that ${\mathbb{E}}_n [ \tau_f]$ is close to $\pi/2$ but
also that
$\tau_f$ itself is concentrated about $\pi/2$.
\begin{thm}
\lambdaabel{th:fasttime}
Let $n \in{\mathbb{N}}$. For any $\delta>0$, as $n \to\infty$,
\begin{eqnarray}
\lambdaabel{fast1} {\mathbb{E}}_n [ \tau_f ] &=& \pi/2 + O\bigl( n^{\delta-(1/2)} \bigr), \\
\lambdaabel{fast2}
{\mathbb{E}}_n [ | \tau_f - (\pi/2) |^2 ] &=& O \bigl(n^{\delta-(1/2)}\bigr) .
\end{eqnarray}
\end{thm}
In the continuous-time fast embedding
the paths of the simple harmonic urn are a discrete stochastic approximation
to continuous circular motion at angular velocity $1$, with the radius
of the motion
growing approximately linearly in line with the transience of the process.
Therefore, a natural quantity to examine is the area enclosed by a path
of the urn across the
first quadrant, together with the two co-ordinate axes. For a typical
path starting at $(n,0)$,
we would expect this to be roughly $\pi n^2 / 4$, this being the area
enclosed by a quarter-circle
of radius $n$ about the origin. We use the percolation model to obtain
an exact relation between the
expected area enclosed and the expected time taken for the urn to
traverse the first quadrant.
\begin{thm}\lambdaabel{th:timearea} For $n \in{\mathbb{N}}$, for any $\delta>0$,
\[
{\mathbb{E}}_n [ \mbox{Area enclosed by a single traversal} ] = \sum_{m=1}^n
m {\mathbb{E}}_m [ \tau_f ] = \frac{\pi n^2}{4} + O \bigl(n^{(3/2)+\delta}\bigr) .
\]
\end{thm}
In view of the first equality in Theorem {e}f{th:timearea} and
Conjecture {e}f{conj:fasttime}, we suspect a~sharp version of the asymptotic
expression for the expected area enclosed to be
\[
{\mathbb{E}}_n [ \mbox{Area enclosed by a single traversal}] = \frac{\pi
n(n+1)}{4} + c + O \bigl(\sqrt{n} {e}^{\alpha_1 n} \bigr)
\]
for some constant $c \in{\mathbb{R}}$.
\subsection{Outline of the paper and related literature}
The outline of the remainder the the paper is as follows. We begin
with a study of the discrete-time embedded process $Z_k$ in the
original urn model. In Section {e}f{secRubin}, we use a decoupling
argument to obtain an explicit formula, involving the Eulerian
numbers, for the transition probabilities of $Z_k$. In
Section {e}f{S: drift}, we study the drift of the process $Z_k$ and
prove Theorem {e}f{th:zinc}. We make use of an attractive coupling
with the renewal process based on the uniform distribution. Then
in Section {e}f{S: short proof}, we give a short, stand-alone proof
of our basic result, Theorem~{e}f{th:main1}. In
Section {e}f{S:zinc}, we study the increments of the process~$Z_k$,
obtaining tail bounds and moment estimates. As a by-product of our
results we obtain (in Lemma {e}f{lem: asymptotic} and
Corollary {e}f{uniform}) sharp expressions for the first two
moments of the counting function of the uniform renewal process,
improving on existing results in the literature. In Section
{e}f{sec:tildez}, we study the asymptotic behavior of the noisy
urn embedded process~${\tilde Z}_k$, building on our results on $Z_k$.
Here, we make use of powerful results of Lamperti and others on
processes with asymptotically zero drift, which we can apply to
the process~${\tilde Z}^{1/2}_k$. Then in Section~{e}f{sec:proofs}, we
complete the proofs of
Theorems {e}f{th:leaky}--{e}f{thm:moments2}, {e}f{thm:difflimit},~{e}f{thm:area} and {e}f{thm:perc}.
In Section~{e}f{sec:continuous},
we study the continuous-time fast embedding described in
Section~{e}f{S: fast}, and in Sections~{e}f{S:fasttime}
and {e}f{S:areatime} present proofs of
Theorems~{e}f{thm:areamean}, {e}f{th:fasttime}
and~{e}f{th:timearea}. In Section~{e}f{S:polys}, we give some
curious exact formulae for the expected area and time described in
Section~{e}f{S: fast}. Finally, in Section~{e}f{sec:misc} we
collect some results on several models that are not directly
relevant to our main theorems but that demonstrate further some of
the surprising richness of the phenomena associated with the
simple harmonic urn and its generalizations.
We finish this section with some brief remarks on modeling applications
related to the simple
harmonic urn. The simple harmonic urn model has some similarities to
R. F. Green's urn model for
cannibalism (see, e.g., \cite{pittel}). The cyclic nature of the model
is similar to that of various
stochastic or deterministic models of certain planar systems with
feedback: see for instance~\cite{fgg} and references therein. Finally, one may view the simple
harmonic urn as a gated polling
model with two queues and a single server. The server serves one queue,
while new arrivals are
directed to the other queue. The service rate is proportional to the
ratio of the numbers of
customers in the two queues. Customers arrive at the unserved queue at
times of a Poisson process of
constant rate. Once the served queue becomes empty, the server switches
to the other queue, and a
new secondary queue is started. This model gives a third
continuous-time embedding of the simple
harmonic urn, which we do not study any further in this paper. This
polling model differs from
typical polling models studied in the literature (see, e.g., \cite{mm}) in that the service rate
depends upon the current state of the system. One possible
interpretation of this unusual service
rate could be that the customers in the primary queue are\vadjust{\goodbreak} in fact
served by the waiting customers in
the secondary queue.
The crucial importance of
the behavior at the boundaries for the recurrence classification of certain such processes
was demonstrated already in \cite{mm}.\lambdaooseness=-1
\section{Transition probabilities for $Z_k$}
\lambdaabel{secRubin}
In this section, we derive an exact formula for the transition probabilities
of the Markov chain $(Z_k)_{k \in{\mathbb{Z}}_+}$ (see Lemma~{e}f{Lemma:
transition probabilities}
below).
We use a coupling (or rather ``decoupling'')
idea that is sometimes attributed to Samuel Karlin
and Herman Rubin.
This construction was used
in \cite{KV}
to study the OK Corral gunfight model, and is
closely related to the embedding of a generic generalized P\'olya\ urn in a
multi-type branching process \cite{AK,AN}. The construction yields
another continuous-time
embedding of the urn process, which, by way of contrast to the
embedding described in Section
{e}f{S: fast}, we refer to as the \textit{slow} embedding of the urn.
We couple the segment of the urn process $(X_n,Y_n)$
between
times $\nu_k+1$ and $\nu_{k+1}$ with certain birth and death
processes, as follows. Let $\lambdaambda_k :=1/k$.
Consider two {\it independent} ${\mathbb{Z}}_+$-valued continuous-time Markov
chains, $U(t)$
and $V(t)$, $t\in{\mathbb{R}}_+$, where $U(t)$ is a pure death process with
the transition rate
\[
{\mathbb{P}}\bigl(U(t+ {d} t)=U(t)-1 \mid U(t)=a\bigr)=\lambdaambda_a\,{d} t ,
\]
and $V(t)$ is a pure birth process with
\[
{\mathbb{P}}\bigl(V(t+ {d} t)=V(t)+1 \mid V(t)=b\bigr)=\lambdaambda_b\, {d} t.
\]
Set $U(0)=z$ and
$V(0)=1$.
From the standard exponential holding-time characterization
for conti\-nuous-time Markov chains and
the properties of independent exponential random\vadjust{\goodbreak}
variables, it follows that the embedded process $(U(t),V(t))$
considered at the times when either of its coordinates changes
has the same distribution as the simple harmonic urn $(X_n,Y_n)$
described above when $(X_n,Y_n)$ is traversing the first quadrant.
More precisely, let $\theta_0:=0$ and define the jump times
of the process $V(t)-U(t)$ for $n \in{\mathbb{N}}$:
\[
\theta_n :=\inf\{t>\theta_{n-1}\dvtx U(t)< U( \theta_{n-1} )\mbox{ or }V(t)>
V( \theta_{n-1})\}.
\]
Since $\lambdaambda_b \lambdaeq1$ for all $b$, the processes $U(t)$, $V(t)$
a.s. do not explode in finite time, so $\theta_n \to\infty$
a.s. as $n \to\infty$. Define $\eta:= \min\{ n \in{\mathbb{N}}:
U(\theta_n) = 0\}$ and set
\[
T := \theta_{\eta} = \inf\{ t > 0 \dvtx U(t) = 0 \} ,
\]
the extinction time of $U(t)$.
The coupling yields the following result (cf. \cite{AN}, Section V.9.2).
\begin{lemma}
\lambdaabel{RubinLemma}
Let $k \in{\mathbb{Z}}_+$ and $z \in{\mathbb{N}}$.
The sequence $(U(\theta_n),V(\theta_n))$, $n =0,1,\lambdadots,\eta$,
with $(U(0),V(0))=(z,1)$,
has the same
distribution as each of the following two sequences:
\begin{itemize}[(ii)]
\item[(i)] $(|X_n|,|Y_n|)$, $n = \nu_{k}+1, \lambdadots, \nu_{k+1}$,
conditioned on $Z_{k} = z$ and $Y_{\nu_k} =0$;
\item[(ii)] $(|Y_n|,|X_n|)$, $n = \nu_{k}+1, \lambdadots, \nu_{k+1}$,
conditioned on $Z_{k} = z$ and\vadjust{\goodbreak} $X_{\nu_k} =0$.
\end{itemize}
\end{lemma}
Note that we set
$V(0)=1$ since
$(X_{\nu_k+1},Y_{\nu_k+1})$ is always one step in the
``anticlockwise'' lattice direction away from $(X_{\nu_k},Y_{\nu_k})$.
Let
\[
T_w':=\inf\{t>0 \dvtx V(t)=w\}.
\]
We can represent the times $T$ and $T'_w$ as
sums of exponential random variables. Write
\begin{equation}
\lambdaabel{E: times as sums of exponential rvs}
T_z = \sum_{k=1}^{z} k \xi_k \quad \mbox{and}\quad
T'_w = \sum_{k=1}^{w-1} k \zeta_k,
\end{equation}
where $\xi_1,\zeta_1, \xi_2, \zeta_2, \lambdadots$ are independent
exponential random variables with mean~$1$. Then setting
$T = T_{U(0)}$, ({e}f{E: times as sums of exponential rvs})
gives useful representations of $T$ and~$T'_w$.
As an immediate illustration of the power of this embedding,
observe that $Z_{k+1} \lambdae Z_k$ if and only if
$V$ has not reached $U(0)+1$ by the
time of the extinction of $U$, that is, $T_{U(0)+1}' > T$.
But ({e}f{E: times as sums of exponential rvs})
shows that $T_{z+1}'$ and $T_z$ are identically distributed continuous
random variables, entailing the following result.
\begin{lemma}
For $z \in{\mathbb{N}}$, \mbox{${\mathbb{P}}(Z_{k+1} \lambdae Z_k \mid Z_k = z )
= {\mathbb{P}}(T_{U(0)+1}' > T_{U(0)} ) = \frac{1}{2}$}.
\end{lemma}
We now proceed to derive from the coupling
described in Lemma {e}f{RubinLemma}
an exact formula for the transition probabilities
of the Markov chain $(Z_k)_{k \in{\mathbb{Z}}_+}$.
Define $p(n,m) = {\mathbb{P}}(Z_{k+1}=m\mid Z_k=n )$. It turns out that
$p(n,m)$ may be expressed in terms of the \textit{Eulerian numbers}
$A(n,k)$, which are the positive integers defined
for $n \in{\mathbb{N}}$ by
\[
A(n,k) = \sum_{i=0}^k (-1)^i \pmatrix{n+1\cr i}(k-i)^n ,\qquad k \in\{1,\lambdadots,n \} .
\]
The Eulerian numbers have several combinatorial interpretations and
have many interesting
properties; see, for example, B\'{o}na \cite{Bona}, Chapter 1.
\begin{lemma}
\lambdaabel{Lemma: transition probabilities}
For $n,m \in{\mathbb{N}}$, the transition probability $p(n,m)$ is given by
\begin{eqnarray*}
p(n,m) & = & m \sum_{r=0}^m (-1)^r \frac{(m-r)^{n+m-1}}{r! (n+m-r)!}\\
& = & \frac{m}{(m+n)!} A(n+m-1,n) .
\end{eqnarray*}
\end{lemma}
We give two proofs of Lemma {e}f{Lemma: transition probabilities},
both using the coupling of
Lemma {e}f{RubinLemma} but in quite different ways. The first uses
moment generating
functions and is similar to calculations in \cite{KV}, while the
second involves a
time-reversal of the death process and makes use of the recurrence
relation satisfied by the Eulerian numbers.
Each proof uses ideas that will be useful later\vadjust{\goodbreak} on.
\begin{pf*}{First proof of Lemma {e}f{Lemma: transition probabilities}}
By Lemma {e}f{RubinLemma},
the conditional distribution of $Z_{k+1}$ on $Z_k=n$ coincides with the
distribution
conditional of $V(T)$ on $U(0) =n$. So
\begin{equation}
\lambdaabel{Zs and Ts}
{\mathbb{P}}(Z_{k+1}> m \mid Z_k=n ) =
{\mathbb{P}}\bigl( V(T ) > m \mid U(0) = n\bigr) =
{\mathbb{P}}(T_n > T'_{m+1} ) ,\hspace*{-20pt}
\end{equation}
using the representations in ({e}f{E: times as sums of exponential rvs}).
Thus, from ({e}f{E: times as sums of exponential rvs}) and ({e}f{Zs
and Ts}), writing
\[
R_{n,m} = \sum_{i=1}^{n} i \xi_i -\sum_{j=1}^{m} j
\zeta_j,
\]
we have that ${\mathbb{P}}(Z_{k+1}> m \mid Z_k=n ) = {\mathbb{P}}( R_{n,m} > 0)$.
The density of $R_{n,m}$ can be calculated using the moment
generating function and partial fractions; for $t \geq0$,
\begin{eqnarray*}
{\mathbb{E}}[ {e}^{t R_{n,m}} ] &=&\prod_{i=1}^n \frac{1/i}{1/i-t} \times
\prod_{j=1}^m \frac{1/j}{1/j+t}
=\prod_{i=1}^n \frac{1}{1-it} \times\prod_{j=1}^m \frac{1}{1+jt}
\\&=&\sum_{i=1}^n \frac{a_i}{1-it} +\sum_{j=1}^m \frac{b_j}{1+jt}
\end{eqnarray*}
for some coefficients $a_i = a_{i;n,m}$ and $b_j = b_{j;n,m}$.
Multiplying both sides of the last displayed
equality by $\prod_{i=1}^n (1- it) \prod_{j=1}^m (1 + jt)$
and setting $t = 1/i$, we obtain
\begin{eqnarray*}
a_i &=& \prod_{\stackrel{j=1}{j \neq i}}^n \frac{1}{1-(j/i)} \prod
_{k=1}^m \frac{1}{1+(k/i)}\\
&=& (-1)^{n-i} i^{n+m -1} \prod_{j=1}^{i-1} \frac{1}{i-j} \prod
_{j=i+1}^n \frac{1}{j-i} \prod_{k=1}^m \frac{1}{k+i} .
\end{eqnarray*}
Simplifying, and then proceeding similarly but taking $t = -1/j$ to
identify~$b_j$, we obtain
\[
a_i = \frac{(-1)^{n-i} i^{n+m} }{ (n-i)! (m+i)! } \quad \mbox{and}\quad
b_j = \frac{(-1)^{m-j} j^{n+m} }{ (m-j)! (n+j)! }.
\]
Consequently, the density of $R_{n,m}$ is
\[
r(x)=
\cases{
\displaystyle \sum_{i=1}^n a_i i^{-1} {e}^{-x/i}, &\quad if
$x\ge0$,\cr
\displaystyle \sum_{j=1}^m b_j j^{-1} {e}^{x/j}, &\quad if $ x< 0$.
}
\]
Thus, we obtain
\begin{eqnarray}
\lambdaabel{eq:cdfofZ}
{\mathbb{P}}(Z_{k+1}>m \mid Z_k=n)&=& {\mathbb{P}}(Z_{k+1}\ge m+1 \mid Z_k=n)\nonumber \\
&=& {\mathbb{P}}(R_{n,m} \geq0)= \sum_{k=1}^n a_{k;n,m}
\nonumber\\[-8pt]\\[-8pt]
&=&\sum_{k=0}^n \frac{(-1)^{n-k} k^{n+m} }{
(n-k)! (m+k)! }=\sum_{i=0}^n \frac{(-1)^i (n-i)^{n+m}}{i! (m+n-i)!}\nonumber
\\
&=&\frac{1}{(m+n)!}\sum_{i=0}^n (-1)^i \pmatrix{m+n\cr i} (n-i)^{n+m}.\nonumber
\end{eqnarray}
It follows that
\begin{eqnarray*}
p(n,m) &=& {\mathbb{P}}(Z_{k+1}\ge m \mid Z_k=n)
- {\mathbb{P}}(Z_{k+1}\ge m+1 \mid Z_k=n)
\\
&=& \sum_{i=0}^n \frac{(-1)^i (n-i)^{n+m-1}}{i! (m-1+n-i)!}
-\sum_{i=0}^n \frac{(-1)^i (n-i)^{n+m}}{i! (m+n-i)!}\\
&=&\sum_{i=0}^n \frac{(-1)^i (n-i)^{n+m-1}} {i! (m+n-i)!}
[(m+n-i)-(n-i) ]
\\
&=&\frac m{(m+n)!}\sum_{i=0}^n (-1)^i \pmatrix{m+n\cr i}
(n-i)^{n+m-1}\\
&=&\frac{m}{(n+m)!}A(m+n-1,n)
\end{eqnarray*}
as required.
\end{pf*}
\begin{pf*}{Second proof of Lemma {e}f{Lemma: transition probabilities}}
Consider the birth process $W(t)$ defined by $W(0) = 1$ and
\[
W(t) = \min\{z \in{\mathbb{Z}}_+ \dvtx T_z > t \} \qquad (t >0),
\]
where $T_z$ is defined as in ({e}f{E: times as sums of exponential rvs}).
The inter-arrival times of $W(t)$ are $(i \xi_i)_{i=1}^z$ and, given
$U(0) = z$,
the death process $U(t)$ has the same inter-arrival
times but taken in the reverse order. The processes
$V(t)$ and $W(t)$ are independent and identically distributed.
Define for $n , m \in{\mathbb{N}}$,
\[
r(n,m) = {\mathbb{P}}\bigl( \exists t> 0 \dvtx W(t) = n, V(t) = m \mid V(0) = W(0) = 1 \bigr).
\]
If $Z_k = n$, then $Z_{k+1}$ is the value of $V$ when $W$
first reaches the value \mbox{$n+1$}; $Z_{k+1} = m$ if and only if the
process $(W,V)$ reaches $(n,m)$ \textit{and} then makes the transition to
$(n+1,m)$. Since\vadjust{\goodbreak} $(W,V)$ is Markov, this occurs with
probability $r(n,m) \frac{m}{n+m}$. So for $n,m \in{\mathbb{N}}$,
\begin{equation}\lambdaabel{E: p and r}
p(n,m) = \frac{m}{n+m} r(n,m) .
\end{equation}
Conditioning on the site from which $(W, V)$ jumps to $(n,m)$,
we get, for $n,m \in{\mathbb{N}}$, $n+m \geq3$,
\begin{equation}\lambdaabel{E: recurrence for r}
r(n,m) = \frac{m}{n+m-1} r(n-1,m) + \frac{n}{n+m-1} r(n,m-1),
\end{equation}
where $r(0,m)= r(n,0) = 0$.
It is easy to check that $r(k,1) = r(1,k) = 1/k!$ for all $k \in{\mathbb{N}}$.
It will be helpful to define
\[
s(n,m) = (n+m-1)! r(n,m) .
\]
Then we have for $n, m \in{\mathbb{N}}$, $n+m \geq3$,
\[
s(n,m) = m s(n-1,m) + n s(n,m-1) ,
\]
\[
s(k,1) = s(1,k) = 1 \qquad \mbox{for all }k \in{\mathbb{N}}.
\]
These constraints completely determine the \textit{positive integers}
$s(n,m)$ for all $m,n \in{\mathbb{N}}$.
Since the Eulerian numbers $A(n+m-1,m)$ satisfy the same initial
conditions and recurrence relation (\cite{Bona}, Theorem 1.7), we have
$s(n,m) = A(m+n -1, m)$,
which together with ({e}f{E: p and r}) gives the desired formula for $p(n,m)$.
\end{pf*}
It is evident from ({e}f{E: recurrence for r}) and its initial
conditions that $r(n,m) = r(m,n)$ for all $n,m \in{\mathbb{N}}$. So
\begin{equation}\lambdaabel{E: p and Z}
n p(n,m) = m p(m,n) .
\end{equation}
Therefore, the $\sigma$-finite measure $\pi$ on $\mathbb{N}$ defined
by $\pi(n) = n$
satisfies the detailed balance equations and hence is invariant for
$p(\cdot,\cdot)$.
In fact there is a pathwise relation of the same type, which we now describe.
We call a sequence
$\omega=(x_j,y_j)_{j=0}^k$ ($k \geq2$)
of points in ${\mathbb{Z}}_+^2$ an
\textit{admissible traversal} if
$y_0 = x_k =0$, $x_0 \geq1$, $y_k \geq1$,
each
point $(x_j,y_j)$, $2 \lambdaeq j \lambdaeq k-1$, is
one of $(x_{j-1}-1,y_{j-1})$, $(x_{j-1},y_{j-1}+1)$, and $(x_1,y_1) =
(x_0,y_0 +1)$,
$(x_k, y_k) = (x_{k-1} - 1,y_{k-1})$. If $\omega$ is an admissible
traversal, then so is the time-reversed and reflected path
$\omega' = (y_{k-j},x_{k-j})_{j=0}^k$. In fact, conditioning on the endpoints,
$\omega$ and $\omega'$ have the same probability of being realized by
the simple harmonic urn.
\begin{lemma}
\lambdaabel{lem:reverse}
For any admissible traversal
$(x_j,y_j)_{j=0}^k$ with $x_0 = n \in{\mathbb{N}}$, $y_k =m \in{\mathbb{N}}$,
\begin{eqnarray*}
&&{\mathbb{P}}\bigl( ( X_j, Y_j )_{j=0}^{\nu_1} =
(x_j,y_j)_{j=0}^k \mid Z_0 = n, Z_1 = m \bigr) \\
&&\qquad =
{\mathbb{P}}\bigl( ( X_j ,Y_j )_{j=0}^{\nu_1} =
(y_{k-j},x_{k-j})_{j=0}^k \mid Z_0 = m, Z_1 = n \bigr) .
\end{eqnarray*}
\end{lemma}
\begin{pf}
Let $\omega=(x_j,y_j)_{j=0}^k$ be an admissible traversal,
and define
\[
p = p(\omega) = {\mathbb{P}}\bigl( ( X_j, Y_j )_{j=0}^{\nu_1} =
(x_j,y_j)_{j=0}^k , Z_1 = m \mid Z_0 = n \bigr)
\]
and
\[
p' = p'(\omega) = {\mathbb{P}}\bigl( ( X_j, Y_j )_{j=0}^{\nu_1} =
(y_{k-j},x_{k-j})_{j=0}^k , Z_1 = n \mid Z_0 = m \bigr) ,
\]
so that
$p'$ is the probability of the reflected and time-reversed path.
To prove the lemma, it suffices to show that for any $\omega$ with
$(x_0,y_0) = (n,0)$ and $(x_k,y_k)=(0,m)$, $p(\omega) /p(n,m)=
p'(\omega)/p(m,n)$. In light of ({e}f{E: p and Z}), it therefore
suffices to show that
$n p = m p'$.
To see this, we use the Markov property along the path $\omega$ to obtain
\[
p = \prod_{j=0}^{k-1} (x_j + y_j )^{-1} \bigl( x_j {\mathbf{1}}_{\{ x_{j+1} = x_j \}}
+ y_j {\mathbf{1}}_{\{ y_{j+1} = y_j \}} \bigr) ,
\]
while, using the Markov property along the reflection and reversal of
$\omega$,
\begin{eqnarray*}
p' & = &\prod_{j=0}^{k-1} (x_{k-j}+y_{k-j})^{-1}
\bigl( x_{k-j} {\mathbf{1}}_{\{ x_{{k-j-1}} = x_{k-j} \}}
+ y_{k-j} {\mathbf{1}}_{\{ y_{k-j-1} = y_{k-j} \}} \bigr) \\
& = &\prod_{i=0}^{k-1} (x_{i+1}+y_{i+1})^{-1}
\bigl( x_{i} {\mathbf{1}}_{\{ x_{{i+1}} = x_{i} \}}
+ y_i {\mathbf{1}}_{\{ y_{i+1} = y_{i} \}} \bigr) ,
\end{eqnarray*}
making the change of variable $i = k-j-1$. Dividing the two products
for $p$ and $p'$ yields, after
cancellation, $p/p' = (x_k+y_k)/ (x_0 +y_0 ) = m/n$,
as required.
\end{pf}
\begin{remarks*}
Of course by summing over paths in the equality $n p(\omega) = m
p'(\omega)$,
we could use the argument in the last proof to prove ({e}f{E: p and Z}).
The reversibility and the invariant measure exhibited in Lemma {e}f
{lem:reverse}
and ({e}f{E: p and Z})
will appear naturally in terms of a stationary model in Section {e}f
{S: stationary model}.
\end{remarks*}
\section{\texorpdfstring{Proof of Theorem \protect{e}f{th:zinc} via the uniform renewal process}
{Proof of Theorem 2.2 via the uniform renewal process}}
\lambdaabel{S: drift}
In this section, we study the
asymptotic behavior of ${\mathbb{E}}[ Z_{k+1} \mid Z_k = n]$
as $n \to\infty$.
The explicit expression for the distribution of $Z_{k+1}$
given $Z_k =n$ obtained in Lemma
{e}f{Lemma: transition probabilities}
turns out not to be very convenient to use directly.
Thus, we proceed somewhat indirectly and exploit
a connection with a renewal process
whose inter-arrival times are uniform on $(0,1)$.
Here and subsequently, we use $U(0,1)$ to denote the uniform
distribution on $(0,1)$.
Let $\chi_1, \chi_2, \chi_3, \lambdadots$ be an i.i.d. sequence of
$U(0,1)$ random variables. Consider the renewal sequence $S_i$, $i \in
{\mathbb{Z}}_+$,
defined by $S_0 := 0$ and, for $i \geq1$, $S_i := \sum_{j=1}^i \chi_j$.
For $t \geq0$, define the counting process
\begin{equation}
\lambdaabel{eq:renewproc}
N(t) := \min\{ i \in{\mathbb{Z}}_+ \dvtx S_i > t \} = 1+\max\{i \in{\mathbb{Z}}_+ \dvtx S_i
\lambdae t\} ,
\end{equation}
so a.s., $N(t) \geq t+1$.
In the language of classical renewal theory,
${\mathbb{E}}[N(t)]$ is a renewal function (note that we
are counting the renewal at time $0$).
The next result establishes the connection between the uniform renewal
process and the simple harmonic urn.
\begin{lemma}\lambdaabel{Lemma: same distribution}
For each $n \in{\mathbb{N}}$, the conditional distribution of $Z_{k+1}$ on $Z_k
= n$ equals the distribution of $N(n)-n$.
In particular, for $n \in{\mathbb{N}}$, ${\mathbb{E}}[ Z_{k+1} \mid Z_k = n] = {\mathbb{E}}[N(n) ]-n$.
\end{lemma}
The proof of Lemma {e}f{Lemma: same distribution} amounts to
showing that ${\mathbb{P}}(N(n) = n+m ) = p (n,m)$ as given by Lemma
{e}f{Lemma: transition probabilities}. This equality is Theorem 3
in \cite{sparac}, and it may be verified combinatorially
using the interpretation of $A(n,k)$ as the number of permutations of
$\{1,\dots,n\}$ with exactly $k-1$ falls, together with the
observation that for $n \in{\mathbb{N}}$,
$N(n)$ is the position of the $n$th fall in
the sequence $\psi_1,\psi_2,\dots,$ where $\psi_k= S_k \bmod1$,
another sequence of i.i.d.\ $U(0,1)$
random variables. Here, we will give a neat proof of Lemma {e}f{Lemma:
same distribution} using the coupling
exhibited above in Section {e}f{secRubin}.
\begin{pf*}{Proof of Lemma {e}f{Lemma: same distribution}}
Consider a doubly-infinite sequence $(\xi_i)_{i \in\mathbb{Z}}$
of independent exponential random variables with mean $1$. Taking
$\zeta_k = \xi_{-k}$, we can write $R_{n,m}$ (as defined
in the first proof of Lemma {e}f{Lemma: transition probabilities})
as $\sum_{i= -m}^n i
\xi_i$. Define $S_{n,m} = \sum_{i = -m}^n \xi_i$. For fixed $n
\in{\mathbb{N}}$, $m \in{\mathbb{Z}}_+$, we consider normalized partial sums
\[
\chi_j' = \Biggl(\sum_{i = -m}^{j-1-m} \xi_i \Biggr) \Big/ S_{n,m} ,\qquad j \in\{ 1,
\lambdadots, n+m \}.
\]
Since $(S_{j-1-m,m})_{j=1}^{n+m}$
are the first $n+m$ points of a unit-rate Poisson process on
${\mathbb{R}}_+$, the vector $( \chi_1' , \chi_2', \lambdadots, \chi_{n+m}') $ is
distributed as the vector of increasing order statistics of the
$n+m$ i.i.d.\ $U(0,1)$ random variables $\chi_1,\lambdadots,\chi_{n+m}$. In
particular,
\[
{\mathbb{P}}\bigl(N(n) > n+m \bigr) = {\mathbb{P}}\Biggl(\sum_{i=1}^{n+m} \chi_i \lambdaeq n \Biggr)
= {\mathbb{P}}\Biggl(\sum_{i=1}^{n+m} \chi_i' \lambdaeq n \Biggr) ,
\]
using the fact that, by ({e}f{eq:renewproc}),
$\{ N(n) > r\} = \{ S_r \lambdaeq n \}$ for $r \in{\mathbb{Z}}_+$ and $n > 0$.
But
\[
n -\sum_{i=1}^{n+m} \chi_i' = \sum_{i=m+1}^{m+n} (1 - \chi_i') -
\sum_{i=1}^m \chi_i'
= \Biggl(\sum_{i =-m}^n i \xi_i \Biggr)\Big/ S_{n,m}
= R_{n,m} / S_{n,m} .
\]
So, using the equation two lines above ({e}f{eq:cdfofZ}),
\[
{\mathbb{P}}\bigl(N(n) - n > m \bigr) = {\mathbb{P}}(R_{n,m} \geq0 ) = {\mathbb{P}}(Z_{k+1} > m \mid Z_k = n ) .
\]
Thus, $N(n)-n$ has the same distribution as $Z_{k+1}$ conditional\vadjust{\goodbreak} on $Z_k=n$.~
\end{pf*}
In view of Lemma {e}f{Lemma: same distribution},
to study ${\mathbb{E}}[Z_{k+1} \mid Z_k = n]$
we need
to study ${\mathbb{E}}[N(n)]$.
\begin{lemma}\lambdaabel{lem:conv} As $n \to\infty$,
\[
{\mathbb{E}}[N(n)] - \bigl(2n+\tfrac23 \bigr) \to0.
\]
\end{lemma}
\begin{pf}
This is a consequence of the renewal theorem.
For a general nonarithmetic renewal process whose inter-arrival times have
mean $\mu$ and variance $\sigma^2$, let $U(t)$ be the expectation of
the number of arrivals up to
time~$t$, including the initial arrival at time $0$. Then
\begin{equation}
\lambdaabel{renewalthm}
U(t) - \frac{t}{\mu} \to
\frac{\sigma^2 + \mu^2}{2 \mu^2} \qquad \mbox{as } t \to\infty.
\end{equation}
We believe this is due to Smith \cite{Smith}. See, for example, Feller
\cite{Feller2}, Section~XI.3, Theorem 1, Cox \cite{Cox}, Section 4, or
Asmussen \cite{Asmussen}, Section V, Proposition~6.1. When the
inter-arrival distribution is $U(0,1)$, we have $U(t) =
\mathbb{E}[N(t)]$ with the notation of ({e}f{eq:renewproc}), and in
this case $\mu= 1/2$ and $\sigma^2 = 1/12$.
\end{pf}
Together with Lemma {e}f{Lemma: same distribution},
Lemma {e}f{lem:conv} gives the following result.
\begin{corollary}\lambdaabel{cor:Ezk+1}
${\mathbb{E}}[Z_{k+1}\mid Z_k=n]-n\to\frac23$ as $n\to\infty$.
\end{corollary}
To obtain the exponential error bound in
({e}f{zinc}) above, we need to know more about the rate of
convergence in Corollary {e}f{cor:Ezk+1} and hence in
Lemma~{e}f{lem:conv}. The existence of a bound like ({e}f{zinc})
for \textit{some} $\alpha_1 < 0$ follows from known results:
Stone \cite{Stone} gave an exponentially small error bound in the
renewal theorem~({e}f{renewalthm}) for inter-arrival distributions
with exponentially decaying tails, and
an exponential bound also
follows from the coupling proof of the renewal theorem (see, e.g.,
Asmussen \cite{Asmussen}, Section VII, Theorem 2.10 and Problem~2.2). However, in this particular case
we can solve the renewal
equation exactly and deduce the asymptotics more precisely, identifying
a (sharp) value for~$\alpha_1$ in ({e}f{zinc}). The first step is the
following result.
\begin{lemma}
Let $\chi_1, \chi_2, \lambdadots$ be an i.i.d. sequence of $U(0,1)$
random variables. For $t \in{\mathbb{R}}_+$,
\[
{\mathbb{P}}\Biggl( \sum_{i=1}^k \chi_i \lambdaeq t \Biggr) =
\sum_{i=0}^{k \wedge\lambdafloor t\rfloor} \frac{(t-i)^k (-1)^i}{i!
(k-i)!} ,
\]
and
\begin{eqnarray}\lambdaabel{eq:robwaters}
\mathbb{E} [N(t) ] = U(t) = \sum_{k=0}^{\infty} {\mathbb{P}}\Biggl(\sum_{i=1}^k
\chi_i \lambdaeq t \Biggr) = \sum_{i=0}^{\lambdafloor t \rfloor} \frac{(i-t)^i {e}
^{t-i}}{i!} .
\end{eqnarray}
\end{lemma}
\begin{pf}
The first formula is classical (see, e.g., \cite{Feller2}, page 27);
according to Feller \cite{Feller1}, page 285, it is due to Lagrange.
The second formula follows from observing (with an empty sum being $0$)
\[
U (t) = {\mathbb{E}}\sum_{k=0}^\infty\mathbf{1} \Biggl\{ \sum_{i=1}^k \chi_i \lambdaeq t
\Biggr\} =
\sum_{k=0}^\infty{\mathbb{P}}\Biggl( \sum_{i=1}^k \chi_i \lambdaeq t \Biggr),
\]
and exchanging the order in the consequent double sum
(which is absolutely convergent).
\end{pf}
We next obtain a more tractable explicit formula for the expression
in~({e}f{eq:robwaters}).
Define for $t \geq0$
\[
f(t) := \sum_{i=0}^{\lambdafloor t \rfloor} \frac{(i-t)^i {e}^{t-i}}{i!} .
\]
It is easy to verify (see also \cite{Asmussen}, page 148)
that $f$ is continuous on $[0,\infty)$ and satisfies
\begin{eqnarray} \lambdaabel{E: differential-delay}
f(t) &=& {e}^t \qquad \hspace*{62.5pt} ( 0 \lambdae t \lambdae1),
\nonumber\\[-8pt]\\[-8pt]
f'(t) &=& f(t) - f(t-1)\qquad (t \geq1) .\nonumber
\end{eqnarray}
\begin{lemma}\lambdaabel{lem: asymptotic}
For all $t > 0$,
\begin{equation}\lambdaabel{E: asymptotic expansion} f(t) = 2t + \frac
{2}{3} + \mathop{\sum_{
\gamma\in\mathbb{C} \dvtx \gamma\neq0,}}_{ \gamma= 1-
\exp(-\gamma)
} \frac{1}{\gamma} {e}^{\gamma t} .
\end{equation}
The sum is absolutely convergent, uniformly for $t$ in $(\varepsilon,\infty
)$ for any $\varepsilon> 0$.
\end{lemma}
\begin{pf}
The Laplace transform $\mathcal{L}f(\lambdaambda)$ of $f$ exists
for $\textup{Re}(\lambdaambda) > 0$ since $f(t) = 2t+2/3 + o(1)$ as $t \to
\infty$, by
({e}f{eq:robwaters}) and Lemma
{e}f{lem:conv}. Using the differential-delay equation ({e}f{E:
differential-delay}), we obtain
\[
\mathcal{L}f(\lambdaambda) = \frac{1}{\lambdaambda- 1 + {e}^{-\lambdaambda}} .
\]
The principal part of $\mathcal{L}f$ at $0$ is $\frac{2}{\lambdaambda^2}
+ \frac{2}{3\lambdaambda}$.\vspace*{-1pt}
There are simple poles at the nonzero roots of $\lambdaambda- 1 + {e}
^{-\lambdaambda}$,
which occur in complex conjugate pairs $\alpha_n \pm i\beta_n$, where
$\alpha= \alpha_1 > \alpha_2 > \cdots$ and $0 < \beta_1 < \beta_2
< \cdots.$\vspace*{1pt}
In fact, $\alpha_n = -\lambdaog(2\pi n) + o(1)$ and $\beta_n = (2n +
\frac{1}{2})\pi+ o(1)$ as $n \to\infty$.
For $\gamma= \alpha_n + i \beta_n$, the absolute value of the term
${e}^{\gamma t}/\gamma$ in the right-hand side of ({e}f{E: asymptotic
expansion}) is $1/(|\gamma| |1-\gamma|^t)$, hence the sum converges
absolutely, uniformly on any interval $(\varepsilon, \infty)$, $\varepsilon> 0$.
To establish ({e}f{E: asymptotic expansion}), we will
compute the Bromwich integral (inverting the Laplace\vadjust{\goodbreak}
transform), using a carefully chosen sequence of rectangular contours:
\[
f(t) = \lambdaim_{R \to\infty}
\int_{\varepsilon- i R}^{\varepsilon+ i R} \frac{{e}^{\lambdaambda t}}{\lambdaambda- 1 +
\exp(-\lambdaambda)} \,d\lambdaambda.
\]
To evaluate this limit for a particular value of $t > 0$, we
will take $\varepsilon= 1/t$ and integrate around a sequence $C_n$
of rectangular contours, with vertices at
$(1/t) \pm(2n - \frac{1}{2})\pi i$ and $-2\lambdaog n \pm(2n - \frac
{1}{2})\pi i$.
The integrand along the vertical segment at real part $-2 \lambdaog n$\vspace*{1pt}
is bounded by $(1+o(1))/n^2$ and the integrand along the horizontal
segments is bounded by ${e}/(2n - \frac{1}{2})\pi$ because the
imaginary parts of $\lambdaambda$ and ${e}^{-\lambdaambda}$ have the same
sign there, so $|\lambdaambda- 1 + {e}^{-\lambdaambda}| \ge\textup
{Im}(\lambdaambda)$.
It follows that the integrals along these three arcs all tend to
zero as $n \to\infty$. Each pole lies inside all but finitely many of
the contours $C_n$, so the principal value of the Bromwich
integral is the sum of the residues of ${e}^{\lambdaambda t}/(\lambdaambda- 1 +
\exp(-\lambdaambda))$.
The residue at $0$ is $2t + 2/3$, and the residue at $\gamma= \alpha
_n + i \beta_n$ is ${e}^{\gamma t}/\gamma$.
Thus, we obtain ({e}f{E: asymptotic expansion}).
\end{pf}
\begin{pf*}{Proof of Theorem {e}f{th:zinc}}
The statement of the theorem follows
from Lemma~{e}f{lem: asymptotic}, since by Lemma {e}f{Lemma: same
distribution} and ({e}f{eq:robwaters})
we have \mbox{${\mathbb{E}}[ Z_{k+1} \mid Z_k = n] = f(n) -n$} for $n \in{\mathbb{N}}$.
\end{pf*}
\begin{remarks*}
According to Feller \cite{Feller2}, Problem 2, page 385,
equation ({e}f{eq:robwaters})
``is frequently rediscovered in
queuing theory, but it reveals little about the nature of $U$.'' We
have not
found the formula ({e}f{E: asymptotic expansion}) in the literature.
The dominant term in $f(t) - 2t - 2/3$ as $t \to\infty$ is ${e}
^{\gamma_1 t}/\gamma_1 + {e}^{\overline{\gamma_1}t}/\overline
{\gamma_1}$, that is,
\[
\frac{1}{\alpha_1^2+\beta_1^2} {e}^{\alpha_1 t} \bigl(\beta_1\sin
(\beta_1 t) + \alpha_1 \cos(\beta_1 t)\bigr) ,
\]
which changes sign infinitely often. After subtracting this term,
the remainder is $O ({e}^{\alpha_2 t} )$. The method that
we have used for analyzing the asymptotic behavior of solutions
to the renewal equation was proposed by A. J. Lotka and was put on
a firm basis by Feller \cite{Feller3}; Laplace
transform inversions of this kind were dealt with by Churchill
\cite{Churchill}.
\end{remarks*}
\section{\texorpdfstring{Proof of Theorem \protect{e}f{th:main1}}{Proof of Theorem 2.1}}\lambdaabel{S: short proof}
The recurrence relation ({e}f{E: recurrence for r}) for $r (n,m)$
permits a direct proof of Theorem
{e}f{th:main1} (transience), without appealing to the more general
Theorem {e}f{thm:mixed}, via standard
martingale arguments applied\break to~$h (Z_k)$ for a judicious choice of
function $h$. This is
the subject of this section.
Rewriting ({e}f{E: recurrence for r}) in terms of $p$
yields the following recurrence relation, which does not seem simple to
prove by
conditioning on a step in the urn model; for $n, m \in{\mathbb{N}}$, $n + m
\geq3$,
\begin{equation}\lambdaabel{E: recurrence for p}
\biggl(\frac{n+m}{m} \biggr) p(n,m) = p(n-1,m) +
\biggl(\frac{n}{m-1} \biggr) p(n,m-1) ,
\end{equation}
where if $m=1$ we interpret the right-hand side of ({e}f{E: recurrence
for p})
as just $p(n-1,1)$, and where $p(0,m)=p(n,0) = 0$. Note $p(1,1) = 1/2$.
For ease of notation, for any function $F$ we will write ${\mathbb{E}}_n[F(Z)]$
for ${\mathbb{E}}[ F(Z_{k+1}) \mid Z_k = n]$,
which, by the Markov property, does not depend on $k$.
\begin{lemma}
\lambdaabel{L: short}
Let $\alpha_1 \approx- 2.0888$ be as in Theorem {e}f{th:zinc}. Then
for $n \geq2$,
\begin{eqnarray*}
{\mathbb{E}}_n \biggl[\frac{1}{Z} \biggr] & =&
\frac{{\mathbb{E}}_n[Z] - {\mathbb{E}}_{n-1}[Z]}{n} = \frac{1}{n} + O ({e}^{\alpha_1
n} ) ,\\
{\mathbb{E}}_n \biggl[\frac{1}{Z^2(Z+1)} \biggr] & =&
\frac{{\mathbb{E}}_{n-1}[ 1/Z ] - {\mathbb{E}}_n [ 1/Z] }{n} = \frac{1}{n^2(n-1)} + O
({e}^{\alpha_1 n} ) ,
\end{eqnarray*}
where the asymptotics refer to the limits as $n \to\infty$.
\end{lemma}
\begin{pf}
We use the recurrence relation ({e}f{E: recurrence for p}) satisfied
by the transition probabilities of $Z_k$.
First, multiply both sides of ({e}f{E: recurrence for p}) by $m$, to
get for $n,m \in{\mathbb{N}}$, $m +n \geq3$,
\[
(n+m)p(n,m)
= m p(n-1,m) + n p(n,m-1) + \frac{n}{m-1} p(n, m-1) ,
\]
where $p(n,0)=p(0,m)=0$. Summing over $m \in{\mathbb{N}}$ we obtain for $n
\geq2$,
\[
n + {\mathbb{E}}_n[Z] = {\mathbb{E}}_{n-1}[Z] + n + n{\mathbb{E}}_n [ 1/Z ],
\]
which yields the first equation of the lemma after an application of
({e}f{zinc}).
For the second equation, divide ({e}f{E: recurrence for p}) through by
$m$ to get for $n, m \in{\mathbb{N}}$, $m+ n \geq3$,
\[
\frac{(n+m)}{m^2}p(n,m) = \frac{1}{m} p(n-1,m) + \frac{n}{m(m-1)}
p(n,m-1) .
\]
On summing over $m \in{\mathbb{N}}$ this gives, for $n \geq2$,
\[
n {\mathbb{E}}_n [ 1/Z^2 ] + {\mathbb{E}}_n [ 1/Z ] = {\mathbb{E}}_{n-1} [1/Z] + n {\mathbb{E}}_n \biggl[\frac
{1}{(Z+1)Z} \biggr] ,
\]
which gives the second equation when we apply the asymptotic part of
the first equation.
\end{pf}
\begin{pf*}{Proof of Theorem {e}f{th:main1}}
Note that $h(x) = \frac{1}{x} - \frac{1}{x^2(x+1)}$ satisfies $h(n) >
0$ for all $n \in{\mathbb{N}}$
while $h(n) \to0$ as $n \to\infty$. By Lemma {e}f{L: short}, we have
\[
{\mathbb{E}}_n [h(Z)] = {\mathbb{E}}_n [ 1/Z ]
- {\mathbb{E}}_n \biggl[ \frac{1}{Z^2 (Z+1) } \biggr] =
\frac{1}{n} - \frac{1}{n^2(n-1)} + O ({e}^{\alpha_1 n} ) ,
\]
which is less than $h(n)$ for $n$ sufficiently large. In particular,
$h(Z_k)$ is a~positive supermartingale for $Z_k$ outside a finite set. Hence,
a standard result such as \cite{Asmussen}, Proposition 5.4, page 22,
implies that the Markov chain $(Z_k)$ is transient.
\end{pf*}
\section{Moment and tail estimates for $Z_{k+1} - Z_k$}
\lambdaabel{S:zinc}
In order to study the asymptotic behavior
of $(Z_k)_{k \in{\mathbb{Z}}_+}$, we build on the analysis of Section
{e}f{S: drift} to obtain more information
about the increments $Z_{k+1} - Z_k$.
We write $\Delta_k := Z_{k+1} - Z_k$ ($k \in{\mathbb{Z}}_+)$.
From the relation to the uniform renewal process, by
Lemma {e}f{Lemma: same distribution},
we have that
\begin{equation}
\lambdaabel{eq:coupleDelta}
{\mathbb{P}}( \Delta_k > x \mid Z_k = n) = {\mathbb{P}}\bigl( N(n) > 2n + x \bigr)
= {\mathbb{P}}\Biggl( \sum_{i=1}^{2n+x} \chi_i \lambdaeq n \Biggr) ,
\end{equation}
where $\chi_1, \chi_2, \lambdadots$ are i.i.d. $U(0,1)$ random variables,
using the notation at~({e}f{eq:renewproc}).
Lemma {e}f{lem:expotail} below gives a tail bound for $|\Delta_k|$ based
on ({e}f{eq:coupleDelta}) and a sharp bound for the moment
generating function of a $U(0,1)$ random variable,
for which we have not been able to find a reference and
which we state first since it may be of interest
in its own right.
\begin{lemma}\lambdaabel{lem:inequality}
For $\chi$ a $U(0,1)$ variable with moment generating function given
for $\lambda\in{\mathbb{R}}$ by
\begin{equation}
\lambdaabel{phidef}
\phi(\lambda) = {\mathbb{E}}[ {e}^{\lambda\chi} ] = \frac{{e}^\lambda- 1}{\lambda} ,
\end{equation}
we have
\[
\lambdaog\phi( -\lambda) \lambdaeq-\frac{\lambda}{2} + \frac{\lambda^2}{24}\qquad ( \lambda\geq0);\qquad
\lambdaog\phi(\lambda) \lambdaeq\frac{\lambda}{2} + \frac{\lambda^2}{24}\qquad ( \lambda\geq0)
.
\]
\end{lemma}
\begin{pf}
Consider the first of the two stated inequalities. Exponentiating
and multiplying both sides by $\lambda{e}^{\lambda/2}$, this is equivalent
to
\begin{equation}
\lambdaabel{shine}
2 \sinh(\lambda/2) \lambdaeq\lambda\exp( \lambda^2 /24 )
\end{equation}
for all $\lambda\geq0$. Inequality ({e}f{shine}) is easily verified
since both sides are entire functions with
nonnegative Taylor coefficients and the right-hand
series dominates the left-hand series term by term,
because $6^n n! \lambdaeq(2n +1)!$ for all $n \in{\mathbb{N}}$.
The second stated inequality reduces to ({e}f{shine})
also on exponentiating and multiplying through
by $\lambda{e}^{-\lambda/2}$.
\end{pf}
Now we can state our tail bound for $|\Delta_k|$.
The bound in Lemma {e}f{lem:expotail} is a slight improvement
on that provided by Bernstein's inequality in this particular case; the latter
yields a weaker bound with $4x$ instead of $2x$ in the denominator of the
exponential.
\begin{lemma}\lambdaabel{lem:expotail}
For $n \in{\mathbb{N}}$ and any integer $x \geq0$, we have
\[
{\mathbb{P}}( | \Delta_k | >x \mid Z_k=n)\lambdae2 \exp\biggl\{ -\frac
{3x^2}{4n+2x}\biggr \}.
\]
\end{lemma}
\begin{pf}
From ({e}f{eq:coupleDelta}) and Markov's
inequality, we obtain for $x \geq0$ and any \mbox{$\lambda\ge0$},
\begin{eqnarray*}
{\mathbb{P}}(\Delta_k >x \mid Z_k=n)
&=&{\mathbb{P}}\Biggl( \exp\Biggl\{ -\lambda\sum_{i=1}^{2n+x} \chi_i \Biggr\} \geq{e}^{-\lambda n}
\Biggr)\\
&\lambdae&\exp\{ \lambda n+(2n+x) \lambdaog\phi(-\lambda) \},
\end{eqnarray*}
where $\phi$ is given by ({e}f{phidef}).
With $\lambda=6x/(2n+x)$, the first inequality of Lemma~{e}f{lem:inequality} yields
\[
{\mathbb{P}}(\Delta_k >x \mid Z_k=n) \lambdae\exp\biggl\{ - \frac{x \lambdaambda}{4} \biggr\} =
\exp\biggl\{ -\frac{3x^2}{4n+2x}\biggr\}.
\]
On the other hand,
for $x\in[0,n-1]$, from ({e}f{eq:coupleDelta}) and Markov's
inequality once more,
\begin{eqnarray*}
{\mathbb{P}}(\Delta_k \lambdae-x \mid Z_k=n)
& = & {\mathbb{P}}\Biggl(\sum_{i=1}^{2n-x} \chi_i > n \Biggr)
= {\mathbb{P}}\Biggl( \exp\Biggl\{ \lambda\sum_{i=1}^{2n-x} \chi_i \Biggr\} > {e}^{\lambda n} \Biggr)\\
&\lambdae& \exp\{ -\lambda n+(2n-x) \lambdaog\phi(\lambda) \}.
\end{eqnarray*}
On setting $\lambda=6x/(2n-x)$, the second inequality in
Lemma {e}f{lem:inequality} yields, for any $x \in[0,n-1]$,
\[
{\mathbb{P}}(\Delta_k < - x \mid Z_k=n) \lambdae\exp\biggl\{ -\frac{3x^2}{4n-2x} \biggr\} \lambdae
\exp\biggl\{ -\frac{3x^2}{4n+2x} \biggr\},
\]
while ${\mathbb{P}}( \Delta_k < -n \mid Z_k = n) =0$.
Combining the left and right tail bounds completes the proof.
\end{pf}
Next, from Lemma {e}f{lem:expotail},
we obtain the following large deviation and moment bounds
for $\Delta_k$.
\begin{lemma}
\lambdaabel{cor:Etails}
Suppose that $\varepsilon>0$. Then for some $C < \infty$ and all $n \in{\mathbb{N}}$,
\begin{equation}
\lambdaabel{deltatail}
{\mathbb{P}}\bigl( | \Delta_k | > n^{(1/2)+\varepsilon} \mid Z_k=n \bigr)
\lambdae C \exp\{ -n^{\varepsilon} \}.
\end{equation}
Also for each $r \in{\mathbb{N}}$, there exists $C(r) < \infty$
such that for any $n \in{\mathbb{N}}$,
\begin{equation}
\lambdaabel{deltamoms}
{\mathbb{E}}[ |\Delta_k|^r \mid Z_k=n ] \lambdae C(r) n^{r/2} .
\end{equation}
\end{lemma}
\begin{pf}
The bound ({e}f{deltatail})
is straightforward
from Lemma {e}f{lem:expotail}. For $r \in{\mathbb{N}}$,
\begin{eqnarray}
\lambdaabel{gammas}
{\mathbb{E}}[ |\Delta_k|^r \mid Z_k=n ] & \lambdaeq&\int_{0}^\infty{\mathbb{P}}( | \Delta
_k | \geq
\lambdafloor x^{1/r} \rfloor\mid Z_k = n)\, {d} x\nonumber\hspace*{-30pt}\\[-8pt]\\[-8pt]
& \lambdaeq & C \int_{0}^{n^r} \exp\biggl\{ - \frac{x^{2/r}}{2n}\biggr \} \,{d} x
+ C \int_{n^r}^{\infty} \exp\biggl\{ - \frac{x^{1/r}}{2} \biggr\} \,{d}
x\nonumber\hspace*{-30pt}
\end{eqnarray}
for some $C < \infty$, by Lemma {e}f{lem:expotail}. With the
substitution $y = x^{1/r}$,
the second integral on the last line of ({e}f{gammas})
is seen to\vadjust{\goodbreak} be $O ( n^{r-1} {e}^{-n})$ by asymptotics for the
incomplete Gamma function. The first integral
on the last line of ({e}f{gammas}), with
the substitution $y = (2n)^{-1} x^{2/r}$, is equal to
\[
\frac{(2n)^{r/2} r}{2} \int_0^{n/2} {e}^{-y} y ^{(r/2) -1} \,{d} y
\lambdaeq\Gamma(r/2) (2n)^{r/2} r /2.
\]
Combining the last two upper bounds we verify ({e}f{deltamoms}).
\end{pf}
The next result gives sharp asymptotics for the first
two moments of
$\Delta_k = Z_{k+1} - Z_k$.
\begin{lemma}
\lambdaabel{lem:Zincrements}
Let
$\alpha_1 \approx-2.0888$ be as in Theorem {e}f{th:zinc}. Then as $n
\to\infty$,
\begin{eqnarray}
\lambdaabel{zinc1}
{\mathbb{E}}[ \Delta_k \mid Z_k=n ] & =& \tfrac23+O({e}^{\alpha_1 n}),\\
\lambdaabel{zinc2}
{\mathbb{E}}[ \Delta_k ^2 \mid Z_k=n ] & =& \tfrac23 n+ \tfrac23 + O (n {e}
^{\alpha_1 n}) .
\end{eqnarray}
\end{lemma}
\begin{pf}
Equation ({e}f{zinc1}) is immediate from ({e}f{zinc}).
Now we observe that $J_n := X_n^2 + Y_n^2 -n$ is a martingale. Indeed,
for any $(x,y) \in{\mathbb{Z}}^2$,
\begin{eqnarray*}
&&{\mathbb{E}}[ J_{n+1} - J_n \mid(X_n, Y_n ) = (x,y) ]\\
&&\qquad = \frac{|x|}{|x|+|y|} \bigl(2 y \operatorname{sgn}(x) + 1\bigr) + \frac{|y|}{|x|+|y|} \bigl(-2 x
\operatorname{sgn}(y) + 1\bigr) -1 = 0 .
\end{eqnarray*}
Between times $\nu_k$ and $\nu_{k+1}$, the urn takes $Z_k+Z_{k+1}$ steps,
so $\nu_{k+1} - \nu_k = Z_k+Z_{k+1}$. Moreover,
$J_{\nu_k} = Z_k^2 - \nu_k$.
Applying the optional stopping theorem at $\nu_k$
and $\nu_{k+1}$, we have that
\begin{eqnarray*}
J_{\nu_k} &=& Z_k^2 - \nu_k =
{\mathbb{E}}[ J_{\nu_{k+1}} \mid Z_k ] = {\mathbb{E}}[ Z_{k+1}^2 -\nu_{k+1} \mid Z_k ]\\
&=& {\mathbb{E}}[ Z_{k+1}^2 - Z_{k+1} \mid Z_k ] - \nu_k - Z_k.
\end{eqnarray*}
The optional stopping theorem is applicable here since a.s. $J_n \lambdaeq C
n^2$ for some $C< \infty$
and all $n$, while there is an exponential tail-bound for $\nu_{k+1}
-\nu_k$
(see Lemma~{e}f{lem:time} below).
Rearranging the equation in the last display, it follows that for $n
\in{\mathbb{N}}$,
\begin{equation} \lambdaabel{eq:sq}
{\mathbb{E}}[ Z_{k+1}^2 \mid Z_k=n ] = n^2+n+ {\mathbb{E}}[ Z_{k+1} \mid Z_k=n ].
\end{equation}
Writing $\Delta_k =Z_{k+1}-Z_k$,
we have that
\[
{\mathbb{E}}[ \Delta_k^2 \mid Z_k = n ] = {\mathbb{E}}[ Z_{k+1}^2 \mid Z_k=n ] -2n {\mathbb{E}}[
Z_{k+1} \mid Z_k=n ] + n^2,
\]
which with ({e}f{eq:sq}) and ({e}f{zinc}) gives ({e}f{zinc2}).
\end{pf}
\begin{remark*}In view of Lemma {e}f{Lemma: same distribution}, we
could have
used renewal
theory (e.g., \cite{smith2}) to estimate ${\mathbb{E}}[ \Delta_k^2 \mid Z_k
=n]$. However,
no result we could find in the literature would yield a bound as sharp
as that in ({e}f{zinc2}).
\end{remark*}
Lemma {e}f{Lemma: same distribution} with ({e}f{zinc1}) and ({e}f{zinc2})
implies an ancillary result on the $U(0,1)$ renewal\vadjust{\goodbreak} process.
\begin{corollary}
\lambdaabel{uniform}
Let $N(t)$ be the counting function of
the uniform renewal process, as defined by ({e}f{eq:renewproc}). Then with
$\alpha_1 \approx-2.0888$ as in Theorem~{e}f{th:zinc}, as $t \to
\infty$,
\[
{\mathbb{E}}[ N(t)^2 ] = 4t^2 + \tfrac{10}{3} t + \tfrac{2}{3} + O (t {e}
^{\alpha_1 t} );\qquad
\operatorname{\mathbb{V}\mathrm{ar}}[ N(t) ] = \tfrac{2}{3} t + \tfrac{2}{9} + O (t {e}^{\alpha_1 t}
) .
\]
\end{corollary}
These asymptotic results are both sharper than any we have seen
in the literature; see, for example, \cite{jensen,sparac} in the particular
case of the uniform renewal process or \cite{smith2} for the
general case. We remark that the formula given
in \cite{sparac}, page 231, for ${\mathbb{E}}[N(t)^2]$ contains an error
(in \cite{sparac} the renewal at $0$ is not counted, so the
notation $m_k(\cdot)$ there is equivalent to our ${\mathbb{E}}[ (N(\cdot)
-1)^k]$).
\section{Asymptotic analysis of the noisy urn}
\lambdaabel{sec:tildez}
\subsection{Connection to Lamperti's problem}
\lambdaabel{sec:lamperti}
In this section, we study the noisy urn model described in Section {e}f
{S:noisy}.
To study the asymptotic behavior of $({\tilde Z}_k)_{k \in{\mathbb{Z}}_+}$, it
turns out to be more convenient to work with\vspace*{-1.5pt}
the process $(W_k)_{k \in{\mathbb{Z}}_+}$ defined by $W_k = {\tilde Z}_k^{1/2}$,
since the latter
process has asymptotically-zero
drift, in a sense to be made precise shortly,
and such processes have been well-studied
in the literature.
Let $(W_k)_{k \in{\mathbb{Z}}_+}$ be an irreducible time-homogeneous Markov
chain whose
state-space is an unbounded
countable subset of ${\mathbb{R}}_+$. Define the increment moment functions
\begin{equation}
\lambdaabel{mudef}
\mu_r(x) := {\mathbb{E}}[ (W_{k+1} - W_k)^r \mid W_k = x ];
\end{equation}
by the Markov property, when the corresponding moments exist
the $\mu_r(x)$ are genuine functions of $x$. Given a reasonable choice
of scale for the process~$W_k$, it is common that $\mu_2(x)$ be uniformly
bounded away from $0$ and $\infty$. In this case, under some mild additional
regularity conditions, the regime where $x| \mu_1 (x)| = O(1)$ is critical
from the point of view of the recurrence classification of $W_k$.
For a nearest-neighbor random walk on ${\mathbb{Z}}_+$ this fact had been
known for a long time (see \cite{harris}), but a study of this and
many other aspects of the problem, in much greater generality
(with absence of the Markovian and countable state-space
assumptions), was carried out by Lamperti \cite{lamp1,lamp2,lamp3}
using martingale techniques. Thus, the analysis of processes with
asymptotically zero drift [i.e., $\mu_1(x) \to0$] is sometimes
known as \textit{Lamperti's problem}.
We will next state some consequences
of Lamperti's results that we will use.
For convenience, we impose conditions that are stronger than
Lamperti's.
We suppose that for each $r \in{\mathbb{N}}$,
\begin{equation}
\lambdaabel{mubound}
\sup_{x } | \mu_r(x) | < \infty.
\end{equation}
The recurrence and transience properties of $W_k$
were studied by Lamperti \cite{lamp1,lamp3} and his results were
refined by Menshikov, Asymont and Iasnogorodskii \cite{mai}. Parts (i)
and (ii) of the following\vadjust{\goodbreak}
result are consequences of Theorems 3.1 and 3.2 of \cite{lamp1} with
Theorem 2.1 of \cite{lamp3},
while
part (iii) is a~consequence of
Theorem 3 of \cite{mai} (which is in fact a much sharper result).
\begin{prop}[(\cite{lamp1,lamp3,mai})]
\lambdaabel{prop:lampclass}
Let $( W_k )$ be an irreducible Markov chain on a countable unbounded
subset of ${\mathbb{R}}_+$.
Suppose that
({e}f{mubound}) holds, and that there exists $v > 0$ such that
$\mu_2(x) > v$ for all $x$ sufficiently large.
Then the following recurrence criteria are valid:
\begin{itemize}[(iii)]
\item[(i)] $W_k$ is transient if there exist $\delta, x_0 \in
(0,\infty)$ such that
for all $x>x_0$,
\[
2x \mu_1 (x) - \mu_2 (x) > \delta.
\]
\item[(ii)] $W_k$ is positive-recurrent if there exist $\delta, x_0
\in(0,\infty)$ such that
for all $x>x_0$,
\[
2x \mu_1 (x) + \mu_2(x) < - \delta.
\]
\item[(iii)] $W_k$ is null-recurrent if there exists $x_0 \in
(0,\infty)$ such that for all $x>x_0$,
\[
2x | \mu_1 (x) | \lambdaeq\biggl( 1 + \frac{1}{\lambdaog x} \biggr) \mu_2 (x) .
\]
\end{itemize}
\end{prop}
In \cite{lamp2},
Lamperti proved the existence of weak-sense limiting diffusions for certain
processes satisfying parts (i) or (iii) of Proposition {e}f{prop:lampclass}.
To state Lamperti's result,
we need some more notation.
To describe the time-homogeneous
diffusions on ${\mathbb{R}}_+$
that arise here, it will suffice to describe the
infinitesimal mean $\mu(x)$ and infinitesimal variance
$\sigma^2 (x)$; see, for example, \cite{kt2}, Chapter~15. The transition
functions $p$
of our diffusions will then satisfy the Kolmogorov backward
equation
\[
\frac{ \partial p}{\partial t} = \mu(x) \frac{\partial p}{\partial x}
+\frac{1}{2} \sigma^2 (x) \frac{ \partial^2 p}{\partial x^2} .
\]
Let $(H^{\alpha,\beta}_t)_{t \in[0,1]}$
denote a diffusion process
on ${\mathbb{R}}_+$ with
infinitesimal mean and variance
\begin{equation}
\lambdaabel{diffdef}
\mu(x) = \frac{\alpha}{x},\qquad \sigma^2 (x) = \beta.
\end{equation}
The particular case of a diffusion satisfying ({e}f{diffdef}) with
$\beta=1$ and
$\alpha= (\gamma-1)/2$ for some $\gamma\in{\mathbb{R}}$ is a Bessel process
with parameter $\gamma$; in this case we use the
notation $V^\gamma_t = H^{(\gamma-1)/2,1}_t$.
Recall that for
$\gamma\in{\mathbb{N}}$, the law of $(V^\gamma_t)_{t \in[0,1]}$
is the same as that of $\| B_t \|_{t \in[0,1]}$ where $(B_t)_{t \in[0,1]}$
is standard $\gamma$-dimensional Brownian motion. In fact, any
$H^{\alpha,\beta}_t$
is related to a Bessel process via simple scaling, as the next result shows.
\begin{lemma}
\lambdaabel{lem:bessel}
Let $\alpha\in{\mathbb{R}}$ and $\beta>0$.
The diffusion process $H_t^{\alpha,\beta}$ is a scaled Bessel process:
\[
(H_t^{\alpha,\beta})_{t \in[0,1]}
\mbox{ has the same law as }
( \beta^{1/2} V_t^\gamma)_{t \in[0,1]} \qquad \mbox{with } \gamma= 1 +
\frac{2\alpha}{\beta} .\vadjust{\goodbreak}
\]
\end{lemma}
\begin{pf}
By the It\^o transformation formula
(cf. page 173 of \cite{kt2}), for any $\beta>0$
the process $( \beta^{1/2} V^\gamma_t)_{t \in[0,1]}$
is a diffusion process on $[0,1]$ with infinitesimal mean $\mu(x) =
\beta(\gamma-1)/(2x)$
and infinitesimal variance $\sigma(x) = \beta$,
from which we obtain the result.
\end{pf}
We need the following form
of Lamperti's invariance principle (\cite{lamp2}, Theorems~2.1, 5.1 and A.2).
\begin{prop}[(\cite{lamp2})]
\lambdaabel{prop:lampinv}
Let $(W_k)$ be an irreducible Markov chain on a~countable unbounded
subset of ${\mathbb{R}}_+$.
Suppose that
({e}f{mubound}) holds, and that
\[
\lambdaim_{x \to\infty} \mu_2 (x) = \beta>0,\qquad \lambdaim_{x \to\infty} x
\mu_1 (x) = \alpha> - (\beta/2) .
\]
Let $(H^{\alpha,\beta}_t)_{t\geq0}$ be a diffusion process
as defined at ({e}f{diffdef}). Then as $k \to\infty$,
\[
( k^{-1/2} W_{kt} )_{t \in[0,1]} \to( H^{\alpha,\beta}_t )_{t \in
[0,1]}
\]
in the sense of
convergence of finite-dimensional distributions.
Marginally,
\begin{eqnarray*}
\lambdaim_{k \to\infty} {\mathbb{P}}( k^{-1/2} W_k \lambdaeq y ) &=& \frac{2}{(2 \beta
)^{(\alpha/\beta)+(1/2)} \Gamma((\alpha/\beta) + (1/2)) }\\
&&{}\times\int_0^y r^{2\alpha/\beta} \exp\bigl( - r^2/(2 \beta) \bigr) \,{d} r .
\end{eqnarray*}
\end{prop}
\subsection{Increment moment estimates for $W_k$}
\lambdaabel{sec:increments}
Now consider the\vspace*{-1pt} process $(W_k)_{k \in{\mathbb{Z}}_+}$ where $W_k = {\tilde Z}_k^{1/2}$;
this is a Markov chain with a countable state space (since
${\tilde Z}_k$ is), so fits into the framework described in Section {e}f
{sec:lamperti}
above. Lemma {e}f{lem:sqrtincrements} below
shows that indeed $W_k$ is an instance of Lamperti's problem in the critical
regime. First we need some simple properties of the random variable
$\kappa$.
\begin{lemma}
\lambdaabel{kappalem}
If $\kappa$ satisfies ({e}f{kappabound}) for $\lambdaambda>0$, then
\begin{equation}
\lambdaabel{kappatail} {\mathbb{P}}( | \kappa| \geq x) \lambdaeq\exp\{ - \lambdaambda x
\}\qquad (x \geq0)
\end{equation}
and
\begin{equation}
\lambdaabel{kappamoms}
{\mathbb{E}}[ | \kappa|^r ] < \infty\qquad ( r \geq0).
\end{equation}
\end{lemma}
\begin{pf}
({e}f{kappatail}) is immediate from Markov's inequality and ({e}f
{kappabound}),
and ({e}f{kappamoms}) is also straightforward.
\end{pf}
Now we can start our analysis of the noisy urn and the associated process~$\tilde Z_k$.
Recall that ${\tilde Z}_k$ is defined as $\max\{ |\tilde X_{\tilde\nu_k
+1}| , |\tilde Y_{\tilde\nu_k +1 } | \}$.
By definition of the noisy urn process, if we start at unit distance
away from an
axis (in the anticlockwise sense), the path of the noisy urn until it
hits the next axis
has the
same distribution as the corresponding path in the original simple
harmonic urn. Since we refer
to this fact often, we state it as a lemma.
\begin{lemma}
\lambdaabel{noisylem}
Given $\tilde Z_k = z$, the path $(\tilde X_n, \tilde Y_n)$ for $n =
\tilde\nu_k +1,\lambdadots, \tilde\nu_{k+1}$
has the same distribution as the
path $(X_n, Y_n)$ for $n = \nu_k +1,\lambdadots, \nu_{k+1}$ given \mbox{$Z_k = z$}.
In particular,
${\tilde Z}_{k+1}$ conditioned
on ${\tilde Z}_k = z$ has the same
distribution as $Z_{k+1} - \min\{ \kappa, Z_{k+1} - 1 \}=Z_{k+1} -
\kappa+ (\kappa+1-Z_{k+1}) {\mathbf{1}} \{ \kappa\ge Z_{k+1} \}$ conditioned
on \mbox{$Z_k = z$}.
\end{lemma}
Recall that $\Delta_k = Z_{k+1} - Z_k$, and write ${\tilde\Delta}_k = {\tilde Z}_{k+1}
- {\tilde Z}_k$. The next result
is an analogue of Lemmas {e}f{cor:Etails} and {e}f{lem:Zincrements}
for ${\tilde\Delta}_k$.
\begin{lemma}
\lambdaabel{lem:tdelta}
Suppose that ({e}f{kappabound}) holds.
Let $\varepsilon>0$. Then for some $C < \infty$ and all $n \in{\mathbb{N}}$,
\begin{equation}
\lambdaabel{Anbound}
{\mathbb{P}}\bigl( | {\tilde\Delta}_k | > n^{(1/2)+\varepsilon} \mid{\tilde Z}_k = n \bigr) \lambdaeq C \exp\{ -
n^{\varepsilon/3} \} .
\end{equation}
Also, for any $r \in{\mathbb{N}}$, there exists $C <\infty$ such that for any
$n \in{\mathbb{N}}$,
\begin{equation}
\lambdaabel{tdeltamoms} {\mathbb{E}}[ | {\tilde\Delta}_k |^r \mid{\tilde Z}_k = n ] \lambdaeq C
n^{r/2} .
\end{equation}
Moreover, there exists $\gamma>0$ for
which, as $n \to\infty$,
\begin{eqnarray}
\lambdaabel{tzinc1}
{\mathbb{E}}[ {\tilde\Delta}_k \mid{\tilde Z}_k = n ]
& =& \tfrac{2}{3} - {\mathbb{E}}[\kappa] + O ({e}^{ - \gamma n}) ,\\
\lambdaabel{tzinc2}
{\mathbb{E}}[ {\tilde\Delta}_k^2 \mid{\tilde Z}_k = n ] & =& \tfrac{2}{3} n + O(1).
\end{eqnarray}
\end{lemma}
\begin{pf}
By the final statement in Lemma {e}f{noisylem},
for any $r \geq0$,
\[
{\mathbb{P}}( | \tilde\Delta_k | > r \mid{\tilde Z}_k = n )
\lambdaeq
{\mathbb{P}}( | \Delta_k - \kappa| >r \mid Z_k = n ) .
\]
We have for any $\varepsilon>0$,
\begin{eqnarray*}
&&{\mathbb{P}}\bigl( | \Delta_k - \kappa| > n^{(1/2)+\varepsilon} \mid Z_k =
n \bigr)\\
&&\qquad \lambdaeq{\mathbb{P}}\bigl( | \Delta_k | > n^{(1+\varepsilon)/2} \mid Z_k = n\bigr)
+ {\mathbb{P}}\bigl( | \kappa| > n^{(1+\varepsilon)/2} \bigr)
\end{eqnarray*}
for all $n$ large enough. Using the bounds in ({e}f{deltatail}) and
({e}f{kappatail}), we obtain ({e}f{Anbound}).
For $r \in{\mathbb{N}}$,
\[
{\mathbb{E}}[ | {\tilde\Delta}_k |^r \mid{\tilde Z}_k = n ] \lambdaeq{\mathbb{E}}[ ( | \Delta_k | + |\kappa
| )^r \mid Z_k = n ] .
\]
Then with Minkowski's inequality, ({e}f{deltamoms})
and ({e}f{kappamoms}) we obtain ({e}f{tdeltamoms}).
Next, we have from Lemma {e}f{noisylem} and ({e}f{zinc1}) that
\begin{eqnarray*}
&&{\mathbb{E}}[ {\tilde\Delta}_k \mid{\tilde Z}_k = n ]\\
&&\qquad = {\mathbb{E}}[ \Delta_k -\kappa
+ (\kappa+1-Z_{k+1}) {\mathbf{1}} \{ \kappa\ge Z_{k+1}\} \mid Z_k = n ] \\
&&\qquad = \tfrac{2}{3} + O ({e}^{\alpha_1 n}) -{\mathbb{E}}[\kappa] + {\mathbb{E}}[
(\kappa+1-Z_{k+1}) {\mathbf{1}} \{ \kappa\ge Z_{k+1}\} \mid Z_k = n ].
\end{eqnarray*}
By the Cauchy--Schwarz inequality, ({e}f{kappamoms}) and the
bound $0\lambdae\kappa+1-Z_{k+1}\lambdae\kappa$, the last term here is
bounded by a constant times the square-root of
\begin{eqnarray*}
{\mathbb{P}}( \kappa\geq Z_{k+1} \mid Z_k =n ) &\lambdaeq&{\mathbb{P}}( | \Delta_k | \geq n/2
\mid Z_k = n ) + {\mathbb{P}}( | \kappa| > n/2 )\\
&=& O ( \exp\{ - \lambdaambda n /2 \} ),
\end{eqnarray*}
using the bounds ({e}f{deltatail}) and ({e}f{kappatail}). Hence, we
obtain ({e}f{tzinc1}).
Similarly, from~({e}f{zinc2}), we obtain ({e}f{tzinc2}).
\end{pf}
Now, we can give the main result of this section on the increments
of the process $(W_k)_{k \in{\mathbb{Z}}_+}$.
\begin{lemma}
\lambdaabel{lem:sqrtincrements}
Suppose that $\kappa$ satisfies ({e}f{kappabound}).
With $\mu_r(x)$ as defined by ({e}f{mudef}), we have
\begin{equation}
\lambdaabel{wlem1}
\sup_x | \mu_r (x)| < \infty
\end{equation}
for each $r \in{\mathbb{N}}$. Moreover as $x \to\infty$,
\begin{equation}
\lambdaabel{wlem2}
\mu_1 (x) = \frac{1-2{\mathbb{E}}[\kappa] }{4x} + O(x^{-2} );\qquad \mu_2 (x) =
\frac{1}{6} + O(x^{-1}) .
\end{equation}
\end{lemma}
\begin{pf}
For the duration of this proof, we write ${\mathbb{E}}_{x^2} [ \cdot]$
for ${\mathbb{E}}[ \cdot\mid{\tilde Z}_k = x^2 ] = {\mathbb{E}}[ \cdot\mid W_k = x ]$.
For $r \in{\mathbb{N}}$ and $x \geq0$, from ({e}f{mudef}),
\begin{equation}
\lambdaabel{murbound}
| \mu_r(x) | \lambdaeq{\mathbb{E}}_{x^2} [ | {\tilde Z}_{k+1}^{1/2} - {\tilde Z}_k^{1/2} |^r ]
= x^r {\mathbb{E}}_{x^2} [ | ( 1+ x^{-2} {\tilde\Delta}_k )^{1/2} - 1 |^r ].
\end{equation}
Fix $\varepsilon>0$ and write
$A(n) := \{ | {\tilde\Delta}_k | > n^{(1/2)+\varepsilon}\}$ and $A^c(n)$ for the
complementary event.
Now for some $C < \infty$ and all $x \geq1$, by Taylor's theorem,
\[
| ( 1+ x^{-2} {\tilde\Delta}_k )^{1/2} - 1 |^r {\mathbf{1}}_{A^c(x^2)}
\lambdaeq C x^{-2r} | {\tilde\Delta}_k |^r {\mathbf{1}}_{A^c(x^2)} .
\]
Hence,
\begin{equation}
\lambdaabel{murbound1}
\quad{\mathbb{E}}_{x^2} \bigl[ | ( 1+ x^{-2} {\tilde\Delta}_k )^{1/2} - 1 |^r {\mathbf{1}}_{A^c(x^2)} \bigr]
\lambdaeq C x^{-2r} {\mathbb{E}}_{x^2} [ | {\tilde\Delta}_k |^r ] = O( x^{-r} )
\end{equation}
by ({e}f{tdeltamoms}).
On the other hand, using the fact that for $y \geq-1$,
$0 \lambdaeq(1+y)^{1/2} \lambdaeq1 + (y/2)$, we have
\begin{eqnarray*}
&&{\mathbb{E}}_{x^2} \bigl[ | ( 1+ x^{-2} {\tilde\Delta}_k )^{1/2} - 1 |^r {\mathbf{1}}_{A(x^2)} \bigr]\\
&&\qquad \lambdaeq{\mathbb{E}}_{x^2} \bigl[ \bigl(1 + (1/2) x^{-2} | {\tilde\Delta}_k | \bigr)^r {\mathbf{1}}_{A(x^2)} \bigr]
\\
&&\qquad \lambdaeq\bigl( {\mathbb{E}}_{x^2} [ (1 + | {\tilde\Delta}_k |)^{2r} ] \bigr)^{1/2}
\bigl( {\mathbb{P}}\bigl( A(x^2) \mid{\tilde Z}_k = x^2 \bigr) \bigr)^{1/2}
\end{eqnarray*}
for $x \geq1$,
by Cauchy--Schwarz. Using ({e}f{tdeltamoms})
to bound the expectation
here and ({e}f{Anbound})
to bound the probability, we obtain, for any $r \in{\mathbb{N}}$,
\begin{equation}
\lambdaabel{murbound2}
{\mathbb{E}}_{x^2} \bigl[ | ( 1+ x^{-2} {\tilde\Delta}_k )^{1/2} - 1 |^r {\mathbf{1}}_{A(x^2)} \bigr]
= O ( \exp\{ - x^{\varepsilon/2} \} ).
\end{equation}
Combining ({e}f{murbound1}) and ({e}f{murbound2}) with ({e}f{murbound}),
we obtain ({e}f{wlem1}).
Now, we prove ({e}f{wlem2}).
We have that for $x \geq0$,
\begin{eqnarray}
\lambdaabel{zmu0}
\mu_1 (x) & =& {\mathbb{E}}_{x^2} [ W_{k+1} - W_k ]
= x {\mathbb{E}}_{x^2} [ (1 + x^{-2} {\tilde\Delta}_k )^{1/2} -1 ]\nonumber\\[-8pt]\\[-8pt]
& =& x {\mathbb{E}}_{x^2} \bigl[ \bigl( (1 + x^{-2} {\tilde\Delta}_k )^{1/2} -1 \bigr) {\mathbf{1}}_{A^c (x^2)} \bigr] +
O ( \exp\{ - x^{\varepsilon/3} \} ) ,\nonumber
\end{eqnarray}
using ({e}f{murbound2}).
By Taylor's theorem with Lagrange form for the remainder, we have
\begin{eqnarray}
\lambdaabel{zmu1}
&&\quad x {\mathbb{E}}_{x^2} \bigl[ \bigl( (1 + x^{-2} {\tilde\Delta}_k )^{1/2} -1 \bigr) {\mathbf{1}}_{A^c(x^2)} \bigr]
\nonumber\\[-8pt]\\[-8pt]
&&\quad\qquad = \frac{1}{2x} {\mathbb{E}}_{x^2} \bigl[ {\tilde\Delta}_k {\mathbf{1}}_{A^c(x^2)} \bigr] -
\frac{1}{8x^3} {\mathbb{E}}_{x^2} \bigl[ {\tilde\Delta}_k^2 {\mathbf{1}}_{A^c(x^2)} \bigr]
+ O ( x^{-5} {\mathbb{E}}_{x^2} [ | {\tilde\Delta}_k|^3 ] ) .\nonumber
\end{eqnarray}
Here we have that
$x^{-5} {\mathbb{E}}_{x^2} [ | {\tilde\Delta}_k|^3 ] = O( x^{-2} )$,
by ({e}f{tdeltamoms}),
while for $r \in{\mathbb{N}}$, we obtain
\[
{\mathbb{E}}_{x^2} \bigl[ {\tilde\Delta}_k^r {\mathbf{1}}_{A^c(x^2)} \bigr] = {\mathbb{E}}_{x^2} [ {\tilde\Delta}_k^r ]
+ O \bigl( \bigl({\mathbb{E}}_{x^2} [ | {\tilde\Delta}_k |^{2r} ] {\mathbb{P}}\bigl( A (x^2) \mid{\tilde Z}_k = x^2 \bigr)
\bigr)^{1/2} \bigr)
\]
by Cauchy--Schwarz. Using ({e}f{Anbound}) again and combining
({e}f{zmu0}) with ({e}f{zmu1}), we obtain
\[
\mu_1 (x) = \frac{1}{2x} {\mathbb{E}}_{x^2} [ {\tilde\Delta}_k ] -
\frac{1}{8x^3} {\mathbb{E}}_{x^2} [ {\tilde\Delta}_k^2 ]
+ O ( x^{-2} ) .
\]
Thus, from ({e}f{tzinc1}) and ({e}f{tzinc2}),
we obtain
the expression for $\mu_1$ in ({e}f{wlem2}).
Now, we use the fact that
\begin{eqnarray*}
(W_{k+1} - W_k)^2 &=& W_{k+1}^2 - W_k^2 - 2 W_k (W_{k+1} - W_k) \\
&=& {\tilde Z}_{k+1} - {\tilde Z}_k - 2W_k (W_{k+1} - W_k )
\end{eqnarray*}
to obtain
$\mu_2 (x) = {\mathbb{E}}_{x^2} [ {\tilde\Delta}_k ] - 2 x \mu_1 (x)$,
which with ({e}f{tzinc1})
yields the expression for $\mu_2$ in ({e}f{wlem2}).
\end{pf}
\section{Proofs of theorems}
\lambdaabel{sec:proofs}
\subsection{\texorpdfstring{Proofs of Theorems \protect{e}f{th:leaky}, \protect{e}f{thm:mixed}, \protect{e}f
{thm:moments} and \protect{e}f{thm:difflimit}}
{Proofs of Theorems 2.3, 2.4, 2.5, and 2.8}}
First, we work with the noisy urn model of Section {e}f{S:noisy}.
Given the moment estimates of Lemma {e}f{lem:sqrtincrements}, we
can now apply the general results described in Section
{e}f{sec:lamperti} and \cite{aim}.
\begin{pf*}{Proof of Theorem {e}f{thm:mixed}}
First, observe that $({\tilde Z}_k)_{k \in{\mathbb{Z}}_+}$ is transient, null-, or
positive-recurrent
exactly when $(W_k)_{k \in{\mathbb{Z}}_+}$ is.
From Lemma {e}f{lem:sqrtincrements}, we have that
\begin{eqnarray*}
2x \mu_1(x) - \mu_2 (x) &=& \tfrac{1}{3} - {\mathbb{E}}[\kappa]+ O(x^{-1});\\
2 x \mu_1 (x) + \mu_2 (x) &=& \tfrac{2}{3} - {\mathbb{E}}[\kappa] + O(x^{-1}).
\end{eqnarray*}
Now apply Proposition {e}f{prop:lampclass}.
\end{pf*}
\begin{pf*}{Proof of Theorem {e}f{thm:moments}}
By the definition of
$\tau_q$ at ({e}f{returntime}),
$\tau_q$ is also the first hitting time of $1$
by $(W_k)_{k \in{\mathbb{N}}}$.
Then with Lemma {e}f{lem:sqrtincrements} we can
apply results of Aspandiiarov, Iasnogorodski and Menshikov \cite{aim}, Propositions 1 and 2, which generalize
those of Lamperti \cite{lamp3} and give
conditions on $\mu_1$ and $\mu_2$ for existence and nonexistence of
passage-time moments,
to obtain the stated result.
\end{pf*}
\begin{pf*} {Proof of Theorem {e}f{thm:difflimit}}
First, Proposition {e}f{prop:lampinv} and Lemma {e}f{lem:sqrtincrements}
imply that, as $n \to\infty$,
\[
( n^{-1/2} W_{nt} )_{t \in[0,1]} \to( H^{\alpha,\beta}_t )_{t \in[0,1]}
\]
in the sense of finite-dimensional distributions,
where $\alpha= (1-2{\mathbb{E}}[\kappa])/4$ and $\beta=1/6$, provided ${\mathbb{E}}
[\kappa] < 2/3$.
By the It\^o transformation formula
(cf. page~173 of \cite{kt2}), with $H^{\alpha,\beta}_t$ as defined at
({e}f{diffdef}),
$(H^{\alpha,\beta}_t)^2$ is a diffusion process
with infinitesimal mean $\mu(x) = \beta+2\alpha$ and infinitesimal
variance $\sigma^2 (x) = 4 \beta x$. In particular,
$(H^{\alpha,\beta}_t)^2$ has the same law as the process
denoted~$D_t$ in the statement of Theorem {e}f{thm:difflimit}.
Convergence of finite-dimensional distributions for
$( n^{-1} W_{nt}^2 )_{t \in[0,1]}
= ( n^{-1} {\tilde Z}_{nt} )_{t \in[0,1]}$ follows. The final statement in
the theorem
follows from Lemma~{e}f{lem:bessel}.
\end{pf*}
Next, consider the leaky urn model of Section {e}f{S:leaky}.
\begin{pf*} {Proof of Theorem {e}f{th:leaky}}
This is an immediate
consequence of the ${\mathbb{P}}(\kappa=1)=1$ cases of Theorems {e}f
{thm:mixed} and {e}f{thm:moments2}.
\end{pf*}
\begin{remark*}
There is a short proof of the first part of Theorem
{e}f{th:leaky} due to the existence of a particular martingale.
Consider the process
$Q'_n$ defined by
$Q'_n = Q(X'_n, Y'_n)$, where
\[
Q(x,y) := \bigl(x + \tfrac{1}{2} \operatorname{sgn}(y ) - \tfrac{1}{2} \mathbf{1}_{\{y =
0\}} \operatorname{sgn}(x ) \bigr)^2
+ \bigl(y - \tfrac{1}{2} \operatorname{sgn}(x ) - \tfrac{1}{2} {\mathbf{1}}_{ \{x = 0 \}} \operatorname{sgn}(y )
\bigr)^2 .
\]
It turns out that $Q'_n$ is a (nonnegative) martingale.
Thus, it converges a.s. as $n \to\infty$. But since
$Q(x,y) \to\infty$
as $\| (x,y)\| \to\infty$, we must have
that eventually $(X'_n,Y'_n)$ gets trapped
in the closed class $\mathcal{C}$. So ${\mathbb{P}}(\tau< \infty)=1$.
\end{remark*}
\subsection{\texorpdfstring{Proofs of Theorems \protect{e}f{thm:moments2} and \protect{e}f{thm:area}}
{Proofs of Theorems 2.6 and 2.9}}
The proofs of Theorems~{e}f{thm:moments2} and~{e}f{thm:area}
that we give in this section both rely on the
good estimates we have for the embedded process $\tilde Z_k$ to
analyze the noisy urn $(\tilde X_n, \tilde Y_n)$.
The main additional ingredient is to
relate the two different time-scales. The first result
concerns the time to traverse a quadrant.
\begin{lemma}
\lambdaabel{lem:time}
Let $k \in{\mathbb{Z}}_+$.
The distribution of $\tilde\nu_{k+1} - \tilde\nu_k$ given ${\tilde Z}_k =n$
coincides with that of $Z_{k+1} + Z_k$ given $Z_k =n$.
In addition,
\begin{equation}
\lambdaabel{noisytime}
\tilde\nu_{k+1} - \tilde\nu_k = | \tilde X_{\tilde\nu_{k+1}} |
+ | \tilde Y_{\tilde\nu_{k+1}} | + \tilde Z_k .
\end{equation}
Moreover,
\begin{equation}
\lambdaabel{timetail}
{\mathbb{P}}( \tilde\nu_{k+1} - \tilde\nu_k > 3n \mid\tilde Z_k = n ) = O (
\exp\{ - n^{1/2} \} ) .
\end{equation}
\end{lemma}
\begin{pf}
Without loss of generality, suppose that we are traversing the first
quadrant. Starting at time $\tilde\nu_k + 1$, Lemma {e}f{noisylem}
implies that the time until hitting the next axis,
$\tilde\nu_{k+1} - \tilde\nu_k -1$, has the same distribution as
the time
taken for the original simple harmonic urn to hit the next axis, starting
from $(Z_k, 1)$.
In this time, the simple harmonic urn must
make $Z_k$ horizontal jumps and $Z_{k+1} -1$ vertical jumps. Thus,
$\tilde\nu_{k+1} - \tilde\nu_k -1$ has the same
distribution as $Z_{k+1} + Z_k -1$, conditional
on $Z_k = \tilde Z_k$. Thus, we obtain the first statement in the lemma.
For equation ({e}f{noisytime}), note that between times $\tilde\nu_k
+1$ and
$\tilde\nu_{k+1}$ the noisy urn must make ${\tilde Z}_k$ horizontal steps and
(in this case) $|\tilde Y_{\tilde\nu_{k+1}} |-1$ vertical steps.
Finally, we have from the first statement of the lemma that
\[
{\mathbb{P}}( \tilde\nu_{k+1} - \tilde\nu_k > 3n \mid\tilde Z_k = n )
= {\mathbb{P}}( Z_{k+1} > 2n \mid Z_k = n ) ,
\]
and then ({e}f{timetail}) follows from ({e}f{deltatail}).
\end{pf}
Very roughly speaking, the key to our Theorems {e}f{thm:moments2}
and {e}f{thm:area} is the fact that
$\tau\approx\sum_{k=0}^{\tau_q} W^2_k$ and $A \approx\sum
_{k=0}^{\tau_q} W^4_k$.
Thus, to study $\tau$ and $A$ we need to look at sums of powers of
$W_k$ over a \textit{single excursion}. First, we will give results
for $S_\alpha:= \sum_{k=0}^{\tau_q} W^\alpha_k$, $\alpha\geq0$.
Then we quantify the
approximations
``$\approx$'' for $\tau$ and $A$ by a series of bounds.
Let $M := \max_{0 \lambdaeq k \lambdaeq\tau_q} W_k$ denote the maximum of
the first excursion of~$W_k$. For ease of notation, for the
rest of this section we set $r := 6 {\mathbb{E}}[ \kappa] - 3$.
\begin{lemma}
\lambdaabel{max} Suppose that $r > -1$. Then for any $\varepsilon>0$, for all
$x$ sufficiently large
\[
x^{-1-r-\varepsilon} \lambdaeq{\mathbb{P}}( M \geq x ) \lambdaeq x^{-1-r+\varepsilon} .
\]
In particular, for any $\varepsilon>0$, ${\mathbb{E}}[ M^{1+r+\varepsilon} ]= \infty$ but
${\mathbb{E}}[ M^{1+r-\varepsilon} ]< \infty$.
\end{lemma}
\begin{pf}
It follows from Lemma {e}f{lem:sqrtincrements} and
some routine Taylor's theorem computations
that for any $\varepsilon>0$ there exists $w_0 \in[1, \infty)$ such that for
any $x \geq w_0$,
\begin{eqnarray*}
{\mathbb{E}}[ W_{k+1}^{1+r+\varepsilon} - W_k ^{1+r+\varepsilon} \mid W_k = x] & \geq&0, \\
{\mathbb{E}}[ W_{k+1}^{1+r-\varepsilon} - W_k ^{1+r-\varepsilon} \mid W_k = x] & \lambdaeq&0.
\end{eqnarray*}
Let $\eta:= \min\{k \in{\mathbb{Z}}_+ \dvtx W_k \lambdaeq w_0 \}$ and $\sigma_x :=
\min\{ k \in{\mathbb{Z}}_+ \dvtx W_k \geq x \}$. Recall that $(W_k)_{k \in
{\mathbb{Z}}_+}$ is an irreducible time-homogeneous Markov chain on a
countable subset of $[1,\infty)$. It follows that to prove the
lemma it suffices to show that, for some $w \geq2w_0$, for any
$\varepsilon>0$,
\begin{equation}
\lambdaabel{eq0}
x^{-1-r-\varepsilon} \lambdaeq{\mathbb{P}}( \sigma_x < \eta\mid W_0 = w )
\lambdaeq x^{-1-r+\varepsilon}
\end{equation}
for all $x$ large enough.
We first prove the lower bound in ({e}f{eq0}). Fix $x > w$. We
have that $W_{k \wedge\eta\wedge\sigma_x}^{1+r+\varepsilon}$\vspace*{2pt} is a
submartingale, and, since $W_k$ is an irreducible Markov chain,
$\eta< \infty$ and $\sigma_x < \infty$ a.s. Hence
\[
{\mathbb{P}}( \sigma_x < \eta) {\mathbb{E}}[ W_{\sigma_x}^{1+r+\varepsilon} ]
+ \bigl(1 - {\mathbb{P}}( \sigma_x < \eta) \bigr) {\mathbb{E}}[ W_{\eta}^{1+r+\varepsilon} ] \geq
w^{1+r+\varepsilon} .
\]
Here $W_\eta\lambdaeq w_0$ a.s., and for some $C \in(0,\infty)$ and all
$x > w$,
\[
{\mathbb{E}}[ W_{\sigma_x}^{1+r+\varepsilon} ] \lambdaeq
{\mathbb{E}}\bigl[ \bigl( x + ( W_{\sigma_x} - W_{\sigma_x -1} )
\bigr)^{1+r+\varepsilon} \bigr] \lambdaeq C x^{1+r+\varepsilon} ,
\]
since ${\mathbb{E}}[ (
W_{\sigma_x} - W_{\sigma_x -1} )^{1+r+\varepsilon} ]$ is uniformly
bounded in $x$, by equation ({e}f{wlem1}). It follows that
\[
{\mathbb{P}}( \sigma_x < \eta) [ C x^{1+r+\varepsilon} - w_0^{1+r+\varepsilon} ] \geq
w^{1+r+\varepsilon} - w_0^{1+r+\varepsilon} > 0 ,
\]
which yields the lower bound
in ({e}f{eq0}). The upper bound follows by a similar argument
based on the supermartingale property of $W_{k \wedge\eta\wedge
\sigma_x}^{1+r-\varepsilon}$.
\end{pf}
The next result gives the desired moment bounds for $S_\alpha$.
\begin{lemma}
\lambdaabel{slem} Let $\alpha\geq0$ and $r > -1$. Then ${\mathbb{E}}[ S_\alpha^p] <
\infty$ if $p< \frac{1+r}{\alpha+2}$ and ${\mathbb{E}}[ S_\alpha^p] =
\infty$ if $p> \frac{1+r}{\alpha+2}$.
\end{lemma}
\begin{pf}
First we prove the upper bound. Clearly, $S_\alpha\lambdaeq
(1+\tau_q) M^\alpha$. Then, by H\"older's inequality,
\[
{\mathbb{E}}[ S_\alpha^p ] \lambdaeq\bigl( {\mathbb{E}}\bigl[ (1+\tau_q)^{(2+\alpha)p/2} \bigr] \bigr)^{\bfrac
{2}{2+\alpha}} \bigl( {\mathbb{E}}\bigl[ M^{(2+\alpha)p} \bigr] \bigr)^{\bfrac{\alpha}{2+\alpha
}} .
\]
For $p < \frac{1+r}{\alpha+2}$ we have $(2+\alpha)p/2 < (1+r)/2 =
3{\mathbb{E}}[\kappa] -1$
and $(2+\alpha)p < 1+r$ so that Lemma {e}f{max} and Theorem
{e}f{thm:moments} give the upper bound.
For the lower bound, we claim that there exists $C \in(0,\infty)$
such that
\begin{equation}
\lambdaabel{claim3}
{\mathbb{P}}( S_\alpha\geq x )
\geq\tfrac{1}{2} {\mathbb{P}}\bigl( M \geq C x^{\bfrac{1}{\alpha+2}} \bigr)
\end{equation}
for all $x$ large enough.
Given the claim ({e}f{claim3}), we have, for any $\varepsilon>0$,
\[
{\mathbb{E}}[ S_\alpha^p ] \geq\frac{p}{2} \int_1^\infty x^{p-1}{\mathbb{P}}\bigl( M \geq
C x^{\bfrac{1}{\alpha+2}}
\bigr)\, {d} x
\geq\frac{p}{2} \int_1^\infty x^{p-1} x^{-\afrac{1+r}{\alpha+2}
-\varepsilon} \,{d} x
\]
by Lemma {e}f{max}. Thus, ${\mathbb{E}}[ S_\alpha^p]=\infty$ for $p > \frac
{1+r}{\alpha+2}$.
It remains to verify ({e}f{claim3}).
Fix $y > 2$. Let ${\mathcal{F}}_k=\sigma(W_1,\dots,W_k)$, and define stopping times
\[
\sigma_1 = \min\{ k \in{\mathbb{N}}\dvtx W_k \geq y \} ;\qquad
\sigma_2 = \min\{ k \geq\sigma_1 \dvtx W_k \lambdaeq y/2 \} .
\]
Then $\{ \sigma_1 < \tau_q \}$, that is, the event that
$W_k$ reaches $y$ before $1$, is ${\mathcal{F}}_{\sigma_1}$-measurable.
Now
\begin{equation}
\lambdaabel{eq5}
\qquad {\mathbb{P}}( \{ \sigma_1 < \tau_q \} \cap\{ \sigma_2 \geq
\sigma_1 + \delta y^2 \} )
= {\mathbb{E}}[ {\mathbf{1}} \{ \sigma_1 < \tau_q \} {\mathbb{P}}( \sigma_2 \geq\sigma_1 +
\delta y^2 \mid{\mathcal{F}}_{\sigma_1} ) ] .
\end{equation}
We claim that there exists $\delta>0$ so that
\begin{equation}
\lambdaabel{claim1a} {\mathbb{P}}( \sigma_2 \geq\sigma_1 + \delta y^2 \mid{\mathcal{F}}
_{\sigma_1} )
\geq\tfrac{1}{2} \qquad \mbox{a.s.}
\end{equation}
Let $D_k = (y - W_k )^2 {\mathbf{1}} \{ W_k < y\}$.
Then, with $\Delta_k = W_{k+1} - W_k$,
\[
{\mathbb{E}}[ D_{k+1} - D_k \mid{\mathcal{F}}_k ] \lambdaeq2(W_k-y) {\mathbb{E}}[ \Delta_k \mid{\mathcal{F}}_k
] + {\mathbb{E}}[ \Delta_k^2 \mid{\mathcal{F}}_t ] .
\]
Lemma {e}f{lem:sqrtincrements} implies that on $\{W_k > y/2\}$ this
last display is bounded above by some
$C <\infty$ not depending on $y$.
Hence, an appropriate maximal inequality (\cite{mvw}, Lemma 3.1), implies
(since $D_{\sigma_1} =0$)
that
${\mathbb{P}}( \max_{0 \lambdaeq s \lambdaeq k} D_{(\sigma_1+s) \wedge\sigma_2} \geq w
) \lambdaeq C k/w$.
Then, since $D_{\sigma_2} \geq y^2 /4$, we have
\begin{eqnarray*}
{\mathbb{P}}( \sigma_2 \lambdaeq\sigma_1 + \delta y^2 \mid{\mathcal{F}}_{\sigma_1} )
&\lambdaeq&{\mathbb{P}}\Bigl( \max_{1 \lambdaeq s \lambdaeq\delta y^2} D_{(\sigma_1 + s) \wedge
\sigma_2} \geq(y^2/4) \bigm|{\mathcal{F}}_{\sigma_1} \Bigr)\\
&\lambdaeq&\frac{C \delta y^2}{(y^2/4) } \lambdaeq\frac{1}{2}\qquad \mbox{a.s.}
\end{eqnarray*}
for
$\delta>0$ small enough. Hence, ({e}f{claim1a}) follows. Combining
({e}f{eq5}) and ({e}f{claim1a}), we get
\[
{\mathbb{P}}( \{ \sigma_1 < \tau_q \} \cap\{ \sigma_2 \geq\sigma_1 +
\delta y^2 \} )
\geq\tfrac{1}{2} {\mathbb{P}}( \sigma_1 < \tau_q ) = \tfrac{1}{2} {\mathbb{P}}( M
\geq y ) .
\]
Moreover, on $\{ \sigma_1 < \tau_q \} \cap\{
\sigma_2 \geq\sigma_1 + \delta y^2 \}$ we have that $W_s \geq
y/2$ for all $\sigma_1 \lambdaeq s < \sigma_2$, of which there are at least
$\delta y^2$ values; hence $S_\alpha\geq\delta y^2 \times
(y/2)^\alpha$. Now taking
$x = 2^{-\alpha} \delta y^{2+\alpha}$, we obtain ({e}f{claim3}), and
so complete the proof.
\end{pf}
Next, we need a technical lemma.
\begin{lemma}
\lambdaabel{kappasums}
Let $p \geq0$. Then for any $\varepsilon>0$ there exists $C<\infty$ such that
\begin{equation} \lambdaabel{claim2}
{\mathbb{E}}\Biggl[ \Biggl( \sum_{k=1}^{\tau_q} |\kappa_{\tilde\nu_{k}}| \Biggr)^p \Biggr]
\lambdaeq C {\mathbb{E}}[ \tau_q^{p+\varepsilon} ] .
\end{equation}
\end{lemma}
\begin{pf}
For any $s \in(0,1)$,
\[
{\mathbb{P}}\Biggl( \sum_{k=1}^{\tau_q} |\kappa_{\tilde\nu_{k}}| > x \Biggr)
\lambdaeq{\mathbb{P}}( \tau_q > x^s ) + {\mathbb{P}}\Biggl( \sum_{k=1}^{x^s} |\kappa_{\tilde
\nu_{k}}| > x \Biggr) .
\]
For any random variable $X$,
\[
{\mathbb{E}}[ X^p] = p \int_0^\infty x^{p-1} {\mathbb{P}}(
X > x)\,{d} x
\lambdaeq1 + p \int_1^\infty x^{p-1} {\mathbb{P}}( X >x ) \,{d} x;
\]
so
\begin{eqnarray}
\lambdaabel{eq3}
{\mathbb{E}}\Biggl[ \Biggl( \sum_{k=1}^{\tau_q} |\kappa_{\tilde\nu_{k}}| \Biggr)^p \Biggr]
&\lambdaeq&1 + p \int_1^\infty x^{p-1} {\mathbb{P}}( \tau_q > x^s ) \,{d} x \nonumber\\[-8pt]\\[-8pt]
&&{}+ p
\int_1^\infty x^{p-1} {\mathbb{P}}\Biggl( \sum_{k=1}^{x^s} |\kappa_{\tilde\nu
_{k}}| > x \Biggr)\, {d} x .\nonumber
\end{eqnarray}
Here, we have that
\[
{\mathbb{P}}\Biggl( \sum_{k=1}^{x^s} |\kappa_{\tilde\nu_{k}}| > x \Biggr) \lambdaeq
{\mathbb{P}}\Biggl( \bigcup_{k=1}^{x^s} \{ |\kappa_{\tilde\nu_{k}}| > x^{1-s} \} \Biggr)
\lambdaeq\sum_{k=1}^{x^s} {\mathbb{P}}( |\kappa| > x^{1-s} )
\]
by Boole's inequality. Then Markov's inequality and the
moment bound ({e}f{kappabound}) yield
\begin{equation}
\lambdaabel{eq4} {\mathbb{P}}\Biggl( \sum_{k=1}^{x^s} |\kappa_{\tilde\nu_{k}}| > x \Biggr)
\lambdaeq x^s {\mathbb{P}}\bigl( {e}^{\lambdaambda| \kappa|} > {e}^{x^{1-s}} \bigr)
\lambdaeq x^s {\mathbb{E}}\bigl[ {e}^{\lambdaambda| \kappa| } \bigr] {e}^{-x^{1-s}} .
\end{equation}
It follows that, since $s<1$, the final integral in ({e}f{eq3})
is finite for any $p$. Also, from Markov's inequality,
for any $\varepsilon>0$,
\[
\int_1^\infty x^{p-1} {\mathbb{P}}( \tau_q > x^s ) \,{d} x \lambdaeq{\mathbb{E}}[ \tau
_q^{p+\varepsilon} ] \int_1^\infty
x^{p-1-s(p+\varepsilon)} \,{d} x;
\]
taking $s$ close to $1$ this last integral is finite, and ({e}f
{claim2}) follows
(noting $\tau_q \geq1$ by definition).
\end{pf}
\begin{pf*}{Proof of Theorem {e}f{thm:moments2}}
By the definitions of $\tau$ and $\tau_q$, we have that
$\tau= \tilde\nu_{\tau_q} = -1+ \sum_{k=1}^{\tau_q} ( \tilde\nu
_k - \tilde\nu_{k-1} )$, recalling $\tilde\nu_0 = -1$.
Hence, by Lemma {e}f{lem:time},
\[
\tau= -1 + \sum_{k=1}^{\tau_q} ( W_{k-1}^2 + W_{k}^2 ) + R
\]
for $R$ a random variable such that $| R | \lambdaeq\sum_{k=1}^{\tau_q}
|\kappa_{\tilde\nu_{k}}|$.
It follows that
\begin{equation}
\lambdaabel{bounds1} -1 + \sum_{k=0}^{\tau_q} W_k^2 - | R | \lambdaeq\tau
\lambdaeq2 \sum_{k=0}^{\tau_q} W_k^2 + |R| .
\end{equation}
Lemma {e}f{kappasums} implies that for any $\varepsilon>0$ there exists $C<
\infty$ such that
${\mathbb{E}}[ |R|^p ] \lambdaeq C {\mathbb{E}}[ \tau_q^{p+\varepsilon} ]$.
The ${\mathbb{E}}[ \kappa] > 1/3$ case of the theorem now follows from ({e}f
{bounds1}) with
Theorem {e}f{thm:moments}, Lemma {e}f{slem} and Minkowski's
inequality. In the ${\mathbb{E}}[\kappa] =1/3$ case,
it is required to prove that ${\mathbb{E}}[\tau^p] = \infty$ for any $p>0$;
this follows from the
${\mathbb{E}}[\kappa] = 1/3$ case of Theorem {e}f{thm:moments} and the fact
that $\tau\geq\tau_q$ a.s.
\end{pf*}
\begin{pf*}{Proof of Theorem {e}f{thm:area}}
First, note that we can write
\[
A = \sum_{n=1}^\tau T_n = \sum_{n=1}^{\tilde\nu_{\tau_q}} T_n =
\sum_{k=1}^{\tau_q} A_k ,
\]
where $A_1 = \sum_{n=1}^{\tilde\nu_1} T_n$ and
$A_k = \sum_{n= \tilde\nu_{k-1} +1}^{\tilde\nu_k} T_n$ ($k \geq2$)
is the area swept
out in traversing a quadrant for the $k$th time. Since $A_k \geq1/2$,
part (i) of the
theorem is immediate from part (i) of Theorem {e}f{thm:mixed}. For
part (ii), we have
that
\begin{eqnarray*}
A_k & \lambdaeq&( \tilde Z_k + | \kappa_{\tilde\nu_k} | )
( \tilde Z_{k-1} + | \kappa_{\tilde\nu_{k-1}} | ) \\
& \lambdaeq& W_k^4 + W_{k-1}^4 + W_{k-1}^2 | \kappa_{\tilde\nu_{k}} | +
W_{k}^2 | \kappa_{\tilde\nu_{k-1}} |
+ | \kappa_{\tilde\nu_{k-1}} || \kappa_{\tilde\nu_{k}} | .
\end{eqnarray*}
Thus,
\[
A \lambdaeq2 \sum_{k=0}^{\tau_q} W_k^4 + R_1 + R_2 + R_3 ,
\]
where $R_1\!=\!\sum_{k=1}^{\tau_q}
W_{k-1}^2 | \kappa_{\tilde\nu_{k}} |$,
$R_2\!=\!\sum_{k=1}^{\tau_q}
W_{k}^2 | \kappa_{\tilde\nu_{k-1}} |$ and
$R_3\!=\!\sum_{k=1}^{\tau_q}
| \kappa_{\tilde\nu_{k-1}} || \kappa_{\tilde\nu_{k}} |$.
Here $\sum_{k=0}^{\tau_q} W_k^4$ has finite $p$th moment
for $p< \frac{3 {\mathbb{E}}[ \kappa] -1}{3}$, by Lemma {e}f{slem}.
Next we deal with the terms $R_1, R_2$ and $R_3$.
Consider $R_1$. We have that, by H\"older's inequality, $ {\mathbb{E}}[ |R_1 |^p
]$ is at most
\begin{eqnarray*}
&&{\mathbb{E}}\Biggl[ \Biggl(\sum_{k=1}^{\tau_q}
W_{k-1}^2 \Biggr)^{3p/2} \Biggr]^{2/3} {\mathbb{E}}\Biggl[ \Biggl(
\sum_{k=0}^{\tau_q} | \kappa_{\tilde\nu_{k}} | \Biggr)^{3p}
\Biggr]^{1/3}\\
&&\qquad \lambdaeq C' {\mathbb{E}}\Biggl[ \Biggl(\sum_{k=0}^{\tau_q}
W_{k}^2 \Biggr)^{3p/2} \Biggr]^{2/3} {\mathbb{E}}[ \tau_q^{3p+\varepsilon} ]^{1/3}
\end{eqnarray*}
for any $\varepsilon>0$, by ({e}f{claim2}). Lemma {e}f{slem} and Theorem
{e}f{thm:moments}
show that this is finite\vspace*{1pt} provided $p < \frac{3 {\mathbb{E}}[ \kappa] -1}{3}$
(taking $\varepsilon$ small enough).
A similar argument holds for~$R_2$. Finally,
\[
{\mathbb{E}}[ |R_3|^p ] \lambdaeq{\mathbb{E}}\Biggl[ \Biggl( \sum_{k=0}^{\tau_q}
| \kappa_{\tilde\nu_{k}} | \Biggr)^{2p} \Biggr]
\lambdaeq C'' {\mathbb{E}}[ \tau_q^{2p+\varepsilon} ]
\]
for any $\varepsilon>0$, by ({e}f{claim2}). For $\varepsilon$ small enough, this
is also finite when $p < \frac{3 {\mathbb{E}}[ \kappa] -1}{3}$
by Theorem {e}f{thm:moments}. These estimates and Minkowski's
inequality then complete
the proof.
\end{pf*}
\subsection{\texorpdfstring{Proof of Theorem \protect{e}f{thm:perc}}{Proofs of Theorem 2.11}}
We now turn to the percolation model
described in Section {e}f{S: percolation}.
\begin{lemma}
\lambdaabel{lem:perc1}
Let $v$ and $v'$ be any two vertices of $G$. Then with probability~1
there exists a vertex $w \in G$ such that the unique semi-infinite
oriented paths in $H$ from $v$ and $v'$ both pass through $w$.
\end{lemma}
\begin{pf}
Without loss of generality,
suppose $v, v'$ are distinct vertices in~$G$
on the positive $x$-axis on the same sheet of $\mathcal{R}$. Let $Z_0 =
|v| < Z'_0 = |v'|$.
The two paths in $H$ started at $v$ and $v'$, call them
$P$ and $P'$, respectively, lead to instances
of processes $Z_k$ and $Z'_k$, each a copy
of the simple harmonic urn embedded process $Z_k$.
Until $P$ and $P'$ meet,
the urn processes they instantiate are independent. Thus, it suffices
to take $Z_k$, $Z'_k$ to be independent and show that they
eventually cross with probability 1, so that the
underlying paths must meet. To do this, we consider the
process $(H_k)_{k \in{\mathbb{Z}}_+}$ defined by $H_k := \sqrt{Z'_k} - \sqrt
{Z_k}$ and show that it is eventually less than
or equal to $0$.
For convenience, we use the notation $W_k = (Z_k)^{1/2}$ and $W'_k =
(Z'_k)^{1/2}$. Since $H_{k+1} - H_k = (W'_{k+1}-W'_k) - (W_{k+1} -
W_k)$, we have that for $x<y$,
\[
{\mathbb{E}}[ H_{k+1} - H_k \mid W_k = x, W'_k = y] = \frac{1}{4y} - \frac
{1}{4x} + O( x^{-2} )
= -\frac{(y-x)}{4xy} + O(x^{-2} )
\]
by the ${\mathbb{E}}[\kappa] =0$ case of
({e}f{wlem2}). Similarly,
\[
{\mathbb{E}}[ (H_{k+1} - H_k)^2 \mid W_k = x, W'_k = y] = \tfrac{1}{3} + O(
x^{-1} ) ,
\]
from ({e}f{wlem2}) again. Combining these, we see that
\begin{eqnarray*}
&&2 (y-x) {\mathbb{E}}[ H_{k+1} - H_k \mid W_k = x, W'_k = y] - {\mathbb{E}}[ (H_{k+1} -
H_k)^2 \mid W_k = x, W'_k = y] \\
&&\qquad \lambdaeq- \tfrac{1}{3} + O (x^{-1} ) < 0
\end{eqnarray*}
for $x > C$, say.
However, we know from Theorem {e}f{th:main1}
that $W_k$ is transient, so in particular
$W_k > C$ for all $k > T$ for some finite $T$.
Let
$\tau= \min\{ k \in{\mathbb{Z}}_+ : H_k \lambdaeq0 \}$. Then we have that
$H_{k} {\mathbf{1}}\{ k < \tau\}$, $k >T$, is a process on ${\mathbb{R}}_+$
satisfying Lamperti's recurrence criterion (cf. Proposition {e}f
{prop:lampclass}). Here
$H_{k \wedge\tau}$ is not a Markov process but
the general form of Proposition {e}f{prop:lampclass} applies (see~\cite{lamp1}, Theorem 3.2)
so we can conclude that
${\mathbb{P}}( \tau< \infty) = 1$.
\end{pf}
\begin{lemma}
\lambdaabel{lem:perc2}
The in-graph of any individual vertex in $H$ is almost surely finite.
\end{lemma}
\begin{pf}
We work in the dual percolation model $H'$.
As we have seen, the oriented paths through $H'$ simulate the leaky
simple harmonic urn via the mapping~${\mathbb{P}}hi$. The path in $H'$ that
starts from a vertex over $(n+1/2,1/2)$
explores the outer boundary of the in-graph in $H$ of a lift of the set
$\{(i,0)\dvtx 1 \lambdae i \lambdae n\}$.
The leaky urn a.s. reaches the
steady state with one ball, so every oriented path in $H'$ a.s.
eventually joins the infinite path cycling immediately around the origin.
It follows that the in-graph of any vertex over a~co-ordinate axis is
a.s. finite. For any vertex $v$ of $H$, the oriented path
from $v$ a.s. contains a vertex $w$ over an axis, and the in-graph
of~$v$ is contained in the in-graph of $w$, so it too is a.s. finite.
\end{pf}
All that remains to complete the proof of Theorem {e}f{thm:perc} is to
establish the two statements about the moments of $I(v)$. For $p < 2/3$,
${\mathbb{E}}[I(v)^p]$ is bounded above by ${\mathbb{E}}[A^p]$, where $A$ is the area
swept out by a path of the leaky simple harmonic urn, or equivalently
by a path of the noisy simple harmonic urn with ${\mathbb{P}}(\kappa= 1) = 1$
up to the hitting time $\tau$.
${\mathbb{E}}[A^p]$ is finite, by Theorem~{e}f{thm:area}(ii).
The final claim ${\mathbb{E}}[I(v)] = \infty$ will be proven in the next
section as equation ({e}f{in-graph}), using a connection with expected
exit times from quadrants.
\section{Continuous-time models}
\lambdaabel{sec:continuous}
\subsection{\texorpdfstring{Expected traversal time: Proof of Theorem \protect{e}f{th:fasttime}}
{Expected traversal time: Proof of Theorem 2.14}}
\lambdaabel{S:fasttime}\mbox{}
\begin{pf*}{Proof of Lemma {e}f{complexmartingale}}
A consequence of Dynkin's formula for a con\-tin\-u\-ous-time
Markov chain $X(t)$ on a countable state-space $S$ with infinitesimal
(generator) matrix $Q = (q_{ij})$
is that for a function $g\dvtx {\mathbb{R}}_+ \times S \to{\mathbb{R}}$ with continuous
time-derivative to be such that $g ( t, X(t) )$ is a local martingale,
it suffices that
\begin{equation}
\lambdaabel{dynkin}
\frac{ \partial g (t , x) }{\partial t} + Q ( g(t, \cdot) ) (t,x) =
0
\end{equation}
for all $x \in S$ and $t \in{\mathbb{R}}_+$: see, for example, \cite{robert}, page 364. In our case, $S = \mathbb{C} \setminus\{ 0 \}$,
$X(t) = A(t) + i B(t)$,
and for $z = x+ i y \in\mathbb{C}$,
\begin{eqnarray*}
Q ( f) (z ) &=& \sum_{w \in S, w \neq z} q_{z w}[ f(w) - f(z) ]\\
&=& | x| \bigl[ f \bigl( z + \operatorname{sgn}(x) i \bigr) - f(z) \bigr] + |y| \bigl[ f\bigl ( z - \operatorname{sgn}(y) \bigr) -
f(z) \bigr] .
\end{eqnarray*}
Taking $f(x+iy) = g(t, x+iy)$ to be first $x \cos t + y \sin t$
and second $y \cos t -x \sin t$, we verify the identity ({e}f{dynkin})
in each case.
Thus, the real and imaginary parts of $M_t$ are local martingales, and hence
martingales since it is not hard to see that ${\mathbb{E}}| A(t) + B(t) | <
\infty$.
\end{pf*}
To prove Theorem {e}f{th:fasttime}, we need the following bound on the
deviations of $\tau_f$ from $\pi/2$.
\begin{lemma}
\lambdaabel{lem:fastdev}
Suppose $\varepsilon_n>0$ and $\varepsilon_n \to0$ as $n \to\infty$. Let $\phi
_n \in[0,\pi/2]$.
Then as $n \to\infty$,
\[
{\mathbb{P}}\biggl( \biggl| \tau_f - \frac{\pi}{2} + \phi_n \biggr|
\geq\varepsilon_n \bigm| A(0) =n \cos\phi_n, B(0) = n \sin\phi_n \biggr) = O(
n^{-1} \varepsilon_n^{-2})
\]
uniformly in $(\phi_n)$.
\end{lemma}
\begin{pf}
First note that $M_0 = n {e}^{i \phi_n}$ and, by the martingale property,
\[
{\mathbb{E}}[ |M_t - M_0 |^2 ] = {\mathbb{E}}[ |M_t |^2 ] - |M_0|^2 =
{\mathbb{E}}[ A(t)^2 + B(t)^2 ] - n^2.
\]
We claim that for all $n \in{\mathbb{N}}$ and $t \in{\mathbb{R}}_+$,
\begin{equation}
\lambdaabel{claim1}
{\mathbb{E}}[ A(t)^2 + B(t)^2 ] - n^2 \lambdaeq\frac{t^2}{2} + 2^{1/2} n t .
\end{equation}
Since $M_t-M_0$ is a (complex) martingale,
$|M_t-M_0|^2$ is a submartingale. Doob's maximal
inequality therefore implies that, for any $r>0$,
\[
{\mathbb{P}}\Bigl( \sup_{0 \lambdaeq s \lambdaeq t} | M_s - M_0 | \geq r \Bigr) \lambdaeq r^{-2}
{\mathbb{E}}[ |M_t - M_0|^2 ] \lambdaeq2 t (t+n) r^{-2}
\]
by ({e}f{claim1}).
Set $t_0 = (\pi/2) -\phi_n + \theta$ for $\theta\in(0,\pi/2)$.
Then on $\{ t_0 < \tau_f \}$, $A(t_0) + i B(t_0)$
has argument in $[ \phi_n , \pi/2]$, so that
$M_{t_0}$ has argument in $[ 2 \phi_n\,{-}\,(\pi/2)\,{-}\,\theta,\allowbreak \phi_n -
\theta]$.
All points with argument in the latter interval
are at distance at least $n \sin\theta$ from $M_0$.
Hence, on $\{ t_0 < \tau_f \}$,
\[
\sup_{0 \lambdaeq s \lambdaeq t_0} | M_s - M_0 |
\geq| M_{t_0} - M_0 | \geq n \sin\theta.
\]
It follows that for $\varepsilon_n >0$ with $\varepsilon_n \to0$,
\begin{eqnarray*}
{\mathbb{P}}\bigl( \tau_f > (\pi/2) -\phi_n +\varepsilon_n \bigr) & \lambdaeq&{\mathbb{P}}\Bigl( \sup_{0 \lambdaeq
s \lambdaeq(\pi/2) -\phi_n +\varepsilon_n } | M_s - M_0 | \geq n
\sin\varepsilon_n \Bigr) \\
& =& O( n^{-1} (\sin\varepsilon_n)^{-2} ) = O( n^{-1} \varepsilon_n^{-2} ).
\end{eqnarray*}
A similar argument yields
the same bound for ${\mathbb{P}}_n ( \tau_f < (\pi/2) - \phi_n - \varepsilon_n )$.
It remains to prove the claim ({e}f{claim1}). First, note that
\begin{eqnarray*}
&&{\mathbb{E}}\bigl[ A(t+\Delta t)^2 + B(t +\Delta t)^2 - \bigl( A (t)^2 + B(t)^2 \bigr) \mid
A(t)=x, B(t)=y \bigr]\\
&&\qquad = (|x| + |y|) \Delta t + O ((\Delta t)^2 ),
\end{eqnarray*}
and $(|x| + |y|)^2 \lambdaeq2 (x^2 + y^2)$.
Writing $g(t) = {\mathbb{E}}[ A (t)^2 + B(t)^2 ]$, it follows that
\[
\frac{{d}}{{d} t} g(t) \lambdaeq\sqrt{2} g(t) ^{1/2}
\]
with $g(0) = n^2$. Hence, $g(t)^{1/2} \lambdaeq n + 2^{-1/2} t$. Squaring
both sides
yields ({e}f{claim1}).
\end{pf}
A consequence of Lemma {e}f{lem:fastdev} is that $\tau_f$
has finite moments of all orders, uniformly in the initial point.
\begin{lemma}
\lambdaabel{taufmoms}
$\!\!\!\!\!$For any $r\!>\!0$, there exists $C\!<\!\infty$ such that \mbox{$\max_{n \in{\mathbb{N}}}
{\mathbb{E}}_n [ \tau_f^r ]\!\lambdaeq\!C$}.
\end{lemma}
\begin{pf}
By Lemma {e}f{lem:fastdev}, we have that there exists $n_0 < \infty$
for which
\begin{equation}
\lambdaabel{taufeq}
\sup_{x >0, y>0 \dvtx | x+ i y | \geq n_0 }
{\mathbb{P}}\bigl( \tau_f - t > 2n_0 \mid A(t) + i B(t) = x + iy \bigr) \lambdaeq1/2
.
\end{equation}
On the other hand, if $| A(t) + iB(t) | < n_0$, we
have that $\tau_f - t$ is stochastically dominated by a sum of
$n_0$ exponential random variables with mean~$1$. Thus, by Markov's
inequality, the bound ({e}f{taufeq}) holds for \textit{all} \mbox{$x>0,
y>0$}. Then, for $t > 1$, by conditioning on the path of the
process at times $2n_0, 4n_0, \lambdadots, 2n_0 (t-1)$ and using the
strong Markov property we have
\begin{eqnarray*}
&&{\mathbb{P}}_n ( \tau_f > 2n_0 t )\\
&&\qquad \lambdaeq\prod_{j=1}^{t-1} \sup_{x_j>0, y_j>0 } {\mathbb{P}}\bigl( \tau_f - 2n_0 j >
2n_0 \mid A(2n_0 j) + i B(2n_0 j) = x_j + i y_j \bigr)\\
&&\qquad \lambdaeq2^{1-t}
\end{eqnarray*}
by ({e}f{taufeq}). Hence, ${\mathbb{P}}_n ( \tau_f > t )$ decays faster than
any power of
$t$, uniformly in~$n$.
\end{pf}
\begin{pf*}{Proof of Theorem {e}f{th:fasttime}}
For now fix $n \in{\mathbb{N}}$.
Suppose $A(0) =Z_0 = n$, $B(0) = 0$. Note that $A(\tau_f) = 0$,
$B(\tau_f) = Z_1$.
The stopping time $\tau_f$ has all moments, by Lemma {e}f{taufmoms},
while ${\mathbb{E}}_n [ | M_t |^2 ] = O(t^2)$ by ({e}f{claim1}),
and ${\mathbb{E}}_n [ | M_{\tau_f} |^2 ] = {\mathbb{E}}_n [ Z_1^2 ] < \infty$.
It follows that the real and imaginary parts of the
martingale~$M_{t \wedge\tau_f}$ are
uniformly integrable. Hence,
we can apply the optional stopping theorem to
any linear combination of the real and imaginary parts of~$M_{t \wedge
\tau_f}$ to obtain
\[
{\mathbb{E}}_n [ Z_1 ( \alpha\sin\tau_f + \beta\cos\tau_f ) ] = \alpha n
\]
for any $\alpha, \beta\in{\mathbb{R}}$. Taking $\alpha= \cos\theta$,
$\beta= \sin\theta$ this says
\begin{eqnarray*}
n \cos\theta&=& {\mathbb{E}}_n [ Z_1 \sin(\theta+\tau_f) ] \\
&=& {\mathbb{E}}_n [ (Z_1 -
{\mathbb{E}}_n Z_1 ) \sin(\theta+\tau_f) ]
+ {\mathbb{E}}_n [ Z_1 ] {\mathbb{E}}_n [ \sin(\theta+ \tau_f) ]
\end{eqnarray*}
for any $\theta$.
By Cauchy--Schwarz,
the first term on the right-hand side here is bounded in absolute value
by $\sqrt{\operatorname{\mathbb{V}\mathrm{ar}}_n(Z_1)}$, so on rearranging we have
\[
\biggl| {\mathbb{E}}_n [ \sin(\theta+\tau_f) ] - \frac{n\cos\theta}{{\mathbb{E}}_n [Z_1]}
\biggr| \lambdae\frac{( \operatorname{\mathbb{V}\mathrm{ar}}_n(Z_1) )^{1/2} }{{\mathbb{E}}_n [Z_1 ]}
\lambdae\frac{( {\mathbb{E}}_n [ \Delta_1^2] )^{1/2} }{{\mathbb{E}}_n [Z_1 ]} ,
\]
and then using ({e}f{zinc1}) and ({e}f{zinc2}) we obtain, as $n \to
\infty$,
\begin{equation}
\lambdaabel{eqtheta}
| {\mathbb{E}}_n [ \sin(\theta+\tau_f) ] - \cos\theta| = O (n^{-1/2})
\end{equation}
uniformly in $\theta$. This strongly suggests that $\tau_f$ is concentrated
around $\pi/2,\allowbreak 5\pi/2, \lambdadots.$ To rule out the larger values, we need
to use Lemma {e}f{lem:fastdev}. We proceed as follows.
Define the event $E_n := \{ | \tau_f - (\pi/2) | < \varepsilon_n \}$ where
$\varepsilon_n \to0$.
From the $\theta=-\pi/2$ case of ({e}f{eqtheta}) we have that
${\mathbb{E}}_n [\sin( \tau_f - (\pi/2)) ] = O(n^{-1/2})$.\vadjust{\goodbreak} Since $\sin x = x
+ O(x^3 )$
as $x \to0$ we have
\begin{eqnarray*}
&&{\mathbb{E}}_n \bigl[ {\mathbf{1}}_{E_n} \sin\bigl( \tau_f - (\pi/2)\bigr) \bigr]\\
&&\qquad = {\mathbb{E}}_n [ \tau_f {\mathbf{1}}_{E_n} ] - \frac{\pi}{2} + O(\varepsilon_n^3) + O ( {\mathbb{P}}
_n (E_n^c) ) \\
&&\qquad = {\mathbb{E}}_n [ \tau_f ] - \frac{\pi}{2} + O(\varepsilon_n^3)+ O \bigl( ({\mathbb{E}}_n [
\tau_f^r] )^{1/r} ({\mathbb{P}}_n (E_n^c))^{1-(1/r)} \bigr)
\end{eqnarray*}
for any $r>1$, by H\"older's inequality. Here ${\mathbb{E}}_n [ \tau_f^r ] =
O(1)$, by Lemma {e}f{taufmoms},
so that for any $\delta>0$, choosing $r$ large enough we see that the
final term in the last display is $O(n^{\delta-1} \varepsilon_n^{-2} )$ by Lemma
{e}f{lem:fastdev}. Hence, for any $\delta>0$,
\[
O(n^{-1/2}) = {\mathbb{E}}_n [ \tau_f ] - \frac{\pi}{2} + O (n^{\delta-1}
\varepsilon_n^{-2} ) + O(\varepsilon_n^3)
+ {\mathbb{E}}_n \bigl[ {\mathbf{1}}_{E^c_n} \sin\bigl( \tau_f - (\pi/2)\bigr) \bigr] ,
\]
and this last expectation is $O( n^{-1} \varepsilon_n^{-2} )$ by Lemma {e}f
{lem:fastdev}
once more. Taking $\varepsilon_n = n^{-1/4}$ yields ({e}f{fast1}).
Next, from the $\theta=0$ case of ({e}f{eqtheta}) we have that
${\mathbb{E}}_n | 1- \cos( \tau_f - (\pi/2)) | = O(n^{-1/2})$.
This time
\[
{\mathbb{E}}_n \bigl[
\bigl| 1- \cos\bigl( \tau_f - (\pi/2)\bigr) \bigr| {\mathbf{1}}_{E_n} \bigr]
= {\mathbb{E}}_n [ | \tau_f - (\pi/2) |^2 {\mathbf{1}}_{E_n} ] + O( \varepsilon_n^4) .
\]
Following a similar argument to that for ({e}f{fast1}),
we obtain ({e}f{fast2}).
\end{pf*}
\subsection{\texorpdfstring{Traversal time and area enclosed: Proofs of Theorems
\protect{e}f{thm:areamean} and \protect{e}f{th:timearea}}
{Traversal time and area enclosed: Proofs of Theorems 2.10 and 2.15}}
\lambdaabel{S:areatime}
Our proofs of Theorems {e}f{thm:areamean} and {e}f{th:timearea}
both use the percolation model of Section {e}f{S: percolation}.
\begin{pf*}{Proof of Theorem {e}f{th:timearea}}
The asymptotic statement in the theorem is a consequence of Theorem
{e}f{th:fasttime}. Thus it remains to prove
the exact formula.
For $x > 0$, and $y \ge0$, let $T(x,y)$ denote ${\mathbb{E}}[ \tau_f \mid A(0)
= x, B(0) = y ]$.
Also, set $T(0,y) = 0$ for $y > 0$. Note that $T(n,0) = {\mathbb{E}}_n [ \tau_f
] $. Conditioning on the first step shows that for $x > 0$ and $y \ge0$,
\[
T(x,y) = \frac{1}{x+y} + \frac{x}{x+y} T(x,y+1) + \frac{y}{x+y}
T(x-1, y) .
\]
For fixed $x$, $T(x,y) \to0$ as $y \to\infty$. Indeed, for $y \ge1$
the time to make $x$ horizontal jumps is
stochastically dominated by the sum of $x$ exponential random variables
with mean $1/y$.
We now consider the percolation model restricted to the first quadrant.
More precisely, we consider the induced graph on the set of sites
$(x,y)$ with $x \ge0$ and $y > 0$, on a single sheet of $\mathcal{R}$.
Let $I(x,y)$ denote the expected number of sites in the in-graph of
$(x,y)$ in this restricted model. This count includes the site $(x,y)$ itself.
For $x > 0$, we also set $I(x,0) = 0$.
Considering the two possible directed edges into the site $(x,y)$, we obtain
\[
I(x,y) = 1 + \frac{y}{x+y+1} I(x+1, y) + \frac{x}{x+ y-1} I(x, y-1) .
\]
Dividing through by $(x+y)$, we have
\[
\frac{I(x,y)}{x+y} = \frac{1}{x+y} + \frac{y}{x+y} \biggl(\frac
{I(x+1,y)}{(x+1) + y} \biggr) + \frac{x}{x+y}\biggl (\frac{I(x,y-1)}{x+(y-1)} \biggr)
.\vadjust{\goodbreak}
\]
We now claim that for each fixed $y$, $I(x,y)$ is bounded as $x \to
\infty$. Indeed, the number of sites in the in-graph of $(x,y)$ is at
most $y$ plus $y$ times the number of horizontal edges in this
in-graph. The number of horizontal edges may be stochastically bounded
above by the sum of $y$ geometric random variables with mean $1/x$, so
its mean tends to $0$ as $x \to\infty$.
We see that $I(y,x)/(x+y)$ and $T(x,y)$ satisfy the same recurrence
relation with the same boundary conditions; their difference satisfies
a homogeneous recurrence relation with boundary condition $0$ at $x=0$
and limit $0$ as $y \to\infty$ for each fixed $x$. An induction with
respect to $x$ shows that the difference is identically zero.
In particular, taking $x = m$ and $y=0$, for any $m \ge1$, we find
\[
\ I(0,m) = m T(m,0) .
\]
The union of the in-graphs of the sites $(0,m)$, for $1 \lambdae m \lambdae n$,
is the set of all sites $(x,y)$ with $x \ge0$ and $y > 0$ that lie
under the oriented path of the dual percolation graph $H'$ that starts
at $(-1/2, n+1/2)$. Each of these sites lies at the center of a unit
square with vertices $(x \pm1/2, y \pm1/2)$, and the union of these
squares is the region bounded by the dual percolation path and the
lines $x = -1/2$ and $y = 1/2$. Reflecting this region in the line $y =
x+ 1/2$, we obtain a sample of the region bounded by a simple harmonic
urn path and the coordinate axes.
The expected number of unit squares in this region is therefore $\sum
_{m=1}^n I(0,m)$, so we are done.
\end{pf*}
\begin{pf*}{Proof of Theorem {e}f{thm:areamean}}
The argument uses a similar idea to the proof of Theorem {e}f
{th:timearea}, this time
for the percolation model on the whole of~$\mathcal{R}$. Choose a
continuous branch of the argument function on $\mathcal{R}$.
Let $I_+(v)$ denote the expected number of points $w$ with $\arg(w) >
0$ in the in-graph of~$v$ in $H$, including~$v$ itself if $\arg(v) >
0$. Arguing as before, if the projection of~$v$ to $\mathbb{Z}^2$ is
$(x,y)$, then $I_+(v)$ satisfies the boundary condition $I_+(v) = 0$
for $\arg(v) \lambdae0$, and the recurrence relation
\begin{eqnarray*}
I_+(v) &=& 1 + \frac{|y|}{|x +\operatorname{sgn}(y) | + |y| }
I_+\bigl( v + (\operatorname{sgn}(y),0) \bigr)\\
&&{} + \frac{|x|}{|x| + |y -\operatorname{sgn}(x)| }
I_+\bigl( v + ( 0 , -\operatorname{sgn}(x))\bigr) ,
\end{eqnarray*}
where on the right-hand side $I_+$ is evaluated at two of the
neighbors of $v$ in the graph $G$.
Setting $J_+(v) := I_+(v) / ( |x(v)| + |y(v)| )$, we have a recurrence
relation for $J_+$:
\begin{eqnarray*}
J_+(v) &=& \frac{1}{|x| + |y|}
+ \frac{|y|}{|x|+ |y| }
J_+\bigl(v + (\operatorname{sgn}(y) , 0)\bigr) \\
&&{}+ \frac{|x|}{|x| + |y| }
J_+\bigl( v + ( 0 , -\operatorname{sgn}(x))\bigr) .
\end{eqnarray*}
The same recurrence relation and boundary conditions hold for
$T_+(\overline{v})$, where $T_+(w)$ is the expected time to hit the
set $\arg z \ge0$ in $\mathcal{R}$ in the fast embedding, starting
from a vertex $w$. Here, $\overline{v}$ is the vertex of $G$ at the
same distance from the origin as $v$, satisfying $\arg(\overline{v})
= -\arg(v)$. The reasoning of the previous proof shows that
$T_+(\overline{v}) = J_+(v)$ for all vertices $v$ with $\arg v \lambdae\pi
/2$, and the argument may be repeated on the subsequent quadrants to
show by induction that $T_+(\overline{v}) = J_+(v)$ for all vertices $v$.
We therefore have the lower bound
\[
J_+(v) = T_+(\overline{v}) \geq\biggl\lambdafloor\frac{\arg(v)}{\pi/2}
\biggr\rfloor\inf_n {\mathbb{E}}_n [ \tau_f ] .
\]
The asymptotic expression ({e}f{fast1}),
together with trivial lower bounds for small~$n$,
implies that $\inf_n {\mathbb{E}}_n [ \tau_f] > 0$. Therefore, as $v$ varies
over the set of vertices of~$G$ with a given projection $(x,y)$, both
$J_+(v)$ and $I_+(v)$ tend to infinity with $\arg(v)$.
Note that $I_+(v)$ is a lower bound for $I(v)$, and $I(v)$ depends only
on the projection $(x,y)$. It follows that
\begin{equation}
\lambdaabel{in-graph}
{\mathbb{E}}[I(v) ] = \infty.
\end{equation}
Recall that the oriented path of $H'$ starting at $(m+\frac{1}{2},
-\frac{1}{2})$ explores the outer boundary of the in-graph of the set
$S$ of vertices with $\arg(v) = 0$ and $ x \lambdae m$, and that it can be
mapped via ${\mathbb{P}}hi$ onto a path of the leaky simple harmonic urn. Let $A$
denote the area swept out by this path up until time $\tau$ (the
hitting time of $\{(x,y) : |x| + |y| = 1\}$). The mapping ${\mathbb{P}}hi$ from
the vertices of $G'$ to $\mathbb{Z}^2$ can be extended by affine
interpolation to a locally area-preserving map from $\mathcal{R}$ to
$\mathbb{R}^2 \setminus(0,0)$. So $A$ is equal to the area swept out
by the dual percolation path until its projection hits the set $\{(\pm
\frac{1}{2}, \pm\frac{1}{2})\}$. Since the expected number of points
in the in-graph that it surrounds is infinite, we have ${\mathbb{E}}[A]
=\infty$.
\end{pf*}
\subsection{Exact formulae for expected traversal time and enclosed area}
\lambdaabel{S:polys}
In this section, we present some explicit, if mysterious, formulae for
the expected area enclosed
by a quadrant-traversal of the urn process and the expected
quadrant-traversal time in the fast embedding.
We obtain these formulae in a similar way to our first proof of Lemma
{e}f{Lemma: transition probabilities},
and they are reminiscent, but more involved than, the formulae for the
Eulerian numbers. There is thus
some hope that the asymptotics of these formulae can be handled as in
the proof
of Lemma {e}f{lem: asymptotic}, which gives a possible approach to the
resolution
of Conjecture {e}f{conj:fasttime}.
\begin{lemma}
${\mathbb{E}}_n[\mbox{Area enclosed} ]$ and ${\mathbb{E}}_n [ \tau_f]$ are
rational polynomials of degree $n$ evaluated at ${e}$:
\begin{eqnarray}
\lambdaabel{poly:area}
{\mathbb{E}}_n [\mbox{Area enclosed} ] & =& \sum_{i=1}^n \sum_{x = 1}^i
\frac{i^{n-x-i} i! (-1)^{n-i}}{(n-i)! (i-x)!} \Biggl({e}^i - \sum
_{k=0}^{i-1} \frac{i^k}{k!} \Biggr), \\
\lambdaabel{poly:time}
{\mathbb{E}}_n [ \tau_f] & =& \sum_{i=1}^n
\sum_{x=1}^i \frac{ i^{n-x-i-1} i! (-1)^{n-i}}{(n-i)!(i-x)!} \Biggl({e}^i
- \sum_{k=0}^{i-1} \frac{i^k}{k!} \Biggr) .
\end{eqnarray}
\end{lemma}
\begin{pf}
The expected area enclosed can be obtained by summing the probabilities
that each unit square of the first quadrant is enclosed; that is,
\[
{\mathbb{E}}_n [ \mbox{Area enclosed} ] = \sum_{x = 1}^{n} \sum_{y =
1}^\infty{\mathbb{P}}_n\bigl((x,y) \mbox{ lies on or below the urn path}\bigr) .
\]
In terms of the slow continuous-time embedding of Section {e}f
{secRubin}, $(x,y)$ lies on or below the urn path if and only if
$\sum_{j=1}^{y-1} j \zeta_j < \sum_{i=x}^n i \xi_i$.
Let
\[
R_{n,x,y} = \sum_{i=x}^n i \xi_i - \sum_{j=1}^{y-1} j \zeta_j ,
\]
so that
\[
{\mathbb{E}}_n [\mbox{Area enclosed} ] = \sum_{x=1}^n \sum_{y = 1}^\infty
{\mathbb{P}}(R_{n,x,y} > 0 ) .
\]
The moment generating function of $R_{n,x,y}$ is
\[
{\mathbb{E}}[\exp(\theta R_{n,x,y} ) ] = \prod_{i=x}^n \frac{1}{1 - i \theta
} \prod_{j=1}^{y-1} \frac{1}{1 + j \theta} = \sum_{i=x}^n \frac
{\alpha_i}{1 - i \theta} + \sum_{j=1}^{y-1} \frac{\beta_j}{1 + j
\theta} ,
\]
where
\[
\alpha_i = \frac{i^{n-x+y-1} i! (-1)^{(n-i)}}{(i+y-1)! (i-x)! (n-i)!} .
\]
Now the density of $R_{n,x,y}$ at $w > 0$ is
\[
\sum_{i=x}^n \alpha_i \frac{\exp(w/i)}{i} ,
\]
so that ${\mathbb{P}}(R_{n,x,y} > 0 ) = \sum_{i = x}^n \alpha_i$.
Therefore
\[
{\mathbb{E}}_n [\mbox{Area enclosed} ] = \sum_{x=1}^n \sum_{y = 1}^\infty
\sum_{i = x}^n \frac{i^{n-x+y-1} i! (-1)^{(n-i)}}{(i+y-1)! (i-x)!
(n-i)!} .
\]
The series converges absolutely so we can rearrange to obtain ({e}f
{poly:area}).
By the first equality in Theorem {e}f{th:timearea}, we find that
${\mathbb{E}}_n[\tau_f]$ is also a rational polynomial of degree $n$ evaluated at
${e}$. After some simplification, we obtain~({e}f{poly:time}).
\end{pf}
A remarkable simplification occurs in the derivation of
({e}f{poly:time}) from ({e}f{poly:area}), so it is natural to try
the same step again, obtaining
\[
\frac{1}{n}({\mathbb{E}}_n [ \tau_f] - {\mathbb{E}}_{n-1} [ \tau_f ]) = \sum_{i=1}^n
\sum_{x=1}^i \frac{ i^{n-x-i-2} i! (-1)^{n-i}}{(n-i)!(i-x)!} \Biggl({e}^i
- \sum_{k=0}^{i-1} \frac{i^k}{k!} \Biggr) .
\]
In light of Theorem {e}f{th:fasttime} and Conjecture {e}f
{conj:fasttime}, we would like to prove that this expression decays
exponentially as $n \to\infty$.
Let us make one more observation that might be relevant
to Conjecture {e}f{conj:fasttime}. Define $F(i) = \sum_{x=1}^i \frac
{i!}{(i-x)! i^x}$, which\vspace*{1pt}
can be interpreted as the expected number of distinct balls drawn
if we draw from an urn containing $i$ distinguishable balls, with
replacement, stopping when we first draw some ball for the second time.
We have already seen, in equation ({e}f{poly:time}), that
\[
{\mathbb{E}}_n [\tau_f] = \sum_{i=1}^n F(i) \frac{(-1)^{n-i}
i^{n-i-1}}{(n-i)!} \Biggl({e}^i - \sum_{k=0}^{i-1}\frac{i^k}{k!} \Biggr) ;
\]
perhaps one could exploit the resemblance to the formula
\[
{\mathbb{E}}_n [ 1 / Z_{k+1} ] = \sum_{i=1}^n \frac{(-1)^{n-i}
i^{n-i-1}}{(n-i)!} \Biggl({e}^i - \sum_{k=0}^{i}\frac{i^k}{k!} \Biggr),
\]
but we were unable to do so.
\section{Other stochastic models related to the simple harmonic urn}
\lambdaabel{sec:misc}
\subsection{A stationary model: The simple harmonic flea circus}
\lambdaabel{S: stationary model}
In Section {e}f{secRubin}, we saw that
the Markov chain $Z_k$ has an infinite invariant measure $\pi(n) = n$.
We can understand this in the probabilistic setting by considering the
formal sum of infinitely many independent copies of the
fast embedding.
Here is an informal description of the model.
At time $0$, populate each vertex of $\mathbb{Z}^2$ with
an independent Poisson-distributed number of fleas with mean $1$.
Each flea performs a copy of the
process $(A(t),B(t))$, independently of all the other fleas. Let
$N_t(m,n)$ denote the number of fleas at location $(m,n)$ at time $t$.
As we make no further use of this process in this paper, we do not
define it more formally. Instead we
just state the following result and sketch the proof: compare the lemma
in \cite{hoffrose}, Section 2, which the authors
attribute to Doob.
\begin{lemma}
The process $\{N_t(m,n)\dvtx m,n \in\mathbb{Z}\}$ is stationary. That is,
for each fixed
time $t > 0$, the array $N_t(m,n)$, $m,n \in\mathbb{Z}$ consists of
independent Poisson(1) random variables.
The process is reversible in the sense that the ensemble of random
variables $N_t(m,n), 0 \lambdae t \lambdae c$,
has the same law as the ensemble $N_{c-t}(m, -n)$, $0 \lambdae t \lambdae c$, for
any $c > 0$.
\end{lemma}
This skew-reversibility allows us to extend the stationary process to
all times $t \in{\mathbb{R}}$.
To see that the process has stationary means, note that the
expectations ${\mathbb{E}}[N_t(m,n)]$ satisfy a system of coupled differential equations:
\begin{eqnarray*}
\frac{{d}}{{d} t} {\mathbb{E}}[N_t(m,n)] &=& -(|m| + |n|){\mathbb{E}}[N_t(m,n)] + |m|{\mathbb{E}}
\bigl[N_t\bigl(m,n-\operatorname{sgn}(m)\bigr)\bigr] \\
&&{}+ |n|{\mathbb{E}}\bigl[N_t\bigl(m+\operatorname{sgn}(n),n\bigr)\bigr],
\end{eqnarray*}
the solution to which is simply ${\mathbb{E}}[N_t(m,n)] = 1$ for all $t$, $m$
and $n$.
To establish the independence of the variables $N_t(a,b)$ when $t > 0$
is fixed, we use a Poisson thinning argument. That is, we construct
each variable
$N_0(m,n)$ as an infinite sum of independent Poisson random variables
$N(m,n,a,b)$ with means
\[
{\mathbb{E}}[N(m,n,a,b)] = {\mathbb{P}}\bigl( (A(t),B(t)) = (a,b) \mid(A(0),B(0)) = (m,n) \bigr) .
\]
The variable $N(m,n,a,b)$ gives the number of fleas that start at
$(m,n)$ at time $0$
and are at $(a,b)$ at time $t$. Then $N_t(a,b)$ is also a sum of
infinitely many independent
Poisson random variables, whose means sum to $1$, so it is a Poisson
random variable with mean $1$.
Moreover, for $(a,b) \neq(a',b')$, the corresponding sets of summands
are disjoint, so $N_t(a,b)$ and $N_t(a',b')$ are independent.
\subsection{The Poisson earthquakes model}
We saw how the percolation model of Section {e}f{S: percolation} gives
a static grand coupling of
many instances of (paths of) the simple harmonic urn. In this section,
we describe a model, based on ``earthquakes,''
that gives a dynamic grand coupling of many instances of simple
harmonic urn processes with particularly
interesting geometrical properties.
The earthquakes model is defined as a continuous-time Markov chain
taking values in the group of area-preserving homeomorphisms of the
plane, which we will write as
\[
\mathfrak{S}_t \dvtx \mathbb{R}^2 \to\mathbb{R}^2 ,\qquad t \in\mathbb{R}.
\]
It will have the properties:
\begin{itemize}
\item$\mathfrak{S}_0$ is the identity,
\item$\mathfrak{S}_t(0,0) = (0,0)$,
\item$\mathfrak{S}_t$ acts on $\mathbb{Z}^2$ as a permutation,
\item$\mathfrak{S}_s \circ\mathfrak{S}_t^{-1}$ has the same
distribution as $\mathfrak{S}_{s-t}$, and
\item for each pair $(x_0, y_0) \neq(x_1, y_1) \in\mathbb{Z}^2$,
the displacement vector
\[
\mathfrak{S}_t (x_1,y_1) - \mathfrak{S}_t (x_0, y_0)
\]
has the distribution of the continuous-time
fast embedding of the simple harmonic urn, starting at $(x_1-x_0, y_1 - y_0)$.
\end{itemize}
In order to construct $\mathfrak{S}_t$, we associate a unit-rate
Poisson process to each horizontal
strip $ H_n: = \{(x,y) \in\mathbb{R}^2 \dvtx n < y < n + 1\}$, and to
each vertical strip $V_n: = \{(x,y) \in\mathbb{R}^2 \dvtx n < x < n + 1\}$
(where $n$ ranges over $\mathbb{Z}$). All these Poisson processes
should be independent.
Each Poisson process determines the sequence of times at which an \emph
{earthquake} occurs along the corresponding strip. An earthquake is a
homeomorphism of the plane that translates one of the complementary
half-planes of the given strip through a unit distance parallel to the
strip, fixes the other complementary half-plane, and shears the strip
in between them. The fixed half-plane is always the one containing the
origin, and the other half-plane always moves in the anticlockwise
direction relative to the origin.
Consider a point $(x_0,y_0) \in\mathbb{R}^2$. We wish to define
$\mathfrak{S}_t(x_0,y_0)$
for all \mbox{$t \ge0$}. We will define inductively a sequence of stopping
times $\varepsilon_i$, and points $(x_i,\allowbreak y_i) \in\mathbb{R}^2$,
for $i \in{\mathbb{Z}}_+$. First, set $\varepsilon_0 = 0$. For $i \in{\mathbb{N}}$, suppose
we have defined $(x_{i-1},y_{i-1})$
and $\varepsilon_{i-1}$. Let $\varepsilon_i$ be the least point greater than $\varepsilon
_{i-1}$ in
the union of the Poisson processes associated to those strips for which
$(x_{i-1},y_{i-1})$ and $(0,0)$ do not both lie in one or other
complementary half-plane. This is a.s. well defined since there are
only finitely many such strips, and a.s. there
is only one strip for which an earthquake occurs at time $\varepsilon_i$.
That earthquake moves $(x_{i-1},y_{i-1})$ to
$(x_i,y_i)$. Note that $\varepsilon_i - \varepsilon_{i-1}$ is an exponential random
variable with
mean $1/(\lambdaceil|x_{i-1}|\rceil+ \lambdaceil|y_{i-1}|\rceil)$,
conditionally independent of all previous jumps, given this mean. Since
each earthquake increases the distance between any two points by at
most $1$, it follows that a.s. the process does not explode in finite
time. That is, $\varepsilon_i \to\infty$ as $i \to\infty$. Define
$\mathfrak{S}_t (x_0,y_0)$ to be $(x_i,y_i)$, where $\varepsilon_i \lambdae t <
\varepsilon_{i+1}$. The construction of $\mathfrak{S}_t$ for $t < 0$ is
similar, using the inverses of the earthquakes.
Note that we cannot simply define $\mathfrak{S}_t$ for $t > 0$ to be
the composition of all the earthquakes that occur between times $0$ and
$t$, because almost surely infinitely many earthquakes occur during
this time; however, any bounded subset of the plane will only be
affected by finitely many of these, so the composition makes sense locally.
The properties listed above follow directly from the construction. For
$(x_0,y_0)$, $(x_1, y_1) \in\mathbb{Z}^2$, the displacement vector $
(\Delta x_t, \Delta y_t) = \mathfrak{S}_t (x_1,y_1) - \mathfrak{S}_t
(x_0, y_0)$ only changes when an earthquake occurs along a strip that
separates the two endpoints; the waiting time after $t$ for this to
occur is exponentially distributed with mean $1/(|\Delta x_t| + |\Delta
y_t| )$, and conditionally independent of $\mathfrak{S}_t$ given
$(\Delta x_t, \Delta y_t)$.
\begin{figure}
\caption{A simulation of $\mathfrak{S}
\end{figure}
The model is spatially homogeneous in the following sense. Fix some
$(a,b) \in\mathbb{Z}^2$ and define
\[
\tilde{\mathfrak{S}}_t(x,y) = \mathfrak{S}_t (x+a,y+b) - \mathfrak
{S}_t(a,b) .
\]
Then $\tilde{\mathfrak{S}}_t$ has the same distribution as $\mathfrak{S}_t$.
\begin{lemma}
Define an oriented polygon $\Gamma$ by the cyclic sequence of vertices
\[
((x_1, y_1), \dots, (x_{n-1}, y_{n-1}), (x_n, y_n), (x_1,y_1)) ,\qquad (x_i,
y_i) \in\mathbb{Z}^2 .
\]
The signed area enclosed by the polygon $\Gamma_t$, given by
\[
(\mathfrak{S}_t(x_1,y_1), \mathfrak{S}_t(x_2, y_2), \dots, \mathfrak
{S}_t(x_n, y_n), \mathfrak{S}_t(x_1, y_1) ) ,
\]
is a martingale.
\end{lemma}
\begin{pf}
For convenience, we write $(x_i(t), y_i(t)) = \mathfrak
{S}_t((x_i,y_i))$. The area enclosed by the oriented polygon $\Gamma
_t$ is
given by the integral $\frac{1}{2}\int_{\Gamma_t} x \,{d}y - y
\,{d}x$, which we can write as
\[
\frac{1}{2}\sum_{i=1}^n \bigl(x_i(t) y_{i+1}(t) - x_{i+1}(t) y_i(t) \bigr) ,
\]
where $(x_{n+1},y_{n+1})$ is taken to mean $(x_1, y_1)$. So it suffices
to show that each term in this sum is itself a martingale; let us
concentrate on the term $x_1(t) y_2(t) - x_2(t) y_1(t)$, considering
the first positive time at which either of $(x_1(t),y_1(t))$ or
$(x_2(t),y_2(t))$ jumps. There appear to be at least $36$ cases to
consider, depending on the ordering of $\{0, x_1, x_2\}$ and $\{0, y_1,
y_2\}$, but we can reduce this to four by taking advantage of the
spatial homogeneity of the earthquakes model, described above. By
choosing $(a,b)$ suitably, and replacing $\mathfrak{S}$ by
$\mathfrak{S}'$, we can assume that $x_i, y_i > 0$, for $i = 1, \dots,
n$. Furthermore, swapping the indices $1$ and $2$ only changes the sign
of $x_1(t) y_2(t) - x_2(t) y_1(t)$, so we may also assume that $x_1 \lambdae
x_2$. Suppose that the first earthquake of interest is along a vertical
line. Then with probability $x_1/x_2$ it increments both $y_1$ and
$y_2$ and otherwise it increments only $y_2$. The expected jump in
$x_1(t) y_2(t) - x_2(t) y_1(t)$ conditional on the first relevant
earthquake being parallel to the $y$-axis is therefore
\begin{eqnarray*}
&&\frac{x_1}{x_2} \bigl(\bigl(x_1 (y_2+1) - x_2(y_1 +1) \bigr) - (x_1 y_2 - x_2 y_1)
\bigr)\\
&&\qquad {}+ \frac{x_2 - x_1}{x_2} \bigl( \bigl(x_1 (y_2 + 1) - x_2 y_1\bigr) - (x_1 y_2 - x_2
y_1) \bigr) = 0 .
\end{eqnarray*}
A similar argument shows that the expected jump in $x_1(t) y_2(t) -
x_2(t) y_1(t)$ conditional on the first relevant earthquake being
parallel to the $x$-axis is also zero.
\end{pf}
\subsection{Random walks across the positive quadrant}
In this section, we describe another possible generalization of the simple
harmonic urn that has some independent interest.
We define a discrete-time process $(A_n,B_n)_{n \in{\mathbb{Z}}_+}$ on ${\mathbb{R}}^2$
based on the distribution of an underlying nonnegative,
nonarithmetic
random variable $X$
with ${\mathbb{E}}[ X ] =\mu\in(0,\infty)$ and $\operatorname{\mathbb{V}\mathrm{ar}}[ X ] = \sigma^2 \in
(0,\infty)$.
Let $X_1, X_2, \lambdadots$ and $X'_1, X'_2, \lambdadots$ be independent copies
of $X$.
Roughly speaking, the walk starts on the horizontal axis and takes
jumps $(-X_i',X_i)$
until its first component is negative. At this point, suppose the walk
is at $(-r,s)$. Then the
walk starts again at $(s,0)$ and the process repeats.
We will see (Lemma {e}f{quadshu}) that in the case when $X \sim U(0,1)$,
this process is closely related to the simple harmonic urn and is
consequently transient.
It is natural to study the same question for general distributions $X$.
It turns out that the
recurrence classification depends only on $\mu$ and $\sigma^2$. Our
proof uses
renewal theory.
We now formally define the model. With $X, X_n, X'_n$ as above,
we suppose that ${\mathbb{E}}[X^4] < \infty$.
Let
$(A_0,B_0)=(a,0)$, for $a>0$. Define the random process for $n \in{\mathbb{Z}}
_+$ by
\[
(A_{n+1},B_{n+1})=
\cases{
( A_n-X'_n, B_n+X_n ), &\quad if $A_n \geq0$,\cr
(B_n , 0), &\quad if $A_n < 0 $.
}
\]
\begin{thm}
\lambdaabel{th:quadwalks}
Suppose ${\mathbb{E}}[X^4]<\infty$.
The walk $(A_n,B_n)$ is transient if and only if $\mu^2 > \sigma^2$.
\end{thm}
Set
$\tau_0 :=-1$ and for $k \in{\mathbb{N}}$,
\[
\tau_k :=\min\{ n > \tau_{k-1} \dvtx A_n<0\}.
\]
Define $T_k:= \tau_k - (\tau_{k-1} + 1)$. That is,
$T_k$ is the number of steps that the random walk takes to cross the
positive quadrant for the $k$th time.
\begin{lemma}
\lambdaabel{quadshu}
If $X \sim U(0,1)$
and the initial value $a$ is distributed as
the sum of $n$ independent $U(0,1)$ random variables,
independent of the $X_i$ and~$X'_i$, then the distribution of the process
$(T_k)_{k \in{\mathbb{N}}}$ coincides with that of the embedded simple harmonic urn
process $(Z_k)_{k \in{\mathbb{N}}}$ conditional on $Z_0 = n$.
\end{lemma}
\begin{pf}
It suffices to show that $T_1 = \tau_1$ has the distribution of $Z_1$
conditional on $Z_0 = n$ and that conditional on $\tau_1$ the new
starting point $A_{1+\tau_1}$, which is $B_{\tau_1}$, has the
distribution of the sum of
$\tau_1$ independent $U(0,1)$ random variables.
Then the lemma will follow since the two processes $(\tau_k, B_{\tau
_k})$ and $(Z_k)$ are both Markov.
To achieve this, we couple the process $(A_n, B_n)$ up to time $\tau
_1$ with the renewal process described
in Section {e}f{S: drift}. To begin,
identify $a$ with the sum $(1-\chi_1) + \cdots+ (1-\chi_n)$.
Then for $k \in\{ 1, \lambdadots, N(n)-n\}$, where $N(n) > n$
is as defined at ({e}f{eq:renewproc}),
we identify $X'_{k}$
with $\chi_{n+k}$. For $m \lambdae\tau_1$ we have
\[
A_m = a - \sum_{i=1}^m X'_i = n - \sum_{i=1}^{n+m} \chi_i ,
\]
so in particular we have $A_{N(n) -n-1} \geq0$
and $A_{N(n)-n} < 0$ by definition of~$N(n)$. Hence, $\tau_1 = N(n)-n$
has the distribution
of $Z_1$ by Lemma {e}f{Lemma: same distribution}.
Moreover, $A_{1+\tau_1} = B_{\tau_1}$ is the sum of
the independent $U(0,1)$ random variables~$X_i$, $i = 1, \lambdadots, \tau_1$.
\end{pf}
Thus, by Theorem {e}f{th:main1}, in the case where $X$ is
$U (0,1)$, the process
$(A_n,B_n)$ is transient, which is consistent with Theorem {e}f{th:quadwalks}
since in the uniform case $\mu= 1/2$ and $\sigma^2 = 1/12$.
To study the general case, it is helpful to
rewrite the definition of $(A_n,B_n)$ explicitly
in the language of renewal theory.
Let $S_0 = S'_0 = 0$ and for $n \in{\mathbb{N}}$ set $S_n = \sum_{i=1}^n X_i$, $S'_n
= \sum_{i=1}^n X'_i$. Define the renewal counting function for $S'_n$
for $a>0$ as
\[
N(a) := \min\{ n \in{\mathbb{Z}}_+ \dvtx S'_n > a \} = 1 + \max\{ n \in{\mathbb{Z}}_+\dvtx
S'_n \lambdaeq a \}.
\]
Then starting at $(A_0,B_0) = (a,0)$, $a>0$, we see $\tau_1 =
N(a)$ so that $B_{\tau_1} = S_{N(a)}$. To study the recurrence and
transience of $(A_n,B_n)$, it thus suffices to study the process
$(R_n)_{n \in{\mathbb{Z}}_+}$ with $R_0 := a$ and $R_n$ having the
distribution of $S_{N(x)}$ given $R_{n-1} = x$. The increment of
the process $R_n$ starting from~$x$ thus is distributed as
$\Delta(x) := S_{N(x)} -x$. It is this random quantity that we
need to analyze.
\begin{lemma}
\lambdaabel{quadjumps}
Suppose that ${\mathbb{E}}[ X^4] < \infty$. Then as $x \to\infty$, ${\mathbb{E}}[
|\Delta(x)|^4 ] = O(x^{2})$ and
\begin{eqnarray*}
{\mathbb{E}}[ \Delta(x) ] & = &\frac{\sigma^2 +\mu^2}{2 \mu}
+ O(x^{-1}), \\
{\mathbb{E}}[ \Delta(x)^2 ] & =& \frac{2 x \sigma^2}{\mu} + O(1) .
\end{eqnarray*}
\end{lemma}
\begin{pf}
We make use of results on higher-order renewal theory
expansions due to Smith \cite{smith2} (note that in \cite{smith2}
the renewal at $0$ is not counted). Conditioning on $N(x)$ and
using the independence of the $X_i$, $X'_i$, we obtain the Wald
equations:
\[
{\mathbb{E}}\bigl[ S_{N(x)} \bigr] = \mu{\mathbb{E}}[ N(x) ] ;\qquad \operatorname{\mathbb{V}\mathrm{ar}}\bigl[ S_{N(x)} \bigr] = \sigma^2 {\mathbb{E}}[
N(x) ] + \mu^2 \operatorname{\mathbb{V}\mathrm{ar}}[ N(x) ] .
\]
Assuming ${\mathbb{E}}[X^3]<\infty$, \cite{smith2}, Theorem 1, shows that
\begin{eqnarray*}
{\mathbb{E}}[ N(x) ] & = &\frac{x}{\mu} + \frac{\sigma^2+\mu
^2}{2 \mu^2} + O(x^{-1}) ,\\
\operatorname{\mathbb{V}\mathrm{ar}}[ N(x) ] & =& \frac{x \sigma^2}{\mu^3} + O(1).
\end{eqnarray*}
The expressions in the lemma for ${\mathbb{E}}[ \Delta(x)]$ and ${\mathbb{E}}[ \Delta
(x)^2]$ follow.
It remains to prove the bound for ${\mathbb{E}}[ |\Delta(x)|^4 ]$. Write
$\Delta(x)$ as
\begin{equation}
\lambdaabel{divideandconkout}
\qquad S_{N(x)} - x = \bigl( S_{N(x)} - \mu N(x) \bigr) + \bigl( \mu N(x) - \mu{\mathbb{E}}[ N(x) ] \bigr)
+ \bigl(\mu{\mathbb{E}}[ N(x) ] - x\bigr) .
\end{equation}
Assuming ${\mathbb{E}}[X^2]<\infty$, a result of Smith \cite{smith2}, Theorem 4,
implies that the
final bracket on the right-hand side of ({e}f{divideandconkout})
is $O(1)$. For the first bracket
on the right-hand side of ({e}f{divideandconkout}), it follows
from the Marcinkiewicz--Zygmund inequalities
(\cite{gut}, Corollary 8.2, page 151), that
\[
{\mathbb{E}}\bigl[ \bigl( S_{N(x)} - \mu N(x) \bigr)^4\bigr] \lambdaeq C {\mathbb{E}}[ N(x)^2 ] ,
\]
provided ${\mathbb{E}}[X^4] < \infty$. This last upper bound is $O(x^2)$ by the
computations in the first part of this proof. It remains to deal
with the second bracket on the right-hand side of ({e}f{divideandconkout}).
By the algebra relating central moments to cumulants, we have
\[
{\mathbb{E}}\bigl[ \bigl( \mu N(x) - \mu{\mathbb{E}}[ N(x) ] \bigr)^4 \bigr]
= \mu^4 \bigl( k_4(x) + 3 k_2(x)^2 \bigr) ,
\]
where $k_r(x)$ denotes the
$r$th cumulant of $N(x)$. Again appealing to a result of Smith
(\cite{smith2}, Corollary 2, page 19), we have that $k_2(x)$ and $k_4(x)$
are both~$O(x)$ assuming ${\mathbb{E}}[X^4]<\infty$. (The fact
that \cite{smith2} does not count the renewal at $0$ is
unimportant here, since the $r$th cumulant of $N(x) \pm1$ differs
from $k_r(x)$ by a constant depending only on $r$.) Putting these
bounds together, we obtain from ({e}f{divideandconkout}) and
Minkowski's inequality that ${\mathbb{E}}[ (S_{N(x)} - x)^4 ] = O(x^2)$.
\end{pf}
To prove Theorem {e}f{th:quadwalks}, we basically need to compare ${\mathbb{E}}
[ \Delta(x)]$ to
${\mathbb{E}}[ \Delta(x)^2]$.
As in our analysis of ${\tilde Z}_k$, it is most convenient to work on the square-root
scale. Set $V_n := R_n^{1/2}$.
\begin{lemma}
\lambdaabel{vmoms}
Suppose that ${\mathbb{E}}[ X^4] < \infty$. Then there exists $\delta>0$ such
that as $y \to\infty$,
\begin{eqnarray*}
{\mathbb{E}}[ V_{n+1} - V_n \mid V_n = y ] & = &\frac{{\mathbb{E}}[ \Delta
(y^2) ]}{2y} - \frac{{\mathbb{E}}[\Delta(y^2)^2 ]}{8y^3}
+ O(y^{-1 -\delta}) ,\\
{\mathbb{E}}[ ( V_{n+1} - V_n)^2 \mid V_n = y ] & =& \frac{{\mathbb{E}}[\Delta(y^2)^2
]}{4y^2} + O (y^{-\delta} ),
\\
{\mathbb{E}}[ | V_{n+1} - V_n |^{3} \mid V_n = y ] & =& O ( 1 ) .
\end{eqnarray*}
\end{lemma}
\begin{pf}
The proof is similar to the proof of Lemma {e}f{lem:sqrtincrements},
except that here we must work a little harder as we have weaker tail bounds
on $\Delta(x)$. Even so, the calculations will be familiar, so we
do not give all the details. Write ${\mathbb{E}}_x [ \cdot]$
for ${\mathbb{E}}[ \cdot\mid R_n = x ]$ and similarly for ${\mathbb{P}}_x$. From
Markov's inequality
and the fourth moment bound in Lemma {e}f{quadjumps}, we have for $\varepsilon
\in(0,1)$ that
\begin{equation}
\lambdaabel{bigjumptail}
{\mathbb{P}}_x \bigl( | \Delta(x) | > x^{1-\varepsilon} \bigr) = O ( x^{4 \varepsilon- 2} ) .
\end{equation}
We have that for $x \geq0$,
\[
{\mathbb{E}}[ V_{n+1} - V_n \mid V_n = x^{1/2} ] = {\mathbb{E}}_x [ R_{n+1}^{1/2} -
R_n^{1/2} ] = {\mathbb{E}}_x \bigl[ \bigl(x + \Delta(x)\bigr)^{1/2} - x^{1/2} \bigr] .
\]
Here we can write
\begin{eqnarray}
\lambdaabel{bigjump}
\bigl(x + \Delta(x)\bigr)^{1/2} - x^{1/2} &=& \bigl[ \bigl(x + \Delta(x)\bigr)^{1/2} - x^{1/2} \bigr]
{\mathbf{1}} \{ | \Delta(x) | \lambdaeq x^{1-\varepsilon} \}\nonumber\\[-8pt]\\[-8pt]
&&{}+ R_1 + R_2 \nonumber
\end{eqnarray}
for remainder terms $R_1$, $R_2$ that we define shortly.
The main term on the right-hand side admits a Taylor expansion
and analysis
(whose details we omit) in a similar
manner to the proof of Lemma {e}f{lem:sqrtincrements}, and
contributes to the main terms in the statement of the present lemma.
The remainder terms in~({e}f{bigjump}) are
\begin{eqnarray*}
R_1 &=& \bigl[ \bigl(x + \Delta(x)\bigr)^{1/2} - x^{1/2} \bigr] {\mathbf{1}} \{ \Delta(x) > x^{1-\varepsilon
} \} ,\\
R_2 &=& \bigl[ \bigl(x + \Delta(x)\bigr)^{1/2} - x^{1/2} \bigr] {\mathbf{1}} \{ \Delta(x) < -
x^{1-\varepsilon} \} .
\end{eqnarray*}
For the second of these, we have $| R_2 | \lambdaeq x^{1/2} {\mathbf{1}} \{ \Delta
(x) < - x^{1-\varepsilon} \}$,
from which we obtain, for $r <4$, ${\mathbb{E}}_x [ |R_2 |^r ] = O( x^{4\varepsilon+(r-4)/2})$,
by ({e}f{bigjumptail}). Taking $\varepsilon$ small enough,
this term contributes only to the negligible
terms in our final expressions. For $R_1$, we have the bound
\[
| R_1 | \lambdaeq C \bigl( 1 + | \Delta(x) | \bigr)^{(1/2) + \varepsilon} {\mathbf{1}} \{ \Delta(x)
> x^{1-\varepsilon} \}
\]
for some $C \in(0,\infty)$ not depending on $x$, again for $\varepsilon$
small enough. An application of H\"older's inequality and the
bound ({e}f{bigjumptail}) implies that, for $r<4$, for any
$\varepsilon>0$,
\begin{eqnarray*}
{\mathbb{E}}_x [ | R_1 |^r] &\lambdaeq& C \bigl( {\mathbb{E}}_x \bigl[ \bigl( 1+ | \Delta(x) |\bigr)^{4} \bigr] \bigr)^{\gfrac
{r(1+ 2\varepsilon)}{8}}
\bigl( {\mathbb{P}}_x \bigl( \Delta(x) > x^{1-\varepsilon} \bigr) \bigr)^{1-\gfrac{r(1+2\varepsilon)}{8}}\\
& = & O\bigl(x^{6\varepsilon+(r-4)/2 } \bigr) .
\end{eqnarray*}
It is now routine to complete the proof.
\end{pf}
\begin{pf*}{Proof of Theorem {e}f{th:quadwalks}}
For the recurrence classification,
the crucial quantity is
\begin{eqnarray*}
&&2 y {\mathbb{E}}[ V_{n+1} - V_n \mid V_n = y ] - {\mathbb{E}}[ ( V_{n+1} - V_n)^2 \mid
V_n = y ] \\
&&\qquad = {\mathbb{E}}[ \Delta(y^2) ] - \frac{{\mathbb{E}}
[ \Delta(y^2)^2]}{2 y^2} + O(y^{-\delta} )
\end{eqnarray*}
by Lemma {e}f{vmoms}. Now by Lemma {e}f{quadjumps}, this last
expression is seen to be equal to
\[
\frac{\mu^2 - \sigma^2}{2 \mu} + O (y^{-\delta} ).
\]
Now \cite{lamp1}, Theorem 3.2, completes the proof.
\end{pf*}
\begin{remarks*}(i) To have some examples, note that if $X$ is
exponential, the
process is recurrent, while
if $X$ is the sum of two independent exponentials, it is transient. We
saw that if $X$
is $U(0,1)$ the process is transient; if $X$ is the square-root of a
$U(0,1)$ random variable,
it is recurrent.
(ii)
Another special case of the model that has some interesting features is
the case where $X$
is exponential with mean $1$.
In this particular case, a calculation shows that the distribution of
$T_{k+1}$ given $T_k=m$ is negative binomial
$(m+1,1/2)$, that is,
\[
{\mathbb{P}}(T_{k+1}=j \mid T_k=m) = \pmatrix{j+m\cr m}2^{-m-j-1}\qquad ( j \in{\mathbb{Z}}_+).
\]
Since $\mu^2 =\sigma^2$, this case is in some sense
critical, a fact supported by the following branching process interpretation.
\end{remarks*}
Consider a version of the gambler's ruin problem. The gambler begins
with an initial stake, a pile of $m_0$ chips. A sequence of
independent
tosses of a fair coin is made; when the coin comes\vadjust{\goodbreak} up heads, a chip is
removed from the gambler's pile, but when it comes up tails, a chip is
added to a second pile by the casino.
The game ends when the gambler's original pile of chips is exhausted;
at this point the gambler receives the second pile of chips as his prize.
The total number of chips in play is a martingale;
by the optional stopping theorem, the expectation of the prize equals
the initial stake.
As a loss leader, the casino announces that it will add one extra chip
to each gambler's initial stake, so that the game is now in favour of
the gambler.
Suppose a gambler decides to play this game repeatedly, each time
investing his prize as the initial stake of the next game.
If the casino were to allow a zero stake (which of course it does not),
then the sequence of augmented stakes would form an irreducible Markov
chain $S_k$ on ${\mathbb{N}}$.
Conditional on $S_k = m$, the distribution of $S_{k+1}$ is negative
binomial $(m+1/2, 1/2)$.
So by the above results, this chain is recurrent. It follows that with
probability one the gambler will eventually lose everything.
We can interpret the sequence of prizes as a Galton--Watson process in
which each generation corresponds to one game, and individuals in the
population correspond to chips
in the gambler's pile at the start of the game. Each individual has a
Geo($1/2$) number of offspring
(i.e., the distribution that puts mass $2^{-1-k}$ on each $k \in{\mathbb{Z}}
_+$), being the chips that are added to the prize pile while that individual
is on top of the gambler's pile, and at each generation there is
additionally a Geo($1/2$) immigration, corresponding to the chips added
to the prize pile while the casino's bonus chip is
on top of the gambler's pile. This is a critical case of the
Galton--Watson process with immigration. By a result of Zubkov \cite{Zub}, if we start at time $0$ with population $0$, the time $\tau$ of
the next visit to $0$ has pgf
\[
{\mathbb{E}}[ s^\tau] = \frac{1}{s} + \frac{1}{\lambdaog(1-s)} .
\]
Since this tends to $1$ as $ s \nearrow1$, we have $\mathbb{P}(\tau<
\infty) = 1$. In fact, this can be deduced in an elementary way as follows.
The pgf of the Geo($1/2$) distribution is $f(s) = 1/(2-s)$, and its
$n$th iterate, the pgf of the $n$th
generation starting from one individual, is $f(s) =
(n-(n-1)s)/((n+1)-ns)$. In particular, the probability that an
individual has no descendants at the $n$th generation is
$n/(n+1)$. If $S_0 = 1$, then $S_k = 1$ if and only if for each $j=0,
\dots,k-1$ the bonus chip from game $j$ has no descendants at the
$(k-j)$th generation. These events are independent, so
\[
{\mathbb{P}}(S_k = 1 \mid S_0 = 1) = \prod_{j=0}^{k-1} \frac{k-j}{k-j-1} =
\frac{1}{k+1} ,
\]
which sums to $\infty$ over $k \in{\mathbb{N}}$ so that
the Markov chain is recurrent (see, e.g.,~\cite{Asmussen}, Proposition 1.2, Section I).
The results of Pakes \cite{Pakes} on the critical Galton--Watson
process with immigration show that the casino should certainly not
add two bonus chips to each stake, for then the process becomes
transient, and gambler's ruin will no longer apply.
\section*{Acknowledgments}
The authors are grateful to John Harris, Sir John Kingman, Iain MacPhee and
James Norris for helpful discussions, and to an anonymous referee for
constructive comments.
Most of this work was
carried out while Andrew Wade was at the Heilbronn Institute for
Mathematical Research, Department of Mathematics, University of
Bristol.
\printaddresses
\end{document} |
\begin{document}
\title{Strong edge-colorings for $k$-degenerate graphs}
\begin{abstract}
We prove that the strong chromatic index for each $k$-degenerate graph with maximum degree $\Delta$ is at most $(4k-2)\Delta-k(2k-1)+1$.
\end{abstract}
A {\em strong edge-coloring} of a graph $G$ is an edge-coloring so that no edge can be adjacent to two edges with the same color. So in a strong edge-coloring, every color class gives an induced matching. The strong chromatic index $\chi_s'(G)$ is the minimum number of colors needed to color $E(G)$ strongly. This notion was introduced by Fouquet and Jolivet (1983, \cite{FJ83}). Erd\H{o}s and Ne\v{s}et\v{r}il during a seminar in Prague in 1985 proposed some open problems, one of which is the following
\begin{conjecture}[Erd\H{o}s and Ne\v{s}et\v{r}il, 1985]
If $G$ is a simple graph with maximum degree $\Delta$, then $\chi_s'(G)\le 5\Delta^2/4$ if $\Delta$ is even, and $\chi_s'(G)\le (5\Delta^2-2\Delta+1)/4$ if $\Delta$ is odd.
\end{conjecture}
This conjecture is true for $\Delta\le 3$ (\cite{A92, HHT93}). Cranston \cite{C06} showed that $\chi_s'(G)\le 22$ for $\Delta=4$. Chung, Gy\'arf\'as, Trotter, and Tuza (1990, \cite{CGTT90}) showed that the upper bounds are exactly the numbers of edges in $2K_2$-free graphs. Molloy and Reed \cite{MR97} proved that graphs with sufficient large maximum degree $\Delta$ has strong chromatic index at most $1.998\Delta^2$. For more results see \cite{SSTM} (Chapter 6, problem 17).
A graph is {\em $k$-degenerate} if every subgraph has minimum degree at most $k$. Chang and Narayanan (2012, \cite{CN12}) recently proved that a $2$-degenerate graph with maximum degree $\Delta$ has strong chromatic index at most $10\Delta-10$. Luo and the author in \cite{LY12} improved the upper bound to $8\Delta-4$.
In~\cite{CN12}, the following conjecture was made
\begin{conjecture}[Chang and Narayanan, \cite{CN12}]
There exists an absolute constant $c$ such that for any $k$-degenerate graphs $G$ with maximum degree $\Delta$, $\chi_s'(G)\le ck^2\Delta$. Furthermore, the $k^2$ may be replaced by $k$.
\end{conjecture}
In this paper, we prove a stronger form of the conjecture. Unlike the priming processes in\cite{CN12, LY12}, we find a special ordering of the edges and by using a greedy coloring obtain the following result.
\begin{theorem}
The strong chromatic index for each $k$-degenerate graph with maximum degree $\Delta$ is at most $(4k-2)\Delta-k(2k-1)+1$.
\end{theorem}
Thus, $2$-degenerate graphs have strong chromatic index at most $6\Delta-5$.
\begin{proof}
By definition of $k$-degenerate graphs, after the removal of all vertices of degree at most $k$, the remaining graph has no edges or has new vertices of degree at most $k$, thus we have the following simple fact on $k$-degenerate graphs (see also \cite{CN12}).
{\em Let $G$ be a $k$-degenerate graph. Then there exists $u\in V(G)$
so that $u$ is adjacent to at most $k$ vertices of degree more than $k$. Moreover, if $\Delta(G)>k$, then the vertex $u$ can be selected with degree more than $k$.}
We call a vertex $u$
a {\em special vertex} if $u$ is adjacent to at most $k$ vertices of degree more than $k$. An edge is a {\em special edge} if it is incident to a special vertex and a vertex with degree at most $k$. The above fact implies that every $k$-degenerate graph has a special edge, and if $\Delta\le k$, then every vertex and every edge are special.
We order the edges of $G$ as follows. First we find in $G$ a special edge, put it at the beginning of the list, and then remove it from $G$. Repeat the above step in the remaining graph. When the process ends, we have an ordered list of the edges in $G$, say $e_1, e_2, \ldots, e_m$, where $m=|E(G)|$. So $e_m$ is the special edge we first chose and placed in the list.
Let $G_i$ be the graph induced by the first $i$ edges in the list, $i=1,2,\ldots, m$. Then $e_i$ is a special edge in $G_i$.
We now count the edges of $G_i$ within distance one to $e_i$ in $G$. We may call the edges in $G_i$ blue edges and the edges in $G-G_i$ yellow edges. Let $u_i,v_i$ be the endpoints of $e_i$ with $u_i$ being a special vertex in $G_i$.
We first count the blue edges incident to $u_i$ and its neighbors. The vertex $u_i$ has three kinds of neighbors: the neighbors in $X_1$ sharing blue edges with $u_i$ and having degree more than $k$, the neighbors in $X_2$ sharing blue edges with $u_i$ and having degree at most $k$ (thus $v_i\in X_2$), and the neighbors in $X_3$ sharing yellow edges with $u_i$. By definition, $|X_1|\le k$, so at most $|X_1|\Delta+k(|X_2|-1)$ blue edges are incident to $X_1\cup (X_2-\{v_i\})$. For each vertex $u$ in $X_3$, $uu_i$ is a yellow edge in $G_i$ but will be a special edge in $G_j$ for some $j>i$. So either $u$ or $u_i$ has degree at most $k$ in $G_j$ (thus also in $G_i$), and if $u_i$ has degree at least $k$ in $G_m$ for some $m$, then all yellow edges incident to $u_i$ in $G_m$ should have degree at most $k-1$ in $G_m$, in order for the yellow edges to be special later. Then among vertices in $X_3$, at most $x=\max\{0,k-|X_1|-|X_2|\}$ vertices have degree more than $k$ in $G_i$, and all other vertices have degree at most $k-1$ in $G_i$. Therefore at most $x\Delta+(|X_3|-x)(k-1)$ blue edges are incident to $X_3$. Note that $d(u_i)\le \Delta, |X_2|\le \Delta$ and $|X_1|+x\le k$, then at most
$$|X_1|\Delta+k(|X_2|-1)+x\Delta+(|X_3|-x)(k-1)=(|X_1|+x)\Delta+(k-1)(d(u_i)-|X_1|-x-1)+|X_2|-1\le 2k\Delta-k^2$$
blue edges are within distance one to $e_i$ from $u_i$ side (not including the edges incident to $v_i$).
We also count the blue edges incident to $v_i$ and its neighbors. Similarly, $v_i$ has two kinds of neighbors: the neighbors in $Y_1$ sharing blue edges with $v_i$, and the neighbors in $Y_2$ sharing yellows edges with $v_i$. From the fact that $e_i$ is a special edge, $|Y_1|\le k$, so at most $(|Y_1|-1)\Delta$ blue edges are incident to $Y_1-\{u_i\}$. For each vertex $v$ in $Y_2$, $vv_i$ is a yellow edge in $G_i$ but will be a special edge in $G_s$ for some $s>i$. Similar to above, at most $k-|Y_1|$ vertices in $Y_2$ have degree more than $k$ in $G_i$, and all other vertices in $Y_2$ have degree at most $k-1$ in $G_i$. So at most $(k-|Y_1|)(\Delta-1)+(|Y_2|-(k-|Y_1|))(k-1)$ blued edges are incident to $Y_2$. In total, at most
$$(|Y_1|-1)\Delta+(k-|Y_1|)(\Delta-1)+(|Y_2|-(k-|Y_1|))(k-1)\le (2k-2)\Delta-k(k-1)$$
So in $G_i$, the number of blue edges within distance one to $e_i$ is at most
$$2k\Delta-k^2+(2k-2)\Delta-k(k-1)\le (4k-2)\Delta-k(2k-1)$$
Now color the edges in the list one by one greedily. For each $i$, when it is the turn to color $e_i$, only the edges in $G_i$ (the blue edges) have been colored. Since there are at least $(4k-2)\Delta-k(2k-1)+1$ colors, we are able to color the edges so that edges within distance one get different colors.
\end{proof}
We shall note that the above result is not only true for simple graphs, but also for multigraphs.
\end{document} |
\begin{document}
\date{\empty}
\newenvironment{proof}{\begin{trivlist}
\item[\hspace{\labelsep}{\bf\noindent Proof. }]}
{\end{trivlist}}
\title{
\Large\bf
Random motion with gamma-distributed\\
alternating velocities in biological modeling\thanks{Paper appeared in: \newline
R.\ Moreno-D\'{\i}az et al.\ (Eds.): EUROCAST 2007,
Lecture Notes in Computer Science, Vol.\ 4739, pp.\ 163--170, 2007.
Springer-Verlag, Berlin, Heidelberg. ISBN: 978-3-540-75866-2.}
}
\author{\bf Antonio Di Crescenzo \rm and \bf Barbara Martinucci\\
\\
\rm Dipartimento di Matematica e Informatica, Universit\`a di Salerno\\
Via Ponte don Melillo, I-84084 Fisciano (SA), Italy\\
Email: \{adicrescenzo,bmartinucci\}@unisa.it
}
\pagestyle{plain}
\maketitle
\begin{abstract}
Motivated by applications in mathematical biology concerning randomly
alternating motion of micro-organisms, we analyze a generalized
integrated telegraph process. The random times between consecutive
velocity reversals are gamma-distributed, and perform an alternating
renewal process. We obtain the probability law and the mean
of the process.
\end{abstract}
\section{Introduction}
\label{sec1}
The telegraph random process describes the motion of a particle on the real line,
traveling at constant speed, whose direction is reversed at the arrival epochs
of a Poisson process. After some initial works, such as \cite{Go51}, \cite{Ka74}
and \cite{Or90a}, numerous efforts have been made by numerous authors and through
different methods to analyze this process. Various results on the telegraph
process, including the first-passage-time density and the distribution of motion
in the presence of reflecting and absorbing barriers have been obtained
in \cite{Fo92}, \cite{FoKa94} and \cite{Or95}. A wide and comprehensive review
devoted to this process has recently been offered by Weiss \cite{Weiss02},
who also emphasized its relations with some physical problems.
\par
In various applications in biomathematics the telegraph process arises as a
stochastic model for systems driven by dichotomous noise (see \cite{DiCrMa2006},
for instance). Two stochastic processes modeling the major
modes of dispersal of cells or organisms in nature are introduced in \cite{Othmer};
under certain assumptions, the motion consisting of sequences of runs
separated by reorientations with new velocities is shown to be governed
by the telegraph equation. Moreover, the discrete analog of
the telegraph process, i.e.\ the correlated random walk, is usually
used as a model of the swarming behavior of myxobacteria
(see \cite{Erdetal2004}, \cite{Hill}, \cite{Komin} and \cite{Lutscher}).
Processes governed by hyperbolic equations are also used to describe movement
and interaction of animals \cite{Lu2002} and chemotaxis \cite{HiSt2000}.
Moreover, the integrated telegraph process has been also used to model
wear processes \cite{DiPe2000} and to describe the dynamics of the price
of risky assets \cite{DiPe2002}.
\par
Many authors proposed suitable generalizations of the telegraph process,
such as the 1-dimensional cases with three cyclical velocities \cite{Or90b},
or with $n$ values of the velocity \cite{Ko98}, or with random
velocities \cite{StZa04}. See also the paper by Lachal \cite{Latchal}, where
the cyclic random motion in $\mathbb{R}^d$ with $n$ directions is studied.
\par
A generalized integrated telegraph process whose random times separating
consecutive velocity reversals have a general distribution and perform an
alternating renewal process has been studied in \cite{DiCrescenzo2001} and \cite{Za03}.
Along the line of such articles, in this paper we study a stochastic model
for particles motion on the real line with two alternating velocities $c$
and $-v$. The random times between consecutive reversals of direction perform
an alternating renewal process and are gamma distributed, which extends
the Erlang-distribution case treated in \cite{DiCrescenzo2001}.
\par
In Section~\ref{sec2} we introduce the stochastic process $\{(X_t,V_t); t\geq 0\}$,
with $X_t$ and $V_t$ denoting respectively position and velocity of the particle
at time $t$. In Section~\ref{sec3} we obtain a series-form of the random motion
probability law for gamma-distributed random inter-renewal times, whereas the
mean value of $X_t$ conditional on initial velocity is finally obtained in
Section~\ref{sec4}.
\section{The random motion}
\label{sec2}
We consider a random motion on $\mathbb{R}$ with two alternating velocities
$c$ and $-v$, with $c,v>0$. The direction of motion is forward or backward
when the velocity is $c$ or $-v$, respectively. Velocities change according
to the alternating counting process $\{N_t; t\geq 0\}$ characterized by
renewal times $T_1,T_2,\ldots$, so that $T_n$ is the $n$-th random instant
in which the motion changes velocity. Hence,
$$
N_0=0, \qquad N_t=\sum_{n=1}^{\infty}{\bf 1}_{\{T_n\leqslant t\}},\quad t>0.
$$
Let $\{(X_t,V_t); t\geq 0\}$ be a stochastic process on $\mathbb{R}\times\{-v,c\}$,
where $X_t$ and $V_t$ give respectively position and velocity of the motion at
time $t$. Assuming that $X_0=0$ and $v_0\in\{-v,c\}$, for $t>0$ we have:
\begin{equation}
X_t =\int_0^t V_s \,{\rm d}s,
\qquad
V_t=\frac{1}{2}(c-v)+\mathop{\rm sgn}(V_0)\,\frac{1}{2}(c+v)\,(-1)^{N_t}.
\label{eq:36}
\end{equation}
Denoting by $U_k$ ($D_k$) the duration of the $k$-th time interval during
which the motion goes forward (backward), we assume that $\{U_k; k=1,2,\ldots\}$
and $\{D_k; k=1,2,\ldots\}$ are mutually independent sequences of independent copies
of non-negative and absolutely continuous random variables $U$ and $D$.
\par
If the motion does not change velocity in $[0,t]$, then $X_t=V_0\,t$.
Otherwise, if there is at least one velocity change in $[0,t]$, then
$-vt<X_t<ct$ w.p.\ 1. Hence, the conditional law of
$\{(X_t,V_t);t\geq 0\}$ is characterized by a discrete component
$$
\hbox{${\bf P}$}\{X_t=yt, V_t=y \,|\,X_0=0, V_0=y\},
$$
and by an absolutely continuous component
\begin{equation}
p(x,t\,|\,y)= f(x,t\,|\,y) + b(x,t\,|\,y),
\label{equation:p}
\end{equation}
where
$$
f(x,t\,|\,y) = {\partial \over \partial x}\hbox{${\bf P}$}\{X_t\leq x,V_t=c\,|\,X_0=0, V_0=y\},
$$
$$
b(x,t\,|\,y) = {\partial \over \partial x}\hbox{${\bf P}$}\{X_t\leq x,V_t=-v\,|\,X_0=0, V_0=y\},
$$
with $t>0$, $-vt<x<ct$ and $y\in\{-v,c\}$.
\par
The formal conditional law of $\{(X_t,V_t);t\geq 0\}$ has been given in Theorem 2.1
of \cite{DiCrescenzo2001} for $V_0=c$. Case $V_0=-v$ can be treated by
symmetry.
\par
Explicit results for the probability law have been obtained in Theorem 3.1 of
\cite{DiCrescenzo2001} when the random times $U$ and $D$ separating consecutive
velocity reversals have Erlang distribution. This case describes the random
motion of particles subject to collisions arriving according to a Poisson
process with rate $\lambda$ if the motion is forward and rate $\mu$ it is
backward. When the motion has initial velocity $c$ ($-v$), then the first $n-1$
($r-1$) collisions have no effect, whereas the $n$th ($r$th) collision causes
a velocity reversal. In the following section we shall treat the more general
case in which the random inter-renewal times are gamma distributed.
\section{\bf Gamma-distributed random times}
\label{sec3}
We assume that the random times $U$ and $D$ are gamma distributed with
parameters ($\lambda$,$\alpha$) and ($\mu$,$\beta$), respectively, where
$\lambda, \mu>0$ and $\alpha, \beta>0$. Hereafter we obtain the probability
law of $\{(X_t,V_t); t\geq 0\}$ for this case.
\begin{theorem}
If $U$ and $D$ are gamma-distributed with parameters $(\lambda,\alpha)$
and $(\mu,\beta)$, respectively, for $t>0$ it is
\begin{equation}
\hbox{${\bf P}$}\{X_t=ct, V_t=c\,|\,X_0=0, V_0=c\}=
\frac{\Gamma(\alpha,\lambda t)}{\Gamma(\alpha)},
\label{discreta}
\end{equation}
and, for $-vt<x<ct$,
\begin{equation}
\hspace*{-0.2cm}
f(x,t\,|\,c)=\frac{1}{c+v}\left\{e^{-\mu\overline{x}}
\sum_{k=1}^{+\infty}\frac{\mu^{k\beta}(\overline{x})^{k\beta-1}}{\Gamma(k\beta)}\bigg[P(k\alpha,\lambda x^*)
-P(k\alpha+\alpha,\lambda x^*)\bigg]\right\},
\label{gammaf}
\end{equation}
\begin{eqnarray}
&& \hspace*{-1.cm}
b(x,t\,|\,c)=\frac{1}{c+v}\Biggl\{\frac{\lambda^{\alpha}e^{-\lambda x^*}
(x^*)^{\alpha-1}}{\Gamma(\alpha)}\frac{\Gamma(\beta,\mu\overline{x})}{\Gamma(\beta)}
\nonumber \\
&& \hspace*{0.4cm}
+ e^{-\lambda x^*} \sum_{k=1}^{+\infty}\frac{\lambda^{(k+1)\alpha}(x^*)^{(k+1)\alpha-1}}
{\Gamma((k+1)\alpha)}
\biggl[P(k\beta,\mu\overline{x})-P(k\beta+\beta,\mu\overline{x})\biggr]\Biggr\},
\label{gammab}
\end{eqnarray}
where
$$
\overline{x}=\overline{x}(x,t)=\frac{ct-x}{c+v},
\qquad
x^{*}=x^{*}(x,t)=\displaystyle\frac{vt+x}{c+v},
$$
and
\begin{equation}
\Gamma(a,u)=\int_u^{\infty} t^{a-1}e^{-t} {\rm d}t,
\qquad
P(a,u)=\frac{1}{\Gamma(a)}\int_0^{u} t^{a-1}e^{-t} {\rm d}t,
\qquad a>0.
\label{gammaP}
\end{equation}
\label{teorema3}
\end{theorem}
\begin{proof}
Making use of (2.4) of \cite{DiCrescenzo2001} and noting that for $k\geq 1$
the pdfs of $U^{(k)}=U_1+\ldots+U_k$ e $D^{(k)}=D_1+\ldots+D_k$ are given by
\begin{eqnarray}
f_U^{(k)}(x)=\frac{\lambda^{k\alpha}x^{k\alpha-1}e^{-\lambda x}}{\Gamma(k\alpha)},
\qquad f_D^{(k)}(x)=\frac{\mu^{k\beta}x^{k\beta-1}e^{-\mu x}}{\Gamma(k\beta)},
\qquad x>0,
\label{eq:319}
\end{eqnarray}
we have
\begin{equation}
f(x,t\,|\,c)=\frac{1}{c+v}\,e^{-\mu\overline{x}}
e^{\lambda\overline{x}}\sum_{k=1}^{+\infty}\frac{\mu^{k\beta}(\overline{x})^{k\beta-1}}{\Gamma(k\beta)}
\frac{\lambda^{k\alpha}}{\Gamma(k\alpha)\Gamma(\alpha)} \, {\cal I}_k,
\label{gammaf2}
\end{equation}
where
\begin{equation}
{\cal I}_k:=\int_{\overline{x}}^t e^{-\lambda s}
(s-\overline{x})^{k\alpha-1}\Gamma(\alpha,\lambda(t-s))\,{\rm d}s,
\qquad k\geq 1.
\label{integI}
\end{equation}
Noting that, due to (\ref{gammaP}), $\Gamma(a,u)=\Gamma(a)\,\big[1-P(a,u)\big]$
we obtain
\begin{equation}
{\cal I}_k={\cal I}_{1,k}-{\cal I}_{2,k},
\label{integI1I2}
\end{equation}
where, for $k\geq 1$
\begin{equation}
{\cal I}_{1,k}:=
\Gamma(\alpha)\int_{\overline{x}}^t e^{-\lambda s}
(s-\overline{x})^{k\alpha-1}\,{\rm d}s
=\Gamma(k\alpha)\,\Gamma(\alpha)\,
e^{-\lambda\overline{x}}\lambda^{-k\alpha} P(k\alpha,\lambda x^*),
\label{eq:323}
\end{equation}
\begin{eqnarray}
&& \hspace{-1.2cm}
{\cal I}_{2,k}:= \Gamma(\alpha)\int_{\overline{x}}^t e^{-\lambda s}
(s-\overline{x})^{k\alpha-1}\,P(\alpha,\lambda(t-s))\,{\rm d}s
\nonumber \\
&& \hspace{-0.4cm}
= e^{-\lambda\overline{x}}\Gamma(\alpha)\int_0^{x^*} e^{-\lambda y}
y^{k\alpha-1}\,P(\alpha,\lambda(x^*-y))\,{\rm d}y
= e^{-\lambda\overline{x}} \lambda^{-k\alpha} \,G(\lambda x^*),
\label{eq:324}
\end{eqnarray}
with
\begin{equation}
G(\lambda x^*):=\int_0^{\lambda x^*} e^{-u}u^{k\alpha-1}
\Biggl(\int_0^{\lambda x^*-u}e^{-\tau}\tau^{\alpha-1} \,{\rm d}\tau\Biggr){\rm d}u.
\label{defG}
\end{equation}
Making use of the Laplace transform of $(\ref{defG})$ it follows
$$
{\cal L}_z\{G(\lambda x^*)\}
={\cal L}_z\left\{ e^{-{\lambda x^*}}\left(\lambda x^*\right)^{k\alpha-1}\right\}
{\cal L}_z\Biggl\{\int_0^{\lambda x^*} e^{-\tau} \tau^{\alpha-1}{\rm d}\tau\Biggr\}
=\frac{\Gamma(k\alpha)\,\Gamma(\alpha)}{z(z+1)^{k\alpha+\alpha}}.
$$
Hence, from identity
$$
{\cal L}_z\left\{P(k\alpha+\alpha,\lambda x^*)\right\}
={\cal L}_z\left\{\int_0^{\lambda x^*} {u^{k\alpha+\alpha-1}e^{-u}\over \Gamma(k\alpha+\alpha)}\,{\rm d}u\right\}
=\frac{1}{z(z+1)^{k\alpha+\alpha}},
$$
we have
$$
G(\lambda x^*)= \Gamma(k\alpha)\,\Gamma(\alpha)\,P(k\alpha+\alpha,\lambda x^*).
$$
Eqs.\ (\ref{integI1I2})$\div$(\ref{defG}) thus give
\begin{equation}
{\cal I}_k
=\Gamma(k\alpha)\,\Gamma(\alpha)\, e^{-\lambda\overline{x}}
\lambda^{-k\alpha}\,\big[P(k\alpha,\lambda x^*) - P(k\alpha+\alpha,\lambda x^*)\big].
\label{valIk}
\end{equation}
Eq. (\ref{gammaf}) then follows from (\ref{gammaf2}) and (\ref{defG}).
In order to obtain $b(x,t\,|\,c)$, we recall (2.5) of \cite{DiCrescenzo2001}
and make use of (\ref{eq:319}) to obtain
\begin{eqnarray}
b(x,t\,|\,c)&=& \frac{1}{c+v}\bigg\{\frac{\lambda^{\alpha}e^{-\lambda x^*}
(x^*)^{\alpha-1}}{\Gamma(\alpha)}\frac{\Gamma(\beta,\mu\overline{x})}{\Gamma(\beta)}
\nonumber \\
&+& \left. e^{-\lambda x^*}e^{\mu x^*}\sum_{k=1}^{+\infty}\frac{\lambda^{(k+1)\alpha}
(x^*)^{(k+1)\alpha-1}}{\Gamma((k+1)\alpha)} \right.
\nonumber \\
&\times& \frac{\mu^{k\beta}}{\Gamma(\beta)\Gamma(k\beta)}
\int_{x^*}^t e^{-\mu s}(s-x^*)^{k\beta-1} \Gamma(\beta,\mu(t-s)){\rm d}s\bigg\}.
\label{eq:328}
\end{eqnarray}
Due to (\ref{integI}), the integral in (\ref{eq:328}) can be calculated from
(\ref{valIk}) by interchanging $x^*$, $\beta$, $\mu$ with $\overline{x}$, $\alpha$,
$\lambda$, respectively. Eq.\ (\ref{gammab}) then follows after some calculations.
$\Box$
\end{proof}
\begin{figure}
\caption{Density (\ref{equation:p}
\end{figure}
\par
Figure 1 shows density $p(x,t\,|\,c)$ as $x$ varies in $(-vt,ct)$ for various
choices of $t$, $\alpha$ and $\beta$.
Hereafter we analyze the obtain the limits of densities $f(x,t\,|\,c)$ and
$b(x,t\,|\,c)$ at the extreme points of interval $(-vt, ct)$, for any fixed $t$.
\begin{proposition}\label{proposizione1}
Under the assumptions of Theorem \ref{teorema3} we have
$$
\lim_{x \downarrow -vt} f(x,t|c)=0,
\quad\!
\lim_{x \uparrow ct} f(x,t|c)
= \cases{+\infty, & $0<\beta<1$ \cr
\displaystyle\frac{\mu}{c+v}\big[P(\alpha,\lambda t)-P(2 \alpha, \lambda t)\big],
& $\beta=1$ \cr
0, & $\beta>1$,}
$$
$$
\lim_{x \uparrow ct} b(x,t|c)
=\frac{\lambda^{\alpha} {\rm e}^{-\lambda t} t^{\alpha-1}}{(c+v) \Gamma(\alpha)},
\quad\!
\lim_{x \downarrow -vt} b(x,t|c)
= \cases{+\infty, & $0<\alpha<1$\cr
\displaystyle\frac{\lambda\,\Gamma(\beta, \mu t)}{(c+v) \Gamma(\alpha) \Gamma (\beta)},
& $\alpha=1$ \cr
0, & $\alpha>1$. \cr}
$$
\end{proposition}
\par
From Proposition \ref{proposizione1} we note that if $\alpha<1$ $(\beta<1)$,
i.e.\ the gamma inter-renewal density has a decreasing hazard rate, then the
backward (forward) density is divergent when $x$ approaches $-vt$ $(ct)$.
This is very different from the behavior exhibited in the case of
Erlang-distributed inter-renewals (see Corollary $3.1$
of \cite{DiCrescenzo2001}), when the limits are finite.
\section{\bf Mean value}
\label{sec4}
In this Section we obtain the mean value of $X_t$
when random times $U$ and $D$ are identically gamma distributed.
\begin{theorem}\label{teorema4}
Let $U$ and $D$ have gamma distribution with parameters $(\lambda, \alpha)$.
For any fixed $t\in(0,+\infty)$, we have
\begin{equation}
E\big[X_t\,\big|\,V_0\big]
= V_0\,t+ \frac{c+v}{\lambda} \mathop{\rm sgn}(V_0)
\sum_{k=1}^{+\infty} (-1)^k \Big[\lambda t \, P(k \alpha, \lambda t)
- k \alpha P(k \alpha+1, \lambda t) \Big].
\label{media_}
\end{equation}
\end{theorem}
\begin{proof}
Due to Eqs.\ (\ref{eq:36}) and recalling that $\hbox{${\bf P}$}(T_{k}\leq s)=P(k\alpha,\lambda s)$,
$s\geq 0$, it is
\begin{eqnarray}
E\big[X_t\,\big|\,V_0\big] \!\!\!
&=& \!\!\! \frac{1}{2}(c-v)t
+ \frac{1}{2}(c+v)\mathop{\rm sgn}(V_0)\int_0^t E\left[(-1)^{N_s}\right]{\rm d}s
\label{medsum} \\
&=& \!\!\! \frac{1}{2}(c-v)t
+ \frac{1}{2}(c+v)\mathop{\rm sgn}(V_0)\int_0^t
\left\{1+2\sum_{k=1}^{+\infty} (-1)^k \,P(k\alpha,\lambda s)\right\}{\rm d}s
\nonumber \\
&=& \!\!\! V_0\,t +(c+v)\mathop{\rm sgn}(V_0)\sum_{k=1}^{+\infty} (-1)^k
\int_0^t P(k\alpha,\lambda s)\,{\rm d}s.
\nonumber
\end{eqnarray}
(Note that the above series is uniformly convergent.)
Moreover, recalling (\ref{gammaP}) it is not hard to see that
$$
\int_0^t P(k\alpha,\lambda s)\,{\rm d}s
= t\,P(k\alpha,\lambda t)-{k \alpha \over \lambda}\,P(k \alpha +1,\lambda t).
$$
Eq.\ (\ref{media_}) then immediately follows.
$\Box$
\end{proof}
\begin{figure}
\caption{Mean value $E[X_t\,|\,V_0=c]$, for $c=v=1$ and $\alpha=0.5$ (dotted line),
$\alpha=1$ (dash-dot line), $\alpha=1.5$ (dash line), $\alpha=2$ (solid line),
with (a) $\lambda=1$ and (b) $\lambda=2$.}
\label{figmedia}
\end{figure}
\par
The graphs given in Figure \ref{figmedia} show the mean value of $X_t$
conditional on $V_0=c$ for some choice of the involved parameters.
We note that, being
$P(\alpha,t) \sim {t^{\alpha-1}/\Gamma(\alpha)}$ as $t\to 0$,
under the assumptions of Theorem \ref{teorema4} from (\ref{media_}) we have
$$
E\big[X_t\,\big|\,V_0\big]\sim V_0\,t
\qquad
\hbox{as }t\to 0.
$$
\par
We remark that when $\alpha=n$ is integer, i.e.\ the random times $U$ and $D$
are Erlang-distributed with parameters $(\lambda, n)$, then
$E\big[X_t\,\big|\,V_0\big]$ can be computed making use of
(\ref{medsum}) and noting that
$$
E\big[(-1)^{N_s}\big]
= 1-2{\rm e}^{-\lambda s}\sum_{k=0}^{+\infty}\,
\sum_{j=2nk+n}^{2nk+2n-1}{(\lambda s)^j\over j!}.
$$
For instance, in this case the following expressions hold for $t>0$:
\begin{center}
\begin{tabular}{ll}
\hline
$n$ & $E\big[X_t\,\big|\,V_0\big]$ \\
\hline
$1\quad$ &
$\displaystyle\frac{(c-v)t}{2}+\frac{(c+v)}{4 \lambda}\mathop{\rm sgn}(V_0)\,[1-{\rm e}^{-2 \lambda t}]$ \\
$2\quad$ &
$\displaystyle\frac{(c-v)t}{2}+\frac{(c+v)}{2 \lambda}\mathop{\rm sgn}(V_0)\,[1-{\rm e}^{- \lambda t} \cos(\lambda t)]$ \\
$3\quad$ &
$\displaystyle\frac{(c-v)t}{2}+\frac{(c+v)}{2 \lambda}\mathop{\rm sgn}(V_0)\left\{\frac{1-{\rm e}^{-2 \lambda t}}{6}
+\frac{4}{3}[1-{\rm e}^{-\frac{\lambda t}{2}}\cos(\frac{\sqrt{3}}{2}\lambda t)]\right\}$ \\
$4\quad$ &
$\displaystyle\frac{(c-v)t}{2}+\frac{(c+v)}{2 \lambda}\mathop{\rm sgn}(V_0)\,\left\{
\big[1-(1+\frac{\sqrt{2}}{2}) {\rm e}^{-\lambda t (1-\frac{\sqrt{2}}{2})}
\cos(\frac{\sqrt{2}}{2}\lambda t)\big] \right.$ \\
{} &
$\left. + \big[1-(1-\displaystyle\frac{\sqrt{2}}{2}) {\rm e}^{-\lambda t (1+\frac{\sqrt{2}}{2})}
\cos(\frac{\sqrt{2}}{2}\lambda t)\big]\right\}$ \\
\hline
\end{tabular}
\end{center}
\end{document} |
\begin{document}
\title[Resolutions for monomial curves defined by Arithmetic Sequences]
{Minimal graded Free Resolutions for Monomial Curves defined by Arithmetic Sequences}
\author{Philippe Gimenez}
\address{Department of Algebra, Geometry and
Topology, Faculty of Sciences, University of Valladolid, 47005
Valladolid, Spain.}
\email{[email protected]}
\thanks{The first author is partially supported by MTM2010-20279-C02-02,
{\it Ministerio de Educaci\'on y Ciencia~-~Espa\~na}.}
\author{Indranath Sengupta}
\address{Department of Mathematics, Jadavpur University, Kolkata,
WB 700 032, India.} \email{[email protected]}
\thanks{The second author thanks DST, Government of India for financial support for the
project ``Computational Commutative Algebra", reference no.
SR/S4/MS: 614/09.
}
\author{Hema Srinivasan}
\address{Mathematics Department, University of
Missouri, Columbia, MO 65211, USA.}
\email{[email protected]}
\subjclass[2000]{Primary 13D02; Secondary 13A02, 13C40.}
\date{}
\begin{abstract}
Let ${\bf m}=(m_0,\ldots,m_n)$ be an arithmetic sequence, i.e., a
sequence of integers $m_0<\cdots<m_n$ with no common factor that
minimally generate the numerical semigroup $\sum_{i=0}^{n}m_i\mathbb N$ and
such that $m_i-m_{i-1}=m_{i+1}-m_i$ for all $i\in\{1,\ldots,n-1\}$.
The homogeneous coordinate ring $\Gamma_{\bf m}$ of the affine monomial
curve parametrically defined by $X_0=t^{m_0},\ldots,X_n=t^{m_n}$ is
a graded $R$-module where $R$ is the polynomial ring
$k[X_0,\ldots,X_n]$ with the grading obtained by setting
$\deg{X_i}:=m_i$. In this paper, we construct an explicit minimal
graded free resolution for $\Gamma_{\bf m}$ and show that its Betti
numbers depend only on the value of $m_0$ modulo $n$. As a consequence, we prove a conjecture of
Herzog and Srinivasan on the eventual periodicity of the Betti numbers of semigroup rings under
translation for the monomial curves defined by an arithmetic sequence.
\end{abstract}
\maketitle
\section*{Introduction}
The study of affine and projective monomial curves has a long
history beginning with the classification of space monomial curves in \cite{herzog}.
One of the most dramatic results in the subject is the
fact that the number of generators for the defining ideal of these
curves in the affine space $\mathbb A^{n+1}$ is unbounded, \cite{bresinski}. Much of the
study to date has focussed on determining the generators and the
first Betti number of the defining ideal for many different
classes of monomial curves. In this paper, we study the later Betti numbers as
well as the structure of the resolution. Exact generators in the
case of curves defined by an arithmetic sequence or an almost
arithmetic sequences are known; see \cite {patil}, \cite{malooseng},
\cite{lipatroberts}. In the case of arithmetic sequences, these ideals
have another interesting structure as sum of two determinantal
ideals; see \cite {hip}. This provides the main impetus for
understanding the resolution of these ideals. In this article, we
construct the minimal resolution explicitly for these ideals and
compute all the Betti numbers.
The main goal of this article is to prove the following conjecture
that states that in codimension $n$, there are exactly $n$ distinct
paterns for the minimal graded free resolution of a monomial curve
defined by an arithmetic sequence:
\begin{center}
\begin{tabular}{p{11cm}}\it
Curves in affine $(n+1)$-space defined by a monomial parametrization $X_0=t^{m_0}$,
\ldots, $X_n=t^{m_n}$ where $m_0 < \ldots < m_n$ are positive integers in arithmetic
progression have the property that their Betti numbers are determined solely by $m_0$ modulo $n$.
\end{tabular}
\end{center}
Basis for this conjecture came from the observations in \cite
{seng}, where an explicit minimal free resolution was constructed
for $n=3$, using Gr\"{o}bner basis techniques. Subsequently, this
property was verified to hold for several examples in \cite{eaca},
and it was proved for certain special cases in \cite{hip}, leading
us to frame it as a conjecture. We give a complete proof of the
conjecture in this article.
Let $k$ denote an arbitrary field and $R$ be the polynomial ring
$k[X_0,\ldots,X_n]$. Associated to a sequence of positive integers ${\bf m} = (m_0, \ldots, m_n)$,
we have the $k$-algebra homomorphism $\varphi:R\rightarrow k[t]$ given
by $\varphi(X_i)=t^{m_i}$ for all $i=0,\ldots,n$. The ideal
${\mathcal P}:=\ker{\varphi}\subset R$ is the defining ideal of the monomial
curve in $\mathbb A_k^{n+1}$ given by the parametrization $X_0=t^{m_0}$,
\ldots, $X_n=t^{m_n}$ which we denote by ${C}_{{\bf m}}$ . The $k$-algebra of the numerical semigroup
$\Gamma ({\bf m})$ generated by ${\bf m} = (m_0,\ldots,m_n)$ is the semigroup ring
$k[\Gamma({\bf m})]:=k[t^{m_0},\ldots,t^{m_n}]\simeq R/{\mathcal P}$ which is
one-dimensional. Moreover, ${\mathcal P}$ is a perfect ideal of codimension $n$ and it
is well known that it is minimally generated by binomials.
Since for any positive integer $t$ the curve ${C}_{t{\bf m}}$ is
isomorphic to the curve $ C_{{\bf m}}$ for they have the same defining
ideal, we may as well assume without loss of generality that the
integers $ m_0, m_1, \ldots, m_n $ have {\it no common factor}.
Further, if the semigroup generated by a proper subset of ${\bf m}$
equals the semigroup $\Gamma ({\bf m})$, then the curve ${C}_{{\bf m}}$ is
degenerate and
the resolution of its coordinate ring can be studied in a polynomial
ring with less variables. Hence we can reduce to the case where the
integers in ${\bf m}$ {\it minimally generate} the numerical semigroup
$\Gamma ({\bf m})$.
If, in addition, the integers $m_i$ are in {\it arithmetic
progression}, i.e., $m_i-m_{i-1}=m_{i+1}-m_i$ for all
$i\in\{1,\ldots,n-1\}$, ${\bf m}$ is said to be an {\bf arithmetic
sequence}.
An interesting feature that was revealed in \cite{hip} is that when
${\bf m} = (m_0,\ldots,m_n)$ is an arithmetic sequence, the ideal ${\mathcal P}$
can be written as a sum of two determinantal ideals, ${\mathcal P}=I_2(A) +
I_2(B)$, as we shall recall in Section~\ref{defidealsection}. Here,
$I_2(A)$ is in fact the defining ideal of the rational normal curve
in $\mathbb P^n$. Let us write $m_0$ as $m_0=an+b$ where $a,b$ are
positive integers and $b\in[1,n]$. When $b=1$, the sum $I_2(A) +
I_2(B)$ is again a determinantal ideal and its resolution is
described in \cite[Theorem~2.1 \& Corollary~2.3]{hip}. When $b=n$,
it is just one generator away from a determinantal ideal which is
again simple; see \cite[Theorem~2.4 \& Corollary~2.5]{hip}. The case
$b=2$ corresponds to the case where $k[\Gamma({\bf m})]$ is Gorenstein.
The resolution was described in this case when $n=4$ in
\cite[Theorem~2.6]{hip}. This result will be completed in
Section~\ref{secgorenstein} where the resolution will be given for
an arbitrary $n$.
The observation that the ideals $I_2(A)$ and $ I_2(B)$ are related
by linear quotients (Lemma~\ref{colon}) holds the key for the
construction of the resolution of ${\mathcal P}$ in general. We construct a
tower of mapping cones each of which is a cone over an inclusion of
a shifted graded Koszul complex into a graded Eagon-Northcott
complex. Unfortunately, the above
construction of iterated mapping cone does not yield a minimal free
resolution for ${\mathcal P}$ and therefore we will have to get rid of the redundancy and make
the resolution minimial. The complete graded description of the
resolution is given in the main Theorem \ref {mainThm}. As a
consequence, we compute the total Betti numbers $\beta _j$ in
Theorem~\ref{ThmIndraConj} as follows: if $m_0\equiv b\mod n$ with
$b\in [1,n]$, then
$$
\beta_j=j{n\choose j+1}+
\left\{\begin{array}{ll}
\displaystyle{(n-b+2-j){n\choose j-1}}&\hbox{if}\ 1\leq j\leq n-b+1,\\
\displaystyle{(j-n+b-1){n\choose j}}&\hbox{if}\ n-b+1<j\leq n.
\end{array}\right.
$$
These numbers clearly depend only on the reminder of $m_0$ modulo
$n$, as conjectured in \cite[Conjecture~1.2]{hip}.
As another application of Theorem~\ref{mainThm}, we prove the following conjecture of Herzog and Srinivasan for monomial curves defined by an arithmetic sequence. The strong form of the conjecture says that if ${\bf m}$ is any increasing sequence of non negative integers and
${\bf m} +(j)$ denotes the sequence translated by $j$, then the Betti numbers of the semigroup ring
$k[\Gamma({\bf m} +(j))]$
are eventually periodic in $j$.
We prove in Theorem \ref{ThmPeriodic} that if ${\bf m}$ is an arithnetic sequence, the strong form of the conjecture holds by showing that the betti numbers are periodic with period $m_n- m_0=nd$ where $d$ is the common difference.
We will begin with some preliminaries on the defining ideal of monomial curves associated to arithmetic sequences, followed by some facts from mapping cones that we need in the paper. Section~\ref{secgorenstein} is entirely on Gorenstein monomial curves defined by an arithmetic sequence where we construct the minimal graded free resolution of its coordinate ring as a direct sum of a resoltution and its dual. Section~\ref{secmain} contains the construction of the minimal graded free resolution of the monomial curves defined by an arithmetic sequence in general. The last section has consequences of the main theorem (Theorem~\ref{mainThm}) for some relation between the regularity of the semigroup ring and the Frobenius number of the semigroup, an independent proof of the characterization of Gorenstein curves defined by an arithmetic sequence as well as the proof of the periodicity conjecture of Herzog and Srinivasan for the semigroup rings defined by an arithmetic sequence.
\section{Preliminaries}\label{secprelim}
Let $({\bf m})= (m_0, \ldots, m_n)$ be an {\it arithmetic sequence},
i.e., a sequence of nonnegative integers such that $m_i=m_0+id$ for
some $d\geq 1$ and all $i\in [0,n]$, and such that $m_0, \ldots, m_n$
are relatively prime and minimally generate the numerical semigroup
$\displaystyle{\Gamma({\bf m}):=\sum_{0\leq i\leq n}m_i\mathbb N}$. Note that
one can always write $m_0$ uniquely as $$m_0=an+b$$ with $a,b$ positive integers and $b\in [1,n]$. The
integer $a$ is non-zero because the sequence $({\bf m})= (m_0, \ldots, m_n)$ is minimal.
\subsection{Defining ideal of monomial curves associated to arithmetic sequences}\label{defidealsection}
One knows by \cite[Theorem~2.1]{hip} that the defining ideal ${\mathcal P}$
of the affine monomial curve $C_{\bf m} \subset \mathbb A_k^{n+1}$ is $I_2(A)+I_2(B)$, the sum of two determinantal
ideals of maximal minors with
$$A= {\left(\begin{array}{ccc}
\begin{array}{c}
X_{0}\\[1.5mm]
X_{1}\\[1.5mm]
\end{array} &
\begin{array}{c}
\cdots\\[1.5mm]
\cdots\\[1.5mm]
\end{array} &
\begin{array}{c}
X_{n-1}\\[1.5mm]
X_{n}\\[1.5mm]
\end{array}
\end{array}\right)},
\ B= {\left(\begin{array}{cccc}
\begin{array}{c}
X_{n}^{a}\\[1.5mm]
X_{0}^{a+d}\\[1.5mm]
\end{array} &
\begin{array}{c}
X_{0}\\[1.5mm]
X_{b}\\[1.5mm]
\end{array} &
\begin{array}{c}
\cdots\\[1.5mm]
\cdots\\[1.5mm]
\end{array} &
\begin{array}{c}
X_{n-b}\\[1.5mm]
X_{n}\\[1.5mm]
\end{array}
\end{array}\right)}.$$
It is well-known that the ideal $I_2(A)$ is the defining ideal of
the rational normal curve in $\mathbb P_k^n$ of degree $n$; see, e.g.,
\cite[Proposition~6.1]{eis2}. The fact that $I_2(A)$ is contained in
${\mathcal P}$ says that the affine monomial curve $C_{\bf m}$ is lying on the
affine cone over the rational normal curve. Indeed, the following
easy lemma states that arithmetic sequences are precisely the ones
whose associated monomial curve lies on this cone. They are also the
only sequences that make the ideal $I_2(A)$ homogeneous with respect
to the gradation obtained by setting $\deg{X_i}=m_i$ for all $i\in
[0,n]$.
\begin{lemma}\label{equivGrad}
Let ${\mathcal P}\subset k[X_0,\ldots,X_n]$ be the defining ideal of the
non-degenerate monomial curve $C_{\bf m} \subset \mathbb A_k^{n+1}$ associated
to a strictly increasing sequence of integers
${\bf m}=(m_0,\ldots,m_n)$. The following are equivalent:
\begin{enumerate}
\item
$\exists\, d\in\mathbb Z,$ such that $ m_i=m_0+id,\ \forall i\in [0,n]$;
\item
$I_2(A)\subset {\mathcal P}$;
\item
$I_2(A)$ is homogeneous w.r.t. the weighted gradation on $R$ given
by $\deg{X_i}=m_i$ for all $i\in[0,n]$.
\end{enumerate}
\end{lemma}
\begin{proof}
As we already recalled, if $m_i=m_0+id$ for some $d\geq 1$ then
$I_2(A)\subset {\mathcal P}$ and $I_2(A)$ is homogeneous w.r.t. the weighted
gradation, so (1) $\Rightarrow$ (2) and (1) $\Rightarrow$ (3).
Conversely, assuming that either (2) or (3) holds, one has that the
integers in the sequence $({\bf m})$ satisfy that
$m_i+m_{j+1}=m_{i+1}+m_j$ for all $i,j$ such that $0\leq i<j\leq
n-1$. In particular, for all $j\in [1,n-1]$, one has that
$m_0+m_{j+1}=m_{1}+m_j$, i.e., $m_{j+1}=m_j+d$ if one sets
$d:=m_1-m_0\geq 1$.
\end{proof}
In this paper, homogeneous and graded will mean homogeneous and
graded with respect to the weighted gradation on $R$ given by
$\deg{x_i}=m_i$ for all $i\in[0,n]$. By Lemma~\ref{equivGrad},
$I_2(A)$ is homogeneous and ${\mathcal P}$ is also homogeneous since
${\mathcal P}:=\ker{\varphi}$ and the map $\varphi:R\rightarrow k[t]$ given by
$\varphi(X_i)=t^{m_i}$ is graded of degree 0.
\subsection{The weighted graded version of the Eagon-Northcott
complex}\label{secENgraded}
The minimal resolution of $R/I_2(A)$ is given by the Eagon-Northcott
complex of the matrix $A$ because the height of $I_2(A)$ is $n$.
Let $F= \oplus _{i=1}^n R e_i$ be a free $R$-module of rank $n$ with
basis $e_1, \ldots, e_n$ and $G = Rg_1\oplus Rg_2$ be a free
$R$-module of rank 2. The $2\times n$ matrix $A$ represents a map
$\varphi: F\longrightarrow G^*$. The Eagon-Northcott complex ${\bf E}$ of the
matrix $A$ is
$$
{\bf E}:\ 0\longrightarrow E_{n-1} \stackrel{d_{n-1}}{\longrightarrow} E_{n-2}
\stackrel{d_{n-2}}{\longrightarrow} \cdots\longrightarrow E_1\stackrel{d_{1}}{\longrightarrow} E_0\,,
$$
with $E_0=R$ and, for all $i\in [1,n-1]$, $E_i :=\wedge
^{i+1}F\otimes D_{i-1}G$ and $d_i$ is given by the diagonalization
of $\wedge F$ and multiplication of $DG$.
\begin{remark}\label{dualENrmk}
Note that $R/I_2(A)$ is Cohen-Macaulay and the complex ${\bf E }^*$
is exact. Thus, we get
$$
{\bf E}^*:\ 0\longrightarrow R \stackrel {d_1^*}{\longrightarrow} E_1^* \longrightarrow \cdots
\stackrel { d_{n-2}^* }{\longrightarrow}\wedge ^{n-1}F^* \otimes
D_{n-3}G^*\stackrel { d_{n-1}^* }{\longrightarrow} \wedge^nF^*\otimes
D_{n-2}G^*.
$$
One has that $d_1^*:R\to \wedge ^2 F^*$ is the map $\wedge ^2
(\varphi^*)$, and for $i>1$, the map $d_i^*: \wedge ^iF^*\otimes
D_{i-2}G^* \longrightarrow \wedge ^{i+1}F^*\otimes D_{i-1}G^*$ is given by
$$d_i^*(x\otimes y) = \sum _{t=1}^n x\wedge e_t^* \otimes \varphi(e_t)y.$$
After identifying $\wedge ^nF^*$ with $R$ and $\wedge ^{n-1}F^*$
with $F$, we see that $$d_{n-1}^*(e_i \otimes
g_1^{*(r)}g_2^{*(s)})=(-1)^{n-i}[
X_{i-1}g_1^{*(r+1)}g_2^*{(s)}+X_{i}g_1^{*(r)}g_2^{*(s+1)}].
$$
This will be useful in Section~\ref{secgorenstein}.
\end{remark}
The ideal $I_2(A)$ is homogeneous with respect to the usual grading
on $R$ and the Eagon-Northcott complex is indeed a minimal graded
free resolution of $R/I_2(A)$. This minimal graded resolution
is 2- linear and it is as follows:
$$
{\bf E}:\ 0\rightarrow R^{n-1}(-n)\stackrel{d_{n-1}}{\longrightarrow}
R^{(n-2){n\choose n-1}}(-n+1) \stackrel{d_{n-2}}{\longrightarrow} \cdots
\stackrel{d_{s}}{\longrightarrow} R^{(s-1){n\choose s}}(-s)
\stackrel{d_{s-1}}{\longrightarrow} \cdots
$$
\begin{flushright}
$\displaystyle{\cdots \stackrel{d_{2}}{\longrightarrow} R^{n\choose 2}(-2)
\stackrel{d_{1}}{\longrightarrow} R \rightarrow R/I_2(A)\rightarrow0. }$
\end{flushright}
But $I_2(A)$ is also homogeneous with respect to our weighted
gradation on $R$ as observed in Lemma~\ref{equivGrad} and the
Eagon-Northcott complex is also a minimal graded free resolution of
$R/I_2(A)$ with respect to this weighted gradation. Of course,
syzygies are no longer concentrated in one single degree at each
step of the resolution as before. As observed in \cite{hip}, the
successive graded free modules in this resolution are $E_0=R$ and,
for all $s\in [2,n]$,
\begin{eqnarray*}
E_{s-1}&=&\bigoplus_{1\le r_1<\ldots < r_s \le n }
\left(\bigoplus_{k=1}^{s-1} R(- sm_0 +kd
-\sum r_i d)\right)\\
&=&
\bigoplus_{k=1}^{s-1}
\left(\bigoplus_{1\le r_1<\ldots < r_s \le n} R(- (sm_0 -kd
+\sum r_i d))\right)\\
&=&
\bigoplus_{k=1}^{s-1}
\left(\bigoplus_{0\le r_1<\ldots < r_s \le n-1} R(- (sm_0 +(s-k)d
+\sum r_i d))\right)\\
&=&
\bigoplus_{k=1}^{s-1}
\left(\bigoplus_{0\le r_1<\ldots < r_s \le n-1} R(- (sm_0 +kd
+\sum r_i d))\right)\\ \,.
\end{eqnarray*}
\begin{notation}
Given two integers $m\geq t\geq 1$, it will be useful to denote by
$\sigma(m,t)$ the collection (with repetitions) of all possible sums of $t$ distinct
nonnegative integers which are all strictly smaller than $m$, i.e.,
$$
\sigma (m,t) = \{ \sum _ {0\le r_1<r_2<\cdots <r_t \le m-1} r_i
\}\,.
$$
For instance, $\sigma (4,2) =
\{1,2,3,3,4,5\}$. Note that for all $t$ and $m$ with $1\leq t\leq
m$, $\#\sigma (m,t)={m\choose t}$.
\end{notation}
The weighted graded free resolution of $R/I_2(A)$ given by the
Eagon-Northcott complex can now be written as follows:
{\small
$$
0\rightarrow
\bigoplus_{k=1}^{n-1}R(- (nm_0+kd + {n\choose 2} d ))
\longrightarrow\cdots\longrightarrow
\bigoplus_{k=1}^{s-1}
\left(\bigoplus_{r\in\sigma(n,s)}
R(- (sm_0 +kd+rd))\right)\longrightarrow\cdots
$$
\begin{flushright}
$ \cdots\longrightarrow\displaystyle{
\bigoplus_{r\in\sigma(n,2)}R(-(2m_0+d+r d))
\longrightarrow R \longrightarrow R/I_2(A)\rightarrow 0\,. }$
\end{flushright}
}
\subsection{Mapping cone}\label{subsecMappingCone}
In this section, we will establish our notation for mapping cones,
complexes and some facts on mapping cones that we will need.
For any complex ${\bf F} = \oplus F_i$, denote by $(\delta_{\bf
F})_i:F_i\rightarrow F_{i-1}$ the boundary maps of ${\bf F}$, and for any
$t\geq 1$, let ${\bf F^t}$ be the complex whose $i$th term is
$(F^t)_i := F_{i-t}$. Now if $\mu: {\bf F}\rightarrow {\bf G}$ is a map of
complexes, the {\it mapping cone} (or {\it cone}) over $\mu$ is the
complex ${\bf G}\oplus {\bf F^1}$ and it is denoted by $\cone{\mu}$.
The boundary maps of this complex are
$$
\begin{array}{rccl}
\left(
\begin{array}{cc}
(\delta_{\bf G})_i&(-1)^{i}\mu_{i-1}\\
0&(\delta_{\bf F})_{i-1}
\end{array}
\right)
:&
G_i\oplus F_{i-1}&\rightarrow &G_{i-1}\oplus
F_{i-2}\,,
\end{array}
$$
\noindent
i.e., $(\delta _{\cone {\mu}})_i(g_i,f_{i-1})=((\delta_{\bf G})_i(g_i)+(-1)^i \mu
_{i-1}(f_{i-1}),(\delta_{\bf
F})_{i-1}(f_{i-1}))$.
Let's recall a few well known facts on mapping cones that we will
need in the sequel. If ${\bf G}$ is acyclic, i.e., $H_i({\bf G})= 0$
for $i\ge 1$, then $\cone {\mu}$ is exact up to degree 1, i.e.,
$H_i(\cone {\mu}) = 0$ for all $i\ge 2$. When ${\bf G}$ is exact
and moreover $\mu_0$ is injective, then $\cone {\mu}$ is acyclic. A
situation of special interest is when $\bf F$ is a resolution of
$R/J$ and $\bf G$ a resolution of $R/I$ for two ideals $I$ and $J$
in $R$. Then, given a map of complexes $\mu: {\bf F}\rightarrow {\bf G}$,
$\cone {\mu}$ resoves $R/I+\mu_0(R)$ provided $\mu_0 (J)$ is
contained in $I$. In particular, consider the following situation:
let $I$ be an ideal in $R$ and take an element $z\in R$. Then,
$$
0\rightarrow R/(I:z) \stackrel {\mu} \longrightarrow R/I \longrightarrow R/I+(z)\rightarrow 0
$$
is exact, where $\mu$ is the map given by multiplication by $z$. Now if
${\bf F}$ resolves $R/(I:z)$, ${\bf G}$ resolves $R/I$, and $\mu :{\bf
F} \rightarrow {\bf G}$ is a map of complexes induced by $\mu$, then
$\cone{\mu}$ resolves $R/I+(z)$.
Let's consider now the graded version of the previous statements. Assume
that $I$ and $J$ are homogeneous ideals in $R$, and consider $\bf F$
a graded resolution of $R/J$, $\bf G$ a graded resolution of
$R/I$, and $\mu: {\bf F}\rightarrow {\bf G}$ a graded map of complexes
with $\mu (J)\subset I$. Then, the exact complex $\cone {\mu}$ is
the graded resolution of $R/I+\mu_0(R)$. In the particular case where $J=(I:z)$ for some homogeneous element $z\in R$ of degree $\delta$, the degree zero map $\mu: R(-\delta)\rightarrow R$ given by multiplication by $z$ induces a graded map of complexes $\mu: {\bf F}(-\delta)\rightarrow {\bf G}$ and $\cone{\mu}$ is a graded free resolution of $R/I+(z)$.
Recall that a resolution
$\bf F$ of $R/I$ is {\it minimal} if the ranks of the $F_i$'s are
minimal or, equivalently, if $(\delta _{\bf F}({\bf F}))\subset
{\bf m}{\bf F}$ where ${\bf m} = (x_0, \ldots, x_n)$ is the irrelevant
maximal ideal. Thus, in a minimal graded resolution, there are no
degree zero components in the resolution unless they are identically
zero. If $\bf F$ and $\bf G$ are minimal graded free resolutions of $R/I:z$ and $R/I$ respectively, then
the only possible obstructions for $\cone{\mu}$ to be minimal are
the degree zero components in $\mu$.
In other words, if $\bf F = \oplus F_i $, $\bf G = \oplus G_i$ with
$F_i = \bigoplus _{j} R(-d_{ij})$ and $G_i = \bigoplus _j
R(-c_{ij})$, and $\mu: {\bf F}\rightarrow {\bf G}$ is the graded map of
complexes induced by multiplication by $z$, $R(-\delta)\rightarrow R$, then $\cone {\mu}$ is a minimal graded free resolution of $R/I+(z)$ provided whenever $d_{ij} = c_{ij}$ for
some $i$ and $j$, the projection of the restriction map
$\mu _i|_{R(-d_{ij})}$
onto $R(-c_{ij})$ is identically zero.
If it is not zero, we can split off or cancel the
same two summands $R(- d_{ij})$ from both $F_i$ and $G_i$ in the
mapping cone construction to achieve minimality.
\begin{definition}
The {\it minimized mapping cone} of the map of complexes $\mu:{\bf F}\rightarrow {\bf G}$, denoted by
$\mincone{\mu}$, is the complex obtained from $\cone{\mu}$ after splitting off all possible summands.
\end{definition}
If $z\in R$ is homogeneous element of degree $\delta$, $\bf F$ and $\bf G$ are minimal graded free resolutions of $R/I:z$ and $R/I$ respectively, and $\mu: {\bf F}(-\delta)\rightarrow {\bf G}$ is the graded map of complexes induced by multiplication by $z$, then
$\mincone{\mu}$ is a minimal graded free resolution of $R/I+(z)$.
We finally mention the following result that we will need later on.
\begin {proposition}\label{sumcomplex}
Let
$${\bf F} : 0 \rightarrow F_s\stackrel{f_s}{\longrightarrow} F_{s-1}
\stackrel{f_{s-1}}{\longrightarrow} \cdots \longrightarrow F_t\stackrel{f_t}{\longrightarrow} \cdots \longrightarrow F_1
\stackrel{f_1}{\longrightarrow} F_0$$ and
$${\bf G }: 0 \rightarrow G_s\stackrel{g_s}{\longrightarrow} G_{s-1}
\stackrel{g_{s-1}}{\longrightarrow} \cdots \longrightarrow G_t\stackrel{g_t}{\longrightarrow} \cdots \longrightarrow G_1
\stackrel{g_1}{\longrightarrow} G_0$$ be two exact complexes of free modules and $\varphi = \oplus
\varphi_t : \bf F \to \bf G$ be a map of complexes. Suppose that the
dual $\bf F^*$ is exact. If $\varphi_s = 0$ then we have a homotopy to
$\varphi$ given by $\psi_i:F_{i-1}\to G_{i}$ for $1\le i\le s$ with
$\psi_s = 0$ and $\varphi_i = g_{i+1}\circ
\psi_{i+1} +(-1)^{s-i}\psi_i \circ f_i $. In particular, $\varphi_0(F_0) \subset g_1(G_1)$.
\end{proposition}
\begin{proof}
Let $\psi_s:F_{s-1}\to G_s$ be the zero map. Since, $\varphi_s= 0$, we get, $\varphi_{s-1}\circ f_s=0$ and hence $f_s^*(\varphi _{s-1}^*)=0$. By the exactness of the dual of ${\bf F}$, there exist a map $\psi_{s-1}^*:G_{s-1}^*\to F_{s-2}^*$ such that $f_{s-1}^*\circ \psi_{s-1}^*= \varphi_{s-1}^*$. So, we get, $\varphi_{s-1} = \psi _{s-1}\circ f_{s-1}-g_s\circ \psi_s$. Suppose that we have constructed $\psi_t$ such that, $\varphi_i = g_{i+1}\circ\psi{i+1} +(-1)^{s-i}\psi_i \circ f_i $ for all $i \ge t$. Now, since $\varphi$ is a map of complexes, we have
$ \varphi_{t-1}\circ f_t= g_{t}\circ \varphi_{t}$. Substituting, we get $ \varphi_{t-1}\circ f_t = g_{t}\circ (g_{t+1}\circ \psi{t+1}+(-1)^{s-t} \psi_{t}\circ f_{t} )= (-1)^{s-t}g_t \circ \psi_t \circ f_t$. Thus $(\varphi_{t-1}+(-1)^{s-t+1}g_t\circ \psi_t)\circ f_t=0$. That is, $f_t^*((\varphi_{t-1}^*+(-1)^{s-t+1} \psi_t^*\circ g_t^* )(G_{t-1}^*)=0$.
By the exactness of the dual ${\bf F^*}$, we get a map $\psi_{t-1}^*:G_{t-1}^*\to F_{t-2}^*$ such that $\varphi_{t-1}^*+(-1)^{s-t+1} \psi_t^*\circ g_t^* = f_{t-1}^*\circ \psi_{t-1}^*$ and hence
$\varphi_{t-1}= g_t \circ \psi_t +(-1)^{s-t+1} \psi_{t-1} \circ f_{t-1} $. Thus we prove the existence of $\psi_i$ for all $i$. Looking at $\psi_1:F_0\to G_1$, we get, $\varphi_0(F_0) = g_1 \circ \psi_1(F_0) \subset g_1(G_1)$.
\end{proof}
\section{Gorenstein ideals}\label{secgorenstein}
In this section we deal with the case when $k[\Gamma({\bf m})]$ is Gorenstein separately and see that an explicit construction of the minimal free resolution is obtained from one single mapping cone construction using the fact observed in Section~\ref{defidealsection} that ${\mathcal P}=I_2(A)+I_2(B)$.
Note that the Gorenstein case will also fit into the general construction of iterated mapping cone given in Section~\ref{secmain} as we shall see in Remark~\ref{rmkGor}. The proof we provide in this section completes the argument presented in \cite[Theorem~2.6]{hip} for the case $n=4$.
The explicit computation of the Cohen-Macaulay type of $k[\Gamma({\bf m})]$ in
\cite[Corollary~6.2]{patseng} under the more general assumption of almost arithmetic sequences implies that if
${\bf m}$ is an arithmetic sequence then $k[\Gamma({\bf m})]$ is Gorenstein if and only if $b=2$.
So let's assume that $m_0 \equiv 2 \mod n$ and write $m_0 = an+2$. Then,
$$B= {\left(\begin{array}{cccc}
\begin{array}{c}
X_{n}^{a}\\[1.5mm]
X_{0}^{a+d}\\[1.5mm]
\end{array} &
\begin{array}{c}
X_{0}\\[1.5mm]
X_{2}\\[1.5mm]
\end{array} &
\begin{array}{c}
\cdots\\[1.5mm]
\cdots\\[1.5mm]
\end{array} &
\begin{array}{c}
X_{n-2}\\[1.5mm]
X_{n}\\[1.5mm]
\end{array}
\end{array}\right)}.$$
The ideals $I_2(A)$ and $I_2(B)$ are both of height $n-1$ and the interesting fact is that
the ideal ${\mathcal P} = I_2(A)+I_2(B)$ is Gorenstein of height $n$.
We will construct a resolution for $R/{\mathcal P}$ as follows. We start with
the following preliminary lemma. Consider the map
\begin{equation}\label{alpha}
\alpha: D_{n-2}G^* \cong R^{n-1}\longrightarrow R\,,\quad
g_1^{(i)}g_2^{(n-2-i)}\mapsto ( -1)^i [
X_n^{a}X_{i+2}-X_0^{a+d}X_i]\,.
\end{equation}
\begin{lemma}\label{Alpha1}
Let ${\bf E}$ be the Eagon-Northcott complex which is a minimal
resolution of $R/I_2(A)$ and $\bf {E}^*$ be as in
Section~\ref{secENgraded}. Then the map $\alpha:\,E_{n-1}^* \longrightarrow R$
defined in {\rm(}\ref{alpha}{\rm)} induces a map of complexes
$\alpha:\ {\bf E}^*\to \bf E$.
\end{lemma}
\begin{proof}
Consider the basis element $e_t\otimes g_1^{(i)}g_2^{(n-3-i)}$ in
$\wedge ^{n-1}F^*\otimes D_{n-3}G^*$ where $e_t$ denotes
($(-1)^{n-i} e_1^*\wedge \cdots e_{t-1}^*\wedge e_{t+1}^*\wedge
\cdots e_{n}^*$. Then
$${\small
\begin{array}{rcl}
\alpha (d_{n-1}^* (e_i \otimes g_1^{(r)}g_2^{(s)}))
& = & ( -1)^{i+1} X_{t-1}[ X_n^{a}X_{i+3}-X_0^{a+d}X_{i+1}]+(-1)^iX_t [ X_n^{a}X_{i+2}-X_0^{a+d}X_i] \\
& = & (-1)^{i+1} (X_n^a [X_{t-1}X_{i+3}-X_{t }X_{i+2}] +X_0^{a+d}[X_{t-1}X_{i+1}-X_tX_{i}]) \\
& \in & I_2(A)\,.\\
\end{array}}$$
So, the composition $\wedge ^{n-1}F^*\otimes D_{n-3}G^* \longrightarrow
D_{n-2}G^* \stackrel{\alpha}{\longrightarrow}R \longrightarrow R/I_2(A)$ is zero. Since
$E$ is exact, we can lift the map $\alpha$ to $\alpha: E^*\to E$ as
a map of complexes.
\end{proof}
\begin {theorem}{\label {minresgor}}
Let ${\bf m}=(m_0, \ldots , m_n)$ be an arithmetic sequence with $m_0
\equiv 2 \mod n$. If ${\mathcal P}\subset R$ is the defining ideal of the
monomial curve $C_{\bf m}\subset\mathbb A_k^{n+1}$ associated to the sequence
${\bf m}$, then $R/{\mathcal P}$ is Gorenstein of codimension $n$ with minimal
resolution given by ${\bf E} \oplus ({\bf E}^*)^{\bf 1}$ where $E$
is the Eagon-Northcott resolution of $R/I_2(A)$.
\end {theorem}
\begin{proof}
As we already recalled in Section~\ref{secprelim}, ${\mathcal P}
=I_2(A)+I_2(B)$, and ${\bf E}\to R / I_2(A)\to 0$
is a minimal resolution of $R/I_2(A)$. Let ${\bf E}: 0\to R \to
E_1^* \to \cdots \to E_{n-2}^*\to E_{n-1}^*$ be its dual. If
$\alpha: R^{n-1} = E_{n-1}^* \to R$ is the map defined in
(\ref{alpha}), one has by Lemma~\ref{Alpha1} that $\alpha$ induces a
map of complexes ${\bf \alpha} :\bf E^* \longrightarrow \bf E$. Hence the
mapping cone ${\bf E} \oplus ({\bf E}^*)^{\bf 1}$ is exact. Note
that the image of $\alpha$ is the ideal generated by the $n-1$
principal $2\times 2$ minors of $B$ and $H_0({\bf E} \oplus {\bf
E}^*) = \hbox{Im} (d_1)+\hbox{Im}(\alpha) = I_2(A)+I_2(B)$ because
the rest of the minors of $B$ are already in the ideal $I_2(A)$. So,
${\bf E}\oplus ({\bf E}^*)^{\bf 1}$ resolves $R/{\mathcal P}$ and it is
minimal because $\alpha $ has positive degree.
\end{proof}
Since for $i\in[1,n-1]$, the rank of the free module $E_i$ is
$i{n\choose {i+1}}$, we immediately get the Betti numbers of
$R/{\mathcal P}$:
\begin{corollary}\label{bettiGor}
Let ${\bf m} = (m_0, \ldots , m_n)$ be an arithmetic sequence with $m_0
\equiv 2 \mod n$. Then the Betti numbers of $R/{\mathcal P}$, where
${\mathcal P}\subsetk[X_0,\ldots,X_n]$ is the defining ideal of the monomial curve $C_{\bf m}$
associated to the sequence ${\bf m}$, are $\beta_0=1$, $\beta_i =
i{n\choose {i+1}}+(n-i){n\choose {i-1}}$ for $i\in [1,n-1]$, and
$\beta_n=1$.
\end{corollary}
\section{Explicit construction of a minimal graded resolution}\label{secmain}
Let's go back to the general situation: $m_0=an+b$ with $a,b$
positive integers and $b\in [1,n]$. In this section, we will not really use that
${\mathcal P} = I_2(A)+I_2(B)$ as in the Gorenstein case but essentially that
${\mathcal P}$ is
minimally generated by the 2 by 2 minors of $A$ and the principal
minors of $B$, {\it i.e.}, $\Delta_i:=\Delta_{1,i-b+2}(B)=X_n^aX_{i}-X_0^{a+d}X_{i-b}$ for
$i\in [b,n]$. In other words, if one sets
$$
\forall\,i\in [b,n],\ I_i:=I_2(A)+ (\Delta _b, \ldots , \Delta _{i})\,,
$$
then ${\mathcal P}=I_n$. For all $i\in [b,n]$, a graded free resolution of $R/I_i$
will be obtained by a series of iterated mapping cones.
Set
$$
\delta _i := \deg \Delta _{i} = m_0(a+d+1)+(i-b)d,\ \forall i\in
[b,n]
$$
(of course the degrees are with respect to the weighted grading).
The following lemma is key to the construction of the minimal resolutions.
\begin{lemma}\label {colon}
\begin{enumerate}
\item\label{firstcolon}
$I_2(A):\Delta _ b= I_2(A)$.
\item\label{coloninclusion}
$\forall\, i\in [b,n-1]$,
$(X_0, X_1, \ldots , X_{n-1})\subseteq I_i:
\Delta _{i+1}$.
\end{enumerate}
\end{lemma}
\begin{proof}
(\ref{firstcolon}) holds because $I_2(A)$ is prime and
$\Delta_b\notin I_2(A)$. Moreover, for any $i\in [b,n-1]$ and $j\in
[0,n-1]$, one has that
$$
\begin{array}{rclcl}
X_j\Delta_{i+1}
&\equiv& X_n^aX_{i}X_{j+1}-X_0^{a+d}X_{i-b+1}X_j&\mod&
(X_jX_{i+1}-X_{j+1}X_i)\\
&\equiv& X_0^{a+d}X_{i-b}X_{j+1}-X_0^{a+d}X_{i-b+1}X_j&\mod&\Delta_i
\\
&=&X_0^{a+d}(X_{i-b}X_{j+1}-X_{i-b+1}X_j)&&
\end{array}
$$
which implies that $X_j\Delta_{i+1}\in I_2(A)+ (\Delta _{i})$
because $X_kX_{l+1}-X_{k+1}X_l\in I_2(A)$ for all $k,l\in [0,n-1]$, and we are done.
\end{proof}
\begin{remark}\label{rmkcolon}
Observe that Lemma~\ref{colon}~ (\ref{coloninclusion}) implies that, for all $i\in [b,n-1]$, either
$I_i:\Delta _{i+1}=(X_0, X_1, \ldots , X_{n-1})$ or $I_i:\Delta _{i+1}=(X_0, X_1, \ldots , X_{n-1}, X_n^\ell)$
for some $\ell\ge 1$. Indeed, we will see in (2) of Inductive Step~\ref{induction} that the latter never occurs.
\end{remark}
We are now ready for our iterated mapping cone construction.
Recall from Section~\ref{secENgraded} that the minimal graded free
resolution ${\bf E }=
\oplus _{i=0}^{n-1} E_i$ of $R/I_2(A)$ given by the Eagon-Northcott complex of the
matrix $A$ is
$$
0\rightarrow E_{n-1}\longrightarrow\cdots\longrightarrow E_1\longrightarrow E_0\longrightarrow R/I_2(A)\rightarrow 0
$$
with $E_0=R$ and
$\displaystyle{E_{s-1}=
\bigoplus_{k=1}^{s-1}
\left(\bigoplus_{r\in\sigma(n,s) } R(- (sm_0 +kd
+r d))\right)}$
for all $s\in [2,n]$.
Let ${\bf C}_b = {\bf E} \oplus {\bf E^1}(-\delta _b)$ denote the mapping cone of the map
${\bf \Delta} _b: {\bf E} \rightarrow {\bf E}$ which is induced by multiplication by
$\Delta _b$ and $\delta_b = \deg(\Delta_b)$. By Lemma~\ref{colon}~(\ref{firstcolon})
together with the fact that all the individual maps in ${\bf \Delta} _b$ are multiplication
by $\Delta _b$ and hence not zero (in fact injective), we get that ${\bf C}_b$ is the
minimal resolution of $R/I_ b$.
\begin{notation}\label{notationL}
Set
$
L(s,k):=\bigoplus_{r\in\sigma(n,s)}R(-[m_0(a+d+s+1)+kd+rd])
$
for all $s\in [1,n]$ and $k\ge 1$. Then, for all $s\in [2,n]$,
$({\bf C}_{b})_s=E_s\oplus\left(\bigoplus_{k=1}^{s-1}\left(L(s,k)\right)\right)$, and
the free modules in ${\bf C}_b$ are
\begin{equation}\label{b}
\begin{array}{rcl}
({\bf C}_{b})_0&=&E_0=R,\\
({\bf C}_{b})_1&=&E_1\oplus E_{0}(-\delta_b)=E_1\oplus R(-(m_0(a+d+1))),\\
({\bf C}_{b})_s&=& E_s\oplus E_{s-1}(-\delta_b)\\
&=&\displaystyle{
E_s\oplus\left(\bigoplus_{k=1}^{s-1}\left(L(s,k)\right)\right),\ \forall s\in[2,n-1]
} \\
({\bf C}_{b})_n&=& \displaystyle{\bigoplus _{k=1}^ {n-1}
R(-[m_0(a+d+n+1)+kd+{n\choose 2}d])}\,.
\end{array}
\end{equation}
\end{notation}
\begin{remark}
If $b = n$, ${\bf C}_b$ is the minimal resolution of
$R/{\mathcal P}$. This is indeed the resolution obtained in
\cite[Theorem~3.4 and Corollary~3.5]{hip} in the case $b=n$.
\end{remark}
Let ${\bf K }= \oplus _{i=0}^n K_i$ be the Koszul resolution of
$R/(X_0, \ldots , X_{n-1})$, i. e.,
$$
0\rightarrow K_{n}\longrightarrow\cdots\longrightarrow K_1\longrightarrow K_0\longrightarrow R/(X_0,\ldots , X_{n-1})\rightarrow
0
$$
with $K_0=R$,
$
K_1=\bigoplus_{k=0}^{n-1}R(-(m_0+kd))
$,
and $K_{s}=
\bigoplus_{r\in\sigma(n,s) }R(-(sm_0+r d))
$ for all $s\in [2,n]$.
Note that for all $s\in [2,n]$ and $i\in [b,n]$,
$$
K_s(-\delta_i)=K_s(-(m_0(a+d+1)+(i-b)d))=L(s,i-b)\,.
$$
For $i\in [b,n]$, we construct inductively two sequences of complexes ${\bf C}_{i}$ and ${\bf M}_i$
both resolving $R/I_i$ with ${\bf M}_i$ being a minimal resolution. For $i=b$,
${\bf C}_{b}={\bf M}_b={\bf E} \oplus {\bf E^1}(-\delta _b)$ is given in (\ref{b}).
We will now prove the following sequence of steps that forms the $i$-th step of our construction.
\begin{inductive}\label{induction}
\begin{enumerate}
\item
The minimal resolution of $R/I_{i-1}$ has length $n$.
\item
$I_{i-1}:\Delta _{i}=(X_0, X_1, \ldots , X_{n-1})$.
\item
Multiplication by $\Delta_i$ on $R$ induces a map of complexes
${\bf \Delta}_i:{\bf K}(-\delta_i)\rightarrow {\bf M}_{i-1}$.
\item
${\bf C}_{i}=\cone{{\bf \Delta}_i}$ resolves $R/I_i$.
\item
${\bf M}_{i}$ is the minimized mapping cone of ${\bf \Delta}_i$ and is given by
\begin{itemize}
\item
$({\bf M}_{i})_0=R$,
\item
$({\bf M}_{i})_1=E_1\oplus
\displaystyle{\bigoplus_{k=0}^{i-b}R(-[m_0(a+d+1)+kd])}$,
\item
$({\bf M}_{i})_s=
\left\{\begin{array}{ll}
E_s \oplus
\left(\bigoplus_{k=s-1}^{i-b} L(s-1,k)\right)&
\hbox{ for }s\in [2,i-b+1],\\
E_{s} \oplus
\left(\bigoplus_{k=i-b+1}^{s-1} L(s,k)\right)&
\hbox{ for }s\in [i-b+2,n].
\end{array}\right.$
\end{itemize}
\item
The minimal resolution of $R/I_{i}$ has length $n$.
\end{enumerate}
\end{inductive}
\noindent
(1) $\Rightarrow$ (2).
As observed in Remark~\ref{rmkcolon}, by Lemma~\ref{colon}~ (\ref{coloninclusion}) one has that
$I_{i-1}:\Delta _{i}$ is either $(X_0, X_1, \ldots , X_{n-1})$ or $(X_0, X_1, \ldots , X_{n-1}, X_n^\ell)$ for some $\ell\ge 1$,
and it is well-known that the Koszul complex provides minimal graded free resolutions for $R/(X_0, \ldots , X_{n-1})$ and
$R/(X_0, \ldots , X_{n-1}, X_n^\ell)$. Consider the Koszul resolution ${\bf K'}= \oplus _{i=0}^{n+1}K_i'$ of
$R/(X_0, \ldots , X_{n-1}, X_n^\ell)$:
$$
0\rightarrow K'_{n+1}\longrightarrow\cdots\longrightarrow K'_1\longrightarrow K'_0\longrightarrow R/(X_0, \ldots , X_{n-1}, X_n^\ell)\rightarrow 0\,.
$$
Assume that $I_{i-1}:\Delta_{i}=(X_0, X_1, \ldots , X_{n-1}, X_n^\ell)$ for some $\ell\ge 1$ and consider the complex map ${\bf\Delta}'_{i}:{\bf K'}(-\delta_{i})\rightarrow {\bf M}_{i-1}$ induced by multiplication by $\Delta_{i}$. Then the mapping cone of ${\bf\Delta}'_{i}$ provides a free resolution of $R/I_{i}$ and since ${\bf K'}(-\delta_{i})$ and ${\bf M}_{i-1}$ have length $n+1$ and $n$ respectively, $\cone{{\bf\Delta}'_{i}}$ has length $n+2$. It may not be minimal nevertheless, since $({\bf\Delta}'_{i})_{n+1}:K'_{n+1}(-\delta_{i})\rightarrow 0$ is the zero map, no cancelation can occur at the last step of $\cone{{\bf\Delta}'_{i}}$ and $R/I_{i}$ would have a minimal resolution of length $n+2$ which is impossible since
$R$ is a regular ring of length $n+1$. Thus, $I_{i-1}:\Delta _{i}=(X_0, X_1, \ldots , X_{n-1})$.
\noindent
(2) $\Rightarrow$ (3) $\Rightarrow$ (4) and (5) $\Rightarrow$ (6) are straightforward. It remains to show that (4) $\Rightarrow$ (5).
\noindent
Set $i=b+t$. The complex map ${\bf \Delta} _{b+t}: {\bf K}(-\delta
_{b+t})\rightarrow {\bf M}_{b+t-1}$
induced by multiplication by
$\Delta_{b+t}$ (which is of degree $\delta_{b+t}=m_0(a+d+1)+td$) is
given by the following diagram (the left column is the shifted
Koszul complex ${\bf K}(-\delta_{b+t})$ that resolves
$R(-[m_0(a+d+1)+td])/(X_0, \ldots , X_{n-1})$ minimally,
and the column on the right hand side is ${\bf M}_{b+t-1}$ that resolves $R/I_{b+t-1}$ minimally):
$$
\begin{CD}
0 @. 0\\
@VV V @VV V\\
L(n,t)
@> ({\bf \Delta}_{b+t})_n>>
\bigoplus_{k=t}^ {n-1} L(n,k)
\\
@VV V @VV V\\
L(n-1,t)
@>({\bf \Delta}_{b+t})_{n-1}>>
E_{n-1}\oplus
\left(\bigoplus_ {k={t}}^{n-2}
L(n-1,k)\right)
\\
@VV V @VV V\\
\vdots @. \vdots \\
@VV V @VV V\\
L(t+2,t)
@> ({\bf \Delta} _{b+t})_{t+2}>>
E_{t+2} \oplus
\left(\bigoplus_{k=t}^{t+1} L(t+2,k)\right)
\\
@VV V @VV V\\
L(t+1,t)
@> ({\bf \Delta} _{b+t})_{t+1}>>
E_{t+1} \oplus
L(t+1,t)
\\
@VV V @ VV V\\
L(t,t)
@>({\bf\Delta} _{b+t})_{t}>>
E_{t}\oplus L(t-1,t-1)
\\
@VV V @ VV V\\
\vdots @. \vdots \\
@VV V @ VV V\\
L(3,t)
@>({\bf\Delta}_{b+t})_3>>
E_3 \oplus
\left(\bigoplus_{k=2}^{t-1} L(2,k)\right)
\\
@VV V @ VV V\\
L(2,t)
@>({\bf \Delta} _{b+t})_{2}>>
E_{2} \oplus
\left(\bigoplus_{k=1}^{t-1}
L(1,k) \right)
\\
@VV V @ VV V\\
L(1,t)
@>({\bf \Delta}_{b+t})_1>>
E_1\oplus
\bigoplus_{k=0}^{t-1}R(-[m_0(a+d+1)+kd])
\\
@VV V @ VV V\\
R(-[m_0(a+d+1)+td])
@>
({\bf \Delta}_{b+t})_0
>>
R
\end{CD}
$$
Now observe in the previous diagram that, for $s\in [t+1,n]$, the left side $({\bf
K}(-\delta_{b+t}))_s$ corresponds to the summand $k=t$ on the
right hand side and we will show that it splits off entirely.
The following lemma shows that none of these maps are identically zero.
\begin {lemma}\label {positive}
Let ${\bf K}$, ${\bf M}$ and ${\bf \Delta} _{b+t}: {\bf K}(-\delta
_{b+t})\rightarrow {\bf M}_{b+t-1}$ be as before. Then, $({\bf \Delta} _{b+t})_i\neq 0$ for all
$0\le i\le n$. In fact, if we
choose a basis for the free modules $K_i$, then we can pick a map
of complexes ${\bf \Delta} _{b+t}$ such that $({\bf \Delta} _{b+t})_i $ is not zero on any of the
chosen basis elements of $K_i$.
\end{lemma}
\begin{proof}
Since $\bf M$ is minimal of length $n$ and $({\bf \Delta} _{b+t})_0$ given by
multiplication by $\Delta_{b+t}$ is injective (not zero), all the
maps $({\bf \Delta} _{b+t})_i, 0\le i\le {n-1}$ are not zero and can be so chosen to
be not zero on any chosen basis elements of $K_t$. The only
question is for the last map $({\bf \Delta} _{b+t})_n$. This we will take care by the
Proposition~\ref{sumcomplex}. Since
$\Delta _{b+t}$ is not contained in the ideal $I_{b+t-1}$, we get that
$({\bf \Delta} _{b+t})_0(R) = \Delta _{b+t}R$ is not contained in the image of ${\bf M}_1$.
Thus by Proposition~\ref{sumcomplex}, we see that $({\bf \Delta} _{b+t})_n$ is not
equal to zero. Since $K_n= R$, therefore, we get that $({\bf \Delta} _{b+t})_n$ is not
equal to zero on a basis for the free module $K_n$.
\end{proof}
Next we show that the projection of $({\bf \Delta} _{b+t})_s$ onto $L(s,t)$ is not zero for any $s\geq t+1$.
By degree consideration, the projection of $({\bf \Delta} _{b+t})_s$ onto $L(s,k)$ for $k>t$ is certainly zero. What is left follows from the next lemma.
\begin {lemma}\label {projection}
For all $1\le t\le n-b$,
the projection of $({\bf \Delta}_{b+t})_k(R(-[m_0(a+d+k+1)+td+rd])$ onto
$R(-[m_0(a+d+k+1)+td+rd])$ is not zero, and hence is a multiplication by a unit,
for every $k\in [t+1,n]$ and $r\in \sigma(n,k)$.
\end{lemma}
\begin{proof}
It suffices to show that
\begin{equation}\label{p}
({\bf \Delta}_{b+t})_k(R(-[m_0(a+d+k+1)+td+rd])\not\subset E_k
\end{equation}
that is, the projection of $({\bf \Delta}_{b+t})_k(L(k,t))$ onto
$L(k,t)$ is not zero. If this projection is not zero for some $k$ and $t$, then the projection of $({\bf \Delta}_{b+t})_k(R(-[m_0(a+d+k+1)+td+rd])$ onto $R(-[m_0(a+d+k+1)+td+rd]$
is not zero for the lowest $r\in\sigma(n,k)$. Then we can split it off and go to the next smallest $r$ and so we get the lemma.
Now the rest of the proof is by descending induction on $k$.
None of these maps $\Delta$ are identically zero by Lemma~\ref{positive}.
If $k=n$, $E_n=0$ and (\ref{p}) holds, hence the lemma is true for all $t$.
Assume that the lemma holds for all $k\in [s+1,n]$ and for some $s$. Let $k=s$. Suppose that there
is an $r= r_1+\cdots +r_s$ such that $(\Delta _{b+t})_{s}(R(-[m_0(a+d+s+1)+td+rd]) $ is
entirely in $E_s$. Pick $r_0$ to be the smallest non negative integer not in $\{r_1, \ldots,
r_s\}$. Consider the commutative diagram:
$$
\begin{array}{ccc}
R(-[m_0(a+d+s+2)+td+r+r_0d] &\longrightarrow& ({\bf M}_{b+t-1})_{s+1}\\
\downarrow && \downarrow\\
L(s,t) &\longrightarrow & ({\bf M}_{b+t-1})_s
\end{array}
$$
Since the lemma is true for $s+1$, we can take for every $r\in \sigma(n,s+1)$,
$$(\Delta _{b+t})_{s+1}(R(-[m_0(a+d+s+2)+td+rd+r_0d]) =
R(-[m_0(a+d+s+2)+td+rd+r_0d])+...$$
Continuing with the vertical arrow on the right, we see that
$R(-[m_0(a+d+s+2)+td+rd+r_0d])$
maps onto
$$\Delta _{b+t-1}E_S+\sum _{i\ge
0}(\pm)X_{r_i}R(-[m_0(a+d+s+2)+td+(r+r_0-r_i)d]).$$
Thus
every one of the $r_i$ and in particular $X_{r_0}$
that makes up the sum $r+r_0$ will appear as part of the the image in $(M_{b+t-1})_s/E_s$. Thus this
image is not
contained in $E_s\bigoplus (X_{r_1}, X_{r_2},\ldots , X_{r_s})\bigoplus
R(-[m_0(a+d+s+2)+(t-1)d+(r+r_0-r_i)d])$.
If we first come down
and then apply $(\Delta _{b+t-1})_s$, the image is
$(\Delta_{b+t-1})_s(\sum _{i\ge 0} \pm X_{r_i} 1 (-[m_0(a+d+s+2)+td+(r+r_0-r_i)d])$ which
is contained in $(E_s\bigoplus (X_{r_1}, X_{r_2},\ldots , X_{r_s})\bigoplus
R(-[m_0(a+d+s+2)+td+(r+r_0-r_i)d])$ - a contradiction. This completes the induction and the proof.
\end{proof}
For $s\in [t+1,n]$, the left side $({\bf
K}(-\delta_{b+t}))_s$ splits off entirely with the summand $k=t$ on the
right hand side.
On the other
hand, for $s\in [0,t]$, no cancelation occurs. So the minimal free resolution of $R/I_i$ is as described in (5) of Inductive Step~\ref{induction}. This completes our inductive construction and we have proved the following:
\begin{theorem}\label{resIi}
Let ${\bf m}=(m_0, \ldots, m_n)$ be an arithmetic sequence with common difference $d$ and write
$m_0=an+b$ for $a,b$ two positive integers with $b\in [1,n]$.
Consider the polynomial ring $R=k[X_0,\ldots , X_n]$ with $\deg{X_i}=m_i$. Let $J\subset R$ be the defining ideal
of the rational normal curve and ${\bf E}$ be its minimal resolution given by the Eagon-Northcott complex. Set $\Delta_i:=X_n^aX_{i}-X_0^{a+d}X_{i-b}$ for all
$i\in [b,n]$, $\delta_i:=\deg{\Delta_i}=m_0(a+d+1)+(i-b)d$, and consider the ideal
$I_i:=J+ (\Delta _b, \ldots, \Delta _{i})\subset R$. Then for all $i\in [b,n]$, $R/I_i$ is Cohen-Macaulay of codimension $n$ with
minimal graded free resolution ${\bf M}_i$ given by
$$
\begin{array}{rcl}
{\bf M}_b&=&\mincone{{\bf \Delta}_b:{\bf E}(-\delta_b)\rightarrow {\bf E}}=\cone{{\bf \Delta}_b:{\bf E}(-\delta_b)\rightarrow {\bf E}}
\\
{\bf M}_i&=&\mincone{{\bf \Delta}_i:{\bf K}(-\delta_i)\rightarrow {\bf M}_{i-1}}\,,\ \forall i\in [b+1,n]
\end{array}
$$
where ${\bf K}$ is the Koszul resolution of $(X_0,\ldots,X_{n-1})$.
The free modules in ${\bf M}_i$ are explicitly given by
$$
0\rightarrow ({\bf M}_i)_n \longrightarrow ({\bf M}_i)_{n-1} \longrightarrow \cdots \longrightarrow ({\bf M}_i)_1\longrightarrow R\longrightarrow R/I_i\rightarrow 0
$$
where
\begin{itemize}
\item
$({\bf M}_{i})_0=R$,
\item
$({\bf M}_{i})_1=E_1\oplus
\displaystyle{\bigoplus_{k=0}^{i-b}R(-[m_0(a+d+1)+kd])}$,
\item
$({\bf M}_{i})_s=
\left\{\begin{array}{ll}
E_s \oplus
\left(\bigoplus_{k=s-1}^{i-b} L(s-1,k)\right)&
\hbox{ for }s\in [2,i-b+1],\\
E_{s} \oplus
\left(\bigoplus_{k=i-b+1}^{s-1} L(s,k)\right)&
\hbox{ for }s\in [i-b+2,n].
\end{array}\right.$
\end{itemize}
In particular,
$({\bf M}_{i})_n=
\left\{\begin{array}{ll}
L(n-1,n-1)&
\hbox{ if $i=n$ and $b=1$},\\
\left(\bigoplus_{k=i-b+1}^{n-1} L(n,k)\right)&
\hbox{ otherwise.}
\end{array}\right.$
\end{theorem}
\begin{corollary}\label{CMtypeIi}
Using notations in Theorem~\ref{resIi}, the Cohen-Macaulay of type of $R/I_i$ is
$\left\{
\begin{array}{ll}
n &\hbox{ if } i=n \hbox{ and } b=1,\\
n-1+b-i &\hbox{ otherwise.}
\end{array}\right.$
\end{corollary}
In particular, since $I_n={\mathcal P}$ is the defining ideal of the monomial curve $C_{\bf m}$, we get the following theorem.
\begin{theorem}\label{mainThm}
Let ${\bf m}=(m_0, \ldots, m_n)$ be an arithmetic sequence and write
$m_0=an+b$ for $a,b$ two positive integers with $b\in [1,n]$. If
${\mathcal P}\subset R:=k[X_0,\ldots,X_n]$ is the defining ideal of the monomial curve
$C_{\bf m}\subset\mathbb A_k^{n+1}$ associated to ${\bf m}$ then $R/{\mathcal P}$ is
Cohen-Macaulay of codimension $n$ and its minimal graded free resolution ${\bf M}_n$
is
$$
0\rightarrow F_n \longrightarrow E_{n-1}\oplus F_{n-1} \longrightarrow \cdots \longrightarrow E_1\oplus
F_1\longrightarrow R\longrightarrow R/{\mathcal P}\rightarrow 0
$$
where, for all $s\in[2,n]$, $\displaystyle{E_{s-1}=
\bigoplus_{k=1}^{s-1}
\left(\bigoplus_{r\in\sigma(n,s) } R(- (sm_0+kd+rd))\right)}$, and
$$
\begin{array}{rcl}
F_1 &=& \displaystyle{ \left(\bigoplus _{k=0}^{n-b}R(-[m_0(a+d+1)+kd])\right)}\\
F_2 &=& \displaystyle{ \left(\bigoplus_{k=1}^{n-b}\left(
\bigoplus_{r=0}^{n-1}R(-[(m_0(a+d+2)+kd +rd])\right)\right)}
\\
F_s &=& \left\{
\begin{array}{ll}
\displaystyle{
\left(\bigoplus_{k=0}^{n-b+1-s}\left( \bigoplus_{r\in \sigma (n,s-1)}
R(-[m_0(a+d+s)+kd+rd])\right)\right)}& \hbox{\rm if}\ s\in[3,n-b+1],
\\
\displaystyle{
\left(\bigoplus_{k=1}^{s-n+b-1}\left( \bigoplus_{r\in \sigma (n,s)}
R(-[m_0(a+d+s+1)+kd+rd])\right)\right)}& \hbox{\rm if}\
s\in[n-b+2,n].
\end{array}\right.
\end{array}
$$
\end{theorem}
\section{Consequences}\label{secconsequences}
The first direct consequence of Theorem~\ref{mainThm} shows that
\cite[Conjecture~2.2]{hip}, which was first stated in \cite{eaca},
holds:
\begin{theorem}\label{ThmIndraConj}
With notations as in Theorem~\ref{mainThm}, the Betti numbers of
$R/{\mathcal P}$ only depend on $n$ and on the value of $m_0$ modulo $n$.
\end{theorem}
\begin{proof}
If one reads the Betti numbers $\{\beta_j,\ j\in [0,n]\}$ in the
minimal graded free resolution in Theorem~\ref{mainThm}, one gets
that $\beta_0=1$ and
\begin{equation}\label{bettiGeneral}
\beta_j=j{n\choose j+1}+
\left\{\begin{array}{ll}
\displaystyle{(n-b+2-j){n\choose j-1}}&\hbox{if}\ 1\leq j\leq n-b+1,\\
\displaystyle{(j-n+b-1){n\choose j}}&\hbox{if}\ n-b+1<j\leq n,
\end{array}\right.
\end{equation}
where $m_0\equiv b\mod n$ and $b\in [1,n]$ and hence the statement
holds.
\end{proof}
\begin{example}
For $n=4$, if ${\mathcal P}\subset R=k[X_0,\ldots , X_4]$ is the defining ideal of the monomial curve associated
to an arithmetic sequence ${\bf m}=(m_0,\ldots , m_4)$, the $4$ different patterns for the global Betti numbers of
$R/{\mathcal P}$ are as follows. They correspond respectively to $b=1$, 2, 3 and 4:
$$
\begin{array}{lllllllllllllll}
0 & \rightarrow & R^4 & \longrightarrow & R^{15} & \longrightarrow & R^{20} & \longrightarrow & R^{10} & \longrightarrow & R & \longrightarrow & R/{\mathcal P} & \rightarrow & 0 \\
0 & \rightarrow & R & \longrightarrow & R^9 & \longrightarrow & R^{16} & \longrightarrow & R^9 & \longrightarrow & R & \longrightarrow & R/{\mathcal P} & \rightarrow & 0 \\
0 & \rightarrow & R^2 & \longrightarrow & R^7 & \longrightarrow & R^{12} & \longrightarrow & R^8 & \longrightarrow & R & \longrightarrow & R/{\mathcal P} & \rightarrow & 0 \\
0 & \rightarrow & R^3 & \longrightarrow & R^{11} & \longrightarrow & R^{14} & \longrightarrow & R^7 & \longrightarrow & R & \longrightarrow & R/{\mathcal P} & \rightarrow & 0
\end{array}
$$
For example, the two arithmetic sequences ${\bf m}_1=(11,13,15,17,19)$ and ${\bf m}_2=(7,12,17,22,27)$ fit into the third pattern because $11\equiv 7\equiv 3\mod 4$. Denoting by ${\mathcal P}_1$ and ${\mathcal P}_2$ the defining ideal of $C_{{\bf m}_1}$ and $C_{{\bf m}_2}$ respectively, the minimal graded free resolutions of $R/{\mathcal P}_1$ and $R/{\mathcal P}_2$ are given by Theorem~\ref{mainThm}, and the result can easily be checked
using the softwares CoCoA, Macaulay2 or Singular:
{\footnotesize
\begin{eqnarray*}
0\rightarrow
R(-115)\oplus R(-117)
\longrightarrow
\begin{array}{r}
R(-58)\oplus R(-60)\\
\oplus R(-62)\oplus R(-98)\\
\oplus R(-100)\oplus R(-102)\\
\oplus R(-104)
\end{array}
\longrightarrow
\begin{array}{r}
R(-41)\oplus R(-43)^2\\
\oplus R(-45)^2\oplus R(-47)^2\\
\oplus R(-49)\oplus R(-68)\\
\oplus R(-70)\oplus R(-72)\\
\oplus R(-74)
\end{array}\\
\longrightarrow
\begin{array}{r}
R(-26)\oplus R(-28)\\
\oplus R(-30)^2\oplus R(-32)\\
\oplus R(-34)\oplus R(-55)\\
\oplus R(-57)
\end{array}
\longrightarrow
R\longrightarrow R/{\mathcal P}_1\rightarrow 0.
\end{eqnarray*}
\begin{eqnarray*}
0\rightarrow
R(-117)\oplus R(-122)
\longrightarrow
\begin{array}{r}
R(-63)\oplus R(-68)\\
\oplus R(-73)\oplus R(-95)\\
\oplus R(-100)\oplus R(-105)\\
\oplus R(-110)
\end{array}
\longrightarrow
\begin{array}{r}
R(-41)\oplus R(-46)^2\\
\oplus R(-51)^2\oplus R(-56)^2\\
\oplus R(-61)^2\oplus R(-66)\\
\oplus R(-71)\oplus R(-76)
\end{array}\\
\longrightarrow
\begin{array}{r}
R(-24)\oplus R(-29)\\
\oplus R(-34)^2\oplus R(-39)\\
\oplus R(-44)\oplus R(-49)\\
\oplus R(-54)
\end{array}
\longrightarrow
R\longrightarrow R/{\mathcal P}_2\rightarrow 0.
\end{eqnarray*}
}
\end{example}
\begin{remark}
It is important to note that the phenomenon described in
Theorem~\ref{ThmIndraConj} is something special about arithmetic
sequences only. In general, if $0 < m_{0} < \cdots < m_{n}$ is a sequence of
integers then the Betti numbers of the monomial curve defined by
$x_{i} = t^{m_{i}}$ do not depend only on $n$ and the remainder of
$m_{0}$ upon division by $n$. It does not hold even for almost
arithmetic sequences, even in dimension 3.
In dimension 3, a monomial curve is either a complete
intersection with $\beta_1=2, \beta_2=1$ or an ideal of $ 2\times 2$ minors
of a $3\times 2$ matrix with $\beta_1=3$ and $\beta_2=2$. Now, it is easy
to see that for ${\bf m} = (7, 10, 15)$, $C_{\bf m}$ is a complete intersection with
${\mathcal P} = (X_1^3-X_2^2, X_0^5-X_1^2X_2)$ where as for ${\bf m} =(13,16, 21)$, $C_{\bf m}$ is
not a complete intersection. However both $7$ and $13 $ are odd and
hence equal 1 modulo 2.
\end {remark}
\begin{corollary}
If $I_i\subset R$ is as in Theorem~\ref{resIi}, $R/I_i$ is Gorenstein if and only if
\begin{itemize}
\item
$b=2$, $i=n$,
\item
$b=1$, $i=n-1$, or
\item
$n=1$.
\end{itemize}
In particular, we recover the result of \cite[Corollary~6.2]{patseng} that $R/{\mathcal P}$ is Gorenstein if and only if $b=2$.
Moreover, $R/I_i$ are never level unless they are Gorenstein.
\end{corollary}
\begin{proof}
If $n=1$, $I_1={\mathcal P}$ is principal and therefore Gorenstein. By Corollary~\ref{CMtypeIi}, the type of $R/I_i$ is $n-1+b-i$. Since $i\neq n$,
$n-1+b-i\geq b-1>1$ if $b>2$ so $R/I_i$ is Gorenstein if and only if either $b=2$, $i=n$ or $b=1$, $i=n-1$.
The non-levelness of $R/I_i$ when it is not Gorenstein follows directly from the degrees in the resolution.
\end{proof}
\begin{remark}\label{rmkGor}
If $m_0\equiv 2\mod n$, i.e., if $R/{\mathcal P}$ is Gorenstein, then by
(\ref{bettiGeneral}) one has that $\beta_0=\beta_n=1$ and
$\beta_j=j{n\choose j+1}+(n-j){n\choose j-1}$ for all $j\in [1,n-1]$
which is Corollary~\ref{bettiGor}. Note that this result is obtained
here by making a resolution minimal which is obtained through iterated
mapping cone construction, while it had been obtained in
Section~\ref{secgorenstein} by a direct argument.
\end{remark}
\begin{remark}
If $m_0\equiv 1\mod n$, one gets by (\ref{bettiGeneral}) that for
all $j\in [1,n]$, $\beta_j=j{n\choose j+1}+(n+1-j){n\choose j-1}$.
Note that in this case the Betti numbers had already been obtained
in \cite[Theorem~3.1]{hip} where we show that for all $j\in [1,n]$,
$\beta_j=j{n+1\choose j+1}$. One can easily check that both numbers
coincide.
\end{remark}
\begin{theorem}\label{CMtype}
With notations as in Theorem~\ref{mainThm}, the Cohen
Macaulay type of $R/{\mathcal P}$ is the unique integer $c$ in $[1,n]$
such that $c\equiv m_0-1 \mod n$.
\end{theorem}
\begin{proof}
The Cohen-Macaulay type of $R/{\mathcal P}$, $\beta_n$, is computed by the
first formula in (\ref{bettiGeneral}) when $b=1$, and by the second
otherwise. Thus, $\beta_n={n\choose n-1}=n$ if $b=1$, and
$\beta_n=(b-1){n\choose n}=b-1$ otherwise.
\end{proof}
Putting together Theorem~\ref{ThmIndraConj} and
Theorem~\ref{CMtype}, one gets the following:
\begin{corollary}\label{CMtypeAndBetti}
With notations as in Theorem~\ref{mainThm}, the Cohen-Macaulay type
of $R/{\mathcal P}$ determines all its Betti numbers.
\end{corollary}
Note that in the previous corollary, the same result holds if one substitutes the minimal number of generators of ${\mathcal P}$ for the Cohen-Macaulay type of $R/{\mathcal P}$.
\begin {remark}
One can also deduce from the minimal graded free resolution in Theorem~\ref{mainThm}, the value of the (weighted) Castelnuovo-Mumford regularity of $R/{\mathcal P}$ which is indeed
$$
\hbox{reg}(R/{\mathcal P})=
\left\{\begin{array}{ll}
{n\choose 2}d+m_0(a+d)+n(m_0-1)&\hbox{if }b=1,
\\
({n\choose 2}+b-1)d+m_0(a+d+1)+n(m_0-1)&\hbox{if }b\geq 2.
\end{array}\right.
$$
On the other hand, the Frobenius number $g({\bf m})$ of the numerical semigroup $\Gamma({\bf m})$ can be computed using \cite[Theorem~3.2.2]{ramirezbook} and one gets that $g({\bf m})=(a-1)m_0+d(m_0-1)$ if $b=1$, and
$g({\bf m})=am_0+d(m_0-1)$ if $b\geq 2$. Thus
$
\hbox{reg}(R/{\mathcal P})-g({\bf m})=({n\choose 2}+b)d+m_0+n(m_0-1)\,.
$
In particular, the regularity is always an upper bound for the conductor $g({\bf m})+1$ of the numerical semigroup $\Gamma({\bf m})$ and it is, in general, much bigger. The above relation between the regularity of the semigroup ring and the Frobenius number of the semigroup
can be nicely expressed as follows:
$$
\hbox{reg}(R/{\mathcal P})=g({\bf m})
+\sum_{i=0}^{n}m_i-(n-b)d-n\,.
$$
\end{remark}
Finally, we use Theorem \ref{ThmIndraConj} to prove a conjecture of Herzog and
Srinivasan on eventual periodicity of Betti numbers of semigroup
rings in our context.
Given a sequence of positive integers ${\bf m}= (m_0, \ldots, m_n)$ and a positive integer $j$,
denote by ${\bf m}+(j) = {\bf m} +(j,j, \ldots, j)$. Herzog and
Srinivasan have conjectured the following:
\begin{Conjecture}[Herzog and Srinivasan] Let ${\bf m}$ and ${\bf m}+(j)$ be as above.
\begin{itemize}
\item[HS1] The Betti numbers of the semigroup ring $k[\Gamma({\bf m} +(j))]$
are eventually periodic in $j$.
\item[HS2] The number of minimal generators of the defining ideal of the monomial curve $C_{{\bf m}+(j)}$
is eventually periodic in $j$ with period $m_n-m_0$.
\item[HS3] The number of minimal generators of the defining ideal of the monomial curve $C_{{\bf m}+(j)}$
is bounded for all $j$.
\end{itemize}
\end{Conjecture}
They prove the conjecture for $n=2$
and A. Marzullo proves it for some cases when $n=4$ in \cite {Ad}.
Our Theorem~\ref{ThmPeriodic} proves this periodicity
conjecture in its strong form (HS1) for arithmetic sequences.
\begin{theorem}\label{ThmPeriodic}
If ${\bf m}=(m_0,\ldots,m_n)$ is an arithmetic sequence and ${\bf m}+(j)={\bf m} +(j,j, \ldots, j)$, then the Betti numbers of $C_{{\bf m}+(j)}$ are eventually periodic in $j$ with period $m_n-m_0$.
\end{theorem}
\begin{proof}
Let ${\bf m}=(m_0,\ldots,m_n)$ be an arithmetic sequence and $j\in\mathbb N$. Since $(m_0+j, \ldots, m_n+j)$ is in arithmetic progression, $\gcd{(m_0+j,\ldots,m_n+j)}=\gcd{(m_0+j,d)}$ where $d$ is the common difference in ${\bf m}$.
We denote by $\widetilde{.}$ division by $\gcd{(m_0+j,d)}$. Then $\widetilde{m_i+j}=\widetilde{m_0+j}+i\widetilde{d}$ and
$\widetilde{{\bf m}+(j)}=(\widetilde{m_0+j},\ldots,\widetilde{m_n+j})$ has gcd 1. Moreover, $C_{{\bf m}+(j)}=C_{\widetilde{{\bf m}+(j)}}$.
We claim that for $j\geq nd-m_0$, $\widetilde{{\bf m}+(j)}$ is always an arithmetic sequence. If
$\widetilde{m_k+j}=\sum_{i\neq k}r_i(\widetilde{m_i+j})$ for $r_i\in\mathbb N$, then
$k\widetilde{d}+\widetilde{m_0+j}=\sum_{i\neq k}r_i(\widetilde{m_0+j}+i\widetilde{d})$ and hence
$\widetilde{d}(k-\sum_{i\neq k}ir_i)=(\widetilde{m_0+j})(\sum_{i\neq k}r_i-1)\geq n\widetilde{d}$ and this is not possible since $k-\sum_{i\neq k}ir_i<n$.
Now,
\begin{eqnarray*}
\widetilde{m_0+j+nd}&=&\widetilde{m_0+j}+n\widetilde{d}\\
&\equiv&\widetilde{m_0+j}\mod n.
\end{eqnarray*}
Note that $\gcd{(m_0+j,d)}$ is periodic with period $d$. So the Betti numbers of $C_{{\bf m}+(j)}$ are periodic with period $nd$ for $j$ large enough.
\end{proof}
\end{document} |
\begin{document}
\title[Weighted Integral Means of Mixed Areas and Lengths]{Weighted Integral Means of Mixed\\
Areas and Lengths under Holomorphic Mappings}
\author{Jie Xiao and Wen Xu}
\address{Department of Mathematics and Statistics,
Memorial University,
NL A1C 5S7, Canada}
\email{[email protected]; [email protected]}
\thanks{JX and WX were in part supported by NSERC of Canada and the Finnish Cultural Foundation, respectively.}
\begin{abstract}
This note addresses monotonic growths and logarithmic convexities of
the weighted ($(1-t^2)^\alpha dt^2$, $-\infty<\alpha<\infty$,
$0<t<1$) integral means $\mathsf{A}_{\alpha,\beta}(f,\cdot)$ and
$\mathsf{L}_{\alpha,\beta}(f,\cdot)$ of the mixed area $(\pi
r^2)^{-\beta}A(f,r)$ and the mixed length $(2\pi r)^{-\beta}L(f,r)$
($0\le\beta\le 1$ and $0<r<1$) of $f(r\mathbb D)$ and $\partial
f(r\mathbb D)$ under a holomorphic map $f$ from the unit disk
$\mathbb D$ into the finite complex plane $\mathbb C$.
\end{abstract}
\maketitle
\section{Introduction}
From now on, $\mathbb D$ represents the unit disk in the finite
complex plane $\mathbb C$, $H(\mathbb D)$ denotes the space of
holomorphic mappings $f: \mathbb D\to\mathbb C$, and $U(\mathbb D)$
stands for all univalent functions in $H(\mathbb D)$. For any real
number $\alpha$, positive number $r\in (0,1)$ and the standard area
measure $dA$, let
$$
dA_\alpha(z)=(1-|z|^2)^\alpha dA(z);\quad r\mathbb D=\{z\in\mathbb
D: |z|<r\};\quad r\mathbb T=\{z\in\mathbb D: |z|=r\}.
$$
In their recent paper \cite{XZ}, Xiao and Zhu have discussed the
following area $0<p<\infty$-integral means of $f\in H(\mathbb D)$:
$$
{M}_{p,\alpha}(f,r)=\left[\frac{1}{A_\alpha(r\mathbb
D)}\int_{r\mathbb D}|f|^p\,dA_\alpha\right]^{\frac1p},
$$
proving that $r\mapsto M_{p,\alpha}(f,r)$ is strictly increasing
unless $f$ is a constant, and $\log r\mapsto\log M_{p,\alpha}(f,r)$
is not always convex. This last result suggests a conjecture that
$\log r\mapsto\log M_{p,\alpha}(f,r)$ is convex or concave when
$\alpha\le 0$ or $\alpha>0$. But, motivated by \cite[Example 10,
(ii)]{XZ} we can choose $p=2$, $\alpha=1$, $f(z)=z+c$ and $c>0$ to
verify that the conjecture is not true. At the same time, this
negative result was also obtained in Wang-Zhu's manuscript
\cite{WZ}. So far it is unknown whether the conjecture is generally
true for $p\not=2$.
The foregoing observation has actually inspired the following
investigation. Our concentration is the fundamental case $p=1$. To
understand this approach, let us take a look at
$M_{1,\alpha}(\cdot,\cdot)$ from a differential geometric viewpoint.
Note that
$$
{M}_{1,\alpha}(f',r)=\frac{\int_{r\mathbb
D}|f'|\,dA_\alpha}{A_\alpha(r\mathbb D)}=\frac{\int_0^r \big[(2\pi
t)^{-1}\int_{t\mathbb
T}|f'(z)||dz|\big](1-t^2)^\alpha\,dt^2}{\int_0^r
(1-t^2)^\alpha\,dt^2}.
$$
So, if $f\in U(\mathbb D)$, then
$$
(2\pi t)^{-1}\int_{t\mathbb T}|f'(z)|\,|dz|
$$
is a kind of mean of the length of $\partial f(t\mathbb D)$, and
hence the square of this mean dominates a sort of mean of the area
of $f(t\mathbb D)$ in the isoperimetric sense:
$$
\Phi_{A}(f,t)=(\pi t^2)^{-1}\int_{t\mathbb D}|f'(z)|^2\,dA(z)\le
\left[(2\pi t)^{-1}\int_{t\mathbb
T}|f'(z)|\,|dz|\right]^2=\big[\Phi_{L}(f,t)\big]^2.
$$
According to the P\'olya-Szeg\"o monotone principle \cite[Problem
309]{PS} (or \cite[Proposition 6.1]{BMM}) and the area Schwarz's
lemma in Burckel, Marshall, Minda, Poggi-Corradini and Ransford
\cite[Theorem 1.9]{BMM}, $\Phi_{L}(f,\cdot)$ and $\Phi_{A}(f,\cdot)$
are strictly increasing on $(0,1)$ unless $f(z)=a_1z$ with
$a_1\not=0$. Furthermore, $\log\Phi_{L}(f,r)$ and
$\log\Phi_{A}(f,r)$, equivalently, $\log L(f,r)$ and $\log A(f,r)$,
are convex functions of $\log r$ for $r\in (0,1)$, due to the
classical Hardy's convexity and \cite[Section 5]{BMM}. Perhaps, it
is worth-wise to mention that if $c>0$ is small enough then the
universal cover of $\mathbb D$ onto the annulus
$\{e^{-\frac{c\pi}{2}}<|z|< e^{\frac{c\pi}{2}}\}$:
$$
f(z)=\exp\Big[ic\log\Big(\frac{1+z}{1-z}\Big)\Big]
$$
enjoys the property that $\log r\mapsto \log A(f,r)$ is not convex;
see \cite[Example 5.1]{BMM}.
In the above and below, we have used the following convention:
$$
\Phi_{A}(f,r)=\frac{A(f,r)}{\pi r^2}\quad\&\quad
\Phi_{L}(f,r)=\frac{L(f,r)}{2\pi r},
$$
where under $r\in (0,1)$ and $f\in H(\mathbb D)$, $A(f,r)$ and
$L(f,r)$ stand respectively for the area of $f(r\mathbb D)$ (the
projection of the Riemannian image of $r\mathbb D$ by $f$) and the
length of $\partial f(r\mathbb D)$ (the boundary of the projection
of the Riemannian image of $r\mathbb D$ by $f$) with respect to the
standard Euclidean metric on $\mathbb C$. For our purpose, we choose
a shortcut notation
$$
d\mu_\alpha(t)=(1-t^2)^\alpha dt^2\quad\&\quad
\nu_\alpha(t)=\mu_\alpha([0,t])\quad\forall\quad t\in (0,1),
$$
and for $0\le\beta\le 1$ define
$$
\Phi_{A,\beta}(f,t)=\frac{A(f,t)}{(\pi t^2)^\beta}\quad\&\quad
\Phi_{L,\beta}(f,t)=\frac{L(f,t)}{(2\pi t)^\beta},
$$
and then
$$
\mathsf{A}_{\alpha,\beta}(f,r)=\frac{\int_0^r \Phi_{A,\beta}(f,t)
\,d\mu_\alpha(t)}{\int_0^r d\mu_\alpha(t)}\quad\&\quad
\mathsf{L}_{\alpha,\beta}(f,r)=\frac{\int_0^r \Phi_{L,\beta}(f,t)\,
d\mu_\alpha(t)}{\int_0^r d\mu_\alpha(t)}
$$
which are called the weighted integral means of the mixed area and
the mixed length for $f(r\mathbb D)$ and $\partial f(r\mathbb D)$,
respectively.
In this note, we consider two fundamental properties: monotonic
growths and logarithmic convexities of both
$\mathsf{A}_{\alpha,\beta}(f,r)$ and
$\mathsf{L}_{\alpha,\beta}(f,r)$, thereby producing two
specialities: (i) if $r\mapsto \Phi_{L}(f,r)$ is monotone increasing
on $(0,1)$, then so is the isoperimetry-induced function:
$$
r\mapsto\frac{\int_0^r
\big[\Phi_{L,1}(f,t)\big]^2\,d\mu_\alpha(t)}{\int_0^r
d\mu_\alpha(t)}\ge \mathsf{A}_{\alpha,1}(f,r);
$$
(ii) the log-convexity for $\mathsf{L}_{\alpha,\beta=1}(f,r)$
essentially settles the above-mentioned conjecture. The details
(results and their proofs) are arranged in the forthcoming two
sections.
\section{Monotonic Growth}
In this section, we deal with the monotonic growths of
$\mathsf{A}_{\alpha,\beta}(f,r)$ and
$\mathsf{L}_{\alpha,\beta}(f,r)$, along with their associated
Schwarz type lemmas. In what follows, $\mathbb N$ is used as the set
of all natural numbers.
\subsection{Two Lemmas} The following two preliminary results are
needed.
\begin{lemma}\cite[Theorems 1 \& 2]{Ma}\label{l2} Let $f\in H(\mathbb D)$ be of the form
$f(z)=a_0+\sum_{k=n}^\infty a_kz^k$ with $n\in\mathbb N$. Then:
\item{\rm(i)} $\pi r^{2n}\Big[\frac{|f^{(n)}(0)|}{n!}\Big]^2\le A(f,r)\quad\forall\quad r\in
(0,1)$.
\item{\rm(ii)} $2\pi r^n \Big[\frac{|f^{(n)}(0)|}{n!}\Big]\le L(f,r)\quad\forall\quad r\in
(0,1)$.
\noindent Moreover, equality in (i) or (ii) holds if and only if
$f(z)=a_0+a_nz^n$.
\end{lemma}
\begin{proof} This may be viewed as the higher order Schwarz type lemma for area and length.
See also the proofs of Theorems 1 \& 2 in \cite{Ma}, and their
immediate remarks on equalities. Here it is worth noticing three
matters: (a) $\frac{f^{(n)}(0)}{n!}$ is just $a_n$; (b)
\cite[Corollary 3]{J} presents a different argument for the area
case; (c) $L(f,r)$ is greater than or equal to the length $l(r,f)$
of the outer boundary of $f(r\mathbb D)$ (defined in \cite{Ma})
which is not less than the length $l^\#(r,f)$ of the exact outer
boundary of $f(r\mathbb D)$ (introduced in \cite{Y}).
\end{proof}
\begin{lemma}\label{l1} Let $0\le\beta\le 1$.
\item{\rm(i)} If $f\in H(\mathbb D)$, then $r\mapsto \Phi_{A,\beta}(f,r)$ is strictly
increasing on $(0,1)$ unless
\[
f=\left\{\begin{array} {r@{\;}l}
constant &\quad \hbox{when}\quad \beta<1\\
linear\ map &\quad \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
\item{\rm(ii)} If $f\in U(\mathbb D)$ or $f(z)=a_0+a_nz^n$ with $n\in\mathbb N$, then $r\mapsto
\Phi_{L,\beta}(f,r)$ is strictly increasing on $(0,1)$ unless
\[
f=\left\{\begin{array} {r@{\;}l}
constant &\quad \hbox{when}\quad \beta<1\\
linear\ map & \quad \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
\end{lemma}
\begin{proof} It is enough to handle $\beta<1$ since the case $\beta=1$ has been treated in \cite[Theorem 1.9 \& Proposition 6.1]{BMM}.
The monotonic growths in (i) and (ii) follow from
$$
\Phi_{A,\beta}(f,r)=(\pi r^2)^{1-\beta}\Phi_{A,1}(f,r)\quad\&\quad
L(f,r)=(2\pi r)^{1-\beta}\Phi_{L,1}(f,r).
$$
To see the strictness, we consider two cases.
(i) Suppose that $\Phi_{A,\beta}(f,\cdot)$ is not strictly
increasing. Then there are $r_1,r_2\in (0,1)$ such that $r_1<r_2$,
and $\Phi_{A,\beta}(f,\cdot)$ is a constant on $[r_1,r_2]$. Hence
$$
\frac{d}{dr}\Phi_{A,\beta}(f,r)=0\quad\forall\quad r\in [r_1,r_2].
$$
Equivalently,
$$
2\beta A(f,r)=r\frac{d}{dr}A(f,r)\quad\forall\quad r\in [r_1,r_2].
$$
But, according to \cite[(4.2)]{BMM}:
$$
2A(f,r)\le r\frac{d}{dr} A(f,r)\quad\forall\quad r\in (0,1).
$$
Since $\beta<1$, we get $A(f,r)=0$ for all $r\in [r_1,r_2]$, whence
finding that $f$ is constant.
(ii) Now assume that $\Phi_{L,\beta}(f,\cdot)$ is not strictly
increasing. There are $r_3,r_4\in (0,1)$ such that $ r_3<r_4$ and
$$
0=\frac{d}{dr}\Phi_{L,\beta}(f,r)=(2\pi
r)^{-\beta}\Big[\frac{d}{dr}L(f,r)-\frac{\beta}{r}L(f,r)\Big]=0\quad\forall\quad
r\in [r_3,r_4].
$$
If $f\in U(\mathbb D)$ then
$$
L(f,r)=\int_{r\mathbb T}|f'(z)|\,|dz|
$$
and hence one has the following ``first variation formula"
$$
\frac{d}{dr}L(f,r)=\int_0^{2\pi}|f'(re^{i\theta})|d\theta+r\frac{d}{dr}\int_0^{2\pi}|f'(re^{i\theta})|d\theta\quad\forall\quad
r\in [r_3,r_4].
$$
The previous three equations yield
$$
0=(1-\beta)\int_0^{2\pi}|f'(re^{i\theta})|d\theta+r\frac{d}{dr}\int_0^{2\pi}|f'(re^{i\theta})|d\theta\quad\forall\quad
r\in [r_3,r_4]
$$
and so
$$
\int_0^{2\pi}|f'(re^{i\theta})|d\theta=0\quad\forall\quad r\in
[r_3,r_4].
$$
This ensures that $f$ is a constant, contradicting $f\in U(\mathbb
D)$. Therefore, $f(z)$ is of the form $a_0+a_nz^n$. But, since
$L(z^n,r)=2\pi r^n$ is strictly increasing, $f$ must be constant.
\end{proof}
\subsection{Monotonic Growth of $\mathsf{A}_{\alpha,\beta}(f,\cdot)$} This
aspect is essentially motivated by the following Schwarz type lemma.
\begin{proposition}\label{pr1} Let $-\infty<\alpha<\infty$, $0\le\beta\le 1$, and $f\in H(\mathbb D)$
be of the form $f(z)=a_0+\sum_{k=n}^\infty a_k z^k$ with $n\in\mathbb N$.
Then
$$
\pi^{1-\beta}\Big[\frac{|f^{(n)}(0)|}{n!}\Big]^2\le
\mathsf{A}_{\alpha,\beta}(f,r)\left[\frac{\nu_\alpha(r)}{\int_0^rt^{2(n-\beta)}\,d\mu_\alpha(t)}\right]\quad\forall\quad
r\in (0,1)
$$
with equality if and only if $f(z)=a_0+a_nz^n$.
\end{proposition}
\begin{proof} The inequality follows from Lemma \ref{l2} (i) right away. When $f(z)=a_0+a_nz^n$,
the last inequality becomes equality due to the equality case of
Lemma \ref{l2} (i). Conversely, suppose that the last inequality is
an equality. If $f$ does not have the form $a_0+a_nz^n$, then the
equality in Lemma \ref{l2} (i) is not true, then there are
$r_1,r_2\in (0,1)$ such that $r_1<r_2$ and
$$
A(f,t)>\pi
t^{2n}\Big[\frac{|f^{(n)}(0)|}{n!}\Big]^2\quad\forall\quad t\in
[r_1,r_2].
$$
This strict inequality forces that for $r\in [r_1,r_2]$,
\begin{eqnarray*}
\pi^{1-\beta}\Big[\frac{|f^{(n)}(0)|}{n!}\Big]^2\int_0^r
t^{2(n-\beta)}\,d\mu_\alpha(t)&=&\int_0^r
(\pi t^2)^{-\beta}A(f,t)\,d\mu_\alpha(t)\\
&=&\left(\int_0^{r_1}+\int_{r_1}^{r_2}+\int_{r_2}^{r}\right)(\pi t^2)^{-\beta} A(f,t)\,d\mu_\alpha(t)\\
&>&\pi^{1-\beta} \Big[\frac{|f^{(n)}(0)|}{n!}\Big]^2 \int_0^{r}
t^{2(n-\beta)}\,d\mu_\alpha(t),
\end{eqnarray*}
a contradiction. Thus $f(z)=a_0+a_nz^n$.
\end{proof}
Based on Proposition \ref{pr1}, we find the monotonic growth for
$\mathsf{A}_{\alpha,\beta}(\cdot,\cdot)$ as follows.
\begin{theorem}\label{th1}
Let $-\infty<\alpha<\infty$, $0\le\beta\le 1$, and $f\in H(\mathbb
D)$. Then $r\mapsto\mathsf{A}_{\alpha,\beta}(f,r)$ is strictly
increasing on $(0,1)$ unless
\[
f=\left\{\begin{array} {r@{\;}l}
constant &\quad \hbox{when}\quad \beta<1\\
linear\ map &\quad \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
Consequently,
\item{\rm(i)}
\[
\lim_{r\to 0}\mathsf{A}_{\alpha,\beta}(f,r)=\left\{\begin{array}
{r@{\;}l}
0\quad & \hbox{when}\quad \beta<1\\
|f'(0)|^2\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
\item{\rm(ii)} If
$$
\Phi_{A,\beta}(f,0):=\lim_{r\to
0}\Phi_{A,\beta}(f,r)\quad\&\quad\Phi_{A,\beta}(f,1):=\lim_{r\to
1}\Phi_{A,\beta}(f,r)<\infty,
$$
then
$$
0<r<s<1\Rightarrow 0\le
\frac{\mathsf{A}_{\alpha,\beta}(f,s)-\mathsf{A}_{\alpha,\beta}(f,r)}{\log\nu_\alpha(s)-\log\nu_\alpha(r)}\leq
\Phi_{A,\beta}(f,s)-\Phi_{A,\beta}(f,0)
$$
with equality if and only if
\[
f=\left\{\begin{array} {r@{\;}l}
\hbox{constant}\quad & \hbox{when}\quad \beta<1\\
\hbox{linear\ map}\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
In particular, $t\mapsto \mathsf{A}_{\alpha,\beta}(f,t)$ is
Lipschitz with respect to $\log\nu_\alpha(t)$ for $t\in (0,1)$.
\end{theorem}
\begin{proof} Note that $\nu_\alpha(r)=\int_0^r d\mu_\alpha(t)$. So $d\nu_\alpha(r)$, the differential of $\nu_\alpha(r)$ with respect to $r\in (0,1)$, equals
$d\mu_\alpha(r)$. By integration by parts we have
$$
\Phi_{A,\beta}(f,r)\nu_\alpha(r)-\int_0^r
\Phi_{A,\beta}(f,t)\,d\mu_\alpha(t)=\int_0^r\big[\frac{d}{dt}\Phi_{A,\beta}(f,t)\big]
\nu_\alpha(t)\,dt.
$$
Differentiating the function $\mathsf{A}_{\alpha,\beta}(f,r)$ with
respect to $r$ and using Lemma \ref{l1} (i), we get
\begin{align*}
\frac{d}{dr}\mathsf{A}_{\alpha,\beta}(f,r)&=\frac{\Phi_{A,\beta}(f,r)2r(1-r^2)^\alpha \nu_\alpha(r)-\Big[\int_0^r\Phi_{A,\beta}(f,t)\, d\mu_\alpha(t)\Big]2r(1-r^2)^\alpha}{\nu_\alpha(r)^2}\\
&=\frac{2r(1-r^2)^\alpha \left[\Phi_{A,\beta}(f,t)\nu_\alpha(r)- \int_0^r \Phi_{A,\beta}(f,t)\, d\mu_\alpha(t)\right]}{\nu_\alpha(r)^2}\\
&=\frac{2r(1-r^2)^\alpha\int_0^r\big[\frac{d}{dt}\Phi_{A,\beta}(f,t)\big]
\nu_\alpha(t)\, dt}{\nu_\alpha(r)^2}\geq 0.
\end{align*}
As a result,
$r\mapsto\mathsf{A}_{\alpha,\beta}(f,r)$ increases on $(0,1)$.
Next suppose that the just-verified monotonicity is not strict. Then
there exist two numbers $r_1,r_2\in (0,1)$ such that $r_1<r_2$ and
$$
\mathsf{A}_{\alpha,\beta}(f,r_1)=\mathsf{A}_{\alpha,\beta}(f,r)=\mathsf{A}_{\alpha,\beta}(f,r_2)\quad
\forall\quad r\in [r_1,r_2].
$$
Consequently,
$$
\frac{d}{dr}\mathsf{A}_{\alpha,\beta}(f,r)=0\quad\forall\quad
r\in[r_1,r_2]
$$
and so
$$
\int_0^r \big[\frac{d}{dt}\Phi_{A,\beta}(f,t)\big]\nu_\alpha(t)\,
dt=0\quad\forall\quad r\in [r_1,r_2].
$$
Then we must have
$$
\frac{d}{dt}\Phi_{A,\beta}(f,t)=0\quad\forall\quad t\in
(0,r)\quad\hbox{with}\quad r\in [r_1,r_2],
$$
whence getting that if $\beta<1$ then $f$ must be constant or if
$\beta=1$ then $f$ must be linear, thanks to the argument for the
strictness in Lemma \ref{l1} (i).
It remains to check the rest of Theorem \ref{th1}.
(i) The monotonic growth of $\mathsf{A}_{\alpha,\beta}(f,\cdot)$
ensures the existence of the limit. An application of
L'H\^{o}pital's rule gives
$$
\lim_{r\to 0}\mathsf{A}_{\alpha,\beta}(f,r)=\lim_{r\to
0}\Phi_{A,\beta}(f,r)= \left\{\begin{array} {r@{\;}l}
0\quad & \hbox{when}\quad \beta<1\\
|f'(0)|^2\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
$$
(ii) Again, the above monotonicity formula of
$\mathsf{A}_{\alpha,\beta}(f,\cdot)$ plus the given condition yields
that for $s\in (0,1)$,
$$
\sup_{r\in
(0,s)}\mathsf{A}_{\alpha,\beta}(f,r)=\mathsf{A}_{\alpha,\beta}(f,s)<\infty.
$$
Integrating by parts twice and using the monotonicity of
$\Phi_{A,\beta}(f,\cdot)$, we obtain that under $0<r<s<1$,
\begin{eqnarray*}
0&\le&\mathsf{A}_{\alpha,\beta}(f,s)-\mathsf{A}_{\alpha,\beta}(f,r)\\
&=&\int_r^s\frac{d}{dt}\mathsf{A}_{\alpha,\beta}(f,t)\,dt\\
&=&\int_r^s\left(\int_0^t\big[\frac{d}{d\tau}\Phi_{A,\beta}(f,\tau)\big]\nu_\alpha(\tau)\,d\tau\right)\,\Big[\frac{d\nu_\alpha(t)}{\nu_\alpha(t)^2}\Big]\\
&=&\int_r^s\left(\nu_\alpha(t)\Phi_{A,\beta}(f,t)-\int_0^t\Phi_{A,\beta}(f,\tau)\,d\nu_\alpha(\tau)\right)\,\Big[\frac{d\nu_\alpha(t)}{\nu_\alpha(t)^2}\Big]\\
&\le&\Big[\Phi_{A,\beta}(f,s)-\Phi_{A,\beta}(f,0)\Big]\int_r^s\frac{d\nu_\alpha(t)}{\nu_\alpha(t)}.
\end{eqnarray*}
This gives the desired inequality right away. Furthermore, the above
argument plus Lemma \ref{l1} (i) derives the equality case.
\end{proof}
As an immediate consequence of Theorem \ref{th1}, we get a sort of
``norm" estimate associated with $\Phi_{A,\beta}(f,\cdot)$.
\begin{corollary}\label{pr2} Let $-\infty<\alpha<\infty$, $0\le\beta\le 1$, and $f\in H(\mathbb D)$.
\item{\rm(i)} If $-\infty<\alpha\le -1$, then
$$
\int_0^1 \Phi_{A,\beta}(f,t)\,d\mu_\alpha(t)=\sup_{r\in
(0,1)}\int_0^r \Phi_{A,\beta}(f,t)\,d\mu_\alpha(t)<\infty
$$
if and only if $f$ is constant. Moreover, $\sup_{r\in
(0,1)}\mathsf{A}_{\alpha,\beta}(f,r)=\Phi_{A,\beta}(f,1).$
\item{\rm(ii)} If $-1<\alpha<\infty$, then
$$
\mathsf{A}_{\alpha,\beta}(f,r)\le\mathsf{A}_{\alpha,\beta}(f,1):=\sup_{s\in
(0,1)}\mathsf{A}_{\alpha,\beta}(f,s)\quad\forall\quad r\in (0,1),
$$
where the inequality becomes an equality for all $r\in (0,1)$ if and
only if
\[
f=\left\{\begin{array} {r@{\;}l}
\hbox{constant}\quad & \hbox{when}\quad \beta<1\\
\hbox{linear\ map}\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
\item{\rm(iii)} The following function $\alpha\mapsto\mathsf{A}_{\alpha,\beta}(f,1)$ is strictly decreasing on
$(-1,\infty)$ unless
\[
f=\left\{\begin{array} {r@{\;}l}
\hbox{constant}\quad & \hbox{when}\quad \beta<1\\
\hbox{linear\ map}\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
\end{corollary}
\begin{proof} (i) By Theorem \ref{th1}, we have
$$
\mathsf{A}_{\alpha,\beta}(f,r)\leq \frac{\int_0^s
\Phi_{A,\beta}(f,t)\,d\mu_\alpha(t)}{\nu_\alpha(s)}\quad\forall\quad
r\in (0,s).
$$
Note that
$$\lim_{s\to 1}\nu_\alpha(s)=\infty\quad\&\quad\lim_{s\to 1}\int_0^s\Phi_{A,\beta}(f,t)\, d\mu_\alpha(t)=\int_0^1
\Phi_{A,\beta}(f,t)\,d\mu_\alpha(t).
$$
So, the last integral is finite if and only if
$$
\Phi_{A,\beta}(f,r)=0\quad\forall\quad r\in (0,1),
$$
equivalently, $A(f,r)=0$ holds for all $r\in (0,1)$, i.e., $f$ is
constant.
For the remaining part of (i), we may assume that $f$ is not a
constant map. Due to $\lim_{r\to 1}\nu_\alpha(r)=\infty$, we obtain
$$
\lim_{r\to 1}\int_0^r \Phi_{A,\beta}(f,t)\,d\mu_\alpha(t)=\int_0^1
\Phi_{A,\beta}(f,t)\,d\mu_\alpha(t)=\infty.
$$
So, an application of L'H\^{o}pital's rule yields
$$
\sup_{0<r<1}\mathsf{A}_{\alpha,\beta}(f,r)=\lim_{r\to
1}\frac{\int_0^r \Phi_{A,\beta}(f,t)\,
d\mu_\alpha(t)}{\nu_\alpha(r)}=\lim_{r\to
1}\frac{\Phi_{A,\beta}(f,r)r(1-r^2)^\alpha}{
r(1-r^2)^\alpha}=\Phi_{A,\beta}(f,1).
$$
(ii) Under $-1<\alpha<\infty$, we have
$$
\lim_{r\to 1}\nu_\alpha(r)=\nu_\alpha(1)\quad\&\quad \lim_{r\to
1}\int_0^r\Phi_{A,\beta}(f,t)\,d\mu_\alpha(t)=\int_0^1
\Phi_{A,\beta}(f,t)\,d\mu_\alpha(t).
$$
Thus, by Theorem \ref{th1} it follows that for $r\in (0,1)$,
$$
\mathsf{A}_{\alpha,\beta}(f,r)\le\lim_{s\to
1}\mathsf{A}_{\alpha,\beta}(f,s)=\big[\nu_\alpha(1)\big]^{-1}\int_0^1
\Phi_{A,\beta}(f,t)\, d\mu_\alpha(t)=\sup_{s\in
(0,1)}\mathsf{A}_{\alpha,\beta}(f,s).
$$
The equality case just follows from a straightforward computation
and Theorem \ref{th1}.
(iii) Suppose $-1<\alpha_1<\alpha_2<\infty$ and
$\mathsf{A}_{\alpha_1,\beta}(f,1)<\infty$, then integrating by parts
twice, we obtain
\begin{align*}
\mathsf{A}_{\alpha_2,\beta}(f,1)&=
\big[\nu_{\alpha_2}(1)\big]^{-1}\int_0^1\Phi_{
A,\beta}(f,r)\,d\mu_{\alpha_2}(r)\\
&= \big[\nu_{\alpha_2}(1)\big]^{-1}\int_0^1 (1-r^2)^{\alpha_2-\alpha_1}\frac{d}{dr}\left[\int_0^r \Phi_{A,\beta}(f,t)\, d\mu_{\alpha_1}(t)\right]\, dr\\
&= \big[\nu_{\alpha_2}(1)\big]^{-1}\left[-\int_0^1\left(\int_0^r\Phi_{A,\beta}(f,t)\,d\mu_{\alpha_1}(t)\right)\, d(1-r^2)^{\alpha_2-\alpha_1}\right]\\
&\leq \big[\nu_{\alpha_2}(1)\big]^{-1}\mathsf{A}_{\alpha_1,\beta}(f,1)\int_0^1 \nu_{\alpha_1}(r)\, d\big[-(1-r^2)^{\alpha_2-\alpha_1}\big]\\
&=\mathsf{A}_{\alpha_1,\beta}(f,1)\big[\nu_{\alpha_2}(1)\big]^{-1}\left[\int_0^1
(1-r^2)^{\alpha_2-\alpha_1}\,d\mu_{\alpha_1}(r)\right]
\\
&=\mathsf{A}_{\alpha_1,\beta}(f,1),
\end{align*}
thereby establishing $\mathsf{A}_{\alpha_2,\beta}(f,1)\le
\mathsf{A}_{\alpha_1,\beta}(f,1)$. If this last inequality becomes
equality, then the above argument forces
$$
\int_0^r\Phi_{A,\beta}(f,t)\,d\mu_{\alpha_1}(t)=\mathsf{A}_{\alpha_1,\beta}(f,1)
\nu_{\alpha_1}(r)\quad\forall\quad r\in (0,1),
$$
whence yielding (via the just-verified (ii))
\[ f=\left\{\begin{array} {r@{\;}l}
\hbox{constant}\quad & \hbox{when}\quad \beta<1\\
\hbox{linear\ map}\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
\end{proof}
\subsection{Monotonic Growth of $\mathsf{L}_{\alpha,\beta}(f,\cdot)$} Correspondingly, we first have
the following Schwarz type lemma.
\begin{proposition}\label{co1} Let $-\infty<\alpha<\infty$, $0\le\beta\le 1$, and $f\in H(\mathbb D)$ be of the form $f(z)=a_0+\sum_{k=n}^\infty a_kz^k$ with $n\in\mathbb N$.
Then
$$
(2\pi)^{1-\beta}\Big[\frac{|f^{(n)}(0)|}{n!}\Big]\le
\mathsf{L}_{\alpha,\beta}(f,r)\left[\frac{\nu_\alpha(r)}{\int_0^rt^{n-\beta}\,d\mu_\alpha(t)}\right]\quad\forall\quad
r\in (0,1)
$$
with equality when and only when $f=a_0+a_nz^n$.
\end{proposition}
\begin{proof} This follows from Lemma \ref{l2} (ii) and its equality
case.
\end{proof}
The coming-up-next monotonicity contains a hypothesis stronger than
that for Theorem \ref{th1}.
\begin{theorem}\label{th2}
Let $-\infty<\alpha<\infty$, $0\le\beta\le 1$, and $f\in U(\mathbb
D)$ or $f(z)=a_0+a_nz^n$ with $n\in\mathbb N$. Then
$r\mapsto\mathsf{L}_{\alpha,\beta}(f,r)$ is strictly increasing on
$(0,1)$ unless
\[
f=\left\{\begin{array} {r@{\;}l}
constant &\quad \hbox{when}\quad \beta<1\\
linear\ map &\quad \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
Consequently,
\item{\rm(i)}
\[
\lim_{r\to 0}\mathsf{L}_{\alpha,\beta}(f,r)=\left\{\begin{array}
{r@{\;}l}
0\quad & \hbox{when}\quad \beta<1\\
|f'(0)|\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
\item{\rm(ii)} If
$$
\Phi_{L,\beta}(f,0):=\lim_{r\to
0}\Phi_{L,\beta}(f,r)\quad\&\quad\Phi_{L,\beta}(f,1):=\lim_{r\to
1}\Phi_{L,\beta}(f,r)<\infty,
$$
then
$$
0<r<s<1\Rightarrow 0\le
\frac{\mathsf{L}_{\alpha,\beta}(f,s)-\mathsf{L}_{\alpha,\beta}(f,r)}{\log\nu_\alpha(s)-\log\nu_\alpha(r)}\leq
\Phi_{L,\beta}(f,s)-\Phi_{L,\beta}(f,0)
$$
with equality if and only if
\[
f=\left\{\begin{array} {r@{\;}l}
\hbox{constant}\quad & \hbox{when}\quad \beta<1\\
\hbox{linear\ map}\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
In particular, $t\mapsto \mathsf{L}_{\alpha,\beta}(f,t)$ is
Lipschitz with respect to $\log\nu_\alpha(t)$ for $t\in (0,1)$.
\end{theorem}
\begin{proof} Similar to that for Theorem \ref{th1}, but this time
by Lemma \ref{l1} (ii).
\end{proof}
Naturally, we can establish the so-called ``norm" estimate
associated to $\Phi_{L,\beta}(f,\cdot)$.
\begin{corollary}\label{co2} Let $0\le\beta\le 1$ and $f\in U(\mathbb D)$ or $f(z)=a_0+a_nz^n$ with $n\in\mathbb N$.
\item{\rm(i)} If $-\infty<\alpha\le -1$, then
$$
\int_0^1 \Phi_{L,\beta}(f,t)\,d\mu_\alpha(t)=\sup_{r\in
(0,1)}\int_0^r \Phi_{L,\beta}(f,t)\,d\mu_\alpha(t)<\infty
$$
if and only if $f$ is constant. Moreover, $\sup_{r\in
(0,1)}\mathsf{L}_{\alpha,\beta}(f,r)=\Phi_{L,\beta}(f,1).$
\item{\rm(ii)} If $-1<\alpha<\infty$, then
$$
\mathsf{L}_{\alpha,\beta}(f,r)\le\mathsf{L}_{\alpha,\beta}(f,1):=\sup_{s\in
(0,1)}\mathsf{L}_{\alpha,\beta}(f,s)\quad\forall\quad r\in (0,1),
$$
where the inequality becomes an equality for all $r\in (0,1)$ if and
only if
\[
f=\left\{\begin{array} {r@{\;}l}
\hbox{constant}\quad & \hbox{when}\quad \beta<1\\
\hbox{linear\ map}\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
\item{\rm(iii)} $\alpha\mapsto\mathsf{L}_{\alpha,\beta}(f,1)$
is strictly decreasing on $(-1,\infty)$ unless
\[
f=\left\{\begin{array} {r@{\;}l}
\hbox{constant}\quad & \hbox{when}\quad \beta<1\\
\hbox{linear\ map}\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
\end{corollary}
\begin{proof} The argument is similar to that for Corollary \ref{pr2},
but via Lemma \ref{l1} (ii).
\end{proof}
\section{logarithmic convexity}
In this section, we treat the convexities of the functions: $\log
r\mapsto \log\mathsf{A}_{\alpha,\beta}(f,r)$ and $\log r\mapsto
\log\mathsf{L}_{\alpha,\beta}(f,r)$ for $r\in (0,1)$.
\subsection{Two More Lemmas} The following are two technical preliminaries.
\begin{lemma}\cite[Corollaries 2-3 \& Proposition 7]{WZ}\label{wz}
Suppose $f(x)$ and $\{h_k(x)\}_{k=0}^\infty$ are positive and twice
differentiable for $x\in (0,1)$ such that the function
$H(x)=\sum_{k=0}^\infty h_k(x)$
is also twice differentiable for $x\in (0,1)$. Then:
\item{\rm(i)} $\log x\mapsto\log f(x)$ is convex if and only if $\log x\mapsto\log f(x^2)$ is convex.
\item{\rm(ii)} The function $\log x\mapsto \log f(x)$ is convex if and only
if the $D$-notation of $f$
$$
D(f(x)):=\frac{f'(x)}{f(x)}+ x\left(\frac{f'(x)}{f(x)}\right)'\ge
0\quad\forall\quad x\in (0,1).
$$
\item{\rm(iii)} If for each $k$ the function $\log x\mapsto \log h_k(x)$ is convex, then $\log x\mapsto \log H(x)$ is also convex.
\end{lemma}
\begin{lemma}\label{uni} Let $f\in H(\mathbb D)$. Then $f$ belongs to $U(\mathbb D)$
provided that one of the following two conditions is valid:
\item{\rm(i)} \cite{Nu} or \cite[Lemma 2.1]{AlD}
$$
f(0)=f'(0)-1=0\quad\&\quad
\left|\frac{z^2f'(z)}{f^2(z)}-1\right|<1\quad\forall\quad z\in
\mathbb D.
$$
\item{\rm(ii)} \cite[Theorem 1]{Ne} or \cite[Theorem 8.12]{Du}
$$
\left|\left[\frac{f''(z)}{f'(z)}\right]'-\frac{1}{2}\left[\frac{f''(z)}{f'(z)}\right]^2\right|\leq
2(1-|z|^2)^{-2}\quad\forall\quad z\in \mathbb D.
$$
\end{lemma}
\subsection{Log-convexity for $\mathsf{A}_{\alpha,\beta}(f,\cdot)$}
Such a property is given below.
\begin{theorem}\label{th3} Let $0\le\beta\le 1$ and $0<r<1$.
\item{\rm(i)} If $\alpha\in (-\infty,-3)$, then there exist $f, g\in H(\mathbb D)$ such that $\log r\mapsto\log\mathsf{A}_{\alpha,\beta}(f,r)$ is not
convex and $\log r\mapsto\log \mathsf{A}_{\alpha,\beta}(g,r)$ is not
concave.
\item{\rm(ii)} If $\alpha\in [-3,0]$, then $\log r\mapsto \log\mathsf{A}_{\alpha,1}(a_nz^n,r\big)$ is
convex for $a_n\not=0$ with $n\in\mathbb N$. Consequently,
$$
\log r\mapsto \log\mathsf{A}_{\alpha,1}\big(f,r\big)
$$
is convex for all $f\in U(\mathbb D)$.
\item{\rm(iii)} If $\alpha\in (0,\infty)$, then $\log r\mapsto\log\mathsf{A}_{\alpha,\beta}(a_nz^n,r)$ is
not convex for $a_n\not=0$ and $n\in \mathbb N$.
\end{theorem}
\begin{proof} The key issue is to check whether or not $\log r\mapsto
\log\mathsf{A}_{\alpha,\beta}(z^n,r)$ is convex for $r\in
(0,1)$.
To see this, let us borrow some symbols from \cite{WZ}. For
$\lambda\ge 0$ and $0<x<1$ we define
$$
f_\lambda (x)=\int_0^x t^\lambda(1-t)^\alpha dt
$$
and
$$
\Delta (\lambda, x)=\frac{f_\lambda'(x)}{f_\lambda
(x)}+x\left(\frac{f_\lambda'(x)}{f_\lambda(x)}\right)'-\left[\frac{f_0'(x)}{f_0(x)}+x\left(\frac{f_0'(x)}{f_0(x)}\right)'\right].
$$
Given $n\in\mathbb N$. A simple calculation shows
$\Phi_{A,\beta}(z^n,t)=\pi^{1-\beta} t^{2(n-\beta)}$, and then a
change of variable derives
\begin{eqnarray*}
\mathsf{A}_{\alpha,\beta}(z^n,r)&=&\frac{\int_0^r
\Phi_{A,\beta}(z^n,t)\,d\mu_\alpha(t)}{\nu_\alpha(r)}\\
&=&\frac{\pi^{1-\beta}\int_0^{r^2}t^{n-\beta}(1-t)^\alpha \,dt
}{\int_0^{r^2} (1-t)^\alpha\, dt}\\
&=& \pi^{1-\beta}\left[\frac{f_{n-\beta}(r^2)}{f_{0}(r^2)}\right].
\end{eqnarray*}
In accordance with Lemma \ref{wz} (i)-(ii), it is readily to work
out that $\log r\mapsto\log\mathsf{A}_{\alpha,\beta}(z^n,r)$ is
convex for $r\in (0,1)$ if and only if $\Delta (n-\beta, x)\ge 0$
for any $x\in (0,1)$.
(i) Under $\alpha\in (-\infty,-3)$, we follow the argument for
\cite[Proposition 6]{WZ} to get
$$
\lim_{x\to 1}\Delta(\lambda,x)=\frac{\lambda (\alpha+1)(\lambda+2+\alpha)}{(\alpha+2)^2(\alpha+3)}.
$$
Choosing
\[
f(z)=z^n=\left\{\begin{array} {r@{\;}l}
z\quad & \hbox{when}\quad \beta<1\\
z^2\quad & \hbox{when}\quad \beta=1
\end{array}
\right.
\]
and $\lambda=n-\beta$, we find $\lim_{x\to 1}\Delta(\lambda,x)<0$,
whence deriving that $\log r\mapsto \log A_\alpha(f,r)$ is not
convex.
In the meantime, picking $n\in \mathbb N$ such that
$n>\beta-(2+\alpha)$ and putting $g(z)=z^n$, we obtain
$$
\lim_{x\to
1}\Delta(n-\beta,x)=\frac{(n-\beta)(\alpha+1)(n-\beta+2+\alpha)}{(\alpha+2)^2(\alpha+3)}>0,
$$
whence deriving that $\log r\mapsto
\log\mathsf{A}_{\alpha,\beta}(g,r)$ is not concave.
(ii) Under $\alpha\in [-3,0]$, we handle the two situations.
{\it Situation 1}: $f\in U(\mathbb D)$. Upon writing
$f(z)=\sum_{n=0}^\infty a_n z^n$, we compute
$$
\Phi_{A,1}\big(f(z),t\big)=(\pi t^2)^{-1}A(f,t)=\sum_{n=0}^\infty
n|a_n|^2 t^{2(n-1)},
$$
and consequently,
$$
\mathsf{A}_{\alpha,1}(f,r)=\frac{\sum_{n=0}^\infty n|a_n|^2\int_0^r
(\pi t^2)^{-1}A(z^n,t)\,
d\mu_\alpha(t)}{\nu_\alpha(r)}=\sum_{n=0}^\infty n |a_n|^2
\mathsf{A}_{\alpha,1}(z^n,r).
$$
So, by Lemma \ref{wz} (iii), we see that the convexity of
$$
\log r\mapsto\log\mathsf{A}_{\alpha,1}(f,r)\quad\hbox{under}\quad
f\in U(\mathbb D)
$$
follows from the convexity of
$$
\log r\mapsto\log\mathsf{A}_{\alpha,1}(z^n,r)\quad\hbox{under}\quad
n\in\mathbb N.
$$
So, it remains to verify this last convexity via the coming-up-next
consideration.
{\it Situation 2}: $f(z)=a_nz^n$ with $a_n\not=0$. Three cases are
required to control.
{\it Case 1}: $\alpha=0$. An easy computation shows
$$
\mathsf{A}_{0,1}(z^n,r)=n^{-1}{r^{2(n-1)}}
$$
and so $\log r\mapsto\log\mathsf{A}_{0,1}(z^n,r)$ is convex.
{\it Case 2}: $-2\le\alpha<0$. Under this condition, we see from the
arguments for \cite[Propositions 4-5]{WZ} that
$$
\Delta(n-1,x)\geq 0\quad\forall\quad n-1\geq 0\ \ \&\ \ 0<x<1,
$$
and so that $\log r\mapsto\log\mathsf{A}_{\alpha,1}(z^n,r)$ is
convex.
{\it Case 3}: $-3\leq \alpha<-2$. With the assumption, we also get
from the arguments for \cite[Propositions 4-5]{WZ} that
$$
\Delta (n-1,x)\geq \Delta(-2-\alpha,x)>0\quad\forall\quad x\in
(0,1)\ \ \&\ \ n-1\in [-2-\alpha,\infty)
$$
and so that $\log r\mapsto\log\mathsf{A}_{\alpha,1}(z^n,r)$ is
convex when $n\ge 2$. Here it is worth noting that the convexity of
$\log r\mapsto\log\mathsf{A}_{\alpha,1}(z,r)=0$ is trivial.
(iii) Under $0<\alpha<\infty$, from the argument for
\cite[Proposition 6]{WZ} we know that $\Delta(n-\beta,x)<0$ as $x$
is sufficiently close to $1$. Thus $\log r\mapsto
\log\mathsf{A}_{\alpha,\beta}(a_n z^n,r)$ is not convex under
$a_n\not=0$.
\end{proof}
The following illustrates that the function $\log r\mapsto
\log\mathsf{A}_{\alpha,\beta}(f,r)$ is not always concave for
$\alpha>0$, $0\le\beta\le 1$, and $f\in U(\mathbb D)$.
\begin{example} Let $\alpha=1$, $\beta\in\{0,1\}$, and $f(z)=z+\frac{z^2}{2}$. Then the function $\log r\mapsto \log\mathsf{A}_{\alpha,\beta}(f,r)$ is neither convex nor
concave for $r\in (0,1)$.
\end{example}
\begin{proof} A direct computation shows
$$
\left|\frac{z^2f'(z)}{f^2(z)}-1\right|=\left|\frac{z^2(1+z)}{(z+\frac{z^2}{2})^2}-1\right|=\frac{|z|^2}{|z+2|^2}<1
$$
since
$$
|z|<1<2-|z|\leq|z+2|\quad\forall\quad z\in \mathbb D.
$$
So, $f\in U(\mathbb D)$ owing to Lemma \ref{uni} (i). By $f'(z)=z+1$
we have
$$
A(f,t)=\int_{t\mathbb D}|z+1|^2\, dA(z)=\pi
\Big(t^2+\frac{t^4}{2}\Big),
$$
plus
\[
\int_0^r \Phi_{A,\beta}(f,t)\,d\mu_1(t)=\left\{\begin{array}
{r@{\;}l}
\frac{\pi}{2}\Big(r^4-\frac{r^6}{3}-\frac{r^8}{4}\Big)\quad & \hbox{when}\quad \beta=0\\
r^2-\frac{r^4}{4}-\frac{r^6}{6}\quad & \hbox{when}\quad \beta=1
\end{array}
\right.
\]
Meanwhile,
$$
\nu_1(r)=\int_0^r (1-t^2)dt^2=r^2-\frac{r^4}{2}.
$$
So, we get
\[
\mathsf{A}_{1,\beta}(f,r)=\left\{\begin{array} {r@{\;}l}
\frac{\pi(12r^2-4r^4-3r^6)}{12(2-r^2)}
\quad & \hbox{when}\quad \beta=0\\
\frac{12-3r^2-2r^4}{6(2-r^2)}\quad & \hbox{when}\quad \beta=1
\end{array}
\right.
\]
and in turn consider the logarithmic convexities of the following
function
\[
h_\beta(x)=\left\{\begin{array} {r@{\;}l}
\frac{12x-4x^2-3x^3}{2-x}\quad & \hbox{when}\quad \beta=0\\
\frac{12-3x-2x^2}{2-x}\quad & \hbox{when}\quad \beta=1
\end{array}
\right.
\]
for $x\in (0,1)$.
Using the so-called D-notation in Lemma \ref{wz}, we have
\[
D(h_\beta(x))=\left\{\begin{array} {r@{\;}l}
D(12x-4x^2-3x^3)-D(2-x)\quad & \hbox{when}\quad \beta=0\\
D(12-3x-2x^2)-D(2-x)\quad & \hbox{when}\quad \beta=1
\end{array}
\right.
\]
for $x\in (0,1)$. By an elementary calculation, we get
\[
\left\{\begin{array} {r@{\;}l}
D(12x-4x^2-3x^3)=\frac{-48-144x+12x^2}{(12-4x-3x^2)^2}\\
D(2-x)=\frac{-2}{(2-x)^2}\\
D(12-3x-2x^2)=\frac{-36-96x+6x^2}{(12-3x-2x^2)^2}.
\end{array}
\right.
\]
Consequently,
\[
D(h_\beta(x))=\left\{\begin{array} {r@{\;}l}
\frac{2g_\beta(x)}{(12-4x-3x^2)^2(2-x)^2}\quad & \hbox{when}\quad \beta=0\\
\frac{2g_\beta(x)}{(12-3x-2x^2)^2(2-x)^2}\quad & \hbox{when}\quad
\beta=1,
\end{array}
\right.
\]
where
\[
g_\beta(x)=\left\{\begin{array} {r@{\;}l}
48-288x+232x^2-72x^3+15x^4\quad & \hbox{when}\quad \beta=0\\
72-192x+147x^2-48x^3+7x^4\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
Now, under $x\in (0,1)$ we find
$$
g_0'(x)=-288+464x-216x^2+60x^3\quad \&\quad
g_0''(x)=464-432x+180x^2.
$$
Clearly, $g_0''(x)$ is an open-upward parabola with the axis of
symmetry $x=\frac{6}{5}>1$. By $g_0''(1)=212>0$ and the monotonicity
of $g_0''$ on $(0,1)$, we have $g_0''(x)>0$ for all $x\in (0,1)$.
Thus $g_0'$ is increasing on $(0,1)$. The following condition
$$
g_0'(0)=-288<0\quad \&\quad g_0'(1)=20>0
$$
yields an $x_1\in (0,1)$ such that $g_0'(x)<0$ for $x\in(0,x_1)$ and
$g_0'(x)>0$ for $x\in (x_1,1)$. Since $g_0(0)=48$ and $g_0(1)=-65$,
there exists an $x_0\in (0,1)$ such that $g_0(x)>0$ for $x\in
(0,x_0)$ and $g_0(x)<0$ for $x\in (x_0,1)$. Thus the function $\log
x\mapsto\log h_0(x)$ is neither convex nor concave.
Similarly, under $x\in (0,1)$ we have
$$
g_1'(x)=-192+294x-144x^2+28x^3\quad \&\quad g_1''(x)=294-288x+84x^2.
$$
Obviously, $g_1''(x)$ is an open-upward parabola with the axis of
symmetry $x=\frac{12}{7}>1$. By $g_1''(1)=90>0$ and the monotonicity
of $g_1''$ on $(0,1)$, we have $g_1''(x)>0$ for all $x\in (0,1)$.
Thus $g_1'$ is increasing on $(0,1)$. The following condition
$$
g_1'(0)=-192<0\quad \&\quad g_1'(1)=-14<0
$$
yields $g_1'(x)<0$ for $x\in(0,1)$. Since $g_1(0)=72$ and
$g_1(1)=-14$, there exists an $x_0\in (0,1)$ such that $g_1(x)>0$
for $x\in (0,x_0)$ and $g_1(x)<0$ for $x\in (x_0,1)$. Thus the
function $\log x\mapsto\log h_1(x)$ is neither convex nor concave.
\end{proof}
\subsection{Log-convexity for $\mathsf{L}_{\alpha,\beta}(f,\cdot)$} Analogously, we can
establish the expected convexity for the mixed lengths.
\begin{theorem}\label{th4} Let $0\le\beta\le 1$ and $0<r<1$.
\item{\rm(i)} If $\alpha\in (-\infty,-3)$, then there exist $f, g\in H(\mathbb D)$ such that $\log r\mapsto\log\mathsf{L}_{\alpha,\beta}(f,r)$ is not
convex and $\log r\mapsto\log \mathsf{L}_{\alpha,\beta}(g,r)$ is not
concave.
\item{\rm(ii)} If $\alpha\in [-3,0]$, then $\log r\mapsto \log\mathsf{L}_{\alpha,1}(a_nz^n,r\big)$ is
convex for $a_n\not=0$ with $n\in\mathbb N$. Consequently, $\log
r\mapsto \log\mathsf{L}_{\alpha,1}(f,r)$ is convex for $f\in
U(\mathbb D)$.
\item{\rm(iii)} If $\alpha\in (0,\infty)$, then $\log r\mapsto\log\mathsf{L}_{\alpha,\beta}(a_nz^n,r)$ is not convex
for $a_n\not=0$ and $n\in \mathbb N$.
\end{theorem}
\begin{proof} The argument is similar to that for Theorem \ref{th3}
except using the following statement for $\alpha\in [-3,0]$ -- If
$f\in U(\mathbb D)$, then there exists $g(z)=\sum_{n=0}^\infty b_n
z^n$ such that $g$ is the square root of the zero-free derivative
$f'$ on $\mathbb D$ and $f'(0)=g^2(0)$, and hence
\begin{eqnarray*}
\Phi_{L,1}(f,t)&=&(2\pi t)^{-1}\int_{t\mathbb
T}|f'(z)||dz|\\
&=&(2\pi t)^{-1} \int_{t\mathbb T} |g(z)|^2|dz|\\
&=&\sum_{n=0}^\infty |b_n|^2 t^{2n}.
\end{eqnarray*}
\end{proof}
Our concluding example shows that under $0<\alpha<\infty$ and
$0\le\beta\le 1$ one cannot get that
$\log\mathsf{L}_{\alpha,\beta}(f,r)$ is convex or concave in $\log
r$ for all functions $f\in U(\mathbb D)$.
\begin{example} Let $\alpha=1$, $\beta\in\{0,1\}$, and $f(z)=(z+2)^3$. Then the function $\log r\mapsto\log\mathsf{L}_{\alpha,\beta}(f,r)$ is neither convex nor concave for $r\in (0,1)$.
\end{example}
\begin{proof} Clearly, we have
$$
f'(z)=3(z+2)^2\ \ \&\ \ f''(z)=6(z+2)
$$
as well as the Schwarizian derivative
$$
\left[\frac{f''(z)}{f'(z)}\right]'-\frac{1}{2}\left[\frac{f''(z)}{f'(z)}\right]^2=\frac{-4}{(z+2)^2}.
$$
It is easy to see that
$$
\sqrt{2}(1-|z|^2)\leq 2-|z|\quad\forall\quad z\in \mathbb D.
$$
So,
$$
\left|\left[\frac{f''(z)}{f'(z)}\right]'-\frac{1}{2}\left[\frac{f''(z)}{f'(z)}\right]^2\right|=\frac{4}{|z+2|^2}\leq
\frac{4}{(2-|z|)^2}\leq \frac{2}{(1-|z|^2)^2}.
$$
By Lemma \ref{uni} (ii), $f$ belongs to $U(\mathbb D)$.
Consequently,
$$
L(f,t)=\int_0^{2\pi}|f'(te^{i\theta})|t\, d\theta=6\pi t(t^2+4)
$$
and
\[
\int_0^r\Phi_{L,\beta}(f,t)\,d\mu_1(t)=\left\{\begin{array}
{r@{\;}l}
12\pi\Big(\frac{4}{3}r^3-\frac{3}{5}r^5-\frac{1}{7}r^7\Big)
\quad & \hbox{when}\quad \beta=0\\
12r^2-\frac{9}{2}r^4-r^6\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
Note that $\nu_1(r)=r^2-\frac{r^4}{2}$. So,
\[
\mathsf{L}_{1,\beta}(f,r)=\left\{\begin{array} {r@{\;}l}
\frac{24\pi(140r-63r^3-15r^5)}{105(2-r^2)}
\quad & \hbox{when}\quad \beta=0\\
\frac{24-9r^2-2r^4}{2-r^2} \quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
To gain our conclusion, we only need to consider the logarithmic
convexity of the function
\[
h_\beta(x)=\left\{\begin{array} {r@{\;}l}
\frac{140x-63x^3-15x^5}{2-x^2}\quad & \hbox{when}\quad \beta=0\\
\frac{24-9x-2x^2}{2-x}\quad & \hbox{when}\quad \beta=1.
\end{array}
\right.
\]
{\it Case 1}: $\beta=0$. Applying the definition of $D$-notation, we
obtain
$$
D(140x-63x^3-15x^5)=\frac{-35280 x-33600
x^3+3780x^5}{(140-63x^2-15x^4)^2}
$$
and
$$
D(2-x^2)=\frac{-8x}{(2-x^2)^2},
$$
whence reaching
$$
D\big(h_0(x)\big)=D(140x-63x^3-15x^5)-D(2-x^2)=\frac{4xg_0(x)}{(140-63x^2-15x^4)^2(2-x^2)^2},
$$
where
$$
g_0(x)=3920-33600x^2+28098x^4-8400x^6+1395x^8.
$$
Obviously,
$$
g_0(0)=3920>0\quad\&\quad g_0(1)=-8587<0.
$$
Now letting $s=x^2$, we get
$$
g_0(x)=G_0(s)=3920-33600s+28098s^2-8400s^3+1395s^4,
$$
and
$$
G'_0(s)=-33600+56196s-25200s^2+5580s^3\ \&\
G''_0(s)=56196-50400s+16740s^2.
$$
Since the axis of symmetry of $G''_0$ is $s=\frac{140}{93}>1$,
$G''_0$ is decreasing on $(0,1)$. Due to $G''_0(1)=22536>0$, we have
$G''_0(s)>0$ for all $s\in (0,1)$, i.e., $G'_0(s)$ is increasing on
$(0,1)$. By
$$
G'_0(0)=-33600<0\quad\&\quad G'_0(1)=2976>0,
$$
we conclude that there exists an $s_0\in(0,1)$ such that $G'_0(s)<0$
for $s\in(0,s_0)$ and $G'_0(s)>0$ for $s\in (s_0,1)$. Then there
exists an $x_0\in (0,1)$ such that $g_0(x)$ is decreasing for $x\in
(0,x_0)$ and $g_0(x)$ is increasing for $x\in(x_0,1)$. Thus there
exists an $x_1\in (0,1)$ such that $g_0(x)>0$ for $x\in(0,x_1)$ and
$g_0(x)<0$ for $x\in(x_1,1)$. As a result, we find that $\log
r\mapsto\log\mathsf{L}_{\alpha,0}(f,r)$ is neither concave nor
convex.
{\it Case 2}: $\beta=1$. Again using the $D$-notation, we obtain
$$
D(24-9x-2x^2)=\frac{-216-192x+18x^2}{(24-9x-2x^2)^2}
$$
and
$$
D(2-x)=\frac{-2}{(2-x)^2},
$$
whence deriving
$$
D\big(h_1(x)\big)=D(24-9x-2x^2)-D(2-x)=\frac{2g_1(x)}{(24-9x-2x^2)^2(2-x)^2},
$$
where
$$
g_1(x)=144-384x+297x^2-96x^3+13x^4.
$$
Now we have
$$
g'_1(x)=-384+594x-288x^2+52x^3\quad \&\quad
g''_1(x)=594-576x+156x^2.
$$
Since the axis of symmetry of $g''_1(x)$ is $x=\frac{24}{13}>1$,
$g''_1(x)$ is decreasing on $(0,1)$. Due to $g''_1(1)=174>0$, we
have $g''_1(x)>0$ for all $x\in (0,1)$, i.e., $g'_1(x)$ is
increasing on $(0,1)$. By
$$
g'_1(0)=-384<0\quad\&\quad g'_1(1)=-26<0,
$$
we conclude that $g'_1(x)<0$ for $x\in (0,1)$. Obviously,
$$
g_1(0)=144>0\quad \&\quad g_1(1)=-26<0.
$$
Hence there exists an $x_0\in (0,1)$ such that $g_1(x)>0$ for $x\in
(0,x_0)$ and $g_1(x)<0$ for $x\in (x_0,1)$. Consequently, we find
that $\log r\mapsto\log\mathsf{L}_{\alpha,\beta=1}(f,r)$ is neither
concave nor convex.
\end{proof}
\end{document} |
\begin{document}
\title{Finite difference schemes on quasi-uniform grids for BVPs on infinite intervals}
\begin{abstract}
The classical numerical treatment of boundary value problems defined on infinite intervals is to replace the boundary conditions at infinity by suitable boundary conditions at a finite point, the so-called truncated boundary.
A truncated boundary allowing for a satisfactory accuracy of the numerical solution has to be determined by trial and errors and this seems to be the weakest point of the classical approach.
On the other hand, the free boundary approach overcomes the need for a priori definition of the truncated boundary.
In fact, in a free boundary formulation the unknown free boundary can be identified with a truncated boundary and being unknown it has to be found as part of the solution.
In this paper we consider a different way to overcome the introduction of a truncated boundary, namely non-standard finite difference schemes defined on quasi-uniform grids.
A quasi-uniform grid allows us to describe the infinite domain by a finite number of intervals.
The last node of such grid is placed on infinity so that right boundary conditions
are taken into account exactly.
We apply the proposed approach to the Falkner-Skan model and to a problem of interest in foundation engineering.
The obtained numerical results are found in good agreement with those available in literature.
Moreover, we provide a simple way to improve the accuracy of the numerical results using Richardson's extrapolation.
Finally, we indicate a possible way to extend the proposed approach to boundary value problems defined on the whole real line.
\epsilonnd{abstract}
\noindent {\bf Key Words.} non linear boundary value problems,
infinite intervals, quasi-uniform grid, non-standard finite difference methods.
\noindent
{\bf AMS Subject Classifications.} 65L10, 65L12, 34B40.
\section{Introduction}\label{S:intro}
The classical numerical treatment of
boundary value problems (BVPs) on infinite intervals is to replace the original problem by one defined
on a finite interval, where a finite value, the so-called
truncated boundary, is used instead of infinity (see, for instance, Collatz \cite[pp. 150-151]{Collatz} or Fox \cite[p. 92]{Fox}).
For the accepted numerical solution, the value of the truncated boundary is varied until the computed results stabilize, at least, to a prefixed number of significant digits.
However, a truncated boundary allowing for a satisfactory accuracy
of the numerical solution has to be determined by trial and errors
and this seems to be the weakest point of the classical approach.
Hence, a priori definition of the truncated boundary
was indicated by Lentini and Keller \cite{Lentini:BVP:1980} as an important area of research.
A theory for defining asymptotic boundary conditions to be imposed
at the truncated boundary has been developed by de Hoog and Weiss
\cite{deHoog:1980:ATB}, Lentini and Keller \cite{Lentini:BVP:1980} and Markowich
\cite{Markowich:TAS:1982,Markowich:ABV:1983}.
The asymptotic boundary conditions have been applied successfully to the numerical approximation of the so-called \lq \lq connecting orbits\rq \rq \ problems of dynamical systems, see Beyn \cite{Beyn:1990:GBN,Beyn:1990:NCC,Beyn:1992:NMD}.
Those problems are of interest, not only in connection with dynamical systems, but also in the study of traveling wave solutions of partial differential equations of parabolic and hyperbolic type as shown by Beyn \cite{Beyn:1990:NCC}, Friedman and Doedel \cite{Friedman:1991:NCC}, Bai et al. \cite{Bai:1993:NCH}, and Liu et al. \cite{Liu:1997:CCH}.
A free boundary formulation was proposed by Fazio
\cite{Fazio:1992:BPF} where the unknown free boundary was identified with a truncated boundary.
In this approach the free boundary is unknown and has to be found as part of the solution.
This free boundary approach overcomes the need for a priori definition of the truncated boundary.
The free boundary formulation has been applied to: the Blasius
problem \cite{Fazio:1992:BPF}, the Falkner-Skan model \cite{Fazio:1994:FSEb}, a model describing the flow of an incompressible fluid over a slender parabola of revolution \cite{Fazio:1996:NAN}, a connecting orbit problem \cite{Fazio:2002:SFB}, and a problem in foundation engineering \cite{Fazio:2003:FBA}.
A different way to avoid the definition of a truncated boundary is to apply coordinate transforms.
The idea of mapping an infinite geometry into a finite one is not original.
For example, van de Vooren and Dijkstra \cite{vandeVooren:1970:NSS} applied coordinate transformations to the numerical solution of laminar flow past a flat plate, Botta et al. \cite{Botta:1972:NSN} and Davis \cite{Davis:1972:NSN} applied similar techniques to laminar flow past a parabola.
Coordinate transforms have been applied to the numerical solution of ordinary and partial differential equations on unbounded domains, see Grosch and Orszag \cite{Grosch:NSP:1977}, Boyd \cite{Boyd:2001:CFS} or Koleva \cite{Koleva:NSH:2006}.
Here we consider finite difference schemes on quasi-uniform grids, defined by coordinate transforms, applied to the numerical solution of BVPs defined on infinite intervals.
The novelty of our approach is that we define non-standard finite differences for the original problem on the infinite domain, whereas Grosch and Orszag transform the governing model and apply the classical finite difference or shooting methods on the transformed finite domain.
In the following sections we consider two test problems.
The first is the Falkner-Skan model of boundary layer theory.
The last one is a problem of interest in foundation engineering.
The obtained numerical results are found in good agreement with those available in literature.
Moreover, we have applied Richardson's extrapolation in order to improve the accuracy of the numerical results.
Preliminary numerical results on the main topic of this paper were presented at the ENUMATH 2013 conference \cite{Fazio:2014:QUG}.
We notice that the free boundary approach or a quasi-uniform grid strategy are as simple as the classical truncated boundary one in contrast with the asymptotic boundary approach, in this context see also the opinion expressed by J. R. Ockendon \cite{Ockendon}.
In the last section, we point out some conclusions supported by the
evidences of the present work and indicate a possible way to
extend the proposed approach to BVPs defined on the whole real
line.
\section{Finite differences on quasi-uniform grids}\label{S:quniform}
Let us consider the smooth strict monotone quasi-uniform maps $x = x(\xi)$, the so-called grid generating functions,
\begin{equation}\label{eq:qu1}
x = -c \cdot \ln (1-\xi) \ ,
\epsilonnd{equation}
and
\begin{equation}\label{eq:qu2}
x = c \frac{\xi}{1-\xi} \ ,
\epsilonnd{equation}
where $ \xi \in \left[0, 1\right] $, $ x \in \left[0, \infty\right] $, and $ c > 0 $ is a control parameter.
So that, a family of uniform grids $\xi_n = n/N$ defined on interval $[0, 1]$ generates one parameter family of quasi-uniform grids $x_n = x (\xi_n)$ on the interval $[0, \infty]$.
The two maps (\ref{eq:qu1}) and (\ref{eq:qu2}) are referred as logarithmic and algebraic map, respectively.
As far as the authors knowledge is concerned, van de Vooren and Dijkstra \cite{vandeVooren:1970:NSS} were the first to use these kind of maps.
We notice that more than half of the intervals are in the domain with length approximately equal to $c$ and
$x_{N-1} = c \ln N$ for (\ref{eq:qu1}),
while $ x_{N-1} \approx c N $ for (\ref{eq:qu2}).
For both maps, the equivalent mesh in $x$ is nonuniform with the
most rapid variation occurring with $c \ll x$.
The logarithmic map (\ref{eq:qu1}) gives slightly better resolution near $x = 0$ than the
algebraic map (\ref{eq:qu2}), while the algebraic map gives much better resolution than the
logarithmic map as $x \rightarrow \infty$.
In fact, it is easily verified that
\[
-c \cdot \ln (1-\xi) < c \frac{\xi}{1-\xi} \ ,
\]
for all $\xi$, see figure \ref{fig:m1N20} below.
The problem under consideration can be discretized by introducing a uniform grid $ \xi_n $ of $N+1$ nodes in $ \left[0, 1\right] $ with $\xi_0 = 0$ and $ \xi_{n+1} = \xi_n + h $ with $ h = 1/N $, so that $ x_n $ is a quasi-uniform grid in $ \left[0, \infty\right] $.
The last interval in (\ref{eq:qu1}) and (\ref{eq:qu2}),
namely $ \left[x_{N-1}, x_N\right] $, is infinite but the point $ x_{N-1/2} $ is finite, because the non integer nodes are defined by
\[
x_{n+\alpha} = x\left(\xi=\frac{n+\alpha}{N}\right) \ ,
\]
with $ n \in \{0, 1, \dots, N-1\} $ and $ 0 < \alpha < 1 $.
This maps allow us to describe the infinite domain by a finite number of intervals.
The last node of such grid is placed on infinity so right boundary condition
is taken into account correctly.
Figure \ref{fig:m1N20} shows the two quasi-uniform meshes $x=x_n$, $n = 0, 1, \dots , N$ defined by (\ref{eq:qu1}) and by (\ref{eq:qu2}) with $c=5$ and $N=20$.
\begin{figure}[!hbt]
\centering
\psfrag{I}[][]{$x_0$}
\psfrag{xxx}[l][]{$x \rightarrow \infty$}
\psfrag{P}[][]{$x_{19}$}
\psfrag{A}[][]{}
\framebox{\includegraphics[width=.9\textwidth]{mappa1N20}
} \\
\framebox{\includegraphics[width=.9\textwidth]{mappa2N20}
}
\caption{\it Quasi-uniform meshes: top frame for (\ref{eq:qu1}) and bottom frame for (\ref{eq:qu2}). We notice that, in both cases, the last mesh-point is $x_N = \infty$.}
\label{fig:m1N20}
\epsilonnd{figure}
We can define the values of $u(x)$ on the mid-points of the grid
\begin{equation}
u_{n+1/2} \approx \frac{x_{n+1}-x_{n+1/2}}{x_{n+1}-x_n} u_n + \frac{x_{n+1/2}-x_{n}}{x_{n+1}-x_n} u_{n+1} \ .
\label{eq:u}
\epsilonnd{equation}
To get (\ref{eq:u}) we can apply Taylor formula, at $x_{n+1/2}$, to both $u_{n+1}$ and $u_{n}$.
A simpler way to obtain (\ref{eq:u}) is to consider the method of undefined coefficients, for $u_{n+1/2}$ as a linear combination of $u_{n}$ and $u_{n+1}$, and to require that the formula is exact for constant and linear functions.
In this way we end up with a linear system of two equations in two unknowns where the coefficient matrix is a Vandermonde matrix.
As far as the first derivative is concerned we can apply the following approximation
\begin{equation}
\frac{du}{dx}(x_{n+1/2}) \approx \frac{u_{n+1}-u_n}{2\left(x_{n+3/4} - x_{n+1/4}\right)} \ .
\label{eq:du}
\epsilonnd{equation}
These formulae use the value $ u_N = u_\infty $, but not $ x_N = \infty $.
In order to justify non-standard finite difference formula (\ref{eq:du}) we note that, by considering $u=u(\xi(x))$, we can write
\begin{equation}
\left. \frac{du}{dx}\right|_{n+1/2} = \left. \frac{du}{d\xi}\right|_{n+1/2}\left. \frac{d\xi}{dx}\right|_{n+1/2} \approx \frac{u_{n+1}-u_n}{\xi_{n+1} - \xi_{n}}
\frac{2\left(\xi_{n+3/4}-\xi_{n+1/4}\right)}{2\left(x_{n+3/4} - x_{n+1/4}\right)} \ .
\label{eq:duder}
\epsilonnd{equation}
The last formula on the right hand side of equation (\ref{eq:duder}) reduces to the right hand side of equation (\ref{eq:du}) because we are using a uniform grid for $\xi$ and therefore $2\left(\xi_{n+3/4}-\xi_{n+1/4}\right)=\xi_{n+1} - \xi_{n}$.
The two finite difference approximations (\ref{eq:u}) and (\ref{eq:du}) have order of accuracy $O(N^{-2})$.
For a system of differential equations, formulae (\ref{eq:u}) and (\ref{eq:du}) can be applied component-wise.
\section{BVPs on infinite intervals}
Let us consider the class of BVPs defined on an infinite interval
\begin{eqnarray}
&& {\displaystyle \frac{d{\bf u}}{dx}} = {\bf f} \left(x, {\bf u}\right)
\ , \quad x \in [0, \infty) \ , \nonumber \\[-1.5ex]
\label{p} \\[-1.5ex]
&& {\bf g} \left( {\bf u}(0), {\bf u} (\infty) \right) = {\bf 0}
\ , \nonumber
\epsilonnd{eqnarray}
where $ {\bf u}(x) $ is a $ d-$dimensional vector with $ u_{\epsilonll}
(x) $ for $ \epsilonll =1, \dots , d $ as components, $ {\bf f}:[0,
\infty) \times \hbox{I\kern-.2em\hbox{R}}^d \rightarrow~\hbox{I\kern-.2em\hbox{R}}^d $, and $ {\bf g}:
\hbox{I\kern-.2em\hbox{R}}^d \times \hbox{I\kern-.2em\hbox{R}}^d \rightarrow \hbox{I\kern-.2em\hbox{R}}^d $.
A non-standard finite difference scheme on a quasi-uniform grid for the class of BVPs
(\ref{p}) can be defined by using the approximations given by (\ref{eq:u}) and (\ref{eq:du}).
We denote by the $d-$dimensional vector $ {\bf U}_n $ the numerical approximation to the solution $ {\bf u} (x_n) $ of (\ref{p}) at the points of the mesh, that is for $ n = 0, 1, \dots , N $ .
A second order finite difference scheme for
(\ref{p}) can be written as follows:
\begin{eqnarray}
& {\bf U}_{n+1} - {\bf U}_{n} - a_{n+1/2} {\bf f} \left( x_{n+
1/2}, b_{n+1/2}{\bf U}_{n+1} + c_{n+1/2}{\bf U}_{n} \right) = {\bf 0}
\ , \nonumber\\
& \mbox{for} \quad n=0, 1, \dots , N-1
\label{boxs} \\
& {\bf g} \left( {\bf U}_0,{\bf U}_N \right) = {\bf 0} \ , \nonumber
\epsilonnd{eqnarray}
where
\begin{eqnarray}\label{eq:abc}
a_{n+1/2} &=& 2\left(x_{n+3/4} - x_{n+1/4}\right) \ , \nonumber \\
b_{n+1/2} &=& \frac{x_{n+1/2}-x_{n}}{x_{n+1}-x_n} \ , \\
c_{n+1/2} &=& \frac{x_{n+1}-x_{n+1/2}}{x_{n+1}-x_n} \nonumber \ ,
\epsilonnd{eqnarray}
for $n=0, 1, \dots , N-1$.
It is evident that (\ref{boxs}) is a nonlinear system of $ d \cdot (N+1)$ equations in the $ d \cdot (N+1)$ unknowns $ {\bf U} = ({\bf U}_0,
{\bf U}_1, \dots , {\bf U}_N)^T $.
We notice that $b_{n+1/2} \approx c_{n+1/2} \approx 1/2$ for all $n=0, 1, \dots , N-2$, but when $n=N-1$, then $b_{N-1/2} = 0$ and $c_{N-1/2} = 1$.
This means that $u_{n-1/2} = u_{n-1}$ and the value of $u_N$ plays no role in defining the middle node value.
In order to avoid a sudden jump for the coefficients of (\ref{boxs}), and to make use of $u_N$, we choose to set $b_{N-1/2} = b_{N-3/2}$ and $c_{N-1/2} = c_{N-3/2}$ .
As it will be clear from the results reported in the next section this choice produces a much smaller error in the numerical solution of the system at $x_N$.
For the solution of (\ref{boxs}) we can apply the classical Newton's method along with the simple termination criterion
\begin{eqnarray*}
{\displaystyle \frac{1}{d(N+1)} \sum_{\epsilonll =1}^{d} \sum_{n=0}^{N}
|\mbox{D}elta U_{n \epsilonll}| \leq {\rm TOL}} \ ,
\epsilonnd{eqnarray*}
where $ \mbox{D}elta U_{n \epsilonll} $, for $ n = 0,1, \dots, N $ and $ \epsilonll
= 1, 2, \dots , d $, is the difference between two successive
iterate components and $ {\rm TOL} $ is a fixed tolerance. The
results listed in the next sections were computed by
setting $ {\rm TOL} = 1\mbox{E}-6 $.
\section{The Falkner-Skan model}
The Falkner-Skan model \cite{Falkner:1931:SAS} of boundary layer theory
\cite{Schlichting:2000:BLT}, defined by
\begin{eqnarray}
& {\displaystyle \frac{d^3 u}{d x^3}} + u {\displaystyle \frac{d^{2}u}{dx^2}} +
P \left[ 1 - \left( {\displaystyle \frac{du}{dx}}\right)^2 \right] = 0 \nonumber \\ [-1ex]
\label{eq:Falkner} \\[-1.5ex]
& u(0) = {\displaystyle \frac{du}{dx}}(0) = 0, \qquad
{\displaystyle \frac{du}{dx}}(\infty) = 1 \ , \nonumber
\epsilonnd{eqnarray}
is a BVP defined on a semi-infinite interval.
As a first step we rewrite the Falkner-Skan equation in (\ref{eq:Falkner}) as a first order system by setting
\begin{eqnarray*}
u_{i+1}(x) = \displaystyle\frac{d^{i} u}{dx^{i}} (x) \ , \quad
\mbox{for} \ i = 0, 1, 2 \ .
\epsilonnd{eqnarray*}
In this way the original BVP (\ref{eq:Falkner}) specializes to
\begin{eqnarray}\label{eq:Falkner:system}
&& {\displaystyle \frac{du_1}{dx}} = u_2 \nonumber \\[0.5ex]
&& {\displaystyle \frac{du_2}{dx}} = u_3 \nonumber \\[-1ex]
&& \\[-1ex]
&& {\displaystyle \frac{du_3}{dx}} = -u_1 u_3 - P (1 - {u_2}^2) \nonumber \\[0.5ex]
&& u_1(0) = u_2(0) = 0 \ ,
\qquad u_2(\infty) = 1 \ , \nonumber
\epsilonnd{eqnarray}
that is,
\begin{eqnarray*}
& {\bf u} = (u_1,u_2,u_3)^T \\
& {\bf f} (x, {\bf u}) =
\left(u_2,u_3,u_4, -u_1 u_3 - P (1 - {u_2}^2) \right)^T \\
& {\bf g} \left( {\bf u}(0), {\bf u} (\infty) \right) =
\left(u_1(0),u_2(0),u_2(\infty) -1\right)^T
\epsilonnd{eqnarray*}
in (\ref{p}).
\subsection{Numerical results}
Table \ref{tab:Falkner} lists the numerical approximations of $\displaystyle{\frac{d^2u}{dx^2}}(\infty)$, $\displaystyle{\frac{d^2u}{dx^2}(0)}$ and order of accuracy for increasing values of $N$.
Here, and in the following, the order of accuracy $p$ is defined by
\begin{equation}\label{eq:p}
p = {\displaystyle \frac{\log(|T_N-T_{1280}|)-\log(|T_{2 N}-T_{1280}|)}{\log(2)}} \ ,
\epsilonnd{equation}
where $T_N$ and $T_{2 N}$ are numerical approximations of our missing initial condition, namely $\displaystyle{\frac{d^2u}{dx^2}}(0)$.
For all values of $N$ we used the initial iterate
\begin{equation}
u_1(x) = u_2(x) = 1/2 \ , \quad u_3(x) = 10^{-2} \ .
\epsilonnd{equation}
Let us remark here that the same values of $\displaystyle{\frac{d^2u}{dx^2}}(0)$ were obtained by using $b_{N-1/2} = 0$ and $c_{N-1/2} = 1$ in (\ref{boxs}),
but, on the contrary, larger values of $\displaystyle{\frac{d^2u}{dx^2}}(\infty)$, of the order $O(10^{-3})$, were computed.
In table \ref{tab:Falkner:comp} we compare the obtained numerical results with those available in literature: the agreement is really good.
\begin{table}[!hbt]
\caption{\it Numerical approximation of $\displaystyle{\frac{d^2u}{dx^2}}(\infty)$, $\displaystyle{\frac{d^2u}{dx^2}}(0)$ and order of accuracy.}
\begin{center}
{\begin{tabular}{rcr@{.}lcc}
\hline \\[-1.5ex]
$N$ & iter & \multicolumn{2}{c}
{$\displaystyle{\frac{d^2u}{dx^2}(\infty)}$}
& $\displaystyle{\frac{d^2u}{dx^2}(0)}$ & {$p$} \\[1.5ex]
\hline
20 & 6 & $-0$ & $21 \cdot 10^{-7}$ & $1.238724$ & \\
40 & 5 & $ 0$ & $24 \cdot 10^{-7}$ & $1.234124$ & 1.998825 \\
80 & 5 & $-0$ & $33 \cdot 10^{-7}$ & $1.232972$ & 2.002822 \\
160 & 5 & $ 0$ & $14 \cdot 10^{-7}$ & $1.232684$ & 2.011345 \\
320 & 5 & $-0$ & $25 \cdot 10^{-7}$ & $1.232612$ & 2.046294 \\
640 & 5 & $ 0$ & $39 \cdot 10^{-7}$ & $1.232594$ & 2.201634 \\
1280 & 5 & $ 0$ & $33 \cdot 10^{-7}$ & $1.232589$ & \\
\hline
\epsilonnd {tabular}}
\epsilonnd{center}
\label{tab:Falkner}
\epsilonnd{table}
Figure \ref{fig:Falkner} shows the numerical solution of Falkner-Skan model (\ref{eq:Falkner}) with $P =1$ and $N = 80$.
\begin{figure}[!hbt]
\centering
\psfrag{x}[][]{$x$}
\psfrag{y}[][]{}
\includegraphics[width=.9\textwidth]{FKc_QUM_P1}
\put(-275,250){${\displaystyle u_1(x)}$}
\put(-200,205){${\displaystyle u_2(x)}$}
\put(-200,40){${\displaystyle u_3(x)}$}
\caption{\it Numerical solution of Falkner-Skan model with $P=1$ obtained with the map $x=x(\xi)$ defined by (\ref{eq:qu1}) with $c=5$ for $N=80$. }
\label{fig:Falkner}
\epsilonnd{figure}
\begin{table}[!htb]
\caption{\it Comparison of $\displaystyle{\frac{d^2 u}{dx^2}}(0)$
and free or truncated boundary ($ x_\epsilonpsilon $ and $x_\infty $ respectively) for the Homann ($P=1/2$) and Hiemenz ($P=1$) flows.}
\begin{center}{ \renewcommand\arraystretch{1.3}
\begin{tabular}{l|r@{.}lccccccc}
\hline
& \multicolumn{3}{c}
{Nasr et al. \cite{Nasr:1990:CSL}}
& \multicolumn{2}{c}
{Fazio \cite{Fazio:1994:FSEb}}
& \multicolumn{2}{c}
{Asaithambi \cite{Asaithambi:1998:FDM}}
& \multicolumn{2}{c}
{This paper}\\
& \multicolumn{3}{c}
{Chebyshev method} & \multicolumn{2}{c}
{Free BF} & \multicolumn{2}{c}
{Finite difference}
& \multicolumn{2}{c}
{Quasi-uniform} \\
\hline\\[-2.5ex]
{$P$} & \multicolumn{2}{c}
{$\ \ \ x_\infty $} &
{$ {\displaystyle \frac{d^2 u}{dx^2}} (0) $} &
{$ x_\epsilonpsilon $} &
{$ {\displaystyle \frac{d^2 u}{dx^2}} (0) $} &
{$ x_\infty $} &
{$ {\displaystyle \frac{d^2 u}{dx^2}} (0) $}
& {$x_N$} & {${\displaystyle \frac{d^2 u}{dx^2}} (0) $} \\[1.5ex] \hline
0.5 & \ \ \ 3 & 7 & 0.927805 & & & & & & \\
0.5 & \ \ \ 7 & 4 & 0.927680 & 5.09 & 0.927680 & 5.67 & 0.927682 & $\infty$ & 0.927681 \\
1 & \ \ \ 3 & 5 & 1.232617 & & & & & & \\
1 & \ \ \ 7 & & 1.232588 & 5.19 & 1.232588 & 5.14 & 1.232589 & $\infty$ & 1.232589 \\
\hline
\epsilonnd{tabular}}
\label{tab:Falkner:comp}
\epsilonnd{center}
\epsilonnd{table}
The Falkner-Skan model (\ref{eq:Falkner}) was solved by Grosch and Orszag \cite{Grosch:NSP:1977}, but only for small values of $P$ (namely $-0.1$, $0$, and $0.1$).
Moreover, these authors used the truncated boundary and the two maps (\ref{eq:qu1}) and (\ref{eq:qu2}) with a shooting method using only $11$ mesh point but, unfortunately, reported only the values of the first derivative at $x=1$ instead of the missing initial condition.
Moreover, for $P=-0.1$ they missed the inverse flow solution of Stewartson \cite{Stewartson:1954:FSF,Stewartson:1964:TLB}, see also Asaithambi \cite{Asaithambi:1997:NMS}, Auteri et al. \cite{Auteri:2012:GLS} and Fazio \cite{Fazio:2013:BPF}.
\section{A pile in soil}
Here we consider a problem that was already used by
Lentini and Keller \cite{Lentini:BVP:1980}
to test the asymptotic boundary
conditions approach.
This problem is of special interest here because none of
the solution components is a monotone function,
on $ [0, \infty) $, see \cite{Fazio:2002:SFB,Fazio:2003:FBA}.
Let $ u(x) $ be the deflection of a semi-infinite pile
embedded in soft soil at a distance $ x $ below the surface of the
soil. The governing differential equation for the movement of the
pile, in dimensionless form, is given by:
\begin{eqnarray}
{\displaystyle \frac{d^4 u}{dx^4}} = - P_1 \left(1 - e^{-P_2 u} \right) \ ,
\qquad x \in \left[0,\infty\right) \ , \nonumber \\[-1.5ex]
\label{eq:pile} \\[-1.5ex]
{\displaystyle \frac{d^2 u}{dx^2}} (0) = 0 \ , \qquad {\displaystyle
\frac{d^3 u}{dx^3}} (0) = P_3 \ , \qquad u(\infty) = {\displaystyle\frac{du}{dx} (\infty)} = 0 \ , \nonumber
\epsilonnd{eqnarray}
where $ P_1 $ and $ P_2 $ are positive material constants.
As far as the boundary conditions are concerned, at the
origin a zero moment and a positive shear $P_3$ are assumed and
from physical considerations it follows that $ u(x) $
and all its derivatives go to zero at infinity,
so that, the zero asymptotic boundary conditions can be imposed.
This problem is of interest in foundation
engineering: for instance, in the design of drilling rigs above
the ocean floor.
The governing differential equation in (\ref{eq:pile}) can be rewritten as a first order system by setting:
\begin{eqnarray*}
u_{i+1}(x) = \displaystyle\frac{d^{i} u}{dx^{i}} (x) \ , \quad
\mbox{for} \ i = 0, 1, 2, 3 \ .
\epsilonnd{eqnarray*}
In this way the original BVP (\ref{eq:pile}) specializes to
\begin{eqnarray}
&& {\displaystyle \frac{du_1}{dx}} = u_2 \ , \nonumber \\
&& {\displaystyle \frac{du_2}{dx}} = u_3 \ , \nonumber \\ && {\displaystyle
\frac{du_3}{dx}} = u_4 \ , \label{bvp2} \\ && {\displaystyle
\frac{du_4}{dx}} = -P_1 \left(1 - {\displaystyle e^{-P_2 u_1}} \right) \ ,
\nonumber \\[0.5ex]
&& u_3(0) = 0 \ , \qquad u_4(0) = P_3 \ , \qquad u_1(\infty) = 0 \ ,
\qquad u_2(\infty) = 0 \ , \nonumber
\epsilonnd{eqnarray}
that is,
\begin{eqnarray*}
& {\bf u} = (u_1,u_2,u_3,u_4)^T \\
& {\bf f} (x, {\bf u}) =
\left(u_2,u_3,u_4, -P_1
\left(1 - e^{-P_2 u_1} \right)\right)^T \\
& {\bf g} \left( {\bf u}(0), {\bf u} (\infty) \right) =
\left(u_3(0),u_4(0) - P_3,u_1(\infty),u_2(\infty) \right)^T
\epsilonnd{eqnarray*}
in (\ref{p}).
\subsection{Numerical results}
In order to be able to compare our numerical results we used the same parameter values employed by Lentini and Keller \cite{Lentini:BVP:1980}
\begin{eqnarray*}
P_1 = 1, \quad P_2 = \frac{1}{2} \quad \mbox{and}
\quad P_3 = \frac{1}{2} \ .
\epsilonnd{eqnarray*}
Moreover, we choose to consider the values of the missing initial
conditions $ u_1 (0) $ and $ u_2 (0) $ as representative results.
Figure \ref{fig:pile} shows the numerical solution of the BVP (\ref{eq:pile}) using the map (\ref{eq:qu1}) with $c=5$ for $N=80$.
\begin{figure}[!hbt]
\centering
\psfrag{x}[][]{$x$}
\psfrag{y}[][]{}
\includegraphics[width=.9\textwidth]{Pilec_QUM}
\put(-300,250){${\displaystyle u_1(x)}$}
\put(-290,50){${\displaystyle u_2(x)}$}
\put(-270,150){${\displaystyle u_3(x)}$}
\put(-325,100){${\displaystyle u_4(x)}$}
\caption{\it Numerical solution of pile model (\ref{eq:pile}) obtained with the map $x=x(\xi)$ defined by (\ref{eq:qu1}) with $c=5$ for $N=80$. }
\label{fig:pile}
\epsilonnd{figure}
Table \ref{tab:pile} lists the numerical approximations of $u(0)$, $\displaystyle{\frac{du}{dx}(0)}$ and the corresponding order of accuracy for increasing values of $N$.
The numerical accuracy values $p$, once again, are computed by formula (\ref{eq:p}),
where $T_N$ and $T_{2 N}$ are numerical approximations of our missing initial conditions, namely $u(0)$ and $\displaystyle{\frac{du}{dx}}(0)$, respectively.
For all values of $N$ we used the initial iterate
\[
u_1(x) = u_2(x) = u_3(x) = u_4(x) = 1 \ .
\]
\begin{table}[!hbt]
\caption{\it Numerical approximation of $u(0)$, $\displaystyle{\frac{du}{dx}}(0)$ and the corresponding order of accuracy. {\rm NaN} means not a number.}
\begin{center}
{\begin{tabular}{rccccc}
\hline \\[-1.5ex]
$N$ & iter & $u(0)$ & {$p$} & $-\displaystyle{\frac{du}{dx}(0)}$ & {$p$} \\[1.5ex]
\hline
20 & 5 & $1.420337$ & & $0.807289$ & \\
40 & 5 & $1.421243$ & $2.003590$ &$0.807934$ & $2.020368$ \\
80 & 5 & $1.421469$ & $2.004801$ & $0.808094$ & $2.048674$ \\
160 & 5 & $1.421526$ & $2.058894$ & $0.808135$ & $2.350497$ \\
320 & 5 & $1.421540$ & $2.169925$ & $0.808145$ & {$\infty$} \\
640 & 5 & $1.421544$ & {$\infty$} & $0.808145$ & {\rm NaN} \\
1280 & 5 & $1.421544$ & & $0.808145$ & \\
\hline
\epsilonnd {tabular}}
\epsilonnd{center}
\label{tab:pile}
\epsilonnd{table}
\begin{table}[!htb]
\caption{\it Comparison of $u(0)$, $\displaystyle{\frac{du}{dx}}(0)$
and free or truncated boundary ($ x_\epsilonpsilon $ and $x_\infty $ respectively) for the pile problem.}
\begin{center}{ \renewcommand\arraystretch{1.3}
\begin{tabular}{cccccc}
\hline
\multicolumn{2}{c}
{Lentini and Keller \cite{Lentini:BVP:1980}}
&\multicolumn{2}{c}
{Fazio \cite{Fazio:1994:FSEb}}
& \multicolumn{2}{c}
{This paper} \\
\multicolumn{2}{c}
{Asymptotic BCs {$ x_\infty = 10$}}
& \multicolumn{2}{c}
{Free BF {$ x_\epsilonpsilon = 17.75$}}
&\multicolumn{2}{c}
{Quasi-uniform {$x_N=\infty$}}
\\
\hline\\[-2.5ex]
{$u(0) $} & {$ {\displaystyle \frac{du}{dx}}(0) $}
& {$u(0) $}
& {$ {\displaystyle \frac{du}{dx}}(0) $}
& {$u(0) $}
& {$ {\displaystyle \frac{du}{dx}}(0) $}
\\[1.5ex] \hline
$1.4215$ & $-0.80814$ & 1.42154 & $ -0.808144 $ & $1.421544$ & $-0.808145$ \\
\hline
\epsilonnd{tabular}}
\label{tab:pile:comp}
\epsilonnd{center}
\epsilonnd{table}
As far as the missing initial conditions for the pile problem are
concerned, a comparison of the obtained values has been considered in table \ref{tab:pile:comp}.
It is easily seen that our results are in good agreement with those available in literature.
Lentini and Keller \cite{Lentini:BVP:1980} used the mentioned asymptotic boundary conditions and employed PASVAR, a routine based upon the
trapezoidal difference scheme with automatic mesh refinement and
deferred corrections as described by Lentini and Pereyra \cite{Lentini:AFD:1977}.
That software is rather sophisticated because it
adjusts automatically the mesh and the order of accuracy of the method employed.
Fazio \cite{Fazio:1994:FSEb} used a free boundary formulation of the pile problem as mentioned in the introduction and the Keller's box finite difference method.
\subsection{Improving the accuracy via Richardson's extrapolation}
There are two possible strategies for improving the accuracy of the obtained numerical results.
The first one is to study higher order finite difference schemes on quasi-uniform grids, whereas the second one is to apply Richardson's extrapolation.
As we shall see shortly the simplest of these two strategies is to apply Richardson's extrapolation, and we will explain in full details its application to the numerical results obtained by our scheme on a quasi-uniform grid.
However, let us indicate first the steps necessary to develop higher order schemes.
First of all, we have to define the values of $u(x)$ on the mid-points of the grid using a wider stencil
\begin{equation}
u_{n+1/2} \approx \alpha_{-1} u_{n-1} + \alpha_0 u_n + \alpha_1 u_{n+1} +\alpha_2 u_{n+2} \ ,
\label{eq:u4ord}
\epsilonnd{equation}
where $\alpha_j$ for $j = -1, 0, 1, 2$ are constants to be determined.
To get the coefficients in (\ref{eq:u4ord}) we require that the formula is exact for constant, linear, quadratic and cubic functions.
In this way we end up with the linear system of four equations in four unknowns $V\alpha = d$, where
\begin{equation}\label{eq:matrix}
V =
\begin{bmatrix}
1 & 1 & 1 & 1 \\
x_{n-1} & x_{n} & x_{n+1} & x_{n+2} \\
x^2_{n-1} & x^2_{n} & x^2_{n+1} & x^2_{n+2} \\
x^3_{n-1} & x^3_{n} & x^3_{n+1} & x^3_{n+2}
\epsilonnd{bmatrix}
\epsilonnd{equation}
$\alpha = [\alpha_{-1},\alpha_0, \alpha_{1}, \alpha_2]^T$ and $d = [1,x_{n+1/2}, x^2_{n+1/2}, x^3_{n+1/2}]^T$.
We notice that the grid-points are all distinct, hence the coefficient matrix is a Vandermonde matrix and the solution of this system exists and is unique.
However, Vandermonde systems might be ill-conditioned, see Gautschi \cite{Gautschi:IVC:1962,Gautschi:NEI:1975} and his review paper \cite{Gautschi:HUA:1990}.
In fact, the condition number of a Vandermonde matrix may be large, causing large errors when computing the solution of the system numerically.
As far as the higher order finite difference formula for the first derivative is concerned, we have to set, again, a wider stencil
\begin{equation}
\frac{du}{dx}(x_{n+1/2}) \approx \beta_{-1} u_{n-1} + \beta_0 u_n + \beta_1 u_{n+1} +\beta_2 u_{n+2} \ ,
\label{eq:du4ord}
\epsilonnd{equation}
where $\beta_j$ for $j = -1, 0, 1, 2$ are constants to be determined.
Once again, in order to find the coefficients in (\ref{eq:du4ord}) we require that the formula is exact for constant, linear, quadratic and cubic functions.
In this way we end up with the linear system of four equations in four unknowns $V\beta = r$, where $V$ is the same matrix defined above in (\ref{eq:matrix}),
$\beta = [\beta_{-1},\beta_0, \beta_{1}, \beta_2]^T$ and $r = [0,1,2 x_{n+1/2}, 3 x^2_{n+1/2}]^T$.
Of course, both systems can be solved exactly, for instance by using a general purpose Computer Algebra system, like the free software AXIOM \cite{AXIOM} or Derive \cite{Derive}.
However, the application of higher order finite difference schemes is not straightforward because we have to get and implement also boundary finite difference formulas of the same order.
Since the computational stencil is wider than in the second order case we need to introduce ghost cells at the boundary and this means that we need a ghost cell greater than infinity.
If we apply lower order boundary conditions, then a reduction of the overall accuracy results, see Fazio and Russo \cite{Fazio:2010:LCS} for numerical results related to the numerical implementation of higher order boundary conditions in the study the dynamics of two gases in a piston problem.
At this stage we prefer to indicate a different strategy to get higher accuracy.
In fact, this is a further advantage in using a family of quasi-uniform grids in calculations.
The algorithm is based on Richardson's extrapolation, introduced by Richardson in \cite{Richardson:1927:DAL}, and it is the same for many
finite difference methods: for numerical differentiation or integration, solving systems of ordinary or partial differential equations.
To apply Richardson's extrapolation, we carry on two calculations on embedded uniform or quasi-uniform grids with total number of nodes $N$ and $2N$.
All nodes of largest steps are identical to even nodes of denser grid due to uniformity.
Let us suppose that we use a numerical method with order of accuracy $p$ to find an approximation of a scalar value $T$.
For smooth enough solutions the error can be decomposed into a sum of inverse powers of $N$.
The Richardson's formula
\begin{equation}
R_{2N} = \frac{T_{2N}-T_N}{2^p-1}
\label{eq:Retra}
\epsilonnd{equation}
defines the main term of such sum.
This formula is asymptotically exact in the limit as $N$ goes to infinity if we use uniform or quasi-uniform grids.
Hence, it gives the real value of numerical solution error without knowledge of exact solution.
We can apply (\ref{eq:Retra}) as a single-step correction formula
\begin{equation}
T = T_{2N} + R_{2N} + O(N^{-p-s})
\label{eq:conextra}
\epsilonnd{equation}
and increase the order of accuracy of our approximation.
In general, we have $s = 2$ in (\ref{eq:conextra}) for a symmetrical finite difference scheme, but only $s = 1$ for non-symmetrical ones.
Such enlargement of accuracy requires to solve the same problem twice and only few further arithmetical operations and so it is very cheap.
It could be possible to apply Richardson's extrapolation to the results reported in table \ref{tab:Falkner} and in table \ref{tab:pile}.
This has been done in table \ref{tab:Falkner:extra} and in tables \ref{tab:pile:extra1} and \ref{tab:pile:extra2}, respectively.
\begin{table}[!hbt]
\caption{\it Richardson's extrapolation for the first line of table \ref{tab:Falkner}.}
\begin{center}
{\begin{tabular}{rcc}
\hline \\[-1.5ex]
$N$ & $T^{(0)}$ & $T^{(1)}$ \\[1.5ex]
\hline
40 & $1.234124$ & \\
80 & $1.232972$ & $ 1.232588 $ \\
160 & $1.232684$ & $ 1.232588 $ \\
\hline
\epsilonnd {tabular}}
\epsilonnd{center}
\label{tab:Falkner:extra}
\epsilonnd{table}
\begin{table}[!hbt]
\caption{\it Richardson's extrapolation of $u(0)$ for the first line of table \ref{tab:pile}.}
\begin{center}
{\begin{tabular}{rccc}
\hline \\[-1.5ex]
$N$ & $T^{(0)}$ & $T^{(1)}$ & $T^{(2)}$ \\[1.5ex]
\hline
40 & $1.421243$ & & \\
80 & $1.421469$ & $1.421544$ & \\
160 & $1.421526$ & $1.421545$ & $1.421545$ \\
\hline
\epsilonnd {tabular}}
\epsilonnd{center}
\label{tab:pile:extra1}
\epsilonnd{table}
\begin{table}[!hbt]
\caption{\it Richardson's extrapolation of $\displaystyle{\frac{du}{dx}}(0)$ for the first line of table \ref{tab:pile}.}
\begin{center}
{\begin{tabular}{rccc}
\hline \\[-1.5ex]
$N$ & $T^{(0)}$ & $T^{(1)}$ & $T^{(2)}$ \\[1.5ex]
\hline
40 & $-0.807934$ & $ $ & \\
80 & $-0.808094$ & $-0.808147$ & \\
160 & $-0.808135$ & $-0.808149$ & $-0.808149$\\
\hline
\epsilonnd {tabular}}
\epsilonnd{center}
\label{tab:pile:extra2}
\epsilonnd{table}
In this way we can avoid to solve the given problem with a large number of grid-points.
A simple extrapolation improves the numerical accuracy obtained for the results to a given problem.
Of course, it is also possible to iterate the extrapolation until $T^{(k)}_{2N} = T^{(k)}_{N}$, for $k = 1,2,\dots$, as in table \ref{tab:Falkner:extra}.
We also stop to apply a nested extrapolation as soon as $T^{(k)}_{N} = T^{(k-1)}_N$ as in tables \ref{tab:pile:extra1} and \ref{tab:pile:extra2}.
Starting with the computed data $T^{(0)}_{}$, we proceed with the extrapolated values
\begin{equation}
T^{(k)} = \frac{2^{k+1} T^{(k-1)}_{2N}-T^{(k-1)}_N}{2^{k+1}-1} \ ,
\label{eq:stpRetra1}
\epsilonnd{equation}
for $k = 1, 2, \dots $.
The extrapolated values reported in tables \ref{tab:Falkner:extra}, \ref{tab:pile:extra1} and \ref{tab:pile:extra2} can be compared with the corresponding values
listed in tables \ref{tab:Falkner} and \ref{tab:pile}, respectively.
It is clear that to get accurate numerical results, for both the considered problems, we can apply few extrapolations with nested quasi-uniform grids involving a small number of grid points.
\section{Concluding remarks}
The numerical results for the test problems reported in the previous sections show that non-standard finite difference schemes on quasi-uniform grids are an effective way to solve BVPs defined on infinite intervals.
The application of non-standard finite difference schemes on quasi-uniform grids overcomes the need for a priori definition of the truncated boundary.
For both examples we used the logarithmic map (\ref{eq:qu1}) because in both cases the largest variation of the solution components occur near the origin.
Let us discuss, at the end of this work, a possible way to extend
the non-standard finite difference schemes on quasi-uniform grids to the numerical solutions of problems
defined on the whole real line, for instance, the connecting orbits
problems mentioned in the introduction.
For these problems, all boundary conditions are imposed at plus or minus infinity.
In such a case it is possible to use the tangential quasi-uniform grid
\[
x_n = c \cdot \tan\left(\frac{n \pi}{2N}\right) \ ,
\]
where $c > 0$ is a control parameter.
In fact, if $ n = -N,-N+1,\dots -1, 0, 1, \dots , N-1, N$, then this tangential grid cover the whole infinite line, and in particular we have that $x_{-N} = -\infty$.
\noindent {\bf Acknowledgement.} {The research of this work was
supported, in part, by the University of Messina and by the GNCS of INDAM.}
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\begin{document}
\setcounter{page}{1} \thispagestyle{empty}
\begin{abstract}
We use localization method to understand the rational equivariant cohomology rings of real Grassmannians and oriented Grassmannians, then relate this to the Leray-Borel description which says the ring generators are equivariant Pontryagin classes, Euler classes in even dimension, and one more new type of classes in odd dimension, as stated by Casian and Kodama. We give additive basis in terms of equivariant characteristic polynomials and equivariant Schubert/canonical classes. We also calculate Poincar\'e series, equivariant Littlewood-Richardson coefficients and equivariant characteristic numbers. Since all these Grassmannians with torus actions are equivariantly formal, many results for equivariant cohomology have similar statements for ordinary cohomology.
\end{abstract}
\title{Localization of equivariant cohomology rings of real Grassmannians}
\section{Introduction}
\vskip 15pt
The study of homology and cohomology of real and complex Grassmannian was initially based on Ehresmann's work \cite{Eh37} in terms of Schubert cells. The relations between Schubert classes and characteristic classes were given by Chern \cite{Ch48} for real Grassmannians in $\mathbb{Z}/2$ coefficients. Using the techniques of spectral sequences of fibre bundles, Leray \cite{Le46,Le49} and Borel \cite{Bo53A,Bo53B} gave a unified way to describe the cohomology rings of certain homogeneous spaces in terms of characteristic classes.
These pioneering work applies not only to ordinary cohomology but also to equivariant cohomology. For compact connected Lie group $G$ and its connected closed subgroup $H$ of the same rank, the maximal torus $T$ acts from the left of the homogeneous space $G/H$ with the Leray-Borel description of its equivariant cohomology in $\mathbb{Q}$ coefficients:
\[
H^*_T(G/H)=\mathbb{S}\mathfrak{t}^*\otimes_{(\mathbb{S}\mathfrak{t}^*)^{W_G}} (\mathbb{S}\mathfrak{t}^*)^{W_H}
\]
where $\mathbb{S}\mathfrak{t}^*$ is the symmetric algebra of the dual Lie algebra $\mathfrak{t}^*$, and $W_G,W_H$ are the Weyl groups of $G$ and $H$.
For example, if we consider complex flag varieties as quotients of unitary groups $U(n)$, then the Weyl groups are products of symmetric groups and the Weyl group invariants $(\mathbb{S}\mathfrak{t}^*)^{W_G}, (\mathbb{S}\mathfrak{t}^*)^{W_H}$ consist of symmetric polynomials in appropriate variables. In topological terminology, these invariants are polynomials of equivariant Chern classes of canonical bundles, with relations from Whitney product formula.
Similarly, for even dimensional oriented Grassmannians viewed as quotients of $SO(n)$, whose Weyl groups act by permutations and sign changes on polynomials in appropriate variables, the Leray-Borel description means theirs equivariant cohomology rings are generated by equivariant Pontryagin classes and equivariant Euler classes of canonical bundles and complementary bundles, with relations from Whitney product formula and square of Euler classes as top Pontryagin classes. Notice that oriented Grassmannians are natural $2$-fold covers of real Grassmannians, then we can identify the equivariant cohomology of real Grassmannians as $\mathbb{Z}/2$-invariant of the equivariant cohomology of oriented Grassmannians. Due to the lack of preferred orientations for subspaces in $\mathbb{R}^n$, there is no Euler class of canonical bundle or complementary bundle over real Grassmannians. These facts give even dimensional real Grassmannians their Leray-Borel description of equivariant cohomology rings generated by equivariant Pontryagin classes, with relations from Whitney product formula. This special case of Leray-Borel description for real Grassmannians are stated in Casian\&Kodama \cite{CK}. For odd dimensional oriented Grassmannians $SO(2n+2)/SO(2k+1)\times SO(2n-2k+1)$ in which $SO(2n+2)$ and $SO(2k+1)\times SO(2n-2k+1)$ have different ranks of maximal tori, similar Leray-Borel description was obtained by Takeuchi \cite{Ta62}.
In this paper, we will use localization methods to understand the rational equivariant cohomology rings of real and oriented Grassmannians and re-derive the Leray-Borel description. We will give additive basis both in characteristic polynomials and in canonical classes, compute Poincar\'e series and characteristic numbers, and try to relate these results with Schubert calculus on real Grassmannians.
Alternatively, Sadykov \cite{Sa17}, Carlson \cite{Carl} and He \cite{HeB} have given short derivations of the Leray-Borel-Takeuchi descriptions of the rational cohomology rings of real and oriented Grassmannians.
\textbf{Acknowledgement} This work was part of the author's PhD thesis. The author would like to thank Victor Guillemin and Jonathan Weitsman for guidance. The author would also like to thank Jeffrey Carlson for many useful discussions and for the references of Leray and Takeuchi.
\vskip 20pt
\section{Torus actions, equivariant cohomology}
\vskip 15pt
In this section, we recall some basics of equivariant cohomology for torus actions.
\subsection{Torus actions and isotropy weights at fixed points}\label{subsec:T-action}
Throughout the paper, a manifold $M$ is always assumed to be smooth, compact, but not necessarily oriented nor connected. Let torus $T$ act on a manifold $M$, we will denote $M^T$ as the fixed-point set. For any point $p$ in a connected component $C$ of $M^T$, there is the \textbf{isotropy representation} of $T$ on the tangent space $T_p M$, which splits into weighted spaces $T_p M = V_0 \oplus V_{[\alpha_1]} \oplus \cdots \oplus V_{[\alpha_r]}$ where the non-zero distinct weights $[\alpha_1],\ldots,[\alpha_r] \in \mathfrak{t}^*_\mathbb{Z}/{\pm 1}$ are determined only up to signs (If $M$ has a $T$-invariant stable almost complex structure, then those weights are determined without ambiguity of signs). Comparing with the tangent-normal splitting $T_p M = T_p C \oplus N_p C$, we get that $T_p C = V_0$ and $N_p C = V_{[\alpha_1]} \oplus \cdots \oplus V_{[\alpha_r]}$. Since $N_p C = V_{[\alpha_1]} \oplus \cdots \oplus V_{[\alpha_r]}$ is of even dimension, the dimensions of $M$ and of the components of $M^T$ will be of the same parity. If $\mathrm{dim}\, M$ is even, the smallest possible components of $M^T$ could be isolated points. If $\mathrm{dim}\, M$ is odd, the smallest possible components of $M^T$ could be isolated circles. Moreover, since $T$ acts on the normal space $N_p C$ by rotation, this gives the normal space $N_p C$ an orientation.
For any subtorus $K$ of $T$, we get two more actions automatically: the \textbf{sub-action} of $K$ on $M$ and the \textbf{residual action} of $T/K$ on $M^K$.
\subsection{Equivariant cohomology}
Let torus $T=(S^1)^n$ act on a manifold $M$. The $T$-equivariant cohomology of $M$ is defined using the Borel construction $H^*_{T} (M) = H^*((ET\times M) / T)$, where $ET=(S^\infty)^{n}$ with $BT=ET/T=(\mathbb{C} P^\infty)^{n}$ and the coefficient ring will usually be $\mathbb{Q}$ throughout the paper, unless otherwise mentioned to be $\mathbb{Z}$. By this definition, if we denote $\mathfrak{t}^*$ as the dual Lie algebra of $T$, then $H^*_T (pt)=H^*(ET / T)=H^*((\mathbb{C} P^\infty)^{n})=\mathbb{S}\mathfrak{t}^*$ is a polynomial ring $\mathbb{Q}[\alpha_1,\dots,\alpha_n]$ or $\mathbb{Z}[\alpha_1,\ldots,\alpha_n]$ under the identification that $c_1(\gamma_i\rightarrow BT)=\alpha_i$, where $\gamma_i\rightarrow BT$ is the canonical complex line bundle on the $i$-th $\mathbb{C} P^\infty$-component of $BT$ and $\alpha_i \in \mathfrak{t}^*_\mathbb{Z}$ is the integral weight dual to the $i$-th $S^1$-component of $T$. The trivial map $\iota: M \rightarrow pt$ induces a homomorphism $\iota^*: H^*_T(pt)\rightarrow H^*_T(M)$ and hence makes $H^*_T(M)$ a $H^*_T (pt)$-module.
For a $T$-equivariant complex, oriented or real vector bundle $V$ over $M$, we can define the equivariant Chern, Euler or Pontryagin classes respectively as $c^T(V)=c((ET\times V) / T)\,,e^T(V)=e((ET\times V) / T)\,,p^T(V)=p((ET\times V) / T)\in H^*_T(M,\mathbb{Z})$.
The famous Atiyah-Bott-Berline-Vergne(ABBV) localization formula says:
\begin{thm}[ABBV Localization Formula, \cite{BV83,AB84}]\label{ABBV}
On an oriented $T$-manifold $M$, an equivariant cohomology class $\omega \in H^*_T (M)$ can be integrated as
\[
\int_M \omega = \sum_{C \subseteq M^T} \int_C \frac{\omega|_C}{e^T(NC)}
\]
where the summation is taken for every component $C \subseteq M^T$ with normal bundle $NC$ and equivariant Euler class $e^T(NC)$.
\end{thm}
Inspired by this localization theorem, one can hope for more connections between the manifold $M$ and its fixed-point set $M^T$, if $H^*_T(M)$ is actually a free $H^*_T (pt)$-module.
\begin{dfn}
An action of $T$ on $M$ is \textbf{equivariantly formal} if $H^*_T (M)$ is a free $H^*_T (pt)$-module.
\end{dfn}
Using the techniques of spectral sequences, equivariant formality has various equivalent expressions.
\begin{thm}[Equivalences of equivariant formality, \cite{AP93} pp.\,210 Thm\,3.10.4, \cite{GGK02} pp.\,206-207] \label{thm:formal}
Let torus $T$ act on a manifold $M$, the following conditions about equivariant cohomology are equivalent:
\begin{enumerate}
\item The $T$-action is equivariantly formal, i.e. $H^*_T (M)$ is a free $H^*_T (pt)$-module
\item The Leray-Serre sequence of the fibration $M \hookrightarrow (M \times ET) /T \rightarrow BT$ collapses with $E_\infty = E_2 = H^*(BT)\otimes H^*(M)$
\item $H^*_T(M)\cong H^*_T (pt)\otimes H^*(M)$ as $H^*_T (pt)$-module
\item $H^*_T(M)\rightarrow H^*(M)$, defined as the restriction to the fibre $M$, is surjective
\item Any additive basis of $H^*(M)$ can be lifted to $H^*_T(M)$, hence give an additive $H^*_T (pt)$-basis for $H^*_T(M)$
\item $\sum \mathrm{dim}\,H^*(M^T) = \sum \mathrm{dim}\,H^*(M)$.
\end{enumerate}
\end{thm}
\begin{rmk}
The equivalences among (2)(3)(4)(5) are direct applications of the Leray-Hirsch theorem (which works not only for $\mathbb{Q}$ coefficients, but also for $\mathbb{Z}$ coefficients if $H^*(M,\mathbb{Z})$ is a free $\mathbb{Z}$-module). For the equivalence to the remaining conditions (1)(6) in $\mathbb{Q}$ coefficients, see the cited references.
\end{rmk}
\begin{rmk}
When the Betti numbers of $M$ and $M^T$ are known, the equality $\sum \mathrm{dim}\,H^*(M^T) = \sum \mathrm{dim}\,H^*(M)$ is a handy way to verify the equivariant formality.
\end{rmk}
\begin{rmk}
The fibre inclusion $M \hookrightarrow (M \times ET) /T$ induces a homomorphism $H^*_T(M) \rightarrow H^*(M)$ factoring through $\mathbb{Q}\otimes_{H^*_T (pt)} H^*_T(M)$, where $\mathbb{Q}$ has a $H^*_T (pt)=\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra structure from the constant-term morphism $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]\rightarrow \mathbb{Q}: f(\alpha_1,\ldots,\alpha_n)\mapsto f(0)$. When the $T$-action is equivariantly formal, i.e. $H^*_T(M)\cong H^*_T (pt)\otimes_\mathbb{Q} H^*(M)$, then we can recover $H^*(M)$ as $\mathbb{Q}\otimes_{H^*_T (pt)} H^*_T(M)$.
\end{rmk}
\vskip 20pt
\section{GKM-type theorems}
\vskip 15pt
In this section, we recall the Chang-Skjelbred lemma, the GKM theorem and the author's recent work on GKM-type theorems for possibly non-orientable manifolds in odd dimensions.
\subsection{Chang-Skjelbred lemma and GKM condition}
Given an action $T\curvearrowright M$, for every point $p\in M$, its stabilizer is defined as $T_p=\{t \in T \mid t\cdot p = p\}$, and its orbit is $\mathcal{O}_p\cong T/T_p$. Let's set the $1$-skeleton $M_1=\{p \mid \textup{dim}\,\mathcal{O}_p\leq 1\}$, the union of $1$-dimensional orbits and fixed points. If $H^*_T (M)$ is a free $H^*_T (pt)$-module, i.e. the action is equivariantly formal, Chang and Skjelbred \cite{CS74} proved that $H^*_T (M)$ only depends on the fixed-point set $M^T$ and the $1$-skeleton $M_1$.
\begin{thm}[Chang-Skjelbred Lemma, \cite{CS74}]\label{Chang}
If an action $T\curvearrowright M$ is equivariantly formal, then
\[
H^*_T(M) \cong H^*_T(M_1)\hookrightarrow H^*_T(M^T).
\]
\end{thm}
This Lemma enables one to describe the equivariant cohomology $H^*_T(M)$ as a sub-ring of $H^*_T(M^T)$, subject to certain algebraic relations determined by the $1$-skeleton $M_1$. To apply the Chang-Skjelbred Lemma, we will follow Goresky, Kottwitz and MacPherson's idea to start with the smallest fixed-point set $M^T$ and $1$-skeleton $M_1$.
\begin{dfn}[GKM condition in either even or odd dimensions]\label{GKMCond}
An action $T\curvearrowright M$ is \textbf{GKM} if
\begin{itemize}
\item[(1)] The fixed-point set $M^T$ consists of isolated points or isolated circles.
\item[(2)] At each fixed point $p\in M^T$, the non-zero weights $[\alpha_1],\ldots,[\alpha_n] \in \mathfrak{t}^*_\mathbb{Z}/{\pm 1}$ of the isotropy $T$-representation $T\curvearrowright T_pM$ are pair-wise independent.
\end{itemize}
\end{dfn}
\begin{rmk}
As mentioned in the Subsection\,\ref{subsec:T-action}, the dimensions of $M$ and of $M^T$ have the same parity. Condition\,(1) in Definition\,\ref{GKMCond} means that $M^T$ consists of isolated points when $M$ is even dimensional or $M^T$ consists of isolated circles when $M$ is odd dimensional.
\end{rmk}
\begin{rmk}
The GKM condition is equivalent to requiring the $1$-skeleton $M_1$ to be 2-dimensional when $M$ is even dimensional or 3-dimensional when $M$ is odd dimensional.
\end{rmk}
\subsection{GKM theorem: the even dimensional case}
Goresky, Kottwitz and MacPherson \cite{GKM98} considered torus actions on algebraic varieties where the fixed-point set $M^T$ is finite and the $1$-skeleton $M_1$ is a union of spheres $S^2$. They proved that the cohomology $H^*_T(M)$ can be described in terms of congruence relations on a graph determined by the $1$-skeleton $M_1$. Goertsches and Mare \cite{GM14} observed that GKM theory also works in non-orientable case by adding $\mathbb{R} P_{[\alpha]}^2$ components to the $1$-skeleton $M_1$.
\begin{dfn}[GKM graph in even dimension]
If an action $T\curvearrowright M^{2m}$ is GKM, then its \textbf{GKM graph} consists of
\begin{description}
\item[Vertices] A $\bullet$ for each fixed point in $M^T$
\item[Edges$\,\&\,$Weights] A solid edge with weight $[\alpha]$ for each $S^1_{[\alpha]}$ joining two $\bullet$'s representing its two fixed points, and a dotted edge with weight $[\beta]$ for each $\mathbb{R} P^2_{[\beta]}$ joining a $\bullet$ to an empty vertex.
\end{description}
\end{dfn}
\begin{thm}[GKM theorem in even dimension, \cite{GKM98} pp.\,26 Thm\,1.2.2, \cite{GM14} pp.\,7 Thm\,3.6]\label{thm:EvenGKM}
If the action of a torus $T$ on a (possibly non-orientable) manifold $M^{2m}$ is equivariantly formal and GKM, then we can construct its GKM graph $\Gamma$, with vertex set $V=M^T$ and weighted edge set $E$, moreover the equivariant cohomology has a graphic description
\[
H^*_T(M) = \big\{ f: V\rightarrow \mathbb{S}\mathfrak{t}^* \mid f_p \equiv f_q \mod{\alpha} \quad \mbox{for each solid edge $\overline{pq}$ with weight $\alpha$
in $E$}\big\}.
\]
\end{thm}
\begin{rmk}
Since $H^*_T(\mathbb{R} P_{[\beta]}^2)=H^*_T(pt)$ does not contribute to congruence relations, we can erase all the dotted edges of $\mathbb{R} P_{[\beta]}^2$ and only keep the solid edges of $S^2_{[\alpha]}$ to construct an \textbf{effective} GKM graph, which does not necessarily have the same number of edges for every vertex.
\end{rmk}
\subsection{GKM-type theorem: the odd dimensional case}
In odd dimensions, we can also consider smallest-dimensional $1$-skeleton, i.e. $M_1$ is $3$-dimensional. Then according to Chang-Skjelbred lemma, the localization boils down to understanding $S^1$-actions on $3$-manifolds, studied by the author \cite{He17}.
Similar to the even dimensional case, we can construct a graph for each odd dimensional $T$-manifold under the GKM condition\,\ref{GKMCond}.
\begin{dfn}[GKM graph in odd dimension]
If an action $T\curvearrowright M^{2m+1}$ (possibly non-orientable) is GKM, then its \textbf{GKM graph} consists of
\begin{description}
\item[Vertices] There will be two types of vertices.
\begin{itemize}
\item [$\circ$] for each fixed circle $C \subset M^T$.
\item [$\Box$] for each 3d connected component $N^3_{[\alpha]}$ in $M^{T_{[\alpha]}}$ of some codimension-$1$ subtorus $T_{[\alpha]}$ which has Lie algebra $\mathfrak{t_\alpha}\subset \mathfrak{t}$ annihilated by $\alpha$. The $\Box$ is then weighted with $[\alpha]$.
\end{itemize}
\item[Edges] An edge joins a $(\Box,N)$ to a $(\circ,C)$, if the 3d manifold $N$ contains the fixed circle $C$ and hence is a connected component of $M^{T_{[\alpha]}}$ for an isotropy weight $[\alpha]$ of $C$. There are no edges directly joining $\circ$ to $\circ$, nor $\Box$ to $\Box$.
\end{description}
\end{dfn}
In order to derive a GKM-type graphic description of $H_{T}^*(M^{2m+1})$, we need to fix in advance an orientation $\theta_i$ for each fixed circle $C_i \subseteq M^T$, and also fix an orientation for each orientable $M^{T_{[\alpha]}} \subseteq M_1$.
\begin{thm}[A GKM-type theorem in odd dimension, \cite{HeA}]\label{thm:OddGKM}
If the action of a torus $T$ on (possibly non-orientable) manifold $M^{2n+1}$ is equivariantly formal and GKM, then we can construct its GKM graph $\Gamma$, with two types of vertex sets $V_\circ$ and $V_\Box$ and edge set $E$. An element of the equivariant cohomology $H^*_T(M)$ can be written as:
\[
(P,Q\theta): V_\circ \longrightarrow \mathbb{S}\mathfrak{t}^* \oplus \mathbb{S}\mathfrak{t}^* \theta
\]
where $\theta$ is the generator of $H^1(S^1)$,
under the relations that for each $\Box$ representing a 3d component $N$ of some $M^{T_{[\alpha]}}$ and the neighbour $\circ$'s representing the fixed circles $C_1,\ldots,C_k$ on this component,
\begin{itemize}
\item if $N$ is non-orientable,
\begin{equation*}
P_{C_1}\equiv P_{C_2}\equiv \cdots\equiv P_{C_k} \mod{\alpha}
\end{equation*}
\item if $N$ is orientable,
\begin{equation*}
P_{C_1}\equiv P_{C_2}\equiv \cdots\equiv P_{C_k} \mbox{ and } \sum_{i=1}^k \pm Q_{C_i}\equiv 0 \mod{\alpha}
\end{equation*}
where the sign for each $Q_{C_i}$ is specified by comparing the prechosen orientation $\theta_i$ with the induced orientation of $N$ on $C_i$.
\end{itemize}
\end{thm}
\begin{rmk}
As discussed in \cite{HeA}, different choices of orientations from $C_i \subseteq M^T$ and from orientable $M^{T_{[\alpha]}} \subseteq M_1$ give the isomorphic equivariant cohomology. When $M$ has a $T$-invariant stable almost complex structure, then the isotropy weights $\alpha$ can be determined without ambiguity of signs. Moreover, $M^T$ and $M^{T_{\alpha}} \subseteq M_1$ are equipped with induced stable almost complex structures, hence are oriented canonically.
\end{rmk}
\begin{rmk}
To describe the $\mathbb{S}\mathfrak{t}^*$-algebra structure of $H^*_T(M^{2m+1})$, it is convenient to write an element $(P,Q\theta)$ as $(P_C+Q_C\theta)_{C\subset M^T}$. Note $\theta^2=0$, then $(P_C+Q_C\theta)_{C\subset M^T} + (\bar{P}_C+\bar{Q}_C\theta)_{C\subset M^T}=([P_C+\bar{P}_C]+[Q_C+\bar{Q}_C]\theta)_{C\subset M^T}$, and $(P_C+Q_C\theta)_{C\subset M^T} \mathbb{C}dot (\bar{P}_C+\bar{Q}_C\theta)_{C\subset M^T}=([P_C \bar{P}_C]+[P_C\bar{Q}_C+\bar{P}_C Q_C]\theta)_{C\subset M^T}$. For any polynomial $R \in \mathbb{S}\mathfrak{t}^*$, we have $R \mathbb{C}dot (P_C+Q_C\theta)_{C\subset M^T} = (R P_C+R Q_C\theta)_{C\subset M^T}$.
\end{rmk}
\vskip 20pt
\section{Equivariant cohomology rings of complex Grassmannians}
\vskip 15pt
In this section, we recall the GKM description and Leray-Borel description of equivariant cohomology rings of complex Grassmannians, together with the characteristic basis and canonical basis of the additive structure. We use the notation $G_k(\mathbb{C}^n)$ for the Grassmannian of $k$-dimensional complex subspaces in $\mathbb{C}^n$.
\subsection{GKM description of complex Grassmannians}
As shown by Guillemin, Holm and Zara \cite{GHZ06}, for compact connected group $G$ and its closed connected subgroup $H$ of the same rank as $G$, the homogeneous space $G/H$ is GKM and equivariantly formal under the left action of maximal torus $T$, hence has a graphic description for its equivariant cohomology. For example, the GKM graph of $T^n \curvearrowright U(n)/(U(k)\times U(n-k))$ is the Johnson graph $J(n,k)$, of which each vertex is a $k$-element subset $S \subseteq \{1,\ldots,n\}$ and two vertices $S,S'$ are joined by an edge if they differ by one element. For later use in the case of real Grassmannians, we will give an explicit description for the $1$-skeleta of complex Grassmannians.
\begin{prop}[$1$-skeleta of complex Grassmannians]
The $T^n$-action on $G_k(\mathbb{C}^n)$ has $\binom{n}{k}$ fixed points of the form $\oplus_{i\in S} \mathbb{C}_i$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$ and $\mathbb{C}_i$ is the $i$-th component of $\mathbb{C}^n$. The isotropy weights at $\oplus_{i\in S} \mathbb{C}_i$ are $\{\alpha_j-\alpha_i \mid i\in S,j\not \in S\}$, and join $\oplus_{i'\in S} \mathbb{C}_{i'}$ to $\oplus_{i'\in (S\smallsetminus\{i\})\cup \{j\}} \mathbb{C}_{i'}$ via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{C}_{i'})\oplus L \mid L \in \mathbb{P}(\mathbb{C}_i\oplus\mathbb{C}_j)\}\cong \mathbb{C} P^1$ in the $1$-skeleton.
\end{prop}
\begin{proof}
$T^n$ acts on $\mathbb{C}^n$ linearly by $(t_1,\ldots,t_n)\cdot (z_1,\ldots,z_n)=(t_1 z_1,\ldots,t_1 z_n)$ and hence induces an action on $G_k(\mathbb{C}^n)$ by mapping every $k$-dimensional subspace $V$ to $t\cdot V$ for each $t\in T^n$. A fixed point $V$ of the $T$-action on $G_k(\mathbb{C}^n)$ is exactly a $k$-dimensional sub-representation of $\mathbb{C}^n= \oplus_{i=1}^n \mathbb{C}_i$. Since $\oplus_{i=1}^n \mathbb{C}_i$ has distinct weights $\alpha_1,\ldots,\alpha_n$, a $k$-dimensional sub-representation is of the form $\oplus_{i\in S} \mathbb{C}_i$ for $S$, a $k$-element subset of $\{1,2,\ldots,n\}$.
To understand the isotropy weights at $\oplus_{i \in S} \mathbb{C}_i \in G_k(\mathbb{C}^n)$, notice that the tangent space of $G_k(\mathbb{C}^n)$ at $\oplus_{i \in S} \mathbb{C}_i$ is $Hom_\mathbb{C}(\oplus_{i \in S} \mathbb{C}_i, \oplus_{j \not \in S} \mathbb{C}_j)\cong (\oplus_{i \in S} \mathbb{C}_i)^* \otimes_\mathbb{C} (\oplus_{j \not \in S} \mathbb{C}_j)\cong\oplus_{i \in S} \oplus_{j \not \in S} (\mathbb{C}_i^* \otimes_\mathbb{C} \mathbb{C}_j)$ with pair-wise independent weights $\{\alpha_j-\alpha_i \mid i \in S \text{ and } j \not \in S\}$.
The finiteness of fixed points and pair-wise independence of isotropy weights at every fixed point verifies that the $T^n$ action on $G_k(\mathbb{C}^n)$ is GKM. Moreover, for a $k$-element subset $S\subseteq \{1,2,\ldots,n\}$ and a weight $\alpha_j-\alpha_i$ with $j\not \in S,\, i \in S$, the $2$-sphere
\[
\big\{(\underset{i'\in S\smallsetminus \{i\}}{\oplus} \mathbb{C}_{i'}) \oplus L \mid L \in \mathbb{P}(\mathbb{C}_i\oplus \mathbb{C}_j) \big\} \cong \mathbb{C} P^1
\]
connects the $T$-fixed point $\oplus_{i'\in S} \mathbb{C}_{i'}$ with the $T$-fixed point $\oplus_{i'\in (S\smallsetminus\{i\})\cup \{j\}} \mathbb{C}_{i'}$, and is fixed by the corank-$1$ subtorus torus $T_{\alpha_j-\alpha_i}$ whose Lie algebra is $\textup{Ker}(\alpha_j-\alpha_i)$.
\end{proof}
Since $G_k(\mathbb{C}^n)$ doesn't have odd-degree cells, the canonical $T^n$ action on $G_k(\mathbb{C}^n)$ is equivariantly formal in $\mathbb{Z}$ coefficients and of course in $\mathbb{Q}$ coefficients. Then we can apply the even dimensional GKM theorem to $G_k(\mathbb{C}^n)$ using congruence relations on the Johnson graph $J(n,k)$.
\begin{thm}[GKM description of complex Grassmannians, \cite{GZ01}]\label{thm:GKMcplxGrass}
Let $\mathcal{S}$ be the collection of $k$-element subsets of $\{1,2,\ldots,n\}$, then the equivariant cohomology of the $T^n$ action on $G_k(\mathbb{C}^n)$ is
\[
H^*_{T}(G_k(\mathbb{C}^n),\mathbb{Q})=\big\{f:\mathcal{S}\rightarrow \mathbb{Q}[\alpha_1,\ldots,\alpha_n] \mid f_{S} \equiv f_{S'} \mod \alpha_j-\alpha_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}\big\}.
\]
\end{thm}
Using Morse theory on graphs, Guillemin and Zara analysed additive basis for equivariant cohomology of GKM manifolds.
\begin{thm}[Canonical basis of complex Grassmannians, \cite{GZ03}]\label{thm:CplxSchub}
There is a self-indexing Morse function on $\mathcal{S}$
\[
\phi: \mathcal{S} \longrightarrow \mathbb{R} : S \longmapsto 2(\sum_{i\in S} i) -k(k+1)
\]
and a canonical class $\tau_S \in H^{\phi(S)}_{T}(G_k(\mathbb{C}^n),\mathbb{Q})$ for each $S\in \mathcal{S}$ such that
\begin{enumerate}
\item $\tau_S$ is supported upward, i.e. $\tau_S(S')=0$ if $\phi(S')\leq \phi(S)$
\item $\tau_S(S)=\prod' (\alpha_j - \alpha_i)$ where the product is taken over the weights at $S$ connecting to $S'$ with $\phi(S')<\phi(S)$
\end{enumerate}
Moreover, $\{\tau_S, S\in \mathcal{S}\}$ give a $H^*_{T}(pt,\mathbb{Q})$-additive basis of $H^*_{T}(G_k(\mathbb{C}^n),\mathbb{Q})$.
\end{thm}
\begin{rmk}
The canonical classes $\tau_S$ are exactly the equivariant Schubert classes via the relation that if $S$ consists of the elements $i_1<i_2<\cdots<i_k$, then the corresponding Schubert symbol is $(i_1-1,i_2-2,\ldots,i_k-k)$.
\end{rmk}
\begin{rmk}
In this paper, we only need the existence of canonical classes $\tau_S$. The general formula of $\tau_S$ restricted at each fixed point was given by Guillemin\&Zara \cite{GZ03} and simplified by Goldin\&Tolman \cite{GT09}.
\end{rmk}
\subsection{Leray-Borel description of complex Grassmannians}
Besides the equivariant Schubert basis, equivariant characteristic classes and characteristic polynomials on complex Grassmannians will give ring generators and additive basis for their equivariant cohomology rings.
\begin{thm}[Leray-Borel description of complex Grassmannians, see \cite{BT82} pp.\,293 Prop\,23.2]
For the complex Grassmannians $G_k(\mathbb{C}^n)$, let $c_1,c_2,\ldots,c_k$ and $\bar{c}_1,\bar{c}_2,\ldots,\bar{c}_{n-k}$ be the Chern classes of the canonical bundle on $G_k(\mathbb{C}^n)$ and its complementary bundle respectively, then
\[
H^*(G_k(\mathbb{C}^n),\mathbb{Z})=\frac{\mathbb{Z}[c_1,c_2,\ldots,c_k;\bar{c}_1,\bar{c}_2,\ldots,\bar{c}_{n-k}]}{(1+c_1+c_2+\ldots+c_k)(1+\bar{c}_1+\bar{c}_2+\ldots+\bar{c}_{n-k})=1}.
\]
\end{thm}
The relation $(1+c_1+c_2+\ldots+c_k)(1+\bar{c}_1+\bar{c}_2+\ldots+\bar{c}_{n-k})=1$ makes either $c_1,c_2,\ldots,c_k$ or $\bar{c}_1,\bar{c}_2,\ldots,\bar{c}_{n-k}$ as sets of ring generators of the cohomology $H^*(G_k(\mathbb{C}^n),\mathbb{Z})$. Certain monomials of $c_1,c_2,\ldots,c_k$ actually give an additive basis of the cohomology $H^*(G_k(\mathbb{C}^n),\mathbb{Z})$, stated by Carrell \cite{Ca78} for complex Grassmannians in $\mathbb{C}$ coefficients as a result of ``standard combinatorial reasoning''. Later, details of proof were supplied by Jaworowski \cite{Ja89} for real Grassmannians in $\mathbb{Z}/2$ coefficients which can be adapted to complex Grassmannians in $\mathbb{Z}$ coefficients.
\begin{thm}[Characteristic basis of complex Grassmannians, \cite{Ca78, Ja89}]
The set of monomials $c_1^{r_1}c_2^{r_2}\cdots c_k^{r_k}$ of cohomological degree $2d=\sum_{i=1}^{k} 2ir_i$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ forms an additive basis for $H^{2d}(G_k(\mathbb{C}^n),\mathbb{Z}),0\leq d\leq k(n-k)$.
\end{thm}
Notice that the cohomology $H^*(G_k(\mathbb{C}^n),\mathbb{Z})$ does not have odd-degree elements, the Leray-Serre sequence of $G_k(\mathbb{C}^n)\hookrightarrow ET\times_T G_k(\mathbb{C}^n) \rightarrow BT$ collapses at $E_2=H^*(BT,\mathbb{Z})\otimes_\mathbb{Z} H^*(G_k(\mathbb{C}^n),\mathbb{Z})$ with $H^*_T(G_k(\mathbb{C}^n),\mathbb{Z})\cong H^*(BT,\mathbb{Z})\otimes_\mathbb{Z} H^*(G_k(\mathbb{C}^n),\mathbb{Z})$ as $H^*(BT,\mathbb{Z})$-modules. Therefore, the action $T^n \curvearrowright G_k(\mathbb{C}^n)$ is equivariantly formal in $\mathbb{Z}$ coefficients, and of course in $\mathbb{Q}$ coefficients. However, $H^*_T(G_k(\mathbb{C}^n),\mathbb{Z})\cong H^*(BT,\mathbb{Z})\otimes_\mathbb{Z} H^*(G_k(\mathbb{C}^n),\mathbb{Z})$ is only a $H^*(BT,\mathbb{Z})$-module isomorphism which
\begin{enumerate}
\item neither gives the $H^*(BT,\mathbb{Z})$-algebra structure of $H^*_T(G_k(\mathbb{C}^n),\mathbb{Z})$.
\item nor specifies a map $H^*_T(G_k(\mathbb{C}^n),\mathbb{Z}) \rightarrow H^*(BT,\mathbb{Z})\otimes_\mathbb{Z} H^*(G_k(\mathbb{C}^n),\mathbb{Z})$.
\end{enumerate}
The above two problems can be resolved using the equivariant version of the Leray-Borel description which is usually given for a compact connected Lie group $G$ with maximal torus $T$ and a closed connected subgroup $H$ containing $T$ as
\[
H^*_T(G/H)=\mathbb{S}\mathfrak{t}^*\otimes_{(\mathbb{S}\mathfrak{t}^*)^{W_G}} (\mathbb{S}\mathfrak{t}^*)^{W_H}
\]
where $W_G$ and $W_H$ are the Weyl groups of $G$ and $H$. For the complex Grassmannian $G_k(\mathbb{C}^n)=U(n)/(U(k)\times U(n-k))$, we have the Weyl group $W_G = S_n$, the symmetric group of $n$ elements, and the Weyl group $W_H = S_k \times S_{n-k}$. Under these Weyl group actions, it is well known that the invariant elements in $\mathbb{S}\mathfrak{t}^*$ are symmetric polynomials, or topologically the equivariant Chern classes:
\begin{thm}[Equivariant Leray-Borel description of complex Grassmannians, \cite{Tu10} pp.\,21]
For the complex Grassmannian $G_k(\mathbb{C}^n)=U(n)/(U(k)\times U(n-k))$, let $T^n$ be the maximal torus of $U(n)$ which acts on the left of $G_k(\mathbb{C}^n)$, and $\alpha_1,\alpha_2,\ldots,\alpha_n$ be the integral basis for its Lie dual algebra $\mathfrak{t}^*$, also let $c^T_1,c^T_2,\ldots,c^T_k$ and $\bar{c}^T_1,\bar{c}^T_2,\ldots,\bar{c}^T_{n-k}$ be the equivariant Chern classes of the canonical bundle on $G_k(\mathbb{C}^n)$ and its complementary bundle respectively, then
\[
H^*_T(G_k(\mathbb{C}^n),\mathbb{Z})=\frac{\mathbb{Z}[\alpha_1,\alpha_2,\ldots,\alpha_n][c^T_1,c^T_2,\ldots,c^T_k;\bar{c}^T_1,\bar{c}^T_2,\ldots,\bar{c}^T_{n-k}]}{c^T\bar{c}^T = \prod_{i=1}^{n}(1+\alpha_i)}.
\]
\end{thm}
Since the equivariant Chern classes $c^T_1,c^T_2,\ldots,c^T_k$ and $\bar{c}^T_1,\bar{c}^T_2,\ldots,\bar{c}^T_{n-k}$ lift the ordinary Chern classes $c_1,c_2,\ldots,c_k$ and $\bar{c}_1,\bar{c}_2,\ldots,\bar{c}_{n-k}$, we get
\begin{thm}[Equivariant characteristic basis of complex Grassmannians]
The set of monomials $(c^T_1)^{r_1}(c^T_2)^{r_2}\cdots (c^T_k)^{r_k}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ form an additive $H^*_T(pt)$-basis for $H_T^*(G_k(\mathbb{C}^n),\mathbb{Z})$.
\end{thm}
\begin{proof}
Combine the ordinary characteristic basis with the equivalence (5) of Theorem \ref{thm:formal}.
\end{proof}
\subsection{Relations between the Leray-Borel and GKM descriptions}\label{subsec:BGKM}
Since the characteristic monomials $(c^T)^I=(c_1^T)^{i_1}\cdots(c_k^T)^{i_k},\,\sum_{j=1}^{k} i_j \leq n-k$ in Leray-Borel description and the canonical classes $\tau_{S}$ in GKM description are both basis for the free $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-module $H^*_{T}(G_k(\mathbb{C}^n),\mathbb{Q})$, there will be transformations $K,\bar{K}$ between them such that
\begin{align*}
(c^T)^I &= \sum_S K^I_S \tau_S\\
\tau_S &= \sum_I \bar{K}_I^S (c^T)^I
\end{align*}
where $K^I_S,\bar{K}_I^S \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$.
\begin{rmk}
The idea of considering the transformations between Schubert classes and characteristic classes dates back to Bernstein, Gelfand and Gelfand \cite{BGG73}. In this paper, we only need the existence of the transformations $K,\bar{K}$. For the complete flag manifold $Fl(\mathbb{C}^n)$, Kaji \cite{Ka} gave explicit algorithms on how to decide the polynomials $K^I_S,\bar{K}_I^S$. It would also be interesting to know what the $K^I_S,\bar{K}_I^S$ explicitly are for complex Grassmannians.
\end{rmk}
The Littlewood-Richardson rule for equivariant Schubert classes is that there are polynomials $N_{S,S'}^{S''}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ (see Knutson\&Tao \cite{KT03}) for the multiplication of Schubert classes
\[
\tau_{S}\tau_{S'} =\sum_{S''}N_{S,S'}^{S''}\tau_{S''}.
\]
On the other hand, the multiplication of the equivariant characteristic classes can be localized. We can express the total equivariant Chern classes $c^T,\bar{c}^T$ of the canonical bundle $\gamma$ and its complementary bundle $\bar{\gamma}$ in GKM description at each fixed point $\oplus_{i\in S} \mathbb{C}_i \in G_k(\mathbb{C}^n)$. Note that the canonical bundle $\gamma$, complementary bundle $\bar{\gamma}$ and tangent bundle $TG_k(\mathbb{C}^n)$ restricted at $\oplus_{i\in S} \mathbb{C}_i \in G_k(\mathbb{C}^n)$ for a $k$-element subset $S \subset \{1,\ldots,n\}$ are the vector spaces $\oplus_{i\in S} \mathbb{C}_i$, $\oplus_{j\not\in S} \mathbb{C}_j $ and $\oplus_{i \in S} \oplus_{j \not \in S} (\mathbb{C}_i^* \otimes_\mathbb{C} \mathbb{C}_j)$ respectively, we get
\begin{align*}
c^T|_S &= c^T(\gamma|_S)=\prod_{i\in S} (1+\alpha_i)\\
\bar{c}^T|_S &= c^T(\bar{\gamma}|_S)=\prod_{j\not\in S} (1+\alpha_j)\\
e^T|_S &=e^T(T_SG_k(\mathbb{C}^n))=\prod_{i\in S}\prod_{j\not\in S} (\alpha_j - \alpha_i).
\end{align*}
Since $\gamma\oplus \bar{\gamma} = \sum_{i=1}^{n}\mathbb{C}_i$, this also shows why there is the relation $c^T\bar{c}^T = \prod_{i=1}^{n}(1+\alpha_i)$.
If we denote $e_l(x_1,\ldots,x_m)$ as the $l$-th elementary symmetric polynomial in variables $x_1,\ldots,x_m$, then $c^T_l|_S = e_l(\alpha_{i\in S}), \bar{c}^T_l|_S = e_l(\alpha_{j\not\in S})$.
\begin{thm}[Equivariant Chern numbers of complex Grassmannians, \cite{Tu10} pp.\,21 Prop\,23]\label{thm:EquivChernNum}
Using the ABBV localization formula \ref{ABBV}, equivariant Chern numbers of complex Grassmannians can be given as
\begin{align*}
\int_{G_k(\mathbb{C}^n)} (c^T)^I
& = \sum_{S} \frac{((c_1^T)^{i_1}\cdots(c_k^T)^{i_k})|_S}{e^T_S} \\
& = \sum_{S} \frac{e^{i_1}_1(\alpha_{i\in S})\cdots e^{i_k}_k(\alpha_{i\in S})}{\prod_{i\in S}\prod_{j\not\in S} (\alpha_j - \alpha_i)}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n].
\end{align*}
where the sum is taken for all $k$-element subsets $S \subset \{1,\ldots,n\}$. When the cohomological degree of a characteristic polynomial matches with the dimension of a Grassmannian, then its equivariant integration results in a constant, i.e. an ordinary Chern number.
\end{thm}
\begin{rmk}
By substituting any $\alpha_i=a_i \in \mathbb{R}$ such that $a_i\not =0, a_i \not = a_j$ into the above fraction, we can evaluate the equivariant Chern numbers. For example, a good choice will be $\alpha_i=i,\forall i$. If the cohomological degree of a characteristic polynomial does not match with the dimension of a Grassmannian, then such evaluation will be zero. The interesting case is when the degree matches the dimension.
\end{rmk}
\begin{cor}\label{thm:OrdChernNum}
When the cohomological degree of a characteristic polynomial matches with the dimension of a Grassmannian, we then get a formula for the ordinary Chern numbers :
\[
\int_{G_k(\mathbb{C}^n)} c^I = \sum_{S} \frac{e^{i_1}_1(S)\cdots e^{i_k}_k(S)}{\prod_{i\in S}\prod_{j\not\in S} (j - i)}\in \mathbb{Q}
\]
where the sum is taken for all $k$-element subsets $S \subset \{1,\ldots,n\}$.
\end{cor}
\begin{rmk}
The above characteristic numbers are with respect to the characteristic classes of the canonical bundle and complementary bundle, not the tangent bundle. However,
\[
c^T(T(G_k(\mathbb{C}^n)))|_S=c^T(\oplus_{i\in S} \oplus_{j\not\in S} (\mathbb{C}_i^* \otimes \mathbb{C}_j))=\prod_{i\in S}\prod_{j\not\in S} (1+\alpha_j - \alpha_i).
\]
We can also use the ABBV formula to calculate the equivariant (and ordinary) characteristic numbers of the tangent bundle.
\end{rmk}
\begin{rmk}
The equivariant Chern classes $(c^T)^I$ are integral. Moreover, the canonical classes $\tau_S$ are actually the equivariant Schubert classes, hence also integral. Therefore, the coefficients $N,K,\bar{K}$ and characteristic numbers $\int (c^T)^I,\int c^I$ are all integral.
\end{rmk}
\vskip 20pt
\section{Equivariant cohomology rings of real Grassmannians}
\vskip 15pt
In this section, we give the GKM description and Leray-Borel description of equivariant cohomology rings of real Grassmannians, together with the canonical basis and characteristic basis of the additive structure. We use the notation $G_k(\mathbb{R}^n)$ for the Grassmannian of $k$-dimensional real subspaces in $\mathbb{R}^n$.
The dimension of $G_k(\mathbb{R}^n)$ is $k(n-k)$. Therefore, $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$ are even dimensional, but $G_{2k+1}(\mathbb{R}^{2n+2})$ is odd dimensional. Moreover, the real Grassmannians $G_k(\mathbb{R}^n)$ differ from each other on Poincar\'e series and orientability according to the parities of $k$ and $n$, as shown by Casian and Kodama:
\begin{thm}[Poincar\'e series of real Grassmannians, \cite{CK} pp.\,11, Thm\,5.1]\label{thm:Poinc}
The relations between Poincar\'e series of real Grassmannians and complex Grassmannians are given as:
\begin{align*}
P_{G_{2k}(\mathbb{R}^{2n})}(t)=P_{G_{2k}(\mathbb{R}^{2n+1})}(t)&=P_{G_{2k+1}(\mathbb{R}^{2n+1})}(t)=P_{G_{k}(\mathbb{C}^{n})}(t^2)\\
P_{G_{2k+1}(\mathbb{R}^{2n+2})}(t)&=(1+t^{2n+1})P_{G_{k}(\mathbb{C}^{n})}(t^2).
\end{align*}
\end{thm}
\begin{rmk}
The Poincar\'e series of complex Grassmannian is (see \cite{BT82} pp.\,292 Prop\,23.1)
\[
P_{G_{k}(\mathbb{C}^{n})}(t)=\frac{(1-t^2)\cdots(1-t^{2n})}{(1-t^2)\cdots(1-t^{2k})(1-t^2)\cdots(1-t^{2(n-k)})}.
\]
Using the relations between Poincar\'e series, we see that $G_{2k}(\mathbb{R}^{2n})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$ have non-zero top Betti numbers, hence orientable; however, $G_{2k}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+1})$ have zero top Betti numbers, hence non-orientable.
\end{rmk}
\subsection{GKM description of real Grassmannians}
Similar to the case of complex Grassmannians, we will show the real Grassmannians also have appropriate torus actions that are equivariantly formal and GKM.
First, we specify the torus actions on real Grassmannians. Write the coordinates on $\mathbb{R}^{2n}$ as $(x_1,y_1,\ldots,x_n,y_n)$. Let $T^n$ act on $\mathbb{R}^{2n},\mathbb{R}^{2n+1},\mathbb{R}^{2n+2}$ so that the $i$-th $S^1$-component of $T^n$ exactly rotates the $i$-th pair of real coordinates $(x_{i},y_{i})$ and leaves the remaining coordinates free, hence we can write $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, $\mathbb{R}^{2n+1}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ and $\mathbb{R}^{2n+2}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}^2_0$ for their decompositions into weighted subspaces, where $[\alpha_i] \in \mathfrak{t}_\mathbb{Z}^*/\pm 1$. These actions induce $T^n$ actions on $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$, $G_{2k+1}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$.
Since there are natural $T^n$-diffeomorphisms $G_{2k}(\mathbb{R}^{2n+1})\cong G_{2n-2k+1}(\mathbb{R}^{2n+1})$ identifying the second and the third types of real Grassmannians, in many discussions we will only consider the three cases of $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$.
\subsubsection{Fixed points}
Similar to the observation in the case of complex Grassmannians, the $T^n$-fixed points of real Grassmannians are exactly some appropriate dimensional sub-representations of the ambient representations. The verification of sub-representations of $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, $\mathbb{R}^{2n+1}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ for the real Grassmannians $G_{2k}(\mathbb{R}^{2n})$ and $G_{2k}(\mathbb{R}^{2n+1})$ are straightforward. Let's focus on the $2k+1$ dimensional sub-representations of $\mathbb{R}^{2n+2}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}^2_0$. Notice that the sub-representation is odd dimensional, hence must have exactly one dimension in the part of trivial representation, therefore has the form $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0$ where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$ and $L_0\in \mathbb{P}(\mathbb{R}^2_0)$. For each $k$-element subset $S$, the connected component $C_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \mathbb{P}(\mathbb{R}^2_0)\}\cong \mathbb{R} P^1$ gives a fixed circle isolated from the other fixed circles. This gives all the fixed points of the $T^n$ action on $G_{2k+1}(\mathbb{R}^{2n+2})$.
\begin{rmk}
Because of the one-to-one correspondence between a $k$-element subset $S \subset \{1,\ldots,n\}$ with a fixed point or circle, sometimes we will use $S$ directly to mean a fixed point or circle.
\end{rmk}
\subsubsection{Isotropy weights}\label{subsubsec:IsoWeights}
Fixing a $k$-element subset $S$, let's describe the tangent spaces at the fixed points in the three cases of real Grassmannians.
\begin{enumerate}
\item The tangent space at $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]} \in G_{2k}(\mathbb{R}^{2n})$ is
\[
Hom_\mathbb{R}\Big(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}, \oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]}\Big) \cong (\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} (\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]})\cong \oplus_{i \in S} \oplus_{j \not \in S} \big((\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}\big).
\]
\item The tangent space at $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]} \in G_{2k}(\mathbb{R}^{2n+1})$ is
\[
Hom_\mathbb{R}\Big(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}, (\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]})\oplus \mathbb{R}_0\Big) \cong \Big(\oplus_{i \in S} \oplus_{j \not \in S} \big((\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}\big)\Big)\oplus \oplus_{i \in S} (\mathbb{R}^2_{[\alpha_i]})^*.
\]
\item The tangent space at $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \in G_{2k+1}(\mathbb{R}^{2n+2})$, where $L_0 \in \mathbb{P}(\mathbb{R}^2_0)$ has a $L_0^\bot \in \mathbb{P}(\mathbb{R}^2_0)$ such that $L_0 \oplus L_0^\bot \cong \mathbb{R}^2_0$, is
\begin{align*}
&Hom_\mathbb{R}\Big((\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0, (\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]})\oplus L_0^\bot\Big)\\
\cong& \Big(\oplus_{i \in S} \oplus_{j \not \in S} \big((\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}\big)\Big) \oplus \Big( \oplus_{i \in S} (\mathbb{R}^2_{[\alpha_i]})^*\otimes_\mathbb{R} L_0^\bot \Big) \oplus \Big( \oplus_{j \not\in S} L_0^*\otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]} \Big) \\
&\oplus\Big( L_0^* \otimes_\mathbb{R} L_0^\bot \Big)
\end{align*}
among which the first three terms and the fourth term give respectively the normal space and tangent space of the fixed circle $C_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \mathbb{P}(\mathbb{R}^2_0)\}$.
\end{enumerate}
The isotropy weights are then determined by the following simple lemma:
\begin{lem}
The weights of the tensor product $(\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}$ are $[\alpha_j-\alpha_i],[\alpha_j+\alpha_i] \in \mathfrak{t}_\mathbb{Z}^*/\pm 1$.
\end{lem}
\begin{proof}
The $T$-action on the dual space $(\mathbb{R}^2_{[\alpha_i]})^*$ is defined in an invariant way so that for $t \in T$, $l \in (\mathbb{R}^2_{[\alpha_i]})^*$, $v \in \mathbb{R}^2_{[\alpha_i]}$, and if we denote $\langle l,v\rangle$ as the natural pairing, we should have
$\langle t\cdot l,t\cdot v\rangle=\langle l,v\rangle$ or equivalently, $(t\cdot l)(v) = l(t^{-1}\cdot v)$. Notice that only the $i$-th and $j$-th $S^1$-component of $T^n$ have non-trivial actions on $(\mathbb{R}^2_{[\alpha_i]})^*$ or $\mathbb{R}^2_{[\alpha_j]}$, let $e^{\sqrt{-1}\theta_i} \in S^1_i$, $e^{\sqrt{-1}\theta_j} \in S^1_j$, and write elements of $(\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}$ as $2 \times 2$ matrices, then the $S^1_i \times S^1_j$ action on $(\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}$ can be given as:
\[
(e^{\sqrt{-1}\theta_i},e^{\sqrt{-1}\theta_j})\cdot
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}=
\begin{pmatrix}
\cos \theta_j & -\sin \theta_j\\
\sin \theta_j & \cos \theta_j
\end{pmatrix}
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\begin{pmatrix}
\cos \theta_i & \sin \theta_i\\
-\sin \theta_i & \cos \theta_i
\end{pmatrix}.
\]
Consider the following new basis of $(\mathbb{R}^2_{[\alpha_i]})^* \otimes_\mathbb{R} \mathbb{R}^2_{[\alpha_j]}$
\[
M_1 =
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix} \quad
M_2 =
\begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix} \quad
M_3 =
\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix} \quad
M_4 =
\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}.
\]
We get
\begin{align*}
(e^{\sqrt{-1}\theta_j}\cdot M_1,e^{\sqrt{-1}\theta_j}\cdot M_2) &= (M_1,M_2)
\begin{pmatrix}
\cos \theta_j & -\sin \theta_j\\
\sin \theta_j & \cos \theta_j
\end{pmatrix}\\
(e^{\sqrt{-1}\theta_i}\cdot M_1,e^{\sqrt{-1}\theta_i}\cdot M_2) &= (M_1,M_2)
\begin{pmatrix}
\cos \theta_i & \sin \theta_i\\
-\sin \theta_i & \cos \theta_i
\end{pmatrix}.
\end{align*}
In other words, $S^1_i \times S^1_j$ acts on $\mathbb{R} M_1 \oplus \mathbb{R} M_2$ with weight $[\alpha_j-\alpha_i]$.
Similarly, $S^1_i \times S^1_j$ acts on $\mathbb{R} M_3 \oplus \mathbb{R} M_4$ with weight $[\alpha_j+\alpha_i]$.
\end{proof}
\subsubsection{$1$-skeleta}
To begin with, let's work out the $1$-skeleton of the $T^2$-action on $G_2(\mathbb{R}^4)$. From the previous discussions, we know that there are two fixed points $\mathbb{R}^2_{[\alpha_1]},\mathbb{R}^2_{[\alpha_2]} \in G_2(\mathbb{R}^4)$, both have the same isotropy weights $[\alpha_2-\alpha_1]$ and $[\alpha_2+\alpha_1]$. Let $T_{\alpha_2-\alpha_1}$ be the subtorus of $T^2$ with Lie algebra annihilated by $\alpha_2-\alpha_1$, i.e. $T_{\alpha_2-\alpha_1}$ is the diagonal $\{(t,t) \in T^2\}$. Similarly, $T_{\alpha_2+\alpha_1}$, the subtorus with Lie algebra annihilated by $\alpha_2+\alpha_1$, is the anti-diagonal $\{(t,t^{-1}) \in T^2\}$.
Note that there is a natural diffeomorphism $\mathcal{F}:\mathbb{C}^2 \rightarrow \mathbb{R}^4$ by forgetting the complex structure. This induces an embedding $\mathbb{C} P^1 \hookrightarrow G_2(\mathbb{R}^4):L \mapsto \mathcal{F}(L)$ where $L$ is a complex line in $\mathbb{C}^2$ and $\mathcal{F}(L)$ its two dimensional real image in $\mathbb{R}^4$. Let $J:\mathbb{C}^2 \rightarrow \mathbb{C}^2: (z_1,z_2) \mapsto (z_1,\bar{z}_2)$ be the diffeomorphism with conjugation on the second variable. This also induces an embedding $\mathbb{C} P^1 \hookrightarrow G_2(\mathbb{R}^4):L \mapsto \mathcal{F}(J(L))$. We will denote the images of the two embeddings as $\mathbb{C} P^1$ and $\overline{\mathbb{C} P^1}$.
\begin{lem}
The fixed-point sets of $T_{\alpha_2-\alpha_1}$ and $T_{\alpha_2+\alpha_1}$ in $G_2(\mathbb{R}^4)$ are $\mathbb{C} P^1$ and $\overline{\mathbb{C} P^1}$ respectively, i.e. the $1$-skeleton of the $T^2$-action on $G_2(\mathbb{R}^4)$ is $\mathbb{C} P^1 \cup \overline{\mathbb{C} P^1}$ glued at the two $T^2$-fixed points $\mathbb{R}^2_{[\alpha_1]},\mathbb{R}^2_{[\alpha_2]} \in G_2(\mathbb{R}^4)$.
\end{lem}
\begin{proof}
Let $L_0=\mathbb{C} \oplus 0$ and $L_\infty = 0 \oplus \mathbb{C}$ be the two complex lines in $\mathbb{C}^2$, they are the two poles of both $\mathbb{C} P^1$ and $\overline{\mathbb{C} P^1}$, and are exactly the two $T^2$-fixed points $\mathbb{R}^2_{[\alpha_1]},\mathbb{R}^2_{[\alpha_2]} \in G_2(\mathbb{R}^4)$. The diagonal circle $T_{\alpha_2-\alpha_1}=\{(t,t) \in T^2\}$ fixes $\mathbb{C} P^1$ because $(t,t)\cdot [z_1,z_2]=[tz_1,tz_2]=[z_1,z_2]$ trivially, hence $\mathbb{C} P^1$ joins $\mathbb{R}^2_{[\alpha_1]}$ to $\mathbb{R}^2_{[\alpha_2]}$ with weight $[\alpha_2-\alpha_1]$. Similarly, $\overline{\mathbb{C} P^1}$ joins $\mathbb{R}^2_{[\alpha_1]}$ to $\mathbb{R}^2_{[\alpha_2]}$ with weight $[\alpha_2+\alpha_1]$. The $2$-spheres $\mathbb{C} P^1$ and $\overline{\mathbb{C} P^1}$ exhaust all the $T^2$-fixed points and the isotropy weights, therefore give the $1$-skeleton of the $T^2$-action on $G_2(\mathbb{R}^4)$.
\end{proof}
Generally, let $T_{\alpha_j-\alpha_i}$ and $T_{\alpha_j+\alpha_i}$ be the subtori of $T^n$ with Lie algebras annihilated by $\alpha_j-\alpha_i$ and $\alpha_j+\alpha_i$ respectively. For the $T^n$-action on $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, the fixed-point sets of $T_{\alpha_j-\alpha_i}$ and $T_{\alpha_j+\alpha_i}$ on $G_2(\mathbb{R}^2_{[\alpha_i]}\oplus \mathbb{R}^2_{[\alpha_j]})$ are two $2$-spheres sharing the poles which are exactly the two $T^n$-fixed points $\mathbb{R}^2_{[\alpha_i]},\mathbb{R}^2_{[\alpha_j]} \in G_2(\mathbb{R}^2_{[\alpha_i]}\oplus \mathbb{R}^2_{[\alpha_j]})$. We will denote the $2$-spheres as $S^2_{[\alpha_j-\alpha_i]}$ and $S^2_{[\alpha_j+\alpha_i]}$ and keep in mind that every element $V$ in $S^2_{[\alpha_j-\alpha_i]}$ or $S^2_{[\alpha_j+\alpha_i]}$ is a $2$-plane in $\mathbb{R}^2_{[\alpha_i]}\oplus \mathbb{R}^2_{[\alpha_j]}$.
Now we are ready to describe the $1$-skeleta of the $T^n$ actions on the three types of real Grassmannians. Let $S$ be a $k$-element subset of $\{1,2,\ldots,n\}$, and $i\in S$, $j\not\in S$.
\begin{enumerate}
\item For $G_{2k}(\mathbb{R}^{2n})$, the $T^n$-fixed point $V_S=\oplus_{i'\in S} \mathbb{R}^2_{[\alpha_{i'}]}$ is joined to $V_{(S\smallsetminus\{i\})\cup \{j\}}$ via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in S^2_{[\alpha_j-\alpha_i]}\}\cong S^2$ of weight $[\alpha_j-\alpha_i]$ and also via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in S^2_{[\alpha_j+\alpha_i]}\}\cong S^2$ of weight $[\alpha_j+\alpha_i]$.
\item For $G_{2k}(\mathbb{R}^{2n+1})$, the $T^n$-fixed point $V_S=\oplus_{i'\in S} \mathbb{R}^2_{[\alpha_{i'}]}$ is joined to $V_{(S\smallsetminus\{i\})\cup \{j\}}$ via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in S^2_{[\alpha_j-\alpha_i]}\}\cong S^2$ of weight $[\alpha_j-\alpha_i]$ and also via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in S^2_{[\alpha_j+\alpha_i]}\}\cong S^2$ of weight $[\alpha_j+\alpha_i]$. Moreover, $V_S$ is contained in $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V\mid V\in G_2(\mathbb{R}^2_{[\alpha_i]}\oplus \mathbb{R}_0)\}\cong \mathbb{R} P^2$ of weight $[\alpha_i]$ without other fixed points.
\item For $G_{2k+1}(\mathbb{R}^{2n+2})$, the $T^n$-fixed circle $C_S=\{(\oplus_{i'\in S} \mathbb{R}^2_{[\alpha_{i'}]})\oplus L_0 \mid L_0\in \mathbb{P}(\mathbb{R}^2_0)\}\cong \mathbb{R} P^1$ is joined to $C_{(S\smallsetminus\{i\})\cup \{j\}}$ via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V \oplus L_0 \mid V\in S^2_{[\alpha_j-\alpha_i]}, L_0\in \mathbb{P}(\mathbb{R}^2_0)\}\cong S^2\times \mathbb{R} P^1$ with weight $[\alpha_j-\alpha_i]$ and also via $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus V \oplus L_0 \mid V\in S^2_{[\alpha_j+\alpha_i]}, L_0\in \mathbb{P}(\mathbb{R}^2_0)\}\cong S^2\times \mathbb{R} P^1$ with weight $[\alpha_j+\alpha_i]$. Moreover, $C_S$ is contained in $\{(\oplus_{i'\in S\smallsetminus\{i\}} \mathbb{R}^2_{[\alpha_{i'}]})\oplus W \mid W\in G_3(\mathbb{R}^2_{[\alpha_{i}]}\oplus \mathbb{R}^2_0)\}\cong \mathbb{R} P^3$ and $\{(\oplus_{i'\in S} \mathbb{R}^2_{[\alpha_{i'}]})\oplus L \mid L\in \mathbb{P}(\mathbb{R}^2_{[\alpha_j]}\oplus\mathbb{R}^2_0)\}\cong \mathbb{R} P^3$ of weights $[\alpha_i]$ and $[\alpha_j]$ respectively without other fixed points.
\end{enumerate}
\subsubsection{GKM graphs of real Grassmannians}
Since $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$ are even dimensional, but $G_{2k+1}(\mathbb{R}^{2n+2})$ is odd dimensional, we will construct GKM graphs according to the parity of dimensions.
\begin{exm}
We give some examples of GKM graphs for $G_k(\mathbb{R}^n)$ when $k$ or $n$ is small.
\begin{enumerate}
\item $\mathbb{R} P^{2n}$ as $G_{1}(\mathbb{R}^{2n+1})$ or $G_{2n}(\mathbb{R}^{2n+1})$
\begin{figure}
\caption{Complete GKM graph for $\mathbb{R}
\caption{Effective GKM graph for $\mathbb{R}
\caption{GKM graphs for $\mathbb{R}
\end{figure}
\item $\mathbb{R} P^{2n+1}$ as $G_{1}(\mathbb{R}^{2n+2})$ or $G_{2n+1}(\mathbb{R}^{2n+2})$
\begin{figure}
\caption{GKM graph for $\mathbb{R}
\caption{Condensed GKM graph for $\mathbb{R}
\caption{GKM graphs for $\mathbb{R}
\end{figure}
\item $G_{2}(\mathbb{R}^{4}),G_{2}(\mathbb{R}^{5}),G_{3}(\mathbb{R}^{5})$ as $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$ when $k=1,n=2$.
\begin{figure}
\caption{GKM graph for $G_{2}
\caption{Condensed GKM graph for $G_{2}
\caption{GKM graphs for $G_{2}
\end{figure}
\begin{figure}
\caption{Complete GKM graph}
\caption{Effective GKM graph}
\caption{Condensed GKM graph}
\caption{GKM graphs for $G_{2}
\end{figure}
\begin{figure}
\caption{Complete GKM graph}
\caption{Effective GKM graph}
\caption{Condensed GKM graph}
\caption{GKM graphs for $G_{3}
\end{figure}
\item $G_{3}(\mathbb{R}^{6})$ as $G_{2k+1}(\mathbb{R}^{2n+2})$ when $k=1,n=2$.
\begin{figure}
\caption{GKM graph for $G_{3}
\caption{Condensed GKM graph for $G_{3}
\caption{GKM graphs for $G_{3}
\end{figure}
\end{enumerate}
\end{exm}
\begin{rmk}
The graphs of $\mathbb{R} P^{2n}$ and $G_2(\mathbb{R}^5)$ have appeared in Goertsches\&Mare \cite{GM14}.
\end{rmk}
\subsubsection{Formality, cohomology and canonical basis of real Grassmannians}
We have given the $1$-skeleta and GKM graphs for real Grassmannians $G_k(\mathbb{R}^n)$ under appropriate torus actions. To apply the GKM-type theorems in even and odd dimensions, we still need to verify that those torus actions on $G_k(\mathbb{R}^n)$ are equivariantly formal.
\begin{prop}[Equivariant formality of torus actions on real Grassmannians]
The total Betti numbers of $G_k(\mathbb{R}^n)$ and of its fixed-point set are equal:
\begin{enumerate}
\item For the $T^n$-actions on $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$, the isolated fixed points $V_S$ are all parametrized by $\mathcal{S}=\{S\subseteq\{1,2,\ldots,n\} \mid \#S=k\}$, and we have
\[
\sum \mathrm{dim}\,H^*(G_{2k}(\mathbb{R}^{2n})) = \sum \mathrm{dim}\,H^*(G_{2k}(\mathbb{R}^{2n+1})) = \sum \mathrm{dim}\,H^*(G_{2k+1}(\mathbb{R}^{2n+1})) = \# \mathcal{S} = \binom{n}{k}.
\]
\item For the $T^n$-action on $G_{2k+1}(\mathbb{R}^{2n+2})$, the isolated fixed circles $C_S$ are also indexed on $\mathcal{S}=\{S\subseteq\{1,2,\ldots,n\} \mid \#S=k\}$, and we have
\[
\sum\mathrm{dim}\,H^*(G_{2k+1}(\mathbb{R}^{2n+1})) = \# \mathcal{S} \cdot \sum\mathrm{dim}\,H^*(S^1) = 2\binom{n}{k}.
\]
\end{enumerate}
Therefore, the torus actions on $G_k(\mathbb{R}^n)$ are equivariantly formal.
\end{prop}
\begin{proof}
The verification is based on the equivalence (6) of Theorem \ref{thm:formal}. The total Betti numbers of $G_k(\mathbb{R}^n)$ can be calculated from the Casian-Kodama formula in Theorem \ref{thm:Poinc} by substituting $t=1$ in the Poincar\'e series.
\begin{align*}
\sum \mathrm{dim}\,H^*(G_{2k}(\mathbb{R}^{2n})) = \sum \mathrm{dim}\,H^*(G_{2k}(\mathbb{R}^{2n+1})) &= \sum \mathrm{dim}\,H^*(G_{2k+1}(\mathbb{R}^{2n+1})) = \sum \mathrm{dim}\,H^*(G_{k}(\mathbb{C}^{n}))\\
\sum \mathrm{dim}\,H^*(G_{2k+1}(\mathbb{R}^{2n+2})) &= 2\sum \mathrm{dim}\,H^*(G_{k}(\mathbb{C}^{n})).
\end{align*}
On the other hand, by the formality of the $T^n$-action on $G_{k}(\mathbb{C}^{n})$, which also has isolated points parametrized by $\mathcal{S}$, we have
\[
\sum \mathrm{dim}\,H^*(G_{k}(\mathbb{C}^{n})) = \# \mathcal{S} = \binom{n}{k}.
\]
Therefore, total Betti numbers of $G_k(\mathbb{R}^n)$ and of its fixed-point set are equal, and the torus actions on $G_k(\mathbb{R}^n)$ are equivariantly formal.
\end{proof}
With the verifications of GKM conditions and equivariant formality, we can give the GKM description of the torus actions on $G_k(\mathbb{R}^n)$ by applying the generalized GKM-type Theorems \ref{thm:EvenGKM} and \ref{thm:OddGKM} in even and odd dimensions.
\begin{thm}[GKM description of equivariant cohomology of real Grassmannians]\label{thm:GKMrealGrass}
Let $\mathcal{S}$ be the collection of $k$-element subsets of $\{1,2,\ldots,n\}$.
\begin{enumerate}
\item For even dimensional Grassmannians $G_{2k}(\mathbb{R}^{2n}), G_{2k}(\mathbb{R}^{2n+1}), G_{2k+1}(\mathbb{R}^{2n+1})$ with $T^n$-actions, they have the same equivariant cohomology
\[
\big\{f:\mathcal{S}\rightarrow \mathbb{Q}[\alpha_1,\ldots,\alpha_n] \mid f_{S} \equiv f_{S'} \mod \alpha^2_j-\alpha^2_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}\big\}.
\]
\item For odd dimensional Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$ with $T^n$-action, an element of the equivariant cohomology is a set of polynomial pairs $(f_S, g_S \theta)$ to each $\circ$-vertex $S$ where $\theta$ is the unit volume form of $S^1$ such that
\begin{enumerate}
\item $g_{S} \equiv 0 \mod \prod_{i=1}^{n}\alpha_i$\quad for every $S$
\item $f_{S} \equiv f_{S'} , \quad g_{S} \equiv g_{S'}\mod \alpha^2_j-\alpha^2_i$\quad for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$.
\end{enumerate}
\end{enumerate}
\end{thm}
\begin{rmk}
For convenience, we will write an element $f \in H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$ as $(f_S)_{S \in \mathcal{S}}$ and an element $(f,g\theta) \in H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2}))$ as $(f_S+g_S\theta)_{S \in \mathcal{S}}$, which are understood as tuples indexed with respect to $S \in \mathcal{S}$.
\end{rmk}
\begin{rmk}
In the $1$-skeleton of the odd dimensional Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$, every $\mathbb{R} P^3_{[\alpha_i]}$ containing a unique fixed circle $C_S$ contributes a relation $g_{S} \equiv 0 \mod \alpha_i$; every $S^2\times \mathbb{R} P^1$ with weight $\alpha_j\pm \alpha_i$ and two fixed circles $C_S,C_{S'}$ contributes two relations $f_{S} \equiv f_{S'} , g_{S} \equiv g_{S'}\mod \alpha_j\pm\alpha_i$. These simple components in $1$-skeleton resolve the sign issues in odd dimensional GKM-type Theorem\,\ref{thm:OddGKM}.
\end{rmk}
\begin{rmk}
Note that in the above description, we have condensed some congruence relations because $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ is a unique-factorization domain.
\begin{align*}
\begin{cases}
f_{S} \equiv f_{S'} \mod \alpha_j-\alpha_i\\
f_{S} \equiv f_{S'} \mod \alpha_j+\alpha_i
\end{cases}
&\Longleftrightarrow \quad f_{S} \equiv f_{S'} \mod \alpha^2_j-\alpha^2_i\\
\begin{cases}
g_{S} \equiv 0 \mod \alpha_1\\
\vdots \quad \qquad \vdots \quad \qquad \vdots\\
g_{S} \equiv 0 \mod \alpha_n
\end{cases}
&\Longleftrightarrow \quad g_{S} \equiv 0 \mod \prod_{i=1}^{n}\alpha_i.
\end{align*}
\end{rmk}
Notice the similarity among the GKM descriptions of the even and odd dimensional real Grassmannians and the complex Grassmannians, we have
\begin{thm}[Relations among equivariant cohomology of real and complex Grassmannians]\label{thm:AllGrass}
The relations between the equivariant cohomology of even, odd dimensional real Grassmannians and complex Grassmannians are
\begin{enumerate}
\item There are a series of $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra isomorphisms:
\[
H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))\cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n+1}))\cong H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+1})).
\]
\item There is an element $r^T \in H^{2n+1}_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2}))$ such that $(r^T)^2=0$, and there is a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra isomorphism
\[
H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2})) \cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))[r^T]/(r^T)^2.
\]
\item There is a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra monomorphism:
\[
H^*_{T^n}(G_{2k}(\mathbb{R}^{2n})) \hookrightarrow H^*_{T^n}(G_{k}(\mathbb{C}^{n})).
\]
\end{enumerate}
\end{thm}
\begin{proof}
All the Grassmannians with $T^n$-action are modelled on the same Johnson graph $J(n,k)$ with slightly different congruence relations.
\begin{enumerate}
\item This is the part\,(1) of Theorem\,\ref{thm:GKMrealGrass}.
\item From Theorem\,\ref{thm:GKMrealGrass}, the GKM descriptions of even and odd dimensional real Grassmannians have the same congruence relations on the $f_S$ polynomials:
\[
f_{S} \equiv f_{S'}\mod \alpha^2_j-\alpha^2_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}.
\]
But the odd dimensional real Grassmannian has extra part of $g_S \theta$ with congruence relations:
\begin{enumerate}
\item $g_{S} \equiv 0 \mod \prod_{i=1}^{n}\alpha_i$\quad for every $S$
\item $g_{S} \equiv g_{S'}\mod \alpha^2_j-\alpha^2_i$\quad for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$.
\end{enumerate}
The first set of congruence relations means that
\[g_{S}=\big(\prod_{i=1}^{n}\alpha_i\big) \cdot h_{S}\]
for a polynomial $h_{S}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ and for every $S$. Substitute into the second set of congruence relations, and note that $\prod_{i=1}^{n}\alpha_i$ is coprime with $\alpha^2_j-\alpha^2_i$, then we get
\[
h_{S} \equiv h_{S'}\mod \alpha^2_j-\alpha^2_i \quad \text{ for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}
\]
exactly the same as the congruence relations on the $f_S$ polynomials. Denote
\[
r^T=\big((\prod_{i=1}^{n}\alpha_i) \theta\big)_{S\in \mathcal{S}}
\]
which has $(r^T)^2=0$ because $\theta$ is the unit volume form of $S^1$, and has degree $2n+1$ because each $\alpha_i$ is of degree $2$ in cohomology. Then we can write
\[
(f_S+g_S\theta)_{S\in \mathcal{S}} = (f_S)_{S\in \mathcal{S}} + r^T\cdot (h_S)_{S\in \mathcal{S}}.
\]
This establishes the bijection
\[
H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2})) \cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))[r^T]/(r^T)^2
\]
which can be easily verified to be a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra isomorphism.
\item From Theorem \ref{thm:GKMrealGrass}, the GKM description of even dimensional real Grassmannians has the congruence relations on the $f_S$ polynomials:
\[
f_{S} \equiv f_{S'}\mod \alpha^2_j-\alpha^2_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}
\]
which automatically satisfy the congruence relations on the $f_S$ polynomials for the complex Grassmannians in Theorem \ref{thm:GKMcplxGrass}:
\[
f_{S} \equiv f_{S'}\mod \alpha_j-\alpha_i \quad \text{for $S,S' \in \mathcal{S}$ with $S\cup\{j\}=S'\cup\{i\}$}.
\]
This establishes the injection
\[
H^*_{T^n}(G_{2k}(\mathbb{R}^{2n})) \hookrightarrow H^*_{T^n}(G_{k}(\mathbb{C}^{n}))
\]
which is also easy to verify as a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra monomorphism.
\end{enumerate}
\end{proof}
\begin{rmk}
Those $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra isomorphisms in Theorem\,\ref{thm:AllGrass} give ring isomorphisms among ordinary cohomology of real Grassmannians. But the ordinary version $H^*(G_{2k}(\mathbb{R}^{2n})) \rightarrow H^*(G_{k}(\mathbb{C}^{n}))$ is not injective simply due to fact that $G_{2k}(\mathbb{R}^{2n})$ is of dimension $4k(n-k)$, twice the real dimension of $G_{k}(\mathbb{C}^{n})$.
\end{rmk}
\begin{thm}[Canonical basis of even dimensional real Grassmannians]\label{thm:RealSchub}
There is a self-indexing Morse function on $\mathcal{S}$
\[
\psi: \mathcal{S} \longrightarrow \mathbb{R} : S \longmapsto 4(\sum_{i\in S} i) -2k(k+1)
\]
and a canonical class $\sigma_S \in H^{\psi(S)}_{T^n}(G_{2k}(\mathbb{R}^{2n}),\mathbb{Q})$ for each $S\in \mathcal{S}$ such that
\begin{enumerate}
\item $\sigma_S$ is supported upward, i.e. $\sigma_S(S')=0$ if $\psi(S')\leq \psi(S)$
\item $\sigma_S(S)=\prod' (\alpha^2_j - \alpha^2_i)$ where the product is taken over the weights at $S$ connecting to $S'$ with $\psi(S')<\psi(S)$
\end{enumerate}
Moreover, $\{\sigma_S, S\in \mathcal{S}\}$ give an additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-basis of $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}),\mathbb{Q})$.
\end{thm}
\begin{proof}
By Theorem \ref{thm:AllGrass}, we can identify $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$ with its embedded image in $H^*_{T^n}(G_{k}(\mathbb{C}^{n}))$. Recall from Theorem \ref{thm:CplxSchub} on the canonical classes of complex Grassmannian $G_{k}(\mathbb{C}^{n})$, we used the function $\phi=\frac{\psi}{2}$, hence both $\psi$ and $\phi$ define the same partial order on $\mathcal{S}$. Moreover, there is a basis $\tau_S$ of $H^*_{T^n}(G_{k}(\mathbb{C}^{n}))$, such that
\begin{enumerate}
\item $\tau_S$ is supported upward, i.e. $\tau_S(S')=0$ if $\phi(S')\leq \phi(S)$
\item $\tau_S(S)=\prod' (\alpha_j - \alpha_i)$ where the product is taken over the weights at $S$ connecting to $S'$ with $\phi(S')<\phi(S)$
\end{enumerate}
Let's introduce the ring homomorphism:
\[
Sq: \mathbb{Q}[\alpha_1,\ldots,\alpha_n] \rightarrow \mathbb{Q}[\alpha_1,\ldots,\alpha_n] : f(\alpha_1,\ldots,\alpha_n) \mapsto f(\alpha^2_1,\ldots,\alpha^2_n).
\]
If $f_S \equiv f_{S'} \mod \alpha_j-\alpha_i$, i.e. $f_S - f_{S'}$ is a multiple of $\alpha_j-\alpha_i$, then $Sq(f_S) - Sq(f_{S'})=Sq(f_S - f_{S'})$ is a multiple of $Sq(\alpha_j-\alpha_i)=\alpha^2_j-\alpha^2_i$, i.e. $Sq(f_S) \equiv Sq(f_{S'}) \mod \alpha^2_j-\alpha^2_i$. The homomorphism $Sq$ not only refines the congruence relations of $H^*_{T^n}(G_{k}(\mathbb{C}^{n}))$, but also has image in $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$, i.e. $Sq(H^*_{T^n}(G_{k}(\mathbb{C}^{n}))) \subseteq H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$.
Now we can define $\sigma_S = Sq(\tau_S) \in H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$, and we see that this collection of classes satisfies the required properties of being supported upward and $\sigma_S(S)=\prod' (\alpha^2_j - \alpha^2_i)$ over weights at $S$ connecting to $S'$ with $\psi(S')<\psi(S)$.
According to Guillemin\&Zara (\cite{GZ03} pp.\,125, Remark of Thm\,2.1), $\{\sigma_S\}$ give an additive basis of $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$.
\end{proof}
Since we have proved $H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2})) \cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))[r^T]/(r^T)^2$ in Theorem \ref{thm:AllGrass}, then
\begin{cor}[Canonical basis of odd dimensional real Grassmannians]
$\sigma_S$ and $r^T\sigma_S$ give an additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-basis of $H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2}))$.
\end{cor}
\begin{rmk}
In the case of complex Grassmannian $G_{k}(\mathbb{C}^{n})$, a subset $S\subseteq \{1,2,\ldots,n\}$ with elements $ i_1<i_2<\cdots<i_k$ corresponds to Schubert symbol $(i_1-1,i_2-2,\ldots,i_k-k)$; there could be correspondences for real Grassmannians
\begin{enumerate}
\item For even dimensional Grassmannians $G_{2k}(\mathbb{R}^{2n}),G_{2k}(\mathbb{R}^{2n+1})$, let $S$ consist of $ i_1<i_2<\cdots<i_k$, then the $T^n$-fixed point $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$, with pivot positions $(2i_1-1,2i_1, 2i_2-1,2i_2,\ldots,2i_k-1,2i_k)$ in its reduced echelon form, will correspond to Schubert symbol $(2i_1-2,2i_1-2, 2i_2-4,2i_2-4,\ldots,2i_k-2k,2i_k-2k)$.
\item For even dimensional Grassmannian $G_{2k+1}(\mathbb{R}^{2n+1})$, the $T^n$-fixed point $\mathbb{R}_0 \oplus (\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})$, with pivot positions $(1,2i_1,2i_1+1, 2i_2,2i_2+1,\ldots,2i_k,2i_k+1)$ in its reduced echelon form, will also correspond to Schubert symbol $(2i_1-2,2i_1-2, 2i_2-4,2i_2-4,\ldots,2i_k-2k,2i_k-2k)$.
\item For odd dimensional Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$, besides the above Schubert symbols $(2i_1-2,2i_1-2, 2i_2-4,2i_2-4,\ldots,2i_k-2k,2i_k-2k)$, there is the class $r^T$, which is conjectured by Casian\&Kodama \cite{CK} to be the Schubert class with the hook Young diagram $1^{2k}\times (2(n-k)+1)$ of symbol $(1,\ldots,1,2(n-k)+1)$ where there are $2k$ copies of $1$. Following this conjecture, we can guess that a class $r^T\sigma_S$ with $S$ given by $i_1<i_2<\cdots<i_k$, corresponds to the Schubert symbol $(2i_1-1,2i_1-1, 2i_2-3,2i_2-3,\ldots,2i_k-2k+1,2i_k-2k+1,2(n-k)+1)$.
\end{enumerate}
\end{rmk}
Recall from Subsection \ref{subsec:BGKM} of the equivariant Littlewood-Richardson rule for complex Grassmannian $G_k(\mathbb{C}^n)$
\[
\tau_{S}\tau_{S'} =\sum_{S''}N_{S,S'}^{S''}\tau_{S''}
\]
where $N_{S,S'}^{S''}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$. If we apply the ring homomorphism $Sq$ on both sides, then we get:
\begin{thm}[Equivariant Littlewood-Richardson coefficients for real Grassmannians]
The equivariant Littlewood-Richardson coefficients for real Grassmannian $G_{2k}(\mathbb{R}^{2n})$ satisfy
\[
\sigma_{S}\sigma_{S'} =\sum_{S''}Sq(N_{S,S'}^{S''})\sigma_{S''}
\]
where $Sq(N_{S,S'}^{S''}) \in \mathbb{Q}[\alpha^2_1,\ldots,\alpha^2_n]$ is obtained from $N_{S,S'}^{S''}$ by replacing $\alpha_i$ to be $\alpha_i^2$.
\end{thm}
\begin{rmk}
Since $Sq$ keeps constant term unchanged, the Littlewood-Richardson rules for ordinary cohomology of complex Grassmannian $G_k(\mathbb{C}^n)$ and real Grassmannian $G_{2k}(\mathbb{R}^{2n})$ are the same.
\end{rmk}
\subsection{Leray-Borel description of real Grassmannians}
Similar to Leray-Borel description of equivariant (ordinary) cohomology of complex Grassmannians using equivariant (ordinary) Chern classes, we will show there is a Leray-Borel description of equivariant (ordinary) cohomology of real Grassmannians using equivariant (ordinary) Pontryagin classes.
\subsubsection{Equivariant Pontryagin classes}
The $T^n$ actions on $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, $\mathbb{R}^{2n+1}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ and $\mathbb{R}^{2n+2}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}^2_0$ induce actions on $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$, $G_{2k+1}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$. These actions induce further actions on the canonical bundles $\gamma$ and complementary bundles $\bar{\gamma}$ over those Grassmannians. Then we can consider their equivariant Pontryagin classes $p^T=p^T(\gamma)$ and $\bar{p}^T=p^T(\bar{\gamma})$.
First, let's compute a warm-up example of equivariant Pontryagin classes.
\begin{lem}\label{lem:Pont}
The total equivariant Pontryagin class of the vector space $\mathbb{R}^2_{[\alpha]}$ with weight $[\alpha] \in \mathfrak{t}^*_\mathbb{Z}/\pm 1$ over a point is $1+\alpha^2$.
\end{lem}
\begin{proof}
Think of the elements of $\mathbb{R}^2_{[\alpha]}$ as $2\times 1$ column vectors. For a Lie algebra element $\xi \in \mathfrak{t}$, the action of its group element $\exp(\xi) \in T$ on $\mathbb{R}^2_{[\alpha]}$ is given as a $2\times 2$ real matrix
\[
\begin{pmatrix}
\cos(\alpha(\xi)) & -\sin(\alpha(\xi))\\
\sin(\alpha(\xi)) & \cos(\alpha(\xi))
\end{pmatrix}
\qquad
\textup{or}
\qquad
\begin{pmatrix}
\cos(\alpha(\xi)) & \sin(\alpha(\xi))\\
-\sin(\alpha(\xi)) & \cos(\alpha(\xi))
\end{pmatrix}.
\]
Tensoring $\mathbb{R}^2_{[\alpha]}$ over $\mathbb{R}$-coefficients with $\mathbb{C}$ means that we can treat the above real matrices as complex matrices. Since both of them have the same characteristic function $\lambda^2-2\cos(\alpha(\xi))\lambda+1=(\lambda-e^{\sqrt{-1}\alpha(\xi)})(\lambda-e^{-\sqrt{-1}\alpha(\xi)})$, the two real matrices have the same diagonalization over $\mathbb{C}$-coefficients:
\[
\begin{pmatrix}
e^{\sqrt{-1}\alpha(\xi)} & 0\\
0 & e^{-\sqrt{-1}\alpha(\xi)}
\end{pmatrix}
\]
i.e. the $T$-action on the complex vector space $\mathbb{R}^2_{[\alpha]}\otimes_\mathbb{R} \mathbb{C}$ has weights $\alpha$ and $-\alpha$. Therefore, $c^T(\mathbb{R}^2_{[\alpha]}\otimes_\mathbb{R} \mathbb{C})=(1-\alpha)(1+\alpha)=1-\alpha^2$. Following Milnor-Stasheff's convention of signs, we get $p^T(\mathbb{R}^2_{[\alpha]})=1+\alpha^2$.
\end{proof}
Second, let's specify the equivariant Pontryagin classes of canonical bundles, complementary bundles and tangent bundles of real Grassmannians in GKM description at each fixed point or circle of the real Grassmannians.
\begin{prop}\label{prop:Pont}
For all the four real Grassmannians $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$, $G_{2k+1}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$ with $T^n$-actions, the equivariant Pontryagin classes $p^T=p^T(\gamma)$ and $\bar{p}^T=p^T(\bar{\gamma})$ of the canonical bundle and complementary bundle localized at each fixed point or circle indexed as a $k$-element subset $S \in \{1,\ldots,n\}$ are
\begin{align*}
p^T|_S &= p^T(\gamma|_S)=\prod_{i\in S} (1+\alpha^2_i)\\
\bar{p}^T|_S &= p^T(\bar{\gamma}|_S)=\prod_{j\not\in S} (1+\alpha^2_j)
\end{align*}
with the relation $p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i)$. The equivariant Pontryagin classes of the tangent bundles are given at each fixed point or circle $S$ as
\begin{align*}
p^T(TG_{2k}(\mathbb{R}^{2n}))|_S&=\prod_{i \in S}\prod_{j \not \in S} \big[(1+(\alpha_j-\alpha_i)^2)(1+(\alpha_j+\alpha_i)^2)\big]\\
p^T(TG_{2k}(\mathbb{R}^{2n+1}))|_S&=\prod_{i \in S}\prod_{j \not \in S} \big[(1+(\alpha_j-\alpha_i)^2)(1+(\alpha_j+\alpha_i)^2)\big] \prod_{i \in S} (1+\alpha_i^2)\\
p^T(TG_{2k+1}(\mathbb{R}^{2n+1}))|_S&=\prod_{i \in S}\prod_{j \not \in S} \big[(1+(\alpha_j-\alpha_i)^2)(1+(\alpha_j+\alpha_i)^2)\big] \prod_{j \not \in S} (1+\alpha_j^2)\\
p^T(TG_{2k+1}(\mathbb{R}^{2n+2}))|_S&=\prod_{i \in S}\prod_{j \not \in S} \big[(1+(\alpha_j-\alpha_i)^2)(1+(\alpha_j+\alpha_i)^2)\big] \prod_{i \in S} (1+\alpha_i^2) \prod_{j \not \in S} (1+\alpha_j^2).
\end{align*}
\end{prop}
\begin{proof}
For $G_{2k}(\mathbb{R}^{2n})$, at each fixed point $S$, we have $\gamma|_S=\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$ and $\bar{\gamma}|_S=\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]}$, and furthermore $\gamma|_S\oplus\bar{\gamma}|_S=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$. The claimed expressions of the localized Pontryagin classes then follow from the Lemma \ref{lem:Pont}. The cases of $G_{2k}(\mathbb{R}^{2n+1})$, $G_{2k+1}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$ are similar. For the Pontryagin classes of the tangent bundles localized at each fixed point or circle, we can apply Lemma \ref{lem:Pont} to the weight decompositions (see Subsubsection \ref{subsubsec:IsoWeights}) of tangent bundles at each fixed point or circle.
\end{proof}
\subsubsection{Characteristic basis of real Grassmannians}
Think of $p^T$ and $\bar{p}^T$ as elements of the embedded image of $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$ in $H^*_{T^n}(G_{k}(\mathbb{C}^{n}))$ using GKM description on the Johnson graph $J(n,k)$. If we compare the localized expressions of $p^T$ and $\bar{p}^T$ with $c^T$ and $\bar{c}^T$ in Subsection \ref{subsec:BGKM}, we get the formula
\[
p^T=Sq(c^T) \qquad \bar{p}^T=Sq(\bar{c}^T)
\]
where the homomorphism $Sq$ is defined in Theorem \ref{thm:RealSchub}.
Recall in Subsection \ref{subsec:BGKM}, we discussed the transformations $K,\bar{K}$ between the characteristic monomials $(c^T)^I=(c_1^T)^{i_1}\cdots(c_k^T)^{i_k}$ in Leray-Borel description and the canonical classes $\tau_S$ in GKM description:
\begin{align*}
(c^T)^I &= \sum_S K^I_S \tau_S\\
\tau_S &= \sum_I \bar{K}_I^S (c^T)^I
\end{align*}
where $K^I_S,\bar{K}_I^S \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$. Apply the homomorphism $Sq$ and recall $\sigma_S=Sq(\tau_S)$ from Theorem \ref{thm:RealSchub}, then
\begin{align*}
(p^T)^I &= \sum_S Sq(K^I_S) \sigma_S\\
\sigma_S &= \sum_I Sq(\bar{K}_I^S) (p^T)^I
\end{align*}
where $Sq(K^I_S),Sq(\bar{K}_I^S) \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$.
Since $\{\sigma_S\}$ give a basis of $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))$, the above transformations imply:
\begin{thm}[Equivariant characteristic basis of real Grassmannians]\label{thm:EquivCharReal}
The set of monomials $(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ forms an additive $H^*_T(pt)$-basis for $H^*_{T^n}(G_{2k}(\mathbb{R}^{2n}))\cong H^*_{T^n}(G_{2k}(\mathbb{R}^{2n+1}))\cong H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+1}))$. Together with the set of monomials $r^T\cdot(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}$, they form an additive $H^*_T(pt)$-basis for $H^*_{T^n}(G_{2k+1}(\mathbb{R}^{2n+2}))$.
\end{thm}
Now we can give the Leray-Borel description for equivariant cohomology of real Grassmannians.
\begin{thm}[Equivariant Leray-Borel description of even dimensional real Grassmannians]\label{thm:EquivBReal}
For the even dimensional real Grassmannians $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+1})$ with $T^n$-actions, their equivariant cohomology is the same:
\[
H^*_T(G_{2k}(\mathbb{R}^{2n}),\mathbb{Q})\cong\frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k}]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i)}.
\]
\end{thm}
\begin{proof}
Apply the $Sq$ to the generators and relations in the Leray-Borel description of $H^*_T(G_k(\mathbb{C}^n))$.
\end{proof}
\begin{thm}[Equivariant Leray-Borel description of odd dimensional real Grassmannians]\label{thm:EquivBReal2}
For the odd dimensional real Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$ with $T^n$-actions, the equivariant cohomology is:
\[
H^*_T(G_{2k+1}(\mathbb{R}^{2n+2}),\mathbb{Q})\cong\frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};r^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\,(r^T)^2=0}.
\]
\end{thm}
For a $T^n$-equivariantly formal space $M$, we can recover the ordinary cohomology from the equivariant cohomology by $H^*(M,\mathbb{Q})=H_T^*(M,\mathbb{Q})\otimes_{\mathbb{Q}[\alpha_1,\ldots,\alpha_n]} \mathbb{Q}$ where $\mathbb{Q}$ has a $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra structure from the constant-term morphism $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]\rightarrow \mathbb{Q}: f(\alpha_1,\ldots,\alpha_n)\mapsto f(0)$. Therefore, the above Theorem \ref{thm:EquivCharReal}, \ref{thm:EquivBReal}, \ref{thm:EquivBReal2} have ordinary versions by ignoring the $\alpha_i$.
Let $r\in H^{2n+1}(G_{2k+1}(\mathbb{R}^{2n+2}),\mathbb{Q})$ be the ordinary image of the $r^T\in H^{2n+1}_T(G_{2k+1}(\mathbb{R}^{2n+2}),\mathbb{Q})$.
\begin{cor}[Ordinary characteristic basis of real Grassmannians]
The set of monomials $(p_1)^{r_1}(p_2)^{r_2}\cdots (p_k)^{r_k}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ forms an additive basis for $H^*(G_{2k}(\mathbb{R}^{2n}))\cong H^*(G_{2k}(\mathbb{R}^{2n+1}))\cong H^*(G_{2k+1}(\mathbb{R}^{2n+1}))$. Together with the set of monomials $r\cdot(p_1)^{r_1}(p_2)^{r_2}\cdots (p_k)^{r_k}$, they form an additive basis for $H^*(G_{2k+1}(\mathbb{R}^{2n+2}))$.
\end{cor}
\begin{cor}[Ordinary Leray-Borel description of real Grassmannians]
For the even dimensional real Grassmannians $G_{2k}(\mathbb{R}^{2n})$, $G_{2k}(\mathbb{R}^{2n+1})$ and $G_{2k+1}(\mathbb{R}^{2n+1})$, their cohomology is the same:
\[
\frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k}]}{p\bar{p} = 1}.
\]
For the odd dimensional real Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$, the cohomology is:
\[
\frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};r]}{p\bar{p} = 1,\, r^2=0}
\]
where $r\in H^{2n+1}(G_{2k+2}(\mathbb{R}^{2n+2}),\mathbb{Q})$ is the ordinary image of the $r^T\in H^{2n+1}_T(G_{2k+2}(\mathbb{R}^{2n+2}),\mathbb{Q})$.
\end{cor}
\begin{rmk}
This explicit Leray-Borel description for the real Grassmannians is stated in Casian\&Kodama \cite{CK}. The even dimensional case is a special case of the Leray-Borel description, and the odd dimensional case is due to Takeuchi \cite{Ta62}.
\end{rmk}
\begin{rmk}
For $n\leq 7$, the ordinary cohomology groups of $G_k(\mathbb{R}^n)$ in $\mathbb{Z}$ coefficients were computed by Jungkind \cite{Ju79}.
\end{rmk}
\vskip 20pt
\section{Equivariant cohomology rings of oriented Grassmannians}
\vskip 15pt
In this section, we give the GKM description and Leray-Borel description of equivariant cohomology rings of oriented Grassmannians, together with the characteristic basis of the additive structure. We use the notation $\tilde{G}_k(\mathbb{R}^n)$ for the Grassmannian of $k$-dimensional oriented subspaces in $\mathbb{R}^n$.
The Pl\"{u}cker embedding of an oriented Grassmannian can be given as follows: for $V\in \tilde{G}_k(\mathbb{R}^n)$, we can choose an ordered orthonormal basis $v_1,\ldots,v_k$ of $V$, then the well-defined wedge product $v_1 \wedge \cdots \wedge v_k \in \tilde{G}_1(\wedge^k \mathbb{R}^n)=S(\wedge^k \mathbb{R}^n)$ in the unit sphere of $\wedge^k \mathbb{R}^n$ gives the embedding $\tilde{G}_k(\mathbb{R}^n) \hookrightarrow S(\wedge^k \mathbb{R}^n)$.
Similar to the case of real Grassmannians, we can consider the $T^n$-action on $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$, $\mathbb{R}^{2n+1}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ and $\mathbb{R}^{2n+2}=(\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}^2_0$ for their decompositions into weighted subspaces, where $\alpha_1,\ldots,\alpha_n$ are the standard basis of $\mathfrak{t}_\mathbb{Z}^*$. These actions induce $T^n$ actions on $\tilde{G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$. More specifically, each $t \in T$ maps $v_1 \wedge \cdots \wedge v_l$ where $l=2k,2k+1$, to $t\cdot v_1 \wedge \cdots \wedge t \cdot v_l$, and it is easy to check the map is independent from the choice of a positive orthonormal basis $v_1,\ldots,v_k$.
Also since there are natural $T^n$-diffeomorphisms $\tilde{G}_{2k}(\mathbb{R}^{2n+1})\cong \tilde{G}_{2n-2k+1}(\mathbb{R}^{2n+1})$ identifying the second and the third types of real Grassmannians, in some discussions we will only consider the three cases of $\tilde{G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$.
\subsection{Oriented Grassmannians as $2$-covers over real Grassmannians}
There are natural $2$-coverings of oriented Grassmannians over real Grassmannians $\pi:\tilde{G}_k(\mathbb{R}^n)\rightarrow G_k(\mathbb{R}^n): v_1 \wedge \cdots \wedge v_k \mapsto \mathrm{Span}_\mathbb{R}(v_1,\ldots,v_k)$ which induces a pull-back morphism $\pi^*: H^*(G_k(\mathbb{R}^n))\rightarrow H^*(\tilde{G}_k(\mathbb{R}^n))$ between their cohomology. The non-trivial deck transformation is defined by reversing orientations $\rho: \tilde{G}_k(\mathbb{R}^n)\rightarrow\tilde{G}_k(\mathbb{R}^n):v_1 \wedge \cdots \wedge v_k \mapsto -(v_1 \wedge \cdots \wedge v_k)$ which induces an isomorphism $\rho^*: H^*(\tilde{G}_k(\mathbb{R}^n)) \rightarrow H^*(\tilde{G}_k(\mathbb{R}^n))$. Both $\pi$ and $\rho$ commute with the $T$-actions that we introduced on the oriented Grassmannians and real Grassmannians.
For covering maps between compact spaces, or equivalently for free actions of finite groups, there is a well-known fact relating their cohomology in rational coefficients:
\begin{lem}
Let $\pi:X\rightarrow Y$ be a covering between compact topological spaces with a finite deck transformation group $G$ which also acts on the cohomology $H^*(X,\mathbb{Q})$. Then $\pi^*: H^*(Y,\mathbb{Q}) \rightarrow H^*(X,\mathbb{Q})$ is injective with image $H^*(X,\mathbb{Q})^G$. This conclusion is also true for equivariant cohomology if a torus $T$ acts on $X$ and commutes with the action of $G$.
\end{lem}
\begin{proof}
For a cocycle $c$ of $X$, the averaged cocycle $\frac{1}{|G|}\sum_{g\in G} gc$ is invariant under $G$-action, hence comes from a cocycle of $Y$. Consider the averaging map $\pi_*: H^*(X,\mathbb{Q}) \rightarrow H^*(Y,\mathbb{Q}):[c]\mapsto \frac{1}{|G|}[\sum_{g\in G} gc]$, then the composition $\pi_* \pi^*$ is the identity map on $H^*(Y,\mathbb{Q})$, hence $\pi^*$ is injective. Note that every cohomology class in $H^*(X,\mathbb{Q})^G$ can be represented by a $G$-invariant cocycle using the averaging method. This proves the image of $\pi^*$ is exactly $H^*(X,\mathbb{Q})^G$.
For the $T^n$-equivariant version, though the Borel construction $X\times_{T^n} (S^\infty)^n,Y\times_{T^n} (S^\infty)^n$ is not compact, we can apply the ordinary version of current Lemma to the compact approximations $X\times_{T^n} (S^N)^n, Y\times_{T^n} (S^N)^n$ for $N\rightarrow \infty$.
\end{proof}
\begin{rmk}
For the averaging method to work, we can relax the $\mathbb{Q}$ coefficients to be any coefficient ring that contains $\frac{1}{|G|}$. In $\mathbb{R}$ coefficients, the ordinary and equivariant de Rham theory together with the averaging method give a proof without using compact approximations.
\end{rmk}
Applying this Lemma to the oriented Grassmannians as $T$-equivariant $2$-covers over real Grassmannians, we get
\begin{prop}\label{prop:OrientVSReal}
The pull-back morphisms of ordinary and equivariant cohomology
\[
\pi^*: H^*(G_k(\mathbb{R}^n)) \hookrightarrow H^*(\tilde{G}_k(\mathbb{R}^n)) \qquad \text{and} \qquad H^*_T(G_k(\mathbb{R}^n)) \hookrightarrow H^*_T(\tilde{G}_k(\mathbb{R}^n))
\]
are both injective. Moreover,
\[
\pi^*(H^*(G_k(\mathbb{R}^n))) = H^*(\tilde{G}_k(\mathbb{R}^n))^{\mathbb{Z}/2} \qquad \text{and} \qquad \pi^*(H^*_T(G_k(\mathbb{R}^n))) = H^*_T(\tilde{G}_k(\mathbb{R}^n))^{\mathbb{Z}/2}
\]
identifies cohomology of real Grassmannians as the ${\mathbb{Z}/2}$-invariant subrings of cohomology of oriented Grassmannians, or equivalently as the $+1$-eigenspaces of $\rho^*$ on cohomology of oriented Grassmannians.
\end{prop}
For odd dimensional oriented Grassmannian $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$, the deck transformation $\rho:\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}) \rightarrow \tilde{G}_{2k+1}(\mathbb{R}^{2n+2}): v_1 \wedge \cdots \wedge v_{2k+1} \mapsto -(v_1 \wedge \cdots \wedge v_{2k+1})=(-v_1) \wedge \cdots \wedge (-v_{2k+1})$ is induced from the antipodal map $A:\mathbb{R}^{2n+2}\rightarrow \mathbb{R}^{2n+2}: v\mapsto -v$ which is homotopic to the identity map on $\mathbb{R}^{2n+2}$ via
\[
\begin{pmatrix}
\begin{matrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{matrix}
& &\\
&
\ddots
&\\
& &
\begin{matrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{matrix}
\end{pmatrix}
\]
which is actually $T^n$-equivariant and further induces $T^n$-equivariant homotopy between $\rho$ and $id$ on $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$.
\begin{thm}[Relations between odd dimensional oriented and real Grassmannians]\label{thm:OddGrass}
Since $\rho^*=id$ on ordinary and equivariant cohomology of odd dimensional oriented Grassmannians, we have
\begin{align*}
H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})) &= H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))^{\mathbb{Z}/2} \cong H^*(G_{2k+1}(\mathbb{R}^{2n+2}))\\ H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})) &= H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))^{\mathbb{Z}/2} \cong H^*_T(G_{2k+1}(\mathbb{R}^{2n+2})).
\end{align*}
\end{thm}
\begin{cor}
The Poincar\'e series of $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are
\[
P_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}(t)=P_{G_{2k+1}(\mathbb{R}^{2n+2})}(t)=(1+t^{2n+1})P_{G_{k}(\mathbb{C}^{n})}(t^2).
\]
\end{cor}
The relations between cohomology of oriented and real Grassmannians in even dimensions are more delicate. In next two subsections, we will try to understand the $\rho^*$-action on finer structures of the cohomology rings of oriented Grassmannians.
\subsection{GKM description of oriented Grassmannians}
The GKM description of oriented Grassmannians is very similar to that of real Grassmannians in previous section. Hence most of the details will be omitted but referred to those of real Grassmannians.
\subsubsection{Orientations and Euler classes of canonical bundle and complementary bundle}\label{subsub:Euler}
The preferred orientation on every oriented $k$-dimensional subspace in $\mathbb{R}^n$ brings new invariants.
For example, the $T^n$-fixed points of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ as $2k$-dimensional $T^n$-subrepresentation of $\mathbb{R}^{2n}=\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]}$ are of the form $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$ where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. Though an orientation on $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$ can not specify the signs of the individual weights $\alpha_i,i\in S$, it does specify the sign of the product of weights as either $\prod_{i\in S}\alpha_i$ or $-\prod_{i\in S}\alpha_i$, which is exactly the equivariant Euler class of an oriented $T$-representation over a point. We will denote $V_{S_+},V_{S_-}$ for the $T$-subrepresentation $\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]}$ with equivariant Euler classes $e^T$ as $\prod_{i\in S}\alpha_i,-\prod_{i\in S}\alpha_i$ respectively. Similarly, for $\tilde{G}_{2k}((\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0)$ we can introduce the same notations with the fixed points $V_{S_\pm}=(\oplus_{i\in S}\mathbb{R}^2_{[\alpha_i]},\pm \prod_{i\in S}\alpha_i)$.
For $\tilde{G}_{2k+1}((\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0)$, the fixed points are of the form $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ which has equivariant Euler class $0$ because of the $0$-weight space $\mathbb{R}_0$. But an orientation on $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0$ gives the complementary $T^n$-subrepresentation $\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]}$ an orientation hence an equivariant Euler class $\bar{e}^T$ either $\prod_{j \not\in S}\alpha_j$ or $-\prod_{j \not\in S}\alpha_j$. We will denote these fixed points as $V_{S_\pm}=((\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0,\pm \prod_{j \not\in S}\alpha_j)$.
For $\tilde{G}_{2k+1}((\oplus_{i=1}^{n} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0^2)$, a fixed component is of the form $\tilde{C}_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \tilde{G}_1(\mathbb{R}^2_0)\}\cong S^1$. Since both $(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0$ and its complement $(\oplus_{j \not\in S} \mathbb{R}^2_{[\alpha_j]})\oplus L_0^\perp$ have a $0$-weight part, the equivariant Euler classes of both $T^n$-subrepresentations are $0$.
\subsubsection{$1$-skeleta}
We can describe the $1$-skeleta of oriented Grassmannians:
\begin{prop}[$1$-skeleta of oriented Grassmannians]\label{prop:OrientSkeleton}
The fixed points, isotropy weights and $1$-skeletons of oriented Grassmannians can be given as
\begin{enumerate}
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n})$, there are $2\binom{n}{k}$ fixed points of the form $V_{S_\pm}=(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]},\pm \prod_{i\in S}\alpha_i)$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. The isotropy weights at both $V_{S_\pm}$ are $\{[\alpha_j\pm\alpha_i] \mid i\in S,j\not \in S\}$, among which $[\alpha_j-\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\pm}$, and $[\alpha_j+\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\mp}$.
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, there are $2\binom{n}{k}$ fixed points of the form $V_{S_\pm}=(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]},\pm \prod_{i\in S}\alpha_i)$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. The isotropy weights at both $V_{S_\pm}$ are $\{[\alpha_j\pm\alpha_i] \mid i\in S,j\not \in S\}\cup \{[\alpha_i]\mid i \in S\}$, among which $[\alpha_j-\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\pm}$, and $[\alpha_j+\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\mp}$, and $[\alpha_i]$ joins $V_{S_+}$ via a $2$-sphere to $V_{S_-}$.
\item For $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, there are $2\binom{n}{k}$ fixed points of the form $V_{S_\pm}=((\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus \mathbb{R}_0,\pm \prod_{j \not\in S}\alpha_j)$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. The isotropy weights at both $V_{S_\pm}$ are $\{[\alpha_j\pm\alpha_i] \mid i\in S,j\not \in S\} \cup \{[\alpha_j]\mid j \not \in S\}$, among which $[\alpha_j-\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\pm}$ and $[\alpha_j+\alpha_i]$ joins $V_{S_\pm}$ via a $2$-sphere to $V_{((S\smallsetminus\{i\})\cup \{j\})_\mp}$, and $[\alpha_j]$ joins $V_{S_+}$ via a $2$-sphere to $V_{S_-}$.
\item For $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$, there are $\binom{n}{k}$ fixed circles of the form $\tilde{C}_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \tilde{G}_1(\mathbb{R}^2_0)\}\cong S^1$, where $S$ is a $k$-element subset of $\{1,2,\ldots,n\}$. The isotropy weights at $\tilde{C}_S$ are $\{[\alpha_j\pm\alpha_i] \mid i\in S,j\not \in S\} \cup \{[\alpha_i]\mid i\in S\}\cup \{[\alpha_j]\mid j \not \in S\}$, among which both $[\alpha_j+\alpha_i]$ and $[\alpha_j-\alpha_i]$ join $\tilde{C}_S$ via a $S^2\times S^1$ to $\tilde{C}_{ (S\smallsetminus\{i\})\cup \{j\}}$, and $[\alpha_i],[\alpha_j]$ join $\tilde{C}_S$ via a $S^3$ to no other fixed circles.
\end{enumerate}
\end{prop}
\begin{proof}
Similar to the case of real Grassmannians.
\end{proof}
\subsubsection{GKM graphs of oriented Grassmannians}
Using the $1$-skeleta of oriented Grassmannians, we can construct their GKM graphs:
\begin{exm}
We will give some examples of GKM graphs for $\tilde{G}_k(\mathbb{R}^n)$ when $k$ or $n$ is small.
\begin{enumerate}
\item $S^{2n}$ as $\tilde{G}_{1}(\mathbb{R}^{2n+1})$ or $\tilde{G}_{2n}(\mathbb{R}^{2n+1})$
\begin{figure}
\caption{GKM graph for $S^{2n}
\caption{Condensed GKM graph for $S^{2n}
\caption{GKM graphs for $S^{2n}
\end{figure}
\item $S^{2n+1}$ as $\tilde{G}_{1}(\mathbb{R}^{2n+2})$ or $\tilde{G}_{2n+1}(\mathbb{R}^{2n+2})$
\begin{figure}
\caption{GKM graph for $S^{2n+1}
\caption{Condensed GKM graph for $S^{2n+1}
\caption{GKM graphs for $S^{2n+1}
\end{figure}
\item $\tilde{G}_{2}(\mathbb{R}^{4}),\tilde{G}_{2}(\mathbb{R}^{5}),\tilde{G}_{3}(\mathbb{R}^{5})$ as $\tilde{G}_{2k}(\mathbb{R}^{2n}), \tilde{G}_{2k}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ when $k=1,n=2$.
\begin{figure}
\caption{GKM graph for $\tilde{G}
\caption{GKM graph for $\tilde{G}
\caption{GKM graph for $\tilde{G}
\caption{GKM graphs for $\tilde{G}
\end{figure}
\item $\tilde{G}_{3}(\mathbb{R}^{6})$ as $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ when $k=1,n=2$.
\begin{figure}
\caption{GKM graph for $\tilde{G}
\caption{Condensed GKM graph for $\tilde{G}
\caption{GKM graphs for $\tilde{G}
\end{figure}
\end{enumerate}
\end{exm}
\subsubsection{Formality, cohomology and canonical basis of oriented Grassmannians}
\begin{prop}[Equivariant formality of torus actions on oriented Grassmannians]
The $T^n$-actions on all the four types of oriented Grassmannians $\tilde{G}_{2k}(\mathbb{R}^{2n}), \tilde{G}_{2k}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are equivariantly formal. All the four oriented Grassmannians have the same total Betti number $2\binom{n}{k}$.
\end{prop}
\begin{proof}
The even dimensional oriented Grassmannians $\tilde{G}_{2k}(\mathbb{R}^{2n}), \tilde{G}_{2k}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ can be viewed as homogeneous spaces of the form $G/H$ with $H$ a compact connected Lie subgroup and of the same rank as the connected compact Lie group $G$. As shown in \cite{GHZ06}, the actions of maximal tori on these homogeneous spaces are equivariantly formal. Their total Betti numbers are the same as their Euler characteristic numbers $|W_G/W_H|$ where $W_G,W_H$ are the Weyl groups of $G$ and $H$. Alternatively, we can compute the total Betti number as the number of fixed points given in Theorem \ref{prop:OrientSkeleton}, namely $2\binom{n}{k}$. The odd dimensional oriented Grassmannian $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ has the same equivariant cohomology as the real Grassmannian $G_{2k+1}(\mathbb{R}^{2n+2})$, hence is also equivariantly formal with total Betti number $2\binom{n}{k}$.
\end{proof}
After verifying GKM conditions and equivariant formality, we can give the GKM description of the torus actions on $\tilde{G}_k(\mathbb{R}^n)$ by applying the even dimensional GKM Theorem as in Guillemin, Holm and Zara \cite{GHZ06} and the odd dimensional GKM-type Theorem \ref{thm:OddGKM}.
\begin{thm}[GKM description of equivariant cohomology of oriented Grassmannians]\label{thm:GKMorientGrass}
The following congruence relations are given for any two $k$-element subsets $S,S'\subset \{1,\ldots,n\}$ differed by one element with $S\cup\{j\}=S'\cup\{i\}$.
\begin{enumerate}
\item For the oriented Grassmannian $\tilde{G}_{2k}(\mathbb{R}^{2n})$, an element of equivariant cohomology is a set of polynomials $f_{S_\pm} \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ to each vertex $S_\pm$ such that
\begin{enumerate}
\item $f_{S_+} \equiv f_{S'_+}, \quad f_{S_-} \equiv f_{S'_-} \mod \alpha_j-\alpha_i$
\item $f_{S_+} \equiv f_{S'_-}, \quad f_{S_-} \equiv f_{S'_+} \mod \alpha_j+\alpha_i$.
\end{enumerate}
\item For the oriented Grassmannian $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, an element of equivariant cohomology is a set of polynomials $f_{S_\pm} \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ to each vertex $S_\pm$ such that
\begin{enumerate}
\item $f_{S_+} \equiv f_{S'_+} , \quad f_{S_-} \equiv f_{S'_-}\mod \alpha_j-\alpha_i$
\item $f_{S_+} \equiv f_{S'_-} , \quad f_{S_-} \equiv f_{S'_+}\mod \alpha_j+\alpha_i$
\item $f_{S_+} \equiv f_{S_-} \mod \prod_{i'\in S}\alpha_{i'}$.
\end{enumerate}
\item For the oriented Grassmannian $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, an element of equivariant cohomology is a set of polynomials $f_{S_\pm} \in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ to each vertex $S_\pm$ such that
\begin{enumerate}
\item $f_{S_+} \equiv f_{S'_+} , \quad f_{S_-} \equiv f_{S'_-}\mod \alpha_j-\alpha_i$
\item $f_{S_+} \equiv f_{S'_-} , \quad f_{S_-} \equiv f_{S'_+}\mod \alpha_j+\alpha_i$
\item $f_{S_+} \equiv f_{S_-} \mod \prod_{j'\not\in S}\alpha_{j'}$.
\end{enumerate}
\item For the odd dimensional oriented Grassmannian $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ with $T^n$-action, an element of equivariant cohomology is a set of polynomial pairs $(f_S, g_S \theta)$ to each $\circ$-vertex $S$ where $\theta$ is the unit volume form of $S^1$ such that
\begin{enumerate}
\item $g_{S} \equiv 0 \mod \prod_{i=1}^{n}\alpha_i$
\item $f_{S} \equiv f_{S'} , \quad g_{S} \equiv g_{S'}\mod \alpha^2_j-\alpha^2_i$.
\end{enumerate}
\end{enumerate}
\end{thm}
For odd dimensional oriented Grassmannians, the induced deck transformation $\rho^*: H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))\rightarrow H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ is the identity map hence acts trivially on the GKM description. Solving the same set of congruence equations as in Theorem \ref{thm:AllGrass} of $H^*_T({G}_{2k+1}(\mathbb{R}^{2n+2}))$, we will also get an element $\tilde{r}^T \in H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ in GKM description localized at a fixed circle $\tilde{C}_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in \tilde{G}_1(\mathbb{R}^2_0)\}\cong S^1$ to be $\tilde{r}^T_S = (\prod_{i=1}^{n}\alpha_i) \theta_{S^1}$ similar to the ${r}^T \in H^*_T({G}_{2k+1}(\mathbb{R}^{2n+2}))$ localized at ${C}_S=\{(\oplus_{i\in S} \mathbb{R}^2_{[\alpha_i]})\oplus L_0 \mid L_0\in {G}_1(\mathbb{R}^2_0)\}\cong \mathbb{R} P^1$ to be ${r}^T_S = (\prod_{i=1}^{n}\alpha_i) \theta_{\mathbb{R} P^1}$, where $\theta_{S^1}$ and $\theta_{\mathbb{R} P^1}$ are the unit volume forms of $S^1$ and $\mathbb{R} P^1$ respectively.
\begin{prop}[Canonical basis of $H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$]\label{prop:CanoBaseIII}
Let $\sigma_{S\in \mathcal{S}}$ be the canonical basis of $H^*_T({G}_{2k}(\mathbb{R}^{2n}))$ from Theorem \ref{thm:RealSchub} and $\tilde{r}^T_S = \prod_{i=1}^{n}\alpha_i \theta_{S^1}$ be the odd-degree generator. Then $\sigma_{S}, \tilde{r}^T \cdot \sigma_{S}$ give additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-basis of $ H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$.
\end{prop}
However, there is a subtlety for the pullback $\pi^*:H^*_T({G}_{2k+1}(\mathbb{R}^{2n+2}))\rightarrow H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ though this is an isomorphism. The $2$-fold covering $\pi: \tilde{G}_{2k+1}(\mathbb{R}^{2n+2}) \rightarrow {G}_{2k+1}(\mathbb{R}^{2n+2})$ restricts to a $2$-fold covering of fixed circles $\pi: (\tilde{C}_S\cong S^1) \rightarrow ({C}_S\cong \mathbb{R} P^1)$ which will give the localized pullback $\pi^*(\theta_{\mathbb{R} P^1})=2\theta_{S^1}$. Hence we get $\pi^*({r}^T)=2\tilde{r}^T$.
\begin{prop}[The explicit pullback of cohomology between odd dimensional Grassmannians]\label{prop:Pullr}
In the canonical basis, the pullback of cohomology of odd dimensional Grassmannian is
\begin{align*}
\pi^*:H^*_T({G}_{2k+1}(\mathbb{R}^{2n+2}))&\longrightarrow H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))\\
\sigma_{S} &\longmapsto \sigma_{S}\\
{r}^T \mathbb{C}dot \sigma_{S} &\longmapsto 2\tilde{r}^T \mathbb{C}dot \sigma_{S}.
\end{align*}
\end{prop}
For the even dimensional oriented Grassmannians, the deck transformation $\rho: \tilde{G}_k(\mathbb{R}^n)\rightarrow \tilde{G}_k(\mathbb{R}^n)$ switches any fixed point ${S_+}$ with its twin fixed point ${S_-}$ by reversing orientations. Then the induced deck transformation $\rho^*:H^*_T(\tilde{G}_k(\mathbb{R}^n))\rightarrow H^*_T(\tilde{G}_k(\mathbb{R}^n))$ in GKM description will switch any polynomial $f_{S_+}$ with $f_{S_-}$. Notice the symmetry in the GKM descriptions, we see that the switch of polynomials preserves the congruence relations.
Since $(\rho^*)^2=id$, both cohomology $H^*(\tilde{G}_k(\mathbb{R}^n)),H^*_T(\tilde{G}_k(\mathbb{R}^n))$ decompose
into $\pm 1$-eigenspaces of $\rho^*$.
\begin{prop}\label{prop:RhoEigen}
For the even dimensional oriented Grassmannians $\tilde{G}_{2k}(\mathbb{R}^{2n}),\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, the elements of $+1$-eigenspace of $\rho^*$ on the equivariant cohomology can be identified as those sets of polynomials $\{f_{S_\pm}, S \in \mathcal{S}\}$ where $\mathcal{S}$ is the collection of $k$-element subsets of $\{1,\ldots,n\}$ such that
\begin{align*}
f_{S_+}=f_{S_-}
\end{align*}
and the elements of $-1$-eigenspace of $\rho^*$ are those with
\[
f_{S_+}=-f_{S_-}.
\]
\end{prop}
\begin{rmk}
As we have proved before, the $+1$-eigenspaces of $\rho^*$ on the equivariant cohomology of oriented Grassmannians are exactly the equivariant cohomology of real Grassmannians.
\end{rmk}
Recall that we defined equivariant Euler classes at each fixed point $S$ for $\tilde{G}_{2k}(\mathbb{R}^{2n}),\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ to be $e^T_{S_{\pm}}=\pm \prod_{i\in S} \alpha_i$ and for $\tilde{G}_{2k}(\mathbb{R}^{2n}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ to be $\bar{e}^T_{S_{\pm}}=\pm \prod_{j\not\in S} \alpha_j$. It is easy to check that $\{e^T_{S_\pm}, S \in \mathcal{S}\}$ and $\{\bar{e}^T_{S_\pm}, S \in \mathcal{S}\}$ are elements of the GKM description of the corresponding equivariant cohomology. Since $\rho$ changes the signs of orientations, $\rho^*$ changes the signs of the equivariant Euler classes. Therefore, $\{e^T_{S_\pm}, S \in \mathcal{S}\}$ and $\{\bar{e}^T_{S_\pm}, S \in \mathcal{S}\}$ are in the $-1$-eigenspaces of $\rho^*$. Topologically, the localized classes $e^T,\bar{e}^T$ in GKM description are exactly the equivariant Euler classes of the canonical oriented bundles and complementary oriented bundles over the oriented Grassmannians.
\begin{prop}[Equivariant Euler class and top equivariant Pontryagin class]\label{prop:EulerPont}
Similar to the relations between ordinary Euler class and top ordinary Pontryagin class,
\begin{enumerate}
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n})$ and $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, we have $(e^T)^2 = p^T_k $
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, we have $(\bar{e}^T)^2 = \bar{p}^T_{n-k}$
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n})$, we have $e^T\bar{e}^T= \prod_{i=1}^{n}\alpha_i$
\end{enumerate}
\end{prop}
\begin{proof}
Let's prove this for $\tilde{G}_{2k}(\mathbb{R}^{2n})$ which covers the remaining cases of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}), \tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$. In Proposition \ref{prop:Pont}, we have given the localized top equivariant Pontryagin classes of real Grassmannians as
\[
p^T_k|_S=\prod_{i \in S} \alpha_i^2 \qquad \qquad \bar{p}^T_{n-k}|_S=\prod_{j \not \in S} \alpha_j^2.
\]
Via the pullback $\pi^*: H^*_T({G}_{2k}(\mathbb{R}^{2n})) \rightarrow H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n}))$, the equivariant Pontryagin classes of ${G}_{2k}(\mathbb{R}^{2n})$ are identified as those of $\tilde{G}_{2k}(\mathbb{R}^{2n})$, and are in the $+1$-eigenspaces of $\rho^*$. Therefore
\[
p^T_k|_{S_\pm}=\prod_{i \in S} \alpha_i^2 \qquad \qquad \bar{p}^T_{n-k}|_{S_\pm}=\prod_{j \not \in S} \alpha_j^2.
\]
Comparing them with
\[
e^T_{S_{\pm}}=\pm \prod_{i\in S} \alpha_i \qquad \qquad \bar{e}^T_{S_{\pm}}=\pm \prod_{j\not\in S} \alpha_j
\]
we get the stated relations.
\end{proof}
The induced deck transformation $\rho^*$ is a ring homomorphism, therefore the multiplication of an element in the $-1$-eigenspace with an element in the $+1$-eigenspace results in the $-1$-eigenspace.
\begin{prop}\label{prop:EulerMult}
Multiplication with the equivariant Euler classes $e^T,\bar{e}^T$ maps $+1$-eigenspaces of $\rho^*$ to $-1$-eigenspaces.
\begin{enumerate}
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, the multiplication with $e^T$ is an isomorphism between $+1$-eigenspace of $\rho^*$ to its $-1$-eigenspace.
\item For $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, the multiplication with $\bar{e}^T$ is an isomorphism between $+1$-eigenspace of $\rho^*$ to its $-1$-eigenspace.
\end{enumerate}
\end{prop}
\begin{proof}
For $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, denote $V_{+1}$ and $V_{-1}$ be the $+1$ and $-1$-eigenspaces of $\rho^*$ on $H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1}))$. The fact that $e^T$ is in the $-1$-eigenspace gives the multiplication $\times e^T: V_{+1}\rightarrow V_{-1}$. On the other hand, every element $\{f_{S_\pm}, S \in \mathcal{S}\}$ of $V_{-1}$ has the form $f_{S_+}=-f_{S_-}$ by Prop \ref{prop:RhoEigen}. Plug this into the congruence relation between $S_+$ and $S_-$ in Theorem \ref{thm:GKMorientGrass}, we get
\[
f_{S_+} \equiv f_{S_-}=-f_{S_+} \mod \prod_{i\in S}\alpha_{i}
\]
or equivalently, both $f_{S_+}$ and $f_{S_-}$ are multiples of $e^T_{S_{\pm}}=\pm \prod_{i\in S} \alpha_i$. Therefore, the localized quotients $f_{S_+}/e^T_{S_+}, f_{S_-}/e^T_{S_-}\in \mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ are polynomials, and this defines a unique element $f/e^T \in V_{+1}$.
The case of $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ is similar.
\end{proof}
\begin{rmk}
For $\tilde{G}_{2k}(\mathbb{R}^{2n})$, neither the multiplication by $e^T$ nor by $\bar{e}^T$ are isomorphisms between the $+1$ and $-1$-eigenspaces of $\rho^*$. We will try to understand the equivariant cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ in next subsection.
\end{rmk}
The above isomorphism between eigenspaces of $\rho^*$, together with the canonical basis $\sigma_{S}$ of $H^*_T({G}_{2k}(\mathbb{R}^{2n+1}))$ and $H^*_T({G}_{2k+1}(\mathbb{R}^{2n+1}))$, give
\begin{prop}[Canonical basis of $H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1})),H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}))$]\label{prop:CanoBaseII}
Let $\sigma_{S\in \mathcal{S}}$ be the canonical basis of $H^*_T({G}_{2k}(\mathbb{R}^{2n+1}))$ and $H^*_T({G}_{2k+1}(\mathbb{R}^{2n+1}))$ from Theorem \ref{thm:RealSchub}. Then $\sigma_{S}, e^T \cdot \sigma_{S}$ and $\sigma_{S}, \bar{e}^T \cdot \sigma_{S}$ give additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-basis of $ H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1}))$ and $H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}))$ respectively.
\end{prop}
\begin{cor}
The Poincar\'e series of $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ are
\begin{align*}
P_{\tilde{G}_{2k}(\mathbb{R}^{2n+1})}(t)&=(1+t^{2k})P_{{G}_{2k}(\mathbb{R}^{2n+1})}(t)=(1+t^{2k})P_{G_{k}(\mathbb{C}^{n})}(t^2)\\
P_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})}(t)&=(1+t^{2n-2k})P_{{G}_{2k+1}(\mathbb{R}^{2n+1})}(t)=(1+t^{2n-2k})P_{G_{k}(\mathbb{C}^{n})}(t^2).
\end{align*}
\end{cor}
\begin{cor}[Relations between some oriented and real Grassmannians]\label{thm:Type2Grass}
The equivariant cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ are $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-algebra extensions by $e^T,\bar{e}^T$ of the equivariant cohomology of ${G}_{2k}(\mathbb{R}^{2n+1})$ and ${G}_{2k+1}(\mathbb{R}^{2n+1})$, i.e.
\begin{align*}
H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1}))&\cong \frac{H^*_T({G}_{2k}(\mathbb{R}^{2n+1}))[e^T]}{(e^T)^2 = p^T_k} \\
H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}))&\cong \frac{H^*_T({G}_{2k+1}(\mathbb{R}^{2n+1}))[\bar{e}^T]}{(\bar{e}^T)^2 = \bar{p}^T_{n-k}}.
\end{align*}
\end{cor}
\begin{proof}
Using Prop \ref{prop:EulerPont}, \ref{prop:CanoBaseII} and dimension counting.
\end{proof}
\subsection{Leray-Borel description of oriented Grassmannians}
In this subsection, we will confirm the ring generators of equivariant cohomology of oriented Grassmannians to be characteristic classes, then determine the complete relations among them, and also give additive basis.
\subsubsection{Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$}
From Theorem \ref{thm:OddGrass} and Theorem \ref{thm:Type2Grass}, we have seen that equivariant cohomology rings of $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$, $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are ring extensions of the equivariant cohomology of their real counterparts. Hence the equivariant Leray-Borel descriptions and equivariant characteristic basis of those oriented Grassmannians can be extended from the related real Grassmannians.
\begin{thm}[Equivariant Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$]
The equivariant cohomology rings of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are generated by equivariant Pontryagin and Euler classes, and an odd-degree class $\tilde{r}^T$:
\begin{align*}
H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1})) &\cong \frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};e^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\, (e^T)^2=p^T_k}\\
H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})) &\cong \frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};\bar{e}^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\, (\bar{e}^T)^2=\bar{p}^T_{n-k}}\\
H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))&\cong\frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};\tilde{r}^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\,(\tilde{r}^T)^2=0}.
\end{align*}
\end{thm}
\begin{thm}[Equivariant characteristic basis of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$]\label{thm:EquivCharOrient}
The sets of monomials $\{(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k},\,e^T\cdot(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}\}$, $\{(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k},\,\bar{e}^T\cdot(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}\}$ and $\{(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k},\,\tilde{r}^T\cdot(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}\}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ form additive $H^*_T(pt)$-basis for $H^*_T(\tilde{G}_{2k}(\mathbb{R}^{2n+1})),H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})),H^*_T(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ respectively.
\end{thm}
The above two theorems of equivariant ring generators and equivariant additive basis both have their ordinary versions by replacing $\alpha_i$ with $0$.
\begin{cor}[Ordinary Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$]
The ordinary cohomology rings of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ are generated by Pontryagin and Euler classes, and an odd-degree class $\tilde{r}$:
\begin{align*}
H^*(\tilde{G}_{2k}(\mathbb{R}^{2n+1})) &\cong \frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};e]}{p\bar{p} = 1,\, e^2=p_k}\\
H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})) &\cong \frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};\bar{e}]}{p\bar{p} = 1,\, \bar{e}^2=\bar{p}_{n-k}}\\
H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))&\cong\frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};\tilde{r}]}{p\bar{p} = 1,\, \tilde{r}^2=0}.
\end{align*}
\end{cor}
\begin{cor}[Ordinary characteristic basis of $\tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$]
The sets of monomials $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k},\,e\cdot p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\}$, $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k},\,\bar{e}\cdot p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\}$ and $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k},\,\tilde{r}\cdot p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\}$ satisfying the condition $\sum_{i=1}^{k} r_i \leq n-k$ form additive basis for $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n+1})),H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})),H^*(\tilde{G}_{2k+1}(\mathbb{R}^{2n+2}))$ respectively.
\end{cor}
\subsubsection{Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n})$}
Now let's turn to the remaining type of oriented Grassmannian $\tilde{G}_{2k}(\mathbb{R}^{2n})$. As we remarked in previous subsection, neither the multiplication by $e^T$ nor by $\bar{e}^T$ are isomorphisms between eigenspaces of $\rho^*$. However, we will show the multiplications by $e^T$ and $\bar{e}^T$, restricted on certain carefully chosen subspaces, do give isomorphism between $+1$ and $-1$-eigenspaces of $\rho^*$.
Notice the equivariant diffeomorphism ${G}_{2k}(\mathbb{R}^{2n}) \cong {G}_{2n-2k}(\mathbb{R}^{2n})$ by mapping an oriented $2k$-dimensional subspace to its perpendicular oriented $(2n-2k)$-dimensional subspace. Then the complementary characteristic monomials $(\bar{p}^T_1)^{r_1}(\bar{p}^T_2)^{r_2}\cdots (\bar{p}^T_{n-k})^{r_{n-k}}$ and $\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}$, satisfying the condition $\sum_{i=1}^{n-k} r_i \leq k$, give additive basis for the equivariant and respectively ordinary cohomology of ${G}_{2k}(\mathbb{R}^{2n})$. Also recall from Prop \ref{prop:EulerPont} on the relations among top Pontryagin classes and Euler classes of the oriented Grassmannian $\tilde{G}_{2k}(\mathbb{R}^{2n})$ that $(e^T)^2=p^T_k,(\bar{e}^T)^2=\bar{p}^T_{n-k},e^T\bar{e}^T=\prod_{i=1}^{n}\alpha_i$ and $e^2=p_k,\bar{e}^2=\bar{p}_{n-k},e\bar{e}=0$.
\begin{prop}[Eigenspaces of $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$]\label{prop:OrientEigen}
Let $\rho$ be the non-trivial deck transformation of the covering $\pi: \tilde{G}_{2k}(\mathbb{R}^{2n})\rightarrow {G}_{2k}(\mathbb{R}^{2n})$ and identify $H^*({G}_{2k}(\mathbb{R}^{2n}))$ as the $+1$-eigenspace of $\rho^*$ on $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$.
\begin{enumerate}
\item The multiplications by $e^T$ and $\bar{e}^T$ are isomorphisms restricted on the following subspaces
\begin{align*}
e \times: \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) & \overset{\cong}{\longrightarrow}
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)\\
\bar{e} \times: \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1) & \overset{\cong}{\longrightarrow}
\bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1).
\end{align*}
\item $e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \oplus \bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ is the $(-1)$-eigenspace of $\rho^*$
\item $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) \cap \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))=0$ and $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) \oplus \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ is the $(-1)$-eigenspace of $\rho^*$
\item The kernels of $e \times$ and $\bar{e} \times$ on $H^*({G}_{2k}(\mathbb{R}^{2n}))$ are $\bar{p}_{n-k}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ and $p_k \cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)$ respectively
\item The following spaces are identical
\begin{align*}
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) =
e\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k) = e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))\\
\bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1) =
\bar{e}\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k) =
\bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n})).
\end{align*}
\end{enumerate}
\end{prop}
\begin{proof}
Note that the total Betti numbers of $H^*({G}_{2k}(\mathbb{R}^{2n}))$ and $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$ are $\binom{n}{k}$ and $2\binom{n}{k}$ respectively, hence the dimension of the $-1$-eigenspace of $\rho^*$ is $\binom{n}{k}$.
\begin{enumerate}
\item The composition of the surjective linear maps
\begin{align*}
&e \times: \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \longrightarrow
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)\\
&e \times: e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \longrightarrow e^2\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)
\end{align*}
is
\[
p_k\times: \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \longrightarrow
p_k\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)
\]
where we have used the relation $e^2=p_k$. The composition maps a sub-basis of $H^*({G}_{2k}(\mathbb{R}^{2n}))$ onto another sub-basis without common vectors, hence is a bijection. Therefore, each individual surjection is a bijection. Similarly, we get the bijection for the restricted $\bar{e} \times$.
\item
We have seen from the above that
\[
e \times: e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \longrightarrow p_k\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)
\]
is a bijection. However, $e \times$ takes $\bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ to zero, because $e\bar{e}=0$. Hence
\[
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \cap \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n})) = 0.
\]
Similarly,
\[
\bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1) \cap e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) = 0.
\]
Combine these two, we get
\[
e\cdot \mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \cap \bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1) = 0.
\]
However, as a subspace in $-1$-eigenspace of $\rho^*$, the sum $e\cdot\mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1) \oplus \bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ has dimension $\binom{n-1}{k}+\binom{n-1}{n-k}=\binom{n}{k}$ the same as dimension of the entire $-1$-eigenspace of $\rho^*$, hence is exactly the $-1$-eigenspace of $\rho^*$.
\item
The above series of zero intersections force $e\cdot\mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)=e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ and $\bar{e}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)=\bar{e}\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$. Hence we get $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) \cap \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))=0$ and $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n})) \oplus \bar{e} \cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ is the $(-1)$-eigenspace of $\rho^*$.
\item
We have proved $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))=e\cdot\mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)$ and they are of dimension $\binom{n-1}{k}$. Since $H^*({G}_{2k}(\mathbb{R}^{2n}))$ is of dimension $\binom{n}{k}$, the kernel of $e\times$ on $H^*({G}_{2k}(\mathbb{R}^{2n}))$ is then of dimension $\binom{n}{k}-\binom{n-1}{k}=\binom{n-1}{n-k}$. Because $e\cdot \bar{p}_{n-k}=e\cdot \bar{e}^2=0$, the subspace $\bar{p}_{n-k}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ of dimension $\binom{n-1}{n-k}$ is clearly in the kernel of $e\times$ on $H^*({G}_{2k}(\mathbb{R}^{2n}))$, hence is exactly the kernel. Similarly, we obtain the kernel of $\bar{e}\times$ on $H^*({G}_{2k}(\mathbb{R}^{2n}))$.
\item
The $e\times$-kernel subspace $\bar{p}_{n-k}\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k}^{r_{n-k}}\mid \sum_{i=1}^{n-k} r_i \leq k-1)$ of $H^*({G}_{2k}(\mathbb{R}^{2n}))$ has complementary subspace $\mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k)$. Hence the restriction
\[
e\times: \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k) \longrightarrow e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))
\]
is bijection, therefore $e\cdot \mathrm{Span}(\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k) = e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$. The identification $e\cdot\mathrm{Span}(p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k-1)=e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ is proved in (3). Similarly, we get the identifications for $\bar{e}\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$.
\end{enumerate}
\end{proof}
The detailed discussion of $e\times$ and $\bar{e}\times$ between the eigenspaces of $\rho^*$ gives:
\begin{cor}[Ordinary characteristic basis of $\tilde{G}_{2k}(\mathbb{R}^{2n})$]
The ordinary cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ is generated by Pontryagin classes and Euler classes with an additive basis $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k\}$ for the $+1$-eigenspace of $\rho^*$ and $\{e\cdot\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}\cup \{ \bar{e}\cdot p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ for the $-1$-eigenspace.
\end{cor}
\begin{rmk}
Using the various identifications of $e\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ and $\bar{e}\cdot H^*({G}_{2k}(\mathbb{R}^{2n}))$ in Theorem \ref{prop:OrientEigen}, we can also give the additive basis of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ in other forms.
\end{rmk}
\begin{cor}
The Poincar\'e series of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ are
\begin{align*}
P_{\tilde{G}_{2k}(\mathbb{R}^{2n})}(t)
&=P_{{G}_{2k}(\mathbb{R}^{2n})}(t)+t^{2k}P_{{G}_{2k}(\mathbb{R}^{2n-2})}(t)+t^{2n-2k}P_{{G}_{2k-2}(\mathbb{R}^{2n-2})}(t)\\
&=P_{G_{k}(\mathbb{C}^{n})}(t^2)+t^{2k}P_{G_{k}(\mathbb{C}^{n-1})}(t^2)+t^{2n-2k}P_{G_{k-1}(\mathbb{C}^{n-1})}(t^2).
\end{align*}
\end{cor}
\begin{proof}
Notice that the $-1$-eigenbasis $\{e\cdot\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}\cup \{ \bar{e}\cdot p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ has factors $\{\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}$ and $ \{p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ which also appear as the additive basis of $H^*({G}_{2k}(\mathbb{R}^{2n-2}))$ and $H^*({G}_{2k-2}(\mathbb{R}^{2n-2}))$ respectively.
\end{proof}
\begin{rmk}
The Poincar\'e series of even dimensional oriented Grassmannians $\tilde{G}_{2k}(\mathbb{R}^{2n}), \tilde{G}_{2k}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ were already computed by H. Cartan \cite{Car50}.
\end{rmk}
\begin{thm}[Ordinary Leray-Borel description of $\tilde{G}_{2k}(\mathbb{R}^{2n})$]
The ordinary cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ is a ring extension of the ordinary cohomology of ${G}_{2k}(\mathbb{R}^{2n})$:
\[
H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))\cong \frac{H^*({G}_{2k}(\mathbb{R}^{2n}))[e,\bar{e}]}{e^2=p_k,\,\bar{e}^2=\bar{p}_{n-k},\,e\bar{e}=0} \cong \frac{\mathbb{Q}[p_1,p_2,\ldots,p_k;\bar{p}_1,\bar{p}_2,\ldots,\bar{p}_{n-k};e,\bar{e}]}{p\bar{p} = 1,\, e^2=p_k,\,\bar{e}^2=\bar{p}_{n-k},\,e\bar{e}=0}.
\]
\end{thm}
\begin{proof}
Consider the ring homomorphism
\[
\frac{H^*({G}_{2k}(\mathbb{R}^{2n}))[e,\bar{e}]}{e^2=p_k,\,\bar{e}^2=\bar{p}_{n-k},\,e\bar{e}=0} \longrightarrow H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))
\]
which sends Pontryagin classes of ${G}_{2k}(\mathbb{R}^{2n})$ to the corresponding Pontryagin classes of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ and sends the abstract symbols $e,\bar{e}$ to the actual Euler classes of the oriented canonical bundle and complementary bundle over $\tilde{G}_{2k}(\mathbb{R}^{2n})$. Since we have proved that $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$ is generated by Pontryagin classes and Euler classes, the above morphism is surjective. It is easy check that $H^*({G}_{2k}(\mathbb{R}^{2n}))[e,\bar{e}]/\{e^2=p_k,\,\bar{e}^2=\bar{p}_{n-k},\,e\bar{e}=0\}$ also has the same additive basis $\{p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k\}\cup\{e\cdot\bar{p}_1^{r_1}\bar{p}_2^{r_2}\cdots \bar{p}_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}\cup \{ \bar{e}\cdot p_1^{r_1}p_2^{r_2}\cdots p_{k-1}^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ as $H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$. Hence, we get a ring isomorphism.
\end{proof}
Since $T^n$ acts on $\tilde{G}_{2k}(\mathbb{R}^{2n})$ equivariantly formal, i.e. $H_T^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))\cong\mathbb{Q}[\alpha_1,\dots,\alpha_n]\otimes_\mathbb{Q} H^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))$ as $\mathbb{Q}[\alpha_1,\dots,\alpha_n]$-modules, we can lift the ordinary basis, characteristic classes and relations to be equivariant, then obtain the equivariant versions of characteristic basis and Leray-Borel description:
\begin{cor}[Equivariant Leray-Borel description and characteristic basis of $\tilde{G}_{2k}(\mathbb{R}^{2n})$]
The equivariant cohomology of $\tilde{G}_{2k}(\mathbb{R}^{2n})$ is a ring extension of the equivariant cohomology of ${G}_{2k}(\mathbb{R}^{2n})$:
\begin{align*}
H_T^*(\tilde{G}_{2k}(\mathbb{R}^{2n}))
&\cong \frac{H_T^*({G}_{2k}(\mathbb{R}^{2n}))[e^T,\bar{e}^T]}{(e^T)^2=p^T_k,\,(\bar{e}^T)^2=\bar{p}^T_{n-k},\,e^T\bar{e}^T=\prod_{i=1}^{n}\alpha_i} \\
&\cong \frac{\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n][p^T_1,p^T_2,\ldots,p^T_k;\bar{p}^T_1,\bar{p}^T_2,\ldots,\bar{p}^T_{n-k};e^T,\bar{e}^T]}{p^T\bar{p}^T = \prod_{i=1}^{n}(1+\alpha^2_i),\,(e^T)^2=p^T_k,\,(\bar{e}^T)^2=\bar{p}^T_{n-k},\,e^T\bar{e}^T=\prod_{i=1}^{n}\alpha_i}
\end{align*}
with additive $\mathbb{Q}[\alpha_1,\alpha_2,\ldots,\alpha_n]$-basis $\{(p^T_1)^{r_1}(p^T_2)^{r_2}\cdots (p^T_k)^{r_k}\mid \sum_{i=1}^{k} r_i \leq n-k\}$ for the $+1$-eigenspace of $\rho^*$ and $\{e^T\cdot(\bar{p}^T)_1^{r_1}(\bar{p}^T)_2^{r_2}\cdots (\bar{p}^T)_{n-k-1}^{r_{n-k-1}}\mid \sum_{i=1}^{n-k-1} r_i \leq k\}\cup \{ \bar{e}^T\cdot (p_1^T)^{r_1}(p_2^T)^{r_2}\cdots (p_{k-1}^T)^{r_{k-1}}\mid \sum_{i=1}^{k-1} r_i \leq n-k\}$ for the $-1$-eigenspace.
\end{cor}
\begin{rmk}
The ordinary cohomology groups of $\tilde{G}_k(\mathbb{R}^n)$ in $\mathbb{Z}$ coefficients for $n\leq 8$ were computed by Jungkind \cite{Ju79}. The ordinary cohomology rings of $\tilde{G}_k(\mathbb{R}^n)$ in $\mathbb{R}$ coefficients for $k=2$ were computed by Shi\&Zhou \cite{SZ14}.
\end{rmk}
\subsubsection{Characteristic numbers of orientable Grassmannians}
All the oriented Grassmannians are canonically oriented. Among the real Grassmannians, only $G_{2k}(\mathbb{R}^{2n})$ and $G_{2k+1}(\mathbb{R}^{2n+2})$ have nonzero top Betti numbers and hence are orientable. We can integrate equivariant cohomology classes on these Grassmannians using the Atiyah-Bott-Berline-Vergne(ABBV) localization formula \ref{ABBV}. According to the additive $\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$-module structures of equivariant cohomology of theses Grassmannians, we shall need to understand the integration of equivariant characteristic classes in various cases for any multi-index $I=(i_1,\ldots,i_k)$: $(p^T)^I,\,e^T\cdot(p^T)^I,\,\bar{e}^T\cdot(p^T)^I,\,r^T\cdot(p^T)^I,\,\tilde{r}^T\cdot(p^T)^I$.
The equivariant Pontryagin classes of canonical bundles, complementary bundles and tangent bundles are given in Prop \ref{prop:Pont}. The equivariant Euler classes of canonical bundles and complementary bundles are given in Subsubsection \ref{subsub:Euler}. The $r^T,\tilde{r}^T$ are given in Theorem \ref{thm:AllGrass} and Prop \ref{prop:CanoBaseIII}. In order to apply the ABBV formula, we need a localized expression for the equivariant Euler class of normal bundle at each fixed point or fixed circle.
\begin{prop}
Let $S$ be a $k$-element subset of $\{1,\ldots,n\}$, the equivariant Euler class of normal bundle at a fixed point or fixed circle associated to $S,S_\pm$ is
\begin{enumerate}
\item For $G_{2k}(\mathbb{R}^{2n}),\tilde{G}_{2k}(\mathbb{R}^{2n})$,
\[
e^N_{S_\pm} = e^N_S = \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i).
\]
\item For $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$,
\[
e^N_{S_\pm} = \pm \prod_{l\in S}\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i).
\]
\item For $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$,
\[
e^N_{S_\pm} = \pm \prod_{l \not\in S}\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i).
\]
\item For ${G}_{2k+1}(\mathbb{R}^{2n+1}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$,
\[
e^N_S = \prod_{l=1}^n\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i).
\]
\end{enumerate}
\end{prop}
\begin{proof}
The tangent spaces with weight decomposition at each fixed point or fixed circle of real Grassmannians (hence also oriented Grassmannians) are given in Subsubsection \ref{subsubsec:IsoWeights}, therefore we get the equivariant Euler classes of normal bundles up to signs as the expressions claimed in current Proposition. To resolve the sign ambiguity, we just need to note that the claimed expressions are invariant under the Weyl groups of the oriented Grassmannians as homogeneous spaces $G/H$, and also invariant under the deck transformation $\rho^*$.
\end{proof}
Next, we will compute and relate equivariant characteristic numbers of different Grassmannians.
\begin{thm}[Equivariant characteristic numbers of real\&oriented Grassmannians]
Let $I=(i_1,\ldots,i_k)$ be a multi-index and $\mathcal{S}$ be the collection of all $k$-element subsets of $\{1,\ldots,n\}$, then
\begin{align*}
&\int_{\tilde{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I=\int_{\tilde{G}_{2k}(\mathbb{R}^{2n+1})}e^T \cdot (p^T)^I=\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})}\bar{e}^T \cdot (p^T)^I\\
=&2\int_{{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I=2\int_{{G}_{2k+1}(\mathbb{R}^{2n+2})}r^T\cdot (p^T)^I=2\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}\tilde{r}^T\cdot (p^T)^I\\
=& 2 \sum_{S \in \mathcal{S}}\frac{\big((p_1^T)^{i_1}\cdots(p_k^T)^{i_k}\big)|_S}{\prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}.
\end{align*}
\end{thm}
\begin{proof}
When applying the ABBV localization formula \ref{ABBV}, besides the localized Pontryagin classes, we just need to observe that for ${G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$, they respectively have
\begin{align*}
\frac{e^T_{S}}{e^N_{S}}&=\frac{1}{\prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)} &
\frac{e^T_{S_\pm}}{e^N_{S_\pm}}&=\frac{1}{\prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}\\
\frac{e^T_{S_\pm}}{e^N_{S_\pm}}&=\frac{\pm \prod_{l\in S}\alpha_l}{\pm \prod_{l\in S}\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)} &
\frac{\bar{e}^T_{S_\pm}}{e^N_{S_\pm}}&=\frac{\pm \prod_{l\not \in S}\alpha_l}{\pm \prod_{l\not\in S}\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}
\end{align*}
for ${G}_{2k+1}(\mathbb{R}^{2n+2}),\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$, they respectively have
\begin{align*}
\frac{\int r^T_S}{e^N_S} &=\frac{\prod_{l=1}^n\alpha_l \int_{S^1}\theta_{S^1}}{\prod_{l=1}^n\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)} &
\frac{\int \tilde{r}^T_S}{e^N_S} &=\frac{\prod_{l=1}^n\alpha_l \int_{\mathbb{R} P^1}\theta_{\mathbb{R} P^1}}{\prod_{l=1}^n\alpha_l \prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}.
\end{align*}
All these fractions are equal to $\frac{1}{\prod_{i\in S}\prod_{j \not \in S}(\alpha^2_j-\alpha^2_i)}$. The difference by factor of $2$ comes from the fact that $\tilde{G}_{2k}(\mathbb{R}^{2n})$, $\tilde{G}_{2k}(\mathbb{R}^{2n+1})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})$ have fixed points indexed by $S_\pm$, while ${G}_{2k}(\mathbb{R}^{2n})$, ${G}_{2k+1}(\mathbb{R}^{2n+2})$ and $\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})$ have fixed points or circles indexed by $S$.
\end{proof}
\begin{rmk}
When the cohomological degree of a characteristic polynomial matches with the dimension of a Grassmannian, or equivalently $\sum_{j=1}^{k}j\cdot i_j=k(n-k)$, then the equivariant characteristic number will be a constant, i.e. an ordinary characteristic number. Moreover, we then get a formula of the ordinary characteristic numbers by substituting any $\alpha_i=a_i \in \mathbb{R}$ such that $a_i\not =0, a_i \not = \pm a_j$ into the localized expression of ABBV formula. For instance, we can choose $\alpha_i=i,\forall i$, or $\alpha_i=\sqrt{i},\forall i$. Moreover, we have the relations between ordinary Pontryagin characteristic numbers:
\begin{align*}
&\int_{\tilde{G}_{2k}(\mathbb{R}^{2n})}p^I=\int_{\tilde{G}_{2k}(\mathbb{R}^{2n+1})}e \cdot p^I=\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+1})}\bar{e} \cdot p^I\\
=&2\int_{{G}_{2k}(\mathbb{R}^{2n})}p^I=2\int_{{G}_{2k+1}(\mathbb{R}^{2n+2})}r\cdot p^I=2\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}\tilde{r}\cdot p^I.
\end{align*}
\end{rmk}
\begin{rmk}
Recall that localized equivariant Pontryagin class can be obtained from localized equivariant Chern class by replacing $\alpha_i$ with $\alpha^2_i$. Let $Sq:\mathbb{Q}[\alpha_1,\ldots,\alpha_n]$ be the ring homomorphism by sending $\alpha_i$ to $\alpha^2_i$ for every $i$. Then we have
\[
\int_{{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I = Sq\big(\int_{{G}_{k}(\mathbb{C}^{n})}(c^T)^I\big).
\]
Since the ring homomorphism $Sq$ keeps rational numbers unchanged, we have the relation of ordinary Pontryagin numbers and Chern numbers
\[
\int_{{G}_{2k}(\mathbb{R}^{2n})}p^I = \int_{{G}_{k}(\mathbb{C}^{n})}c^I = \sum_{S\in \mathcal{S}} \frac{e^{i_1}_1(S)\cdots e^{i_k}_k(S)}{\prod_{i\in S}\prod_{j\not\in S} (j - i)}
\]
where the second identity is given in Cor \ref{thm:OrdChernNum}.
\end{rmk}
\begin{rmk}
Consider the $2$-covers of orientable Grassmannians $\pi: \tilde{G}_{2k}(\mathbb{R}^{2n}) \rightarrow {G}_{2k}(\mathbb{R}^{2n})$ and $\pi: \tilde{G}_{2k+1}(\mathbb{R}^{2n+2}) \rightarrow {G}_{2k+1}(\mathbb{R}^{2n+2})$. By the naturality of equivariant Pontryagin classes and the above relations among equivariant characteristic numbers, we see
\[
\int_{\tilde{G}_{2k}(\mathbb{R}^{2n})}\pi^*\big((p^T)^I\big)=\int_{\tilde{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I = 2\int_{{G}_{2k}(\mathbb{R}^{2n})}(p^T)^I.
\]
From Prop \ref{prop:Pullr}, we have $ \pi^*{r}^T=2\tilde{r}^T$ , then
\[
\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}\pi^*\big({r}^T\cdot (p^T)^I\big) = 2\int_{\tilde{G}_{2k+1}(\mathbb{R}^{2n+2})}\tilde{r}^T\cdot (p^T)^I = 2\int_{{G}_{2k+1}(\mathbb{R}^{2n+2})}r^T\cdot (p^T)^I.
\]
\end{rmk}
\vskip 20pt
\end{document} |
\betaegin{document}
\tauhispagestyle{plain}
\betaegin{center}
{\langlembdaarge \betaf \!\!\uparrowpercase{Reversible sequences of cardinals, reversible\\[1mm]
equivalence relations, and similar structures}}
\etand{center}
\betaegin{center}
{\betaf Milo\v s S.\ Kurili\'c\varphiootnote{Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad,
Trg Dositeja Obradovi\'ca 4, 21000 Novi Sad, Serbia.
email: [email protected]}
and Nenad Mora\v ca\varphiootnote{Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad,
Trg Dositeja Obradovi\'ca 4, 21000 Novi Sad, Serbia.
email:
[email protected]}}
\etand{center}
\betaegin{abstract}
\noindent
A relational structure ${\mathbb X}$ is said to be reversible iff every bijective endomorphism $f:X\rhoightarrow X$ is an automorphism.
We define a sequence of non-zero cardinals $\langlembdaa \kappa _i :i\in I\rhoa$ to be reversible
iff each surjection $f :I\rhoightarrow I$ such that
$\kappa _j =\sigmaum _{i\in f^{-1}[\{ j \}]}\kappa_i$, for all $j\in I $, is a bijection, and characterize such sequences:
either $\langlembdaa \kappa _i :i\in I\rhoa$ is a finite-to-one sequence, or $\kappa _i\in {\mathbb N}$, for all $i\in I$,
$K:=\{ m\in {\mathbb N} : \kappa _i =m ,\mubox{ for infinitely many } i\in I \}$
is a non-empty independent set,
and $\gammacd (K)$ divides at most finitely many elements of the set $\{ \kappa _i :i\in I \}$.
We
isolate a class of binary structures such that a structure from the class is reversible iff
the sequence of cardinalities of its connectivity components is reversible.
In particular, we characterize reversible equivalence relations, reversible posets which are
disjoint unions of cardinals $\langlembdaeq \omega$, and some similar structures. In addition, we show that
a poset with linearly ordered connectivity components is reversible, if the corresponding
sequence of cardinalities is reversible and, using this fact, detect a wide class of examples of reversible posets and topological spaces.
{\sigmal 2010 MSC}:
03C50,
03C07,
03E05,
06A06,
05C20,
05C40.
{\sigmal Key words}: reversible sequence of cardinals, reversible sequence of natural numbers, reversible equivalence relation, digraph, poset.
\etand{abstract}
\sigmaection{Introduction}\langlembdaabel{S1}
A structure is called reversible iff all its bijective endomorphisms are automorphisms
and the class of reversible structures contains, for example, Euclidean, compact and many other relevant topological spaces {\mathfrak{c}}ite{RajWil,DoyHoc,Dow},
linear orders, Boolean lattices, well founded posets with finite levels {\mathfrak{c}}ite{Kuk,Kuk1},
tournaments, Henson graphs {\mathfrak{c}}ite{KuMoExtr}, and Henson digraphs {\mathfrak{c}}ite{KRet}.
In addition, reversible structures have several distinguished properties; for example, the Cantor-Schr\"{o}der-Bernstein property for condensations
(bijective homomorphisms).
It seems that the property of reversibility of relational structures is more of set-theoretical or combinatorial,
than of model-theoretical nature--it is an invariant of isomorphism and condensational equivalence, while it is not preserved under
bi-embeddability, bi-definability and elementary equivalence {\mathfrak{c}}ite{KDef,KuMoSim}.
But it is an invariant of some forms of bi-interpretability {\mathfrak{c}}ite{KRet},
extreme elements of $L_{\infty \omega}$-definable classes of structures are reversible under some syntactical restrictions {\mathfrak{c}}ite{KuMoExtr},
and all structures first-order definable in linear orders by quantifier-free formulas without parameters (i.e., monomorphic or chainable structures)
are reversible {\mathfrak{c}}ite{KDef}.
In this article we continue the investigation of reversibility in the class of disconnected binary structures initiated in {\mathfrak{c}}ite{KuMoDiscI}.
If ${\mathbb X}$ is a binary structure and ${\mathbb X} _i$, $i\in I$, are its connectivity components, then, clearly, the sequence of cardinal numbers $\langlembdaa |X_i|:i\in I\rhoa$
is an isomorphism-invariant of the structure and in some classes of structures (for example, in the class of equivalence relations)
that cardinal invariant characterizes the structure up to isomorphism.
In such classes the reversibility of a structure, being an isomorphism-invariant as well,
can be regarded as a property of the corresponding sequence of cardinals.
So, using the characterization of reversible disconnected binary structures from {\mathfrak{c}}ite{KuMoDiscI} (see Fact \rhoef{TB044})
we easily isolate the following property of sequences of cardinals (called reversibility as well)
which characterizes reversibility in the class of equivalence relations:
If $I$ is a non-empty set, an $I$-sequence of non-zero cardinals $\langlembdaa \kappa _i :i\in I\rhoa$ will be called
{\it reversible} iff there is no non-injective surjection $f :I\rhoightarrow I$ such that
\betaegin{equation}\langlembdaabel{EQB032}\tauextstyle
\varphiorall j\in I \;\;\kappa _j =\sigmaum _{i\in f^{-1}[\{ j \}]}\kappa_i .
\etand{equation}
The first main result of this paper is the following characterization of reversible sequences of cardinals.
In order to state it we recall some definitions.
For a subset $K$ of the set of natural numbers, ${\mathbb N}$, let $\langlembdaa K\rhoa$ denote the subsemigroup of the semigroup $\langlembdaa {\mathbb N} , +\rhoa$
generated by $K$. A set $K$ is called {\it independent} iff
\betaegin{equation}\langlembdaabel{EQB003}
\varphiorall n\in K \;\; n\not\in \langlembdaa K\sigmaetminus \{ n \}\rhoa.
\etand{equation}
So, $\etamptyset$ is an independent set. If $K \neq\etamptyset$, by $\gammacd (K)$ we denote the greatest common divisor of the numbers from $K$.
\betaegin{te}\langlembdaabel{TB042}
A sequence of non-zero cardinals $\langlembdaa \kappa _i :i\in I\rhoa$ is reversible iff
\betaegin{itemize}
\item[-] either $\langlembdaa \kappa _i :i\in I\rhoa$ is a finite-to-one sequence,
\item[-] or $\kappa _i\in {\mathbb N}$, for all $i\in I$,\\
$K:=\{ m\in {\mathbb N} : |\{ i\in I : \kappa _i =m\}|\gammaeq \omega\}$
is a non-empty independent set, \\
and $\gammacd (K)$ divides at most finitely many elements of the set $\{ \kappa _i :i\in I \}$.\varphiootnote{For example, if $I$ is a non-empty set of any size
and $\langlembdaa n_i :i\in I\rhoa \in {}^{I}{\mathbb N}$, then by Theorem \rhoef{TB042} we have:
if $K=\etamptyset$ (which is possible if $|I|\langlembdaeq \omega$), then $\langlembdaa n_i \rhoa $ is a reversible sequence;
if $K=\{ 2,5\}$, then $\langlembdaa n_i \rhoa $ is a reversible sequence iff the set $\{ n_i :i\in I \}$ is finite;
if $K=\{ 4,10\}$, then $\langlembdaa n_i \rhoa $ is a reversible sequence iff the set $\{ n_i :i\in I \}$
contains at most finitely many even numbers.
}
\etand{itemize}
\etand{te}
A proof of Theorem \rhoef{TB042} is given in the last (and the largest) Section \rhoef{S4}, where,
in addition, we show that the set of reversible sequences of natural numbers is a dense $F_{\sigma \delta \sigma}$-subset of the Baire space,
and that it is not a subsemigroup of $\langlembdaa {\mathbb N} ^{\mathbb N} ,{\mathfrak{c}}irc\rhoa$.
Section \rhoef{S2} contains definitions and facts making the paper self-contained.
In Section \rhoef{S3}, generalizing the situation with equivalence relations, we isolate a wider class of
structures with the same property--that the reversibility of a structure from the class is equivalent
to the reversibility of the corresponding sequence of sizes of its components--the class of structures having
the sequence of components rich for monomorphisms. We also study the class $\rhofm$ of such sequences of structures,
compare it with some relevant classes, detect some classes of structures such that the reversibility of a structure from the class
follows from the reversibility of the corresponding cardinal sequence and in this way detect
wide classes of reversible digraphs, posets, and topological spaces.
\sigmaection{Preliminaries}\langlembdaabel{S2}
\psiaragraph{Reversible structures}
If $L=\langlembdaa R_i :i\in I\rhoa$ is a relational language, where $\alphar (R _i)=n_i\in {\mathbb N}$, for $i\in I$, and ${\mathbb X}$ and ${\mathbb Y}$ are $L$-structures, then by ${\mathcal I}so ({\mathbb X} ,{\mathbb Y} )$, ${\mathbb C}ond ({\mathbb X} ,{\mathbb Y} )$ and ${\mathcal M}ono ({\mathbb X} ,{\mathbb Y} )$
we denote the set of all isomorphisms, condensations (bijective homomorphisms) and monomorphisms (injective homomorphisms) from ${\mathbb X}$ to ${\mathbb Y}$ respectively.
Clearly, ${\mathcal I}so ({\mathbb X} ,{\mathbb X} )$ is the set of automorphisms, ${\mathbb A}ut ({\mathbb X} )$, of ${\mathbb X}$, instead of ${\mathbb C}ond ({\mathbb X} ,{\mathbb X} )$ we will write ${\mathbb C}ond ({\mathbb X} )$ etc.
For a set $X$ by ${\mathbb S}ym (X)$ (resp.\ ${\mathbb S}ur (X)$) we denote the set of all bijections (resp.\ surjections) $f:X\rhoightarrow X$.
The {\it condensational preorder} $\psireccurlyeq _c $ on the class of $L$-structures is defined by ${\mathbb X} \psireccurlyeq _c {\mathbb Y}$ iff ${\mathbb C}ond ({\mathbb X} ,{\mathbb Y} )\neq\etamptyset$,
the {\it condensational equivalence} is the equivalence relation defined on the same class by ${\mathbb X} \sigmaim _c {\mathbb Y}$ iff
${\mathbb X} \psireccurlyeq _c {\mathbb Y}$ and ${\mathbb Y} \psireccurlyeq _c {\mathbb X}$
and it determines the antisymmetric quotient of the condensational preorder, the {\it condensational order}, in the usual way.
An $L$-structure ${\mathbb X} =\langlembdaa X,\rho\rhoa$ is called {\it reversible} iff ${\mathbb C}ond ({\mathbb X} )={\mathbb A}ut ({\mathbb X} )$.
Clearly, $\rho=\langlembdaa \rho _i :i\in I\rhoa$ is an element of the set ${\mathcal I}nt _L (X)=\psirod _{i\in I}P(X^{n_i})$
of all interpretations of the language $L$ over the domain $X$ and defining the partial order $\sigmaubset$ on ${\mathcal I}nt _L (X)$ by
$\rho\sigmaubset \sigma$ iff $\rho_i \sigmaubset \sigma _i$, for all $i\in I$, it is easy to obtain the following simple characterizations
of reversible $L$-structures (see {\mathfrak{c}}ite{KuMoVar}).
\betaegin{fac}\langlembdaabel{TB045}
For an $L$-structure ${\mathbb X} =\langlembdaa X,\rho\rhoa$ the following conditions are equivalent
(a) ${\mathbb X} $ is a reversible structure,
(b) $\varphiorall\sigma\in{\mathcal I}nt _L (X)\;\;(\sigma\varsubsetneq \rho \Rightarrow \sigma\not{\mathfrak{c}}ong \rho )$,
(c) $\varphiorall\sigma\in{\mathcal I}nt _L (X)\;\;(\rho\varsubsetneq \sigma \Rightarrow \sigma\not{\mathfrak{c}}ong \rho )$,
(d) $\varphiorall f\in {\mathbb S}ym (X)\;\; ( f[\rho ]\sigmaubset \rho \Rightarrow f[\rho ]= \rho)$.
\etand{fac}
Reversible $L$-structures
have the Cantor-Schr\"{o}der-Bernstein property for condensations. Moreover we have (see {\mathfrak{c}}ite{KuMoVar})
\betaegin{fac}\langlembdaabel{TA011}
Let ${\mathbb X}$ and ${\mathbb Y}$ be $L$-structures. If ${\mathbb X}$ is a reversible structure and ${\mathbb Y} \sigmaim _c {\mathbb X}$, then
${\mathbb Y} {\mathfrak{c}}ong {\mathbb X}$ (thus ${\mathbb Y}$ is reversible too) and ${\mathbb C}ond ({\mathbb X} ,{\mathbb Y} )={\mathcal I}so ({\mathbb X} ,{\mathbb Y} )$.
\etand{fac}
\psiaragraph{Disconnected binary structures}
Let $L_b$ be the binary language, that is,
$L_{b}=\langlembdaa R\rhoa$ and $\alphar (R)=2$. If ${\mathbb X}=\langlembdaa X,\rhoho \rhoa$ is an $L_b$-structure, then
the transitive closure $\rhoho _{rst}$ of the relation $\rhoho _{rs} ={\mathbb D}elta _X {\mathfrak{c}}up \rhoho {\mathfrak{c}}up \rhoho ^{-1}$ (given by $x \,\rhoho _{rst} \,y$ iff there are $n\in {\mathbb N}$ and
$z_0 =x , z_1, \dots ,z_n =y$ such that $z_i \;\rhoho _{rs} \;z_{i+1}$, for each $i<n$)
is the minimal equivalence relation on $X$ containing $\rhoho $. The corresponding equivalence classes are called the {\it components} of ${\mathbb X}$ and
the structure ${\mathbb X}$ is called {\it connected} iff $|X/\rhoho _{rst} |=1$.
If ${\mathbb X} _i=\langlembdaa X_i, \rhoho _i \rhoa$, $i\in I$, are connected $L_b$-structures and $X_i {\mathfrak{c}}ap X_j =\etamptyset$, for
different $i,j\in I$, then the structure $\betaigcup _{i\in I} {\mathbb X} _i =\langlembdaa \betaigcup _{i\in I} X_i , \betaigcup _{i\in I} \rhoho _i\rhoa$ is
the {\it disjoint union} of the structures ${\mathbb X} _i$, $i\in I$, and the structures ${\mathbb X} _i$, $i\in I$, are its components.
\betaegin{fac}[{\mathfrak{c}}ite{KuMoDiscI}]\langlembdaabel{TB044}
Let ${\mathbb X} _i$, $i\in I$, be pairwise disjoint and connected $L_b$-structures.
\betaegin{itemize}
\item[(a)] If $\;\betaigcup _{i\in I} {\mathbb X} _i$ is reversible, then all structures $\muathbb{X}_i$, $i\in I$, are reversible.
\item[(b)] $\betaigcup _{i\in I} {\mathbb X} _i$ is a reversible structure iff
whenever $f:I\rhoightarrow I$ is a surjection,
$g_i \in {\mathcal M}ono({\mathbb X} _i ,{\mathbb X} _{f(i)})$, for $i\in I$, and
\betaegin{equation}\langlembdaabel{EQB035}
\varphiorall j\in I \;\; {\mathbb B}ig({\mathbb B}ig\{g_i[X_i]: i\in f^{-1}[\{ j\}] {\mathbb B}ig\} \mubox{ is a partition of }X_j{\mathbb B}ig),
\etand{equation}
we have
\betaegin{equation}\langlembdaabel{EQB034}
f\in {\mathbb S}ym (I) \;\langlembdaand \;\varphiorall i\in I \;\; g_i \in {\mathcal I}so ({\mathbb X} _i,{\mathbb X} _{f(i)}) .
\etand{equation}
\etand{itemize}
\etand{fac}
\sigmaection{Sequences of structures rich for monomorphisms}\langlembdaabel{S3}
We will say that a sequence of $L$-structures $\langlembdaa {\mathbb X}_i :i\in I\rhoa$ is
{\it rich for monomorphisms} iff
\betaegin{equation}\langlembdaabel{EQA027}
\varphiorall i,j\in I \;\; \varphiorall A\in [X_j ]^{|X_i|} \;\;\etaxists g\in {\mathcal M}ono ({\mathbb X} _i , {\mathbb X} _j)\;\; g[X_i]=A.
\etand{equation}
By Fact \rhoef{TB044}(a), a necessary condition for the reversibility of a disconnected binary structure is the reversibility of its components.
Hence, and in order to simplify notation, in the sequel we work under the following assumption:
\betaegin{itemize}
\item[($\alphast$)] ${\mathbb X} _i$, $i\in I$, are pairwise disjoint, connected and reversible $L_b$-structures.
\etand{itemize}
Let $\rhofm$ denote the class of sequences of $L_b$-structures $\langlembdaa {\mathbb X}_i :i\in I\rhoa$ (where $I$ is any non-empty set)
satisfying $(\alphast)$ and which are rich for monomorphisms.
\sigmaubsection{Reversible equivalence relations and similar structures}
First we show that the reversibility of a structure having the sequence of components in $\rhofm$ depends only on the corresponding cardinal sequence.
\betaegin{te}\langlembdaabel{TA021}
If $\langlembdaa {\mathbb X}_i :i\in I\rhoa \in \rhofm$, then
(a) The structures of the same size are isomorphic,
(b) $\betaigcup _{i\in I}{\mathbb X}_i$ is reversible ${\mathcal L}eftrightarrow$ $\langlembdaa |X_i|: i\in I \rhoa$ is a reversible sequence of cardinals.
\etand{te}
\noindent{\betaf Proof. }
(a) If $|X_i|=|X_j|$, then by (\rhoef{EQA027}) there are $g\in {\mathbb C}ond ({\mathbb X} _i , {\mathbb X} _j)$ and $g'\in {\mathbb C}ond ({\mathbb X} _j , {\mathbb X} _i)$. So
${\mathbb X} _i \sigmaim _c {\mathbb X} _j$, which by Fact \rhoef{TA011} implies that ${\mathbb X} _i {\mathfrak{c}}ong {\mathbb X} _j$.
(b) ($\Rightarrow$) Suppose that the sequence $\langlembdaa |X_i|: i\in I \rhoa$ is not reversible and that
$f:I\rhoightarrow I$ is a noninjective surjection such that for each $j\in I$ we have $|X_j| =\sigmaum _{i\in f^{-1}[\{ j \}]}|X_i|$.
Then for $j\in I$ there is a partition $\{ A^j_i:i\in f^{-1}[\{ j \}]\}$ of $X_j$ such that $|A^j_i|=|X_i|$, for all $i\in f^{-1}[\{ j \}]$
and, by (\rhoef{EQA027}), there are monomorphisms $g_i : {\mathbb X} _i \rhoightarrow {\mathbb X}_j ={\mathbb X} _{f(i)}$ satisfying $g_i [X_i]=A^j_i$.
By Fact \rhoef{TB044}(b) the structure $\betaigcup _{i\in I}{\mathbb X}_i $ is not reversible.
(${\mathcal L}eftarrow$) Let $\langlembdaa |X_i|: i\in I \rhoa$ be a reversible sequence of cardinals.
In order to use Fact \rhoef{TB044}(b), assuming that $f:I\rhoightarrow I$ is a surjection,
$g_i \in {\mathcal M}ono({\mathbb X} _i ,{\mathbb X} _{f(i)})$, for $i\in I$, and that (\rhoef{EQB035}) holds,
we prove (\rhoef{EQB034}).
First, for $i\in I$, since the function $g_i$ is injection we have $|X_i|=|g_i[X_i]|$.
So, by (\rhoef{EQB035}) for each $j\in I$ we have $|X_j| =\sigmaum _{i\in f^{-1}[\{ j \}]}|X_i|$ and,
since the sequence $\langlembdaa |X_i|: i\in I \rhoa$ is reversible, $f\in {\mathbb S}ym (I)$.
Consequently, for $i\in I$ we have $g_i[X_i]=X_{f(i)}$ and, hence, $|X_i|=|X_{f(i)}|$,
which by (a) implies ${\mathbb X} _i {\mathfrak{c}}ong {\mathbb X} _{f(i)}$ and, in addition, $g_i\in {\mathbb C}ond ({\mathbb X} _i, {\mathbb X} _{f(i)})$.
Since the structures ${\mathbb X}_i$ are reversible, by Fact \rhoef{TA011} we have
${\mathbb C}ond ({\mathbb X} _i, {\mathbb X} _{f(i)})={\mathcal I}so ({\mathbb X} _i, {\mathbb X} _{f(i)})$; so $g_i\in {\mathcal I}so ({\mathbb X} _i, {\mathbb X} _{f(i)})$, for all $i\in I$,
and (\rhoef{EQB034}) is true indeed.
${\mathbb B}ox$
\betaegin{te}\langlembdaabel{TB048}
Let $\sigmaim$ be an equivalence relation on a set $X$, ${\mathbb X} =\langlembdaa X ,\sigmaim\rhoa$, and $\{ X_i :i\in I\}$ the corresponding partition.
Then the structure ${\mathbb X}$ is reversible iff $\langlembdaa |X_i| :i\in I \rhoa$ is a reversible sequence of cardinals.
The same holds for the graphs (resp.\ posets) of the form ${\mathbb X} =\betaigcup _{i\in I}{\mathbb X}_i$, where ${\mathbb X}_i$, $i\in I$, are pairwise disjoint
complete graphs (resp.\ ordinals $\langlembdaeq \omega$).
\etand{te}
\noindent{\betaf Proof. }
It is clear that any sequence of disjoint $L_b$-structures with full relations, or complete graphs, or well orders $\langlembdaeq \omega$ belongs to
$\rhofm$; so Theorem \rhoef{TA021} applies.
${\mathbb B}ox$
\betaegin{rem}\langlembdaabel{RB001}\rhom
There are ${\muathfrak c}$-many non-isomorphic countable reversible equivalence relations
(and the same holds for the classes of graphs and posets from Theorem \rhoef{TB048}).
By Theorems \rhoef{TB048} and \rhoef{TB042}, if $\langlembdaa n_i :i\in {\mathbb N}\rhoa \in {}^{{\mathbb N}}{\mathbb N}$ is an increasing sequence,
then the structure ${\mathbb X}_{\langlembdaa n_i\rhoa}$ with the equivalence relation on ${\mathbb N}$
determined by a partition $\{ C_i :i\in {\mathbb N} \}$, where $|C_i|=n_i$, for all $i\in {\mathbb N}$, is reversible.
Also, if $\langlembdaa n_i :i\in {\mathbb N}\rhoa\neq \langlembdaa n'_i :i\in {\mathbb N}\rhoa $, then the corresponding structures are non-isomorphic.
For $A\in [{\mathbb N} ]^\omega$ let $\langlembdaa n^A_i :i\in {\mathbb N}\rhoa$ be the increasing enumeration of the set $A$. Then the structures
${\mathbb X} _{\langlembdaa n^A_i\rhoa}$, $A\in [{\mathbb N} ]^\omega$, are non-isomorphic, countable and reversible.
\etand{rem}
\sigmaubsection{More reversible digraphs, posets, and topological spaces}
In the following theorem we detect a class of structures such that the reversibility of a structure belonging to the class {\it follows} from the
reversibility of the sequence of cardinalities of its components.
\betaegin{te}\langlembdaabel{TB047}
If ${\mathbb X}_i$, $i\in I$, are disjoint tournaments and the sequence of cardinals $\langlembdaa |X_i|: i\in I\rhoa$ is reversible,
then the digraph $\betaigcup _{i\in I}{\mathbb X}_i$ is reversible.
This statement holds if, in particular, ${\mathbb X}_i$, $i\in I$, are disjoint linear orders. Then $\betaigcup _{i\in I}{\mathbb X}_i$ is a reversible
disconnected partial order.
\etand{te}
\noindent{\betaf Proof. }
In order to apply Fact \rhoef{TB044}(b) we suppose that $f:I\rhoightarrow I$ is a surjection,
$g_i \in {\mathcal M}ono({\mathbb X} _i ,{\mathbb X} _{f(i)})$, for $i\in I$, and that (\rhoef{EQB035}) holds.
Then, since $|g_i[X_i]|=|X_i|$, for $i\in I$, for each $j\in I$ by (\rhoef{EQB035}) we have $|X_j|=\sigmaum _{i\in f^{-1}[\{ j\} ]}|X_i|$, which,
since the sequence $\langlembdaa |X_i|: i\in I\rhoa$ is reversible, implies that $f\in {\mathbb S}ym (I)$.
Thus for each $i\in I$ we have $g_i \in {\mathbb C}ond({\mathbb X} _i ,{\mathbb X} _{f(i)})$, and, since the structures ${\mathbb X} _i$, $i\in I$,
are tournaments, ${\mathbb C}ond({\mathbb X} _i ,{\mathbb X} _{f(i)})= {\mathcal I}so({\mathbb X} _i ,{\mathbb X} _{f(i)})$. Thus (\rhoef{EQB034}) is true and the digraph $\betaigcup _{i\in I}{\mathbb X}_i$ is reversible
indeed.
${\mathbb B}ox$
\betaegin{ex}\langlembdaabel{EXB011}\rhom
The converse of Theorem \rhoef{TB047} is not true. Let $I={\mathbb N}$ and
${\mathbb X} _i {\mathfrak{c}}ong \omega i$, for $i\in {\mathbb N}$. By Theorem \rhoef{TB042} the sequence of cardinals $\langlembdaa \omega, \omega , \dots\rhoa$
is not reversible. Using Fact \rhoef{TB044}(b) we show that ${\mathbb X} =\betaigcup _{i\in {\mathbb N}}{\mathbb X}_i$ is a reversible structure.
Let $f:{\mathbb N}\rhoightarrow {\mathbb N}$ be a surjection,
$g_i \in {\mathcal M}ono({\mathbb X} _i ,{\mathbb X} _{f(i)})$, for $i\in {\mathbb N}$, and let (\rhoef{EQB035}) hold.
First, by induction we show that $f(i)=i$, for all $i\in {\mathbb N}$.
If $i\in {\mathbb N}$ and $f(i)=1$, then $g_i \in {\mathcal M}ono({\mathbb X} _i , {\mathbb X} _1 )$ and,
since monomorphisms between linear orders are embeddings, $\omega i \hookrightarrow \omega$
and, hence, $i=1$. Thus $f^{-1}[\{ 1\}]\sigmaubset \{ 1\}$ and, since $f$ is a surjection, $f^{-1}[\{ 1\}]= \{ 1\}$.
Let $j\in {\mathbb N}$ and $f(k)=k$, for all $k<j$. If $i\in {\mathbb N}$ and $f(i)=j$, then $g_i \in {\mathcal M}ono({\mathbb X} _i , {\mathbb X} _j )$ and,
as above, $\omega i \hookrightarrow \omega j$, which means that $i\langlembdaeq j$. By the induction hypothesis we have $i\gammaeq j$,
so $i=j$ and, thus, $f^{-1}[\{ j\}]\sigmaubset \{ j\}$ and, since $f$ is a surjection, $f^{-1}[\{ j\}]= \{ j\}$.
So, $f=\mathop{\mathrm{id}}\nolimits _{\mathbb N}\in {\mathbb S}ym ({\mathbb N} )$, which by (\rhoef{EQB035}) implies that for each $i\in {\mathbb N}$ we have $g_i \in {\mathbb C}ond({\mathbb X} _i ,{\mathbb X} _i)={\mathcal I}so({\mathbb X} _i ,{\mathbb X} _i)$
and (\rhoef{EQB034}) is proved.
\etand{ex}
\betaegin{ex}\langlembdaabel{EXB015}\rhom
More reversible posets and topological spaces.
The reversible posets constructed in Examples \rhoef{EXB011} and \rhoef{EXB014} are well-founded and with infinite levels.
More generally, by Theorem \rhoef{TB047}, if $\langlembdaa \kappa _i : i\in I\rhoa$ is {\it any} reversible sequence of cardinals
(e.g., if it is finite-to-one, if we would like infinite components) and $L_i$, $i\in I$, are {\it any} linear orders, where $|L_i|=\kappa _i$, then the poset $\betaigcup _{i\in I}L_i$ is reversible.
Recalling that if ${\mathbb P} =\langlembdaa P, \langlembdaeq \rhoa$ is a partial order and ${\mathcal O}$ the topology on the set $P$
generated by the base consisting of the sets of the form $B_p:= \{ q\in p: q\langlembdaeq p\}$, then endomorphisms of ${\mathbb P}$
are exactly the continuous self mappings of the space $\langlembdaa P,{\mathcal O}\rhoa$, we conclude that
the poset ${\mathbb P}$ is reversible iff $\langlembdaa P,{\mathcal O}\rhoa$ is a reversible topological space (i.e., each continuous bijection is an automorphism).
So, Examples \rhoef{EXB011}, \rhoef{EXB014} and Theorem \rhoef{TB047} generate a large class of reversible topological spaces.
\etand{ex}
\sigmaubsection{More sequences from RFM}
We recall that a relational structure ${\mathbb X}$ is called {\it monomorphic}
iff each two finite substructures of ${\mathbb X}$ of the same size are isomorphic,
and that, by the well-known theorems of Fra\"{\i}ss\'{e} (for finite languages) and Pouzet (for languages and structures of any size), see {\mathfrak{c}}ite{Fra},
an infinite structure ${\mathbb X}$ is monomorphic iff it is {\it chainable} i.e.\ there is a linear order $\psirec$ on its domain, $X$,
such that the relations of ${\mathbb X}$ are definable in the structure $\langlembdaa X,\psirec\rhoa$ by quantifier-free formulas without parameters.
Then it is said that $\psirec$ {\it chains} ${\mathbb X}$, or that ${\mathbb X}$ is {\it chainable} by $\psirec$.
For convenience, a structure ${\mathbb X}$ will be called {\it copy-maximal} (resp.\ {\it mono-range-maximal}) iff
for each $A\in [X ]^{|X|}$ there is an embedding (resp.\ a monomorphism) $g:{\mathbb X} \rhoightarrow {\mathbb X}$ satisfying $g[X]=A$.
By (\rhoef{EQA027}), Theorem \rhoef{TA021}(a) and since each set of cardinals is well ordered,
a sequence $\langlembdaa {\mathbb X} _i :i\in I\rhoa\in \rhofm$ can be described in the following way.
There are an ordinal $\etata$ and a sequence of connected reversible $L_b$-structures $\langlembdaa {\mathbb Y} _\xi :\xi <\etata\rhoa$ (the {\it range}) such that, defining $\kappa _\xi := |Y_\xi|$, we have
(r1) $\xi < \zetaeta <\etata \Rightarrow \kappa_\xi<\kappa _\zetaeta$,
(r2) ${\mathbb Y} _\xi$ is a mono-range-maximal structure, for each $\xi <\etata$,
(r3) $\xi < \zetaeta <\etata \Rightarrow \varphiorall A\in [Y_\zetaeta ]^{\kappa _\xi } \;{\mathbb C}ond ({\mathbb Y} _\xi ,A)\neq\etamptyset$,
\noindent
and there is a surjection $h:I\rhoightarrow \etata$ such that for each $\xi <\etata $ and $i\in h^{-1}[\{ \xi \}]$ we have ${\mathbb X}_i{\mathfrak{c}}ong {\mathbb Y} _\xi$,
and $X_i {\mathfrak{c}}ap X_j =\etamptyset$, for $i\neq j$.
So, by Theorem \rhoef{TA021}(b), the structure
$\betaigcup _{i\in I}{\mathbb X} _i$ is reversible iff $\langlembdaa \kappa _{h(i)}:i\in I \rhoa$ is a reversible sequence of cardinals.
Here we consider conditions (r2) and (r3).
\psiaragraph{Condition (r2)}
Clearly, condition (r2) will be satisfied if the structures ${\mathbb Y} _\xi$ are finite or copy-maximal.
From more general results of Gibson, Pouzet and Woodrow {\mathfrak{c}}ite{Gib} it follows that a structure ${\mathbb X}$ of size $\kappa\gammaeq \omega$
is copy-maximal iff it is $\kappa$-chainable, that is, there is a linear order $\psirec$ on $X$ which chains ${\mathbb X}$ and $\langlembdaa X ,\psirec\rhoa{\mathfrak{c}}ong \langlembdaa \kappa ,<\rhoa$.
On the other hand, a simple application of Ramsey's theorem shows that, up to isomorphism,
there are only eight countable binary copy-maximal structures and the same
holds for uncountable binary structures (see also {\mathfrak{c}}ite{Ktow,KZb}). The six connected of them are
$\langlembdaa \kappa , \kappa ^2\rhoa$, $\langlembdaa \kappa , \kappa ^2 \sigmaetminus {\mathbb D}elta _\kappa \rhoa$, $\langlembdaa \kappa , <\rhoa$, $\langlembdaa \kappa , \langlembdaeq \rhoa$ $\langlembdaa \kappa , >\rhoa$, and $\langlembdaa \kappa , \gammaeq \rhoa$, and
they are reversible.
In addition, since in the class of linear orders monomorphisms are embeddings, mono-range-maximal linear orders are copy-maximal
thus the only four mono-range-maximal linear orders of size $\kappa$ are mentioned above.
The following example shows that the class of mono-range-maximal posets is not so restrictive.
\betaegin{ex}\langlembdaabel{EXB009}\rhom
The posets of the form ${\mathbb X}_{\langlembda ,\kappa}:={\mathbb A}_\langlembda +{\mathbb B}L_\kappa$, where $2\langlembdaeq \langlembda <\kappa\gammaeq \omega$, ${\mathbb A}_\langlembda$ is an antichain of size $\langlembda$,
and ${\mathbb B}L _\kappa {\mathfrak{c}}ong \langlembdaa \kappa , <\rhoa$, are not copy-maximal and, moreover, if $\langlembda \gammaeq \omega$, ${\mathbb X}_{\langlembda ,\kappa}$ is not almost chainable (see {\mathfrak{c}}ite{Fra, Gib} for details).
But ${\mathbb X}_{\langlembda ,\kappa}$ is mono-range-maximal (if $S \in [X]^\kappa$, then $S{\mathfrak{c}}ong {\mathbb A}_\muu +{\mathbb B}L_\kappa$, for some $\muu \langlembdaeq \langlembda$, and it is easy
to construct a monomorphism from ${\mathbb X}_{\langlembda ,\kappa}$ onto $S$). If $\langlembda <\omega$, then ${\mathbb X}_{\langlembda ,\kappa}$ is a well-founded poset with finite levels so, by
{\mathfrak{c}}ite{Kuk}, it is reversible.
\etand{ex}
\psiaragraph{Condition (r3)}
All the structures considered in Theorem \rhoef{TB048} -
disjoint unions of
(a) structures with full relations,
(b) complete graphs, and
(c) ordinals $\langlembdaeq \omega$,
give examples of sequences satisfying (r3) and all of them have monomorphic components. The following examples show that this condition
is not necessary for application of Theorem \rhoef{TA021}(b).
\betaegin{ex}\langlembdaabel{EXB013}\rhom
Structures from $\rhofm$ with non-monomorphic components.
Let
- ${\mathbb B}T _3$ be the three-element tree $\langlembdaa \{ 0,1,2\}, \{ \langlembdaa 0,1\rhoa , \langlembdaa 0,2\rhoa\}\rhoa$,
- ${\mathbb B}L _5$ the five-element linear order,
- ${\mathbb B}K _6^*$ a complete graph with 6 nodes and 3 of them reflexified (loops),
- ${\mathbb F} _8$ the eight-element structure with the full relation.
\noindent
Now, if $\kappa$ and $\langlembda$ are infinite cardinals, $m,n\in \omega$ and ${\mathbb X}$ is the (pairwise disjoint) union of
$\kappa$-many copies of ${\mathbb B}T _3$, $\langlembda$-many copies of ${\mathbb B}L _5$, $m$ copies of ${\mathbb B}K _6^*$ and $n$ copies of ${\mathbb F} _8$, then
the sequence $\langlembdaa {\mathbb B}T _3 ,{\mathbb B}L _5,{\mathbb B}K _6^* ,{\mathbb F} _8\rhoa $ satisfies (r1)-(r3),
the corresponding sequence of components of ${\mathbb X}$ belongs to $\rhofm$
and ${\mathbb X}$ is reversible because,
in notation of Proposition \rhoef{TB037}, $K=\{ 3,5\}$ and the set $\{ n_i : i\in I\}=\{3,5,6,8\}$ is finite and we apply Theorem \rhoef{TA021}(b).
\etand{ex}
\betaegin{ex}\langlembdaabel{EXB014}\rhom
A structure from $\rhofm$ having all components non-monomorphic.
Let ${\mathbb X}_{2,\kappa}={\mathbb A} _2 +{\mathbb B}L _\kappa$, for $1\langlembdaeq \kappa \langlembdaeq \omega$, be the posets defined as in Example \rhoef{EXB009}.
It is easy to see that $\langlembdaa {\mathbb X} _{2,\kappa} : 1\langlembdaeq \kappa \langlembdaeq \omega\rhoa \in \rhofm$ .
Since the corresponding sequence of cardinals
$\langlembdaa 3,4,5, \dots ,\omega\rhoa$ is one-to-one and, thus, reversible, the structure ${\mathbb X} =\betaigcup _{1\langlembdaeq \kappa \langlembdaeq \omega}{\mathbb X} _{2,\kappa}$ is reversible.
Clearly, its components, ${\mathbb X} _{2,\kappa}$, are not 2-monomorphic.
\etand{ex}
\sigmaubsection{The classes RFM, RC, and RU}
If by $\rhoc$ (resp.\ $\rhou$)
we denote the class of sequences $\langlembdaa {\mathbb X}_i :i\in I\rhoa$
satisfying $(\alphast)$ and such that $\langlembdaa |{\mathbb X}_i | :i\in I\rhoa$ is a reversible sequence of cardinals,
(resp.\ the structure $\betaigcup _{i\in I}{\mathbb X}_i$ is reversible),
then by Theorem \rhoef{TA021}(b) we have $\rhofm {\mathfrak{c}}ap \rhou =\rhofm {\mathfrak{c}}ap \rhoc$. The following
example shows that this equality is the only constraint, regarding the relationship between the classes $\rhofm$, $\rhoc$ and $\rhou$.
\betaegin{ex}\langlembdaabel{EXB010}\rhom
(a) $\rhofm \sigmaetminus (\rhou {\mathfrak{c}}up \rhoc )\neq\etamptyset$.
If ${\mathbb X} _i {\mathfrak{c}}ong \langlembdaa \omega , <\rhoa$, for $i\in \omega$, then by Theorem \rhoef{TB042} the sequence of cardinals $\langlembdaa \omega, \omega , \dots\rhoa$
is not reversible but, since ($\langlembdaa A, < \,\!\!\uparrowharpoonright \!A\rhoa {\mathfrak{c}}ong \langlembdaa \omega ,< \rhoa$, for each $A\in [\omega ]^\omega$,
the sequence $\langlembdaa {\mathbb X}_i :i\in I\rhoa$ is rich for monomorphisms. It is easy to see that the structure $\betaigcup _{i\in I}{\mathbb X}_i$ is not reversible.
(b) $\rhoc \sigmaetminus (\rhofm {\mathfrak{c}}up \rhou)\neq\etamptyset$. Let ${\mathbb X} = \langlembdaa {\mathbb Z} ,\rho \rhoa$, where $\rho =\{ \langlembdaa i,i \rhoa :i\gammaeq 0 \}$. Then
${\mathbb X}=\betaigcup _{i\in {\mathbb Z}}{\mathbb X}_i$, where ${\mathbb X}_i=\langlembdaa \{ i\} , \etamptyset \rhoa$, for $i<0$, and ${\mathbb X}_i=\langlembdaa \{ i\} , \{ \langlembdaa i,i \rhoa \} \rhoa$, for $i\gammaeq 0$.
The corresponding sequence of cardinals $\langlembdaa \dots ,1, 1 , \dots\rhoa$ is reversible and, since ${\mathbb X} {\mathfrak{c}}ong \langlembdaa {\mathbb Z} ,\rho \sigmaetminus \{ \langlembdaa 0,0\rhoa\}\rhoa$, by Fact \rhoef{TB045} the structure $\betaigcup _{i\in {\mathbb Z}}{\mathbb X}_i$ is not reversible. Since ${\mathbb X} _{-1}\not{\mathfrak{c}}ong {\mathbb X} _0$, by Theorem \rhoef{TA021}(a)
the sequence of structures $\langlembdaa {\mathbb X}_i :i\in {\mathbb Z}\rhoa$
is not rich for monomorphisms.
(c) $\rhou \sigmaetminus (\rhofm {\mathfrak{c}}up \rhoc)\neq\etamptyset$. Let ${\mathbb X} = \langlembdaa {\mathbb Z} ,\rho \rhoa$, where
$
\rho =\{ \langlembdaa i,i \rhoa :i< 0 \}{\mathfrak{c}}up \{ \langlembdaa 2i, 2i+1 \rhoa :i\gammaeq 0\}.
$
Then we have
${\mathbb X}=\betaigcup _{i\in {\mathbb Z}}{\mathbb X}_i$, where ${\mathbb X}_i=\langlembdaa \{ i\} , \{ \langlembdaa i,i \rhoa \} \rhoa$, for $i<0$,
and ${\mathbb X}_i=\langlembdaa \{ 2i, 2i+1 \} , \{ \langlembdaa 2i, 2i+1 \rhoa \} \rhoa$, for $i\gammaeq 0$.
Now, the corresponding sequence of cardinals $\langlembdaa \dots ,1,1,2,2, \dots\rhoa$ is not reversible, because the set $K=\{ 1,2\}$ is not independent ($1+1=2$).
Since ${\mathcal M}ono ({\mathbb X} _{-1},{\mathbb X} _0)=\etamptyset$ we have $\langlembdaa {\mathbb X}_i :i\in {\mathbb Z}\rhoa \not\in \rhofm$.
But, by Fact \rhoef{TB045}, the structure $\betaigcup _{i\in {\mathbb Z}}{\mathbb X}_i$ is reversible,
namely, if $\sigma \varsubsetneq \rho$, then the structure $\langlembdaa {\mathbb Z} ,\sigma \rhoa$ has an one-element component with the empty relation and, hence,
it is not isomorphic to ${\mathbb X}$.
(d) $(\rhou {\mathfrak{c}}ap \rhoc)\sigmaetminus \rhofm\neq\etamptyset$. Let $I$ be the ordinal $\omega +2 =\omega {\mathfrak{c}}up \{ \omega ,\omega +1\}$ and let ${\mathbb X} =\betaigcup _{i\in \omega +2}{\mathbb X}_i$,
where ${\mathbb X}_i$ are pairwise disjoint linear orders such that ${\mathbb X}_i {\mathfrak{c}}ong i+1$, for $i\in\omega$, ${\mathbb X} _\omega {\mathfrak{c}}ong \omega$, and ${\mathbb X} _{\omega +1}{\mathfrak{c}}ong {\mathbb Q}$.
The corresponding sequence of cardinals $\langlembdaa 1,2,\dots , \omega ,\omega\rhoa$ is finite-to-one and, by Theorem \rhoef{TB042}, reversible. By Theorem
\rhoef{TB047} the union $\betaigcup _{i\in I}{\mathbb X}_i$ is reversible too. Since $\omega \not{\mathfrak{c}}ong {\mathbb Q}$ by Theorem \rhoef{TA021}(a) we have $\langlembdaa {\mathbb X}_i :i\in I\rhoa \not\in \rhofm$.
\etand{ex}
Let $\rhofm _{LO}$, $\rhoc_{LO}$ and $\rhou_{LO}$ denote the classes of sequences of linear orders $\langlembdaa {\mathbb X}_i :i\in I\rhoa$
belonging to classes $\rhofm$, $\rhoc$ and $\rhou$. Here, by Theorem \rhoef{TB047} we obtain one more constraint:
$\rhoc_{LO} \sigmaubset \rhou_{LO}$, and the following example shows that, in general, there are no more constraints.
\betaegin{ex}\langlembdaabel{EXB012}\rhom
$\rhofm _{LO}\sigmaetminus \rhou_{LO}\neq\etamptyset$ is witnessed by the poset $\betaigcup _\omega \omega$, from Example \rhoef{EXB010}(a).
The poset $\betaigcup _{n\in {\mathbb N}}n {\mathfrak{c}}up \omega {\mathfrak{c}}up {\mathbb Q}$ from Example \rhoef{EXB010}(d) belongs to the class $\rhoc_{LO}\sigmaetminus\rhofm _{LO}$,
while the poset $\betaigcup _{n\in {\mathbb N}}\omega n$ (see Example \rhoef{EXB011}) belongs to the class $\rhou_{LO}\sigmaetminus \rhoc_{LO}$.
\etand{ex}
\sigmaection{Reversible cardinal sequences -- a proof of Theorem \rhoef{TB042}}\langlembdaabel{S4}
Theorem \rhoef{TB042} follows from Propositions \rhoef{TA020} and \rhoef{TB037} given in the sequel.
If $\langlembdaa \kappa _i :i\in I\rhoa$ is a sequence of cardinals and $\kappa$ a cardinal, let
$$
I_\kappa := \{ i\in I : \kappa _i =\kappa\} .
$$
\sigmaubsection{Reduction to the case when the cardinals are finite}
\betaegin{prop}\langlembdaabel{TA020}
A sequence of non-zero cardinals $\langlembdaa \kappa _i :i\in I\rhoa$ is reversible iff it is a finite-to-one sequence or a reversible sequence in ${\mathbb N}$.
\etand{prop}
\noindent{\betaf Proof. }
The implications ``$ {\mathcal L}eftarrow $" and ``$ \Rightarrow $" follow from Claims \rhoef{TB040} and \rhoef{TB041} respectively.
\betaegin{cla}\langlembdaabel{TB040}
If $\langlembdaa \kappa _i :i\in I\rhoa$ is a finite-to-one sequence, it is reversible.
\etand{cla}
\noindent{\betaf Proof. }
Let $|I_\kappa |<\omega $, for all $\kappa \in {\mathbb C}ard$. The {\it set} $\{ \kappa_i:i\in I \}$ is well-ordered
and, hence, there is an ordinal $\zeta$
and an enumeration $\{ \kappa_i:i\in I \} =\{ \kappa _\xi : \xi <\zeta \}$
such that $ \xi <\xi'$ implies $ \kappa _\xi <\kappa _{\xi'}$.
Assuming that $f:I\rhoightarrow I$ is a surjection satisfying (\rhoef{EQB032}) we show that $f$ is a bijection.
First, by induction we prove that
\betaegin{equation}\langlembdaabel{EQA026}
\varphiorall \xi <\zeta \;\; f[I _{\kappa _\xi}]=I _{\kappa _\xi}.
\etand{equation}
If $j\in I_{\kappa _0}$, then, by (\rhoef{EQB032}),
for $i\in f^{-1}[\{ j \}]$ we have $\kappa_i \langlembdaeq \kappa_j =\kappa _0$, which, by the minimality of $\kappa_0$,
implies that $\kappa_i=\kappa _0$, that is, $i\in I_{\kappa _0}$.
Thus $f^{-1}[\{ j \}]\sigmaubset I_{\kappa _0}$, for all $j\in I_{\kappa _0}$, and, hence,
$f^{-1}[I_{\kappa _0}]\sigmaubset I_{\kappa _0}$. Since $f$ is onto we have $I_{\kappa _0} =f[f^{-1}[I_{\kappa _0}]]\sigmaubset f[I_{\kappa _0}]$ thus
$|I_{\kappa _0}|\langlembdaeq |f[I_{\kappa _0}]|\langlembdaeq |I_{\kappa _0}|$ and, hence, $|f[I_{\kappa _0}]|= |I_{\kappa _0}|$, which, since the set $I_{\kappa _0} $
is finite and $I_{\kappa _0}\sigmaubset f[I_{\kappa _0}]$, implies that $f[I_{\kappa _0}]= I_{\kappa _0}$.
Assuming that $\etata <\zeta$ and $f[I _{\kappa _\xi}]=I _{\kappa _\xi}$, for all $\xi <\etata$, we prove $f[I _{\kappa _\etata}]=I _{\kappa _\etata}$.
If $j\in I_{\kappa _\etata}$,
then, by (\rhoef{EQB032}), for $i\in f^{-1}[\{ j \}]$ we have
$\kappa_i\langlembdaeq \kappa_j=\kappa _\etata$.
The inequality $\kappa_i<\kappa _\etata$ would imply that $\kappa_i=\kappa _\xi$, for some $\xi <\etata$,
and, hence, $i\in I _{\kappa _\xi}$
and, by the induction hypothesis, $f(i)=j\in I _{\kappa _\xi}$, which is not true.
Thus $\kappa_i=\kappa _\etata$
and, hence, $i\in I_{\kappa _\etata}$.
Thus $f^{-1}[\{ j \}]\sigmaubset I_{\kappa _\etata}$, for all $j\in I_{\kappa _\etata}$, and, hence,
$f^{-1}[I_{\kappa _\etata}]\sigmaubset I_{\kappa _\etata}$.
Now, as above we show that $f[I_{\kappa _\etata}]= I_{\kappa _\etata}$ and (\rhoef{EQA026}) is proved.
By (\rhoef{EQA026}) and since the sets $I_{\kappa _\xi}$ are finite,
the restrictions $f\!\!\uparrowharpoonright I_{\kappa _\xi}: I_{\kappa _\xi}\rhoightarrow I_{\kappa _\xi}$, $\xi <\zeta$, are bijections
and, since $\{ I _{\kappa _\xi}: \xi <\zeta \}$ is a partition of the set $I$, $f$ is a bijection as well.
${\mathbb B}ox$
\betaegin{cla}\langlembdaabel{TB041}
If $\langlembdaa \kappa _i :i\in I\rhoa$ is a sequence of cardinals and some of them is infinite, then
\betaegin{equation}\langlembdaabel{EQB033}
\langlembdaa \kappa _i :i\in I\rhoa \mubox{ is reversible } {\mathcal L}eftrightarrow \langlembdaa \kappa _i :i\in I\rhoa \mubox{ is finite-to-one}.
\etand{equation}
\etand{cla}
\noindent{\betaf Proof. }
Let $i^*\in I$, where $\kappa_{i^*} \gammaeq \omega$.
By Claim \rhoef{TB040} the implication ``$\Rightarrow$" remains to be checked and we prove its contrapositive.
Suppose that $|I_{\kappa _0}|\gammaeq \omega$, for some cardinal $\kappa_0$.
If $\kappa_0 \langlembdaeq \kappa_{i^*}$, then we choose different $i_n\in I_{\kappa _0}\sigmaetminus \{ i^*\}$, $n\in \omega$,
and define a surjection $f:I\rhoightarrow I$ by:
$$
f (i)=\langlembdaeft\{ \betaegin{array}{cl}
i^*, & \mubox{ if } i\in \{ i^*, i_0\} ,\\
i_{n-1} , & \mubox{ if } i=i_n, \mubox{ and } n\gammaeq 1, \\
i , & \mubox{ if } i\in I \sigmaetminus (\{ i^*\} {\mathfrak{c}}up \{ i_n :n\in \omega\}).
\etand{array}
\rhoight.
$$
Now, for $j\in I \sigmaetminus (\{ i^*\} {\mathfrak{c}}up \{ i_n :n\in \omega\}) $ we have $f^{-1}[\{ j\}]=\{ j\}$;
for $n\in {\mathbb N}$ we have $f^{-1}[\{ i_{n-1}\}]=\{ i_{n}\}$ and $\kappa _{i_n}=\kappa _{i_{n-1}}=\kappa _0$;
finally $f^{-1}[\{ i^*\}]=\{ i^*, i_0\}$ and $\kappa _{i^*}=\kappa _{i^*}+ \kappa _0=\kappa _{i^*}+ \kappa _{i_0}$.
So (\rhoef{EQB032}) is true and, since $f$ is not a bijection,
the sequence $\langlembdaa \kappa _i :i\in I\rhoa$ is not reversible.
If $\kappa_0 > \kappa_{i^*}$, then
we choose different $i_n\in I_{\kappa _0}$, for $n\in \omega$,
and define a non-injective surjection $f:I\rhoightarrow I$ by:
$$
f (i)=\langlembdaeft\{ \betaegin{array}{cl}
i_0, & \mubox{ if } i\in \{ i_0, i_1\} ,\\
i_{n-1} , & \mubox{ if } i=i_n, \mubox{ and } n\gammaeq 2, \\
i , & \mubox{ if } i\in I \sigmaetminus \{ i_n :n\in \omega\}.
\etand{array}
\rhoight.
$$
Since $f^{-1}[\{ i_0\}]=\{ i_0, i_1\}$ and $\kappa_0$ is an infinite cardinal, we have
$\kappa _{i_0}=\kappa _0 =\kappa _{0}+\kappa _{0}=\kappa _{i_0}+\kappa _{i_1}$.
So (\rhoef{EQB032}) is true and
$\langlembdaa \kappa _i :i\in I\rhoa$ is not reversible again.
${\mathbb B}ox$
\sigmaubsection{Reversible sequences of natural numbers}
Here we characterize reversible sequences of the form $\langlembdaa n_i :i\in I\rhoa \in {}^{I}{\mathbb N}$, where $I\neq \etamptyset$.
Clearly, $I=\betaigcup _{m\in {\mathbb N}}I_m$, where
$$
I_m =\{ i\in I : n_i =m\},\; \mubox{ for } m\in {\mathbb N} ,
$$
and the following statement is the main result of this paragraph.
\betaegin{prop}\langlembdaabel{TB037}
A sequence $\langlembdaa n_i :i\in I\rhoa \in {}^{I}{\mathbb N}$ is reversible if and only if the set $K:=\{ m\in {\mathbb N} : |I_m|\gammaeq \omega\}$ is independent
and, if $K$ is a non-empty set,
then at most finitely many elements of the set $\{ n_i :i\in I \}$ are divisible by the $\gammacd (K)$.
\etand{prop}
A proof of Proposition \rhoef{TB037} is given in the sequel.
First for $d\in {\mathbb N}$ we define $d{\mathbb N} :=\{ dk:k\in {\mathbb N}\}$ and recall some facts from elementary number theory (giving their proofs for reader's convenience).
\betaegin{fac}\langlembdaabel{TB032}
Let $K$ be a nonempty subset of ${\mathbb N}$ and $d=\gammacd (K)$. Then we have:
(a) If $|K|=\omega $, then $\gammacd(K^\psirime)=d$, for some finite $K^\psirime\sigmaubset K$;
(b) If $d=1$, then there is $M\in {\mathbb N}$ such that $[M,\infty )\sigmaubset \langlembdaa K \rhoa$;
(c) If $d>1$, then there is $M\in {\mathbb N}$ such that $[dM ,\infty ){\mathfrak{c}}ap d{\mathbb N}\sigmaubset \langlembdaa K \rhoa\sigmaubset d{\mathbb N}$;
(d) Each independent set is finite.
\etand{fac}
\noindent{\betaf Proof. }
(a) Let $K=\{ n_r:r\in {\mathbb N}\}$ and $d_r =\gammacd \{ n_1,\dots ,n_r\}$, for $r\in {\mathbb N}$. Then $d_1 \gammaeq d_2 \gammaeq \dots$
and, hence, there is $s\in {\mathbb N}$ such that $d_r=d_s$, for all $r\gammaeq s$. Clearly we have
$d\langlembdaeq d_{s}$ and, since $d_{s}$ divides all $n_r$'s, $d\gammaeq d_{s}$, by the maximality of $d$.
Now we take $K'=\{ n_1,\dots ,n_{s}\}$.
(b) By (a) there is $K^\psirime=\{ n_1,\dots ,n_{s}\}\sigmaubset K$ such that $\gammacd(K^\psirime)=1$.
By B\'ezout's lemma
there are $a_r\in\muathbb{Z}$, for $1\langlembdaeq r\langlembdaeq s$, such that
$\sigmaum_{r=1}^s a_r n_r=1$, which for $M:=n_1\sigmaum_{r=1}^s|a_r|n_r$, and for any $m\in\{0,1,\langlembdadots,n_1-1\}$,
implies $M+m=\sigmaum_{r=1}^s(n_1|a_r|+m a_r)n_r\in\langlembdaangle K^\psirime\rhoangle$; so, $[M,M+n_1)\sigmaubset\langlembdaangle K^\psirime\rhoangle$.
Since $k n_1\in\langlembdaangle K^\psirime\rhoangle$, we also have that $[M+k n_1,M+(k+1)n_1)\sigmaubset\langlembdaangle K^\psirime\rhoangle$, for any $k\in\muathbb{N}$. Hence, $[M,\infty)\sigmaubset\langlembdaangle K^\psirime\rhoangle\sigmaubset\langlembdaangle K\rhoangle$.
(c) It is clear that $\langlembdaa K \rhoa\sigmaubset d{\mathbb N}$.
By (a) there is $K^\psirime=\{ n_1,\dots ,n_{s}\}\sigmaubset K$ such that $\gammacd(K^\psirime)=d$
and, hence, $K^\psirime=\{d m_1, \dots ,d m_s\}$, where $\gammacd(\{m_1 ,\dots ,m_s\})=1$.
By (b) there is $M\in\muathbb{N}$ such that $[M,\infty)\sigmaubset\langlembdaangle\{m_1, \dots ,m_s\}\rhoangle$, so $[dM,\infty){\mathfrak{c}}ap d{\mathbb N}\sigmaubset\langlembdaangle K^\psirime\rhoangle\sigmaubset\langlembdaangle K\rhoangle$.
(d) If $K$ is an infinite set, then by (a) there is a finite $K^\psirime\sigmaubset K$ such that $\gammacd(K^\psirime)=\gammacd(K)=d$. Since $K\sigmaetminus K^\psirime\sigmaubset d{\mathbb N}$ is infinite, for every $M\in\muathbb{N}$ we have $(K\sigmaetminus K^\psirime){\mathfrak{c}}ap[dM,\infty){\mathfrak{c}}ap d{\mathbb N}\neq\etamptyset$. By (c) there is $M\in\muathbb{N}$ such that $[dM,\infty){\mathfrak{c}}ap d{\mathbb N}\sigmaubset\langlembdaangle K^\psirime\rhoangle$. Then $(K\sigmaetminus K^\psirime){\mathfrak{c}}ap\langlembdaangle K^\psirime\rhoangle\sigmaupset(K\sigmaetminus K^\psirime){\mathfrak{c}}ap[dM,\infty){\mathfrak{c}}ap d{\mathbb N}\neq\etamptyset$. Take $n\in(K\sigmaetminus K^\psirime){\mathfrak{c}}ap\langlembdaangle K^\psirime\rhoangle$. Then $n\in K$ and $n\in\langlembdaangle K^\psirime\rhoangle\sigmaubset\langlembdaangle K\sigmaetminus\{n\}\rhoangle$, which means that the set $K$ is not independant.
${\mathbb B}ox$
\psiaragraph{Proof of ``$\Rightarrow$" of Proposition \rhoef{TB037}}
Let $\langlembdaa n_i :i\in I\rhoa$ be a reversible sequence.
First, suppose that the set $K$ is not independent.
Then for some $m\in K$ there are $s>0$, $k_r \in {\mathbb N}$ and different $m_r\in K\sigmaetminus \{ m \}$, for $0\langlembdaeq r<s$, such that
\betaegin{equation}\langlembdaabel{EQB013}\tauextstyle
m=\sigmaum _{0\langlembdaeq r<s}k_r m_r .
\etand{equation}
We take countable subsets with 1-1 enumerations
$$
I'_m =\{ j_l :l\in \omega\} \sigmaubset I_m
$$
$$
I'_{m _r} =\{ i^r_l :l\in \omega\} \sigmaubset I_{m_r}, \mubox{ for } r<s,
$$
and define $f : I\rhoightarrow I$ by
$$
f (i)=\langlembdaeft\{ \betaegin{array}{cl}
j_0, & \mubox{ if } i=i^r_l, \mubox{ where } r<s \mubox{ and } l<k_r ,\\
i^r_{l-k_r} , & \mubox{ if } i=i^r_l, \mubox{ where } r<s \mubox{ and } l\gammaeq k_r, \\
j_{l+1}, & \mubox{ if } i=j_l , \mubox{ where } l\in \omega , \\
i , & \mubox{ if } i\in I \sigmaetminus (I'_m {\mathfrak{c}}up \betaigcup _{r<s} I'_{m_r}).
\etand{array}
\rhoight.
$$
It is easy to see that $f [I'_m {\mathfrak{c}}up \betaigcup _{r<s} I'_{m_r}] =I'_m {\mathfrak{c}}up \betaigcup _{r<s} I'_{m_r}$ so $f$
is a surjection, satisfies (\rhoef{EQB004}) and it is not 1-1, which gives a contradiction.
So the set $K$ is independent and, by Fact \rhoef{TB032}(d), $|K |<\omega$.
Second, suppose that $K\neq\etamptyset$, $d=\gammacd (K)$ and $|\{ n_i :i\in I \} {\mathfrak{c}}ap d{\mathbb N} | =\omega$.
\betaegin{cla}\langlembdaabel{TB046}
There is a sequence $\langlembdaa q_r :r\in \omega\rhoa$ in $\{ n_i :i\in I\} {\mathfrak{c}}ap \langlembdaa K\rhoa \sigmaetminus K$
such that
\betaegin{equation}\langlembdaabel{EQB021}
\varphiorall r\in \omega \;\; q_{r+1}-q _r \in \langlembdaa K\rhoa.
\etand{equation}
\etand{cla}
\noindent{\betaf Proof. }
Since $K$ is a finite set, by Fact \rhoef{TB032}(c)
there is $M\in {\mathbb N}$ such that $M>\muax K$ and
\betaegin{equation}\langlembdaabel{EQB022}
\langlembdaa K\rhoa {\mathfrak{c}}ap [dM ,\infty )=d{\mathbb N} {\mathfrak{c}}ap [dM ,\infty )=\{ dm: m\gammaeq M\}.
\etand{equation}
So $\{ n_i :i\in I \} {\mathfrak{c}}ap d{\mathbb N} {\mathfrak{c}}ap [dM ,\infty )=\{ n_i :i\in I \} {\mathfrak{c}}ap \langlembdaa K\rhoa {\mathfrak{c}}ap [dM ,\infty )$ is an infinite set.
Let $\{ n_i :i\in I \} {\mathfrak{c}}ap \langlembdaa K\rhoa {\mathfrak{c}}ap [dM ,\infty )=\{ n_{i_k}: k\in \omega\}$,
where $n_{i_0}< n_{i_1}< n_{i_2}<\dots$.
By recursion we easily construct a sequence $\langlembdaa k_r :r\in \omega\rhoa$ in $\omega$
such that $n_{i_{k_{r+1}}} - n_{i_{k_r}} \gammaeq dM$, which implies that $n_{i_{k_r}}\in \langlembdaa K \rhoa\sigmaetminus K$ and
$n_{i_{k_{r+1}}} - n_{i_{k_r}} \in \langlembdaa K \rhoa$. Defining $q_r=n_{i_{k_r}}$, for $r\in \omega$, we finish the proof of Claim \rhoef{TB046}.
${\mathbb B}ox$ \psiar \vspace*{2mm}
For $r\in \omega$ we choose $i_r\in I$ such that
\betaegin{equation}\langlembdaabel{EQB024}
q_r=n_{i_r}\in\langlembdaa K\rhoa \sigmaetminus K.
\etand{equation}
Then by (\rhoef{EQB021}) and (\rhoef{EQB024}), $\{ I_m : m\in K \} {\mathfrak{c}}up \{ I_{n_{i_r}}: r\in \omega \}$ is a family of pairwise disjoint
subsets of $I$.
For each $m\in K$ we choose a countably infinite, co-infinite subset $I'_m$ of $I_m$ and an 1-1 enumeration of $I'_m$, that is
\betaegin{equation}\langlembdaabel{EQB025}
I'_m =\{ i^m_l:l\in \omega \}\sigmaubset I_m \;\;\langlembdaand \;\;|I'_m|=\omega \;\;\langlembdaand \;\; |I_m \sigmaetminus I'_m|\gammaeq \omega,
\etand{equation}
and in this way we obtain an ``one-to-one matrix indexing" $\{ i^m_l : \langlembdaa m,l\rhoa \in K\tauimes \omega\}$
of the set $\betaigcup _{m\in K}I'_m$.
Now, by (\rhoef{EQB021}), (\rhoef{EQB024}) and since the sets $I'_m$ are infinite, we can choose non-empty sets $L_r$, for $r\in \omega$, such that
(l1) $L_r \in [K\tauimes \omega]^{<\omega}$,
(l2) $r_1\neq r_2 \Rightarrow L_{r_1} {\mathfrak{c}}ap L_{r_2}=\etamptyset$,
(l3) $q_0=n_{i_0}=\sigmaum _{\langlembdaa m,l\rhoa\in L_0}n_{i^m_l}$,
(l4) $q_{r+1}-q_r = n_{i_{r+1}}-n_{i_r}=\sigmaum _{\langlembdaa m,l\rhoa\in L_{r+1}}n_{i^m_l}$, for $r\in \omega$.
\noindent
First, defining for each $r\in \omega$
(g1) $g(i_r)=i_{r+1}$,
(g2) $g(i^m_l)=i_r$, for all $\langlembdaa m,l \rhoa\in L_r$,
\noindent
by (l2) we obtain a surjection
\betaegin{equation}\langlembdaabel{EQB026}\tauextstyle
g: \{ i^m_l : \langlembdaa m,l \rhoa\in \betaigcup _{r\in \omega }L_r \} {\mathfrak{c}}up \{ i_r :r\in \omega\}\rhoightarrow \{ i_r :r\in \omega\}.
\etand{equation}
Since $g^{-1}[\{ i_0\}]=\{ i^m_l:\langlembdaa m,l \rhoa\in L_0\}$ by (l3) we have
\betaegin{equation}\langlembdaabel{EQB027}\tauextstyle
n_{i_0}=\sigmaum _{\langlembdaa m,l\rhoa\in L_0}n_{i^m_l}= \sigmaum _{i\in g^{-1}[\{ i_0\}]}n_i .
\etand{equation}
Since $g^{-1}[\{ i_{r+1}\}]=\{ i_r\} {\mathfrak{c}}up \{ i^m_l:\langlembdaa m,l \rhoa\in L_{r+1}\}$ by (l4) we have
\betaegin{equation}\langlembdaabel{EQB028}\tauextstyle
n_{i_{r+1}}=n_{i_r}+\sigmaum _{\langlembdaa m,l\rhoa\in L_{r+1}}n_{i^m_l}= \sigmaum _{i\in g^{-1}[\{ i_{r+1}\}]}n_i .
\etand{equation}
By (\rhoef{EQB024}) we have $n_{i_0}\not\in K$ so, by (\rhoef{EQB027}) we have $|L_0|>1$ and, hence, $g$ is a surjection but not a bijection.
In addition, by (\rhoef{EQB027}) and (\rhoef{EQB028})
\betaegin{equation}\langlembdaabel{EQB029}\tauextstyle
\varphiorall j\in \{ i_r :r\in \omega\} \;\; n_{j}= \sigmaum _{i\in g^{-1}[\{ j\}]}n_i .
\etand{equation}
For each $m\in K$ we have $I_m {\mathfrak{c}}ap \{ i^{m'}_{l'} : \langlembdaa m',l' \rhoa\in \betaigcup _{r\in \omega }L_r \}\sigmaubset I_m '$
so by (\rhoef{EQB025}) we have $|I_m|=|I_m \sigmaetminus \{ i^{m'}_{l'} : \langlembdaa m',l' \rhoa\in \betaigcup _{r\in \omega }L_r \} |$ and, hence,
there are bijections
\betaegin{equation}\langlembdaabel{EQB030}\tauextstyle
g_m : I_m \sigmaetminus \{ i^{m'}_{l'} : \langlembdaa m',l' \rhoa\in \betaigcup _{r\in \omega }L_r \} \rhoightarrow I_m.
\etand{equation}
So, for $j\in I_m$ we have $g_m ^{-1}[\{ j\}]=\{ i_j\}$, for some $i_j\in \mathop{\mathrm{dom}}\nolimits g_m$ and, since $i,i_j\in I_m$,
\betaegin{equation}\langlembdaabel{EQB031}\tauextstyle
\varphiorall j\in I_m \;\;n_j =n_{i_j}= \sigmaum _{i\in g_m^{-1}[\{ j\}]}n_i .
\etand{equation}
By (\rhoef{EQB026}) and (\rhoef{EQB030}) the function $g{\mathfrak{c}}up \betaigcup _{m\in K}g_m$ maps the set
$\betaigcup _{m\in K}I_m {\mathfrak{c}}up \{ i_r :r\in \omega\}$ onto itself and, defining
$$\tauextstyle
f=g{\mathfrak{c}}up \betaigcup _{m\in K}g_m {\mathfrak{c}}up \mathop{\mathrm{id}}\nolimits _{I\sigmaetminus (\betaigcup _{m\in K}I_m {\mathfrak{c}}up \{ i_r :r\in \omega\})}
$$
by (\rhoef{EQB029}) and (\rhoef{EQB031}) we obtain a surjection $f:I\rhoightarrow I$ which is not a bijection and satisfies (\rhoef{EQB004}),
which contradicts our assumption that the sequence $\langlembdaa n_i :i\in I\rhoa$ is reversible.
The implication ``$\Rightarrow$" of Proposition \rhoef{TB037} is proved.
${\mathbb B}ox$
\psiaragraph{Proof of ``${\mathcal L}eftarrow$" of Proposition \rhoef{TB037}}
Let $K$ be an independent set and, if $K\neq\etamptyset$, let $|\{ n_i :i\in I \} {\mathfrak{c}}ap d{\mathbb N} |<\omega$, where $d=\gammacd (K)$.
Suppose that the sequence $\langlembdaa n_i :i\in I\rhoa$ is not reversible.
Then by Claim \rhoef{TB040} we have $K\neq\etamptyset$ and, hence, $|\{ n_i :i\in I \} {\mathfrak{c}}ap d{\mathbb N} |<\omega$.
Let $f:I\rhoightarrow I$ be a surjection such that
\betaegin{equation}\langlembdaabel{EQB004}\tauextstyle
\varphiorall j\in I \;\; n_j=\sigmaum _{i\in f^{-1}[\{ j \}]}n_i.
\etand{equation}
\betaegin{equation}\langlembdaabel{EQB005}\tauextstyle
J:= \{ j\in I : |f^{-1}[\{ j \}]|>1\} \neq \etamptyset.
\etand{equation}
\betaegin{cla}\langlembdaabel{TB031}
(a) For each $i\in I$ we have $n_i\langlembdaeq n_{f(i)}$.
(b) For each $j\in I$ there is a sequence
$\langlembdaa i^j _k :k\in {\mathbb N}\rhoa$ in $I$ such that
\betaegin{equation}\langlembdaabel{EQB007}\tauextstyle
f(i^j_1)=j \;\;\langlembdaand \;\; \varphiorall k\in {\mathbb N} \;\; f(i^j_{k+1})=i^j_k ,
\etand{equation}
\betaegin{equation}\langlembdaabel{EQB008}\tauextstyle
\dots n_{i^j_{k+1}}\langlembdaeq n_{i^j_k} \langlembdaeq \dots n_{i^j_{3}}\langlembdaeq n_{i^j_2} \langlembdaeq n_{i^j_{1}}\langlembdaeq n_{j}.
\etand{equation}
(c) If, in addition, $n_{i^j_{1}}< n_{j}$ in (\rhoef{EQB008}), then $i^j_k\neq i^j_l$, whenever $k\neq l$.
\etand{cla}
\noindent{\betaf Proof. }
(a) follows from (\rhoef{EQB004}).
(b) If $j\in I$,
then, since $f$ is an onto mapping, there is $i^j_1\in I$ such that $f(i^j_1)=j$,
there is $i^j_2\in I$ such that $f(i^j_2)=i^j_1$,
there is $i^j_3\in I$ such that $f(i^j_3)=i^j_2$,
and so on.
So in this way we obtain a sequence $\langlembdaa i^j _k :k\in {\mathbb N}\rhoa \in {}^{{\mathbb N}}I$
satisfying (\rhoef{EQB007})
which, together with (a), gives (\rhoef{EQB008}).
(c) If $n_{i^j_{1}}< n_{j}$ then, by (\rhoef{EQB008}), $n_{i^j_{k}}< n_{j}$, for all $k\in {\mathbb N}$ and, hence,
\betaegin{equation}\langlembdaabel{EQB010}\tauextstyle
\varphiorall k\in {\mathbb N} \;\; i^j_k\neq j .
\etand{equation}
On the contrary, let $k$ be the minimal element of ${\mathbb N}$ such that $i^j_k= i^j_l$, for some $l>k$.
Then by (\rhoef{EQB007}), for $k=1$ we would have $i^j_{l-1}= f(i^j_l)=f(i^j_k)=f(i^j_1)=j$, which is impossible by (\rhoef{EQB010}).
For $k>1$ we would have $i^j_{l-1}= f(i^j_l)=f(i^j_k)=i^j_{k-1}$, which is false by the minimality of $k$.
${\mathbb B}ox$
\betaegin{cla}\langlembdaabel{TB029}
There is a sequence $\langlembdaa p_r :r\in \omega \rhoa$ in ${\mathbb N}$
such that, defining for convenience $p_{-1}:=0$, for each $r\in \omega$ we have:
(i) $p_r=\muin \{ n_j : j\in J \langlembdaand n_j >p_{r-1}\}$,
(ii) $\varphiorall j\in I_{p_r}{\mathfrak{c}}ap J\;\; \varphiorall i\in f^{-1}[\{ j \}] \;\; n_i \in K {\mathfrak{c}}up \{ p_s : 0\langlembdaeq s<r\}$,
(iii) $p_r\in \langlembdaa K\rhoa\sigmaetminus K$,
(iv) $\etaxists i\in I_{p_r} \;\; ( f(i)\in J \langlembdaand n_{f(i)}>p_r)$,
(v) $\{ n_j : j\in J\}{\mathfrak{c}}ap [1,p_r]=\{ p_s : 0\langlembdaeq s\langlembdaeq r\}$.
\etand{cla}
\noindent{\betaf Proof. }
We construct the sequence by recursion.
First, by (\rhoef{EQB005}) we have $J\neq \etamptyset$ so
$\etamptyset \neq \{ n_j : j\in J\}=\{ n_j : j\in J\langlembdaand n_j >0\}\sigmaubset {\mathbb N}$ and defining
\betaegin{equation}\langlembdaabel{EQB011}\tauextstyle
p_0=\muin \{ n_j : j\in J\}
\etand{equation}
we see that the sequence $\langlembdaa p_0\rhoa$ satisfies (i).
(ii) Let $j\in I_{p_0}{\mathfrak{c}}ap J$ and $i\in f^{-1}[\{ j \}]$.
Then, since $j\in J$, by (\rhoef{EQB005}) we have $|f^{-1}[\{ j \}]|>1$
and, by (\rhoef{EQB004}), $n_j =\sigmaum _{i'\in f^{-1}[\{ j \}]}n_{i'}$, so $n_i <n_{j}$.
As in Claim \rhoef{TB031} we define $i^{j}_k\in I$, for $k\in {\mathbb N}$, satisfying $i^{j}_1:=i$, (\rhoef{EQB007}) and (\rhoef{EQB008})
and so we obtain $\dots n_{i^{j}_{3}}\langlembdaeq n_{i^{j}_2} \langlembdaeq n_{i^{j}_{1}}< n_{j} $.
Assuming that $n_{i^{j}_{k+1}}< n_{i^{j}_k}$ for some $k\in {\mathbb N}$, since $f(i^{j}_{k+1})=i^{j}_k$ by (\rhoef{EQB004})
we would have $i^{j}_k\in J$ and $n_{i^{j}_k}<n_{j}=p_0$, which is, by (\rhoef{EQB011}), impossible.
Thus there is $m\in {\mathbb N}$ such that $n_{i^{j}_k}=m$, for all $k\in {\mathbb N}$.
By Claim \rhoef{TB031}(c) we have $i^{j}_k\neq i^{j}_l$, whenever $k\neq l$, thus $|I_m|\gammaeq \omega$. So $n_i =n_{i^{j}_1}=m\in K$.
(iii) By the previous item and (\rhoef{EQB004}) we have $p_0=n_{j} \in \langlembdaa K\sigmaetminus \{ p_0\}\rhoa$ and, since the set $K$ is independent, $p_0\not \in K$.
(iv) By (iii) we have $p_0\not\in K$, that is $|I_{p_0}|<\omega$. Suppose that $f[I_{p_0}]\sigmaubset I_{p_0}$.
Then by (\rhoef{EQB004}) $f\!\!\uparrowharpoonright I_{p_0}$ is an injection and, since the set $I_{p_0}$ is finite, $f[I_{p_0}]= I_{p_0}$.
By (\rhoef{EQB011}) there is $j\in I_{p_0}{\mathfrak{c}}ap J$ and by the previous conclusion, $j=f(i)$, for some $i\in I_{p_0}$, which implies that
$n_i=n_j=p_0$. But this contradicts the fact that $j\in J$. So, there is $i\in I_{p_0} $ such that $f(i)\not\in I_{p_0}$
and, hence, $n_{f(i)}> n_i=p_0$ and $f(i)\in J$.
(v) By (\rhoef{EQB011}) we have $\{ n_j : j\in J\}{\mathfrak{c}}ap [1,p_0]=\{ p_0\}$.
Suppose that $\langlembdaa p_0, \dots ,p_r\rhoa$ is a sequence satisfying (i)--(v). By (iv) there is $j\in J$ such that $n_j >p_r$
and defining
\betaegin{equation}\langlembdaabel{EQB012}\tauextstyle
p_{r+1}=\muin \{ n_j : j\in J \langlembdaand n_j >p_r\} .
\etand{equation}
we have (i).
(ii) Let $j\in I_{p_{r+1}}{\mathfrak{c}}ap J$ and $i\in f^{-1}[\{ j \}]$. Then, since $j\in J$, by (\rhoef{EQB005}) we have $|f^{-1}[\{ j \}]|>1$
and, by (\rhoef{EQB004}), $n_j =\sigmaum _{i'\in f^{-1}[\{ j \}]}n_{i'}$, so $n_i <n_{j}$.
Again, as in Claim \rhoef{TB031} we define $i^{j}_k\in I$, for $k\in {\mathbb N}$, satisfying $i^{j}_1:=i$, (\rhoef{EQB007}) and (\rhoef{EQB008})
and so we obtain $\dots n_{i^{j}_{3}}\langlembdaeq n_{i^{j}_2} \langlembdaeq n_{i^{j}_{1}}< n_{j} $.
If $n_{i^{j}_{k+1}}< n_{i^{j}_k}$ for some $k\in {\mathbb N}$, let $k$ be the minimal such $k$.
Then
\betaegin{equation}\langlembdaabel{EQB014}
n_{i^{j}_{k+1}}<n_{i^{j}_k} = \dots = n_{i^{j}_2} = n_{i^{j}_{1}}=n_i< n_{j}=p_{r+1} .
\etand{equation}
In addition, since $f(i^{j}_{k+1})=i^{j}_k$, by (\rhoef{EQB004}) we have $i^{j}_k\in J$
which implies that $n_{i^{j}_k}\in \{ n_j:j\in J\} {\mathfrak{c}}ap [1,p_{r+1})$ and, by (\rhoef{EQB012}),
$n_{i^{j}_k}\in \{ n_j:j\in J\} {\mathfrak{c}}ap [1,p_r]$.
So, by (v), there is $s_0\langlembdaeq r$ such that $n_{i^{j}_k}=p_{s_0}$
and, by (\rhoef{EQB014}), $n_i=p_{s_0}\in \{ p_s : 0\langlembdaeq s<r+1\}$.
Otherwise, there is $m\in {\mathbb N}$ such that $n_{i^{j}_k}=m$, for all $k\in {\mathbb N}$.
By Claim \rhoef{TB031}(c) we have $i^{j}_k\neq i^{j}_l$, whenever $k\neq l$,
thus $|I_m|\gammaeq \omega$,
and, hence, $m\in K$.
So $n_i =n_{i^{j}_1}=m\in K$ and (ii) is true indeed.
(iii) By (\rhoef{EQB012}) there is $j\in J$ such that $p_{r+1}=n_j>p_r$.
Thus $j\in I_{p_{r+1}}{\mathfrak{c}}ap J$ and, by (ii) and (\rhoef{EQB004}), $n_j$
is a sum of at least two integers from $K{\mathfrak{c}}up \{ p_s : 0\langlembdaeq s\langlembdaeq r\}$.
By (iii) of the induction hypothesis we have $p_s\in \langlembdaa K\rhoa$, for $0\langlembdaeq s\langlembdaeq r$, and, hence, $p_{r+1}\in \langlembdaa K\sigmaetminus \{p_{r+1} \}\rhoa$.
Since the set $K$ is independent we have $p_{r+1}\not\in K$.
(iv) Since $p_{r+1}\not\in K$ we have $|I_{p_{r+1}}|<\omega$. Suppose that $f[I_{p_{r+1}}]\sigmaubset I_{p_{r+1}}$.
Then by (\rhoef{EQB004}) $f\!\!\uparrowharpoonright I_{p_{r+1}}$ is an injection and, since the set $I_{p_{r+1}}$ is finite, $f[I_{p_{r+1}}]= I_{p_{r+1}}$.
By (\rhoef{EQB012}) there is $j\in I_{p_{r+1}}{\mathfrak{c}}ap J$ and, since $f[I_{p_{r+1}}]= I_{p_{r+1}}$, $j=f(i)$, for some $i\in I_{p_{r+1}}$, which implies that
$n_i=n_j=p_{r+1}$. But this contradicts the fact that $j\in J$. So, there is $i\in I_{p_{r+1}} $ such that $f(i)\not\in I_{p_{r+1}}$
and, hence, $n_{f(i)}> n_i=p_{r+1}$ and $f(i)\in J$.
(v) By (\rhoef{EQB012}) and the induction hypothesis we have $\{ n_j : j\in J\}{\mathfrak{c}}ap [1,p_{r+1}]=\{ p_s : 0\langlembdaeq s\langlembdaeq r+1\}$.
Thus the recursion works.
${\mathbb B}ox$ \psiar \vspace*{2mm}
Now, by Claim \rhoef{TB029}(v), (iii) and (i),
$\{ n_j :j\in J\} =\{ p_r :r\in \omega\} \sigmaubset \langlembdaa K\rhoa \sigmaetminus K$ and $p_0 <p_1 <\dots < p_r < \dots $,
which implies that $| \{ n_i :i\in I \}{\mathfrak{c}}ap\langlembdaa K \rhoa |=\omega$. Since, by Fact \rhoef{TB032}(c), $\langlembdaa K\rhoa \sigmaubset d{\mathbb N}$,
we have $|\{ n_i :i\in I \} {\mathfrak{c}}ap d{\mathbb N} | =\omega$ and we obtain a contradiction.
${\mathbb B}ox$
\psiaragraph{Reversible functions in the Baire space}
Each countable sequence of natural numbers $\langlembdaa n_i :i\in {\mathbb N}\rhoa \in {}^{{\mathbb N}}{\mathbb N}$ can be regarded
as a function $\varphi :{\mathbb N} \rhoightarrow {\mathbb N}$, where $\varphi(i)=n_i$, for $i\in {\mathbb N}$, and, hence, as an element
of the Baire space ${\mathbb N}^{\mathbb N}$ with the standard topology (see {\mathfrak{c}}ite{Kech}). So we can consider the set of reversible functions
belonging to ${\mathbb N}^{\mathbb N}$,
$$
{\mathbb N}rev \!:= {\mathbb B}ig\{ \varphi \in {\mathbb N}^{\mathbb N} : \neg \etaxists f\in {\mathbb S}ur({\mathbb N})\sigmaetminus{\mathbb S}ym({\mathbb N} ) \;\varphiorall j\in {\mathbb N} \;\;\varphi (j)=\!\sigmaum _{i\in f^{-1}[\{ j \}]} \varphi (i) {\mathbb B}ig\} .
$$
\betaegin{te}
${\mathbb N}rev $ is a dense $F_{\sigmaigma\delta\sigmaigma}(={\mathbb S}igma ^0_4)$ subset of ${\mathbb N}^{\mathbb N}$ of size ${\muathfrak c}$.
\etand{te}
\noindent{\betaf Proof. }
If $B=\betaigcap _{k\langlembdaeq n}\psii ^{-1}_{i_k}[\{ j_k\}]$ is a basic open set, then, since the finite function $p=\{ \langlembdaa i_k ,j_k\rhoa : k\langlembdaeq n\}$
can be extended to an finite-to-one function $\varphi \in {\mathbb N}^{\mathbb N}$ and by Proposition \rhoef{TB037} we have $\varphi\in {\mathbb N}rev$, it follows that
$B{\mathfrak{c}}ap {\mathbb N}rev \neq \etamptyset$ so ${\mathbb N}rev $ is dense in ${\mathbb N}^{\mathbb N}$. $|{\mathbb N}rev|={\muathfrak c}$ follows from the fact that ${\mathbb N}^{\mathbb N}$ contains
${\muathfrak c}$-many injections.
Let ${\mathcal I}$ be the set of non-empty independent subsets of ${\mathbb N}$ and, for $K\in {\mathcal I}$, let $d_K:=\gammacd (K)$.
Then by Proposition \rhoef{TB037}
\betaegin{equation}\langlembdaabel{EQB023}\tauextstyle
{\mathbb N}rev = A{\mathfrak{c}}up \betaigcup _{K\in {\mathcal I}}\;B_K {\mathfrak{c}}ap C_K {\mathfrak{c}}ap D_K,
\etand{equation}
where
\betaegin{eqnarray*}
A & := & {\mathbb B}ig\{ \varphi\in {\mathbb N}^{\mathbb N} : \varphiorall m\in {\mathbb N} \;\; ( \varphi(i)=m \mubox{ for } <\omega\mubox{-many } i\in {\mathbb N}){\mathbb B}ig\} , \\
& = &\tauextstyle \betaigcap _{m\in {\mathbb N}} \betaigcup _{k\in {\mathbb N}} \betaigcap _{i\gammaeq k} \psii ^{-1}_i[{\mathbb N} \sigmaetminus \{ m\}] , \\
B_K & := & {\mathbb B}ig\{ \varphi\in {\mathbb N}^{\mathbb N} : \varphiorall m\in K \;\; ( \varphi(i)=m \mubox{ for } \omega\mubox{-many } i\in {\mathbb N}){\mathbb B}ig\} ,\\
& = &\tauextstyle \betaigcap _{m\in K} \betaigcap _{k\in {\mathbb N}} \betaigcup _{i\gammaeq k} \psii ^{-1}_i[\{ m\}] , \\
C_K & := & {\mathbb B}ig\{ \varphi\in {\mathbb N}^{\mathbb N} : \varphiorall m\in {\mathbb N} \sigmaetminus K \;\; ( \varphi(i)=m \mubox{ for } <\omega\mubox{-many } i\in {\mathbb N}){\mathbb B}ig\} , \\
& = &\tauextstyle \betaigcap _{m\in {\mathbb N}\sigmaetminus K} \betaigcup _{k\in {\mathbb N}} \betaigcap _{i\gammaeq k} \psii ^{-1}_i[{\mathbb N} \sigmaetminus \{ m\}] , \\
D_K & := & {\mathbb B}ig\{ \varphi\in {\mathbb N}^{\mathbb N} : \varphi(i)\in d{\mathbb N} \mubox{ for } <\omega\mubox{-many } i\in {\mathbb N}{\mathbb B}ig\} \\
& = &\tauextstyle \betaigcup _{m\in {\mathbb N}} \betaigcap _{i\gammaeq m} \betaigcap _{k\in {\mathbb N}} \psii ^{-1}_i[{\mathbb N} \sigmaetminus \{ dk\}] .
\etand{eqnarray*}
So, for $K\in {\mathcal I}$ we have $B_K \in G_\delta$, $D_K\in F_\sigmaigma$ and $C_K\in F_{\sigmaigma\delta}$, which implies that
$B_K {\mathfrak{c}}ap C_K {\mathfrak{c}}ap D_K \in F_{\sigmaigma\delta}$ and, since by Fact \rhoef{TB032}(d) we have ${\mathcal I} \sigmaubset [{\mathbb N} ]^{<\omega}$, it follows that
$\betaigcup _{K\in {\mathcal I}}\;B_K {\mathfrak{c}}ap C_K {\mathfrak{c}}ap D_K \in F_{\sigmaigma\delta\sigmaigma}$. Since $A\in F_{\sigmaigma\delta}\sigmaubset F_{\sigmaigma\delta\sigmaigma}$,
by (\rhoef{EQB023}) we have ${\mathbb N}rev \in F_{\sigmaigma\delta\sigmaigma}={\mathbb S}igma ^0_4$.
${\mathbb B}ox$
\betaegin{rem}\langlembdaabel{RB000}\rhom
Let the equivalence relation $\sigmaim$ on ${\mathbb N}^{\mathbb N}$ be defined by $\varphi \sigmaim \psi$ iff there is $f\in {\mathbb S}ym ({\mathbb N})$ such that
$\varphi=\psi{\mathfrak{c}}irc f$. It is evident that the set ${\mathbb N}rev$ is $\sigmaim$-invariant, that is $\psi\sigmaim \varphi\in {\mathbb N}rev$ implies $\psi\in {\mathbb N}rev$.
But ${\mathbb N}rev$ is not a subsemigroup of $\langlembdaa {\mathbb N} ^{\mathbb N} ,{\mathfrak{c}}irc \rhoa$ (it is not closed under composition). Let ${\mathbb N} \sigmaetminus \{ 2\}=A {\mathfrak{c}}up B$ and ${\mathbb N} =C{\mathfrak{c}}up D {\mathfrak{c}}up E$ be partitions,
where $A,B,C,D,E \in [{\mathbb N} ]^\omega$ and $|A{\mathfrak{c}}ap (2{\mathbb N} +1)|=|B{\mathfrak{c}}ap (2{\mathbb N} +1)|=\omega$. Then, by Proposition \rhoef{TB037},
$
\varphi =\{ \langlembdaa 2,2 \rhoa\}{\mathfrak{c}}up (A\tauimes \{ 3 \}) {\mathfrak{c}}up (B\tauimes \{ 5 \})\in {\mathbb N}rev.
$
If $\psi_{DA}:D\rhoightarrow A{\mathfrak{c}}ap (2{\mathbb N} +1)$ and $\psi_{EB}:E\rhoightarrow B{\mathfrak{c}}ap (2{\mathbb N} +1)$ are bijections then, by Proposition \rhoef{TB037} again,
$
\psi = (C\tauimes \{ 2\}) {\mathfrak{c}}up \psi_{DA}{\mathfrak{c}}up\psi_{EB}\in {\mathbb N}rev .
$
But $\varphi {\mathfrak{c}}irc \psi \not\in {\mathbb N}rev$, because the set $\{ 2,3,5\}$ is not independent.
\etand{rem}
\varphiootnotesize
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\etand{document} |
\begin{document}
\title{Learning to Predict: A Fast Re-constructive Method to Generate Multimodal Embeddings}
\author{Guillem Collell\inst{1} \and Ted Zhang\inst{2} \and Marie-Francine Moens\inst{3}}
\institute{Computer Science Department, KU Leuven, Belgium\inst{1}\inst{2}\inst{3}\\
\email{[email protected], [email protected], [email protected]} }
\maketitle
\begin{abstract}
Integrating visual and linguistic information into a single multimodal representation is an unsolved problem with wide-reaching applications to both natural language processing and computer vision. In this paper, we present a simple method to build multimodal representations by learning a language-to-vision mapping and using its output to build multimodal embeddings. In this sense, our method provides a cognitively plausible way of building representations, consistent with the inherently re-constructive and associative nature of human memory. Using seven benchmark concept similarity tests we show that the mapped vectors not only implicitly encode multimodal information, but also outperform strong unimodal baselines and state-of-the-art multimodal methods, thus exhibiting more ``human-like" judgments---particularly in zero-shot settings.
\end{abstract}
\section{Introduction}
\label{sect:intro}
Convolutional neural networks (CNN) and distributional-semantic models have provided breakthrough advances in representation learning in computer vision (CV) and natural language processing (NLP) respectively \cite{lecun2015deep}. Lately, a large body of research has shown that using rich, multimodal representations created from combining textual and visual features instead of unimodal representations (a.k.a. embeddings) can improve the performance of semantic tasks. In other words, a single multimodal representation that captures information from two modalities (vision and language) is semantically richer than those from a single modality or unimodal (either vision or language). Building multimodal representations has become a popular problem in NLP that has yielded a wide variety of methods \cite{lazaridou2015combining,kiela2014learning,silberer2014learning}. Additionally, the use of a mapping to bridge vision and language has also been explored, typically with the goal of zero-shot image classification \cite{lazaridou2014wampimuk,socher2013zero}.
\begin{figure}
\caption{Overview of our multimodal method.}
\label{diagram}
\end{figure}
Here, we propose a cognitively plausible approach to concept representation that consists of: (1) learning a language-to-vision mapping; and (2) using the outputs of the mapping as multimodal representations---with the second step being the main novelty of our approach. By re-constructing visual knowledge from textual input, our method behaves similarly as human memory, namely in an associative \cite{anderson2014human,reijmers2007localization} and re-constructive manner \cite{vernon2014artificial,hawkins2007intelligence,loftus1981reconstructive}. Concretely, our method does not seek the perfect recall of visual representations but rather its re-construction and association with language. We leverage the intuitive fact that, by learning to predict, the mapping necessarily encodes information from both modalities---and in turn discards noise and irrelevant information from the visual vectors during the learning phase. Thus, given a word embedding as input, the mapped output is not purely a visual representation but rather a multimodal one.
By using seven concept similarity benchmarks, we show that our representations not only are multimodal but they improve performance over strong unimodal baselines and state-of-the-art multimodal approaches---inclusive in a zero-shot setting. In turn, the fact that our evaluation tests are composed of human ratings of similarity supports our claim that our method provides more ``human-like" judgments. Further details and insight can be found at the extended version of the present paper \cite{collell2017imagined}.
The rest of the paper is organized as follows. In the next section, we introduce related work. Next, we describe and provide insight on our method. Afterwards, we describe our experimental setup. Finally, we discuss our results, followed by conclusions.
\section{Related work and background}
\label{sect:related}
\subsection{Cognitive grounding}
\label{sect:cognitive}
A large body of research evidences that human memory is inherently re-constructive \cite{vernon2014artificial,hawkins2007intelligence,loftus1981reconstructive}. That is, memories are not ``static" exact copies of reality, but are rather re-constructed from their essential elements each time they are retrieved, triggered by either internal or external stimuli. Arguably, this mechanism is, in turn, what endows humans with the capacity to imagine themselves in yet-to-be experiences and to re-combine existing knowledge into new plans or structures of knowledge \cite{hawkins2007intelligence}. Moreover, the associative nature of human memory is also a widely accepted theory in experimental psychology \cite{anderson2014human} with identifiable neural correlates involved in both learning and retrieval processes \cite{reijmers2007localization}.
In this respect, our method employs a retrieval process analogous to that of humans, in which the retrieval of a visual output is triggered and mediated by a linguistic input (Fig. \ref{diagram}). Effectively, visual information is not only retrieved (i.e., mapped), but also associated to the textual information thanks to the learned cross-modal mapping---analogous to a mental model that associates semantic and visual components of concepts, acquired through lifelong experience. Since the retrieved (mapped) visual information is often insufficient to completely describe a concept, it is of interest to preserve the linguistic component. Thus, we consider the concatenation of the ``imagined" visual representations to the text representations as a comprehensive way of representing concepts.
\subsection{Multimodal representations}
\label{sect:multimodal}
It has been shown that visual and textual features capture complementary attributes \cite{collell2016is}, and the advantages of combining both modalities have been largely demonstrated in a number of linguistic tasks \cite{lazaridou2015combining,kiela2014learning,silberer2014learning}. Based on current literature, we suggest a classification of the existing strategies to build multimodal embeddings. Broadly, multimodal representations can be built by learning from raw input enriched with both modalities (\textbf{\textit{simultaneous}} learning), or by learning each modality separately and integrating them afterwards (\textbf{\textit{a posteriori}} combination).
\begin{enumerate}
\item \textbf{\textit{A posteriori}} combination.
\begin{itemize}
\item \emph{Concatenation}. That is, the fusion of pre-learned visual and text features by concatenating them \cite{kiela2014learning}. Concatenation has been proven effective in concept similarity tasks \cite{bruni2014multimodal,kiela2014learning}, yet suffers from an obvious limitation: multimodal features can only be generated for those words that have images available.
\item \emph{Autoencoders} form a more elaborated approach that do not suffer from the above problem. Encoders are fed with pre-learned visual and text features, and the hidden representations are then used as multimodal embeddings. This approach has shown to perform well in concept similarity tasks and categorization (i.e., grouping objects into categories such as ``fruit", ``furniture", etc.) \cite{silberer2014learning}.
\item A \emph{mapping} between visual and text modalities (i.e., our method). The outputs of the mapping themselves are used to build multimodal representations.
\end{itemize}
\item \textbf{\textit{Simultaneous}} learning. Distributional semantic models are extended into the multimodal domain \cite{lazaridou2015combining,hill2014learning} by learning in a skip-gram manner from a corpus enriched with information from both modalities and using the learned parameters of the hidden layer as multimodal representations. Multimodal skip-gram methods have been proven effective in similarity tasks \cite{lazaridou2015combining,hill2014learning} and in zero-shot image labeling \cite{lazaridou2015combining}.
\end{enumerate}
With this taxonomy, the gap that our method fills becomes more clear, with it being aligned with a re-constructive and associative view of knowledge representation. Furthermore, in contrast to other multimodal approaches such as skip-gram methods \cite{lazaridou2015combining,hill2014learning}, our method directly learns from pre-trained embeddings instead of training from a large multimodal corpus, rendering it thus simpler and faster.
\subsection{Cross-modal mappings}
\label{sect:crossmodalmap}
Several studies have considered the use of mappings to bridge modalities. For instance, \cite{socher2013zero} and \cite{lazaridou2014wampimuk} use a linear vision-to-language projection in zero-shot image classification. Analogously, language-to-vision mappings have been considered, generally to generate missing perceptual information about abstract words \cite{hill2014learning,johns2012perceptual} and in zero-shot image retrieval \cite{lazaridou2015combining}. In contrast to our approach, the methods above do not aim to build multimodal representations to be used in natural language processing tasks.
\section{Proposed method}
In this section we describe the three main steps of our method (Fig. \ref{diagram}): (1) Obtain visual representations of concepts; (2) Build a mapping from the linguistic to the visual space; and (3) Generate multimodal representations.
\subsection{Obtaining visual representations}
\label{sect:visrep}
We employ raw, labeled images from ImageNet \cite{russakovsky2015imagenet} as the source of visual information, although alternatives such as the ESP game data set \cite{von2004labeling} can be considered. To extract visual features from each image, we use the forward pass of a pre-trained CNN model. The hidden representation of the last layer (before the softmax) is taken as a feature vector, as it contains higher level features. For each concept $w$, we \textit{average} the extracted visual features of individual images to build a single visual representation $\overrightarrow{v_w}$.
\subsection{Learning to map language to vision}
\label{sect:mapping}
Let $\mathcal{L} \subset \mathbb{R}^{d_l}$ be the linguistic space and $\mathcal{V} \subset \mathbb{R}^{d_v}$ the visual space of representations, where $d_l$ and $d_v$ are their respective dimensionalities. Let $\overrightarrow{l_w} \in \mathcal{L}$ and $\overrightarrow{v_w} \in \mathcal{V}$ denote the text and visual representations for the concept $w$ respectively. Our goal is thus to learn a mapping (regression) $f:\mathcal{L}\rightarrow \mathcal{V}$. The set of $N$ visual representations along with their corresponding text representations compose the training data $\{ (\overrightarrow{l_i},\overrightarrow{v_i})\}^{N}_{i=1}$ used to learn $f$. In this work, we consider two different mappings $f$.
\textbf{(1) Linear:} A simple perceptron composed of a $d_l$-dimensional input layer and a linear output layer with $d_v$ units.
\textbf{(2) Neural network:} A network composed of a $d_l$-unit input layer, a single hidden layer of $d_h$ Tanh units and a linear output layer of $d_v$ units.
For both mappings, a mean squared error (MSE) loss function is employed: $Loss(y,\hat{y}) = \frac{1}{2} ||\hat{y} - y||^2_2 $, where $y$ is the actual output and $\hat{y}$ the model prediction.
\subsection{Generating multimodal representations}
\label{sect:mapped}
Finally, the mapped representation $\overrightarrow{m_w}$ of each concept $w$ is calculated as the image $f(\overrightarrow{l_w})$ of its linguistic embedding $\overrightarrow{l_w}$. For instance, $\overrightarrow{m_{dog}} = f(\overrightarrow{l_{dog}})$. We henceforth refer to the mapped representations as \textit{MAP}$_{f}$, where $f$ indicates the mapping function employed ($lin$ = linear, $NN$ = neural network). As argued below, the mapped representations are effectively multimodal. However, since $f(\overrightarrow{l_w})$ formally belongs to the visual domain, we also consider the concatenation of the $\ell_2$-normalized mapped representations $f(\overrightarrow{l_w})$ with the textual representations $\overrightarrow{l_w}$, namely $\overrightarrow{l_w}\oplus f(\overrightarrow{l_w})$, where $\oplus$ denotes the concatenation operator. We denote these concatenated representations as \textit{MAP-C}$_{f}$.
Since the outputs of a text-to-vision mapping are strictly speaking, ``visual predictions", it might not seem readily obvious that they are also grounded with textual knowledge. To gain insight, it is instructive to refer to the training phase where the parameters $\theta$ of $f$ are learned as a function of the training data $\{(\overrightarrow{l_i},\overrightarrow{v_i})\}^{N}_{i=1}$. E.g., in gradient descent, $\theta$ is updated according to: $\theta \leftarrow \theta - \eta \frac{\partial}{\partial \theta } Loss( \theta ; \{(\overrightarrow{l_i},\overrightarrow{v_i})\}^{N}_{i=1}).$ Hence, the parameters $\theta$ of $f$ are effectively a function of the training data points $\{(\overrightarrow{l_i},\overrightarrow{v_i})\}^{N}_{i=1}$ and it is therefore expected that the outputs $f(\overrightarrow{l_w})$ are grounded with properties of the input data $\{\overrightarrow{l_i}\}^{N}_{i=1}$. It can be additionally noted that the output of the mapping $f(\overrightarrow{l_w})$ is a (continuous) transformation of the input vector $\overrightarrow{l_w}$. Thus, unless the mapping is completely uninformative (e.g., constant or random), the input vector $\overrightarrow{l_w}$ is still ``present"---yet transformed. Thus, the output of the mapping necessarily contains information from both modalities, vision and language, which is essentially the core idea of our method. Further insight is provided at the extended version of the article \cite{collell2017imagined}.
\section{Experimental setup}
\subsection{Word embeddings}
We use 300-dimensional GloVe\footnote{\tt http://nlp.stanford.edu/projects/glove} vectors \cite{pennington2014glove} pre-trained on the Common Crawl corpus consisting of 840B tokens and a 2.2M words vocabulary.
\subsection{Visual data and features}
We use ImageNet \cite{russakovsky2015imagenet} as our source of labeled images. ImageNet covers 21,841 WordNet synsets (or meanings) \cite{fellbaum1998wordnet} and has 14,197,122 images. We only keep synsets with more than 50 images, and an upper bound of 500 images per synset is used to reduce computation time. With this selection, we cover 9,251 unique words.
To extract visual features from each image, we use a pre-trained VGG-m-128 CNN \cite{Chatfield14} implemented with the Matlab MatConvNet toolkit \cite{vedaldi15matconvnet}. We take the 128-dimensional activation of the last layer (before the softmax) as our visual features.
\subsection{Evaluation sets}
We test the methods in seven benchmark tests, covering three tasks: \textbf{(i) General relatedness}: \textit{MEN} \cite{bruni2014multimodal} and \textit{Wordsim353-rel} \cite{agirre2009study}; \textbf{(ii) Semantic or taxonomic similarity}: \textit{SemSim} \cite{silberer2014learning}, \textit{Simlex999} \cite{hill2015simlex}, \textit{Wordsim353-sim} \cite{agirre2009study} and \textit{SimVerb-3500} \cite{gerz2016simverb}; \textbf{(iii) Visual similarity}: \textit{VisSim} \cite{silberer2014learning} which contains the same word pairs as \textit{SemSim}, rated for visual instead of semantic similarity. All tests contain word pairs along with their human similarity rating. The tests \textit{Wordsim353-sim} and \textit{Wordsim353-rel} are the similarity and relatedness subsets of \textit{Wordsim353} \cite{finkelstein2001placing} proposed by \cite{agirre2009study} who noted that the distinction between similarity (e.g., ``tiger" is similar to ``cat") and relatedness (e.g., ``stock" is related to ``market") yields different results. Hence, for being redundant with its subsets, we do not count the whole \textit{Wordsim353} as an extra test set.
A large part of words in our tests do not have a visual representation $\overrightarrow{v_w}$ available, i.e., they are not present in our training data. We refer to these words as zero-shot (ZS).
\subsection{Evaluation metric and prediction}
We use Spearman correlation $\rho$ between model predictions and human similarity ratings as evaluation metric. The prediction of similarity between two concept representations, $\overrightarrow{u_1}$ and $\overrightarrow{u_2}$, is computed by their cosine similarity: $\cos(\overrightarrow{u_1},\overrightarrow{u_2}) = \frac{\overrightarrow{u_1} \cdot \overrightarrow{u_2}}{\| \overrightarrow{u_1} \|\cdot\| \overrightarrow{u_2} \|}$.
\subsection{Model settings}
Both, neural network and linear models are learned by stochastic gradient descent and nine parameter combinations are tested (learning\_rate = [0.1, 0.01, 0.005] and dropout\_rate = [0.5, 0.25, 0.1]). We find that the models are not very sensitive to parameter variations and all of them perform reasonably well. We report a linear model with learning rate of 0.1 and dropout rate of 0.1. For the neural network we use 300 hidden units, dropout rate of 0.25 and learning of 0.1. All mappings are implemented with the scikit-learn toolkit \cite{scikit-learn} in Python 2.7.
\section{Results and discussion}
\label{sect:results}
In the following we summarize our main findings. For clarity, we refer to the concatenation of \textit{CNN}$_{avg}$ and GloVe as \textit{CONC}.
\textbf{\textit{Overall}}, a post-hoc Nemenyi test including all disjoint regions (ZS and VIS) shows that both \textit{MAP-C} methods (\textit{lin} and \textit{NN}) perform significantly better than GloVe (p $\approx$ 0.03) and than \textit{CNN}$_{avg}$ (p $\approx$ 0.06). Hence, our multimodal representations \textit{MAP-C} clearly accomplish one of their foremost goals, namely to improve the unimodal representations of GloVe and \textit{CNN}$_{avg}$.
Clearly, the consistent improvement of \textit{MAP}$_{lin}$ and \textit{MAP}$_{NN}$ over \textit{CNN}$_{avg}$ in all seven test sets supports our claim that the \textit{imagined} visual representations are more than purely visual representations and contain multimodal information---as argued in subsection \ref{sect:mapped}. Moreover, the \textit{MAP-C} method generally performs better than the\textit{MAP} vectors alone, implying that even though the \textit{MAP} vectors are indeed multimodal, they are still predominantly visual and their concatenation with textual representations helps.
Using the \textit{\textbf{concreteness}} ratings of \cite{brysbaert2014concreteness} in a 1-5 scale (with 5 being the most concrete and 1 the most abstract) we find that the average concreteness is larger than 4.4 in all VIS regions, while it is lower than 3.3 in all ZS regions except in \textit{MEN} and \textit{VisSim}/\textit{SemSim} test sets which average 4.2 and 4.8 respectively. Therefore, with the exceptions of \textit{MEN}, \textit{VisSim} and \textit{SemSim}, the inclusion of multimodal information in the ZS regions is arguably less beneficial than in the VIS regions, given that visual information can only sensibly enrich representations of words that are to some extent visual.
\begin{table*}[ht]
\footnotesize
\centering
\caption{Spearman correlations between model predictions and human ratings. For each test set, ALL is the whole set of word pairs, VIS are those pairs with both visual representations available, and ZS denotes its complement, i.e., zero-shot words. Boldface indicates best results per column and \# inst. the number of word pairs in ALL, VIS or ZS. It must be noted that the VIS region of the compared methods is only approximated, as they do not report the exact evaluated instances.}
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{lcccccccccccc}
\multicolumn{1}{l|}{} & \multicolumn{3}{c|}{Wordsim353} & \multicolumn{3}{c|}{MEN} & \multicolumn{3}{c|}{SemSim} & \multicolumn{3}{c|}{VisSim} \\ \cline{2-13}
\multicolumn{1}{l|}{} & ALL & VIS & \multicolumn{1}{c|}{ZS} & ALL & VIS & \multicolumn{1}{c|}{ZS} & ALL & VIS & \multicolumn{1}{c|}{ZS} & ALL & VIS & \multicolumn{1}{c|}{ZS} \\ \hline
\multicolumn{1}{l|}{Silberer \& Lapata 2014} & & & \multicolumn{1}{c|}{} & & 0.7 & \multicolumn{1}{c|}{} & & 0.64 & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} \\
\multicolumn{1}{l|}{Lazaridou et al. 2015} & & & \multicolumn{1}{c|}{} & 0.75 &0.76 & \multicolumn{1}{c|}{} & 0.72 & 0.72 & \multicolumn{1}{c|}{} & 0.63 & 0.63 & \multicolumn{1}{c|}{} \\
\multicolumn{1}{l|}{Kiela \& Bottou 2014} & & 0.61 & \multicolumn{1}{c|}{} & & 0.72 & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} \\ \hline
\multicolumn{1}{l|}{GloVe} & \textbf{0.712} & 0.632 & \multicolumn{1}{c|}{\textbf{0.705}} & 0.805 & 0.801 & \multicolumn{1}{c|}{0.801} & 0.753 & 0.768 & \multicolumn{1}{c|}{0.701} & 0.591 & 0.606 & \multicolumn{1}{c|}{0.54} \\
\multicolumn{1}{l|}{\textit{CNN}$_{avg}$} & - & 0.448 & \multicolumn{1}{c|}{-} & - & 0.593 & \multicolumn{1}{c|}{-} & - & 0.534 & \multicolumn{1}{c|}{-} & - & 0.56 & \multicolumn{1}{c|}{-} \\
\multicolumn{1}{l|}{\textit{CONC}} & - & 0.606 & \multicolumn{1}{c|}{-} & - & 0.8 & \multicolumn{1}{c|}{-} & - & 0.734 & \multicolumn{1}{c|}{-} & - & 0.651 & \multicolumn{1}{c|}{-} \\ \hline
\multicolumn{1}{l|}{\textit{MAP}$_{NN}$} & \multicolumn{1}{l}{0.443} & \multicolumn{1}{l}{0.534} & \multicolumn{1}{l|}{0.391} & \multicolumn{1}{l}{0.703} & \multicolumn{1}{l}{0.761} & \multicolumn{1}{l|}{0.68} & \multicolumn{1}{l}{0.729} & \multicolumn{1}{l}{0.732} & \multicolumn{1}{l|}{0.718} & \multicolumn{1}{l}{\textbf{0.658}} & \multicolumn{1}{l}{\textbf{0.659}} & \multicolumn{1}{l|}{\textbf{0.655}} \\
\multicolumn{1}{l|}{\textit{MAP}$_{lin}$} & \multicolumn{1}{l}{0.402} & \multicolumn{1}{l}{0.539} & \multicolumn{1}{l|}{0.366} & \multicolumn{1}{l}{0.701} & \multicolumn{1}{l}{0.774} & \multicolumn{1}{l|}{0.674} & \multicolumn{1}{l}{0.738} & \multicolumn{1}{l}{0.738} & \multicolumn{1}{l|}{0.74} & \multicolumn{1}{l}{0.646} & \multicolumn{1}{l}{0.644} & \multicolumn{1}{l|}{0.651} \\
\multicolumn{1}{l|}{\textit{MAP-C}$_{NN}$} & 0.687 & 0.644 & \multicolumn{1}{c|}{0.673} & \textbf{0.813} & \textbf{0.82} & \multicolumn{1}{c|}{\textbf{0.806}} & 0.783 & \textbf{0.791} & \multicolumn{1}{c|}{0.754} & 0.65 & 0.657 & \multicolumn{1}{c|}{0.626} \\
\multicolumn{1}{l|}{\textit{MAP-C}$_{lin}$} & 0.694 & \textbf{0.649} & \multicolumn{1}{c|}{0.684} & 0.811 & 0.819 & \multicolumn{1}{c|}{0.802} & \textbf{0.785} & \textbf{0.791} & \multicolumn{1}{c|}{\textbf{0.764}} & 0.641 & 0.647 & \multicolumn{1}{c|}{0.623} \\ \hline
\multicolumn{1}{l|}{\# inst.} & 353 & 63 & \multicolumn{1}{c|}{290} & 3000 & 795 & \multicolumn{1}{c|}{2205} & 6933 & 5238 & \multicolumn{1}{c|}{1695} & 6933 & 5238 & \multicolumn{1}{c|}{1695} \\
\multicolumn{13}{l}{} \\
\multicolumn{1}{l|}{} & \multicolumn{3}{c|}{Simlex999} & \multicolumn{3}{c|}{Wordsim353-rel} & \multicolumn{3}{c|}{Wordsim353-sim} & \multicolumn{3}{c|}{SimVerb-3500} \\ \cline{2-13}
\multicolumn{1}{l|}{} & ALL & VIS & \multicolumn{1}{c|}{ZS} & ALL & VIS & \multicolumn{1}{c|}{ZS} & ALL & VIS & \multicolumn{1}{c|}{ZS} & ALL & VIS & \multicolumn{1}{c|}{ZS} \\ \hline
\multicolumn{1}{l|}{Silberer \& Lapata 2014} & & & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} \\
\multicolumn{1}{l|}{Lazaridou et al. 2015} & 0.4 & \textbf{0.53} & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} \\
\multicolumn{1}{l|}{Kiela \& Bottou 2014} & & & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} & & & \multicolumn{1}{c|}{} \\ \hline
\multicolumn{1}{l|}{GloVe} & 0.408 & 0.371 & \multicolumn{1}{c|}{\textbf{0.429}} & \textbf{0.644} & 0.759 & \multicolumn{1}{c|}{\textbf{0.619}} & \textbf{0.802} & 0.688 & \multicolumn{1}{c|}{\textbf{0.783}} & 0.283 & 0.32 & \multicolumn{1}{c|}{0.282} \\
\multicolumn{1}{l|}{\textit{CNN}$_{avg}$} & - & 0.406 & \multicolumn{1}{c|}{-} & - & 0.422 & \multicolumn{1}{c|}{-} & - & 0.526 & \multicolumn{1}{c|}{-} & - & 0.235 & \multicolumn{1}{c|}{-} \\
\multicolumn{1}{l|}{\textit{CONC}} & - & 0.442 & \multicolumn{1}{c|}{-} & - & 0.665 & \multicolumn{1}{c|}{-} & - & 0.664 & \multicolumn{1}{c|}{-} & - & 0.437 & \multicolumn{1}{c|}{-} \\ \hline
\multicolumn{1}{l|}{\textit{MAP}$_{NN}$} & \multicolumn{1}{l}{0.322} & \multicolumn{1}{l}{0.451} & \multicolumn{1}{l|}{0.296} & \multicolumn{1}{l}{0.33} & \multicolumn{1}{l}{0.606} & \multicolumn{1}{l|}{0.267} & \multicolumn{1}{l}{0.536} & \multicolumn{1}{l}{0.599} & \multicolumn{1}{l|}{0.475} & \multicolumn{1}{l}{0.213} & \multicolumn{1}{l}{\textbf{0.513}} & \multicolumn{1}{l|}{0.21} \\
\multicolumn{1}{l|}{\textit{MAP}$_{lin}$} & \multicolumn{1}{l}{0.322} & \multicolumn{1}{l}{0.412} & \multicolumn{1}{l|}{0.286} & \multicolumn{1}{l}{0.28} & \multicolumn{1}{l}{0.553} & \multicolumn{1}{l|}{0.243} & \multicolumn{1}{l}{0.505} & \multicolumn{1}{l}{0.569} & \multicolumn{1}{l|}{0.477} & \multicolumn{1}{l}{0.212} & \multicolumn{1}{l}{0.338} & \multicolumn{1}{l|}{0.21} \\
\multicolumn{1}{l|}{\textit{MAP-C}$_{NN}$} & 0.405 & 0.404 & \multicolumn{1}{c|}{0.417} & 0.623 & 0.778 & \multicolumn{1}{c|}{0.589} & 0.769 & 0.696 & \multicolumn{1}{c|}{0.745} & \textbf{0.286} & 0.49 & \multicolumn{1}{c|}{0.284} \\
\multicolumn{1}{l|}{\textit{MAP-C}$_{lin}$} & \textbf{0.41} & 0.388 & \multicolumn{1}{c|}{0.422} & 0.629 & \textbf{0.797} & \multicolumn{1}{c|}{0.601} & 0.781 & \textbf{0.698} & \multicolumn{1}{c|}{0.766} & \textbf{0.286} & 0.371 & \multicolumn{1}{c|}{\textbf{0.285}} \\ \hline
\multicolumn{1}{l|}{\# inst.} & 999 & 261 & \multicolumn{1}{c|}{738} & 252 & 28 & \multicolumn{1}{c|}{224} & 203 & 45 & \multicolumn{1}{c|}{158} & 3500 & 41 & \multicolumn{1}{c|}{3459}
\end{tabular}
\end{adjustbox}
\label{tab:results}
\end{table*}
Both \textit{MAP}$_{NN}$ and \textit{MAP}$_{lin}$ exhibit an overall gain in \textit{MEN} and in the VIS region of \textit{Wordsim353-rel}. It might seem counter-intuitive that vision can help to improve \textbf{\textit{relatedness}} understanding. However, a closer look reveals that visual features generally account for object co-occurrences, which is often a good indicator of their relatedness (e.g., between ``car" and ``garage" in Fig. \ref{car-garage}). For instance, in \textit{MEN}, the human relatedness rating between ``car" and ``garage" is 8.2 while GloVe's score is only 5.4. However, \textit{CNN}$_{avg}$'s rating is 8.7 and that of \textit{MAP}$_{lin}$ is 8.4---closer to the human score.
\begin{figure}
\caption{Sample images from ``car" (top row) and ``garage" (bottom row) synsets of ImageNet.}
\label{car-garage}
\end{figure}
Crucially, \textit{MAP-C}$_{NN}$ and \textit{MAP-C}$_{lin}$ significantly improve the performance of GloVe in all seven \textit{\textbf{VIS regions}} (p $\approx$ 0.008), with an average improvement of 4.6$\%$ for \textit{MAP-C}$_{NN}$. Conversely, the concatenation of GloVe with the original visual vectors (\textit{CONC}) does not improve GloVe (p $\approx$ 0.7)---worsening it in 4 out of 7 test sets---suggesting that simple concatenation without seeking the association between modalities might be suboptimal. Moreover, the concatenation of the mapped visual vectors with GloVe (\textit{MAP-C}$_{NN}$) outperforms the concatenation of the original visual vectors with GloVe (\textit{CONC}) in 6 out of 7 test sets (p $\approx$ 0.06), which supports our claim that the mapped visual vectors are semantically richer than the original visual vectors.
\section{Conclusions}
We have presented a cognitively-inspired method capable of generating multimodal representations in a fast and simple way. In a variety of similarity tasks and seven benchmark tests, our method generally outperforms unimodal baselines and state-of-the-art multimodal methods. Moreover, the performance gain in zero-shot settings indicates that the method generalizes well and learns relevant cross-modal associations. Finally, the overall performance supports the claim that our approach builds more ``human-like" concept representations. Ultimately, the present work sheds light on fundamental questions of natural language understanding such as whether the nature of the knowledge representation obtained by the fusion of vision and language should be static and additive (e.g., concatenation without associating modalities) or rather re-constructive and associative.
\end{document} |
\begin{document}
\begin{abstract}
Tilting mutation is a way of producing new tilting complexes from old ones replacing only one indecomposable summand. In this paper, we give a purely combinatorial procedure for performing tilting mutation of suitable algebras.
As an application, we recreate a result due to Ladkani, which states that the path algebra of a quiver shaped like a line (with certain relations) is derived equivalent to the path algebra of a quiver shaped like a rectangle. We will do this by producing an explicit series of tilting mutations going between the two algebras.
\end{abstract}
\title{A combinatorial procedure for tilting mutation}
\section{Introduction}
Tilting theory is central to the study of derived equivalences, and among the most important results in the field is Rickard's Morita theorem for derived categories.\cite{Rickard1989}
The theorem states that two $k$-algebras are derived equivalent if and only if there exists a certain tilting complex over one of them, which means that for a given $k$-algebra, every tilting complex gives a $k$-algebra which is derived equivalent to it. Thus if we can find a way to easily generate tilting complexes, we get a simple way to generate derived equivalent $k$-algebras. This is where tilting mutation comes in.
Tilting mutation was first developed in \cite{RS1991} as a way of obtaining new tilting complexes by modifying known tilting complexes. This idea was generalised in \cite{BMRRT2004} to define cluster tilting mutation, and in \cite{AI2012} to define silting mutation.
A purely combinatorial approach to perform silting mutation was developed in \cite{Oppermann2015}.
The goal of this paper is to develop a similar combinatorial approach for tilting mutation, which relies only on modifying the quiver with relations of the algebra we are mutating. Furthermore, we will through an example show how this approach can be used to easily generate a chain of derived equivalent algebras, which in turn could be used as a way to examine whether two given algebras are derived equivalent. This example will be inspired by the work of Ladkani in \cite{Ladkani2013}.
Note that in this paper we only consider right tilting mutation. An entirely dual result can be found by instead using left tilting mutation.
In that case, the combinatorial procedure will be exactly the same, except that all arrows will be flipped around.
\section{Notation}\label{section:setup}
Throughout this paper, $(Q,I)$ will be a quiver with admissible relations, $k$ will be a field, and $\Lambda=kQ/I$ is the corresponding path algebra. For ease of use, we will refer to $(Q,I)$ simply as $Q$. Let $P_i$ denote the indecomposable projective $\Lambda$-module of all paths starting in vertex $i$.
If there is a path $\beta$ in $Q$ from vertex $i$ to vertex $j$, then there is a morphism $P_j\xrightarrow{\beta} P_i$ given by precomposing with the path $i\xrightarrow{\beta} j$.
Note that we use the same notation for a path between two vertices and the corresponding morphism between projective modules. This is to keep the notation simple, and it will be clear from context when we are referring to one or the other. Composition of paths in $Q$ will be written from right to left, while composition of morphisms between $\Lambda$-modules will be written from left to right.
For an arrow $\alpha$, we denote by $s(\alpha)$ and $t(\alpha)$ the vertices where the arrow begins and ends, respectively (similarly for paths and relations).
If $\alpha\colon i\rightarrow j$ is an arrow, and $r\colon i \rightarrow k$ is a linear combination of paths in $Q$ (e.g. a relation), then $\sfrac{r}{\alpha}$ will denote the result of taking the linear combination of all paths in $r$ that begin with the arrow$ \alpha$, and then removing $\alpha$ from each of them. So for each arrow $\alpha$ starting in $i$, there is a linear combination of paths $\sfrac{r}{\alpha}\colon t(\alpha)\rightarrow j$, and together they satisfy $\sum\limits_{\alpha\colon i\rightarrow ?} \sfrac{r}{\alpha}\cdot\alpha = r$ (note that $\sfrac{r}{\alpha}=0$ if $r$ contains no path beginning with $\alpha$). Similarly, $\bsfrac{r}{\beta}\colon h\rightarrow s(\beta)$ refers to the linear combination of paths obtained by removing the arrow $\beta\colon s(\beta)\rightarrow t(r)$ from the end of $r$.
When we write something like $[\beta\alpha]_{\alpha\in A}$ this refers to a vector where each term consists of the expression in brackets, iterating over whatever is in the subscript (in this case it means the vector consisting of each arrow $\alpha\in A$, all followed by the same arrow $\beta$).
We usually only write $[\beta\alpha]_{\alpha}$, omitting the set we are iterating over, as it is clear from context which set that is.
When we use double subscript we mean the matrix consisting of the expression the first variable along the rows and the second variable down the columns. So for example $[\beta\alpha]_{\alpha,\beta}$ will be a matrix where the $i$-th column consists of the arrow $\alpha_i$ composed with each of the arrows $\beta_j$.
For the vectors it will be clear from context whether we view them as row or column vectors.
\section{Statement of main theorem}\label{section:mutationsteps}
In this section we state our main result, and give an example to illustrate how it can be used in practice.
The main result of this paper is a purely combinatorial procedure for quiver mutation, which gives us an easy way to perform tilting mutation of a path algebra. Given a quiver $Q$ with relations and a specified vertex $i$ in that quiver, we will construct a new quiver with relations whose path algebra is isomorphic to the algebra obtained by performing tilting mutation of the path algebra of $Q$ at vertex $i$.
Tilting mutation of $\Lambda$ at $i$ is performed by replacing the indecomposable projective direct summand $P_i$ with another $\Lambda$-module, which we will denote by $P_i^*$. Specifically, $P_i^*$ the cocone of a right approximation of $P_i$ by the other $P_j$ (see \cref{obs:mutationTriangle} for more details).
Inspired by this, the quiver mutation we define here is performed by taking the quiver $Q$ and replacing one of its vertices $i$ with a new vertex $i^*$, and changing the arrows and relations accordingly.
Unfortunately, tilting mutation is not always possible, in the sense that mutating an arbitrary tilting complex with respect to an arbitrary indecomposable projective module does not always yield a new tilting complex.
However, under the following assumptions on the path algebra, mutation of $\Lambda$ with respect to $P_i$ is always possible.
\begin{itemize}
\item Any non-iso endomorphism of $P_i$ factors through another indecomposable projective module. This is equivalent to the quiver having no cycles of length one on vertex $i$ (i.e. no arrows $i\rightarrow i$).
\item We have that $\text{Hom}_{\Lambda}(P_i^*[1], \Lambda) = 0$. This is a sufficient condition for the tilting mutation to work (that is, the result of the mutation will still be a tilting complex).
\end{itemize}
As we will see later, in practice we can use the second point as an easy check to identify some cases where mutation is not possible. For each nonzero path ending in vertex $i$, there must be at least one arrow out of $i$ such that the composition of the path with that arrow is nonzero. If not, then mutation is not possible at vertex $i$.
In particular, if there is only one arrow $\alpha$ starting in $i$, then mutation is not possible in $i$ if there is a minimal zero relation whose last arrow is $\alpha$.
\begin{theorem}\label{theorem:mainResult}
Let $\Lambda=kQ/I$ be the path algebra of a quiver $Q$ with relations $I$, and let $i$ be a vertex of $Q$ such that there are no arrows $i\rightarrow i$. Let $\mu_i^R(\Lambda)\simeq\Lambda/P_i\oplus P_i^*$ denote the right tilting mutation of $\Lambda$ at the indecomposable projective $\Lambda$-module $P_i$. We assume that $\text{Hom}_\Lambda(P_i^*[1], \Lambda)=0$ for all $i\neq 0$.
Then, mutating the quiver $Q$ at vertex $i$ according to the mutation procedure stated below yields a quiver, $m_i(Q)$, whose path algebra is isomorphic to $\text{End}_\Lambda(\mu_i^R(\Lambda))^{\text{op}}$.
\end{theorem}
The following list of steps shows exactly which arrows and relations are to be added and removed during the quiver mutation. Note that, rather than simply removing vertex $i$ and adding vertex $i^*$, it can be helpful to visualise the procedure as $i^*$ actually replacing $i$. That way we can talk about ``flipping an arrow'', and ``changing a relation into an arrow'', rather than having to say we remove an arrow/relation to or from vertex $i$ and add an arrow/relation to or from $i^*$.
\emph{The procedure}
\begin{itemize}
\item \textbf{Step 1: Add arrows for compositions through $\boldsymbol{i}$}\\
When we remove vertex $i$ we lose all arrows $\beta\colon h\rightarrow i$ and $\alpha\colon i\rightarrow j$, but the compositions $\alpha\beta\colon h\rightarrow j$ are still paths (which are now minimal). Thus we add them as arrows.\\
\item \textbf{Step 2: Flip arrows out of $\boldsymbol{i}$}\\
Any arrow $\alpha\colon i\rightarrow j$ is replaced by an arrow $\alpha^*\colon j\rightarrow i^*$.\\
\item \textbf{Step 3: Relations out of $\boldsymbol{i}$ become arrows}\\
A minimal relation $r\colon i\dashrightarrow k$ is replaced by an arrow $\overline{r}\colon i^*\rightarrow k$. If the quiver contains a cycle on $i$, and there is a minimal relation $r\colon i\dashrightarrow i$, we get one arrow $\alpha\overline{r}\colon i^*\rightarrow t(\alpha)$ for each arrow $\alpha\colon i \rightarrow t(\alpha)$.\\
\item \textbf{Step 4: Arrows into $\mathbf{i}$ become relations}\\
For each arrow $\beta\colon h\rightarrow i$ we get a new relation $h\dashrightarrow i^*$ given by ${\sum\limits_{\alpha\colon i\rightarrow ?} \alpha^*\alpha\beta=0}$, where $\alpha$ runs over all arrows in $Q$ starting in $i$. \\
\item \textbf{Step 5: Add relations for new compositions}\\
We add a relation for every composition through $i^*$.
Any compositon of arrows through vertex $i^*$ consists of arrows $\alpha^*$ and $\overline{r}$ as defined in step 2 and 3. Each such composition gives rise to a relation given by $\overline{r}\alpha^*=\sfrac{r}{\alpha}$, where $\sfrac{r}{\alpha}\colon t(\alpha)\rightarrow k$ is the (linear combination of) path(s) in the relation $r\colon i\dashrightarrow k$ whose first arrow is $\alpha$, with $\alpha$ removed ($\sfrac{r}{\alpha}=0$ if no such path exists in $r$).\\
\item \textbf{Step 6: Extend relations into $\boldsymbol{i}$}\\
Any relation in $Q$ which ends in $i$ will give relations ending in $t(\alpha)$ for each arrow $\alpha$ starting in $i$. A relation ending in vertex $i$ can be written as $r=\sum\limits_{\beta: ?\rightarrow i} \beta\bsfrac{r}{\beta}$, where $\beta$ runs over all arrows in $Q$ ending in $i$. Thus postcomposing with an arrow $\alpha\colon i \rightarrow j$ gives a relation $\sum\limits_{\beta: ?\rightarrow i} \alpha\beta\bsfrac{r}{\beta}$ ending in $j$.\\
\item \textbf{Step 7: Add relations out of $\boldsymbol{i^*}$}\\
A linear combination of paths from $i^*$ to some vertex $l$ defines a relation in $m_i(Q)$ if and only if precomposing it with each arrow ${\alpha^*\colon t(\alpha)\rightarrow i^*}$ gives a relation in the quiver $Q$. Specifically, if $R$ is a subset of the set of relations in $Q$ which begin in $i$, and $\{\varepsilon_r\}_{r\in R}$ is a collection of (linear combinations of) paths $t(r)\rightarrow l$, then $\sum\limits_{r\in R} \varepsilon_r \overline{r}=0$ is a relation $i^*\dashrightarrow l$ in $m_i(Q)$ if and only if $\sum\limits_{r\in R} \varepsilon_r \sfrac{r}{\alpha}=0$ is a relation in $Q$ for each $\alpha$ starting in $i$.
\end{itemize}
\textbf{Note:} It can happen that we after mutation obtain a relation which is not admissible, i.e. containing a path of length one. Such a relation can be interpreted as setting the corresponding arrow in the quiver equal to some other linear combination of paths (or zero), and thus, removing both the arrow and the relation from the quiver will give an equivalent path algebra.
In other words, whenever we get a relation which contains a path of length one, we can ``cancel'' the relation against the corresponding arrow, and remove both from the quiver.
\newgeometry{margin=1.1in}
The following table is meant to help visualise how each step of the mutation procedure works in practice.
It shows the relevant parts of a quiver before and after each step has been applied.
Be aware that these by no means cover all possible cases for how each step behaves.
The arrows/relations that are directly affected by a given step are marked by green before mutation (only where applicable), and red after mutation.
\begin{tabular}{|c|c|c|l|}
\hline
Step & Before mutation & After mutation & Relations \\ \hline
1 &
\begin{tikzcd}
h \arrow[dr, "\beta"'] & & j\\
& i \arrow[ur, "\alpha"'] &
\end{tikzcd}
&
\begin{tikzcd}
h \arrow[r, color=ForestGreen, "{\color{black}\alpha\beta}"] & j
\end{tikzcd}
& \\ \hline
2&
\begin{tikzcd}
i \arrow[r, color=red, "{\color{black}\alpha}"] & j
\end{tikzcd}
&
\begin{tikzcd}
i^* & \arrow[l, color=ForestGreen, "{\color{black}\alpha^*}"'] j
\end{tikzcd}
& \\ \hline
3&
\begin{tikzcd}
i \arrow[dr, "\alpha"'] \arrow[rr, dashed, color=red, "{\color{black}r}"] & & k \\
& j \arrow[ur, "\gamma"'] &
\end{tikzcd}
&
\begin{tikzcd}
i^* \arrow[rr, color=ForestGreen, "{\color{black}\overline{r}}"] & & k \\
& j \arrow[ul, "{\alpha^*}"] \arrow[ur, "\gamma"'] &
\end{tikzcd}
& \breakcell{$r=\gamma\alpha=0$ \\} \\ \hline
4&
\begin{tikzcd}
h \arrow[dr, color=red, "{\color{black}\beta}"'] & & j \\
& i \arrow[ur, "\alpha"'] &
\end{tikzcd}
&
\begin{tikzcd}
h \arrow[rr, "{\alpha\beta}"] \arrow[dr, dashed, color=ForestGreen] & & j \arrow[dl, "{\alpha^*}"] \\
& i^* &
\end{tikzcd}
& \breakcell{$\alpha^*\alpha\beta=0$ \\} \\ \hline
5&
\begin{tikzcd}
& j_1 & \\
i \arrow[ur, "\alpha_1"] \arrow[r, "\alpha_2"] \arrow[rr, bend right, dashed, "r"'] & j_2 \arrow[r, "\gamma"] & k
\end{tikzcd}
&
\begin{tikzcd}
j_1 \arrow[dr, "\alpha_1^*"'] \arrow[drr, dashed, color=ForestGreen] & & \\
& i^* \arrow[r, "\overline{r}" near start] & k\\
j_2 \arrow[ur, "\alpha_2^*"] \arrow[urr, bend right, "\gamma"'] \arrow[urr, dashed, color=red] & &
\end{tikzcd}
& \breakcell{$r=\gamma\alpha_2=0$ \\ \\ $\overline{r}\alpha_1^*=0$ \\ $\overline{r}\alpha_2^*=\gamma$ \\}\\ \hline
6&
\begin{tikzcd}
& h \arrow[r, "\beta"] & i \arrow[dr, "\alpha"] &\\
g \arrow[ur, "\delta"] \arrow[urr, dashed] & & & j
\end{tikzcd}
&
\begin{tikzcd}
& h \arrow[r, dashed] \arrow[drr, "\alpha\beta"'] & i^* &\\
g \arrow[ur, "\delta"] \arrow[rrr, dashed, color=ForestGreen] & & & j \arrow[ul, "\alpha^*"']
\end{tikzcd}
& \breakcell{$\alpha^*\alpha\beta=0$ \\ $\alpha\beta\delta=0$ \\} \\ \hline
7&
\begin{tikzcd}
& j \arrow[r, "\gamma"] \arrow[drr, dashed] & k \arrow[dr, "\varepsilon"]&\\
i \arrow[ur, "\alpha"] \arrow[urr, dashed, "r"'] & & & l
\end{tikzcd}
&
\begin{tikzcd}
& j \arrow[dl, "\alpha^*"'] & k \arrow[dr, "\varepsilon"] &\\
i^* \arrow[urr, "\overline{r}"'] \arrow[rrr, dashed, color=ForestGreen] & & & l
\end{tikzcd}
& \breakcell{$\varepsilon\overline{r}=0$}\\ \hline
\end{tabular}
\restoregeometry
Here are a couple of quick examples to show the steps in action, a more comprehensive example is given in \cref{section:example}.
\begin{example}
Let $Q$ be the following quiver and let $\Lambda=kQ/I$, where $I$ is the set of relations as indicated by the dashed arrows. \begin{equation*}
\begin{tikzcd}
& & & 4 \arrow[dr, "\delta"] \arrow[drr, dashed, bend left] & & \\
Q = 1 \arrow[r, "\alpha"] \arrow[rr, bend left, dashed] & 2 \arrow[r, "\beta"] & 3 \arrow[ur, "\gamma"] \arrow[dr, "\varepsilon"] \arrow[rr, dashed] & & 6 \arrow[r, "\eta"] & 7 \\
& & & 5 \arrow[ur, "\zeta"] \arrow[urr, dashed, bend right] & &
\end{tikzcd}
\end{equation*}
Explicitly, the set of relations is given by
$$\beta\alpha=0,\quad \delta\gamma+\zeta\varepsilon = 0,\quad \eta\delta = 0,\quad \zeta\eta = 0.$$
Suppose we want to use the mutation procedure to determine the resulting quiver from performing right tilting mutation at vertex $3$ in $Q$. Then we simply replace vertex $3$ by $3^*$, and then apply the steps in order, starting with step $1$.
\begin{enumerate}
\item We add arrows $\gamma\beta\colon 2\rightarrow 4$ and $\varepsilon\beta\colon 2\rightarrow 5$, corresponding to the compositions through $3$.
\item We flip the arrows $\gamma\colon 3\rightarrow 4$ and $\varepsilon\colon 3\rightarrow 5$, to get $\gamma^*$ and $\varepsilon^*$.
\item We change the relation $\delta\gamma+\zeta\varepsilon\colon3\dashrightarrow 6$ into an arrow $\overline{\delta\gamma+\zeta\varepsilon}$.
\item We change the arrow $\beta\colon 2\rightarrow 3$ into a relation, defined by $\gamma^*\gamma\beta+\varepsilon^*\varepsilon\beta=0$.
\item We add relations $4\dashrightarrow 6$ and $5\dashrightarrow 6$, corresponding to to the new compositions through $3^*$. Explicitly, they are defined as $\delta+\Big[\overline{\delta\gamma+\zeta\varepsilon}\Big]\gamma^* = 0$ and $\zeta + \Big[\overline{\delta\gamma+\zeta\varepsilon}\Big]\varepsilon^*=0$.
\item We remove the relation $\beta\alpha\colon1\dashrightarrow 3$, and in its place add relations $\gamma\beta\alpha\colon 1\dashrightarrow 4$ and $\epsilon\beta\alpha\colon 1\dashrightarrow 5$.
\item Since there are relations $4\dashrightarrow 7$ and $5\dashrightarrow 7$ we add a relation $3^*\dashrightarrow 7$, given by $\eta\Big[\overline{\delta\gamma+\zeta\varepsilon}\Big]=0$
\end{enumerate}
Thus we obtain the following quiver
\begin{equation*}
\begin{tikzcd}
& & & 4 \arrow[dr, "\delta"] \arrow[dl, "{\gamma^*}"] \arrow[dr, bend right, dashed] \arrow[drr, dashed, bend left] & & \\
1 \arrow[r, "\alpha"] \arrow[urrr, bend left, dashed] \arrow[drrr, bend right, dashed] & 2 \arrow[r, dashed] \arrow[urr, "{\gamma\beta}"] \arrow[drr, "{\varepsilon\beta}"'] & 3^* \arrow[rr, "{\overline{\delta\gamma+\zeta\varepsilon}}"] \arrow[rrr, bend right, looseness=0.5, dashed] & & 6 \arrow[r, "\eta"] & 7 \\
& & & 5 \arrow[ur, "{\zeta}"'] \arrow[ul, near start, "{\varepsilon^*}"'] \arrow[ur, bend left, dashed] \arrow[urr, dashed, bend right] & & .
\end{tikzcd}
\end{equation*}
The relation $4\dashrightarrow 6$ gives that the arrow $\delta$ is equal (up to sign) to the path $\overline{\delta\gamma+\zeta\varepsilon}$, hence we can remove both the arrow and the relation. The same is true for the arrow $\zeta$ and the relation $5\dashrightarrow 6$. We also remove the relations $4\dashrightarrow 7$ and $5\dashrightarrow 7$, since they both now factor through the relation $3^*\dashrightarrow 7$, and hence are superfluous. Cleaning up a bit, this leaves us with the following quiver as the result of the mutation of $Q$ at vertex $3$.
\begin{equation*}
\begin{tikzcd}
& & 4 \arrow[dr, "{\gamma^*}"] & & & \\
1 \arrow[r, "\alpha"] \arrow[urr, bend left, dashed] \arrow[drr, bend right, dashed] & 2 \arrow[ur, "{\gamma\beta}"] \arrow[dr, "{\varepsilon\beta}"] \arrow[rr, dashed] & & 3^* \arrow[r, "{\overline{\delta\gamma+\zeta\varepsilon}}"'] \arrow[rr, bend left, dashed] & 6 \arrow[r, "\eta"'] & 7 \\
& & 5 \arrow[ur, "{\varepsilon^*}"] & & &
\end{tikzcd}
\end{equation*}
\end{example}
The following example is meant to show that the mutation procedure can be applied to quivers which contain cycles, although the process becomes slightly more complicated.
\begin{example}
Consider the following quiver, with the relation $\beta\alpha = 0$, and suppose we want to mutate it at vertex $1$.
\begin{equation*}
\begin{tikzcd}
1 \arrow[rr, shift left, "\alpha"] \arrow[out=30,in=330,loop,looseness=32, dashed, near start, "{\beta\alpha}"] & & 2 \arrow[ll, shift left, "\beta"] &
\end{tikzcd}
\end{equation*}
We replace vertex $1$ by $1^*$, and then we apply the steps of the mutation procedure in order. Especially take note of step 3, since there is a cycle on vertex $1$.
\begin{enumerate}
\item We add an arrow $\alpha\beta\colon 2\rightarrow 2$, corresponding to the compostition through vertex $1$.
\item We flip the arrow $\alpha$, resulting in the arrow $\alpha^*\colon 2\rightarrow 1^*$.
\item Because the relation $\beta\alpha$ both begins and ends in $1$, we add an arrow $\alpha\overline{\beta\alpha}\colon 1^*\dashrightarrow 2$, corresponding to the composition of the relation $\beta\alpha$ with the arrow $\alpha$.
\item We change the arrow $2\rightarrow 1$ into a relation $2\dashrightarrow 1^*$, given by $\alpha^*\alpha\beta=0$.
\item We add a relation $2\dashrightarrow 2$, corresponding to the new composition through $1^*$. This composition is given by $\alpha\overline{\beta\alpha}\alpha^*=\alpha\beta$
\item We remove the relation $\beta\alpha\colon1\dashrightarrow 1$, and in its place add a relation $1^*\dashrightarrow 2$ given by $\alpha\beta\alpha\overline{\beta\alpha}=0$.
\item Since there is only one minimal relation in the unmutated quiver, step $7$ doesn't come into play.
\end{enumerate}
Thus we obtain the following quiver
\begin{equation*}
\begin{tikzcd}
1^* \arrow[rr, shift left, "\alpha\overline{\beta\alpha}"] \arrow[rr, out=40,in=320,loop,looseness=5, dashed, very near start, "{\alpha\beta\alpha\overline{\beta\alpha}}"] & & 2 \arrow[ll, shift left, "\alpha^*"] \arrow[out=40,in=320,loop, "\alpha\beta"] \arrow[ll, out=40,in=320,loop,looseness=5, dashed, very near end, "{\alpha^*\alpha\beta}"] \arrow[out=45,in=135,loop,looseness=5, dashed, "\circlearrowleft"] &
\end{tikzcd}
\end{equation*}
The relation $\alpha\beta = \alpha\overline{\beta\alpha}\alpha^*$ shows that the arrow $\alpha\beta$ factors through the composition $\alpha\overline{\beta\alpha}\alpha^*$, which means that $\alpha\beta$ is a superfluous arrow in the quiver. This also shows that the relation $\alpha^*\alpha\beta = 0$ can be rewritten as $\alpha^*\alpha\beta = \alpha^* (\alpha\overline{\beta\alpha}) = 0$, and the relation $\alpha\beta\alpha\overline{\beta\alpha}$ can be rewritten as $\alpha\overline{\beta\alpha}\alpha^*\alpha\overline{\beta\alpha} = 0$, which is redundant because of the previous relation. Thus we end up with the following quiver, where the relation is given by the composition $1^*\rightarrow 2\rightarrow 1^*$ being zero (this quiver with relation is actually isomorphic to the one we started with).
\begin{equation*}
\begin{tikzcd}
1^* \arrow[rr, shift left, "\alpha\overline{\beta\alpha}"] \arrow[out=30,in=330,loop,looseness=25, dashed] & & 2 \arrow[ll, shift left, "\alpha^*"]
\end{tikzcd}
\end{equation*}
\end{example}
This was meant as two simple examples to see how the mutation procedure can be used in practice.
\section{Tilting mutation}
In this section we will define the right tilting mutation of an algebra, and present some results which we will need in order to prove the main theorem.
\begin{lemma}\label{lemma:minrightapprox}
Let $P_i$ be an indecomposable projective direct summand of $\Lambda=\bigoplus\limits_{j\in Q_0} P_j$, such that $i$ has no cycle of length $1$. Then $\bigoplus\limits_{\alpha:i \rightarrow ?} P_{t(\alpha)} \rightarrow P_i$ is a minimal right $\Lambda/P_i$-approximation of $P_i$ (the sum runs over all arrows starting in vertex $i$).
\end{lemma}
\begin{observation}\label{obs:mutationTriangle}
Since the derived category of $\Lambda$ is triangulated, we can complete the minimal right approximation $\left[ \bigoplus\limits_{\alpha:i\rightarrow?} P_{t(\alpha)} \xrightarrow{[\alpha]_{\alpha}} P_i \right]$ to a distinguished triangle. We denote by $P_i^*$ the cocone of the morphism $[\alpha]_{\alpha}$, and thus we get a triangle $P_i^*\rightarrow \bigoplus\limits_{\alpha:i\rightarrow?} P_{t(\alpha)} \rightarrow P_i $.
\end{observation}
\begin{definition}
Let $P_i$ be the indecomposable projective direct summand of $\Lambda=kQ/I$ corresponding to vertex $i$. The \emph{right tilting mutation of $\Lambda$ with respect to $P_i$} is defined as $\mu_i^R (\Lambda) :=\Lambda/P_i \oplus P_i^*$.
\end{definition}
One unfortunate property of tilting mutation is that it isn't always possible. When we take $\Lambda$, which is a tilting object, and replace the direct summand $P_i$ by $P_i^*$, it's not guaranteed that the result is again a tilting object.
However, under certain assumptions tilting mutation is always possible, as the following theorem shows.
\begin{theorem}\label{thm:mutationAssumptions}
Let $P_i$ be a projective indecomposable direct summand of $\Lambda=kQ/I$. If $\text{Hom}_\Lambda(P_i^*[1], \Lambda) = 0$, then the tilting mutation $\mu_i^R (\Lambda) = \Lambda/P_i \oplus P_i^*$ is a tilting complex.
\end{theorem}
\begin{proof}
To show that $\Lambda/P_i \oplus P_i^*$ is a tilting complex there are two things we need to check, namely that $\Lambda/P_i \oplus P_i^*$ has no shifted endomorphisms, and that it generates per$(\Lambda)$ as a triangulated category. The module $\Lambda/P_i \oplus P_i^*$ can be viewed as a complex as follows
$$\dots\rightarrow 0 \rightarrow \Lambda/P_i \oplus \bigoplus\limits_{\alpha:i\rightarrow ?} P_{t(\alpha)} \xrightarrow{[0 \ \alpha]_\alpha} P_i \rightarrow 0 \rightarrow \dots$$
Since it only has two nonzero components, any morphism from the complex shifted by two or more in either direction to itself must be zero. The fact that $[\alpha]$ is a right approximation of $P_i$ ensures that zero is the only morphism from the complex shifted by $-1$ to itself. In general, there could be nonzero morphisms from the complex shifted by $1$ to itself, which is why we need the assumption that $\text{Hom}(P_i^*[1], \Lambda)=0$.
So given that this assumption is true, $\Lambda/P_i\oplus P_i^*$ will have no shifted endomorphisms.
To see that it also generates per$(\Lambda)$ as a triangulated category, notice that $\bigoplus_{\alpha}P_{t(\alpha)}$ is a direct summand in $\text{add}(\Lambda/P_i)$, and that it, as well as $P_i$ and $P_i^*$ appear in the triangle
\begin{equation*}
P_i^*\rightarrow \bigoplus\limits_{\alpha: i\rightarrow ?} P_{t(\alpha)} \rightarrow P_i \rightarrow P_i^*[1].
\end{equation*}
This means that $\Lambda/P_i\oplus P_i^*$ and $\Lambda/P_i \oplus P_i\simeq \Lambda$ generate the same triangulated subcategory of the derived category of $\Lambda$. Since $\Lambda$ is a tilting complex, that subcategory is per$(\Lambda)$, which is what we need.
\end{proof}
In the theorem above, the condition is on the algebra $\Lambda$, rather than on the quiver $Q$.
Still, in some cases it will be enough to look at the quiver in order to conclude that mutation is impossible at a certain vertex.
Clearly, there exists a nonzero morphism ending in $P_i$ if and only if there is at least one arrow starting in vertex $i$ in the quiver. In other words, there is an equivalence
\begin{equation*}
\text{Hom}_\Lambda(\Lambda/P_i, P_i) \neq 0 \iff \exists \alpha \in Q_1 \colon s(\alpha) = i.
\end{equation*}
Notice now that if there exists a nonzero morphism
In other words, condition $1$ is satisfied if and only if there exists at least one arrow in $Q$ starting in vertex $i$. So when we consider a quiver to determine which vertices allow for mutation, we can immediately discard any vertex which has no arrows going out of it.
Condition $2$, on the other hand, is in general not equivalent to a condition on the quiver. It can, however, be used to give an easy criterion on the quiver for when mutation is not possible, as we can see by the following corollary.
\begin{theorem}
Let $(Q,I)$ be a quiver with relations, and let $i$ be a vertex in $Q$. Two cases where tilting mutation of $\Lambda=kQ/I$ with respect to $P_i$ is not possible are:
\begin{itemize}
\item If there are no arrows in $Q$ going out of $i$.
\item If there is at least one arrow out of $i$ in $Q$, and there exists a nonzero path in $Q$ ending in $i$ such that composing that path with each arrow out of $i$ gives zero.
\end{itemize}
\end{theorem}
\begin{proof}
If there are no arrows $\alpha$ going out of $i$ then there will be no modules $P_{t(\alpha)}$, and the direct sum $\bigoplus\limits_{\alpha:i \rightarrow ?} P_{t(\alpha)}$ will be equal to zero. This means that there is no nonzero right approximation $\bigoplus\limits_{\alpha:i \rightarrow ?} P_{t(\alpha)}\rightarrow P_i$, so we can't construct $P_i^*$. Thus, mutation is not possible.
Assume now that there is at least one arrow out of $i$, and that there exists a nonzero path $\beta\colon j\rightarrow i$ for some vertex $j$, such that $\alpha\beta = 0$ for each arrow $\alpha\colon i\rightarrow t(\alpha)$. If we view the module $\Lambda/P_i \oplus P_i^*$ as a complex in the same way as before, then this gives us the following diagram.
\begin{equation*}
\begin{tikzcd}
\cdots \arrow[r] & 0 \arrow[d] \arrow[r] & \bigoplus\limits_{\alpha:i\rightarrow ?} P_{t(\alpha)} \arrow[rr, "{[\alpha]_{\alpha}}"] \arrow[d] & & P_i \arrow[d, "{\beta}"] \arrow[r] & 0 \arrow[r] \arrow[d] & \cdots \\
\cdots \arrow[r] & 0 \arrow[r] & 0 \arrow[rr] & & P_j \arrow[r] & 0 \arrow[r] & \cdots
\end{tikzcd}
\end{equation*}
Notice now that since $\alpha\beta = 0$ for each $\alpha$, the middle square will commute. Thus $\beta$ defines a morphism of complexes between $[\bigoplus_\alpha P_{t(\alpha)}\rightarrow P_i]\simeq P_i^*[1]$ and $P_j$. And composing this with the inclusion of $P_j$ into $\Lambda/P_i$ gives a nonzero morphism $P_i^*[1] \rightarrow \Lambda$, hence $\text{Hom}(P_i^*[1], \Lambda)\neq 0$.
\end{proof}
In particular, if $\alpha$ is the only arrow out of $i$, then mutation is impossible if there is a minimal zero relation whose last arrow is $\alpha$.
Let's make the following observation about relations in a quiver. Any relation from vertex $i$ to some vertex $k$ is given as some linear combination of paths from $i$ to $k$ being equal to zero. One consequence of this is that, since each of those paths begin with an arrow $i \xrightarrow{\alpha} t(\alpha)$ for some vertex $t(\alpha)$, we can view a relation starting in $i$ as a collection of paths starting in such vertices $t(\alpha)$, satisfying that the path starting in $i$ obtained by precomposing with the arrows $\alpha$ and taking the sum is zero.
Roughly speaking, this means that a relation starting in $i$ corresponds to a combination of paths starting in the vertices that are hit by arrows from $i$, that become zero if you compose them with the arrows from $i$ to get a collection of paths starting in $i$. We can use this to give a projective resolution of the simple module corresponding to vertex $i$, which will be useful later.
\begin{lemma}\label{lemma:projres}
The sequence
\begin{equation*}
\begin{tikzcd}
\bigoplus\limits_{r:i\dashrightarrow ?} P_{t(r)} \arrow[r, "{[ \sfrac{r}{\alpha} ]_{r,\alpha}}"] & \bigoplus\limits_{\alpha: i\rightarrow ?} P_{t(\alpha)} \arrow[r, "{[\alpha]_\alpha}"] & P_i \arrow[r] & S_i \arrow[r] & 0
\end{tikzcd}
\end{equation*}
defines a projective resolution for $S_i$, the simple $\Lambda$-module at $i$.
\end{lemma}
\begin{proof}
We start by showing that it is in fact a complex. Each component of the map $[ \sfrac{r}{\alpha}]_{r,\alpha}$ is given by the paths $t(\alpha)\rightarrow t(r)$ which are obtained by removing the arrow $\alpha\colon i \rightarrow t(\alpha)$ from the paths $i\rightarrow t(r)$ corresponding to the relations $r\colon i\dashrightarrow t(r)$ passing through $t(\alpha)$.
For each relation $r$, composing $[\sfrac{r}{\alpha}]_\alpha\colon P_{t(r)}\rightarrow \bigoplus_\alpha P_{t(\alpha)}$ with $[\alpha]_\alpha\colon \bigoplus_\alpha P_{t(\alpha)}\rightarrow P_i$ will yield
$$[\sfrac{r}{\alpha}]_\alpha\cdot[\alpha]_\alpha=\sum_{\alpha:i\rightarrow ?} \sfrac{r}{\alpha}\circ\alpha=r=0.$$
We see that the composition is equal to the relation $r$ (we simply recreate each path in the linear combination by adding the first arrows), which is equal to zero by definition. This is true for all relations $r\colon i\dashrightarrow ?$, hence the composition of $[\sfrac{r}{\alpha}]_{r,\alpha}$ and $[\alpha]_\alpha$ is zero, and we indeed have a complex.
Now, we'll show that the complex is exact. Recall that $P_i$ consists of all paths that start in vertex $i$, and $S_i$ contains only the trivial path in vertex $i$. Since $P_i$ is a projective cover of $S_i$, it is clearly exact in the right-hand term. The kernel of the map $P_i\rightarrow S_i$ will correspond to all paths that start in vertex $i$, except the trivial path. In other words, the kernel consists of all paths starting in vertex $i$ of length $\geq 1$, which is precisely the same as the image of the map $[\alpha]_\alpha$. So the complex is exact in the middle term. Finally, the kernel of $[\alpha]_\alpha$ consists of all paths starting in the $j$-vertices such that precomposing with the arrows $\alpha$ makes (the corresponding linear combination of) them zero. But this is just another way to describe the relations starting in vertex $i$, which is the image of the map $[\sfrac{r}{\alpha}]_{r,\alpha}$. Thus, the complex is exact, which means that the projective terms form a resolution.
\end{proof}
With these definitions and results, we are ready to start working towards proving the main result of the paper.
\section{Proof of main result}
In this section we will prove that applying the steps given in \cref{section:mutationsteps} to a quiver corresponds to right tilting mutation of its path algebra.
Assume we are given a quiver with relations $(Q,I)$ and a vertex $i$ in $Q$, such that $\Lambda= kQ/I$ satisfies the conditions in \cref{thm:mutationAssumptions}.
We will show that we can construct a new quiver, $Q_i^{i^*}$, which is isomorphic to the quiver we get when we apply the mutation steps to $(Q,I)$, and whose path algebra is isomorphic to $\text{End}_{\Lambda}(\Lambda/P_i\oplus P_i^*)$.
In order to define the quiver $Q_i^{i^*}$, we must first construct two other quivers, $Q_i$ and $Q^{i^*}$. Let $Q$ be a quiver, let $i$ be a vertex in $Q$ and let $i^*$ not be a vertex in $Q$. Then:
\begin{itemize}
\item $Q_i$ is equal to $Q$ except that vertex $i$ has been removed, and the arrows and relations have been modified so that the path algebra of $Q_i$ is isomorphic to $\text{End}_\Lambda(\Lambda/P_i)$.
\item $Q^{i^*}$ is equal to $Q$ except that a new vertex $i^*$ has been added, along with certain arrows and relations, so that the path algebra of $Q^{i^*}$ is isomorphic to $\text{End}_\Lambda(\Lambda\oplus P_i^*)$.
\end{itemize}
The exact constructions of these two quivers are given in \cref{lemma:QuiverRemoveVertex} and \cref{lemma:Qi*}, respectively.
In terms of these two constructions, the quiver we are after can be given as $Q_i^{i^*}:= (Q^{i^*})_i$, meaning that we first add the vertex $i^*$ to $Q$, and then we remove vertex the $i$ from the resulting quiver. Then, the path algebra of $Q_i^{i^*}$ will be isomorphic to $\text{End}_\Lambda(\Lambda/P_i\oplus P_i^*)$, which is what we want.
What we will do now, is examine the structure of $\text{End}_{\Lambda}(\Lambda/P_i\oplus P_i^*)$ by using the known structure of
$\text{End}_{\Lambda}(\Lambda)\simeq \Lambda$, and we will do this in two steps.
First we show how $\text{End}_\Lambda(\Lambda)$ changes when we add the direct summand $P_i^*$, meaning we look at $\text{End}_{\Lambda}(\Lambda\oplus P_i^*)$. Then we show how $\text{End}_\Lambda(\Lambda\oplus P_i^*)$ changes when we remove the summand $P_i$.
\begin{equation*}
\begin{tikzcd}
\text{End}_{\Lambda}(\Lambda) \arrow[rr, dotted] \arrow[dr, "{\text{Add } P_i^*}"', sloped] & & \text{End}_{\Lambda}(\Lambda/P_i\oplus P_i^*) \\
& \text{End}_{\Lambda}(\Lambda \oplus P_i^*) \arrow[ur, "{\text{Remove } P_i}"', sloped]&
\end{tikzcd}
\end{equation*}
The first step corresponds to adding a new vertex $i^*$ to the quiver $Q$, and changing/adding arrows and relations in accordance with $\text{End}_{\Lambda}(\Lambda\oplus P_i^*)$. We denote this quiver by $Q^{i^*}$. The next step corresponds to removing the vertex $i$, along with any arrow and relation starting or ending in it (but keeping compositions through $i$). We will denote this quiver by $Q_i^{i^*}$.
In practice what we will do is use the mutation triangle defined in \cref{obs:mutationTriangle}, together with the known structure of $\text{End}_\Lambda(\Lambda)$ to determine the structure of $\text{End}_\Lambda(\Lambda\oplus P_i^*)$, and then we'll show how the resulting path algebra is affected by the removal of vertex $i$, giving us $\Lambda/P_i\oplus P_i^*$. For this last part we will use the following lemma, which states what happens to a path algebra when we remove a vertex from it.
\begin{lemma}\label{lemma:QuiverRemoveVertex}
Let $\Lambda = kQ/I$ be a path algebra. Let $i$ be a vertex in $Q$ with no cycles of length one, and with no minimal relations from $i$ to $i$. The quiver associated to $\mathrm{End}_\Lambda(\Lambda/P_i)\simeq (1-e_i)\Lambda(1-e_i)$ is obtained by removing vertex $i$ from the quiver $Q$. We will denote this quiver by $Q_i$, and its arrows and relations are explicitly given by the following:
\begin{itemize}
\item Any arrow in $Q$ that neither starts nor ends in vertex $i$ is also an arrow in $Q_i$. The same is true for relations.
\item If there are arrows $\beta\colon h\rightarrow i$ and $\alpha\colon i\rightarrow j$ in $Q$, then there is an arrow $\alpha\beta\colon h\rightarrow j$ in $Q_i$.
\item If there is an arrow $\alpha\colon i\rightarrow j$ and a relation $r\colon g\dashrightarrow i$ in $Q$, then $\alpha r\colon g\dashrightarrow j$ is a relation in $Q_i$.
\item If there is an arrow $\beta\colon h\rightarrow i$ and a relation $r'\colon i\dashrightarrow k$ in $Q$, then $r'\beta\colon h\dashrightarrow k$ is a relation in $Q_i$.
\end{itemize}
\end{lemma}
\textbf{Note:} There is no ambiguity when we denote arrows and relations in $Q_i$ as the compositions of arrows/relations in $Q$, because any time we do so those arrows/relations are not themselves in $Q_i$.
\begin{proof}[Proof of lemma]
Because the only paths in $Q$ which are not paths in $Q_i$ are those that begin or end in vertex $i$, any path that does neither will be preserved. And since arrows are paths of length one, all arrows that neither begin nor end in vertex $i$ will still be arrows in $Q_i$. This also applies to relations.
Assume now that we have arrows $\beta\colon h\rightarrow i$ and $\alpha\colon i\rightarrow j$ in $Q$. The composition $\alpha\beta$ is a path from $h$ to $j$, and will therefore be preserved. However, since neither $\alpha$ nor $\beta$ will appear in $Q_i$, we can't decompose $\alpha\beta$ in $Q_i$. This means that the length of $\alpha\beta$ is $1$ in $Q_i$, so it is an arrow. Similarly, for a relation into (resp. out of) vertex $i$, postcomposing with an arrow out of $i$ (resp. precomposing with an arrow into $i$) will give a relation which is preserved in $Q_i$.
\end{proof}
We will now turn our attention to the first part of the proof of the main result, namely determining how the quiver $Q$ changes when we add the vertex $i^*$ to it.
Combined with the above lemma, this will allow us to prove the main result.
\subsection*{The structure of \texorpdfstring{$\text{End}(\Lambda\oplus P_i^*)$}{End(Λ⊕Pi*)}}
We begin by considering $\text{End}_\Lambda(\Lambda\oplus P_i^*)$. We want to find a generating subset (which will correspond to the arrows in the quiver $Q^{i^*}$). Notice that we can split this into four direct summands, which can be considered individually:
$$\text{End}_\Lambda(\Lambda\oplus P_i^*) \simeq \begin{bmatrix}\text{End}_{\Lambda}(\Lambda) & \text{Hom}_{\Lambda}(\Lambda, P_i^*) \\ \text{Hom}_{\Lambda}(P_i^*, \Lambda) & \text{End}_{\Lambda}(P_i^*)\end{bmatrix}.$$
We already know the structure of $\text{End}_\Lambda(\Lambda)$, so we only need to examine the other three direct summands. The sets $\text{Hom}_\Lambda(\Lambda, P_i^*)$ and $\text{Hom}_\Lambda(P_i^*,\Lambda)$ correspond to paths out of and into $i^*$ in the quiver $Q^{i^*}$, respectively (recall that a path $i\rightarrow j$ gives a map $P_i\leftarrow P_j$). $\text{End}_{\Lambda}(P_i^*)$ corresponds to paths that both start and end in $i^*$. This always contains the trivial path $e_i^*$, but could also contain cycles of length $\geq 1$.
Recall that $P_i^*$ is defined by completing the right $\Lambda/P_i$-approximation $\bigoplus\limits_{\alpha: i\rightarrow ?} P_{t(\alpha)}\rightarrow P_i$ of $P_i$ to the distinguished triangle
$$P_i^*\xrightarrow{[\alpha^*]_\alpha} \bigoplus\limits_{\alpha:i\rightarrow ?} P_{t(\alpha)}\xrightarrow{[\alpha]_\alpha} P_i.$$
Throughout this proof, whenever we apply the functor $\text{Hom}_{\Lambda}(\Lambda, - )$ to a map $P_j\xrightarrow{\alpha} P_i$, we will denote the induced map as $\alpha$ as well (same for the corresponding contravariant $\text{Hom}$-functor). This abuse of notation is to make the proof easier to read. We will also write $\text{Hom}_{\Lambda}(-,-)$ as $(-,-)$, to save space.
\\
\begin{lemma}\label{lemma:EndPi*}
Any morphism from $P_i^*$ to itself which is not an isomorphism will factor through $P_i^*\xrightarrow{[\alpha^*]_\alpha}\bigoplus_\alpha P_{t(\alpha)}$.
\end{lemma}
\begin{proof}
Assume that $\gamma$ is a morphism from $P_i^*$ to itself which is not an isomorphism. We will now show that this can be completed to a morphism of distinguished triangles from the mutation triangle to itself. Consider the solid part of the following diagram.
\begin{equation*}
\begin{tikzcd}
P_i[-1] \arrow[r, "c"] \arrow[d, dashed, "{\beta[-1]}"] & P_i^* \arrow[r, "{[\alpha^*]_{\alpha}}"] \arrow[d, "{\gamma}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"] \arrow[d, dotted, "{\delta}"] & P_i \arrow[d, dashed, "{\beta}"] \\
P_i[-1] \arrow[r, "c"] & P_i^* \arrow[r, "{[\alpha^*]_{\alpha}}"] & \bigoplus\limits_{\alpha:i\rightarrow ?} P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"] & P_i
\end{tikzcd}
\end{equation*}
The composition $c \gamma\cdot[\alpha^*]_{\alpha}$ is a morphism between projective objects in different shifted degrees, and is therefore zero. Since $P_i[-1]$ is a weak kernel of $[\alpha^*]_\alpha$, this implies that $c \gamma $ factors through $P_i[-1]$, and thus there exists a morphism $\beta[-1]$ making the square commute. The $2$-out-of-$3$ property for triangulated categories now ensures the existence of $\delta$, completing the morphism of triangles.
We will now show that $\beta$ can't be an isomorphism. If $\beta$ is an isomorphism then it has an inverse $\beta^{-1}\colon P_i\rightarrow P_i$.
The morphism $\bigoplus_{\alpha}P_{t(\alpha)} \xrightarrow{[\alpha]_{\alpha}} P_i$ is a right $\Lambda/P_i$-approximation, so the composition $[\alpha]_\alpha\cdot\beta^{-1}$ factors through $\bigoplus_{\alpha}P_{t(\alpha)}$. Thus there exists a morphism $\delta'$ making the following diagram commute.
\begin{equation*}
\begin{tikzcd}
\bigoplus\limits_{\alpha:i\rightarrow ?}P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"] \arrow[d, "{\delta}"] & P_i \arrow[d, "{\beta}"] \\
\bigoplus\limits_{\alpha:i\rightarrow ?}P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"] \arrow[d, dashed, "{\delta'}"] & P_i \arrow[d, "{\beta^{-1}}"] \\
\bigoplus\limits_{\alpha:i\rightarrow ?} P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"] & P_i
\end{tikzcd}
\end{equation*}
We now have that $\delta\delta'\cdot[\alpha]_\alpha=[\alpha]_\alpha\cdot\beta^{-1}\beta=[\alpha]_\alpha$, and since $[\alpha]_\alpha$ is right minimal, this implies that $\delta\delta'$ is an isomorphism.
This means that $\delta$ has a right inverse, explicitly given by $\left[\delta'(\delta\delta')^{-1}\right]$.
A similar argument shows that $\delta$ has a left inverse, and hence it is an isomorphism. In other words, $\beta$ being an isomorphism implies $\delta$ is an isomorphism. But then, by the five-lemma, $\gamma$ would also be an isomorphism, which contradicts our assumption. So $\beta$ can't be an isomorphism.
This means that $\beta\colon i\rightarrow i$ is a linear combination of paths whose length is bigger than $1$, so it factors through $\bigoplus_{\alpha}P_{t(\alpha)} $, since the direct sum is taken over all arrows $\alpha$ that start in $i$ (and we assume the quiver has no cycles of length one at the vertex i). Using this, we will now show that $\gamma$ factors through $[\alpha^*]_{\alpha}$.
For each $\alpha$, let $d_\alpha$ be a path $t(\alpha)\rightarrow i$ such that $\beta=\sum_\alpha \alpha d_\alpha$. This gives a morphism $[d_\alpha]_\alpha\colon P_i\rightarrow \bigoplus_\alpha P_{t(\alpha)}$ satisfying $\beta=[\alpha]_{\alpha}\cdot [d_\alpha]_\alpha$. Inserting this into the diagram from before we get the following, where the lower right triangle commutes, in addition to the squares.
\begin{equation*}
\begin{tikzcd}
P_i[-1] \arrow[r, "c"] \arrow[d, "{\beta[-1]}"] & P_i^* \arrow[r, "{[\alpha^*]_{\alpha}}"] \arrow[d, "{\gamma}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"] \arrow[d, "{\delta}"] \arrow[dr, phantom, "\circlearrowleft", near end ] & P_i \arrow[d, "{\beta}"] \arrow[dl, pos=0.4, "{[d_\alpha]_\alpha}"] \\
P_i[-1] \arrow[r, "c"'] & P_i^* \arrow[r, "{[\alpha^*]_{\alpha}}"'] & \bigoplus\limits_{\alpha:i\rightarrow ?} P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"'] & P_i
\end{tikzcd}
\end{equation*}
Commutativity of the rightmost square and triangle gives that
$$\delta\cdot[\alpha]_{\alpha}=[\alpha]_{\alpha}\cdot\beta=[\alpha]_{\alpha}\cdot [d_\alpha]_\alpha\cdot[\alpha]_{\alpha},$$
and by rearranging this, we get $\left( \delta - [\alpha]_\alpha\cdot [d_\alpha]_\alpha \right)\cdot [\alpha]_\alpha=0$. Thus, since $P_i^*$ is a weak kernel of $[\alpha]_{\alpha}$ there exists a morphism $[d'_\alpha]_\alpha\colon \bigoplus_{\alpha}P_{t(\alpha)} \rightarrow P_i^*$ such that $[d'_\alpha]_\alpha\cdot [\alpha^*]_{\alpha}= \delta - [\alpha]_\alpha\cdot [d_\alpha]_\alpha$. We insert $[d'_\alpha]_\alpha$ into the diagram
\begin{equation*}
\begin{tikzcd}
P_i[-1] \arrow[r, "c"] \arrow[d, "{\beta[-1]}"] & P_i^* \arrow[r, "{[\alpha^*]_{\alpha}}"] \arrow[d, "{\gamma}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"] \arrow[d, "{\delta}"] \arrow[dl, dashed, "{[d'_\alpha]_\alpha}"] \arrow[dr, phantom, "\circlearrowleft", near end ] & P_i \arrow[d, "{\beta}"] \arrow[dl, pos=0.4, "{[d_\alpha]_\alpha}"] \\
P_i[-1] \arrow[r, "c"'] & P_i^* \arrow[r, "{[\alpha^*]_{\alpha}}"'] & \bigoplus\limits_{\alpha:i\rightarrow ?} P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"'] & P_i
\end{tikzcd}
\end{equation*}
Observe that we can rewrite the last equation as $\delta=[d'_\alpha]_\alpha\cdot[\alpha^*]_{\alpha} + [\alpha]_\alpha\cdot[d_\alpha]_\alpha$, so by the commutativity of the middle square we get
\begin{align*}
\gamma\cdot[\alpha^*]_{\alpha} &= [\alpha^*]_{\alpha}\cdot\gamma \\
&= [\alpha^*]_{\alpha}\cdot\left([d'_\alpha]_\alpha\cdot[\alpha^*]_{\alpha} + [\alpha]_{\alpha}\cdot[d_\alpha]_\alpha\right) \\
&= [\alpha^*]_{\alpha}\cdot [d'_\alpha]_\alpha\cdot[\alpha^*]_{\alpha} + \underbrace{[\alpha^*]_{\alpha}\cdot[\alpha]_{\alpha}}_{0}\cdot[d_\alpha]_\alpha = [\alpha^*]_{\alpha}\cdot [d'_\alpha]_\alpha\cdot[\alpha^*]_{\alpha}
\end{align*}
This means that $\left( \gamma - [\alpha^*]_{\alpha}\cdot[d'_\alpha]_\alpha\right)\cdot[\alpha^*]_{\alpha}=0$, and since $P_i[-1]$ is a weak kernel of $[\alpha^*]_{\alpha}$ there exists a morphism $d''\colon P_i^*\rightarrow P_i[-1]$ such that $d'' c=\gamma - [\alpha^*]_{\alpha}\cdot [d'_\alpha]_\alpha$. Inserting $d''$ into the diagram, we end up with
\begin{equation*}
\begin{tikzcd}
P_i[-1] \arrow[r, "c"] \arrow[d, "{\beta[-1]}"] & P_i^* \arrow[r, "{[\alpha^*]_{\alpha}}"] \arrow[d, "{\gamma}"] \arrow[dl, dotted, "{d''}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"] \arrow[d, "{\delta}"] \arrow[dl, dashed, "{[d'_\alpha]_\alpha}"] \arrow[dr, phantom, "\circlearrowleft", near end ] & P_i \arrow[d, "{\beta}"] \arrow[dl, pos=0.4, "{[d_\alpha]_\alpha}"] \\
P_i[-1] \arrow[r, "c"'] & P_i^* \arrow[r, "{[\alpha^*]_{\alpha}}"'] & \bigoplus\limits_{\alpha:i\rightarrow ?} P_{t(\alpha)} \arrow[r, "{[\alpha]_{\alpha}}"'] & P_i
\end{tikzcd}
\end{equation*}
Again we can rewrite the last equation into $\gamma=d'' c + [\alpha^*]_{\alpha}\cdot [d'_\alpha]_\alpha$. Thus, if we can show that $d''=0$, we see that $\gamma$ factors through $\bigoplus_{\alpha}P_{t(\alpha)}$. Note that $d''$ is an element of $(P_i^*, P_i[-1])\simeq(P_i^*[1], P_i)$, which is zero by the assumption that $(P_i^*[1], \Lambda)=0$.
Thus, we have that $d''=0$, which means that $\gamma=[\alpha^*]_{\alpha}\cdot [d'_\alpha]_\alpha$, so $\gamma$ factors through $\bigoplus_\alpha P_{t(\alpha)}$. This is true for any non-isomorphism $P_i^*\rightarrow P_i^*$.
\end{proof}
\begin{lemma}\label{lemma:HomPiLambda}
As a right $\text{End}_\Lambda(\Lambda \oplus P_i^*)$-module, $(P_i^*, \Lambda \oplus P_i^*)$ fits in the following exact sequence
\begin{equation*}
\begin{tikzcd}[column sep = small]
(P_i,\Lambda \oplus P_i^*) \arrow[r, "{[\alpha]_{\alpha}}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}(P_{t(\alpha)},\Lambda \oplus P_i^*) \arrow[r, "{[\alpha^*]_{\alpha}}"] & (P_i^*,\Lambda \oplus P_i^*) \arrow[r] & \frac{(P_i^*,\Lambda\oplus P_i^*)}{\Rad((P_i^*,\Lambda\oplus P_i^*))} \arrow[r] & 0,
\end{tikzcd}
\end{equation*}
where $\frac{(P_i^*,\Lambda\oplus P_i^*)}{\Rad((P_i^*,\Lambda\oplus P_i^*))}\simeq S_i^*$, the simple module at vertex $i^*$.
\end{lemma}
\begin{proof}
If we apply $(-,\Lambda \oplus P_i^*)$ to the triangle
\begin{equation*}
P_i^*\xrightarrow{[\alpha^*]_{\alpha}} \bigoplus\limits_{\alpha: i\rightarrow ?} P_{t(\alpha)} \xrightarrow{[\alpha]_{\alpha}} P_i \rightarrow P_i^*[1],
\end{equation*}
we get a long exact sequence, which contains the following part
\begin{equation*}
\begin{tikzcd}[column sep = small]
\cdots \arrow[r] & (P_i,\Lambda \oplus P_i^*) \arrow[r, "{[\alpha]_{\alpha}}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}(P_{t(\alpha)},\Lambda \oplus P_i^*) \arrow[r, "{[\alpha^*]_{\alpha}}"] & (P_i^*,\Lambda \oplus P_i^*) \arrow[dll, out = 0, in = 180, looseness=2, overlay] \\
& (P_i[-1], \Lambda \oplus P_i^*) \arrow[r] & \cdots. &
\end{tikzcd}
\end{equation*}
We have that $(P_i[-1], \Lambda \oplus P_i^*)\simeq (P_i[-1], \Lambda)\oplus(P_i[-1], P_i^*)$, because the contravariant Hom functor preserves products. Observe now that
$(P_i[-1], \Lambda)=0$, since $P_i[-1]$ and $\Lambda$ are projective modules in different shifted degrees.
It follows that $[\alpha^*]_\alpha$ is surjective on all vertices except $i^*$. For $i^*$, \cref{lemma:EndPi*} implies that all radical endomorphisms factor through $[\alpha^*]_\alpha$, which means that the image of $[\alpha^*]_\alpha$ is $\Rad(P_i^*,\Lambda\oplus P_i^*)$. This completes the proof of the lemma.
\end{proof}
The lemma shows that any element of $(P_i^*, \Lambda\oplus P_i^*)$ is either an isomorphism from $P_i^*$ to itself, or given by some linear combiantion of elements in $\bigoplus\limits_{\alpha:i\rightarrow ?}(P_{t(\alpha)},\Lambda\oplus P_i^*)$, composed with the map $[\alpha^*]_\alpha$.
This means that any linear combination of paths in $Q^{i^*}$ from some vertex $k$ to $i^*$ is given by some linear combination of paths $k\rightarrow t(\alpha)$ in $Q$ (for some subset of the arrows $\alpha$), each followed by the corresponding arrow $\alpha^*$.
From the lemma we also see that $\Ker[\alpha^*]_{\alpha} = \Img[\alpha]_\alpha$, meaning that an element of $(\bigoplus\limits_{\alpha: i\rightarrow ?} P_{t(\alpha)}, \Lambda \oplus P_i^*)$ is sent to zero by $[\alpha^*]_\alpha$ if and only if it comes from an element of $(P_i, \Lambda\oplus P_i^*)$. So the composition $[\alpha^*]_\alpha\cdot[\alpha]_\alpha=\sum_{\alpha}\alpha^*\alpha$ is zero, which defines a relation $i\dashrightarrow i^*$ in $Q^{i^*}$, and any relation in $Q^{i^*}$ ending in $i^*$ factors through this relation.
We summarize these observations in the following remark.
\begin{remark}\label{remark:intoi}
There is a one-to-one correspondence between arrows out of $i$ in $Q$ and arrows into $i^*$ in $Q^{i^*}$.
There is a minimal relation $i\dashrightarrow i^*$ in $Q^{i^*}$ given by $\sum_{\alpha}\alpha^*\alpha=0$, and any relation ending in $i^*$ factors through this relation.
\end{remark}
Before we state the next lemma, which describes the structure of $(\Lambda, P_i^*)$, we will define some morphisms which we will need in order to do so. Recall from \cref{section:setup} that if $r$ is a relation and $\alpha$ is an arrow, both starting in $i$, then $\sfrac{r}{\alpha}$ denotes the linear combination of paths in $Q$ given by taking all paths in $r$ which begin with the arrow $\alpha$, and removing $\alpha$ from them. So $\sfrac{r}{\alpha}$ is a linear combination of paths from vertex $t(\alpha)$ to vertex $t(r)$.
As usual, $\sfrac{r}{\alpha}$ will for simplicity also refer to the induced maps $P_{t(r)}\xrightarrow{\sfrac{r}{\alpha}} P_{t(\alpha)}$ and $(\Lambda, P_{t(r)})\xrightarrow{\sfrac{r}{\alpha}} (\Lambda, P_{t(\alpha)})$.
If we take the mutation triangle
\begin{equation*}
P_i^*\xrightarrow{[\alpha^*]_{\alpha}} \bigoplus\limits_{\alpha: i\rightarrow ?} P_{t(\alpha)} \xrightarrow{[\alpha]_{\alpha}} P_i \rightarrow P_i^*[1]
\end{equation*}
and apply the functor $(P_{t(r)}, - )$ to it, we get a long exact sequence containing the following part
\begin{equation*}
\begin{tikzcd}[column sep = small, row sep = tiny]
\cdots \arrow[r] & (P_{t(r)}, P_i[-1]) \arrow[r] & (P_{t(r)}, P_i^*) \arrow[r, "{[\alpha^*]_{\alpha}}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}(P_{t(r)}, P_{t(\alpha)}) \arrow[r, "{[\alpha]_{\alpha}}"] & (P_{t(r)}, P_i) \arrow[r] & \cdots, \\
& 0 \arrow[u, equals] & & &
\end{tikzcd}
\end{equation*}
where $(P_{t(r)}, P_i[-1]) = 0$ because the modules are projective in different shifted degrees. Hence, by exactness $[\alpha^*]_\alpha$ is a monomorphism.
Notice that for fixed $r$, $[\sfrac{r}{\alpha}]_{\alpha}$ is an element of $\bigoplus\limits_{\alpha:i\rightarrow ?}(P_{t(r)}, P_{t(\alpha)})$, and when composed with $[\alpha]_\alpha$ this gives $[\sfrac{r}{\alpha}]_\alpha\cdot[\alpha]_\alpha=\sum_\alpha \sfrac{r}{\alpha}\circ\alpha=r$.
Since $r$ is a relation, and hence equal to zero in $\Lambda$, this means that $[\sfrac{r}{\alpha}]_\alpha\in\Ker([\alpha]_\alpha)$. And since $\Ker([\alpha]_\alpha)\simeq\Img([\alpha^*]_\alpha)$ by exactness, we can find an element of $(P_{t(r)}, P_i^*)$ which is sent to $[\sfrac{r}{\alpha}]_\alpha$ by $[\alpha^*]_\alpha$. We call this element $\overline{r}$. In other words, we have that $\overline{r}\in (P_{t(r)}, P_i^*)$, and $\overline{r}$ satisfies $\overline{r}\cdot \alpha^*=\sfrac{r}{\alpha}$ for all $\alpha$.
We are now ready to state the lemma.
\begin{lemma}\label{lemma:homLambdaPi}
The map
$$(\Lambda, [\overline{r}]_r)\colon \bigoplus\limits_{r:i\dashrightarrow ?}\big(\Lambda, P_{t(r)}\big) \rightarrow \big(\Lambda, P_{i}^*\big)$$
is surjective. Moreover its kernel coincides with the kernel of $\big( \Lambda, [\sfrac{r}{\alpha}]_{r,\alpha} \big)$.
\end{lemma}
\begin{proof}
We again consider the mutation triangle
\begin{equation*}\label{eq:mutationTriangle}
P_i^*\xrightarrow{[\alpha^*]_{\alpha}} \bigoplus\limits_{\alpha: i\rightarrow ?} P_{t(\alpha)} \xrightarrow{[\alpha]_{\alpha}} P_i \rightarrow P_i^*[1],
\end{equation*}
and we apply the functor $(\Lambda, -)$. Like before, this gives a long exact sequence, containing the following part
\begin{equation*}
\begin{tikzcd}[column sep = small]
\cdots \arrow[r] & (\Lambda, P_i[-1]) \arrow[r] & (\Lambda, P_i^*) \arrow[r, "{[\alpha^*]_{\alpha}}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}(\Lambda, P_{t(\alpha)}) \arrow[r, "{[\alpha]_{\alpha}}"] & (\Lambda, P_i) \arrow[r] & \cdots,
\end{tikzcd}
\end{equation*}
where $(\Lambda, P_i[-1])=0$ because the modules are projective and in different shifted degrees. So we get that the following sequence is exact
\begin{equation*}
\begin{tikzcd}
0 \arrow[r] & (\Lambda, P_i^*) \arrow[r, "{[\alpha^*]_{\alpha}}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}(\Lambda, P_{t(\alpha)}) \arrow[r, "{[\alpha]_{\alpha}}"] & (\Lambda, P_i).
\end{tikzcd}
\end{equation*}
By applying $(\Lambda, - )$ to the projective resolution given in \cref{lemma:projres}, we also get that the sequence
\begin{equation*}
\begin{tikzcd}
\Ker([\sfrac{r}{\alpha}]_{r,\alpha}) \arrow[r] & \bigoplus\limits_{r:i\dashrightarrow ?}(\Lambda, P_{t(r)}) \arrow[r, "{[\sfrac{r}{\alpha}]_{r, \alpha}}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}(\Lambda, P_{t(\alpha)}) \arrow[r, "{[\alpha]_{\alpha}}"] & (\Lambda, P_i)
\end{tikzcd}
\end{equation*}
is exact.
Combining these two exact sequences, we get the following diagram
\begin{equation*}
\begin{tikzcd}
0 \arrow[r] & (\Lambda, P_i^*) \arrow[r, "{[\alpha^*]_{\alpha}}"] & \bigoplus\limits_{\alpha:i\rightarrow ?}(\Lambda, P_{t(\alpha)}) \arrow[r, "{[\alpha]_{\alpha}}"] & (\Lambda, P_i)\\
\Ker([\sfrac{r}{\alpha}]_{r,\alpha}) \arrow[r] \arrow[u, "0"] & \bigoplus\limits_{r:i\dashrightarrow ?} (\Lambda, P_{t(r)}) \arrow[r, "{[\sfrac{r}{\alpha}]_{r,\alpha}}"] \arrow[u, "{[\overline{r}]_r}"] & \bigoplus\limits_{r:i\dashrightarrow ?} (\Lambda, P_{t(\alpha)}) \arrow[u, equals] \arrow[r, "{[\alpha]_\alpha}"]& (\Lambda, P_i). \arrow[u, equals]
\end{tikzcd}
\end{equation*}
Here, the rows are exact and the squares commute. The first vertical map is zero, and hence an epimorphism, and the equalities are both monomorphisms and epimorphisms. Thus, by the four lemma, the map $[\overline{r}]_r$ is an epimorphism, and hence surjective.
Now, let's consider the relations in $(\Lambda, P_i^*)$, that is, elements in $(\Lambda, P_i^*)$ which are equivalent to zero.
Because $[\alpha^*]_\alpha$ is a monomorphism, an element of $(\Lambda, P_i^*)$ is equivalent to zero if and only if it is in $\Ker([\alpha^*]_\alpha)$.
Any element of $(\Lambda,P_i^*)$ is equal to some element of $\bigoplus\limits_{r:i\dashrightarrow ?}\big(\Lambda, P_{t(r)}\big)$ composed with $[\overline{r}]_{r}$, since $[\overline{r}]_r$ is a surjection.
In particular, this means that any element of $\Ker([\alpha^*]_\alpha)$ is generated by some element of $\bigoplus\limits_{r:i\dashrightarrow ?}\big(\Lambda, P_{t(r)}\big)$, which by commutativity must be sent to zero by $[\sfrac{r}{\alpha}]_{r,\alpha}$. Hence, $\Ker([\sfrac{r}{\alpha}]_{r,\alpha})$ generates $\Ker([\alpha^*]_\alpha)$, and consequently it generates all relations in $(\Lambda, P_i^*)$.
\end{proof}
This lemma shows that any morphism $P_k\rightarrow P_i^*$ can be written as some morphsim $P_k \rightarrow \bigoplus\limits_{r:i\dashrightarrow ?} P_{t(r)}$ composed with $[\overline{r}]_{r}$. In terms of the quivers, this means that any path $i^*\rightarrow k$ is given as $\gamma\overline{r}$, where $\gamma$ is some path $t(r)\rightarrow k$ and $\overline{r}$ is the arrow $i^*\rightarrow t(r)$ satisfying $\overline{r}\alpha^*=\sfrac{r}{\alpha}$ for each arrow $\alpha\colon i\rightarrow ?$.
The lemma also shows that for a vertex $l$ in $Q$, any relation $P_l\dashrightarrow P_i^*$ is of the form $[\varepsilon_r]_r\cdot[\overline{r}]_r = 0$, for certain $\varepsilon_r\colon P_l\rightarrow P_{t(r)}$.
Since $[\alpha^*]_\alpha$ is a monomorphism, we get that $[\varepsilon_r]_r\cdot[\overline{r}]_r = 0$ if and only if $[\varepsilon_r]_r\cdot[\overline{r}]_r\cdot[\alpha^*]_\alpha = 0$. This can be rewritten as
$$[\varepsilon_r]_r\cdot[\overline{r}]_r\cdot[\alpha^*]_\alpha = \left(\sum\limits_{r\in R} \varepsilon_r \overline{r} \right)\cdot [\alpha^*]_\alpha = \left[\sum\limits_{r\in R} \varepsilon_r \overline{r}\alpha^* \right]_\alpha = \left[\sum\limits_{r\in R} \varepsilon_r \sfrac{r}{\alpha} \right]_\alpha =0,$$
which means that $[\varepsilon_r]_r\cdot[\overline{r}]_r = 0$ in $(\Lambda, P_i^*)$ if and only if $\sum_r \varepsilon_r\sfrac{r}{\alpha}=0$ for each $\alpha$ starting in $i$. Thus, $[\varepsilon_r]_r\cdot[\overline{r}]_r$ defines a relation in $(\Lambda, P_i^*)$ if and only if $[\varepsilon_r]_r\cdot[\sfrac{r}{\alpha}]_r$ defines a relation in $(\Lambda, P_{t(\alpha)})$ for each $\alpha$.
In terms of the quivers, this means $\sum\limits_{r\in R} \varepsilon_r \overline{r}=0$ will define a relation in $Q^{i^*}$ if and only if $\sum\limits_{r\in R} \varepsilon_r \sfrac{r}{\alpha} =0$ defines a relation in $Q$ for each $\alpha\colon i\rightarrow ?$.
\begin{remark}\label{remark:outofi}
There is a one-to-one correspondence between arrows in $Q^{i^*}$ starting in $i^*$ and minimal relations in $Q$ starting in $i$, given by $\overline{r}\leftrightarrow r$.
This means that there are no minimal relations in $Q^{i^*}$ from $i$ to itself. Because if $r$ is any relation in $Q$ starting in $i$, then
$$r=\sum\limits_{\alpha:i\rightarrow ?} \sfrac{r}{\alpha} \alpha=\sum\limits_{\alpha:i\rightarrow ?}\overline{r}\alpha^*\alpha=\overline{r}\sum\limits_{\alpha:i\rightarrow ?} \alpha^*\alpha,$$
so $r$ factors through $\sum_\alpha \alpha^* \alpha$ in $Q^{i^*}$, and hence is not minimal.
Also, any linear combination of paths in $Q^{i^*}$ defines a relation $i^*\dashrightarrow l$ if and only if precomposing it with each of the arrows $\alpha^*$ gives a linear combination of paths which is equal to a relation in $Q$.
\end{remark}
We are now ready to define the quiver $Q^{i^*}$, whose path algebra (by construction) is isomorphic to $\text{End}_\Lambda(\Lambda\oplus P_i^*)^{\text{op}}$.
\begin{lemma}\label{lemma:Qi*}
Let $(Q^{i^*}, I^{i^*})$ be the following quiver with relations:
\begin{enumerate}\addtocounter{enumi}{-1}
\item The set of vertices in $Q^{i^*}$ consists of all vertices in $Q$, and an additional vertex $i^*$.
\item All arrows in $Q$ are also arrows in $Q^{i^*}$, and all relations in $Q$ are also relations in $Q^{i^*}$.
\item There is an arrow $\alpha^*\colon t(\alpha)\rightarrow i^*$ in $Q^{i^*}$ for each arrow $\alpha\colon i\rightarrow t(\alpha)$ in $Q$, and all arrows ending in $i^*$ are given as such.
\item There is an arrow $\overline{r}\colon i^*\rightarrow t(r)$ for each minimal relation $r\colon i \dashrightarrow t(r)$ in $Q$, and all arrows starting in $i^*$ are given as such.
\item There is a relation $i\dashrightarrow i^*$ in $Q^{i^*}$, given by $\sum\limits_{\alpha:i\rightarrow ?}\alpha^*\alpha=0$, and no other minimal relations end in $i^*$.
\item For each arrow $\alpha$ starting in $i$, and each relation $r$ starting in $i$, there is a relation $t(\alpha)\dashrightarrow t(r)$ in $Q^{i^*}$, given by $\overline{r}\alpha^*=\sfrac{r}{\alpha}$.
\item There is a relation defined by $\sum\limits_{r:i\dashrightarrow ?} \varepsilon_r \overline{r}=0$ in $Q^{i^*}$ if and only $\sum\limits_{r:i\dashrightarrow ?} \varepsilon_r \sfrac{r}{\alpha}=0$ defines a relation in $Q$ for each arrow $\alpha\colon i\rightarrow ?$.
\end{enumerate}
Then the path algebra $kQ^{i^*}/I^{i^*}$ is isomorphic to $\text{End}_\Lambda(\Lambda\oplus P_i^*)^{\text{op}}$.
\end{lemma}
\begin{proof}
Clearly, the primitive idempotents in $\text{End}_\Lambda(\Lambda\oplus P_i^*)^{\text{op}}$ are the same as those in $\text{End}_\Lambda(\Lambda)^{\text{op}}$, plus the one in $\text{End}_\Lambda(P_i^*)^{\text{op}}$. Hence, $Q^{i^*}$ has the same vertices as $Q$, plus the additional vertex $i^*$. This proves point $0$.
Since $\text{End}_\Lambda(\Lambda)\subseteq \text{End}_\Lambda(\Lambda\oplus P_i^*)$, any arrow and relation in $Q$ will still exist in $Q^{i^*}$, proving point $1$.
From \cref{remark:intoi} we know that there is a one-to-one correspondence between arrows in $Q$ starting in $i$, and arrows in $Q^{i^*}$ ending in $i^*$, which proves point $2$. The remark also tells us that there is one, and only one, minimal relation ending in $i^*$, namely $\sum\limits_{\alpha:i\rightarrow ?}\alpha^*\alpha=0$. This proves point $4$.
\Cref{remark:outofi} tells us that $\overline{r} \leftrightarrow r$ defines a one-to-one correspondence between arrows in $Q^{i^*}$ starting in $i^*$ and relations in $Q$ starting in $i$, thus proving point $3$.
By construction, the arrows $\overline{r}$ in $Q^{i^*}$ satisfy $\overline{r}\alpha^*=\sfrac{r}{\alpha}$ for each arrow $\alpha$ in $Q$ starting in $i$. This defines a set of relations on $Q^{i^*}$, which proves point $5$.
Also from \cref{remark:outofi}, we get that a linear combination of paths in $Q^{i^*}$ defines a relation $i^*\dashrightarrow l$ if and only if precomposing it with each of the arrows $\alpha^*$ gives a linear combination of paths which is equal to a relation in $Q$. In other words, we have that
$$\sum\limits_{r:i\dashrightarrow ?} \varepsilon_r \overline{r}=0 \text{ in } Q^{i^*} \quad \iff \quad \sum\limits_{r:i\dashrightarrow ?} \varepsilon_r \sfrac{r}{\alpha}=0 \text{ for all } \alpha\colon i\rightarrow ? \text{ in } Q,$$
which proves point $6$.
This concludes the proof of the lemma.
\end{proof}
Now we are ready to prove the main result, namely that mutating a quiver according to the mutation procedure stated in \cref{section:mutationsteps} yields a quiver whose path algebra is isomorphic to to the right tilting mutation of the path algebra of the original quiver.
More precisely, the statement is (in the notation of \cref{theorem:mainResult}) that $m_i(Q)\simeq Q_i^{{i^*}}$, where $m_i(Q)$ is the quiver obtained by mutating $Q$ at vertex $i$, and $Q_i^{i^*}$ is the quiver obtained by removing vertex $i$ from the quiver $Q^{i^*}$.
And by combining \cref{lemma:Qi*} and \cref{lemma:QuiverRemoveVertex}, we know that the path algebra of $Q_i^{i^*}$ is isomorphic to $\text{End}_\Lambda(\mu_i^R(\Lambda))^{\text{op}}$, so this implies that the path algebra of $m_i(Q)$ is isomorphic to the right tilting mutation of $\Lambda$ at $P_i$.
\begin{proof}[Proof of Theorem \ref{theorem:mainResult}]
We want to show that $m_i(Q)\simeq Q_i^{i^*}$. The structure of $m_i(Q)$ is defined in \cref{section:mutationsteps}, so we already know it. Hence, we must determine the structure of $Q_i^{i^*}$, and show that it is the same as the known structure of $m_i(Q)$.
In order to do so, we will apply \cref{lemma:QuiverRemoveVertex} to the quiver obtained in \cref{lemma:Qi*}. Recall that $\mu_i^R(\Lambda)\simeq\Lambda/P_i\oplus P_i^*$, so we have that
$$\text{End}_\Lambda(\mu_i^R(\Lambda))^{\text{op}}\simeq\text{End}_\Lambda(\Lambda/P_i\oplus P_i^*)^{\text{op}}.$$
Starting with a quiver $Q$ whose endomorphism ring is the algebra $\Lambda$, \cref{lemma:Qi*} yields the quiver $Q^{i^*}$ whose endomorphism ring is isomorphic to $\text{End}_\Lambda(\Lambda\oplus P_i^*)$.
What we want to do now is apply \cref{lemma:QuiverRemoveVertex} to $Q^{i^*}$, but in order to do so we must check that the assumptions in the lemma are satisfied. The assumptions we need to check are that there are no cycles of length $1$ on vertex $i$, and that there are no minimal relations from $i$ to itself.
By assumption $i$ has no cycles of length $1$ in $Q$, and because we don't add any such cycles in \cref{lemma:Qi*}, the same is true for $Q^{i^*}$. Also, we know from \cref{remark:outofi} that $Q^{i^*}$ has no minimal relations from $i$ to itself.
Thus, the assumtions needed for using \cref{lemma:QuiverRemoveVertex} are satisfied.
Now, applying \cref{lemma:QuiverRemoveVertex} to the quiver $Q^{i^*}$ will then yield the quiver $Q_i^{i^*}$, corresponding to $\text{End}_\Lambda(\Lambda/P_i\oplus P_i^*)$, which is precisely what we want.
As we can see from \cref{lemma:QuiverRemoveVertex}, most parts of the quiver $Q^{i^*}$ will remain unchanged when we transform it into $Q_i^{i^*}$. It keeps all vertices except $i$, as well as all arrows and relations which neither start nor end in $i$. Basically the only difference between the quivers is that vertex $i$ is removed, that minimal paths through $i$ are replaced with arrows between the same vertices, and that relations starting or ending in $i$ are extended by one step in the necessary direction.
Note that the assumption that there are no arrows $i\rightarrow i$ in $Q$ ensures that there are no arrows $i\rightarrow i^*$ in $Q^{i^*}$, which means that none of the arrows ending in $i^*$ will be removed in $Q_i^{i^*}$.
Looking at \cref{lemma:Qi*}, we can go through each step and make the necessary change to $Q^{i^*}$ as implied by \cref{lemma:QuiverRemoveVertex}, ending up with the quiver $Q_i^{i^*}$ having the following structure.
\begin{enumerate}\addtocounter{enumi}{-1}
\item The set of vertices in $Q^{i^*}_i$ consists of $i^*$ and all vertices in $Q$ except $i$.
\item All arrows in $Q$ which neither start nor end in $i$ are also arrows in $Q_i^{i^*}$. The same is true for relations.
\item There is an arrow $\alpha\beta\colon h\rightarrow j$ in $Q_i^{i^*}$ for each pair of arrows $\beta\colon h \rightarrow i$ and $\alpha\colon i \rightarrow j$ in $Q$.
\item There is an arrow $\alpha^*\colon t(\alpha)\rightarrow i^*$ in $Q_i^{i^*}$ for each arrow $\alpha\colon i\rightarrow t(\alpha)$ in $Q$, and all arrows ending in $i^*$ are given as such.
\item There is an arrow $\overline{r}\colon i^*\rightarrow t(r)$ in $Q_i^{i^*}$ for each minimal relation $r\colon i \dashrightarrow t(r)$ in $Q$, and all arrows starting in $i^*$ are given as such.
\item For each arrow $\beta\colon h\rightarrow i$ in $Q$ there is a relation $h\dashrightarrow i^*$ in $Q_i^{i^*}$, given by $\sum\limits_{\alpha:i\rightarrow ?}\alpha^*\alpha\beta=0$.
\item For each arrow $\alpha$ in $Q$ starting in $i$, and each relation $r$ starting in $i$, there is a relation $t(\alpha)\dashrightarrow t(r)$ in $Q^{i^*}$, given by $\overline{r}\alpha^*=\sfrac{r}{\alpha}$.
\item If $r\colon h\dashrightarrow i$ is a relation in $Q$, and $B$ is the set of arrows ending in $i$ (which means that $r$ can be written as $\sum\limits_{\beta\in B} \beta \bsfrac{r}{\beta}=0$), then there is a relation $h\dashrightarrow t(\alpha)$ in $Q_i^{i^*}$ for each arrow $\alpha$, defined by $\sum\limits_{\beta\in B} \alpha\beta \bsfrac{r}{\beta}=0$.
\item There is a relation defined by $\sum\limits_{r:i\dashrightarrow ?} \varepsilon_r \overline{r}=0$ in $Q_i^{i^*}$ if and only $\sum\limits_{r:i\dashrightarrow ?} \varepsilon_r \sfrac{r}{\alpha}=0$ defines a relation in $Q$ for each arrow $\alpha\colon i\rightarrow ?$.
\end{enumerate}
As we can see, this gives precisely the same quiver as the mutation procedure we defined in section \cref{section:mutationsteps}, which means that $Q_i^{i^*}\simeq m_i(Q)$. Hence, mutating a quiver $Q$ at vertex $i$ according to the mutation procedure yields a quiver whose path algebra is isomorphic to the right tilting mutation of the path algebra of $Q$ at the indecomposable projective summand $P_i$. This concludes the proof of the main theorem.
\end{proof}
\section{Example}\label{section:example}
Let us now consider an example to see how the mutation procedure can be used in practice. This example is inspired by Ladkani's work in \cite{Ladkani2013}, specifically the following result. Let $k$ be a field, and let $A_{n,m}$ denote the algebra obtained by taking the path algebra over $k$ of the linear quiver $A_{n}$ modulo the ideal generated by all paths of length $m$. Then the algebra $A_{r\cdot n ,r+1}$ is derived equivalent to $kA_r\otimes_k kA_n$ (which is isomorphic to the path algebra of a commutative $r\times n$-rectangle).
We will now show how we can use our mutation procedure to not only recreate this result, but also give an explicit series of derived equivalencies between the two algebras.
Take the quiver corresponding to the algebra $A_{2n,3}$, which is the line quiver $A_{2n}$ with a relation for every composition of 3 consecutive arrows. We will now show that by preforming a series of mutations, we can transform this quiver into the quiver given by a column of connected commutative squares, with relations for every com position of two consecutive arrows (this is essentially the quiver of $kA_2 \otimes_k kA_n$). Note, throughout this section we will occasionally renumber the vertices in the quivers, in order to highlight the relevant part of the quiver for each mutation.
\begin{equation*}
\begin{tikzcd}
_1 \arrow[r] \arrow[rrr, bend right, dashed] & _2 \arrow[r] \arrow[rrr, bend right, dashed] & _3 \arrow[r] \arrow[rrr, bend right, dashed] & _4 \arrow[r] \arrow[rrr, bend right, dashed] & _5 \arrow[r] \arrow[rrr, bend right, dashed] & _6 \arrow[r] \arrow[r] \arrow[rrr, bend right, dashed] & _7 \arrow[r] & _8 \arrow[r] & \cdots
\end{tikzcd}
\end{equation*}
We begin by mutating at the first (leftmost) vertex.
To do so, we apply the mutation procedure at vertex $1$. Since there are no arrows or relations into vertex $1$, we see that steps 1, 3 and 6 don't apply in this case. By step 2 we flip the arrow $1\rightarrow 2$ to an arrow $1^*\leftarrow 2$, and by step 4 the relation $1\dashrightarrow 4$ becomes an arrow $1^*\rightarrow 4$. Thus we get a new composition through the vertex $1^*$, and by step 5 we add a relation corresponding to that composition. this relation is what makes the commutative square. Finally, by step 7 we get a relation $1^*\dashrightarrow 5$, since there is a zero relation $2\dashrightarrow 5$ which is hit by the relation $1\dashrightarrow 4$. Thus, we end up with the following.
\begin{equation*}
\begin{tikzcd}
_2 \arrow[d] \arrow[r] \arrow[dr, dashed] & _{1^*} \arrow[d] \arrow[dr, dashed] & & & & & & &\\
_3 \arrow[r] \arrow[rrr, bend right, dashed] & _4 \arrow[r] \arrow[rrr, bend right, dashed] & _5 \arrow[r] \arrow[rrr, bend right, dashed] & _6 \arrow[r] \arrow[rrr, bend right, dashed] & _7 \arrow[r] \arrow[rrr, bend right, dashed] & _8 \arrow[r] \arrow[r] \arrow[rrr, bend right, dashed] & _9 \arrow[r] & _{10} \arrow[r] & \cdots &
\end{tikzcd}
\end{equation*}
We proceed by mutating at vertex $3$. By mutation step $1$ we get a new arrow $2\rightarrow 4$, corresponding to the composition through $3$, and this arrow is cancelled against the commutativity relation $2\dashrightarrow 4$. Step $2$ works just as in the previous mutation, by flipping the arrow $3\rightarrow 4$, and step $3$ says we should add a relation $2\dashrightarrow 3$ corresponding to the composition of these two arrows. Steps $4, 5$ and $7$ work just as in the previous mutation, and step $6$ isn't used since there are no relations ending in vertex $3$.
Thus, by mutating the above quiver at vertex $3$ according to the mutation rules, we get the following quiver.
\begin{equation*}
\begin{tikzcd}
_2 \arrow[r] \arrow[rrr, dashed, bend left] & _{1^*} \arrow[r] \arrow[dr, dashed] & _4 \arrow[d] \arrow[r] \arrow[dr, dashed] & _{3^*} \arrow[d] \arrow[dr, dashed] & & & & &\\
& & _5 \arrow[r] \arrow[rrr, bend right, dashed] & _6 \arrow[r] \arrow[rrr, bend right, dashed] & _7 \arrow[r] \arrow[rrr, bend right, dashed] & _8 \arrow[r] \arrow[r] \arrow[rrr, bend right, dashed] & _9 \arrow[r] & _{10} \arrow[r] & \cdots &
\end{tikzcd}
\end{equation*}
As we can see, the resulting quiver is more or less equal to the one we started with, except the commutative square has ''moved'' two steps to the right.
Next, we mutate at vertex $5$. Notice that locally, the quiver around vertex $5$ is quite similar to the quiver around vertex $3$ before the last mutation. The only meaningful difference is that there is a relation ending in $5$. This means that each step in the mutation is the same as in the previous mutation, except step $6$. Mutation step $6$ tells us that the relation $1^*\dashrightarrow 5$ is extended to a relation $1^*\dashrightarrow 6$.
Thus, mutating the above quiver at vertex $5$ results in the following quiver.
\begin{equation*}
\begin{tikzcd}
_2 \arrow[r] \arrow[rrr, dashed, bend left] & _{1^*} \arrow[r] \arrow[rrr, bend left, dashed] & _4 \arrow[r] \arrow[rrr, bend left, dashed] & _{3^*} \arrow[r] \arrow[dr, dashed] & _6 \arrow[d] \arrow[r] \arrow[dr, dashed] & _{5^*} \arrow[d] \arrow[dr, dashed] & & &\\
& & & & _7 \arrow[r] \arrow[rrr, bend right, dashed] & _8 \arrow[r] \arrow[r] \arrow[rrr, bend right, dashed]& _9 \arrow[r] & _{10} \arrow[r] & \cdots &
\end{tikzcd}
\end{equation*}
Agian, we see that the mutation yields a quiver which is similar, but where the commutative square has been moved two steps to the right.
By using the mutation procedure, it is easy to check that this pattern continues, i.e. that mutating at vertex $7$ will now move the square two more steps to the right. For the remainder of this example, we will no longer explicitly mention which step of the mutation procedure contributes what to a mutated quiver. We will simply show a quiver, state which vertex we mutate at, and then show the resulting mutated quiver.
If we take the last quiver and keep mutating at the vertex in the lower left corner of the commutative square, we can push the square as far to the right as we want.
Eventually, the commutative square is pushed all the way to the end of the quiver, and using the mutation procedure it is easy to check that nothing strange happens with the square at the right end of the quiver.
We end up with the following quiver (recall that we are looking at $A_{2n,3}$).
\begin{equation*}
\begin{tikzcd}
\cdots \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-6} \arrow[r] \arrow[rrr, bend left, dashed] & _{(2n-7)^*} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-4} \arrow[r] \arrow[rrr, bend left, dashed] & _{(2n-5)^*} \arrow[r] \arrow[dr, dashed] & _{2n-2} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{(2n-3)^*} \arrow[d] \\
& & & & & _{2n-1} \arrow[r] & _{2n}
\end{tikzcd}
\end{equation*}
Since the entire left part of this quiver (everything but the commutative square on the end) has the same form as the quiver we started with, we can repeat the same construction to push another commutative square along the quiver. When the new square gets close to the existing square at the right end of the quiver, it is not clear what will happen when we mutate. Mutating at a vertex doesn't affect parts of the quiver that are sufficiently far away (in terms of arrows and relations), but when the two squares are close they might interact in unexpected ways.
So let's see what happens.
Pushing the commutative square to the right as descirbed above, eventually the quiver will look like this (note that we have renumbered the vertices)
\begin{equation*}
\begin{tikzcd}
\cdots \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-10} \arrow[r] \arrow[dr, dashed]& _{2n-9} \arrow[r] \arrow[d] \arrow[dr, dashed] & _{2n-8} \arrow[d] \arrow[dr, dashed] & & & & \\
& & _{2n-7} \arrow[r] \arrow[rrr, bend right, dashed] & _{2n-6} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-5} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-4} \arrow[r] \arrow[dr, dashed] & _{2n-3} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{2n-2} \arrow[d] \\
& & & & & & _{2n-1} \arrow[r] & _{2n} \\
& & & & & & &
\end{tikzcd}
\end{equation*}
We mutate at vertex $2n-7$ using the mutation procedure, and see that we get the following quiver
\begin{equation*}
\begin{tikzcd}
\cdots \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-10} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-9} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-8} \arrow[r] \arrow[dr, dashed] & _{2n-6} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{(2n-7)^*} \arrow[d] \arrow[dr, dashed] & & \\
& & & & _{2n-5} \arrow[r] \arrow[rrr, out=-40, in=150, dashed] & _{2n-4} \arrow[r] \arrow[dr, dashed] & _{2n-3} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{2n-2} \arrow[d] \\
& & & & & & _{2n-1} \arrow[r] & _{2n}
\end{tikzcd}
\end{equation*}
Finally, applying the mutation procedure at vertex $2n-5$ gives us
\begin{equation*}
\begin{tikzcd}
\cdots \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-10} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-9} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-8} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-6} \arrow[r] \arrow[rrr, bend left, dashed] & _{(2n-7)^*} \arrow[r] \arrow[dr, dashed] & _{2n-4} \arrow[d] \arrow[r] \arrow[dr, dashed] \arrow[dd, bend right, dashed] & _{(2n-5)^*} \arrow[d] \arrow[dd, bend left, dashed] \\
& & & & & & _{2n-3} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{2n-2} \arrow[d] \\
& & & & & & _{2n-1} \arrow[r] & _{2n}
\end{tikzcd}
\end{equation*}
So the new commutative square slides in on top of the old one.
Again, the left part of the quiver is like when we started, so as before we can repeat this construction. Now, we only need to check the last mutation, because the process of moving the square to the right is identical to before.
\begin{equation}
\begin{tikzcd}
\cdots \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-12} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-11} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-10} \arrow[r] \arrow[dr, dashed] & _{2n-9} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{2n-8} \arrow[d] \arrow[dr, dashed] & & \\
& & & & _{2n-7} \arrow[r] \arrow[rrr, out=-40, in=150, dashed] & _{2n-6} \arrow[r] \arrow[dr, dashed] & _{2n-5} \arrow[d] \arrow[r] \arrow[dr, dashed] \arrow[dd, bend right, dashed] & _{2n-4} \arrow[d] \arrow[dd, bend left, dashed] \\
& & & & & & _{2n-3} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{2n-2} \arrow[d] \\
& & & & & & _{2n-1} \arrow[r] & _{2n}
\end{tikzcd}
\end{equation}
\begin{equation*}
\begin{tikzcd}
\cdots \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-12} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-11} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-10} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-9} \arrow[r] \arrow[rrr, bend left, dashed] & _{2n-8} \arrow[r] \arrow[dr, dashed] & _{2n-6} \arrow[d] \arrow[r] \arrow[dd, bend right, dashed] \arrow[dr, dashed] & _{(2n-7)^*} \arrow[d] \arrow[dd, bend left, dashed] \\
& & & & & & _{2n-5} \arrow[d] \arrow[r] \arrow[dr, dashed] \arrow[dd, bend right, dashed] & _{2n-4} \arrow[d] \arrow[dd, bend left, dashed] \\
& & & & & & _{2n-3} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{2n-2} \arrow[d] \\
& & & & & & _{2n-1} \arrow[r] & _{2n}
\end{tikzcd}
\end{equation*}
Observe that the mutation doesn't affect the bottom part of the column. This means that, from the perspective of a new square that has been pushed along the quiver to the position right before the column, the situation is identical to before the last mutation.
Because one mutation has a limited range of influence in the quiver, the same is true no matter how tall the "commutative column" is. Hence we can move all the vertices in the quiver into the column part, by first creating a commutative square at the beginning of the quiver, pushing it to the right until it becomes part of the column, and repeat. When we get to the beginning of the quiver, we can see that it all wraps up nicely:
\begin{multicols}{2}
\begin{equation*}
\begin{tikzcd}
_{1} \arrow[r] \arrow[rrr, bend left, dashed] & _{2} \arrow[r] \arrow[dr, dashed] & _3 \arrow[d] \arrow[r] \arrow[dr, dashed] \arrow[dd, bend right, dashed] & _{4} \arrow[d] \arrow[dd, bend left, dashed] \\
& & _5 \arrow[d] \arrow[r] \arrow[dd, bend right, dashed] \arrow[dr, dashed] & _{6} \arrow[d] \arrow[dd, bend left, dashed] \\
& & _{7} \arrow[d] \arrow[r] \arrow[dr, dashed] \arrow[dd, bend right, dashed] & _{8} \arrow[d] \arrow[dd, bend left, dashed] \\
& & _{9} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{10} \arrow[d] \\
& & \vdots & \vdots
\end{tikzcd}
\end{equation*}
\begin{equation*}
\begin{tikzcd}
_{2} \arrow[r] \arrow[d] \arrow[dd, bend right, dashed] \arrow[dr, dashed] & _{1^*} \arrow[d] \arrow[dd, bend left, dashed] \\
_3 \arrow[d] \arrow[r] \arrow[dr, dashed] \arrow[dd, bend right, dashed] & _{4} \arrow[d] \arrow[dd, bend left, dashed] \\
_5 \arrow[d] \arrow[r] \arrow[dd, bend right, dashed] \arrow[dr, dashed] & _{6} \arrow[d] \arrow[dd, bend left, dashed] \\
_{7} \arrow[d] \arrow[r] \arrow[dr, dashed] \arrow[dd, bend right, dashed] & _{8} \arrow[d] \arrow[dd, bend left, dashed] \\
_{9} \arrow[d] \arrow[r] \arrow[dr, dashed] & _{10} \arrow[d] \\
\vdots & \vdots
\end{tikzcd}
\end{equation*}
\end{multicols}
Thus, we have an explicit chain of right tilting mutations which allows us to transform the quiver $A_{n,3}$ into the quiver consisting of a column of commutative squares with relations of length $2$ along the outsides.
This in turn tells us that their respective path algebras are derived equivalent.
\printbibliography
\Addresses
\end{document} |
\begin{document}
\title{Anti-Parity-Time Symmetry in a Single Damping Mechanical Resonator}
\author{Xun-Wei Xu}
\email{[email protected]}
\affiliation{Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Key Laboratory for Matter Microstructure and Function of Hunan Province, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China}
\author{Jie-Qiao Liao}
\email{[email protected]}
\affiliation{Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Key Laboratory for Matter Microstructure and Function of Hunan Province, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China}
\author{Hui Jing}
\email{[email protected]}
\affiliation{Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Key Laboratory for Matter Microstructure and Function of Hunan Province, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China}
\author{Le-Man Kuang}
\email{[email protected]}
\affiliation{Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Key Laboratory for Matter Microstructure and Function of Hunan Province, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China}
\date{\today}
\begin{abstract}
A damping mechanical resonator undergos a phase transition from a oscillatory motion with damping amplitude (under-damping) to a monotonically damping motion without oscillation (over-damping) across a critical-damping state.
However, what kind of symmetry is broken for this phase transition in the damping mechanical resonator is still unclear.
Here we discover a hidden symmetry, i.e., anti-parity-time (anti-$\mathcal{PT}$) symmetry, in the effective Hamiltonian of a damping mechanical resonator.
We show that the broken of anti-$\mathcal{PT}$ symmetry with a exceptional point (EP) yields the phase transition between different damping behaviors, i.e., the over-damping and under-damping across a critical-damping.
We propose that the mechanical anti-$\mathcal{PT}$ phase transition can be induced by the optical spring effect in a quadratic optomechanical system with a strong driving field, and highly sensitive optomechanical sensing can be realized around the EPs for anti-$\mathcal{PT}$ phase transition.
\end{abstract}
\maketitle
\section{Introduction}
Damping oscillation is one of the most fundamental and important physical processes~\cite{Waves2014}, for such behavior appears in almost all kinds of systems, such as electronic, atomic, mechanical (acoustic) and optical (photonic) resonators.
It is well known that a damping resonator undergos a phase transition from the oscillatory behavior with a damping amplitude (under-damping) to a monotonically damping without oscillation (over-damping) across a critical-damping state with the damping rate as much as twice the frequency of the resonator.
But, there is no discussion on whether there is a kind of symmetry that is broken for this phase transition in the damping resonator.
According to the quantum theory in open systems~\cite{Breuer2007}, the damping effect can be described simply with a damping rate by eliminating the reservoir coupling to the resonator adiabatically, thus the symmetry in the damping oscillator should be discussed based on a non-Hermitian Hamiltonian with damping rate included.
Non-Hermitian physics has attracted intense interest in the past decades due to the entirely real spectrum in systems with Parity-time ($\mathcal{PT}$) symmetry ~\cite{BenderPRL98,BenderRPP07,Konotop2016RMP,El-Ganainy2018NatPh,WangHF2021JOpt} and the phase transition with an exceptional point (EP) in the parameter space~\cite{MiriSCI19}. With a significant progress in experiment, $\mathcal{PT}$ symmetry has been demonstrated in photonic~\cite{GuoAPRL09,Christian2010NatPh,OzdemirNM2019,MiriSCI19,Klauck2019NaPho}, acoustic~\cite{Bender2013AmJPh,ZhuXFPRX14,FleuryNC15,DingKPRX16}, electronic~\cite{Schindler2011PRA,SchindlerJPA2012,YangXPRL2022}, magnonic~\cite{LiuHL2019SciA}, atomic~\cite{WangWCPRA21,DingLYPRL21}, and single-spin~\cite{WuY2019Sci} systems.
Non-Hermitian systems showcase a variety of features that may not be available in Hermitian counterparts, such as directional invisibility~\cite{LinZPRL11,Peng014NatPh,Chang2014NaPho,LiuTPRL18}, giant enhancement of mechanical gain~\cite{JingPRL14,LvXYPRL15}, parameter sensing~\cite{LiuZPPRL16,ChenWNature17,HodaeiNat17}, single-mode lasing~\cite{Feng2014Sci}, loss-induced suppression and revival of lasing~\cite{PengB2014Sci}, coherent perfect absorption~\cite{SunY2014PRL}, and robust wireless power transfer~\cite{Assawaworrarit2017Nat,DongZNE19}.
However, the requirement of balanced gain and loss has hindered the possibilities of exhibiting $\mathcal{PT}$ symmetry in a single damping resonator.
Anti-$\mathcal{PT}$ symmetry is another non-Hermitian symmetry~\cite{LiGPRA13} has been a subject of considerable recent theoretical and experimental interest for it can be used to achieve exotic functionalities in pure dissipative systems, such as unidirectional reflectionless~\cite{WuJHPRL14}, nanoparticle sensing~\cite{ZhangNanoLett20}, topological energy transfer~\cite{XuHTNat2016}.
A tremendous effort has also been witnessed in achieving
anti-$\mathcal{PT}$ symmetry, which has been demonstrated by
using dissipative coupled systems, including atomic systems~\cite{WuJHPRA15,PengP2016NatPh,JiangYPRL19}, electrical circuits~\cite{Choi2018NatCo}, thermal materials~\cite{LiY2019Sci}, optical devices~\cite{ZhangLight2019LSA,LiQOpt2019Optic,LaiYH2019Natur,PengZH2020PRA,Bergman2021NatCo}, mechanical~\cite{Zhang2021arXiv} and magnonic systems~\cite{ZhaoJPRAPP20}. Theoretically, anti-$\mathcal{PT}$ symmetry can also be realized in a single mode system containing parametric (nonlinear) driving~\cite{WangYXPRA19,ZhangXHPRA21}.
Nevertheless, the possibility of achieving anti-$\mathcal{PT}$ symmetry in a single damping linear resonator has not been reported yet.
Here, taking mechanical system as a simple example, we find that the effective Hamiltonian of a damping mechanical resonator without any nonlinearity is anti-$\mathcal{PT}$ symmetric, and the broken of anti-$\mathcal{PT}$ symmetry induces the phase transition from the over-damping to under-damping states, and the EP corresponds to the critical-damping state.
This result is general, which can be applied to any resonant systems.
We propose that anti-$\mathcal{PT}$ phase transition can be observed in a single damping mechanical resonator by optical string effect in a optomechanical system, which provides us a ideal platform to modify the the spring constant as well as the resonant frequency of a damping mechanical resonator through radiation pressure or gradient force~\cite{Aspelmeyer2014RMP}.
Moreover, we show that anti-$\mathcal{PT}$ symmetry in a damping mechanical resonator can be used for ultra-sensitive sensing for the frequency splitting around the EP is highly sensitive to the external perturbation, such as the change of mass induced by small nano-particles. Our work provides an alternative method to achieve ultra-sensitive sensing based on the EPs for the anti-$\mathcal{PT}$ phase transition in a single damping mechanical resonator.
\section{Mechanical anti-$\mathcal{PT}$ symmetry}
\begin{figure}
\caption{(Color online) Three states of a damping mechanical resonator: (a) under-damping ($\gamma _{m}
\label{fig1}
\end{figure}
We consider a mechanical resonator with mass $m$ and spring constant $k$,
which can be described by a Hamiltonian $H_{m}=p^2/(2m)+kx^2/2$,
with the displacement $x$ and momentum $p$.
The resonant frequency of the mechanical resonator is $\omega _{m}=\sqrt{k/m}$, and $H_{m}$ can be rewritten as $H_{m}=\hbar\omega _{m}\left( Q^{2}+P^{2}\right)/2$ with the dimensionless displacement $Q=x/\sqrt{\hbar/(m\omega _{m})}$ and momentum $P=p/\sqrt{\hbar m \omega _{m}}$.
The equations of motion for the mean values of dimensionless displacement $\langle Q \rangle$ and
momentum $\langle P \rangle$ are given by $d\langle Q \rangle /dt=\omega _{m}\langle P \rangle$ and $d\langle P \rangle/dt=-\omega _{m}\langle Q \rangle-\gamma _{m}\langle P \rangle$, where $\gamma _{m}$ is damping rate induced by the coupling to the thermal reservoir~\cite{Bowen2015Book}.
It is well known that the damping mechanical resonator exhibits under-damping, critical-damping, and over-damping states, corresponding to $\omega _{m}>\gamma _{m}/2$, $\omega _{m}=\gamma _{m}/2$, and $\omega _{m}<\gamma _{m}/2$.
The dynamic behaviors for these three different states are shown in Figs.~\ref{fig1}(a)-\ref{fig1}(c), respectively.
Here, we find that the effective Hamiltonian of a damping mechanical resonator is anti-$\mathcal{PT}$ symmetric, and the over-, under- and critical-damping states are associated with the phases of anti-$\mathcal{PT}$ symmetry, anti-$\mathcal{PT}$ broken, and EP, respectively.
To reveal the anti-$\mathcal{PT}$ symmetry in a damping mechanical resonator, we rewrite the mechanical displacement $Q$ and
momentum $P$ as $Q=\left( b^{\dag }+b\right) /\sqrt{2}$ and $P=i\left( b^{\dag }-b\right) /\sqrt{2}$, then the equations of motion are rewritten as $i d \left (\langle b\rangle, \langle b^{\dag }\rangle\right)^{T}/dt =H_{\mathrm{eff}}\left( \langle b\rangle, \langle b^{\dag }\rangle\right)^{T}$, with the effective Hamiltonian
\begin{equation}\label{eq1}
\frac{H_{\mathrm{eff}}}{\hbar}=\left(
\begin{array}{cc}
\omega _{m}-i\frac{\gamma _{m}}{2} & i\frac{\gamma _{m}}{2}
\\
i\frac{\gamma _{m}}{2} & -\omega _{m}-i\frac{\gamma _{m}}{2}
\end{array}
\right).
\end{equation}
It is easy to verify that this Hamiltonian Eq.~(\ref{eq1}) is anti-$\mathcal{PT}$-symmetric, i.e., $(\mathcal{PT})H_{\mathrm{eff}}(\mathcal{PT})^{-1}=-H_{\mathrm{eff}}$, with the parity operation $\mathcal{P}\equiv \sigma_x$ for switching $b \leftrightarrow b^{\dag }$, and the time-reversal operation $\mathcal{T}$ for complex conjugation.
The eigenvalues of the effective Hamiltonian $H_{\mathrm{eff}}/\hbar$ are given by
\begin{equation}\label{eq2}
\lambda _{\pm }=-i\frac{\gamma _{m}}{2}\pm i\sqrt{\left( \frac{\gamma _{m}
}{2}\right) ^{2}-\omega _{m}^{2}},
\end{equation}
corresponding to eigenstate $\Psi _{\pm }=(\beta _{\pm },\beta _{\pm }^{\prime
})^{T}$, with coefficients
\begin{equation}\label{eq3}
\frac{\beta _{\pm }}{\beta _{\pm }^{\prime }}=-i\frac{2\omega _{m}}{\gamma
_{m}}\pm \sqrt{1-\left( \frac{2\omega _{m}}{\gamma _{m}}\right) ^{2}}.
\end{equation}
As shown in Figs.~\ref{fig1}(d) and \ref{fig1}(e), there are two different phases for the damping mechanical resonator with phase transition occurring at $\gamma _{m}/\omega _{m}=2$.
In the anti-$\mathcal{PT}$-symmetric phase with $\gamma _{m}/\omega _{m}>2$, $\Psi _{\pm }$ is also the eigenstates of the parity-time operator $\mathcal{PT}$, corresponding to the well known over-damping state for $\omega _{m}<\gamma _{m}/2$.
In the anti-$\mathcal{PT}$ broken phase with $\gamma _{m}/\omega _{m}<2$, $\Psi _{\pm }$ is no longer the eigenstates of the parity-time operator $\mathcal{PT}$, corresponding to the well known under-damping state for $\omega _{m}>\gamma _{m}/2$.
Moreover, the critical point $\gamma _{m}/\omega _{m}=2$ for anti-$\mathcal{PT}$ phase transition, i.e., EP, corresponds to the well known critical-damping state for $\omega _{m}=\gamma _{m}/2$.
The correspondence between the dynamic states and the sorts of symmetry for a damping mechanical resonator is demonstrated in Fig.~\ref{fig1}(f).
As the symmetry of the damping resonator depends on the rate of $\gamma _{m}/\omega _{m}$, anti-$\mathcal{PT}$ phase transition can be realized by adjusting the decay rate $\gamma_m$ or the resonant frequency $\omega_{m}$ of the damping mechanical resonator.
\section{Optomechanical induced anti-$\mathcal{PT}$ phase transition}
Optomechanical system, that a mechanical resonator coupling to a optical mode through radiation pressure or gradient force~\cite{Aspelmeyer2014RMP}, provides us a ideal platform to observe anti-$\mathcal{PT}$ phase transition via regulating the resonant frequency $\omega_{m}$ of a mechanical resonator based on optical spring effect.
Specifically, we consider a mechanical resonator quadratically coupled to an optical mode ($A$ and $A^{\dag }$, with frequency $\omega _{c}$) through optomechanical interaction, and the optical mode is driven resonantly by an external field with the strength $\Omega $ and frequency $\omega _{L}=\omega _{c}$. In the
rotating reference frame with frequency $\omega _{L}$ of the external
optical driving field, the system can be described by a Hamiltonian
\begin{equation}
H_{\rm OM}=\frac{p}{2m}+\frac{1}{2}(k+ 4m \omega_m g A^{\dag }A)x^2+\hbar\Omega \left( A^{\dag }+A\right),
\end{equation}
where $ 4m \omega_m g A^{\dag }A$ is the spring effect induced by the optical mode with the quadratic optomechanical coupling strength $g$.
The quadratic optomechanical interaction has been demonstrated in various cavity-optomechanical systems, including
mechanical resonator (membrane~\cite{ThompsonNat08,Sankey2010NatPh,Karuza2012JOP}, nanosphere~\cite{Fonseca2016PRL,Uros2019PRL,Bullier2021PRR}, cold atoms~\cite{Purdy2010PRL}) trapped in Fabry-Perot cavities,
or coupled to whispering-gallery-mode~\cite{Hill2013PHD,Doolin2014PRA,Brawley2016NatCo} or photonic crystal cavities~\cite{Kaviani2015Optica,TaofiqPRX15}.
Without loss of generality, in calculations, we will take the experimental parameters in a planar silicon photonic crystal optomechanical cavity~\cite{TaofiqPRX15} with quadratic optomechanical coupling strength $g/2\pi=245$ Hz, mechanical resonance frequency $\omega _{m}/2\pi=8.7$ MHz (quality factor $Q_{m} = 10^4$), and optical damping rates $\gamma _{c}/2\pi = 5$ GHz, i.e., the system works in the sideband unresolved regime with $\gamma _{c}\gg \omega _{m}$.
Radiation pressure-induced buckling transitions have been reported in the optomechanical system with a dielectric membrane in the
middle of a symmetrical optical cavity~\cite{XuHT2017NatCo}.
These transitions can be understood from the effective potential
energy of the mechanical mode which changes smoothly from a single-well
to a multi-well potential with the increasing of the optical driving power.
The mean values in the steady states (i.e., evolution time $t \gg 1/\gamma_c$) $\left\langle A\right\rangle =\alpha_s $, $\left\langle P\right\rangle =P_{s}$, and $\left\langle Q\right\rangle =Q_{s}$ can be obtained analytically from the dynamic equations (See Ref.~\cite{Seok2013PRA} for more details). Two different results appear when the
coupling $g$ is positive $g>0$ or negative $g<0$. If $g>0$, we always have $\alpha_s =-i2\Omega /\gamma _{c}$ and $ Q_{s}=0$. If $g$ is negative $g<0$, then we have $\alpha_s =-i2\Omega /\left( \gamma _{c}+i4gQ_{s}^{2}\right) $ and
\begin{equation}
Q _{s}^{2}=\left\{
\begin{array}{cc}
0 & \Omega \leq \Omega _{c} ,\\
\sqrt{-\frac{1}{g\omega _{m}}\left( \Omega ^{2}-\Omega_{c} ^{2}\right) } & \Omega >\Omega _{c},
\end{array}
\right.
\end{equation}
with the appearance of spontaneous symmetry broken (SPB) phase transition at the critical driving strength
$\Omega _{c}=\sqrt{-\gamma _{c}^{2}\omega _{m}/(16g)}$, as shown Fig.~\ref{fig2}(a).
The effective potential energy of the mechanical mode can be obtained analytically by eliminated the optical freedom adiabatically under the conditions that $\omega _{m}\ll \gamma _{c}$, as $
U_{_{\mathrm{eff}}}=\frac{1}{2}\omega _{m}\left\langle Q\right\rangle ^{2}+
\frac{2\Omega ^{2}}{\gamma _{c}}\arctan \left( \frac{4g\left\langle
Q\right\rangle ^{2}}{\gamma _{c}}\right)$.
Here the second term describes the optical spring effect induced by the quadratic optomechanical interaction, which is negative for negative $g$.
Then under the critical driving for $0<\Omega \leq \Omega _{c}$, the effective spring constant $k_{\rm eff}=k(1-\Omega^2 /\Omega _{c}^2)$ monotonously decreases (i.e., the spring softens) with the driving strength $\Omega$.
When the driving strength $\Omega$ exceeds the critical point (CP) $\Omega _{c}$, the effective potential
energy of the mechanical mode becomes a double well, i.e., the parity symmetry of the ground states of the mechanical resonator is broken when $\Omega > \Omega _{c}$.
Apart from SPB phase transition, here we propose the observation of mechanical anti-$\mathcal{PT}$ phase transitions in both the regimes of under and over the critical driving $\Omega_c$.
\begin{figure}
\caption{(Color online) (a) Mean value of the mechanical displacement operator in the steady states $Q_s$ versus the driving strength $\Omega$. Eigenvalues of the effective mechanical Hamiltonian $\lambda_{m,\pm}
\label{fig2}
\end{figure}
Under the critical driving for $0<\Omega \leq \Omega _{c}$, as the effective frequency $\omega _{{\rm eff}}=\sqrt{k_{\rm eff}/m}$ with the effective spring constant $k_{\rm eff}=k(1-\Omega^2 /\Omega _{c}^2)$,
the first EP (EP1) appears around the point $\omega _{\rm eff}=\gamma_m/2$ with the optical driving strength
\begin{equation}
\Omega _{\mathrm{EP1}}^{2}=\Omega _{c}^2\left[1-\left(\frac{\gamma_m}{2\omega_m}\right)^2\right].
\end{equation}
EP1 corresponds to the transition from anti-$\mathcal{PT}$-broken to anti-$\mathcal{PT}$-symmetric phase.
EP1 can also be obtained from the eigenvalues of the effective Hamiltonian Eq.~(\ref{eq2}) with $\omega_m$ replaced by $\omega _{{\rm eff}}$, as
\begin{equation}
\lambda _{\pm }=-i\frac{\gamma _{m}}{2}\pm i\sqrt{\left( \frac{\gamma _{m}
}{2}\right) ^{2}-\omega _{m}^{2}\left(1-\frac{\Omega^2}{\Omega _{c}^2}\right)}.
\end{equation}
The eigenvalues are shown in Figs.~\ref{fig2}(b) and \ref{fig2}(c), which is consistent well with the analytical results.
Over the critical driving for $\Omega > \Omega _{c}$, the position of $Q_{s}=0$ become unstable [see the dashed curves in Fig.~\ref{fig2}(c)], and two new stable positions $Q_{s} = \pm [\left( \Omega_{c} ^{2}-\Omega ^{2}\right)/(g\omega _{m})]^{1/4}$ appear. Around the new stable positions, we have the effective spring constant $k_{\rm eff}=4 k(1-\Omega _{c}^2 /\Omega^2)$, and the eigenvalues of the effective Hamiltonian
\begin{equation}
\lambda _{\pm }=-i\frac{\gamma _{m}}{2}\pm i\sqrt{\left( \frac{\gamma _{m}
}{2}\right) ^{2}-4\omega _{m}^{2}\left(1-\frac{\Omega_{c}^2}{\Omega^2}\right)}.
\end{equation}
So the driving strength for the second EP (EP2) in the SPB regime can be given by
\begin{equation}
\Omega _{\mathrm{EP2}}^{2}=\Omega _{c}^2\left[1-\left(\frac{\gamma_m}{2\omega_m}\right)^2\right]^{-1}.
\end{equation}
Different from the EP1, EP2 is corresponding to the transition from anti-$\mathcal{PT}$-symmetric to
anti-$\mathcal{PT}$-broken phase as the driving power increases.
The frequency bifurcates around the EPs in the mechanical anti-$\mathcal{PT}$ system provides us a sensitive way to detect the small variations of the parameters, such as the frequency shift of the mechanical resonator $\omega_{m}$ caused by external perturbation, which can be used for ultra-sensitive sensing, such as the mass sensing of small nano-particles.
\begin{figure}
\caption{(Color online) (a) The sensitivity $|d\omega_{m,\pm}
\label{fig3}
\end{figure}
\section{Optomechanical anti-$\mathcal{PT}$ sensor}
High sensitivity is a long-term pursue goal due to the vital importance in both fundamental and applied physics.
We noted that ultrasensitive sensing based on the EP have been studied both theoretically~\cite{ZhangNanoLett20,MaoX2020NJPh,LiT2021PRA,Djorwe2019PAAPP} and experimentally~\cite{ChenWNature17,LaiYH2019Natur}. However, the proposal of ultrasensitive nanoparticle sensing based on the EPs in a single damping mechanical resonator has not reported yet.
Without loss of generality, we consider ultrasensitive sensing on the frequency perturbation of the mechanical resonator $\omega_m^{\prime}=\omega_m+\delta$, which is one of the most important parameters for optical sensing.
The sensitivity of the frequency splitting $\omega_{m,\pm }$ on the mechanical resonator $\omega_m$ can be described by the derivative of $ \omega_{m,\pm }$ with respect to $\omega_m$ as
\begin{equation}
\frac{d\omega _{m,\pm }}{d\omega _{m}}=\left\{
\begin{array}{ll}
\pm \frac{\omega _{m}}{\omega _{m,\pm }}\left( 1-\frac{\Omega ^{2}}{2\Omega
_{c}^{2}}\right) & \Omega <\Omega _{\mathrm{EP1}}, \\
0 & \Omega _{\mathrm{EP1}}<\Omega <\Omega _{\mathrm{EP2}}, \\
\pm \frac{4\omega _{m}}{\omega _{m,\pm }}\left( 1-\frac{3}{2}\frac{\Omega
_{c}^{2}}{\Omega ^{2}}\right) & \Omega >\Omega _{\mathrm{EP2}}.
\end{array}
\right.
\end{equation}
The sensitivity $|d\omega _{m,\pm }/d\omega _{m}|$ are shown in Fig.~\ref{fig3}(a). It is clear that the sensitivity is enhanced sharply as the driving strength coming close to the EPs, and it becomes divergent at the two EPs as $|\omega _{m,\pm }|\rightarrow 0$.
As the frequency splitting $\omega_{m,\pm }$ dependencies on the value of the frequency perturbation $\delta$, we show the dependence of frequency splitting $\omega _{m,\pm }$ on the frequency perturbation $\delta$ in Fig.~\ref{fig3}(b), corresponding to EP1 (dashed curves) and EP2 (solid curves). Around the two EPs, the frequency splitting $\omega _{m,\pm }$ is given by
\begin{equation}
\frac{\omega _{m,\pm }}{\gamma _{m}}\approx \left\{
\begin{array}{cc}
\pm \sqrt{\frac{\omega _{m}}{\gamma _{m}}}\sqrt{\frac{\delta }{\gamma _{m}}}
& \Omega =\Omega _{\mathrm{EP1}}, \\
\pm 2\sqrt{\frac{\omega _{m}}{\gamma _{m}}}\sqrt{\frac{-\delta }{\gamma _{m}}
} & \Omega =\Omega _{\mathrm{EP2}}.
\end{array}
\right.
\end{equation}
As $\omega _{m,\pm }/\gamma _{m}$ depends on the square root of $\delta/\gamma _{m}$, the sensitivity can be enhanced sharply when $|\delta/\gamma _{m}| \ll 1$.
Moreover, there is an enhancement factor $\sqrt{\omega _{m}/\gamma _{m}}$ in front of the frequency perturbation $\delta$, so that the sensitivity becomes even higher for a mechanical resonator with higher quality.
Making the effect of EP enhancement more clearly, we can compare the frequency splitting $\omega _{m,\pm }$ for the driving strength around the EPs with the results for the driving strength far away from the EPs. The frequency splitting $\omega _{m,\pm }$ for the driving strength far away from the EPs are given approximately as
\begin{equation}
\omega _{m,\pm }\approx \left\{
\begin{array}{cc}
\pm \left( \omega _{m}+\delta \right) & \Omega \ll \Omega _{\mathrm{EP1}},
\\
\pm 2\left( \omega _{m}+\delta \right) & \Omega \gg \Omega _{\mathrm{EP2}}.
\end{array}
\right.
\end{equation}
There are two differences in expressions of the frequency splitting $\omega _{m,\pm }$ for the driving strength close to and far away from the EPs. First, $ \omega _{m,\pm }$ depends on the square root of the frequency perturbation $\delta$ for the driving strength close to the EPs, while $\omega _{m,\pm }$ is linearly dependent on the frequency perturbation $\delta$ for the driving strength far away from the EPs. Second, there is an enhancement factor $\sqrt{\omega _{m}/\gamma _{m}}$ in the expressions of $\omega _{m,\pm }$ for the driving strength close to the EPs.
Based on these two points, ultra-sensitive sensing can be realized based on the frequency splitting around the EP for the driving strength close to the EPs.
As a specific application, let us consider the detection of the mass of a nanoparticle as a simple example for ultra-sensitive sensing based on the EPs.
Consider a nanoparticle with mass $m_p$ adheres to the mechanical resonator, then the frequency of the mechanical resonator is shifted from $\omega_m$ to $\omega'_m = \omega_m+\delta$ with $\delta \approx -\frac{\omega _{m}}{2}\frac{m_{p}}{m}$.
With a driven strength of the driving field close to the EP2, i.e., $\Omega \approx \Omega _{\mathrm{EP2}}$, we have
\begin{equation}
\frac{\omega _{m,\pm }}{\gamma _{m}}\approx \pm \sqrt{2}\frac{\omega _{m}}{
\gamma _{m}}\sqrt{\frac{m_{p}}{m}}.
\end{equation}
The frequency splitting can be resolved, i.e., $|\omega _{m,+}-\omega _{m,-}|>\gamma _{m}$, when the mass of the nanoparticle $m_{p}/m \geq 1/(8 Q_m^{2})$ for $Q_m\equiv\omega_m/\gamma _{m}$.
Taking the experimental parameters~\cite{TaofiqPRX15}: mechanical mass $m=3.6$ pg and Q-factor $Q_m=7\times 10^5$, we can detect a nanoparticle with mass as small as $m_{p} \approx 1\times10^{-24}$ g, which is much smaller than the mass sensitivity ever reported in the optomechanical systems~\cite{LI2013223PR,Liu2013OE,HeY2015ApPhL,YuW2016NatCo,LinQ2017PhRvA,LiuSP2019PRA,Sansa2020NatCo}.
\section{Discussions and conclusions}
In summary, we discovered the inherent anti-$\mathcal{PT}$ symmetry in a damping mechanical resonator, which provided new perspectives on the dynamic behaviors of a damping mechanical resonator, and similar results can be observed in any resonant systems.
Our work indicates that anti-$\mathcal{PT}$ symmetry is commonly presented and can be realized in a simple damping resonator, without deliberate design as done in previous studies.
Damping resonator with anti-$\mathcal{PT}$ symmetry could be a useful element for more complex applications, such as high-order EPs~\cite{Jing2017NatSR,LiuYL2017PRA,esee8c365,ZhangSM2020PRA} and non-Hermitian topological phases~\cite{Bergholtz2021RMP}.
In addition, we proposed to observe mechanical anti-$\mathcal{PT}$ phase transition by optical spring effect in a quadratic optomechanical system, and the frequency
bifurcation around the EPs in the mechanical anti-$\mathcal{PT}$ symmetric system provides us a effective way for ultra-sensitive sensing, such as mass sensing.
\emph{{\color{blue}Acknowledgement}.---}X.-W.X. was supported by the National Natural Science Foundation of China (NSFC)
(Grant No.~12064010), and Natural Science Foundation of Hunan Province of
China (Grant No.~2021JJ20036).
J.-Q.L. was supported in part by the NSFC (Grants No. 12175061
and No. 11935006) and the Science and Technology Innovation Program of Hunan Province (Grants No.
2021RC4029, No. 2017XK2018, and No. 2020RC4047).
H.J. was supported by the NSFC (Grants No. 11935006 and No. 11774086) and the Science and Technology Innovation
Program of Hunan Province (Grant No. 2020RC4047).
L.-M.K. was supported by the NSFC (Grants No. 1217050862, 11935006 and No. 11775075) and the Science and Technology Innovation
Program of Hunan Province (Grant No. 2020RC4047).
\end{document} |
\begin{document}
\title{{f Centers of reversible cubic perturbations of the symmetric 8-loop Hamiltonian }
\begin{abstract}
We show that the center set of reversible cubic systems, close to the symmetric Hamiltonian system $x'=y, y'= x-x^3$ has two irreducible components of co-dimension two in the parameter space. One of them corresponds to the Hamiltonian stratum, the other to systems which are polynomial pull back of an appropriate linear system
\\
{\it Keywords:} center-focus problem, cubic vector field.
\end{abstract}
\section{Introduction}
\label{section1}
The paper is a contribution to the study of the center set of plane polynomial cubic vector fields, in a neighbourhood of the symmetric cubic vector field, see Fig.\ref{Fig1},
\begin{eqnarray}\label{nonperturbed}
X_{0}:
\left\{\begin{array}{ccl} \dot{x}&=&y \\
\dot{y}&=& x-x^{3}
\end{array}\right.
\end{eqnarray}
$X_0$ is Hamiltonian, and has a first integral
\begin{equation*}
H(x,y)= \frac{1}{2}y^{2}- \frac{1}{2}x^{2} +\frac{1}{4}x^{4}.
\end{equation*}
Under a small analytic perturbation the centres of $X_0$ near $(1,0)$ and $(-1,0)$ are either simultaneously destroyed, or simultaneously persistent. The set of cubic vector fields close to $X_0$ and having
a center near $(\pm1,0)$ is
the \emph{center set} $\mathcal C_0$. It is known that the center set is an algebraic set in the space of parameters, but even the number of its irreducible components is unknown.
Recently Iliev, Li and Yu \cite{Iliev} studied special one-parameter families of perturbations of the form
\begin{eqnarray*}
X_{\varepsilon}:
\left\{\begin{array}{ccl} \dot{x}&=&y + \varepsilon P(x,y) \\
\dot{y}&=& x-x^{3} + \varepsilon Q(x,y)
\end{array}\right.
\end{eqnarray*}
where $P,Q$ are arbitrary \emph{fixed} real cubic polynomials. The displacement function near the singular points $(\pm1,0)$ has an analytic expansion
\begin{align}
d(h,\varepsilon)= \varepsilon M_1(h)+ \varepsilon^2 M_2(h) + \varepsilon^3 M_3(h) + \dots
\end{align}
where as usual $h$ is the restriction of the Hamiltonian $H$ on a cross-section to the vector field. The so called Melnikov functions $M_k$ vanish if and only if the displacement map is the zero map, that is to say the the centers $(1,0)$ and $(-1,0)$ are simultaneously persistent.
It was shown then in \cite[Theorem 1]{Iliev} that if $M_1=M_2=M_3=M_4=0$, then the displacement map is identically zero and therefore $X_\varepsilon$ has a center near $(1,0)$ and $(-1,0)$. The set of such system is an algebraic set $\mathcal C^l$ contained in $\mathcal C$. It turns out that $\mathcal C^l$ is a union of vector spaces, and a vector field $X_\varepsilon$ which belongs to an irreducible component of $\mathcal C^l$ is either Hamiltonian, or $y$-reversible, or $x$-reversible.
It is clear that when a vector field $X_\varepsilon$ is Hamiltonian, or $y$-reversible, then it has a center near $(\pm1,0)$. If, however, the vector field is $x$-reversible, it does not follow that it has a center near $(\pm1,0)$. Therefore, it makes a sense to consider the case of $x$-reversible systems separately.
The purpose of this paper is to give complete description of the center set $\mathcal C$ under the restriction that the vector field is $x$-reversible, that is to say, the associated foliation by orbits is invariant under the involution $(x,y)\to (-x,y)$. Our approach is the following. The invariance under $x \to -x$ suggests to introduce the quotient vector field which is stil polynomial and cubic. It turns out, that this new vector field is of Li\'enard type, whose centers were extensively studied. We apply a classical result of \cite[Cherkas] {cherkas} revisited recently by \cite[Chrystopher]{chri99}. As a result we obtain that in this case the center set has two co-dimension two smooth irreducible components which correspond either to Hamiltonian systems, or to systems obtained as polynomial pull back from linear systems. From this, the result of the paper follows.
\begin{figure}
\caption{The phase portrait of (\ref{nonperturbed}
\label{Fig1}
\end{figure}
\begin{figure}
\caption{The phase portrait of the perturbed reversible system \eqref{reversible}
\label{Fig2}
\end{figure}
\section{Statement of the results }
\label{section2}
Consider the following perturbed cubic system
\begin{eqnarray}\label{perturbed}
X_{\lambda}:
\left\{\begin{array}{ccl} \dot{x}&=&y + P(x,y) \\
\dot{y}&=& x-x^{3} + Q(x,y)
\end{array}\right.
\end{eqnarray}
where
$$
P(x,y)= \sum_{i+j\leq 3} a_{ij} x^iy^j, \; Q(x,y)= \sum_{i+j\leq 3} b_{ij} x^iy^j
$$
$\lambda = \{ a_{ij},b_{ks} \}$ are small parameters. For $\lambda=0$ the system $X_0$ has a first integral
\begin{equation*}
H(x,y)= \frac{1}{2}y^{2}- \frac{1}{2}x^{2} +\frac{1}{4}x^{4}
\end{equation*}
and two centers at $(x,y)=(\pm1,0)$ shown on Fig 1.
The perturbed vector field $X_\lambda$ has therefore a saddle, close to the origine $(0,0)$, as well two anti-saddles (centers or foci) close to $(\pm1,0)$.
The anti-saddles near $(\pm1,0)$ are either simultaneously centers, or saddles\footnote{this fact is obvious if the vector field $X_\lambda$ is $x$-reversible, but holds true for arbitrary analytic perturbations too}.
The center set $\mathcal C_0$ of \emph{small} parameters $\lambda$ for which the vector filed has a center near $(\pm1,0)$ is a germ of analytic set in the space of parameters $\lambda$.
The set $\mathcal C_0$ is in fact algebraic, it is globally defined as the zero set of a finite family of polynomials in $\lambda$, but its number of irreducible components is not known in general.
The purpose of the present paper is to describe $\mathcal C_0$ in the particular case, when $X_\lambda$ is reversible in $x$. More precisely
\begin{defi}
The vector field (\ref{perturbed}) is said to be reversible with respect to $x$ provided that the involution $x\mapsto -x$ sends $X_\lambda$ to $-X_\lambda$, or equivalently
$P(-x,y)=P(x,y) $, $Q(-x,y)=-Q(x,y)$. The set of reversible cubic systems (\ref{reversible}) having a center near $(\pm 1, 0)$ is denoted $\mathcal C^R_0$.
\end{defi}
The set of reversible in $x$ vector fields $X_\lambda^R$
\begin{eqnarray}\label{reversible}
X_{\lambda}^R:
\left\{\begin{array}{ccl} \dot{x}&=&y + P^R(x,y) \\
\dot{y}&=& x-x^{3} + Q^R(x,y)
\end{array}\right.
\end{eqnarray}
is therefore parameterised by the space of special cubic polynomials $P^R,Q^R$ of the form
\begin{align}
\label{p}
P^R(x,y) &= a_{00}+a_{20}x^2+a_{21}x^{2}y +a_{01}y+a_{02}y^{2}+ a_{03}y^{3}, \\
\label{q}
Q^R(x,y)&= b_{10}x+b_{11}xy+b_{12}xy^{2}+b_{30}x^{3}
\end{align}
where $\lambda = \{ a_{ij},b_{ks} \}$ are small real parameters.
The possible phase portraits of
the perturbed vector field $X_\lambda^R$ in the finite plane (that is to say, in a disc of finite radius) is shown on Fig.1, where there is some unknown number of limit cycles. We note the orbits which belong to the exterior period annulus are always closed, because the vector field $X_\lambda^R$ in consideration is reversible. We have a strict inclusion $\mathcal C^R_0 \subset \mathcal C_0$ and $\mathcal C^R_0$ is an algebraic set in the parameter space.
We shall prove
\begin{thm}
\label{th0}
The center set $\mathcal C^R_0$ of reversible cubic systems $X_\lambda^R$ with a center near $(\pm1,0)$ has two irreducible components of co-dimension two in the set of polynomials (\ref{p}), (\ref{q}). The components correspond either to Hamiltonian systems or to systems which are obtained as polynomial pull back from an appropriate linear system.
\end{thm}
To describe \emph{explicitly } $\mathcal C^R_0$ we shall normalise first $X_\lambda$ as follows. Note that the affine transformations
$$
(x,y) \mapsto (\alpha x, y + \beta)
$$
transform a reversible cubic system to a reversible cubic system of the same form, and therefore act on the parameter space and the center set $\mathcal C^R_0$.
Therefore, performing an appropriate affine change of $x,y$, we may assume that $X_\lambda^R$ has a singular point at $(1,0)$ for all sufficiently small $\lambda$ and by the $x$-reversibility, it will have another singular point at $(-1,0)$. The normalised vector field (\ref{reversible}) takes the form
\begin{eqnarray}\label{normal}
X_{\lambda}^R:
\left\{\begin{array}{ccl} \dot{x}&=&y +a_{20}(x^{2}-1)+ a_{21}x^{2}y + a_{01}y + a_{02}y^{2}+ a_{03}y^{3} \\
\dot{y}&=& x-x^{3} +b_{30}(x^{3} -x) + b_{11}xy +b_{12}xy^{2}
\end{array}\right.
\end{eqnarray}
(we denote the coefficients of this normalised reversible vector field by the same letters $a_{ij}$).
Theorem \ref{th0} is an obvious consequence of the following
\begin{thm} \label{thm1}
The system (\ref{normal})
has a non-degenerate real center at $(\pm1,0) $ if and only if
\begin{align*} 2a_{20}+b_{11} = 0
\end{align*}
and either
\begin{itemize}
\item $a_{21} + b_{12}=0$ (Hamiltonian case), or
\item $(1-b_{30})a_{02}=a_{20}(2a_{21}- b_{12})$ (pull back case).
\end{itemize}
In the second case the system (\ref{normal}) is a polynomial pull back of a linear system under the map
\begin{align*}
(x,y) \to ( x^2-1+P_2(y), y^2), P_{2}(y)= \frac{1}{b_{30}-1}( b_{11}y +b_{12}y^{2}) .
\end{align*}
\end{thm}
Note that the trace of $X_\lambda^R$ at $(\pm1,0) $ equals $2a_{20}+b_{11}$.
To determine the center conditions of (\ref{normal})
we shall use the Cherkas-Christopher theorem which we explain in the next section.
\section{The Cherkas-Christopher Theorem}
\label{section3}
Consider the plane cubic differential system
\begin{eqnarray}\label{cubic1}
\left\{\begin{array}{ccl} \dot{\xi}&=&P_{3}(y)+ P_{1}(y) \xi \\
\dot{y}&=& -\xi -P_{2}(y)
\end{array}\right.
\end{eqnarray}
where
\begin{align}
P_{1}(y)= a_{0}+ a_{1}y,\; P_{2}(y)=b_{1}y+ b_{2}y^{2}, \; P_{3}(y)=c_{1}y+ c_{2}y^{2}+ c_{3}y^{3} \in \mathbb{R}[y] .
\end{align}
Upon substituting $\xi \to R(x,y)$ in (\ref{cubic1}), where $R$ is a quadratic polynomial, we get a new cubic system ($\ref{cubic1}^*$), which is pull back of (\ref{cubic1}) under the polynomial map $(\xi,y)= (R(x,y),y)$. In such a way centers of (\ref{cubic1}) produce new centers of more general cubic systems.
In this section we determine the necessary and sufficient conditions so that the cubic system (\ref{cubic1})
has a non-degenerate singular point at the origin of center type. We assume therefore through this section, that (\ref{cubic1}) has already a linear center at the origin, that is to say
\begin{align}
a_0-b_1 = 0, c_1 - a_0b_1 >0 .
\end{align}
Substituting $\xi = u-P_{2}(y) $ in (\ref{cubic1}) we get:
\begin{equation*}
udu+\left[ u \left( (a_{0}-b_{1}) + (a_{1}-2b_{2})y \right) +(c_{1}-a_{0}b_{1})y +(c_{2}-a_{0}b_{2}-a_{1}b_{1})y^{2}+(c_{3}-a_{1}b_{2})y^{3} \right] dy=0,
\end{equation*}
which is the polynomial foliation of the Li\'enard equation
\begin{eqnarray}\label{cubic2}
\left\{\begin{array}{ccl} \dot{y}&=&u \\
\dot{u}&=& -q(y)-up(y)
\end{array}\right.
\end{eqnarray}
where
\begin{align}
p(y)&=a_{0}-b_{1} + (a_{1}-2b_{2})y , \;\\ q(y)&=
(c_{1}-a_{0}b_{1})y +(c_{2}-a_{0}b_{2}-a_{1}b_{1})y^{2}+(c_{3}-a_{1}b_{2})y^{3} .
\end{align}
The center set of the above Li\'enard system is well known, see \cite{lubo} and \cite{cherkas}.
It has a center at the origin if and only if the primitives
$P(y)= \int p(y) dy $ and $ Q(y)=\int q(y)dy$ have a common composition factor $W(y)$ with a Morse critical point at the origin. Therefore
$a_{0}-b_{1}=0$, $W=P$ and there exists a degree two polynomial $\tilde Q$, such that $Q(y)= \tilde Q (W(y))$.The latter is equivalent to the condition that
$Q$ is even, when $p\neq 0$. In the case $p=0$ the system is obviously Hamiltonian. We resume the result in the following
\begin{thm}
\label{th3}
The system of Li\'enard type (\ref{cubic1}) has a non-degenerate center at the origin if and only if
\begin{align*}
a_{0}-b_{1}=0, \; c_1 - a_0 b_1>0
\end{align*}
and either
\begin{itemize}
\item $a_1-2b_2=0$ (Hamiltonian case), or
\item $c_{2} -a_{0}(a_{1}+b_{2}) =0 $ (pull back case) .
\end{itemize}
\end{thm}
\section{Proof of Theorem \ref{thm1}}
\label{section4}
The substitution $\xi= x^{2}-1$ takes the system (\ref{normal})
\begin{align*}
(y +a_{20}(x^{2}-1)+ a_{21}x^{2}y + a_{01}y + a_{02}y^{2}+ a_{03}y^{3}) dy
-(x-x^{3} +b_{30}(x^{3} -x) + b_{11}xy +b_{12}xy^{2})dx = 0
\end{align*}
to the Li\'enard form (\ref{cubic1})
\begin{align*}
(\xi +P_{2}(y)) d\xi+ (P_{3}(y)+ P_{1}(y) \xi )dy =0
\end{align*}
where
\begin{align*}
P_{1}(y)&= \frac{2}{1-b_{30}}(a_{20}+ a_{21}y) \\
P_{2}(y)&= \frac{1}{b_{30}-1}( b_{11}y +b_{12}y^{2}) \\
P_{3}(y) &= \frac{2}{1-b_{30}}((1+a_{01}+a_{21})y+a_{02}y^{2}+a_{03}y^{3})
\end{align*}
Applying Theorem \ref{th3} with
\begin{align*}
&a_0= \frac{2a_{20}}{1-b_{30}}, &a_1&=\frac{2a_{21}}{1-b_{30}} & \\
&b_1= -\frac{b_{11} }{1-b_{30}}, &b_2 &= -\frac{b_{12} }{1-b_{30}} & \\
&c_1= \frac{2(1+a_{01}+a_{21})}{1-b_{30}} , &c_2&= \frac{2 a_{02} }{1-b_{30}}, &c_3 = \frac{2 a_{03} }{1-b_{30}}
\end{align*}
we obtain the equation for the center set $\mathcal C_0^R$ for the normalised reversible vector field (\ref{normal}).
Finally, to find the first integral in the logarithmic case, we recall that (\ref{normal}) is a polynomial pull back
under $ u=x^2-1 +P_2(y)$
of the Li\'enard system
\begin{eqnarray}\label{cubic3}
\left\{\begin{array}{ccl} \dot{y}&=&u \\
\dot{u}&=& -
q(y)-up(y)
\end{array}\right.
\end{eqnarray}
where (in the pull back case)
\begin{align*}
p(y)&= (a_{1}-2b_{2})y , \;\\ q(y)&=
(c_{1}-a_{0}b_{1})y +(c_{3}-a_{1}b_{2})y^{3} .
\end{align*}
As (\ref{cubic3}) is also $y$-reversible then it is a pull back of the following linear system
\begin{eqnarray}\label{linear}
\left\{\begin{array}{ccl} \dot{v}&=&2u \\
\dot{u}&=& -(c_{1}-a_{0}b_{1}) - (c_{3}-a_{1}b_{2})v
-u ( a_{1} -2b_{2})
\end{array}\right.
\end{eqnarray}
under the map $y\to v =y^2$. This completes the proof of Theorem \ref{thm1}.
We note finally that he system (\ref{linear}) (as any non-degenerate linear system) has a logarithmic first integral of the form $l_1^{\alpha}l_2^{\beta}$ where $l_1,l_2$ are linear functions in $\lambda,u$ and $\alpha, \beta$ are suitable complex numbers. Although (\ref{linear}) has a Darboux type first integral, it has no center, except in the Hamiltonian case $a_{1} -2b_{2}=0$.
The reversible vector field $X_\lambda^R$ has also a first integral of Darboux type (pull back of the Darboux first integral of the linear system), but its centers near $(\pm1,0)$ are of pull back type. An explicit computation of this integral in some cases can be found in \cite[section 5]{Iliev}.
$\Box$
\end{document} |
\begin{document}
\contourlength{1pt}
\contournumber{32}
\title{The Ribaucour families of discrete R-congruences}
\begin{center}
\begin{minipage}{11cm}\small
\textbf{Abstract.} While a generic smooth Ribaucour sphere congruence admits exactly two envelopes, a discrete R-congruence gives rise to a 2-parameter family of discrete enveloping surfaces. The main purpose of this paper is to gain geometric insights into this ambiguity. In particular, discrete R-congruences that are enveloped by discrete channel surfaces and discrete Legendre maps with one family of spherical curvature lines are discussed.
\end{minipage}
\vspace*{0.5cm}\\\begin{minipage}{11cm}\small
\textbf{MSC 2010.} 53A40 · 53B25 · 37K25 · 37K35
\end{minipage}
\vspace*{0.5cm}\\\begin{minipage}{11cm}\small
\textbf{Keywords.} discrete differential geometry; Lie sphere geometry; Ribaucour transformation; discrete Legendre maps; Lie inversions;
\end{minipage}
\end{center}
\section{Introduction}
\mathfrak{n}oindent Classically, a smooth 2-parameter family of spheres is called \emph{Ribaucour sphere congruence} if its two enveloping surfaces have corresponding curvature lines. Since a smooth sphere congruence admits at most two envelopes, each such sphere congruence provides a \emph{Ribaucour pair of surfaces}. Over the decades, this transformation concept was extended to submanifolds, partly also with singularities, \cite{MR2103311, MR2320656, MR1901374,saji2020behavior, MR2251001} and various (integrable) approaches for the construction were discussed \cite{MR2254053, MR2028680, MR2743444}. In these developments, special interest was paid to constrained Ribaucour transformations that allow to generate envelopes with special geometric properties, for example, the Darboux transformation of isothermic surfaces and Ribaucour transformations that preserve classes of $O$-surfaces \cite{O_surface}.
A characteristic feature in the theory of smooth Ribaucour transformations are permutability theorems. In discrete differential geometry, those became crucial as underlying concept of integrable discretizations of curvature-line parametrized surfaces and orthogonal coordinate systems \cite{ddg_book, org_principles}.
\mathfrak{n}oindent Discrete equivalents of Ribaucour sphere congruences were developed in \cite{org_principles} and \cite{DOLIWA1999169}: \emph{discrete R-congruences} are provided by $Q$-nets in the Lie quadric, that is, a discrete congruence of spheres with planar faces. However, more flexibility in the discrete setup results in a 2-parameter family of discrete envelopes; we call it the \emph{Ribaucour family} of a discrete R-congruence (cf.\,Definition \ref{cor_initial_contact_el}).
In the last years, in a variety of works, discrete Ribaucour transformations between general discrete surfaces were discussed \cite{rib_coord, zbMATH01721625, zbMATH01327008}. Furthermore, restricted transformations for particular classes of discrete surfaces as discrete (special) isothermic surfaces \cite{ddg_book, discrete_cmc, zbMATH06406008, MR2004958, MR1676683} and discrete O-surfaces \cite{MR1997461} were investigated, to name just a few.
\mathfrak{n}oindent However, to the best of the authors' knowledge, there are no systematic studies of the geometry of the entire Ribaucour families in the literature, since usually pairs of discrete envelopes are considered as the main objects.
Geometric investigations of this 2-parameter family of envelopes is the main contribution of this work. In most parts we consider the discrete R-congruences as the primary objects of interest and explore the corresponding Ribaucour families by using data provided by the sphere congruence.
\mathfrak{n}oindent This paper is organized as follows.
We recall basic principles and constructions in Lie sphere geometry in Section \ref{section_prel}.
Section \ref{section_r_congruences} is devoted to the construction of envelopes of discrete R-congruences.
As a crucial result, for any face of a discrete R-congruence, we introduce two involutive Lie inversions that map adjacent contact elements of discrete envelopes onto each other (Proposition~\ref{prop_inversions_for_r}).
Using these inversions, we obtain a construction of a unique envelope from one prescribed initial contact element.
Hence, in this way, we can parametrize the entire Ribaucour family of a discrete R-congruence.
In Section \ref{section_special_congr} we restrict to discrete R-congruences with special properties and analyse their corresponding Ribaucour families.
This section also reveals relations to recent works on discrete Ribaucour coordinates \cite{rib_coord} and discrete $\Omega$-surfaces \cite{discrete_omega}.
Since we consider the discrete R-congruence as primary object, our approach sheds light on these situations from a different point of view.
Focusing again on general discrete R-congruences, in Subsection \ref{subsec_rib_pairs}, we discuss how the choice of a facewise constant Lie inversion decomposes a Ribaucour family into pairs of envelopes.
Those Lie inversions also interact well with face-cyclides of these Ribaucour pairs and, therefore, induce a Ribaucour transformation of cyclidic nets as pointed out in Subsection \ref{subsec_cyclidic_nets}.
As an application of the developed framework for discrete R-congruences, we investigate envelopes with one family of spherical curvature lines in Section \ref{section_spherical}. In particular, we characterize sphere congruences of Ribaucour families containing at least two discrete channel surfaces.
\mathfrak{n}oindent\textbf{Acknowledgements.} The authors would like to thank Fran Burstall for helpful discussions about isothermic Q-nets. Moreover, financial support by JSPS Grant-in-Aid (as part of the FY2017 JSPS Postdoctoral fellowship) and TU Wien (Hörbiger Award) is gratefully acknowledged.
This research was supported by the DFG Collaborative Research Center TRR
109 ``Discretization in Geometry and Dynamics''.
\section{Preliminaries}\label{section_prel}
\mathfrak{n}oindent In this section we briefly summarize some basic principles of Lie sphere geometry and recall some concepts that will be crucial for the rest of this work. For more details on this topic, the interested reader is referred to \cite{blaschke} and \cite{book_cecil}.
\mathfrak{n}oindent Throughout the paper we use the hexaspherical model introduced by Lie and work in the vector space $\mathbb{R}^{4,2}$ endowed with the inner product $\lspan{\cdot\, , \cdot}$ given by
\begin{equation*}
\lspan{ x , y } = -x_1y_1 + x_2y_2 +x_3y_3 +x_4y_4 +x_5y_5 -x_6y_6.
\end{equation*}
Homogeneous coordinates of elements in the projective space $\mathbb{P}(\mathbb{R}^{4,2})$ will be denoted by the corresponding black letter (i.e., oriented spheres $r,s \in \mathbb{P}(\mathcal{L})$ have homogeneous coordinates $\mathfrak{r}, \mathfrak{s} \in \mathbb{R}^{4,2}$ such that $\lspan{\mathfrak{r}} = r$ and $\lspan{\mathfrak{s}} = s$).
If statements hold for arbitrary homogeneous coordinates we do this without explicitly mentioning it.
The projective light cone is denoted by $\mathbb{P}(\mathcal{L})$ and represents the set of oriented 2-spheres in $\mathbb{S}^3$ in this model. Two spheres $r,s \in \mathbb{P}(\mathcal{L})$ are in oriented contact if and only if $\lspan{\mathfrak{r}, \mathfrak{s}}=0$. Thus, the set $\mathcal{Z}$ of lines in $\mathbb{P}(\mathcal{L})$ corresponds to contact elements.
\mathfrak{n}oindent By breaking symmetry, we can recover various subgeometries of Lie sphere geometry. Thus, let $\mathfrak{p} \in \mathbb{R}^{4,2}$ be a vector that is not lightlike, i.e., $\lspan{\mathfrak{p},\mathfrak{p}} \mathfrak{n}eq 0$. If $\mathfrak{p}$ is timelike, then $\langle\mathfrak{p}\rangle^\perp \cong
\mathbb{R}^{4,1}$ defines a Riemannian conformal geometry resp.\,M\"obius geometry.
In this case, elements in $\mathbb{P}(\mathcal{L})\cap \langle \mathfrak{p} \rangle^\perp$ are considered points and will be called point spheres.
If $\mathfrak{p}$ is spacelike, it determines a Lorentzian conformal geometry $\langle\mathfrak{p}\rangle^\perp \cong \mathbb{R}^{3,2}$ or planar Lie geometry.
\subsection{Linear systems and linear sphere complexes}
Let $s_1, s_2, s_3 \in \mathbb{P}(\mathcal{L})$ be three spheres that are mutually not in oriented contact, that is, $\lspan{\mathfrak{s_i}, \mathfrak{s_j}} \mathfrak{n}eq 0$ for $i\mathfrak{n}eq j \in \{1, 2, 3\}$. The space
\begin{equation}
\mathfrak{n}onumber
\spann{s_1, s_2, s_3} \cap \mathbb{P}(\mathcal{L})
\end{equation}
is called a \emph{linear system}\footnote{\cite[\S 53]{blaschke}: Lineare Kugelscharen und Kugelkomplexe} and the Lie invariant
\begin{equation}
\mathfrak{n}onumber
\delta:=\text{sign} \{ \langle \mathfrak{s}_1, \mathfrak{s}_2 \rangle \langle \mathfrak{s}_2, \mathfrak{s}_3 \rangle \langle \mathfrak{s}_3, \mathfrak{s}_1 \rangle \}
\end{equation}
provides information about its geometry:
\begin{enumerate}[label=$\bullet$]
\item If $\delta=-1$, then $s_1, s_2, s_3$ span a $(2,1)$-plane. Hence, the spheres of the linear system are curvature spheres of a Dupin cyclide and there exists a 1-parameter family of spheres in oriented contact with all spheres of the linear system -- the spheres of the orthogonal $(2,1)$-plane.
\item If $\delta=1$, then $s_1, s_2, s_3$ span a $(1,2)$-plane. Thus, there does not exist a sphere that is in oriented contact with all spheres of the linear system, since the orthogonal complement is a $(3,0)$-plane that does not contain any spheres.
\end{enumerate}
\mathfrak{n}oindent Any element $a \in \mathbb{P}(\mathbb{R}^{4,2})$ defines a \emph{linear sphere complex} $\mathbb{P}(\mathcal{L}) \cap a^\perp$, a 3-dimensional family of 2-spheres. Depending on the type of the vector $a$, we distinguish three cases:
\begin{enumerate}[label=$\bullet$]
\item \emph{parabolic} linear sphere complex, $\lspan{ \mathfrak{a}, \mathfrak{a}}=0$: all spheres in the complex are in oriented contact with the sphere defined by $a$, hence the linear sphere complex consists of all contact elements containing the sphere $a$.
\item \emph{hyperbolic} linear sphere complex, $\lspan{ \mathfrak{a}, \mathfrak{a}} < 0$: if we fix a M\"obius geometry by choosing $a$ as the point sphere complex, then the linear sphere complex consists of all point spheres.
\item \emph{elliptic} linear sphere complex, $\lspan{ \mathfrak{a}, \mathfrak{a}} > 0$: in a M\"obius geometry, these linear sphere complexes then consist of all contact elements that intersect a fixed sphere at a fixed angle (see Fig.~\ref{fig:sphere_complex}, \emph{left}).
\end{enumerate}
\mathfrak{n}oindent Since elliptic linear sphere complexes will be essential in this paper, we will discuss their geometric construction illustrated in Fig.~\ref{fig:sphere_complex} in more detail. Assume that $a \in \mathbb{P}(\mathbb{R}^{4,2})$ determines an elliptic linear sphere complex. Additionally fix a parabolic complex $q$, as well as a hyperbolic complex $p$ to distinguish a M\"obius geometry modelled on $p^\perp$.
Without loss of generality, we choose $\mathfrak{q}=(1,-1,0,0,0,0)$, $\mathfrak{p}=(0,0,0,0,0,1)$ and homogeneous coordinates $\mathfrak{a} = (\mathfrak{a}_i)_i \in \mathbb{R}^{4,2}$ of $a$ such that $\lspan{\mathfrak{a}, \mathfrak{q}}=1$. By defining
\begin{equation*}
c:=(\mathfrak{a}_3, \mathfrak{a}_4, \mathfrak{a}_5), \
R:= \left\vert\sqrt{1 + \|c\|^2 - 2\mathfrak{a}_1}\right\vert
\text{, and }
r:= \lspan{\mathfrak{a}, \mathfrak{p}},
\end{equation*}
we obtain two concentric spheres $s_r$ and $s_R$ with center $c$ and radii $r$ and $R$, respectively (see Fig.~\ref{fig:sphere_complex}, \emph{left}). Then a straightforward computation shows that any sphere $s$ in the elliptic linear sphere complex $\lspan{\mathfrak{a}}^\perp$ intersects the sphere $s_R$ under the constant oriented angle $\cos \gamma= \frac{r}{R}$.
If $\lspan{\mathfrak{a}, \mathfrak{q}}=0$, the sphere $s_R$ corresponds to a plane and the sphere $s_r$ becomes the point at infinity.
\begin{figure}
\caption{
\emph{Left:}
\label{fig:sphere_complex}
\end{figure}
\subsection{Lie inversions}
Let $a \in \mathbb{P}(\mathbb{R}^{4,2})$, $\lspan{ \mathfrak{a}, \mathfrak{a}} \mathfrak{n}eq 0$, then the \emph{Lie inversion with respect to the linear sphere complex $a$} is given by
\begin{equation}\label{equ_lie_inversion}
\sigma_\mathfrak{a}:\mathbb{R}^{4,2} \rightarrow \mathbb{R}^{4,2}, \ \ \sigma_\mathfrak{a}(\mathfrak{r}):=\mathfrak{r}-\frac{2\lspan{\mathfrak{r}, \mathfrak{a}}}{\lspan{\mathfrak{a}, \mathfrak{a}}}\mathfrak{a}.
\end{equation}
Lie inversions are involutive linear maps that preserve oriented contact between spheres and, therefore, also contact elements. Moreover, from the definition, we directly conclude that spheres contained in the linear sphere complex $a^\perp$ are fixed by the Lie inversion $\sigma_\mathfrak{a}$.
For any pair of spheres that are not contained in the linear sphere complex $a^\perp$, we obtain the following property:
\begin{lem}\label{lem_inversion_lin_system}
Let $r, s \in \mathbb{P}(\mathcal{L})$ be two spheres that do not lie in the linear sphere complex $a^\perp$ with $\lspan{\mathfrak{a}, \mathfrak{a}} \mathfrak{n}eq 0$. Then, the $\spann{r, s, \sigma_{\mathfrak{a}}(r), \sigma_{\mathfrak{a}}(s)} \cap \mathbb{P}(\mathcal{L})$ is a linear system.
\end{lem}
\mathfrak{n}oindent The following two constructions give Lie inversions that map prescribed spheres onto each other:
\begin{lem}\label{lem_inversion_family}
Let $\lspan{\mathfrak{r}} = r$ and $\lspan{\bar{\mathfrak{r}}}= \bar{r}$ be fixed homogeneous coordinates of two spheres that are not in oriented contact and let $\sigma_\lambda$ denote the Lie inversion with respect to the linear sphere complex $\mathfrak{n}_\lambda:= \mathfrak{r} - \lambda\bar{\mathfrak{r}}$, where $\lambda \in \mathbb{R}^\times$. Then the following properties hold:
\begin{enumerate}[label=(\roman*)]
\item For any $\lambda \in \mathbb{R}^\times$, the Lie inversion $\sigma_\lambda$ maps the sphere $r$ to the sphere $\bar{r}$.
\item All spheres in oriented contact with $r$ \emph{and} $\bar{r}$ are fixed by any Lie inversion $\sigma_\lambda$.
\item A contact element that contains $r$ is mapped by any Lie inversion~$\sigma_\lambda$, $\lambda \in \mathbb{R}$, to the same contact element; however, different Lie inversions induce different correspondences between the spheres in the two contact elements. \label{lem_inversion_two}
\end{enumerate}
\end{lem}
\begin{proof}
The properties (i) and (ii) follow directly from equation (\ref{equ_lie_inversion}). To prove the third statement, we suppose that $v \in \mathbb{P}(\mathcal{L})$ is a sphere in oriented contact with the sphere $r$ and $\lspan{\mathfrak{v}, \mathfrak{\bar{r}}}\mathfrak{n}eq 0$. Then, the sphere
\begin{equation*}
\tilde{\mathfrak{s}}:={\lspan{\mathfrak{v}, \bar{\mathfrak{r}}}}\mathfrak{r} - {\lspan{\mathfrak{r}, \bar{\mathfrak{r}}}}\mathfrak{v}
\end{equation*}
lies in the contact element $f:=\spann{r, v}$ and, furthermore, for any $\lambda \in \mathbb{R}$ in the linear sphere complex $\langle \mathfrak{n}_\lambda \rangle^\perp$. Hence, any Lie inversion $\sigma_\lambda$ preserves the sphere $\tilde{s}$ and maps the contact element~$f$ to the contact element $\sigma_\lambda(f)=\spann{\tilde{s}, \bar{r}}$.
\end{proof}
\begin{lem}\label{lem_unique_inversion_four_spheres}
Let $f$ and $\bar{f}$ be two contact elements sharing a common sphere $s \in \mathbb{P}(\mathcal{L})$. Then, for four spheres $r, t \in f$ and $\bar{r}, \bar{t} \in \bar{f}$ that do not coincide with $s$, there exists a unique Lie inversion $\sigma$ satisfying
\begin{equation*}
\sigma(\mathfrak{r})=\bar{\mathfrak{r}} \ \ \text{and} \ \ \sigma(\mathfrak{t})=\bar{\mathfrak{t}}.
\end{equation*}
\end{lem}
\begin{proof}
By assumption, there exist constants $\lambda, \mu, \bar{\lambda}, \bar{\mu} \in \mathbb{R}^\times$ such that
\begin{equation*}
\mathfrak{s}= \lambda \mathfrak{r} + \mu \mathfrak{t} \ \ \text{and} \ \ \mathfrak{s}= \bar{\lambda} \bar{\mathfrak{r}} + \bar{\mu} \bar{\mathfrak{t}}.
\end{equation*}
Then, the Lie inversion with respect to the linear sphere complex $a \in \mathbb{P}(\mathbb{R}^{4,2})$ determined by
\begin{equation*}
\mathfrak{a}:=\lambda \mathfrak{r} - \bar{\lambda} \bar{\mathfrak{r}} = \bar{\mu} \bar{\mathfrak{t}} - \mu \mathfrak{t}
\end{equation*}
provides the sought-after map.
Moreover, $a$ is the intersection of the lines $\spann{r,\bar{r}}$ and $\spann{t,\bar{t}}$ and hence unique.
\end{proof}
\mathfrak{n}oindent For later reference, we remark the following property for the composition of two Lie inversions that follows from straightforward computations:
\begin{lem}\label{lem_inversions_commute}
Two Lie inversions commute, $\sigma_{\mathfrak{a}} \circ \sigma_{\mathfrak{b}}=\sigma_{\mathfrak{b}} \circ \sigma_{\mathfrak{a}}$, if and only if the corresponding linear sphere complexes are involutive, that is, $\lspan{\mathfrak{a},\mathfrak{b}}=0$.
\end{lem}
\subsection{Cross-ratio of four spheres}
Similar to the cross-ratio of four concircular points, one defines the cross-ratio of four spheres in a linear system. We recall this definition and different ways to compute it.
\begin{defi}
Let $r_1, r_2, r_3, r_4 \in \mathbb{P}(\mathcal{L})$ be four spheres in a common contact element, then the \emph{cross-ratio} is defined by
\begin{equation*}
cr(\mathfrak{r}_1,\mathfrak{r}_2,\mathfrak{r}_3,\mathfrak{r}_4)=\frac{\alpha \bar{\beta}}{\beta \bar{\alpha}}, \ \text{ where} \ \mathfrak{r}_3=\alpha \mathfrak{r}_1 + \beta \mathfrak{r}_2 \text{ and } \mathfrak{r}_4= \bar{\alpha} \mathfrak{r}_1 + \bar{\beta} \mathfrak{r}_2.
\end{equation*}
\end{defi}
\mathfrak{n}oindent It is independent of the choice of homogeneous coordinates and can be equivalently described by the four radii of the spheres
\begin{equation*}
cr(\mathfrak{r}_1,\mathfrak{r}_2,\mathfrak{r}_3,\mathfrak{r}_4)=\frac{(R_1-R_4)(R_2-R_3)}{(R_1-R_3)(R_2-R_4)},
\end{equation*}
where $R_i$ denotes the radius of the sphere $r_i$ (for point spheres the radius is assumed to be $0$ and for a plane we define the radius as $\infty$).
Moreover, one can define the \emph{cross-ratio of four spheres in a linear system} by tracking it back to the cross-ratio of four spheres in a contact element: assume that $r_1, r_2, r_3$ and $r_4$ are four spheres in a linear system and choose an arbitrary contact element $f \mathfrak{n}i r_1$. Then there exist three unique spheres $s_i$ in $f$ such that $s_i \perp r_i$ for $i=2,3,4$ and we define
\begin{align*}
cr(r_1, r_2, r_3, r_4)&:= cr(r_1, s_2, s_3, s_4).
\end{align*}
\mathfrak{n}oindent In particular, the cross-ratio of four spheres that are pairwise related by a Lie inversion is given by the following formula (cf.\,\cite[\S 53]{blaschke}):
\begin{lem}\label{lem_cr_J}
Let $a \in \mathbb{P}(\mathbb{R}^{4,2}) \setminus \mathbb{P}(\mathcal{L})$ and $s_1, s_2 \in \mathbb{P}(\mathcal{L})$ be two spheres that do not lie in the linear sphere complex $a^\perp$, then
\begin{equation*}
cr(\mathfrak{s}_1, \mathfrak{s}_2, \sigma_{\mathfrak{a}}(\mathfrak{s}_2) ,\sigma_{\mathfrak{a}}(\mathfrak{s}_1))=\frac{\phantom{2}\langle \mathfrak{s}_1, \mathfrak{s}_2 \rangle \langle \mathfrak{a}, \mathfrak{a} \rangle}{2\langle \mathfrak{s}_1, \mathfrak{a} \rangle \langle \mathfrak{s}_2, \mathfrak{a} \rangle}.
\end{equation*}
\end{lem}
\subsection{Discrete Legendre maps}
Discrete surfaces in this paper will be represented by discrete Legendre maps from a connected quadrilateral cell complex $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{F})$ of degree $4$ to the space of contact elements $\mathcal{Z}$. The set of vertices (0-cells), edges (1-cells) and faces (2-cells) of the cell complex $\mathcal{G}$ are denoted by $\mathcal{V}$, $\mathcal{E}$ and $\mathcal{F}$, respectively.
The directions in the cell complex will be labelled by upper indices $(1)$ and $(2)$ and we obtain two distinguished sets of edges $\mathcal{E}^{(1)} \overset{.}{\cup} \mathcal{E}^{(2)}=\mathcal{E}$. A $(1)$- resp.\,$(2)$-coordinate ribbon is then the sequence of faces bounded by two adjacent $(1)$- resp.\,$(2)$-coordinate lines.
\begin{defi}[\cite{ddg_book, lin_weingarten_discrete}]
A discrete line congruence $f: \mathcal{V} \rightarrow \mathcal{Z}, i \mapsto f_i$, is a \emph{discrete Legendre map} if two adjacent contact elements $f_i$ and $f_j$ share a common \emph{curvature sphere} $s_{ij}:= f_i \cap f_j$.
\end{defi}
\mathfrak{n}oindent Note that, for any discrete Legendre map, we therefore obtain two \emph{curvature sphere congruences} $s^{(1)}:\mathcal{E}^{(1)} \rightarrow \mathbb{P}(\mathcal{L})$ and $s^{(2)}:\mathcal{E}^{(2)} \rightarrow \mathbb{P}(\mathcal{L})$.
Moreover, for any fixed point sphere complex $\mathfrak{p} \in \mathbb{R}^{4,2}$, $\lspan{\mathfrak{p}, \mathfrak{p}}=-1$, the point sphere congruence $p_i:= f_i \cap p^\perp$ of a discrete Legendre map provides a circular net, that is, any four point spheres of an elementary quadrilateral are concircular.
\section{Discrete R-congruences}\label{section_r_congruences}
\mathfrak{n}oindent Classically, a smooth sphere congruence is called \emph{Ribaucour} if the curvature lines of its two envelopes correspond. Representing discrete surfaces by discrete Legendre maps, the analogous discrete problem, was studied in \cite{org_principles}: discrete sphere congruences enveloped by at least two generic discrete Legendre maps are given by \emph{$Q$-nets} in the Lie quadric, i.\,e.\,, sphere congruences $q:\mathcal{V} \rightarrow \mathbb{P}(\mathcal{L})$ such that the four spheres of any elementary quadrilateral are coplanar.
In this paper, we will exclude degenerate faces of a $Q$-net and will assume that the spheres of a quadrilateral are not in oriented contact:
\begin{defi}
A map $r:\mathcal{V} \rightarrow \mathbb{P}(\mathcal{L})$ is called a \emph{discrete R-congruence} if the four spheres of any elementary quadrilateral lie in a unique linear system.
\end{defi}
\mathfrak{n}oindent As pointed out in \cite[\S 5]{org_principles}, discrete R-congruences with spheres lying in linear systems of signature $(2,1)$ provide the natural discrete counterparts to smooth Ribaucour sphere congruences. Furthermore, since in this case any four spheres of a quadrilateral are curvature spheres of a Dupin cyclide, these $Q$-nets often allow for elegant and straightforward geometric interpretations. Thus, in this work, we will often focus on these \emph{discrete $(2,1)$-R-congruences}.
\begin{remark*}
In the smooth case, Blaschke gave the following characterization of smooth Ribaucour sphere congruences \cite[\S 77]{blaschke}: a smooth sphere congruence $r:U \rightarrow \mathbb{P}(\mathcal{L})$ is a Ribaucour sphere congruence if and only if for any choice of coordinates $(u,v)$ there exists a map of \emph{osculating complexes}~$t:~U \rightarrow~\mathbb{P}(\mathbb{R}^{4,2})$ such that
\begin{equation*}
\spann{\mathfrak{r}, \ \partial_u \mathfrak{r}, \ \partial_v \mathfrak{r}, \ \partial_{uu} \mathfrak{r}, \ \partial_{vv} \mathfrak{r}, \ \partial_{uv} \mathfrak{r}} \subset \mathfrak{t}^\perp.
\end{equation*}
Similar osculating complexes also exist at vertices of a discrete R-congruence: by definition, the nine R-spheres of the four adjacent quadrilateral lie in a common linear sphere complex. This complex is spanned by the vertex sphere and its four neighbours.
\end{remark*}
\mathfrak{n}oindent Any face of a discrete R-congruence induces two special Lie inversions that will turn out to be crucial in the construction of its envelopes (see Fig.~\ref{fig:schematic_quad} \textit{right}):
\begin{prop}\label{prop_inversions_for_r}
A discrete R-congruence induces a unique map of linear sphere complexes determined by
\begin{equation*}
(\one{n}{},\two{n}{}):\mathcal{F} \rightarrow \mathbb{P}(\mathbb{R}^{4,2})\setminus \mathbb{P}(\mathcal{L}) \times \mathbb{P}(\mathbb{R}^{4,2})\setminus \mathbb{P}(\mathcal{L})
\end{equation*}
such that for any quadrilateral $(ijkl)$ the corresponding Lie inversions $\sigma_{\mathfrak{n}^{(1)}}$ and $\sigma_{\mathfrak{n}^{(2)}}$ satisfy
\begin{equation}
\begin{aligned}\label{eq:prop_n}
&\sigma_{\mathfrak{n}^{(1)}}(\mathfrak{r}_i)=\mathfrak{r}_j, \ \ \sigma_{\mathfrak{n}^{(1)}}(\mathfrak{r}_l)=\mathfrak{r}_k \ \ \text{ and }
\\&\sigma_{\mathfrak{n}^{(2)}}(\mathfrak{r}_i)=\mathfrak{r}_l, \ \ \sigma_{\mathfrak{n}^{(2)}}(\mathfrak{r}_j)=\mathfrak{r}_k;
\end{aligned}
\end{equation}
these induced Lie inversions will be denoted by
\begin{equation*}
\one{\sigma}:=\sigma_{\mathfrak{n}^{(1)}} \ \ \text{and } \ \ \two{\sigma}:=\sigma_{\mathfrak{n}^{(2)}}.
\end{equation*}
\end{prop}
\begin{proof}
Let $r_i, r_j, r_k$ and $r_l$ be four spheres of an elementary quadrilateral of a discrete R-congruence (see Fig.~\ref{fig:schematic_quad}). Since the four spheres lie in a linear system and are therefore linearly dependent, we can choose homogeneous coordinates such that
\begin{equation}\label{choice_coord_spheres}
\mathfrak{r}_i - \mathfrak{r}_j + \mathfrak{r}_k - \mathfrak{r}_l =0.
\end{equation}
Then, by Lemma \ref{lem_inversion_family} (i), the vectors
\begin{equation}\label{equ_coord_compl}
\one{\mathfrak{n}}:= \mathfrak{r}_i-\mathfrak{r}_j=\mathfrak{r}_l-\mathfrak{r}_k \ \text{ and } \ \two{\mathfrak{n}}:=\mathfrak{r}_i - \mathfrak{r}_l = \mathfrak{r}_j-\mathfrak{r}_k
\end{equation}
define two linear sphere complexes satisfying~Equation \eqref{eq:prop_n}.
\end{proof}
\begin{figure}
\caption{\emph{Left}
\label{fig:schematic_quad}
\end{figure}
The choice of homogeneous coordinates in Equation \eqref{choice_coord_spheres} is local and depends on the quadrilateral under consideration. However, for special discrete R-congruences there exist global homogeneous coordinates inducing the Lie inversions $\one{\sigma}$ and $\two{\sigma}$ (see Subsection \ref{section_iso}).
\begin{lem}\label{lem_involutive}
For any quadrilateral, the two linear sphere complexes determined by $n^{(1)}$ and $n^{(2)}$ are involutive.
\end{lem}
\begin{proof}
Without loss of generality, for any quadrilateral we choose homogeneous coordinates as given in Equation~\eqref{equ_coord_compl}. Then
\begin{align*}
2 \langle \mathfrak{n}^{(2)} , \mathfrak{n}^{(1)}\rangle &= \langle \mathfrak{r}_i-\mathfrak{r}_l, \mathfrak{r}_i - \mathfrak{r}_j \rangle + \langle \mathfrak{r}_j-\mathfrak{r}_k , \mathfrak{r}_l - \mathfrak{r}_k \rangle
\\&= \langle \mathfrak{r}_j , -\mathfrak{r}_i - \mathfrak{r}_k + \mathfrak{r}_l \rangle + \langle \mathfrak{r}_l, - \mathfrak{r}_i+ \mathfrak{r}_j - \mathfrak{r}_k \rangle =0.
\end{align*}
\end{proof}
\begin{lem}
Let $(r_i, r_j, r_k, r_l)$ be four spheres of an elementary quadrilateral of a discrete R-congruence.
\begin{enumerate}[label=(\roman*)]
\item $cr(r_i, r_j, r_k, r_l)< 0$ if and only if $\one{n}$ and $\two{n}$ determine two elliptic or two hyperbolic linear sphere complexes.
\item $cr(r_i, r_j, r_k, r_l)> 0$ if and only if the linear sphere complexes determined by $\one{n}$ and $\two{n}$ are of different type, that is, one linear sphere complex is elliptic and the other one is hyperbolic.
\end{enumerate}
\end{lem}
\begin{proof}
Suppose that $(r_i, r_j, r_k, r_l)$ are four spheres of an elementary quadrilateral of a discrete R-congruence and choose homogeneous coordinates such that $0=\mathfrak{r}_i-\mathfrak{r}_j+\mathfrak{r}_k-\mathfrak{r}_l$. Then, by Lemma~\ref{lem_cr_J}, we obtain that
\begin{equation}\label{equ_cr}
cr(r_i, r_j, r_k, r_l)=
\frac{
\phantom{2}
\langle \mathfrak{r}_i, \mathfrak{r}_j \rangle
\langle \mathfrak{\two{n}}, \mathfrak{\two{n}} \rangle
}
{
2\langle \mathfrak{r}_i, \mathfrak{\two{n}} \rangle
\langle \mathfrak{r}_j, \mathfrak{\two{n}} \rangle
}
=
-\frac{
\lspan{\one{\mathfrak{n}},\one{\mathfrak{n}}}
}{
\lspan{\two{\mathfrak{n}},\two{\mathfrak{n}}}
}
\end{equation}
and therefore conclude that
\begin{align*}
cr(r_i, r_j, r_k, r_l) < 0 \ &\Leftrightarrow \ \text{sign} \{{\lspan{\mathfrak{\one{n}}, \mathfrak{\one{n}}}}\} = \text{sign} \{{\lspan{\mathfrak{\two{n}}, \mathfrak{\two{n}}}}\}.
\end{align*}
The points $\one{n}$ and $\two{n}$ lie in the plane spanned by the spheres ${r}_i, {r}_j, {r}_k$ and ${r}_l$. Therefore, if the plane has signature $(2,1)$, the linear sphere complexes are elliptic; if the spheres lie in a $(1,2)$-plane, then the linear sphere complexes are hyperbolic.
Moreover, $cr(r_i, r_j, r_k, r_l) > 0$ if and only if
\begin{equation*}
\text{sign} \{{\lspan{\mathfrak{\two{n}}, \mathfrak{\two{n}}}}\}=-\text{sign} \{{\lspan{\mathfrak{\one{n}}, \mathfrak{\one{n}}}}\},
\end{equation*}
that is, if and only if the two linear sphere complexes are of different type (see Fig.~\ref{fig:embedded_quad} for the M\"obius geometric picture).
\end{proof}
This proposition also includes the special case of four concircular point spheres. It is well-known that a circular quadrilateral is embedded if and only if the cross-ratio is negative. In this case, the Lie inversions $\one{\sigma}$ and $\two{\sigma}$ become M\"obius inversions (see Figure 3).
\begin{figure}
\caption{
Whether a circular quadrilateral is embedded or non-embedded depends on the sign of the cross-ratio of the circular quadrilateral. The sign also determines the type of the M\"obius inversion (resp.\ linear sphere complex in Lie geometry) associated to the quadrilateral.
}
\label{fig:embedded_quad}
\end{figure}
\mathfrak{n}oindent With the Lie inversions at hand, we are now prepared to discuss the main objects of interest, namely the envelopes of a discrete R-congruence:
\begin{defi}
A discrete Legendre map $f:\mathcal{V}\rightarrow \mathcal{Z}$ is the \emph{envelope} of a discrete R-congruence $r:\mathcal{V}\rightarrow \mathbb{P}(\mathcal{L})$, if $r_i \in f_i$ for all $i \in \mathcal{V}$.
\end{defi}
\mathfrak{n}oindent Envelopes of a discrete R-congruence can be constructed from one prescribed initial contact element using the inversions defined in Proposition \ref{prop_inversions_for_r}. Thus, suppose that $f_i:=\spann{s_0, r_i}$ is an arbitrary initial contact element at the vertex $i \in \mathcal{V}$. Then, the contact elements
\begin{equation*}
\mathfrak{f}_j:=\one{\sigma}(\mathfrak{f}_i) , \ \ \mathfrak{f}_k:=\two{\sigma}(\mathfrak{f}_j), \ \ \mathfrak{f}_l:=\one{\sigma}(\mathfrak{f}_k)=\two{\sigma}(\mathfrak{f}_i)
\end{equation*}
define an envelope for the face $(ijkl)$ of the discrete R-congruence $r$: firstly, observe that by Lemmas~\ref{lem_inversions_commute} and \ref{lem_involutive}, the contact elements $\one{\sigma}(\mathfrak{f}_k)$ and $\two{\sigma}(\mathfrak{f}_i)$ indeed coincide. Furthermore, from Proposition~\ref{prop_inversions_for_r} and Lemma~\ref{lem_inversion_family}, we deduce that the contact elements envelop the discrete R-congruence,
\begin{equation*}
r_j \in f_j, \ \ r_k \in f_k \ \ \text{and } \ r_l \in f_l,
\end{equation*}
and two adjacent ones intersect.
Moreover, due to Lemma~\ref{lem_inversion_family}\ref{lem_inversion_two}, this construction uniquely extends to all vertices $\mathcal{V}$ of the quadrilateral cell complex.
\mathfrak{n}oindent Conversely, any envelope of a discrete R-congruence arises in this way:
\begin{lem}\label{transport_contact}
Let $f:\mathcal{V} \rightarrow \mathcal{Z}$ be a discrete Legendre map enveloping the discrete R-congruence $r:\mathcal{V}\rightarrow \mathbb{P}(\mathcal{L})$, then for any quadrilateral $(ijkl)$ we obtain that
\begin{align*}
&\one{\sigma}(\mathfrak{f}_i)=\mathfrak{f}_j, \ \ \one{\sigma}(\mathfrak{f}_l)=\mathfrak{f}_k \ \ \text{ and } \ \ \two{\sigma}(\mathfrak{f}_i)=\mathfrak{f}_l, \ \ \two{\sigma}(\mathfrak{f}_j)=\mathfrak{f}_k.
\end{align*}
\end{lem}
\begin{proof}
Assume that $f$ is an envelope, then we know that $r_i \in f_i$ and $r_j \in f_j$ (see Fig.~\ref{fig:contact_elements} \emph{Left} for the labelling of a quadrilateral). Therefore, the Lie inversion $\one{\sigma}$ preserves the curvature sphere $s_{ij}=f_i \cap f_j$.
Hence, for any sphere $\mathfrak{r}_i + \lambda\mathfrak{s}_{ij} \in \mathfrak{f}_i$ in the contact element, it follows that
\begin{align*}
\one{\sigma}(\mathfrak{r}_i + \lambda\mathfrak{s}_{ij})= \one{\sigma}(\mathfrak{r}_i) + \lambda \one{\sigma}(\mathfrak{s}_{ij})= \mathfrak{r}_j + \lambda \mathfrak{s}_{ij} \in \mathfrak{f}_j,
\end{align*}
which completes the proof.
\end{proof}
\mathfrak{n}oindent Thus, we have reproven the following existence result for envelopes of a discrete R-congruence that was already given in \cite[Theorem~3.37]{ddg_book}:
\begin{corollaryand}\label{cor_initial_contact_el}
Any envelope of a discrete R-congruence is uniquely determined by the choice of an initial contact element and, therefore, any discrete R-congruence admits a 2-parameter family of envelopes.
This family of envelopes is said to be the \emph{Ribaucour family of a discrete R-congruence}. If the discrete R-congruence consists of faces with signature $(2,1)$, the family is also called a \emph{$(2,1)$-Ribaucour family}.
\end{corollaryand}
\mathfrak{n}oindent
For later reference we remark on some relations between the envelopes and the Lie inversions associated to the discrete R-congruence (see Fig.~\ref{fig:contact_elements}, \emph{Right}):
\begin{lem}\label{lem_curv_spheres_swapped}
Let $f:\mathcal{V} \rightarrow \mathcal{Z}$ be a discrete Legendre map enveloping the discrete R-congruence $r:\mathcal{V}\rightarrow \mathbb{P}(\mathcal{L})$. Then, for any quadrilateral, the Lie inversions $\one{\sigma}$ and $\two{\sigma}$ swap the curvature spheres on opposite edges:
\begin{equation*}
\two{\sigma}(\mathfrak{s}_{ij})=\mathfrak{s}_{kl} \ \ \text{and} \ \ \one{\sigma}(\mathfrak{s}_{jk})=\mathfrak{s}_{li}.
\end{equation*}
\end{lem}
\begin{figure}
\caption{
\emph{Left:}
\label{fig:contact_elements}
\end{figure}
\subsection{M\"obius geometric point of view}\label{subsection_moebius}
In this subsection we briefly discuss the interplay between the construction of Ribaucour families in Lie sphere and M\"obius geometry. Thus, we fix a point sphere complex $\mathfrak{p} \in \mathbb{R}^{4,2}$, $\lspan{\mathfrak{p},\mathfrak{p}}=-1$, to distinguish a M\"obius geometry modelled on $\lspan{\mathfrak{p}}^\perp$.
Since a Ribaucour family consists of discrete Legendre maps, the choice of a point sphere complex reveals a 2-parameter family of enveloping circular principal contact element nets. The underlying circular nets are called the \emph{point sphere envelopes in $\lspan{\mathfrak{p}}^\perp$} (see Fig.~\ref{fig:circular_envelopes}).
\begin{figure}
\caption{
A discrete R-congruence with various
envelopes (plotted as circular quadrilaterals) of the Ribaucour family.
}
\label{fig:circular_envelopes}
\end{figure}
\mathfrak{n}oindent Note that a Lie inversion $\sigma_{\mathfrak{a}}$ is a M\"obius transformation, that is, it maps point spheres onto point spheres, if and only if $\lspan{\mathfrak{a},\mathfrak{p}}=0$. Therefore, generically, the Lie inversions $\one{\sigma}$ and $\two{\sigma}$ that map adjacent enveloping contact elements onto each other, do not transport point spheres along the discrete R-congruence.
However, for each edge there exists a M\"obius transformation uniquely determined by data from the discrete R-congruence that relates point spheres of adjacent contact elements in the Ribaucour family:
\begin{prop}\label{prop_moebius_inversion_edge}
Let $r: \mathcal{V} \rightarrow \mathbb{P}(\mathcal{L})$ be a discrete R-congruence. Then the Lie inversions $\sigma_\mathfrak{m}$ with respect to the linear sphere complexes determined by
\begin{equation*}
m:\mathcal{E} \rightarrow \mathbb{R}^{4,2}, \ m_{ij}=\lspan{\mathfrak{p}, r_j}r_i - \lspan{\mathfrak{p}, r_i}r_j
\end{equation*}
are M\"obius transformations and satisfy, for any edge $(ij)$,
\begin{equation*}
\sigma_\mathfrak{m_{ij}}(\mathfrak{r}_i)=\mathfrak{r}_j.
\end{equation*}
Moreover, those M\"obius transformations transport the contact elements of envelopes along the discrete R-congruence and map adjacent point spheres of the point sphere envelope onto each other.
\end{prop}
\subsection{Construction of discrete R-congruences}
\label{subsection_construction}
Discrete R-congruences can be constructed from two arbitrary initial curves of spheres defined on two intersecting coordinate lines $\one{I}=(\one{\mathcal{V}}, \one{\mathcal{E}})$ and $\two{I}=(\two{\mathcal{V}}, \two{\mathcal{E}})$ of a quadrilateral cell complex $\mathcal{G}$ by the following iteration:
\begin{itemize}
\item fix two initial sphere curves $\one{c}: \one{\mathcal{V}} \rightarrow \mathbb{P}(\mathcal{L})$ and $\two{c}: \two{\mathcal{V}} \rightarrow \mathbb{P}(\mathcal{L})$ that intersect at one vertex
\item choose for each face $(ijkl)$ that contains an edge $(ij)$ of the curve $\one{c}$ a Lie inversion that maps the prescribed spheres $c_i$ and $c_j$ onto each other; for each face this amounts to the choice of a $\lambda \in \mathbb{R}$:
\begin{equation*}
\one{n}_{(ijkl)}:=\one{\mathfrak{c}}_i - \lambda\one{\mathfrak{c}}_j
\end{equation*}
\item starting at a face $(ijkl)$ with three prescribed spheres $\one{c}_i$, $\one{c}_j$ and $\two{c}_l$, we complete the face by defining
\begin{equation*}
\mathfrak{r}_k:=\one{\sigma}_{(ijkl)}(\two{\mathfrak{c}}_l)
\end{equation*}
\item iteratively this procedure gives a coordinate ribbon of a discrete R-con\-gruence including the spheres of the curve $\one{c}$ and two spheres of the curve $\two{c}$
\item to obtain the next coordinate ribbon we choose again suitable Lie inversions along the just constructed sphere curve and proceed as described above.
\end{itemize}
\begin{remark*}
The choice of the parameter $\lambda$ in the construction above is equivalent to choosing the cross-ratios for the quadrilaterals. The relation is given by Lemma~\ref{lem_cr_J}:
\begin{align*}
cr(\mathfrak{c}_i, \mathfrak{c}_l, \one{\sigma}_{(ijkl)}(\mathfrak{c}_l), \one{\sigma}_{(ijkl)}(\mathfrak{c}_i))
= \frac
{\lspan{\mathfrak{c}_i, \mathfrak{c}_l}}
{\lspan{\mathfrak{c}_i, \mathfrak{c}_l} - \lambda\lspan{\mathfrak{c}_j, \mathfrak{c}_l}}\,.
\end{align*}
\end{remark*}
Conversely, given a nowhere umbilic discrete Legendre map $f: \mathcal{V} \rightarrow \mathbb{P}(\mathbb{R}^{4,2})$, that is, opposite curvature spheres do not coincide, then any choice of spheres in the contact elements along two intersecting coordinate lines uniquely determines a discrete R-congruence. The construction relies on the following simple observation:
\begin{lem}
\label{lem:r_congruence_from_legendre}
Given four contact elements $(f_i, f_j, f_k, f_l)$ of an elementary quadrilateral of a nowhere umbilic discrete Legendre map and three spheres $r_i \in f_i$, $r_j \in f_j$ and $r_l \in f_l$, then there exists a unique sphere $r_k \in f_k$ such that $(r_i, r_j, r_k, r_l)$ provides a quadrilateral of a discrete R-congruence.
\end{lem}
\begin{proof}
By Lemma \ref{lem_unique_inversion_four_spheres}, there exists a unique Lie inversion $\sigma$ satisfying
\begin{equation*}
\sigma(\mathfrak{r}_i)=\mathfrak{r}_l \ \text{ and } \ \sigma(\mathfrak{s}_{ij})=\mathfrak{s}_{kl}.
\end{equation*}
Then, $\mathfrak{r}_k:=\sigma(\mathfrak{r}_j) \in \spann{\mathfrak{s}_{jk}, \mathfrak{s}_{kl}}$ provides the sought-after sphere that completes the elementary quadrilateral of a discrete R-congruence.
\end{proof}
As a consequence of this Lemma, we obtain the following construction for discrete R-congruences of a discrete Legendre map: we fix two intersecting coordinate lines $\one{I}=(\one{\mathcal{V}}, \one{\mathcal{E}})$ and $\two{I}=(\two{\mathcal{V}}, \two{\mathcal{E}})$ of the quadrilateral cell complex $\mathcal{G}$ and suppose that $\one{c}: \one{\mathcal{V}} \rightarrow \mathbb{P}(\mathcal{L})$ and $\two{c}: \two{\mathcal{V}} \rightarrow \mathbb{P}(\mathcal{L})$ are chosen such that
\begin{equation*}
\one{c}_i \in f_i \text{ for any } i \in \one{\mathcal{V}} \ \text{and } \ \one{c}_j \in f_j \text{ for any } j \in \two{\mathcal{V}}.
\end{equation*}
Then, by Lemma~\ref{lem:r_congruence_from_legendre} above, these choices uniquely determine a discrete R-congruence of the discrete Legendre map $f$.
This is another instance of the interplay between line congruences and Q-nets as studied in~\cite{DoliwaSantiniManas:2000:TransformationsOfQnets, bobenko_schief-discrete_line_complexes}.
\section{Special discrete R-congruences}\label{section_special_congr}
\mathfrak{n}oindent This section is devoted to discrete R-congruences with special properties. We discuss their geometry and consequences for the corresponding Ribaucour families.
\subsection{Discrete R-congruences in a fixed linear sphere complex}\label{subsection_fixed_complex}
Suppose that $r: \mathcal{V} \rightarrow \mathbb{P}(\mathcal{L})$ is a discrete R-congruence lying in a fixed linear sphere complex determined by $a \in \mathbb{P}(\mathbb{R}^{4,2})$, that is, $\lspan{\mathfrak{r}_i, \mathfrak{a}}=0$ for all $i \in \mathcal{V}$. Since the spheres all lie in the complex, so do $\one{n}$ and $\two{n}$ and the linear sphere complex $a^\perp$ is invariant under $\one{\sigma}$ and $\two{\sigma}$.
Depending on the type of the linear sphere complex we obtain three geometrically different situations.
\subsubsection*{Hyperbolic complex}
\mathfrak{n}oindent In the case of a {hyperbolic} linear sphere complex we can fix $a \in \mathbb{P}(\mathbb{R}^{4,2})$, $\lspan{\mathfrak{a}, \mathfrak{a}} < 0$, to be the point sphere complex and consider the M\"obius geometry modelled on $\lspan{\mathfrak{a}}^\perp$. Then, on each quadrilateral the R-spheres become point spheres lying in a linear system with signature $(2,1)$. Therefore, the R-spheres are curvature spheres of a Dupin cyclide consisting of point spheres and the discrete R-congruence becomes a circular net. In this case, the Lie inversions $\one{\sigma}$ and $\two{\sigma}$ become M\"obius inversions mapping the point spheres of the circular net onto each other.
For each circular net there exists a 2-parameter choice of contact elements that yield principal contact element nets. This ambiguity in the choice of the contact elements is reflected in the Ribaucour family of $r$.
\subsubsection*{Parabolic complex}
If $a \in \mathbb{P}(\mathcal{L})$ determines a {parabolic} linear sphere complex, then all R-spheres of the congruence are in oriented contact with the sphere determined by $a$.
According to Corollary \ref{cor_initial_contact_el}, the choice of an initial contact element determines a unique envelope.
Let us choose a particular initial contact element at the sphere $r_i$ given by $f_i:=\spann{r_i, a}$. Since the sphere determined by $a$ is preserved by the Lie inversions $\one{\sigma}$ and $\two{\sigma}$, this envelope is totally umbilic, that is, all curvature spheres of the discrete Legendre map coincide.
Moreover, observe that, for any projection to a M\"obius geometry, the point sphere map of a totally umbilic envelope lies on a fixed sphere, which, in this case, is the sphere determined by $a$.
Hence, we deduce the following classification of discrete R-congruences with spheres in a fixed parabolic complex:
\begin{corollary}\label{cor_parabolic_complex}
A discrete R-congruence lies in a fixed parabolic linear sphere complex if and only if its Ribaucour family contains a totally umbilic envelope.
\end{corollary}
\mathfrak{n}oindent Generically, any discrete Legendre map $f:\mathcal{V} \rightarrow \mathcal{Z}$ admits discrete R-congruences of this type: the intersection of a fixed parabolic complex $a \in \mathbb{P}(\mathcal{L})$ with the discrete Legendre map, $r_i:=f_i \cap a^\perp$, provides a Q-net in the Lie quadric $\mathbb{P}(\mathcal{L})$.
Therefore, any discrete Legendre map admits totally umbilic Ribaucour transforms (see also \cite[Theorem 3.9]{rib_coord}).
\subsubsection*{Elliptic complex}
Suppose that a discrete R-congruence $r: \mathcal{V} \rightarrow \mathbb{P}(\mathcal{L})$ lies in a fixed elliptic linear sphere complex $a^\perp$, $\lspan{\mathfrak{a}, \mathfrak{a}} > 0$, and fix an arbitrary R-sphere $r_{i_0}$ of the congruence. Then, there exists a 1-parameter family of contact elements $f_{i_0} \mathfrak{n}i r_{i_0}$ with spheres that are also contained in the elliptic linear sphere complex $a^\perp$. Hence, by Lemma~\ref{transport_contact}, any envelope $f$ that is constructed from such an initial contact element $f_{i_0}$, satisfies $\lspan{\mathfrak{f}_i, \mathfrak{a}}=0$ for any $i \in \mathcal{V}$. Therefore, this envelope is spherical, that is, for any projection to a M\"obius geometry, the point sphere map lies on a fixed sphere. However, note that in this case the envelope is not totally umbilic.
Conversely, the contact elements of a spherical and not totally umbilic discrete Legendre map lie in a fixed elliptic linear complex. Thus, in summary, we have proven:
\begin{corollary}\label{cor_elliptic_complex}
A discrete R-congruence lies in a fixed elliptic linear sphere complex if and only if there exists a spherical and not totally umbilic envelope in the Ribaucour family.
\end{corollary}
\mathfrak{n}oindent We remark that the special envelopes obtained in the Corollaries \ref{cor_parabolic_complex} and \ref{cor_elliptic_complex} are Ribaucour coordinates as discussed in \cite{rib_coord}.
\subsection{Discrete isothermic R-congruences}\label{section_iso}
A discrete $Q$-net in the Lie quadric $\mathbb{P}(\mathcal{L})$ is called \emph{isothermic} if any diagonal vertex star of the $Q$-net lies in a projective 3-dimensional subspace of $\mathbb{P}(\mathbb{R}^{4,2})$ that does not contain the four outer spheres of the vertex star. Equivalently, isothermicity of a $Q$-net is characterized by the existence of a Moutard lift (see \cite{ddg_book, discrete_omega, lin_weingarten_discrete}):
\begin{defi}
A lift $\mu: \mathcal{V} \rightarrow \mathcal{L} \subset \mathbb{R}^{4,2}$ of a discrete Q-net $s: \mathcal{V} \rightarrow \mathbb{P}(\mathcal{L})$ is called a \emph{Moutard lift} if opposite diagonals are parallel, that is,
\begin{equation*}
\delta \mu_{ik} \ || \ \delta \mu_{jl},
\end{equation*}
where $\delta \mu_{ik}:= \mu_k - \mu_i$ and $\delta \mu_{jl}:= \mu_l - \mu_j$.
\end{defi}
\mathfrak{n}oindent If the Q-net is not degenerate, hence a discrete R-congruence in the sense of this paper, the special global choice of homogeneous coordinates of a Moutard lift gives rise to additional Lie inversions that diagonally swap the R-spheres on each quadrilateral:
\begin{prop}
A lift $\mu: \mathcal{V} \rightarrow \mathcal{L} \subset \mathbb{R}^{4,2}$ of a discrete R-congruence $r: \mathcal{V} \rightarrow \mathbb{P}(\mathcal{L})$ is a Moutard lift if and only if for each quadrilateral the Lie inversion $\sigma^\delta$ with respect to the linear sphere complex $\langle \delta\mu_{ik} \rangle^\perp$ diagonally interchanges the R-spheres:
\begin{equation*}
\sigma^\delta(\mu_i)=\mu_k \ \ \text{and } \ \ \sigma^\delta(\mu_j)=\mu_l.
\end{equation*}
\end{prop}
\begin{proof}
Suppose that $\mu$ is a Moutard lift, then the linear sphere complex is determined by
\begin{equation*}
\langle \delta\mu_{ik} \rangle = \langle \delta\mu_{jl} \rangle
\end{equation*}
and we therefore obtain
\begin{align*}
\sigma^\delta(\mu_i)=\mu_k \ \ \text{and } \ \ \sigma^\delta(\mu_j)=\mu_l.
\end{align*}
Conversely, if we have
\begin{align*}
\mu_k &= \mu_i - \frac{2\lspan{\mu_i,\delta\mu_{ik}}}{\lspan{\delta\mu_{ik},\delta\mu_{ik}}}\delta\mu_{ik},
\\\mu_l &= \mu_j - \frac{2\lspan{\mu_j,\delta\mu_{ik}}}{\lspan{\delta\mu_{ik},\delta\mu_{ik}}}\delta\mu_{ik},
\end{align*}
it follows that $\delta \mu_{ik} \ || \ \delta \mu_{jl}$ and $\mu$ is indeed a Moutard lift.
\end{proof}
\begin{figure}
\caption{
The Lie inversion $\sigma^\delta$ interchanges opposite R-spheres $r_i \leftrightarrow r_k$ and $r_j \leftrightarrow r_l$, curvature spheres $s_{ij}
\label{fig_sigma_delta}
\label{fig:diagonal_inversion}
\end{figure}
\mathfrak{n}oindent The interaction of these Lie inversions $\sigma^\delta$ induced by a Moutard lift with the geometric data of the envelopes is illustrated in Figure \ref{fig_sigma_delta}:
\begin{corollary}
The Lie inversions $\sigma^\delta$ map facewise opposite curvature spheres onto each other and diagonally interchanges the contact elements of an enveloping discrete Legendre map (see Fig.~\ref{fig:diagonal_inversion}):
\begin{align*}
\sigma^\delta(\mathfrak{s}_{ij})=\sigma^\delta(\mathfrak{s}_{kl}), \ \ \sigma^\delta(\mathfrak{s}_{jk})=\sigma^\delta(\mathfrak{s}_{il}) \ \ \text{and} \ \
\sigma^\delta(\mathfrak{f}_i)=\sigma^\delta(\mathfrak{f}_{k}), \ \ \sigma^\delta(\mathfrak{f}_{j})=\sigma^\delta(\mathfrak{f}_{l}).
\end{align*}
\end{corollary}
\mathfrak{n}oindent A Moutard lift of a discrete isothermic R-congruence $r$ also defines the aforementioned Lie inversions $\one{\sigma}$ and $\two{\sigma}$: let $\mu$ denote the Moutard lift of $r$ and define the function $\lambda: \mathcal{F}\rightarrow \mathbb{R}$ by
\begin{equation*}
\lambda_{F}\, \delta \mu_{jl} = \delta \mu_{ik}
\end{equation*}
for a face $F=(ijkl)$.
Then, in terms of the homogeneous coordinates of a Moutard lift, the points $\one{n}$ and $\two{n}$ are given by
\begin{align*}
\one{\mathfrak{n}}: &= \mu_i - \lambda_{F} \mu_j = \mu_k - \lambda_{F} \mu_l,
\\
\two{\mathfrak{n}}: &= \mu_i + \lambda_{F} \mu_l = \mu_k + \lambda_{F} \mu_j.
\end{align*}
\mathfrak{n}oindent Moreover, since the homogeneous coordinates of a Moutard lift are global, these maps also induce an edge labelling: for any face $F = (ijkl)$, we define the edge function $e: \mathcal{E} \rightarrow \mathbb{R}$ by
\begin{align*}
\one{e}_{ij}=\one{e}_{kl}
&:=\frac{\lspan{\one{\mathfrak{n}}_F, \one{\mathfrak{n}}_F}}{2\lambda_F}=
-\lspan{\mu_i, \mu_j} = -\lspan{\mu_k, \mu_l}
\quad \text{and} \\
\two{e}_{il}=\two{e}_{jk}
&:=-\frac{\lspan{\two{\mathfrak{n}}_F, \two{\mathfrak{n}}_F}}{2\lambda_F}=
\lspan{\mu_i, \mu_l} = \lspan{\mu_j, \mu_k}.
\end{align*}
Then, according to Lemma \ref{lem_cr_J}, the edge function $e$ factorizes the cross-ratio of the discrete isothermic R-congruence (cf.\,\cite[Lemma 3.5]{lin_weingarten_discrete}):
\begin{equation*}
cr(r_i, r_j, r_k, r_l)=
-\frac
{\lspan{\one{\mathfrak{n}}_F, \one{\mathfrak{n}}_F}}
{\lspan{\two{\mathfrak{n}}_F, \two{\mathfrak{n}}_F}}
=
\frac{e_{ij}^{(1)}}{e_{jk}^{(2)}}.
\end{equation*}
\mathfrak{n}oindent Envelopes of discrete isothermic R-congruences belong to the special class of discrete $\Omega$-surfaces introduced in \cite{lin_weingarten_discrete}. Those are discrete Legendre maps enveloped by a pair of sphere congruences that admit K\"onigs dual lifts. In \cite{discrete_omega}, it is shown that K\"onigs dual lifts can be constructed even from one enveloping isothermic sphere congruence of a discrete Legendre map. Therefore, to summarize:
\begin{corollary}\label{cor_omega}
The Ribaucour family of a discrete isothermic R-congruence consists of discrete $\Omega$-surfaces.
\end{corollary}
\subsection{Multi-R-congruences}
If we impose the characteristic property of a discrete net not only on each quadrilateral but also on arbitrary parameter rectangles, we obtain a so-called multi-net. Various multi-nets were investigated in \cite{multinet}.
In particular, a multi-Q-net is a Q-net such that arbitrary parameter rectangles are planar. In this subsection we are interested in multi-Q-nets in the Lie-quadric:
\begin{defi}
A \emph{discrete multi-R-congruence} is a discrete R-congruence $r: \mathcal{V}\rightarrow \mathbb{P}(\mathcal{L})$ such that also the spheres of any rectangular parameter quadrilateral lie in a unique linear system.
\end{defi}
\mathfrak{n}oindent Special examples of multi-R-congruences are given by the point sphere maps of multi-circular nets; hence, geometrically, point sphere maps of discrete isothermic channel surfaces (see \cite{multinet, discrete_channel}).
According to \cite[Theorem 2.4]{multinet}, any discrete multi-R-congruence $r:\mathcal{V} \rightarrow \mathbb{P}(\mathcal{L})$ admits a global choice of homogeneous coordinates $\mathfrak{r}:\mathcal{V} \rightarrow \mathcal{L} \subset \mathbb{R}^{4,2}$ such that we obtain
\begin{equation}\label{equ_cond_multi_lift}
\mathfrak{r}_i - \mathfrak{r}_j + \mathfrak{r}_{j'} - \mathfrak{r}_{i'}=0
\end{equation}
for any parameter rectangle $(ijj'i')$. Thus, the associated linear sphere complexes determined by $\one{n}$ and $\two{n}$ are constant along each coordinate ribbon and we obtain (see Fig.~\ref{fig:multi_isothermic_cs_proof} \emph{left})
\begin{corollary}
For any multi-R-congruence the Lie inversions $\one{\sigma}$ and $\two{\sigma}$ are constant along each coordinate ribbon.
\end{corollary}
\mathfrak{n}oindent As a second immediate consequence of the existence of the lift $\mathfrak{r}$, we obtain a Moutard lift for any discrete R-congruence by swapping the signs of the homogeneous coordinates $\mathfrak{r}$ along every second coordinate line in one family. Therefore, we extended the already known fact for multi-circular nets to multi-R-congruences:
\begin{prop}
A discrete multi-R-congruence is an isothermic sphere congruence.
\end{prop}
\mathfrak{n}oindent Therefore, by Corollary \ref{cor_omega}, the envelopes of a discrete multi-R-congruence are discrete $\Omega$-surfaces. In this special case we even obtain particular discrete $\Omega$-surfaces:
\begin{corollary}
The Ribaucour family of a multi-R-congruence consists of discrete $\Omega$-surfaces with curvature spheres that lie in a fixed linear sphere complex along each coordinate ribbon.
\end{corollary}
\begin{figure}
\caption{In case of a multi-R-congruence the complexes $\one{n}
\label{fig:multi_isothermic_cs_proof}
\end{figure}
\section{Ribaucour families}\label{section_rib_family}
\subsection{Geometric properties of envelopes in the Ribaucour family} In the following paragraphs we discuss various special properties of the 2-parameter family of envelopes in the Ribaucour family of a discrete R-congruence.
\subsubsection*{Curvature spheres in a Ribaucour family}
In Lemma \ref{transport_contact} we saw that the Lie inversions $\one{\sigma}$ and $\two{\sigma}$ transport the contact elements of the envelopes along the discrete R-congruence. Since two adjacent contact elements share a common sphere, namely the curvature sphere, this sphere is fixed by the corresponding Lie inversion. Therefore, we obtain the following property for all curvature spheres in a Ribaucour family:
\begin{prop}\label{prop_curv_spheres}
The curvature spheres of opposite edges of any envelope in a Ribaucour family lie in the same linear sphere complex.
\end{prop}
\subsubsection*{Totally umbilic faces} If the four curvature spheres of a face of a discrete Legendre map coincide, the face is called \textit{umbilic}. While in the smooth case umbilic points are rather special, in any $(2,1)$-Ribaucour family we obtain envelopes with umbilic faces:
\begin{lem}\label{lem_totally_umbilic}
Let $f$ be a discrete Legendre map in a $(2,1)$-Ribaucour family. A face $(ijkl)$ of $f$ is umbilic if and only if the four contact elements $f_i, f_j, f_k$ and $f_l$ are also contact elements of the Dupin cyclide generated by the R-spheres of this face.
In this case, for any projection to a M\"obius geometry, the four point spheres of $f$ lie on a curvature circle of the Dupin cyclide.
\end{lem}
\begin{proof}
Let $f$ be a discrete Legendre map in a $(2,1)$-Ribaucour family and assume that the contact elements $f_i, f_j, f_k$ and $f_l$ of a face $(ijkl)$ coincide with contact elements of the Dupin cyclide $d= D_1 \oplus_\perp D_2$, where the R-spheres $r_i, r_j, r_k$ and $r_l$ lie in the $(2,1)$-plane $D_1$. Then, $f_i=\spann{r_i, s_i}$, where $s_i \in D_2$. But, since $s_i$ is also in oriented contact with the R-spheres $r_j, r_k$ and $r_l$, the sphere $s_i$ is the constant curvature sphere of the envelope $f$ at this face. Hence this face of $f$ is umbilic.
Conversely, suppose that an envelope $f$ of a discrete $(2,1)$-R-congruence has a totally umbilic face, then the constant curvature sphere has to be in oriented contact with all four R-spheres of this face. Since these four R-spheres determine a Dupin cyclide, the constant curvature sphere of $f$ is also a curvature sphere of this Dupin cyclide (in the other curvature sphere family than the R-spheres). This completes the proof.
\end{proof}
\mathfrak{n}oindent Recall that, by Corollary \ref{cor_initial_contact_el}, any choice of an initial contact element at an initial vertex provides a unique envelope in the Ribaucour family. Therefore, with the help of Lemma \ref{lem_totally_umbilic}, we can construct umbilic faces at any face of a discrete $(2,1)$-R-congruence:
\begin{corollary}
For any face of a discrete $(2,1)$-R-congruence there exists a 1-parameter family of envelopes in the Ribaucour family that are umbilic at this face.
\end{corollary}
\subsubsection*{Permutability theorems} A key property in the theory of transformations is the existence of permutability theorems: given two Ribaucour transforms $f_1$ and $f_2$ of a Legendre map $f$, then there exists a 1-parameter family of Legendre maps that are simultaneous transforms of $f_1$ and $f_2$. Moreover, corresponding points of these Legendre maps are concircular. This result, generically, holds for smooth, as well as, for discrete Legendre maps (cf.\,\cite[Theorem 3.6]{ddg_book}, \cite{MR2254053}, \cite[\S 8]{MR2004958}).
However, contrary to the smooth case, for any discrete Legendre map circularity of corresponding points in the permutability theorem can fail: four envelopes in a Ribaucour family obviously satisfy the permutability theorem. But, if we project to a M\"obius geometry, corresponding point spheres of the four envelopes do not have to lie on a circle. The point spheres lie on the corresponding R-sphere and are therefore in general only cospherical.
\begin{prop}\label{prop_permutability_unusual}
Let $f$ be a discrete Legendre map, then there exist three Ribaucour transforms $f_1$, $f_2$ and $f_{12}$ of $f$ such that $f_{12}$ is a simultaneous Ribaucour transform of $f_1$ and $f_2$ and, for any projection to a M\"obius geometry, corresponding point spheres of these four nets are not circular.
\end{prop}
We emphasize that if corresponding contact elements of $f$, $f_1$, and $f_2$ span a 3-dimensional projective subspace of signature $(2,2)$, then there only exists a 1-parameter family of simultaneous Ribaucour transforms $f_{12}$. The points of the contact elements will then be circular. Hence in this case, the usual permutability theorem holds.
We remark that the significant difference to the smooth transformation theory pointed out in Proposition \ref{prop_permutability_unusual} leads to examples of discrete nets that have no counterparts in the smooth surface theory (see for example \cite[\S 1.1]{discrete_channel}).
\subsubsection*{Deformations of envelopes in the Ribaucour family} The ambiguity of envelopes in the Ribaucour family of a discrete R-congruence $r:\mathcal{V} \rightarrow \mathbb{P}(\mathcal{L})$ provides a possibility to smoothly deform two envelopes $f$ and $\hat{f}$ into each other.
Let $r_0$ be an R-sphere of the congruence $r$ at an initial vertex $v_0 \in \mathcal{V}$. Then, by Corollary~\ref{cor_initial_contact_el}, any choice of a smooth initial Legendre curve
\begin{equation*}
\gamma_0:[0,1] \rightarrow \mathcal{Z} \ \ \text{with }
\begin{cases}
r_0 \in \gamma_0(t) \ \text{for any } t \in [0,1],
\\[3pt]\gamma_0(0)=f_0 \ \ \text{and} \ \ \gamma_0(1)=\hat{f}_0
\end{cases}
\end{equation*}
gives rise to a 1-parameter family $\{f^{\gamma_0(t)}\}_{t \in [0,1] }$ of envelopes in the Ribaucour family (see Fig.~\ref{fig:ribaucour_curve}).
\begin{figure}
\caption{Two envelopes $f$ and $\hat{f}
\label{fig:ribaucour_curve}
\end{figure}
In particular, this construction yields a smooth Legendre curve $t \mapsto f^{\gamma_0(t)}_{v}$ for each vertex $v \in \mathcal{V}$ that lies in the fixed parabolic linear sphere complex determined by the corresponding R-sphere $r_v$. Moreover, according to Lemma \ref{transport_contact}, two adjacent Legendre curves are related by a Lie inversion. Thus, they envelop a 1-parameter family of spheres and form a smooth Ribaucour pair of curves.
\mathfrak{n}oindent If projecting to a M\"obius geometry, the point sphere envelopes of $\{f^{\gamma_0(t)}\}_{t \in [0,1] }$ provide smooth deformations of circular nets, where the vertices move along spherical curves. In particular, if the initial curve $\gamma_0$ is a (part of a) circle, then circularity of it is preserved along the entire discrete R-congruence (cf.\,Proposition \ref{prop_moebius_inversion_edge}).
\mathfrak{n}oindent The construction pointed out in this section provides a way of obtaining discrete and semi-discrete triply orthogonal systems with special vertical coordinate surfaces having one family of spherical curvature lines. For particular choices of the initial curve $\gamma_0$ we even obtain semi-discrete cyclic systems. A deeper analysis of these systems will be given in a future work.
\subsection{Discrete Ribaucour pairs in the Ribaucour family}\label{subsec_rib_pairs}
To gain further geometric insights into the Ribaucour family of a discrete R-congruence, in this subsection, we fix two envelopes of a discrete R-congruence, classically called a \emph{Ribaucour pair} of discrete Legendre maps.
The contact elements of a Ribaucour pair form a fundamental line system in the sense of~\cite{bobenko_schief-discrete_line_complexes, DoliwaSantiniManas:2000:TransformationsOfQnets} since R-congruences are Q-nets.
\mathfrak{n}oindent Thus, suppose that $f, \hat{f}:\mathcal{V}\rightarrow \mathcal{Z}$ are two discrete Legendre maps enveloping the discrete R-congruence $r:\mathcal{V}\rightarrow \mathbb{P}(\mathcal{L})$. Then, for any edge $(ij)$ the contact elements $(f_i, f_j, \hat{f}_j, \hat{f}_i)$ provide a quadrilateral of a discrete Legendre map, that is, adjacent contact elements share a common sphere. Thus, along each coordinate line of a Ribaucour pair the contact elements of $f$ and $\hat{f}$ yield a coordinate ribbon of a discrete Legendre map; these coordinate ribbons will be called \emph{vertical ribbons of the Ribaucour pair}. The ``curvature spheres'' of the vertical ribbons are given by the R-spheres and the curvature spheres of $f$ and $\hat{f}$ along the corresponding coordinate line of the Ribaucour pair.
Moreover, note that any two consecutive vertical ribbons also form a Ribaucour pair enveloping curvature spheres of $f$ and $\hat{f}$.
In the following paragraphs we demonstrate how the structures of the various Ribaucour pairs interact with each other.
\begin{prop}\label{prop_three_n}
Let $f, \hat{f}:\mathcal{V}\rightarrow \mathcal{Z}$ be two envelopes of a discrete R-congruence. Then there exists a map
\begin{equation*}
\three{n}:\mathcal{F}\rightarrow \mathbb{P}(\mathbb{R}^{4,2}) \setminus \mathbb{P}(\mathcal{L})
\end{equation*}
such that for any quadrilateral $(ijkl)$ the induced Lie inversion $\three{\sigma}:=\sigma_{\three{\mathfrak{n}}}$ interchanges the curvature spheres of $f$ and $\hat{f}$,
\begin{align*}
\sigma_{\three{\mathfrak{n}}}(\mathfrak{s}_{ij})=\hat{\mathfrak{s}}_{ij}, \
\sigma_{\three{\mathfrak{n}}}(\mathfrak{s}_{jk})=\hat{\mathfrak{s}}_{jk}, \
\sigma_{\three{\mathfrak{n}}}(\mathfrak{s}_{kl})=\hat{\mathfrak{s}}_{kl}, \
\sigma_{\three{\mathfrak{n}}}(\mathfrak{s}_{il})=\hat{\mathfrak{s}}_{il}.
\end{align*}
Moreover, for any quadrilateral the Lie inversion $\three{\sigma}$ preserves the R-spheres, maps corresponding contact elements of the Ribaucour pair $(f, \hat{f})$ onto each other and is involutive to $\one{\sigma}$ and $\two{\sigma}$ (see Fig.~\ref{fig:sigma3} for notation).
\end{prop}
\begin{proof}
Firstly observe that, since $\mathfrak{s}_{kl}=\two{\sigma}(\mathfrak{s}_{ij})$ and $\hat{\mathfrak{s}}_{kl}=\two{\sigma}(\hat{\mathfrak{s}}_{ij})$, we can choose homogeneous coordinates such that
\begin{align*}
0=\mathfrak{s}_{ij} - \hat{\mathfrak{s}}_{ij} - \mathfrak{s}_{kl} +\hat{\mathfrak{s}}_{kl},
\end{align*}
as well as homogeneous coordinates
\begin{equation*}
0=\mathfrak{s}_{jk} - \hat{\mathfrak{s}}_{jk} - \mathfrak{s}_{il} +\hat{\mathfrak{s}}_{il}
\end{equation*}
for the other four curvature spheres. Moreover, we define two vectors $n^{(3)}$ and $\tilde{n}^{(3)}$ by
\begin{align*}
\mathfrak{n}^{(3)}&:=\mathfrak{s}_{ij} - \hat{\mathfrak{s}}_{ij} = \mathfrak{s}_{kl} - \hat{\mathfrak{s}}_{kl},
\\\tilde{\mathfrak{n}}^{(3)}&:= \mathfrak{s}_{il} - \hat{\mathfrak{s}}_{il}=\mathfrak{s}_{jk} - \hat{\mathfrak{s}}_{jk}.
\end{align*}
Then, the R-spheres $r_i, r_j, r_k$ and $r_l$ lie in the induced linear sphere complexes $n^{(3)}$ and $\tilde{n}^{(3)}$, which are therefore involutive to $\one{n}$ and $\two{n}$.
Furthermore, since
\begin{align*}
\mathfrak{s}_{il}&= \lspan{\mathfrak{s}_{ij}, \two{\mathfrak{n}}} \mathfrak{r}_i - \lspan{\mathfrak{r}_{i}, \two{\mathfrak{n}}} \mathfrak{s}_{ij},
\\\hat{\mathfrak{s}}_{il}&= \lspan{\hat{\mathfrak{s}}_{ij}, \two{\mathfrak{n}}} \mathfrak{r}_i - \lspan{\mathfrak{r}_{i}, \two{\mathfrak{n}}} \hat{\mathfrak{s}}_{ij},
\end{align*}
the two linear sphere complexes induced by $n^{(3)}$ and $\tilde{n}^{(3)}$ coincide:
\begin{align*}
\tilde{\mathfrak{n}}^{(3)}&=\lspan{\two{\mathfrak{n}},\three{\mathfrak{n}}}\mathfrak{r}_i - \lspan{\two{\mathfrak{n}},\mathfrak{r}_i}\three{\mathfrak{n}}=- \lspan{\two{\mathfrak{n}},\mathfrak{r}_i}\three{\mathfrak{n}}.
\end{align*}
\end{proof}
\begin{figure}
\caption{Two envelopes $f$ and $\hat{f}
\label{fig:sigma3}
\end{figure}
Thus, on each hexahedron of a Ribaucour pair $(f, \hat{f})$ we obtain the following symmetric configuration of the curvature spheres and the R-spheres:
\begin{equation}\label{symmetric_configuration}
\begin{aligned}
\one{s}_{ij}, \one{s}_{kl}, \one{\hat{s}}_{kl}, \one{\hat{s}}_{ij} \perp \one{n},
\\\two{s}_{jk}, \two{s}_{il}, \two{\hat{s}}_{il}, \two{\hat{s}}_{jk} \perp \two{n},
\\r_i, r_j, r_k, r_l \perp \three{n}.
\end{aligned}
\end{equation}
In particular, for $\mathfrak{n}u=1,2,3$, the Lie inversion $\sigma^{(\mathfrak{n}u)}$ fixes the four spheres assigned to the $(\mathfrak{n}u)$-edges and interchanges the spheres within the other two linear sphere complexes.
Thus, the Lie inversions $\three{\sigma}$ given in Proposition \ref{prop_three_n} reveal a crucial property between two envelopes of a discrete R-congruence:
\begin{theorem}
Two envelopes of a discrete R-congruence are related by a facewise constant Lie inversion.
\end{theorem}
Furthermore, as a consequence of Lemma \ref{lem_inversions_commute}, the Lie inversions $\three{\sigma}$ constructed in Proposition~\ref{prop_three_n} induce pairings in the entire Ribaucour family:
\begin{corollary}\label{cor_rib_pairs}
Let $(f, \hat{f})$ be a discrete Ribaucour pair of a discrete R-congruence. Then the corresponding Lie inversions $\three{\sigma}$ decompose the Ribaucour family into discrete Ribaucour pairs.
\end{corollary}
\mathfrak{n}oindent We emphasize that the Lie inversions $\three{\sigma}$ and the induced decomposition of the Ribaucour family depend on the initial choice of the Ribaucour pair $(f, \hat{f})$. Hence, the decomposition given in Corollary \ref{cor_rib_pairs} is not unique.
\mathfrak{n}oindent The Ribaucour family of a discrete R-congruence in a fixed elliptic linear sphere complex $a^\perp$, $\lspan{\mathfrak{a}, \mathfrak{a}} > 0$, as discussed in the Subsection \ref{subsection_fixed_complex} admits a special decomposition into Ribaucour pairs: let $f:\mathcal{V}\rightarrow \mathcal{Z}$ be an envelope that is not spherical, then the Legendre map $\hat{\mathfrak{f}}_i:=\sigma_\mathfrak{a}(\mathfrak{f}_i)$ also lies in the Ribaucour family. Hence, we obtain Ribaucour pairs related by the constant Lie inversion $\sigma_a$.
This fact coincides with the smooth case, where a smooth Ribaucour pair enveloping a sphere congruence with a constant osculating complex is also related by a fixed Lie inversion (\cite[\S 89]{blaschke}).
\subsection{Cyclidic nets in the Ribaucour family}\label{subsec_cyclidic_nets}
For any face of a discrete Legendre map there exists a 1-parameter family of \emph{face-cyclides}, Dupin cyclides that share the four curvature spheres assigned to a face with the discrete Legendre map (cf.\,\cite{paper_cyclidic, discrete_channel}). Note that any face-cyclide has four distinguished curvature lines, namely the curvature lines that join two adjacent contact elements of the discrete Legendre map and lie on the corresponding curvature sphere.
We will denote the space of $(2,1)$-planes in $\mathbb{R}^{4,2}$ by $G_{(2,1)}(\mathbb{R}^{4,2})$ and Dupin cyclides by two complementary $(2,1)$-planes $\one{D}$ and $\two{D}$ with $\one{D} \oplus_\perp \two{D} = \mathbb{R}^{4,2}$.
To begin with, we shall point out how the Lie inversions $\one{\sigma}$ and $\two{\sigma}$ corresponding to a discrete R-congruence interact with the face-cyclides of its envelopes.
\begin{lem}
A congruence of face-cyclides of an envelope is preserved by the Lie inversions $\one{\sigma}$ and $\two{\sigma}$; however, the contact elements of opposite curvature lines going through the vertices of the discrete Legendre map are interchanged by the corresponding Lie inversion.
\end{lem}
\begin{proof}
Let $d=\one{D} \oplus_\perp \two{D}$ be a face-cyclide of the face $(ijkl)$ with $s_{ij}, s_{kl} \in \one{D}$ and $s_{il}, s_{jk} \in \two{D}$. Moreover, let $\tilde{s} \in \one{D}$ be a curvature sphere of the face-cyclide. Then $\lspan{\tilde{\mathfrak{s}}, \mathfrak{s}_{il}}=\lspan{\tilde{\mathfrak{s}}, \mathfrak{s}_{jk}}=0$ and we conclude that $\tilde{s}\perp \one{n}$. Thus, the spheres in $\one{D}$ are fixed by the Lie inversion $\one{\sigma}$ and, therefore, also the face-cyclide as unparametrized surface.
Furthermore, any contact element along the curvature line going through $f_i$ and $f_l$ is given by $\lspan{s_{il},\tilde{s}}$ for a sphere $\tilde{s}\in \one{D}$. Since
\begin{equation*}
\one{\sigma}(\mathfrak{s}_{il})= \mathfrak{s}_{jk} \ \ \text{and } \ \one{\sigma}(\tilde{\mathfrak{s}})=\tilde{\mathfrak{s}},
\end{equation*}
these contact elements are mapped to the contact elements of the opposite curvature line of the face-cyclide. Analogous arguments for the other pair of curvature lines complete the proof.
\end{proof}
This lemma also provides a construction for a special congruence
\begin{equation*}
d:\mathcal{F}\rightarrow G_{(2,1)}(\mathbb{R}^{4,2}) \times G_{(2,1)}(\mathbb{R}^{4,2})
\end{equation*}
of face-cyclides for $f$ from a given face-cyclide $d_{\alpha}=\one{D}_{\alpha} \oplus_{\perp} \two{D}_{\alpha}$ of an initial face~$(ijj'i')$ (notations see Fig.\,\ref{fig_labels}): let
\begin{equation}\label{equ_d_beta}
\one{D}_{\beta}:= (s_{jj'} \oplus \one{D}_{\alpha}) \cap (\one{n}_{\beta})^{\perp} \ \ \text{and } \ \two{D}_\beta:=(\one{D}_{\beta})^\perp,
\end{equation}
then $d_\beta= \one{D}_{\beta} \oplus \two{D}_\beta$ is a face-cyclide for the face~$(jkk'j')$. Furthermore, defining
\begin{equation}\label{equ_d}
\begin{aligned}
\two{D}_\gamma&:= (s_{j'k'} \oplus \two{D}_\beta) \cap (\two{n}_{\gamma})^{\perp},
\\\one{D}_\delta&:=(s_{j'j''} \oplus \one{D}_\gamma) \cap (\one{n}_{\delta})^{\perp},
\end{aligned}
\end{equation}
yields unique face-cyclides $d_\gamma$ and $d_\delta$ for the four faces of the vertex-star. This construction consistently extends on all faces and provides a face-cyclide congruence for the discrete Legendre map.
\begin{figure}
\caption{
Notation used for face cyclides on four quadrilaterals at a common vertex.
}
\label{fig_labels}
\end{figure}
Projecting to any M\"obius geometry $\langle \mathfrak{p} \rangle^\perp$, reveals a remarkable property of the just constructed face-cyclide congruence $d$.
Firstly, observe that the face-cyclides $d_\alpha$ and $d_\beta$ share a common curvature line $c_{\alpha\beta}$, namely the circle of point spheres joining the point spheres $p_j \in f_j$ and $p_{j'} \in f_{j'}$. The contact elements along $c_{\alpha\beta}$ coincide due to Equation \eqref{equ_d_beta}, and hence the two face-cyclides define a piecewise smooth surface that is $C^1$ across the common curvature line.
In the same way the pairs $\{d_\beta, d_\gamma\}$ and $\{d_\gamma, d_\delta\}$ meet at a common circular curvature line $c_{\beta\gamma}$ and $c_{\gamma\delta}$, respectively. Note that the circles $c_{\alpha\beta}$ and $c_{\beta\gamma}$, as well as $c_{\beta\gamma}$ and $c_{\gamma\delta}$, intersect orthogonally. Therefore, since the circular curvature lines $c_{\alpha\beta}, c_{\beta\gamma}$ and $c_{\gamma\delta}$ of the face-cyclides around the vertex-star go through the common contact element $f_{j'}$, also the face-cyclides $d_\delta$ and $d_\alpha$ meet at a common curvature line $c_{\delta\alpha}$. Since all contact elements of $d_\delta$ and $d_\alpha$ along the common circular curvature line $c_{\delta\alpha}$ contain the curvature sphere $s_{i'j'}$ of $f$, also the contact elements of $d_\delta$ and $d_\alpha$ along $c_{\delta\alpha}$ coincide.
Thus, with the help of the Lie inversions $\one{\sigma}$ and $\two{\sigma}$, we have constructed a particular face-cyclide congruence, discussed in \cite{cas} and \cite{paper_cyclidic}:
\begin{defi}
Let $f: \mathcal{V} \rightarrow \mathcal{Z}$ be a discrete Legendre map, then a \emph{cyclidic net of $f$} is a congruence of face-cyclides $d: \mathcal{F} \rightarrow G_{(2,1)}(\mathbb{R}^{4,2}) \times G_{(2,1)}(\mathbb{R}^{4,2})$ such that two face-cyclides adjacent along the edge $(ij)$ share the same contact elements along the curvature direction going through $f_i$ and $f_j$.
\end{defi}
Note that the construction given above only depends on the choice of an initial face-cyclide and is independent of the discrete R-congruence $r$; Lie inversions $\one{\sigma}$ and $\two{\sigma}$ determined by any other discrete R-congruence of $f$ lead to the same cyclidic net for $f$.
In particular, this construction gives a cyclidic net for a principal net in a conformal geometry $\lspan{\mathfrak{p}}^\perp$: since its point sphere congruence is a special discrete R-congruence, the corresponding Lie inversions $\one{\sigma}$ and $\two{\sigma}$ descend to M\"obius transformations and can be used to determine the Dupin cyclide patches of a cyclidic net (cf.\,(\ref{equ_d_beta}) and (\ref{equ_d})).
The symmetries described in (\ref{symmetric_configuration}) can be exploited to relate cyclidic nets of a Ribaucour pair:
\begin{theorem}\label{thm_cyclidic_for_Rib}
Let $(f, \hat{f})$ be a discrete Ribaucour pair related by the facewise constant Lie inversion $\three{\sigma}$. If $d=(\one{D}, \two{D}):\mathcal{F}\rightarrow G_{(2,1)}(\mathbb{R}^{4,2}) \times G_{(2,1)}(\mathbb{R}^{4,2})$ is a cyclidic net of $f$, then
\begin{equation*}
\hat{d}:\mathcal{F} \rightarrow G_{(2,1)}(\mathbb{R}^{4,2}) \times G_{(2,1)}(\mathbb{R}^{4,2}), \ \ \hat{d}=(\three{\sigma}(\one{D}), \ \three{\sigma}(\two{D}) )
\end{equation*}
provides a cyclidic net for $\hat{f}$.
\end{theorem}
\begin{proof}
Since corresponding curvature spheres of $f$ and $\hat{f}$ are mapped onto each other by the Lie inversions $\three{\sigma}$
and Dupin cyclides are invariant under Lie inversions, $\hat{d}_{ijkl}$ defines a face-cyclide for the face $(ijkl)$ of $\hat{f}$.
Moreover, to prove that $\hat{d}$ indeed provides a cyclidic net of $\hat{f}$, we consider two adjacent faces along an edge $(ij)$: firstly, observe that the Lie inversions $\three{\sigma}_{n}$ and $\three{\sigma}_{\bar{n}}$ belonging to the two adjacent faces are determined by the two linear sphere complexes
\begin{equation*}
\three{\mathfrak{n}}:=\mathfrak{s}_{ij}-\lambda \hat{\mathfrak{s}}_{ij} \ \text{and } \ \three{\bar{\mathfrak{n}}}:=\mathfrak{s}_{ij}-\bar{\lambda} \hat{\mathfrak{s}}_{ij},
\end{equation*}
where $\mathfrak{s}_{ij} \in s_{ij}$, $\hat{\mathfrak{s}}_{ij} \in \hat{s}_{ij}$ and $\lambda, \bar{\lambda} \in \mathbb{R}$ are appropriately chosen. Thus, by Lemma~\ref{lem_inversion_family}~\ref{lem_inversion_two}, the contact elements of the face-cyclides of $f$ along the common curvature line passing through $f_i$ and $f_j$ are mapped to the same contact elements by $\three{\sigma}_{n}$ and $\three{\sigma}_{\bar{n}}$. Thus, two adjacent face-cyclides of $\hat{d}$ share common contact elements along the curvature line through $\hat{f}_i$ and $\hat{f}_j$.
\end{proof}
\mathfrak{n}oindent In \cite[Definition 4.4]{trafo_channel}, the existence of two special Dupin cyclide congruences for a smooth Ribaucour pair of Legendre maps was pointed out. We report on a similar construction in the discrete case:
\begin{defi}\label{def_rib_cyclides}
Let $f, \hat{f}:\mathcal{V}\rightarrow \mathcal{Z}$ be two envelopes of a discrete R-congruence. Face-cyclides along a vertical ribbon will be called \emph{R-cyclides of the Ribaucour pair $(f, \hat{f})$}, that is, for a vertical face, a Dupin cyclide $\delta=R \oplus_\perp \tilde{R} \subseteq \mathbb{R}^{4,2}$ satisfying
\begin{equation*}
\mathfrak{s}_{ij}, \hat{\mathfrak{s}}_{ij} \in R \ \ \text{and} \ \ \mathfrak{r}_i, \mathfrak{r}_j \in \tilde{R}.
\end{equation*}
\end{defi}
In Theorem \ref{thm_cyclidic_for_Rib}, we have learned that cyclidic nets for a Ribaucour pair arise in distinguished pairs, where the face-cyclides are related by the Lie inversions $\three{\sigma}$. For these cyclidic nets there exists a canonical choice for the R-cyclides on the vertical ribbons:
\begin{corollary}\label{cor_induced_R_cyclides}
Suppose that $d$ and $\hat{d}$ are cyclidic nets of a Ribaucour pair $(f, \hat{f})$ related by the Lie inversions $\three{\sigma}$. Then for an edge $(ij)$ of the Ribaucour pair, the contact elements along two corresponding curvature lines of $d$ and $\hat{d}$ passing through $f_i$ and $f_j$, as well as $\hat{f}_i$ and $\hat{f}_j$, uniquely determine an R-cyclide for the corresponding vertical face.
\end{corollary}
\begin{proof}
Since the contact elements along the curvature lines under consideration are related by the Lie inversion $\three{\sigma}$, two corresponding contact elements share a common sphere lying in $(\three{n})^\perp$. In this way, we obtain a 1-parameter family of spheres that are in oriented contact with the spheres $s_{ij}$ and $\hat{s}_{ij}$ and are therefore curvature spheres of a face-cyclide for the vertical face.
\end{proof}
We deduce that, by construction, the induced R-cyclides investigated in Corollary \ref{cor_induced_R_cyclides} provide (one ribbon of) a cyclidic net along each vertical ribbon. However, observe that two adjacent induced R-cyclides belonging to two vertical ribbons from different coordinate directions do not share a common curvature line. In particular, these Dupin cyclides do not give a 3D cyclidic net as introduced in \cite[Section 3.2]{paper_cyclidic}.
\section{Envelopes with spherical curvature lines}\label{section_spherical}
In this section, we will draw attention to the Ribaucour transformation of discrete channel surfaces as discussed in \cite{discrete_channel} and the wider class of discrete Legendre maps with a family of spherical curvature lines.
\subsection{Discrete spherical curvature lines}
Inspired by the classification of spherical curvature lines in the smooth case (see \cite{blaschke, spherical_curv_lines}), we introduce the notion of osculating complexes for discrete Legendre maps.
To obtain uniqueness of the osculating complexes, we suppose a mild genericity condition on the discrete Legendre map. Note that also in the smooth case, uniqueness fails for the class of channel surfaces.
Thus, let $f: \mathcal{V} \rightarrow \mathcal{Z}$ be a discrete Legendre map and fix a point sphere complex $\mathfrak{p}\in \mathbb{R}^{4,2}$, $\lspan{\mathfrak{p},\mathfrak{p}} < 0$. Furthermore, suppose that four consecutive contact elements $f_{i'}, f_i, f_j$ and $f_{j'}$ along a coordinate line are nowhere circular, that is, the four point spheres $p_{i'}, p_i, p_j$ and $p_{j'}$ do not lie on a circle.
Then the spheres of these four contact elements lie in a unique elliptic linear sphere complex: let $s$ and $\tilde{s}$ be the two oriented spheres that contain the four point spheres $p_{i'}, p_i, p_j$ and $p_{j'}$. Then the sought-after linear sphere complex is given by
\begin{equation*}
\mathfrak{t}:={\lspan{\tilde{\mathfrak{s}}, \mathfrak{s}_{ij}}}\mathfrak{s} - {\lspan{\mathfrak{s}, \mathfrak{s}_{ij}}}\tilde{\mathfrak{s}},
\end{equation*}
where $s_{ij}$ denotes the curvature sphere belonging to the edge $(ij)$. Clearly, the spheres of the contact elements $\mathfrak{f}_i:=\spann{\mathfrak{s}_{ij}, \mathfrak{p}_i}$ and $\mathfrak{f}_j:=\spann{\mathfrak{s}_{ij}, \mathfrak{p}_j}$ lie in $t^\perp$. Moreover, since $s_{ii'}\in f_i$ and $s_{jj'}\in f_j$, also the spheres of the contact elements
\begin{equation*}
f_{i'}=\spann{s_{ii'}, p_i} \ \ \text{and} \ \ f_{j'}=\spann{s_{jj'}, p_j}
\end{equation*}
are contained in $t^\perp$.
\begin{defi}
Let $f:\mathcal{V} \rightarrow \mathcal{Z}$ be a nowhere circular discrete Legendre map, then
\begin{equation*}
t:\mathcal{E}\rightarrow \mathbb{P}(\mathbb{R}^{4,2}), \ (ij) \mapsto t_{ij}:= \spann{ f_{i'}, f_{i}, f_{j} ,f_{j'}}^\perp
\end{equation*}
are called the \emph{osculating complexes of $f$}.
\end{defi}
\mathfrak{n}oindent As an immediate consequence of the above considerations, we can characterize spherical curvature lines of a discrete Legendre map:
\begin{prop}\label{spherical_osculating}
The $(1)$-coordinate lines of a nowhere circular discrete Legendre map are spherical if and only if the osculating complexes along each $(1)$-coordinate line are constant.
\end{prop}
\subsection{Ribaucour transformations of discrete channel surfaces}
Curvature spheres of discrete channel surfaces are constant in the circular direction and hence the corresponding contact elements along the circular parameter lines lie in a parabolic linear sphere complex.
We observe the following property if the spheres of an R-congruence lie in a parabolic subcomplex:
\begin{prop}\label{prop_const_curvsphere_envelope}
A discrete R-congruence admits an envelope with constant curvature spheres along each $(1)$-coordinate line if and only if along each $(1)$-coordinate line the R-spheres lie in a parabolic complex and the map $n^{(2)}$ is constant along each $(1)$-coordinate ribbon.
\end{prop}
\begin{proof}
Suppose that $r:\mathcal{V}\rightarrow \mathbb{P}(\mathcal{L})$ is a discrete R-congruence admitting an envelope with a constant curvature sphere along each $(1)$-coordinate line. Then, along each such coordinate line the constant curvature sphere and the R-spheres are in oriented contact and therefore lie in a fixed parabolic linear sphere complex.
Furthermore, let us consider a $(1)$-coordinate ribbon bounded by two $(1)$-coordinate lines $\gamma_{i}$ and $\gamma_{i+1}$. Moreover, without loss of generality, we choose homogeneous coordinates such that the induced Lie inversions $\two{\sigma}_\alpha$ and $\two{\sigma}_\beta$ of two faces adjacent along the edge $(jk)$ of the coordinate ribbon are given by
\begin{equation*}
\mathfrak{n}_\alpha^{(2)}:= \mathfrak{r}_j - \mathfrak{r}_k \ \ \text{and } \ \ \mathfrak{n}_\beta^{(2)}:= \mathfrak{r}_j - \lambda_\beta \mathfrak{r}_k,
\end{equation*}
where $\lambda_\beta \in \mathbb{R}\setminus \{ 0 \}$ is a suitable constant.
Then, denoting the constant curvature spheres along the coordinate line $\gamma_{i}$ by $\one{s}_{i}$, we conclude that the spheres given by $\sigma_{\mathfrak{n}_\alpha^{(2)}}(\one{\mathfrak{s}}_i)$ and $\sigma_{\mathfrak{n}_\beta^{(2)}}(\one{\mathfrak{s}}_i)$ have to coincide. Hence, there exists a constant $c \in \mathbb{R}\setminus \{ 0\}$ such that
\begin{align*}
0&=\sigma_{\mathfrak{n}_\alpha^{(2)}}(\one{\mathfrak{s}}_{i})- c \sigma_{\mathfrak{n}_\beta^{(2)}}(\one{\mathfrak{s}}_{i})
\\&= (1-c)\one{\mathfrak{s}}_{i} + (1-c \lambda_\beta)\frac{\lspan{\one{\mathfrak{s}}_{i},\mathfrak{r}_k}}{\lspan{\mathfrak{r}_k,\mathfrak{r}_j}} \mathfrak{r}_k + (c-1)\frac{\lspan{\one{\mathfrak{s}}_{i},\mathfrak{r}_k}}{\lspan{\mathfrak{r}_k,\mathfrak{r}_j}} \mathfrak{r}_j.
\end{align*}
Therefore, since $r_j, r_k$ and $\one{s}_{i}$ are linearly independent, each scalar factor has to vanish and we conclude that $n^{(2)}_\alpha = \two{n}_\beta$. So it is constant along each $(1)$-coordinate ribbon. Geometrically, the constant $\two{n}$ is the intersection of the lines $\lspan{\one{s}_i, \one{s}_{i+1}}$ and $\lspan{r_j, r_k}$.
Conversely, assume that along each $(1)$-coordinate line the R-spheres lie in a fixed parabolic complex. Then, in particular, along each $(1)$-coordinate line all R-spheres are in oriented contact with the constant curvature sphere along this coordinate line. Furthermore, since the map $n^{(2)}$ is constant along each $(1)$-coordinate ribbon, the choice of an initial contact element containing the corresponding constant curvature sphere reveals the sought-after envelope of the R-congruence (cf.\,Lemma \ref{transport_contact}).
\end{proof}
\mathfrak{n}oindent We recall that a discrete Legendre map is a discrete channel surface in the sense of \cite{discrete_channel}, if it admits a face-cyclide congruence which is constant along one family of coordinate ribbons.
In particular, discrete channel surfaces can be characterized by special properties of their curvature sphere congruences \cite[Proposition 2.4]{discrete_channel}: a discrete Legendre map is a discrete channel surface with circular $(1)$-direction if and only if the curvature sphere congruence $s^{(1)}$ is constant along any $(1)$-coordinate line and the curvature spheres $s^{(2)}$ determine a fixed $(2,1)$-plane along each $(1)$-coordinate ribbon.
\begin{theorem}\label{thm_two_channel}
A discrete R-congruence is enveloped by two discrete channel surfaces with circular $(1)$-direction if and only if the map $n^{(2)}$ is constant along each $(1)$-coordinate ribbon and the R-spheres along each $(1)$-coordinate line are curvature spheres of a Dupin cyclide.
\end{theorem}
\begin{proof}
Let $r:\mathcal{V}\rightarrow \mathbb{P}(\mathcal{L})$ be a discrete R-congruence enveloped by two discrete channel surfaces $f$ and $\hat{f}$ with circular $(1)$-direction. Then, by Proposition \ref{prop_const_curvsphere_envelope}, the map $\two{n}$ is constant along each $(1)$-coordinate ribbon.
To prove the second property of the discrete R-congruence, let us consider a $(1)$-coordinate ribbon and denote the constant curvature spheres of $f$ and $\hat{f}$ along the two boundary $(1)$-coordinate lines by $s_i, s_{i+1}, \hat{s}_i$ and $\hat{s}_{i+1}$, respectively. Furthermore, by \cite[Proposition 2.4]{discrete_channel}, the other family of curvature spheres of $f$ along this $(1)$-coordinate ribbon lies in a $(2,1)$-plane $D_i$.
Defining the projection onto $\lspan{\hat{s}_i}^\perp$
\begin{equation*}
\pi(\tau)=\tau - \frac{\lspan{\tau, \hat{s}_i}}{\lspan{s_i, \hat{s}_i}}s_i,
\end{equation*}
we deduce that all R-spheres along the $(1)$-coordinate line lie in the $(2,1)$-plane $\pi(D_i)$. Therefore, the R-spheres are curvature spheres of a Dupin cyclide.
Conversely, suppose that the R-spheres along a $(1)$-coordinate line $\gamma_{i_0}$ are curvature spheres of a Dupin cyclide, that is, they lie in a $(2,1)$-plane $C_{i_0}$. Then, any choice of two initial contact elements $f_0:=\spann{r_{i_0}, s_{i_0}}$ and $\hat{f}_0:=\spann{r_{i_0}, \hat{s}_{i_0}}$, where $s_{i_0}, \hat{s}_{i_0} \in C_{i_0}^\perp$ provides two enveloping discrete channel surfaces.
\end{proof}
As an immediate consequence of the 1-parameter freedom in the choice of the initial contact element $f_0$ in the proof of Theorem \ref{thm_two_channel}, we obtain the following corollary:
\begin{corollary}\label{cor_channel_family}
If a discrete R-congruence admits two discrete channel surfaces as envelopes, then there exists a 1-parameter family of enveloping discrete channel surfaces.
\end{corollary}
\mathfrak{n}oindent Furthermore, we remark that Theorem \ref{thm_two_channel} reveals how the constructions given in Subsection \ref{subsection_construction} yield discrete R-congruences admitting a 1-parameter family of discrete channel surfaces in their Ribaucour families.
In particular, suppose that a discrete R-congruence consists of point spheres and satisfies the conditions of Theorem \ref{thm_two_channel}. Geometrically, those discrete R-congruences are provided by circular nets where the point spheres of one family of coordinate lines lie on circles such that two adjacent ones are related by a M\"obius inversion. Then, by Corollary \ref{cor_channel_family}, there exists a 1-parameter choice of contact elements such that the constructed principal net provides a discrete channel surface, that is, the principal net has indeed a constant curvature sphere along each coordinate line of one family (for details see also \cite{discrete_channel}).
\mathfrak{n}oindent Using the R-cyclides of a discrete Ribaucour pair given in Definition \ref{def_rib_cyclides}, we observe that the geometric structure of two enveloping discrete channel surfaces is also reflected in the geometry of the vertical faces of the Ribaucour pair:
\begin{corollary}
A Ribaucour pair consists of two discrete channel surfaces with circular $(1)$-direction if and only if along each vertical $(1)$-coordinate ribbon there exists a constant R-cyclide.
\end{corollary}
\begin{proof}
Suppose that a discrete R-congruence is enveloped by two discrete channel surfaces $f$ and $\hat{f}$ with circular $(1)$-direction. Then the contact elements of each $(1)$-vertical coordinate ribbon provide a coordinate ribbon of a discrete channel surface: according to Theorem \ref{thm_two_channel}, the curvature spheres along the vertical ribbon, namely the R-spheres, lie in a $(2,1)$-plane. Furthermore, the other curvature spheres of the vertical Legendre map, given by the curvature spheres of $f$ and $\hat{f}$, are constant (cf.\,\cite[Proposition~2.4]{discrete_channel})
Hence, since the $(1)$-vertical Legendre maps are discrete channel surfaces, there exists a constant face-cyclide along each of these coordinate ribbons, which is then by definition also an R-cyclide of the Ribaucour pair $(f, \hat{f})$.
Conversely, if along each $(1)$-vertical ribbon there exists a constant R-cyclide, the R-spheres along each ribbon lie in a fixed $(2,1)$-plane and, by Proposition \ref{prop_const_curvsphere_envelope}, the map $\one{n}$ is constant along each $(1)$-coordinate ribbon of the Ribaucour pair. Thus, the claim follows from Theorem \ref{thm_two_channel}.
\end{proof}
To conclude this section we remark on a general property of Ribaucour transforms of discrete channel surfaces. This also gives insights into the geometry of the other envelopes of a Ribaucour family containing a 1-parameter family of discrete channel surfaces.
\begin{prop}
The Ribaucour transforms of a discrete channel surface have a family of discrete spherical curvature lines.
\end{prop}
\begin{proof}
Let $f$ be a discrete channel surface with circular $(1)$-direction and denote by $r$ a discrete R-congruence of $f$. Contemplate a $(1)$-coordinate line with an adjacent coordinate ribbon: we denote by $s^{(1)}_i$ the constant curvature sphere of $f$ and by $D^{(2)}_{ij}$ the $(2,1)$-plane containing the curvature spheres of the other curvature sphere congruence along the coordinate ribbon. Then, the contact elements of $f$ along this coordinate line lie in the 3-dimensional space $s^{(1)}_i \oplus D^{(2)}_{ij}$. Hence, in particular, the R-spheres of the discrete R-congruence along this coordinate line, as well as the elements $\one{n}$ along the coordinate ribbon, lie in this space.
Therefore, the contact elements along each $(1)$-coordinate line of any envelope of $r$ lie in a fixed linear sphere complex and, by Proposition \ref{spherical_osculating}, we indeed obtain an envelope with a family of spherical curvature lines.
\end{proof}
\begin{minipage}{7cm}
\textbf{Thilo R\"orig}
\\TU Berlin, Institute of Mathematics
\\Secr. MA 8-4,
\\10623 Berlin, Germany
\\[email protected]
\end{minipage}
\begin{minipage}{6.6cm}
\textbf{Gudrun Szewieczek}
\\TU Wien
\\Wiedner Hauptstra\ss e 8-10/104
\\1040 Vienna, Austria
\\[email protected]
\end{minipage}
\end{document} |
\begin{document}
\title{Algorithms for deletion problems on split graphs}
\author{Dekel Tsur
\thanks{Ben-Gurion University of the Negev.
Email: \texttt{[email protected]}}}
\date{}
\maketitle
\begin{abstract}
In the \emph{Split to Block Vertex Deletion} and
\emph{Split to Threshold Vertex Deletion} problems the input is
a split graph $G$ and an integer $k$, and the goal is to decide whether there
is a set $S$ of at most $k$ vertices such that $G-S$ is a block graph and $G-S$
is a threshold graph, respectively.
In this paper we give algorithms for these problems whose running times are
$O^*(2.076^k)$ and $O^*(2.733^k)$, respectively.
\end{abstract}
\paragraph{Keywords} graph algorithms, parameterized complexity.
\section{Introduction}
A graph $G$ is called a \emph{split graph} if its vertex set can be partitioned
into two disjoint sets $C$ and $I$ such that $C$ is a clique and $I$ is an
independent set.
A graph $G$ is a \emph{block graph} if every biconnected component of $G$
is a clique.
A graph $G$ is a \emph{threshold graph} if there is a $t \in \mathbb{R}$
and a function $f \colon V(G) \to \mathbb{R}$ such that
for every $u,v \in V(G)$, $(u,v)$ is an edge in $G$ if and only if
$f(u)+f(v) \geq t$.
In the \emph{Split to Block Vertex Deletion} (SBVD) problem the input is
a split graph $G$ and an integer $k$,
and the goal is to decide whether there is a set $S$ of at most $k$ vertices
such that $G-S$ is a block graph.
Similarly, in the \emph{Split to Threshold Vertex Deletion} (STVD) problem
the input is a split graph $G$ and an integer $k$,
and the goal is to decide whether there is a set $S$ of at most $k$ vertices
such that $G-S$ is a threshold graph.
The SBVD and STVD problems were shown to be NP-hard by Cao et al.~\cite{cao2018vertex}.
A split graph $G$ is a block graph if and only if $G$ does not contain
an induced diamond, where a diamond is a graph with $4$ vertices and $5$ edges.
Additionally, a split graph $G$ is threshold graph if and only if $G$
does not contain an induced path with 4 vertices.
Therefore, SBVD and STVD are special cases of the 4-Hitting Set problem.
Using the fastest known parameterized algorithm for 4-Hitting Set,
due to Fomin et al.~\cite{fomin2010iterative}, the SBVD and STVD problems
can be solved in $O^*(3.076^k)$ time.
Choudhary et al.~\cite{choudhary2019vertex} gave faster algorithms for
SBVD and STVD whose running times are $O^*(2.303^k)$ and $O^*(2.792^k)$,
respectively.
In this paper we give algorithms for SBVD and STVD whose running times are
$O^*(2.076^k)$ and $O^*(2.733^k)$, respectively.
\section{Preliminaries}
For a graph $G$ and a vertex $v \in V(G)$,
$N(v)$ is the set of vertices that are adjacent to $v$.
For a set $S$ of vertices, $G-S$ is the graph obtained from $G$ by
deleting the vertices of $S$ (and incident edges).
Let $P_4$ denote a graph that is a path on 4 vertices.
In the \emph{3-Hitting Set} problem the input is a family $\mathcal{F}$ of
subsets of size at most 3 of a set $U$ and an integer $k$,
and the goal is to decide whether there is a set $X \subseteq U$
of size at most $k$ such that
$X\cap A \neq \emptyset$ for every $A \in \mathcal{F}$.
For two families of sets $\mathcal{A}$ and $\mathcal{B}$,
$\mathcal{A} \circ \mathcal{B}
= \{A\cup B \colon A \in \mathcal{A}, B \in \mathcal{B} \}$.
\subsection{Branching algorithm}
A \emph{branching algorithm} (cf.~\cite{cygan2015parameterized})
for a parameterized problem is a recursive algorithm that uses \emph{rules}.
Given an instance $(G,k)$ to the problem, the algorithm applies some rule.
In each rule, the algorithm either computes the answer to the instance $(G,k)$,
or performs recursive calls on instances
$(G_1,k-c_1),\ldots,(G_t,k-c_t)$, where $c_1,\ldots,c_t > 0$.
The algorithm returns `yes' if and only if at least one recursive call
returned `yes'.
The rule is called a \emph{reduction rule} if $t = 1$, and
a \emph{branching rule} if $t \geq 2$.
To analyze the time complexity of the algorithm, define $T(k)$ to be the
maximum number of leaves in the recursion tree of the algorithm when the
algorithm is run on an instance with parameter $k$.
Each branching rule corresponds to a recurrence on $T(k)$:
\[ T(k) \leq T(k-c_1)+\cdots+T(k-c_t). \]
The largest real root of $P(x) = 1-\sum_{i=1}^t x^{-c_i}$
is called the \emph{branching number} of the rule.
The vector $(c_1,\ldots,c_t)$ is called the \emph{branching vector} of
the rule.
Let $\gamma$ be the maximum branching number over all branching rules.
Assuming that the application of a rule takes $O^*(1)$ time,
the time complexity of the algorithm is $O^*(\gamma^k)$.
\section{Algorithm for SBVD}
\begin{lemma}\label{lem:block}
Let $G$ be a split graph with a partition $C,I$ of its vertices.
$G$ is a block graph if and only if
(1) A vertex in $I$ with degree at least $2$ is adjacent to all vertices in $C$, and
(2) There is at most one vertex in $I$ with degree at least 2.
\end{lemma}
\begin{proof}
Suppose that $G$ is a block graph.
If a vertex $v \in I$ has degree at least 2, let $a_1,a_2 \in C$ be two neighbors
of $v$. For every $b \in C\setminus \{a_1,a_2\}$, we have that $v$ is adjacent
o $b$, otherwise $v,a_1,a_2,b$ induces a diamond, contradicting the assumption that
$G$ is a block graph.
Now, suppose conversely that there are $u,v \in I$ with degree at least 2.
Let $a_1,a_2$ be two vertices in $C$. From the paragraph above,
$a_1,a_2$ are neighbors of $u$ and of $v$.
Therefore, $u,v,a_1,a_2$ induces a diamond, contradicting the assumption that
$G$ is a block graph.
To prove the opposite direction, suppose that $G$ satisfied (1) and (2).
Suppose conversely that $G$ is not a block graph.
Then there is a set of vertices $X$ that induces a diamond.
Since $C$ is a clique and $I$ is an independent set, $|X\cap I|$ is
equal to either 1 or 2.
If $|X\cap I|=1$ then $G$ does not satisfy (1), and if $|X\cap I|=2$ then
$G$ does not satisfy (2), a contradiction.
Therefore $G$ is a block graph.
\end{proof}
IF $(G,k)$ is a yes instance, let $S$ be a solution for $(G,k)$.
By Lemma~\ref{lem:block}, there is at most one vertex in $I\setminus S$ with
degree at least 2 in $G-S$.
Denote this vertex, if it exists, by $v^*$.
For every $v \in I \setminus (\{v^*\} \cup S)$ we have that
$v$ has degree at most 1 in $G-S$.
The algorithm for SBVD goes over all possible choices for the vertex
$v^* \in I$.
Additionally, the algorithm also inspects the case in which no such vertex
exist.
For every choice of $v^*$, the algorithm deletes from $G$ all the vertices of
$C$ that are not adjacent to $v^*$ and decreases the value of $k$ by the number
of vertices deleted.
When the algorithm inspects the case when $v^*$ does not exists, the graph is
not modified.
For every choice of $v^*$,
the algorithm generates an instance $(\mathcal{F},k)$ of 3-Hitting Set as follows:
For every vertex $v \in I \setminus \{v^*\}$ that has at least two neighbors,
and for every two neighbors $a,b \in C$ of $v$, the algorithm adds
the set $\{v,a,b\}$ to $\mathcal{F}$.
The algorithm then uses the algorithm of
Wahlstr{\"o}m~\cite{wahlstrom2007algorithms} to solve the instance
$(\mathcal{F},k)$ in $O^*(2.076^k)$ time.
If $(\mathcal{F},k)$ is a yes instance of 3-Hitting Set then the algorithm
returns yes.
If all the constructed 3-Hitting Set instances, for all choices of $v^*$, are no
instances, the algorithm returns no.
\section{Algorithm for STVD}
Let $I_0$ be the set of all vertices in $I$ that have minimum degree
(namely, a vertex $u \in I$ is in $I_0$ if $|N(u)|\leq |N(v)|$ for every
$v\in I$).
We say that two vertices $u,v \in I$ are \emph{twins} if $N(u) = N(v)$.
Let $\twins{v}$ be a set containing $v$ and all the twins of $v$.
Recall that a split graph $G$ is a threshold graph if and only if $G$ does not
contain an induced $P_4$.
Note that an induced $P_4$ in $G$ must be of the form $u,a,b,v$ where
$u,v \in I$ and $a,b \in C$.
The algorithm for STVD is a branching algorithm.
At each step, the algorithm applies the first applicable rule from the rules
below.
The reduction rules of the algorithm are as follows.
\begin{rrule}
If $k\leq 0$ and $G$ is not a threshold graph, return `no'.
\end{rrule}
\begin{rrule}
If $G$ is an empty graph, return `yes'.
\end{rrule}
\begin{rrule}
If $v$ is a vertex such that there is no induced $P_4$ in $G$ that contains $v$,
delete $v$.\label{rrule:no-P4}
\end{rrule}
If Rule~(R\ref{rrule:no-P4}) cannot be applied we have that
for every $a\in C$ there is a vertex $v \in I$ such that $a \notin N(v)$.
We now describe the branching rules of the algorithm.
When we say that the algorithm branches on sets $S_1,\ldots,S_p$, we mean
that the algorithm is called recursively on the instances
$(G-S_1,k-|S_1|),\ldots,(G-S_p,k-|S_p|)$.
\begin{brule}
If there are non-twin vertices $u,v \in I$ such that $|N(u)| = |N(v)| = 1$,
branch on $N(u)$ and $N(v)$.\label{brule:degree-1a}
\end{brule}
To show the safeness of Rule~(B\arabic{brule}),
denote $N(u) = \{a\}$ and $N(v) = \{b\}$.
If $S$ is a solution for the instance $(G,k)$ then $S$ must contain at least one
vertex from the induced path $u,a,b,v$.
If $u \in S$ then $S' = (S \setminus \{u\}) \cup \{a\}$ is also a solution
(since every induced $P_4$ that contains $u$ also contains $a$).
Additionally, if $v \in S$ then $(S \setminus \{v\}) \cup \{b\}$ is also a
solution.
Therefore, there is a solution $S$ such that either $a \in S$ or $b \in S$.
Thus, Rule~(B\arabic{brule}) is safe.
The branching vector of Rule~(B\arabic{brule}) is $(1,1)$.
\begin{brule}
If there is a vertex $u \in I$ such that $|N(u)| = 1$, let $v \in I$
be a vertex such that $N(u) \not\subseteq N(v)$.
Branch on $\{v\}$, $N(u)$ and $N(v)$.\label{brule:degree-1}
\end{brule}
Note that the vertex $v$ exists since Rule~(R\ref{rrule:no-P4}) cannot be
applied.
To prove the safeness of Rule~(B\arabic{brule}), note that if $S$ is a solution
for the instance $(G,k)$ then either $u\in S$, $v\in S$,
$N(u) \subseteq S$, or $N(v) \subseteq S$.
If one of the last three cases occurs we are done.
Otherwise (if $u \in S$), $S' = (S \setminus \{u\}) \cup N(u)$ is also
a solution.
It follows that Rule~(B\arabic{brule}) is safe.
Since Rule~(B\ref{brule:degree-1a}) cannot be applied, $|N(v)| \geq 2$.
Therefore, the branching vector of Rule~(B\arabic{brule}) is at least $(1,1,2)$.
Note that if Rule~(B\arabic{brule}) cannot be applied, every vertex in $I$ has
degree at least~2.
\begin{brule}
If there are vertices $u,v \in I$ such that
$|N(u) \setminus N(v)| \geq 2$ and $|N(v) \setminus N(u)| \geq 2$,
branch on $\{u\}$, $\{v\}$, $N(u) \setminus N(v)$,
and $N(v) \setminus N(u)$.\label{brule:2-2}
\end{brule}
If $S$ is a solution for the instance $(G,k)$ then either
$u \in S$, $v\in S$, $N(u) \setminus N(v) \subseteq S$, or
$N(v) \setminus N(u) \subseteq S$
(If neither of the above cases hold, let
$a \in (N(u) \setminus N(v))\setminus S$ and
$b \in (N(v) \setminus N(u))\setminus S$.
Then, $u,a,b,v$ is an induced $P_4$ in $G-S$, a contradiction).
Therefore, Rule~(B\arabic{brule}) is safe.
The branching vector of Rule~(B\arabic{brule}) is at least $(1,1,2,2)$.
\begin{lemma}\label{lem:Nu-Nv}
If Rule~(B\arabic{brule}) cannot be applied and $u,v \in I$ are two
vertices such that $|N(u)| \leq |N(v)|$
then $|N(u) \setminus N(v)| \leq 1$.
\end{lemma}
\begin{proof}
Suppose conversely that $|N(u) \setminus N(v)| \geq 2$.
Then, $|N(v) \setminus N(u)| \geq |N(u) \setminus N(v)| \geq 2$.
Therefore, Rule~(B\arabic{brule}) can be applied on $u,v$, a contradiction.
\end{proof}
We now consider two cases.
\paragraph{Case 1}
In the first case, every two vertices in $I_0$ are twins.
The algorithm picks an arbitrary vertex $u \in I_0$ and vertices
$a_1,a_2 \in N(u)$.
Since Rule~(R\ref{rrule:no-P4}) cannot be applied, there is
a vertex $v_1 \in I$ such that $a_1 \notin N(v_1)$ and
a vertex $v_2 \in I$ such that $a_2 \notin N(v_2)$.
For $i = 1,2$ we have that $v_i \notin I_0$ since $v_i$ is not a twin of $u$.
Since $u \in I_0$, it follows that $|N(v_i)| > |N(u)|$.
Thus, $|N(v_i) \setminus N(u)| > |N(u) \setminus N(v_i)| \geq 1$.
By Lemma~\ref{lem:Nu-Nv} and the fact that $a_i \in N(u) \setminus N(v_i)$
we obtain that $ N(u) \setminus N(v_i) = \{a_i\}$ and thus
$N(u) \setminus \{a_i\} \subseteq N(v_i)$.
In particular, $a_2 \in N(v_1)$ and $a_1 \in N(v_2)$.
Note that this implies that $v_1 \neq v_2$.
\begin{lemma}
$|(N(v_1) \cap N(v_2)) \setminus N(u)| \geq 2$.\label{lem:Nv1Nv2}
\end{lemma}
\begin{proof}
Suppose without loss of generality that $|N(v_1)| \leq |N(v_2)|$.
By Lemma~\ref{lem:Nu-Nv} and the fact that $a_2 \in N(v_1) \setminus N(v_2)$
we have that
$N(v_1) \setminus N(u) \subseteq N(v_2)$.
Therefore, $(N(v_1) \cap N(v_2)) \setminus N(u) = N(v_2) \setminus N(u)$.
We have shown above that $|N(v_2) \setminus N(u)| \geq 2$.
\end{proof}
\begin{brule}
If Case~1 occurs and
$a_1,a_2 \in N(w)$ for every $w \in I \setminus \{v_1,v_2\}$,
branch on $\{u\}$, $(N(v_1) \cap N(v_2)) \setminus N(u)$,
and $\{v_1,v_2\}$.\label{brule:case1-1}
\end{brule}
We now prove the safeness of Rule~(B\arabic{brule}).
In order to delete the paths of the form $u,a_1,b,v_1$ or $u,a_2,b,v_2$
for some $b \in (N(v_1) \cap N(v_2)) \setminus N(u)$,
a solution $S$ must satisfy one of the following
(1) $u \in S$
(2) $(N(v_1) \cap N(v_2)) \setminus N(u) \subseteq S$, or
(3) $S$ contains at least one vertex from $\{a_1,v_1\}$ and at least one vertex
from $\{a_2,v_2\}$.
Suppose that $S$ is a solution that satisfies~(3).
Due to the assumption of Rule~(B\arabic{brule}) and the fact that $a_1 \in N(v_2)$,
we have that every vertex in $I \setminus \{v_1\}$ is adjacent to $a_1$.
Therefore, every induced $P_4$ that contains $a_1$ is of the form $v_1,x,a_1,y$.
Thus, if $v_1 \notin S$ then $S' = (S \setminus \{a_1\}) \cup \{v_1\}$
is also a solution.
Similarly, if $v_2 \notin S$ then $S' = (S \setminus \{a_2\}) \cup \{v_2\}$
is also a solution.
Therefore, if $(G,k)$ is a yes instance, there is a solution $S$ such that
either $S$ satisfies (1) or (2) above, or $\{v_1,v_2\} \subseteq S$.
By Lemma~\ref{lem:Nv1Nv2},
the branching vector of Rule~(B\arabic{brule}) is at least $(1,2,2)$.
\begin{brule}
If Case~1 occurs, let $w \in I \setminus \{v_1,v_2\}$ be a vertex such that
$\{a_1,a_2\} \not\subseteq N(w)$,
and without loss of generality assume that $w_1 \notin N(w)$.
Branch on $\{u\}$, $(N(v_1) \cap N(v_2)) \setminus N(u)$,
and on the sets in
$\{ \{a_1\}, \{v_1\} \cup (N(w) \setminus N(u)), \{v_1,w\} \}
\circ \{ \{a_2\},\{v_2\} \}$.\label{brule:case1-2}
\end{brule}
We now show the safeness of Rule~(B\arabic{brule}).
In order to delete the induced paths of the form $u,a_1,b,v_1$ or $u,a_2,b,v_2$
for $b \in (N(v_1) \cap N(v_2)) \setminus N(u)$,
a solution $S$ must satisfy (1), (2), or (3) above.
Suppose that (1) is not satisfied (namely, $u \notin S$) and that (3) is
satisfied.
Additionally, suppose that $a_1 \notin S$.
Therefore, $v_1 \in S$ and $S$ contains at least one vertex from $\{a_2,v_2\}$.
In order to delete the induced paths of the form $u,a_1,c,w$ for
every $c \in N(w) \setminus N(u)$,
either $w \in S$ or $N(w)\setminus N(u) \subseteq S$.
We have that $w \notin I_0$ since $w$ is not a twin of $u$.
Since $u \in I_0$, it follows that $|N(w)| > |N(u)|$.
Thus, $|N(w) \setminus N(u)| > |N(u) \setminus N(w)| \geq 1$.
From the previous inequality, Lemma~\ref{lem:Nv1Nv2},
and the fact that $v_1,v_2,a_2 \notin N(w) \setminus N(u)$,
it follows that the branching vector of Rule~(B\arabic{brule})
is at least $(1,2,2,4,3,2,4,3)$.
\paragraph{Case 2}
In the second case, there are non-twin vertices in $I_0$.
Suppose that $u_1,u_2 \in I_0$ are non-twin vertices,
where the choice of $u_1,u_2$ will be given later.
By Lemma~\ref{lem:Nu-Nv},
$|N(u_1) \setminus N(u_2)| = |N(u_2) \setminus N(u_1)| = 1$.
Denote $N(u_1) \setminus N(u_2) = \{a_1\}$ and
$N(u_2) \setminus N(u_1) = \{a_2\}$.
Let $I_1 = I_0 \setminus (\twins{u_1} \cup \twins{u_2})$.
\begin{lemma}\label{lem:sunflower}
If $I_1 \neq \emptyset$ then either
(1) for every $u \in I_1$,
$N(u)$ consists of $N(u_1) \cap N(u_2)$ plus an additional vertex
that is not in $\{a_1,a_2\}$,
or
(2) for every $u \in I_1$,
$N(u)$ consists of $a_1$, $a_2$, and all the vertices of $N(u_1) \cap N(u_2)$
except one vertex.
\end{lemma}
\begin{proof}
We first claim that every $u \in I_1$, $|N(u) \cap {a_1,a_2}|$ is either 0 or 2.
Suppose conversely that $N(u)$ contains exactly one vertex from $a_1,a_2$ and
without loss of generality, $a_2 \in N(u)$ and $a_1 \notin N(u)$.
By Lemma~\ref{lem:Nu-Nv} on $u,u_1$ we obtain that $N(u)$ contains all the
vertices in $N(u_1) \setminus \{a_1\} = N(u_2) \setminus \{a_2\}$.
Since we assumed that $a_2 \in N(u)$, we have that $N(u_2) \subseteq N(u)$.
From the fact that $|N(u_2)| = |N(u)|$ we obtain that $N(u_2) = N(u)$,
contradicting the assumption that $u$ is not a twin of $u_2$.
Therefore, $|N(u) \cap {a_1,a_2}|$ is either 0 or 2.
We first assume that there is no vertex $u \in I_1$ such that
$a_1,a_2 \in N(u)$.
From the claim above we have that $a_1 \notin N(u)$.
By Lemma~\ref{lem:Nu-Nv} on $u,u_1$, $N(u)$ contains all the vertices in
$N(u_1) \setminus \{a_1\} = N(u_1)\cap N(u_2)$ plus an additional vertex
that is not in $\{a_1,a_2\}$.
Now suppose that there is a vertex $u_3 \in I_1$ such that $a_1,a_2 \in N(u_3)$.
Consider some $u \in I_1$. We claim that $a_1 \in N(u)$.
Suppose conversely that $a_1 \notin N(u)$.
From the claim above, $a_2 \notin N(u)$.
Therefore, $a_1,a_2 \in N(u_3)\setminus N(u)$,
contradicting Lemma~\ref{lem:Nu-Nv}.
Thus, $a_1 \in N(u)$. From the claim above and Lemma~\ref{lem:Nu-Nv} we conclude
that $N(u)$ contains $a_2$ and all the vertices in $N(u_1) \cap N(u_2)$ except
one vertex.
\end{proof}
If $I_1 = \emptyset$ or the first case of Lemma~\ref{lem:sunflower} occurs,
we say that the vertices of $I_0$ form a \emph{sunflower}.
Note that if the vertices of $I_0$ do not form a sunflower,
for every vertex $a \in N(u_1) \cup N(u_2)$ there are non-twin vertices
$u,u' \in I_0$ that are adjacent to $a$.
\begin{brule}
If there are non-twin vertices $u_1,u_2 \in I_0$ such that $a_1,a_2 \in N(w)$
for every $w \in I \setminus I_0$, then suppose without loss of generality that
$|\twins{u_1}| \leq |\twins{u_2}|$.
Branch on $\twins{u_1}$ and $\{a_1\}$.\label{brule:case2-1}
\end{brule}
To prove the safeness of Rule~(B\arabic{brule}), suppose that $(G,k)$ is a yes
instance and let $S$ be a solution.
If $a_1 \in S$ or $\twins{u_1} \subseteq S$ we are done, so suppose that
that $a_1 \notin S$ and $\twins{u_1} \not\subseteq S$.
We can assume that $S \cap \twins{u_1} = \emptyset$ (otherwise,
$S' = S \setminus \twins{u_1}$ is also a solution).
Since $u'_1,a_1,a_2,u'_2$ is an induced path for every $u'_1 \in \twins{u_1}$
and $u'_2 \in \twins{u_2}$, either $a_2 \in S$ or $\twins{u_2} \subseteq S$.
Note that we can assume that if $S$ contains at least one vertex from
$\twins{u_2}$ then it contains all the vertices of $\twins{u_2}$.
Define a set $S'$ by taking the vertices in
$S \setminus (\{a_2\} \cup \twins{u_2})$.
Additionally, if $a_2 \in S$, add $a_1$ to $S'$, and
if $\twins{u_2} \subseteq S$, add $\twins{u_1}$ to $S'$.
We now show that $S'$ is also a solution.
Since $|\twins{u_1}| \leq |\twins{u_2}|$, we have that $|S'| \leq |S| \leq k$.
Suppose conversely that $G-S'$ contains an induced $P_4$ and denote this path
by $P'$.
Create a path $P$ by taking $P'$ and performing the following steps:
(1) If $a_1$ is in $P'$, replace it with $a_2$.
(2) If $a_2$ is in $P'$, replace it with $a_1$.
(3) If $P'$ contains a vertex $u'_1 \in \twins{u_1}$, replace it with $u_2$.
(4) If $P'$ contains a vertex $u'_2 \in \twins{u_2}$, replace it with $u_1$.
Recall that $a_1,a_2 \in N(w)$ for every $w \in I \setminus I_0$.
Additionally, for every $u \in I_1$, $a_1,a_2 \notin N(u)$ if
the vertices of $I_0$ form a sunflower, and $a_1,a_2 \in N(u)$ otherwise.
Therefore, for every two vertices $x',y'$ in $P'$ and the corresponding vertices
$x,y$ in $P$, we have that $(x,y)$ is an edge if and only if $(x',y')$ is
an edge.
It follows that $P$ is also an induced path in $G$.
From the assumptions that $a_1 \notin S$ and $\twins{u_1} \cap S = \emptyset$
and from the definition of $S'$ we have that $S$ does not contain a vertex of
$P$.
This contradicts the assumption that $S$ is a solution.
Therefore, $S'$ is a solution. The solution $S'$ contains either $\twins{u_1}$
or $\{a_1\}$, and therefore Rule~(B\arabic{brule}) is safe.
The branching vector of Rule~(B\arabic{brule}) is at least $(1,1)$.
Now suppose that Rule~(B\ref{brule:case2-1}) cannot be applied.
We choose non-twin vertices $u_1,u_2 \in I_0$,
a vertex $a \in N(u_1) \cap N(u_2)$, and a vertex $v \in I \setminus I_0$ that
is not adjacent to $a$ as follows.
\begin{enumerate}
\item
If the vertices of $I_0$ form a sunflower,
pick arbitrary non-twin vertices $u_1,u_2 \in I_0$.
Pick $a \in N(u_1) \cap N(u_2)$.
Since Rule~(R\ref{rrule:no-P4}) cannot be applied, there is a vertex $v\in I$
such that $a \notin N(v)$.
Since $a \in N(u)$ for every $u \in I_0$
(as the vertices of $I_0$ form a sunflower)
it follows that $v \in I \setminus I_0$.
\item
Otherwise, since Rule~(B\ref{brule:case2-1}) cannot be applied,
there is a vertex $v \in I \setminus I_0$ such that
$\bigcup_{u\in I_0} N(u) \not\subseteq N(v)$.
Pick $a \in (\bigcup_{u\in I_0} N(u)) \setminus N(v)$.
Since the vertices of $I_0$ do not form a sunflower,
there are non-twin vertices $u_1,u_2 \in I_0$ such that
$a \in N(u_1) \cap N(u_2)$.
\end{enumerate}
Since Rule~(B\ref{brule:case2-1}) cannot be applied, there is a vertex
$w \in I \setminus I_0$ such that, without loss of generality,
$a_1 \notin N(w)$.
\begin{brule}
Branch on $\{u_1\}$, $(N(v) \cap N(w)) \setminus N(u_1)$,
and on the sets in
$\{ \{a\},\{v\} \} \circ \{ \{a_1\},\{w,a_2\},\{w,u_2\} \}$.
\end{brule}
The proof of the safeness of Rule~(B\arabic{brule}) is similar to the proof
for Rule~(B\ref{brule:case1-2}).
To bound the branching vector of Rule~(B\arabic{brule}) we use the following lemma.
\begin{lemma}
$|(N(v) \cap N(w)) \setminus N(u_1)| \geq 2$.\label{lem:NvNw}
\end{lemma}
\begin{proof}
Since $|N(v)| > |N(u_1)|$, we have that
$|N(v) \setminus N(u_1)| > |N(u_1) \setminus N(v)| \geq 1$.
Similarly,
$|N(w) \setminus N(u_1)| > |N(u_1) \setminus N(w)| \geq 1$.
By Lemma~\ref{lem:Nu-Nv} and the fact that $a_1 \in N(u_1) \setminus N(w)$
we have that $a \in N(w)$.
We consider two cases.
If $|N(v)| > |N(w)|$ then by Lemma~\ref{lem:Nu-Nv} and the fact that
$a \in N(w) \setminus N(v)$ we have that
$N(w) \setminus N(u_1) \subseteq N(w) \setminus \{a\} \subseteq N(v)$.
Therefore, $(N(v)\cap N(w)) \setminus N(u_1) = N(w) \setminus N(u_1)$
and the lemma follows since $|N(w) \setminus N(u_1)| \geq 2$.
If $|N(v)| \leq |N(w)|$ then by Lemma~\ref{lem:Nu-Nv} and the fact that
$a_1 \in N(v)\setminus N(w)$ we have that
$N(v) \setminus N(u_1) \subseteq N(v) \setminus \{a_1\} \subseteq N(w)$.
Therefore, $(N(v)\cap N(w)) \setminus N(u_1) = N(v)\setminus N(u_1)$
and the lemma follows since $|N(v) \setminus N(u_1)| \geq 2$.
\end{proof}
By Lemma~\ref{lem:NvNw},
the branching vector of Rule~(B\arabic{brule}) is at least $(1,2,2,2,3,3,3,3)$.
The rule with largest branching number is Rule~(B\ref{brule:2-2}) and
its branching number is at most 2.733.
Therefore, the running time of the algorithm is $O^*(2.733^k)$.
\end{document} |
\begin{document}
\title{Quantified Propositional G\"odel
Logics\protect\footnotetext{\protect\href{http://e-math.ams.org/msc/}\thispagestyle{ref}\pagestyle{headings}
\begin{abstract}
It is shown that \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}, the quantified propositional G\"odel logic based
on the truth-value set $V_\uparrow = \{1 - 1/n : n \ge 1\}\cup\{1\}$,
is decidable. This result is obtained by reduction to B\"uchi's
theory S1S. An alternative proof based on elimination of quantifiers
is also given, which yields both an axiomatization and a
characterization of \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}{} as the intersection of all finite-valued
quantified propositional G\"odel logics.
\end{abstract}
\section{Introduction}
In 1932, G\"odel~\cite{goedel} introduced a family of finite-valued
propositional logics to show that intuitionistic logic does not have a
characteristic finite matrix. Dummett~\cite{dummett} later
generalized these to an infinite set of truth-values, and showed that
the set of its tautologies {\bf LC} is axiomatized by intuitionistic
logic extended by the linearity axiom $(A \impl B) \lor (B \impl A)$.
G\"odel-Dummett logic naturally turns up in a number of different
areas of logic and computer science. For instance, Dunn and
Meyer~\cite{DM} pointed out its relation to relevance logic;
Visser~\cite{visser} employed it in investigations of the provability
logic of Heyting arithmetic; Pearce used it to analyze inference in
extended logic programming \cite{PearceD:99}; and eventually it was
recognized as one of the most important formalizations of fuzzy logic
\cite{hajek}.
The propositional G\"odel logics are well understood: Any infinite set
of truth-values characterizes the same set of tautologies. {\bf
LC}~is also characterized as the intersection of the sets of
tautologies of all finite-valued G\"odel logics
$\mathbf{G}_k$~\cite{dummett}, and as the logic determined either by
linearly ordered Kripke frames or linearly ordered Heyting
algebras~\cite{horn}.
When G\"odel logic is extended beyond pure propositional logic,
however, the situation is more complex. For
the cases of propositional entailment and extension to first-order
validity, infinite truth-value sets with different order types
determine different logics with different properties. There are
infinitely many sets of truth values which give rise to distinct
logics. As an example, consider the truth-value sets
\begin{eqnarray*}
V_\infty & = &[0, 1]\\
V_\downarrow & =& \{0\} \cup \{1/n : n \ge 1\} \\
V_\uparrow & = &\{1\} \cup \{1-1/n : n \ge 1\}\\
V_k & = & \{1\} \cup \{1-1/n : n = 1, \dots, k-1\}
\end{eqnarray*}
Propositional entailment with respect to $V_\infty$ is compact, but
not with respect to $V_\downarrow$ or $V_\up$. If a formula $A$ is
entailed by a set $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ with respect to $V_k$ for every $k$, then
it is also entailed with respect to $V_\up$, but not necessarily with
respect to $V_\infty$ or $V_\down$ \cite{BaazZach}. Similarly, the
first-order logic based on $V_\infty$ is axiomatizable (this is
Takeuti and Titani's intuitionistic fuzzy logic \cite{takeuti}), while
those based on $V_\up$ and $V_\downarrow$ are not
\cite{baaz-leitsch-zach}. The first-order G\"odel logic based on
$V_\up$ is the intersection of all finite-valued
first-order G\"odel logics.
Another interesting generalization of propositional logic is obtained
by adding quantifiers over propositional variables. In classical
logic, propositional quantification does not increase expressive power
per se. It does, however, allow expressing complicated properties
more naturally and succinctly, e.g., satisfiability and validity of
formulas are easily expressible within the logic once such quantifiers
are available. This fact can be used to provide efficient proof
search methods for several non-monotonic reasoning
formalisms~\cite{EETW}.
For G\"odel logic the increase in expressive power is witnessed by the
fact that statements about the topological structure of the set of
truth-values (taken as infinite subsets of the real interval $[0,1]$)
can be expressed using propositional
quantifiers~\cite{baaz-veith-98b}. In~\cite{baaz-veith-98b} it is
also shown that there is an uncountable number of different quantified
propositional infinite-valued G\"odel logics. The same paper
investigates the quantified propositional G\"odel logic
$\mathbf{G}^\mathrm{qp}_\infty$ based on the set of truth-values $[0, 1]$,
which was shown to be decidable. It is of some interest to
characterize the intersection of all finite-valued quantified
propositional G\"odel logics. As was pointed out in
\cite{baaz-veith-98b}, $\mathbf{G}^\mathrm{qp}_\infty$ does not provide such a
characterization.
In this paper we study the quantified propositional G\"odel logic $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}$
based on the truth-value set~$V_\uparrow$. We show that \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}{} is
decidable. In general, it is not obvious that a quantified
propositional logic is decidable or even axiomatizable. For instance,
neither the closely related quantified propositional intuitionistic
logic, nor the set of valid first-order formulas on the truth-value
set $V_\uparrow$ are r.e. Although our result can be obtained by
reduction to B\"uchi's monadic second order theory of one successor
S1S \cite{buechi}, we also give a more informative proof based on
elimination of propositional quantifiers. This proof allows us to
characterize \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}{} as the intersection of all finite-valued quantified
propositional G\"odel logics, and moreover yields an axiomatization
of~\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}.
A remark is in order about the relationship between the approach taken
here using truth-value semantics and Kripke semantics. As was pointed
out above, {\bf LC} is often defined as the propositional logic of
linearly ordered Kripke frames. In Kripke semantics, quantified
propositional \ensuremath{\mathbf{LC}}{} would then result by adding quantifiers over
propositions (subsets of the set of worlds closed under
accessibility). Here different classes of linear Kripke structures
which all define {\bf LC} in the pure propositional case in general do
not define the same quantified propositional logic. In particular,
the logic obtained by just taking Kripke models of order type $\omega$
is not the same as that defined by the class of all finite linear
orders. It follows from the results of this paper that the logic of
all finite linear Kripke structures coincides with~\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}.
\section{G\"odel Logics} \label{sec:goedel}
\paragraph{Syntax.}
We work in the language of propositional logic containing a countably
infinite set $\mathop\mathit{Var} = \{p, q, \ldots \}$ of (propositional)
variables, the constants $\bot, \top$, as well as the connectives
$\land, \lor$, and $\impl$. Propositional variables and constants are
considered atomic formulas. Uppercase letters will serve as
meta-variables for formulas. If $A(p)$ is a formula containing the
variable $p$ free, then $A(X)$ denotes the formula with all
occurrences of the variable $p$ replaced by the formula $X$. $Var(A)$
is the set of variables occurring in the formula $A$.
We use the abbreviations $\neg A$ for $A \supset \bot$
and $A \leftrightarrow B$ for $(A \impl B) \land (B \impl A)$.
\paragraph{Semantics.}
The most important form of G\"odel logic is defined over the real unit
interval $V_\infty = [0,1]$; in a more general framework, the
truth-values are taken from a set $V$ such that $\{0,1\} \subseteq V
\subseteq [0,1]$. In the case of $k$-valued G\"odel logic ${\bf
G}_k$, we take $V_k = \{1 - 1/i : i = 1, \ldots, k-1\} \cup \{1 \}$.
The logic we will be most interested in is based on the set $V_\up =
\{1 -1/i : i \ge 1\} \cup \{1\}$.
A \emph{valuation}~$v\colon \mathop\mathit{Var} \to V$ is an assignment of values in
$V$ to the propositional variables. It can be extended to formulas using
the following truth functions introduced by G\"odel~\cite{goedel}:
\[
\begin{array}{cc}
\begin{array}{rcl}
v(\bot) &=& 0 \\
v(\top) &=& 1 \\
v(A \land B) &=& \min(v(A), v(B))
\end{array} &
\begin{array}{rcl}
v(A \lor B) &=& \max(v(A), v(B))\\
v(A \supset B) &=&
\begin{cases}
1 & {\rm if\ } v(A) \leq v(B) \\
v(B) & {\rm otherwise}
\end{cases}
\end{array}
\end{array}
\]
A formula $A$ is a \emph{tautology} over a truth-value set $V
\subseteq [0,1]$ if for all valuations $v\colon \mathop\mathit{Var} \to V$, $v(A) =
1$. The \emph{propositional logics} $\mathbf{LC}$, $\mathbf{G}_\up$
and $\mathbf{G}_k$ are the sets of tautologies over the corresponding
truth value sets, e.g., $\mathbf{LC} = \mathbf{G}_\infty = \{A : A
\textrm{ a tautology over }V_\infty\}$. We also write $\mathbf{G}
\models A$ for $A \in \mathbf{G}$ ($\mathbf{G} \in \{\mathbf{LC},
\mathbf{G}_\up, \mathbf{G}_k\}$).
It is easily seen that $\mathbf{LC} \supseteq \mathbf{G}_\up \supseteq
\mathbf{G}_k$. Dummett \cite{dummett} showed that $\mathbf{LC} =
\mathbf{G}_\up$ and that $\mathbf{LC} = \bigcap_{k\ge 2}
\mathbf{G}_k$.
The abbreviation $A \prec B$ for $(A \impl B) \land ((B \impl A) \impl
A)$ will be used extensively below. It expresses strict linear order
in the sense that
\[
v(A \prec B) =
\begin{cases}
1 & \mathrm{if\ } v(A) < v(B) \mathrm{\ or\ } v(B) = 1\\
\min(v(A),v(B)) & {\rm otherwise} \end{cases}
\]
\paragraph{Propositional Quantification.}
In {\em classical} propositional logic we define $(\exists p) A(p)$ by
$A(\bot) \lor A(\top)$ and $(\forall p) A(p)$ by $A(\bot) \land
A(\top)$. In other words, propositional quantification is
semantically defined by the supremum and infimum, respectively, of
truth functions (with respect to the usual ordering ``$0 < 1$'' over
the classical truth-values $\{0,1\}$). This can be extended to
G\"odel logic by using {\em fuzzy quantifiers}. Syntactically, this
means that we allow formulas $(\forall p) A$ and $(\exists p) A$ in
the language. Free and bound occurrences of variables are defined in
the usual way. Given a valuation $v$ and $w \in V$, define $v[w/p]$
by $v[w/p](p) = w$ and $v[w/p](q) = v(q)$ for $q \not\equiv p$. The
semantics of fuzzy quantifiers is then defined as follows:
\[
v((\exists p) A) = \sup \{ v[w/p](A) : w \in V \} \hspace{4ex}
v((\forall p) A) = \inf \{ v[w/p](A) : w \in V\}
\]
When we consider quantifiers, $V$ has to be closed under
infima and suprema, since otherwise truth values for quantified
formulas are not defined.
We also add the additional unary connective $\hbox{\Large$\circ$}$ to the
language. The truth function for $\hbox{\Large$\circ$}$ is given by $v(\hbox{\Large$\circ$} A) =
v((\forall p) ((p \impl A) \vee p))$. In \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}, this makes
\[
v(\hbox{\Large$\circ$} A) =\begin{cases} 1 & {\rm if\ } v(A) = 1 \\
1 - \frac{1}{n+1} & {\rm if\ } v(A) = 1 - \frac{1}{n} \cr
\end{cases}
\]
We abbreviate $\hbox{\Large$\circ$} \dots \hbox{\Large$\circ$} A$ ($n$ occurrences of $\hbox{\Large$\circ$}$) by
$\hbox{\Large$\circ$}^n A$.
Using the above definitions, it is straightforward to extend the
notion of tautologyhood to the new language. We write $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}$
($\mathbf{G}_\infty^\mathrm{qp}$, $\mathbf{G}_k^\mathrm{qp}$) for the set of
tautologies in the extended language over $V_\up$ ($V_\infty$, $V_k$).
We will show below that every quantified propositional formula is
equivalent in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}{} to a quantifier-free formula, which in general can
contain~$\hbox{\Large$\circ$}$. $\hbox{\Large$\circ$} A$ itself (or the equivalent formula
$(\forall p)((p \impl A) \lor p)$), however, is not in general
equivalent to a quantifier-free formula not containing~$\hbox{\Large$\circ$}$.
Inspection of the truth tables shows that a quantifier-free formula
containing only the variable~$q$ takes one of $0$, $v(q)$, or 1 as its
value under a given valuation $v$, and thus no such formula can
define~$\hbox{\Large$\circ$} q$.
\section{Hilbert-style Calculi}
All the calculi we consider are based on the following set of axioms:
\[
\begin{array}{l@{\quad}l@{\qquad}l@{\quad}l}
\mathrm{I1} & A \impl (B \impl A) &
\mathrm{I7} & (A \land \neg A) \impl B\\
\mathrm{I2} & (A \land B) \impl A &
\mathrm{I8} & (A \impl \neg A) \impl \neg A\\
\mathrm{I3} & (A \land B) \impl B &
\mathrm{I9} & \bot \impl A\\
\mathrm{I4} & A \impl (B \impl (A \land B)) &
\mathrm{I10} & A \impl \top\\
\mathrm{I5} & A \impl (A \lor B) &
\mathrm{I11} & (A \impl (B \impl C)) \impl ((A \impl B) \impl (A \impl C))\\
\mathrm{I6} & B \impl (A \lor B) &
\mathrm{I12} & ((A \impl C)\land (B \impl C)) \impl ((A \lor B) \impl C)\\
\end{array}
\]
These axioms, together with the rule of modus ponens, define the
system \log{IPC} that is sound and complete for intuitionistic
propositional logic. The system~\log{LC} is obtained by adding
to~\log{IPC} the linearity axiom
\[
\mathrm{LC} \quad (A \impl B) \lor
(B \impl A).
\]
It is well known \cite{dummett} that \log{IPC} and
\log{LC} are sound for all propositional G\"odel logics, and that
\log{LC} is complete for all infinite-valued propositional G\"odel
logics. We will make frequent use of this fact below, and omit
derivations of formulas which are (instances of) quantifier- and
$\hbox{\Large$\circ$}$-free tautologies in $\mathbf{G}_\up$. These omissions are
indicated by pointing out that the formula follows already in \log{LC}
or \log{IPC}. In particular, familiar inference patterns such as the
chain rule or case distinction are derivable in \log{LC} and its
extensions.
When we turn to quantified propositional logics, a natural
system~$\log{IPC}^\mathrm{qp}$ to start with is obtained by adding to~\log{IPC}
the following two axioms:
\[
{\impl}{\exists} \quad A(C) \impl (\exists p) A(p) \qquad\qquad
{\impl}{\forall} \quad (\forall p) A(p) \impl A(C)
\]
and the rules:
\[
\frac{A(p) \impl B^{(p)}}{(\exists p)A(p) \impl B^{(p)}}
\mathrm{R}{\exists}
\quad \quad \quad
\frac{B^{(p)} \impl A(p)}{B^{(p)} \impl (\forall p) A(p)}
\mathrm{R}{\forall}
\]
where for any formula $C$, the notation $C^{(p)}$ indicates that $p$
does not occur free in $C$, i.e., $p$ is a (propositional) {\em
eigenvariable}.
Let $\mathsf{Q}G$ be the system obtained by adding to~$\log{IPC}^\mathrm{qp}$ the
axioms (LC),
\[
{\forall}{\lor}\qquad (\forall p) [A \lor B(p))] \impl [A \lor (\forall p) B(p)]
\]
where $p \notin A$, and the following:
\[
\begin{array}{l@{\quad}l@{\quad}l@{\quad}l}
\mathrm{G1} & \hbox{\Large$\circ$} (A \impl B) \leftrightarrow (\hbox{\Large$\circ$} A \impl \hbox{\Large$\circ$} B) &
\mathrm{G4} & (A \impl \hbox{\Large$\circ$} B) \impl ((A \impl C) \vee (C \impl B)) \\
\mathrm{G2} & A \prec \hbox{\Large$\circ$} A &
\mathrm{G5} & (A \leftrightarrow \bot) \vee (\exists p) (A \leftrightarrow \hbox{\Large$\circ$} p) \\
\mathrm{G3} & (\hbox{\Large$\circ$} A \impl \hbox{\Large$\circ$} B) \impl ((A \impl B) \vee \hbox{\Large$\circ$} B) &
\mathrm{G6} & (A \prec B) \impl (\hbox{\Large$\circ$} A \impl B)
\end{array}
\]
\begin{proposition}
The system $\mathsf{Q}G$ is sound for $\mathbf{G}_k^\mathrm{qp}$ and~$\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}$.
\end{proposition}
\begin{proof} It is easily seen that the rules of inference preserve
validity. For instance, if $B \impl A(p)$ is valid, then, for any
valuation~$v$, $v[w/p](B) \le v[w/p](A(p))$ where $w \in V$. If $p$
does not occur in $B$, then $v(B) = v[w/p](B)$ and we have $v(B) \le
\inf \{v[w/p](A(p)) : w \in V\}$. That $\log{LC}$ is sound for
arbitrary G\"odel logics was shown in~\cite{dummett}. The tedious but
straightforward verification that the remaining axioms
(${\lor}{\forall}$) and (G1)--(G6) are valid is left to the reader.
\end{proof}
\begin{remark}
In \cite{baaz-veith-98b} it was shown that a system sound and complete
for $\mathbf{G}_\infty^\mathrm{qp}$, the quantified propositional G\"odel
logic based on the truth-value set $[0, 1]$, is obtained by
extending~$\log{IPC}^\mathrm{qp}$ with (LC), (${\lor}{\forall}$) and the axiom
\[
(\forall p)[(A^{(p)} \impl p) \lor (p \impl B^{(p)})]
\impl \ (A^{(p)} \impl B^{(p)}).
\]
This schema is not valid in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}{} (it comes out $=0$ under any $v$ with
$v(A) = 1/2$ and $v(B) = 0$). On the other hand, it is easy to see
that $v(\hbox{\Large$\circ$} A) = v(A)$ in $V_\infty$, and hence axiom (G2) is not
valid in $\mathbf{G}_\infty^\mathrm{qp}$. Thus neither of
$\mathbf{G}_\infty^\mathrm{qp}$ and \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}{} is included in the other. This is in
contrast to the situation in propositional entailment and first-order
logic, where $V_\infty$ defines the smallest G\"odel logic and is
included in all others.
\end{remark}
\section{Decidability}
In this section we prove that $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}$ is decidable. This is done by
defining a reduction of tautologyhood in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}{} to S1S, the monadic
theory of one successor, which was shown to be decidable
by~B\"uchi~\cite{buechi}.
S1S is the set of second-order formulas in the language with
second-order quantification restricted to monadic set variables $X$,
$Y$, \dots{} with one unary function $'$ (successor) which are true in
the model~$\langle \omega, '\rangle$. For the purposes of this
section we consider $\hbox{\Large$\circ$} A$ to be an abbreviation of $(\forall p)
((p \impl A) \vee p)$.
Suppose $A$ is a quantified propositional formula, and
$B$ is a formula in the language of S1S with only $x$ free.
Let $TV(B(x))$ abbreviate $(\forall z)(B(z') \impl B(z))$.
We define $A^x$ by:
\begin{eqnarray*}
p^x & = & X_p(x) \\
\bot^x & = & X_\bot(x) \\
\top^x & = & (\forall z)(z=z) \\
(B \land C)^x & = & B^x \land C^x \\
(B \lor C)^x & = & B^x \lor C^x\\
(B \impl C)^x & = & (\forall y)(B^y \impl C^y) \lor
(\exists y)(B^y \land \neg C^y) \land C^x \\
(\forall p)B^x & = & (\forall X_p)(TV(X_p(x)) \impl B^x) \\
(\exists p)B^x & = & (\exists X_p)(TV(X_p(x)) \land B^x)
\end{eqnarray*}
Consider the following reduction:
\[
\Phi(A) = (\forall X_\bot)((\forall x)\neg X_\bot(x) \impl (\forall x)A^x)
\]
The idea behind this is to correlate truth-values in $V_\up$ with
subsets of~$\omega$ which are closed under predecessor, i.e.,
predicates in
\[
TV = \{P \subseteq \omega : \textrm{if\ } n \in P \textrm{\ then\ } m \in P \textrm{\ for all\ } m \le n\}.
\]
Under this correlation, $1$ corresponds to $\omega$, and $1 - 1/n$
corresponds to $\{1, \ldots, n\}$.
Let $s$ be an interpretation of the language of S1S, mapping variables
to elements or subsets of~$\omega$. We denote by $s[n/x]$ the
interpretation which is just like $s$ except that it assigns $n$ to
$x$. Then $TV(A(x))$ obviously expresses the condition that the
predicate~$A(x)[s] = \{n : S1S \models A(x) [s[n/x]]\}$ defined by
$A(x)$ in $s$ is closed under predecessor.
If a monadic predicate~$P$ is closed under predecessor, we define its
truth value by
\[
tv(P) = \sup \{1 - \frac{1}{n} : 1^n \in P\}.
\]
Conversely, every truth-value~$v \in V_\uparrow$ corresponds to a
monadic predicate
\[
mp(v) = \begin{cases}\{k : k \le n\} & \textrm{if\ } v = 1 - 1/n\\
\omega & \textrm{if\ } v = 1.
\end{cases}
\]
Note that for $P, Q \in TV$, $P \subseteq Q$ iff $tv(P)
\le tv(Q)$, and conversely, for $v, w \in V_\uparrow$, $v \le w$ iff
$mp(v) \subseteq mp(w)$.
\begin{lemma}\label{gifs1s}
Let $v$ be a valuation and $s$ be the interpretation defined by
$s(X_p) = mp(v(p))$ and $s(X_\bot) = \emptyset$. Then we have
$tv(A^x[s]) = v(A)$.
\end{lemma}
\begin{proof}
By induction on the complexity of $A$. The claim is obvious for atomic
formulas, conjunction and disjunction. If $A \equiv B \impl C$ we have
to distinguish two cases. Suppose first that $v(B) \le v(C)$.
By induction hypothesis,
$B^x[s] = mp(v(B)) \subseteq mp(v(C)) = C^x[s]$, and hence the first
disjunct in the definition of $(B \impl C)^x$ is true. Thus $(B \impl
C)^x$ defines $\omega$ and $tv((B\impl C)^x[s]) = 1$. Now
suppose that $v(B) > v(C)$. Then $tv(B^x[s]) \supsetneq tv(C^x[s])$,
$S1S \nmodels (\forall y)(B^y \impl C^y)\ [s]$ and $S1S \models
(\exists y)(B^y \land \neg C^y)\ [s]$, and thus $(B\impl C)^x[s] =
C^x[s]$.
If $A \equiv (\exists p)B$, let $v[w/p]$ be the
valuation which is just like $v$ except that $v[w/p](p) = w$, and let
$s[mp(w)/X_p]$ be the corresponding interpretation which is like $s$
except that it assigns $mp(w)$ to $X_p$.
By induction hypothesis, $tv(B^x[s[mp(w)/X_p]]) = v[w/p](B)$. We
again have two cases. Suppose first that $\sup \{v[w/p](B) : w \in
V_\uparrow\} = 1 - 1/n$. For all $m > n$, $S1S \nmodels B^x [m/x,
mp(w)/X_p]$, since $v[w/p](B^x) < 1-1/m$ by induction hypothesis. On
the other hand, $S1S \models TV(P_p) \impl B^x\ [s[k/x,
mp(1-1/n)/P_p]]$ for all $k\le n$, and so $tv((\exists p)B^x[s]) = 1 -
1/n$. Now consider the case where $\sup \{v[w/p](B) : w \in
V_\uparrow\} = 1$. Here there is no bound $n$ on the
the members of sets defined by $B^x[s[mp(w)/X_p]]$ where $w \in
V_\uparrow$. Hence, $mp((\exists p)B)^x[s]) = \omega$ and
$tv((\exists p)B^x[s]) = 1$.
The case $A \equiv (\forall p)B$ is similar.\qed
\end{proof}
\begin{lemma}\label{s1sifg}
Let $s$ be an interpretation with $s(X_\bot) = \emptyset$ and $s(X_p)
\in TV$. Let $v$ be defined by $v(p) = tv(s(X_p))$. Then $A^x[s] \in
TV$, and $v(A) = tv(A^x[s])$.
\end{lemma}
\begin{proof}
By induction on the complexity of $A$. The claim is again trivial for
atomic formulas, conjunctions or disjunctions. If $A \equiv B
\impl C$, two cases occur. If $S1S \models (\forall y)(B^y \impl
C^y)$, then $B^y[s] \subseteq C^y[s]$. By induction hypothesis, $v(B)
\le v(C)$, and hence $v(B \impl C) = 1 = tv((B\impl C)^x[s])$.
Otherwise, for some $n$ we have $n \in B^y[s]$ but $n \notin
C^y[s]$. So $(\exists y)(B^y \land \neg C^y)$ must be true and the
predicate defined is the same as~$C^y[s]$.
Now for the case $A \equiv (\exists p)B$: If $S1S \models (\exists
X_p)(TV(X_p) \impl B^x)[s[n/x]]$, then there is a prefix closed
witness $P$ so that $S1S \models B^x[s[n/x,P/X_p]]$. By induction
hypothesis, $B^x[s[P/X_p]] \in TV$, and hence $S1S \models
TV(X_p) \impl B^x\ [s[m/x,P/X_p]]$ for all $m \le n$, and thus
$((\exists p)B)^x[s] \in TV$ as well.
Consider $N = ((\exists p)B)^x[s]$. First, suppose that $\sup N = k$.
That means that for some $P \in TV$, $1^k \in B^x[s[P/X_p]]$, and for
no $Q \in TV$ and no $j > k$, $j \in B^x[s[Q/X_p]]$. By induction
hypothesis, $v[tv(P)/p](B) = 1 - 1/k$ and for all $w \in V_\uparrow$,
$v[w/p](B) \le 1 - 1/k$. Hence $v((\exists p)B) = 1 - 1/k$.
If $\sup N$ does not exist, for each $k$ there is a witness $Q_k \in
TV$ with $k \in B^x[s[Q_k/X_p]]$. By induction hypothesis, for each
$k$ we have $v[tv(Q_k)/p](B) \ge 1 - 1/k$, and so $v((\exists p)B) =
1$.
The case $A \equiv (\forall p)B$ is similar.\qed
\end{proof}
\begin{theorem} \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}{} is decidable. \end{theorem}
\begin{proof}
If there is a valuation $v$ such that $v(A)
< 1$, then by Lemma~\ref{gifs1s} there is an $s$ with $s(P_\bot) =
\emptyset$ and $n$ so that $n \notin A^x[s]$, and
hence $S1S \nmodels \Phi(A)$.
Conversely, suppose $S1S \nmodels \Phi(A)$. We may assume, without
loss of generality, that all propositional variables in $A$ are bound.
Then there is an interpretation $s$ with $X_\bot(x)[s] = \emptyset$ so
that some $n \notin A^x[s]$. By Lemma~\ref{s1sifg}, $A^x[s] \in TV$.
Hence, if $n \notin A^x[s]$, then $k \notin A^x[s]$ for all $k\ge n$,
and, also by Lemma~\ref{s1sifg}, $v(A) = tv(A^x[s]) < 1$.
Thus a formula $A$ is a tautology in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}{} iff $S1S \models \Phi(A)$.
The claim follows by the decidability of $S1S$.\qed
\end{proof}
\section{Properties and Normal Forms}
In this section we introduce suitable normal forms for formulas of
$\mathsf{Q}G$ and prove some useful properties of $\mathsf{Q}G$. These results will
be crucial in the proof of the elimination of quantifiers.
\begin{proposition}
\label{basicpr}
\begin{enumerate}
\item $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (A \impl B) \impl (\hbox{\Large$\circ$} A \impl \hbox{\Large$\circ$} B)$
\item $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash \hbox{\Large$\circ$} (A \wedge B) \leftrightarrow (\hbox{\Large$\circ$} A \wedge \hbox{\Large$\circ$} B)$
\item $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash \hbox{\Large$\circ$} (A \vee B) \leftrightarrow (\hbox{\Large$\circ$} A \vee \hbox{\Large$\circ$} B)$
\end{enumerate}
\end{proposition}
\begin{proof}
(1) From (G2) we have $(A \impl B) \impl \hbox{\Large$\circ$}(A \impl B)$, which,
together with the left-to-right direction of (G1) yields the
result.
(2) The left-to-right implication immediately follows from
axioms (I2) and (I3) together with Prop.~\ref{basicpr}(1).
For the converse, replace $B$ by $B \impl (A \land B)$
in Prop.~\ref{basicpr}(1) and use (I4) to derive
$\hbox{\Large$\circ$} A \impl \hbox{\Large$\circ$} (B \impl (A \wedge B))$.
Then, using (G1), one has $\hbox{\Large$\circ$} A \impl (\hbox{\Large$\circ$} B \impl
\hbox{\Large$\circ$} (A \wedge B))$. The claim follows by \log{IPC}.
(3) In $\log{LC}$, we have $(A \lor B) \leftrightarrow (A \impl B) \impl B)
\land (B \impl A) \impl A)$. Replacing $A$ by $\hbox{\Large$\circ$} A$ and $B$ by
$\hbox{\Large$\circ$} B$, we have $(\hbox{\Large$\circ$} A \lor \hbox{\Large$\circ$} B) \leftrightarrow (\hbox{\Large$\circ$} A \impl \hbox{\Large$\circ$}
B) \impl \hbox{\Large$\circ$} B) \land (\hbox{\Large$\circ$} B \impl \hbox{\Large$\circ$} A) \impl \hbox{\Large$\circ$} A)$. The
result follows using (G1) and \log{IPC}.\qed
\end{proof}
\begin{proposition}\label{equivsubst}
\begin{enumerate}
\item If $p$ does not occur boind in $C(p)$, then
\[ \ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\forall \bar q)(A \leftrightarrow B) \impl (C(A) \impl C(B)) \]
where $\bar q$ are the propositional variables occurring free in $A$
and $B$.
\item If $C(p)$ is quantifier-free, we also have
\[ \ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (A \leftrightarrow B) \impl (C(A) \impl C(B)) \]
\end{enumerate}
\end{proposition}
\begin{proof}
By induction on the complexity of $C$. Cases for $\land$, $\lor$, and
$\impl$ are easy. If $C(p) \equiv \hbox{\Large$\circ$} D(p)$, we use the induction
hypothesis and Prop.~\ref{basicpr}(1). If $C(p) \equiv (\exists
r)D(p, r)$, we argue:
\begin{eqnarray*}
(1) & (\forall \bar q)(A \leftrightarrow B) \impl (D(A, r) \impl D(B, r)) & \textrm{by IH}\\
(2) & \qquad ((\forall \bar q)(A \leftrightarrow B) \land D(A, r)) \impl D(B, r)) & \textrm{(1), \log{IPC}}\\
(3) & D(B, r) \impl (\exists r)D(B, r) & {\impl}{\exists}\\
(4) & (\forall \bar q)(A \leftrightarrow B) \land D(A, r)) \impl (\exists r)D(B, r) & \textrm{(2), (3)}\\
(5) & D(A, r) \impl ((\forall \bar q)(A \leftrightarrow B) \impl (\exists r)D(B, r)) & \textrm{(4), \log{IPC}}\\
(6) & (\exists r)(D(A, r) \impl ((\forall \bar q)(A \leftrightarrow B) \impl (\exists r)D(B, r))) & \textrm{(5), R}\exists \\
(7) & (\forall \bar q)(A \leftrightarrow B) \impl ((\exists r)D(A, r) \impl (\exists r)D(B, r)) & \textrm{(6), \log{IPC}}
\end{eqnarray*}
The case of $C \equiv (\forall r)D(p, r)$ is handled similarly. \qed
\end{proof}
\begin{defn}
A formula $A$ of $\mathsf{Q}G$ is in $\hbox{\Large$\circ$}$-{\em normal form} if
it is quantifier-free and for all subformulas $\hbox{\Large$\circ$} B$ of $A$, $B
\in \{\bot, \top\} \cup \mathop\mathit{Var}$ or $B \equiv \hbox{\Large$\circ$} B'$.
\end{defn}
\begin{proposition}\label{opernf}
Let $A$ be a quantifier-free formula of $\mathsf{Q}G$. Then there exists a
formula $A'$ of $\mathsf{Q}G$ in $\hbox{\Large$\circ$}$-normal form such that
$\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow A'$.
\end{proposition}
\begin{proof}
Follows from axiom (G1), Prop.~\ref{basicpr}(2) and (3)
using Prop.~\ref{equivsubst}(2).\qed
\end{proof}
\begin{proposition}
For every $n \geq 0$, $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash \hbox{\Large$\circ$}^n \top \leftrightarrow \top.$
\end{proposition}
\begin{proof}
$\hbox{\Large$\circ$}^n \top \impl \top$ is already derivable intuitionistically.
For $\top \impl \hbox{\Large$\circ$}^n \top$, use (G2), Prop.~\ref{basicpr}(1), and
induction on~$n$.\qed
\end{proof}
For propositional G\"odel logic, a normal form similar to the
disjunctive normal form of classical logic has been introduced in
\cite{baz96} (see also \cite{baaz-veith-98,baaz-veith-98b}). This
so-called {\em chain normal form} is based on the fact that, in a
sense, the truth value of a formula only depends on the ordering of
the variables occurring in the formula induced by the valuation under
consideration. The chain normal form can then be constructed by
enumerating all such orderings (using $\prec$ and $\leftrightarrow$ to encode
the ordering) in a way similar to how one constructs a disjunctive
normal form by enumerating all possible truth value assignments. We
extend the notion of chain normal form and the results of
\cite{baaz-veith-98} in order to deal with the $\hbox{\Large$\circ$}$ connective.
This is possible, since by Prop.~\ref{opernf} we can always push the
$\hbox{\Large$\circ$}$ in front of atomic subformulas, so we only need to consider
orderings of subformulas of the form $\hbox{\Large$\circ$}^j B$ with $B$ atomic. Let
$\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ be a finite subset of $\{ \hbox{\Large$\circ$}^j p, \hbox{\Large$\circ$}^j \bot : p \in
\mathop\mathit{Var}, j \in \omega\} \cup \{\top\}$ and $\top, \bot \in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$.
\begin{defn}
A $\hbox{\Large$\circ$}$-{\em chain} over $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ is an expression of the form
\[
(S_1 \star_1 S_2)
\land \cdots \land (S_{n-1} \star_{n-1} S_{n})
\]
such that $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma = \{S_1, \ldots, S_n\}$, $S_1 \equiv \bot$, $S_n
\equiv \top$, and $\star_i \in \{\leftrightarrow, \prec\}$, for all $i=1, \dots
,n$.
\end{defn}
Every $\hbox{\Large$\circ$}$-chain $C$ uniquely determines a partition $\Pi_1^C$,
\dots, $\Pi_k^C$ of $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ so that $\Pi_i^C = \{S_{j_i}, \ldots,
S_{j_{i+1}-1}\}$ where $j_1 = 1$, $j_{k+1} = n+1$, $j_{i} < j_{i+1}$,
$\star_{j_i} = \cdots = \star_{j_{i+1}-2} = {\leftrightarrow}$, and
$\star_{j_{i+1}-1} = {\prec}$. Conversely, every such partition
determines a $\hbox{\Large$\circ$}$-chain up to provable equivalences. It is easily
seen that if $C$ is such a chain, then $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \impl (S_i \leftrightarrow S_j)$
if $S_i, S_j \in \Pi_l^C$ for some $l$, and $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \impl (S_i \prec
S_{i'})$ if $S_i \in \Pi_j^C$, $S_{i'} \in \Pi_{j'}^C$ and~$j <
j'$. Thus $C$ also uniquely corresponds to an ordering of $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$
which we denote $<_C$, defined by $S_{i} <_C S_{i'}$ iff $S_i \in
\Pi^C_j$, $S_{i'} \in \Pi_{j'}^C$ and $j < j'$. This order is total,
the $\Pi_i^C$ are maximal anti-chains, $\bot$ is minimal, and $\top$
is maximal.
Suppose now that $A$ is in $\hbox{\Large$\circ$}$-normal form, and that $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$
contains all the subformulas of $A$ of the form $\hbox{\Large$\circ$}^j p$ or $\hbox{\Large$\circ$}
^j \bot$, as well as $\top$; that $C$ is an $\hbox{\Large$\circ$}$-chain on $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$;
and that the valuation $v$ agrees with $<_C$, i.e., $S_i <_C S_j$ iff
$v(S_i) < v(S_j)$. Using the same idea as in the proof of Lemma~3 in
\cite{baaz-veith-98}, one can find $A^C \in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$, the ``value'' of
$A$ under $C$, so that $v(A^C) = v(A)$, and the choice of $A^C$
depends only on $<_C$, not on $v$ itself. Specifically, $A^C$ can be
constructed as follows: (1) If $A \in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$, then $A^C \equiv
A$. (2) If $A \equiv D \land E$, then $A^C \equiv D^C$ if $D^C <_C
E^C$ and $\equiv E^C$ otherwise. (3) If $A \equiv D \lor E$, then $A^C
\equiv D^C$ if $E^C <_C D^C$, and $\equiv E^C$ otherwise. (4) If $A
\equiv D \impl E$, then $A^C \equiv E^C$ if $E^C <_C D^C$, and $\equiv
\top$ otherwise. This ``evaluation'' of $A$ is provable in the sense
that $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \impl (A \leftrightarrow A^C)$. This follows easily using
the following theorems of~\ensuremath{\mathbf{LC}}:
\[\begin{array}{rcl@{\qquad}rcl}
(D \prec E) & \impl & (D \land E \leftrightarrow D) &
(E \prec D) & \impl & (D \land E \leftrightarrow E) \\
(D \leftrightarrow E) & \impl & (D \land E \leftrightarrow D) &
(D \prec E) & \impl & (D \lor E \leftrightarrow E) \\
(E \prec D) & \impl & (D \lor E \leftrightarrow D) &
(E \leftrightarrow D) & \impl & (D \lor E \leftrightarrow E) \\
(D \prec E) & \impl & (D \impl E \leftrightarrow \top) &
(E \prec D) & \impl & (D \impl E \leftrightarrow E) \\
(E \leftrightarrow D) & \impl & (D \impl E \leftrightarrow \top)
\end{array}
\]
\begin{defn}
Let $A$ be a quantifier free formula in $\hbox{\Large$\circ$}$-normal form,
$\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma_A$ be the set of all subformulas of $A$ of the form
$\hbox{\Large$\circ$}^j p, \hbox{\Large$\circ$}^k \bot, \top$, $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma \supseteq \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma_A$, and
$C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)$ the set of all possible $\hbox{\Large$\circ$}$-chains over $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$. Then
\[
\bigvee_{C \in C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)} C \land A^{C}
\]
is the $\hbox{\Large$\circ$}$-{\em chain normal form} for $A$ over $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$.
\end{defn}
\begin{theorem}\label{chainnf}
Let $A$ and $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ be as above, and $A'$ be the $\hbox{\Large$\circ$}$-chain normal
form for $A$ over $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$. Then $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow A'$.
\end{theorem}
\begin{proof} (See also Thm.~4 of \cite{baaz-veith-98}.)
First note that $\bigvee_{C \in C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)} C$ is a tautology and provable
in~\log{LC}. Since for each $C \in C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)$ we have $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (C \land A^C)
\impl A$, the right-to-left implication $A' \impl A$ follows by case
distinction.
For the left-to-right implication, consider $A \impl (A \land \bigvee_{C \in
C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)} C)$. This is provable, since $\bigvee_{C \in
C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)} C$ is provable. By distributivity of $\land$ over $\lor$,
we have $A \impl \bigvee_{C \in C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)} (A \land C)$. We also have
$(A \land C) \impl (C \land A^C)$ for each $C \in C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)$ from
$\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \impl (A \leftrightarrow A^C)$. Together we get $A \impl \bigvee_{C \in
C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)}(C \land A^C).$\qed
\end{proof}
We now strengthen the $\hbox{\Large$\circ$}$-normal form result so that only
$\hbox{\Large$\circ$}$-chains that are intuitively ``possible'' need to be
considered. For this, we have to verify that we can exclude
chains~$C$ which result in orders which, e.g., have $\hbox{\Large$\circ$} S <_C S$.
\begin{defn}
\label{minnorform}
A formula $A$ is in {\em minimal normal form} over $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ if it is
of the form $\bigvee_{C \in {\cal C} \subseteq C(\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma)} C$, where each
$C$ is a
$\hbox{\Large$\circ$}$-chain over $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$, and so that the corresponding ordered
partition $\Pi_1^{C}, \ldots, \Pi_k^{C}$ satisfies
\begin{enumerate}
\item for no $i <j$ and $S \in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ do we have $\hbox{\Large$\circ$}^{r + s} S \in \Pi_i^C$
and $\hbox{\Large$\circ$}^{r} S \in \Pi_j^{C}$ with $s > 0$;
\item for all $S \in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$, if $\hbox{\Large$\circ$}^s S \in \Pi_i^C$ ($i < k$), then
$\hbox{\Large$\circ$}^r S \notin \Pi_i^C$ if $r \neq s$; and
\item for no $j, j'$ and $S \in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ do we have both $\hbox{\Large$\circ$}^i S \in \Pi_j^C$
and $\hbox{\Large$\circ$}^{i+1} S \in \Pi_{j'}^C$ with $j' > j + 1$.
\end{enumerate}
\end{defn}
\begin{theorem}\label{minnf}
Let $A$ be in $\hbox{\Large$\circ$}$-normal form. There exists a formula $A^\mathrm{nf}$ in
minimal normal form such that $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow A^\mathrm{nf}$.
\end{theorem}
\begin{proof}
By Thm.~\ref{chainnf}, $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow A'$ where $A'$ is a $\hbox{\Large$\circ$}$-chain
normal form over $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$. Consider a disjunct of $A'$ of the form $C
\land A^C$, where $\Pi_1^C$, \dots, $\Pi_k^C$ is the ordered
partition of $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ corresponding to $C$. If $A^C \in \Pi_k^C$, then
$\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (C \land A^C) \leftrightarrow C$, since $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A^C \leftrightarrow
(A^C \leftrightarrow \top)$.
Otherwise, $A^C \in \Pi_i^C$ with $i < k$.
Then the sequence $\Pi_i^C$, \dots, $\Pi_k^C$ corresponds to a conjunction
\[
C' \equiv (A^C \star_{1} S'_1) \land \ldots \land (S'_{j-1} \star_j \top)
\]
where for at least one $l \le j$, $\star_j = \prec$, and $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C
\leftrightarrow C'' \land C'$, where $C''$ is the part of $C$ corresponding to
$\Pi_1^C$, \dots, $\Pi_{i-1}^C$.
Since $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A^C \leftrightarrow (A^C \leftrightarrow \top)$, we have
\begin{equation}
\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (C' \land A^C) \leftrightarrow (C' \land (\top \leftrightarrow A^C)) \label{eqn1}
\end{equation}
As is easily seen, the right-hand side of~(\ref{eqn1}) is provably
equivalent to
\[
C'''\equiv (A^C \leftrightarrow S'_1) \land \ldots \land (S'_{j-1} \leftrightarrow \top)
\]
In sum, $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (C \land A^C) \leftrightarrow (C'' \land C''')$, and
$C'' \land C'''$ is a $\hbox{\Large$\circ$}$-chain.
By induction on the number of disjuncts in $A'$ one shows that
there is $A''$ which is a disjunction of $\hbox{\Large$\circ$}$-chains such that
$\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow A''$. Now we have to prove that there exists a
disjunction of $\hbox{\Large$\circ$}$-chains $A^\mathrm{nf}$ satisfying 1--3 of
Def.~\ref{minnorform} so that $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A'' \leftrightarrow A^\mathrm{nf}$.
Suppose that for some disjunct $C$ in $A''$ we have $\hbox{\Large$\circ$}^{r+s} S \in
\Pi_i^C$ and $\hbox{\Large$\circ$}^r S \in \Pi_j^C$ where $s > 0$ and $i < j$. Then,
since $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\hbox{\Large$\circ$}^{r+s} A \prec \hbox{\Large$\circ$}^r A) \leftrightarrow \hbox{\Large$\circ$}^r A$ we have
$\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \leftrightarrow C'$ where $C'$ is the $\hbox{\Large$\circ$}$-chain corresponding to
$\Pi_1^C, \dots, \Pi_{i - 1}^C, \Pi_i^C \cup \ldots \cup \Pi_k^C$.
Consider a disjunct $C$ of $A''$ where for some $i<k$, both
$\hbox{\Large$\circ$}^r S\in \Pi_i^C$ and $\hbox{\Large$\circ$}^s S \in \Pi_i^C$ where $r < s$. Then
$\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \impl (\hbox{\Large$\circ$}^s S \leftrightarrow \top)$. To see this, recall that $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash
\hbox{\Large$\circ$}^r v \prec \hbox{\Large$\circ$}^s S$ if $r < s$. By definition of $\prec$, that
means that
\begin{equation}
\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash ((\hbox{\Large$\circ$}^s S \impl \hbox{\Large$\circ$}^r S) \impl \hbox{\Large$\circ$}^r S) \land
(\hbox{\Large$\circ$}^r S\impl \hbox{\Large$\circ$}^s S). \label{rsprec}
\end{equation}
Since $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \impl (\hbox{\Large$\circ$}^s S \leftrightarrow \hbox{\Large$\circ$}^r S)$, we have $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C
\impl (\hbox{\Large$\circ$}^s S \impl \hbox{\Large$\circ$}^r S)$ which together with the left
conjunct of (\ref{rsprec}) gives $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \impl \hbox{\Large$\circ$}^r S$. Thus, as
before, $C$ is provably equivalent to the $\hbox{\Large$\circ$}$-chain corresponding to
$\Pi_1^C$, \dots, $\Pi_i^C \cup \ldots \cup \Pi_k^C$.
Lastly, suppose that for a disjunct $C$ of $A''$ we have both $\hbox{\Large$\circ$}^i
S\in \Pi_j^C$ and $\hbox{\Large$\circ$}^{i+1} S \in \Pi_{j'}^C$ for some $j$, $j'$ such
that $j' > j+1 $. Then by axiom (G6) together with transitivity we get
$C \impl (\hbox{\Large$\circ$}^{i+1} S \prec \hbox{\Large$\circ$}^{i+1} S)$, and since $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (B
\prec B) \leftrightarrow B$ we have $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \leftrightarrow C'$ where $C'$ is the
$\hbox{\Large$\circ$}$-chain corresponding to $\Pi_1^C, \dots, \Pi_{j - 1}^C, \Pi_j^C
\cup \ldots \cup \Pi_{j'}^C \ldots \cup \Pi_k^C$.
By induction on the number of disjuncts in $A''$ we
obtain the desired $A^\mathrm{nf}$. \qed
\end{proof}
\section{Quantifier Elimination}
In this section we prove quantifier elimination for $\mathsf{Q}G$. As a
corollary of this result we show that the system $\mathsf{Q}G$ is sound and
complete for $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}$ and that the latter is the intersection of all
finite-valued quantified propositional G\"odel
logics~$\mathbf{G}_k^\mathrm{qp}$.
\begin{proposition}\label{telprop}
\begin{enumerate}
\item $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\forall p)A(p) \leftrightarrow (A(\bot) \land (\forall p)A(\hbox{\Large$\circ$} p))$
\item $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\exists p)A(p) \leftrightarrow (A(\bot) \lor (\exists p)A(\hbox{\Large$\circ$} p)).$
\end{enumerate}
\end{proposition}
\begin{proof}
(1) The left-to-right implication follows easily from the two
instances of (${\impl}{\forall}$)
\[
(\forall p)A(p) \impl A(\bot) \qquad\textrm{and}\qquad (\forall p)A(p)
\impl A(\hbox{\Large$\circ$} p).
\]
For right-to-left, consider
\begin{eqnarray}
(q \leftrightarrow \bot) & \impl & (A(\bot) \land (\forall p)A(\hbox{\Large$\circ$} p)) \impl A(q) \\
(q \leftrightarrow \hbox{\Large$\circ$} p) & \impl & (A(\bot) \land (\forall p)A(\hbox{\Large$\circ$} p)) \impl A(q) \label{eqn2}
\end{eqnarray}
which are derived easily from Prop.~\ref{equivsubst}(2) using $\log{IPC}^\mathrm{qp}$.
Use (R$\exists$) to introduce the existential quantifier in the antecedent of (\ref{eqn2}), and then (I12) to obtain
\begin{equation}
[(q \leftrightarrow \bot) \lor (\exists p)(q \leftrightarrow \hbox{\Large$\circ$} p)] \impl
(A(\bot) \land (\forall p)A(\hbox{\Large$\circ$} p)) \impl A(q) \label{eqn3}
\end{equation}
The antecedent of (\ref{eqn3}) is an instance of (G5), and so
\[
\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (A(\bot) \land (\forall p)A(\hbox{\Large$\circ$} p)) \impl A(q)
\]
from which the right-to-left direction of~(1) follows by
(R$\forall$).
(2) The argument is analogous to the derivation
of~(1).\qed
\end{proof}
\begin{defn}
For $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma \subseteq \mathop\mathit{Var}\cup\{\bot, \top\}$, let $\mathrm\mathit{OP}_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma(A)$ be
the set of formulas inductively defined as follows:
\begin{eqnarray*}
\mathrm\mathit{OP}_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma(A \ast B) &=& \mathrm\mathit{OP}_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma(A) \cup \mathrm\mathit{OP}_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma(B), \quad {\rm where}
\; \ast \in \{\vee, \wedge, \impl\}\\
\mathrm\mathit{OP}_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma ((\mathsf{Q} p)A) & = & \mathrm\mathit{OP}_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma(A), \quad {\rm where} \;
\mathsf{Q} \in\{\forall, \exists\}\\
\mathrm\mathit{OP}_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma(\hbox{\Large$\circ$}^k v)& = &
\begin{cases}
\{\hbox{\Large$\circ$}^k v\} & {\rm if} \; v \in \ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma\\
\emptyset & {\rm otherwise}
\end{cases}
\end{eqnarray*}
Then $\exp_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma(A) = \{k : \hbox{\Large$\circ$}^k q \in \mathrm\mathit{OP}_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma(A)\}$
\end{defn}
\begin{defn} The {\em
quantifier depth}~$\mathop\mathrm{qd}(A)$ of a formula is defined by:
\[\begin{array}{c@{\qquad}c}
\mathop\mathrm{qd}(p) = \mathop\mathrm{qd}(\bot) = 0 & \mathop\mathrm{qd}((\forall p) B) = \mathop\mathrm{qd}((\exists p) B) = \mathop\mathrm{qd}(B) + 1 \\
\multicolumn{2}{c}{
\mathop\mathrm{qd}(B * C) = \max (\mathop\mathrm{qd}(B), \mathop\mathrm{qd}(C)) \textrm{ for } * \in \{\land, \lor, \impl\}}
\end{array}\]
\end{defn}
\begin{lemma}\label{mainlemma}
Let $A$ be a closed formula such that (a) every quantifier free
subformula of~$A$ is in $\hbox{\Large$\circ$}$-normal form and (b) no two quantifier
occurrences bind the same variable. Let $\Delta = \{p_1, \ldots,
p_j\}$ be the set of variables belonging to the innermost quantifiers
in $A$, and $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma = \mathop\mathit{Var}(A) \setminus \Delta$. Then there is a
formula $A^\sharp$ so that \begin{enumerate}
\item $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A\leftrightarrow A^\sharp$,
\item $\max \exp_\Delta(A^\sharp) \le \min \exp_\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma(A^\sharp)$,
\item $\max \exp_{\mathop\mathit{Var}(A^\sharp)}(A^\sharp) \le 2\cdot \max\exp_{\mathop\mathit{Var}(A)}(A)$,
\item $\mathop\mathrm{qd}(A^\sharp) \le \mathop\mathrm{qd}(A)$.
\end{enumerate}
\end{lemma}
\begin{proof} Suppose $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma = \{q_1, \ldots, q_l\}$.
Let $A_0 = A$, $m = \max \exp_\Delta (A)$. At stage $i$, pick the
non-innermost quantified subformula $(\forall q_i) B_i(q_i)$ or
$(\exists q_i) B_i(q_i)$ of $A_i$ corresponding to $q_i$ and replace
\begin{eqnarray*}
(\forall q_i) B_i(q_i) & \textrm{\ by\ } & B_i(\bot) \land \ldots \land
B_i(\hbox{\Large$\circ$}^{{m}-1} \bot) \land (\forall p) B_i(\hbox{\Large$\circ$}^{m} q_i) \\ (\exists q_i)
B_i(p) & \textrm{\ by\ } & B_i(\bot) \lor \ldots \lor
B_i(\hbox{\Large$\circ$}^{{m}-1}\bot ) \lor (\exists q_i) B_i(\hbox{\Large$\circ$}^{m} q_i)
\end{eqnarray*}
to obtain $A_{i+1}$. The procedure terminates with $A_l = A^\sharp$.
At each stage $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A_i \leftrightarrow A_{i+1}$ follows by induction on $m$
from Prop.~\ref{telprop}. The lower bounds are obvious from the
construction of $A^\sharp$.\qed
\end{proof}
\begin{lemma}\label{sublemma}
Suppose $A(p)$ is in $\hbox{\Large$\circ$}$-normal form and
\[
\max \exp_{\{p\}} A \le \min\exp_{\mathop\mathit{Var}(A)\setminus\{p\}}A.
\]
There is a formula $A^\exists$,
with $\mathop\mathit{Var}(A^\exists) \subseteq \mathop\mathit{Var}(A) \setminus \{p\}$ so that
\[
\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\exists p) A \leftrightarrow A^\exists
\]
and $\max\exp_{\mathop\mathit{Var}(A^\exists)\cup\{\bot\}} A^\exists \le
\max\exp_{\mathop\mathit{Var}(A)\cup\{\bot\}}A + 1$.
\end{lemma}
\begin{proof}
Let $m = \max \exp_{\mathop\mathit{Var}(A) \cup \{\bot\}} A$ be the maximal exponent
of a subformula $\hbox{\Large$\circ$}^j S$ and let $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma =
\{\hbox{\Large$\circ$}^i S : S \in \mathop\mathit{Var} \cup \{\bot\}, i \le m\}$.
Theorem~\ref{minnf} provides us with $A^\mathrm{nf}$ in minimal normal form
over $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$ so that $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\exists p) A \leftrightarrow (\exists p) A^\mathrm{nf}$.
Since $\exists$ distributes over $\lor$, we only have to consider
formulas of the form $(\exists p) C$ where $C$ is a $\hbox{\Large$\circ$}$-chain and
satisfies the conditions of Thm.~\ref{minnf}.
$C$ corresponds to an ordered partition $\Pi_1$, \ldots, $\Pi_k$ over
$\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}amma$. We prove that $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\exists p) C \leftrightarrow C'$ for some
quantifier-free $C'$ by induction on $k$.
If $k = 2$, then either $p \in \Pi_1$ or $p \in \Pi_k$. In the first
case, $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\exists p) C(p) \leftrightarrow C(\bot)$, in the second one, $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash
(\exists p) C(p) \leftrightarrow C(\top)$.
Now suppose $k > 2$. Three cases arise, according to how the
equivalence classes containing $p$ are distributed.
(1) The partition corresponding to $C$ is of the form
\[
\Pi_1, \ldots, \Pi_i, \{p\}, \{\hbox{\Large$\circ$} p\}, \ldots, \{\hbox{\Large$\circ$}^j
p\}\cup\Pi_k
\]
Then $C(p)$ is of the form
\[
B \land \underbrace{(v \prec p) \land (p \prec \hbox{\Large$\circ$} p) \land \ldots \land
(\hbox{\Large$\circ$}^j p \leftrightarrow \top)}_{D(p)} \land \, E
\]
Since $D(\top)$ is provable, $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\exists p) C \leftrightarrow B \land v \prec
\top \land E$.
(2) The partition corresponding to $C$ is of the form
\[
\Pi_1, \ldots, \Pi_i, \{p\}, \{\hbox{\Large$\circ$} p\}, \ldots, \{\hbox{\Large$\circ$}^j
p\}, \Pi_{i'}, \ldots, \Pi_k
\]
and $\hbox{\Large$\circ$}^j p \notin \Pi_{i'}$. Then $C(p)$ is of the form
\[
B \land \underbrace{(S \prec p) \land (p \prec \hbox{\Large$\circ$} p) \land \ldots
\land (\hbox{\Large$\circ$}^j p \prec S')}_{D(p)} \land E
\]
We first show that $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\exists p) D(p) \leftrightarrow (\hbox{\Large$\circ$}^{j+1} S \prec S')$.
For the right-to-left direction, observe that
\[
\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\hbox{\Large$\circ$}^{j+1} S \prec S') \impl [(S \prec \hbox{\Large$\circ$} S) \land \ldots
\land (\hbox{\Large$\circ$}^{j}S \prec \hbox{\Large$\circ$}^{j+1}S) \land (\hbox{\Large$\circ$}^{j+1} S \prec S'),
\]
from which the claim follows by~($\mathrm{R}{\exists}$). The
left-to-right direction is proved by induction on $j$, using
axiom~(G6). In sum, we have
\[
\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\exists p) C(p) \leftrightarrow (B \land (\hbox{\Large$\circ$}^{j+1} S \prec S') \land E)
\]
(3) The partition corresponding to $C$ is of the form
\[
\Pi_1, \ldots, \Pi_i, \{p\}, \{\hbox{\Large$\circ$} p\}, \ldots, \{\hbox{\Large$\circ$}^j
p\}\cup\Pi, \Pi_{i'}, \ldots, \Pi_k
\]
with $S \in \Pi$, $S \neq \hbox{\Large$\circ$}^j p$. Because of the condition on $\max
\exp_{\{p\}} A$ we can assume that $S \equiv \hbox{\Large$\circ$}^n q$ with $n \ge
j$.
We proceed by induction on $j$. If $j = 0$, then we have a conjunct
$p \leftrightarrow S$, and $(\exists p) C \equiv C(S)$. Otherwise, we have a
conjunct $\hbox{\Large$\circ$}^j p \leftrightarrow \hbox{\Large$\circ$}^n q$ with $n \ge j$. Using (G3), this
conjunct is provably equivalent to $(\hbox{\Large$\circ$}^{j-1} p \leftrightarrow \hbox{\Large$\circ$}^{n-1} q)
\lor (\hbox{\Large$\circ$}^j p \land \hbox{\Large$\circ$}^n q)$. Hence, $C$ is equivalent to the
disjunction of two $\hbox{\Large$\circ$}$-chains corresponding to
\begin{eqnarray*}
\Pi_1, \ldots, \Pi_i, \{p\}, \{\hbox{\Large$\circ$} p\}, \ldots, & & \{\hbox{\Large$\circ$}^{j-1}
p, \hbox{\Large$\circ$}^{n-1} q\}, \Pi, \Pi_{i'}, \ldots, \Pi_k\\
\Pi_1, \ldots, \Pi_i, \{p\}, \{\hbox{\Large$\circ$} p\}, \ldots, & & \{\hbox{\Large$\circ$}^j
p\}\cup \Pi \cup \Pi_{i'} \cup \ldots\cup \Pi_k
\end{eqnarray*}
For the first $\hbox{\Large$\circ$}$-chain, the maximum exponent of $p$ is smaller
and hence the induction hypothesis of the present subcase applies.
The second $\hbox{\Large$\circ$}$-chain is shorter overall, and hence the induction
hypothesis based on number of equivalence classes applies.\qed
\end{proof}
\begin{lemma}
Let $A(p)$ be in $\hbox{\Large$\circ$}$-normal form, and so that
\[
\max \exp_{\{p\}} A \le \min\exp_{\mathop\mathit{Var}(A)\setminus\{p\}}A.\]
There is a
formula $A^\forall$, with $\mathop\mathit{Var}(A^\forall) \subseteq \mathop\mathit{Var}(A) \setminus
\{p\}$ so that
\[
\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\forall p) A \leftrightarrow A^\forall
\]
and $\max\exp_{\mathop\mathit{Var}(A^\forall) \cup \{\bot\}} A^\forall \le
\max\exp_{\mathop\mathit{Var}(A)\cup\{\bot\}}A + 1$.
\end{lemma}
\begin{proof}
Let $A^\mathrm{nf}$ be the minimal normal form of $A$. It is provably
equivalent to the formula obtained from $A^\mathrm{nf}$ by replacing each
element of a chain $S \prec S'$ by $\hbox{\Large$\circ$} S \impl S'$. By
distributivity then, $A \leftrightarrow A'$ where $A'$ is a conjunction of
disjunctions of implications of the form $\hbox{\Large$\circ$}^i S \impl
\hbox{\Large$\circ$}^j S'$. Any such disjunct of the form $\hbox{\Large$\circ$}^i p \impl
\hbox{\Large$\circ$}^j p$ is provably equivalent to $\top$ if $i \le j$ (in which
case the entire disjunction can be deleted), or to $\top \impl \hbox{\Large$\circ$}^j p$
if $i > j$. The part of a disjunction in $A'$ containing $p$ thus
can be assumed to be of the form
\[
\bigvee_i (D_i \impl \hbox{\Large$\circ$}^{n_i} p) \lor \bigvee_j (\hbox{\Large$\circ$}^{m_j} p \impl E_j)
\]
where $p \notin D_i, E_i$. This, in turn, is equivalent to a
conjunction of disjunctions of the form
\[
\bigvee_i (D \impl \hbox{\Large$\circ$}^{n_i} p) \lor \bigvee_j (\hbox{\Large$\circ$}^{m_j} p \impl E)
\]
This can again be simplified by taking $n = \max \{n_i\}$ and $m =
\min \{m_j\}$, since $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (A \impl B) \lor (A \impl C) \leftrightarrow (A \impl C)$
if $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash B \impl C$.
Since $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\forall p) (A \land B) \leftrightarrow (\forall p) A \land
(\forall p) B$ and $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\forall p)(A(p) \lor B) \leftrightarrow (\forall p)
A(p) \lor B$ if $p \notin B$, it suffices to show that a formula of
the form
\[
F \equiv (\forall p)(D \impl \hbox{\Large$\circ$}^n p) \lor (\hbox{\Large$\circ$}^m p \impl E))
\]
is equivalent to a quantifier free formula. We distinguish three cases:
(1) $E \equiv \hbox{\Large$\circ$}^k \top$, $k \ge 0$. Then $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\hbox{\Large$\circ$}^m p \impl
E)$ and hence $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash F \leftrightarrow \top$.
(2) $E \equiv \hbox{\Large$\circ$}^k \bot$, $k < m$. Then $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\hbox{\Large$\circ$}^m p \impl
E) \leftrightarrow E$, and hence $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash F \leftrightarrow (A \impl \hbox{\Large$\circ$}^n \bot) \lor
E$.
(3) Since $\max \exp_{\{p\}} A \le \min\exp_{\mathop\mathit{Var}(A)\setminus\{p\}}A$
by assumption, this leaves only the case $E \equiv \hbox{\Large$\circ$}^m S$.
Then $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash F \leftrightarrow (A \impl \hbox{\Large$\circ$}^{n+1} S) \lor \hbox{\Large$\circ$}^m S$. The
left-to-right implication is obvious by (${\impl}{\forall}$),
instantiating $p$ by~$\hbox{\Large$\circ$} S$. For the right-to-left implication
two cases arise:
(a) $n \le m$. By (G4), we have $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (A \impl \hbox{\Large$\circ$}^{n+1} S) \impl
[(A \impl \hbox{\Large$\circ$}^{n} p) \lor (\hbox{\Large$\circ$}^n p \impl \hbox{\Large$\circ$}^n S)]$.
Furthermore, $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (\hbox{\Large$\circ$}^n p \impl \hbox{\Large$\circ$}^n S)
\impl (\hbox{\Large$\circ$}^m p \impl \hbox{\Large$\circ$}^m S)$. In sum, we have
\[
[(A \impl \hbox{\Large$\circ$}^{n+1} S) \lor \hbox{\Large$\circ$}^m S] \impl [(A \impl \hbox{\Large$\circ$}^n p)
\lor (\hbox{\Large$\circ$}^m p \impl \hbox{\Large$\circ$}^m S) \lor \hbox{\Large$\circ$}^m S]
\]
Since $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash \hbox{\Large$\circ$}^m S \impl (\hbox{\Large$\circ$}^m p \lor \hbox{\Large$\circ$}^m S)$, we have
$\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash [(A \impl \hbox{\Large$\circ$}^{n+1} S) \lor \hbox{\Large$\circ$}^m S] \impl F$.
(b) $n > m$. By (G2), $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash \hbox{\Large$\circ$}^m S \impl \hbox{\Large$\circ$}^{n+1} S$, and so
$\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash [(A \impl \hbox{\Large$\circ$}^{n+1} S) \lor \hbox{\Large$\circ$}^{m} S] \impl (A \impl
\hbox{\Large$\circ$}^{n+1} S]$. Using induction and (G4), it is easy to show that
\[
\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash (A \impl \hbox{\Large$\circ$}^{n+1} S) \impl [\underbrace{(A \impl \hbox{\Large$\circ$}^n p)
\lor \bigvee_{i=m}^{n-1}(\hbox{\Large$\circ$}^{i+1} p \impl \hbox{\Large$\circ$}^i p)}_D \lor
(\hbox{\Large$\circ$}^m p \impl \hbox{\Large$\circ$}^m S].
\]
Each of the disjuncts $\hbox{\Large$\circ$}^{i+1} p \impl \hbox{\Large$\circ$}^i p$ implies
$\hbox{\Large$\circ$}^i p$, which in turn implies $A \impl \hbox{\Large$\circ$}^n p$, so $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash D
\impl (A \impl \hbox{\Large$\circ$}^n p)$. In sum, we have again $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash
[(A \impl \hbox{\Large$\circ$}^{n+1} S) \lor \hbox{\Large$\circ$}^m S] \impl F$.
The bound on $\max \exp_{\mathop\mathit{Var}(A^\forall) \cup \{\bot\}} A$ follows by
inspection.\qed
\end{proof}
\begin{theorem}
For every closed formula $A$ of $\mathsf{Q}G$ there exists a
variable-free formula $A^\mathrm{qf}$ such that $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow A^\mathrm{qf}$, and
$\max \exp_{\{\bot\}} A^\mathrm{qf}\le 2^{\mathop\mathrm{qd}(A) + l}$ where $l = \max
\exp_{\mathop\mathit{Var}(A) \cup \{\bot\}}$.
\end{theorem}
\begin{proof}
We may assume, renaming variables if necessary, that each variable in
$A$ is bound by only one quantifier occurrence. By induction on
$\mathop\mathrm{qd}(A)$. If $\mathop\mathrm{qd}(A) = 0$, there is nothing to prove. If $\mathop\mathrm{qd}(A) >
0$, let $A^\sharp$ be as in Lemma~\ref{mainlemma}. Replace each
innermost quantified formula $(\exists p) B$, $(\forall p) B$ by
$B^\exists$ or $B^\forall$, respectively. The resulting formula $A'$
satisfies $\mathop\mathrm{qd}(A') \le \mathop\mathrm{qd}(A) - 1$ and $\max \exp_{\mathop\mathit{Var}(A)
\cup\{\bot\}} A' \le 2\max \exp_{\mathop\mathit{Var}(A) \cup \{\bot\}} A+1$.\qed
\end{proof}
\begin{proposition}
Let $A$ be variable-free, and in $\hbox{\Large$\circ$}$-normal
form. Then either $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow \top$ or $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow \hbox{\Large$\circ$}^k
(\bot)$ where $k \le \max \exp_{\{\bot\}} A = n$.
\end{proposition}
\begin{proof}
Consider the minimal normal form $A^\mathrm{nf}$ of $A$ over $\{\hbox{\Large$\circ$}^k(\bot)
: k \le n\}$. Each chain in $A^\mathrm{nf}$ is of one of two forms
\begin{eqnarray*}
C & = & (\bot \prec \hbox{\Large$\circ$}(\bot)) \land (\hbox{\Large$\circ$}(\bot) \prec
\hbox{\Large$\circ$}\hbox{\Large$\circ$}(\bot)) \land \ldots \land (\hbox{\Large$\circ$}^{n-1} \bot \prec
\hbox{\Large$\circ$}^n(\bot)) \\[-2ex]
C_m & = & (\bot \prec \hbox{\Large$\circ$}(\bot)) \land
(\hbox{\Large$\circ$}(\bot) \prec \hbox{\Large$\circ$}\hbox{\Large$\circ$}(\bot)) \land \ldots \land (\hbox{\Large$\circ$}^{m-1}
\bot \prec \hbox{\Large$\circ$}^m(\bot)) \land \bigwedge_{k=m}^n \hbox{\Large$\circ$}^k(\bot)
\end{eqnarray*}
$C$ is provable, so $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C \leftrightarrow \top$, and $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash C_m \leftrightarrow
\hbox{\Large$\circ$}^m(\bot)$. So if $A^\mathrm{nf}$ contains $C$, then $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow
\top$, otherwise $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow \hbox{\Large$\circ$}^k(\bot)$, where $k$ is the
maximum of $C_i$ occurring in $A^\mathrm{nf}$.\qed
\end{proof}
\begin{corollary}
Let $A$ be closed and not containing $\hbox{\Large$\circ$}$. Then either $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A$
or $\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow \hbox{\Large$\circ$}^k(\bot)$, where $k \le 2^{\mathop\mathrm{qd}(A)}$.
\end{corollary}
\begin{corollary}\label{soundcompl}
The calculus $\mathsf{Q}G$ is complete for $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}$.
\end{corollary}
\begin{proof}
If $\mathsf{Q}G \not\vdash A$, then $\mathsf{Q}G \vdash A \leftrightarrow \hbox{\Large$\circ$}^k\bot$ for some
$k$. Since $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow} \nmodels \hbox{\Large$\circ$}^k\bot$ for all~$k$, $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow} \nmodels A$.
\end{proof}
\begin{theorem}
$\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow}$ is the intersection of all finite-valued quantified
propositional G\"odel logics.
\end{theorem}
\begin{proof}
\mathsf{Q}G{} is sound for each finite-valued G\"odel logic, so $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow} \subseteq
{\bf G}_k^\mathrm{qp}$ for each $k$. Conversely, if $\ensuremath{\mathbf{G}^\mathrm{qp}_\uparrow} \not\models A$, then
$\ensuremath{\log{QG}^\mathrm{qp}_\uparrow} \vdash A \leftrightarrow \hbox{\Large$\circ$}^k(\bot)$ for some $k$. Since \mathsf{Q}G{} is sound for
${\bf G}_{k+2}$, we have ${\bf G}_{k+2} \not\models A$ as obviously
${\bf G}_{k+2} \nmodels \hbox{\Large$\circ$}^k\bot$.
\end{proof}
\end{document} |
\begin{document}
\title{On the Spectrum of weighted Laplacian operator and its application to uniqueness of K\"ahler Einstein metrics}
\author{Long Li}
\maketitle{}
\begin{abstract}
The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of K\"ahler Einstein metrics on Fano manifolds, based on the convexity of $Ding$-functional on Chen's weak $\mathcal{C}^{1,\bar{1}}$ geodesic without
using any further regularities. Unlike the smooth case, the lack of regularities on the geodesic forbids us to use spectral formula of the weighed Laplacian operator directly. However, we can use smooth $\epsilon$-geodesics to approximate the weak one, then prove that a sequence of eigenfunctions will converge into the first eigenspace of the weighted Laplacian operator.
\end{abstract}
\section{Introduction}
The study of K\"ahler Einstein metrics on Fano manifolds is an old but lasting subject in complex geometry: on geometrical point of view, it characterizes the manifold with constant Ricci curvature, i.e. the K\"ahler metric satisfies
\[
Ric( \omega) = \omega;
\]
on analytical point of view, the complex Monge-Amp\`ere equations arise from the study of this curvature equation, i.e. the K\"ahler potential $\varphi\in\mathcal{H}$ is the solution of the following equation
\[
(\omega_0 + i\partial\bar{\partial}\varphi)^{n} = e^{h - \varphi} \omega_0^n
\]
where $\mathcal{H}:= \{\omega = \omega_0 + i\partial\bar{\partial}\varphi> 0 \}$. Now as a PDE problem on manifolds, it's natural to ask two questions - existence and uniqueness. After Yau's celebrated work[14] on solving the Calabi Conjecture, Tian's $\alpha$ invariant[13] gives a sufficient condition to solve Monge Amp\`ere
equation on Fano manifolds in 1980's. Then many people contribute to this problem during these years. And quite recently, Chen-Donalson-Sun's work([8], [9], [10]) proves the existence of K\"ahler Einstein metrics on Fano manfolds is equivalent to K-stability condition.This settles a long standing stability conjecture on K\"ahler Einstein metrics which goes back to Yau.
\\
\\
The problem of uniqueness of K\"ahler Einstein metrics on Fano manifolds also keeps attractive during these years. It is first proved by Bando and Mabuchi[1] in 1987, and we will give an alternative proof in this paper. The statement is as follows
\begin{thm}
Let $X$ be a compact complex manifold with $-K_X>0$. Suppose $\omega_1$ and $\omega_2$ are two K\"ahler Einstein metrics on $X$, then there is a holomorphic automorphism $F$, such that
\[
F^*(\omega_2) = \omega_1
\]
where this $F$ is generated by a holomorphic vector field $\mathcal{V}$ on X.
\end{thm}
They solve this problem by considering a special energy(Mabuchi energy) decreasing along certain continuity path. Then the existence of
weak $\mathcal{C}^{1,\bar{1}}$ geodesic between any two smooth K\"ahler potentials is proved by X.X.Chen[7] in 2000, and this idea turns out to be an important tool in proving uniqueness theorems. For instance, Berman[3] gives a new proof of Bando-Mabuchi's theorem by arguing the geodesic connecting two K\"ahler Einstein metrics is actually smooth. And Berndtsson[5] proves the uniqueness of possible singular K\"ahler Einstein metrics along $\mathcal{C}^0$ geodesics. He observes the $Ding$-functional is convex along these geodesics from his curvature formula on the Bergman kernel[6]. Moreover, this curvature formula plays a major role to create a holomorphic vector fields when the functional is affine. This method is used by Berman again to prove the uniqueness of Donaldson's equation[2], and generalized to the $klt-pairs$ in [BBEGZ12].
\\
\\
The idea of this paper is also initiated from the convexity of $Ding$-functional along geodesics from a different perspective. However, instead of using Berndtsson's curvature formula, we are going to use the Futaki's formula(refer to Section 2) of weighted Laplacian operator to derive the holomorphic vector fields. Unlike the former case, here the main difficulty arises from the change of metrics during the convergence of Laplacian operators. Fortunately, we have control on the mixed derivatives $\partial_{\alpha}\partial_{\bar{\beta}}\phi$ on the product manifold, i.e. Chen's existence theorem of weak geodesic[7] guarantees a uniform bound of mixed second derivatives of the potential in both space and time directions on the geodesic. Moreover, we can perturb the weak geodesic to a sequence of nearby smooth metrics $\{g_{\epsilon}\}$ with mixed second derivatives under control[7].
\\
\\
However, this is not quite enough for our purpose, because we are lack of a uniform lower bound of these metrics, and the lower bound of metrics(or the upper bounds of the inverse metric) is crucially involved in the weighted Laplacian operator as
\[
\Box_{\phi_g}u = \partial^{\phi_g} (\omega_g \lrcorner \bar{\partial} u)
\]
where $u $ is a smooth function on $X$. More fundamentally, it plays an important role in the $L^2$ norm of $(0,1)$ forms as
\[
<\xi,\eta >_g = \int_X g^{i\bar{j}}\xi_{i}\overline{\eta_{j}} d\mu_g
\]
where $\xi = \xi_{i}d\bar{z}^{i}$ and $\eta= \eta_{j}d\bar{z}^j$. This forbids us to use standard $L^2$ theorems to get the a uniform control. We will overcome this difficulty in Section 5 by separating the Laplacian equation to two equations, i.e.
\[
\omega_g\wedge v_g = \bar{\partial} u
\]
and
\[
\partial^{\phi_g} v = \Box_{\phi_g}u.
\]
This idea is initiated from solving the equation $\partial^{\phi}v= \pi_{\perp}\phi'$ in Berndtsson's work[5]. Then a crutial $W^{1,2}$ estimate of the sequence of vector fields $v_{g_{\epsilon}}$ shows it converges to some vector fields $v_{\infty}$ in strong $L^2$ norm, and a further $L^1$ estimate on $\bar{\partial}v_{g}$ indicates the vector fields $v_{\infty}$ is in fact holomorphic, under certain conditions(refer to proposition 12). This solves our problem on fiber direction, but on time direction we need to argue the holomorphic vector field keeps to be a constant. This is guaranteed since it corresponds to an eigenfuntion in the first eigenspace of the weighted Laplacian operator and satisfies the geodesic equation.
\\
\\
$\mathbf{Acknowledgement}$: I would like to express my great thanks to Prof. Xiuxiong Chen, who suggested me to do this problem and showed me the way of approach when the geodesic is smooth. I would also thank to Prof. Eric Bedford, Prof. Song Sun, Prof. Weiyong He, and Dr. Kai Zheng for helpful discussion. And especially, I want to thanks Prof. Futaki for pointing out one error in the old version. Finally, the suggestion from Chengjian Yao also helps me to make this paper more clear.
\section{Futaki's formula and Hessian of $Ding$-functional}
The manifolds $X$ in our consideration is Fano, then we can assume the K\"ahler class $[\omega] = c_1(X)$, i.e. for each K\"ahler metric $\omega_g$, there exists a smooth function $F_g$ such that
\[
Ric(\omega_g) - \omega_g = i\partial\bar{\partial}F_g,
\]
hence we can define a weighted volume form as $e^F\det g$(we will write $F_g$ as $F$ when there is no confusion), and a pairing for any $u,v\in \mathcal{C}^{\infty}(X)$
\[
(u,v)_g = \int_X u \bar{v} e^{F}\det g,
\]
then Futaki[9] considers a weighted Laplacian operator
\[
\Delta_{F} u = \Delta_g u - \nabla^j u \nabla_j F.
\]
the reason to do this is because the new Laplacian operator is easy to do integration by parts under the weighted volume form
\[
\int_X (\Delta_{F} u )\bar{u} e^{F} \det g = -\int_X (\nabla_j\nabla^j u + \nabla^j u \nabla_j F ) \bar{u} e^{F}\det g
\]
\[
= \int_X \nabla^j u \nabla_j\bar{u}e^{F}\det g
\]
\[
= \int_X |\bar{\partial} u|^2 e^{F} \det g
\]
where the norm of the 1-form is take with respect to the metric $g$. Hence it's an elliptic operator, and its spectral is discrete as $0 < \lambda_1 <\lambda_2 < \cdots$. Then for each eigenfunction $\Delta_{F} u = \lambda u$, Futaki[11] writes the following formula
\[
\lambda \int_X |\bar{\partial} u|^2 e^{F}\det g = \int_X |\bar{\partial} u |^2 e^{F}\det g + \int_X |L_g u|^2 e^{F}\det g
\]
where $L_g $ is a second order differential operator defined as
\[
L_g u = \nabla_{\bar{j}}\nabla^i u \frac{\partial}{\partial z^i}\otimes d\bar{z}^j.
\]
Now observe the RHS of Futaki's formula is in fact $\int_X |\Delta_{F_g}u|^2 e^F\det g$, we can generalize it to all smooth function as
\begin{lem}
For any smooth function $u$ on $X$, we have
\[
\int_X | \Delta_{F}u|^2 e^F\det g = \int_X |\bar{\partial} u |^2 e^{F}\det g + \int_X |L_g u|^2 e^{F}\det g.
\]
\end{lem}
\begin{pf}
we can decompose $u = \Sigma_0^{\infty} a_i(u) e_i $ into the eigenspace of the operator $\Delta_{F_g}$, and notice that the eigenfunction $e_i$ is orthogonal with respect to each other under the weighted volume form and metric $g$. Then the first two terms in above equation will preserve this orthogonality, i.e. choose eigenfunctions $u$ and $w$ of $\Delta_F$ which are orthogonal to each other, then
\[
\int_X |\bar{\partial}u+ \bar{\partial}w |^2 e^F \det g = \int_X |\bar{\partial}u|^2 e^F \det g + \int_X |\bar{\partial}w|^2e^F\det g
\]
and
\[
\int_X |\Delta_F u+ \Delta_F w |^2 e^F \det g = \int_X |\Delta_F u|^2 e^F \det g + \int_X |\Delta_F w|^2e^F\det g
\]
Moreover, the differential operator $L_g$ keeps this orthogonality of eigenfunctions, but first notice
\[
F_{,\alpha\bar{\beta}} = R_{\alpha\bar{\beta}} - g_{\alpha\bar{\beta}}
\]
from the definition of $F$, then we compute as follows
\[
\int_X \langle L_g u, L_g w \rangle_g e^{F}\det g = \int_X g^{\alpha\bar{\lambda}}g^{\mu\bar{\beta}} u_{,\bar{\lambda}\bar{\beta}} \bar{w}_{,\mu\alpha}e^F\det g
\]
\[
= -\int_X g^{\alpha\bar{\lambda}}g^{\mu\bar{\beta}} u_{,\bar{\lambda}\bar{\beta}\alpha}\bar{w}_{,\mu} e^F\det g - \int_X g^{\alpha\bar{\lambda}}g^{\mu\bar{\beta}} u_{,\bar{\lambda}\bar{\beta}}\bar{w}_{,\mu}
F_{,\alpha} e^F\det g
\]
\[
= - \int_X g^{\alpha\bar{\lambda}}g^{\mu\bar{\beta}} u_{,\bar{\lambda}\alpha\bar{\beta}}\bar{w}_{,\mu}e^F\det g
- \int_X g^{\mu\bar{\beta}} R_{\bar{\beta}}^{\bar{\gamma}} u_{,\bar{\gamma}}\bar{w}_{,\mu} e^F\det g
\]
\[
+ \int_X g^{\alpha\bar{\lambda}}g^{\mu\bar{\beta}} u_{,\bar{\lambda}}\bar{ w}_{,\mu\bar{\beta}} F_{,\alpha} e^F\det g
+ \int_X g^{\mu\bar{\beta}} u_{,\bar{\lambda}}\bar{w}_{,\mu} F^{, \bar{\lambda}}_{\bar{\beta}} e^F\det g
+\int_X g^{\alpha\bar{\lambda}}g^{\mu\bar{\beta}} u_{,\bar{\lambda}}\bar{w}_{,\mu} F_{,\alpha} F_{,\bar{\beta}} e^F\det g
\]
\[
= \int_X g^{\alpha\bar{\lambda}}g^{\mu\bar{\beta}}u_{,\bar{\lambda}\alpha}\bar{w}_{,\mu\bar{\beta}} e^F\det g + \int_X g^{\alpha\bar{\lambda}}g^{\mu\bar{\beta}} u_{\bar{\lambda}\alpha}\bar{w}_{,\mu}
F_{,\bar{\beta}}e^F\det g
\]
\[
+ \int_X g^{\alpha\bar{\lambda}}g^{\mu\bar{\beta}} u_{,\bar{\lambda}}\bar{ w}_{,\mu\bar{\beta}} F_{,\alpha} e^F\det g
+\int_X (g^{\alpha\bar{\lambda}} u_{,\bar{\lambda}}F_{,\alpha} )( g^{\mu\bar{\beta}} \bar{w}_{,\mu} F_{,\bar{\beta}}) e^F\det g - \int_X g^{\mu\bar{\beta}}u_{,\bar{\beta}}\bar{w}_{\mu}e^F\det g
\]
\[
= \int_X ( g^{\alpha\bar{\lambda}}u_{,\alpha\bar{\lambda}} + g^{\alpha\bar{\lambda}}u_{,\bar{\lambda}}F_{,\alpha})(g^{\mu\bar{\beta}}\bar{w}_{\mu\bar{\beta}} + g^{\mu\bar{\beta}}\bar{w}_{\mu}F_{,\bar{\beta}}) e^F \det g
\]
\[
=\int_X (\Delta_F u, \Delta_F w )_g e^F \det g = 0.
\]
\end{pf}
Next let's consider an easy case: according to He[12], the second derivative of $Ding$-functional on a smooth geodesic equals
\[
\frac{\partial^2 \mathcal{D}}{\partial t^2} = (\int_X e^{F_g} \det g)^{-1} \{ \int_X ( |\bar{\partial}\varphi'|^2_g - (\pi_{\perp}\varphi')^2)e^{F_g}\det g \}
\]
where the metric $g$ is induced by the K\"ahler form $\omega_{\varphi}$, and the projection operator is defined as $\pi_{\perp} u = u- \int_X u e^{F_g} \det g / \int_X e^{F_g}\det g $. This implies
$Ding$-functional is convex along smooth geodesics. Now suppose there is a smooth geodesic connecting two K\"ahler Einstein metrics, the $Ding$-functional must keep to be a constant along it. Hence we get
\[
\int_X |\bar{\partial}\varphi'|^2_g e^{F_g}\det g = \int_X (\pi_{\perp}\varphi')^2 e^{F_g}\det g,
\]
then we see the first eigenvalue $\lambda_1$ of the weighted Laplacian operator $\Delta_{F_g}$ is $1$, and $\pi_{\perp}\varphi'$ belong to the first eigenspace, i.e.
\[
\Delta_{F_g}(\pi_{\perp}\varphi') = \pi_{\perp}\varphi'.
\]
Now by Futaki's formula, we see
\[
L_g (\pi_{\perp}\varphi') = 0,
\]
then the induced vector field $V _t = \nabla^i\varphi' \frac{\partial}{\partial z^i}$ is holomorphic on $X$. Moreover, let's differentiate this vector field with respect to $t$ on the geodesic
\[
(g^{j\bar{k}}\varphi'_{\bar{k}})' = g^{j\bar{k}}\varphi''_{\bar{k}} - g^{j\bar{q}} \varphi'_{p\bar{q}} g^{p\bar{k}}\varphi'_{\bar{k}}
\]
\[
= g^{j\bar{k}}(g^{\alpha\bar{\beta}}\varphi'_{\alpha}\varphi'_{\bar{\beta}})_{,\bar{k}} - g^{j\bar{q}} \varphi'_{p\bar{q}} g^{p\bar{k}}\varphi'_{\bar{k}}
\]
\[
= g^{j\bar{k}} g^{\alpha\bar{\beta}} \varphi'_{\alpha}\varphi'_{,\bar{\beta}\bar{k}} = 0
\]
by the holomorphicity of $V_t$. Finally, this gives us a holomorphic vector field $\mathcal{V} = V_t - \partial/\partial t$ on $X\times S$, and its induced automorphism will give the uniqueness of the two K\"ahler Einsteim metrics.
\section{Some $L^2$ theorems }
In this section, we are going to use $L^2$ theorem to investigate the weighted Laplacian operator $\Delta_{F_g}$ and its spectrum, then we shall project our target to the front eigenspace in the proof of uniqueness theorem. First notice that we always have $\lambda_1 \geqslant 1$ by Futaki's formula. Then we are going to introduce some notations.
\\
\\
From now on, we shall assume the manifold $X$ admits non-trivial holomorphic vector fields, and $H^{0,1}(X) =0$. Then fix one $t$ and restrict our attention to this fiber $X\times \{t \}$. Since $-K_X = [\omega]$, we can write
\[
\omega_g = i\partial\bar{\partial}\phi_g
\]
where $\phi_g$ is a plurisubharmonic metric on the line bundle $-K_X$. We claim the measure
\[
e^{F_g}\det g = e^{-\phi_g},
\]
and this is because locally $F_g = -\log\det g - \phi_g$. Then naturally the pairing between functions on $X$ with this weight can be written as
\[
(u,v)_g = \int_X u\bar{v} e^{-\phi_g}.
\]
Here is the $L^2$ theorem coming to play with. Let's consider the space of all $L^2$ bounded $-K_X$ valued $(n,0)$ forms under the metric $\phi_g$, i.e. it consists of every function $u$ on $X$ such that
\[
\int_X |u |^2 e^{-\phi_g} < +\infty,
\]
we denote this space as $L^2_{(n,0)}(-K_X, \phi_g)$, and similarly we can consider all $L^2$ bounded $-K_X$ valued $(n, 1)$ forms under the weighted norm
\[
\int_X g^{\alpha\bar{\beta}}v_{\alpha}\overline{v_{\beta}} e^{-\phi_g} < +\infty,
\]
and we denote this space as $L^2_{(n,1)}(-K_X, \phi_g)$, then we can define an unbounded operator $\bar{\partial}$ between them
\[
\bar{\partial}: L^2_{(n,0)}(-K_X, \phi_g) \dashrightarrow L^2_{(n,1)}(-K_X, \phi_g).
\]
Notice that the domains of these two operator are not the whole $L^2$ spaces. In fact, we can define
\[
dom(\bar{\partial}): = \{ u\in L^2_{(n,0)}(-K_X, \phi_g) ; \ \bar{\partial}u\in L^2_{(n,1)}(-K_X, \phi_g) \},
\]
but it is not densely defined in $L^2$ space when $g_{\phi}$ is a $\mathcal{C}^{1,\bar{1}}$ solution of geodesic equation on a fiber $X\times\{ t\}$. Hence we should consider the Hilbert space $\mathcal{H}_1$ to be the closure of $dom(\bar{\partial})$ in $L^2_{(n,0)}(-K_X, \phi_g)$. We claim that $\mathcal{H}_1$ is not empty.\\
\\
First notice that for any non-trivial holomorphic vector field $v \in L^2_{(n-1,0)}(-K_X)$, we can solve the following equation
\[
\bar{\partial}u = \omega_{g_{\phi}}\wedge v,
\]
since $\bar{\partial}( \omega_{g_{\phi}}\wedge v) = 0$ in the sense of distributions, but $\ker(\bar{\partial}) = Range(\bar{\partial})$ from $H^{0,1}(X) =0$. Next, consider the subspace
$\mathcal{W}$ containing all such $u$, i.e. define
\[
\mathcal{W}: = \{u\in L^2_{(n,0)}(-K_X, \phi_g);\ \bar{\partial}u =\omega_{g_{\phi}}\wedge v, \ \forall v\in L^2_{(n-1,0)}(-K_X) \},
\]
then it is a non-empty subspace in $L^2_{(n,0)}(-K_X, \phi_g)$, and it's easy to check
\[
\mathcal{W} \subset dom(\bar{\partial}),
\]
hence we proved the claim. Now $\bar{\partial}$ is a densely defined, closed operator on the Hilbert space $\mathcal{H}_1$ - it's closed from the continuity property of differential operators in the distribution sense. We can discuss its Hilbert adjoint operator $\bar{\partial}^*_{\phi}$, which is a densely defined, closed operator on $L^2_{(n,1)}(-K_X, \phi_g)$. Moreover, they have closed ranges
\begin{lem}
$\bar{\partial}$ and $\bar{\partial}^{*}_{\phi_g}$ are densely defined, closed operators with closed ranges.
\end{lem}
\begin{pf}
We need to estimate the $L^2$ norm of $\bar{\partial}u$. Take $h$ to be a fixed smooth metric with positive Ricci curvature on $X$, and $u\in dom(\bar{\partial})\cap \ker (\bar{\partial})^{\perp}$, we have
\[
\int_X |\bar{\partial}u|_g^2 e^{-\phi_g} \geqslant \int_X |\bar{\partial}u|_h^2 \det h
\]
\[
\geqslant c \int_X |u|^2 \det h
\]
\[
\geqslant c' \int_X |u|_g^2 e^{-\phi_g}.
\]
this estimate implies $\bar{\partial}$ has closed range, and hence its adjoint $\bar{\partial}^*_{\phi_g}$ by functional analysis reason.
\end{pf}
Then we can define the Laplacian operator as $\Box_{\phi_g} = \bar{\partial}^*_{\phi_g}\bar{\partial}$, where also as an unbounded closed operator, i.e.
\[
\Box_{\phi_g}: L^2_{(n,0)}(-K_X, \phi_g)\dashrightarrow L^2_{(n,0)}(-K_X, \phi_g)
\]
and its domain of definition is
\[
dom(\Box_{\phi_g}): =\{ u\in L^2_{(n,0)}(-K_X, \phi_g);\ u\in dom(\bar{\partial})\ and \ \bar{\partial}u \in dom(\bar{\partial}^*_{\phi_g}) \}.
\]
we claim this operator also has closed range. and
\begin{prop}
we have
\[
\ker \Box_{\phi_g} = coker \Box_{\phi_g},
\]
hence they are both finite dimensional.
\end{prop}
\begin{pf}
First note $\ker \Box_{\phi_g} = \ker \bar{\partial}$ is the 1 dimensional space of constant functions on $X$. In order to prove $coker \Box_{\phi_g}$ also has finite rank, it's enough to prove the weighted Laplacian operator has closed range, since it's self-adjoint
\[
coker \Box_{\phi_g} = R(\Box_{\phi_g})^{\perp} = \ker\Box_{\phi_g}.
\]
Now we are going to prove the closed range property, but this follows from the following estimate for $u\in dom(\Box_{\phi_g})\cap \ker (\bar{\partial})^{\perp}$
\[
|| u ||_g^2 \leqslant C ||\bar{\partial}u ||_g^2
\]
\[
\leqslant C (\Box_{\phi_g} u, u)_g
\]
\[
\leqslant 2C || \Box_{\phi_g} u ||^2_g + \frac{1}{2} || u ||^2_g
\]
and hence
\[
|| u ||^2_g \leqslant C' ||\Box_{\phi_g} u ||^2_g,
\]
which implies the claim.
\end{pf}
Notice that this is not enough to guarantee the existence of discrete spectral, but we have a further estimate,
\begin{lem}
For all $u\in dom(\Box_{\phi_g})\cap \ker(\bar{\partial})^{\perp}$, there is an uniform constant $C$, such that
\[
|| u ||_{W^{1,2}} \leqslant C || \Box_{\phi_g} u ||^2_g.
\]
\end{lem}
\begin{pf}
we still compare it with some fixed smooth weight(metric) $h$,
\[
||\bar{\partial}u ||_h^2 \leqslant C ||\bar{\partial}u ||_g^2
\]
\[
=C (\Box_{\phi_g}u, u )_g
\]
\[
\leqslant C ||\Box_{\phi_g} u||_g ||u||_g
\]
\[
\leqslant C' ||\Box_{\phi_g} u||_g ||u||_h
\]
\[
\leqslant C'' ||\Box_{\phi_g}u||_g ||\bar{\partial}u ||_h,
\]
then
\[
||\bar{\partial}u ||_h^2 \leqslant C'' ||\Box_{\phi_g}u||_g.
\]
finally, an integration by part gives the desired estimate since
\[
\int_X h^{\alpha\bar{\beta}} u_{,\alpha}\overline{u_{,\beta}} \det h = -\int_X h^{\alpha\bar{\beta}} u_{,\alpha\bar{\beta}} \bar{u}\det h
\]
\[
= -\int_X h^{\alpha\bar{\beta}} u_{,\bar{\beta}\alpha} \bar{u}\det h
\]
\[
= \int_X h^{\alpha\bar{\beta}}u_{,\bar{\beta}}\overline{u_{,\bar{\alpha}}} \det h
\]
\end{pf}
Then we can discuss the spectral of $\Box_{\phi_g}$, when $g_{\phi}$ is the $\mathcal{C}^{1,\bar{1}}$ function. Suppose $\lambda$ is an eigenvalue of $\Box_{\phi_g}$, and
let $\Lambda$ be the corresponding eigenspace, we claim
\begin{prop}
$\dim \Lambda < +\infty$
\end{prop}
\begin{pf}
Let $v_i\in \Lambda$ be a sequence of eigenfunctions with bound $L^2$ norm, i.e. $||v_i ||^2_g =1$, then since
\[
|| v_i ||_{W^{1,2}} \leqslant C || \Box_{\phi_g} v_i ||_g
\]
\[
= C \lambda,
\]
hence there exists a $W^{1,2}$ function $v_{\infty}$ such that $v_i\rightarrow v_{\infty}$ in strong $L^2$ norm, by compact embedding theorem. And since
$\Lambda =\ker (\Box_{\phi_g}- \lambda I)$ is a closed subspace of $L^2$
\[
v_{\infty}\in \Lambda.
\]
This implies every bounded sequence in $\Lambda$ has a convergent subsequence, i.e. the unit ball in $\Lambda$ is compact, hence $\dim\Lambda$ is finite.
\end{pf}
Next we are going to discuss some computations when the weight $\phi_g$ is at least $\mathcal{C}^2$. First notice that formally
\[
< \Box_{\phi_g} u, v >_g \ = \ < \bar{ \partial} u, \bar{\partial} v>_g
\]
for any pairing $u, v$. It's easy to see
\[
\Box_{\phi_g}u = \Delta_{\phi_g}u
\]
for all smooth functions $u$, when the metric $\phi_g$ is smooth. If we look closer at these operators, there is a more computable way to express them. For this purpose, let's assume $\phi_g$ is a $\mathcal{C}^2$ metric, then for any $(n,1)$ form $\alpha $ with value in $-K_X$,
\[
\bar{\partial}^*_{\phi_g} \alpha = \partial^{\phi_g} (\omega_g \lrcorner \alpha)
\]
where $\partial^{\phi} v = e^{\phi}\partial (e^{-\phi} v ) = \partial v - \partial\phi\wedge v$ for any $(n-1,0)$ form with value in $-K_X$(that is a vector field on $X$). Hence if we define
\[
v= \omega_g\lrcorner \alpha,
\]
we will have
\[
\bar{\partial}^*_{\phi_g} \alpha = \partial^{\phi_g} v
\]
and the weighted Laplacian operator could be computed as
\[
\Box_{\phi_g} u = \partial^{\phi_g} (\omega_g\lrcorner \bar{\partial} u)
\]
for $u\in dom \Box_{\phi_g}\cap L^2_{(n,0)}(-K_X, \phi_g)$. Notice that there is commutation relation between the new defined operator $\partial^{\phi}$ and $\bar{\partial}$, that is
\begin{equation}
\partial^{\phi}\bar{\partial} + \bar{\partial} \partial^{\phi} = i\partial\bar{\partial}\phi\wedge\cdot
\end{equation}
Now if $u$ is any eigenfunction of the weighted Laplacian operator with eigenvalue $\lambda$, i.e. $\Box_{\phi_g} u = \lambda u$, we can decompose it into two equations
\[
\omega_g \lrcorner \bar{\partial}u = v \ \ \ \partial^{\phi_g} v = \lambda u.
\]
here we can write $v = X\lrcorner 1$, where the constant function $1$ is read as an $(n,0)$ form with value in $-K_X$, and $X= X^{\alpha}\frac{\partial}{\partial z^{\alpha}}$ is a vector field in
$(1,0)$ direction on the manifolds. Next we are going to prove Futaki's formula by the commutation equality.
\begin{lem}(Futaki's formula)
Let $u$ be a eigenfunction of weighted Laplacian with eigenvalue $\lambda$, i.e. $\Box_{\phi_g}u = \lambda u$, then
\[
\lambda \int_X |\bar{\partial}u|^2_g e^{-\phi_g} = \int_X (|L_g u|^2 + |\bar{\partial}u|^2_g)e^{-\phi_g}.
\]
\end{lem}
\begin{pf}
First notice $u$ is pure real or imaginary. Hence here we will give the proof when $u$ is real valued - the case when $u$ is pure imaginary is similar. Now by the commutation relation of $\partial^{\phi_g}$, we compute $\bar{\partial}(\lambda u)$
\[
-\partial^{\phi_g}\bar{\partial} v + i\partial\bar{\partial}\phi_g \wedge v = \lambda\bar{ \partial} u,
\]
notice that $i\partial\bar{\partial} \phi_g = \omega_g$, hence
\[
-\partial^{\phi_g}\bar{\partial} v = (\lambda -1)\bar{\partial} u,
\]
pair it with $\bar{\partial}u$,
\[
(\lambda -1) \int_X |\bar{\partial}u |^2_ge^{-\phi_g}=-\int_X \langle \partial^{\phi_g}\bar{\partial} v, \bar{\partial}u \rangle_g e^{-\phi_g}
\]
\[
= \int_X -g^{\lambda\bar{\mu}}\partial_{\alpha} (e^{-\phi_g} \partial_{\bar{\mu}} X^{\alpha}) \overline {\partial_{\bar{\lambda}}u}
\]
\[
=\int_X \partial_{\bar{\mu}}X^{\alpha} \overline{\partial_{\bar{\alpha}}X^{\mu}} e^{-\phi_g}.
\]
Now notice that $X^{\alpha} = g^{\alpha\bar{\beta}}u_{,\bar{\beta}}$, under the normal coordinate when $g_{i\bar{j}} = \delta_{ij} \Lambda_i$,
\[
\partial_{\bar{\mu}}X^{\alpha}\partial_{\alpha}X^{\bar{\mu}} = g^{\alpha\bar{\beta} }u_{,\bar{\beta}\bar{\mu}} g^{\lambda\bar{\mu}}u_{, \lambda\alpha}
\]
\[
= \frac{1}{\Lambda_{\alpha} \Lambda_{\lambda}}u_{,\bar{\alpha}\bar{\lambda}} u_{,\lambda\alpha}
\]
\[
= \frac{1}{\Lambda_{\alpha}\Lambda_{\lambda}} u_{,\bar{\alpha}\bar{\lambda}} u_{,\alpha\lambda}
\]
\[
= g_{\alpha\bar{\beta}}g^{\lambda\bar{\mu}} \partial_{\bar{\mu}}X^{\alpha} \overline{\partial_{\bar{\lambda}}X^{\beta}},
\]
hence we proved the Futaki's formula
\[
(\lambda -1)\int_{X}|\bar{\partial}u|^2_ge^{-\phi_g} = \int_X |\bar{\partial}X|^2_g e^{-\phi_g}.
\]
\end{pf}
\section{$Ding$-functionals along the approximation geodesics}
Let $X$ be an $n$ dimensional compact complex K\"ahler manifold with K\"ahler metric $\omega$, then we can write the K\"ahler form locally as
\[
\omega = g_{\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta}
\]
where $\alpha,\beta = 1,\cdots,n$. Take $S$ to be a cylinder, and $z^{n+1} = t + \sqrt{-1}s$ be its coordinate. Then $z = (z^1,\cdots, z^n, z^{n+1})$ is a point in $X\times S$, and we can define
\[
\tilde{\omega} = g_{\alpha\bar{\beta}}dz^{\alpha}\wedge d\bar{z}^{\beta} + dz^{n+1}\wedge d\bar{z}^{n+1}
\]
as a K\"ahler metric on $X\times S$. And $\tilde{\varphi} = \varphi - |z^{n+1}|^2$ as the new potential. We shall write $\tilde{\omega}$ as $\omega$ and $\tilde{\varphi}$ as $\varphi$ when there is no fusion. Then Chen[7] proves the following two theorems
\begin{thm}(Existence of weak geodesic)
Let $\varphi_0, \varphi_1\in \mathcal{H}$, then there exists a unique $C^{1,\bar{1}}$ geodesic connecting them, i.e. the following homogenous Monge-Amp\`ere equation has a unique weak solution $\varphi\in \overline{\mathcal{H}}$(the closure is taken under the $C^{1,\bar{1}}$ topology) on $X\times S$
\[
\det (g_{i\bar{j}}+ \partial_{i}\partial_{\bar{j}}\varphi)_{(n+1)(n+1)} = 0
\]
where $i,j =1,\cdots, n+1$, and on the boundary $\partial(X\times S)$
\[
\varphi(0,s,z) = \varphi_0(z),\ \varphi(1,s,z) =\varphi_1(z)
\]
with the following estimate
\[
||\varphi||_{\mathcal{C}^1(X\times S)}+ \max\{|\partial_i\partial_{\bar{j}}\varphi | \} < C
\]
where $C$ is a uniform constant only depending on $\varphi_0$ and $\varphi_1$.
\end{thm}
\begin{thm}($\epsilon$- approximation geodeiscs)
Given $\varphi_0,\varphi_1\in\mathcal{H}$, we can have a sequence of approximation geodesics $\varphi_{\epsilon}(t, z)$ as follows: for each small $\epsilon>0$, there exists a unique solution of the equation
\[
(\varphi_{tt} - |\partial_X \varphi'|_{g_{\varphi}}^2)\det(g_{\varphi}) = \epsilon \det h
\]
such that there exists a uniform constant $C$ with
\[
|\varphi'_t| + |\varphi''_t| + |\varphi|_{\mathcal{C}^1} + \max\{ |\partial_{\alpha}\partial_{\bar{\beta}}\varphi| \} < C,
\]
and $\varphi_{\epsilon}$ converges to the $\mathcal{C}^{1,\bar{1}}$ geodesic $\varphi$ in the weak $\mathcal{C}^{1,\bar{1}}$ topology.
\end{thm}
Notice that for any plurisubharmonic metric $\phi$ on $-K_X$, we can write its potential as $\varphi = \phi - \phi_0$, where $\phi$ and $\phi_0$ are corresponding metrics on the line bundle
$-K_X$. Now suppose $\phi_0, \phi_1$ are two smooth K\"ahler Einstein metrics on $X$, with their K\"ahler forms $\omega_i=i\partial\bar{\partial}\phi_i, i =0,1$ satisfying
\[
\omega_i^n = \frac{e^{-\phi_i}}{\int_X e^{-\phi_i}}.
\]
define the following functionals
\[
\mathcal{F}(\phi):= -\log\int_X e^{-\phi}
\]
and
\[
\mathcal{E}(\phi):= \frac{1}{n+1} \Sigma_{j=0}^n \int_X \varphi \omega_0^j\wedge\omega_{\phi}^{n-j}
\]
where $\omega_{\phi} = i\partial\bar{\partial}\phi$. Then the $Ding$-functional is defined as
\[
\mathcal{D} = -\mathcal{E} + \mathcal{F} = - \frac{1}{n+1} \Sigma_{j=0}^n \int_X (\phi-\phi_0) \omega_0^j\wedge\omega_{\phi}^{n-j}-\log\int_X e^{-\phi}.
\]
Notice the along a curve of metrics $\phi_t$, the derivative of $Ding$-functional is
\[
\frac{\partial\mathcal{D}}{\partial t} = \int_X \phi' ( - \omega_{\phi}^n + \frac{ e^{-\phi}}{\int_X e^{-\phi}}).
\]
we see the critical point of this functional is the K\"ahler Einstein metric, and its second derivative is
\[
\frac{\partial^2 \mathcal{D}}{\partial t^2} = - \int_X (\phi'' - |\partial \phi'|_g^2 )\omega_{\phi}^n+ (\int_X e^{-\phi})^{-1} \{ \int_X(\phi'' - |\partial\phi'|^2_g )e^{-\phi}
+\int_X (|\partial\phi'|_g^2 - (\pi_{\perp}\phi')^2)e^{-\phi} \}
\]
where the metric $g =i\partial\bar{\partial}\phi_t$, and if we denote the term $f = \phi'' - |\partial \phi'|_g^2 $, $c_t = \int_X e^{-\phi}$ and $\delta_t = |\partial\phi'|_g^2 - (\pi_{\perp}\phi')^2$, the equation reads
\[
\frac{\partial^2 \mathcal{D}}{\partial t^2} = -\int_X f \omega_{\phi}^n + \int_X ( f+\delta_t) e^{-\phi} /c_t,
\]
then we are going to consider the behavior of $Ding$-functional on the approximation geodesic. First from Chen's theorem, we can find a $\mathcal{C}^{1,\bar{1}}$ geodesic $\phi_t$ connecting the two K\"ahler Einstein metrics. Moreover for any small $\epsilon>0$, there is the smooth approximation geodesic $\phi_{\epsilon}(t,z)$ connecting the two end points $\phi_0, \phi_1$, which converges weakly to the $\mathcal{C}^{1,\bar{1}}$ geodesic. Now if we consider the $Ding$-functional on these approximation geodesics, we have estimates
\[
\frac{\partial^2 \mathcal{D}}{\partial t^2} \geqslant -\epsilon \int_X \det h
\]
from $f = \epsilon \det h / \det g > 0$ and $\int_X \delta_t e^{-\phi} > 0 $. Let $\epsilon\rightarrow 0$, we see that $Ding$-functional keeps to be convex on $\mathcal{C}^{1,\bar{1}}$ geodesic.
Now we can integrate it back along $t$
\[
\frac{\partial \mathcal{D}}{\partial t}(1) -\frac{\partial \mathcal{D}}{\partial t}(0) = \int_{X\times I} -f\omega^n_{\phi}dt + \int_{X\times I} f e^{-\phi}/c_t dt + \int_{X\times I} \delta_t e^{-\phi}/c_t dt,
\]
notice that at end points $\phi_0,\phi_1$ are both K\"ahler Einstein, hence the first derivative of $Ding$-functionals vanish. And on the approximation geodesic, we have the equation
\[
f \det g = \epsilon \det h
\]
and $f \leqslant \phi'' < C$ uniformly independent of $\epsilon$. Then the equation above reads
\[
A \epsilon = \int_{X\times I} f e^{-\phi}/c_t dt + \int_{X\times I} \delta_t e^{-\phi}/c_t dt
\]
\[
\geqslant \int_{X\times I} f e^{-\phi}dt + \int_{X\times I} \delta_t e^{-\phi}dt,
\]
because we have uniform $\mathcal{C}^0$ estimate on $\phi_{\epsilon}$. Now since we want to discuss the eigenfunctions on each fiber, we need to a lemma to pull back the estimate to fibers.
\begin{lem}
Suppose $F_{\epsilon}(t)$ is a sequence of non-negative function on $[0,1]$, with integration estimate
\[
\int_0^1 F_{\epsilon} dt < A \epsilon,
\]
then for almost everywhere $t\in[0,1]$, we can find a subsequence(depending on $t$) $F_{\epsilon_j}$, such that
\[
F_{\epsilon_j} < C_t\epsilon_j
\]
where $C_t$ is a constant independent of $\epsilon$.
\end{lem}
\begin{pf}
Let $\tilde{F}_{\epsilon} = F_{\epsilon}/ \epsilon$, then by Fatou's lemma
\[
\int_0^1 \liminf_{\epsilon} \tilde{F}_{\epsilon} dt \leqslant \liminf_{\epsilon} \int_0^1 \tilde{F}_{\epsilon}dt \leqslant A,
\]
hence the function $\liminf_{\epsilon}\tilde{F}_{\epsilon} \in L^1$, i.e. for almost everywhere $t$, there is a subsequence $\tilde{F}_{\epsilon_j}$ and a constant $C_t$ such that
\[
\tilde{F}_{\epsilon_j} < C_t,
\]
hence
\[
F_{\epsilon_j} < C_t \epsilon_j.
\]
\end{pf}
Now put $F_{\epsilon} = \int_X f_{\epsilon}e^{-\phi_{\epsilon}} + \int_X\delta_{\epsilon}e^{-\phi_{\epsilon}}$ and notice the two terms on RHS are both non-negative, we have proved
\begin{prop}
Consider the approximation geodesic $\phi_{\epsilon} $ connection two K\"ahler Einstein metrics. For almost everywhere $t$, there is a constant $C_t$, such that for each such $t$, there
exists a subsequence $\epsilon_j$, such that the following estimates
\[
\int_X f e^{-\phi} (\epsilon_j) < C_t \epsilon_j
\]
and
\[
\int_X (|\partial\phi'|^2_g - (\pi_{\perp}\phi')^2)e^{-\phi} (\epsilon_j) < C_t \epsilon_j
\]
hold simutaneouly.
\end{prop}
\section{Convergence in the first eigenspace}
In this section, we shall focus our attention to the one fiber $X\times\{t \}$, and picked up a subsequence $\phi_{\epsilon_j}$ from above section. Then we can consider the sequence of weighted Laplacian operator $\Box_{\phi_{\epsilon}}$(we shall omit the subindex $j$ here). For each $\epsilon$, we can arrange its eigenvalues as $0<\lambda_1^{\epsilon}\leqslant \lambda_2^{\epsilon}\leqslant\cdots$, corresponding with one eigenfunction $e_i(\epsilon)$, i.e.
\[
\Box_{\phi_{\epsilon}} e_i(\epsilon) = \lambda_i^{\epsilon} e_i(\epsilon).
\]
Then let $u_{\epsilon}(z)$ be a sequence of smooth functions on $X$, such that $u_{\epsilon}\perp \ker\bar{\partial}$. Then it decomposes into the eigenspace of weighted Laplacian operator $\Box_{\phi_{\epsilon}}$, i.e.
\[
u_{\epsilon} = \Sigma_{i=1}^{N_{\epsilon}} a_i(\epsilon) e_{i}(\epsilon)
\]
where $e_i\in \Lambda_i$, and in prior, $N_{\epsilon}$ could equal to $+\infty$ in the above notation.
Then we can consider the action by the weighted Laplacian operator on this sequence of functions, i.e. we can write $\Box_{\phi_{\epsilon}} u_{\epsilon}$ as
\[
v_{\epsilon} = \omega_{g_{\epsilon}}\lrcorner \bar{\partial}u
\]
and
\[
\partial^{\phi_{\epsilon}} v_{\epsilon} = \Sigma_{i=1}^{N_{\epsilon}} \lambda^{\epsilon}_i a_i(\epsilon)e_i(\epsilon).
\]
Under certain constraint, we claim these vector fields $v_{\epsilon}$ will converge to a holomorphic one with the same equation satisfied,
\begin{prop}
Let $u_{\epsilon}$ be a sequence of functions as above. Suppose it satisfies the following conditions:
1) $\Sigma_{i=1}^{N_{\epsilon}} |a_i(\epsilon)|^2 < A$ for an uniform constant $A$, and the sums does not converge to zero.
2) there exists a uniform constant $K$, such that $\lambda_{N_{\epsilon}}^{\epsilon} < K$ for each $\epsilon$
3) the following estimate holds
\begin{equation}
\int_X (|\bar{\partial} u_{\epsilon}|^2_{g_{\epsilon}} - (\pi_{\perp}u_{\epsilon})^2)e^{-\phi_{\epsilon}} < C \epsilon.
\end{equation}
then by passing to a subsequence, we have
\[
u_{\epsilon}\rightarrow u_{\infty}
\]
in strong $L^2$ sense, where $u_{\infty}\in W^{1,2}$ is nontrivial. Moreover there exists a nontrivial holomorphic
$(n-1,0)$ form $v_{\infty}$ with value in $-K_X$, such that
\[
v_{\epsilon} \rightarrow v_{\infty}
\]
in strong $L^2$ sense, and the equation
\[
\omega_g \wedge v_{\infty} = \bar{\partial} u_{\infty}
\]
holds in the sense of $L^2$ functions, where $g$ is the metric found on the $\mathcal{C}^{1,\bar{1}}$ geodesic.
\end{prop}
before proving the proposition, we need a lemma
\begin{lem}
Let $f_j, g_j$ be two sequence of $L^2$ functions with $|| f_j g_j ||_{L^p} < C$ for some $p \geqslant 1$. Suppose that $\int_X | f_j |^2 d\mu< C'$ and $g_j\rightarrow g\in L^2$ in $L^2$ norm, then there exists an $L^2$ function $f$ such that
\[
f_j g_j\rightarrow fg \in L^p
\]
in the sense of distributions.
\end{lem}
\begin{pf}
First note there exists an $L^2$ function $f$ such that $f_j\rightarrow f$ in weak $L^2$ topology. Then we check
\[
\int_X ( fg - f_j g_j ) d\mu = \int_X g(f-f_j)d\mu + \int_X f_j(g-g_j)d\mu,
\]
the first term on the RHS of above equation converges to zero from the weak convergence of $f_j$, and the second term converges to zero too, since
\[
|\int_X f (g-g_j)d\mu|^2 \leqslant (\int_X |f|^2d\mu )(\int_X |g-g_j|^2d\mu) \rightarrow 0.
\]
hence $f_jg_j$ converges to $fg$ in the sense of distributions. Moreover, from the $L^p$ bound of $f_jg_j$, we have an $L^p$ function $k$ such that $f_jg_j\rightarrow k$ in weak $L^p$ topology. Then
\[
fg = k
\]
as $L^p$ functions.
\end{pf}
\begin{rmk}
Suppose the sequence $|f_j|$ is uniformly bounded in lemma 13, then the limit $f$ is an $L^{\infty}$ function, then $fg\in L^2$ automatically.
\end{rmk}
\begin{pf}(of proposition 12)
First we can write equation (2) as
\[
\Sigma_{i=1}^{N_{\epsilon}} (\lambda^{\epsilon}_i -1 )|a_i(\epsilon)|^2 < C\epsilon
\]
by Futaki's formula, we know
\[
\int_X |L_{g_{\epsilon}} u_{\epsilon}|^2 e^{-\phi_{\epsilon}} = \Sigma_{i=1}^{N_{\epsilon}} \lambda_{i}^{\epsilon}(\lambda_i^{\epsilon} -1)|a_i(\epsilon)|^2
\]
\[
\leqslant KC\epsilon
\]
from condition (2) and (3). But if we write $v_{\epsilon} = X_{\epsilon}\lrcorner 1$ for some vector field $X_{\epsilon} = X_{\epsilon}^{\alpha}\frac{\partial }{\partial z^{\alpha}}$, then
\[
(L_{g} u )_{\bar{j}}^{\ i}= = g^{i\bar{k}} u_{, \bar{k}\bar{j}} = \frac{\partial X^i}{\partial\bar{z}^j},
\]
hence the $L^2$ norm is
\[
|L_{g} u|^2 = g_{\alpha\bar{\beta}}g^{\mu\bar{\lambda}}\frac{\partial X^{\alpha}}{\partial\bar{z}^{\lambda}}\overline{\frac{\partial X^{\beta}}{\partial\bar{z}^{\mu}}}
= |\frac{\partial X}{\partial\bar{z}}|^2_g.
\]
now we choose a fixed smooth background metric $h$ to estimate
\[
|\frac{\partial X}{\partial\bar{z}}|^2_h = h_{\alpha\bar{\beta}}h^{\mu\bar{\lambda}}\frac{\partial X^{\alpha}}{\partial\bar{z}^{\lambda}}\overline{\frac{\partial X^{\beta}}{\partial\bar{z}^{\mu}}}
\]
\[
= h_{\alpha\bar{\beta}}h^{\mu\bar{\lambda}} g^{\alpha\bar{\eta}}u_{,\bar{\eta}\bar{\lambda}} g^{\gamma\bar{\beta}}u_{, \gamma\mu}
= \frac{1}{\Lambda_{\alpha}^2} |u_{,\bar{\alpha}\bar{\lambda}}|^2
\]
\[
\leqslant \Sigma(\frac{\Lambda_{\lambda}}{\Lambda_{\alpha}}) \Sigma \frac{1}{\Lambda_{\alpha}\Lambda_{\lambda}} |u_{,\bar{\alpha}\bar{\lambda}}|^2
\]
\[
\leqslant C( tr_{g}h) |\frac{\partial X}{\partial\bar{z}}|^2_g
\]
where we compute in some normal coordinate. And correspondingly, the $L^2$ norm of $X$ can be estimated by
\[
|X|^2_h = h_{\alpha\bar{\beta}}g^{\alpha\bar{\lambda}}u_{,\bar{\lambda}} \overline{g^{\beta\bar{\eta}}u_{, \bar{\eta}}}
\]
\[
= \frac{1}{\Lambda_{\alpha}^2}|u_{,\bar{\alpha}}|^2
\]
\[
\leqslant \Sigma(\frac{1}{\Lambda_{\alpha}}) \Sigma \frac{1}{\Lambda_{\alpha}} |u_{,\bar{\alpha}}|^2
\]
\[
\leqslant (tr_g h) |\bar{\partial} u|^2_g.
\]
Recall that $f = \phi'' - |\partial\phi'|^2_g $ is bounded from above, then we can estimate the $L^2$ norm of $\bar{\partial}v$ as
\[
\int_X |\frac{\partial X}{\partial\bar{z}}|^2_h \det h\leqslant C\int_X |\frac{\partial X}{\partial\bar{z}}|^2_h \frac{1}{f} \det h
\]
\[
\leqslant\frac{C}{\epsilon} \int_X |\frac{\partial X}{\partial\bar{z}}|^2_g (tr_g h) \det g
\]
\[
\leqslant \frac{C'}{\epsilon}\int_X |\frac{\partial X}{\partial\bar{z}}|^2_g e^{-\phi_g} \leqslant C''.
\]
note $X$ is a vector in $(1,0)$ direction, which means locally its coefficients are functions. Hence its full gradient is uniformly bounded in $L^2$ norm, i.e.
\[
\int_X |\nabla X_{\epsilon} |^2_h\det h < C
\]
for some constant independent of $\epsilon$. We claim it's also $L^1$ bounded. Recall from our choice of $\epsilon$, we have
\[
\int_X f e^{-\phi_{\epsilon}} < C_1 \epsilon,
\]
then we can estimate
\[
\int_X e^{F_{\epsilon}} \det h = \frac{1}{\epsilon} \int_X f e^{-\phi_{\epsilon}} < C_1,
\]
hence
\[
(\int_X |X|_h \det h )^2 \leqslant C ( \int_X |X|^2_h e^{F_{g}}\det g)^2
\]
\[
\leqslant C (\int_X |X|^2_h (\det g)^2 e^{F_g})(\int_X e^{F_g})
\]
\[
\leqslant C' (\int_X |\bar{\partial}u |^2_g e^{-\phi_g}) < C''.
\]
Hence it's uniformly $L^1$ bounded, then by Poinc\'are inequality, we know $||X||_{L^2} < C $ for some uniform constant. These together imply the sequence of vector fields $X_{\epsilon}$
are uniformly $W^{1,2}$ bounded. Now by compact imbedding theorem, there exists a vector field
$X=X^{\alpha}\frac{\partial}{\partial z^{\alpha}} \in W^{1,2}$ such that $X_{\epsilon}\rightarrow X$ in strong $L^2$ norm.
Moreover, observe that
\[
(\int_X |\frac{\partial X}{\partial\bar{z}}|_h e^{-\phi_g} )^2= (\int_X |\frac{\partial X}{\partial\bar{z}}|_h e^{F_g}\det g)^2
\]
\[
\leqslant ( \int_X |\frac{\partial X}{\partial\bar{z}}|^2_h (\det g)^2 e^{F_g} )(\int_X e^{F_g})
\]
\[
\leqslant (C \int_X |\frac{\partial X}{\partial\bar{z}}|^2_g e^{-\phi_g} )(\int_X e^{F_g})
\]
\[
\leqslant C' \epsilon \int_X e^{F_g} = C' \int_X f e^{-\phi_g}
\]
\[
< C'' \epsilon \rightarrow 0
\]
from our choice of sequence $\epsilon$. Hence $\bar{\partial}X \rightarrow 0$ in weak $L^1$ sense, but this is enough to imply $\bar\partial X = 0 $ in the sense of distributions.
Then $X$ is in fact a holomorphic $(1,0)$ vector field on the manifolds, and we can define $v_{\infty} = X\lrcorner 1$, which is a $-K_X$ valued holomorphic $(n-1,0)$ form.
On the other hand,
for the function $u_{\epsilon}$ itself, we have
\[
\int_X |\bar{\partial}u_{\epsilon}|^2_h \det h \leqslant C \int_X |\bar{\partial} u_{\epsilon}|^2_{g_{\epsilon}} e^{-\phi_{\epsilon}}
\]
\[
= C\Sigma_{i=1}^{N_{\epsilon}} \lambda_i^{\epsilon}|a_i(\epsilon)|^2 \leqslant C',
\]
hence $u_{\epsilon}$ has a uniform $W^{1,2}$ bound, and it converges to a function $u_{\infty}\in W^{1,2}$ in strong $L^2$ norm. Then by condition (1), the $L^2$ norm of $u_{\infty}$ is
non-trivial. Moreover, we know the equation
\[
g^{\epsilon}_{\alpha\bar{\beta}}X_{\epsilon}^{\alpha} = u(\epsilon)_{,\bar{\beta}}
\]
holds for every $\epsilon$. Now $g^{\epsilon}_{\alpha\bar{\beta}}$ is uniformly bounded from above, hence converges to $g_{\alpha\bar{\beta}}$ in weak $L^{\infty}$, where $g_{\alpha\bar{\beta}}$ is the weak $\mathcal{C}^{1,\bar{1}}$ solution of the geodesic equation. And $X_{\epsilon} \rightarrow X$ in strong $L^2$, hence by the Remark after lemma 13, we see that
the equation
\[
g_{\alpha\bar{\beta}} X^{\alpha} =\partial_{\bar{\beta}} u_{\infty}
\]
holds in the sense of $L^2$ functions. In particular, they are equal almost everywhere.
Finally, observe that $u_{\infty}\perp \ker\bar{\partial}$, since
\[
\int_X u_{\infty} e^{-\phi} = \lim_{\epsilon\rightarrow 0} \int_X u_{\epsilon} e^{-\phi_{\epsilon}} = 0.
\]
Hence if $v_{\infty} $ is trivial, then $\bar{\partial}u = 0$, i.e. $u\in \ker\bar{\partial}$, which implies $u = 0$, a contradiction. So $v_{\infty}$ is non-trivial too.
\end{pf}
Notice that before taking the limits, the vector field $v_{\epsilon}$ also satisfies another equation, i.e.
\[
\partial^{\phi_{\epsilon}} v_{\epsilon} = \Sigma_{i=1}^{N_{\epsilon}} \lambda_{i}^{\epsilon} a_i(\epsilon) e_i(\epsilon).
\]
the LHS converges weakly to $\partial^{\phi}v_{\infty}$, since for any smooth testing $(n,0)$ form $W$,
\[
\int_X v_{\epsilon}\wedge\overline{\bar{\partial}W} e^{-\phi_{\epsilon}} \rightarrow \int_X v_{\infty}\wedge\overline{\bar{\partial}W}e^{-\phi}
\]
and the RHS converges to $u_{\infty}$ since condition (3). And the RHS
\[
|| \Sigma_{i=1}^{N_{\epsilon}} \lambda^{\epsilon}_i a_i(\epsilon) e_i(\epsilon) - u_{\epsilon} ||^2 \leqslant K \Sigma_{i=1}^{N_{\epsilon}} (\lambda^{\epsilon}_i -1) |a_i(\epsilon)|^2
\]
converges to zero. We have equality
\[
\partial^{\phi}v_{\infty} = u_{\infty}
\]
holds in the weak sense. But since both sides of above equation are $L^2$ functions, the equation actually holds as $L^2$ functions.
This reminds us that $u_{\infty}$ might be the eigenfunction of the operator $\Box_{\phi}$ with eigenvalue $1$. In fact, we have
\begin{cor}
Let $u_{\epsilon}$ be a sequence of functions satisfying condition (1) - (3) in proposition 7, then there exists a function $u_{\infty}\in W^ {1,2}$ such that
\[
u_{\epsilon}\rightarrow u_{\infty}
\]
in strong $L^2$ sense, and $u_{\infty}$ is a nontrivial eigenfunction of the operator $\Box_{\phi_g}$ with eigenvalue $1$.
\end{cor}
\begin{pf}
First notice $u_{\infty}\in dom(\Box_{\phi_g})$. This is because $\bar{\partial} u = \omega_g\wedge v_{\infty}$, hence $u\in \mathcal{W} \subset dom(\bar{\partial})$, and $\bar{\partial}u \in dom(\bar{\partial}^*_{\phi_g})$ since $v_{\infty}$ is holomorphic. Now for any smooth testing $(n,0)$ form $W$ with value in $-K_X$, we compute
\[
\int_X \bar{\partial}^*_{\phi_g}\bar{\partial}u_{\infty} \wedge\overline{W} e^{-\phi_g} = ( \bar{\partial}^*_{\phi_g}\bar{\partial}u_{\infty}, W )_g
\]
\[
= \langle \bar{\partial}u_{\infty}, \bar{\partial}W \rangle_g
\]
\[
= \langle \omega_g\wedge v_{\infty}, \bar{\partial}W \rangle_g
\]
\[
= \int_X v_{\infty}^{\alpha}\overline{\partial_{\bar{\alpha}}W} e^{-\phi_g}
\]
\[
= ( \partial^{\phi_g}v_{\infty}, W )_g
\]
\[
= \int_X u_{\infty} \wedge\overline{W} e^{-\phi_g}.
\]
hence $\Box_{\phi_g}u_{\infty} = u_{\infty}$ as $L^2$ functions.
\end{pf}
\section{the eigenspace decomposition of $\phi'$ (the easy case)}
In this section, we shall construct a sequence of functions $u_{\epsilon}$, which could satisfy the condition $(1) - (3)$ in proposition 12 from $\phi'_{\epsilon}$, then construct a holomorphic vector
field from there. However, we need to discuss case by case this time, i.e. let
\[
\pi_{\perp}\phi'_{\epsilon} = \Sigma_{i=1}^{+\infty} a_i(\epsilon) e_i(\epsilon),
\]
then
\[
\Box_{\phi_{\epsilon}}(\pi_{\perp}\phi'_{\epsilon}) = \Sigma_{i=1}^{+\infty} \lambda_i^{\epsilon} a_i(\epsilon) e_i(\epsilon).
\]
Note the restriction from the vanishing of $Ding$-functional gives
\begin{equation}
\Sigma_{i=1}^{+\infty} ( \lambda_i^{\epsilon}-1) | a_i(\epsilon)|^2 < C\epsilon
\end{equation}
by passing to the chosen subsequence $\epsilon_j$. And notice that
\[
\int_X |\bar{\partial}\phi'_{\epsilon}|^2_h \leqslant C \int_X |\bar{\partial}\phi'_{\epsilon}|^2_{g_{\epsilon}} e^{-\phi_{\epsilon}}
\]
\[
\leqslant C \int_X \phi''_{\epsilon} e^{-\phi_{\epsilon}} < C',
\]
then there exists a function $\psi \in W^{1,2}$ such that $\phi'_{\epsilon}\rightarrow \psi$ in strong $L^2$ norm. Hence we can assume
\begin{equation}
\frac{1}{2} < \Sigma_{i=1}^{+\infty} | a_i(\epsilon)|^2 < 2
\end{equation}
for $\epsilon$ small enough.
\begin{rmk}
In fact ,we have $|\phi_{\epsilon} |_{\mathcal{C}^1 } < C$, hence $|| \phi_{\epsilon} ||_{W^{1,p}} < C$ for any $p$ large. Then by compact imbedding theorem, we can assume
\[
\phi_{\epsilon}\rightarrow \phi
\]
in $\mathcal{C}^{0,\alpha}$ norm.
\end{rmk}
In fact, we are going to prove
\begin{thm}
There is a holomorphic vector field $v$ on the manifolds, such that
\[
\omega_g\wedge v = \bar{\partial}\psi
\]
where $\psi$ is the $L^2$ limit of $\phi'_{\epsilon}$ and $g$ is the $\mathcal{C}^{1,\bar{1}}$ solution of geodesic equation. Moreover, $\psi$ is a eigenfunction of the operator $\Box_{\phi_g}$ with eigenvalue $1$, i.e.
\[
\Box_{\phi_g}\psi = \psi.
\]
\end{thm}
In order to prove this theorem, we shall discuss case by case. First there are two possibilities for the convergence of eigenvalue $\lambda_i^{\epsilon}$:
\\
\\
$Case\ 1$, there exist a finite integer $k$ such that the following two things hold
i) for each $1\leqslant i \leqslant k$, $\lambda_i^{\epsilon}\rightarrow 1$ as $\epsilon\rightarrow 0$;
ii) $\lambda_{k+1}^{\epsilon}$ does not converges to $1$.
\\
\\
$Case\ 2$, for each $1\leqslant i < +\infty$, $\lambda_i^{\epsilon}\rightarrow 1$ as $\epsilon\rightarrow 0$.
\\
\\
Let's discuss $Case\ 1$ first in this section. In this case, we shall define
\[
u_{\epsilon}: = \Sigma_{i=1}^k a_i(\epsilon)e_i(\epsilon).
\]
Notice that the divergence of $\lambda_i^{\epsilon}$ implies $\lambda_i^{\epsilon} > 1+\delta$ for some small $\delta>0$, by passing to a subsequence. Then since $\lambda_i^{\epsilon}$ is
a non-decreasing sequence in $i$, we have for all $i>k$
\[
\lambda_i^{\epsilon} > 1+\delta
\]
for the same subsequence. Now by equation (3), we see
\[
C\epsilon > \Sigma_{i=k+1}^{+\infty} (\lambda_i^{\epsilon} - 1) |a_i(\epsilon)|^2
\]
\[
\geqslant \Sigma_{i=k+1}^{+\infty} \delta |a_i(\epsilon)|^2,
\]
hence $\Sigma_{i=k+1}^{+\infty} |a_i(\epsilon)|^2 \rightarrow 0$ when $\epsilon\rightarrow 0$. This gives condition (1), i.e.
\[
\Sigma_{i=1}^k |a_i(\epsilon)|^2 > 1/4.
\]
condition (2) is satisfied because $\lambda_{k}^{\epsilon} \rightarrow 1$ by the assumption, and condition (3) is automatically satisfied by equation (3). Hence we can generate a holomorphic
vector field $v_{\infty}$ from proposition (12).
Moreover, we could see $ ||\pi_{\perp}\phi'_{\epsilon} - u_{\epsilon} ||_{L^2} $ converges to zero in above argument, hence we actually have
\[
\psi = u_{\infty}
\]
after taking the limit. And hence it's the eigenfunction of $\Box_{\phi_g}$ with eigenvalue $1$, by corollary (14). Hence we proved theorem 15 in this case.
\section{the hard case}
Now we are going to deal with $Case\ 2$, i.e. we assume
\[
\lambda_i^{\epsilon} \rightarrow 1
\]
for each $1\leqslant i < +\infty$. Here we still subdivide it into two subcases as follows:
\\
\\
$subCase\ 1$, for any $1< k<\infty$, the partial sum $\Sigma_{i=1}^{k-1} |a_{i}(\epsilon)|^2 \rightarrow 0$, when $\epsilon\rightarrow 0$.
\\
\\
$subCase\ 2$, there exists a finite number $K$, such that $\Sigma_{i=1}^{K-1} |a_{i}(\epsilon)|^2$ does not converge to zero.
\\
\\
Before going to the subcases, we need a lemma first
\begin{lem}
Let $e_i(\epsilon)$ be the eigenfunction of the weighted Laplacian $\Box_{\phi_{\epsilon}}$ with eigenvalue $\lambda_i^{\epsilon}$, i.e.
\[
\Box_{\phi_{\epsilon}}e_i(\epsilon) = \lambda_i^{\epsilon} e_i(\epsilon).
\]
Suppose there exists an uniform constant $C$, such that $\lambda_i^{\epsilon} <1+ C\epsilon$, then $e_i(\epsilon)$ converges to a non-trivial eigenfunction $e_i$ of the operator
$\Box_{\phi_g}$ with eigenvalue $1$. Moreover, suppose there is another $j\neq i$, such that $\lambda_j$ satisfies the same condition, then $e_i, e_j$ are mutually orthogonal to each other.
\end{lem}
\begin{pf}
we define $u_{\epsilon}= e_i(\epsilon)$, then condition $(1)$ and $(2)$ hold automatically. And condition (3) is also satisfied because
\[
\int_X (|\bar{\partial} u_{\epsilon}|^2_{g_{\epsilon}} - (\pi_{\perp}u_{\epsilon})^2)e^{-\phi_{\epsilon}} = (\lambda_i^{\epsilon}-1) < C\epsilon,
\]
hence by proposition (12) and corollary (14), we get
\[
e_i(\epsilon)\rightarrow e_i
\]
in strong $L^2$ sense, where $e_i\in W^{1,2}$ is a eigenfunction of $\Box_{\phi_g}$ with eigenvalue $1$. Now for $j\neq i$, we have similar convergence and eigenfunction $e_j$, but
\[
\int_X e_i \bar{e}_j e^{-\phi_{g}} = \lim_{\epsilon\rightarrow 0}\int_X e_i(\epsilon)\overline {e_j(\epsilon)} e^{-\phi_{\epsilon}} =0
\]
by the strong $L^2$ convergence of $e_i(\epsilon)$, and $L^{\infty}$ convergence of $\phi_{\epsilon}$.
\end{pf}
Now let's begin to discuss the $subCase\ 1$. For any fixed $k$, by equation (4), we can find a large integer $N_{\epsilon, k}$ such that
\[
\Sigma_{i=1}^{N_{\epsilon,k}} |a_i(\epsilon)|^2 \geqslant 1/4
\]
by the assumption in this subcase, for $\epsilon$ small
\[
\Sigma_{i=k}^{N_{\epsilon,k}}|a_i(\epsilon)|^2 \geqslant 1/8.
\]
but then by equation (3),
\[
\frac{1}{8}(\lambda_k^{\epsilon} -1)\leqslant \Sigma_{i=k}^{N_{\epsilon,k}} (\lambda_i^{\epsilon} -1)|a_i(\epsilon)|^2 < C\epsilon,
\]
because the sequence $\lambda_i^{\epsilon}$ is non-decreasing. Hence we proved for each $k$,
\[
\lambda_k^{\epsilon} < 1+ 8C\epsilon
\]
for $\epsilon$ small enough. Now by lemma 16, we get an eigenfunction $e_k$ for each $1\leqslant i<\infty$, and they are orthogonal to each other. However, this is impossible since the eigenspace with eigenvalue $1$ of an elliptic operator $\Box_{\phi_g}$ has only finite rank. Hence the $subCase\ 1$ actually never happens.
\section{the final case}
Let's discuss $subCase\ 2$. Under the assumption in this case, we can find $K_1$, a finite integer, to be the first number such that $\Sigma_{i=1}^{K_1 -1} |a_i(\epsilon)|^2$ does not converge to zero. Then by passing to a subsequence, we can assume $\Sigma_{i=1}^{K_1 -1} |a_i(\epsilon)|^2 > \delta_1 $ for some fixed positive number $\delta_1$. Now consider the truncated sequence
\[
\Lambda_1(\phi') = \Sigma_{i=K_1}^{+\infty} a_i(\epsilon) e_i(\epsilon).
\]
suppose there exists another integer $K_2 > K_1$, such that $\Sigma_{i=K_1}^{K_2 -1} |a_i(\epsilon)|^2$ does not converge to zero, and then we can assume
$\Sigma_{i=K_1}^{K_2 -1} |a_i(\epsilon)|^2 > \delta_2 $. We can repeat this argument, to find $0< K_1<K_2<K_3<\cdots$, but we claim this process will terminate in finite steps.
\begin{lem}
There exists an finite integer $n$, such that
\[
\Sigma_{i=K_n}^{+\infty} |a_i(\epsilon)|^2 \rightarrow 0.
\]
\end{lem}
\begin{pf}
Let's define a sequence of sequence of functions $u^{(j)}_{\epsilon}$ as
\[
u^{(0)}_{\epsilon}: =\Sigma_{i=1}^{K_1-1}a_i(\epsilon)e_i(\epsilon)
\]
\[
u^{(1)}_{\epsilon}: =\Sigma_{i=K_1}^{K_2-1}a_i(\epsilon)e_i(\epsilon)
\]
\[
\cdots
\]
\[
u^{(j)}_{\epsilon}: = =\Sigma_{i=K_{j}}^{K_{j+1}-1}a_i(\epsilon)e_i(\epsilon)
\]
and so on. We now claim $u_{\epsilon}^{(j)}$ satisfying all the conditions (1) - (3) in proposition (12). Condition (1) is satisfied automatically by assumption, and condition (2) is satisfied since
$\lambda_k^{\epsilon}\rightarrow 1$ for any fixed $k$. Condition (3) is satisfied too because of equation (3), i.e.
\[
\Sigma_{i=K_j}^{K_{j+1}-1}(\lambda_i^{\epsilon} - 1)|a_i(\epsilon)|^2< C\epsilon,
\]
then by proposition (12) and corollary (14), we see there exists an non-trivial $W^{1,2}$ function $u^{(j)}$ such that
\[
u_{\epsilon}^{(j)}\rightarrow u^{(j)}
\]
in strong $L^2$ norm. And $u^{(j)}$ is a eigenfunction of operator $\Box_{\phi_g}$ with eigenvalue $1$. However, notice that $u^{j}_{\epsilon}$ and $u_{\epsilon}^{(k)}$ are mutually orthogonal,
and by the same argument used in lemma 16, this implies
\[
u^{(j)}\perp u^{(k)}
\]
for all different $j$ and $k$. Now we can find finite many such $u^{(j)}$ since they are all in the eigenspace with eigenvalue $1$ of the weighted
Laplacian operator $\Box_{\phi_g}$, hence we proved the lemma.
\end{pf}
Next we are going to complete the proof of theorem 15. Now let's define
\[
u_{\epsilon}: =\Sigma_{i=1}^{K_n-1}a_i(\epsilon) e_i(\epsilon)
\]
where $K_n$ is the number appearing in lemma 17. Now people can check the three conditions in proposition 12 are satisfied, and hence there exists a $W^{1,2}$ function $u$ such that
\[
u_{\epsilon}\rightarrow u
\]
in $L^2$ sense, and $u$ is a eigenfunction with eigenvalue $1$ of operator $\Box_{\phi_g}$, and there is a holomorphic vector field $v$ such that
\[
\omega_g\wedge v = \bar{\partial}u.
\]
Moreover, the difference of the $L^2$ norm is
\[
||\pi_{\perp}\phi'_{\epsilon} - u_{\epsilon} ||_{L^2} = \Sigma_{i=K_n}^{+\infty}|a_i(\epsilon)|^2 \rightarrow 0
\]
by our choice of $K_n$, hence we have
\[
\psi = u.
\]
And we complete the proof.
\begin{rmk}
If there is no any non-trivial holomorphic vector field on $X$, then {\em proposition 12} directly implies $\phi'=0$ almost everywhere on $X\times I$ from above case by case discussion. Without using {\em corollary 14}, we don not need to invoke any eigenfunction of the first eigenspace of the weighted Laplacian operator in the limit. Hence we proved uniqueness in this case.
\end{rmk}
\section{Time direction}
Up to now, we construct a holomorphic vector field $v_t$ on a fiber $X\times{t}$ for almost everywhere $t\in [0,1]$.
And this vector field can be computed as
\[
v_t = \omega_g \lrcorner \bar{\partial}\psi
\]
where $\phi'_{\epsilon}\rightarrow \psi$ in strong $L^2$ norm at time $t$. Notice that there are more information to use for the convergence of
$\phi'_{\epsilon}$. In fact, we know $|\phi'|, |\phi_{t\bar{z}}|$ and $|\phi_{z\bar{t}}|$ are all uniformly bounded on $X\times I$, i.e.
\[
|\phi'|_{\mathcal{C}^1}< C,
\]
then we can assume $\phi'_{\epsilon}\rightarrow \phi' \in \mathcal{C}^1(X\times I)$, in $\mathcal{C}^{0,\alpha}$ norm. Hence the two limits actually agree with each other, i.e.
\[
\psi = \phi'
\]
as $L^2$ functions on $X$. Now the holomorphic vector field can be written as
\[
v_t = \omega_g \lrcorner \bar{\partial} \phi'.
\]
Then we can define the following subset of the unit interval
\[
S:= \{ t\in I ; \ there\ is\ a\ holomorphic\ vector\ field\ v_t\ on\ X\times\{ t\}\ satisfying\ \omega_{g}\wedge v_t =\bar{\partial}\phi' \}
\]
we know the set $I - S$ has measure zero. Next we are going to prove a stronger result
\begin{prop}
The subset $S$ coincides with the whole unit interval, i.e.
\[
S = I.
\]
\end{prop}
\begin{pf}
First recall that $\phi_{\epsilon}\rightarrow \phi$ in $\mathcal{C}^{0,\alpha}(X\times I)$ norm, by the uniform bound on $\mathcal{C}^1$ norm of $\phi$. Then on each fiber $X\times\{ t\}$, the convergence still holds, i.e.
\[
\phi_{\epsilon}\rightarrow \phi
\]
in $\mathcal{C}^{0,\alpha}(X)$, and this implies
\[
g_{\epsilon,\alpha\bar{\beta}} \rightarrow g_{\alpha\bar{\beta}}
\]
in the sense of distribution on the fiber $X\times\{t \}$. Pick up a point $\underline{t}\in I-S$, and a sequence $t_i \in S$ such that $t_i \rightarrow \underline{t}$. Observe that the space of all
holomorphic vector fields is finite dimensional, i.e. let
\[
\Gamma(X): = H^0(TX),
\]
then $\Gamma$ is a finite dimensional vector space. Write $v_{t_i} = X_i\lrcorner 1 $, where $v_{t_i}\in \Gamma$ is the vector field satisfying the equation in the definition of $S$.
Observe that $v_t$ is the unique solution to the following equation
\[
\partial^{\phi_t}v_t = \Box_{\phi_t}\phi' = \pi_{\perp}\phi'
\]
under the condition $H^{0,1}(X)=0$, then the standard $L^2$ estimate(Berndtsson[5]) gives us
\[
|| v_t ||_h \leqslant C || \pi_{\perp} \phi' ||_h
\]
for some fixed metric $h$ and uniform constant $C$ independent of time $t$. Consider the sequence $\{X_i \}\in H^0(TX)$, the uniform bounds on the $L^2$ norm of $X_i$ shows it must converges under the fixed metric $h$, i.e. there exists a vector field $X\in \Gamma$ such that
\[
|| X - X_i ||_h^2 \rightarrow 0.
\]
Let's write $g_{\alpha\bar{\beta}} = g_{ \alpha\bar{\beta}}(\underline{t})$ and $g_{i, \alpha\bar{\beta}} = g_{\alpha\bar{\beta}}(t_i )$, then
\[
|| X - X_i ||_g^2 \leqslant C || X -X_i ||_h^2,
\]
hence converges to zero too. Now we claim the equation
\[
\omega_g\wedge X = \bar{\partial}\phi'
\]
holds in the sense of distribution. Put $\chi(z)$ be any smooth compact supported testing function on $X$(we can further assume $\chi$ is supported in some coordinate chart), we fix a pair of index $\alpha, \beta$, and compute
\[
\int_X (g_{\alpha\bar{\beta}}X^{\alpha} - g_{i,\alpha\bar{\beta}}X_i^{\alpha} )\chi(z)\det h
\]
\[
= \int_X \chi (g_{\alpha\bar{\beta}} -g_{i, \alpha\bar{\beta}})X^{\alpha} \det h + \int_X\chi (X^{\alpha} - X_i^{\alpha} )g_{i,\alpha\bar{\beta}} \det h,
\]
since $g_{i, \alpha\bar{\beta}}$ is uniformly bounded, the second term in above equation converges to zero in strong $L^2$ sense. And the first term, we can decompose it into
\[
\int_X \chi (g_{\alpha\bar{\beta}} - g_{i, \alpha\bar{\beta}} ) X^{\alpha}\det h
\]
\[
= \int_X\chi( g_{\alpha\bar{\beta}} - g_{\alpha\bar{\beta}}^{\epsilon} ) X^{\alpha}\det h -\int_X\chi (g_{i, \alpha\bar{\beta}} - g_{i, \alpha\bar{\beta}}^{\epsilon}) X^{\alpha}\det h
+ \int_X \chi (g_{i, \alpha\bar{\beta}}^{\epsilon} - g_{\alpha\bar{\beta}}^{\epsilon}) X^{\alpha}\det h,
\]
the first and second terms converge to zero as $\epsilon\rightarrow 0$, and for the third term, we integration by parts
\[
\int_X \chi (g_{i, \alpha\bar{\beta}}^{\epsilon} - g_{\alpha\bar{\beta}}^{\epsilon}) X^{\alpha}\det h = \int_X\chi_{,\bar{\beta}} (\phi^{\epsilon}_{i, \alpha}- \phi^{\epsilon}_{\alpha})X^{\alpha}\det h
\]
\[
= \int_X\chi_{,\bar{\beta}} (t_i - \underline{t})\phi'_{,\alpha}(t)X^{\alpha}\det h
\]
\[
\leqslant A |t_i - \underline{t}|
\]
where $A$ is a constant independent of $\epsilon$. Hence
\[
\int_X \chi (g_{\alpha\bar{\beta}} - g_{i, \alpha\bar{\beta}} ) X^{\alpha}\det h \rightarrow 0
\]
as $t_i \rightarrow \underline{t}$, and we proved
\[
g_{i,\alpha\bar{\beta}}X_i^{\alpha} \rightarrow g_{i,\alpha\bar{\beta}}X_i^{\alpha}
\]
in the sense of distributions. But we know $\phi'_i\rightarrow \phi'$ in $\mathcal{C}^{0,\alpha}$ norm, hence $\bar{\partial}\phi'_i\rightarrow \bar{\partial}\phi'$ in the sense of distribution too.
Finally, the limit equation
\[
g_{\alpha\bar{\beta}}X^{\alpha} = \phi'_{,\bar{\beta}}
\]
holds in distribution sense on $X\times \{\underline{t} \}$. Now since both sides in above equation are $L^{\infty}$ functions, we see the equation actually holds in the sense of $L^2$ functions
by the same argument in $Remark\ 1$.
\end{pf}
Now it makes sense to talk about the time derivative of vector fields $v_t $ in distribution sense, i.e. on the $\mathcal{C}^{1,\bar{1}}$ geodesic, we compute in the sense of distributions
\[
\phi''_{,\bar{\beta}} = ( g_{\alpha\bar{\beta}} X^{\alpha} )',
\]
and computation implies
\[
(g^{\alpha\bar{\lambda}}\phi'_{,\alpha}\phi'_{,\bar{\lambda}})_{,\bar{\beta}} = \phi'_{\alpha\bar{\beta}}X^{\alpha} + g_{\alpha\bar{\beta}}(X^{\alpha})'.
\]
note the RHS is in fact equal to
\[
\nabla_{\bar{\beta}}(\phi'_{,\alpha}X^{\alpha}) = \phi'_{,\alpha\bar{\beta}}X^{\alpha}+ \phi'_{,\alpha}X^{\alpha}_{,\bar{\beta}} = \phi'_{,\alpha\bar{\beta}}X^{\alpha},
\]
here Leibniz rule makes sense since $X$ is holomorphic. Hence we get
\[
g_{\alpha\bar{\beta}}(X^{\alpha})' = 0
\]
which is equivalent to the vanishing of $\frac{\partial}{\partial t}v_t = 0$, i.e. we have an unchanged holomorphic vector field $v$ on the geodesic.
\\
\\
We finished the proof of uniqueness theorem by taking the holomorphic vector field
\[
\mathcal{V}:= \frac{\partial}{\partial t} - V,
\]
then it's easy to check $\mathcal{L}_{\mathcal{V}} (i\partial\bar{\partial}\phi_t) = 0$ during the flow, hence the induced the automorphism $F$ preserves the metric along the geodesic.
\\
\\
\end{document} |
\begin{document}
\title[Linkage of modules over Cohen-Macaulay rings]
{Linkage of modules over Cohen-Macaulay rings}
\author[M. T. Dibaei]{Mohammad T. Dibaei$^{1}$}
\author[M. Gheibi]{Mohsen Gheibi$^2$}
\author[S. H. Hassanzadeh]{S. H. Hassanzadeh$^{3}$}
\author[A. Sadeghi]{Arash Sadeghi$^4$}
\address{$^{1, 2, 3, 4}$ Faculty of Mathematical Sciences and Computer,
Tarbiat Moallem University, Tehran, Iran.}
\address{$^{1, 2, 4}$ School of Mathematics, Institute for Research in Fundamental Sciences
(IPM), P.O. Box: 19395-5746, Tehran, Iran.} \email{[email protected]}
\email{[email protected]} \email{[email protected]}
\address{$^{3}$ Departamento de Mate\'{m}atica,CCEN,
Universidade Federal de Pernambuco,50740-540 Recife, PE
Brazil.} \email{[email protected]}
\keywords{Linkage of modules, Sliding Depth of Extention modules, modules with Cohen-Macaulay extension,
sequentially Cohen-Macaulay \\
1. M.T. Dibaei was supported in part by a grant from IPM (No. 89130114)\\
3. S.H.Hassanzadeh was partially supported by a grant from CNPq (Brazil).}
\subseteqbjclass[2000]{13C40, 13D45,13C14}
\begin{abstract} Inspired by the works in linkage theory of ideals, the concept of sliding depth of extension modules is defined to prove the Cohen-Macaulyness of linked module if the base ring is merely Cohen-Macaulay. Some relations between this new condition and other module-theory conditions such as G-dimension and sequentially Cohen-Macaulay are established. By the way several already known theorems in linkage theory are improved or recovered by new approaches.
\end{abstract}
\maketitle
\section{introduction}
Classification is one of the main perspective in any field of
mathematics. Among rare theories deliberate this viewpoint in
commutative algebra and Algebraic Geometry, linkage theory is a
well-developed theory during decades. Classically it refers to
Halphen (1870) and M. Noether \cite{No}(1882) who worked to classify
space curves. During forties and fifties in twenty century Apery and
Ga\'{e}ta started the new contributions to classify curves in
$\mathbb{P}^3$, the reason one always feels that the twisted cubic
curve is that smooth is behind the fact that it is linked to a line.
In 1974 the significant work of Peskine and Szpiro\cite{PS} brought
breakthrough to this theory and stated it in the modern algebraic
language; two closed subschemes $V_1$ and $V_2$ in $\mathbb{P}^n$
are said to be linked if they are unmixed with no common component
and their union is complete intersection. More precisely two ideals
$I$ and $J$ in Cohen-Macaulay local ring $R$ is said to be linked if
there is a regular sequence $\alpha$ in their intersection such that
$I=\alpha:J$ and $J=\alpha:I$. The first main theorem in the theory
of linkage is the following \cite{PS}.
{\mbox{q}\,uote\it {Theorem A. If $(R,\mathfrak{m})$ is a Gorenstein local ring, $I$
and $J$ are two linked ideals of R then $R/I$ is Cohen-Macaulay if and only if $R/J$ is so.}}
Attempts to generalize this theorem lead to several development in
linkage theory, especially the works by C. Huneke and B. Ulrich \cite{Hu},\cite{HU1}. A
counterexample given by Peskine and Szpiro in the same article shows
that {\it Theorem A} is no longer true if the base ring $R$ is only
Cohen-Macaulay (from now on CM). Trying to determine the accurate
condition for an ideal $I$, in a CM local ring, such that any ideal
which is linked to $I$ be CM, Huneke \cite{Hu} introduced the
concept of Strongly Cohen-Macaulay (SCM) condition, in the sense
that an ideal $I$ is SCM if all of the Koszul homology modules with
respect to some generating set of $I$ are Cohen-Macaulay. He stated
that {\it in a Cohen-Macaulay local ring any ideal linked to a
strongly Cohen-Macaulay ideal is Cohen-Macaulay.} Herzog,
Vasconcelos and Villarreal \cite{HVV} replaced the SCM condition by
the so called Sliding Depth condition; namely we say that an ideal
$I$ satisfies sliding depth condition if $\mbox{depth}\, H_i \geq
\mbox{dim}\,(R)-r+i$, for $i > 0$, where $H_i$ is $i$th homology of Koszul
with respect to some generating set of $I$ and $r$ is the number of
elements of this generating set.
The new progress in the linkage theory is the recent work of Martsinkovsky and Strooker
\cite{MS} which established the concept of linkage of modules.
This paper began to attract some interest, they not only could recover some
of the known theorems in the theory of linked ideals such as the ones in \cite{Sc} but also present
new conceptual ideas that only exist in module theory.
In this paper, inspired by the works in the ideal case, we extend the strongly Cohen-Macaulay and sliding depth condition
for modules; so that we can state Theorem A for linked modules in
CM local rings.
In section 2, as mentioned above, we define the new sliding depth
conditions for modules so called SDE (Sliding Depth on Ext´s) or CME
(Cohen-Macaulay Ext´s). Some sufficient condition for being SDE or
CME is given, for example in Proposition \ref{B0} it is shown that
Cohen-Macaulay $R$--modules with finite G--dimension are CME. As
well it is proven that over Cohen-Macaulay local ring $R$, if $M$
is SDE, then $\lambda M$ is maximal Cohen-Macaulay(see Corollary
\ref{B1}).
Trying to detect the module theory invariants that have no ideal inscription in the theory of linkage of ideals, we encounter to the combinatorial conception \emph{sequentially Cohen-Macaulay}. We first present a computational criterion for this concept involving the ideas from linkage of module in Corollary \ref{CC}. Finally in this section we pose an extension to a theorem of Foxby \cite{F} for the class of CME modules and answer this question; so that
in Cohen-Macaulay local ring
with canonical module $\omega_R$, it is shown that $M$ is CME if and
only if $M\otimes_R\omega_R$ is sequentially Cohen-Macaulay and
$\mbox{Tor}\,_i^R(M,\omega_R)=0$ for $i>0$ ( Theorem \ref{A3}).
In section 3, for a finite $R$--module $M$ over Cohen-Macaulay local
ring $R$ of dimension $d\geq 2$ with canonical module $\omega_R$,
we establish a duality between local cohomology modules of $M\otimes_R\omega_R$
and those of $\lambda M$ (Theorem \ref{A4}) provided $M\otimes_R\omega_R$ be generalized
Cohen-Macaulay. This theorem is a generalization to \cite[Theorem 10]{MS} and
also \cite{Sc} while for its proof instead of techniques in derived category we appeal to Spectral sequences.
Also whenever $M$ is generalized Cohen-Macaulay, under some vanishing assumption on Tor-modules of $M$ and $\omega_R$ we show
that $\mbox{H}^i_\mathfrak{m}(\lambda M)\cong \mbox{Ext}\,^i_R(M,R)$ for $i=1,\ldots ,d-1$,
(Corollary \ref{B2}).
\section{SDE and CME modules}
Throughout, $R$ is a Noetherian ring and $M$ is a finite generated
$R$--module. Assume that $M$ is a stable $R$--module (i.e. $M$ has
no projective summand). Let
$P_1\overset{f}{\rightarrow}P_0\rightarrow M\rightarrow 0$ be a
finite projective presentation of $M$. The transpose $\mbox{Tr}\, M$ of $M$
is defined to be $\mbox{Coker}\, f^*$ where $(-)^* := \mbox{Hom}\,_R(-,R),$ which is
unique up to projective equivalence. Thus the minimal projective
presentations of $M$ represent isomorphic transposes of $M$ and it
is also stable $R$--module (see \cite[Theorem 32.13]{AF}). Let
$P\overset{\alpha}{\rightarrow}M$ be an epimorphism such that $P$ is
a projective. The syzygy module of $M$, denoted by $\Omega M$, is
the kernel of $\alpha$ which is unique up to projective equivalence.
Thus $\Omega M$ is determined uniquely up to isomorphism if
$P\rightarrow M$ is a projective cover. The operator $\lambda =
\Omega\mbox{Tr}\,$, introduced by Martsinkovsky and Strooker, enabled them
to define linkage for modules: Two finitely generated $R$--modules $M$ and $N$ are said to
be\emph{ horizontally linked} if $M\cong \lambda N$ and
$N\cong\lambda M$. Thus, $M$ is horizontally linked (to $\lambda M$)
if and only if $M\cong\lambda^2M$. It is
shown
in \cite[Proposition 8 in section 4]{MS} that, over a Gorenstein local ring $R$, a stable $R$--module
$M$ with $\mbox{dim}\, M= \mbox{dim}\, R$ is maximal Cohen-Macaulay if and only if
$\lambda M$ is maximal Cohen-Macaulay and $M$ is unmixed.
If the ring $R$ is merely a Cohen-Macaulay local ring this statements is not true \cite[section 6]{MS}.
The following definition is the module theory version of strongly Cohen-Macaulay and Sliding depth conditions.
\begin{defn}\label{D1}
\emph{Let $R$ be a local ring of dimension $d$ and let $M$ be a finitely generated $R$--module.
The module $M$ is called to be SDE (having Sliding Depth of Extension modules) if either $\mbox{Ext}\,^i_R(M,R)=0$ or
$\mbox{depth}\,_R(\mbox{Ext}\,^i_R(M,R))\geq d-i$ for all $i=1,\ldots ,d-1$. Also $M$ is called to be CME (having Cohen-Macaulay
Extension modules) if either $\mbox{Ext}\,^i_R(M,R)=0$ or $\mbox{Ext}\,^i_R(M,R)$ is
Cohen-Macaulay of dimension $d-i$ for all $i=1,\ldots ,d-1$.}
\end{defn}
To see the ambiguity of CME-modules, it is shown in the next proposition that any Cohen-Macaulay module with finite
G-dimension is a CME. Clearly any CME module is SDE.
For the definition of G-dimension we refer to \cite{C}.
\begin{prop}\label{B0}
Let $R$ be a Cohen-Macaulay local ring of dimension $d$. Then any Cohen-Macaulay
$R$--module with finite G--dimension is
\emph{CME}.
\end{prop}
\begin{proof} Let $M$ be Cohen-Macaulay
$R$--module with finite G--dimension. The Auslander-Bridger formula
$\mbox{G--dim}\,_R(M)+ \mbox{depth}\,_R(M)=\mbox{depth}\, R$ \cite[Theorem 1.4.8]{C} in
conjunction with the Cohen-Macaulayness of $M$, imply that
$\mbox{grade}\,_R(M)=\mbox{G--dim}\,_R(M)=:g$. So that $\mbox{Ext}\,^i_R(M, R)= 0$ for all $i\neq
g$. Choose $\underline{x}:=x_1, \ldots , x_g$ to be a maximal
$R$--sequence contained in $\mbox{Ann}\,_R(M)$. We have
$\mbox{Ext}\,^{g}_R(M,R)\cong \mbox{Hom}\,_{R/(\underline{x})}(M,R/(\underline{x}))$
and $\mbox{Ext}\,^i_{R/(\underline{x})}(M,R/(\underline{x}))=0$ for all $i>
0$. Since $M$ is a maximal Cohen-Macaulay
$R/(\underline{x})$--module, by \cite[Proposition 3.3.3]{BH},
$\mbox{Hom}\,_{R/(\underline{x})}(M,R/(\underline{x}))$ is maximal
Cohen-Macaulay $R/(\underline{x})$--module. Therefore
$\mbox{Ext}\,^{g}_R(M,R)$ is Cohen-Macaulay of dimension $d-g$.
\end{proof}
Determining the depth of linked ideals is in the center of the questions on the arithmetic properties of ideals. About linkage of modules the depth of modules linked to SDE modules is rather under control.
\begin{prop}\label{A1}
Let $(R,\mathfrak{m})$ be a local ring and let $M$ be a \emph{SDE} $R$--module. Then $\emph\mbox{depth}\,_R(\lambda M)
\geq\min\{\emph\mbox{depth}\,_R(M),\emph\mbox{depth}\, R\}$.
\end{prop}
\begin{proof} Set $t=\min\{\mbox{depth}\,_R(M),\mbox{depth}\,_R(R)\}$. For $t=0$ it is trivial. Suppose that $t>0$.
Set $X:=\displaystyle\cup^{d-1}_{i=1}\mbox{Ass}\,_R(\mbox{Ext}\,^i_R(M,R))\cup
\mbox{Ass}\,_R(M)\cup \mbox{Ass}\,_R(R)$. As $M$ is SDE and $t>0$, there is
$x\in\mathfrak{m}\setminus\underset{\mathfrak{p}\in X}{\cup}$$\mathfrak{p}$. Set
$\overline{M}=M/xM$ and $\overline{R}=R/xR$. The exact sequence
$0\rightarrow R\overset{x}{\longrightarrowngrightarrow}$$R\longrightarrowngrightarrow
\overline R\longrightarrowngrightarrow 0$
implies the exact sequence\\
\centerline{$0\longrightarrowngrightarrow M^* \overset{x}{\longrightarrowngrightarrow}$$ M^*
\longrightarrowngrightarrow\mbox{Hom}\,_R(M,\overline{R}) \longrightarrowngrightarrow
\mbox{Ext}\,^1_R(M,R)\overset{x}{\longrightarrowngrightarrow}$$\mbox{Ext}\,^1_R(M,R)\longrightarrowngrightarrow
\mbox{cd}\,ots.$} As each map $\mbox{Ext}\,^i_R(M,R)\overset{x}{\longrightarrowngrightarrow}$
$\mbox{Ext}\,^i_R(M,R)$ is an injection for all $i=0, \mbox{cd}\,ots, d-1$, we have
standard isomorphisms $\mbox{Ext}\,^i_{\overline{R}}(
\overline{M},\overline{R})\cong \mbox{Ext}\,^i_R(M,\overline{R})\cong
\mbox{Ext}\,^i_R(M,R)/x\mbox{Ext}\,^{i}_R(M,R)$ for all $i=0,1,\ldots ,d-2$.
Therefore $\overline{M}$ is $\mbox{SDE}\,$ as $\overline{R}$--module. Let
$P_1\longrightarrowngrightarrow P_0\longrightarrowngrightarrow M\longrightarrowngrightarrow 0$
be a minimal projective presentation of $M$ and consider the exact sequence $0\longrightarrowngrightarrow M^*\longrightarrowngrightarrow
P^*_0\longrightarrowngrightarrow \lambda M\longrightarrowngrightarrow 0$. As $\lambda M$ is a syzygy module, $x$ is also a non-zero-divisor on
$\lambda M$. Thus there is a commutative diagram
with exact rows
$$\begin{CD}
&&&&&&&&\\
\ \ &&&& 0 @>>> M^*/{xM^*} @>>>P^*_0/{xP^*_0} @>>> {\lambda M}/{x\lambda M} @>>>0& \\
&&&&&& @VV{\cong}V @VV{\cong}V \\
\ \ &&&& 0 @>>>\mbox{Hom}\,_{\overline R}(\overline M,\overline R) @>>> \mbox{Hom}\,_{\overline R}(\overline P_0,\overline R)
@>>>\lambda_{\overline{R}}\overline{M} @>>>0&\\
\end{CD}$$\\
which implies that $\lambda M/{x\lambda M} \cong \lambda_{\overline{R}}\overline{M}$.\\
By induction
$\mbox{depth}\,_{\overline{R}}(\lambda_{\overline{R}}\overline{M})\geq
\min\{\mbox{depth}\,_{ \overline{R}}(
\overline{M}),\mbox{depth}\,_{\overline{R}}(\overline{R})\}=t-1$. Thus
$\mbox{depth}\,_R(\lambda M)\geq t$.
\end{proof}
As a corollary of the above general proposition, we have the following generalization of\emph{ Theorem A} which is in fact the module theory version of \cite[Proposition1.1]{Hu}.
\begin{cor}\label{B1}
Let $R$ be a Cohen-Macaulay local ring of dimension, and let $M$ be
a maximal Cohen-Macaulay and $\emph{SDE}$ $R$--module.
Then $\lambda M$ is maximal Cohen-Macaulay.
\end{cor}
The composed
functors $\mathcal{T}_i:=\mbox{Tr}\, \Omega ^{i-1}$ for $i>0$ have been already
introduced by Auslander and Bridger \cite{AB} and recently used by
Nishida \cite{N} to relate linkage and duality. In the following
result, over Cohen-Macaulay
local ring, we characterize an SDE module $M$ in terms of depths of the $R$-modules $\mathcal{T}_iM$ and
$\lambda\Omega^iM$. Moreover, it follows that for $\lambda M$ to be
maximal Cohen-Macaulay we only need $M$ to be SDE.
\begin{thm}\label{A2}
Let $R$ be a Cohen-Macaulay local ring of dimension $d\geq2$, $M$ a finitely generated $R$--module.
The following statements are equivalent.
\begin{itemize}
\item[(i)]{$M$ is $\emph{SDE}$.}
\item[(ii)]{$\emph{depth}_R(\mathcal{T}_iM)\geq d-i$ for all $i=1,\ldots ,d-1$.}
\item[(iii)]{$\emph{depth}_R(\lambda\Omega^iM)\geq d-i$ for all $i=0,\ldots ,d-2$.}
\end{itemize}
\end{thm}
\begin{proof}For a stable finite $R$--module $N$, there is an exact sequence (\cite[section 5]{MS}),
\centerline{$ 0\longrightarrowngrightarrow \mbox{Ext}\,^1_R(\mbox{Tr}\, N,R)\longrightarrowngrightarrow
N\longrightarrowngrightarrow \lambda ^2N\longrightarrowngrightarrow 0$.}
Note that since the
transpose of every finite $R$--module is either stable or zero,
$\mbox{Tr}\,\mathcal{T}_iM$ is stably isomorphic to $\Omega^{i-1}M$ for
$i>0$, and $\mbox{Ext}\,^1_R(\Omega^{i-1}M,R)\cong \mbox{Ext}\,^i_R(M,R)$, so we
have the exact sequence
\begin{equation}\label{e} 0\longrightarrowngrightarrow \mbox{Ext}\,^i_R(M,R)\longrightarrowngrightarrow \mathcal{T}_iM\longrightarrowngrightarrow \lambda^2
\mathcal{T}_iM\longrightarrowngrightarrow 0.
\end{equation} Also, since $\lambda ^2\mathcal{T}_iM$ is stably isomorphic to $\Omega\mathcal{T}_{i+1}M$ and $R$ is
Cohen-Macaulay, we have \begin{equation}\label{e2} \mbox{depth}\,_R(\lambda ^2\mathcal{T}_iM)=\mbox{depth}\,_R
(\Omega\mathcal{T}_{i+1}M).
\end{equation}
(i)$\Longrightarrow$(ii). We proceed by induction on $i$. From the
exact sequence (\ref{e}) we have $\mbox{depth}\,_R(\mathcal{T}_{d-1}M)\geq
1$. Now suppose that $i\leq d-2$ and
$\mbox{depth}\,_R(\mathcal{T}_{i+1}M)\geq d-i-1$, accordingly,
$\mbox{depth}\,_R(\Omega\mathcal{T}_{i+1}M)\geq d-i$ which in turn implies
$\mbox{depth}\,_R(\mathcal{T}_iM)\geq d-i$, using (\ref{e2}) and (\ref{e}).
(ii)$\Longrightarrow$(i). By (\ref{e2}) and the assumption,
$\mbox{depth}\,_R(\lambda ^2\mathcal{T}_iM)=\mbox{depth}\,_R
(\Omega\mathcal{T}_{i+1}M)\geq d-i$
for all $i=1,\ldots ,d-1$. Using (\ref{e}), we get either $\mbox{Ext}\,^i_R(M,R)=0$ or $\mbox{depth}\,_R(\mbox{Ext}\,^i_R(M,R))
\geq d-i$ for all $i=1,\ldots ,d-1$.\\
(ii)$\Longleftrightarrow$(iii) Note that
$\Omega\mathcal{T}_{i+1}M=\lambda\Omega^iM$ for each $i$. Thus
$\mbox{depth}\,_R(\mathcal{T}_iM)\geq d-i$
for all $i=1,\ldots ,d-1$ if and only if $\mbox{depth}\,_R(\lambda\Omega^iM)=\mbox{depth}\,_R(\Omega\mathcal{T}_{i+1}M)
\geq d-i$ for all $i=0,\ldots ,d-2$.
\end{proof}
A shellable simplicial complex is a special kind of Cohen-Macaulay complex
with a simple combinatorial definition. Shellability is a simple but
powerful tool for proving the Cohen-Macaulay property. A simplicial complex $\mbox{D}elta$ is pure if each facet (= maximal
face) has the same dimension(cf. \cite[Section II]{S} ).
The concept of \emph{Sequentially Cohen-Macaulay} was defined by combinatorial commutative algebraists (\emph{loc. cit. 3.9} ) to answer
a basic question to find a
"nonpure" generalization of the concept of a Cohen-Macaulay module,
so that the face ring of a shellable (nonpure) simplicial complex has this
property.
This concept was then applied by commutative algebraists to study some algebraic invariants or special algebras come from graphs(c.f. \cite{AH}).
In following propositions we see the relation between \emph{Sequentially Cohen-Macaulay}, SDE and CME as well as a way to construct a family of modules with these properties.
\begin{defn}
\emph{Let $(R,\mathfrak{m})$ be a local Noetherian ring and let $M$ be a finitely generated
$R$--module. A finite filtration
$0=M_0\subseteqbset M_1\subseteqbset M_2\subseteqbset\ldots \subseteqbset M_r=M $ of
submodules of $M$ is called a} Cohen-Macaulay filtration, \emph{if
each quotient $M_i/M_{i-1}$ is Cohen-Macaulay, and
$\mbox{dim}\,_R(M_1/M_0)<\mbox{dim}\,_R(M_2/M_1)<\ldots <\mbox{dim}\,_R(M_r/M_{r-1})$. The
module $M$ is called} Sequentially Cohen-Macaulay \emph{if $M$
admits a Cohen-Macaulay filtration.}
\end{defn}
A basic fact about Sequentially Cohen-Macaulay modules is the following theorem of Herzog and Popescu
\cite[Theorem 2.4]{HP}.
\begin{thm}\label{SQ}
Let $R$ be Cohen-Macaulay local of dimension $d$ with canonical
module $\omega_R$. The following conditions are equivalent.
\begin{itemize}
\item[(i)] $M$ is Sequentially Cohen-Macaulay.
\item[(ii)] $\emph\mbox{Ext}\,^{d-i}_R(M,\omega _R)$ are either 0
or Cohen-Macaulay of dimension i for all $i\geq 0$.
\end{itemize}
\end{thm}
Thus one observes that over a Gorenstein local ring the conditions SDE, CME and Sequentially
Cohen-Macaulay are equivalent. Hence Theorem \ref{A2} provides the following computable characterization of
sequentially Cohen-Macaulay modules.
\begin{cor}\label{CC}
Let $R$ be Gorenstein local ring of dimension $d\geq 2$ and $M$ be a finitely generated
$R$--module. The following conditions are equivalent.
\begin{itemize}
\item[(i)] $M$ is Sequentially Cohen-Macaulay.
\item[(ii)] $\emph\mbox{depth}\,_R(\lambda{\Omega^i M})\geq d-i
$ for all $i$, $0\leq i\leq d-2$.
\end{itemize}
\end{cor}
To prove Proposition \ref{P}, we need to recall the next generalization of the definition of linkage of modules, \cite[Definition 4]{MS}.
\begin{defn}
\emph{Let $M$ and $N$ be two finitely generated
$R$--modules. The module $M$
is said to be} linked \emph{to $N$ by an ideal $\mathfrak{c}$ of $R$, if $\mathfrak{c} \subseteqbseteq \mbox{Ann}\,_R(M) \cap \mbox{Ann}\,_R(N)$
and $M$ and $N$ are horizontally linked as $R/\mathfrak{c}$--modules.}
\end{defn}
The following result shows that, over Gorenstein local ring, each of
the properties Sequentially Cohen-Macaulay, (or equivalently SDE or
CME) is preserved under evenly linkage.
\begin{prop}\label{P}
Let $R$ be a Gorenstein local ring.
Then the condition sequentially Cohen-Macaulay (or equivalently,
\emph{SDE} or \emph{CME}) is preserved under evenly linkage by
ideals.
\end{prop}
\begin{proof} Set $d:=\mbox{dim}\, R$. Let $\mathfrak{c}_1$ and $\mathfrak{c}_2$ be Gorenstein ideals.
Assume that $M_1$, $M$, and $M_2$ are $R$--modules such that $M_1$
is linked to $M$ by $\mathfrak{c}_1$ and $M$ is linked to $M_2$ by $\mathfrak{c}_2$.
For each $i
> 0$, by \cite[Lemma 11 and Proposition 16]{MS}, we have
\[\begin{array}{rl}
\mbox{Ext}\,^{i+g}_R(M_1,R)&\cong \mbox{Ext}\,^i_{R/\mathfrak{c}_1}(M_1,R/\mathfrak{c}_1)\\
&\cong\mbox{Ext}\,^i_{R/\mathfrak{c}_2}(M_2,R/\mathfrak{c}_2)\\
&\cong \mbox{Ext}\,^{i+g}_R(M_2,R)
\end{array}\]
where $g=\mbox{ht}\,{\mathfrak{c}_1}=\mbox{ht}\,{\mathfrak{c}_2}=\mbox{grade}\,_R{M_1}=\mbox{grade}\,_R{M_2}$.
Suppose that $M_1$ is Sequentially Cohen-Macaulay. By Theorem 2.9,
$\mbox{Ext}\,^{d-i}_R(M_1,R)$ is either zero or Cohen-Macaulay of dimension $i$ for each $i$. Hence
$\mbox{Hom}\,_{R/{\mathfrak{c}_1}}(M_1,{R/{\mathfrak{c}_1}})(\cong\mbox{Ext}\,^g_R(M_1,R)) $ is Cohen-Macaulay $R$--module of dimension $d-g$,
and so it is maximal Cohen-Macaulay $R/{\mathfrak{c}_1}$--module.
Note that, for $j=1, 2$ there are exact sequences\\
\centerline{$ 0 \rightarrow \mbox{Hom}\,_{R/{\mathfrak{c}_j}}(M_j,{R/{\mathfrak{c}_j}}) \rightarrow P_j \rightarrow
{\lambda_{R/{\mathfrak{c}_j}}M_j} \rightarrow 0 $} where $P_j$ is a
projective ${R/{\mathfrak{c}_j}}$--module. Thus we have $\mbox{depth}\,_{R/\mathfrak{c}_2}(\lambda_{R/{\mathfrak{c}_2}}M_2)= \mbox{depth}\,_R(M)=
\mbox{depth}\,_R(\lambda_{R/{\mathfrak{c}_1}}M_1)\geq d-g-1$.
Again $\mbox{Hom}\,_{R/\mathfrak{c}_2}(M_2,{R/\mathfrak{c}_2})$
is maximal Cohen-Macaulay $R/{\mathfrak{c}_2}$--module, and so that $\mbox{Ext}\,^g_R(M_2,R)$
is Cohen-Macaulay $R$--module of dimension $d-g$.
Hence $M_2$ is Sequentially Cohen-Macaulay $R$--module by Theorem \ref{SQ}.
\end{proof}
As mentioned just after Theorem \ref{SQ}, over Gorenstein local
rings, CME modules are exactly sequentially Cohen-Macaulay modules.
On the other hand , when $(R,\mathfrak{m})$ is Cohen-Macaulay ring with
canonical module $\omega_R$, it follows from the result
\cite[Theorem 2.5]{F} of Foxby that $\mbox{Tor}\,_i^R(M,\omega_R)=0$ for all
$i>0$, whenever $\mbox{G--dim}\,_R M <\infty$.
Moreover, Khatami and Yassemi in \cite[Theorem 1.11]{KY} prove that whenever $(R,\mathfrak{m})$ is Cohen-Macaulay ring with
canonical module $\omega_R$ and $M$ is an $R$-module with finite Gorenstein dimension then
$M\otimes_R \omega_R$ is Cohen-Macaulay if and
only if $M$ is Cohen-Macaulay. Note that by Lemma \ref{B0}, if
$\mbox{G--dim}\,_R M <\infty$
and $M$ is Cohen-Macaulay then $M$ is CME, i.e. the class of CME module contains
the class of Cohen-Macaulay modules of finite G-dimensions.
Hence the following question is naturally posed.
{\it What does happen if in results of Foxby, Yassemi and Khatami one replace
finite G-dimension and Cohen-Macaulay conditions of $M$ with the condition
that $M$ is \emph{CME}?}
The following Theorem provides an answer to this question.
\begin{thm}\label{A3}
Let $R$ be a Cohen-Macaulay local ring with the canonical
module $\omega_R$, and let $M$ be a finitely generated $R$--module.
Then the following two statements are equivalent.
\begin{itemize}
\item[(i)] $M$ is \emph{CME}.
\item[(ii)]{$M\otimes_R\omega_R$ is sequentially Cohen-Macaulay
and $\emph{Tor}^R_i(M,\omega_R)=0$ for all $i>0$.}
\end{itemize}
\end{thm}
\begin{proof} (i)$\mathbb{R}ightarrow$(ii). Let $P_\bullet: \mbox{cd}\,ots\rightarrow P_1\rightarrow
P_0\rightarrow 0$ be a projective resolution of $M$,
and let $I^\bullet: 0\rightarrow I^0\rightarrow I^1\rightarrow
\mbox{cd}\,ots$ be an injective resolution of $\omega_R$ and construct the
third quadrant double complex
$F:=\mbox{Hom}\,_R(\mbox{Hom}\,_R(P_\bullet,R),I^\bullet).$ Let $^v\mbox{E}$ (resp.
$^h\mbox{E}$) denote the vertical (resp. horizontal) spectral sequence
associated to the double complex $F$. Then $^v\mbox{E}^{i,j}_2
\cong\mbox{Ext}\,^i_R(\mbox{Ext}\,^j_R(M,R),\omega_R)$. Since $\mbox{Ext}\,^i_R(M,R)$ is
either zero or is Cohen-Macaulay of dimension $d-i$, we have
$$^v\mbox{E}^{i,j}_2\cong \left\lbrace
\begin{array}{c l}
\mbox{Ext}\,^i_R(\mbox{Ext}\,^j_R(M,R),\omega_R)\ \ & \text{ \ \ $i=j$,}\\
0\ \ & \text{ \ \ $\textrm{otherwise}$.}
\end{array}
\right.$$\\%
By using the equivalence of functors $\mbox{Hom}\,_R(\mbox{Hom}\,_R(X,R),Y)$ and
$X\otimes_RY$, when $X$(resp. $Y$) belongs to the subcategory of
projective (respectively injective)$R$--modules, we find that the
double complex $\mbox{Hom}\,_R(\mbox{Hom}\,_R(P_\bullet,R),I^\bullet)$ is isomorphic
to the third quadrant double complex $P_\bullet\otimes_RI^\bullet$.
Now we may use this double complex to find that
$$^h\mbox{E}^{i,j}_2\cong \left\lbrace
\begin{array}{c l}
\mbox{Tor}\,^R_i(M,\omega_R)\ \ & \text{ \ \ $j=0$,}\\
0\ \ & \text{ \ \ $\textrm{otherwise}$.}
\end{array}
\right.$$\\%
It follows that $^h\mbox{E}_{\infty}=\ ^h\mbox{E}_2$ and $^v\mbox{E}_{\infty}=\
^v\mbox{E}_2$. By comparing the two spectral sequences $^h\mbox{E}$ and $^v\mbox{E}$
we get $\mbox{Tor}\,^R_i(M,\omega_R)= 0$ for all $i>0$. Thus there is a
filtration $0=\Phi_{d+1}\subseteqbset\Phi_d\subseteqbset\ldots
\subseteqbset\Phi_0=M\otimes_R\omega_R$ of $M\otimes_R\omega_R$ such that
$\mbox{Ext}\,^i_R(\mbox{Ext}\,^i_R(M,R),\omega_R)\cong\Phi_i/\Phi_{i+1}$ for $i=0,\ldots ,d$. Note that, by
\cite[Theorem 3.3.10]{BH}, $\mbox{Ext}\,^i_R(\mbox{Ext}\,^i_R(M,R),\omega_R)$
is either zero or Cohen-Macaulay of dimension $d-i$. In other words $M\otimes_R\omega_R$ is
sequentially Cohen-Macaulay.\\
(ii)$\mathbb{R}ightarrow$(i). Consider the third quadrant double complex
$\mbox{Hom}\,_R(P_\bullet \otimes_R \omega_R , E^\bullet )$. Using the same
notation as before, let $^vE$ (resp. $^hE$) be the vertical (resp.
horizontal) spectral sequences associated
to the double complex
$\mbox{Hom}\,_R(P_\bullet \otimes_R \omega_R , E^\bullet ).$ Then $^v\mbox{E}^{i,j}_2\cong\mbox{Ext}\,^i_R
(\mbox{Tor}\,^R_j(M,\omega_R),\omega_R)\cong0$, for all $j>0$,
by our assumption. By using the equivalence of functors $\mbox{Hom}\,_R(X\otimes_R\omega_R,Y)$
and $\mbox{Hom}\,_R(X,\mbox{Hom}\,_R(\omega_R,Y))$
in the category of $R$--modules,
we find the following isomorphism of double complexes $\mbox{Hom}\,_R(P_\bullet\otimes_R\omega_R,E^\bullet)
\cong\mbox{Hom}\,_R(P_\bullet,\mbox{Hom}\,_R(\omega_R,E^\bullet)).$
Thus, we get $^hE^2_{i,j}\cong\mbox{Ext}\,^i_R(M,\mbox{Ext}\,^j_R(\omega_R,\omega_R))$, for all $i,j\geq0$.
As $\mbox{Ext}\,^i_R(\omega_R,\omega_R)=0$ for $i>0$
and $\mbox{Hom}\,_R(\omega_R,\omega_R)\cong R$, we get
$$^h\mbox{E}^2_{i,j}\cong \left\lbrace
\begin{array}{c l}
\mbox{Ext}\,^i_R(M,R)\ \ & \text{ \ \ $j=0$,}\\
0\ \ & \text{ \ \ $\textrm{otherwise}$.}
\end{array}
\right.$$\\%
As the two spectral sequences $^vE$ and $^hE$ collapse, we have
$^hE^\infty=\ ^hE^2$ and $^vE^\infty=\ ^vE^2$ and so that
$\mbox{Ext}\,^i_R(M,R)\cong\mbox{Ext}\,^i_R(M\otimes_R\omega_R,\omega_R)$. Since
$M\otimes_R\omega_R$ is sequentially Cohen-Macaulay,
$\mbox{Ext}\,^i_R(M,R)=0$ or Cohen-Macaulay of dimension $d-i$
(see\cite[Theorem 1.9]{BH}), i.e. $M$ is $\mbox{CME}\,$.
\end{proof}
\begin{cor}
Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring and let $M$ be a finitely generated
$R$--module. Set $\omega_{\widehat{R}}$ as the canonical module of
$\widehat{R}$, the completion of $R$ with respect to the $\mathfrak{m}$--adic
topology. Then the following are equivalent.
\begin{itemize}
\item[(i)]{$M$ is $\emph{CME}$.}
\item[(ii)]{$\widehat{M}\otimes_{\widehat{R}}\omega_{\widehat{R}}$ is sequentially Cohen-Macaulay and
$\emph{Tor}^{\widehat{R}}_i(\widehat{M},\omega_{\widehat{R}})=0$ for all $i>0$.}
\end{itemize}
\end{cor}
\section{local cohomology and linkage}
The main purpose of this section is to give a generalization of
\cite[Theorem 10]{MS} which states that $\mbox{H}^i_\mathfrak{m}(\lambda M)\cong
\mbox{D}(\mbox{H}^{d-1}_\mathfrak{m}(M))$ for $i=1,\ldots ,d-1$, whenever $M$ is a
generalized Cohen-Macaulay module over Gorenstein local ring $R$,
where $\mbox{D}(-)$ is the Matlis duality functor. Here we assume that $R$
is Cohen-Macaulay with canonical module $\omega_R$ such that
$M\otimes_R\omega_R$ is generalized Cohen-Macaulay; it is then shown
that for each $i=1,\ldots ,d-1$,
$\mbox{H}^i_{\mathfrak{m}}(M\otimes_R\omega_R)\cong \mbox{D}(\mbox{H}^{d-i}_{\mathfrak{m}}(\lambda M))$.
Also whenever $M$ is generalized Cohen-Macaulay, under some
vanishing assumption on Tor-modules of $M$ and $\omega_R$, we show
that $\mbox{H}^i_\mathfrak{m}(\lambda M)\cong \mbox{Ext}\,^i_R(M,R)$ for $i=1,\ldots ,d-1$
(see Corollary \ref{B2}).
The next proposition will lead to a "cohomologic criterion" for generalized Cohen-Macaulay modules to be linked (Corollary \ref{Clink}). This proposition
has its own interest as it shows the exactness of the sequence \ref{mes}.
Although this exact sequence may be already known, but for the sake of a detailed proof and statement we mention it.
\begin{prop}\label{A}
Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ with
the canonical module $\omega_R$. Assume that $M$ is a finitely
generated $R$--module, $\emph\mbox{Ass}\,_R(M)\subseteqbseteq \emph\mbox{Ass}\,
R\cup\{\mathfrak{m}\}$ and that M satisfies the Serre condition $(S_2)$ on
the punctured spectrum. Set $M^\upsilon =\emph\mbox{Hom}\,_R(M, \omega_R)$.
Let $\phi :M\longrightarrowngrightarrow M^{\upsilon\upsilon}$ be the natural
map, $K:=\emph\mbox{Ker}\,(\phi)$ and $C:=\emph\mbox{Coker}\,(\phi)$. The following
statements holds true.
\begin{itemize}
\item[(i)] If $d=0$ then $K=0$.
\item[(ii)] If $d\leq 1$ then $C= 0$.
\item[(iii)] If $d\geq 1$ then $K\cong\Gammaamma_\mathfrak{m}(M)$.
\item[(iv)] If $d\geq 2$ then $C\cong\emph\mbox{H}_\mathfrak{m}^1(M)$ and so there is an exact sequence
\begin{equation}\label{mes}
0\longrightarrowngrightarrow \Gammaamma_\mathfrak{m}(M)\longrightarrowngrightarrow M\longrightarrowngrightarrow
M^{\upsilon\upsilon}
\longrightarrowngrightarrow \emph\mbox{H}^1_\mathfrak{m}(M)\longrightarrowngrightarrow 0.
\end{equation}
\end{itemize}
\end{prop}
\begin{proof} If $d= 0$, it is clear by \cite[Theorem
3.3.10]{BH} that $C=0 $ and $K= 0$. Assume that $d\geq 1$. One has
$\mbox{depth}\,_R(M^{\upsilon\upsilon})\geq
\mbox{Min}\,\{2,\mbox{depth}\,_R(\omega_R)\}\geq 1 $ and so
$\Gammaamma_{\mathfrak{m}}(M^{\upsilon\upsilon})=0$. By applying
$\Gammaamma_{\mathfrak{m}}(-)$ on the exact sequence
\begin{equation}\label{c} 0\longrightarrowngrightarrow K\longrightarrowngrightarrow
M\overset{\phi}{\longrightarrowngrightarrow}
M^{\upsilon\upsilon}\longrightarrowngrightarrow C\longrightarrowngrightarrow
0,\end{equation} it follows that $\Gammaamma_{\mathfrak{m}}(K)=\Gammaamma_{\mathfrak{m}}(M)$.
Taking $d=1$, for each $\mathfrak{p}\in \mbox{Spec}\, R\setminus \{\mathfrak{m}\}$,
$M_{\mathfrak{p}}\cong (M^{\upsilon\upsilon})_{\mathfrak{p}}\cong
(M_{\mathfrak{p}})^{\upsilon\upsilon}$, which implies that
$\mbox{Supp}\,_R(K)\subseteqbseteq \{\mathfrak{m}\}$ i.e. $K=\Gammaamma_\mathfrak{m}(M)$. Hence we get
the exact sequence
\begin{equation}\label{a} 0\longrightarrowngrightarrow M/{\Gammaamma_{\mathfrak{m}}(M)}\longrightarrowngrightarrow
M^{\upsilon\upsilon}\longrightarrowngrightarrow C\longrightarrowngrightarrow0,\end{equation}
from which, by applying $\Gammaamma_{\mathfrak{m}}(-)$, we obtain the exact
sequence
\begin{equation}\label{b} 0\longrightarrowngrightarrow \Gammaamma_\mathfrak{m}(C)\longrightarrowngrightarrow
\mbox{H}^1_{\mathfrak{m}}(M)\longrightarrowngrightarrow
\mbox{H}^1_{\mathfrak{m}}(M^{\upsilon\upsilon})\longrightarrowngrightarrow
\mbox{H}^1_{\mathfrak{m}}(C).\end{equation}
As $\mbox{depth}\,_R(M^\upsilon)\geq\min\{2, \mbox{depth}\,_R(\omega_R)\}\geq 1$, $M^\upsilon$ is maximal Cohen-Macaulay.
Therefore the natural
map $M^\upsilon\longrightarrowngrightarrow
M^{\upsilon\upsilon\upsilon}$ is isomorphism. Using
the local duality theorem functorially gives the commutative diagram
$$\begin{CD}
&&&&\\
\ \ &&&&0 @> >>\Gammaamma_\mathfrak{m}(C)@>>>\mbox{H}^1_{\mathfrak{m}}(M)@>>> \mbox{H}^1_{\mathfrak{m}}(M^{\upsilon\upsilon})& \\
&&&&&&&& @VV{\cong}V@VV{\cong}V \\
\ \ &&&&&&&& D(M^\upsilon)@>{\cong}>>D(M^{\upsilon\upsilon\upsilon}),&&\\
\end{CD}$$\\
where $D(-)=\mbox{Hom}\,_R(-,E(R/\mathfrak{m}))$. Thus we get $\Gammaamma_\mathfrak{m}(C)=0$. Note
that if $\mathfrak{p}\in\mbox{Spec}\, R\setminus\{\mathfrak{m}\}$, then $\mbox{dim}\, R_\mathfrak{p}=0$
and so $C_\mathfrak{p}=0=K_\mathfrak{p}$ by \cite[Theorem
3.3.10]{BH}. Hence $C=\Gammaamma_\mathfrak{m}(C)=0$.
In case $d\geq 2$,
$\mbox{depth}\,_R(M^{\upsilon\upsilon})\geq\min\{2,
\mbox{depth}\,_R(\omega_R)\}\geq 2$ and (\ref{b}) implies that
\begin{equation}\label{d}\Gammaamma_\mathfrak{m}(C)\cong\mbox{H}_\mathfrak{m}^1(M).\end{equation}
Finally, we prove by induction
on $d\geq 2$ that $K=\Gammaamma_\mathfrak{m}(M)$ and $C=\mbox{H}_\mathfrak{m}^1(M)$. Assume that
the statement is settled for rings with
dimension smaller than $d$. Let $\mathfrak{p}\in\mbox{Supp}\,_R(M)\setminus\{\mathfrak{m}\}$. We first show that
$\mathfrak{p}\not\in\mbox{Supp}\,_R(K)\cup\mbox{Supp}\,_R(C)$.
If $\mbox{ht}\,\mathfrak{p}=0$, the claim holds true as before. Assume that $\mbox{ht}\,\mathfrak{p}\geq1$. As $\mbox{dim}\, R_\mathfrak{p}<d$,
induction hypothesis for
$R_\mathfrak{p}$ implies that $K_{\mathfrak{p}}=\Gammaamma_{\mathfrak{p} R_{\mathfrak{p}}}(M_{\mathfrak{p}})$ and
$C_\mathfrak{p}=\mbox{H}_{\mathfrak{p} R_\mathfrak{p}}^1(M_\mathfrak{p})$. Since $\mathfrak{p}\not\in \mbox{Ass}\,_R(R)$ and so
$\mathfrak{p}\not\in \mbox{Ass}\,_R(M)$, we get $\mbox{depth}\,_{R_\mathfrak{p}}(M_{\mathfrak{p}})\geq 1$ and
thus $K_{\mathfrak{p}}=0$, i.e. $\mathfrak{p}\not\in\mbox{Supp}\,_R(K)$. For the case
$\mbox{ht}\,\mathfrak{p}=1$, we already have, $C_{\mathfrak{p}}=0$. Assume that $\mbox{ht}\,\mathfrak{p}\geq2$.
Again from the exact sequence \ref{c} and the fact that
$\mbox{depth}\,_R(M^{\upsilon\upsilon})>1$, we get
$K=\Gammaamma_{\mathfrak{m}}(K)=\Gammaamma_{\mathfrak{m}}(M)$. As $R$ is Cohen-Macaulay and
$\mbox{Ass}\,_R(M)\subseteqbseteq\mbox{Ass}\,(R)\cup\{\mathfrak{m}\}$,
$\mbox{dim}\,_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})=\mbox{dim}\,_{R_{\mathfrak{p}}}(R_{\mathfrak{p}})=\mbox{ht}\,{\mathfrak{p}}\geq 2$ and
so $\mbox{depth}\,_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\geq
\mbox{Min}\,\{2,\mbox{dim}\,_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\}=2$ because $M$ satisfies $(S_2)$.
Hence $\mbox{H}^1_{{\mathfrak{p}}R_{\mathfrak{p}}}(M_{\mathfrak{p}})=0$. Hence $C_{\mathfrak{p}}=0$, i.e.
$\mathfrak{p}\not\in\mbox{Supp}\,_R(C)$. In particular, $K=\Gammaamma_\mathfrak{m}(K)$ and
$C=\Gammaamma_\mathfrak{m}(C)$. Now $K=\Gammaamma_\mathfrak{m}(M)$ by (\ref{c}), and
$C=\mbox{H}_\mathfrak{m}^1(M)$ by (\ref{d}).
\end{proof}
\begin{cor}\label{Clink}
Let $R$ be a Gorenstein local ring and let $M$ be a generalized Cohen-Macaulay stable $R$--module
with $\emph\mbox{dim}\,_R(M)=\emph\mbox{dim}\, R$. A necessary and sufficient
condition for $M$ to be horizontally linked is that
$\Gammaamma_\mathfrak{m}(M)=0$.
\end{cor}
\begin{proof} Note that, by \cite[Exescise 9.5.6]{BS} , $\mbox{Ass}\,_R(M)\subseteqbseteq \mbox{Ass}\,_R(R)\cup \{\mathfrak{m}\}$ and $M$
satisfies $(S_2)$ on the punctured spectrum. As $M$ is linked if
and only if the natural map $M\rightarrow M^{**}$ is one to one, the
result follows by Proposition \ref{A}.
\end{proof}
In the following result, we extend \cite[Theorm 10 in section
10]{MS} for Cohen-Macaulay rings with canonical module.
\begin{thm}\label{A4}
Let $(R,\mathfrak{m})$ be local Cohen-Macaulay ring of dimension $d\geq 2$ with
canonical module $\omega_R$. Let $M$ be a finitely
generated $R$--module of dimension $d$, such that
$M\otimes_R\omega_R$ is generalized Cohen-Macaulay. Then for each
$i=1,\ldots ,d-1$, $\emph{H}^i_{\mathfrak{m}}(M\otimes_R\omega_R)\cong
\emph{Hom}_R(\emph{H}^{d-i}_{\mathfrak{m}}(\lambda M),E(R/{\mathfrak{m}}))$.
\end{thm}
\begin{proof} First we examine the general situation for an $R$--module $N$ which is a generalized
Cohen-Macaulay of dimension
$d$. Let $0\rightarrow I^0\rightarrow I^1\rightarrow \mbox{cd}\,ots$ be an injective resolution
of $\omega_R$ and
$\mbox{cd}\,ots\rightarrow P_1\rightarrow P_0\rightarrow 0$ be a projective resolution of $N$
and construct the third quadrant double complex
$F:=\mbox{Hom}\,_R(\mbox{Hom}\,_R(P_{\bullet},\omega_R),I^{\bullet})$. Let $^v\mbox{E}$ (resp. $^h\mbox{E}$)
denote the vertical (resp. horizontal) spectral
sequence associated to the double complex $F$. Then $^v\mbox{E}^{i,j}_2
\cong\mbox{Ext}\,^i_R(\mbox{Ext}\,^j_R(N,\omega_R),\omega_R)$. As $N$ is generalized
Cohen-Macaulay, by local duality theorem, $\mbox{Ext}\,^i_R(N,\omega_R)$ is
of finite length, for all $i=1,\ldots ,d$. Therefore
$$^v\mbox{E}^{i,j}_2\cong \left\lbrace
\begin{array}{c l}
\mbox{Ext}\,^i_R(N^\upsilon ,\omega_R) \ \ & \text{if \ \ $j=0$,} \\
\mbox{H}^{d-j}_{\mathfrak{m}}(N) \ \ & \text{if \ \ $j\neq 0 ,i= d$,}\\
0\ \ & \text{if \ \ $j\neq 0\ ,i\neq d$.}
\end{array}
\right.$$\\
As the map $d^r$ is of bidegree $(r,1-r)$, one can observe that
$^v\mbox{E}^{r,0}_r\cong \mbox{Ext}\,^{d-r}_R(N^\upsilon,\omega_R)$, for
$r\geq 2$. Thus we have the following diagram:
$$
\xymatrix{
\mbox{Ext}\,^0_R(N^\upsilon,\omega_R)\ar@{..>}^{d^d}[rrrrddd]& \mbox{cd}\,ots &\mbox{Ext}\,^{d-2}_R(N^\upsilon,\omega_R)\ar^{d^2}[rrd] &\mbox{Ext}\,^{d-1}_R(N^\upsilon,\omega_R) &\mbox{Ext}\,^{d}_R(N^\upsilon,\omega_R) \\
0 & & &0 & \mbox{H}^{d-1}_{\mathfrak{m}}(N)\\
\vdots & & & & \vdots\\
0 & \mbox{cd}\,ots & &0 & \mbox{H}^{1}_{\mathfrak{m}}(N)\\
0 & \mbox{cd}\,ots & &0 & \mbox{H}^{0}_{\mathfrak{m}}(N)\\
}
$$
To compute $^h\mbox{E}_2$, we change our double complex with the functorial isomorphisms
$\mbox{Hom}\,_R(\mbox{Hom}\,_R(P_i,\omega_R),I^j)\cong P_i\otimes_R\mbox{Hom}\,_R(\omega_R,I^j).$ Thus we get
$$^h\mbox{E}_2\cong\left\lbrace
\begin{array}{c l}
N \ \ & \text{if \ \ $i=0, j=0$,} \\
0\ \ & \text{otherwise}.
\end{array}
\right.$$\\
As $\mbox{Ker}\, d^r$ and $\mbox{Coker}\, d^r$ are isomorphic to $^v\mbox{E}_\infty$,
comparing the two spectral sequences, one get isomorphisms
$d^r:\mbox{Ext}\,^{d-r}_R(N^\upsilon,\omega_R)\longrightarrowngrightarrow
\mbox{H}^{d-r+1}_\mathfrak{m}(N)$ for $r=2,\ldots ,d-1$. Therefore
$\mbox{Ext}\,^{d-r}_R(N^\upsilon,\omega_R)$ is of finite length and so,
by local duality theorem,
$\mbox{Ext}\,^{d-r}_R(N^\upsilon,\omega_R)\cong
\mathrm{D}(\mbox{H}^r_\mathfrak{m}(N^\upsilon)).$ Hence one obtains the
isomorphisms $\mbox{H}^{d-r+1}_\mathfrak{m}(N)\cong
\mathrm{D}(\mbox{H}^r_\mathfrak{m}(N^\upsilon))$, for all $r=2,\ldots ,d-1$.
Replacing $N$ by $M\otimes_R\omega_R$, gives
$$\mbox{H}^{d-i+1}_\mathfrak{m}(M\otimes_R\omega_R)\cong \mathrm{D} \big( \mbox{H}^i_\mathfrak{m}(
\mbox{Hom}\,_R( M\otimes_R\omega_R,\omega_R) ) \big) \cong \mathrm{D}(
\mbox{H}^i_\mathfrak{m}(M^*)),$$ for all $i=2,\ldots ,d-1.$ Consider the exact
sequence $0\rightarrow M^* \rightarrow P_0^* \rightarrow \lambda M
\rightarrow 0.$ Applying $\Gammaamma_\mathfrak{m}(-)$ we get
$\mbox{H}^{i+1}_\mathfrak{m}(M^*)\cong \mbox{H}^i_\mathfrak{m}(\lambda M)$ for $i=0,\ldots ,d-2$.
Therefore we have isomorphisms $\mbox{H}^i_\mathfrak{m}(M\otimes_R\omega_R)\cong
\mathrm{D}(\mbox{H}^{d-i}_\mathfrak{m}(\lambda M))$, for $i=2,\ldots ,d-1$. Now it
remains to prove the claim for $i=1$. Applying Theorem \ref{A} to
$M\otimes_R\omega_R$ and applying the functor $\mbox{Hom}\,_R(-,\omega_R)$
on the exact sequence $0\rightarrow M^* \rightarrow P_0^*
\rightarrow \lambda M \rightarrow 0,$ we get the following
commutative diagram with exact rows and columns.
$$\begin{CD}
&&&&&&&&\\
\ \ &&&& P_0\otimes_R\omega_R @>>>M\otimes_R\omega_R @>>>0& \\
&&&& @V{\cong}VV @VVV \\
\ \ &&&& \mbox{Hom}\,_R(P_0^*,\omega_R) @>>> {(M\otimes_R\omega_R)}^{\upsilon\upsilon}
@>>> \mbox{Ext}\,^1_R(\lambda M,\omega_R)@>>>0&\\
&&&&&&@VVV\\
\ \ &&&&&& \mbox{H}^1_{\mathfrak{m}}(M\otimes_R\omega_R)&\\
&&&&&&@VVV \\
&&&&&&0 \\
\end{CD}$$\\
which implies that $\mbox{H}^1_\mathfrak{m}(M\otimes_R\omega_R)\cong \mbox{Ext}\,^1_R(\lambda M,\omega_R)\cong
\mathrm{D}(\mbox{H}^{d-1}_\mathfrak{m}(\lambda M) ).$
\end{proof}
As the final result, we state the next corollary of Theorem \ref{A4}.
\begin{cor}\label{B2}
Let $R$ be a Cohen-Macaulay ring of dimension $d\geq2$ with canonical
module $\omega_R$, and let $M$ be a finitely
generated $R$--module. Suppose that $\emph{Ext}^i_R(M,R)$ is of finite length for $i=0,\ldots
,d-1$ and $\emph{Tor}^R_i(M,\omega_R)=0$
for $i>0$. Then $\emph{H}^i_{\mathfrak{m}}(\lambda M)\cong \emph{Ext}^i_R(M,R)$, $i=1,\ldots ,d-1$,
and so $\lambda M$ is generalized Cohen-Macaulay.
\end{cor}
\begin{proof} Since $\mbox{Tor}\,^R_i(M,\omega_R)=0$ for all $i>0$, as in the proof of Theorem
\ref{A3} ((ii)$\mathbb{R}ightarrow$(i)),
$\mbox{Ext}\,^i_R(M\otimes_R\omega_R,\omega_R)\cong \mbox{Ext}\,^i_R(M,R)$ for
$i=0,\ldots ,d-1$. Hence $\mbox{Ext}\,^i_R(M\otimes_R\omega_R,\omega_R)$ is
of finite length for $i=1,\ldots ,d-1$ and so that
$M\otimes_R\omega_R$ is generalized Cohen-Macaulay. Therefore the
result follows from local duality theorem and Theorem \ref{A4}.
\end{proof}
\end{document} |
\begin{document}
\title{Cubic graphical regular representations of $\PSL_2(q)$}
\author[Xia]{Binzhou Xia}
\address{Beijing International Center for Mathematical Research\\
Peking University\\
Beijing, 100871\\
P. R. China}
\email{[email protected]}
\author[Fang]{Teng Fang}
\address{Beijing International Center for Mathematical Research\\
Peking University\\
Beijing, 100871\\
P. R. China}
\email{[email protected]}
\maketitle
\begin{abstract}
We study cubic graphical regular representations of the finite simple groups $\PSL_2(q)$. It is shown that such graphical regular representations exist if and only if $q\neq7$, and the generating set must consist of three involutions.
\end{abstract}
\textbf{\noindent Keywords: Cayley graph; cubic graph; graphical regular representation; projective special linear group.}
\section{Introduction}
Given a group $G$ and a subset $S\subset G$ such that $1\notin S$ and $S=S^{-1}:=\{g^{-1}\mid g\in S\}$,
the \emph{Cayley graph} $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ of $G$ is the graph with vertex set $G$ such that two vertices $x,y$ are adjacent
if and only if $yx^{-1}\in S$. It is easy to see that $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ is connected if and only if $S$ generates the group $G$. If one identifies $G$ with its right regular representation, then $G$ is a subgroup of $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S))$. We call $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ a \emph{graphical regular representation} (\emph{GRR} for short) of $G$ if $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S))=G$. The problem of seeking graphical regular representations for given groups has been investigated for a long time. A major accomplishment for this problem is the determination of finite groups without a GRR, see~\cite[16g]{Biggs1993}. It turns out that most finite groups admit at least one GRR. For instance, every finite unsolvable group has a GRR~\cite{Godsil1981}.
In contrast to unrestricted GRRs, the question of whether a group has a GRR of prescribed valency is largely open. Research on this subject have been focusing on small valencies~\cite{FLWX2002,Godsil1983,XX2004}. In~2002, Fang, Li, Wang and Xu~\cite{FLWX2002} issued the following conjecture.
\begin{conjecture}\label{conj1}
(\cite[Remarks on Theorem~1.3]{FLWX2002}) Every finite nonabelian simple group has a cubic GRR.
\end{conjecture}
Note that any GRR of a finite simple group must be connected, for otherwise its full automorphism group would be a wreath product. Hence if $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ is a GRR of a finite simple group $G$, then $S$ is necessarily a generating set of $G$. Apart from a few small groups, Conjecture~\ref{conj1} was only known to be true for the alternating groups~\cite{Godsil1983} and Suzuki groups~\cite{FLWX2002}, while no counterexample was found yet. In this paper, we study cubic GRRs for finite projective special linear groups of dimension two. In particular, Theorem~\ref{thm4} shows that Conjecture~\ref{conj1} fails for $\PSL_2(7)$ but holds for all $\PSL_2(q)$ with $q\neq7$.
For any subset $S$ of a group $G$, denote by $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,S)$ the group of automorphisms of $G$ fixing $S$ setwise. Each element in $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,S)$ is an automorphism of $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ fixing the identity of $G$. Hence a necessary condition for $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ to be a GRR of $G$ is that $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,S)=1$. In~\cite{FLWX2002}, the authors showed that this condition is also sufficient for many cubic Cayley graphs of finite simple groups. We state their result for simple groups $\PSL_2(q)$ as follows, which is the starting point of the present paper.
\begin{theorem}\label{thm3}
\emph{(\cite{FLWX2002})} Let $G=\PSL_2(q)$ be a simple group, where $q\neq11$ is a prime power, and $S$ be a generating set of $G$ with $S^{-1}=S$ and $|S|=3$. Then $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ is a GRR of $G$ if and only if $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,S)=1$.
\end{theorem}
The following are our three main results.
\begin{theorem}\label{thm4}
For any prime power $q\geqslant5$, $\PSL_2(q)$ has a cubic GRR if and only if $q\neq7$.
\end{theorem}
\begin{theorem}\label{thm2}
For each prime power $q$ there exist involutions $x$ and $y$ in $\PSL_2(q)$ such that the probability for a randomly chosen involution $z$ to make
$$
\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(\PSL_2(q),\{x,y,z\})
$$
a cubic GRR of $\PSL_2(q)$ tends to $1$ as $q$ tends to infinity.
\end{theorem}
\begin{proposition}\label{thm1}
Let $q\geqslant5$ be a prime power and $G=\PSL_2(q)$. If $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ is a cubic GRR of $G$, then $S$ is a set of three involutions.
\end{proposition}
Theorem~\ref{thm2} shows that it is easy to make GRRs for $\PSL_2(q)$ from three involutions. On the other hand, Proposition~\ref{thm1} says that one can only make GRRs for $\PSL_2(q)$ from three involutions, which is a response to~\cite[Problem~1.2]{Godsil1983} as well. (Note that for a cubic Cayley graph $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$, the set $S$ either consists of three involutions, or has the form $\{x,y,y^{-1}\}$ with $o(x)=2$ and $o(y)>2$.) The proof of Theorem~\ref{thm2} is at the end of Section~\ref{sec1}, and the proofs of Theorem~\ref{thm4} and Proposition~\ref{thm1} are in Section~\ref{sec2}. We also pose two problems concerning cubic GRRs for other families of finite simple groups at the end of this paper.
\section{Preliminaries}
The following result is well known, see for example~\cite[II~\S7 and~\S8]{Huppert1967}.
\begin{lemma}\label{lem3}
Let $q\geqslant5$ be a prime power and $d=\gcd(2,q-1)$. Then $\PGL_2(q)$ has a maximal subgroup $M=\mathrm{D}} \def\di{\,\big|\,_{2(q+1)}$. Moreover, $M\cap\PSL_2(q)=\mathrm{D}} \def\di{\,\big|\,_{2(q+1)/d}$, and for $q\notin\{7,9\}$ it is maximal in $\PSL_2(q)$.
\end{lemma}
The next lemma concerns facts about involutions in two-dimensional linear groups which is needed in the sequel.
\begin{lemma}\label{lem1}
Let $q=p^f\geqslant5$ for some prime $p$ and $G=\PSL_2(q)$. Then the following hold.
\begin{itemize}
\item[(a)] There is only one conjugacy class of involutions in $G$.
\item[(b)] For any involution $g$ in $G$,
$$
\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(g)=
\begin{cases}
\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}_2^f,\quad\textup{if $p=2$},\\
\mathrm{D}} \def\di{\,\big|\,_{q-1},\quad\textup{if $q\equiv1\pmod{4}$},\\
\mathrm{D}} \def\di{\,\big|\,_{q+1},\quad\textup{if $q\equiv3\pmod{4}$}.
\end{cases}
$$
\item[(c)] If $p>2$, then for any involution $\alpha$ in $\PGL_2(q)$, the number of involutions in $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha)$ is at most $(q+3)/2$.
\end{itemize}
\end{lemma}
\begin{proof}
Parts~(a) and~(b) can be found in~\cite[Lemma~A.3]{GZ2010}. To prove part~(c), assume that $p>2$ and $\alpha$ is an involution in $\PGL_2(q)$. By \cite[Lemma~A.3]{GZ2010} we have $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha)=\mathrm{D}} \def\di{\,\big|\,_{q+\varepsilon}$ with $\varepsilon=\pm1$. As a consequence, the number of involutions in $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha)$ is at most $1+(q+\varepsilon)/2\leqslant(q+3)/2$. This completes the proof.
\end{proof}
\section{GRRs from three involutions}\label{sec1}
Recall from Lemma~\ref{lem3} that $\PSL_2(q)$ has a maximal subgroup $\mathrm{D}} \def\di{\,\big|\,_{2(q+1)/d}$, where $d=\gcd(2,q-1)$. The following proposition plays the central role in this paper.
\begin{proposition}\label{prop1}
Let $q=p^f\geqslant11$ for some prime $p$, $d=\gcd(2,q-1)$, $G=\PSL_2(q)$, and $H=\mathrm{D}} \def\di{\,\big|\,_{2(q+1)/d}$ be a maximal subgroup of $G$. Then for any two involutions $x,y$ with $\langle x,y\rangle=H$, there are at least
$$
\frac{q^2-4d^2fq-(d+2)q-4d^2f-3d^2+2d-1}{d}
$$
involutions $z\in G$ such that $\langle x,y,z\rangle=G$ and $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,\{x,y,z\})=1$.
\end{proposition}
\begin{proof}
Fix involutions $x,y$ in $H$ such that $\langle x,y\rangle=H$. Identify the elements in $G$ with their induced inner automorphisms of $G$. In this way, $G$ is a normal subgroup of $A:=\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G)$, and the elements of $A$ act on $G$ by conjugation. Denote by $V$ the set of involutions in $G$, and
\begin{eqnarray*}
L&=&\{y^\alpha\mid\alpha\in A,x^\alpha=x\}\cup\{y^\alpha\mid\alpha\in A,x^\alpha=y\}\\
&&\cup\ \{x^\alpha\mid\alpha\in A,y^\alpha=x\}\cup\{x^\alpha\mid\alpha\in A,y^\alpha=y\}.
\end{eqnarray*}
By Lemma~\ref{lem1}, $x$ and $y$ are conjugate in $A$. Hence
\begin{eqnarray}\label{eq1}
|L|&\leqslant&|\{\alpha\in A\mid x^\alpha=x\}|+|\{\alpha\in A\mid x^\alpha=y\}|\\
\nonumber&&+\ |\{\alpha\in A\mid y^\alpha=x\}|+|\{\alpha\in A\mid y^\alpha=y\}|\\
\nonumber&=&4|\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_A(x)|\leqslant4|\mathrm{Out}(G)||\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(x)|=4df|\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(x)|\leqslant4df(q+1)
\end{eqnarray}
by virtue of Lemma~\ref{lem1}. Denote by $I$ the set of involutions $\alpha\in A$ such that $x^\alpha=y$ and $y^\alpha=x$.
Take an arbitrary $\alpha\in I$. Then $H^\alpha=\langle x^\alpha,y^\alpha\rangle=\langle y,x\rangle=H$, i.e., $\alpha\in\mathrm{N}or_A(H)$. Write $\mathrm{N}or_A(H)=\langle a\rangle\rtimes\langle b\rangle$ with $\langle a\rangle=\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}_{q+1}<\PGL_2(q)$ and $\langle b\rangle=\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}_{2f}$, so that $H=\mathrm{N}or_A(H)\cap G=\langle a^d\rangle\rtimes\langle b^f\rangle$. Since $x$ and $y$ are two involutions generating $H$, we have $x=a^{di}b^f$ and $y=a^{dj}b^f$ for some integers $i$ and $j$. Moreover, either $\alpha=a^kb^f$ for some integer $k$, or $\alpha=a^{(q+1)/2}$ with $q$ odd, because $\alpha$ is an involution in $\mathrm{N}or_A(H)$. However, if $\alpha=a^{(q+1)/2}$ with $q$ odd, then $\alpha$ will fix $x$ and $y$, respectively. Thus $\alpha=a^kb^f$ for some integer $k$, and in particular, $\alpha\in\PGL_2(q)$. In view of $a^{b^f}=a^{-1}$, one computes that
$$
x^\alpha=(a^{di}b^f)^{a^kb^f}=(a^{di})^{a^kb^f}(b^f)^{b^fa^{-k}}=(a^{di})^{b^f}(b^f)^{a^{-k}}=a^{2k-di}b^f.
$$
This together with the assumption $x^\alpha=y=a^{dj}b^f$ gives $(a^k)^2=a^{d(i+j)}$. As a consequence, $|I|\leqslant d$. If $p=2$, then $\alpha\in\PGL_2(q)=G$, and we see from Lemma~\ref{lem1} that $|V\cap\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha)|=q-1$. If $p$ is odd, then Lemma~\ref{lem1} asserts that $|V\cap\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha)|\leqslant(q+3)/2$. To sum up, we have
\begin{equation*}
|\bigcup\limits_{\alpha\in I}(V\cap\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha))|\leqslant|I|\cdot\frac{q+3d-3}{d}\leqslant d\cdot\frac{q+3d-3}{d}=q+3d-3.
\end{equation*}
Due to Lemma~\ref{lem1} and the orbit-stabilizer theorem, $|V|\geqslant|G|/(q+1)=q(q-1)/d$, whence
\begin{eqnarray}\label{eq2}
&&|(V\setminus H)\setminus\bigcup\limits_{\alpha\in I}\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha)|\geqslant|V|-|V\cap H|-|\bigcup\limits_{\alpha\in I}(V\cap\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha))|\\
\nonumber&\geqslant&\frac{q(q-1)}{d}-\left(\frac{q+1}{d}+1\right)-(q+3d-3)=\frac{q^2-(d+2)q-3d^2+2d-1}{d}.
\end{eqnarray}
Suppose that $z$ is an involution of $G$ outside $H$. It follows that $\langle x,y,z\rangle=G$. If $\alpha$ is a nonidentity in $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,\{x,y,z\})$, then either $z^\alpha=z$ or $z\in L$. In the former case, $\alpha$ is an involution as $\alpha$ interchanges $x$ and $y$, and hence $z\in\bigcup_{\alpha\in I}\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha)$. This implies that for any $z$ in $(V\setminus H)\setminus\bigcup_{\alpha\in I}\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha)$ but not in $L$, one has $\langle x,y,z\rangle=G$ and $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,\{x,y,z\})=1$. Now combining~\eqref{eq1} and~\eqref{eq2} we deduce that the number of choices of such $z$ is at least
\begin{eqnarray*}
|(V\setminus H)\setminus\bigcup\limits_{\alpha\in I}\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}en_G(\alpha)|-|L|&\geqslant&\frac{q^2-(d+2)q-3d^2+2d-1}{d}-4df(q+1)\\
&=&\frac{q^2-4d^2fq-(d+2)q-4d^2f-3d^2+2d-1}{d},
\end{eqnarray*}
as the lemma asserts.
\end{proof}
As an immediate consequence of Proposition~\ref{prop1}, we give a proof for Theorem~\ref{thm2}.
\vskip0.1in
\noindent\textbf{Proof of Theorem~\ref{thm2}:} Assume without loss of generality that $q\geqslant11$, and take $x,y$ to be any two involutions generating the maximal subgroup $\mathrm{D}} \def\di{\,\big|\,_{2(q+1)/d}$ of $\PSL_2(q)$, where $d:=\gcd(2,q-1)$. Let $J=J_{q,x,y}$ be the set of involutions $z$ such that $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(\PSL_2(q),\{x,y,z\})$ is a GRR of $\PSL_2(q)$. By Theorem~\ref{thm3}, $J$ equals the set of involutions $z$ such that $\langle x,y,z\rangle=\PSL_2(q)$ and $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(\PSL_2(q),\{x,y,z\})=1$. Hence applying Proposition~\ref{prop1} we obtain
\begin{eqnarray*}
|J|&\geqslant&\frac{q^2-4d^2fq-(d+2)q-4d^2f-3d^2+2d-1}{d}\\
\nonumber&\geqslant&\frac{q^2-16fq-4q-16f-9}{d}\geqslant\frac{q^2-16q\log_2q-4q-16\log_2q-9}{d}.
\end{eqnarray*}
Moreover, combining Lemma~\ref{lem1} and the orbit-stabilizer theorem shows that the number of involutions in $\PSL_2(q)$ is at most $|\PSL_2(q)|/(q-1)=q(q+1)/d$. Thus for a randomly chosen involution $z$, the probability such that $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(\PSL_2(q),\{x,y,z\})$ is
a cubic GRR of $\PSL_2(q)$ is at least
$$
\frac{|J|}{q(q+1)/d}\geqslant\frac{q^2-16q\log_2q-4q-16\log_2q-9}{q(q+1)},
$$
which tends to $1$ as $q$ tends to infinity.
\qed
\section{Conclusion}\label{sec2}
\begin{lemma}\label{lem5}
Let $q=p^f$ for some prime $p$, and $d=\gcd(2,q-1)$. If $q\geqslant29$ or $q=23$, then
$$
q^2-4d^2fq-(d+2)q-4d^2f-3d^2+2d-1>0.
$$
\end{lemma}
\begin{proof}
It is direct to verify the conclusion for $q\in\{32,64,81,128,256\}$. Hence we only need to prove the lemma with $q\notin\{32,64,81,128,256\}$. Note that under this assumption, $q>22f$. Since $q>25/2\geqslant(16f+9)/(6f-4)$, or equivalently $4q+16f+9<6fq$, we have
\begin{eqnarray*}
&&q^2-4d^2fq-(d+2)q-4d^2f-3d^2+2d-1\\
&\geqslant& q^2-16fq-4q-16f-9>q^2-16fq-6fq=q(q-22f)>0
\end{eqnarray*}
as desired.
\end{proof}
\begin{lemma}\label{lem6}
Let $q$ be an odd prime power, and $x$ be an involution of $\PSL_2(q)$. If $q\equiv3\pmod{4}$, then for any $y\in\PSL_2(q)$, there exists $\alpha\in\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(\PSL_2(q))$ such that $x^\alpha=x$ and $y^\alpha=y^{-1}$.
\end{lemma}
\begin{proof}
Appeal to the isomorphism $\PSL_2(q)\cong\PSU_2(q)$. Let $i$ be an element of order four in $\mathbb{F}_{q^2}^\times$, $G=\PSU_2(q)$, $A=\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G)$ and $\psi$ be the homomorphism from $\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}U_2(q)$ to $G$ modulo $\mathbf{Z}} \def\ZZ{\mathrm{C}(\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}U_2(q))$. In view of Lemma~\ref{lem1} we may assume that
$$
x=
\begin{pmatrix}
i&0\\
0&-i
\end{pmatrix}
^\psi
$$
($x$ is indeed an element of $\PSL_2(q)$ since $i^2=-1$). By~\cite[II~8.8]{Huppert1967} one has
$$
y=
\begin{pmatrix}
a&b\\
-b^q&a^q
\end{pmatrix}
^\psi,
$$
where $a,b\in\mathbb{F}_{q^2}$ such that $a^{q+1}+b^{q+1}=1$. Then set $c=b^{q-1}$ if $b\neq0$, and $c=1$ if $b=0$. Define
$$
\alpha:g\mapsto
\begin{pmatrix}
0&c^q\\
1&0
\end{pmatrix}
^\psi g
\begin{pmatrix}
0&1\\
c&0
\end{pmatrix}
^\psi
$$
for any $g\in G$. It is straightforward to verify that $\alpha\in A$, $x^\alpha=x$ and $y^\alpha=y^{-1}$. Thus the lemma is true.
\end{proof}
We remark that if $q\not\equiv3\pmod{4}$ then the conclusion of Lemma~\ref{lem6} may not hold. For example, when $q=8$ or $13$, respectively, there exist $x,y\in\PSL_2(q)$ with $o(x)=2$ and $o(y)>2$ such that $\{x,y\}$ does not generate $\PSL_2(q)$ and $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(\PSL_2(q),\{x,y,y^{-1}\})=1$.
\vskip0.1in
Now we are able to prove Theorem~\ref{thm4} and Proposition~\ref{thm1}.
\vskip0.1in
\noindent\textbf{Proof of Theorem~\ref{thm4}:} For $q\in\{8,9,11,13,16,17,19,25,27\}$, computation in \magma~\cite{magma} shows that there exist involutions $x,y,z\in\PSL_2(q)$ such that $\langle x,y,z\rangle=\PSL_2(q)$ and $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(\PSL_2(q),\{x,y,z\})=1$ (one can further take $x,y$ to be the generators of $\mathrm{D}} \def\di{\,\big|\,_{2(q+1)/\gcd(2,q-1)}$). This together with Proposition~\ref{prop1} and Lemma~\ref{lem5} indicates that whenever $q\geqslant8$, there exist involutions $x,y,z\in\PSL_2(q)$ such that $\langle x,y,z\rangle=\PSL_2(q)$ and $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(\PSL_2(q),\{x,y,z\})=1$. Hence by Theorem~\ref{thm3}, $\PSL_2(q)$ has a cubic GRR if $q\geqslant8$ and $q\neq11$. Moreover, the existence of cubic GRRs for $\PSL_2(5)$ and $\PSL_2(11)$ was proved in~\cite[Remarks on Theorem~1.3]{FLWX2002}. Therefore, $\PSL_2(q)$ has a cubic GRR provided $q\neq7$.
It remains to show that $\PSL_2(7)$ has no cubic GRR. Suppose on the contrary that $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(\PSL_2(7),S)$ is a cubic GRR of $\PSL_2(7)$. Then according to Lemma~\ref{lem6}, $S$ must be a set of three involutions. However, computation in \magma~\cite{magma} shows that for any involutions $x,y,z$ with $\langle x,y,z\rangle=\PSL_2(7)$, there exists an involution $\alpha\in\PGL_2(7)$ such that $\{x^\alpha,y^\alpha,z^\alpha\}=\{x,y,z\}$. This contradiction completes the proof.
\qed
\vskip0.1in
\noindent\textbf{Proof of Proposition~\ref{thm1}:} By contradiction, suppose that $S$ does not consist of three involutions. Then since $1\notin S$ and $S=S^{-1}$, we deduce that $S=\{x,y,y^{-1}\}$ with $o(x)=2$ and $o(y)>2$. It follows that $\langle x,y\rangle=G$.
First assume that $q$ is even. By virtue of Lemma~\ref{lem1}, $x$ can be taken as any involution in $G=\mathrm{SL}} \def\SO{\mathrm{SO}} \def\Soc{\mathrm{Soc}} \def\Sp{\mathrm{Sp}} \def\Stab{\mathrm{Stab}} \def\SU{\mathrm{SU}} \def\Suz{\mathrm{Suz}} \def\Sy{\mathrm{S}} \def\Sym{\mathrm{Sym}} \def\Sz{\mathrm{Sz}_2(q)$, whence we may assume that
$$
x=
\begin{pmatrix}
1&1\\
0&1
\end{pmatrix}.
$$
Suppose
$$
y=
\begin{pmatrix}
a&b\\
c&d
\end{pmatrix},
$$
where $a,b,c,d\in\mathbb{F}_q$ such that $ad-bc=1$. If $c=0$, then $\langle x,y\rangle$ is contained in the group of upper triangular matrices in $\mathrm{SL}} \def\SO{\mathrm{SO}} \def\Soc{\mathrm{Soc}} \def\Sp{\mathrm{Sp}} \def\Stab{\mathrm{Stab}} \def\SU{\mathrm{SU}} \def\Suz{\mathrm{Suz}} \def\Sy{\mathrm{S}} \def\Sym{\mathrm{Sym}} \def\Sz{\mathrm{Sz}_2(q)$, impossible because $\langle x,y\rangle=G$. Consequently, $c\neq0$. Set
$$
h=
\begin{pmatrix}
c&a+d\\
0&c
\end{pmatrix},
$$
and define $\alpha:g\mapsto h^{-1}gh$ for any $g\in G$. Then $\alpha\in A$, and one sees easily that $x^\alpha=x$ and $y^\alpha=y^{-1}$. This implies $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,S)\neq1$, contrary to the condition that $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ is a GRR of $G$.
Next assume that $q\equiv1\pmod{4}$. Let $\omega$ be an element of order four in $\mathbb{F}_q^\times$, and $\varphi$ be the homomorphism from $\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}L_2(q)$ to $\PGL_2(q)$ modulo $\mathbf{Z}} \def\ZZ{\mathrm{C}(\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}L_2(q))$. Due to Lemma~\ref{lem1}, we may assume that
$$
x=
\begin{pmatrix}
\omega&0\\
0&-\omega
\end{pmatrix}
^\varphi
$$
($x$ is indeed an element of $\PSL_2(q)$ since $\omega^2=-1$). Suppose
$$
y=
\begin{pmatrix}
a&b\\
c&d
\end{pmatrix}
^\varphi,
$$
where $a,b,c,d\in\mathbb{F}_q$ such that $ad-bc=1$. If $bc=0$, then the preimage of $\langle x,y\rangle$ under $\varphi$ is contained either in the group of upper triangular matrices or in the group of lower triangular matrices in $\mathrm{SL}} \def\SO{\mathrm{SO}} \def\Soc{\mathrm{Soc}} \def\Sp{\mathrm{Sp}} \def\Stab{\mathrm{Stab}} \def\SU{\mathrm{SU}} \def\Suz{\mathrm{Suz}} \def\Sy{\mathrm{S}} \def\Sym{\mathrm{Sym}} \def\Sz{\mathrm{Sz}_2(q)$. Hence $bc\neq0$ as $\langle x,y\rangle=G$. Set
$$
h=
\begin{pmatrix}
0&b\\
-c&0
\end{pmatrix}
^\varphi,
$$
and define $\alpha:g\mapsto h^{-1}gh$ for any $g\in G$. Then $\alpha\in A$, and it is direct to verify that $x^\alpha=x$ and $y^\alpha=y^{-1}$. This implies $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,S)\neq1$, contrary to the condition that $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ is a GRR of $G$.
Finally assume that $q\equiv3\pmod{4}$. Then we derive from Lemma~\ref{lem6} that $\mathrm{A}} \def\AGL{\mathrm{AGL}} \def\AGaL{\mathrm{A\mathrm{G}} \def\GaL{\mathrm{\Gamma L}} \def\GF{\mathrm{GF}} \def\GL{\mathrm{GL}} \def\GO{\mathrm{GO}} \def\GU{\mathrm{GU}amma L}} \def\Aut{\mathrm{Aut}ut(G,S)\neq1$, again a contradiction to the condition that $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,S)$ is a GRR of $G$. The proof is thus completed.
\qed
We conclude with two problems on cubic GRRs for other families of finite simple groups. First, in view of Theorem~\ref{thm4}, a natural problem is to determine which finite nonabelian simple groups have no cubic GRR. Such groups would be vary rare, and we conjecture that there are only finitely many of them.
\begin{conjecture}
There are only finitely many finite nonabelian simple groups that have no cubic GRR.
\end{conjecture}
\begin{problem}
Classify the finite nonabelian simple groups that have no cubic GRR.
\end{problem}
Second, as it is shown in Proposition~\ref{thm1} that for finite simple groups $\PSL_2(q)$, a GRR can only be made from three involutions, one would ask what is the situation for other finite nonabelian simple groups, which is the problem below.
\begin{problem}
Classify the finite nonabelian simple groups $G$ having no GRR of form $\mathrm{C}} \def\calB{\mathcal{B}} \def\Cay{\mathrm{Cay}} \def\Cen{\mathbf{C}} \def\Co{\mathrm{Co}} \def\Cos{\mathsf{Cos}ay(G,\{x,y,y^{-1}\})$, where $o(x)=2$ and $o(y)>2$.
\end{problem}
\noindent\textsc{Acknowledgements.}
The first author acknowledges the support of China Postdoctoral Science Foundation Grant 2014M560838 and National Science Foundation of China grant 11501011. The authors would like to thank the anonymous referees for their comments to improve the presentation.
\end{document} |
\begin{enumerate}gin{document}
\title{Higher topological Hochschild Homology of periodic complex K-theory}
\def\mathcal{C}{\mathbb{C}}
\begin{enumerate}gin{abstract} We describe the topological Hochschild homology of the periodic complex $K$-theory spectrum, $THH(KU)$, as a commutative $KU$-algebra: it is equivalent to $KU[K(\mathbb{Z},3)]$ and to $F(\mathbb{S}igma KU_\mathbb{Q})$, where $F$ is the free commutative $KU$-algebra functor on a $KU$-module. Moreover, $F(\mathbb{S}igma KU_\mathbb{Q})\simeq KU \vee \mathbb{S}igma KU_\mathbb{Q}$, a square-zero extension. In order to prove these results, we first establish that topological Hochschild homology commutes, as an algebra, with localization at an element.
Then, we prove that $THH^n(KU)$, the $n$-fold iteration of $THH(KU)$, i.e. $T^n\otimes KU$, is equivalent to $KU[G]$ where $G$ is a certain product of integral Eilenberg-Mac Lane spaces, and to a free commutative $KU$-algebra on a rational $KU$-module.
We prove that $S^n \otimes KU$ is equivalent to $KU[K(\mathbb{Z},n+2)]$ and to $F(\mathbb{S}igma^n KU_\mathbb{Q})$. We describe the topological André-Quillen homology of $KU$ as $KU_\mathbb{Q}$.
\end{abstract}
\maketitle
\section{Introduction}
Topological Hochschild homology ($THH$) of structured ring spectra was introduced by Bökstedt \cite{bokstedt} and Breen \cite{breen}; for an introduction to the subject, see \cite[Chapter 4]{dgm}, \cite[Chapter IX]{ekmm} and \cite{shipley-thh}. It is the generalization to structured ring spectra of classical Hochschild homology ($HH$) of rings.
It was realized in \cite{mc-schw-vo} that the $THH$ of a commutative ring spectrum $R$ can be expressed as $S^1\otimes R$, where $\otimes$ denotes the tensor of the category of commutative ring spectra over unbased spaces. Tensors with other spaces also give interesting information \cite{bcd}, \cite{cdd}, and we refer to them as giving ``higher $THH$'' of $R$. For example, tori $T^n$ give $n$-fold iterated $THH$.
Spheres $S^n$ give a topological version of Pirashvili's higher order Hochschild homology of commutative rings \cite{pirashvili}. Complete calculations of these invariants for a given $R$ are scarce: see for example \cite{schlichtkrull-higher} for the case of spectra, \cite{veen} and \cite{blprz} for partial computations for the Eilenberg-Mac Lane ring spectrum $H\mathbb{F}_p$ of the field with $p$ elements and other related ring spectra, and \cite{dlr} for Eilenberg-Mac Lane spectra of some rings of integers.
In this paper, we present complete descriptions of the commutative $KU$-algebras $T^n\otimes KU$ and $S^n\otimes KU$ for $n\geq 1$, where $KU$ is the ring spectrum of periodic complex topological $K$-theory. Close results to the $n=1$ case were known, as we explain below, but only additively. The classical André-Quillen homology of commutative rings also has a topological analogue, denoted $TAQ$ \cite{basterra}: we determine the $KU$-module $TAQ(KU)$.
Our computations showcase some interesting phenomena. The formulas for the (higher) topological Hochschild homology and topological André-Quillen homology of $KU$ which we obtain are the ones we would get if $KU$ was somehow a Thom spectrum (which it isn't),
see Remarks \ref{thh-thom}, \ref{x-thom} and \ref{jurs}.\ref{taq-ku-thom}. Also, our results show that the conclusion of McCarthy-Minasian's adaptation of the Hochschild-Kostant-Rosenberg theorem \cite{mccarthy-minasian}, which applies only to connective ring spectra, holds for $KU$ (Remark \ref{jurs}.\ref{hokr}).
Finally, another remarkable phenomenon highlighted by our computations involves invariance under stable equivalences of spaces: let $R$ be a commutative ring spectrum and $X$ and $Y$ be spaces such that $\mathbb{S}igma X \simeq \mathbb{S}igma Y$. One may ask the question of whether $X\otimes R\simeq Y\otimes R$, i.e. of whether $-\otimes R$ is a \emph{stable invariant}. This turns out not to be true in general \cite{dundas-tenti}, but the computations of \cite{veen} show that, in a certain range relating $n$ and $p$, $X\otimes H\mathbb{F}_p \simeq Y\otimes H\mathbb{F}_p$ when $X=T^n$ and $Y=\begin{itemize}gvee\limits_{i=1}^n (S^i)^{\vee {n \choose i}}$. Our results show that the same is true for $R=KU$, see Remark \ref{stabeq}. It would be interesting to know whether $-\otimes KU$ is a stable invariant, and to try to characterize the commutative ring spectra $R$ such that $-\otimes R$ is a stable invariant.\\
\paragrafo{Summary of results} We work in the context of $\mathbb{S}$-modules and commutative $\mathbb{S}$-algebras from \cite{ekmm}. We use an adaptation of the model for $KU$ given by Snaith \cite{snaith79}, \cite{snaith81}, namely $\mathbb{S}igma^\infty_+ K(\mathbb{Z},2)[x^{-1}]$, to this context. If we are to use this model to compute $THH(KU)$, we first need to prove that $THH$ commutes with localizations: this is done in \textbf{Corollary \ref{cor-thhloc}}, taking care of the multiplicative structure.
Our first expression for $THH(KU)$ as a commutative $KU$-algebra is obtained in \textbf{Theorem \ref{thhku1}}:
\[THH(KU)\simeq KU[K(\mathbb{Z},3)],\]
where the underlying $KU$-module of $KU[K(\mathbb{Z},3)]$ is $KU \wedge K(\mathbb{Z},3)_+$. The second one is given in \textbf{Theorem \ref{thhku2}}: there are weak equivalences of commutative $KU$-algebras
\[\xymatrix{ KU \vee \mathbb{S}igma KU_\mathbb{Q} & \ar[l]_-\sim F(\mathbb{S}igma KU_\mathbb{Q}) \ar[r]^-\sim & THH(KU).}\]
Here $F$ denotes the free commutative $KU$-algebra on a $KU$-module functor, and the $KU$-algebra structure on $KU\vee \mathbb{S}igma KU_\mathbb{Q}$ is that of a square-zero extension.
Note that, previously, McClure and Staffeldt \cite[8.1]{mc-st} established that $THH(L)\simeq L \vee \mathbb{S}igma L_\mathbb{Q}$ as spectra, where $L$ is the $p$-adic completion of the Adams summand of $KU$ for a given odd prime $p$. In \cite[7.9]{thhko}, the authors show that $THH(KO)\simeq KO\vee \mathbb{S}igma KO_\mathbb{Q}$ as $KO$-modules; here $KO$ is the periodic real complex $K$-theory ring spectrum. Note that these results (and others closely related, see Remark \ref{antecedentes}) are about the additive structure and do not involve the multiplicative structure, which we take into consideration. Finally, note that a lot of effort was devoted to describing $THH(ku)$ \cite{ausoni-thhku}, where $ku$ denotes the connective complex $K$-theory spectrum: that case is markedly harder.\\
We consider the iterated $THH$ of $KU$. The first expression we gave above for $THH(KU)$ directly generalizes: one replaces $K(\mathbb{Z},3)$ by a suitable product of integral Eilenberg-Mac Lane spaces. See \textbf{Theorem \ref{thhnku1cor}}: there is a zig-zag of weak equivalences of commutative $KU$-algebras
\[THH^n(KU)\simeq KU\left[\prod\limits_{i=1}^n K(\mathbb{Z},i+2)^{\times {n \choose i}}\right].\]
The second expression for $THH(KU)$ also generalizes: this is \textbf{Theorem \ref{thhnkufree}}, where we get a zig-zag of weak equivalences of commutative $KU$-algebras
\[F\left(\begin{itemize}gvee\limits_{i=1}^n (S^i)^{\vee {n\choose i}} \wedge KU_\mathbb{Q}\right) \simeq T^n\otimes KU.\]
The expression $KU \vee \mathbb{S}igma KU_\mathbb{Q}$ for $THH(KU)$ also generalizes to $THH^n(KU)$.
In this case, the augmentation ideal $\overline{THH}^n(KU)$ is still rational, but it has a non-trivial non-unital commutative $KU$-algebra structure. We describe the non-unital commutative $\mathbb{Q}[t^{\pm 1}]$-algebra $\overline{THH}^n_*(KU)$ as iterated Hochschild homology. See \textbf{Theorem \ref{high-hoch}}.
\\
We then shift our attention to $X\otimes KU$, where $X$ is a based CW-complex which is a reduced suspension, e.g. a sphere $S^n$. In this case, the first description for $THH(KU)$ generalizes as a zig-zag of weak equivalences of commutative $KU$-algebras:
\[S^n\otimes KU \simeq KU[K(\mathbb{Z},n+2)].\]
This is \textbf{Theorem \ref{holo}}. The second description for $THH(KU)$ generalizes as
\[F(S^n \wedge KU_\mathbb{Q}) \simeq S^n\otimes KU.\]
This is a particular case of \textbf{Theorem \ref{xku}}. Finally, we establish that
\[TAQ(KU)\simeq KU_\mathbb{Q}\]
as $KU$-modules, where $TAQ(KU)$ denotes the topological André-Quillen $KU$-module of $KU$. This is \textbf{Corollary \ref{cortaq}}.\\
\paragrafo{Remark on trace methods} One reason for the importance of $THH$ is its relation to algebraic $K$-theory. If $R$ is a (discrete) ring, then the trace map $K(R)\to HH(R)$ factors through the topological Hochschild homology of the Eilenberg-Mac Lane ring spectrum of $R$. Moreover, the trace map $K(A)\to THH(A)$ exists for any ring spectrum $A$. Out of topological Hochschild homology one can build topological cyclic homology, $TC(A)$, and the trace map further factors through it. The spectrum $TC(A)$ is closely related to algebraic $K$-theory: see \cite{dgm}. We might thus see $THH$ as a more easily approachable stepping stone on the way to the more fundamental algebraic $K$-theory.
It is important to note, however, that $THH(KU)$ is unlikely to be of assistance in the determination of $K(KU)$ via the methods we pointed out in the previous paragraph. First of all, note that one of the most useful theorems for computing algebraic $K$-theory via trace maps, namely, the theorem of Dundas-Goodwillie-McCarthy \cite[7.0.0.2]{dgm} only applies to connective ring spectra. Therefore, one may wish to get to $K(KU)$ by noting that $KU$ is the localization of $ku$, and by applying trace methods for $ku$. Indeed, in \cite{bm-localization-08}, Blumberg and Mandell prove that $K(KU)$ sits in a localization cofiber sequence $K(\mathbb{Z})\to K(ku)\to K(KU)\to \mathbb{S}igma K(\mathbb{Z})$. In \cite{bm-localization-14}, they establish an analogous cofiber sequence for $THH$, but the term involving $KU$ is \emph{not} $THH(KU)$ but an appropriate modification of it which receives a trace map from $K(KU)$; Ausoni shows in \cite[8.3]{ausoni-2010} how to compute the $V(1)$-homotopy ($p$ odd) of $K(KU)$ using this approach, and in \cite[3.6]{ausoni-rognes-rational} him and Rognes determine $K(KU)$ rationally. Note that these computations do not involve $THH(KU)$.
On the other hand, an interesting question to ask is if there are any elements in $K(KU)$ which survive to $THH(KU)$ via the trace.
We know that that $V(1)_*THH(KU)=0$ and $V(0)_*THH(KU)\cong V(0)_*KU$, but rationally, the trace $K(KU)\to THH(KU)$ is non-zero (see \cite[Paragraph 5.3]{ausoni-rognes-rational}). Therefore, the trace $K(KU)\to THH(KU)$ might well be useful in studying the integral homotopy type of $K(KU)$.
More generally, it would be interesting to detect elements in $THH^n(KU)$ that survive from the $n$-fold iterated algebraic $K$-theory of $KU$ via the iterated trace. See \cite{cdd}: they propose $THH^n$ as ``a computationally tractable cousin of $n$-fold iterated algebraic $K$-theory''.\\
\paragrafo{Comment on cofibrancy} At the heart of the computation of $THH(KU)$ lies the isomorphism $THH(\mathbb{S}[G])\cong \mathbb{S}[B^\mathrm{cy} G]$ where $G$ is a topological commutative monoid. The core of this result was already known; we prove it in Proposition \ref{conmutarTHH}, taking care of the multiplicative structures. This is a point-set result. However, the procedure of localization of a commutative $\mathbb{S}$-algebra at a homotopy element takes a \emph{cofibrant} commutative $\mathbb{S}$-algebra as input, so if we are to exploit the isomorphism just stated for the computation of $THH(KU)$ via Snaith's theorem, we first need to prove that $THH$ preserves the weak equivalence to $\mathbb{S}[G]$ from a cofibrant commutative $\mathbb{S}$-algebra replacement of it. This is obtained in Section \ref{sect:cofibrancy}. Assuming $G$ is a CW-complex with unit a 0-cell, the key property that $\mathbb{S}[G]$ satisfies is that it is \emph{flat}, i.e. smashing an $\mathbb{S}$-module with it preserves weak equivalences. This, along with the fact that the simplicial cyclic bar construction $B^\mathrm{cy}_\bullet\mathbb{S}[G]$ is a \emph{proper} simplicial $\mathbb{S}$-module, proves to be enough.\\
\paragrafo{Outline of the paper} In Section \ref{sect-cof}, we review some model categorical aspects of \cite{ekmm}, particularly those pertaining to commutative $\mathbb{S}$-algebras. In Section \ref{inversion}, we prove some elementary properties of localization of a commutative $\mathbb{S}$-algebra at an element. In Section \ref{sect-thh}, we review some needed aspects of topological Hochschild homology, and we prove that $THH$ commutes with localization at an element. Section \ref{sect:thhku} contains the results pertaining to $THH(KU)$,
and in Sections \ref{sect-iterated} and \ref{section:snku} we prove our results about $T^n\otimes KU$ and $S^n\otimes KU$. Finally, in Section \ref{section:taq}, we determine the topological André-Quillen homology of $KU$.\\
\paragrafo{Conventions} By \emph{space} we will mean ``compactly generated weakly Hausdorff topological space'', and we will denote the cartesian closed category they form by $\ensuremath{\mathbf{Top}}$, which we endow with the Quillen model structure. The corresponding model category of based spaces will be denoted by $\ensuremath{\mathbf{Top}}_*$.
We will work with the categories of \cite{ekmm}: our main objects are $\mathbb{S}$-modules, commutative $\mathbb{S}$-algebras $R$, $R$-modules and commutative $R$-algebras $A$.\\
\paragrafo{Acknowledgments} I would like to thank Christian Ausoni, my PhD supervisor, for suggesting this project, sharing his ideas and his support; Geoffroy Horel for his very useful suggestions; Eva Höning for our many engaging and fruitful discussions; Christian Schlichtkrull for his careful reading and his corrections; and Bj\o rn Dundas for so warmly and selflessly sharing so much of his time, ideas and insights at the University of Bergen. I would also like to thank Tobias Barthel and Magdalena Zielenkiewicz for their assistance with later revisions.
Parts of the content of this article are part of the author's PhD dissertation at Université Paris 13. Research partially supported by the \emph{ANR-16-CE40-0003 project ChroK} and the \emph{Fondation Sciences Mathématiques de Paris} (FSMP). The author would like to thank the Max Planck Institute for Mathematics at Bonn for their hospitality.
\section{Model structures} \label{sect-cof}
We will freely use the language of (enriched, monoidal) model categories as expounded in e.g. \cite{mayponto}.
The category $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$ of $\mathbb{S}$-modules has a $\ensuremath{\mathbf{Top}}_*$-enriched symmetric monoidal cofibrantly generated model structure \cite[VII.4]{ekmm}.
A commutative $\mathbb{S}$-algebra is, by definition, a commutative monoid in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$. The category they form, $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$, can also be described as the category of $\mathbb P$-algebras where $\mathbb P$ is the commutative monoid monad. The forgetful functor $U:\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}\to \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$ creates a model structure on $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$\footnote{A functor $U:\mathcal C\to \mathcal M$ \emph{creates a model structure} on $\mathcal C$ if $\mathcal M$ is a model category and $\mathcal C$ is a model category such that $f$ is a fibration (resp. weak equivalence) in $\mathcal C$ if and only if $Uf$ is a fibration (resp. weak equivalence) in $\mathcal M$. We say that $U$ \emph{strongly creates} the model structure of $\mathcal C$ if, in addition, $f$ is a cofibration in $\mathcal C$ if and only if $Uf$ is a cofibration in $\mathcal M$. We are following the nomenclature of \cite[15.3.5]{mayponto}.}. In particular, there is a Quillen adjunction $\xymatrix{\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}\ar@<.2pc>[r]^-F & \ar@<.2pc>[l]^-U \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}}$. The category $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ has a $\ensuremath{\mathbf{Top}}$-enriched symmetric monoidal cofibrantly generated model structure. In both $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$ and $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$, the monoidal product is the smash product $\wedge$ and the monoidal unit is the sphere spectrum $\mathbb{S}$.\footnote{Note that the pushout-product axiom is satisfied in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$: indeed, the smash product of commutative $\mathbb{S}$-algebras $R$ and $T$ is their coproduct, i.e. the pushout of $T \leftarrow \mathbb{S} \to R$ \cite[II.3.7]{ekmm}. Therefore, a ``pushouts commute with pushouts'' argument proves that the pushout-product map is an isomorphism. This proves, more generally, that any model category with the cocartesian monoidal structure (i.e. the monoidal product is the coproduct and the unit is the initial object) satisfies the pushout-product axiom and hence is a monoidal model category.}
Let $R\in \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$, and consider the category of $R$-modules, $\mathbb{R}Mod$. The forgetful functor $\mathbb{R}Mod\to \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$ creates a model structure on $\mathbb{R}Mod$, and $\mathbb{R}Mod$ acquires a $\ensuremath{\mathbf{Top}}_*$-enriched symmetric monoidal cofibrantly generated model category structure. The forgetful functor $U:\mathbb{R}CAlg\to \mathbb{R}Mod$ creates a model structure on $\mathbb{R}CAlg$, and thus $\mathbb{R}CAlg$ has a $\ensuremath{\mathbf{Top}}$-enriched symmetric monoidal cofibrantly generated model category structure. In both $\mathbb{R}CAlg$ and $\mathbb{R}Mod$, the monoidal product is the smash product relative to $R$, $\wedge_R$, and the monoidal unit is $R$. In all these model categories, all objects are fibrant.
Cofibrancy is more delicate. The sphere $\mathbb{S}$-module $\mathbb{S}$ is not cofibrant as an $\mathbb{S}$-module, but it is cofibrant as a commutative $\mathbb{S}$-algebra. More generally, the underlying $R$-module of a cofibrant commutative $R$-algebra is generally not cofibrant as an $R$-module.
Let $R$ be a commutative $\mathbb{S}$-algebra. We record the following useful properties:
\begin{enumerate}
\item \label{ujin} \label{moguil} If $M$ is a cofibrant $R$-module, then $M\wedge_R -$ preserves all weak equivalences of $R$-modules, so if $X$ is any $R$-module, then $M\wedge_R X$ represents the derived smash product \cite[III.3.8]{ekmm}. Note that $X\wedge_R-$ preserves weak equivalences between cofibrant $R$-modules. Indeed: let $f:M\to N$ be such a weak equivalence. Let $\gamma_X:\Gamma X\to X$ be a cofibrant replacement of $X$. We have a commutative diagram
\begin{enumerate}gin{equation}\label{moguil-diag}\xymatrix@C+1pc{X\wedge_R M \ar[r]^-{\mathrm{id} \wedge f} & X\wedge_R N \\ \Gamma X \wedge_R M \ar[r]^-\sim_-{\mathrm{id} \wedge f} \ar[u]_-\sim^-{\gamma_X \wedge \mathrm{id}} & \Gamma X \wedge_R N \ar[u]^-\sim_-{\gamma_X \wedge \mathrm{id}}}\end{equation}
where the two vertical maps and the bottom horizontal map are weak equivalences by the result just quoted, so the top vertical map is a weak equivalence, too.
\item \label{es} Suppose $R$ is cofibrant as a commutative $\mathbb{S}$-algebra.
Let $A$ and $B$ be cofibrant commutative $R$-algebras. Let $\gamma_A:\Gamma A \to A$ and $\gamma_B: \Gamma B \to B$ be cofibrant replacements of $A$ and $B$ in the category of $R$-modules. Then \[\gamma_A \wedge \gamma_B: \Gamma A \wedge_R \Gamma B \to A\wedge_R B\] is a weak equivalence of $R$-modules \cite[VII.6.4, 6.5, 6.7]{ekmm}. This tells us that $A\wedge_R B$ computes the derived smash product of $A$ and $B$ as $R$-modules. As a consequence of this and of (\ref{ujin}), by the 2-out-of-3 property we deduce that \[\gamma_A \wedge \mathrm{id}_B: \Gamma A \wedge_R B \to A \wedge_R B\] is a weak equivalence, since $\gamma_A \wedge \gamma_B = (\gamma_A \wedge \mathrm{id}_B) \circ (\mathrm{id}_{\Gamma A} \wedge \gamma_B)$. Similarly, $\mathrm{id}_A \wedge \gamma_B$ is also a weak equivalence.
\item As in any model category, the coproduct of cofibrant objects is cofibrant. Hence, if $A$ and $B$ are cofibrant commutative $R$-algebras, then $A\wedge_R B$ is a cofibrant commutative $R$-algebra \cite[VII.6.8]{ekmm}.
\item Let $\mathbb{S}\to A\to B$ be cofibrations of commutative $\mathbb{S}$-algebras. Then the functor $B\wedge_A-:A\mbox{-}\mathbf{CAlg}\to B\mbox{-}\mathbf{CAlg}$ preserves weak equivalences between commutative $A$-algebras which are cofibrant as commutative $\mathbb{S}$-algebras \cite[VII.7.4]{ekmm}.
\item \label{cinco} The category $\mathbb{R}CAlg$ can also be described as the category of objects of $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ under $R$. As such, the forgetful functor $\mathbb{R}CAlg \to \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ strongly creates a model structure on $\mathbb{R}CAlg$ \cite[Theorem 15.3.6]{mayponto}. This model structure coincides with the one described above \cite[Remark 2.4.1]{eva-thesis}. In conclusion, a map $f:A\to B$ is a cofibration in $\mathbb{R}CAlg$ if and only if it is a cofibration in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$. In particular, if $R$ is a cofibrant commutative $\mathbb{S}$-algebra and $A$ is a cofibrant commutative $R$-algebra, then $A$ is cofibrant as a commutative $\mathbb{S}$-algebra.
\end{enumerate}
Note: in \cite{ekmm} they call \emph{q-cofibration} what we call a cofibration. We will have no use for what they call a cofibration. We use the term ``homotopy category'' in the model categorical sense \cite[14.4.1]{mayponto}: in \cite{ekmm} these were called \emph{derived categories}.
\begin{obs}\label{funct-fact} The following remarks will be used below. In a cofibrantly generated model category constructed via Quillen's small object argument, the factorization of an arrow as a cofibration followed by an acyclic fibration is \emph{functorial} \cite[12.2.2]{riehl}, a concept carefully defined e.g. in \cite[12.1.1]{riehl}, \cite[14.1.10]{mayponto}. As an example, the category $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ admits such a functorial factorization. We can factor the unit maps in order to obtain a functorial cofibrant replacement functor $Q$.
Let $R$ be a commutative $\mathbb{S}$-algebra. If $A$ is a commutative $R$-algebra, then we may factor the unit map $\eta:R\to A$ with the functorial factorization in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ described above. Denote by $Q_RA$ the object appearing in the factorization, i.e. $\eta$ is factored as $R\to Q_RA\to A$. From item (\ref{cinco}) above we obtain that the first arrow is a cofibration in $\mathbb{R}CAlg$ and the second arrow is an acyclic fibration in $\mathbb{R}CAlg$. The functoriality of the factorization proves that this defines a functor $Q_R$ of cofibrant replacement in the category $\mathbb{R}CAlg$.
\end{obs}
We end this section on model structures with the following general lemma which will be used in the proof of Theorem \ref{xku}. We thank Eva Höning for explaining this lemma to us.
\begin{enumerate}gin{lema} \label{hompus} Let \begin{enumerate}gin{equation}\label{tef}\xymatrix@C+1pc{B \ar[d]_-u^-\sim & A \ar[l]_-f \ar[d]_-\sim^-v \ar[r]^-g & C \ar[d]^-w_-\sim \\ B' & A' \ar[l]^-{f'} \ar[r]_ -{g'} & C'}\end{equation}
be a diagram in a model category where the vertical arrows are weak equivalences, all objects are cofibrant and one map in each horizontal line is a cofibration. Suppose both squares are homotopy commutative. Then there is a natural zig-zag of weak equivalences between the pushouts of both horizontal lines.
\begin{proof}
Let $(\textup{Cyl}(A),i_0,i_1)$ denote a cylinder object for $A$. Let $H:\textup{Cyl}(A)\to B'$ denote a homotopy from $uf$ to $f'v$, and $G:\textup{Cyl}(A)\to C'$ denote a homotopy from $wg$ to $g'v$. We have the following (strictly) commutative diagram:
\[\xymatrix{
B \ar[d]_-u^-\sim & A \ar[l]_-f \ar[r]^-g \ar[d]^-\mathrm{id} & C \ar[d]^-w_-\sim \\
B' \ar[d]_-\mathrm{id} & A \ar[l]_-{uf} \ar[r]^-{wg} \ar[d]^-{i_0}_-\sim & C' \ar[d]^-\mathrm{id} \\
B' & \textup{Cyl}(A) \ar[l]_-H \ar[r]^-G & C' \\
B' \ar[d]_-\mathrm{id} \ar[u]^-\mathrm{id} & A \ar[d]^-v_-\sim \ar[l]_-{f'v} \ar[r]^-{g'v} \ar[u]_-{i_1}^-\sim & C' \ar[d]^-\mathrm{id} \ar[u]_-\mathrm{id} \\
B' & A' \ar[l]^-{f'} \ar[r]_-{g'} & C' }\]
By a repeated application of the homotopy invariance of homotopy pushouts \cite[Dual of 13.3.4]{hirschhorn}, we obtain a zig-zag of weak equivalences between the homotopy pushout of $(f,g)$ and the homotopy pushout of $(f',g')$. But these homotopy pushouts are computed by the (categorical) pushouts, since in (\ref{tef}) all objects are cofibrant and one map in each line is a cofibration \cite[Dual of 13.3.8]{hirschhorn}.
\end{proof}
\end{lema}
\section{Inversion of an element}\label{inversion}
In this section, we recall the procedure of inverting a homotopy element in a commutative $\mathbb{S}$-algebra following \cite{ekmm} and prove some properties which will be needed below.
\begin{teo} \label{rect} \cite[VIII.2.2, VIII.4.2]{ekmm} Let $R$ be a cofibrant commutative $\mathbb{S}$-algebra and $x\in \pi_*R$. There exists a cofibrant
commutative $R$-algebra $R[x^{-1}]$ with unit $j:R\to R[x^{-1}]$ satisfying that $\pi_*(R[x^{-1}])=\pi_*(R)[x^{-1}]$, and if $f:R\to T$ is a map in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ such that $(\pi_*f)(x)\in \pi_*T$ is invertible, then there exists a map $\tilde f:R[x^{-1}]\to T$ in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ making the following diagram commute:
\[\xymatrix{R\ar[r]^-f \ar[d]_-j & T \\ R[x^{-1}] \ar@{.>}[ru]_{\tilde{f}} }.\]
The map $\tilde f$ is unique up to homotopy of commutative $\mathbb{S}$-algebras.
Moreover, if the morphism $\pi_*(R)[x^{-1}]\to \pi_*T$ coming from the universal property for localizations of commutative $\pi_*(R)$-algebras is an isomorphism, then $\tilde f$ is a weak equivalence.
\end{teo}
The previous theorem is valid, \emph{mutatis mutandis}, if $\mathbb{S}$ is replaced by some cofibrant commutative $\mathbb{S}$-algebra.
\begin{enumerate}gin{lema} \label{idem} The multiplication map $\mu:R[x^{-1}]\wedge_R R[x^{-1}]\to R[x^{-1}]$ is a weak equivalence of commutative $R[x^{-1}]$-algebras.
\begin{proof} The Tor spectral sequence \cite[IV.4.1]{ekmm} here takes the form
\[E^2_{*,*}=\mathrm{Tor}_{*,*}^{\pi_*R}(\pi_* R[x^{-1}], \pi_* R[x^{-1}]) \mathbb{R}ightarrow \pi_*(R[x^{-1}]\wedge_R R[x^{-1}]). \]
Since the localization morphism $\pi_*R\to \pi_*R[x^{-1}]$ is flat, the spectral sequence is concentrated in the 0-th column and thus the edge homomorphism
\begin{enumerate}gin{equation}\label{edge} \nabla: \pi_*R[x^{-1}]\otimes_{\pi_*R}\pi_*R[x^{-1}] \to \pi_*(R[x^{-1}]\wedge_R R[x^{-1}])\end{equation}
is an isomorphism. Since $\wedge_R$ is the coproduct in the category of commutative $R$-algebras, we can consider the canonical maps $i_1,i_2:R[x^{-1}]\to R[x^{-1}]\wedge_R R[x^{-1}]$. The edge homomorphism $\nabla$ coincides with the map $(\pi_*i_1, \pi_*i_2)$ defined via the universal property of the coproduct of commutative $\pi_*R$-algebras.
We have the following commutative diagram of commutative $\pi_*R$-algebras:
\[\xymatrix{
\pi_*R[x^{-1}] \ar@/_2pc/[ddr]_-\mathrm{id} \ar[r]^-{\iota_1} \ar[rd]_-{\pi_*i_1}
& \pi_*R[x^{-1}]\otimes_{\pi_*R} \pi_*R[x^{-1}] \ar[d]^-\nabla
& \pi_*R[x^{-1}] \ar[l]_-{\iota_2} \ar[ld]^-{\pi_*i_2} \ar@/^2pc/[ddl]^-\mathrm{id}
\\
& \pi_*(R[x^{-1}] \wedge_R R[x^{-1}]) \ar[d]^-{\pi_*\mu}
\\
& \pi_*R[x^{-1}] }\]
where $\iota_1,\iota_2$ are the canonical inclusions into a coproduct of commutative $\pi_*R$-algebras. Again, by the universal property of the coproduct of commutative $\pi_*R$-algebras, there is a unique arrow $\pi_*R[x^{-1}]\otimes_{\pi_*R} \pi_*R[x^{-1}] \to \pi_*R[x^{-1}]$ making the outer diagram commute. One such arrow is the canonical isomorphism that one has for any such algebraic localization, i.e. $h:S^{-1}A\otimes_A S^{-1}A \stackrel{\cong}{\to} S^{-1}A$ for any commutative ring $A$ and multiplicative subset $S\subset A$. Another such arrow is $\pi_*\mu \circ \nabla$. Therefore, $h=\pi_*\mu \circ \nabla$. Since $\nabla$ and $h$ are isomorphisms, so is $\pi_*\mu$.
\end{proof}
\end{lema}
If $f:R\to T$ is a morphism between cofibrant commutative $\mathbb{S}$-algebras and $x\in \pi_*R$, then Theorem \ref{rect} gives us a map of cofibrant commutative $\mathbb{S}$-algebras
\[\xymatrix{R\ar[r]^-f \ar[d]_-{j_R} & T \ar[d]^-{j_T} \\ R[x^{-1}] \ar@{.>}[r]_-{f[x^{-1}]} & T[(\pi_*f)(x)^{-1}]}\]
such that if $f$ is a weak equivalence, then $f[x^{-1}]$ is a weak equivalence.
Note that $f[x^{-1}]$ turns $T[(\pi_*f)(x)^{-1}]$ into a commutative $R[x^{-1}]$-algebra.
The previous square induces an arrow from the pushout $R[x^{-1}]\wedge_R T$ in $R\mbox{-}\mathbf{CAlg}$. The following theorem tells us that it is a weak equivalence. Compare with \cite[V.1.15]{ekmm} which handles the case where $T$ is replaced by an $R$-module.
\begin{prop}[Base change for localization]\label{base-loc} Let $f:R\to T$ be a morphism of cofibrant commutative $\mathbb{S}$-algebras and $x\in \pi_*R$. The morphism of commutative $R$-algebras \begin{enumerate}gin{equation}\label{base-loceq}(f[x^{-1}],j_T): R[x^{-1}] \wedge_R T \to T[(\pi_*f)(x)^{-1}]\end{equation}
is a weak equivalence.
\end{prop}
Note that (\ref{base-loceq}) is also a weak equivalence in $R[x^{-1}]\mbox{-}\mathbf{CAlg}$ and in $T\mbox{-}\mathbf{CAlg}$. Note as well that if $\varepsilon \mbox{-}ilon:T\to R$ is a morphism of $\mathbb{S}$-algebras such that $\varepsilon \mbox{-}ilon \circ f=\mathrm{id}_R$ so that $T$ becomes an augmented commutative $R$-algebra, then in (\ref{base-loceq}) both sides are naturally augmented over $R[x^{-1}]$ and the morphism commutes with the augmentations.
\begin{proof} Denote the morphism $(f[x^{-1}],j_T)$ by $h$, for simplicity. Like in the proof of Lemma \ref{idem}, the Tor spectral sequence that computes the homotopy groups of $R[x^{-1}]\wedge_R T$ from those of $R[x^{-1}]$ and $T$ collapses, since $\pi_*R\to \pi_*R[x^{-1}]= (\pi_*R)[x^{-1}]$ is flat.
Therefore, the map $\pi_*h$, fitting in a commutative diagram
\[\xymatrix{\pi_*(R[x^{-1}]\wedge_R T) \ar[r]^-{\pi_*h} & (\pi_*T)[(\pi_*f)(x)^{-1}]\\ \ar[u]^-\cong (\pi_*R)[x^{-1}] \otimes_{\pi_*R} \pi_*T, \ar[ru]_-\cong }\]
is an isomorphism, since the diagonal map is an isomorphism. Indeed, this is the map appearing in the analogous statement in commutative algebra of the theorem we are proving, applied to $\pi_*f: \pi_*R\to \pi_*T$. But this statement of commutative algebra is not hard to prove: it follows from the universal properties and the extension-restriction of scalars adjunction.
\end{proof}
\begin{prop} \label{sepa} Let $R$ and $T$ be cofibrant commutative $\mathbb{S}$-algebras, $x\in \pi_nR$ and $y\in \pi_mT$. Denote by $x\wedge y$ the image of $x\otimes y$ under the morphism
\[\xymatrix{\pi_*R \otimes_{\pi_*\mathbb{S}} \pi_*T \ar[r] & \pi_*(R\wedge T)}.\]
There is a weak equivalence of commutative $\mathbb{S}$-algebras
\[R[x^{-1}]\wedge T[y^{-1}] \to (R\wedge T)[(x\wedge y)^{-1}]\]
which is natural on $R$ and $T$.
\end{prop}
Note that this is is also a map of commutative $R[x^{-1}]$ and $T[y^{-1}]$-algebras.
\begin{proof} Let $i_1:R\to R\wedge T$, $i_2:T\to R\wedge T$ be the canonical maps into the coproduct. There exists a map $f$ making the following diagram commute.
\[\xymatrix{R \ar[d]_-{j_R} \ar[r]^{i_1} & R\wedge T \ar[r]^-{j_{R\wedge T}} & (R\wedge T)[(x\wedge y)^{-1}] \\ R[x^{-1}] \ar@{.>}[rru]_-f}\]
Indeed, applying $\pi_*$ to the horizontal composition, we get the map \[\pi_*(j_{R\wedge T}\circ i_1): \pi_*R \to \pi_*(R\wedge T)[(x\wedge y)^{-1}]\]which maps $x$ to $x\wedge 1$. This is an invertible element with inverse $(1\wedge y)(x\wedge y)^{-1}$, since the map $(\pi_*i_1,\pi_*i_2): \pi_*R \otimes_{\pi_*\mathbb{S}} \pi_* T \to \pi_*(R\wedge T)$
is multiplicative. Therefore, the property of Theorem \ref{rect} provides us with the arrow $f$ in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$. Similarly, we get a map $g:T[y^{-1}] \to (R\wedge T)[(x\wedge y)^{-1}]$. We assemble $f$ and $g$ into the coproduct map in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$
\[(f,g):R[x^{-1}]\wedge T[y^{-1}] \to (R\wedge T)[(x\wedge y)^{-1}].\]
Now recall from \cite[Section V.1]{ekmm} that $R[x^{-1}]$ is weakly equivalent, in $\mathbb{R}Mod$, to the homotopy colimit of the tower
\begin{enumerate}gin{equation}\label{telescope}\xymatrix{R\ar[r]^-x & \mathbb{S}igma^{-n} R \ar[r]^-x & \mathbb{S}igma^{-2n} R\ar[r]^-x & \dots}.\end{equation}
The $T$-module $T[y^{-1}]$ is described similarly. The $R\wedge T$-module $(R\wedge T)[(x\wedge y)^{-1}]$ is weakly equivalent to the homotopy colimit of the tower
\[\xymatrix{R \wedge T\ar[r]^-{x\wedge y} & \mathbb{S}igma^{-n-m} R\wedge T \ar[r]^-{x\wedge y} & \mathbb{S}igma^{-2n-2m} R\wedge T\ar[r]^-{x\wedge y} & \dots}.\]
Smashing the homotopy colimit computing $R[x^{-1}]$ with the one computing $T[y^{-1}]$ we obtain the homotopy colimit computing $(R\wedge T)[(x\wedge y)^{-1}]$, since the diagonal map $\mathbb{N}\to \mathbb{N}\times \mathbb{N}$ is homotopy cofinal. The map $(f,g)$ is compatible with these identifications, hence it is a weak equivalence.
\end{proof}
\begin{enumerate}gin{notation} Let $f:R\to T$ be a weak equivalence of cofibrant commutative $\mathbb{S}$-algebras. If $x\in \pi_*R$, we denote the algebra $T[(\pi_*f)(x)^{-1}]$ by $T[x^{-1}]$. Similarly, if $y\in \pi_*T$ we define $R[y^{-1}]$ as $T[(\pi_*f)^{-1}(y)^{-1}]$. In particular, if $R$ and $T$ are connected by a zig-zag of weak equivalences of cofibrant commutative $\mathbb{S}$-algebras, then a homotopy element $x\in \pi_*R$ defines $T[x^{-1}]$ and conversely.
\end{notation}
We now turn to the inversion of an element in a non-cofibrant commutative $\mathbb{S}$-algebra $A$. Let $Q$ be a cofibrant replacement functor in the category of commutative $\mathbb{S}$-algebras, as obtained in Remark \ref{funct-fact}.
\begin{defn} \label{def-nocof} Let $A\in \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ and $x\in \pi_*A$. We define $A[x^{-1}]_h$ to be the cofibrant commutative $\mathbb{S}$-algebra $(QA)[x^{-1}]$.
\end{defn}
\begin{obs} \label{lifto} If $\tilde A$ is a cofibrant commutative $\mathbb{S}$-algebra and $\tilde A\to A$ is a weak equivalence, then there is a weak equivalence of commutative $\mathbb{S}$-algebras $\tilde A\to QA$, and hence a weak equivalence $\tilde A[x^{-1}]\to QA[x^{-1}]$ of cofibrant commutative $\mathbb{S}$-algebras. Indeed, the existence of this weak equivalence follows from the lifting properties for the model category $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ and the 2-out-of-3 property for weak equivalences:
\[\xymatrix{\mathbb{S} \ar[r] \ar@{>->}[d] & QA \ar@{->>}[d]^-{\sim} \\ \tilde A \ar@{.>}[ru] \ar[r]_-\sim & A.}\]
In particular, the object $A[x^{-1}]_h$ in the homotopy category of commutative $\mathbb{S}$-algebras does not depend on the choice of $Q$, up to isomorphism. This explains the choice of the letter $h$ for the subscript.
\end{obs}
\section{Topological Hochschild homology} \label{sect-thh}
\subsection{Symmetric monoidal categories}
Let $(\mathcal{V},\otimes, \ensuremath{\mathbbm{1}})$ be a symmetric monoidal category. Denote by $\ensuremath{\mathbf{Mon}}(\mathcal{V})$ and $\ensuremath{\mathbf{CMon}}(\mathcal{V})$ the corresponding symmetric monoidal categories of monoids and commutative monoids in $\mathcal{V}$, respectively. Denote by $s\mathcal{V}$ the category of simplicial objects in $\mathcal{V}$; it is a symmetric monoidal category with levelwise monoidal product.
Suppose $\mathcal{V}$ is closed and cocomplete. Then if $A\in \ensuremath{\mathbf{CMon}}(\mathcal{V})$, the category $A\mbox{-}\mathbf{Mod}$ of $A$-modules gets a relative monoidal product $\otimes_A$ such that $(A\mbox{-}\mathbf{Mod}, \otimes_A, A)$ is a symmetric monoidal category. One can thus speak of commutative $A$-algebras, which are the commutative monoids in $A\mbox{-}\mathbf{Mod}$. We denote by $\mathcal{A}CAlg$ the category they form.
An \emph{augmented} commutative $A$-algebra $B$ has an \emph{augmentation} map $B\to A$ which is a morphism of commutative monoids.
Let $F:\mathcal{V}\to \mathcal{W}$ be a strong symmetric monoidal functor between cocomplete closed symmetric monoidal categories, and suppose $F$ preserves colimits. Then $F$ induces a functor on commutative monoids, on modules over commutative monoids, and on commutative algebras. More specifically, there is an induced functor $F:\mathcal{A}CAlg \to F(A)\mbox{-}\mathbf{CAlg}$.
We will need the following
\begin{enumerate}gin{lema} \label{lemext} Let $F:\mathcal{V}\to \mathcal{W}$ be a functor as above, and let $A\in \ensuremath{\mathbf{CMon}}(\mathcal{V})$. Then there is a natural isomorphism
\[\xymatrix@C+1pc{\ensuremath{\mathbf{CMon}}(\mathcal{V}) \ar[r]^-{A\otimes -} \ar[d]_-F & \mathcal{A}CAlg \ar[d]^-F \\ \ensuremath{\mathbf{CMon}}(\mathcal{W}) \ar[r]_-{F(A) \otimes -} & F(A)\mbox{-}\mathbf{CAlg}. \xtwocell[-1,-1]{}\omit{<1>} }\]
\begin{proof} First, note that there is a strong symmetric monoidal functor $A\otimes -:\mathcal{V}\to \mathcal{A}Mod$, which therefore induces the functor at the top of the diagram, and similarly for the one in the bottom. The isomorphism \[\nabla: F(A)\otimes F(B) \to F(A\otimes B)\] natural in $B\in \ensuremath{\mathbf{CMon}}(\mathcal{V})$ is given by the structure isomorphism of $F$. The only thing one needs to check is that $\nabla$ is a map of $F(A)$-commutative algebras, but this is a straightforward verification.
\end{proof}
\end{lema}
\subsection{Simplicial cyclic bar construction in general}
The results in this section are similar to the ones in \cite[Section 1]{stonek-graded} which are done for the simplicial \emph{reduced} bar construction.
\begin{defn} The \emph{simplicial cyclic bar construction} is a functor \[B^\mathrm{cy}_\bullet:\ensuremath{\mathbf{Mon}}(\mathcal{V}) \to s\mathcal{V}\]
defined as follows. If $A\in \ensuremath{\mathbf{Mon}}(\mathcal{V})$ with multiplication $\mu:A\otimes A\to A$ and unit $\eta:\ensuremath{\mathbbm{1}}\to A$, then $B^\mathrm{cy}_n(A)=A^{\otimes n+1}$. The faces $d_i:A^{\otimes n+1}\to A^{\otimes n}$, $i=0,\dots,n$ are defined as
\[d_i= \mathrm{id}^{\otimes i} \otimes \mu \otimes \mathrm{id}^{\otimes n-i-1} \hspace{.5cm} \text{if } i=0,\dots,n-1, \textup{ and}\]
\[d_n=(\mu \otimes \mathrm{id}^{\otimes (n-1)}) \circ \sigma_{n+1}\]
where $\sigma_{n+1}:A^{\otimes n+1}\to A^{\otimes n+1}$ is the isomorphism that puts the last $A$ term at the beginning. The degeneracies $s_i:A^{\otimes n+1}\to A^{\otimes n+2}$ are \[s_i=\mathrm{id}^{\otimes i+1} \otimes \eta \otimes \mathrm{id}^{\otimes n-i} \hspace{.5cm} \text{ for all }i=0,\dots,n.\]
\end{defn}
This is a strong symmetric monoidal functor, so it induces a functor between the categories of commutative monoids. But a version of the Eckmann-Hilton argument (see e.g. \cite[6.29]{aguiar}) says that $\ensuremath{\mathbf{CMon}}(\ensuremath{\mathbf{Mon}}(\mathcal{V}))\cong \ensuremath{\mathbf{CMon}}(\mathcal{V})$, so we get a functor
\[B_\bullet^\mathrm{cy}:\ensuremath{\mathbf{CMon}}(\mathcal{V})\to s\ensuremath{\mathbf{CMon}}(\mathcal{V}).\]
For a commutative monoid $A$ in $\mathcal{V}$, we have that $B_\bullet^\mathrm{cy}(A)\in s\mathcal{A}CAlg$. Indeed, there is a natural morphism $cA\to B_\bullet^\mathrm{cy} A$ in $s\ensuremath{\mathbf{CMon}}(\mathcal{V})$, where $cA$ denotes the constant simplicial object at $A$. In simplicial level $n$, it is $\mathrm{id} \otimes \eta^{\otimes n}:A\to A^{\otimes n+1}$. We could specify the simplicial commutative $A$-algebra structure of $B^\mathrm{cy}_\bullet(A)$ more explicitely: the $A$-module structure on $A^{\otimes n+1}\cong A\otimes A^{\otimes n}$ is given by acting on the first factor, and the multiplication over $A$ is given by
\[\xymatrix{A^{\otimes n+1}\otimes_A A^{\otimes n+1}\cong A \otimes (A^{\otimes n} \otimes A^{\otimes n}) \ar[r]^-{\mathrm{id} \otimes \mu} & A\otimes A^{\otimes n}}\]
where $\mu$ denotes the product of $A^{\otimes n}\in \ensuremath{\mathbf{CMon}}(\mathcal{V})$.
Note moreover that $B_\bullet^\mathrm{cy}(A)$ admits a map of simplicial commutative monoids to $cA$: in simplicial level $n$, it is the multiplication of $n$ elements of $A$, which makes sense by commutativity of $A$. So $B_\bullet^\mathrm{cy}(A)$ is a simplicial augmented commutative $A$-algebra.\\
There is a relative version of this construction: if $M$ is an $A$-bimodule, then one can define $B_\bullet^\mathrm{cy}(A,M)$ which has $B_n^\mathrm{cy}(A,M)=M\otimes A^{\otimes n}$ with similar faces and degeneracies. If $M$ is a commutative $A$-algebra, then $B_\bullet^\mathrm{cy}(A,M)$ is a simplicial augmented commutative $M$-algebra.\\
Let $F:\mathcal{V}\to \mathcal{W}$ be a strong symmetric monoidal functor between cocomplete closed symmetric monoidal categories which preserves colimits. Since it takes commutative $A$-algebras to commutative $F(A)$-algebras, then $F(B^\mathrm{cy}_\bullet A)$ is a simplicial commutative $F(A)$-algebra. The structure morphisms of $F$ provide an
isomorphism \begin{enumerate}gin{equation} \label{bf} B_\bullet^\mathrm{cy}(FA)\stackrel{\cong}{\to} F(B_\bullet^\mathrm{cy} A)\end{equation} of simplicial augmented commutative $F(A)$-algebras.
\subsection{Geometric realization} \label{sect:geom} Consider $F=\mathbb{S}igma^\infty_+:\ensuremath{\mathbf{Top}}\to \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$: it is a strong symmetric monoidal left adjoint functor \cite[II.1.2]{ekmm}. If $G$ is a topological commutative monoid, we denote by $\mathbb{S}[G]$ the $\mathbb{S}$-module $\mathbb{S}igma^\infty_+ G$ together with the commutative $\mathbb{S}$-algebra structure induced by the monoid structure of $G$. Note that the map $G\to *$ gives $\mathbb{S}[G]$ an augmentation $\mathbb{S}[G]\to \mathbb{S}$.
Endow the cartesian category $\ensuremath{\mathbf{Top}}$ with the standard cosimplicial space $\mathcal{D}elta_\mathrm{top}^\bullet$ and the symmetric monoidal category $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$ with the cosimplicial spectrum $\mathbb{S}igma^\infty_+ \mathcal{D}elta^\bullet_\mathrm{top}$. By \cite[2.9]{stonek-graded},
these beget strong symmetric monoidal functors of geometric realization
\[|-|:s\ensuremath{\mathbf{Top}} \to \ensuremath{\mathbf{Top}} \hspace{1cm} \textup{and} \hspace{1cm} |-|:s\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}} \to \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}.\]
If $A$ is a topological commutative monoid or a commutative $\mathbb{S}$-algebra, define \[B^\mathrm{cy}(A)\coloneqq|B^\mathrm{cy}_\bullet(A)|.\]
It is an augmented commutative $A$-algebra. In the $\mathbb{S}$-module case, this object defines the topological Hochschild homology of $A$, denoted $THH(A)$ (which coincides with the derived smash product $A \wedge^L_{A^e} A$ when $A$ is a cofibrant commutative $\mathbb{S}$-algebra \cite[IX.2.7]{ekmm}),
and if $M$ is an $A$-bimodule or commutative $A$-algebra then $|B_\bullet^\mathrm{cy}(A,M)|$ defines $THH(A,M)$. Note that if $f:A\to B$ is a weak equivalence of cofibrant commutative $\mathbb{S}$-algebras, then $THH(A)\to THH(B)$ is a weak equivalence. First note that $B_\bullet^\mathrm{cy}(A)\to B_\bullet^\mathrm{cy}(B)$ is a weak equivalence in each level. Indeed, $f^{\wedge p+1}$ is a weak equivalence since $f$ is a weak equivalence and $A$ and $B$ are cofibrant.
Then, since the simplicial cyclic bar construction gives a proper simplicial $\mathbb{S}$-module \cite[IX.2.8]{ekmm}, we can apply \cite[X.2.4]{ekmm} to get the conclusion.
Since the functors $|-|$ are strong symmetric monoidal, by realizing the isomorphism (\ref{bf}) we obtain the following
\begin{prop} \label{conmutarTHH} Let $G$ be a topological commutative monoid. There is an isomorphism of augmented commutative $\mathbb{S}[G]$-algebras
\[\xymatrix{THH(\mathbb{S}[G])\ar[r]^-\cong & \mathbb{S}[B^\mathrm{cy}(G)].} \]
\end{prop}
Versions of the previous proposition have appeared in \cite[Remark 4.4]{svw}, in \cite[Theorem 7.1]{hesselholt-madsen97} in the setting of functors with smash product, and in \cite[Example 4.2.2.7]{dgm} in the setting of $\Gamma$-spaces. Note that we take care to note that this isomorphism respects the commutative $\mathbb{S}[G]$-algebra structures.\\
The following proposition is obtained by applying Lemma \ref{lemext} to the functor $\mathbb{S}igma^\infty_+$; the isomorphism commutes with the augmentations by inspection.
\begin{prop} \label{Smono} Consider $G, H\in \ensuremath{\mathbf{CMon}}(\ensuremath{\mathbf{Top}})$. There is an isomorphism of augmented commutative $\mathbb{S}[G]$-algebras
\[\xymatrix{\mathbb{S}[G] \wedge \mathbb{S}[H] \ar[r]^-\cong &\mathbb{S}[G \times H]} \]
natural in $H$.
\end{prop}
\subsection{Cyclic bar construction of a topological abelian group} \label{section:cyclictop}
Let $G$ be a topological abelian group with unit $0$.
Denote by $BG$ the model for the classifying space of $G$ which is given by the geometric realization of the reduced bar construction $B_\bullet(0,G,0)$ of $G$: it is a topological abelian group. Moreover, if $G$ is a CW-complex with $0$ a 0-cell and addition is a cellular map, then the same can be said of $BG$. All of this is due to Milgram \cite{milgram}.
The space $G\times BG$ gets the structure of a commutative $G$-algebra, via the inclusion of the first factor $G\to G\times BG$, which is a morphism of topological abelian groups. It has an augmentation given by projection to the first factor.
\begin{prop} \label{BcyG} Let $G$ be a topological abelian group. There is a homeomorphism of augmented commutative $G$-algebras
\[B^\mathrm{cy} G \cong G \times BG.\]
\begin{proof}
Let $G_\bullet$ denote the constant simplicial commutative $G$-algebra on $G$. Consider the maps $r_\bullet:B^\mathrm{cy}_\bullet G \to G_\bullet$, $(g_0,\dots,g_p)\mathrm{map}sto g_0+\dots+g_p$, and $p_\bullet:B^\mathrm{cy}_\bullet G \to B_\bullet G$, $(g_0,\dots,g_p)\mathrm{map}sto (g_1,\dots,g_p)$. They assemble to a map
\[\xymatrix{B^\mathrm{cy}_\bullet G \ar[r]^-{(r_\bullet, p_\bullet)} & G_\bullet \times B_\bullet G, \hspace{.5cm} (g_0,\dots,g_p)\mathrm{map}sto (g_0+\dots+g_p,g_1,\dots,g_p).}\]
We also have maps $i_\bullet:G_\bullet \to B_\bullet^\mathrm{cy} G$, $g\mathrm{map}sto (g,0,\dots,0)$ and $s_\bullet:B_\bullet G \to B^\mathrm{cy}_\bullet G$, $(g_1,\dots,g_p)\mathrm{map}sto (-g_1-\dots-g_p,g_1,\dots,g_p)$. We sum them up to a map
\[\xymatrix{ G_\bullet \times B_\bullet G \ar[r]^-{i_\bullet+s_\bullet} & B_\bullet^\mathrm{cy} G, \hspace{.5cm} (g,g_1,\dots,g_p)\mathrm{map}sto (g-g_1-\dots-g_p,g_1,\dots,g_p).}\]
The maps $(r_\bullet,p_\bullet)$ and $i_\bullet+s_\bullet$ are morphisms of simplicial augmented commutative $G$-algebras which are inverse to one another. (Note that the obvious isomorphisms $G\times G^p \cong G^{p+1}$ are not good, because they do not commute with the last face map.) Applying geometric realization we obtain the result.
\end{proof}
\end{prop}
A classical result (which we will not use) states that $B^\mathrm{cy} G$ is homotopy equivalent to the free loop space of $BG$ (see e.g. \cite[Section 2]{bhm}).\index{Free loop space}
\subsection{Inverting an element in \texorpdfstring{$THH$}{THH}}
Let $R$ be a cofibrant commutative $\mathbb{S}$-algebra and $x\in \pi_*R$. Denote by $\eta:R\to THH(R)$ the unit. Since $THH(R)=B^\mathrm{cy}(R)$ is cofibrant as a commutative $\mathbb{S}$-algebra \cite[Lemma 3.6]{svw}, Proposition \ref{base-loc} gives a weak equivalence of augmented commutative $R[x^{-1}]$-algebras
\begin{enumerate}gin{equation}\label{comienzo} \xymatrix{THH(R,R[x^{-1}]) \cong R[x^{-1}]\wedge_R THH(R) \ar[r]^-\sim & THH(R)[\pi_*\eta(x)^{-1}].}\end{equation}
For simplicity, we denote the codomain of this arrow by $THH(R)[x^{-1}]$.
We now aim to prove that $THH(R,R[x^{-1}])$ and $THH(R[x^{-1}])$ are weakly equivalent commutative $R[x^{-1}]$-algebras. We will obtain this as a consequence of the following more general theorem, by taking the sequence (\ref{seqc}) to be $\mathbb{S}\to R\to R[x^{-1}]$.
\begin{teo} \label{thh-base} Let \begin{enumerate}gin{equation}\label{seqc}\mathbb{S}\to A\stackrel{f}{\to} B\end{equation} be a sequence of cofibrations of commutative $\mathbb{S}$-algebras. Suppose that the multiplication map $\mu:B\wedge_A B\to B$ is a weak equivalence. Then the map of augmented commutative $B$-algebras
\begin{enumerate}gin{equation}\label{thh-base-eq}\xymatrix{ B\wedge_A THH(A)\cong THH(A,B) \ar[rr]^-{THH(f,\mathrm{id})} && THH(B,B)=THH(B)}\end{equation}
is a weak equivalence.
\end{teo}
This theorem is valid \emph{mutatis mutandis} when $\mathbb{S}$ is replaced by some cofibrant commutative $\mathbb{S}$-algebra.
We draw inspiration from \cite[Lemma 2.4.10]{eva-thesis}. For $R\in \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$, denote $R^e\coloneqq R\wedge R$.
\begin{proof}
Consider $A$ (resp. $B$) as a commutative $A^e$-algebra (resp. $B^e$-algebra) via the multiplication map $A^e\to A$ (resp. $B^e\to B$).
Recall that $THH(A,B)\cong B\wedge_{A^e} B(A,A,A)$ and similarly for $THH(B)$ (see \cite[IV.7.2, IX.2.4]{ekmm} for a definition of the two-sided bar construction $B(A,A,A)$ and a proof of the isomorphism).
Let $\tilde B \stackrel{\sim}{\to} B$ be a cofibrant replacement of $B$ in the category of commutative $B^e$-algebras. There is a commutative diagram of $\mathbb{S}$-modules
\begin{enumerate}gin{equation}\label{sorw}\xymatrix{THH(A,B) \ar[d]_-{THH(f,\mathrm{id})} & \tilde B \wedge_{A^e} B(A,A,A) \ar[l]_-\sim \ar[r]^-\sim \ar[d]_-{(\mathrm{id}, f)} & \tilde B \wedge_{A^e} A \ar[d]_-{\overline f} \\ THH(B) & \tilde B \wedge_{B^e} B(B,B,B) \ar[r]^-\sim \ar[l]_-\sim & \tilde B \wedge_{B^e} B.}\end{equation}
Indeed, recall that there is a weak equivalence of commutative $A^e$-algebras $B(A,A,A)\to A$ \cite[IV.7.5]{ekmm} and a cofibration in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ $A^e\to B(A,A,A)$ given by inclusion of the first and last smash factors. See \cite[Proof of Lemma 2.4.8]{eva-thesis} for a proof of this last fact.
The arrow $(\mathrm{id},f)$ in the middle is defined via the universal property for the coproduct in commutative $A^e$-algebras, using the canonical map $\tilde B \to \tilde B \wedge_{B^e} B(B,B,B)$ to the first factor, and the map $B(A,A,A)\to B(B,B,B)$ defined by smash powers of $f$ at the simplicial level followed by the canonical map to the second factor.
The arrow $\overline{f}$ is described as follows. First note that there are isomorphisms
\[\tilde B \wedge_{A^e} A \cong \tilde B \wedge_{B^e} (B^e \wedge_{A^e} A)\cong \tilde B \wedge_{B^e} (B\wedge_A B).\]
The last step comes from the isomorphism of commutative $B^e$-algebras $B^e \wedge_{A^e} A \cong B\wedge_A B$
which appears e.g. in \cite[Lemma 2.1]{lindenstrauss}.
Then $\overline{f}$ is defined to be the composition
\[\xymatrix{\tilde B \wedge_{A^e} A \cong \tilde B \wedge_{B^e} (B\wedge_A B) \ar[r]^-{\mathrm{id}\wedge \mu} & \tilde B \wedge_{B^e} B.}\]
The previous diagram appears as the geometric realization of a diagram in simplicial $\mathbb{S}$-modules. The arrows in this latter diagram are very explicitely defined, and it is immediate that they make the diagram commute.
Therefore, to see that $THH(f,\mathrm{id})$ is a weak equivalence, it suffices to see that $\mathrm{id}\wedge \mu$ is a weak equivalence. This is the case: indeed, the functor $\tilde B \wedge_{B^e}-$ preserves weak equivalences between cofibrant commutative $\mathbb{S}$-algebras because $\tilde B$ is cofibrant as a commutative $B^e$-algebra. Now note that $B\wedge_A B$ is a cofibrant commutative $\mathbb{S}$-algebra because it is a cofibrant commutative $A$-algebra (it is a coproduct of two cofibrant commutative $A$-algebras).
\end{proof}
Lemma \ref{idem} allows us to apply Theorem \ref{thh-base} to $\mathbb{S}\to R\to R[x^{-1}]$. Putting this together with the weak equivalence (\ref{comienzo}), we obtain:
\begin{corolario} \label{cor-thhloc} Let $R$ be a cofibrant commutative $\mathbb{S}$-algebra, and let $x\in \pi_*R$. There are weak equivalences of augmented commutative $R[x^{-1}]$-algebras
\[\xymatrix{THH(R)[x^{-1}] & THH(R,R[x^{-1}]) \ar[l]_-\sim \ar[r]^-\sim & THH(R[x^{-1}]). }\]
\end{corolario}
\begin{obs} We have recently become aware that, in \cite[Page 353]{svw}, the authors state that ``one can prove that $THH$ commutes with localizations'', pointing to an article in preparation which never appeared.
\end{obs}
\begin{obs} We know three proofs of the fact that Hochschild homology commutes with localizations. Weibel \cite[9.1.8(3)]{weibel} proves it using the fact that Tor behaves well under flat base change. Brylinski \cite{brylinski} (see also \cite[1.1.17]{loday}) proves it by comparing the homological functors defined on $A$-bimodules $S^{-1}HH_n(A,-)$ and $HH_n(S^{-1}A,S^{-1}-)$, where $S$ is a multiplicative subset of the commutative algebra $A$. In \cite{weibel-geller}, Geller and Weibel prove the more general result that Hochschild homology behaves well with respect to étale maps of commutative algebras $A\to B$, of which a localization map is an example. Our proof of Theorem \ref{thh-base} is closer to the first of these approaches.
\end{obs}
\begin{obs} For a map $f:A\to B$ of commutative $\mathbb{S}$-algebras as in Theorem \ref{thh-base}, the question of under what conditions is (\ref{thh-base-eq}) a weak equivalence has been considered before. For example, in \cite[Lemma 5.7]{mccarthy-minasian} the authors prove that it holds when $A$ and $B$ are connective and the unit $B\to THH^A(B)$ is a weak equivalence. Mathew \cite[Theorem 1.3]{mathew-thh}, working in the context of the $E_\infty$-rings of Lurie, proved that a map $A\to B$ of $E_\infty$-rings satisfies that (\ref{thh-base-eq}) is an equivalence provided $f$ is étale, with no hypotheses on connectivity. There is a notion of localization of $E_\infty$-rings, and Lurie proved that localization maps are étale \cite[7.5.1.13]{ha}. This gives a short proof of Theorem \ref{thh-base} applied to $\mathbb{S}\to R\to R[x^{-1}]$ in the context of $E_\infty$-rings: this is the point of view adopted for the more general result of \cite[Corollary 7.4]{rsv-thom}.
\end{obs}
\section{Topological Hochschild homology of \texorpdfstring{$KU$}{KU}} \label{sect:thhku}
\subsection{Flatness} \label{sect:cofibrancy}
Let $G$ be a topological commutative monoid. We cannot prove that the commutative $\mathbb{S}$-algebra $\mathbb{S}[G]$ is cofibrant (and we believe it is not, in general), even if $G$ satisfies good cofibrancy hypotheses. Instead, we remark that, when $G$ is a CW-complex and the unit is a 0-cell, $\mathbb{S}[G]$ is flat (to be defined below) and $B^\mathrm{cy}_\bullet(\mathbb{S}[G])$ is a proper simplicial $\mathbb{S}$-module: these properties ensure that $THH(\mathbb{S}[G])$ has homotopical meaning. We thank Cary Malkiewich and Michael Mandell for helping us realize this.
\begin{defn} \label{def-flat}An $\mathbb{S}$-module $M$ is \emph{flat} if $M\wedge-:\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}\to \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$ preserves all weak equivalences.
\end{defn}
Note that if $M$ is flat and $N$ is any $\mathbb{S}$-module, then $N\wedge M$ computes the derived smash product.
\begin{obs} Flat $\mathbb{S}$-modules satisfy the following properties: \label{pseudo-rem}
\begin{itemize}
\item Any cofibrant $\mathbb{S}$-module is flat, and $\mathbb{S}$ is flat.
\item Smash products of flat $\mathbb{S}$-modules are flat.
\item Coproducts of flat $\mathbb{S}$-modules are flat.
\item Weak equivalences between flat $\mathbb{S}$-modules are closed under finite smash products.
\end{itemize}
We do not know whether the underlying $\mathbb{S}$-module to a cofibrant commutative $\mathbb{S}$-algebra is automatically flat.
\end{obs}
\begin{enumerate}gin{lema} \label{cofcofsemi} Let $A$ and $A'$ be cofibrant commutative $\mathbb{S}$-algebras, $N$ be any $\mathbb{S}$-module and $M$ be a flat $\mathbb{S}$-module. Let $f:A\to N$, $g:A'\to M$ be weak equivalences of $\mathbb{S}$-modules. Then
\[f \wedge g : A \wedge A' \to N \wedge M\]
is a weak equivalence of $\mathbb{S}$-modules.
\begin{proof} Let $\gamma_A:\Gamma A \to A$ and $\gamma_{A'}:\Gamma A' \to A'$ be cofibrant replacements of $A$ and $A'$ in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$. Consider the following commutative diagram:
\[\xymatrix@C+1pc{A \wedge A' \ar[r]^-{f \wedge g} & N \wedge M \\ \Gamma A \wedge \Gamma A' \ar[u]^-{\gamma_A \wedge \gamma_{A'}} \ar[r]_-{\mathrm{id} \wedge (g\circ \gamma_{A'})} & \Gamma A \wedge M. \ar[u]_-{(f\circ \gamma_A) \wedge \mathrm{id}} }\]
By Properties (\ref{ujin}) and (\ref{es}) of Section \ref{sect-cof}, the left vertical map and the bottom horizontal map are weak equivalences. The right vertical map is a weak equivalence because $M$ is flat. In conclusion, $f\wedge g$ is a weak equivalence.
\end{proof}
\end{lema}
\begin{enumerate}gin{lema} \label{isflat} Let $Y$ be a based CW-complex. Then $\mathbb{S}igma^\infty Y$ is a flat $\mathbb{S}$-module. Also, the functor $\mathbb{S}igma^\infty Y \wedge -:\mathbb{S}\mbox{-}\mathbf{Mod}\to \mathbb{S}\mbox{-}\mathbf{Mod} $ is left Quillen,
so if $M$ is a cofibrant $\mathbb{S}$-module, then $\mathbb{S}igma^\infty Y \wedge M$ is so too.
\begin{proof} The first statement is \cite[4.11(i)]{mandell-may}. Recall that $\mathbb{S}igma^\infty Y \wedge-$ is isomorphic to $Y\wedge -$, where the latter $\wedge$ denotes the tensor of $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$ over $\ensuremath{\mathbf{Top}}_*$ \cite[II.1.4]{ekmm}. Since $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$ is a $\ensuremath{\mathbf{Top}}_*$-model category and $Y$ is cofibrant in $\ensuremath{\mathbf{Top}}_*$, this implies that $\mathbb{S}igma^\infty Y \wedge -$ is left Quillen, so it takes the cofibrant $M$ to a cofibrant $\mathbb{S}$-module.
\end{proof}
\end{lema}
\begin{prop} \label{pseudocof-thh} Let $G$ be a CW-complex which is a topological commutative monoid with unit a 0-cell. Let $f:\widetilde{\mathbb{S}[G]}\to \mathbb{S}[G]$ be a weak equivalence in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$, where $\widetilde{\mathbb{S}[G]}$ is a cofibrant commutative $\mathbb{S}$-algebra. Then \[THH(f):THH(\widetilde{\mathbb{S}[G]})\to THH(\mathbb{S}[G])\] is a weak equivalence of commutative $\widetilde{\mathbb{S}[G]}$-algebras.
\begin{proof} First, note that $B_\bullet^\mathrm{cy}(\mathbb{S}[G])$ is a proper simplicial $\mathbb{S}$-module. Indeed, by Proposition \ref{conmutarTHH}, $B_\bullet^\mathrm{cy}(\mathbb{S}[G])\cong \mathbb{S}[B_\bullet^\mathrm{cy} G]$. Now, the functor $\mathbb{S}igma^\infty_+:\ensuremath{\mathbf{Top}}\to \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$ preserves properness of simplicial objects, as was observed in \cite[IV.7.8]{ekmm}. But $B^\mathrm{cy}_\bullet(G)$ is a proper simplicial space, since it is good \cite[3.2]{thh-thom}
and
any good simplicial space is proper \cite[Proof of 2.4(b)]{lewis-lillig}. Therefore, by \cite[X.2.4]{ekmm}, it suffices to see that $B^\mathrm{cy}_\bullet(f):B^\mathrm{cy}_\bullet(\widetilde{\mathbb{S}[G]})\to B^\mathrm{cy}_\bullet(\mathbb{S}[G])$ is levelwise a weak equivalence.
The map $f\wedge f: \widetilde{\mathbb{S}[G]} \wedge \widetilde{\mathbb{S}[G]} \to \mathbb{S}[G]\wedge \mathbb{S}[G]$ is a weak equivalence by the previous two lemmas. For higher smash powers of $f$, the statement is proven by an analogous argument and induction.
\end{proof}
\end{prop}
\begin{obs} More generally, we have just proven that if $A\to B$ in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ is a weak equivalence where $A$ is a cofibrant commutative $\mathbb{S}$-algebra and $B$ is a flat $\mathbb{S}$-module such that $B_\bullet^\mathrm{cy}(B)$ is a proper simplicial $\mathbb{S}$-module, then $THH(A)\to THH(B)$ is a weak equivalence of commutative $A$-algebras.
\end{obs}
\subsection{Topological Hochschild homology of \texorpdfstring{$\mathbb{S}[G][x^{-1}]$}{S[G][1/x]}}
Let $G$ be a CW-complex which is a topological abelian group with unit a 0-cell and cellular addition map. As remarked in Section \ref{section:cyclictop}, these assumptions guarantee that $BG$ is again a CW-complex, so that $\mathbb{S}[BG]$ is a flat $\mathbb{S}$-module by Lemma \ref{isflat}. This will be useful in the proof of Theorem \ref{teoloc}.
We first isolate a result that does not involve inverting an element.
\begin{enumerate}gin{lema} \label{sinloc} There is an isomorphism of augmented commutative $\mathbb{S}[G]$-algebras
\[THH(\mathbb{S}[G]) \cong \mathbb{S}[G] \wedge \mathbb{S}[BG] = \mathbb{S}[G][BG].\]
\begin{proof} It is an application of Propositions \ref{conmutarTHH}, \ref{BcyG} and \ref{Smono}:
\[\xymatrix{THH(\mathbb{S}[G]) \ar[r]^-\cong & \mathbb{S}[B^\mathrm{cy} G] \ar[r]^-\cong & \mathbb{S}[G\times BG] & \mathbb{S}[G] \wedge \mathbb{S}[BG] \ar[l]_-\cong.}\qedhere\]
\end{proof}
\end{lema}
Let $x\in \pi_* \mathbb{S}[G]$. Recall from Definition \ref{def-nocof} that $\mathbb{S}[G][x^{-1}]_h$ is defined as $(Q\mathbb{S}[G])[x^{-1}]$ where $Q$ is a cofibrant replacement functor in the category of commutative $\mathbb{S}$-algebras coming from a functorial factorization. It factors the unit $e_R:\mathbb{S}\to R$ of a commutative $\mathbb{S}$-algebra $R$ as
\begin{enumerate}gin{equation}\label{unitrep1} \xymatrix{\mathbb{S} \ar[rr]^-{e_R} \ar@{>->}[rd]_-{e_{QR}} && R \\ & QR. \ar@{->>}[ru]^-\sim_-{q_R}}\end{equation}
\begin{teo} \label{teoloc} The commutative $\mathbb{S}[G][x^{-1}]_h$-algebras $THH(\mathbb{S}[G][x^{-1}]_h)$ and $\mathbb{S}[G][x^{-1}]_h[BG]$ are weakly equivalent as $\mathbb{S}[G][x^{-1}]_h$-algebras.
\end{teo}
For any commutative $\mathbb{S}$-algebra $A$, the notation $A[BG]$ stands for the commutative $A$-algebra $A\wedge \mathbb{S}[BG]$: thus, its underlying $A$-module is $A\wedge (BG)_+$.
No confusion should arise from the usage of square brackets for two different notions.
\begin{proof} For ease of notation, let us denote $\mathbb{S}[G]$ by $A$ and $\mathbb{S}[BG]$ by $B$. As in Remark \ref{funct-fact} for $R=QA$, the functor $Q$ begets a cofibrant replacement functor $Q_{QA}$ in the category of commutative $QA$-algebras. For ease of notation we denote it by $Q_A$. This functor factors the unit $u_X:QA\to X$ of a $QA$-commutative algebra $X$ as
\begin{enumerate}gin{equation} \label{unitrep2}\xymatrix{QA \ar[rr]^-{u_X} \ar@{>->}[rd]_-{u_{Q_AX}} && X \\ & Q_A X. \ar@{->>}[ru]^-\sim_-{q^A_X}}\end{equation}
Using Corollary \ref{cor-thhloc}, we obtain a zig-zag of two weak equivalences of commutative $QA[x^{-1}]$-algebras
\[THH(QA[x^{-1}])\simeq THH(QA)[x^{-1}].\]
By Proposition \ref{pseudocof-thh}, the map of commutative $QA$-algebras $THH(QA)\to THH(A)$ is a weak equivalence. We apply $Q_A$ and obtain a weak equivalence of commutative $QA$-algebras \[\xymatrix{Q_A(THH(QA)) \ar[r]^-\sim & Q_A THH(A).}\]
Note that if we had applied $Q$ instead of $Q_A$, we would not be able to guarantee that the above morphism be a morphism of $QA$-algebras.
We also have a weak equivalence of commutative $QA$-algebras $Q_A THH(QA) \to THH(QA)$. After inverting $x$, we obtain a zig-zag of weak equivalences of commutative $QA[x^{-1}]$-algebras:
\begin{enumerate}gin{equation}\label{minu}\xymatrix{THH(QA)[x^{-1}] & (Q_ATHH(QA))[x^{-1}] \ar[l]_-\sim \ar[r]^-\sim & (Q_A THH(A))[x^{-1}].}\end{equation}
Now, by Lemma \ref{sinloc}, $THH(A)\cong A \wedge B$ as commutative $A$-algebras, so we obtain an isomorphism of commutative $QA[x^{-1}]$-algebras
\[(Q_A THH(A))[x^{-1}] \cong (Q_A(A \wedge B))[(x\wedge 1)^{-1}].\]
We now construct a weak equivalence of commutative $QA$-algebras \[ r: Q_AA \wedge QB \to Q_A(A \wedge B)\]
where $Q_AA\wedge QB$ is a $QA$-algebra by means of the map $\xymatrix{QA\ar[r]^-{u_{Q_AA}} & Q_AA \ar[r]^-{\mathrm{id} \wedge e_{QB}} & Q_AA \wedge QB.}$ Since $\wedge$ is the coproduct in the category of commutative $\mathbb{S}$-algebras, we define the map $r$ to be the morphism in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ given as follows. It is $Q_A(\mathrm{id} \wedge e_B):Q_AA \to Q_A(A\wedge B)$ on the first component. On the second component, it is the map $t:QB\to Q_A(A\wedge B)$ gotten from the functorial factorization on $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ applied to the vertical arrows in the following commutative diagram:
\[\xymatrix@C+1pc{\mathbb{S} \ar[r]^-{e_{QA}} \ar[d]_-{e_B} & QA \ar[d]^-{q_A \wedge e_B} \\ B\ar[r]_-{e_A\wedge \mathrm{id}_B} & A \wedge B. }\]
This defines the map $r$ as a morphism of commutative $\mathbb{S}$-algebras. It is a morphism of commutative $QA$-algebras: the composition $\xymatrix@C+1pc{QA \ar[r]^-{u_{Q_AA}} & Q_AA \ar[r]^-{Q_A(\mathrm{id}\wedge e_B)} & Q_A(A\wedge B)}$ coincides with $u_{Q_A(A\wedge B)}$ by functoriality of the factorization. We will now prove that $r$ is a weak equivalence. First, consider the following diagram:
\[\xymatrix@C+2pc{Q_AA \ar[r] \ar[rd]_-{Q_A(\mathrm{id}\wedge e_B)} \ar@/_2pc/[rdd]_-{q^A_A\wedge e_B} & Q_AA \wedge QB \ar[d]^-r & QB \ar[l] \ar[ld]^-t \ar@/^2pc/[ldd]_-{\ \ e_A\wedge q_B} \\ & Q_A(A\wedge B) \ar[d]^-{q^A_{A\wedge B}} \\ & A\wedge B }\]
The two upper triangles commute by definition of $r$. The two lower triangles also commute: the left one by definition of $Q_A(\mathrm{id}\wedge e_B)$, the right one by definition of $t$. By the universal property of the coproduct in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$, this proves that $q^A_{A\wedge B} \circ r=q_A^A \wedge q_B$.
Lemma \ref{cofcofsemi} applies to prove that $q_A^A \wedge q_B$ is a weak equivalence. Therefore, since $q^A_{A\wedge B}$ is also a weak equivalence, so is $r$.
Inverting $x\wedge 1$ in $r$, we obtain a weak equivalence of commutative $QA[x^{-1}]$-algebras
\[
\xymatrix{Q_A(A \wedge B)[(x\wedge 1)^{-1}] & \ar[l]_-\sim (Q_AA \wedge Q B)[(x\wedge 1)^{-1}].}\]
By Proposition \ref{sepa}, there is a weak equivalence of commutative $QA[x^{-1}]$-algebras
\begin{enumerate}gin{equation}\label{vers}\xymatrix{(Q_AA \wedge Q B)[(x\wedge 1)^{-1}] & \ar[l]_-\sim Q_AA[x^{-1}] \wedge QB. }\end{equation}
From (\ref{unitrep2}) applied to $X=A$, we get that $q_A=q_A^A\circ u_{Q_AA}$, so $u_{Q_AA}:QA\to Q_AA$ is a weak equivalence of commutative $QA$-algebras.
We can invert $x$ to obtain the weak equivalence of commutative $QA[x^{-1}]$-algebras $QA[x^{-1}]\to Q_AA[x^{-1}]$, which after smashing with $QB$ becomes
\[\xymatrix{Q_A A[x^{-1}] \wedge QB & QA[x^{-1}] \wedge QB, \ar[l]_-\sim }\]
a weak equivalence of commutative $QA[x^{-1}]$-algebras.
Now consider the weak equivalence $q_B:QB\to B$ of commutative $\mathbb{S}$-algebras. Lemma \ref{cofcofsemi} applies to prove that
\[\xymatrix{QA[x^{-1}]\wedge QB \ar[r]^-\sim & QA[x^{-1}] \wedge B }\]
is a weak equivalence of commutative $QA[x^{-1}]$-algebras.
Putting together all these weak equivalences, we obtain a zig-zag of weak equivalences of commutative $(Q\mathbb{S}[G])[x^{-1}]$-algebras \linebreak $THH((Q\mathbb{S}[G])[x^{-1}])\simeq ((Q\mathbb{S}[G])[x^{-1}])[BG]$.
\end{proof}
\begin{obs} \label{augment-complicado} In this remark, we explain how compatible is the zig-zag of weak equivalences just obtained with the augmentations.
Ignoring the last weak equivalence in the zig-zag of weak equivalences obtained in the proof, we have obtained
\begin{enumerate}gin{equation}\label{zaq} THH(QA[x^{-1}])\simeq QA[x^{-1}]\wedge QB. \end{equation}
The left-hand side is naturally augmented over $QA[x^{-1}]$. We now give the right-hand side an augmentation over $QA[x^{-1}]$ defined from the universal property of the coproduct in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$. On the first factor, it is the identity. On the second factor, it is the arrow:
\[\xymatrix@C+1pc{QB \ar[r]^-{Q(e_A \circ \varepsilon \mbox{-}ilon_B)} & QA \ar[r] & QA[x^{-1}] }\]
where $\varepsilon \mbox{-}ilon_B$ is the augmentation of $B=\mathbb{S}[BG]$ coming from $BG\to *$ and the second map is the localization map.
If one goes through the proof of the previous theorem, one can prove that the zig-zag (\ref{zaq}) commutes with the augmentations, in the following sense: this zig-zag fits as one long horizontal side of a ladder diagram, where the other side starts and ends with $QA[x^{-1}]$ and in the middle has as objects $QA[x^{-1}]$, $Q_AQA[x^{-1}]$ or $Q_AA[x^{-1}]$, connected to each other via the obvious weak equivalences or via identities when possible. The ingredients used in the proof of this are: the fact that Corollary \ref{cor-thhloc} and Lemma \ref{sinloc} are compatible with the augmentations, the naturality of Proposition \ref{sepa} and of $THH$, and the functoriality of the factorization (cofibrations, acyclic fibrations).
The last weak equivalence
\begin{enumerate}gin{equation}\label{xlo} QA[x^{-1}]\wedge QB \to QA[x^{-1}]\wedge B\end{equation}
is more problematic. The only sensible augmentation over $QA[x^{-1}]$ to define on the codomain would be $\mathrm{id} \wedge \varepsilon \mbox{-}ilon_B$, but then the weak equivalence (\ref{xlo}) commutes with the augmentations only up to homotopy of commutative $QA[x^{-1}]$-algebras. Indeed, in the following diagram
\[\xymatrix@C+2pc{QB\ar[r]^-{Q(e_A \circ \varepsilon \mbox{-}ilon_B)} \ar[d]_-{q_B} & QA \ar[d]^-{q_A}_-\sim \\ B \ar[ru]^-{e_{QA} \circ \varepsilon \mbox{-}ilon_B} \ar[r]_-{e_A \circ \varepsilon \mbox{-}ilon_B} & A}\]
the square commutes, the bottom triangle commutes, but the upper triangle does not seem to commute. However, since $q_A$ is a weak equivalence, it commutes in the homotopy category of $QA\mbox{-}\mathbf{CAlg}$.
\end{obs}
\subsection{Snaith's theorem and first description of $THH(KU)$} \label{construction-ku} \index{KU} \index{Topological $K$-theory}
There is a cofibrant commutative $\mathbb{S}$-algebra $KU$ of complex topological $K$-theory \cite[VIII.4.3]{ekmm}. It is obtained by applying the localization theorem we reviewed in Theorem \ref{rect} to the cofibrant commutative $\mathbb{S}$-algebra $ku$ of connective complex $K$-theory and its Bott element. Here $ku$ is constructed by multiplicative infinite loop space theory.
The presentation for $KU$ which we will use relies on the following theorem of Snaith \cite{snaith79}, \cite{snaith81}.
\begin{teo} $KU$ is weakly equivalent as a homotopy commutative ring spectrum to \linebreak $\mathbb{S}[\mathbb C P^\infty][x^{-1}]_{\textup{tel}}$, where $x\in \pi_2(\mathbb{S}[\mathcal{C} P^\infty])$ is represented by the map induced from the inclusion $\mathcal{C} P^1\to \mathcal{C} P^\infty,$ i.e.
\begin{enumerate}gin{equation}\label{xproj} \mathbb{S}igma^\infty S^2\cong \mathbb{S}igma^\infty \mathcal{C} P^1 \to \mathbb{S} \vee \mathbb{S}igma^\infty \mathcal{C} P^\infty \simeq \mathbb{S}igma^\infty_+ \mathcal{C} P^\infty. \end{equation}
Here $\mathbb{S}[\mathcal{C} P^\infty][x^{-1}]_{\textup{tel}}$ means the homotopy commutative ring spectrum obtained with a telescope construction, as in (\ref{telescope}).
\end{teo}
As remarked in \cite[VIII.4]{ekmm}, the more structured version of the inversion of a homotopy element described in Section \ref{inversion} is weakly equivalent to this telescope construction as homotopy commutative ring spectra (i.e. commutative monoids in the homotopy category of spectra).
Indeed, the technology of $\mathbb{S}$-algebras did not exist at the time Snaith's theorem got published, but this is not a problem:
\begin{teo} \label{bari} \cite[6.2]{baker-richter} Let $A$ be an $E_\infty$-ring spectrum which is weakly equivalent to $KU$ as a homotopy commutative ring spectrum. Then $A$ is weakly equivalent to $KU$ as an $E_\infty$-ring spectrum.
\end{teo}
Any $E_\infty$-ring spectrum $A$ admits a weak equivalence of $E_\infty$-ring spectra from the commutative $\mathbb{S}$-algebra $\mathbb{S}\wedge_\mathcal{L} A$, and this construction is functorial \cite[II.3.6]{ekmm}; moreover, the morphisms of commutative $\mathbb{S}$-algebras are exactly the morphisms of the underlying $E_\infty$-ring spectra. So if $A$ in the statement of Theorem \ref{bari} was a commutative $\mathbb{S}$-algebra to begin with, then it is weakly equivalent to $KU$ as a commutative $\mathbb{S}$-algebra.
Let $K(\mathbb{Z},2)$ denote the topological abelian group given by $B(B\mathbb{Z})$, where $B$ is the classifying space construction reviewed in Section \ref{section:cyclictop}.
The homotopy commutative ring spectrum $\mathbb{S}[\mathcal{C} P^\infty]$ is weakly equivalent to the cofibrant commutative $\mathbb{S}$-algebra $Q\mathbb{S}[K(\mathbb{Z},2)]$.
Therefore, \[\mathbb{S}[\mathcal{C} P^\infty][x^{-1}]_{\textup{tel}} \simeq Q\mathbb{S}[K(\mathbb{Z},2)][x^{-1}]\]
as homotopy commutative ring spectra. By the results above, we obtain that
\begin{enumerate}gin{equation}\label{defku} KU \simeq Q\mathbb{S}[K(\mathbb{Z},2)][x^{-1}]\end{equation}
as commutative $\mathbb{S}$-algebras. This is the description of $KU$ that we shall be using, so from now on we let $KU$ denote the cofibrant commutative $\mathbb{S}$-algebra $Q\mathbb{S}[K(\mathbb{Z},2)][x^{-1}]$ that we have also denoted by $\mathbb{S}[K(\mathbb{Z},2)][x^{-1}]_h$.
\begin{obs} We thank Christian Schlichtkrull for pointing out the article \cite{arthan} to us. In Theorems 5.1 and 5.2 therein, it is proven that if $t\in \pi_n(\mathbb{S}[K(\mathbb{Z},n)])$ is a generator,
then the spectrum $\mathbb{S}[K(\mathbb{Z},n)][t^{-1}]_{\textup{tel}}$ is contractible for $n$ odd and is equivalent to $H\mathbb{Q}[t^{\pm 1}]$ for $n\geq 4$ even. So the case $n=2$ which we treat here is the only interesting localization of $\mathbb{S}[K(\mathbb{Z},n)]$.
\end{obs}
Note that $\mathbb{Z}$ is a CW-complex (with only 0-cells) and a (discrete) topological abelian group with cellular addition map, so this guarantees that $K(\mathbb{Z},2)$ satisfies the same hypotheses, as recalled in Section \ref{section:cyclictop}. Therefore, we can apply Theorem \ref{teoloc} to obtain:
\begin{teo} \label{thhku1} The commutative $KU$-algebras $THH(KU)$ and $KU[K(\mathbb{Z},3)]$ are weakly equivalent as commutative $KU$-algebras.
\end{teo}
Remark \ref{augment-complicado} tells us that the zig-zag of weak equivalences is compatible up to homotopy with the augmentations.
\begin{obs} \label{thh-thom} Compare with what happens to $THH(MU)$: in \cite{thh-thom}, the authors establish a weak equivalence of $\mathbb{S}$-modules $THH(MU)\simeq MU \wedge SU_+$. They actually prove the following more general result. Let $BF$ denote a classifying space for stable spherical fibrations.
If $f:X\to BF$ is a 3-fold loop map and $T(f)$ is its Thom spectrum, then there is a weak equivalence of $\mathbb{S}$-modules \begin{enumerate}gin{equation}\label{thhtf} THH(T(f))\simeq T(f)\wedge BX_+.\end{equation}Note that this result was improved to a weak equivalence of $E_\infty$ $\mathbb{S}$-algebras by Schlichtkrull \cite[Corollary 1.2]{schlichtkrull-higher} in the case where $X$ is a grouplike $E_\infty$-space and $f$ is an $E_\infty$-map.
Our Theorem \ref{thhku1} gives in particular a weak equivalence of $\mathbb{S}$-modules \[THH(KU)\simeq KU \wedge K(\mathbb{Z},3)_+.\]By comparing this formula to (\ref{thhtf}), one is naturally led to conjecture that $KU$ is the Thom spectrum of an $\infty$-loop map $K(\mathbb{Z},2)\simeq BU(1)\to BU$. However, this is not possible, since such Thom spectra are connective. In the last decade, more general Thom spectra which can be non-connective have been introduced, and in \cite[Example 4.23]{rsv-thom} the authors remarked that $KU$ cannot be such a Thom spectrum of a map from $K(\mathbb{Z},2)$, as a consequence of the Thom isomorphism theorem. One possible explanation of why does $KU$ behave like a Thom spectrum to the eyes of topological Hochschild homology is to be found in that paper, where the above expression for $THH(KU)$ is obtained by considering $KU$ as an étale extension of $\mathbb{S}[K(\mathbb{Z},2)]$, which is a trivial Thom spectrum.
\end{obs}
\subsection{Rationalization} \index{Rationalization}In this section we review some facts about rationalization of $\mathbb{S}$-modules and of based spaces that we will be using, often without explicit mention.\\
\label{rationalization}
Consider a model for the Eilenberg-Mac Lane spectrum of $\mathbb{Q}$ which is a commutative $\mathbb{S}$-algebra \cite[II.4]{ekmm}. Let $H\mathbb{Q}$ be a cofibrant $\mathbb{S}$-module equivalent
to it in the category of $\mathbb{S}$-modules. In particular, $H\mathbb{Q}$ is a homotopy commutative ring spectrum. Denote by $\iota:\mathbb{S}\to H\mathbb{Q}$ its unit and by $\mu:H\mathbb{Q}\wedge H\mathbb{Q}\to H\mathbb{Q}$ its multiplication map, which is a weak equivalence.
We consider Bousfield localization \cite[VIII.1]{ekmm}, \cite[19.2]{mayponto} of $\mathbb{S}$-modules with respect to $H\mathbb{Q}$, and we call this process \emph{rationalization}. A map $X\to Y$ of $\mathbb{S}$-modules is an \emph{$H\mathbb{Q}$-equivalence} (or \emph{rational equivalence}) if it is a weak equivalence after smashing it with $H\mathbb{Q}$. An $\mathbb{S}$-module $W$ is \emph{$H\mathbb{Q}$-acyclic} if $H\mathbb{Q} \wedge W\simeq *$. An $\mathbb{S}$-module $X$ is \emph{$H\mathbb{Q}$-local} (or \emph{rational}) if, for every $H\mathbb{Q}$-acyclic $\mathbb{S}$-module $W$, the only map $W\to X$ in the homotopy category of $\mathbb{S}$-modules is the trivial one. A \emph{rationalization map} for $X$ is an $H\mathbb{Q}$-equivalence $X\to Y$ where $Y$ is rational; rationalizations are unique up to homotopy.
Note that $H\mathbb{Q}\wedge X$ is rational since it is an $H\mathbb{Q}$-module \cite[1.17]{ravenel-loc}.
The following diagram is homotopy commutative:
\[\xymatrix@C+1pc{H\mathbb{Q}\wedge X \ar[r]^-{\mathrm{id}\wedge \iota \wedge \mathrm{id}} \ar[rd]_-{\mathrm{id}} & H\mathbb{Q} \wedge H\mathbb{Q} \wedge X\ar[d]^-{\mu \wedge \mathrm{id}} \\ & H\mathbb{Q} \wedge X,}\]
so we only need $\mu\wedge \mathrm{id}$ to be a weak equivalence in order to assert that $\iota \wedge \mathrm{id}: X\to H\mathbb{Q} \wedge X$ is a rationalization of $X$. This is true because, as we have seen in Property (\ref{moguil}) of Section \ref{sect-cof}, all $\mathbb{S}$-modules $X$ satisfy that $X\wedge -$ preserves weak equivalences between cofibrant $\mathbb{S}$-modules. Therefore, $\iota \wedge \mathrm{id}: X\to H\mathbb{Q}\wedge X$ is a rationalization map for $X$, and from now on we let $X\to X_\mathbb{Q}$ mean this map. This construction is functorial in $X\in\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}$.
From this construction one deduces some properties: if $f:X\to Y$ is a weak equivalence of $\mathbb{S}$-modules then it is a rational equivalence, since $H\mathbb{Q}$ is cofibrant. The homotopy groups of $X_\mathbb{Q}$ are isomorphic to $\mathbb{Q} \otimes \pi_*X$.
An $\mathbb{S}$-module is rational if and only if $X\to X_\mathbb{Q}$ is a weak equivalence,
if and only if the homotopy groups of $X$ are rational (i.e. $\mathbb{Q}$-vector spaces).
If $f:X\to Y$ is a rational equivalence between rational $\mathbb{S}$-modules, then it is a weak equivalence.
If $X$ is the suspension spectrum of a based CW-complex, then $X_\mathbb{Q}=H\mathbb{Q}\wedge X$ is cofibrant (Lemma \ref{isflat}).
Let $Y$ be another $\mathbb{S}$-module.
We have a map \[X_\mathbb{Q} \wedge Y_\mathbb{Q} \to (X\wedge Y)_\mathbb{Q}\] given by applying the multiplication map of $H\mathbb{Q}$, and it is a weak equivalence.
In particular, the smash product of the rationalization maps, $X\wedge Y \to X_\mathbb{Q} \wedge Y_\mathbb{Q}$, is a rationalization of $X\wedge Y$. Note that if $X$ and $Y$ are rational and one of them is cofibrant or flat, so that $X\wedge Y$ computes the derived smash product,
then $\pi_*(X \wedge Y)\cong \pi_*X \otimes_\mathbb{Q} \pi_*Y$ by a Künneth spectral sequence argument computing $H\mathbb{Q}_*(X\wedge Y)$ \cite[IV.4.7]{ekmm}.
Let $n$ be any integer. The degree $n$ map $n:\mathbb{S}\to \mathbb{S}$ induces a map $n:X\to X$ on any $\mathbb{S}$-module $X$ by smashing with it. If $p:X\to X$ is a weak equivalence for every prime $p$
then the homotopy groups of $X$ are rational, since $p$ induces the multiplication by $p$ map on homotopy groups.
Therefore, in this case, $X$ is rational.\\%, i.e. the rationalization map $X\to X_\mathbb{Q}$ is a weak equivalence.\\
Based spaces $X$ also admit rationalizations $q:X\to X_\mathbb{Q}$ (see \cite[9.(b)]{fht-rational} or \cite[6.5]{mayponto} for the simply-connected case which is the one we shall be using).
We will need the following fact concerning the rationalization of integral Eilenberg-Mac Lane spaces \cite[Page 202]{fht-rational}: for $n\geq 2$, there are homotopy equivalences
\[K(\mathbb{Z},n)_\mathbb{Q} \stackrel{\sim}{\leftarrow} \begin{enumerate}gin{cases} S^n_\mathbb{Q} & \textup{if } n \textup{ is odd}, \\ \Omega S^{n+1}_\mathbb{Q} & \textup{if } n \textup{ is even.} \end{cases}\]
Actually, the authors prove that, for $n$ even, there is a rational homotopy equivalence $\Omega S^{n+1} \to K(\mathbb{Z},n)$, so we get a homotopy equivalence $(\Omega S^{n+1})_\mathbb{Q} \stackrel{\sim}{\to} K(\mathbb{Z},n)_\mathbb{Q}$, which is not exactly what we wrote. But more generally, we have that $(\Omega X)_\mathbb{Q}$ is homotopy equivalent to $\Omega X_\mathbb{Q}$. Indeed,
by taking homotopy groups, we quickly see that $\Omega X_\mathbb{Q}$ is rational and that $\Omega q:\Omega X\to \Omega X_\mathbb{Q}$ is a rational homotopy equivalence. Since rationalizations are unique up to homotopy, this gives the result.\\
If $X$ is a based simply-connected space with rationalization map $q:X\to X_\mathbb{Q}$, then $\mathbb{S}igma^\infty q:\mathbb{S}igma^\infty X \to \mathbb{S}igma^\infty X_\mathbb{Q}$ is immediately seen to be a rationalization of $\mathbb{S}igma^\infty X$.
Therefore, $(\mathbb{S}igma^\infty X)_\mathbb{Q}$ is homotopy equivalent to $\mathbb{S}igma^\infty X_\mathbb{Q}$, i.e. rationalization of based spaces and of $\mathbb{S}$-modules are compatible under the $\mathbb{S}igma^\infty$ functor.
\subsection{$THH(KU)$, continuation} \label{sect-thhku2}
We will now describe the commutative $KU$-algebra $THH(KU)$ as the free commutative $KU$-algebra on the $KU$-module $\mathbb{S}igma KU_\mathbb{Q}$,
and we will prove this algebra is weakly equivalent to the split square-zero extension of $KU$ by $\mathbb{S}igma KU_\mathbb{Q}$. Let us first define this concept.
Let $R$ be a commutative $\mathbb{S}$-algebra, let $A$ be a commutative $R$-algebra and let $M$ be a non-unital commutative $A$-algebra. Then $A\vee M$ (coproduct of $A$-modules) has a commutative $A$-algebra structure. Indeed, after distributing, a multiplication map
\[(A\vee M) \wedge_A (A\vee M) \to A\vee M\]
looks like
\begin{enumerate}gin{equation}\label{sqz}(A\wedge_A A) \vee (A\wedge_A M) \vee (M\wedge_A A) \vee (M\wedge_A M) \to A\vee M.\end{equation}
We may define a map like (\ref{sqz}) by defining maps from each of the wedge summands to $A\vee M$.
Define the maps to $A\vee M$ from $A\wedge_A A$, $A\wedge_A M$ and $M\wedge_A A$ to be the canonical isomorphisms followed by the canonical maps into the respective factor. Finally, consider the map $M\wedge_A M\to A\vee M$ given by the multiplication map of $M$ followed by the canonical map to $A\vee M$. We have thus defined a multiplication map (\ref{sqz}) such that $A\vee M$ is a commutative $A$-algebra with unit given by the canonical map $A\to A\vee M$. We say that $A\vee M$ is a \emph{split extension of $A$ by $M$}. Note that $A\vee M$ is augmented over $A$: the augmentation is the identity on $A$ and the trivial map on $M$.
If the multiplication of $M$ is trivial, then $A\vee M$ is a \emph{split square-zero extension of $A$ by $M$}; in this case, $M$ is no more than an $A$-module.
The rest of this section is devoted to the proof of the following
\begin{teo} \label{thhku2} There are weak equivalences of commutative $KU$-algebras
\[\xymatrix{KU \vee \mathbb{S}igma KU_\mathbb{Q} & F(\mathbb{S}igma KU_\mathbb{Q}) \ar[l]_-h^-\sim \ar[r]^-{\tilde f}_-\sim & THH(KU)}\]
where $KU \vee \mathbb{S}igma KU_\mathbb{Q}$ is a split square-zero extension. Here $f:\mathbb{S}igma KU_\mathbb{Q}\to THH(KU)$ is a morphism of $KU$-modules to be constructed in (\ref{defff}), and the morphism $\tilde f$ is induced from $f$ by the free commutative algebra functor $F:KU\mbox{-}\mathbf{Mod}\to KU\mbox{-}\mathbf{CAlg}$. The map $h$ is adjoint to the wedge inclusion of $KU$-modules $\mathbb{S}igma KU_\mathbb{Q} \to KU \vee \mathbb{S}igma KU_\mathbb{Q}$.
\end{teo}
\begin{obs} \label{f-monadic} The functor $F$, or more generally, the free commutative algebra functor $F_R:\mathbb{R}Mod\to R\mbox{-}\mathbf{CAlg}$ where $R$ is a commutative $\mathbb{S}$-algebra, is the left adjoint of the forgetful functor $U_R: R\mbox{-}\mathbf{CAlg}\to\mathbb{R}Mod$, or alternatively, the free algebra functor for the monad $\mathbb P_R$ on $\mathbb{R}Mod$ defined as
\begin{enumerate}gin{equation}\label{monadwedge} \mathbb P_R(M)=\begin{itemize}gvee_{n\geq 0} \begin{itemize}gslant{M^{\wedge_R n}}{\mathbb{S}igma_n} = R \vee M \vee \begin{itemize}gvee_{n\geq 2} \begin{itemize}gslant{M^{\wedge_R n}}{\mathbb{S}igma_n},\end{equation} where $\mathbb{S}igma_n$ is the symmetric group on $n$ elements (see e.g. \cite[II.7.1]{ekmm} or \cite[Section 1]{basterra}). Note that $F_RM$ is augmented over $R$: the augmentation is the identity on the $0$-th term and the trivial map on the other terms.
As explained in Section \ref{sect-cof}, the functor $U_R:\mathbb{R}CAlg\to \mathbb{R}Mod$ is a right Quillen functor, so $F_R:\mathbb{R}Mod \to \mathbb{R}CAlg$ is a left Quillen functor. In particular, it preserves weak equivalences between cofibrant $R$-modules.
Note as well that, if $R$ is a cofibrant commutative $R$-algebra and $M\in \mathbb{R}Mod$ is
cofibrant, then the arrow $\begin{itemize}gvee_{n\geq 0} (M^{\wedge_R n})_{h\mathbb{S}igma_n}\to F_R(M)$ induced from the canonical arrows from the homotopy orbits to the orbits is a
weak equivalence \cite[III.5.1]{ekmm}. This is a step in the proof of the determination of the model structure on $\mathbb{R}CAlg$.
\end{obs}
\begin{obs} \label{antecedentes} A spectrum-level result related to Theorem \ref{thhku2} was obtained by McClure and Staffeldt in \cite[Theorem 8.1]{mc-st}: they showed that $THH(L)\simeq L \vee \mathbb{S}igma L_\mathbb{Q}$ as spectra, where $L$ is the $p$-adic completion of the Adams summand of $KU$ for a given odd prime $p$; the result was extended to $p=2$ by Angeltveit, Hill and Lawson in \cite[2.3]{thhko}. Ausoni \cite[Proposition 7.13]{ausoni-thhku} formulated without proof the analogous theorem (for an odd $p$) for $KU$ completed at $p$ in place of $L$. In Corollary 7.9 of \cite{thhko}, the authors show that $THH(KO)\simeq KO\vee \mathbb{S}igma KO_\mathbb{Q}$ as $KO$-modules. The methods used in the proofs of the results just cited are different from ours.
\end{obs}
We first prove some results needed for the proof. Note that in the following statement we are considering $K(\mathbb{Z},3)$ as a based space: we are not adding a disjoint basepoint.
\begin{enumerate}gin{prop} \label{kurat} There is a zig-zag of weak equivalences of $KU$-modules \[KU\wedge K(\mathbb{Z},3) \simeq \mathbb{S}igma KU_\mathbb{Q}.\]
\begin{proof} Let $p$ be a prime and consider the cofiber sequence of $KU$-modules
\begin{enumerate}gin{equation}\label{cofib}\xymatrix{KU \ar[r]^-p & KU \ar[r] & KU/p
.}\end{equation}
If $p>2$, then $KU/p$ is equivalent to $\begin{itemize}gvee\limits_{i=0}^{p-2} \mathbb{S}igma^{2i} K(1)$ (see \cite[Lecture 4]{adams-lect}), where $K(1)\simeq L/p$ is the first Morava $K$-theory at $p$. If $p=2$, then $K(1) \simeq KU/2$.
The homology $K(1)_* K(\mathbb{Z},3)$ is trivial: see \cite[Theorem 12.1]{rw80} for the $p>2$ case, and \cite[Appendix]{jw85} for the $p=2$ case.
Therefore, after smashing (\ref{cofib}) with $K(\mathbb{Z},3)$, we get a weak equivalence of $KU$-modules \[\xymatrix{KU \wedge K(\mathbb{Z},3) \ar[r]_-{p \wedge \mathrm{id}}^\sim & KU \wedge K(\mathbb{Z},3)}\]
for all primes $p$. This means that $KU\wedge K(\mathbb{Z},3)$ is rational, and so we have weak equivalences
\begin{enumerate}gin{align*}
\xymatrix{
KU \wedge K(\mathbb{Z},3) \ar[r]^-\sim &
(KU \wedge K(\mathbb{Z},3))_\mathbb{Q} \\
&
\ar[u]_-\sim KU_\mathbb{Q} \wedge K(\mathbb{Z},3)_\mathbb{Q} &
\ar[l]_-\sim KU_\mathbb{Q} \wedge S^3_\mathbb{Q} \ar[r]^-\sim &
(KU \wedge S^3)_\mathbb{Q} \ar[r]^-\sim &
\mathbb{S}igma KU_\mathbb{Q}}\end{align*}
by the results quoted in Section \ref{rationalization}, plus Bott periodicity for the last step.
\end{proof}
\end{prop}
\begin{enumerate}gin{lema} \label{lema-htpyorbits} The $\mathbb{S}$-modules $\begin{itemize}gslant{(\mathbb{S}igma H\mathbb{Q})^{\wedge n}}{\mathbb{S}igma_n}$ are weakly contractible for all $n\geq 2$.
\begin{enumerate}gin{proof}
First, note that $\mathbb{S}igma H\mathbb{Q}$ is a cofibrant $\mathbb{S}$-module. Indeed, $H\mathbb{Q}$ is a cofibrant $\mathbb{S}$-module and $S^1$ is a CW-complex, so by Lemma \ref{isflat} $\mathbb{S}igma H\mathbb{Q}=\mathbb{S}igma^\infty S^1 \wedge H\mathbb{Q}$ is a cofibrant $\mathbb{S}$-module. Therefore, we can apply \cite[III.5.1]{ekmm} to deduce that the map from the homotopy orbits \[((\mathbb{S}igma H\mathbb{Q})^{\wedge n})_{h\mathbb{S}igma_n} \to \begin{itemize}gslant{(\mathbb{S}igma H\mathbb{Q})^{\wedge n}}{\mathbb{S}igma_n}\] is a weak equivalence. We will prove that the homotopy orbits form a weakly contractible $\mathbb{S}$-module, thus finishing the proof.
The homotopy orbits spectral sequence \cite[3.2]{tsalidis} here looks like this:
\[E^2_{*,*} \cong H_*(\mathbb{S}igma_n; \pi_*((\mathbb{S}igma H\mathbb{Q})^{\wedge n})) \mathbb{R}ightarrow \pi_*(((\mathbb{S}igma H\mathbb{Q})^{\wedge n})_{h\mathbb{S}igma_n}).\]
Remark that $\pi_*((\mathbb{S}igma H\mathbb{Q})^{\wedge n}) \cong \widetilde{H}_*((S^1)^{\wedge n}; \mathbb{Q})$ as $\mathbb{S}igma_n$-modules, since indeed rearranging the factors in $(\mathbb{S}igma H\mathbb{Q})^{\wedge n}=(\mathbb{S}igma^\infty S^1 \wedge H\mathbb{Q})^{\wedge n}$ is $\mathbb{S}igma_n$-equivariant and so is the iterated multiplication map $(H\mathbb{Q})^{\wedge n}\to H\mathbb{Q}$ (at least up to homotopy).
Since the homology of $\mathbb{S}igma_n$ with rational coefficients vanishes in positive degrees,
the only group in the $E^2$-page of the spectral sequence which could be non-trivial is $H_0(\mathbb{S}igma_n; \widetilde{H}_n((S^1)^{\wedge n}; \mathbb{Q}))$. The action of $\mathbb{S}igma_n$ on $(S^1)^{\wedge n}$ permutes the factors. Under the homeomorphism $(S^1)^{\wedge n}\cong S^n$, each $\sigma\in \mathbb{S}igma_n$ acts on $S^n$ by a map $S^n\to S^n$ whose degree is the sign of $\sigma$. Indeed, one can decompose $\sigma$ as a composition of transpositions, which reduces the problem to the determination of the degree of the map $S^2\to S^2$ determined by permuting the two smash factors. This map does have degree $-1$, since it is a reflection of $S^2$ along a plane cutting the sphere in two identical parts.
In conclusion, the action of $\mathbb{S}igma_n$ on $\widetilde{H}_n(S^n;\mathbb{Q})\cong \mathbb{Q}$ is the rational sign representation of $\mathbb{S}igma_n$, so $H_0(\mathbb{S}igma_n;\widetilde{H}_n(S^n;\mathbb{Q}))$ vanishes.
Therefore, the $E^2$-page of the spectral sequence is trivial, and thus $\pi_*(((\mathbb{S}igma H\mathbb{Q})^{\wedge n})_{h\mathbb{S}igma_n})=0$, finishing the proof.
\end{proof}
\end{lema}
For the next results, recall that $F_R$ denotes the free commutative $R$-algebra on an $R$-module functor described in Remark \ref{f-monadic}.
\begin{corolario} \label{gunb} The map of commutative augmented $\mathbb{S}$-algebras
\[h_\mathbb{S}: F_\mathbb{S}(\mathbb{S}igma H\mathbb{Q}) \to \mathbb{S} \vee \mathbb{S}igma H\mathbb{Q}\]
defined as the adjoint to the wedge inclusion of $\mathbb{S}$-modules $\mathbb{S}igma H\mathbb{Q} \to \mathbb{S} \vee \mathbb{S}igma H\mathbb{Q}$ is a weak equivalence,
where $\mathbb{S}\vee \mathbb{S}igma H\mathbb{Q}$ is a split square-zero extension.
\begin{proof} We have that $F_\mathbb{S}(\mathbb{S}igma H\mathbb{Q})= \mathbb{S} \vee \mathbb{S}igma H\mathbb{Q} \vee \begin{itemize}gvee\limits_{n\geq2} \begin{itemize}gslant{(\mathbb{S}igma H\mathbb{Q})^{\wedge n}}{\mathbb{S}igma_n}$. By construction, $h_\mathbb{S}$ is the identity on the first two wedge summands and it is a trivial map on the $n\geq 2$ summands, so it is a weak equivalence by the previous lemma.
\end{proof}
\end{corolario}
\begin{enumerate}gin{prop} \label{split} Let $R$ be a cofibrant commutative $\mathbb{S}$-algebra.
The map of commutative augmented $\mathbb{S}$-algebras
\[h: F_R(\mathbb{S}igma R_\mathbb{Q}) \to R \vee \mathbb{S}igma R_\mathbb{Q}.\]
defined as the adjoint to the wedge inclusion of $R$-modules $\mathbb{S}igma R_\mathbb{Q} \to R \vee \mathbb{S}igma R_\mathbb{Q}$ is a weak equivalence, where $R\vee \mathbb{S}igma R_\mathbb{Q}$ is a split square-zero extension.
\end{prop}
\begin{obs} \label{f-cof}Note that we are applying $F_R$ to a cofibrant $R$-module, and so in particular $F_R(\mathbb{S}igma R_\mathbb{Q})$ is a cofibrant commutative $R$-algebra. Indeed, as observed in Lemma \ref{lema-htpyorbits}, $\mathbb{S}igma H\mathbb{Q}$ is a cofibrant $\mathbb{S}$-module. Now, the extension of scalars functor $R\wedge-: \ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}\to \mathbb{R}Mod$ is left Quillen: indeed, its right adjoint, the restriction of scalars functor, is right Quillen since the model structure in $\mathbb{R}Mod$ is created through it. Therefore, $R\wedge(\mathbb{S}igma H\mathbb{Q})\cong\mathbb{S}igma R_\mathbb{Q}$ is a cofibrant $R$-module.
\end{obs}
\begin{proof} Note that for an $\mathbb{S}$-module $X$, we have a natural isomorphism $F_R(R\wedge X)\cong R\wedge F_\mathbb{S}(X)$. Indeed,
\[F_R(R\wedge X)= \begin{itemize}gvee_{n\geq 0} \begin{itemize}gslant{(R\wedge X)^{\wedge_R n}}{\mathbb{S}igma_n}\cong R\wedge \begin{itemize}gvee_{n\geq 0} \begin{itemize}gslant{X^{\wedge n}}{\mathbb{S}igma_n} = R\wedge F_\mathbb{S}(X)\]
since the functor $R\wedge-:\mathbb{S}\mbox{-}\mathbf{Mod}\to R\mbox{-}\mathbf{Mod}$ is a left adjoint and strong symmetric monoidal.
Therefore,
\begin{enumerate}gin{equation}\label{furi} F_R(\mathbb{S}igma R_\mathbb{Q})\cong F_R(R\wedge \mathbb{S}igma H\mathbb{Q})\cong R\wedge F_\mathbb{S}(\mathbb{S}igma H\mathbb{Q}).\end{equation}
From Corollary \ref{gunb} we get a weak equivalence of commutative augmented $\mathbb{S}$-algebras $h_\mathbb{S}: F_\mathbb{S}(\mathbb{S}igma H\mathbb{Q}) \to \mathbb{S}\vee \mathbb{S}igma H\mathbb{Q}$, and it is readily verified that the following diagram commutes:
\[\xymatrix@C+1pc{R\wedge F_\mathbb{S}(\mathbb{S}igma H\mathbb{Q}) \ar[r]^-{\mathrm{id} \wedge h_\mathbb{S}} & R \wedge (\mathbb{S} \vee \mathbb{S}igma H\mathbb{Q}) \ar[d]_-\cong\\ F_R(\mathbb{S}igma R_\mathbb{Q}) \ar[u]^-\cong \ar[r]_-h & R \vee \mathbb{S}igma R_\mathbb{Q}.}\]
Therefore, $h$ is a weak equivalence if and only if $\mathrm{id} \wedge h_\mathbb{S}$ is a weak equivalence, which follows from an application of Lemma \ref{cofcofsemi}.
\end{proof}
\begin{obs} \label{remark-i} Let $i:R\vee \mathbb{S}igma R_\mathbb{Q}\to F_R(\mathbb{S}igma R_\mathbb{Q})$ be the $R$-module map given by the wedge inclusion. By construction of $h$, we have that $h\circ i=\mathrm{id}$. In particular, $i$ is a weak equivalence.
\end{obs}
\begin{proof}[Proof of Theorem \ref{thhku2}] The map $h$ is the one gotten in Proposition \ref{split} for the case $R=KU$. We now aim to establish the equivalence of commutative $KU$-algebras $\tilde f:F(\mathbb{S}igma KU_\mathbb{Q})\to THH(KU)$. First, we work additively, and then we will determine the multiplicative structure.
Recall that for any well-based space $X$, there is a homotopy equivalence of based spaces $\mathbb{S}igma(X_+)\simeq S^1 \vee \mathbb{S}igma X$.
It makes the following diagram of based spaces commute:
\[ \xymatrix{& S^1 \ar[ld] \ar[rd]^-{i_1} \\ \mathbb{S}igma(X_+) \ar[rr]^-{\simeq} && S^1\vee \mathbb{S}igma X}\]
where the left diagonal map is $\mathbb{S}igma u:\mathbb{S}igma (*_+) \to \mathbb{S}igma(X_+)$; here $u:*\to X$ is the basepoint. The map $i_1$ is the wedge inclusion in the first factor. Applying $\mathbb{S}igma^\infty_1$ (the left adjoint to the 1-st space functor from spectra to based spaces)
gives a homotopy equivalence of $\mathbb{S}$-modules $\mathbb{S}igma^\infty_+ X \simeq \mathbb{S} \vee \mathbb{S}igma^\infty X$, and the previous diagram becomes the following commutative diagram of $\mathbb{S}$-modules:
\begin{enumerate}gin{equation}\label{rew} \xymatrix{& \mathbb{S} \ar[ld]_-e \ar[rd]^-{i_1} \\ \mathbb{S}igma^\infty_+ X \ar[rr]^-{\simeq} && \mathbb{S} \vee \mathbb{S}igma^\infty X.}\end{equation}
Here $i_1$ is the wedge inclusion in the first factor.
Applying this to $X=K(\mathbb{Z},3)$ and combining it with Theorem \ref{thhku1} and Proposition \ref{kurat}, we obtain weak equivalences of $KU$-modules
\begin{enumerate}gin{align}\label{goba} THH(KU) &\simeq KU \wedge \mathbb{S}igma^\infty_+ K(\mathbb{Z},3) \stackrel{\simeq}{\to} KU \wedge (\mathbb{S} \vee \mathbb{S}igma^\infty K(\mathbb{Z},3)) \cong \\ &\cong KU \vee (KU \wedge K(\mathbb{Z},3)) \simeq KU \vee \mathbb{S}igma KU_\mathbb{Q}. \nonumber \end{align}
Note that each of the $KU$-modules in that chain has a map of $\mathbb{S}$-modules from $KU$, namely: $\eta: KU\to THH(KU)$ is the unit, $KU\to KU\wedge \mathbb{S}igma^\infty_+ K(\mathbb{Z},3)$ is $\mathrm{id}\wedge e$, $KU\to KU\wedge (\mathbb{S} \vee \mathbb{S}igma^\infty K(\mathbb{Z},3))$ is $\mathrm{id}\wedge i_1$, $KU\to KU \vee (KU \wedge K(\mathbb{Z},3))$ is the inclusion in the first factor and the same goes for $KU \vee \mathbb{S}igma KU_\mathbb{Q}$. The weak equivalences above are compatible with these maps: the first one because it is a zig-zag of weak equivalences of $KU$-algebras, then we use the commutativity of (\ref{rew}), and then it follows from an inspection of how the distributivity isomorphism works.
In the homotopy category of $KU\mbox{-}\mathbf{Mod}$, we consider the map $\mathbb{S}igma KU_\mathbb{Q}\to THH(KU)$ which is the inclusion into $KU \vee \mathbb{S}igma KU_\mathbb{Q}$ followed by the isomorphism obtained from (\ref{goba}). Since $\mathbb{S}igma KU_\mathbb{Q}$ is a cofibrant $KU$-module, we can represent this map by a morphism of $KU$-modules
\begin{enumerate}gin{equation}\label{defff} f:\mathbb{S}igma KU_\mathbb{Q}\to THH(KU).\end{equation}
After passing to the homotopy category of $KU$-modules, the map of $KU$-modules \begin{enumerate}gin{equation}\label{etaf} (\eta,f):KU \vee \mathbb{S}igma KU_\mathbb{Q}\to THH(KU)\end{equation}
coincides with the isomorphism obtained from (\ref{goba}), by construction and by the remarks right after (\ref{goba}).
In particular, $(\eta,f)$ is a weak equivalence of $KU$-modules.
The morphism of $KU$-modules $f$ induces a map of commutative $KU$-algebras
\[\tilde f:F(\mathbb{S}igma KU_\mathbb{Q}) \to THH(KU).\]
To see that it is a weak equivalence, note that, by definition of $\tilde f$, it is such that $\tilde f\circ i=(\eta,f)$, where the weak equivalence of $KU$-modules $i:KU \vee \mathbb{S}igma KU_\mathbb{Q}\to F(\mathbb{S}igma KU_\mathbb{Q})$ was introduced in Remark \ref{remark-i}.
Since $(\eta,f)$ and $i$ are weak equivalences, then $\tilde f$ is a weak equivalence, too.
\end{proof}
\begin{obs} \label{augment-final} In this remark, we prove that $\tilde f:F(\mathbb{S}igma KU_\mathbb{Q})\to THH(KU)$ is compatible with the augmentations in a sense to be made explicit below.
Each of the steps in the zig-zag of weak equivalences of $KU$-modules (\ref{goba}) between $KU\wedge K(\mathbb{Z},3)_+$ and $KU \vee \mathbb{S}igma KU_\mathbb{Q}$ is augmented over $KU$ and the maps in the zig-zag commute with the augmentations. Here $KU \wedge K(\mathbb{Z},3)_+$ has the augmentation over $KU$ given by $\mathrm{id} \wedge \varepsilon \mbox{-}ilon$, as in the last part of Remark \ref{augment-complicado}, and $KU\vee \mathbb{S}igma KU_\mathbb{Q}$ has the augmentation over $KU$ given by $\mathrm{id} \vee *$, where $*$ denotes the trivial map. As for the intermediate steps, $KU\wedge (\mathbb{S}\vee \mathbb{S}igma^\infty K(\mathbb{Z},3))$ is augmented by $\mathrm{id} \wedge (\mathrm{id} \vee *)$, and $KU\vee (KU \wedge K(\mathbb{Z},3))$ is augmented by $\mathrm{id} \vee (\mathrm{id} \wedge *)=\mathrm{id}\vee *$. The only non-trivial part in this verification is in the first step, the one obtained from the homotopy equivalence $\mathbb{S}igma^\infty_+ K(\mathbb{Z},3) \simeq \mathbb{S} \vee \mathbb{S}igma^\infty K(\mathbb{Z},3)$ used in (\ref{rew}). More generally, for any well-based space $X$ the following diagram commutes:
\[\xymatrix{\mathbb{S}igma^\infty_+ X \ar[rr]^-\simeq \ar[rd]_-\varepsilon \mbox{-}ilon && \mathbb{S} \vee \mathbb{S}igma^\infty X \ar[ld]^-{\mathrm{id}\vee *} \\ & \mathbb{S}}\]
where $\varepsilon \mbox{-}ilon:\mathbb{S}igma^\infty_+ X \to \mathbb{S}$ is obtained from $X\to *$. This follows from the commutativity of the following diagram of based spaces:
\[\xymatrix{\mathbb{S}igma(X_+) \ar[rr]^-{\simeq} \ar[rd] && S^1\vee \mathbb{S}igma X \ar[ld] \\ & S^1}\]
where the left diagonal map is $\mathbb{S}igma(X\to *)_+$ and the right diagonal map is the identity map on $S^1$ and the suspension of the trivial based map $X\to S^0$ on $\mathbb{S}igma X$.
Combining this with Remark \ref{augment-complicado}, we obtain a ``diagram''
\[\xymatrix{THH(KU) \ar@{-}[r]^-\sim \ar[d] & KU \wedge \mathbb{S}igma^\infty_+ K(\mathbb{Z},3) \ar@{-}[r]^-\sim \ar[d] & KU \vee \mathbb{S}igma KU_\mathbb{Q} \ar[ld]^-{\mathrm{id} \vee *} \\ KU \ar@{-}[r]^-\sim & KU }\]
understood to mean that the left square is actually a ladder diagram of commutative $KU$-algebras and the right triangle is a ladder diagram (with the lower side collapsed to $KU$) of $KU$-modules. The triangle commutes (in the sense that all the triangles it hides commute), and the square commutes up to a homotopy of commutative $KU$-algebras. Similarly to how we constructed $f:\mathbb{S}igma KU_\mathbb{Q}\to KU$, we can obtain a morphism of commutative $KU$-algebras $g:KU\to KU$ such that the following diagram is homotopy commutative in $KU$-modules:
\[\xymatrix{THH(KU) \ar[d] & KU \vee \mathbb{S}igma KU_\mathbb{Q} \ar[l]_-{(\eta,f)}^-\sim \ar[d]^-{\mathrm{id}\vee *} \\ KU & KU. \ar[l]^-g_-\sim}\]
Precomposing with the canonical map into the second factor $\mathbb{S}igma KU_\mathbb{Q} \to KU \vee \mathbb{S}igma KU_\mathbb{Q}$, we obtain the homotopy commutative diagram in $KU$-modules:
\[\xymatrix{THH(KU) \ar[d] & \mathbb{S}igma KU_\mathbb{Q} \ar[l]_-{f}^-\sim \ar[d]^-{*} \\ KU & KU. \ar[l]^-g_-\sim}\]
Since $F$ is left Quillen and $\mathbb{S}igma KU_\mathbb{Q}$ is a cofibrant $KU$-module, this implies that the diagram
\begin{enumerate}gin{equation}\label{terf} \xymatrix{THH(KU) \ar[d] & F(\mathbb{S}igma KU_\mathbb{Q}) \ar[l]_-{\tilde f}^-\sim \ar[d] \\ KU & KU. \ar[l]^-g_-\sim}\end{equation}
is homotopy commutative in $KU\mbox{-}\mathbf{CAlg}$, by an application of \cite[8.5.16]{hirschhorn}.
\end{obs}
\subsection{The morphism \texorpdfstring{$\sigma$}{sigma}} If $R$ is a commutative $\mathbb{S}$-algebra, there is a natural transformation of $\mathbb{S}$-modules \cite[Section 3]{mc-st}, \cite[IX.3.8]{ekmm}, \cite[3.12]{angeltveit-rognes}\[\sigma: \mathbb{S}igma R\to THH(R).\]
Consider the map of $\mathbb{S}$-modules
\[(\eta,\sigma): KU \vee \mathbb{S}igma KU \to THH(KU).\]
It is tempting to conjecture that its rationalization
\[(\eta_\mathbb{Q},\sigma_\mathbb{Q}):KU_\mathbb{Q} \vee \mathbb{S}igma KU_\mathbb{Q} \to THH(KU)_\mathbb{Q}\]
is a weak equivalence, since by the results of the previous section, the $\mathbb{S}$-modules $KU_\mathbb{Q} \vee \mathbb{S}igma KU_\mathbb{Q}$ and $THH(KU)_\mathbb{Q}$ are weakly equivalent.
However, this is not the case. I thank Geoffroy Horel and Thomas Nikolaus for pointing out this fact and the following proof to me. We will prove that $\sigma:\mathbb{S}igma KU \to THH(KU)$ is zero in $\pi_1$, therefore it is still zero after rationalization.
By naturality of $\sigma$, we have a commutative diagram
\begin{enumerate}gin{equation}\label{cuadku}\xymatrix{\mathbb{S}igma \mathbb{S} \ar[r]^-\sigma \ar[d]_{\mathbb{S}igma \iota} &THH(\mathbb{S}) \simeq \mathbb{S} \ar[d]^-{THH(\iota)} \\ \mathbb{S}igma KU \ar[r]_-\sigma & THH(KU)}\end{equation}
where $\iota:\mathbb{S}\to KU$ is the unit of $KU$. After taking $\pi_1$, we obtain a commutative diagram of abelian groups
\begin{enumerate}gin{equation}\label{cuad}\xymatrix{\mathbb{Z} \ar[r] \ar[d]_-\mathrm{id} & \mathbb{Z}/2 \ar[d] \\ \mathbb{Z} \ar[r] & \mathbb{Q}.}\end{equation}
Therefore, $\mathbb{Z}\to \mathbb{Q}$ must be the zero map, since only the abelian group map $\mathbb{Z}/2\to \mathbb{Q}$ is the zero map.
Note that the same proof works for $L$ (the $p$-adic completion of the Adams summand of $KU$, $p$ a prime) instead of $KU$. Recall that $\pi_*L \cong \mathbb{Z}_{(p)}[(v_1)^{\pm 1}]$, with $v_1$ in degree $2p-2$.
After replacing $KU$ with $L$ in (\ref{cuadku}) and taking $\pi_1$, we obtain a square which looks like (\ref{cuad}) except with a $\mathbb{Z}_{(p)}$ on the lower left corner. The vertical map $\mathbb{Z}\to \mathbb{Z}_{(p)}$ is the unit of $\mathbb{Z}_{(p)}$: this still forces $\pi_1\sigma: \pi_1(\mathbb{S}igma L)\to \pi_1(THH(L))$ to be zero. \index{L}
This corrects an error in \cite[8.4]{mc-st} where it is claimed that there is a weak equivalence $L_\mathbb{Q} \vee \mathbb{S}igma L_\mathbb{Q} \stackrel{\sim}{\to} THH(L)_\mathbb{Q}$ induced by $(\eta,\sigma)$. As a positive result, we have Theorem \ref{thhku2} and the weak equivalence (\ref{etaf}) instead.
\section{Iterated topological Hochschild homology of \texorpdfstring{$KU$}{KU}} \label{sect-iterated}
Let $A$ be a commutative $\mathbb{S}$-algebra. We denote by $THH^n(A)$ the \emph{iterated topological Hochschild homology of $A$}, i.e. $THH(\dots(THH(A)))$ where $THH$ is applied $n$ times. Other expressions for $THH^n(A)$ include $T^n \otimes A$ or $\mathcal{L}ambda_{T^n}(A)$, where $T^n$ is an $n$-torus and $\mathcal{L}ambda$ is the Loday functor \cite{cdd}. Note that $THH^n(A)$ is an augmented commutative $A$-algebra.
We will now give two different descriptions of $THH^n(KU)$ for $n\geq 2$. The first one, given in Theorem \ref{thhnku1cor}, generalizes Theorem \ref{thhku1} which describes $THH(KU)$ via Eilenberg-Mac Lane spaces. The second one, given in Theorem \ref{thhnkufree}, generalizes Theorem \ref{thhku2} which describes $THH(KU)$ as a free commutative $KU$-algebra on a $KU$-module.
We have also given a description of the commutative $KU$-algebra $THH(KU)$ as the split square-zero extension $KU \vee \mathbb{S}igma KU_\mathbb{Q}$ in Theorem \ref{thhku2}. For $n\geq 2$, $THH^n(KU)$ is not a split square-zero extension of $KU$, as we shall see. However, it is a split extension: we will describe the non-unital commutative algebra structure of the homotopy groups of its augmentation ideal, which is rational as in the $n=1$ case.
\subsection{Description via Eilenberg-Mac Lane spaces} Let $G$ be a CW-complex which is a topological abelian group with unit a 0-cell and a cellular addition map. Applying Lemma \ref{sinloc} and Proposition \ref{Smono}, we obtain isomorphisms of commutative $\mathbb{S}[G]$-algebras:
\begin{enumerate}gin{align*}THH^2(\mathbb{S}[G])& \cong THH(\mathbb{S}[G] \wedge \mathbb{S}[BG]) \cong THH(\mathbb{S}[G\times BG]) \\ & \cong \mathbb{S}[G\times BG] \wedge \mathbb{S}[B(G\times BG)] \cong \mathbb{S}[G] \wedge \mathbb{S}[BG \times BG \times B^2G] \end{align*}
which we have written as $\mathbb{S}[G][BG \times BG \times B^2G]$.
For general $n\geq 2$, the same type of computation gives a description of $THH^n(\mathbb{S}[G])$: we obtain an isomorphism of commutative $\mathbb{S}[G]$-algebras
\begin{enumerate}gin{equation}\label{ane}THH^n(\mathbb{S}[G]) \cong \mathbb{S}[G][B^{a_1}G\times \dots\times B^{a_{2^n-1}}G].\end{equation} The numbers $a_i$ can be described as follows. Let $v_0=0$. Define by induction \begin{enumerate}gin{equation}\label{defai} v_n=(v_{n-1},v_{n-1}+(1,\dots,1)) = (a_0, \dots,a_{2^n-1})\in \mathbb{N}^{2^n}\end{equation}
for $n\geq 1$. For example, $v_1=(0,1)$, $v_2=(0,1,1,2)$ and $v_3=(0,1,1,2,1,2,2,3)$.
This sequence of integers can be found in the On-Line Encyclopedia of Integer Sequences \cite{oeis}. We can give an easier description. Let $I_n$ be the multiset having as elements the numbers $i$ with multiplicity ${n\choose i}$, for $i=1,\dots,n$. Denote the multiplicity of an element $x$ of a multiset by $|x|$. Now note that the multiset underlying the sequence $(a_1,\dots,a_{2^n-1})$ defined in $(\ref{defai})$ coincides with $I_n$, by Pascal's rule.
Therefore, the isomorphism (\ref{ane}) can be reformulated as
\begin{enumerate}gin{equation}\label{thhnsg-ref} THH^n(\mathbb{S}[G])\cong \mathbb{S}[G]\left[\prod\limits_{i=1}^n (B^iG)^{\times {n \choose i}}\right].\end{equation}
The following theorem generalizes Theorem \ref{teoloc} to higher iterations of $THH$.
\begin{teo} \label{thhnloc} Let $x\in \pi_*\mathbb{S}[G]$ and $n\geq 1$. There is a zig-zag of weak equivalences of commutative $\mathbb{S}[G][x^{-1}]_h$-algebras
\begin{enumerate}gin{equation}\label{gtre}THH^n(\mathbb{S}[G][x^{-1}]_h) \simeq \mathbb{S}[G][x^{-1}]_h[B^{a_1}G\times \dots\times B^{a_{2^n-1}}G],\end{equation}
or alternatively,
\begin{enumerate}gin{equation}\label{gtre2}THH^n(\mathbb{S}[G][x^{-1}]_h)\simeq \mathbb{S}[G][x^{-1}]_h\left[\prod\limits_{i=1}^n (B^iG)^{\times {n \choose i}}\right].\end{equation}
\begin{proof} The proof is by induction. The base case is Theorem \ref{teoloc}. We do the induction step for $n=2$ for simplicity: for higher $n$ it is analogous, only more cumbersome to write down.
We use the notations introduced for the proof of Theorem \ref{teoloc}, namely $A=\mathbb{S}[G]$, $B=\mathbb{S}[BG]$, $Q$ is the cofibrant replacement functor in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ obtained from a given functorial factorization in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$, and $Q_A$ is the cofibrant replacement functor in $QA\mbox{-}\mathbf{CAlg}$ obtained by factoring the unit of a $QA$-commutative algebra in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$. From the proof of that theorem, we obtain a zig-zag of weak equivalences of cofibrant commutative $QA[x^{-1}]$-algebras which we follow by an isomorphism gotten by applying Proposition \ref{Smono}:
\begin{enumerate}gin{equation}\label{gohj} THH(QA[x^{-1}]) \simeq Q_A(A\wedge B)[(x\wedge 1)^{-1}] \cong Q_A \mathbb{S}[G \times BG][(x\times 1)^{-1}].\end{equation}
We apply $THH$ to obtain weak equivalences of commutative $QA[x^{-1}]$-algebras
\begin{enumerate}gin{align*}THH^2(QA[x^{-1}]) &\simeq THH(Q_A\mathbb{S}[G\times BG][(x\times 1)^{-1}]) \\
&\simeq (Q_A \mathbb{S}[G \times BG])[(x\times 1)^{-1}] \wedge Q\mathbb{S}[B(G\times BG)].\end{align*}
The second line comes from the zig-zag of weak equivalences obtained as follows. First, we observe that if $C$ is a $QA$-commutative algebra, then $Q_AC\to C$ defines a functorial cofibrant replacement of $C$ in the category of commutative $\mathbb{S}$-algebras.
We apply this remark to $C=\mathbb{S}[G\times BG]$. As a consequence, analogous steps to the ones taken in the proof of Theorem \ref{teoloc} from the beginning up to (\ref{vers}) apply \emph{mutatis mutandis} and get us the result.
We continue:
\begin{enumerate}gin{align*}
(Q_A \mathbb{S}[G \times BG])[(x\times 1)^{-1}] \wedge Q\mathbb{S}[B(G\times BG)] &\cong Q_A(A\wedge B)[(x\wedge 1)^{-1}] \wedge Q\mathbb{S}[BG\times B^2G] \\
&\simeq QA[x^{-1}]\wedge QB \wedge Q\mathbb{S}[BG\times B^2G]\\
&\simeq (QA[x^{-1}])[BG\times BG \times B^2G].
\end{align*}
Here, the first step is gotten from the isomorphism in (\ref{gohj}). In the course of the proof of Theorem \ref{teoloc} we obtained a zig-zag of weak equivalences of commutative $QA[x^{-1}]$-algebras $Q_A(A\wedge B)[(x\wedge 1)^{-1}] \simeq QA[x^{-1}]\wedge QB$. The steps in this zig-zag are all cofibrant commutative $\mathbb{S}$-algebras and $Q\mathbb{S}[BG\times B^2G]$ is a cofibrant commutative $\mathbb{S}$-algebra: this explains the second step. The third step is an application of Lemma \ref{cofcofsemi} together with the isomorphism from Proposition \ref{Smono}.
\end{proof}
\end{teo}
\begin{obs} \label{augment-complicado2} As in Remark \ref{augment-complicado}, the zig-zag in the previous theorem is compatible up to homotopy of commutative $QA[x^{-1}]$-algebras with the augmentations, where the right-hand side of (\ref{gtre}) is augmented by means of the map $B^{a_1}G\times \dots\times B^{a_{2^n-1}}G\to *$ and similarly in (\ref{gtre2}).
\end{obs}
As a corollary, we obtain:
\begin{teo} \label{thhnku1cor}
There is a zig-zag of weak equivalences of commutative $KU$-algebras
\begin{enumerate}gin{equation}\label{thhnku1}THH^n(KU)\simeq KU[K(\mathbb{Z},a_1+2) \times \dots \times K(\mathbb{Z},a_{2^n-1}+2)],\end{equation}
or alternatively,
\begin{enumerate}gin{equation}\label{thhnku2}THH^n(KU)\simeq KU\left[\prod\limits_{i=1}^n K(\mathbb{Z},i+2)^{\times {n \choose i}}\right].\end{equation}
\end{teo}
For example,
\begin{enumerate}gin{equation}\label{thh2kua} THH^2(KU)\simeq KU[K(\mathbb{Z},3)\times K(\mathbb{Z},3)\times K(\mathbb{Z},4)].\end{equation}
The previous theorem generalizes the expression of Theorem \ref{thhku1} for $THH(KU)$ as \linebreak $KU[K(\mathbb{Z},3)]$ to $THH^n(KU)$. We can also generalize the expression for $THH(KU)$ as $F(\mathbb{S}igma KU_\mathbb{Q})$ of Theorem \ref{thhku2}. Note that the proof uses results from Section \ref{section:snku} below.
\begin{teo} \label{thhnkufree} Let $n\geq 2$. There is a zig-zag of weak equivalences of commutative $KU$-algebras
\[F\left(\begin{itemize}gvee\limits_{i=1}^n (S^i)^{\vee {n\choose i}} \wedge KU_\mathbb{Q}\right)
\simeq THH^n(KU).\]
\begin{proof} Since $- \wedge KU_\mathbb{Q}:\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\textbf{Mod}}\to KU\mbox{-}\mathbf{Mod}$ and $F:KU\mbox{-}\mathbf{Mod}\to KU\mbox{-}\mathbf{CAlg}$ are left adjoints, they preserve coproducts, so:
\[F\left(\begin{itemize}gvee\limits_{i=1}^n (S^i)^{\vee {n\choose i}} \wedge KU_\mathbb{Q}\right)
\cong F\left(\begin{itemize}gvee_{i=1}^n (\mathbb{S}igma^i KU_\mathbb{Q})^{\vee {n\choose i} } \right) \cong \begin{itemize}gwedge_{\substack{KU \\ i=1}}^n F(\mathbb{S}igma^i KU_\mathbb{Q})^{\wedge_{KU} {n\choose i}}.\]
From (\ref{snfree}), we obtain a zig-zag of weak equivalences of commutative $KU$-algebras
\[\begin{itemize}gwedge_{\substack{KU \\ i=1}}^n F(\mathbb{S}igma^i KU_\mathbb{Q})^{\wedge_{KU} {n \choose i}} \simeq \begin{itemize}gwedge_{\substack{KU \\ i=1}}^n (S^i\otimes KU)^{\wedge_{KU} {n \choose i}}. \]
Using Theorem \ref{holo},
\[\begin{itemize}gwedge_{\substack{KU \\ i=1}}^n (S^i\otimes KU)^{\wedge_{KU} {n \choose i}} \simeq \begin{itemize}gwedge_{\substack{KU \\ i=1}}^n KU[K(\mathbb{Z},i+2)]^{\wedge_{KU} {n \choose i}} \cong KU\left[ \prod_{i=1}^n K(\mathbb{Z},i+2)^{\times {n\choose i}}\right]\]
which is weakly equivalent to $THH^n(KU)$ by Theorem \ref{thhnku1cor}.
\end{proof}
\end{teo}
\begin{obs} \label{stabeq} We might be tempted to prove the previous theorem more directly, arguing from the weak equivalence of based spaces
\begin{enumerate}gin{equation}\label{stabletorus}\mathbb{S}igma T^n\simeq \mathbb{S}igma \begin{itemize}gvee\limits_{i=1}^n (S^i)^{\vee {n \choose i}}.\end{equation} However, we do not know a priori whether this guarantees that $THH^n(KU)= T^n\otimes KU$ is weakly equivalent to $\left( \begin{itemize}gvee\limits_{i=1}^n (S^i)^{\vee {n \choose i}}\right) \otimes KU$ (which we can easily compute using the description from Theorem \ref{holo} of $S^i \otimes KU$ for all $i\geq 1$ and the fact that $-\otimes KU$ preserves coproducts). Indeed, there are counterexamples to the statement that if $A$ is a commutative $\mathbb{S}$-algebra, then $X\otimes A \simeq Y\otimes A$ provided $\mathbb{S}igma X \simeq \mathbb{S}igma Y$ \cite{dundas-tenti}. After having proved the theorem, though, we have that $KU$ does satisfy this for the special case of (\ref{stabletorus}). The partial results of \cite{veen} (extended by \cite{blprz}) prove that $A=H\mathbb{F}_p$ also satisfy it for (\ref{stabletorus}), at least in a certain range relating $n$ and $p$.
We are led to ask ourselves the question, as \cite[4.1]{dundas-tenti} did for $A=H\mathbb{F}_p$, of whether more generally $KU$ is such that $X\otimes KU \simeq Y\otimes KU$ provided $\mathbb{S}igma X\simeq \mathbb{S}igma Y$. More ambitiously, it would be interesting to find conditions on any commutative $\mathbb{S}$-algebra $A$ that guarantee this property.\end{obs}
\subsection{The augmentation ideal} \label{second-desc}In this section, we investigate the augmentation ideal of \linebreak $THH^n(KU)$. Let us define this concept.
Let $R$ be a commutative $\mathbb{S}$-algebra and $A$ be a commutative $R$-algebra with augmentation $\varepsilon \mbox{-}ilon:A\to R$. Denote by $\overline A$ the fiber of $\varepsilon \mbox{-}ilon$, i.e. it is the $R$-module obtained as the pullback
\[\xymatrix{\overline A \ar[d] \ar[r]^-i & A \ar[d]^-\varepsilon \mbox{-}ilon \\ {*} \ar[r] & R}\]
in $\mathbb{R}Mod$. It gets a non-unital multiplication from the universal property of pullbacks, by considering the following commutative diagram in $\mathbb{R}Mod$. See \cite[Section 2]{basterra} for further elaboration.
\[\xymatrix{\overline A \wedge_R \overline A \ar[d] \ar[r]^{i\wedge i} & A \wedge_R A \ar[d]^-{\varepsilon \mbox{-}ilon \wedge \varepsilon \mbox{-}ilon} \ar[r]^-\mu & A \ar[d]^-\varepsilon \mbox{-}ilon \\ {*} \ar[r] & R \wedge_R R \ar[r]_-\cong & R}\]
Consider the augmentation $\varepsilon \mbox{-}ilon:THH^n(KU)\to KU$. To ensure that we compute its \emph{homotopy} fiber, we first replace $\varepsilon \mbox{-}ilon$ by a fibration in the category of commutative $KU$-algebras, i.e. we replace $\varepsilon \mbox{-}ilon$ by the fibration appearing in its factorization by an acyclic cofibration followed by a fibration. We denote the fiber of this new fibration by $\overline{THH}^n(KU)$.
We first need a generalization of Proposition \ref{kurat}:
\begin{enumerate}gin{prop} \label{lemgen} Let $r\geq 3$. There are zig-zags of weak equivalences of $KU$-modules
\[KU \wedge K(\mathbb{Z},r) \simeq \begin{enumerate}gin{cases} \mathbb{S}igma KU_\mathbb{Q} & \text{if } r \text{ is odd}, \\ \begin{itemize}gvee\limits_{m\geq 1} KU_\mathbb{Q} & \text{if } r \text{ is even}.\end{cases}\]
\begin{proof} When $r$ is odd, the proof of Proposition \ref{kurat} works just as well, and when $r$ is even it gives us
\[KU \wedge K(\mathbb{Z},r) \simeq KU_\mathbb{Q} \wedge K(\mathbb{Z},r)_\mathbb{Q}.\]
So let $r$ be even. As noted in Section \ref{rationalization}, $K(\mathbb{Z},r)_\mathbb{Q} \simeq \Omega S^{r+1}_\mathbb{Q}$. Now we use the James splitting which says that, for $X$ a connected based CW-complex, $\mathbb{S}igma \Omega \mathbb{S}igma X \simeq \mathbb{S}igma \begin{itemize}gvee_{m\geq 1} X^{\wedge m}$.
Therefore,
$\mathbb{S}igma^\infty \Omega \mathbb{S}igma X \simeq \mathbb{S}igma^\infty \begin{itemize}gvee_{m\geq 1} X^{\wedge m}$. Rationalizing it and applying it to $X=S^r$, we obtain
\[\mathbb{S}igma^\infty K(\mathbb{Z},r)_\mathbb{Q} \simeq \mathbb{S}igma^\infty \Omega S^{r+1}_\mathbb{Q} \simeq \mathbb{S}igma^\infty \begin{itemize}gvee_{m\geq 1} S^{rm}_\mathbb{Q}.\]
Since $r$ is even, Bott periodicity gives the result.
\end{proof}
\end{prop}
\begin{corolario} \label{hocs} The augmentation ideal $\overline{THH}^n(KU)$ is rational.
\begin{proof} The expression (\ref{thhnku1}) gives, after splitting off the units of the spherical group rings like in (\ref{goba}), a zig-zag of weak equivalences of $KU$-modules
\begin{enumerate}gin{equation} \label{kol} THH^n(KU) \simeq KU \wedge (\mathbb{S} \vee \mathbb{S}igma^\infty K(\mathbb{Z},a_1+2)) \wedge \dots \wedge (\mathbb{S} \vee \mathbb{S}igma^\infty K(\mathbb{Z},a_{2^n-1}+2)). \end{equation}
Distributing the terms in the right-hand side and applying Proposition \ref{lemgen} proves that the homotopy fiber of the augmentation of the right-hand side is rational.
Since the zig-zag is compatible up to homotopy with the augmentations (Remark \ref{augment-complicado2}), this implies that the homotopy fibers are weakly equivalent, so in particular $\overline{THH}^n(KU)$ is rational.
\end{proof}
\end{corolario}
Recall that $H\mathbb{Q}$ is a homotopy commutative ring spectrum whose multiplication map is a weak equivalence, and it is a cofibrant $\mathbb{S}$-module. In particular, $KU_\mathbb{Q}$ is a homotopy commutative ring spectrum, and from (\ref{thhnku1}) we obtain a zig-zag of weak equivalences of commutative $KU_\mathbb{Q}$-ring spectra:
\[THH^n(KU)_\mathbb{Q} \simeq KU_\mathbb{Q} \wedge K(\mathbb{Q},a_1+2)_+ \wedge \dots \wedge K(\mathbb{Q},a_{2^n-1}+2)_+.\]
By using the identification of the rationalized Eilenberg-Mac Lane spaces of Section \ref{rationalization} and the computation of the rational homology of loop spaces of odd-dimensional spheres \cite[Page 225]{fht-rational}, we obtain
\begin{prop} There is an isomorphism of commutative $\mathbb{Q}[t^{\pm 1}]$-algebras
\begin{enumerate}gin{equation} \label{thhnkuq} H\mathbb{Q}_*(THH^n(KU)) \cong \mathbb{Q}[t^{\pm1}] \otimes \begin{itemize}gotimes_{a_i \textup{ odd}} E(\sigma^it) \otimes \begin{itemize}gotimes_{a_j \textup{ even}} \mathbb{Q}[\sigma^j t] \end{equation}
where $|\sigma^rt|=a_r+2$ and $i,j \in\{1,\dots, 2^n-1\}$. \end{prop}
For example,
\[H\mathbb{Q}_*(THH^2(KU)) \cong \mathbb{Q}[t^{\pm 1}] \otimes E(\sigma t) \otimes E(\sigma t) \otimes \mathbb{Q}[\sigma^2t]\]
with $|\sigma t|=3$ and $|\sigma^2t|=4$.\\
We can recognize the right-hand side of the expression (\ref{thhnkuq}) as an iterated Hochschild homology algebra: \index{Hochschild homology!iterated}
\begin{enumerate}gin{equation} H\mathbb{Q}_*(THH^n(KU)) \cong HH^{\mathbb{Q},n}_*(\mathbb{Q}[t^{\pm 1}]).\end{equation}
Indeed, $HH^\mathbb{Q}_*(\mathbb{Q}[t^{\pm1}])\cong \mathbb{Q}[t^{\pm 1}] \otimes E(\sigma t)$, and $HH^\mathbb{Q}_*(E(\sigma t))\cong E(\sigma t)\otimes \mathbb{Q}[\sigma^2t]$. These Hochschild homology calculations are classical and can be found e.g. in \cite[Section 2]{mc-st} and \cite[2.4]{angeltveit-rognes}. We use that localization commutes with Hochschild homology \cite[Theorem 9.1.8(3)]{weibel}. Also note that in general, the Hochschild homology of an exterior algebra is isomorphic to the tensor product of this same exterior algebra with a divided power algebra, but over $\mathbb{Q}$ such algebras are polynomial.\\
Denote by $\overline{HH}_*^{\mathbb{Q},n}(B)$ the kernel of the augmentation $HH_*^{\mathbb{Q},n}(B)\to B$. From these remarks and the proof of Corollary \ref{hocs}, we obtain:
\begin{teo} \label{high-hoch} There is an isomorphism of non-unital commutative $\mathbb{Q}[t^{\pm 1}]$-algebras
\[\overline{THH}^n_*(KU) \cong \overline{HH}_*^{\mathbb{Q},n}(\mathbb{Q}[t^{\pm 1}]).\]
\end{teo}
\section{\texorpdfstring{$\mathbb{S}igma Y\otimes KU$}{Sigma Y tensor KU}} \label{section:snku}
In this section, we evaluate the commutative $KU$-algebra $\mathbb{S}igma Y \otimes KU$ when $Y$ is a based CW-complex, by comparing it with $Y\otimes_{KU}(S^1\otimes KU)$. We are very grateful to Bj{\o}rn Dundas for suggesting this line of argument.\\
Recall that if $R$ is a commutative $\mathbb{S}$-algebra, the category $\mathbb{R}CAlg$ is tensored over $\ensuremath{\mathbf{Top}}$ \cite[VII.2.9]{ekmm}. If $A\in \mathbb{R}CAlg$, then the tensor $S^1\otimes_R A$ is naturally isomorphic to $THH^R(A)$ as a commutative augmented $A$-algebra \cite{mc-schw-vo}, \cite[IX.3.3]{ekmm}, \cite[Section 3]{angeltveit-rognes}. Therefore, in this section we will identify $S^1\otimes_R A$ and $THH^R(A)$ without further notice.
\subsection{The morphism \texorpdfstring{$\nu$}{nu}}
\def\mathcal{C}{\mathcal{C}}
Let $\mathcal{C}$ be a category enriched and tensored over $\ensuremath{\mathbf{Top}}$. Denote its tensor by $\otimes$. Fix a based space $(Z,z_0)$. We denote by $\nu^Z$ the natural transformation
\begin{enumerate}gin{equation}\label{nuu}\xymatrix@C+2pc{\mathcal{C} \rtwocell<5>^{\mathrm{id}}_{Z\otimes -}{\;\;\nu^Z} & \mathcal{C}}\end{equation}
whose component in $C\in \mathcal{C}$ is given by \begin{enumerate}gin{equation}\label{defnu}\nu^Z_C\coloneqq\eta_Z^C(z_0):C\to Z\otimes C.\end{equation} Here $\eta_Z^C:Z\to \mathcal{C}(C,Z\otimes C)$ is the unit at $Z$ of the adjunction
\begin{enumerate}gin{equation} \label{adjo} \xymatrix@C+2pc{\ensuremath{\mathbf{Top}} \ar@/^.8pc/[r]^(.5){-\otimes C} & \mathcal{C}. \ar@/^.8pc/[l]^-{\mathcal{C}(C,-)}}
\end{equation}
Let us now highlight the naturality properties of $\nu^Z_C$ at $C$ and at $Z$. Let $\varphi:C\to C'$ be a morphism in $\mathcal{C}$. The naturality of the isomorphism \[\mathcal{C}(Z\otimes C,Z\otimes -)\cong \ensuremath{\mathbf{Top}}(Z,\mathcal{C}(C,Z\otimes -))\] gives the commutativity of the following diagram.
\begin{enumerate}gin{equation}\label{cuadradonu}\xymatrix{C\ar[d]_-\varphi \ar[r]^-{\nu^Z_C} & Z\otimes C \ar[d]^-{\mathrm{id}\otimes \varphi} \\ C' \ar[r]_-{\nu_{C'}^Z} & Z\otimes C'}\end{equation}
Let $u:Z\to Z'$ be a morphism of based spaces. The naturality of $\eta^C$ gives the commutativity of the following diagram.
\begin{enumerate}gin{equation}\label{nuu2}\xymatrix{C\ar[r]^-{\nu_C^Z} \ar[rd]_-{\nu_C^{Z'}} & Z\otimes C \ar[d]^-{u\otimes \mathrm{id}} \\ & Z'\otimes C }\end{equation}
\begin{enumerate}j The category $\ensuremath{\mathbf{Top}}_*$ is tensored over $\ensuremath{\mathbf{Top}}$: if $X\in \ensuremath{\mathbf{Top}}_*$ and $Y\in \ensuremath{\mathbf{Top}}$, then $Y\otimes X$ is defined as $Y_+\wedge X$. When $(Y,y_0)$ is based, we denote by \begin{enumerate}gin{equation}\label{nyx} n^Y_X:X\to Y_+\wedge X\end{equation} the map $\nu_X^Y$ of (\ref{defnu}) applied to $\mathcal{C}=\ensuremath{\mathbf{Top}}_*$. More explicitely, the map $n^Y_X$ takes $X$ to the copy of $X$ lying over $y_0$ in $Y_+\wedge X$.
\end{enumerate}j
\subsection{In commutative algebras}
Let $R$ be a commutative $\mathbb{S}$-algebra. Let $A$ be a commutative $R$-algebra and $(X,x_0)$ be a based space. The map (\ref{defnu}) in this scenario is a map of commutative $R$-algebras \[\nu^X_A:A\to X\otimes_R A\] which gives $X\otimes_R A$ the structure of a commutative $A$-algebra. In particular, when $X=S^1$, this is the usual structure of an $A$-algebra of $THH^R(A)$.
Now, take $R=\mathbb{S}$ and $A=KU$. Let $(Y,y_0)$ be a based space. We use the symbol $\otimes$ to denote the tensor of $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ over $\ensuremath{\mathbf{Top}}$. Consider the following diagram in $KU\mbox{-}\mathbf{CAlg}$. Here the map $e:S^1\to *$ collapses the circle into its basepoint, and we have identified $F(*\wedge KU_\mathbb{Q})$ and $*\otimes KU$ with $KU$.
\begin{enumerate}gin{equation}\label{diagorra}\xymatrix@C+2pc{
Y\otimes_{KU} F(S^1\wedge KU_\mathbb{Q}) \ar[d]^-{\mathrm{id} \otimes \tilde f}_\sim & F(S^1\wedge KU_\mathbb{Q}) \ar[l]_-{\nu^Y_{F(S^1\wedge KU_\mathbb{Q})}} \ar[d]^-{\tilde f}_\sim \ar[r]^-{F(e\wedge \mathrm{id})} & KU \ar[d]^-g_-\sim \\
Y\otimes_{KU} (S^1\otimes KU) & S^1\otimes KU \ar[l]^-{\nu^Y_{S^1\otimes KU}} \ar[r]_{e\otimes \mathrm{id}} & KU }\end{equation}
The weak equivalence $\tilde f$ comes from Theorem \ref{thhku2}. The map $g:KU\to KU$ comes from (\ref{terf}): in that remark we proved that the right square commutes up to a homotopy of commutative $KU$-algebras. The left square commutes as an application of the commutativity of (\ref{cuadradonu}). Note that $\mathrm{id}\otimes \tilde f$ is a weak equivalence because $Y\otimes_{KU}-$ is a left Quillen functor, assuming $Y$ is a based CW-complex.
We will now identify the members of the left column.
\begin{enumerate}gin{prop} Let $(X,x_0)$ and $(Y,y_0)$ be based spaces, and let $A$ be a commutative $R$-algebra.
\begin{enumerate}
\item There is an isomorphism of commutative $A$-algebras
\[Y \otimes_A (X \otimes_R A) \cong (Y_+ \wedge X) \otimes_R A\]
where $\otimes_R$ (resp. $\otimes_A$) denotes the tensoring of $\mathbb{R}CAlg$ (resp. $\mathcal{A}CAlg$) over $\ensuremath{\mathbf{Top}}$.
Moreover, the isomorphism makes the following diagram in $A\mbox{-}\mathbf{CAlg}$ commute. The morphism $n^Y_X:X\to Y_+\wedge X$ was defined in (\ref{nyx}).
\begin{enumerate}gin{equation}\label{ndiag}\xymatrix@C+1pc{X\otimes_R A \ar[r]^-{\nu^Y_{X\otimes_RA}} \ar[rd]_-{n^Y_X\otimes \mathrm{id}} & Y\otimes_A (X\otimes_R A) \ar[d]^-\cong \\ & (Y_+\wedge X) \otimes_R A}\end{equation}
\item Let $M$ be an $A$-module. Let $F:A\mbox{-}\mathbf{Mod}\to A\mbox{-}\mathbf{CAlg}$ be the free commutative algebra functor. There is an isomorphism
\[Y\otimes_A F(X\wedge M) \cong F(Y_+ \wedge X\wedge M)\]
making the following diagram commute.
\[\xymatrix@C+1pc{F(X\wedge M) \ar[rd]_-{F(n^Y_X \wedge \mathrm{id})} \ar[r]^-{\nu^Y_{F(X\wedge M)}} & Y\otimes_A F(X\wedge M) \ar[d]^-\cong \\ & F(Y_+\wedge X\wedge M)}\]
In the expression $Z\wedge M$ for a based space $Z$ we are using the tensor of $\mathcal{A}Mod$ over $\ensuremath{\mathbf{Top}}_*$.
\end{enumerate}
\begin{proof} (1) Let $B$ be a commutative $A$-algebra with unit $\varphi: A\to B$. Using the defining adjunction for $Y\otimes_A-$, we get a homeomorphism
\begin{enumerate}gin{equation}\label{pro1}A\mbox{-}\mathbf{CAlg}(Y\otimes_A (X \otimes_R A),B) \cong \ensuremath{\mathbf{Top}}(Y,A\mbox{-}\mathbf{CAlg}(X\otimes_R A,B)).\end{equation}
The morphisms of commutative $A$-algebras $X\otimes_R A\to B$ are the morphisms of commutative $R$-algebras $g:X\otimes_R A\to B$ making the following diagram commute:
\[\xymatrix{& A \ar[rd]^-\varphi \ar[ld]_-{\nu_A^{X}} \\ X\otimes_R A \ar[rr]_-g && B.}\]
Recalling the definition of $\nu$, this means that
\begin{enumerate}gin{equation}\label{hoji}g\circ \eta^A_{X}(x_0)=\varphi.\end{equation}
The adjoint map of $g$ by the defining adjunction of $-\otimes_R A$ is the map in $\ensuremath{\mathbf{Top}}$
\begin{enumerate}gin{equation}\label{hojiadj}\xymatrix{X\ar[r]^-{\eta_{X}^A} & \mathbb{R}CAlg(A,X\otimes_R A) \ar[r]^-{g_*} & \mathbb{R}CAlg(A,B).}\end{equation}
Let the space $\mathbb{R}CAlg(A,B)$ be pointed by $\varphi:A\to B$. The condition (\ref{hoji}) on the map $g$ is then translated to the adjoint (\ref{hojiadj}) by stating that it is a based map, i.e. it takes $x_0$ to $\varphi$. Thus, continuing (\ref{pro1}),
\begin{enumerate}gin{equation}\label{prox}\ensuremath{\mathbf{Top}}(Y,\mathcal{A}CAlg(X\otimes_R A,B)) \cong \ensuremath{\mathbf{Top}}(Y,U \ensuremath{\mathbf{Top}}_*(X,\mathbb{R}CAlg(A,B))),\end{equation}
where $U:\ensuremath{\mathbf{Top}}_*\to \ensuremath{\mathbf{Top}}$ is the functor forgetting the basepoint. It is the right adjoint to the functor $(-)_+:\ensuremath{\mathbf{Top}}\to \ensuremath{\mathbf{Top}}_*$ which adds a disjoint basepoint, so we continue:
\[\ensuremath{\mathbf{Top}}(Y,U \ensuremath{\mathbf{Top}}_*(X,\mathbb{R}CAlg(A,B))) \cong U\ensuremath{\mathbf{Top}}_*(Y_+,\ensuremath{\mathbf{Top}}_*(X,\mathbb{R}CAlg(A,B))).\]
Since $\ensuremath{\mathbf{Top}}_*(X,-):\ensuremath{\mathbf{Top}}_*\to \ensuremath{\mathbf{Top}}_*$ is the right adjoint to $-\wedge X$, we get:
\[U\ensuremath{\mathbf{Top}}_*(Y_+,\ensuremath{\mathbf{Top}}_*(X,\mathbb{R}CAlg(A,B))) \cong U\ensuremath{\mathbf{Top}}_*(Y_+\wedge X,\mathbb{R}CAlg(A,B)).\]
By the same argument proving (\ref{prox}), we get
\[U\ensuremath{\mathbf{Top}}_*(Y_+\wedge X,\mathbb{R}CAlg(A,B)) \cong \mathcal{A}CAlg((Y_+ \wedge X) \otimes_R A,B).\]
In conclusion, we have a homeomorphism
\[A\mbox{-}\mathbf{CAlg}(Y\otimes_A (X \otimes_R A),B) \cong \mathcal{A}CAlg((Y_+ \wedge X) \otimes_R A,B),\]
and the Yoneda lemma finishes the proof.
The isomorphism was established using a chain of adjunctions. Following this chain, one observes that both $n^Y_X$ and $\nu_{X\otimes_RA}^Y$, which are defined via units of adjunctions by analogous procedures, make the diagram (\ref{ndiag}) commute.\\
(2) The functor $F$ is defined via a \emph{continuous} monad in $\mathcal{A}Mod$ (i.e. it is enriched over $\ensuremath{\mathbf{Top}}$), see \cite[proof of VII.2.9]{ekmm}. Therefore, the functor $F$ preserves tensors over $\ensuremath{\mathbf{Top}}$, so we get the desired isomorphism.
\end{proof}
\end{prop}
Applying the previous proposition to $R=\mathbb{S}$, $A=KU$, $X=S^1$ and $M=KU_\mathbb{Q}$, the diagram (\ref{diagorra}) can be replaced with the following one.
\begin{enumerate}gin{equation}\label{diagorra2}\xymatrix@C+2pc{
F(Y_+ \wedge S^1\wedge KU_\mathbb{Q}) \ar[d]^-\sim & F(S^1 \wedge KU_\mathbb{Q}) \ar[l]_-{F(n^Y_{S^1}\wedge \mathrm{id})} \ar[r]^-{F(e\wedge \mathrm{id})} \ar[d]^-{\tilde f}_\sim & KU \ar[d]^-g_-\sim \\
(Y_+\wedge S^1) \otimes KU & S^1\otimes KU \ar[l]^-{n^Y_{S^1}\otimes \mathrm{id}} \ar[r]_-{e\otimes \mathrm{id}} & KU
}\end{equation}
We suppose that $Y$ is a based CW-complex, so that the vertical map on the left is a weak equivalence.
Now, note that the following is a pushout square of based or unbased spaces.
\[\xymatrix{S^1\ar[r]^-e \ar[d]_-{n^Y_{S^1}} & \ast \ar[d] \\ Y_+\wedge S^1 \ar[r] & Y\wedge S^1}\]
Since the functors $-\otimes KU:\ensuremath{\mathbf{Top}}\to KU\mbox{-}\mathbf{CAlg}$ and $F(-\wedge KU_\mathbb{Q}):\ensuremath{\mathbf{Top}}_*\to KU\mbox{-}\mathbf{CAlg}$ are left adjoints, they preserve pushouts, so the pushout of the top line of (\ref{diagorra2}) is $F(Y\wedge S^1\wedge KU_\mathbb{Q})$ and the pushout of the bottom line is $(Y\wedge S^1) \otimes KU.$
Now, the three vertical maps of (\ref{diagorra2}) are weak equivalences. The horizontal maps pointing left are cofibrations: indeed, $n^Y_{S^1}$ is a cofibration,
$KU_\mathbb{Q}$ is a cofibrant $KU$-module (similarly as in Remark \ref{f-cof}) so $- \wedge KU_\mathbb{Q}$ is left Quillen,
$F$ is left Quillen and $-\otimes KU$ is left Quillen. Moreover, all the objects in the diagram are cofibrant in $KU\mbox{-}\mathbf{CAlg}$. Since the left square commutes and the right square commutes up to a homotopy of commutative $KU$-algebras, an application of Lemma \ref{hompus} plus the naturality of $n^Y_{S^1}$ in $Y$ (\ref{nuu2}) proves the following
\begin{teo} \label{xku}There is a zig-zag of weak equivalences of commutative $KU$-algebras \[F(Y\wedge S^1\wedge KU_\mathbb{Q}) \simeq (Y\wedge S^1)\otimes KU\]
natural in the based CW-complex $Y$.
\end{teo}
This determines $\mathbb{S}igma Y\otimes KU$ as the free commutative $KU$-algebra on the $KU$-module $\mathbb{S}igma Y \wedge KU_\mathbb{Q}$, up to weak equivalence. In particular, we have a zig-zag of weak equivalences of commutative $KU$-algebras
\begin{enumerate}gin{equation} \label{snfree} F(\mathbb{S}igma^n KU_\mathbb{Q}) \simeq S^n \otimes KU\end{equation}
for every $n\geq 1$.
As in Remark \ref{f-cof}, the $KU$-modules $\mathbb{S}igma^n KU_\mathbb{Q}$ are cofibrant for $n\geq 0$. Since $F$ is a left Quillen functor, Bott periodicity implies that we have zig-zags of weak equivalences of commutative $KU$-algebras
\begin{enumerate}gin{equation}S^n\otimes KU \simeq \begin{enumerate}gin{cases} F(\mathbb{S}igma KU_\mathbb{Q}) & \text{if } n \text{ is odd,} \\ F(KU_\mathbb{Q}) & \text{if } n \text{ is even} \end{cases}\end{equation}
for every $n\geq 1$.
The line (\ref{snfree}) generalizes the expression of Theorem \ref{thhku2} for $THH(KU)$ as the free commutative $KU$-algebra on $\mathbb{S}igma KU_\mathbb{Q}$. The following generalizes the expression of Theorem \ref{thhku1} for $THH(KU)$ via Eilenberg-Mac Lane spaces.
\begin{teo} \label{holo}Let $n\geq 1$. Then $S^n \otimes KU \simeq KU[K(\mathbb{Z},n+2)]$ as commutative $KU$-algebras.
\end{teo}
\begin{proof}
We learned of results similar to the following from \cite{veen}: if $\mathbb{S}\to A\to B$ are cofibrations of commutative $\mathbb{S}$-algebras, then there is a weak equivalence \[S^{n+1}\otimes_A B \stackrel{\sim}{\leftarrow} B^A(B,S^n\otimes_A B,B)\]
where the term on the right side is a two-sided bar construction. Here $\otimes_A$ denotes the tensor of commutative $A$-algebras over $\ensuremath{\mathbf{Top}}$. Let us give a proof. Since the functor $-\otimes_A B$ is left Quillen, it preserves pushouts and cofibrations, so we have a pushout of commutative $A$-algebras where the arrows $S^n\otimes_A B \to D^{n+1}\otimes_A B$ are cofibrations:
\[\xymatrix{S^n\otimes_A B \ar@{>->}[r] \ar@{>->}[d] & D^{n+1}\otimes_A B \ar[d] \\ D^{n+1}\otimes_A B \ar[r] & S^{n+1}\otimes_A B.}\]
Therefore,
\begin{enumerate}gin{align*}
S^{n+1}\otimes_A B & \cong (D^{n+1}\otimes_A B) \wedge_{S^n\otimes_A B} (D^{n+1}\otimes_A B) \\ &\stackrel{\sim}{\leftarrow} B^A(D^{n+1}\otimes_A B,S^n \otimes_A B,D^{n+1}\otimes_A B)\\
&\stackrel{\sim}{\leftarrow} B^A(B, S^n\otimes_A B, B)\end{align*}
where the weak equivalence in the middle is an application of \cite[VII.7.3]{ekmm},
and the last one comes from two applications of \cite[VII.7.2]{ekmm}.
We use this to prove the result by induction. The result is true for $n=1$ (Theorem \ref{thhku1}); suppose it is true for some $n\geq1$. Then
\begin{enumerate}gin{align*}
S^{n+1}\otimes KU &\simeq B^\mathbb{S}(KU,KU[K(\mathbb{Z},n+2)],KU) \\
&\simeq B^\mathbb{S}(KU,KU,KU) \wedge B^\mathbb{S}(\mathbb{S},\mathbb{S}[K(\mathbb{Z},n+2)],\mathbb{S}) \\
&\simeq KU \wedge \mathbb{S}[K(\mathbb{Z},n+3)] = KU[K(\mathbb{Z},n+3)].
\end{align*}
Here we have used that $B^\mathbb{S}(\mathbb{S},\mathbb{S}[G],\mathbb{S})\cong \mathbb{S}[BG]$ for $G$ a topological commutative monoid. This result is proven in the same fashion as Proposition \ref{conmutarTHH}, which deals with the analogous result for the \emph{cyclic} bar construction.
\end{proof}
\begin{obs} \label{x-thom} In Remark \ref{thh-thom} we observed that $KU$ behaves like a Thom spectrum to the eyes of topological Hochschild homology. Comparing Theorems \ref{thhnku1cor} and \ref{holo} with \cite[1.1]{schlichtkrull-higher} or \cite[4.11]{rsv-thom}, we see that, more generally, $KU$ behaves like a Thom spectrum to the eyes of $X\otimes -$ when $X$ is an $n$-torus or an $n$-sphere, $n\geq 1$. See also Remark \ref{jurs}.\ref{taq-ku-thom} for a similar observation about $TAQ$.\end{obs}
\section{Topological André-Quillen homology of \texorpdfstring{$KU$}{KU}} \label{section:taq}
If $A\to B$ is a morphism of commutative $\mathbb{S}$-algebras, one can define its \emph{cotangent complex} $\Omega_{B|A}\in B\mbox{-}\mathbf{Mod}$, also known as its \emph{topological André-Quillen} $B$-module, $TAQ(B|A)$: see \cite{basterra}.
We adopt the latter notation. When $A=\mathbb{S}$, we delete it from the notation.
\begin{teo} \label{taqku1} The $KU$-modules $TAQ(KU)$ and $KU \wedge \mathbf{K(\Z,2)}$ are weakly equivalent.
\end{teo}
Here $\mathbf{K(\Z,2)}$ is the $\mathbb{S}$-module associated to the topological abelian group $K(\mathbb{Z},2)$: it is a model for $\mathbb{S}igma^2 H \mathbb{Z}$.
More generally, as explained in \cite{basterra-mandell} before Theorem 5, for a topological abelian group $G$ there is an $\mathbb{S}$-module associated to $G$ whose zeroth space is $G$. We denote it by $\mathbf{G}$. More generally, we denote by $\mathbf{X}$ the $\mathbb{S}$-module associated to an $E_\infty$-space $X$ whose zeroth space is the group completion of $X$.
In the next proof we will use the localization of a module, which we have not used before. For the purposes of this section, if $R$ is a cofibrant commutative $\mathbb{S}$-algebra, $x\in \pi_*R$ and $M$ is an $R$-module, then we define the $R[x^{-1}]$-module $M[x^{-1}]$ by $R[x^{-1}]\wedge_R M$ \cite[VII.4]{ekmm}.
\begin{proof} Basterra \cite[Proposition 4.2]{basterra} proved that, if $A\to B\to C$ are maps of cofibrant commutative $\mathbb{S}$-algebras, then
\[TAQ(B|A) \wedge_B C \to TAQ(C|A) \to TAQ(C|B)\]
is a homotopy cofiber sequence of $C$-modules. Recall from (\ref{defku}) that we defined $KU$ as $Q\mathbb{S}[K(\mathbb{Z},2)][x^{-1}]$, where $Q$ is a cofibrant replacement functor in $\ensuremath{\mathbb{S}}\mbox{-}\ensuremath{\mathbf{CAlg}}$ and $x\in \pi_2\mathbb{S}[K(\mathbb{Z},2)]$. The following sequence of cofibrant commutative $\mathbb{S}$-algebras
\[\mathbb{S}\to Q\mathbb{S}[K(\mathbb{Z},2)]\to Q\mathbb{S}[K(\mathbb{Z},2)][x^{-1}]\]
begets a homotopy cofiber sequence of $KU$-modules
\begin{enumerate}gin{equation}\label{hocotaq}TAQ(Q\mathbb{S}[K(\mathbb{Z},2)]) \wedge_{Q\mathbb{S}[K(\mathbb{Z},2)]} KU \to TAQ(KU) \to TAQ(KU|Q\mathbb{S}[K(\mathbb{Z},2)]).\end{equation}
Now,
$TAQ(KU|Q\mathbb{S}[K(\mathbb{Z},2)])$ is contractible, since $Q\mathbb{S}[K(\mathbb{Z},2)]\to KU$ is a localization map \cite[Remark 3.4]{mccarthy-minasian}. Since by definition the leftmost factor of (\ref{hocotaq}) is $TAQ(Q\mathbb{S}[K(\mathbb{Z},2)])[x^{-1}]$,
the sequence (\ref{hocotaq}) gives a weak equivalence of $KU$-modules
\begin{enumerate}gin{equation}\label{turka}TAQ(Q\mathbb{S}[K(\mathbb{Z},2)])[x^{-1}]\stackrel{\sim}{\to} TAQ(KU).\end{equation}
But \cite[Theorem 5]{basterra-mandell} gives that if $G$ is a topological abelian group, then the $Q\mathbb{S}[G]$-modules $TAQ(Q\mathbb{S}[G])$ and $Q\mathbb{S}[G] \wedge \mathbf{G}$ are weakly equivalent.
Taking $G=K(\mathbb{Z},2)$, localizing this equivalence at $x$ and combining it with (\ref{turka}), we get weak equivalences of $KU$-modules
\[KU \wedge \mathbf{K(\Z,2)} \simeq TAQ(Q\mathbb{S}[K(\mathbb{Z},2)])[x^{-1}] \stackrel{\sim}{\to} TAQ(KU). \qedhere\]
\end{proof}
We thank the anonymous referee for pointing us in the direction of the proof of the following fact, of which our previous proof was less simple.
\begin{corolario} \label{cortaq} The $KU$-modules $TAQ(KU)$ and $KU_\mathbb{Q}$ are weakly equivalent.
\begin{proof} From Bott periodicity and the comment following the statement of Theorem \ref{taqku1}, we get that $TAQ(KU)$ and $KU\wedge H\mathbb{Z}$ are weakly equivalent. But the map $KU \wedge H\mathbb{Z}\to KU \wedge H\mathbb{Q}$ induced from the inclusion $\mathbb{Z}\subset \mathbb{Q}$ is a weak equivalence \cite[16.25]{switzer}, hence the result.
\end{proof}
\end{corolario}
\begin{obs} \phantomsection \label{jurs} \begin{enumerate}
\item \label{taq-od} The topological André-Quillen $B$-module $TAQ(B|A)$ can be computed as a stabilization, as follows from the work of \cite{basterra-mandell} and as made more explicit e.g. in \cite[Page 164]{schlichtkrull-higher}. More precisely, there is a tower with $\Omega^n (S^n\tilde\otimes_A B)$ in level $n$ whose homotopy colimit is weakly equivalent to $TAQ(B|A)$; here $S^n\tilde\otimes_A B$ is the $B$-module which is the cofiber of the map $B\to S^n\otimes_A B$ given by the inclusion of the basepoint in $S^n$. The symbol $\otimes_A$ denotes the tensor over $\ensuremath{\mathbf{Top}}$ of the category of commutative $A$-algebras. From Theorem \ref{holo} we deduce that $S^n\tilde\otimes KU \simeq KU \wedge K(\mathbb{Z},n+2)$: we have identified these in Proposition \ref{lemgen}. This indicates a different way of computing $TAQ(KU)$.
\item \label{taq-ku-thom} Compare Theorem \ref{taqku1} with the reformulation
found e.g. in \cite[Page 164]{schlichtkrull-higher} of a result of \cite{basterra-mandell}.
It states that if $f:X\to BF$ is a map of $\infty$-loop spaces where $BF$ is a classifying space for stable spherical fibrations, then $TAQ(T(f))\simeq T(f)\wedge \mathbf{X}$. Just as in Remark \ref{x-thom}, the result for $TAQ(KU)$ coincides with the result we would obtain if we knew that $KU$ was somehow the Thom spectrum of a map $K(\mathbb{Z},2)\to BU$.
\item \label{hokr} Consider the version of the Hochschild-Kostant-Rosenberg theorem in \cite[Theorem 1.1]{mccarthy-minasian}: if $A$ is a connective smooth commutative $\mathbb{S}$-algebra, there is a weak equivalence of commutative $A$-algebras $F(\mathbb{S}igma TAQ(A))\stackrel{\sim}{\to}THH(A)$, where $F:A\mbox{-}\mathbf{Mod}\to A\mbox{-}\mathbf{CAlg}$ is the free commutative algebra functor.\footnote{Note, however, that the proof contains a gap: see \cite[Footnote 4]{antieau-vezzosi}. We thank Benjamin Antieau for pointing this out to us.} This statement does not apply to $KU$ since $KU$ is not connective (we have not checked the smoothness condition), but the conclusion is true (Theorem \ref{thhku2} and Corollary \ref{cortaq}). Just as in Remarks \ref{thh-thom}, \ref{x-thom} and the one just above, here is an example of a theorem that does not apply to $KU$ because $KU$ is not connective, but whose conclusion is nonetheless true. We speculate that there should be a version of the HKR theorem for $E_\infty$-ring spectra which dispenses with the connectiveness hypothesis. I would like to thank Tomasz Maszczyk for asking me about the HKR theorem in relation to Theorem \ref{thhku2}, thus inciting me to make these reflections.
\end{enumerate}
\end{obs}
\begin{itemize}bliographystyle{alpha}
\begin{itemize}bliography{../../../INCLUDE/main.bib}
\end{document} |
\begin{document}
\title{Universal Stacky Semistable Reduction}
\maketitle
\begin{abstract}
Given a log smooth morphism $f: X \rightarrow S$ of toroidal embeddings, we perform a Raynaud-Gruson type operation on $f$ to make it flat and with reduced fibers. We do this by studying the geometry of the associated map of cone complexes $C(X) \rightarrow C(S)$. As a consequence, we show that the toroidal part of semistable reduction of Abramovich-Karu can be done in a canonical way.
\end{abstract}
\begin{section}{Introduction}
The semistable reduction theorem of \cite{KKMS} is one of the foundations of the study of compactifications of moduli problems. Roughly, the main result of \cite{KKMS} is that given a flat family $X \rightarrow S = \operatorname{Spec} R$ over a discrete valuation ring, smooth over the generic point of $R$, there exists a finite base change $\operatorname{Spec} R' \rightarrow \operatorname{Spec} R$ and a modification $X'$ of the fiber product $X \times_{\operatorname{Spec} R} \operatorname{Spec} R'$ such that the central fiber of $X'$ is a divisor with normal crossings which is reduced. \\
Extensions of this result to the case where the base of $X \rightarrow S$ has higher dimension are explored in the work \cite{AK} of Abramovich and Karu. Over a higher dimensional base, semistable reduction in the strictest sense is not possible; nevertheless, the authors prove a version of the result, which they call weak semistable reduction. The strongest possible version of semistable reduction for a higher dimensional base was proven recently in \cite{ALT}. In all cases, the statement is proven in two steps. In the first, one reduces to the case where $X \rightarrow S$ is toroidal. This step requires resolution of singularities, and is proven in characteristic $0$. The second step is performing semistable reduction for the toroidal morphism; this is essentially a combinatorial problem, independent of characteristic. In \cite{AK}, this step has the following form: given a morphism $X \rightarrow S$ of projective toroidal embeddings, then one can find an alteration $S' \rightarrow S$ and a modification $X'$ of the normalization of the main component of $X \times_S S'$ which is weakly semistable, rather than semistable -- in other words, such that the family $X' \rightarrow S'$ is flat and has reduced fibers (and where $S'$ is non-singular). \\
The main result of this paper is that if we relax the hypotheses to allow families $X \rightarrow S$ of stacks rather than schemes, the toroidal part of weak semistable reduction can be done ``universally". Specifically, we show
\begin{theorem}[Universal Weak Semistable Reduction]
\label{theorem:main}
Let $X \rightarrow S$ be any proper, surjective, log smooth morphism of toroidal embeddings. Then, there exists a commutative diagram
\begin{align*}
\xymatrix{ \mathcal{X} \ar[r] \ar[d] & X \ar[d] \\ \mathcal{S} \ar[r] & S }
\end{align*}
where $\mathcal{X} \rightarrow \mathcal{S}$ is a representable morphism of tame toroidal algebraic stacks, such that for any diagram
\begin{align*}
\xymatrix{ Y \ar[r] \ar[d] & X \ar[d] \\ T \ar[r] & S}
\end{align*}
where $T \rightarrow S$ a toroidal alteration; $Y$ is a modification of the normalization of the main component of the fiber product $X \times_S T$; and $Y \rightarrow T$ is weakly semistable, the morphism $Y \rightarrow T$ factors uniquely through $\mathcal{X} \rightarrow \mathcal{S}$. Furthermore, $\mathcal{X} \times_\mathcal{S} T \rightarrow T$ is weakly semistable.
\end{theorem}
The construction can be thought of as a logarithmic version of the Raynaud-Gruson flattening theorem. The stack structure ensures that the fibers of the morphism are reduced. However, it is not clear that the steps in our construction can be done separately -- first flattening the map without using stacks, then reducing the fibers. A similar idea of using stacks to perform semistable reduction universally in the case where the base is one dimensional was pursued by Olsson in \cite{Oss}, but our constructions are different.
\end{section}
\begin{section}{The Toric Case}
We begin by studying the toric case first. We do this because the exposition is simpler in this case, yet all the essential ideas and proofs are already present. For simplicity, we work throughout over an algebraically closed field $k$. \\
Let $\mathcal{T}$ denote the category of toric varieties. We may identify a toric variety by a pair $(F,N)$ of a lattice $N$ and a fan $F$ in $N_\mathbf R = N \otimes_\mathbf Z \mathbf R$. We usually denote the toric variety associated to $(F,N)$ by $\mathbb A(F,N)$, and refer to it as the geometric realization of $(F,N)$. We will blur the distinction between $(F,N)$ and $\mathbb A(F,N)$ and refer to either as a toric variety depending on context. A morphism of toric varieties $(F,N) \rightarrow (G,Q)$ is a homomorphism of lattices $p:N \rightarrow Q$, such that $p_\mathbf R:N_\mathbf R \rightarrow Q_\mathbf R$ takes each cone $\sigma \in F$ into a cone $\kappa \in G$. We recall the following notions, stated on the level of fans rather than on the level of varieties:
\begin{defn}
The support $\Supp F$ of a fan $F$ is the set of vectors in $N_\mathbf R$ that belong to some cone in $F$.
\end{defn}
\begin{defn}
A morphism of toric varieties $p:(F,N) \rightarrow (G,Q)$ is called proper if $p^{-1}(\Supp G) = \Supp F$.
\end{defn}
\begin{defn}
A morphism of toric varieties $i:(F',N) \rightarrow (F,N)$ is called a modification or subdivision if $i:N \rightarrow N$ is the identity and $\Supp{F'} = \Supp F$.
\end{defn}
\begin{defn}
A morphism of toric varieties $j:(G',Q') \rightarrow (G,Q)$ is called an alteration if $j:Q' \rightarrow Q$ is a finite index injection and $\Supp {G'} = \Supp G$.
\end{defn}
\begin{rem}
Note that if $j:Q' \rightarrow Q$ is an injection with finite cokernel, then the condition $\Supp {G'} = \Supp G$ is equivalent to saying the morphism $(G',Q') \rightarrow (G,Q)$ is proper. Furthermore, for any homomorphism $j:Q' \rightarrow Q$ we get a ``pull-back" fan $j_{\mathbf R}^{-1}(G)$ which is isomorphic to $G$, since $j_\mathbf R$ is an isomorphism. Thus, any toric alteration can be factored as a modification $(G',Q') \rightarrow (G,Q')$ composed with a finite index inclusion $(G,Q') \rightarrow (G,Q)$.
\end{rem}
\begin{rem}
The pullback fan of remark 2.0.6 is a special case of the following more general construction:
\begin{subsection}{Minimal Modification}
Let $p:N \rightarrow Q$ be a fixed homomorphism of lattices, and suppose $F,G$ are fans in the lattices $N,Q$ respectively.
\begin{lemma}
\label{lem:minmod}
There exists a minimal modification $(F',N)$ of $(F,N)$ which maps to $(G,Q)$ inducing $p:N \rightarrow Q$.
\end{lemma}
\begin{proof}
The fan $F'$ is defined as $F':=\{p^{-1}(\kappa) \cap \sigma: \kappa \in G, \sigma \in F\}$. The proof that this is a fan and satisfies the universal property is straightforward and can be found in \cite{AM}.
\end{proof}
\end{subsection}
\end{rem}
\begin{defn}
\label{def: weaklysemistable}
We call a morphism $p:(F,N) \rightarrow (G,Q)$ of toric varieties \emph{weakly semistable} if
\begin{enumerate}
\item Every cone $\sigma$ in $F$ surjects onto a cone $\kappa \in G$ \\
\item Whenever we have $p(\sigma) = \kappa$, we have an equality of monoids $p(N \cap \sigma) = Q \cap \kappa$
\end{enumerate}
\end{defn}
We remark here that our definition of weak semistability is slightly weaker than the one introduced in \cite{AK}: our base is not required to be smooth. Our definition is for practical purposes very close to the one of \cite{AK}. First, we have
\begin{lemma}
If $p:(F,N) \rightarrow (G,Q)$ is weakly semistable and $F,G$ are smooth, then $p$ is semistable.
\end{lemma}
Next, it is proven in my appendix to \cite{Wmin} that the central fiber of such a morphism resembles the central fiber of a semistable morphism as in \cite{KKMS}, at least in codimension one. Furthermore, it is shown in \cite{AK} that a morphism that satisfies $(1)$ is equidimensional and a morphism that satisfies $(2)$ has reduced fibers. In \cite{AK}, equidimensionality is combined with smoothness of the base to deduce flatness of the morphism. Nevertheless, even over a singular base, we still have:
\begin{theorem}
\label{theorem:flat}
A morphism of toric varieties is weakly semistable if, and only if, it is flat and has reduced fibers.
\end{theorem}
\begin{proof}
To begin, we will show that a weakly semistable morphism is saturated in the terminology of \cite{Ts}. From \cite[Theorem II.4.2]{Ts}, it will then follow that the map is flat with reduced fibers. The statement is local, so we may assume we are in the situation where a single cone $\sigma$ in $N$ maps into a single cone $\kappa$ in $Q$. By assumption, we have that faces of $\sigma$ map onto faces of $\kappa$, and whenever $\tau$ maps onto $\lambda$, we have $N \cap \tau$ mapping onto $\lambda \cap Q$. \\
Consider the dual monoids $Q_\kappa^{\vee}=\kappa^{\vee} \cap Q^{\vee}$ and $N_{\sigma}^{\vee} = \sigma^{\vee} \cap N^{\vee}$ in the dual lattices. Since $N \cap \sigma$ surjects onto $Q \cap \kappa$, the dual map $Q_\kappa^{\vee} \rightarrow N_{\sigma}^{\vee}$ is injective, and its cokernel has no torsion. To see flatness, we will verify that this dual map is an integral map of monoids in the sense of Kato. We use Kato's equational criterion for integrality \cite{K}. Suppose we are given
\begin{align*}
p_1 + q_1 = p_2 + q_2
\end{align*}
\noindent where $p_i \in N_\sigma^{\vee}$ and $q_i \in Q_\kappa^{\vee}$. We want to show that $p_1 = w + r_1$, $p_2 = w + r_2$, where $w \in N_\sigma^{\vee}$, $r_i \in Q_\kappa^{\vee}$, and $q_1+r_1 = q_2 + r_2$. Since the map $Q_\kappa^{\vee} \rightarrow N_{\sigma}^{\vee}$ is injective, we certainly have a (non-canonical) splitting of lattices $N_{\sigma}^{\vee,\textrm{gp}} = Q_{\kappa}^{\vee,\textrm{gp}}\oplus L$. So, we may identify any $p_1,p_2$ with $(w,r_1),(w,r_2)$ and we must have $q_1+r_1=q_2+r_2$. The point however is that this splitting may not respect the monoids, i.e $w$ may not be positive on $\sigma \cap N$. To fix this, we will carefully choose a particular splitting. Pick the face $\tau$ of $\sigma$ which maps isomorphically onto $\kappa$ and on which $p_1$ is minimal. To see that this is possible, let $v_1,\cdots,v_m$ be the extremal rays of $\kappa$, and let $u_k$ denote lifts of the rays $v_i$ in $\sigma$. Among the $u_k$, choose $u_1,u_2, \cdots u_m$ such that $u_i \mapsto v_i$ and such that $p_1(u_i)$ is minimal along all possible lifts of $v_i$ to an extremal ray of $\sigma$. The face $\tau$ of $\sigma$ generated by the $u_i$ is the desired face. By assumption, we have $\tau \cap N = Q \cap \kappa$. Using this splitting $N_\sigma^{\vee,\textrm{gp}} = N_\tau^{\vee,\textrm{gp}} \oplus L = Q_\kappa^{\vee,\textrm{gp}} \oplus L$, we see that we may write $p_1+q_1=p_2+q_2$ in the form $(w,r_1+q_1)=(w,r_2+q_2)$. We may identify every $\kappa$ with $\tau$, and thus the projection $\sigma \rightarrow \kappa$ gives us a map $p:\sigma \rightarrow \tau$. Every element $x$ of $\sigma$ can be written uniquely as $x=p(x)+v,v\in \Ker p$. Note that by construction, $w(x)=p_1(v)$, and $r_1(x)=p_1(p(x))$. To check that $w$ is non-negative on $\sigma$, it suffices to check it is non-negative on its extremal rays. For such a ray $x$ in $\sigma \cap \Ker p$, the result is clear since then $w(x) =p_1(x) \ge 0 $ by assumption. For an extremal ray not in $\Ker p$, we write $x = p(x)+v$, where $p(x)$ is an extremal ray on $\tau$. We then have that $p_1(p(x))\le p_1(x)$ by choice of $\tau$; hence $w(x) = p_1(v) = p_1(x-p(x)) \ge 0$, which completes the proof of integrality. The fact that the morphism of monoids is saturated then follows by \cite[Remark I.4.4]{Ts}.
For the converse, suppose that the morphism $p: (F,N) \rightarrow (G,Q)$ is flat has reduced fibers. From \cite[Lemma 4.1]{AK}, it follows that all cones of $F$ map onto cones of $G$. Suppose then that the cone $\sigma$ in $N$ maps onto the cone $\kappa$ in $Q$, but that the map of lattices $N_\sigma:= N \cap \Span \sigma \rightarrow Q_\kappa = Q \cap \Span \kappa$ is not surjective. For a face $\tau < \sigma$ which maps isomorphically onto $\kappa$ (such a face must exist for dimension reasons), the map $N_\tau \rightarrow Q_\kappa$ is also not surjective, as the image of $N_\tau$ is contained in the image of $N_\sigma$. As the question is local, we may thus assume without loss of generality that $F$ consists of a single cone $\sigma$ which maps isomorphically to $\kappa$, and that $N_\sigma \rightarrow Q_\kappa$ therefore has finite cokernel. Choose a splitting $N = N_\sigma \oplus L$, $Q = Q_\kappa \oplus K$, and consider the diagram
\begin{align*}
\xymatrix{\mathbb A(\sigma, N) \cong \mathbb A(\sigma,N_\sigma) \times \mathbb{G}_m^l \ar[r]^p \ar[d] & \mathbb A(\kappa,Q) \cong \mathbb A(\kappa,Q_\kappa) \times \mathbb{G}_m^l \ar[d] \\ \mathbb A(\sigma, N_\sigma) \ar[r] & \mathbb A(\kappa,Q_\kappa)}
\end{align*}
As $p$ and the two projections are flat with reduced fibers, the map $\mathbb A(\sigma,N_\sigma) \rightarrow \mathbb A(\kappa,Q_\kappa)$ must be flat with reduced fibers as well. So, as $\sigma \cong \kappa$ and $(N_\sigma)_\mathbf R$ is isomorphic to $(Q_\kappa)_\mathbf R$ we may further reduce to the case where $(F,N) \rightarrow (G,Q)$ is of the form $(\sigma, N) \rightarrow (\sigma, Q)$, with $\sigma$ full dimensional and $N \rightarrow Q$ a finite index inclusion, with non-trivial cokernel. But such a map is never reduced at the torus fixed point, and we obtain a contradiction. Therefore, $N_\sigma$ must surject onto $Q_\tau$, and $p$ is weakly semistable.
\end{proof}
\begin{remark}
A consequence of \ref{theorem:flat} is that our definition of weak semistability, phrased in terms of the morphism of fans $(F,N) \rightarrow (G,Q)$, is equivalent to the definition of a saturated morphism of \cite{Ts}, a condition a priori phrased in terms of the log structures of $\mathbb A(F,N),\mathbb A(G,Q)$\footnote{This kind of translation between notions on a log scheme, which have more apparent functoriality properties, to notions on the fan/cone complex of the log scheme, which, at least to the author, provide better geometric intuition, is one of the themes of this paper}. The key point of the proof is that weak semistability implies that the map of log structures is integral, which implies flatness, and then use condition $2$ in \ref{def: weaklysemistable} to show that the fibers are reduced. Note however that we are using both conditions of \ref{def: weaklysemistable} to prove integrality, not just condition $1$.
\end{remark}
The advantage of this definition of weak semistability is that it is stable under pullbacks -- this is lemma \ref{lem:fiberproduct}. As the property of being flat with reduced fibers is clearly stable under pullback, in order to understand the meaning of the lemma, we need to discuss fiber products of toric varieties.
\begin{subsection}{Fiber Products}\label{subsection: fiberproducts}
The category of toric varieties posseses fiber products:
\begin{defn}[Toric Fiber Products]
The toric fiber product of
\begin{align*}
\xymatrix{ & (F,N) \ar[d] \\ (H,L) \ar[r] & (G,Q) }
\end{align*}
is the toric variety with fan $F \times_G H = \{ \sigma \times_{\kappa} \lambda: \sigma \in F, \lambda \in H, \kappa \in G, \sigma \rightarrow \kappa,\lambda \rightarrow \kappa \}$ in the lattice $N \times_Q L $.
\end{defn}
It is straightforward to verify that this collection forms a fan and that it satisfies the universal property of the fiber product with respect to toric maps. However, the toric fiber product can be ill-behaved, as it does not in general agree with the fiber product of the associated toric varieties in the category of schemes:
\begin{align*}
\mathbb A(F \times_G H) \ne \mathbb A(F) \times_{\mathbb A(G)} \mathbb A(H)
\end{align*}
When considering the toric varieties as fine saturated log schemes, we can take yet another fiber product, the fiber product in the category of fine saturated log schemes, which we will denote by $(\mathbb A(F) \times_{\mathbb A(G)} \mathbb A(H))_{\textup{tor}}$. This fiber product is closely related to the toric fiber product, though it is not exactly the same. We will explain the connection between $\mathbb A(F \times_G H), \mathbb A(F) \times_{\mathbb A(G)} \mathbb A(H)$ and $(\mathbb A(F) \times_{\mathbb A(G)} \mathbb A(H))_\textup{tor}$ momentarily, after some preparation.
\begin{example}
Consider the diagram
\begin{align*}
\xymatrix{ & (\mathbf R_+^2, \mathbf Z^2) \ar[d] \\ (\mathbf R_+^2, \mathbf Z^2) \ar[r] & (\mathbf R_+^2, \mathbf Z^2) }
\end{align*}
\noindent where the morphisms are $(a,b) \mapsto (a,a+b)$ and $(c,d) \mapsto (c+d,d)$ respectively (these are the two charts of the blowup of $\mathbb A^2$ at the origin). We have $\mathbf R_+^2 \times_{\mathbf R_+^2} \mathbf R_+^2 = \{(a,b,c,d); a+b=d, a=c+d\} \cong \mathbf R_+ \subset \mathbf R^2$. On the other hand, the fiber product of these two morphisms in the category of schemes is the variety $\{(x,y,z,w): xy = z, x = zw\}$ which is reducible.
\end{example}
\begin{example}
What fails in the previous example is that the morphisms considered are not flat. However, flatness does not suffice to ensure toric fiber products agree with schematic fiber products. For instance, given the diagram
\begin{align*}
\xymatrix{ & (\mathbf R_+,\mathbf Z) \ar[d] \\ (\mathbf R_+,\mathbf Z) \ar[r] & (\mathbf R_+,\mathbf Z) }
\end{align*}
where the two morphisms are $a \mapsto 2a$, $b \mapsto 3b$ respectively, the toric fiber product is $\{(a,b):2a=3b\} \cong \mathbf R_+(3,2) \subset (\mathbf Z(3,2))_\mathbf R$ whose geometric realization is $\mathbb A^1$, whereas the schematic fiber product is $\{(x,y):x^2=y^3\}$.
\end{example}
\begin{rem}[Colimits of Lattices]
Given a diagram of lattices, we may take the limit or colimit of the diagram in the category of abelian groups. The limit of such a diagram is always a lattice, hence coincides with the limit in the category of lattices as well. In general, colimits of lattices are not lattices. However, given a finitely generated abelian group $L$, we can form the associated lattice $L/\textup{Torsion}$. The functor $L \mapsto L/\textup{Torsion}$ is a left adjoint, and thus, the colimit of the $L_i$ in the category of lattices coincides with $\displaystyle \lim_{ \longrightarrow } \, L_i/\textup{Torsion}$, where the colimit is understood in the category of abelian groups.
\end{rem}
\begin{rem}[Double Dual of a Lattice]
Note that for any finitely generated abelian group $L$, we have a natural isomorphism $L/\textup{Torsion} \cong (L^{\vee})^{\vee}$, where $L^{\vee} = \operatorname{Hom}(L,\mathbf Z)$ as usual. Since we have
\begin{align*}
(\displaystyle \lim_{ \longrightarrow } \, L_i)^{\vee}:=\operatorname{Hom}(\displaystyle \lim_{ \longrightarrow } \, L_i,\mathbf Z) = \displaystyle \lim_{ \longleftarrow } \, L_i^{\vee}
\end{align*}
by the defining property of a colimit, it follows that
\begin{align*}
(\displaystyle \lim_{ \longleftarrow } \, L_i)^{\vee} = (\displaystyle \lim_{ \longrightarrow } \, L_i^{\vee})^{\vee})^{\vee} \cong \displaystyle \lim_{ \longrightarrow } \, L_i^{\vee}/\textup{Torsion}
\end{align*}
\noindent In particular, we have that for lattices $Q,N,L$
\begin{align*}
(N \times_Q L)^{\vee} = N^{\vee} \oplus_{Q^{\vee}} L^{\vee}/\textup{Torsion}
\end{align*}
In the situations we are interested, the map $N \rightarrow Q$ will arise from a log smooth morphism. Therefore, the map $Q^{\vee} \rightarrow N^{\vee}$ will be injective, and in characteristic $p$ its cokernel will not have $p$-torsion. We claim that in this situation, the pushout has no $p$-torsion either. This is essentially because log smoothness is stable under pullbacks, but one can see it directly by applying the functor $\operatorname{Hom}(\mathbf Z/p\mathbf Z,?)$ to the exact sequence
\begin{align*}
\begin{xymatrix}
{0 \ar[r] & L^{\vee} \ar[r] & N^{\vee} \oplus_{Q^{\vee}} L^{\vee} \ar[r] & N^{\vee}/Q^{\vee} \ar[r] & 0}
\end{xymatrix}
\end{align*} \\
\end{rem}
In general, for cones $\sigma \in F, \kappa \in G, \lambda \in H$, we get maps $\sigma^{\vee} \cap N^{\vee} = N_{\sigma}^{\vee} \rightarrow N^{\vee}$, $Q_{\kappa}^{\vee} \rightarrow Q^{\vee}, L_{\lambda}^{\vee} \rightarrow L^{\vee}$ and so a map $N_{\sigma}^{\vee} \oplus_{Q_{\kappa}^{\vee}} L_{\lambda}^{\vee} \rightarrow N^{\vee} \oplus_{Q^{\vee}}L^{\vee} \rightarrow (N \times_Q L)^{\vee} = N^{\vee} \oplus_{Q^{\vee}}L^{\vee}/\textup{Torsion}$. It is clear that vectors in the image of $N_{\sigma}^{\vee} \oplus_{Q_{\kappa}^{\vee}} L_{\lambda}^{\vee}$ are non-negative on $\sigma \times_{\kappa} \lambda \subset N \times_Q L$, and so we get a map $N_{\sigma}^{\vee} \oplus_{Q_{\kappa}^{\vee}} L_{\lambda}^{\vee} \rightarrow (\sigma \times_{\kappa} \lambda)^{\vee}$. On the one hand,
\begin{align*}
k[(\sigma \times_{\kappa} \lambda)^{\vee} \cap (N \times_Q L)^{\vee}]
\end{align*}
are the affine charts for the fiber product of $(F,N),(G,Q),(H,L)$ in the category of toric varieties. On the other hand, since the functor \textbf{Mon} $\rightarrow$ \textbf{$k$-alg}, $M \rightarrow k[M]$ is left adjoint to the inclusion of $k$-algebras into monoids, the functor preserves colimits, so
\begin{align*}
k[N_{\sigma}^{\vee} \oplus_{Q_{\kappa}^{\vee}} L_{\lambda}^{\vee}] \cong k[N_{\sigma}^{\vee}] \otimes_{k[Q_{\kappa}^{\vee}]} k[L_{\lambda}^{\vee}]
\end{align*}
\noindent which are the affine charts of the fiber product in the category of schemes. Thus, we see that the toric fiber product and the usual schematic fiber product coincide if and only if
\begin{align*}
N_{\sigma}^{\vee} \oplus_{Q_{\kappa}^{\vee}} L_{\lambda}^{\vee} \rightarrow (\sigma \times_{\kappa} \lambda)^{\vee} \cap (N \times_Q L)^{\vee}
\end{align*}
is an isomorphism for all cones $\{\sigma \times_{\kappa} \lambda\}$ in $F \times_G H$. The monoid $(\sigma \times_{\kappa} \lambda)^{\vee} \cap (N \times_Q L)^{\vee}$ can be obtained from $N^{\vee} \oplus_{Q^{\vee}}L^{\vee}$ as a composition of three functors: It is the composition of the integralization functor $P \mapsto P^{\textup{int}}$ which replaces $P$ with its image in $P^\textrm{gp}$, followed by the saturation functor $P \mapsto P^{\textup{sat}} = \{x \in P^\textrm{gp}: nx \in P^{\textup{int}}\}$, followed by the functor $P \mapsto P/\textup{Torsion}$. As $P \rightarrow P^{\textup{int}}$ is surjective, $\operatorname{Spec} k[P^{\textup{int}}] \rightarrow \operatorname{Spec} {k[P]}$ is a closed immersion, and as $\operatorname{Spec} k[P^\textrm{gp}]$ is open in $\operatorname{Spec} k[P^{\textup{int}}]$, the geometric realization $\operatorname{Spec} k[P^{\textup{int}}]$ of $P^{\textup{int}}$ is the closure of $\operatorname{Spec} k[P^\textrm{gp}]$ in $\operatorname{Spec} k[P]$. Similarily, given an integral monoid $P \subset P^\textrm{gp}$, the algebra $k[P^{\textup{sat}}] \subset k[P^\textrm{gp}]$ is the integral closure of $k[P]$, and hence $\operatorname{Spec} k[P^{\textup{sat}}] $ is the normalization of $\operatorname{Spec} k[P]$. Performing these two functors produces the fiber product in the category of f.s. log schemes. Therefore, the fiber product $(\mathbb A(F) \times_{\mathbb A(G)} \mathbb A(H))_\textup{tor}$ is the normalization of the main component of $\mathbb A(F) \times_{\mathbb A(G)} \mathbb A(H)$, that is, of the component containing the fiber product of the tori. It is also almost the toric fiber product $\mathbb A(F \times_G H)$, but it is not quite as $\operatorname{Spec} k[P^\textrm{gp}]$ is not a torus if there is torsion. However, in the situation we are interested in, where the torsion is prime to the characteristic of $k$, the process of killing torsion is very mild, and exhibits $\operatorname{Spec} k[P]$ as a disjoint union of schemes isomorphic to $\operatorname{Spec} k[P/\textup{Torsion}]$. Therefore, the toric fiber product $\mathbb A(F \times_G H)$ is obtained from the log fiber product $(\mathbb A(F) \times_{\mathbb A(G)} \mathbb A(H))_{\textup{tor}}$ by keeping only the component that contains the identity of the fiber product of tori. \\
With these observations at hand, we find:
\begin{lemma}
\label{lem:fiberproduct}
Suppose
\begin{align*}
\xymatrix {(F,N) \ar[d]_p \\ (G,Q)}
\end{align*}
is weakly semistable. Then, for a map $(G',Q') \rightarrow (G,Q)$ of toric varieties, the geometric realization of the diagram
\begin{align*}
\xymatrix{ F_\tau = F \times_G G' \ar[r] \ar[d]_{p_G'} & F \ar[d]^p \\ G' \ar[r] & G}
\end{align*}
is cartesian in the category of schemes, and $p_{\tau}$ is also weakly semistable.
\end{lemma}
\begin{proof}
The property of being flat with reduced fibers is stable under pullbacks. Thus, the interesting point is showing that the diagram is cartesian in the category of schemes. Both properties are local on $F$, so we may replace $F$ by a single cone $\sigma$, $G$ by a single cone $\kappa$, and $G'$ by a single cone $\lambda$.
Since $p$ is flat with reduced fibers, theorem II.4.2 in \cite{Ts} implies that the pushout
\begin{align*}
N_{\sigma}^{\vee} \oplus_{Q_{\kappa}^{\vee}} L_{\lambda}^{\vee}
\end{align*}
is saturated in its associated group. This associated group is actually a lattice, since by choosing a face of $\tau$ of $\sigma$ which maps isomorphically to $\kappa$, and with $N \cap \tau \cong Q \cap \kappa$, we obtain a splitting $(N_{\sigma}^{\vee})^\textrm{gp} \cong (Q_{\kappa}^{\vee})^\textrm{gp} \oplus N'$ -- c.f the proof of \ref{theorem:flat}. Thus,
\begin{align*}
N_{\sigma}^{\vee} \oplus_{Q_{\kappa}^{\vee}} L_{\lambda}^{\vee}
\end{align*}
is identified with the intersection of a cone $C$ in $(N \times_Q L)_{\mathbf R}^{\vee}$ with $(N \times_Q L)^{\vee}$. The same is true with $(\sigma \cap \lambda) \cap (N \times_Q L)$. However, the dual of each of these cones is isomorphic to $\sigma \times_{\kappa} \lambda$, in the first case by the defining property of the colimit, and in the second by the relation $(C^{\vee})^{\vee}=C$ for the double dual of a cone in a fixed lattice. Applying the dual again, we see that $N_{\sigma}^{\vee} \oplus_{Q_{\kappa}^{\vee}} L_{\lambda}^{\vee}$ must be isomorphic to $(\sigma \cap \lambda) \cap (N \times_Q L)$, and the result follows by the discussion preceeding the lemma.
\end{proof}
\end{subsection}
\begin{subsection}{Toric Stacks}
In what follows, we will need the notion of a \emph{toric stack}. For us, a toric stack will be always given by the data of a ``KM" fan, i.e a triple $(F,N,\{N_{\sigma}\}_{\sigma \in F})$, where $(F,N)$ is the usual data of a toric variety, and $N_{\sigma}$ is a collection of sublattices of $N$, one for each $\sigma \in F$, with the properties
\begin{itemize}
\item $N_{\sigma} \subset N \cap \Span(\sigma)$ is a finite index inclusion. \\
\item $N_{\sigma} \cap \Span \tau = N_{\tau}$ for a face $\tau$ of $\sigma$.
\end{itemize}
A morphism of KM fans $(F,N,\{N_{\sigma}\}) \rightarrow (G,Q,\{Q_{\kappa}\})$ is a morphism $(F,N) \rightarrow (G,Q)$ such that whenever $\sigma \mapsto \kappa$, $N_{\sigma} \rightarrow N \rightarrow Q$ factors through $Q_{\kappa}$. \\
The data of a KM fan $(F,N,N_{\sigma})$ has a geometric realization into a stack $\mathbb A(F,N,\{N_{\sigma}\})$. The stack is constructed as follows: the finite index sublattice $N_\sigma \rightarrow N \cap \Span \sigma$ lifts to a finite index sublattice $L \rightarrow N$ with $L \cap \Span{\sigma} = N_\sigma$, which induces a finite map $T(L) \rightarrow T(N)$ of tori, with kernel a finite group $K_\sigma$. Then $\mathbb A(F,N,\{N_{\sigma}\})$ is the colimit of the local pieces $\mathbb A(\sigma,N,N_\sigma):=[\mathbb A(\sigma,L)/K_\sigma]$\footnote{this quotient is independent of the choice of the lift $L$}, with a map $\mathbb A(\tau,N,N_\tau) \rightarrow \mathbb A(\sigma, N, N_\sigma)$ for every face $\tau <\sigma$. The coarse moduli space of the stack is the toric variety $(F,N)$. The stack is a tame Artin stack, and is Deligne-Mumford whenever the index of $N_\sigma$ in $N$ is prime to the characteristic for all cones $\sigma$ -- so in particular, this is always the case in characteristic $0$. The data of a morphism of KM fans has a geometric realization into a morphism of the geometric realizations of the fans, and the associated morphism between coarse spaces is simply the geometric realization of $(F,N) \rightarrow (G,Q)$. The notion of a KM fan was first introduced in \cite{Ttor}; their main properties and the connection of the geometric realization with intrinsically defined toric stacks are developed in \cite{GMtor}. In the language of log geometry, the data of a stacky fan determines a Kummer extension of the log structure of $\mathbb A(F,N)$, and $\mathbb A(F,N,\{N_\sigma\})$ is exactly the stack one obtains from the construction of \cite{BV} -- we will use this observation later to define toroidal stacks, but we will phrase everything using the fan formalism as it is better suited for doing geometry on the combinatorial level. \\
Any toric variety can be regarded as a toric stack, by taking $N_{\sigma} = N \cap \Span \sigma$ for each cone $\sigma$ -- note that there is no additional information in the $N_{\sigma}$ in this case. Under this identification, the category of toric varieties becomes a full subcategory of the category of toric stacks. We will use this identification in what follows and keep denoting a toric variety $(F,N)$ by $(F,N)$ even when the context makes it clear that it is considered as a toric stack. The proof of the following lemma is simple, and can be found in \cite{GMtor}:
\begin{lemma}
\label{lem:representable}
Let $p:(F,N,\{N_{\sigma}\}) \rightarrow (G,Q,\{Q_{\kappa}\})$ be a morphism of KM fans. If $p^{-1}(Q_{\kappa}) = N_{\sigma}$ whenever $\sigma \mapsto \kappa$, the geometric realization of $p$ is representable.
\end{lemma}
\end{subsection}
\begin{subsection}{The Main Construction}
We now fix the morphism $p:(F,N) \rightarrow (G,Q)$ which is \emph{surjective and proper}.
\begin{defn}
\label{defn:main}
Let $\mathcal{C}$ be the category whose objects are diagrams
\begin{align*}
\xymatrix{(\Phi,N') \ar[r]^j \ar[d]_\pi & (F,N) \ar[d]^p \\ (\Gamma,Q') \ar[r]_i & (G,Q)}
\end{align*}
such that
\begin{itemize}
\item The map $i$ is an alteration. \\
\item $N'$ is the fiber product $N \times_Q Q'$. \\
\item $\Phi$ is a modification of $j^{-1}(F).$\\
\item $\pi$ is weakly semistable. \\
\end{itemize}
A morphism in $\mathcal{C}$ is a commutative diagram
\begin{align*}
\xymatrix{(\Phi'',N'') \ar[r] \ar[d] & (\Phi',N') \ar[d] \\ (\Gamma'',Q'') \ar[r] & (\Gamma',Q')}
\end{align*}
\noindent which commutes with the morphisms to $p:(F,N) \rightarrow (G,Q)$.
\end{defn}
\begin{theorem}
\label{theorem:maintoric}
The category $\mathcal{C}$ has a terminal object which is a toric stack. In other words, there is a diagram
\begin{align*}
\xymatrix{(F',N,\{N_\sigma\}) \ar[r] \ar[d] & (F,N) \ar[d] \\ (G',Q,\{Q_\kappa\}) \ar[r] & (G,Q)}
\end{align*}
such that every diagram
\begin{align*}
\xymatrix{(\Phi,N') \ar[r]^j \ar[d]_\pi & (F,N) \ar[d]^p \\ (\Gamma,Q') \ar[r]_i & (G,Q)}
\end{align*}
factors uniquely as
\begin{align*}
\xymatrix{(\Phi,N') \ar[r] \ar[d]_\pi & (F',N,\{N_\sigma\}) \ar[r] \ar[d] & (F,N) \ar[d]^p \\ (\Gamma,Q') \ar[r] & (G',Q,\{Q_\kappa\}) \ar[r]& (G,Q)}
\end{align*}
\end{theorem}
\begin{proof}
We first construct $G'$. The idea of the construction comes from \cite{KSZ}. Let $p(F)$ denote the collection of images of cones of $F$. Note that though every cone $p(\sigma)$ is contained in a cone of $G$, thus is convex, $p(F)$ is in general not a fan, as cones may not intersect along faces. We define $G'$ as the subdivision of $G$ determined by the cones in $p(F)$. Explicitly, this means the following: For every vector $w$ in $G$, we look at the collection
\begin{align*}
N_0(w) = \{\sigma \in F: p(v)=w \textup{ for some } v \textup{ in the interior } of \sigma\}
\end{align*}
The cones $\kappa$ of $G'$ are precisely the cones such that for any two $w,w'$ in the interior of $\kappa$, we have $N_0(w)=N_0(w')$. Next, for a cone $\kappa \in G'$, we take
\begin{align*}
Q_\kappa = \cap_{\sigma \in N_0(\kappa)} p(N \cap \sigma)
\end{align*}
In the interior of $\kappa$, this has the following description: $w \in Q_\kappa$ if and only if there exist $v_i \in \sigma_i$ with $p(v_i)=w$ for every cone $\sigma_i \in N_0(\kappa)$. This completes the construction of the base $(G',Q,Q_\kappa)$. \\
At this point we need to verify that this construction actually yields a fan. The difficult part is verifying that the cones are strictly convex. So fix a cone $\kappa \in G'$, and pick two interior vectors $w,w' \in \kappa$. We will show that the whole line segment connecting $w$ to $w'$ must also be in the interior of $\kappa$. Suppose there exists a $t \in (0,1)$ for which $N_0(tw+(1-t)w')$ is different from $N_0(w)=N_0(w')$. Take for simplicity the smallest such $t$ -- this makes sense since the condition $N_0(w)=N_0(u)$ is an open condition on $u$-- and denote the point $tw+(1-t)w'$ by $w''$ to ease the notation. Certainly, since every cone $\sigma_i$ in $N_0(w)$ is strictly convex, the line segment between two lifts of $w,w'$ in $\sigma_i$ is also in $\sigma_i$, so $N_0(w) \subset N_0(w'')$. So take a cone $\sigma \in N_0(w'')-N_0(w)$, and a lift $v''$ of $w''$ in $N_0(w'')$. We look at the fiber of the map of vector space $N_{\mathbf R} \rightarrow Q_{\mathbf R}$ over the interval $[w,w']$ in $Q_{\mathbf R}$. Call this fiber $N_{[w,w']}$, and let $F_{[w,w']}$ be the intersection of $N_{[w,w']}$ with the fan $F$. Then $F_{[w,w']}$ is a polyhedral decomposition of $N_{[w,w']}$. The cone $\sigma$ intersects $F_{[w,w']}$ into a one-dimensional union of cells, since the relative dimension of $\sigma$ under $F \rightarrow G$ is $0$ by assumption, and contains $v''$ as an extreme point. Similarily, cones in $N_0(w)$ correspond to edges in $F_{[w,w']}$. Since $(F,N) \rightarrow (G,Q)$ is surjective and proper, the support of $F_{[w,w']}$ is all of $N_{[w,w']}$. In particular, the star of $v''$ in $F_{[w,w']}$ must intersect $F_{[w,w'')}$ non-trivially; so, in particular, there is an edge in the star of $v''$ in $F_{[w,w']}$ which maps to a vector $sw+(1-s)w'$ with $s<t$. By assumption on $t$, we have $N_0(sw+(1-s)w')=N_0(w)$, so in fact the edge corresponds to a cone in $N_0(w)$ and thus contains a lift of $w$. Since $\sigma$ is by choice not in $N_0(w)$, $v''$ is an extreme point of the edge. But this is a contradiction, since the edge must extend to contain a lift of $w'$ as well, as we assumed $N_0(w)=N_0(w')$. Thus we must have $N_0(w)=N_0(sw+(1-s)w')$ for all $s \in [0,1]$, and convexity follows.
To construct $(F',N,\{N_\sigma\})$, we simply take the minimal subdivision of $(F,N)$ that maps to $(G',Q,Q_\kappa)$, as in \ref{lem:minmod}. This means that $F'$ is the fan $\{p^{-1}(\kappa) \cap \sigma: \kappa \in G', \sigma \in F\}$ and the sublattice corresponding to $\sigma' := p^{-1}(\kappa) \cap \sigma$ is $L_{\sigma'} := p^{-1}(Q_\kappa \cap \sigma')$. Furthermore, the morphism $(F',N,\{N_\sigma\})$ is weakly semistable: cones of $F'$ map onto cones of $G'$ by construction, and $N_{\sigma}$ maps onto $Q_{\kappa}$ whenever $\sigma \mapsto \kappa$ by construction again. \\
Suppose now we are given a diagram
\begin{align*}
\xymatrix{(\Phi,N') \ar[r]^j \ar[d]_\pi & (F,N) \ar[d]^p \\ (\Gamma,Q') \ar[r]_i & (G,Q)}
\end{align*}
\noindent where $i$ is an alteration, $N'$ the fiber product $N \times_Q Q'$, and $\Phi$ a subdivision of $j^{-1}F$. Assume furthermore that $\pi$ is semistable. Let $w,w'$ be two lattice points in the interior of a cone $\gamma$ of the fan $\Gamma$. Suppose that $w$ maps into a cone $\kappa \in G'$; we show that $w'$ maps to the same cone as well. Consider lifts $v_1,\cdots,v_n$ of $v=i(w)$ to cones $\sigma_i \in N_0(w) \subset F$. Since $\Phi$ subdivides $j^{-1}F$, there are cones $g_1,\cdots,g_n$ in $\Phi$ such that $j(g_i) \subseteq \sigma_i$; so we may find lifts $w_1,\cdots,w_n$ of $w$ in $g_i$. But then each cone $g_i$ maps to $\gamma$ under the projection $\pi$, and hence maps onto $\gamma$ and $\gamma \cap Q' = \pi(g_i \cap N)$ from conditions $(1),(2)$ in the definition of semistability. Since $w'$ is in $\gamma \cap Q'$ as well, this means that there exists $w_1',\cdots,w_n' \in g_i \cap N$ that map to $w'$ as well -- and hence there are $v_1'=j(w_1'),\cdots,v_n'=j(w_n')$ in the cones $\sigma_i \in N_0(w)$ that map to $w'$ as well. It follows that $N_0(w) \subset N_0(w')$, thus, by symmetry, $N_0(w)=N_0(w')$; hence $w,w'$ belong to the same cone $\kappa$ of $G'$. Furthermore, they are in the image of the lattice $N_{\sigma_i}$ for each cone in $N_0(w)$, thus in fact in the monoid $Q_\kappa$. Thus $(\Gamma,Q')$ factors through $(G',Q,Q_\kappa)$. The fact that $(\Phi,N)$ must factor through $(F,N,N_\sigma)$ factors through the universal property defining $(F,N,N_\sigma)$ automatically.
\end{proof}
\begin{rem}
It is worth pointing out that this proof goes through without assuming that the map $i:(\Gamma,Q') \rightarrow (G,Q)$ is an alteration. All that is required is that the kernel of $N \rightarrow Q$ and that the kernel of $N' \rightarrow Q'$ coincide.
\end{rem}
Using the notation of definition \ref{defn:main}, we have as a corollary:
\begin{corollary}
The minimal modification $\Phi$ of $j^{-1}(F)$ such that
\begin{align*}
\xymatrix{(\Phi,N') \ar[r]^j \ar[d]_\pi & (F,N) \ar[d]^p \\ (\Gamma,Q') \ar[r]_i & (G,Q)}
\end{align*}
\noindent commutes and $\pi$ is weakly semistable is given by the fiber product of
\begin{align*}
\xymatrix{& (F,N,\{N_{\sigma}\}) \ar[d]^p \\ (\Gamma,Q') \ar[r]_i & (G',Q,\{Q_{\kappa}\})}
\end{align*}
\noindent Its geometric realization coincides with $\mathbb A(\Gamma,Q') \times_{\mathbb A(G',Q,\{Q_{\kappa}\})} \mathbb A(F,N,\{N_{\sigma}\})$.
\end{corollary}
\end{subsection}
\begin{proof}
This follows immediately by combining \ref{theorem:maintoric}, \ref{lem:fiberproduct}, \ref{lem:representable}.
\end{proof}
\end{section}
\begin{section}{Globalizing}
We are now ready to discuss the changes necessary to generalize the above construction to the toroidal case. Recall the relevant definitions from \cite{KKMS}. To any toroidal embedding $(X,U)$ there is associated a stratification, the strata being determined by the irreducible components of the divisor $X-U$. For each stratum $Y$, we set
\begin{align*}
M^Y = \textup{Divisors on } \textup{Star} Y \\
M^Y_+ = \textup{Effective Divisors on } \textup{Star} Y \\
N^Y = \operatorname{Hom}(M_Y,\mathbf Z)\\
\sigma^Y = \{v \in N^Y \otimes_\mathbf Z \mathbf R: v \textup{ is non-negative on } M^Y_+\}
\end{align*}
The collection of the cones $\sigma^Y$ together with their integral structure $N^Y$ is a cone complex, which we denote by $C(X)$; the only contrast with the toric theory is that the cones do not all inhabit a single (canonical) lattice $N$. Subdivisions of this cone complex correspond to birational modifications of $(X,U)$ which are the identity on $U$. \\
At this point, it is useful to connect the theory of toroidal embeddings with the language of logarithmic geometry\footnote{We will not use logarithmic geometry in a deep way. However, to avoid the language completely would make the presentation of what follows unnecessarily cumbersome. The reader who wishes to circumvent the use of logarithmic geometry entirely should be able to prove all relevant statements by using analogous results in \cite{KKMS} without much difficulty.}. A toroidal embedding $(X,U)$ carries a canonical structure of a log scheme $(X,M_X)$, by setting
\begin{align*}
M_X(V) = \{f \in \mathcal{O}_X(V): f \in \mathcal{O}^*_X(V \cap U)\}
\end{align*}
Conversely, by the chart criterion of log smoothness of \cite{K}, it follows that a toroidal embedding without self intersection is the same thing as a fine, saturated (f.s.) log smooth log scheme $(X,M_X)$ with log structure defined on the Zariski site of $X$. The ``characteristic monoid'' $\ov{M}_X:= M_X/\mathcal{O}^*_X$ is a constructible sheaf, constant on the strata of the associated stratification of $(X,U)$; its value at the generic point $\mathrm{\acute{e}t}a_Y$ of a stratum $Y$ is
\begin{align*}
\ov{M}_{X,\mathrm{\acute{e}t}a_Y} = (\sigma^{Y})^{\vee} \cap M^Y
\end{align*}
By basic results in the theory of log smoothness, or, as shown directly in \cite{KKMS}, a toroidal embedding admits \'etale locally an \'etale map to a toric variety, and, on a stratum $Y$, even a smooth map to $\operatorname{Spec} k[M^Y]$.
By a morphism of toroidal embeddings $(X,U) \rightarrow (Y,V)$ we will simply mean a morphism of the associated log schemes; concretely, this means a morphism $X \rightarrow Y$ that takes the subsheaf $M_Y$ of $\c{O}_Y$ defined above into $M_X$. The \emph{toroidal} morphisms of \cite{KKMS},\cite{AK} are the same as log smooth morphisms, defined by Kato in \cite{K}. We tend to prefer the latter terminology, as it has become more standard. The reader who prefers to avoid using logarithmic geometry can find a thorough treatment of toroidal morphisms in \cite[Section 1]{AK}. \\
We need the following observation, explained in \cite{KKMS}. We consider morphisms
\begin{align*}
\lambda: \operatorname{Spec} k[[\mathbf N]] \rightarrow X
\end{align*}
\noindent which take the generic point of $\operatorname{Spec} k[[\mathbf N]]$ to $U$, and $\lambda(0)$ to $Y$. Here, $k[[\mathbf N]]$ denotes the completion of $k[\mathbf N] := k[t]$ with respect to its natural valuation, which we will denote by $\textup{ord}_0$. Then, for a divisor $D$ in $M^Y$, we get a pairing
\begin{align*}
\langle \lambda,D \rangle = \textup{ord}_{0}\lambda^{*}D
\end{align*}
This way we obtain a map $\operatorname{Hom}{(\operatorname{Spec} k[[\mathbf N]],X)} \rightarrow \sigma^Y$, which is actually surjective. So we may identify an integral point $v \in \sigma^Y \cap N^Y$ with an equivalence class of maps $\operatorname{Spec} k[[\mathbf N]]$ to $X$, two maps being equivalent if and only if their order of intersection with each divisor is the same. We will abbreviate this equivalence class of maps by $v$ as well. Similarily, the cone $\sigma^Y$ itself can be identified with the image of $\operatorname{Hom}{(\operatorname{Spec} k[[\mathbf R_+]], X)}$. Here, the ring $k[[\mathbf R_+]]$ is the completion of $k[\mathbf R_+]$ with respect to the valuation on $k[\mathbf R_+]$ which takes a polynomial $\sum_{\alpha \in \mathbf R_+} c^{\alpha}x^{\alpha}$ to $\textup{inf}\{\alpha: c_\alpha \ne 0\}$. The completion $k[[\mathbf R_+]]$ is naturally a valuation ring as well; we denote its fraction field by $k[[\mathbf R]]$. The relationship with $\sigma^Y$ is best explained through the following three observations.
\begin{rem}
Suppose $V$ is an affine toric variety, corresponding to the cone $\sigma$ in the lattice $N$. Denote the dual lattice of $N$ by $M$ as usual, and denote by $\sigma^{\vee}$ the dual cone of $\sigma$, i.e $\{u \in M_\mathbf R: \left \langle u,v \right \rangle \ge 0 \textup{ for all } v \in \sigma\}$, so that $V = \operatorname{Spec} k[ \sigma^{\vee} \cap M]$. An element $v \in \sigma \cap N$ is the same thing as a homomorphism of monoids $\sigma^{\vee} \cap M \rightarrow \mathbf N$. We have
\begin{align*}
\operatorname{Hom}_{\textup{Mon}} (\sigma^{\vee} \cap M,\mathbf N) & = \operatorname{Hom}_{k-\textup{alg}} (k[\sigma^{\vee} \cap M], k[\mathbf N]) = \\
& = \operatorname{Hom}_{\textup{Schemes}}(\operatorname{Spec} k[\mathbf N] = \mathbb A^1, V)
\end{align*}
If $v$ in in the interior of $\sigma$, the image of $0$ under $\mathbb A^1 \rightarrow V$ is precisely the torus fixed point of $V$. Composing $k[\sigma^{\vee} \cap M] \rightarrow k[\mathbf N]$ with the completion $k[\mathbf N] \rightarrow k[[\mathbf N]]$ gives a morphism $\operatorname{Spec} k[[\mathbf N]] \rightarrow V$ which defines precisely the same homomorphism $\sigma^{\vee} \cap M \rightarrow \mathbf N$ as $v$. Thus, for an affine toric variety each equivalence class of morphisms $\operatorname{Spec} k[[\mathbf N]] \rightarrow V$ has a canonical representative, obtained by completing the homomorphism $k[\mathbf N] \rightarrow V$ correpsonding to $v$.
\end{rem}
\begin{rem}
\label{rem:Rrepresentatives}
Suppose $V=V(\sigma)$ is the affine toric variety associated to the cone $\sigma$ in $N$, as in the preceeding remark. A similar description as the one given in the preceeding remark can in fact be given for any vector $v \in \sigma$ rather than just the integral ones. A vector $v$ in $\sigma$ (which we may well assume to be in the interior of $\sigma)$ corresponds to a homomorphism $\sigma^{\vee} \rightarrow \mathbf R_+$, which induces (and is determined by) by restriction to a homomorphism $\sigma^{\vee} \cap M \rightarrow \mathbf R_+$. This is the same data as a $k$-algebra homomorphism $k[\sigma^{\vee} \cap M] \rightarrow k[\mathbf R_+]$ as above, which induces a map $k[\sigma^{\vee} \cap M] \rightarrow k[[\mathbf R_+]]$, i.e a map $\lambda: \operatorname{Spec} k[[\mathbf R_+]] \rightarrow V(\sigma)$. When $v$ is in the interior of $\sigma$, we have that under this morphism the closed point maps to the torus fixed point of $V(\sigma)$. Note that for any toric divisor $D$ of $V(\sigma)$, i.e any element $u \in M$, we have $\left \langle v,u \right \rangle = ord_{t=0} \lambda^{*} u$ by construction. We may thus identify vectors $v \in \sigma$ with morphisms
\begin{align*}
\lambda: \operatorname{Spec} k[[\mathbf R_+]] \rightarrow V(\sigma)
\end{align*}
such that $\lambda(\mathrm{\acute{e}t}a) \in \textup{ torus}$, and $\lambda(0) = \textup{torus fixed point}$, up to the equivalence relation that $ord_{t=0} \lambda^*$ induces the same homomorphism on $M$. We also have the analogue of the observation in the preceeding remark, that every equivalence class of a morphism $\operatorname{Spec} k[[\mathbf R_+]] \rightarrow V(\sigma)$ has a unique representative obtained by completing $\operatorname{Spec} k[\mathbf R_+] \rightarrow V(\sigma)$.
\end{rem}
\begin{rem}
We can now combine the two remarks above with the fact that every toroidal embedding is \'etale locally (hence formally locally) isomorphic to a toric variety, to obtain that every interior vector $v \in \sigma^Y$, not necessarily integral, corresponds to an equivalence class of morphisms
\begin{align*}
\lambda: \operatorname{Spec} k[[\mathbf R_+]] \rightarrow V
\end{align*}
such that $\lambda(\mathrm{\acute{e}t}a) \in U$, $\lambda(0) \in Y$. Of course, there is no longer a canonical representative for a morphism in this equivalence class.
\end{rem}
\begin{subsection}{Toroidal Stacks}
We would now like to transport the main points of the theory of toric stacks to toroidal embeddings. Though it is possible to give analogous definitions by working \'etale locally and modifying the appropriate results of \cite{GMtor}, we prefer not to work from scratch, and use the construction of the ``root stack" of Borne and Vistoli, \cite[Section 4]{BV}. We thus use the log scheme associated to a toroidal embedding.
Given a log scheme $(X,M)$, the construction of Borne and Vistoli produces for any map of monoids $\ov{M} \rightarrow \ov{M}'$ which is \emph{Kummer}, i.e injective with finite cokernel, a log stack $(\mathcal{X},M')$ (which is tame, and Deligne-Mumford if the order of the cokernel is prime to the characteristic) mapping to $(X,M)$, with the morphism $M \rightarrow M'$ inducing the given map $\ov{M} \rightarrow \ov{M'}$. Furthremore, $(\mathcal{X},M')$ is the terminal object of the category of log schemes $(Y,N) \rightarrow (X,M)$ such that $M \rightarrow N$ factors through $M \rightarrow M'$. In the toroidal case, where we can choose \'etale locally on $X$ a chart $X \rightarrow \operatorname{Spec} k[\ov{M}]$, the stack $\mathcal{X}$ is described \'etale locally as a fiber product $X \times_{\operatorname{Spec} k[\ov{M}]} [\operatorname{Spec} k[\ov{M}']/K]$, where $K$ is the kernel of tori $\operatorname{Spec} k[\ov{M}^\textrm{gp}] \rightarrow \operatorname{Spec} k[\ov{M}'^\textrm{gp}]$. Put otherwise, if $\sigma$ is the dual monoid of $\ov{M}$ in $N=(\ov{M}^\textrm{gp})^\vee$, then $N \rightarrow N' = (\ov{M'}^\textrm{gp})^\vee$ is a finite index inclusion, and $\mathcal{X}$ is \'etale locally isomorphic to $X \times_{\mathbb A(\sigma,N)} \mathbb A(\sigma,N,N')$.
Starting from the log scheme $(X,M_X)$ associated to a toroidal embedding, and a Kummer extension $\ov{M}_X \rightarrow \ov{M'}_X$, the stack $(\mathcal{X},M'_\mathcal{X})$ is log smooth with Zariski log structure. We will refer to it as a toroidal stack (without self intersection). To translate the Borne-Vistoli formalism to a formalism more analogous to the toric formalism of the previous section, given a toroidal embedding $(X,U)$, we will assign for each cone $\sigma^{Y}$ in $C(X)$ a sublattice $N_{\sigma^Y} \subset N^Y \cap \Span(\sigma^Y)$ which is injective with finite cokernel, and with the property that $N_{\tau} = \Span \tau \cap N_{\sigma}$ for a face $\tau$ of $\sigma$. Recall that the sheaf $\ov{M}_X$ is constructible, constant on the strata of $(X,U)$, with value $M^Y$ on $Y$; therefore, dualizing the inclusions $N_{\sigma^Y} \rightarrow N^Y$ gives exactly a Kummer extension $\ov{M}'_X$ of $\ov{M}_X$. Thus, a compatible triple $(C(X),\{N^Y\},\{N_{\sigma^Y}\})$, as $Y$ ranges through the strata of $(X,U)$ produces a toroidal stack over $X$ by the \cite{BV} construction.
Suppose now $C(X')$ is a subdivision of $C(X)$. The process outlined in \cite{KKMS} produces a log blowup $X' \rightarrow X$ with cone complex $C(X')$. Combining such subdivisions with Kummer extensions produces triples $(C(X'),\{N^Z\},\{N_{\sigma^Z}\})$, with $Z$ running through the strata of $(X',U)$, and this data yields a stack $\c{X'}$ mapping to $X$. We call such a triple a stacky subdivision of $C(X)$, and refer to it as the cone complex of $\c{X'}$. The notation $(C(X'),\{N^Z\},\{N_{\sigma^Z}\})$ may be cumbersome, but we find it very useful, as doing geometry on this stacky polyhedral complex is much more intuitive than doing geometry on the dual sheaf of monoids. The following lemma is a combination of \cite{KKMS} and \cite{BV}.
\begin{lemma}
Let $X$ be a toroidal embedding, $\c{X'}$ the log stack corresponding to a triple $(C(X'),\{N^Z\},\{N_{\sigma^Z}\})$ which is a stacky subdivision of $C(X)$. Let $T \rightarrow X$ be a map from a toroidal embedding $T$, and assume that $C(T) \rightarrow C(X)$ factors through $(C(X'),\{N^Z\},\{N_{\sigma^Z}\})$. Then there exists a unique lift of $T \rightarrow X$ to $T \rightarrow \c{X'}$ which induces the given map $C(T) \rightarrow (C(X'),\{N^Z\},\{N_{\sigma^Z}\})$.
\end{lemma}
\begin{proof}
Note first that by \cite{KKMS}, the map $T \rightarrow X$ uniquely factors through $X'$ where $X'$ is the toroidal embedding with cone complex $C(X')$ constructed in \cite{KKMS}. Then $\c{X'} \rightarrow X'$ is the log stack associated to the Kummer extension $(C(X'),N^Z,N_{\sigma^Z})$ of $C(X')$ and the result follows from \cite{BV}.
\end{proof}
The combinatorial definition of weak semistability \ref{def: weaklysemistable} for toric fans does not make use of the global lattices in which the fans live, so it applies without change to maps of cone complexes. The connection with algebraic geometry is as follows:
\begin{lemma}
Let $f: X \rightarrow S$ be a log smooth morphism of toroidal embeddings, and assume that the map of cone complexes $C(X) \rightarrow C(S)$ is weakly semistable. Then $X \rightarrow S$ is flat with reduced fibers.
\end{lemma}
\begin{proof}
For the proof, we use some basic results from the theory of logarithmic geometry. It suffices to check that the conditions hold around every point $x \in X$; as the question is \'etale local on $X$, we may assume that $X$ and $S$ have a single closed stratum, and that $x$ and $f(x)$ belong to the closed strata. In terms of cone complexes this means that $C(X)$ and $C(S)$ consist of a single cone, and the assumption of weak semistability translates to the fact that the map $\sigma^X \cap N^X \rightarrow \sigma^S \cap N^S$ is surjective; this in turns implies that the dual map $\overline{M}_{S,f(x)}^\textrm{gp} \rightarrow \overline{M}_{X,x}^\textrm{gp}$ is injective with torsion-free cokernel. Then, by replacing $X$ with an \'etale cover, we can choose a neat chart for $f$, which we can even take to be a characteristic chart (see for instance by \cite[Theorem 2.4.4]{oguslog}); then, we have a diagram
\begin{align*}
\begin{xymatrix}
{X \ar[r] \ar[d] & \operatorname{Spec} \mathbf Z[\overline{M}_{X,x}] \ar[d] \\ S \ar[r] & \operatorname{Spec} \mathbf Z[\overline{M}_{S,f(x)}]}
\end{xymatrix}
\end{align*}
From the toric case, we know that $\operatorname{Spec} \mathbf Z[\overline{M}_{X,x}] \rightarrow \operatorname{Spec} \mathbf Z[\overline{M}_{S,f(x)}]$ is flat and has reduced fibers, and, as the chart is neat, the map $X \rightarrow S \times_{\operatorname{Spec} \mathbf Z[\overline{M}_{S,f(x)}]} \operatorname{Spec} \mathbf Z[\overline{M}_{X,x}]$ is smooth. Therefore, the composition $X \rightarrow S$ is also flat with reduced fibers.
\end{proof}
By the fiber product of morphisms of toroidal embeddings we will mean the fiber product \emph{in the category of} f.s. log schemes. We will denote this by $(X \times_S T)_{\textup{tor}}$ to avoid confusion, and write $X \times_S T$ for the schematic fiber product. For log smooth morphisms, this fiber product is a toroidal embedding, and the underlying scheme of this fiber product is connected to the fiber product of the underlying schemes in the way explained in \ref{subsection: fiberproducts} -- it is the normalization of the closure of the main component of the underlying fiber product. This follows from the existence of the \'etale local maps to toric varieties, as the statement can be checked \'etale locally. A consequence of weak semistability is therefore that the fiber product in the category of f.s. log schemes has the same underlying scheme as the fiber product of the underlying schemes. The cone complex of the fiber product $(X \times_S T)_{\textup{tor}}$ is the fiber product of cone complexes $C(X) \times_{C(S)} C(T)$. This again follows from the local case.
\end{subsection}
\begin{subsection}{The Main Construction}
The toric construction expalined in section $2$ carries over to the toroidal case with minimal changes, by replacing the fans of $X$ and $S$ with the cone complexes $C(X),C(S)$. The morphism $X \rightarrow S$ induces a morphism $p:C(X) \rightarrow C(S)$ by composing a map $k[[ \mathbf R_+]] \rightarrow X$ with $X \rightarrow S$. So suppose a cone $\kappa \in C(S)$, and a point $w \in \kappa$ are given. We consider
\begin{align*}
N_0(w) = \{\sigma: \exists ! v \in \sigma^o \textup{ such that } p(v) = w\}
\end{align*}
\begin{lemma}
\label{lem:convex}
The cones $\{w: N_0(w) = \textup{ fixed} \}$ are strictly convex, and form a subdivision of $\kappa$.
\end{lemma}
\begin{proof}
We adapt the proof of the toric case \ref{theorem:maintoric}. The question is local on $S$, so we may assume that $C(S)$ is a single cone $\kappa$. We take two vectors $w,w'$ in $\kappa$ for which $N_0(w)=N_0(w')$, and try to show that we have $N_0(tw+(1-t)w')=N_0(w)$ for all $t \in [0,1]$. As above, we may assume that this condition fails for some $t$ and derive a contradiction, and even take for $t$ the minimal element of $[0,1]$ for which the condition fails. So we may replace $\kappa$ by the interval $[w,w']$, and $C(X)$ by its fiber over $[w,w']$, which we will denote by $C(X)$ as well for simplicity. Put $w''=tw+(1-t)w'$. As above, we can choose an element in $N_0(w'')-N_0(w)$, which corresponds to a vertex $v''$ in $C(X)$. The key step in the toric proof is that the star of $v''$ in $C(X)$ intersects the fiber of $C(X)$ over $[w,w'')$, which follows from the properness of the map. This statement is not immediately clear in the toroidal situation, but we claim it is nevertheless still correct. To see this, pick a family of maps
\begin{align*}
\operatorname{Spec} k[[\mathbf R_+]] \times [0,1] \rightarrow S
\end{align*}
corresponding to the interval $[w,w']$ in $\kappa$, and a lift
\begin{align*}
\operatorname{Spec} k[[\mathbf R_+]] \rightarrow X
\end{align*}
corresponding to $v''$. We abusively denote the lift by $v''$ as well. Let $x=v''(0) \in X$. Since $X \rightarrow S$ is log smooth, we may choose a chart
\begin{align*}
\xymatrix{ (X,x) \ar[r]^f \ar[d]_p & (V,v) \ar[d]^\pi \\ (S,p(x)) \ar[r]_g & (W,\pi(v))}
\end{align*}
\noindent for the morphism $p$, where: the horizontal morphisms $f,g$ are \'etale; $V,W$ are toric varieties, $\pi$ is a toric morphism, and $v,\pi(v)$ are special points in the torus orbits; and the morphism $N_\mathbf R \rightarrow Q_\mathbf R$ is surjective, where $N,Q$ are the lattices of $V$ and $W$ respectively. Let $[z,z']$ denote the interval corresponding to $[w,w']$ in $Q_\mathbf R$ under $g$, and let $y'' \in N_\mathbf R$ denote the element corresponding to $v''$ under $f$. Since $N_\mathbf R \rightarrow Q_\mathbf R$ is surjective, we may lift $[z,z']$ to an interval $[y,y'] \in N_\mathbf R$ lying over $[z,z']$, with $y'' \in [y,y']$. In other words, if we denote by $T(V),T(W)$ the tori of $V$ and $W$ respectively, we get that the family
\begin{align*}
\operatorname{Spec} k[[\mathbf R]] \times [0,1] \rightarrow T(W) \subset W
\end{align*}
\noindent corresponding to $[z,z']$ lifts to a family of maps
\begin{align*}
\operatorname{Spec} k[[\mathbf R]] \times [0,1] \rightarrow T(V) \subset V
\end{align*}
\noindent which under $\pi$ projects to $[z,z']$, and with $y''$ the map over $z''$, i.e such that the morphism at $t \in [0,1]$ is $y''$. Composing with the inverse of the isomorphism $\mathcal{\hat{O}}_{X,x} \cong \mathcal{\hat{O}}_{V,v}$ we get a family of maps
\begin{align*}
\operatorname{Spec} k[[\mathbf R]] \times [0,1] \rightarrow X
\end{align*}
\noindent which under $p$ compose to the original family $\operatorname{Spec} k[[\mathbf R]] \rightarrow S$ corresponding to $[w,w']$. Now, for each $s \in [0,1]$, the map $\operatorname{Spec} k[[\mathbf R]] \rightarrow X$ extends to a map $\operatorname{Spec} k[[\mathbf R_+]] \rightarrow X$ by the properness of $X \rightarrow S$. But such a map corresponds to an element of $C(X)$, so we get a family of elements in $C(X)$ which at $s=t$ specialize to $v''$. Thus, the star of $v''$ in $C(X)$ contains a lift of the line segment $[w,w'')$, and the same argument as in the toric case goes through.
\end{proof}
Theorem \ref{theorem:maintoric} now carries through without any change in the proof, once we consider the appropriate generalization of the category $\mathcal{C}$ in the toroidal setting. We fix a proper, surjective log smooth morphism $X \rightarrow S$, which gives a cone complex morphism $C(X) \rightarrow C(S)$. We consider the subdivision of $S$ determined by the cones $\{w \in C(S): N_0(w) = \textup{ constant}\}$, and the subdivision of $C(X)$ given by $p^{-1}(\{w \in C(S) :N_0(w) = \textup{ constant}\}) \cap \sigma$. For a cone $\kappa$ whose interior is given by the collection $\{w: N_0(w)=\{\sigma_i\}_1^n\}$, we take the sublattice $Q_\kappa'$of $Q_\kappa$ to be the lattice generated by the elements in $\cap_{i=1}^{n} p(\sigma_i \cap N_{\sigma_i})$, and for a cone $\sigma_i \cap p^{-1}(\kappa)$ we take the sublattice $N'_{\sigma_i \cap p^{-1}(\kappa)} = p^{-1}(Q'_\kappa)$. This construction produces a log smooth morphism of toroidal stacks $\mathcal{X} \rightarrow \mathcal{S}$. \\
\begin{defn}
Let $\mathcal{C}_t$ be the category whose objects are diagrams
\begin{align*}
\xymatrix{Y \ar[r]^j \ar[d]_\pi & X \ar[d]^p \\ T \ar[r]_i & S}
\end{align*}
such that
\begin{itemize}
\item $Y,T$ are toroidal embeddings and $j,\pi,i$ are log smooth morphisms. \\
\item The map $i$ is an alteration. \\
\item $Y$ is a modification of the fiber product $(X \times_S T)_{\textup{tor}}$. \\
\item $\pi$ is weakly semistable. \\
\end{itemize}
A morphism in $\mathcal{C}_t$ is a commutative diagram
\begin{align*}
\xymatrix{Y' \ar[r] \ar[d] & Y \ar[d] \\ T '\ar[r] & T}
\end{align*}
\noindent which commutes with the morphisms to $p:X \rightarrow S$.
\end{defn}
Then we have
\begin{theorem}
The family $\mathcal{X} \rightarrow \mathcal{S}$ is the terminal object of $C_t$.
\end{theorem}
\begin{proof}
The proof of the toric case in \ref{theorem:maintoric} carries through in exactly the same way.
\end{proof}
As a corollary, we get
\newtheorem*{theorem:main}{Theorem \ref{theorem:main}}
\begin{theorem:main}[Universal Weak Semistable Reduction]
Let $X \rightarrow S$ be any proper, surjective, log smooth morphism of toroidal embeddings. Then, there exists a commutative diagram
\begin{align*}
\xymatrix{ \mathcal{X} \ar[r] \ar[d] & X \ar[d] \\ \mathcal{S} \ar[r] & S }
\end{align*}
where $\mathcal{X} \rightarrow \mathcal{S}$ is a representable morphism of tame algebraic stacks, such that for any diagram
\begin{align*}
\xymatrix{ Y \ar[r] \ar[d] & X \ar[d] \\ T \ar[r] & S}
\end{align*}
where $T \rightarrow S$ a toroidal alteration; $Y$ is a modification of the fiber product $(X \times_S T)_{\textup{tor}}$; and $Y \rightarrow T$ is weakly semistable, the morphism $Y \rightarrow T$ factors uniquely through $\mathcal{X} \rightarrow \mathcal{S}$. Furthermore, $\mathcal{X} \times_\mathcal{S} T \rightarrow T$ is weakly semistable.
\end{theorem:main}
\end{subsection}
\begin{subsection}{Generalizations}
We note first of all that the assumption that $T \rightarrow S$ is a toroidal alteration is not important. The theorem holds for an arbitrary log map $T \rightarrow S$, with the same proof. The reason to state the theorem with $T \rightarrow S$ an alteration is to emphasize that $\mathcal{S} \rightarrow S$ is terminal in the category of alterations so as to make contact with the statement of \cite{AK}. The assumption that $k$ is algebraically closed is also not needed, with the caveat that in the non-algebraically closed case we work with toric varieties with a split torus. \\
The assumption that $X$ and $S$ are toroidal is more serious, though it is important only in the following ways: (1) A toroidal embedding has a ``fan'', i.e., its cone complex $C(X)$; (2) a map $X \rightarrow Y$ induces a map of fans $C(X) \rightarrow C(Y)$; (3) weak semistability of $C(X) \rightarrow C(S)$ implies weak semistability of $X \rightarrow S$. In short, for toroidal embeddings there is a functorial theory of fans, and the fan captures enough local information for $X \rightarrow S$. In general, there are many constructions of fans of logarithmic schemes of varying levels of generality -- for instance, in \cite{KKMS},\cite{Ktor}, or \cite{ACMUW}; all constructions are essentially equivalent, and they all satisfy (3) for log smooth morphisms; the constructions however are not functorial. It is unclear for what classes of morphisms a theory of fans satisfying (1),(2),(3) can exist. \\
Since we could not see any reasonable generalization to a wider class of log schemes, we decided to keep the statements in the language of toroidal embeddings as much as possible, as it is more widely known. However, it may be worth noting that given any context for which assumptions (1), (2), and (3) do hold, the main theorem \ref{theorem:main} also holds. For instance, the theorem holds for log curves over a log point. Starting with a log curve $X \rightarrow S$ over a log point, one often needs to perform a log blowup $Y \rightarrow X$; this produces a curve $Y \rightarrow S$, but this is not a log curve (in the sense of F.Kato \cite{fK}), as $Y \rightarrow S$ is not weakly semistable. However, the construction still applies and gives a universal log curve $\c{Y} \rightarrow \c{S}$ over a stacky blowup of the base.
\end{subsection}
\end{section}
{}
\end{document} |
\begin{document}
\title{An incomplete variant of Wilson's congruence}
\begin{abstract}
\noindent
This article examines the nontrivial solutions of the congruence \[ (p-1)\cdots(p-r) \equiv -1 \pmod p. \] We discuss heuristics for the proportion of primes $p$ that have exactly $N$ solutions to this congruence. We supply numerical evidence in favour of these conjectures, and discuss the algorithms used in our calculations.
\end{abstract}
\section{Heuristics and conjectures}
Wilson's Theorem \cite[Theorems 80 and 81]{HW} states that
\[ (p-1)! \equiv -1 \pmod p \]
if and only if $p$ is a prime. Now truncate the factorial after $r$ terms. For which primes $p$ is there an $r$ for which
\begin{equation}\label{wit}
(p-1)\cdots(p-r) \equiv -1 \pmod p ?
\end{equation}
Certainly $r=1$ is trivial; $r=p-1$ follows from Wilson's Theorem, whence $r=p-2$ follows trivially. Henceforth we consider only $2 \leq r \leq p - 3$.
Initially we proceeded as follows. The congruence \eqref{wit} has no solutions if and only if none of the $p - 4$ integers $(p-1)\cdots(p-r) + 1$, for $2\leq r \leq p-3$, are divisible by $p$. The probability that a prime $p$ divides a `random' integer $N$ is $1/p$. Given $m$ random integers chosen independently, the probability that $p$ does not divide any of them is then $(1 - 1/p)^m$. Thus, under heroic randomness and independence assumptions, we expect the proportion of $p$ for which \eqref{wit} has no solutions to be roughly
\[ (1-1/p)^{p-4}\rightarrow e^{-1} \approx 0.36788 \]
when $p$ is large.
Turning to numerical experiment, we find that 429 of the 1229 primes less than $10^4$ have no solutions to \eqref{wit}. The proportion is 0.349, reasonably close to our initial guess.
It turns out that this guess is almost certainly wrong. The remainder of this paper may serve as yet another cautionary tale about the dangers of heuristic probabilistic reasoning in number theory.
First, our independence assumption is not justified. Denoting by $T_r$ the partial product $T_r = (p - 1) \cdots (p - r)$, we have:
\begin{Lem}\label{Lemma1}
For any $2 \leq r \leq p - 3$,
\[ T_r T_{p-r-1} \equiv (-1)^{r+1} \pmod p. \]
\end{Lem}
\begin{proof}
Observe that $T_r \equiv (-1)^r r! \pmod p$, and apply Wilson's Theorem.
\end{proof}
Thus the cases $r$ and $s = p - r - 1$ are not independent. For odd $r$, we see that \eqref{wit} holds for $r$ if and only if it holds for $s$. For even $r$, we see that \eqref{wit} holds for either $r$ or $s$, but not both; and it holds for one of them if and only if $T_r \equiv \pm 1 \pmod p$.
Taking these observations into account, we should posit $p/4 + O(1)$ independent events with probability $1 - 1/p$ corresponding to the odd $r < p/2$, and $p/4 + O(1)$ independent events with probability $1 - 2/p$ corresponding to the even $r < p/2$. Our revised estimate for the proportion of primes for which \eqref{wit} has no solutions is thus
\[ (1 - 1/p)^{p/4} (1 - 2/p)^{p/4} \rightarrow e^{-3/4} \approx 0.47237. \]
This `improved' heuristic is an even worse match for the observed data!
In fact we are on the right track. We have simply forgotten the following arithmetic gem.
\begin{Lem}\label{Lemma2}
Let $p \equiv 3 \pmod 4$. Then
\begin{equation*}
\left(\frac{p-1}2\right)! \equiv (-1)^{\nu_p} \pmod p,
\end{equation*}
where $\nu_p$ is the number of quadratic non-residues $1 < x < p/2$.
\end{Lem}
\begin{proof}
See \cite[Theorem 114]{HW}.
\end{proof}
When $\nu_p$ is even, the congruence \eqref{wit} automatically has the solution $r = (p-1)/2$. To incorporate this into our model, we must address the question as to how often $\nu_p$ is even. Numerical evidence (see Section \ref{sec:computations}) suggests the following conjecture:
\begin{con}\label{con1}
For $p \equiv 3 \pmod 4$, the proportion of $p$ for which $\left(\frac{p-1}2\right)! \equiv 1 \pmod p$ approaches $\frac{1}{2}$ as $p \to \infty$.
\end{con}
We are not aware of this conjecture having appeared before in print, but it has been raised on the Mathoverflow discussion forum \cite{MO}. The problem has been recast by Mordell \cite{Mordell1961} in terms of the class number $h(-p)$ of $\mathbb Q(\sqrt{-p})$. Namely, for $p > 3$ we have
\[ \nu_p = \begin{cases} 0 \pmod 2 & \text{if $h(-p) \equiv 3 \pmod 4$}, \\ 1 \pmod 2 & \text{if $h(-p) \equiv 1 \pmod 4$}. \end{cases} \]
We do not know if this interpretation sheds any light on Conjecture \ref{con1}.
We now revise our model a second time, taking into account Lemma \ref{Lemma2} and Conjecture \ref{con1}. Of those primes satisfying $p \equiv 1 \pmod 4$, asymptotically half the primes, our estimate for the proportion of primes for which \eqref{wit} has no solution is still $e^{-3/4}$. For those primes $p \equiv 3 \pmod 4$ with $\nu_p$ odd, the estimate is again $e^{-3/4}$. According to Conjecture \ref{con1} this accounts for another quarter of the primes. However, for the remaining primes, where $\nu_p$ is even, our estimate is zero. This leads to our main conjecture.
\begin{con}\label{Conmain}
The proportion of primes for which \eqref{wit} has no nontrivial solutions is
\[ \frac{3}{4} e^{-3/4} \approx 0.3542749. \]
\end{con}
Using the same model, we may develop a more refined conjecture that estimates the proportion of primes $p$ for which there are exactly $N$ values of $r$ satisfying \eqref{wit}.
\begin{con}\label{ConN}
Let $N \geq 0$. The proportion of primes $p$ for which \eqref{wit} has exactly $N$ nontrivial solutions is
\[
\frac{e^{-3/4}}{2^{N+1}}
\left( \frac{3}{2} \sum_{k=0}^{\lfloor N/2 \rfloor} \frac{1}{k! (N-2k)!} + \sum_{k=0}^{\lfloor (N-1)/2 \rfloor} \frac{1}{k! (N-1-2k)!}\right).
\]
\end{con}
This formula is derived as follows. For $k \geq 0$, denote by $P_k$ the probability that \eqref{wit} has exactly $k$ odd solutions in the range $3 \leq r < (p-1)/2$. By the discussion following Lemma \ref{Lemma1}, and the usual properties of the binomial distribution, for large $p$ our model suggests that
\[ P_k = \binom{p/4 + O(1)}{k} \left(\frac1p\right)^k \left(1 - \frac1p\right)^{p/4 - k + O(1)} \rightarrow \frac{e^{-1/4}}{4^k k!}. \]
Similarly, for $\ell \geq 0$ denote by $Q_\ell$ the probability that $T_r \equiv \pm 1 \pmod p$ has exactly $\ell$ even solutions in the range $2 \leq r < (p-1)/2$. Then
\[ Q_\ell = \binom{p/4 + O(1)}{\ell} \left(\frac2p\right)^\ell \left(1 - \frac2p\right)^{p/4 - \ell + O(1)} \rightarrow \frac{e^{-1/2}}{2^\ell \ell!}. \]
Assuming that the behaviour for odd and even $r$ is independent, the probability of observing exactly $N$ solutions for $2 \leq r \leq p - 3$, $r \neq (p-1)/2$, should be
\[ \sum_{2k + \ell = N} P_k Q_\ell = \sum_{k=0}^{\lfloor N/2\rfloor} \frac{e^{-1/4}}{4^k k!} \cdot \frac{e^{-1/2}}{2^{N-2k} (N-2k)!} = \frac{e^{-3/4}}{2^N} \sum_{k=0}^{\lfloor N/2\rfloor} \frac1{k!(N-2k)!}. \]
Finally, for $p \equiv 1 \pmod 4$, and for $p \equiv 3 \pmod 4$ with $\nu_p$ odd, the probability that \eqref{wit} has exactly $N$ solutions is given by the above formula (the exceptional value $r = (p-1)/2$ makes a negligible contribution asymptotically). For $p \equiv 3 \pmod 4$ with $\nu_p$ even, we must replace $N$ by $N-1$ to account for the automatic solution $r = (p-1)/2$. Our final estimated probability is thus
\[ \frac34 \left(\frac{e^{-3/4}}{2^N} \sum_{k=0}^{\lfloor N/2\rfloor} \frac1{k!(N-2k)!} \right) + \frac 14 \left(\frac{e^{-3/4}}{2^{N-1}} \sum_{k=0}^{\lfloor (N-1)/2\rfloor} \frac1{k!(N-1-2k)!}\right). \]
\section{Algorithms and computations}
\label{sec:computations}
We first consider the motivating problem, counting the number of nontrivial solutions to \eqref{wit}. For this the na\"ive algorithm appears to be the best available. For each prime $p$ up to some bound, we compute $T_2, T_3, \ldots$, by successive multiplication modulo $p$, and count how many times we see $-1$.
We wrote a simple C implementation of this algorithm, paying some attention to efficient modular arithmetic. We ran it for all primes up to $10^8$. The running time was 22 hours on a 16-core 2.6 GHz Intel Xeon server. Table \ref{tab1} summarises the results. The last column shows the probabilities for each $N$ proposed in Conjecture \ref{ConN}; they are a superb fit for the observed proportions in the previous column.
It is difficult to push the search bound higher. The running time for each prime is essentially linear in $p$, so the cost of handling all $p < x$ grows essentially quadratically in $x$. For example, to increase the search bound to $10^9$ would take about three months on the same hardware. We do not know of any asymptotically faster algorithms for this problem.
\begin{table}[h]
\centering
\caption{Statistics of nontrivial solutions to \eqref{wit} for $p < 10^8$}
\begin{tabular}{rrrr}
\toprule
$N$ & \# Primes with $N$ solutions & Proportion & Conjecture \ref{ConN} \\
\midrule
0 & 2041117 & 0.3542711 & 0.3542749 \\
1 & 1701240 & 0.2952796 & 0.2952291 \\
2 & 1104376 & 0.1916835 & 0.1918989 \\
3 & 553921 & 0.0961426 & 0.0959495 \\
4 & 232308 & 0.0403211 & 0.0402865 \\
5 & 87019 & 0.0151037 & 0.0151612 \\
6 & 29037 & 0.0050399 & 0.0050358 \\
7 & 8887 & 0.0015425 & 0.0015638 \\
8 & 2631 & 0.0004567 & 0.0004423 \\
9 & 692 & 0.0001201 & 0.0001190 \\
10 & 165 & 0.0000286 & 0.0000298 \\
11 & 42 & 0.0000073 & 0.0000071 \\
12 & 17 & 0.0000030 & 0.0000016 \\
13 & 3 & 0.0000005 & 0.0000004 \\
\midrule
Total & 5761455 & 1.0000000 & 1.0000000 \\
\bottomrule
\end{tabular}
\label{tab1}
\end{table}
Next we consider the problem of computing $((p-1)/2)! \pmod p$ for $p \equiv 3 \pmod 4$, in order to test Conjecture \ref{con1}. For this there is a greater variety of algorithms available. The na\"ive approach leads to an $O(p)$ algorithm as above (with a comparable implied constant). An algorithm with complexity $p^{1/2+o(1)}$ can be deduced from \cite{BGS}. We opted to implement an algorithm with average complexity only $(\log p)^{4 + o(1)}$ per prime, using the ``accumulating remainder tree'' technique introduced in \cite{CGH}.
We give a brief sketch of this algorithm. Suppose that we wish to compute $r_p = ((p-1)/2)! \bmod p$ for all primes $p \equiv 3 \pmod 4$ in some interval $2M < p < 2N$, where $M$ and $N$ are positive integers. Consider the binary tree, with nodes indexed by pairs $(a, b)$, where $b > a > 0$ are integers, defined as follows. The root node is $(M, N)$. The children of a given node $(a, b)$ are $(a, c)$ and $(c, b)$, where $c = \lfloor (a + b)/2\rfloor$. For each node let
\[ I_{a,b} = \{k \in \mathbb Z : \text{$k$ odd, } 2a < k < 2b\}. \]
Thus at level $d$, the intervals $I_{a,b}$ partition $I_{M,N}$ into $2^d$ subintervals of roughly equal size. We stop at level $\ell = \lceil \log_2(N - M)\rceil$; at this level each $I_{a,b}$ has cardinality either zero or one.
The algorithm now proceeds as follows. First, for each node let
\[ P_{a,b} = \prod_{\substack{p \in I_{a,b} \\ p \equiv 3 \bmod 4 \\ \text{$p$ prime}}} p, \qquad V_{a,b} = \prod_{k \in I_{a,b}} \frac{k+1}2. \]
Compute $V_{a,b}$ and $P_{a,b}$ for each node, working from the bottom of the tree to the top, using the identities $V_{a,b} = V_{a,c} V_{c,b}$ and $P_{a,b} = P_{a,c} P_{c,b}$. Second, for each node let
\[ X_{a,b} = a! \bmod {P_{a,b}}. \]
Compute $X_{M,N} = M! \bmod P_{M,N}$ using the method of Sch\"onhage (see for example \cite[Prop.~2.3]{CGH}). Then compute $X_{a,b}$ for each node, now working from the top of the tree downwards, using the formulae $X_{a,c} = X_{a,b} \bmod P_{a,c}$ and $X_{c,b} = X_{a,b} V_{a,c} \bmod P_{c,b}$ to descend from each node to its children. Finally, for each $p \equiv 3 \pmod 4$ in the interval $2M < p < 2N$, there is a unique node $(a, b)$ at level $\ell$ such that $p \in I_{a,b}$; for this node we have $I_{a,b} = \{p\}$, $P_{a,b} = p$ and $X_{a,b} = ((p-1)/2)! \pmod p$. For more details, including a complexity analysis, see \cite{CGH}. We mention here only that the complexity bound depends essentially on asymptotically fast algorithms for multiplication and division of large integers.
Using a straightforward implementation of the above algorithm in the Sage computer algebra system \cite{sage}, we computed $r_p$ for all $p < 10^{10}$. To keep memory usage under control, we split the work into intervals $(M, N)$ of size $1.5 \times 10^8$. The total CPU time expended was 4.4 days. The results, shown in Table \ref{tab2}, are in excellent agreement with Conjecture \ref{con1}.
\begin{table}[h]
\centering
\caption{Statistics of $r_p$ for $p < 10^{10}$, $p \equiv 3 \pmod 4$}
\begin{tabular}{lrrr}
\toprule
$X$ & $\# \{p < X\}$ & $\# \{ p < X: r_p = 1 \}$ & proportion \\
\midrule
$10^1$ & 2 & 1 & 0.5000000000 \\
$10^2$ & 13 & 6 & 0.4615384615 \\
$10^3$ & 87 & 43 & 0.4942528736 \\
$10^4$ & 619 & 310 & 0.5008077544 \\
$10^5$ & 4\,808 & 2\,418 & 0.5029118136 \\
$10^6$ & 39\,322 & 19\,704 & 0.5010935354 \\
$10^7$ & 332\,398 & 166\,270 & 0.5002135994 \\
$10^8$ & 2\,880\,950 & 1\,440\,268 & 0.4999281487 \\
$10^9$ & 25\,424\,042 & 12\,713\,329 & 0.5000514474 \\
$10^{10}$ & 227\,529\,235 & 113\,772\,462 & 0.5000344769 \\
\bottomrule
\end{tabular}
\label{tab2}
\end{table}
\end{document} |
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\title{Grothendieck Rings of Theories of Modules}
\keywords{Grothendieck ring; model theory; module; positive primitive formula; abstract simplicial complex, monoid ring}
\subjclass[2010]{03C60, 55U05, 16Y60, 20M25, 06A12}
\maketitle
\begin{center}
AMIT KUBER\footnote{Email \texttt{[email protected]}.
Research partially supported by a School of Mathematics, University of Manchester Scholarship.},
School of Mathematics, \ University of Manchester, \\
Manchester M13 9PL, \ England.
\end{center}
\begin{abstract}
The model-theoretic Grothendieck ring of a first order structure, as defined by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, $K_0(M_\mathcal R)$, of a right $R$-module $M$, where $\mathcal R$ is any unital ring. As a corollary we prove a conjecture of Prest that $K_0(M)$ is non-trivial, whenever $M$ is non-zero. The main proof uses various techniques from the homology theory of simplicial complexes.
\end{abstract}
\section{Introduction}
In \cite{Kra}, Krajic\v{e}k and Scanlon introduced the concept of the model-theoretic Grothendieck ring of a structure. Amongst many other results, they proved that such a Grothendieck ring is nontrivial if and only if the definable subsets of the structure satisfy a version of the combinatorial pigeonhole principle, called the \textquotedblleft onto pigeonhole principle'' ($onto PHP$). Grothendieck rings have been studied for various rings and fields considered as models of a first order theory (see \cite{Kra}, \cite{Clu}, \cite{CluHask}, \cite{DenLoes1} and \cite{DenLoes2}) and they are found to be trivial in many cases (see \cite{Clu},\cite{CluHask}).
Prest conjectured that in stark contrast to the case of rings, for any ring $\mathcal R$, the Grothendieck ring of a nonzero right $\mathcal R$ module $M_\mathcal R$, denoted $K_0(M_\mathcal R)$, is nontrivial. Perera (\cite{Perera}) investigated the problem in his doctoral thesis and found that elementarily equivalent modules have isomorphic Grothendieck rings, which is not the case for general structures. He computed the Grothendieck ring for modules over semisimple rings and showed that they are polynomial rings in finitely many variables over the ring of integers.
In this paper we compute the Grothendieck ring for arbitrary modules and show that they are quotients of monoid rings $\mathbb Z[\mathcal X]$, where $\mathcal X$ is the multiplicative monoid of isomorphism classes of fundamental definable subsets of the module - the $pp$-definable subgroups. This is the content of the main theorem, theorem \ref{FINALgeneral}, which also describes the `invariants ideal' - the ideal of the monoid ring that codes indices of pairs of $pp$-definable subgroups. We further show (corollary \ref{MAINRESULTgeneral}) that there is a split embedding $\mathbb Z\rightarrow K_0(M)$, whenever the module $M$ is nonzero, proving Prest's conjecture.
The proof of the main theorem uses inputs from various mathematical areas like model theory, algebra, combinatorics and algebraic topology. A special case of the main theorem (theorem \ref{FINAL}) is proved at the end of section \ref{spcasemult}. The special case assumes that the theory $T$ of the module $M$ satisfies the model theoretic condition $T=T^{\aleph_0}$. This condition is equivalent to the statement that the invariants ideal is trivial. The reader should note that the proof of the general case of the main theorem is not given in full detail since it develops along lines similar to the special case and uses only a few modifications to incorporate the invariants ideal.
The fundamental theorem of the model theory of modules (theorem \ref{PPET}) states that every definable set is a boolean combination of $pp$-definable sets, but such a boolean combination is far from being unique. We achieve a `uniqueness' result as a by-product of the theory we develop. We call this result the `cell decomposition theorem' (Theorem \ref{CDT1}) which states that definable sets can be represented uniquely using $pp$-definable sets provided the theory $T$ of the module satisfies $T=T^{\aleph_0}$. Though this theorem is not used directly in any other proof, its underlying idea is one of the most important ingredients of the main proof. Based on this idea, we define various classes of definable sets of increasing complexity, namely $pp$-sets, convex sets, blocks and cells. Our strategy to prove every result about a general definable set is to prove it first for convex sets, then blocks and then cells.
An important theme of the paper is the use of geometric and topological ideas in the setting of definable sets. We use the idea of a `neighbourhood' and `localization' to understand the structure of definable sets. We develop a notion of `connectedness' of a definable set in \ref{C} and prove theorem \ref{topconn} which clearly shows the analogy with its topological counterpart.
The main proof takes place at two different levels, which we name `local' and `global' following geometric intuition. We try to describe the ``shape'' of each definable set in terms of integer valued functions called `local characteristics'. These numbers are computed using Euler characteristics of various abstract simplicial complexes which code the ``local geometry'' of the given set. The local data is combined to get a family of integer valued functions, each of which is called a `global characteristic'. The global characteristics enjoy the property of being preserved under definable bijections. The family of such functions is indexed by the elements of the monoid $\mathcal X$ and the functions collate to give the necessary monoid ring.
The rest of the paper is organized as follows. Section \ref{prelim} contains the background material on Grothendieck rings and the model theory of modules. It also describes some important theorems in the homology theory of simplicial complexes. The core part of the proof of the special case is the content of section \ref{spcaseadd}. It introduces the terminology that we use and proves important facts about local and global characteristics. The highlights of this section are theorems \ref{t1} and \ref{t4}. Section \ref{spcasemult} contains proofs of the multiplicative properties of the global characteristics, completing the proof of the special case. Section \ref{gencase} introduces new terminology and the modifications in the proof of the special case necessary to handle the general case. Some applications of the main theorem are discussed in section \ref{appl}. The maps between modules which fit with model theory are called pure embeddings. We study their effect on Grothendieck rings in \ref{pure}. We also show the existence of Grothendieck rings containing nontrivial torsion elements in \ref{tors}. The cell decomposition theorem is proved in \ref{CDT}, whereas the discussion on connectedness is included in \ref{C}. We conclude the paper with section \ref{rmkcom} which contains further remarks on the technique of the proof and mentions some directions for further research in this area.
\section{Preliminaries}\label{prelim}
\subsection{Semirings and Grothendieck rings}\label{SGR}
We recall the notion of a semiring and how to construct a ring in a canonical fashion from a given semiring. A detailed exposition on this material can be found in \cite{Lee}.
Let $L_{ring}=\langle0,1,+,\cdotp\rangle$ be the language of rings.
\begin{definitions}
Any $L_{ring}$ structure $S$ satisfying the following conditions is a commutative \textbf{semiring} with unity.
\begin{itemize}
\item $(S,+,0)$ is a commutative monoid
\item $(S,\cdotp,1)$ is a commutative monoid
\item $a\cdotp 0=0$ for all $a\in S$
\item multiplication ($\cdotp$) distributes over addition ($+$)
\end{itemize}
A \textbf{semiring homomorphism} is an $L_{ring}$-homomorphism.
A semiring $S$ is said to be \textbf{cancellative} if $a+c=b+c\ \Rightarrow\ a=b$ for all $a,b,c\in S$.
\end{definitions}
In \cite{Lee}, a cancellative semiring is called a \emph{halfring}. All the semirings considered here are commutative semirings with unity, allowing the possibility $0=1$.
\begin{definition}
A binary relation $\thicksim$ on a semiring $S$ is said to be a \textbf{congruence relation} if the following properties hold.
\begin{itemize}
\item $\thicksim$ is an equivalence relation
\item $a\thicksim b,c\thicksim d$ for $a,b,c,d\in S$ $\Rightarrow (a+c)\thicksim(b+d),a\cdotp c\thicksim b\cdotp d$
\end{itemize}
\end{definition}
There is a canonical way of constructing a cancellative semiring from any semiring $S$ as stated in the following theorem.
\begin{theorem}\label{QUOCONST}\textbf{Quotient construction}:
Let $S$ be a semiring and let $\thicksim$ be the binary relation defined as follows.
\begin{equation}\label{CANCEL}
For\ a,b\in S,\ a\thicksim b\ \Leftrightarrow\ \exists c\in S,\ a+c=b+c
\end{equation}
Then $\thicksim$ is a congruence relation. If $\tilde{a}$ denotes the $\thicksim$ equivalence class of $a\in S$, then $\tilde{S}:=\{\tilde{a}:a\in S\}$ is a cancellative semiring with respect to the induced addition and multiplication operations. There is a surjective semiring homomorphism $q:S\rightarrow\tilde{S}$ given by $a\mapsto \tilde{a}$. Furthermore, given any cancellative semiring $T$ and a semiring homomorphism $f:S\rightarrow T$, there exists a unique semiring homomorphism $\tilde{f}:\tilde{S}\rightarrow T$ such that the diagram $\xymatrix{{S}\ar[rr] ^{q}\ar[dr]^f & & {\tilde{S}}\ar@{->}[dl]^{\exists !\tilde{f}} \\ & {T}}$ commutes.
\end{theorem}
One can embed a cancellative semiring in a ring in a canonical fashion as stated in the following theorem.
\begin{theorem}\label{GRCONSTR}
\textbf{Ring of Differences for a Cancellative Semiring}:
Let $R$ denote a cancellative semiring and let $E$ denote the binary relation on the set $R\times R$ of ordered pairs of elements from $R$ defined as follows.
\begin{equation}\label{NEGADD}
For\ (a,b),(c,d)\in R\times R,\ (a,b) E (c,d)\ \Leftrightarrow\ a+d=b+c
\end{equation}
Then $R$ is an equivalence relation. If $(a,b)_E$ denotes the $E$-equivalence class of $(a,b)$, then the quotient structure $(R\times R)/E:=\{(a,b)_E:(a,b)\in R\times R\}$ is a ring with respect to the operations given by
\begin{eqnarray}\label{RINGOPER}
(a,b)_E+(c,d)_E&:=&(a+c,b+d)_E\\
(a,b)_E\cdotp(c,d)_E&:=&(a\cdotp c+b\cdotp d,a\cdotp d+b\cdotp c)_E\\
-(a,b)_E&:=&(b,a)_E
\end{eqnarray}
for $(a,b)_E,(c,d)_E\in(R\times R)/E$. We denote the ring $(R\times R)/E$ by $K_0(R)$ following the conventions of K-theory. The semiring $R$ can be embedded into the ring $K_0(R)$ by the semiring homomorphism $i$ given by $a\mapsto (a,0)$. Furthermore, given any ring $T$ and a semiring homomorphism $g:R\rightarrow T$, there exists a unique ring homomorphism $\overline g:K_0(R)\rightarrow T$ such that the diagram $\xymatrix{{R}\ar[rr] ^{i}\ar[dr]^g & & {K_0(R)}\ar@{->} [dl]^{\exists !\overline{g}} \\ & {T}}$ commutes.
\end{theorem}
Note that each of the $E$-equivalence classes of the elements from $R\times R$, as constructed in the previous theorem, contains a pair of the form $(a,0)$ or $(0,a)$ for some $a\in R$.
For a semiring $S$, let $K_0(S)$ denote the ring $K_0(\tilde S)$ for simplicity, where $\tilde S$ is the cancellative semiring obtained from $S$ as stated in the theorem \ref{QUOCONST} and let the canonical map $S\rightarrow K_0(S)$ be denoted by $\eta_S$. We finally note the following result which combines the previous two theorems.
\begin{cor}\label{GrRngAdj}
A semiring $S$ can be embedded in a ring if and only if $S$ is cancellative. Given any ring $T$ and a semiring homomorphism $g:S\rightarrow T$, there exists a unique ring homomorphism $\overline g:K_0(S)\rightarrow T$ such that the diagram $\xymatrix{S\ar[rr] ^{\eta_S}\ar[dr]^g & & {K_0(S)}\ar@{->}[dl]^{\exists !\overline{g}} \\ & {T}}$ commutes.
\end{cor}
This result can be stated in category theoretic language as follows. Let $\mathbf{CSemiRing}$ denote the category of commutative semirings with unity and semiring homomorphisms preserving unity. Let $\mathbf{CRing}$ denote its full subcategory consisting of commutative rings with unity and let $I:\mathbf{CRing}\rightarrow\mathbf{CSemiRing}$ be the inclusion functor. Then $I$ admits a left adjoint, namely $K_0:\mathbf{CSemiRing}\rightarrow\mathbf{CRing}$. For each semiring $S$, the ring $K_0(S)$ is called the Grothendieck Ring constructed from $S$. If $\eta$ is the unit of the adjunction, the diagram in the previous corollary represents the universal property of the adjunction.
\subsection{Grothendieck rings of first order structures}\label{GRFOS}
We aim to introduce the notion of the model theoretic Grothendieck ring of a first order structure in this section. This account is based on \cite{Kra}. After setting some background in model theory, we state how to construct the semiring of definable isomorphism classes of definable subsets of finite cartesian powers of the given structure $M$. Following the method described in the previous section, we then construct the Grothendieck ring $K_0(M)$.
Let $L$ denote any language and $M$ denote any first order $L$-structure. The term definable will always mean definable with parameters from $M$.
\begin{definitions}
For every $n\geq 1$, we define $\mathrm{Def}(M^n)$ to be the collection of all definable subsets of $M^n$. We define $\overline{\mathrm{Def}}(M):=\bigcup_{n\geq 1}\mathrm{Def}(M^n)$.
\end{definitions}
\begin{definition}\label{defiso}
We say that two definable sets $A,B\in\overline{\mathrm{Def}}(M)$ are \textbf{definably isomorphic} if there exists a definable bijection between them, i.e., a bijection $f:A\rightarrow B$ such that the graph $Gr(f)\in\overline{\mathrm{Def}}(M)$. This is an equivalence relation on $\overline{\mathrm{Def}}(M)$ and the equivalence class of a set $A$ is denoted by $[A]$. We use $\widetilde{\mathrm{Def}}(M)$ to denote the set of all equivalence classes with respect to this relation. We use $[-]:\overline{\mathrm{Def}}(M)\rightarrow\widetilde{\mathrm{Def}}(M)$ to denote the surjective map defined by $A\mapsto[A]$.
\end{definition}
We can regard $\widetilde{\mathrm{Def}}(M)$ as an $L_{ring}$-structure. In fact, it is a semiring with respect to the operations defined as follows.
\begin{itemize}
\item $0 := [\emptyset]$
\item $1 := [\{*\}]$ for any singleton subset $\{*\}$ of $M$
\item $[A]+[B] := [A'\sqcup B']$ for $A'\in[A],B'\in[B]$ such that $A'\cap B'=\emptyset$
\item $[A]\cdotp[B] := [A\times B]$
\end{itemize}
(NB: We use $\sqcup$ to denote disjoint unions.)
Now we are ready to give the most important definition.
\begin{definition}
We define the \textbf{model-theoretic Grothendieck ring of the first order structure} $M$, denoted by $K_0(M)$, to be the ring $K_0(\widetilde{\mathrm{Def}}(M))$ obtained from corollary \ref{GrRngAdj}, where the semiring structure on $\widetilde{\mathrm{Def}}(M)$ is as defined above.
\end{definition}
This ring captures the definable combinatorics of the structure $M$. We are interested to know whether $K_0(M)=\{0\}$. It is useful to consider some definable combinatorial aspects to tackle this problem.
\begin{definition}
We say that an infinite structure $M$ satisfies the \textbf{pigeonhole principle} if for each $A\in\overline{\mathrm{Def}}(M)$, each definable injection $f:A\rightarrowtail A$ is an isomorphism. We write this as $M\vDash PHP$.
\end{definition}
This condition is very strong to be true for many structures. As an example, consider the additive group of integers $\mathbb Z$ in the language of abelian groups. The function $\mathbb Z\xrightarrow{(-)\times 2}\mathbb Z$ is a definable injection but not an isomorphism. So it is useful to consider some weaker forms. Though there are several of them (see \cite{Kra}), we note the one important for us.
\begin{definition}
We say that an infinite structure $M$ satisfies the \textbf{onto pigeonhole principle} if for each $A\in\overline{\mathrm{Def}}(M)$ and each definable injection $f:A\rightarrowtail A$, we have $f(A)\neq A\setminus\{a\}$ for any $a\in A$. We write this as $M\vDash ontoPHP$.
\end{definition}
The following proposition gives the necessary and sufficient condition for $K_0(M)$ to be nontrivial (i.e. $0\neq 1$ in $K_0(M)$). We include a proof for the sake of completeness.
\begin{pro}
Given any infinite structure $M$, $K_0(M)\neq\{0\}$ if and only if $M\vDash ontoPHP$.
\end{pro}
\begin{proof}
Recall the construction of the cancellative semiring from (\ref{CANCEL}). The condition $0=1$ in $K_0(M)$ is thus equivalent to the statement that for some $A\in\overline{\mathrm{Def}}(M)$, we have $0+[A]=1+[A]$. This is precisely the statement that $M\nvDash ontoPHP$.
\end{proof}
\textbf{A brief survey of known Grothendieck Rings:} Very few examples of Grothendieck rings are known in general. If $M$ is a finite structure, then $K_0(M)\cong\mathbb Z$. Kraji\v{c}ek and Scanlon have shown in \cite[Example\,3.6]{Kra} that $K_0(\mathbb R)\cong\mathbb Z$ using the dimension theory and cell decomposition theorem for o-minimal structures, where $\mathbb R$ denotes the real closed field. Cluckers and Haskell (\cite{Clu}, \cite{CluHask}) proved that the fields of p-adic numbers have trivial Grothendieck rings, by constructing definable bijections from a set to the same set minus a point. Denef and Loeser (\cite{DenLoes1},\cite{DenLoes2}) have found that the Grothendieck ring $K_0(\mathbb C)$ of the field $\mathbb C$ of complex numbers regarded as an $L_{ring}$-structure admits the ring $\mathbb Z[X;Y]$ as a quotient. Kraji\v{c}ek and Scanlon have strengthened this result and shown that $K_0(\mathbb C)$ contains an algebraically independent set of size continuum, and hence the ring $\mathbb Z[X_i:i\in\mathfrak{c}]$ embeds into $K_0(\mathbb C)$. Perera showed in \cite[Theorem\,4.3.1]{Perera} that $K_0(M)\cong\mathbb Z[X]$ whenever $M$ is an infinite module over an infinite division ring. Prest conjectured \cite[Ch.\,8,\,Conjecture A]{Perera} that $K_0(M)$ is nontrivial for all nonzero right $\mathcal R$-modules $M$. We prove that $K_0(M)$ is actually a quotient of a monoid ring and furthermore it is nontrivial. Most of the paper is devoted to the proof of this statement.
\subsection{Euler characteristic of simplicial complexes}\label{ECSC}
We introduce the concept of an abstract simplicial complex and a couple of ways to calculate its Euler characteristic. We also state some important results in the homology theory of simplicial complexes. The material on homology and relative homology presented in this section is taken from \cite[II.4]{FerPic}. This theory provides the basis for the analysis of `local characteristics' in \ref{LC}.
\begin{definition}
An \textbf{abstract simplicial complex} is a pair $(X,\mathcal{K})$ where $X$ is a finite set and $\mathcal{K}$ is a collection of subsets of $X$ satisfying the following properties.
\begin{itemize}
\item $\emptyset\notin\mathcal{K}$
\item $\{x\}\in\mathcal{K}$ for each $x\in X$
\item if $F\in\mathcal{K}$ and $\emptyset\neq F'\subsetneq F$, then $F'\in\mathcal{K}$
\end{itemize}
\end{definition}
We usually identify the simplicial complex $(X,\mathcal{K})$ with $\mathcal{K}$. The elements $F\in\mathcal{K}$ are called the \textbf{faces} of the complex and the singleton faces are called the \textbf{vertices} of the complex. We use $\mathcal V(\mathcal K)$ to denote the set of vertices of $\mathcal K$.
Let $\Delta^k:=\mathbb{P}([k+1])\setminus\{\emptyset\}$ denote the \textbf{standard $k$-simplex}, where $\mathbb P$ denotes the power set operator and $[k+1]=\{1,2,\hdots,k+1\}$ for $k\geq 0$. We define the \textbf{geometric realization} of the standard $k$-simplex, denoted $|\Delta^k|$, to be the set of all points of $\mathbb R^{k+1}$ which can be expressed as a convex linear combination of the standard basis vectors of $\mathbb R^{k+1}$. In fact we can associate to every abstract simplicial complex a topological space $|\mathcal K|$, called its geometric realization. This topological space is constructed by `gluing together' the geometric realizations of its simplices.
We assign dimension to every face $F\in\mathcal{K}$ by stating $\mathrm{dim} F:=|F|-1$ and we say that the \textbf{dimension of the complex} is the maximum of the dimensions of its faces.
\begin{definition}\label{Euler}
We define the \textbf{Euler characteristic} of the complex $\mathcal{K}$, denoted $\chi(\mathcal{K})$, to be the integer $\Sigma_{n=0}^{\mathrm{dim}\mathcal{K}}(-1)^n v_n$ where $v_n$ is the number of faces in $\mathcal{K}$ with dimension $n$.
\end{definition}
It is easy to check that $\chi(\Delta^k)=1$ for each $k\geq 0$. Since we also allow our complex to be empty, we define $\chi(\emptyset):=0$ though $\mathrm{dim}\emptyset$ is undefined.
There is yet another way to obtain the Euler characteristics of simplicial complexes, via homology. The word homology will always mean simplicial homology with integer coefficients in this paper. If $b_n$ denotes the $n^{th}$ Betti number of the simplicial complex $\mathcal{K}$ (i.e. the rank of the $n^{th}$ homology group $H_n(\mathcal{K})$), then we have the identity $\chi(\mathcal{K})=\Sigma_{n=0}^\infty(-1)^n b_n$ where the sum on the right hand side is finite. We use the notation $C_*(\mathcal K)$ to denote the chain complex $C_n(\mathcal K)_{n\geq 0}$ and $H_*(\mathcal K)$ to denote the chain complex $(H_n(\mathcal K))_{n\geq0}$, where $C_n(\mathcal K)$ is the free abelian group generated by the set of $n$-simplices in $\mathcal K$.
The following result states that homology is a homotopy invariant. It will be useful in proving a key result (proposition \ref{p1}).
\begin{theorem}\label{HTPYINV}
If $\mathcal K_1$ and $\mathcal K_2$ (meaning, their geometric realizations) are homotopy equivalent, then $H_*(\mathcal K_1)\cong H_*(\mathcal K_2)$.
\end{theorem}
The definition of Euler characteristic in terms of Betti numbers gives the following corollary.
\begin{cor}\label{EULHTPY}
If $\mathcal K_1$ and $\mathcal K_2$ are homotopy equivalent, then $\chi(\mathcal K_1)=\chi(\mathcal K_2)$.
\end{cor}
The homology groups $H_n(\mathcal K)$, for $n\geq 1$, calculate the number of ``$n$-dimensional holes'' in the geometric realization of the complex $\mathcal K$. But sometimes it is important to ignore the data present in a smaller part of the given structure. This can be done in two ways, viz. using the cone construction for a subcomplex or by using relative homology. Given a complex $\mathcal{K}$ and a subcomplex $\mathcal{Q}\subseteq\mathcal{K}$, we write $\mathcal K\cup\mathrm{Cone}(\mathcal{Q})$ for the simplicial complex whose vertex set is $\mathcal V(\mathcal K)\cup\{x\}$, where $x\notin\mathcal V(\mathcal K)$, and the faces are $\mathcal{K}\cup\{\{x\}\cup F:F\in\mathcal{Q}\}$. We say that $x$ is the \textbf{apex} of the cone. In the same situation, we use the notation $H_n(\mathcal K;\mathcal{Q})$ to denote the $n^{th}$ homology of $\mathcal K$ relative to $\mathcal{Q}$.
The following theorem connects the relative homologies with the homologies of the original complexes.
\begin{theorem}(see \cite[Theorem\,2.16]{Hatcher})\label{LONGEXACT}
Given a pair of simplicial complexes $\mathcal{Q}\subset\mathcal K$, we have the following long exact sequence of homologies.\\
\begin{equation*}
\cdots\rightarrow H_n(\mathcal{Q})\rightarrow H_n(\mathcal{K})\rightarrow H_n(\mathcal K;\mathcal{Q})\rightarrow H_{n-1}(\mathcal{Q})\rightarrow\cdots\rightarrow H_0(\mathcal K;\mathcal{Q})\rightarrow 0
\end{equation*}
\end{theorem}
We shall also make use of the following result.
\begin{theorem}
Given a pair of simplicial complexes $\mathcal{Q}\subseteq\mathcal{K}$, we have $H_n(\mathcal K;\mathcal{Q}) \cong H_n(\mathcal K\cup\mathrm{Cone}(\mathcal{Q}))$ for $n\geq 1$ and $H_0(\mathcal K\cup \mathrm{Cone}(\mathcal{Q}))\cong H_0(\mathcal K;\mathcal{Q})\oplus \mathbb Z$.
\end{theorem}
\begin{illust}
Let $\mathcal K=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\}\}$ and $\mathcal Q$ denote the subcomplex $\{\{1\},\{3\}\}$. Then
\begin{eqnarray*}
H_n(\mathcal K)&=&\begin{cases}
\mathbb Z, &\mbox{if } n=0,\\
0, &\mbox{otherwise}
\end{cases}\\
H_n(\mathcal Q)&=&\begin{cases}
\mathbb Z\oplus\mathbb Z, &\mbox{if } n=0,\\
0, &\mbox{otherwise}
\end{cases}\\
H_n(\mathcal K;\mathcal Q)&=&\begin{cases}
\mathbb Z, &\mbox{if } n=1,\\
0, &\mbox{otherwise}
\end{cases}\\
H_n(\mathcal K\cup\mathrm{Cone}(\mathcal Q))&=&\begin{cases}
\mathbb Z, &\mbox{if } n=0,1,\\
0, &\mbox{otherwise}
\end{cases}
\end{eqnarray*}
\end{illust}
Combining the previous two results with the definition of Euler characteristic, we get
\begin{cor}\label{HMLGCONE}
For a pair of simplicial complexes $\mathcal{Q}\subseteq\mathcal K$, $\chi(\mathcal K\cup \mathrm{Cone}(\mathcal{Q}))+\chi(\mathcal{Q})=\chi(\mathcal K)+1$.
\end{cor}
\subsection{Products of simplicial complexes}\label{PRODSIMPCOMP}
We define various products of simplicial complexes and study their interrelations. The inclusion-exclusion principle stated in lemma \ref{ecdpl} is equivalent to the statement that `local characteristics are multiplicative' (lemma \ref{localcharmult}).
Let $\mathcal K$ and $\mathcal Q$ be two simplicial complexes with vertex sets $\mathcal{V}(\mathcal K)$ and $\mathcal{V}(\mathcal Q)$ respectively and let $\pi_1:\mathcal{V}(\mathcal K)\times\mathcal{V}(\mathcal Q)\rightarrow \mathcal{V}(\mathcal K)$ and $\pi_2:\mathcal{V}(\mathcal K)\times\mathcal{V}(\mathcal Q)\rightarrow \mathcal{V} (\mathcal Q)$ denote the projection maps. We define two simplicial complexes with the vertex set $\mathcal V(\mathcal K)\times\mathcal V(\mathcal Q)$. The following product is defined in \cite[\S 3]{EilZil}.
\begin{definition}
The \textbf{simplicial product} $\mathcal K\vartriangle\mathcal Q$ of two simplicial complexes $\mathcal K$ and $\mathcal Q$ is a simplicial complex with vertex set $\mathcal V(\mathcal K)\times\mathcal V(\mathcal Q)$ where a nonempty set $F\subseteq\mathcal V(\mathcal K)\times\mathcal V(\mathcal Q)$ is a face of $\mathcal K\vartriangle\mathcal Q$ if and only if $\pi_1(F)\in\mathcal K$ and $\pi_2(F)\in\mathcal Q$.
\end{definition}
\begin{definition}
The \textbf{disjunctive product} $\mathcal K\boxtimes\mathcal Q$ of two simplicial complexes $\mathcal K$ and $\mathcal Q$ is a simplicial complex with vertex set $\mathcal V(\mathcal K)\times\mathcal V(\mathcal Q)$ where a nonempty set $F\subseteq\mathcal V(\mathcal K)\times\mathcal V(\mathcal Q)$ is a face of $\mathcal K\boxtimes\mathcal Q$ if and only if $\pi_1(F)\in\mathcal K$ or $\pi_2(F)\in\mathcal Q$.
\end{definition}
Observe that the previous two definitions are identical except for the word `and' in the former is replaced by the word `or' in the latter. Thus the simplicial product $\mathcal K\vartriangle\mathcal Q$ is always contained in the disjunctive product $\mathcal K\boxtimes\mathcal Q$.
\begin{illust}\label{spdp}
Let $\mathcal K=\{\{1\},\{2\}\}$ denote the complex consisting precisely of two vertices. Then $\mathcal K\vartriangle\mathcal K$ contains only the vertices of the `square' $\mathcal K\boxtimes\mathcal K$ given by $\{\{(1,1)\},\{(1,2)\},\{(2,1)\},\{(2,2)\},\{(1,1),(1,2)\},\{(2,1),(2,2)\},\{(1,1),(2,1)\},\\\{(1,2),(2,2)\}\}$. For each $k\geq 0$ the complex $\mathcal K\vartriangle\Delta^k$ is the union of two disjoint copies of $\Delta^k$, whereas the complex $\mathcal K\boxtimes\Delta^k$ is a copy of $\Delta^{2k+1}$.
\end{illust}
The main aim of this section is to prove the following lemma about the Euler characteristic of the disjunctive product.
\begin{lemma}\label{ecdpl}
The Euler characteristics of two simplicial complexes $\mathcal K$ and $\mathcal Q$ satisfy
\begin{equation}\label{ecdp}
\chi(\mathcal K\boxtimes\mathcal Q)=\chi(\mathcal K)+\chi(\mathcal Q)-\chi(\mathcal K)\chi(\mathcal Q).
\end{equation}
\end{lemma}
\begin{illust}
Let $\mathcal K$ be as defined in \ref{spdp}. Then we observe that $\chi(\mathcal K)=2$. Since $\mathcal K\boxtimes\mathcal K$ contains $4$ vertices and $4$ edges, we get $\chi(\mathcal K\boxtimes\mathcal K)=0=2\chi(\mathcal K)-\chi(\mathcal K)^2$ verifying equation (\ref{ecdp}) in this case.
\end{illust}
The proof of the lemma uses tensor products of chain complexes.
\begin{definition}
Let $C_*=\{C_n,\partial_n\}_{n\geq 0}$ and $D_*=\{D_n,\delta_n\}_{n\geq 0}$ denote two bounded chain complexes of abelian groups. The \textbf{tensor product complex} $C_*\otimes D_*=\{(C_*\otimes D_*)_n,d_n\}_{n\geq 0}$ is defined by
\begin{eqnarray*}
(C_*\otimes D_*)_n&=&\bigoplus_{i+j=n}C_i\otimes D_j,\\
d_n(a_i\otimes b_j)&=&\partial_i(a_i)\otimes b_j+(-1)^{i}a_i\otimes\delta_j(b_j).
\end{eqnarray*}
\end{definition}
\begin{illust}\label{tensor}
We compute the tensor product $C_*(\partial\Delta^2)\otimes C_n(\Delta^1)$ as an example, where $\partial\Delta^2$ denotes the boundary of $\Delta^2$.
\begin{eqnarray*}
C_n(\partial\Delta^2)&=&\begin{cases}
\mathbb Z\oplus\mathbb Z\oplus\mathbb Z, &\mbox{if } n=0,1,\\
0, &\mbox{otherwise}
\end{cases}\\
C_n(\Delta^1)&=&\begin{cases}
\mathbb Z\oplus\mathbb Z, &\mbox{if } n=0,\\
\mathbb Z, &\mbox{if } n=1,\\
0, &\mbox{otherwise}
\end{cases}\\
C_n(\partial\Delta^2)\otimes C_n(\Delta^1)&=&\begin{cases}
\oplus_{i=1}^6\mathbb Z, &\mbox{if } n=0,\\
\oplus_{i=1}^9\mathbb Z, &\mbox{if } n=1,\\
\oplus_{i=1}^3\mathbb Z, &\mbox{if } n=2,\\
0, &\mbox{otherwise}
\end{cases}
\end{eqnarray*}
\end{illust}
There is yet one more product of simplicial complexes, viz., the cartesian product, defined in the literature (see \cite{EilZil}). We avoid its use by dealing with the product of geometric realizations (with the product topology). The homology of such (finite) product spaces is easily computed using triangulation. We first note that the Euler characteristic is multiplicative.
\begin{pro}\label{euprod}(see \cite[p.205,\,Ex.\,B.4]{Spa})
Let $\mathcal K$ and $\mathcal Q$ be any simplicial complexes. Then
\begin{equation*}\chi(|\mathcal K|\times|\mathcal Q|)=\chi(\mathcal K)\chi(\mathcal Q).\end{equation*}
\end{pro}
A famous theorem of Eilenberg and Zilber (see \cite{EilZil}) connects the homologies of two semi-simplicial complexes (a term used in 1950 that includes the class of simplicial complexes) with that of their cartesian product. We state this result below using the cartesian product of their geometric realizations. More details can be found in \cite[\S 2.1]{Hatcher} and \cite[\S III.6]{FerPic}.
\begin{theorem}\label{EZT}(see. \cite[\S III.6.2]{FerPic})
Let $\mathcal K$ and $\mathcal Q$ be any two simplicial complexes. Then we have \\
$H_*(|\mathcal K|\times|\mathcal Q|)\cong H_*(C_*(\mathcal K)\otimes C_*(\mathcal Q))$.
\end{theorem}
Furthermore, Eilenberg and Zilber state the following corollary of the previous theorem in \cite{EilZil}.
\begin{cor}\label{simpprod}
Let $\mathcal K$ and $\mathcal Q$ be any two simplicial complexes. Then\\
$H_*(\mathcal K\vartriangle\mathcal Q)\cong H_*(C_*(\mathcal K)\otimes C_*(\mathcal Q))$.
\end{cor}
\begin{illust}
We continue the example in \ref{tensor}. The computation of the boundary operators yields
\begin{equation*}
H_n(C_*(\partial\Delta^2)\otimes C_*(\Delta^1))=\begin{cases}
\mathbb Z, &\mbox{if } n=0,1,\\
0, &\mbox{otherwise}
\end{cases}
\end{equation*}
The space $|\partial\Delta^2|\times|\Delta^1|$ is a cylinder which is homotopy equivalent to $S^1$. Hence $H_n(|\partial\Delta^2|\times|\Delta^1|)=\mathbb Z$ for $n=0,1$ and is zero for other values of $n$. This completes the illustration of theorem \ref{EZT}.
Furthermore the complex $\partial\Delta^2\vartriangle\Delta^1$ is the union of three copies of $\Delta^3$ each of which shares exactly one edge (i.e. a copy of $\Delta^1$) with every other copy and these three edges are pairwise disjoint. It can be easily see that this complex (i.e. its geometric realization) is homotopy equivalent to the circle and hence the conclusions of the corollary \ref{simpprod} hold.
\end{illust}
\begin{proof} (Lemma \ref{ecdpl})
We first observe that there is an embedding of simplicial complexes $\iota_1:\mathcal K\vartriangle(\Delta^{|\mathcal V(\mathcal Q)|-1})\rightarrow\mathcal K\boxtimes\mathcal Q$ induced by some fixed enumeration of $\mathcal V(\mathcal Q)$. Similarly there is an embedding $\iota_2:(\Delta^{|\mathcal V(\mathcal K)|-1})\vartriangle\mathcal Q\rightarrow\mathcal K\boxtimes\mathcal Q$ induced by some fixed enumeration of $\mathcal V(\mathcal K)$. Furthermore, the intersection $\iota_1(\mathcal K\vartriangle(\Delta^{|\mathcal V(\mathcal Q)|-1}))\cap\iota_2((\Delta^{|\mathcal V(\mathcal K)|-1})\vartriangle\mathcal Q)$ is precisely the complex $\mathcal K\vartriangle\mathcal Q$.
This gives us, using the counting definition of the Euler characteristics, that the identity
\begin{equation}\label{ecintermediate}
\chi(\mathcal K\boxtimes\mathcal Q)=\chi(\mathcal K\vartriangle(\Delta^{|\mathcal V(\mathcal Q)|-1}))+\chi((\Delta^{|\mathcal V(\mathcal K)|-1})\vartriangle\mathcal Q)-\chi(\mathcal K\vartriangle\mathcal Q)
\end{equation}
holds.
If we can prove that $\chi(\mathcal K\vartriangle\mathcal Q)=\chi(\mathcal K)\chi(\mathcal Q)$ for any simplicial complexes $\mathcal K$ and $\mathcal Q$, then (\ref{ecintermediate}) will yield (\ref{ecdp}) since $\chi(\Delta^k)=1$ for each $k\geq 0$.
Now we have
$H_*(\mathcal K\vartriangle\mathcal Q)\cong H_*(C_*(\mathcal K)\otimes C_*(\mathcal Q))\cong H_*(C_*(|\mathcal K|\times|\mathcal Q|))$, where the first isomorphism is by \ref{simpprod} and the second by \ref{EZT}.
Hence we have $\chi(\mathcal K\vartriangle\mathcal Q)=\chi(|\mathcal K|\times|\mathcal Q|)=\chi(\mathcal K)\chi(\mathcal Q)$ by \ref{euprod} as required. This completes the proof.
\end{proof}
\subsection{Model theory of modules}\label{MTM}
We introduce the terminology and some basic results in the model theory of modules in this section. A detailed exposition can be found in \cite{PreBk}. Instead of working with formulas all the time, we fix a structure and work with the definable subsets of finite cartesian powers of that structure.
Let $\mathcal{R}$ be a fixed ring with unity. Then every right $\mathcal{R}$-module $M$ is a structure for the first order language $L_\mathcal{R}=\langle 0,+,-,m_r:r\in\mathcal{R}\rangle$, where each $m_r$ is a unary function symbol representing the action of right multiplication by the element $r$. When we are working in a fixed module $M$, we usually write the element $m_r(a)$ in formulas as $ar$ for each $a\in M$.
First we note the following result of Perera which states that the Grothendieck ring of a module is an invariant of its theory. A proof of this proposition can be found at the end of section \ref{gencase} as a corollary of theorem \ref{FINALgeneral}.
\begin{pro}(see \cite[Corollary\,5.3.2]{Perera})\label{eleequivmod}
Let $M$ and $N$ be two right $\mathcal R$-modules such that $M\equiv N$, then $K_0(M)\cong K_0(N)$.
\end{pro}
Let us fix a right $\mathcal{R}$-module $M$. Then every definable subset of $M^n$ for any $n\geq 1$ can be expressed in terms of certain fundamental definable subsets of $M^n$. In order to state this partial quantifier elimination result, we first define the formulas which define these fundamental subsets.
\begin{definition}
A \textbf{positive primitive formula} (\textbf{pp-formula} for short) is a formula in the language $L_\mathcal{R}$ which is equivalent to one of the form
\begin{equation*}
\phi(x_1,x_2,\hdots,x_n)=\exists y_1\exists y_2\hdots\exists y_m\bigwedge_{i=1}^t\left(\sum_{j=1}^n x_j r_{ij}+\sum_{k=1}^m y_ks_{ik}+c_i=0\right),
\end{equation*}
where $r_{ij},s_{ik}\in\mathcal R$ and the $c_i$ are parameters from $M$.
\end{definition}
A subset of $M^n$ which is defined by a $pp$-formula (with parameters) will be called a $pp$-set. If a subgroup of $M^n$ is $pp$-definable, then its cosets are also $pp$-definable. The following lemma is well known and a proof can be found in \cite[Corollary\,2.2]{PreBk}.
\begin{lemma}
Every parameter-free $pp$-formula $\phi(\overline x)$ defines a subgroup of $M^n$, where $n$ is the length of $\overline x$. If $\phi(\overline x)$ contains parameters from $M$, then it defines either the empty set or a coset of a $pp$-definable subgroup of $M^n$. Furthermore, the conjunction of two $pp$-formulas is (equivalent to) a $pp$-formula.
\end{lemma}
Let $\mathcal{L}_n$ denote the meet-semilattice of all $pp$-subsets of $M^n$ ordered by the inclusion relation $\subseteq$. We will use the notation $\mathcal{L}_n(M_\mathcal{R})$, specifying the module, when we work with more than one module at a time.
\begin{definition}
Let $M$ be a right $\mathcal R$-module and let $A,B\in\mathcal L_n$ be subgroups. We define the invariant $\mathrm{Inv}(M,A,B)$ to be the index $[A:A\cap B]$ if this if finite or $\infty$ otherwise.
\end{definition}
An \textbf{invariants condition} is a statement that a given invariant is greater than or equal to or less than a certain number. These invariant conditions can be expressed as sentences in $L_\mathcal R$. An \textbf{invariants statement} is a finite boolean combination of invariants conditions.
We are now ready to state the promised fundamental theorem of the model theory of modules.
\begin{theorem}(see \cite{Baur})\label{PPET}
Let $T$ be the theory of right $\mathcal R$-modules and $\phi(\overline x)$ be an $L_\mathcal R$ formula (possibly with parameters). Then we have
\begin{equation*}
T\vDash \forall \overline x (\phi(\overline x)\leftrightarrow\bigvee_{i=1}^m\left(\psi_i(\overline x)\wedge\bigwedge_{j=1}^{l_i}\neg\chi_{ij}(\overline x)\right)\wedge I),
\end{equation*}
where $I$ is an invariants statement and $\psi_i(\overline x),\chi_{ij}(\overline x)$ are $pp$-formulas.
\end{theorem}
We may assume that $\chi_{ij}(M)\subseteq\psi_i(M)$ for each value of $i$ and $j$, otherwise we redefine $\chi_{ij}$ as $\chi_{ij}\wedge\psi_i$. When we work in a complete theory, the invariants statements will vanish and hence we get the following form.
\begin{theorem}
For each $n\geq 1$, every definable subset of $M^n$ can be expressed as a finite boolean combination of $pp$-subsets of $M^n$.
\end{theorem}
Using this result together with the meet-semilattice structure of $\mathcal{L}_n$, we can express each definable subset of $M^n$ in a ``disjunctive normal form'' of $pp$-sets. Expressing a definable set as a disjoint union helps to break it down to certain low complexity fragments, each of which has a specific shape given by the normal form. A proof of this result can be found in \cite[Lemma\,3.2.1]{Perera}.
\begin{lemma}\label{REP}
Every definable subset of $M^n$ can be written as $\bigsqcup_{i=1}^t (A_i\setminus(\bigcup_{j=1}^{s_i} B_{ij}))$ for some $A_i,B_{ij}\in\mathcal{L}_n$.
\end{lemma}
The following lemma is one of the important tools in our analysis.
\begin{lemma}\label{NL}
\textbf{Neumann's Lemma} (see \cite[Theorem\,2.12]{PreBk})\\
If $H$ and $G_i$ are subgroups of some group $(K,+)$ and a coset of $H$ is covered by a finite union of cosets of the $G_i$, then this coset of $H$ is in fact covered by the union of just those cosets of $G_i$ where $G_i$ is of finite index in $H$, i.e. where $[H:G_i]:=[H:H\cap G_i]$ is finite.
\begin{equation*}
c+H\subseteq \bigcup_{i\in I}c_i+G_i\ \ \ \Rightarrow\ \ \ c+H\subseteq \bigcup_{i\in I_0}c_i+G_i,
\end{equation*}
where $I_0=\{i\in I:[H:G_i]<\infty\}$.
\end{lemma}
\section{Special Case: Additive Structure}\label{spcaseadd}
\subsection{The condition $\mathrm{T=T^{\aleph_0}}$}\label{TATA}
Let $M$ be a fixed right $\mathcal R$-module. For brevity we denote $Th(M)$ by $T$. We work with this fixed module throughout this section. If $X,Y\subseteq M^n$, $X,Y\neq\emptyset$, then we use the Minkowski sum notation $X+Y$ to denote the set $\{x+y:x\in X,y\in Y\}$. In case $X=\{x\}$, we use $x+Y$ to denote $X+Y$.
\begin{pro}\label{EC}
The following conditions are equivalent for a module $M$.
\begin{enumerate}
\item $\mathrm{Inv}(M;A,B)$ is either equal to $1$ or $\infty$ for each $A,B\in\mathcal L_n$ such that $0\in A\cap B$, for each $n\geq 1$,
\item $M\equiv M\oplus M$,
\item $M\equiv M^{(\aleph_0)}$.
\end{enumerate}
\end{pro}
\begin{definition}
The theory $T=Th(M)$ is said to satisfy the condition $T=T^{\aleph_0}$ if either (and hence all) of the conditions of proposition \ref{EC} hold.
\end{definition}
We wish to add yet one more condition to the list. The rest of this section is devoted to formulating the condition and deriving its consequences.
We need to introduce some new notation to do this. Let us denote the set of all finite subsets of $\mathcal L_n\setminus\{\emptyset\}$ by $\mathcal P_n$ and the set of all finite antichains in $\mathcal L_n\setminus\{\emptyset\}$ by $\mathcal A_n$. Clearly $\mathcal A_n\subseteq\mathcal P_n$ for each $n\geq 1$. We use lowercase Greek letters to denote elements of $\mathcal A_n$ and $\mathcal P_n$.
\begin{definition}
A definable subset $A$ of $M^n$ will be called $pp$-\textbf{convex} if there is some $\alpha\in\mathcal P_n$ such that $A=\bigcup\alpha$.
\end{definition}
Neumann's lemma (\ref{NL}) takes the following simple form if we add the equivalent conditions of \ref{EC} to our hypotheses.
\begin{cor}\label{NLU}
Suppose $T=T^{\aleph_0}$. If $A\in\mathcal L_n$ and $\mathcal F\subseteq\mathcal L_n$ such that $A\subseteq\bigcup\mathcal F$, then $A\subseteq F$ for at least one $F\in\mathcal F$.
\end{cor}
Under the same hypotheses, we want to show that for every $\alpha\in\mathcal P_n$ the $pp$-convex set $\bigcup\alpha$ is uniquely determined by the antichain $\beta\subseteq\alpha$ of all maximal elements in $\alpha$.
\begin{pro}\label{UNIQUE1}
Suppose that $T=T^{\aleph_0}$ holds. Let $A\subseteq M^n$ be a $pp$-convex set for some $n\geq 1$. Then there is a unique $\beta\in\mathcal{A}_n$ such that $A=\bigcup\beta$.
\end{pro}
\begin{proof}
Let $\alpha_1,\alpha_2\in\mathcal P_n$ be such that $A=\bigcup\alpha_1=\bigcup\alpha_2$. Without loss of generality we may assume $\alpha_1,\alpha_2\in\mathcal{A}_n$. Let $\alpha_1=\{C_1,C_2,\hdots,C_l\}$ and $\alpha_2=\{D_1,D_2,\hdots,D_m\}$.
We have $D_j\subseteq\bigcup_{i=1}^lC_i$ for each $1\leq j\leq m$. Then by \ref{NLU}, we have $D_j\subseteq C_i$ for at least one $i$. By symmetry we also get that each $C_i$ is contained in a $D_j$. Using that both $\alpha_1$ and $\alpha_2$ are antichains with the same union, the proof is complete.
\end{proof}
This proposition shows that under the hypothesis $T=T^{\aleph_0}$ the set of $pp$-convex subsets of $M^n$ is in bijection with $\mathcal A_n$ for each $n\geq 1$. We shall often use this correspondence without mention. For $\alpha\in\mathcal A_n$, we define the \textbf{rank} of the $pp$-convex set $\bigcup\alpha$ to be the integer $|\alpha|$.
The set $\mathcal{A}_n$ can be given the structure of a poset by introducing the relation $\prec_n$ defined by $\beta\prec_n\alpha$ if and only if for each $B\in\beta$, there is some $A\in\alpha$ such that $B\subsetneq A$.
\begin{deflem}\label{UNIQUE2}
Assume that $T=T^{\aleph_0}$. We say that a definable subset $C$ of $M^n$ is a \textbf{cell} if there are $\alpha,\beta\in\mathcal A_n$ with $\beta\prec_n\alpha$ such that $C=\bigcup\alpha\setminus\bigcup\beta$. We denote the set of all cells contained in $M^n$ by $\mathcal C_n$. The antichains $\alpha$ and $\beta$, denoted by $P(C)$ and $N(C)$ respectively, are uniquely determined by the cell $C$. In other words, there is a bijection between the set $\mathcal C_n$ and the set of pairs of antichains strictly related by $\prec_n$. In case $|P(C)|=1$, we say that $C$ is a \textbf{block}. We denote the set of all blocks in $\mathcal C_n$ by $\mathcal B_n$.
\end{deflem}
\begin{proof}
Given any $\alpha,\beta\in\mathcal A_n$ such that $\beta\prec_n\alpha$ and $C=\bigcup\alpha\setminus\bigcup\beta$, the $pp$-convex set $\bigcup(\alpha\cup\beta)$ is determined by $C$. But this set is uniquely determined by the set of maximal elements in $\alpha\cup\beta$ by \ref{UNIQUE1}. Since $\beta\prec_n\alpha$, the required set of maximal elements is precisely $\alpha$. Furthermore, the set $\bigcup\alpha\setminus C=\bigcup\beta$ is $pp$-convex and thus is uniquely determined by $\beta$ by \ref{UNIQUE1} and this finishes the proof.
\end{proof}
We know from \ref{REP} that any definable subset of $M^n$ can be represented as a disjoint union of blocks. So it will be important for us to understand the structure of blocks in detail. A block is always nonempty since any finite union of proper $pp$-subsets cannot cover the given $pp$-set by \ref{UNIQUE1}. For each $B\in\mathcal B_n$, we use the notation $\overline B$ to denote the unique element of $P(B)$.
\begin{theorem}\label{MINK}
Let $M$ be an $\mathcal{R}$-module. Then $Th(M)=Th(M)^{\aleph_0}$ if and only if for each $B\in\mathcal B_n,\ n\geq 1$, we have $B+B-B=\overline B$. Under these conditions, we also get $B-B=\overline{B}$ whenever $\overline{B}$ is a subgroup.
\end{theorem}
\begin{proof}
Assume that $Th(M)=Th(M)^{\aleph_0}$ holds. Let $B\in\mathcal B_n$ be such that $N(B)=\{D_1,D_2,\hdots,D_l\}\prec P(B)=\{A\}$. Let $D=\bigcup N(B)$. We want to show that $B+B-B=A$. But clearly $B\subseteq B+B-B$. So it suffices to show that $D\subseteq B+B-B$.
First assume that $A$ is a subgroup of $M^n$. Let $d\in D$. Since $A\setminus (D-d)\neq\emptyset$, we can choose some $x\in A\setminus (D-d)$ by \ref{NLU}. Then $x+d\in (A+d)\setminus D=A\setminus D=B$, since $A$ is a subgroup. Again choose some $y\in A\setminus((D-d)\cup(D-d-x))$. Then $y+d\in (A+d)\setminus(D\cup(D-x))$ for similar reasons. Thus $y+d, y+x+d\in A\setminus D=B$. Now $d=(d+x)+(d+y)-(d+x+y)\in B+B-B$ and hence the conclusion follows.
In the case when $A$ is a coset of a $pp$-definable subgroup $G$, say $A=a+G$, let $C=D-a$. Then, by the first case, $G=C+C-C$. Now if $d\in A$, then $d-a\in G$. Hence there are $x,y,z\in C$ such that $d-a=x+y-z$. Thus $d=(x+a)+(y+a)-(z+a)\in B+B-B$ and this completes the proof in one direction.
For the converse, suppose that $Th(M)\neq Th(M)^{\aleph_0}$. Then there are two $pp$-definable subgroups $G,H$ of $M^n$ for some $n\geq 1$ such that $H\leq G$ and $1<[G:H]<\infty$. Let $[G:H]=k$ and let $H_1, H_2,\cdots, H_k$ be the distinct cosets of $H$ in $G$. Since $H$ is a $pp$-set, all the cosets $H_i$ are $pp$-sets as well. Now let $B=H_k=G\setminus\bigcup_{i=1}^{k-1}H_i$. Then $B$ is a nonempty block since $k>1$. But, since $B$ is a coset, $B+B-B=B\neq G$ which proves the result in the other direction.
Now we prove the last statement under the hypothesis $Th(M)=Th(M)^{\aleph_0}$. Let $B,A,D$ be as defined in the first paragraph of the proof and assume that $A$ is a subgroup of $M^n$. Given any $a\in A$, we choose $x\in A\setminus (D\cup(D-a))$, which is possible by \ref{NLU}. Then $x,x+a\in B$ and hence $a=(x+a)-x\in (B-B)$. This shows the inclusion $A\subseteq B-B$. We clearly have $(B-B)\subseteq (A-A)$ and $A-A=A$ since $A$ is a subgroup. This completes the proof.
\end{proof}
A map $f:B\rightarrow M^n$ is \textbf{linear} if $f(x+y-z)=f(x)+f(y)-f(z)$ for all $x,y,z\in B$ such that $x+y-z\in B$. We use the previous theorem to show that any linear map on $B$ can be extended uniquely to a linear map on $\overline B$.
\begin{lemma}\label{COLOURINJ}
Suppose that $T=T^{\aleph_0}$ holds. Then for each $n\geq 1$, each $B\in\mathcal B_n$ and each injective linear map $f:B\rightarrowtail M^n$, there exists a unique injective linear extension $\overline{f}:\overline B\rightarrowtail M^n$.
\end{lemma}
\begin{proof}
Let $B=A\setminus\bigcup_{i=1}^mD_i$ be a block and assume that $A$ is a $pp$-definable subgroup. Let $D=\bigcup_{i=1}^mD_i$ and, for each $i$, let $H_i$ denote the subgroup of $A$ whose coset is $D_i$. Let $H=\bigcup_{i=1}^m H_i$. We choose and fix elements $x_i\in B$ sequentially depending on the earlier choices as follows. We choose $x_1\in A\setminus(D\cup H)$ and for each $1<i\leq m$, choose $x_i\in A\setminus(D\cup H\cup \bigcup_{j=1}^{i-1}(D+x_j))$. We can choose $x_i$ at each step by \ref{NLU}. Then we define $\overline{f}(d_i)=f(x_i+d_i)-f(x_i)$ for $d_i\in D_i$ and $\overline{f}(b)=f(b)$ for $b\in B$.
$\overline{f}$ is well-defined: Let $d\in D_i\cap D_j$ for some $j<i$. Then by the choice of $x_i$, $(x_i-x_j)\in B$. Also $x_i,x_i+d,x_j,x_j+d\in B$. Hence $f(x_i-x_j)$ is defined and is equal to both $f(x_i)-f(x_j)$ and $f(x_i+d)-f(x_j+d)$. Hence we see that $f(x_i+d)-f(x_i)=f(x_j+d)-f(x_j)$, which proves that $\overline{f}(d)$ is well-defined for each $d\in D$.
$\overline{f}$ is linear: Let $b\in B$ and $d\in D_i$. Then there are two possibilities namely, $b+d\in B$ or $b+d\in D_j$ for some $j$. In the former case we have $\overline{f}(b+d)=f(b+d)=f(b+x_i+d-x_i)=f(b)+f(x_i+d)-f(x_i)=\overline{f}(b)+\overline{f}(d)$ since $x_i+d,x_i,b\in B$, while in the latter case we have $\overline{f}(b+d)=f(b+d+x_j)-f(x_j)=f(b+d+x_j-x_i+x_i)-f(x_j)=f(b)+f(x_i+d)-f(x_i)+f(x_j)-f(x_j)= f(b)+f(x_i+d)-f(x_i)=\overline{f}(b)+\overline{f}(d)$ since $b,x_i,x_j,x_i+d,x_j+d\in B$.
Let $b\in D_k$ and $d\in D_i$. Then there are two possibilities namely, $b+d\in B$ or $b+d\in D_j$ for some $j$. In the former case we have $\overline{f}(b+d)=f(b+d)=f(b+x_k-x_k+d+x_i-x_i)=
f(b+x_k)-f(x_k)+f(d+x_i)-f(x_i)=\overline{f}(b)+\overline{f}(d)$ since $b+x_k,x_k,x_i,x_i+d\in B$, while in the latter case we have $\overline{f}(b+d)=f(b+d+x_j)-f(x_j)=f(b+x_k-x_k+d+x_i-x_i+x_j)-f(x_j)=
f(b+x_k)-f(x_k)+f(d+x_i)-f(x_i)+f(x_j)-f(x_j)=\overline{f}(b)+\overline{f}(d)$ since $b+x_k,x_k,x_i,x_i+d,x_j\in B$.
In the case when $b,d\in B$ and $b+d\in B$, the linearity of $\overline{f}$ follows from the linearity of $f$. When $b+d\in D_i$, $\overline{f}(b+d)=f(b+d+x_i)-f(x_i)=f(b)+f(d)+f(x_i)-f(x_i)=
\overline{f}(b)+\overline{f}(d)$ since $b,d,x_i\in B$. So we have showed in each case that $\overline{f}$ is linear.
$\overline{f}$ is injective: Without loss we may assume that $\overline f(0)=0$, otherwise we may consider the function $\overline f(-)-\overline f(0)$. Let $a\in A$ be such that $\overline{f}(a)=0$. Then if $a\in B$, then $f(a)=0$ and hence $a=0$ by injectivity of $f$. If $a\in D_i$, then $f(x_i+a)-f(x_i)=0$ and hence $f(x_i+a)=f(x_i)$. But then $x_i+a=x_i$ by injectivity of $f$ since both $x_i+a,x_i\in B$. This again implies that $a=0$.
$\overline{f}$ is unique: Let $h:A\rightarrow M^n$ be any linear injective extension of $f$. Then $h(b)=f(b)=\overline{f}(b)$ for each $b\in B$ and hence, for $d\in D_i$, we have $\overline{f}(d)=f(x_i+d)-f(x_i)=h(x_i+d)-h(x_i)=h(d)$ since $x_i+d,x_i\in B$ and $h$ is linear.
If $A$ is a nontrivial coset of some $pp$-definable subgroup $G$ of $M^n$, $D\subsetneq A$, $B=A\setminus D$ and we are given some linear map $f:B\rightarrowtail M^n$, we choose and fix some $b\in B$. Then clearly $A-b=G$ and we take $C=B-b$. Define $g:C\rightarrow M^n$ by setting $g(c)=f(c+b)-f(b)$. Now whenever $x,y\in C$ such that $x+y\in C$, we have $g(x+y)=f(x+y+b)-f(b)=f((x+b)-b+(y+b))-f(b)= f(x+b)-f(b)+f(y+b)-f(b)=g(x)+g(y)$ since $x+b,y+b,b\in B$ and $f$ is linear. Hence $g$ is linear on $C$. Therefore by the subgroup case, we have a unique linear injective extension of $g$ to $G$, say $\overline{g}$. Then we define $\overline{f}:A\rightarrow M^n$ by setting $\overline{f}(a)=\overline{g}(a-b)+f(b)$. It can be easily seen that $\overline{f}$ is indeed an extension of $f$. The uniqueness, linearity and injectivity of $\overline{f}$ follows from the uniqueness of $\overline{g}$. This argument completes the proof of this case and hence that of the lemma.
\end{proof}
\subsection{Local characteristics}\label{LC}
We fix some $n\geq 1$ and drop all the subscripts $n$. We also assume hereafter that $Th(M)=Th(M)^{\aleph_0}$ holds for some fixed right $\mathcal R$-module $M$. For brevity, we denote the sets $\mathcal L\setminus\{\emptyset\},\mathcal A\setminus\{\emptyset\},\hdots$ by $\mathcal L^*,\mathcal A^*,\hdots$ respectively.
\begin{definition}
Let $\mathcal D$ be a finite subset of $\mathcal L^*$. The smallest sub-meet-semilattice of $\mathcal L$ containing $\mathcal D$ will be called the $pp$-nest (or simply nest) corresponding to $\mathcal D$ and will be denoted by $\hat{\mathcal D}$. Note that $\hat{\mathcal D}$ is finite. In general, any finite sub-meet-semilattice of $\mathcal L$ will also be referred to as a $pp$-nest.
\end{definition}
\begin{definition}
For each finite subset $\mathcal F$ of $\mathcal L^*$ and $F\in\mathcal F$, we define the \textbf{$\mathcal F$-core} of $F$ to be the block $\mathrm{Core}_\mathcal F(F):=F\setminus\bigcup\{G: G\in\mathcal F, G\cap F\subsetneq F\}$.
\end{definition}
Let $D\subseteq M^n$ be definable. Then $D=\bigsqcup_{i=1}^m B_i$ for some $B_i\in\mathcal B$ by \ref{REP}. We say that $\mathcal D$ is the nest corresponding to this partition of $D$ if it is the nest corresponding to the finite family $\bigcup_{i=1}^m(P(B_i)\cup N(B_i))$. Every definable set can be partitioned canonically given a suitable nest, which is the content of the following lemma whose proof is omitted.
\begin{deflem}\label{CH1}
Suppose $D\subseteq M^n$ is definable and $\mathcal D$ is the nest corresponding to a given partition $D=\bigsqcup_{i=1}^m B_i$. For every nonempty $F\in\mathcal D$, $\mathrm{Core}_\mathcal D(F)\cap~D\neq\emptyset$ if and only if $\mathrm{Core}_\mathcal D(F)\subseteq D$. We define the \textbf{characteristic function} of the nest $\mathcal D$, $\delta_\mathcal D:\mathcal D\rightarrow \{0,1\}$, by $\delta_\mathcal D(F)=1$ if and only if $F\neq\emptyset$ and $\mathrm{Core}_\mathcal D(F)\subseteq D$. We denote the sets $\delta_\mathcal D^{-1}(1)$ and $\delta_\mathcal D^{-1}(0)$ by $\mathcal D^+$ and $\mathcal D^-$ respectively. Then $D=\bigcup_{F\in\mathcal D^+}\mathrm{Core}_\mathcal D(F)$.
\end{deflem}
\begin{illust}
If $B$ is a block with $P(B)=A$ and $N(B)=\{D_1,D_2\}$ such that $D_1\cap D_2\neq\emptyset$, then $\mathcal D=\{A,D_1,D_2,D_1\cap D_2\}$ is the corresponding nest. Clearly $B=\mathrm{Core}_\mathcal D(A)$ and hence $\mathcal D^+=\{A\}$.
\end{illust}
We will sometimes use another family of characteristic functions stated in the following definition.
\begin{definition}\label{CH2}
Given any $C\in\mathcal C$, we define the \textbf{characteristic function} of the cell $C$, $\delta(C):\mathcal L^*\rightarrow\{0,1\}$, as $\delta(C)(P)=1$ if $P\subseteq C$ and $\delta(C)(P)=0$ otherwise, for each $P\in\mathcal{L}^*$. When $P=\{a\}$, we write the expression $\delta(C)(a)$ instead of $\delta(C)(\{a\})$.
\end{definition}
The set $\mathcal A$ of antichains is ordered by the relation $\prec$ but can also be considered as a poset with respect to the natural inclusion ordering on the set of all $pp$-convex sets. For $\alpha,\beta\in\mathcal{A}$, we define $\alpha\wedge\beta$ to be the antichain corresponding (in the sense of \ref{UNIQUE1}) to $(\bigcup\alpha)\cap(\bigcup\beta)$ and $\alpha\vee\beta$ to be the antichain corresponding to $(\bigcup\alpha)\cup(\bigcup\beta)$. Since the intersection and union of two $pp$-convex sets are again $pp$-convex, the binary operations $\wedge,\vee:\mathcal A\times\mathcal A\rightarrow\mathcal A$ are well-defined. It can be easily seen that $\mathcal{A}$ is a distributive lattice with respect to these operations.
We want to understand the structure of any definable set ``locally'' in a neighbourhood of a point in $M^n$. The following lemma defines a class of sub-lattices of $\mathcal A$ which provides the necessary framework to define the concept of localization. The proof is an easy verification of an adjunction and is not given here.
\begin{deflem}
Fix some $a\in M^n$. Let $\mathcal{L}_a:=\{A\in\mathcal{L}:a\in A\}$ and $\mathcal{A}_a$ denote the set of all antichains in the meet-semilattice $\mathcal{L}_a$. Then $\mathcal{A}_a$ is a sub-lattice of $\mathcal{A}$. We denote the inclusion $\mathcal{A}_a\rightarrow\mathcal{A}$ by $\mathcal{I}_a$.
We also consider the map $\mathcal{N}_a:\mathcal{A}\rightarrow\mathcal{A}_a$ defined by $\alpha\mapsto \alpha\cap\mathcal{L}_a$. We call the antichain $\mathcal{N}_a(\alpha)$ the \textbf{localization} of $\alpha$ at $a$. Then $\mathcal{N}_a$ is a right adjoint to $\mathcal{I}_a$ if we consider the posets $\mathcal A$ and $\mathcal A_a$ as categories in the usual way, and the composite $\mathcal{N}_a\circ\mathcal{I}_a$ is the identity on $\mathcal{A}_a$. This in particular means that $\mathcal{A}_a$ is a reflective subcategory of $\mathcal{A}$. Furthermore, the map $\mathcal{N}_a$ not only preserves the meets of antichains, being a right adjoint, but it also preserves the joins of antichains.
\end{deflem}
Fix some $a\in M^n$. Let us denote the set of all finite subsets of $\mathcal{L}_a$ by $\mathcal P_a$ and let $\alpha\in\mathcal{P}_a$. We construct a simplicial complex $\mathcal{K}^a(\alpha)$ which determines the ``geometry'' of the intersection of elements of $\alpha$ around $a$. This construction is similar to the construction of the nerve of an open cover, except for the meaning of the ``triviality'' of the intersection. We know that a $pp$-set is finite if and only if it has at most $1$ element. We also know that $\bigcap\alpha\supseteq\{a\}$.
\begin{definition}\label{SIMPCOMP}
We associate an abstract simplicial complex $\mathcal K^a(\alpha)$ to each $\alpha\in\mathcal P_a$ by taking the vertex set $\mathcal V(\mathcal K^a(\alpha)):=\alpha\setminus\{a\}$. We say that a nonempty set $\beta\subseteq\alpha$ is a face of $\mathcal{K}^a(\alpha)$ if and only if $\bigcap\beta$ is infinite (i.e., strictly contains $a$). If the only element of $\alpha$ is $\{a\}$ or if $\alpha=\emptyset$, then we set $\mathcal{K}^a(\alpha)=\emptyset$, the empty complex.
\end{definition}
\begin{illust}
Consider the real vector space $\mathbb R_{\mathbb R}$. The theory of this vector space satisfies the condition $\rm T=T^{\aleph_0}$. We consider subsets of $\mathbb R^3$. If $\alpha$ denotes the antichain corresponding to the union of $3$ coordinate planes and $a$ is the origin, then $\mathcal K^a(\alpha)$ is a copy of $\partial\Delta^2$. The $2$-dimensional face of $\Delta^2$ is absent since the intersection of the coordinate planes does not contain the origin properly.
\end{illust}
Since $\beta_1\subseteq\beta_2\ \Rightarrow\ \bigcap\beta_2\subseteq\bigcap\beta_1$, $\mathcal{K}^a(\alpha)$ is indeed a simplicial complex. We tend to drop the superscript $a$ when it is clear from the context. To extend this definition to arbitrary elements of $\mathcal P$, we extend the notion of localization operator (at $a$) to $\mathcal{P}$ by setting $\mathcal{N}_a(\alpha)=\alpha\cap\mathcal{L}_a$ for each $\alpha\in\mathcal P$. Now we are ready to define a family of numerical invariants for convex subsets of $M^n$, which we call ``local characteristics''.
\begin{definition}
We define the function $\kappa_a:\mathcal{P}\rightarrow\mathbb{Z}$ by setting $\kappa_a(\alpha):= \chi(\mathcal{K}(\mathcal{N}_a(\alpha)))- \delta(\alpha)(a)$, where $\chi(\mathcal{K})$ denotes the Euler characteristic of the complex $\mathcal{K}$ as defined in \ref{Euler} and $\delta(\alpha)$ is the characteristic function of the set $\bigcup\alpha$ as defined in \ref{CH2}. The value $\kappa_a(\alpha)$ will be called the \textbf{local characteristic} of the antichain $\alpha$ at $a$.
\end{definition}
If we view the local characteristic $\kappa_a(\alpha)$ as a function of $a$ for a fixed antichain $\alpha$, the correction term $\delta(\alpha)(a)$ makes sure that $\kappa_a(\alpha)=0$ for all but finitely many values of $a$. This fact will be useful in the next section.
We want to show that the local characteristic satisfies the inclusion-exclusion principle for antichains.
\begin{theorem}\label{t1}
For $\alpha,\beta\in\mathcal{A}$, we have $\kappa_a(\alpha\vee\beta)+\kappa_a(\alpha\wedge\beta)=\kappa_a(\alpha)+\kappa_a(\beta)$.
\end{theorem}
The rest of this section is devoted to the proof of this theorem. First we observe that it is sufficient to prove this theorem for $\alpha,\beta\in\mathcal{A}_a$. We also observe that it is sufficient to prove this theorem in the case when $\kappa_a$ is replaced by the function $\chi(\mathcal K(-))$ because $\kappa_a(\alpha)=\chi(\mathcal{K}(\alpha))-1$ whenever $a\in\bigcup\alpha$ and the cases when either $a\notin\bigcup\alpha$ or $a\notin\bigcup\beta$ are trivial. We write $\kappa_a$ as $\kappa$ for simplicity of notation.
The following proposition is the first step in this direction, which states that $\kappa(\alpha)$ is actually determined by the $pp$-convex set $\bigcup\alpha$.
\begin{pro}\label{p1}
Let $\alpha\in\mathcal{A}_a$ and $\beta\in\mathcal{P}_a$. If $\bigcup\alpha=\bigcup\beta$, then $\kappa(\alpha)=\kappa(\beta)$.
\end{pro}
\begin{proof}
It is clear that $\beta\supseteq\alpha$ since $\beta$ is finite. Hence $\mathcal{K}(\alpha)$ is a full sub-complex of $\mathcal{K}(\beta)$ (i.e. if $\beta'\in \mathcal K(\beta)$ and $\beta'\subseteq\alpha$, then $\beta'\in\mathcal K(\alpha)$). We can also assume that $\{a\}\notin\beta$. Note that every element $\beta\setminus\alpha$ is properly contained in at least one element of $\alpha$. We use induction on the size of $\beta\setminus\alpha$ to prove this result.
If $\beta\setminus\alpha=\emptyset$, then the conclusion is trivially true. For the inductive case, suppose $\alpha\subseteq\beta'\subsetneq\beta$ and the result has been proved for $\beta'$. Let $B\in\beta\setminus\beta'$. Since $\alpha$ is the set of maximal elements of $\beta$, there is some $A\in\alpha$ such that $A\supsetneq B$.
Consider the complex $\mathcal{K}_1=\{F\in\mathcal{K}(\beta'):(F\cup\{B\})\in\mathcal{K}(\beta'\cup\{B\})\}$ as a full sub-complex of $\mathcal{K}(\beta')$. Observe that whenever $B\in F\in\mathcal{K}(\beta'\cup\{B\})$, we have $(F\cup\{A\})\setminus\{B\}\in\mathcal{K}(\beta')$. As a consequence, $\mathcal K_1=\mathrm{Cone}(\mathcal K(\beta'\setminus\{A\}))$ where the apex of the cone is $A$. In particular, $\mathcal K_1$ is contractible.
Also note that $\mathcal{K}(\beta'\cup\{B\})=\mathcal{K}(\beta')\cup\mathrm{Cone}(\mathcal{K}_1)$, where the apex of the cone is $B$. Now we compare the pair $\mathcal K_1\subseteq\mathcal{K}(\beta')$ with another pair $\mathrm{Cone}(\mathcal K_1)\subseteq\mathcal K(\beta'\cup\{B\})$ of simplicial complexes. Observe the set equality $\mathcal{K}(\beta')\setminus\mathcal K_1=\mathcal K(\beta'\cup\{B\})\setminus \mathrm{Cone}(\mathcal K_1)$. Also both $\mathcal K_1$ and $\mathrm{Cone}(\mathcal K_1)$ are contractible. Thus we conclude that $\mathcal{K}(\beta'\cup\{B\})$ and $\mathcal{K}(\beta')$ are homotopy equivalent. Finally, an application of \ref{EULHTPY} completes the proof.
\end{proof}
Note that this result is very helpful for the computation of local characteristics as we get the equalities $\kappa(\alpha\vee\beta)=\kappa(\alpha\cup\beta)$ and $\kappa(\alpha\wedge\beta)=\kappa(\alpha\circ\beta)$ for all $\alpha,\beta\in\mathcal{A}_a$, where $\alpha\circ\beta=\{A\cap B: A\in\alpha, B\in\beta\}$. The vertices of $\mathcal{K}(\alpha\circ\beta)$ will be denoted by the elements from $\alpha\times\beta$.
We use induction twice, first on $|\beta|$ and then on $|\alpha|$, to prove the main theorem of this section. The following lemma is the first step of this induction.
\begin{lemma}
For $\alpha,\beta\in\mathcal{A}_a$ and $\left|\alpha\right|\leq 1$, we have $\kappa(\alpha\vee\beta)+\kappa(\alpha\wedge\beta)=\kappa(\alpha)+\kappa(\beta)$.
\end{lemma}
\begin{proof}
The cases $\left|\alpha\right|=0$ and $\alpha=\{\{a\}\}$ are trivial. So we assume that $\alpha=\{A\}$ where $A$ is infinite. We can make similar non-triviality assumptions on $\beta$, namely there is at least one element in $\beta$ and all the elements of $\beta$ are infinite.
There are only two possible cases when $|\beta|=1$ and the conclusion holds true in both these cases. For example when $\beta=\{B\}$ and $A\cap B=\{a\}$, we have $\mathcal K(\alpha)\cong\mathcal K(\beta)\cong\Delta^0$, $\mathcal K(\alpha\circ\beta)$ is empty and $\mathcal K(\alpha\cup\beta)$ is disjoint union of two copies of $\Delta^0$. Hence the identity in the statement of the lemma takes the form $1+(-1)=0+0$.
Suppose for the inductive case that the result is true for $\beta$ i.e. $\kappa(\alpha\vee\beta)+\kappa(\alpha\wedge\beta)=\kappa(\alpha)+\kappa(\beta)$ holds. We want to show that the result holds for $\beta\cup\{B\}$ i.e. $\kappa(\alpha\vee(\beta\cup\{B\}))+\kappa(\alpha\wedge(\beta\cup\{B\}))=\kappa(\alpha)+\kappa(\beta\cup\{B\})$.
We introduce some superscript and subscript notations to denote new simplicial complexes obtained from the original. The following list describes them and also explains the rules to handle two or more scripts at a time.
\begin{itemize}
\item Let $\mathcal K_0$ denote the complex $\mathcal K(\alpha)$, i.e. the complex consisting of only one vertex and $\mathcal{K}$ denote the complex $\mathcal{K}(\beta)$.
\item Let $\mathcal{K}^S$ denote the complex $\mathcal{K}(\beta\cup S)$ for any finite $S\subseteq\mathcal{L}_a$ which contains only infinite elements. Also, $\mathcal K^{A,B}$ is a short hand for $\mathcal K^{\{A,B\}}$.
\item Whenever $C$ is a vertex of $\mathcal{Q}$, the notation $\mathcal{Q}_C$ denotes the sub-complex $\{F\in\mathcal{Q}:C\notin F, F\cup\{C\}\in\mathcal{Q}\}$ of $\mathcal{Q}$.
\item If $\mathcal Q=\mathcal K(\gamma)$ for some antichain $\gamma$ and $A\notin\gamma$, then the notation $^A\mathcal{Q}$ denotes the complex $\mathcal{K}(\{A\}\circ\gamma)$.
\item The notation $^C\mathcal K^S_B$ means $^C((\mathcal K^S)_B)$. This describes the order of the scripts.
\item The Euler characteristic of $^C\mathcal K^S_B$ will be denoted by $^C\chi^S_B$.
\end{itemize}
Using this notation, the inductive hypothesis is
\begin{equation}\label{1}\chi^A+\,^A\chi=\chi_0+\chi\end{equation} and our claim is \begin{equation}\label{2}\chi^{B,A}+\,^A\chi^B=\chi_0+\chi^B.\end{equation}
\textbf{Case I: $(A\cap B)=\{a\}.$} In this case, the faces of $\mathcal{K}^{A,B}$ not present in $\mathcal{K}^A$ are the faces of $\mathcal{K}^B$. Hence $b_n(\mathcal{K})-b_n(\mathcal{K}^B)=b_n(\mathcal{K}^A)-b_n(\mathcal{K}^{A,B})$ for all $n\geq 0$, where $b_n$ denotes the $n^{th}$ Betti number. Hence we get
\begin{equation*}\chi^{B,A}-\chi^A=\chi^B-\chi\end{equation*}
Also note that the hypothesis $(A\cap B)=\{a\}$ yields $H_*(^A\mathcal{K})=H_*(^A\mathcal{K}^B)$ since only infinite elements matter for the computations. It follows that equation (\ref{2}) holds in this case.
\textbf{Case II: $A\cap B\supsetneq\{a\}.$} Note that whenever $C$ is not a vertex of $\mathcal Q$, we have $\mathcal Q^C_C\subseteq\mathcal Q$ and $\mathcal Q\cup\mathrm{Cone}(\mathcal Q^C_C)=\mathcal Q^C$, where the apex of the cone is $C$. Hence corollary \ref{HMLGCONE} can be restated in this notation as the following identity. \begin{equation}\label{3}\chi(\mathcal{Q})+1=\chi(\mathcal{Q}^C)+\chi(\mathcal{Q}^C_C)\end{equation}
As particular cases of (\ref{3}), we get the following equalities.
\begin{equation}\label{4}\chi+1=\chi^B+\chi^B_B.\end{equation} \begin{equation}\label{5}\chi^A+1=\chi^{A,B}+\chi^{A,B}_B\end{equation} \begin{equation}\label{6}\chi^B_B+1=\chi^{A,B}_B+\chi^{A,B}_{A,B}\end{equation}
It can be checked that $\mathcal{K}^{A,B}_{A,B}\cong\ ^A\mathcal{K}^B_B$ via the map $F\mapsto \{\{C,A\}: C\in F\}$. This gives us the following equation.
\begin{equation}\label{8}\chi^{A,B}_{A,B}=\ ^A\chi^B_B\end{equation}
If we combine equations (\ref{1}),(\ref{4}),(\ref{5}),(\ref{6}) and (\ref{8}), it remains to prove the following to get equation (\ref{2}) in the claim.
\begin{equation}\label{7}^A\chi+1=\,^A\chi^B+\,^A\chi^B_B\end{equation}
Observe that the natural inclusion maps $i_1:\mathcal{K}_0\rightarrow\mathcal{K}^A$ and $i_2:\mathcal{K}\rightarrow\mathcal{K}^A$ are inclusions of sub-complexes and their images are disjoint. Furthermore, the set theoretic map $g:\mathcal{K}^A\setminus(Im(i_1)\sqcup Im(i_2))\rightarrow\,^A\mathcal{K}$ defined by $F\mapsto\{\sigma\subseteq F:A\in\sigma,|\sigma|=2\}$ is a bijection. Now consider the composition $^A\mathcal{K}^B\cong\mathcal{K}^{A,B}\setminus(i_1(\mathcal{K}_0)\sqcup i_2(\mathcal{K}^B))\xrightarrow{\pi_B}\mathcal{K}^A\setminus(i_1(\mathcal{K}_0)\sqcup i_2(\mathcal{K}))\cong\, ^A\mathcal{K}$, where $\pi_B(F)=F\setminus\{B\}$. The union of images (under this composition of maps) of those faces in $^A\mathcal K^B$ which contain $A\cap B$ is the sub-complex $^A\mathcal{K}^B_B$ of $^A\mathcal{K}$. Hence $(^A\mathcal{K}\cup \mathrm{Cone}(^A\mathcal{K}^B_B))\cong\,^A\mathcal{K}^B$, where the apex of the cone is $\{A,B\}$. An application of \ref{HMLGCONE} gives the required identity in equation (\ref{7}).
\end{proof}
We use definition \ref{Euler} of Euler characteristic to prove the second step in the proof of the main theorem since we do not have a proof using homological techniques. In this step, we allow the size of $\beta$ to be an arbitrary fixed positive integer and we use induction on the size of $\alpha$. The lemma just proved is the base case for this induction. Let $A$ be a new element of $\mathcal L_a$ to be added to $\alpha$ and assume the result is true for $\alpha$. Again we may assume that $A$ is infinite.
We construct the complex $\mathcal{K}(\alpha\cup\beta\cup\{A\})$ in steps starting with the complex $\mathcal{K}(\alpha\cup\beta)$ and the conclusion of the theorem holds for the latter by the inductive hypothesis. We do this in such a way that at each step $\mathcal{K}_1$ of the construction, the following identity is satisfied.
\begin{equation}\label{count}
\chi(\mathcal K(\alpha\cup\{A\})\cap\mathcal K_1)+\chi(\mathcal K(\beta))=\chi(\mathcal K(\alpha\cup\{A\}\cup\beta)\cap\mathcal K_1)+\chi(\mathcal K((\alpha\cup\{A\})\circ\beta)\cap\mathcal K_1)
\end{equation}
In this expression, $\mathcal K((\alpha\cup\{A\})\circ\beta)\cap\mathcal K_1$ denotes the subcomplex of $\mathcal K((\alpha\cup\{A\})\circ\beta)$ whose faces are appropriate projections of the faces of $\mathcal K_1$.
For the first step, we construct all the elements in $\mathcal{K}(\alpha\cup\{A\})$ not in $\mathcal K(\alpha)$. Let $\mathcal K_1'$ denote the resulting complex. No new faces of the complex $\mathcal K((\alpha\cup\{A\})\circ\beta)$ are constructed in this process. Hence, for each $n\geq 0$, we have
\begin{equation*}
v_n(\mathcal K_1')-v_n(\mathcal K(\alpha\cup\beta\cup\{A\}))=v_n(\mathcal K_1'\cap\mathcal{K}(\alpha\cup\{A\}))-v_n(\mathcal{K}(\alpha\cup\{A\})),
\end{equation*}
where $v_n(\mathcal Q)$ denotes the number $n$-dimensional faces of $\mathcal Q$. Hence equation (\ref{count}) is satisfied for $\mathcal K_1'$.
For the second step, we further construct all the faces corresponding to $\{A\}\circ\beta$. The conclusion in this case follows from the previous lemma.
Finally we construct the faces containing $A$ and intersecting both $\alpha$ and $\beta$. We construct a face $F$ of size $m+k$ whenever all the proper sub-faces of $F$ have already been constructed, where $F\cap(\alpha\cup\{A\})$ and $F\cap\beta$ have sizes $m$ and $k$ respectively. It is clear that $m,k\geq 1$.
Let the sub-complex of $\mathcal{K}(\alpha\cup\beta\cup\{A\})$ consisting of the already constructed faces be denoted by $\mathcal{K}$. We assume, for induction, that equation (\ref{count}) is true for $\mathcal K$. Let $g(F')=\{\sigma\subseteq F':|\sigma\cap(\alpha\cup\{A\})|=1,|\sigma\cap\beta|=1\}$ for $F'\in\mathcal{K}$. Let $\mathcal{K}_3=\bigcup_{F'\subsetneq F}g(F')$ and $\mathcal{K}_2=\bigcup_{F'\in\mathcal{K}}g(F')$. Note the inclusions $\mathcal K\subseteq\mathcal K_2\subseteq\mathcal{K}((\alpha\cup\{A\})\circ\beta)$.
The construction of the face $F$ changes $\chi(\mathcal{K})$ by $(-1)^{\mathrm{dim} F}=(-1)^{m+k-1}$, while the numbers $\chi(\mathcal{K}(\alpha\cup\{A\}))$ and $\chi(\mathcal{K}(\beta))$ are unaltered.
We calculate the change in the value of $\chi(\mathcal{K}_3)$ after the addition of $F$. The complex $g(F)$ is contractible. Hence its Euler characteristic is equal to $1$ by \ref{EULHTPY}.
Let $w_n$ denote the number of $n$-dimensional faces of $\mathcal{K}_3$. Recall that $\mathcal V(\mathcal K_3)= (\alpha\cup\{A\})\times\beta$. If $\mathrm{dim} F'=n+2$ for some $F'\in\mathcal K(\alpha\cup\beta\cup\{A\})$ such that $|F'\cap(\alpha\cup\{A\})|\geq 1,|F'\cap\beta|\geq 1$, then $\mathrm{dim}(g(F'))=n$. Therefore to calculate $w_n$, we choose total $n+2$ vertices from $F$, making sure that we choose at least one vertex from both $\alpha\cup\{A\}$ and $\beta$. Hence $w_n=\Sigma_{j=1}^{n+1}\binom{m}{j}\binom{k}{n+2-j}$. This number can be easily shown to be equal to $\binom{m+k}{n+2}-\binom{m}{n+2}-\binom{k}{n+2}$.
The maximum dimension of the face of $\mathcal{K}_3$ is equal to $m+k-3$. Hence the required change in the value of $\chi(\mathcal{K}_3)$ is $1-\Sigma_{n=0}^{m+k-3}(-1)^n[\binom{m+k}{n+2}-\binom{m}{n+2}-\binom{k}{n+2}]$.
To obtain equation (\ref{count}) for $\mathcal K\cup\{F\}$, we need to show that there is no change in the value of $\chi(\mathcal{K})+\chi(\mathcal K_3)$ after construction of $F$.
But we know that $\Sigma_{n=0}^{m+k}(-1)^n[\binom{m+k}{n}-\binom{m}{n}-\binom{k}{n}]=0$ since each of the three alternating sums is zero. This equation rearranges to give the required cancelation equation and completes the proof.
\subsection{Global characteristic}\label{GCDS}
Let the function $\kappa:\mathcal{A}\times M^n\rightarrow\mathbb{Z}$ be defined by $\kappa(\alpha,a)=\kappa_a(\alpha)$. Suppose $\alpha$ is a singleton. If $\bigcup\alpha$ is infinite, then $\kappa(\alpha,-)$ is the constant $0$ function and if $\alpha=\{a\}$, then $\kappa(\alpha,b)=0$ for all $b\neq a$ and $\kappa(\alpha,a)=-1$. For arbitrary $\alpha\in\mathcal{A}$, if $a\notin\bigcup\alpha$, then $\kappa(\alpha,a)=0$.
\begin{definitions}
For $\alpha\in\mathcal{A}$, we define the \textbf{set of singular points} of $\alpha$ to be the set $\mathrm{Sing}(\alpha):=\{a\in M^n: \kappa(\alpha,a)\neq 0\}$. $\mathrm{Sing}(\alpha)$ is always finite since all the singular points appear as singletons in the nest corresponding to $\alpha$. Using finiteness of $\mathrm{Sing}(\alpha)$, we define the \textbf{global characteristic} of $\alpha$ to be the sum $\Lambda(\alpha):=-\Sigma_{a\in M^n} \kappa(\alpha,a)$, which in fact is equal to the finite sum $\Lambda(\alpha)=-\Sigma_{a\in \mathrm{Sing}(\alpha)}\kappa(\alpha,a)$.
\end{definitions}
Fix some $a\in M^n$. Let $\alpha,\beta\in\mathcal{A}$ be such that $\beta\prec\alpha$. Then either $\mathcal{N}_a(\alpha)=\mathcal{N}_a(\beta)=\emptyset$ or $\mathcal{N}_a(\beta)\prec\mathcal{N}_a(\alpha)$. If $C:=\bigcup\alpha\setminus\bigcup\beta$ is a cell, we define the homology $H_*(C)$ to be the relative homology $H_*(\mathcal{K}(\mathcal{N}_a(\alpha\cup\beta));\mathcal{K}(\mathcal{N}_a(\beta)))$. In particular, the alternating sum of the Betti numbers of $H_*(C)$, denoted by $\chi_a(C)$, is equal to the difference $\chi(\mathcal{K}(\mathcal{N}_a(\alpha)))-\chi(\mathcal{K}(\mathcal{N}_a(\beta)))$ by \ref{p1} and \ref{LONGEXACT}. We also have the equation $\delta(C)=\delta(\alpha)-\delta(\beta)$. Hence if we define the local characteristic of $C$ as $\kappa_a(C):=\chi_a(C)-\delta(C)(a)$, we get the identity $\kappa_a(C)=\kappa_a(P(C))-\kappa_a(N(C))$. We define the extension of the function $\kappa$ to include all cells by setting $\kappa(C,a):=\kappa_a(C)$ for $a\in M^n,C\in\mathcal{C}$.
\begin{definitions}
We define the set of singular points $\mathrm{Sing}(C)$ for $C\in\mathcal{C}$ analogously by setting $\mathrm{Sing}(C):=\{a\in M^n:\kappa_a(C)\neq 0\}$. This set is finite since $\mathrm{Sing}(C)\subseteq \mathrm{Sing}(P(C))\cup \mathrm{Sing}(N(C))$. We also extend the definition of global characteristic for cells by setting $\Lambda(C):=-\Sigma_{a\in M^n} \kappa(C,a)$.
\end{definitions}
It is immediate that $\Lambda(C)=\Lambda(P(C))-\Lambda(N(C))$ for every $C\in\mathcal{C}$.
The main aim of this section is to prove that the global characteristic is additive in the following sense.
\begin{theorem}\label{t2}
If $\{B_i:1\leq i\leq l\}, \{B_j':1\leq j\leq m\}$ are two finite families of pairwise disjoint blocks such that $\bigsqcup_{i=1}^l B_i=\bigsqcup_{j=1}^m B_j'$, then $\Sigma_{i=1}^l\Lambda(B_i)=\Sigma_{j=1}^m\Lambda(B_j')$.
\end{theorem}
The proof of this theorem follows at once from the following local version.
\begin{lemma}\label{l1}
If $a\in M^n$ and $\{B_i:1\leq i\leq l\}, \{B_j':1\leq j\leq m\}$ are two finite families of pairwise disjoint blocks such that $\bigsqcup_{i=1}^l B_i=\bigsqcup_{j=1}^m B_j'$, then $\Sigma_{i=1}^l\kappa_a(B_i)=\Sigma_{j=1}^m\kappa_a(B_j')$.
\end{lemma}
\begin{proof}
It will be sufficient to show that both these numbers are equal to the sum $\Sigma_{B\in\mathcal{F}}\,\kappa_a(B)$ where $\mathcal{F}$ is any finite family of blocks finer than both the given families. We can in particular choose a finite $pp$-nest $\mathcal{D}$ containing all the elements in $\bigcup_{i=1}^l(P(B_i)\cup N(B_i))\cup\bigcup_{j=1}^m(P(B_j)\cup N(B_j))$ and set $\mathcal{F}=\{\mathrm{Core}_\mathcal{D}(D):D\in\mathcal{D}^+\}$. This involves partitioning every $B_i$ and $B_j'$ into smaller blocks of the form $\mathrm{Core}_\mathcal D(D)$ for $D\in\mathcal D^+$.
Thus it will be sufficient to show that if $\mathcal{F}$ is a finite family of blocks corresponding to cores of a $pp$-nest $\mathcal{D}$ such that $B=\bigcup \mathcal{F}\in\mathcal{B}$, then $\kappa_a(B)=\Sigma_{F\in\mathcal{F}}\kappa_a(F)$. Consider the sub-poset $\mathcal{H}$ of $\mathcal{L}$ containing all the elements of $\bigcup_{F\in\mathcal{F}}(P(F)\cup N(F))$. Then we construct the antichains $\{\alpha_s\}_{s\geq 0}$ in such a way that $\alpha_s$ is the set of all minimal elements of $\mathcal{H}\setminus\bigcup_{0\leq t<s}\alpha_t$. Then this process stops, say $\alpha_v$ is $P(B)$. Then we have a chain of antichains $\alpha_0\prec\alpha_1\prec\cdots\prec\alpha_v$. Now $\kappa_a(B)=\kappa_a(\alpha_v)-\kappa_a(\alpha_0)= \Sigma_{t=1}^v\kappa_a(\alpha_t)-\kappa_a(\alpha_{t-1})$. In other words, if $C_t$ denotes the cell $\bigcup\alpha_t\setminus\bigcup\alpha_{t-1}$ for $1\leq t\leq v$, then $\kappa_a(B)=\Sigma_{t=1}^v\kappa_a(C_t)$.
Now it remains to show that for each $1\leq t\leq v$, $\kappa_a(C_t)=\Sigma_{F\in\alpha_t} \kappa_a(\mathrm{Core}_\mathcal{D}(F))$. This follows from the following proposition by first choosing $A_j$ to consist of elements of $\alpha_t$ and then choosing $A_j$ to consist of elements of $\alpha_{t-1}$. Then by our construction of the chain and the definition of $\kappa_a(C_t)$, we get the required result.
\end{proof}
\begin{pro}\label{pr2}
For any $\alpha_j\in\mathcal{A},A_j=\bigcup\alpha_j,j\in[k]=\{1,2,\hdots,k\}$ where $k\geq 2$, we have $\kappa_a(\bigcup_{j\in[k]}A_j)=\Sigma_{S\subseteq[k],S\neq\emptyset}\kappa_a(\bigcap_{s\in S}A_s\setminus\bigcup_{t\notin S}A_t)$.
\end{pro}
\begin{proof}
We observe that all the arguments on the right hand side of the above expression are cells or possibly empty sets and they form a partition of the cell in the argument of the left hand side. Then we restate theorem \ref{t1} as $\kappa_a((\bigcup\alpha)\cup(\bigcup\beta))= \kappa_a((\bigcup\alpha)\setminus(\bigcup\beta))+ \kappa_a((\bigcup\beta)\setminus(\bigcup\alpha))+ \kappa_a((\bigcup\alpha)\cap(\bigcup\beta))$. Since the set of $pp$-convex sets is closed under taking unions and intersections, a simple induction proves the proposition with \ref{t1} being the base case.
\end{proof}
Theorem \ref{t2} allows us to define the global characteristic for arbitrary definable sets.
\begin{definition}
Let $D\subseteq M^n$ be definable. Then we define the \textbf{global characteristic} $\Lambda(D)$ as the sum of global characteristics of any finite family of blocks partitioning $D$.
\end{definition}
\subsection{Preservation of global characteristics}\label{PTGC}
The aim of this section is to show that the global characteristic is preserved under definable isomorphisms.
\begin{theorem}\label{t3}
Suppose $D\in \mathrm{Def}(M^n)$ and $f:D\rightarrow M^n$ is a definable injection. Then $\Lambda(D)=\Lambda(f(D))$.
\end{theorem}
\begin{proof}
We first prove the local version which states that for any $a\in M^n$ and $B\in\mathcal{B}$ if $g:B\rightarrow M$ is a $pp$-definable injection, then $\kappa_a(B)=\kappa_{g(a)}(g(B))$. We observe that $\delta(B)(a)=\delta(g(B))(g(a))$. Lemma \ref{COLOURINJ} gives that the complex $\mathcal{K}(\mathcal{N}_a(\alpha))$ is isomorphic to the complex $\mathcal{K}(\mathcal{N}_{g(a)}(g[\alpha]))$ where $g[\alpha]=\{g(A):A\in\alpha\}$ and $\alpha$ is either $P(B)$ or $N(B)$. We conclude that $g(\mathrm{Sing}(B))=\mathrm{Sing}(g(B))$. Hence $\Lambda(B)=\Sigma_{a\in \mathrm{Sing}(B)}\kappa_a(B)=\Sigma_{a\in \mathrm{Sing}(B)}\kappa_{g(a)}(g(B))=\Sigma_{a\in \mathrm{Sing}(g(B))}\kappa_a(g(B))=\Lambda(g(B))$.
To prove the theorem, we consider any partition of $D$ into finitely many blocks $B_i,1\leq i\leq m$ such that $f\upharpoonright B_i$ is $pp$-definable. This is possible by an application of lemma \ref{REP}) to the set $Graph(f)$ followed projection of the finitely many blocks onto the first $n$ coordinates. Note that $D=\bigsqcup_{i=1}^m B_i \Rightarrow f(D)=\bigsqcup_{i=1}^m f(B_i)$ since $f$ is injective. Hence $\Lambda(f(D))=\Sigma_{i=1}^m\Lambda(f(B_i))=\Sigma_{i=1}^m\Lambda(B_i)=\Lambda(D)$, where the first and third equality follows by theorem \ref{t2} and the second equality follows from the previous paragraph.
\end{proof}
Now we are ready to prove a special case of the result promised at the end of section \ref{GRFOS}, which states that the Grothendieck ring of a right $\mathcal R$-module $M$ satisfying $M\equiv M^{(\aleph_0)}$ contains $\mathbb Z$ as a subgroup. This shows, in particular, that $K_0(M)$ is nontrivial in this case.
\begin{cor}\label{MAINRESULT}
Suppose $D\subseteq M^n$ is definable and $f:D\rightarrowtail D$ is a definable injection whose image is cofinite in the codomain, then $f$ is an isomorphism.
\end{cor}
\begin{proof}
We extend the function $f$ to an injective function $g:M^n\rightarrowtail M^n$ by setting $g(a)=f(a)$ if $a\in D$ and $g(a)=a$ otherwise. Now $F:=M^n\setminus Im(g)$ is finite; say it has $p$ elements. Further $\Lambda(Im(g))=\Lambda(M^n\setminus F)=\Lambda(M^n)-\Lambda(F)=-p$.
By theorem \ref{t3}, we get $\Lambda(M^n)=\Lambda(Im(g))$ since $g$ is definable injective. Hence $p=0$ and thus $g$ is an isomorphism. Since $g$ is the identity function outside $D$, we conclude that $f$ is a definable isomorphism.
\end{proof}
\subsection{Coloured global characteristics}\label{LCCC}
Let $P\in\mathcal{L}^*$ be fixed for this section. We develop the notion of localization at $P$ and local characteristic at $P$; we have developed these ideas earlier when $P$ is a singleton. After stating what we mean by a colour, we define the notion of a ``coloured global characteristic'' and outline the proof that these invariants are preserved under definable isomorphisms.
\begin{definition}
We use $\mathcal{L}_P$ to denote the meet-semilattice of all upper bounds of $P$ in $\mathcal{L}$, i.e. $\mathcal{L}_P:=\{A\in\mathcal{L}:A\supseteq P\}$. As usual, we denote the set of all finite antichains in this semilattice by $\mathcal{A}_P$.
\end{definition}
Since every element of $\mathcal{L}_P$ contains $P$, we may as well quotient out $P$ from each such element. Such a process is consistent with our earlier definition of localization since taking quotient with respect to a singleton set gives an isomorphic copy of the original set.
\begin{definitions}
We define the operator $\mathcal{Q}_P$ on the elements of $\mathcal{L}_P$ by setting $\mathcal{Q}_P(A):=p+\frac{A-p}{P-p}=\{a+(P-p):a\in A\}$ for any $p\in P$. We can clearly extend this operator to finite subsets of $\mathcal{L}_P$. Now let $\mathcal{L}_{(P)}:=\mathcal{Q}_P[\mathcal{L}_P]$. We use $\mathcal{A}_{(P)}$ to denote the set of all finite antichains in this semilattice.
\end{definitions}
It is easy to see that $\mathcal{A}_{(P)}=\mathcal{Q}_P[\mathcal{A}_P]$.
The appropriate analogue of the localization operator $\mathcal N_a:\mathcal A\rightarrow\mathcal A_a$ is a function $\mathcal{N}_P:\mathcal{A}\rightarrow\mathcal{A}_{(P)}$.
\begin{deflem}
For $\alpha\in\mathcal{A}$, we define $\mathcal{N}_P(\alpha):=\mathcal{Q}_P(\alpha\cap\mathcal{L}_P)$. As an operator on $pp$-convex sets, $\mathcal{N}_P$ preserves both unions and intersections.
\end{deflem}
The proof is easy and thus omitted.
Recall from definition \ref{SIMPCOMP} of $\mathcal K^a(\alpha)$ that the ``trivial intersections'' were precisely those which were empty or a singleton. On the other hand, ``nontrivial intersections'' were precisely those which contained the $pp$-set $\{a\}$ properly. As $\mathcal N_P$ takes values in $\mathcal A_{(P)}$, we get the correct notion of non-trivial intersections followed by the quotient operation so that the techniques developed for a singleton $P$ still remain valid. Now we are ready to state the analogue of definition \ref{SIMPCOMP}.
\begin{definition}
For $\alpha\in\mathcal{A}$, we define the \textbf{simplicial complex} of $\alpha$ \textbf{in the neighbourhood of} $P$ as the complex $\mathcal{K}(\mathcal{N}_P(\alpha))=\{\beta\subseteq\mathcal{N}_P(\alpha):|\bigcap\beta|=\infty\}$. For simplicity of notation, we denote this complex by $\mathcal{K}^P(\alpha)$.
\end{definition}
We can easily extend the notion of local characteristic at $P$ as follows.
\begin{definition}
We define the \textbf{local characteristic} of $\alpha$ at $P$ by $\kappa_P(\alpha):=\chi(\mathcal{K}^P(\alpha))-\delta(\alpha)(P)$.
\end{definition}
It can be observed that we recover the definition of the local characteristic at a point $a\in M$ by choosing $P=\{a\}$. The proofs of theorem \ref{t1} and lemma \ref{l1} go through if we replace $\kappa_a$ by $\kappa_P$. Thus we can define $\kappa_P(D)$ for arbitrary definable sets $D\subseteq M^n$.
We define the function $\kappa:\mathrm{Def}(M^n)\times\mathcal{L}^*\rightarrow\mathbb{Z}$ by setting $\kappa(D,P):=\kappa_P(D)$.
\begin{definition}
The \textbf{set of $\mathcal{L}$-singular elements} of a definable set $D\subseteq M^n$ is defined as the set $\mathrm{Sing}_\mathcal{L}(D):=\{P\in\mathcal{L}:\kappa(D,P)\neq0\}$.
\end{definition}
Fixing any partition of $D$ into blocks, it can be checked that the set $\mathrm{Sing}_\mathcal L(D)$ is contained in the nest corresponding to that partition and hence is finite. This finiteness will be used to define analogues of the global characteristic, which we call ``coloured global characteristics''.
\begin{definition}
For a given $P\in\mathcal{L}$, we define the \textbf{colour} of $P$ to be the set $\{A\in L:$ there is a bijection $f:A\cong P$ such that $Graph(f)$ is $pp$-definable $\}$. We denote the colour of $P$ by $[[P]]$.
\end{definition}
Note the significance of this definition. Theorem \ref{PPET} describes the $pp$-sets as fundamental definable sets and we are trying to classify definable sets up to definable isomorphism (definition \ref{defiso}). In fact it is sufficient to classify $pp$-sets up to $pp$-definable isomorphisms, which is the motivation behind the definition of a colour.
Let $\mathcal{X}$ denote the set of colours of elements from $\mathcal L$. We use letters $\mathfrak{A},\mathfrak{B},\mathfrak{C}$ etc. to denote the colours. It can be observed that $[[\emptyset]]$ is a singleton. We denote the colour of any singleton by $\mathfrak{U}$. We use $\mathcal{X}^*$ to denote $\mathcal{X}\setminus\{[[\emptyset]]\}$.
The global characteristic $\Lambda(D)$ is equal to $-\Sigma_{P\in\mathfrak{U}}\kappa_P(D)$ for each definable set $D$. This observation can be used to extend the notion of global characteristic.
\begin{definition}
For $\mathfrak{A}\in\mathcal{X}^*$, we define the \textbf{coloured global characteristic} with respect to $\mathfrak A$ for a definable set $D$ to be the integer $\Lambda_{\mathfrak{A}}(D):=-\Sigma_{P\in\mathfrak{A}}\kappa_P(D)$. This integer is well defined as it is equal to the finite sum $-\Sigma\{\kappa_P(D):P\in(\mathfrak{A}\cap \mathrm{Sing}_\mathcal{L}(D))\}$.
\end{definition}
The property of coloured global characteristics that we are looking for is stated in the following analogue of theorem \ref{t3}. The proof is analogous to that of \ref{t3} and thus is omitted.
\begin{theorem}\label{t4}
If $f:D\rightarrow D'$ is a definable bijection between definable sets $D,D'$, then $\Lambda_\mathfrak{A}(D)=\Lambda_\mathfrak{A}(D')$ for each $\mathfrak{A}\in\mathcal{X}^*$.
\end{theorem}
\section{Special Case: Multiplicative Structure}\label{spcasemult}
\subsection{Monoid rings}\label{MRSK}
We need the notion of an algebraic structure called a \emph{monoid ring}.
\begin{definition}
Let $(A,\star,1)$ be a commutative monoid and $S$ be a commutative ring with unity. Then we define an $L_{ring}$-structure $(S[A],0,1,+,\cdotp)$ called a \textbf{monoid ring} as follows.
\begin{itemize}
\item $S[A]:=\{\phi:A\rightarrow S :$ the set $\mathrm{Supp}(\phi)=\{a:\phi(a)\neq 0\}$ is finite$\}$
\item $(\phi+\psi)(a):=\phi(a)+\psi(a)$ for $a\in A$
\item $(\phi\cdotp\psi)(a):=\Sigma_{b\star c=a}\phi(b)\psi(c)$ for $a\in A$
\end{itemize}
An element $\phi$ of $S[A]$ can be represented as a formal sum $\Sigma_{a\in A}s_a a$ where $s_a=\phi(a)$.
\end{definition}
As an example, let $A=\mathbb N$ ,equivalently the monoid $\{X^n\}_{n\geq 0}$ considered multiplicatively. Then the monoid ring $S[A]=S[\mathbb N]\cong S[X]$, the polynomial ring in one variable with coefficients from $S$.
Let the symbols $\overline{\mathcal L},\overline{\mathcal A},\overline{\mathcal X},\hdots$ etc. denote the unions $\bigcup_{n=1}^\infty\mathcal L_n$, $\bigcup_{n=1}^\infty\mathcal A_n$, $\bigcup_{n=1}^\infty\mathcal X_n,\hdots$ respectively. We shall be especially concerned with the sets $\overline{\mathcal L}^*:=\overline{\mathcal L}\setminus\{\emptyset\}$ and $\overline{\mathcal X}^*:=\overline{\mathcal X}\setminus\{[[\emptyset]]\}$.
There is a binary operation $\times:\overline{\mathcal L}^*\times\overline{\mathcal L}^*\rightarrow \overline{\mathcal L}^*$ which maps a pair $(A,B)$ to the cartesian product $A\times B$. This map commutes with the operation $[[-]]$ of taking colour i.e., whenever $[[A_1]]=[[A_2]]$ and $[[B_1]]=[[B_2]]$, we have $[[A_1\times B_1]]=[[A_2\times B_2]]$. This allows us to define a binary operation $\star:\overline{\mathcal X}^* \times \overline{\mathcal X}^* \rightarrow \overline{\mathcal X}^*$ which takes a pair of colours $(\mathfrak A,\mathfrak B)$ to $[[A\times B]]$ for any $A\in\mathfrak A,B\in\mathfrak B$. The colour $\mathfrak U$ of singletons acts as the identity element for the operation $\star$. Hence $(\overline{\mathcal X}^*,\star,\mathfrak U)$ is a monoid.
Consider the maps $\Lambda_\mathfrak A:\widetilde{\mathrm{Def}}(M)\rightarrow\mathbb Z$ for $\mathfrak A\in\overline{\mathcal X}^*$ defined by $[D]\mapsto\Lambda_\mathfrak A(D')$ for any $D'\in[D]$. These maps are well defined due to theorem \ref{t4}. We can fix some $[D]\in\widetilde{\mathrm{Def}}(M)$ and look at the set $\mathrm{Supp}([D]):=\{\mathfrak A\in\overline{\mathcal X}^*:\Lambda_\mathfrak A(D)\neq 0\}$. This set is finite since it is contained in the finite set $\{[[P]]:P\in \mathrm{Sing}_{\overline{\mathcal L}}(D)\}$. This shows that the evaluation map $ev_{[D]}:\overline{\mathcal X}^*\rightarrow\mathbb Z$ defined by $\mathfrak A\mapsto\Lambda_{\mathfrak A}([D])$ for each $[D]\in\widetilde{\mathrm{Def}}(M)$ is an element of the monoid ring $\mathbb Z[\overline{\mathcal X}^*]$.
Let us consider an example. We take $\mathcal R$ to be an infinite skew-field (i.e. a (possibly non-commutative) ring in which every nonzero element has two-sided multiplicative inverse) and $M$ to be any nonzero $\mathcal R$-vector space. This example has been studied in detail in \cite{Perera}. In this case, we have $Th(M)=Th(M)^{\aleph_0}$. Using the notion of affine dimension, it can be shown that $\overline{\mathcal X}^*\cong\mathbb N$. It has been shown that $K_0(M)\cong\mathbb Z[X]\cong\mathbb Z[\mathbb N]$. The proof in \cite{Perera} explicitly shows that the semiring $\widetilde{\mathrm{Def}}(M)$ is cancellative and is isomorphic to the semiring of polynomials in $\mathbb Z[X]$ with non-negative leading coefficients.
We will prove that a similar fact holds for an arbitrary module $M$, i.e., the structure of the Grothendieck ring $K_0(M)$ is entirely determined by the monoid $\overline{\mathcal X}^*$.
\begin{theorem}\label{FINAL}
Let $M$ be a right $\mathcal R$-module satisfying $Th(M)=Th(M)^{\aleph_0}$. Then $K_0(M)\cong\mathbb Z[\overline{\mathcal X}^*]$. In particular, $K_0(M)$ is nontrivial for every nonzero module $M$.
\end{theorem}
The proof of this theorem will occupy the next two sections.
\subsection{Multiplicative structure of $\widetilde{\mathrm{Def}}(M)$}\label{MULT}
Given $D_1\in \mathrm{Def}(M^n)$ and $D_2\in \mathrm{Def}(M^m)$, their cartesian product $D_1\times D_2\in \mathrm{Def}(M^{(n+m)})$. This shows that $\overline{\mathrm{Def}}(M)$ is closed under cartesian products. We want to show that the sets $\overline{\mathcal L}$, $\overline{\mathcal A}$, $\overline{\mathcal B}$ and $\overline{\mathcal C}$ are all closed under multiplication.
Let $P\in\mathcal L_n$ and $Q\in\mathcal L_m$. Then there are $pp$ formulas $\phi(\overline{x})$ and $\psi(\overline{y})$ defining those sets respectively. Without loss, we may assume that $\overline{x}\cap\overline{y}=\emptyset$. Now the formula $\rho(\overline{x},\overline{y})=\phi(\overline{x})\wedge\psi(\overline{y})$ is again a $pp$-formula and it defines the set $P\times Q\in\mathcal L_{n+m}$. This shows that the set $\overline{\mathcal L}$ is closed under multiplication.
Now we want to show that the product of two antichains $\alpha\in\mathcal A_n$ and $\beta\in\mathcal A_m$ is again an antichain in $\mathcal A_{n+m}$. We have natural projection maps $\pi_1:M^{n+m}\rightarrow M^n$ and $\pi_2:M^{n+m}\rightarrow M^m$ which project onto the first $n$ and the last $m$ coordinates respectively. First we observe that $(\bigcup\alpha)\times(\bigcup\beta)=\bigcup_{A\in\alpha}\bigcup_{B\in\beta}A\times B$. If either $A_1,A_2\in\alpha$ are distinct or $B_1,B_2\in\beta$ are distinct, then all the distinct elements from $\{A_i\times B_j\}_{i,j=1}^2$ are incomparable with respect to the inclusion ordering since at least one of their projections is so. Hence $\bigcup\alpha\times\bigcup\beta$ is indeed an antichain of the rank $|\alpha|\times|\beta|$. We will denote this antichain by $\alpha\times\beta$.
Given $C_1,C_2\in\overline{\mathcal C}$, we have $C_1\times C_2=\bigcup(\alpha_1\times\alpha_2)\setminus (\bigcup(\alpha_1\times\beta_2)\cup\bigcup(\beta_1\times\alpha_2))$ where $\alpha_i=P(C_i)$ and $\beta_i=N(C_i)$ for $i=1,2$. This shows that $C_1\times C_2\in\overline{\mathcal C}$ since $\overline{\mathcal A}$ is closed under both products and unions. Furthermore, we observe that $P(C_1\times C_2)=P(C_1)\times P(C_2)$. This in particular shows that the set $\overline{\mathcal B}$ of blocks is also closed under products.
\begin{lemma}\label{localcharmult}
Let $P,Q\in\overline{\mathcal L}$ and $\alpha,\beta\in\overline{\mathcal A}$. Then $\kappa_{P\times Q}(\alpha\times \beta)=-\kappa_P(\alpha)\kappa_Q(\beta)$.
\end{lemma}
\begin{proof}
First assume that $\delta(\alpha)(P)=\delta(\beta)(Q)=1$. Then observe that
\begin{equation}\label{discomp}\mathcal K^{P\times Q}(\alpha\times\beta)\cong\mathcal K^P(\alpha)\boxtimes\mathcal K^Q(\beta).\end{equation}
Hence we have
\begin{eqnarray*}
\kappa_{P\times Q}(\alpha\times\beta)&=&\chi(\mathcal K^{P\times Q}(\alpha\times\beta))-1\\
&=&\chi(\mathcal K^P(\alpha))+\chi(\mathcal K^Q(\beta))-\chi(\mathcal K^P(\alpha))\chi(\mathcal K^Q(\beta))-1\\
&=&(\kappa_P(\alpha)+1)+(\kappa_Q(\beta)+1)-(\kappa_P(\alpha)+1)(\kappa_Q(\beta)+1)-1\\
&=&-\kappa_P(\alpha)\kappa_Q(\beta)
\end{eqnarray*}
The first and third equality is by definition of the local characteristic and the second is by equation (\ref{ecdp}) of lemma \ref{ecdpl} applied to (\ref{discomp}).
In the remaining case when either $\delta(\alpha)(P)$ or $\delta(\beta)(Q)$ is $0$, we have $\delta(\alpha\times\beta)(P\times Q)=0$. Hence $\kappa_{P\times Q}(\alpha\times\beta)=0$ and either $\kappa_P(\alpha)$ or $\kappa_Q(\beta)$ is $0$. This gives the necessary identity and thus completes the proof in all cases.
\end{proof}
The aim of this section is to prove the following theorem.
\begin{theorem}\label{t5}
The map $ev:\widetilde{\mathrm{Def}}(M)\rightarrow\mathbb Z[\overline{\mathcal X}^*]$ defined by $[D]\mapsto ev_{[D]}$ is a semiring homomorphism.
\end{theorem}
\begin{proof} We have already seen that $ev$ is additive, since each $\Lambda_{\mathfrak A}$ is. So it remains to show that it is multiplicative.
We have observed that the set $[\overline{\mathcal A}]$ is a monoid with respect to cartesian product, the isomorphism class of a singleton being the identity for the multiplication. So we will first show that $ev:[\overline{\mathcal A}]\rightarrow\mathbb Z[\overline{\mathcal X}^*]$ is a multiplicative monoid homomorphism.
Let $\alpha,\beta\in\overline{\mathcal A}$ be fixed. Note that
\begin{equation}\label{singincl}S:=\mathrm{Sing}_{\overline{\mathcal L}}(\alpha\times\beta)\subseteq\{P\times Q:P\in \mathrm{Sing}_{\overline{\mathcal L}}(\alpha),Q\in \mathrm{Sing}_{\overline{\mathcal L}}(\beta)\}.\end{equation}
We need to show that $ev_{[\alpha]}\cdotp ev_{[\beta]}=ev_{[\alpha\times\beta]}$ as maps on $\overline{\mathcal X}^*$. This is equivalent to $ev_{[\alpha\times\beta]}(\mathfrak C)=\sum_{\mathfrak A\star\mathfrak B=\mathfrak C}ev_{[\alpha]}(\mathfrak A) ev_{[\beta]}(\mathfrak B)$ for each $\mathfrak C\in\overline{\mathcal X}^*$. Using the definition of the evaluation map, it is enough to check that $\Lambda_{\mathfrak C}([\alpha\times\beta])=\sum_{\mathfrak A\star\mathfrak B=\mathfrak C}\Lambda_{\mathfrak A}([\alpha]) \Lambda_{\mathfrak B}([\beta])$ for each $\mathfrak C\in\overline{\mathcal X}^*$.
The left hand side of the above equation is
\begin{eqnarray*}
\Lambda_{\mathfrak C}([\alpha\times\beta]) &=& -\sum_{R\in\mathfrak C}\kappa_R(\alpha\times\beta)\\
&=& -\sum_{R\in(\mathfrak C\cap S)}\kappa_R(\alpha\times\beta)\\
&=& \sum_{R\in(\mathfrak C\cap S)}\kappa_{\pi_1(R)}(\alpha)\kappa_{\pi_2(R)}(\beta)
\end{eqnarray*}
The last equality is given by the lemma \ref{localcharmult} since, by (\ref{singincl}), every $R\in\mathfrak C\cap S$ can be written as $R=\pi_1(R)\times\pi_2(R)$. The right hand side is
\begin{eqnarray*}
\sum_{\mathfrak A\star\mathfrak B=\mathfrak C}\Lambda_{\mathfrak A}([\alpha])\Lambda_{\mathfrak B}([\beta]) &=& \sum_{\mathfrak A\star\mathfrak B=\mathfrak C}\left(-\sum_{P\in\mathfrak A}\kappa_P(\alpha)\right)\left(-\sum_{Q\in\mathfrak B}\kappa_Q(\beta)\right) \\
\ &=& \sum_{\mathfrak A\star\mathfrak B=\mathfrak C}\sum_{P\in\mathfrak A,Q\in\mathfrak B}\kappa_P(\alpha)\kappa_Q(\beta)
\end{eqnarray*}
Using the definition of $\mathrm{Sing}_{\overline{\mathcal L}}(-)$, we observe that the final expressions on both sides are equal. This completes the proof that $ev$ is a multiplicative monoid homomorphism on $[\overline{\mathcal A}]$.
Now we will show that $ev$ is also multiplicative on the monoid $[\overline{\mathcal C}]$. Let $C_1,C_2$ be cells with $\alpha_i=P(C_i)$ and $\beta_i=N(C_i)$ for each $i=1,2$. Then $C_1\times C_2=\bigcup(\alpha_1\times\alpha_2)\setminus (\bigcup(\alpha_1\times\beta_2)\cup\bigcup(\beta_1\times\alpha_2))$. We also know that $ev_{[C]}=ev_{P(C)}-ev_{N(C)}$ for each cell $C$.
We need to show that $\Lambda_{\mathfrak C}(C_1\times C_2)=\sum_{\mathfrak A\star\mathfrak B=\mathfrak C}\Lambda_{\mathfrak A}([C_1])\Lambda_{\mathfrak B}([C_2])$ for each $\mathfrak C\in\overline{\mathcal X}^*$. Now we have
\begin{equation*}
\Lambda_{\mathfrak C}(C_1\times C_2)=\Lambda_{\mathfrak C}(\alpha_1\times\alpha_2)-\Lambda_{\mathfrak C}((\alpha_1\times\beta_2)\vee(\beta_1\times\alpha_2))
\end{equation*}
and we also have
\begin{eqnarray*}
\sum_{\mathfrak A\star\mathfrak B=\mathfrak C}\Lambda_{\mathfrak A}([C_1])\Lambda_{\mathfrak B}([C_2]) &=& \sum_{\mathfrak A\star\mathfrak B=\mathfrak C}(\Lambda_{\mathfrak A}([\alpha_1])-\Lambda_{\mathfrak A}([\beta_1]))(\Lambda_{\mathfrak B}([\alpha_2])-\Lambda_{\mathfrak B}([\beta_2])) \\
&=& \Lambda_{\mathfrak C}(\alpha_1\times\alpha_2)+\Lambda_{\mathfrak C}(\beta_1\times\beta_2)-\Lambda_{\mathfrak C}(\beta_1\times\alpha_2)-\Lambda_{\mathfrak C}(\alpha_1\times\beta_2)
\end{eqnarray*}
Therefore we need to show
\begin{equation*}
\Lambda_{\mathfrak C}((\alpha_1\times\beta_2)\vee(\beta_1\times\alpha_2))+\Lambda_{\mathfrak C}(\beta_1\times\beta_2)=\Lambda_{\mathfrak C}(\alpha_1\times\beta_2)+\Lambda_{\mathfrak C}(\beta_1\times\alpha_2).
\end{equation*}
This is true by theorem \ref{t1} since we have $(\alpha_1\times\beta_2)\wedge(\beta_1\times\alpha_2)=(\beta_1\times\beta_2)$.
In the last step, we show that $ev_{[D_1\times D_2]}=ev_{[D_1]}\cdotp ev_{[D_2]}$ for arbitrary definable sets $D_1,D_2$. Let $[D_1]=\sum_{i=1}^k[B_{1i}]$ and $[D_2]=\sum_{j=1}^l[B_{2j}]$ be obtained from any decompositions of $D_1$ and $D_2$ into blocks. Then $[D_1\times D_2]=\sum_{i=1}^{k}\sum_{j=1}^{l}[B_{1i}\times B_{2j}])$. For each $\mathfrak C\in\overline{\mathcal X}^*$, we have
\begin{eqnarray*}
ev_{[D_1]}\cdotp ev_{[D_2]}(\mathfrak C) &=& \sum_{\mathfrak A\star\mathfrak B=\mathfrak C}\Lambda_{\mathfrak A}([D_1])\Lambda_{\mathfrak B}([D_2]) \\
&=& \sum_{\mathfrak A\star\mathfrak B=\mathfrak C}\left(\sum_{i=1}^{k}\Lambda_{\mathfrak A}([B_{1i}])\right)\left(\sum_{j=1}^{l}\Lambda_{\mathfrak B}([B_{2j}])\right) \\
&=& \sum_{i=1}^{k}\sum_{j=1}^{l}\sum_{\mathfrak A\star\mathfrak B=\mathfrak C}\Lambda_{\mathfrak A}([B_{1i}])\Lambda_{\mathfrak B}([B_{2j}]) \\
&=& \sum_{i=1}^{k}\sum_{j=1}^{l}\Lambda_{\mathfrak C}([B_{1i}\times B_{2j}]) \\
&=& \Lambda_{\mathfrak C}(\sum_{i=1}^{k}\sum_{j=1}^{l}[B_{1i}\times B_{2j}]) \\
&=& ev_{[D_1\times D_2]}(\mathfrak C).
\end{eqnarray*}
This completes the proof showing $ev$ is a semiring homomorphism.
\end{proof}
\subsection{Computation of the Grothendieck ring}\label{COMPUTATION}
In the previous section, we showed that $ev:\widetilde{\mathrm{Def}}(M)\rightarrow\mathbb Z[\overline{\mathcal X}^*]$ is a semiring homomorphism. Since the codomain of this map is a ring, it factorizes through the unique homomorphism of cancellative semirings $\widetilde{ev}:\widetilde{\widetilde{\mathrm{Def}}(M)}\rightarrow\mathbb Z[\overline{\mathcal X}^*]$ where $\widetilde{\widetilde{\mathrm{Def}}(M)}$ is the quotient semiring of $\widetilde{\mathrm{Def}}(M)$ obtained as in theorem \ref{QUOCONST}. Our next aim is to prove the following lemma.
\begin{lemma}\label{INJEV}
The map $\widetilde{ev}:\widetilde{\widetilde{\mathrm{Def}}(M)}\rightarrow\mathbb Z[\overline{\mathcal X}^*]$ is injective.
\end{lemma}
\begin{proof}
We will prove this lemma in several steps. First we will identify a subset of $\overline{\mathrm{Def}}(M)$ where the restriction of the evaluation function is injective.
Let $\mathcal U=\{\alpha\in\overline{\mathcal A}:A_1\cap A_2=\emptyset$ for all distinct $A_1,A_2\in\alpha\}$. Then it can be easily checked that $\Lambda_{\mathfrak A}(\alpha)=|\alpha\cap\mathfrak A|$ for each $\mathfrak A\in\overline{\mathcal X}^*$ and $\alpha\in\mathcal U$. Hence if $ev_{[\alpha]}=ev_{[\beta]}$ for any $\alpha,\beta\in\mathcal U$, then we have $[\alpha]=[\beta]$. This proves that the map $ev$ is itself injective on $\mathcal U$.
Given any $[D_1],[D_2]\in\widetilde{\mathrm{Def}}(M)$ such that $ev_{[D_1]}=ev_{[D_2]}$, we will find some $[X]\in\widetilde{\mathrm{Def}}(M)$ such that $[D_1]+[X]=[\alpha']$ and $[D_2]+[X]=[\beta']$ for some $\alpha',\beta'\in\mathcal U$. Then we get $ev_{[\alpha']}=ev_{[D_1]}+ev_{[X]}=ev_{[D_2]}+ev_{[X]}=ev_{[\beta']}$ and hence we will be done by the previous paragraph.
\textbf{Claim:} It is sufficient to assume $[D_1],[D_2]\in[\overline{\mathcal A}]$.
Let $[D_1]=\sum_{i=1}^k[B_{1i}]$ and $[D_2]=\sum_{j=1}^l[B_{2j}]$ be obtained from any decompositions of $D_1$ and $D_2$ into blocks. We have $[P(B)]=[B]+[N(B)]$ for any $B\in\overline{\mathcal B}$. Therefore if we choose $[Y]=\sum_{i=1}^k[N(B_{1i})]+\sum_{j=1}^l[N(B_{2j})]$, we get $[D_1]+[Y]=\sum_{i=1}^l[P(B_{1i})]+ \sum_{j=1}^l[N(B_{2j})]$ and $[D_2]+[Y]=\sum_{i=1}^l[N(B_{1i})]+\sum_{j=1}^l[P(B_{2j})]$. Hence both $[D_1]+[Y],[D_2]+[Y]\in[\overline{\mathcal A}]$. This finishes the proof of the claim.
Now let $\alpha,\beta\in\overline{\mathcal A}$ be such that $ev_{[\alpha]}=ev_{[\beta]}$. We describe an algorithm which terminates in finitely many steps and yields some $[X]$ such that $[\alpha]+[X],[\beta]+[X]\in[\mathcal U]$. Before stating the algorithm, we define a \textbf{complexity function} $\Gamma:\overline{\mathcal A}\rightarrow\mathbb N$. For each antichain $\alpha$, the complexity $\Gamma(\alpha)$ is defined to be the maximum of the lengths of chains in the smallest nest corresponding to $\alpha$, where the length of a chain is the number of elements in it. Note that $\Gamma(\alpha)\leq 1$ if and only if $\alpha\in\mathcal U$.
Let $\alpha=\{A_1,A_2,\hdots,A_k\}$ be any enumeration and let $\alpha_i=\{A_1,A_2,\hdots,A_i\}$ for each $1\leq i\leq k$. Similarly choosing an enumeration $\beta=\{B_1,B_2,\hdots,B_l\}$, we define $\beta_j$ for each $1\leq j\leq l$. Then we observe that $\bigcup\alpha=\bigsqcup_{i=1}^k \mathrm{Core}_{\alpha_i}(A_i)$ and $\bigcup\beta=\bigsqcup_{j=1}^l \mathrm{Core}_{\beta_j}(B_j)$. Now each $\mathrm{Core}_{\alpha_i}(A_i)$ is a block, which can be completed to a $pp$-set if we take its (disjoint) union with $N(\mathrm{Core}_{\alpha_i}(A_i))$. This can be written as the equation $[A_i]=[\mathrm{Core}_{\alpha_i}(A_i)]+ [N(\mathrm{Core}_{\alpha_i}(A_i))]$. If $\bigcup\alpha\subseteq M^n$, we consider $M^{nk}$ and inject $\mathrm{Core}_{\alpha_i}(A_i)$ in the obvious way into the $i^{th}$ copy of $M^n$ in $M^{nk}$ for each $i$. This gives us a definable set definably isomorphic to $\bigcup\alpha$. The advantage of this decomposition is that we can also add an isomorphic copy of $N(\mathrm{Core}_{\alpha_i}(A_i))$ at the appropriate place for each $i$ and obtain a new antichain representing $\sum_{i=1}^k [A_i]$.
Repeating the same procedure for $\beta$ yields a representative of $\sum_{j=1}^l [B_j]$. In order to maintain the evaluation function on both sides, we add disjoint copies of the antichains $N(\mathrm{Core}_{\alpha_i}(A_i))$, $N(\mathrm{Core}_{\beta_j}(B_j))$ to both sides. So we choose $[W]=\sum_{i=1}^k[N(\mathrm{Core}_{\alpha_i}(A_i))]+ \sum_{j=1}^l [N(\mathrm{Core}_{\beta_j}(B_j))]$, hence $[\alpha]+[W],[\beta]+[W]$ are both in $[\overline{\mathcal A}]$ so that the particular antichains $\alpha',\beta'$ in these classes we constructed above satisfy $\Gamma((\bigcup\alpha')\sqcup(\bigcup\beta'))<\Gamma((\bigcup\alpha)\sqcup(\bigcup\beta))$. The inequality holds since we isolate the maximal elements of the nest corresponding to $(\bigcup\alpha)\cup(\bigcup\beta)$ in the process.
We repeat this process, inducting on the complexity of the antichains, till the disjoint union of the pair of antichains in the output lies in $\mathcal U$. Since the complexity decreases at each step, this algorithm terminates in finitely many steps. The required $[X]$ is the sum of the $[W]$'s obtained at each step. This finishes the proof of the injectivity of the map $\widetilde{ev}$.
\end{proof}
Finally we are ready to prove theorem \ref{FINAL} regarding the structure of the Grothendieck ring $K_0(M)$.
\begin{proof} (of Theorem \ref{FINAL}) It is easy to observe that the image of $\mathcal U$ under the evaluation map is the monoid semiring $\mathbb N[\overline{\mathcal X}^*]$. The Grothendieck ring $K_0(\mathbb N[\overline{\mathcal X}^*])$ is clearly isomorphic to the monoid ring $\mathbb Z[\overline{\mathcal X}^*]$.
Since the map $\widetilde{ev}$ is injective by lemma \ref{INJEV} and $\mathbb N[\overline{\mathcal X}^*]\subseteq Im(\widetilde{ev})\subseteq\mathbb Z[\overline{\mathcal X}^*]$, we have $K_0(M)=K_0(Im(\widetilde{ev}))\cong\mathbb Z[\overline{\mathcal X}^*]$ by the universal property of $K_0$ in theorem \ref{GRCONSTR}.
\end{proof}
\section{General Case}\label{gencase}
\subsection{Finite indices of $pp$-pairs}\label{TNTA}
So far we have considered the Grothendieck ring of a right $\mathcal R$-module $M$ whose theory $T:=Th(M)$ satisfies $T=T^{\aleph_0}$. From this section onwards we remove this condition and work with an arbitrary right $\mathcal R$-module $M$.
We continue to use the notations $\mathcal L_n,\mathcal P_n,\mathcal A_n,\mathcal X_n$ to denote the set of all $pp$-subsets of $M^n$, the set of all finite subsets of $\mathcal L_n$, the set of all finite antichains in $\mathcal L_n$ and the set of all $pp$-isomorphism classes (colours) in $\mathcal L_n$ respectively. We still use the representation theorem \ref{REP}, but lemma \ref{NLU} is unavailable to obtain the uniqueness - proposition \ref{UNIQUE1}. As a result we do not have a bijection between the set of all $pp$-convex sets, which we denote by $\mathcal O_n$, and the set $\mathcal A_n$. The elements of the set $\mathcal C_n:=\{(\bigcup\alpha)\setminus(\bigcup\beta)| \alpha,\beta\in\mathcal A_n,\ \bigcup\beta\subsetneq\bigcup\alpha\}$ will be called cells. The cells allowing a representation of the form $P\setminus\bigcup\beta$ for some $P\in\mathcal L_n$ and $\beta\in\mathcal A$ such that $P\subsetneq\bigcup\beta$ will be called blocks and the set of all blocks in $\mathcal C_n$ is denoted by $\mathcal B_n$.
Let $(-)^\circ:\mathcal L_n\rightarrow \mathcal L_n$ denote the function which takes a coset $P$ to the subgroup $P^\circ:=P-p$, where $p\in P$ is any element. We use $\mathcal L_n^\circ$ to denote the image of this function, i.e. the set of all $pp$-definable subgroups. Let $\sim_n$ denote a relation on $\mathcal L_n^\circ$ defined by $P\sim_nQ$ if and only if $[P:P\cap Q]+[Q:P\cap Q]<\infty$. This is the \textbf{commensurability relation} and it can be easily checked to be an equivalence relation. We can extend this relation to all elements of $\mathcal L_n$ using the same definition if we set the index $[P:Q]:=[P^\circ:P^\circ\cap Q^\circ]$ for all $P,Q\in\mathcal L_n$. Let $\mathcal Y_n$ denote the set of all commensurability equivalence classes of $\mathcal L_n$ (\textbf{bands} for short). We use capital bold letters $\mathbf{P},\mathbf{Q},\cdots$ etc. to denote bands. The equivalence class (band) of $P$ will be denoted by the corresponding bold letter $\mathbf{P}$.
Now we fix some $n\geq 1$ and drop all the subscripts as usual. Note that, in the special case, a band is just the collection of all cosets of a $pp$-subgroup. In particular any two distinct elements of a band are disjoint. This `discreteness' has been exploited heavily in all the proofs for the special case. We need to work hard to set up the technical machinery for defining the local characteristics. The proofs for the general case will be similar to those for the special case once we obtain the required discreteness condition.
Let $\mathbf{P}\in\mathcal Y$. It can be easily checked that if $P,Q\in\mathbf{P}$ and $P\cap Q\neq\emptyset$ then $P\cap Q\in\mathbf{P}$ i.e. $\mathbf{P}$ is closed under intersections which are nonempty. By definition of the index, it is also clear that if $P\in\mathbf{P}$ and $a\in M^n$, then $a+P\in\mathbf{P}$. Let $\mathcal A(\mathbf{P}),\mathcal P(\mathbf{P})$ and $\mathcal O(\mathbf{P})$ denote the sets of all finite antichains in $\mathbf{P}$, finite subsets of $\mathbf{P}$ and unions of finite subsets of $\mathbf{P}$ respectively.
We have the following analogue of proposition \ref{UNIQUE1} for $pp$-convex sets. The proof is omitted as it is similar to the $\mathrm{T=T^{\aleph_0}}$ case.
\begin{pro}
Let $X\in\mathcal O$. Then the set $S(X):=\{\mathbf{P}\in\mathcal Y: \exists \alpha\in\mathcal A\ (P\in~\alpha,$ $ \bigcup\alpha=X)\}$ is finite. Furthermore for any two $\alpha,\beta\in\mathcal A$ such that $\bigcup\alpha=\bigcup\beta=X$ and each $\mathbf{P}\in S(X)$, we have $\bigcup(\alpha\cap\mathbf{P})=\bigcup(\beta\cap\mathbf{P})$. Thus $X$ is uniquely determined by the family $\{X_\mathbf{P}:=\bigcup(\alpha\cap\mathbf{P}) \in\mathcal O(\mathbf{P})\mid\mathbf{P}\in S(X)\}$ for any $\alpha\in\mathcal A$ such that $\bigcup\alpha=X$.
\end{pro}
Given some $X\in\mathcal O(\mathbf{P})$ there could be two different $\alpha,\beta\in\mathcal A(\mathbf{P})$ such that $\bigcup\alpha=\bigcup\beta=X$. The nests corresponding to such antichains could have entirely different (semilattice) structures. The following proposition gives us a way to obtain an antichain $\alpha$ representing $X$ such that if $A,B\in\alpha$ and $A\neq B$, then $A\cap B=\emptyset$.
\begin{pro}
Let $X\in\mathcal O(\mathbf{P})$. Then for any $\alpha\in\mathcal A(\mathbf{P})$ such that $\bigcup\alpha=X$, there is some $\mathbf{P}(\alpha)\in\mathbf{P}^\circ$ such that $X$ is a finite union of distinct cosets of $\mathbf{P}(\alpha)$.
\end{pro}
\begin{proof}
Choose $\mathbf{P}(\alpha)=\bigcap\{Q^\circ:Q\in\alpha\}$ and observe that $\mathbf{P}(\alpha)\in\mathbf{P}$ since $\mathbf{P}$ is closed under finite nonempty intersections.
\end{proof}
The previous two propositions together imply that we can always find a `nice' antichain representing the given $pp$-convex set. The following definition describes what we mean by this.
\begin{definition}
A finite set $\alpha\in\mathcal P$ is said to be in \textbf{discrete form} if $\alpha\cap\mathbf{P}$ consists of finitely many cosets of a fixed element of $\mathbf{P}^\circ$, denoted $\mathbf{P}(\alpha)$, for each $\mathbf{P}\in\mathcal Y$. The set of all finite sets $\alpha\in\mathcal P$ in discrete form will be denoted by $\mathcal P^d$ and the set of all antichains in discrete form will be denoted by $\mathcal A^d$.
\end{definition}
We would like to define the local characteristics for the elements of $\mathcal P^d$ as before and show that they satisfy the conclusion of theorem \ref{t1}. We will restrict our attention only to those $\alpha\in\mathcal P^d$ such that $\alpha=\hat{\alpha}$ (i.e. the nest corresponding to $\alpha$ is $\alpha$ itself). We denote the set of all such finite sets by $\hat{\mathcal P}^d$. Since we will deal with finite index subgroup pairs in $\mathcal L^\circ$, we will need more conditions on compatibility of $P$ and $\alpha$ as stated in the following definition.
\begin{definition}
A finite family $\mathcal F$ of elements of $\mathcal P$ is called \textbf{compatible} if $\mathcal F\subseteq\hat{\mathcal P}^d$ and for all $\alpha,\beta\in\mathcal F$ and $\mathbf{P}\in\mathcal Y$, we have $\mathbf{P}(\alpha)=\mathbf{P}(\beta)$ whenever $\mathbf{P}\cap\alpha,\mathbf{P}\cap\beta\neq\emptyset$. Furthermore, we say that $P\in\mathcal L$ is \textbf{compatible with} a finite family $\mathcal F$ of elements of $\mathcal P$ if $\mathcal F$ is compatible and $P\in\bigcup\mathcal F$.
\end{definition}
It is very easy to observe that given any finite family $\{X_1,X_2,\hdots,X_k\}$ of $pp$-convex sets, we can obtain a compatible family $\{\alpha_1,\alpha_2,\hdots,\alpha_k\}$ of antichains such that $\bigcup\alpha_i=X_i$ for each $i$. Finally we are ready to define the local characteristics in this set-up.
\begin{definition}
Let $P\in\mathcal L$ be compatible with a family $\mathcal F$ and let $\alpha\in\mathcal F$. We associate an abstract simplicial complex $\mathcal K^P(\alpha)$ with the pair $(\alpha,P)$ by setting $\mathcal K^P(\alpha):=\{\beta\subseteq\alpha: \beta\neq\emptyset,\,\bigcap\beta\supsetneq P\}$. We define the \textbf{local characteristic} $\kappa_P$ by the formula $\kappa_P(\alpha):=\chi(\mathcal K^P(\alpha))-\delta(\alpha)(P)$.
\end{definition}
Now we are ready to state the analogue of theorem \ref{t1} and it has essentially the same proof. The previous statement is justified because we have carefully developed the idea of a compatible family to avoid finite index pairs of $pp$-subgroups. Since we achieve discreteness simultaneously for any finite family of antichains, no changes in the proof of theorem \ref{t1} are necessary.
\begin{theorem}\label{t1general}
Let $X,Y\in\mathcal O$. Then $X\cup Y,X\cap Y\in\mathcal O$. For any compatible family $\mathcal F:=\{\alpha_1,\alpha_2,\beta_1,\beta_2\}$ such that $\bigcup\alpha_1=X$, $\bigcup\alpha_2=Y$, $\bigcup\beta_1=X\cup Y$ and $\bigcup\beta_2=X\cap Y$ and any $P\in\mathcal L$ compatible with $\mathcal F$, we have
\begin{equation*}
\kappa_P(\alpha_1)+\kappa_P(\alpha_2)=\kappa_P(\beta_1)+\kappa_P(\beta_2).
\end{equation*}
\end{theorem}
We observe that the set $\overline{\mathcal A^d}$ is closed under cartesian products and thus we have the following analogue of lemma \ref{localcharmult} with the same proof.
\begin{lemma}\label{localcharmultgeneral}
Let $P,Q\in\overline{\mathcal L}$ be compatible with $\{\alpha,\beta\}\subseteq\overline{\mathcal A^d}$. Then
\begin{equation*}
\kappa_{P\times Q}(\alpha\times \beta)=-\kappa_P(\alpha)\kappa_Q(\beta).
\end{equation*}
\end{lemma}
\subsection{The invariants ideal}\label{II}
Once again, we use the notations $\overline{\mathcal L},\overline{\mathcal X}$ to denote the unions $\bigcup_{n=1}^{\infty}\mathcal L_n,$ $\bigcup_{n=1}^{\infty}\mathcal X_n$ etc. and set $\overline{\mathcal L}^*=\overline{\mathcal L}\setminus\{\emptyset\}, \overline{\mathcal X}^*=\overline{\mathcal X}\setminus\{[[\emptyset]]\}$ where $[[-]]:\overline{\mathcal L}\rightarrow\overline{\mathcal X}$ is the map taking a $pp$-set to its colour. Now, $\overline{\mathcal X}^*$ is a multiplicative monoid and we consider the monoid ring $\mathbb Z[\overline{\mathcal X}^*]$.
In the case when $\mathrm{T\neq T^{\aleph_0}}$, there are $P,Q\in\mathcal L_n$ such that $1<\mathrm{Inv}(M;P,Q)<\infty$ for each $n\geq 1$. We can assume without loss that $0\in Q\subseteq P$. Now we define an ideal of the monoid ring, called \textbf{the invariants ideal}, which encodes these invariants. The following proposition is the motivation.
\begin{pro}\label{partitionfurther}
Let $\mathbf{P}\in\mathcal Y_n$ and $X\in\mathcal O(\mathbf{P})$. For any $\alpha,\beta\in\mathcal A^d_n$ with $\bigcup\alpha=\bigcup\beta=X$, we have
\begin{equation*}
[\mathbf{P}(\alpha):\mathbf{P}(\beta)]|\alpha\cap\mathbf{P}| =[\mathbf{P}(\beta):\mathbf{P}(\alpha)]|\beta\cap\mathbf{P}|
\end{equation*}
\end{pro}
\begin{proof}
Partition those cosets of both $\mathbf{P}(\alpha)$ and of $\mathbf{P}(\beta)$ which are contained in $X$ into cosets of $\mathbf{P}(\alpha)\cap\mathbf{P}(\beta)$ to get the required equality.
\end{proof}
\begin{definition}
Let $\delta_{\mathfrak A}:\overline{\mathcal X}^*\rightarrow\mathbb Z$ denote the characteristic function of the colour $\mathfrak A$ for each $\mathfrak A\in\overline{\mathcal X}^*$. We define \textbf{the invariants ideal $\mathcal J$} of the monoid ring $\mathbb Z[\overline{\mathcal X}^*]$ to be the ideal generated by the set
\begin{equation*}
\{\delta_{[[P]]}=[P:Q]\delta_{[[Q]]}: P,Q\in\overline{\mathcal L},\ P\supseteq Q,\ \mathrm{Inv}(M;P,Q)<\infty\}.
\end{equation*}
\end{definition}
The main aim of this section is to prove the following theorem.
\begin{theorem}\label{FINALgeneral}
For every right $\mathcal R$-module $M$, we have
\begin{center}
$K_0(M)\cong\mathbb Z[\overline{\mathcal X}^*]/\mathcal J$.
\end{center}
\end{theorem}
We have proved this theorem when $\mathrm{T=T^{\aleph_0}}$ since the invariants ideal is trivial in that case.
Let $\overline{\mathcal Y}=\bigcup_{n=1}^\infty\mathcal Y_n$. Given $\mathfrak A\in\overline{\mathcal X}^*$, we define $\mathcal Y(\mathfrak A):=\{\mathbf{P}\in\overline{\mathcal Y}:\mathbf{P}\cap\mathfrak A\neq\emptyset\}$. In order to define the global characteristics in this case, we need to find the set over which they vary. Let $\mathfrak A,\mathfrak B\in\overline{\mathcal X}^*$. We say that $\mathfrak A\approx\mathfrak B$ if and only if $\mathcal Y(\mathfrak A)\cap\mathcal Y(\mathfrak B)\neq\emptyset$. This relation is reflexive and symmetric. We use $\approx$ again to denote its transitive closure. The $\approx$-equivalence class of $\mathfrak A$ will be denoted by $\widetilde{\mathfrak A}$.
\begin{definition}
Let $\mathfrak A\in\overline{\mathcal X}^*$. Define the \textbf{colour class group} $\mathcal R(\widetilde{\mathfrak A})$ as the quotient of the free abelian group $\mathbb Z\langle \delta_{\mathfrak A}:\mathfrak A\in\widetilde{\mathfrak A}\rangle$ by the subgroup $\mathcal J(\widetilde{\mathfrak A})$ generated by the relations $\{\delta_{[[P]]}=[P:Q]\delta_{[[Q]]}: P,Q\in\bigcup\widetilde{\mathfrak A},\ P\supseteq Q\}$.
\end{definition}
It can be observed that the underlying abelian group of the monoid ring $\mathbb Z[\overline{\mathcal X}^*]$ is formed by taking the quotient of the direct sum of the free abelian groups $\mathbb Z\langle \delta_{\mathfrak A}:\mathfrak A\in\widetilde{\mathfrak A}\rangle$, one for each equivalence class of colours, by the multiplicative relations of the monoid $\overline{\mathcal X}^*$. Furthermore, the set $\bigcup\{\mathcal J(\widetilde{\mathfrak A}):\widetilde{\mathfrak A}\in\overline{\mathcal X}^*\}$ generates the ideal $\mathcal J$ in this ring.
The discussion in the previous paragraph suggests to us to isolate the information in the evaluation map into different global characteristics, one for each colour class. These maps take values in the corresponding colour class group. We define the \textbf{global characteristic} $\Lambda_{\widetilde{\mathfrak A}}$ corresponding to $\widetilde{\mathfrak A}$ as the function $\overline{\hat{\mathcal P}^d}\rightarrow\mathcal R(\widetilde{\mathfrak A})$ given by $\alpha\mapsto-\sum_{\mathfrak A\in\widetilde{\mathfrak A}}\left(\sum_{P\in\mathfrak A}\kappa_P(\alpha)\right)\delta_{\mathfrak A}$.
The following result is an easy corollary of proposition \ref{partitionfurther}. It states that the global characteristics depend only on the $pp$-convex sets and not on their representations as antichains.
\begin{cor}\label{glogeneralconvex}
Let $X\in\overline{\mathcal O}$ and $\alpha,\beta\in\overline{\hat{\mathcal P}^d}$ be such that $\bigcup\alpha=\bigcup\beta=X$. Then $\Lambda_{\widetilde{\mathfrak A}}(\alpha)=\Lambda_{\widetilde{\mathfrak A}}(\beta)$ for each $\mathfrak A\in\overline{\mathcal X}^*$.
\end{cor}
This finishes the technical setup for the general case when the theory $T$ of the module $M$ does not necessarily satisfy $T=T^{\aleph_0}$. The antichains in discrete form behave as if the theory satisfies $T=T^{\aleph_0}$, the bands allow us to go down (via intersections) so that any finite family can be converted to a compatible family and the notion of compatibility allows us to do appropriate local analysis. The local data can be pasted together using the information coded in the colour class groups.
Now we give some important definitions and state results from the special case $\mathrm{T=T^{\aleph_0}}$ in a form compatible with the general case. The proofs of these results are omitted since they are similar to their special counterparts; the basic ingredients are provided by lemma \ref{NLU}, theorem \ref{t1general}, lemma \ref{localcharmultgeneral} and corollary \ref{glogeneralconvex}. The necessary change is to deal only with antichains which are in discrete form.
Since cells are the difference sets of two $pp$-convex sets, we can obtain a compatible family $\{\alpha,\beta\}$ for any $C\in\overline{C}$ such that $C=\bigcup\alpha\setminus\bigcup\beta$.
\begin{definition}
Let $C\in\overline{\mathcal C}$ and $\mathfrak A$ be a colour. We define the global characteristic $\Lambda_{\widetilde{\mathfrak A}}(C):=\Lambda_{\widetilde{\mathfrak A}}(\alpha)-\Lambda_{\widetilde{\mathfrak A}}(\beta)\in\mathcal R(\widetilde{\mathfrak A})$ for any compatible family $\{\alpha,\beta\}$ representing $C$.
\end{definition}
The following theorem is the analogue of theorem \ref{t2} and uses the inductive version of \ref{t1general} in its proof.
\begin{theorem}\label{t2general}
If $\{B_i:1\leq i\leq l\}, \{B_j':1\leq j\leq m\}$ are two finite families of pairwise disjoint blocks such that $\bigsqcup_{i=1}^l B_i=\bigsqcup_{j=1}^m B_j'$, then $\Sigma_{i=1}^l\Lambda_{\widetilde{\mathfrak A}} (B_i)=\Sigma_{j=1}^m\Lambda_{\widetilde{\mathfrak A}}(B_j')$ for every $\mathfrak A\in\overline{\mathcal X}^*$.
\end{theorem}
This theorem allows us to extend the definition of global characteristics to all sets in $\overline{\mathrm{Def}}(M)$. Moreover the following theorem, the proof of which is an easy adaptation of that of theorem \ref{t3}, states that each of them is preserved under definable bijections.
\begin{theorem}\label{t3general}
Suppose $D\in \mathrm{Def}(M^n)$ and $f:D\rightarrow M^n$ is a definable injection. Then $\Lambda_{\widetilde{\mathfrak A}}(D)=\Lambda_{\widetilde{\mathfrak A}}(f(D))$ for each colour class $\widetilde{\mathfrak A}$.
\end{theorem}
Let $ev:\overline{\mathrm{Def}}(M)\rightarrow\mathbb Z[\overline{\mathcal X}^*]/\mathcal J$ be the map defined by $D\mapsto \sum\{\Lambda_{\widetilde{\mathfrak A}}(D):\widetilde{\mathfrak A}\in\overline{\mathcal X}^*/\approx\}$. This map is well defined since the sum is finite for every $D$ for reasons similar to those for the special case. Furthermore $ev_{D_1}=ev_{D_2}$ whenever $D_1$ and $D_2$ are definably isomorphic since $\Lambda_{\widetilde{\mathfrak A}}(D_1)=\Lambda_{\widetilde{\mathfrak A}}(D_2)$ for each colour class $\widetilde{\mathfrak A}$. In fact $ev$ is a semiring homomorphism. The proof of the following theorem is analogous to that of theorem \ref{t5}.
\begin{theorem}\label{t5general}
The map $ev:\widetilde{\mathrm{Def}}(M)\rightarrow\mathbb Z[\overline{\mathcal X}^*]/\mathcal J$ defined by $[D]\mapsto ev_{[D]}$ is a semiring homomorphism.
\end{theorem}
The final step in the proof of \ref{FINALgeneral} is the following analogue of lemma \ref{INJEV}.
\begin{lemma}\label{INJEVgeneral}
The map $\widetilde{ev}:\widetilde{\widetilde{\mathrm{Def}}(M)}\rightarrow\mathbb Z[\overline{\mathcal X}^*]/\mathcal J$ is injective.
\end{lemma}
\begin{proof} The proof of this lemma needs some modification of the first paragraph of the proof of lemma \ref{INJEV} in order to incorporate the invariants ideal. Let $\mathcal U:= \{\alpha\in\overline{\mathcal A^d}: A_1\cap A_2=\emptyset$ for all distinct $A_1,A_2\in\alpha\}$.
If $ev_{[\alpha]}=ev_{[\beta]}$ for some $\alpha,\beta\in\mathcal U$, then we can obtain two antichains $\alpha'\in[\alpha]\cap\mathcal U,\beta'\in[\beta]\cap\mathcal U$ such that $\bigcup\alpha=\bigcup\alpha',\bigcup\beta=\bigcup\beta'$ and $\{\alpha',\beta'\}$ is compatible. Hence we have $\Lambda_{\widetilde{\mathfrak A}}(\alpha)=\Lambda_{\widetilde{\mathfrak A}}(\alpha')$, $\Lambda_{\widetilde{\mathfrak A}}(\beta)=\Lambda_{\widetilde{\mathfrak A}}(\beta')$ for each colour class $\widetilde{\mathfrak A}$. Observe that the equalities, if considered in the codomain ring, are modulo the invariants ideal. Now $\Lambda_{\widetilde{\mathfrak A}}(\alpha')=|\alpha'\cap(\bigcup\widetilde{\mathfrak A})|\delta_{[[\mathbf{P}(\alpha')]]}$, where $\mathbf{P}$ is the only band (if exists) such that $\mathbf{P}\cap\alpha'\cap(\bigcup\widetilde{\mathfrak A})\neq\emptyset$. Since $\mathbf{P}(\alpha')=\mathbf{P}(\beta')$ for each such colour class by the definition of compatibility, we get $|\alpha\cap(\bigcup\widetilde{\mathfrak A})|=|\beta\cap(\bigcup\widetilde{\mathfrak A})|$ for each colour class $\widetilde{\mathfrak A}$.
A definable isomorphism can be easily constructed between the $pp$-convex sets represented by $\alpha'$ and $\beta'$, which are the sets represented by $\alpha$ and $\beta$ respectively. The rest of the proof is similar to the proof of \ref{INJEV}.
\end{proof}
\begin{proof} (Theorem \ref{FINALgeneral}) We have shown that the map $\widetilde{ev}$ is injective in the previous lemma. Then we observe that the sets of the form $\bigcup\alpha$ for some $\alpha\in\mathcal U$ are capable of producing every element of the quotient ring $\mathbb Z[\overline{\mathcal X}^*]/\mathcal J$ of the form $\sum n_{\mathfrak A}\delta_{\mathfrak A}+\mathcal J$, where the nonzero coefficients are positive. This completes the proof by an argument similar to the proof of theorem \ref{FINAL}.
\end{proof}
Since the Grothendieck ring is a quotient ring, we do not necessarily know if it is nontrivial. But the following corollary of theorem \ref{FINALgeneral} shows this result, proving Prest's conjecture in full generality.
\begin{cor}\label{MAINRESULTgeneral}
If $M$ is a nonzero right $\mathcal R$-module, then there is a split embedding $\mathbb Z\rightarrowtail K_0(M)$.
\end{cor}
\begin{proof}
Consider the colour class $\widetilde{\mathfrak U}$, where $\mathfrak U$ is the identity element of the monoid $\overline{\mathcal X}^*$. A $pp$-set $P$ is an element of $\bigcup\widetilde{\mathfrak U}$ if and only if $P$ is finite. Finite sets enjoy the special property that two finite sets are isomorphic to each other if and only their cardinalities are equal. Furthermore, every such isomorphism is definable. In particular, $\mathcal R(\widetilde{\mathfrak U})\cong\mathbb Z$ if $M$ is a nonzero module. Next we observe that the set $\bigcup\widetilde{\mathfrak U}$ is closed under multiplication and hence the colour class group $\mathcal R(\widetilde{\mathfrak U})$ can be given the structure of a quotient of the monoid ring $\mathbb Z[\bigcup\widetilde{\mathfrak U}]$ with certain relations, where the multiplicative relations of the monoid ring are finitary and hence already present in the relations for $\mathcal R(\widetilde{\mathfrak U})$. We have thus described the ring structure of $\mathcal R(\widetilde{\mathfrak U})$ and this ring is naturally a subring of $K_0(M)$.
To complete the proof, we show that the map $\pi_0:K_0(M)\rightarrow\mathcal R(\widetilde{\mathfrak U})$ given by $\sum_{\widetilde{\mathfrak A}\in(\overline{\mathcal X}^*/\approx)} n_{\widetilde{\mathfrak A}}\delta_{\widetilde{\mathfrak A}}\mapsto n_{\widetilde{\mathfrak U}}\delta_{\widetilde{\mathfrak U}}$ is a surjective ring homomorphism.
The map $\pi_0$ is clearly an additive group homomorphism. Note that the multiplicative monoid $\bigcup\widetilde{\mathfrak U}$ is a sub-monoid of $\overline{\mathcal X}^*$. Also note that $\mathcal J(\widetilde{\mathfrak A})\cap\mathcal J(\widetilde{\mathfrak B})=\emptyset$ if $\widetilde{\mathfrak A}\neq\widetilde{\mathfrak B}$. Furthermore, $\mathfrak A\star\mathfrak B\in\bigcup\widetilde{\mathfrak U}$ if and only if $\mathfrak A,\mathfrak B\in\bigcup\widetilde{\mathfrak U}$. Thus the coefficient of $\delta_{\widetilde{\mathfrak U}}$ in the product of two elements of $K_0(M)$ is determined by the coefficient of $\delta_{\widetilde{\mathfrak U}}$ of the individual elements. Hence $\pi_0$ is also multiplicative. The surjectivity is clear. This completes the proof.
\end{proof}
Now we can give a proof that the Grothendieck ring of a module is an invariant of its theory.
\begin{proof} (Proposition \ref{eleequivmod})
Elementarily equivalent modules have isomorphic lattices of $pp$-sets and they also satisfy the same invariant conditions (see \cite[Corollary\,2.18]{PreBk}). Hence theorem \ref{FINALgeneral} yields the result.
\end{proof}
\section{Applications}\label{appl}
\subsection{Pure embeddings and Grothendieck rings}\label{pure}
We will investigate some categorical properties of Grothendieck rings of modules in this section. The main aim is to prove the following theorem.
\begin{theorem}\label{puresurj}
Let $i:N\rightarrow M$ be a pure embedding of right $\mathcal R$-modules such that the theory of $M$ satisfies $Th(M)=Th(M)^{\aleph_0}$. Then $i$ induces a surjective ring homomorphism $I:K_0(M)\twoheadrightarrow K_0(N)$.
\end{theorem}
This theorem will be proved using a series of results of functorial nature. We begin with the definition of a pure embedding.
\begin{definition}
Let $M$ be a right $\mathcal R$-module. A submodule $N\leq M$ is called a \textbf{pure submodule} if, for each $n$, $A\cap N^n\in\mathcal L_n^\circ(N)$ for every $A\in\mathcal L_n^\circ(M)$.\\
A monomorphism $i:N\rightarrow M$ is said to be a \textbf{pure monomorphism} if $iN$ is a pure submodule of $M$.
\end{definition}
The following lemma states that a pure embedding induces a map of lattices of $pp$-formulas.
\begin{lemma}\label{purelat}(see \cite[Lemma\,3.2.2]{PrePSL})
If $i:N\rightarrow M$ is a pure embedding then, for each $n$, the natural map $\overline{i}:\mathcal L_n^\circ(M)\rightarrow\mathcal L_n^\circ(N)$ given by $\overline{i}(A)=A\cap N^n$ is a surjection of lattices.
\end{lemma}
Now we state the following result about integral monoid rings.
\begin{pro}\label{monringfunc}(see \cite[II,\,Proposition\,3.1]{Lang})
Let $\Phi:A\rightarrow B$ be a homomorphism of monoids. Then there exists a unique homomorphism $h:\mathbb Z[A]\rightarrow\mathbb Z[B]$ such that $h(x)=\Phi(x)$ for all $x\in A$ and $h(1)=1$. Furthermore, $h$ is surjective if $\Phi$ is so.
\end{pro}
\begin{cor}
A pure embedding $i:N\rightarrow M$ induces a surjective homomorphism $\mathfrak i:\mathbb Z[\overline{\mathcal X}^*(M)]\twoheadrightarrow\mathbb Z[\overline{\mathcal X}^*(N)]$ of rings.
\end{cor}
\begin{proof}
Observe that every colour $\mathfrak A\in\overline{\mathcal X}^*$ has a representative in $\overline{\mathcal L}^\circ:=\bigcup_{n=1}^\infty\mathcal L_n^\circ$. Thus we get an induced surjective homomorphism $\overline{\mathcal X}^*(M)\twoheadrightarrow\overline{\mathcal X}^*(N)$ of the colour monoids using lemma \ref{purelat}. Then proposition \ref{monringfunc} yields the required surjective map of the integral monoid rings.
\end{proof}
\begin{proof} (Theorem \ref{puresurj})
Observe that since $Th(M)=Th(M)^{\aleph_0}$ holds, theorem \ref{FINAL} gives $K_0(M)\cong\mathbb Z[\overline{\mathcal X}^*(M)]$. By theorem \ref{FINALgeneral}, we have $K_0(N)\cong\mathbb Z[\overline{\mathcal X}^*(N)]/\mathcal J(N)$. Let $\pi:\mathbb Z[\overline{\mathcal X}^*(N)]\twoheadrightarrow K_0(N)$ denote the natural quotient map. Take $I=\pi\circ\mathfrak i$, where $\mathfrak i$ is the map from the previous corollary, to finish the proof.
\end{proof}
We will see an example at the end of the next section to see that theorem \ref{puresurj} fails if $Th(M)\neq Th(M)^{\aleph_0}$.
Recall that the notation $M^{(\aleph_0)}$ denotes the direct sum of countably many copies of a module $M$. It follows immediately from \cite[Lemma\,2.23(c)]{PreBk}) that the lattices $\mathcal L_1(M)$ and $\mathcal L_1(M^{(\aleph_0)})$ are isomorphic and $T:=Th(M^{(\aleph_0)})$ satisfies $T=T^{\aleph_0}$. We summarize these observations in the following corollary of theorem \ref{puresurj}.
\begin{cor}
Let $i_n:M\rightarrow M^{(\aleph_0)}$ denote the natural embedding of $M$ onto the $n^{th}$ component of $M^{(\aleph_0)}$. Then $i_n$ induces the natural quotient map $K_0(M^{(\aleph_0)})=\mathbb Z[\overline{\mathcal X}^*(M)]\twoheadrightarrow\mathbb Z[\overline{\mathcal X}^*(M)]/\mathcal J(M)=K_0(M)$.
\end{cor}
For a ring $\mathcal R$, let $\rm Mod\mbox{-}\mathcal R$ denote the category of right $\mathcal R$-modules. The theory $Th({\rm Mod\mbox{-}\mathcal R})$ is not a complete theory. But we may take a canonical complete theory extending it as follows. Recall that Grothendieck rings of elementarily equivalent modules are isomorphic by proposition \ref{eleequivmod}. Equivalently, $K_0(M)$ is determined by $Th(M)$ which, in turn, is determined by its invariants conditions (theorem \ref{FINALgeneral}).
\begin{definition}
Let $P$ be a direct sum of one model of each complete theory of right $\mathcal R$-modules. Then $T^*=Th(P)$ is referred to as \textbf{the largest complete theory of right $\mathcal R$-modules}.
\end{definition}
Thus every right $\mathcal R$-module is elementarily equivalent to a direct summand of some model of $Th(P)$. Now we note the following result without proof and define the Grothendieck ring of the module category.
\begin{deflem}(see \cite[6.1.1,\,6.1.2]{Perera})\label{GrRngModCat}
Let $T^*$ denote the largest complete theory of right $\mathcal R$-modules. Then $T^*=(T^*)^{\aleph_0}$. Furthermore if $P_1$ and $P_2$ are both direct sums of one model of each complete theory of right $\mathcal R$-modules, then $K_0(P_1)\cong K_0(P_2)$. We define the \textbf{Grothendieck ring of the module category}, denoted $K_0({\rm Mod\mbox{-}\mathcal R})$, to be the Grothendieck ring of the largest complete theory of right $\mathcal R$-modules.
\end{deflem}
As a consequence of theorem \ref{puresurj}, we state a result connecting Grothendieck rings of individual modules with that of the module category.
\begin{cor}
Let $M$ be a right $\mathcal R$-module. Then $K_0(M)$ is a quotient of $K_0({\rm Mod\mbox{-}\mathcal R})$.
\end{cor}
\begin{proof} Let $T^*$ be the largest complete theory of right $\mathcal R$-modules. Then lemma \ref{GrRngModCat} gives that, for any $P\models T^*$, $Th(P)=T^*$ satisfies $T^*=(T^*)^{\aleph_0}$ and we also have $K_0(P)\cong K_0({\rm Mod\mbox{-}\mathcal R})$.
By the definition of $T^*$, there is a module $M'$ elementarily equivalent to $M$ such that $M'$ is a direct summand of $P$. Since the embedding $M'\rightarrowtail P$ is pure, we get a surjective homomorphism $K_0(P)\twoheadrightarrow K_0(M')$. Thus the required quotient map is the composite $K_0(\mathrm{Mod\mbox{-}\mathcal R})\cong K_0(P)\twoheadrightarrow K_0(M')\cong K_0(M)$, where the last isomorphism is obtained from proposition \ref{eleequivmod}.
\end{proof}
\subsection{Torsion in Grothendieck rings}\label{tors}
As an application of the structure theorem for Grothendieck rings, theorem \ref{FINALgeneral}, we provide an example of a module whose Grothendieck ring contains a nonzero torsion element (i.e. a nonzero element $a$ such that $na=0$ for some $n\geq 1$). We also calculate the Grothendieck ring $K_0(\mathbb Z_\mathbb Z)$.
\begin{definition}
The \textbf{ring of $p$-adic integers}, denoted $\mathbb Z_p$, is the inverse limit of the system $\hdots\twoheadrightarrow\mathbb Z/p^n\mathbb Z\twoheadrightarrow\hdots\twoheadrightarrow\mathbb Z/p^2\mathbb Z\twoheadrightarrow\mathbb Z/p\mathbb Z\twoheadrightarrow 0$.\\
\end{definition}
The ring $\mathbb Z_p$ is a commutative local PID with the ideal structure given by
\begin{equation*}
\mathbb Z_p\supsetneq p\mathbb Z_p\supsetneq\hdots\supsetneq p^n\mathbb Z_p\supsetneq\hdots\supsetneq 0.
\end{equation*}
In particular, $\mathbb Z_p$ is a commutative noetherian ring and hence satisfies the hypothesis of the following proposition.
\begin{pro}(see \cite[p.19,\,Ex.\,2(ii)]{PreBk})
If $\mathcal R$ is a commutative noetherian ring then the $pp$-definable subgroups of the module $\mathcal R_\mathcal R$ are precisely the finitely generated ideals of $\mathcal R$.
\end{pro}
It can be observed that the maps $t_n:\mathbb Z_p\rightarrow p^n\mathbb Z_p$ which are `multiplication by $p^n$' are $pp$-definable isomorphisms for each $n\geq 1$. Thus a simple computation shows that the monoid of colours, $\overline{\mathcal X}^*(\mathbb Z_p)$, is isomorphic to the monoid $\mathbb N$.
If $X$ denotes the class of $\mathbb Z_p$ in $K_0(\mathbb Z_p)$, then the invariants ideal $\mathcal J(\mathbb Z_p)$ is generated by the relations $\{X=p^nX:n\geq 1\}$. The relation $(p^n-1)X=0$ is an integral multiple of the relation $(p-1)X=0$ for each $n\geq 1$. Thus $\mathcal J(\mathbb Z_p)$ is principal and generated by the single relation $(p-1)X=0$. We summarize this discussion as the following corollary to theorem \ref{FINALgeneral}.
\begin{cor}\label{p-adic}
Let $\mathbb Z_p$ denote the ring $p$-adic integers. Then \center{$K_0(\mathbb Z_p)\cong\mathbb Z[X]/\langle(p-1)X\rangle$.}
\end{cor}
Consider the split (hence pure) embedding $i:\mathbb Z_p^{(2)}\rightarrowtail\mathbb Z_p^{(3)}$ of $\mathbb Z_p$-modules given by $(a,b)\mapsto(a,b,0)$, where $M^{(k)}$ denotes the direct sum of $k$ copies of $M$. We want to show that this embedding witnesses the failure of theorem \ref{puresurj} since the theory $T:=Th(\mathbb Z_p^{(3)})$ of the target module doesn't satisfy the condition $T=T^{\aleph_0}$. The following proposition is helpful for the calculation of Grothendieck rings.
\begin{pro}(see \cite[Lemma\,2.23]{PreBk})
If $\phi(x)$ and $\psi(x)$ denote $pp$-formulas, then
\begin{enumerate}
\item $\phi(M\oplus N)=\phi(M)\oplus\phi(N)$,
\item $\mathrm{Inv}(M\oplus N;\phi,\psi)=\mathrm{Inv}(M;\phi,\psi)\mathrm{Inv}(N;\phi,\psi)$.
\end{enumerate}
\end{pro}
It is clear that the induced map $\mathfrak{i}:\mathbb Z[\overline{\mathcal X}^*(\mathbb Z_p^{(3)})]\rightarrow\mathbb Z[\overline{\mathcal X}^*(\mathbb Z_p^{(2)})]$ is the identity map on $\mathbb Z[X]$ since $\mathbb Z[\overline{\mathcal X}^*(\mathbb Z_p^{(k)})]\cong K_0(\mathbb Z_p^{(\aleph_0)})\cong\mathbb Z[X]$ for any $k\geq 1$. Further the previous proposition shows that $\mathcal J(\mathbb Z_p^{(k)})=\langle(p^k-1)X\rangle$ for any $k\geq 1$. Since $\mathcal J(\mathbb Z_p^{(3)})\nsubseteq\mathcal J(\mathbb Z_p^{(2)})$, there is no surjective map $K_0(\mathbb Z_p^{(3)})\twoheadrightarrow K_0(\mathbb Z_p^{(2)})$.
\textbf{The abelian group of integers}: Since the ring $\mathbb Z$ is a commutative PID, the $pp$-definable subgroups of the module $\mathbb Z_\mathbb Z$ are precisely the ideals $n\mathbb Z$ for $n\geq 0$. Thus the monoid $\overline{\mathcal X}^*(\mathbb Z)$ is isomorphic to $\mathbb N$. Furthermore if $X$ denotes the class of $\mathbb Z$ in $K_0(\mathbb Z)$, the invariants ideal is generated by the relations $X=nX$ for each $n\geq 1$. This forces $\mathcal J(\mathbb Z)=\langle X\rangle$ and thus $K_0(\mathbb Z_\mathbb Z)\cong\mathbb Z$.
\subsection{Representing definable sets uniquely}\label{CDT}
We fix some $\mathcal R$-module $M$ whose theory $T$ satisfies the condition $T=T^{\aleph_0}$ and some $n\geq 1$. As usual we drop all the subscripts $n$ and write $\mathcal L\setminus\{\emptyset\},\mathcal A\setminus\{\emptyset\},\hdots$ as $\mathcal L^*,\mathcal A^*,\hdots$ respectively.
The $pp$-elimination theorem for the model theory of modules (theorem \ref{PPET}) states that every definable set can be written as a finite disjoint union of blocks. But this representation is far from being unique in any sense. On the other hand we have unique representations for $pp$-convex sets (proposition \ref{UNIQUE1}) and cells (lemma \ref{UNIQUE2}). We exploit these ideas to achieve a unique representation for every definable set - an expression as a disjoint union of cells. This result will be called the `cell decomposition theorem'.
We begin by defining some terms useful to describe the cell decomposition theorem.
\begin{definition}
Let $\mathcal F=\{C_j\}_{j=1}^l\subseteq\mathcal C$ be a family of pairwise disjoint cells. If there is a permutation $\sigma$ of $[l]$ such that $P(C_{\sigma(j+1)})\prec N(C_{\sigma(j)})$ for $1\leq j\leq l-1$, then we say that the family $\mathcal F$ is a \textbf{tower of cells}. We call the number $l$ the \textbf{height} of the tower. We denote the set of all finite towers of cells by $\mathcal T$. We define a function $\zeta:\mathcal T\rightarrow \mathbb N$ which assigns its height to a tower.
\end{definition}
\begin{definition}
Let $\alpha_i\in\mathcal A^*$ for $1\leq i\leq k$. If $\alpha_{i+1}\prec\alpha_{i}$ for each $1\leq i\leq k-1$, we say that $\overline\alpha=\{\alpha_i\}_{i=1}^k$ is a $\prec$-\textbf{chain}. We denote the set of all finite $\prec$-chains in $\mathcal A^*$ by $\mathcal W$. We define a function $\omega:\mathcal W\rightarrow \mathbb N$, which assigns \textbf{height} to each $\prec$-chain, by $\omega(\overline\alpha)=\lceil(\frac{|\overline\alpha|}{2})\rceil$ where $\lceil q\rceil$ is the smallest integer larger than or equal to $q$.
\end{definition}
The following proposition states that towers and chains are two different ways of expressing the same kind of object.
\begin{pro}
There is a bijection $\Phi:\mathcal T\rightarrow\mathcal W$ preserving height i.e., $\omega(\Phi(\mathcal F))=\zeta(\mathcal F)$ for every $\mathcal F\in\mathcal T$.
\end{pro}
\begin{proof} Let $\mathcal F=\{C_j\}_{j=1}^l$ be a tower of cells with height $l$. Without loss, we may assume that $P(C_j)\prec N(C_{j+1})$, i.e., the associated permutation is the identity. We first define a non-negative integer $k$ as follows.
\begin{math}
k=\begin{cases}
0, & \mbox{if } l=0, \\
2l+1, &\mbox{if } l>0\mbox{ and } N(C_l)=\emptyset,\\
2l+2, &\mbox{if } l>0\mbox{ and } N(C_l)\neq\emptyset.
\end{cases}
\end{math}
For each $1\leq i\leq k$, we define an antichain $\alpha_i\in\mathcal A^*$ as follows.
\begin{math}
\alpha_i=\begin{cases}
P(C_j), & \mbox{if } i=2j+1,\\
N(C_j), &\mbox{if } i=2j+2.
\end{cases}
\end{math}
Then $\overline\alpha=\{\alpha_i\}_{i=1}^k$ is clearly a $\prec$-chain and the map $\Phi(\mathcal F):=\overline{\alpha}$ can be easily checked to be injective.
To prove surjectivity, let $\overline\beta\in\mathcal W$. We modify $\overline\beta$ to obtain a $\prec$-chain $\overline\beta'=\{\beta_i'\}_{i=1}^{2\omega(\overline\beta)}$ in $\mathcal A$ as follows.
\begin{math}
\beta_i'=\begin{cases}
\beta_i, & \mbox{if } 1\leq i\leq |\overline\beta|,\\
\emptyset, &\mbox{if } |\overline\beta|\neq 2\omega(\overline\beta)\mbox{ and } i=2\omega(\overline\beta).
\end{cases}
\end{math}
Then $|\overline\beta'|$ is an even integer. We define $C'_j=\bigcup\beta_{2j+1}\setminus\bigcup\beta_{2j+2}$ for $1\leq j\leq |\overline\beta'|/2$. The family $\mathcal F':=\{C'_j\}_{j=1}^{|\overline\beta'|/2}$ clearly satisfies $\Phi(\mathcal F')=\overline{\beta}$.
The height preservation property is easy to check from the explicit constructions above.
\end{proof}
\begin{pro}
Let $\{A_i\}_{i=1}^m\in\mathcal P$ and $B\in\mathcal B$ be such that $B\subseteq\bigcup_{i=1}^m A_i$. Then $\overline B\subseteq\bigcup_{i=1}^m A_i$.
\end{pro}
\begin{proof}
We have $\overline B=B\cup\bigcup N(B)$. Hence $\overline B\subseteq\bigcup_{i=1}^m A_i\cup\bigcup N(B)$. By \ref{NLU}, $\overline B\subseteq A_i$ for some $i$, or $\overline B\subseteq D$ for some $D\in N(B)$. The latter case is not possible since $N(B)\prec P(B)=\{\overline B\}$. Hence the result.
\end{proof}
\begin{lemma}\label{CHAINANTITOWER}
Let $D\in \mathrm{Def}(M^n)$. Then there is a unique $pp$-convex set $\overline D$ which satisfies $D\subseteq\bigcup\alpha\ \Rightarrow\ \overline D\subseteq\bigcup\alpha$ for every $\alpha\in\mathcal A$.
\end{lemma}
\begin{proof} Let $D=\bigsqcup_{i=1}^m B_i=\bigsqcup_{j=1}^l B'_j$ be any two representations of $D$ as disjoint unions of blocks.
\textbf{Claim}: $\bigcup_{i=1}^m \overline{B_i}=\bigcup_{j=1}^l \overline{B'_j}$
Proof of the claim: We have $B_i\subseteq\bigsqcup_{i=1}^m B_i=\bigsqcup_{j=1}^l B'_j\subseteq\bigcup_{j=1}^l \overline{B'_j}$ for each $i$. Hence $\overline{B_i}\subseteq\bigcup_{j=1}^l \overline{B'_j}$ by the previous proposition. Therefore $\bigcup_{i=1}^m \overline{B_i}\subseteq\bigcup_{j=1}^l \overline{B'_j}$. The reverse containment is by symmetry and hence the claim.
Now we define $\overline D=\bigcup_{i=1}^m \overline{B_i}$. By the claim, this $pp$-convex set is uniquely defined.
Let $\alpha\in\mathcal A$ be such that $D\subseteq\bigcup\alpha$. But $D=\bigsqcup_{i=1}^m B_i$. Hence $B_i\subseteq\bigcup\alpha$ for each $i$. By arguments similar to the proof of the claim, we get $\bigcup_{i=1}^m \overline{B_i}\subseteq\bigcup\alpha$ i.e., $\overline D\subseteq\bigcup\alpha$.
\end{proof}
The assignment $D\mapsto\overline{D}$, where $\overline{D}$ is the $pp$-convex set obtained from the lemma, defines a closure operator $\mathrm{Def}(M^n)\rightarrow\mathcal A_n$. This closure operation is extremely useful in proving the cell decomposition theorem.
\begin{theorem}\label{CDT1}
\textbf{Cell Decomposition Theorem}: There is a bijection between the set $\mathrm{Def}(M^n)$ of all definable subsets of $M^n$ and the set $\mathcal T$ of towers of cells.
\end{theorem}
\begin{proof}
Let $D\in \mathrm{Def}(M^n)$. We construct a tower $\mathcal F$ of cells by defining a nested sequence $\{D_j\}_{j\geq 0}$ of definable subsets of $D$ as follows.
We set $D_0:=D$ and, for each $j>0$, we set $D_j:=D_{j-1}\setminus C_j$, where $C_j:=\overline{D_{j-1}}\setminus (\overline{\overline{D_{j-1}}\setminus D_{j-1}})$ is a cell. We stop this process when we obtain $D_j=\emptyset$ for the first time. This process must terminate because the elements of the antichains involved in this process belong to some finite nest containing a fixed decomposition of $D$ into blocks.
In the converse direction, we assign $\bigcup\mathcal F\in \mathrm{Def}(M^n)$ to $\mathcal F\in\mathcal T$.
It is easy to verify that the two assignments defined above are actually inverses of each other.
\end{proof}
The following corollary combines theorem \ref{CDT1} with lemma \ref{CHAINANTITOWER} and gives a combinatorial representation theorem for $\mathrm{Def}(M^n)$, which roughly states that every definable subset of $M^n$ can be represented uniquely as a finite $\prec$-chain in the free distributive lattice $\mathcal A$ over the meet semilattice $\mathcal L$.
\begin{cor}
There is a bijection between the set $\mathcal W_n$ of finite chains in $\mathcal A_n^*$ and $\mathrm{Def}(M^n)$.
\end{cor}
\subsection{Connectedness}\label{C}
We fix a right $\mathcal R$-module $M$ satisfying $Th(M)=Th(M)^{\aleph_0}$ and some $n\geq 1$. We drop all the subscripts $n$ as usual.
Recall that every global characteristic of a definable set is preserved under definable isomorphisms (Theorem \ref{t4}). In this section we describe what we mean by the statement that a definable subset of a (finite power of a) module is connected. The property of being connected is not preserved under definable isomorphisms. We prove a (topological) property of connected sets which states that a definable connected set $A$ contained in another definable set $B$ is in fact contained in a connected component of $B$.
Let $\mathcal{F},\mathcal{F}'\subseteq\mathcal{B}$ be two finite families of disjoint blocks such that $\bigcup\mathcal{F}=\bigcup\mathcal{F}'$. Then we say that $\mathcal{F}'$ is a \textbf{refinement} of $\mathcal{F}$ if for each $F'\in\mathcal{F}'$, there is a unique $F\in\mathcal{F}$ such that $F'\subseteq F$. Recall from \ref{CH1} that if $\bigcup\mathcal{F}\in\mathcal{B}$ and if $\mathcal{D}$ is the corresponding nest, then $\{\mathrm{Core}_\mathcal{D}(D)\}_{D\in\mathcal{D}^+}$ is a refinement of $\mathcal{F}$, where $\mathcal{D}^+$ is the set $\delta_{\mathcal{D}}^{-1}\{1\}$. We use this property of nests to attach a digraph with each of them.
\begin{definition}
Let $\mathcal D$ be a nest corresponding to a fixed finite family of pairwise disjoint blocks. We define a \textbf{digraph structure} $\mathcal{H}(\mathcal{D}^+)$ on the set $\mathcal{D}^+$. The pair $(F_1, F_2)$ of elements of $\mathcal{D}^+$ will be said to constitute an arrow in the digraph if $F_1\subsetneq F_2$ and $F_1\subseteq F\subseteq F_2$ for some $F\in\mathcal{D}^+$ if and only if $F=F_1$ or $F=F_2$.
\end{definition}
If $\bigcup_{F\in\mathcal{D}^+}\mathrm{Core}_\mathcal{D}(F)\in\mathcal{B}$, then $\mathcal{D}^+$ is an upper set and in particular $\mathcal{H}(\mathcal{D}^+)$ is \textbf{weakly connected} i.e., its underlying undirected graph is connected. It seems natural to use this property to define the connectedness of a definable set.
\begin{definition}\label{conn}
Let $D\in\mathrm{Def}(M^n)$ be represented as $D=\bigcup\mathcal F$, where $\mathcal F\subseteq\mathcal B$ be a finite family of pairwise disjoint blocks and let $\mathcal{D}$ denote the nest corresponding to $\mathcal F$. We say that $D$ is \textbf{connected} if and only if the digraph $\mathcal{H}(\mathcal{D}'^+)$ is weakly connected for some nest $\mathcal{D}'$ containing $\mathcal{D}$.
\end{definition}
Note the existential clause in this definition. Let $\mathcal F,\mathcal F'$ be two finite families of pairwise disjoint blocks with $\bigcup\mathcal F=\bigcup\mathcal F'$ and let $\mathcal D,\mathcal D'$ denote the nests corresponding to them. If $\mathcal F'$ refines $\mathcal F$, then the number of weakly connected components of $\mathcal H(\mathcal D'^+)$ is bounded between $0$ and the number of weakly connected components of $\mathcal H(\mathcal D^+)$. This observation allows us to define the following invariant.
\begin{definition}
We define the \textbf{number of connected components} of $D$, denoted $\lambda(D)$, for every nonempty definable set $D$ to be the least number of weakly connected components of $\mathcal H(\mathcal D^+)$, where $\mathcal D$ varies over nests refining a fixed partition of $D$ into disjoint blocks. We set $\lambda(\emptyset)=0$.
\end{definition}
In the discussion on connectedness, we have treated blocks as if they are the basic connected sets. Note that a definable set $D$ is connected if and only if $\lambda(D)=1$. We denote the set of all connected definable subsets of $M^n$ by $\mathbf{Con}_n$. We tend to drop the suffix $n$ if it is clear from the context. We have $\mathbf B_n\subseteq\mathbf{Con}_n$ as expected.
\begin{illust}
Consider the vector space $\mathbb R_\mathbb R$. The $pp$-definable subsets of the plane, $\mathbb R^2$, are points and lines and the plane.
Note that if a definable subset of $\mathbb R^2$ is topologically connected, then it is connected according to definition \ref{conn}. But the converse is not true. The set $B=\{(x,0):x\neq 0\}$ is not topologically connected, but $B\in\mathbf{Con}$ since $B$ is a block.
If $D$ denotes the union of two coordinate axes with the origin removed, then the number of topologically connected components of $D$ is $4$, whereas $\lambda(D)=2$.
\end{illust}
\begin{rmk}
If $X$ is a `nice' topological space (e.g., a manifold), then the rank $\beta_0$ of the homology group $H_0(X)$ is the number of (path) connected components of $X$. To note the analogy, consider $P\in \mathcal L_n$ and $\alpha\in\mathcal L_P$. If $\alpha\neq\emptyset$, then $\beta_0(\mathcal K^P(\alpha))=\lambda(\bigcup\alpha\setminus P)$. Note that the `deleted neighbourhood' of $P$ in $\alpha$, i.e., the set $\bigcup\alpha\setminus P$, occurs in this correspondence since the `non-deleted neighbourhood' $\bigcup\alpha$ is connected.
\end{rmk}
Topologically connected sets satisfy the following property. If a connected set $A$ is contained in another set $B$, then $A$ is actually contained in a connected component of $B$. We have a similar result here.
\begin{theorem}\label{topconn}
Let $A,B_i\in\mathbf{Con}$ for $1\leq i\leq m$ be such that $\lambda(\bigcup_{i=1}^mB_i)=m$. If $A\subseteq\bigcup_{i=1}^mB_i$, then $A\subseteq B_i$ for a unique $i$.
\end{theorem}
\begin{proof}
Let $\mathcal D$ be a nest containing the nests corresponding to some fixed families of blocks partitioning $A$ and all the $B_i$. The restriction of the digraph $\mathcal H(\mathcal D^+)$ to $A$ is a subdigraph of $\mathcal H(\mathcal D^+)$. Since the former is weakly connected, it is a sub-digraph of exactly one of the $m$ weakly connected components of the latter.
\end{proof}
\subsection{Remarks and questions}\label{rmkcom}
Consider the structure of the proof of the special case of the main theorem. Manipulation of different lattice-like structures is one of the important themes in this paper. The partial quantifier elimination result for theories of modules (theorem \ref{PPET}) makes the meet-semi-lattice $\mathcal L_n$, of $pp$-definable sets, the basic object of study. The lattice of antichains $\mathcal A_n$ is the free distributive lattice on $\mathcal L_n$ and simplicial methods are natural for studying the `set-theoretic geometry' associated with antichains. The local processes in $\mathrm{Def}(M^n)$ are similar to, but independent from, the local processes in $\mathrm{Def}(M^m)$ when $n\neq m$ and these different `dimensions' start to interact with each other only when we are concerned with the multiplicative structure. The fact that the $pp$-sets are closed under projections is not directly relevant to the technique.
Note that the model-theoretic condition $\rm T=T^{\aleph_0}$ is equivalent to the lattice-theoretic statement that every element of $\mathcal L_n$ considered as an element of the lattice $\mathcal A_n$ is `join-irreducible'. The unique representation theorem (theorem \ref{CDT1}) relies solely on this fact and in particular this is a statement about lattices of sets. We would like to know if this idea can be expressed in some more abstract setting.
The algebraic K-theory functor ${\rm K_0:Ring\rightarrow Ab}$ is covariant, whereas the model-theoretic Grothendieck ring functor $K_0$ is contravariant on pure embeddings (theorem \ref{puresurj}). Note that $K_0(M)$ depends on $\overline{\mathcal L}(M)$ in a covariant way and the assignment $M\mapsto\overline{\mathcal L}(M)$ is contravariant. This strongly suggests that the answer to the following question is positive.
\begin{que}
Is there a way to define the Grothendieck ring for a sequence $(L_n)_{n\geq 0}$ of meet-semi-lattices (with inclusion and projection maps) under certain conditions in a way that is abstractly similar to the technique used in the proof of theorem \ref{FINAL}?
\end{que}
A more specific question could be asked for model-theoretic Grothendieck rings.
\begin{que}
Are there any structures admitting some form of quantifier elimination, whose Grothendieck rings can be computed using a similar technique?
\end{que}
Though there are modules with additive torsion elements in Grothendieck rings (corollary \ref{p-adic}), we believe that there are no examples with non-trivial multiplicative torsion elements (i.e. elements $a\in K_0(M)$ such that $a^n=1$ for some $n>1$).
\begin{conj}
There are precisely two units (namely $\pm 1$) in the Grothendieck ring $K_0(M)$ of a nonzero module $M$.
\end{conj}
\end{document} |
\begin{document}
\title[Brunnian links, claspers and Goussarov-Vassiliev
invariants]{Brunnian links, claspers and Goussarov-Vassiliev finite
type invariants}
\author{Kazuo Habiro}
\address{Research Institute for Mathematical Sciences\\Kyoto
University\\Kyoto\\606-8502\\Japan}
\email{[email protected]}
\dedicatory{Dedicated to the memory of Mikhail Goussarov}
\begin{abstract}
We prove that if $n\ge 1$, then an $(n+1)$-component Brunnian link $L$
in a connected, oriented $3$-manifold is $C_n$-equivalent to an
unlink. We also prove that if $n\ge 2$, then $L$ can not be
distinguished from an unlink by any Goussarov-Vassiliev finite type
invariant of degree\;$<2n$.
\end{abstract}
\date{October 21, 2005}
\thanks{This research was partially supported by the Japan Society for
the Promotion of Science, Grant-in-Aid for Young Scientists (B),
16740033.}
\keywords{Brunnian links, Goussarov-Vassiliev finite type link
invariants, claspers}
\maketitle
\section{Introduction}
\label{sec:introduction-1}
Goussarov \cite{Gusarov:91-1,Gusarov:91-2} and Vassiliev
\cite{Vassiliev} independently introduced the notion of finite type
invariants of knots, which provides a beautiful,
unifying view over the quantum link invariants
\cite{Birman:93,Birman-Lin,Kontsevich,Bar-Natan}.
For each oriented, connected $3$-manifold $M$, there is a filtration
\begin{equation*}
\modZ \modL = J_0\supset J_1\supset \cdots
\end{equation*}
of the free abelian group $\modZ \modL $ generated by the set $\modL =\modL (M)$ of
ambient isotopy classes of oriented, ordered links in $M$, where for
$n\ge 0$, the subgroup $J_n$ is generated by all the $n$-fold
alternating sums of links defined by `singular links' with $n$ double
points. An abelian-group-valued link invariant is said to be of
degree$\le n$ if it vanishes on $J_{n+1}$.
Goussarov \cite{Goussarov:finite,Gusarov:variations} and the author
\cite{H} independently introduced theories of surgery along embedded
graphs in $3$-manifolds, which are called {\em $Y$-graphs} or {\em
variation axes} by Goussarov, and {\em claspers} by the author. For
links, one has the notion of {\em $n$-variation equivalence} (simply
called {\em $n$-equivalence} in \cite{Gusarov:variations}) or {\em
$C_n$-equivalence}, which is generated by {\em $n$-variation}
\cite{Gusarov:variations} or {\em $C_n$-moves} \cite{H}, respectively.
As proved by Goussarov \cite[Theorem 9.3]{Gusarov:variations}, for
string links and knots in $S^3$, the $n$-variation (or $C_n$-)
equivalence is the same as the Goussarov-Ohyama $n$-equivalence
\cite{Gusarov:91-2,Ohyama}. The $C_n$-equivalence is generated by the
local move depicted in Figure \ref{F02}, i.e., band-summing Milnor's
link of $(n+1)$-components \cite[Figure 7]{Milnor}, see
Figure~\ref{F01}.
One of the main achievements of these theories is the following
characterization of the topological information carried by
Goussarov-Vassiliev finite type invariants.
\FI{F02}{A special $n$-variation or a special $C_n$-move ($n=5$).}
\FI{F01}{Milnor's link of $6$-components.}
\begin{theorem}[\cite{Gusarov:variations,H}]
\label{r13}
Two knots $K$ and $K'$ in $S^3$ are $n$-variation (or
$C_n$-)equivalent if and only if we have $K-K'\in J_n$ (i.e., $K$ and
$K'$ are not distinguished by any Goussarov-Vassiliev invariants of
degree$<n$.)
\end{theorem}
The variant of Theorem \ref{r13}, with $n$-variation equivalence
replaced by Goussarov-Ohyama $n$-equivalence, is proved previously by
Goussarov \cite{Gusarov:93}.
In \cite[Proposition 7.4]{H}, we observed that for links in $S^3$ there
is a certain difference between the notion of $C_n$-equivalence and
the notion of the Goussarov-Vassiliev finite type invariants of
degree\;$<n$, i.e., Theorem \ref{r13} does not extend to links in
$S^3$. More specifically, we showed that if $n\ge 2$, then Milnor's
link $L_{n+1}$ of $(n+1)$-components is $C_n$-equivalent but {\em not}
$C_{n+1}$-equivalent to the unlink $U_{n+1}$, but we have
$L_{n+1}-U_{n+1}\in J_{2n}$. (For $2$-component links, one can easily
observe a similar facts for the Whitehead link $W_2$: $W_2$ is
$C_2$- but not $C_3$-equivalent to the unlink $U_2$, but we have
$W_2-U_2\in J_3$, $\not\in J_4$.)
Note that Milnor's links are examples of {\em Brunnian links}. Here,
a link $L$ is {\em Brunnian} if any proper sublink of $L$ is an
unlink. The purpose of this paper is to prove the following results,
which are generalizations of the above-mentioned facts about Milnor's
links to Brunnian links.
Let $M$ be a connected, oriented $3$-manifold.
\begin{theorem}[{Announced in \cite[Remark 7.5]{H} for $M=S^3$}]
\label{r10}
For $n\ge 1$, every $(n+1)$-component Brunnian link in $M$ is
$C_n$-equivalent to an unlink.
\end{theorem}
\begin{theorem}[{Announced in \cite[Remark 7.5]{H} for $M=S^3$}]
\label{r3}
Let $n\ge 2$, and let $U$ denote $(n+1)$-component unlink in $M$. For
every $(n+1)$-component Brunnian link $L$ in $M$, we have
$L-U\in J_{2n}$. (Consequently, $L$ and $U$ can not be distinguished
by any Goussarov-Vassiliev invariant of degree\;$<2n$ with values in
any abelian group.)
\end{theorem}
We remark that Theorem \ref{r10} follows from a stronger, but more
technically stated, result (see Theorem \ref{r9} below), which is
proved also by Miyazawa and Yasuhara \cite{Miyazawa-Yasuhara} for
$M=S^3$, independently to the present paper.
\section{Preliminaries}
\label{sec:preliminaries-1}
\subsection{Preliminaries}
\label{sec:preliminaries}
In the rest of this paper, we freely use the definitions, notations
and conventions in \cite{H}.
Throughout the paper, let $M$ denote a connected, oriented
$3$-manifold (possibly noncompact, possibly with boundary).
By a {\em tangle} $\gamma $ in $M$, we mean a proper embedding
$\gamma \colon\thinspace\alpha \rightarrow M$ of a compact, oriented $1$-manifold $\alpha $ into
$M$. By a {\em link}, we mean a tangle consisting only of circle
components. (In \cite{H}, tangles are called `links'.) We sometimes
confuse $\gamma $ and the image $\gamma (\alpha )\subset M$.
Two tangles $\gamma $ and $\gamma '$ in $M$ are {\em equivalent}, denoted by
$\gamma \cong\gamma '$, if $\gamma $ and $\gamma '$ are ambient isotopic fixing the
endpoints.
\subsection{Claspers and tree claspers}
\label{sec:clasp-tree-clasp}
Here we recall some definition of claspers and tree claspers. See
\cite[\S2, \S3]{H} for the details.
A {\em clasper} $C$ for a tangle $\gamma $ in a $3$-manifold $M$ is a
(possibly unorientable) compact surface $C$ in $\int M$ with some
structure. $G$ is decomposed into finitely many subsurfaces called
{\em edges}, {\em leaves}, {\em disk-leaves}, {\em nodes} and {\em
boxes}.
We do not repeat here all the rules that should be satisfied by the
subsurfaces. For the details, see \cite[Definition 2.5]{H}. We
follow the drawing convention for claspers \cite[Convention 2.6]{H},
in which we draw an edge as a line instead of a band.
Given a clasper $C$, there is defined a way to associate a framed link
$L_C$, see \cite[\S2.2]{H}. {\em Surgery along $C$} is defined to be
surgery along $L_C$. A clasper $C$ is called {\em tame} if surgery
along $C$ preserves the homeomorphism type of a regular neighborhood
of $C$ relative to the boundary. All the clasper which appear in the
present paper are tame, and thus surgery along a clasper can be
regarded as a move of tangle in a fixed $3$-manifold. The result from
a tangle $\gamma $ of surgery along a clasper $C$ is denoted by $\gamma ^C$.
A {\em strict tree clasper} $T$ is a simply-connected clasper $T$
consisting only of disk-leaves, nodes and edges. The degree of $T$ is
defined to be the number of nodes plus $1$, which is equal to the
number of disk-leaves minus $1$. For $n\ge 1$, a {\em $C_n$-tree} will
mean a strict tree clasper of degree $n$. A {\em $C_n$-move} is
surgery along a $C_n$-tree, which may be regarded as a local move of
tangle since the regular neighborhood of $T$ is a $3$-ball. The {\em
$C_n$-equivalence} of tangles is the equivalence relation generated by
$C_n$-moves and equivalence of tangles.
A disk-leaf in a clasper is said to be {\em simple} if it intersects
the tangle by one point. A strict tree clasper is {\em simple} if all
its leaves are simple.
A {\em forest} $F$ will mean `strict forest clasper' in the sense of
\cite[Definition 3.2]{H}, i.e., a clasper consisting of finitely many
disjoint strict tree claspers. $F$ is said to be simple if all the
components of $F$ are simple. A {\em $C_n$-forest} is a forest
consisting only of $C_n$-trees.
\section{Brunnian links and $C^a_n$-moves}
\label{sec:brunnian-links-cn}
\subsection{Definition of $C^a_n$-moves}
\label{sec:can-moves}
\begin{definition}
\label{r7}
For $k\ge 1$, a {\em $C^a_k$-tree} for a tangle $\gamma $ in $M$ is a
$C_k$-tree $T$ for $\gamma $ in $M$, such that
\begin{enumerate}
\item for each disk-leaf $A$ of $T$, all the strands intersecting
$A$ are contained in one component of $\gamma $, and
\item each component of $\gamma $ intersects at least one disk-leaf of
$T$. (In other words, $T$ intersects {\em all} the components of
$\gamma $; this explains `$a$' in `$C^a_k$'.)
\end{enumerate}
Note that such a tree exists only when $k\ge l-1$, where $l$ is the
number of components in $\gamma $. Note also that the condition (1) is
vacuous if $T$ is simple.
A {\em $C^a_k$-move} on a link is surgery along a $C^a_k$-tree. The
{\em $C^a_k$-equivalence} is the equivalence relation on tangles
generated by $C^a_k$-moves. A {\em $C^a_k$-forest} is a forest
consisting only of $C^a_k$-trees.
\end{definition}
What makes the notion of $C^a_k$-move useful in the study of Brunnian
links is the following.
\begin{proposition}
\label{r16}
A $C^a_k$-move on a tangle preserves the types of the proper
subtangles. In particular, if a link $L'$ is $C^a_k$-equivalent to
a Brunnian link $L$, then $L'$ also is a Brunnian link.
\end{proposition}
\begin{proof}
Let $T$ be a $C^a_k$-tree for a tangle $\gamma $. For any proper
subtangle $\gamma '$, $T$ viewed as a clasper for $\gamma '$ has at least one
disk-leaf which intersects no components of $\gamma '$. Hence, by
\cite[Proposition 3.4]{H}, we have $\gamma '_T\cong \gamma '$.
\end{proof}
Obviously, $C^a_k$-equivalence implies $C_k$-equivalence. But the
converse does not hold in general, since a $C_k$-move can transform an
unlink into a non-Brunnian link (e.g., a link with a knotted
component).
The following result gives a characterization of Brunnian
links in terms of clasper moves.
Theorem \ref{r10} follows from Theorem \ref{r9} below.
\begin{theorem}
\label{r9}
An $(n+1)$-component link $L$ in $M$ ($n\ge 1$) is Brunnian if and
only if $L$ is $C^a_n$-equivalent to an $n$-component unlink $U$ in
$M$.
\end{theorem}
As mentioned in the introduction, Theorem \ref{r9} is proved
independently by Miyazawa and Yasuhara \cite{Miyazawa-Yasuhara} for
$M=S^3$.
The rest of this subsection is devoted to proving Theorem \ref{r9}.
The following two lemmas easily follow from the proof of the
corresponding results in \cite{H}.
\begin{lemma}[{$C^a$-version of \cite[Theorem 3.17]{H}}]
\label{r18}
For two tangles $\gamma $ and $\gamma '$ in $M$, and an integer $k\ge 1$, the
following conditions are equivalent.
\begin{enumerate}
\item $\gamma $ and $\gamma '$ are $C^a_k$-equivalent.
\item There is a simple $C^a_k$-forest $F$ for $\gamma $ in $M$ such
that $\gamma ^F\cong \gamma '$.
\end{enumerate}
\end{lemma}
\begin{lemma}[{$C^a$-version of \cite[Proposition 4.5]{H}}]
\label{r4}
Let $\gamma $ be a tangle in $M$, and let $\gamma _0$ be a component of $\gamma $.
Let $T_1$ and $T_2$ be $C_k$-trees for a tangle $\gamma $ in $M$,
differing from each other by a crossing change of an edge with the
component $\gamma _0$. Suppose that $T_1$ and $T_2$ are $C^a_k$-trees
for either $\gamma $ or $\gamma \setminus \gamma _0$. Then $\gamma ^{T_1}$ and $\gamma ^{T_2}$ are
related by one $C^a_{k+1}$-move.
\end{lemma}
Now we prove Theorem \ref{r9}.
\begin{proof}[Proof of Theorem \ref{r9}]
Let $L=L_0\cup L_1\cup \cdots\cup L_n$.
The `if' part follows since a $C^a_n$-move for an
$(n+1)$-component link $L$ preserves each proper sublinks of $L$ up
to isotopy.
The proof of the `only if' part is by induction on $n$.
Suppose $n=1$. Since $L=L_0\cup L_1$ is Brunnian, it follows that both
$L_0$ and $L_1$ are unknotted in $M$. In $M$ we can homotop $L_1$
into an unknot $U_1$, such that $L_0\cup U_1$ is an unlink. This
homotopy can be done by ambient isotopy and crossing changes between
distinct components, i.e., (simple) $C^a_1$-moves. This shows the assertion.
Suppose $n>1$. Since $L$ is Brunnian in $M$, it follows that
$L'=L\setminus L_0$ is an $n$-component Brunnian link in $M\setminus L_0$. By
induction hypothesis, it follows that $L'$ is $C^a_{n-1}$-equivalent
in $M\setminus L_0$ to an $n$-component unlink $U'$ in $M\setminus L_0$. By Lemma
\ref{r18}, there is a $C^a_{n-1}$-forest $F$ for $U'$ in $M\setminus L_0$
satisfying $(U')^F\cong L'$ in $M\setminus L_0$. Since $L_0\cup U'$ is an
unlink, there is a disk $D_0$ in $M$ disjoint from $U'$.
We may assume that $D_0$ intersects $F$
only by finitely many transverse intersections with the edges of
$F$. By crossing changes between $L_0$ and edges of $F$
intersecting $D$, we obtain from $L_0$ an unknot $U_0$ in $M$ which
bounds a disk disjoint from $L'$ and $F$. By Lemma \ref{r4}, it
follows that these crossing changes do not change the
$C^a_n$-equivalence class of the result of surgery. Hence we have
\begin{equation*}
L=L_0\cup L'
\cong L_0\cup (U')^F
\cong (L_0\cup U')^F
\underset{C^a_n}{\sim} (U_0\cup U')^F
\cong U_0\cup (U')^F
\cong U_0\cup L'.
\end{equation*}
Since $U_0\cup L'$ is an unlink, the assertion follows.
\end{proof}
\subsection{Generalization to tangles}
\label{sec:bottom-tangles}
One can generalize Theorem \ref{r9} to tangles as follows.
Let $c_0,\ldots,c_n\subset \partial M$ be disjoint arcs, and set $c=c_0\cup \cdots\cup c_n$. A
$(n+1)$-component {\em tangle in $M$ with arc basing $c$} is a tangle
$\gamma $ consisting of $n+1$ properly embedded arcs $\gamma _0,\ldots,\gamma _n$ in $M$
such that $\partial \gamma _i=\partial c_i$ for $i=0,\ldots,n$. A tangle $\gamma $ with arc
basing $c$ is called {\em trivial} (with respect to $c$) if simple
closed curves $\gamma _i\cup c_i$ for $i=0,\ldots,n$ bounds disjoint disks in $M$.
A tangle $\gamma $ with arc basing $c$ is {\em Brunnian} if every proper
subtangle of $\gamma $ is trivial with respect to the corresponding
$1$-submanifold of $c$.
\begin{theorem}
\label{r5}
If $\gamma =\gamma _0\cup \cdots\cup \gamma _n$ ($n\ge 1$) is an $(n+1)$-component tangle in
$M$ with arc basing $c=c_1\cup \cdots\cup c_n$. Then $\gamma $ is Brunnian if and
only if $\gamma $ is $C^a_n$-equivalent to an $(n+1)$-component trivial
tangle with respect to $c$.
\end{theorem}
\begin{proof}
Similar to the proof of Theorem \ref{r9}.
\end{proof}
\begin{remark}
\label{r15}
The case $M=B^3$ of Theorem \ref{r5} is independently proved by
Miyazawa and Yasuhara \cite[Proposition 4.1]{Miyazawa-Yasuhara}.
\end{remark}
\begin{remark}
\label{r26}
Taniyama \cite{Taniyama} (see also Stanford \cite{Stanford}) proved
that an $(n+1)$-component Brunnian link is {\em $n$-trivial}, or
{\em $n$-equivalent} to an unlink. Here, by `$n$-triviality' and
`$n$-equivalence' we mean the notion introduced independently by
Goussarov \cite{Gusarov:91-2} and Ohyama \cite{Ohyama} (see also
\cite{Taniyama,Gusarov:variations}). It is well known that
$C_n$-equivalence implies $n$-equivalence, but the converse seems
open for links with at least $2$-components. However, Goussarov
\cite{Gusarov:variations} proved that $C_n$-equivalence (or
$n$-variation equivalence) and $n$-equivalence are the same for
string links in $D^2\times [0,1]$, and hence the case $M=B^3$ of
Theorem~\ref{r5} follows from the fact (which seems to be well
known) that $(n+1)$-component Brunnian tangle of arcs in $B^3$ is
$n$-trivial.
\end{remark}
Using Theorems \ref{r9} and \ref{r5}, we can prove the following fact,
which means that {\em a Brunnian link in $S^3$ is the closure of a
Brunnian tangle in $B^3$.} (It is clear that, conversely, the closure
of a Brunnian tangle is Brunnian.)
\begin{proposition}
\label{r2}
Let $n\ge 2$. Given an $n$-component Brunnian link $L=L_1\cup \cdots\cup L_n$
in $S^3$, there is an $n$-component Brunnian tangle
$\gamma =\gamma _1\cup \cdots\cup \gamma _n$ in a $3$-ball $B^3$ with respect to a basing
$c=c_1\cup \cdots\cup c_n\subset \partial B^3$ such that the union
$\bigcup_{i=1}^n\gamma _i\cup c_i\subset B^3\subset S^3$ viewed as a link in $S^3$ is
equivalent to $L$.
\end{proposition}
\begin{proof}
By Theorem \ref{r9} and Lemma \ref{r18}, there is a
simple $C^a_{n-1}$-forest $F$ for an $n$-component unlink
$U=U_1\cup \cdots\cup U_n$ such that $U^F\cong L$. Let $D_1,\ldots,D_n$ be
disjoint discs in $S^3$ bounded by $U_1,\ldots,U_n$, and set
$D=D_1\cup \cdots\cup D_n$. Choose a point $p_0\in S^3$ disjoint from $F\cup D$.
For each $i=1,\ldots,n$, let $p_i\in U_i\setminus F$ and
let $g_i$ be a simple arc in $M\setminus F$ from $p_0$ to
$p_i$ such that $g_i\cap D=p_i$. Here we may assume that
$g_i\cap g_j=p_0$ if $i\neq j$.
Let $N$ be a small
regular neighborhood of $g_1\cup \cdots\cup g_n$, which is a $3$-ball. Set
$B^3=\overline{S^3\setminus N}$. For $i=1,\ldots,n$, set $c_i=\partial B^3\cap D_i$, and
set $\gamma ^0_i=U_i\cap B^3$. Then, by Theorem \ref{r5} the result of surgery
$\gamma =(\gamma ^0_1\cup \cdots\cup \gamma ^0_n)^F$ is Brunnian with respect to
$c_1\cup \cdots\cup c_n$, and satisfies the assertion.
\end{proof}
\section{Brunnian links and the Goussarov-Vassiliev filtration}
\label{sec:vass-gouss-finite}
\subsection{Definition of the Goussarov-Vassiliev filtration}
\label{sec:defin-gouss-vass}
Here we recall the definition of the Goussarov-Vassiliev filtration for
links using strict tree claspers. For the details, see \cite[\S 6]{H}.
Let $\modL (M)$ denote the set of equivalence classes of tangles in
$M$. For $n\ge 0$, define $J_n=J_n(M)\subset \modZ \modL (M)$ as follows.
By a {\em forest scheme} for a tangle $\gamma $ in $M$, we mean a `strict
forest scheme' in the sense of \cite[Definition 6.6]{H}, i.e., a set
$S=\{T_1,\ldots,T_p\}$ of disjoint, strict tree claspers $T_1,\ldots,T_p$ for
a tangle $\gamma $ in $M$. The {\em degree} of $S$ is defined to be the
sum of the degrees of $T_1,\ldots,T_p$. Set
\begin{equation*}
[\gamma ,S]=[\gamma ;T_1,\ldots,T_p]
=\sum_{S'\subset S}(-1)^{p-|S'|}\gamma ^{\bigcup S'}\in \modZ \modL (M),
\end{equation*}
where the sum is over all subsets $S'$ of $S$, $|S'|$ denotes the
cardinality of $S'$, and $\bigcup S'$ denote the clasper consisting of
the elements of $S'$.
For $n\ge 0$, let $J_n=J_n(M)$ denote the $\modZ $-submodule of $\modZ \modL (M)$
spanned by the elements $[\gamma ,S]$ for any pair $(\gamma ,S)$ of a link $\gamma $
in $M$ and a forest scheme $S$ for $\gamma $ in $M$ of degree $n$.
This defines a descending filtration of $\modZ \modL (M)$:
\begin{equation*}
\modZ \modL (M)=J_0(M)\supset J_1(M)\supset \cdots,
\end{equation*}
which is the same as the Goussarov-Vassiliev filtration in the usual
sense, defined using singular tangles.
\subsection{Proof of Theorem \ref{r3}}
\label{sec:theorem-proof}
We need some lemmas before proving Theorem \ref{r3}.
\begin{lemma}[{A variant of \cite[Lemma 3.20]{H}}]
\label{r8}
Let $\gamma $ be a tangle in $M$, and let $T$ be a strict tree clasper
for $\gamma $ in $M$. Let $N$ be a small regular neighborhood of $T$ in
$M$. Then the pair $(N,(\gamma \cap N)^T)$ is homeomorphic to
$(D^2,(\text{$p$ points}))\times [0,1]$, where $p$ is the number of
points in $T\cap \gamma $.
\end{lemma}
\begin{proof}
The case where $T$ is simple is a part of \cite[Lemma 3.20]{H}.
The general case immediately follows from this case.
\end{proof}
\begin{lemma}
\label{r12}
Let $1\le n\le r$, and let $L$ be an $(n+1)$-component Brunnian link in
$M$. Then there is a forest $F$ for an $(n+1)$-component unlink $U$
in $M$ satisfying the following properties.
\begin{enumerate}
\item $F$ consists of $C^a_l$-trees with $n\le l<r$.
\item $U$ bounds $n+1$ disjoint disks $D_1\cup \cdots\cup D_{n+1}$ in $M$
which are disjoint from edges and trivalent vertices of $F$.
\item $L$ is $C^a_r$-equivalent to $U^F$.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is by induction on $r$. The case $r=n$ follows
immediately from Theorem \ref{r9} by setting $F=\emptyset$. Suppose
that the result is true for $r\ge n$ and let us verify the case for
$r+1$. Let $F$ be as in the statement of the lemma. Let $N$ be a
small regular neighborhood of $F$ in $M$. Then $U^F$ is obtained
from $U$ by replacing the part $U\cap N$ by $(U\cap N)^F$. Since $L$ is
$C^a_r$-equivalent to $U^F$, it follows from Lemma \ref{r18} that
there is a $C^a_r$-forest $F'$ for $U^F$ such that
\begin{equation}
\label{e1}
(U^F)^{F'}\cong L.
\end{equation}
Using Lemma \ref{r8}, we may assume that $F'$ is disjoint from
$N$, and thus can be regarded as a forest for $U$ disjoint from
$F$. Hence we have
\begin{equation}
\label{e2}
(U^F)^{F'}\cong U^{F\cup F'}.
\end{equation}
Now $F'$ may intersects $D=D_1\cup \cdots\cup D_{n+1}$. We may assume that $F'$
intersects $D$ only by disk-leaves and finitely many transverse
intersection of $D$ and edges of $F'$. By Lemma \ref{r4}, without
changing the result of surgery up to $C^a_{r+1}$-equivalence, we can
remove the intersection of $D$ and the edges of $F'$ by crossing
changes between components of $U$ and edges of $F'$ intersecting
$D$. Let $F''$ denote the forest obtained from this operation. Now
$D$ is disjoint from the edges and trivalent vertices of $F''$, and
$U^{F\cup F''}$ and $U^{F\cup F'}$ are $C^a_{r+1}$-equivalent. From this,
\eqref{e1} and \eqref{e2}, it follows that $F\cup F''$ is a forest with
the desired properties.
\end{proof}
\begin{definition}
\label{r14}
Let $C$ be a clasper for a tangle $\gamma $ in $M$. We say that a simple
disk-leaf $A$ of $C$ {\em monopolizes} a circle component $K$ of
$\gamma $ in $(C,\gamma )$ if there is a $3$-ball $B\subset M$ such that
$(\gamma \cup C)\cap B$ looks as depicted in Figure \ref{F03}. We call the
pair $(A,K)$ a {\em monopoly} in $(C,\gamma )$. The monopolized
component $K$ bounds a disk $D$ in $\int M$ which intersect $C$ by
an arc $A\cap D$. We call $D$ a {\em monopoly disk} for $K$.
\FI{F03}{A monopoly.}
\end{definition}
\begin{lemma}[Monopoly Lemma]
\label{r11}
Suppose $l\ge 1$ and $0\le k\le l+1$ be integers. Let $T$ be a $C_l$-tree
for a tangle $\gamma $ in $M$ with $k$ distinct monopolies in $(T,\gamma )$.
Then we have
\begin{equation}
\label{e7}
\gamma ^T - \gamma \in J_{d(l,k)}(M),
\end{equation}
where
\begin{equation*}
d(l,k) =
\begin{cases}
1&\text{if $l=1$, $0\le k\le 2$},\\
l+k&\text{if $l\ge 2$, $0\le k\le l$},\\
l+k-1&\text{if $l\ge 2$, $k=l+1$}.
\end{cases}
\end{equation*}
\end{lemma}
\begin{proof}
The case $l=1$ is trivial. Also, the case $k=l+1$ and $l\ge 2$
follows from the case $k=l\ge 2$ by ignoring one monopoly. Hence it
suffices to prove the case $l\ge 2$, $0\le k\le l$. Note that if
$(l,k)=(1,0)$, then we have $d=l+k$. We will prove by induction on
$l+k$ that the assertion is true if either $(l,k)=(1,0)$ or $l\ge 2$
and $0\le k\le l$.
As we have seen, the case $(l,k)=(1,0)$ is trivial. Assume
$l+k\ge 2$. Let $(A_1,K_1),\ldots,(A_k,K_k)$ be the $k$ monopolies in
$(T,\gamma )$ with monopoly disks $D_1,\ldots,D_k$, respectively. Since
$k\le l$, we can choose one disk-leaf $A_0$ of $T$ distinct from
$A_1,\ldots,A_k$. Since $l\ge 2$, $A_0$ is adjacent to a node $Y$. Let
$E$ denote the edge between $A_0$ and $Y$. Let $P'$ and $P''$ be
the two components of $T\setminus (Y\cup E\cup A_0)$, which are two subtrees in
$T$.
Let $l',l''\ge 1$ denote the number of disk-leaves in $P'$ and $P''$,
respectively. Let $k'\le l'$ and $k''\le l''$ denote the numbers of the
monopolizing disk-leaves from $A_1,\ldots,A_k$ contained in $P'$, and
$P''$, respectively. We have $l'+l''=l$ and $k'+k''=k$.
The proof is divided into two cases.
{\em Case 1. Either $(l',k')$ or $(l'',k'')$ is $(1,1)$.} We assume
that $(l',k')=(1,1)$; the other case is proved by the same argument.
Then $P'$ consists of a monopolizing disk-leaf $A_i$,
$i\in \{1,\ldots,k\}$, and the incident edge $E'$. Without loss of
generality, we may assume that $i=1$. See Figure \ref{F05} (a).
\FI{F05}{Here the lines labeled $\gamma $ depicts a parallel family of
strands of $\gamma $.} Let $C$ be a $C_1$-tree for $\gamma $ disjoint from
$T$, as depicted in Figure \ref{F05} (b). Figure \ref{F06} and
\cite[Proposition 3.4]{H} imply that $\gamma ^{T\cup C}\cong\gamma ^C$. (This fact
is implicit in the proof of \cite[Proposition 7.4]{H}.)
\FI{F06}{Here we use moves 1, 2, 10 of \cite[Proposition 2.7]{H}.
The orientations given to the circle components are `temporary' and
may possibly be the opposite to the actual orientations
simultaneously for all the four figures.} Hence we have
\begin{equation}
\label{e8}
\gamma ^T-\gamma =-(\gamma ^{T\cup C}-\gamma ^T-\gamma ^C+\gamma )=-[\gamma ;T,C].
\end{equation}
Let $N$ be a small regular neighborhood of $T\cup D_2\cup D_3\cup \cdots\cup D_k$,
which is a $3$-ball. Then, by the induction hypothesis, we have
\begin{equation}
\label{e9}
(\gamma \cap N)^T - \gamma \cap N \in J_{l+k-1}(N).
\end{equation}
Since $C$ is a $C_1$-tree, \eqref{e8} and \eqref{e9} implies
\eqref{e7}.
{\em Case 2. Otherwise.} Apply move 9 in \cite[Proposition 2.9]{H}
at the disk-leaf $A_0$, see Figure \ref{F04}. \FI{F04}{} The result
is a union $T'\cup T''$ of a $C_{l'}$-tree $T'$ and a $C_{l''}$-tree
$T''$ for $\gamma $ such that $\gamma ^T\cong \gamma ^{T'\cup T''}$. Let $N'$ be a
small regular neighborhood of the union of $T'$ and the monopoly
disks intersecting $T'$. Similarly, let $N''$ be a small regular
neighborhood of the union of $T''$ and the monopoly disks
intersecting $T''$. Since $(l',k'),(l'',k'')\neq(1,1)$ and
$l'+k',l''+k''<l+k$, it follows by induction hypothesis that we have
\begin{gather*}
(\gamma \cap N')^{T'}-\gamma \cap N' \in J_{l'+k'}(N'),\\
(\gamma \cap N'')^{T''}-\gamma \cap N'' \in J_{l''+k''}(N'').
\end{gather*}
Using \cite[Proposition 3.4]{H}, we see that
$\gamma ^{T'}\cong\gamma ^{T''}\cong\gamma $. Hence it follows that
\begin{equation*}
\gamma ^T-\gamma =\gamma ^{T'\cup T''}-\gamma ^{T'}-\gamma ^{T''}+\gamma \in J_{l+k}(M)
\end{equation*}
\end{proof}
\begin{remark}
\label{r20}
In \cite[Lemma 7.1]{Conant}, a result similar to Lemma \ref{r11} is
proved, but it is not strong enough for our purpose.
\end{remark}
Now we prove Theorem \ref{r3}.
\begin{proof}[Proof of Theorem \ref{r3}]
By Theorem \ref{r9} and Lemma \ref{r12} for $r=2n$, there is a forest
$F$ for $U$ in $M$ consisting of simple $C^a_l$-trees with $n\le l<2n$ such
that
\begin{itemize}
\item[(a)] $U$ bounds $n+1$ disjoint disks $D_1,\ldots,D_{n+1}$ in $M$,
disjoint from edges and trivalent vertices of $F$, and
\item[(b)] $L$ is $C^a_{2n}$-equivalent to $U^F$.
\end{itemize}
By the condition (b), we have
\begin{equation}
\label{e3}
L- U^F \in J_{2n}.
\end{equation}
Let $S=\{T_1,\ldots,T_p\}$, $p\ge 0$, be a forest scheme for $U$ in $M$
consisting of the tree claspers $T_1,\ldots,T_p$ contained in $F$. By an
easy calculation, we have
\begin{equation}
\label{e5}
U^F = \sum_{S'\subset S} [U,S'],
\end{equation}
where $S'$ runs over all subsets of $S$. Since $\deg T_i\ge n$ for all $i$,
we have $\deg S'\ge n|S'|$, where $|S'|$ denotes the number of elements in
$S'$. Since $|S'|\ge 2$ implies $[U,S']\in J_{2n}$, it follows from
\eqref{e5} that
\begin{equation}
\label{e4}
U^F-U \equiv \sum_{i=1}^p [U;T_i] \pmod {J_{2n}}.
\end{equation}
Hence, by \eqref{e3} and \eqref{e4}, it suffices to prove the case
$F=T$ is a $C^a_l$-tree with $n\le l<2n$. By assumption, there are at
least $k=2n+1-l$ monopolies in $(T,U)$. Hence by Lemma \ref{r11}, we
have $U^T-U \in J_{d(l,k)}$, where $d(l,k)$ is defined in Lemma
\ref{r11}. Since $l\ge n\ge 2$, we have $d(l,k)\ge l+k-1\ge 2n$. Hence we
have $U^T-U\in J_{2n}$. This completes the proof.
\end{proof}
\subsection{Remarks}
\label{sec:remarks}
\begin{remark}
\label{r17}
Przytycki and Taniyama \cite{Przytycki-Taniyama01} proved a
conjecture by Kanenobu and Miyazawa \cite{Kanenobu-Miyazawa} about
the homfly polynomial of Brunnian links, and also announced a
similar result for the Kauffman polynomial. These results follow
from Theorem~\ref{r3}.
\end{remark}
\begin{remark}
\label{r1}
Yasuhara pointed out to the author that Theorem \ref{r3} implies the
following generalization.
{\it Let $n\ge 2$, $m\ge 1$, and let $M$ be a connected, oriented
$3$-manifold. Let $L$ and $L'$ be two $(n+1)$-component links in
$M$ such that
\begin{enumerate}
\item both $L$ and $L'$ are $C_m$-equivalent to an $(n+1)$-component
unlink $U$,
\item $L$ and $L'$ are $C^a_n$-equivalent to each other.
\end{enumerate}
Then we have $L'-L\in J_{l}$, where $l=\min(2n,n+m)$.}
The proof is as follows. We may assume that $L=U^F$, where $F$ is a
$C_m$-forest for $U$. We may assume also that $L'=U^{F\cup F'}$, where
$F'$ is a $C^a_n$-forest for $U$, disjoint from $F$. Then we have
\begin{equation*}
L'-L
=U^{F\cup F'}-U^F
=(U^{F\cup F'}-U^F-U^{F'}+U)+(U^{F'}-U).
\end{equation*}
Here we have $U^{F\cup F'}-U^F-U^{F'}+U\in J_{n+m}$. We also have
$U^{F'}-U\in J_{2n}$ by Theorem \ref{r3}. Hence the assertion.
\end{remark}
\begin{acknowledgments}
I thank Akira Yasuhara for helpful discussions and comments and for
asking me about the proof of Theorem \ref{r3} (in the case of
Brunnian links in $S^3$), which motivated me to write this paper.
Also, I thank Jean-Baptiste Meilhan for many helpful discussions and
comments.
\end{acknowledgments}
\end{document} |
\begin{document}
\title{Mechanizing Matching Logic In Coq}
\defMechanizing Matching Logic In Coq{Mechanizing Matching Logic In Coq}
\author{P\'{e}ter Bereczky\institute{E\"{o}tv\"{o}s Lor\'{a}nd University, Hungary} \and
Xiaohong Chen\institute{University of Illinois Urbana-Champaign, USA} \and
D\'{a}niel Horp\'{a}csi\institute{E\"{o}tv\"{o}s Lor\'{a}nd University, Hungary} \and
Lucas Pe\~{n}a\institute{University of Illinois Urbana-Champaign, USA} \and
Jan Tu\v{s}il\institute{Masaryk University, Brno, Czech Republic}
}
\defP. Bereczky et al.{P. Bereczky et al.}
\maketitle
\begin{abstract}
Matching logic is a formalism for specifying, and reasoning about, mathematical structures, using patterns and pattern matching. Growing in popularity, it has been used to define many logical systems such as separation logic with recursive definitions and linear temporal logic. In addition, it serves as the logical foundation of the K semantic framework, which was used to build practical verifiers for a number of real-world languages. Despite being a fundamental formal system accommodating substantial theories, matching logic lacks a general-purpose, machine-checked formalization. Hence, we formalize matching logic using the Coq proof assistant. Specifically, we create a new representation of matching logic that uses a locally nameless encoding, and we formalize the syntax, semantics, and proof system of this representation in the Coq proof assistant. Crucially, we prove the soundness of the formalized proof system and provide a means to carry out interactive matching logic reasoning in Coq. We believe this work provides a previously unexplored avenue for reasoning about matching logic, its models, and the proof system.
\end{abstract}
\section{Introduction}
Matching logic~\cite{Rosu17,ChenRosu19Mml}
is a unifying logical
framework for defining the formal semantics of programming languages
and specifying their language tools.
Given a programming language $L$,
its formal semantics is defined by a \emph{matching logic theory} $\Gamma^L$,
i.e., a set of axioms.
Many language tools, such as parsers, interpreters, compilers, and even deductive
program verifiers, are best-effort implementation
of matching logic reasoning.
The correctness of language tools is justified by matching logic proof objects,
checkable by a 240-line proof checker~\cite{ml-checker}.
The formal semantics of many real-world programming languages
have been \emph{completely defined} as matching logic theories.
These languages include
C~\cite{HER15},
Java~\cite{k-java},
JavaScript~\cite{k-js},
Python~\cite{python-semantics},
Rust~\cite{krust-singapore,krust-shanghai},
Ethereum bytecode~\cite{HSZ+18},
x86-64~\cite{DPK+19},
and LLVM~\cite{k-llvm}.
The $\mathbb{K}$\xspace framework (\url{https://kframework.org})
is a best-effort implementation of matching logic.
From the formal semantics of these real-world languages,
$\mathbb{K}$\xspace automatically generates implementations and formal analysis tools,
some of which have been commercialized~\cite{rv-match}.
Ultimately, $\mathbb{K}$\xspace can be used as a tool for formally defining languages,
and as a tool for formally reasoning about properties of programming languages and programs.
$\mathbb{K}$\xspace currently provides the most comprehensive support for
\emph{automated reasoning} for matching logic
by means of various algorithms that are specifically targeted at
the (automatic) generation of language tools
such as interpreters and deductive verifiers, which, respectively,
\emph{are} specific forms of matching logic reasoning,
and an integration with the state-of-the-art SMT solvers such as
Z3~\cite{DMB08}.
However, there is no ``exit solution'' in $\mathbb{K}$\xspace when
these automatic algorithms and external solvers fail,
in which case \emph{lemmas}, whose correctness
is justified externally, often informally,
need to be manually added to fill in the reasoning gaps.
In this paper, we aim to bridge the aforementioned reasoning gaps
by bringing \emph{interactive reasoning} to matching logic.
Specifically, we give, for the first time, a complete mechanization
of matching logic in Coq~\cite{CoqArt}.
\subsection*{Contributions}
In this work, we investigate the formal definition of matching logic in an interactive theorem prover, with the aim of enabling computer-aided reasoning \emph{about} and \emph{within} matching logic:
\begin{itemize}
\item \emph{Mechanize the soundness of the matching logic proof system}---while matching logic has been proven sound on paper, its soundness has not been formalized yet. This soundness proof is crucial in providing the highest level of assurance for mechanized reasoning.
\item \emph{Enable interactive, mechanically verified reasoning about matching logic theories}---the formalization needs to leverage the full power of the theorem prover to encode matching logic theories in a modular way and do reasoning at the highest abstraction level possible.
\end{itemize}
Following upon similar formalization efforts (such as the solutions to the POPLMark Challenge~\cite{poplmark}), we have decided to encode the logic in a widely used and mature system, the Coq proof assistant~\cite{CoqArt}. Furthermore, we carry out the embedding in a locally nameless~\cite{ChargueraudLocallyNameless,Leroy07alocally,McBrideM04,Gordon93} representation, which, in contrast to named approaches, is more amenable to computer-aided verification. The entire formalization is open, and available online~\cite{ml-formalization}. The paper shows the following main contributions:
\begin{itemize}\setlength\itemsep{0em}
\item A locally nameless definition of the matching logic syntax, semantics, and proof system;
\item The design of the embedding of the locally nameless matching logic in the Coq proof assistant;
\item The first \emph{mechanized} proof of the soundness of the matching logic proof system;
\item Example theories with proofs about their semantic and proof-theoretical properties in the mechanized matching logic;
\item A preliminary implementation of a matching logic proof mode to simplify interactive reasoning.
\end{itemize}
The paper is structured as follows.
Section~\ref{sec:ml-intro} introduces matching logic, in
a locally nameless representation. In Section~\ref{sec:formalization}, we outline the Coq
formalization, including some technical challenges faced.
Section~\ref{sec:soundness} discusses examples of meta-level reasoning and the soundness proof in particular.
Section~\ref{sec:proofmode} presents an example interactive proof in our formalization.
Finally, Section~\ref{sec:related-work} discusses related
work, and
Section~\ref{sec:conclusion} presents areas for future work and
concludes.
\section{Introduction to Matching Logic}\label{sec:ml-intro}
In this section, we present the syntax, semantics, and proof
system of matching logic.
Unlike in previous work \cite{ChenRosu19Mml,ChenRosu20Binders,ChenLucanuRosuInitAlgebraTR},
here we introduce a
\emph{locally nameless} presentation of matching logic.
This new presentation is more convenient to be formalized in proof assistants,
which is discussed in detail in \Cref{sec:formalization}.
\subsection{Matching Logic Locally Nameless Syntax}
\label{sec:matching-logic-syntax}
The syntax we present in this section is in literature known
as that of a \emph{locally nameless} one~\cite{ChargueraudLocallyNameless,Leroy07alocally,McBrideM04,Gordon93}.
A locally nameless syntax is a combination of the traditional
named representation and the entirely nameless de Bruijn encoding,
using named variables if they occur free and de Bruijn encoding
if they are bound.
In particular, it distinguishes free and bound variables on the syntactic level,
enabling capture-avoiding substitutions without variable renaming.
This design decision eliminates the need for reasoning about $\alpha$-equivalence.
Firstly, we define matching logic \emph{signatures}.
A signature provides us infinitely many named variables and a set of
(user-defined) constant symbols.
\begin{definition}[Signatures]\label{def:ml-signature}
A \emph{matching logic signature} is a tuple
$(\mathsf{EVar},\mathsf{SVar},\Sigma)$ where
\begin{itemize}
\item $\mathsf{EVar}$ is a set of \emph{element variables},
denoted $x$, $y$, \dots;
\item $\mathsf{SVar}$ is a set of \emph{set variables},
denoted $X$, $Y$, \dots;
\item $\Sigma$ is a set of \emph{(constant) symbols}, denoted
$\sigma$, $f$, $g$, \dots
\end{itemize}
Both $\mathsf{EVar}$ and $\mathsf{SVar}$ are countably infinite sets.
$\Sigma$ is a countable set, possibly empty.
When the sets of variables are understood from the context,
we feel free to use $\Sigma$ to denote a matching logic signature.
\end{definition}
Given a signature $\Sigma$, the syntax of matching logic
generates a set of well-formed formulas, called \emph{patterns}.
In the following, we first define pseudo-patterns.
\begin{definition}[Locally Nameless Representation of Pseudo-Patterns]\label{def:ml-pseudopattern}
Given a signature $\Sigma$, the set of \emph{pseudo-patterns}
is inductively defined by the following grammar:
$$
\varphi \Coloneqq
x \mid X \mid \uline{n} \mid \se{N} \mid \sigma \mid
\varphi_1 \, \varphi_2 \mid \bot \mid \varphi_1 \to \varphi_2
\mid \exists \,.\, \varphi \mid \mu \,.\, \varphi
$$
\noindent
In the above grammar,
$x$ and $X$ are element and set variables, respectively;
$\uline{n}$ and $\se{N}$ are de Bruijn indices that represent
the bound element and set variables, respectively,
where $n,N \in \mathbb{N}$;
$\sigma$ is any symbol in the given signature $\Sigma$;
$\varphi_1 \, \varphi_2$ is called \emph{application},
where $\varphi_1$ is applied to $\varphi_2$;
$\bot$ and $\varphi_1 \to \varphi_2$ are propositional operations;
$\exists \,.\, \varphi$ is the FOL (first-order logic)-style quantification;
and $\mu \,.\, \varphi$ is the least fixpoint pattern.
Both $\exists$ and $\mu$
use the nameless de Bruijn encoding for their bound variables.
\end{definition}
\begin{definition}[Locally Nameless Representation of Patterns]\label{def:ml-pattern}
We say that a pseudo-pattern $\psi$ is \emph{well-formed},
if (1) for any subpattern $\mu \,.\, \varphi$, the nameless variable bound by $\mu$ has no negative occurrences in $\varphi$; and
(2) all of its nameless variables (i.e., de Bruijn indices)
are correctly bound by the quantifiers $\exists$ and $\mu$, that is,
for any de Bruijn indices $n$ and $N$ that occur in $\psi$,
\begin{itemize}
\item $\uline{n}$ is in the scope of at least $n+1$ $\exists$-quantifiers;
\item $\se{N}$ is in the scope of at least $N+1$ $\mu$-quantifiers.
\end{itemize}
A well-formed pseudo-pattern is called a \emph{pattern}. For example, the definition of transitive closure in \Cref{sec:example_eq} is a pattern, but $\exists \,.\, \uline{0} \to \uline{1}$ is not, since $\uline{1}$ is not bound by any quantifier.
\end{definition}
\paragraph{Opening quantified patterns.}
In a locally nameless representation, we use both
named and unnamed variables.
Named variables are for free variables,
while unnamed variables are for bound variables.
Therefore, when we have a pattern $\exists \,.\, \varphi$
(similarly for $\mu \,.\, \varphi$) and want to extract its body,
we need to assign a fresh named variable to the unnamed variable
that corresponds to the topmost quantifier $\exists$ (resp. $\mu$).
This operation is called \emph{opening} a (quantified) pattern~\cite{ChargueraudLocallyNameless}.
We use $\mathsf{open}e(\varphi,x)$ and $\mathsf{open}s(\varphi,X)$
to denote the opening of the bodies of the quantified patterns
$\exists \,.\, \varphi$ and $\mu \,.\, \varphi$, respectively,
where $x$ and $X$ are the corresponding new named variables.
These opening operations are special instances of the general case of substitution,
which is defined in the usual way\footnote{We denote substitution with $\phi[\psi/x]$, where $x$ (a \textit{bound} or \textit{free}, \textit{set} or \textit{element} variable) is replaced by $\psi$ in $\phi$.}.
\subsection{Matching Logic Models and Semantics}
\label{sec:matching-logic-semantics}
In this section, we formally define the models and semantics of matching logic.
Intuitively speaking, matching logic has a \emph{pattern matching semantics}.
A matching logic pattern is interpreted as the \emph{set}
of elements that \emph{match} it.
Firstly, we define the notion of matching logic models.
\begin{definition}[Matching Logic Models]\label{def:ml-model}
Let $\Sigma$ be a signature.
A \emph{$\Sigma$-model}, or simply a \emph{model},
is a tuple $(M, @_M, \{\sigma_M\}_{\sigma \in \Sigma})$
where:
\begin{itemize}
\item $M$ is a nonempty carrier set;
\item $@_M \colon M \times M \to \mathcal{P}(M)$ is a binary application function,
where $\mathcal{P}(M)$ denotes the powerset of $M$;
\item $\sigma_M \subseteq M$ is the interpretation of $\sigma$,
for each $\sigma \in \Sigma$.
\end{itemize}
We use the same letter $M$ to denote the model defined as above.
\end{definition}
Next, we define valuations of variables.
Note that matching logic has both element and set variables.
As expected, a valuation assigns element variables to elements
and set variables to subsets of the underlying carrier set.
Formally,
\begin{definition}[Variable Valuations]\label{def:ml-valuation}
Given a signature $\Sigma$ and a $\Sigma$-model $M$,
a variable valuation $\rho$ is a mapping such that:
\begin{itemize}
\item $\rho(x) \in M$ for all $x \in \mathsf{EVar}$;
\item $\rho(X) \subseteq M$ for all $X \in \mathsf{SVar}$.
\end{itemize}
\end{definition}
We are now ready to define the semantics of matching
logic patterns.
\begin{definition}[Matching Logic Semantics]\label{def:ml-semantics}
Given a matching logic model $M$ and a variable valuation $\rho$,
we define the semantics of any (well-formed) pattern $\psi$,
written $\ev{\psi}{M,\rho}$, inductively as follows:
\vspace*{-0.4cm}
\begin{table}[h]
\renewcommand{1}{1.5}
\begin{tabular}{c c}
$\ev{x}{M,\rho} = \{ \rho(x) \}$; &
$\ev{X}{M,\rho} = \rho(X)$; \\
$\ev{\sigma}{M,\rho} = \sigma_M$; &
$\ev{\bot}{M,\rho} = \emptyset$; \\
$\ev{\varphi_1 \to \varphi_2}{M,\rho} = M \setminus ( \ev{\varphi_1}{M,\rho} \setminus \ev{\varphi_2}{M,\rho})$; &
\hspace{1cm}
$\ev{\varphi_1 \, \varphi_2}{M,\rho} = \bigcup_{a_1 \in \ev{\varphi_1}{M,\rho}}\bigcup_{a_2 \in \ev{\varphi_2}{M,\rho}}
a_1 \mathbin{@_M} a_2 $; \\
\multicolumn{2}{c}{$\ev{\exists \,.\, \varphi}{M,\rho}
= \bigcup_{a \in M} \ev{\mathsf{open}e(\varphi,x)}{M,\rho[a/x]}$ for fresh $x \in \mathsf{EVar}$;} \\
\multicolumn{2}{c}{$\ev{\mu \,.\, \varphi}{M,\rho} = \mathbf{lfp} \ \mathcal{F}^\rho_{\varphi,X}$,
where $\mathcal{F}^\rho_{\varphi,X}(A) = \ev{\mathsf{open}s(\varphi,X)}{M,\rho[A/X]}$
for fresh $X \in \mathsf{SVar}$.}
\end{tabular}
\end{table}
\noindent The above definition is well-defined ($\mathsf{open}e$ and $\mathsf{open}s$ are defined at the end of
\Cref{sec:matching-logic-syntax}). For examples, we refer to the formalization~\cite{ml-formalization} and to~\cite[Section~4]{ml-explained}.
\end{definition}
In the following, we define validity and the semantic entailment relation
in matching logic.
\begin{definition}\label{def:ml-satisfaction}
For $M$ and $\varphi$, we write $M \models \varphi$ iff
$\ev{\varphi}{M,\rho} = M$ for all valuations $\rho$.
For a pattern set $\Gamma$, called a \emph{theory}, we write $M \models \Gamma$ iff
$M \models \varphi$ for all $\varphi \in \Gamma$.
We write $\Gamma \models \varphi$ iff for any $M$,
$M \models \Gamma$ implies $M \models \varphi$.
\end{definition}
\subsection{Matching Logic Proof System}
In this section, we present the proof system of matching logic.
Matching logic has a Hilbert-style proof system with
19 simple proof rules, making it small and easy to implement.
The proof system defines the \emph{provability relation}
written as $\Gamma \vdash \varphi$, where
$\Gamma$ is a set of patterns
(often called a \emph{theory} and the patterns are called \emph{axioms})
and $\varphi$ is a pattern that is said to be \emph{provable}
from the axioms in $\Gamma$.
\begin{table*}[t]
\renewcommand{1}{1.35}
\caption{Matching Logic Proof System under Locally Nameless Representation\\
($C_1,C_2$ are application contexts, $FV(\varphi)$ denotes the set of free variables in $\varphi$)}\label{tab:ps}
\begin{tabular}{llll}
\hline
\emph{\textbf{Proof Rule Names}} & \emph{\textbf{Proof Rules}} &
\emph{\textbf{Proof Rule Names}} & \emph{\textbf{Proof Rules}}
\\
\hline
\prule{Proposition 1} & $\varphi_1 \to (\varphi_2 \to \varphi_1)$ &
\prule{Proposition 2} &
\renewcommand{1}{1}
\begin{tabular}{@{}l@{}}
$(\varphi_1 \to (\varphi_2 \to \varphi_3)) \to$\\\kern1em$(\varphi_1 \to \varphi_2)\to (\varphi_1 \to \varphi_3)$
\end{tabular}\\[15pt]
\prule{Proposition 3} & $((\varphi \to \bot) \to \bot) \to \varphi$ &
\prule{Modus Ponens} &
{
\begin{prooftree}
\hypo{\varphi_1}
\hypo{\varphi_1 \to \varphi_2}
\infer2{\varphi_2}
\end{prooftree}
} \\
\prule{$\exists$-Quantifier} &
$\mathsf{open}e(\varphi,x) \to \exists \,.\, \varphi$ with $x \in \mathsf{EVar}$ \kern-8em\\[5pt]
\prule{$\exists$-Generalization} &
{
\begin{prooftree}
\hypo{\mathsf{open}e(\varphi_1,x) \to \varphi_2}
\infer1[with $x \not\in \FV{\varphi_2}$]{(\exists \,.\, \varphi_1) \to \varphi_2}
\end{prooftree}
} \kern-8em \\[10pt]
\hline
\prule{Propagation Left$_\bot$} & $\bot \, \varphi \to \bot$
& \prule{Propagation Right$_\bot$} & $\varphi \, \bot \to \bot$ \\
\prule{Propagation Left$_\vee$} & $(\varphi_1 \vee \varphi_2) \, \varphi_3 \to (\varphi_1 \, \varphi_3) \vee (\varphi_2 \, \varphi_3)$ \kern-10em \\
\prule{Propagation Right$_\vee$} & $\varphi_1 \, (\varphi_2 \lor \varphi_3) \to (\varphi_1 \, \varphi_2) \vee (\varphi_1 \, \varphi_3)$ \kern-10em \\
\prule{Propagation Left$_\exists$}
& $ (\exists \,.\, \varphi_1) \, \varphi_2 \to \exists \,.\, \varphi_1 \, \varphi_2$ &
\prule{Propagation Right$_\exists$}
& $ \varphi_1 \, (\exists \,.\, \varphi_2) \to \exists \,.\, \varphi_1 \, \varphi_2$ \\[3pt]
\prule{Framing Left} &
{
\begin{prooftree}
\hypo{\varphi_1 \to \varphi_2}
\infer1{\varphi_1 \, \varphi_3 \to \varphi_2 \, \varphi_3}
\end{prooftree}
}
&
\prule{Framing Right} &
{
\begin{prooftree}
\hypo{\varphi_2 \to \varphi_3}
\infer1{\varphi_1 \, \varphi_2 \to \varphi_1 \, \varphi_3}
\end{prooftree}
}
\\[8pt]
\hline
\prule{Substitution} &
{
\begin{prooftree}
\hypo{\varphi}
\infer1{\varphi[\psi/X]}
\end{prooftree}
}
& \prule{Pre-Fixpoint} &\kern-1em
$\varphi[(\mu \,.\, \varphi) / \se{0}]\to \mu \,.\, \varphi$ \\[10pt]
\prule{Knaster-Tarski} &
{
\begin{prooftree}[rule margin=0pt]
\hypo{\varphi_1[\varphi_2/\se{0}] \to \varphi_2}
\infer1{(\mu \,.\, \varphi_1) \to \varphi_2}
\end{prooftree}
}
\\[8pt]
\hline
\prule{Existence} & $\exists \,.\, \uline{0}$ &
\prule{Singleton} &
\kern-2em$\neg(C_1[x \land \varphi] \wedge C_2[x \land \neg \varphi])$
\\\hline
\end{tabular}
\renewcommand{1}{1}
\end{table*}
We present the proof system of matching logic
in Table~\ref{tab:ps}.
To understand it, we first need to define the notion of
\emph{contexts} and a particular type of contexts called
\emph{application contexts}.
\begin{definition}
A \emph{context} $C$ is simply a pattern with one unique placeholder denoted $\square$.
We write $C[\varphi]$ to mean the result of plugging $\varphi$ in $\square$ in the context $C$.
We call $C$ an \emph{application context} if from the root of $C$ to $\square$
there are only applications; that is, $C$ is (inductively) constructed as follows:
\begin{itemize}
\item $C$ is $\square$ itself, called the identity context; or
\item $C\equiv C_1 \, \varphi$, where $C_1$ is an application context; or
\item $C\equiv\varphi \, C_2$, where $C_2$ is an application context.
\end{itemize}
\end{definition}
\noindent
The proof rules in Table~\ref{tab:ps} can be divided into four categories:
\begin{itemize}
\item \textbf{FOL reasoning} containing the standard proof rules as in FOL;
\item \textbf{Frame reasoning} consisting of six propagation rules and two framing rules, which allow us to propagate formal reasoning that is carried out within an application context throughout the context. Note that we proved these rules equivalent to the ones in~\cite{ChenRosu19Mml}, where application contexts are not splitted into applications to the left and right;
\item \textbf{Fixpoint reasoning} containing the fixpoint rules as in modal $\mu$-calculus;
\item \textbf{Miscellaneous rules}.
\end{itemize}
Next, we state the soundness theorem of the proof system, which has been proved by induction over the structure of the proof $\Gamma \vdash \varphi$ in~\cite{ChenRosu19Mml}. We elaborate on the mechanization of this proof in \Cref{sec:soundness}.
\begin{theorem}[Soundness Theorem]
$\Gamma \vdash \varphi$ implies $\Gamma \models \varphi$.
\end{theorem}
\subsection{Example Matching Logic Theories}
\paragraph{First-order logic.}
It is easy to see that matching logic is at least as expressive as classical first-order logic, and Chen et al.~\cite{ml-explained} show that it is indeed more expressive than FOL. At the same time, they describe a direct and natural connection between FOL and matching logic.
A FOL term $t$ is interpreted as an element in the underlying carrier set.
From the matching logic point of view, $t$ is a pattern
that is matched by \emph{exactly one element}.
We use the terminology \emph{functional patterns} to refer to
patterns whose valuations are singleton sets.
Intuitively, FOL terms \emph{are} functional patterns in matching logic.
FOL formulas are interpreted as two logical
values: true and false.
In matching logic, there is a simple analogy
where we use the empty set $\emptyset$ to represent the logical false
and the total carrier set to represent the logical true.
A pattern whose interpretation is always $\emptyset$ or the full set
is called a \emph{predicate pattern}.
Intuitively, FOL formulas \emph{are} predicate patterns in matching logic.
\paragraph{Equality.}
\label{sec:example_eq}
Although matching logic has no built-in notion of equality, it can be easily defined using a construct called \emph{definedness}.
Formally, we define $\Sigma^\mathsf{DEF}=\{ \defined{\_} \}$, $\Gamma^\mathsf{DEF}=\{ \defined{x} \}$ ($\defined{x}$ denotes $\defined{\_}\, x$);
this axiom ensures that whenever a pattern $\varphi$ is matched by some model element, the pattern $\defined{\varphi}$ is matched by all elements of the model, and vice versa.
Equality is then defined as a notation $\varphi_1 = \varphi_2 \equiv \neg \defined{\neg (\varphi_1 \leftrightarrow \varphi_2 )}$,
intuitively saying that there is no element that would match only one of the two formulas.
It is easy to see that for any pattern $\varphi$, the pattern $\defined{\varphi}$ is a predicate pattern, and that equality of two patterns is a predicate.
With this, we can similarly define other notations, such as membership, subset, totality, etc. seen below.
To make things easier, we use the notion of matching logic \emph{specification} introduced in~\cite{ml-explained}.
The signature $\Sigma^\mathsf{DEF}$ and theory $\Gamma^\mathsf{DEF}$ are then defined by Spec.~\ref{spec:definedness}. For more details on definedness, we refer to \Cref{Sec:formequality}.
\begin{specification}[th]
\begin{tcolorbox}
\begin{lstlisting}[mathescape=true,language=AML]
spec $\mathsf{DEF}$
${\textsf{Symbol}}$ $\defined{\_}$
${\textsf{Notation}}$
$\arraycolsep=1.0pt\begin{array}{rclrcl}
\defined{\varphi} &\equiv& \defined{\_} \, \varphi&
\quad \total{\varphi} &\equiv& \neg \defined{ \neg \varphi} \\
\varphi_1 = \varphi_2 &\equiv& \total{\varphi_1 \leftrightarrow
\varphi_2} &
\quad \varphi_1 \neq \varphi_2 &\equiv& \neg (\varphi_1 = \varphi_2) \\
x \in \varphi &\equiv& \defined{x \wedge \varphi}&
\quad \varphi_1 \subseteq \varphi_2 &\equiv& \total{\varphi_1 \rightarrow
\varphi_2} \\
x \not\in \varphi &\equiv& \neg(x \in \varphi)&
\quad \varphi_1 \not\subseteq \varphi_2 &\equiv& \neg(\varphi_1
\subseteq \varphi_2)
\end{array}$
${\textsf{Axiom}}$
$\axname{Definedness}$ $\defined{x}$
endspec
\end{lstlisting}
\end{tcolorbox}
\caption{Definedness and related notions}
\label{spec:definedness}
\end{specification}
\paragraph{Induction and transitive closure.}
Chen et al. show how matching logic, by using the application and least fixpoint operators, can not only axiomatize equality, but also product types ($\langle \_ , \_ \rangle$), and inductive types~\cite{ml-explained}.
Hence, another notable example of matching logic's expressiveness is that it can specify the transitive closure of a binary relation $R$ the following way (where $\in$ is defined by Spec.~\ref{spec:definedness}):
$$
\mu X \,.\, R \lor \exists x \,.\, \exists y \,.\, \exists z \,.\, \langle x,z \rangle \land \langle x, y \rangle \in X \land \langle y, z \rangle \in X
$$
\noindent
Or rather, the same matching logic pattern expressed in the locally nameless representation:
$$
\mu \,.\, R \lor \exists \,.\, \exists \,.\, \exists \,.\, \langle \uline{2},\uline{0} \rangle \land \langle \uline{2}, \uline{1} \rangle \in \se{0} \land \langle \uline{1}, \uline{0} \rangle \in \se{0}
$$
\begin{comment}
\subsection{Example: Theory of Sorts}
As in the case of equality, matching logic has no built-in support for sorts, but they can be axiomatized as in~\cite{ml-explained}.
One defines a symbol $\mathit{inh}$, a notation $\brac{s} \equiv \mathit{inh}\ s$, and calls the pattern $\brac{s}$
the \emph{inhabitant set} of $s$ (for any pattern $s$ that represents a sort).
Spec.~\ref{spec:naivesorts} then shows a straightforward way of encoding sorted quantification,
denoted as $\forall\left( s\right).\ \varphi$ (where $s$ represents a sort).
\begin{specification}[t]
\begin{tcolorbox}
\begin{lstlisting}[mathescape=true,language=AML]
spec $\mathsf{NAIVESORTS}$
${\textsf{Import}}$ $\mathsf{DEF}$
${\textsf{Symbol}}$ $\mathit{inh}$
${\textsf{Notation}}$
$\arraycolsep=1.0pt\begin{array}{rclrcl}
\brac{s} &\equiv& \mathit{inh} \ s &
\quad \forall\left( s\right).\ \varphi &\equiv& \forall.\ \uline{0} \in \brac{s} \rightarrow \varphi \\
\neg_s \varphi &\equiv& (\neg \varphi) \land \brac{s} &
\quad \exists\left( s\right).\ \varphi &\equiv& \exists.\ \uline{0} \in \brac{s} \land \varphi
\end{array}$
endspec
\end{lstlisting}
\vspace*{-2ex}
\end{tcolorbox}
\caption{A naive specification of sorts}
\label{spec:naivesorts}
\end{specification}
That mostly works for many use-cases.
However, consider the mathematical (nominal-style) formula
$\forall x : s,\, P(x)$.
In this case we think of $s$ as an argument of the quantifier; $x$ is not yet bound in $s$.
We would want the matching logic pattern $\forall\left( s\right).\ \varphi$ to behave the same.
Why?
For example, we may have a dependent sort $\mathit{BV}\ n$ of \emph{bitvectors} of lenght $n$,
and want to express that the symbol $\mathit{BV\_inc}$ is a function on this sort.
A natural way to express this in the locally nameless matching logic is
\begin{equation}\label{eqn:BVincfunc}
\forall (\mathit{nat}),\ \forall( \mathit{BV}\ \uline{0}),\ \exists( \mathit{BV}\ \uline{1}),\ \mathit{BV\_inc}\ \uline{1} = \uline{0} \, .
\end{equation}
But when using the notations from $\mathsf{NAIVESORTS}$, the above intuition breaks, because the sorts that are passed to the notations contain dangling bound variables.
The problem becomes visible when we expand the notations.
The expansion of the above formula looks like this:
\begin{align*}
&\forall.\ \uline{0} \in \brac{\mathit{nat}} \rightarrow
\forall.\ \uline{0} \in \brac{\mathit{BV}\ \uline{\bm{0}}} \rightarrow \exists.\ \uline{0} \in \brac{\mathit{BV}\ \uline{\bm{1}}} \land \mathit{BV\_inc}\ \uline{1} = \uline{0} \, .
\end{align*}
while we wanted to say rather something like
\begin{align*}
&\forall.\ \uline{0} \in \brac{\mathit{nat}} \rightarrow
\forall.\ \uline{0} \in \brac{\mathit{BV}\ \uline{\bm{1}}} \rightarrow \exists.\ \uline{0} \in \brac{\mathit{BV}\ \uline{\bm{2}}} \land \mathit{BV\_inc}\ \uline{1} = \uline{0} \, .
\end{align*}
(notice the bold de Bruijn indices).
There are multiple possible solutions to this problem: for example, to allow only well-formed patterns as sorts, or to require user to use higher indices in quantifier parameters.
However, the first limits the usefullness of $\mathsf{NAIVESORTS}$, while the second is unnatural and brittle.
We avoid this misunderstanding by defining a pattern-transforming function $\mathit{nest}_\exists$ (and similarly $\mathit{nest}_\mu$)
that increments dangling element (or set, respectively) de Bruijn indices, and
notations for sorted quantification as
in Spec.~\ref{spec:sorts}.
\begin{specification}[t]
\begin{tcolorbox}
\begin{lstlisting}[mathescape=true,language=AML]
spec $\mathsf{SORTS}$
${\textsf{Import}}$ $\mathsf{DEF}$
${\textsf{Symbol}}$ $\mathit{inh}$
${\textsf{Notation}}$
$\arraycolsep=1.0pt\begin{array}{rclrcl}
\brac{s} &\equiv& \mathit{inh} \ s &
\quad \forall\left( s\right).\ \varphi &\equiv& \forall.\ \uline{0} \in \brac{\mathit{nest}_\exists(s)} \rightarrow \varphi \\
\neg_s \varphi &\equiv& (\neg \varphi) \land \brac{s} &
\quad \exists\left( s\right).\ \varphi &\equiv& \exists.\ \uline{0} \in \brac{\mathit{nest}_\exists(s)} \land \varphi
\end{array}$
endspec
\end{lstlisting}
\vspace*{-2ex}
\end{tcolorbox}
\caption{A specification of sorts}
\label{spec:sorts}
\end{specification}
Intuitively, $\mathsf{open}e(\varphi,x)$ (occuring in the semantics of quantifiers) and $\mathit{nest}_\exists$
cancels each other out.
Formally, we proved a lemma
\begin{lemma}
For any pattern $\varphi$, any element variable $x$ fresh in $\varphi$,
\begin{equation*}
\mathsf{open}e(\mathit{nest}_\exists(\varphi),x) = \varphi \, .
\end{equation*}
\end{lemma}
ensuring that a pattern nested inside a quantifier behaves as if it were not nested inside the quantifier,
but as if it were one quantifier level above its occurrence.
That way, we can use the pattern from~\cref{eqn:BVincfunc} to specify the desired property.
\end{comment}
\section{Matching Logic Formalization in Coq}\label{sec:formalization}
In this section, we describe how the locally nameless matching logic (as defined in \Cref{sec:ml-intro}) has been encoded in Coq.
The formalization is distributed as a library including the definition of the
logic as well as that of some standard theories, with the two dependencies being the Std++ library~\cite{stdpp} and the Equations plugin~\cite{coq-equations}.
Some of our proofs rely on classical and extensionality axioms (namely, functional extensionality, propositional extensionality,
and the axiom of excluded middle), but these are known to be compatible with Coq's logic~\cite{carneiro_master}.
We implement matching logic in a \emph{deep embedding} style; that is, formulas, models, and proofs are represented as data in Coq.
Presumably, a shallow embedding could provide a more lightweight implementation, but deep embedding has couple of advantages for our use cases; mainly, it allows us to inspect matching logic proofs without reflection, reason about them directly (for instance, when checking the side conditions of the \emph{deduction theorem} from~\cite{chen-rosu-2019-tr-mml}) and to extract them from Coq (see \Cref{sec:conclusion})
\footnote{It is beyond the scope of this work to tell if a shallow embedding would be more suitable for object-level reasoning.}
.
\subsection{Syntax}
We represent a matching logic signature $(\mathsf{EVar},\mathsf{SVar},\Sigma)$,
defined in \Cref{def:ml-signature}, as an instance of the \coqinline{Class Signature} that encapsulates the sets of variables and the set of symbols. The sets for variables are required to be countable and infinite, and in addition, it is also required that equality on variables and symbols is decidable.
We also provide an off-the-shelf, string-based instance of the type class as a default option.
Although this instance will suffice for most applications, to prove the completeness of the matching logic proof system, a more general carrier set will be needed\footnote{We refer to the proof of the completeness of matching logic without $\mu$ and the extension lemma~\cite{chen-rosu-2019-tr-mml}.}.
The way we represent free variable names has some features in common with the concept of \emph{atoms} used in \emph{nominal} approaches~\cite{nominalcoq} (e.g., any countably infinite set with decidable equality can be used for names), but in the locally nameless approach we do not rely on permutative renaming when implementing capture-avoidance.
We formalize (pseudo-)patterns (Definition~\ref{def:ml-pseudopattern}) as the inductive definition
\coqinline{Inductive Pattern : Set}.
Due to using a locally nameless embedding,
we have separate constructors for free variables (named
variables) and bound variables (de Bruijn indices).
The main advantage this provides is in the equivalence
of quantified patterns. Specifically, in a fully named representation,
the patterns $\exists x \,.\, x$ and $\exists y \,.\, y$ are equivalent,
yet not syntactically equal.
To formally prove their equivalence, we would need a notion of
$\alpha$-equivalence of patterns that is constantly applied.
However, in our locally nameless representation, both these can only
be represented by the pattern $\exists \,.\, \uline{0}$, and thus are
syntactically equal, so no additional notions of equality need to be
supplied.
This is implicitly used throughout our soundness proof and in proving
properties of specifications.
\vspace*{-2ex}
\paragraph{Well-formedness.}\label{sec:wf}
For practical reasons, the type \coqinline{Pattern} corresponds to the definition
of pseudo-patterns, while the restrictions on non-negativity and scoping defined for patterns (\Cref{def:ml-pattern}) are
implemented as a pair of \coqinline{bool}-valued auxiliary functions:
\begin{coqcode}
well_formed_positive : Pattern -> bool
well_formed_closed : Pattern -> bool
\end{coqcode}
\noindent where the first function performs a check for the positivity constraint and the second one checks the scoping requirements. The predicate \coqinline{well_formed_closed} is constructed from two parts:
\begin{coqcode}
well_formed_closed_ex_aux : Pattern -> nat -> bool
well_formed_closed_mu_aux : Pattern -> nat -> bool
\end{coqcode}
\noindent
These predicates return \coqinline{true}, when the parameter pattern contains only smaller unbound de Bruijn indices for element and set variables than the given number.
A pattern is \coqinline{well_formed} if it satisfies both \coqinline{well_formed_closed} and \coqinline{well_formed_positive}.
Most of our functions operate on the type \lstinline{Pattern} without the well-formedness constraint;
we use the constraint mainly for theorems.
This way we separate proofs from data.
\paragraph{Substitution and opening.}
In the locally nameless representation, there are separate substitution functions for bound and free variables (both for element and set variables). In our formalization, we followed the footsteps of Leroy~\cite{Leroy07alocally}, so the bound variable substitution decrements the indices of those (bound) variables that are greater than the substituted index.
We define
\begin{coqcode}
Definition evar_open (k : db_index) (x : evar) (p : Pattern) : Pattern.
Definition svar_open (k : db_index) (X : svar) (p : Pattern) : Pattern.
\end{coqcode}
\noindent
which correspond to $\mathsf{open}e(\varphi,x)$ and
$\mathsf{open}s(\varphi,X)$ from Section~\ref{sec:matching-logic-syntax}, with a difference that this version of opening allows for
substitution of any de Bruijn index, not only the one corresponding
to the topmost quantifier (\uline{0} or \se{0}).
\paragraph{Derived notations.}
Matching logic is intentionally minimal.
As a consequence, any non-trivial theory is likely to heavily rely on notations that abbreviate common operations.
Besides basic notations for boolean operations, universal quantification and greatest fixpoint,
one also can define equality, subset and membership relations on top of the definedness symbol (as we did in Spec.~\ref{spec:definedness}),
which is not part of matching logic, but is usually considered as a part of
the ``standard library'' for the core logic.
Coq provides (at least) two ways to extend the syntax of the core logic with derived notations.
The first option is to use Coq's \coqinline{Notation} mechanism. For example, the following would define the notations for negation, disjunction, and conjunction:
\begin{coqcode}
Notation "! p" := (p ---> ?$\bot$?).
Notation "p or q" := (! p ---> q).
Notation "p and q" := ! (! p or ! q).
\end{coqcode}
The problem with this is that the pattern
\begin{coqcode}
(x ---> ?$\bot$?) ---> ?$\bot$?
\end{coqcode}
\noindent could be interpreted either as \coqinline{! (! x)}, or as \coqinline{x or ?$\bot$?},
which would be confusing to the user, having no control on which interpretation Coq chooses to display.
Therefore, we decided to opt for the other option, representing each derived notation as a Coq \coqinline{Definition}, as in the following snippet:
\begin{coqcode}
Definition patt_not p := p ---> ?$\bot$?.
Definition patt_or p q := patt_not p ---> q.
Definition patt_and p q := patt_not (patt_or (patt_not p) (patt_not q))
\end{coqcode}
We can define the notations on top of these definitions. This way, the user can fold/unfold derived notations as needed.
However, this representation of notations poses another problem: many functions, especially the substitutions
such as \coqinline{bevar_subst}, preserve the structure of the given formula, but since they build a \coqinline{Pattern}
from the low-level primitives, the information about derived notations is lost whenever such function is called.
We solve this by defining for each kind of syntactical construct (e.g., for unary operation, binary operation, element variable binder)
a type class containing a rewriting lemma for \coqinline{bevar_subst} such as this one:
\begin{coqcode}
Class Binary (binary : Pattern -> Pattern -> Pattern) :=
{
binary_bevar_subst :
forall ?$\psi$?, well_formed_closed ?$\psi$? ->
forall n ?$\phi_1$? ?$\phi_2$?,
bevar_subst (binary ?$\phi_1$? ?$\phi_2$?) ?$\psi$? n
= binary (bevar_subst ?$\phi_1$? ?$\psi$? n) (bevar_subst ?$\phi_2$? ?$\psi$? n) ;
(* ... *)
}.
\end{coqcode}
\noindent The user of our library then can instantiate the class for their derived operations
and use the tactic \coqinline{simpl_bevar_subst} to simplify the expressions containing \coqinline{bevar_subst} and \coqinline{evar_open}.
\paragraph{Fresh variables.}
We say that a variable is fresh in a pattern $\varphi$ if it does not occur among the free variables of $\varphi$.
Sometimes (for example, in the semantics of existential and fixpoint patterns) it is necessary
to find a variable that does not occur in the given pattern.
We required the variable types (\coqinline{evar} and \coqinline{svar}) to be infinite, thus we can use the solution of the Coq Std++ library~\cite{stdpp} for fresh variable generation.
We then have a function \coqinline{fresh_evar : Pattern -> evar} and the following lemma:
\begin{coqcode}
Lemma set_evar_fresh_is_fresh ?$\varphi$? : fresh_evar ?$\varphi$? ?$\notin$? free_evars ?$\varphi$?.
\end{coqcode}
\subsection{Semantics}\label{sec:semantics}
On the semantics side
we have a \coqinline{Record Model}.
We do not require the carrier set of the model to have decidable equality.
We represent the variable valuation function defined in \Cref{def:ml-valuation} as a record of two separate functions, one mapping element variables to domain elements
and another mapping set variables to sets of domain elements.
With this, we define the interpretation of patterns ($\ev{\_}{M,\rho}$ in
\Cref{def:ml-semantics}) as expected, with two points worth noting:
\begin{enumerate}
\item The interpretation of patterns cannot be defined using structural recursion on the formula,
because in the $\mu$ (and $\exists$) case,
one calls \coqinline{eval} on \coqinline{svar_open 0 X p'} (and \coqinline{evar_open 0 x p'}), which is not a structural subformula
of \coqinline{p}. Therefore, we do recursion over the size of the formula,
implemented as an \coqinline{Equation}:
\begin{coqcode}
Equations eval (?$\rho$? : @Valuation M) (?$\varphi$? : Pattern) : propset (@Domain M) by wf (size ?$\varphi$?) :=
(* ... *)
eval ?$\rho$? (patt_mu ?$\varphi'$?) := let X := fresh_svar ?$\varphi'$? in
@LeastFixpointOf _ OS L (fun S =>
let ?$\rho'$? := (update_svar_val X S ?$\rho$?) in
eval ?$\rho'$? (svar_open 0 X ?$\varphi'$?)).
\end{coqcode}
\item We decided to give semantics to patterns that are not well-formed, including arbitrary $\mu$ patterns.
This way, we do not have to supply the \coqinline{eval} function
with the well-formedness constraint, which makes it easier to use.
We did that by
(1) defining the function \coqinline{LeastFixpointOf} to return
the intersection of all prefixpoints;
(2) mechanizing the relevant part of the Knaster-Tarski fixpoint theorem~\cite{tarski1955}, and
(3) proving that the function associated to a well-formed $\mu$ pattern is monotone.
\end{enumerate}
\subsection{Proof System}
We formalize the proof system of matching logic as an inductive definition:
\begin{coqcode}
Inductive ML_proof_system theory : Pattern -> Set := (* ... *) .
\end{coqcode}
The proof system is defined as expected; however, one may ask why the proof system lives in \coqinline{Set}
and not in \coqinline{Prop}.
The answer is that in our \emph{deep embedding} we care about the internal structure of matching logic proofs, not only about provability.
We do not want two matching logic proofs to be considered identical only because they prove the same formula,
for at least two reasons.
First, some theorems (e.g. the deduction theorem from~\cite{chen-rosu-2019-tr-mml}) can only be applied to a proof if that proof satisfies particular
conditions regarding its internal structure.
Second, when extracting OCaml or Haskell programs from this Coq formalization,
we need the manipulation of the proof system terms to be preserved;
that will allow us to extract Metamath proof objects in the future (see \Cref{sec:conclusion}).
Another point worth mentioning is that the rules in the formalization of the proof system
often require some well-formedness constraints.
A consequence of this is that only well-formed patterns are provable:
\begin{coqcode}
Lemma proved_impl_wf ?$\Gamma$? ?$\phi$?: ?$\Gamma$? ?$\vdash$? ?$\phi$? -> well_formed ?$\phi$?.
\end{coqcode}
We discuss the soundness of the proof system in
Section~\ref{sec:soundness}.
Crucially, we rely on the $\mu$ patterns to be interpreted as
least fixpoints, as explained in Section~\ref{sec:semantics}.
\section{Reasoning about Matching Logic}\label{sec:soundness}
After encoding matching logic in Coq, we overview some results it allows for concerning meta-level reasoning. In particular, we highlight some challenges we faced when proving the mechanized matching logic proof system sound, and we demonstrate semantics-based reasoning about the theory of equality.
\subsection{Soundness of The Proof System}
The most crucial result of the mechanization of matching logic is the proof of the soundness of its proof system (\Cref{tab:ps}). Even though this theorem has already been investigated in related publications, we have developed the first complete, machine-checked proof, which verifies the prior paper-based results.
We state our soundness theorem below:
\begin{coqcode}
Theorem Soundness : forall phi : Pattern, forall theory : Theory,
well_formed phi -> theory ?$\vdash$? phi -> theory ?$\vDash$? phi.
\end{coqcode}
The proof of soundness begins via induction on the hypothesis
\coqinline{theory ?$\vdash$? phi}, meaning we consider all cases from
the proof system which may have produced this hypothesis.
Many cases, such as the propositional proof rules, were straightforwardly discharged
using set reasoning. Other cases, like Modus Ponens, were discharged via
the application of the induction hypothesis.
The most involved cases were the proof rules involving quantification,
specifically the $\exists$-Quantifier, Prefixpoint, and
Knaster-Tarski rules.
For these rules, the key steps were to establish complex substitution lemmas (separate
lemmas for existential quantification and for $\mu$-quantification).
The proofs of these lemmas were very involved, and we note that the set substitution lemma was not proved in related work previously. For existential quantification, we adapt the
``element substitution lemma'' which appears
in~\cite[Lemma 41]{chen-rosu-2019-tr-mml}.
For the soundness of the Pre-fixpoint and Knaster-Tarski rules with $\mu$-quantification,
we introduce a new similar lemma called
\emph{set substitution lemma}, which
links syntactic substitution with semantic substitution,
stating that the
following two ways of plugging a pattern $\varphi_2$ into a pattern
$\varphi_1$ are equivalent:
\begin{enumerate}
\item syntactically substitute $\varphi_2$ for a free set variable $X$ in $\varphi_1$ and interpret the resulting pattern;
\item interpret $\varphi_2$ separately, then interpret $\varphi_1$ in a valuation where $X$ is mapped to the value of $\varphi_2$.
\end{enumerate}
\subsection{Theory of Equality}\label{Sec:formequality}
The formalization also allows for reasoning about matching logic models. We implemented the theory of definedness and equality as presented in~\cite{ml-explained,Rosu17}, and \Cref{sec:example_eq}. Then, we established some results about models that satisfy the definedness axiom, which provides support for common cases of semantic reasoning. This branch of the development showcases applications of our mechanization for reasoning about matching logic models.
\paragraph{Definedness and totality.}
Definedness has the important property that, applied to any formula $\varphi$ which matches at least one model element, the result matches all elements of the model (represented by $\top$):
\begin{coqcode}
Lemma definedness_not_empty_iff : forall (M : @Model ?$\Sigma$?),
M ?$\vDash^T$? theory ->
forall (?$\phi$? : Pattern) (?$\rho_e$? : @EVarVal ?$\Sigma$? M) (?$\rho_s$? : @SVarVal ?$\Sigma$? M),
(@eval ?$\Sigma$? M ?$\rho_e$? ?$\rho_s$? ?$\phi$?) <> ?$\emptyset$? <-> (@eval ?$\Sigma$? M ?$\rho_e$? ?$\rho_s$? ?$\lceil$? ?$\phi$? ?$\rceil$? ) = ?$\top$?.
\end{coqcode}
This is why it is called \emph{definedness}: $\lceil \phi \rceil $ evaluates to full set if and only if $\phi$ is \emph{defined};
that is, matched by at least one element.
Note that one needs the definedness axiom only for the ``if'' part;
the ``only if'' part is guaranteed by the definition of the extension of application:
anything applied to the empty set results in the empty set.
The dual of definedness is called \emph{totality}: a pattern $\phi$ is considered \emph{total} iff it is matched by all elements of the model, and totality of a pattern ($\lfloor \phi \rfloor$) \emph{holds} (is matched by all elements of the model) only in that case:
\begin{coqcode}
Lemma totality_not_full_iff : forall (M : @Model ?$\Sigma$?),
M ?$\vDash^T$? theory ->
forall (?$\phi$? : Pattern) (?$\rho_e$? : @EVarVal ?$\Sigma$? M) (?$\rho_s$? : @SVarVal ?$\Sigma$? M),
@eval ?$\Sigma$? M ?$\rho_e$? ?$\rho_s$? ?$\phi$? <> ?$\top$? <-> @eval ?$\Sigma$? M ?$\rho_e$? ?$\rho_s$? ?$\lfloor$? ?$\phi$? ?$\rfloor$? = ?$\emptyset$?.
\end{coqcode}
\paragraph{Equality.}
As we have seen in \Cref{sec:ml-intro}, equality is defined using totality.
We proved that equality defined this way indeed has the intended property, i.e., equality of two patterns holds iff the two patterns are interpreted to equal sets.
\begin{coqcode}
Lemma equal_iff_interpr_same : forall (M : @Model ?$\Sigma$?),
M ?$\vDash^T$? theory ->
forall (?$\phi1$? ?$\phi2$? : Pattern) (?$\rho_e$? : @EVarVal ?$\Sigma$? M) (?$\rho_s$? : @SVarVal ?$\Sigma$? M),
@eval ?$\Sigma$? M ?$\rho_e$? ?$\rho_s$? (?$\phi$?1 =ml ?$\phi$?2) = ?$\top$? <->
@eval ?$\Sigma$? M ?$\rho_e$? ?$\rho_s$? ?$\phi1$? = @eval ?$\Sigma$? M ?$\rho_e$? ?$\rho_s$? ?$\phi2$?.
\end{coqcode}
Our formalization also demonstrates that in matching logic, one cannot simply use equivalence ($\leftrightarrow$) instead of equality.
As argued in \cite{Rosu17}, one could expect that the pattern
$\exists \,.\, f\ x \leftrightarrow \uline{0}$ specifies that $f$ behaves like a function;
however, there exists a model in which that is not the case.
In the model whose domain is \coqinline{exampleDomain} and whose interpretation of
application is defined as \coqinline{example_app_interp} below:
\begin{coqcode}
Inductive exampleDomain : Set := one | two | f.
Definition example_app_interp (d1 d2 : exampleDomain) : Power exampleDomain :=
match d1, d2 with
| f, one => ?$\top$?
| f, two => ?$\emptyset$?
| _, _ => ?$\emptyset$?
end.
\end{coqcode}
\noindent the pattern
$\exists.\, f\ x \leftrightarrow \uline{0}$ holds (in every interpretation of $x$),
even though the model does not implement $f$ as a function. For more technical details, we refer to the formalization~\cite{ml-formalization}.
\section{Reasoning in Matching Logic}\label{sec:proofmode}
In~\Cref{sec:formalization}, we encoded matching logic and its proof system in Coq. With the minimal proof system, one can already reason about syntactic consequence by using Coq's \coqinline{apply} tactic.
However, only using the presented rules to reason about derived operations and complex theories is not really productive: it is easy to get lost when facing a complex proof obligation expressed in vanilla matching logic after unfolding the derived notations.
\paragraph{Derived rules.}
To support object-level reasoning, we proved several derived axioms and rules, essentially enriching the proof system with rules about derived constructs (commonly used operations that are not included in the syntax of matching logic) and common theories (such as definedness). These alone can shorten a typical matching logic proof script significantly. For example, think of destructing a disjunctive premise into two premises, which is naturally one step in an informal proof, but takes a couple of steps with the matching logic proof system. However, writing proofs with the derived rules is still cumbersome, because now we get lost in the details of combining and applying these theorems with the correct parameters. We tackle this problem with a dedicated Coq proof mode.
\subsection{Matching Logic Proof Mode}
To further simplify reasoning in the embedded logic, we conceptually separate the matching logic proof state from the Coq proof state, introduce a local proof context, and define a set of special Coq tactics that manipulate the dedicated proof state. We call these concepts together \emph{the matching logic proof mode}\footnote{We borrow the term \emph{proof mode} and the approach from the authors of the Iris proof mode~\cite{IrisProofMode}, and the Coq reference manual~\cite{coq-manual}.}. The ultimate goal with the proof mode is to make matching logic proofs simple to read and write, especially for users familiar with Coq.
The contents of this section are work-in-progress, but nicely demonstrate the potential that lies in carrying out interactive matching logic proofs in Coq.
\paragraph{Matching logic proof state.}
The concept of the proof state allows us to nicely mimic Coq-style reasoning in matching logic by rendering matching logic proof goals as a list of named hypotheses and a goal pattern. Behind the scenes, the goal on provability is turned into a record that stores the list of the premises along with the conclusion.
The proof mode allows for moving left-hand sides of implication conclusions to the list of premises (the local context), which is essential in matching logic as the deduction theorem~\cite{chen-rosu-2019-tr-mml} can be applied to totality patterns only.
The proof mode provides a better overview on the state of the proof in the interactive mode. In particular, it contains the following sections:
\begin{itemize}
\item A meta-level context (such as hypotheses on the well-formedness of patterns)
\item A global matching logic context (a set of patterns known to be valid);
\item A local matching logic context (a named list of patterns assumed to be valid);
\item A matching logic goal (a single pattern, the conclusion).
\end{itemize}
We provide a notation for this proof state, which also resembles the proof state in Coq (see \Cref{fig:proofstate}). In this
example, $\phi_1, \dots, \phi_n$ form the global matching logic context, while $\psi_1, \dots, \psi_n$ form the local one, and $\chi$ is the goal.
A matching logic proof state can automatically be converted to a syntactic provability statement as presented in \Cref{fig:proofstateconv}
which describes this conversion of the proof state in \Cref{fig:proofstate}.
\begin{figure}
\caption{Notation}
\caption{Conversion}
\caption{Matching logic proof state}
\label{fig:proofstate}
\label{fig:proofstateconv}
\end{figure}
The mapping between matching logic proof states and matching logic proofs of syntactic consequence is not injective: there can be multiple proof states that represent the same matching logic proof obligation when the the conclusion is an implication pattern.
\paragraph{Matching logic proof tactics.}
To create proof tactics, we first lift the derived proof rules to work with matching logic proof states. By lifting,
we actually mean proving the derived rules for the new proof state. The created tactics can be divided into three main groups:
\begin{itemize}
\item Tactics that restructure the local context (e.g., \coqinline{mlIntro}, \coqinline{mlRevertLast}, \coqinline{mlClear}).
\item Tactics that apply lifted derived rules to the proof state (e.g., \coqinline{mlApply}, \coqinline{mlApplyMeta}, \coqinline{mlDestructOr}).
\item Miscellaneous tactics (e.g., \coqinline{mlRewrite} which replaces parts of the matching logic goal, \coqinline{mlTauto} which is a preliminary
tautology solver).
\end{itemize}
We can use these tactics in a similar way as their Coq counterparts (e.g., \coqinline{mlIntro} mimics the effect of \coqinline{intro}), and create matching logic syntactic proofs conveniently. For the sake of brevity, we do not go into details about the implementation of the tactics, but in the background, they expand to applications of the proof system rules, therefore they construct valid matching logic proofs.
\subsection{An Interactive Proof}
In this section, we show an interactive proof outline (\Cref{fig:example}) with the matching logic proof mode. The complete proof is available in the
formalization~\cite{ml-formalization} (moreover, there is also a short tutorial about the currently formalized tactics). We show an example proof state transformation from each tactic category, but first, we present two lemmas that
are essential to construct the proof.
\begin{figure}
\caption{Case Study for the Proof Mode}
\label{fig:example}
\end{figure}
The first lemma is about the connection of conjunction and totality.
\begin{coqcode}
Lemma patt_total_and {?$\Sigma$? : Signature} {syntax : Syntax}:
forall ?$\Gamma$? ?$\varphi$? ?$\psi$?, theory ?$\subseteq$? ?$\Gamma$? -> well_formed ?$\varphi$? -> well_formed ?$\psi$? ->
?$\Gamma$? ?$\vdash$? ?$\lfloor$? ?$\varphi$? and ?$\psi$? ?$\rfloor$? <---> ?$\lfloor$? ?$\varphi$? ?$\rfloor$? and ?$\lfloor$? ?$\psi$? ?$\rfloor$?.
\end{coqcode}
The second lemma is the congruence lemma, which states that one can replace equivalent subpatterns in \emph{any} context results in equivalent patterns.
\begin{coqcode}
Lemma prf_equiv_congruence ?$\Gamma$? p q C:
PC_wf C -> ?$\Gamma$? ?$\vdash$? (p <---> q) -> ?$\Gamma$? ?$\vdash$? ((C [p]) <---> (C [q])).
\end{coqcode}
We implemented \coqinline{mlRewrite} based on the congruence lemma. At line \ref{l11} (\Cref{fig:example}) we can use this tactic with the first lemma, since it states the equivalence of two patterns. With this, we are able to propagate totality to the subpatterns of the conjunction for our concrete patterns.
\noindent\begin{minipage}{.49\textwidth}
\begin{coqcode}
______________________________________(1/1)
?$\Gamma$? ?$\vdash$?
?$\lceil$? pY and pX ?$\rceil$? --->
?$\lfloor$? (pY ---> pX) and (pX ---> pY) ?$\rfloor$?
\end{coqcode}
\end{minipage}
\begin{minipage}{.49\textwidth}
\begin{coqcode}
______________________________________(1/1)
?$\Gamma$? ?$\vdash$?
?$\lceil$? pY and pX ?$\rceil$? --->
?$\lfloor$? (pY ---> pX) ?$\rfloor$? and ?$\lfloor$? (pX ---> pY) ?$\rfloor$?
\end{coqcode}
\end{minipage}
Next, in line \ref{l15}, we reshape the structure of the matching logic proof state by using \coqinline{mlIntro} twice. This tactic
moves the left-hand side of the implication in the goal to the local matching logic context (note that conjunction is a syntactic sugar).
\noindent\begin{minipage}{.49\textwidth}
\begin{coqcode}
______________________________________(1/1)
?$\Gamma$? ?$\vdash$?
?$\lceil$? pY and pX ?$\rceil$? --->
?$\lfloor$? (pY ---> pX) ?$\rfloor$? and ?$\lfloor$? (pX ---> pY) ?$\rfloor$?
\end{coqcode}
\end{minipage}
\begin{minipage}{.49\textwidth}
\begin{coqcode}
______________________________________(1/1)
?$\Gamma$? ?$\vdash$?
"H0" : ?$\lceil$? pY and pX ?$\rceil$?,
"H1" : ! ?$\lfloor$? pY ---> pX ?$\rfloor$? or ! ?$\lfloor$? pX ---> pY ?$\rfloor$?,
--------------------------------------
?$\bot$?
\end{coqcode}
\end{minipage}
Finally, we also show the usage of \coqinline{mlApply} in line \ref{l16}. The conclusion of \coqinline{"H1'"} matches the goal,
thus we can apply it, and show its premise.
\noindent\begin{minipage}{.49\textwidth}
\begin{coqcode}
______________________________________(1/1)
?$\Gamma$? ?$\vdash$?
"H0" : ?$\lceil$? pY and pX ?$\rceil$?,
"H1'" : ! ?$\lfloor$? pY ---> pX ?$\rfloor$?,
--------------------------------------
?$\bot$?
\end{coqcode}
\end{minipage}
\begin{minipage}{.49\textwidth}
\begin{coqcode}
______________________________________(1/1)
?$\Gamma$? ?$\vdash$?
"H0" : ?$\lceil$? pY and pX ?$\rceil$?,
"H1'" : ?$\lfloor$? pY ---> pX ?$\rfloor$? ---> ?$\bot$?,
--------------------------------------
?$\lfloor$? pY ---> pX ?$\rfloor$?
\end{coqcode}
\end{minipage}
It can be observed that we used a number of standard Coq tactics, and explicit parameters during the proof (in \Cref{fig:example}). It is ongoing work to continue refining the proof mode and adding new tactics on demand to formalize as many paper-based matching logic proofs in Coq as possible.
\section{Related Work}\label{sec:related-work}
\subsection{Embedding Logical Languages in Coq}\label{sec:embedding}
Ideally, different sorts of problems are specified in different logical languages which fit the problem domain best. For instance, separation logics excel at describing algorithms manipulating shared and mutable states, temporal logics provide abstractions for specifying systems properties qualified in terms of time, whereas matching logic gives a fairly generic basis for reasoning about programming language semantics and program behavior. It is highly desired to carry out proofs in these domain-specific logical systems interactively and mechanically verified, but these logics tend to significantly diverge from the logics of general-purpose proof assistants such as Coq, leading to an abstraction gap.
To use a proof assistant to formalize reasoning in a specific logic, one needs to encode the logic as a theory in the proof assistant and then carry out reasoning at the meta-level with considerable overhead. Related works have been investigating different ways of embedding with the aim of reducing the overhead and facilitate productive object-level reasoning in various logical languages. To name a few, (focused) linear logic~\cite{power1999working,xavier2018mechanizing}, linear temporal logic~\cite{8133459}, different dialects of separation logic~\cite{krebbers2018mosel,mccreight2009practical,appel2007separation} and differential dynamic logic~\cite{10.1145/3018610.3018616} have been addressed in the past. Note that some of these encodings are full-featured proof modes, which create a properly separated proof environment and tactic language for the object logic. A slightly different idea worth mentioning is encoding one theorem prover’s logic in another to make the proofs portable, such as taking HOL proofs to Coq~\cite{wiedijk2007encoding,keller2010importing}.
The existing approaches show significant differences depending upon whether the formalization is aimed at proving the properties of the logic or at advocating reasoning in the logic. One particular consideration is to variable representation and the level of embedding. The majority of the cited formalizations apply a so-called shallow embedding, where they reuse core elements of the meta-logic; for instance, names are realized by using higher-order abstract syntax or the parametrized variant thereof, and exploit the binding and substitution mechanism built-in the proof assistant. With this, name binding, scopes, and substitution come for free, but the formalization is tied to the meta-logic’s semantics in this aspect, which may not be suitable in all cases. In fact, one of the main design decisions in our work was to use deep embedding facilitated by notations and locally nameless variable representation.
\subsection{Matching Logic Implementations}
This paper is not the only attempt that tries to formalize matching logic
using a formal system.
In \cite{ChenTrustworthyK}, the authors propose a matching logic formalization based on Metamath \cite{metamath}, a formal language used to encode abstract mathematical axioms and theorems.
The syntax and proof system of matching logic are defined in Metamath
in a few hundreds lines of code \cite{ml-checker}.
Matching logic (meta)theorems can be formally stated in Metamath, and
their formal proofs can be encoded in Metamath as machine-checkable proof objects.
While the Metamath implementation is simpler with a smaller trust base, the
Coq formalization of matching logic is more versatile; in Coq one can
express a larger variety of metatheorems such as the deduction theorem.
In general, the Metamath formalization focuses only on proofs of explicit
matching logic theories, while our formalization, despite a larger trust-base,
focuses mostly on models, semantics, and metatheorems of matching logic.
This dichotomy allows for potential integration with the Metamath formalization (see~\ref{sec:conclusion}).
Another matching logic implementation is through the $\mathbb{K}$\xspace framework.
The $\mathbb{K}$\xspace framework is a very robust engine for formalizing programming
language syntax and semantics. A version of matching logic, called
Kore, is used by $\mathbb{K}$\xspace to represent processed formal semantics. This is
how full large-scale languages can be simply represented as matching
logic theories. Admittedly, defining matching logic theories on the
scale of programming languages directly in Coq is not currently feasible.
However, our formalization brings more interactivity to matching logic reasoning,
which is currently missing in $\mathbb{K}$\xspace.
\section{Conclusion and Future Work}\label{sec:conclusion}
In this work, we defined a locally nameless representation of matching logic.
We also presented the first formal
definition of \emph{any} version of matching logic using an interactive
theorem prover, namely Coq.
We mechanized the soundness theorem of matching logic, and presented some nontrivial matching logic theories and
interactive proofs with a preliminary matching logic proof mode.
We believe this paves the way for Coq and interactive theorem provers to be
used more frequently with matching logic. We discuss some areas for
future work below.
\begin{itemize}\setlength\itemsep{0em}
\item
\emph{Complete Coq Proof Mode for Matching Logic. }
Proving formulas using the matching logic Hilbert-style proof system is not always convenient,
especially when compared to the way one can prove theorems in Coq.
For this reason we are working on the presented proof mode for matching logic in Coq,
that allows users to prove matching logic theorems using tactics that manipulate the goal
and local context.
We took the inspiration mainly from the Iris project, where the authors built a proof mode
for a variant of separation logic \cite{IrisProofMode}.
\item
\emph{Create Tactics for Type Class Instantiation. } While using the formalization with actual signatures or new derived notations,
the user needs to instantiate certain simple type classes. We plan to create tactics to carry out this work automatically.
\item
\emph{Exporting Metamath Proof Objects. }
An interesting way of combining advantages of both our Coq
formalization and the Metamath formalization
in~\cite{ChenTrustworthyK} would be the ability to convert matching
logic proofs in Coq to matching logic proofs in Metamath.
One challenge here is posed by the fact that Metamath uses the traditional named
representation of matching logic patterns, which is different from the
locally nameless representation used in our Coq development.
\item
\emph{Importing $\mathbb{K}$\xspace Definitions. }
As mentioned in Section~\ref{sec:related-work}, the $\mathbb{K}$\xspace framework
is a matching logic (specifically Kore) implementation with the
advantage of being able to naturally define real large-scale
programming languages.
As future work, we plan to formalize Kore as a matching logic theory inside Coq
and write a translator from Kore files to Coq files using this theory,
thus giving $\mathbb{K}$\xspace framework a Coq-based backend.
This would allow languages defined in $\mathbb{K}$\xspace and properties of those
languages proved in $\mathbb{K}$\xspace to be \emph{automatically} translated to Coq
definitions and theorems.
\item
\emph{Completeness. }
For the fragment of matching logic without the $\mu$ operator, the proof system is complete.
We would like to formalize the proof of completeness from~\cite{chen-rosu-2019-tr-mml};
however, we expect the proof to be non-constructive, which implies we would not be able
to compute proof terms (and extract Metamath proofs) from proofs of semantic validity.
\end{itemize}
\paragraph{Acknowledgements.} We warmly thank Runtime Verification Inc. for their generous funding support. Supported by the ÚNKP-21-4 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund.
\pagebreak
\end{document} |
\begin{document}
\title{Efficient tree-structured categorical retrieval}
\begin{abstract}
We study a document retrieval problem in the new framework where $D$ text documents are organized in a {\em category tree} with a pre-defined number $h$ of categories. This situation occurs e.g. with taxomonic trees in biology or subject classification systems for scientific literature. Given a string pattern $p$ and a category (level in the category tree), we wish to efficiently retrieve the $t$ \emph{categorical units} containing this pattern and belonging to the category. We propose several efficient solutions for this problem. One of them uses $n(\log\sigma(1+o(1))+\log D+O(h)) + O(\Delta)$ bits of space and $O(|p|+t)$ query time, where $n$ is the total length of the documents, $\sigma$ the size of the alphabet used in the documents and $\Delta$ is the total number of nodes in the category tree. Another solution uses $n(\log\sigma(1+o(1))+O(\log D))+O(\Delta)+O(D\log n)$ bits of space and $O(|p|+t\log D)$ query time. We finally propose other solutions which are more space-efficient at the expense of a slight increase in query time.
\end{abstract}
\keywords{pattern matching, document retrieval, category tree, space-efficient data structures}
\section{Introduction}
Data is often structured using {\em category hierarchies} represented by trees. In many applications, such hierarchies play a crucial guiding role: for example, the International Classification of Diseases (ICD) provides a hierarchical classification of all human disesases and constitues a common reference for diagnostics. In this paper, we are interested in sequence data, such as biological sequences or text documents, that are linked to a given hierarchy. More precisely, in our framework sequences are associated to leaves of a hierarchy, and tree nodes are mapped to several fixed levels, also called {ranks}.
This situation is common and occurs in several important applications. One is biology where species are classified according to the famous Linnaean taxonomy including eight common {\em taxonomic ranks}: species, genus, family, order, class, phylum, kingdom, domain. Then, given a set of sequences (DNA, RNA or protein) belonging to known species, one can associate them to the corresponding leaves of the taxonomic tree. Such a structure is used, for example, for phylogeny-based metagenomic classification where one considers the tree of known genomic sequences as a reference for classifying sequences of a metagenomic sample, see e.g. \cite{Wood2014}. A classification procedure may involve queries asking for the taxonomic units (i.e. internal nodes of the tree) of a certain rank whose sequences contain a given pattern, or similar type of queries.
Another example is provided by text documents such as scientific papers. The latter are usually annotated by subjects belonging to a fixed hierarchical nomenclature, such as ACM Computing Classification System (CCS) or Mathematics Subject Classification (MSC). Those subject hierarchies have a predefined number of levels: four levels for CCS and three for MSC. Given a corpus of scientific papers, one could ask about subject categories at a certain level whose documents contain a given pattern. This is a natural information retrieval scenario.
Here we study this problem from the stringology perspective (see e.g. \cite{KucherovNekrichStarikovskayaSPIRE12,GawrychowskiEtAlSPIRE13}). Assume we are given
a set of $D$ documents of total length $n$ over an alphabet of size $\sigma$, organized in
a tree of height $h$. The tree has $D$ leaves, each associated with a distinct document, and the leaves are all at level $h$ of the tree. The total number of nodes in the tree is denoted by $\Delta$.
The tree specifies a hierarchy of categories: each level of the tree corresponds to a category, and each internal node corresponds to a {\em categorical unit}.
The basic type of query we study in this paper is the following.
\begin{quote}
Given a pattern $p$, and a tree level (rank) $i\in [1..h]$, return all nodes (categorical units) $d_1,\cdots, d_t$ at level $i$ that have at least one leaf (document) in their subtree that contains pattern $p$.
\end{quote}
For example, given a large collection of genomic sequences organized in a taxonomic tree (for example, all known animal genomes), one may ask which animal families have a given sequence in the genomes of their members. Or, given a large hierarchy of documents (for example, all Computer Science papers), one may wonder in which subfields of Computer Science (corresponding to a certain level of the hierarchy) the term '{\em suffix tree}' is used.
This basic type of queries can be further extended in different ways. For example, one may impose an additional requirement of the mimimum number of documents of the categorical unit containing the given pattern. In this first study, we focus on the basic query type.
In this work, we propose several algorithms for this problem. Our first solution (Section~\ref{muthu-solution}) is based on the approach of Muthukrishnan \cite{muthukrishnan2002efficient} to the document retrieval problem. By combining several algorithmic tools - efficient text index, colored range reporting queries, and level ancestor queries - we obtain a solution with $n(\log\sigma(1+o(1))+\log D+O(h)) + O(\Delta)$ bits of space and $O(|p|+t)$ query time, where $t$ is the output size, i.e. the number of retrieved categorical units. To improve the space bound, in particular to get rid of the $O(nh)$ term which can be as big as $O(nD)$, we then develop a solution based on a wavelet tree built on top of the input category tree (Section~\ref{wavelet-tree-based}). On this way, we first obtain a solution taking $n(\log\sigma+\log D)+O(D\log n)$ bits and $O(|p|+t\cdot h\log D)$ query time. We further improve it using the technique of heavy path decomposition, to obtain a solution in $n(\log\sigma(1+o(1))+\log D)+O(\Delta)$ bits of space and $O(|p|+t\log D)$ query time. In the final part of the paper (Section~\ref{succinct}), we focus on solutions using succinct and compressed data structures, on top of the input data. That is, our main goal here is to replace the $n\log D$ bits by respectively $n\log \sigma$ or by $nH_0+o(n\log\sigma)$ in representing the document array. We obtain memory-time trade-offs showing how this goal can be achieved at the price of a slight increase of query time.
We summarize our main results in the following table.
\begin{center}
\begin{tabular}{ |c|c|c| }
\hline
algorithm & space (bits) & query time \\ \hline
based on colored & $n(\log\sigma(1+o(1))+\log D+O(h))$ & $O(|p|+t)$ \\
range queries (Sect.~\ref{muthu-solution}) & $+ O(\Delta)$ & \\\hline
based on wavelet & $n(\log\sigma(1+o(1))+O(\log D))$ & $O(|p|+t\log D)$ \\
tree (Sect.~\ref{wavelet-tree-based}) & $+O(\Delta)+O(D\log n)$ & \\\hline
compact space (Sect.~\ref{succinct}) & $O(n\log\sigma)$& $O(|p|+(t+1)\cdot\log^\epsilon n(1+\frac{h}{\log\sigma}))$\\\hline
compressed space (Sect.~\ref{succinct}) &$nH_k+o(n\log\sigma)+O(D\log n)$ &$O(|p|+t\cdot h\log n(\log\log n)^2)$\\
\hline
\end{tabular}
\end{center}
\section{Preliminaries}
We first briefly present main algorithmic tools used by our algorithms.
\subsection{Level ancestor queries on trees}
Consider a rooted tree. To each node in the tree we associate its {\em level} so that the level of the root is $1$, and the level of a child node is 1 more than the level of its parent.
The height of a tree is defined as the maximal level of any node in the tree. We denote by $\ell_\alpha$ the level of a node $\alpha$.
We will use the implementation of level ancestor queries specified by the following lemma.
\begin{lemma}[\cite{navarro2014fully}]\label{lemma:laq}
There exists a data structure that represents a tree with $n$ nodes within space $2n+o(n)$ and allows answering the following queries in constant time:
\begin{enumerate}
\item given a level $\ell$ and a node $\alpha$ at level at least $\ell$, return the ancestor node $\beta$ of $\alpha$ at level $\ell$,
\item given an integer $i$, return the node $\alpha$ where $\alpha$ is the leaf number $i$ in left-to-right order.
\end{enumerate}
\end{lemma}
We denote by $\mathtt{LAQ}(\alpha,i)$ the query which asks for the ancestor at level $i$ of node $\alpha$.
We denote by $\mathtt{leafselect}(i)$ the query which returns the $i$-th leaf of the tree in left to right order.
\subsection{$\mathtt{rank}$/$\mathtt{select}$ queries and wavelet trees}
\label{bitvecs}
$\mathtt{rank}$ and $\mathtt{select}$ queries on sequences constitute basic building blocks of many succinct data structures \cite{Jacobson89}. Given a string $S[1..n]$ on an alphabet $\Sigma$, a query $\mathtt{rank}_c(S,i)$, with $c\in\Sigma$ and $i\in[1..n]$, asks for the number of occurrences of $c$ in $S[1..i]$ and
$\mathtt{select}_c(S,j)$ asks for the unique position $i$ such that $S[i]=c$ and $\mathtt{rank}_c(S,i)=j$.
Consider first the important case of binary sequences (bitvectors).
The following result is well-known, see \cite{navarro_compact_2016}.
\begin{lemma}\label{lemma:bitvector}
A bitvector $B[1..n]$ can be represented using $n+o(n)$ bits of space, so that queries $\mathtt{rank}$ and $\mathtt{select}$ are answered in constant time.
\end{lemma}
In the case of non-binary alphabet, $\mathtt{rank}$/$\mathtt{select}$ queries can be efficiently answered using {\em wavelet trees}.
The wavelet tree has been formally introduced in~\cite{grossi2003high}, but a similar structure has been used
earlier \cite{chazelle1988functional}.
Suppose we are given a sequence $S$ of length $n$ over an alphabet $\Sigma$.
The {\em (binary) wavelet tree} is a binary tree representation of $S$ that is defined recursively as follows.
Let $\Sigma_0\neq \emptyset$ and $\Sigma_1\neq \emptyset$ form a partition of $\Sigma$ (that is, $\Sigma=\Sigma_0\cup \Sigma_1$ and $\Sigma_0 \cap \Sigma_1=\emptyset$). Then the root of the binary wavelet tree will contain a binary vector $B$, such that $B[i]=0$ iff $S[i]\in \Sigma_0$.
Let the sequence $S_0$ (resp., $S_1$) be formed by keeping only the elements of $S$ that belong to $\Sigma_0$ (resp., $\Sigma_1$), in the same order. Then, the left (resp., right) child is defined recursively using $S_0$ (resp., $S_1$) and a binary partition of $\Sigma_0$ (resp., $\Sigma_1$).
The recursion stops whenever we reach a leaf that corresponds to a singleton subset of $\Sigma$. Such nodes will form the leaves of the wavelet tree. We refer the reader to the survey~\cite{navarro2014wavelet} for more details about wavelet trees.
We will make use of the following lemma:
\begin{lemma}[\cite{grossi2003high}]\label{lemma:WT}
The wavelet tree over the alphabet $[1..\sigma]$ can be represented using $n(\log\sigma+o(1))+O(\sigma\log n)$ bits of space, supporting
$\mathtt{rank}$ and $\mathtt{select}$ queries in $O(\log\sigma)$ time.
\end{lemma}
\mathchardef\mhyphen="2D
\newcommand{\mathtt{range}\mhyphen\mathtt{distinct}}{\mathtt{range}\mhyphen\mathtt{distinct}}
The definition of binary wavelet tree can be readily generalized to the non-binary case. As in the binary case, to any node $\alpha$ labeled by an interval $\Sigma_\alpha$ is (implicitly) associated the sequence $S_\alpha$ which is the subsequence of $S[1..n]$ consisting of all characters belonging to $\Sigma_\alpha$. If a node $\alpha$ of a wavelet tree has $d$ children, then the alphabet interval $\Sigma_\alpha \subseteq[1..\sigma]$ assigned to $\alpha$ is partitioned into $d$ disjoint subintervals instead of two, and $\alpha$ stores a sequence $C_\alpha$ over alphabet $[1..d]$ of length $|S_\alpha|$ such that $C_\alpha[i]=j$ iff $S_\alpha[j]\in \Sigma_{\alpha_j}$.
\subsection{Text indexes}
We assume familiarity with main text indexing structures: suffix trees, suffix arrays and BWT-indexes. Here we only recall some basic facts about them.
Given a text $T$ over an alphabet $\Sigma=[1..\sigma]$, a suffix tree~\cite{weiner1973linear} is a tree data structure that stores in its leaves the suffixes of $T\$$, where $\$$ is a special character that does not appear in $T$ and is lexicographically smaller than any character of $T$. Each suffix is associated with its starting position in $T\$$. Suffix tree allows answering basic string pattern matching queries: given a pattern $p$, return the set of starting positions of $p$ in $T$.
The suffix array of $T$ is a related but more space-efficient data structure defined as the array $\mathtt{SA}[1..n+1]$ obtained by sorting all the suffixes of $T\$$ in lexicographic order and setting $\mathtt{SA}[i]=j$ if and only if the suffix $T[j..n]\$$ has lexicographic rank $i$ among all suffixes of $T\$$.
A suffix tree occupies $O(n\log n)$ bits of space and a matching query needs access to the original text $T$ in addition to the suffix tree. The query time is $O(|p|\log\sigma)$. The suffix array~\cite{manber1993suffix} is an alternative to the suffix tree which occupies the same $O(n\log n)$ bits of space, but has lower constant factors in space and supports matching queries in $O(|p|+\log n)$ time.
The BWT-index (FM-index) is a space-efficient alternative to suffix arrays and suffix trees which uses $O(n\log\sigma)$ bits of space only. It was originally proposed in~\cite{ferragina2005indexing} and has seen many improvements. We will use the following version of BWT-index with alphabet-independent query time.
\begin{lemma}[\cite{belazzougui2014alphabet}]~\label{lemma:textindex}
Given a text $T$ of length $n$ over alphabet $[1..\sigma]$, we can build a BWT-index which occupies $n\log\sigma(1+o(1))$ bits of space and supports computing the range of suffixes prefixed by a pattern $p$ in time $O(|p|)$.
\end{lemma}
Note that computing the range of suffixes answers also whether the pattern occurs in the text at all, and if so, reports the number of its occurrences (the size of the lexicographic order interval).
For this reason, the query presented in the lemma above is usually refered to as a $\mathtt{count}$ query.
The BWT-index is usually augmented with position information so that it becomes able to report the location of each occurrence of the pattern in addition to the number of occurrences. This can be achieved using fo the example the compressed suffix array representation:
\begin{lemma}[\cite{grossi2005compressed}]~\label{lemma:CSA}
Given a text $T$ of length $n$ over alphabet $[1..\sigma]$ and a constant $\epsilon>0$, we can build a data structure which occupies $O(n\log\sigma)$ bits of space and that returns $\mathtt{SA}[i]$ for any $i\in[1..n]$ in time $O(\log^\epsilon n)$.
\end{lemma}
All the above-mentioned text indexes can trivially be extended to support the same type of queries on a collection of documents instead of a single document. More precisely, given a collection of texts $T_1,T_2,\ldots,T_D$ over the same alphabet $\Sigma$, the same queries can be supported by constructing an index of the string $T_1\$T_\$\ldots T_D\$$.
\subsection{Colored range reporting and document retrieval}
Muthukrishnan~\cite{muthukrishnan2002efficient} was the first to study the problem of efficiently retrieving documents containing a given string pattern. Through the use of a text index, he reduced the problem to the one of {\em color range reporting}, i.e. reporting all {\em distinct} values (``colors'') occuring in a given interval of an array. His data structure relies on the use of {\em range minimum query} data structures -- a data structure that can find in constant time the smallest element in an sub-range of an array.
His algorithm was subsequently improved in terms of space (Theorem 4 in \cite{sadakane2007succinct}).
We will use the following result on color range reporting, which can be obtained by using the optimal range-minimum query data structure~\cite{FH11} in the method of \cite{sadakane2007succinct}:
\begin{lemma}\label{lemma:doc_retrieval}
Given an array $A[1..n]\in [1..\sigma]^n$, we can build a static data structure that occupies $2n+o(n)$ bits that allows reporting all $d$ distinct values occurring in a query interval $A[i..j]$ in time $O(d)$ ($O(1)$ time per reported value). The query will make read-only access to the data structure, read-only random access to elements of the array $A$
and read-write access to a bitvector $B$ of size $\sigma$. The bitvector
needs to be initalized to zero before the first query and is reset to zero at the end of each query.
\end{lemma}
In combination with text indexing, colored range reporting allows supporting document retrieval queries. More precisely, define the {\em document array} as follows: given a collection of $D$ documents $T_1,T_2\ldots T_D$ of total length $n$, lexicographically sort all the suffixes of the text $T^*=T_1\$T_2\$\ldots T_D\$$, and set $A[i]=j$ iff the suffix of $T^*$ of lexicographic rank $i$ starts inside $T_j$
(if the suffix starts with $\$$, then set $A[i]=0$). Document array $A$ can be easily obtained from a text index of $T^*=T_1\$T_2\$\ldots T_D\$$.
For this, one can construct a bitmap of length $|T^*|$ with $1$'s at positions of $\$$ in $T^*$ and $0$'s otherwise. Then $A[i]=\mathtt{rank}_1(A,\mathtt{SA}[i])+1$ for $i>D$ and $A[i]=0$ for $i\leq D$.
It is then immediate that using these data structures, Lemmas~\ref{lemma:textindex},~\ref{lemma:CSA}, and \ref{lemma:doc_retrieval} lead to solving the document retrieval problem in time $O(|p|+d\log^\epsilon n)$, where $d$ is the number of resulting documents. For this, we can use the document alphabet-independent BWT index to compute the range $[i..j]$ of occurrences of $p$ in $O(|p|)$ time and then report the $d$ distinct documents that appear in the range $A[i..j]$ in $O(d\log^\epsilon n)$ time.
\section{Solution based on Muthukrishnan's data structure}
\label{muthu-solution}
Our first solution will be
a combination of tools presented in the previous section.
We first build a text index for the concatenation of documents $T_1\$T_2\ldots T_D\$$. More specifically,
we build an instance of the text index of Lemma~\ref{lemma:textindex} which occupies $n\log\sigma(1+o(1))$ bits
and allows to locate the interval of all suffixes
of the documents that start with $p$ in time $O(|p|)$. We also build the document array $A[1..n]$, of size $n\log D$, indexed by the document suffixes sorted in lexicographic order and storing the documents each of the suffixes belongs to.
We further store $h$ instances $C_1,\ldots C_h$ of the data structure of Lemma~\ref{lemma:doc_retrieval},
one instance per level of the tree, defined as follows.
Consider $d$ (virtual) arrays $A_i[1..n]$, one per level $i\in[1..h]$ of the tree, such that
$A_i[j]$ stores the ancestor at level $i$ of document $A[j]$. Then, each $C_i$ is the data structure of Lemma~\ref{lemma:doc_retrieval} for supporting range reporting queries on array $A_i$.
Thus, $C_i$ allows to return, for any interval $[r..\ell]$, all distinct elements in $A_i[r..\ell]$ in constant time per element provided that a random-access to each element in $A_i$ is supported in constant time.
Note that according to Lemma~\ref{lemma:doc_retrieval}, a query will need to use $D$ bits of working space\footnote{We define the working space as a writable space that is only used during queries and is restored to its initial state at the end of the query}
since it will need to use a temporary bitvector $B$ of size $D_i\leq D$ where $D_i$ is the number of nodes at level $i$ of the tree\footnote{We can use the same bitvector $B$ (Lemma~\ref{lemma:doc_retrieval}) of size $D$ for all $h$ levels: for a query on level $i$, the first $D_i$ bits of $B$ are initally set to zero and are reset to zero at the end of the query}. By Lemma~\ref{lemma:doc_retrieval}, each $C_i$ occupies only $2n+o(n)$.
Finally, in order to simulate constant-time random access to entries of arrays $A_i$, $1\leq i\leq h$, we build a data structure for constant-time level ancestor queries on the category tree (Lemma~\ref{lemma:laq}). Notice that we can access cell $A_i[j]$ using the formula $A_i[j]=\mathtt{LAQ}(\mathtt{leafselect}(A[j]),i)$.
The data structure will occupy $2\Delta+o(\Delta)$ bits of space, where $\Delta$ is the total number of nodes in the tree.
To answer a query consisting of a pattern $p$ and level $i$, we proceed as follows. We first compute, in time $O(|p|)$, the interval $[\ell..r]$ of suffixes using the BWT-index (Lemma~\ref{lemma:textindex}). The documents containing $p$ are then those contained in $A[\ell..r]$. We then have to output all distinct ancestors at level $i$ of documents $A[\ell..r]$, i.e. all distinct elements of $A_i[\ell..r]$. This is done in constant time per reported element using $C_i$, as follows from Lemma~\ref{lemma:doc_retrieval} and constant-time access to elements of $A_i$ using $\mathtt{LAQ}$ and $\mathtt{leafselect}$ queries.
The document array occupies $n\log D$ bits of space. The text index is built on top of the $n\log\sigma(1+o(1))$ bits.
Each of the $h$ instances of the data structure of Lemma~\ref{lemma:doc_retrieval} will occupy $2n+o(n)$ bits of space each for a total space of $2nh+o(hn)$ bits of space. The data structure built on top of the category tree occupies $2\Delta+o(\Delta)$ bits of space.
We thus have proved the following theorem:
\begin{theorem}
\label{muthu-theorem}
Given a collection of $D$ documents of total length $n$ over alphabet $[1..\sigma]$ so that the documents
are organized in a hierarchy of documents represented by a tree of total size $\Delta$
and of height $h$, we can build a data structure
of size $n(\log\sigma(1+o(1))+\log D+O(h)) + O(\Delta)$ bits of space that, given a pattern $p$, can find all $t$ categories of documents at a given level $i$ that have at least one document that contains the pattern in total time $O(|p|+t)$.
\end{theorem}
This data structure will be good enough whenever $h$ is small, for example, when $h=\log D$, which holds for example when each internal node in the tree has at least two children.
\section{Wavelet-tree-based solution}
\label{wavelet-tree-based}
If each node of our tree is branching, i.e. has two or more children, then $h=O(\log D)$ and the solution of Secton~\ref{muthu-solution} takes $O(n(\log\sigma+\log D))$ bits of space. (Recall that all leaves of our tree occur at level $h$) However, this may not be the case as the tree may have many non-branching (unary) nodes. In the extreme case, we may have $h=\Omega(D)$ and the space of Theorem~\ref{muthu-theorem} will become $\Omega(nD)$ which can be too large if $D$ is large.
In this section, we deal with this issue and present solutions based on wavelet trees.
As in Secton~\ref{muthu-solution}, we assume that we first located an interval $[\ell..r]$ in the document array $A$ that corresponds to the occurrences of the query pattern $p$. The goal is then to return
all internal nodes at level $i$ containing documents from $A[\ell..r]$ in their subtree.
In Section~\ref{basic-wavelet}, we present the first ''warm-up'' solution that we subsequently improve in Section~\ref{heavy-path-solution}.
\input{Wavelet_tree_based_solution1.tex}
\input{Wavelet_tree_based_solution2.tex}
\section{Compact and compressed data structures for categorical data queries}
\label{succinct}
In this section we explore more space-effcient versions of the problem. More in detail, we are interested
in studying the problem under succinct and compressed-space constraints. Namely, our aim is to use
$O(n\log\sigma)$ bits for the succinct case and $nH_0+o(n\log\sigma)+O(D\log n)$ bits of space for the compressed case.
To achieve this, we will improve the solution of Section~\ref{muthu-solution}. More precisely, we avoid the storage of the document array and simulate direct access
to the document array using Lemma~\ref{lemma:CSA}. As a consequence, we can achieve time $O(\log^\epsilon n)$
to get the
given document index $A[i]$ for any $i\in[1..n]$. This will reduce the space to represent the document array from $O(n\log D)$ to $O(n\log\sigma)$ bits. Now the space used by the range minimum query data structures will become the bottleneck. To reduce the space usage we will make use of sparsification. More precisely, we will divide the document array into blocks and sample just the values of the $A$ array that are the smallest in each block. The space becomes $O(n/\alpha)$ bits where $\alpha$ is the sparsification factor. For details on how the sparsification is used to simulate the reporting of distinct documents that appear in interval $A[i..j]$, we refer the reader to~\cite{belazzougui2013improved,hon2014space}. Here we just mention that the time per reported document becomes $O(\alpha\log^\epsilon n)$ and entails $O(\alpha)$ accesses to the document array, each of which requires $O(\log^\epsilon n)$ time.
We thus have the following result.
\begin{theorem}
Given a parameter $\alpha\geq 1$ and a collection of $D$ documents of total length $n$ over alphabet $[1..\sigma]$ and so that the documents
are organized in a hierarchy of documents represented by a tree of height $h$, we can build a data structure
of size $O(n\log\sigma)+O(nh/\alpha)$ bits of space that can, given a pattern $p$, find all $t$ categories of documents at level $i$ that have at least one document that contains the pattern in total time $O(|p|+t\cdot\alpha\log^\epsilon n)$.
\end{theorem}
By setting $\alpha=\lceil\frac{h}{\log\sigma}\rceil$ we get space $O(n\log\sigma)$ bits and query time $O(|p|+(t+1)\log^\epsilon n\cdot(1+\frac{h}{\log\sigma}))$.
We thus have the following corollary.
\begin{corollary}
Given a parameter $\alpha$ and collection of $D$ documents of total length $n$ over alphabet $[1..\sigma]$ and so that the documents
are organized in a hierarchy of documents represented by a tree of height $h$, we can build a data structure
of size $O(n\log\sigma)$ bits of space that can, given a pattern $p$, find all $t$ categories of documents at level $i$ that have at least one document that contains the pattern in total time $O(|p|+(t+1)\cdot\log^\epsilon n(1+\frac{h}{\log\sigma}))$.
\end{corollary}
Whenever $h=\log D$ (e.g. every internal node is branching), the query time simplifies to $O(|p|+(t+1)\cdot\log_\sigma D\cdot\log^\epsilon n)\in O(|p|+(t+1)\log^{1+\epsilon}n)$.
We can also get compressed space. Namely, we can use a compressed suffix array~\cite{grossi2003high} with query time $\log n\log\log n$ and space $nH_k+o(n)$ to represent the document array. We will combine the compressed suffix array with the alphabet-independent variant of BWT-index presented in~\cite{belazzougui2014alphabet}. We then get an index that uses space $nH_k+o(n\log\sigma)$ with query time $O(|p|)$ to find the suffix array interval of a pattern and $O(\log n\log\log n)$ time to access an element of the suffix array. Notice that we can translate access to a suffix array element to an access to a document array element using $O(D\log n)$ bits of space. Summing up, we get the following theorem.
\begin{theorem}
Given a parameter $\alpha$ and a collection of $D$ documents of total length $n$ over alphabet $[1..\sigma]$ and so that the documents
are organized in a hierarchy of documents represented by a tree of height $h$, we can build a data structure
of size $nH_k+o(n\log\sigma)+O(D\log n)+O(nh/\alpha)$ bits of space that can, given a pattern $p$, find all $t$ categories of documents at level $i$ that have at least one document that contains the pattern in total time $O(|p|+t\cdot\alpha\log n\log\log n)$.
\end{theorem}
By setting $\alpha=h\cdot \log\log n$, we get space $nH_k+o(n\log\sigma)+O(D\log n)$ bits and query time $O(|p|+t\cdot h\log n(\log\log n)^2)$. The latter becomes $O(|p|+t\log D\log n(\log\log n)^2)$ whenever $h=O(\log D)$.
\section{Conclusions}
In this paper, we proposed several solutions for the problem of categorical retrieval. Possible extensions of our work include the case when the document hierarchy is a DAG rather than a tree. This situation occurs, for example, with phylogenetic networks.
The solution in Section~\ref{muthu-solution} could easily be extended to DAG structured categories if there was an efficient support for level ancestor queries on DAGs. Other possible extensions includes top-k queries in which categories are either ordered by a static order or by the total frequency of the pattern in the documents that belong to the reported categories.
\end{document} |
\begin{document}
\title{ extsc{Counterexamples for percolation on unimodular random graphs}
\begin{abstract}
We construct an example of a bounded degree, nonamenable, unimodular random rooted graph with $p_c=p_u$ for Bernoulli bond percolation, as well as an example of a bounded degree, unimodular random rooted graph with $p_c<1$ but with an infinite cluster at criticality. These examples show that two well-known conjectures of Benjamini and Schramm are false when generalised from transitive graphs to unimodular random rooted graphs.
\end{abstract}
\mathscrection{Introduction}
In \textbf{Bernoulli bond percolation}, each edge of a connected, locally finite graph $G$ is chosen to be deleted randomly with probability $1-p$, independently of all other edges, to obtain a random subgraph $G[p]$ of $G$. When $G$ is infinite, the \textbf{critical parameter} is defined to be
\[
p_c(G) = \inf\{ p\in [0,1] : G[p] \text{ contains an infinite connected component almost surely}\}
\]
and the \textbf{uniqueness threshold} is defined to be
\[
p_u(G) = \inf\{p \in [0,1] : G[p] \text{ contains a unique infinite connected component almost surely}\}.
\]
Traditionally, percolation was studied primarily on the hypercubic lattice $\mathbb Z^d$ and other Euclidean lattices.
In their seminal paper \cite{bperc96}, Benjamini and Schramm proposed a systematic study of percolation on more general graphs, and
posed many questions. They were particularly interested in \textbf{quasi-transitive} graphs, that is, graphs whose automorphism groups have only finitely many orbits.
Two central questions concern the existence or non-existence of infinite clusters at $p_c$, and the equality or inequality of $p_c$ and $p_u$.
They made the following conjectures. Specific instances of these conjectures, such as those concerning $\mathbb Z^d$, are much older.
\begin{conjecture}
\label{conj1}
Let $G$ be a quasi-transitive graph, and suppose that $p_c(G)<1$. Then $G[p_c]$ does not contain an infinite cluster almost surely.
\end{conjecture}
\begin{conjecture}
\label{conj2}
Let $G$ be a quasi-transitive graph. Then $p_c(G)<p_u(G)$ if and only if $G$ is nonamenable.
\end{conjecture}
Given a set $K$ of vertices in a graph $G$, we define $\partial_E K$ to be the set of edges of $G$ that have exactly one endpoint in $K$.
A graph is said to be \textbf{nonamenable} if
\[
\inf\left\{ \mathfrakrac{|\partial_E K|}{\mathscrum_{v\in K} \deg(v)} : K \mathscrubseteq V \text{ finite}\right\}>0,
\]
and \textbf{amenable} otherwise. It follows from the work of Burton and Keane \cite{burton1989density} and Gandolfi, Keane and Newman \cite{gandolfi1992uniqueness} that $p_c(G)=p_u(G)$ for every amenable quasi-transitive graph, so that only the `if' direction of Conjecture \ref{conj2} remains open. It was also proven by H\"aggstr\"om, Peres, and Schonmann \cite{MR1676835,MR1676831,haggstrom1999percolation} that there is a unique infinite cluster for every $p>p_u$ when $G$ is quasi-transitive.
We refer the reader to \cite{grimmett2010percolation} for an account of what is known in the Euclidean case $G=\mathbb Z^d$, and to \cite{LP:book} for percolation on more general graphs.
Substantial progress on Conjecture \ref{conj1} was made in 1999 by Benjamini, Lyons, Peres, and Schramm~\cite{BLPS99b}, who proved that the conjecture is true for any \emph{nonamenable, unimodular} quasi-transitive graph. Here, a graph is unimodular if it satisfies the \emph{mass-transport principle}, see \cite[Chapter 8]{LP:book}. (More recently, the conjecture has been verified for all quasi-transitive graphs of \emph{exponential growth} \cite{timar2006percolation,Hutchcroft2016944}, and in particular for all nonamenable quasi-transitive graphs, without the assumption of unimodularity.)
In the mid 2000's, Aldous and Lyons \cite{AL07} showed that this result, as well as several other important results such as those of \cite{newman1981infinite,burton1989density,gandolfi1992uniqueness,BLS99,MR1676835,MR1676831,haggstrom1999percolation,LS99}
can be generalized, with minimal changes to the proofs,
to \emph{unimodular random rooted graphs}.
These graphs appear naturally in many applications: For example, the connected component at the origin in percolation on a unimodular transitive graph is itself a unimodular random rooted graph.
An important caveat is that when working with unimodular random rooted graphs one should consider a different, weaker notion of nonamenability than the classical one, which we call \emph{invariant nonamenability} \cite[\S 8]{AL07}.
In this note, we construct examples to show that, in contrast to the situation for the classical results mentioned in the previous paragraph, Conjectures \ref{conj1} and \ref{conj2} are in fact both \emph{false} when generalized to unimodular random rooted graphs, even with the assumption of bounded degrees.
\begin{theorem}
\label{thm:critical}
There exists a bounded degree unimodular random rooted graph $(G,\rho)$ such that $p_c(G)<1$ but there is an infinite cluster $G[p_c]$ almost surely.
\end{theorem}
\begin{theorem}
\label{thm:pcpu}
There exists a unimodular random rooted graph $(G,\rho)$ such that $G$ has bounded degrees, is nonamenable, and has $p_c(G)=p_u(G)$ for Bernoulli bond percolation almost surely.
\end{theorem}
We stress that the example in Theorem \ref{thm:pcpu} is nonamenable in the classical sense (which is a stronger property than being invariantly nonamenable).
Thus, any successful approach to Conjectures 1 and 2 cannot rely solely on mass-transport arguments. See \cite{beringer2016percolation} for some further examples of unimodular random rooted graphs with unusual properties for percolation, and \cite{1105.2638} for another related example.
\mathscrection{Basic constructions}
\mathscrubsection{Unimodularity and normalizability of unrooted graphs}
We assume that the reader is familiar with the basic notions of unimodular random rooted graphs, referring them to \cite{AL07} otherwise. Since it will be important to us and is perhaps less widely known, we quickly recall the theory of unimodular random rooted graphs with fixed underlying graph from \cite[Section 3]{AL07}.
Let $G$ be a graph, let $\Gamma \mathscrubseteq \operatorname{Aut}(G)$ be a group of automorphisms of $G$, and for each $v\in V$ let $\operatorname{Stab}_v = \{\gamma \in \Gamma : \gamma v = v\}$ be the stabilizer of $v$ in $\Gamma$. The group $\Gamma$ is said to be \textbf{unimodular} if
\[
|\operatorname{Stab}_v \gamma v|=|\operatorname{Stab}_{\gamma v} v|
\]
for every $v \in V$ and $\gamma \in \Gamma$, where $\operatorname{Stab}_v u$ is the orbit of $u$ under $\operatorname{Stab}_v$. The graph $G$ is said to be unimodular if $\operatorname{Aut}(G)$ is unimodular.
Let $G$ be a connected, locally finite, unimodular graph and let $\mathcal O$ be a set of \textbf{orbit representatives} of $\Gamma$. That is, $\mathcal O \mathscrubseteq V$ is such that for every vertex $v\in V$, there exists exactly one vertex $o \in \mathcal O$ such that $\gamma v = o$ for some $\gamma \in \Gamma$.
We say that $(G,\Gamma)$ is \textbf{normalizable} if there exists a measure $\mu_G$ on $\mathcal O$ such that if $\rho$ is distributed according to $\mu_G$ then the random rooted graph $(G,\rho)$ is unimodular. It is easily seen that the measure $\mu_G$ is unique when it exists.
It is proven in \cite[Theorem 3.1]{AL07} that a connected, locally finite, unimodular graph $G$ is normalizable if and only if
\[
Z_v(G)=\mathscrum_{ o \in \mathcal O} |\operatorname{Stab}_o(v)|^{-1} < \infty
\]
for some (and hence every) vertex $v \in V$, and moreover the measure $\mu_G$ can be expressed as
\[
\mu_G(\{o\})= Z_v(G)^{-1}|\operatorname{Stab}_o(v)|^{-1} \qquad o \in \mathcal O.
\]
\mathscrubsection{Building new examples from old via replacement}
We will frequently make use of the following construction, which allows us to construct one normalizable unimodular graph from another.
Constructions of this form are well-known, see \cite{khezeli2017shift} and \cite{beringer2016percolation} for further background.
Let $G=(V,E)$ be a connected, locally finite graph, let $\Gamma \mathscrubseteq \operatorname{Aut}(G)$ be a unimodular subgroup of automorphisms, and let $G'=(V',E')$ be a connected, locally finite graph. Let $M_1(V')$ be the set of functions $m: V' \to [0,1]$ with $|m|:=\mathscrum_{v\in V'}m(v)<\infty$, and suppose that there exists a function $m:V\to M(V')$, $m : v \mapsto m_v$ such that
\begin{enumerate}
\item The functions $\{m_v : v\in V\}$ are a partition of unity on $V'$ in the sense that $\mathscrum_{v\in V} m_v(u) = 1$ for every $u \in V'$.
\end{enumerate}
and
\begin{enumerate}
\item[2.] $m$ is automorphism-equivariant on $V^2$ in the following sense: If $u,v,w,x\in V$ are such that $(w,x)=(\gamma u, \gamma v)$ for some $\gamma \in \operatorname{Aut}(G)$, then there exists an automorphism $\gamma'$ of $G'$ such that $(m_w,m_x) = (\gamma m_u, \gamma m_v)$.
\end{enumerate}
Then $G'$ is also unimodular. If furthermore $G$ is normalizable and
\begin{equation}
\label{eq:m}
\mathscrum_{o\in \mathcal O(G)} \mu_G(\{o\}) |m_o|<\infty,
\end{equation}
then $G'$ is normalizable with
\[
\mu_{G'}(\{o'\}) = \mathscrum_{v\in V'} \mathbbm{1}\left[o' \in \operatorname{Aut}\left(G'\right) v\right] \mathscrum_{o\in \mathcal O(G)} \mathfrakrac{\mu_G(\{o\})m_o(v)}{\mathscrum_{o\in \mathcal O(G)} \mu_G(\{o\}) |m_o|} \qquad o' \in \mathcal O(G').
\]
Following \cite{beringer2016percolation}, we call this method of constructing new normalizable unimodular graphs from old ones \textbf{replacement}.
To give a simple example of replacement, suppose that $G$ is a connected, locally finite, normalizable unimodular graph, and let $G'$ be the graph in which each edge of $G$ is replaced with a path of length two. Define $m: V\to M_1(v')$ by setting $m_v(u)$ to be $1$ if $u$ is equal to $v$, and to be $1/2$ if $u$ is the midpoint of a path of length $2$ emanating from $v$ in $G'$ that was formerly an edge of $G$. It is easily verified that $m$ satisfies conditions $1$ and $2$ above. If furthermore $G$ has finite expected degree in the sense that $\mathscrum_{o\in \mathcal O(G)}\mu_G(\{o\})\deg(o)<\infty$, then \eqref{eq:m} is satisfied and $G'$ is normalizable.
One can also consider a variation of this procedure allowing for randomization:
Let $G=(V,E)$ be a connected, locally finite, unimodular graph, let $V'$ be a set, and let $G'=(V',E')$ be a random connected, locally finite graph with vertex set $V'$, which we consider to be a random element of $\{0,1\}^{V^2}$.
Suppose that there exists a function $m:V\to M(V')$, $m : v \mapsto m_v$ such that
\begin{enumerate}
\item The functions $\{m_v : v\in V\}$ are a partition of unity on $V'$ in the sense that $\mathscrum_{v\in V} m_v(u) = 1$ for every $u \in V'$.
\end{enumerate}
and
\begin{enumerate}
\item[2.] $m$ is automorphism-equivariant on $V^2$ in the following sense: If $u,v,w,x\in V$ are such that $(w,x)=(\gamma u, \gamma v)$ for some $\gamma \in \operatorname{Aut}(G)$, then there exists a
bijection
$\gamma' : V' \to V'$ such that $(m_w,m_x) = (\gamma m_u, \gamma m_v)$ and the \emph{law} of $G'$ is invariant under the action of $\gamma'$ on $V^2$.
\end{enumerate}
Let $\tilde \rho$ be a random element of $\mathcal O(G)$ drawn from biased measure $\tilde \mu_G$ defined by
\[
\tilde \mu_G(\{o\}) = \mathfrakrac{\mu_G(\{o\})|m(o)|}{\mathscrum_{o\in \mathcal O(G)} \mu_G(\{o\}) |m(o)|}, \qquad o \in \mathcal O(G)
\]
and, conditional on $\tilde \rho$, let $\rho'\in V'$ be chosen according to the conditional distribution
\[
\mathbb P(\rho'=u \mid \tilde \rho) = \mathfrakrac{m_\rho(u)}{|m_\rho|}.
\]
Then the random rooted graph $(G',\rho')$ is unimodular.
Fixing the vertex set of $G'$ in advance is of course rather unnecessary and restrictive, but it is sufficient for the examples we consider here.
\mathscrection{A discontinuous phase transition}
\mathscrubsection{Trees of tori}
Let $d \geq 2$.
The \textbf{$d$-ary canopy tree} $T_d$ is the tree with vertex set $\mathbb Z \times \mathbb N$ and edge set
\[\left\{ \left\{ (i,j) , (k,j-1) \right\} : j \geq 1,\, d i \leq k \leq d (i+1)-1 \right\}.\]
In other words, $T_d$ is the tree that has infinitely many leaves (that have no children), and such that every vertex that is not a leaf has exactly $d$ children, that is, neighbours that are closer to the leaves than it is.
Note that the isomorphism class of $(T_d,v)$ depends only on the distance between $v$ and the leaves, called the \textbf{height} of $v$, and denoted $|v|$. We also say that vertices with height $k$ for $k\geq 0$ are in \textbf{level} $k$. It is well known and easily verified that $T_d$ is unimodular and normalizable, with
$\mu_{T_d}(\{o\}) = d^{-|o|+1}/(d-1).$
Let $n \geq 1$, and let $d,r \geq 2$.
We define the \textbf{tree of tori} $\mathbb{T}^n(d,r)$ to be the connected, locally finite graph with vertex set
\[ V\left(\mathbb{T}^n(d,r)\right) = \left\{ (v,x) : v \in V(T_d),\, x \in \mathbb Z^n / r^{|v|} \mathbb Z^n \right\}, \]
and where we connect two vertices $(v,x)$ and $(u,y)$ of $\mathbb{T}^n(d,r)$ by an edge if and only if either
\begin{enumerate}
\item $v=u$ and $x$ and $y$ are adjacent in the torus, or else
\item $u$ is adjacent to $v$ in $T_d$, and either $|v| \geq |u|$ and
$x$ is mapped to $y$ by the quotient map $\mathbb Z^n / r^{|v|} \mathbb Z^n \to \mathbb Z^n / r^{|u|} \mathbb Z^n$ or, symmetrically, $|u| \geq |v|$ and
$y$ is mapped to $x$ by the quotient map $\mathbb Z^n / r^{|u|} \mathbb Z^n \to \mathbb Z^n / r^{|v|} \mathbb Z^n$.
\end{enumerate}
See Figure \ref{fig:treeoftori} for an illustration. (Note that removing the torus edges from this graph yields the horocyclic product of the $d$-ary canopy tree with the $r^n$-ary tree, which also arises as a half-space of the Diestel-Leader graph $DL(d,r^n)$ \cite{MR1856226}.)
\begin{figure}
\caption{The canopy tree of one-dimensional tori $\mathbb{T}
\label{fig:treeoftori}
\end{figure}
The following is an easy consequence of replacement.
\begin{proposition}
\label{prop:treesoftori}
If $r^n < d$ then $\mathbbT^n(d,r)$ is unimodular and normalizable, and
\[\mu_{\mathbbT^n(d,r)}(\{o\}) = \mathfrakrac{d^{-|o|+1}r^{n|o|}}{d-r^n}.\]
\end{proposition}
It will also be useful to consider a more general version of this construction, in which we let the sizes of the tori grow as a specified function of the height. Let $d\geq 2$, let $n \geq 1$, and let $r:\mathbb N \to \mathbb N$ be an increasing function. We define the tree of tori $\tilde \mathbbT^n(d,r)$ similarly to above, with vertex set
\[
V\left(\tilde{\mathbb{T}}^n(d,r)\right)
= \left\{ (v,x) : v \in V(T_d),\, x \in \mathbb Z^n /
2^{ r(|v|)} \mathbb Z^n
\right\}, \]
and where we connect two vertices $(v,x)$ and $(u,y)$ of $\mathbb{T}^n(d,r)$ by an edge if and only if either
\begin{enumerate}
\item $v=u$ and $x$ and $y$ are adjacent in the torus, or else
\item $u$ is adjacent to $v$ in $T_d$, and either $|v| \geq |u|$ and
$x$ is mapped to $y$ by the quotient map $\mathbb Z^n / 2^{ r(|v|)} \mathbb Z^n \to \mathbb Z^n / 2^{r(|u|)} \mathbb Z^n$ or, symmetrically, $|u| \geq |v|$ and
$y$ is mapped to $x$ by the quotient map $\mathbb Z^n / 2^{r(|u|)} \mathbb Z^n \to \mathbb Z^n / 2^{r(|v|)} \mathbb Z^n$.
\end{enumerate}
We now have, by replacement, that $\tilde\mathbbT^n(d,r)$ is unimodular and is normalizable if and only if
\[
\mathscrum_{\ell \geq 0} d^{-\ell} 2^{n r(\ell)}<\infty, \quad \text{ in which case }\quad \mu_{\tilde{\mathbbT}^n(d,r)}(\{o\})= \mathfrakrac{d^{-|o|+1}2^{nr(|o|)}}{\mathscrum_{\ell \geq 0} d^{-\ell} 2^{n r(\ell)}}.
\]
\mathscrubsection{Proof of Theorem \ref{thm:critical}}
In this section we prove Theorem \ref{thm:critical}.
We begin with unbounded degree example, and then show how it can be modified to obtain a bounded degree example.
Let $d \geq 2$ and let $T_d$ be $d$-ary canopy tree. We write
\[
\log^+ x = \begin{cases}
1 & x \leq e\\
\log x & x > e,
\end{cases}
\]
and write $\asymp$ for equalities that hold up to positive multiplicative constants.
For each $\gamma \in \mathbb R$, let $G_{d,\gamma}$ be obtained from $T_d$ by replacing each edge connecting a vertex at height $n$ to a vertex at height $n+1$ with
\[m_\gamma(n) := \left\lceil \mathfrakrac{\log^+ n + \gamma \log^+ \log^+ n}{\log 2} \right\rceil\] parallel edges, which is chosen so that
\[\left(\mathfrakrac{1}{2}\right)^{m_\gamma(n)} \asymp \mathfrakrac{1}{n \log^\gamma n}.\]
It follows by replacement that $G_{d,\gamma}$ is unimodular.
The basic idea behind this construction is that the coefficient of $\log^+ n$ above determines the value of $p_c$ (set here to be $1/2$), while the coefficient of $\log^+ \log^+ n$ determines the behaviour of percolation \emph{at} $p_c$.
\begin{proposition}
$p_c(G_{d,\gamma})=1/2$ for every $\gamma \in \mathbb R$. If $\gamma>1$, then critical percolation on $G_{d,\gamma}$ contains an infinite cluster almost surely.
\end{proposition}
\begin{proof}
If $v$ is a vertex of $G_{d,\gamma}$, then the cluster of $v$ in $G_{d,\gamma}[p]$ is infinite if and only if every ancestor of $v$ has an open edge connecting it to its parent. This event occurs with probability
\[ \theta_\gamma(v,p) = \prod_{n \geq |v|} \left(1-(1-p)^{m_{\gamma}(n)}\right).\]
In particular, $\theta_\gamma(v,p)>0$ if and only if
\[
\mathscrum_{n \geq 0} (1-p)^{m_\gamma(n)} < \infty.
\]
Since
\[
n^\mathfrakrac{\log p}{\log 2} (\log n)^\mathfrakrac{\gamma \log p}{\log 2}
\leq (1-p)^{m_\gamma(n)}
\leq (1-p) n^\mathfrakrac{\log p}{\log 2} (\log n)^\mathfrakrac{\gamma \log p}{\log 2},
\]
it follows that $\theta_\gamma(v,p)>0$ if and only if
\[\mathscrum_{n \geq 0} n^\mathfrakrac{\log p}{\log 2} (\log n)^\mathfrakrac{\gamma \log p}{\log 2} <\infty.
\]
Recall that the series
\[
\mathscrum_{n \geq 0} \mathfrakrac{1}{n^\alpha (\log^+ n)^\beta}
\]
converges if and only if either $\alpha>1$ or $\alpha=1$ and $\beta>1$.
Thus, $\theta_\gamma(v,p)>0$ for some (and hence every) vertex $v$ of $G_m$ if and only if either $p>1/2$, or $p=1/2$ and $\gamma>1$.
In particular, $p_c(G_{d,\gamma})=1/2$ for every $d$ and every value of $\gamma$, while if $\gamma>1$ then $\theta_{\gamma}(v,1/2)>0$ for every vertex $v$ of $G$ as desired.
\end{proof}
We now build a bounded degree variation on this example using trees of tori. Let $d,r \geq 2$, be such that $d > r^2$, and let
$m : \mathbb N \to \mathbb N \mathscretminus\{0\}$ be a function.
Let $\mathbb{G}(d,r,m)$ be the graph obtained by replacing each edge connecting two vertices of height $\ell$ and $\ell+1$ in $\mathbb{T}^2(d,r)$ with a path of length $m(\ell)$. It follows by replacement that $G$ is unimodular, and is normalizable if
\begin{equation}
\label{eq:pcnormalization}
\mathscrum_{\ell \geq 0} d^{-\ell} r^{2 \ell} m(\ell) <\infty.
\end{equation}
Thus, Theorem \ref{thm:critical} follows immediately from the following proposition.
\begin{proposition}
\label{prop:criticalbdddegree}
Let $0<q<1$ be sufficiently large that $\theta_q(\mathbb Z^2)>3/4$, and
let $m:\mathbb N \mathscretminus\{0\}\to\mathbb N \mathscretminus \{0\}$ be such that there exists a positive constant $c$ such that
\[
c\, 4^{-\ell}(\ell+1)^2 \leq q^{m(\ell)} \leq 4^{-\ell}(\ell+1)^2
\]
for every $\ell \geq 1$.
Then $\mathbbG=\mathbbG(5,2,m)$ is a normalizable, bounded degree, unimodular graph, $p_c(\mathbbG)=q$, and $\mathbbG[q]$ contains an infinite cluster almost surely.
\end{proposition}
For an example of a function $m$ of the form required by Proposition \ref{prop:criticalbdddegree}, we can take
\[
m (\ell) \equiv \left\lceil
\mathfrakrac{(\ell+2) \log 4 - 2 \log^+ \ell}{\log (1/q)}
\right\rceil.
\]
\begin{proof}
It is clear that $\mathbbG=\mathbb{G}(5,2,m)$ has bounded degrees, and we have already established that it is unimodular and normalizable.
We now prove the statements concerning percolation on $\mathbbG$. Suppose that $v$ is a vertex of the canopy tree $T_d$ with $|v|=\ell$, and suppose that $u$ is the parent of $v$ in $T_d$. Thus, the torus $\{v\}\times \left(\mathbb Z^2/2^\ell \mathbb Z^2\right)$ is connected in $\mathbbG$ to the torus $\{u\}\times \left(\mathbb Z^2/2^{\ell+1} \mathbb Z^2\right)$ by
$4^{\ell+1}$ paths of length $m(\ell)$.
If $p<q$ then $p=q^{1+\delta}$ for some $\delta>0$, and so
the expected number of these paths that are open in $\mathbbG[p]$ is
\[ 4^{\ell+1} p^{m(\ell)}
= 4^{\ell+1}q^{(1+\delta)m(\ell)}
\asymp \ell^2 4^{-\delta \ell}.\]
Since this expectation converges to zero, it follows that $\mathbbG[p]$ does not contain an infinite cluster almost surely, and we conclude that $p_c(\mathbbG) \geq q$.
It remains to prove that $\mathbbG[q]$ contains an infinite cluster almost surely. Broadly speaking, the idea is that, since $\theta_q(\mathbb Z^2)>3/4$, each torus in $\mathbbG$ has a high probability to contain a giant open component which contains at least three quarters of its vertices, which is necessarily unique. The logarithmic correction in the definition of $m$ then ensures that the giant component in each torus is very likely to be connected by an open path to the giant component in its parent torus, which implies that an infinite open component exists as claimed.
To make this argument rigorous, we will apply the following rather crude estimate.
\begin{proposition}
\label{lem:supercriticaltoruspercolation}
Consider Bernoulli bond percolation on the $n \times n$ torus, $\mathbb Z^2/n\mathbb Z^2$, for $p>p_c(\mathbb Z^2)$ supercritical. There exist positive constants $c_1$ and $c_2$ depending on $p$ such that for every $\varepsilon>0$, the probability that $\mathbb Z^2/n\mathbb Z^2$ does \emph{not} contain an open cluster $C$ with $|C| \geq (\theta_{p}(\mathbb Z^2)-\varepsilon)n^2$ is at most
\[\mathfrakrac{c_1 }{\varepsilon^2 n^2} + n^2e^{-c_2 \varepsilon n}.\]
\end{proposition}
\begin{proof}
It suffices to prove the analogous statement for the box $[1,n]^2$, which we consider as a subgraph of $\mathbb Z^2$.
It follows from \cite[Theorem 1.1]{AntalPisztora96} that if $p>p_c(\mathbb Z^2)$, $\delta>0$, and $x,y \in [\delta n , (1-\delta)n]^2$, then there exists a positive constant $c_p$ such that
\[
\mathbb P\left( x \leftrightarrow \infty \text{ and } y \leftrightarrow \infty, \text{ but } x \nleftrightarrow y \text{ in } [0,n]^2 \cap \mathbb Z^2 \right)\leq e^{-c_p \delta n}.
\]
Thus, it follows by a union bound that the probability that the largest cluster in $[1,n]^2$ has size at most $(\theta_p(\mathbb Z^2)-\varepsilon)n^2$ is at most
\[
\mathbb P_p\left(\mathscrum_{x,y \in [\delta n , (1-\delta)n]^2 } \mathbbm{1}\left(x\leftrightarrow \infty, y \leftrightarrow \infty \right) \leq
(\theta_p(\mathbb Z^2)-\varepsilon)n^2 \right) + n^2 e^{-c_p \delta n}.
\]
On the other hand, we have that \cite[Section 11.6]{grimmett2010percolation}
\[
\operatorname{Var}\left[ \mathscrum_{x\in [\delta n, (1-\delta) n]^2} \mathbbm{1}\left( x \leftrightarrow \infty \right)\right] \leq Cn^2
\]
for some constant $C=C_p$, and it follows by Chebyshev's inequality that
\[
\mathbb P\left(\mathscrum_{x\in [\varepsilon n/2 , (1-\varepsilon/2)n]^2} \mathbbm{1}\left( x \leftrightarrow \infty \right) \leq (\theta_p(\mathbb Z^2)-\varepsilon)n^2\right) \leq C\left[(1-\delta)^2 \theta_p(\mathbb Z^2)-\theta_p(\mathbb Z^2) +\varepsilon\right]^{-2} n^{-2}
\]
when the right hand side is positive.
We conclude by taking $\delta>0$ so that $(1-\delta)^2 \theta_p(\mathbb Z^2)-\theta_p(\mathbb Z^2) +\varepsilon = \varepsilon/2$.
\end{proof}
We now apply Lemma \ref{lem:supercriticaltoruspercolation} to complete the proof of Proposition \ref{prop:criticalbdddegree}.
Let $v_0$ be a leaf of $T_5$, let $v_1,v_2,\ldots$ be its sequence of ancestors, and let $\Lambda_i$ be the torus $\{v_i\} \times \left(\mathbb Z^2/ 2^i \mathbb Z^2\right)$ in $\mathbbG$. It follows from Lemma \ref{lem:supercriticaltoruspercolation} and the Borel-Cantelli Lemma that $\Lambda_i[q]$ contains a (necessarily unique) giant open cluster of size at least $(3/4)4^i$ for every $i \geq i_0$ for some random, almost surely finite $i_0$. Thus, for each $i \geq i_0$, there exist at least $4^i/2$ vertices of $\Lambda_i$ that are both contained in the giant open cluster of $\Lambda_i[q]$, and have a parent in $\Lambda_{i+1}[q]$ that is contained in the giant open cluster of $\Lambda_{i+1}[q]$. Thus, conditional on this event, for each $i$ sufficiently large, the probability that the giant open cluster of $\Lambda_i[q]$ is \emph{not} connected by an open path to the giant open cluster of $\Lambda_{i+1}[q]$ is at most
\[
\left(1-q^{m(i)}\right)^{4^{i}/2}
\leq \left(1-q i^2 4^{-i}\right)^{4^{i}/2} \leq e^{-qi^2/2},
\]
where we have used
the inequality $(1-x) \leq e^{-x}$, which holds for all $x \geq 1$,
to obtain the second inequality. Since these probabilities are summable, it follows by Borel-Cantelli that there exists a random, almost surely finite $i_1 \geq i_0$ such that the giant open cluster of $\Lambda_i[q]$ is connected to the giant open cluster of $\Lambda_{i+1}[q]$ for every $i\geq i_1$. It follows that $\mathbbG[q]$ contains an infinite cluster almost surely. \qedhere
\end{proof}
\mathscrection{Nonamenability and uniqueness}
In this section we prove Theorem \ref{thm:pcpu} by constructing a nonamenable, unimodular, normalizable, bounded degree graph $G$ for which $p_c(G)=p_u(G)$ for Bernoulli bond percolation. We begin by constructing a family of partitions of the four regular tree.
\mathscrubsection{Isolated, invariantly defined partitions of the tree}
\begin{figure}
\caption{Recursively constructing the class containing the origin in the hierarchical partition of the $4$-regular tree. $(1)$ shows the decomposition of the tree into its two bipartite classes. $(2)$ shows the subdivision of one of the two classes appearing in $(1)$ classes into four classes as occurs in $\mathbbV_2(S)$. $(3)$ shows the bipartite tree corresponding to the class of the origin in $\mathbbV_2(S)$, in which the two bipartite classes are `red' and `everything else'.
$(4)$ shows the outcome of applying the same procedure another time, splitting the class of the origin in $\mathbbV_2(S)$ into four further subclasses and obtaining an associated bipartite tree for each of these classes. $(5)$ shows the classes in $(4)$ as they appear in the original $4$-regular tree.}
\end{figure}
Let $S$ be a $4$-regular tree. If we draw $S$ in the plane, then for each vertex $v$ of $S$ we obtain a cyclic ordering of the edges emanating from $v$ that encodes the clockwise order that the edges appear around $v$ in the drawing. We fix one such family of cyclic orderings, and let $\Gamma$ be the group of automorphisms of $S$ that fix this family of cyclic orderings. (In other words, we consider $S$ as a \textbf{plane tree}.) It is well known that $\Gamma$ is unimodular.
We define the \textbf{isolation} of a subset $W$ of $V(S)$ to be the minimal distance between distinct points of $S$.
If $A$ and $B$ are partitions of a set, we say that $A$ \textbf{refines} $B$ if every set in $W\in A$ is contained in a set of $B$.
We say further that $A$ is a $k$\textbf{-fold refinement} of $B$ if every set in $B$ is equal to the union of exactly $k$ sets in $A$. Similarly, we say that the fold of the refinement is \textbf{bounded by $k$} if every set in $B$ is equal to the union of at most $k$ sets in $A$.
\begin{prop}
\label{prop:partition}
There exists a random sequence of partitions $(\tilde \mathbbV_k)_{k\geq0}$ of $V(S)$ with the following properties.
\begin{enumerate}
\item The law of $(\tilde \mathbbV_k)_{k\geq 0}$ is invariant under $\Gamma$.
\item $\mathbbV_0 = \{V(S)\}$, and for each $n\geq 0$, $\tilde \mathbbV_{n+1}$ is a refinement of $\tilde \mathbbV_n$ with fold bounded by $4$.
\item Each $W \in \tilde \mathbbV_k$ has isolation at least $2\lfloor \log_2 k -1 \rfloor$.
\end{enumerate}
\end{prop}
The statement `the law of $(\tilde \mathbbV_k)_{k\geq 0}$ is invariant under $\Gamma$' should be interpreted as follows: $Gamma$ naturally acts pointwise on subsets of $V(S)$, and hence also on partitions of $V(S)$.
Then for any $\gamma\in\Gamma$, the image of $\gamma$ on the partition has the same law on the partition.
(A partition is described as a subset of $V(S)^2$, with the product $\mathscrigma$-algebra.)
We define a tree $\mathbbD$, such that $V(\mathbbD) = \bigcup_k \tilde \mathbbV_k$.
The root of $\mathbbD$ is the trivial partition $V(S)$, and the children of a vertex in $\tilde\mathbbV_k$ are the included parts of $\tilde\mathbbV_{k+1}$.
Thus $\mathbbD$ has bounded degrees.
\begin{proof}
We begin by constructing a deterministic sequence of partitions which have isolation growing linearly and fold growing exponentially.
We then construct a random sequence of partitions that intermediate between these partitions, which will satisfy the conclusions of \cref{prop:partition}.
Let $(F_n)_{n \geq 0}$ be the sequence defined recursively by $F_0=F_1=4$ and
\[F_{n+1}=F_n(F_{n-1}-1) \qquad n\geq 1.\]
Note that this sequence grows doubly-exponentially in $n$. In particular, $F_n \leq 4^{2^n}$ for every $n\geq 0$.
We construct a sequence of partitions $(\mathbbV_k)_{k\geq1}$ of $V(S)$, which we call the \textbf{hierarchical partition}, with the following properties:
\begin{enumerate}
\item $(\mathbbV_k)_{k\geq1}$ is $\Gamma$-invariant in the sense that for any two vertices $u,v \in V(S)$, any $\gamma \in \Gamma$ and any $k\geq 1$, if $u,v \in V$ are in the same piece of the partition $\mathbbV_k$ (i.e., there exists $W\in \mathbbV_k$ such that $u,v\in W$) then $\gamma u,\gamma v$ are also in the same piece of the partition $\mathbbV_k$ (i.e., there exists $W' \in \mathbbV_k$ such that $\gamma u, \gamma v \in W$).
\item
$\mathbbV_1(S)$ is the partition of $V(S)$ into its two bipartite classes.
\item For each $k\geq 1$, the partition $\mathbbV_{k+1}(S)$ is an $F_{k-1}$-fold refinement of the partition $\mathbbV_k$.
\item Each $W \in \mathbbV_k$ has isolation at least $2k$.
\end{enumerate}
The hierarchical partition may be constructed recursively as follows.
Suppose that $n\geq 1$, and that $S_n$ is the plane tree whose vertices are separated into bipartite classes $V_1$ and $V_2$ such that every vertex in $V_1$ has degree $F_n$ and that every vertex in $V_2$ has degree $F_{n-1}$. We call vertices in $V_1$ \textbf{primary} and vertices in $V_2$ \textbf{secondary}. Consider a coloring of the primary vertices $V_1$ with the property that for every secondary vertex $v\in V_2$, the vertices $u_1,\ldots,u_{F_{n-1}}$ appearing in clockwise order adjacent to $v$ in $T$ have colors $1,\ldots,F_{n-1}$ up to a cyclic shift. Such a coloring is easily seen to exist and is unique up to a cyclic shifts of the colors.
For each color $1 \leq i \leq d_2$, let $S_{n,i}$ be the tree with vertex set $V_1$ in which two vertices are connected by an edge if and only if their distance in $S_n$ is $2$ and one of them has color $i$. This tree inherits a plane structure from $S_{n}$. Let $V_1(T_i)$ be the subset of $V(S_i)=V_1(S)$ containing the color $i$ vertices and let $V_2(S_i)$ be the subset of $V(S_i)=V(S)$ containing the vertices with color other than $i$. It is easily verified that $V_1(S_i)$ and $V_2(S_i)$ are the two bipartite classes of $S_{n,i}$ and that vertices in these classes have degrees $F_n(F_{n-1}-1)=F_{n+1}$ and $F_n$ respectively, so that $S_{n,i}$ is isomorphic to $S_{n+1}$.
Moreover, the distance in $S_{n,i}$ between any two vertices in $V_1(S_{n,i})$ is at least two less than the distance of the corresponding vertices in $S_n$:
Indeed, it is easily verified that the distance between $u,v \in V_1(S_{n,i})$ in $S_{n,i}$ is equal to their distance in $S_n$, minus the number of vertices in $V_1(S_{n,i})$ that are included in the geodesic between $u$ and $v$ in $S_n$ (which is at least $2$ due to the endpoints being in $V_1(S_{n,i})$). See Figure \ref{fig:geodesic}.
\begin{figure}
\caption{If $u,v \in V_1(S_{n,i}
\label{fig:geodesic}
\end{figure}
We apply this construction recursively, beginning with the $4$-regular tree $S=S_1$ separated into its two bipartite classes. When we start step $n$ of the recursion, we
have constructed the sequence of partitions $(\mathbbV_k)_{k \leq n}$ and have given each $W \in \mathbbV_{n}$ the structure of the bipartite plane tree $S_n$ in such a way that the distance between any two vertices of $W$ in the associated copy of $S_n$ is at most their distance in $S_1$ minus $2(n-1)$. Given this data, we apply the above procedure to each of these copies of $S_n$ to complete the next stage of the recursion, obtaining a $F_{n-1}$-fold refinement $\mathbbV_{k+1}$ of $\mathbbV_k$.
This is the hierarchical partition: the above discussion implies inductively that it has the properties required above.
It remains to modify this hierarchical partition to have bounded fold.
This is achieved by randomly adding further partitions that intermediate between $\mathbbV_n$ and $\mathbbV_{n+1}$.
This is necessary, so that the construction in the next subsection gives a graph with unbounded degrees.
We define the \textbf{randomly intermediated hierarchical partition} $(\tilde \mathbbV_{k})_{k\geq0}$ of $S$ as follows.
For each $n \geq 0$, let $a_n = \lceil \log_4 F_n \rceil$, and let $b_n = \mathscrum_{i=0}^n a_n$.
Note that $a_n \leq 2^n$ and hence $b_n \leq 2^{n+1}$ for every $n\geq 0$.
\begin{enumerate}
\item Let $\tilde \mathbbV_0=\mathbbV_0=\{V(S)\}$.
\item For each $n \geq 1$, let $\tilde \mathbbV_{b_n} = \mathbbV_n$.
\item We construct the partitions $(\tilde \mathbbV_{b_n-k})_{k=1}^{a_n-1}$ recursively as follows: Given $\tilde \mathbbV_{b_n-k}$, for each set $W \in \tilde \mathbbV_{b_{n-1}}=\mathbbV_{n-1}$, choose uniformly at random a partition of the set $\{
W' \in \tilde \mathbbV_{b_n-k} : W' \mathscrubseteq W
\}$
into sets that all have size four except possibly for one of the sets.
These random choices are made independently of each other, and independent of all other randomness used in the construction.
\end{enumerate}
The definition of $a_n$ and $b_n$ ensure that $\tilde \mathbbV_{n+1}$ is a refinement of $\tilde \mathbbV_{n}$ with fold bounded by $4$ for every $n\geq 1$.
Moreover, the sequence of random variables $(\tilde \mathbbV_k)_{k \geq0}$ is invariant \emph{in distribution} under $\Gamma$.
Finally, note that if we define $c_k$ to be maximal such that $b_{c_k} \leq k$ for each $k\geq 1$, then every set in $\tilde \mathbbV_k$ is $2c_k$ isolated for every $k\geq 0$, since every such set is contained in a set in $\mathbbV_{c_k}$. Moreover, we have that
\[
c_k \geq \lfloor \log_2k -1 \rfloor
\]
for every $k\geq 1$.
\end{proof}
\mathscrubsection{Proof of \cref{thm:pcpu}}
We now use the randomly intermediated hierarchical partition whose existence is stipulated by \cref{prop:partition} to construct the example required by Theorem \ref{thm:pcpu}.
Let $q$ be such that $\theta_q(\mathbb Z^2)>3/4$, and let $m: \mathbb N \to \mathbb N\mathscretminus \{0\}$ and $r:\mathbb N\to \mathbb N\mathscretminus\{0\}$ be increasing functions such that
\begin{align*}
2^{r(\ell)} \asymp \ell+1, \qquad
q^{m(\ell)} \asymp (\ell+1)^{-2} \quad \text{ and } \quad 2^{2r(\ell)}q^{m(\ell)} \leq 10^{-4}.
\end{align*}
Suppose further that $r$ and $m$ have bounded increments.
For example, we can take
\[
r(\ell) = \left\lceil \mathfrakrac{\log^+\ell}{\log 2}\right\rceil \qquad \text{ and } \qquad m(\ell) = \left \lceil \mathfrakrac{2\log^+\ell + 4\log 10}{\log(1/q)}\right \rceil.
\]
Let $\mathbbT= \tilde{\mathbbT}^2(100,r)$ be the canopy tree of tori, and let $\mathbbG$ be obtained from $\mathbbT$ by replacing edges between different levels of $\mathbbT$ with paths of length $m$, similarly to the construction in the previous section.
For each vertex $v$ of $\mathbbG$, we write $|v|$ for the unique $\ell \geq 0$ such that either $v$ is a level-$\ell$ vertex of $\mathbbT$, or $v$ lies on the interior of one of the paths of length $m(\ell)$
connecting level $\ell$ of $\mathbbT$ to level $\ell+1$ of $\mathbbT$ that is added when constructing $\mathbbG$ from $\mathbbT$.
Let $(\tilde \mathbbV_k)_{k\geq 0}$ be a random sequence of partitions of $V(S)$ satisfying the conclusions of \cref{prop:partition}, and $\mathbbD$ the associated tree.
Conditional on $(\tilde \mathbbV_k)_{k\geq 0}$, we define the graph $\mathbbH$ as follows.
The vertex set $V(\mathbbH)$ is a subset of $\mathbbG\times S \times \mathbbD$ given by
\[
\left \{(v_1,v_2,W) \in \mathbbG \times S \times \mathbbD \text{\ such that } |W|=|v_1| \right\}.
\]
Here, $|W|=k$ if $W\in\tilde\mathbbV_k$.
This construction is somewhat similar to Diestel-Leader graphs (more precisely, half of the Diestel-Leader graph), since the tree structure of $\mathbbG$ branches towards level $0$, and the tree $\mathbbD$ branches away from level $0$.
We call a vertex $(v_1,v_2,W)$ of $\mathbbH$ \textbf{type-1} if $v_2\in W$ and \textbf{type-2} otherwise.
We connect two vertices $(v_1,v_2,W_1)$ and $(v'_1,v'_2,W')$ of $\mathbbH$ by an edge if and only if one of the following hold:
\begin{enumerate}
\item[(1).]
$(v_1,v_2,W_1)$ and $(v'_1,v'_2,W')$ are both type $1$, $v_2=v_2'$, and $v_1,v'_1$ are adjacent in $\mathbbG$, and $W,W'$ are adjacent in $\mathbbD$, or
\item[(2).] $v_1=v_1'$, $W=W'$, and $v_2$ and $v_2'$ are adjacent in $S$.
\end{enumerate}
We call an edge of $\mathbbH$ a \textbf{$\mathbbG$-edge} if its endpoints have the same $S$-coordinate (in which case they must both be type-$1$ vertices), and an $S$\textbf{-edge} otherwise (in which case its endpoints have the same $\mathbbG$-coordinate, and at least one of the vertices must be type-$2$).
Note that every connected component of the subgraph of $\mathbbH$ induced by the type-1 vertices (equivalently, spanned by the $\mathbbG$-edges) is isomorphic to $\mathbbG$. We call these \textbf{type-1 copies of $\mathbbG$ in $\mathbbH$.} Similarly, the type-$2$ edges span $\mathbbH$, and every connected component of the associated subgraph is isomorphic to the $4$-regular tree $S$.
Let $\rho_1$ be a random root for the deterministic graph $\mathbbG$ chosen from the law $\mu_{\mathbbG}$, let $\rho_2$ be a fixed root vertex of the deterministic graph $S$, and let $W_\rho$ be chosen uniformly from $\tilde \mathbbV_{|\rho_1|}$. Let $\rho=(\rho_1,\rho_2,W_\rho)$.
It follows by replacement (applied to the product $\mathbbG \times S$, which is unimodular and normalizable by e.g.\ \cite[Proposition 4.11]{AL07}) that $(\mathbbH,\rho)$ is a unimodular random rooted graph.
We call a vertex $(v_1,v_2,W)$ of $\mathbbH$ \textbf{type-1} if $v_2\in W$ and \textbf{type-2} otherwise.
We call an edge of $\mathbbH$
a \textbf{$\mathbbG$-edge} if its endpoints have the same $S$-coordinate (in which case they must both be type-$1$ vertices), and an $S$\textbf{-edge} otherwise (in which case its endpoints have the same $\mathbbG$-coordinate, and at least one of the vertices must be type-$2$).
Note that every connected component of the subgraph of $\mathbbH$ induced by the type-1 vertices (equivalently, spanned by the $\mathbbG$-edges) is isomorphic to $\mathbbG$. We call these \textbf{type-1 copies of $\mathbbG$ in $\mathbbH$.}
Finally, given a constant $M \geq 1$, we define the graph $\tilde \mathbbH(M)$ by replacing each of the $S$-edges of $\mathbbH$ with a path of length $M$. It follows by replacement that $\tilde \mathbbH(M)$ can be rooted in such a way that it is a unimodular random rooted graph. Theorem \ref{thm:pcpu} therefore follows immediately from the following proposition.
\begin{proposition}
\label{prop:pcpu}
The random graph $\tilde\mathbbH(M)$ described has bounded degrees and is nonamenable. If $M$ is sufficiently large then
$p_c(\tilde \mathbbH)=p_u(\tilde \mathbbH)=q$.
\end{proposition}
The proof of Proposition \ref{prop:pcpu} will apply the notion of disjoint occurrence and the BK inequality~\cite{van1985inequalities}, see \cite[Section 2.3]{grimmett2010percolation} for background.
\begin{proof}
Write $\tilde \mathbbH=\tilde \mathbbH(M)$.
Moreover, it is immediate from the assumption that $r$ and $m$ have bounded increments that $\tilde \mathbbH$ has bounded degrees.
It is easily seen that stretching some edges by a bounded amount preserves nonamenability (indeed, nonamenability is stable under rough isometry), and so to prove that $\tilde \mathbbH$ is nonamenable it suffices to prove that $\mathbbH$ is nonamenable.
Observe that we can partition the vertex set of $\mathbbH$ into sets $\{V_i :i \in I\}$ whose induced subgraphs are copies of the $4$-regular tree $S$.
Thus, given any finite set of vertices $K$ in $\mathbbH$, we can write $K = \bigcup_{i \in I} K_i$ where $K_i = K \cap V_i$. Since the subgraph of $\mathbbH$ induced by $V_i$ is a $4$-regular tree for every $i \in I$, it follows that the external edge boundary of $K_i$ in the subgraph induced by $V_i$ has size at least $|K_i|$ for every $i \in I$, and so we have that
\[
|\partial_E K| \geq \mathscrum_{i \in I}|K_i| = |K|,
\]
and hence that
\[
\mathfrakrac{|\partial_E K|}{\mathscrum_{v\in K} \deg(v)} \geq \mathfrakrac{1}{\max_{v\in \mathbbH} \deg(v)}
\]
for every finite set of vertices $K$, so that $\mathbbH$ is nonamenable as claimed.
We now prove the statements concerning percolation on $\tilde \mathbbH$.
It follows similarly to the proof of \cref{prop:criticalbdddegree} that $p_c(\mathbbG)=q$.
\begin{lemma}
$p_u(\tilde \mathbbH) \leq p_c(\mathbbG)=q$.
\end{lemma}
\begin{proof}
The proof is an an easy modification of the argument of Lyons and Schramm \cite[Theorem 6.12]{LS99}, and applies the main theorem of that paper as generalised to unimodular random graphs by Aldous and Lyons \cite[6.15]{AL07}; see also Theorem 6.17 of that paper.
The sketch of the argument is as follows: It is easily verified that $\mathbbG$ is invariantly amenable (see e.g.\ \cite[Section 8]{AL07} and \cite{unimodular2}), so that $p_u(\mathbbG)=p_c(\mathbbG)=q$ by \cite[Corollary 6.11, 8.13]{AL07}.
Thus, for every $p>q$, every type-$1$ copy of $\mathbbG$ in $\tilde \mathbbH$ contains a unique infinite open cluster almost surely.
It is easy to deduce using insertion tolerance and the mass-transport principle that these clusters must all be connected to each other by open edges in $\tilde \mathbbH$.
Finally, indistinguishability implies that there cannot be any \emph{other} infinite open cluster in $\tilde \mathbbH$, since the $\mathbbG$-edges within any such cluster would only have finite connected components.
\end{proof}
To complete the proof, it suffices to show that $p_c(\tilde \mathbbH)\geq q$ when $M$ is sufficiently large.
Let $V_1(\mathbbH)$ be the set of type-1 vertices of $\mathbbH$.
Define $\chi(\ell,k)$ and $\tilde \chi(\ell,k)$ for $\ell,k \geq 0$ by
\[\chi(\ell,k) = \mathscrup_{v \in V(\mathbbG), |v|=\ell} \mathscrum_{u \in V(\mathbbG), |u|=k} \mathbb P(v \leftrightarrow u \text{ in } \mathbbG[q]).\]
and
\[\tilde \chi(\ell,k) = \mathscrup_{v \in V_1( \mathbbH), |v|=\ell} \mathscrum_{u \in V_1(\mathbbH), |u|=k} \mathbb P(v \leftrightarrow u \text{ in } \tilde \mathbbH[q]).\]
It follows easily by mass transport and insertion tolerance that every infinite cluster of $\tilde \mathbbH[q]$ contains infinitely many type-1 vertices of $\mathbbH$, and so to prove that $p_c(\tilde \mathbbH) \geq q$ it suffices to prove that $\mathscrum_{k\geq0} \tilde \chi(\ell,k)<\infty$ for some (and hence every) $\ell \geq 0$.
As a first step we bound the susceptibility in $\mathbbG$.
\begin{lemma}\label{lem:C}
There exists a constant $C$ such that
\[\chi(\ell,k) \leq C (k+1)^2\]
for every $\ell,k\geq 0$.
\end{lemma}
Note that the choice of $M$ does not affect the definition of $\chi(\ell,k)$, and so the constant $C$ here does not depend on the choice of $M$.
A vertex of $\mathbbG$ at level at most $k$ has a good chance of being connected in $\mathbbG[q]$ to the giant component in a torus at level $k$, and therefore the dependence on $k$ cannot be improved here.
\begin{proof}
For each vertex $u$ in $\mathbbG$, let $t(u)$ be the associated vertex of the canopy tree $T$. Similarly, for each $x\in T$ let $\Lambda_x$ be the associated torus in $\mathbbT$, and let $V_x=\{ v \in V(\mathbbG) : x(v)=x\}$ be the associated set of vertices of $\mathbbG$.
As in the proof of \cref{prop:criticalbdddegree}, if $y$ is the parent of $x$ in $T$, then the probability that $\Lambda_x$ is connected to $\Lambda_y$ by an open path in $\mathbbG[q]$ is at most
\[
2^{2r(|x|)} q^{m(|x|)} \leq 10^{-4}.
\]
Thus, if $x$ and $y$ are vertices of $T$ whose most recent common ancestor has height $n$, then the probability $\Lambda_x$ is connected to $\Lambda_y$ in $\mathbbG[q]$ is at most
$10^{-8n+4|x|+4|y|}$.
Let $x \in V(T)$, let $y$ be the parent of $x$ in $T$, and let $u \in V_x$. Then
\begin{align*}
\mathscrum_{u \in V(\mathbbG): |u|=k} \mathbb P\left(v \leftrightarrow u \text{ in }\mathbbG[q]\right)
&\leq \mathscrum_{n \geq \ell \varepsilone k} \mathscrum_{|w|=k, |w \wedge x| =n} \mathbb P\left(\Lambda_x \cup \Lambda_{\mathscrigma(x)} \leftrightarrow \Lambda_w \cup \Lambda_{\mathscrigma(w)} \text{ in }\mathbbG[q]\right) |V_w|\\
&\leq C
\mathscrum_{n \geq \ell \varepsilone k} \mathscrum_{|w|=k, |w \wedge x| =n} 10^{-8n+4k+4\ell} |V_w|\\
&\leq C' \mathscrum_{n\geq \ell \varepsilone k} 10^{-8n+4\ell+4k} 10^{2n-2k} (k+1)^2\\
&\leq 2C' (k+1)^2
\end{align*}
as claimed, where $C,C'$ are constants.
\end{proof}
We now apply \cref{lem:C} to prove that $\mathscrum_{k \geq0} \tilde \chi(\ell,k)<\infty$.
Observe that we may consider percolation on $\tilde \mathbbH$ as an inhomogeneous percolation on $\mathbbH$ in which every $\mathbbG$-edge of $\mathbbH$ is open with probability $q$, and every $S$-edge of $\mathbbH$ is open with probability $q^M$. We will work with this equivalent model for the rest of the proof.
We define a \textbf{traversal} in $\tilde \mathbbH$ to be a simple path in $\tilde \mathbbH$ that starts and ends at type-1 vertices, while every vertex in its interior is a type-2 vertex.
Observe that every traversal uses only $S$-edges, and that every simple path in $\tilde \mathbbH$ that starts and ends at type-1 vertices can be written uniquely as a concatenation of traversals and $\mathbbG$-edges.
For each two type 1 vertices $u,v$ in $\tilde \mathbbH$ let $\tau(u,v)$ be the probability that $u$ and $v$ are connected by an open path. Let $\mathscrA_i(u,v)$ be the event that $u$ and $v$ are connected by a simple open path containing exactly $i$ traversals, let $\tau_i(u,v)$ be the probability of this event, and let
\[
\tilde \chi_i(\ell,k) = \mathscrup_{|u|=\ell} \mathscrum_{|v|=k} \tau_i(u,v).
\]
We have that
$\tau(u,v) \leq \mathscrum_{i\geq0} \tau_i(u,v)$ and hence that
\[
\tilde \chi(\ell,k) \leq \mathscrum_{i\geq0} \tilde \chi_i (\ell,k).
\]
Furthermore, $\tau_0(u,v)$ is positive if and only if $u$ and $v$ are in the same type-$1$ copy of $\mathbbG$, and in this case it is equal to the probability that they are connected by an open path in this copy.
Let $u,v$ be vertices of $\tilde \mathbbH$ with $|u|=\ell$, $|v|=k$, and let $i\geq 1$. For each type 1 vertex $w$, let $\operatorname{Tr}_w$ be the set of traversals starting at $w$. Given a traversal $t\in \operatorname{Tr}_w$, we write $t^+$ for the type-$1$ vertex at the other end of $t$.
Summing over possible choices of the $i$th traversal along a simple open path from $u$ to $v$ and applying the BK inequality, we obtain that
\begin{align}
\tau_i(u,v) &\leq \mathscrum_{j \geq 0} \mathscrum_{w \in V_1(\mathbbH), |w|=j} \mathscrum_{t\in \operatorname{Tr}_w } \mathbb P(\mathscrA_{i-1}(u,w) \circ \{ t \text{ open} \} \circ \mathscrA_{0}(t^+,v))
\nonumber
\\
&\leq \mathscrum_{j \geq0} \mathscrum_{w \in V_1(\mathbbH), |w|=j} \mathscrum_{t\in \operatorname{Tr}_w } \tau_{i-1}(u,w)\tau_0(t^+,v) \mathbb P(t \text{ open}).
\label{eq:BK}
\end{align}
In $q^M$-percolation on the $4$-regular tree, the expected number of vertices that have distance at least $k$ from the root and are connected to the root by an open path is equal to
\[
\mathscrum_{\ell \geq k} 3 \cdot 4^{k-1} (q^M)^k \leq \mathfrakrac{(4q^M)^k}{1-4q^M}.
\]
Furthermore, by the isolation property of the hierarchical partition, for each type-$1$ vertex $w$ of $\mathbbH$, every traversal in $\operatorname{Tr}_w$ has length at least $2 \log^+|w|/\log 2$. Thus, we deduce that
\begin{equation}
\label{eq:trav}
\mathscrum_{t\in \operatorname{Tr}_w}
\mathbb P(t \text{ open}) \leq \mathfrakrac{(4q^M)^{2 \log^+|w|/\log 2}}{1-4q^{M}}.\end{equation}
Thus, substituting \eqref{eq:trav} into \eqref{eq:BK} and summing over $v$, we obtain that
\begin{align*}
\mathscrum_{|v|=k} \tau_i(u,v)
& \leq 4 \mathscrum_{j \geq0} \tilde \chi_{i-1}(\ell,j) \chi (j,k) \mathfrakrac{(4q^M)^{2 \log^+j/\log 2}}{1-4q^{M}}
\\&\leq \mathfrakrac{C'}{1-4q^{M}} \mathscrum_{j \geq0} \tilde \chi_{i-1}(\ell,j) \chi (j,k)
(j \varepsilone e)^{-\alpha(M)}
\end{align*}
and hence that
\begin{equation}
\label{eq:induction}
\tilde \chi_i(\ell,k) \leq \mathfrakrac{C'}{1-4q^{M}}\mathscrum_{j \geq0} (j\varepsilone e)^{-\alpha(M)} \tilde \chi_{i-1}(\ell,j) \chi (j,k),
\end{equation}
where \[\alpha(M) = \mathfrakrac{2M\log(1/q)}{\log2}-4\] and $C'$ is a constant.
Take $M$ sufficiently large that
\[
\mathfrakrac{2 C \cdot C'}{1-4q^{M}} \mathscrum_{j\geq0} (j\varepsilone e)^{-\alpha(M)}(j+1)^2
\leq 1/2,\] where $C'$ is the constant above, $C$ is the constant from \cref{lem:C}.
We now prove by induction on $i$ that, for this choice of $M$,
\begin{equation}
\label{eq:induction2}
\tilde \chi_i(\ell,k) \leq C 2^{-i} (k+1)^2
\end{equation}
for every $i\geq 0$ and $\ell,k\geq0$. The case $i=0$ follows from Lemma \ref{lem:C}. If $i\geq 1$, then \eqref{eq:induction} and the induction hypothesis yield that
\begin{align*}
\tilde \chi_i (\ell,k) &\leq \mathfrakrac{2^{-i+1} C^2\cdot C'}{1-4q^{M}} \mathscrum_{j\geq0} (k+1)^2 (j\varepsilone e)^{-\alpha(M)}(j+1)^2,
\end{align*}
and our choice of $M$ yields that
\[
\tilde \chi_{i}(\ell,k) \leq C 2^{-i} (k+1)^2
\]
as claimed. This completes the proof of \eqref{eq:induction2}.
We conclude the proof by summing over $i$ and $k$ to deduce that $\tilde \chi(\ell,k)<\infty$ for every $\ell,k\geq 0$ as claimed.
\end{proof}
\mathscrubsection*{Acknowledgments}
This was was carried out while TH was a PhD student at the University of British Columbia, during which time he was supported by a Microsoft Research PhD Fellowship.
\mathscretstretch{1}
\end{document} |
\begin{document}
\begin{frontmatter}
\title {The simple exclusion process on finite \\ connected graphs}
\runtitle {The simple exclusion process on finite connected graphs}
\author {Shiba Biswal and Nicolas Lanchier}
\runauthor {Shiba Biswal and Nicolas Lanchier}
\address {School for Engineering of Matter, \\ Transport and Energy, \\ Arizona State University, \\ Tempe, AZ 85287, USA. \\ [email protected]}
\address {School of Mathematical and Statistical Sciences, \\ Arizona State University, \\ Tempe, AZ 85287, USA. \\ [email protected]}
\maketitle
\begin{abstract} \ \
Consider a system of~$K$ particles moving on the vertex set of a finite connected graph with at most one particle per vertex.
If there is one, the particle at~$x$ chooses one of the~$\deg (x)$ neighbors of its location uniformly at random at rate~$\rho_x$, and jumps
to that vertex if and only if it is empty.
Using standard probability techniques, we identify the set of invariant measures of this process to study the occupation time at each vertex.
Our main result shows that, though the occupation time at vertex~$x$ increases with~$\deg (x) / \rho_x$, the ratio of the occupation times at
two different vertices converges \emph{monotonically} to one as the number of particles increases to the number of vertices.
The occupation times are also computed explicitly for simple examples of finite connected graphs: the star and the path.
\end{abstract}
\begin{keyword}[class=AMS]
\kwd[Primary ]{60K35}
\end{keyword}
\begin{keyword}
\kwd{Interacting particle systems, simple exclusion process, occupation time, reversibility.}
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{sec:intro}
The simple exclusion process, introduced by Spitzer in~\cite{spitzer_1970}, is one of the most popular interacting particle systems with the voter
model~\cite{clifford_sudbury_1973, holley_liggett_1975} and the contact process~\cite{harris_1974}.
These three models can be viewed as spatial stochastic models of diffusion, competition, and invasion, respectively.
In particular, the simple exclusion process consists of a system of symmetric random walks that move independently on a connected graph except that
jumps onto already occupied vertices are suppressed (exclusion rule) so that each vertex is occupied by at most one particle. \\
\mathbf{1}ent All three models have been studied extensively on infinite lattices, and we refer to~\cite{liggett_1985, liggett_1999} for a review
of their main properties.
The voter model and the contact process have also been studied on the torus in~$\mathbb{Z}^d$ (the time to consensus of the voter model on the torus
is studied in~\cite{cox_1989} and the time to extinction of the contact process on the torus is studied in~\cite{durrett1, durrett2, durrett3})
as well as various finite deterministic and random graphs.
In contrast, to the best of our knowledge, there is no work about the simple exclusion process on finite graphs in the probability literature with
the notable exception of the asymmetric nearest neighbor exclusion process on the finite path (each particle jumps to its immediate left or right
with different probabilities) introduced in~\cite{liggett_1975} and reviewed in~\cite[chapter~III.3]{liggett_1999}.
The primary motivation, however, was to study the properties of the stationary distribution on a large path and relate them to those of the
infinite system. \\
\mathbf{1}ent The main objective of this paper is to initiate the study of simple exclusion processes on general finite connected graphs:
a particle at vertex~$x$ now jumps to a vertex chosen uniformly at random among the~$\deg (x)$ neighbor(s) of~$x$.
We also assume that the rate at which particles jump depend on their location, with the particle at~$x$ jumping at rate~$\rho_x$.
This modeling approach is motivated by engineering applications such as robotic swarms where particles represent robots moving on a finite
graph and avoiding each other.
An important question in this field is how should we choose the rates~$\rho_x$ so that the system converges to a fixed desired distribution?
But regardless of its potential applications, our model is natural and of interest mathematically, and as a first step we study the invariant
measures and prove a monotonicity property for the occupation times at different vertices that holds for all finite connected
graphs and all choices of the rates~$\rho_x$.
\section{Main results}
\label{sec:results}
Letting~$\mathscr{G} = (\mathscr{V},\mathscr{E})$ be a finite connected graph on~$N$ vertices, with vertex set~$\mathscr{V}$ and edge set~$\mathscr{E}$, the process is a continuous-time Markov
chain whose state at time~$t$ is a configuration
$$ \eta_t : \mathscr{V} \to \{0, 1 \} \quad \hbox{where} \quad \eta_t (x) =
\left\{\hspace*{-3pt} \begin{array}{rl} 0 & \hbox{if vertex~$x$ is empty} \vspace*{2pt} \\
1 & \hbox{if vertex~$x$ is occupied by a particle.} \end{array} \right. $$
Motivated by potential applications in robotic swarms, we assume that particles may jump at a rate that depends on their location, and
denote by~$\rho_x$ the rate attached to vertex~$x$.
To describe the dynamics, for all~$x, y \in \mathscr{V}$, we let~$\tau_{x, y} \,\eta$ be the configuration
$$ (\tau_{x, y} \,\eta) (z) = \eta (x) \,\mathbf{1} \{z = y \} + \eta (y) \,\mathbf{1} \{z = x \} + \eta (z) \,\mathbf{1} \{z \hspace*{-5pt}eq x, y \} $$
obtained from~$\eta$ by exchanging the states at~$x$ and~$y$.
Then, for all~$\eta, \xi \in \{0, 1 \}^{\mathscr{V}}$, the process jumps from configuration~$\eta$ to configuration~$\xi$ at rate
$$ q (\eta, \xi) = \frac{\rho_x}{\deg (x)} \ \mathbf{1} \{\eta_t (x) = 1 \ \hbox{and} \ \xi = \tau_{x, y} \,\eta \ \hbox{for some} \ (x, y) \in \mathscr{E} \}. $$
In words, the particle at~$x$, if there is one, chooses one of the neighbors of~$x$ uniformly at random at rate~$\rho_x$, and jumps to this vertex
if and only if it is empty.
It will be convenient later to identify each configuration~$\eta$ with the subset of vertices occupied by a particle:
$$ \eta \equiv \{x \in \mathscr{V} : \eta (x) = 1 \} \subset \mathscr{V}. $$
This defines a natural bijection between the set of configurations and the subsets of the vertex set, and it will be obvious from
the context whether~$\eta$ refers to a configuration or a subset. \\
\mathbf{1}ent The main objective of this paper is to study the fraction of time each vertex is occupied in the long run.
We can prove that these limits exist and only depend on the initial configuration through its number of particles, so we will write from now on
\begin{equation}
\label{eq:limits}
p_K (x) = \lim_{t \to \infty} P (x \in \eta_t \,| \,\card (\eta_0) = K) \quad \hbox{for all} \quad 0 < K \leq N \ \hbox{and} \ x \in \mathscr{V}.
\end{equation}
To state our results, for all~$0 < K \leq N$ and~$B \subset \mathscr{V}$, we define
$$ \Lambda_K^+ (B) = \{\eta \in \Lambda_K : B \cap \eta = B \} \quad \hbox{and} \quad
\Lambda_K^- (B) = \{\eta \in \Lambda_K : B \cap \eta = \varnothing \} $$
where~$\Lambda_K$ is the set of configurations with~$K$ particles.
In addition, for each vertex~$z \in \mathscr{V}$, each subset~$\eta \subset \mathscr{V}$, and each collection~$\mathscr{C}$ of subsets of~$\mathscr{V}$, we let
$$ D (z) = \frac{\deg (z)}{\rho_z}, \quad D (\eta) = \prod_{z \in \eta} \,D (z) \quad \hbox{and} \quad \Sigma (\mathscr{C}) = \sum_{\eta \in \mathscr{C}} \,D (\eta). $$
Using that the sets~$\Lambda_K$ are closed communication classes as well as reversibility to identify the stationary distribution on each~$\Lambda_K$,
we prove that the limits in~\eqref{eq:limits} are characterized as follows.
\begin{theorem}[occupation time] --
\label{th:limit}
For all~$0 < K \leq N$ and~$x \in \mathscr{V}$,
\begin{equation}
\label{eq:OccProb}
p_K (x) = \frac{\Sigma (\Lambda_K^+ (x))}{\Sigma (\Lambda_K)}.
\end{equation}
In particular, the limits in~\eqref{eq:limits} are characterized by
\begin{equation}
\label{eq:ratio}
\frac{p_K (x)}{p_K (y)} = \frac{\Sigma (\Lambda_K^+ (x))}{\Sigma (\Lambda_K^+ (y))} \quad \hbox{and} \quad \sum_{z \in \mathscr{V}} \,p_K (z) = K.
\end{equation}
\end{theorem}
Even though the right-hand side of~\eqref{eq:OccProb} cannot be simplified in general, some interesting properties can be deduced from this expression
for arbitrary finite connected graphs.
It is intuitively clear that, the graph being fixed, the probability~$p_K (x)$ increases with~$K$.
It can also be proved that this probability increases with~$D (x)$ while a more precise and challenging analysis shows that, though the occupation
time at~$x$ is smaller than the occupation time at~$y$ when~$D (x) < D (y)$,
the ratio of the two occupation times converges monotonically to one as the number of particles increases.
\begin{theorem}[monotonicity] --
\label{th:monotonicity}
For all~$1 < K < N - 1$,
$$ \frac{D (x)}{D (y)} = \frac{p_1 (x)}{p_1 (y)} < \frac{p_K (x)}{p_K (y)} < \frac{p_{K + 1} (x)}{p_{K + 1} (y)} < \frac{p_N (x)}{p_N (y)} = 1
\quad \hbox{when} \quad D (x) < D (y). $$
\end{theorem}
It follows from (the proof of) the theorem that, when all the~$D (x)$ are equal, all the vertices are equally likely to be occupied at equilibrium.
In particular, assuming for simplicity that the particles always jump at rate~$\rho_x \equiv 1$, all the vertices are occupied with the same
probability~$K/N$ for the process on finite regular graphs.
Along these lines, the probabilities in~\eqref{eq:OccProb} can be computed explicitly when~$\rho_x \equiv 1$ and most of the vertices have the
same degree.
For instance, for the star graph in which all the vertices have degree one except the center, we have the following result.
\begin{theorem}[star] --
\label{th:star}
For the star graph with~$N$ vertices and center~0,
$$ \begin{array}{rcl}
\displaystyle p_K (0) & \hspace*{-5pt} = \hspace*{-5pt} & \displaystyle \frac{(N - 1) K}{(N - 1) K + (N - K)} \vspace*{8pt} \\
\displaystyle P_K (x) & \hspace*{-5pt} = \hspace*{-5pt} & \displaystyle \bigg(\frac{K}{N - 1} \bigg) \frac{(N - 1) K - (K - 1)}{(N - 1) K + (N - K)}
\quad \hbox{for} \ \ x \hspace*{-5pt}eq 0. \end{array} $$
\end{theorem}
For the path graph in which all the vertices have degree two except the two endpoints, the algebra is a little more complicated but we can prove the
following result.
\begin{theorem}[path] --
\label{th:path}
For the path graph with vertex set~$\mathscr{V} = \{0, 1, \ldots, N - 1 \}$,
$$ \begin{array}{rcl}
\displaystyle p_K (x) & \hspace*{-5pt} = \hspace*{-5pt} & \displaystyle \frac{(2N - K - 1) K}{(K - 1) K + 4 (N - K)(N - 1)} \quad \hbox{for} \ \ x = 0, N - 1 \vspace*{8pt} \\
\displaystyle p_K (x) & \hspace*{-5pt} = \hspace*{-5pt} & \displaystyle \bigg(\frac{K}{N - 2} \bigg) \frac{(K - 2)(K - 1) + 4 (N - K)(N - 2)}{(K - 1) K + 4 (N - K)(N - 1)}
\quad \hbox{for} \ \ 0 < x < N - 1. \end{array} $$
\end{theorem}
Taking the ratio of the probabilities in both theorems~(see~\eqref{eq:star-5} and~\eqref{eq:path-5} below) gives results in agreement
with Theorem~\ref{th:monotonicity}.
Figure~\ref{fig:examples} shows~\eqref{eq:OccProb} obtained from numerical simulations of the simple exclusion process on the star, the path and the
two-dimensional grid with~25 vertices, along with our analytical results for the first two graphs.
\begin{figure}
\caption{\upshape{Limiting behavior of the simple exclusion process on three specific graphs with~25 vertices and~$\rho_x \equiv 1$.
The horizontal axis represents the number of particles~($K$), and the vertical axis represents the fraction of time vertices with
degree~$d$ are occupied ($p_K (x)$).
The squares are obtained from simulating the process for~$10^8$ units of time, while the curves in the first two pictures show the analytical
results from Theorems~\ref{th:star}
\label{fig:examples}
\end{figure}
\section{Proof of Theorem~\ref{th:limit}}
\label{sec:limit}
Note that the simple exclusion process is not irreducible because configurations with different numbers of particles do not communicate.
However, the set of configurations with~$K$ particles forms a closed communication class, so the process restricted to~$\Lambda_K$ is irreducible and
converges to a unique stationary distribution.
To find this stationary distribution and prove the theorem, we will also use reversibility.
To show that~$\Lambda_K$ is a communication class, we first prove that any two configurations in~$\Lambda_K$ that only differ in two vertices communicate.
\begin{lemma} --
\label{lem:x-y}
For all~$ \eta \in \{0, 1 \}^{\mathscr{V}}$ and~$x, y \in \mathscr{V}$,
$$ P ( \eta_t = \tau_{x, y} \, \eta \,| \, \eta_0 = \eta) > 0 \quad \hbox{for all} \quad t > 0. $$
\end{lemma}
\begin{proof}
The result is obvious when~$ \eta (x) = \eta (y)$ because in this case~$\tau_{x, y} \, \eta = \eta$.
The result is also clear when~$x$ and~$y$ are not in the same state but connected by an edge because
\begin{equation}
\label{eq:x-y-1}
q (\eta, \tau_{x, y} \,\eta) = \lim_{\epsilon \downarrow 0} \ \frac{P ( \eta_{t + \epsilon} = \tau_{x, y} \, \eta \,| \, \eta_t = \eta)}{\epsilon}
= \frac{\rho_x}{\deg (x)} \quad \hbox{for all} \quad (x, y) \in \mathscr{E}.
\end{equation}
To deal with the nontrivial case when the vertices are neither in the same state nor connected by an edge, we may assume without loss of generality
that, in configuration~$ \eta$, vertex~$x$ is occupied and vertex~$y$ empty.
Because the graph~$\mathscr{G}$ is connected, there exists a self-avoiding path
$$ (z_1, z_2, \ldots, z_l) \subset \mathscr{V} \quad \hbox{with} \quad (z_i, z_{i + 1}) \in \mathscr{E}, \ z_1 = x \ \hbox{and} \ z_l = y $$
connecting~$x$ and~$y$.
Due to the absence of cycles, we have also~$l \leq N$.
To remove the particle at~$x$ and put a particle at~$y$ without changing the state of the other vertices, we let
$$ \{i : z_i \in \eta \} = \{z_{i (1)}, z_{i (2)}, \ldots, z_{i (k)} \} \quad \hbox{with} \quad 1 = i (1) < i (2) < \cdots < i (k) < l $$
be the set of vertices along the self-avoiding path that are occupied in configuration~$ \eta$.
It is also convenient to set~$i (k + 1) = l$.
To obtain configuration~$\tau_{x, y} \, \eta$ from~$ \eta$, the basic idea, which is illustrated in Figure~\ref{fig:move}, is to move the particle
at~$z_{i (k)}$ to~$z_{i (k + 1)}$, then the particle at~$z_{i (k - 1)}$ to~$z_{i (k)}$, and so on.
To prove that this sequence of events indeed occurs with positive probability, note that, because the
vertices~$z_{i (k) + 1}, z_{i (k) + 2}, \ldots, z_{i (k + 1)}$ are empty,
$$ \tau_{z_{i (k)}, z_i (k + 1)} \,\eta =
(\tau_{z_{i (k + 1) - 1}, z_{i (k + 1)}} \circ \cdots \circ \tau_{z_{i (k) + 1}, z_{i (k) + 2}} \circ \tau_{z_{i (k)}, z_{i (k) + 1}})(\eta). $$
This, together with~\eqref{eq:x-y-1}, implies that
$$ P (\eta_t = \tau_{z_{i (k)}, z_{i (k + 1)}} \,\eta \,| \,\eta_0 = \eta) > 0 \quad \hbox{for all} \quad t > 0. $$
\begin{figure}
\caption{\upshape{Evolution along the self-avoiding path~$(z_1, z_2, \ldots, z_l)$.}
\label{fig:move}
\end{figure}
Similarly, we prove by induction that, for~$j = 1, 2, \ldots, k$,
\begin{equation}
\label{eq:x-y-2}
\begin{array}{l}
P ( \eta_t = \tau_{z_{i (j)}, z_{i (k + 1)}} \, \eta \,| \, \eta_0 = \tau_{z_{i (j + 1)}, z_{i (k + 1)}} \, \eta) \vspace*{4pt} \\ \hspace*{40pt} =
P ( \eta_t = \tau_{z_{i (j)}, z_{i (j + 1)}} (\tau_{z_{i (j + 1)}, z_{i (k + 1)}} \, \eta) \,| \, \eta_0 = \tau_{z_{i (j + 1)}, z_{i (k + 1)}} \, \eta) > 0
\end{array}
\end{equation}
for all~$t > 0$.
In addition, for~$j = 1, 2, \ldots, k$,
\begin{equation}
\label{eq:x-y-3}
\tau_{z_{i (j)}, z_{i (k + 1)}} \, \eta =
(\tau_{z_{i (j)}, z_{i (j + 1)}} \circ \cdots \circ \tau_{z_{i (k - 1)}, z_{i (k)}} \circ \tau_{z_{i (k)}, z_{i (k + 1)}})( \eta).
\end{equation}
Using~\eqref{eq:x-y-2} and~\eqref{eq:x-y-3}, and that~$z_{i (1)} = x$ and~$z_{i (k + 1)} = y$, we deduce that
$$ \begin{array}{l}
\displaystyle P ( \eta_{kt} = \tau_{x, y} \, \eta \,| \, \eta_0 = \eta) \geq
\displaystyle \prod_{j = 1}^k \,P ( \eta_{(k - j + 1) t} = \tau_{z_{i (j)}, z_{i (k + 1)}} \, \eta \,| \, \eta_{(k - j) t} = \tau_{z_{i (j + 1)}, z_{i (k + 1)}} \, \eta) \vspace*{-8pt} \\ \hspace*{60pt} =
\displaystyle \prod_{j = 1}^k \,P ( \eta_t = \tau_{z_{i (j)}, z_{i (k + 1)}} \, \eta \,| \, \eta_0 = \tau_{z_{i (j + 1)}, z_{i (k + 1)}} \, \eta) > 0 \end{array} $$
for all~$t > 0$.
This completes the proof.
\end{proof}
\begin{lemma} --
\label{lem:class}
For all~$K = 0, 1, \ldots, N$, the set~$\Lambda_K$ is a closed communication class.
\end{lemma}
\begin{proof}
The fact that~$\Lambda_K$ is closed is an immediate consequence of the fact that the number~$K$ of particles is preserved by the dynamics.
To prove that this set is also a communication class, fix two configurations~$ \eta$ and~$ \xi$ with~$K$ particles, and define the sets
$$ S = \{z \in \mathscr{V} : \eta (z) = 1, \, \xi (z) = 0 \} \quad \hbox{and} \quad T = \{z \in \mathscr{V} : \eta (z) = 0, \, \xi (z) = 1 \} $$
that we call respectively the source set and the target set.
Because~$ \eta$ and~$ \xi$ have the same number of particles, these sets have the same number of vertices, and we write
$$ S = \{x_1, x_2, \ldots, x_k \} \subset \mathscr{V} \quad \hbox{and} \quad T = \{y_1, y_2, \ldots, y_k \} \subset \mathscr{V}. $$
By definition of the source and target sets,
$$ \xi = (\tau_{x_1, y_1} \circ \tau_{x_2, y_2} \circ \cdots \circ \tau_{x_k, y_k})( \eta). $$
In particular, letting~$\sigma_j = \tau_{x_1, y_1} \circ \cdots \circ \tau_{x_j, y_j}$ and using Lemma~\ref{lem:x-y}, we get
$$ \begin{array}{l}
\displaystyle P ( \eta_{kt} = \xi \,| \, \eta_0 = \eta) \geq
\displaystyle \prod_{j = 1}^k P ( \eta_{jt} = \sigma_j \, \eta \,| \, \eta_{(j - 1) t} = \sigma_{j - 1} \, \eta) \vspace*{-8pt} \\ \hspace*{140pt} =
\displaystyle \prod_{j = 1}^k P ( \eta_t = \tau_{x_j, y_j} (\sigma_{j - 1} \, \eta) \,| \, \eta_0 = \sigma_{j - 1} \, \eta) > 0 \end{array} $$
for all~$t > 0$.
Since this holds for any two configurations in~$\Lambda_K$ and since configurations outside~$\Lambda_K$ cannot be reached from~$\Lambda_K$,
the result follows.
\end{proof} \\ \\
We now use reversibility to identify the stationary distributions.
\begin{lemma} --
\label{lem:reversible}
For all~$K = 0, 1, \ldots, N$, the distribution
$$ \pi_K ( \eta) = \frac{D (\eta)}{\Sigma (\Lambda_K)} \quad \hbox{for all} \quad \eta \in \Lambda_K $$
is a reversible distribution concentrated on~$\Lambda_K$.
\end{lemma}
\begin{proof}
Let~$ \eta, \xi \in \Lambda_K$, $ \eta \hspace*{-5pt}eq \xi$. Then,
$$ q ( \eta, \xi) = q ( \xi, \eta) = 0 \quad \hbox{when} \quad \xi \hspace*{-5pt}eq \tau_{x, y} \, \eta \quad \hbox{for all} \quad (x, y) \in \mathscr{E} $$
in which case it is clear that
\begin{equation}
\label{eq:reversible-1}
\pi_K ( \eta) \,q ( \eta, \xi) = \pi_K ( \xi) \,q ( \xi, \eta) = 0.
\end{equation}
When~$\xi = \tau_{x, y} \,\eta$ for some~$(x, y) \in \mathscr{E}$, with say~$ \eta (x) = \xi (y) = 1$,
$$ q (\eta, \xi) = \frac{\rho_x}{\deg (x)} \quad \hbox{and} \quad q ( \xi, \eta) = \frac{\rho_y}{\deg (y)}. $$
In this case, because~$\eta \setminus \{x \} = \xi \setminus \{y \}$,
\begin{equation}
\label{eq:reversible-2}
\begin{array}{rcl}
\displaystyle \pi_K ( \eta) \,q ( \eta, \xi) & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle \frac{D (\eta)}{\Sigma (\Lambda_K)} \ q ( \eta, \xi) =
\displaystyle \frac{D (\eta \setminus \{x \})}{\Sigma (\Lambda_K)} \bigg(\frac{\deg (x)}{\rho_x} \bigg) q ( \eta, \xi) \vspace*{8pt} \\ & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle \frac{D (\eta \setminus \{x \})}{\Sigma (\Lambda_K)} =
\displaystyle \frac{D (\xi \setminus \{y \})}{\Sigma (\Lambda_K)} =
\displaystyle \pi_K ( \xi) \,q ( \xi, \eta). \end{array}
\end{equation}
Combining~\eqref{eq:reversible-1} and~\eqref{eq:reversible-2} gives the result.
\end{proof} \\ \\
\begin{proofof}{Theorem~\ref{th:limit}}
According to Lemma~\ref{lem:reversible}, $\pi_K$ is a reversible distribution so this is also a stationary distribution.
See e.g.~\cite[Sec.~10.3]{lanchier_2017} for a proof.
Now, according to~Lemma~\ref{lem:class}, the set~$\Lambda_K$ is a finite closed communication class for the simple exclusion process therefore there
is a unique stationary distribution that concentrates on~$\Lambda_K$, and this distribution is the limit of the process starting
from~$\eta_0 \in \Lambda_K$.
See e.g.~\cite[Sec.~10.4]{lanchier_2017} for a proof.
In particular,
$$ \lim_{t \to \infty} P (\eta_t = \eta \,| \,\eta_0 = \xi) = \pi_K (\eta) = \frac{D (\eta)}{\Sigma (\Lambda_K)} \quad
\hbox{for all} \quad \eta, \xi \in \Lambda_K $$
from which it follows that
$$ p_K (x) = \sum_{\eta \in \Lambda_K : x \in \eta} \pi_K (\eta)
= \sum_{\eta \in \Lambda_K : x \in \eta} \frac{D (\eta)}{\Sigma (\Lambda_K)}
= \sum_{\eta \in \Lambda_K^+ (x)} \frac{D (\eta)}{\Sigma (\Lambda_K)}
= \frac{\Sigma (\Lambda_K^+ (x))}{\Sigma (\Lambda_K)}. $$
This shows the second part of the theorem, and
$$ \frac{p_K (x)}{p_K (y)} = \frac{\Sigma (\Lambda_K^+ (x))}{\Sigma (\Lambda_K)} \ \frac{\Sigma (\Lambda_K)}{\Sigma (\Lambda_K^+ (x))}
= \frac{\Sigma (\Lambda_K^+ (x))}{\Sigma (\Lambda_K^+ (y))}. $$
Finally, using that the expected value is linear, we get
$$ \sum_{z \in \mathscr{V}} \,p_K (z) = \lim_{t \to \infty} \sum_{z \in \mathscr{V}} \,E (\mathbf{1} \{z \in \eta_t \})
= \lim_{t \to \infty} E \bigg(\sum_{z \in \mathscr{V}} \,\mathbf{1} \{z \in \eta_t \} \bigg) = E (K) = K. $$
This completes the proof.
\end{proofof}
\section{Proof of Theorem~\ref{th:monotonicity}}
\label{sec:monotonicity}
In the presence of one or~$N$ particles, we have
\begin{equation}
\label{eq:monotonicity-0}
\frac{\Sigma (\Lambda_1^+ (x))}{\Sigma (\Lambda_1^+ (y))} = \frac{D (x)}{D (y)} = \frac{\rho_y \deg (x)}{\rho_x \deg (y)} \quad \hbox{and} \quad
\frac{\Sigma (\Lambda_N^+ (x))}{\Sigma (\Lambda_N^+ (y))} = \frac{D (\mathscr{V})}{D (\mathscr{V})} = 1.
\end{equation}
In all the other cases, however, the ratios above become much more complicated.
Also, the theorem cannot be proved using direct calculations.
The main ingredient is given by the next lemma whose proof relies on a somewhat sophisticated construction.
\begin{lemma} --
\label{lem:injection}
For all~$0 < K < N$, we have~$\Sigma (\Lambda_{K + 1}) \,\Sigma (\Lambda_{K - 1}) < (\Sigma (\Lambda_K))^2$.
\end{lemma}
\begin{proof}
The key is to find partitions~$\mathcal P$ of $\Lambda_{K + 1} \times \Lambda_{K - 1}$ and~$\mathcal Q$ of~$\Lambda_K \times \Lambda_K$ such that
\begin{enumerate}
\item partition~$\mathcal P$ has less elements than partition~$\mathcal Q$, \vspace*{4pt}
\item each~$A_i \in \mathcal P$ can be paired with a~$B_i \in \mathcal Q$ such that~$\card (A_i) \leq \card (B_i)$, \vspace*{4pt}
\item for all~$(\eta, \eta') \in A_i$ and~$(\xi, \xi') \in B_i$, we have~$D (\eta) \,D (\eta') = D (\xi) \,D (\xi')$.
\end{enumerate}
Let $x_1, x_2, \ldots, x_N $ denote the $N$ vertices.
To construct the partitions, let
$$ S_{2K} = \{(u_1, u_2, \ldots, u_N) \in \{0, 1, 2 \}^N : u_1 + \cdots + u_N = 2K \} $$
and~$\phi : \Lambda_{K + 1} \times \Lambda_{K - 1} \to S_{2K}$ and~$\psi : \Lambda_K \times \Lambda_K \to S_{2K}$ defined as
\begin{equation}
\label{eq:injection-0}
\begin{array}{rclcrcl}
\phi (\eta, \eta') & \hspace*{-5pt} = \hspace*{-5pt} & u = (u_1, u_2, \ldots, u_N) & \hbox{where} & u_i & \hspace*{-5pt} = \hspace*{-5pt} & \mathbf{1} \{x_i \in \eta \} + \mathbf{1} \{x_i \in \eta' \} \vspace*{4pt} \\
\psi (\xi, \xi') & \hspace*{-5pt} = \hspace*{-5pt} & u = (u_1, u_2, \ldots, u_N) & \hbox{where} & u_i & \hspace*{-5pt} = \hspace*{-5pt} & \mathbf{1} \{x_i \in \xi \} + \mathbf{1} \{x_i \in \xi' \}. \end{array}
\end{equation}
The two functions have the same expression but differ in that they are not defined on the same sets of configurations. The two partitions are then given by
$$ \begin{array}{rcl}
\mathcal P & \hspace*{-5pt} = \hspace*{-5pt} & \{\phi^{-1} (u) : u \in S_{2K} \ \hbox{and} \ \phi^{-1} (u) \hspace*{-5pt}eq \varnothing \} \vspace*{4pt} \\
\mathcal Q & \hspace*{-5pt} = \hspace*{-5pt} & \{\psi^{-1} (u) : v \in S_{2K} \ \hbox{and} \ \psi^{-1} (u) \hspace*{-5pt}eq \varnothing \}. \end{array} $$
See Figure~\ref{fig:injection} for a schematic representation of the two partitions and an illustration of the proof presented below.
The function~$\psi$ is surjective.
In contrast, $\phi (\eta, \eta')$ has~$\card (\eta \setminus \eta') \geq 2$ coordinates equal to one therefore it is not surjective:
$$ S_{2K}^* = \{u \in S_{2K} : \phi^{-1} (u) \hspace*{-5pt}eq \varnothing \} \hspace*{-5pt}eq S_{2K} $$
from which it follows that
\begin{equation}
\label{eq:injection-1}
\card (\mathcal P) = \card (S_{2K}^*) < \card (S_{2K}) = \card (\mathcal Q).
\end{equation}
\begin{figure}
\caption{\upshape{Construction in the proof of Lemma~\ref{lem:injection}
\label{fig:injection}
\end{figure}
This proves the first item above. Now, let
$$ K_1 = \card \{i : u_i = 1 \} \quad \hbox{and} \quad K_2 = \card \{i : u_i = 2 \}, $$
with~$K_1 \geq 2$.
To count the number of preimages~$(\eta, \eta')$ and~$(\xi, \xi')$ in~\eqref{eq:injection-0}, note that the vertices that are either empty
or occupied in all four configurations are uniquely determined by~$u$.
This leaves~$K_1$ vertices that are occupied in two of the four configurations and
\begin{itemize}
\item the number of choices for~$\eta$ is the number of choices of~$K + 1 - K_2$ vertices among~$K_1$ vertices to be occupied in configuration~$\eta$ but not~$\eta'$, \vspace*{4pt}
\item the number of choices for~$\xi$ is the number of choices of~$K - K_2$ vertices among~$K_1$ vertices to be occupied in configuration~$\xi$ but not~$\xi'$.
\end{itemize}
Using also that~$K_1 + 2K_2 = 2K < 2K + 1$, we get
\begin{equation}
\label{eq:injection-2}
\begin{array}{rcl}
\displaystyle \card (\phi^{-1} (u)) & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle {K_1 \choose K + 1 - K_2} = \bigg(\frac{K_1 - K + K_2}{K + 1 - K_2} \bigg) {K_1 \choose K - K_2} \vspace*{8pt} \\ & \hspace*{-5pt} < \hspace*{-5pt} &
\displaystyle {K_1 \choose K - K_2} = \card (\psi^{-1} (u)), \end{array}
\end{equation}
which shows the second item above.
Finally, for all~$(\eta, \eta') \in \phi^{-1} (u)$,
$$ \begin{array}{rcl}
D (\eta) \,D (\eta') & \hspace*{-5pt} = & \displaystyle \prod_{z \in \eta} \,\frac{\deg (z)}{\rho_z} \prod_{z \in \eta'} \,\frac{\deg (z)}{\rho_z} =
\displaystyle \prod_{z \in \eta \Delta \eta'} \frac{\deg (z)}{\rho_z}
\displaystyle \prod_{z \in \eta \cap \eta'} \bigg(\frac{\deg (z)}{\rho_z} \bigg)^2 \vspace*{4pt} \\
& \hspace*{-5pt} = & \displaystyle \prod_{i = 1}^N \ \bigg(\frac{\deg (x_i)}{\rho_{x_i}} \bigg)^{u_i} = \widehat D (u) \end{array} $$
is a function~$\widehat D (u)$ of the vector~$u$ only.
The same holds for~$(\xi, \xi') \in \psi^{-1} (u)$.
This implies that the third item above is also satisfied in the sense that
\begin{equation}
\label{eq:injection-3}
D (\eta) \,D (\eta') = D (\xi) \,D (\xi') = \widehat D (u) \quad \hbox{for all} \quad (\eta, \eta') \in \phi^{-1} (u), (\xi, \xi') \in \psi^{-1} (u).
\end{equation}
Combining~\eqref{eq:injection-1}--\eqref{eq:injection-3}, we conclude that
$$ \begin{array}{rcl}
\displaystyle \Sigma (\Lambda_{K + 1}) \,\Sigma (\Lambda_{K - 1}) & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle \sum_{u \in S_{2K}^*} \ \sum_{(\eta, \eta') \in \phi^{-1} (u)} D (\eta) \,D (\eta') \ \overset{\eqref{eq:injection-3}}{=}
\displaystyle \sum_{u \in S_{2K}^*} \ \sum_{(\eta, \eta') \in \phi^{-1} (u)} \widehat D (u) \vspace*{8pt} \\ & \hspace*{-5pt} \overset{\eqref{eq:injection-2}}{<} \hspace*{-5pt} &
\displaystyle \sum_{u \in S_{2K}^*} \ \sum_{(\xi, \xi') \in \psi^{-1} (u)} \widehat D (u) \ \overset{\eqref{eq:injection-3}}{=}
\displaystyle \sum_{u \in S_{2K}^*} \ \sum_{(\xi, \xi') \in \psi^{-1} (u)} D (\xi) \,D (\xi') \vspace*{8pt} \\ & \hspace*{-5pt} \overset{\eqref{eq:injection-1}}{<} \hspace*{-5pt} &
\displaystyle \sum_{u \in S_{2K}} \ \sum_{(\xi, \xi') \in \psi^{-1} (u)} D (\xi) \,D (\xi') =
\displaystyle (\Sigma (\Lambda_K))^2. \end{array} $$
This completes the proof.
\end{proof} \\ \\
With the previous technical lemma, we can now prove the theorem. \\ \\
\begin{proofof}{Theorem~\ref{th:monotonicity}}
To simplify the notations, we write
$$ A_K = \Sigma (\Lambda_K^+ (\{x, y \})) \quad \hbox{and} \quad B_K = \Sigma (\Lambda_{K - 1}^- (\{x, y \})). $$
Then, we can rewrite
$$ \begin{array}{rcl}
\Sigma (\Lambda_K^+ (x)) & \hspace*{-5pt} = \hspace*{-5pt} &
\Sigma (\Lambda_K^+ (x) \cap \Lambda_K^+ (y)) + \Sigma (\Lambda_K^+ (x) \setminus \Lambda_K^+ (y)) \vspace*{4pt} \\ \hspace*{80pt} & \hspace*{-5pt} = \hspace*{-5pt} &
\Sigma (\Lambda_K^+ (\{x, y \})) + D (x) \,\Sigma (\Lambda_{K - 1}^- (\{x, y \})) = A_K + D (x) \,B_K. \end{array} $$
Using some obvious symmetry, we deduce that
\begin{equation}
\label{eq:monotonicity-1}
\frac{p_K (x)}{p_K (y)} = \frac{\Sigma (\Lambda_K^+ (x))}{\Sigma (\Lambda_K^+ (y))} = \frac{A_K + D (x) \,B_K}{A_K + D (y) \,B_K}.
\end{equation}
Now, applying Lemma~\ref{lem:injection} to the configurations on~$\mathscr{V} - \{x, y \}$, we get
$$ \begin{array}{rcl}
A_K B_{K + 1} & \hspace*{-5pt} = \hspace*{-5pt} & \Sigma (\Lambda_K^+ (\{x, y \})) \,\Sigma (\Lambda_K^- (\{x, y \})) \vspace*{4pt} \\
& \hspace*{-5pt} = \hspace*{-5pt} & D (\{x, y \}) \,\Sigma (\Lambda_{K - 2}^- (\{x, y \})) \,\Sigma (\Lambda_K^- (\{x, y \})) \vspace*{4pt} \\
& \hspace*{-5pt} < \hspace*{-5pt} & D (\{x, y \}) \,\Sigma (\Lambda_{K - 1}^- (\{x, y \})) \,\Sigma (\Lambda_{K - 1}^- (\{x, y \})) \vspace*{4pt} \\
& \hspace*{-5pt} = \hspace*{-5pt} & \Sigma (\Lambda_{K + 1}^+ (\{x, y \})) \,\Sigma (\Lambda_{K - 1}^- (\{x, y \})) = A_{K + 1} B_K. \end{array} $$
This, together with~$D (x) < D (y)$, implies that
$$ D (x) \,A_{K + 1} B_K + D (y) \,A_K B_{K + 1} < D (x) \,A_K B_{K + 1} + D (y) \,A_{K + 1} B_K $$
which is equivalent to
\begin{equation}
\label{eq:monotonicity-2}
\begin{array}{l}
(A_K + D (x) \,B_K)(A_{K + 1} + D (y) \,B_{K + 1}) \vspace*{4pt} \\ \hspace*{80pt} < \
(A_{K + 1} + D (x) \,B_{K + 1})(A_K + D (y) \,B_K). \end{array}
\end{equation}
Combining~\eqref{eq:monotonicity-1} and~\eqref{eq:monotonicity-2} gives
\begin{equation}
\label{eq:monotonicity-3}
\frac{P_K (x \in \eta_t)}{P_K (y \in \eta_t)} = \frac{A_K + D (x) \,B_K}{A_K + D (y) \,B_K}
= \frac{A_{K + 1} + D (x) \,B_{K + 1}}{A_{K + 1} + D (y) \,B_{K + 1}}
= \frac{p_{K + 1} (x)}{p_{K + 1} (y)}.
\end{equation}
The theorem is then a combination of~\eqref{eq:monotonicity-0} and~\eqref{eq:monotonicity-3}.
\end{proofof}
\section{Star and path graphs}
\label{sec:star-path}
We now use Theorem~\ref{th:limit} to find the limiting fraction of time each vertex is occupied in the simple exclusion process on the star and the path
graphs with~$\rho_x \equiv 1$.
The reason why an explicit calculation is possible in these cases is because most of the vertices have the same degree. \\ \\
\begin{proofof}{Theorem~\ref{th:star}}
The center~0 is connected to all the other~$N - 1$ vertices therefore its degree is~$N - 1$.
The other vertices~$1, 2, \ldots, N - 1$, called leaves, are only connected to the center and therefore have degree one.
For the center, we compute
\begin{equation}
\label{eq:star-1}
\Sigma (\Lambda_K^+ (0)) = \deg (0) \ \Sigma (\Lambda_{K - 1}^- (0)) = (N - 1) {N - 1 \choose K - 1},
\end{equation}
while for all~$x = 1, 2, \ldots, N - 1$, we have
$$ \begin{array}{rcl}
\Sigma (\Lambda_K^+ (x)) & \hspace*{-5pt} = \hspace*{-5pt} & \deg (x) \ \Sigma (\Lambda_{K - 1}^- (x)) = \Sigma (\Lambda_{K - 1}^- (x)) \vspace*{4pt} \\ & \hspace*{-5pt} = \hspace*{-5pt} &
\Sigma (\Lambda_{K - 1}^- (x) \cap \Lambda_{K - 1}^+ (0)) + \Sigma (\Lambda_{K - 1}^- (x) \cap \Lambda_{K - 1}^- (0)) \vspace*{4pt} \\ & \hspace*{-5pt} = \hspace*{-5pt} &
\deg (0) \ \Sigma (\Lambda_{K - 2}^- (0, x)) + \Sigma (\Lambda_{K - 1}^- (0, x)) \end{array} $$
from which it follows that
\begin{equation}
\label{eq:star-2}
\Sigma (\Lambda_K^+ (x)) = (N - 1) {N - 2 \choose K - 2} + {N - 2 \choose K - 1}.
\end{equation}
Note also that the total mass is
\begin{equation}
\label{eq:star-3}
\Sigma (\Lambda_K) = \Sigma (\Lambda_K^+ (0)) + \Sigma (\Lambda_K^- (0)) = (N - 1) {N - 1 \choose K - 1} + {N - 1 \choose K}.
\end{equation}
Combining~\eqref{eq:star-1}--\eqref{eq:star-3}, using the identities
$$ {N - 1 \choose K - 1} = \frac{N - 1}{K - 1} \ {N - 2 \choose K - 2} = \frac{N - 1}{N - K} \ {N - 2 \choose K - 1} = \frac{K}{N - K} \ {N - 1 \choose K}, $$
applying Theorem~\ref{th:limit}, and simplifying, we deduce that
\begin{equation}
\label{eq:star-4}
\begin{array}{rcl}
\displaystyle p_K (0) = \frac{\Sigma (\Lambda_K^+ (0))}{\Sigma (\Lambda_K)} & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle \frac{(N - 1) K}{(N - 1) K + (N - K)} \vspace*{8pt} \\
\displaystyle p_K (x) = \frac{\Sigma (\Lambda_K^+ (x))}{\Sigma (\Lambda_K)} & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle \bigg(\frac{K}{N - 1} \bigg) \frac{(N - 1) K - (K - 1)}{(N - 1) K + (N - K)}. \end{array}
\end{equation}
Using again~\eqref{eq:star-1} and~\eqref{eq:star-2}, or alternatively~\eqref{eq:star-4}, we also get
\begin{equation}
\label{eq:star-5}
\frac{p_K (x)}{p_K (0)} = \frac{\Sigma (\Lambda_K^+ (x))}{\Sigma (\Lambda_K^+ (0))} = \frac{(N - 1)(K - 1) + (N - K)}{(N - 1)^2}.
\end{equation}
The right-hand side is equal to~$1 / (N - 1)$ when~$K = 1$ and one when~$K = N$, and is increasing with respect to the number of particles, in accordance
with Theorem~\ref{th:monotonicity}.
Note also that, for the star graph, the ratio above is linear in the number~$K$ of particles.
\end{proofof} \\ \\
\begin{proofof}{Theorem~\ref{th:path}}
The end vertices~0 and~$N - 1$ each have degree one while the other vertices~$1, 2, \ldots, N - 2$, each have degree two.
For the end nodes, we compute
$$ \begin{array}{rcl}
\Sigma (\Lambda_K^+ (0)) & \hspace*{-5pt} = \hspace*{-5pt} & \deg (0) \ \Sigma (\Lambda_{K - 1}^- (0)) = \Sigma (\Lambda_{K - 1}^- (0)) \vspace*{4pt} \\ & \hspace*{-5pt} = \hspace*{-5pt} &
\Sigma (\Lambda_{K - 1}^- (0) \cap \Lambda_{K - 1}^+ (N - 1)) + \Sigma (\Lambda_{K - 1}^- (0) \cap \Lambda_{K - 1}^- (N - 1)) \vspace*{4pt} \\ & \hspace*{-5pt} = \hspace*{-5pt} &
\deg (N - 1) \ \Sigma (\Lambda_{K - 2}^- (0, N - 1)) + \Sigma (\Lambda_{K - 1}^- (0, N - 1)) \end{array} $$
from which it follows that
\begin{equation}
\label{eq:path-1}
\Sigma (\Lambda_K^+ (0)) = \Sigma (\Lambda_K^+ (N - 1)) = 2^{K - 2} {N - 2 \choose K - 2} + 2^{K - 1} {N - 2 \choose K - 1}.
\end{equation}
Similarly, for all~$0 < x < N - 1$,
$$ \Sigma (\Lambda_K^+ (x)) = \deg (x) \ \Sigma (\Lambda_{K - 1}^- (x)) = 2 \,\Sigma (\Lambda_{K - 1}^- (x)), $$
and including and/or excluding~0 and/or~$N - 1$, we get
\begin{equation}
\label{eq:path-2}
\begin{array}{rcl}
\Sigma (\Lambda_K^+ (x)) & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle 2^{K - 2} {2 \choose 2}{N - 3 \choose K - 3} + 2^{K - 1} {2 \choose 1}{N - 3 \choose K - 2} + 2^K {2 \choose 0}{N - 3 \choose K - 1} \vspace*{8pt} \\ & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle 2^{K - 2} {N - 3 \choose K - 3} + 2^K {N - 3 \choose K - 2} + 2^K {N - 3 \choose K - 1} \vspace*{8pt} \\ & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle 2^{K - 2} {N - 3 \choose K - 3} + 2^K {N - 2 \choose K - 1}. \end{array}
\end{equation}
The total mass in this case is
\begin{equation}
\label{eq:path-3}
\begin{array}{rcl}
\Sigma (\Lambda_K) & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle 2^{K - 2} {2 \choose 2}{N - 2 \choose K - 2} + 2^{K - 1} {2 \choose 1}{N - 2 \choose K - 1} +
2^K {2 \choose 0}{N - 2 \choose K} \vspace*{8pt} \\ & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle 2^{K - 2} {N - 2 \choose K - 2} + 2^K {N - 1 \choose K}. \end{array}
\end{equation}
Combining~\eqref{eq:path-1}--\eqref{eq:path-3}, using the identities
$$ \begin{array}{rcl}
\displaystyle {N - 2 \choose K - 1} & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle \frac{N - K}{K - 1} \ {N - 2 \choose K - 2} \vspace*{8pt} \\ & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle \frac{(N - 2)(N - K)}{(K - 2)(K - 1)} \ {N - 3 \choose K - 3} = \frac{K}{N - 1} \ {N - 1 \choose K}, \end{array} $$
applying Theorem~\ref{th:limit}, and simplifying, we deduce that
\begin{equation}
\label{eq:path-4}
\begin{array}{rcl}
\displaystyle p_K (0) = \frac{\Sigma (\Lambda_K^+ (0))}{\Sigma (\Lambda_K)} & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle \frac{(2N - K - 1) K}{(K - 1) K + 4 (N - K)(N - 1)} \vspace*{8pt} \\
\displaystyle p_K (x) = \frac{\Sigma (\Lambda_K^+ (x))}{\Sigma (\Lambda_K)} & \hspace*{-5pt} = \hspace*{-5pt} &
\displaystyle \bigg(\frac{K}{N - 2} \bigg) \frac{(K - 2)(K - 1) + 4 (N - K)(N - 2)}{(K - 1) K + 4 (N - K)(N - 1)}. \end{array}
\end{equation}
Using again~\eqref{eq:path-1} and~\eqref{eq:path-2}, or alternatively~\eqref{eq:path-4}, we also get
\begin{equation}
\label{eq:path-5}
\frac{p_K (0)}{p_K (x)} = \frac{\Sigma (\Lambda_K^+ (0))}{\Sigma (\Lambda_K^+ (x))} = \frac{(N - 2)(2N - K - 1)}{(K - 2)(K - 1) + 4 (N - K)(N - 2)}.
\end{equation}
The right-hand side is equal to one-half when~$K = 1$ and one when~$K = N$, and is increasing with respect to the number of particles, in accordance
with Theorem~\ref{th:monotonicity}.
\end{proofof}
\end{document} |
\begin{document}
\mainmatter
\title{Non-monotone Behavior of the Heavy Ball Method}
\titlerunning{Non-monotone Behavior of the Heavy Ball Method}
\author{Marina Danilova\inst{1}, Anastasiya Kulakova\inst{2} and Boris Polyak\inst{3}}
\authorrunning{M. Danilova, A. Kulakova, B. Polyak}
\institute{
Institute of Control Sciences RAS,
Laboratory of Adaptive and Robust Systems,
Profsoyuznaya, 65,
Moscow 117342,
Russia \\
\email{[email protected]}
\and
Moscow Institute of Physics and Technology,
Department of Control and Applied Mathematics,
Institutskiy per., 9,
Dolgoprudniy 141700,
Russia \\
\email{[email protected]}
\and
Institute of Control Sciences RAS,
Laboratory of Adaptive and Robust Systems,
Profsoyuznaya, 65,
Moscow 117342,
Russia \\
\email{[email protected]}}
\maketitle
\begin{abstract}
We focus on the solutions of second-order stable linear difference equations and demonstrate that their behavior can be non-monotone and exhibit peak effects depending on initial conditions. The results are applied to the analysis of the accelerated unconstrained optimization method --- the Heavy Ball method. We explain non-standard behavior of the method discovered in practical applications. In addition, such non-monotonicity complicates the correct choice of the parameters in optimization methods. We propose to overcome this difficulty by introducing new Lyapunov function which should decrease monotonically. By use of this function convergence of the method is established under less restrictive assumptions (for instance, with the lack of convexity). We also suggest some restart techniques to speed up the method's convergence.
\keywords{difference equations, optimization methods, non-monotone behavior, the Heavy Ball method, Lyapunov function, global convergence}
\end{abstract}
\section{Introduction}
It is well known that $n$-th order scalar linear difference equations
\[
x_k + a_1x_{k-1} + \dots + a_n x_{k-n} = 0,\ k = n,\ n+1,\ \dots;\ a_i\in \mathbb{R}
\]
with initial conditions
$$x^{(0)} = (x_0, \dots, x_{n-1})\in \mathbb{R}^n$$
are stable, i.e. $\lim\limits_{k\rightarrow\infty} x_k=0$, if and only if the moduli of the roots $\lambda_i$
of the characteristic polynomial
\begin{equation*}\label{poly}
p(\lambda) = \lambda^n + a_1 \lambda^{n-1} + \dots + a_{n-1} \lambda + a_n
\end{equation*}
are less than 1 \cite{El}.
However, the convergence to zero can be non-monotone. This effect can be described by the following quantity:
\begin{equation*}\label{eta}
\eta(x^{(0)}) = \max_{k=n, n+1, \dots}|x_k|,
\end{equation*}
which will be referred to as peak of the solution (provided that $\eta(x^{(0)}) > 1$) for a given root location $\lambda$ and initial condition $x^{(0)}$. Without loss of generality we assume that $\|x^{(0)}\|_{\infty}\leq 1$. The estimates of $\eta(x^{(0)})$ are demonstrated in the recent paper \cite{shch2018}.
The main objective of the present paper is to link the peak effects in linear difference equations with the non-monotone behavior of such unconstrained optimization methods as the Heavy Ball method~\cite{Polyak_1,polyak1987}, Nesterov's accelerated gradient method~\cite{nest2004} and, for example, the recently proposed triple momentum method~\cite{fast2018}.
When applied to a quadratic function, momentum methods are described by second-order linear matrix difference equations. In numerous simulations it was found out that the methods demonstrate a non-monotone convergence to a minimum~\cite{boyd2014}.
Below we restrict ourselves with the analysis of the Heavy Ball method, but similar techniques can be extended to other accelerated optimization methods. It is worth mentioning that such methods are currently widespread and implemented to minimization problems occurring in neural networks. That is why a detailed study of these methods is of a great importance.
In addition, the numerous restart techniques~\cite{Restart_2} gain their popularity. To design the restart technique it is important to know how large the deviations from the solution should be to make an assumption about incorrect parameters' choice and start the method from the beginning at the current point.
The paper
is organized as follows. In Section \ref{lde_sec} we provide the results on peak effects in second-order linear difference equations. Next Section 3 contains applications of these results to the Heavy Ball method and demonstrates the dependence of peak value on initial conditions. In Section 4 a new Lyapunov function
is constructed; it decreases monotonically in contrast with the objective function or distance to the solution. By use of this function we are also able to prove global convergence for non-convex functions (under less restrictive Polyak-$\L$ojasiewicz condition). Conclusions and future directions for research are summarized in Section 5.
\section{Peak effect for second-order difference equations}\label{lde_sec}
We consider the case of second-order difference equation ($n=2$), which can be written down in the form
\begin{equation}\label{lde_2}
x_k = a_1x_{k-1} + a_2 x_{k-2},\ k = 2,\ 3,\ \dots;\ a_1,\ a_2 \in \mathbb{R}
\end{equation}
with initial conditions
$$x^{(0)} = (x_0, x_1)\in \mathbb{R}^2$$
and the characteristic polynomial
\begin{equation}\label{poly_2}
p(\lambda) = \lambda^2 - a_1 \lambda - a_2.
\end{equation}
For the second-order difference equation the following stability domain (Fig.~\ref{domain}) in the space $(a_1, a_2)$ is well known \cite{El}.
\begin{figure}
\caption{The stability domain of the second-order difference equation, the shaded area corresponds to peak effect.}
\label{domain}
\end{figure}
Our aim is to describe possible non-monotone behavior of stable solutions and, in particular, to measure $\eta(x^{(0)})$.
Two cases can be considered: the first corresponding to equal roots of the characteristic polynomial and the second with different roots.
In the case of equal roots, i.e. $\lambda^2 - a_1 \lambda - a_2 = (\lambda - \rho)^2 = \lambda^2 - 2\lambda\rho + \rho^2,\ 0<\rho<1$, and (\ref{lde_2}) reads:
\begin{equation*}\label{rho_root}
x_k = 2\rho x_{k-1} - \rho^2x_{k-2}.
\end{equation*}
The following expression for solution can be easily derived:
\begin{equation}\label{xk_rho}
x_k = x_1k\rho^{k-1} - x_0(k-1)\rho^k.
\end{equation}
In this case the non-monotone behavior can be observed (Fig.~\ref{double}).
\begin{figure}
\caption{The iterative process with $\rho = 0.6$ and $x^{(0)}
\label{double}
\end{figure}
We conclude that for all $k\geq 2$
\begin{equation}\label{peak_2}
\max\limits_{\parallel x^{(0)}\parallel_{\infty}\leq1}x_k = k\rho^{k-1} + (k-1)\rho^k.
\end{equation}
This maximum is achieved for $x^{(0)} = (-1, 1)$.
Now let's derive $k_{max} = \mathrm{argmax}\left(k\rho^{k-1} + (k-1)\rho^k \right) $ and $\eta(x^{(0)})$.
By differentiation and setting the derivative value equal to zero we obtain an expression for $k_{max}$:
\begin{equation}\label{k_max}
k_{max} = \left\lceil \frac{\rho \ln \rho - \rho - 1}{\ln \rho (1 + \rho)} \right\rceil.
\end{equation}
The value of the peak can be obtained by a substitution (\ref{k_max}) in the formula~(\ref{peak_2}).
For $\rho \to 1$ we get
\begin{equation*}
\eta(x^{(0)})\approx \frac{2}{e(1-\rho)}.
\end{equation*}
Thus large deviations can arise for some initial conditions. However, some pairs $(x_0, x_1)$ do not imply peak effects. For instance, for $x_0=1, x_1=1$ we obtain that $|x_k|< 1, \ k=2, \dots $.
Considering the case of different real roots $\lambda_1, \lambda_2$ of the characteristic polynomial~(\ref{poly_2}), we notice the following:
$\bullet$ If $\lambda_2 > \lambda_1 > \rho, \ 0<\rho<1$, than there exist such initial conditions $x^{(0)}$, that the trajectory $x_k$ will behave non-monotonically and its peak will be located higher than in the case of equal roots (Fig.~\ref{more}).
\begin{figure}
\caption{The trajectories of the iterative processes with initial conditions $x^{(0)}
\label{more}
\end{figure}
$\bullet$ If $\lambda_1 < \lambda_2 < \rho, \ 0<\rho<1$, the peak of $x_k$ (if any) will be located lower than in the case of equal roots $\rho$ for all initial conditions (Fig.~\ref{less}).
\begin{figure}
\caption{The trajectories of the iterative processes with initial conditions $x^{(0)}
\label{less}
\end{figure}
The proof of both statements is given in~\cite{shch2018}.
\section{Analysis of the Heavy Ball method }
\subsection{The Heavy Ball method}
We consider the simplest unconstrained optimization problem
\begin{equation}\label{opt_pr}
\min\limits_{x\in \mathbb{R}^n} f(x),
\end{equation}
where $f(x): \mathbb{R}^n \to \mathbb{R}$ is a smooth objective function, $x^*$~--- the minimum point. We restrict our analysis with the case of
quadratic objective function:
\begin{equation*}\label{opt_qu}
\min\frac{1}{2}(Ax,x)-(b,x),\ x,\ b \in \mathbb{R}^n
\end{equation*}
with $A\in \mathbb{R}^{n \times n}$ being positive-definite matrix $A\succ 0$, $\ \nabla f(x) = Ax - b$, $\ x^* = A^{-1}b$, $L$ and $\mu>0$~--- the maximum and minimum eigenvalues of $A$ respectively. It means that we focus on strongly convex, smooth case of the objective function.
Without loss of generality we can assume that after substitution $\hat{x} = x - x^*$ objective function $f \ $ has the
following form:
\begin{equation*}\label{opt_qu1}
f(x) = \frac{1}{2}(A\hat{x},\hat{x})
\end{equation*}
So, the optimal point is $x^* = 0$, $f^* = 0$.
There are numerous iterative methods to solve this problem; gradient methods are among the most popular, see e.g. \cite{polyak1987,nest2004}. The behavior of gradient methods is simple enough: they exhibit monotone convergence both for objective function and distance to the minimum point. The situation with accelerated first-order methods is much more complicated. We will focus on one of them --- so called Heavy Ball method proposed in \cite{Polyak_1}:
\begin{equation}\label{mtsh}
x_{k+1}=x_k-\alpha \nabla f( x_k)+\beta (x_k-x_{k-1})=x_k-\alpha A x_k+\beta (x_k-x_{k-1})
\end{equation}
Here a momentum term was added to the classic gradient method, which accelerated the convergence and made the trajectory look like a smooth descent to the bottom of the ravine, rather than zigzag. The traditional choice of initial condition is
\begin{equation}\label{x0}
x_1=x_0,
\end{equation}
i.e. the first iteration is just a gradient step.
It is known from \cite{Polyak_1} that for
\begin{equation*}\label{al_bet}
0\le \beta <1, \quad 0< \alpha < \frac{2(1+\beta)}{L}
\end{equation*}
there is a convergence to the solution with linear rate, $||x_k||= O(q^k), \ q<1$.
The optimal parameters $\alpha$ and $\beta$ providing the fastest convergence $q=\frac{\sqrt{L}-\sqrt{\mu}}{\sqrt{L}+\sqrt{\mu}}$ are also known:
\begin{equation}\label{al_mtsh}
\alpha=\frac{4}{(\sqrt{L}+\sqrt{\mu})^2},
\quad
\beta= \left(\frac{\sqrt{L}-\sqrt{\mu}}{\sqrt{L}+\sqrt{\mu}}\right)^2 = q^2.
\end{equation}
However, the non-asymptotic behavior of the method with optimal parameters for a simple example is shown on Fig. \ref{oscill_o} and with parameters very close to optimal --- on Fig. \ref{oscill}. In all examples below matrix $A$ is taken diagonal.
\begin{figure}
\caption{Non-monotone behavior of the Heavy Ball method with $x_0 = [0, 0, 0, 1]$,\ $\mu=1$,\ $L=10^4$,\ $\alpha,\ \beta$~--- optimal.}
\label{oscill_o}
\end{figure}
\begin{figure}
\caption{Non-monotone behavior of the Heavy Ball method with $x_0 = [0, 0, 0, 1]$,\ $\mu=1$,\ $L=10^4$,\ $\alpha,\ \beta$~--- close to optimal.}
\label{oscill}
\end{figure}
We conclude that the method exhibits strongly non-monotone
behavior.
\subsection{Convergence analysis}
To explain the behavior of the Heavy Ball method (\ref{mtsh}) with optimal $\alpha$ and $\beta$ (\ref{al_mtsh}) written in the form of second-order difference equation we consider it component-wise.
Let's start with the coordinate $x^1= (x,e_1)$, which corresponds to the minimal eigenvalue $\lambda_{\min} = \mu, \ A e_1=\mu e_1$. The method for this coordinate in the form of scalar linear difference equation along with its characteristic polynomial is written down below:
\begin{equation*}\label{mtsh_x1}
x^1_{k+1}=(1-\alpha \mu + \beta) x^1_k - \beta x^1_{k-1},
\end{equation*}
\begin{equation*}\label{poly_x1}
\rho^2 - (1-\alpha \mu + \beta) \rho + \beta = 0.
\end{equation*}
It is easily determined that a characteristic polynomial has both roots equal to $q$, meaning that
$\rho=q = \left( \frac{\sqrt{L} - \sqrt{\mu}}{\sqrt{L} + \sqrt{\mu}}\right).$ So the general solution is given by expression~(\ref{xk_rho}), while maximum provided by formula (\ref{peak_2}) with obvious change of notation.
Moving on to the coordinate $x^n=(x,e_n), \ A e_n=L e_n$, which corresponds to the maximum eigenvalue $\lambda_{\max} = L$, we notice that the only difference from the previous case is in the sign of roots of the characteristic polynomial, i.e. $\rho=-q$. However this implies different behavior of solutions, even for $x_0=x_1=1$ the trajectory is oscillating with possible large deviations. From formula (\ref{xk_rho}) we conclude that initial conditions $x_0=x_1=1$ cause the largest (in absolute value) peak effect equal to (\ref{peak_2}).
Now we consider a more general case of the coordinate $x^i=(x,e_i), \ 2\le i\le n-1, \ Ae_i=\lambda e_i, \ \mu<\lambda<L$. The characteristic polynomial
\begin{equation*}\label{poly_x2}
\rho^2 - (1-\alpha \lambda + \beta) \rho + \beta = 0
\end{equation*}
has complex roots
\begin{equation*}\label{x2_roots}
\rho_{1,2}=\frac{\left( \sqrt{L-\lambda} \pm i \sqrt{\lambda - \mu} \right)^2}{(\sqrt{L} + \sqrt{\mu})^2},\quad
|\rho|=\frac{\sqrt{L} - \sqrt{\mu}}{\sqrt{L} + \sqrt{\mu}} = q.
\end{equation*}
The general solution is written down below:
\begin{equation*}\label{x2_sol}
x^i_k=\left[C_1 \cos(\omega k) + C_2 \sin(\omega k) \right]q^k,
\end{equation*}
where
\[
\sin(\omega) = \frac{2 \sqrt{\lambda - \mu} \sqrt{L - \lambda}}{(L - \mu)}, \quad \cos(\omega) = \frac{L + \mu - 2 \lambda}{(L - \mu)}
\]
\[
C_1=x^2_0,\quad
C_2=\frac{x^2_1(\sqrt{L}+\sqrt{\mu})^2 - x^2_0 (L+\mu-2\lambda)}{2\sqrt{\lambda-\mu}\sqrt{L-\lambda}}.
\]
The trajectory (for $n=3$) demonstrates oscillations (Fig. \ref{icdea_6}).
\begin{figure}
\caption{Dependence of the coordinate $|x^2| \ $ on the number of iterations $k$ under conditions $x_0 = 0_n, \ x_1 = 1_n$.}
\label{icdea_6}
\end{figure}
\subsection{Peak effect}
From our previous observations in Section \ref{lde_sec} we note that peak effects in linear difference equations are common and depend on initial conditions. For the worst case the following proposition holds (we provide the estimates for the most important case of large condition number $\kappa=L/\mu$).
\begin{Th}
Assume that $f(x) = \frac12 \left( Ax, x\right),$ where $A \in \mathbb{R}^{n \times n}, \ A = A^{\top} \succ 0$. We have strong convexity parameter $\mu = \lambda_{\min} > 0$ and $L=\lambda_{\max}$, where $\lambda_{\min}$ and $\lambda_{\max}$ are the minimum and maximum eigenvalues of $A$, respectively. Then initial conditions $x_0 =-e_1, \ x_1 =e_1\in \mathbb{R}^n, \ \|x_0\| = \|x_1\|= 1$ cause peak effect in the Heavy Ball method with optimal parameters $\ \alpha^*, \beta^*$:
\begin{equation*}
\max\limits_{k}||x_k|| \geq \frac{\sqrt{\kappa}}{2 e},
\end{equation*}
while standard (\ref{x0}) initial conditions $x_0 = x_1 =e_n\in \mathbb{R}^n, \ \|x_0\| = \|x_1\|= 1$ cause the same peak effect combined with oscillating behavior.
\end{Th}
The proof follows from the estimates obtained in the previous section. Of course, other initial conditions also may lead to non-monotone behavior of iterations; we indicate the ones which provide the largest deviations from the minimum point.
The figures below show that the Heavy Ball method exhibits the non-monotone behavior on the test problem $n=3, \ \kappa =10^4$, namely, a sharp increase in the function under various initial conditions.
\begin{figure}
\caption{Dependence of the objective function $f(x_k) \ $ and $\|x_k\|$ on the number of iterations $k$ under the initial conditions $x_0 = 0_n, \ x_1 = 1_n$.}
\end{figure}
\begin{figure}
\caption{Dependence of the objective function $f(x_k) \ $ and $\|x_k\|$ on the number of iterations $k$ under the initial conditions $x_0 = x_1 = 1_n$.}
\end{figure}
To sum up, we applied the results on linear difference equations to the Heavy Ball method analysis. It was shown that even the choice of optimal parameters and standard initial conditions can not guarantee the monotone convergence.
\section{Lyapunov function for the Heavy ball method}
In this section we extend the analysis of the Heavy Ball method (both for continuous and discrete versions) to non-quadratic objective functions via Lyapunov function technique. Thus we treat the unconstrained optimization problem
\begin{equation*}\label{min}
\min\limits_{x\in \mathbb{R}^n} f(x),
\end{equation*}
where $f(x)$ is differentiable function bounded from below: $f(x)\ge f^*$. Notice that here we do not assume neither convexity nor strong convexity of $f(x)$.
\subsection{Construction of the Lyapunov function}
Lyapunov functions is a common tool for proving the stability of nonlinear systems described by differential or difference equations. The Lyapunov function is a scalar function that decreases monotonically in stable system.
The Heavy Ball method, as we verified above, does not exhibit monotone behavior even for the simplest case of quadratic function. Thus neither $f(x)$ nor $\|x-x^*\|$ can be used as the Lyapunov function. We suggest new Lyapunov function that can help to select the parameters of the method and overcome the difficulties related to its non-monotone transient process.
\subsubsection{Continuous case.}
Before moving to the construction of the Lyapunov function for discrete case, we would like to consider continuous case from \cite{Polyak_1}:
\begin{equation}
\label{eq4}
\ddot{x} + a\dot{x}+b\nabla f(x)=0.
\end{equation}
In mechanical interpretation, $x(t)\in \mathbb{R}^n$ is the trajectory of the body (''heavy ball''), $\dot{x}, \ddot{x}$ are its velocity
and acceleration, $a>0, \ b>0$ are scalar parameters while $f(x)\ge f^*$ is the potential energy and $\nabla f(x)$ is its gradient. It is known from \cite{Polyak_1,Polyak_Sherbakov} that $\nabla f(x(t))$ tends to zero, however the convergence can be non-monotone. Our goal is to obtain the upper bounds for the convergence.
Firstly, let's rewrite (\ref{eq4}):
\begin{eqnarray*}
&&\dot{x}=y,\\
&&\dot{y}=-ay-bf'(x) \nonumber
\end{eqnarray*}
with some initial conditions $x(0), \ y(0)$.
According to paper \cite{Polyak_Sherbakov}, $V(x,y)$ can be chosen from mechanical analogies. Consequently, it is possible to represent the function $V(x,y)$ in the form of the total energy of the system:
\begin{equation*}
V(x,y)=f(x)+\frac{1}{2b}\|y\|^2.
\end{equation*}
For the time derivative we have
\begin{equation*}
\dot{V}(x,y) = \bigl(f'(x), y \bigr) + \frac{1}{2b}2\bigl(y, -ay-bf'(x) \bigr) = -\frac{a}{b}\|y\|^2\le 0.
\end{equation*}
Thus we get an upper bound for $f(x(t))$:
\begin{equation*}\label{1}
f(x(t))-f^*\le f(x(0))-f^*+\frac{1}{2b}\|y(0)\|^2.
\end{equation*}
In particular, for zero initial velocity $ f(x(t))-f^*\le f(x(0))-f^*$.
\subsubsection{Discrete case.}
Now we proceed to discrete-time version of the Heavy Ball method
\begin{equation}
\label{discrete}
x_{k+1}= x_{k} -\alpha \nabla f(x_{k}) + \beta (x_{k} - x_{k-1}),
\end{equation}
and assume additionally that $f$ is $L$-smooth:
\begin{equation*}
\label{L}
||\nabla f(x)-\nabla f(y)||\le L||x-y||.
\end{equation*}
\begin{Th} \label{th2}
Assume that
\begin{equation}
\label{ab}
0< \alpha < \frac{1}{L}, \; 0\le \beta \le \sqrt{1 - \alpha L}.
\end{equation}
Then for any initial conditions $x_0, \ x_1 \in \mathbb{R}^n$ the function
\begin{equation}
\label{Lyap_function}
V_k = f(x_k)-f^* + \frac{1-\alpha L}{2 \alpha} \|x_k - x_{k-1}\|^2
\end{equation}
is the Lyapunov function for the discrete case of the Heavy Ball method, that is $V_k\le V_{k-1}$.
\end{Th}
\begin{proof}
For iterations (\ref{discrete}), we have
\begin{equation}
\label{11}
x_k - x_{k-1} = -\alpha \nabla f(x_{k-1}) + \beta (x_{k-1} - x_{k-2}),
\end{equation}
\begin{eqnarray}\label{12}
\|x_k - x_{k-1}\|^2 = \alpha^2 \|\nabla f(x_{k-1})\|^2 + \beta^2 \|x_{k-1} - x_{k-2}\|^2 \\ \nonumber
- 2 \alpha \beta \left( \nabla f(x_{k-1}), x_{k-1} - x_{k-2}\right).
\end{eqnarray}
Since function $f \in \mathscr F^{1,1}_{L} \ $, it has the Lipschitz gradient, so following equation from \cite{nest2004} can be applied
\begin{equation}
\label{13}
f(x_{k}) \leq f(x_{k-1}) + \left\langle \nabla f(x_{k-1}), x_{k} - x_{k-1}\right\rangle + \frac{L}{2} \|x_{k} - x_{k-1}\|^2.
\end{equation}
Adding (\ref{11}), (\ref{12}) to (\ref{13}), we get
\begin{eqnarray}\label{14}
\lefteqn{f(x_{k}) \leq f(x_{k-1}) - \alpha \|\nabla f(x_{k-1})\|^2 +} \\ \nonumber
& & \beta \left\langle \nabla f(x_{k-1}), x_{k-1} - x_{k-2}\right\rangle
+ \frac{\alpha^2 L}{2} \|\nabla f(x_{k-1})\|^2 + \\ \nonumber
& & \frac{\beta^2 L}{2} \|x_{k-1} - x_{k-2}\|^2 - L \alpha \beta \left( \nabla f(x_{k-1}), x_{k-1} - x_{k-2}\right).
\end{eqnarray}
Multiplying (\ref{12}) by $\frac{1-\alpha L}{2 \alpha}$
and adding to (\ref{14}) we obtain
\[
f(x_{k}) + \frac{1-\alpha L}{2 \alpha} \|x_k - x_{k-1}\|^2 \; \leq \; f(x_{k-1}) - \alpha \|\nabla f(x_{k-1})\|^2 +
\]
\[
\beta \left\langle \nabla f(x_{k-1}), x_{k-1} - x_{k-2}\right\rangle +
\frac{\alpha^2 L}{2} \|\nabla f(x_{k-1})\|^2 + \frac{\beta^2 L}{2} \|x_{k-1} - x_{k-2}\|^2 -
\]
\[
L \alpha \beta \left( \nabla f(x_{k-1}), x_{k-1} - x_{k-2}\right) + \frac{1-\alpha L}{2 \alpha}\cdot
\]
\[
\left( \alpha^2 \|\nabla f(x_{k-1})\|^2 +
\beta^2 \|x_{k-1} - x_{k-2}\|^2 -
2 \alpha \beta \left( \nabla f(x_{k-1}), x_{k-1} - x_{k-2}\right)\right).
\]
Collecting terms yields
\[
f(x_{k}) + \frac{1-\alpha L}{2 \alpha} \|x_k - x_{k-1}\|^2 \; \; \leq \; \; f(x_{k-1}) + \left( \frac{\beta^2 L}{2} + \frac{(1-\alpha L)\beta^2}{2 \alpha} \right)\cdot
\]
\[
\|x_{k-1} - x_{k-2}\|^2 +
\left( \frac{\alpha^2 L}{2} - \alpha + \frac{(1- \alpha L) \alpha}{2} \right) \|\nabla f(x_{k-1})\|^2 +
\]
\[
\left(\beta - L \alpha \beta -\beta + \alpha L \beta \right) \left( \nabla f(x_{k-1}), x_{k-1} - x_{k-2}\right).
\]
As a result,
\[
f(x_{k}) + \frac{1-\alpha L}{2 \alpha} \|x_k - x_{k-1}\|^2 \; \; \leq \; \; f(x_{k-1}) + \left( \frac{\beta^2 L}{2} + \frac{(1-\alpha L)\beta^2}{2 \alpha} \right)\cdot
\]
\[
\|x_{k-1} - x_{k-2}\|^2 +
\left( \frac{\alpha^2 L}{2} - \alpha + \frac{(1- \alpha L) \alpha}{2} \right) \|\nabla f(x_{k-1})\|^2
\]
\begin{itemize}
\item
\[ \frac{\beta^2 L}{2} + \frac{(1-\alpha L)\beta^2}{2 \alpha} \; = \; \frac{\beta^2 L \alpha + \beta^2 - \beta^2 \alpha L}{2 \alpha} \; = \; \frac{\beta^2}{2 \alpha} \; > \; 0
\]
\item
\[
\frac{\alpha^2 L}{2} - \alpha + \frac{(1- \alpha L) \alpha}{2} = \frac{\alpha^2 L - 2 \alpha + \alpha - \alpha^2 L}{2} = - \frac{\alpha}{2} \; < \; 0
\]
\end{itemize}
\begin{eqnarray}\label{15}
f(x_{k}) + \frac{1-\alpha L}{2 \alpha} \|x_k - x_{k-1}\|^2 \; \; \leq \; \; f(x_{k-1}) + \left( \frac{\beta^2}{2 \alpha}\right) \|x_{k-1} - x_{k-2}\|^2 + \\ \nonumber
\left( - \frac{\alpha}{2} \right) \|\nabla f(x_{k-1})\|^2
\end{eqnarray}
Since $\left( - \frac{\alpha}{2} \right) \|\nabla f(x_{k-1})\|^2 \; < \; 0 \ $
and having (\ref{ab}) we arrive to the desired inequality $V_k\le V_{k-1}$.
\end{proof}
Theorem \ref{th2} provides the upper bound of $f(x_k)$ for the Heavy Ball method:
\begin{equation*}\label{f_k}
f(x_k)-f^*\le f(x_0)-f^*+\frac{1-\alpha L}{2\alpha}||x_1-x_0||^2
\end{equation*}
and for standard initial condition $x_1=x_0$ we obtain
\begin{equation*}\label{f_kst}
f(x_k)-f^*\le f(x_0)-f^*.
\end{equation*}
Notice the lack of such bound for more narrow class of strongly convex quadratic functions, on the other hand this estimate holds for more restrictive conditions on parameters $\alpha, \ \beta$, see (\ref{ab}).
\subsubsection{Numerical experiments.}
We have obtained the following Lyapunov function:
\[
V_k = f(x_k) + \frac{1-\alpha L}{2 \alpha} \|x_k - x_{k-1}\|^2
\]
with the restriction on the parameters (\ref{ab}).
We would like to show numerically that the function proposed above is indeed the Lyapunov function. As a particular example a quadratic function $f(x), \ n = 20, \ \kappa = 10^5 \ $ is taken along with the Heavy Ball method with a set of parameters $\alpha, \ \beta$. It can be observed (Fig. \ref{mono}) that the Lyapunov function decreases monotonically on the trajectory of the method.
\begin{figure}
\caption{\label{mono}
\label{mono}
\end{figure}
\subsection{Global convergence}
In the statement above we did not prove neither convergence of $f(x_k)$ to $f^*$ nor convergence $V_k$ to zero. To obtain such results further assumptions on $f(x)$ are needed. The least restrictive condition is
\begin{equation}\label{PL}
||\nabla f(x)||^2\ge 2\mu (f(x)-f^*), \mu>0
\end{equation}
for all $x\in \mathbb{R}^n$. This inequality is satisfied for strongly convex functions \cite{polyak1987,nesterov_1}, but in general it does not require convexity. The condition has been proposed in \cite{pol63}, sometimes it is called Polyak-$\L$ojasiewicz condition.
\begin{Th}
If (\ref{PL}) holds and
\[\; \alpha \in \left( 0, \frac{1}{L}\right), \; \beta \in \left[ 0, \sqrt{(1 - \alpha L)(1 - \alpha \mu)} \right].
\]
then for any initial conditions $x_0, x_1 \in \mathbb{R}^n$ Lyapunov function converges linearly
\begin{equation*}
\label{rate}
V_k\le V_0 q^k,\quad q=1-\alpha\mu <1,
\end{equation*}
while for $x_0 =x_1 $ objective function converges linearly
\begin{equation*}
\label{rate}
f(x_k)-f^*\le (f(x_0)-f^*) q^k.
\end{equation*}
\end{Th}
\begin{proof}
It was shown above (\ref{15}), that
\[
f(x_k) - f^* + \frac{1-\alpha L}{2 \alpha} \|x_k - x_{k-1}\|^2 \; \leq \; f(x_{k-1}) - f^* + \frac{\beta^2}{2 \alpha} \|x_{k-1} - x_{k-2}\|^2 -
\]
\[\frac{\alpha}{2} \|\nabla f(x_{k-1})\|^2.
\]
Due to (\ref{PL}) we get
\[
f(x_k) - f^* + \frac{1-\alpha L}{2 \alpha} \|x_k - x_{k-1}\|^2 \; \leq \; f(x_{k-1}) - f^* + \frac{\beta^2}{2 \alpha} \|x_{k-1} - x_{k-2}\|^2 -
\]
\[\alpha \mu (f(x_{k-1}) - f^*) =
\left(1 - \alpha \mu \right) \left((f(x_{k-1}) - f^*) + \frac{\beta^2}{2 \alpha (1 - \alpha \mu)} \|x_{k-1} - x_{k-2}\|^2 \right).
\]
Provided that
\[
\frac{\beta^2}{2 \alpha (1 - \alpha \mu)} \; \leq \; \frac{1-\alpha L}{2 \alpha};
\quad \quad
\beta \; \leq \; \sqrt{(1 - \alpha L)(1 - \alpha \mu)} ,
\]
we obtain
\[
V(x_{k+1}) \; \leq \; q V(x_{k}) \; \leq \; q^k V(x_1), \quad q = (1 - \alpha \mu) < 1.
\]
For convenience, we denote $\ \gamma = \frac{1-\alpha L}{2 \alpha}\ $ and thus
\[
f(x_{k+1}) - f^* \; \leq \; f(x_{k+1}) - f^* + \gamma \|x_{k+1} - x_{k}\|^2 = V_{k+1} \; \leq
\]
\[q^k V_1 = q^k \left( f(x_1) - f^* + \gamma \|x_1 - x_0\|^2 \right) = q^k \left( f(x_0) - f^* \right).
\]
\end{proof}
Global convergence of the Heavy Ball method is established here for function under condition (\ref{PL}) through Lyapunov function (\ref{Lyap_function}). The similar results on global convergence for more narrow class of strongly convex functions were obtained in paper \cite{Global}.
\subsection{Adaptive algorithm}
In this section we will consider a general idea of choosing the optimal parameters for the accelerated gradient method.
We consider the smooth and strongly convex problem (\ref{opt_pr}) focusing on the Heavy Ball method (\ref{mtsh}). In order to ensure the convergence of the method, it is necessary to select the parameters (\ref{al_mtsh}) correctly. Therefore, the values of strong convexity constant $\mu$ and the Lipschitz constant $L$ are required. Unfortunately, these constants are difficult and time-consuming to compute for a real problem. Moreover, as already discussed earlier, accelerated first-order methods are not guaranteed to be monotonic.
To sum up, the following situations should be distinguished:
\begin{itemize}
\item non-monotone behavior (natural situation arising in first-order methods);
\item mistake in parameter values.
\end{itemize}
Wide range of papers offer different options for adaptive restarting schemes dedicated to avoiding non-monotone behavior. By restart we denote starting the method from the very beginning with new parameters, where new initial condition is the current iteration. It is suggested to restart after a fixed number of iterations (“fixed restart”) or more efficiently after checking certain conditions of restart (“adaptive restart”). The papers \cite{boyd2014}, \cite{Restart_2}, \cite{Restart_1} propose the different adaptive restart schemes and analysis of these techniques. For example, \cite{Restart_2} offers the following two adaptive restart techniques. They restart whenever:
\begin{enumerate}
\item function scheme
\[
f(x_k) > f(x_{k-1});
\]
\item gradient scheme
\[
\nabla f(x_{k-1})^T(x_k - x_{k-1}) > 0.
\]
\end{enumerate}
In this work, we offer an alternative to restart techniques in the form of the Lyapunov function $\ V(x) \ $. This function strictly decreases at each iteration. It assures that the method is stable. The algorithm converges to optimal value. We propose to monitor the values of the Lyapunov function $\ V(x) \ $ instead of the objective function $\ f(x) \ $ at each iteration.
We will restart the algorithm with new parameters, only if the value of the Lyapunov function starts to increase.
Lyapunov scheme
\[
V(x_k) > V(x_{k-1}).
\]
We propose construction of the Lyapunov function (\ref{Lyap_function}) for discrete case:
\[
V(x_k) = f(x_k) + \frac{1-\alpha L}{2 \alpha} \|x_k - x_{k-1}\|^2.
\]
Note that the correct choice of the parameters implies only the knowledge of Lipschitz constant, which can be determined iteratively according to \cite{Nesterov_3_L}. It means, that we will check inequalities (\ref{13}) with additional term $\frac{\epsilon}{2} \ $, where $\epsilon > 0$ --- the required accuracy.
\[
f(x_{k}) \leq f(x_{k-1}) + \left\langle \nabla f(x_{k-1}), x_{k} - x_{k-1}\right\rangle + \frac{L}{2} \|x_{k} - x_{k-1}\|^2 + \frac{\epsilon}{2}
\]
As a result, we obtain the following adaptive algorithm:
\begin{algorithm}[H]
\caption{Adaptive Heavy ball method with Lyapunov function}
\begin{algorithmic}[1]
\State \textbf{Input:} $f \in \mathscr F^{1,1}_{L} \, , \ x_0=y_0 \in R^n\, , \ L_0 > 0, \
\alpha_0 \in \left( 0, \frac{1}{L_0} \right), \
\beta_0 \in \left( 0, \sqrt{1 - \alpha_0 L_0}\right)$.
\State \textbf{for} $k \geq 0$ \textbf{do}
\State $ \qquad x_{k+1} = x_k - \alpha_k \nabla f(x_k) + \beta_k \left(x_{k+1} - x_k \right)$
\State $ \qquad $ \textbf{if} $ \ V(x_{k+1}) \ > \ V(x_k) $
\State $ \qquad \qquad $ \textbf{then} $ \ L_{k+1} = 2L_k, \ \alpha_{k+1} \in \left( 0, \frac{1}{L_{k+1} } \right), \
\beta_{k+1} \in \left( 0, \sqrt{1 - \alpha_{k+1} L_{k+1} }\right)$
\State $ \qquad \qquad $ \textbf{else} $ \ L_{k+1} = L_k, \ \alpha_{k+1} = \alpha_k, \ \beta_{k+1} = \beta_k$
\end{algorithmic}
\end{algorithm}
\section{Conclusion}
In this work, attention has been paid to behavior of second-order difference equations' solutions and their connection with accelerated gradient methods' convergence. Firstly, considering linear difference equations we derived the initial conditions causing peaking effects. In the next section these results were applied to the analysis of the Heavy Ball method in case of a quadratic objective function and optimal parameters $\alpha$ and $\beta$. Then, we moved on to discussing the non-monotonic behaviour of the method for strongly convex and smooth functions $ f(x) \in \mathscr F^{1,1}_{L,\mu} \ $. Finally, the concept of the Lyapunov function for method's control was suggested.
The future work implies expanding the notion of Lyapunov function to other classes of objective functions and developing an adaptive algorithm with a better convergence rate. The obtained results are supposed to be applied to numerous unconstrained optimization problems, arising in power system engineering, deep-learning and other fields.
\section*{Funding}
Financial support for this work was provided by the Russian Science Foundation, project no. 16-11-10015.
\end{document} |
\begin{document}
\begin{abstract}
We introduce an anisotropic global wave front set of Gelfand--Shilov ultradistributions with different indices
for regularity and decay at infinity.
The concept is defined by the lack of super-exponential decay along power type curves in the phase space of
the short-time Fourier transform.
This wave front set captures the phase space behaviour of oscillations of power monomial type,
a k a chirp signals.
A microlocal result is proved with respect to pseudodifferential operators
with symbol classes that give rise to continuous operators on Gelfand--Shilov spaces.
We determine the wave front set of certain series of derivatives of the Dirac delta,
and exponential functions.
\end{abstract}
\keywords{Ultradistributions, Gelfand--Shilov spaces, pseudodifferential operators, wave front sets, microlocal analysis, phase space, anisotropy}
\subjclass[2010]{46F05, 46F12, 35A27, 47G30, 35S05, 35A18, 81S30, 58J47}
\maketitle
\section{Introduction}\langlebel{sec:intro}
Gelfand--Shilov spaces, for $t > 0$ and $s > 0$, are defined by
\begin{equation}\langlebel{eq:GelfandShilovest}
|x^\alpha D ^\beta f(x)|
\leqslant C h^{|\alpha + \beta |} \alpha !^t \, \beta !^s
\end{equation}
which we assume to be valid for every $h > 0$ and a suitable $C > 0$ depending on $h$
(spaces of Beurling type $\Sigma_t^s(\rr d)$),
or else for some $h > 0$ and some $C > 0$ (Roumieu type $\mathcal S_t^s(\rr d)$).
The ultradistributions $(\Sigma_t^s)'(\rr d)$, $(\mathcal S_t^s)'(\rr d)$ are defined as their respective topological duals.
Attention in our paper will be limited to the Beurling case
under the assumption $t + s > 1$ granting $\Sigma_t^s(\rr d) \neq \{ 0\}$.
The definition was introduced in \cite{Gelfand2}, and then analyzed in various contexts,
with application to linear and nonlinear partial differential equations, in connection also
with problems in Mathematical Physics.
The literature on the subject is extremely wide, see for example \cite{Pilipovic1,Debrouwere1,Teofanov1}
for recent contributions to the general theory, and \cite{Cappiello0a,Carypis1,Morimoto1,Morimoto2}
concerning travelling waves, Boltzmann and Schr\"odinger equations.
In particular, Gelfand--Shilov spaces have been considered in the framework of
pseudodifferential operators.
Namely, classes of pseudodifferential operators were introduced,
with symbols satisfying suitable factorial and exponential estimates,
acting continuously on Gelfand--Shilov spaces,
see for example \cite{Abdeljawad1,Cappiello2}.
In our paper we shall refer to the class of symbols satisfying
\begin{equation}\langlebel{eq:symbolest}
|\pdd x \alpha \pdd \xi\beta a(x,\xi)|
\leqslant C h^{|\alpha + \beta |} \alpha !^s \, \beta !^t
e^{\mu \left( |x|^{\frac1t} + |\xi|^{\frac1s} \right)}
\end{equation}
for some $\mu > 0$ and all $h > 0$, with $C > 0$ depending on $h$.
This symbol class was introduced in \cite{Abdeljawad1}.
The corresponding Weyl operators $a^w(x,D)$ were proved to act continuously on $\Sigma_t^s(\rr d)$
and on $(\Sigma_t^s)'(\rr d)$
in \cite[Theorem~3.15]{Abdeljawad1}.
Our attention will be actually addressed to another ingredient of the microlocal analysis: the wave front set.
The classical definition of H\"ormander \cite{Hormander0} in the setting of Schwartz distributions
was extended in different ways.
In particular H\"ormander \cite{Hormander1} introduced for $u \in \mathscr{S}'(\rr d)$ the notion of $\mathrm{WF}g(u)$
adapted to the study of global regularity in $T^* \rr d \setminus 0$.
Let us recall the definition by using the short-time Fourier transform (Gabor transform)
with window $\varphi \in \mathscr{S}(\rr d) \setminus 0$, cf. \cite{Rodino2}:
\begin{equation*}
V_\varphi u (x,\xi) = (2\pi )^{-\frac d2} \int_{\rr d} e^{- i \scal y \xi} u(y) \overline{\varphi(y-x)} \mathrm {d} y.
\end{equation*}
We have $z_0 = (x_0,\xi_0) \mathbf Ntin \mathrm{WF}g(u)$, $z_0 \neq 0$, if
\begin{equation}\langlebel{eq:conicdecay1}
\sup_{z \in {G(X|Y)}amma} \eabs{z}^N |V_\varphi u (z)| < \infty \quad \forall N \geqslant 0
\end{equation}
for a suitable conic neighborhood ${G(X|Y)}amma$ of $z_0$ in $\rr {2d} \setminus 0$.
Looking for a counterpart of \eqref{eq:conicdecay1} in the Gelfand--Shilov setting,
we may start with the equivalent definition of the $\Sigma_t^s(\rr d)$ regularity
of $u \in (\Sigma_t^s)'(\rr d)$ given by the estimates, with window $\varphi \in \Sigma_t^s(\rr d) \setminus 0$,
\begin{equation}\langlebel{eq:STFTGFstequiv}
| V_\varphi u (x,\xi)| \lesssim e^{-r (|x|^{\frac1t} + |\xi|^{\frac1s})} \quad \forall r > 0.
\end{equation}
For the equivalence with \eqref{eq:GelfandShilovest} see for example \cite{Toft1}.
Hence in the case $s = t$ we may define as $\Sigma_s^s(\rr d)$ regularity at $z_0 \in T^* \rr d \setminus 0$
\begin{equation}\langlebel{eq:conicdecay2}
\sup_{z \in {G(X|Y)}amma} e^{r |z|^{\frac1s}} |V_\varphi u (z)| < \infty \quad \forall r > 0
\end{equation}
where again ${G(X|Y)}amma$ is a conic neighborhood of $z_0$ in $\rr {2d} \setminus 0$.
Based on \eqref{eq:conicdecay2}, the Gelfand--Shilov wave front set for $s = t$ was recently defined and used
in applications to partial differential equations \cite{Boiti1,Cappiello1,Carypis1}.
Let us address for some early ideas to \cite{Hormander1}, and to the theory of Fourier hyperfunctions
\cite{Kaneko1,Kaneko2}.
If $s \neq t$, cones ${G(X|Y)}amma \subseteq T^* \rr d \setminus 0$ are not anymore appropriate to micro-localize the decay of the Gabor transform
in \eqref{eq:STFTGFstequiv}.
The natural idea is to replace the standard cones with anisotropic cones,
namely we replace the straight lines through $(x_0,\xi_0) \in T^* \rr d \setminus 0$
with the curves $\{x = \langlembda^t x_0, \ \xi = \langlembda^s \xi_0, \ \langlembda > 0 \}$
and we define the anisotropic cone as the union of such curves through a neighborhood $U \subseteq T^* \rr d \setminus 0$ of $(x_0,\xi_0)$.
The required decay to define $(x_0,\xi_0) \mathbf Ntin \mathrm{WF}^{t,s} (u)$ can then be expressed by
\begin{equation*}
\sup_{\langlembda > 0, \ (x,\xi) \in U} e^{r \langlembda} |V_\varphi u(\langlembda^t x, \langlembda^s \xi)| < \infty, \quad \forall r > 0.
\end{equation*}
Let us describe in short the contents of the paper.
Section {\rm Re}f{sec:prelim} is devoted to some preliminaries.
We give in particular a new proof of the celebrated Peetre inequality;
the optimality of the constant in our formula seems new in the literature, surprisingly.
The definition of $\mathrm{WF}^{t,s} (u)$ is reported in Section {\rm Re}f{sec:GelfandShilovWF}.
We give there examples about $\mathrm{WF}^{s,s} (u)$, i.e. the case $s=t$,
and then prove invariance properties under change of window and the action
of certain metaplectic operators.
Section {\rm Re}f{sec:chirp} is devoted to chirp signals, providing an interesting example
of anisotropic wave front set.
Namely in dimension $d=1$, for
\begin{equation}\langlebel{eq:chirp1}
u (x) = e^{i c x^{m}}, \quad m \in \mathbf N \setminus \{ 0,1 \}, \quad c \in \mathbf R \setminus 0,
\end{equation}
we obtain if $t (m-1) > 1$
\begin{equation}\langlebel{eq:WFchirp1}
\mathrm{WF}^{t, t (m-1)}(u) = \{ (x, \xi = c m x^{m-1} ) \in \rr 2, \ x \neq 0 \}.
\end{equation}
Section {\rm Re}f{sec:GSGevrey} is addressed to the relations between the Gelfand--Shilov wave front set
and the Gevrey wave front set $\mathrm{WF}_s (u)$ for $u \in (\Sigma_t^s)'(\rr d)$, $s > 1$.
We shall refer to \cite{Rodino1}, results given there for the Roumieu case being easily translated
to the present Beurling framework.
The main result of the paper is in Section {\rm Re}f{sec:microlocal}, where we prove the microlocal inclusion
\begin{equation}\langlebel{eq:microlocal0}
\mathrm{WF}^{t,s}( a^w(x,D) u ) \subseteq \mathrm{WF}^{t,s}(u), \quad u \in (\Sigma_t^s)'(\rr d),
\end{equation}
for symbols satisfying \eqref{eq:symbolest}.
Several examples are then given.
Namely in Section {\rm Re}f{sec:polynomials} we compute $\mathrm{WF}^{t,s} (u)$ for polynomials
and finite linear combinations of derivatives of the delta distribution $\delta_0$.
The analysis extends to ultradistributions of the form
\begin{equation*}
u = \sum_{\alpha \in \nn d} c_\alpha D^\alpha \delta_0
\end{equation*}
under suitable bounds on the coefficients $c_\alpha \in \mathbf C$, and their Fourier transforms.
In Section {\rm Re}f{sec:exponential} we first consider $e^{\langle \, \cdot \, t, z \rangle} \in (\Sigma_t^s)'(\rr d)$,
with $z \in \cc d$ fixed, $t \leqslant 1$.
From \eqref{eq:microlocal0} we obtain
\begin{equation*}
\mathrm{WF}^{t,s} ( e^{\langle \, \cdot \, t, z \rangle} ) = (\rr d \setminus 0) \times \{ 0 \}.
\end{equation*}
Combining with the example \eqref{eq:chirp1} in dimension $d = 1$, we then consider
\begin{equation*}
u(x) = e^{z x + i c x^{m}}
\end{equation*}
and we deduce for $\mathrm{WF}^{t,s} (u)$ the same identity \eqref{eq:WFchirp1}.
In conclusion, we would like to observe that anisotropic cones are not a novelty in microlocal analysis.
They were used as a partition of the space $\rr d$ of the dual variables by \cite{Lascar1} and \cite{Parenti1},
soon followed by other authors, see for more recent contributions \cite{Garello1} and its references.
In these papers the anisotropic cones in $\rr d$ are used as a suitable option in the microlocal study
of equations of parabolic type, whereas in our case the anisotropy in $T^* \rr d$ is forced by the very structure of the function spaces.
As background we mention recent works of ours (written after this paper) concerning anisotropic global wave front sets and their propagation for certain evolution equations \cite{Rodino3,Wahlberg4,Wahlberg5}.
Applications to partial differential equations will be given in a sequel of this paper.
We are then inspired by \cite{Cappiello0b}, where the authors prove Gelfand--Shilov regularity for operators of the type
\begin{equation*}
P = - \Delta + |x|^{2m}, \quad m \in \mathbf N \setminus 0.
\end{equation*}
We aim for microlocal versions of this result, as well as
propagation of singularities for Schr\"odinger operators of the form
\begin{equation*}
Q = i \partial_t - P = i \partial_t + \Delta - |x|^{2m}.
\end{equation*}
\section{Preliminaries}\langlebel{sec:prelim}
An open ball in $\rr d$ of radius $r > 0$ centered at $x \in \rr d$ is denoted $\operatorname{B}_r(x)$, and $\operatorname{B}_r(0) = \operatorname{B}_r$.
The unit sphere is denoted $\sr {d-1} \subseteq \rr d$.
The group of invertible matrices in $\rr {d \times d}$ is $\operatorname{GL}(d,\mathbf R)$,
and the determinant of $A \in \rr {d \times d}$ is $|A|$.
The transpose of $A \in \rr {d \times d}$ is denoted $A^T$ and the inverse transpose
of $A \in \operatorname{GL}(d,\mathbf R)$ is $A^{-T}$.
The derivative $D_j = - i \partial_j$ is used extended to multi-indices.
We write $f (x) \lesssim g (x)$ provided there exists $C>0$ such that $f (x) \leqslant C \, g(x)$ for all $x$ in the domain of $f$ and of $g$.
We use the bracket $\eabs{x} = (1 + |x|^2)^{\frac12}$ for $x \in \rr d$.
Peetre's inequality is usually stated as
\begin{equation*}
\eabs{x+y}^s \leqslant 2^{\frac{|s|}{2}} \eabs{x}^s\eabs{y}^{|s|}\qquad x,y \in \rr d,
\qquad s \in \mathbf R,
\end{equation*}
but in fact the constant can be improved as follows.
\begin{lem}\langlebel{Peetresharp}
We have
\begin{equation*}
\eabs{x+y}^s \leqslant \left( \frac{2}{\sqrt{3}} \right)^{|s|} \eabs{x}^s\eabs{y}^{|s|}\qquad x,y \in \rr d, \quad s \in \mathbf R,
\end{equation*}
where the constant is optimal.
\end{lem}
\begin{proof}
It suffices to show
\begin{equation*}
\sup_{x,y \in \rr d} \frac{1 + |x + y|^2}{ (1 + |x|^2) (1 + |y|^2) } = \frac43.
\end{equation*}
If $|x| = 2^{- \frac12}$ and $y = x$ then
\begin{equation*}
\frac{1 + |x + y|^2}{ (1 + |x|^2) (1 + |y|^2) } = \frac{1 + 4 |x|^2}{ (1 + |x|^2)^2} = \frac43
\end{equation*}
so it remains to show
\begin{equation*}
3 (1 + |x + y|^2 ) \leqslant 4 (1 + |x|^2) (1 + |y|^2), \quad x, y \in \rr d.
\end{equation*}
The latter inequality can be written
\begin{equation*}
4 \langle x, y \rangle \leqslant 1 + | x - y |^2 + 4 |x|^2 |y|^2
\end{equation*}
whose truth is a consequence of $(2 |x| |y| - 1)^2 \geqslant 0$ and the Cauchy--Schwarz inequality.
\end{proof}
The normalization of the Fourier transform is
\begin{equation*}
\mathscr{F} f (\xi )= \widehat f(\xi ) = (2\pi )^{-\frac d2} \int _{\rr
{d}} f(x)e^{-i\scal x\xi }\, dx, \qquad \xi \in \rr d,
\end{equation*}
for $f\in \mathscr{S}(\rr d)$ (the Schwartz space), where $\scal \, \cdot \, \, \cdot \, $ denotes the scalar product on $\rr d$.
The conjugate linear action of a (ultra-)distribution $u$ on a test function $\phi$ is written $(u,\phi)$, consistent with the $L^2$ inner product $(\, \cdot \, ,\, \cdot \, ) = (\, \cdot \, ,\, \cdot \, )_{L^2}$ which is conjugate linear in the second argument.
Denote translation by $T_x f(y) = f( y-x )$ and modulation by $M_\xi f(y) = e^{i \scal y \xi} f(y)$
for $x,y,\xi \in \rr d$ where $f$ is a function or distribution defined on $\rr d$.
The composition is denoted $\Pi(x,\xi) = M_\xi T_x$.
Let $\varphi \in \mathscr{S}(\rr d) \setminus \{0\}$.
The short-time Fourier transform (STFT) \cite{Cordero1} of a tempered distribution $u \in \mathscr{S}'(\rr d)$ is defined by
\begin{equation*}
V_\varphi u (x,\xi) = (2\pi )^{-\frac d2} (u, M_\xi T_x \varphi) = \mathscr{F} (u T_x \overline \varphi)(\xi), \quad x,\xi \in \rr d.
\end{equation*}
Then $V_\varphi u$ is smooth and polynomially bounded \cite[Theorem~11.2.3]{Grochenig1}.
When $u \in \mathscr{S}(\rr d)$ it is instead superpolynomially decreasing, that is
\begin{equation*}
|V_\varphi u (x,\xi)| \lesssim \eabs{(x,\xi)}^{-N}, \quad (x,\xi) \in T^* \rr d, \quad \forall N \geqslant 0.
\end{equation*}
The inverse transform is given by
\begin{equation}\langlebel{eq:STFTinverse}
u = (2\pi )^{-\frac d2} \iint_{\rr {2d}} V_\varphi u (x,\xi) M_\xi T_x \varphi \, \mathrm {d} x \, \mathrm {d} \xi
\end{equation}
provided $\| \varphi \|_{L^2} = 1$, with action under the integral understood, that is
\begin{equation}\langlebel{eq:moyal}
(u, f) = (V_\varphi u, V_\varphi f)_{L^2(\rr {2d})}
\end{equation}
for $u \in \mathscr{S}'(\rr d)$ and $f \in \mathscr{S}(\rr d)$, cf. \cite[Theorem~11.2.5]{Grochenig1}.
\subsection{Spaces of functions and ultradistributions}
Let $s,t, h > 0$.
The space denoted $\mathcal S_{t,h}^s(\rr d)$
is the set of all $f\in C^\infty (\rr d)$ such that
\begin{equation}\langlebel{eq:seminormSigmas}
\nm f{\mathcal S_{t,h}^s}\equiv \sup \frac {|x^\alpha D ^\beta
f(x)|}{h^{|\alpha + \beta |} \alpha !^t \, \beta !^s}
\end{equation}
is finite, where the supremum is taken over all $\alpha ,\beta \in
\mathbf N^d$ and $x\in \rr d$.
The function space $\mathcal S_{t,h}^s$ is a Banach space which increases
with $h$, $s$ and $t$, and $\mathcal S_{t,h}^s \subseteq \mathscr{S}$.
The topological dual $(\mathcal S_{t,h}^s)'(\rr d)$ is
a Banach space such that $\mathscr{S}'(\rr d) \subseteq (\mathcal S_{t,h}^s)'(\rr d)$.
The Beurling type \emph{Gelfand--Shilov space}
$\Sigma _t^s(\rr d)$ is the projective limit
of $\mathcal S_{t,h}^s(\rr d)$ with respect to $h$ \cite{Gelfand2}. This means
\begin{equation}\langlebel{GSspacecond1}
\Sigma _t^s(\rr d) = \bigcap _{h>0} \mathcal S_{t,h}^s(\rr d)
\end{equation}
and the Fr{\'e}chet space topology of $\Sigma _t^s (\rr d)$ is defined by the seminorms $\nm \, \cdot \, t{\mathcal S_{t,h}^s}$ for $h>0$.
If $s + t > 1$ then $\Sigma _t^s(\rr d)\neq \{ 0\}$ \cite{Petersson1}.
The topological dual of $\Sigma _t^s(\rr d)$ is the space of (Beurling type) \emph{Gelfand--Shilov ultradistributions} \cite[Section~I.4.3]{Gelfand2}
\begin{equation}\tag*{({\rm Re}f{GSspacecond1})$'$}
(\Sigma _t^s)'(\rr d) =\bigcup _{h>0} (\mathcal S_{t,h}^s)'(\rr d).
\end{equation}
The dual space $(\Sigma _t^s)'(\rr d)$ may be equipped with several topologies: the weak$^*$ topology, the strong topology, the Mackey topology, and the topology defined by the union \eqref{GSspacecond1}$'$ as an inductive limit topology \cite{Schaefer1}. The latter topology is the strongest topology such that the inclusion $(\mathcal S_{t,h}^s)'(\rr d) \subseteq (\Sigma _t^s)'(\rr d)$ is continuous for all $h > 0$.
The Roumieu type Gelfand--Shilov space is the union
\begin{equation*}
\mathcal S_t^s(\rr d) = \bigcup _{h>0}\mathcal S_{t,h}^s(\rr d)
\end{equation*}
equipped with the inductive limit topology \cite{Schaefer1}, that is
the strongest topology such that each inclusion $\mathcal S_{t,h}^s(\rr d) \subseteq\mathcal S_t^s(\rr d)$
is continuous.
Then $\mathcal S _t^s(\rr d)\neq \{ 0\}$ if and only if $s+t \geqslant 1$ \cite{Gelfand2}.
The corresponding (Roumieu type) Gelfand--Shilov ultradistribution space is
\begin{equation*}
(\mathcal S_t^s)'(\rr d) = \bigcap _{h>0} (\mathcal S_{s,h}^t)'(\rr d).
\end{equation*}
For every $s,t > 0$ such that $s+t > 1$, and for any $\varepsilon > 0$ we have
\begin{equation*}
\Sigma _t^s (\rr d)\subseteq \mathcal S_t^s(\rr d)\subseteq
\Sigma _{t+\varepsilon}^{s+\varepsilon}(\rr d).
\end{equation*}
We will not use the Roumieu type spaces in this article but mention them as a service to a reader interested in a wider context.
We write $\Sigma _s^s (\rr d) = \Sigma_s (\rr d)$ and $(\Sigma _s^s)' (\rr d) = \Sigma_s' (\rr d)$.
Then $\Sigma_s(\rr d) \neq \{ 0 \}$ if and only if $s > \frac12$.
The Gelfand--Shilov (ultradistribution) spaces enjoy invariance properties, with respect to
translation, dilation, tensorization, coordinate transformation and (partial) Fourier transformation.
The Fourier transform extends
uniquely to homeomorphisms on $\mathscr S'(\rr d)$, from $(\mathcal
S_t^s)'(\rr d)$ to $(\mathcal
S_s^t)'(\rr d)$, and from $(\Sigma _t^s)'(\rr d)$ to $(\Sigma _s^t)'(\rr d)$, and restricts to
homeomorphisms on $\mathscr S(\rr d)$, from $\mathcal S_t^s(\rr d)$ to $\mathcal S_s^t(\rr d)$,
and from $\Sigma _t^s(\rr d)$ to $\Sigma _s^t(\rr d)$, and to a unitary operator on $L^2(\rr d)$.
Likewise \eqref{eq:moyal} holds when $u \in (\Sigma_t^s)'(\rr d)$, $f \in \Sigma_t^s(\rr d)$, $\varphi \in \Sigma_t^s(\rr d)$ and $\| \varphi \|_{L^2} = 1$.
At one occasion we will need Gelfand--Shilov spaces defined on $\rr {2d}$ which has possibly different behavior with respect to the two $\rr d$ coordinates \cite{Abdeljawad1,Cappiello2,Gelfand2}. Then the seminorms \eqref{eq:seminormSigmas}
are generalized into
\begin{equation}\langlebel{eq:seminormSigmas2}
\nm f{\mathcal S_{t_1,t_2,h}^{s_1,s_2}}\equiv \sup \frac {|x_1^{\alpha _1} x_2^{\alpha _2} D_{x_1} ^{\beta_1} D_{x_2} ^{\beta_2}
f( x_1, x_2 )|}{h^{|\alpha_1 + \alpha_2 + \beta_1 + \beta_2 |} \alpha_1 !^{t_1} \, \alpha_2 !^{t_2} \beta_1 !^{s_1} \beta_2 !^{s_2}}
\end{equation}
for $t_j, s_j > 0$, $j = 1,2$. The spaces $\Sigma_{t_1, t_2}^{s_1,s_2}(\rr {2d})$ and $(\Sigma_{t_1, t_2}^{s_1,s_2}) '(\rr {2d})$ are defined as above.
Working with Gelfand--Shilov spaces we will often need the inequality (cf. \cite{Cappiello2})
\begin{equation*}
|x+y|^{\frac1s} \leqslant \kappa(s^{-1} ) ( |x|^{\frac1s} + |y|^{\frac1s}), \quad x,y \in \rr d, \quad s > 0,
\end{equation*}
where
\begin{equation*}
\kappa (t)
=
\left\{
\begin{array}{ll}
1 & \mbox{if} \quad 0 < t \leqslant 1 \\
2^{t-1} & \mbox{if} \quad t > 1
\end{array}
\right. ,
\end{equation*}
which implies
\begin{equation*}
\begin{aligned}
e^{r |x+y|^{\frac1s} } & \leqslantlant e^{ \kappa(s^{-1} ) r |x|^{\frac1s}} e^{ \kappa(s^{-1} )r |y|^{\frac1s}}, \quad x,y \in \rr d, \quad r >0, \\
e^{- r \kappa(s^{-1} ) |x+y|^{\frac1s} } & \leqslantlant e^{- r |x|^{\frac1s}} e^{ \kappa(s^{-1} ) r |y|^{\frac1s}}, \quad x,y \in \rr d, \quad r >0.
\end{aligned}
\end{equation*}
We will often use the following estimate where we use $|\alpha|! \leqslant \alpha! d^{|\alpha|}$ for $\alpha \in \nn d$ \cite[Eq.~(0.3.3)]{Nicola1}. For any $s > 0$, $h > 0$ and any $\alpha \in \nn d$ we have
\begin{equation}\langlebel{eq:expestimate0}
\alpha!^{-s} h^{- |\alpha|}
= \left( \frac{h^{- \frac{|\alpha|}{s}}}{\alpha!} \right)^s
\leqslant \left( \frac{ \left( d h^{- \frac{1}{s}} \right)^{|\alpha|}}{|\alpha|!} \right)^s
\leqslant e^{s d h^{- \frac{1}{s}}}.
\end{equation}
\subsection{Weyl pseudodifferential operators}
Finally we need some elements from the calculus of pseudodifferential operators \cite{Folland1,Hormander0,Nicola1,Shubin1}.
Let $a \in C^\infty (\rr {2d})$ and $m \in \mathbf R$. Then $a$ is a \emph{Shubin symbol} of order $m$, denoted $a\in {G(X|Y)}amma^m$, if for all $\alpha,\beta \in \nn d$ there exists a constant $C_{\alpha,\beta}>0$ such that
\begin{equation}\langlebel{eq:shubinineq}
|\partial_x^\alpha \partial_\xi^\beta a(x,\xi)| \leqslant C_{\alpha,\beta} \langlengle (x,\xi)\operatorname{Ran}gle^{m-|\alpha + \beta|}, \quad x,\xi \in \rr d.
\end{equation}
The Shubin symbols ${G(X|Y)}amma^m$ form a Fr\'echet space where the seminorms are given by the smallest possible constants in \eqref{eq:shubinineq}.
For $a \in {G(X|Y)}amma^m$ a pseudodifferential operator in the Weyl quantization is defined by
\begin{equation}\langlebel{eq:weylquantization}
a^w(x,D) f(x)
= (2\pi)^{-d} \int_{\rr {2d}} e^{i \langlengle x-y, \xi \operatorname{Ran}gle} a \left(\frac{x+y}{2},\xi \right) \, f(y) \, \mathrm {d} y \, \mathrm {d} \xi, \quad f \in \mathscr{S}(\rr d),
\end{equation}
when $m<-d$. The definition extends to general $m \in \mathbf R$ if the integral is viewed as an oscillatory integral.
The operator $a^w(x,D)$ then acts continuously on $\mathscr{S}(\rr d)$ and extends uniquely by duality to a continuous operator on $\mathscr{S}'(\rr d)$.
By Schwartz's kernel theorem the Weyl quantization procedure may be extended to a weak formulation which yields continuous linear operators $a^w(x,D):\mathscr{S}(\rr{d}) \to \mathscr{S}'(\rr{d})$, even if $a$ is only an element of $\mathscr{S}'(\rr{2d})$.
Likewise $a^w(x,D):\Sigma_s(\rr{d}) \to \Sigma_s'(\rr{d})$ if $a \in \Sigma_s'(\rr {2d})$ and $s > \frac12$.
If $s > \frac12$ and $a \in \Sigma_s'(\rr {2d})$ the Weyl quantization extends a continuous operator $\Sigma_s(\rr d) \rightarrow \Sigma_s'(\rr d)$
that satisfies
\begin{equation}\langlebel{eq:wignerweyl}
(a^w(x,D) f, g) = (2 \pi)^{-d} (a, W(g,f) ), \quad f, g \in \Sigma_s(\rr d),
\end{equation}
where the cross-Wigner distribution is defined as
\begin{equation*}
W(g,f) (x,\xi) = \int_{\rr d} g (x+y/2) \overline{f(x-y/2)} e^{- i \langle y, \xi \rangle} \mathrm {d} y, \quad (x,\xi) \in \rr {2d}.
\end{equation*}
We have $W(g,f) \in \Sigma_s(\rr {2d})$ when $f,g \in \Sigma_s(\rr d)$.
The real phase space $T^* \rr d \simeq \rr d \operatorname{Op}lus \rr d$ is a real symplectic vector space equipped with the
canonical symplectic form
\begin{equation*}
\sigma((x,\xi), (x',\xi')) = \langlengle x' , \xi \operatorname{Ran}gle - \langlengle x, \xi' \operatorname{Ran}gle, \quad (x,\xi), (x',\xi') \in T^* \rr d.
\end{equation*}
This form can be expressed with the inner product as $\sigma(X,Y) = \langle \mathcal{J} X, Y \rangle$ for $X,Y \in T^* \rr d$
where
\begin{equation}\langlebel{eq:Jdef}
\mathcal{J} =
\left(
\begin{array}{cc}
0 & I_d \\
-I_d & 0
\end{array}
\right) \in \rr {2d \times 2d}.
\end{equation}
The real symplectic group $\operatorname{Sp}(d,\mathbf R)$ is the set of matrices in $\operatorname{GL}(2d,\mathbf R)$ that leaves $\sigma$ invariant.
Hence $\mathcal{J} \in \operatorname{Sp}(d,\mathbf R)$.
To each symplectic matrix $\chi \in \operatorname{Sp}(d,\mathbf R)$ is associated an operator $\mu(\chi)$ that is unitary on $L^2(\rr d)$, and determined up to a complex factor of modulus one, such that
\begin{equation}\langlebel{symplecticoperator}
\mu(\chi)^{-1} a^w(x,D) \, \mu(\chi) = (a \circ \chi)^w(x,D), \quad a \in \mathscr{S}'(\rr {2d})
\end{equation}
(cf. \cite{Folland1,Hormander0}).
The operator $\mu(\chi)$ is a homeomorphism on $\mathscr S$ and on $\mathscr S'$.
The same conclusions hold if $a \in \Sigma_s'(\rr {2d})$ in the functional framework $\Sigma_s$, $\Sigma_s'$ if $s > \frac12$.
In fact $\mu(\chi)$ is a homeomorphism on $\Sigma_s(\rr d)$ which extends uniquely to a homeomorphism on $\Sigma_s'(\rr d)$ \cite[Proposition~4.4]{Carypis1}.
The mapping $\operatorname{Sp}(d,\mathbf R) \ni \chi \rightarrow \mu(\chi)$ is called the metaplectic representation \cite{Folland1}.
It is in fact a representation of the so called $2$-fold covering group of $\operatorname{Sp}(d,\mathbf R)$, which is called the metaplectic group.
The metaplectic representation satisfies the homomorphism relation modulo a change of sign:
\begin{equation*}
\mu( \chi \chi') = \pm \mu(\chi ) \mu(\chi' ), \quad \chi, \chi' \in \operatorname{Sp}(d,\mathbf R).
\end{equation*}
\section{The Gabor and the $t,s$-Gelfand--Shilov wave front sets}\langlebel{sec:GelfandShilovWF}
First we define the Gabor wave front set $\mathrm{WF}g$ introduced in \cite{Hormander1} and further elaborated in \cite{Rodino2}.
\begin{defn}\langlebel{def:Gaborwavefront}
Let $\varphi \in \mathscr{S}(\rr d) \setminus 0$, $u \in \mathscr{S}'(\rr d)$ and $z_0 \in T^* \rr d \setminus 0$.
Then $z_0 \mathbf Ntin \mathrm{WF}g(u)$ if there exists an open conic set ${G(X|Y)}amma \subseteq T^* \rr d \setminus 0$ such that $z_0 \in {G(X|Y)}amma$ and
\begin{equation}\langlebel{eq:conedecay}
\sup_{z \in {G(X|Y)}amma} \eabs{z}^N | V_\varphi u(z)| < \infty, \quad N \geqslant 0.
\end{equation}
\end{defn}
This means that $V_\varphi u$ decays rapidly (super-polynomially) in ${G(X|Y)}amma$.
The condition \eqref{eq:conedecay} is independent of $\varphi \in \mathscr{S}(\rr d) \setminus 0$, in the sense that super-polynomial decay will hold also for $V_\psi u$ if $\psi \in \mathscr{S}(\rr d) \setminus 0$, in a possibly smaller cone containing $z_0$.
The Gabor wave front set is a closed conic subset of $T^*\rr d \setminus 0$.
By \cite[Proposition~2.2]{Hormander1} it is symplectically invariant in the sense of
\begin{equation}\langlebel{eq:metaplecticWFG}
\mathrm{WF}g( \mu(\chi) u) = \chi \mathrm{WF}g (u), \quad \chi \in \operatorname{Sp}(d, \mathbf R), \quad u \in \mathscr{S}'(\rr d).
\end{equation}
The Gabor wave front set is naturally connected to the definition of the $C^\infty$ wave front set \cite[Chapter~8]{Hormander0}, often called just the wave front set and denoted $\mathrm{WF}$.
For $u \in \mathscr{D}'(\rr d)$ a point in the phase space $(x_0,\xi_0) \in T^* \rr d$ such that $\xi_0 \neq 0$ satisfies $(x_0,\xi_0) \mathbf Ntin \mathrm{WF}(u)$
if there exists $\varphi \in C_c^\infty(\rr d)$ such that $\varphi(0) \neq 0$,
an open conical set ${G(X|Y)}amma_2 \subseteq \rr d \setminus 0$ such that $\xi_0 \in {G(X|Y)}amma_2$, and
\begin{equation*}
\sup_{\xi \in {G(X|Y)}amma_2} \eabs{\xi}^N | V_\varphi u(x_0,\xi)| < \infty, \quad N \geqslant 0.
\end{equation*}
The difference compared to $\mathrm{WF}g (u)$ is that the $C^\infty$ wave front set $\mathrm{WF}(u)$ is defined in terms of super-polynomial decay in the frequency variable, for $x_0 \in \rr d$ fixed, instead of super-polynomial decay in an open cone in the phase space $T^* \rr d$ containing the point of interest.
Pseudodifferential operators with Shubin symbols are microlocal with respect to the Gabor wave front set.
In fact we have by \cite[Proposition~2.5]{Hormander1}
\begin{equation*}
\mathrm{WF}g (a^w(x,D) u) \subseteq \mathrm{WF}g (u)
\end{equation*}
provided $a \in {G(X|Y)}amma^m$ and $u \in \mathscr{S}'(\rr d)$.
Let $u \in (\Sigma_t^s)' (\rr d)$ with $s + t > 1$.
If $\psi \in \Sigma_t^s (\rr d) \setminus 0$ then
\begin{equation}\langlebel{eq:STFTGFstdistr}
| V_\psi u (x,\xi)| \lesssim e^{r (|x|^{\frac1t} + |\xi|^{\frac1s})}
\end{equation}
for some $r > 0$. We have $u \in \Sigma_t^s (\rr d)$ if and only if
\begin{equation}\langlebel{eq:STFTGFstfunc}
| V_\psi u (x,\xi)| \lesssim e^{-r (|x|^{\frac1t} + |\xi|^{\frac1s})}
\end{equation}
for all $r > 0$. See e.g. \cite[Theorems~2.4 and 2.5]{Toft1}.
For $u \in (\Sigma_t^s)' (\rr d)$ we define the $t,s$-Gelfand--Shilov wave front set $\mathrm{WF}^{t,s} (u)$ as a closed subset of the phase space $T^* \rr d \setminus 0$ as follows.
\begin{defn}\langlebel{def:wavefrontGFst}
Let $s,t > 0$ satisfy $s + t > 1$, and suppose $\psi \in \Sigma_t^s(\rr d) \setminus 0$ and $u \in (\Sigma_t^s)'(\rr d)$.
Then $(x_0,\xi_0) \in T^* \rr d \setminus 0$ satisfies $(x_0,\xi_0) \mathbf Ntin \mathrm{WF}^{t,s} (u)$ if there exists an open set $U \subseteq T^*\rr d \setminus 0$ containing $(x_0,\xi_0)$ such that
\begin{equation}\langlebel{eq:notinWFGFst1}
\sup_{\langlembda > 0, \ (x,\xi) \in U} e^{r \langlembda} |V_\psi u(\langlembda^t x, \langlembda^s \xi)| < \infty, \quad \forall r > 0.
\end{equation}
\end{defn}
Due to \eqref{eq:STFTGFstdistr} it is clear that it suffices to check \eqref{eq:notinWFGFst1} for $\langlembda \geqslant L$
where $L > 0$ can be arbitrarily large, for each $r > 0$.
A consequence of Definition {\rm Re}f{def:wavefrontGFst} is that we have the scaling invariance (here we assume
$(x,\xi) \in T^* \rr d \setminus 0$)
\begin{equation}\langlebel{eq:WFstscalinv}
(x,\xi) \in \mathrm{WF}^{t,s} (u) \quad \Longleftrightarrow \quad (\langlembda^t x, \langlembda^s \xi) \in \mathrm{WF}^{t,s} (u) \quad \forall \langlembda > 0.
\end{equation}
Another immediate consequence of Definition {\rm Re}f{def:wavefrontGFst} is
\begin{equation}\langlebel{eq:WFstsublinear}
\mathrm{WF}^{t,s} (u + v) \subseteq \mathrm{WF}^{t,s} (u) \cup \mathrm{WF}^{t,s} (v), \quad u,v \in (\Sigma_t^s)'(\rr d).
\end{equation}
If $t = s > \frac12$ and $u \in \Sigma_s' (\rr d)$ then $\mathrm{WF}^{s,s} (u) = \mathrm{WF}^s(u)$, that is we recapture
the $s$-Gelfand--Shilov wave front set $\mathrm{WF}^s (u)$ (which is a slightly modified version of Cappiello's and Schulz's \cite[Definition~2.1]{Cappiello1}),
as defined originally in \cite[Definition~4.1]{Carypis1}:
\begin{defn}\langlebel{def:wavefrontGFs}
Let $s > 1/2$, $\psi \in \Sigma_s(\rr d) \setminus 0$ and $u \in \Sigma_s'(\rr d)$.
Then $z_0 \in T^*\rr d \setminus 0$ satisfies $z_0 \mathbf Ntin \mathrm{WF}^s (u)$ if there exists an open conic set ${G(X|Y)}amma_{z_0} \subseteq T^*\rr d \setminus 0$ containing $z_0$ such that
\begin{equation*}
\sup_{z \in {G(X|Y)}amma_{z_0}} e^{r | z |^{\frac1s}} |V_\psi u(z)| < \infty, \quad \forall r > 0.
\end{equation*}
\end{defn}
In Definition {\rm Re}f{def:wavefrontGFst} we ask for exponential decay with arbitrary parameter $r > 0$ (super-exponential) of $V_\psi u$ along the curve $C_{x,\xi} \in T^* \rr d$
defined by $\mathbf R_+ \ni \langlembda \to (\langlembda^t x, \langlembda^s \xi)$ which passes through $(x,\xi) \in T^* \rr d \setminus 0$.
This power type curve reduces to a straight line if $t=s$.
By \eqref{eq:STFTGFstdistr} a generic point $(x,\xi) \in T^* \rr d \setminus 0$ has an exponential growth upper bound along the curve $C_{x,\xi}$.
Due to \eqref{eq:STFTGFstfunc} we have $\mathrm{WF}^{t,s} (u) = \emptyset$ if and only if $u \in \Sigma_t^s(\rr d)$.
Thus $\mathrm{WF}^{t,s} (u) \subseteq T^* \rr d \setminus 0$ can be seen as a measure of singularities of $u \in (\Sigma_t^s)'(\rr d)$: It records the phase space points $(x,\xi) \in T^* \rr d \setminus 0$ such that $V_\psi u$ does not decay super-exponentially along the curve $C_{x,\xi}$,
that is, does not behave like an element in $\Sigma_t^s(\rr d)$ there.
We will soon show that Definition {\rm Re}f{def:wavefrontGFst} does not depend on the window function $\psi \in \Sigma_t^s(\rr d) \setminus 0$ (see Proposition {\rm Re}f{prop:windowinvariance}).
If $\check u(x) = u(-x)$ then
\begin{equation}\langlebel{eq:evensteven0}
V_{\check \psi} \check u(x,\xi)
= V_\psi u(-x,-\xi).
\end{equation}
If $u$ is even or odd we thus have the following symmetry:
\begin{equation}\langlebel{eq:evensteven1}
\check u = \pm u \quad \Longrightarrow \quad \mathrm{WF}^{t,s} (u) = - \mathrm{WF}^{t,s} (u).
\end{equation}
We also have
\begin{equation}\langlebel{eq:evensteven2}
V_{\psi} \overline{u}(x,\xi)
= \overline{V_{\overline \psi} u(x,-\xi)}.
\end{equation}
\begin{rem}\langlebel{rem:WFinclusion}
Suppose $s_j,t_j > 0$, $j=1,2$, $s_1 + t_1 > 1$,
and $t_2/t_1 = s_2/s_1 = a \geqslant 1$.
Then we have for $u \in (\Sigma_{t_2}^{s_2})'(\rr d) \subseteq (\Sigma_{t_1}^{s_1})'(\rr d)$
\begin{equation*}
\mathrm{WF}^{t_2,s_2} (u) \subseteq \mathrm{WF}^{t_1,s_1} (u).
\end{equation*}
In fact this follows directly from
Definition {\rm Re}f{def:wavefrontGFst} with $\psi \in \Sigma_{t_1}^{s_1}(\rr d) \setminus 0$, and
$\langlembda^{t_2} = (\langlembda^ a)^{t_1}$,
$\langlembda^{s_2} = (\langlembda^ a)^{s_1}$,
and $\langlembda \leqslant \langlembda^a$ for $\langlembda \geqslant 1$.
\end{rem}
\subsection{Examples of Gabor and $s$-Gelfand--Shilov wave front sets}
In this subsection we compile known and deduce a few new results on the $t,s$-Gelfand--Shilov wave front set.
We have
\begin{equation}\langlebel{eq:GaborGSinclusion}
\mathrm{WF}g ( u ) \subseteq \mathrm{WF}^s(u), \quad \forall s > \frac12, \quad u \in \mathscr{S}'(\rr d).
\end{equation}
If $\frac12 < s_1 < s_2$ then
\begin{equation}\langlebel{eq:strictinclusion1}
\Sigma_{s_1}(\rr d) \subsetneq \Sigma_{s_2}(\rr d),
\end{equation}
\begin{equation}\langlebel{eq:strictinclusion2}
\Sigma_{s_2}'(\rr d) \subsetneq \Sigma_{s_1}'(\rr d)
\end{equation}
and
\begin{equation*}
\mathrm{WF}^{s_2}(u) \subseteq \mathrm{WF}^{s_1}(u), \quad u \in \Sigma_{s_2}'(\rr d).
\end{equation*}
The strictness of the inclusions \eqref{eq:strictinclusion1} and \eqref{eq:strictinclusion2}
can be seen for instance from the Hilbert sequence space characterizations of $\Sigma_{s}(\rr d)$ and $\Sigma_{s}'(\rr d)$
for series expansions in Hermite functions (cf. e.g. \cite{Wahlberg3}).
If $u \in \Sigma_{s_2}(\rr d) \setminus \Sigma_{s_1}(\rr d)$ then $u \in \Sigma_{s_2}'(\rr d)$ and
\begin{equation*}
\mathrm{WF}^{s_2}(u) = \emptyset \neq \mathrm{WF}^{s_1}(u).
\end{equation*}
So given $s_2 > s_1 > \frac12$ there exists $u \in \Sigma_{s_2}'(\rr d)$ such that $\mathrm{WF}^{s_2}(u) \neq \mathrm{WF}^{s_1}(u)$.
This gives some motivation for the interest of the scale of wave front sets $\mathrm{WF}^{s}(u)$ for $s > \frac12$.
In the given example it is a measure of very fine singularities within $\mathscr{S}$.
If on the other hand $u \in \Sigma_{s_1}'(\rr d) \setminus \Sigma_{s_2}'(\rr d)$ then
$\mathrm{WF}^{s_1}(u)$ is well defined, and $\mathrm{WF}^{s_1}(u) \neq \emptyset$ since $u \in \Sigma_{s_1}$ would imply
$u \in \Sigma_{s_2}'$. But $\mathrm{WF}^{s_2}(u)$ is not well defined so we cannot compare $\mathrm{WF}^{s_1}(u)$ and $\mathrm{WF}^{s_2}(u)$.
It is also clear that if $u \in \mathscr{S}(\rr d) \setminus \Sigma_s(\rr d)$ for some $s > \frac12$
then $u \in \Sigma_s'(\rr d)$ and
\begin{equation*}
\emptyset = \mathrm{WF}g (u) \neq \mathrm{WF}^s(u).
\end{equation*}
Nevertheless it seems that for most ultradistributions $u$ for which $\mathrm{WF}^s(u)$ can be determined we have
\begin{equation*}
\mathrm{WF}g (u) = \mathrm{WF}^s(u) \quad \mbox{for all} \quad s > \frac12
\end{equation*}
(cf. \cite{Carypis1,PRW1}).
We collect a few examples.
For any $x \in \rr d$ we have
\begin{equation}\langlebel{eq:diracWF}
\mathrm{WF}g (\delta_x) = \mathrm{WF}^s(\delta_x) = \{ 0 \} \times (\rr d \setminus 0) \quad \forall s > \frac12.
\end{equation}
For any $\xi \in \rr d$ we have
\begin{equation}\langlebel{eq:planewaveWF}
\mathrm{WF}g ( e^{i \langle \, \cdot \, t, \xi \rangle} ) = \mathrm{WF}^s( e^{i \langle \, \cdot \, t, \xi \rangle} ) = (\rr d \setminus 0) \times \{ 0 \} \quad \forall s > \frac12.
\end{equation}
For any $A \in \rr {d \times d}$ symmetric we have
\begin{equation}\langlebel{eq:chirpWF1}
\mathrm{WF}g ( e^{i \langle x, A x \rangle/2} ) = \mathrm{WF}^s( e^{i \langle x, A x \rangle/2} ) = \{ (x,Ax): \ x \in \rr d \setminus 0 \} \quad \forall s > \frac12.
\end{equation}
The latter formula can be generalized, by combining \cite[Example~7.1]{PRW1} (generalized to the Gelfand--Shilov framework) and \cite[Corollary~9.2]{Carypis1}.
This gives the following formula when $A \in \cc {d \times d}$ is symmetric and ${\rm Im} A \geqslant 0$:
\begin{equation}\langlebel{eq:chirpWF2}
\mathrm{WF}g ( e^{i \langle x, A x \rangle/2} ) = \mathrm{WF}^s( e^{i \langle x, A x \rangle/2} ) = \{ (x, {\rm Re} A \, x): \ x \in \rr d \cap \operatorname{Op}eratorname{Ker} ({\rm Im} A) \setminus 0 \} \quad \forall s > \frac12.
\end{equation}
If $d = 1$ then (cf. \cite[Section~8]{PRW1}) for $k \geqslant 2$ we have
\begin{equation}\langlebel{eq:evenosc1}
\mathrm{WF}g ( e^{i x^{2k}} ) = \{ 0 \} \times (\mathbf R \setminus 0)
\end{equation}
and for $k \geqslant 1$ we have
\begin{equation}\langlebel{eq:oddosc1}
\mathrm{WF}g ( e^{i x^{2k+1}} ) = \{ 0 \} \times \mathbf R_+.
\end{equation}
It also follows from the proof of \cite[Proposition~8.2]{PRW1} that
\begin{equation}\langlebel{eq:modulusosc1}
\mathrm{WF}g ( e^{i |x|^p} ) = \{ 0 \} \times (\mathbf R \setminus 0)
\end{equation}
if $p > 2$.
We obtain from \eqref{eq:GaborGSinclusion} if $k \geqslant 2$
\begin{equation*}
\{ 0 \} \times (\mathbf R \setminus 0)
\subseteq \mathrm{WF}^s( e^{i x^{2k}} )
\quad \forall s > \frac12
\end{equation*}
In the same way we obtain from \eqref{eq:oddosc1} if $k \geqslant 1$
\begin{equation*}
\{ 0 \} \times \mathbf R_+
\subseteq \mathrm{WF}^s( e^{i x^{2k+1}} ) \quad \forall s > \frac12
\end{equation*}
and from \eqref{eq:modulusosc1}
\begin{equation*}
\{ 0 \} \times (\mathbf R \setminus 0)
\subseteq \mathrm{WF}^s( e^{i |x|^p} ) \quad \forall s > \frac12
\end{equation*}
if $p > 2$.
In \cite[Theorem~6.1]{Schulz1} we prove that given any closed conic set ${G(X|Y)}amma \subseteq T^*\rr d \setminus 0$
there exists $u \in \mathscr{S}'(\rr d)$ such that $\mathrm{WF}g (u) = {G(X|Y)}amma$.
By a careful examination of the proof it follows that $\mathrm{WF}g (u) = \mathrm{WF}^s(u) = {G(X|Y)}amma$ for all $s > \frac12$.
A similar result is given in \cite[Proposition~3.5]{Cappiello1} for a wave front set that is similar to $\mathrm{WF}^s(u)$ albeit with the Roumieu choice of behaviour instead of Beurling.
\subsection{Invariances of the $t,s$-Gelfand--Shilov wave front set}
In \cite[Proposition~4.3]{Carypis1} it is shown that $\mathrm{WF}^s (u)$ does not depend on the chosen window function $\psi \in \Sigma_s(\rr d) \setminus 0$.
The following result generalizes this statement to $\mathrm{WF}^{s,t} (u)$ with $t \neq s$.
\begin{prop}\langlebel{prop:windowinvariance}
Let $s,t > 0$ satisfy $s + t > 1$,
and let
$u \in (\Sigma_t^s)'(\rr d)$.
Suppose $z_0 \in T^* \rr d \setminus 0$.
If $\psi \in \Sigma_t^s(\rr d) \setminus 0$ and \eqref{eq:notinWFGFst1} holds for an open set $U \subseteq T^*\rr d \setminus 0$ containing $z_0$,
and $\varphi \in \Sigma_t^s(\rr d) \setminus 0$ then there exists an open set $V \subseteq U$ such that $z_0 \in V$ and
\begin{equation}\langlebel{eq:notinWFGFst2}
\sup_{\langlembda > 0, \ (x,\xi) \in V} e^{r \langlembda } |V_\varphi u(\langlembda^t x, \langlembda^s \xi)| < \infty, \quad \forall r > 0.
\end{equation}
\end{prop}
\begin{proof}
Since $z_0 \in U \subseteq \rr {2d}$ where $U$ is open we may pick an open set $V \subseteq U$
such that $z_0 \in V$ and $V + \operatorname{B}_\varepsilon \subseteq U$ for some $0 < \varepsilon \leqslant 1$, and we may assume
\begin{equation}\langlebel{eq:Vbounds}
\sup_{z \in V} |z| \leqslant |z_0| + 1 := \mu.
\end{equation}
By \eqref{eq:STFTGFstdistr} we have
\begin{equation}\langlebel{eq:STFT1}
| V_\varphi u (x,\xi)| \lesssim e^{r_1 (|x|^{\frac1t} + |\xi|^{\frac1s})}
\end{equation}
for some $r_1 > 0$.
By \cite[Lemma~11.3.3]{Grochenig1} we have
\begin{equation*}
|V_\varphi u (z)| \leqslant (2 \pi)^{-\frac{d}{2}} \| \psi \|_{L^2}^{-2} \, |V_\psi u| * |V_\varphi \psi | (z), \quad z \in \rr {2d},
\end{equation*}
and according to \eqref{eq:STFTGFstfunc} we have
\begin{equation}\langlebel{eq:STFTGFstwindow1}
| V_\varphi \psi (x,\xi)| \lesssim e^{-r_2 (|x|^{\frac1t}+ |\xi|^{\frac1s})}
\end{equation}
for any $r_2 > 0$.
Let $r > 0$ and $\langlembda > 0$. We have
\begin{align*}
& e^{r \langlembda} |V_\varphi u (\langlembda^t x, \langlembda^s \xi)| \\
& \lesssim \iint_{\rr {2d}} e^{r \langlembda } |V_\psi u ( \langlembda^t (x- \langlembda^{-t} y), \langlembda^s (\xi - \langlembda^{-s} \eta))| \ |V_\varphi \psi (y,\eta) | \, \mathrm {d} y \, \mathrm {d} \eta \\
& = I_1 + I_2
\end{align*}
where we split the integral into the two terms
\begin{align*}
I_1 = & \iint_{\rr {2d} \setminus \Omega_\langlembda} e^{r \langlembda} |V_\psi u ( \langlembda^t (x- \langlembda^{-t} y), \langlembda^s (\xi - \langlembda^{-s} \eta))| \ |V_\varphi \psi (y,\eta) | \, \mathrm {d} y \, \mathrm {d} \eta, \\
I_2 = & \iint_{\Omega_\langlembda} e^{r \langlembda} |V_\psi u ( \langlembda^t (x- \langlembda^{-t} y), \langlembda^s (\xi - \langlembda^{-s} \eta))| \ |V_\varphi \psi (y,\eta) | \, \mathrm {d} y \, \mathrm {d} \eta
\end{align*}
where
\begin{equation*}
\Omega_\langlembda = \{(y,\eta) \in \rr {2d}: |y|^{\frac1t} + |\eta|^{\frac1s} < 2^{-\frac{1}{2v}} \varepsilon^{\frac1v} \langlembda \}
\subseteq \rr {2d}
\end{equation*}
with $v = \min(s,t)$.
First we estimate $I_1$ when $(x,\xi) \in V$.
Set $\kappa = \max(\kappa(t^{-1}), \kappa(s^{-1}))$.
From \eqref{eq:Vbounds}, \eqref{eq:STFT1} and \eqref{eq:STFTGFstwindow1} we obtain for some $r_1 > 0$ and any $r_2 > 0$
\begin{equation}\langlebel{eq:estimateI1a}
\begin{aligned}
I_1
& \lesssim e^{r \langlembda } \iint_{\rr {2d} \setminus \Omega_\langlembda} e^{r_1 \langlembda |x- \langlembda^{-t} y|^{\frac1t} + r_1 \langlembda |\xi- \langlembda^{-s} \eta|^{\frac1s}} \ |V_\varphi \psi (y,\eta) | \, \mathrm {d} y \, \mathrm {d} \eta \\
& \leqslant e^{ r \langlembda+ \kappa r_1 \langlembda |x|^{\frac1t} + \kappa r_1 \langlembda |\xi|^{\frac1s} } \iint_{\rr {2d} \setminus \Omega_\langlembda} e^{ r_1 \kappa (| y|^{\frac1t} + | \eta|^{\frac1s}) } \ |V_\varphi \psi (y,\eta) | \, \mathrm {d} y \, \mathrm {d} \eta \\
& \lesssim e^{ \langlembda \left( r + 2 r_1 \kappa \mu^{\frac1v} \right) } \iint_{\rr {2d} \setminus \Omega_\langlembda} e^{ (\kappa r_1 - \kappa r_1 - 1 - r_2) (|y|^{\frac1t} + |\eta|^{\frac1s})} \, \mathrm {d} y \, \mathrm {d} \eta \\
& \leqslant e^{ \langlembda \left( r + 2 r_1 \kappa \mu^{\frac1v} \right) - \langlembda \, r_2 2^{-\frac{1}{2v}}\varepsilon^{\frac1v} } \iint_{\rr {2d}} e^{ - (|y|^{\frac1t} + |\eta|^{\frac1s})} \, \mathrm {d} y \, \mathrm {d} \eta \\
& \lesssim e^{ \langlembda \left( r + 2 r_1 \kappa \mu^{\frac1v} - r_2 2^{-\frac{1}{2v}} \varepsilon^{\frac1v } \right) }\leqslant C_{r}
\end{aligned}
\end{equation}
for any $\langlembda > 0$,
provided we pick $r_2 \geqslant 2^{\frac{1}{2v}} \varepsilon^{- \frac1v} \left( r + 2 r_1 \kappa \mu^{\frac1v}\right)$.
Here $C_{r} > 0$ is a constant that depends on $r > 0$ but not on $\langlembda > 0$.
Thus we have obtained the requested estimate for $I_1$.
It remains to estimate $I_2$.
From $|y|^{\frac1t} + |\eta|^{\frac1s} < 2^{-\frac{1}{2v}} \varepsilon^{\frac1v} \langlembda$
we obtain
\begin{align*}
& \langlembda^{-t} |y| < \varepsilon^{\frac{t}{v}} \, 2^{-\frac{t}{2v}} \leqslant \varepsilon \, 2^{-\frac{1}{2}}, \\
& \langlembda^{-s} |\eta| < \varepsilon^{\frac{s}{v}} \, 2^{-\frac{s}{2v}} \leqslant \varepsilon \, 2^{-\frac{1}{2}}
\end{align*}
which gives $(\langlembda^{-t} y, \langlembda^{-s} \eta ) \in \operatorname{B}_{\varepsilon}$.
Hence if $(x,\xi) \in V$ then $( x- \langlembda^{-t} y, \xi - \langlembda^{-s} \eta) \in U$ and we may use the estimate
\eqref{eq:notinWFGFst1}.
This gives for a constant $C_{r} > 0$, using \eqref{eq:STFTGFstwindow1}
\begin{equation}\langlebel{eq:estimateI2a}
\begin{aligned}
I_2 = & \iint_{\Omega_\langlembda} e^{r \langlembda} |V_\psi u ( \langlembda^t (x- \langlembda^{-t} y), \langlembda^s (\xi - \langlembda^{-s} \eta))| \ |V_\varphi \psi (y,\eta) | \, \mathrm {d} y \, \mathrm {d} \eta \\
& \leqslant C_{r} \iint_{\rr {2d}} |V_\varphi \psi (y,\eta) | \, \mathrm {d} y \, \mathrm {d} \eta \\
& \leqslant C_{r}'
\end{aligned}
\end{equation}
for all $\langlembda > 0$,
Thus we have obtained the requested estimate for $I_2$.
The statement follows from \eqref{eq:estimateI1a} and \eqref{eq:estimateI2a}.
\end{proof}
\subsection{Metaplectic properties}
The $s$-Gelfand--Shilov wave front set is symplectically invariant as (cf. \cite[Corollary~4.5]{Carypis1})
\begin{equation}\langlebel{eq:metaplecticWFs}
\mathrm{WF}^s( \mu(\chi) u) = \chi \mathrm{WF}^s(u), \quad \chi \in \operatorname{Sp}(d, \mathbf R), \quad u \in \Sigma_s'(\rr d), \quad s > \frac12.
\end{equation}
When $t \neq s$ the $t,s$-Gelfand--Shilov wave front set $\mathrm{WF}^{t,s} (u)$ is not symplectically invariant.
Nevertheless, two of the generators of the symplectic group behave invariantly in certain individual senses which we now describe.
By \cite[Proposition~4.10]{Folland1} each matrix $\chi \in \operatorname{Sp}(d,\mathbf R)$ is a finite product of matrices in $\operatorname{Sp}(d,\mathbf R)$ of the form
\begin{equation*}
\mathcal{J}, \quad
\left(
\begin{array}{cc}
A^{-1} & 0 \\
0 & A^{T}
\end{array}
\right),
\quad
\left(
\begin{array}{cc}
I & 0 \\
B & I
\end{array}
\right),
\end{equation*}
for $A \in \operatorname{GL}(d,\mathbf R)$ and $B \in \rr {d \times d}$ symmetric.
The corresponding metaplectic operators are
$\mu(\mathcal{J}) = \mathscr{F}$,
\begin{equation*}
\mu \left(
\begin{array}{cc}
A^{-1} & 0 \\
0 & A^{T}
\end{array}
\right) f(x)
= |A|^{\frac12} f(Ax),
\end{equation*}
if $A \in \operatorname{GL}(d,\mathbf R)$, and
\begin{equation*}
\mu
\left(
\begin{array}{cc}
I & 0 \\
B & I
\end{array}
\right) f(x)
= e^{\frac{i}{2} \langle B x, x \rangle} f(x),
\end{equation*}
if $B \in \rr {d \times d}$ is symmetric.
\begin{prop}\langlebel{prop:WFstsymplectic}
Let $s,t > 0$ satisfy $s + t > 1$, and suppose
$u \in (\Sigma_t^s)'(\rr d)$. Then we have
\begin{enumerate}[\rm(i)]
\item
\begin{equation*}
\mathrm{WF}^{s,t} (\widehat u) = \mathcal{J} \mathrm{WF}^{t,s} (u);
\end{equation*}
\item if $A \in \operatorname{GL}(d,\mathbf R)$ and $u_A (x) = |A|^{\frac12} u(Ax)$ then
\begin{equation*}
\mathrm{WF}^{t,s} (u_A) =
\left(
\begin{array}{cc}
A^{-1} & 0 \\
0 & A^{T}
\end{array}
\right)
\mathrm{WF}^{t,s} (u).
\end{equation*}
\end{enumerate}
\end{prop}
\begin{proof}
Let $\psi \in \Sigma_t^s(\rr d) \setminus 0$.
We have from the proof of \cite[Corollary~4.5]{Carypis1}
\begin{equation}\langlebel{eq:STFTmetaplectic}
|V_{\mu (\chi) \psi} (\mu(\chi) u)( \chi(x,\xi))| = |V_{\psi} u( x, \xi)|
\end{equation}
for all $\chi \in \operatorname{Sp}(d, \mathbf R)$. If $\chi = \mathcal{J}$ we obtain
\begin{equation*}
|V_{\widehat \psi} \widehat u(\mathcal{J} (x,\xi))| = |V_{\widehat \psi} \widehat u(\xi, - x)| = |V_{\psi} u( x,\xi )|.
\end{equation*}
Note that $\widehat \psi \in \Sigma_s^t(\rr d) \setminus 0$ and $\widehat u \in (\Sigma_s^t)'(\rr d)$.
From this it follows that $(x,\xi) \mathbf Ntin \mathrm{WF}^{t,s} (u)$ if and only if $\mathcal{J}(x,\xi) \mathbf Ntin \mathrm{WF}^{s,t} (\widehat u)$ which proves claim (i).
Next we insert $u_A$ for $A \in \operatorname{GL}(d,\mathbf R)$ into \eqref{eq:STFTmetaplectic} which gives
\begin{equation*}
|V_{\psi_A} u_A ( A^{-1}x, A^T \xi )| = |V_\psi u( x,\xi) |.
\end{equation*}
Note that $\psi_A \in \Sigma_t^s(\rr d) \setminus 0$ and $u_A \in (\Sigma_t^s)'(\rr d)$.
We obtain $(x,\xi) \mathbf Ntin \mathrm{WF}^{t,s} (u)$ if and only if $(A^{-1}x, A^T \xi) \mathbf Ntin \mathrm{WF}^{t,s} (u_A)$ which shows claim (ii).
\end{proof}
\begin{rem}\langlebel{rem:propagator}
Proposition {\rm Re}f{prop:WFstsymplectic} implies that $\mathrm{WF}^{v,s}$ when $s \neq v$ does not behave as $\mathrm{WF}^{s}$ with respect to Schr\"odinger type propagators, in the case of quadratic potential.
In fact let $Q \in \rr {2d \times 2d}$ be symmetric, let
\begin{equation*}
q(x,\xi) = \langle (x,\xi), Q (x,\xi) \rangle, \quad x, \ \xi \in \rr d,
\end{equation*}
and consider the initial value Cauchy problem
\begin{equation}\langlebel{eq:schrodeq}
\left\{
\begin{array}{rl}
\partial_t u(t,x) + i q^w(x,D_x) u (t,x) & = 0, \\
u(0,\, \cdot \, t) & = u_0,
\end{array}
\right.
\end{equation}
where $q^w(x,D_x)$ acts on the $x \in \rr d$ variable.
If $u_0 \in D (q^w(x,D)) \subseteq L^2(\rr d)$, the domain of the closure of $q^w(x,D)$ considered as an unbounded operator in $L^2(\rr d)$, the equation is solved by
\begin{equation*}
u(t,x) = e^{- i t q^w(x,D)} u_0
\end{equation*}
where $e^{- i t q^w(x,D)}$ is the propagator one-parameter group of unitary operators indexed by $t \in \mathbf R$ (cf. e.g.\cite{Carypis1,Hormander2}).
The propagator is the metaplectic operator $e^{- i t q^w(x,D)} = \mu( e^{2 t \mathcal{J} Q})$ \cite{Folland1},
which extends to a continuous operator on $\Sigma_s'(\rr d)$ for $s > \frac12$ and the equation \eqref{eq:schrodeq} admits initial datum $u_0 \in \Sigma_s'(\rr d)$ \cite{Carypis1,Wahlberg3}.
By the metaplectic invariance \eqref{eq:metaplecticWFs} we thus have the propagation of singularities equality
\begin{equation}\langlebel{eq:propschrodWFs}
\mathrm{WF}^s ( e^{- i t q^w(x,D)} u_0) = e^{2 t \mathcal{J} Q} \mathrm{WF}^s(u_0), \quad t \in \mathbf R, \quad u_0 \in \Sigma_s'(\rr d), \quad s > \frac12.
\end{equation}
If $Q = I_{2d}$ then
\begin{equation*}
e^{2 t \mathcal{J} Q} =
\left(
\begin{array}{ll}
\mathbf Cs 2t & \sin 2t \\
- \sin 2t & \mathbf Cs 2t
\end{array}
\right)
\end{equation*}
so
\begin{equation}\langlebel{eq:WFsFourier}
\mathrm{WF}^s ( e^{- i \frac{\pi}{4} q^w(x,D)} u_0)
= \mathrm{WF}^s ( \widehat u_0)
= \mathcal{J} \mathrm{WF}^s(u_0).
\end{equation}
If $s \neq v$ then the equality \eqref{eq:propschrodWFs} cannot hold for $\mathrm{WF}^{v,s}$,
since \eqref{eq:WFsFourier} for $\mathrm{WF}^{v,s} (u)$ would contradict Proposition {\rm Re}f{prop:WFstsymplectic} (i).
\end{rem}
The next result reveals
that if $\mathrm{WF}^t (u)$ has empty intersection with the frequency axis $\{ 0 \} \times (\rr d \setminus 0)$
then $\mathrm{WF}^{t,s}$ is contained in the space axis $(\rr d \setminus 0) \times \{ 0 \}$ if $s > t$.
\begin{prop}\langlebel{prop:WFsvsWFst1}
If $s > t > \frac12$, $u \in (\Sigma_t^s)'(\rr d)$ and
\begin{equation}\langlebel{eq:WFsfreqaxis}
\mathrm{WF}^t (u) \cap \{ 0 \} \times (\rr d \setminus 0) = \emptyset
\end{equation}
then
\begin{equation}\langlebel{eq:WFstconclusion1}
\mathrm{WF}^{t,s} (u) \subseteq (\rr d \setminus 0) \times \{ 0 \}.
\end{equation}
\end{prop}
\begin{proof}
We have $(\Sigma_t^s)'(\rr d) \subseteq \Sigma_t'(\rr d)$ since $\Sigma_t(\rr d) \subseteq \Sigma_t^s(\rr d)$.
By the assumption \eqref{eq:WFsfreqaxis} there exists $C > 0$ such that
for the open conic set
\begin{equation*}
{G(X|Y)}amma = \{ (x,\xi) \in T^* \rr d \setminus 0: |\xi| > C |x| \} \subseteq T^* \rr d
\end{equation*}
we have
\begin{equation*}
\sup_{z \in {G(X|Y)}amma} e^{r | z |^{\frac1t}} |V_\psi u(z)| < \infty \quad \forall r > 0
\end{equation*}
where $\psi \in \Sigma_t(\rr d) \setminus 0 \subseteq \Sigma_t^s (\rr d) \setminus 0$.
Let $(x_0,\xi_0) \in T^* \rr d \setminus 0$ where $\xi_0 \neq 0$.
If $x_0 = 0$ we pick $U \subseteq {G(X|Y)}amma$ as an open set containing $(0,\xi_0)$.
Then if $(x,\xi) \in U$ we have $(\langlembda^t x, \langlembda^s \xi ) \in {G(X|Y)}amma$ for $\langlembda \geqslant 1$,
since $|\xi| > C |x|$ implies $\langlembda^{s-t} |\xi| > C |x|$.
If instead $x_0 \neq 0$ then we pick as $U \subseteq \rr {2d}$ an open set containing $(x_0,\xi_0)$ such that $\varepsilon < |x| < 2 |(x_0,\xi_0)|$
and $\varepsilon < |\xi| < 2 |(x_0,\xi_0)|$ when $(x,\xi) \in U$ where $\varepsilon > 0$.
If $(x,\xi) \in U$ then
\begin{equation*}
C |x| |\xi|^{-1}
< 2 | (x_0,\xi_0) | C \varepsilon^{-1} \leqslant \langlembda^{s-t}
\end{equation*}
if $\langlembda \geqslant L > 0$ provided $L$ is sufficiently large.
This gives $(\langlembda^t x, \langlembda^s \xi ) \in {G(X|Y)}amma$ for $\langlembda \geqslant L$.
If necessary we increase $L > 0$ such that $|(x, \langlembda^{s-t} \xi)| \geqslant 1$ when $\langlembda \geqslant L$ and $(x,\xi) \in U$.
This gives for any $r > 0$
\begin{align*}
\sup_{\langlembda \geqslant L, \ (x,\xi) \in U} e^{r \langlembda } |V_\psi u(\langlembda^t x, \langlembda^s \xi)|
& \leqslant \sup_{\langlembda \geqslant L, \ (x,\xi) \in U} e^{r \langlembda |(x, \langlembda^{s-t} \xi)|^{\frac1t}} |V_\psi u(\langlembda^t x, \langlembda^s \xi)| \\
& \leqslant \sup_{\langlembda \geqslant L, \ (x,\xi) \in U} e^{r |(\langlembda^t x, \langlembda^s \xi)|^{\frac1t}} |V_\psi u(\langlembda^t x, \langlembda^s \xi)| \\
& \leqslant \sup_{z \in {G(X|Y)}amma} e^{r | z |^{\frac1t}} |V_\psi u(z)| < \infty.
\end{align*}
We have shown $(x_0, \xi_0) \mathbf Ntin \mathrm{WF}^{t,s} (u)$ which proves \eqref{eq:WFstconclusion1}.
\end{proof}
\section{The $t,s$-Gelfand--Shilov wave front set of oscillatory functions}\langlebel{sec:chirp}
A main reason for the introduction of the wave front set $\mathrm{WF}^{t,s} (u)$ is that it describes accurately the phase space singularities of oscillatory functions of the form
\begin{equation}\langlebel{eq:chirpdef1}
u(x) = e^{i c x^m}, \quad x \in \mathbf R, \quad m \in \mathbf N \setminus \{0, 1\}
\end{equation}
or
\begin{equation}\langlebel{eq:chirpdef2}
u(x) = e^{i c |x|^\alpha}, \quad x \in \mathbf R, \quad \alpha \in \mathbf R \setminus 2 \mathbf N, \quad \alpha > 1
\end{equation}
where
$c \in \mathbf R \setminus 0$ in both cases.
These functions are known as chirp signals.
Here we work in dimension $d = 1$.
In \eqref{eq:chirpdef2} we ask $\alpha \mathbf Ntin 2 \mathbf N$ since $\alpha \in 2 \mathbf N$ is covered by \eqref{eq:chirpdef1}.
If $u$ is defined by \eqref{eq:chirpdef1},
and $s$ is chosen adapted to $t$ and $m$,
we will see that $\mathrm{WF}^{t,s} (u)$ is the curve in phase space described by the instantaneuos frequency of $u$, that is the
derivative of the phase function.
We will need a lemma.
\begin{lem}\langlebel{lem:chirplemma}
Suppose $s, t, \varepsilon > 0$, $U \subseteq \rr {2d} \setminus 0$ is open
and $f \in C^\infty(\rr {2d})$.
If the estimate
\begin{equation*}
\sup_{(x,\xi) \in U} \langlembda^{s k} \varepsilon^{2k} | f (\langlembda^t x, \langlembda^s \xi) |
\leqslant C_h \langlembda^t h^k k!^s
\end{equation*}
holds for all $h > 0$, all $\langlembda \geqslant 1$ and all $k \in \mathbf N$, then
for any $r > 0$ and any $\langlembda \geqslant 1$ we have
\begin{equation*}
\sup_{(x,\xi) \in U} e^{r \langlembda} \left| f (\langlembda^t x,\langlembda^s\xi) \right| \leqslant C_{r,\varepsilon,t}.
\end{equation*}
\end{lem}
\begin{proof}
Let $r > 0$. We have if $(x,\xi) \in U$
\begin{align*}
e^{\frac{r \langlembda}{s} \varepsilon^{\frac{2}{s}}} \left| f (\langlembda^t x,\langlembda^s \xi) \right|^{\frac{1}{s}}
& =
\sum_{k=0}^{\infty} 2^{-k} k!^{-1} \left( \frac{2 r}{s} ( \langlembda^s \varepsilon^2 )^{\frac{1}{s}} \right)^k
\left| f (\langlembda^t x,\langlembda^s \xi) \right|^{\frac{1}{s}} \\
& \leqslant
2 \left( \sup_{k \geqslant 0} k!^{-s} \left( \left( \frac{2 r}{ s} \right)^s \langlembda^{s} \varepsilon^2 \right)^k \left| f ( \langlembda^t x,\langlembda^s \xi ) \right| \right)^{\frac{1}{s}} \\
& \leqslant 2 \, C_h^{\frac1s} \langlembda^{\frac{t}{s}} \sup_{k \geqslant 0} \left( \left( \frac{2r}{s} \right)^{s} h \right)^{\frac{k}{s}} \\
& \leqslant C_r^{\frac1s} \langlembda^{\frac{t}{s}}
\end{align*}
for all $\langlembda \geqslant 1$,
provided we pick
\begin{equation*}
0 < h \leqslant \left( \frac{s}{2r} \right)^{s}.
\end{equation*}
Thus
for any $r > 0$, $(x,\xi) \in U$ and $\langlembda \geqslant 1$
\begin{equation*}
e^{r \langlembda \varepsilon^{\frac{2}{s}}} \left| f ( \langlembda^t x , \langlembda^s \xi) \right|
\leqslant C_r \langlembda^t
\end{equation*}
which gives finally
\begin{equation}\langlebel{eq:WFstnonmembership1q}
\begin{aligned}
e^{r \langlembda} \left| f (\langlembda^t x,\langlembda^s\xi) \right|
& = e^{- r \langlembda } e^{ 2 r \varepsilon^{-\frac2s} \langlembda \varepsilon^{\frac2s}} \left| f (\langlembda^t x,\langlembda^s \xi) \right| \\
& \leqslant C_{r,\varepsilon} \langlembda^t e^{- r \langlembda } \\
& \leqslant C_{r,\varepsilon,t}
\end{aligned}
\end{equation}
for all $\langlembda \geqslant 1$ and $(x,\xi) \in U$.
\end{proof}
The next result generalizes \eqref{eq:chirpWF1} for $d=1$.
\begin{thm}\langlebel{thm:chirpWFst}
Suppose
$c \in \mathbf R \setminus 0$.
\begin{enumerate}[\rm (i)]
\item If $u$ is defined by \eqref{eq:chirpdef1} and $t > \frac1{m-1}$ then
\begin{equation}\langlebel{eq:conclusion1a}
\mathrm{WF}^{t,t(m-1)} (u)
= \{ (x, c m x^{m-1} ) \in \rr 2: \ x \neq 0 \}.
\end{equation}
\item If $u$ is defined by \eqref{eq:chirpdef2} and $t > \frac1{\alpha-1}$ then
\begin{equation}\langlebel{eq:conclusion2}
\{ 0 \} \times (\mathbf R \setminus 0) \subseteq
\mathrm{WF}^{t,t(\alpha-1)} (u)
\subseteq \{ (x, c \alpha \operatorname{sgn}(x) |x|^{\alpha-1} ) \in \rr 2: \ x \neq 0 \}
\cup \{ 0 \} \times (\mathbf R \setminus 0) .
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}
Case (i):
Set $s = t(m-1) > 1$.
This implies that there are compactly supported Gevrey functions \cite{Rodino1} of order $s$ in the space $\Sigma_t^s(\mathbf R)$.
Set
\begin{align*}
W
& = \{ (x, c m x^{m-1} ) \in \rr 2: \ x \neq 0 \} \subseteq \rr 2 \setminus 0.
\end{align*}
Suppose $(x_0, \xi_0) \in \rr 2 \setminus 0$ and $(x_0, \xi_0) \mathbf Ntin W$.
Then there exists an open set $U$ such that $(x_0,\xi_0) \in U$, and $0 < \varepsilon \leqslant 1$, $\delta > 0$,
such that
\begin{align*}
(x,\xi) \in U, \quad |x-y| \leqslant \delta & \quad \Longrightarrow \quad |\xi - c m x^{m-1} | \geqslant 2 \varepsilon, \quad
m \, |c| \, | x^{m-1} - y^{m-1}| \leqslant \varepsilon.
\end{align*}
Then if $(x,\xi) \in U$ and $|x-y| \leqslant \delta$
\begin{equation}\langlebel{eq:lowerboundfas0}
|\xi - c m y^{m-1}|
\geqslant |\xi - c m x^{m-1}| - m \, |c| \, | y^{m-1} - x^{m-1} \,|
\geqslant \varepsilon.
\end{equation}
Let $\psi \in \Sigma_t^s(\mathbf R) \setminus 0$ be such that $\operatorname{supp} \psi \subseteq \operatorname{B}_\delta$.
From the stationary phase theorem \cite[Theorem~7.7.1]{Hormander0}
this gives for any $k \in \mathbf N$, any $h > 0$ and any $\langlembda \geqslant 1$, if $(x,\xi) \in U$, using \eqref{eq:lowerboundfas0} and \eqref{eq:expestimate0},
\begin{equation}\langlebel{eq:estimateWF0}
\begin{aligned}
|V_\psi u ( \langlembda^t x, \langlembda^s \xi)|
& = (2 \pi)^{-\frac12} \left| \int_{\mathbf R} e^{i (c y^{m} - y \langlembda^s \xi )} \overline{\psi( \langlembda^t (\langlembda^{-t} y-x) )} \, \mathrm {d} y \right| \\
& = (2 \pi)^{-\frac12} \langlembda^t \left| \int_{\mathbf R} e^{i \langlembda^{m t} (c y^{m} - y \xi ) )} \overline{\psi( \langlembda^t (y-x) )} \, \mathrm {d} y \right| \\
& \leqslant C \langlembda^t \sum_{n = 0}^k \langlembda^{n t} \sup_{|x-y| \leqslant \delta} |(D^n\psi)( \langlembda^t (y-x) )| \, |\xi - c m y^{m-1}|^{n - 2k} \langlembda^{m t (n-2k)} \\
& \leqslant C \langlembda^t \varepsilon^{- 2 k} \sum_{n = 0}^k \sup_{|x-y| \leqslant \delta} |(D^n\psi)( \langlembda^t (y-x) )| \langlembda^{- t k (m-1)} \langlembda^{t(1+m) (n-k)} \\
& \leqslant C \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} \sum_{n = 0}^k \sup_{|x-y| \leqslant \delta} |(D^n\psi)( \langlembda^t (y-x) )| \\
& \leqslant C_h \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} \sum_{n = 0}^k h^n n!^s \\
& = C_h \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} h^k \sum_{n = 0}^k
h^{-(k-n)} n!^s \\
& \leqslant C_h \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} h^k e^{s h^{-\frac1s}} \sum_{n = 0}^k (n! (k-n)!)^s \\
& \leqslant C_{s,h} \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} h^k k!^s \sum_{n = 0}^k \\
& \leqslant C_{s,h} \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} (2 h)^k k!^s .
\end{aligned}
\end{equation}
Since $h > 0$ is arbitrary
we obtain
\begin{equation}\langlebel{eq:STFTestimate0}
\langlembda^{s k} \varepsilon^{2k} | V_\psi u (\langlembda^t x, \langlembda^s \xi) |
\leqslant C_h \langlembda^t h^k k!^s, \quad (x,\xi) \in U,
\end{equation}
for all $h > 0$, all $\langlembda \geqslant 1$ and all $k \in \mathbf N$.
Applying Lemma {\rm Re}f{lem:chirplemma}
it follows that
\begin{equation*}
(x_0, \xi_0) \mathbf Ntin \mathrm{WF}^{t,t(m-1)} (u)
\end{equation*}
and we may conclude
\begin{equation}\langlebel{eq:chirpinclusion1}
\mathrm{WF}^{t,t(m-1)} (u) \subseteq W.
\end{equation}
In order to prove \eqref{eq:conclusion1a} for Case (i) it hence remains to strengthen the above inclusion into an equality.
If $m$ is even then $u$ is even and $W = -W$, so by \eqref{eq:evensteven1} we have either $\mathrm{WF}^{t,t(m-1)} (u) =\emptyset$ or $\mathrm{WF}^{t,t(m-1)} (u) = W$.
The former is not true since $u \mathbf Ntin \Sigma_t^s(\mathbf R)$. Thus we have proved \eqref{eq:conclusion1a} for Case (i) and $m$ even.
If $m$ is odd then $\check u (x) = \overline{ u(x) } = e^{- i c x^m}$.
Again $\mathrm{WF}^{t,t(m-1)} (u) =\emptyset$ cannot hold since $u \mathbf Ntin \Sigma_t^s(\mathbf R)$.
If we assume that the inclusion \eqref{eq:chirpinclusion1} is strict we get a contradiction from
\eqref{eq:evensteven0} and \eqref{eq:evensteven2}.
Indeed suppose e.g.
\begin{equation*}
\mathrm{WF}^{t,t(m-1)} (u)
= \{ (x, c m x^{m-1} ) \in \rr 2: \ x > 0 \}.
\end{equation*}
By \eqref{eq:evensteven0} and \eqref{eq:evensteven2} we then get the contradiction
\begin{align*}
\mathrm{WF}^{t,t(m-1)} ( \check u)
& = \{ (x, - c m x^{m-1} ) \in \rr 2: \ x < 0 \} \\
& = \{ (x, - c m x^{m-1} ) \in \rr 2: \ x > 0 \}
= \mathrm{WF}^{t,t(m-1)} ( \overline{u} ).
\end{align*}
This proves \eqref{eq:conclusion1a} for Case (i) when $m$ is odd.
Case (ii):
In this case $u(x) = e^{i c |x|^\alpha}$ is not smooth at $x=0$ which causes some problems.
Set again $s = t(\alpha-1) > 1$, and
\begin{align*}
W = \{ (x, c \alpha \operatorname{sgn}(x) |x|^{\alpha-1} ) \in \rr 2: \ x \neq 0 \} \subseteq \rr 2 \setminus 0.
\end{align*}
Suppose $(x_0, \xi_0) \in \rr 2 \setminus 0$, $(x_0, \xi_0) \mathbf Ntin W$ and $(x_0, \xi_0) \neq \{ 0 \} \times (\mathbf R \setminus 0)$.
There exists an open set $U$ such that $(x_0,\xi_0) \in U$, and $0 < 2 \delta \leqslant \varepsilon \leqslant 1$,
such that
\begin{align*}
(x,\xi) \in U, \quad |x-y| \leqslant \delta \quad \Longrightarrow
& \quad |\xi - c \alpha \operatorname{sgn}(x) |x|^{\alpha-1}| \geqslant 2 \varepsilon, \quad |x| \geqslant \varepsilon, \quad \\
& \quad \alpha \, |c| \, | \, \operatorname{sgn}(y) |y|^{\alpha-1} - \operatorname{sgn}(x) |x|^{\alpha-1}| \leqslant \varepsilon.
\end{align*}
Then if $(x,\xi) \in U$ and $|x-y| \leqslant \delta$
\begin{equation}\langlebel{eq:lowerboundfas1}
|\xi - c \alpha \operatorname{sgn}(y) |y|^{\alpha-1}|
\geqslant |\xi - c \alpha \operatorname{sgn}(x) |x|^{\alpha-1}| - \alpha \, |c| \, | \operatorname{sgn}(y) |y|^{\alpha-1} - \operatorname{sgn}(x) |x|^{\alpha-1}| \geqslant \varepsilon.
\end{equation}
Let $\psi \in \Sigma_t^s(\mathbf R) \setminus 0$ be such that $\operatorname{supp} \psi \subseteq \operatorname{B}_\delta$.
Then if $\langlembda \geqslant 1$, $\langlembda^t(y-x) \in \operatorname{supp} \psi$ and $|x| \geqslant \varepsilon$ we have $|y| \geqslant \varepsilon/2$.
From the stationary phase theorem \cite[Theorem~7.7.1]{Hormander0}
this gives for any $k \in \mathbf N$, any $h > 0$ and any $\langlembda \geqslant 1$, if $(x,\xi) \in U$, using \eqref{eq:lowerboundfas1}
and the final estimates in \eqref{eq:estimateWF0},
\begin{equation*}
\begin{aligned}
|V_\psi u ( \langlembda^t x, \langlembda^s \xi)|
& = (2 \pi)^{-\frac12} \left| \int_{\mathbf R} e^{i (c |y|^\alpha - y \langlembda^s \xi )} \overline{\psi( \langlembda^t (\langlembda^{-t}y-x) )} \, \mathrm {d} y \right| \\
& = (2 \pi)^{-\frac12} \langlembda^t \left| \int_{|y| \geqslant \varepsilon/2} e^{i \langlembda^{t \alpha} (c |y|^\alpha - y \xi ) )} \overline{\psi( \langlembda^t (y-x) )} \, \mathrm {d} y \right| \\
& \leqslant C \langlembda^t \sum_{n = 0}^k \langlembda^{n t} \sup_{|x-y| \leqslant \delta} |(D^n\psi)( \langlembda^t (y-x) )| \, |\xi - c \alpha \operatorname{sgn}(y) |y|^{\alpha-1}|^{n - 2k} \langlembda^{t \alpha(n-2k)} \\
& \leqslant C \langlembda^{t} \varepsilon^{- 2 k} \sum_{n = 0}^k \sup_{|x-y| \leqslant \delta} |(D^n\psi)( \langlembda^t (y-x) )| \langlembda^{- t k (\alpha-1)} \langlembda^{t(1+\alpha) (n-k)} \\
& \leqslant C \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} \sum_{n = 0}^k \sup_{|x-y| \leqslant \delta} |(D^n\psi)( \langlembda^t (y-x) )| \\
& \leqslant C_{s,h} \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} (2 h)^k k!^s .
\end{aligned}
\end{equation*}
Appealing to Lemma {\rm Re}f{lem:chirplemma} it follows that
\begin{equation*}
(x_0, \xi_0) \mathbf Ntin \mathrm{WF}^{t,t(\alpha-1)} (u)
\end{equation*}
and we may conclude
\begin{equation*}
\mathrm{WF}^{t,t(\alpha-1)} (u) \subseteq W \cup \{ 0 \} \times (\mathbf R \setminus 0)
\end{equation*}
which is the right inclusion in \eqref{eq:conclusion2} for Case (ii).
It remains to show the left inclusion in \eqref{eq:conclusion2}, that is
\begin{equation}\langlebel{eq:conclusion2a}
\{ 0 \} \times (\mathbf R \setminus 0) \subseteq
\mathrm{WF}^{t,t(\alpha-1)} (u).
\end{equation}
We have for $\xi > 0$
\begin{equation*}
|V_\psi u ( 0, \pm \langlembda^s \xi)|
= (2 \pi)^{-\frac12} \left| \int_{\mathbf R} e^{i (c |y|^\alpha \mp y \langlembda^s \xi )} \overline{\psi( y )} \, \mathrm {d} y \right|
= |\mathscr{F}( \psi \, e^{ - i c |\, \cdot \, t|^\alpha } )(\mp \langlembda^s \xi)|.
\end{equation*}
Let $\psi$ be even and satisfy $\psi(0) \neq 0$.
Then $\mathscr{F}( \psi \, e^{ - i c |\, \cdot \, t|^\alpha } )$ is also even.
If we assume $(0, \xi ) \mathbf Ntin \mathrm{WF}^{t,t(\alpha-1)} (u)$ or $(0, -\xi ) \mathbf Ntin \mathrm{WF}^{t,t(\alpha-1)} (u)$ then
\begin{equation*}
|\mathscr{F}( \psi \, e^{ - i c |\, \cdot \, t|^\alpha } )(\xi)|
\lesssim e^{- r |\xi|^{\frac1s}}, \quad \xi \in \mathbf R,
\end{equation*}
for all $r > 0$. But this implies $\psi \, e^{ - i c |\, \cdot \, t|^\alpha } \in C^\infty$
which is a contradiction as $\alpha \mathbf Ntin 2 \mathbf N \setminus 0$ and $\psi(0) \neq 0$.
This shows \eqref{eq:conclusion2a} and thus \eqref{eq:conclusion2} for Case (ii) has been proved.
\end{proof}
\begin{rem}\langlebel{rem:weakassumption}
The wave front set $\mathrm{WF}^{t,t(\alpha-1)} (u)$
is well defined if $t + t(\alpha-1) = t \alpha > 1$ for $u \in (\Sigma_t^{t(\alpha-1)})' (\mathbf R)$.
If we weaken the assumption $t > \frac{1}{m-1}$ ($t > \frac{1}{\alpha-1}$) into $t > \frac1m$ ($t > \frac{1}{\alpha}$)
in Theorem {\rm Re}f{thm:chirpWFst},
then we obtain from Theorem {\rm Re}f{thm:chirpWFst} and Remark {\rm Re}f{rem:WFinclusion}
if $m \in \mathbf N \setminus \{ 0, 1 \}$
\begin{equation}\langlebel{eq:conclusion1c}
\{ (x, c \, m x^{m-1} ) \in \rr 2: \ x \neq 0 \}
\subseteq \mathrm{WF}^{t,t(m-1)} (u)
\end{equation}
and if $\alpha \in \mathbf R \setminus 2 \mathbf N$, $\alpha > 1$
\begin{equation}\langlebel{eq:conclusion2c}
\{ 0 \} \times (\mathbf R \setminus 0) \subseteq \mathrm{WF}^{t,t(\alpha-1)} (u).
\end{equation}
Thus \eqref{eq:conclusion1c} has been weakened into an inclusion instead of the equality \eqref{eq:conclusion1a},
and \eqref{eq:conclusion2c} gives a lower bound only as compared to \eqref{eq:conclusion2}.
\end{rem}
\begin{rem}\langlebel{rem:fourierchirp}
The Fourier transform $\widehat u$ of a chirp \eqref{eq:chirpdef1} with $m \in \mathbf N \setminus \{ 0, 1 \}$ is known explicitly for $m = 2$.
It is $\widehat u(\xi) = (2 |c|)^{-\frac12} e^{i \frac{\pi}{4} \operatorname{sgn}(c)} e^{- \frac{i}{4c} \xi^2}$
\cite[Theorem~7.6.1]{Hormander0}.
For larger $m$ one has $\widehat u \in \mathscr{S}'(\mathbf R)$.
From the discussion concerning the Airy function ($m=3$, $c = \frac13$) \cite[Chapter~7.6]{Hormander0}
it can be seen that $\widehat u$ is actually real analytic provided $m$ is odd, and extends to an entire function on $\mathbf C$.
But if $m$ is even it seems difficult to obtain
explicit information about $\widehat u$.
Nevertheless, combining Theorem {\rm Re}f{thm:chirpWFst} with Proposition {\rm Re}f{prop:WFstsymplectic},
we obtain the following identity for its anisotropic Gelfand--Shilov wave front set when $t > \frac{1}{m-1}$:
\begin{equation*}
\mathrm{WF}^{t(m-1),t} (\widehat u)
= \{ ( (-1)^{m-1} c m x^{m-1}, x ) \in \rr 2: \ x \neq 0 \}.
\end{equation*}
If $m = 3$ and $c = 1/3$ then $u (x) = e^{i x^3/3}$ and
$v (\xi) = (2 \pi)^{\frac12} \mathscr{F}^{-1} u(\xi) = (2 \pi)^{\frac12} \widehat u (-\xi)$ is the Airy function \cite{Hormander0}.
Using \eqref{eq:evensteven0} we conclude
\begin{equation*}
\mathrm{WF}^{2t,t} (v)
= -\mathrm{WF}^{2t,t} ( \widehat u )
= \{ ( - x^2, x ) \in \rr 2: \ x \neq 0 \}
\end{equation*}
when $t > \frac12$.
\end{rem}
We would also like to determine $\mathrm{WF}^{t,s} (u)$ when $s \neq t (\alpha-1)$ for the chirp functions.
The following two results treat this question and show that $\mathrm{WF}^{t,s} (u)$ does not give a meaningful result then.
\begin{prop}\langlebel{prop:chirpnegative1}
Suppose
$c \in \mathbf R \setminus 0$.
\begin{enumerate}[\rm (i)]
\item If $u$ is defined by \eqref{eq:chirpdef1} and $s > t (m-1) > 1$ then
\begin{equation}\langlebel{eq:conclusion3a}
\mathrm{WF}^{t,s} (u) = (\mathbf R \setminus 0) \times \{ 0 \}.
\end{equation}
\item If $u$ is defined by \eqref{eq:chirpdef2} and $s > t (\alpha-1) > 1$ then
\begin{equation}\langlebel{eq:conclusion4}
\{ 0 \} \times (\mathbf R \setminus 0) \subseteq
\mathrm{WF}^{t,s} (u)
\subseteq (\mathbf R \setminus 0) \times \{ 0 \}
\cup \{ 0 \} \times (\mathbf R \setminus 0) .
\end{equation}
\end{enumerate}
\end{prop}
\begin{proof}
Case (i):
Suppose $(x_0, \xi_0) \in \rr 2$ and $\xi_0 \neq 0$.
There exists $U \subseteq \rr {2}$ such that $(x_0, \xi_0) \in U$,
and $0 < \varepsilon \leqslant 1$,
$L \geqslant 1$ such that
\begin{equation*}
| \xi - c m \langlembda^{ t (m-1) - s} y^{m-1}| \geqslant \varepsilon
\end{equation*}
when $(x,\xi) \in U$, $| x - y| \leqslant 1$ and $\langlembda \geqslant L$, due to the assumption $t(m-1) - s < 0$.
Let $\psi \in \Sigma_t^s(\mathbf R) \setminus 0$ be such that $\operatorname{supp} \psi \subseteq \operatorname{B}_1$.
From the stationary phase theorem \cite[Theorem~7.7.1]{Hormander0}
we have for any $k \in \mathbf N$, any $h > 0$ and any $\langlembda \geqslant L$, if $(x,\xi) \in U$,
again using \eqref{eq:estimateWF0},
\begin{align*}
|V_\psi u ( \langlembda^t x, \langlembda^s \xi)|
& = (2 \pi)^{-\frac12} \left| \int_{\mathbf R} e^{i (c y^{m} - y \langlembda^s \xi )} \overline{\psi( \langlembda^t (\langlembda^{-t} y-x) )} \, \mathrm {d} y \right| \\
& = (2 \pi)^{-\frac12} \langlembda^t \left| \int_{\mathbf R} e^{i \langlembda^{t+s} ( \langlembda^{t (m - 1)-s} c y^{m} - y \xi ) )} \overline{\psi( \langlembda^t (y-x) )} \, \mathrm {d} y \right| \\
& \leqslant C \langlembda^t \sum_{n = 0}^k \langlembda^{nt} \sup_{|x-y| \leqslant 1} |(D^n\psi)( \langlembda^t (y-x) )| \, |\xi - c m \langlembda^{t ( m - 1)-s} y^{m-1}|^{n - 2k} \langlembda^{(t+s)(n-2k)} \\
& \leqslant C \langlembda^{t} \varepsilon^{-2 k} \sum_{n = 0}^k \sup_{|x-y| \leqslant 1} |(D^n\psi)( \langlembda^t (y-x) )| \langlembda^{- s k} \langlembda^{ s (n-k) + 2t( n - k)} \\
& \leqslant C \langlembda^{t} \varepsilon^{-2 k} \langlembda^{- s k} \sum_{n = 0}^k \sup_{|x-y| \leqslant 1} |(D^n\psi)( \langlembda^t (y-x) )| \\
& \leqslant C_{s,h} \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} (2 h)^k k!^s .
\end{align*}
Lemma {\rm Re}f{lem:chirplemma} gives
\begin{equation*}
\mathrm{WF}^{t,s} (u) \subseteq (\mathbf R \setminus 0) \times \{ 0 \}
\end{equation*}
which shows the inclusion ``$\subseteq$'' in \eqref{eq:conclusion3a}.
Equality in \eqref{eq:conclusion3a}
again follows from \eqref{eq:evensteven0}, \eqref{eq:evensteven2}, $u \mathbf Ntin \Sigma_t^s(\mathbf R)$,
and $\check u = \overline u$ if $m$ is odd.
Case (ii):
Suppose $(x_0, \xi_0) \in \rr 2$, $x_0 \neq 0$ and $\xi_0 \neq 0$.
Then there exists $U \subseteq \rr {2d}$ such that $(x_0, \xi_0) \in U$,
and $0 < \varepsilon \leqslant 1$,
$L \geqslant 1$,
such that
\begin{equation*}
\inf_{(x,\xi) \in U} |x| = \varepsilon
\end{equation*}
and
\begin{equation*}
| \xi - c \alpha \operatorname{sgn}(y) \langlembda^{ t (\alpha-1) - s} |y|^{\alpha-1}| \geqslant \varepsilon
\end{equation*}
when $(x,\xi) \in U$, $|x - y| \leqslant \varepsilon/2$ and $\langlembda \geqslant L$.
Pick $\psi \in \Sigma_t^s(\mathbf R) \setminus 0$ such that $\operatorname{supp} \psi \subseteq \operatorname{B}_{\varepsilon/2}$.
Then if $\langlembda \geqslant L$, $\langlembda^t(y-x) \in \operatorname{supp} \psi$ and $|x| \geqslant \varepsilon$ we have $|y| \geqslant \varepsilon/2$.
From the stationary phase theorem \cite[Theorem~7.7.1]{Hormander0}
this gives for any $k \in \mathbf N$, any $h > 0$ and any $\langlembda \geqslant L$, if $(x,\xi) \in U$,
using \eqref{eq:estimateWF0},
\begin{align*}
|V_\psi u ( \langlembda^t x, \langlembda^s \xi)|
& = (2 \pi)^{-\frac12} \left| \int_{\mathbf R} e^{i (c |y|^\alpha - y \langlembda^s \xi )} \overline{\psi( \langlembda^t (\langlembda^{-t}y-x) )} \, \mathrm {d} y \right| \\
& = (2 \pi)^{-\frac12} \langlembda^t \left| \int_{|y| \geqslant \varepsilon/2} e^{i \langlembda^{t+s} (c \langlembda^{t (\alpha-1) - s} |y|^\alpha - y \xi ) )} \overline{\psi( \langlembda^t (y-x) )} \, \mathrm {d} y \right| \\
& \leqslant C \langlembda^t \sum_{n = 0}^k \langlembda^{n t} \sup_{|x-y| \leqslant \varepsilon/2} |(D^n\psi)( \langlembda^t (y-x) )| \, |\xi - c \alpha \operatorname{sgn}(y) \langlembda^{t (\alpha-1) - s} |y|^{\alpha-1}|^{n - 2k} \\
& \qquad \qquad \times \langlembda^{(t+s) (n-2k)} \\
& \leqslant C_{s,h} \langlembda^{t} \varepsilon^{- 2k} \langlembda^{- s k} (2 h)^k k!^s.
\end{align*}
Using Lemma {\rm Re}f{lem:chirplemma} we obtain
\begin{equation*}
\mathrm{WF}^{t,s} (u) \subseteq (\mathbf R \setminus 0) \times \{ 0 \} \cup \{ 0 \} \times (\mathbf R \setminus 0).
\end{equation*}
Finally $\{ 0 \} \times (\mathbf R \setminus 0) \subseteq \mathrm{WF}^{t,s} (u)$ follows recycling the argument at the end of the proof of
Theorem {\rm Re}f{thm:chirpWFst}.
\end{proof}
\begin{prop}\langlebel{prop:chirpnegative2}
Suppose
$c \in \mathbf R \setminus 0$.
\begin{enumerate}[\rm (i)]
\item If $u$ is defined by \eqref{eq:chirpdef1} and $t (m-1) > s > 1$ then
\begin{equation}\langlebel{eq:conclusion5a}
\mathrm{WF}^{t,s} (u) \subseteq \{ 0 \} \times (\mathbf R \setminus 0)
\end{equation}
and if $m$ is even then
\begin{equation}\langlebel{eq:conclusion5b}
\mathrm{WF}^{t,s} (u) = \{ 0 \} \times (\mathbf R \setminus 0).
\end{equation}
\item If $u$ is defined by \eqref{eq:chirpdef2} and $t (\alpha-1) > s > 1$ then
\begin{equation}\langlebel{eq:conclusion6}
\{ 0 \} \times (\mathbf R \setminus 0) \subseteq
\mathrm{WF}^{t,s} (u)
\subseteq (\mathbf R \setminus 0) \times \{ 0 \}
\cup \{ 0 \} \times (\mathbf R \setminus 0) .
\end{equation}
\end{enumerate}
\end{prop}
\begin{proof}
Case (i):
Suppose $(x_0, \xi_0) \in \rr 2$ and $x_0 \neq 0$.
There exists $U \subseteq \rr {2d}$ such that $(x_0, \xi_0) \in U$,
and $0 < \varepsilon \leqslant 1$,
$L \geqslant 1$,
such that
\begin{equation*}
| c m y^{m-1} - \langlembda^{ s - t (m-1)} \xi| \geqslant \varepsilon
\end{equation*}
when $(x,\xi) \in U$, $| x - y| \leqslant \varepsilon$ and $\langlembda \geqslant L$, due to the assumption $s - t(m-1) < 0$.
If $0 \leqslant n \leqslant k$ we have
\begin{equation*}
s k + n t + t m(n-2k)
< t ( k(m-1) + n - m k ) \\
\leqslant 0.
\end{equation*}
Let $\psi \in \Sigma_t^s(\mathbf R) \setminus 0$ be such that $\operatorname{supp} \psi \subseteq \operatorname{B}_\varepsilon$.
From the stationary phase theorem \cite[Theorem~7.7.1]{Hormander0}
we have for any $k \in \mathbf N$,
any $h > 0$ and any $\langlembda \geqslant L$, if $(x,\xi) \in U$,
again reusing \eqref{eq:estimateWF0},
\begin{align*}
|V_\psi u ( \langlembda^t x, \langlembda^s \xi)|
& = (2 \pi)^{-\frac12} \left| \int_{\mathbf R} e^{i (c y^{m} - y \langlembda^s \xi )} \overline{\psi( \langlembda^t (\langlembda^{-t} y-x) )} \, \mathrm {d} y \right| \\
& = (2 \pi)^{-\frac12} \langlembda^t \left| \int_{\mathbf R} e^{i \langlembda^{t m} ( c y^{m} - \langlembda^{s - t (m - 1)} y \xi ) )} \overline{\psi( \langlembda^t (y-x) )} \, \mathrm {d} y \right| \\
& \leqslant C \langlembda^t \sum_{n = 0}^k \langlembda^{n t} \sup_{|x-y| \leqslant \varepsilon} |(D^n\psi)( \langlembda^t (y-x) )| \, |c m y^{m-1} - \langlembda^{s - t (m - 1)} \xi |^{n - 2k} \langlembda^{t m(n-2k) } \\
& \leqslant C \langlembda^{t} \varepsilon^{-2 k} \langlembda^{- s k} \sum_{n = 0}^k \sup_{|x-y| \leqslant \varepsilon} |(D^n\psi)( \langlembda^t (y-x) )| \\
& \leqslant C_{s,h} \langlembda^{t} \varepsilon^{- 2 k} \langlembda^{- s k} (2h)^k k!^s .
\end{align*}
Lemma {\rm Re}f{lem:chirplemma} gives
\begin{equation*}
\mathrm{WF}^{t,s} (u) \subseteq \{ 0 \} \times (\mathbf R \setminus 0)
\end{equation*}
which is \eqref{eq:conclusion5a}.
The equality \eqref{eq:conclusion5b} when $m$ is even follows from \eqref{eq:evensteven1}
and $u \mathbf Ntin \Sigma_t^s(\mathbf R)$.
Case (ii):
Suppose $(x_0, \xi_0) \in \rr 2$, $x_0 \neq 0$ and $\xi_0 \neq 0$.
Then there exists $U \subseteq \rr {2d}$ such that $(x_0, \xi_0) \in U$,
and $0 < \varepsilon \leqslant 1$, $L \geqslant 1$, such that
\begin{equation*}
\inf_{(x,\xi) \in U} |x| = \varepsilon
\end{equation*}
and
\begin{equation*}
| \xi - c \alpha \operatorname{sgn}(y) \langlembda^{ t (\alpha-1) - s} |y|^{\alpha-1}| \geqslant \varepsilon
\end{equation*}
when $(x,\xi) \in U$, $|x - y| \leqslant \varepsilon/2$ and $\langlembda \geqslant L$.
If $n \leqslant k$ then
\begin{equation*}
s k + n t + (t+s) (n-2k)
\leqslant s k + n t - (t+s) k
\leqslant 0.
\end{equation*}
Let $\psi \in \Sigma_t^s(\mathbf R) \setminus 0$ be such that $\operatorname{supp} \psi \subseteq \operatorname{B}_{\varepsilon/2}$.
Then if $\langlembda \geqslant L$, $\langlembda^t(y-x) \in \operatorname{supp} \psi$ and $|x| \geqslant \varepsilon$ we have $|y| \geqslant \varepsilon/2$.
From the stationary phase theorem \cite[Theorem~7.7.1]{Hormander0}
this gives for any $k \in \mathbf N$, any $h > 0$ and any $\langlembda \geqslant L$, if $(x,\xi) \in U$
and the final estimates in \eqref{eq:estimateWF0},
\begin{align*}
|V_\psi u ( \langlembda^t x, \langlembda^s \xi)|
& = (2 \pi)^{-\frac12} \left| \int_{\mathbf R} e^{i (c |y|^\alpha - y \langlembda^s \xi )} \overline{\psi( \langlembda^t (\langlembda^{-t}y-x) )} \, \mathrm {d} y \right| \\
& = (2 \pi)^{-\frac12} \langlembda^t \left| \int_{|y| \geqslant \varepsilon/2} e^{i \langlembda^{t+s} (c \langlembda^{t (\alpha-1) - s} |y|^\alpha - y \xi ) )} \overline{\psi( \langlembda^t (y-x) )} \, \mathrm {d} y \right| \\
& \leqslant C \langlembda^t \sum_{n = 0}^k \langlembda^{n t} \sup_{|x-y| \leqslant \varepsilon/2} |(D^n\psi)( \langlembda^t (y-x) )| \, |\xi - c \alpha \operatorname{sgn}(y) \langlembda^{t (\alpha-1) - s} |y|^{\alpha-1}|^{n - 2k} \\
& \qquad \qquad \times \langlembda^{(t+s) (n-2k)} \\
& \leqslant C_{s,h} \langlembda^{t} \varepsilon^{- 2k} \langlembda^{- s k} (2 h)^k k!^s.
\end{align*}
Lemma {\rm Re}f{lem:chirplemma} gives again
\begin{equation*}
\mathrm{WF}^{t,s} (u) \subseteq (\mathbf R \setminus 0) \times \{ 0 \} \cup \{ 0 \} \times (\mathbf R \setminus 0).
\end{equation*}
Finally $\{ 0 \} \times (\mathbf R \setminus 0) \subseteq \mathrm{WF}^{t,s} (u)$ follows again using the argument at the end of the proof of
Theorem {\rm Re}f{thm:chirpWFst}.
\end{proof}
\begin{rem}\langlebel{rem:schrodingernonquadratic}
By using Theorem {\rm Re}f{thm:chirpWFst}, Propositon {\rm Re}f{prop:chirpnegative1} and Propositon {\rm Re}f{prop:chirpnegative2}
we may now give a counterpart of Remark {\rm Re}f{rem:propagator},
showing that the anisotropic wave front set turns out to be needed
when treating Schr\"odinger propagators in the case of non-quadratic potentials.
Consider the Cauchy problem for the anisotropic free particle equation
in dimension $d = 1$
\begin{equation}\langlebel{eq:schrodeq2}
\left\{
\begin{array}{rl}
\partial_t u(t,x) + i D_x^{m} u (t,x) & = 0, \quad m \in \mathbf N \setminus \{ 0, 1 \}, \\
u(0,\, \cdot \, t) & = u_0.
\end{array}
\right.
\end{equation}
The Hamilton flow, along which we expect propagation of microlocal singularities, is given by
\begin{equation}\langlebel{eq:hamiltonflow1}
(x,\xi) = \chi_t (x_0, \xi_0)
= (x_0 + m t \xi_0^{m-1}, \xi_0), \quad t \in \mathbf R,
\end{equation}
and we are looking for parameters $v,s > 0$ such that $v + s > 1$ and
\begin{equation}\langlebel{eq:propagationschrodnonq1}
\mathrm{WF}^{v,s} (e^{- i t D_x^{m}} u_0 ) = \chi_t (\mathrm{WF}^{v,s} (u_0) ) .
\end{equation}
The explicit solution to \eqref{eq:schrodeq2} is given by
\begin{equation}\langlebel{eq:solutionschrod2}
u (t,x)
= e^{- i t D_x^{m}} u_0
= (2 \pi)^{- \frac12} \int_{\mathbf R} e^{i x \xi - i t \xi^{m}} \widehat u_0 (\xi) \mathrm {d} \xi.
\end{equation}
For simplicity let us test \eqref{eq:propagationschrodnonq1} on the case $u_0 = \delta_0$,
and denote by $w_t$ the solution to \eqref{eq:schrodeq2}.
It is easy to prove that
\begin{equation*}
\mathrm{WF}^{v,s} ( w_0 ) = \mathrm{WF}^{v,s} ( \delta_0 ) = \{ 0 \} \times (\mathbf R \setminus 0)
\end{equation*}
for any $v,s > 0$ with $v + s > 1$, cf. \eqref{eq:diracWF} for $v = s$ and Proposition {\rm Re}f{prop:WFstelementary},
and from \eqref{eq:solutionschrod2}
\begin{equation}\langlebel{eq:solutionschrod3}
\widehat w_t (\xi) = (2 \pi)^{- \frac12} e^{-i t \xi^{m}}.
\end{equation}
Hence from \eqref{eq:hamiltonflow1} and \eqref{eq:propagationschrodnonq1} we expect
\begin{equation}\langlebel{eq:propagationschrodnonq2}
\mathrm{WF}^{v,s} ( w_t ) = \chi_t ( \{ 0 \} \times (\mathbf R \setminus 0) ) = \{ (m t \xi^{m-1}, \xi) \in \rr 2, \ \xi \neq 0 \}.
\end{equation}
This shows that the correct choice is $v = s (m-1) > 1$, $s > 0$.
In fact from Theorem {\rm Re}f{thm:chirpWFst} applied to \eqref{eq:solutionschrod3} we have for $v (m-1) > 1$ and $t \neq 0$
\begin{equation}\langlebel{eq:solutionschrod4}
\mathrm{WF}^{v, v (m-1)} (\widehat w_t) = \{ (\xi, - m t \xi^{m-1} ) \in \rr 2, \ \xi \neq 0 \}
\end{equation}
and hence, in view of Proposition {\rm Re}f{prop:WFstsymplectic}, swapping the roles of $s$ and $v$,
\begin{equation}\langlebel{eq:solutionschrod5}
\mathrm{WF}^{s(m-1),s} (w_t) = \{ (m t \xi^{m-1}, \xi ) \in \rr 2, \ \xi \neq 0 \}
\end{equation}
if $s (m-1) > 1$,
as expected from \eqref{eq:propagationschrodnonq2}.
Other choices of $v > 0$ do not work. In fact by applying Proposition {\rm Re}f{prop:chirpnegative1}
to \eqref{eq:solutionschrod3} we have if $v > s(m-1) > 1$
\begin{equation*}
\mathrm{WF}^{s, v} ( \widehat w_t) = (\mathbf R \setminus 0) \times \{ 0 \}
\end{equation*}
and hence
\begin{equation*}
\mathrm{WF}^{v,s} ( w_t) = \{ 0 \} \times (\mathbf R \setminus 0)
\end{equation*}
for every $t \in \mathbf R$.
Whereas by applying Proposition {\rm Re}f{prop:chirpnegative2}
to \eqref{eq:solutionschrod3} we have if $1 < v < s(m-1)$,
in particular if $v = s > 1$, we obtain
\begin{equation*}
\mathrm{WF}^{s, v} ( \widehat w_t ) \subseteq \{ 0 \} \times (\mathbf R \setminus 0),
\end{equation*}
hence
\begin{equation*}
\mathrm{WF}^{v,s} ( w_t ) \subseteq (\mathbf R \setminus 0) \times \{ 0 \}, \quad t \neq 0.
\end{equation*}
(These inclusions are equalities if $m$ is even.)
This is not consistent with \eqref{eq:propagationschrodnonq2}.
\end{rem}
\begin{rem}\langlebel{rem:comment}
\textit{Addendum at revision.}
After finishing this work we have proved a generalization of the conjecture \eqref{eq:propagationschrodnonq2} with $v = s(m-1) > 1$,
see \cite[Theorem~7.1]{Wahlberg4}.
\end{rem}
\section{Relations between the $t,s$-Gelfand--Shilov wave front set and the $s$-Gevrey wave front set}\langlebel{sec:GSGevrey}
Next we show a few results that are valid when $s > 1$.
Then Gevrey functions of order $s$ and of compact support exist \cite{Rodino1}.
We define Gevrey functions of order $s > 1$ slightly differently from \cite{Rodino1}, using again Beurling instead of Roumieu type.
Let $\Omega \subseteq \rr d$ be open. Then $f \in G^s(\Omega)$ provided $f \in C^\infty(\Omega)$
and for each compact $K \subseteq \Omega$ we have
\begin{equation*}
|\pd \alpha f (x)| \leqslant C_{K,h} h^{|\alpha|} \alpha!^s, \quad x \in K, \quad \alpha \in \nn d, \quad \forall h > 0.
\end{equation*}
The topology on $G^s(\Omega)$ is defined first as the projective limit with respect to $h > 0$, and then as the inductive limit with respect to an exhaustive increasing sequence of compact sets $K \subseteq \Omega$.
In the sequel we limit attention to $\Omega = \rr d$.
The space of compactly supported Gevrey functions is embedded in the usual test function space as
$G_c^s(\rr d) \subseteq C_c^\infty(\rr d)$.
The topological duals therefore satisfy the embedding $\mathscr{D}'(\rr d) \subseteq \mathscr{D}_s'(\rr d)$
where $\mathscr{D}_s'(\rr d)$ is the space of Gevrey ultradistributions of order $s > 1$.
With small modifications of the proof of \cite[Theorem~1.6.1]{Rodino1} we obtain that for $f \in C_c^\infty(\rr d)$ we have $f \in G_c^s(\rr d)$ if and only if the Fourier transform satisfies
\begin{equation*}
|\widehat f(\xi)| \lesssim e^{- r |\xi|^{\frac1s}} \quad \forall r > 0.
\end{equation*}
Denoting $\mathscr{E}_s' (\rr d)$
the subspace of $\mathscr{D}_s'(\rr d)$ of ultradistributions of compact support,
we also have $f \in \mathscr{E}_s' (\rr d)$ if and only if
\begin{equation*}
\exists r > 0: \quad |\widehat f(\xi)| \lesssim e^{r |\xi|^{\frac1s}}
\end{equation*}
cf. \cite{Komatsu1,Sobak1} and \cite[Theorems~1.6.1 and 1.6.7]{Rodino1}.
This is the basis of the definition of the Gevrey wave front set $\mathrm{WF}_s( u)$ of $u \in \mathscr{D}_s'(\rr d)$ \cite{Rodino1}.
A phase space point $(x_0, \xi_0) \in \rr d \times (\rr d \setminus 0)$
satisfies $(x_0, \xi_0) \mathbf Ntin \mathrm{WF}_s( u)$ if there exists $\varphi \in G_c^s(\rr d)$ such that $\varphi(x_0) = 1$
and an open conical neighborhood ${G(X|Y)}amma \subseteq \rr d \setminus 0$ containing $\xi_0$ such that
\begin{equation*}
\sup_{\xi \in {G(X|Y)}amma} e^{r |\xi|^{\frac1s}} |\widehat{ u \varphi} (\xi)| < \infty \quad \forall r > 0.
\end{equation*}
Hence $\mathrm{WF}_s( u) = \emptyset$ if and only if $u \in G^s(\rr d)$.
Note that for every $s > 1$ and any $t > 0$ we have
\begin{align*}
& G_c^s(\rr d) \subseteq \Sigma_t^s(\rr d) \subseteq G^s(\rr d), \\
& \mathscr{E}_s' (\rr d) \subseteq ( \Sigma_t^s )' (\rr d) \subseteq \mathscr{D}_s'(\rr d).
\end{align*}
Inspired by the proofs in \cite{Wahlberg2} we obtain the following results.
Here $\pi_2(x,\xi) = \xi$ for $(x,\xi) \in T^* \rr d$.
\begin{prop}\langlebel{prop:WFGevreyWFst}
If $t \geqslant s > 1$ and
$u \in (\Sigma_t^s)'(\rr d)$ then
\begin{equation*}
\{ 0 \} \times \pi_2 \mathrm{WF}_s(u) \subseteq \mathrm{WF}^{t,s}(u).
\end{equation*}
\end{prop}
\begin{proof}
Suppose $\xi_0 \in \rr d \setminus 0$ and $(0,\xi_0) \mathbf Ntin \mathrm{WF}^{t,s}(u)$.
By \eqref{eq:WFstscalinv} we may assume that $|\xi_0| = 1$.
Let $\varphi \in G_c^s(\rr d) \subseteq \Sigma_t^s (\rr d)$ satisfy $\varphi (0) = 1$.
We have for some $\varepsilon > 0$, for any $r > 0$
\begin{equation*}
e^{r \langlembda} |V_\varphi u ( \langlembda^t x, \langlembda^s (\xi_0 + \xi) )| \leqslant C_r < \infty
\end{equation*}
if $(x,\xi) \in \operatorname{B}_\varepsilon$ and $\langlembda > 0$.
Define the open set
\begin{equation*}
{G(X|Y)}amma = \{ ( \langlembda^t x, \langlembda^s (\xi_0 + \xi) ) \in \rr {2d}: \, (x,\xi) \in \operatorname{B}_\varepsilon, \, \langlembda > 0 \} \subseteq \rr {2d}.
\end{equation*}
We have to show that $(x_0,\xi_0) \mathbf Ntin \mathrm{WF}_s(u)$ for all $x_0 \in \rr d$.
Let $x_0 \in \rr d$.
Define for $\delta > 0$ the open conic set containing $\xi_0$
\begin{equation*}
{G(X|Y)}amma_{\delta} = \left\{ \xi \in \rr d \setminus 0: \, \left| \frac{\xi}{|\xi|} - \xi_0 \right| < \delta \right\} \subseteq \rr d \setminus 0.
\end{equation*}
Pick $\delta > 0$ sufficiently small so that $\delta (1 + |x_0|) \leqslant 1$ and
\begin{equation*}
\delta^2 \left(1 + \frac{| x_0 |^2}{(1 - \delta |x_0| )^2} \right) \leqslant \varepsilon^2.
\end{equation*}
Then we have
\begin{equation}\langlebel{eq:gammainclusion1}
(\{ x_0 \} \times {G(X|Y)}amma_{\delta}) \setminus \operatorname{B}_{\delta^{-1}} \subseteq {G(X|Y)}amma.
\end{equation}
In fact let $\eta \in {G(X|Y)}amma_{\delta}$ and $|(x_0, \eta)| \geqslant \delta^{-1}$.
Then $|\eta| \geqslant \delta^{-1} - |x_0| \geqslant 1$.
We write for $\langlembda > 0$
\begin{equation*}
(x_0, \eta) = ( \langlembda^t x, \langlembda^s (\xi_0 + \xi) )
\end{equation*}
that is $x = \langlembda^{-t} x_0$ and $\xi = \langlembda^{-s} \eta - \xi_0$.
In order to show \eqref{eq:gammainclusion1} we have to show that $(x,\xi) \in \operatorname{B}_\varepsilon$ for some $\langlembda > 0$.
If we set $\langlembda = |\eta|^{\frac1s} > 0$ then $|\xi| < \delta$ and we obtain using the assumption $t \geqslant s$
\begin{align*}
| x |^2 + | \xi |^2
& < \langlembda^{-2t} |x_0|^2 + \delta^2
= |\eta|^{- \frac{2t}s} |x_0|^2 + \delta^2
\leqslant |\eta|^{-2} |x_0|^2 + \delta^2 \\
& \leqslant ( \delta^{-1} - |x_0| )^{-2} |x_0|^2 + \delta^2
= \delta^2 \left( 1 + \frac{|x_0|^2}{(1 - \delta |x_0|)^2} \right)
\leqslant \varepsilon^2.
\end{align*}
Thus $(x,\xi) \in \operatorname{B}_\varepsilon$ and we have shown \eqref{eq:gammainclusion1}.
Finally let $\eta \in {G(X|Y)}amma_{\delta}$ and
$|\eta| \geqslant \delta^{-1} + |x_0|$, which implies $|(x_0, \eta)| \geqslant \delta^{-1}$.
By \eqref{eq:gammainclusion1} we have $(x_0, \eta) \in {G(X|Y)}amma$, that is
$(x_0, \eta) = (\langlembda^t x, \langlembda^s (\xi_0 + \xi) )$ for some $\langlembda > 0$ and some $(x,\xi) \in \operatorname{B}_\varepsilon$.
Since
\begin{equation*}
|\eta|^{\frac1s} =
\langlembda |\xi_0 + \xi|^{\frac1s}
\leqslant \langlembda \kappa (s^{-1}) \left( |\xi_0|^{\frac1s} + \varepsilon^{\frac1s} \right)
\end{equation*}
we obtain for any $r > 0$
\begin{align*}
\sup_{\eta \in {G(X|Y)}amma_{\delta}, \ |\eta| \geqslant \delta^{-1} + |x_0|} e^{r | \eta |^{\frac1s}} |V_\varphi u(x_0,\eta)|
& \leqslant \sup_{(x,\xi) \in \operatorname{B}_\varepsilon, \ \langlembda > 0} e^{ \langlembda r \kappa (s^{-1}) \left( |\xi_0|^{\frac1s} + \varepsilon^{\frac1s} \right)} |V_\varphi u( \langlembda^t x, \langlembda^s (\xi_0 + \xi) )| \\
\leqslant C_{ r \kappa (s^{-1}) ( |\xi_0|^{\frac1s} + \varepsilon^{\frac1s})}
\end{align*}
which shows that $(x_0,\xi_0) \mathbf Ntin \mathrm{WF}_s(u)$.
\end{proof}
The following result gives a sufficient condition for the opposite inclusion.
\begin{prop}\langlebel{prop:WFsfreqaxis}
If $s > 1$, $t > 0$
and $u \in
\mathscr{E}_s'(\rr d) + \Sigma_t^s(\rr d)$ then
\begin{equation}\langlebel{eq:WFstinclusion}
\mathrm{WF}^{t,s}(u) \subseteq \{ 0 \} \times \pi_2 \mathrm{WF}_s(u).
\end{equation}
\end{prop}
\begin{proof}
We may assume $u \in \mathscr{E}_s'(\rr d) \subseteq ( \Sigma_t^s )' (\rr d)$.
We start with the less precise inclusion
\begin{equation}\langlebel{eq:subinclusion1}
\mathrm{WF}^{t,s} (u) \subseteq \{ 0 \} \times (\rr d \setminus 0).
\end{equation}
Suppose $(x_0,\xi_0) \in \rr {2d}$ with $x_0 \neq 0$.
We pick a neighborhood $U \subseteq \rr {2d}$ such that $(x_0,\xi_0) \in U$ and
\begin{equation*}
\inf_{(x,\xi) \in U} |x| = \delta > 0.
\end{equation*}
If we pick $\varphi \in G_c^s(\rr d) \subseteq \Sigma_t^s(\rr d)$ we have $V_\varphi u(x,\xi) = 0$ if $|x| \geqslant r$ for $r>0$ sufficiently large due to $u \in \mathscr{E}_s'(\rr d)$. This implies that $V_\varphi u(\langlembda^t x,\langlembda^s \xi) = 0$ if $\langlembda^t \geqslant r \delta^{-1}$, for all $(x,\xi) \in U$.
Hence $(x_0, \xi_0) \mathbf Ntin \mathrm{WF}^{t,s} (u)$ and we have shown \eqref{eq:subinclusion1}.
In order to show the sharper inclusion \eqref{eq:WFstinclusion}, suppose $0 \neq (x_0,\xi_0) \mathbf Ntin \{ 0 \} \times \pi_2 \mathrm{WF}_s (u)$.
Then either $x_0 \neq 0$ or $\xi_0 \mathbf Ntin \pi_2 \mathrm{WF}_s (u)$. If $x_0 \neq 0$ then by \eqref{eq:subinclusion1} we have $(x_0,\xi_0) \mathbf Ntin \mathrm{WF}^{t,s}(u)$.
Therefore we may assume that $x_0=0$, $\xi_0 \mathbf Ntin \pi_2 \mathrm{WF}_s (u)$ and $\xi_0 \neq 0$, and our goal is to show $(0,\xi_0) \mathbf Ntin \mathrm{WF}^{t,s}(u)$, which will prove \eqref{eq:WFstinclusion}.
By a slight modification to the Gevrey framework of the proof of \cite[Proposition~8.1.3]{Hormander0} we have $\pi_2 \mathrm{WF}_s(u) = V_s(u)$, where $V_s(u) \subseteq \rr d \setminus 0$ is a closed conic set defined as follows for $u \in \mathscr{E}_s '(\rr d)$.
A point $\eta \in \rr d \setminus 0$ satisfies $\eta \mathbf Ntin V_s (u)$ if $\eta \in {G(X|Y)}amma_2$ where ${G(X|Y)}amma_2 \subseteq \rr d \setminus 0$ is open and conic, and
\begin{equation}\langlebel{eq:frequencydecay0}
\sup_{\xi \in {G(X|Y)}amma_2} e^{r |\xi|^{\frac1s}} |\widehat u(\xi)| < \infty \quad \forall r > 0.
\end{equation}
Thus we have $\xi_0 \mathbf Ntin V_s(u)$, so there exists an open conic set ${G(X|Y)}amma_2 \subseteq \rr d \setminus 0$ such that $\xi_0 \in {G(X|Y)}amma_2$, and \eqref{eq:frequencydecay0} holds.
Let $\varepsilon > 0$ be small enough so that $\xi_0 + \operatorname{B}_{2 \varepsilon} \subseteq {G(X|Y)}amma_2$.
We assume $\varepsilon \leqslant \frac12 |\xi_0|$ which gives $| \xi_0 + \xi | > \frac12 |\xi_0|$ when $|\xi| < \varepsilon$.
We have
\begin{equation*}
V_\varphi u (x,\xi) = \widehat{u T_x \overline{\varphi}} (\xi)
= (2 \pi)^{-\frac{d}2} \widehat{u} * \widehat{T_x \overline{\varphi}} (\xi)
\end{equation*}
which gives
\begin{equation}\langlebel{eq:STFTconvolution1}
|V_\varphi u (x,\xi)|
\lesssim |\widehat u| * |g| (\xi), \quad x, \ \xi \in \rr d,
\end{equation}
where $g (\xi)= \widehat \varphi(-\xi) \in
\Sigma_s^t (\rr d)$.
Since
$u \in \mathscr{E}_s' (\rr d)$
we obtain from the Paley--Wiener--Schwartz theorem (Gevrey version cf. \cite{Komatsu1,Sobak1} and \cite[Theorems~1.6.1 and 1.6.7]{Rodino1}) for some $a > 0$
\begin{equation}\langlebel{eq:PWSGevrey}
|\widehat{u} (\xi)| \lesssim e^{a |\xi|^{\frac1s}}, \quad \xi \in \rr d,
\end{equation}
and we have
\begin{equation}\langlebel{eq:PWSGevrey2}
|g(\xi)|
\lesssim e^{-r |\xi|^{\frac1s}},
\quad \xi \in \rr d, \quad \forall r > 0.
\end{equation}
Let $(x,\xi) \in \operatorname{B}_\varepsilon$, $r > 0$ and $\langlembda > 0$.
We have
\begin{align*}
e^{r \langlembda} |V_\varphi u ( \langlembda^t x,\langlembda^s (\xi_0 + \xi ))|
\lesssim e^{r \langlembda} \int_{\rr d} | \widehat u ( \langlembda^s (\xi_0 + \xi - \langlembda^{-s} \eta) ) | \, | g (\eta) | \, \mathrm {d} \eta
= I_1 + I_2
\end{align*}
where we split the integral into the two terms
\begin{align*}
I_1 = & e^{r \langlembda} \int_{\rr {d} \setminus \Omega_\langlembda} | \widehat u ( \langlembda^s (\xi_0 + \xi - \langlembda^{-s} \eta) ) | \, | g (\eta) | \, \mathrm {d} \eta, \\
I_2 = & e^{r \langlembda} \int_{\Omega_\langlembda} | \widehat u ( \langlembda^s (\xi_0 + \xi - \langlembda^{-s} \eta) ) | \, | g (\eta) | \, \mathrm {d} \eta
\end{align*}
where
\begin{equation*}
\Omega_\langlembda = \{ \eta \in \rr d: \, |\eta|^{\frac1s} < \langlembda \varepsilon^{\frac1s} \} \subseteq \rr d.
\end{equation*}
For $I_1$ we use \eqref{eq:PWSGevrey} which together with \eqref{eq:PWSGevrey2} give for any $r_1 > 0$
\begin{align*}
I_1 \lesssim & e^{r \langlembda} \int_{\rr {d} \setminus \Omega_\langlembda} e^{a| \langlembda^s (\xi_0 + \xi) - \eta) |^{\frac1s}} \, | g (\eta) | \, \mathrm {d} \eta \\
& \leqslant e^{\langlembda (r + a \kappa(s^{-1}) |\xi_0 + \xi |^{\frac1s}) } \int_{\rr {d} \setminus \Omega_\langlembda} e^{ \kappa(s^{-1}) a | \eta |^{\frac1s}} \, | g (\eta) | \, \mathrm {d} \eta \\
& \leqslant e^{\langlembda (r + a \kappa(s^{-1})(|\xi_0| + \varepsilon)^{\frac1s})} \int_{\rr {d} \setminus \Omega_\langlembda} e^{ ( \kappa(s^{-1}) a - \kappa(s^{-1}) a - r_1 - 1) | \eta |^{\frac1s}} \, \mathrm {d} \eta \\
& \leqslant e^{\langlembda (r + a \kappa(s^{-1})(|\xi_0|+\varepsilon)^{\frac1s} - r_1 \varepsilon^{\frac1s})} \int_{\rr d} e^{ - | \eta |^{\frac1s}} \, \mathrm {d} \eta \\
& \leqslant C_r
\end{align*}
provided we pick $r_1 \geqslant \varepsilon^{-\frac1s}( r+ a \kappa(s^{-1})(|\xi_0|+\varepsilon)^{\frac1s} )$.
It remains to estimate $I_2$.
If $\eta \in \Omega_\langlembda$ then $\langlembda^{-s} | \eta| < \varepsilon$ which implies $\xi - \langlembda^{-s} \eta \in \operatorname{B}_{2 \varepsilon}$, and thus $\xi_0 + \xi - \langlembda^{-s} \eta \in {G(X|Y)}amma_2$.
Since ${G(X|Y)}amma_2$ is conic we have $\langlembda^s (\xi_0 + \xi - \langlembda^{-s} \eta) \in {G(X|Y)}amma_2$.
Thus we may use \eqref{eq:frequencydecay0}, which together with \eqref{eq:PWSGevrey2} give for any $r_1, r_2 > 0$
\begin{align*}
I_2 & \lesssim e^{r \langlembda} \int_{\Omega_\langlembda} e^{- \kappa(s^{-1}) r_1 | \langlembda^s (\xi_0 + \xi - \langlembda^{-s} \eta) |^{\frac1s} } \, | g (\eta) | \, \mathrm {d} \eta \\
& \leqslant e^{r \langlembda} \int_{\Omega_\langlembda} e^{- r_1 \langlembda | \xi_0 + \xi |^{\frac1s} + \kappa(s^{-1}) r_1 |\eta |^{\frac1s} } \, | g (\eta) | \, \mathrm {d} \eta \\
& \leqslant e^{\langlembda (r - r_1 2^{- \frac1s} | \xi_0 |^{\frac1s} )} \int_{\rr d} e^{ \kappa(s^{-1}) r_1 |\eta |^{\frac1s} } \, | g (\eta) | \, \mathrm {d} \eta \\
& \lesssim e^{\langlembda (r - r_1 2^{- \frac1s} | \xi_0 |^{\frac1s} )} \int_{\rr d} e^{(\kappa(s^{-1}) r_1 - r_2)|\eta |^{\frac1s} } \mathrm {d} \eta \\
& \leqslant C_r
\end{align*}
if we first pick $r_1 \geqslant 2^{ \frac1s} | \xi_0 |^{-\frac1s} r$ and then pick $r_2 > \kappa(s^{-1}) r_1$.
We have shown $(0,\xi_0) \mathbf Ntin \mathrm{WF}^{t,s} (u)$.
\end{proof}
\begin{cor}\langlebel{cor:WFsfreqaxis}
If $t \geqslant s > 1$ and $u \in
\mathscr{E}_s' (\rr d)
+ \Sigma_t^s(\rr d)$ then
\begin{equation*}
\mathrm{WF}^{t,s}(u) = \{ 0 \} \times V_s(u).
\end{equation*}
\end{cor}
The following result is a sort of converse to Corollary {\rm Re}f{cor:WFsfreqaxis}.
\begin{prop}\langlebel{prop:WFsnotfreqaxis}
Let $s,t > 0$ satisfy $s + t > 1$, and let
$u \in (\Sigma_t^s)'(\rr d)$.
If
\begin{equation*}
\mathrm{WF}^{t,s}(u) \cap \{ 0 \} \times (\rr d \setminus 0) = \emptyset
\end{equation*}
then $u \in C^\infty(\rr d)$ and there exist $C,r > 0$ such that
\begin{equation}\langlebel{eq:exponentialbound1}
|\partial^\alpha u(x)| \leqslant C^{1+|\alpha|} \alpha!^s e^{r |x|^\frac1t}, \quad x \in \rr d, \quad \alpha \in \nn d.
\end{equation}
\end{prop}
\begin{proof}
Let $\varphi \in \Sigma_t^s (\rr d)$ satisfy $\| \varphi \|_{L^2} = 1$.
Using the compactness of $\sr {d-1} \subseteq \rr d$ we obtain the following conclusion from the assumption.
There exists $\varepsilon > 0$ such that
\begin{equation}\langlebel{eq:decayGamma1}
\sup_{(x,\xi) \in \operatorname{B}_\varepsilon, \ \xi_0 \in \sr {d-1}, \ \langlembda > 0} e^{r \langlembda} |V_\varphi u (\langlembda^t x, \langlembda^s (\xi_0 + \xi))| < \infty \quad \forall r > 0.
\end{equation}
Set
\begin{equation*}
{G(X|Y)}amma = \{ (\langlembda^t x, \langlembda^s (\xi_0 + \xi) ) \in \rr {2d} \setminus 0: \, \xi_0 \in \sr {d-1}, \, (x,\xi) \in \operatorname{B}_\varepsilon, \, \langlembda > 0 \}.
\end{equation*}
If $(y,\eta) \in {G(X|Y)}amma$ then $\eta = \langlembda^s (\xi_0 + \xi)$ and $y = \langlembda^t x$ for some $\xi_0 \in \sr {d-1}$, $(x,\xi) \in \operatorname{B}_\varepsilon$, and $\langlembda > 0$,
so $|\eta|^{\frac1s} = \langlembda |\xi_0 + \xi|^{\frac1s} < \langlembda ( 1 + \varepsilon )^{\frac1s}$ and $|y|^\frac1t = \langlembda |x|^\frac1t < \langlembda \varepsilon^\frac1t$.
Thus from \eqref{eq:decayGamma1} it follows that we have
\begin{equation}\langlebel{eq:decayGamma2}
\sup_{(x,\xi) \in {G(X|Y)}amma} e^{r ( |x|^{\frac1t} + |\xi|^{\frac1s} )} |V_\varphi u (x,\xi)| < \infty \quad \forall r > 0.
\end{equation}
We claim that if $(y,\eta) \in \rr {2d} \setminus 0$ then
\begin{equation}\langlebel{eq:implicationGamma}
|y|^{\frac1t} < \varepsilon^{\frac1t} | \eta |^{\frac1s} \quad \Longrightarrow \quad (y, \eta) \in {G(X|Y)}amma.
\end{equation}
In fact suppose $|y|^{\frac1t} < \varepsilon^{\frac1t} | \eta |^{\frac1s}$. Since $\eta \neq 0$ we may define $\langlembda = |\eta|^\frac1s > 0$
and $\xi_0 = \langlembda^{-s} \eta \in \sr {d-1}$, whence $\eta = \langlembda^s \xi_0$. Set $x = \langlembda^{-t} y$ so that $y = \langlembda^{t} x$. We have
\begin{equation*}
|x|^\frac1t = |\eta|^{-\frac1s} |y|^\frac1t < \varepsilon^{\frac1t}
\end{equation*}
so $x \in \operatorname{B}_\varepsilon$ which proves that $(y, \eta) \in {G(X|Y)}amma$.
From \eqref{eq:implicationGamma} we may conclude
\begin{equation}\langlebel{eq:cover1}
{G(X|Y)}amma \cup \Omega = \rr {2d} \setminus 0
\end{equation}
where
\begin{equation*}
\Omega = \{ (y,\eta) \in \rr {2d} \setminus 0: \, | \eta |^{\frac1s} \leqslant C |y|^{\frac1t} \}
\end{equation*}
for some $C > 0$.
We use \eqref{eq:STFTinverse} for $u \in ( \Sigma_t^s )' (\rr d)$ and $\varphi \in \Sigma_t^s (\rr d)$ with $\| \varphi \|_{L^2} = 1$, cf. \cite{Toft1},
and show that the integral for $\partial^\alpha u$ is absolutely convergent for any $\alpha \in \nn d$.
Thus we write formally
\begin{equation}\langlebel{eq:STFTreconstruction}
\partial^\alpha u (y) = (2\pi)^{-\frac{d}{2}} \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} \int_{\rr {2d}} V_\varphi u(x,\xi) \, (i\xi)^\beta e^{i \langlengle \xi,y \operatorname{Ran}gle} \partial^{\alpha-\beta} \varphi (y-x) \, \mathrm {d} x \, \mathrm {d} \xi.
\end{equation}
We will need the estimate for any $r > 0$
\begin{align*}
|\xi|^{\beta}
& =
\left( \frac{d s}{r} \right)^{s |\beta|} \beta!^s
\left(
\frac{ \left( \frac{r}{d s} |\xi|^{\frac1s} \right)^{|\beta|}}{\beta!}
\right)^s
\leqslant
\left( \frac{d s}{r} \right)^{s |\beta|} \beta!^s
\left(
\frac{ \left( \frac{r}{s} |\xi|^{\frac1s} \right)^{|\beta|}}{|\beta|!}
\right)^s \\
& \leqslant \left( \frac{d s}{r} \right)^{s |\beta|} \beta!^s e^{r |\xi|^{\frac1s}}
\end{align*}
as well as
\begin{equation}\langlebel{eq:Sstderivativeestimate1}
|D^\beta \varphi (x)| \leqslant C_{r,h} h^{|\beta|} \beta!^s e^{- r |x|^{\frac1t}}, \quad \beta \in \nn d, \quad x \in \rr d,
\end{equation}
for any $h, r > 0$.
In order to prove \eqref{eq:Sstderivativeestimate1} we may use the seminorms \eqref{eq:seminormSigmas} with $h^{|\alpha+\beta|}$ replaced by $h_1^{|\alpha|} h_2^{|\beta|}$ for two different arbitrary $h_1, h_2 > 0$.
The argument is known but we repeat it for the benefit of the reader.
If $r > 0$ then we obtain from \eqref{eq:seminormSigmas} for any $h_1, h_2 > 0$
\begin{align*}
e^{\frac{r}{t} |x|^{\frac1t}} |D^\beta \varphi (x)|^{\frac1t}
& = \sum_{n=0}^\infty 2^{-n} \left( \frac{\left( \frac{2r}{t} \right)^{tn} }{n!^t} |x|^n \, |D^\beta \varphi (x)| \right)^{\frac1t} \\
& \leqslant 2 \left( \sup_{n \geqslant 0} \frac{\left( \frac{2r}{t} \right)^{t n} d^{\frac{n}{2}}}{n!^t} \max_{|\alpha|=n} |x^{\alpha} D^\beta \varphi (x)| \right)^{\frac1t} \\
& \leqslant \left( C_{h_1,h_2} h_2^{|\beta|} \beta!^s
\sup_{n \geqslant 0} \left( \left( \frac{2r}{t} \right)^{t} d^{\frac12} h_1 \right)^n
\right)^{\frac1t} \\
& \leqslant \left( C_{h_2,r} \, h_2^{|\beta|} \beta!^s \right)^{\frac1t}
\end{align*}
provided $h_1 \leqslant \left( \frac{t}{2r} \right)^{t} d^{-\frac12}$.
We have proved \eqref{eq:Sstderivativeestimate1} for any $h, r > 0$.
We split the integral \eqref{eq:STFTreconstruction} in two parts.
We obtain using \eqref{eq:decayGamma2} for any $r_1, r_2, r_3 > 0$ and $0 < h \leqslant 1$
\begin{equation}\langlebel{eq:integralGamma}
\begin{aligned}
& \left| \int_{{G(X|Y)}amma} V_\varphi u(x,\xi) \, (i\xi)^\beta e^{i \langlengle \xi,y \operatorname{Ran}gle} \partial^{\alpha-\beta} \varphi(y-x) \, \mathrm {d} x \, \mathrm {d} \xi \right| \\
& \leqslantlant \int_{{G(X|Y)}amma} |V_\varphi u(x,\xi)| \, |\xi|^{|\beta|} \, |\partial^{\alpha-\beta} \varphi(y-x)| \, \mathrm {d} x \, \mathrm {d} \xi \\
& \lesssim h^{|\alpha-\beta|} \left( \frac{d s}{r_2} \right)^{s |\beta|} (\alpha-\beta)!^s \beta!^s \int_{{G(X|Y)}amma} e^{-r_1 (|x|^\frac1t + |\xi|^\frac1s) + r_2 |\xi|^\frac1s - \kappa(t^{-1}) r_3 |y-x|^\frac1t} \, \mathrm {d} x \, \mathrm {d} \xi \\
& \lesssim \left( h^{-1} \left( \frac{d s}{r_2} \right)^s \right)^{|\beta|} \alpha!^s e^{ -r_3 |y|^{\frac1t}}
\int_{\rr {2d}} e^{-r_1 (|x|^\frac1t + |\xi|^\frac1s) + r_2 |\xi|^{\frac1s} + \kappa(t^{-1}) r_3 |x|^{\frac1t}} \, \mathrm {d} x \, \mathrm {d} \xi \\
& \lesssim \left( h^{-1} \left( \frac{d s}{r_2} \right)^s \right)^{|\alpha|} \alpha!^s e^{ -r_3 |y|^{\frac1t}}
\end{aligned}
\end{equation}
provided $h \leqslant \left( \frac{d s}{r_2} \right)^s$ and $r_1 > \max(r_2, \kappa(t^{-1}) r_3)$.
For the remaining part of the integral we may by \eqref{eq:cover1} assume that $(x,\xi) \in \Omega$.
Using \eqref{eq:STFTGFstdistr} we obtain for some $r_1 > 0$ and any $r_2, r_3 > 0$ and $0 < h \leqslant 1$
\begin{equation}\langlebel{eq:integralOmega}
\begin{aligned}
& \left| \int_{\Omega} V_\varphi u(x,\xi) \, (i\xi)^\beta e^{i \langlengle \xi,y \operatorname{Ran}gle} \partial^{\alpha-\beta} \varphi(y-x) \, \mathrm {d} x \, \mathrm {d} \xi \right| \\
& \lesssim \int_{| \xi |^{\frac1s} \leqslant C |x|^{\frac1t}} e^{r_1 ( |x|^\frac1t +|\xi|^\frac1s}) |\xi|^{|\beta|} \, |\partial^{\alpha-\beta} \varphi(y-x)| \, \mathrm {d} x \, \mathrm {d} \xi \\
& \lesssim h^{|\alpha-\beta|} \left( \frac{d s}{r_2} \right)^{s |\beta|} (\alpha-\beta)!^s \beta!^s \int_{| \xi |^{\frac1s} \leqslant C |x|^{\frac1t} } e^{ r_1 ( |x|^\frac1t +|\xi|^\frac1s) + r_2 |\xi|^{\frac1s} - \kappa(t^{-1}) r_3 |y-x|^\frac1t } \, \mathrm {d} x \, \mathrm {d} \xi \\
& \lesssim
\left( h^{-1} \left( \frac{d s}{r_2} \right)^s \right)^{|\beta|} \alpha!^s e^{ \kappa(t^{-1}) r_3 |y|^{\frac1t} }
\int_{| \xi |^{\frac1s} \leqslant C |x|^{\frac1t}} e^{ r_1 ( |x|^\frac1t +|\xi|^\frac1s) + r_2 |\xi|^{\frac1s} - r_3 |x|^\frac1t } \, \mathrm {d} x \, \mathrm {d} \xi \\
& \leqslant
\left( h^{-1} \left( \frac{d s}{r_2} \right)^s \right)^{|\alpha|} \alpha!^s e^{ \kappa(t^{-1}) r_3 |y|^{\frac1t} }
\int_{| \xi |^{\frac1s} \leqslant C |x|^{\frac1t}} e^{ - |\xi|^\frac1s + (r_1 - r_3) |x|^\frac1t + (1 + r_1 + r_2) |\xi|^\frac1s } \, \mathrm {d} x \, \mathrm {d} \xi \\
& \leqslant
\left( h^{-1} \left( \frac{d s}{r_2} \right)^s \right)^{|\alpha|} \alpha!^s e^{ \kappa(t^{-1}) r_3 |y|^{\frac1t} }
\int_{| \xi |^{\frac1s} \leqslant C |x|^{\frac1t}} e^{ - |\xi|^\frac1s + (r_1 + C (1 + r_1 + r_2) - r_3) |x|^\frac1t } \, \mathrm {d} x \, \mathrm {d} \xi \\
& \lesssim \left( h^{-1} \left( \frac{d s}{r_2} \right)^s \right)^{|\alpha|} \alpha!^s e^{ \kappa(t^{-1}) r_3 |y|^{\frac1t} }
\end{aligned}
\end{equation}
provided $h \leqslant \left( \frac{d s}{r_2} \right)^s$ and $r_3 > r_1 + C (1 + r_1 + r_2)$.
Combining \eqref{eq:integralGamma} and \eqref{eq:integralOmega} shows in view of \eqref{eq:STFTreconstruction} that $u \in C^\infty(\rr d)$ and the estimate \eqref{eq:exponentialbound1} follows.
\end{proof}
\section{Microlocality}\langlebel{sec:microlocal}
The next result concerns microlocality with respect to $\mathrm{WF}^{t,s}$ of pseudodifferential operators.
We use a space of smooth symbols originally introduced in \cite[Definition~1.8]{Abdeljawad1} and denoted ${G(X|Y)}amma_{t,s}^{s,t; 0} (\rr {2d})$.
For $s,t > 0$ such that $s + t > 1$, $a \in {G(X|Y)}amma_{t,s}^{s,t; 0} (\rr {2d})$ means that $a \in C^\infty(\rr {2d})$ and
\begin{equation}\langlebel{eq:symbolvillkor2}
|\partial_{x}^{\alpha} \partial_{\xi}^{\beta} a(x,\xi)| \lesssim h^{|\alpha + \beta|} \alpha!^s \beta!^t e^{\mu (|x|^{\frac1t} + |\xi|^{\frac1s} )}, \quad \alpha, \beta \in \nn d, \quad x, \xi \in \rr d,
\end{equation}
for some $\mu > 0$ and for all $h > 0$.
The space ${G(X|Y)}amma_{t,s}^{s,t; 0} (\rr {2d})$ is characterized in \cite[Proposition~2.3]{Abdeljawad1} using the STFT as follows. Let $\Phi \in \Sigma_{t,s}^{s,t}(\rr {2d}) \setminus 0$ be arbitrary.
Then $a \in {G(X|Y)}amma_{t,s}^{s,t; 0} (\rr {2d})$ if and only if
\begin{equation}\langlebel{eq:STFTvillkor1}
| V_\Phi a(z_1, z_2,\zeta_1, \zeta_2) | \lesssim e^{\mu (|z_1|^{\frac1t} + |z_2|^{\frac1s})- b (|\zeta_1|^{\frac1s} + |\zeta_2|^{\frac1t})}, \quad z_1, z_2, \zeta_1, \zeta_2 \in \rr d,
\end{equation}
for some $\mu > 0$ and all $b > 0$.
If $a \in {G(X|Y)}amma_{t,s}^{s,t; 0} (\rr {2d})$ then $a^w(x,D): \Sigma_t^s (\rr d) \to \Sigma_t^s (\rr d)$ is continuous
and extends uniquely to a continuous operator $a^w(x,D): (\Sigma_t^s)' (\rr d) \to (\Sigma_t^s)' (\rr d)$ according to \cite[Theorem~3.15]{Abdeljawad1}.
By the following result
it is also microlocal with respect to the $t,s$-Gelfand--Shilov wave front set.
\begin{thm}\langlebel{thm:microlocalWFs}
If $s,t > 0$ satisfy $s + t > 1$
and $a \in {G(X|Y)}amma_{t,s}^{s,t; 0} (\rr {2d})$ then
\begin{equation}\langlebel{eq:microlocal1}
\mathrm{WF}^{t,s}( a^w(x,D) u ) \subseteq \mathrm{WF}^{t,s}(u), \quad u \in (\Sigma_t^s)'(\rr d).
\end{equation}
\end{thm}
\begin{proof}
Pick $\varphi \in \Sigma_t^s(\rr d)$ such that $\| \varphi \|_{L^2}=1$.
Recall the notation $\Pi(x,\xi) = M_\xi T_x$ for $(x,\xi) \in \rr {2d}$.
Denoting the formal adjoint of $a^w(x,D)$ by $a^w(x,D)^*$, \eqref{eq:moyal} gives for $u \in (\Sigma_t^s)'(\rr d)$ and $z \in \rr {2d}$
\begin{align*}
(2 \pi)^{\frac{d}{2}} V_\varphi (a^w(x,D) u) (z)
& = ( a^w(x,D) u, \Pi(z) \varphi ) \\
& = ( u, a^w(x,D)^* \Pi(z) \varphi ) \\
& = \int_{\rr {2d}} V_\varphi u(w) \, ( \Pi(w) \varphi,a^w(x,D)^* \Pi(z) \varphi ) \, \mathrm {d} w \\
& = \int_{\rr {2d}} V_\varphi u(w) \, ( a^w(x,D) \, \Pi(w) \varphi,\Pi(z) \varphi ) \, \mathrm {d} w \\
& = \int_{\rr {2d}} V_\varphi u(z-w) \, ( a^w(x,D) \, \Pi(z-w) \varphi,\Pi(z) \varphi ) \, \mathrm {d} w.
\end{align*}
By e.g. \cite[Lemma 3.1]{Grochenig2}, or a direct computation involving \eqref{eq:wignerweyl}, we have
\begin{equation*}
|( a^w(x,D) \, \Pi(z-w) \varphi,\Pi(z) \varphi )|
= \left| V_\Phi a \left( z-\frac{w}{2}, \mathcal{J} w \right) \right|
\end{equation*}
where $\Phi$ is the Wigner distribution $\Phi = W(\varphi,\varphi)$.
We have $\Phi \in \Sigma_{t,s}^{s,t}(\rr {2d})$.
In fact we have $\varphi \otimes \overline \varphi \in \Sigma_{t,t}^{s,s}(\rr {2d})$
and therefore also $(\varphi \otimes \overline \varphi) \circ \kappa \in \Sigma_{t,t}^{s,s}(\rr {2d})$
where $\kappa(x,y) = (x+y/2, x- y/2)$.
Since $W(\varphi,\varphi) = (2 \pi)^{\frac{d}{2}} \mathscr{F}_2 ( (\varphi \otimes \overline \varphi) \circ \kappa )$
we obtain from \cite[Proposition~1.1]{Abdeljawad1}
the conclusion $\Phi \in \Sigma_{t,s}^{s,t}(\rr {2d})$.
Combining the preceding identities we deduce
\begin{align*}
|V_\varphi (a^w(x,D) u) (z)|
& \lesssim \int_{\rr {2d}} |V_\varphi u(z-w)| \, \left| V_\Phi a \left( z-\frac{w}{2}, \mathcal{J} w \right) \right| \, \mathrm {d} w.
\end{align*}
Suppose $z_0 \in \rr {2d} \setminus 0$ and $z_0 \mathbf Ntin \mathrm{WF}^{t,s}(u)$.
There exists an open set $V$ such that $z_0 \in V$ and
\eqref{eq:notinWFGFst2} holds.
We pick an open set $U$ such that
$z_0 \in U$ and $U + \operatorname{B}_\varepsilon \subseteq V$ for some $0 < \varepsilon \leqslant 1$, and we may assume
\begin{equation}\langlebel{eq:Ubound}
\sup_{z \in U} |z| \leqslant |z_0| + 1 := \alpha.
\end{equation}
Let $r > 0$ and $\langlembda > 0$. We have
\begin{align*}
& e^{r \langlembda} |V_\varphi (a^w(x,D) u) (\langlembda^t x, \langlembda^s \xi)| \\
& \lesssim \iint_{\rr {2d}} e^{r \langlembda } |V_\varphi u ( \langlembda^t (x- \langlembda^{-t} y), \langlembda^s (\xi - \langlembda^{-s} \eta))| \, \left| V_\Phi a \left( \langlembda^t x-\frac{y}{2}, \langlembda^s \xi-\frac{\eta}{2}, \eta, -y \right) \right| \, \mathrm {d} y \, \mathrm {d} \eta \\
& = I_1 + I_2
\end{align*}
where we split the integral into the two terms
\begin{align*}
I_1 = & \iint_{\rr {2d} \setminus \Omega_\langlembda} e^{r \langlembda} |V_\varphi u ( \langlembda^t (x- \langlembda^{-t} y), \langlembda^s (\xi - \langlembda^{-s} \eta))| \, \left| V_\Phi a \left( \langlembda^t x-\frac{y}{2}, \langlembda^s \xi-\frac{\eta}{2}, \eta, -y \right) \right| \, \mathrm {d} y \, \mathrm {d} \eta, \\
I_2 = & \iint_{\Omega_\langlembda} e^{r \langlembda} |V_\varphi u ( \langlembda^t (x- \langlembda^{-t} y), \langlembda^s (\xi - \langlembda^{-s} \eta))| \, \left| V_\Phi a \left( \langlembda^t x-\frac{y}{2}, \langlembda^s \xi-\frac{\eta}{2}, \eta, -y \right) \right| \, \mathrm {d} y \, \mathrm {d} \eta
\end{align*}
where
\begin{equation*}
\Omega_\langlembda = \{(y,\eta) \in \rr {2d}: |y|^{\frac1t} + |\eta|^{\frac1s} < 2^{-\frac{1}{2v}} \varepsilon^{\frac1v} \langlembda \}
\end{equation*}
with $v = \min(s,t)$.
First we estimate $I_1$ when $(x,\xi) \in U$.
Set $\kappa = \max(\kappa(t^{-1}), \kappa(s^{-1}))$.
From
\eqref{eq:STFTGFstdistr}, \eqref{eq:STFTvillkor1} and \eqref{eq:Ubound}
we obtain for some $r_1, \mu > 0$ and any $b > 0$
\begin{equation}\langlebel{eq:estimateI1b}
\begin{aligned}
I_1
& \lesssim e^{r \langlembda } \iint_{\rr {2d} \setminus \Omega_\langlembda} e^{r_1 \langlembda |x- \langlembda^{-t} y|^{\frac1t} + r_1 \langlembda |\xi- \langlembda^{-s} \eta|^{\frac1s}} \, \left| V_\Phi a \left( \langlembda^t x-\frac{y}{2}, \langlembda^s \xi-\frac{\eta}{2}, \eta, -y \right) \right| \, \mathrm {d} y \, \mathrm {d} \eta \\
& \leqslant e^{ r \langlembda+ \kappa r_1 \langlembda |x|^{\frac1t} + \kappa r_1 \langlembda |\xi|^{\frac1s} } \iint_{\rr {2d} \setminus \Omega_\langlembda} e^{ r_1 \kappa (| y|^{\frac1t} + | \eta|^{\frac1s} )} \, \left| V_\Phi a \left( \langlembda^t x-\frac{y}{2}, \langlembda^s \xi-\frac{\eta}{2}, \eta, -y \right) \right| \, \mathrm {d} y \, \mathrm {d} \eta \\
& \lesssim e^{ r \langlembda+ r_1 \langlembda \kappa (\alpha^{\frac1t} + \alpha^{\frac1s} ) }
\iint_{\rr {2d} \setminus \Omega_\langlembda}
e^{ r_1 \kappa (| y|^{\frac1t} + | \eta|^{\frac1s})
+ \mu \left( \left| \langlembda^t x-\frac{y}{2}\right|^{\frac1t} + \left| \langlembda^s \xi-\frac{\eta}{2}\right|^{\frac1s} \right)
- (b+1) \left( \left| \eta \right|^{\frac1s} + \left| y \right|^{\frac1t} \right) } \, \mathrm {d} y \, \mathrm {d} \eta \\
& \lesssim e^{ \langlembda \left( r + (r_1+\mu) \kappa (\alpha^{\frac1t} + \alpha^{\frac1s}) \right)} \iint_{\rr {2d} \setminus \Omega_\langlembda} e^{ \kappa (r_1 + 2^{-\frac1t} \mu - b) | y|^{\frac1t} + \kappa ( r_1 + 2^{-\frac1s} \mu - b) | \eta|^{\frac1s}
- \left( \left| \eta \right|^{\frac1s} + \left| y \right|^{\frac1t} \right) } \, \mathrm {d} y \, \mathrm {d} \eta \\
& \leqslant e^{ \langlembda \left( r + 2 (r_1+\mu) \kappa \alpha^{\frac1v} \right)} \iint_{\rr {2d} \setminus \Omega_\langlembda} e^{ \kappa (r_1 + \mu - b) (| y|^{\frac1t} + | \eta|^{\frac1s} )
- \left( \left| \eta \right|^{\frac1s} + \left| y \right|^{\frac1t} \right) } \, \mathrm {d} y \, \mathrm {d} \eta \\
& \leqslant e^{ \langlembda \left( r + 2 (r_1+\mu) \kappa \alpha^{\frac1v}
+ \kappa (r_1 + \mu - b) 2^{-\frac{1}{2v}} \varepsilon^{\frac1v} \right)}
\iint_{\rr {2d}} e^{-\left( \left| \eta \right|^{\frac1t} + \left| y \right|^{\frac1s} \right) } \, \mathrm {d} y \, \mathrm {d} \eta \\
& \lesssim e^{ \langlembda \left( r + 2 (r_1+\mu) \kappa \alpha^{\frac1v}
+ \kappa (r_1 + \mu - b) 2^{-\frac{1}{2v}} \varepsilon^{\frac1v} \right)}
\leqslant C_{r}
\end{aligned}
\end{equation}
for any $\langlembda > 0$,
provided we pick $b \geqslant r_1 + \mu + \kappa^{-1} 2^{\frac{1}{2v}} \varepsilon^{- \frac1v} \left( r + 2 (r_1+\mu) \kappa \alpha^{\frac1v} \right)$.
Here $C_{r} > 0$ is a constant that depends on $r > 0$ but not on $\langlembda > 0$.
Thus we have obtained the requested estimate for $I_1$.
It remains to estimate $I_2$.
From $|y|^{\frac1t} + |\eta|^{\frac1s} < 2^{-\frac{1}{2v}} \varepsilon^{\frac1v} \langlembda$
we obtain
\begin{align*}
& \langlembda^{-t} |y| < \varepsilon^{\frac{t}{v}} \, 2^{-\frac{t}{2v}} \leqslant \varepsilon \, 2^{-\frac{1}{2}}, \\
& \langlembda^{-s} |\eta| < \varepsilon^{\frac{s}{v}} \, 2^{-\frac{s}{2v}} \leqslant \varepsilon \, 2^{-\frac{1}{2}}
\end{align*}
which gives $(\langlembda^{-t} y, \langlembda^{-s} \eta ) \in \operatorname{B}_{\varepsilon}$.
Hence if $(x,\xi) \in U$ then $( x- \langlembda^{-t} y, \xi - \langlembda^{-s} \eta) \in V$ and we may use the estimate
\eqref{eq:notinWFGFst2}.
This gives for some $\mu > 0$, any $b > 0$ and a constant $C_r = C_{r,\mu,s,t} > 0$, using
\eqref{eq:STFTvillkor1} and \eqref{eq:Ubound}
\begin{equation}\langlebel{eq:estimateI2b}
\begin{aligned}
I_2 & = \iint_{\Omega_\langlembda} e^{r \langlembda} |V_\varphi u ( \langlembda^t (x- \langlembda^{-t} y), \langlembda^s (\xi - \langlembda^{-s} \eta))| \, \left| V_\Phi a \left( \langlembda^t x-\frac{y}{2}, \langlembda^s \xi-\frac{\eta}{2}, \eta, -y \right) \right| \, \mathrm {d} y \, \mathrm {d} \eta \\
& = e^{-\langlembda \kappa \mu 2 \alpha^{\frac1v} } \\
& \quad \times \iint_{\Omega_\langlembda} e^{ (r+ \kappa \mu 2 \alpha^{\frac1v} ) \langlembda} |V_\varphi u ( \langlembda^t (x- \langlembda^{-t} y), \langlembda^s (\xi - \langlembda^{-s} \eta))| \, \left| V_\Phi a \left( \langlembda^t x-\frac{y}{2}, \langlembda^s \xi-\frac{\eta}{2}, \eta, -y \right) \right| \, \mathrm {d} y \, \mathrm {d} \eta \\
& \leqslant C_r e^{-\langlembda \kappa \mu 2 \alpha^{\frac1v}} \iint_{\Omega_\langlembda} \left| V_\Phi a \left( \langlembda^t x-\frac{y}{2}, \langlembda^s \xi-\frac{\eta}{2}, \eta, -y \right) \right| \, \mathrm {d} y \, \mathrm {d} \eta \\
& \lesssim C_r e^{- \langlembda \kappa \mu 2 \alpha^{\frac1v}}
\iint_{\rr {2d}}
e^{ \mu \left( \left| \langlembda^t x-\frac{y}{2}\right|^{\frac1t} + \left| \langlembda^s \xi-\frac{\eta}{2}\right|^{\frac1s} \right)
- b \left( \left| \eta \right|^{\frac1s} + \left| y \right|^{\frac1t} \right) }
\, \mathrm {d} y \, \mathrm {d} \eta \\
& \leqslant C_r e^{-\langlembda \kappa \mu 2 \alpha^{\frac1v} + \langlembda \kappa \mu 2 \alpha^{\frac1v}}
\iint_{\rr {2d}}
e^{ \mu \kappa \left( \left|y\right|^{\frac1t} + \left| \eta \right|^{\frac1s} \right)
- b \left( \left| \eta \right|^{\frac1s} + \left| y \right|^{\frac1t} \right) }
\, \mathrm {d} y \, \mathrm {d} \eta \\
& = C_r
\iint_{\rr {2d}}
e^{ (\kappa \mu- b) \left( \left| \eta \right|^{\frac1s} + \left| y \right|^{\frac1t} \right) }
\, \mathrm {d} y \, \mathrm {d} \eta \\
& \lesssim C_r
\end{aligned}
\end{equation}
provided $b > \kappa \mu$,
for all $\langlembda > 0$.
Thus we have obtained the requested estimate for $I_2$.
Combining \eqref{eq:estimateI1b} and \eqref{eq:estimateI2b}
we may conclude that $z_0 \mathbf Ntin \mathrm{WF}^{s,t}( a^w(x,D) u )$
and hence we have proved \eqref{eq:microlocal1}.
\end{proof}
As a corollary we obtain the following generalization of \cite[Proposition~4.10]{Carypis1}.
Here we use a space of smooth symbols originally introduced in \cite[Definition~2.4]{Cappiello2} and denoted ${G(X|Y)}amma_{0,s}^\infty(\rr {2d})$, and which is identical to ${G(X|Y)}amma_{s,s}^{s,s; 0}(\rr {2d})$.
For $s > \frac12$, $a \in {G(X|Y)}amma_{0,s}^\infty(\rr {2d})$ means that $a \in C^\infty(\rr {2d})$ and
\begin{equation}\langlebel{eq:symbolvillkor1}
|\pd \alpha a(z)| \lesssim h^{|\alpha|} \alpha!^s e^{\mu |z|^{\frac1s}}, \quad \alpha \in \nn {2d}, \quad z \in \rr {2d},
\end{equation}
for some $\mu > 0$ and for all $h > 0$.
The space ${G(X|Y)}amma_{0,s}^\infty(\rr {2d})$ is characterized in \cite[Proposition~3.2]{Cappiello2} using the STFT as follows. Let $\Phi \in \Sigma_s(\rr {2d}) \setminus 0$ be arbitrary.
Then $a \in {G(X|Y)}amma_{0,s}^\infty(\rr {2d})$ if and only if
\begin{equation}\langlebel{eq:STFTvillkor2}
| V_\Phi a(z,\zeta) | \lesssim e^{\mu |z|^{\frac1s} - b |\zeta|^{\frac1s}}, \quad z,\zeta \in \rr {2d},
\end{equation}
for some $\mu > 0$ and all $b > 0$.
If $a \in {G(X|Y)}amma_{0,s}^\infty(\rr {2d})$ then $a^w(x,D): \Sigma_s (\rr d) \to \Sigma_s (\rr d)$ is continuous
and extends uniquely to a continuous operator $a^w(x,D): \Sigma_s' (\rr d) \to \Sigma_s' (\rr d)$ according to \cite[Proposition~4.10]{Cappiello2}.
\begin{cor}\langlebel{cor:microlocalWFs}
If $s > \frac12$ and $a \in {G(X|Y)}amma_{0,s}^\infty(\rr {2d})$ then
\begin{equation*}
\mathrm{WF}^s ( a^w(x,D) u ) \subseteq \mathrm{WF}^s (u), \quad u \in \Sigma_s'(\rr d).
\end{equation*}
\end{cor}
\begin{rem}\langlebel{rem:microlocal}
It is interesting to compare the assumption $a \in {G(X|Y)}amma_{0,s}^\infty (\rr {2d})$, which is equivalent to
the STFT estimates
\begin{equation}\langlebel{eq:STFTvillkor1b}
| V_\Phi a(z,\zeta) | \lesssim e^{\mu |z|^{\frac1s} - b |\zeta|^{\frac1s}}
\end{equation}
for some $\mu > 0$ and all $b > 0$,
with the estimates
\begin{equation}\langlebel{eq:STFTvillkor2}
|V_\Phi a (z,\zeta) | \lesssim e^{\frac{b}{4} |z|^{\frac1s} - b |\zeta|^{\frac1s}}
\end{equation}
for all $b > 0$.
Condition \eqref{eq:STFTvillkor2} for all $b > 0$ has been shown to imply continuity $a^w(x,D): \Sigma_s(\rr d) \to \Sigma_s(\rr d)$ \cite[Lemma~6.5 and Proposition~6.6]{Wahlberg3}, but it does not imply microlocality with respect to $\mathrm{WF}^s$.
In fact microlocality for operators of this type is contradicted by \cite[p.~556]{Carypis1} with $Q = i I_{2d}$ and $t \mathbf Ntin \pi \mathbf Z$.
\end{rem}
The next result is another consequence of Theorem {\rm Re}f{thm:microlocalWFs}.
\begin{cor}\langlebel{cor:translationmodulationinvar}
Suppose $s,t > 0$ satisfy $s + t > 1$.
For any $z \in \rr {2d}$ and any $u \in (\Sigma_t^s)'(\rr d)$ we have
\begin{equation*}
\mathrm{WF}^{t,s}( \Pi(z) u ) = \mathrm{WF}^{t,s}(u).
\end{equation*}
\end{cor}
\begin{proof}
By a calculation it is verified that $\Pi(x,\xi) = a_{x,\xi}^w(x,D)$ where
\begin{equation*}
a_{x,\xi} (y,\eta) = e^{ \frac{i}{2} \langle x, \xi \rangle + i \left( \langle y, \xi \rangle - \langle x, \eta \rangle \right)}, \quad (y,\eta) \in \rr {2d}.
\end{equation*}
Using \eqref{eq:expestimate0}
we may estimate
\begin{align*}
\left| \partial_y^\alpha \partial_\eta^\beta a_{x,\xi} (y,\eta) \right|
& = |\xi^\alpha x^\beta|
\leqslant e^{s d h^{- \frac{1}{s}} + t d h^{- \frac{1}{t}}} ( |(x,\xi)| h)^{|\alpha+\beta|} \alpha!^{s} \beta!^{t} \\
& = C_{t,s,h,d} ( |(x,\xi)| h)^{|\alpha+\beta|} \alpha!^{s} \beta!^{t}
\end{align*}
for any $h > 0$ and $\alpha, \beta \in \nn d$.
This implies that $a_{x,\xi} \in {G(X|Y)}amma_{t,s}^{s,t; 0}(\rr {2d})$.
Thus we may apply Theorem {\rm Re}f{thm:microlocalWFs} which gives
\begin{equation*}
\mathrm{WF}^{t,s}( \Pi(z) u ) \subseteq \mathrm{WF}^{t,s}(u).
\end{equation*}
The opposite inclusion follows from $u = e^{- i \langle x, \xi \rangle} \Pi(-(x,\xi)) \Pi(x,\xi) u$.
\end{proof}
\section{Global wave front sets of polynomials and generalizations}\langlebel{sec:polynomials}
\begin{prop}\langlebel{prop:WFstelementary}
If $s,t > 0$ satisfy $s + t > 1$ then:
\begin{enumerate}[\rm (i)]
\item for any $x \in \rr d$ and any $\alpha \in \nn d$
\begin{equation}\langlebel{eq:diracWFst}
\mathrm{WF}g ( \pd \alpha \delta_x ) = \mathrm{WF}^{t,s}( \pd \alpha \delta_x ) = \{ 0 \} \times ( \rr d \setminus 0 );
\end{equation}
\item for any $\alpha \in \nn d$
\begin{equation}\langlebel{eq:monomialWFst}
\mathrm{WF}g ( x^\alpha ) = \mathrm{WF}^{t,s}( x^\alpha ) = ( \rr d \setminus 0 ) \times \{ 0 \};
\end{equation}
\item for any $\xi \in \rr d$
\begin{equation}\langlebel{eq:planewaveWFst}
\mathrm{WF}g (e^{i \langle \, \cdot \, t, \xi \rangle} ) = \mathrm{WF}^{t,s}( e^{i \langle \, \cdot \, t, \xi \rangle} ) = ( \rr d \setminus 0 ) \times \{ 0 \}.
\end{equation}
\end{enumerate}
\end{prop}
Proposition {\rm Re}f{prop:WFstelementary} follows from the arguments in Section {\rm Re}f{sec:GelfandShilovWF}, the details of the proof are left to the reader.
We fix attention on the following generalizations
of Proposition {\rm Re}f{prop:WFstelementary}.
Consider a polynomial on $\rr d$
\begin{equation}\langlebel{eq:polynomial1}
p(x) = \sum_{\alpha \in \nn d, \, |\alpha| \leqslant m} c _\alpha x^{\alpha}, \quad x \in \rr d,
\end{equation}
with $c_\alpha \in \mathbf C$ and $m \in \mathbf N \setminus 0$.
\begin{prop}\langlebel{prop:polynomial}
Suppose $s,t > 0$ satisfy $s + t > 1$,
let $p$ be the polynomial \eqref{eq:polynomial1}
and define
\begin{equation*}
u = \sum_{\alpha \in \nn d, \, |\alpha| \leqslant m} c _\alpha D^\alpha \delta_0 \in \mathscr{S}'(\rr d).
\end{equation*}
Then
\begin{equation}\langlebel{eq:diracpolynomial1}
\mathrm{WF}g( u )
= \mathrm{WF}^{t,s}( u )
= \{ 0 \} \times ( \rr d \setminus 0 )
\end{equation}
and
\begin{equation}\langlebel{eq:polynomial2}
\mathrm{WF}g( p )
= \mathrm{WF}^{t,s}( p )
= ( \rr d \setminus 0 ) \times \{ 0 \}.
\end{equation}
\end{prop}
\begin{proof}
Fourier transformation gives $\widehat u = (2 \pi)^{- \frac{d}{2} } p$ so \eqref{eq:polynomial2} is a consequence of \eqref{eq:diracpolynomial1} and the Fourier invariances \eqref{eq:metaplecticWFG} and Proposition {\rm Re}f{prop:WFstsymplectic} (i).
Thus it suffices to show \eqref{eq:diracpolynomial1}.
From Proposition {\rm Re}f{prop:WFstelementary} (i) and \eqref{eq:WFstsublinear} we obtain
\begin{equation*}
\mathrm{WF}^{t,s}( u )
\subseteq \{ 0 \} \times ( \rr d \setminus 0 )
\end{equation*}
and \eqref{eq:GaborGSinclusion} gives
\begin{equation*}
\mathrm{WF}g( u )
\subseteq \mathrm{WF}^{v,v}( u )
\subseteq \{ 0 \} \times ( \rr d \setminus 0 )
\end{equation*}
where $v = \max(t,s) > \frac12$.
Hence it suffices to show
\begin{equation}\langlebel{eq:WFstdiracsum}
\{ 0 \} \times ( \rr d \setminus 0 ) \subseteq \mathrm{WF}^{t,s}( u )
\end{equation}
and
\begin{equation}\langlebel{eq:WFgdiracsum}
\{ 0 \} \times ( \rr d \setminus 0 ) \subseteq \mathrm{WF}g ( u ).
\end{equation}
Let $\varphi \in \Sigma_t^s (\rr d) \setminus 0$ satisfy $\varphi(0) \neq 0$.
We have
\begin{align*}
& V_\varphi u (0,\xi) \\
& = (2 \pi)^{- \frac{d}{2}} \sum_{|\alpha| \leqslant m} c _\alpha \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} \xi^\beta \overline{D^{\alpha-\beta} \varphi(0)} \\
& = (2 \pi)^{- \frac{d}{2}} \left( \sum_{|\alpha| = m} c _\alpha \xi^\alpha \overline{\varphi (0)}
+ \sum_{|\alpha| = m} c _\alpha \sum_{\beta < \alpha} \binom{\alpha}{\beta} \xi^\beta \overline{ D^{\alpha-\beta} \varphi(0)}
+ \sum_{|\alpha| < m} c _\alpha \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} \xi^\beta \overline{ D^{\alpha-\beta} \varphi(0)}
\right).
\end{align*}
Define the principal part of $p$ as
\begin{equation*}
p_m(x) = \sum_{|\alpha| = m} c _\alpha x^{\alpha}.
\end{equation*}
If $\xi \in \rr d \setminus 0$, $p_m(\xi) \neq 0$ and $\langlembda > 0$ then
\begin{align*}
& (2 \pi)^{\frac{d}{2}} V_\varphi u (0,\langlembda^s \xi) \\
& = \langlembda^{s m} p_m(\xi) \overline{\varphi (0)}
+ \underbrace{\sum_{|\alpha| = m} c _\alpha \sum_{\beta < \alpha} \binom{\alpha}{\beta} \langlembda^{s |\beta|} \xi^\beta \overline{ D^{\alpha-\beta} \varphi(0)}
+ \sum_{|\alpha| < m} c _\alpha \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} \langlembda^{s |\beta|} \xi^\beta \overline{ D^{\alpha-\beta} \varphi(0)}}_{:=R}.
\end{align*}
Since $R$ contains terms $\langlembda^{s k}$ where $k < m$
this implies that $(0,\xi) \in \mathrm{WF}g( u)$ and $(0,\xi) \in \mathrm{WF}^{t,s}( u)$.
If instead $\xi \in \rr d \setminus 0$ and $p_m(\xi) = 0$, then for any $\varepsilon > 0$ the ball $\operatorname{B}_\varepsilon(\xi)$ contains $\eta \in \rr d \setminus 0$ such that $p_m(\eta) \neq 0$.
In fact $p_m$ extends to an entire function on $\cc d$ whose zeros are isolated.
From the argument above it follows that $(0,\eta) \in \mathrm{WF}g( u)$ and $(0,\eta) \in \mathrm{WF}^{t,s}( u)$.
It follows that
$(0,\xi) \in \mathrm{WF}g( u)$ and $(0,\xi) \in \mathrm{WF}^{t,s}( u)$.
We have now shown \eqref{eq:WFstdiracsum} and \eqref{eq:WFgdiracsum}.
\end{proof}
In order to generalize Propositions {\rm Re}f{prop:WFstelementary} and {\rm Re}f{prop:polynomial} we would like to study series of the form
\begin{equation}\langlebel{eq:diracseries1}
u = \sum_{\alpha \in \nn d} c _\alpha D^\alpha \delta_0
\end{equation}
containing infinitely many nonzero terms $c_\alpha \in \mathbf C$,
and the corresponding power series
\begin{equation}\langlebel{eq:maclaurinseries1}
f(x) = \sum_{\alpha \in \nn d} c _\alpha x^\alpha,
\end{equation}
under suitable hypotheses on the coefficients $c_\alpha \in \mathbf C$.
First we note that $u \mathbf Ntin \mathscr{S}'(\rr d)$.
In fact we have for $\varphi \in \mathscr{S}(\rr d)$
\begin{equation}\langlebel{eq:series1}
(u,\varphi) = \sum_{\alpha \in \nn d} c _\alpha i^{|\alpha|} \pd \alpha \overline{\varphi (0)}
\end{equation}
and it is known that a smooth function $\varphi$ may have arbitrary growth of $\alpha \mapsto \pd \alpha \varphi (0)$
(Borel's lemma \cite[Theorem~1.2.6]{Hormander0}).
Thus the sum \eqref{eq:series1} is not guaranteed to converge for $\varphi \in \mathscr{S}(\rr d)$, unless the series is finite.
The series \eqref{eq:diracseries1} does not converge in $\mathscr{S}'(\rr d)$, and $u \mathbf Ntin \mathscr{S}'(\rr d)$ if the series is infinite.
For the same reason \eqref{eq:maclaurinseries1} does not converge in $\mathscr{S}'(\rr d)$, and $f \mathbf Ntin \mathscr{S}'(\rr d)$.
(Note that $\widehat u = (2 \pi)^{-\frac{d}{2}} f$ when the series is finite.)
Nevertheless it is possible to state conditions on $\{ c_\alpha \}_{\alpha \in \nn d}$ that are sufficient for $u \in (\Sigma_t^s)'(\rr d)$.
Suppose $s > 0$ and
\begin{equation}\langlebel{eq:seriescondition1}
\sum_{\alpha \in \nn d} |c _\alpha | \, r^{|\alpha|} \alpha!^s < \infty
\end{equation}
for some $r > 0$.
Then for $t > 0$ such that $s + t > 1$, and $\varphi \in \Sigma_t^s(\rr d)$, we have
\begin{align*}
|(u,\varphi)| \leqslant \sum_{\alpha \in \nn d} |c _\alpha | \, |\pd \alpha \varphi (0)|
\leqslant \nm \varphi{\mathcal S_{t,h}^s} \sum_{\alpha \in \nn d} |c _\alpha | \alpha!^s h^{|\alpha|}
\lesssim \nm \varphi{\mathcal S_{t,h}^s}
\end{align*}
provided $h \leqslant r$. Thus the series \eqref{eq:diracseries1} converges in $(\Sigma_t^s)'(\rr d)$ and $u \in (\Sigma_t^s)'(\rr d)$.
We may also conclude that \eqref{eq:maclaurinseries1} converges in $(\Sigma_s^t)'(\rr d)$, $f \in (\Sigma_s^t)'(\rr d)$, and
the Fourier transform acts termwise as $\widehat u = (2 \pi)^{-\frac{d}{2}} f \in (\Sigma_s^t)'(\rr d)$.
We may distinguish two rather different situations
under condition \eqref{eq:seriescondition1}.
Namely, if $s > 1$ then $u \in \mathscr{E}_s'(\rr d)$, with support in the origin, cf. \cite[Example~1.5.3 and 1.6.5]{Rodino1}.
The absolutely convergent series $f$ satisfies
\begin{equation*}
|f(x)| \lesssim e^{a |x|^{\frac1s}}, \quad x \in \rr d,
\end{equation*}
for some $a > 0$ in $\rr d$, cf. \eqref{eq:PWSGevrey},
and more precise bounds in $\cc d$ can be deduced from the Paley--Wiener--Schwartz theorem in $\mathscr{E}_s'(\rr d)$,
cf. \cite[Theorem~1.6.7]{Rodino1}, \cite{Komatsu1,Sobak1}.
If instead $0 < s \leqslant 1$
the series \eqref{eq:maclaurinseries1} also converges absolutely for any $x \in \rr d$,
and is an entire function.
In fact
\begin{align*}
\sum_{\alpha \in \nn d} \left| c _\alpha x^\alpha \right|
& \leqslant \sum_{\alpha \in \nn d} | c _\alpha| \, r^{|\alpha|} \alpha!^s \left(\frac{ (r^{-1} |x|)^{\frac{|\alpha|}{s}} }{\alpha!} \right)^s \\
& \leqslant \sum_{\alpha \in \nn d} | c _\alpha| \, r^{|\alpha|} \alpha!^s \left(\frac{ \left( d (r^{-1} |x|)^{\frac{1}{s}} \right)^{|\alpha|}}{|\alpha|!} \right)^s \\
& \leqslant e^{ s d r^{-\frac1s} |x|^{\frac1s} } \sum_{\alpha \in \nn d} | c _\alpha| \, r^{|\alpha|} \alpha!^s \\
& \lesssim e^{ s d r^{-\frac1s} |x|^{\frac1s} }
\end{align*}
which also reveals the growth bound
\begin{equation*}
|f(x)| \lesssim e^{ s d r^{-\frac1s} |x|^{\frac1s} }, \quad x \in \rr d.
\end{equation*}
But the definition of support of $u \in (\Sigma_t^s)'(\rr d)$ breaks down if $s \leqslant 1$.
Consider as an example for $z \in \cc d$
\begin{equation*}
u = \sum_{\alpha \in \nn d} \frac{(-\overline{z})^\alpha}{\alpha!} D^\alpha \delta_0.
\end{equation*}
Condition \eqref{eq:seriescondition1} is satisfied if $r < |z|^{-1}$, and thus $u \in (\Sigma_t^s)'(\rr d)$.
The corresponding test functions $\varphi \in \Sigma_t^s(\rr d)$ extend to entire functions on $\cc d$.
From Maclaurin expansion we have
\begin{equation*}
(u, \varphi)
= \sum_{\alpha \in \nn d} \frac{(-\overline{z})^\alpha}{\alpha!} \overline{D^\alpha \varphi (0)}
= \overline{ \sum_{\alpha \in \nn d} \frac{(iz)^\alpha}{\alpha!} \pd \alpha \varphi (0)}
= \overline{\varphi(iz)}.
\end{equation*}
Thus $u$ may be regarded as a delta distribution at the point $i z \in \cc d$.
In the following result we require that \eqref{eq:seriescondition1} holds for all $r > 0$
which precludes the preceding example.
\begin{prop}\langlebel{prop:diracseries1}
Let $s,t > 0$ satisfy $s + t > 1$,
suppose that \eqref{eq:seriescondition1} holds for all $r > 0$,
and define $u \in (\Sigma_t^s)'(\rr d)$ and $f \in (\Sigma_s^t)'(\rr d)$ by \eqref{eq:diracseries1} and \eqref{eq:maclaurinseries1} respectively.
Then
\begin{equation}\langlebel{eq:diracseries2}
\mathrm{WF}^{t,s}( u )
\subseteq \{ 0 \} \times ( \rr d \setminus 0 )
\end{equation}
and
\begin{equation}\langlebel{eq:taylorseries2}
\mathrm{WF}^{s,t}( f )
\subseteq ( \rr d \setminus 0 ) \times \{ 0 \}.
\end{equation}
\end{prop}
\begin{proof}
Since $\widehat u = (2 \pi)^{-\frac{d}{2}} f \in (\Sigma_s^t)'(\rr d)$ it again suffices to show \eqref{eq:diracseries2} by the Fourier invariance Proposition {\rm Re}f{prop:WFstsymplectic} (i).
If $s > 1$ the result follows from Proposition {\rm Re}f{prop:WFsfreqaxis}, cf. \eqref{eq:subinclusion1}.
Consider the general case $s > 0$.
Let $\varphi \in \Sigma_t^s(\rr d) \setminus 0$,
let $(x_0,\xi_0) \in T^* \rr d \setminus 0$ satisfy $x_0 \neq 0$, and let $(x_0,\xi_0) \in U$ where $U \subseteq \rr {2d}$ is open and satisfies
\begin{equation*}
\sup_{(x,\xi) \in U} |\xi| \leqslant |\xi_0| + 1 := a, \quad \inf_{(x,\xi) \in U} |x| \geqslant \varepsilon > 0.
\end{equation*}
If $(x,\xi) \in U$ then we obtain,
using the estimates \eqref{eq:Sstderivativeestimate1},
for any $h, r, \langlembda > 0$
\begin{align*}
(2 \pi)^{\frac{d}{2}} |V_\varphi u( \langlembda^t x, \langlembda^s \xi )|
& = \left| \sum_{\alpha \in \nn d} c _\alpha \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} \langlembda^{s |\beta|} \xi^\beta \overline{D^{\alpha-\beta} \varphi(- \langlembda^t x)} \right| \\
& \leqslant C_{r,h} \sum_{\alpha \in \nn d} | c _\alpha | \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} \langlembda^{s |\beta|} |\xi|^{|\beta|}
h^{|\alpha-\beta|} (\alpha-\beta)!^s e^{- 2 r \varepsilon^{-\frac1t} \langlembda |x|^{\frac1t}} \\
& \leqslant C_{r,h} e^{- 2 r \langlembda} \sum_{\alpha \in \nn d} | c _\alpha | \, h^{|\alpha|} \alpha!^s \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} a^{|\beta|}
\left( \frac{ \left( \langlembda h^{-\frac1s} \right)^{|\beta|}}{\beta!} \right)^{s} \\
& \leqslant C_{r,h} e^{- 2 r \langlembda} \sum_{\alpha \in \nn d} | c _\alpha | h^{|\alpha|} \alpha!^s \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} a^{|\beta|}
\left( \frac{ \left( d \langlembda h^{-\frac1s} \right)^{|\beta|}}{|\beta|!} \right)^{s} \\
& \leqslant C_{r,h} e^{- 2 \langlembda r + \langlembda s d h^{-\frac1s} } \sum_{\alpha \in \nn d} | c _\alpha | h^{|\alpha|} \alpha!^s \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} a^{|\beta|} \\
& = C_{r,h} e^{- 2 \langlembda r + \langlembda s d h^{-\frac1s} } \sum_{\alpha \in \nn d} | c _\alpha | ( (a+1) h) ^{|\alpha|} \alpha!^s.
\end{align*}
If we pick $h = r^{-s} s^{s} d^{s}$ and use \eqref{eq:seriescondition1} then
\begin{align*}
|V_\varphi u( \langlembda^t x, \langlembda^s \xi )|
& \leqslant C_{r} e^{- \langlembda r } \sum_{\alpha \in \nn d} | c _\alpha | ( (a+1) h) ^{|\alpha|} \alpha!^s \\
& \leqslant C_{r}' e^{- \langlembda r }
\end{align*}
for a new constant $C_r' > 0$. Since $(x,\xi) \in U$ and $r > 0$ are arbitrary we have shown $(x_0,\xi_0) \mathbf Ntin \mathrm{WF}^{t,s} (u)$ which proves \eqref{eq:diracseries2}.
\end{proof}
\begin{rem}\langlebel{rem:suffequality1}
In dimension $d=1$ we can state conditions that are sufficient for equality in \eqref{eq:diracseries2} and
\eqref{eq:taylorseries2}.
In fact suppose
\begin{equation*}
u = \sum_{k = 0}^\infty c _k D^k \delta_0
\end{equation*}
where \eqref{eq:seriescondition1} is satisfied for all $r > 0$, and
either $c_{2k} = 0$ for all $k \geqslant 0$ or $c_{2k+1} = 0$ for all $k \geqslant 0$.
Then for $\varphi \in \Sigma_t^s(\mathbf R)$
\begin{equation*}
(\check u, \varphi)
= \sum_{k = 0}^\infty c _k ( D^k \delta_0, \check \varphi)
= \sum_{k = 0}^\infty c _k (-1)^k ( D^k \delta_0, \varphi)
= \pm (u,\varphi)
\end{equation*}
which means that $u$ is either even or odd.
By \eqref{eq:evensteven1} we have $\mathrm{WF}^{t,s}(u) = - \mathrm{WF}^{t,s}(u)$, and since $\mathrm{WF}^{t,s}(u) \neq \emptyset$ due to $u \mathbf Ntin \Sigma_t^s(\mathbf R)$, we must have
\begin{equation*}
\mathrm{WF}^{t,s}( u )
= \{ 0 \} \times ( \mathbf R \setminus 0 ).
\end{equation*}
Equality in \eqref{eq:taylorseries2} follows.
\end{rem}
We can also get equalities for $\mathrm{WF}^{t,s}(u)$ and $\mathrm{WF}^{s,t}(f)$ in terms of the subset
$V_s(u)$ defined in \eqref{eq:frequencydecay0}.
Using $\widehat u = (2 \pi)^{-\frac{d}{2}} f \in (\Sigma_s^t)'(\rr d)$ we may rephrase \eqref{eq:frequencydecay0} as follows:
$x_0 \in \rr d \setminus 0$ satisfies $x_0 \mathbf Ntin V_s(u)$ if there exists an open set $U \subseteq \rr d \setminus 0$ such that $x_0 \in U$ and
\begin{equation}\langlebel{eq:Scomplement}
\sup_{x \in U, \ \langlembda > 0} e^{ r \langlembda } |f( \langlembda^s x)| < \infty \quad \forall r > 0.
\end{equation}
Thus $V_s(u)$ consists of the directions in $\rr d \setminus 0$ in which $\widehat u( x)$ does not decay like $e^{- r |x|^{\frac1s}}$ for all $r > 0$.
Note that we assume $s > 1$ in the following result. This depends on the fact that we need a window function with certain properties.
\begin{prop}\langlebel{prop:diracseries2}
Let $s > 1$ and $t > 0$.
Suppose that \eqref{eq:seriescondition1} holds for all $r > 0$
and define $u \in (\Sigma_t^s)'(\rr d)$ and $f \in (\Sigma_s^t)'(\rr d)$ by \eqref{eq:diracseries1} and \eqref{eq:maclaurinseries1} respectively.
Then
\begin{equation}\langlebel{eq:diracseries3}
\mathrm{WF}^{t,s}( u )
= \{ 0 \} \times V_s(u)
\end{equation}
and
\begin{equation}\langlebel{eq:taylorseries3}
\mathrm{WF}^{s,t}( f )
= V_s(u) \times \{ 0 \}.
\end{equation}
\end{prop}
\begin{proof}
Again Fourier transformation gives $\widehat u = (2 \pi)^{-\frac{d}{2}} f$ so again by Proposition {\rm Re}f{prop:WFstsymplectic} (i) it suffices to show \eqref{eq:diracseries3}.
As for the inclusion
\begin{equation*}
\mathrm{WF}^{t,s}( u ) \subseteq \{ 0 \} \times V_s(u)
\end{equation*}
it is a direct consequence of Proposition {\rm Re}f{prop:WFsfreqaxis} since $V_s(u) = \pi_2 \mathrm{WF}_s(u)$.
The opposite inclusion cannot be deduced from Proposition {\rm Re}f{prop:WFGevreyWFst} and Corollary {\rm Re}f{cor:WFsfreqaxis},
because of the restrictive assumption $t \geqslant s$ there.
Instead we argue as follows.
We know from Proposition {\rm Re}f{prop:diracseries1} that
$\mathrm{WF}^{t,s}( u ) \subseteq \{ 0 \} \times ( \rr d \setminus 0 )$.
Assume $\xi_0 \in \rr d \setminus 0$ and $(0,\xi_0) \mathbf Ntin \mathrm{WF}^{t,s}( u )$.
Let $\varphi \in \Sigma_t^s (\rr d)$ satisfy $\varphi(0) = 1$ and $\pd \alpha \varphi(0) = 0$ for all $\alpha \neq 0$,
which is possible since $s > 1$.
If we fix $x = 0$ in \eqref{eq:notinWFGFst1} and assume there $U = A \times B \subseteq \rr {2d}$ where $A \subseteq \rr d$ is a neighborhood of $0$ and $B \subseteq \rr d$ is a neighborhood of $\xi_0$, we obtain
\begin{equation*}
\sup_{\langlembda > 0, \ \xi \in B}
e^{r \langlembda} |V_\varphi u(0, \langlembda^s \xi)| < + \infty \quad \forall r > 0.
\end{equation*}
Since
\begin{align*}
V_\varphi u (0,\xi)
& = (2 \pi)^{- \frac{d}{2}} \sum_{\alpha \in \nn d} c_\alpha \sum_{\beta \leqslant \alpha} \binom{\alpha}{\beta} \xi^\beta \overline{ D^{\alpha-\beta} \varphi (0)} \\
& = (2 \pi)^{- \frac{d}{2}} \sum_{\alpha \in \nn d} c_\alpha \xi^\alpha = (2 \pi)^{- \frac{d}{2}} f(\xi),
\end{align*}
\eqref{eq:Scomplement} is satisfied with $U = B$
and we conclude $\xi_0 \mathbf Ntin V_s(u)$.
Thus $\{ 0 \} \times V_s(u) \subseteq \mathrm{WF}^{t,s}( u )$.
\end{proof}
\section{The $t,s$-Gelfand--Shilov wave front set of an exponential function}\langlebel{sec:exponential}
For $z \in \cc d$ fixed consider the exponential function $\rr d \ni x \mapsto a(x) = e^{\langle x, z \rangle}$.
If $s > 0$, $0 < t \leqslant 1$, $s + t > 1$ and $\varphi \in \Sigma_t^s(\rr d)$ then by \eqref{eq:Sstderivativeestimate1}
we have for some $h > 0$
\begin{equation*}
\left| \int_{\rr d} a(x) \overline{ \varphi (x)} \mathrm {d} x \right|
\leqslant \| \varphi \|_{\mathcal S_{t,h}^s} \int_{\rr d} e^{|z| |x| - (|z|+1) |x|^{\frac{1}{t}}} \mathrm {d} x
\lesssim \| \varphi \|_{\mathcal S_{t,h}^s}
\end{equation*}
which implies $a \in (\Sigma_t^s)'(\rr d)$.
We consider $a$ as the multiplier operator $T f = a f$.
Then $T = a^w(x,D)$ with $a(x,\xi) = a(x) = e^{\langle x, z \rangle}$.
From \eqref{eq:expestimate0} for any $h > 0$
we obtain for any $\alpha \in \nn d$
\begin{equation*}
|\pd \alpha a(x)| = |z^\alpha | e^{{\rm Re} \langle x,z \rangle}
\leqslant C_{s,d,h} ( h |z|)^{|\alpha|} \alpha!^s e^{|{\rm Re} z| |x|}.
\end{equation*}
This means that $a \in {G(X|Y)}amma_{t,s}^{s,t; 0} (\rr {2d})$ for all $0 < t \leqslant 1$, $s > 0$, $s + t > 1$.
Theorem {\rm Re}f{thm:microlocalWFs}
combined with Proposition {\rm Re}f{prop:WFstelementary}
now gives
\begin{equation}\langlebel{eq:exponentialWF1}
\mathrm{WF}^{t,s}( e^{\langle \, \cdot \, t , z \rangle} ) \subseteq ( \rr d \setminus 0 ) \times \{ 0 \}
\end{equation}
for any $z \in \cc d$.
By considering the operator
$T^{-1}$ with symbol $e^{-\langle x, z \rangle} \in {G(X|Y)}amma_{t,s}^{s,t; 0} (\rr {2d})$,
so that
$T^{-1} ( e^{ \langle \, \cdot \, t, z \rangle} ) = 1$,
we deduce the opposite inclusion.
We have obtained:
\begin{prop}\langlebel{prop:exponentialWFs}
If $0 < t \leqslant 1$, $s > 0$, $s + t > 1$, and $z \in \cc d$ then
\begin{equation*}
\mathrm{WF}^{t,s} ( e^{\langle \, \cdot \, t ,z \rangle} ) = ( \rr d \setminus 0 ) \times \{ 0 \}.
\end{equation*}
\end{prop}
\begin{cor}\langlebel{cor:diracseriesexp}
If $0 < t \leqslant 1$, $s > 0$, $s + t > 1$,
$z \in \cc d$ and
\begin{equation*}
u = \sum_{\alpha \in \nn d} \frac{z^\alpha}{\alpha!} (-D)^\alpha \delta_0
\end{equation*}
then $u \in (\Sigma_s^t)'(\rr d)$ and
\begin{equation*}
\mathrm{WF}^{s,t} ( u ) = \{ 0 \} \times ( \rr d \setminus 0 ) .
\end{equation*}
\end{cor}
\begin{proof}
We have the Maclaurin series
\begin{equation}\langlebel{eq:maclaurinseries0}
f(x) = e^{\langle x ,z \rangle} = \sum_{\alpha \in \nn d} \frac{z^\alpha}{\alpha!} x^\alpha, \quad x \in \rr d,
\end{equation}
which converges in $(\Sigma_t^s)'(\rr d)$ to $f \in (\Sigma_t^s)'(\rr d)$.
We apply the Fourier transform termwise with convergence in $(\Sigma_s^t)'(\rr d)$ which gives
\begin{align*}
\widehat f & = \sum_{\alpha \in \nn d} \frac{z^\alpha}{\alpha!} \mathscr{F} (x^\alpha) \\
& = (2 \pi)^{\frac{d}{2}} \sum_{\alpha \in \nn d} \frac{z^\alpha}{\alpha!} (-D)^\alpha \delta_0
= (2 \pi)^{\frac{d}{2}} u \in (\Sigma_s^t)'(\rr d).
\end{align*}
Proposition {\rm Re}f{prop:WFstsymplectic} (i)
and Proposition {\rm Re}f{prop:exponentialWFs}
now give
\begin{equation*}
\mathrm{WF}^{s,t}(u) = \mathcal{J} \mathrm{WF}^{t,s}(f) = \{ 0 \} \times ( \rr d \setminus 0 ) .
\end{equation*}
\end{proof}
\begin{rem}\langlebel{rem:fourierexponential}
Note that $u$ may be considered as the Dirac distribution
\begin{equation*}
(u,\varphi) = \overline{\varphi( i \overline z)}, \quad \varphi \in \Sigma_s^t (\rr d),
\end{equation*}
which makes sense since $\varphi$ extends to an entire function on $\cc d$ as $t \leqslant 1$.
If $z = i \xi$ with $\xi \in \rr d$ we recapture the well known identity
\begin{equation*}
\mathscr{F} ( e^{i \langle \, \cdot \, t, \xi \rangle} ) = \widehat f = (2 \pi)^{\frac{d}{2}} u = (2 \pi)^{\frac{d}{2}} \delta_{\xi} \in \mathscr{D}'(\rr d).
\end{equation*}
\end{rem}
\begin{rem}\langlebel{rem:exponential}
Proposition {\rm Re}f{prop:diracseries1} contains the ``$\subseteq$'' inclusion of
Proposition {\rm Re}f{prop:exponentialWFs} and Corollary {\rm Re}f{cor:diracseriesexp}, under the restriction $0 < t < 1$ (that is avoiding $t=1$), as a particular case.
In fact comparing \eqref{eq:maclaurinseries0} with \eqref{eq:maclaurinseries1} we can identify the Maclaurin coefficients for $f = e^{\langle \, \cdot \, t, z \rangle}$ where $z \in \cc d$. They are $c_\alpha = z^\alpha/\alpha!$.
If $0 < s < 1$ we have for any $r > 0$, and $0 < a < 1$
\begin{align*}
\sum_{\alpha \in \nn d} |c_\alpha| \, r^{|\alpha|} \alpha!^{s}
& \leqslant \sum_{\alpha \in \nn d} (|z| r)^{|\alpha|} \alpha!^{s-1}
= \sum_{\alpha \in \nn d} a^{|\alpha|} \left( \frac{(|z| r a^{-1})^{\frac{|\alpha|}{1-s}}}{\alpha!} \right)^{1-s} \\
& \leqslant \sum_{\alpha \in \nn d} a^{|\alpha|}
\left( \frac{ \left( d (|z| r a^{-1})^{\frac{1}{1-s}} \right)^{|\alpha|} }{|\alpha|!} \right)^{1-s} \\
& \leqslant (1 -a )^{-d} \exp \left( (1-s) d (|z| r a^{-1})^{\frac{1}{1-s}} \right).
\end{align*}
By Proposition {\rm Re}f{prop:diracseries1} we may conclude
\begin{equation*}
\mathrm{WF}^{s,t}( e^{\langle \, \cdot \, t, z \rangle} )
\subseteq ( \rr d \setminus 0 ) \times \{ 0 \}
\end{equation*}
and
\begin{equation*}
\mathrm{WF}^{t,s}( u )
\subseteq \{ 0 \} \times ( \rr d \setminus 0 )
\end{equation*}
where
\begin{equation*}
u = \sum_{\alpha \in \nn d} \frac{z^\alpha}{\alpha!} (- D)^\alpha \delta_0.
\end{equation*}
\end{rem}
By combining with the results of Section {\rm Re}f{sec:chirp} we finally consider in dimension $d = 1$
\begin{equation}\langlebel{eq:expchirp}
v (x) = e^{ z x + i c x^{m} }
\end{equation}
with $z \in \mathbf C$, $c \in \mathbf R \setminus 0$, $m \in \mathbf N$, $m \geqslant 2$.
Then $v \in (\Sigma_t^s)'(\rr d)$ if $0 < t \leqslant 1$, $s > 0$, $s + t > 1$.
\begin{prop}\langlebel{prop:expchirp}
If $\frac1{m-1} < t \leqslant 1$ then for $v$ defined by \eqref{eq:expchirp} we have
\begin{equation}\langlebel{eq:WFexpchirp}
\mathrm{WF}^{t,t(m-1)} (v) = \{ (x, c m x^{m-1}) \in \rr 2, \ x \neq 0 \}.
\end{equation}
\end{prop}
\begin{proof}
As before define $T = a^w(x,D)$ with $a(x,\xi) = e^{zx}$ regarded as a symbol in
${G(X|Y)}amma_{t,s}^{s,t;0}(\rr 2)$, for any $s > 0$ such that $s + t > 1$ and $t \leqslant 1$.
Set
\begin{equation*}
w (x) = e^{i c x^{m} } \in \mathscr{S}'(\mathbf R) \subseteq (\Sigma_t^s)'(\mathbf R).
\end{equation*}
We have $v = T w$.
From Theorem {\rm Re}f{thm:microlocalWFs} we deduce
\begin{equation*}
\mathrm{WF}^{t,s}( v ) \subseteq \mathrm{WF}^{t,s}(w).
\end{equation*}
By considering the operator $T^{-1}$ we deduce the opposite inclusion,
hence $\mathrm{WF}^{t,s}( v ) = \mathrm{WF}^{t,s}(w)$.
Under the assumption $t > \frac1{m-1}$ we may apply
Theorem {\rm Re}f{thm:chirpWFst}, and obtain \eqref{eq:WFexpchirp}.
\end{proof}
\section*{Acknowledgment}
Work partially supported by the MIUR project ``Dipartimenti di Eccellenza 2018-2022'' (CUP E11G18000350001).
\end{document} |
\mathsf {b}egin{document}
\title[phantom stable category of $n$-Frobenius categories]
{phantom stable category of $n$-Frobenius categories}
\mathsf {a}uthor[Bahlekeh, Fotouhi, Salarian and Sartipzadeh]{Abdolnaser Bahlekeh, Fahimeh Sadat Fotouhi, Shokrollah Salarian and Atousa Sartipzadeh}
\operatorname{\mathsf{add}}ress{Department of Mathematics, Gonbade-Kavous University, Postal Code:4971799151, Gonbad-e-Kavous, Iran}
\email{[email protected]}
\operatorname{\mathsf{add}}ress{ School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O.Box: 19395-5746, Tehran, Iran}\email{[email protected]}
\operatorname{\mathsf{add}}ress{Department of Pure Mathematics, Faculty of Mathematics and Statistics,
University of Isfahan, P.O.Box: 81746-73441, Isfahan,
Iran and \\ School of Mathematics, Institute for Research in Fundamental Science (IPM), P.O.Box: 19395-5746, Tehran, Iran}
\email{[email protected]}
\operatorname{\mathsf{add}}ress{Department of Pure Mathematics, Faculty of Mathematics and Statistics,
University of Isfahan, P.O.Box: 81746-73441, Isfahan, Iran}
\email{[email protected]}
\subjclass[2010]{18E10, 18G15, 18G65, 14F08}
\mathsf {K}eywords{$n$-Frobenius category; phantom stable category; $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism; semi-separated noetherian scheme}
\mathsf {b}egin{abstract}
Let $n$ be a non-negative integer. An exact category $\mathfrak{m}athscr{C} $ is said to be an $n$-Frobenius category, provided that
it has enough $n$-projectives and $n$-injectives and the $n$-projectives coincide with the $n$-injectives. It is proved that any abelian category with non-zero $n$-projective objects, admits a non-trivial $n$-Frobenius subcategory. In particular, we explore several examples of $n$-Frobenius categories. Also, as a far reaching generalization of the stabilization of a Frobenius category, we define and study phantom stable category of an $n$-Frobenius category $\mathfrak{m}athscr{C} $. Precisely,
assume that $\mathcal P\subseteq\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }$ is the subfunctor consisting of all conflations of length $n$ factoring through $n$-projective objects. A couple $(\mathfrak{m}athscr{C} _{\mathcal P}, T)$, where $\mathfrak{m}athscr{C} _{\mathcal P}$ is an additive category and $T$ is a covariant additive functor from $\mathfrak{m}athscr{C} $ to $\mathfrak{m}athscr{C} _{\mathcal P}$, is a phantom stable category of $\mathfrak{m}athscr{C} $, provided that for any morphism $f$ in $\mathfrak{m}athscr{C} $, $T(f)=0$, whenever $f$ is an $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism and $T(f)$ is an isomorphism in $\mathfrak{m}athscr{C} _{\mathcal P}$, if $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$, and $T$ has the universal property with respect to these conditions. The main focus of this paper is to show that the phantom stable category of an $n$-Frobenius category always exists.
Some properties of phantom stable categories that reveal the efficiency of these categories are studied.
\end{abstract}
\mathfrak{m}aketitle
\tableofcontents
\section{Introduction}
Assume that $\mathcal A$ is an abelian category and $\mathfrak{m}athscr{C} $ a full additive subcategory of $\mathcal A$ which is closed under extensions. It is known that, the exact strucure of $\mathcal A$ is inherited by $\mathfrak{m}athscr{C} $; see \operatorname{\underline{\mathscr{C}}}ite[Lemma 10.20]{buh}. {Assume that $n$ is a non-negative integer.} An extension which is obtained by splicing of $n$ conflations in $\mathfrak{m}athscr{C} $, will be called a {\em conflation of length $n$}. For arbitrary objects $A,B\in\mathfrak{m}athscr{C} $, the equivalence classes of all conflations of length $n$ in $\mathfrak{m}athscr{C} $, $\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(A, B)$, forms an abelian group with respect to the Baer sum operation. {In the case $n=0$, we set $\operatorname{{\mathsf{Ext}}}^0_{\mathfrak{m}athscr{C} }(A, B):=\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(A, B)$.} Moreover, the notion of $n$-projective and $n$-injective objects in $\mathfrak{m}athscr{C} $, are defined in terms of the vanishing of $\operatorname{{\mathsf{Ext}}}^{n+1}$ functor. Now we call $\mathfrak{m}athscr{C} $ an {\it $n$-Frobenius category}, provided that it has enough $n$-projectives and $n$-injectives and the $n$-projectives coincide with the $n$-injectives. Assume that $\mathfrak{m}athscr{C} $ is an $n$-Frobenius category. Then, for any $k\mathbf geq 1$, a given object $N\in\mathfrak{m}athscr{C} $ fits into conflations of length $k$, say $\mathsf{\Omega}^kN\mathbf hookrightarrowghtarrow P_{k-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_0\mathbf hookrightarrowghtarrow N$, where $P_i$'s are $n$-projective, which will be called {\it unit conflations}. Also $\mathsf{\Omega}^kN$ is said to be a $k$-th syzygy of $N$. We set $\mathfrak{m}athsf{H}:=\mathsf {b}igcup_{M, N\in\mathfrak{m}athscr{C} }\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(M, N)$ and $\operatorname{{\mathsf{Ext}}}^n:=\mathsf {b}igcup_{M, N\in\mathfrak{m}athscr{C} }\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^nN)$, where $\mathsf{\Omega}^nN$ runs over all the $n$-th syzygies of $N$. For any $f,g\in\operatorname{\mathsf{H}}$, the pull-back and push-out of any conflation of length $n$ along $f$ and $g$ are again conflations of length $n$. These operations, which abbreviately are denoted by $\operatorname{{\mathsf{Ext}}}^nf$ and $g\operatorname{{\mathsf{Ext}}}^n$, respectively, induce an $\operatorname{\mathsf{H}}$-bimodule structure on $\operatorname{{\mathsf{Ext}}}^n$. We denote by $\mathcal P$ the subfunctor of $\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }$ consisting of all conflatons of the form $\operatorname{{\mathsf{Ext}}}^nf$, for some $f:M\mathbf hookrightarrowghtarrow P$ of $\operatorname{\mathsf{H}}$, where $P$ is an $n$-projective object of $\mathfrak{m}athscr{C} $. Particularly, Proposition \twoheadrightarrowf{equal} indicates that $\mathcal P$ is a submodule of $\operatorname{{\mathsf{Ext}}}^n$, and then, $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$ will be an $\operatorname{\mathsf{H}}$-bimodule. It is proved in Section 6 that if $f$ annihilates $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$ from the left, then it annihilates this module from the right and vice versa. Similarly, we stablish that an element $g\in\operatorname{\mathsf{H}}$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$ from the left and the right, simultaneously; see Corollaries \twoheadrightarrowf{lr} and \twoheadrightarrowf{qo}. Now consider two classes of morphisms in $\operatorname{\mathsf{H}}$, as follows:
\mathsf {b}egin{itemize}\item The class of all morphisms in $\operatorname{\mathsf{H}}$ annihilating $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$, that is called $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphisms. For the historical remark on phantom morphisms; see \twoheadrightarrowf{s100}. \item The class of all morphisms in $\operatorname{\mathsf{H}}$ acting as invertible on $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$, which will be called quasi-invertible morphisms.
\end{itemize}
By a {\it phantom stable category of $\mathfrak{m}athscr{C} $}, we mean an additive category $\mathfrak{m}athscr{C} _{\mathcal P}$, together with a covariant additive functor $T:\mathfrak{m}athscr{C} \mathsf {L}ongrightarrow\mathfrak{m}athscr{C} _{\mathcal P}$ such that:\\ (1) $T(s)$ is an isomorphism in $\mathfrak{m}athscr{C} _{\mathcal P}$, for any quasi-invertible morphism $s$. \\(2) For any $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism $\operatorname{\mathsf{V}}arphi$, $T(\operatorname{\mathsf{V}}arphi)=0$ in $\mathfrak{m}athscr{C} _{\mathcal P}$. \\(3) Any covariant additive functor $T':\mathfrak{m}athscr{C} \mathsf {L}ongrightarrow\mathfrak{m}athbb{D} $ satisfying the conditions (1) and (2), factors in a unique way through $T$.
In this paper, first we provide some important examples of $n$-Frobenius categories, and then we show that for any $n$-Frobenius category $\mathfrak{m}athscr{C} $, the phantom stable category $(\mathfrak{m}athscr{C} _{\mathcal P}, T)$ exists; see Theorem \twoheadrightarrowf{thmst}. Our formalism reveals that the phantom stabilization of an $n$-Frobenius category, is an efficient and natural extension of the classical stable category of a Frobenius category. Indeed, in the case $n=0$, morphisms factoring through projective objects are $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphisms, which is an ideal of $\operatorname{\mathsf{H}}$. Particularly, in order to examine the efficiency of phantom stable categories, it is proved that for given two objects $M,N\in\mathfrak{m}athscr{C} $ and arbitrary syzygies $\mathsf{\Omega} M$ and $\mathsf{\Omega} N$ (with respect to $n$-projectives), there is an induced map $\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} _{\mathcal P}}(M, N)\mathsf {L}ongrightarrow\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} _{\mathcal P}}(\mathsf{\Omega} M, \mathsf{\Omega} N)$, which is an isomorphism; see Theorem \twoheadrightarrowf{syziso}.
Our motivation in studing the $n$-Frobenius category and then introducing the concept of phantom stable category, comes from the fact that there are categories rarely have enough projectives or even, they have no projective objects. However, they have often enough $n$-projectives or their subcategories of $n$-projective objects are non-trivial, for some integer $n\mathbf geq 1$. For instance, there are no projective objects in the category of quasi-coherent sheaves over the projective line $\mathfrak{m}athbf{P^1}(R)$, where $R$ is a commutative ring with identity; see \operatorname{\underline{\mathscr{C}}}ite[Corollary 2.3]{ee} and also \operatorname{\underline{\mathscr{C}}}ite[Exercise III 6.2(b)]{har}. However, as proved by Serre, the category of coherent sheaves over a projective scheme, has enough locally free sheaves; see \operatorname{\underline{\mathscr{C}}}ite[Corollary 5.18]{har}. More generally, the argument given in the proof of \operatorname{\underline{\mathscr{C}}}ite[Lemma 1.12]{or} reveals that for a semi-separated noetherian scheme $\mathsf {X}$ of finite Krull dimension, there exists a non-negative integer $n$ such that locally free sheaves of finite rank, are $n$-projective objects in the category of coherent sheaves, ${\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$, and in particular, the subcategory of $n$-projective objects of ${\rm{op}}eratorname{\mathsf{cone}}h(X)$, is non-trivial.
Assume that $\mathcal A$ is an abelian category such that its subcategory of $n$-projective objects is non-trivial, for some integer $n\mathbf geq 1$. Then it is proved that $\mathcal A$ admits an $n$-Frobenius subcategory $\mathfrak{m}athscr{C} $; see Theorem \twoheadrightarrowf{subcat}. We should stress that this fact is conceivable, as it is known that any abelian category with non-zero projective objects admits a 0-Frobenius subcategory. Assume that $\mathsf {X}$ is a semi-separated noetherian scheme of finite Krull dimension. As we have already mentioned, the category of locally free sheaves of finite rank, $\mathcal L$, is a subcategory of $n$-$\mathcal Proj{\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$, for some non-negative integer $n$. We will see that the subcategory consisting of all syzygies of complete resolutions of locally frees of finite rank, $\mathfrak{m}athscr{C} (\mathcal L)$, is an $n$-Frobenius subcategory of ${\rm{op}}eratorname{\mathsf{cone}}h\mathsf {X}$, and in particular, $n$-$\mathcal Proj\mathfrak{m}athscr{C} (\mathcal L)=\mathcal L$; see Proposition \twoheadrightarrowf{locally}.
In order to explore more examples of phantom stable categories, we consider the category of complexes of flat $\mathcal{O}_{\mathsf {X}}$-modules, $\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$, over the scheme $\mathsf {X}$. Pursuing the argument given in \operatorname{\underline{\mathscr{C}}}ite[page 28]{ha1}, yields that this is an exact category, with exact structure being short exact sequences of complexes. We will see in Theorem \twoheadrightarrowf{fp}, that the aforementioned category is $n$-Frobenius, for some integer $n\mathbf geq 0$, and in particular, its $n$-projective objects (and then $n$-injective objects), are exactly the flat complexes, i.e., those acyclic complexes with flat kernels; see \operatorname{\underline{\mathscr{C}}}ite[Definition 2.5]{e}.
The paper is organized as follows. In Section 2, we study $n$-Frobenius categories and explore some examples of such categories. {We will observe that semi-separated noetherian schemes of finite Krull dimension are good venue for searching such examples.} It is shown that, any abelian category with non-zero $n$-projective objects admits an $n$-Frobenius subcategory. Assume that $\mathsf {X}$ is a semi-separated noetherian scheme of finite Krull dimension. Then we will see that $\mathfrak{m}athscr{C} (\mathcal L)$ is an $n$-Frobenius subcategory of ${\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$, for some integer $n$. Also, we prove that the category $\mathfrak{m}athscr{C} (\mathsf {Flat}\mathsf {X})$ consisting of all syzygies of complete resolution of flat sheaves, is an $n$-Frobenius subcategory of $\operatorname{\mathsf{Qcoh}}(\mathsf {X})$, for some non-negative integer $n$, and $n$-$\mathcal Proj\mathfrak{m}athscr{C} (\mathsf {Flat}\mathsf {X})=\mathsf {Flat}\mathsf {X}$. In Section 3, we study those morphisms in $\operatorname{\mathsf{H}}$ acting as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathfrak{m}athscr{C} }$. It is proved that a given morphism $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathfrak{m}athscr{C} }$ from the left if and only if it acts as invertible from the right. These morphisms will be called quasi-invertible morphisms.
Section 4 is devoted to study conflations factoring through $n$-projective objects, namely, those arising from pull-back along morphisms ending at $n$-projectives. This class of conflations forms a subfunctor of $\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }$ and will be denoted by $\mathcal P$. It is proved that a given conflation lies in $\mathcal P$ if and only if it is obtained from push-out along a morphism starting at an $n$-projective object, and so, $\mathcal P$ is an $\operatorname{\mathsf{H}}$-bisubmodule of $\operatorname{{\mathsf{Ext}}}^n$. In Section 5, we show that any conflation $\mathbf ga$ in $\mathfrak{m}athscr{C} $ can be represented as a pull-back as well as push-out of unit conflations. These representations will be called a right (left) unit factorization of $\mathbf ga$. In Section 6, we investigate those morphisms annihilating the module $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathfrak{m}athscr{C} }$. It is shown that a given object $f\in\operatorname{\mathsf{H}}$ annihilates $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathfrak{m}athscr{C} }$ from the left if and only if it annihilates from the right. We call such a morphism $f$, as an $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism. In Section 7, we introduce a composition operator $``\operatorname{\underline{\mathscr{C}}}irc"$ on $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$. It is shown that this operator is associative and distributive over the Baer sum on both sides. In particular, for any object $M$, $\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^nM)/{\mathcal P}$ has a ring structure with identity element, and also, there exists a ring homomorphism $\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(M, M)\mathsf {L}ongrightarrow\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^nM)/{\mathcal P}$ sending quasi-invertible morphisms to invertible elements. In Section 8, we consider an equivalence relation on $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$. It is observed that this relation is compatible with the composition $``\operatorname{\underline{\mathscr{C}}}irc"$ as well as the Baer sum operation. In the paper's final section, we show that the phantom stable category $(\mathfrak{m}athscr{C} _{\mathcal P}, T)$ of an $n$-Frobenius category $\mathfrak{m}athscr{C} $, always exists.
\section{$n$-Frobenius categories}
Let $\mathcal A$ be an abelian category and let $n$ be a non-negative integer. In this section, we study subcategories of $\mathcal A$ having enough $n$-projectives and $n$-injectives and the class of $n$-projectives coincides with the class of $n$-injectives, which we call $n$-Frobenius categories. It is shown that if $\mathcal A$ has non-zero $n$-projective objects, then it admits a non-trivial $n$-Frobenius subcategory. We also explore several examples of $n$-Frobenius categories. Let us begin this section with stating our convention.
\mathsf {b}egin{conv}Throughout the paper, $\mathcal A$ is an abelian category and $\mathfrak{m}athscr{C} $ is a full additive subcategory of $\mathcal A$ which is closed under extensions. So, as we have mentioned in the introduction, $\mathfrak{m}athscr{C} $ becomes an exact category. We also assume that $(\mathsf {X}, \mathcal{O}_{\mathsf {X}})$ is a semi-separated, noetherian scheme of finite Krull dimension and all locally free sheaves are assumed to be of finite rank.
\end{conv}
\mathsf {b}egin{dfn} We say that an extension of length $t\mathbf geq 1$, $0\mathbf hookrightarrowghtarrow B\mathbf hookrightarrowghtarrow X_{t-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow X_0\mathbf hookrightarrowghtarrow A\mathbf hookrightarrowghtarrow 0$ in $\mathcal A$, is a {\it conflation} of length $t$ in $\mathfrak{m}athscr{C} $, provided that it is obtained by splicing $t$ conflations of length 1 in $\mathfrak{m}athscr{C} $ and it will be denoted by $B\mathbf hookrightarrowghtarrow X_{t-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow X_{0}\mathbf hookrightarrowghtarrow A$. The set of all equivalence classes of such conflations of lenght $t$, will be depicted by $\operatorname{{\mathsf{Ext}}}^t_{\mathfrak{m}athscr{C} }(A, B)$. We also set $\operatorname{{\mathsf{Ext}}}^0_{\mathfrak{m}athscr{C} }(A, B):=\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(A, B)$. It is easily seen that for any $i\mathbf geq 0$, $\operatorname{{\mathsf{Ext}}}^i_{\mathfrak{m}athscr{C} }(-, -):\mathfrak{m}athscr{C} ^{{\rm{op}}}\times\mathfrak{m}athscr{C} \mathsf {L}ongrightarrow\mathfrak{m}athbf{Ab}$ is a bifunctor.
Recall that two conflations $\operatorname{\boldsymbol{\epsilon}}, \operatorname{\boldsymbol{\epsilon}}'\in\operatorname{{\mathsf{Ext}}}^t_{\mathfrak{m}athscr{C} }(A, B)$ are equivalent, provided that there is a chain of conflations of length $t$, $\operatorname{\boldsymbol{\epsilon}}=\operatorname{\boldsymbol{\epsilon}}_0, \operatorname{\boldsymbol{\epsilon}}_1\operatorname{\underline{\mathscr{C}}}dots, \operatorname{\boldsymbol{\epsilon}}_k=\operatorname{\boldsymbol{\epsilon}}'$ such that for any $0\mathsf {L}eq i\mathsf {L}eq k-1$, we have either a morphism $\operatorname{\boldsymbol{\epsilon}}_i\mathbf hookrightarrowghtarrow\operatorname{\boldsymbol{\epsilon}}_{i+1}$ or a morphism $\operatorname{\boldsymbol{\epsilon}}_{i+1}\mathbf hookrightarrowghtarrow\operatorname{\boldsymbol{\epsilon}}_i$ with fixed ends; see \operatorname{\underline{\mathscr{C}}}ite[Proposition 3.1]{mit}.
Following Keller \operatorname{\underline{\mathscr{C}}}ite{ke,ke1} conflations of length 1, will be called just conflations. In particular, if $B\stackrel{f}\mathbf hookrightarrowghtarrow C\stackrel{g}\mathbf hookrightarrowghtarrow A$ is a conflation, then $f$ (resp. $g$) is called an inflation (resp. a deflation). So if $\operatorname{\boldsymbol{\epsilon}}:B\mathbf hookrightarrowghtarrow X_{t-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow X_0\mathbf hookrightarrowghtarrow A$ is a conflation of length $t$, then $\operatorname{\boldsymbol{\epsilon}}=\operatorname{\boldsymbol{\epsilon}}_{t-1}\operatorname{\underline{\mathscr{C}}}dots\operatorname{\boldsymbol{\epsilon}}_0$, where $\operatorname{\boldsymbol{\epsilon}}_i^,s$ are conflations with compatible ends.
\end{dfn}
\mathsf {b}egin{dfn}Let $n$ be a non-negative integer.\\
(1) A given object $P\in\mathfrak{m}athscr{C} $ (resp. $I\in\mathfrak{m}athscr{C} $) is said to be an {\it $n$-projective} (resp. {\it $n$-injective}) object of $\mathfrak{m}athscr{C} $, if $\operatorname{{\mathsf{Ext}}}^i_{\mathfrak{m}athscr{C} }(P, X)=0$ (resp. $\operatorname{{\mathsf{Ext}}}^i_{\mathfrak{m}athscr{C} }(X, I)=0$) for all integers $i>n$ and all objects $X\in\mathfrak{m}athscr{C} $. The class of all $n$-projective (resp. $n$-injective) objects will be denoted by $n$-$\mathcal Proj\mathfrak{m}athscr{C} $ (resp. $n$-$\operatorname{\mathsf{inj}}\mathfrak{m}athscr{C} $). \\
(2) The category $\mathfrak{m}athscr{C} $ is said to have enough $n$-projectives, provided that each object $M$ in $\mathfrak{m}athscr{C} $ fits into a deflation $P\mathbf hookrightarrowghtarrow M$ with $P$ $n$-projective. Dually one has the notion of having enough $n$-injectives.\\
(3) We say that the exact category $\mathfrak{m}athscr{C} $ is {\it $n$-Frobenius}, if
$\mathfrak{m}athscr{C} $ has enough $n$-projectives and $n$-injectives and $n$-$\mathcal Proj\mathfrak{m}athscr{C} $ coincides with $n$-$\operatorname{\mathsf{inj}}\mathfrak{m}athscr{C} $.
\end{dfn}
\mathsf {b}egin{rem}\mathsf {L}abel{remexam}(1) It follows from the definition that $0$-Frobenius categories are indeed the usual notion of Frobenius categories.\\ (2) If $\mathfrak{m}athscr{C} $ is an $n$-Frobenius category, then it will be an $i$-Frobenius category, for any $i\mathbf ge n$. In particular, $n$-$\mathcal Proj\mathfrak{m}athscr{C} = i$-$\mathcal Proj\mathfrak{m}athscr{C} $.
\end{rem}
\mathsf {b}egin{s} Assume that $\mathfrak{m}athscr{C} $ is an $n$-Frobenius category. Then, for any $k\mathbf geq 1$, a given object $N\in\mathfrak{m}athscr{C} $ fits into conflations of length $k$;
$\mathsf{\Omega}^kN\mathbf hookrightarrowghtarrow P_{k-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_0\mathbf hookrightarrowghtarrow N$ and $N\mathbf hookrightarrowghtarrow P^1\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P^k\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-k}N$
such that $P_i, P^i$'s are $n$-projective, which will be called {\it unit conflations}. Also $\mathsf{\Omega}^kN$ is said to be a $k$-th syzygy of $N$. Clearly, unit conflations are not uniquely determined. We denote the class of all unit conflations of length $k$ ending at $N$ (resp. beginning with $N$) by $\mathcal U_k(N)$ (resp. $\mathcal U^k(N)$). Unit conflations, usually will be depicted by $\mathsf {d}elta$.
\end{s}
\mathsf {b}egin{rem}\mathsf {L}abel{pp1} (1) Assume that $k\mathbf geq 1$ and there exists a morphism of conflations;
{\mathbf footnotesize \[\mathsf {X}ymatrix{\mathsf {a}lpha:N \ ~\mathsf {a}r[r] \ \ \mathsf {a}r[d]_f& \ \ X_{k-1}\mathsf {a}r[r] \ \ \mathsf {a}r[d]& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& \ \ X_0\mathsf {a}r[r] \ \ \mathsf {a}r[d] & \ M\mathsf {a}r[d]_g\\ \mathsf {b}eta:N' \ ~\mathsf {a}r[r] \ \ &\ \ Y_{k-1}\mathsf {a}r[r] \ \ &\operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r]& \ \ Y_0\mathsf {a}r[r] \ \ & M'.}\]}
It is evident from the definition of push-out and pull-back that $\mathsf {a}l\mathsf {L}ongrightarrow f\mathsf {a}l$ and $\mathsf {b}e g\mathsf {L}ongrightarrow\mathsf {b}e$ are morphisms of conflations with the right and the left fixed ends, respectively.
Now using the universal properties of push-out and pull-back diagrams, one may find the morphism of conflations $f\mathsf {a}l\mathsf {L}ongrightarrow\mathsf {b}e g$ with fixed ends. Consequently, applying \operatorname{\underline{\mathscr{C}}}ite[Proposition 3.1]{mit} yields that $f\mathsf {a}l=\mathsf {b}e g$. In particular, any morphism of conflations of length $k$ with the left (resp. right) fixed ends, is a pull-back (resp. push-out) diagram.\\ (2) Assume that $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} }(C, A) $ and take morphisms $h:C'\mathbf hookrightarrowghtarrow C$ and $l:A\mathbf hookrightarrowghtarrow A'$ in $\mathfrak{m}athscr{C} $. It can be easily seen that there is a morphism
of conflations;
{\mathbf footnotesize \[\mathsf {X}ymatrix{\mathbf gamma h: A \ \ ~\mathsf {a}r[r] \ \ \mathsf {a}r[d]_l& \ \ X_{k-1}\mathsf {a}r[r] \ \ \mathsf {a}r[d]& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& \ \ X_0\mathsf {a}r[r] \ \ \mathsf {a}r[d] & \ C'\mathsf {a}r[d]_h\\ \ l\mathbf gamma:A' \ \ ~\mathsf {a}r[r] \ \ \ &\ \ Y_{k-1}\mathsf {a}r[r] \ \ &\operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r]& \ \ Y_0\mathsf {a}r[r] \ \ & C.}\]}
So, as we have observed just above, $l(\mathbf ga h)=(l\mathbf ga)h$; see also \operatorname{\underline{\mathscr{C}}}ite[Page 171, (2)]{mit}.
\end{rem}
\mathsf {b}egin{s}
We set $\operatorname{\mathsf{H}}:=\mathsf {b}igcup_{M,N\in\mathfrak{m}athscr{C} }\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(M, N)$ and $\operatorname{{\mathsf{Ext}}}^n:=\mathsf {b}igcup_{M,N\in\mathfrak{m}athscr{C} }\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^nN)$, where $\mathsf{\Omega}^nN$ runs over all the $n$-th syzygies of $N$. \\
Assume that $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^n N)$, $a\in\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(M', M)$ and $b\in\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(\mathsf{\Omega}^nN, {\mathsf{\Omega}'}^nN)$. According to Remark \twoheadrightarrowf{pp1}, we have $b(\mathbf ga a)=(b\mathbf ga)a$. So $\operatorname{{\mathsf{Ext}}}^n$ has an $\operatorname{\mathsf{H}}$-bimodule structure.
\end{s}
\mathsf {b}egin{rem}\mathsf {L}abel{zero}Assume that $f:M\mathbf hookrightarrowghtarrow N$ is a morphism in $\mathfrak{m}athscr{C} $. So, for any $X\in\mathfrak{m}athscr{C} $ and $k\mathbf geq 0$, one has the induced morphism $\operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} }(X, M)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} }(X, N)$ mapping each $\mathbf ga$ to $f\mathbf ga$, the push-out of $\mathbf ga$ along $f$. Similarly, $\operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} }(N, X)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} }(M, X)$ sends each object $\mathsf {a}l$ to $\mathsf {a}l f$, the pull-back along $f$. Indeed, these morphisms can be interpreted as multiplication by $f$ from the left and the right, respectively. From this point of view, we will denote these morphisms again by $\mathbf f$. One should note that, in the case $k=0$, since $\operatorname{{\mathsf{Ext}}}^0_{\mathfrak{m}athscr{C} }(-, -)=\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(-, -)$, $f\mathbf ga$ and $\mathsf {a}l f$ are indeed composition morphisms.
\end{rem}
In the sequel, we will see that any abelian category with non-zero $n$-projective objects, admits a non-trivial $n$-Frobenius subcategory. First we state a definition.\\
\mathsf {b}egin{dfn} An acyclic complex of $n$-projective objects $\mathfrak{m}athbf{P^{\mathsf {b}ullet}}:\operatorname{\underline{\mathscr{C}}}dots\mathsf {L}ongrightarrow P^{i-1}\stackrel{d^{i-1}}\mathsf {L}ongrightarrow P^i\stackrel{d^i}\mathsf {L}ongrightarrow P^{i+1}\mathsf {L}ongrightarrow\operatorname{\underline{\mathscr{C}}}dots$ is said to be {\it a complete resolution of $n$-projective objects}, if $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(\mathfrak{m}athsf{im} d^i, Q)=0$, for any $i\in\mathfrak{m}athbb{Z} $ and $Q\in n$-$\mathcal Proj\mathcal A$.\\
Assume that $\mathfrak{m}athcal{I} $ is a resolving subcategory of $n$-$\mathcal Proj\mathcal A$, that is, in any conflation $P'\mathbf hookrightarrowghtarrow P\mathbf hookrightarrowghtarrow P''$, with $P''\in\mathfrak{m}athcal{I} $, we have $P'\in\mathfrak{m}athcal{I} $ if and only if $P\in\mathfrak{m}athcal{I} $. Assume that $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ is the subcategory of $\mathcal A$ consisting of all objects $M$ which is a syzygy of a complete resolution of objects in $\mathfrak{m}athcal{I} $, i.e., an acyclic complex $\mathfrak{m}athbf{P^{\mathsf {b}ullet}}:\operatorname{\underline{\mathscr{C}}}dots\mathsf {L}ongrightarrow P^{i-1}\stackrel{d^{i-1}}\mathsf {L}ongrightarrow P^i\stackrel{d^i}\mathsf {L}ongrightarrow P^{i+1}\mathsf {L}ongrightarrow\operatorname{\underline{\mathscr{C}}}dots,$ in $\mathfrak{m}athcal{I} $ such that $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(\mathfrak{m}athsf{im} d^i, Q)=0$, for any $i\in\mathfrak{m}athbb{Z} $ and $Q\in\mathfrak{m}athcal{I} $.
If $\mathfrak{m}athcal{I} =n$-$\mathcal Proj\mathcal A$, instead of $\mathfrak{m}athscr{C} (n$-$\mathcal Proj\mathcal A)$, we write $\mathfrak{m}athscr{C} (\mathcal A).$ It is evident that $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ is a full subcategory of $\mathcal A$ containing all objects in $\mathfrak{m}athcal{I} $ and it is closed under finite direct sums. Moreover, as the next proposition indicates, $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ is an exact category. First, we need a couple of preliminary lemmas. In the rest of this section, we assume that the abelian category $\mathcal A$ has non-zero $n$-projective objects.
\end{dfn}
\mathsf {b}egin{lem}\mathsf {L}abel{000}{The following statements are satisfied:
\mathsf {b}egin{enumerate}\item Let $Q\mathbf hookrightarrowghtarrow M\stackrel{f}\mathbf hookrightarrowghtarrow N$ be a conflation in $\mathcal A$ such that $Q\in\mathfrak{m}athcal{I} $. Then for any $X\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ and $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n_{\mathcal A}(X, N)$, there exists an object $\mathbf ga'\in\operatorname{{\mathsf{Ext}}}^n_{\mathcal A}(X, M)$ such that $f\mathbf ga'=\mathbf ga$. \item Let $M\stackrel{f}\mathbf hookrightarrowghtarrow N\mathbf hookrightarrowghtarrow Q$ be a conflation in $\mathcal A$ with $Q\in n$-$\mathcal Proj\mathcal A$. Then for any $X\in\mathcal A$ and $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n_{\mathcal A}(M, X)$, there exists an object $\mathbf ga'\in\operatorname{{\mathsf{Ext}}}^n_{\mathcal A}(N,X)$ such that $\mathbf ga'f=\mathbf ga$.
\end{enumerate}}
\end{lem}
\mathsf {b}egin{proof}We only prove the statement (1), the other one is obtained dually. Since $Q\in\mathfrak{m}athcal{I} $ and $X\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(X, Q)=0$, and so, making use of \operatorname{\underline{\mathscr{C}}}ite[Chapter VII, Theorem 5.1]{mit} forces
$\operatorname{{\mathsf{Ext}}}^n_{\mathcal A}(X, M)\stackrel{\mathbf f}\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^n_{\mathcal A}(X, N)$ to be an epimorphism. Hence, there exists $\mathbf ga'\in\operatorname{{\mathsf{Ext}}}^ n_{\mathcal A}(X, M)$ such that $f\mathbf ga'=\mathbf ga$, as needed.
\end{proof}
\mathsf {b}egin{lem}\mathsf {L}abel{cok}The following assertions hold: \mathsf {b}egin{enumerate}\item
Let $Q\mathbf hookrightarrowghtarrow M\stackrel{f}\mathbf hookrightarrowghtarrow N$ be a conflation in $\mathcal A$ such that $Q\in\mathfrak{m}athcal{I} $. If $N\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} ),$ then so does $M$.\item Let $M\stackrel{f}\mathbf hookrightarrowghtarrow N\mathbf hookrightarrowghtarrow Q$ be a conflation in $\mathcal A$ such that $Q\in\mathfrak{m}athcal{I} $. If $M\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, then so does $N$.\end{enumerate}
\end{lem}
\mathsf {b}egin{proof}(1) Since $N\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, by the definition, there exists a complete resolution of objects in $\mathfrak{m}athcal{I} $; $\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow Q^{-1}\stackrel{d^{-1}}\mathbf hookrightarrowghtarrow Q^0\stackrel{d^0}\mathbf hookrightarrowghtarrow Q^{1}\stackrel{d^{1}}\mathbf hookrightarrowghtarrow Q^{2}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots,$ such that $N=\mathfrak{m}athsf{im} d^0$. Take the conflation $\mathbf ga:N\mathbf hookrightarrowghtarrow Q^{1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow Q^{n}\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-n}N$ in $\mathcal A$. {Clearly, $\mathsf{\Omega}^{-n}N\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. So} applying Lemma \twoheadrightarrowf{000}(1), gives us an object $\mathbf ga'\in\operatorname{{\mathsf{Ext}}}^n_{\mathcal A}(\mathsf{\Omega}^{-n}N, M)$ such that $f\mathbf ga'=\mathbf ga$. Namely, we have the following push-out diagram;
{\mathbf footnotesize\[\mathsf {X}ymatrix{\mathbf ga':M~ \ \ \mathsf {a}r[r]\mathsf {a}r[d]_{f} & \ \ T\mathsf {a}r[d]\mathsf {a}r[r] \ \ & \ \ Q^{2}\mathsf {a}r[r]\ \ \mathsf {a}r@{=}[d] &\operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& \ \ Q^{n}\mathsf {a}r@{=}[d]\mathsf {a}r[r] \ \ & \ \ \mathsf{\Omega}^{-n}N\mathsf {a}r@{=}[d]\\ \mathbf ga:N~\mathsf {a}r[r] \ \ & \ \ Q^{1}\mathsf {a}r[r]\ \ & \ \ Q^{2}\mathsf {a}r[r]\ \ & \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r] &\ \ Q^{n}\mathsf {a}r[r] \ \ & \ \ \mathsf{\Omega}^{-n}N.}\]} {Next take the following pull-back diagram; \[\mathsf {X}ymatrix{L~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& T'\mathsf {a}r[r]\mathsf {a}r[d]& M\mathsf {a}r[d]_{f}\\ L~\mathsf {a}r[r] & Q^0\mathsf {a}r[r]& N.}\]
Since $\mathfrak{m}athcal{I} $ is closed under extensions, one may get that $T$ and $T'$ belong to $\mathfrak{m}athcal{I} $. In particular, we obtain the complete resolution of {objects in $\mathfrak{m}athcal{I} $;} $\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow Q^{-1}\mathbf hookrightarrowghtarrow T'\mathbf hookrightarrowghtarrow T\mathbf hookrightarrowghtarrow Q^{2}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots$ such that $M=\mathfrak{m}athsf{im}(T'\mathbf hookrightarrowghtarrow T)$, and then, $M\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$.}\\ (2) This is obtained by dualizing the argument given in the first assertion, so we skip it. Thus the proof is completed.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{proj1}The category $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ is closed under extensions and kernels of epimorphisms.
\end{prop}
\mathsf {b}egin{proof}Let us first prove that $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ is closed under extensions. To do this, assume that $M\stackrel{f}\mathbf hookrightarrowghtarrow N\stackrel{g}\mathbf hookrightarrowghtarrow K$ is a conflation in $\mathcal A$ such that $M, K\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. We shall prove that $N\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, as well. By the hypothesis, we may take a conflation $M\mathbf hookrightarrowghtarrow Q^{1}\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-1} M$ in $\mathcal A$ such that $Q^{1}\in\mathfrak{m}athcal{I} $ and $\mathsf{\Omega}^{-1}M\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. So considering the push-out diagram; \[\mathsf {X}ymatrix{M~\mathsf {a}r[r]\mathsf {a}r[d]_{f}& Q^{1}\mathsf {a}r[r]\mathsf {a}r[d]& \mathsf{\Omega}^{-1}M\mathsf {a}r@{=}[d]\\ N \mathsf {a}r[r] & T\mathsf {a}r[r] & \mathsf{\Omega}^{-1}M,}\] we obtain the conflation $Q^{1}\mathbf hookrightarrowghtarrow T\mathbf hookrightarrowghtarrow K$ in $\mathcal A$. As $K\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, Lemma \twoheadrightarrowf{cok}(1) implies that $T\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. This enables us to have the following commutative diagram; \[\mathsf {X}ymatrix{N~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& T\mathsf {a}r[r]\mathsf {a}r[d]& \mathsf{\Omega}^{-1}M\mathsf {a}r[d]\\ N~\mathsf {a}r[r] & P^{1}\mathsf {a}r[r]\mathsf {a}r[d]& G^1\mathsf {a}r[d]\\ & \mathsf{\Omega}^{-1}T~\mathsf {a}r@{=}[r] & \mathsf{\Omega}^{-1}T,}\] in which $P^{1}\in\mathfrak{m}athcal{I} $ and $\mathsf{\Omega}^{-1}T\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. As $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(\mathsf{\Omega}^{-1}M, Q)=0=\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(\mathsf{\Omega}^{-1}T, Q)$ for any object $Q\in\mathfrak{m}athcal{I} $, we have that $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(G^1, Q)=0$. Now consider the conflation $\mathsf{\Omega}^{-1}M\mathbf hookrightarrowghtarrow G^1\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-1}T$ in $\mathcal A$. Since $\mathsf{\Omega}^{-1}M, \mathsf{\Omega}^{-1}T\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, applying the above argument, will give us conflations $G^1\mathbf hookrightarrowghtarrow P^{2}\mathbf hookrightarrowghtarrow G^2$ and $\mathsf{\Omega}^{-2}M\mathbf hookrightarrowghtarrow G^2\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-2}T$ in $\mathcal A$ with $\mathsf{\Omega}^{-2}M, \mathsf{\Omega}^{-2}T\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, $P^{2}\in\mathfrak{m}athcal{I} $ and $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(G^2, Q)$ vanishes, for any object $Q\in\mathfrak{m}athcal{I} $. In particular, repeating this manner, gives rise to the existence of an acyclic complex $0\mathbf hookrightarrowghtarrow N\stackrel{\operatorname{\boldsymbol{\epsilon}}silon}\mathbf hookrightarrowghtarrow P^{1}\stackrel{d^{1}}\mathbf hookrightarrowghtarrow P^{2}\stackrel{d^{2}}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots,$ where each $P^i$ belongs to $\mathfrak{m}athcal{I} $ and $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(\mathfrak{m}athsf{im} d^i, Q)=0$ for any object $Q\in\mathfrak{m}athcal{I} $ and $i\mathbf geq 1$. {Moreover, a dual argument gives us an acyclic complex, $\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P^{-1}\stackrel{d^{-1}}\mathbf hookrightarrowghtarrow P^0\stackrel{\operatorname{\boldsymbol{\epsilon}}silon'}\mathbf hookrightarrowghtarrow N\mathbf hookrightarrowghtarrow 0$, where $P^i\in\mathfrak{m}athcal{I} $ and $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(\mathfrak{m}athsf{im} d^i, Q)=0$, for all $i\mathsf {L}eq -1$. }Thus $N\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, as required. Next we show that $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ is closed under kernels of epimorphisms. So assume that $M\stackrel{f}\mathbf hookrightarrowghtarrow N\stackrel{g}\mathbf hookrightarrowghtarrow K$ is a conflation in $\mathcal A$ such that $N, K\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. We have to show that $M\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, as well. Since $N\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, we may obtain the following commutative diagram; \[\mathsf {X}ymatrix{M~\mathsf {a}r[r]\mathsf {a}r[d]_{f}& Q^{1}\mathsf {a}r[r]\mathsf {a}r@{=}[d]& T\mathsf {a}r[d]\\ N \mathsf {a}r[r] & Q^{1}\mathsf {a}r[r] & \mathsf{\Omega}^{-1}N,}\] with $\mathsf{\Omega}^{-1}N\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ and $Q^{1}\in\mathfrak{m}athcal{I} $. So applying the snake lemma, gives us the conflation $K\mathbf hookrightarrowghtarrow T\mathbf hookrightarrowghtarrow \mathsf{\Omega}^{-1}N$ in $\mathcal A$. Since $\mathsf{\Omega}^{-1}N, K\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, as we have already seen, $T\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. Thus, by taking the right half of a complete resolution $0\mathbf hookrightarrowghtarrow T\mathbf hookrightarrowghtarrow P^1\stackrel{d^2}\mathbf hookrightarrowghtarrow P^2\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots$ of $T$, we will get an acyclic complex $0\mathsf {L}ongrightarrow M\mathsf {L}ongrightarrow Q^{1}\stackrel{d^{1}}\mathsf {L}ongrightarrow P^{1}\stackrel{d^{2}}\mathsf {L}ongrightarrow\operatorname{\underline{\mathscr{C}}}dots$, whenever all objects, except $M$, lie in $\mathfrak{m}athcal{I} $ and $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(\mathfrak{m}athsf{im} d^i, \mathfrak{m}athcal{I} )=0$, for all $i$.
{Next take the following pull-back diagram; \[\mathsf {X}ymatrix{& \mathsf{\Omega} K\mathsf {a}r@{=}[r]\mathsf {a}r[d]& \mathsf{\Omega} K\mathsf {a}r[d]\\ M~\mathsf {a}r[r]\mathsf {a}r@{=}[d] & L\mathsf {a}r[r]\mathsf {a}r[d]& P\mathsf {a}r[d]\\ M\mathsf {a}r[r] & N~\mathsf {a}r[r] & K,}\] where $P\in\mathfrak{m}athcal{I} $. Since $N, \mathsf{\Omega} K\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, by the first assertion, the same is true for $L$. So, considering the pull-back diagram; \[\mathsf {X}ymatrix{\mathsf{\Omega} L~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& G\mathsf {a}r[r]\mathsf {a}r[d]& M\mathsf {a}r[d]\\ \mathsf{\Omega} L~\mathsf {a}r[r] & P'\mathsf {a}r[r]\mathsf {a}r[d]& L\mathsf {a}r[d]\\ & P~\mathsf {a}r@{=}[r] & P,}\] and using the fact that $\mathfrak{m}athcal{I} $ is resolving, we infer that $G\in\mathfrak{m}athcal{I} $. This, in conjunction with $\mathsf{\Omega} L$ being in $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, would guarantee the existence of a left resolution of objects in $\mathfrak{m}athcal{I} $ for $M$; $\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P^{-1}\stackrel{d^{-1}}\mathbf hookrightarrowghtarrow P^0\stackrel{d^0}\mathbf hookrightarrowghtarrow G\mathbf hookrightarrowghtarrow M\mathbf hookrightarrowghtarrow 0$, such that $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathcal A}(\mathfrak{m}athsf{im} d^i, \mathfrak{m}athcal{I} )=0$ for all $i\mathsf {L}eq 0$. Consequently, $M\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$,} as required.
\end{proof}
\mathsf {b}egin{rem}\mathsf {L}abel{exact}Assume that $A\mathbf hookrightarrowghtarrow B\mathbf hookrightarrowghtarrow C$ is a conflation in $\mathfrak{m}athscr{C} $. Then the same argument given in the proof of \operatorname{\underline{\mathscr{C}}}ite[Chapter VII, Theorem 5.1]{mit}, yields that for any object $X$ in $\mathfrak{m}athscr{C} $
and any $n\mathbf geq 1$, there exists an exact sequence; $$\operatorname{{\mathsf{Ext}}}^{n-1}_{\mathfrak{m}athscr{C} }(X, C)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(X, A)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(X, B)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(X, C)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathfrak{m}athscr{C} }(X, A).$$ Also, a dual argument gives us the exact sequence;
$$\operatorname{{\mathsf{Ext}}}^{n-1}_{\mathfrak{m}athscr{C} }(A, X)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(C, X)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(B, X)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(A, X)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathfrak{m}athscr{C} }(C, X).$$
\end{rem}
\mathsf {b}egin{theorem}\mathsf {L}abel{subcat}$\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ is an $n$-Frobenius subcategory of $\mathcal A$.
\end{theorem}
\mathsf {b}egin{proof}First one should note that by Proposition \twoheadrightarrowf{proj1}, $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ is an exact category. In view of the definition of $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, we only need to show that any $n$-injective object of $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ is also $n$-projective over $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ and vice versa. Take an object $N\in n$-$\operatorname{\mathsf{inj}}\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ and consider unit conflations $\mathsf {d}elta_N:N\mathbf hookrightarrowghtarrow P^{1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P^{n}\mathsf {L}ongrightarrow \mathsf{\Omega}^{-n}N$ and $\mathsf{\Omega}^{-n}N\stackrel{h}\mathbf hookrightarrowghtarrow P^{n+1}\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-n-1} N$, where $P^i\in\mathfrak{m}athcal{I} $, for any $i$. Since $\mathsf{\Omega}^{-n-1}N\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )}(\mathsf{\Omega}^{-n-1}N,N)=0$,
implying that, there exists an object $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )}( P^{n+1},N)$ such that $\mathbf ga h=\mathsf {d}elta_N$. Namely, we have the following pull-back diagram;
{\mathbf footnotesize\[\mathsf {X}ymatrix{\mathsf {d}elta_N \ :N~ \ \ \ \mathsf {a}r[r] \ \ \mathsf {a}r@{=}[d] & \ \ P^{1}\mathsf {a}r@{=}[d]\mathsf {a}r[r]\ \ & \ \ \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]\ \ &P^{n-1}\mathsf {a}r@{=}[d] \mathsf {a}r[r]& \ \ P^{n}\mathsf {a}r[d]\mathsf {a}r[r] \ \ & \ \ \mathsf{\Omega}^{-n}N\mathsf {a}r[d]^{h}\\ \mathbf ga : \ \ \ N~ \ \ \mathsf {a}r[r] \ \ \ & \ \ P^{1}\mathsf {a}r[r]\ \ & \ \ \operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r] \ \ & P^{n-1} \mathsf {a}r[r] & \ \ H\mathsf {a}r[r] &P^{n+1}\ \ .}\]} As $H\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, there is an inflation $H\mathbf hookrightarrowghtarrow Q$ with $Q\in\mathfrak{m}athcal{I} $. So by taking the conflation $L\mathbf hookrightarrowghtarrow H\mathbf hookrightarrowghtarrow P^{n+1}$, one gets the following commutative diagram; \[\mathsf {X}ymatrix{L~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& H\mathsf {a}r[r]\mathsf {a}r[d]&P^{n+1} \mathsf {a}r[d]^{g}\\ L~\mathsf {a}r[r] & Q\mathsf {a}r[r]& \mathsf{\Omega}^{-1}L.}\] In particular, we have the following pull-back diagram of unit conflations;
\mathsf {b}egin{equation}\mathsf {L}abel{xy}
{\mathbf footnotesize \mathsf {X}ymatrix{\mathsf {d}elta_N : N~ \ \ \ \mathsf {a}r[r] \ \ \mathsf {a}r@{=}[d] & \ \ P^{1}\mathsf {a}r@{=}[d] \ \ \mathsf {a}r[r] \ \ & \ \ \operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r] \ \ &P^{n-1} \mathsf {a}r[r] \mathsf {a}r@{=}[d]& \ \ P^{n}\mathsf {a}r[d]\mathsf {a}r[r] \ \ & \mathsf{\Omega}^{-n}N\mathsf {a}r[d]^{gh}\\ \mathsf {b}eta : \ \ \ N~ \ \ \mathsf {a}r[r]\ \
\ & \ \ P^{1}\mathsf {a}r[r]\ \ & \operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r]\ \ & P^{n-1} \mathsf {a}r[r] & \ \ Q\mathsf {a}r[r] \ \ &{\mathsf{\Omega}}^{-1}L}.}
\end{equation}
One should note that, according to our notation, $\mathsf{\Omega}^{-1}L={\mathsf{\Omega}'}^{-n}N$.
Hence, for any object $X\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, we will have the following square;{\mathbf footnotesize \[\mathsf {X}ymatrix{\operatorname{{\mathsf{Ext}}}_{\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )}^{n+1}(N, X)~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng}\mathsf {a}r@{=}[d] & \operatorname{{\mathsf{Ext}}}_{\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )}^{2n+1}({\mathsf{\Omega}'}^{-n}N, X)\mathsf {a}r[d]^{\mathbf g\mathbf h}\\ \operatorname{{\mathsf{Ext}}}_{\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )}^{n+1}(N, X)~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng} & \operatorname{{\mathsf{Ext}}}_{\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )}^{2n+1}({\mathsf{\Omega}}^{-n}N, X).}\]}As $gh$ factors through $P^{n+1}$, the right column is zero, and then, $\operatorname{{\mathsf{Ext}}}_{\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )}^{n+1}(N, X)=0$, meaning that $N\in n$-$\mathcal Proj\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. Since the converse is obtained dually, we skip it. So the proof is finished.
\end{proof}
Assume that $R$ is a commutative noetherian ring. A finitely generated $R$-module $M$ is said to be of $G$-dimension zero, if it is a syzygy of a complete resolution of projectives. The $G$-dimension of a finitely generated $R$-module $N$, $G$-$\mathsf {d}im_RN$, is the length of a shortest resolution of $N$ by $G$-dimension zero modules. If there is no such a resolution of finite length, then we write $G$-$\mathsf {d}im_RN=\infty$. This invariant has been defined by Auslander and Bridger \operatorname{\underline{\mathscr{C}}}ite{ab} and provides a refinement of the projective dimension of a module. The category of all modules of $G$-dimension zero (resp. of finite $G$-dimension), is denoted by $\mathfrak{m}athcal{G}$ (resp. $\mathfrak{m}athcal{G}^{<\infty}$).
\mathsf {b}egin{example}\mathsf {L}abel{ex1} Assume that $(R, \mathfrak{m})$ is a $d$-dimensional commutative noetherian local ring. {In view of the Auslander-Buchsbaum formula, any module of finite projective dimension, is $d$-projective. Set $\mathfrak{m}athcal{I} :=d$-$\mathcal Proj\mathfrak{m}d R$. Assume that $M\in\mathfrak{m}athcal{G}^{<\infty}$ is arbitrary. Then by \operatorname{\underline{\mathscr{C}}}ite[Proposition 1.2]{et} (see also \operatorname{\underline{\mathscr{C}}}ite[Lemma 2.17]{cfh}), there is a short exact sequence $0\mathbf hookrightarrowghtarrow M\mathbf hookrightarrowghtarrow P\mathbf hookrightarrowghtarrow X\mathbf hookrightarrowghtarrow 0$ in which $P\in\mathfrak{m}athcal{I} $ and $X\in\mathfrak{m}athcal{G}$. All of these facts, would imply that $M\in\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. Moreover, it is standard to see that any object in $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$ lies in $\mathfrak{m}athcal{G}^{<\infty}$. Hence by Theorem \twoheadrightarrowf{subcat}, $\mathfrak{m}athcal{G}^{<\infty}$ is a $d$-Frobenius subcategory of $\mathfrak{m}d R$. In particular, if $R$ is Gorenstein, (i.e. $\id_RR$ is finite), then $\mathfrak{m}d R$ is indeed a $d$-Frobenius category. We also note that the category $\mathfrak{m}athcal{G}$, is a $0$-Frobenius subcategory of $\mathfrak{m}d R$.}
\end{example}
In the sequel, we explore more examples of categories which admit $n$-Frobenius subcategories, for some non-negative integer $n$.
\mathsf {b}egin{example}\mathsf {L}abel{ex123}
(1) According to \operatorname{\underline{\mathscr{C}}}ite[Lemma 1.12]{or}, {the category of locally free sheaves of finite rank, $\mathcal L$, is a subcategory of $n$-$\mathcal Proj{\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$, for some integer $n\mathbf geq 0$. So the category $n$-$\mathcal Proj{\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$ is non-trivial.} Thus Theorem \twoheadrightarrowf{subcat} yields that ${\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$ has a non-trivial $n$-Frobenius subcategory, that we denote it by $\mathfrak{m}athscr{C} (\mathsf {X})$.\\
(2) Assume that $\operatorname{\mathsf{Qcoh}}(\mathsf {X})$ is the category of quasi-coherent sheaves over $\mathsf {X}$ and $\mathsf {Flat}\mathsf {X}$ is its subcategory of flats. Following the argument given in the proof of \operatorname{\underline{\mathscr{C}}}ite[Lemma 1.12]{or}, gives rise to the existence of an integer $n\mathbf geq 0$ such that for any object $\mathcal{F}\in\mathsf {Flat}\mathsf {X}$, $\operatorname{{\mathsf{Ext}}}^{n+1}(\mathcal{F}, \mathfrak{m}athcal{Q} )=0$, for all quasi-coherent sheaves $\mathfrak{m}athcal{Q} $. Namely, $\mathsf {Flat}\mathsf {X}$ is a subcategory of $n$-$\mathcal Proj\operatorname{\mathsf{Qcoh}}(\mathsf {X})$. Thus $\operatorname{\mathsf{Qcoh}}(\mathsf {X})$ has enough $n$-projective objects, as it has enough flat sheaves; see \operatorname{\underline{\mathscr{C}}}ite[Corollary 3.21]{mu}. Hence, Theorem \twoheadrightarrowf{subcat} implies that $\operatorname{\mathsf{Qcoh}}(\mathsf {X})$ admits a non-trivial $n$-Frobenius subcategory $\mathfrak{m}athscr{C} (\mathsf {X})$. In addition, as $\mathsf {Flat}\mathsf {X}$ is a resolving subcategory of $n$-$\mathcal Proj\operatorname{\mathsf{Qcoh}}(\mathsf {X})$, another use of Theorem \twoheadrightarrowf{subcat} yields that $\mathfrak{m}athscr{C} (\mathsf {Flat}\mathsf {X})$ is also an $n$-Frobenius subcategory of $\operatorname{\mathsf{Qcoh}}(\mathsf {X})$.\\
\end{example}
Assume that $\mathfrak{m}athcal{I} $ is a resolving subcategory of $n$-$\mathcal Proj\mathcal A$. In view of the definition of $\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$, we have $\mathfrak{m}athcal{I} \subseteq n$-$\mathcal Proj\mathfrak{m}athscr{C} (\mathfrak{m}athcal{I} )$. The next couple of results, provide examples in which the equality holds.
\mathsf {b}egin{prop}\mathsf {L}abel{cf}Keeping the notation above, the equality $n$-$\mathcal Proj\mathfrak{m}athscr{C} (\mathsf {Flat}\mathsf {X})=\mathsf {Flat}\mathsf {X}$ holds.
\end{prop}
\mathsf {b}egin{proof}Since $\mathfrak{m}athscr{C} (\mathsf {Flat}\mathsf {X})$ is an $n$-Frobenius subcategory of $\operatorname{\mathsf{Qcoh}}(\mathsf {X})$, it suffices to prove that any object of $n$-$\operatorname{\mathsf{inj}}\mathfrak{m}athscr{C} (\mathsf {Flat}\mathsf {X})$ is flat. Take an arbitrary object $N\in n$-$\operatorname{\mathsf{inj}}\mathfrak{m}athscr{C} (\mathsf {Flat}\mathsf {X})$. Consider the unit conflations; $\mathsf {d}elta:N\mathbf hookrightarrowghtarrow P^{1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P^{n}\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-n}N$ and $\mathsf{\Omega}^{-n}N\stackrel{h}\mathbf hookrightarrowghtarrow P^{n+1}\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-n-1}N$, where $P^i\in\mathsf {Flat}\mathsf {X}$, for any $i$. So, {according to the proof of Theorem \twoheadrightarrowf{subcat}, we get the diagram \twoheadrightarrowf{xy}.} In particular, for any object $X\in\operatorname{\mathsf{Qcoh}}(\mathsf {X})$, we will have the following commutative square; {\mathbf footnotesize\[\mathsf {X}ymatrix{\mathfrak{m}athcal{T}or_{n+1}^{\mathsf {Ch}}(X, \mathsf{\Omega}^{-n}N)~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng}\mathsf {a}r[d]_{\mathfrak{m}athcal{T}or_1(1_X, gh)} & \mathfrak{m}athcal{T}or_{1}^{\mathsf {Ch}}(X, N)\mathsf {a}r@{=}[d]\\ \mathfrak{m}athcal{T}or_{n+1}^{\mathsf {Ch}}(X, {\mathsf{\Omega}'}^{-n}N)~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng} & \mathfrak{m}athcal{T}or_{1}^{\mathsf {Ch}}(X, N).}\]} As $gh$ factors through the flat sheaf $P^{n+1}$, the left column is zero, implying that $\mathfrak{m}athcal{T}or_1^{\mathsf {Ch}}(X, N)=0$, and then $N$ is flat, as needed.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{locally}For an integer $n\mathbf geq 0$, $\mathfrak{m}athscr{C} (\mathcal L)$ is an $n$-Frobenius subcategory of ${\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$ with $n$-$\mathcal Proj\mathfrak{m}athscr{C} (\mathcal L)=\mathcal L$.
\end{prop}
\mathsf {b}egin{proof} As we have mentioned in Example \twoheadrightarrowf{ex123}(1), $\mathcal L$ is a subcategory of $n$-$\mathcal Proj{\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$, which is evidently resolving. So by Theorem \twoheadrightarrowf{subcat}, $\mathfrak{m}athscr{C} (\mathcal L)$, is an $n$-Frobenius subcategory of ${\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$. Moreover, Proposition \twoheadrightarrowf{cf} and the fact that every coherent flat sheaf is locally free, leads us to deduce that $n$-$\mathcal Proj\mathfrak{m}athscr{C} (\mathcal L)=\mathcal L$. So the proof is completed.
\end{proof}
{
\mathsf {b}egin{rem}\mathsf {L}abel{chf}According to Example \twoheadrightarrowf{ex123}(2), there exists an integer $t\mathbf geq 0$ such that for any $F\in\mathsf {Flat}\mathsf {X}$, we have $F\in t$-$\mathcal Proj\mathsf {Flat}\mathsf {X}$. On the other hand, it is known that there is an integer $k\mathbf geq 0$ such that for any object $F\in\mathsf {Flat}\mathsf {X}$, one has an exact sequence $0\mathbf hookrightarrowghtarrow F\mathbf hookrightarrowghtarrow C^0\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow C^k\mathbf hookrightarrowghtarrow 0$, where each $C^i$ is cotorsion flat, implying that $F\in k$-$\operatorname{\mathsf{inj}}\mathsf {Flat}\mathsf {X}$, because each $C^i$ lies in $\operatorname{\mathsf{inj}}\mathsf {Flat}\mathsf {X}$. So, using the same method appeared in \operatorname{\underline{\mathscr{C}}}ite[page 28]{ha1}, we may deduce that any complex of the form $\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow 0\mathbf hookrightarrowghtarrow F\stackrel{1}\mathbf hookrightarrowghtarrow F\mathbf hookrightarrowghtarrow 0\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots$ with $F$ flat, is an $n$-projective and an $n$-injective object of $\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$, for some integer $n$. Here $\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$ stands for the category of complexes of flats. We let $\mathcal{J} $ denote the subcategory consisting of all contractible complexes of flats. {One should note that any object of $\mathcal{J} $ is a direct sum of complexes of the form $\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow 0\mathbf hookrightarrowghtarrow F\stackrel{1}\mathbf hookrightarrowghtarrow F\mathbf hookrightarrowghtarrow 0\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots$ with $F$ flat, and so, it belongs to $n$-$\mathcal Proj\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$ as well as to $n$-$\operatorname{\mathsf{inj}}\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$.} Assume that $F^{\mathsf {b}ullet}:\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow F^{-1}\mathbf hookrightarrowghtarrow F^0\mathbf hookrightarrowghtarrow F^{1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots$ is an arbitrary object of $\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$. So, according to the short exact sequence of complexes, $0\mathbf hookrightarrowghtarrow F^{\mathsf {b}ullet}\mathbf hookrightarrowghtarrow{\rm{op}}eratorname{\mathsf{cone}}(1_{F^{\mathsf {b}ullet}})\mathbf hookrightarrowghtarrow F^{\mathsf {b}ullet}[1]\mathbf hookrightarrowghtarrow 0$, we infer that $F^{\mathsf {b}ullet}$ is embeded in (and also a homomorphic image of) an object in $\mathcal{J} $. It is evident that $\mathcal{J} $ is a subcategory of $\mathsf {Ch}_{\mathcal Pp}(\mathsf {Flat}\mathsf {X})$, the category of flat complexes. The result below indicates that $\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$ is an $n$-Frobenius category with $n$-$\mathcal Proj\mathsf {Ch}(\mathsf {Flat}\mathsf {X})=\mathsf {Ch}_{\mathcal Pp}(\mathsf {Flat}\mathsf {X})$.
\end{rem}
\mathsf {b}egin{theorem}\mathsf {L}abel{fp}For a non-negative integer $n$, $\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$ is an $n$-Frobenius category, with $n$-$\mathcal Proj\mathsf {Ch}(\mathsf {Flat}\mathsf {X})=\mathsf {Ch}_{\mathcal Pp}(\mathsf {Flat}\mathsf {X})$.
\end{theorem}
\mathsf {b}egin{proof}{According to Remark \twoheadrightarrowf{chf}, $\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$ has enough $n$-projective and $n$-injective objects. So it remains to examine the validity of the equalities; $n$-$\mathcal Proj\mathsf {Ch}(\mathsf {Flat}\mathsf {X})=\mathsf {Ch}_{\mathcal Pp}(\mathsf {Flat}\mathsf {X})=n$-$\operatorname{\mathsf{inj}}\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$. In this direction, first we show that $n$-$\mathcal Proj\mathsf {Ch}(\mathsf {Flat}\mathsf {X})=n$-$\operatorname{\mathsf{inj}}\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$.
Take an arbitrary object $N^\mathsf {b}ullet\in n$-$\operatorname{\mathsf{inj}}\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$. By using Remark \twoheadrightarrowf{chf}, we may have conflations; $N^{\mathsf {b}ullet}\mathbf hookrightarrowghtarrow {P^{\mathsf {b}ullet}}^1\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow {P^{\mathsf {b}ullet}}^n\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-n}N^{\mathsf {b}ullet}$ and $\mathsf{\Omega}^{-n}N^{\mathsf {b}ullet}\stackrel{h^{\mathsf {b}ullet}}\mathbf hookrightarrowghtarrow {P^{\mathsf {b}ullet}}^{n+1}\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-n-1}N^{\mathsf {b}ullet}$, where $P_i^{\mathsf {b}ullet}\in\mathcal{J} $, for any $i$. Since $\mathsf{\Omega}^{-n-1}N^{\mathsf {b}ullet}\in\mathsf {Ch}(\mathsf {Flat}\mathsf {X}),$ we have $\operatorname{{\mathsf{Ext}}}^{n+1}_{\mathsf {Ch}}(\mathsf{\Omega}^{-n-1}N^{\mathsf {b}ullet}, N^{\mathsf {b}ullet})=0$.
So, the argument given in the proof of Theorem \twoheadrightarrowf{subcat}, gives us the diagram similar to \twoheadrightarrowf{xy}. Take an arbitrary object $X^{\mathsf {b}ullet}\in\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$. As for any $i>n$, $\operatorname{{\mathsf{Ext}}}^i_{\mathsf {Ch}}(P_j^{\mathsf {b}ullet}, X^{\mathsf {b}ullet})=0$, applying the functor $\operatorname{{\mathsf{Ext}}}_{\mathsf {Ch}}(-, X^{\mathsf {b}ullet})$, gives rise to the following commutative square; {\mathbf footnotesize \[\mathsf {X}ymatrix{\operatorname{{\mathsf{Ext}}}_{\mathsf {Ch}}^{i}(N^{\mathsf {b}ullet}, X^{\mathsf {b}ullet})~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng}\mathsf {a}r@{=}[d] & \operatorname{{\mathsf{Ext}}}_{\mathsf {Ch}}^{n+i}({\mathsf{\Omega}'}^{-n}N^{\mathsf {b}ullet}, X^{\mathsf {b}ullet})\mathsf {a}r[d]^{\mathbf g^{\mathsf {b}ullet}\mathbf h^{\mathsf {b}ullet}}\\ \operatorname{{\mathsf{Ext}}}_{\mathsf {Ch}}^{i}(N^{\mathsf {b}ullet}, X^{\mathsf {b}ullet})~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng} & \operatorname{{\mathsf{Ext}}}_{\mathsf {Ch}}^{n+i}({\mathsf{\Omega}}^{-n}N^{\mathsf {b}ullet}, X^{\mathsf {b}ullet}).}\]}Since $g^{\mathsf {b}ullet}h^{\mathsf {b}ullet}$ factors through an object of $\mathcal{J} $, the right column will be zero, and then, $\operatorname{{\mathsf{Ext}}}_{\mathsf {Ch}}^{i}(N^{\mathsf {b}ullet}, X^{\mathsf {b}ullet})=0$ for any $i>n$, meaning that $N^{\mathsf {b}ullet}\in n$-$\mathcal Proj\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$. The dual method indicates that if $N^{\mathsf {b}ullet}\in n$-$\mathcal Proj\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$, it will belong to $n$-$\operatorname{\mathsf{inj}}\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$. Next assume that $N^\mathsf {b}ullet\in n$-$\operatorname{\mathsf{inj}}\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$ and consider the latter conflations. Take an arbitrary object $X\in\operatorname{\mathsf{Qcoh}}(\mathsf {X})$. As ${P^{\mathsf {b}ullet}}^j$, for any $j$, is contractible, $\operatorname{\mathsf{H}}_i({P^{\mathsf {b}ullet}}^j\otimes_{\mathcal{O}_{\mathsf {X}}}X)=0$ for all $i\in \mathfrak{m}athbb{Z} $. Hence, one may obtain the following commutative square; \[\mathsf {X}ymatrix{\operatorname{\mathsf{H}}_{n+i}(\mathsf{\Omega}^{-n}N^\mathsf {b}ullet\otimes_{\mathcal{O}_{\mathsf {X}}} X)~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng}\mathsf {a}r[d] & \operatorname{\mathsf{H}}_{i}(N^\mathsf {b}ullet\otimes_{\mathcal{O}_{\mathsf {X}}} X)\mathsf {a}r@{=}[d]\\ \operatorname{\mathsf{H}}_{n+i}({\mathsf{\Omega}'}^{-n}N^\mathsf {b}ullet\otimes_{\mathcal{O}_{\mathsf {X}}} X)~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng} & \operatorname{\mathsf{H}}_{i}(N^\mathsf {b}ullet\otimes_{\mathcal{O}_{\mathsf {X}}} X).}\] Since $g^{\mathsf {b}ullet}h^{\mathsf {b}ullet}$ factors through ${P^{\mathsf {b}ullet}}^{n+1}$, the left column will be zero, implying that $\operatorname{\mathsf{H}}_i(N^\mathsf {b}ullet\otimes_{\mathcal{O}_{\mathsf {X}}}X)=0$, for all $i\in\mathfrak{m}athbb{Z} $. Since $X$ was arbitrary, we conclude that $N^\mathsf {b}ullet\in\mathsf {Ch}_{\mathcal Pp}(\mathsf {Flat}\mathsf {X})$.
Conversely, assume that $N^\mathsf {b}ullet\in\mathsf {Ch}_{\mathcal Pp}(\mathsf {Flat}\mathsf {X})$. By \operatorname{\underline{\mathscr{C}}}ite[Proposition 2.6]{hs}, there is a short exact sequence, $0\mathsf {L}ongrightarrow N^\mathsf {b}ullet\mathsf {L}ongrightarrow {C^\mathsf {b}ullet}^0\mathsf {L}ongrightarrow {P^\mathsf {b}ullet}^1\mathsf {L}ongrightarrow 0$ in $\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$ with ${P^\mathsf {b}ullet}^1\in\mathsf {Ch}_{\mathcal Pp}(\mathsf {Flat}\mathsf {X})$ and
${C^\mathsf {b}ullet}^0$ is dg-cotorsion, that is, ${C^\mathsf {b}ullet}^0\in\mathsf {Ch}_{\mathcal Pp}(\mathsf {Flat}\mathsf {X})^{\mathsf {b}ot}$.
Thus ${C^\mathsf {b}ullet}^0\in\mathsf {Ch}_{\mathcal Pp}(\mathsf {Flat}\mathsf {X})$, as well. Repeating this manner, one may obtain an exact sequence, $0\mathsf {L}ongrightarrow N^\mathsf {b}ullet\mathsf {L}ongrightarrow {C^\mathsf {b}ullet}^0\mathsf {L}ongrightarrow {C^\mathsf {b}ullet}^1\mathsf {L}ongrightarrow\operatorname{\underline{\mathscr{C}}}dots\mathsf {L}ongrightarrow {C^\mathsf {b}ullet}^{k-1}\mathsf {L}ongrightarrow {C^\mathsf {b}ullet}^k\mathsf {L}ongrightarrow 0$ in $\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$, with $k=\mathsf {d}im\mathsf {X}$. Since ${C^\mathsf {b}ullet}^i$, for any $i$, is a pure acyclic complex of cotorsion flats, it is contractible by \operatorname{\underline{\mathscr{C}}}ite[Corollary 3.1.2]{hs}, and in particular, it will belong to $n$-$\mathcal Proj\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$. Hence, it is easily seen that $N^{\mathsf {b}ullet}\in n$-$\mathcal Proj\mathsf {Ch}(\mathsf {Flat}\mathsf {X})$. So the proof is finished.}
\end{proof}
\mathsf {b}egin{rem}\mathsf {L}abel{remfin}As we have mentioned in Example \twoheadrightarrowf{ex123}(1), $\mathcal L$ is a subcategory of $n$-$\mathcal Proj{\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$, for some integer $n$. On the other hand, since any locally free sheaf is flat and, as observed in Remark \twoheadrightarrowf{chf}, each flat sheaf lies in $n$-$\operatorname{\mathsf{inj}}\mathcal L$, we conclude that $\mathcal L$ will be also a subcategory of $n$-$\operatorname{\mathsf{inj}}\mathcal L$. So, similar to Remark \twoheadrightarrowf{chf}, we infer that any complex of the form $\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow 0\mathbf hookrightarrowghtarrow L\stackrel{1}\mathbf hookrightarrowghtarrow L\mathbf hookrightarrowghtarrow 0\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots$, with $L$ locally free, is an $n$-projective and $n$-injective over $\mathsf {Ch}(\mathcal L)$, the category of complexes of locally free sheaves of finite rank. Assume that $\mathcal{J} $ is the subcategory consisting of all contractible complexes of locally free sheaves of finite rank. Again, similar to Remark \twoheadrightarrowf{chf}, one may observe that any object of $\mathsf {Ch}(\mathcal L)$ can be embedded in an object of $\mathcal{J} $, as well as, it is a homomorphic image of an objects in $\mathcal{J} $. The same argument given in the proof of Theorem \twoheadrightarrowf{fp}, clarifies that the subcategory $\mathsf {Ch}_{\mathcal Pp}(\mathcal L)$ consisting of all acyclic complexes of locally free sheaves with locally free kernels, forms $n$-projective objects of $\mathsf {Ch}(\mathcal L)$. Precisely, we have the next interesting result.
\end{rem}
\mathsf {b}egin{theorem}$\mathsf {Ch}(\mathcal L)$ is an $n$-Frobenius category, for some integer $n$. Moreover, $n$-$\mathcal Proj\mathsf {Ch}(\mathcal L)=\mathsf {Ch}_{\mathcal Pp}(\mathcal L)$.
\end{theorem}
\section{Quasi-invertible morphisms}
Assume that $\mathfrak{m}athscr{C} $ is an $n$-Frobenius category. In this section, we will show that a morphism $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}_{\mathfrak{m}athscr{C} }^{n+1}$ from the left and from the right, simultaneously. A morphism satisfying this condition will be called a quasi-invertible morphism in $\mathfrak{m}athscr{C} $.
In the remainder of this paper, unless otherwise specified, by a conflation of length $t$, we mean a conflation of length $t$ in $\mathfrak{m}athscr{C} $. Also, if there is no ambiguity, we drop the ``of length $t$''. Furthermore, instead of $\operatorname{{\mathsf{Ext}}}^i_{\mathfrak{m}athscr{C} }(-,-)$, we write $\operatorname{{\mathsf{Ext}}}^i(-,-)$.
\mathsf {b}egin{lem}\mathsf {L}abel{101}(1) Let $N\stackrel{f}\mathbf hookrightarrowghtarrow X$ and $N\stackrel{g}\mathbf hookrightarrowghtarrow X'$ be two morphisms in $\mathfrak{m}athscr{C} $ such that $f$ or $g$ is an inflation. Then $N\stackrel{[f~~g]^t}\mathsf {L}ongrightarrow X{\rm{op}}lus X'$ is also an inflation.\\ (2) Let $X\stackrel{f}\mathbf hookrightarrowghtarrow N$ and $X'\stackrel{g}\mathbf hookrightarrowghtarrow N$ be two morphisms in $\mathfrak{m}athscr{C} $ such that $f$ or $g$ is a deflation. Then $X{\rm{op}}lus X'\stackrel{[f~~g]}\mathsf {L}ongrightarrow N$ is a deflation.
\end{lem}
\mathsf {b}egin{proof}Let us prove the first assertion. The second one is obtained dually.
Without loss of generality, we may assume that $f$ is an inflation. Consider the following commutative diagram; {\mathbf footnotesize\[\mathsf {X}ymatrix{ & X'\mathsf {a}r@{=}[r]\mathsf {a}r[d]& X'\mathsf {a}r[d]\\ N~\mathsf {a}r[r]^{{{\tiny {\mathsf {L}eft[\mathsf {b}egin{array}{ll} f \\ g \end{array} \mathbf hookrightarrowght]}}}}\mathsf {a}r@{=}[d] & X{\rm{op}}lus X' \mathsf {a}r[r]\mathsf {a}r[d]_{[1~0]}&L'\mathsf {a}r[d]_{h}\\ N\mathsf {a}r[r]^{f}&X\mathsf {a}r[r]& L.}\]}Since the bottom row is a conflation and $\mathfrak{m}athscr{C} $ is closed under extensions, we infer that $L'\in\mathfrak{m}athscr{C} $, and so, $h$ is a morphism in $\mathfrak{m}athscr{C} $. Now, as $\mathfrak{m}athscr{C} $ is closed under pull-back, the middle row will be a conflation, giving the desired result.
\end{proof}
\mathsf {b}egin{dfn}Assume that $f:M\mathbf hookrightarrowghtarrow N$ is a morphism in $\mathfrak{m}athscr{C} $ and $i\mathbf geq 0$ is an integer. We say that $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^i$ from the left (resp. right), if for any $X\in\mathfrak{m}athscr{C} $, $\operatorname{{\mathsf{Ext}}}^i(X, M)\stackrel{\mathbf f}\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^i(X, N)$ (resp. $\operatorname{{\mathsf{Ext}}}^i(N, X)\stackrel{\mathbf f}\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^i(M, X)$) is an isomorphism.
\end{dfn}
{
\mathsf {b}egin{lem}\mathsf {L}abel{conf}
Let $M\stackrel{f}\mathbf hookrightarrowghtarrow N$ be a morphism in $\mathfrak{m}athscr{C} $ acting as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$ from the left or from the right. Assume that $Q\stackrel{\mathcal Pi}\mathbf hookrightarrowghtarrow N$ is a deflation and $M\stackrel{l}\mathbf hookrightarrowghtarrow P$ is an inflation such that $P, Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Then
\mathsf {b}egin{enumerate}\item $M{\rm{op}}lus Q\stackrel{[f~~\mathcal Pi]}\mathbf hookrightarrowghtarrow N$ is a deflation such that its kernel is $n$-projective.\item $M\stackrel{[f~~l]^{t}}\mathbf hookrightarrowghtarrow N{\rm{op}}lus P$ is an inflation such that its cokernel is $n$-projective.\end{enumerate}
\end{lem}
\mathsf {b}egin{proof}We only deal with the case $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$ from the left. The other case can be treated by the similar way. Since $M\stackrel{l}\mathbf hookrightarrowghtarrow P$ is an inflation, by Lemma \twoheadrightarrowf{101}(1), we have the conflation $M\stackrel{h}\mathbf hookrightarrowghtarrow N{\rm{op}}lus P\mathbf hookrightarrowghtarrow L$ in $\mathfrak{m}athscr{C} $, where $h={{{\tiny {\mathsf {L}eft[\mathsf {b}egin{array}{ll} f \\ l \end{array} \mathbf hookrightarrowght]}}}}$. {We shall prove that $L\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Take an arbitrary object $X\in\mathfrak{m}athscr{C} $. By applying the functor $\operatorname{{\mathsf{Ext}}}(X, -)$ to this conflation and using Remark \twoheadrightarrowf{exact}, } we obtain the long exact sequence;{\mathbf footnotesize $$\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{i}(X, M)\stackrel{\mathbf h}\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{i}(X, N{\rm{op}}lus P)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{i}(X, L)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{i+1}(X, M)\stackrel{\mathbf h}\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{i+1}(X, N{\rm{op}}lus P)\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots.$$}{As $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$, it is easily seen that it acts as invertible on $\operatorname{{\mathsf{Ext}}}^i$ for all $i>n$. Moreover,} since $P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, $l\operatorname{{\mathsf{Ext}}}^{i}=0$ for all $i>n$, implying that $f\operatorname{{\mathsf{Ext}}}^{i}=h\operatorname{{\mathsf{Ext}}}^{i}$. In particular, $h$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^{i}$, and then, $\operatorname{{\mathsf{Ext}}}^{i}(X, L)=0$ for all $i>n$, meaning that $L\in n$-$\operatorname{\mathsf{inj}}\mathfrak{m}athscr{C} $, and so, it belongs to $n$-$\mathcal Proj\mathfrak{m}athscr{C} $, {giving the first assertion. For the second one,}
one may apply Lemma \twoheadrightarrowf{101}(2) to obtain the conflation $L'\mathbf hookrightarrowghtarrow M{\rm{op}}lus Q\stackrel{[f~~\mathcal Pi]}\mathbf hookrightarrowghtarrow N$. So, repeating the above method, we deduce that $\operatorname{{\mathsf{Ext}}}^{i}(X, L')=0$ for all $i>n+1$. Now, as $\mathfrak{m}athscr{C} $ is $n$-Frobenius, it is easily seen that $L'\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. So, the proof is finished.
\end{proof}
\mathsf {b}egin{cor}\mathsf {L}abel{lr}Let $f:M\mathbf hookrightarrowghtarrow N$ be a morphism in $\mathfrak{m}athscr{C} $. Then $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$ from the left if and only if it acts as invertible from the right.
\end{cor}
\mathsf {b}egin{proof}Assume that $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$ from the left. In view of Lemma \twoheadrightarrowf{conf}, there exists a conflation $M\stackrel{h}\mathbf hookrightarrowghtarrow N{\rm{op}}lus P\mathbf hookrightarrowghtarrow P'$ in $\mathfrak{m}athscr{C} $, where $P, P'\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Suppose that $X\in\mathfrak{m}athscr{C} $ is arbitrary. So by applying the functor $\operatorname{{\mathsf{Ext}}}(-, X)$ to this conflation, we get the following long exact sequence; $$\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{i}(P', X)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{i}(N{\rm{op}}lus P, X)\stackrel{\mathbf h}\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{i}(M, X)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{i+1}(P', X)\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots.$$ As $\operatorname{{\mathsf{Ext}}}^{i}(P', X)=0$, for any $i>n$, $h$ (and so, $f$) will act as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$ from the right. Since the sufficiency can be shown in a dual manner, we ignore it. So the proof is finished.
\end{proof}
}
\mathsf {b}egin{lem}\mathsf {L}abel{epi}
Let $M\stackrel{f}\mathbf hookrightarrowghtarrow N$ be a morphism in $\mathfrak{m}athscr{C} $ such that $\operatorname{\mathsf{ker}} f, {\rm{op}}eratorname{\mathsf{cone}}k f\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Assume that $Q\stackrel{\mathcal Pi}\mathbf hookrightarrowghtarrow N$ is a deflation and $M\stackrel{i}\mathbf hookrightarrowghtarrow P$ is an inflation such that $P, Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Then
\mathsf {b}egin{enumerate}\item
$M{\rm{op}}lus Q\stackrel{[f~~\mathcal Pi]}\mathsf {L}ongrightarrow N$ is a deflation such that its kernel is $n$-projective.\item $M\stackrel{[f~~i]^{t}}\mathsf {L}ongrightarrow N{\rm{op}}lus P$ is an inflation such that its cokernel is $n$-projective.
\end{enumerate}
\end{lem}
\mathsf {b}egin{proof}Let us prove only the first assertion. Then second one is obtained dually.
By the hypothesis, there exist conflations $P'\mathbf hookrightarrowghtarrow M\stackrel{h}\mathbf hookrightarrowghtarrow L$ and $L\stackrel{g}\mathbf hookrightarrowghtarrow N\mathbf hookrightarrowghtarrow P''$ such that $gh=f$ and $P', P''\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Consider the following commutative diagram; {\mathbf footnotesize\[\mathsf {X}ymatrix{T~\mathsf {a}r[r]\mathsf {a}r[d]_{\operatorname{\mathsf{V}}arphi}& Q\mathsf {a}r[r]^{l\mathcal Pi}\mathsf {a}r[d]_{\mathcal Pi}& P''\mathsf {a}r@{=}[d]\\ L~\mathsf {a}r[r]^{g} & N\mathsf {a}r[r]^l& P'',}\]}where $\operatorname{\mathsf{V}}arphi$ is an induced map. One should note that since $l$ and $\mathcal Pi$ are deflation, $l\mathcal Pi$ is so. Thus, the top row is also a conflation. Since $Q, P''\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, the same is true for $T$.
In particular, we have the conflation $T\stackrel{[-\operatorname{\mathsf{V}}arphi~~\mathsf {a}lpha]^t}\mathsf {L}ongrightarrow L{\rm{op}}lus Q\stackrel{[g~~\mathcal Pi]}\mathsf {L}ongrightarrow N$. Now consider the following pull-back diagram; {\mathbf footnotesize\[\mathsf {X}ymatrix{P'~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& T'\mathsf {a}r[r]\mathsf {a}r[d]& T\mathsf {a}r[d]\\ P'~\mathsf {a}r[r] & M{\rm{op}}lus Q\mathsf {a}r[r]^{u}\mathsf {a}r[d]& L{\rm{op}}lus Q\mathsf {a}r[d]\\ & N~\mathsf {a}r@{=}[r] & N,}\]}where $u={\tiny {\mathsf {L}eft[\mathsf {b}egin{array}{ll} h & 0 \\ 0 & {1} \end{array} \mathbf hookrightarrowght]}}$. Evidently, $T'$ is $n$-projective, because
$P',T\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Consequently, $T'\mathbf hookrightarrowghtarrow M{\rm{op}}lus Q\stackrel{[f~~\mathcal Pi]}\mathbf hookrightarrowghtarrow N$ is the desired conflation. So the proof is finished.
\end{proof}
As a consequence of Lemma \twoheadrightarrowf{epi} and the proof of Corollary \twoheadrightarrowf{lr}, we include the next result.
\mathsf {b}egin{cor}\mathsf {L}abel{is}Let $f:M\mathbf hookrightarrowghtarrow N$ be a morphism in $\mathfrak{m}athscr{C} $ such that $\operatorname{\mathsf{ker}} f, {\rm{op}}eratorname{\mathsf{cone}}k f\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Then $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$.
\end{cor}
\mathsf {b}egin{dfn}We say that a given morphism $f$ in $\mathfrak{m}athscr{C} $ is quasi-invertible, provided that $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$. The class of all quasi-invertible morphisms will be denoted by $\mathsf{\Sigma}$.
\end{dfn}
\mathsf {b}egin{rem}\mathsf {L}abel{rems}Assume that $f:M\mathbf hookrightarrowghtarrow N$ is a morphism in $\mathsf{\Sigma}$. As we have seen in the proof of Lemma \twoheadrightarrowf{conf}, by taking an inflation (resp. a deflation) $g:M\mathbf hookrightarrowghtarrow P$ (resp. $g:P\mathbf hookrightarrowghtarrow N$), we will obtain an inflation (resp. a deflation) $M\stackrel{[f~~g]^t}\mathsf {L}ongrightarrow N{\rm{op}}lus P$ (resp. $M{\rm{op}}lus P\stackrel{[f~~g]}\mathsf {L}ongrightarrow N$) such that its cokernel (resp. kernel) is $n$-projective. Moreover, since $\operatorname{{\mathsf{Ext}}}^{n+1}(-, Q)=0=\operatorname{{\mathsf{Ext}}}^{n+1}(Q, -)$, for any $Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, we may deduce that the maps $f$, ${\tiny {\mathsf {L}eft[\mathsf {b}egin{array}{ll} f \\ g \end{array} \mathbf hookrightarrowght]}}$ and $[f~~g]$ act identically on $\operatorname{{\mathsf{Ext}}}^{n+1}$. So, if $f$ acts as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$, without loss of generality, we may further assume that $f$ is an inflation or a deflation with cokernel and kernel $n$-projective, respectively.
\end{rem}
The result below reveals that being a unit conflation is stable under the pull-back and push-out along morphisms in $\mathsf{\Sigma}$.
\mathsf {b}egin{lem}\mathsf {L}abel{unit}Let $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^k(N, \mathsf{\Omega}^k N)$ with $k\mathbf geq 1$ and let $a:X\mathbf hookrightarrowghtarrow N$ and $b:\mathsf{\Omega}^kN\mathbf hookrightarrowghtarrow Y$ be two morphisms in $\mathsf{\Sigma}$. Then the following assertions hold:
\mathsf {b}egin{enumerate}\item $\mathbf ga$ is a unit conflation if and only if $\mathbf ga a$ is so. \item $\mathbf ga$ is a unit conflation if and only if $b\mathbf ga$ is so.
\end{enumerate}
\end{lem}
\mathsf {b}egin{proof}We only prove the first assertion. The second one is obtained dually. First one should note that, by the definition of pull-back diagram, without loss of generality, we may assume that $k=1$. As $a\in\mathsf{\Sigma}$ , by Lemma \twoheadrightarrowf{conf}, there exists a conflation $Q\mathbf hookrightarrowghtarrow X{\rm{op}}lus P\stackrel{[a~~\mathcal Pi]}\mathbf hookrightarrowghtarrow N$, where $P, Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. So taking the following pull-back diagram; {\mathbf footnotesize \[\mathsf {X}ymatrix{\mathbf ga [a~~\mathcal Pi]: \mathsf{\Omega} N~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& H\mathsf {a}r[r]\mathsf {a}r[d]& X{\rm{op}}lus P\mathsf {a}r[d]_{[a~~\mathcal Pi]}\\ \mathbf ga: \mathsf{\Omega} N \mathsf {a}r[r] & T\mathsf {a}r[r] & N,}\]}gives rise to the conflation, $Q\mathbf hookrightarrowghtarrow H\mathbf hookrightarrowghtarrow T$. We show that $H$ and $T$ are $n$-projective, simultaneously. If $T$ is $n$-projective, then the same is true for $H$, because $n$-$\mathcal Proj\mathfrak{m}athscr{C} $ is closed under extensions. Conversely, assume that $H$ is $n$-projective. As $\mathfrak{m}athscr{C} $ is an $n$-Frobenius category, it is easily seen that the class $n$-$\mathcal Proj\mathfrak{m}athscr{C} $ is closed under cokernels of monomorphisms, implying that $T$ is $n$-projective. This means that the conflation $\mathbf ga$ is unit if and only if $\mathbf ga[a~~\mathcal Pi]$ is so. Next considering the following pull-back diagram; {\mathbf footnotesize\[\mathsf {X}ymatrix{\mathsf {b}e:\mathsf{\Omega} N~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& L\mathsf {a}r[r]\mathsf {a}r[d]& X\mathsf {a}r[d]_{{{\tiny {\mathsf {L}eft[\mathsf {b}egin{array}{ll} 1\\ 0 \end{array} \mathbf hookrightarrowght]}}}}\\ \mathbf ga[a~~\mathcal Pi]:\mathsf{\Omega} N~\mathsf {a}r[r] & H\mathsf {a}r[r]\mathsf {a}r[d]& X{\rm{op}}lus P\mathsf {a}r[d]\\ & P~\mathsf {a}r@{=}[r] & P,}\]}we conclude that $\mathbf ga[a~~\mathcal Pi]$ and $\mathsf {b}e$ are unit conflation, simultaneously. Now the equality $\mathsf {b}e=(\mathbf ga[a~~\mathcal Pi]){{{\tiny {\mathsf {L}eft[\mathsf {b}egin{array}{ll} 1\\ 0 \end{array} \mathbf hookrightarrowght]}}}}=\mathbf ga a$, completes the proof.
\end{proof}
We close this section with the following result.
\mathsf {b}egin{lem}\mathsf {L}abel{sig}
Let $f:M\mathbf hookrightarrowghtarrow N$ be an inflation or a deflation in $\mathfrak{m}athscr{C} $. Then the following assertions hold:
\mathsf {b}egin{enumerate}
\item If there exists $\mathsf {d}elta_N\in\mathcal U_n(N)$ such that $\mathsf {d}elta_Nf\in\mathcal U_n(M)$, then $f$ lies in $\mathsf{\Sigma}$.
\item If there exists $\mathsf {d}elta_M\in\mathcal U^n(M)$ such that $f\mathsf {d}elta_M\in\mathcal U^n(N)$, then $f$ belongs to $\mathsf{\Sigma}$.
\end{enumerate}
\end{lem}
\mathsf {b}egin{proof}
We only prove the first assertion. The second one is obtained dually. Without loss of generality,
we assume that $f$ is an inflation. By the definition of pull-back diagram, we may assume that $n=1$. By the hypothesis, there exists a pull-back diagram;
{\mathbf footnotesize\[\mathsf {X}ymatrix{\mathsf{\Omega} N~\mathsf {a}r[r] \mathsf {a}r@{=}[d]& T\mathsf {a}r[r]\mathsf {a}r[d]_{h}& M\mathsf {a}r_{f}[d]\\ \mathsf {d}elta_N:\mathsf{\Omega} N~\mathsf {a}r[r] & P\mathsf {a}r[r]& N,}\]}where $P,T\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Now since $h$ is an inflation, we have that ${\rm{op}}eratorname{\mathsf{cone}}k h$ is $n$-projective, and so, the same will be true for ${\rm{op}}eratorname{\mathsf{cone}}k f$. Now Corollary \twoheadrightarrowf{is} completes the proof.
\end{proof}
\section{$\mathcal P$-subfunctor of $\operatorname{{\mathsf{Ext}}}^n$}
Assume that $\mathfrak{m}athscr{C} $ is an $n$-Frobenius category. The aim of this section is to study a subfunctor of $\operatorname{{\mathsf{Ext}}}^n$ consisting of all conflations arising as a pull-back along morphisms ending at $n$-projective objects that we call a $\mathcal P$-subfunctor of $\operatorname{{\mathsf{Ext}}}^n$. We begin with the following useful observation.
\mathsf {b}egin{rem}\mathsf {L}abel{use}Assume that $X, Z$ are arbitrary objects of $\mathfrak{m}athscr{C} $. Consider the unit conflations $Z\mathbf hookrightarrowghtarrow P\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-1}Z$ and $\mathsf{\Omega} X\mathbf hookrightarrowghtarrow Q\mathbf hookrightarrowghtarrow X$. So, we will have the following commutative diagram with exact rows and columns;
\[\mathsf {X}ymatrix{
\operatorname{{\mathsf{Ext}}}^n(P, Q)~\mathsf {a}r[r]\mathsf {a}r[d]& \operatorname{{\mathsf{Ext}}}^n(Z, Q)\mathsf {a}r[r]\mathsf {a}r[d]_{\mathsf {b}eta}& \operatorname{{\mathsf{Ext}}}^{n+1}(\mathsf{\Omega}^{-1}Z, Q)\mathsf {a}r[d]\mathsf {a}r[r] &0\\ \operatorname{{\mathsf{Ext}}}^n(P, X)~\mathsf {a}r[r]^{\mathsf {a}lpha}\mathsf {a}r[d]& \operatorname{{\mathsf{Ext}}}^n(Z, X)\mathsf {a}r[r]^{\mathcal Psi}\mathsf {a}r[d]_{\operatorname{\mathsf{V}}arphi}& \operatorname{{\mathsf{Ext}}}^{n+1}(\mathsf{\Omega}^{-1}Z, X)\mathsf {a}r[d]_{\operatorname{\boldsymbol{\eta}}a}\mathsf {a}r[r] &0\\ \operatorname{{\mathsf{Ext}}}^{n+1}(P, \mathsf{\Omega} X) \mathsf {a}r[r] & \operatorname{{\mathsf{Ext}}}^{n+1}(Z, \mathsf{\Omega} X)\mathsf {a}r[r]^{\operatorname{\boldsymbol{\theta}}eta} & \operatorname{{\mathsf{Ext}}}^{n+2}(\mathsf{\Omega}^{-1}Z, \mathsf{\Omega} X)\mathsf {a}r[r]& 0.}\]
Since $P,Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, $\operatorname{{\mathsf{Ext}}}^{n+1}(P, \mathsf{\Omega} X)=0=\operatorname{{\mathsf{Ext}}}^{n+1}(\mathsf{\Omega}^{-1}Z, Q)$, and so, we may deduce that $\mathfrak{m}athsf{im}\mathsf {a}lpha=\mathfrak{m}athsf{im}\mathsf {b}eta$.
\end{rem}
\mathsf {b}egin{s}\mathsf {L}abel{use1}Assume that $M,N\in\mathfrak{m}athscr{C} $ and $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(M,N)$ such that there is a morphism $f:M\mathbf hookrightarrowghtarrow P$ with $P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $ and a conflation $\operatorname{\boldsymbol{\eta}}\in\operatorname{{\mathsf{Ext}}}^n(P, N)$ such that $\mathbf ga=\operatorname{\boldsymbol{\eta}} f$. Since $\mathfrak{m}athscr{C} $ is $n$-Frobenius, there is an inflation $i:M\mathbf hookrightarrowghtarrow P'$, where $P'\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $.
According to Lemma \twoheadrightarrowf{101}(1), $[f~~i]^t:M\mathbf hookrightarrowghtarrow P{\rm{op}}lus P'$ is also an inflation. Now letting $\operatorname{\boldsymbol{\eta}}'=\operatorname{\boldsymbol{\eta}}{\rm{op}}lus(0\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow 0\mathbf hookrightarrowghtarrow P'\mathbf hookrightarrowghtarrow P')$, we have
$\mathbf ga=\operatorname{\boldsymbol{\eta}}'[f~~i]^t$. Consequently, without loss of generality, we may assume that $f$ is an inflation. Dually, if $\mathbf ga=g\mathsf {b}e$, for some morphism
$g:Q\mathbf hookrightarrowghtarrow N$, with $Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $ and $\mathsf {b}e\in\operatorname{{\mathsf{Ext}}}^n(M, Q)$, one may assume that $g$ is a deflation.
\end{s}
The result below is an immediate consequence of Remark \twoheadrightarrowf{use} and \twoheadrightarrowf{use1}. So we omit its proof.
\mathsf {b}egin{prop}\mathsf {L}abel{equal}Let $X, Z$ be arbitrary objects of $\mathfrak{m}athscr{C} $ and $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(Z, X)$. Then the following statements are equivalent:
\mathsf {b}egin{enumerate}\item There is an object $\operatorname{\boldsymbol{\eta}}\in\operatorname{{\mathsf{Ext}}}^n(P, X)$, with $P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, and a morphism $Z\stackrel{f}\mathbf hookrightarrowghtarrow P$ such that $\mathbf ga=\operatorname{\boldsymbol{\eta}} f$.
\item There is an object $\operatorname{\boldsymbol{\eta}}'\in\operatorname{{\mathsf{Ext}}}^n(Z, Q)$, with $Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, and a morphism $Q\stackrel{f'}\mathbf hookrightarrowghtarrow X$ such that $\mathbf ga=f'\operatorname{\boldsymbol{\eta}}'$.
\end{enumerate}
\end{prop}
\mathsf {b}egin{s}\mathsf {L}abel{pp}
{\sc $\mathcal P$-subfunctor.} For every pair $X, Y$ of objects $\mathfrak{m}athscr{C} $, we let $\mathcal P(X, Y)$ denote the additive subgroup of $\operatorname{{\mathsf{Ext}}}^n(X, Y)$ satisfying one of the equivalent conditions in Proposition \twoheadrightarrowf{equal}. It is easily seen that for given morphisms $A\stackrel{f}\mathbf hookrightarrowghtarrow X$ and $Y\stackrel{g}\mathbf hookrightarrowghtarrow B$ in $\mathfrak{m}athscr{C} $, the natural transformation $\operatorname{{\mathsf{Ext}}}(f, g):\operatorname{{\mathsf{Ext}}}^n(X, Y)\mathsf {L}ongrightarrow\operatorname{{\mathsf{Ext}}}^n(A, B)$ respects $\mathcal P$. Namely, for any $\mathbf ga\in\mathcal P(X, Y)$, $g(\mathbf ga f)=(g\mathbf ga)f\in\mathcal P(A, B)$. Consequently, $\mathcal P$ is a subfunctor of $\operatorname{{\mathsf{Ext}}}^n$; see \operatorname{\underline{\mathscr{C}}}ite{as, fght}. Indeed, from our point of view, $\mathcal P$ is a submodule of $\operatorname{{\mathsf{Ext}}}^n$. A given conflation $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(X, Y)$ will be called a {\em $\mathcal P$-conflation,} whenever $\mathbf ga$ belongs to $\mathcal P(X, Y)$. It is worth noting that in the case $n=0$, $\mathcal P$-conflations are those morphisms in $\mathfrak{m}athscr{C} $ factoring through projective objects. If there is no ambiguity, we denote $\mathcal P(-, -)$ by $\mathcal P$.
\end{s}
The next result is also a direct consequence of Remark \twoheadrightarrowf{use} and \twoheadrightarrowf{use1}. So we ignore its proof.
\mathsf {b}egin{cor}\mathsf {L}abel{lem2}Let $f:Z\mathbf hookrightarrowghtarrow P$ with $P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, be an inflation. Then any $\mathcal P$-conflation $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(Z, X)$, factors through $f$.
\end{cor}
{
\mathsf {b}egin{prop}\mathsf {L}abel{pprop}Let $M\stackrel{f}\mathbf hookrightarrowghtarrow N\stackrel{g}\mathbf hookrightarrowghtarrow K$ be a conflation in $\mathfrak{m}athscr{C} $. Then, for any object $X\in\mathfrak{m}athscr{C} $, there exists an exact sequence;
$$\operatorname{{\mathsf{Ext}}}^n(K, X)/{\mathcal P}\stackrel{\mathsf {b}ar{\mathbf g}}\mathsf {L}ongrightarrow\operatorname{{\mathsf{Ext}}}^n(N, X)/{\mathcal P}\stackrel{\mathsf {b}ar{\mathbf f}}\mathsf {L}ongrightarrow\operatorname{{\mathsf{Ext}}}^n(M, X)/{\mathcal P}.$$
\end{prop}
\mathsf {b}egin{proof}
Take inflations $N\stackrel{h}\mathbf hookrightarrowghtarrow P$ and $K\stackrel{h'}\mathbf hookrightarrowghtarrow P'$, where $P, P'\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $.
So, we will have the following commutative diagram; {\mathbf footnotesize
\[\mathsf {X}ymatrix{M~\mathsf {a}r[r]^{hf} \mathsf {a}r[d]_{f}& P\mathsf {a}r[r]\mathsf {a}r[d]_{[1~~0]^t}& \mathsf{\Omega}^{-1}M\mathsf {a}r[d]_{f'}\\ N~\mathsf {a}r[r]^{[h~~h'g]^t} \mathsf {a}r[d]_{g} & P{\rm{op}}lus P'\mathsf {a}r[r] \mathsf {a}r[d]_{[0~~1]}& \mathsf{\Omega}^{-1}N \mathsf {a}r[d]_{g'} \\ K\mathsf {a}r[r]^{h'} & P' \mathsf {a}r[r] & \mathsf{\Omega}^{-1}K,}\]}where rows and columns are conflation. Indeed, $hf$ is an inflation, because $f$ and $h$ are so. Moreover, $u:=[h~~h'g]^t$ is an inflation, thanks to Lemma \twoheadrightarrowf{101}. So, by applying the functor $\operatorname{{\mathsf{Ext}}}(-, X)$ to this diagram, gives us the following commutative diagram;
{\mathbf footnotesize
\[\mathsf {X}ymatrix{\operatorname{{\mathsf{Ext}}}^n(P', X)~\mathsf {a}r[r] \mathsf {a}r[d]& \operatorname{{\mathsf{Ext}}}^n(K, X)\mathsf {a}r[r]\mathsf {a}r[d]_{\mathbf g}& \operatorname{{\mathsf{Ext}}}^{n+1}(\mathsf{\Omega}^{-1}K, X)\mathsf {a}r[d]\mathsf {a}r[r]&0\\ \operatorname{{\mathsf{Ext}}}^n(P{\rm{op}}lus P', X)~\mathsf {a}r[r] \mathsf {a}r[d] & \operatorname{{\mathsf{Ext}}}^n(N, X)\mathsf {a}r[r] \mathsf {a}r[d]_{\mathbf f}& \operatorname{{\mathsf{Ext}}}^{n+1}(\mathsf{\Omega}^{-1}N, X) \mathsf {a}r[d]\mathsf {a}r[r]&0 \\ \operatorname{{\mathsf{Ext}}}^n(P, X)\mathsf {a}r[r] & \operatorname{{\mathsf{Ext}}}^n(M, X) \mathsf {a}r[r] & \operatorname{{\mathsf{Ext}}}^{n+1}(\mathsf{\Omega}^{-1}M, X)\mathsf {a}r[r]&0,}\]}where rows and columns are exact. Now, by applying Corollary \twoheadrightarrowf{lem2}, we may obtain the exact sequence; $\operatorname{{\mathsf{Ext}}}^n(K, X)/{\mathcal P}\stackrel{\mathsf {b}ar{\mathbf g}}\mathsf {L}ongrightarrow\operatorname{{\mathsf{Ext}}}^n(N, X)/{\mathcal P}\stackrel{\mathsf {b}ar{\mathbf f}}\mathsf {L}ongrightarrow\operatorname{{\mathsf{Ext}}}^n(M, X)/{\mathcal P}.$ So the proof is finished.
\end{proof}
\mathsf {b}egin{s}\mathsf {L}abel{ccor}Let $M\mathbf hookrightarrowghtarrow P\mathbf hookrightarrowghtarrow\mathsf{\Omega}^{-1}M$ and $\mathsf{\Omega} N\mathbf hookrightarrowghtarrow Q\mathbf hookrightarrowghtarrow N$ be two arbitrary unit conflations in $\mathfrak{m}athscr{C} $. According to the proof of Proposition \twoheadrightarrowf{pprop}, we may get the natural isomorphism $\operatorname{{\mathsf{Ext}}}^{n+1}(\mathsf{\Omega}^{-1}M, N){\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{{\mathsf{Ext}}}^n(M, N)/{\mathcal P}$. Also, a dual argument, gives us the natural isomorphism $\operatorname{{\mathsf{Ext}}}^{n+1}(M, \mathsf{\Omega} N){\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{{\mathsf{Ext}}}^n(M, N)/{\mathcal P}$. These facts, would imply the result below.
\end{s}
\mathsf {b}egin{cor}\mathsf {L}abel{qo}A given morphism $f:M\mathbf hookrightarrowghtarrow N$ is quasi-invertible if and only if it acts as invertible on $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$.
\end{cor}
The next result indicates that being $\mathcal P$-conflation behaves well with respect to the pull-back and push-out along morphisms in $\mathsf{\Sigma}$.
\mathsf {b}egin{prop}\mathsf {L}abel{nul} Let $a:N\mathbf hookrightarrowghtarrow X$ and $b:X'\mathbf hookrightarrowghtarrow X$ be morphisms in $\mathsf{\Sigma}$. Then
\mathsf {b}egin{enumerate}\item a given conflation $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(M, N)$ is a $\mathcal P$-conflation if and only if $a\mathbf ga$ is so.
\item a given object $\mathsf {b}e\in\operatorname{{\mathsf{Ext}}}^n(X, N)$ is a $\mathcal P$-conflation if and only if $\mathsf {b}e b$ is so.
\end{enumerate}
\end{prop}
\mathsf {b}egin{proof}By the similarity, we prove only the first assertion. Since the `only if' part follows from the fact the subfunctor $\mathcal P$ is closed under push-outs, we only prove the `if' part. To this end, assume that $a\mathbf ga$ is a $\mathcal P$-conflation. As $a\in\mathsf{\Sigma}$, Corollary \twoheadrightarrowf{qo}, the morphism $\operatorname{{\mathsf{Ext}}}^n(M, N)/{\mathcal P}\mathsf {L}ongrightarrow\operatorname{{\mathsf{Ext}}}^n(M, X)/{\mathcal P}$ sending $\mathbf ga+\mathcal P$ to $a\mathbf ga+\mathcal P$ is an isomorphism. Now since $a\mathbf ga$ is $\mathcal P$-conflation, injectivity of this morphism yields that $\mathbf ga$ is a $\mathcal P$-conflation, as needed.
\end{proof}
As a direct consequence of Proposition \twoheadrightarrowf{nul}, we include the next result.
\mathsf {b}egin{cor}\mathsf {L}abel{div}Let $a:X\mathbf hookrightarrowghtarrow X'$ be a morphism in $\mathsf{\Sigma}$ and $Y\in\mathfrak{m}athscr{C} $. Then
\mathsf {b}egin{enumerate}\item for any $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(X, Y)$, there exists $\mathbf ga'\in\operatorname{{\mathsf{Ext}}}^n(X', Y)$ such that $\mathbf ga-\mathbf ga'a$ is a $\mathcal P$-conflation. \item for a given $\mathsf {b}e\in\operatorname{{\mathsf{Ext}}}^n(Y, X')$, there exists $\mathsf {b}e'\in\operatorname{{\mathsf{Ext}}}^n(Y, X)$ such that $\mathsf {b}e -a\mathsf {b}e'$is a $\mathcal P$-conflation.
\end{enumerate}
\end{cor}
\section{Unit factorizations of conflations}
In this section, we show that any conflation in $\mathfrak{m}athscr{C} $ can be represented as a pull-back, as well as, push-out of unit conflations. We begin with the following lemma.
\mathsf {b}egin{lem}\mathsf {L}abel{gencog}Let $k\mathbf geq 1$ and $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^k(M, N)$. Then the following assertions hold:
\mathsf {b}egin{enumerate}\item There exists an object $\mathbf ga'\in\operatorname{{\mathsf{Ext}}}^k(M, N)$ and a morphism of conflations with fixed ends;
{\mathbf footnotesize \[\mathsf {X}ymatrix{\mathbf ga:N~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& X_{k-1}\mathsf {a}r[r]\mathsf {a}r[d]_{a_{k-1}}& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& X_1\mathsf {a}r[r]\mathsf {a}r[d]_{a_1} &X_0\mathsf {a}r[r]\mathsf {a}r[d]_{a_0}& M\mathsf {a}r@{=}[d]\\ \mathbf ga':N~\mathsf {a}r[r] &Q_{k-1}\mathsf {a}r[r]&\operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r]& Q_1\mathsf {a}r[r]& H\mathsf {a}r[r] & M,}\]}
such that ${Q_i}^{,}$s are $n$-projective and each $a_i$ is an inflation.
\item There exists an object $\mathbf ga''\in\operatorname{{\mathsf{Ext}}}^k(M, N)$ and a morphism of conflations with fixed ends;
{\mathbf footnotesize \[\mathsf {X}ymatrix{\mathbf ga'':N~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& H'\mathsf {a}r[r]\mathsf {a}r[d]_{b_{k-1}}& P_{k-2}\mathsf {a}r[r]\mathsf {a}r[d]_{b_{k-2}}& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r] &P_0\mathsf {a}r[r]\mathsf {a}r[d]_{b_0}& M\mathsf {a}r@{=}[d]\\ \mathbf ga:N~\mathsf {a}r[r] &{X_{k-1}}\mathsf {a}r[r]& {X_{k-2}}\mathsf {a}r[r]& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& X_0\mathsf {a}r[r] & M,}\]}
such that all $P_i^,$s are $n$-projective and each $b_i$ is a deflation.
\end{enumerate}
\end{lem}
\mathsf {b}egin{proof}Let us prove only the statement (1), since the other will be gained dually. To this end, we argue by induction on $k$. If $k=1$, then there is nothing to prove. So assume that $k\mathbf geq 2$ and the result has been proved for all integers smaller than $k$. Assume that $\mathbf ga=N\mathbf hookrightarrowghtarrow X_{k-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow X_0\mathbf hookrightarrowghtarrow M$. So letting $\mathbf ga_k=N\mathbf hookrightarrowghtarrow X_{k-1}\mathbf hookrightarrowghtarrow L$ and $\mathbf ga^{k-1}=L\mathbf hookrightarrowghtarrow X_{k-2}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow X_0\mathbf hookrightarrowghtarrow M$, we have $\mathbf ga=\mathbf ga_k\mathbf ga^{k-1}$. Take the following commutative diagram;
\[\mathsf {X}ymatrix{\mathbf ga_k:N\mathsf {a}r@{=}[d]\mathsf {a}r[r]&X_{k-1}\mathsf {a}r[r]\mathsf {a}r[d]_{a_{k-1}}&L\mathsf {a}r[d]_{a} \\ \mathsf {d}elta':N\mathsf {a}r[r]& Q_{k-1} \mathsf {a}r[r]& T,}\]
where $a_{k-1}$ is an inflation with $Q_{k-1}\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Since $a\mathbf ga^{k-1}\in\operatorname{{\mathsf{Ext}}}^{k-1}(M, T)$, by the induction hypothesis, there is a conflation $\mathbf ga_1:T\mathbf hookrightarrowghtarrow Q_{k-2}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow Q_1\mathbf hookrightarrowghtarrow H\mathbf hookrightarrowghtarrow M$ and a morphism of conflations $a\mathbf ga^{k-1}\mathbf hookrightarrowghtarrow\mathbf ga_1$ with fixed ends. Hence, by setting $\mathbf ga':=\mathsf {d}elta'\mathbf ga_1$ and using the fact that $\mathbf ga=(\mathsf {d}elta' a)\mathbf ga^{k-1}$, we will have a morphism of conflations $\mathbf ga\mathbf hookrightarrowghtarrow\mathbf ga'$ with fixed ends. So the proof is finished.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{102}Let
$M, N\in\mathfrak{m}athscr{C} $ and let $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^k(M, \mathsf{\Omega}^kN)$ with $k\mathbf geq 1$. Then the following assertions hold:
\mathsf {b}egin{enumerate}
\item There exists a unit conflation $\mathsf {d}elta\in\mathcal U^k(\mathsf{\Omega}^kN)$ and $f\in\operatorname{\mathsf{H}}$ such that $\mathbf ga=\mathsf {d}elta f$.
\item There exists a unit conflation $\mathsf {d}elta \in\mathcal U_k(M)$ and $g\in\operatorname{\mathsf{H}}$ such that $\mathbf ga=g\mathsf {d}elta$
\end{enumerate}
\end{prop}
\mathsf {b}egin{proof}We only prove the first assertion. The second one is obtained dually. By Lemma \twoheadrightarrowf{gencog} together with \operatorname{\underline{\mathscr{C}}}ite[Proposition 3.1]{mit}, we may assume that $\mathbf ga$ has the form $\mathsf{\Omega}^kN\mathbf hookrightarrowghtarrow P_{k-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_1\mathbf hookrightarrowghtarrow H\mathbf hookrightarrowghtarrow M$, where $P_i\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, for any $i$. Taking the conflation $L\mathbf hookrightarrowghtarrow H\mathbf hookrightarrowghtarrow M$ and an inflation $H\mathbf hookrightarrowghtarrow P_0$ with $P_0\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, one gets the following commutative diagram; \[\mathsf {X}ymatrix{L~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& H\mathsf {a}r[r]\mathsf {a}r[d]&M \mathsf {a}r[d]_{f}\\ L~\mathsf {a}r[r] & P_0\mathsf {a}r[r]& N'.}\]
Now letting $\mathsf {d}elta:=\mathsf{\Omega}^kN\mathbf hookrightarrowghtarrow P_{k-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_1\mathbf hookrightarrowghtarrow P_0\mathbf hookrightarrowghtarrow N'$, one has the equality $\mathbf ga=\mathsf {d}elta f$, as desired.
\end{proof}
\mathsf {b}egin{dfn}Assume that $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^k(M, \mathsf{\Omega}^kN)$ with $k\mathbf geq 1$, is given. Assume that there is a unit conflation $\mathsf {d}elta\in\mathcal U^k(\mathsf{\Omega}^kN)$ and $f\in\operatorname{\mathsf{H}}$ such that $\mathbf ga=\mathsf {d}elta f$. Then we say $\mathbf ga$ factors through $f$ by the unit conflation $\mathsf {d}elta$, and the equality $\mathbf ga=\mathsf {d}elta f$ is said to be {\it a right unit factorization} (abb. $\operatorname{\mathsf{RUF}}$) of $\mathbf ga$. Dually, if there exists $\mathsf {d}elta\in\mathcal U_k(M)$ and $g\in\operatorname{\mathsf{H}}$ such that $\mathbf ga=g\mathsf {d}elta$, then we call this a {\it left unit factorization} (abb. $\mathsf {L}uf$) of $\mathbf ga$.\\ In view of Proposition \twoheadrightarrowf{102}, every conflation admits an $\operatorname{\mathsf{RUF}}$, as well as, an $\mathsf {L}uf$.
\end{dfn}
}
\mathsf {b}egin{dfn}Assume that $a$ is a morphism in $\mathsf{\Sigma}$ and $k\mathbf geq 1$. We say that $a$ is
\mathsf {b}egin{enumerate}\item {\it induced by identity over $N$}, provided that there are unit conflations $\mathsf {d}elta, \mathsf {d}elta'\in\mathcal U_k(N)$ such that $\mathsf {d}elta=a\mathsf {d}elta'$ and $a$ is inflation.\item {\it co-induced by identity over $N$}, if there exist $\mathsf {d}elta, \mathsf {d}elta'\in\mathcal U^k(N)$ such that $\mathsf {d}elta=\mathsf {d}elta'a$ and $a$ is deflation.
\end{enumerate}
\end{dfn}
\mathsf {b}egin{prop}\mathsf {L}abel{pro100}Let $k\mathbf geq 1$. Then the following statements hold:
\mathsf {b}egin{enumerate}\item For given $\mathsf {d}elta_1, \mathsf {d}elta_2,\mathsf {d}elta_3\in\mathcal U^k(N)$, there exist deflations {$a_1,a_2,a_3\in\mathsf{\Sigma}$} such that $\mathsf {d}elta_1 a_1=\mathsf {d}elta_2a_2=\mathsf {d}elta_3a_3$. In particular, $a_1, a_2, a_3$ are co-induced by identity over $N$.
\item For given $\mathsf {d}elta_1, \mathsf {d}elta_2, \mathsf {d}elta_3 \in\mathcal U_k(N)$, there exist inflations {$a_1,a_2,a_3\in\mathsf{\Sigma}$} such that $a_1\mathsf {d}elta_1=a_2\mathsf {d}elta_2=a_3\mathsf {d}elta_3$. In particular, $a_1, a_2,a_3$ are induced by identity over $N$.
\end{enumerate}
\end{prop}
\mathsf {b}egin{proof}We deal only with the first assertion, the second one is obtained dually.
Assume that $\mathsf {d}elta_i=N\stackrel{j_i^0}\mathsf {L}ongrightarrow P_i^1\mathsf {L}ongrightarrow P_i^2\mathsf {L}ongrightarrow\operatorname{\underline{\mathscr{C}}}dots\mathsf {L}ongrightarrow P_i^k\mathsf {L}ongrightarrow N_i$, for any $1\mathsf {L}eq i\mathsf {L}eq 3$. According to the proof of Lemma \twoheadrightarrowf{101}, one gets the conflation $N\stackrel{j^0}\mathsf {L}ongrightarrow{\rm{op}}lus_{i=1}^{3}P_i^1\mathsf {L}ongrightarrow L^1$, where $j^0=[j_1^0~~j_2^0~~j_3^0]^t$. So we may have the following commutative diagram; \[\mathsf {X}ymatrix{N~\mathsf {a}r[r]\mathsf {a}r@{=}[d]&{\rm{op}}lus_{i=1}^{3} P_i^1\mathsf {a}r[r]\mathsf {a}r[d]_{e_i^1}& L^1\mathsf {a}r[d]_{b_i^1}\\ N~\mathsf {a}r[r] & P_i^1\mathsf {a}r[r]& L_i^1,}\] where $e_i^1$ is the projection. Now taking an inflation $L^1\stackrel{u^1}\mathsf {L}ongrightarrow Q^2$, we obtain a conflation $L^1\stackrel{j^1}\mathsf {L}ongrightarrow{\rm{op}}lus_{i=1}^3P_i^2{\rm{op}}lus Q^2\mathsf {L}ongrightarrow L^2$, where $j^1=[j_1^1b_1^1~~j_2^1b_2^1~~j_3^1b_3^1~~u^1]^t$ and $j_i^1:L_i^1\mathbf hookrightarrowghtarrow P_i^2$ is inclusion, for any $1\mathsf {L}eq i\mathsf {L}eq 3$. Consequently, we have the following commutative diagram; \[\mathsf {X}ymatrix{L^1~\mathsf {a}r[r]\mathsf {a}r[d]_{b_i^1}&{\rm{op}}lus_{i=1}^{3} P_i^2{\rm{op}}lus Q^2\mathsf {a}r[r]\mathsf {a}r[d]_{e_i^2}& L^2\mathsf {a}r[d]_{b_i^2}\\ L_i^1~\mathsf {a}r[r] & P_i^2\mathsf {a}r[r]& L_i^2,}\] where $e_i^2$ is the projection. Thus, continuing this procedure and splicing the diagrams, gives us the following commutative diagram;{\mathbf footnotesize \[\mathsf {X}ymatrix{ \mathsf {d}elta_4:N~\mathsf {a}r[r]^{j^0}\mathsf {a}r@{=}[d] & {\rm{op}}lus_{i=1}^3P_i^1\mathsf {a}r[d]_{e_i^1}\mathsf {a}r[r] & {\rm{op}}lus_{i=1}^3P_i^2{\rm{op}}lus Q^2\mathsf {a}r[d]_{e_i^2}\mathsf {a}r[r]&\operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& {\rm{op}}lus_{i=1}^3P_i^k{\rm{op}}lus Q^k\mathsf {a}r[d]_{e_i^k}\mathsf {a}r[r]\mathsf {a}r[d]_{e_i^k} & N_4\mathsf {a}r[d]_{a_i}\\ \mathsf {d}elta_i:N~\mathsf {a}r[r]^{j_i^0} & P_i^1\mathsf {a}r[r]& P_i^2\mathsf {a}r[r]& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r] & P_i^k\mathsf {a}r[r] & N_i,}\]}where each $e^j_i$ is the projection. Thus each $a_i$ is a deflation with kernel $n$-projective, and so, it belongs to $\mathsf{\Sigma}$, thanks to Corollary \twoheadrightarrowf{is}. In particular, by Remark \twoheadrightarrowf{pp1}(1), we have that $\mathsf {d}elta_4=\mathsf {d}elta_i a_i$, for any $1\mathsf {L}eq i\mathsf {L}eq 3$. So the proof is finished.
\end{proof}
\mathsf {b}egin{dfn}(1) Assume that $\mathsf {d}elta, \mathsf {d}elta' \in \mathcal U^k(N)$ with $k\mathbf geq 1$ and $a, a'$ are co-induced morphisms by identity over $N$ such that $\mathsf {d}elta a=\mathsf {d}elta'a'$. Then we say that $[\mathsf {d}elta a, \mathsf {d}elta'a']$
is a {\it co-angled pair}. In some cases, based on our need, we denote it by $\mathsf {d}elta\stackrel{a}\mathsf {L}ongleftarrow\mathsf {d}elta''\stackrel{a'}\mathsf {L}ongrightarrow\mathsf {d}elta'$, where $\mathsf {d}elta''=\mathsf {d}elta a$. One should note that by Proposition \twoheadrightarrowf{pro100} such a co-angled pair exists.\\
(2) Assume that $\mathsf {d}elta_1, \mathsf {d}elta_2 \in\mathcal U_k(N)$ with $k\mathbf geq 1$ and $a_1,a_2$ are induced morphisms by identity over $N$. Then we say that $[a_1\mathsf {d}elta_1, a_2\mathsf {d}elta_2]$ is {\it an angled pair}, if $a_1\mathsf {d}elta_1=a_2\mathsf {d}elta_2$. Sometimes we display it by $\mathsf {d}elta_1\stackrel{a_1}\mathsf {L}ongrightarrow\mathsf {d}elta_3\stackrel{a_2}\mathsf {L}ongleftarrow\mathsf {d}elta_2$, whenever $\mathsf {d}elta_3=a_1\mathsf {d}elta_1$.
\end{dfn}
\mathsf {b}egin{prop}\mathsf {L}abel{coin}Let $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^k(M, \mathsf{\Omega}^k N)$ with $k\mathbf geq 1$ and let $\mathbf ga=\mathsf {d}elta f=\mathsf {d}elta'f'$ be two $\operatorname{\mathsf{RUF}}$s of $\mathbf ga$. Assume that
$\mathsf {d}elta\stackrel{a}\mathsf {L}ongleftarrow\mathsf {d}elta''\stackrel{a'}\mathsf {L}ongrightarrow\mathsf {d}elta'$ is a co-angled pair which is obtained in Proposition \twoheadrightarrowf{pro100}.
Then there is a morphism $h\in\mathfrak{m}athscr{C} $ such that $f=ah$ and $f'=a'h$. Particularly, $\mathbf ga=\mathsf {d}elta''h$ is an $\operatorname{\mathsf{RUF}}$ of $\mathbf ga$.
\end{prop}
\mathsf {b}egin{proof}
By the hypothesis, we have the following commutative diagrams;{\mathbf footnotesize \[\mathsf {X}ymatrix{\mathbf ga:\mathsf{\Omega}^kN~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& X_{k-1}\mathsf {a}r[r]\mathsf {a}r[d]_{f_{k-1}}& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& X_0\mathsf {a}r[r]\mathsf {a}r[d]_{f_0}& M\mathsf {a}r[d]_{f}\\ \mathsf {d}elta:\mathsf{\Omega}^kN~\mathsf {a}r[r] &P_{k-1}\mathsf {a}r[r]&\operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r]& P_0\mathsf {a}r[r]& N,}\] \[\mathsf {X}ymatrix{\mathbf ga:\mathsf{\Omega}^kN~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& X_{k-1}\mathsf {a}r[r]\mathsf {a}r[d]_{f'_{k-1}}& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& X_0\mathsf {a}r[r]\mathsf {a}r[d]_{f'_0}& M\mathsf {a}r[d]_{f'}\\ \mathsf {d}elta':\mathsf{\Omega}^kN~\mathsf {a}r[r] &P'_{k-1}\mathsf {a}r[r]&\operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r]& P'_0\mathsf {a}r[r]& N'.}\]}So using the proof of Proposition \twoheadrightarrowf{pro100}, one may obtain the following commutative diagram; {\mathbf footnotesize \[\mathsf {X}ymatrix{\mathbf ga:\mathsf{\Omega}^kN~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& X_{k-1}\mathsf {a}r[r]\mathsf {a}r[d]_{{{\tiny {\mathsf {L}eft[\mathsf {b}egin{array}{ll} f_{k-1} \\ f'_{k-1} \end{array} \mathbf hookrightarrowght]}}}}& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& X_0\mathsf {a}r[r]\mathsf {a}r[d]_{{{{\tiny {\mathsf {L}eft[\mathsf {b}egin{array}{ll} f_0 \\ f'_0 \\ 0 \end{array} \mathbf hookrightarrowght]}}}}} & M\mathsf {a}r[d]_{h}\\ \mathsf {d}elta'':\mathsf{\Omega}^kN~\mathsf {a}r[r] &P_{k-1}{\rm{op}}lus P'_{k-1}\mathsf {a}r[r]&\operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r]& P_0{\rm{op}}lus P'_0{\rm{op}}lus Q_0\mathsf {a}r[r]& N'',}\]}namely $\mathsf {d}elta''h=\mathbf ga$. Now since $\mathsf {d}elta''=\mathsf {d}elta a=\mathsf {d}elta'a'$, we have $\mathsf {d}elta(ah)=\mathsf {d}elta'(a'h)=\mathbf ga$. Particularly, one has the commutative diagram;
{\mathbf footnotesize\[\mathsf {X}ymatrix{N~& N''\mathsf {a}r[l]_{a}\mathsf {a}r[r]^{a'}& N'\\ & M\mathsf {a}r[u]^{h}\mathsf {a}r[ul]^{f}\mathsf {a}r[ur]_{f'} ,& }\]}
meaning that $f=ah$ and $f'=a'h$. Thus the proof is completed.
\end{proof}
The result below can be obtained by dualizing the argument given in the proof of Proposition \twoheadrightarrowf{coin}. So its proof will be omitted.
\mathsf {b}egin{prop}\mathsf {L}abel{lif}Let $k\mathbf geq 1$ and $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^k(M, \mathsf{\Omega}^k N)$ and let $\mathbf ga=f\mathsf {d}elta=f'\mathsf {d}elta'$ be two $\mathsf {L}uf$s of $\mathbf ga$. Assume that $\mathsf {d}elta\stackrel{a}\mathsf {L}ongrightarrow\mathsf {d}elta''\stackrel{a'}\mathsf {L}ongleftarrow\mathsf {d}elta'$ is an angled pair which is obtained in Proposition \twoheadrightarrowf{pro100}. Then there exists $h\in\operatorname{\mathsf{H}}$ such that $\mathbf ga=h\mathsf {d}elta''$ is also an $\mathsf {L}uf$ of $\mathbf ga$. In particular, $f=ha$ and $f'=ha'$.
\end{prop}
\section{$n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphisms}
Assume that $\mathfrak{m}athscr{C} $ is an $n$-Frobenius category. In this section, we will see that an object $f\in\operatorname{\mathsf{H}}$ annihilates $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$ from the left and from the right, simultaneously. We call such a morphism $f$, an {\em $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism}. This notion has connection with many branches of mathematics; see \twoheadrightarrowf{s100}. We begin with the following easy observation.
\mathsf {b}egin{s}\mathsf {L}abel{cof}Assume that $f:X\mathbf hookrightarrowghtarrow N$ is a morphism in $\mathfrak{m}athscr{C} $ factoring through an $n$-projective object. Then $f$ annihilates $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$ from the both sides. To see this, take $P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $ and morphisms $f_1, f_2$ such that $f:X\stackrel{f_1}\mathsf {L}ongrightarrow P\stackrel{f_2}\mathsf {L}ongrightarrow N$. So for a given object $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(N, Y)$, $\mathbf ga f=(\mathbf ga f_2)f_1$ will be a $\mathcal P$-conflation, because $\mathbf ga f_2\in\operatorname{{\mathsf{Ext}}}^n(P, Y)$. Namely, $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})f=0$. The same method reveals that $f(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})=0$.
\end{s}
The next interesting result says that a given morphism $f$ annihilates $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$, whenever it annihilates some unit conflation $\mathsf {d}elta$.
\mathsf {b}egin{prop}\mathsf {L}abel{three}
Let $n\mathbf geq 1$ and $f:X\mathbf hookrightarrowghtarrow N$ be a morphism in $\mathfrak{m}athscr{C} $. Then the following are equivalent: \mathsf {b}egin{enumerate} \item There exists a unit conflation $\mathsf {d}elta\in\mathcal U_n(N)$ such that $\mathsf {d}elta f$ is a $\mathcal P$-conflation.\item
For any unit conflation $\mathsf {d}elta'\in\mathcal U_n(N)$, $\mathsf {d}elta' f$ is a $\mathcal P$-conflation.\item $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})f=0$. \end{enumerate}
\end{prop}
\mathsf {b}egin{proof}
$(1)\mathfrak{m}athcal{R} ightarrow (2)$ Take an arbitrary unit conflation $\mathsf {d}elta'\in\mathcal U_n(N)$. We intend to show that $\mathsf {d}elta'f$ is a $\mathcal P$-conflation. To see this, consider an angled pair
$\mathsf {d}elta\stackrel{a}\mathbf hookrightarrowghtarrow\mathsf {d}elta''\stackrel{b}\mathsf {L}eftarrow \mathsf {d}elta'$. Since $\mathsf {d}elta f$ is a $\mathcal P$-conflation, by applying Proposition \twoheadrightarrowf{nul}(1), we deduce that the same is true for $a(\mathsf {d}elta f)=(a\mathsf {d}elta)f$.
Moreover, as $a\mathsf {d}elta=b\mathsf {d}elta'$, another use of Proposition \twoheadrightarrowf{nul}(1) enables us to infer that $\mathsf {d}elta'f$ is also a $\mathcal P$-conflation, as needed.\\
$(2)\mathfrak{m}athcal{R} ightarrow (3)$ Assume that $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(N, M)$ is an arbitrary conflation and $\mathbf ga=g\mathsf {d}elta'$ is an $\mathsf {L}uf$ of $\mathbf ga$, where $\mathsf {d}elta'\in\mathcal U_n(N)$. By the hypothesis, $\mathsf {d}elta'f$ is a $\mathcal P$-conflation. So using the fact that being a $\mathcal P$-conflation is preserved under push-out, we infer that $\mathbf ga f=g(\mathsf {d}elta'f)$ is also a $\mathcal P$-conflation, that is, $\mathbf ga f\in\mathcal P$.
Consequently, $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})f=0$ .\\
$(3)\mathfrak{m}athcal{R} ightarrow (1)$ This implication is obvious. So the proof is finished.
\end{proof}
As an immediate consequence of Proposition \twoheadrightarrowf{three}, we include the next result.
\mathsf {b}egin{cor}\mathsf {L}abel{ccoo}Let $n\mathbf geq 1$ and ${\mathbf ga}\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^n
N)$ be a conflation and let $\mathbf ga=\mathsf {d}elta f$ be an $\operatorname{\mathsf{RUF}}$
of $\mathbf ga$. Then $\mathbf ga$ is a $\mathcal P$-conflation if and only if $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})f=0$.
\end{cor}
\mathsf {b}egin{lem}\mathsf {L}abel{ruf}Let $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)$ be a $\mathcal P$-conflation. Then there is an $\operatorname{\mathsf{RUF}}$ $\mathbf ga=\mathsf {d}elta f$ of $\mathbf ga$ with $f$ factoring through an $n$-projective object.
\end{lem}
\mathsf {b}egin{proof}Since $ \mathbf ga $ is a $ \mathcal P $-conflation, there exist an object $\operatorname{\boldsymbol{\eta}}\in \operatorname{{\mathsf{Ext}}}^n( P , \mathsf{\Omega}^n N)$ with $P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, and a morphism $ h: M\mathbf hookrightarrowghtarrow P $ such that $\mathbf ga= \operatorname{\boldsymbol{\eta}} h$. Taking an $\operatorname{\mathsf{RUF}}$ $\operatorname{\boldsymbol{\eta}}=\mathsf {d}elta g$ of $\operatorname{\boldsymbol{\eta}}$, we get $\mathbf ga=\mathsf {d}elta(gh)$ is an $\operatorname{\mathsf{RUF}}$ of $\mathbf ga$, in which $gh$ factors through the $n$-projective object $P$. So we are done.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{ph}Let $f:M\mathbf hookrightarrowghtarrow N$ be a morphism in $\mathfrak{m}athscr{C} $ and $n\mathbf geq 1$. If $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})f=0$, then there are morphisms $N\stackrel{a}\mathsf {L}eftarrow N''\stackrel{b}\mathbf hookrightarrowghtarrow N'$ with $a, b\in\mathsf{\Sigma}$ and $h:M\mathbf hookrightarrowghtarrow N''$ such that $f=ah$ and $bh$ factors through an $n$-projective object.
\end{prop}
\mathsf {b}egin{proof}Fix a unit conflation $\mathsf {d}elta_N\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nN)$. By our hypothesis, $\mathsf {d}elta_Nf$ is a $\mathcal P$-conflation.
So, using Lemma \twoheadrightarrowf{ruf}, we may find an $\operatorname{\mathsf{RUF}}$ $\mathsf {d}elta_Nf=\mathsf {d}elta g$ of $\mathsf {d}elta_Nf$ such that $g$ factors through an $n$-projective object. Consider a co-angled pair $\mathsf {d}elta_N\stackrel{a}\mathsf {L}eftarrow \mathsf {d}elta''\stackrel{b}\mathbf hookrightarrowghtarrow \mathsf {d}elta$, as the one obtained in Proposition \twoheadrightarrowf{pro100}. Indeed, we have a pair of morphisms $N\stackrel{a}\mathsf {L}eftarrow N''\stackrel{b}\mathbf hookrightarrowghtarrow N'$, where $N'$ (resp. $N''$) is the right end term of the unit conflation $\mathsf {d}elta$ (resp. $\mathsf {d}elta''$). So applying Proposition \twoheadrightarrowf{coin} ensures the existence of a morphism $h:M\mathbf hookrightarrowghtarrow N''$ such that $\mathsf {d}elta_Nf=\mathsf {d}elta'' h$ is also an $\operatorname{\mathsf{RUF}}$ of $\mathsf {d}elta_Nf$. Particulary, $f=ah$ and $bh$ factors through an $n$-projective object, because of the equality $bh=g$. So the proof is completed.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{sif}Let $a:X\mathbf hookrightarrowghtarrow N$ be a morphism in $\mathsf{\Sigma}$. Then the following are satisfied:
\mathsf {b}egin{enumerate}\item A given morphism $f:N\mathbf hookrightarrowghtarrow M$ annihilates $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$ from the right if and only if so does $fa$. \item A given morphism $g:M\mathbf hookrightarrowghtarrow X$ annihilates $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$ from the left if and only if so does $ag$.
\end{enumerate}
\end{prop}
\mathsf {b}egin{proof}By the similarity, we only prove the first statement.
If $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})f=0$ , evidently the same is true for$(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})(fa)$, because being $\mathcal P$-conflation is closed under pull-back. Now assume that $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})(fa)=0$. So, by Proposition \twoheadrightarrowf{three}, there exists a unit conflation $\mathsf {d}elta_M\in\mathcal U_k(M)$ such that $\mathsf {d}elta_M(fa)$ is a $\mathcal P$-conflation. Hence, invoking Proposition \twoheadrightarrowf{nul}(2) yields that $\mathsf {d}elta_Mf$ is a $\mathcal P$-conflation, as well. Consequently, another use of Proposition \twoheadrightarrowf{three} forces $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})f$ to be zero, as required.
\end{proof}
Now we are ready to state the main result of this section.
\mathsf {b}egin{theorem}\mathsf {L}abel{main}
Let $f:M\mathbf hookrightarrowghtarrow N$ be a morphism in $\mathfrak{m}athscr{C} $. Then $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})f=0$ if and only if $f(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})=0$.
\end{theorem}
\mathsf {b}egin{proof}Since the result in the case $n=0$ holds obviously, we may assume that
$n\mathbf geq 1$. Suppose that $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})f=0$. So, in view of Proposition \twoheadrightarrowf{ph}, there are morphisms $N\stackrel{a}\mathsf {L}eftarrow N''\stackrel{b}\mathbf hookrightarrowghtarrow N'$ with $a, b{\in\mathsf{\Sigma}}$ and $h:M\mathbf hookrightarrowghtarrow N''$ such that $f=ah$ and $bh$ factors through an $n$-projective object. So, as we have observed in \twoheadrightarrowf{cof}, $(bh)(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})=0$. As $a, b\in\mathsf{\Sigma}$, by applying Proposition \twoheadrightarrowf{sif} successively, one may get that $f(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})=0$. Since the reverse implication is obtained analogously, we ignore it. So, the proof is finished.
\end{proof}
\mathsf {b}egin{dfn} A morphism $f$ in $\mathfrak{m}athscr{C} $ is called an {\em $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism}, if it annihilates $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$.
\end{dfn}
The result below follows directly from \twoheadrightarrowf{ccor}. So we skip its proof.
\mathsf {b}egin{cor}A given morphism $f$ in $\mathfrak{m}athscr{C} $ is an $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism if and only if $\operatorname{{\mathsf{Ext}}}^{n+1}f=0$.
\end{cor}
\mathsf {b}egin{rem}It should be noted that our notion of $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphisms, is indeed $(n+1)$-$\operatorname{{\mathsf{Ext}}}$-phantom morphisms in the sense of Mao \operatorname{\underline{\mathscr{C}}}ite{mao}. However, due to the harmony with $n$-Frobenius category, we call them $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphisms. We also emphasize that a 1-$\operatorname{{\mathsf{Ext}}}$-phantom morphism is exactly a $\mathcal P$-phantom morphism in the sense of Fu et. al. \operatorname{\underline{\mathscr{C}}}ite{fght}.
\end{rem}
\mathsf {b}egin{s}\mathsf {L}abel{s100}The concept of phantom morphisms has its roots in topology in the study of maps between CW-complexes \operatorname{\underline{\mathscr{C}}}ite{mc}. A map $f : X \mathbf hookrightarrowghtarrow Y$ between CW-complexes is said to be a phantom map, if its restriction to each skeleton $X_n$ is null homotopic. Later, this notion has been used in various settings of mathematics. In the context of triangulated categories, phantom morphisms were first studied by Neeman \operatorname{\underline{\mathscr{C}}}ite{ne}. The notion of phantom morphisms also was developed in the stable category of a finite group ring in a series of works of Benson and Gnacadja \operatorname{\underline{\mathscr{C}}}ite{gn, be2, be1, be}. The definition of a phantom morphism was generalized to the category of $R$-modules over an associative ring $R$ by Herzog \operatorname{\underline{\mathscr{C}}}ite{he}. Precisely, a morphism $f:M\mathbf hookrightarrowghtarrow N$ of left $R$-modules is called a phantom morphism, if the natural transformation $\mathfrak{m}athcal{T}or^R_1(-, f):\mathfrak{m}athcal{T}or^R_1(-, M)\mathbf hookrightarrowghtarrow\mathfrak{m}athcal{T}or^R_1(-, N)$ is zero, or equivalently, the pullback of any short exact sequence along $f$ is pure exact. Similarly, a morphism $g:M\mathbf hookrightarrowghtarrow N$ of left $R$-modules is said to be an $\operatorname{{\mathsf{Ext}}}$-phantom morphism \operatorname{\underline{\mathscr{C}}}ite{hext}, if the induced morphism $\operatorname{{\mathsf{Ext}}}^1_R(B, g):\operatorname{{\mathsf{Ext}}}^1_R(B,M)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^1_R(B,N)$ is 0, for every finitely presented left $R$-module $B$. For any integer $n\mathbf geq 1$, the concepts of $n$-phantom morphism and $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphisms, which are higher dimensional generalization of phantom morphisms and $\operatorname{{\mathsf{Ext}}}$-phantom morphisms, respectively, have been introduced and studied by Mao \operatorname{\underline{\mathscr{C}}}ite{mao, mao1}.
\end{s}
\section{A composition operator on $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$}
Assume that $\mathfrak{m}athscr{C} $ is an $n$-Frobenius category. In this section, we introduce a composition operator $``\operatorname{\underline{\mathscr{C}}}irc"$ on $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$. It is proved that the operator $``\operatorname{\underline{\mathscr{C}}}irc"$ is associative and distributive over Baer sum on both sides. These facts enable us to see that for any object $M\in\mathfrak{m}athscr{C} $, $(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)/{\mathcal P}, +, \operatorname{\underline{\mathscr{C}}}irc)$ is a ring with identity, where $+$ stands for the Baer sum operation. Surprisingly, we find a ring homomorphism $\operatorname{\mathsf{V}}arphi:\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(M, M)\mathsf {L}ongrightarrow\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)/{\mathcal P}$ such that for any quasi-invertible morphism $f$, $\operatorname{\mathsf{V}}arphi(f)$ is invertible and $ \operatorname{\mathsf{V}}arphi(f)=0$, if $f$ is an $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism. We begin with the following notation.
\mathsf {b}egin{s}Let $a:X\mathbf hookrightarrowghtarrow X'$ be a quasi-invertible morphism and $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(X, Y)$. In view of Corollary \twoheadrightarrowf{div}, there exists $\mathbf ga'\in\operatorname{{\mathsf{Ext}}}^n(X', Y)$ such that $\mathbf ga-\mathbf ga'a$ is a $\mathcal P$-conflation. In this case, for the simplicity, we write $\mathbf ga'=_{_{\mathcal P}}\mathbf ga a^{-1}$. Also, {for a given object} $\mathsf {b}e\in\operatorname{{\mathsf{Ext}}}^n(Y, X')$, there exists $\mathsf {b}e'\in\operatorname{{\mathsf{Ext}}}^n(Y, X)$ such that $\mathsf {b}e -a\mathsf {b}e'$ is a $\mathcal P$-conflation. Then we write $\mathsf {b}e'=_{_{\mathcal P}}a^{-1}\mathsf {b}e$. {From now on, a given object $\mathbf ga+\mathcal P\in\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$, will be denoted by $\mathsf {b}ar{\mathbf ga}$.}
\end{s}
\mathsf {b}egin{dfn}\mathsf {L}abel{compo}Assume that $M, N, K\in\mathfrak{m}athscr{C} $ and fix a unit conflation $\mathsf {d}elta_N\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nN)$. We define the composition
$$\operatorname{\underline{\mathscr{C}}}irc: \operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nK)/{\mathcal P}\times\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}\mathsf {L}ongrightarrow\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nK)/{\mathcal P},$$ $(\mathsf {b}ar{\mathsf {b}e}, \mathsf {b}ar{\mathbf ga})\mathsf {L}ongrightarrow\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}$ as follows: \\
Assume that $\mathbf ga=\mathsf {d}elta f$ is an $\operatorname{\mathsf{RUF}}$ of $\mathbf ga$, where $\mathsf {d}elta\in\mathcal U^n(\mathsf{\Omega}^nN)$ and $f\in\operatorname{\mathsf{H}}$.
Now we set $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}:=\overline{((\mathsf {b}e a_1)b^{-1}_1)f}$, where $\mathsf {d}elta_N\stackrel{a_1}\mathsf {L}ongleftarrow\mathsf {d}elta_1\stackrel{b_1}\mathsf {L}ongrightarrow\mathsf {d}elta$
is a co-angled pair.
\end{dfn}
The result below allows us to assume that the co-angled pair in the above definition, as the one obtained in Proposition \twoheadrightarrowf{pro100}(1).
\mathsf {b}egin{lem}\mathsf {L}abel{ds}With the notation above, assume that $\mathsf {d}elta_N\stackrel{a}\mathsf {L}ongleftarrow\mathsf {d}elta''\stackrel{b}\mathsf {L}ongrightarrow\mathsf {d}elta$ is the co-angled pair which is obtained in Proposition \twoheadrightarrowf{pro100}(1). Then $((\mathsf {b}e a_1)b^{-1}_1)f=_{\mathcal P}((\mathsf {b}e a)b^{-1})f$.
\end{lem}
\mathsf {b}egin{proof} Since $\mathsf {d}elta_N\stackrel{a_1}\mathsf {L}ongleftarrow\mathsf {d}elta_1\stackrel{b_1}\mathsf {L}ongrightarrow\mathsf {d}elta$ is a co-angled pair, we have $\mathsf {d}elta_1=\mathsf {d}elta_Na_1=\mathsf {d}elta b_1$. So by Proposition \twoheadrightarrowf{coin}, $\mathsf {d}elta_1=\mathsf {d}elta''h$, for some morphism $h$ in $\mathfrak{m}athscr{C} $. Indeed, the argument given in the proof of Proposition \twoheadrightarrowf{coin}, gives rise to the following commutative diagram;
{\mathbf footnotesize\[\mathsf {X}ymatrix{N~& N''\mathsf {a}r[l]_{a}\mathsf {a}r[r]^{b}& N'\\ & N_1\mathsf {a}r[u]^{h}\mathsf {a}r[ul]^{a_1}\mathsf {a}r[ur]_{b_1} ,& }\]}where $N', N''$ and $N_1$ stand for the right end terms of $\mathsf {d}elta, \mathsf {d}elta''$ and $\mathsf {d}elta_1$, respectively. This, in particular, implies that {$((\mathsf {b}e a_1)b^{-1}_1)f=_{\mathcal P}((\mathsf {b}e ah)(bh)^{-1})f=_{\mathcal P}((((\mathsf {b}e a)h)h^{-1})b^{-1})f=_{\mathcal P}((\mathsf {b}e a)b^{-1})f$}, giving the desired result.
\end{proof}
\mathsf {b}egin{theorem}\mathsf {L}abel{welldef}The definition of $``\operatorname{\underline{\mathscr{C}}}irc"$ is independent of the choice of an $\operatorname{\mathsf{RUF}}$ of $\mathbf ga$.
\end{theorem}
\mathsf {b}egin{proof}
Assume that $\mathbf ga=\mathsf {d}elta'f'$ is another $\operatorname{\mathsf{RUF}}$ of $\mathbf ga$ and suppose that $\mathsf {d}elta_N\stackrel{a_2}\mathsf {L}ongleftarrow\mathsf {d}elta_2\stackrel{b_2}\mathsf {L}ongrightarrow\mathsf {d}elta'$
is a co-angled pair. We shall prove that $((\mathsf {b}e a_1)b_1^{-1})f=_{\mathcal P}((\mathsf {b}e a_2)b_2^{-1})f'$. Take a co-angled pair
$\mathsf {d}elta\stackrel{a_3}\mathsf {L}ongleftarrow\mathsf {d}elta_3\stackrel{b_3}\mathsf {L}ongrightarrow\mathsf {d}elta'$. According to Proposition \twoheadrightarrowf{pro100}(1), there exist $a_4,b_4,c_4\in\mathsf{\Sigma}$ such that $\mathsf {d}elta_1a_4=\mathsf {d}elta_2b_4=\mathsf {d}elta_3c_4$ and $a_4,b_4,c_4$ are co-induced by identity over $\mathsf{\Omega}^nN$. In particular, denoting the latter equalities by $\mathsf {d}elta_4$, we will get the following diagram of co-angled pairs; {\mathbf footnotesize \[\mathsf {X}ymatrix{&\mathsf {d}elta_N& \\ \mathsf {d}elta_1~\mathsf {a}r[d]_{b_1}\mathsf {a}r[ur]^{a_1}& \mathsf {d}elta_4\mathsf {a}r[l]_{a_4}\mathsf {a}r[r]^{b_4}\mathsf {a}r[d]_{c_4}& \mathsf {d}elta_2\mathsf {a}r[d]_{b_2}\mathsf {a}r[ul]_{a_2}\\ \mathsf {d}elta~ & \mathsf {d}elta_3\mathsf {a}r[l]_{a_3}\mathsf {a}r[r]^{b_3} & \mathsf {d}elta'.}\]} {Since by Lemma \twoheadrightarrowf{ds}, the co-angled pair $\mathsf {d}elta\stackrel{a_3}\mathsf {L}ongleftarrow\mathsf {d}elta_3\stackrel{b_3}\mathsf {L}ongrightarrow\mathsf {d}elta'$ can be considered as the one obtained in Proposition \twoheadrightarrowf{pro100}(1), }
by virtue of Proposition \twoheadrightarrowf{coin}, there is a morphism $h$ such that $\mathbf ga=\mathsf {d}elta_3h$, $f=a_3h$ and $f'=b_3h$. In particular, one may have the following diagram; {\mathbf footnotesize\[\mathsf {X}ymatrix{\mathsf {d}elta~& \mathsf {d}elta_3\mathsf {a}r[l]_{a_3}\mathsf {a}r[r]^{b_3}& \mathsf {d}elta'\\ & \mathbf ga\mathsf {a}r[u]^{h}\mathsf {a}r[ul]^{f}\mathsf {a}r[ur]_{f'} .& }\]}
Thus we have the following equalities modulo $\mathcal P$; $$((\mathsf {b}e a_2)b_2^{-1})f'=_{\mathcal P}((\mathsf {b}e a_2)b_2^{-1})b_3h=_{\mathcal P}(((\mathsf {b}e a_2)b_2^{-1})b_3)h=_{\mathcal P}((((\mathsf {b}e a_2)b_2^{-1})b_3)a_3^{-1})f$$ $$=_{\mathcal P}((((\mathsf {b}e a_2)b_4){c_4}^{-1})a_3^{-1})f=_{\mathcal P} (((\mathsf {b}e(a_2b_4){a_4}^{-1})b_1^{-1})f=_{\mathcal P}((\mathsf {b}e a_1)b_1^{-1})f.$$
Here the first and third equalities hold, because the latter diagram is commutative, and the second one is clear.
The validity of the forth and fifth equalities come from the fact that $b_3c_4=b_2b_4$ and $b_1a_4=a_3c_4$, respectively. Finally, the last one holds true, because $a_2b_4=a_1a_4$. Thus
$((\mathsf {b}e a_1)b_1^{-1})f=_{\mathcal P}((\mathsf {b}e a_2)b_2^{-1})f'$, as needed.
\end{proof}
Let $M, N$ be two objects of $\mathfrak{m}athscr{C} $. For given two objects $\mathsf {b}ar{\mathbf ga}, \mathsf {b}ar{\mathbf ga'}\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P
}$, we define the addition $\mathsf {b}ar{\mathbf ga}+\mathsf {b}ar{\mathbf ga'}:=\overline{\mathbf ga+\mathbf ga'}$, where $\mathbf ga+\mathbf ga'$ is the usual Baer sum operation. Evidently, this definition is well-defined.
\mathsf {b}egin{prop}\mathsf {L}abel{srt1}The operator $``\operatorname{\underline{\mathscr{C}}}irc"$ is distributive over $``+"$ on both sides.
\end{prop}
\mathsf {b}egin{proof} Assume that $ K, M,N\in \mathfrak{m}athscr{C} $ and fix unit conflations $\mathsf {d}elta_N\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nN)$ and $\mathsf {d}elta_K\in\operatorname{{\mathsf{Ext}}}^n(K, \mathsf{\Omega}^nK)$. Assume that $\mathsf {b}ar{\mathsf {a}l}\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^n N)/{\mathcal P}$, $\mathsf {b}ar{\mathbf ga}\in \operatorname{{\mathsf{Ext}}}^n(K, \mathsf{\Omega}^n L)/{\mathcal P}$ and $\mathsf {b}ar{\mathsf {b}e}, \mathsf {b}ar{\mathsf {b}e'}\in \operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^n K)/{\mathcal P}$. First, we show that
$(\mathsf {b}ar{\mathsf {b}e}+\mathsf {b}ar{\mathsf {b}e'})\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}=\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}+ \mathsf {b}ar{\mathsf {b}e'}\operatorname{\underline{\mathscr{C}}}irc \mathsf {b}ar{\mathsf {a}l}$. In this direction, assume that
$\mathsf {a}l=\mathsf {d}elta f$ is an $\operatorname{\mathsf{RUF}}$ of $\mathsf {a}l$ and suppose that $[\mathsf {d}elta_Na, \mathsf {d}elta b]$ is a co-angled pair. So, $(\mathsf {b}ar{\mathsf {b}e}+\mathsf {b}ar{\mathsf {b}e'})\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}=\overline{(((\mathsf {b}e+\mathsf {b}e')a)b^{-1})f}=\overline{((\mathsf {b}e a)b^{-1})f}+\overline{((\mathsf {b}e' a)b^{-1})f}=\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}+ \mathsf {b}ar{\mathsf {b}e'}\operatorname{\underline{\mathscr{C}}}irc \mathsf {b}ar{\mathsf {a}l}$, where the second equality follows from the fact that pull-backs distributes over the Baer sum; see \operatorname{\underline{\mathscr{C}}}ite[Chapter VII, Lemma 3.2]{mit}. Next we would like to show that $\mathsf {b}ar{\mathbf ga }\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathsf {b}e}+\mathsf {b}ar{ \mathsf {b}e'})=\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e}+\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e'}$. Assume that $\mathsf {b}e=\mathsf {d}elta f$ and $\mathsf {b}e'=\mathsf {d}elta' f'$ are $\operatorname{\mathsf{RUF}}$s of $\mathsf {b}e$ and $\mathsf {b}e'$, respectively. Take an $\operatorname{\mathsf{RUF}}$, $\mathfrak{n}abla(\mathsf {d}elta{\rm{op}}lus\mathsf {d}elta')=\mathsf {d}elta''g$, where $\mathfrak{n}abla:\mathsf{\Omega}^nK{\rm{op}}lus\mathsf{\Omega}^nK\stackrel{[1~~1]}\mathsf {L}ongrightarrow\mathsf{\Omega}^nK$.
Namely, we have the following commutative diagram; \[\mathsf {X}ymatrix{\mathfrak{n}abla(\mathsf {d}elta{\rm{op}}lus\mathsf {d}elta'):\mathsf{\Omega}^n K~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& T_{n-1}\mathsf {a}r[r]\mathsf {a}r[d] &\operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r] &T_0\mathsf {a}r[r]\mathsf {a}r[d] & K'{\rm{op}}lus K''\mathsf {a}r[d]_{g}\\ \mathsf {d}elta'':\mathsf{\Omega}^nK ~\mathsf {a}r[r] & {Q}_{n-1}\mathsf {a}r[r]& \operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r]& Q_0\mathsf {a}r[r]& K_1,}\] where each $Q_i$ is $n$-projective. Now setting $g:=[g'~~g'']$,
one may easily deduce that $\mathsf {b}e=\mathsf {d}elta''(g'f)$ and $\mathsf {b}e'=\mathsf {d}elta''(g''f')$ are also $\operatorname{\mathsf{RUF}}$s of $\mathsf {b}e$ and $\mathsf {b}e'$, respectively. We claim that $\mathsf {b}e+\mathsf {b}e'=\mathsf {d}elta''h$, for some morphism $h$ in $\mathfrak{m}athscr{C} $. Since there is a morphism of conflations $\mathsf {b}e{\rm{op}}lus\mathsf {b}e'\mathsf {L}ongrightarrow\mathfrak{n}abla(\mathsf {d}elta{\rm{op}}lus\mathsf {d}elta')$, by the universal property of push-out diagram, there exists a unique morphism $\mathfrak{n}abla(\mathsf {b}e{\rm{op}}lus\mathsf {b}e')\mathsf {L}ongrightarrow\mathfrak{n}abla(\mathsf {d}elta{\rm{op}}lus\mathsf {d}elta')$. Indeed, we have $\mathfrak{n}abla(\mathsf {b}e{\rm{op}}lus\mathsf {b}e')=\mathfrak{n}abla(\mathsf {d}elta{\rm{op}}lus\mathsf {d}elta')(f{\rm{op}}lus f')$. As $\mathsf {b}e+\mathsf {b}e'=\mathfrak{n}abla(\mathsf {b}e{\rm{op}}lus\mathsf {b}e')\mathfrak{m}athbb{D} elta$, where $\mathfrak{m}athbb{D} elta:N\stackrel{{{\tiny {\mathsf {L}eft[\mathsf {b}egin{array}{ll} 1 \\ 1 \end{array} \mathbf hookrightarrowght]}}}}\mathsf {L}ongrightarrow N{\rm{op}}lus N$, by lettting $h=g(f{\rm{op}}lus f')\mathfrak{m}athbb{D} elta$, we have $\mathsf {b}e+\mathsf {b}e'=\mathsf {d}elta''h$, as claimed. Therefore, considering the co-angled pair $[\mathsf {d}elta_Ka_1, \mathsf {d}elta''b_1]$, we have $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathsf {b}e}+\mathsf {b}ar{ \mathsf {b}e'})=\overline{((\mathbf ga a_1)b_1^{-1})h}$. Now since $h=[g'~~g''](f{\rm{op}}lus f')\mathfrak{m}athbb{D} elta=g'f+g''f'$, we infer that $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathsf {b}e}+\mathsf {b}ar{ \mathsf {b}e'})=\overline{((\mathbf ga a_1)b_1^{-1})g'f}+\overline{((\mathbf ga a_1)b_1^{-1})g''f'}=\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e}+\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e'}$. So the proof is completed.
\end{proof}
\mathsf {b}egin{lem}\mathsf {L}abel{corwell}
Let ${\mathbf ga}\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)$ and ${\mathsf {b}e}\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nK)$. If $\mathbf ga$ or $\mathsf {b}e$ is a $\mathcal P$-conflation, then $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=0$. In particular, if $\mathsf {b}e'$ and $\mathbf ga'$ are two conflations such that $\mathsf {b}ar{\mathsf {b}e}=\mathsf {b}ar{\mathsf {b}e'}$ and $\mathsf {b}ar{\mathbf ga}=\mathsf {b}ar{\mathbf ga'}$, then $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=\mathsf {b}ar{\mathsf {b}e'}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga'}$.
\end{lem}
\mathsf {b}egin{proof} First assume that $ \mathsf {b}e$ is a $ \mathcal P $-conflation. Assume that $ \mathbf ga= \mathsf {d}elta f $ is an $\operatorname{\mathsf{RUF}}$ of $ \mathbf ga $ and suppose that $[\mathsf {d}elta_N a , \mathsf {d}elta b]$ is a co-angled pair, where $\mathsf {d}elta_N\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nN)$ is a unit conflation.
As $\mathsf {b}e$ is a $\mathcal P$-conflation, applying Proposition \twoheadrightarrowf{nul} ensures that $(\mathsf {b}e a)b^{-1}$ is a $\mathcal P$-conflation, and so, the same will be true for $((\mathsf {b}e a)b^{-1})f$, meaning that $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=\mathsf {b}ar{0}$.\\
Next assume that $ \mathbf ga $ is a $ \mathcal P $-conflation. So, in view of Lemma \twoheadrightarrowf{ruf}, there is an $\operatorname{\mathsf{RUF}}$ $\mathbf ga=\mathsf {d}elta f$ of $\mathbf ga$ such that $f$ factors through an $n$-projective object.
Consider a co-angled pair $[\mathsf {d}elta_N a, \mathsf {d}elta b]$. According to the proof of Theorem \twoheadrightarrowf{welldef}, $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=\overline{((\mathsf {b}e a) b^{-1})f}$.
Now since $f$ factors through an $n$-projective object, it is evident that $((\mathsf {b}e a) b^{-1})f$ is a $\mathcal P$-conflation, and then, $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=\mathsf {b}ar{0}$. The second assertion follows by combining the first assertion with Proposition \twoheadrightarrowf{srt1}. Thus the proof is finished
\end{proof}
\mathsf {b}egin{rem}\mathsf {L}abel{bimod}Let $X, Y$ be two arbitrary objects in $\mathfrak{m}athscr{C} $ and $\mathsf {b}e, \mathsf {b}e'\in\operatorname{{\mathsf{Ext}}}^n(X, Y)$. Assume that there is a morphism of conflations $\mathsf {b}e\mathbf hookrightarrowghtarrow\mathsf {b}e'$ with fixed ends. Then any $\operatorname{\mathsf{RUF}}$ of $\mathsf {b}e'$, will be an $\operatorname{\mathsf{RUF}}$ of $\mathsf {b}e$, as well. So, for any conflation $\mathbf ga$, we will have $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e}=\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e'}$, where the composition makes sense. On the other hand,
since for a given morphism $f$, the equality $\mathsf {b}e f=\mathsf {b}e'f$ holds, one may infer that $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}=\mathsf {b}ar{\mathsf {b}e'}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}$, for all conflations $\mathsf {a}l\in\operatorname{{\mathsf{Ext}}}^n(Z, X)$.
These facts, in conjunction with \operatorname{\underline{\mathscr{C}}}ite[Proposition 3.1]{mit} guarantees that the
operator $``\operatorname{\underline{\mathscr{C}}}irc^{,,}$ is compatible with the equivalence classes in $\operatorname{{\mathsf{Ext}}}^n(X, Y)$. On the other hand, as we have mentioned in \twoheadrightarrowf{pp}, $\mathcal P$ is an $\operatorname{\mathsf{H}}$-bisubmodule of $\operatorname{{\mathsf{Ext}}}^n$. Hence $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$ will be an $\operatorname{\mathsf{H}}$-bimodule.
\end{rem}
Taking all of the previous results together, leads us to deduce that the composition operator $``\operatorname{\underline{\mathscr{C}}}irc"$, introduced in Definition \twoheadrightarrowf{compo}, is well-defined. Indeed we have the next result.
\mathsf {b}egin{theorem}\mathsf {L}abel{circ}The composition $``\operatorname{\underline{\mathscr{C}}}irc"$ is well-defined
\end{theorem}
\mathsf {b}egin{prop}\mathsf {L}abel{ass}The composition operator $``\operatorname{\underline{\mathscr{C}}}irc"$ is associative.
\end{prop}
\mathsf {b}egin{proof}
Assume that $M, N, K, L\in\mathfrak{m}athscr{C} $. Fix unit conflations $\mathsf {d}elta_N\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nN)$ and $\mathsf {d}elta_K\in\operatorname{{\mathsf{Ext}}}^n(K, \mathsf{\Omega}^nK)$ and let $\mathsf {b}ar{\mathsf {a}l}\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}$, $\mathsf {b}ar{\mathsf {b}e}\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^n K)/{\mathcal P}$ and $\mathsf {b}ar{\mathbf ga}\in\operatorname{{\mathsf{Ext}}}^n(K, \mathsf{\Omega}^nL)/{\mathcal P}$. We would like to show that $(\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e})\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}=\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l})$. Let us first compute $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l})$. In this direction, assume that $\mathsf {a}l=\mathsf {d}elta f$ is an $\operatorname{\mathsf{RUF}}$ of $\mathsf {a}l$ and suppose that $[\mathsf {d}elta_Na_1, \mathsf {d}elta b_1]$ is a co-angled pair. So by definition, $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}=\overline{((\mathsf {b}e a_1)b_1^{-1})f}$. Set, for the simplicity, $\mathsf {b}e':=(\mathsf {b}e a_1)b_1^{-1}$ and take an $\operatorname{\mathsf{RUF}}$, $\mathsf {b}e'=\mathsf {d}elta' g'$ of $\mathsf {b}e'$, and then, $\mathsf {b}e'f=\mathsf {d}elta'(g'f)$ is an $\operatorname{\mathsf{RUF}}$ of $\mathsf {b}e'f$. Now, assuming $[\mathsf {d}elta_Ka_2, \mathsf {d}elta'b_2]$ is a co-angled pair, we have $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l})=\overline{((\mathbf ga a_2)b_2^{-1})g'f}$. Next we calculate $(\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e})\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}$. Suppose that $\mathsf {b}e=\mathsf {d}elta''g$ is an $\operatorname{\mathsf{RUF}}$ of $\mathsf {b}e$ and $[\mathsf {d}elta_Ka_3, \mathsf {d}elta''b_3]$ is a co-angled pair, and so, $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e}=\overline{((\mathbf ga a_3)b_3^{-1})g}$. By our assumption, $\mathsf {b}e a_1=_{\mathcal P}\mathsf {b}e'b_1$. {So applying Lemma \twoheadrightarrowf{corwell}, we have $0=\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\overline{(\mathsf {b}e a_1-\mathsf {b}e'b_1)}=\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\overline{\mathsf {b}e a_1}-\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\overline{\mathsf {b}e'b_1}$. As $\mathsf {b}e=\mathsf {d}elta''g$, we have $\mathsf {b}e a_1=\mathsf {d}elta''(ga_1)$. Thus, considering the co-angled pair $[\mathsf {d}elta_Ka_3, \mathsf {d}elta''b_3]$, one has $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\overline{\mathsf {b}e a_1}=\overline{((\mathbf ga a_3)b_3^{-1})g a_1}$. Similarly, since $\mathsf {b}e'b_1=\mathsf {d}elta'(g'b_1)$, we obtain that $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\overline{\mathsf {b}e' b_1}=\overline{((\mathbf ga a_2)b_2^{-1})g' b_1}$.}
Consequently, $((\mathbf ga a_3)b_3^{-1})g a_1=_{\mathcal P}{((\mathbf ga a_2)b_2^{-1})g'b_1}$, meaning that $(\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e})a_1=\overline{(((\mathbf ga a_2)b_2^{-1})g')b_1}$. Thus, by considering the co-angled pair $[\mathsf {d}elta_Na_1, \mathsf {d}elta b_1]$, one may deduce that $(\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e})\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}=\overline{(((\mathbf ga a_2)b_2^{-1})g')f}$. Therefore, $(\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {b}e})\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}=\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l})$, as needed.
\end{proof}
\mathsf {b}egin{cor}\mathsf {L}abel{cor100}Let $M\in\mathfrak{m}athscr{C} $ and fix a unit conflation $\mathsf {d}elta_M\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)$. Then $(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)/{\mathcal P}, +, \operatorname{\underline{\mathscr{C}}}irc)$ has a ring structure with the identity element $\mathsf {b}ar{\mathsf {d}elta}_M$. Moreover, for any unit conflation $\mathsf {d}elta\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)$, $\mathsf {b}ar{\mathsf {d}elta}$ is invertible.
\end{cor}
\mathsf {b}egin{proof}According to Theorem \twoheadrightarrowf{circ} and Propositions \twoheadrightarrowf{ass} and \twoheadrightarrowf{srt1}, we only need to show that $\mathsf {b}ar{\mathsf {d}elta}_M$ is the unit element of $\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)/{\mathcal P}$. To see this, take $\mathsf {b}ar{\mathbf ga}\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^n M)/{\mathcal P}$. Since $\mathsf {d}elta_M=\mathsf {d}elta_M 1_M$ is an $\operatorname{\mathsf{RUF}}$ of $\mathsf {d}elta_M$, one has $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {d}elta}_M =\mathsf {b}ar{\mathbf ga}$. Now suppose that $\mathbf ga=\mathsf {d}elta f $ is an $\operatorname{\mathsf{RUF}}$ of $ \mathbf ga $ and $[\mathsf {d}elta_Ma, \mathsf {d}elta b]$ is a co-angled pair. Consequently, $\mathsf {b}ar{\mathsf {d}elta}_M\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=\overline{((\mathsf {d}elta_Ma)b^{-1})f}=\overline{\mathsf {d}elta f}= \mathsf {b}ar{ \mathbf ga} $, giving the first assertion.
For the second assertion, assume that $\mathsf {d}elta\in\mathcal U_n(M)\operatorname{\underline{\mathscr{C}}}ap\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)$. Consider a co-angled pair $\mathsf {d}elta_M\stackrel{a}\mathsf {L}ongleftarrow\mathsf {d}elta'\stackrel{b}\mathsf {L}ongrightarrow\mathsf {d}elta$. Take $\mathsf {a}l\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)$ such that $(\mathsf {a}l a)b^{-1}=_{\mathcal P}\mathsf {d}elta_M$, i.e., $\mathsf {a}l a=\mathsf {d}elta_Mb$. According to Lemma \twoheadrightarrowf{unit}, $\mathsf {a}l$ is a unit conflation. As $\mathsf {a}l=\mathsf {a}l 1_M$ is an $\operatorname{\mathsf{RUF}}$ of $\mathsf {a}l$, $\mathsf {b}ar{\mathsf {d}elta}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {a}l}=\overline{((\mathsf {d}elta b)a^{-1})1}_M=\mathsf {b}ar{\mathsf {d}elta}_M$. Similarly, since $\mathsf {d}elta=\mathsf {d}elta 1_M$ is an $\operatorname{\mathsf{RUF}}$ of $\mathsf {d}elta$, by the definition, we have $\mathsf {b}ar{\mathsf {a}l}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathsf {d}elta}=\overline{((\mathsf {a}l a)b^{-1})1}_M=\mathsf {b}ar{\mathsf {d}elta}_M$. Hence $\mathsf {b}ar{\mathsf {a}l}$ is the inverse of $\mathsf {b}ar{\mathsf {d}elta}$, and so, the proof is finished.
\end{proof}
We close this section with the following interesting result.
\mathsf {b}egin{prop}\mathsf {L}abel{ringhom} Let $M\in\mathfrak{m}athscr{C} $ and fix a unit conflation $\mathsf {d}elta_M\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)$. Then there exists a ring homomorphism $\operatorname{\mathsf{V}}arphi :\operatorname{\mathsf{Hom}}_{ \mathfrak{m}athscr{C} }(M,M) \mathbf hookrightarrowghtarrow \operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^n M)/{\mathcal P} $ with $ \operatorname{\mathsf{V}}arphi(f)=\overline{ \mathsf {d}elta_Mf}$ such that for any quasi-invertible morphism $f$, $\operatorname{\mathsf{V}}arphi(f)$ is an invertible element of $\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^n M)/{\mathcal P}$, and $\operatorname{\mathsf{V}}arphi(f)=0$, if $f$ is an $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism.
\end{prop}
\mathsf {b}egin{proof} Assume that $f, g\in\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(M, M).$ According to \operatorname{\underline{\mathscr{C}}}ite[Chapter VII, Lemma 3.2]{mit}, $ \mathsf {d}elta_M (f+g)= \mathsf {d}elta_Mf + \mathsf {d}elta_Mg$, meaning that $ \operatorname{\mathsf{V}}arphi$ is a morphism of abelian groups. Moreover, we have $\operatorname{\mathsf{V}}arphi(g)\operatorname{\underline{\mathscr{C}}}irc\operatorname{\mathsf{V}}arphi(f)=\overline{\mathsf {d}elta_Mg}\operatorname{\underline{\mathscr{C}}}irc\overline{\mathsf {d}elta_Mf}=\overline{(\mathsf {d}elta_Mg)f}=\overline{\mathsf {d}elta_M(gf)}=\operatorname{\mathsf{V}}arphi(gf)$. Note that the second equality follows from the fact that, one may choose $[\mathsf {d}elta_M1_M, \mathsf {d}elta_M 1_M]$ as a co-angled pair. Finally, $\operatorname{\mathsf{V}}arphi(1_M)=\overline{\mathsf {d}elta_M1}_M=\mathsf {b}ar{\mathsf {d}elta}_M$, and then, $\operatorname{\mathsf{V}}arphi$ is a ring homomorphism. Next assume that $f\in\mathsf{\Sigma}$. Since $\mathsf {d}elta_M\in\mathcal U_n(M)$, by Lemma \twoheadrightarrowf{unit}, $\mathsf {d}elta_Mf \in \mathcal U_n(M)$. Indeed, $\mathsf {d}elta_Mf\in\mathcal U_n(M)\operatorname{\underline{\mathscr{C}}}ap\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)$. So Corollary \twoheadrightarrowf{cor100} implies that $\operatorname{\mathsf{V}}arphi(f)$ is invertible. Moreover, the last assertion holds true, because of Proposition \twoheadrightarrowf{three}. Hence the proof is finished.
\end{proof}
\section{An equivalence relation on $\operatorname{{\mathsf{Ext}}}^n/{\mathcal P}$}
Let $M$ and $N$ be objects of $\mathfrak{m}athscr{C} $. This section is devoted to define an equivalence relation on the class $\mathsf {b}igcup_{ \mathsf {d}elta_N}(\operatorname{{\mathsf{Ext}}}^n( M, \mathsf{\Omega}^n N)/{\mathcal P}, \mathsf {d}elta_N)$.
We will see that this relation is compatible with the composition operator $``\operatorname{\underline{\mathscr{C}}}irc"$, as well as the operator $``+"$.
\mathsf {b}egin{dfn}\mathsf {L}abel{rel}
Assume that $M,N\in\mathfrak{m}athscr{C} $ and $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N), (\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_N)$ are arbitrary objects of $\mathsf {b}igcup_{ \mathsf {d}elta_N\in\mathcal U_n(N)}(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^n N)/{\mathcal P}, \mathsf {d}elta_N)$. We write $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_N)$, if $a\mathbf ga-b\mathbf ga'$ is a $\mathcal P$-conflation, where $[a\mathsf {d}elta_N, b\mathsf {d}elta'_N]$ is an angled pair, which exists by Proposition \twoheadrightarrowf{pro100}.
\end{dfn}
In the sequel, we show that $``\operatorname{\mathsf{thick}}sim"$ is an equivalence relation.
\mathsf {b}egin{theorem}
$``\operatorname{\mathsf{thick}}sim"$ is an equivalence relation.
\end{theorem}
\mathsf {b}egin{proof}
Since the reflexivity and symmetry hold trivially, we only need to show the transitivity. To this end,
assume that $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta)\mathsf{\Sigma}m (\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta')$ and $(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta')\mathsf{\Sigma}m (\mathsf {b}ar{\mathbf ga''}, \mathsf {d}elta'')$. We shall prove that $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta)\operatorname{\mathsf{thick}}sim (\mathsf {b}ar{\mathbf ga''}, \mathsf {d}elta'')$. By our assumption, there are angled pairs $\mathsf {d}elta'\stackrel{a_1}\mathsf {L}ongrightarrow\mathsf {d}elta_1\stackrel{b_1}\mathsf {L}ongleftarrow\mathsf {d}elta$ and $\mathsf {d}elta'\stackrel{a_2}\mathsf {L}ongrightarrow\mathsf {d}elta_2\stackrel{b_2}\mathsf {L}ongleftarrow\mathsf {d}elta''$ such that $a_1\mathbf ga'-b_1\mathbf ga$ and $a_2\mathbf ga'-b_2\mathbf ga''$
are $\mathcal P$-conflations. Now considering an angled pair $\mathsf {d}elta_1\stackrel{a_3}\mathsf {L}ongrightarrow\mathsf {d}elta_3\stackrel{b_3}\mathsf {L}ongleftarrow\mathsf {d}elta_2$, which exists by
part (2) of Proposition \twoheadrightarrowf{pro100}, we will get the following commutative diagram of angled pairs; \[\mathsf {X}ymatrix{&&\mathsf {d}elta'\mathsf {a}r[ld]_{a_1}\mathsf {a}r[rd]^{a_2}&& \\ \mathsf {d}elta\mathsf {a}r[r]^{b_1}&\mathsf {d}elta_1~\mathsf {a}r[r]^{a_3}& \mathsf {d}elta_3& \mathsf {d}elta_2\mathsf {a}r[l]_{b_3}&\mathsf {d}elta''\mathsf {a}r[l]_{b_2} .}\] Since $(a_3a_1-b_3a_2)\mathsf {d}elta'=0$, dualizing the argument given in the proof of Proposition \twoheadrightarrowf{three}, would imply that $(a_3a_1-b_3a_2)\mathbf ga'$ is a $\mathcal P$-conflation. Hence, in view of our hypothesis, we may deduce that $(a_3b_1)\mathbf ga-(b_3b_2)\mathbf ga''$ is a $\mathcal P$-conflation. Since the class $\mathsf{\Sigma}$ is closed under composition, $[(a_3b_1)\mathsf {d}elta, (b_3b_2)\mathsf {d}elta'']$ will be also an agled pair. Consequently, $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta)\operatorname{\mathsf{thick}}sim (\mathsf {b}ar{\mathbf ga''}, \mathsf {d}elta'')$, as desired.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{ind}Let $M, N$ be objects of $\mathfrak{m}athscr{C} $ and fix a unit conflation $\mathsf {d}elta_N\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nN)$. Then for any $(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_{N})\in(\operatorname{{\mathsf{Ext}}}^n( M, {\mathsf{\Omega}'}^nN)/{\mathcal P}, \mathsf {d}elta'_N)$, there is a unique object $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$ which is equivalent to $(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_N)$.
\end{prop}
\mathsf {b}egin{proof}
Take an angled pair $\mathsf {d}elta_N\stackrel{a}\mathsf {L}ongrightarrow\mathsf {d}elta''\stackrel{b}\mathsf {L}ongleftarrow\mathsf {d}elta'_N$. Since $b\mathbf ga'\in\operatorname{{\mathsf{Ext}}}^n(M, {\mathsf{\Omega}''}^nN)$ and $(\mathsf{\Omega}^nN\stackrel{a}\mathbf hookrightarrowghtarrow{\mathsf{\Omega}''}^nN)\in\mathsf{\Sigma}$, in view of Corollary \twoheadrightarrowf{div}(2), there exists $\mathbf ga\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)$ such that $a\mathbf ga-b\mathbf ga'$ is a $\mathcal P$-conflation. Moreover, uniqueness of $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)$ follows from Proposition \twoheadrightarrowf{nul}. So the proof is finished.
\end{proof}
\mathsf {b}egin{rem}
Assume that $M,N\in\mathfrak{m}athscr{C} $ and fix a unit conflation $\mathsf {d}elta_N\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nN)$. Then it follows from Proposition \twoheadrightarrowf{ind} that $ (\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^n N)/{\mathcal P}, \mathsf {d}elta_N)$ is equal to the set of all equivalence classes of $\mathsf {b}igcup_{{\mathsf {d}elta'}_{N}}(\operatorname{{\mathsf{Ext}}}^n(M, {\mathsf{\Omega}'}^n N)/{\mathcal P}, \mathsf {d}elta'_N)$ modulo the equivalence relation $``\operatorname{\mathsf{thick}}sim"$.
\end{rem}
The following easy observation is needed in the next proposition.
\mathsf {b}egin{lem}\mathsf {L}abel{pushco}Let $\mathsf {d}elta, \mathsf {d}elta'\in\mathcal U^n(\mathsf{\Omega}^nM)$ and $\mathsf {d}elta\stackrel{a}\mathsf {L}ongleftarrow\mathsf {d}elta''\stackrel{a'}\mathsf {L}ongrightarrow\mathsf {d}elta'$ be a co-angled pair. Let $b:\mathsf{\Omega}^nM\mathbf hookrightarrowghtarrow{\mathsf{\Omega}'}^nM$ be a morphism in $\mathsf{\Sigma}$.
Then $b\mathsf {d}elta\stackrel{a}\mathsf {L}ongleftarrow b\mathsf {d}elta''\stackrel{a'}\mathsf {L}ongrightarrow b\mathsf {d}elta'$ is also a co-angled pair.
\end{lem}
\mathsf {b}egin{proof}Since $b\in\mathsf{\Sigma}$, by Lemma \twoheadrightarrowf{unit}, the push-out of any unit conflation along $b$ is also a unit conflation.
By the hypothesis, $\mathsf {d}elta a=\mathsf {d}elta''=\mathsf {d}elta'a'$, and so, applying Remark \twoheadrightarrowf{pp1}, we obtain the co-angled pair $b\mathsf {d}elta\stackrel{a}\mathsf {L}ongleftarrow b\mathsf {d}elta''\stackrel{a'}\mathsf {L}ongrightarrow b\mathsf {d}elta'$. So we are done.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{comequ}The equivalence relation $``\operatorname{\mathsf{thick}}sim"$ is compatible with the composition $``\operatorname{\underline{\mathscr{C}}}irc"$.
\end{prop}
\mathsf {b}egin{proof}Let $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_M)\in(\operatorname{{\mathsf{Ext}}}^n(K, \mathsf{\Omega}^nM)/{\mathcal P}, \mathsf {d}elta_M)$ and $(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_M)\in(\operatorname{{\mathsf{Ext}}}^n(K, {\mathsf{\Omega}'}^nM)/{\mathcal P}, \mathsf {d}elta'_M)$ such that $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_M)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_M)$. Suppose that $(\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$. We must show that $(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta_N)$. By our hypothesis, there is an angled pair $\mathsf {d}elta_M\stackrel{a}\mathsf {L}ongrightarrow\mathsf {d}elta''_M\stackrel{b}\mathsf {L}ongleftarrow\mathsf {d}elta'_M$
such that $a\mathbf ga-b\mathbf ga'$ is a $\mathcal P$-conflation. Set $\mathbf ga'':=a\mathbf ga$. So $(\mathsf {b}ar{\mathbf ga''}, \mathsf {d}elta''_M)\in(\operatorname{{\mathsf{Ext}}}^n(K, {\mathsf{\Omega}''}^nM)/{\mathcal P}, \mathsf {d}elta''_M)$. Hence in order to obtain the result, it suffices to show that $(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga''}, \mathsf {d}elta_N)$. To do this, assume that $\mathbf ga=\mathsf {d}elta_1f$ is an $\operatorname{\mathsf{RUF}}$ of $\mathbf ga$. Take a co-angled pair $[\mathsf {d}elta_M a_1, \mathsf {d}elta_1 b_1]$. As $a\in\mathsf{\Sigma}$, by Lemma \twoheadrightarrowf{pushco}, $[(a\mathsf {d}elta_M) a_1, (a\mathsf {d}elta_1) b_1]$ is also a co-angled pair. Since $\mathbf ga=\mathsf {d}elta_1f$, one has $\mathbf ga''=a\mathbf ga=a(\mathsf {d}elta_1f)=(a\mathsf {d}elta_1)f$, meaning that $\mathbf ga''=(a\mathsf {d}elta_1)f$ is an $\operatorname{\mathsf{RUF}}$ of $\mathbf ga''$. Consequently, $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=\overline{((\mathsf {b}e a_1){b_1}^{-1})f}=\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga''}$, and so, $(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga''}, \mathsf {d}elta_N)$, as desired.
Next consider $(\mathsf {b}ar{\mathsf {b}e'}, \mathsf {d}elta'_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, {\mathsf{\Omega}'}^nN)/{\mathcal P}, \mathsf {d}elta'_N)$ such that $(\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e'}, \mathsf {d}elta'_N)$. We would like to show that $(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e'}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta'_N)$.
By the hypothesis, there is an angled pair $[a\mathsf {d}elta_N, a'\mathsf {d}elta'_N]$ such that $a\mathsf {b}e-a'\mathsf {b}e'$ is a $\mathcal P$-conflation, and so, $a\mathsf {b}ar{\mathsf {b}e}=a'\mathsf {b}ar{\mathsf {b}e'}$. Now from the defnition of the composition $``\operatorname{\underline{\mathscr{C}}}irc"$, one may deduce that $a(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga})=a'(\mathsf {b}ar{\mathsf {b}e'}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga})$, meaning that $(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e'}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta'_N)$. Thus, the proof is finished.
\end{proof}
\mathsf {b}egin{lem}The equivalence relation $``\operatorname{\mathsf{thick}}sim"$ is compatible with $``+"$.
\end{lem}
\mathsf {b}egin{proof}Assume that $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N), (\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$ and $(\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta'_N), (\mathsf {b}ar{\mathsf {b}e'}, \mathsf {d}elta'_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, {\mathsf{\Omega}'}^n N)/{\mathcal P}, \mathsf {d}elta'_N)$ such that $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta'_N)$ and $(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e'}, \mathsf {d}elta'_N)$. We must show that $(\overline{\mathbf ga+\mathbf ga'}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\overline{\mathsf {b}e+\mathsf {b}e'}, \mathsf {d}elta'_N)$.
Since $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta'_N)$ and $(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathsf {b}e'}, \mathsf {d}elta'_N)$, by the definition, $a\mathbf ga-b\mathsf {b}e$ and $a\mathbf ga'-b\mathsf {b}e'$ are $\mathcal P$-conflations, where $[a\mathsf {d}elta_N, b\mathsf {d}elta'_N]$ is an angled pair. Hence, $a(\mathbf ga+\mathbf ga')-b(\mathsf {b}e+\mathsf {b}e')$ is a $\mathcal P$-conflation, as well. Consequently, $(\overline{\mathbf ga+\mathbf ga'}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim (\overline{\mathsf {b}e+\mathsf {b}e'}, \mathsf {d}elta'_N)$, as required.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{val}Let $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_N)$. Then $\mathbf ga$ is $\mathcal P$-conflation if and only if $\mathbf ga'$ is so.
\end{prop}
\mathsf {b}egin{proof}Assume that $\mathbf ga$ is a $\mathcal P$-conflation. We show that $\mathbf ga'$ is a $\mathcal P$-conflation, as well. By the hypothesis, $a\mathbf ga-b\mathbf ga'$ is a $\mathcal P$-conflation, where $[a\mathsf {d}elta_N, b\mathsf {d}elta'_N]$ is an angled pair. Since $\mathbf ga$ is a $\mathcal P$-conflation, applying Proposition \twoheadrightarrowf{nul} yields that $a\mathbf ga$ is also a $\mathcal P$-conflation. So using the fact that $a\mathbf ga-b\mathbf ga'$ is a $\mathcal P$-conflation, we infer that the same is true for $b\mathbf ga'$. Hence, another use of Proposition \twoheadrightarrowf{nul}, guarantees that $\mathbf ga'$ is also a $\mathcal P$-conflation. Since the reverse implication is obtained similarly, we skip it. So the proof is finished.
\end{proof}
{
\section{Phantom stable categories}
Inspired by the stabilization of a Frobenius category, we introduce and study the notion of phantom stable category of an $n$-Frobenius category. We begin with the following motivating observation.
\mathsf {b}egin{s}\mathsf {L}abel{s1s1}
Let $\mathfrak{m}athscr{C} '$ be a Frobenius category and let $\mathfrak{m}athcal{I}$ be the ideal consisting of all morphisms factoring through projective objects. Assume that $\mathfrak{m}athscr{C} '/{\mathfrak{m}athcal{I}}$ is the stable category of $\mathfrak{m}athscr{C} '$. So we have the natural functor $\mathcal Pi:\mathfrak{m}athscr{C} '\mathsf {L}ongrightarrow\mathfrak{m}athscr{C} '/{\mathfrak{m}athcal{I}}$ such that for any morphism $f$ that its kernel and cokernel are both projective, $\mathcal Pi(f)$ is an isomorphism and for any $g\in\mathfrak{m}athcal{I}$, $\mathcal Pi(g)=0$. It is easily seen that the pair $(\mathfrak{m}athscr{C} '/{\mathfrak{m}athcal{I}}, \mathcal Pi)$ has universal property with respect to these conditions. This fact is our idea to introduce the notion of {\it phantom stable category} of an $n$-Frobenius category, for any $n\mathbf geq 0$. To be precise, let $\mathfrak{m}athscr{C} $ be an $n$-Frobenius category and let $\operatorname{{\mathsf{Ext}}}^n$ be all equivalence classes of conflations of length $n$. Assume that $\mathcal P$ is a subfunctor of $\operatorname{{\mathsf{Ext}}}^n$ consisting of all
conflations of length $n$ which are obtained as pull-back of conflations along morphisms of the form $M\mathbf hookrightarrowghtarrow P$, for some $M\in\mathfrak{m}athscr{C} $ and $P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Assume that $\mathsf{\Sigma}$ is the class of all morphisms acting as invertible on $\operatorname{{\mathsf{Ext}}}^{n+1}$. We introduce the additive category $\mathfrak{m}athscr{C} _{\mathcal P}$ and an additive functor $T:\mathfrak{m}athscr{C} \mathbf hookrightarrowghtarrow\mathfrak{m}athscr{C} _{\mathcal P}$ with $T(s)$ is an isomorphism, for any quasi-invertible morphism $s$ and $T(f)=0$ for all $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphisms $f$, which has universal property with respect to these conditions. Indeed, we have the following definition.
\end{s}
\mathsf {b}egin{dfn}We say that a couple $(\mathfrak{m}athscr{C} _{\mathcal P}, T)$, where $\mathfrak{m}athscr{C} _{\mathcal P}$ is an additive category and $T:\mathfrak{m}athscr{C} \mathsf {L}ongrightarrow\mathfrak{m}athscr{C} _{\mathcal P}$ is a covariant additive functor, is the {\it phantom stable category of $\mathfrak{m}athscr{C} $}, if:\\ (1) $T(s)$ is an isomorphism in $\mathfrak{m}athscr{C} _{\mathcal P}$, for any quasi-invertible morphism $s$.\\
(2) For any $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism $\operatorname{\mathsf{V}}arphi$, $T(\operatorname{\mathsf{V}}arphi)=0$ in $\mathfrak{m}athscr{C} _{\mathcal P}$.\\(3) Any covariant additive functor $T':\mathfrak{m}athscr{C} \mathsf {L}ongrightarrow\mathfrak{m}athbb{D} $ satisfying the conditions (1) and (2), factors in a unique way through $T$.
\end{dfn}
In the following, we show the existence of the phantom stable category of $\mathfrak{m}athscr{C} $. First we qoute a couple of results.
\mathsf {b}egin{lem}\mathsf {L}abel{iso}Let $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$ such that $\mathbf ga$ is a unit conflation. Then for any $\mathsf {d}elta_M\in\mathcal U_n(M)$, there exists $(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta_M)\in(\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nM)/{\mathcal P}, \mathsf {d}elta_M)$ such that $\mathsf {b}ar{\mathbf ga'}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=\mathsf {b}ar{\mathsf {d}elta}_M$ and $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga'}=\mathsf {b}ar{\mathsf {d}elta}_N$.
\end{lem}
\mathsf {b}egin{proof}Take a co-angled pair $[\mathbf ga a, \mathsf {d}elta_Nb]$ and set $\mathbf ga':=(\mathsf {d}elta_Ma)b^{-1}$. So considering an $\operatorname{\mathsf{RUF}}$ $\mathbf ga=\mathbf ga 1_M$ of $\mathbf ga$, we have $\mathsf {b}ar{\mathbf ga'}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=\overline{(\mathbf ga'b)a^{-1}}=\overline{(((\mathsf {d}elta_Ma)b^{-1})b)a^{-1}}=\mathsf {b}ar{\mathsf {d}elta}_M$. Next we show that $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga'}=\mathsf {b}ar{\mathsf {d}elta}_N$. Since $\mathbf ga'=(\mathsf {d}elta_Ma)b^{-1}$, {by Lemma \twoheadrightarrowf{unit}, $\mathbf ga'$ is a unit conflation and so} we may take the co-angled pair $[\mathbf ga'b, \mathsf {d}elta_Ma]$. Now considering $\mathbf ga'=\mathbf ga' 1_N$ as an $\operatorname{\mathsf{RUF}}$ of $\mathbf ga'$, and applying Theorem \twoheadrightarrowf{welldef}, $\mathsf {b}ar{\mathbf ga}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga'}=\overline{(\mathbf ga a)b^{-1}}=\mathsf {b}ar{\mathsf {d}elta}_N$, giving the desired result.
\end{proof}
\mathsf {b}egin{rem}\mathsf {L}abel{00}Assume that $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_N)$. So, by the definition, $a\mathbf ga-b\mathbf ga'$ is a $\mathcal P$-conflation, where $[a\mathsf {d}elta_N, b\mathsf {d}elta'_N]$ is an angled pair. Thus for any morphism $f$, $(a\mathbf ga-b\mathbf ga')f=a(\mathbf ga f)-b(\mathbf ga'f)$ will be also a $\mathcal P$-conflation, meaning that $(\overline{\mathbf ga f}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\overline{\mathbf ga'f}, \mathsf {d}elta'_N)$.
\end{rem}
The result below is the main theorem of this section.
\mathsf {b}egin{theorem}\mathsf {L}abel{thmst}The phantom stable category $(\mathfrak{m}athscr{C} _{\mathcal P}, T)$ of $\mathfrak{m}athscr{C} $ exists.
\end{theorem}
\mathsf {b}egin{proof}We define the category $\mathfrak{m}athscr{C} _{\mathcal P}$ as follows; the objects of $\mathfrak{m}athscr{C} _{\mathcal P}$ are the same as objects of $\mathfrak{m}athscr{C} $. Moreover, for any two objects $M, N\in\mathfrak{m}athscr{C} $, first we fix a unit conflation $\mathsf {d}elta_N\in\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nN)$ and set $\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} _{\mathcal P}}(M, N):=(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$ modulo the equivalence relation $``\operatorname{\mathsf{thick}}sim"$.
Assume that $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$ and $(\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta_K)\in(\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nK)/{\mathcal P}, \mathsf {d}elta_K)$. We define $(\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta_K)\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N):=(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_K)$, which is well-defined, thanks to Proposition \twoheadrightarrowf{comequ}. According to Proposition \twoheadrightarrowf{ass}, the composition operator $``\operatorname{\underline{\mathscr{C}}}irc"$ is associative. Moreover, for a given object $M\in\mathfrak{m}athscr{C} _{\mathcal P}$, we fix a unit conflation $\mathsf {d}elta_M\in\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nM)$. We claim that $1_M=(\mathsf {b}ar{\mathsf {d}elta}_M, \mathsf {d}elta_M)$.
Indeed, it is evident that for any $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$, $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathsf {d}elta}_M, \mathsf {d}elta_M)=(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)$. Moreover, take a morphism
$(\mathsf {b}ar{\mathsf {a}l}, \mathsf {d}elta'_M)\in(\operatorname{{\mathsf{Ext}}}^n(N, {\mathsf{\Omega}'}^nM)/{\mathcal P}, \mathsf {d}elta'_M)$. Applying Propositions \twoheadrightarrowf{comequ} and \twoheadrightarrowf{ind}, allows us to assume that $\mathsf {d}elta'_M=\mathsf {d}elta_M$, and then, one may easily see that $(\mathsf {b}ar{\mathsf {d}elta}_M, \mathsf {d}elta_M)\operatorname{\underline{\mathscr{C}}}irc(\mathsf {b}ar{\mathsf {a}l}, \mathsf {d}elta_M)=(\mathsf {b}ar{\mathsf {a}l}, \mathsf {d}elta_M)$. Thus $1_M=(\mathsf {b}ar{\mathsf {d}elta}_M, \mathsf {d}elta_M),$ as claimed. So $\mathfrak{m}athscr{C} _{\mathcal P}$ is a category. Clearly, $\mathfrak{m}athscr{C} _{\mathcal P}$ is closed under finite direct sums, because the same is true for $\mathfrak{m}athscr{C} $, and for any two objects $M, N$, $\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} _{\mathcal P}}(M, N)$ is an abelian group. Moreover, Propositions \twoheadrightarrowf{srt1} and \twoheadrightarrowf{comequ} guarantee that the composition is bilinear, that is, the composition distributes over addition. Consequently, $\mathfrak{m}athscr{C} _{\mathcal P}$ is an additive category.
Now let us define the functor $T:\mathfrak{m}athscr{C} \mathsf {L}ongrightarrow\mathfrak{m}athscr{C} _{\mathcal P}$. For any object $M\in\mathfrak{m}athscr{C} $, we let $T(M)=M$. Moreover, for given $M, N\in\mathfrak{m}athscr{C} $, we consider a unit conflation $\mathsf {d}elta_N$, and then we define the morphism $T_{M,N}:\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(M, N)\mathsf {L}ongrightarrow\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} _{\mathcal P}}(M, N)$, as $T(f):=(\overline{\mathsf {d}elta_Nf}, \mathsf {d}elta_N)$, for any morphism $f\in\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} }(M, N)$. It should be pointed out that, Proposition \twoheadrightarrowf{ind} together with Remark \twoheadrightarrowf{00}, insure that $T$ is well-defined.
By our definition, $T(1_N)=(\mathsf {b}ar{\mathsf {d}elta}_N, \mathsf {d}elta_N)=1_{T(N)}$. Furthermore, it is easily seen that for any two composable morphisms $M\stackrel{f}\mathbf hookrightarrowghtarrow N\stackrel{g}\mathbf hookrightarrowghtarrow K$ in $\mathfrak{m}athscr{C} $, the equalities $\overline{\mathsf {d}elta_Kg}\operatorname{\underline{\mathscr{C}}}irc\overline{\mathsf {d}elta_Nf}=\overline{(\mathsf {d}elta_Kg)f}=\overline{\mathsf {d}elta_K(gf)}$ hold true, implying that $T(gf)=T(g)\operatorname{\underline{\mathscr{C}}}irc T(f)$. Thus $T$ is a covariant functor. Since $\mathfrak{m}athscr{C} $ and $\mathfrak{m}athscr{C} _{\mathcal P}$ are additive categories and $T$ preserves finite direct sums, it will be an additive functor. Assume that $f:M\mathbf hookrightarrowghtarrow N$ lies in $\mathsf{\Sigma}$. By virtue of Lemma \twoheadrightarrowf{unit}, $\mathsf {d}elta_Nf$ is a unit conflation, and so, Lemma \twoheadrightarrowf{iso} forces $T(f)$ to be an isomorphism.
Suppose that $h:X\mathbf hookrightarrowghtarrow N$ is an $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism. So, by Proposition \twoheadrightarrowf{three}, $\mathsf {d}elta_Nh$ will be a $\mathcal P$-conflation, implying that $T(h)=0$.
Finally, assume that $T':\mathfrak{m}athscr{C} \mathsf {L}ongrightarrow\mathfrak{m}athbb{D} $ is a covariant additive functor such that $T'(f)$ is an isomorphism, for any $f\in\mathsf{\Sigma}$ and for any $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism $g$, $T'(g)=0$. In order to complete the proof, we must prove that there exists a unique additive functor $F:\mathfrak{m}athscr{C} _{\mathcal P}\mathsf {L}ongrightarrow\mathfrak{m}athbb{D} $ such that $FT=T'$. To do this, we shall define the functor $F:\mathfrak{m}athscr{C} _{\mathcal P}\mathsf {L}ongrightarrow\mathfrak{m}athbb{D} $ as follows; for any object $X\in\mathfrak{m}athscr{C} _{\mathcal P}$, write $F(X)=T'(X)$. Also, for any morphism $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\in\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} _{\mathcal P}}(M,N)=(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$, we take an $\operatorname{\mathsf{RUF}}$ $\mathbf ga=\mathsf {d}elta' f'$ of $\mathbf ga$. So, considering a co-angled pair $[\mathsf {d}elta_Na_1, \mathsf {d}elta' b_1]$, we define $F((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N))=T'(a_1)T'(b_1)^{-1}T'(f')$, with $(T'(b_1))^{-1}$ being the inverse of $T'(b_1)$. We would like to show that $F$ is well-defined. In this direction, first we prove that $F((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N))$ is independent of the choice of $\operatorname{\mathsf{RUF}}$ of $\mathbf ga$. Assume that $\mathbf ga=\mathsf {d}elta'' f''$ is also another $\operatorname{\mathsf{RUF}}$ of $\mathbf ga$. Now taking a co-angled pair $[\mathsf {d}elta_N a_2, \mathsf {d}elta''b_2]$, we shall show that $T'(a_1)T'(b_1)^{-1}T'(f')=T'(a_2)T'(b_2)^{-1}T'(f'')$. According to the proof of Theorem \twoheadrightarrowf{welldef} and applying the notation used there, we obtain the following commutative diagram in $\mathfrak{m}athscr{C} $;
{\mathbf footnotesize \[\mathsf {X}ymatrix{&N& \\ N_1~\mathsf {a}r[d]_{b_1}\mathsf {a}r[ur]^{a_1}& N_4\mathsf {a}r[l]_{a_4}\mathsf {a}r[r]^{b_4}\mathsf {a}r[d]_{c_4}& N_2\mathsf {a}r[d]_{b_2}\mathsf {a}r[ul]_{a_2}\\ N'~ & N_3\mathsf {a}r[l]_{a_3}\mathsf {a}r[r]^{b_3} & N'',}\]}where $N'$ (resp. $N''$) is the right end term of the unit conflation $\mathsf {d}elta'$ (resp. $\mathsf {d}elta''$), $N_i$ for any $i$, is the right end term of $\mathsf {d}elta_i$ and all morphisms are co-induced by identity over $\mathsf{\Omega}^nN$ . Moreover, by virtue of Proposition \twoheadrightarrowf{coin} and combining with Lemma \twoheadrightarrowf{ds}, there is a morphism $h:M\mathbf hookrightarrowghtarrow N_3$ such that $f'=a_3h$ and $f''=b_3h$. Namely, one may have the following commutative diagram in $\mathfrak{m}athscr{C} $; {\mathbf footnotesize\[\mathsf {X}ymatrix{N'~& N_3\mathsf {a}r[l]_{a_3}\mathsf {a}r[r]^{b_3}& N''\\ & M\mathsf {a}r[u]^{h}\mathsf {a}r[ul]^{f'}\mathsf {a}r[ur]_{f''} .& }\]}Since $T'$ is an additive functor, applying to the above diagrams, gives us commutative diagrams in $\mathfrak{m}athbb{D} $. This, in conjunction with the fact that $T'(a)$ is an isomorphism, for any $a\in\mathsf{\Sigma}$, one may deduce that $T'(a_1)T'(b_1)^{-1}T'(f')=T'(a_2)T'(b_2)^{-1}T'(f'')$, as claimed. Finally, we show that if $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\operatorname{\mathsf{thick}}sim(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_N)$, then $F((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N))=F((\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_N))$. Assume that $\mathbf ga=\mathsf {d}elta f$ and $\mathbf ga'=\mathsf {d}elta'f'$ are $\operatorname{\mathsf{RUF}}$s of $\mathbf ga$ and $\mathbf ga'$, respectively. So, taking co-angled pairs $[\mathsf {d}elta_Na_1, \mathsf {d}elta b_1]$ and $[\mathsf {d}elta'_Na_2, \mathsf {d}elta'b_2]$, we have to show that $T'(a_1)T'(b_1)^{-1}T'(f)=T'(a_2)T'(b_2)^{-1}T'(f')$. By our hypothesis, there is an angled pair $\mathsf {d}elta_N\stackrel{a}\mathsf {L}ongrightarrow\mathsf {d}elta''_N\stackrel{b}\mathsf {L}ongleftarrow\mathsf {d}elta'_N$
such that $a\mathbf ga-b\mathbf ga'$ is a $\mathcal P$-conflation. Set $\mathbf ga'':=a\mathbf ga=_{\mathcal P}b\mathbf ga'$. So considering $(\mathsf {b}ar{\mathbf ga''}, \mathsf {d}elta''_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, {\mathsf{\Omega}''}^nN)/{\mathcal P}, \mathsf {d}elta''_N)$, we infer that $\mathbf ga''=(a\mathsf {d}elta)f=(b\mathsf {d}elta')f'$ are two $\operatorname{\mathsf{RUF}}$s of $\mathbf ga''$. According to Lemma \twoheadrightarrowf{pushco}, we may have the co-angled pairs, $[\mathsf {d}elta''_Na_1,(a\mathsf {d}elta)b_1]$ and $[\mathsf {d}elta''_Na_2, (b\mathsf {d}elta')b_2]$. Since, as we have seen just above, the definition of $F$ is independent of the choice of $\operatorname{\mathsf{RUF}}$ of $\mathbf ga''$, we infer that $T'(a_1)T'(b_1)^{-1}T'(f)=T'(a_2)T'(b_2)^{-1}T'(f')$, as desired. \mathfrak{n}ew{For a given object $M\in\mathfrak{m}athscr{C} _{\mathcal P}$, we clearly have the equalities, $F(1_M)=(\mathsf {b}ar{\mathsf {d}elta}_M, \mathsf {d}elta_M)=T'(1_M)=1_{T'(M)}=1_{F(M)}$. Next assume that $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\in(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$ and $(\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta_K)\in(\operatorname{{\mathsf{Ext}}}^n(N, \mathsf{\Omega}^nK)/{\mathcal P}, \mathsf {d}elta_K)$. We have to show that $F((\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta_K))\operatorname{\underline{\mathscr{C}}}irc F((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N))=F((\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_K))$. Suppose that $\mathbf ga=\mathsf {d}elta_{N'}f$ and $\mathsf {b}e=\mathsf {d}elta_{K'}$ are $\operatorname{\mathsf{RUF}}$s of $\mathbf ga$ and $\mathsf {b}e$, respectively. Thus, taking co-angled pairs $[\mathsf {d}elta_Na, \mathsf {d}elta_{N'}b]$ and $[\mathsf {d}elta_Kc, \mathsf {d}elta_{K'}e]$, we get the equalities; $F((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N))=T'(a)T'(b)^{-1}T'(f)$ and $F((\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta_K))=T'(c)T'(e)^{-1}T'(g)$, and so, $F((\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta_K))\operatorname{\underline{\mathscr{C}}}irc F((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N))=T'(c)T'(e)^{-1}T'(g)T'(a)T'(b)^{-1}T'(f)$. On the other hand, $\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}=\overline{((\mathsf {b}e a)b^{-1})f}.$ Set $\operatorname{\boldsymbol{\eta}}:=(\mathsf {b}e a)b^{-1}$. So one has that $(\overline{\mathsf {b}e a}, \mathsf {d}elta_K)=(\overline{\operatorname{\boldsymbol{\eta}} b}, \mathsf {d}elta_K)$. Taking an $\operatorname{\mathsf{RUF}}$ $\operatorname{\boldsymbol{\eta}}=\mathsf {d}elta_{K''}h$ of $\operatorname{\boldsymbol{\eta}}$, one gets that $\operatorname{\boldsymbol{\eta}} f=\mathsf {d}elta_{K''}(hf)$ is an $\operatorname{\mathsf{RUF}}$ of $\operatorname{\boldsymbol{\eta}} f$. Therefore, considering a co-angled pair$[\mathsf {d}elta_K s, \mathsf {d}elta_{K''}s]$, we have that $F(\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_K)=F(\overline{\operatorname{\boldsymbol{\eta}} f}, \mathsf {d}elta_K)=T'(s)T'(t)^{-1}T'(h)T'(f)$. Moreover, the well-definedness ofyeilds that $F((\overline{\mathsf {b}e a}, \mathsf {d}elta_K))=F((\overline{\operatorname{\boldsymbol{\eta}} b}, \mathsf {d}elta_K))$, and so, $T'(c)T'(e)^{-1}T'(g)T'(a)=T'(s)T'(t)^{-1}T'(h)T'(b)$. This would imply that $F((\mathsf {b}ar{\mathsf {b}e}, \mathsf {d}elta_K))\operatorname{\underline{\mathscr{C}}}irc F((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N))=F((\mathsf {b}ar{\mathsf {b}e}\operatorname{\underline{\mathscr{C}}}irc\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_K))$, as desired.} It is clear that $FT=T'$ and also uniqueness of $F$ is obvious. So the proof is completed.
\end{proof}
Assume that $\mathfrak{m}athscr{C} '$ is a Frobenius category (or 0-Frobenius category in our sense). Then it follows from the proof of the above theorem the phantom stable category $(\mathfrak{m}athscr{C} '_{\mathcal P}, T)$ is indeed $(\mathfrak{m}athscr{C} '/{\mathfrak{m}athcal{I} }, \mathcal Pi)$, which has been mentioned in \twoheadrightarrowf{s1s1}. Namely, in the case $n=0$, the phantom stable category is actually the classical stable category of a Frobenius category.
From now on, to simplify the notation, we shall denote $\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} _{\mathcal P}}(-, -)$ by $\mathfrak{m}athscr{C} _{\mathcal P}(-, -)$.
\mathsf {b}egin{lem}\mathsf {L}abel{lem1}Let $\mathfrak{m}athscr{C} '$ be a full subcategory of $\mathfrak{m}athscr{C} $ which is {closed under extensions and kernels of epimorphisms}. Assume that:
\mathsf {b}egin{enumerate}\item $\mathfrak{m}athscr{C} '$ is an $n$-Frobenius category. \item $n$-$\mathcal Proj\mathfrak{m}athscr{C} '\subseteq n$-$\mathcal Proj\mathfrak{m}athscr{C} $. \item $\mathfrak{m}athscr{C} $ has enough $n$-$\mathcal Proj\mathfrak{m}athscr{C} '$.
\end{enumerate}Then for any two objects $M, N$ in $\mathfrak{m}athscr{C} '$, ${\mathfrak{m}athscr{C} _{\mathcal P}}(M, N)={\mathfrak{m}athscr{C} '_{\mathcal P}}(M, N)$.
\end{lem}
\mathsf {b}egin{proof}
Assume that $M, N$ are arbitrary objects of $\mathfrak{m}athscr{C} '$ and take a morphism $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\in{\mathfrak{m}athscr{C} _{\mathcal P}}(M, N)=(\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)$. {Since $n$-$\mathcal Proj\mathfrak{m}athscr{C} '\subseteq n$-$\mathcal Proj\mathfrak{m}athscr{C} $, we may assume that $\mathsf{\Omega}^nN\in\mathfrak{m}athscr{C} '$. Moreover}, using the fact that $\mathfrak{m}athscr{C} $ has enough $n$-$\mathcal Proj\mathfrak{m}athscr{C} '$, one may follow the argument given in the proof of Lemma \twoheadrightarrowf{gencog}, and get the following commutative diagram;
{\mathbf footnotesize \[\mathsf {X}ymatrix{\mathbf ga':\mathsf{\Omega}^nN~\mathsf {a}r[r]\mathsf {a}r@{=}[d]& H'\mathsf {a}r[r]\mathsf {a}r[d]_{b_{n-1}}& P_{n-2}\mathsf {a}r[r]\mathsf {a}r[d]_{b_{n-2}} &\operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& P_0\mathsf {a}r[r]\mathsf {a}r[d]_{b_0} & M\mathsf {a}r@{=}[d]\\ \mathbf ga:\mathsf{\Omega}^nN~\mathsf {a}r[r] &X_{n-1}\mathsf {a}r[r]& X_{n-2}\mathsf {a}r[r]&\operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r]& X_0\mathsf {a}r[r] & M,}\]}such that all $P_i^,$s belong to $n$-$\mathcal Proj\mathfrak{m}athscr{C} '$. Since $\mathfrak{m}athscr{C} '$ is a full subcategory of $\mathfrak{m}athscr{C} $ which is closed under extensions and kernels of epimorphisms, we infer that $\mathbf ga'$ is a conflation in $\mathfrak{m}athscr{C} '$. This, in turn, implies that $\mathbf ga$ can be considered as a conflation in $\mathfrak{m}athscr{C} '$, because $\mathbf ga=\mathbf ga'$.
Now, we should prove that if $\mathbf ga$ is a $\mathcal P$-conflation in $\mathfrak{m}athscr{C} '$, then it is a $\mathcal P$-conflation in
$\mathfrak{m}athscr{C} $ and vice versa. First, assume that $\mathbf ga$ is a $\mathcal P$-conflation in $\mathfrak{m}athscr{C} '$. Since $\mathfrak{m}athscr{C} '\subseteq\mathfrak{m}athscr{C} $ and any $n$-projective of $\mathfrak{m}athscr{C} '$ lies in $n$-$\mathcal Proj\mathfrak{m}athscr{C} $, we infer that $\mathbf ga$ is a $\mathcal P$-conflation in $\mathfrak{m}athscr{C} $. Next, we assume that $\mathbf ga$ is a $\mathcal P$-conflation in $\mathfrak{m}athscr{C} $.
Thus, there is a morphism $h:P\mathbf hookrightarrowghtarrow\mathsf{\Omega}^nN$ with $P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, and $\operatorname{\boldsymbol{\epsilon}}\in\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, P)$ such that $\mathbf ga=h\operatorname{\boldsymbol{\epsilon}}$. Take a conflation $\mathsf{\Omega} P\mathbf hookrightarrowghtarrow Q\stackrel{g}\mathbf hookrightarrowghtarrow P$, where $Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} '$. Evidently, $\mathsf{\Omega} P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. This, in turn, yields that the morphism $\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, Q)\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, P)$ is an epimorphism, and so, there exists $\operatorname{\boldsymbol{\eta}}\in\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, Q)$ such that $g\operatorname{\boldsymbol{\eta}}=\operatorname{\boldsymbol{\epsilon}}$. As $M, Q\in\mathfrak{m}athscr{C} ',$ similar to the above diagram, we may assume that $\operatorname{\boldsymbol{\eta}}\in\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} '}(M, Q)$. Consequently, $\mathbf ga=(hg)\operatorname{\boldsymbol{\eta}}$, i.e., $\mathbf ga$ is a $\mathcal P$-conflation in $\mathfrak{m}athscr{C} '$.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{l}Let $\mathfrak{m}athscr{C} '$ be a full subcategory of $\mathfrak{m}athscr{C} $ which is closed under extensions and kernels of epimorphisms and let $k\mathsf {L}eq n$ be non-negative integers. Assume that:
\mathsf {b}egin{enumerate}\item $\mathfrak{m}athscr{C} '$ is a $k$-Frobenius category. \item $k$-$\mathcal Proj\mathfrak{m}athscr{C} '\subseteq n$-$\mathcal Proj\mathfrak{m}athscr{C} $. \item $\mathfrak{m}athscr{C} $ has enough $k$-$\mathcal Proj\mathfrak{m}athscr{C} '$.
\end{enumerate}Consider the composition functor $T'':\mathfrak{m}athscr{C} '\stackrel{i}\mathbf hookrightarrowghtarrow\mathfrak{m}athscr{C} \stackrel{T}\mathbf hookrightarrowghtarrow\mathfrak{m}athscr{C} _{\mathcal P}$, with $i$ the inclusion functor.
Then there is a unique induced fully faithful functor $F:\mathfrak{m}athscr{C} '_{\mathcal P}\mathsf {L}ongrightarrow\mathfrak{m}athscr{C} _{\mathcal P}$
such that $FT'=T''$, where $(\mathfrak{m}athscr{C} _{\mathcal P}, T)$ and $(\mathfrak{m}athscr{C} '_{\mathcal P}, T')$ are phantom
stable categories.
\end{prop}
\mathsf {b}egin{proof} For any $M, N\in \mathfrak{m}athscr{C} '$, we consider a unit conflation $\mathsf {d}elta_N:\mathsf{\Omega}^nN\mathbf hookrightarrowghtarrow P_{n-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_0\mathbf hookrightarrowghtarrow N$, with $P_i\in k$-$\mathcal Proj\mathfrak{m}athscr{C} '$, for any $i$, and then we have $T''(f)=(\overline{\mathsf {d}elta_Nf}, \mathsf {d}elta_N)$, for any morphism $f:M\mathbf hookrightarrowghtarrow N$, because the definition of $T$ is independent of the choice of the unit conflation $\mathsf {d}elta_N$. Assume that $f\in\operatorname{\mathsf{Hom}}_{\mathfrak{m}athscr{C} '}(M, N)$ which belongs to $\mathsf{\Sigma}$. {In view of Lemma \twoheadrightarrowf{conf}, there is a conflation $M\mathbf hookrightarrowghtarrow N{\rm{op}}lus Q\mathbf hookrightarrowghtarrow P$, in which $P, Q\in k$-$\mathcal Proj\mathfrak{m}athscr{C} '$. Since by our assumption, $P, Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, applying Corollary \twoheadrightarrowf{is} and Remark \twoheadrightarrowf{rems} together, we deduce that $f\in\mathsf{\Sigma},$ as a morphism in $\mathfrak{m}athscr{C} $, and then,} $T''(f)=T(f)$ will be an isomorphism in $\mathfrak{m}athscr{C} _{\mathcal P}$. Now suppose that $f:M\mathbf hookrightarrowghtarrow N$ is an$n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism in $\mathfrak{m}athscr{C} '$. We shall prove that $T''(f)=0$ in $\mathfrak{m}athscr{C} _{\mathcal P}$. Since $f$ is an$n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism, in view of Proposition \twoheadrightarrowf{ph}, there are morphisms $N\stackrel{a}\mathsf {L}eftarrow N''\stackrel{b}\mathbf hookrightarrowghtarrow N'$ with $a, b\in\mathsf{\Sigma}$ and $h:M\mathbf hookrightarrowghtarrow N''$ such that $f=ah$ and $bh$ factors through an object $Q\in k$-$\mathcal Proj\mathfrak{m}athscr{C} '$. By our assumption, $Q\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, and so $bh$ will be an $n$-$\operatorname{{\mathsf{Ext}}}$-phantom morphism in $\mathfrak{m}athscr{C} $, implying that $T''(bh)=T(bh)=0$. Therefore, as $T''(b)$ is an isomorphism, we have $T''(h)=0$. Consequently, $T''(f)=T''(a)T''(h)=0$ in $\mathfrak{m}athscr{C} _{\mathcal P}$, as needed. Therefore, the universal property of the phantom stable category, gives rise to the existence of a unique functor $F:\mathfrak{m}athscr{C} '_{\mathcal P}\mathbf hookrightarrowghtarrow\mathfrak{m}athscr{C} _{\mathcal P}$ such that $FT'=T''$.
Now we prove that $F$ is faithful. The result for the case $k=n$, follows from Lemma \twoheadrightarrowf{lem1}. So assume that $k<n$.
Take a morphism $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta)\in{\mathfrak{m}athscr{C} '_{\mathcal P}}(M, N)=(\operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} '}(M, \mathsf{\Omega}^kN)/{\mathcal P}, \mathsf {d}elta)$ such that $F((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta))=0$ in $\mathfrak{m}athscr{C} _{\mathcal P}$. Suppose that $\mathbf ga=\mathsf {d}elta' f$ is an $\operatorname{\mathsf{RUF}}$ of $\mathbf ga$ and take a co-angled pair $[\mathsf {d}elta a, \mathsf {d}elta' b]$. Since, by the definition $F((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta))=T''(a)T''(b)^{-1}T''(f)$ and $T''(a)$ and $T''(b)^{-1}$ are isomorphisms, we get that $T''(f){\rm{op}}eratorname{\mathsf{cone}}ng 0$ in $\mathfrak{m}athscr{C} _{\mathcal P}$.
So taking a unit conflation $\mathsf {d}elta_1:\mathsf{\Omega}^nN\mathsf {L}ongrightarrow P_{n-1}\mathsf {L}ongrightarrow\operatorname{\underline{\mathscr{C}}}dots\mathsf {L}ongrightarrow P_k\mathsf {L}ongrightarrow\mathsf{\Omega}^kN$ with $P_i^,s\in k$-$\mathcal Proj\mathfrak{m}athscr{C} '$, and letting $\mathsf {d}elta'_N:=\mathsf {d}elta_1\mathsf {d}elta'$, we have $T''(f)=(\overline{\mathsf {d}elta'_Nf}, \mathsf {d}elta'_N)$. So one has the following push-out diagram; {\mathbf footnotesize\[\mathsf {X}ymatrix{\operatorname{\boldsymbol{\eta}}:P~\mathsf {a}r[r]\mathsf {a}r[d]_{g} & L\mathsf {a}r[d]\mathsf {a}r[r] &\operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r]& P_1\mathsf {a}r[r]\mathsf {a}r@{=}[d]\mathsf {a}r[r]& H\mathsf {a}r@{=}[d]\mathsf {a}r[r] & M\mathsf {a}r@{=}[d]\\ \mathsf {d}elta'_Nf:\mathsf{\Omega}^nN~\mathsf {a}r[r] & P_{n-1}\mathsf {a}r[r]& \operatorname{\underline{\mathscr{C}}}dots \mathsf {a}r[r] & P_1\mathsf {a}r[r]& H\mathsf {a}r[r] & M,}\]}where $P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $. Take a unit conflation $\mathsf{\Omega} P\mathbf hookrightarrowghtarrow Q\mathbf hookrightarrowghtarrow P$, where $Q\in k$-$\mathcal Proj\mathfrak{m}athscr{C} '$. Since $k<n$ and $\mathsf{\Omega} P\in n$-$\operatorname{\mathsf{inj}}\mathfrak{m}athscr{C} $, it is easily seen that $\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(-, P)=0$ over $\mathfrak{m}athscr{C} '$. Consequently, $\operatorname{\boldsymbol{\eta}}=0$, and then, the same is true for $\mathsf {d}elta'_Nf$. Now decomposing the unit conflation $\mathsf{\Omega}^nN\mathbf hookrightarrowghtarrow P_{n-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_k\mathbf hookrightarrowghtarrow\mathsf{\Omega}^kN$ into conflations of length 1, one may obtain the isomorphisms; $\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} '}(M, \mathsf{\Omega}^nN){\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{{\mathsf{Ext}}}^{n-1}_{\mathfrak{m}athscr{C} '}(M, \mathsf{\Omega}^{n-1}N){\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{\underline{\mathscr{C}}}dots{\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{{\mathsf{Ext}}}^{k+1}_{\mathfrak{m}athscr{C} '}(M, \mathsf{\Omega}^{k+1}N)$. So, letting $\mathsf {d}elta_2:=\mathsf{\Omega}^{k+1}N\mathbf hookrightarrowghtarrow P_k\stackrel{h}\mathbf hookrightarrowghtarrow\mathsf{\Omega}^kN$, we will have $\mathsf {d}elta_2\mathbf ga=0$. Now by considering the exact sequence $\operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} '}(M, P_k)\stackrel{\mathbf h}\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} '}(M, \mathsf{\Omega}^kN)\stackrel{}\mathbf hookrightarrowghtarrow\operatorname{{\mathsf{Ext}}}^{k+1}_{\mathfrak{m}athscr{C} '}(M, \mathsf{\Omega}^{k+1}N)$, we infer that there exists $\mathsf {a}l\in\operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} '}(M, P_k)$ such that $h\mathsf {a}l=\mathbf ga$, that is, $\mathbf ga$ is a $\mathcal P$-conflation in $\mathfrak{m}athscr{C} '$, and then, $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta)=0$.
Finally, we show that the functor $F$ is full. Assume that $M, N\in\mathfrak{m}athscr{C} '$ and fix a unit conflation $\mathsf {d}elta_N:\mathsf{\Omega}^nN\mathsf {L}ongrightarrow P_{n-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_0\mathbf hookrightarrowghtarrow N$, where $P_i\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, for any $i$. Suppose that $(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)\in{\mathfrak{m}athscr{C} _{\mathcal P}}(M, N)=
(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/\mathcal P, \mathsf {d}elta_N)$.
In the case $k=n$, the fullness of $F$ follows from Lemma \twoheadrightarrowf{lem1}.
So assume that $k<n$. Since any object of $k$-$\mathcal Proj\mathfrak{m}athscr{C} '$ is $n$-projective over $\mathfrak{m}athscr{C} $, we may further
assume that all terms of the unit conflation $\mathsf {d}elta_N$ lie in $\mathfrak{m}athscr{C} '$. So one may deduce that for any $Q\in k$-$\mathcal Proj\mathfrak{m}athscr{C} '$ and $i>k$, $\operatorname{{\mathsf{Ext}}}^i_{\mathfrak{m}athscr{C} }(-, Q)=0$ over $\mathfrak{m}athscr{C} '$. Now, decomposing the unit conflation
$\mathsf {d}elta_1:\mathsf{\Omega}^nN\mathbf hookrightarrowghtarrow P_{n-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_k\mathbf hookrightarrowghtarrow\mathsf{\Omega}^kN$ into conflations of length 1, one may obtain the isomorphisms;
$$\operatorname{{\mathsf{Ext}}}^n_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^nN){\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{{\mathsf{Ext}}}^{n-1}_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^{n-1}N){\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{\underline{\mathscr{C}}}dots{\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{{\mathsf{Ext}}}^{k+1}_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^{k+1}N){\rm{op}}eratorname{\mathsf{cone}}ng \operatorname{{\mathsf{Ext}}}^k_{\mathfrak{m}athscr{C} }(M, \mathsf{\Omega}^kN)/\mathcal P.$$ Hence, there exists $(\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_N)\in{\mathfrak{m}athscr{C} '_{\mathcal P}}(M, N)$ such that $F((\mathsf {b}ar{\mathbf ga'}, \mathsf {d}elta'_N))=(\overline{\mathsf {d}elta_1\mathbf ga'}, \mathsf {d}elta_1\mathsf {d}elta'_N)=(\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)$, where $\mathsf {d}elta'_N:\mathsf{\Omega}^kN\mathbf hookrightarrowghtarrow P_{k-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_0\mathbf hookrightarrowghtarrow N$. So the proof is finished.
\end{proof}
\mathsf {b}egin{cor}\mathsf {L}abel{p}Let $n>k$ and $\mathfrak{m}athscr{C} $ be also a $k$-Frobenius category. Then the phantom stable categories of $\mathfrak{m}athscr{C} $ as $k$ and $n$-Frobenius, are equivalent.
\end{cor}
\mathsf {b}egin{proof}Assume that $\mathfrak{m}athscr{C} '_{\mathcal P}$ and $\mathfrak{m}athscr{C} _{\mathcal P}$ denote the phantom stable categories of $\mathfrak{m}athscr{C} $, as a $k$ and an $n$-Frobenius category, respectively. According to Proposition \twoheadrightarrowf{l}, there is a fully faithful functor $F:\mathfrak{m}athscr{C} '_{\mathcal P}\mathsf {L}ongrightarrow\mathfrak{m}athscr{C} _{\mathcal P}$. Evidently, $F$ is also dense, and then, the proof is finished.
\end{proof}
\mathsf {b}egin{example}\mathsf {L}abel{exfaith}
(1) With the notation of Example \twoheadrightarrowf{ex1}, we set $\mathfrak{m}athscr{C} :=\mathfrak{m}athcal{G}^{<\infty}$ and $\mathfrak{m}athscr{C} ':=\mathfrak{m}athcal{G}$. Then, by Proposition \twoheadrightarrowf{l}, there exists a fully faithful functor $\mathfrak{m}athscr{C} '_{\mathcal P}\mathbf hookrightarrowghtarrow\mathfrak{m}athscr{C} _{\mathcal P}$. \\
(2) Assume that $\mathfrak{m}athscr{C} $ (resp. $\mathfrak{m}athscr{C} '$) is the category consisting of all syzygies of complete resolutions of $n$-projectives (resp. locally free) sheaves over $\mathsf {X}$. If $\mathfrak{m}athscr{C} $ has enough locally free sheaves, then Proposition \twoheadrightarrowf{l} ensures the existence of
a fully faithful functor $\mathfrak{m}athscr{C} '_{\mathcal P}\mathbf hookrightarrowghtarrow\mathfrak{m}athscr{C} _{\mathcal P}$.
\end{example}
}
\mathsf {b}egin{prop}
For a given object $M\in\mathfrak{m}athscr{C} _{\mathcal P}$, the following are equivalent:
\mathsf {b}egin{enumerate}
\item ${\mathfrak{m}athscr{C} _{\mathcal P}}(-, M)=0$.
\item ${\mathfrak{m}athscr{C} _{\mathcal P}}(M, -)=0$.
\item ${\mathfrak{m}athscr{C} _{\mathcal P}}(M, M)=0$.
\item $M\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $.
\end{enumerate}
\end{prop}
\mathsf {b}egin{proof}
$(1\mathfrak{m}athcal{R} ightarrow 2)$: As ${\mathfrak{m}athscr{C} _{\mathcal P}}(-, M)=0$, we have ${\mathfrak{m}athscr{C} _{\mathcal P}}(M, M)=0$. So $\mathsf {d}elta_M$ is a $\mathcal P$-conflation, and in particular, the same is true for $\mathsf {d}elta_M1_M$. So applying Corollary \twoheadrightarrowf{ccoo}, we infer that $(\operatorname{{\mathsf{Ext}}}^n/{\mathcal P})1_M=0$, and particularly, $(\operatorname{{\mathsf{Ext}}}^n(M, -)/{\mathcal P})1_M=0$, namely, $\mathfrak{m}athscr{C} _{\mathcal P}(M, -)=0$.\\
$(2\mathfrak{m}athcal{R} ightarrow 3)$: This is obvious.\\
$(3\mathfrak{m}athcal{R} ightarrow 4)$: Take the identity morphism $(\mathsf {b}ar{\mathsf {d}elta}_M, \mathsf {d}elta_M)\in\mathfrak{m}athscr{C} _{\mathcal P}(M, M)$. By our assumption $\mathsf {d}elta_M$ is a $\mathcal P$-conflation. Thus there exist a morphism $h:P\mathbf hookrightarrowghtarrow\mathsf{\Omega}^nM$ with
$P\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $ and $\operatorname{\boldsymbol{\eta}}\in\operatorname{{\mathsf{Ext}}}^n(M, P)$ such that $\mathsf {d}elta_M=h\operatorname{\boldsymbol{\eta}}$. Taking an $\mathsf {L}uf$ $\operatorname{\boldsymbol{\eta}}=g\mathsf {d}elta'_M$ of $\operatorname{\boldsymbol{\eta}}$, we obtain the following push-out diagram;
\[\mathsf {X}ymatrix{{\mathsf {d}elta'_M:\mathsf{\Omega}'}^nM ~\mathsf {a}r[r] \mathsf {a}r[d]_{hg}& Q\mathsf {a}r[r]\mathsf {a}r[d]& \operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r] & P_0\mathsf {a}r[r] \mathsf {a}r@{=}[d] & M\mathsf {a}r@{=}[d]\\ \mathsf {d}elta_M:\mathsf{\Omega}^n M~\mathsf {a}r[r] &P_{n-1}~\mathsf {a}r[r] & \operatorname{\underline{\mathscr{C}}}dots\mathsf {a}r[r] & P_0 \mathsf {a}r[r] & M.}\] Hence, for a given object $X\in\mathfrak{m}athscr{C} $, we will have the following commutative square;
{\mathbf footnotesize\[\mathsf {X}ymatrix{\operatorname{{\mathsf{Ext}}}^{n+1}(X, M)~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng} \mathsf {a}r@{=}[d]& \operatorname{{\mathsf{Ext}}}^{2n+1}(X, {\mathsf{\Omega}'}^nM)\mathsf {a}r[d]^{ \mathbf h\mathbf g}\\ \operatorname{{\mathsf{Ext}}}^{n+1}(X, M)~\mathsf {a}r[r]^{{\rm{op}}eratorname{\mathsf{cone}}ng} &\operatorname{{\mathsf{Ext}}}^{2n+1}(X, \mathsf{\Omega}^nM).}\]}As $hg$ factors through the $n$-projective object $P$, the right column is zero, and then $\operatorname{{\mathsf{Ext}}}^{n+1}(X, M)=0$, meaning that $M\in n$-$\operatorname{\mathsf{inj}}\mathfrak{m}athscr{C} $. Therefore, $M\in n$-$\mathcal Proj\mathfrak{m}athscr{C} $, because $\mathfrak{m}athscr{C} $ is $n$-Frobenius.\\
$(4\mathfrak{m}athcal{R} ightarrow 1)$: This implication is clear. So the proof is finished.
\end{proof}
\mathsf {b}egin{prop}\mathsf {L}abel{cp}(1) Let $f:M\mathbf hookrightarrowghtarrow N$ be a morphism in $\mathsf{\Sigma}$. Then for any $X\in\mathfrak{m}athscr{C} $, ${\mathfrak{m}athscr{C} _{\mathcal P}}(N, X)\stackrel{\mathsf {b}ar{\mathbf f}}\mathbf hookrightarrowghtarrow{\mathfrak{m}athscr{C} _{\mathcal P}}(M, X)$ is an isomorphism.\\ (2) Let $M\stackrel{f}\mathbf hookrightarrowghtarrow N\stackrel{g}\mathbf hookrightarrowghtarrow K$ be a conflation in $\mathfrak{m}athscr{C} $. Then, for any object $X\in\mathfrak{m}athscr{C} $, there exists an exact sequence;
$${\mathfrak{m}athscr{C} _{\mathcal P}}(K, X)\stackrel{\mathsf {b}ar{\mathbf g}}\mathsf {L}ongrightarrow{\mathfrak{m}athscr{C} _{\mathcal P}}(N, X)\stackrel{\mathsf {b}ar{\mathbf f}} \mathsf {L}ongrightarrow{\mathfrak{m}athscr{C} _{\mathcal P}}(M, X).$$
\end{prop}
\mathsf {b}egin{proof}The first statement is clear. The second one follows from Corollary \twoheadrightarrowf{qo} and Proposition \twoheadrightarrowf{pprop}.
\end{proof}
\mathsf {b}egin{s}Let $M,N\in\mathfrak{m}athscr{C} $ and let $\mathsf{\Omega} M\mathbf hookrightarrowghtarrow Q\mathbf hookrightarrowghtarrow M$ and $\mathsf{\Omega} N\mathbf hookrightarrowghtarrow P\mathbf hookrightarrowghtarrow N$ be two syzygy sequences of $M$ and $N$, respectively. We would like to define an induced map $\mathsf{\Omega}:\mathfrak{m}athscr{C} _{\mathcal P}(M, N)\mathbf hookrightarrowghtarrow\mathfrak{m}athscr{C} _{\mathcal P}(\mathsf{\Omega} M, \mathsf{\Omega} N)$. In this direction, we must define a map; $$\mathsf{\Omega}:(\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}, \mathsf {d}elta_N)\mathsf {L}ongrightarrow(\operatorname{{\mathsf{Ext}}}^n(\mathsf{\Omega} M, \mathsf{\Omega}^{n+1}N)/{\mathcal P}, \mathsf {d}elta_{\mathsf{\Omega} N}).$$ One should note that, Proposition \twoheadrightarrowf{ind} allows us to take $\mathsf {d}elta_N:=\mathsf{\Omega}^nN\mathbf hookrightarrowghtarrow P_{n-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_0\mathbf hookrightarrowghtarrow N$ and $\mathsf {d}elta_{\mathsf{\Omega} N}:=\mathsf{\Omega}^{n+1}N\mathbf hookrightarrowghtarrow P_n\mathbf hookrightarrowghtarrow P_{n-1}\mathbf hookrightarrowghtarrow\operatorname{\underline{\mathscr{C}}}dots\mathbf hookrightarrowghtarrow P_0\mathbf hookrightarrowghtarrow N.$ Consider the natural isomorphisms; $\operatorname{{\mathsf{Ext}}}^n(M, \mathsf{\Omega}^nN)/{\mathcal P}{\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{{\mathsf{Ext}}}^{n+1}(M, \mathsf{\Omega}^{n+1}N){\rm{op}}eratorname{\mathsf{cone}}ng\operatorname{{\mathsf{Ext}}}^n(\mathsf{\Omega} M, \mathsf{\Omega}^{n+1}N)/{\mathcal P}$, where the first isomorphism holds true, because of Proposition \twoheadrightarrowf{pprop} and the second one comes from \twoheadrightarrowf{ccor}. Denoting the composition of these isomorphisms by $\operatorname{\boldsymbol{\theta}}eta$, we define $\mathsf{\Omega}((\mathsf {b}ar{\mathbf ga}, \mathsf {d}elta_N)):=(\operatorname{\boldsymbol{\theta}}eta(\mathsf {b}ar{\mathbf ga}), \mathsf {d}elta_{\mathsf{\Omega} N})$. Indeed, we have the following result, which is analogue to the well-known result in the classical stable category of a Frobenius category.
\end{s}
\mathsf {b}egin{theorem}\mathsf {L}abel{syziso}With the notation above, there is an induced map $\mathsf{\Omega}:\mathfrak{m}athscr{C} _{\mathcal P}(M, N)\mathbf hookrightarrowghtarrow\mathfrak{m}athscr{C} _{\mathcal P}(\mathsf{\Omega} M, \mathsf{\Omega} N)$ which is isomorphism.
\end{theorem}
\mathsf {b}egin{rem}Let $\mathcal L$ be the subcategory of ${\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$ consisting of all locally free sheaves. As observed in Proposition \twoheadrightarrowf{locally}, $\mathfrak{m}athscr{C} (\mathcal L)$ is an $n$-Frobenius subcategory of ${\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$ with $n$-$\mathcal Proj\mathfrak{m}athscr{C} (\mathcal L)=\mathcal L$. So considering the phantom stable category $\mathfrak{m}athscr{C} (\mathcal L)_{\mathcal P}$, we have that an object $\mathcal{F}\in\mathfrak{m}athscr{C} (\mathcal L)$ is locally free if and only if $\mathfrak{m}athscr{C} (\mathcal L)_{\mathcal P}(\mathcal{F}, -)=0=\mathfrak{m}athscr{C} (\mathcal L)_{\mathcal P}(-, \mathcal{F})$. Next assume that $\mathsf {X}$ is a Grenstein scheme, i.e. all its local rings are Gorenstein local rings. Then, for any $\mathcal{F}\in{\rm{op}}eratorname{\mathsf{cone}}h(\mathsf {X})$, $\mathsf{\Omega}^d\mathcal{F}\in\mathfrak{m}athscr{C} (\mathcal L)$, where $d=\mathsf {d}im\mathsf {X}$; see \operatorname{\underline{\mathscr{C}}}ite[Theorem 2.2.3]{ajs}. Thus, we may infer that $\mathsf {X}$ is singular if and only if $\mathfrak{m}athscr{C} (\mathcal L)_{\mathcal P}=0$.
\end{rem}
{\mathsf {b}f{Acknowledgements.}} The authors are grateful to Sergio Estrada and Rasool Hafezi for their reading and suggestions on the first draft of the paper.
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\end{document} |
\begin{document}
\date{}
\author{Thabet ABDELJAWAD\footnote{\c{C}ankaya University, Department of
Mathematics, 06530, Ankara, Turkey} , Duran T\"{U}RKO\~{G}LU \footnote{ Department of Mathematics, Faculty of Science and Arts, Gazi University, 06500, Ankara-Turkey. [email protected].}}
\title{Locally Convex Valued Rectangular Metric Spaces and The Kannan's Fixed Point Theorem }
\maketitle
\begin{abstract}
Rectangular TVS-cone metric spaces are introduced and Kannan's fixed point theorem is proved in these spaces. Two approaches are followed for the proof. At first we prove the theorem by a direct method using the structure of the space itself. Secondly, we use the nonlinear scalarization used recently by Wei-Shih Du in [A note on cone metric fixed point theory and its equivalence, {Nonlinear Analysis},72(5),2259-2261 (2010).] to prove the equivalence of the Banach contraction principle in cone metric spaces and usual metric spaces. The proof is done without any normality assumption on the cone of the locally convex topological vector space, and hence generalizing several previously obtained results.
\end{abstract}
\emph{Keywords}: TVS-cone metric space, rectangular TVS-cone metric space, Kannan's fixed point theorem.
\section{Introduction and Preliminaries } \label{s:1}
Many authors attempted to generalize the notion of the metric space.
In 2007, Huang and Zhang \cite{HZ} announced the notion of cone metric spaces
(CMS) by using the same idea, namely, by replacing real numbers with an ordering real Banach space.
In that paper, they also discussed some properties of convergence of sequences and proved the
fixed point theorems of contractive mapping for cone metric spaces:
Any mapping $T$ of a complete cone metric space $X$ into itself that
satisfies, for some $0\leq k<1$, the inequality $d(Tx,Ty)\leq k
d(x,y)$, for all $x,y \in X$, has a unique fixed point. Lately, many
results on fixed point theorems have been extended to cone metric
spaces (see e.g.\cite{HZ},\cite{RH},\cite{Ishak},\cite{TAA},\cite{TAA2},\cite{K},\cite{T},\cite{AK}, \cite{TA}). For Kannan's fixed point theorem in rectangular metric spaces (R-MS) we refer to \cite{Das} and for the contraction principle and Kannan's fixed point theorem in rectangular cone metric space (R-CMS) see \cite{Akbar} and \cite{Beg}, respectively.
Recently, Du \cite{D_2009} gave the definition of generalized cone
metric space, namely topological vector space-cone metric space
(TVS-CMS), and proved some fixed point theorems on that class.
The author showed also that Banach contraction principles in
usual metric spaces and in TVS-CMS are equivalent.
In this manuscript, we first introduce the notion of rectangular TVS-cone metric spaces (R-TVS-CMS) and then prove Kannan's fixed point theorem in this class of spaces. The obtained result generalizes those in \cite{Beg} and \cite{Das} and hence the classical Kannan's fixed point theorem. Two proofs are presented and the proofs are done without any normality assumption.
Throughout this paper, $(E,S)$ stands for real Hausdorff locally convex topological vector space (t.v.s.) with $S$ its generating system of seminorms.
A non-empty subset $P$ of $E$ is called cone if $P+P \subset P$, $\lambda P \subset P$
for $\lambda \geq 0$ and $P \cap (-P) =\{0\}$. The cone $P$ will be assumed to be closed and has nonempty interior as well.
For a given cone $P$, one can define a partial ordering (denoted by $\leq$ or $\leq_P$) with respect to $P$
by $x\leq y$ if and only if $y-x \in P$. The notation $x<y$ indicates that $x\leq y$ and $x\neq y$ while
$x<<y$ will show $y-x\in intP$, where $intP$ denotes the interior of $P$.
Continuity of the algebric operations in a topological vector space and the properties of the cone imply the relations:
$$intP+intP\subseteq intP ~\emph{and}~\lambda intP \subseteq intP~(\lambda > 0).$$
We appeal to these relations in the following.
\begin{definition} \cite{Ali}
A cone $P$ of a topological vector space $(X,\tau)$ is said to be normal whenever $\tau$ has a base of zero consisting of $P-$ full sets. Where a subset of $A$ of an order vector space via a cone $P$ is said to be $P-$full if for each $x, y \in A$ we have $\{a \in E: x\leq a \leq y\}\subset A$.
\end{definition}
\begin{theorem} \cite{Ali}
(a) A cone $P$ of a topological vector space $(X,\tau)$ is normal if and only if whenever $\{x_\alpha\}$ and $\{y_\alpha\}$, $\alpha \in \Delta$ are two nets in $X$ with $0\leq x_\alpha \leq y_\alpha$ for each $\alpha \in \Delta$ and $y_\alpha \rightarrow 0$, then $x_\alpha \rightarrow 0$.
(b) The cone of an ordered locally convex space $(X,\tau)$ is normal if and only if $\tau$ is generated by a family of monotone $\tau-$ continuous seminorms. Where a seminorm $q$ on $X$ is called monotone if $q(x)\leq q(y)$ for all $x, y \in X$ with $0\leq x \leq y$.
\end{theorem}
In particular, if $P$ is a cone of a real Banach space $E$, then it is called
\textit{normal} if there is a number $K \geq 1$ such that for
all $x,y \in E$:\ $
0\leq x \leq y\Rightarrow \|x\|\leq K \|y\|.$ The least positive integer $K$, satisfying this inequality,
is called the normal constant of $P$.
Also, $P$ is said to be \textit{regular} if every increasing sequence which is bounded
from above is convergent. That is, if $\{x_n\}_{n\geq 1}$ is a sequence such
that $x_1 \leq x_2\leq \cdots\leq y$ for some $y \in E$, then there is $x \in E$
such that $\lim_{n\rightarrow\infty} \|x_n-x\|=0$. For more details about cones in locally convex topological vector spaces we may refer the reader to \cite{Ali}.
\
\begin{definition} (See \cite{CHY}, \cite{D_2008}, \cite{D_2009})
For $e \in intP$, the nonlinear scalarization function $\xi_e:E\rightarrow \mathbb R$ is defined by
\[\xi_e(y)=\inf\{t \in \mathbb R: y \in te-P\}, \ \mbox{for all} \ y \in E.\]
\end{definition}
\begin{lemma} (See \cite{CHY}, \cite{D_2008}, \cite{D_2009})
For each $t\in \mathbb R$ and $y \in E$, the following are satisfied:
\begin{itemize}
\item[$(i)$] $\xi_e(y)\leq t\Leftrightarrow y \in te-P$,
\item[$(ii)$] $\xi_e(y)> t\Leftrightarrow y \notin te-P$,
\item[$(iii)$] $\xi_e(y)\geq t\Leftrightarrow y \notin te-intP$,
\item[$(iv)$] $\xi_e(y)< t\Leftrightarrow y \in te-intP$,
\item[$(v)$] $\xi_e(y)$ is positively homogeneous and continuous on $E$,
\item[$(vi)$] if $y_1\in y_2+P$, then $\xi_e(y_2)\leq \xi_e(y_1)$,
\item[$(vii)$] $\xi_e(y_1+y_2)\leq \xi_e(y_1)+\xi_e(y_2)$, for all $y_1,y_2 \in E$.
\end{itemize}
\label{lemma_scalarization}
\end{lemma}
\begin{definition} Let $X$ be a non-empty set and $E$ as usual a Hausdorff locally convex topological space. Suppose a vector-valued function $p:X\times X\rightarrow E$ satisfies:
\begin{enumerate}
\item[$(M1)$] $0\leq p(x,y)$ for all $x,y \in X$,
\item[$(M2)$] $p(x,y)=0$ if and only if $x=y$,
\item[$(M3)$] $p(x,y)=p(y,x)$ for all $x,y \in X$
\item[$(M4)$] $p(x,y) \leq p(x,z)+p(z,y)$, for all $x,y,z \in X$.
\end{enumerate}
Then, $p$ is called TVS-cone metric on $X$, and the pair $(X,p)$ is called
a TVS-cone metric space (in short, TVS-CMS).
\end{definition}
Note that in \cite{HZ}, the authors considered $E$ as a real Banach space
in the definition of TVS-CMS. Thus, a cone metric space (in short, CMS) in the sense of Huang and
Zhang \cite{HZ} is a special case of TVS-CMS.
\begin{lemma} (See \cite{D_2009})
Let $(X,p)$ be a TVS-CMS. Then, $d_p:X \times X\rightarrow [0,\infty)$ defined by $d_p=\xi_e\circ p$ is a metric.
\label{lemma_usual_metric}
\end{lemma}
\begin{remark}
Since a cone metric space $(X,p)$ in the sense of Huang and Zhang \cite{HZ}, is a special case of TVS-CMS, then
$d_p:X \times X\rightarrow [0,\infty)$ defined by $d_p=\xi_e\circ p$ is also a metric.
\label{remark_CMS_usual_ms}
\end{remark}
\begin{definition}(See \cite{D_2009})
Let $(X,p)$ be a TVS-CMS, $x\in X$ and $\{x_n\}_{n=1}^{\infty}$ a sequence in $X$.
\label{definition_convergence}
\begin{itemize}
\item[($i$)] $\{x_n\}_{n=1}^{\infty}$ TVS-cone converges to $x\in X$ whenever
for every $0<<c\in E$, there is a natural number $M$ such that $p(x_n,x)<<c$
for all $n\geq M$ and denoted by $cone-\lim_{n\rightarrow \infty}x_n=x$ (or
$x_n\stackrel{cone}{\rightarrow} x$ as $n\rightarrow \infty$),
\item[($ii$)] $\{x_n\}_{n=1}^{\infty}$ TVS-cone Cauchy sequence in $(X,p)$ whenever
for every $0<<c\in E$, there is a natural number $M$ such that $p(x_n,x_m)<<c$
for all $n,m \geq M$,
\item[($iii$)] $(X,p)$ is TVS-cone complete if every sequence TVS-cone Cauchy sequence in $X$ is a TVS-cone
convergent.
\end{itemize}
\end{definition}
\begin{lemma} (See \cite{D_2009})
Let $(X,p)$ be a TVS-CMS, $x\in X$ and $\{x_n\}_{n=1}^{\infty}$ a sequence in $X$.
Set $d_p=\xi_e \circ p$. Then the following statements hold:
\begin{itemize}
\item[($i$)] If $\{x_n\}_{n=1}^{\infty}$ converges to $x$ in TVS-CMS $(X,p)$, then
$d_p(x_n,x)\rightarrow 0$ as $n\rightarrow \infty,$
\item[($ii$)] If $\{x_n\}_{n=1}^{\infty}$ is a Cauchy sequence in TVS-CMS $(X,p)$, then
$\{x_n\}_{n=1}^{\infty}$ is a Cauchy sequence (in usual sense) in $(X,d_p)$,
\item[($iii$)] If $(X,p)$ is a complete TVS-CMS, then $(X,d_p)$
is a complete metric space.
\end{itemize}
\label{lemma_eq_statements}
\end{lemma}
\begin{proposition}(See \cite{D_2009})
Let $(X,p)$ be a complete TVS-CMS and $T:X\rightarrow X$ satisfy the contractive
condition
\begin{equation}
p(Tx,Ty)\leq k p(x,y)
\label{contraction}
\end{equation}
for all $x,y \in X$ and $0 \leq k <1$. Then, $T$ has a unique fixed
point in $X$. Moreover, for each $x\in X$, the iterative sequence
$\{T^nx\}_{n=1}^{\infty}$ converges to fixed point.
\label{Du_thm22}
\end{proposition}
\begin{definition} \label{defn of rec TVS-cone}
Let $X$ be a nonempty set. A vector-valued function $p:X \times X \rightarrow E$ is said to be a rectangular $TVS-$ cone metric, if the following conditions hold:
\begin{itemize}
\item[($RC1$)] $0 \leq p(x,y)$ for all $x,y \in X$ and $p(x,y)=0$ if and only if $x=y$,
\item[($RC2$)]$p(x,y)=p(y,x)$ for all $x,y \in X$ ,
\item[($RC3$)] $p(x,z)\leq p(x,y)+p(y,w)+p(w,z)$ for all $x,y \in X$ and for all distinct points $z,w \in X$ each of them different from $x$ and $y$.
\end{itemize}
The pair $(X,p)$ is then called a rectangular TVS-cone metric space (R-TVS-CMS). When $E$ is Banach space $(X,p)$ is called rectangular cone metric space (R-CMS). When $E=\mathbb{R}$ and $P=[0,\infty)$, $(X,p)$ is called rectangular metric space (R-MS).
\end{definition}
Every TVS-CMS is R-TVS-CMS. However, the converse need not be true.
\begin{example} \label{not} ( \cite{Akbar}, see also \cite{Branciari})
Let $X=\{1,2,3,4\}$, $E=\mathbb{R}^2$ and $P=\{(x,y):x, y \geq 0\}$. Define $d:X \times X\rightarrow E$ as follows:
$$d(1,2)=d(2,1)=(3,6),~~d(2,3)=d(3,2)=d(1,3)=d(3,1)=(1,2),~~$$
$$d(1,4)=d(4,1)=d(2,4)=d(4,2)=d(3,4)=d(4,3)=(2,4).$$
Then $(X,d)$ is a R-CMS which is not a CMS, because
$$(3,6)=d(1,2)>d(1,3)+d(3,2)=(1,2)+(1,2)=(2,4).$$
\end{example}
\begin{definition} \label{conver}
Let $(X,p)$ be a rectangular TVS-cone metric space, $x \in X$ and $\{x_n\}$ a sequence in $X$.
(i) $\{x_n\}$ is said to be a Cauchy sequence if for any $0\ll c$ there exists $n_0\in \mathbb{N}$ such that for all $m,n\in \mathbb{N}$, $n\geqslant n_0$, one has $p(x_n,x_{n+m})\ll c$.
(ii)$\{x_n\}$ is said to converge to $x$ if for any $0\ll c$ there exists $n_0\in \mathbb{N}$ such that for all $n\geqslant n_0$, one has $p(x_n,x)\ll c$.
(iii) $(X,p)$ is called \textbf{complete} if every Cauchy sequence in $X$ is convergent in $X$.\\
\end{definition}
Let $T:X\rightarrow X$ be a mapping where $X$ is a R-TVS-CMS. For each $x\in X$, let
\begin{displaymath}
\textbf{O}(x)=\{x,Tx,T^2x,T^3x,\dotso\}.
\end{displaymath}
\begin{definition}
A cone metric space $X$ is said to be $T$-orbitally complete if every Cauchy sequence which is contained in $\textbf{O}(x)$ for some $x\in X$ converges in $X$.
\end{definition}
\section{Kannan's Fixed Point Theorem in R-TVS-CMS}
In order to realize the difference between TVS-CMS and R-TVS-CMS, we first prove Kannan's fixed point theorem in TVS-CMS.
\begin{theorem} \label{SA}
Let $(X,d)$ be a TVS-CMS and the mapping $T:X\rightarrow X$ satisfy the contractive condition
\begin{equation} \label{K}
d(Tx,Ty)\leqslant \beta [d(x,Tx)+d(y,Ty)]
\end{equation}
holds for all $x,y\in X$ where $\displaystyle 0<\beta<\frac{1}{2}$. If $X$ is $T$-orbitally complete then $T$ has a unique fixed point in $X$.
\end{theorem}
\textbf{Proof}
Let $x\in X$.
\begin{displaymath}
\begin{array}{r c l}
d(Tx,T^2x)&\leqslant& \beta [d(x,Tx)+d(Tx,T^2x)]\\[3mm]
i.e.,~~d(Tx,T^2x)&\leqslant& \frac{\beta}{1-\beta}d(x,Tx)
\end{array}
\end{displaymath}
Again,
\begin{displaymath}
\begin{array}{r c l}
d(T^2x,T^3x)&\leqslant& \beta [d(Tx,T^2x)+d(T^2x,T^3x)]\\[3mm]
i.e.,~~d(T^2x,T^3x)&\leqslant& \frac{\beta}{1-\beta}d(Tx,T^2x)\leqslant {\left(\frac{\beta}{1-\beta}\right)}^2d(x,Tx)
\end{array}
\end{displaymath}
Similarly,
\begin{displaymath}
d(T^3x,T^4x)\leqslant {\left(\frac{\beta}{1-\beta}\right)}^3d(x,Tx)
\end{displaymath}
Thus in general, if $n$ is a positive integer, then
\begin{equation}
d(T^nx,T^{n+1}x)\leqslant r^nd(x,Tx)
\end{equation}
where $\displaystyle r=\frac{\beta}{1-\beta}$. Since $\displaystyle 0<\beta< \frac{1}{2}$, clearly $0<r<1$.\\
Now, our aim is to show that $\{T^nx\}$ is a Cauchy sequence.
Assume $m\in \mathbb{N}$ and $n>m$, then we have
\begin{displaymath}
\begin{array}{r c l}
d(T^nx,T^mx)&\leqslant& d(T^nx,T^{n-1}x)+d(T^{n-1}x,T^{n-2}x)+\dotso+d(T^{m+1}x,T^mx)\\[3mm]
&\leqslant& (r^{n-1}+r^{n-2}+\dotso+r^m)d(x,Tx)\\[3mm]
&\leqslant& \frac{r^m}{1-r}d(x,Tx)
\end{array}
\end{displaymath}
Let $0\ll c$ be given. Find $\delta >0$ and $q \in S$ such that $q(b) < \delta$ implies $b\ll c$.\\
Now, since
\begin{displaymath}
\frac{r^m}{1-r}d(x,Tx) \to 0 \hspace{2 mm} as \hspace{2 mm} m\to \infty
\end{displaymath}
then find $n_0$ such that :
\begin{displaymath}
q( {\frac{r^m}{1-r}d(x,Tx)}) < \delta \hspace{3 mm} \forall m\geqslant n_0
\end{displaymath}
Hence, $\displaystyle \frac{r^m}{1-r}d(x,Tx)\ll c$, $\forall m\geqslant n_0$.\\
Thus, $d(T^nx,T^mx)\ll c$ for $n>m \geq n_0$. Therefore, $\{T^nx\}$ is a Cauchy sequence in $(X,d)$. Since $(X,d)$ is $T$-orbitally complete, there exists $x^*\in X$ such that $T^nx \to x^*$.\\
Choose a natural number $n_1$ such that $d(T^{n-1}x,T^nx)\ll \frac{c}{2}$ and $\displaystyle d(T^nx,x^*)\ll \frac{c}{2}$, for all $n\geqslant n_1$. Hence, for $n>n_1$ we have
\begin{displaymath}
\begin{array}{r c l}
d(Tx^*,x^*)&\leqslant& d(TT^{n-1}x,Tx^*)+d(T^nx,x^*)\\[3mm]
&\leqslant& \beta[d(T^{n-1}x,T^nx)+d(x^*,Tx^*)]+d(T^nx,x^*)\\[3mm]
&=&\beta d(T^{n-1}x,T^nx)+\beta d(x^*,Tx^*)+d(T^nx,x^*)\\[3mm]
&\leqslant& \frac{c}{2}+ \beta d(x^*,Tx^*)+\frac{c}{2}
\end{array}
\end{displaymath}
So,
\begin{displaymath}
(1-\beta)d(Tx^*,x^*)\ll c
\end{displaymath}
Hence,
\begin{displaymath}
(1-\beta)d(Tx^*,x^*)\ll \frac{c}{m}\hspace{3 mm}\forall m\geqslant1
\end{displaymath}
Hence, $\displaystyle \frac{c}{m}-(1-\beta)d(Tx^*,x^*)\in P$, for all $m\geqslant1$. Since $\displaystyle \frac{c}{m}\to 0$ as $m\to \infty$ and $P$ is closed;
\begin{displaymath}
-(1-\beta)d(Tx^*,x^*)\in P\hspace{3 mm}and \hspace{3 mm}(1-\beta)d(Tx^*,x^*)\in P
\end{displaymath}
from the cone properties, $(1-\beta)d(Tx^*,x^*)=0$. Since $(1-\beta)$ never be equal to zero, then $d(Tx^*,x^*)=0$. Thus $Tx^*=x^*$.\\
Now, if $y^*$ is another fixed point of $T$ then $Tx^*=x^*$ and $Ty^*=y^*$. Then, we have
\begin{displaymath}
0\leqslant d(x^*,y^*)=d(Tx^*,Ty^*)\leqslant \beta d(x^*,Tx^*)+\beta d(y^*,Ty^*)=0
\end{displaymath}
Hence, $d(x^*,y^*)=0$ and so $x^*=y^*$. Therefore, the fixed point of $T$ is unique.\\
Now, we prove Kannan's fixed point theorem in R-TVS-CMS.
\begin{theorem} \label{AS}
Let $T:X\rightarrow X$ be a mapping where $(X,d)$ is a $T$-orbitally complete R-TVS-CMS such that
\begin{equation} \label{Ka}
d(Tx,Ty)\leqslant \beta [d(x,Tx)+d(y,Ty)]
\end{equation}
holds for all $x,y\in X$ and $\displaystyle 0<\beta< \frac{1}{2}$. Then, $T$ has a unique fixed point in $X$.
\end{theorem}
\begin{proof}
As in the proof of Theorem \ref{SA}, for a fixed $x\in X$, we have for all $n \in \mathbb{N}$
\begin{equation}
d(T^nx,T^{n+1}x)\leqslant r^nd(x,Tx) \label{d}
\end{equation}
where $\displaystyle r=\frac{\beta}{1-\beta}$. Since $\displaystyle 0<\beta< \frac{1}{2}$, clearly $0<r<1$.\\
Since we are not able to use the triangle inequality, we divide the proof into two cases so that we can make use of the rectangle inequality.
\textbf{Case I}:
First assume that $T^mx\neq T^nx$ for $m,n\in N,m\neq n$. Then, for $n \in N$. Clearly,
\begin{displaymath}
\begin{array}{r c l}
d(T^nx,T^{n+1}x)&\leqslant& r^nd(x,Tx)<\left(\frac{r^n}{1-r}\right)d(x,Tx)\\[3mm]
\texttt{and}~~
d(T^nx,T^{n+2}x)&\leqslant& \beta [d(T^{n-1}x,T^nx)+d(T^{n+1}x,T^{n+2}x)]\\[3mm]
&\leqslant&\beta\left[{\left(\frac{\beta}{1-\beta}\right)}^{n-1}d(x,Tx)+{\left(\frac{\beta}{1-\beta}\right)}^{n+1}d(x,Tx)\right] \textbf{\hspace{1cm}(by\hspace{2mm}\ref{d})}\\[4mm]
&\leqslant& {\left(\frac{\beta}{1-\beta}\right)}^nd(x,Tx)+{\left(\frac{\beta}{1-\beta}\right)}^{n+1}d(x,Tx)\\[4mm]
&\leqslant& \left(\frac{r^n}{1-r}\right)d(x,Tx)
\end{array}
\end{displaymath}
since $\displaystyle 0<\beta< \frac{1}{2}$, $\displaystyle \beta \leqslant \frac{\beta}{1-\beta}$.\\
Now if $m>2$ is odd then writing $m=2k+1$, $k\geqslant 1$ and using the fact that $T^px\neq T^rx$ for $p,r\in N$, $p\neq r$ we can easily show that by the rectangular inequality
\begin{equation}
\begin{array}{r c l}
d(T^nx,T^{n+m}x)&\leqslant& d(T^nx,T^{n+1}x)+d(T^{n+1}x,T^{n+2}x)+\dotso+d(T^{n+2k}x,T^{n+2k+1}x)\\[3mm]
&\leqslant& r^nd(x,Tx)+r^{n+1}d(x,Tx)+\dotso+r^{n+2k}d(x,Tx) \textbf{\hspace{1cm}(by\hspace{2mm}\ref{d})}\\[3mm]
&\leqslant& \left(\frac{r^n}{1-r}\right)d(x,Tx) \label{e}
\end{array}
\end{equation}
Again if $m>2$ is even then writing $m=2k$, $k\geqslant2$ and using the same arguments as before we can get by the rectangular inequality
\begin{displaymath}
\begin{array}{r c l}
d(T^nx,T^{n+m}x)&\leqslant& d(T^nx,T^{n+2}x)+d(T^{n+2}x,T^{n+3}x)+\dotso+d(T^{n+2k-1}x,T^{n+2k}x)\\[3mm]
&\leqslant&[r^n+r^{n+1}+r^{n+3}+\dotso+r^{n+2k-1}]d(x,Tx) \hspace{1cm}\textbf{(by\hspace{2mm}\ref{d},\hspace{2mm}\ref{e})}
\end{array}
\end{displaymath}
Thus combining all the cases we have
\begin{equation}
d(T^nx,T^{n+m}x)\leqslant \left(\frac{r^n}{1-r}\right)d(x,Tx)\label{f}
\end{equation}
for all $m,n\in N$. Since $0<r<1$, $r^n \to 0$ as $n\to\infty $ and so by following a similar argument as in the proof of Theorem \ref{SA}, $\{T^nx\}$ is a Cauchy sequence. Since $X$ is $T$-orbitally complete, $\{T^nx\}$ is convergent. Let $u$ is defined as:
\begin{equation}
u=\lim_{n \to \infty}{T^nx}
\end{equation}
We shall now show that $Tu=u$. Without any loss of generality we assume that $T^nx\neq u$ and $T^nx\neq Tu$ for any $n\in N$. Then by (\ref{Ka}) and the rectangular inequality, we obtain
\begin{displaymath}
\begin{array}{r c l}
d(u,Tu)&\leqslant& d(u,T^nx)+d(T^nx,T^{n+1}x)+d(T^{n+1}x,Tu)\\[3mm]
&\leqslant& d(u,T^nx)+d(T^nx,T^{n+1}x)+\beta [d(T^nx,T^{n+1}x)+d(u,Tu)]\\[3mm]
i.e., ~~d(u,Tu)&\leqslant& \frac{1}{1-\beta}[d(u,T^nx)+(1+\beta)d(T^nx,T^{n+1}x)]
\end{array}
\end{displaymath}
Since $T^nx\rightarrow u$ and $\{T^nx\}$ is Cauchy then we obtain $0\leq d(u,Tu)\ll c$ for all $c\gg 0$. Then closeness of the cone $P$ implies that $u=Tu$.\\
\textbf{Case II}:
Let $T^mx=T^nx$ for some $m,n\in N$, $m\neq n$. Let $m>n$. Then $T^{m-n}(T^nx)=T^nx$ i.e.,$T^ky=y$ where $k=m-n,y=T^nx$. Now if $k>1$
\begin{displaymath}
d(y,Ty)=d(T^ky,T^{k+1}y)\leqslant {\left(\frac{\beta}{1-\beta}\right)}^kd(y,Ty) \hspace{1cm}\textbf{(by\hspace{2 mm}\ref{d})}
\end{displaymath}
Since $\displaystyle 0<\frac{\beta}{1-\beta}<1$, $d(y,Ty)=0$ i.e., $Ty=y$. That the fixed point of $T$ is unique easily follows from (\ref{Ka}). This completes the proof of the theorem.
\end{proof}
Theorem \ref{AS} above generalizes the results obtained in \cite{Beg}, where Kannan's fixed point theorem was proved in CMS and under the normality assumption. However, the proofs in this article are done without any normality type assumption.
\section{ The nonlinear scalarization and Kannan's fixed point theorem}
In this section, we use the nonlinear scalarization function to obtain a simpler shorter proof for the Kannan's fixed point theorem in R-TVS-CMS.
\begin{theorem} \label{rectangular}
Let $(X,p)$ be a rectangular TVS-cone metric space. Then $(X,d_p)$, where $d_p:=\xi_e \circ p$, is a rectangular metric space (R-MS).
\end{theorem}
\begin{proof}
By $RC1$, the definition of $\xi_e$ and that $P\cap -P=\{0\}$ we have $d_p(x,y)\geq 0$ for all $x, y \in X$. By $RC2$, $d_p(x,y)=d_p(y,x)$ for all $x, y \in X$. If $x=y$, then by $RC1$ $d_p(x,y)=\xi_e(0)=0$. Conversely, if $d_p(x,y)=0$, then by Lemma \ref{lemma_scalarization}, $RC1$ and that $P\cap -P=\{0\}$, we conclude that $p(x,y)=0$ and hence by $RC2$, $x=y$. Finally the rectangular inequality follows by Lemma \ref{lemma_scalarization} $(vi)$, $(vii)$ and $RC3$. \end{proof}
\begin{lemma} \label{TH}
Let $(X,p)$ be a R-TVS-CMS, $x\in X$ and $\{x_n\}_{n=1}^{\infty}$ a sequence in $X$.
Set $d_p=\xi_e \circ p$. Then the following statements hold:
\begin{itemize}
\item[($i$)] $\{x_n\}_{n=1}^{\infty}$ converges to $x$ in the R-TVS-CMS $(X,p)$ if and only if
$d_p(x_n,x)\rightarrow 0$ as $n\rightarrow \infty,$
\item[($ii$)] $\{x_n\}_{n=1}^{\infty}$ is Cauchy sequence in the R-TVS-CMS $(X,p)$ if and only if
$\{x_n\}_{n=1}^{\infty}$ is a Cauchy sequence in the rectangular metric space $(X,d_p)$,
\item[($iii$)] $(X,p)$ is a complete R-TVS-CMS if and only if $(X,d_p)$
is a complete rectangular metric space.
\end{itemize}
\end{lemma}
\begin{proof}
Applying Theorem \ref{rectangular}, $d_p$ is a rectangular metric on $X$. Regarding (i) First, assume $\{x_n\}$ converges to $x$ in the R-TVS-CMS $(X,p)$ and let $\epsilon > 0$ be given. Find $n_0$ such that $p(x_n,x)\ll \epsilon e$ for all $n>n_0$. Therefore, by Lemma \ref{lemma_scalarization} (iv), $d_p(x_n,x)=\xi_e \circ p(x_n,x)<\epsilon $, for all $n>n_0$. Conversely, we prove that if $x_n\rightarrow x$ in $(X,d_p)$ then $x_n\rightarrow x$ in the R-TVS-CMS $(X,p)$. To this end, let $c>>0$ be given, then find $q \in S$ and $\delta >0$ such that $q(b)<\delta$ implies that $b<< c$. Since $\frac{e}{n}\rightarrow 0$ in $(E,S)$ find $\epsilon = \frac{1}{n_0}$ such that $\epsilon q(e)=q(\epsilon e)<\delta $ and hence $\epsilon e << c$. Now, find $n_0$ such that $d_p (x_n,x)=\xi_e\circ p (x_n,x)< \epsilon$ for all $n\geq n_0$. Hence, by Lemma \ref{lemma_scalarization} (iv) $p (x_n,x)<< \epsilon e<< c$ for all $n\geq n_0$. The proof of (ii) is similar to the proof of (i). Finally, (iii) is immediate from (i) and (ii).
\end{proof}
Now the proof of Theorem \ref{SA} can be achieved by Lemma \ref{TH}, Theorem \ref{rectangular} and by Kannan's fixed point theorem (see \cite{Das}) applied to the R-MS $(X,d_p)$.
\end{document} |
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